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+[    {        "title": "1010.0464v1.Probing_Lorentz_Symmetry_with_Gravitationally_Coupled_Matter.pdf",        "content": "arXiv:1010.0464v1  [hep-ph]  4 Oct 2010October 30, 2018 17:33 Proceedings Trim Size: 9in x 6in paper\nPROBING LORENTZ SYMMETRY WITH\nGRAVITATIONALLY COUPLED MATTER\nJAY D. TASSON\nPhysics Department, Indiana University\nBloomington, IN 47405, USA\nE-mail: jtasson@indiana.edu\nMethods for obtaining additional sensitivities to Lorentz violation in the fermion\nsector ofthe Standard-Model Extension usinggravitationa l couplings arediscussed.\n1. Introduction\nThe Standard Model of particle physics together with Einstein’s Gen eral\nRelativity provide a remarkably successful description of known ph enom-\nena. General Relativity describes gravitation at the classical level, while all\nother interactionsare described down to the quantum level by the Standard\nModel. However, a single quantum-consistent theory at the Planck scale\nremains elusive.\nIdeally, experimental information would guide the development of th e\nunderlying theory; however, directly probing the Planckscale is impr actical\nat present. A feasible alternative is to search for suppressed effe cts arising\nfrom Planck-scale physics in sensitive experiments that can be perf ormed\nat presently accessible energies. Relativity violations arising from Lo rentz-\nsymmetryviolationintheunderlyingtheoryprovideacandidatesupp ressed\neffect.1,2The Standard-Model Extension (SME) is an effective field theory\nthat describes Lorentz violation at our present energies.3,4\nA large number of experimental tests of Lorentz symmetry have b een\nperformed in the context of the minimal SME. Those test include, in\nthe Minkowski-spacetime limit, experiments with electrons,5protons and\nneutrons,6photons,7mesons,8muons,9neutrinos,10and the Higgs.11The\npure-gravity sector has is also being investigated in the post-Newt onian\nlimit.12,13Although no compelling experimental evidence for Lorentz vi-\nolation has been found to date, much remains unexplored. For exam ple,\nonly about half of the coefficients for Lorentz violation involving light a nd\n1October 30, 2018 17:33 Proceedings Trim Size: 9in x 6in paper\n2\nordinary matter (protons, neutrons, and electrons) have been investigated\nexperimentally, and other sectors remain nearly unexplored.\nIn the remainder of this proceedings, a theoretical basis for exte nding\n(SME) studies with ordinary matter into the post-Newtonian regime , de-\nveloped with Alan Kosteleck´ y, will be discussed.14The goal of this work is\nto obtain new sensitivities to Lorentz violation in the fermion sector u sing\ncouplings to gravity. These couplings introduce new operator stru ctures\nthat provide sensitivities to coefficients for Lorentz violation that a re un-\nobservable in Minkowski spacetime.\n2. Relativistic Theory\nGravitational effects are incorporated into the SME action in Ref. 4 . The\ngeneralgeometric frameworkassumed is Riemann-Cartanspacet ime, which\nallows for a nonzero torsion tensor Tλ\nµνas well as the Riemann curvature\ntensorRκ\nλµν. Due to the need to incorporate spinor fields, the vierbein\nformalism is adopted. The vierbein also allows one to easily distinguish\ngeneral coordinate and local Lorentz transformations, a featu re convenient\nin studying Lorentz violation.4The spin connection ωab\nµalong with the\nvierbeineµ\naare taken as the fundamental gravitational objects, while the\nbasic non-gravitational fields are the photon Aµand the Dirac fermion ψ.\nThe minimal-SME action can be expanded in the following way:\nS=SG+Sψ+S′. (1)\nHere, first term, SGis the action of the pure-gravity sector, which contains\nthe dynamics of the gravitational field and can also contain coefficien ts for\nLorentz violation in that sector.4,12The Einstein-Hilbert action of General\nRelativity is recovered in the limit of zero torsion and Lorentz invarian ce.\nThe action for the fermion sector is provided by the term Sψin Eq. (1).\nSψ=/integraldisplay\nd4x(1\n2ieeµ\naψΓa↔\nDµψ−eψMψ). (2)\nHere, ΓaandMtake the form shown in the following definitions.\nΓa≡γa−cµνeνaeµ\nbγb−dµνeνaeµ\nbγ5γb\n−eµeµa−ifµeµaγ5−1\n2gλµνeνaeλ\nbeµ\ncσbc. (3)\nM≡m+aµeµ\naγa+bµeµ\naγ5γa+1\n2Hµνeµ\naeν\nbσab. (4)\nThe symbols aµ,bµ,cµν,dµν,eµ,fµ,gµνλ,Hµνare the coefficient fields for\nLorentz violation of the minimal fermion sector. In general, they va ry withOctober 30, 2018 17:33 Proceedings Trim Size: 9in x 6in paper\n3\nposition and differ for each species of particle. For additional discus sion of\nthe fermion-sector action, see Ref. 4.\nThe final portion, S′, of the action (1) contains the dynamics associ-\nated with the coefficient fields for Lorentz violation and is responsible for\nspontaneous breaking of Lorentz symmetry. Through symmetry breaking,\nthe coefficient fields for Lorentz violation are expected to acquire v acuum\nvalues. Thus, it is possible to write\ntλµν...=tλµν...+/mapsto/tildewidetλµν..., (5)\nwheretλµν...represents an arbitrary coefficient field for Lorentz violation,\ntλµν...is the corresponding vacuum value, and/mapsto/tildewidetλµν...is the fluctuations\nabout that vacuum value. Note that a subset of these fluctuation s are\nthe massless Nambu-Goldstone modes associated with Lorentz-sy mmetry\nbreaking.15It is possible to develop the necessary tools to analyze fermion\nexperiments in the presence of gravity and Lorentz violation withou t spec-\nifyingS′.14Those results are summarized here.\nThe relativistic quantum-mechanical hamiltonian obtained from actio n\n(2) provides a first step toward the goal of obtaining experimenta l access\nto Lorentz violation. The hamiltonian my be obtained pertrubatively s ince\ngravitational and Lorentz-violating effects are small in the regimes of inter-\nest. Gravity may be considered perturbatively small in the laborato ry and\nsolar-system tests that will be of interest, and Lorentz violation is assumed\nsmall since it has not been observed in nature. In what follows, orde rs\nin these small quantities will be denoted O(m,n), wheremandnare the\norder of the given term in coefficients for Lorentz violation (vacuum values)\nand the metric fluctuation respectively.\nAfter incorporating relevant effects to order m+n= 2 and making a\nfield redefinition required to define the hamiltonian,14,16a hamiltonian of\nthe form,\nH=H(0,0)+H(0,1)+H(1,0)+H(1,1)+H(0,2), (6)\nis found. Here, the Lorentz invariant contributions consist of the conven-\ntional Minkowski-spacetime hamiltonian, H(0,0), and the first and second\norder gravitational corrections denoted H(0,1)andH(0,2)respectively. The\nfirst order correction to the hamiltonian due to Lorentz violation, H(1,0), is\nthe same as that found in the flat-spacetime SME.16TheO(1,1) correction\nto the hamiltonian is the term of interest since it contains coefficients for\nLorentz violation coupled to gravity. It can be written as follows:\nH(1,1)=H(1,1)\nh+H(1,1)\na+H(1,1)\nb+···+H(1,1)\ng+H(1,1)\nH,(7)October 30, 2018 17:33 Proceedings Trim Size: 9in x 6in paper\n4\nwhereH(1,1)\nhis the perturbation to the hamiltonian from Lorentz-violating\ncorrections to the metric. The relevant corrections to the metric can be ob-\ntained though investigations of the classical limit in Sec. 4. The terms de-\nnotedH(1,1)\ntare perturbations to the hamiltonian due to Lorentz-violating\neffects on the test particle at O(1,1). As a sample of O(1,1) effects,H(1,1)\na\ntakes the following form:\nH(1,1)\na=/mapsto/tildewidea0−ajhj0+/bracketleftBig/mapsto/tildewideaj−1\n2ajh00−1\n2akhjk/bracketrightBig\nγ0γj. (8)\nThe explicit form of the relativistic hamiltonian including all of the coef-\nficients of the minimal fermion sector of the SME, along with additiona l\ndetails of its derivation, can be found in Ref. 14.\n3. Non-relativistic Theory\nThe relativistic hamiltonian above is most useful as a tool to derive th e\nnon-relativistic hamiltonian, rather than to analyze experiments dir ectly,\nbecause experiments most sensitive to the position operator are n on-\nrelativistic. A Foldy-Wouthuysen transformation can be applied to o btain\nthe non-relativistic hamiltonian taking care to maintain the desired or der\nin the small quantities.\nAfter performing the transformation, the non-relativistic hamilto nian,\nHNR, can be written,\nHNR=H(0,0)\nNR+H(0,1)\nNR+H(1,0)\nNR+H(1,1)\nNR+H(0,2)\nNR. (9)\nHere, as in the relativistic case, H(0,0)\nNRis the conventional Minkowski-\nspacetime hamiltonian, H(0,1)\nNRandH(0,2)\nNRcontain the leading and sub-\nleading gravitationalcorrections, and H(1,0)\nNRare the leading corrections due\nto Lorentz violation, which match Ref. 16. Again, the leading coupling s of\nLorentz violation to gravitational effects, H(1,1)\nNR, are the contributions of\ninterest. In a manner analogous to the relativistic case, this term c ontains\ncontributions from the modified metric fluctuation, written H(1,1)\nNR,h, as well\nas perturbations from each of the coefficients for the test partic le,H(1,1)\nNR,t.\nA result of the Foldy-Wouthuysen transformation is that at leading\norder in the coefficients for Lorentz violation at the non-relativistic level,\ncoefficients aµandeµalways appear in the combination ( aeff)µ≡aµ−meµ.\nThis result is consistent with those of Ref. 4 indicating that eµcan be\nredefined into aµat leading order in the coefficients for Lorentz violation\nvia an appropriate field redefinition. As a sample of O(1,1) contributionsOctober 30, 2018 17:33 Proceedings Trim Size: 9in x 6in paper\n5\ntoHNR, the correction to the hamiltonian due to aµandeµcan be written\nH(1,1)\nNR,aeff= (/mapsto/tildewideaeff)0+(aeff)kh0k−1\nm(aeff)jhjkpk+1\nm/bracketleftBig\n(/mapsto/tildewideaeff)j−1\n2(aeff)jh00/bracketrightBig\npj,\n(10)\nto second order in momentum. Additional details of the derivation of HNR,\nalong with an explicit form for the remaining spin-independent contrib u-\ntions toHNRcan be found in Ref. 14.\n4. Classical Theory\nThe classical action can be written as the sum of partial actions\nS=SG+Su+S′, (11)\njust as in the relativistic case. Here, SGandS′are as in Eq. (1). The\npartial action Suis the point-particle limit of Sψappearing in Eq. (1).\nThe classical theory is useful for several applications including obt aining\nthe equations of motion for a classical particle, analyzing non-relat ivistic\nquantum-mechanical experiments via the path integral approach , and ob-\ntaining the modified metric.\nUpon inspection of the non-relativistic hamiltonian discussed in Sec. 3 ,\na point-particle action which corresponds to this hamiltonian can be f ound.\nThe spin-independent contributions to this action action can be writ ten\nSu=/integraldisplay\ndτ/parenleftbigg\n−m/radicalBig\n−(gµν+2cµν)uµuν−(aeff)µuµ/parenrightbigg\n,(12)\nwhereuµis the four velocity as usual. The validity of this action has been\nestablishedhereonlytowithintheassumptionsmadeuptotheprese ntation\nof the non-relativistic hamiltonian, HNR, and in performing calculations\nwith this action one should not exceed the order in small quantities or the\norder in momentum discussed in the last section. In addition to the ma tch\nbetween the classical action above and the non-relativistic hamilton ian, the\nfact that the dispersion relation generated by the classical action matches\nthe relativistic theory confirms the validity of this action. Note also t hat\nthecµνcontributions to this action have been discussed previously in the\ncontext of work done in the photon sector.17After some consideration,14\nthis action can be extended to addressthe casein which particlesar e bound\nwithin macroscopic matter as well.\n5. Experimental Tests\nWith the theory developed above, experiments in any regime, from r ela-\ntivistic quantum mechanics to classical mechanics, can be analyzed, pro-October 30, 2018 17:33 Proceedings Trim Size: 9in x 6in paper\n6\nvided the contributions from S′are known. These contributions can be\nestablished directly within a model of spontaneous Lorentz violation or de-\ntermined for a large class of models by examining the general form of the\ncontributions along with the constraints available from conservatio n laws.\nUpon a general analysis, effects are found in a number of experimen ts.\nThese effects include annual and sidereal variations in the newtonia n gravi-\ntational accelerationaswell as variationsin the gravitationalforc ebased on\nthe proton, neutron, electron content of the bodies involved. Th ese effects\nlead to signals in gravimeter tests as well as in some experiments desig ned\nto test the weak equivalence principle. These tests are described in de-\ntail in Ref. 14 and will provide the first direct sensitivities to the ( aeff)µ\ncoefficients for the proton, neutron, and electron.\nReferences\n1. See, for example, Data Tables for Lorentz and CPT Violation, V.A. Kost-\neleck´ y and N. Russell, 2010 edition, arXiv:0801.0287v3; R . 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D 70, 076006 (2004)."    },    {        "title": "1301.3139v2.Universal_Properties_of_the_Higgs_Resonance_in__2_1__Dimensional_U_1__Critical_Systems.pdf",        "content": "Universal Properties of the Higgs Resonance in (2+1)-Dimensional U(1)Critical\nSystems\nKun Chen1;2, Longxiang Liu1, Youjin Deng1;2,\u0003Lode Pollet3,yand Nikolay Prokof'ev2;4z\n1National Laboratory for Physical Sciences at Microscale and Department of Modern Physics,\nUniversity of Science and Technology of China, Hefei, Anhui 230026, China\n2Department of Physics, University of Massachusetts, Amherst, Massahusetts 01003, USA\n3Department of Physics and Arnold Sommerfeld Center for Theoretical Physics,\nLudwig-Maximilians-Universit at M unchen, D-80333 M unchen, Germany and\n4Russian Research Center \\Kurchatov Institute\", 123182 Moscow, Russia\n(Dated: March 28, 2022)\nWe present spectral functions for the magnitude squared of the order parameter in the scaling limit\nof the two-dimensional super\ruid to Mott insulator quantum phase transition at constant density,\nwhich has emergent particle-hole symmetry and Lorentz invariance. The universal functions for\nthe super\ruid, Mott insulator, and normal liquid phases reveal a low-frequency resonance which\nis relatively sharp and is followed by a damped oscillation (in the \frst two phases only) before\nsaturating to the quantum critical plateau. The counterintuitive resonance feature in the insulating\nand normal phases calls for deeper understanding of collective modes in the strongly coupled (2+1)-\ndimensional relativistic \feld theory. Our results are derived from analytically continued correlation\nfunctions obtained from path-integral Monte Carlo simulations of the Bose-Hubbard model.\nPACS numbers: 05.30.Jp, 74.20.De, 74.25.nd, 75.10.-b\nField theories of a complex scalar order parameter, \t,\ncan have two types of collective excitations. The \frst\none originates from \ructuations of the phase of \t and\ndescribes a Bogoliubov sound mode. The second one,\nif present, describes amplitude \ructuations and is associ-\nated with a Higgs mode. In super\ruids, sound excitations\nare gapless while the Higgs mode, if present, is gapped\nbut the gap may go to zero under special circumstances\nsuch as an emergent particle-hole symmetry and Lorentz\ninvariance. This is what happens in the vicinity of the\nsuper\ruid (SF) to Mott insulator (MI) quantum critical\npoint (QCP) of the Bose-Hubbard model when the phase\ntransition is crossed at constant density.\nMean-\feld theory predicts a stable Higgs particle. In\n(3+1) dimensions, where the QCP is a Gaussian \fxed\npoint (with logarithmic UV corrections), there is com-\npelling experimental evidence for the existence of a Higgs\nmode, most beautifully illustrated for the TlCuCl 3com-\npound [1] (see Ref. [2] for the latest results with cold\ngases). In (2+1) dimensions, where scaling theory is ex-\npected to apply, the massive Higgs particle is strongly\ncoupled to sound modes and it was argued for a long\ntime, on the basis of a 1 =Nexpansion to leading order\n(N= 2 corresponds to our case), that it cannot survive\nnear criticality [3{6]. Moreover, since the longitudinal\nsusceptibility diagram has an IR divergence going as !\u00001,\nit may well dominate any possible Higgs peak. However,\nit was recently emphasized that the type of the probe is\nimportant [7{9]: for scalar susceptibility (i.e., the correla-\ntion function ofj j2) the spectral function S(!) vanishes\nas!3at low frequencies [3, 7], and this o\u000bers better con-\nditions for revealing the Higgs peak. In the scaling limit\nthe theory predicts that S(!) in the SF phase takes theform\nSSF(!)/\u00013\u00002=\u0017\bSF(!\n\u0001); (1)\nwhere \u0001 is the MI gap for the same amount of detuning\nfrom the QCP, and \u0017= 0:6717 is the correlation length\nexponent for the U(1)\u0011O(2) universality in (2 + 1)\ndimensions [10, 11]. The universal function \b SF(x) starts\nas \b SF(x!0)/x3and saturates to a quasiplateau\n\bSF(x\u001d1)/x3\u00002=\u0017\u0019x0:0225. The Higgs resonance (at\nx\u00181) can be seen right before the incoherent quantum\ncritical continuum with weak !dependence.\nWe are not aware of solid state studies of the Higgs\nmode in two-dimensional (2D) super\ruids near the QCP.\nRecently, the cold atom experiment [12], where a 2D\nBose-Hubbard system was gently \"shaken\" by modulat-\ning the lattice laser intensity and probed by in situ single\nsite density measurements, saw a broad spectral response\nwhose onset softened on approach to the QCP, in line\nwith the scaling law (1), and no Higgs resonance. This\noutcome can be explained by tight con\fnement, \fnite\ntemperature, and detuning from the QCP, as shown by\nquantum Monte Carlo (MC) simulations [14] performed\nfor the experimental setup \"as is\" in the spirit of the\nquantum simulation paradigm [13]. On the other hand,\nsimulations for the homogeneous Bose-Hubbard model\n(below,J,U, and\u0016stand for the tunneling amplitude,\non-site interaction, and chemical potential, respectively;\nin what follows energy and frequency are measured in\nunits ofJ)\nH=\u0000JX\n<ij>by\nibj+U\n2X\nini(ni\u00001)\u0000\u0016X\nini;(2)\nin the vicinity of the SF-MI point featuring emergentarXiv:1301.3139v2  [cond-mat.stat-mech]  24 Apr 20132\nparticle-hole symmetry and Lorentz invariance [15] un-\nambiguously revealed a well-de\fned Higgs resonance\nwhich becomes more pronounced on approach to the\nQCP [14]. However, its universal properties, i.e., the pre-\ncise structure of \b SF(x), were not answered in Ref. [14].\nThe Higgs mode is not discussed in the MI phase since\nthe order parameter is zero in the thermodynamic limit.\nLikewise, no resonance is expected in the normal quan-\ntum critical liquid (NL), i.e., at \fnite temperature for\ncritical parameters ( U;\u0016) = (Uc;\u0016c). However, simula-\ntions reveal a resonance in the MI phase right after the\ngap threshold [14] suggesting that \fnite-energy probes\nare primarily sensitive to local correlations at length\nscales where MI and SF are indistinguishable. The uni-\nversality of the MI response was likewise never clari\fed.\nIn their most recent calculation, Podolsky and Sachdev\n[16] found that including next-order corrections in a 1 =N\nexpansion in the scaling limit radically changes previ-\nous conclusions in that S(!) does contain an oscillatory\ncomponent, in line with MC simulations. However, the\nprecise shape of the \b SF(x) function could not be estab-\nlished within the approximations used.\nIn this Letter, we aim to determine the universal scal-\ning spectral functions when approaching the QCP from\nthe SF, MI, and NL phases. We rely on the worm algo-\nrithm [18{20] in the path integral representation to per-\nform the required large-scale simulations. By collapsing\nspectral functions evaluated along the trajectories spec-\ni\fed by the dashed lines in Fig 1, we extract universal\nfeatures for all three phases. They are summarized in\nFig. 2, which is our main result. Surprisingly, all of them\ninclude a universal resonance peak (relatively sharp in SF\nand MI phases), followed by a broad secondary peak (in\nSF and MI phases only) before merging with the incoher-\nent critical quasiplateau (the plateau value is the same\nin all cases, as expected). Our results are in agreement\nwith scaling theory, and \frmly establish that the damped\nresonance is present in all three phases. (The integrated\nspectral weight of the !3law at low frequencies is too\nsmall to be resolved reliably by analytical continuation\nmethods [14].)\nThe phase diagram of the 2D Bose-Hubbard model,\nshown in Fig. 1, is known with high accuracy [21{23] at\nboth zero and \fnite temperature. The QCP is located\natUc= 16:7424(1),\u0016c= 6:21(2). When the system is\nslightly detuned from the QCP, either by changing the\nchemical potential or the interaction strength, we de\fne\nthe corresponding characteristic energy scale \u0001 using the\nenergy gap in the MI phase, Egap(g), by the rule illus-\ntrated in Fig. 1: For positive g= (U\u0000Uc)=Jit is half the\ngap, \u0001(g>0) =Egap(g)=2, whereEgap=\u0016(+)\nc\u0000\u0016(\u0000)\ncis\ndeduced from the upper and lower critical chemical po-\ntentials for a given g. Forg<0 along the trajectory i in\nthe SF phase it is \u0001( g <0) =Egap(\u0000g)=2. ForU=Uc\nand negative g\u0016= (\u0016\u0000\u0016c)=Jalong the trajectory ii\nSF MIMI SFNL\n5678\n0 -1 -0.5 0.5 1µ/J\n(U-Uc)/J00.30.60.9\n-1.5-0.50.51.5T/J\n(U-Uc)/JFIG. 1. (Color online) Ground state phase diagram of the\nBose-Hubbard model in the vicinity of the QCP marked by a\nlarge (blue) dot (based on Ref. [22] data). The (blue) dashed\ncurves specify trajectories in parameter space used to detune\nthe system away from the QCP (trajectories i and iii corre-\nspond to unity \flling factor n= 1, trajectory ii has constant\ninteraction strength). The (black) lines with arrows explain\nhow the characteristic energy scale \u0001 is obtained for these\nparameters (see text). The inset shows the phase diagram at\n\fnite temperature, and the trajectory taken in the NL phase.\nin the SF phase we \frst \fnd gsuch that\u0016(\u0000)\nc(g) =\u0016\nand then de\fne \u0001( g\u0016) =CEgap(g)=2 where the constant\nC= 1:2 (see below) is \fxed by demanding that the uni-\nversal function is the same along both SF trajectories.\nNote thatEgap(g) in the thermodynamic limit can be\ndetermined accurately from the imaginary time Green\nfunction data [22] and \fnite-size scaling analysis.\n 0 0.5 1 1.5 2 2.5\n 0 8 16 24 32Φ(ω/∆)\nω/∆(-g)SF\n            \n 0 8 16 24\nω/∆(g)MI\n            \n 0 6 12 18 24\nω/TNL\nFIG. 2. Universal spectral functions for scalar response in the\nsuper\ruid, Mott insulator, and normal liquid phases. For SF,\nthe Higgs peak is at !H=\u0001 = 3:3(8), for MI, !H=\u0001 = 3:2(8),\nand for NL, !H=T= 6(1). There is a secondary peak around\n!=\u0001\u001915 in the SF and MI phases, and all responses reach a\nquasiplateau at the same height 0 :6(1) at higher frequencies.\nThe error bars on \b SF;MIcome from the spread of collapsed\ncurves, while the ones on \b NLare based on the variance of\nthe analytical continuation results [14].\nTo study the scalar response, we can imagine adding a3\nsmall uniform modulation term to the Hamiltonian\n\u000eH(t) =\u0000\u000eJcos(!t)X\n<ij>by\nibj\u0011\u000eJ\nJK(t);(3)\nwhere\u000eJ=J\u001c1. The imaginary time correlation func-\ntion for kinetic energy, \u001f(\u001c) =hK(\u001c)K(0)i\u0000hKi2, is\nrelated toS(!) through the spectral integral with the\n\fnite-temperature kernel, N(\u001c;!) =e\u0000!\u001c+e\u0000!(1=T\u0000\u001c):\n\u001f(\u001c) =Z+1\n0N(\u001c;!)S(!): (4)\nWe employ the same protocol of collecting and analyzing\ndata as in Ref. [14]. More speci\fcally, in the MC simu-\nlation we collect statistics for the correlation function at\nMatsubara frequencies !n= 2\u0019Tn with integer n\n\u001f(i!n) =hK(\u001c)K(0)ii!n+hKi (5)\nwhich is related to \u001f(\u001c) by a Fourier transform. In the\npath integral representation, \u001f(i!n) has a direct unbi-\nased estimator,jP\nkei!n\u001ckj2, where the sum runs over\nall hopping transitions in a given con\fguration, i.e. there\nis no need to add term (3) to the Hamiltonian explicitly.\nOnce\u001f(\u001c) is recovered from \u001f(i!n), the analytical con-\ntinuation methods described in Ref. [14] are applied to\nextract the spectral function S(!). A discussion on the\nreproducibility of the analytically continued results for\nthis type of problem can also be found in Ref. [14].\nWe consider system sizes signi\fcantly larger than the\ncorrelation length by a factor of at least 4 to ensure that\nour results are e\u000bectively in the thermodynamic limit.\nFurthermore, for the SF and MI phases, we set the tem-\nperatureT= 1=\fto be much smaller than the charac-\nteristic Higgs energy, so that no details in the relevant\nenergy part of the spectral function are missed.\nWe consider two paths in the SF phase to approach the\nQCP: by increasing the interaction U!Ucat unity \fll-\ning factorn= 1 (trajectory i perpendicular to the phase\nboundary in Fig 1), and by increasing \u0016!\u0016cwhile keep-\ningU=Ucconstant (trajectory ii tangential to the phase\nboundary in Fig 1). We start with trajectory i by consid-\nering three parameter sets for ( jgj;L;\f ): (0:2424;20;10),\n(0:0924;40;20), and (0 :0462;80;40). The prime data in\nimaginary time domain are shown in Fig. 3 using scaled\nvariables to demonstrate collapse of \u001f(\u001c) curves at large\ntimes. Analytically continued results are shown in the in-\nset of Fig. 4. After rescaling results according to Eq. (1),\nwe observe data collapse shown in the main panel of\nFig.4. This de\fnes the universal spectral function in the\nsuper\ruid phase \b SF.\nWhen approaching the QCP along trajectory ii,\nwith (jg\u0016j;L;\f ) = (0:40;25;15), (0:30;30;15), and\n(0:20;40;20) we observe a similar data collapse and arrive\nat the same universal function \b SF; see Fig. 5. The \f-\nnal match is possible only when the characteristic energy\nscale \u0001(g\u0016) =C\u0001(g(g\u0016)) involves a factor of C= 1:2.\n 0.1 1 10 100\n 0 0.2 0.4 0.6 0.8  1 1.2 1.4[∆(-g)/J]2/ν−4χSF(τ)/J2\n∆(-g)τ|g|=0.2424\n|g|=0.0924\n|g|=0.0462FIG. 3. (Color online) Collapse of correlation functions in\nimaginary time domain for di\u000berent values of Ualong trajec-\ntory i in the SF phase, labeled by the detuning g= (U\u0000Uc)=J.\n 0 0.5 1 1.5 2 2.5 3\n 0 5 10 15 20 25 30 35 40[∆(-g)/J]2/ν−3SSF(ω)/J\nω/∆(-g)|g|=0.2424\n|g|=0.0924\n|g|=0.0462\n 0.5 1 1.5 2\n 2  4  6  8SSF(ω)/J\nω/J\nFIG. 4. (Color online) Collapse of spectral functions for dif-\nferent values of Ualong trajectory i in the SF phase, labeled\nby the detuning g= (U\u0000Uc)=J. Inset: Original data for\nSSF(!).\n 0 0.5 1 1.5 2 2.5\n 0 5 10 15 20 25 30 35 40[(∆(gµ)/J]2/ν−3SSF(ω)/J\nω/∆(gµ)|gµ|=0.40\n|gµ|=0.30\n|gµ|=0.20\n 0.5 1 1.5\n 2  4  6  8SSF(ω)/J\nω/J\nFIG. 5. (Color online) Collapse of spectral functions for dif-\nferent\u0016along trajectory ii in the SF phase, labeled by the\ndetuningg\u0016= (\u0016\u0000\u0016c)=J. Inset: original data for SSF(!).4\nThe universal spectral function \b SFhas three distinct\nfeatures: a) A pronounced peak at !H=\u0001\u00193:3, which\nis associated with the Higgs resonance. Since the peak's\nwidth\r=\u0001\u00191 is comparable to its energy, the Higgs\nmode is strongly damped. It behaves as a well-de\fned\nparticle only in a moving reference frame; b) A mini-\nmum and another broad maximum between !=\u00012[5;25]\nwhich may originate from multi-Higgs excitations [14];\nc) the onset of the quantum critical quasiplateau, in\nagreement with the scaling hypothesis (1), starting at\n!=\u0001\u001925. These features are captured by an approxi-\nmate analytic expression with normalized \u001f2\u00181,\n\bSF(x) =0:65x3\n35 +x2=\u0017\u0014\n1 +7 sin(0:55x)\n1 + 0:02x3\u0015\n(6)\nWe only claim that a plateau is consistent with our imag-\ninary time data and emerges from the analytic continu-\nation procedure which seeks smooth spectral functions;\ni.e.,other analytic continuation methods may produce an\noscillating behavior in the same frequency range within\nthe error bar in Fig. 2\nIn the MI phase we approach the QCP along trajec-\ntory iii in Fig 1. The scaling hypothesis for the spectral\nfunction has a similar structure to the one in Eq. (1),\nSMI(!)/\u00013\u00002=\u0017\bMI(!\n\u0001): (7)\nThe low-energy behavior of \b MIstarts with the thresh-\nold singularity at the particle-hole gap value, \b MI(x)\u0019\n1=log2(4=(x\u00002))\u0012(x\u00002), see Ref. [16]. At high frequen-\ncies \b MI(x\u001d1) has to approach the universal quantum\ncritical quasiplateau (same as in the SF phase). Our\nresults for the spectral functions at g= 0:2576 (with\nL= 20;\f= 10) and g= 0:1276 (with L= 40;\f= 20)\nare presented in Fig.6. The universal scaling spectral\nfunction shows an energy gap (this is also fully pro-\nnounced in the imaginary time data). The left-hand side\nof the \frst peak is much steeper than in the SF phase, in\nagreement with the theoretical prediction for the thresh-\nold singularity.\nThe universal spectral function in the MI is remarkably\nsimilar to its SF counterpart featuring a sharp resonance\npeak. (Since MI and SF are separated by a critical line\ntheir scaling functions \b MIand \b SFremain fundamen-\ntally di\u000berent at energies smaller than !H). This obser-\nvation is rather counterintuitive given that the super\ruid\norder parameter is zero and raises a number of theoretical\nquestions regarding the nature and properties of collec-\ntive excitations in the MI phase at \fnite energies. In\nparticular, can it be linked to the established picture of\nrenormalized free-energy functional for the order param-\neter \feld [17] at distances under the correlation length?\nIf \fnite energy excitations probe system correlations\npredominantly in a \fnite space-time volume, one would\nexpect that some resonant feature may survive even in\n 0 0.5 1 1.5 2 2.5\n 0 4 8 12 16 20 24 28 32[∆(g)/J]2/ν−3SMI(ω)/J\nω/∆(g)g=0.2576\ng=0.1276\n 0.5 1 1.5\n 2  4  6  8SMI(ω)/J\nω/JFIG. 6. (Color online) Collapse of the spectral functions for\ndi\u000berentUalong trajectory iii in the MI phase, labeled by the\ndetuningg= (U\u0000Uc)=J. Inset: Original data for SMI(!).\nthe NL phase at su\u000eciently low, but \fnite tempera-\ntureT < J (atg=g\u0016= 0, the super\ruid transi-\ntion temperature is zero) In this quantum critical region,\ntemperature determines the characteristic energy scale,\nthusSNL(!)/T3\u00002=\u0017\bNL(!=T), and all excitations are\nstrongly damped. Simulations performed at T=J = 0:5\non the trajectory iv in the inset of Fig. 1 indeed \fnd a\npeak at low energies before the critical quasiplateau, see\nFig. 2, but it is much less pronounced and the oscillatory\ncomponent (second peak) is lost. Unfortunately, numer-\nical complexity does not allow us to verify the scaling\nlaw directly by collapsing simulations at lower tempera-\ntures and bigger system sizes. Our case for universality\nof \b NL(x) is thus much weaker and rests solely on the\ntheoretical consideration that the plateau (at the same\nvalue as in the SF and MI phases) separates universal\nphysics from model speci\fc behavior.\nIn conclusion, we have constructed the universal spec-\ntral functions \b for the kinetic energy correlation func-\ntion for all three phases in the vicinity of the interaction\ndriven QCP of the 2D Bose-Hubbard model. Although\nthe nature of excitations in these phases is fundamen-\ntally di\u000berent at low temperature, their \b functions all\nfeature a resonance peak which in the SF and MI phases\nis followed by a broad second peak and evolve then to a\nquasiplatform at higher energy in agreement with scaling\npredictions. In the SF phase, the \frst peak is interpreted\nas a damped Higgs mode. In the MI and NL phase, the\nexistence of a resonance is unexpected and requires fur-\nther theoretical understanding of amplitude oscillations\nat mesoscopic length scales. Experimental veri\fcation\nwith cold gases requires \ratter traps and lower temper-\natures and is accessible within current technology. It\nwould signify a new hallmark, going beyond the previous\nstudies of criticality near Gaussian \fxed points.\nWe wish to thank I. Bloch, M. Endres, D. Podol-\nsky, and B. V. Svistunov for valuable discussions. This5\nwork was supported by the National Science Foundation\nGrant No. PHY-1005543, by a grant from the Army Re-\nsearch O\u000ece with funding from DARPA, and partially\nby NNSFC Grant No. 11275185, CAS, and NKBRSFC\nGrant No. 2011CB921300.\nNote added { During the \fnal stage of this work, the\nauthors of Ref. [24] shared with us their results for the\nSF phase based on MC simulations of a classical model\nbelonging to the same universality class. We agree on\nthe existence and the position of the Higgs resonance.\n\u0003yjdeng@ustc.edu.cn\nylode.pollet@physik.uni-muenchen.de\nzprokofev@physics.umass.edu\n[1] Ch. R uegg, B. Normand, M. Matsumoto, A. Furrer,\nD. F. McMorrow, K. W. Kr amer, H. -U. G udel, S. N.\nGvasaliya, H. Mutka, and M. Boehm, Phys. Rev. Lett.\n100, 205701 (2008).\n[2] U. Bissbort, S. G otze, Y. Li, J. Heinze, J. S. Krauser, M.\nWeinberg, C. Becker, K. Sengstock, and W. Hofstetter,\nPhys. Rev. Lett. 106, 205303 (2011).\n[3] A. V. Chubukov, S. Sachdev, and J. Ye, Phys. Rev. B\n49, 11919 (1994).\n[4] S. Sachdev, Phys. Rev. B 59, 14054 (1999).\n[5] W. Zwerger, Phys. Rev. Lett. 92, 027203 (2004).\n[6] S. Sachdev, Quantum Phase Transitions , 2nd ed. (Cam-\nbridge University Press, Cambridge, 2011).\n[7] D. Podolsky, A. Auerbach, and D. P. Arovas, Phys. Rev.\nB84, 174522 (2011).\n[8] S. D. Huber, E. Altman, H. P. B uchler, and G. Blatter,Phys. Rev. B 75, 085106 (2007).\n[9] S. D. Huber, B. Theiler, E. Altman, and G. Blatter, Phys.\nRev. Lett. 100, 050404 (2008).\n[10] E. Burovski, J. Machta, N.V. Prokof'ev, and B.V. Svis-\ntunov, Phys. Rev. B 74132502 (2006).\n[11] M. Campostrini, M. Hasenbusch, A. Pelissetto, and E.\nVicari, Phys. Rev. B 74, 144506 (2006).\n[12] M. Endres, T. Fukuhara, D. Pekker, M. Cheneau, P.\nSchau\f, C. Gross, E. Demler, S. Kuhr, and I. Bloch,\nNature 487, 454-458 (2012).\n[13] L. Pollet, Rep. Prog. Phys. 75, 094501 (2012).\n[14] L. Pollet and N. Prokof'ev, Phys. Rev. Lett. 109, 010401\n(2012).\n[15] M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D.\nS. Fisher, Phys. Rev. B 40, 546 (1989).\n[16] D. Podolsky and S. Sachdev, Phys. Rev. B 86, 054508\n(2012).\n[17] J. Berges, N. Tetradis, and C. Wetterich, Phys. Rept.\n363, 223 (2002).\n[18] N. V. Prokof'ev, B. V. Svistunov, and I. S. Tupitsyn,\nPhys. Lett. A, 238, 253 (1998);\n[19] N. V. Prokof'ev, B. V. Svistunov, and I. S. Tupitsyn,\nSov. Phys. - JETP 87, 310 (1998).\n[20] L. Pollet, K. Van Houcke, and S. Rombouts, Comp. Phys.\n225, 2249 (2007).\n[21] N. Elstner, and H. Monien, Phys. Rev. B 59, 12184\n(1999).\n[22] B. Capogrosso-Sansone, S. G. S oyler, N. V. Prokof'ev,\nand B. V. Svistunov, Phys. Rev. A 77, 015602 (2008).\n[23] S. G. S oyler, M. Kiselev, N. V. Prokof'ev, and B.V. Svis-\ntunov, Phys. Rev Lett. 107, 185301, (2011).\n[24] S. Gazit, D. Podolsky, and A. Auerbach, Phys. Rev. Lett.\n110, 140401 (2013)."    },    {        "title": "0911.0867v2.Conformally_flat_Lorentz_manifolds_and_Fefferman_Lorentz_metrics.pdf",        "content": "arXiv:0911.0867v2  [math.DG]  24 Nov 2010CONFORMALLY LORENTZ PARABOLIC STRUCTURE\nAND FEFFERMAN LORENTZ METRICS\nYOSHINOBU KAMISHIMA\nAbstract. We study conformal Fefferman-Lorentz manifolds of\n(2n+2)-dimension. In order to do so, we introduce Lorentz para-\nbolic structure on (m+2)-dimensional manifolds as a G-structure.\nBy using causal conformal vector fields preserving that structu re,\nwe shall establish two theorems on compact Fefferman-Lorentz\nmanifolds: One is the coincidence of vanishing curvature between\nWeyl conformal curvature tensor ofFeffermanmetricson aLorentz\nmanifold S1×NandChern-Moser curvature tensor on a strictly\npseudoconvex CR-manifold N. Another is the analogue of the\nconformal rigidity theorem of Obata and Ferrand to the compact\nFefferman-Lorentz manifolds admitting noncompact closed causal\nconformal transformations.\n1.Introduction\nA pseudo-Riemannian metric gof signature ( m+ 1,1) is called a\nLorentz metric on an m+2-dimensional smooth manifold. An m+2-\ndimensional Lorentz manifold Mis a smooth manifold equipped with a\nLorentz metric. Two Lorentz metrics g,g′are conformal if there exists\na positive function uonMsuch that g′=u·g. The equivalence class\n[g] ofgis a conformal class and ( M,[g]) is called a conformal Lorentz\nmanifold. Ontheotherhand, givena(2 n+1)-dimensionalstrictlypseu-\ndoconvex CR-manifold N, C. Fefferman constructed a Lorentz metric\non the product S1×Nwhich has the following properties (cf .[21]):\n•The conformal class of this metric is a CR-invariant.\n•S1acts aslightlikeisometries.\nItisinteresting toknowhowtheFefferman-Lorentzmetricof S1×Nin-\nteracts on the CR-structure of N. Compare [21] and general references\ntherein for the relation between Fefferman metrics and the Cartan con-\nnection.\nDate: July 22, 2021.\n2000Mathematics Subject Classification. 53C55, 57S25, 51M10.\nKey words and phrases. Conformal structure, Lorentz structure, G-structure,\nConformally flat structure, Weyl curvature tensor, Fefferman m etric, Integrability,\nUniformization, Transformation groups.\n12\nIn this paper, we shall take a different approach to the Fefferman-\nLorentz metrics on (2 n+ 2)-dimensional manifolds by introducing a\nGC-structure called Fefferman-Lorentz parabolic structure. (Compare\nSection 3.1.)\nRecall that conformal Lorentz structure on anm+ 2-dimensional\nmanifold MisanO(m+1,1)×R+-structure[18]. AnintegrableO( m+\n1,1)×R+-structure is conformally flat Lorentz structure onM. (See\nalso [18, p.10].) We focus on parabolic subgroup of O(m+1,1) which is\ndefined as follows. Let PO( m+1,1) be the real hyperbolic group, then\nthe minimal parabolic subgroup is isomorphic to either O( m+1) or the\nsimilarity subgroup Sim( Rm), which lifts to the parabolic subgroup of\nO(m+1,1) as O(m+1)×Z2or Sim(Rm)×Z2respectively.\nForm= 2n, we define a subgroup GCof Sim(R2n)×R+≤O(2n+\n1,1)×R+. If U(n+ 1,1) denotes the unitary Lorentz group which\nembeds into O(2 n+2,2), thenGCis characterized as the intersection\nU(n+ 1,1)∩(Sim(R2n)×R+) where Sim( R2n) is identified with the\nstabilizer PO(2 n+1,1)∞at the point at infinity.\nThen we see that a Fefferman-Lorentz manifold admits a Fefferman-\nLorentz parabolic structure by reinterpreting the proof of [21, (5.17)\nTheorem].\nOne of our results concerns the relation between a CR-manifold N\nand a Fefferman-Lorentz manifold S1×N. We provide a geometric\nproof to the coincidence of vanishing between Weyl conformal curva-\nture tensor andChern-Moser curvature tensor . This result may be\nobtained as a special case of more general calculations by Fefferma n\n[9].\nTheorem A (Theorem 7.4.2) .A Fefferman-Lorentz manifold S1×N\nis conformally flat if and only if Nis a spherical CR-manifold.\nBy our definition, we obtain the following classes of conformally flat\nLorentz parabolic manifolds. We shall give compact examples in each\nclass. (See Section 5.3.)\n•Lorentz flat space forms.\n•Fefferman-Lorentz parabolic manifolds (locally modelled on\n(ˆU(n+1,1),S2n+1,1)).\n•Fefferman-Lorentz manifolds S1×NwhereNis a spherical\nCR-manifold.\nIn the second part, we study the Vague Conjecture [7] that the\nexistence of a global geometric flow determines a compact geomet-\nric manifold uniquely, i.e .isomorphic to the standard model with flat\nG-structure . The celebrated theorem of Obata and Ferrand provides a\nsupporting example for this, i.e .if a closed group Racts conformally on\na compact Riemannian manifold, then it is conformal to the standard\nsphereSn.3\nWe study the analogue of the theorem of Obata and Ferrand to\ncompact Lorentz manifolds. In general, it is not true only by the ex-\nistence of a noncompact conformal closed subgroup R. It is a problem\nwhich conformal group gives rise to an affirmative answer to the com -\npact Lorentz case, i.e .a compact Lorentz manifold is conformal to the\nLorentz model Sn−1,1. C. Frances and K. Melnick gave a sufficient\ncondition on the nilpotent dimension of a nilpotent Lie group acting\nconformally on a compact Lorentz manifold. (Compare [11] more gen -\nerally for pseudo-Riemannian manifolds.)\nWe prove affirmatively the theorem of Obata and Ferrand to the\ncompact Fefferman-Lorentz manifolds under the existence of two di-\nmensional causal abelian Lie groups . LetM=S1×Nbe a compact\nFefferman-Lorentzmanifoldonwhich S1actsasLorentzlightlikeisome-\ntries. Denote by CConf(M,g)(S1) the centralizer of S1in Conf(M,g).\nTheorem B (Theorem 8.1.1) .Suppose that CConf(M,g)(S1)contains a\nclosed noncompact subgroup of dimension 1at least. Then Mis con-\nformally equivalent to the two-fold cover S1×S2n+1of the standard\nLorentz manifold S2n+1,1.\nDimension 2 for the dimension of CConf(M,g)(S1) is rather small rela-\ntivetothenilpotentdimensionin[11]. However thefactthat S1ofMis\nthe group of lightlike isometries with respect to our Fefferman-Lore ntz\nmetric is more geometric. In fact, let N−be a Lorentz hyperbolic 3-\nmanifold PSL(2 ,R)/Γ. Then M=S1×N−admits a conformally flat\nLorentz metric on which S1acts asspacelike isometries. (See Remark\n8.1.1). The metric is not a Fefferman-Lorentz metric but there exist s a\ntwo dimensional noncompact abelian Lie group S1×Racting isometri-\ncally onM. On the other hand, if we note that N−admits a spherical\nCR-structure, then Mdoes admit a Fefferman-Lorentz metric whose\nconformalLorentzgroupiscompact (Lorentzisometrygroup). Inaddi-\ntion, the above group S1×Ris not a conformal group of the underlying\nFefferman-Lorentz metric.\nContents\n1. Introduction 1\n2. Preliminaries 4\n2.1. Cayley-Klein model 4\n2.2. Causality 5\n3. Conformally Lorentz parabolic geometry 5\n3.1. Conformally Lorentz parabolic structure 5\n3.2. Lorentz similarity geometry ( Lsim(Rm+2),Rm+2) 7\n3.3. Fefferman-Lorentz manifold 8\n4. Conformally flat Fefferman-Lorentz manifold 11\n4.1. Confomally flat Fefferman-Lorentz model 114\n4.2. Examples of Fefferman-Lorentz manifolds I 20\n5. Uniformization 22\n5.1. Uniformization of Fefferman-Lorentz parabolic manifolds 22\n5.2. Examples of 4-dimensional Lorentz flat parabolic\nmanifolds II 24\n5.3. Examples of conformally flat Fefferman-Lorentz parabolic\nmanifolds III 25\n6. Conformally flat Fefferman-Lorentz parabolic geometry 27\n6.1. One-parameter subgroups in ˆU(n+1,1) 28\n7. Coincidence of curvature flatness 34\n7.1. Causal vector fields on S2n+1,134\n7.2. Nilpotent group case 34\n7.3. Compact torus case 35\n7.4. Curvature equivalence 38\n8. Application to Obata & Ferrand’s theorem 43\n8.1. Noncompact conformal group actions 43\nReferences 49\n2.Preliminaries\n2.1.Cayley-Klein model. We start with the Cayley-Klein projec-\ntive model for K=R,CorH. (Compare [1],[3],[10],[13],[26],[31] for\ninstance.) There is an equivariant principal bundle:\nK∗→(GL(n+2,K)·K∗,Kn+2−{0})P−→(PGL(n+2,K),KPn+1).\nFix nonnegative integers p,qsuch that n=p+q. The nondegenerate\nHermitian form on Kn+2is defined by\n(2.1.1)B(x,y) = ¯x1y1+···+ ¯xp+1yp+1−¯xp+2yp+2−···−¯xn+2yn+2\nDenote by O( p+1,q+1;K) the subgroup of GL( n+2,K):\n{A∈GL(n+2,K)| B(Ax,Ay) =B(x,y),x,y∈Kn+2}.\nThe group O( p+1,q+1;K) leaves invariant the K-cone inKn+2−{0}:\nV0={x∈Kn+2−{0} | B(x,x) = 0}. (2.1.2)\nPut\nP(V0) =\n\nSp,q≈ Sq×Sp/Z2\nS2p+1,2q≈S2q+1×S2p+1/S1\nS4p+3,4q≈S4q+3×S4p+3/S3(2.1.3)\nLet PO(p+1,q+1;K)denote theimageof O( p+1,q+1;K)inPGL( n+\n2,K). According to K=R,CorH, we have the nondegenerate flat5\ngeometry of signature ( p,q).\n/braceleftigg(PO(p+1,q+1),Sp,q) Conformally flat geometry\n(PU(p+1,q+1),S2p+1,2q) Spherical CRgeometry\n(PSp(p+1,q+1),S4p+3,4q) Flat pseudo-conformal q CRgeometry\n(Compare [13].) In particular, when p=n,q= 0, i.e.positive definite\ncase, wehave theusualhorospherical geometry, i.e .thegeometryonthe\nboundaryofthereal, complexorquaternionichyperbolicspaces. W hen\np=n−1,q= 1, (PO( n,2),Sn−1,1) is said to be the n-dimensional\nconformally flat Lorentz geometry .\n2.2.Causality. Letξbe a vector field on a Lorentz manifold ( M,g).\nWe recall causality of vector fields (cf.[26]).\nDefinition 2.2.1 (Causal vector fields) .Letx∈M.\n(2.2.1)\n\nξis spacelike g(ξx,ξx)>0whenever ξx/\\e}atio\\slash= 0.\nξis lightlike g(ξx,ξx) = 0whenever ξx/\\e}atio\\slash= 0.\nξis timelike g(ξx,ξx)<0whenever ξx/\\e}atio\\slash= 0.\nEach vector ξxis called a causal vector . Suppose that ξis a vector\nfield defined on a domain Ω of M. Ifξx/\\e}atio\\slash= 0 and ξxis a causal vector\nat each point x∈Ω, thenξis said to be a causal vector field on Ω.\n3.Conformally Lorentz parabolic geometry\n3.1.Conformally Lorentz parabolic structure. Let{e1,...,e m}\nbe the standard orthonormal basis of Rm+2with respect to the Lorentz\ninner product B;B(ei,ej) =δij(1≤i,j≤m+1),B(em+2,em+2) =−1.\n(See (2.1.1) for K=R.) Set\nℓ1=e1+em+2/√\n2, ℓm+2=e1−em+2/√\n2.\nPuttingB=/a\\}b∇acketle{t,/a\\}b∇acket∇i}htonRm+2,F={ℓ1,e2,...,e m+1,ℓm+2}is a new basis\nsuch that /a\\}b∇acketle{tℓ1,ℓ1/a\\}b∇acket∇i}ht=/a\\}b∇acketle{tℓm+2,ℓm+2/a\\}b∇acket∇i}ht= 0,/a\\}b∇acketle{tℓ1,ℓm+2/a\\}b∇acket∇i}ht= 1. The symmetric\nmatrixI1\nm+1with respect to this basis Fis described as\n(3.1.1) I1\nm+1=\n00···01\n0 0\n...Im...\n0 0\n10···00\n.\nNote that\nO(m+1,1) ={A∈GL(m+2,R)|AI1\nm+1tA=I1\nm+1}.\nThe similarity subgroup is described in O( m+1,1) as follows:6\n(3.1.2) Sim( Rm) =\n\n\nλ x−|x|2\n2λ\nB−Btx\nλ\n01\nλ\n|λ∈R+, B∈O(m)\nx∈Rm\n\n.\nWe introduce the following subgroup in GL( m+2,R):\n(3.1.3) GR= Sim(Rm)×R+.\nLetP=√u·Q∈GR(Q∈Sim(Rm),√u∈R+). For the Lorentz inner\nproduct:\n/a\\}b∇acketle{tx,y/a\\}b∇acket∇i}ht=xI1\nm+1ty,\n(x= (x1,...,x m+2),y= (y1,...,y m+2)∈Rm+2) as above,\n/a\\}b∇acketle{txP,yP/a\\}b∇acket∇i}ht=u·xQI1\nm+1tQty=u·xI1\nm+1ty=u·/a\\}b∇acketle{tx,y/a\\}b∇acket∇i}ht,\ni.e.aGR-structure defines a conformal class of Lorentz metrics on an\n(m+2)-manifold.\nLetGC≤GL(2n+2,R) be a subgroup defined by\n(3.1.4) GC=\n\n\nu√ux−|x|2\n2 √uB−Btx\n0 1\n|u∈R+,x∈Cn,B∈U(n)\n\n.\nIfPis an element of GCas above, then\nP=√u·Q,\nQ=\n√u x−|x|2\n2√u\nB−1√uBtx\n01√u\n.(3.1.5)\nIt is easy to see that Qis an element of Sim( R2n)≤O(2n+1,1). So\nGCis a subgroup of GRform= 2n. A homomorphism P/ma√sto→Qgives\nrise to an isomorphism of GConto Sim( Cn) =Cn⋊(U(n)⋊ R+) of\nSim(R2n).\nDefinition 3.1.1.\n•AGR-structure on an (m+ 2)-manifold is called conformally\nLorentz parabolic structure. An (m+ 2)-manifold is said to\nbe a conformally Lorentz parabolic manifold if it admits a GR-\nstructure.7\n•AGC-structure on a (2n+ 2)-manifold is Fefferman-Lorentz\nparabolicstructure . In other words, the Fefferman-Lorentz par-\nabolic structure is a reduction of GRtoGC. A Fefferman-\nLorentz parabolic manifold is a (2n+ 2)-dimensional manifold\nequipped with a Fefferman-Lorentz parabolic structure.\nDefinition 3.1.2. LetMbe a Fefferman-Lorentz parabolic manifold.\nThenConfFLP(M)is the group of conformal transformations preserving\nthe Fefferman-Lorentz parabolic structure.\n3.2.Lorentz similarity geometry (Lsim(Rm+2),Rm+2).LetRm+2\nbe the (m+ 2)-dimensional euclidean space equipped with a Lorentz\ninner product (cf .Section 3.1). Form the Lorentz similarity subgroup\nLsim(Rm+2) =Rm+2⋊(O(m+ 1,1)×R+) from the affine group\nAff(Rm+2) =Rm+2⋊GL(m+2,R). If an (m+2)-manifold Mis locally\nmodelled on Rm+2with coordinate changes lying in Lsim(Rm+2), then\nMis said to be a Lorentz similarity manifold.\nProposition 3.2.1. LetMbe an(m+2)-dimensional compact Lorentz\nsimilarity manifold with virtually solvable fundamental g roup. Then M\nis either a Lorentz flat parabolic manifold or finitely covere d by a Hopf\nmanifold Sm+1×S1, or anm+2-torusTm+2.\nProof.Given a compact Lorentz similarity manifold M, there exists a\ndeveloping pair ( ρ,dev) : (π1(M),˜M)→(Lsim(Rm+2),Rm+2). Suppose\nthatπ1(M)isvirtuallysolvable. Let L:Lsim(Rm+2)→O(m+1,1)×R+\nbe the linear holonomy homomorphism. Then a subgroup of finite\nindex in L(ρ(π1(M)) is solvable in O( m+1,1)×R+so it belongs to a\nmaximal amenable subgroup which is either isomorphic to (O( m+1)×\nO(1))×R+or to (Sim( Rm)×Z2)×R+up to conjugate. By Definition\n3.1.1, the latter case implies that M(or two fold-cover) is a Lorentz\nflat parabolic manifold. If a subgroup of finite index in L(ρ(π1(M))\nlies in (O( m+1)×O(1))×R+, thenMis a similarity manifold where\nO(m+ 1)×O(1)≤O(m+ 2). It follows from the result by Fried\nthatMis covered finitely by an m+2-torus Tm+2or a Hopf manifold\nSm+1×S1.\n/square\nThe Lorentz similarity geometry contains Lorentz flat geometry\n(E(m+1,1),Rm+2) where E( m+1,1) =Rm+2⋊O(m+1,1). IfMis an\nm+2-dimensional compact Lorentz flat manifold, then it is known that\nMis geodesically complete and the fundamental group of a compact\ncompleteLorentzflatmanifoldisvirtuallysolvable. Applyingtheabove\nproposition, we have\nCorollary 3.2.1. Anym+2-dimensional compact Lorentz flat mani-\nfold is finitely covered by an m+2-torus or an infrasolvmanifold.8\nProof.The holonomy homomorphism ρ:π1(M)→Rm+2⋊O(m+1,1)\nreduces to a homomorphism: ρ:π1(M)→Rm+2⋊Sim(Rm)×Z2. Put\nΓ =ρ(π1(M)) which is a virtually solvable discrete subgroup. As\nSim(Rm) =Rm⋊(O(m)×R+), a subgroup Γ′of finite index in Γ is\nconjugate to a discrete subgroup of a solvable Lie group G=Rm+2⋊\n(Rm⋊(Tm×R+)). LetM′=ρ−1(Γ′)\\˜Mbe a finite covering of M. As\ndev :˜M→Rm+2is a diffeomorphism, it follows that M′∼=Γ′\\Rm+2=\nΓ′\\G/HwhereH=Rm⋊(Tm×R+).\n/square\n3.3.Fefferman-Lorentz manifold. In [8] Fefferman has shown that\nwhenNis a (2n+1)-dimensional strictly pseudoconvex CR-manifold,\nS1×Nadmits a Lorentz metric gon which S1acts as lightlike isome-\ntries. We recall the construction of the metric from [21]. Let (Ker ω,J)\nbe aCR-structure on Nwith characteristic (Reeb) vector field ξfor\nsome contact form ω. The circle S1generates the vector field Son\nS1×N(extending trivially on N). Note that\n(3.3.1) T(S1×N) =/a\\}b∇acketle{tS/a\\}b∇acket∇i}ht⊕/a\\}b∇acketle{tξ/a\\}b∇acket∇i}ht⊕Kerω.\nLet (Ker ω)⊗C={Y1,...,Y n} ⊕ {¯Y1,...,¯Yn}be the canonical de-\ncomposition for Jfor which we choose such as dω(Yi,¯Yj) =−iδij. As\nusual, letting Xi=Yi+¯Yi/√\n2,Xn+i=i(Yi−¯Yi)/√\n2, it implies that\n(Kerω) ={X1,...,X 2n}such that JXi=Xn+iand\n(3.3.2) dω(JXi,Xj) =δij(i= 1,...,n).\nThis gives a (real) frame {S,ξ,X1,...,X 2n}at a neighborhood of S1×\nN. Letθibe the dual frame to Xi(i= 1,...,2n). From (3.3.2), note\nthat\n(3.3.3) dω(J−,−) =2n/summationdisplay\ni=1θi·θion Kerω.\nLetdtbe a 1-form on S1×Nsuch that\n(3.3.4) dt(S) = 1, dt(V) = 0 (∀V∈TN).\nPutη=ω1∧···∧ωnwhereωα=θα+iθn+α. By Proposition (3.4) of\n[21] there exists a unique real 1-form σonS1×Nsatisfying that\nd(ω∧η) =i(n+2)σ∧ω∧η,\nσ∧dη∧¯η= Tr(dσ)iσ∧ω∧η∧¯η.(3.3.5)\nThe explicit form of σis obtained from [21, (5.1) Theorem] that\n(3.3.6) σ=1\nn+2/parenleftbigg\ndt+iP∗ωα\nα−1\n2(n+1)ρ·P∗ω/parenrightbigg\n.9\nHereP:S1×N→Nisthecanonical projectionand ωβ\nαisaconnection\nform ofωsuch that\ndω=iδαβωα∧ω¯β,\ndωα=ωβ∧ωα\nβ+ω∧τβ.\nThe function ρis the Webster scalar curvature on N. (Since we chose\nhα¯β=δαβ, note that −i\n2hα¯βdhα¯β= 0 in the equation (5.3) of [21, (5.1)\nTheorem].) It follows that\n(3.3.7) σ(S) =1\nn+2.\nDefine a symmetric 2-form\nσ⊙ω=σ·ω+ω·σ.\nExtending θi(S) = 0 and ω(S) = 0, we have a Fefferman - Lorentz\nmetric on S1×Nby\ng(X,Y) =σ(X)·ω(Y)+ω(X)·σ(Y)+dω(JXh,Yh)\n=σ⊙ω(X,Y)+2n/summationdisplay\ni=1θi·θi(X,Y).(3.3.8)\nHereXhstands for the horizontal part of X, i.e.Xh∈Kerω. By\n(3.3.7) we have that\n(3.3.9) g(ξ,S) =1\nn+2.\nSinceg(S,S) = 0,gbecomes a Lorentz metric on S1×N. In particular\nS1acts as lightlike isometries of g.\nThe following result has been achieved by Lee [21]. We shall give an\nelementary proof of invariance from the viewpoint of G-structure.\nTheorem 3.3.1. A strictly pseudoconvex CR-structure on Ngives a\nconformal class of Lorentz metrics on S1×N. Indeed S1×Nadmits\na Fefferman-Lorentz parabolic structure (GC-structure ).\nProof.Suppose that (Ker ω′,J) represents the same CR-structure on\nN. Then it follows that ω′=u·ωfor some positive function uon\nN. Let{S′,ξ′,X′\n1,...,X′\n2n}be another frame on the neighborhood of\nS1×Nforω′. SinceS′generates the same S1, note that\n(3.3.10) S=S′.\nThere exist xi∈R(i= 1,...,2n) for which the characteristic vector\nfieldξ′is described as\n(3.3.11) ξ=u·ξ′+x1√uX′\n1+···+x2n√uX′\n2n.10\nAsu·dω=dω′anddω(J−,J−) =dω(−,−) on Ker ω, there exists a\nB= (bij)∈U(n) such that\n(3.3.12) Xi=√u/summationdisplay\nkbikX′\nk.\nTwo frames {S,ξ,X1,...,X 2n},{S′,ξ′,X′\n1,...,X′\n2n}are uniquely de-\ntermined each other by the equations (3.3.10), (3.3.11), (3.3.12).\nIt suffices to prove that another Lorentz metric g′is conformal to g:\ng′=σ′⊙ω′+2n/summationdisplay\ni=1θ′i·θ′i.\nThe equations (3.3.10), (3.3.11), (3.3.12) determine the relation be-\ntween the dual frames {ω,θ1,...,θ2n,σ},{ω′,θ′1,...,θ′2n,σ′}.\nω′=u·ω,\nθ′i=√u/summationdisplay\njbjiθj+√uxi·ω,(3.3.13)\nMoreover, by the uniqueness property of σfrom (3.3.5), σ′is trans-\nformed into the following form (cf .[21, (5.16) Proposition]):\n(3.3.14) σ′=σ−/summationdisplay\ni,jbjixiθj−|x|2\n2ω.\nUsing (3.1.5), the above equations show that\n(ω′,θ′1,...,θ′2n,σ′) = (ω,θ1,...,θ2n,σ)\nu√ux−|x|2\n2 √uB−Btx\n0 1\n\n= (ω,θ1,...,θ2n,σ)√u·Q.(3.3.15)\nAsB∈U(n) andx∈Cnbecause the basis {X1,...,X 2n}invari-\nant under J, note that√u·Q∈GC(i.e.Q∈SimC(n)). Hence the\nCR-structure defines a Fefferman-Lorentz parabolic structure on M\n(cf.Definition 3.1.1). Moreover, a calculation shows\ng′=σ′⊙ω′+dω′(J−,−) =σ′·ω′+ω′·σ′+/summationdisplay\nθ′i·θ′i\n= (ω′,θ′1,...,θ′2n,σ′)I1\n2n+1t(ω′,θ′1,...,θ′2n,σ′)\n= (√u)2(ω,θ1,...,θ2n,σ)QI1\n2n+1tQt(ω,θ1,...,θ2n,σ)\n=u·(ω,θ1,...,θ2n,σ)I1\n2n+1t(ω,θ1,...,θ2n,σ)\n=u(σ⊙ω+/summationdisplay\nθi·θi) =u·g,(3.3.16)11\nHence the CR-structure determines a conformal class of Fefferman-\nLorentz metric g.\n/square\n4.Conformally flat Fefferman-Lorentz manifold\n4.1.Confomally flat Fefferman-Lorentz model. Let\nV0={x= (x1,...,x 2n+4)∈R2n+4−{0} | B(x,x) = 0}\nbe as in (2.1.2) for K=R. In this case, when the Hermitian bilinear\nform is defined by\n(4.1.1) /a\\}b∇acketle{tz,w/a\\}b∇acket∇i}ht= ¯z1w1+···+ ¯zn+1wn+1−¯zn+2wn+2onCn+2,\nV0is identified with\n{z= (z1,...,z n+2)∈Cn+2−{0} | /a\\}b∇acketle{tz,z/a\\}b∇acket∇i}ht= 0}.\nLet U(n+1,1) be the unitary Lorentz group with the center S1. Obvi-\nouslythetwo-foldcoverof S2n+1,1iscontainedin V0, i.e.S1×S2n+1⊂V0\nbut not invariant under U( n+ 1,1). Consider the commutative dia-\ngram.\n(4.1.2)(Z2,R∗) (Z2,R∗)\n/arrowbt/arrowbt\n(S1,C∗)− −−− → Cn+2−{0}PC− −−− → CPn+1\n||/uniontext /uniontext\n(S1,C∗)− −−− →S1×S2n+1⊂V0PC− −−− → S2n+1=PC(V0)\n/arrowbt PR/arrowbt ||\nS1/Z2=S1− −−− →S2n+1,1=PR(V0)P− ���−− →S2n+1=P(S2n+1,1)\n||/intersectiontext /intersectiontext\nS1− −−− → RP2n+3 P− −−− → CPn+1.\nPut\nˆU(n+1,1) = U(n+1,1)/Z2\nwhereZ2is a cyclic group of order two in S1. The natural embedding\nU(n+1,1)→O(2n+2,2) induces an embedding of Lie groups:\nˆU(n+1,1)→PO(2n+2,2).\nLet\nω0=−in+1/summationdisplay\nj=1¯zjdzj, ξ=n+1/summationdisplay\nj=1(xjd\ndyj−yjd\ndxj) (4.1.3)12\nbe the standard contact form on S2n+1with the characteristic vector\nfieldξ. Note that ω0(ξ) =n+1/summationdisplay\nj=1|zj|2= 1 onS2n+1. As in Section 4.2, ω0\nis the connection form on the principal bundle : S1→S2n+1π−→CPn.\nThe spherical CR-structure (Ker ω0,J0) defines a Lorentz metric on\nS1×S2n+1:\n(4.1.4) g0(X,Y) =σ0⊙P∗\nCω0(X,Y)+dω0(J0PC∗X,PC∗Y)\nwhereσ0is obtained from (3.3.6) (cf .(4.2.2)).\nProposition 4.1.1. The group U(n+1,1)acts conformally on S1×\nS2n+1with respect to g0. Especially, so does ˆU(n+1,1)on(S2n+1,1,ˆg0).\nProof.Let U(n+1,1) = (U(n+1)×U(1))·(N ×R+) be the Iwasawa\ndecomposition in which there is the equivariant projection:\n(4.1.5) (U(n+1,1),V0)(P,PC)−−−→(PU(n+1,1),S2n+1).\nIftθ=eiθ∈ZU(n+1,1) which is the center S1of U(n+1,1), then by\nthe form (3.3.6) it follows that\n(4.1.6) t∗\nθdt=dt, t∗\nθσ0=σ0.\nLet U(n+ 1) be the maximal compact subgroup of PU( n+ 1,1). If\nγ=P(˜γ)∈U(n+ 1), then γ∗ω0=ω0from (4.1.3). Then ˜ γ∗g0=g0,\ni.e.˜γacts as an isometry of S1×S2n+1.\nSuppose that γ=P(˜γ)∈ N ×R+whereNis the Heisenberg Lie\ngroup such that N ∪{∞} =S2n+1. Recall from Section 2 of [14] that\nif (t,(z1,···,zn)) is the coordinate of N=R×Cn, then the contact\nformωNonNis described as:\n(4.1.7) ωN=dt+n/summationdisplay\nj=1(xjdyj−yjdxj) =dt+Im/a\\}b∇acketle{tz,dz/a\\}b∇acket∇i}ht.\n(Here/a\\}b∇acketle{tz,w/a\\}b∇acket∇i}ht=n/summationdisplay\ni=1¯ziwiand Imxis the imaginary part of x.) An\nelementg= ((a,z),λ·A)∈ N⋊(U(n)×R+) acts on ( t,w)∈ Nas\n(4.1.8) g·(t,w) = (a+λ2·t−Im/a\\}b∇acketle{tz,λ·A·w/a\\}b∇acket∇i}ht,z+λ·A·w).\nIn particular, when γ= (a,z)∈ N, thenγ∗ωN=ωN, whileR+=/a\\}b∇acketle{tγθ/a\\}b∇acket∇i}ht\nsatisfies that γθ(t,z) = (e2θ·t,eθ·z) onNso\nγθ∗ωN=d(e2θ·t)+Im/a\\}b∇acketle{teθ·z,d(eθ·z)/a\\}b∇acket∇i}ht=e2θ·ωN.13\nAs (ω0,J) and (ωN,J) define the same spherical CR-structure on\nN, there exists a smooth function uonNsuch that ωN=u·ω0. Let\ngN=σN⊙P∗ωN+dωN(J0P∗−,P∗−)\nbetheLorentzmetricon S1×NwhereP:S1×N→N istheprojection.\nThen it follows from Theorem 3.3.1 that\n(4.1.9) gN=u·g0.\nAs above, it is easy to check that if P(˜γ) =γ∈ N, thenγ∗gN=\ngNand ifP(γθ) =γθ∈R+, thenγ∗\nθgN=e2θ·gN. (Note that the\nequation γθ∗ωN=e2θ·ωNimplies that γθ∗dωN=e2θ·dωNbecause\ne2θis constant.) As a consequence, if γ=P(˜γ)∈ N ×R+, then there\nexists a positive constant τsuch that ˜ γ∗gN=τ·gN. Letting a positive\nfunction v=γ∗u−1·τ·uonN, it is easy to see that\n˜γ∗g0=v·g0onS1×N.\nSince this is true on a neighborhood at any point in S1×S2n+1, ˜γ\nacts conformally on S1×S2n+1with respect to g0. For either ˜ γ∈\nU(n+1)×U(1) or ˜γ∈ N×R+, the above observation shows that every\nelement of U( n+1,1) acts as conformal transformation on S1×S2n+1\nwith respect to the Lorentz metric g0.\n/square\nNote from (4.1.9) that gN=u·g0. The Weyl conformal curvature\ntensor satisfies that W(g0) =W(gN) onS1× N. In order to prove\nthat the Fefferman-Lorentz metric g0is a conformally flat metric, we\ncalculatethe Weyl conformal curvature tensor of gNonS1×Ndirectly.\nIn view of the contact form ωNonN(cf.(4.1.7)), dωN(J0−,−) is\nthe euclidean metric ˆ gC= 2n/summationdisplay\nj=1|dzj|2onCn. AsωNis the connection\nform of the principal bundle: R → Nπ−→Cn, it follows from (3.3.6)\nthat\nσN=1\nn+2/parenleftbigg\ndt+iP∗π∗ϕα\nα−1\n2(n+1)π∗ρ·P∗ωN/parenrightbigg\n.\nSince (Cn,ˆgC) is flat, it follows that ϕα\nα= 0,ρ=π∗s= 0, which shows\n(4.1.10) σN=1\nn+2dt.\nThe metric gNreduces to the following:\ngN=1\nn+2dt⊙P∗ωN+P∗ˆgC.\nIt is easy to check that the following are equivalent:14\n(i)X∈C⊥whereC=/a\\}b∇acketle{tS,ξ/a\\}b∇acket∇i}htinduced by S1×R.\n(ii)gN(X,S) = 0 and gN(X,ξ) = 0.\n(iii)X∈P∗KerωN,dt(X) = 0.\nAs a consequence, C⊥= KerωN. Putting C=S1× R, there is a\npseudo-Riemannian submersion:\n(4.1.11) C − −− → (S1×N,gN)π− −− →(Cn,ˆgC).\nThe Riemannian curvature tensor ˆRon the flat space Cnis zero,\nˆgC(ˆR(π∗X,π∗Y)π∗Z,π∗W) = 0.\nIfX,Y∈P∗KerωN, then [X,Y]V∈ /a\\}b∇acketle{tξ/a\\}b∇acket∇i}htwhereξisthe characteristic\nvector field for ωN. (HereXVstands for the fiber component of the\nvectorX.) Sincedt(ξ) = (n+2)σN(ξ) = 0 from (4.1.10),\ngN(ξ,ξ) = 0,i.e. ξis lightlike .\nWe apply the O’Neill’s formula (cf .[5, (3.30)] for example) to the\npseudo-Riemannian submersion of (4.1.11):\ngN(R(X,Y)Z,W) = ˆgC(ˆR(π∗X,π∗Y)π∗Z,π∗W)\n+1\n4gN([X,Z]V,[Y,W]V)−1\n4gN([Y,Z]V,[X,W]V)\n+1\n2gN([Z,W]V,[X,Y]V]).(4.1.12)\nThis shows that\nLemma 4.1.1.\nRXYZW=gN(R(X,Y)Z,W) = 0 (∀X,Y,Z,W ∈P∗KerωN).\nLemma 4.1.2.\nRξABC= 0 (∀A,B,C∈T(S1×N)).\nProof.Putω=ωN,g=gNand Ker ω=P∗KerωN. Let∇be a\ncovariant derivative for gonS1×N;\n2g(∇XY,Z) =Xg(Y,Z)+Yg(X,Z)−Zg(X,Y)\n+g([X,Y],Z)+g([Z,X],Y)+g([Z,Y],X).\nAs KerωisC-invariant, we note the following.\n(4.1.13) [ X,ξ] = [X,S] = 0 (∀X∈Kerω).\nPutσ=σN=1\nn+2dt. By(iii),\n(4.1.14) σ([X,Y]) = 0 (∀X,Y∈Kerω).\nWe may choose X,Y,Zto be orthonormal vector fields in Ker ω. Then\n2g(∇Xξ,Z) =g([Z,X],ξ) =σ([Z,X]) = 0 (∀X,Z∈Kerω).15\nSimilarly from (4.1.13),\n2g(∇Xξ,S) =Xg(ξ,S) =X(1\nn+2) = 0,\n2g(∇Xξ,ξ) = 0.\nThis implies that\n(4.1.15) ∇Xξ= 0.\nIt follows similarly that\n(4.1.16) 2 g(∇Sξ,Z) = 0,2g(∇Sξ,S) = 0,2g(∇Sξ,ξ) = 0.\nThis shows that\n(4.1.17) ∇Sξ= 0.\nIt is easy to see that ∇SS=∇ξξ= 0, i.e.the orbits of S1andRare\ngeodesics. From these, we obtain that\n(4.1.18) ∇ξA= 0 (∀A∈T(S1×N)).\nThis implies that RξABC= 0 (∀A,B,C∈T(S1×N)).\n/square\nWe set formally\n(4.1.19) JS= 0, Jξ= 0\nso thatJis defined on T(S1×N) ={S,ξ}⊕Kerω.\nLemma 4.1.3.\n∇SA=−1\nn+2JA.(∀A∈T(S1×N)).\nProof.For the vector field S, we see that\n(4.1.20) 2 g(∇XS,ξ) = 0,2g(∇XS,S) = 0.\nAs we assumed that g(X,Z) = 0 and g(X,X) =g(Z,Z) = 1, calculate\n2g(∇XS,Z) =ω([Z,X])·σ(S) =1\nn+2·ω([Z,X])\n=−2\nn+2·dω(Z,X) =−2\nn+2·ˆgC(Z,ˆJX)\n=−2\nn+2·g(JX,Z).\nUsing (4.1.20), we obtain that\n(4.1.21) ∇XS=−1\nn+2JX.16\nAs [S,X] = 0, note that\n∇SX=−1\nn+2JX.\nSince∇Sξ=∇SS= 0 from (4.1.17), we have that ∇SA=−1\nn+2JA\n(∀A∈T(S1×N)).\n/square\nLemma 4.1.4. The remaining curvature tensor RABCDonS1× N\nbecomes as follows.\n(1)RSXSY=−1\n(n+2)2·gN(X,Y) (∀X,Y∈P∗KerωN).\n(2)RSXYZ= 0 (∀X,Y,Z∈P∗KerωN).\nProof.Using Lemma 4.1.3,\nR(X,S)S=∇X∇S(S)−∇S∇X(S)−∇[S,X]S\n=−∇S(−1\nn+2JX) =1\n(n+2)2X.\nIt follows that\nRSXSY=−RXSSY\n=−g(R(X,S)S,Y) =−1\n(n+2)2g(X,Y)\nAsX,Yare orthonormal and σ([X,Y]) = 0 by (4.1.14), it follows that\n2g(∇XY,ξ) =g([X,Y],ξ) = 0,\nso there exists a function a(X,Y) such that\n∇XY≡a(X,Y)ξmodKer ω.\nIn particular it follows that\n(4.1.22) ∇XJY−J∇XY≡a(X,JY)ξmodKer ω.\nLetˆ∇be a covariant derivative for the K¨ ahler metric ˆ gConCnas\nbefore. Recall from [5, (3.23) p.67] that\n(4.1.23) g(∇XY,Z) = ˆgC(ˆ∇π∗Xπ∗(Y),π∗Z) (∀X,Y,Z∈Kerω).17\nIf we note that the complex structure ˆJis parallel with respect to ˆ gC,\ni.e.ˆ∇ˆJ=ˆJˆ∇, then\ng(∇XJY,Z) = ˆgC(ˆ∇π∗Xπ∗(JY),π∗Z)\n= ˆgC(ˆ∇π∗XˆJπ∗(Y),π∗Z)\n= ˆgC(ˆJˆ∇π∗Xπ∗(Y),π∗Z)\n=−ˆgC(ˆ∇π∗Xπ∗(Y),π∗JZ)\n=−g(∇XY,JZ) =g(J∇XY,Z).\n(4.1.22) implies that\n(4.1.24) ∇XJY−J∇XY=a(X,JY)ξ.\nAs [X,S] = 0, Lemma 4.1.3 shows that\nR(X,S)Y=∇X∇SY−∇S∇XY\n=−1\nn+2(∇XJY−J∇XY)\n=−1\nn+2a(X,JY)ξ.\nBy (1) of Lemma 4.1.4, it follows that\ng(R(X,S)Y,S) =−1\n(n+2)2a(X,JY)\n=RXSYS=−1\n(n+2)2g(X,Y).\nHencea(X,JY) =g(X,Y) so that we obtain\n(4.1.25) R(X,S)Y=−1\nn+2g(X,Y)ξ.\nIt follows that\nRSXYZ=−RXSYZ=−g(R(X,S)Y,Z) = 0.\n/square\nProposition 4.1.2. Let(S1×N,gN)be a Fefferman-Lorentz nilman-\nifold of dimension 2n+2. Then the following hold.\n(1)The scalar curvature function S= 0.\n(2)The Ricci tensor has the following form.\n(i)RYZ= 0 (∀Y,Z∈Kerω).\n(ii)RξA= 0 (∀A∈T(S1×N)).\n(iii)RSY= 0 (∀Y∈Kerω).\n(iv)RSS=−2n/(n+2)2.18\nProof.Note that g(ξ,ξ) =g(S,S) = 0. Then\nS=RABCDgACgBD\n=RXYZWgXZgYW+RξBCDgξCgBD+RSBCDgSCgBD\nwhereX,Y,Z,W ∈KerωandB,C,D∈T(S1× N). The first term\nis zero;RXYZW= 0 by Lemma 4.1.1. The second term RξBCD= 0 by\nLemma 4.1.2. According to whether gSC= 0 orgSξ=n+2, the third\nterm becomes RSBCDgSCgBD=RSBξDgSξgBD= 0 because RSBξD= 0\nby Lemma 4.1.2 again. Hence the scalar curvature S= 0.\nThe Ricci tensor satisfies that\nRSY=RASBYgAB\n=RXSZYgXZ+RξSBYgξB+RASξYgAξ+RASSYgAS.\nThenRXSZY=−RSXZY= 0 by (2) of Lemma 4.1.4. RξSBY=\n0,RASξY=RξYAS= 0 by Lemma 4.1.2. When gξS=n+2,RξSSY= 0\nas above. According to whether gAS= 0 orgξS=n+2, the third term\nRASSYgAS= 0. So RSY= 0, (iii) follows. Similarly (i), (ii) follow by\ncalculations:\nRYZ=RAYBZgAB=RξYBZgξB+RSYBZgSB\n=RSYξZgSξ= 0.\nRξA=RBξCAgBC= 0 (∀A∈T(S1×N)).\nAs we chose g(X,X) = 1, (1) of Lemma 4.1.4 implies that\nRSS=RASBSgAB\n=RXSXSgXX+RXSξSgXξ+RξSBSgξB+RSSBSgSB\n= 2n·RXSXS=−2n·1\n(n+2)2.\nThis shows (iv).\n/square\nLetWABCDbe the Weyl conformal curvature tensor of gNonS1×N\nof dimension 2 n+2. Recall that\nWABCD=RABCD+1\n2n(RBCgAD−RBDgAC−RACgBD+RADgBC)\n+S\n2n(2n+1)(gBDgAC−gBCgAD).\nProposition 4.1.3. The Fefferman-Lorentz manifold (S1×N,gN)is\na conformally flat Lorentz nilmanifold.19\nProof.We shall prove that all the Weyl conformal curvature tensors\nvanish. By (1) of Proposition 4.1.2, this reduces to\nWABCD=RABCD+1\n2n(RBCgAD−RBDgAC−RACgBD+RADgBC).\nIt follows from (i) of Proposition 4.1.2 and Lemma 4.1.1,\n(4.1.26) WXYZW= 0 (∀X,Y,Z,W ∈Kerω).\nIt follows from (i), (ii), (iii) of Proposition 4.1.2 and Lemma 4.1.2,\nWξYZW= 0 (∀Y,Z,W∈Kerω),\nWξYξW= 0, WξYξS= 0, WξYSW= 0.(4.1.27)\nSimilarly by (ii), (iii) of Proposition 4.1.2,\nWξSZW= 0,\nWξSξW=−WξWξS= 0.(4.1.28)\nLet\nWξSξS=1\n2n(RSξgξS−RSSgξξ−RξξgSS+RξSgSξ).\nSinceRSS/\\e}atio\\slash= 0 butgξξ= 0, it follows that\n(4.1.29) WξSξS= 0.\nAsWξSSW=1\n2n(RSSgξW−RSWgξS−RξSgSW+RξWgSS) butgξW=\n0, it follows that\n(4.1.30) WξSSW= 0, WξSSξ= 0.\nFrom (4.1.27), (4.1.28), (4.1.29), (4.1.30), the Weyl tensors conta in-\ningξare zero. It follows similarly\nWSξZW=−WξSZW= 0,\nWSξξW=−WξSξW= 0,\nWSξSW=−WξSSW= 0,\nWSξξS= 0, WSξSξ= 0,\nWSYξW=WξWSY= 0,\nWSYξS=WξSSY= 0.(4.1.31)20\nBy (1) of Lemma 4.1.4 and (iv) of Proposition 4.1.2, we calculate\nWSYSW=RSYSW\n+1\n2n(RYSgSW−RYWgSS−RSSgYW+RSWgYS)\n=RSYSW−1\n2nRSSgYW\n=−1\n(n+2)2g(Y,W)−1\n2n/parenleftbigg−2n\n(n+2)2g(Y,W)/parenrightbigg\n= 0.(4.1.32)\nSo all the terms containing Sare zero. We thus conclude that the Weyl\nconformal curvature tensors of ( S1×N,gN) vanish.\n/square\nProposition 4.1.4. The Fefferman-Lorentz metric g0is a conformally\nflat Lorentz metric on S1×S2n+1.\nProof.If we note that gN=u·g0as before, the Weyl conformal cur-\nvature tensor satisfies that W(g0) =W(gN) = 0 on S1× N. Since\nU(n+1,1) acts conformally and transitively on S1×S2n+1,W(g0) = 0\nonS2n+1,1. /square\n4.2.Examples of Fefferman-Lorentz manifolds I. We shall give\nexamples of conformally flat Fefferman-Lorentz manifolds which adm it\ncausal Killing fields. Consider principal S1-bundles as a connection\nbundle over a K¨ ahler manifold W.\nS1→N2n+1π−→W.\nThere exists a 1-form ωsuch that dω=π∗Ω for which Ω = iδαβωα∧ω¯β\nis the K¨ ahler form on W. As Ker ωis isomorphic to TWat each point\nofW. LetJbe a complex structure on Ker ωobtained from that of W\nby the pullback of π∗. Then (Ker ω,J) is a strictly pseudoconvex CR-\nstructure on N. Letξbe a characteristic vector field induced by S1.\nLetP:S1×N→Nbe the projection as before. We have a Lorentz\nmetric on S1×N:\n(4.2.1) g=σ⊙P∗ω+dω(JP∗−,P∗−).\nLetdωα=ωβ∧ϕα\nβbe the structure equation on Wfor the K¨ ahler form\nΩ. Asdω=iδαβπ∗ωα∧π∗ω¯β, the structure equation for ωbecomes\ndπ∗ωα=π∗ωβ∧π∗ϕα\nβ.\nLetsbe the scalar curvature of W. Then the Webster scalar function\nis defined as ρ=π∗s. By the definition,\n(4.2.2) σ=1\nn+2/parenleftbigg\ndt+iP∗π∗ϕα\nα−1\n2(n+1)π∗s·P∗ω/parenrightbigg\n.21\nAsξis characteristic, ω(ξ) = 1, and P∗ξ=ξ,π∗(ξ) = 0 on W. From\n(4.2.2), we obtain that\n(4.2.3) σ(ξ) =−1\n2(n+1)(n+2)s.\nLetcbe a positive constant. When Wis the complex projective space\nCPn, a complex torus Tn\nCor a complex hyperbolic manifold Hn\nC/Γ of\nconstant holomorphic sectional curvature c, 0,−crespectively, the\nscalar curvature s=n(n+1)c\n2,0,−n(n+1)c\n2respectively. It follows\nfrom (4.2.3) that\nσ(ξ) =−nc\n4(n+2),0,nc\n4(n+2)\nrespectively. We obtain that\n(4.2.4) g(ξ,ξ) =\n\n−nc\n2(n+2)forCPn,\n0 for Tn\nC,\nnc\n2(n+2)forHn\nC/Γ.\nNote that there are principal S1-bundles as a connection bundle:\n(4.2.5)S1− −− →S2n+1π− −− →CPn,\nR− −− → Nπ− −− →Cn,\nS1− −− →V2n,1\n−1π− −− →Hn\nC.\nHereNis the Heisenberg Lie group with isometry group Isom( N) =\nN⋊U(n). There is a discrete subgroup ∆ ≤ N⋊U(n) whose quo-\ntient has a principal fibration: S1→N/∆−→Tn\nC. Using the complex\ncoordinates, V2n,1\n−1is defined as\n(4.2.6) {(z1,...,z n+1)∈Cn+1| |z1|2+···+|zn|2−|zn+1|2=−1}.\nMoreover, the complement S2n+1−S2n−1is identified with V2n,1\n−1. Then\nthe group U( n,1) acts transitively on V2n,1\n−1whose stabilizer at a point\nis isomorphic to U( n). Moreover, there exists a discrete cocompact\nsubgroup Γ of U( n,1) such that the image π(Γ) is a torsionfree discrete\ncocompact subgroup of U( n,1) = Isom( Hn\nC).\nNote that if the first summand S1ofS1×Ngenerates a vector field\nS, thenσ(S) =1\nn+2dt(S) = 1 and P∗ω(S) =ω(P∗S) = 0. Then\ng(S,S) = 0, i.e .S1is lightlike.22\nProposition 4.2.1. LetXbe one of the compact conformally flat\nFefferman-Lorentz manifolds S1×S2n+1,S1×N/∆orS1×V2n,1\n−1/Γ.\nThenXadmits a lightlike Killing vector field Sand a timelike (resp.\nlightlike, spacelike )Killing vector field ξ.\n5.Uniformization\n5.1.Uniformization of Fefferman-Lorentz parabolic manifolds.\nWe prove the following uniformization concerning Fefferman-Lorent z\nparabolic manifolds.\nProposition5.1.1. LetMbe a(2n+2)-dimensionalFefferman-Lorentz\nparabolic manifold (i.e.admits a GC-structure. )IfMis conformally\nflat, then Mis uniformizable with respect to (ˆU(n+1,1),S2n+1,1).\nProof.Suppose that Mis a conformally flat Lorentz (2 n+2)-manifold.\nBy the definition, there exists a collection of charts {Uα,ϕα}α∈Λ. Let\nϕα:Uα→S2n+1,1,ϕβ:Uβ→S2n+1,1be charts with Uα∩Uβ/\\e}atio\\slash=∅. The\ncoordinate change ϕα◦ϕ−1\nβ:ϕβ(Uα∩Uβ)→ϕα(Uα∩Uβ) extends to a\ntransformation gαβ∈PO(2n+2,2) ofS2n+1,1.\nBytheexistenceof GC-structure, wehavetheprincipalframebundle:\nGC→P−→MwhereGC≤O(2n+1,1)×R+. This bundle restricted\nto each neighborhood Uαgives the trivial principal bundle on each\nneighborhood of S2n+1,1:\nGC→ϕα∗(P|Uα)−→ϕα(Uα).\nThere is the commutative diagram:\n(5.1.1)GC− −− → GC/arrowbt/arrowbt\nϕβ∗(P|Uβ)gαβ∗− −− → ϕα∗(P|Uα)/arrowbt/arrowbt\nϕβ(Uα∩Uα)gαβ− −− →ϕα(Uα∩Uα).\nSince the subgroup ˆU(n+1,1) acts transitively on S2n+1,1, we choose\nan element h∈ˆU(n+ 1,1) for which g=h·gαβ∈PO(2n+ 2,2)\nsatisfies that gx=xfor some point x∈S2n+1,1. Then the differential\nmapg∗:TxS2n+1,1→TxS2n+1,1satisfies that g∗∈GC.\nSuppose that His a subgroup of PO(2 n+2,2) containing ˆU(n+1,1)\nwhich preserves the GC-structure. As above, note that g∈Hx. If\nτ:Hx→Aut(TxS2n+1,1) is the tangential representation, then it follows\nthat\nτ(Hx)≤GC∼=R2n⋊(U(n)×R+).23\nSinceτis injective for any connected compact subgroup of Hx, this\nimplies that a maximal compact subgroup K′ofHxis isomorphic to\nU(n). LetKbe a maximal compact subgroup of Hcontaining K′. By\nthe Iwasawa-Levi decomposition,\nK/K′∼=H/Hx=S2n+1,1=S1×S2n+1/Z2,\nKmust be isomorphic to U( n+1)·U(1).\nOn the other hand, U( n+1)·U(1)≤(O(2n+2)·O(2)) is the maximal\ncompact unitary subgroup of ˆU(n+ 1,1). AsˆU(n+ 1,1)≤H, we\nobtain that ˆU(n+ 1,1) =H. In particular, g=h·gαβ∈Hx≤\nˆU(n+1,1). It follows that gαβ∈ˆU(n+1,1). Therefore the maximal\ncollection of charts {Uα,ϕα}α∈Λgives a uniformization with respect to\n(ˆU(n+1,1),S2n+1,1). /square\nRemark 5.1.1. Ifˆ∞is the infinity point of S2n+1,1(which maps to\nthe point at infinity {∞}ofS2n+1) (cf.(6.1)of Section 6.1.1 ), then it\nis noted that the stabilizer (up to conjugacy )is\nPO(2n+2,2)ˆ∞=R2n+2⋊(O(2n+1,1)×R+).\nNote that the intersection ˆU(n+1,1)∩PO(2n+2,2)ˆ∞is\nˆU(n+1,1)ˆ∞=N⋊(U(n)×R+).\nIn fact,O(2n+1,1)contains the similarity subgroup R2n⋊(O(2n)×R+)\nso that\nN⋊U(n)⊂(R2n+2⋊R2n)⋊O(2n).\nThe conformally flat Lorentz geometry (PO(2 n+ 2,2),S2n+1,1) re-\nstricts a subgeometry ( ˆU(n+ 1,1),S2n+1,1). It is noted that the full\nsubgroup of PO(2 n+ 2,2) preserving the G-structure on S2n+1,1is\nˆU(n+1,1).\nDefinition 5.1.1. The pair (ˆU(n+1,1),S2n+1,1)is said to be confor-\nmally flat Fefferman-Lorentz parabolic geometry . A smooth (2n+2)-\ndimensional manifold Mis aconformally flat Fefferman-Lorentz par-\nabolic manifold ifMis locally modelled on ˜S2n+1,1with local changes\nlying inU(n+1,1)∼.\nHere U(n+1,1)∼is a lift of ˆU(n+1,1) to PO(2 n+1,2)∼which has\nthe central group extension:\n1→Z→U(n+1,1)∼ˆQZ−→ˆU(n+1,1)→1.\nWe close this section by showing the following examples of compact\nconformally flat Lorentz parabolic manifolds .24\n•Lorentz flat space forms which admit Lorentz parabolic struc-\nture but not Fefferman-Lorentz parabolic structure.\n•Conformally flat Fefferman-Lorentz parabolic manifold which\ndo not admit Fefferman-Lorentz structure.\n•Conformally flat Fefferman-Lorentz manifolds S1× N3/∆ on\nwhichS1acts as lightlike isometries. (This is shown in Section\n4.2 and Proposition 4.1.3.)\n5.2.Examples of 4-dimensional Lorentz flat parabolic mani-\nfolds II. LetN3=R×Cbe the 3-dimensional Heisenberg group with\ngroup law:\n(a,z)(b,w) = (a+b−Im¯zw,z+w).\nRecall that Lorentz flat geometry (E(2 ,1),R3) where E(2 ,1) =R3⋊\nO(2,1). Let O(2 ,1)∞be the stabilizer at the point at infinity in S1=\n∂H2\nR. It is isomorphic to Sim( R1) =R⋊(O(1)×R+) as in (3.1.2). Put\nz=x+it,x,t∈R. We define a continuous homomorphism:\nρ:N3−→R3⋊O(2,1)∞,\nρ(/parenleftbigg\na\nx/parenrightbigg\n) = (\na\nx\n0\n,I),\nρ/parenleftbigit/parenrightbig\n= (\n−t3\n6\n−t2\n2\nt\n,\n1t−t2\n2\n0 1−t\n0 0 1\n).(5.2.1)\nIt is easy to see that ρis a simply transitive representation of N3onto\nthe Lorentz flat space R3. (Compare [16].)\nProposition 5.2.1. There is a 4-dimensional compact Lorentz flat\nspace form S1×N3/∆which admits a Lorentz parabolic structure but\nnot admit Fefferman-Lorentz parabolic structure.\nProof.TakingRas timelike parallel translations, we extend the repre-\nsentation ρnaturally to a simply transitive 4-dimensional representa-\ntion:\n˜ρ:R×N3−→R×R3⋊O(2,1)∞⊂E(3,1)\n(5.2.2) ˜ ρ(R×N3) =R4.\nHere note that E(3 ,1)R4⋊O(3,1)⊂O(4,2)∞. If we choose a discrete\nuniform subgroup ∆ ⊂ N3, then a compact aspherical manifold S1×\nN3/∆ admits a (complete) flat Lorentz structure such that\nS1×N3/∆∼=R4/˜ρ(Z×∆).25\nWe check that S1×N3/∆ cannot admit a Fefferman-Lorentz parabolic\nstructure. For this, if so, by Proposition 5.1.1, the group R× N3is\nconjugate to a subgroup of U(2 ,1) up to an element of O(4 ,2). Since\nR× N3is nilpotent, it belongs to U(2 ,1)∞=S1· N3⋊(U(1)×R+)\nup to conjugate. This is impossible because S1is lightlike.\n/square\n5.3.Examples of conformally flat Fefferman-Lorentz parabolic\nmanifolds III. We shall give compact conformally flat Fefferman-\nLorentz parabolic manifolds which are not equivalent to the product of\nS1with spherical CR-manifold, i.e .not a Fefferman-Lorentz manifold.\nConsider the commutative diagram.\n(5.3.1)Z Z/arrowbt/arrowbt\nR− −−− →(U(n+1,1)∼,˜S2n+1,1)ց(˜P,˜P)\n/arrowbt (ˆQZ,ˆQ)/arrowbt (PU(n+1,1),S2n+1)\nS1− −−− →(ˆU(n+1,1),S2n+1,1)ր(ˆP,P)\nWestartwithadiscrete subgroupΓ ⊂ˆU(n+1,1)such that S1∩Γ =Zp\nfor some integer p. If we let π=ˆQ−1\nZ(Γ), then there is the nontrivial\ngroup extensions:\n(5.3.2)1− −−− →1\npZ− −−− → π− −−− → ˜P(Γ)− −−− →1\n∩ ∩ ∩\n1− −−− →R− −−− →U(n+1,1)∼˜P− −−− →PU(n+1,1)− −−− →1\nThe group πdefines a cocycle [ f]∈H2(˜P(Γ),1\npZ). Suppose that a\nis an irrational number. Then [ a·f]∈H2(˜P(Γ),R) which induces a\ngroup extension:\n1→a\npZ→π(a)−→˜P(Γ)→1.\nHereπ(a) is viewed as the producta\npZ×P(Γ) with group law:\n(a\npm,α)(a\npℓ,β) = (a\np(m+ℓ)+a·f(α,β),αβ) (∀α,β∈˜P(Γ)).\n(Refer to [24] and references therein for a construction of grou p actions\nby group extensions.)\nAsRis the center of U( n+1,1)∼, it follows that\n(a\npm,α) = (a\npm,1)(1,α)∈R·U(n+1,1)∼= U(n+1,1)∼.26\nThis shows that\n(5.3.3) π(a)⊂U(n+1,1)∼.\nAs˜P(Γ) is discrete, so is π(a) in U(n+1,1)∼. LetL(˜P(Γ)) be the limit\nset of˜P(Γ) inS2n+1. Then it is known that ˜P(Γ) acts properly discon-\ntinuously on the domain Ω = S2n+1−L(˜P(Γ)) (cf.[17],[12]). If Ω /\\e}atio\\slash=∅,\nthenthequotient Ω /˜P(Γ)isaspherical CR-orbifold. Since S1=R/a\npZ\nis compact, it is easy to see that π(a) acts properly discontinuously on\n˜S2n+1,1−˜P−1(L(˜P(Γ))). Putting\nM(a) =˜S2n+1,1−˜P−1(L(˜P(Γ)))/π(a),\nM(a) is a smooth compact conformally flat Fefferman-Lorentz para-\nbolic manifold which supports a fibration:\nS1→M(a)ˆP−→Ω/˜P(Γ).\nOn the other hand, as ˆQZ(π(a)) =ˆQZ(a\npZ)·Γ from (5.3.1), the closure\ninˆU(n+1,1) becomes\n(5.3.4) ˆQZ(π(a)) =S1·Γ.\nWhenever ais irrational, M(a) cannot descend to a locally smooth\norbifold modelled on ( ˆU(n+1,1),S2n+1,1). SoM(a) is not equivalent\nto the product manifold. Hence we have\nProposition 5.3.1. Letabe an irrational number. There exists a\ncompact (2n+2)-dimensional conformally flat Fefferman-Lorentz par-\nabolic manifold M(a)which is a nontrivial S1-bundle over a spherical\nCR-manifold. Moreover, M(a)is not equivalent to the product mani-\nfold.\nFor example, such π(a) is obtained as follows. PU( n+ 1,1) has\nthe subgroup U( n,1) = P(U( n,1)×U(1)) which acts transitively on\nS2n+1−S2n−1=V2n,1\n−1. (See (4.2.6).) Since the stabilizer at a point is\nisomorphic to U( n), there exists a U( n,1)-invariant Riemannian metric\nonV2n,1\n−1. If Γ is a discrete cocompact subgroup of U( n,1), thenL(Γ) =\nL(U(n,1)) =S2n−1. Chasing the diagram\n(5.3.5)S1− −− → ˆU(n+1,1)ˆP− −− →PU(n+1,1)\n/uniontext /uniontext /uniontext\nS1− −− →U(n,1)×U(1)− −− → U(n,1),27\nwe start with Γ ⊂U(n,1)× {1} ⊂U(n,1)×U(1). Then we get a\nFefferman Lorentz manifolds M(a) =R×V2n+1\n−1/π(a) where˜S2n+1,1−\n˜S2n−1,1=R×V2n+1\n−1.\nPutπ=π(1) fora= 1. In this case, ˆQZ(π) = Γ. The previous\nconstruction shows that\nM(1) =˜S2n+1,1−˜S2n−1,1/π=S2n+1,1−S2n−1,1/Γ =S1×\nZ2V2n+1\n−1/Γ.\nM(1) is a conformally flat Fefferman Lorentz manifold M. Varying a,\nwe see that M(a) is nonequivalent with M(1) as a Fefferman-Lorentz\nmetric.\nRemark 5.3.1. We have also O(m+ 1)×R+-structure on (m+ 2)-\nmanifolds as a parabolic structure. Similar to the proof of P roposition\n5.1.1, we can show that\nProposition 5.3.2. LetMbe a smooth (m+ 2)-manifold with an\nO(m+ 1)×R+-structure. If Mis conformally flat Lorentz such that\nO(m+ 1)≤O(m+ 2) (not maximal) , thenMis uniformized with\nrespect to (O(2)×O(m+2),S1×Sm+1). In particular, if Mis compact,\nthenMcoversSm+1,1.\nLetMbe a Riemannian manifold of dimension m+1. ThenS1×M\nadmits a natural Lorentz metric gfor which S1acts as timelike isome-\ntries. Even if Mis conformally flat, S1×Mneed not be a con-\nformally flat Lorentz manifold. For example, S1×Hm+1\nR. However,\nS1×Hm+1\nR/Γis covered by (P(O(1,1)×O(m+ 1,1)),R×Hm+1\nR)for\nwhichP(O(1,1)×O(m+ 1,1))≤PO(m+ 2,2). SoS1×Hm+1\nR/Γ\nis conformally flat Lorentz but S1(orR)is not a group of timelike\nisometries.\n6.Conformally flat Fefferman-Lorentz parabolic\ngeometry\nRecall that Conf FLP(M) is the group of conformal transformations\npreserving the Fefferman-Lorentz parabolic structure (cf .(3.1.2)). We\nshall consider the representations of one-parameter subgroup sH≤\nConfFLP(M).28\n6.1.One-parameter subgroups in ˆU(n+1,1).The following com-\nmutative diagrams are obtained.\n(6.1.1)Z Z\n/arrowbt/arrowbt\nR− −−− →(U(n+1,1)∼,˜S2n+1,1)(˜P,˜P)− −−− →(PU(n+1,1),S2n+1)\n/arrowbt (ˆQZ,ˆQ)/arrowbt ||\nS1− −−− → (ˆU(n+1,1),S2n+1,1)(ˆP,P)− −−− →(PU(n+1,1),S2n+1).\nHere˜S2n+1,1=R×S2n+1.\n(6.1.2)(U(n+1,1)∼,R×S2n+1)\n(QZ,˜Q)/arrowbt ց(ˆQZ,ˆQ)\nZ2− −−− →(U(n+1,1),S1×S2n+1)(Q2,PR)− −−−− → (ˆU(n+1,1),S2n+1,1).\nBy (4.1.2), there is the projection:\n(6.1.3) PC=P◦PR:V0(⊃S1×S2n+1)PR− −−− →S2n+1,1P− −−− →S2n+1.\nAs usual the following points {∞,0}are defined on the conformal Rie-\nmannian sphere:\n∞=PC(f1) =/bracketleftbigg1√\n2,0,...,0,1√\n2/bracketrightbigg\n= (0,...,0,1)∈S2n+1,\n0 =PC(fn+2) =/bracketleftbigg1√\n2,0,...,0,−1√\n2/bracketrightbigg\n= (0,...,0,−1)∈S2n+1.(6.1.4)\nWe put\nˆ∞=PR(f1)∈S2n+1,1,ˆ0 =PR(fn+2)∈S2n+1,1\nsuch that\nP( ˆ∞) =∞, P(ˆ0) = 0.\nSuppose that His a one-parameter subgroup {φt}t∈Rof Conf FLP(M)\nand˜H={˜φt}t∈Ris its lift to Conf FLP(˜M). Let ˜ρ:˜H→U(n+1,1)∼be\na homomorphism. For simplicity write ˜ ρ(˜φt) = ˜ρ(t) (t∈R) and put\nρ=QZ◦˜ρ:˜H→U(n+1,1),\nˆρ=Q2◦ρ=ˆQZ◦˜ρ:˜H→ˆU(n+1,1).(6.1.5)\nAsP: U(n+ 1,1)→PU(n+ 1,1) is the projection, it follows from\n(6.1.1), (6.1.5) that P◦QZ=ˆP◦ˆQZ=˜Pfor which\n˜P(˜ρ(t)) =Pρ(t). (6.1.6)29\nSince˜P◦˜ρ(˜H) =P◦ρ(˜H), we put\n(6.1.7) G=P◦ρ(˜H)≤PU(n+1,1).\nWe determine the connected closed subgroup Gby using the results\nof [14]. First recall that {e1,...,e n+2}is the standard complex ba-\nsis ofCn+2equipped with the Lorentz Hermitian inner product /a\\}b∇acketle{t,/a\\}b∇acket∇i}ht\n(cf.(4.1.1)); /a\\}b∇acketle{tei,ej/a\\}b∇acket∇i}ht=δij(2≤i,j≤n+ 2),/a\\}b∇acketle{ten+2,en+2/a\\}b∇acket∇i}ht=−1. Set-\ntingf1=e1+en+2/√\n2, fn+2=e1−en+2/√\n2 as before, the frame\n{f1,e2,...,e n+1,fn+2}is the new basis such that\n/a\\}b∇acketle{tf1,f1/a\\}b∇acket∇i}ht=/a\\}b∇acketle{tfn+2,fn+2/a\\}b∇acket∇i}ht= 0,/a\\}b∇acketle{tf1,fn+2/a\\}b∇acket∇i}ht=/a\\}b∇acketle{tfn+2,f1/a\\}b∇acket∇i}ht= 1.\n6.1.1.Case I:Gis noncompact. It follows from [14, §3] thatP(ρ(˜H))\nitself is closed. We may put\n(6.1.8) G=P◦ρ(˜H) ={Pρ(t)}t∈R.\nMoreover, Gbelongs to N⋊(U(n)×R+) = PU(n+1,1)∞up to conju-\ngate. (SeeRemark5.1.1.)Moreover, theexplicitformof {Pρ(t)}canbe\ndescribed with respect to the basis {f1,e2,···,en+1,fn+2}. (Compare\n[15].) It has the following form\n(6.1.9) Pρ(t) =\n10ti\n0At0\n001\n\nwhereAt= (eita1,...,eitan)∈Tn≤U(n).\n(6.1.10) Pρ(t) =\n1t0t2/2+1\n0 10t\n0 0Bt0\n0 001\n\nwhereBt= (eitb1,...,eitbn−1)∈Tn−1≤U(n−1).\n(6.1.11) Pρ(t) =\net00\n0At0\n00e−t\n\nwhereAt= (eita1,...,eitan)∈Tn≤U(n).\nLetCtbe the matrix accordingly as whether [ Ct] is (6.1.9), (6.1.10)\nor (6.1.11). Noting that the center of U( n+ 1,1) isS1={eit}, the\nholonomy map ρ:˜H→U(n+1,1) has the following form:30\n(6.1.12) ρ(t) =/braceleftbigg\nCt (i),\neit·Ct(ii).\nFor (6.1.9), (6.1.10), Ghas the unique fixed point {∞}inS2n+1. As\nPC(f1) =∞(cf.(6.1.4)), ρ(t)f1=λ·f1for some λ∈C∗. Ifρ(t) =Ct\nfor (6.1.12), then ρ(t)f1=f1so that ˆρ(t) ˆ∞= ˆ∞by (6.1.2). Hence\n(6.1.13) ˆ ρ(˜H) has the fixed point set {S1·ˆ∞}inS2n+1,1.\nFor (6.1.11), Ghas two fixed points {0,∞}inS2n+1. Ifρ(t) =Ct,\nthenρ(t)f1=et·f1,ρ(t)fn+2=e−t·fn+2. SincePR(s·v) =PR(v) for\n∀s∈R∗,v∈V0, it follows that ˆ ρ(t) ˆ∞= ˆ∞and ˆρ(t)ˆ0 =ˆ0 inS2n+1,1.\nSimilarly as above,\n(6.1.14) ˆ ρ(˜H) has the fixed point set {S1·ˆ0,S1·ˆ∞}inS2n+1,1.\n6.1.2.Case II: Gis compact. Using (6.1.6),\nP(ρ(t)) = (eita1,...,eitak,1,...,1)∈Tn+1\nforsomenonzeronumbers a1,...,a k. PutEt= (eita1,...,eitak,1,...,1).\nWe may assume that the g.c.m of all aiis 1 (up to scale factor of pa-\nrametert). Thenρ(t) has one of the following forms:\n(6.1.15) ρ(t) =/braceleftbigg\nEt∈Tn+1·S1(i),\nEt·eit∈Tn+1·S1(ii).\nProposition6.1.1. Let(M,g)bea conformallyflatFefferman-Lorentz\nparabolicmanifoldwhichadmitsa one-parametersubgroup H≤ConfFLP(M)\nacting without fixed points on M. Suppose that\n(6.1.16) (˜ ρ,dev) : (˜H,˜M)→(U(n+1,1)∼,˜S2n+1,1)\nis the developing pair. If G/\\e}atio\\slash={1}, then either one of Case A, Case\nB, Case C orCase D holds:\nCase A. The action of ρ(˜H)is of type (i)of(6.1.12).\n(1)WhenCthas the form of either (6.1.9)or(6.1.10),\n(6.1.17) ˆQ(dev(˜M))⊂S2n+1,1−S1·ˆ∞=S1×N,\nwhere the centralizer C(ˆρ(˜H))ofˆρ(˜H)inˆU(n+ 1,1)is\ncontained in S1×(N⋊U(n)).\n(2)WhenCthas the form (6.1.11),\n(6.1.18) ˆQ(dev(˜M))⊂S2n+1,1−S1·{ˆ0,ˆ∞}= (S2n×R+)×S1,\nwhereC(ˆρ(˜H))is contained in S1×(U(n)×R+).31\nCase B. The action of ρ(˜H)is of type (ii)of(6.1.12). Thenˆρ(˜H)has\nno fixed point set on S2n+1,1. In this case, C(ˆρ(˜H))is either\ncontained in S1×(N⋊U(n))orS1×(U(n)×R+).\nCase C. The action of ˆρ(˜H)is of type (i)of(6.1.15).\n(6.1.19) ˆQ(dev(˜M))⊂S2n+1,1−S1·S2(n−k)+1\non which the subgroup S1×(U(n−k+1,1)/hatwide×U(k))acts tran-\nsitively with compact stabilizer. The centralizer of ˆρ(˜H)in\nˆU(n+1,1)isS1×(U(n−k+1,1)/hatwide×Tk)whereˆρ(˜H)⊂Tk.\nCase D. The action of ˆρ(˜H)is of type (ii)of(6.1.15). Thenˆρ(˜H)has\nno fixed point set on S2n+1,1and its centralizer in ˆU(n+1,1)is\nS1×(U(n−k+1,1)/hatwide×Tk)whereS1׈ρ(˜H)⊂Tk.\nProof.By the hypothesis, Hhas no fixed point so does ˜Hon˜M. Since\ndev is an immersion, the image dev( ˜M) misses the fixed point set of\n˜ρ(˜H) in˜S2n+1,1. Recall that there is the covering space from (6.1.1):\nZ→(U(n+1,1)∼,˜S2n+1,1)(ˆQZ,ˆQ)− −−− →(ˆU(n+1,1),S2n+1,1).\nNoting that ˜ ρ(˜H) is connected, the image ˆQ(dev(˜M)) also misses the\nfixed point set of ˆQZ(˜ρ(˜H)) = ˆρ(˜H) ofˆU(n+ 1,1) (cf.(6.1.5)). Then\n(1), (2) of Case A follow from (6.1.13) and (6.1.14) respectively and\nCase B follows easily because the center S1ofˆU(n+1,1) acts freely\nonS2n+1,1.\nFor the case (i) of (6.1.15), the fixed point set of P(ρ(˜H)) is\nS2(n−k)+1={(0,...,0,zk+1,...,z n+1)∈S2n+1}\nin which the subgroup of PU( n+1,1) preserving S2n+1−S2(n−k)+1is\nP(U(n−k+1,1)×U(k)). Since ˆQ(dev(˜M)) misses the fixed point set\nof ˆρ(˜H),Case C follows that\nˆρ(t) = (eita1,...,eitak,1,...,1)∈Tk≤S1·Tn+1,\nˆQ(dev(˜M))⊂S2n+1,1−S1·S2(n−k)+1.\nFor the case (ii) of (6.1.15), it follows that\nˆρ(t) = (eit(a1+1),...,eit(ak+1),eit,...,eit)∈S1·Tn+1.\nSimilarly as Case B,\nC(ˆρ(˜H))≤S1·U(n−k+1,1)/hatwide×Tk)\nwhere ˆρ(˜H)⊂S1·Tkwhich shows Case D.\n/square32\nDenotebyIsom FLP(M)thegroupofisometriespreservingtheFefferman-\nLorentz parabolic structure such that Isom FLP(M)≤ConfFLP(M).\nProposition6.1.2. Let(M,g)bea conformallyflatFefferman-Lorentz\nparabolicmanifoldadmittinga 1-parametersubgroup H≤IsomFLP(M).\nIfG={1}, then the following hold.\n(i)The lift˜Hacts properly and freely on ˜Mas lightlike isometries.\n(ii)˜M/˜His a simply connected spherical CR-manifold on which\nthe quotient group C(˜H)/˜Hacts asCR-transformations. Here\nC(˜H)is the centralizer of ˜HinConfFLP(˜M).\n(iii)The conformal developing pair (˜ρ,dev)forMinduces a CR-\ndeveloping pair:\n(6.1.20) (ˆ ρ,ˆdev): (C(˜H)/˜H,˜M/˜H)→(PU(n+1,1),S2n+1).\nProof.Supposethat G={1}. SinceG=˜P◦˜ρ(˜H)from(6.1.6), (6.1.7),\nit follows that ˜ ρ(˜H) =R≤U(n+1,1)∼which is lightlike with respect\ntog0where\n(6.1.21) g0=σ0⊙˜P∗ω0+dω0(J0˜P∗−,˜P∗−)\nis the standard Lorentz metric on ˜S2n+1,1=R×S2n+1induced from\n(4.2.1). Inparticular, ˜ ρ:˜H→Risanisomorphism. As Ractsproperly\non˜S2n+1,1=R×S2n+1,˜Hacts properly on ˜M. On the other hand,\nthere exists a function usuch that\n(6.1.22) dev∗g0=u·g.\nLetHbe the vector field induced by ˜Hon˜M. AsSis the vector\nfield induced by R, we have that dev ∗(H) =S. Sinceu·g(H,H) =\ng0(S,S) = 0, noting the hypothesis that ˜H≤IsoFLP(˜M),˜Hacts as\nlightlike isometries. This shows (i).\nThere is the commutative diagram:\n(6.1.23)˜Mdev− −− →˜S2n+1,1\nP/arrowbt ˜P/arrowbt\n˜M/˜Hˆdev− −− →S2n+1.\nWe put\n(6.1.24) P∗ω(X) =g(H,X) (∀X∈T˜M).\nThenωis a well-defined 1-form on ˜M/˜Hby the fact that g(H,H) = 0.\nNote from (4.2.2), (3.3.6) that\n(6.1.25) σ0=1\nn+2/parenleftig\ndt+i˜P∗π∗ϕα\nα−nc\n4·˜P∗ω0/parenrightig\n.33\nFor ˜ρ(˜H) =R, it follows that ˜ ρ(h)∗σ0=σ0(∀h∈˜H). This implies\nthat ˜ρ(h)∗g0=g0. Applying dev∗to this,h∗dev∗g0= dev∗g0=u·g\nby (6.1.22). As h∗dev∗g0=h∗(u·g) =h∗u·g, it follows h∗u=uand\nsoufactors through a map ˆ u:˜M/˜H→R+such that\n(6.1.26) P∗ˆu=u.\nMoreover σ0(S) =1\nn+2from(3.3.4)and(6.1.25),so(6.1.21)implies\nthat\ng0(S,−) =1\nn+2˜P∗ω0(−).\nUsing (6.1.26), the equation dev∗g0=u·gyields that\nP∗ˆu·P∗ω(X) =u·g(H,X) =g0(S,dev∗X)\n=1\nn+2˜P∗ω0(dev∗X) =1\nn+2dev∗˜P∗ω0(X)\n=1\nn+2P∗ˆdev∗ω0(X),\nhence\n(6.1.27) ( n+2)ˆu·ω=ˆdev∗ω0.\nAsˆdev:˜M/˜H→S2n+1is an immersion, ˆdev∗: kerω→kerω0is an\nisomorphism. Define ˆJon kerωto be\n(6.1.28) ˆdev∗(ˆJX) =J0ˆdev∗(X).\nIf we note that J0is a complex structure on ker ω0,ˆJturns out to be a\ncomplex structure on ker ω. Hence (ker ω,ˆJ) gives a CR-structure on\n˜M/˜Hfor which ˆdev is aCR-immersion.\nLetC(˜H) be the centralizer of ˜Hin Conf FLP(˜M). Fors∈ C(˜H) with\nˆs∈ C(˜H)/˜H, there is a positive function von˜Msuch that s∗g=v·g.\nNoting that ˜H≤IsoFLP(˜M), we can check that ˜h∗v=v(∀h∈˜H),\ni.e.there exists a function ˆ von˜M/˜Hsuch that P∗ˆv=v. Then it\nis easy to see that ˆ s∗ω= ˆv·ωon˜M/˜H. Using (6.1.28), it follows\nthat ˆs∗◦ˆJ=ˆJ◦ˆs∗on kerω. Hence the group C(˜H)/˜Hpreserves the\nCR-structure (ker ω,ˆJ) on˜M/˜H. This shows (ii).\nAs ˜ρ(˜H) =Ris the center of U( n+1,1)∼,ρ: Conf FLP(˜M)→U(2n+\n1,1)∼induces a homomorphism ˆ ρ:C(˜H)/˜H→PU(n+1,1). Using the\nabove commutative diagram, it follows that\nˆdev(ˆαˆx) = ˆρ(ˆα)ˆdev(ˆx) (∀ˆα∈ C(˜H)/˜H,ˆx∈˜M/˜H).\nHence (iii) is proved.\n/square34\nRemark 6.1.1. A Fefferman-Lorentz manifold M=S1×Nis obvi-\nously an example satisfying the hypothesis of Proposition 6 .1.2.\n7.Coincidence of curvature flatness\nIn this section we shall prove the equivalence between conformally\nflatness of Fefferman-Lorentz manifolds and (spherical) flatness of un-\nderlying CR-manifolds. (Compare Theorem 7.4.2.)\n7.1.Causal vector fields on S2n+1,1.LetH≤ConfFLP(M) be as\nbefore and ρ:˜H→U(n+ 1,1) the representation. Denote by ξ\nthe vector field on S1×S2n+1induced by the orbit ρ(˜H)·zfor some\nz∈S1×S2n+1. Recall from (6.1.1) that\n(7.1.1) PR(ρ(t)(λ·z)) =Q2(ρ(t))·PR(λ·z) = ˆρ(t)·PR(z)\nso we put the vector field ˆξonS2n+1,1by\n(7.1.2) ( PR)∗(ξz) =ˆξPR(z).\nNote that P∗(ˆξ) =PC∗(ξz) is a vector field on S2n+1by using PC=\nP◦PR(cf.(6.1.3)). From (4.2.1), let\ng0(X,Y) = (σ0⊙PC∗ω0)(X,Y)+dω0(J0PC∗X,PC∗Y)\nbe the standard Lorentz metric on S1×S2n+1(X,Y∈T(S1×S2n+1)).\nUsing theclassificationofone-parametersubgroups ρ(˜H) ={ρ(t)} ≤\nU(n+1,1) of Section 6.1, we examine the causality of the vector field\nξinduced by H.\n7.2.Nilpotent group case. (Compare Section 6.1.1,(6.1.12).)\nAsgN=u·g0onS1×N, usinggNinstead of g0, it suffices to check\nthe causality of ξ. Note that σN=1\nn+2dt.\nFor the vector field ξrestricted to S1×N ⊂S1×S2n+1, letP∗(ξ) be\nthe vector field on Nwhich is induced by the one-parameter subgroup\nP(H)≤PU(n+1,1)asabove. It isnotnecessarily acharacteristic vec-\ntor field except for the case ( i) of (6.1.12), but note that ωN(P∗(ξ))/\\e}atio\\slash= 0\nonN. (Compare [14].) In fact, for the cases (6.1.9), (6.1.10), (6.1.11)\nrespectively,35\n(1)P∗(ξ) =d\ndt+k/summationdisplay\nj=1aj(xjd\ndyj−yjd\ndxj),\nωN(P∗(ξ)) = 1+( a1|z1|2+···+ak|zk|2).\n(2)P∗(ξ) =−y1d\ndt+d\ndx1+n/summationdisplay\nj=2bj(xjd\ndyj−yjd\ndxj),\nωN(P∗(ξ)) =−2y1+(b2|z2|2+···+bn|zn|2).\n(3)P∗(ξ) = 2td\ndt+n/summationdisplay\nj=1((xj−ajyj)d\ndxj+(yj+ajxj)d\ndyj),\nωN(P∗(ξ)) = 2t+(a1|z1|2+···+an|zn|2).\nAs above, σN(ξ) =1\nn+2dt(ξ) =1\nn+2δwhereδ= 0,1 according\nto the case (i) or (ii) of (6.1.12) respectively. We obtain that\ngN(ξ,ξ) =2δ\nn+2ωN(P∗(ξ))+dωN(JP∗(ξ),P∗(ξ)).\nMoreover, it follows from (4.1.7) that\n(7.2.1) dωN(JP∗X,P∗Y) =n/summationdisplay\nj=1(dx2\nj+dy2\nj)(P∗X,P∗Y).\nCalculating dωN(JP∗(ξ),P∗(ξ)) for the above P∗(ξ) respectively, we see\nthatgN(ξ,ξ) = 0 if and only if δ= 0 and P∗(ξ) is the characteristic\nvector field for ωNin (1). As a consequence, we obtain that\nCausality (1). Suppose that Gis noncompact. Let ξbe a lightlike\nconformal vector field on S2n+1,1induced by G. Thenξis a nonzero\nlightlike vector field of gNonS1×Nif and only if ˜H=R,P∗(ξ) =d\ndtis the characteristic vector field for ωN.\n7.3.Compact torus case. (cf.Section 6.1.2, (6.1.15)). It is possible\nto calculate g0by making use of σ0, however it is difficult to see ωα\nα.\nSo we consider a different approach. Let R∗→V0PR−→S2n+1,1be the\nprojection for which ˆ g0is the standard Lorentz metric on S2n+1,1with\ng0=P∗\nRˆg0. If we choose c=2\nnin (4.2.3), then\n(7.3.1) ˆ g0(ξ,ξ) =−1\nn+2onS2n+1,1.36\nRecall from (4.1.1) that\nV0={z= (z1,...,z n+2)∈Cn+2−{0} | /a\\}b∇acketle{tz,z/a\\}b∇acket∇i}ht= 0}\nin which /a\\}b∇acketle{tz,w/a\\}b∇acket∇i}ht= ¯z1w1+···+ ¯zn+1wn+1−¯zn+2wn+2onCn+2. For an\narbitrary point PR(¯p) = [¯p]∈S2n+1,1(cf.(4.1.2)), choose a point ¯ q∈V0\nsuch that\n/a\\}b∇acketle{t¯p,¯q/a\\}b∇acket∇i}ht=r(∃r∈R−{0}). (7.3.2)\nNote that /a\\}b∇acketle{t¯p,¯p/a\\}b∇acket∇i}ht=/a\\}b∇acketle{t¯q,¯q/a\\}b∇acket∇i}ht= 0. From (4.1.2), we have the decomposition:\nT¯pR∗=R¯p→T¯pV0=C¯p+iR¯q+W0\nPR∗−→T[¯p]S2n+1,1=C¯p+iR¯q+W0/R¯p\n≈iR¯p+iR¯q+W0(7.3.3)\nwhereT¯pV0={v∈Cn+2|Re/a\\}b∇acketle{t¯p,v/a\\}b∇acket∇i}ht= 0}andW0=/a\\}b∇acketle{t¯p,¯q/a\\}b∇acket∇i}ht⊥. The\ndecomposition is independent of the choice of ¯ qwith respect to /a\\}b∇acketle{t¯p,¯q/a\\}b∇acket∇i}ht /\\e}atio\\slash=\n0. Note that Re /a\\}b∇acketle{t,/a\\}b∇acket∇i}htis the metric on V0of dimension 2 n+3.\nProposition 7.3.1. Whenc=2\nn,\nˆg0(PR∗X,PR∗Y) = Re/a\\}b∇acketle{tX,Y/a\\}b∇acket∇i}ht(X,Y∈T¯pV0).\nProof.For an arbitrary point x∈S2n+1,1, we can choose\n¯p= (a1,...,a n+1,1√n+2z) (∃z∈S1)\nsuch that PR(¯p) = [¯p] =x. Thenξxis induced by the S1-orbit at ¯ p:\nc(θ) = (a1,...,a n+1,1√n+2z·e−iθ)∈V0.\nSince ˙c(0) = (0 ,···,0,−1√n+2zi), it follows that PR∗(˙c(0)) =ξx.\nSimilarly, Sxis induced by the S1-orbit at ¯ p:\ns(θ) = (a1·eiθ,...,a n+1·eiθ,1√n+2z·eiθ)∈V0.\nIt follows that ˙ s(0) = (a1i,...,a n+1i,1√n+2zi) for which PR∗(˙s(0)) =\nSx. Then we check that\nRe/a\\}b∇acketle{t˙c(0),˙c(0)/a\\}b∇acket∇i}ht=−1\nn+2,\nRe/a\\}b∇acketle{t˙s(0),˙c(0)/a\\}b∇acket∇i}ht=1\nn+2,\nRe/a\\}b∇acketle{t˙s(0),˙s(0)/a\\}b∇acket∇i}ht=/a\\}b∇acketle{t¯p,¯p/a\\}b∇acket∇i}ht= 0.37\nThus (7.3.1) or (3.3.9) respectively shows that\nˆg0(ξ,ξ) = ˆg0(PR∗(˙c(0)),PR∗(˙c(0))) = Re /a\\}b∇acketle{t˙c(0),˙c(0)/a\\}b∇acket∇i}ht.\nˆg0(S,ξ) = ˆg0(PR∗(˙s(0)),PR∗(˙c(0))) = Re /a\\}b∇acketle{t˙s(0),˙c(0)/a\\}b∇acket∇i}ht.\nSince it is easy to see that ker ω0=PR∗(W0) from (4.1.3), we have\nthat\ndω0(JPR∗X,PR∗Y) =|dz1|2+···+|dzn+1|2(X,Y)\n= Re/a\\}b∇acketle{tX,Y/a\\}b∇acket∇i}ht(∀X,Y∈W0).\nHence we obtain that\nˆg0(PR∗X,PR∗Y) = Re/a\\}b∇acketle{tX,Y/a\\}b∇acket∇i}ht(X,Y∈T(V0)).\n/square\nLemma 7.3.1. If the one-parameter group has the form (cf.Case C)\nˆρ(t) = (eita1,...,eitak,1,...,1)∈Tn+1·S1,\nthenˆξis spacelike on S2n+1,1−S1·S2(n−k)+1.\nProof.Choose an arbitrary point v= (z1,...,z k,zk+1,...,z n+1,w1)∈\nV0such that\n(7.3.4) |z1|2+···+|zk|2+|zk+1|2+···+|zn+1|2−|w1|2= 0.\nThen\nρ(t)(z1,...,z k,zk+1,...,z n+1,w1)\n= (eita1z1,...,eitakzk,zk+1,...,z n+1,w1).(7.3.5)\nIt follows that ξv= (ia1z1,...,iakzk,0,...,0) for which\n/a\\}b∇acketle{tξv,ξv/a\\}b∇acket∇i}ht=a2\n1|z1|2+···+a2\nk|zk|2≥0.\n/a\\}b∇acketle{tξv,ξv/a\\}b∇acket∇i}ht= 0 if and only if z1=···=zk= 0. In this case such a point\nvsatisfies that\nPR(v) =PR((0,...,0,zk+1,...,z n+1,w1))\n=w1\n|w1|PR((0,...,0,zk+1\nw1,...,zn+1\nw1,1))∈S1·S2(n−k)+1.(7.3.6)\nThis shows the lemma. (Compare (6.1.19).) /square\nLemma 7.3.2. If the one-parameter group has the form (cf.Case D)\nˆρ(t) = (eita1,...,eitak,1,...,1)eit∈Tn+1·S1,\nthen either one of the following holds.\n(1)When all ai>0or allai<−2 (i= 1,...,k),ˆξis spacelike on\nS2n+1,1−S1·S2(n−k)+1.38\n(2)When all −2≤ai<0,ˆξis timelike on S2n+1,1−S1·S2(n−k+ℓ)+1\nfor some ℓ < k.\n(3)Suppose that there exist ai,aj(1≤i,j≤k)with(i)ai>0or\nai<−2, or with (ii)−2< aj<0. Thenˆξis not causal on\nS2n+1,1−S1·S2(n−k)+1.\nProof.For a point vof (7.3.4), it follows similarly as above:\nξv= (i(a1+1)z1,...,i(ak+1)zk,izk+1,...,izn+1,iw1),\nand so\n/a\\}b∇acketle{tξv,ξv/a\\}b∇acket∇i}ht= ((a1+1)2−1)|z1|2+···+((ak+1)2−1)|zk|2. (7.3.7)\nThen the following possibilities occur:\n(1)If allai>0 or allai<−2, then/a\\}b∇acketle{tξv,ξv/a\\}b∇acket∇i}ht>0 and/a\\}b∇acketle{tξv,ξv/a\\}b∇acket∇i}ht= 0 if and\nonly ifz1=···=zk= 0 for which PC(v)∈S2(n−k)+1.\n(2)Ifall−2≤ai<0, then/a\\}b∇acketle{tξv,ξv/a\\}b∇acket∇i}ht ≤0. Supposethat ai1=···=aiℓ=\n−2forsome ℓ. Bytheassumptionthattheg.c.mofall aiis1(cf.Section\n6.1.2), note that ℓ < k. Letv= (z1,...,z k,zk+1,...,z n+1,w1)∈V0\nsuch that ziℓ+1=···=zik= 0. Then /a\\}b∇acketle{tξv,ξv/a\\}b∇acket∇i}ht= 0 if and only if\nPC(v)∈S2(n−k+ℓ)+1.\n(3)If there exist ai,ajsuch that ai>0 (orai<−2), or−2< aj<0\n(i.e.(ai+1)2−1>0, (aj+1)2−1<0), then/a\\}b∇acketle{tξv,ξv/a\\}b∇acket∇i}htcan be taken to\nbe zero, positive or negative. /square\nAs a consequence,\nCausality (2) . Ifρ(t) =e−it·Ctorρ(t) =eit·Ctfor (i), (ii) of (6.1.12),\nξcannot be lightlike.\n7.4.Curvature equivalence. Suppose that ( S1,M,[g]) is a (2n+2)-\ndimensional conformally flat Fefferman-Lorentz parabolic manifold f or\nwhichS1≤ConfFLP(M) and let\n(7.4.1) (˜ ρ,dev) : (˜S1,˜M)→(U(n+1,1)∼,˜S2n+1,1)\nbethedeveloping pair. Here ˜S1istheliftof S1totheuniversal covering\n˜M. It is either S1orR. If˜S1induces the vector field ˜ξon˜M, then we\nnote from Definition 2.2.1 that\n˜gx(˜ξ,˜ξ) = 0,˜ξx/\\e}atio\\slash= 0 (x∈˜M).\nAs dev(t·x) = ˜ρ(t)dev(x) (t∈˜S1), ˜ρ(˜S1) induces the vector field\ndev∗˜ξon the domain dev( ˜M)⊂˜S2n+1,1. Let ˜g(respectively ˜ g0) be the39\nlift ofg(respectively the lift of canonical metric g0to˜S2n+1,1). Since\ndev is a conformal immersion, there is a function u >0 on˜Msuch\nthatu(x)·˜gx(v,w) = ˜g0(dev∗v,dev∗w) In particular,\n(7.4.2) ˜ g0\ndev(x)(dev∗˜ξ,dev∗˜ξ) = 0 (dev( x)∈dev(˜M)).\nTheorem 7.4.1. Let(M,[g])be a(2n+2)-dimensionalconformallyflat\nFefferman-Lorentz parabolic manifold which admits S1≤ConfFLP(M).\nIf theS1-action is lightlike and has no fixed points on M, then one of\nthe following holds.\n(i)Mis a Seifert fiber space over a spherical CRorbifoldM/S1.\n(ii)The developing pair (˜ρ,dev)reduces to\n(C(˜S1),˜M)→(R×(N⋊U(n)),R×N)\nwhereR×S2n+1−R1·∞=R×N.\nProof.Suppose that G/\\e}atio\\slash={1}forH=S1. By the hypothesis, there\nare four possibilities Cases A, B,C, D by Proposition 6.1.1. Among\nthem, as S1is lightlike, Causality (1) andCausality (2) of Section\n7.1 imply Case A (1), which shows (ii).\nWhenG={1}, first note that ˜ ρ(˜S1) =R, the center of U( n+1,1)∼.\nAs˜S1=R, we have a central extension : 1 →Z→R−→S1→1. Let\nZ(π) be the center of π=π1(M). Since ˜S1belongs to the centralizer\nZDiff(˜M)(π), it follows ˜S1∩π⊂ Z(π). This shows that Z=˜S1∩Z(π) =\n˜S1∩π. Then this induces a central extensions:\n(7.4.3)1− −− →Z− −− →π− −− →Q− −− →1\n/intersectiontext /intersectiontext||\n1− −− →R− −− →π·R− −�� →Q− −− →1.\nAsR=˜S1acts properly and freely on ˜M, putW=˜M/˜S1. More-\nover, noting that R/Z=S1andπacts properly discontinuously, the\ngroupπ·Racts properly on ˜M. As a consequence, Qacts properly\ndiscontinuously on Wwith the equivariant fibration:\n(7.4.4) ( R,R)→(π·R,˜M)−→(Q,W).\nOn the other hand, there is the commutative diagram of the holonom y:\n(7.4.5)R− −− →U(n+1,1)∼˜P− −− →PU(n+1,1)\n||/uniontext /uniontext\n˜ρ(R)− −− → ˜ρ(π·R)˜P− −− → ˆρ(Q).40\nThenthedeveloping pair(7.4.1)inducesanequivariant developing map\non the quotient space:\n(ˆρ,ˆdev): (Q,W)→(PU(n+1,1),S2n+1),\nwhereS2n+1=˜S2n+1,1/R. Since (PU( n+1,1),S2n+1) is the spherical\nCR-geometry, Winherits a spherical CR-structure on which Qacts as\nCR-transformations. Taking the quotient of (7.4.4), S1→M→M/S1is\na Seifert fiber space over the CR-orbifold M/S1=Q\\W. /square\nUsing Theorem 7.4.1 we can prove the following equivalence.\nTheorem 7.4.2. Let(S1×N,g)be a Fefferman-Lorentz manifold for a\nstrictly pseudoconvex CR-manifold (N,(ωN,JN))of dimension 2n+1≥\n3in which g=σ⊙P∗ωN+dωN(JNP∗−,P∗−)is a Fefferman metric.\nThen(S1×N,g)is conformally flat if and only if (N,(ωN,JN))is\nspherical CR.\nProof.By Theorem 7.4.1, the case (i) or (ii) occurs. If (i) occurs, then\n(Q,W) = (π1(N),˜N) with˜S1=Rfor which there is a developing map\n(7.4.6) (ˆ ρ,ˆdev) : (Q,˜N)→(PU(n+1,1),S2n+1).\nWe have to check that the spherical CR-structure ( ω,ˆJ) induced by\nˆdev coincides with the original one ( ωN,JN) onN. The contact form\nωis obtained as\nP∗ω=g(S,−)\nfrom (6.1.24) of Proposition 6.1.2 and the complex structure ˆJis de-\nfined by\nˆdev∗ˆJ=J0ˆdev∗\non Kerωfrom (6.1.28).\nLetSbe the vector field induced by S1onS1×N. Sinceσ(S) =\n1\nn+2from (3.3.7), it follows that g(S,−) =1\nn+2P∗ωN(−) and so\n(7.4.7) ωN= (n+2)ω.\nIf we note that S1acts as lightlike isometries of ( S1×N,g), then it\nsatisfies also Proposition 6.1.2 (cf .Remark 6.1.1). Then from (6.1.27),\n(n+2)ˆu·ω=ˆdev∗ω0,\nwhich implies that ˆdev∗(Kerω) = Kerω0.\nLet dev∗g0=u·gas before. If P∗X,P∗Y∈Kerω, then\ng(X,Y) =σ⊙P∗ωN(X,Y)+dωN(JNP∗X,P∗Y) =dωN(JNP∗X,P∗Y)41\nNotingP∗ˆu=u,\n(7.4.8) u·g(X,Y) = ˆu·dωN(JNP∗X,P∗Y).\nOn the other hand, using ˆdev∗ˆJ=J0ˆdev∗with (6.1.23),\ndev∗g0(X,Y) =g0(dev∗X,dev∗Y)\n=dω0(J0P∗(dev∗X),P∗(dev∗Y)\n=dω0(J0ˆdev∗P∗X,ˆdev∗P∗Y)\n=dω0(ˆdev∗ˆJP∗X,ˆdev∗P∗Y)\n=ˆdev∗dω0(ˆJP∗X,P∗Y).\nNoting that ( n+2)ˆu·dω=ˆdev∗dω0on Kerω, it follows by (7.4.7) that\ndev∗g0(X,Y) = (n+2)ˆu·dω(ˆJP∗X,P∗Y)\n= ˆu·dωN(ˆJP∗X,P∗Y).(7.4.9)\nCompared (7.4.8) and (7.4.9), we conclude that\nˆJ=JN.\nHence (Ker ω,ˆJ) = (Ker ωN,JN) so that ( N,(ωN,JN)) is a spherical\nCR-manifold.\nWe have to show that the case (ii) of Theorem 7.4.1 does not occur.\nIf (ii) occurs, then we have a developing pair by Proposition 6.1.1:\n(7.4.10) (˜ ρ,dev) : (˜S1,RטN)→(R×(N⋊U(n)),R×N).\nHere˜S1=R. Let˜Sbe the vector field induced by RonRטNas\nbefore. Put dev ∗˜S=˜S′. Let\ngN=σN⊙P∗ωN+dωN(JP∗−,P∗−)\nbe the Lorentz metric on R× Nwhich is conformal to the standard\nmetricg0. (Compare Section 7.1.) Note that gN(˜S′,˜S′) = 0 because ˜S\nis lightlike. In this case, Causality (1) shows that ˜S′is the character-\nistic vector field, i.e .ωN(P∗˜S′) = 1. Moreover, Proposition 6.1.1 with\nCausality (2) implies that the lightlike vector field ˜S′is of type (i) of\n(6.1.12). As σN=1\nn+2dt(cf.Section 7.2), it follows that σN(˜S′) = 0.42\nAsbefore, thereexistsafunction u >0onR×Nsuchthatdev∗gN=\nu·g. Noting that P∗˜S′is characteristic, a calculation shows that\ngN(dev∗˜S,dev∗V) =gN(˜S′,dev∗V)\n=σN⊙P∗ωN(˜S′,dev∗V)+dωN(JP∗˜S′,P∗dev∗V)\n=σN(dev∗V)\nu·g(˜S,V) =u(σ⊙P∗ωN(˜S,V)+dωN(JP∗˜S,P∗V))\n=u\nn+2·ωN(P∗V) (P∗˜S= 0).\nIt follows that\n(7.4.11) dev∗σN=u\nn+2·P∗ωN.\nIt is easy to see thatun+1\n(n+2)n+1P∗(ωN∧(dωN)n) = dev∗(σN∧(dσN)n)\nonRטN. AsdσN= 0 as above, it follows that P∗(ωN∧(dωN)n) = 0\nso thatωN∧(dωN)n= 0 on˜N, which contradicts that ωNis a contact\nformon ˜N. Therefore thecase (ii) ofTheorem 7.4.1 cannot occur. This\nproves the necessary condition.\nSuppose that Nis spherical CR. There exists a collection of charts\n{Uα,ϕα}α∈Λsuch that ϕα:Uα→ϕα(Uα)⊂S2n+1is a homeomorphism.\nConsider the pullback of the S1-bundle:\n(7.4.12)S1− −− → S1\n/arrowbt/arrowbt\nS1×Uα˜ϕα− −− →S1×S2n+1\n/arrowbt PC/arrowbt\nUαϕα− −− → S2n+1\nin which\n(7.4.13) ˜ ϕα(t,x) = (t,ϕα(x))\nWhenUα∩Uβ/\\e}atio\\slash=∅, the local change ϕα◦ϕ−1\nβextends to an automor-\nphismh∈PU(n+ 1,1) ofS2n+1. PutU=ϕβ(Uα∩Uβ)⊂S2n+1.\nConsider the local diffeomorphism:\n(7.4.14) ˜h= ˜ϕα◦˜ϕ−1\nβ:S1×U→S1×S2n+1\nfor which\n(7.4.15) ˜h(t,z) = (t,hz).43\nIf we note that U( n+1,1)acts invariantly on V0such that S1×S2n+1⊂\nV0, then there exists an element f∈U(n+ 1,1) withPf=hwhich\nsatisfies that\n(7.4.16) ˜h=f|S1×U.\nBy Proposition 4.1.1, U( n+1,1) acts conformally on S1×S2n+1with\nrespect to g0so it follows that\n˜h∗g0=f∗g0=v·g0(∃v >0).\nSince˜his a local conformal diffeomorphism, ˜hextends to a global\nconformal transformation of S1×S2n+1by the Liouville’s theorem. By\nuniqueness,\n˜h=fonS1×S2n+1.\nAs a consequence, the local change ˜ ϕα◦˜ϕ−1\nβextends to an automor-\nphism˜h∈U(n+ 1,1) ofS2n+1×S1. Therefore the charts {Uα×\nS1,˜ϕα}α∈ΛofN×S1gives a uniformization with respect to (U( n+\n1,1),S2n+1×S1). As (U( n+ 1,1),S2n+1×S1) is the lift of ( ˆU(n+\n1,1),S2n+1,1),N×S1is a conformally flat Fefferman-Lorentz mani-\nfold.\n/square\n8.Application to Obata & Ferrand’s theorem\n8.1.Noncompact conformal group actions. LetCbeaclosednon-\ncompact subgroup of Diff( M). Suppose that Cactsanalytically onM.\nContrary to compact group actions, noncompact (analytic) Lie gr oup\nactions on a compact manifold is quite different. For example, there is\na noncompact analytic action ( C,M) such that the set of nonprincipal\norbitsM0={x∈M|dimC ·x <dimC}coincides with M. (See [28],\n[2, 3.1 Theorem, p.79] for instance.) We give a sufficient condition that\nM0is nowhere dense in Mfor noncompact Lorentz groups Cacting\nanalytically on a Lorentz manifold ( M,g). LetC=S1×Rwhich is\nclosed in Diff( M). IfS1does not act as Lorentz isometries with respect\ntog, then we put\n(8.1.1) ˜g=/integraldisplay\nS1h∗gdh\nwheredsis a right-invariant Haar measure on S1. ThenS1acts as\nLorentz isometries with respect to ˜g. On the other hand, as S1acts\nconformally with respect g,h∗g=λh·gfor some function λh>0 on\nM. Letting τ(x) =/integraldisplay\nS1λh(x)dh(x∈M), it follows that ˜g=τ·gon\nM. We obtain a Lorentz metric ˜gconformal to g. So ifC=S1×R44\nacts conformally on ( M,g), we may assume that S1acts as isometries\nwithin the conformal class of the Lorentz metric g.\nProposition 8.1.1. If a Lorentz (n+ 2)-manifold (M,g)admits a\nclosed two dimensional subgroup Cisomorphic to S1×RwhereS1\nconsists of lightlike conformal transformations, then\nV={x∈M|dimC ·x= 2}\nis a dense open subset of M.\nProof.We suppose that S1acts isometries. Let ˜F(respectively F)\nbe the fixed point set of R(respectively S1). Note that S1leaves˜F\ninvariant. If Eis the set of exceptional orbits of S1, thenS1acts freely\non the complement M0=M−(E∪F∪˜F). Note that M0is a dense\nopen subset of M. There is a principal bundle over the orbit space\nN0=M0/S1;\n(8.1.2) S1− −− →M0P− −− →N0.\nSuppose that dim C·x= 1 for some open subset UofM0(∀x∈U). If\nwe set{ϕt}t∈R=R, then it follows that\nC ·x={ϕtx}t∈R=S1·x.\nLetξxbe the vector induced by C ·x(x∈U). By the hypothesis, ξis\na lightlike (Killing) vector field on U. For an arbitrary point y∈U, as\nϕty∈S1·y, there exists an element hy\nt∈S1such that\n(8.1.3) ϕty=hy\nt·y.\nThis implies that P◦ϕt=PonM0. Putz=ϕty=htywhere\nwe letht=hy\ntfor brevity. For a vector vy∈TyM0, we have that\nP∗ϕt∗vy=P∗vy=P∗ht∗vy. Sinceϕt∗vy, ht∗vy∈TzM0, it follows that\nϕt∗vy=ht∗vy+aξz(∃a∈R).\nAsξyis lightlike, we can find a vector ηy∈TyM0such that\ng(ηy,ηy) = 0, g(ξy,ηy) = 1.\nAs above, there exists an element b∈Rsuch that\nϕt∗ηy=ht∗ηy+bξz.\nSinceg(ϕt∗ηy,ϕt∗ηy) =λt(y)·g(ηy,ηy) = 0 and hty=z, a calculation\nshows that\n0 =g(ht∗ηy+bξz,ht∗ηy+bξz)\n= 2bg(ht∗ηy,ξz) = 2bg(ht∗ηy,ht∗ξy)\n= 2bg(ηy,ξy) = 2b,(8.1.4)\nso it follows that ϕt∗ηy=ht∗ηy.45\nNoting that {ξy,ηy}spans a nondegenerate plane of signature (1 ,1),\nthereexistsavector vysuchthat g(vy,vy) = 1, g(ξy,vy) =g(ηy,vy) = 0.\nThere are n-independent such vectors. The set of those vectors with\n{ξy,ηy}constitutes TyM0. As above, let ϕt∗vy=ht∗vy+aξz. Similarly,\nusingϕt∗ηy=ht∗ηy, the equation g(ϕt∗ηy,ϕt∗vy) = 0 shows that a= 0,\ni.e.ϕt∗vy=ht∗vy. From these calculations, we obtain that\n(8.1.5) ϕt∗Xy=ht∗Xy,(∀Xy∈TyM0).\nNow, noting ht∈S1,\n(8.1.6)ϕ∗\ntg(Xy,Yy) =g(ht∗Xy,ht∗Yy) =g(Xy,Yy) (∀Xy,Yy∈TyM0).\nOn the other hand, since Racts conformally, there exists a positive\nfunction λtonMsuch that ϕ∗\ntg=λt·gfor eacht, (8.1.6) implies that\nλt(y) = 1. This is true for an arbitrary point y∈U, soλt= 1 onU.\nIn particular, ϕt(∀t∈R) becomes a Lorentz isometry on U(and so is\nonMby analyticity).\nRecall from (8.1.3) that ϕty=ht·y. SinceR={ϕt}t∈R, there exists\nan element a∈Rsuch that ϕay=y. (In fact, if a(y) = min\nt∈R+{t|ϕty=\ny}, we put a=a(y).) Then it follows that y=ϕay=hay. AsS1acts\nfreely on M0,ha= 1. From (8.1.5), we have that\n(8.1.7) ϕa∗Xy=Xy(∀Xy∈TyM0).\nSinceϕais a Lorentz isometry, if γis any geodesic issuing from y, then\nϕaγis also a geodesic on U. From (8.1.7), the uniqueness of geodesic\nimplies that ϕaγ=γonU. Hence ϕa= id on U. By analyticity,\nϕa= id on M. Letting Z=/a\\}b∇acketle{tna/a\\}b∇acket∇i}htn∈Zso thatS1=R/Z,Cwould\nbe isomorphic to S1×S1. This contradicts our hypothesis that Cis\nnoncompact. Hence the subset {x∈M|dimC ·x= 2}is dense open\ninM.\n/square\nTheorem 8.1.1. LetM=S1×Nbe a compact Fefferman-Lorentz\nmanifold and CConf(M,g)(S1)the centralizer of S1inConf(M,g). Sup-\npose that CConf(M,g)(S1)contains a closed noncompact subgroup of di-\nmension 1at least. Then Mis conformally equivalent to the two-fold\ncoverS1×S2n+1of the standard Lorentz manifold S2n+1,1.\nProof.We can choose a closed subgroup C=S1×RfromCConf(M,g)(S1)\nby the hypothesis. Here recall that the vector field Sgenerated by S1\nofM=S1×Nis lightlike.\nRecall from (4.2.1) that\n(8.1.8) g=σ⊙P∗ω+dω(JP∗−,P∗−)46\nisaLorentzmetriconaFefferman-Lorentzmanifold M=S1×Nwhere\nP:S1×N→Nis the projection. Then Cinduces an action of Ron\nthe quotient Nsuch that Pis equivariant:\n(8.1.9) P: (C,M)→(R,N).\nIf{ϕt}t∈Ris a 1-parameter group of RofC, then there exists a 1-\nparameter group of {ˆϕt}t∈Rsuch that\nP◦ϕt= ˆϕt◦P.\nSinceRacts as conformal transformations with respect to g, there\nexists a function λt:M→R+such that\n(8.1.10) ϕ∗\ntg=λt·g.\nIfh∈S1, sinceh∗ϕ∗\ntg=ϕ∗\nth∗g=ϕ∗\ntgandh∗ϕ∗\ntg=h∗(λt·g) =h∗λt·g,\nit follows that h∗λt=λt(∀h∈S1). Soλtfactors through a function\nˆλt:N→R+(∀t∈R). We note also that ϕt∗S=SandP∗S= 0. Then\nϕ∗\ntg(X,S) =1\nn+2ω(P∗ϕ∗X) =1\nn+2P∗ˆϕ∗ω(X)\n=λt·g(X,S) =1\nn+2P∗ˆλt·P∗ω(X).(8.1.11)\nit follows that\n(8.1.12) ˆ ϕ∗\ntω=ˆλt·ω(∀t∈R).\nThis implies that\n(8.1.13) ˆ ϕt∗Kerω= Kerω.\nRecall that there is a complex structure Jon Kerω. For convenience,\nwe put˜JonP∗Kerωformally such that P∗:P∗Kerω→Kerωis almost\ncomplex, i.e .P∗◦˜J=J◦P∗.\nLetX,Y∈P∗Kerω. Calculate\nϕ∗\ntg(−X,Y) =g(−ϕt∗X,ϕt∗Y)\n=dω(JP∗(−ϕt∗X),P∗(ϕt∗Y)) by (8.1.8)\n=dω(−Jˆϕt∗P∗X,ˆϕt∗P∗Y)\n=dω(ˆϕt∗P∗X,Jˆϕt∗P∗Y).(8.1.14)47\nNoting that ˆλt·dω= ˆϕ∗\ntdωon Kerω, calculate\nϕ∗\ntg(−X,Y) =λt·g(−X,Y)\n=λt·dω(JP∗(−X),P∗Y) =λt·dω(P∗X,JP ∗Y)\n=λt·dω(P∗X,P∗(˜J)Y) =P∗(ˆλt·dω)(X,˜JY)\n=ˆλt·dω(P∗X,P∗(˜JY)) =ˆλt·dω(P∗X,JP ∗Y)\n= ˆϕ∗\ntdω(P∗X,JP ∗Y)\n=dω(ˆϕ∗\ntP∗X,ˆϕ∗\ntJP∗Y).(8.1.15)\nAsdωis nondegenerate on Ker ω, we conclude that\n(8.1.16) ˆ ϕt∗J=Jˆϕt∗on Kerω.\nLet Aut CR(N) be the group of CR-transformations of ( ω,J) onN. By\nthe definition, {ˆϕt}t∈R⊂AutCR(N). Note that {ˆϕt}t∈Ris closed by\nthe hypothesis. Moreover, the action of {ˆϕt}t∈Ris nontrivial on Nby\nProposition 8.1.1. Hence, {ˆϕt}t∈Ris a closed noncompact subgroup in\nAutCR(N). It follows from the CR-analogue of Obata-Ferrand rigidity\n(for example [15], [30], [10], [22], [29]) that NisCR-isomorphic to\nthe standard sphere S2n+1. Then ( S1×N,g) is conformally flat by\nTheorem 7.4.2. Let ˜Cbe a lift of Cto˜M. By Proposition 5.1.1, we\nhave the developing pair:\n(8.1.17) (˜ ρ,dev) : (˜C,˜M)−→(U(n+1,1)∼,˜S2n+1,1).\nRecall that Ris the center of U( n+1,1)∼which is the kernel of pro-\njection˜P: U(n+ 1,1)∼→PU(n+ 1,1) andRis the center of the\nHeisenberg group Nin PU(n+ 1,1) (cf.(6.1.1)). Let ˜S1be a lift of\nlightlike one-parameter subgroup to ˜C. We show that\n(8.1.18) ˜ ρ:˜S1−→R\nis isomorphic. For this, put G=˜P◦˜ρ(˜S1)⊂PU(n+1,1) as in (6.1.7).\nCausality (1) and (2) in Section 7 yield that\n˜ρ(˜H1) =/braceleftbigg\nRifG/\\e}atio\\slash={1},\nRifG={1}.\nIfG/\\e}atio\\slash={1}, then by (ii) of Theorem 7.4.1 the developing pair reduces\nto\n(˜ρ,dev) : (C(˜S1),˜M)→(R×(N⋊U(n)),R×N)\nwhereR×S2n+1−R1·∞=R×N. Since˜C ⊂ C(˜S1)andR×(N⋊U(n))\nis transitive on R×Nwith compact stabilizer U(n), ˜Madmits a π·˜C-\ninvariant Riemannian metric ˜g. Taking the quotient, it follows that\nC ≤Isom(M,g). AsMis compact, Isom( M) is compact. Since Cis a48\nclosed noncompact subgroup (in Diff( M)) by our hypothesis, this case\ncannot occur.\nThenG={1}and so ˜ρ(˜S1) =R. It follows from (i) of Theorem\n7.4.1 that Mis a Seifert fiber space over a spherical CRorbifoldM/S1.\nInour case, M/S1=˜M/˜S1=Nwhich is simply connected. We obtain\nthe following commutative diagram:\n(8.1.19)˜S1˜ρ− −− → R/arrowbt/arrowbt\n˜M˜dev− −− →R×S2n+1\nP/arrowbt ˜P/arrowbt\nNˆdev− −− → S2n+1.\nMoreover, the CR-structure ( ω,J) onNcoincides with the pullback of\nthe standard CR-structure ( ω0,J0) ofS2n+1. In fact, we have shown in\nthe proof of Theorem 7.4.2 that\nˆu·ω=ˆdev∗ω0,\nˆdev∗ˆJ=J0dev∗on Kerω(8.1.20)\nAs (N,(ω,J)) isCR-isomorphic to S2n+1as above, there exists an\nelement h∈PU(n+ 1,1) such that h◦ˆdev :N→S2n+1is aCR-\ndiffeomorphism. (Asaconsequence, ˆdevitselfisa CR-diffeomorphism.)\nBy the diagram (8.1.19), ˜dev :˜M→R×S2n+1is a conformal diffeo-\nmorphism. Taking a quotient, ˜dev induces a conformal diffeomorphism\ndev :M→S1×S2n+1.\n/square\nRemark 8.1.1. LetSL(2,R)/Γbe a Lorentz space form of negative\nconstant curvature where Γ≤O(2,2)0= SL(2,R)·SL(2,R)is a sub-\ngroupacting properlydiscontinuouslyon SL(2,R)(cf.[20]). If wechoose\nΓ≤SO(2)×\nZ2SL(2,R),\nthenSL(2,R)/Γis a spherical CR-space form because there is a canon-\nical identification:\nSO(2)×\nZ2SL(2,R) = U(1,1)<U(1)×U(1,1)<U(2,1),\nSL(2,R) =S3−S1.49\n(1)ThusM=S1×SL(2,R)/Γis a conformally flat Fefferman-\nLorentz4-manifold on which S1acts as lightlike isometries by the def-\ninition.\n(2)Recall that (PO(4,2),S3,1)is the conformally flat Lorentz geom-\netry. Let (O(4,2),S1×S3)be the two fold covering. The subgroup of\nO(4,2)preserving S1×(S3−S1) =S1×SL(2,R)is isomorphic to\nO(2)×O(2,2). When we restrict Γto{1} ×SL(2,R), we obtain a\nconformally flat Lorentz parabolic manifold M=S1×SL(2,R)/Γon\nwhichS1= SO(2) acts as spacelike isometries. 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Kamishima, Nondegenerate conformal, CR, Quaternionic CR structure on\nmanifolds ,Handbook of Pseudo-Riemannian Geometry and Supersymmetry”,\n(ed. Vicente Cortes), EMS in the series IRMA lectures in Mathematic s and\nTheoretical Physics. Vol. 16, 863-896 (2010).\n[14] Y. Kamishima, Heisenberg, Spherical CR geometry and Bochner flat locally\nconformal K¨ ahler manifolds ,International Journal of Geometric Methods in\nModern Physics , Vol. 3, Nos. 5-6, 1089-1116 (2006).\n[15] Y. Kamishima, Geometric flows on compact manifolds and global rigidity ,\nTopology, 35, 439-450 (1996).\n[16] Y. Kamishima, Completeness of Lorentz manifolds of constant curvature ad -\nmitting Killing vector fields ,J. Differential Geom. , vol. 37, 569-601 (1993).\n[17] Y. Kamishima and T. Tsuboi, CR-structures on Seifert manifolds ,Invent.\nMath., vol. 104, 149-163 (1991).\n[18] S. Kobayashi, Transformation Groups in Differential Geometry ,Springer-\nVerlag, Ergebnisse Math., vol. 70, 1970.\n[19] J. L. Koszul, ‘Lectures on groups of transformations,’ Tata Institute of Fun-\ndamental Research, Bombay, Notes by R. R. Simha and R. Sridharan, 1965.\n[20] R.S. Kulkarni, F. Raymond, 3 -dimensional Lorentz space-forms and Seifert\nfiber spaces , J. Differential Geom. 21, no. 2, 231-268 (1985).\n[21] J. M. Lee, The Fefferman metric and pseudoHermitian invariants , Trans.\nA.M.S. 296, no. 1, 411-429 (1986).\n[22] J. M. Lee, CR manifolds with noncompact connected automorphisms grou ps,\nJ. Geom. Anal., (1)6, 79-90 (1996).\n[23] J. Lelong-Ferrand, Transformations conformes et quasi conformes des vari´ et´ es\nriemanniennes compactes , Acad. Roy. Belgique Sci. Mem. Coll., 8, 1-44(1971).\n[24] K.B. Leeand F. Raymond, Seifert manifolds , Handbookofgeometrictopology,\n635-705, North-Holland, Amsterdam, 2002.\n[25] M. Obata, The conjectures on conformal transformations of Riemannia n man-\nifolds, J. Diff. Geom., 6, 247-258 (1971).\n[26] B. O’Neill, Semi-Riemannian geometry ,Academic Press Inc. , 1983.\n[27] R. S. Palais, ‘A global formulation of the Lie theory of transform ation groups,’\nMem. Amer. Math. Soc. , Vol 22 (1957).\n[28] R. W. Richardson, Jr., “Deformation of Lie subgroups and the v ariation of the\nisotropy subgroups,” Acta. Math. 129, 35-73 (1972).\n[29] R. Schoen, On the conformal and CR automorphism groups , Geom. Funct.\nAnal., (2)5, 464-481 (1995).\n[30] S. Webster On the transformation group of a real hypersurfaces ,Trans. Amer.\nMath. Soc . Vol 231 (2), 179-190 (1977).\n[31] J. Wolf, ‘Spaces of constant curvature,’ McGraw-Hill, Inc. , 1967.\nDepartment of Mathematics, Tokyo Metropolitan University ,\nMinami-Ohsawa 1-1, Hachioji, Tokyo 192-0397, Japan.\nE-mail address :kami@tmu.ac.jp"    },    {        "title": "2304.11008v2.Violation_of_the_Landau_Yang_theorem_from_Infrared_Lorentz_Symmetry_Breaking.pdf",        "content": ",\nPrepared for submission to JHEP\nViolation of the Landau-Yang theorem from Infrared\nLorentz Symmetry Breaking\nM. Asorey,aA. P. Balachandran,bM. Arshad Momencand B. Qureshid\naCentro de Astropart´ ıculas y F´ ısica de Altas Energ´ ıas, Departamento de F´ ısica Te´ orica, Universi-\ndad de Zaragoza, E-50009 Zaragoza, Spain\nbPhysics Department, Syracuse University, Syracuse, New York 13244-1130, U.S.A.\ncDept. of Physical Science, Independent University, Bangladesh(IUB), Dhaka 1229, Bangladesh\nand\nCenter for Computational and Data Sciences(CCDS), Independent University, Bangladesh(IUB),\nDhaka 1229, Bangladesh\ndDepartment of Mathematical Sciences, IBA, Karachi, Pakistan\nE-mail: asorey@unizar.es ,apbal1938@gmail.com ,arshad@iub.edu.bd ,\nbabaraqureshi@gmail.com\nAbstract: Lorentz symmetry forbids decays of massive spin-1 particle like the Z0into\ntwo massless photons, a result known as the Landau-Yang theorem. But it is known that\ninfrared effects can break Lorentz invariance. Employing the construction of Mund et.\nal. [1] which incorporated this Lorentz violation, we propose an interaction leading to the\ndecay Z0→2γand study the dependence of the decay on the parameter of this Lorentz\nviolation.\nArXiv ePrint: 2304.11008arXiv:2304.11008v2  [hep-ph]  11 Sep 2023Contents\n1 Introduction 1\n2 The Infrafield 2\n3 A Model Interaction for Z0→2γ 3\n4 Calculation of the Amplitude using the Mode Expansion of the Escort\nField 4\n1 Introduction\nIn Quantum Field Theories (QFTs) with a mass gap, it is assumed that the Poincar´ e group\nhas a unitary action in the Hilbert space of the theory leaving the vacuum invariant. It\nis thus a ‘symmetry’ of the theory acting on the local observables in a bounded spacetime\nregion. Causality can also be formulated by requiring the local observables in spacelike\ncomplements to commute.\nBut there are physically important theories with no mass gap, QED being a prime\nexample. In QED, because of infrared effects, there are uncountably many superselection\nsectors, and Lorentz transformations map one such sector into another. Hence Lorentz\nsymmetry is spontaneously broken [12], just like the U(1) gauge group which is sponta-\nneously broken in superconductivity or ferromagnetism.\nThere is an important theorem of Landau and Yang [2], [3] for Lorentz invariant\ntheories : the decay of a massive spin 1 particle such as Z0to two photons is forbidden.\nThis theorem is remarkable: its proof does not rely on principles of quantum field theory\n(QFT) such as causality. This result is due to the fact that the the Clebsch-Gordan\nseries for the tensor product of two irreducible Poincar´ e group representations of photons,\nwhen Bose symmetrised, does not contain a massive spin one representation, as shown by\nBalachandran, Jo and Marmo [4]. There are generalisations of this result to other decays\n[5] shown there. For example, Z0cannot decay into two massless neutrinos.\nThus, the Landau-Yang and related theorems use only Poincar´ e invariance and statis-\ntics of identical particles and invokes no other principles of quantum field theory. There\nare also experiments in non-linear optics which put limits on the rate of such Landau-Yang\nselection rule violating processes [6–8]. The observation of any one of these decays will\nthus profoundly affect QFT.\n– 1 –On the other hand, it has been known for a long time that infrared effects in QED create\nuncountably many superselection sectors and that the action of Lorentz transformations\ninterchanges these sectors. Hence by definition, Lorentz symmetry is spontaneously broken.\nIn a recent paper, Mund, Rehren and Schroer [1] have reviewed these results and developed\na theory of ‘infrafields’ which can serve as order parameters for this symmetry breaking.\nGiven these results and developments, it is natural to ask if a model for say Z0→2γdecay\ncan be formulated using the infrafields. In this paper, we argue that this can be done and\nan explicit decay rate can be derived. We emphasize that the mechanism of Z0→2γ\ndecay that we are discussing does not invoke any Beyond Standard Model (BSM) physics\nas happens with extra generations of fermions ( see e.g. [9]) or by requiring explicit Lorentz\nviolating new interactions ( see e.g. [10] for a recent review ). Rather it arises because of\nthe infrared structure of QED itself.\nIn the subsequent sections, we review the construction of infrafields and their response\nto gauge transformations. Then we formulate an interaction for Z0→2γwhich is invariant\nunder “small” or Gauss law-generated gauge transformations which is not Lorentz invariant\nand compute the rate for the above decay.\n2 The Infrafield\nSuch fields first acquired a prominent role in the quantum field theory of massless ‘continu-\nous spin’ particles which occur in the work of Wigner[13]. Their momenta are lightlike and\nfuture pointing so that no known physical principle forbids their existence. But Yngvason\n[11] first proved that local quantum fields do not exist for such particles. It was later\nshown that ‘fields localised on cones’ do exist for these particles. These fields were later\nadapted to QED for its formulation entirely on a Hilbert space, avoiding indefinite metric\nconstructions altogether. They relied on the axial gauge and Dirac’s construction of gauge\ninvariant fields.\nThe Wilson line from xto∞in the spacelike direction ηgives the infrafield ϕ, the\nabove Wilson line for charge qbeing eiqϕ(x,η). Then, under a U(1) gauge transformation\neiχ, the Wilson line\nW(x, η) =eiqR∞\n0Aµ(x+τη)ηµdτ≡eiqϕ(x,η)\ntransforms as\nW(x, η)→eiqχ(η∞)W(x, η)e−iqχ(x), η ∞:= lim\nτ→∞τη.\nThis transformation suggests the interpretation that Wcarries a charge, say −qatxand\nqatη∞. Ifψis a charge qfield, then following Dirac[14] and Mandelstam[15], we see that\nW(x, η)ψ(x) is small gauge invariant, but still has a charge qblip at η∞.\n– 2 –Ifψis a field of charge q, then as Dirac observed, the field eiqϕ(x,η)ψ(x) is invariant\nunder local or small gauge transformations. It can be smeared in xwith test functions\nlocalised in a finite region containing xwithout spoiling small gauge invariance. But\nacting on the vacuum, it produces only a charge qblip at infinity. So it produces surface\nexcitations at infinity. That is the case even if xis smeared.\nWe want to choose ηso that Aµηµdoes not produce negative norm states acting on\nthe vacuum. That requires that this field has only spacelike components in µ. Hence ηµis\nchosen to be along a spacelike direction. In the mostly minus metric convention, it follows\nthat η.η=−κ2, with κreal. But unlike in the axial gauge, we do not fix the direction of\nη.\nFor fixed κ, this is a deSitter space with a spherical boundary for finite η. But W(x, η)\nis invariant under the scalings η→cη, c > 0. So we can fix κso that a compactification\nwith a spherical boundary is a preferred choice of spacelike boundaries.\nUnder standard Lorentz transformations Λ of Aµ, not only the argument, but also\nthe index µofAµgets transformed, so that\nΛ :ϕ(x, η)→ϕ(Λx,Λη).\nHence, ηas well gets transformed in the “escort” field ϕ(x, η).\n3 A Model Interaction for Z0→2γ\nWe need the interaction to have the following properties :\n1. It should be gauge invariant under small gauge transformations.\n2. The net charge at infinity should add up to zero so that charge conservation is\nmaintained. So we want to preserve global U(1) invariance.\nFor the escort field of charge q, the operator\n:eiqϕ(x,η): :e−iqϕ(x,η′):\nis invariant under small gauge transformations and has zero net charge. Also Z0\nµ(x) is\nneutral under this U(1). Hence if fµis a vector-valued test function, a Gauss law-invariant\ninteraction with zero net charge is\nd fµ(x)Z0\nµ(x) :eiqϕ(x,η): :e−iqϕ(x,η′): (3.1)\nwith an interaction strength d.\n– 3 –A natural choice for fµ(x) ishµwhere hµis a chosen polarisation vector for Zµin its\nrest system. This choice is suggested as we will calculate the decay in the Zµrest system.\nWe can also smear just Z0\nµwith a scalar function f(x) and use a fixed vector hµfor this\npolarization vector. But our preferred choice in this paper is (3.1). Finally we have also\nthe option of smearing the variables ηandη′, but we will avoid it for now. The process\nwhich lets us avoid the Landau-Yang theorem by means of the escort fields is sumarized in\nFigure 1.\nFigure 1 . Diagram associated to the decay of Z0into two γs in the presence of escort fields.\n4 Calculation of the Amplitude using the Mode Expansion of the Escort\nField\nWe can obtain the mode expansion of eiϕfrom that of Aµ( following Mund et. al. [1]) :\nAµ(x) =Z\n˜dkh\naµ(k)e−ik·x+a†\nµ(k)eik·xi\nwith the Lorentz invariant phase space (LIPS) measure:\n˜dk=d3k\n(2π)32|k|.\nAlthough the operator Aµ(x) is defined only on an indefinite metric (Krein) space ,\nthat is not the case for Aµηµasηµis spacelike.\nOne then defines\nϕ(x, η) =Z∞\n0dτA µ(x+ητ)ηµdτ.\n– 4 –Expanding ϕ(x) ( using (+,-,-,-) metric), one gets, using the “ iϵ” prescription,\nϕ(x, η) =Z\n˜dkηµ\u0014\naµ(k)e−ik·xZ∞\n0dτ e−i[(k.η)−iϵ]τ\n+a†\nµ(k)eik·xZ∞\n0dτ ei[(k.η)+iϵ]τ\u0015\n=iZ\n˜dk ηµ\"\naµ(k)\nη·k−iϵe−ik·x−a†\nµ(k)\nη·k+iϵeik·x#\n=iZ\n˜dkh\nχk(η)e−ik·x−χ†\nk(η)eik·xi\nwhere\nχk(η)≡a(k)·η\nk.e−iϵ\n.\nTherefore\nϕ(x, η) =i\u0014Z\n˜dk\u0010\nχk(η)e−ik·x−χ†\nk(η)eik·x\u0011\u0015\nNow\n:eiqϕ(x,η):=eqR˜dkχ†\nkeik·xe−qR˜dkχke−ik·x\nAccordingly\n:eiqϕ(x,η): :e−iqϕ(x,η′):=\neqR˜dkχ†\nkeik·xe−qR˜dkχke−ik·xe−qR\nd˜k′χ†\nk′eik′·xeqR\nd˜k′χk′e−ik′·x\nAs Baker-Campbell-Haussdorf formula gives us\neMeN=eM+N+1\n2[M,N ]+···⇒eMeN=eNeMe[M,N ]\nwhere in the last step we have assumed [ M, N ] is a c-number,\n:eiqϕ(x,η)::e−iqϕ(x,η′):=\neqR˜dkχ†\nkeik·xe−qR\nd˜k′χ†\nk′eik′·xe−qR˜dkχke−ik·xeqR\nd˜k′χk′e−ik′·xe∆(η,η′,x). (4.1)\nNow\n∆(η, η′, x) =q2Z Z\n˜dkd˜k′[χk, χ†\nk′]e−i(k−k′)·x(4.2)\nBut\n[χk, χ†\nk′] = [χk(η), χ†\nk′(η′)] = ( e·e′)(2π)32Ekδ3(k−k′)\n(k.η+iϵ)(k′.η′−iϵ)(4.3)\n– 5 –Hence,\n∆(η, η′, x) =q2(η·η′)×(Z Z\n˜dk˜dk′(2π)32Ekδ3(⃗k−⃗k′)e−i(k−k′)·x\n(k.η+iϵ)(k′.η′−iϵ))\n=q2Z\n˜dk1\n(k.η+iϵ)(k.η′−iϵ)≡q2Q(η, η′)\nends up being independent of x. Note that in the last step, we have carried out the k′\nintegration.\nThe integral Q(η, η′) can be rewritten ( using Feynman-Schwinger parameterization )\nas\nQ(η, η′) =Z\n˜dkZ1\n0dλ(η·η′)\n[λ(k·η+iϵ) + (1 −λ)(k·η′−iϵ)]2\n=Z\n˜dkZ1\n0dλ(η·η′)\n[k·η′+λk.(η−η′)−iϵ(1−2λ))]2\n=(η·η′)\n2(2π)3Z∞\n0kdkZ1\n0dλZ\ndΩk1\n[k.(η′+λ(η−η′)) +iϵ(1−2λ)]2.\nUsing the identity Z\ndΩˆn1\n(ˆn·⃗ a+b)2=4π\n(b2−⃗ a·⃗ a),\nwe get\nQ(η, η′) =(η.η′)\n(2π)2Z∞\n01\nkdkZ1\n0dλ\n[(2λ2−2λ+ 1) + 2 λ(1−λ)η.η′]2\nwhere the limit ϵ→0+has been employed. The kintegral is logarithmically divergent and\nneeds both an ultraviolet and infrared momentum cutoff, MUVandmIRrespectively:\nQ(η, η′) =(η.η′)\n(2π)2ln\u0012MUV\nmIR\u0013Z1\n0dλ\n[(2λ2−2λ+ 1) + 2 λ(1−λ)η.η′]2\n≡ln\u0010\nMUV\nmIR\u0011\n(2π)2S(η·η′)\nNote that the UV part of the logarithmic divergences ofR1\nkdkcan alternatively be absorbed\nby the renormalization of the electric charge:\n∆ =3\n2\u0012\n1−q2\nq2\nR\u0013\nS(η·η′) =3\n2\u0012\n1−4π\n137q2\nR\u0013\nS(η·η′)\nThus the c-number factor appearing in (4.1) takes the form\ne∆(η,η′)=e3\n2\u0012\n1−4π\n137q2\nR\u0013\nS(η·η′)\n(4.4)\n– 6 –Figure 2 . The plot of the eS(η,η′)vs (η·η′)\nwhere we have dropped the xargument in ∆( η, η′, x) as it is independent of x.\nA plot of the function eS(η,η′)is shown in Figure 2 Note that when η=η′so that η·η′=\n−1 , the factor e∆vanishes. That is, as long as we remain in the same “superselection”\nsector, the Landau-Yang theorem is validated. But as long as η̸=η′there is a none-zero\nprobability for the decay of Z0into two photons.\nLet us recall that the decay width of a particle of mass mAat rest into two identical\nmassless particles is given by the formula:\nΓ =1\n32πm2\nA|A|2. (4.5)\nNow we are interested in the product of matrix elements :\n⟨k1, ε1;k2, ε2|:eiqϕ(x,η)::e−iqϕ(x,η′):|0⟩γ⟨0|f·Z|Z, h⟩Z\n=−q2e∆(η,η′)Z Z\n˜dkd˜k′⟨k1, ε1;k2, ε2|χ†\nkχ†\nk′|0⟩γ× ⟨0|f·Z|Z, h⟩Z\nd4xeik·xeik′·xe−iP.x\n=−q2e∆(η,η′)\u0014(η·ε1)\n(η·k1+iϵ)(η′·ε2)\n(η′·k2+iϵ)+(η·ε2)\n(η·k2+iϵ)(η′·ε1)\n(η′·k1+iϵ)\u0015\n(f·h)\nwhere ε1,2are the polarization vectors of the photons with the momenta k1, k2respectively,\nwhile his the polarization/spin vector associated with the Z0boson.\nThe above result confirms the claim that escort field can allow decays which are forbid-\nden in the Standard Model. In particular the decay of Z0into two photons is now possible\n– 7 –for any value of η·η′except for η·η′=−1 where we recover the standard Landau-Yang\nresult. The reason being that in this case the two branches of escort field cancel out.\nDecay of Z0into two photons could in principle be observable in collider experiments.\nObservation of such an event will have profound consequences and can be interpreted as\nevidence of the novel infrared structure of QED. On the other hand, limits on decay width\nof this process from collider data can constrain the Lorentz violation that we have discussed\nin the paper.\nViolation of the Landau-Yang theorem by the inclusion of the escort field allows for\nstudy of other interesting physics phenomena. Not only one has the possibility of studying\nnovel and rare decays in atomic physics but also its impact in the cosmology for instance\nin the mapping of the 21cm hydrogen line. We hope to address to some of this phenomena\nusing this framework in the future.\nAppendix\nWe give a closed expression for the integral encountered in the text :\nF(η·η′):=Z1\n01\n(2(η·η′)(1−λ)λ+ (2λ2−2λ+ 1))2dλ\nThis integration can be done using the identity\nZ1\n0dλ\n[(2λ2−2λ+ 1) + 2 λ(1−λ)η·η′]2=1\n1 +η·η′+2 tan−1\u0010q\n1−η·η′\n1+η·η′\u0011\n(1 +η·η′)p\n1−(η·η′)2\nParameterizing A=η·η′=−cos Θ (as both of them are spacelike vectors), one gets\na simplified expression\nF(Θ) =1 +Θ\nsin Θ\n1−cos Θ\n“\nAcknowledgements\nM.A. is partially supported by Spanish MINECO/FEDER Grant PGC2022-126078NB-C21\nfunded by MCIN/AEI/10.13039/501100011033, DGA-FSE grant 2020-E21-17R and Plan\nde Recuperaci´ on, Transformaci´ on y Resiliencia - supported European Union – NextGen-\nerationEU. We wish express our thanks for Amilcar Queiroz for discussions on the early\nstages of this investigation.\n————————————–\n– 8 –References\n[1] J. Mund, K. Rehren and B. Schroer, JHEP 04(2022) 083 and references therein.\n[2] L. D. Landau, Dokl. Akad. Nauk SSSR. 60(1948) 207–209\n[3] C. N. Yang, Phys. Rev. 77(1950)242.\n[4] A. P. Balachandran, S. Jo and G. Marmo, Group Theory and Hopf Algebras : Lectures for\nPhysicists World Scientific, Singapore (2010).\n[5] A.P. Balachandran and S.G. Jo, Int. J. Mod. Phys. A22(2007)6133.\n[6] M. G. Kozlov, D. Budker, and D. English, Phys. Rev. A 80(4) (2009) 042504.\n[7] D. English, V. V. Yashchuk, and D. Budker, Phys. Rev. Lett. 104(2010) 253604\n[8] T. A. Zalialiutdinov, D. A. Solovyev, L. N. Labzowsky, G. Plunien, Physics Reports, 737\n(2018),\n[9] M. Peskin, Lectures presented at the Seventeenth SLAC Summer Institute Physics at the 100\nGeV Mass Scale Stanford, California, July 10- 21, 1989. SLAC-PUB-5210.\n[10] Addazi, A. et. al. “Quantum gravity Phenomenology at the dawn of the multi-messenger era\n- a review”, Progress in Particle and Nuclear Physics 125( 2022) :103948.\n[11] J. Yngvason, Commun. in Math. Phys. 18(1970) 195-203.\n[12] J. Fr¨ ohlich , G. Morchio, F. Strocchi,Ann. Phys. 119(1979) 241\n[13] S. Weinberg, The Quantum Theory of Fields. Vol. 2, Cambridge University Press ,\nCambridge U.K. (1995).\n[14] P.A.M. Dirac, Can. J. Phys. 33(1955) 650.\n[15] S. Mandelstam, Ann. Phys. 19(1962)1-24\n– 9 –"    },    {        "title": "1208.5475v1.Optimization_of_the_damped_quantum_search.pdf",        "content": "Optimization of the damped quantum search\nNeris Ilano, Cristine Villagonzalo and Ronald Banzon\nNational Institute of Physics, University of the Philippines,\nDiliman Quezon City, 1101 Philippines\nOctober 13, 2021\nAbstract\nThe damped quantum search proposed in [A. Mizel Phys. Rev. Lett.\n102150501 (2009)] was analyzed by calculating the highest possible prob-\nability of \fnding the target state in each iteration. A new damping pa-\nrameter that depends on the number of iterations was obtained, this was\ncompared to the critical damping parameter for di\u000berent values of target\nto database size ratio. The result shows that the range of the new damp-\ning parameter as a function of the target to database size ratio increases as\nthe number of iterations is increased. Furthermore, application of the new\ndamping parameter per iteration on the damped quantum search scheme\nshows a signi\fcant improvement on some target to database size ratio\n(i.e.\u001550% maximum percentage di\u000berence) over the critically damped\nquantum search.\n1 Introduction\nClassically, unsorted database are searched by inspecting each element and\nchecking if they satisfy the desired property. If Nis the size of the database\nandMis the number of target items, then the time complexity of obtaining the\none of the desired element is O(N=M ). In [1], Grover introduced a quantum\nsearch algorithm which can speed up the process to O(p\nN=M ) operations. A\nproof that Grover's search algorithm is the best possible oracle based search\nalgorithm was developed in [2, 3]. The generality of the search based problem\nmakes Grover's algorithm one of the interesting \feld of research in quantum\ncomputing. Some research are focused on its application [4, 5], while others on\nits extension [6, 8, 7].\nOne notable property of the quantum search algorithm is its oscillatory na-\nture, i.e. the probability of \fnding one of the target states oscillates from zero to\nsome maximum value. The number of queries that will give the maximum prob-\nability of success is dependent on the number of target states in the database\n[9, 10]. Searching a quantum database using Grover's algorithm without a prior\nknowledge of the number of target items poses a problem because the prob-\nability of \fnding a target state successfully does not converge as the number\n1arXiv:1208.5475v1  [physics.comp-ph]  27 Aug 2012of iterations is increased. Several modi\fcations were proposed to address this\ndilemma [11, 12, 13] which, however, cannot preserve the O(N=M ) signature.\nAn alternative way of solving the problem is proposed in [14] where the quan-\ntum search algorithm is damped by attaching an external spin to the quantum\ndatabase. In this new scheme, knowledge of the number of target items in the\ndatabase is not needed.\nIn the damped quantum search algorithm [14] there exists a critical damping\nparameter which is introduced using a physical argument. The critical damping\nparameter suppresses the oscillations and the error probability from increasing.\nBy simulating the behavior of the critical damping parameter, the author was\nable to propose a damping parameter which varies with each iteration. The\nresults were then able to suppress the oscillation of the success probability as\nthe search approaches one of the target states.\nIn this work, we obtained the optimized damping parameter up to the tenth\niteration by calculating the failure probability. This was done by solving the\nabsolute minima of the failure probability for a given number of target states\nand database size as a function of the damping parameter per iteration. We also\nexamined the critical damping parameter by comparing it with the optimized\none. The goal of this work is to analyze if the critical damping is optimized.\nWe will do this by directly probing the behavior of the probability of obtaining\nthe target state as the number of target states is increased.\n2 Damped Quantum Search\nSuppose we have a total number of states N, out of which Mare target states.\nLetj i= cos\u0012=2j\u000bi+sin\u0012=2j\fi, wherej\u000biis the equal superposition of the non-\ntarget states,j\fiis the equal superposition of the target states and sin \u0012=2 =p\nM=N . The action of the Grover's algorithm is encapsulated in the operator\nG. For a detailed discussion of the Grover's algorithm, we refer the reader to\n[9].\nThe damped quantum search introduced in [14] modi\fed the Grover formu-\nlation as follows. First, an external spin j#iis appended to the initial state j i\n(i.e.j i\u0000!j i\nj#i ). Then, the following operator is introduced\nU=\u0014\nG1\u0000Sz\n2+1 +Sz\n2\u0015\u0014\ne\u0000i\u001eSy1\u0000Z\n2+1 +Z\n2\u0015\n(1)\nwhereSyandSzare the external Pauli spin matrices acting in the Hilbert\nspace of the externally appended spin, the oracle Z=j\u000bih\u000bj\u0000j\fih\fjacts on\nthe Hilbert space of the database and \u001eis the damping parameter. The action\nofUis described as follows: (i) the right factor calls the oracle Zand \rips the\nexternal spin ifj i=j\fi, (ii) the left factor utilizes Gonly if the external spin\nhas not \ripped. In this way, the external spin limits the application of Gas the\ntarget state is reached.\nEach application of Uis followed by a measurement of the external spin.\nIf the external spin has \ripped, then the iteration stops. Without knowing\n2the measurement result, the system composite density matrix assumes the form\n(1\u0000Tr\u001a)j\fih\fj\nj\"ih\"j +\u001a\nj#ih#j where\u001ais initiallyj ih j. This means that\nthe probability that the system will collapse to j\fih\fj\nj\"ih\"j , i.e. the target\nstate has been found and the external spin has \ripped, is 1 \u0000Tr\u001a. Furthermore,\neach iteration yields the following:\n2\n4Tr(\u001a0X)\nTr(\u001a0Z)\nTr(\u001a0)3\n5=2\n64cos 2\u0012cos\u001esin 2\u00121+cos2\u001e\n2sin 2\u00121\u0000cos2\u001e\n2\n\u0000sin 2\u0012cos\u001ecos 2\u00121+cos2\u001e\n2cos 2\u00121\u0000cos2\u001e\n2\n01\u0000cos2\u001e\n21+cos2\u001e\n23\n752\n4Tr(\u001aX)\nTr(\u001aZ)\nTr(\u001a)3\n5(2)\nwhereX=j\fih\u000bj+j\u000bih\fjand [ Tr(\u001aX)Tr(\u001aZ)Tr(\u001a)]T= [sin\u0012cos\u00121]T.\nThe combination of U and the measurement of the external spin increases 1 \u0000\nTr\u001aas the number of iterations is increased. The average number of oracle\nqueries to \fnd the target item for small \u001ehas the quantum search signature\nO(p\nN), while for \u001e\u0018\u0019=2, it takes an average of O(N) queries, which is the\nclassical search limit. The existence of a critical damping de\fned by cos \u001ecrit=\n(1\u0000sin\u0012)=(1 + sin\u0012) divides the two regimes. When \u001e=\u001ecrit, the three\neigenvalues of the square matrix in Eq. 2 are equal. It turns out that even\nfor the weakest possible value of \u001e, whereMis set to 1 for arbitrary N, the\ndamping is still evident. However, this smallest value of \u001ecan only damp the\nsearch e\u000bectively if M < N= 2. In the absence of the knowledge of Mand in\ncaseM\u0015N=2, the critical damping varies from iteration to iteration according\nto cos\u001er= (1\u0000sin(\u0019=2r))=(1 + sin(\u0019=2r)), wherercorresponds to the rth\niteration. This proposed variation of \u001estarts with the largest damping and\nweakens with increasing number of iterations. This variation is designed to\nmimic the behavior of the critical damping.\n3 Analytical calculation of Tr \u001a(\u0012;\u001e)\nGiven the size of the database Nand the damping parameter \u001e, we execute n\ncalls toU. The transition of the external spin from j#itoj\"iindicates that the\ntarget has been found. However, if the external spin remains unaltered then\nwe perform an additional query to the oracle to check if the target has been\nfound. If the outcome turns out to be negative, then we start the procedure all\nover again. Fig. 1 shows the minimum expected number of calls before success,\nE(n) = (n+ 1)=P(n), whereP(n) is the probability of success after the nthcall\nas given by\nP(n)/\f\f\f\u0000\nh\fj\nh\"j +h\fj\nh#j\u0001\n\u0001\u0000\nUnj i\nj#i\u0001\f\f\f2\n: (3)\nThe linear behavior in the region of large damping parameter portrays the\nclassical result, while the region of small damping parameter shows the quadratic\nquantum character. The valley separating these two regimes is interpreted as\nthe region that has the critical damping, \u001ecrit, based from the eigenvalues of\nthe square matrix in Eq. (2).\n3Figure 1: Expected number of oracle calls before success as a function of the\ndatabase size, N, and damping parameter, \u001e.\nWe analyze the distinct valley to determine if \u001ecritis the same as the damp-\ning parameter that gives the smallest possible Tr\u001a. We refer to this as the\noptimum damping ,\u001eopt. To start with our analysis, let us de\fne a positive\nnumber that serves as an upper bound to the probability that the external spin\nhas not \ripped, such that\nTr\u001a\u0014\u000e: (4)\nWe call\u000ethefailure tolerance .\nSuppose we perform niterations, what is the smallest possible failure toler-\nance,\u000emin, that we can assign? What is the value of the damping parameter\n(\u001eopt) that will give a \u000emin? Is the computed \u001eoptthe same as the \u001ecrit? In\norder to address these questions, we calculate Tr\u001aas a function of \u0012and\u001eafter\na particular number of iterations via Eq. (2).\nTo obtain the value of the damping parameter that will give \u000emin, we take\nthe derivative of Tr\u001awith respect to \u001eand set the resulting expression to zero.\nWe then compare the behavior of the obtained damping parameter ( \u001eopt) with\nthat of\u001ecritby plotting them both as a function of \u0012.\n3.1 First iteration\nReplace the trace matrix of Eq. (2) by [sin \u0012cos\u00121]T. We will get a column\nmatrix with Tr\u001aX,Tr\u001aYandTr\u001aas the \frst, second and third entry of the\n\fnal column matrix. This process will yield Tr\u001afor the \frst iteration\nTr\u001a(\u0012;x) =\u0012\n\u00001\n2cos\u0012+1\n2\u0013\nx2+\u00121\n2cos\u0012+1\n2\u0013\n; (5)\nwherex= cos\u001e. For a given \u0012,Tr\u001ais a quadratic function of cos \u001e. Since\n0\u0014\u0012\u0014\u0019=2 and 0\u0014\u001e\u0014\u0019=2, then the minimum of Tr\u001ais cos2(\u0012=2) which\nis at\u001e=\u0019=2. The\u000ethat we must choose should not be less than the initial\n4error probability, cos2(\u0012=2). Thus, for the \frst iteration, the optimized damping\nparameter is \u0019=2 and is independent of the number of target states described\nby\u0012.\n3.2 Second iteration\nFor the second iteration, Tr\u001ais a quartic function of xwhich implies that we\nhave to solve the real root of a cubic polynomial in xin order to obtain the\nextremum of Tr\u001a,\nTr\u001a(\u0012;x) =\u0012\n\u00001\n2cos3\u0012+1\n2cos2\u0012\u0013\nx4+\u0000\n\u0000cos3\u0012+ cos\u0012\u0001\nx3\n+\u0000\n\u0000cos2\u0012+ 1\u0001\nx2+\u0000\ncos3\u0012\u0000cos\u0012\u0001\nx+\u00121\n2cos3\u0012+1\n2cos2\u0012\u0013\n:(6)\nThe real root can still be obtained analytically and, as expected, it is no longer\nindependent of \u0012,\nxopt=\u00002\n3\u00121 + cos\u0012\ncos\u0012\u0013\n+[(1 + cos\u0012)[(2\u0000cos\u0012)2+p\nf(cos\u0012)]]1=3\n3 cos\u0012\n+4\n3 \n1 + cos\u0012\n[(1 + cos\u0012)[(2\u0000cos\u0012)2+p\nf(cos\u0012)]]1=3!\n;(7)\nwheref(cos\u0012) =\u000063 cos4\u0012\u000072 cos3\u0012+ 24 cos3\u0012\u000032 cos\u0012+ 16. Using Eq. (7)\ninto Eq. (6) will give us \u000emin. Figure 2 shows the plot of Tr\u001aas a function of \u0012\nfor the \frst and second iteration. There is a slight di\u000berence between the Tr\u001a\nof the critical damped search and the Tr\u001aof the optimum damped search for\nthe \frst and second iteration.\n0 1/8 1/4 3/8 1/200.20.40.60.81\nθ (in units of π rad)Tr ρ\n  Tr ρopt\nTr ρcrit \n(a) 1stiteration\n0 1/8 1/4 3/8 1/200.20.40.60.81\nθ (in units of π rad )Tr ρ\n  \nTr ρopt\nTr ρcrit (b) 2nditeration\nFigure 2: Tr\u001a(\u001eopt;\u0012) and Tr\u001a(\u001ecrit;\u0012) of the 1stand the 2nditerations\nThe analytical calculations of xoptis no longer possible for higher iterations,\ni.e. forn>2, because generally there is no explicit formula for the real root of\n5a 2n\u00001 degree polynomial. So we resort to numerical calculations of the real\nroot of a 2n\u00001 degree polynomial in xwithx2[0;1].\n4 Numerical calculations and results\nFigure 3 is a sample plot of Tr\u001aas a function of \u0012and\u001efor the 3rditeration,\nshowing that for a particular \u0012there exist a unique minimum Tr\u001a. The goal\nof this section is to obtain the value of \u001ethat will give the minimum Tr\u001afor a\ngiven\u0012.\nFigure 3: The probability that the spin will not \rip, Tr\u001a, as a function of \u0012and\n\u001efor the 3rditeration.\nThe calculation of \u001eoptforn > 2 iterations is carried out numerically as\ndescribed in the following. The explicit forms of Tr\u001aas a function of \u001eand\u0012\nfor di\u000berent iterations are calculated by taking the nthpower of Eq. (2) and\nchoosing the third entry after the resulting square matrix is pre-multiplied by\n[Tr(\u001aX)Tr(\u001aZ)Tr(\u001a)]T= [sin\u0012cos\u00121]T. The resulting Tr\u001ais di\u000berentiated\nwith respect to cos \u001eand is equated to zero. This will allow us to \fnd the\noptimum value of \u001ethat will give the minimum possible Tr\u001a. Only the root\nthat is real and lies in the interval [0 ;1] is chosen. The behavior of \u001eoptin each\niteration is analyzed by plotting it as a function of \u0012. This method is limited\nin a sense that the result will depend largely on the resolution of the grid.\nNevertheless, this will provide a useful visual representation of \u001eoptbehavior for\ndi\u000berent\u0012.\nThe plots of \u001eoptas a function of \u0012are obtained for the third up to the\ntenth iterations. Only the fourth and tenth iterations of the optimum and the\ncritical\u001eas function of \u0012plots are shown in Fig. 4 for convenience. The results\nshow that for any number of iterations the critical damping is equal to the\noptimized damping parameter in the region near \u0012=\u0019=2, indicating that the\ncritical damping is optimized for a target with a size comparable to that of the\ndatabase. The leftmost part of the \u001eoptcurve will coincide eventually with the\n60 1/8 1/4 3/8 1/201/81/43/81/2\nθ (in units of π rad)φ (in units of π rad)\n  \nφopt\nφcrit(a) 4thiteration\n0 1/8 1/4 3/8 1/201/81/43/81/2\nθ (in units of π rad)φ (in units of π rad)\n  \nφopt\nφcrit (b) 10thiteration\nFigure 4: Comparison of \u001eopt(o) and\u001ecrit(+) as a function of \u0012for the 4thand\nthe 10thiterations.\n0 1/8 1/4 3/8 1/21/61/31/2\nθ (in units of π rad)φopt (in units of π rad)\n \n1st iteration\n2nd iteration\n3rd iteration\n4th iteration\n(a) 1stto 4thiterations\n0 1/8 1/4 3/8 1/21/61/31/2\nθ (in units of π rad)φopt (in units of π rad)\n  \n5th iteration\n7th iteration\n9th iteration (b) 5th, 7thand 9thiterations\nFigure 5: The changes in the range of \u001eoptas the iteration is increased.\n0 1/8 1/4 3/8 1/200.20.40.60.81\nθ (in units of π radian)Tr ρ 1/8 3/16 1/400.20.4\n(a) 4thiteration\n0 1/8 1/4 3/8 1/200.20.40.60.81\nθ (in units of π radian)Tr ρ 1/16 1/8 3/1600.1250.25 (b) 10thiteration\nFigure 6: Tr\u001a(\u001eopt;\u0012) and Tr\u001a(\u001ecrit;\u0012) of the 4thand the 10thiterations\n7leftmost part of the \u001ecritcurve as we further increase the number of iterations.\nIn Fig. 5, the range of the \u001eoptis observed to increase as the number of iterations\nis increased. This implies that we do not have to use all the possible values of\n\u001ewhen optimizing the search for small number of iterations. Also, jumps begin\nto appear in the \ffth iterations onwards as shown in Fig. 5(b).\nWe have shown that the critical damping is not the same as the optimum\ndamping that will give the smallest probability of failure. To continue with our\nanalysis, we further investigate the di\u000berence of the resulting Tr\u001ausing\u001eoptas\ncompared to that of \u001ecritfor any\u0012.\nUsing the numerically calculated \u001ecrit, the Tr\u001aare evaluated for di\u000berent\nvalues of\u0012and for a di\u000berent number of iterations. In Fig. 6, the values of\nTr\u001aare shown as a function of \u001eoptand\u0012(denoted by\u000e), and as a function of\n\u001ecritand\u0012(denoted by +). The values overlap for M\u0018N=2 or\u0012\u0018\u0019=2 but\ndi\u000ber for small values of \u0012. Notice that the interval of \u0012, where the di\u000berence\ninTr\u001ais evident, goes to smaller values, i.e. smaller values of the target items,\nas the number of iterations is increased. It turns out that the critically damped\nquantum search becomes less optimized for larger number of iterations if the\nnumber of target states is small.\n0 1/8 1/4 3/8 1/200.010.020.030.040.05\nθ (in units of π radian)difference in Tr ρmax. diff. = 0.04875\n% diff = 51.11 %\n(a) 4thiteration\n0 1/8 1/4 3/8 1/200.020.040.06\nθ (in units of π radian)difference in Tr ρmax. diff. = 0.05636\n% diff = 60.88 % (b) 10thiteration\nFigure 7: The plot of the di\u000berence in the probability between the critical\ndamping parameter and the optimum damping parameter. The inset also shows\nthe corresponding percent di\u000berence of the peak.\nIn Fig. 7, the di\u000berence in the probabilities Tr\u001a(\u001eopt;\u0012) and Tr\u001a(\u001ecrit;\u0012)\nincreases as the number of iterations is increased. The percent di\u000berence of\nthe corresponding maximum di\u000berence are also shown in the plot. Table 1\nsummarizes the maximum di\u000berence in the probabilities in each iteration. We\nobserved that the percent di\u000berence with respect to the order of Tr\u001a(\u001ecrit) of the\nmaximum di\u000berence with Tr\u001a(\u001eopt) is more than 50% from the fourth iteration\nonwards.\n8Table 1: The maximum di\u000berence and the corresponding percentage di\u000berence\nbetween Tr \u001aof the optimized and the critical damping parameters.\nIteration di\u000berence % di\u000berence\n1st 0.0077 0.8105\n2nd 0.0194 15.88\n3rd 0.0415 43.52\n4th 0.0487 51.11\n5th 0.0520 57.54\n6th 0.0538 58.93\n7th 0.0547 53.78\n8th 0.0555 64.53\n9th 0.0559 67.11\n10th 0.0564 60.88\n5 Summary and Conclusion\nAn investigation was made on the damped quantum search by appealing di-\nrectly to the global minimum of the failure probability. We have found that the\ncritical damping parameter is generally not equal to the optimum damping pa-\nrameter. There exists some interval in \u0012where the critically damped quantum\nsearch is evidently not optimized. In addition, the resulting failure probability\nis smaller compared to that of the critical damping parameter. However, such\ndi\u000berences vanish readily as the number of target states approaches that of the\ndatabase. Using the result of the optimum quantum search, we could develop\nan alternative way of calculating the damping parameter by solving a certain\nhigh order polynomial depending on the number of iterations.\nAcknowledgement\nN.I. acknowledges support from the Department of Science and Technology SEI-\nASTHRDP.\nReferences\n[1] L.K. Grover (1997), Quantum Mechanics Helps in Searching for a Needle\nin a Haystack , Phys. Rev. Lett. 79 (2), pp 325-328.\n[2] C.H. Bennett, E. Bernstein, G. Brassard, and U. Vazirani (1997), Strength\nand weaknesses of quantum computing , SIAM J. Comput. 26 (5), pp 1510-\n1523\n[3] M. Boyer, G. Brassard, P. Hoyer, and A. Tapp (1998), Tight bounds on\nquantum searching , .Fortsch. Phys. - Prog. Phys., 46 (4-5), pp 493-505.\n9[4] Y. Liu and G. Koehler (2010), Using modi\fcations to Grover's Search\nalgorithm for quantum global optimization , Eur. J. Oper. Res. 207 (2), pp\n620-632.\n[5] S. Aaronson and A. Ambainis (2005), Quantum search of spatial regions ,\nTheory of Computing (1), pp 47-79.\n[6] C. Cafaro and S. Mancini (2012), On Grover's search algorithm from a\nquantum information geometry viewpoint , Phys. A 391 (4), pp 1610-1625.\n[7] E. Farhi and S. Gutmann (1998), Analog analogue of a digital quantum\ncomputation , Phys. Rev. A 391 57 (4) pp 2403-2406.\n[8] A. Childs, E. Deotto, E. Farhi and J. Goldstone (2002), Quantum search\nby measurement , Phys. Rev. A 391 66 (3).\n[9] Nielsen M A and Chuang I L (2000), Quantum Computation and Quantum\nInformation , Cambridge University Press (Cambridge, England).\n[10] Brylinsky R and Chen G (2002), Mathematics of Quantum Computation ,\nChapman and Hall/CRC (Florida).\n[11] L.K. Grover (2005), Fixed Point Quantum Search , Phys. Rev. Lett. 95,\n150501.\n[12] T. Tulsi, L. K. Grover, and A. Patel (2006), A new algorithm for \fxed\npoint quantum search Quantum Inf. Comput. 6, 483.\n[13] Grover L K, Patel A, Tulsi T (2006), Quantum algorithms with \fxed\npoints: The case of database search , quant-ph/0603132.\n[14] A. Mizel (2009), Critically Damped Quantum Search , Phys. Rev. Lett. 102\n(15), 150501.\n10"    },    {        "title": "1601.05227v1.Introduction_to_Landau_Damping.pdf",        "content": "arXiv:1601.05227v1  [physics.acc-ph]  20 Jan 2016Introduction toLandau Damping\nW.Herr\nCERN,Geneva, Switzerland\nAbstract\nThemechanismofLandaudampingisobservedinvarioussyste msfromplasma\noscillations to accelerators. Despite its widespread use, some confusion has\nbeen created, partly because ofthedifferent mechanisms pr oducing thedamp-\ning but also due tothe mathematical subtleties treating the effects. In this arti-\ncle the origin of Landau damping is demonstrated for the damp ing of plasma\noscillations. Inthesecond partitisapplied tothedamping ofcoherent oscilla-\ntions in particle accelerators. The physical origin, the ma thematical treatment\nleading totheconcept ofstability diagrams andtheapplica tions arediscussed.\n1Introduction andhistory\nLandau damping is referred to as the damping of a collective m ode of oscillations in plasmas without\ncollisions of charged particles. These Langmuir [1] oscill ations consist of particles with long-range in-\nteractions and cannot be treated with a simple picture invol ving collisions between charged particles.\nThe damping of such collisionless oscillations was predict ed by Landau [2]. Landau deduced this effect\nfrom a mathematical study without reference to a physical ex planation. Although correct, this deriva-\ntion is not rigorous from the mathematical point of view and r esulted in conceptual problems. Many\npublications and lectures have been devoted to this subject [3–5]. In particular, the search for stationary\nsolutions led to severe problems. This was solved by Case [6] and Van Kampen [7] using normal-mode\nexpansions. For many theorists Landau’s result is counter- intuitive and the mathematical treatment in\nmany publications led to some controversy and is still debat ed. This often makes it difficult to connect\nmathematical structures to reality. It took almost 20 years before dedicated experiments were carried\nout [8] to demonstrate successfully the reality of Landau da mping. In practice, Landau damping plays\na very significant role in plasma physics and can be applied to study and control the stability of charged\nbeams inparticle accelerators [9,10].\nIt is the main purpose of this article to present aphysical pi cture together with some basic mathe-\nmatical derivations, without touching onsome of the subtle problems related to this phenomenon.\nThe plan of this article is the following. First, the Landau d amping in plasmas is derived and the\nphysical picture behind the damping is shown. In the second p art it is shown how the concepts are used\ntostudy thestability of particle beams. Someemphasis ispu t onthederivation of stability diagrams and\nbeamtransferfunctions(BTFs)andtheirusetodetermineth estability. Inanaccelerator thedecoherence\nor filamentation of an oscillating beam due to non-linear fiel ds is often mistaken for Landau damping\nand significant confusion in the community of accelerator ph ysicists still persists today. Nevertheless, it\nbecame a standard tool to stabilize particle beams of hadron s. In this article it is not possible to treat all\nthe possible applications nor themathematical subtleties and the references should be consulted.\n2Landau damping inplasmas\nInitially, Landau damping was derived for the damping of osc illations in plasmas. In the next section,\nweshall follow the steps of this derivation insome detail.\n2.1 Plasmaoscillations\nWe consider an electrically quasi-neutral plasma in equili brium, consisting of positively charged ions\nand negatively charged electrons (Fig. 1). For a small displ acement of the electrons with respect to the-2-1.5-1-0.5 0 0.5 1 1.5 2\n-2 -1.5 -1 -0.5  0  0.5  1  1.5  2\n                                                \nFig.1:Plasma withoutdisturbance(schematic)\n-2-1.5-1-0.5 0 0.5 1 1.5 2\n-2 -1.5 -1 -0.5  0  0.5  1  1.5  2\n                                                \nFig.2:Plasma withdisturbance(schematic)\n-2-1.5-1-0.5 0 0.5 1 1.5 2\n-2 -1.5 -1 -0.5  0  0.5  1  1.5  2\n                                                                                    \nFig.3:Plasmawith disturbanceandrestoringfield(schematic)\n-2-1.5-1-0.5 0 0.5 1 1.5 2\n-2 -1.5 -1 -0.5  0  0.5  1  1.5  2\n                                                             \nFig.4:Plasma withdisturbance\nions (Fig. 2), the electric fields act on the electrons as a res toring force (Fig. 3). Due to the restoring\nforce,standingdensitywavesarepossiblewithafixedfrequ ency[1]ω2\np=ne2\nmǫ0,wherenisthedensityof\nelectrons,ethe electric charge, mthe effective mass of an electron and ǫ0the permittivity of free space.\nTheindividual motionof theelectrons isneglected for this standing wave. Inwhat follows, weallow for\narandom motionoftheelectrons withavelocity distributio n fortheequilibrium stateandevaluate under\nwhich condition waves withawave vector kand afrequency ωarepossible.\nTheoscillating electrons produce fields (modes) of the form\nE(x,t) =E0sin(kx−ωt) (1)\nor, rewritten,\nE(x,t) =E0ei(kx−ωt). (2)\nThe corresponding wave (phase) velocity is then v=ω\nk. In Fig. 4, we show the electron distribution\ntogether withthe produced field. Thepositive ions areomitt ed in thisfigure and areassumed toproduce\n2a stationary, uniform background field. This assumption is v alid when we consider the ions to have\ninfinite mass, which is a good approximation since the ion mas s is much larger than the mass of the\noscillating electrons.\n2.2 Particle interaction with modes\nThe oscillating electrons now interact with the field they pr oduce, i.e. individual particles interact with\nthe field produced by all particles. This in turn changes the b ehaviour of the particles, which changes\nthe field producing the forces. Furthermore, the particles m ay have different velocities. Therefore, a\nself-consistent treatment isnecessary. If weallow ωto be complex ( ω=ωr+iωi), weseparate the real\nand imaginary parts of the frequency ωand rewrite the fields:\nE(x,t) =E0ei(kx−ωt)⇒E(x,t) =E0ei(kx−ωrt)·eωit(3)\nand wehave a dampedoscillation for ωi<0.\nIf we remember that particles may have different velocities , we can consider a much simplified\npicture as follows.\ni) If moreparticles are moving moreslowly than the wave:\nNet absorption of energy fromthe wave →waveis damped.\nii) If moreparticles are moving faster than the wave:\nNet absorption of energy bythe wave →wave isantidamped.\nWe therefore have to assume that the slopeof the particle distribution at the wave velocity is importa nt.\nAlthough this picture is not completely correct, one can ima gine a surfer on a wave in the sea, getting\nthe energy from the wave (antidamping). Particles with very different velocities do not interact with the\nmode and cannot contribute to the damping or antidamping.\n2.3 Liouville theorem\nThe basis for the self-consistent treatment of distributio n functions is the Liouville theorem. It states\nthat the phase-space distribution function is constant alo ng the trajectories of the system, i.e. the density\nof system points in the vicinity of a given system point trave lling through phase space is constant with\ntime, i.e. the density is always conserved. If the density di stribution function is described by ψ(/vector q,/vector p,t),\nthen the probability to find the system in the phase-space vol umedqndpnis defined by ψ(/vector q,/vector p)dqndpn\nwith/integraltext\nψ(/vector q,/vector p,t)dqndpn=N. We have used canonical coordinates qi, i= 1,...,nand momenta\npi, i= 1,...,nsince it is defined for a Hamiltonian system. The evolution in time is described by the\nLiouville equation:\ndψ\ndt=∂ψ\n∂t+n/summationdisplay\ni=1/parenleftbigg∂ψ\n∂qi˙qi+∂ψ\n∂pi˙pi/parenrightbigg\n= 0. (4)\nIf the distribution function is stationary (i.e. does not depend on qandt), thenψ(/vector q,/vector p,t)becomesψ(/vector p).\nFigure5showsschematically thechangeofaphase-space dis tribution undertheinfluenceofalinearand\na non-linear field. The shape of the distribution function is distorted by the non-linearity, but the local\nphase-space densityisconserved. However,the globaldensitymaychange,i.e. the(projected) measured\nbeam size. The Liouville equation will lead us to the Boltzma nn and Vlasov equations. We move again\ntoCartesian coordinates xandv.\n3-15-10-5 0 5 10 15\n-15 -10 -5  0  5  10  15V\nXPhase space density\n                   \n-15-10-5 0 5 10 15\n-15 -10 -5  0  5  10  15X'\nXPhase space density\n                   \n                   \n-15-10-5 0 5 10 15\n-15 -10 -5  0  5  10  15V\nXPhase space density\n                   \n                   \nFig.5:Changeof phasespace. Left, originaldistribution;centre ,distributionwith linearfields; right,distribution\nwithnon-linearfieldsafterelapsedtime t.\n2.4 Boltzmann andVlasov equations\nTimeevolution of ψ(/vector x,/vector v,t)isdescribed by the Boltzmann equation:\ndψ\ndt=∂ψ\n∂t/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\ntime change+/vector v·∂ψ\n∂/vector x/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nspace change+1\nm/vectorF(/vector x,t)·∂ψ\n∂/vector v/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nvchange,forceF+ Ω(ψ)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\ncollision. (5)\nThis equation contains a term which describes mutual collis ions of charged particles in the distribution\nΩ(ψ). TostudyLandaudamping,weignorecollisionsandthecolli sionlessBoltzmannequationbecomes\nthe Vlasov equation:\ndψ\ndt=∂ψ\n∂t+/vector v·∂ψ\n∂/vector x+1\nm/vectorF(/vector x,t)·∂ψ\n∂/vector v= 0. (6)\nHere/vectorF(/vector x,t)isthe force of the field(mode) onthe particles.\nWhyis theVlasov equation useful?\ndψ\ndt=∂ψ\n∂t+/vector v·∂ψ\n∂/vector x+1\nm/vectorF(/vector x,t)·∂ψ\n∂/vector v= 0. (7)\nHere/vectorF(/vector x,t)can be a force caused by impedances, beam–beam effects etc. F rom the solution one\ncan determine whether a disturbance is growing (instabilit y, negative imaginary part of frequency) or\ndecaying (stability, positive imaginary part of frequency ). It isastandard tool tostudy collective effects.\n2.4.1 Vlasov equation for plasma oscillations\nFor our problem weneed the force /vectorF(depending on field /vectorE):\n/vectorF=e·/vectorE (8)\nand the field /vectorE(depending on potential Φ):\n/vectorE=−∇Φ (9)\nfor the potential Φ(depending ondistribution ψ):\n∆Φ =−ρ\nǫ0=−e\nǫ0/integraldisplay\nψdv. (10)\nTherefore,\ndψ\ndt=∂ψ\n∂t+/vector v·∂ψ\n∂/vector x+e\nm/vectorE(/vector x,t)·∂ψ\n∂/vector v= 0. (11)\n4We have obtained a set of coupled equations: the perturbatio n produces a field which acts back on the\nperturbation.\nCanwefindasolution? Assumeasmall non-stationary perturbation ψ1ofthestationary distribu-\ntionψ0(/vector v):\nψ(/vector x,/vector v,t) =ψ0(/vector v)+ψ1(/vector x,/vector v,t). (12)\nThen weget for the Vlasov equation:\ndψ\ndt=∂ψ1\n∂t+/vector v·∂ψ1\n∂/vector x+e\nm/vectorE(/vector x,t)·∂ψ0\n∂/vector v= 0 (13)\nand\n∆Φ =−ρ\nǫ0=−e\nǫ0/integraldisplay\nψ1dv. (14)\nThe density perturbation produces electric fields which act back and change the density perturbation,\nwhich therefore changes with time.\nψ1(/vector x,/vector v,t) =⇒/vectorE(/vector x,t) =⇒ψ′\n1(/vector x,/vector v,t) =⇒ ···. (15)\nHowcanonetreatthisquantitatively andfindasolutionfor ψ1? Wefindtwodifferent approaches,\none due toVlasov and the other due toLandau.\n2.4.2 Vlasov solution anddispersion relation\nTheVlasovequationisapartialdifferential equationandw ecantrytoapplystandardtechniques. Vlasov\nexpanded the distribution and the potential as adouble Four ier transform [11]:\nψ1(/vector x,/vector v,t) =1\n2π/integraldisplay+∞\n−∞/integraldisplay+∞\n−∞˜ψ1(k,/vector v,ω)ei(kx−ωt)dkdω, (16)\nΦ(/vector x,/vector v,t) =1\n2π/integraldisplay+∞\n−∞/integraldisplay+∞\n−∞˜Φ(k,/vector v,ω)ei(kx−ωt)dkdω (17)\nand applied these to the Vlasov equation. Since we assumed th e field (mode) of the form E(x,t) =\nE0ei(kx−ωt), weobtain thecondition (after some algebra)\n1+e2\nǫ0mk/integraldisplay∂ψ0/∂v\n(ω−kv)dv= 0 (18)\nor, rewritten using the plasma frequency ωp,\n1+ω2\np\nk/integraldisplay∂ψ0/∂v\n(ω−kv)dv= 0 (19)\nor, slightly re-arranged for later use,\n1+ω2\np\nk2/integraldisplay∂ψ0/∂v\n(ω/k−v)dv= 0. (20)\nThisisthedispersionrelationforplasmawaves,i.e.itrel atesthefrequency ωwiththewavevector k. For\nthis relation, waves withfrequency ωand wavevector karepossible, answering the previous question.\nLooking at this relation, wefind thefollowing properties.\n5i) It depends on the (velocity) distribution ψ.\nii) It depends on the slope of the distribution ∂ψ0/∂v.\niii) Theeffect isstrongest for velocities close to thewave velocity, i.e. v≈ω/k.\niv) There seems tobe acomplication (singularity) at v≡ω/k.\nCanwedeal withthis singularity? Someproposals have been m adein thepast:\ni) Vlasov’s hand-waving argument [11]: inpractice ωis never real (collisions).\nii) Optimistic argument [3,4]: ∂ψ0/∂v= 0, wherev≡ω\nk.\niii) Alternative approach [6,7]:\na) search for stationary solutions (normal-mode expansion );\nb) results in continuous versus discrete modes.\niv) Landau’s argument [2]:\na) Initial-value problem with perturbation ψ1(/vector x,/vector v,t)att= 0(time-dependent solution with\ncomplexω).\nb) Solution: in time domain use Laplace transformation; in s pace domain use Fourier transfor-\nmation.\n2.4.3 Landau'ssolution anddispersion relation\nLandaurecognizedtheproblemasaninitial-valueproblem( inparticularfortheinitialvalues x= 0,v′=\n0) and accordingly used adifferent approach, i.e. heused aFo urier transform inthe space domain:\n˜ψ1(k,/vector v,t) =1\n2π/integraldisplay+∞\n−∞ψ1(/vector x,/vector v,t)ei(kx)dx, (21)\n˜E(k,t) =1\n2π/integraldisplay+∞\n−∞E(/vector x,t)ei(kx)dx (22)\nand aLaplace transform inthe timedomain:\nΨ1(k,/vector v,p) =/integraldisplay+∞\n0˜ψ1(k,/vector v,t)e(−pt)dt (23)\nE(k,p) =/integraldisplay+∞\n0˜E(k,t)e(−pt)dt. (24)\nInsertedintotheVlasovequationandaftersomealgebra(se ereferences), thisleadstothemodified\ndispersion relation:\n1+e2\nǫ0mk/bracketleftBigg\nP.V./integraldisplay∂ψ0/∂v\n(ω−kv)dv−iπ\nk/parenleftbigg∂ψ0\n∂v/parenrightbigg\nv=ω/k/bracketrightBigg\n= 0 (25)\nor, rewritten using the plasma frequency ωp,\n1+ω2\np\nk/bracketleftBigg\nP.V./integraldisplay∂ψ0/∂v\n(ω−kv)dv−iπ\nk/parenleftbigg∂ψ0\n∂v/parenrightbigg\nv=ω/k/bracketrightBigg\n= 0. (26)\n6 0 0.2 0.4 0.6 0.8 1 1.2 1.4\n 0  0.5  1  1.5  2f(V)\nVslower particles\nfaster particles\nV phase                   \nFig.6:Velocitydistributionanddampingconditions\nHereP.V.refers to‘Cauchy principal value’.\nIt must be noted that the second term appears only in Landau’s treatment as a consequence of the\ninitial conditions and is responsible for damping. The trea tment by Vlasov failed to find this term and\ntherefore did not lead toadamping of the plasma. Evaluating the term\n−iπ\nk/parenleftbigg∂ψ\n∂v/parenrightbigg\nv=ω/k, (27)\nWefinddamping of theoscillations if/parenleftBig\n∂ψ\n∂v/parenrightBig\nv=ω/k<0. Thisisthe condition for Landau damping. How\nthe dispersion relations for both cases are evaluated is dem onstrated in Appendix A. The Maxwellian\nvelocity distribution is used for this calculation.\n2.5 Dampingmechanism in plasmas\nBased onthese findings, wecan giveasimplified picture of thi s condition. InFig. 6,weshow avelocity\ndistribution (e.g. aMaxwellian velocity distribution). A smentioned earlier, the damping depends onthe\nnumber of particles below and above the phase velocity.\ni) More ‘slower’ than ‘faster’ particles =⇒damping.\nii) More ‘faster’ than ‘slower’ particles =⇒antidamping.\nThisintuitive picture reflects the damping condition deriv ed above.\n3Landau damping inaccelerators\nHowtoapply ittoaccelerators? Herewedonot haveavelocity distribution, but afrequency distribution\nρ(ω)(in the transverse plane the tune). It should be mentioned he re thatρ(ω)is the distribution of\nexternal focusing frequencies . Since we deal with a distribution, we can introduce a freque ncy spread\nof the distribution and call it ∆ω. The problem can be formally solved using the Vlasov equatio n, but\nthe physical interpretation is very fuzzy (and still debate d). Here we follow a different (more intuitive)\ntreatment (following [5,12,13]). Although, if not taken wi th the necessary care, it can lead to a wrong\nphysical picture, itdelivers very useful concepts andtool s for thedesign andoperation ofanaccelerator.\nWeconsider now the following issues.\ni) Beam response toexcitation.\nii) BTFand stability diagrams.\niii) Phase mixing.\n7 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4\n-4 -2  0  2  4  6  8ρ(ω)\nωωx\n∆ω\nFig.7:Frequencydistributionin abeamandfrequencyspread(very schematic)\niv) Conditions and tools for stabilization and therelated p roblems.\nThis treatment will lead again to a dispersion relation. How ever, the stability of a beam is in general\nnot studied bydirectly solving thisequation, but byintrod ucing theconcept of stability diagrams, which\nallowusmoredirectlytoevaluatethestability ofabeamdur ingthedesignoroperation ofanaccelerator.\n3.1 Beamresponse toexcitation\nHowdoes abeam respond toan external excitation?\nTostudythedynamicsinaccelerators, wereplacetheveloci tyvby˙xtobeconsistentwiththestan-\ndard literature. Consider a harmonic, linear oscillator wi th frequency ωdriven by an external sinusoidal\nforcef(t)withfrequency Ω. Theequation of motion is\n¨x+ω2x=AcosΩt=f(t). (28)\nFor initial conditions x(0) = 0and˙x(0) = 0, the solution is\nx(t) =−A\n(Ω2−ω2)(cosΩt−cosωt/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nx(0)=0,˙x(0)=0). (29)\nThe term −cosωtis needed to satisfy the initial conditions. Its importance will become clear later. The\nbeam consists of an ensemble of oscillators with different f requenciesωwith a distribution ρ(ω)and a\nspread∆ω,schematically shown inFig. 7.\nRecallthatforatransverse(betatron)motion ωxisthetune. Thenumberofparticlesperfrequency\nband isρ(ω) =1\nNdN/dωwith/integraltext∞\n−∞ρ(ω)dω= 1. Theaveragebeam response (centre of mass) isthen\n∝angb∇acketleftx(t)∝angb∇acket∇ight=/integraldisplay∞\n−∞x(t)ρ(ω)dω, (30)\nand re-written using (29)\n∝angb∇acketleftx(t)∝angb∇acket∇ight=−/integraldisplay∞\n−∞/bracketleftbiggA\n(Ω2−ω2)(cosΩt−cosωt)/bracketrightbigg\nρ(ω)dω. (31)\nFor a narrow beam spectrum around a frequency ωx(tune) and the driving force near this frequency\n(Ω≈ωx),\n∝angb∇acketleftx(t)∝angb∇acket∇ight=−A\n2ωx/integraldisplay∞\n−∞/bracketleftbigg1\n(Ω−ω)(cosΩt−cosωt)/bracketrightbigg\nρ(ω)dω. (32)\n8For the further evaluation, we transform variables from ωtou=ω−Ω(see Ref. [12]) and assume that\nΩis complex: Ω = Ω r+iΩiwhere\n∝angb∇acketleftx(t)∝angb∇acket∇ight=−A\n2ωxcos(Ωt)/integraldisplay∞\n−∞duρ(u+Ω)1−cos(ut)\nu(33)\n+A\n2ωxsin(Ωt)/integraldisplay∞\n−∞duρ(u+Ω)sin(ut)\nu. (34)\nThisavoids singularities for u= 0.\nWeare interested inlong-term behaviour, i.e. t→ ∞,so weuse\nlim\nt→∞sin(ut)\nu=πδ(u), (35)\nlim\nt→∞1−cos(ut)\nu= P.V./parenleftbigg1\nu/parenrightbigg\n, (36)\n∝angb∇acketleftx(t)∝angb∇acket∇ight=A\n2ωx/bracketleftbigg\nπρ(Ω)sin(Ωt)+cos(Ωt)P.V./integraldisplay∞\n−∞dωρ(ω)\n(ω−Ω)/bracketrightbigg\n. (37)\nThisresponse or BTFhas twoparts asfollows.\ni) Resistive part, the first term in the expression is inphase withthe excitation: absorbs energy from\noscillation =⇒damping (would not be there without the term −cosωtin(29)).\nii) Reactive part, the second term inthe expression isout of phase withthe excitation.\nAssuming Ωis complex, we integrate around the pole and obtain a P.V. and a ‘residuum’ (Sokhotski–\nPlemelj formula), astandard technique in complex analysis .\nWecan discuss this expression and find\ni) the ‘damping’ part only appeared because of the initial co nditions;\nii) with other initial conditions, weget additional terms i nthe beam response;\niii) that is, for x(0)∝negationslash= 0and˙x(0)∝negationslash= 0wemayadd\nx(0)/integraldisplay\ndωρ(ω)cos(ωt)+ ˙x(0)/integraldisplay\ndωρ(ω)sin(ωt)\nω. (38)\nWith these initial conditions, we do not obtain Landau dampi ng and the dynamics is very different. We\nhave again:\ni) oscillation of particles withdifferent frequencies (tu nes);\nii) now with different initial conditions, x(0)∝negationslash= 0and˙x(0) = 0orx(0) = 0and˙x(0)∝negationslash= 0;\niii) again weaverage over particles toobtain the centre of m ass motion.\nWe obtain Figs. 8–10. Figure 8 shows the oscillation of indiv idual particles where all particles have the\ninitial conditions x(0) = 0 and˙x(0)∝negationslash= 0. In Fig. 9, we plot again Fig. 8 but add the average beam\nresponse. We observe that although the individual particle s continue their oscillations, the average is\n‘damped’ to zero. The equivalent for the initial conditions x(0)∝negationslash= 0and˙x(0) = 0is shown in Fig. 10.\nWith a frequency (tune) spread the average motion, which can be detected by aposition monitor, damps\nout. However, this is notLandau damping, rather filamentation or decoherence. Contr ary to Landau\ndamping, it leads to emittance growth.\n9-2-1.5-1-0.5 0 0.5 1 1.5 2\n 0  200  400  600  800  1000  1200x(t), <x(t)>\ntime tBeam response\nFig.8:Motionofparticleswithfrequencyspread. Initialconditi ons:x(0) = 0and˙x(0)∝negationslash= 0\n-2-1.5-1-0.5 0 0.5 1 1.5 2\n 0  200  400  600  800  1000  1200x(t), <x(t)>\ntime tBeam response\nFig. 9:Motion of particles with frequency spread and total beam res ponse. Initial conditions: x(0) = 0 and\n˙x(0)∝negationslash= 0.\n3.2 Interpretation of Landaudamping\nComparedwiththeprevious caseofLandaudampinginaplasma , theinterpretation ofthemechanism is\nquitedifferent. Theinitial conditions ofthebeamsareimp ortant andalsothespreadofexternal focusing\nfrequencies. For the initial conditions x(0) = 0 and˙x(0) = 0, the beam is quiet and a spread of\nfrequenciesρ(ω)ispresent. When anexcitation isapplied:\ni) particles cannot organize into collective response (pha se mixing);\nii) average response is zero;\niii) the beam iskept stable, i.e. stabilized.\nIn the case of accelerators the mechanism is therefore not a d issipative damping but a mechanism for\nstabilization. Landau damping should be considered as an ‘a bsence of instability’. In the next step this\nisdiscussed quantitatively and thedispersion relations a rederived.\n3.3 Dispersion relations\nWerewrite (simplify) the response incomplex notation:\n∝angb∇acketleftx(t)∝angb∇acket∇ight=A\n2ωx/bracketleftbigg\nπρ(Ω)sin(Ωt)+cos(Ωt)P.V./integraldisplay∞\n−∞dωρ(ω)\n(ω−Ω)/bracketrightbigg\n(39)\n10-2-1.5-1-0.5 0 0.5 1 1.5 2\n 0  200  400  600  800  1000  1200x(t), <x(t)>\ntime tBeam response\nFig. 10: Motion of particles with frequency spread and total beam res ponse. Initial conditions x(0)∝negationslash= 0and\n˙x(0) = 0.\nbecomes\n∝angb∇acketleftx(t)∝angb∇acket∇ight=A\n2ωxe−iΩt/bracketleftbigg\nP.V./integraldisplay\ndωρ(ω)\n(ω−Ω)+iπρ(Ω)/bracketrightbigg\n. (40)\nThefirst part describes an oscillation with complex frequen cyΩ.\nSince we know that the collective motion is described as e(−iΩt), an oscillating solution Ωmust\nfulfil therelation\n1+1\n2ωx/bracketleftbigg\nP.V./integraldisplay\ndωρ(ω)\n(ω−Ω)+iπρ(Ω)/bracketrightbigg\n= 0. (41)\nThis is again a dispersion relation, i.e. the condition for t he oscillating solution. What do we do with\nthat? We could look where Ωi<0provides damping. Note that no contribution to damping is po ssible\nwhenΩis outside the spectrum. In the following sections, we intro duce BTFs and stability diagrams\nwhich allow us todetermine thestability of abeam.\n3.4 Normalized parametrization andBTFs\nWe can simplify the calculations by moving to normalized par ametrization. Following Chao’s pro-\nposal [12], inthe expression\n∝angb∇acketleftx(t)∝angb∇acket∇ight=A\n2ωxe−iΩt/bracketleftbigg\nP.V./integraldisplay\ndωρ(ω)\n(ω−Ω)+iπρ(Ω)/bracketrightbigg\n(42)\nweuse again u,but normalized to frequency spread ∆ω. Wehave\nu= (ωx−Ω) =⇒u=(ωx−Ω)\n∆ω(43)\nand introduce twofunctions f(u)andg(u):\nf(u) = ∆ωP.V./integraldisplay\ndωρ(ω)\nω−Ω, (44)\ng(u) =π∆ωρ(ωx−u∆ω) =π∆ωρ(Ω). (45)\nTheresponse withthe driving force discussed above isnow\n∝angb∇acketleftx(t)∝angb∇acket∇ight=A\n2ωx∆ωe−iΩt[f(u)+i·g(u)], (46)\n11/0/0/1/1\n/0/0/0/0/0/0\n/1/1/1/1/1/1X0\nFig.11:Bunchwithoffsetina cavity-likeobject\nwhere∆ωisthefrequencyspreadofthedistribution. Theexpression f(u)+i·g(u)istheBTF.Withthis,\nit is easier to evaluate the different cases and examples. Fo r important distributions ρ(ω)the analytical\nfunctionsf(u)andg(u)exist (see e.g. [14]) and will lead usto stability diagrams.\n3.5 Example: response inthepresence of wakefields\nThedrivingforcecomesfromthedisplacementofthebeamasa whole,i.e. ∝angb∇acketleftx∝angb∇acket∇ight=X0,forexampledriven\nby awakefield or impedance (see Fig. 11). Theequation of moti on for aparticle isthen something like\n¨x+ω2x=f(t) =K·∝angb∇acketleftx∝angb∇acket∇ight, (47)\nwhereKisa‘coupling coefficient’. Thecoupling coefficient Kdepends onthenature ofthewakefield.\ni) IfKis purely real: the force isin phase with the displacement, e .g. image space charge inperfect\nconductor.\nii) Forpurely imaginary K: theforceisinphasewiththevelocity andout ofphasewitht hedisplace-\nment.\niii) In practice, wehave both and wecan write\nK= 2ωx(U−iV). (48)\nInterpretation:\na) abeam travelling off centre through animpedance induces transverse fields;\nb) transverse fields act back on all particles inthe beam, via\n¨x+ω2x=f(t) =K·∝angb∇acketleftx∝angb∇acket∇ight; (49)\nc) if thebeam movesas awhole (in phase, collectively), this can grow for V >0;\nd) the coherent frequency Ωbecomes complex and is shifted by (Ω−ωx).\n3.5.1 Beam withoutfrequency spread\nFor a beam without frequency spread (i.e.ρ(ω) =δ(ω−ωx)), we can easily sum over all particles and\nfor the centre of massmotion ∝angb∇acketleftx∝angb∇acket∇ightweget\n¨∝angb∇acketleftx∝angb∇acket∇ight+Ω2∝angb∇acketleftx∝angb∇acket∇ight=f(t) =−2ωx(U−iV)·∝angb∇acketleftx∝angb∇acket∇ight. (50)\nFor theoriginal coherent motion withfrequency Ω,this means that:\ni) in-phase component Uchanges the frequency;\nii) out-of-phase component Vcreates growth ( V >0) or damping ( V <0).\nFor anyV >0, thebeam isunstable (even if very small).\n123.5.2 Beam withfrequency spread\nWhat happens for abeam withafrequency spread ?\nTheresponse (and therefore the driving force) was\n∝angb∇acketleftx(t)∝angb∇acket∇ight=A\n2ωx∆ωe−iΩt[f(u)+i·g(u)]. (51)\nThe(complex) frequency Ωis now determined by the condition\n−(Ω−ωx)\n∆ω=1\n(f(u)+ig(u)). (52)\nAll information about thestability is contained inthis rel ation.\ni) The (complex) frequency difference (Ω−ωx)contains the effects of impedance, intensity, γetc.\n(see the article by G.Rumolo [15]).\nii) The right-hand side contains information about the freq uency spectrum (see definitions for f(u)\nandg(u)in (44) and (45)).\nWithout Landau damping (no frequency spread):\ni) ifIm(Ω−ωx)<0, the beam isstable;\nii) ifIm(Ω−ωx)>0, the beam isunstable (growth rate τ−1).\nWith afrequency spread, wehave a condition for stability (L andau damping)\n−(Ω−ωx)\n∆ω=1\n(f(u)+ig(u)). (53)\nHowdo weproceed tofind thelimits?\nWe could try to find the complex Ωat the edge of stability ( τ−1= 0). In the next section, we\ndevelop amore powerful tool to assess the stability of adyna mic system.\n4Stability diagrams\nTostudythestabilityofaparticlebeam,itisnecessary tod evelopeasytousetoolstorelatethecondition\nfor stability with the complex tune shift due to, e.g., imped ances. We consider the right-hand side first\nand call itD1. BothD1and the tune shift are complex and should beanalysed inthe co mplex plane.\nUsing the (real) parameter uin\nD1=1\n(f(u)+ig(u)), (54)\nif we knowf(u)andg(u)we can scan ufrom−∞to+∞and plot the real and imaginary parts of D1\nina complex plane.\nWhyis this formulation interesting? Theexpression\n(f(u)+ig(u)) (55)\nis actually the BTF, i.e. it can be measured. With its knowled ge (more precisely: its inverse), we have\nconditions on (Ω−ωx)for stability and the limits for intensities and impedances .\n13-2-1.5-1-0.5 0 0.5 1 1.5 2\n-2 -1.5 -1 -0.5  0  0.5  1  1.5  2Imag D1\nReal D1Stability diagram\nFig.12:Stabilitydiagramforrectangularfrequencydistribution\n4.1 Examplesfor bunchedbeams\nAsanexample, weuse arectangular distribution function fo r the frequencies (tunes), i.e.\nρ(ω) =/braceleftbigg1\n2∆ωfor|ω−ωx| ≤∆ω,\n0 otherwise .(56)\nWenow follow some standard steps.\nStep1. Compute f(u)andg(u)(or lookit up,e.g.[14]): weget for therectangular distrib ution function\nf(u) =1\n2ln/vextendsingle/vextendsingle/vextendsingle/vextendsingleu+1\nu−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle, g(u) =π\n2·H(1−|u|). (57)\nStep2. Plot the real and imaginary parts of D1.\nTheresult of this procedure isshown inFig. 12 and the interp retation isas follows.\n•Weplot Re (D1)versus Im (D1)for arectangular distribution ρ(ω).\n•This isastability boundary diagram.\n•It separates stable from unstable regions (stability limit ).\nNow we plot the complex expression of −(Ω−ωx)\n∆ωin the same plane as a point (this point depends on\nimpedances, intensities, etc.). In Fig. 13, weshow the same stability diagram together with examples of\ncomplextuneshifts. Thestableandunstablepointsinthest abilitydiagramareindicatedintheright-hand\nside of Fig.13.\nWecanuseother typesof frequency distributions, for examp leabi-Lorentz distribution ρ(ω). We\nfollow the same procedure as above and the result is shown in F ig. 14. It can be shown that in all cases\nhalf of thecomplex plane is stable without Landau damping, a s indicated inFig. 14.\n4.2 Examplesfor unbunchedbeams\nAsimilartreatmentcanbeappliedtounbunched beams,altho ughsomecarehastobetaken, inparticular\ninthe case of longitudinal stability.\n4.2.1 Transverse unbunchedbeams\nThetechnique applies directly. Frequency (tune) spread is from:\ni) change of revolution frequency withenergy spread (momen tum compaction);\n14-2-1.5-1-0.5 0 0.5 1 1.5 2\n-2 -1.5 -1 -0.5  0  0.5  1  1.5  2Imag D1\nReal D1Stability diagram\n      \n              \n-2-1.5-1-0.5 0 0.5 1 1.5 2\n-2 -1.5 -1 -0.5  0  0.5  1  1.5  2Imag D1\nReal D1Stability diagram\nstable\nunstablestable\nFig.13:Stabilitydiagramsforrectangulardistributiontogether withexamplesforcomplextuneshifts(left);stable\nandunstablepointsareindicated(right).\n-2-1.5-1-0.5 0 0.5 1 1.5 2\n-8 -6 -4 -2  0  2  4  6  8Imag D1\nReal D1Stability diagram\nstable\nunstable\nFig.14:Stabilitydiagramforbi-Lorentzfrequencydistribution. Stableandunstableregionsareindicated\nii) change of betatron frequency with energy spread (chroma ticity).\nTheoscillation depends on the modenumber n(number of oscillations around the circumference C):\n∝exp(−iΩt+in(s/C)) (58)\nand the variable ushould be written\nu= (ωx+n·ω0−Ω)/∆ω. (59)\nThe rest has the same treatment. Transverse collective mode s in an unbunched beam for mode numbers\n4and 6are shown inFig. 15.\n4.2.2 Longitudinal unbunchedbeams\nWhatabout longitudinal instability ofunbunched beams? Th isisaspecial casesince thereisnoexternal\nfocusing, therefore alsonospread ∆ωof focusing frequencies. However,wehave aspread inrevolu tion\nfrequency, whichisdirectlyrelatedtoenergy, andenergye xcitationsdirectlyaffectthefrequencyspread:\n∆ωrev\nω0=−η\nβ2∆E\nE0. (60)\n15-2-1.5-1-0.5 0 0.5 1 1.5 2\n-2 -1.5 -1 -0.5  0  0.5  1  1.5  2 \n Collective mode\n-2-1.5-1-0.5 0 0.5 1 1.5 2\n-2 -1.5 -1 -0.5  0  0.5  1  1.5  2 \n Collective mode\nFig.15:Transversecollectivemodewith modeindex n= 4andn= 6\nThefrequency distribution isdescribed by\nρ(ωrev) and ∆ωrev. (61)\nSince there is noexternal focusing ( ωx= 0), wehave to modify the definition of our parameters:\nu=(ωx+n·ω0−Ω)\n∆ω=⇒u=(n·ω0−Ω)\nn·∆ω(62)\nand introduce twonew functions F(u)andG(u): thevariable nisthe mode number.\nF(u) =n·∆ω2P.V./integraldisplay\ndω0ρ′(ω0)\nn·ω0−Ω, (63)\nG(u) =π∆ω2ρ′(Ω/n) (64)\ntoobtain\n−(Ω−n·ω0)2\nn2∆ω2=1\n(F(u)+iG(u))=D1. (65)\nAsanimportant consequence, theimpedance isnowrelated to thesquareof thecomplexfrequency shift\n(Ω−n·ω0)2. Thishas rather severe implications.\ni) Consequence: no more stable region in one half of the plane .\nii) Landau damping isalways required toensure stability.\nAs an illustration, we show some stability diagrams derived from the new D1. The stability diagram\nfor unbunched beams, for the longitudinal motion, and witho ut spread, is shown in Fig. 16. The stable\nregion isjust an infinitely narrow line for Im (D1) = 0and positive Re (D1).\nIntroducing afrequencyspreadforaparabolicdistributio n, wehavethestabilitydiagramshownin\nFig. 17. The locus of the diagram is now the stable region. Asi n the previous example, wetreat again a\nLorentzdistribution ρ(ω)andweshowbothinFig.18. ThestableregionfromtheLorentz distribution is\nsignificantlylarger. Wecaninvestigatethisbystudyingth edistributions. Bothdistributions (normalized)\n16-0.6-0.4-0.2 0 0.2 0.4 0.6\n-1 -0.5  0  0.5  1Imag D1 (Real Z)\nReal D1 (Imag Z)Stability diagram\nstableNo spread           \nFig.16:Re(D1)versusIm (D1)forunbunchedbeamwithoutspread\n-0.6-0.4-0.2 0 0.2 0.4 0.6\n-1 -0.5  0  0.5  1Imag D1 (Real Z)\nReal D1 (Imag Z)Stability diagram\nstableunstableParabolic           \nFig.17:Re(D1)versusIm (D1)forparabolicdistribution ρ(ω)andunbunchedbeam\n-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8\n-1 -0.5  0  0.5  1Imag D1 (Real Z)\nReal D1 (Imag Z)Stability diagram\n                                     Parabolic\n             Lorentz\nFig.18:Re(D1)versusIm (D1)forparabolicandLorentzdistributions ρ(ω)andunbunchedbeam\nare displayed in Fig. 19. With a little imagination, we can as sume that the difference is due to the very\ndifferent populations of the tails of the distribution.\nA particular problem we encounter is when we do not know the sh ape of the distribution. In\nFig. 20, we show the stability diagrams again together with a circle inscribed inside both distributions.\nThis is a simplified (and pessimistic) criterion for the long itudinal stability of unbunched beams [17].\nSinceD1is directly related to the machine impedance Z, we obtain a criterion which can be readily\n17 0 0.2 0.4 0.6 0.8 1 1.2\n-4 -2  0  2  4ρ\nuDistribution function\nLorentzParabolicParabolic          \nLorentz            \nFig.19:FrequencydistributionforparabolicandLorentzdistribu tionsρ(ω)andunbunchedbeam\n-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8\n-1 -0.5  0  0.5  1Imag D1 (Real Z)\nReal D1 (Imag Z)Stability diagram\n                                     Parabolic\n             Lorentz\n                  Inscribed circle\nFig.20:StabilitydiagramsforparabolicandLorentzdistribution s,acircle inscribed\napplied for estimating thebeam stability. For longitudina l stability/instability, wehave the condition\n|Z/bardbl|\nn≤Fβ2E0|ηc|\nqI/parenleftbigg∆p\np/parenrightbigg2\n. (66)\nThisistheKeil–Schnellcriterion[17]applicabletoother typesofdistributionsandfrequentlyusedforan\nestimateofthemaximumallowedimpedance. Wehavethefrequ encyspreadfrommomentumspreadand\nmomentum compaction ηcrelated to the machine impedance. For given beam parameters , this defines\nthe maximum allowed impedance|Z/bardbl|\nn.\n5Effect ofsimplifications\nWehave used afew simplifications in thederivation.\ni) Oscillators are linear.\nii) Movement of thebeam isrigid (i.e. beam shape and size do n ot change).\nWhat if weconsider the‘real’ cases, i.e. non-linear oscill ators? Consider now abunched beam; because\nofthesynchrotron oscillation, revolution frequencyandb etatronspread(fromchromaticity) averageout.\nAssources for the frequency spread wehave non-linear force s, such asthe following examples.\ni) Longitudinal: sinusoidal RFwave.\nii) Transverse: octupolar or high multipolar field componen ts.\n18Canweuse the sameconsiderations as for an ensemble of linearoscillators?\nThe excited betatron oscillation will change the frequency distribution ρ(ω)(frequency depends\nnowontheoscillation amplitude). Anoscillating bunch cha nges thetune(and ρ(ω)) duetothedetuning\ninthe non-linear fields.\nA complete derivation can be done through the Vlasov equatio n [5], but this is well beyond the\nscope of this article. Theequation\n∝angb∇acketleftx(t)∝angb∇acket∇ight=A\n2ωxe−iΩt/bracketleftbigg\nP.V./integraldisplay\ndωρ(ω)\n(ω−Ω)+iπρ(Ω)/bracketrightbigg\n(67)\nbecomes\n∝angb∇acketleftx(t)∝angb∇acket∇ight=A\n2ωxe−iΩt/bracketleftbigg\nP.V./integraldisplay\ndω∂ρ(ω)/∂ω\n(ω−Ω)+iπ∂ρ(Ω)/∂Ω/bracketrightbigg\n. (68)\nThe distribution function ρ(ω)has to be replaced by the derivative ∂ρ(ω)/∂ω. We evaluate now\nthis configuration for instabilities in the transverse plan e.\nSince the frequency ωdepends now onthe particle’s amplitudes JxandJy(see [16]), theexpression\nωx(Jx,Jy) =∂H\n∂Jx(69)\nisthe amplitude-dependent betatron tune (similar for ωy).\nWethen have to write\nρ(ω) =⇒ρ(Jx,Jy). (70)\nAssuming a periodic force in the horizontal (x)plane and using now the tune (normalized frequency)\nQ=ω/ω0,\nFx=A·exp(−iω0Qt), (71)\nthe dispersion integral can be written as (see also [25] and r eferences therein)\n1 =−∆Qcoh/integraldisplay∞\n0dJx/integraldisplay∞\n0dJyJx∂ρ(Jx,Jy)\n∂Jx\nQ−Qx(Jx,Jy). (72)\nThen weproceed asbefore to get the stability diagram.\nIf the particle distribution changes (often as a function of time), obviously the frequency distribu-\ntionρ(ω)changes aswell.\ni) Examples: higher-order modes; coherent beam–beam modes .\nii) Treatment requires solving the Vlasov equation (pertur bation theory or numerical integration).\niii) Instead, we use a pragmatic approach: unperturbed stability region andperturbed complex tune\nshift.\n6Landau damping asa cure\nIf the boundary of\nD1=1\n(f(u)+ig(u))(73)\ndetermines thestability, can weincrease the stable region by either of the following methods?\n19 0 5e-05 0.0001 0.00015 0.0002 0.00025\n-0.002 -0.0015 -0.001 -0.0005  0Imag ∆Q\nReal ∆Q 0 5e-05 0.0001 0.00015 0.0002 0.00025\n-0.002 -0.0015 -0.001 -0.0005  0Imag ∆Q\nReal ∆Q\nFig. 21:Stability diagrams for an octupolar field (left) and togethe r with the complex tune shifts for a stable and\nanunstablemode(right).\ni) Modifying the frequency distribution Qx(ω), i.e. the distribution of the amplitudes Qx(Jx,Jy)\n(see definition of f(u)andg(u))?\nii) Introducing tune spread artificially (octupoles, other high-order fields)?\nThetune dependence of anoctupole ( k3) can bewritten as[16]\nQx(Jx,Jy) =Q0+a·k3·Jx+b·k3·Jy. (74)\nOther sources to introduce tune spread are for example:\ni) space charge;\nii) chromaticity;\niii) high-order multipole fields;\niv) beam–beam effects (colliders only).\n6.1 Landaudampinginthepresence of non-linear fields\nTherecipe for ‘generating’ Landau damping is:\ni) for amultipole field, compute detuning Q(Jx,Jy);\nii) given the distribution ρ(Jx,Jy);\niii) compute the stability diagram by scanning frequency, i .e. the amplitudes.\n6.2 Stabilization with octupoles\nFigure21(left) showsthestability diagram wecangetforan octupole. InFig. 21(right), wehaveadded\nto complex tune shift for a stable and an unstable mode. The po int is inside the region and the beam is\nstable. An unstable mode (e.g. after increase of intensity o r impedance) is shown in Fig. 21 (right); it\nlies outside the stable region.\nTheoretically, we can increase the octupolar strength to in crease the stable region. However, can\nweincrease the octupole strength aswelike? Thereare some d ownsides as follows.\ni) Octupoles introduce strong non-linearities at large amp litudes and may lead to reduced dynamic\naperture and bad lifetime.\nii) Wedo not have manyparticles at large amplitudes; this re quires large strengths of the octupoles.\n20 0 5e-05 0.0001 0.00015 0.0002 0.00025\n-0.002 -0.0015 -0.001 -0.0005  0Imag ∆Q\nReal ∆Q 0 5e-05 0.0001 0.00015 0.0002 0.00025\n-0.002 -0.0015 -0.001 -0.0005  0Imag ∆Q\nReal ∆Qm = 0,  Q' =  0   \nFig.22:Stabilitydiagramsforanoctupolarfieldand m= 0head–tailmodewith Q′= 0\niii) Cancause reduction of dynamic aperture and lifetime.\niv) Theychange thechromaticity via feed-down effects!\nv) Thelesson: use them if you have no other choice.\nWesee[16]thatthecontributionfromoctupolestothestabi lityregionmainlycomesfromlarge-amplitude\nparticles, i.e. the population of tails. This has some frigh tening consequences: changes in the distribu-\ntionofthetailscansignificantly changethestableregion. Itisrather doubtful whether tailscanbeeasily\nreproduced. Furthermore, inaccelerators requiring small lossesandlonglifetimes(e.g.colliders) thetail\nparticles are unwanted and often caught by the collimation s ystem to protect the machine. A reliable\ncalculation of the stability becomes a gamble. Proposals to artificially increase tails in a hadron collider\nare a bad choice although it had been proposed for fast-cycli ng machines since they have no need for a\nlong beam lifetime.\n6.3 Example: head–tail modes\nWe now apply the scheme to the explicit example of head–tail m odes. They can be due to short-range\nwake fieldsor broadband impedance, etc.\nThegrowth and damping times of head–tail modes are usually c ontrolled withchromaticity Q′:\ni) some modes need positive Q′;\nii) some modes need negative Q′;\niii) some modes can bedamped by feedback ( m= 0).\nIn Figs. 22–24, the stability diagrams are shown for differe nt head–tail modes and different values of\nthe chromaticity Q′. For a zero chromaticity Q′= 0, them= 0mode is unstable (Fig. 22). The\npositive chromaticity stabilizes the m= 0mode by shifting it into the stable area Fig. 23. A negative\nchromaticity makes the beam more unstable by shifting the tu ne shift further into the unstable region.\nHowever,toolargepositivechromaticitymoveshigherorde rhead–tail modestolargerimaginaryvalues,\nuntil they may become unstable (Fig. 24). Increasing the chr omaticity further than in Fig. 24, the mode\nm=−1becomesunstable. Landaudampingisrequiredinthiscase,b utwiththenecessarycaretoavoid\ndetrimental effects if this isachieved with octupoles.\n6.4 Stabilization with othernon-linear elements\nCanwealways stabilize the beam with strong octupole fields?\ni) Would need verylarge octupole strength for stabilizatio n.\n21 0 5e-05 0.0001 0.00015 0.0002 0.00025\n-0.002 -0.0015 -0.001 -0.0005  0Imag ∆Q\nReal ∆Qm = 0,  Q' =  0   \nm = 0,  Q' =  - 3  \n 0 5e-05 0.0001 0.00015 0.0002 0.00025\n-0.002 -0.0015 -0.001 -0.0005  0Imag ∆Q\nReal ∆Qm = 0,  Q' =  0   \nm = 0,  Q' =  - 3  \nm = 0,  Q' =  + 3  \nFig.23:Stabilitydiagramsforan octupolarfieldand m= 0head–tailmodewith negativechromaticity Q′=−3\n(left)andin theright-handfigureaswell ahead–tailmodewi thpositivechromaticity Q′= +3.\n 0 5e-05 0.0001 0.00015 0.0002 0.00025\n-0.002 -0.0015 -0.001 -0.0005  0Imag ∆Q\nReal ∆Qm = 0,  Q' =  0   \nm = 0,  Q' =  - 3  \nm = 0,  Q' =  + 3  \nm = 1,  Q' =  + 3  \nFig. 24:Stability diagrams for an octupolar field and m= 0andm=−1head–tail modes with positive chro-\nmaticityQ′= +3.\nii) Theknown problems:\na) can cause reduction of dynamic aperture and lifetime;\nb) lifetime important when beam stays inthe machine for alon g time;\nc) colliders, lifetime more than 10–20 hneeded.\niii) Is there another option?\nIn the case of colliders, the beam–beam effects provide a lar ge tune spread that can be used for\nLandaudamping(Fig.25). Evenwhenthetunespreadiscompar able, thestabilityregionfromahead-on\nbeam–beam interaction is much increased [18]. We observe a l arge difference between the two stability\ndiagrams. Wheredoesthisdifference comefrom? Thetunedep endence ofanoctupole canbewrittenas\nQx(Jx,Jy) =Q0+aJx+bJy (75)\nand islinear in theaction J(for coefficients, see Appendix B).\nThe tune dependence of a head-on beam–beam collision can be w ritten [19,20] as follows: with\nα=x/σ∗, we get∆Q/ξ= (4/α2)/bracketleftBig\n1−I0(α2/4)·e−α2/4/bracketrightBig\n. As a demonstration, we show in Fig. 26\nthe tune spread from octupoles, long-range beam–beam effec ts [19] and head-on beam–beam effects.\n22 0 5e-05 0.0001 0.00015 0.0002 0.00025\n-0.002 -0.0015 -0.001 -0.0005  0Imag ∆Q\nReal ∆Qm = 0,  Q' =  0   \nm = 0,  Q' =  - 3 \nm = 0,  Q' =  + 3 \nm = 1,  Q' =  + 3 \nFig.25:Stabilitydiagramsforanoctupolarfieldandhead–tailmode s(Fig.24)andstabilitydiagramfrombeam–\nbeameffects.\nFig. 26:Tune spread in two dimensions for octupoles, long-range bea m–beam and head-on beam–beam. Tune\nspreadadjustedtobethe sameforallcases.\nThe parameters are adjusted such that the overall tune sprea d is always the same for the three cases.\nHowever, the head-on beam–beam tune spread has the largest e ffect forsmallamplitudes, i.e. where\nthe particle density is largest and the contribution to the s tability is strong. For the tune spreads shown\nin Fig. 26, we have computed the corresponding stability dia grams [21]. Although the tune spread is\nthe same, the stability region shown in Fig. 27 is significant ly larger. As another and most important\nadvantage, sincethestabilityisensuredbysmall-amplitu de particles, thestableregionisveryinsensitive\nto the presence of tails or the exact distribution function. Special cases such as collisions of ‘hollow\nbeams’ are not noteworthy in this overview.\n6.4.1 Conditions for Landaudamping\nAtune spread isnot sufficient toensure Landau damping. Tosu mmarize, weneed:\n23Fig.27:Stability diagramsforoctupoles,long-rangebeam–beaman dhead-onbeam–beam. Tunespreadadjusted\ntobethe sameforallcases.\nSpectrum of coherent modes, no Landau damping   \n00.10.20.30.40.50.60.70.8\n-2 -1.5 -1 -0.5 0 0.5Spectrum of coherent modes, Landau damping      \n00.10.20.30.40.50.60.70.8\n-2 -1.5 -1 -0.5 0 0.5\nFig. 28:Coherent beam–beam modes with π-mode inside and outside the incoherent tune spread due to be am–\nbeameffects. Decouplingis duetointensitydifferencebet weenthetwobeams.\ni) presence of incoherent frequency (tune) spread;\nii) coherent mode must be inside this spread;\niii) the same particles must beinvolved inthe oscillation a nd the spread.\nThe last item requires some explanation. When the tune sprea d is not provided by the particles actually\ninvolved in the coherent motion, it is not sufficient and resu lts in no damping. For example, this is the\ncaseforcoherent beam–beammodeswheretheeigenmodesofco herentbeam–beam modesdriveneither\nby head-on effects or long range interactions are very diffe rent [22–24]. For modes driven by the head-\non interactions, mainly particles at small amplitudes are p articipating in the oscillation. The tune spread\nproduced by long range interactions produces little dampin g inthis case.\nAnother option to damp coherent beam–beam modes is shown in F ig. 28. It shows the results\nof a simulation for the main coherent beam–beam modes (0-mod e andπ-mode) for slightly different\nintensities. The difference is sufficient to decouple the tw o beams and they cannot maintain the correct\nphase. Wehave:\n24i) coherent mode inside or outside the incoherent spectrum, depending on theintensity difference;\nii) Landau damping restored when the symmetry issufficientl y broken.\n6.5 Landaudampingwithnon-linear fields: are there anyside effects?\nIntroducing non-linear fields into an accelerator is not alw ays a desirable procedure. It may have impli-\ncations for the beam dynamics and weseparate them into three categories.\ni) Good:\na) stability region increased.\nii) Bad:\na) non-linear fieldsintroduced (resonances);\nb) changes optical properties, e.g. chromaticity (feed-do wn).\niii) Special cases:\na) non-linear effects for large amplitudes (octupoles);\nb) much better: head-on beam–beam (but only incolliders).\nLandaudamping withnon-linear fieldsisaverypowerful tool ,but thesideeffectsandimplications have\ntobe taken into account.\n7Summary\nThe collisionless damping of coherent oscillations as pred icted by Landau in 1946 is an important con-\ncept in plasma physics as well as in other applications such a s hydrodynamics, astrophysics and bio-\nphysics to mention a few. It is used extensively in particle a ccelerators to avoid coherent oscillations of\nthebeams andthe instabilities. Despite itsintensive use, it isnot asimple phenomenon andthe interpre-\ntation of the physics behind the mathematical structures is a challenge. Even after many decades after\nitsdiscovery, thereis(increasing) interest inthispheno menon andworkonthetheory(andextensions of\nthe theory) continues.\nReferences\n[1] I. Langmuir and L.Tonks, Phys. Rev. 33(1929) 195.\n[2] L.D.Landau, J. Phys. USSR 10(1946) 26.\n[3] D.Bohm and E.Gross, Phys. Rev. 75(1949) 1851.\n[4] D.Bohm and E.Gross, Phys. Rev. 75(1949) 1864.\n[5] D.Sagan, Onthe physics of Landau damping, CLNS93/1185 ( 1993).\n[6] K.Case, Ann. Phys. 7(1959) 349.\n[7] N.G.VanKampen, Physica21(1955) 949.\n[8] J. Malmberg and C.Wharton, Phys. Rev. Lett. 13(1964) 184.\n[9] V.Neil and A.Sessler, Rev. Sci. Instrum. 36(1965) 429.\n[10] L.Laslett, V.Neil and A. Sessler, Rev. Sci. Instrum. 36(1965) 436.\n[11] A.A.Vlasov, J. Phys. USSR 9(1945) 25.\n[12] A.Chao, Theory of Collective Beam Instabilities in Accelerators (Wiley, NewYork, 1993).\n[13] A.Hofmann, Landau damping, Proc. CERNAccelerator Sch ool (2009).\n[14] A.Chao and M.Tigner, Handbook of Accelerator Physics and Engineering (World Scientific, Sin-\ngapore, 1998).\n25[15] G.Rumolo, Beam instabilities, these proceedings, CER NAccelerator School (2013).\n[16] W. Herr, Mathematical and numerical methods for non-li near beam dynamics, these proceedings,\nCERNAccelerator School (2013).\n[17] E. Keil and W. Schnell, Concerning longitudinal stabil ity in the ISR, CERN-ISR-TH-RH/69-48\n(1969).\n[18] W.HerrandL.Vos, Tunedistributions andeffective tun espread frombeam–beam interactions and\nthe consequences for Landau damping in the LHC,LHCProject N ote316 (2003).\n[19] W.Herr, Beam–beam effects, Proc. CERNAccelerator Sch ool (2003).\n[20] T.Pieloni, Beam–beam effects, these proceedings, CER NAccelerator School (2013).\n[21] X. Buffat, Consequences of missing collisions – beam st ability and Landau damping, Proc. ICFA\nBeam–Beam Workshop 2013, CERN(2013).\n[22] Y. Alexahin, W. Herr et al., Coherent beam–beam effects, Proc. HEACC 2001, Tsukuba, Ja pan,\n2001.\n[23] Y.Alexahin,Astudyofthecoherentbeam–beameffectin theframeworkoftheVlasovperturbation\ntheory, LHCProject Report 461 (2001).\n[24] Y.Alexahin, Nucl. Instrum. Methods A480(2002) 235.\n[25] J. Scott Berg and F.Ruggiero, Landau damping withtwo-d imensional tune spread, CERNSL-AP-\n96-71 (AP)(1996).\nAppendices\nASolvingthe dispersionrelation\nA.1 Dispersion relation from Vlasov’s calculation\nStarting withthe dispersion relation derived using Vlasov ’s approach (20):\n1+ω2\np\nk2/integraldisplay∂ψ0/∂v\n(ω\nk−v)dv= 0, (A.1)\nwehavetomakeafewassumptions. Weassumethatwecanrestri ctourselvestowaveswith ω/k≫vor\nω/k≪v. The latter case cannot occur in Langmuir waves and we can ass ume the case with ω/k≫v.\nThen wecan integrate theintegral byparts and get\n1+ω2\np\nk2/integraldisplayψ0\n(ω/k−v)2dv= 0 (A.2)\nor, rewritten for the next step,\n1+ω2\np\nω2/integraldisplayψ0\n(1−vk/ω)2dv= 0. (A.3)\nWith the assumption ω/k≫v,wecan expand the denominator inseries ofvk\nωand obtain\n1+ω2\np\nω2/integraldisplay\nψ0dv·/parenleftBigg\n1+2·/parenleftbiggvk\nω/parenrightbigg\n+3·/parenleftbiggvk\nω/parenrightbigg2/parenrightBigg\n= 0. (A.4)\nFor thenext step asan explicit example weuse aMaxwellian ve locity distribution, i.e.\nψ(v) =1√\n2π1\nvpe−v2/2v2\np. (A.5)\n26Theindividual integrals arethen\n/integraldisplay\nψ(v)dv= 1,/integraldisplay\nψ(v)·vdv= 0,/integraldisplay\nψ(v)·v2dv=v2\np=ω2\np\nk2(A.6)\ntoobtain finally\n1−ω2\np\nω2−3k2v2\npω2\np\nω4= 0. (A.7)\nThisdispersion relation can now be solved for ωand weget twosolutions:\nω2=1\n2ω2\np±1\n2ω2\np/parenleftBigg\n1+12k2v2\np\nω2p/parenrightBigg1/2\n. (A.8)\nRewritten, weobtain thewell-known dispersion relation fo r Langmuir waves:\nω2=ω2\np/parenleftbig\n1+3k2λ2/parenrightbig\n(λ=vp/ωp). (A.9)\nThefrequency isreal and there isno damping.\nA.2 Dispersion relation from Landau’sapproach\nHerewesolve the dispersion relation using the result obtai ned by Landau (26):\n1+ω2\np\nk/bracketleftBigg\nP.V./integraldisplay∂ψ0/∂v\n(ω−kv)dv−iπ\nk/parenleftbigg∂ψ0\n∂v/parenrightbigg\nv=ω/k/bracketrightBigg\n= 0. (A.10)\nThisshould lead toadamping.\nIntegration by parts leads now to\n1−ω2\np\nω2−3k2v2\npω2\np\nω4−iπ\nk/parenleftbigg∂ψ0\n∂v/parenrightbigg\nv=ω/k= 0. (A.11)\nFor thereal part, wecan use the samereasoning as for Vlasov’ s calculation and findagain\n1−ω2\np\nω2r−3k2v2\npω2\np\nω4r= 0. (A.12)\nNowwehave used ωrto indicate that wecomputed the real part of the complex freq uency\nω=ωr+i·ωi. (A.13)\nWe assume ‘weak damping’, i.e. ωi≪ωr. This leads us to ω2≃ωr+2iωrωi. With the solution (A.9)\nandkλ≪1, wefindfor the imaginary part ωi\nωi=π·ω3\np\n2·k2·/parenleftbigg∂ψ0\n∂v/parenrightbigg\nv=ω/k. (A.14)\nUsing the Maxwell velocity distribution as anexample:\nψ(v) =1√\n2π1\nvpe−v2/2v2\np, (A.15)\n27weget for the derivative\n∂ψ(v)\n∂v=−1√\n2π1\nv3pe−v2/2v2\np. (A.16)\nUsing again λ=vp/ωpto simplify the expression, we can expand in a series (becaus eω/k≫v) and\narrive at\nωi=−ωp·1\nk3λ3/radicalbiggπ\n2·exp/parenleftbigg\n−1\n2k2λ2−3\n2/parenrightbigg\n. (A.17)\nThisis the damping term weobtain using Landau’s result for p lasma oscillations.\nBDetuning with octupoles\nThetune dependence of anoctupole can bewritten as[16]\nQx(Jx,Jy) =Q0+aJx+bJy (B.1)\nfor the coefficients:\n∆Qx=/bracketleftbigg3\n8π/integraldisplay\nβ2\nxK3\nBρds/bracketrightbigg\nJx−/bracketleftbigg3\n8π/integraldisplay\n2βxβyK3\nBρds/bracketrightbigg\nJy (B.2)\nand\n∆Qy=/bracketleftbigg3\n8π/integraldisplay\nβ2\nyK3\nBρds/bracketrightbigg\nJy−/bracketleftbigg3\n8π/integraldisplay\n2βxβyK3\nBρds/bracketrightbigg\nJx. (B.3)\n28This figure \"imp1.png\" is available in \"png\"\n format from:\nhttp://arxiv.org/ps/1601.05227v1This figure \"ps1.png\" is available in \"png\"\n format from:\nhttp://arxiv.org/ps/1601.05227v1This figure \"ps2.png\" is available in \"png\"\n format from:\nhttp://arxiv.org/ps/1601.05227v1This figure \"ps3.png\" is available in \"png\"\n format from:\nhttp://arxiv.org/ps/1601.05227v1This figure \"hoI055.png\" is available in \"png\"\n format from:\nhttp://arxiv.org/ps/1601.05227v1This figure \"hoI065.png\" is available in \"png\"\n format from:\nhttp://arxiv.org/ps/1601.05227v1This figure \"rho.png\" is available in \"png\"\n format from:\nhttp://arxiv.org/ps/1601.05227v1This figure \"velo.png\" is available in \"png\"\n format from:\nhttp://arxiv.org/ps/1601.05227v1"    },    {        "title": "1204.1948v1.Global_MRI_with_Braginskii_viscosity_in_a_galactic_profile.pdf",        "content": "arXiv:1204.1948v1  [astro-ph.GA]  9 Apr 2012Mon. Not. R. Astron. Soc. 000, 1–??(2011) Printed 8 October 2018 (MN L ATEX style file v2.2)\nGlobal MRI with Braginskii viscosity in a galactic profile\nM. S. Rosin1,2⋆and A. J. Mestel3\n1UCLA Department of Math, 520 Portola Plaza, Los Angeles, CA 9 0095, U.S.A.\n2DAMTP, Centre for Mathematical Sciences, University of Cam bridge, Wilberforce Road, Cambridge, CB3 0WA, U.K.\n3Department of Mathematics, Imperial College, South Kensin gton Campus, London, SW7 2AZ, U.K.\nSubmitted to MNRAS April 2012\nABSTRACT\nWe present a global-in-radius linear analysis of the axisymmetric magn etorotational\ninstability (MRI) in a collisional magnetized plasma with Braginskii viscos ity. For a\ngalactic angular velocity profile Ω we obtain analytic solutions for thre e magnetic field\norientations:purelyazimuthal,purelyverticaland slightlypitched( almostazimuthal).\nIn the first two cases the Braginskii viscosity damps otherwise neu trally stable modes,\nand reduces the growth rate of the MRI respectively. In the final case the Bragin-\nskii viscosity makes the MRI up to 2√\n2 times faster than its inviscid counterpart,\neven for asymptotically small pitch angles. We investigate the transition between the\nLorentz-force-dominated and the Braginskii viscosity-dominate d regimes in terms of\na parameter ∼ΩνB/B2whereνBis the viscous coefficient and Bthe Alfv´ en speed. In\nthe limit where the parameter is small and large respectively we recov er the inviscid\nMRI and the magnetoviscous instability (MVI). We obtain asymptotic expressions for\nthe approach to these limits, and find the Braginskii viscosity can ma gnify the effects\nof azimuthal hoop tension (the growth rate becomes complex) by o ver an order of\nmagnitude. We discuss the relevance of our results to the local app roximation, galax-\nies and other magnetized astrophysical plasmas. Our results shou ld prove useful for\nbenchmarking codes in global geometries.\nKeywords: instabilities–accretion,accretiondiscs–Galaxy:disc–MHD–magnet ic\nfields – plasmas.\n1 INTRODUCTION\nIn the formation of compact objects (stars, planets and\nblack holes) from accretion discs, turbulence driven by the\nMRI, and possibly the MVI, offers a promising mechanism\nfor the necessary angular momentum transport (Velikhov\n1959; Chandrasekhar 1960; Balbus & Hawley 1991; Balbus\n2004). It has also been suggested that the observed velocity\nfluctuations ∼6kms−1in parts of the interstellar medium\n(ISM) with low star formation rates may, in part, arise\nfrom this process (Sellwood & Balbus 1999; Tamburro et al.\n2009). The evidence for this comes primarily from numerical\nsimulations andawiderangeofstudiesagreethatweakmag-\nnetic fields and outwardly decreasing angular velocity pro-\nfiles are an unstable combination (Balbus & Hawley 1998;\nBalbus 2003)\nThe most illuminating explanation for this comes from\na shearing sheet analysis in which the mean azimuthal flow\nof the differentially rotating disc is locally approximated by\na constant angular velocity rotation plus a linear velocity\nshear (Goldreich & Lynden-Bell 1965; Umurhan & Regev\n⋆E-mail: msr35@math.ucla.edu2004). In the simplest possible setup, incompressible,\nisothermal, dissipationless, axisymmetric linear pertur ba-\ntions to amagnetic field with aweak vertical component, i.e.\nparallel to the rotation axis of the disc, are unstable when\nthe angular velocity decreases away from the disc’s centre.\nAzimuthal velocity perturbations to fluid elements at differ -\nent heights, tethered to each other by the magnetic field,\nincrease (decrease) their angular momentum. This causes\nthem to move to larger (smaller) radii as dictated by the\ngravitational field which sets the mean flow. In the frame\nrotating at constant angular velocity, this motion deforms\nthe tethering magnetic field, provided it is not too strong,\nand this induces a prograde (retrograde) Lorentz force on\nthe outer (inner) element thus destabilising the system as\nit moves to yet larger (smaller) radii. This mechanism is at\nthe heart of the MRI.\nHowever, although this model and description captures\nmuch of the essential physics, to fully understand the MRI,\nor at least the framework within which the shearing sheet\nshould exist, a more nuanced approach is needed. In part,\nthis is because the shearing sheet is formulated in a Carte-\nsiancoordinate systemwherecurvatureterms thatarise nat -\nurally from the cylindrical geometry of the accretion disc\nc/circleco√yrt2011 RAS2M. S. Rosin, A. J. Mestel\nare neglected. Indeed, in the local approximation, the over -\nstabilising effects of hoop tension (a curvature effect) asso ci-\nated with the azimuthal magnetic field, are totally ignored.\nFurthermore, the model predicts that the fastest growing,\nand therefore most physically relevant, linear MRI modes\nhave a homogeneous radial structure on the scale of the\nshearing sheet in which the local approximation is made\n( Guan et al. (2009) has shown the MRI is well localized\nin the non-linear regime). This means that the global disc\nstructure, including boundary conditions, not captured by\nthe local approximation may have a significant effect on\nthese large scale modes in a way that cannot be deter-\nmined locally. Other limitations exist too (Knobloch 1992;\nRegev & Umurhan 2008).\nThe extent to which these limitations matter\nshould, and under a variety of assumptions have,\nbeen investigated by global analyses that take into\naccount the full radial structure of the disc and\nits boundaries (Dubrulle & Knobloch 1993; Curry et al.\n1994; Curry & Pudritz 1995; Ogilvie & Pringle 1996;\nOgilvie 1998), and specifically for the galaxy by\nKitchatinov & R¨ udiger (2004). The conclusions of these in-\nvestigations largely confirms the local picture of a large ra -\ndial scale instability driven by the differential rotation o f the\ndisc. This suggests that whilst a local MRI analysis is gener -\nally correct, its regime of validity must be checked globall y.\nIt is the purpose of this work to do just that for the MRI\noperating in a collisional, magnetized (the ion cyclotron fre-\nquencyωi≫ion-ion collision frequency νii) plasma (like the\nISM).\nIn such a plasma Braginskii (1965) has shown that to\nlowest order in ωi/νii, the deviatoric stress tensor is diag-\nonal and anisotropic. This leads to different parallel and\nperpendicular viscosities or, more fundamentally, pressu res\n(andthermalconductivities1)with respecttothelocal direc-\ntion of the magnetic field. Of the important physical conse-\nquencesofthis, itwill betheeffect oftheBraginskii viscos ity\nin the presence of a galactic shear flow that will concern us\nhere.\nThis is not a new topic and in recent years the study\nof magnetized accretion discs has attracted attention in\nboth the collisionless (Quataert et al. 2002; Sharma et al.\n2003, 2006, 2007) and collisional regimes (Balbus 2004;\nIslam & Balbus 2005; Ferraro 2007; Devlen & Pek¨ unl¨ u\n2010),andawelldevelopedcodetosimulatethemnowexists\n(Parrish & Stone 2007; Stone et al. 2008). However, a num-\nber of fundamental questions remain unanswered. Primarily ,\nwhat is the non-linear fate of the MRI in a collisional mag-\nnetized plasma? Does it transport angular momentum and\nif so, is the transporting stress primarily viscous, Maxwel l\nor Reynolds2? On what scales do the most unstable modes\nemerge and how does this vary with the system parame-\nters? In what regime will a local analysis become untenable\n1In the presence of temperature gradients, anisotropic ther mal\nconduction can lead to the magnetothermal (Balbus 2000) and\nheat flux buoyancy instabilities (Quataert 2008).\n2Simulations in the collisionless regime by Sharma et al. (20 06)\nshows that there is angular momentum transport and the\nanisotropic pressure constitutes a significant portion of t he to-\ntal stress ( ∼Maxwell and ≫Reynolds).and global effects (either radial or vertical) become impor-\ntant (Gammie & Balbus 1994)? What effect does the pres-\nence, or absence, of a net vertical field have given Cowl-\ning’s anti-dynamo theorem and the dissipative properties\nof the Braginskii viscosity (Moffatt 1978; Lyutikov 2007)?\nCould viscous heating from the Braginskii viscosity lead\nto secondary magnetized or unmagnetized thermal instabil-\nities (Balbus 2001; Quataert 2008; Kunz et al. 2011)? Do\nchannel solutions, or something approaching them, emerge\n(Goodman & Xu 1994)? If they do, the associated field\ngrowth will generate pressure anisotropies that could feed\nnew parasitic instabilities such as the mirror. What would\ntheir effects be at this stage and in the inevitable turbu-\nlence where the mirror and firehose instabilities will both\narise (Schekochihin et al. 2005)?\nAddressing these questions will require a two-pronged\napproach involving both numerical and analytic studies. It\nmay transpire that much of the existing work on the unmag-\nnetized and collisionless MRI is directly applicable, but t his\nneeds determining. As such we conduct a global linear sta-\nbility analysis for three separate background magnetic fiel d\norientations: purely azimuthal, purely vertical and pitch ed\n(magnetic field lines follow helical paths on cylinders of co n-\nstant radius). We embed these in a galactic rotation profile.\nIn agreement with earlier, local studies, we find that when\nthe field has both a vertical and an azimuthal component,\na linear instability with a real part up to 2√\n2 times faster\nthan the MRI emerges (Balbus 2004; Islam & Balbus 2005).\nIn contrast to local studies we also find that it has a trav-\nelling wave component, and its growth rate depends on the\nviscous coefficient.\nIn the presence of a vertical field, we also recover the\nstandard inviscid MRI and show that upon introducing the\nBraginskii viscosity, its growth rate is always reduced. A\nsimilar effect is found for a purely azimuthal field where we\nfind that the Braginskii viscosity damps modes that are,\ninviscidly, neutrally stable.\nThe layout of this paper is as follows. In Section 2, we\nintroduce and perturb a series of equilibrium solutions to\nthe Braginksii-MHD equations and this forms the basis of\nour global stability analysis. Relegating the manipulatio n of\nthe perturbed equations to Appendix A, we obtain a single\nODE governing the perturbed modes. We proceed by solv-\ning this for azimuthal, vertical and pitched field orientati ons\nin Sections 3, 4 and 5 respectively. For each case we contrast\nthe behaviour with and without Braginskii viscosity. In Sec -\ntion 6 we discuss the physical mechanism of the instability,\nwhere it may occur astrophysically, and the relation of our\nresults to the local approximation. Finally, we conclude in\nSection 7 with some thoughts on open questions relating to\nmagnetized astrophysical plasmas.\n2 GLOBAL STABILITY ANALYSIS\n2.1 Governing equations\nThe simplest set of equations that capture the physics of\nthe collisional magnetized MRI are those those of isother-\nmal ideal MHD with the Braginskii viscosity (Lifshitz et al.\n1984). Explicitly these equations are the momentum equa-\ntion, including the Braginskii stress tensor; the inductio n\nc/circleco√yrt2011 RAS, MNRAS 000, 1–??Galactic MRI with Braginskii viscosity 3\nequation that describes the evolution of the magnetic field;\nthe incompressibility condition (because the perturbatio ns\nare linear, the Mach number can always be made small\nenough to ensure this); and the stress tensor itself:\n∂u\n∂t+u·∇u=−∇Π+B·∇B−∇ΦD−∇·T,(1)\n∂B\n∂t+u·∇B=B·∇u, (2)\n∇·u= 0, (3)\nT=νB(I−3bb)bb:∇u, (4)\nwhere the constant mass density has been scaled out of the\nproblem, uis the velocity field, Bis the mass-density scaled\nmagnetic field, i.e. the Alfv´ en velocity, b=B/Bis the direc-\ntion of the magnetic field, Φ Dis the gravitational potential,\nΠ =p+B2/2 is the total (gas plus magnetic) pressure, ‘:’\nis the full inner product, Ithe identity, and Tis the full\nBraginskii stress tensor whose form we now explain.\nIn a magnetized plasma the total pressure tensor P=\npI+T(whose divergence appears in the momentum equa-\ntion) is given by\nP=pI+1\n3(I−3bb)/parenleftbig\np⊥−p/bardbl/parenrightbig\n,\nwherep⊥,p/bardblare the perpendicular and parallel scalar pres-\nsures with respect to the magnetic field direction. When the\nplasma is also collisional, thepressure anisotropy p⊥−p/bardblcan\nbe related to the rate-of-strain by a Chapman-Enskog style\nperturbation theory (Braginskii 1965; Chapman & Cowling\n1970).\nMicrophysically, the anisotropy arises from the conser-\nvation of the magnetic moment (first adiabatic invariant) of\na gyrating particle in a magnetic field. However, collisions\nbreak this conservation and relax the anisotropy by pitch-\nangle scattering particles in velocity space (this dissipa tive\nprocess will turn out to be important in isolated field con-\nfigurations). The competition between these two processes\nis governed by\nd\ndt(p⊥−p/bardbl)≃3pdlnB\ndt−νii(p⊥−p/bardbl),\nwhere we have used the BGK operator to approximate the\nfull collision operator.\nIn the presence of time-varying magnetic fields, the\npressure anisotropy tends to a steady state that tracks the\nfields’ rate of change. It follows\np⊥−p/bardbl= 3νBdlnB\ndt= 3νBbb:∇u,\nwhereνB∼p/νiiis thecoefficient of theBraginskii viscosity,\nand we have used equation (2) in the final equality.Equation\n(4) follows directly.\n2.2 Equilibrium solutions\nWorking in cylindrical polar coordinates ( r,φ,z), we intro-\nduce equilibrium solutions to equations (1)-(4) that descr ibe\na differentially rotating global shear flow constrained by\ngravity and threaded by a magnetic field that lies on cylin-\nders of constant radius. Our equilibrium solutions are uni-\nform inz, and the plasma motion is restricted at an inner\nboundary of finite radius r0. (The validity of these assump-\ntions with be discussed in Section 6.2.)We allow gravity ∇ΦDto dictate the rotation profile\nof the equilibrium flow Ω = Ω 0(r/r0)−qwhere Ω 0is the\nrotation frequency at the inner boundary and qis a dimen-\nsionless measure of the shear. Of interest to us here is the\ncase ofq= 1 that is both analytically treatable and phys-\nically corresponds to a galactic disc where, unlike the Ke-\nplerian case of q= 3/2, the gravitational potential of the\ndark matter halo sets the rotation profile (Rubin & Ford\n1970; Sofue & Rubin 2001). In modelling this we set the\ndark matter mass distribution (whose sole purpose is to ul-\ntimately set the rotation profile) to a Mestel (1963) profile\nso, via Poisson’s equation, ∇ΦD∝1/r. In this case, the flow\nu= Ω(r)rˆeφ= Ω0ˆeφdoes not vary with radius.\nWe decompose the magnetic field into vertical and az-\nimuthal components, (so as to construct a time indepen-\ndent equilibrium, we neglect radial magnetic fields3). That\nis,B=Bzˆez+Bφˆeφ=B(sinθˆez+ cosθˆeφ) where θ=\narctan(Bz/Bφ) is the pitch angle of the magnetic field. We\ndo not specify θyet, however, for mathematical simplicity,\nwe demand that both BφandBzare independent of radius\nsoθremains constant. Whilst this implies a vertical current\n∝1/ris associated with Bφ, a simple super-galactic mag-\nnetic field can account for Bz. In all, our equilibrium fields\ntake the form\nu= Ω0ˆeφ,B=Bzˆez+Bφˆeφ, (5)\nandare constantin space andtime. As is physically relevant ,\nwe restrict B/Ω<1.\nIt is a mathematically convenient feature of our equi-\nlibrium solutions that there is no evolution of the magnetic\nfield strength. From equations (4) and (5), Tis absent from\nthe unperturbed state and the system will be stable to pres-\nsure anisotropy driven microscale instabilities, e.g. fire hose\nandmirror(Schekochihin et al.2005).Incontrasttothecas e\nwhere Laplacian viscosity (or indeed resistivity) is prese nt,\nwe can construct an ideal MHD solution independent of any\nradial flows (Kersal´ e et al. 2004).\n2.3 Perturbed equations\nTo determine the stability of this system we linearise equa-\ntions (1)–(4) about (5) with axisymmetric velocity pertur-\nbationsδu=δu(r)exp[ikz+γt] and similarly for the mag-\nnetic and pressure fields. Here kis the wavenumber in the\nz direction and γthe growth rate. We retain curvilinear\nterms from the cylindrical geometry but neglect self-gravi ty.\nWe non-dimensionalise with respect to time-scales Ω 0and\nlength-scales r0.\nAs detailed in Appendix A, the linearised equations\ncombine into a single complex second order ordinary dif-\nferential equation for δur, the Modified Bessel equation\nd2δur\ndr2+1\nrdδur\ndr−/parenleftbigg\np2−v2\nr2/parenrightbigg\nδur= 0, (6)\n3Although magnetic field configurations vary from galaxy to\ngalaxy, they are commonly found tracing the spiral arms and\ntherefore, in the plane of the disc, predominantly azimutha l.\nBeck et al. (1996) gives values for the mean in-plane field\nBr/Bφ∼0.25 thereby justifying, to some degree, our neglect\nof the radial field.\nc/circleco√yrt2011 RAS, MNRAS 000, 1–??4M. S. Rosin, A. J. Mestel\n1 2 3 4 5r0.20.40.60.81.0∆ur\nFigure 1. Radial mode structure of the fastest growing n = 0\nbranch MRI modes with θ=π/2,z= 0,B= 2.0·10−2and\nSB= 10(solid) ,102(dashes) ,103(dots/dashes).The(purelyreal)\ngrowth rates are γm= 0.39,0.31,0.19 and associated wavenumber\nkm= 36.0,30.0,20.6.\nwhere\np2≡k2\nE1(σ2+γ2)(σ2+γ2+SBγσ2cos2θ), (7)\nv2≡ −1\nE1(γ4+a3γ3+a2γ2+a1γ+a0), (8)\nand\nE1= (σ2+γ2)(σ2+γ2+SBγσ2),\na3=SBσ2[cos2θ(cot2θ−1)+sin2θ],\na2= 2[σ2(1+cot2θ)+k2]+2iSBσ2sin2θk(1−cot2θ),\na1=SBσ4/bracketleftbig\ncos2θ/parenleftbig\n3−2k2σ−2/parenrightbig\n+1−cot2θ/bracketrightbig\n−8iσ2kcotθ,\na0=σ2[σ2(1−2cot2θ)−2k2].\nHereσ≡kBsinθis the vertical Alfv´ en frequency and SB=\n3νB/B2( = 3νBΩ0/(B2nmi) in dimensional form) ≃β/νii\nis a measure of the relative sizes of the anisotropic pressur e\nforce and Lorenz force4. Herenis the number density, mi\nthe ion mass and βthe plasma-beta.\nTo obtain solutions to equation (6), we choose our\nboundary conditions to be an impenetrable wall at r0so\nδur(r0) = 0 and the requirement r1\n2δurdecays at infinity\n(see Furukawa et al. (2007) for an inviscid treatment that\nincludes the shear singularity at the origin).\nIn this case, the eigenfunctions of equation (6) are mod-\nified Bessel functions of the second kind K iv(pr) with argu-\nmentprand order v(Watson 1944; Abramowitz & Stegun\n1964). The spectrum of solutions is discrete and we index\nvwith n,vn. In the special case when p(orγfrom equa-\ntion (7)) is real (in general it is complex) the problem is\nSturm-Liouville and vnis an infinite ordered set of eigenval-\nuesv0< v1< v2... < v ∞. To determine vnand therefore\n4Using a Landau fluid closure and including the effect of colli-\nsions Quataert et al. (2002); Sharma et al. (2003) were the fir st\nto show the dependance of the MRI on Ω ,νiiandβ. Because the\nclosure for the pressure anisotropy differs between the coll ision-\nless and Braginskii regimes, the transition between the pre ssure\nanisotropy driven MRI, and the MHD MRI scales, in dimensiona l\nform, as ( νii/Ω)β1/2there, and ( νii/Ω)βhere.γ, it is necessary to solve\nKivn(p) = 0. (9)\nFrom solutions to this and equations (7) and (8), the full\nset of flow, magnetic and pressure fields can be constructed,\nAppendix A. In Fig. 1 we show the radial structure of δur\nwhenθ=π/2.\nIn general, determining γmust be done numerically.\nHowever, for the most physically relevant magnetic field\nconfigurations, the problem becomes, in part, analytically\ntractable. These cases, a purely azimuthal field θ= 0 (in\nthe galactic plane), a purely vertical field θ=π/2 (a super-\ngalactic field)andaslightly pitchedfield θ≪1(veryslightly\nout of the galactic plane), exhibit categorically different be-\nhaviour such that θ→0,π/2 are singular limits.\nIn the first case, linear perturbations are damped; in\nthe second, the system is unstable to the MRI but the\nBraginskii viscosity reduces the maximum growth rate γm\nbelow the Oort-A value maximum |dlnΩ/dlnr/2|= 1/2,\n(Balbus & Hawley 1992); in the third, the system is unsta-\nble with γm→/radicalbig\n|dlnΩ2/dlnr|=√\n2, even for asymptoti-\ncally small θ(Balbus 2004). We present the details of these\ncalculations now.\n3 AZIMUTHAL FIELD, θ= 0\nThe stability of inviscid axisymmetric perturbations to a\npurely azimuthal magnetic field in the presence of a shear\nflow are well known. When q <2 and the magnetic field\nB=Br−dˆeφhasd >−1 the system is always stable\n(as ours is). When only one criterion is met, depending on\nthe form of the fields, the system may still remain stable\nby the modified Rayleigh criterion (Rayleigh 1916; Michael\n1954; Chandrasekhar 1961). (It is worth noting however\nthat global non-axisymmetric perturbations are unstable\n(Ogilvie & Pringle 1996).)\nDotheseresultsfor ourinviscidlystablesystem, R(γ) =\n0, persist in the presence of the Braginskii viscosity? The\nanswer is no.\nSettingθ= 0 in equations (6) to (8) and making a\nchange of variables, δur→r−1/2δur, we obtain the simple\nexpression\nγ2d2δur\ndr2−/parenleftbig\nγ2Q1+γQ2+Q3/parenrightbig\nδur= 0, (10)\nwhere\nQ1=3\n41\nr2+k2, Q2=SBB2k2\nr2, Q3=2k2\nr2(1+B2).\nWe multiply equation (10) by the complex conjugate of δur,\nδur†and integrate between r0and infinity. Boundary terms\nvanish, so\nγ2/bracketleftBigg/parenleftbiggdδur\ndr/parenrightbigg2\n+Q1|δur|2/bracketrightBigg\n+γQ2|δur|2+Q3|δur|2= 0,(11)\nwhere|δur|2=/integraltext∞\n1drδurδur†/greaterorequalslant0 is non-negative, as is\n(dδur/dr)2.\nEquation (11) is a quadratic in γwhose roots de-\npend crucially on SB. When SB= 0 (the inviscid limit),\nγis purely imaginary (neutrally stable, travelling waves),\nwhereas when SB>0 the result is quite different. In this\nc/circleco√yrt2011 RAS, MNRAS 000, 1–??Galactic MRI with Braginskii viscosity 5\nFigure 2. Left panel: Growth rate of the n=0 (fastest) branch of the inviscid globa l MRI as a function of the vertical Alfv´ en frequency\nkBsinθfor several values of B(time and length are non-dimensionalised with respect to r0and Ω 0). In the weak field limit, we recover\nthe behaviour of the local MRI (dotted line). For all figures, the top-to-bottom order of the varied parameter correspond s to the top-to-\nbottom order of the main-panel curves. Right panel: Maximum growth rate γmas a function of Bsinθ. AsB→0, γmasymptotes to\nthe local Oort-A maximum = 1 /2 (dotted line). Inset: The wavenumber kmat which γmoccurs as a function of Balong with the local\nmaximum/radicalbig\n3/4 (dotted line). Note: In the inviscid weak-field limit, two c onfigurations with the same vertical magnetic field will exhi bit\nidentical behaviour – Section 5.3\ncaseR(γ)<0∀SBand the only question is whether per-\nturbations are purely damped, or damped and travelling.\nIt follows that if the system is stable in the absence of\nthe Braginskii viscosity, it remains so in its presence.\n4 VERTICAL FIELD, θ=π/2\n4.1 Inviscid MRI, SB= 0\nFor simplicity, we start by considering the inviscid limit o f\na purely vertical field B=Bzˆez, i.e.θ=π/2,SB= 0. In\nthis case equations (6) to (8) simplify5:\nKivn(k) = 0, (12)\nwith\nv2\nn≡ −/bracketleftBigg\n2k2\nγ2+σ2/parenleftbig\nγ2−σ2/parenrightbig\n(γ2+σ2)+1/bracketrightBigg\n. (13)\nIfv2\nn<0, Kivn(k) has no nontrivial zeros and it follows\nthat for an instability v2\nn>0 (hence our sign convention in\nequation (6)). This implies γ2is bounded from above by σ2.\nNumerical solutions to equations (12) and (13) are ob-\ntained using Newton’s method (implemented in Mathemat-\nica). There is an unstable solution (and three stable ones)\nwhich is shown in Fig. 2. We find γis real, positive, and of\norder the shear rate. The instability is the global MRI, and\nthe n = 0 ,v0branch has the largest growth rate.\nThe fastest growing modes have k∼1/B, so in the\nstrong field regime B/lessorsimilar1,k/greaterorsimilar1 the mode is large scale\n(physically, smaller scale modes are suppressed by magneti c\ntension) and is peaked away from the inner boundary. The\n5Equations (12) and (13) correspond to equations (3.4) and (3 .5)\nof Curry et al. (1994) with a= 1 and equations (30) and (31) of\nDubrulle & Knobloch (1993) who have already solved this prob -\nlem.result is that γmis reduced below the Oort-Amaximum that\noccurs when B∼1/k≪1 andthe mode is localised at r= 1\n(Curry et al. 1994).\nIn this weak field regime the problem is amenable to\nasymptotic analysis. It has been shown by Cochran (1965)\nand Ferreira & Sesma (1970, 2008) that in this limit, the\nzeros ofKivn(k) are given by\nvn∼k+sn2−1k1\n3+..., (14)\nwheresn=an22\n3,anis the modulus of the nthreal negative\nzero of the Airy function Ai(Abramowitz & Stegun 1964)\nand omitted terms are of the form kbwithb <1/3. This\nresult can be used to find the asymptotic-in- kform ofγby\ninverting equation (13) to form a bi-quadratic\nγ4+2/parenleftbigg\nσ2+k2\n1+v2n/parenrightbigg\nγ2+σ2/parenleftbigg\nσ2−2k2\n1+v2n/parenrightbigg\n= 0,(15)\ninto which we substitute equation (14). Retaining the first\ntwo terms in vn, we solve exactly for γ2\nγ2=−σ2+/parenleftBig\n1−snk−2\n3/parenrightBig/parenleftBigg\n−1±/radicalBigg\n1+4σ2\n1−snk−2\n3/parenrightBigg\n,(16)\nin which the positive root corresponds to the instability.\nNumerically we find the fastest growing mode (for a given\nB,SB),γmand the wavenumber at which it occurs km, obey\n∂γm/∂B,∂(kmB)/∂B <0.\nTo draw an analogy with the local approximation (see\nSection 6.3), n that indexes the number of zeros in the\ndomain is like a radial wavenumber. For large enough n,\nthe solutions are plane waves. To see this, we apply a\nWKB analysis to equation (6) using the small parameter\nkr/vn≡x/vn≪1 (Dubrulle & Knobloch 1993; Ogilvie\n1998). We have\nd2δur(x)\ndx2+v2\nn\nx2δur(x) = 0. (17)\nDemanding that δuris real, this has solutions\nδur=√xcos(vnlnx), (18)\nc/circleco√yrt2011 RAS, MNRAS 000, 1–??6M. S. Rosin, A. J. Mestel\nand to ensure cos( vnlnx) = 0 at r= 1, it must satisfy the\nboundary condition\nvnlnk=/parenleftbigg\nn+1\n2/parenrightbigg\nπ, (19)\nwhich determines the spectrum of solutions.\nCombining equations (13) and (19) we have\nγ2=−σ2+α/parenleftBig\n−1+/radicalbig\n1+4σ2α−1/parenrightBig\n, (20)\nwhereα=k2(lnk)2/n2π2and we have neglected factors of\n1/2≪n.\nIn the large n limit, on small enough scales, the mode\nstructure given byequation (18) is especially simple. Reve rt-\ning toras our radial coordinate, we expand equation (18)\naboutr=r1∼ O(1). To leading order we find\nδur=A/parenleftbigg\n1+1\n2r\nr1/parenrightbigg\ncos/parenleftbiggvn\nr1r+ξ/parenrightbigg\n, (21)\nwhereA=√kr1andξ=vnlnkr1.\nThese solutions describes rapidly oscillating modes with\nfrequency vnand a slowly varying amplitude ∝ A. For suffi-\nciently large vn, the solutions are plane waves whose growth\nrate decreases with n.\n4.2 Viscous MRI, SB∝negationslash= 0\nAllowing for viscosity whilst retaining a vertical field, eq ua-\ntions (7) and (8) again reduce to a simple form:\np2≡k2γ2+σ2\nγ2+σ2+SBγσ2, (22)\nv2\nn≡ −/bracketleftBigg\n2k2\nγ2+σ2+SBγσ2/parenleftbig\nγ2−σ2/parenrightbig\n(γ2+σ2)+1/bracketrightBigg\n. (23)\nUnlike the inviscid case, we can no longer guarantee the re-\nality ofpasγcan, and indeed sometimes does, take complex\nvalues. The problem is generally not of Sturm-Liouville for m\nand so the roots of K ivn(p) are complex.\nNumerically we again find four branches of which three\nare stable, and one unstable. The unstable mode (the only\none of interest) is real, so if ℜ(γ)>0 thenγ∈ ℜ. In this\ncase, the problem is Sturm-Liouville. The unstable branch\ncan be traced from the inviscid MRI (and is maximized for\nn= 0,v0), Figs. 1, 3 and 5.\nAsymptotic expressions for γandkcan be found using\nthe results of Cochran (1965) and Ferreira & Sesma (2008)\nfor the (complex) roots of equation (9):\nvn∼π/(ln2−γEuler−lnp), p ∈C,|p| ≪1,(24)\nvn∼p+an2−1\n3p1\n3+..., p ∈C,|p| ≫1,(25)\nwhereγEuler≃0.58 is the Euler constant.\nForSB≫1 we combine equations (22) and (23) to get\nγ4\nk4+γ2\nk2/parenleftbigg\nRe−1γ+2\n1+v2n+2B2/parenrightbigg\n+B2/parenleftbigg\nRe−1γ−2\n1+v2n+B2/parenrightbigg\n= 0 (26)\nwhere Re−1=SBB2is the inverse of the Reynolds number.We expand equation (26) in γ2/k2∼ǫ≪1 and solve γ=\nγ0+ǫγ1+...order by order. To lowest order we find\nγ0= Re/parenleftbigg2\n1+v2n−B2/parenrightbigg\n,\nand toO(ǫ)\nγ1=−1\nB2/bracketleftbigg\nγ0+2Re/parenleftbigg1\n1+v2n+B2/parenrightbigg/bracketrightbigg\n,\nand so\nγ≃Re/parenleftbigg2\n1+v2n−B2/parenrightbigg\n−γ2\n0\nk2B2/bracketleftbigg\nγ0+2Re/parenleftbigg1\n1+v2n+B2/parenrightbigg/bracketrightbigg\n.(27)\nNumerically we find km≪1 which, combined with\nequations (22) and (24), implies vn≃ −π/lnk. It follows\nthat to lowest order\nγm= Re(2−B), S B≫1,(28)\nand, differentiating equation (27) with respect to k and ne-\nglecting leading order logarithmic variations,\nkm≃2−B2\nBπ/radicalbigg\n2+B2\n2Re/parenleftbig\nlnRe−1/parenrightbig3/2, S B≫1.(29)\nForSB,B≪1, equation (25) applies and so γis gov-\nerned by\n/parenleftbig\nγ2+σ2/parenrightbig2+/bracketleftbig\n2−k−2/parenleftbig\nSBγσ2−2/parenrightbig/bracketrightbig/parenleftbig\nγ2−σ2/parenrightbig\n= 0,\nwhere we have used vn≃k≫1. We expand in 1 /k2≪1\nsoγ=γ0+k−2γ1+..., and again solve order by order. To\nlowest order we find γ0satisfies the inviscid equation\nγ0=/radicalBig\n−(σ2+1)+/radicalbig\n1+4σ2, S B≪1,(30)\nand atO(k−2)\nγ1=−γ2\n0/parenleftbig\n1+SBγ0σ2/parenrightbig\n+σ2/parenleftbig\n1−SBγ0σ2/parenrightbig\n4(γ0+σ)3+γ0, SB≪1.(31)\nTo see the effect of variations in nonγwe perform\na WKB analysis in which we order vn≫k≫1≫SBso\nlnp/vnis a small parameter – see Section 4.1. The boundary\ncondition for the viscous modes is then\nvnln/bracketleftbigg\nk/parenleftbigg\n1−1\n2SBγσ2\nγ2+σ2/parenrightbigg/bracketrightbigg\n=/parenleftbigg\nn+1\n2/parenrightbigg\nπ.\nCombining this with equation (23), we expand in SB≪1\nsoγ=γ0+SBγ1+.... To lowest order γ0is given by the\ninviscid equation (20), and to O(SB)\nγ1=−γ0σ2/radicalbig\nγ2+σ2\n4(γ0+σ)3+2αγ0<0, v n≫1,SB≪1.(32)\nIn the various limits, our asymptotics confirm the nu-\nmerical results that γ∈ ℜ, and∂γ/∂S B<0. Contrasting\ntheresults of this section with theinviscid case when B≪1,\nwe have\nSB→0, γm≃1/2, k m≃B−1/radicalbig\n3/4,\nSB→ ∞, γm≃S−1\nBB−2, km≃S−1\nBB−3(lnSBB2)3/2.\nc/circleco√yrt2011 RAS, MNRAS 000, 1–??Galactic MRI with Braginskii viscosity 7\nFigure 3. Growth rate of the n=0 (fastest) branch of the global MRI with Braginskii viscosity for a two different weak, B= 10−3,\nfield configurations and a range of SB.Left panel: Vertical Field: For fixed k,γdecreases as SBincreases and is bounded above by the\ninviscid growth rate (dotted line), but the critical wavenu mber above which γ <0 is independent of SB.Right panel: Slightly pitched\nfield: In contrast to θ=π/2, for a given k, the viscous case γ(SB/negationslash= 0)> γ(SB= 0) inviscid case (dotted), but is less than the local\nLorentz-force free, or MVI, limit (dashed). As SB→ ∞,km→0 but, unlike the vertical case, γm→√\n2.\n5 PITCHED FIELD, θ≪1\n5.1 Ordering assumptions\nWhen the magnetic field has both a vertical and an az-\nimuthal component, the perturbed Braginskii stress tensor\nexerts an azimuthal ‘tension’ force on separating plasma el e-\nments. For arbitrary θ, thesystem is neither Sturm-Liouville\n(equation (6) is complex) nor is it amenable to the kind of\npolynomial inversion used in Section 4.\nHowever, assuming θ≪1 matters simplify consider-\nably. Physically, this choice of pitch angle represents the\nmost realistic non-isolated galactic magnetic field configu -\nration (Beck et al. 1996). Mathematically, as we now show,\nit is a singular limit that constitutes the stability thresh old\nbetween actively damped modes, Section 3, and an unsta-\nble configurations that grows even faster than the inviscid\nglobal MRI (to which the following stability threshold also\napplies)\nDamped : θ= 0, Unstable : θ∼ǫ.\nHereǫ≪1is asmall parameter with respect to which we or-\nder the pitch angle and the remaining quantities in equation\n(6):γ,k,d/dr,BandSB.\nLettingθ∼ǫwe can retain only the first few terms in\na series expansion of our trigonometric functions\ncosθ= 1−θ2\n2+O(θ4),sinθ=θ−θ3\n6+O(θ5).\nTo include their effects, we assume a strong magnetic\nfieldB∼ O(1) and Braginskii viscosity SB∼ O(1). To\nretain vertical magnetic tensions σ=kBsinθto order 1\n(failing to do so removes the high wavenumber cutoff and\nleaves the equations ill-posed), we set k∼1/ǫ. Then, in\nanticipation of unstable modes that grow at the shear rate,\nwe order γ∼1 too. Balancing terms in equation (6), we find\nd/dr∼p∼k∼1/ǫ.\nIn summary, our orderings are\nθ∼ǫ, γ ∼σ∼B∼SB∼1, k∼d\ndr∼ǫ−1.(33)We apply these scalings to equations (6) to (8) and re-\ntain the lowest order ǫterms. We find δuris still governed\nby Bessel’s equation and, since p≡k,γis determined by\nequation (12) and\nv2≡ −1\nE1(γ4+b3γ3+b2γ2+b1γ+b0), (34)\nwhere\nE1= (σ2+γ2)(σ2+γ2+SBγσ2),\nb3=SBσ2θ−2,\nb2= 2k2/parenleftbig\n1+B2/parenrightbig\n−4iSBσ2kθ−1,\nb1=−SBσ2k2/parenleftbig\n2+B2/parenrightbig\n−8iσk2B,\nb0=−2σ2k2/parenleftbig\n1+B2/parenrightbig\n.\nWe start by solving the inviscid problem.\n5.2 Pitched inviscid MRI. SB= 0\nIn the inviscid weak-field regime, we recover the vertical\nweak-fieldinstability ofSection 4.1andFig. 2. Bappears via\nthe vertical Alfv´ en frequency only so km(θ=π/2)/km(θ=\ntilted) = 1 /θ≫1 so the unstable mode will be confined to\na boundary layer of width ∼1/(Bθ)∼ O(ǫ2).\nIfBis small but finite (implicitly all orderings are sub-\nsidiary to equation (33)) the governing dispersion relatio n is\nthe complex quartic6\n(γ2+σ2)2+2(1+B2)(γ2−σ2)−8iBσγ= 0.\nWriting γas a series in B≪1 we find\nγ≃2iBσ√\n4σ2+1+/radicalbigg/parenleftBig/radicalbig\n4σ2+1−σ2−1/parenrightBig\n, (35)\n6Solutions to this polynomial are considered in detail in\nCurry & Pudritz (1995) and further discussion can be found in\nKnobloch (1992).\nc/circleco√yrt2011 RAS, MNRAS 000, 1–??8M. S. Rosin, A. J. Mestel\nFigure 4. ℜ(γm),ℑ(γm) (if it exists), and the corresponding vertical Alfv´ en fre quencykmBsinθfor two field configurations. Left panel:\nVertical field: γm∈ ℜandkmBsinθvs. Re−1=SBB2for a range of B. These are bounded above by the B≪1 local limits of 1 /2 and/radicalbig\n3/4 (dotted lines), and well matched by the asymptotic results (dashed lines) given by equations (28) and (29). Right Panel : Slightly\npitched, θ= 10−3:ℜ(γm) andkmBsinθare bounded from above by the local B≪1 limit of√\n2 (main panel, top dotted line) and the\ninviscid limit/radicalbig\n3/4 (top inset, dotted line). The imaginary part of the growth r ateℑ(γm) (bottom inset) is well described in the small\nand large SBlimits by equations (36) and (48). Note that ∂γm(B∼1)/∂SB<0 (bottom curve, main panel) demonstrates the combined\neffects of hoop tension and the Braginskii viscosity.\nwith a maximum value\nγm=1\n2+i/radicalbigg\n3\n4B, (36)\nwhich occurs at km=/radicalbig\n3/4B−1, Fig. 4.\n5.3 Pitched MRI with Braginskii viscosity, SB∝negationslash= 0\nIn the presence of the Braginskii viscosity we find a com-\nplex instability whose real part exceeds that of its invis-\ncid counterpart. Numerical solutions described by equa-\ntions (12) and (34) are shown in Figs. 3, 4 and 5. When\nB→0,SB→ ∞the growth rate tends to the shear rate\nℜ(γ)→/radicalbig\n2dlnΩ/dlnr=√\n2 and, as before, the fastest\ngrowing modes occur for n = 0.\nAsymptotic expressions for γandkcan be found by\nsubstituting equation (14) into equation (34). For B≪1,γ\nis governed by\n(γ2+σ2)2+2(γ2−σ2)+SBσ2γ(γ2+σ2−2) = 0.\nDifferentiating this with respect to σ(k,Bandθappear\ntogether only in one combination and so σis treated as a\nsingle independent variable), and setting d γ/dσ= 0, the\nstationary points of γobey:\nSBγ3+2γ2+2SB(σ2−1)γ+2(σ2−1) = 0. (37)\nBy considering the branch σmthat maximizes the stationary\nvalue of γ, and introducing ζm=σ2\nm, we find\nζm=SBγm(2−γ2\nm)+2(1−γm)\n2(γmSB+1), (38)\nand substituting this into equation (37) we obtain a poly-\nnomial whose solutions describe γm:\nγ6\nm−4γ4\nm+4(1−4S−2\nB)γ2\nm+4S−1\nB(2γm+S−1\nB−5γ3\nm) = 0.(39)\nBecause it is a sixth order polynomial, there is no gen-\neral formulae for its roots. However, the presence of theasymptotic parameter SBrecommends aseries solution. The\npower of the expansion parameter SBdepends on whether\nit is large or small.\nTakingSB≪1 first, we write\nγm=γm,0+SBγm,1+O(S2\nB), (40)\nζm=ζm,0+SBζm,1+O(S2\nB). (41)\nSubstituting equation (40) into equation (39), and solv-\ning order by order, we find, to lowest order γm,0= 1/2. To\nnext order γm,1= 3/32 and so\nγm=1\n2/parenleftbigg\n1+SB3\n16/parenrightbigg\n, S B≪1.(42)\nSubstituting this into equations (38) and (41), we find σm=/radicalbig\n3/4(1−SB/48), so the most unstable mode is\nkm=1\nBθ/radicalbigg\n3\n4/parenleftbigg\n1−SB1\n48/parenrightbigg\nSB≪1.(43)\nNow taking SB≫1, we write\nγm=γm,0+S−1/2\nBγm,1+S−1\nBγm,2+O(S−3/2\nB), (44)\nζm=ζm,0+S−1/2\nBζm,1+S−1\nBζm,2+O(S−3/2\nB). (45)\nFollowing the same procedure, to lowest order γm,0=\n±√\n2,0 and we take the positive root corresponding to the\ninstability. At first order we obtain no information, but at\nsecond order we find that γm,1=±23/4/31/2. Taking the\nnegative root (see Fig. 5) we have\nγm=√\n2/parenleftBigg\n1−/parenleftbigg2\n9/parenrightbigg1/4\nS−1/2\nB/parenrightBigg\n, S B≫1.(46)\nSubstituting this and equation (45) into equation (38), to\nlowest order ζm,0= 0, and at next order ζm,1= 25/2/31/2so\nσm= (25/9)1/8S−1/4\nB, and\nkm=1\nBθ/parenleftbigg25\n9/parenrightbigg1/8\nS−1/4\nB, S B≫1.(47)\nc/circleco√yrt2011 RAS, MNRAS 000, 1–??Galactic MRI with Braginskii viscosity 9\nFigure 5. Direct comparison of the γm(main panel) and\nkmBsinθ(insets) behaviour of the unstable mode for θ= 10−3\n(top solidlineand inset)and θ=π/2 (bottom solidlineand inset)\nwithSBfor a weak magnetic field, B= 10−3. The dotted lines\nare the local maxima, and the dashed lines are the asymptotic\nscalings given by equations (28) and (46) for γm, and equations\n(29) and (47) for kmBsinθ.\nNow considering the effect of finite magnetic fields,\nℜ(γm) (andℑ(γm) which is now present) increase with SB.\nGuided by the numerical results, we assume |γ| ∼1and take\nthe leading order balance of terms ∝SB→ ∞in equation\n(34). We find\nγ2/parenleftbig\nσ2+B2/parenrightbig\n−4iσBγ+σ2(σ2−B2−2) = 0,\nand the resultant growth rate is\nγ=2iBσ+σ/radicalbig\n(B2−σ2)−2(B−σ)\nB2+σ2,B/lessorsimilar1,SB≫1,(48)\nwhich agrees well with the numerics, Fig. 4.\nTo see the effects of n on γwe adopt the same WKB\napproach and orderings as Section 4.2 where vnis now given\nby the appropriate limit of equation (34). We assume B≪1\nand expand in SB≪1 soγ=γ0+SBγ1+.... To lowest\norderγ0is given by equation (20), and to O(SB)\nγ1=γ0σ2(2α−γ2\n0−σ2)\n4(γ0+σ)3+2αγ0>0, (49)\nso the Braginskii viscosity increases the growth rate beyon d\nthe inviscid limit.\nContrasting the results of this section with the inviscid\ncase, for B≪1 we have\nSB→0,ℜ(γm)≃1/2, k m≃(θB)−1/radicalbig\n3/4,\nSB→ ∞,ℜ(γm)≃√\n2−S−1/2\nB, km≃(BθS1/4\nB)−1,\nwith the caveat that km→0 only in the formal limit B→0\ntoo (see Fig. 4 for the effect of finite Bat large SB).\nConsidering the effects of hoop tension, we find ℑ(γ)\nis an order of magnitude greater in the viscous limit. For\nexample, we find numerically that ℑ(γ(SB= 108)/γ(SB=\n0))≃40forB= 10−3(whichagrees withequations(36)and\n(48) to within half a percent for this and all other examples\nwithB/lessorsimilar0.5,SB≫1.)6 DISCUSSION\n6.1 Physical mechanism\nWe have seen that the orientation of the magnetic field cat-\negorically determines the behaviour of the system. How is\nthis to be understood physically?\nThe most informative explanation comes in the weak\nfield regime where the role of the magnetic field is twofold.\nFirstly it facilitates the generation of pressure anisotro pies\nproportional to its rate of change, or equivalently (in the c ol-\nlisional regime) ∝δbb:∇u. Because collisions are involved\nthis is a necessarily dissipative process. Equations (1) to (4)\nyield an energy conservation law\nd\ndt/parenleftBigg/angbracketleftbig\nu2/angbracketrightbig\n2+/angbracketleftbig\nB2/angbracketrightbig\n2/parenrightBigg\n=−3νB/angbracketleftbig\n|bb:∇u|2/angbracketrightbig\n,\n=−3νB/angbracketleftBigg/parenleftbiggdlnB\ndt/parenrightbigg2/angbracketrightBigg\n,\nwhere<·>are volume averages.\nThat is, the Braginskii viscosity damps any motions\nthat change the magnetic field strength. This is the first\nrole of the magnetic field.\nThe second role of the magnetic field is a geometric\none. Assuming anisotropies do arise (from changes in the\nmagnetic field strength), the field’s orientation dictates t he\nprojection of the anisotropic stress onto the fluid elements of\nthe plasma, thereby affecting their dynamics. As we now ex-\nplain, this fact is crucial in determining the stability, or lack\nthereof, and the role global effects have on a differentially\nrotating magnetized plasma.\nWhen the field lines have both vertical and azimuthal\ncomponents (it is pitched) fluid elements at different height s\ncan exert an azimuthal stress on each other. The sign of\nthis stress can be either positive or negative. If the mag-\nnetic field is unstable, so its rate of change is positive, the\nanisotropic stress acts tooppose anyazimuthal separation of\nthe two elements (this can be identified as the fluid version\nof the stress responsible for the microscopic mirror force) .\nIn this case, like the MRI, velocity perturbations to fluid el -\nements at different heights increase (decrease) their angul ar\nmomentum causing them to move to larger (smaller) radii.\nIn a system with an outwardly decreasing angular velocity\nprofile, this leads to an azimuthal separation of the fluid el-\nements. The associated magnetic field growth ensures the\nstress is of the right sign to oppose this separation and this\ntransfers angular momentum between them in a way that\nfacilitates further (radial) separation; i.e. an instabil ity (see\nQuataert et al. (2002) for a physical explanation including\na spring analogy). This is the second role of the magnetic\nfield.\nIn conjunction, the two roles explain our results. In iso-\nlated field geometries where there is no projection of the\nstress onto fluid elements at different heights, the Bragin-\nskii viscosity does not give rise to an instability; its only ef-\nfect is dissipative (Kulsrud 2005; Lyutikov 2007; Kunz et al .\n2011). This accounts for the damping of perturbations in\nthe azimuthal configuration. It also accounts for the re-\nduced growth rate of the vertical MRI (that depends on\nthe Maxwell, not the Braginskii, stress), along with the in-\ncreased radial extent of the mode away from the region\nof maximum shear (Curry et al. 1994). When the field is\nc/circleco√yrt2011 RAS, MNRAS 000, 1–??10M. S. Rosin, A. J. Mestel\npitched its dissipative effects persist, but the free energy\ncontribution from the differential shear flow will alwaysbe\ngreater, leading to an instability (the dissipative effects can-\nnot change the stability boundary, just the growth rate).\nOf course the total, equilibrium plus perturbed, field energ y\nwill decrease if viscosity is present (and viscous heating i s\nnot) but, from the perspective of generating turbulence or\ntransporting angular momentum, this is a secondary conse-\nquence.\nNow considering finite magnetic field strengths (and\ntherefore hoop tension and viscous curvature stresses) we\nfind these modes become over-stable. The variation of the\ntravelling wave component of the unstable mode is simply a\ncombination of these two effects, to varying degrees.\n6.2 Astrophysical example\nThe mathematical results in this paper should be appli-\ncable to any collisional magnetized disc with a decreas-\ning angular velocity profile. However, it is helpful to pro-\nvide a physical example where our analysis holds. Be-\ncause our theory is developed for a disc with an angu-\nlar velocity profile ∝r−1we choose the ISM of a spi-\nral galaxy where a range of analytic and numerical stud-\nies already exist (Kim et al. 2003; Kitchatinov & R¨ udiger\n2004; Dziourkevitch et al. 2004; Piontek & Ostriker 2005;\nWang & Abel 2009).\nFurther motivation to study this system comes from\nSellwood & Balbus (1999) who have argued that in the HI\nregion outside the optical disc of a spiral galaxy, the veloc ity\ndispersion measurements of ∼5−7 km s−1may be driven\nbythedifferential rotation of the disc, mediated bythe MRI.\nOn this basis, we consider the warm ISM in the qui-\nescent regions of a typical spiral galaxy where the plasma\nis generally subject to a weak magnetic field and an out-\nwardly decreasing angular velocity profile. It is also mag-\nnetized, and so subject to the effects of the Braginskii vis-\ncosity. This last feature can be seen, and the other rele-\nvant parameters estimated, by adopting the following set of\nfiducial parameters for the ISM (Binney & Tremaine 1988;\nBeck et al. 1996;Ferri` ere2001).These areinagreementwit h\nNGC1058, thewell studiedface-on disc galaxy considered by\nSellwood & Balbus (1999).\nReverting to dimensionalised units for clarity, we have:\n•Particle number density (ion and electron are the same)\nn∼0.3 cm−3.\n•Temperature (we assume ions and electrons are in ther-\nmal equilibrium)\nTi∼5×104K;\nconsequently the ion thermal speed is\nvthi=/parenleftbigg2Ti\nmi/parenrightbigg1/2\n∼3×106cm s−1;\n(Tiis in erg.)\n•The ion-ion collision frequency (in seconds, assuming n\nin cm−3,Tiin K and the Coulomb logarithm lnΛ = 25) is7\nνii∼1.5nT−3/2\ni∼4×10−8s−1;\n7The full expression for the ion-ion collision frequency (io n-consequently the mean free path is\nλmfp=vthi\nνii∼7×1013cm.\n•The typical rotation rate of a spiral galaxy is\nΩ∼5×10−16s−1.\nA typical value for the outer edge of the optical disc where\nthe turbulence cannot be generated by stellar processes is\nr∼3×1022cm,\nand even within the optical disc, at the corotation radius in\nbetween the spiral arms, magnetized shear instabilities ma y\nbe important (A. Shukurov – private communication).\nOutside the optical disc a reasonable value for the vertical\nscale-height is\nH∼1021cm,\nand so the disc is thin. The measured system-scale rotation\n(not turbulent) velocity is\nU∼2×107cm s−1.\n•The observed mean magnetic field strengths vary be-\ntween galaxies but on the lower end of the scale are, at the\npresent time (Beck et al. 1996)\nB∼8×105cms−1−present.\nHowever, if the MRI is the dominant turbulence gener-\nating mechanism in the ISM, this value must represent\nthe saturated state of the magnetic field. Assuming, as\none must, that present field strengths have been ampli-\nfied over time, at some earlier time they were weaker,\ne.g. Malyshkin & Kulsrud (2002); Kitchatinov & R¨ udiger\n(2004).\nTo ensure the most unstable modes exist at scales\n> λmfpso our theory is fluid-like we adopt the follow-\ning value for the historical ‘initial’ field strength and lay\naside the problem of where this came from (Kulsrud 1999;\nBrandenburg & Subramanian 2005),\nB∼80 cms−1−initial.\nIf we considering a plasma in this era, or the ISM in a galaxy\nwhere the magnetic field is not saturated, the plasma beta\nis\nβ=v2\nthi\nB2∼1.3×109;\nthe ion cyclotron frequency is\nωi=/radicalbigg\n4πn\nmieB\nc∼2×10−6s−1;\n(eis the elementary charge, cthe speed of light) and the ion\nLarmor radius is\nρi=vthi\nωi∼1.5×1012cm;\nand so the plasma is magnetized ωi≫νii(and will become\neven more so if the magnetic field becomes stronger).\nelectron collisions are sub-dominant for a thermalised pla sma) is\ngiven by νii= 4√πne4lnΛ/3m1/2\niT3/2whereeis the elementary\ncharge and Tis in erg (Braginskii 1965).\nc/circleco√yrt2011 RAS, MNRAS 000, 1–??Galactic MRI with Braginskii viscosity 11\n•The magnetic Prandtl number Pmis huge (and so we\ncan take the Braginskii viscosity to be the only significant\ndissipative process)8:\nPm=νB\nη≃7.5×10−6T4\nn∼7.5×1013,\nwhereηis the coefficient of resistivity and Tis in K.\n•The dimensionless parameter SB= 3νBΩ/(nmiB2)≃\nβΩ/νiiis\nSB∼1.1×109ΩT5/2\nB2∼42,\nfor the conditions above, so the Braginskii viscosity plays a\nrole.\nOur model of a gravitationally constrained, differen-\ntially rotating disc made up of an isothermal, incompress-\nible, magnetized plasma fluid is consistent. Neglecting str uc-\nture inzrequires us to restrict our analysis to the galactic\nplane by considering vertical scales much less than the scal e\nheight of the disc H∼cs/Ω∼wherecs∼vthiis the isother-\nmal sound speed.\nFormally, both this and our remaining model assump-\ntionscanbeexpressedasahierarchyoftime-scales whichar e\nwell satisfied for our set of parameters (we restrict attenti on\ntoγm),\n1\nωi≪1\nνii≪1\nkcs≪1\nΩ≪1\nΩr0\nH.\nIn order of increasing periods of time, these scalings corre -\nspond to: the plasma being magnetized; the plasma being\ncollisional, i.e. a fluid; the model being uniform in the vert i-\ncal direction; and the disc being thin, i.e. rotationally ra ther\nthan pressure dominated.\nThe parameters in this section describe a physically re-\nalistic regime in which our analysis is valid, and potential ly\nimportant in explaining the gas motions in parts of the ISM.\n6.3 Relation to the local approximation\nBecause of its widespread use, we briefly comment on the\nrelation of our results to the local analysis. In this ap-\nproximation the coordinate system is Cartesian, the shear\nis modelled linearly, and the perturbed quantities are as-\nsumed to vary rapidly in the radial direction (with respect\nto the background variations) so they may be written as\nδu(r) =δuexp[i(lr+kz) +γt],lr≫1. Under this as-\nsumption, a WKB analysis ignores terms proportional to\n1/rand replaces d /drbyil. The local dispersion relation is\n(Islam & Balbus 2005; Ferraro 2007):\nγ4+d1γ3+d2γ2+d3γ+d4= 0, (50)\n8The coefficient of the Braginskii viscosity is given by νB=\n0.96nmiv2\nthi/νiiandη= 3/radicalbig\n2me/πc2lnΛe2T−3/2wheremeis\nthe electron mass and Tis in erg (Spitzer 1962; Braginskii 1965).where\nd1=SBσ2l2+k2cos2θ\nl2+k2,\nd2=2/parenleftbigg\nσ2+k2\nl2+k2/parenrightbigg\n,\nd3=SBσ2/parenleftbigg\nσ2l2+k2cos2θ\nl2+k2−2cos2θk2\nl2+k2/parenrightbigg\n,\nd4=σ2/parenleftbigg\nσ2−2k2\nl2+k2/parenrightbigg\n.\nIn thek≫l,l≫klimits for θ= 0,π/2 the local\nand global results are identical (modulo different boundary\ncondition-dependent restrictions on the spectrum of allow ed\nmodes in the latter case).\nHowever for θ≪1,B/lessorsimilar1 the analyses differ. Locally\nwe find unstable modes have γ∈ ℜ, whilst globally γis\ncomplex. Whilst in the inviscid limit ℑ(γ)∼Bcan often be\nneglected. In the highly viscous limit ℑ(γ)∼σB/(σ2+B2)\nand, as shown in Fig. 4, this can be over an order of magni-\ntude greater. This difference may prove important (from a\nmodelling perspective) for viscous systems which could hav e\nbeen treated locally, were they inviscid.\nFurthermore, inconsistent with the global picture, for\nθ= 0,π/2 andl= 0 the local analysis neglects the Bragin-\nskii viscosity. This can be understood from\nδ(bb:∇u)G=ˆeφ·(ˆeφ·∇(δurˆer)) =δur\nr, θ= 0,\nδ(bb:∇u)G=ˆez·(ˆez·∇(δuzˆez)) = ikδuz, θ=π/2,\nwhereas locally, assuming ∇ ·δu=i(kδuz+lδur) = 0 we\nhave\nδ(bb:∇u)L= 0, θ = 0,π/2.\n(Here the subscripts G and L stand for global and local.)\nPhysically, in the global case, a component of the flow is\nprojected along the magnetic field direction by the curvi-\nlinear geometry (azimuthal case) and the demands of the\nglobalincompressibility condition (vertical case), and so |B|\nchanges at linear order, thereby activating the Braginskii\nviscosity. This is not so in the local case.\nOne final clarification is worth noting. The local de-\nscription predicts that the fastest growing unstable modes\n(θ∝negationslash= 0) occur as l→0. Formally this is inconsistent with the\nassumption l≫1/r∝negationslash= 0 that went into deriving equation\n(50). The local solutions obtained under the WKB approxi-\nmationarenotself-consistentandshouldbedescribedbyth e\ntype of global solution we have constructed here. However,\nhaving determined the global solutions, we can confirm the\nn = 0 branch corresponds to γm, and so the local analysis\nis, at least qualitatively, correct in this respect.\n7 CONCLUSION\nThe nature of the ideal MRI has been well established but\nhow non-ideal modifications affect it remains an active topic\nof research (Blaes & Balbus 1994; Balbus & Terquem 2001;\nKunz & Balbus 2004; Ferraro 2007; Pessah & Chan 2008;\nDevlen & Pek¨ unl¨ u 2010). Of interest to us, and the subject\nof this study, has been the global nature of the Braginskii-\nMHD MRI.\nc/circleco√yrt2011 RAS, MNRAS 000, 1–??12M. S. Rosin, A. J. Mestel\nConsidering an isothermal, magnetized, collisional disc,\nperhaps the early ISM, we have found that for an azimuthal\nmagnetic field with an asymptotically small vertical compo-\nnent, a singular instability emerges - the magnetized MRI.\nThe growth rate of this instability ∼Ω depends on SB, a\ndimensionless combination of the temperature, shear rate\nand Alfv´ en speed, and it is up to a factor of 2√\n2 faster\nthan its unmagnetized counterpart. In the limit where SB\nis large, we determined the asymptotic maximum growth\nrateγm≃√\n2/parenleftBig\n1−S−1/2\nB/parenrightBig\nand corresponding wavenum-\nberkm≃(θB)−1S−1/4\nB. As the field increases in strength\nthe Larmor radius decreases d ρi/dt∝ −Ω (so the plasma\nbecomes more magnetized) and as the azimuthal compo-\nnent ofBincreases, purely growing modes mutate into over-\nstabilities whose imaginary part also depends on SB. Both\nof these factors may be important for the questions posed in\nSection 1, and for the turbulence that will eventually arise .\nIn the early stages of the instability, before fully develop ed\nturbulence has set it, the instability will generate non-li near\npressure anisotropies. What exact effect these will have is\nan open question, and so we finish with some thoughts on\nthe subject.\nOne major uncertainty in the evolution of magne-\ntized accretion discs is the effect of pressure anisotropy\ndriven micro-instabilities whose growth rates are gen-\nerally well in excess of the MRI e.g. mirror9and fire-\nhose (Schekochihin et al. 2005). Although various mod-\nels for treating them exist, a complete first-principles\ntheory remains outstanding (Schekochihin & Cowley 2006;\nSharma et al. 2006, 2007; Rosin et al. 2011). It is crucial for\naccretion theories to determine the fate of these instabili ties\nfluctuations over long (transport) time-scales. This is be-\ncause they determine the Maxwell and Braginskii stresses\nthat, in turn, dictate the angular momentum transport\n(Shakura & Sunyaev 1973). Even if the pressure anisotropy\nis pinned at the marginal value for the microscale instabili -\nties in some self-regularizing way, there are further compl i-\ncations that must be addressed.\nSpecifically, the viscous stress generated by the Bragin-\nskii viscosity can heat the plasma – at a rate ∝(p⊥−p/bardbl)2\n(Kunz et al. 2011). Spatial inhomogeneities in the growth\nrate of the magnetic field, as one would expect in a turbu-\nlentsystem,will leadtoinhomogeneities inthelocal press ure\nanisotropy, both in magnitude and sign. In regions of de-\ncreasing field strength the firehose would pin the anisotropy\nat|(p⊥−p/bardbl)/p|= 2/βand in regions of increasing field\nstrength the mirror would pin it at |(p⊥−p/bardbl)/p|= 1/β.\nThe implication of this is that heating in regions of increas -\ning and decreasing magnetic field strength could differ by a\nfactor of ∼four and would occur on the decorrelation scale\nof the turbulent cascade’s viscous cut-off – where the shear\nis maximized (Schekochihin & Cowley 2006). If differential\nheating of this nature does occur then one might expect\n9In regions where dln B/dt >0 slow-wave polarised modes\n(like the MRI) are unstable to the mirror with γm∼ωi((p⊥−\np/bardbl)/p)2. For the set of parameters listed in Section 6.2 this is\n≃2·10−22s−1, a factor 106slower than the MRI. However in\nmany other contexts, e.g. the intracluster medium, the mirr or\ncan be up to a factor ∼108faster than the macroscopic shear\nrate (Hellinger 2007).the non-linear dynamics to be further complicated by mag-\nnetized (and unmagnetized) temperature gradient instabil -\nities, and the temperature dependance of both νBand the\nmicro-instability thresholds (Balbus 2001; Quataert 2008 ;\nSchekochihin et al. 2010).\nUnderstanding the rich interplay between these realisa-\ntions of magnetized plasma phenomena constitutes an im-\nportant and, as of yet, unsolved issue in astrophysics; spec if-\nically in accretion discs, but also in galaxies and galaxy cl us-\nters. The overall picture is a deeply interconnected one and\nthetypesofprocesses outlinedabove areprobably pertinen t,\nto some degree, to most magnetized astrophysical settings.\nTo address these issues there is a need for both non-\nlinear simulations that include a wide range of magne-\ntized physics, and a more complete theory of the trans-\nport effects of micro-instabilities. For the first task at lea st,\nthis work should be useful for benchmarking global codes\n(Skinner & Ostriker 2010).\nACKNOWLEDGMENTS\nWe thank J. Binney, T. Heinemann, M. Kunz, G. Lesur,\nA. Nahum, G. Ogilvie, A. Schekochihin, A. Shukurov,\nJ. Stone and O. Umurhan for useful discussions and sug-\ngestions at various stages of this project. This work was\nsupported by a STFC studentship (MSR), DOE grant DE-\nFG02-05ER-25710 (MSR), and Leverhulme Trust Interna-\ntional Network for Magnetized Plasmas (MSR).\nREFERENCES\nAbramowitz M., Stegun I. A., 1964, Handbook of Mathe-\nmatical Functions. 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N., 1944, A Treatise on the Theory of Bessel\nFunctions, second edn. Cambridge UniversityPress, Cam-\nbridge\nc/circleco√yrt2011 RAS, MNRAS 000, 1–??14M. S. Rosin, A. J. Mestel\nAPPENDIX A: LINEAR STABILITY ANALYSIS\nThe linearised equations (1) to (4) are, in component form,\nγδur−2δuφ\nr=−dδΠ\ndr+iσδBr−2BcosθδBφ\nr−dδZ\ndr−3cos2θ\nrδZ, (A1)\nγδuφ+δur\nr=BcosθδBr\nr+iσδBφ+3iksin2θ\n2δZ, (A2)\nγδuz=−ikδΠ+iσδBz−ik(1−3sin2θ)δZ, (A3)\nγδBr= iσδur, (A4)\nγδBφ= iσδuφ−δBr\nr+Bcosθδur\nr, (A5)\nγδBz= iσδuz, (A6)\nikδuz=−1\nrd(rδur)\ndr, (A7)\nδZ=SB\n3B2/bracketleftBigg/parenleftbigg\ncos2θ−ik\nγsin2θ\n2/parenrightbiggδur\nr+iksin2θδuz+iksin2θ\n2δuφ/bracketrightBigg\n, (A8)\nwhereZ=SBB2bb:∇u. Equations (A1)-(A8) form a closed set which we can combine i nto a single differential equation.\nEliminating the perturbed magnetic fields, stress, pressur e and the vertical component of the velocity field yields two c oupled\nordinary differential equations for δurandδuφ\nC0dδuφ\ndr+C1δuφ=C2d2δur\ndr2C3dδur\ndr+C4δur, (A9)\nD0δuφ=D1dδur\ndr+D2δur, (A10)\nwhere\nC0=1\nrk22/parenleftbigg\n1−iσ\nγBcosθ/parenrightbigg\n−ikSBσ2cos2θcotθ, (A11)\nC1=−ikSBσ2sin2θ\n2, (A12)\nC2=−γ−1(γ2+σ2)−SBσ2sin2θ, (A13)\nC3=1\nr/bracketleftbigg\n−1\nγ(γ2+σ2)+SBσ2(E0−cot2θ)/bracketrightbigg\n, (A14)\nC4=1\nγ(γ2+σ2)/parenleftbigg\nk2+1\nr2/parenrightbigg\n+1\nr2/bracketleftbigg2k2\nγBcosθ/parenleftbigg\nBcosθ−iσ\nγ/parenrightbigg\n+SBσ2E0(cot2θ−1)/bracketrightbigg\n, (A15)\nD0=γ2+σ2+γSBσ2cos2θ, (A16)\nD1=−ik−1SBγσ2cosθsinθ, (A17)\nD2=1\nr/bracketleftbigg\nσ/parenleftbigg\n2iBcosθ+σ\nγ−γ\nσ/parenrightbigg\n+i\nkSBγσ2cotθE0/bracketrightbigg\n, (A18)\nE0=−ikγ−1sinθcosθ+cos2θ−sin2θ. (A19)\nEquations (A9) and (A10) can then readily be combined into eq uation (6). Solutions to this equation are modified Bessel\nfunctions and so the functional forms of the perturbed fields are:\nδur= K ivn(pr)exp[ikz+γt], (A20)\nδuφ=D1\nD0dδur\ndr+D2\nD0δur, (A21)\nδuz=i\nk1\nrd(rδur)\ndr, (A22)\nδBr= iσ\nγδur, (A23)\nδBφ= iσ\nγD1\nD0dδur\ndr+/bracketleftbigg\niσ\nγD2\nD0+1\nr/parenleftbiggBcosθ\nγ−iσ\nγ2/parenrightbigg/bracketrightbigg\nδur, (A24)\nδBz=−σ\nγ1\nkrd(rδur)\ndr, (A25)\nδΠ =(γ2+σ2)\nk2γ1\nrd(rδur)\ndr−SBB2/parenleftbigg1\n3−sin2θ/parenrightbigg/bracketleftbigg/parenleftbiggD2\nD0−sin2θ/parenrightbiggdδur\ndr+/parenleftbiggE0\nr+iksin2θ\n2D1\nD0/parenrightbigg\nδur/bracketrightbigg\n. (A26)\nc/circleco√yrt2011 RAS, MNRAS 000, 1–??"    },    {        "title": "1703.06135v1.Scaling_of_Relativistic_Shear_Flows_with_Bulk_Lorentz_Factor.pdf",        "content": " 1 Scaling of Relativistic Shear Flows with Bulk Lorentz Factor  Edison Liang1, Wen Fu1, Markus Böttcher2, Parisa Roustazadeh2 1 Rice University, Houston, TX 77005 2 North-West University, Potchefstroom, 2520, South Africa  Abstract We compare Particle-in-Cell simulation results of relativistic electron-ion shear flows with different bulk Lorentz factors, and discuss their implications for spine-sheath models of blazar versus gamma-ray burst (GRB) jets. Specifically, we find that most properties of the shear boundary layer scale with the bulk Lorentz factor: the lower the Lorentz factor, the thinner the boundary layer, and the weaker the self-generated fields.  Similarly, the energized electron spectrum peaks at an energy near the ion drift energy, which increases with bulk Lorentz factor, and the beaming of the accelerated electrons gets narrower with increasing Lorentz factor.  This predicts a strong correlation between emitted photon energy, angular beaming and temporal variability with the bulk Lorentz factor.   Observationally, we expect systematic differences between the high-energy emissions of blazars and GRB jets. Subject Keywords: Shear Flow; Gamma-Ray Bursts; Quasars   1. INTRODUCTION Unveiling the composition of relativistic jets of active galactic nuclei (AGN) and gamma-ray bursts (GRB), and the mechanisms of particle acceleration to ultrarelativistic energies within these jets, is among the prime outstanding issues in gamma-ray astronomy, as probed by the  2 Fermi Gamma-Ray Space Telescope and ground-based Atmospheric Cherenkov Telescopes, such as H.E.S.S., MAGIC, and VERITAS, and the future Cherenkov Telescope Array (CTA). The physics of relativistic jets of AGN is most directly probed by observations of blazars, whose jets are oriented at a small angle with respect to our line of sight. Their broad-band nonthermal continuum emission consists of two broad emission components, and is almost certainly produced in small, localized regions within the relativistic jet.  It is commonly accepted that the radio through optical/UV (and in some cases X-ray) emission from blazars is synchrotron emission from relativistic particles.  Leptonic models for the high-energy emission of blazars propose that the X-rays and gamma-rays from blazars are the result of Compton upscattering of lower-energy photons by the same relativistic electrons (see, e.g. Boettcher 2007 for a review of blazar emission models).   There are several lines of evidence which suggest that the jets in blazars exhibit at least a two-component structure: a mildly relativistic, outer sheath with higher density, carries most of the kinetic energy of the jet, while a fast, highly relativistic inner spine of low co-moving particle density carries most of the angular momentum.  Direct observational evidence for radially structured spine-sheath jets comes from the limb-brightening of blazar and radio galaxy jets revealed in VLBI observations (Giroletti 2004). Prompted by such evidence, Ghisellini (2005) proposed the radiative interaction between a fast, inner spine and a slower sheath in a blazar jet as a way to overcome problems with extreme bulk Lorentz factors required by spectral fits to several TeV BL Lac objects. Hydrodynamic/MHD simulations of spine-sheath jets (Meliani & Keppens 2007, 2009, Mizuno 2007) indicate that the sheath, in combination with a poloidal magnetic field, aids in stabilizing the jet. Although Kelvin-Helmholtz-type instabilities (KHI,  3 Chandrasekhar 1981) may develop at the spine-sheath interface and lead to turbulent mixing of the two phases, they may not disrupt the jet out to large distances from the central engine (Meliani & Keppens 2007, 2009).  The MHD turbulence developing at the spine-sheath interface of relativistic jets (Zhang et al 2009) offers a promising avenue for relativistic particle acceleration in radio-loud AGNs and GRBs.   However, the MHD approximation cannot directly address the creation of magnetic fields from unmagnetized shear flows or the acceleration of nonthermal particles.  The kinetic physics of relativistic shear flows has been successfully simulated using Particle-in-Cell (PIC, Birdshall & Langdon 1991) simulations (Alves et al 2012, 2014, 2015, Grismayer et al 2013, Liang et al 2013ab, Nishikawa et al 2013, 2014, 2016).  In our previous papers (Liang et al 2013b, 2016), we have shown that ion-dominated relativistic shear flows lead to the creation of ordered dc electromagnetic (EM) fields near the shear boundary via the electron counter-current instability (ECCI), and the development of highly relativistic electron distributions peaking near the ion kinetic energy. However, those simulations assumed a high spine Lorentz factor (Γ = 451 in the central engine frame, po=15 in the center-of-momentum (CM) frame).  Hence those results are more relevant to GRBs (Liang 2013b, 2016) than to AGNs.   In this paper, we present new PIC simulation results for a more moderate bulk Lorentz factor (po=5, Γ=51), relevant to radio-loud AGN, in particular blazars, in which bulk Lorentz factors Γ~O(10) are typically inferred from superluminal motion and radio brightness-temperature arguments (Jorstad 2005, Hovatta 2009).  We will systematically compare the po=5 shear  4 boundary with the po=15 shear boundary.  To simplify the comparison, we first focus on pure electron-ion (e-ion) plasmas.  Generalization to mixtures of e-ion and electron-positron (e+e-ion) plasmas does not alter our major conclusions, and will be briefly mentioned at the end of Sec.2.     2. COMPARISON OF po=5 AND po=15 SHEAR BOUNDARIES.   As in our previous shear flow PIC simulations (Liang et al 2013ab, 2016), we use the 2.5D (2D space, 3-momenta) code Zohar-II (Birdsall & Langdon 1991, Langdon & Lasinski 1976) as the primary simulation tool.  Though our Zohar-II simulation box is limited to 1024x2048 cells, this code has high numerical fidelity, and the numerical Cerenkov instability (NCI, Godfrey 1974, 1975) is strongly suppressed (Godfrey and Langdon 1976). Hence it is well suited for simulations with relativistic particle drifts.  In all ion-dominated shear flows, the T-mode (Liang et al 2013a) in the y-z plane (Fig.1) saturates at very low amplitude compared to the P-mode (Liang et al 2013a) in the x-y plane, and has negligible effects on the shear boundary structure (Liang et al 2013b, confirmed by both 2.5D runs in the y-z plane and 3D runs).  Hence we focus on the 2D P-mode (Fig.1) results in the x-y plane in this paper.  All simulations are performed in the CM frame with periodic boundary conditions and initial temperature kT = 2.5 keV for both electrons and ions (mi/me=1836). Throughout this paper and in all figures, distances are measured in units of electron skin depth c/ωe (ωe = electron plasma frequency) and times are measured in units of 1/ωe.  We normalize the initial density n=1 so that the cell size = c/ωe.  The plasmas are initially unmagnetized.  Initially right-moving plasma occupies the central 50% of the y-grid (hereafter called the “spine”), while initially left-moving plasma occupies the top 25%  5 and bottom 25% of the y-grid (hereafter called the “sheath”)(Fig.1).  To increase numerical stability, we used time-step Δt = 0.1/ωe.  Overall energy conservation was better than 1%.   We first compare the main features of po=5 and po=15 shear boundaries.  Figure 2 shows the energy flows between ions, electrons and EM fields for the two runs. We see that in both cases, the electron and ion energies reach equipartition after tωe~ 9000, and EM energy saturates at ~ 12% of total energy, showing that the e-ion equipartition and EM energy saturation are insensitive to po. Figure 3 compares the spatial profiles of Bz Ey, Ex, Jx, and net charge ρ = (n+-n-) at tωe=1000, 3000 and 12000 respectively for the two runs.  While the overall patterns are qualitatively similar, the shear boundary layers of the po=5 case are thinner than those of the po=15 case by ~ factor of two.  This is not unexpected since the thickness of the boundary layer should be related to the relativistic skin depth and relativistic gyroradius, both of which increase with increasing po.  The maximum values of the dc fields (Bz, Ey) are also lower for po=5 than po=15 (Fig.3ab).  Figure 4 compares the x-averaged density profiles of ions, electrons and net charge as functions of y for the two runs.  This shows that the ion vacuum gap created by magnetic expulsion from the shear interface is present in both runs, but the gap is wider for po=15 than for po=5 due to stronger dc fields.  This robust ion vacuum gap is a unique feature of relativistic ion-dominated shear flows, which sustains the separation of the opposing flows and the long-term stability of the laminar boundary layer structure against turbulent mixing of opposing ions.  Electrons are evacuated less than the ions due to their mobility, leading to charge separation and the formation of a triple layer (double capacitor) at the shear boundary and associated Ey fields (cf. Fig.3b).  Inductive Ex fields are generated adjacent to the boundary layer by ∂Bz/∂t (Fig.3c), which accelerates the electrons and decelerates the ions.   6 Figure 5 compares the electron and ion energy distributions for the two runs at late times.  Because the bulk of particle acceleration/deceleration is done by the Ex fields, the artificial periodic y-boundary condition turns out to have little effect on the late-time electron and ion distributions, as we had previously demonstrated using much larger y-grids (Liang 2013b, 2016).  In both runs the electron spectrum exhibits a narrow peak near the (decelerated) ion drift kinetic energy.  In the po=5 case, the electron spectrum peaks at γe~3000, consistent with the ion energy peak at ~2.5mic2 (hence kinetic energy ~1.5mic2).  Similarly, for the po=15 case, the electron spectrum peaks at γe~14000, consistent with the ion energy peak at ~ 7mic2 (Liang et al 2013b).  This confirms the scaling of the electron peak energy γe with po.  As we discuss below in Sec.3, in the context of synchrotron models, the electron peak energy γe can be related to the synchrotron critical frequency (Rybicki & Lightman 1979) via ωcr ~ γe2ωB, where ωB = eB/mc is the electron gyrofrequency (=Lamor frequency).  On the other hand, for Compton models, the inverse Compton peak is located at ωIC ~ γe2ωo, where ωo is the characteristic soft photon energy (ωo~ωcr for SSC models, Boettcher 2007).  Even though pure e-ion shear flows do not accelerate electrons much above the ion kinetic energy (Fig.5), when we add a moderate amount of e+e- plasma into the e-ion plasma, a power-law tail eventually develops above γe, due to the presence of nonlinear EM waves created outside the dc slab fields of Fig.3 (Liang et al 2013b), which scatter the leptons stochastically to form the power-law tail.  We observe power-law tails develop in both the po=5 and po=15 cases, but preliminary results suggest that the power-law slope may vary with both po and e+/ion ratio.  Details remain to be investigated systematically.   3. APPLICATIONS TO BLAZARS AND GRBS   7 Assuming that blazar and GRB jets indeed have a spine-sheath structure, our shear boundary PIC simulations results above should be applicable to the local emission properties of the spine-sheath interface.  To better visualize the differences in particle momentum distribution and radiation characteristics between the po=5 and po=15 cases, it is better to Lorentz boost the particle momenta from the CM frame of Sec.2 back to the “laboratory” frame (LF) in which the sheath is initially at rest, and the spine moves with the bulk Lorentz factor Γ=2po2+1.  Figures 6 & 7 compare various phase plots for the two runs, after Lorentz boosting (in the –x direction) from the CM frame back to the LF.  We see that for po=5 (Figs.6a, 7a), spine electrons are accelerated to peak at γLab ~ 30000 or 15 GeV, whereas for po=15 (Figs.6b,7b), spine electrons are accelerated to peak at γLab ~ 4.4x105 or 220 GeV. The highest-energy spine electron momenta achieve more extreme anisotropy (pxLab >>> py) for po=15 and than for po=5, while the beam angle |py/pxLab| decreases exponentially with increasing energy for both po=5 and po=15 (Fig.8).  In fact, on average both beam angles are much narrower than simple Doppler boosting of an isotropic distribution in the spine rest frame to the LF (1/Γ, red dashed line).  Observationally, we therefore expect GRB jets to emit much harder radiation with narrower beaming and more rapid time variability than blazar jets, and the photon energy should be correlated with time variability and anti-correlated with beam angle.  Such observational predictions should be testable.    In summary, our PIC simulation results show that efficient lepton acceleration up to γe ~ Γi mi/me occurs in relativistic shear boundary layers, and proceeds in a strongly anisotropic manner. The highest-energy leptons are beamed into an angle much narrower than 1/Γ in the laboratory frame.  In the process of Compton scattering by relativistic leptons, the scattered, high-energy  8 photon emerges in the direction of the scattering lepton. Hence jets viewed in the direction tangential to the shear boundary will exhibit very hard radiation spectra, much beyond the usual spectral hardening effect due to bulk Doppler boosting of a co-moving isotropic particle distribution (which is just a shift of the peak frequency by factor Γ).  On the other hand, jets viewed at substantial off-axis angles (assuming that the shear layer is largely parallel to the global jet axis) will exhibit softer spectra.   These beaming effects should become more acute for GRBs (Meszaros 2002, Piran 2004, Preece et al 1998) than blazars, and more extreme for the Compton peak than the synchrotron peak.  The narrow beaming may also explain the minute-scale rapid time-variability of some blazars (Tavecchio and Ghisellini, private communications). The results presented above indicate that relativistic shear layers in ion-dominated plasmas are capable of producing relativistic electron distributions in the CM frame with pronounced peaks at γe ~ few x103 for blazar Lorentz factors Γ~10. In the presence of a magnetic field of B = BG Gauss in the CM frame, this results in an observed synchrotron peak frequency of ν~ 1014 BG Hz in the LF, typically observed in low-synchrotron-peaked (LSP) blazars, i.e. flat-spectrum radio quasars and low-frequency-peaked BL Lac objects.   These same electrons will then also produce gamma-rays via Compton up-scattering of the co-spatially produced synchrotron photons (the synchrotron self-Compton (SSC) process) and possibly photons produced external to the jet (the external-Compton (EC) process), e.g., in the broad-line region or infrared-emitting dusty torus around the central accretion flow (Boettcher 2013).  Synchrotron photons can be up-scattered (SSC) in the Thomson regime, which is expected to be the case for blazars for any plausible magnetic-field value.  This will then result in a peak photon energy of the SSC emission of  ~ few MeV BG.  Gamma-ray emission of LSP  9 blazars is often dominated by SSC emission (Boettcher 2013).  External photons with stationary-frame energy hνo ~ eV will be Compton up-scattered in the Klein-Nishina regime to yield maximum observed photon energies of  ~ 15 GeV in LF.  This is consistent with the gamma-ray peaks in LSP blazars typically being located at 100 MeV to GeV.   These estimates illustrate that for characteristic values of bulk Lorentz factor Γ ~ 10, the shear boundary energization scenario predicts synchrotron peaks in the IR-optical and EC gamma-ray peaks up to the GeV regime, as typically observed in LSP blazars (Abdo 2010), along with SSC-dominated hard X-ray and soft gamma-ray emission, peaking around  ~ few MeV.  However, applying this scenario to high-frequency-peaked BL Lac objects (HBLs) with observed synchrotron peak frequencies of ν ~ 1017 Hz would require bulk Lorentz factors much higher than typical values of Γ ~10 inferred for blazars in general, unless the magnetic field in the CM frame is >> Gauss.  If the jet composition is dominated by ions in both the spine and the sheath as assumed in this paper, one expects a small population of AGNs viewed under very small viewing angles with θobs << 1/Γ, with very hard gamma-ray spectra (such as those detected by LAT onboard Fermi Observatory), while a larger population of off-axis AGNs with θobs > 1/Γ appear to have much softer spectra (cf. Fig.8).  The recently emerging class of extreme BL Lac objects (e.g., Bonnoli et al. 2015) may possibly represent the small population of extremely narrowly beamed, very-hard-spectrum blazars expected in the ion-dominated shear flow scenario.    ACKNOWLEDGMENTS  10 This work was partially supported by NSF AST1313129 and Fermi Cycles 4 & 5 GI grants to Rice University, and NASA Fermi GI Grant no. NNX12AE31G to Ohio University.  The work of M.B. is supported through the South African Research Chairs Initiative (SARChI) of the Department of Science and Technology and the National Research Foundation1 of South Africa under SARChI Chair grant no. 64789.  We thank Drs. Fabrizio Tavecchio and Gabriele Ghisellini for useful discussions.  Simulations with the Zohar-II code were supported by the Lawrence Livermore National Laboratory.    REFERNCES Abdo, A. A., et al., 2010, ApJ, 716, 30.  Alves, E.P., T Grismayer, SF Martins, F Fiúza, RA Fonseca, LO Silva, 2012, Ap. J. Lett. 746, L14. Alves, E. P., Griesmayer, T., Fonseca, R. A., & Silva, L. O. 2014, New J. Phys., 16, 035007. Alves, E. P., Griesmayer, T., Fonseca, R. A., & Silva, L. O. 2015, Phys. Rev. E, 92, 021101. Birdsall, C. and A. B. 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S., & Band, D. L., 1998, ApJ, 506, L23 Rybicki, G. & Lightman, A. 1979, Radiative Processes in Astrophysics (Freeman, SF). Sironi, L., & Spitkovsky, A., 2009, ApJ 707, L92. Tavecchio, F. & Ghsiellini, G., 2008, MNRAS, 385, L98 Tchekhovskoy, A., McKinney, J. C., & Narayan, R., 2008, MNRAS, 388, 551.  Zhang, W., MacFadyen, A., and Wang, P. 2009, ApJ 692, L40.           13 Figure Captions Fig.1 Setup of the (initially unmagnetized) shear flow PIC simulations of the e-ion plasma.  This paper focuses on the longitudinal P-mode evolution in the x-y plane only, since the transverse T-mode saturates at very low level compared to the P-mode. In the present case the plasma consists of right-moving plasma in the central 50% of the y-grid, referred to as the “spine”, sandwiched between left-moving plasmas at the top 25% and bottom 25% of the y-grid, referred to as the “sheath”. The simulation box has periodic boundary conditions on all sides.  Inset: Sketch illustrating d.c. magnetic field creation by the ECCI.  Throughout this paper and in all Figures, spatial scales are in units of the electron skin depth c/ωe.  Fig.2 (a) Evolution of energy components of the 2D e-ion shear flow with po=5: EM field energy (A, black), electron energy (B, red), ion energy (C, blue).  At late times the EM field energy saturates at ~12% of total energy; (b) Evolution of energy components of the 2D shear flow with po=15: EM field energy (A, black), electron energy (B, red), ion energy (C, blue).  At late times the EM field energy also saturates at ~12% of total energy. However, the electron energy approaches the ion energy faster in the po=15 case than in the po=5 case.  Fig.3 Comparison of spatial profiles of (a) Bz, (b) Ey, (c) Ex, (d) jx, (e) ρ = net charge = (ni-ne), between po=5 (left columns) and po=15 (right columns) cases at three different times: tωe = 1000 (top), 3000 (middle), 12000 (bottom).  We note that the shear boundary layer thickness of the po=5 case is ~ half that of the po=15 case.  Color scales show that the maximum Bz, Ey fields of the po=5 case is much lower than those of the po=15 case.   14 Fig.4  Comparison of density profiles n vs. y (averaged over x) between the po=5 and po=15 cases at tωe = 8000. Curve 1 (red) electrons, Curve 2 (blue) ions, Curve 3 (black) net charge (=ni-ne) These density profiles highlight the persistence of the ion vacuum gap, which is wider for the po=15 case than the po=5 case.  Electrons, however, are not fully evacuated from the boundary layer, leading to the formation of the charge triple layer (double capacitor), which plays an important role in electron energization.  Fig.5  Comparison of electron distribution function fe(γ) (particle no. per unit γ) vs. γ for (a) the po=5 case with that of (b) the po=15 case at tωe = 10000; (c)(d) Same as (a)(b) for ion distribution function fi(γ) at tωe = 10000.  We see that the electron energy peaks at γe ~ γimi/me in both cases.  Fig.6 Phase plot (py vs pxLab) of spine electrons for the po=5 case (a) compare to that of the po=15 case (b) at tωe=8000, after Lorentz boosting px to the “laboratory frame” in which the sheath is initially at rest.  By this time, some of the spine electrons have diffused into the sheath region and are decelerated, forming the low-energy bow-shaped population at left.  The arrow-shaped high energy population corresponds to electrons remaining in the spine.  Note that electrons are more concentrated at the highest energy for the po=15 case.    Fig.7 Phase plots (y vs. pxLab) of spine electrons for the po=5 case (a) compare to that of the po=15 case (b) at tωe=8000, after Lorentz boosting px to the “laboratory frame” in which the sheath is initially at rest.  By this time, some of the spine electrons have diffused into the sheath region and are decelerated, forming the low-energy population at left.  The central high energy  15 population corresponds to electrons remaining in the spine.  Note that electrons are more concentrated at the highest energy for the po=15 case.   Fig.8 Distribution of the tangent of the “beam angle” (=|py/pxLab|) of spine electrons vs. Lorentz factor γLab in the “laboratory frame” at tωe=8000 for (a) po=5 and (b) po=15.  We see that in both cases all of the high-energy spine electrons (i.e. those that did not cross over to the sheath and get decelerated) have beam angles much smaller than 1/Γ (red dashed lines).  In both cases there exists an anti-correlation between beam angle and electron energy.    16     \n   \nFig.1    .XB=0  kT=2.5 keV  n=1mi/me=1, 100, 1836       20-40 particles/cell px=-15mcpx=+15mcpx=-15mcxωe/cyωe/c\nzωe/c20485121536010241024Momentum-plane (P)Transverse  plane (T)\npion pe Jnet In ion-dominated shear boundary, fields are generated by electron counter-current (ECC) instability Bin \nmi/me=1836 \n- pomc \n+ pomc \n- pomc  17 (a) po=5 \ntωe  (b) po=15 \n tωe  Fig.2  \n 18 y     po=5   tωe = 1000   y     po=15 \n y         tωe = 3000    y \n        y         tωe = 12000   y \n x            x Fig.3(a) Bz          \n 19 y     po=5   tωe = 1000   y     po=15 \n y         tωe = 3000    y \n y         tωe = 12000   y \n x            x Fig.3(b) Ey          \n 20 y     po=5   tωe = 1000   y     po=15 \n y         tωe = 3000    y \n  y         tωe = 12000   y \n x            x Fig.3(c) Ex         \n 21 y     po=5   tωe = 1000   y     po=15 \n y         tωe = 3000    y \n y         tωe = 12000   y \n x            x Fig.3(d) jx         \n 22 y     po=5   tωe = 1000   y     po=15 \n y         tωe = 3000    y \n y         tωe = 12000   y \n x            x Fig.3(e) ρ          \n 23   n     (a) po=5          n    (b) po=15 \n  y              y       Fig.4   \n 24 fe(γ)       (a) po=5      fe(γ)     (b) po=15 \n        γ             γ    fi(γ)        (c) po=5         fi(γ)    (d) po=15 \n γ             γ   Fig.5    \n 25  py        (a) po=5           py    (b) po=15 \n pxLab             pxLab      Fig.6     \n 26     y         (a) po=5      y     (b) po=15 \n pxLab            pxLab     Fig.7    \n 27   log|py/pxLab|       (a) po=5      log|py/pxLab|   (b) po=15 \n γLab            γLab        Fig.8   \n"    },    {        "title": "2403.10224v1.The_self_dual_Lorentz_violating_model__quantization__scattering_and_dual_equivalence.pdf",        "content": "arXiv:2403.10224v1  [hep-th]  15 Mar 2024The self-dual Lorentz violating model: quantization, scat tering and dual equivalence\nM. A. Anacleto,1,∗F. A. Brito,1,2,†and E. Passos1,‡\n1Departamento de F´ ısica, Universidade Federal de Campina G rande\nCaixa Postal 10071, 58429-900 Campina Grande, Para´ ıba, Br azil\n2Departamento de F´ ısica, Universidade Federal da Para´ ıba ,\nCaixa Postal 5008, 58051-970 Jo˜ ao Pessoa, Para´ ıba, Brazi l\nIn this paper, we analysis the dynamics, at the quantum level , of the self-dual field minimally\ncoupled to bosons with Lorentz symmetry breaking. We quanti ze the model by applying the Dirac\nbracket canonical quantization procedure. In addition, we test the relativistic invariance of the\nmodel by computing the boson-boson elastic scattering ampl itude. Therefore, we show that the\nLorentz symmetry breaking has been restored at the quantum l evel. We finalize our analysis by\ncomputing the dual equivalence between the self-dual model with Lorentz symmetry breaking cou-\npled with bosonic matter and the Maxwell-Chern-Simons with Lorentz invariance violation coupled\nwith bosonic field.\nI. INTRODUCTION\nInphysics,symmetriesplayimportantroles. Inparticular,theviola tionofLorentzsymmetryinitiatedbyKostelecky\nand collaborators [ 1] has been widely investigated in various fields. The motivation comes f rom the theory of\nsuperstrings indicating that the Lorentz symmetry would be broke n at high energies. It is well known that Lorentz\nsymmetry is of fundamental importance in the construction of phy sical theories. Therefore, this symmetry was tested\nwith great precision [ 2]. From a theoretical point of view, it is conjectured that violations o f Lorentz symmetry can\noccur on the Planck scale. In addition, these effects are observed even in the low-energy region. In [ 3], the breaking\nof Lorentz symmetry in field theory was studied. Lorentz symmetr y breaking in QED was investigated in [ 4–6].\nHere, our focus of attention is on the self-dual field model coupled with bosons, in the background violating Lorentz\nsymmetry.\nThe self-dual field model proposed by Townsend, Pilch and Van Nieuw enhuizen [ 7] has received a lot of attention.\nThe equivalence between the free self-dual model and the Maxwell Chern-Simons model was established in [ 8,9].\nAlso, in [ 10], equivalence at the level of the Green function has been establishe d. In [11], Gomes and collaborators\nexplore the equivalence between the self-dual model minimally couple d with a fermion field and the Maxwell Chern-\nSimons model coupled with fermions. In addition, the quantization of the self-dual model coupled with fermions\nand the relativistic invariance test via calculation of the fermion-fer mion scattering amplitude was performed [ 12].\nFurthermore, the quantization of the self-dual model coupled wit h bosons, as well as the relativistic invariance test\ncomputing the boson-boson scattering was also studied in [ 13]. Later, the dual equivalence of the self-dual model\ncoupled with bosons and the Maxwell Chern-Simons model coupled wit h bosons was established [ 14]. An analysis on\nthe coupling of the self-dual field to dynamic U(1) matter and its dua l theory has been carried out in [ 15] Duality in\nthe context of Lorentz symmetry breaking has been investigated in four-dimensional [ 16–18] and three-dimensional\ntheories [ 19]. Besides, different aspects of duality were constructed [ 20–24]. Also see [ 25–32] for other applications.\nRecently, in [ 33] the duality between the Maxwell-Chern-Simons and self-dual mode ls in very special relativity has\nbeen investigated.\nIn this paper, we are interested in studying the canonical quantiza tion of the self-dual field model coupled with\nbosons violating Lorentz symmetry. In this study, we will quantize t he model by applying the canonical quantization\nprocedure of the Dirac brackets. Besides, we will compute the bos on-boson scattering amplitude to test the Lorentz\ninvariance of the model. Therefore, as a result of this analysis, we s how that the combined action of the non-covariant\nparts of the interaction Hamiltonian is replaced by the minimum covaria nt field-current interaction. In addition, we\nverify that Lorentz invariance is preserved at the quantum level. M oreover, we investigate the duality between the\nself-dual and Maxwell-Chern-Simons models with Lorentz symmetry breaking and coupled with bosonic matter.\nThe paper is organized as follows. In Sec. IIwe introduce the Lorentz symmetry breaking effect in the self-dua l\nfield model coupled with bosons and quantize the model via Dirac brac kets. In Sec. IIIwe compute the boson-boson\nelastic scattering amplitude to test the Lorentz invariance of the m odel. In Sec. IVwe explore the dual equivalence\n∗anacleto@df.ufcg.edu.br\n†fabrito@df.ufcg.edu.br\n‡passos@df.ufcg.edu.br2\nbetween the self-dual and Maxwell-Chern-Simons models with Loren tz symmetry breaking and coupled with bosonic\nmatter. Finally in Sec. Vwe present our final considerations.\nII. THE LORENTZ-BREAKING SELF-DUAL MODEL\nIn this section, we consider the self-dual field model coupled to bos ons with Lorentz symmetry breaking, given by\nthe following Lagrangian:\nL=Lsd+Lφf+Lϕ+Lint, (1)\nbeing\nLsd=m\n2ǫµνρ(fµ∂νfρ)−m2\n2fµfµ, (2)\nLφf=1\n2∂µφ∂µφ+2mφvµfµ, (3)\nLϕ=∂µϕ∗∂µϕ−M2ϕ∗ϕ, (4)\nLint=−efµJµ+e2fµfµϕ∗ϕ−e2φ2ϕ∗ϕ, (5)\nwhereJµ=i(ϕ∗∂µϕ−∂µϕ∗ϕ) is the current, mis a parameter with dimensions of mass, ϕis the charged scalar\nfield,Mis the boson mass, fµis the self-dual field and φis the real scalar field interacting with the self-dual field\nfµ. Here, 2 mφvµfµis the term carrying the Lorentz symmetry breaking with the const ant 3-vector introducing a\npreferred frame of reference in spacetime. The model consisting ofLsdandLφfwas considered in [ 19] to examine\nthe self-dual/Maxwell-Chern-Simons duality with Lorentz symmetry -breaking. In the Maxwell-Chern-Simons model\nwith Lorentz symmetry breaking, the mixed term, φǫµνρFνρ, was obtained through the dimensional reduction of the\nJackiw term [ 34–36]. For other applications of this term, see also [ 37–39].\nNow we start the development of the canonical quantization appro ach of the self-dual model coupled to bosons and\nwith Lorentz symmetry breaking. Then, by computing the canonica lly conjugate moments of the model, we find\nπα=∂L\n∂(∂0fα)=−m\n2ǫ0αµfµ, (6)\nΠφ=∂L\n∂(∂0φ)=∂0φ, (7)\nΠ =∂L\n∂(∂0ϕ)=∂0ϕ∗−ief0ϕ∗, (8)\nΠ∗=∂L\n∂(∂0ϕ∗)=∂0ϕ+ief0ϕ. (9)\nHence, we find that the primary constraints are given by\nΦ(1)\n0=π0≈0, (10)\nΦ(1)\ni=πi+m\n2ǫijfj≈0. (11)\nAccording to the Dirac quantization procedure, the symbol ( ≈) introduced above means weak equality and with\nequality occurring on the constraint surface [ 40]. However, the canonical Hamiltonian density is given by\nH= Πφ∂0φ+Π∂0ϕ+Π∗∂0ϕ∗+πi∂0fi−L,\n= Π∗Π+Π2\nφ\n2−1\n2∂iφ∂iφ−∂iϕ∗∂iϕ+M2ϕ∗ϕ+e2φ2ϕ∗ϕ+efiJi−e2fifiϕ∗ϕ\n+m2\n2fµfµ−2mφvifi−f0[mǫij∂ifj+ie(Πϕ−Π∗ϕ∗)+2mφv0]. (12)\nThus, the primary Hamiltonian can be presented as\nHp=/integraldisplay\nd2x(H+U0Φ(1)\n0+UiΦ(1)\ni), (13)3\nbeingU0andUithe Lagrange multipliers.\nNext, we apply the Dirac algorithm by imposing the condition of conser vation in time, such that:\n˙Φ(1)\n0={Φ(1)\n0,Hp}p={π0(/vector x),Hp(/vector y)}p≈0. (14)\nThis condition gives rise to the secondary constraint\nΦ(2)=m2/parenleftbigg\nf0−1\nmǫij∂ifj−ie\nm2(Πϕ−Π∗ϕ∗)−2φv0\nm/parenrightbigg\n≈0. (15)\nThen, to complete the Dirac algorithm we must again check whether t he conservation of the constraints Φ(2)implies\nnewsecondaryconstraints. Itiseasytoseethatnomoreconstr aintsariseandtheLagrangemultipliersaredetermined.\nWe can verify that the system of constraints is of the second class .\nAt this point, we begin the process of quantizing the theory with sec ond-class constraints by introducing the Dirac\nbrackets defined for arbitrary functions of fields and momenta as follows:\n{Λ,Ω}D={Λ,Ω}−3/summationdisplay\nq=03/summationdisplay\np=0/integraldisplay /integraldisplay\nd2ud2v{Λ,ξp(/vector u)}R−1\npq{ξq(/vector v),Ω}, (16)\nwhereξ0≡Φ(1)\n0,ξl≡Φ(1)\nl,l= 1,2 ,ξ3≡Φ(2)andRpq(/vector u,/vector v) ={ξp(/vector u),ξq(/vector v)}is a matrix of the Poisson brackets of\nthe constraints (it is non-degenerate as it should be for the syste m of the second-class constraints).\nIn the sequence to quantize the theory, we define the fields and mo menta as being operators and thus we have the\nfollowing relation involving the commutator and Dirac bracket:\n[Λ,Ω] =i{Λ,Ω}D. (17)\nThus, after computing the Dirac brackets, we find the following com mutation relations at equal times for the dynamic\nvariables\n[f0(/vector x),fj(/vector y)] =i∂j\nxδ(/vector x−/vector y),\n[fk(/vector x),fj(/vector y)] =−imǫkjδ(/vector x−/vector y),\n[f0(/vector x),πk(/vector y)] =−i\n2mǫkj∂j\nxδ(/vector x−/vector y),\n[fj(/vector x),πk(/vector y)] =i\n2gj\nkδ(/vector x−/vector y),\n[πj(/vector x),πk(/vector y)] =−i\n4mǫjkδ(/vector x−/vector y),\n[f0(/vector x),ϕ(/vector y)] =eϕδ(/vector x−/vector y),\n[f0(/vector x),ϕ†(/vector y)] =−eϕ†δ(/vector x−/vector y),\n[φ(/vector x),Πφ(/vector y)] =iδ(/vector x−/vector y),\n[ϕ(/vector x),Π(/vector y)] =iδ(/vector x−/vector y),\n[φ†(/vector x),Π†(/vector y)] =iδ(/vector x−/vector y). (18)\nThe Hamiltonian operator ( 12), with use of the constraint ( 15), can be rewritten as\nH=/integraldisplay\nd2x/parenleftBigg\nΠ†Π+Π2\nφ\n2−1\n2∂iφ∂iφ−∂iϕ†∂iϕ+M2ϕ†ϕ+e2φ2ϕ†ϕ\n+efiJi−e2fifiϕ†ϕ+m2\n2fifi−m2\n2f0f0−2mφvifi/parenrightbigg\n, (19)\nand applying condition ( 15) to eliminate the f0operators, the Hamiltonian takes the form\nHI=HI\n0+HI\nint1++HI\nint2, (20)\nbeing\nHI\n0=/integraldisplay\nd2x/parenleftBigg\nΠI†ΠI+(ΠI\nφ)2\n2−1\n2∂iφI∂iφI−2φI2v2\n0−∂iϕI†∂iϕI+M2ϕI†ϕI\n+m2\n2fI\nifIi−1\n2ǫijǫkl(∂ifI\nj)(∂kfI\nl)/parenrightbigg\n, (21)4\nHI\nint1=−2/integraldisplay\nd2x/parenleftbig\nmviφIfIi+v0φIǫij∂ifI\nj/parenrightbig\n, (22)\nand\nHI\nint2=/integraldisplay\nd2x/parenleftbigg\nefIkJI\nk−ie\nmǫkj(∂kfj)(ΠIϕI−ΠI†ϕI†)−2ie\nmv0φI(ΠIϕI−ΠI†ϕI†)\n+e2\n2m2(ΠIϕI−ΠI†ϕI†)2−e2fI\nkfIkϕI†ϕI+e2φI2ϕI†ϕI/parenrightbigg\n. (23)\nIn the Hamiltonian operator above, we write the superscript Ito indicate that the field operators belong to the\ninteraction picture. Note that in the Hamiltonian HI\n0the Lorentz symmetry breaking term generated a mass term\nfor the scalar field φ. In the Hamiltonian HI\nint1we have the interaction of the scalar field φwith the gauge field f,\nwhereas in the Hamiltonian HI\nint2we have the interaction of the field ϕwith the gauge field fand with the scalar\nfieldφ. Then, the equations of motion that ϕIandϕI†obey are respectively given by:\n˙ϕ(/vector x) =i[HI\n0,ϕI(/vector x)] = ΠI†, (24)\n˙ϕI†(/vector x) =i[HI\n0,ϕI†(/vector x)] = ΠI. (25)\nThus, the propagator of the boson field in the momentum space is\n∆(p) =i\np2−M2+iǫ. (26)\nFor the self-dual field ( fI\ni,i= 1,2), the propagator is given by [ 19]\n1\nm2Dlj(k) =−i\nk2−m2+iǫ/parenleftbigg\nglj−klkj\nm2+i\nmǫljk0−4vlvj\nk2−vlkj\nk2/parenrightbigg\n, (27)\nand for the mixed propagators, we have\n/an}bracketle{tfiφ/an}bracketri}ht=i\nk2−m2+iǫ/parenleftbigg2mvi\nk2/parenrightbigg\n, (28)\nand\n/an}bracketle{tφφ/an}bracketri}ht=−i\np2−m2+iǫ/parenleftbigg4m2\nk2/parenrightbigg\n. (29)\nTherefore, we can now write the Hamiltonian of interactions in terms of the fundamental fields as follows:\nHI\nint2=/integraldisplay\nd2x/parenleftbigg\niefIk(ϕI†∂kϕI−∂kϕI†ϕI)−ie\nmǫlj(∂lfj)( ˙ϕI†ϕI−ϕI†˙ϕI)\n+e2\n2m2( ˙ϕI†ϕI−ϕI†˙ϕI)2−2ie\nmv0φI( ˙ϕI†ϕI−ϕI†˙ϕI)\n−e2fI\nkfIkϕI†ϕI+e2φI2ϕI†ϕI/parenrightbig\n. (30)\nNote that the Hamiltonian of interactions above has six terms. The fi rst term corresponds to the spatial part of\nthe dual field-current interaction. The second term, third term, and fourth term arise when we use the constraint\ncondition to eliminate f0from the Hamiltonian. Being the second term the interaction of the m agnetic field with the\ntime component of the current, while the third is the interaction of t he temporal part of currents. Moreover, the extra\nterms are local in spacetime and not renormalizable by power countin g. The fourth term is the scalar field interacting\nwith the temporal component of the current. The fifth term is the spatial part of the gauge-boson field interaction,\nand the sixth term is the scalar-boson field interaction.\nHere, we mention that the introduced Feynman rules are not manife stly covariant. Therefore, we must clarify\nwhether or not this can lead to a relativistically invariant Smatrix. In the following, we will focus on testing the\nrelativistic invariance of the model in connection with the specific pro cess of elastic boson-boson scattering. Since we\nare dealing with a non-renormalizable theory, we restrict our calcula tions to the tree approximation.5\nIII. BOSON-BOSON SCATTERING\nAt this point, we will investigatethe Lorentz invarianceofthe model. For this purpose, we will compute the e2order\ncontribution to the lowest order elastic boson-boson scattering a mplitude. Here, due to the non-renormalizability of\nthe model, we will restrict the calculation to the tree-level approxim ation. Hence, for the scattering amplitude, we\nhave seven different types of terms grouped as follows:\nS(2)=7/summationdisplay\nµ=1S(2)\nµ (31)\nwhere\nS(2)\n1=e2\n2/integraldisplay /integraldisplay\nd3xd3y/an}bracketle{tΦf|T{: [ϕI†(x)∂kϕI(x)−∂kϕI†(x)ϕI(x)]fk(x) :\n×: [ϕI†(y)∂jϕI(y)−∂jϕI†(y)ϕI(y)]fj(y) :} |Φi/an}bracketri}ht, (32)\nS(2)\n2=−e2\nm/integraldisplay /integraldisplay\nd3xd3y/an}bracketle{tΦf|T{: [ϕI†(x)∂kϕI(x)−∂kϕI†(x)ϕI(x)]fk(x) :\n×: [ǫlj∂lfj(y)( ˙ϕI†(y)ϕI(y)−ϕI†(y) ˙ϕI(y))] :} |Φi/an}bracketri}ht, (33)\nS(2)\n3=e2\n2m2/integraldisplay /integraldisplay\nd3xd3y/an}bracketle{tΦf|T{: [ǫkj∂kfj(x)( ˙ϕI†(x)ϕI(x)−ϕI†(x) ˙ϕI(x))] :\n×: [ǫil∂ifl(y)( ˙ϕI†(y)ϕI(y)−ϕI†(y) ˙ϕI(y))] :} |Φi/an}bracketri}ht, (34)\nS(2)\n4=−ie2\n2m2/integraldisplay /integraldisplay\nd3xd3yδ(x−y)/an}bracketle{tΦf|T{: [ ˙ϕI†(x)ϕI(x)−ϕI†(x) ˙ϕI(x)] :\n×: [ ˙ϕI†(y)ϕI(y)−ϕI†(y) ˙ϕI(y)] :} |Φi/an}bracketri}ht, (35)\nS(2)\n5=−2e2v0\nm/integraldisplay /integraldisplay\nd3xd3y/an}bracketle{tΦf|T{: [ϕI†(x)∂kϕI(x)−∂kϕI†(x)ϕI(x)]fk(x) :\n×: [ ˙ϕI†(y)ϕI(y)−ϕI†(y) ˙ϕI(y)]φ(y) :} |Φi/an}bracketri}ht, (36)\nS(2)\n6=−2e2v0\nm2/integraldisplay /integraldisplay\nd3xd3y/an}bracketle{tΦf|T{: [ ˙ϕI†(x)ϕI(x)−ϕI†(x) ˙ϕI(x)]φ(x) :\n×: [ ˙ϕI†(y)ϕI(y)−ϕI†(y) ˙ϕI(y)]ǫkj∂kfj(y) :} |Φi/an}bracketri}ht, (37)\nS(2)\n7=e2v2\n0\nm2/integraldisplay /integraldisplay\nd3xd3y/an}bracketle{tΦf|T{: [ ˙ϕI†(x)ϕI(x)−ϕI†(x) ˙ϕI(x)]φ(x) :\n×: [ ˙ϕI†(x)ϕI(y)−ϕI†(y) ˙ϕI(y)]φ(y) :} |Φi/an}bracketri}ht. (38)\nBeing|Φi/an}bracketri}htand/an}bracketle{tΦf|the initial and final state of the process and Tthe chronological ordering operator. Here both\n|Φi/an}bracketri}htand/an}bracketle{tΦf|are two-boson states.\nIn thenext step, wecomputepartialamplitudes( 32)-(38)which intermsofinitial( p1,p2) andfinal ( p′\n1,p′\n2)moments6\nbecome\nS(2)\n1=−e2Np(2π)3δ(p′\n1+p′\n2−p1−p2)/bracketleftbigg\n(p′\n1+p1)j(p′\n2+p2)l1\nm2Djl(k)+p1↔p2/bracketrightbigg\n, (39)\nS(2)\n2=−e2Np(2π)3δ(p′\n1+p′\n2−p1−p2)/bracketleftbigg\n(p′\n1+p1)j(p′\n2+p2)01\nm2Γj(k)\n+ (p′\n1+p1)0(p′\n2+p2)j1\nm2Γj(−k)+p1↔p2/bracketrightbigg\n, (40)\nS(2)\n3=−e2Np(2π)3δ(p′\n1+p′\n2−p1−p2)/bracketleftbigg\n(p′\n1+p1)0(p′\n2+p2)01\nm2Λ(k)+p1↔p2/bracketrightbigg\n, (41)\nS(2)\n4=−e2Np(2π)3δ(p′\n1+p′\n2−p1−p2)/bracketleftbigg\n(p′\n1+p1)0(p′\n2+p2)0i\nm2+p1↔p2/bracketrightbigg\n, (42)\nS(2)\n5=−e2Np(2π)3δ(p′\n1+p′\n2−p1−p2)/bracketleftbigg\n(p′\n1+p1)j(p′\n2+p2)0/parenleftbiggv0vj\nm4/parenrightbigg\n+ (p′\n1+p1)0(p′\n2+p2)j/parenleftbiggv0vj\nm4/parenrightbigg\n+p1↔p2/bracketrightbigg\n, (43)\nS(2)\n6=−e2Np(2π)3δ(p′\n1+p′\n2−p1−p2)/bracketleftbigg\n(p′\n1+p1)0(p′\n2+p2)0/parenleftbiggklvl\nm4/parenrightbigg\n+p1↔p2/bracketrightbigg\n, (44)\nS(2)\n7=−e2Np(2π)3δ(p′\n1+p′\n2−p1−p2)/bracketleftbigg\n(p′\n1+p1)0(p′\n2+p2)0/parenleftbiggv2\n0\nm2k2/parenrightbigg\n+p1↔p2/bracketrightbigg\n, (45)\nwhere we have\n1\nm2Djl(k) =i\nk2−m2+iǫ/parenleftbigg\n−gjl+kjkl\nm2−i\nmǫjlk0−4vjvl\nk2−vjkl\nk2/parenrightbigg\n, (46)\n1\nm2Γj(k) =i\nk2−m2+iǫ/parenleftbiggkjk0\nm2+i\nmǫjlkl−4vjv0\nk2−v0kj\nk2/parenrightbigg\n, (47)\n1\nm2Λ(k) =i\nk2−m2+iǫ/parenleftbigg\n−klkl\nm2−4v0v0\nk2−v0k0\nk2/parenrightbigg\n, (48)\n1\nm2Σj(k) =i\nk2−m2+iǫ/parenleftbigg\n−4v0vj\nk2/parenrightbigg\n, (49)\n1\nm2Q(k) =i\nk2−m2+iǫ/parenleftbigg\n−4vlkl\nk2/parenrightbigg\n, (50)\n1\nm2Φ(k) =i\nk2−m2+iǫ/parenleftbigg\n−4v0v0\nk2/parenrightbigg\n, (51)\nwith\nk≡(p′\n1−p1) =−(p′\n2−p2), (52)\nthe momentum transfer. Now, by replacing equations ( 39)-(45) into (31) we get\nS(2)=−e2Np(2π)3δ(p′\n1+p′\n2−p1−p2)/bracketleftbigg\n(p′\n1+p1)µ(p′\n2+p2)ν1\nm2Dµν(k)+p1↔p2/bracketrightbigg\n, (53)\nbeing\n1\nm2Dµν(k) =−i\nk2−m2+iǫ/parenleftbigg\ngµν−kµkν\nm2+i\nmǫµναkα−4vµvν\nk2−vµkν\nk2/parenrightbigg\n, (54)\nthe propagator of the self-dual field fµ. Note that the scattering amplitude above is a Lorentz scalar and t hus the\nmodel under consideration has passed the test of relativistic invar iance at the quantum level. Furthermore, in the\ntree-level approximation we can replace all non-covariant terms in the Hamiltonian of interactions by a minimum\ncovariant interaction given by:\nHI\nint2=eJI\nµfµ, (55)7\nwhere\nJI\nµ=i(ϕI∗∂µϕI−∂µϕI∗ϕI)−efI\nµϕI∗ϕI−2m\neφvµ, (56)\nis the effective bosonic matter current. Moreover, by taking vµ= 0 (without the effect of breaking the Lorentz\nsymmetry), we have\n1\nm2Dµν(k) =−i\nk2−m2+iǫ/parenleftbigg\ngµν−kµkν\nm2+i\nmǫµναkα/parenrightbigg\n, (57)\nwith\nHI\nint=eJI\nµfµ, (58)\nwhere\nJI\nµ=i(ϕI∗∂µϕI−∂µϕI∗ϕI)−efI\nµϕI∗ϕI. (59)\nNotice that in this way we recover the results found in [ 13].\nIV. DUAL EQUIVALENCE\nIn this section, we study the duality between the self-dual and Max well-Chern-Simons models with Lorentz\nsymmetry breaking coupled with bosonic matter. For this we write th e Lagrangian as follows\n˜L=m\n2ǫµνρ(fµ∂νfρ)−˜m2\n2fµfµ+1\n2∂µφ∂µφ+2mφvµfµ\n+∂µϕ∗∂µϕ−M2ϕ∗ϕ−efµJµ, (60)\nwhere ˜m2=m2−2e2ϕ∗ϕandM2=M2+e2φ2. Note that the mass parameters are field dependent, i.e., ˜ mdepends\non the charged scalar field and Mdepends on the real scalar field.\nEquation of motion for the field fµ\nmǫµνρ(∂νfρ)−˜m2fµ−e˜Jµ= 0, (61)\nwhere we have defined ˜Jµ=Jµ−2e−1mφvµ.\nNow, to determine the Euler vector we apply the following gauge tran sformation to the vector field fµ, such that\nδfµ=∂µǫ, beingǫa parameter of gauge transformations. Thus, by varying the Lag rangian with respect to fµ, we\nfindδ˜L=Kµ∂µ, where\nKµ=mǫµνρ(∂νfρ)−˜m2fµ−e˜Jµ= 0, (62)\nis the Euler vector.\nFrom the Euler-Lagrange equation, we obtain the equations for th e fieldsφandϕ, given respectively by\n∂µ∂µφ= 2mvµfµ, (63)\nand\n(∂µ∂µ+M2)ϕ= 2iefµ∂µϕ. (64)\nThe next step is to establish duality between the self-dual and Maxw ell-Chern-Simons models with Lorentz symmetry\nbreaking coupled with bosonic matter. Thus, we proceed by adoptin g the gauge embedding approach as done\nin [14,19,20]. In this way, we add to the original Lagrangian a term of the form F(Kµ), as follows:\n˜L →˜L+F(Kµ), (65)\nbeingF(0) = 0. Now to obtain the form of the function F(Kµ), we define\nL(1)=˜L+BµKµ, (66)8\nwhereBµactsasaLagrangemultiplier. Inaddition, weassumethat δBµ=∂µǫtocancelthevariationof ˜L. Therefore,\nproceeding as in Refs. [ 14,19,20] and after eliminating Bµ, we find the following gauge invariant effective Lagrangian\nLeff=˜L+1\n2˜m2KµKµ. (67)\nHence, renaming fµ→Aµto show the invariant character of the theory, we have\nLeff=1\n2FµFµ−m\n2ǫµνρAµ∂νAρ+1\n2∂µφ∂µφ+∂µϕ∗∂µϕ−M2ϕ∗ϕ−em\n˜m2˜JµFµ+e2\n2˜m2˜Jµ˜Jµ,(68)\nbeing,\nFµ≡1\n2ǫµνρFνρ, (69)\nthe dual of the tensor Fνρ. By replacing ˜Jµwe obtain\nLeff=1\n2FµFµ−m\n2ǫµνρAµ∂νAρ+1\n2∂µφ∂µφ+2m2\n˜m2φ2vµvµ+∂µϕ∗∂µϕ−M2ϕ∗ϕ\n−em\n˜m2JµFµ+2m2φ\n˜m2vµFµ+e2\n2˜m2/bracketleftbigg\nJµ−4m\neφvµ/bracketrightbigg\nJµ. (70)\nNote that a mass term for the field φhas been generated, which has also been previously obtained by the quantization\nprocedure.\nFor this model, the equations for the φfield read\n∂µ∂µφ= 2mvµ/bracketleftbiggm\n˜m2Fµ+2m\n˜m2φvµ−e\n˜m2Jµ/bracketrightbigg\n, (71)\nand for the ϕfield, we have\n/parenleftbig\n∂µ∂µ+M2/parenrightbig\nϕ=−2ie/bracketleftbiggm\n˜m2Fµ+2m\n˜m2φvµ−e\n˜m2Jµ/bracketrightbigg\n∂µϕ. (72)\nFinally, by comparing equations ( 63), (64), (71) and (72), we obtain that the correct map between the self-dual model\nand the Maxwell-Chern-Simons is given by\nfµ→m2\n˜m2/parenleftbiggFµ\nm+2φ\nmvµ−e\nm2Jµ/parenrightbigg\n. (73)\nIn this way we have successfully completed the dual mapping betwee n the two models. The mapping obtained here\nis a generalization of the results obtained in [ 14] and [19]. By setting vµ= 0 (in the absence of Lorentz symmetry\nbreaking) we will recover the results obtained in [ 14].\nfµ→m2\n˜m2/parenleftbiggFµ\nm−e\nm2Jµ/parenrightbigg\n. (74)\nOn the other hand, without the presence of the field ϕ, but maintaining the effect of breaking the Lorentz symmetry,\nwe have the result found in [ 19].\nfµ→/parenleftbiggFµ\nm+2φ\nmvµ/parenrightbigg\n. (75)\nV. SUMMARY\nIn this work, we have considered the dynamics, at the quantum leve l, of the self-dual model coupled to Lorentz\nsymmetry-breaking bosons. Then, after eliminating the degree of freedom f0the equal-time commutators with the\nbosonic fields that are not zero, we have formulated the dynamics o f the interaction framework for the self-dual model\nminimally coupled to the bosons by breaking the Lorentz symmetry. A s a consequence, the Hamiltonian formulation\ndid not retain any relics of relativistic covariance. In addition, we hav e shown that a mass term is generated for\nthe real scalar field φ. However, we prove that the non-covariant parts in HI\nint2are equivalent to the minimum\ncovariant field-current interaction. 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Introduction\nThe Standard-Model Extension (SME) provides a general framew ork for\ntheoretical and experimental studies of Lorentz and CPT violation .1Most\nearly researchfocused on the minimal SME (mSME), which restricts atten-\ntion to operators of renormalizable dimensions d= 3 and 4 in flat space-\ntime.2However, the full SME encompasses curved spacetime3and includes\noperators of arbitrary dimension. Here, we give a brief discussion o f recent\nefforts to classify terms in the photon and neutrino sectors of the SME, in-\ncluding those of nonrenormalizable dimensions d≥5. Details can be found\nin Refs. 4 and 5. A summary of experimental results is given in Ref. 1.\n2. Photons\nThe pure photon sector of the SME is given by the lagrangian4\nL=−1\n4FµνFµν+1\n2ǫκλµνAλ(ˆkAF)κFµν−1\n4Fκλ(ˆkF)κλµνFµν.(1)\nIn addition to the usual Maxwell term, there are two Lorentz-viola ting\nterms involving the CPT-odd four-vector ( ˆkAF)κand CPT-even tensor\n(ˆkF)κλµν. Both (ˆkAF)κand (ˆkF)κλµνare constants in the mSME but are\nmomentum dependent in the full SME, where they each take the for m of a\npower series in photon momentum. The expansion constants in the s eries\ngive coefficients for Lorentz violation.\nThe total number of coefficients for Lorentz violation that appear in\nthe expansions of ( ˆkAF)κand (ˆkF)κλµνgrows rapidly ( ∼d3) as we con-\nsider higher dimensions. To aid in classifying the numerous coefficients , aProceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)\n2\nspherical-harmonic expansion of ( ˆkAF)κand (ˆkF)κλµνis performed, giv-\ning sets of spherical coefficients for Lorentz violation that can be t ested by\ndifferent types of experiment. Several classes of experiment pro vide high\nsensitivity to various subsets of the coefficients, including searche s for as-\ntrophysical birefringence and dispersion, and resonant-cavity t ests. Below\nis a summary of coefficients that have been experimentally tested th us far.\nBirefringent vacuum coefficients. The coefficients k(d)\n(E)jm,k(d)\n(B)jm, and\nk(d)\n(V)jmlead to birefringence of light propagating in vacuo. The result is a\nchange in polarization as light propagates. The effect can depend on the\ndirection of propagation and photon energy. Polarimetry of radiat ion from\ndistant astrophysical sources has led to many constraints on the mSME\ncoefficients6,7and nonminimal coefficients up to dimension d= 9.4,8\nNonbirefringent vacuum coefficients. The coefficients c(d)\n(I)jmaffect the\npropagation of light in a polarization-independent way, implying no bire -\nfringence. However, they do lead to vacuum dispersion for d≥6. Time-of-\nflight tests involving high-energy sources, such as γ-ray bursts, search for\ndifferencesin arrivaltimes ofphotonsat differentenergies.Again, this effect\ncan depend on the photon energy and direction of propagation. Co nstraints\non thec(d)\n(I)jmcoefficients for d= 6 and 8 have been found.4,7,9\nCamouflage coefficients. A large class of Lorentz violation has no ef-\nfect on the propagation of light. These violations are referred to a s vacuum\northogonal. Among them are the so-called camouflage violations, wh ich\ngenerically give polarization-independent effects, in addition to giving con-\nventional light propagation. As a result, the effects of the associa ted cam-\nouflage coefficients ( c¬(d)\nF)(0E)\nnjmare particularly subtle. They can, however,\nbe tested in resonant-cavity experiments, where they lead to tiny shifts in\nresonant frequencies.4,10Numerous cavity searches for d= 4 coefficients\nhave been performed.11A recent experiment placed the first cavity bounds\non nonminimal d= 6 and 8 camouflage coefficients.12\n3. Neutrinos\nNeutrinos in the SME are governed by a 6 ×6 effective hamiltonian that\nacts on the six-dimensional space that includes both neutrinos and antineu-\ntrinos. The Lorentz-violating part of the hamiltonian takes the for m5\nδheff=1\n|/vector p|/parenleftBigg\n/hatwideaeff−/hatwideceff−/hatwidegeff+/hatwideHeff\n−/hatwideg†\neff+/hatwideH†\neff−/hatwideaT\neff−/hatwidecT\neff/parenrightBigg\n. (2)Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)\n3\nThe 3×3 matrices/hatwideaeff,/hatwideceff,/hatwidegeff, and/hatwideHeffare functions of the neutrino\nmomentum vector /vector p, giving rise to unusual direction dependence and an\nunconventional energy-momentum relation. Note that the /hatwidegeff, and/hatwideHeff\nterms cause mixing between neutrinos and antineutrinos. The /hatwideceffand/hatwideHeff\nmatrices contain the CPT-even violations, while the CPT-odd terms a re\nin/hatwideaeffand/hatwidegeff. As with photons, a spherical-harmonic expansion is used\nto enumerate and classify the various effects and coefficients. For example,\nthe/hatwideaeffmatrix has the expansion ( /hatwideaeff)ab=/summationtext|/vector p|d−2Yjm(ˆp)/parenleftbig\na(d)\neff/parenrightbigab\njm, giv-\ning spherical coefficients for Lorentz violation/parenleftbig\na(d)\neff/parenrightbigab\njm. Most research in\nneutrinos, so far, focuses on one of the following cases.\nOscillation coefficients. Except for a small subset of flavor-diagonal co-\nefficients, most of the spherical coefficients,/parenleftbig\na(d)\neff/parenrightbigab\njm,/parenleftbig\nc(d)\neff/parenrightbigab\njm,/parenleftbig\ng(d)\neff/parenrightbigab\njm,\nand/parenleftbig\nH(d)\neff/parenrightbigab\njm, produce oscillations. Signatures of Lorentz violation in os-\ncillations include direction dependence, unconventional energy dep endence,\nneutrino-antineutrino oscillations, and CPT asymmetries. Neutrino oscilla-\ntions are interferometric in nature, so high sensitivity to Lorentz v iolation\nis possible. While some bounds on coefficients up to d= 10 have been de-\nduced from earlier analyses,5most of the research so far has focused on the\nd= 3 and 4 cases,13with many constraints on mSME coefficients.14\nOscillation-free coefficients. Lorentzviolationalsoaffectsthe kinematics\nof neutrino propagation. Kinematic effects are characterized, ind ependent\nof oscillations, in the simple oscillation-free limit, where oscillations are\nneglected, and all neutrinos are treated the same. The energy in t his limit\nis given by\nE=|/vector p|+|ml|2\n2|/vector p|+/summationdisplay\n|/vector p|d−3Yjm(ˆp)/bracketleftBig/parenleftbig\na(d)\nof/parenrightbig\njm−/parenleftbig\nc(d)\nof/parenrightbig\njm/bracketrightBig\n,(3)\nwhere/parenleftbig\na(d)\nof/parenrightbig\njmand/parenleftbig\nc(d)\nof/parenrightbig\njmare oscillation-free coefficients. This provides\na framework for a range of studies, such as time-of-flight tests, analyses\nof meson-decay thresholds, and ˇCerenkov-like decays of neutrinos, each of\nwhich has produced constraints on coefficients up to d= 10.5,15\nReferences\n1.Data Tables for Lorentz and CPT Violation, V.A. Kosteleck´ y and N. Russell,\n2013 edition, arXiv:0801.0287v6.\n2. D. Colladay and V.A. Kosteleck´ y, Phys. Rev. D 55, 6760 (1997); Phys. Rev.\nD58, 116002 (1998).\n3. V.A. Kosteleck´ y, Phys. Rev. D 69, 105009 (2004); Q.G. Bailey and V.A.\nKosteleck´ y, Phys. Rev. D 74, 045001 (2006); V.A. Kosteleck´ y and J.D. Tas-\nson, Phys. Rev. Lett. 102, 010402 (2009); Phys. Rev. D 83, 016013 (2011).Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)\n4\n4. V.A. Kosteleck´ y and M. Mewes, Phys. Rev. D 80, 015020 (2009).\n5. V.A. Kosteleck´ y and M. Mewes, Phys. Rev. D 85, 096005 (2012).\n6. S.M. Carroll et al., Phys. Rev. D 41, 1231 (1990); V.A. Kosteleck´ y and M.\nMewes, Phys. Rev. Lett. 87, 251304 (2001); Phys. Rev. Lett. 97, 140401\n(2006); Phys. Rev. Lett. 99, 011601 (2007); T. Kahniashvili et al., Phys.\nRev. D78, 123009 (2008); E.Y.S. Wu et al., Phys. Rev. Lett. 102, 161302\n(2009); Z. Xiao and B.-Q. Ma, Phys. Rev. D 80, 116005 (2009); G. Gubitosi\net al., JCAP08, 021 (2009); E. Komatsu et al., Ap. J. Suppl. 192, 18 (2011);\nQ. Exirifard, Phys. Lett. B 699, 1 (2011).\n7. V.A. Kosteleck´ y and M. Mewes, Ap. J. 689, L1 (2008).\n8. F.W. Stecker, Astropart. Phys. 35, 95 (2011); V.A. Kosteleck´ y and M.\nMewes, Phys. Rev. Lett. 110, 201601 (2013).\n9. V. Vasileiou, arXiv:1008.2913; V. Vasileiou et al., Phys. Rev. D 87, 122001\n(2013).\n10. V.A. Kosteleck´ y and M. Mewes, Phys. Rev. D 66, 056005 (2002); M. Mewes\nand A. Petroff, Phys. Rev. D 75, 056002 (2007); M. Mewes, Phys. Rev. D\n78, 096008 (2008); Q. Exirifard, arXiv:1010.2057; M. Mewes, P hys. Rev. D\n85, 116012 (2012).\n11. J. Lipa et al., Phys. Rev. Lett. 90, 060403 (2003); H. M¨ uller et al., Phys.\nRev. Lett. 91, 020401 (2003); P. Wolf et al., Gen. Rel. Grav. 36, 2351 (2004);\nP. Wolfet al., Phys. Rev. D 70, 051902 (2004); P. Antonini et al., Phys. Rev.\nA71, 050101 (2005); Phys. Rev. A 72, 066102 (2005); P.L. Stanwix et al.,\nPhys. Rev. Lett. 95, 040404 (2005); S. Herrmann et al., Phys. Rev. Lett.\n95, 150401 (2005); M.E. Tobar et al., Phys. Rev. A 72, 066101 (2005); P.L.\nStanwix et al., Phys. Rev. D 74, 081101 (2006); H. M¨ uller et al., Phys. Rev.\nLett.99, 050401 (2007); Ch. Eisele et al., Phys. Rev. Lett. 103, 090401\n(2009); S. Herrmann et al., Phys. Rev. D 80, 105011 (2009); M.A. Hohensee\net al., Phys. Rev. D 82, 076001 (2010); F. Baynes et al., Phys. Rev. D 84,\n081101 (2011); F.N. Baynes et al., Phys. Rev. Lett. 108, 260801 (2012); Y.\nMichimura et al., Phys. Rev. Lett. 110, 200401 (2013).\n12. S. Parker et al., Phys. Rev. Lett. 106, 180401 (2011).\n13. V.A. Kosteleck´ y and M. Mewes, Phys. Rev. D 69, 016005 (2004); Phys. Rev.\nD70, 031902 (2004); Phys. Rev. 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D 87, 096004 (2013)."    },    {        "title": "0712.2055v1.One_Loop_Renormalization_of_QCD_with_Lorentz_Violation.pdf",        "content": "arXiv:0712.2055v1  [hep-ph]  12 Dec 2007NCF/002\nOne-Loop Renormalization of QCD with Lorentz Violation\nDon Colladay∗and Patrick McDonald†\nNew College of Florida\nSarasota, FL, 34243, U.S.A.\n(Dated: October 30, 2018)\nThe explicit one-loop renormalizability of the gluon secto r of QCD with Lorentz violation is\ndemonstrated. The result is consistent with multiplicativ e renormalization as the required counter\nterms are consistent with a single re-scaling of the Lorentz -violation parameters. In addition, the\nresulting beta functions indicate that the CPT-even Lorent z-violating terms increase with energy\nscale in opposition to the asymptotically free gauge coupli ng and CPT-odd terms. The calculations\nare performed at lowest-order in the Lorentz-violating ter ms as they are assumed small.\nPACS numbers: Valid PACS appear here\nI. INTRODUCTION\nAs is well known, the Standard Model is defined by a\nLagrangianwhich exhibits ultraviolet divergences arising\nfromthestructureofthetheoryatsmalldistances. These\ndivergences can be removed by a singular redefinition of\nthe parameters defining the theory through the process\nof renormalization. The required calculations involve a\nnumberofremarkablecancellationswhicharemosteasily\nobtained by exploiting the varioussymmetries of the the-\nory. The symmetries ofthe conventionalStandardModel\ninclude the Lorentz group.\nThe investigation of the renormalizability properties\nof the SM in the presence of Lorentz violation began in\n[1, 2] where the authors studied one-loop radiative cor-\nrection for QED with Lorentz violation. These calcula-\ntionswerecarriedoutin the frameworkofanexplicitthe-\nory called the Standard Model Extension (SME), which\nhasbeen formulated toinclude possible Lorentz-violating\nbackground couplings to Standard Model fields [3, 4].\nIn this theory, the one-loop renormalizability of general\nLorentz and CPT violating QED has been established\n[1]. The manuscript [1] includes an analysis of the ex-\nplicit one-loop structure of Lorentz-violating QED and\nthe resulting running of the couplings. The authors es-\ntablish that conventional multiplicative renormalization\nsucceeds and they find that the beta functions indicate\na variety of running behaviors, all controlled by the run-\nningofthe charge. Portionsofthisanalysishavebeen ex-\ntended to allow for a curved-space background [5], while\notheranalysisinvolvedfinite, butundetermined radiative\ncorrections due to CPT violation [6, 7, 8, 9, 10, 11, 12].\nTheLorentzviolatingQEDresultsof[1]wereextended\nto non-abelian gauge theories in [13] where the authors\nestablished that the associated pure Yang-Mills theory\nis renormalizable at one-loop. More precisely, conven-\ntional multiplicative renormalization succeeds and the\nbeta functions indicate that CPT-even Lorentz violat-\n∗Electronic address: colladay@ncf.edu\n†Electronic address: mcdonald@ncf.eduing terms increase with energy scales in opposition to\nthe asymptotically free gauge couplings and the CPT-\nodd couplings. The primary purpose of this paper is to\nextend the results of [13] to include fermions as well as\nto remove certain technical restrictions on the trace of\nthe CPT-even terms. The results are directly applica-\nble to one-loop renormalizability of the gluon sector of\nQCD in the presence of Lorentz violation. In addition,\nthe running of the couplings are studied using the associ-\natedbetafunctions. Newmethodsforcarryingoutrenor-\nmalization calculations inside the SME using functional\ndeterminants are provided, giving a second, alternative\nderivation of the results presented in [13] .\nThis work should be viewed as part of an extensive,\nsystematic investigation of Lorentz violation and its pos-\nsible implications for Planck-scale physics [14, 15, 16,\n17, 18, 19, 20, 21, 22]. Extensive calculations using the\nSME have led to numerous experiments (see, for exam-\nple [23]), which have in turn placed stringent bounds\non parameters in the theory associated with electrons,\nphotons, neutrinos, and hadrons. Recent work involv-\ning Lorentz violation and cosmic microwave background\ndata[24]suggestthat the SMEmightplayauseful rolein\ncosmology. In addition to the above, the SME formalism\nhas been extended to include gravity [25, 26, 27], where\nit has been suggested that Lorentz violation provides an\nalternative means of generating General Relativity [28].\nSome other related work includes a study of deformed\ninstantons in the theory [29, 30], an analysis of the\nCasimireffectinthepresenceofLorentzviolation[31], an\nanalysis of gauge invariance of Lorentz-violating QED at\nhigher-orders[32], and possible effects due to nonpolyno-\nmial interactions [33]. Some investigations into possible\nLorentz-violation induced from the ghost sector of scalar\nQED have also been performed [34]. More recently, func-\ntional determinants have been used to compute finite\ncorrections to CPT-violating gauge terms arising from\nfermion violation [35].2\nII. NOTATION AND CONVENTIONS\nTo simplify notation we limit our investigation to the\ncase of a single fermion. The associated Lagrangian with\nLorentz violation is taken to be\nL=LA+Lψ+LG, (1)\nwhereLAis the gauge field contribution, Lψis the\nfermionic contribution, and LGis the ghost contribu-\ntion. In computing the UV divergence, we treat each\nterm in the Lagrangian separately. We begin with the\npure Yang-Mills contribution [3, 4]\nLA=−1\n2tr/bracketleftbig\nFµνFµν+(kF)µναβFµνFαβ+\n(kAF)κǫκλµν(AλFµν−2\n3igAλAµAν)+2λF[A]2/bracketrightbig\n,(2)\nwhere (kF)µναβand (kAF)κare tensors governing the\nLorentz violation in the Yang-Mills sector. For the Stan-\ndard Model Extension the tensor kFis CPT-even, sat-\nisfies a Jacobi identity and is constrained to have the\nsymmetries of the Riemann tensor, while the tensor kAF\nis CPT-odd. The parameter λmultiplies a gauge fixing\ntermF. The generators of the Lie Algebra defined by\nAµ=Aaµtaare taken to satisfy\n[ta,tb] =ifabctc, (3)\nandfabcare totally anti-symmetric structure constants.\nThe product of these generators is normalized to\ntr[tatb] =C(r)δab, (4)\nwhereC(r) depends on the representation r. In the ad-\njoint representationused for the gauge fields, this is writ-\ntenC(G) =C2(G) whereC2(r) is the quadratic Casimir\noperator\ntata=C2(r)·1. (5)\nThe field tensor is defined as\nFµν=−i\ng[Dµ,Dν], (6)\nwhere the covariant derivative is Dµ=∂µ+igAµ.\nThe fermionic contribution contribution to the La-\ngrangian is given by [3, 4]\nLψ=¯ψ(iΓµDµ−M)ψ , (7)\nwhere Γν=γν+Γν\n1, M=m+M1,and Γ1andM1are\nof the form\nΓν\n1=cνµγµ+dµνγ5γµ+eν+ifνγ5+1\n2gλµνσλµ(8)\nM1=aµγµ+bµγ5γµ+1\n2Hµνσµν. (9)\nHere theγµare the standard gamma matrices, σλµare\nthestandardsigmamatrices,andtheremainingsmallpa-\nrameters control Lorentz violation. The parameters cνµanddνµare traceless, Hµνis antisymmetric, and gλµνis\nantisymmetric in the first two components. The param-\netersaµ, bµ, Hµν,have the dimension of mass, while the\nremaining parameters are dimensionless.\nFinally, the ghost Lagrangianis written in terms of the\nscalar, anticommuting field φ\nLG=−φ(M−CµνDµDν)φ , (10)\nwhereMis the variation of the gauge fixing functional\nFwith respect to the gauge transformation and the con-\nstantsCµνparameterize possible Lorentz violation in the\nghost sector [34].\nIII. FUNCTIONAL DETERMINANTS AND\nBACKGROUND FIELDS\nRecall, the one-loop effective action for the theory can\nbe written as a functional integral over fields Ψ:\nexpiΓ[Ψ] =/integraldisplay\nDΨeiR\nd4xL[Ψ]. (11)\nThe effective action is constructed by writing the under-\nlying fields as the sum of a classical background and a\nfluctuating quantum field. The effective action is given\nby a classical term perturbed by terms quadratic in the\nfluctuation. The quadratic term gives rise to a Gaussian\nintegral, which in turn can be described by a functional\ndeterminant [36]. Using Lcl=L0+Lc.t.for the classical\nLagrangian as a function of the background field where\nLc.t.is the counterterm Lagrangian, the expression be-\ncomes\nexpiΓ[Ψ] =eiR\nd4xLcldet(∆A)−1\n2det(∆ψ)1\n2det(∆φ),\n(12)\nwhere the ∆ are operators which are given explicitly be-\nlow. To compute the above determinants, dimensional\nregularization is used. Each determinant is treated sep-\narately, beginning with the pure Yang-Mills gauge field\ncontribution. The calculation is performed to first or-\nder in Lorentz violating parameters. As this is the case,\nthe computations of the various terms decouple and the\nCPT-even and CPT-odd cases can be treated indepen-\ndently.\nWe will write the gauge fields as the sum of a classi-\ncal background field (denoted with an underline) and a\nfluctuating quantum field:\nAµ=Aµ+Aµ. (13)\nWith this convention the curvature can be expressed as\nFaµν=Faµν+(DµAν)a−(DνAµ)a\n−gfabcAbµAcν, (14)\nwhere the underline denotes background curvature and\nthe covariant derivatives are taken with respect to the3\nbackground fields. The gauge fixing functional is chosen\nto be\nF[A] =DµAµ, (15)\nandλis set equal to 1 incorporating Feynman gauge.\nRescaling the vector potential to absorb g, substituting\n(14) into (2) and retaining terms which are quadratic in\nthe perturbation of the background field we have\nLquad\nA=−1\n2g2trAµ/bracketleftbig\n−gµνD2−2iFµν\n−i(kF)αβµνFαβ−2(kF)µανβDαDβ\n−(kAF)κǫκλµνDλ/bracketrightBig\nAν.(16)\nThe trace is performed over the Lie Algebra indices and\nall fields are written in the adjoint representation. The\ntrace is extended to cover the Lorentz indices as well\nusing the matrix notations\n(ταβ)µν=i(gαµgβν−gανgβµ) (17)\n(ǫαβ)µν=ǫαβµν (18)\n(kI\nFαβ)µν= (kF)αβµν (19)\n(kII\nFαβ)µν= (kF)µανβ, (20)\nthe quadratic Lagrangian can be rewritten as\nLquad\nA=−1\n2g2trA/bracketleftBig\n(−gαβ−2kII\nFαβ)DαDβ\n−/parenleftbig\nταβ+ikI\nFαβ/parenrightbig\nFαβ−(kAF)αǫαβDβ/bracketrightBig\nA\n=−1\n2g2trA∆AA, (21)\nwherethetraceisperformedoverbothLorentzandgauge\nspaces, ∆ A=PA+ ∆(1)\nA+ ∆(2)\nA+ ∆(F)\nA,∆(i)\nAis orderi\nin the fields ( i= 1,2), and ∆(F)\nAcontains all curvature\ncontributions:\nPA=−(gαβ+2kII\nFαβ)∂α∂β−kα\nAFǫαβ∂β\n∆(1)\nA=−i(gαβ+2kII\nFαβ)(∂αAβ+Aα∂β)−ikα\nAFǫαβAβ\n∆(2)\nA= (gαβ+2kI\nFαβ)(AαAβ)\n∆(F)\nA=−[ταβ+ikI\nFαβ]Fαβ. (22)\nWecomputeanexpansionforlogdet(∆ A)retainingterms\nwhich are linear in the small parameters kFandkAF.\nlogdet(PA−1∆A) = logdet/bracketleftBig\n1+P−1\nA(∆(1)\nA+∆(2)\nA+∆(F)\nA)/bracketrightBig\n.\n(23)\nNote thatPAis actually independent of the backgroud\nfield and will cancel out of the functional determinant\nwith proper normalization. The determinant is writ-\nten in terms of the logarithm function using the relation\nlogdetS= TrlogS. The logarithm is then expanded as\nTrlog(P−1\nA∆A) = Tr(P−1\nA(∆(1)\nA+∆(2)\nA+∆(F)\nA))\n−1\n2Tr((P−1\nA(∆(1)\nA+∆(2)\nA+∆(F)\nA))2)+ h.o. (24)The analysis of the first term appearing in the expansion\nis straightforward: Since the Lie algebra elements trace\nto zero, the first term reduces to a quadratic divergence:\nTr(P−1\nA(∆(1)\nA+∆(2)\nA+∆(F)\nA)) = Tr(P−1\nA∆(2)\nA)\n=tr/integraldisplay\nd4k\n(2π)4d4p\n(2π)41\np2(gµν+2(kII\nF)µν)Aµ(k)Aν(−k).\n(25)\nwhere, as before, the trace refers to both Lorentz and\ngauge space. To calculate the contribution which arises\nfrom the second order terms in the expansion, note that\ntrace considerations immediately reduce the problem to\nstudying the contribution arising from terms of the form\nP−1\nA∆(1)\nAP−1\nA∆(1)\nAandP−1\nA∆(F)\nAP−1\nA∆(F)\nA.The first of the\nabove terms produces a quadratic divergence that ex-\nactly cancels the quadratic divergence arising from the\nfirst order terms. A lengthy computation employing di-\nmensionalregularizationgivesthe totalLorentz-violating\ndivergent contribution as\nlogdet(P−1\nA∆A) =i\n(4π)2Γ(2−d\n2)tr/integraldisplay\nd4k\n(2π)4\n/bracketleftbig7\n3kλ\nFµλν(kµkνA2−2kµAνk·A+k2AµAν)\n−(12)kFµανβ(kαkβAµAν)/bracketrightbig\n.(26)\nNote that the trace over Lorentz indices has been per-\nformed in the above expression and only the gauge space\ntrace remains. The contribution from the Lorentz violat-\ning CPT-odd terms is finite: there is no corresponding\nUV divergence. This calculation confirms the results ob-\ntained in [13].\nWe can analyze the contribution arising due to Lψin\na similar manner. The mass-term Mdoes not contribute\nany divergences to lowest order, so it is omitted from the\ncalculation. The contribution from the kinetic piece is\nsquared to facilitate computation\n−(ΓµDµ)2=−Pψ(1−P−1\nψ(∆(1)\nψ+∆(2)\nψ+∆(F)\nψ)),(27)\nwhere\nPψ= (gαβ+{γα,Γβ\n1})∂α∂β (28)\n∆(1)\nψ=−i(gαβ+{γα,Γβ\n1})(∂αAβ+Aα∂β) (29)\n∆(2)\nψ= (gαβ+{γα,Γβ\n1})AαAβ (30)\n∆(F)\nψ=−(Sαβ+1\n2[γα,Γβ\n1])Fαβ, (31)\nwhereγare the standard gamma-matrices, Γ 1is as de-\nfined in (8), and Sαβare the generators for the Lorentz\ntransformations in the spinor representation. With these\nadjustments, the expansion (24) holds for the case of Lψ.\nSymmetry considerations imply that, to lowest order in\nthe parameters, terms involving parameters other than\ncµνandbµdo not contribute to the UV divergences in\nthe pure Yang-Millssector. Similarly, explicit calculation\nconfirms that terms involving bµdo not contribute to4\nUV divergences. Thus, for the purpose of computing UV\ndivergences arising from the Lorentz violating fermionic\nLagrangiangivenby(8) and (9), onlythe terms involving\ncµνare relevant, and we can proceed with the computa-\ntion assuming that Γµ\n1is replaced by cµνγνandM1is\nreplaced by 0 in (8) and (9), respectively, and that the\nassociated simplifications are made in (28)-(31). With\nthese simplifications, we proceed as we did in the case of\npure Yang-Mills. The analysis of the first order contri-\nbution leads to a quadratic divergence. More precisely,\nsince ∆(1)\nψis linear in the fields and the ∆(F)\nψterm will\ntrace to zero, we have the Lorentz-violating contribution\nTr(P−1\nψ(∆(1)\nψ+∆(2)\nψ+∆(F)\nψ)) =\ntr/integraldisplay\nd4k\n(2π)4d4p\n(2π)41\np2(2cµν)Aµ(k)Aν(−k).(32)\nUsing dimensional regularization and following the same\ncalculation as was carried out for pure Yang-Mills, the\nquadratic piece of the divergence is exactly cancelled by\na term arising from a second order contribution. The\ntotal contribution associated to the second term in the\nexpansion (24)is\nlogdet(P−1\nψ∆ψ) =−i\n3C(r)\n(4π)2Γ(2−d\n2)cµνQµν.(33)\nwhereC(r) is given by the relation tr(ta\nrtb\nr) =C(r)δab\nandrrefers to the fermion representation. The other\nnew factor is defined as\nQµν=/integraldisplay\nd4k\n(2π)4(kµkνA2−2kµAνk·A+k2AµAν).(34)\nTo treat the ghost we write the quadratic contribution\nLG=−φ(−DµDµ−CµνDµDν)φ , (35)\nand express the functional determinant using\nPφ= (−∂2−Cµν∂µ∂ν) (36)\n∆(1)\nφ=−i(gµν+Cµν)(Aµ∂ν+∂µAν) (37)\n∆(2)\nφ= (gµν+Cµν)(AµAν). (38)\nComputing as above and using the notation introduced\nin (34), the ghost contribution is given by\nlogdet(P−1\nφ∆φ) =−i\n6C2(G)\n(4π)2Γ(2−d\n2)CµνQµν.(39)\nIV. RENORMALIZATION FACTORS\nIn this section we complete the explicit one-loop renor-\nmalizability calculuation. We begin by taking logarithms\nof the expression (12) and substituting for the resulting\ndivergencesusing (26), (33), and(39). Totreatthe diver-\ngencesarisingfromthe pure Yang-Millsterm, weproceed\nby noting that kFcan be treated as the sum of selfd-\nual and anti-selfdual parts [37]. Noting that the selfdualcontribution is trace-free, we match the structure of the\nLagrangian to the structure of the corresponding singu-\nlarity in the expression of the divergence (26). The sum\nof corresponding terms is given by\nL0+δL=−1\n4g2/parenleftBig\n1−6g2\n(4π)2Γ(2−d\n2)/parenrightBig\n(kF)µναβFµνFαβ.\n(40)\nRescaling the bare parameters gandkFvia\ngb=Zggr (41)\n(kF)b=ZkF(kF)r, (42)\nleads to\n(kF)r\ng2r=Z2\ng\nZkF(kF)b\ng2\nb. (43)\nThe calculation of Zgproduces the same result as the\nstandard calculation for renormalizability of standard\nYang-Mills\nZg= 1−g2\n(4π)2Γ(2−d\n2)(11\n6C2(G)−2\n3nfC(r)),(44)\nwherenfis the number of fermion species assumed to\nall be in representation r. WhenkFis selfdual (that is,\nwhen (kF)µναβ=1\n4ǫµνλκ(kF)λκρσǫρσαβ,see [37]). This\nleads immediately to the scaling for kF:\nZkF= 1+g2\n(4π)2Γ(2−d\n2)(7\n3C2(G)+4\n3nfC(r)),(45)\nwhich coincides with the trace-free result given in [13] in\nthe absence of fermions.\nFor the anti-selfdual contribution, we note that kµναβ\nF\ncan be written in terms of Λµν=1\n2kµαν\nFα[37]:\nkµναβ\nF= Λ[µ[αgν]β]. (46)\nIn the absence of fermions and ghosts, term matching\nleads to the expression\nL0+δL=−1\ng2/parenleftBig\n1−11\n3C2(G)g2\n(4π)2Γ(2−d\n2)/parenrightBig\nΛµνQµν,(47)\nwhereQµνis as given in (34). Given the rescaling of g\nin Eq.(44), we see that in this case Λ (and hence kF) is\nunaffected byrenormalizationdue tothe pure Yang-Mills\nsector. Only the fermions and ghosts contribute.\nAdding a fermion and the ghosts we use (33) and (39)\nand match terms to obtain\nL0+δL=−1\ng2r/parenleftBig\n1−4\n3g2\n(4π)2Γ(2−d\n2)nfC(r)/parenrightBig\n⊗/bracketleftBig\nΛµν+1\n6g2\n(4π)2Γ(2−d\n2)(C(r)cµν+C2(G)Cµν)/bracketrightBig\nQµν,\n(48)\nwithnf= 1 for one fermion. Defining the renormalized\nΛ parameter using\nL0+δL=−1\ng2rΛµν\nrQµν (49)5\nand defining\nΛµν\nb= (ZΛ)µν\nαβΛαβ\nr (50)\nyields the relationship\n(ZΛ)µν\nαβ/bracketleftBig\nΛαβ\nb+1\n6g2\n(4π)2Γ(2−d\n2)/parenleftbig\nC(r)cαβ+C2(G)Cαβ/parenrightbig/bracketrightBig\n=ZSΛµν\nb, (51)\nwith\nZS=/parenleftBig\n1+4\n3g2\n(4π)2Γ(2−d\n2)nfC(r)/parenrightBig\n.(52)\nFinally, incorporating terms for arbitrary fermions is\nstraightforward as the contributions simply add. This\nis accomplished in the above formula by letting nfbe\narbitrary and making the replacement c→/summationtext\nfcfas a\nsum over fermion species.\nThe CPT-odd terms kAFcontain no divergent contri-\nbutions, however, they still need to be renormalized as\nthe combination kAF/g2appears in the classical action.\nThis means that ZkAF=Z2\ngat one-loop, in agreement\nwith the result found in [13].\nV. BETA FUNCTIONS\nTacitly assuming for the moment that our renormal-\nization prescription can be extended to all orders, the\nrenormalization constants ZkFandZkAFcan be used to\ndeduce the one-loop beta functions for these parameters.\nFollowing the developments presented in [1], use is made\nof\nβxj= lim\nǫ→0/bracketleftBigg\n−ρxjaj\n1+N/summationdisplay\nk=1ρxkxk∂aj\n1\n∂xk/bracketrightBigg\n,(53)\nwherexjrepresents an arbitrary running coupling in the\ntheory, the parameters ρxjare determined by comparing\nthe mass dimension of the renormalized parameters to\nthe bare parameters in d= 4−2ǫdimensions, and the\naj\n1represent the first order divergent contribution to the\nrescaling factor associated to the variable xj.In more\ndetail, writing\nxjb=µρxjǫZxjxj. (54)\ngives the values\nρg= 1, ρkF=ρkAF= 0, (55)\nand theaj\n1are defined by the expansion\nZxjxj=xj+∞/summationdisplay\nn=1aj\nn\nǫn. (56)\nAs in the QED case [1], the coupling gcompletely con-\ntrols the running of the Lorentz-violating parameters.\nThe resulting beta function for gis given by\nβg=−g3\n(4π)2/parenleftbig11\n3C2(G)−4\n3nfC(r)/parenrightbig\n,(57)the same as the conventional case. The beta function\ncorresponding to kAFis\nβkAF=−g2\n(4π)2/parenleftbig22\n3C2(G)−8\n3nfC(r)/parenrightbig\nkAF.(58)\nwhere the Lorentz indices have been suppressed for sim-\nplicity. The selfdual part of kFhas the beta function\nβkF=g2\n(4π)2/parenleftbig14\n3C2(G)+8\n3nfC(r)/parenrightbig\nkF.(59)\nThe anti-selfdual contributions coupled to the fermions\nand ghosts give\nβΛµν=−1\n3g2\n(4π)2\nC(r)\n/summationdisplay\nfcµν\nf−8nfΛµν\n+C2(G)Cµν\n.\n(60)\nNote that special values cand Λ can lead to cancelation\nin the beta function.\nVI. BRST SYMMETRY\nDue to the preservation of gauge invariance in the\nperturbed theory, the Lorentz-violating action satisfies\na standard Becchi-Rouet-Stora-Tyutin (BRST)[38] sym-\nmetry provided that there is no explicit ghost violation\nintroduced. The ghost violation terms are not invariant\nunder a standard BRST transformation, but a specific\nform for the gauge fixing term can be chosen to maintain\ninvariance. However, this introduces additional violation\ninto the photon propagator which can be absorbed by a\nbetter choice of gauge. This means that explicit Lorentz\nviolation in the ghost sector alone can violate the gauge\nsymmetry as well.\nThis symmetry should ensure that the multiplicative\nrenormalization will be consistent to all orders by fixing\nthe ratios of the relevant counter-terms in the renormal-\nized lagrangian. This implies that all coupling constants\n(g,kF, andkAF) are universal as gis in the conventional\ncase. An explicit proof of this fact to all orders is beyond\nthe scope of the present paper. This fact agrees with the\nexplicit one loop calculations performed in this paper.\nVII. SUMMARY\nThe functional determinant technique is particularly\nwell suited to Lorentz violation loop calculations as the\ntraces conveniently preserve both the observer Lorentz\ninvariance and the gauge invariance throughout the cal-\nculation. Sums over special subclasses of diagrams are\nrequired to maintain a similar invariance using the dia-\ngrammaticapproach[13]. In addition to the ease oforga-\nnization of the calculation, the present approach is well\nsuited to exploring renormalization in more complicated\nversions of the Lorentz-violating standard model.6\nNew results of this paper include the contribution of\nthe trace components of kf, terms that were neglected in\nthe previous paper [13]. These terms in fact renormalize\ndifferently than the trace-free kfcomponents indicating\ntheir fundamentally different properties. Additional new\nresults include the incorporation of fermions and explicit\nghost violation. Both effects are accommodated by a\nrenormalization of the trace components of kf, while the\ntrace-free components are unaffected. These additional\nresults give the entire one-loop gluon sector effective ac-\ntion in Lorentz-violating QCD and demonstrate renor-\nmalizability at this order.In addition, explicit BRST symmetry is present in the\nfull Lorentz violating theory (without explicit ghost vi-\nolation) and should be crucial in eventually establishing\nfull renormalizability of the theory.\nAcknowledgments\nWe wish to acknowledge the support of New College\nof Florida’s faculty development funds that contributed\nto the successful completion of this project.\n[1] V. A. Kostelecky, C. Lane, and A. Pickering, Phys. Rev.\nD65, 056006 (2002).\n[2] V. A. 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D 74, 033016 (2006).\n[32] B. Altschul, Phys. Rev. D 70, 101701 (2004).\n[33] B. Altschul and A. Kostelecky, Phys. Lett. B 628, 106\n(2005).\n[34] B. Altschul, Phys. Rev. D 73, 045004 (2006).\n[35] T. Mariz, J. R. Nascimento, A. Y. Petrov, L. Y. Santos,\nand A. J. de Silva, arXiv:0708.3348.\n[36] M. Peskin and D.Schroeder, An Introduction to Quantum\nField Theory (Perseus Books, 1995).\n[37] D. Colladay and P. McDonald, J. Mat. Phys. 45, 3228\n(2004).\n[38] C. Becchi, A. Rouet, and R. Stora, Commun. Math.\nPhys.42, 127 (1975)."    },    {        "title": "2308.08559v1.High_frequency_homogenization_for_periodic_dispersive_media.pdf",        "content": "High-frequency homogenization for periodic dispersive media\nMarie Touboul∗Benjamin Vial†Raphaël Assier‡Sébastien Guenneau§\nRichard V. Craster¶\nAugust 21, 2023\nAbstract\nHigh-frequency homogenization is used to study dispersive media, containing inclusions placed\nperiodically, for which the properties of the material depend on the frequency (Lorentz or Drude\nmodel with damping, for example). Effective properties are obtained near a given point of the dis-\npersion diagram in frequency-wavenumber space. The asymptotic approximations of the dispersion\ndiagrams, and the wavefields, so obtained are then cross-validated via detailed comparison with\nfinite element method simulations in both one and two dimensions.\n1 Introduction\nDispersive media have material properties that are frequency dependent and form an important and\ncommonly occurring class of materials in physics, material science and engineering, they are of increas-\ning importance in the design of advanced materials particularly in plasmonics [1, 2]. In addition to\npractical relevance, dispersive media have interesting mathematical features for instance arising from\nresonances that lead to unusual material properties (e.g. artificial magnetism and negative refraction\nin metamaterials [3], zero-index band gaps [4], or folded bands [5]).\nIn electromagnetism, e.g. in optics, the electrodynamic properties of metals differ from those of di-\nelectrics and, in particular, the relative permittivity is frequency dependent. The underlying physics\nis that free electrons form a plasma, are free to move, support a current, and create forces. At low\nfrequencies in, say, microwaves, the effect is minimal, but as one approaches the optical regime, at\nTeraHertz (THz) frequencies, the plasma frequency becomes commensurate with the optical frequen-\ncies and the frequency dependence of the properties are important. Indeed the field has a long and\ndistinguishedhistory, thecoloursofmetalglasseswereinvestigatedusingtheDrudemodelbyMaxwell-\nGarnet [6], and the reflectance properties of metals and entire areas of physics such as plasmonics rely\nupon the properties captured within dispersive media. Historically, Drude, Lorentz and Sommerfeld\npioneered the material models that are now commonly used [7], and used in our numerical examples\nin section 2.5. There is also broader usage of the Drude-Lorenz-Sommerfeld model in other fields as\nthey are also well suited to describe resonant media, arising due to high material contrast of inclusions\neither in elasticity [8, 9] or in electromagnetism [10, 11], or due to the presence of geometric resonances\nin acoustics e.g. Helmholtz resonators [12, 13].\n∗Department of Mathematics, Imperial College London, London SW7 2AZ, UK ( m.touboul@imperial.ac.uk ).\n†Department of Mathematics, Imperial College London, London SW7 2AZ, UK ( b.vial@imperial.ac.uk ).\n‡Department of Mathematics, University of Manchester, Manchester, UK ( raphael.assier@manchester.ac.uk ).\n§The Blackett Laboratory, Department of Physics, Imperial College London, London SW7 2AZ, UK ( s.guenneau@\nimperial.ac.uk ).\n¶DepartmentofMathematics, UMI2004AbrahamdeMoivre-CNRS,DepartmentofMechanicalEngineering, Imperial\nCollege London, London SW7 2AZ, UK. ( r.craster@imperial.ac.uk ).\n1arXiv:2308.08559v1  [physics.class-ph]  14 Aug 2023Structuring dispersive media to contain periodically arranged inclusions, or alternating layers [4]\ncreatesadditionaldifficultiesduetotheintroductionofgeometricstructureandtheresultingscattering\nand reflection from inclusions and surfaces; there are applications for photonic crystals studied with\nDrude-like models [14], creating complete electromagnetic band gaps, or even for the more numerically\nchallenging metallo-dielectric photonic crystals [15]. Critical to the study of wave propagation through\nperiodic media is the concept of Bloch waves and of dispersion curves that connect frequency to the\nphase shift across a single cell of the periodic medium [16, 17]; the resulting band diagrams then\nneatly encapsulate the essential physics of the structure, i.e. band-gaps of forbidden frequencies can\narise where waves will not propagate in the structure. Our aim here is to provide homogenization\nmodels that allow us to side-step numerical issues, generate insight, and create effective media for\nwave propagation through dispersive media that contain periodically arranged inclusions, or consist\nof repeating layers.\nClassically, dynamichomogenizationisunderstoodasalow-frequencyapproximationtowaveprop-\nagation in heterogeneous media. A particularly successful method, for periodic media, is the two-scale\nasymptotic expansion method and the notion of slow and fast variables [18, 19, 20, 21]. In the case of\ndispersive media, it has been extended to surfaces recently [22]. The idea of high-frequency homog-\nenization (HFH), introduced in [23], is to use similar asymptotic methods to approximate how the\ndispersion relation, and hence the media behave, near a given point in wavenumber-frequency space\nthat satisfies the dispersion relation; recent work by [24] pushes this asymptotic analysis one order\nfurther. A uniform approximation is also derived, considering the fact that some branches do not\nintersect at the edges of the Brillouin zone but are close to each other. Other works concerned the\ninclusion of a source term [25] and the derivation of the process in the time domain [26]. The method-\nology has also been applied to several other configurations such as discrete lattice media [27, 28], frame\nstructures [29], optics [30], elastic plates [31], full vector wave systems [32], elastic composites [33],\nreticulated structures [34], or imperfect interfaces [35, 36]. The two-scale approach is also connected\nto homogenization near a neighbourhood of an edge gap in the context of approximations of operator\nresolvents [37] and there are connections into spectral theory.\nHere we extend the HFH method to the case of dispersive media where the properties of the material\ndepend on the frequency; this is not a routine extension as the dispersion curves are now complex and\nadditional complications due to the frequency dependence, including resonances, now occur. In Sec-\ntion 2 we consider a one-dimensional (1D) setting of waves though a laminate of alternating layers, and\nhigh-frequency homogenization is applied for different cases: single eigenvalues at the edges, double\neigenvalues at the edges, nearby eigenvalues at the edges, or single eigenvalues outside the edges (when\nno damping is considered); numerical examples are then presented to cross-validate the asymptotic\napproximations developed. In Section 3, the asymptotic results are extended to two-dimensions (2D)\nand then cross-validated via comparisons with finite element simulations for metallic rods in a vac-\nuum. The effective parameter obtained with HFH is also used to investigate properties of the material.\n2 One-dimensional (1D) case\n2.1 Setting\nWe begin with the 1D case and consider linear waves propagating at a given angular frequency ω\nthrough a dispersive periodic medium of periodicity h(>0)and with a macroscopic characteristic\nlengthL(>0); physically this would correspond to a laminate of dispersive medium layers where the\nlayers alternate with different material properties or a bimaterial string constructed from alternating\ndispersive media, see Fig. 1 and for clarity of exposition and notation we will fix one of the media to\nbe non-dispersive. We denote the physical space variable as X; the material parameters ˆah(X,ω)and\n2ˆbh(X,ω)are assumed to be h-periodic in Xand frequency dependent. The governing equation for the\nfieldUhis:\nd\ndX/parenleftbigg\nˆah(X,ω)dUh\ndX)/parenrightbigg\n+ˆbh(X,ω)ω2Uh= 0. (1)\nThis equation is very general and the field Uhdescribes the transverse electric field for s-polarisation,\nthe transverse magnetic field for p-polarisation in electromagnetism, the displacement in elasticity,\nor the pressure in acoustics; the parameter ˆahis then the inverse of the permeability, the inverse of\nthe permittivity, the shear modulus, or the inverse of mass density, respectively, while ˆbhdenotes the\npermittivity, the permeability, the mass density, or the compressibility, respectively. Henceforth we\nwill assume the elastic setting in terms of notation.\nThe unit cell is divided into two parts distinguished by a “volume fraction” ϕ. The left part is\ncharacterized by constant positive physical parameters, while in the right part they are frequency\ndependent and dispersive:\nˆah(X,ω) =/braceleftigg\na0forX∈(0,ϕh)\nˆ𝕒(ω)forX∈(ϕh,h )(2)\nˆbh(X,ω) =/braceleftiggb0forX∈(0,ϕh)\nˆ𝕓(ω)forX∈(ϕh,h ).(3)\nFor the frequency dependence the example of a Lorentz type dependence is given in Appendix A and\nused in the numerical examples in section 2.5 .\nThe edges of the periodic cell are assumed, without loss of generality, to be located at Xn=nhfor\nn∈ℤ, as illustrated in Figure 1 (left). We further assume that the interfaces across the edges of the\nperiodic cells are perfect, implying continuity for the displacement Uhand the stress ˆahdU\ndXthere; the\nsame is assumed within the unit cells at nϕh.\n2.2 Non dimensionalization\nTo non-dimensionalize the physical problem, we introduce a reference wavespeed c0=/radicalbig\na0/b0and the\nfollowing non-dimensional quantities\nx=X\nL, δ =h\nL,Ω =ωh\nc0, κ =Lk, uδ(x) =Uh(X)\nL. (4)\nMoreover, by periodicity\nˆbh(X,ω) =ˆb/parenleftbiggX\nh,ω/parenrightbigg\nand ˆah(X,ω) = ˆa/parenleftbiggX\nh,ω/parenrightbigg\n,\nwhere ˆbandˆaare 1-periodic in their first argument. These physical quantities are non-dimensionalized\nby introducing\nb/parenleftbiggx\nδ,Ω/parenrightbigg\n=ˆb/parenleftig\nX\nh,ω/parenrightig\nb0anda/parenleftbiggx\nδ,Ω/parenrightbigg\n=ˆa/parenleftig\nX\nh,ω/parenrightig\na0.\nUsing these quantities, (1) is rewritten as the non-dimensional governing equation\nδ2d\ndx/parenleftbigg\na/parenleftbiggx\nδ,Ω/parenrightbiggduδ\ndx(x)/parenrightbigg\n+ Ω2b/parenleftbiggx\nδ,Ω/parenrightbigg\nuδ(x) = 0. (5)\nUpon introducing\n𝕒(Ω) = ˆ𝕒/parenleftbiggc0Ω\nh/parenrightbigg\nand𝕓(Ω) = ˆ𝕓/parenleftbiggc0Ω\nh/parenrightbigg\n,\n3Figure 1: The three coordinate systems: (left) Physical coordinates, (middle) non-dimensional long-\nscale, (right) non-dimensional short-scale.\nthe adimensionalized physical parameters depend only on the short scale ξ=x/δand not on the long\nscalexand become\na(ξ,Ω) =/braceleftigg\n1forξ∈(0,ϕ)\n𝕒(Ω)forξ∈(ϕ,1)(6)\nb(ξ,Ω) =/braceleftigg\n1forξ∈(0,ϕ)\n𝕓(Ω)forξ∈(ϕ,1).(7)\nWe still have continuity for uδandaduδ\ndxat the points nδandnϕδforn∈ℤin the geometry setting\nof Figure 1 (centre).\n2.3 Floquet-Bloch analysis\nThe periodicity of the parameters a(ξ,Ω)andb(ξ,Ω)defined in (99)-(100), allows us to write the\nsolution of (5), as uδ(x) =uδ(x)eiκx, for aδ-periodic function uδand Bloch wavenumber κ∈[0,π/δ].\nFor any Bloch wavenumber κ, this implies that uδ/parenleftbigδ+/parenrightbig=eiκδuδ(0+)andu′\nδ(δ+) =u′\nδ(0+)eiκδ, where\nwe use the prime symbol for differentiation. Using perfect contact conditions at x=δandx=δϕ,\nthe whole problem is reduced to the unit cell x∈(0,δ)where (5) must be satisfied together with\nuδ(δ−) =eiκδuδ(0+)and 𝕒(Ω)u′\nδ(δ−) =eiκδu′\nδ(0+), (8)\nas well as\nuδ(δϕ−) =uδ(δϕ+)andu′\nδ(δϕ−) =𝕒(Ω)u′\nδ(δϕ+). (9)\nThe coefficients of (5) are piecewise constant (with respect to ξ) and so we have two different equations\non(0,δϕ)and(δϕ,δ):\n\n\nu′′\nδ(x) +/parenleftig\nΩ\nδ/parenrightig2uδ(x) = 0forx∈(0,δϕ)\nu′′\nδ(x) +/parenleftig\nΩ\nδ/parenrightig2𝕘(Ω)uδ(x) = 0forx∈(δϕ,δ),where 𝕘(Ω) :=𝕓(Ω)\n𝕒(Ω)· (10)\nEquations (10) have solution\n\n\nuδ(x) =C1cos/parenleftig\nΩx\nδ/parenrightig\n+C2sin/parenleftig\nΩx\nδ/parenrightig\nforx∈(0,δϕ)\nuδ(x) =C3cos/parenleftbigg\nΩ√\n𝕘(Ω)x\nδ/parenrightbigg\n+C4sin/parenleftbigg\nΩ√\n𝕘(Ω)x\nδ/parenrightbigg\nforx∈(δϕ,δ)· (11)\nUsing the periodicity and interface conditions (8) and (9), the integration constants Cisatisfy the\nfollowing linear system:\nM(C1,C2,C3,C4)⊺= (0,0,0,0)⊺, (12)\n4where the 4×4matrix Mis given by\n\neiκδ0−cos/parenleftig\nΩ/radicalbig\n𝕘(Ω)/parenrightig\n−sin/parenleftig\nΩ/radicalbig\n𝕘(Ω)/parenrightig\n0eiκδ𝕒(Ω)/radicalbig\n𝕘(Ω) sin/parenleftig\nΩ/radicalbig\n𝕘(Ω)/parenrightig\n−𝕒(Ω)/radicalbig\n𝕘(Ω) cos/parenleftig\nΩ/radicalbig\n𝕘(Ω)/parenrightig\ncos(Ωϕ) sin(Ωϕ)−cos/parenleftig\nΩ/radicalbig\n𝕘(Ω)ϕ/parenrightig\n−sin/parenleftig\nΩ/radicalbig\n𝕘(Ω)ϕ/parenrightig\n−sin(Ωϕ) cos(Ωϕ)𝕒(Ω)/radicalbig\n𝕘(Ω) sin/parenleftig\nΩ/radicalbig\n𝕘(Ω)ϕ/parenrightig\n−𝕒(Ω)/radicalbig\n𝕘(Ω) cos/parenleftig\nΩ/radicalbig\n𝕘(Ω)ϕ/parenrightig\n.\nNote that to get the second and fourth lines of M, we divided through by Ω/δ. The system (12)\nhas non-trivial solutions only when Mis singular. Upon dividing through by eiκδ2𝕒(Ω)/radicalbig\n𝕘(Ω), the\nequation det(M) = 0reduces to the dispersion relation Disp (Ω,κ) = 0, where\nDisp(Ω,κ) = cos(κδ)−cos(Ωϕ) cos/parenleftbigg\nΩ/radicalig\n𝕘(Ω)(ϕ−1)/parenrightbigg\n−𝕕(Ω) sin(Ωϕ) sin/parenleftbigg\nΩ/radicalig\n𝕘(Ω)(ϕ−1)/parenrightbigg\n,(13)\nwhere the function 𝕕has been defined by\n𝕕(Ω) :=1\n2/parenleftigg\n1\n𝕒(Ω)/radicalbig\n𝕘(Ω)+𝕒(Ω)/radicalig\n𝕘(Ω)/parenrightigg\n.\nDispersion relations can usefully be thought of as nonlinear eigenvalue problems and, in this context, it\nis known that if we take any open connected domain in the Ωcomplex plane on which the entries of M\nare holomorphic, then there will be a finite (possibly zero) number of isolated zeros of det(M)within\nthis domain (see e.g. Theorem 2.1 in [38]). Therefore the same is true for the solution of the dispersion\nrelation. One should however be careful about domains that contain points for which the entries of\nMare singular (e.g. branch points or poles). Given the form of M, these potentially problematic\npoints are values of Ωfor which 𝕘(Ω) = 0 or𝕘(Ω) =∞. For spectral properties of absorptive and\ndispersive photonic crystals, we refer the reader to [39] and [40], respectively. More details are given\nin the next two paragraphs for the case of the Lorentz model (see Appendix A for the expression of\nthe physical parameters in this case) as it is representative of issues that arise.\nPoints for which 𝕘(Ω) = 0for the Lorentz model Given that 𝕘(Ω) = 𝕓(Ω)/𝕒(Ω), these are the\npoints for which either 𝕓(Ω) = 0or𝕒(Ω) =∞. So they are points that are solutions of (cf equations\n(99) and (100))\n\n1−/summationdisplay\np≥0Ω2\np,i\nΩ(Ω + iγp,i)−Ω2\nD,p,i\n= 0 fori= 1ori= 2.\nIf there is only one term in each sum, these points can be written down easily explicitly, but otherwise\nforseveraltermsinthesumstheyhavetobefoundnumerically; findingthesepointsisstraightforward.\nPoints for which 𝕘(Ω) =∞for the Lorentz model Given that 𝕘(Ω) = 𝕓(Ω)/𝕒(Ω), these are the\npoints for which either 𝕓(Ω) =∞or𝕒(Ω) = 0and are solutions of\n\n1−/summationdisplay\np≥0Ω2\np,i\nΩ(Ω +iγp,i)−Ω2\nD,p,i\n=∞fori= 1ori= 2.\n5The points are found explicitly by nullifying the denominators of each term in the sums and are given\nfori= 1,2andpbyΩ =R±\np,i, where\nR±\np,i=−iγp,i±/radicalig\n4Ω2\nD,p,i−γ2\np,i\n2·\nIn a neighbourhood of these points the theorem mentioned above does not apply, and some of these\nwill beaccumulation points . In other words, if we take any open connected set containing one of\nthese points, it will contain infinitely many zeros of the dispersion relation; this phenomenom is\nillustrated in Figure 8b. It is interesting to note that these points are independent of the choice\nof Bloch wavenumber κ. In the remaining parts of this paper, we will aim to provide an asymptotic\nhomogenised approximation to the dispersion diagram and the corresponding wave field in the vicinity\nof an exact solution (Ω0,κ)of the dispersion relation; our method works for points Ω0that are not too\nclose to an accumulation point. In the vicinity of accumulation points, another approach is required\nas some form of resonance is expected [41, 42, 43, 44].\n2.4 High-frequency homogenization\nWe assume that δ≪1and we recall that ξ=x/δ. To start with, we pick a frequency-wavenumber\npair (Ω0,κ)∈ℂ×[0,π/δ]that satisfies Disp (Ω0,κ) = 0and is such that we are not too close to an\naccumulation point. Following the two-scale expansion technique, we further assume the usual HFH\nansatz for the wave field uδand the reduced frequency Ω:\nuδ(x) =/summationdisplay\nj⩾0δjuj(x,ξ)and Ω2=/summationdisplay\nℓ⩾0δℓΩ2\nℓ, (14)\nwhere we treat xandξas two independent variables. The latter implies thatd\ndx↔∂\n∂x+1\nδ∂\n∂ξ. We will\nassume that\nuj(x,ξ+ 1) = eiκδuj(x,ξ), (15)\nso we will restrict the analysis to (0,1) (see Figure 1 right).\nUsing this ansatz, and considering that the physical parameters are piecewise constant, the governing\nequation (5) becomes\n/summationdisplay\nj⩾0\na(ξ,Ω)/braceleftigg\nδj∂2uj\n∂ξ2+ 2δj+1∂2uj\n∂x∂ξ+δj+2∂2uj\n∂x2/bracerightigg\n+/summationdisplay\nℓ⩾0δℓ+jΩ2\nℓb(ξ,Ω)uj\n= 0. (16)\nImportantly, aandbdepend implicitly on δthrough Ωin the above expression. Therefore we have to\nwrite their expansion in powers of δ. Up to the second order, we find that for q={a,b}:\nq(ξ,Ω) =D0(q) +δD1(q) +δ2D2(q) (17)\nwith\nD0(q) =q(ξ,Ω0), (18)\nD1(q) =Ω2\n1\n2Ω0∂q\n∂Ω(ξ,Ω0), (19)\nD2(q) =Ω4\n1\n8Ω2\n0∂2q\n∂Ω2(ξ,Ω0) +1\n2Ω0/parenleftigg\nΩ2\n2−Ω4\n1\n4Ω0/parenrightigg\n∂q\n∂Ω(ξ,Ω0). (20)\nWe defineAi=Di(a)andBi=Di(b)fori={0,1,2}, where we have to keep in mind that Aiand\nBidepend onξand{Ω0,···,Ωi}. For the dispersion model we chose their expressions are detailed in\n60\nSingle eigenvalues at the edges\n(Sec \u0001on 2.4.1)Double eigenvalues at the edges\n(Sec \u0001on 2.4.2)\nNearby eigenvalues at the edges\n(Sec\u0001on 2.4.3)\nSingle eigenvalues inside\n(Sec\u0001on 2.4.4)\nLow-frequency case\n(Sec \u0001on 2.4.5)\nFigure 2: Different cases approximated with high-frequency homogenization.\nAppendix A.\nWe also need to introduce the average operator ⟨·⟩defined by\n⟨g⟩=/integraldisplay1\n0g(ξ)dξ\nfor any function g.\nIn the next sections, we will apply high-frequency homogenization to get asymptotic approximations\nof both the wavefields and the dispersion diagrams for all the relevant cases which are represented in\nFigure 2.\n2.4.1 Single eigenvalues at the edges of the Brillouin zone\nWe first take points at the edges of the Brillouin zone, that is at κ= 0orκ=π/δwhich is asso-\nciated to periodic and antiperiodic conditions, respectively, for the fields. In these cases, we expect\nthe dispersion relation to be locally quadratic i.e. we set Ω1= 0. Indeed, the mapping Ωn(k)de-\nnoting the dispersion diagram along a branch is holomorphic [38] except around accumulation points\nand at singular points (when eigenvalues are no longer single ones); combined with reciprocity, i.e.\nΩn(k) = Ωn(−k), this gives that dΩn/dk= 0at edges.\nZeroth-order field Collecting the terms of order δ0, we get in (0,1):\nA0(ξ,Ω0)∂2u0\n∂ξ2(x,ξ) + Ω2\n0B0(ξ,Ω0)u0(x,ξ) = 0 (21)\nwhereA0andB0are piecewise constants defined in (18). We also have continuity for u0andA0∂u0\n∂ξ\natϕand0together with the 1-periodicity/antiperiodicity for u0:\n\n\nu0(ϕ−) =u0(ϕ+)\n∂u0\n∂ξ(ϕ−) =𝕒(Ω0)∂u0\n∂ξ(ϕ+)\nu0(1−) =±u0(0+)\n𝕒(Ω0)∂u0\n∂ξ(1−) =±∂u0\n∂ξ(0+).(22)\n7As discussed in Section 2.3, we will build asymptotic approximations sufficiently far away from the\naccumulation points. We therefore know that there is a discrete set of eigenvalues and then choose Ω0\nwhich is assumed to be a simple eigenvalue associated to the eigenfunction U0(ξ,Ω0). The zeroth-order\nfield is therefore\nu0(x,ξ) =f0(x)U0(ξ,Ω0), (23)\nwhere the slowly varying amplitude f0(x)has to be determined.\nFirst-order field Collecting the terms of order δ, and keeping in mind that Ω1= 0, we get on (0,1):\nA0∂\n∂ξ/parenleftbigg∂u1\n∂ξ+ 2∂u0\n∂x/parenrightbigg\n+ Ω2\n0B0u1= 0 (24)\ntogether with periodicity/antiperiodicity for u1, continuity for u1atϕand 0, and continuity for\nA0/parenleftig\n∂u0\n∂x+∂u1\n∂ξ/parenrightig\natϕand0. Using (23), Equation(24) reduces to:\nA0∂2u1\n∂ξ2+ Ω2\n0B0u1=−2A0U′\n0(ξ)f′\n0(x). (25)\nThen, we write u1as:\nu1(x,ξ) =f1(x)U0(ξ) +f′\n0(x)V(ξ), (26)\nwhere \n\nA0V′′(ξ) + Ω2\n0B0V(ξ) =−2A0U′\n0(ξ)\nVis periodic/antiperiodic\nVandA0V′+A0U0are continuous at ϕand 0.(27)\nSecond-order field Collecting terms of order δ2, we get on (0,1):\nA0/parenleftigg\n∂2u0\n∂x2+ 2∂2u1\n∂x∂ξ+∂2u2\n∂ξ2/parenrightigg\n+A2∂2u0\n∂ξ2+ Ω2\n0(B0u2+B2u0) + Ω2\n2B0u0= 0, (28)\ntogether with periodicity/antiperiodicity for u2, continuity for u2atϕand 0, and continuity for\nA0/parenleftig\n∂u2\n∂ξ+∂u1\n∂x/parenrightig\n+A2∂u0\n∂ξatϕand0. Consider now the equation\n⟨u2×(21)−u0×(28)⟩= 0.\nAfter integration by part, some algebra and dividing through by f0we get:\n⟨−A0U2\n0−A0V′U0+A0VU′\n0⟩f′′\n0+⟨−Ω2\n2B0U2\n0−Ω2\n0B2U2\n0+A2(U′\n0)2⟩f0= 0. (29)\nFurthermore, note that when Ω1= 0,A2andB2defined in (20), regardless of the dispersive model\nchosen, is:\nB2(Ω0,0,Ω2) = Ω2\n2˜B2(Ω0)andA2(Ω0,0,Ω2) = Ω2\n2˜A2(Ω0). (30)\nTherefore, we get the sought-after effective equation for f0andΩ2:\nTf′′\n0+ Ω2\n2f0= 0, (31)\nwhere\nT=⟨A0U2\n0+A0V′U0−A0VU′\n0⟩\n⟨B0U2\n0+ Ω2\n0˜B2U2\n0−˜A2(U′\n0)2⟩=/angbracketleftbiga(·,Ω0)(U2\n0+V′U0−VU′\n0)/angbracketrightbig\nS(Ω0,U0,U0), (32)\n8for any frequency-dependent functions aandb, and where for any two functions fandg, and any\nreduced frequency Ω0,Sis defined by:\nS(Ω0,f,g) =/angbracketleftbigg/parenleftbigg\nb(·,Ω0) +Ω0\n2∂b\n∂Ω(·,Ω0)/parenrightbigg\nfg−1\n2Ω0∂a\n∂Ω(·,Ω0)f′g′/angbracketrightbigg\n. (33)\nTherefore, we get the effective string described by (31) on the long-scale where the complex material\nproperties are now solely concentrated in a single effective parameter T(32).\nApplying the Bloch-Floquet conditions (15) gives the final expression for the quadratic term in the\ndispersion relation\nΩ2\n2=T˜κ2, (34)\nwith ˜κ=κnear 0 and ˜κ=π/δ−κnearπ/δso that the dispersion relation is approximated by\nΩ≈Ω0+T\n2Ω0(˜κδ)2. (35)\n2.4.2 Double eigenvalues at the edges of the Brillouin zone\nWe next consider the case of multiplicity two for the eigenvalue Ω0. In that case, Ω1is no longer 0,\nand the zeroth-order wavefield is now written as\nu0(x,ξ) =f(1)\n0(x)U(1)\n0(ξ) +f(2)\n0(x)U(2)\n0(ξ) (36)\nwhereU(1)\n0(ξ)andU(2)\n0(ξ)are two independent eigenfunctions associated to Ω0, whilef(1)\n0(x)and\nf(2)\n0(x)are the slow modulation functions to be found.\nConsequently, both eigenfunctions satisfy (21) and we denote (21)(i)the equation for the ith eigen-\nfunction. Furthermore, because Ω1is non-zero, the system satisfied by the first-order field becomes\nA0∂\n∂ξ/parenleftbigg∂u1\n∂ξ+ 2∂u0\n∂x/parenrightbigg\n+A1∂2u0\n∂ξ2+ Ω2\n0(B0u1+B1u0) + Ω2\n1B0u0= 0, (37)\ntogether with periodicity/antiperiodicity for u1, continuity for u1atϕand 0, and continuity for\nA0/parenleftig\n∂u0\n∂x+∂u1\n∂ξ/parenrightig\n+A1∂u0\n∂ξatϕand0.\nWe introduce ˜B1and ˜A1so that\nB1(Ω0,Ω1) =Ω2\n1\nΩ2\n0˜B1(Ω0)andA1(Ω0,Ω1) =Ω2\n1\nΩ2\n0˜A1(Ω0). (38)\nThen considering the equations ⟨(21)(i)×u1−(37)×U(i)\n0⟩= 0fori= 1,2, we get the effective equation\nforF= (f(1)\n0,f(2)\n0)⊺:\nF′(x) =Ω2\n1\n⟨A0w0⟩NF, (39)\nwhere w0is the Wronskian defined by\nw0(ξ) =U(1)\n0(ξ)U(2)\n0′(ξ)−U(1)\n0′(ξ)U(2)\n0(ξ), (40)\nNis the matrix defined by\nN=/parenleftigg\nS(Ω0,U(1)\n0,U(2)\n0)S(Ω0,U(2)\n0,U(2)\n0)\n−S(Ω0,U(1)\n0,U(1)\n0)−S(Ω0,U(1)\n0,U(2)\n0)/parenrightigg\n, (41)\n9andSis defined in (33).\nRegarding the dispersion diagram, using Bloch-Floquet conditions (15) we get the two opposite\nslopes (here the upper and lower notation does not stand for left or right edge of the Brillouin zone\nbut for the upper and lower branch starting from Ω0)\nΩ2\n1=±TD˜κ (42)\nso that\nΩ≈Ω0±TD\n2Ω0˜κδ (43)\nwith ˜κ=κnear 0 and ˜κ=π/δ−κnearπ/δ, andTDdefined by\nTD=⟨a(·,Ω0)w0⟩\n/parenleftig\nS(Ω0,U(1)\n0,U(1)\n0)S(Ω0,U(2)\n0,U(2)\n0)−S(Ω0,U(1)\n0,U(2)\n0)2/parenrightig1/2. (44)\n2.4.3 Nearby eigenvalues at the edges of the Brillouin zone\nLet us assume now that we have two simple eigenvalues close to each other following [35, 24]. The\ntwo nearby simple eigenvalues are denoted by Ω(1)\n0andΩ(2)\n0. Their proximity is quantified by writing\n(Ω(2)\n0)2−(Ω(1)\n0)2=αδ, (45)\nfor some constant α>0. We denote byU(1)\n0andU(2)\n0the eigenfunctions associated to Ω(1)\n0andΩ(2)\n0,\nrespectively.\nThe ansatz is considered around the eigenvalue Ω(1)\n0:\nuδ(x) =/summationdisplay\nj⩾0δjuj(x,ξ)and Ω2= (Ω(1)\n0)2+/summationdisplay\nℓ⩾1δℓΩ2\nℓ, (46)\nBy similarity with the double eigenvalue case, and to take into account the coupling between both\neigenvalues we look for the zeroth-order wavefield as\nu0(x,ξ) =f(1)\n0(x)U(1)\n0(ξ) +f(2)\n0(x)U(2)\n0(ξ). (47)\nWe will use the notation D(i)\nj, forD=A, Bandi= 1,2, andj= 0,1so that we get from Taylor\nexpansions\nA(i)\n0∂2U(i)\n0\n∂ξ2+ (Ω(i)\n0)2B(i)\n0U(i)\n0= 0 (48)\nand\nD(2)\n0=D(1)\n0+αδ\n2Ω(1)\n0∂d\n∂Ω(Ω(1)\n0) +O(δ2) (49)\nwhered=a, bforD=A, B, respectively. This leads to\nA(1)\n0∂2u0\n∂ξ2+ (Ω(1)\n0)2B(1)\n0u0=δ/bracketleftigg\nα\n2A(1)\n0Ω(1)\n0B(1)\n0∂a\n∂Ω(ξ,Ω(1)\n0)\n−αB(1)\n0−α\n2Ω(1)\n0∂b\n∂Ω(ξ,Ω(1)\n0)/bracketrightbigg\nu(2)\n0.(50)\n10Consequently, (21) is satisfied by the zeroth-order wavefield up to right-hand side residual term of\n(50) that modifies the equation for the first order field that now becomes:\nA(1)\n0∂\n∂ξ/parenleftbigg∂u1\n∂ξ+ 2∂u0\n∂x/parenrightbigg\n+A(1)\n1∂2u0\n∂ξ2+ (Ω(1)\n0)2(B(1)\n0u1+B(1)\n1u0) + Ω2\n1B(1)\n0u0\n+αu(2)\n0/bracketleftigg\n1\n2A(1)\n0Ω(1)\n0B(1)\n0∂a\n∂Ω(ξ,Ω(1)\n0)−B(1)\n0−1\n2Ω(1)\n0∂b\n∂Ω(ξ,Ω(1)\n0)/bracketrightigg\n= 0,(51)\ntogether with periodicity/antiperiodicity for u1, continuity for u1atϕand 0, and continuity for\nA(1)\n0/parenleftig\n∂u0\n∂x+∂u1\n∂ξ/parenrightig\n+A(1)\n1∂u0\n∂ξatϕand0. As in the double eigenvalue case, considering ⟨(48)(i)×u1−\n(51)×U(i)\n0⟩fori= 1,2allows to obtain the effective equation for F= (f(1)\n0,f(2)\n0)T:\nF′(x) =Ω2\n1\n⟨A(1)\n0w0⟩NαF(x) (52)\nwith w0still given by (40). The matrix Nαis defined by\nNα=/parenleftigg\nS(Ω(1)\n0,U(1)\n0,U(2)\n0)S(Ω(1)\n0,U(2)\n0,U(2)\n0)\n−S(Ω(1)\n0,U(1)\n0,U(1)\n0)−S(Ω(1)\n0,U(1)\n0,U(2)\n0)/parenrightigg\n+α\nΩ2\n1/parenleftigg\n0−G2\n0G1/parenrightigg\n(53)\nwith\nGi=/angbracketleftigg\n(˜B(1)\n1+B(1)\n0)U(i)\n0U(2)\n0−B(1)\n0\nA(1)\n0˜A(1)\n1U(i)\n0U(2)\n0/angbracketrightigg\n=/angbracketleftigg\nb(·,Ω(1)\n0)/parenleftigg\n1 +Ω(1)\n0\n2∂\n∂Ω[log(b/a)](·,Ω(1)\n0)/parenrightigg\nU(i)\n0U(2)\n0/angbracketrightigg\n.(54)\nThe dispersion relation is then obtained by solving\ndet/parenleftigg\nΩ2\n1\n⟨A(1)\n0w0⟩Nα−iκ𝕀2/parenrightigg\n= 0. (55)\nOne notes that Nα=Nwhenα= 0and we then recover the double case.\n2.4.4 Simple eigenvalues inside the Brillouin zone (no damping)\nIn this section, we get a linear approximation for an arbitrary point inside (strictly) the Brillouin\nzone. However, this is possible only if the physical parameters are real. Consequently, we consider\nall the damping terms γp,1andγp,2equal to 0, in the framework of this subsection only, so that the\ncoefficients AiandBi(i= 1,2) are real.\nLet us pick a point (κ⋆,Ω0)withκ⋆/∈{0,π/δ}andΩ0solution of the eigenvalue problem satisfied by\nthe zeroth-order field:\nA0∂2u0\n∂ξ2+ Ω2\n0B0u0= 0in(0,δ)×(0,1), (56)\ntogether with u0(x,ξ+ 1) = eiκ⋆δu0(x,ξ)in(0,δ)×(0,1), continuity for u0atξ=ϕandξ= 0, and\ncontinuity for A0∂u0\n∂ξatξ=ϕandξ= 0.\nFor the first order, we get:\nA0∂\n∂ξ/parenleftbigg∂u1\n∂ξ+ 2∂u0\n∂x/parenrightbigg\n+A1∂2u0\n∂ξ2+ Ω2\n0(B0u1+B1u0) + Ω2\n1B0u0= 0in(0,δ)×(0,1),(57)\n11together with u1(x,ξ+ 1) = eiκ⋆δu1(x,ξ)in(0,δ)×(0,1), continuity for u1atξ=ϕandξ= 0, and\ncontinuity for A0/parenleftig\n∂u0\n∂x+∂u1\n∂ξ/parenrightig\n+A1∂u0\n∂ξatξ=ϕandξ= 0.\nWe consider I=⟨¯u1×(56)−¯(57)×u0⟩= 0. We still have\n/angbracketleftbigg∂\n∂ξ/bracketleftbigg\nA0/parenleftbigg∂¯u0\n∂x+∂¯u1\n∂ξ/parenrightbigg\nu0+A1∂¯u0\n∂ξu0−A0∂u0\n∂ξ¯u1/bracketrightbigg/angbracketrightbigg\n= 0 (58)\nbecause the quantities above are continuous and 1-periodic. Therefore, I= 0reduces to\n/angbracketleftigg\nA0/parenleftigg\n∂2¯u0\n∂ξ∂xu0−∂¯u0\n∂x∂u0\n∂ξ/parenrightigg\n−A1/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂u0\n∂ξ/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n+ Ω2\n0B1|u0|2+ Ω2\n1B0|u0|2/angbracketrightigg\n= 0. (59)\nDividing through by f0and using (56), we end up with a first-order ODE for f0:\nTif′\n0(x)−iΩ2\n1f0(x) = 0 (60)\nwithTidefined by\nTi=2⟨a(·,Ω0)ℑ(U′\n0¯U0)⟩\nS(Ω0,U0,¯U0), (61)\nwhere we remind that Sis defined in (33). Regarding the dispersion relation, applying the Bloch-\nFloquet conditions (15) gives\nΩ2\n1=Ti(κ−κ⋆). (62)\n2.4.5 The low-frequency case\nWe also obtain the classical low-frequency homogenization by considering Ω0= 0in (21), which leads\nto the fact thatU0is uniform, and we, without loss of generality, choose U0= 1. Then, we write\nu1(x,ξ) =f′\n0(x)U1(ξ)whereu1satisfies (24) with Ω0= 0andu0=f0. Therefore⟨A0(1 +U′\n1)⟩=\n⟨1/A0⟩−1. Integration on a unit cell of (28) for Ω0= 0then leads to the usual homogenized equation\n⟨1/A0⟩−1f′′\n0(x) + Ω2\n2⟨B0⟩f0(x) = 0 (63)\nand dispersion relation\nΩ2\n2=⟨1/A0⟩−1⟨B0⟩−1κ2. (64)\n2.5 Numerical investigation\nWe now use two different methods to compute the whole dispersion diagrams: we either track the zeros\nofthedispersionfunction(13)inthecomplexplanealongabranch, orweusethefiniteelementmethod\n(FEM) to directly solve for (10). The details are given in Appendix B for the latter. Hereafter, the\ndispersion diagrams computed either by zero tracking or by FEM from the exact dispersion function\nwillbedenotedasthe exact dispersion diagrams incontrasttotheasymptoticapproximationsobtained\nby HFH with which they will be compared.\n2.5.1 Dielectric and metallic layers (Drude with damping)\nMotivatedbytheconfigurationof[45], weconsiderwavepropagationthroughalternatedlayersofsilver\n(Ag) and Titanium dioxide (TiO 2). Here only the permittivity is frequency-dependent, following a\nDrude law in the metal layer made of silver (Ag). More precisely, we have a= 1in both materials,\nand in silver\n𝕓(Ω) =b∞\nb0/parenleftigg\n1−Ω2\n1,2\nΩ(Ω + iγ1,2)/parenrightigg\n12withb0= 6.2,b∞= 1, and one resonance for bwith Ω1,2= 5.01, andγ1,2= 0.01in (100). The filling\nratio of the dielectric layer is ϕ= 10/11for a periodicity of h= 110nm.\nSince there is damping, the frequency solutions of the dispersion relation are complex and to visualize\nthedispersionfunctionanditszeroswithouthavingtoconsidertherealandimaginarypartsseparately,\nwe plot at a given frequency its phase portrait in the complex plane, see Figure 3a for κ= 0.\nAlternatively, we track the zeros of the function along a branch of the dispersion diagram in the\ncomplex plane, see Figure 3b.\n(a) Phase portrait in complex plane for the dispersion\n(i.e. determinant) function (13) taken at κ= 0\n(b) Dispersion diagram in the complex plane\n(hereκ∈(0π/δ))\nFigure 3: Representation of the dispersion function and its zeros.\nSimple eigenvalue approximations at the edges Firstly, we use the asymptotic approximations\nobtained for simple eigenvalues near the edges of the Brillouin zone, see Section 2.4.1. The resulting\nasymptotic approximations of the dispersion diagrams for both the real part and the imaginary part\nare displayed in Figure 4, where we used the zero tracking method to compute the diagrams for the\nexact dispersion relation. The absolute errors for each of the branches and for both κ= 0andκ=π/δ\nare then shown in Figure 5, where we recover that the quadratic term Ω2is well taken into account\nasymptotically and that in fact the next term is zero so that the error is O((˜κδ)4).\nNearby approximations at the edges Given these numerical solutions, we now compare with\nthe asymptotics and use the nearby approximations developed in Section 2.4.3 for single eigenvalues\nnear the edges of the Brillouin zone. Solutions from the FEM method are shown in Figure 6 for real\nand imaginary parts for four modes, and in Figure 7 for the dispersion diagram in the complex plane.\nIt is clearly seen that the agreement with the dispersion diagram is much longer lived than that of the\nsimple eigenvalue approximation for both real and imaginary parts. The nearby approximation leads\nnotably to a better fit of the imaginary parts, which are quite small due to the fact that the damping\ncoefficients γiare also small in practice.\n2.5.2 Stack of positive and negative index materials (Lorentz with no damping)\nIn a second more challenging example, we now reproduce the results of Li et al. [4], see Figure 2\nof the latter and compare with the asymptotic results. This consists of a 1D system of periodicity\nh= 18mm with alternate layers of air (12 mm thick) and of an effective material which is dispersive\n13(a) Real part\n (b) Imaginary part\nFigure 4: Superposition of the dispersion diagram from zero tracking (solid lines) and the asymptotic\napproximations obtained by HFH in the simple eigenvalue case (35) around ˜κ= 0(dashed lines).\nlog( )log(|( -(02+2\n22)1/2|)\n()4\n()3\n(a) Nearκδ= 0\nlog( -)log(|( -(02+2\n22)1/2|)\n()4 (b) Nearκδ=π\nFigure 5: Absolute error between the exact dispersion diagram and the HFH approximation in the\nsimple eigenvalue case (35) in a log-log scale (solid lines). Dashed lines are reference orders of conver-\ngence.\n(6.0 mm thick). Both parameters in the dispersive medium follow a Lorentz law without damping in\nthe effective layers, see (99) and (100). More precisely, we set ϕ= 2/3,a∞=a0=b∞=b0= 1,\nΩ1,1= 1.131,ΩD,1,1= 0.34,Ω1,2= 1.885,ΩD,1,2= 0.3393,Ω2,2= 3.7699andΩD,2,2= 4.3354resulting\nin\n𝕒(Ω) =/parenleftigg\n1 +Ω2\n1,1\nΩ2−Ω2\nD,1,1/parenrightigg−1\n𝕓(Ω) = 1 +Ω2\n1,2\nΩ2−Ω2\nD,1,2+Ω2\n2,2\nΩ2−Ω2\nD,2,2.(65)\nWe first compute the dispersion relation using Bloch-Floquet analysis (see Section 2.3). The logarithm\nof the dispersion function (13), is plotted in Figure 8a, the dispersion curve therefore corresponds to\nthe dark lines in the map. The main features of Figure 2 in [4] are recovered, together with the\n141.0 1 .5 2 .0 2 .5 3 .0\nκδ123456Re Ωexact\nsingle\nnearby(a) Real part for the first and second mode around\nκδ=π.\n1.0 1 .5 2 .0 2 .5 3 .0\nκδ−0.0020−0.0015−0.0010−0.00050.00000.0005Im Ω\nexact\nsingle\nnearby(b) Imaginary part for the first and second mode\naroundκδ=π.\n0.00 0 .25 0 .50 0 .75 1 .00 1 .25 1 .50\nκδ5678Re Ωexact\nsingle\nnearby\n(c) Real part for the second and third mode around\nκδ= 0.\n0.00 0 .25 0 .50 0 .75 1 .00 1 .25 1 .50\nκδ−0.000125−0.000100−0.000075−0.000050−0.0000250.0000000.000025Im Ωexact\nsingle\nnearby(d) Imaginary part for the second and third mode\naroundκδ= 0.\nFigure 6: Comparison of the exact dispersion relation (solid lines) and the effective one obtained by\nHFH in the simple eigenvalue case (35) (dashed lines) and with nearby approximations (55) (dotted\nlines).\nappearance of accumulation points, see Figure 8b for phase portraits zoomed-in around one of these\npoints. However, we will consider the same range of frequencies as in [4], for which we are away from\nany of these points and able to propose high-frequency homogenized approximations.\nSimple eigenvalue approximations at the edges We then recover this band diagram near the\nedges using the quadratic approximations obtained with HFH in Section 2.4.1. The comparison is\ngiven in Figure 9a where the numerical curves are obtained using zero tracking; the branches near the\nedges are well approximated. For a quantitative validation, we plot the difference between the exact\ndispersion relation and the quadratic approximation on a log-log scale. It validates the approximation\nof the quadratic term and again underlines that there is no third-order term because we get an error\nof orderO((δ˜κ)4), see Figure 9b for the case κ= 0.\nObtaining accurate asymptotics for the dispersion curves is a useful application of the theory, by\nvalidating it and by encapsulating the physics into a coefficient Tthat allows us to tune or design\n152 4 6 8 10 12\nRe Ω−0.0004−0.0003−0.0002−0.00010.00000.0001Im Ω\nexact\nsingle\nnearbyFigure 7: Comparison in the complex plane of the exact dispersion relation (solid lines) and the\neffective one obtained by HFH for the single eigenvalue case (35) (dashed lines) and with nearby\napproximations (55) (dotted lines). The circle markers correspond to κδ= 0.\nfeatures. An equally important application of the theory is to model forcing, that is, to apply a\nsource in a structured medium and then use the effective equations to model the response; we now\nproceed to demonstrate the efficiency of that approach. We introduce a source term and choose a\nfrequency of excitation close to an eigenfrequency at κ= 0, and then compare the wavefields for the\nmicrostructured medium using both numerical simulation and the high-frequency approximations in\nFigure 10. We first use a point source spatially located at ξ= 0.8and repeated periodically, with\nfrequency Ω = 1.01Ω(4)\n0= 1.027, close to the fourth eigenfrequency studied in Fig. (9a); numerically\nthis is modelled by finite elements studying one unit cell and applying periodic boundary conditions.\nAs a comparison we solve the effective equation (31) obtained by HFH to get the envelope function f0\nand then recover the first order field using Eq. (23). As displayed on Fig. (10a), an excellent agreement\nis obtained with the simulations for the microstructured medium (solid lines) and the homogenized\none (dashed lines). Next, we consider a finite stack consisting of 20 periods of the microstructured\nmedium and compare it with the effective medium. The point source is located in the center at\nξ= 10 +ϕ/2with frequency Ω = 1.01Ω(3)\n0= 0.752, and numerically we use Perfectly Matched Layers\n[46] on either side to truncate the simulation domain and damp propagating waves to avoid reflections\nat the computational domain boundary. The dashed lines on Fig. (10b) show the field for the long-\nscale envelope function f0which is in good agreement with the results from the finite multilayer stack,\nalbeit with some minor discrepancies likely due to the finite extent of the stack and boundary effects\nnot taken into account in our model.\nInside the Brillouin zone Finally, we make use of the linear asymptotic approximations inside\nthe Brillouin zone, i.e. equation (62) of Section 2.4.4, on the same example. We see in Figure 11a that\nusing this approximation for only three points inside the Brillouin zone and combining it with the\nquadratic asymptotic approximations at the edges, we almost recover the entire dispersion diagram\n(obtained with FEM). The effective coefficient Tialso gives an insight on the group velocity in Figure\n16(a) Map in (κ,Ω)space of the logarithm\n (b) Phase portrait around an accumulation point at κ= 0\nFigure 8: Dispersion function. (Left) Map of the logarithm of the dispersion function in the\nwavenumber-frequency space. The zeros are the solutions of the dispersion relation, therefore repre-\nsented by the dark lines. (Right) Phase portrait in the complex plan zoomed around the accumulation\npoint which occurs at R+\n1,1= ΩD,1,1= 4.3354, as predicted in Section 2.3\n(a) Superposition of the exact dispersion diagram\n(plain lines) and the asymptotic approximations ob-\ntained by HFH around the edges (dotted lines)\nlog( )log( -(02+2\n22)1/2)\n()4\n()3(b) Absolute error near κδ= 0between both disper-\nsion relations in a log-log scale (plain lines). Dotted\nlines are reference orders of convergence.\nFigure 9: Comparison of the exact dispersion relation and the effective one obtained by HFH for the\nsingle eigenvalue case (35).\n11b since they are proportional.\n2.5.3 Double eigenvalue case\nWe now investigate a double eigenvalue case (asymptotic approximations developed in Section 2.4.2).\nChoosingϕ= 0.5,b= 1and a Drude model with no damping for awith the parameters a0=a∞= 1,\nγ1,1= ΩD,1,1= 0andΩ1,1= 24.3347in (99). This leads to a double eigenvalue Ω0= 25.1322at\nκδ=π, with 𝕒(Ω0)≃m2wheremis an integer ( m= 4here, cf. [23]). We note that the value of\nΩ1,1is close to Ω0and corresponds to a pole of 𝕒, meaning the behaviour of the material properties\n170.0 0 .2 0 .4 0 .6 0 .8 1 .0\nξ1.001.011.021.031.041.051.06Reuexact\nHFH(a) Periodic source located at ξ= 0.8with frequency\nclose to the fourth eigenfrequency in Fig.(9) for κδ= 0\n(Ω = 1.01Ω(4)\n0= 1.027). The effective coefficient is\nT= 1.336.\n0 5 10 15 20\nξ0.00.20.40.60.81.0Reuexact\nHFH(b) Finite stack of 20 periods with a source located at\nξ= 10+ϕ/2with frequency close to the third eigenfre-\nquency in Fig.(9) for κδ= 0(Ω = 1.01Ω(3)\n0= 0.752).\nThe dashed line shows the envelope f0obtained by\nHFH. The effective coefficient is T=−0.573.\nFigure 10: Comparison of the wavefields for a point source forcing. HFH approximation is the one of\nthe simple eigenvalue case (31).\n0 1 2 3\nκδ0.51.01.52.02.53.03.54.0Ω\n0 1 2 3\nκδ\n(a) Exact dispersion diagram (left) and the linear\nasymptotic approximations (right) obtained by HFH\nat three points inside the Brillouin zone (62) (dashed\nlines). We also plot the quadratic HFH approxima-\ntions around κδ= 0andκδ=π(35) (plain lines)\n0.0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0\nκδ−3−2−10123Timode 1\nmode 2\nmode 3\nmode 4\nmode 5(b) Evolution of the coefficient Ti(61) as a function of\nκδfor the five modes studied in Figure 11a.\nFigure 11: Effective properties obtained by HFH inside the Brillouin zone (Section 2.4.4).\naround those frequencies is highly dispersive. Even in this case, our method recovers the expected\nlinear asymptotics with opposite slopes characteristic of a degenerate root, as shown in Figure 12.\nThis is confirmed quantitatively by the curves in Figure 13 representing the errors between the exact\n18dispersion relation and the linear asymptotic approximations on a log-log scale, showing an O((δ˜κ)2)\nconvergence for both branches.\n0 1 2 3\nκδ24.024.525.025.526.026.5Ω\nΩ1,1log10|Disp(Ω , κ)|\nexact\nHFH\n2.6 2 .8 3 .0\nκδ24.925.025.125.225.325.4Ωexact\nHFH\n−2024\nFigure 12: A double eigenvalue case. Comparison of the exact dispersion (green solid lines) with the\nlinear asymptotics obtained by HFH (43) (black dashed lines). The colormap on the left panel shows\nthe logarithm of the determinant whose zeros give the dispersion relation, and the dotted horizontal\nline indicates the position of a pole of 𝕒and of the determinant at Ω1,1. The right panel is a zoom-\nmagnified view close to κδ=π. The computed value of the slope is TD= 17.240\n−2.0 −1.8 −1.6 −1.4 −1.2 −1.0\nlog(π−κδ)−4.0−3.5−3.0−2.5−2.0log(|Ω−(Ω20+δΩ21|1/2)-\n+\n(π−κδ)2\nFigure 13: Absolute error near κδ=πbetween both dispersion relations (the exact dispersion relation\nsolved by FEM and the HFH approximation (43)) in a log-log scale (plain lines, blue for the lowest\nbranch “-” and red for the upper branch “+”). The dotted line is the reference order of convergence.\n193 Two-dimensional (2D) case\nWe now extend the results to the 2D case; as the method is very similar to the one used in 1D, we\nwill highlight only the differences due to the higher dimensions together with the final asymptotic\napproximations obtained.\n3.1 Setting\nWe consider the Helmholtz equation in a doubly periodic structure on a square lattice of size h, see\nFigure 14a,\n∇X·[ˆah(X,ω)∇XUh(X)] +ω2ˆbh(X,ω)Uh(X) = 0 (66)\nThe parameters are frequency-dependent inside ˆYiin the periodic unit cell, and are simply taken to\n(a) 2D configuration with zoom on a unit cell\n (b) The irreducible Brillouin zone, in wavenum-\nber space\nFigure 14: 2D configuration in physical space with zoom on an inclusion in the different coordinate\nsystems (a) and in wavenumber space (b)\nbe constants outside it. As in the 1D case the methodology is developed for any frequency-dependent\nfunction and the Drude-Lorentz model is used for numerical examples (see Appendix A for details on\nthis model).\nAs in 1D, we introduce the two-scales x=X/Landξ=X/h=x/δ; we callYthe unit cell in\nξ-coordinate, with Yithe inclusion where the parameters are frequency-dependent. Except for this\ngeometry difference, the non-dimensionalization step is the same as in 1D and a Bloch-Floquet analysis\nsimilar to Section 2.3 allows us to get the 2D dispersion relation.\n203.1.1 Ansatz\nWe pick a frequency-wavenumber pair (Ω0,κ)∈ℂ×[0,π/δ]2that satisfies the dispersion relation in\nthe irreducible Brillouin zone, see Figure 14b. The ansatz for the non-dimensionalized field uand\nfrequency Ω(3.6), together with the expansions for both aandb(17) are the same in 2D so we get\nthe following non-dimensionalized equation:\n/summationdisplay\nj⩾0/bracketleftig\na(ξ,Ω)/braceleftig\nδj∆ξuj+δj+1[∇ξ·∇xuj+∇x·∇ξuj] +δj+2∆xuj/bracerightig\n+/summationdisplay\nℓ⩾0δℓ+jΩ2\nℓb(ξ,Ω)uj] = 0.(67)\n3.1.2 Zeroth-order field\nCollecting the terms of order δ0, using continuity and periodicity, we get in Y:\n\n\nA0∆ξu0+ Ω2\n0B0u0= 0inY\nu0continuous at ∂Yiand∂Y\nA0∇ξu0·ncontinuous at ∂Yiand∂Y\nu0(x,ξ1+ 1,ξ2) = eiκ1δu0(x,ξ1,ξ2)andu0(x,ξ1,ξ2+ 1) = eiκ2δu0(x,ξ1,ξ2)(68)\n3.2 High-frequency homogenization for single eigenvalues at the edges of the Bril-\nlouin zone\nWestartwiththecaseoftheedgesoftheBrillouinzone Γ,Xand M,forwhichκ= (0,0),(π/δ,0),(π/δ,π/δ ),\nrespectively. We choose Ω0which is assumed to be a simple eigenvalue associated to the eigenfunction\nU0(ξ,Ω0). The zeroth-order field then writes\nu0(x,ξ) =f0(x)U0(ξ,Ω0),\nwheref0(x)has to be determined.\n3.2.1 First-order field\nFor this single eigenvalue case, we assume that Ω1= 0and we are looking for the quadratic term Ω2.\nCollecting the terms of order δ, we get inY:\nA0(∆ξu1+ 2∇ξ·∇xu0) + Ω2\n0B0u1= 0 (69)\ntogether with:\n•u1(x,1,ξ2) =±u1(x,0,ξ2)andu1(x,ξ1,1) =±u1(x,ξ1,0),\n•continuity for u1at∂Yiand∂Y,\n•continuity for [A0(∇xu0+∇ξu1)]·nat∂Yiand∂Y.\nThen, we write u1as:\nu1(x,ξ) =f1(x)U0(ξ) +∇xf0(x)·V(ξ) (70)\nwhereV= (V1,V2)with forj= 1,2:\n\n\nA0∆Vj+ Ω2\n0B0Vj=−2A0∂U0\n∂ξjinY\nVj(1,ξ2) =±Vj(0,ξ2)andVj(ξ1,1) =±Vj(ξ1,0)\ncontinuity for Vjat∂Yiand∂Y\ncontinuity for A0(∇Vj+U0ej)·nat∂Yiand∂Y.(71)\n213.2.2 Second-order field\nCollecting terms of order δ2, we get inY:\nA0(∆xu0+ 2∇ξ·∇xu1+ ∆ξu2) +A2∆ξu0+ Ω2\n0(B0u2+B2u0) + Ω2\n2B0u0= 0 (72)\ntogether with:\n•u2(x,1,ξ2) =±u2(x,0,ξ2)andu2(x,ξ1,1) =±u2(x,ξ1,0)\n•continuity for u2at∂Yand∂Yi\n•continuity for [A0(∇ξu2+∇xu1) +A2∇ξu0]·nat∂Yand∂Yi.\nWe introduce the average operator ⟨·⟩in 2D\n⟨f⟩=/integraldisplay\nYf(ξ)dξ\nand then consider the expression\n⟨u2×(68)−u0×(72)⟩) = 0.\nAfter integration by parts, some algebra and dividing through by f0we get the final effective equation:\nTij∂2f0\n∂xi∂xj+ Ω2\n2f0= 0 (73)\nwith\nTij=/angbracketleftig\nA0/parenleftig\nVi∂U0\n∂ξj−∂Vi\n∂ξjU0−U2\n0δij/parenrightig/angbracketrightig\nS(Ω0,U0,U0)(74)\nwhere we defined in 2D\nS(Ω0,f,g) =/angbracketleftbigg/parenleftbigg\nb(·,Ω0) +Ω0\n2∂b\n∂Ω(·,Ω0)/parenrightbigg\nfg−1\n2Ω0∂a\n∂Ω(·,Ω0)∇f·∇g/angbracketrightbigg\n(75)\nand where we sum over the repeated subscript indexes.\nFrom this effective equation, we also get the quadratic term for the dispersion relation\nΩ2\n2= ˜κiTij˜κj (76)\nwith ˜κi=κi−diwheredi= 0orπ/δdepending on the high-symmetry point we choose.\nThe tensorT(74) encapsulates the effective properties of the periodic structure beyond the quasi-\nstatic, classical homogenization, regime. Typically, this tensor may have eigenvalues of markedly\ndifferent magnitude or of opposite sign, which leads to a change of character of the underlying effective\nequation, from elliptic to parabolic and from elliptic to hyperbolic, respectively. The former appears\nat a frequency near band edges and the latter near a frequency at which a saddle point occurs in the\ncorresponding dispersion curves. This has been used notably to design dielectric photonic crystals\nwith spectacular directive emission in the form of + and x wave patterns for a source placed inside\nin the microwave regime [47]. The present high-frequency algorithm makes possible the extension of\nsuch experiments to the optical wavelengths wherein the periodic assembly of dielectric rods has a\nfrequency dependent refractive index.\n223.3 High-frequency homogenization for repeated eigenvalues at the edges of the\nBrillouin zone\nIn this section, we still consider approximation around the edges of the Brillouin zone, but for the\ncase of repeated eigenvalues that gives rise to a linear approximation of the dispersion diagram.\n3.3.1 Zeroth-order field\nSystem (68) still holds but now we assume repeated eigenvalues of multiplicity N. We introduce the\nassociated eigenfunctions U(j)(j= 1,...,N). The solution for the leading-order problem is now\nu0(x,ξ) =N/summationdisplay\nj=1f(j)\n0(x)U(j)\n0(ξ). (77)\nWe will denote (68)(l)the system (68) satisfied by u(l)\n0=f(l)\n0U(l)\n0forl∈{1,...,N}.\n3.3.2 First-order field\nThe main difference is that now Ω1̸= 0, therefore the system for the first-order field is modified and\ncollecting the terms of order δwe now get inY:\nA0(∆ξu1+ 2∇ξ·∇xu0) + Ω2\n0(B0u1+B1u0) + Ω2\n1B0u0+A1∆ξu0= 0 (78)\ntogether with:\n•u1(x,1,ξ2) =±u1(x,0,ξ2)andu1(x,ξ1,1) =±u1(x,ξ1,0),\n•continuity for u1at∂Yiand∂Y,\n•continuity for [A0(∇xu0+∇ξu1)]·n+A1∇ξu0·nat∂Yiand∂Y.\nLet us pick one l∈{1,...,N}and compute⟨(68)(l)×u1−(78)×u(l)\n0⟩. By integrating by parts, using\ndifferent continuity conditions and dividing through by f(l)\n0we get the effective equation:\n/summationdisplay\nj̸=l⟨A0Wjl\n0⟩·∇xf(j)\n0=−Ω2\n1/summationdisplay\njS(Ω0,U(l)\n0,U(j)\n0)f(j)\n0 (79)\nwith\nWjl\n0=U(l)\n0∇ξU(j)\n0−U(j)\n0∇ξU(l)\n0. (80)\nWe setf(l)\n0=ˆf(l)\n0exp(i˜κjxj)and get the following system of equations:\nCˆF0=0 (81)\nwith the 2×2matrixCdefined by\nClj= i⟨a(·,Ω0)Wjl\n0⟩·˜κ+ Ω2\n1S(Ω0,U(l)\n0,U(j)\n0). (82)\nThe value of Ω2\n1is then obtained by solving det(C) = 0.\nRemark 1 For double eigenvalues we get the following expression for the linear term (opposite slopes)\nΩ2\n1=±˜κ·⟨a(·,Ω0)W12\n0⟩\n/parenleftig\nS(Ω0,U(1)\n0,U(1)\n0)S(Ω0,U(2)\n0,U(2)\n0)−(S(Ω0,U(1)\n0,U(2)\n0))2/parenrightig1/2. (83)\n233.4 High-frequency homogenization for nearby eigenvalues at the\nedges of the Brillouin zone\nIn this section, we again consider asymptotic approximations around the edges of the Brillouin zone,\nand we assume that the eigenvalues are single but close to each other. More precisely, we consider N\neigenvalues close to each other so that the distances between them scale into the small parameter δ\nand write\nδαl= (Ω(l)\n0)2−(Ω(1)\n0)2, (84)\nforl∈{1,...,N}. To take into account their competitive nature, we assume that the leading order\nfield is\nu0(x,ξ) =N/summationdisplay\nj=1f(j)\n0(x)U(j)\n0(ξ). (85)\nAs in 1D, the ansatz is considered around the eigenvalue Ω(1)\n0.\nIn that case, the residual term for the zeroth order equation is\nδ/summationdisplay\nq̸=1αqu(q)\n0/bracketleftigg\n1\n2A(1)\n0Ω(1)\n0B(1)\n0∂a\n∂Ω(·,Ω(1)\n0)−B(1)\n0−1\n2Ω(1)\n0∂b\n∂Ω(·,Ω(1)\n0)/bracketrightigg\n(86)\nthat will in turn modify the equation for the first order field in Yto be:\nA(1)\n0(∆ξu1+ 2∇ξ·∇xu0) + (Ω(1)\n0)2(B(1)\n0u1+B1u0) + Ω2\n1B(1)\n0u0+A1∆ξu0\n+/summationdisplay\nq̸=1αqu(q)\n0/bracketleftigg\n1\n2A(1)\n0Ω(1)\n0B(1)\n0∂a\n∂Ω(·,Ω(1)\n0)−B(1)\n0−1\n2Ω(1)\n0∂b\n∂Ω(·,Ω(1)\n0)/bracketrightigg\n= 0,(87)\ntogether with u1(x,1,ξ2) =±u1(x,0,ξ2)andu1(x,ξ1,1) =±u1(x,ξ1,0), continuity for u1at∂Yiand\n∂Y, and continuity for [A0(∇xu0+∇ξu1)]·n+A1∇ξu0·nat∂Yiand∂Y.\nLet us pick one l∈{1,...,N}and consider⟨(68)(l)×u1−(87)×u(l)\n0⟩= 0. By integrating by parts,\nusing the different continuity conditions, dividing through by f(l)\n0, and neglecting the higher-order\nterms we get the effective equation\nN/summationdisplay\nj=1⟨A0Wjl\nnby⟩·∇xf(j)\n0=N/summationdisplay\nj=1{−Ω2\n1⟨S(Ω(1)\n0,U(l)\n0,U(j)\n0)⟩+αjNjl}f(j)\n0 (88)\nwith\n\nWjl\nnby=U(l)\n0∇ξU(j)\n0−U(j)\n0∇ξU(l)\n0,\nNjl=/angbracketleftigg\nb(·,Ω(1)\n0)/parenleftigg\n1 +Ω(1)\n0\n2∂\n∂Ω[log(b/a)](·,Ω(1)\n0)/parenrightigg\nU(j)\n0U(l)\n0/angbracketrightigg\n.(89)\nThe linear term Ω1of the dispersion relation is given by solving det(Cnby) = 0where theN×N\nmatrixCnbyis defined by\n(Cnby)lj= i/angbracketleftig\na(·,Ω(1)\n0)Wjl\nnby/angbracketrightig\n·˜κ+ Ω2\n1S(Ω(1)\n0,U(l)\n0,U(j)\n0)−αjNjl. (90)\nWe can notice that we recover the repeated eigenvalues case when the distances αjtend to 0.\nRemark 2 In the case of two nearby eigenvalues the dispersion relation is given by\nΩ4\n1/bracketleftig\nS(Ω(1)\n0,U(1)\n0,U(1)\n0)S(Ω(1)\n0,U(2)\n0,U(2)\n0)−(S(Ω(1)\n0,U(1)\n0,U(2)\n0))2/bracketrightig\n+ Ω2\n1α2/bracketleftig\n−N22S(Ω(1)\n0,U(1)\n0,U(1)\n0) +N21S(Ω(1)\n0,U(1)\n0,U(2)\n0)/bracketrightig\n−/parenleftig\n⟨a(·,Ω(1)\n0)W21\nnby⟩·˜κ/parenrightig2−iα2N21⟨a(·,Ω(1)\n0)W21\nnby⟩·˜κ= 0.(91)\n243.5 High-frequency homogenization for simple eigenvalues inside the Brillouin\nzone (no damping)\nThen, when no damping is considered, we are able to get linear asymptotic approximations near κ⋆\nwhich is not one of the high-symmetry points. The method being very similar to the 1D case, we only\ngive the effective equation\nTint·∇f0−iΩ2\n1f0= 0 (92)\nwhereTint= (Tint\n1,Tint\n2)withTint\nigiven, fori= 1,2by:\nTint\ni=2⟨A0ℑ(∂U0\n∂ξi¯U0)⟩\nS(Ω0,U0,¯U0). (93)\nAnd consequently, the dispersion relation reads\nΩ2\n1=Tint·(κ−κ⋆). (94)\n3.6 Low-frequency case\nFinally, we also obtain the classical low-frequency homogenized equation. In that case, we consider\nΩ0= 0in (68), which leads to the fact that U0is uniform, sayU0= 1. Then, we write u1(x,ξ) =\n∇xf0(x)·V(ξ)whereu1satisfies (71) with Ω0= 0andu0=f0. ThereforeVsatisfies (71) with\nΩ0= 0andU0= 1. Integrating (72) on a unit cell for Ω0= 0then leads to the usual homogenized\nequation/angbracketleftbigg\nA0/parenleftbigg\nδij+∂\n∂ξiVj/parenrightbigg/angbracketrightbigg∂2\n∂xi∂xjf0(x) + Ω2\n2⟨B0⟩f0(x) = 0 (95)\ntogether with the dispersion relation\nΩ2\n2=/angbracketleftbigg\nA0/parenleftbigg\nδij+∂\n∂ξiVj/parenrightbigg/angbracketrightbigg\n⟨B0⟩−1κiκj. (96)\nIt is well-known one can identify the homogenized matrix in (95) making use of the acoustic band\nnear the origin [18, 21], albeit for non-dispersive media (see section 3.2 in [21] for a summary of results\npublished back in 1978 in the first edition of [18]).\n3.7 Numerical example\nWe consider here a two-dimensional lattice of dispersive rods as studied by Brûlé et al. [41]. The\nmaterial parameters are the permittivity εgiven by a single resonance Drude model ε(Ω) = 1−\nΩ2\np/(Ω(Ω + iγ)), with Ωp/2π= 1.1,γ/2π= 0.05, and non magnetic material with permeability µ= 1.\nIn the TM polarization (s-polarization) case this corresponds in our notations to 𝕒= 1/µand𝕓=ε,\nwhereas in the TE polarization (p-polarization) case 𝕒= 1/εand𝕓=µ. The dispersion diagrams\nare computed by FEM from the exact dispersion function and here again will be denoted as the exact\ndispersion diagrams by opposition to the asymptotic approximations obtained by HFH.\nSingle eigenvalues at symmetry points The first example is a square array of period hof square\nrods of size L= 0.806hmade of this Drude permittivity in vacuum. Assuming first TM polarization,\nwe plot the dispersion along the edge of the first Brillouin zone on Figure 15 for the first three modes.\nThe results obtained by HFH in Section 3.2 approximate well the dispersion behaviour locally around\nthe symmetry points. The spectral features showing deformed triangles in the complex plane and\nincluding an intertwining of the second and third band is well recovered by the HFH asymptotic\napproximations (see panel (c) in Figure 15).\n25Γ X M Γ5.05.56.06.57.07.58.0Re Ω\nexact\nHFH(a) Real part\nΓ X M Γ−0.12−0.10−0.08−0.06−0.04Im Ω\nexact\nHFH (b) Imaginary part\n5 6 7 8\nRe Ω−0.12−0.10−0.08−0.06−0.04Im Ω\nΓXMΓ\nXΓX\nMexact\nHFH (c) Complex plane\nFigure 15: Simple eigenvalue HFH approximations (76) (dashed lines) for the first three bands of the\nsquare photonic crystal in TM polarization.\nNearby eigenvalues We now make use of the linear asymptotic approximations of Section 3.4 for\nsingle eigenvalues near edges of the Brillouin zone. Our focus is the cluster of four eigenfrequencies\nalong ΓXclose to the point Γ: the results of FEM computations are displayed on Fig. (16). The exact\ndispersion curves (solid lines) are correctly approximated by the single eigenvalue HFH model (dashed\ncurves), and even better so by the nearby case (dotted lines). Indeed the nearby approximation leads\nto a better prediction of the local behaviour of bands, and as in the 1D numerical example this is\nparticularly striking for the imaginary parts. The dispersion diagram in the complex plane and the\ncorresponding asymptotic approximations are reported in Fig. (17).\n0.0 0 .2 0 .4 0 .6 0 .8 1 .0 1 .2\nκxδ9.09.510.010.511.011.512.0Re Ωexact\nsingle\nnearby\n(a) Real part for a cluster of four eigenvalues around\nΓpoint.\n0.0 0 .2 0 .4 0 .6 0 .8 1 .0 1 .2\nκxδ−0.07−0.06−0.05−0.04−0.03Im Ωexact\nsingle\nnearby(b) Imaginary part for a cluster of four eigenvalues\naround Γpoint.\nFigure 16: Comparison of the exact dispersion relation approximated by FEM (solid lines) and the\neffective one obtained by HFH for the single eigenvalue case (76) (dashed lines) and with nearby\napproximations (90) (dotted lines).\nSimple eigenvalues inside the Brillouin zone For eigenvalues not located at symmetry points,\nwe study the same structure but setting γ= 0. Making use of the linear approximation, (94), obtained\n269.5 10 .0 10 .5 11 .0 11 .5\nRe Ω−0.065−0.060−0.055−0.050−0.045−0.040Im Ωexact\nsingle\nnearbyFigure 17: Comparison in the complex plane of the exact dispersion relation computed by FEM (solid\nlines) and the effective one obtained by HFH for the single eigenvalue case (76) (dashed lines) and with\nnearby approximations (90) (dotted lines) around Γpoint. The circle markers correspond to κxδ= 0.\nin Section 3.5, we recover locally the behaviour of the bands (see Figure 18).\nΓ X M Γ01234Ω\nexact\nHFH\n(a) TE polarization\nΓ X M Γ4567891011Ω\nexact\nHFH (b) TM polarization\nFigure 18: Superposition of the exact dispersion diagram (plain lines) and the asymptotic approxima-\ntions obtained by HFH (dashed lines) for points inside the Brillouin zone (94).\nRepeated eigenvalues For certain choices of the dispersive behaviour and material distribution,\nthere exist accidental degeneracies. In this section we choose circular inclusions with R= 0.364hand\n27use the same Drude model for the permittivity with Ωp/2π= 1.434andγ= 0. The band structure\nin TM polarization is represented on Figure 19, where we can see around Γpoint the coalescence of\nfour bands around Ω = 13.4. The repeated eigenvalue approximation obtained by HFH (see Section\n3.3) around this point shows two linear terms with opposite slopes Ω2\n1=±24.64κxand two flat bands\nwith slope close to zero that approximate well the exact curves.\n0.0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0\nκxδ91011121314151617Re Ω\nexact\nrepeated\nFigure 19: Comparison of the exact dispersion relation (red solid lines) and the effective one obtained\nby HFH for the repeated eigenvalue case (82) (black dashed lines) around Γpoint.\nField approximation HFH theory has been developed in this paper for periodic infinite systems.\nHowever, in this paragraph, we use the asymptotic approximations obtained to describe finite-size\nsystems, neglecting therefore the boundary effects: we now consider a 14 by 14 square array of rods\nwith the same Drude permittivity as for the approximation of Figure 15 but with a smaller damping\ntermγ/2π= 0.005. The finite photonic crystal is excited by a line source at the center with frequency\nclose to the real part of an eigenfrequency of the periodic system near symmetry point X. The first\nexampleis an arrayof squarerods ofsize L= 0.806hin TMpolarization(see Figure20). Atthe chosen\nfrequency, the real parts of the coefficients Tiiin the effective tensor of the HFH theory (74) of Section\n3.2 have opposite sign, ℜ(T11) =−2.108andℜ(T22) = 0.876and are of the same order of magnitude,\nleading to an effective hyperbolic behavior. The predicted theoretical wave field distribution is shown\nin Figure 20b: the wave propagation is highly directive and is aligned along the diagonals of the\nsystem. The resulting X-shape effect comes from the superposition of two effective media, for points\nX(π,0)and M(0,π)of the Brillouin zone. The full numerical simulations shown in Figure 20a share\nthis same qualitative feature.\nNext we study the TE polarization case for cylindrical rods of diameter D= 0.91h. In this case, HFH\npredicts a distinct anisotropy aligned along the lattice axis, since |ℜ(T11)|≪|ℜ (T22)|. This is indeed\nwhat we observe on full wave simulations shown in Figure 21a with the same qualitative agreement\nfor the solution of the effective parabolic equation shown in Figure 21b. This directive emission due\nto the excitation of a surface plasmon-like mode, where the field is mostly confined at the interface\nbetweenthedielectricbackgroundandtheDrudemetal, iswellcapturedbythedispersiveHFHtheory,\nincluding the field decay as a result of material losses.\n28−5 0 5\nx/h−6−4−20246y/h|u|, Ω = 5 .496\n0.00.20.40.60.81.0(a) Photonic crystal\n−5 0 5\nx/h−6−4−20246y/h|u|, Ω = 5 .496\n0.00.20.40.60.81.0 (b) HFH\nFigure 20: Field reconstruction for a finite square array of 196 square rods in TM polarization excited\nby a line source at the center with frequency Ω = 5.496, close to the real part of an eigenfrequency\nof the periodic system Ω0= 5.442−0.008iat point X. The dynamic effective tensor (74) is T=\ndiag(−2.108−0.018i,0.876 + 0.003i), indicating hyperbolic behaviour. In both figures the norm of\nthe field has been normalized to its maximum value.\n−5 0 5\nx/h−6−4−20246y/h|u|, Ω = 3 .509\n0.00.20.40.60.81.0\n(a) Photonic crystal\n−5 0 5\nx/h−6−4−20246y/h|u|, Ω = 3 .509\n0.00.20.40.60.81.0 (b) HFH\nFigure 21: Field reconstruction for a finite square array of 196 circular rods in TE polarization excited\nby a line source at the center with frequency Ω = 3.509, close to the real part of an eigenfrequency\nof the periodic system Ω0= 3.506−0.010iat point X. The dynamic effective tensor (74) is T=\ndiag(0.005 + 3×10−5i,0.950−0.012i), indicating parabolic behaviour. In both figures the norm of\nthe field has been normalized to its maximum value.\n294 Conclusion\nIn this paper, we have extended the technique of high-frequency homogenization to dispersive peri-\nodic media, within which the physical properties do depend on the frequency. The work has been\nperformed in both 1D and 2D and we considered different cases depending on the nature of the point\naround which we want to build an effective approximation. Near the edges of the Brillouin zone, we\nperformed high-frequency homogenization for a frequency which is a simple eigenvalue, or a repeated\none, and we also considered the case where several single eigenvalues are close to each other. Far from\nthe edges, we developed an approximation in the case where no damping is considered.\nIn each of these cases, we were able to develop an approximation for both the dispersion diagram and\nthe envelope function which defines the wavefield at the zeroth order. These asymptotic approxima-\ntions come with an effective parameter or tensor that encapsulates the dispersive properties of the\nconsidered material. The results have been validated using comparisons with Finite Element Simula-\ntions for different configurations. We also discussed the interpretation of the effective parameter with\nrespect to the nature of the wavefield.\nPotential extensions of this work include pushing the asymptotics presented in this paper to higher\norders, or extending it to the case of waves in other periodic dispersive media: for example the full\nvector Maxwell’s equations (such as in photonic crystal fibres within which TE and TM waves are\nusually fully coupled in oblique incidence), the Navier equations for fully coupled in-plane pressure\nand shear waves in phononic crystals, the Kirchhoff-Love equations for flexural waves in thin plates,\nor arrays of resonators (such as Helmholtz resonators, high-contrasted inclusions [48, 10], or bubbles\nhosting Minnaert resonant frequencies [49]).\nA Expressions of aand bfor the Lorentz model\nPhysical parameters In the case of a Lorentz model, the physical parameters (2) and (3) are\nˆah(X,ω) =\n\na0 forX∈(0,ϕh)\nˆ𝕒(ω) =a∞\n1−/summationdisplay\np≥0ω2\np,1\nω(ω+ iˆγp,1)−ω2\nD,p,1\n−1\nforX∈(ϕh,h )(97)\nand\nˆbh(X,ω) =\n\nb0 forX∈(0,ϕh)\nˆ𝕓(ω) =b∞\n1−/summationdisplay\np≥0ω2\np,2\nω(ω+ iˆγp,2)−ω2\nD,p,2\nforX∈(ϕh,h )(98)\nrespectively, where ωp,istands for a plasmon frequency, ˆγpifor a damping coefficient, and ωD,p,ifor a\nLorentz resonant frequency. When all the ωD,p,iare zero, the Lorentz model becomes known as the\nDrude model.\nAdimensionalized parameters The adimensionalized physical parameters (6) and (7) for this\nmodel then read\na(ξ,Ω) =\n\n1 forξ∈(0,ϕ)\n𝕒(Ω) =a∞\na0\n1−/summationdisplay\np≥0Ω2\np,1\nΩ(Ω + iγp,1)−Ω2\nD,p,1\n−1\nforξ∈(ϕ,1)(99)\n30and\nb(ξ,Ω) =\n\n1 forξ∈(0,ϕ)\n𝕓(Ω) =b∞\nb0\n1−/summationdisplay\np≥0Ω2\np,2\nΩ(Ω + iγp,2)−Ω2\nD,p,2\nforξ∈(ϕ,1)(100)\nrespectively, with for i= 1,2:\nΩp,i=ωp,ih\nc0,ΩD,p,i=ωD,p,ih\nc0andγp,i=ˆγp,ih\nc0.\nB Finite element formulation\nUsing a Lorentz model, we write 𝕒(Ω) =Na(Ω)\nDa(Ω)and𝕓(Ω) =Nb(Ω)\nDb(Ω), whereNa,Da,Nb,Dbare polyno-\nmials of Ω. We have to solve the following eigenproblem\n∇·[a(ξ,Ω)∇u(ξ)] + Ω2b(ξ,Ω)u(ξ) = 0. (101)\nThe weak formulation of the problem is derived by multiplying Eq. (101) by the complex conjugate\nof a test function vand integrating the first term by part on the unit cell Y:\n−/integraldisplay\nYa(Ω)∇u(ξ)·∇v∗(ξ)dξ+ Ω2/integraldisplay\nYb(Ω)u(ξ)v∗(ξ)dξ= 0 (102)\nwhere the boundary term vanishes because of the quasi-periodic boundary conditions. We define for:\n𝔸0(u,v) =−/integraldisplay\nY\\Yi∇u(ξ)·∇v∗(ξ)dξ, 𝔸1(u,v) =−/integraldisplay\nYi∇u(ξ)·∇v∗(ξ)dξ\nand\n𝔹0(u,v) =/integraldisplay\nY\\Yiu(ξ)v∗(ξ)dξ, 𝔹1(u,v) =/integraldisplay\nYiu(ξ)v∗(ξ)dξ.\nPlugging the expression for aandbin (102) and rearranging we get:\nT(Ω,u,v) =Da(Ω)Db(Ω)/bracketleftig\n𝔸0(u,v) + Ω2𝔹0(u,v)/bracketrightig\n+Na(Ω)Db(Ω)𝔸1(u,v)\n+ Ω2Da(Ω)Nb(Ω)𝔹1(u,v) = 0\nwhich is a polynomial eigenvalue problem solved using the open source FEniCS finite element library\n[50] interfaced with the SLEPc eigensolver [51, 52].\nAcknowledgments\nMT and RA would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge,\nfor support and hospitality during the programme Mathematical theory and applications of multiple\nwave scattering where work on this paper was undertaken. This work was supported by EPSRC grant\nno EP/R014604/. BV and RVC are supported by the H2020 FET-proactive Metamaterial Enabled\nVibration Energy Harvesting (MetaVEH) project under Grant Agreement No. 952039. SG and RVC\nwere funded by UK Research and Innovation (UKRI) under the UK government’s Horizon Europe\nfundingguarantee[grantnumber10033143]. TheauthorsalsothankPingShengforfruitfuldiscussions\nabout the accumulation points.\n31References\n[1] Vasily Klimov. Nanoplasmonics . Taylor Francis, New York, 2014.\n[2] Stefan A. Maier. Plasmonics: Fundamentals and Applications . Springer Netherlands, 2007.\n[3] David R Smith, John B Pendry, and Mike CK Wiltshire. Metamaterials and negative refractive\nindex.science, 305(5685):788–792, 2004.\n[4] Jensen Li, Lei Zhou, C. T. Chan, and P. Sheng. Photonic band gap from a stack of positive and\nnegative index materials. Physical Review Letters , 90(8):083901, feb 2003.\n[5] P Y Chen, C G Poulton, A A Asatryan, M J Steel, L C Botten, C Martijn de Sterke, and\nR C McPhedran. 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SIAM Journal on Scientific Computing , 38(5):S385–S411, January 2016.\n35"    },    {        "title": "1902.09649v1.Resonant_absorption_as_a_damping_mechanism_for_the_transverse_oscillations_of_the_coronal_loops_observed_by_SDO_AIA.pdf",        "content": "arXiv:1902.09649v1  [astro-ph.SR]  25 Feb 2019Resonant absorption as a damping mechanism for the transver se\noscillations of the coronal loops observed by SDO/AIA\nJavad Ganjali1, Nastaran Farhang1, Shahriar Esmaeili2, Mohsen Javaherian3, and Hossein\nSafari1\n1 Department of Physics, University of Zanjan, University B lvd., Zanjan, 45371-38791,\nIran,\n2 Department of Physics and Astronomy, Texas A&M University , 4242 TAMU, University\nDr, College Station, TX 77840, USA,\n3 Research Institute for Astronomy and Astrophysics of Mara gha (RIAAM), Maragha,\n55134-441, Iran.\nAbstract\nSolar coronal loops represent the variety of fast, intermed iate, and slow normal mode oscil-\nlations. In this study, the transverse oscillations of the l oops with a few-minutes period and\nalso with damping caused by the resonant absorption were ana lyzed using extreme ultraviolet\n(EUV) images of the Sun. We employed the 171 data recorded by S olar Dynamic Observatory\n(SDO)/Atmospheric Imaging Assembly (AIA) to analyze the pa rameters of coronal loop oscil-\nlations such as period, damping time, loop length, and loop w idth. For the loop observed on 11\nOctober 2013, the period and the damping of this loop are obta ined to be 19 and 70 minutes,\nrespectively. The damping quality, the ratio of the damping time to the period, is computed\nabout 3.6. The period and damping time for the extracted loop recorded on 22 January 2013 are\nabout 81 and 6.79 minutes, respectively. The damping qualit y is also computed as 12. It can be\nconcluded that the damping of the transverse oscillations o f the loops is in the strong damping\nregime, so resonant absorption would be the main reason for t he damping.\nKey words: Sun: corona – Sun: magnetic fields – Sun: oscillations.\n1 Introduction\nThe field of coronal seismology has been developing during l ast decades. It seems to be\nheading toward revolution in the physics of the Sun. It means a very efficient instrument\nis achieved to explore the basic intrinsic physical paramet ers of the solar corona including\nmagnetic field, temperature, and density of plasma. The slo w, intermediate, and fast oscilla-\ntions of the solar coronal loops were detected by various typ es of space telescopes Abedini\n(2016, 2018); Aschwanden et al. (1999); Aschwanden, Schrij ver (2011); Dadashi et al. (2009);\nErdélyi, Taroyan (2008); Moortel, Brady (2007); Nakariako v (1999); Ofman, Wang (2002); Safari et al.\n(2007); Taran et al. (2014); Verth et al. (2007); Verwichte e t al. (2009); Wang, Solanki (2004);\nWang et al. (2003).\nFrom the theoretical point of view, the coronal seismology w as proposed by Uchida (1970)\nand Roberts et al. (1984) for the flux tube standing waves. Th e coronal seismology is based on\nthe dispersion relation defined for a plasma cylinder; a cyl inder with non-uniform plasma struc-\nture formed by a magnetic field Abedini et al. (2012); Andrie s et al. (2005); Edwin, Roberts\n(1983); Erdelyi, Fedun (2007); Esmaeili et al. (2015, 2017) ; Esmaeili et al. (2016); Farahani et al.\n1/10(2017); Goossens et al. (2002); Karami et al. (2002); Nakari akov (1999); Ofman, Aschwanden\n(2002); Ruderman, Roberts (2002); Safari et al. (2006); van Doorsselaere et al. (2004a,b).\nWith the advancement of technology, due to high-resolution spatial imaging, the new coro-\nnal seismological field entered the golden age of explorati ons. The first evidence for fast MHD\nkink mode was achieved from TRACE observation, which was bas ed on detecting the transverse\nloop movement oscillations with the theoretically expecte d periods of kink mode Schrijver et al.\n(2002). Some kink modes seem to be majorly in the transverse d irection Aschwanden et al.\n(1999); while other modes clearly oscillate in perpendicul ar direction Wang, Solanki (2004).\nThe second-order geometrical and physical effects of coron al oscillation have been theoreti-\ncally studied; nevertheless, the effect of curvature of loo ps on the oscillation period van Doorsselaere et al.\n(2004b), the impact of the elliptic transverse cross sectio ns on damping of oscillation Ruderman\n(2003), and also the effect of density stratification on the loop oscillation have been carefully\nspecified Andries et al. (2005); Dymova, Ruderman (2005); F arahani et al. (2014); Fathalian, Safari\n(2010); Grant et al. (2015); Mendoza-Briceno et al. (2004); Pascoe et al. (2018, 2017); Safari et al.\n(2007); Shukhobodskiy et al. (2018); Soler et al. (2011); Ve rth et al. (2007, 2010).\nThe observation of the first two harmonics of the horizontal ly polarized kink waves excited\nin the coronal loop system was reported by Guo et al. (2015). Z hang et al. (2016) also inves-\ntigated the evolution of two prominences (P1,P2)and two bundles of coronal loops (L1,L2).\nAnother development concerning the nondamping oscillatio ns at flaring loops was recently pub-\nlished by Li et al. (2018). As a seismological application, p eriods and damping rates of the fast\nsausage oscillations in multishelled coronal loops were in vestigated by Chen et al. (2015). Also,\nJin et al. (2018) recently studied the damping of two-fluid M HD Waves in stratified solar atmo-\nsphere.\nAnalysis of the transverse oscillations of loops (kink mode ) in an active flaring region in-\ndicates that the initiation of these oscillations is caused by a disturbance, movements from the\ncenter of a flare towards the outside at the velocity of 700 km /s, which can produce a shock\nwave Aschwanden (2006). The understanding of coronal loop o scillations and the mechanism\nunderlying their damping has been subjected to a vast studie s. Various damping mechanisms\nfor oscillations of the coronal loops have been discussed by Roberts (2000).\nHollweg, Yang (1988) were the first ones who investigated an d discussed the damping of\nkink oscillations caused by resonant absorption. A method t o analyze the dissipative processes\nin the regimes with the vicinity of the singularity has been d eveloped by Sakurai et al. (1991a),\nSakurai et al. (1991b), Goossens et al. (1992), and Goossens et al. (1995). Ruderman, Roberts\n(2002) rebuilt this idea. They considered this problem for a straight magnetic flux tube disturbed\nin cold plasma. The tube had a homogeneous core and a thin laye r of thickness lwhose density\nuniformly reduced from center to the tube boundary.\nResonant absorption occurs when the waves entered the flux t ube from the footpoints area\nare frequently reflected and the kink oscillation frequenc y of the tube becomes equal to the local\nAlfvén frequency in a place within the resonant layer of the t ube. Thus, resonantwill occur for\nthe standing waves, and the energy of these waves will be conv erted to the thermal energy of\nthe environment through the ohmic resisitivity and viscous dissipation.\nConsidering the fact that the damping time caused by resonan t absorption is about the\norder of (a/l)P, where l,a, and Pare length, radius, and period of the loop respectively,\nRuderman, Roberts (2002) employed a new proposed mechanism for the data observed by\nNakariakov (1999), and concluded that l/a=0.23. Goossens et al. (2006) used the mecha-\nnism proposed by Hollweg, Yang (1988), and Ruderman, Robert s (2002) in order to estimate\nthe amount of (l/a)for 11 damped loops. They obtained this value in the range 0 .16 to\n0.491. These answers were obtained by assuming l≪a. These results were inspired by\nvan Doorsselaere et al. (2004a) for elimination of the limit ation of l≪aas well as numeri-\ncally solving the damping of the loop. They came to the conclu sion that the difference between\nnumerical and analytical values for l/a≤1/3 is very small. Even for l≃a, the difference\nwas not more than 0 .25. Recently, Su et al. (2018) investigated the strength of t he magnetic\n2/10field using the densities obtained by the differential emis sion measure (DEM) method and they\nconcluded that the magnetic field decays during the oscilla tion.\nGoossens et al. (1992) studied the resonant absorption and o btained the ratio of the damping\nrate to the oscillation frequency for the long wavelengths i n case the magnetic field is constant\nand parallel to the axis of the tube everywhere. The ratio of t he damping time to the oscillation\nperiod is obtained 4.97, which has a value of about 3 to 5 based on observations. Eventually, we\nconclude that the resonant absorption is an acceptable mech anism for explaining the damping\nobserved in the transverse kink oscillations of coronal loo ps.\nThis paper is organized as follows, the method applied for ex tracting the oscillation is intro-\nduced in Section 2. In Section 3, we present the extracted res ults for frequencies and damping\ntimes of the loops recorded on 22 January 2013 and 11 October 2 013. The main conclusions is\nalso presented in Section 4.\n2 Method\nThe coronal loops of the Sun are curved and bright structures . The hot plasma trapped around\nthe magnetic field lines inside the loop leads to seem bright er than their surrounding environ-\nment. Due to factors such as fast oscillating waves caused by flares, coupling with oscillating\nmodes enforced by pressure, and pulses of driven hot plasma, these magnetic loops represent\nthe normal oscillation modes. The coronal loops have length s from a few mega meters to several\nhundred mega meters. The limitation of the spatial resoluti on capability of solar observatories\nin the EUV and X-ray pass bands has made it almost impossible t o observe and analyze the\ninternal structure of the thin loops Esmaeili et al. (2017); Esmaeili et al. (2016). Coronal seis-\nmology provides an alternative method for understanding th e physical and geometrical structure\nof the loops.\nPursuing higher spatial resolution, the Solar Dynamic Obse rvatory (SDO) spacecraft was\nlaunched in February 2010 to study the Sun interior, solar ma gnetic field, solar coronal hot\nplasma, and effects of photospheric phenomena on space weat her. In this study, we used the\nsolar data provided by the Atmospheric Imaging Assembly (AI A) instrument on-board the SDO\nwhich provides full-disk images of the Sun’s atmosphere wit h a time cadence of 12 seconds in\nvarious EUV pass bands. Therefore, the damping of loops are w e investigated using successive\nEUV images at 171 Å.\nTo this end, we will address the creation of space-time image s from consecutive EUV images\nof the coronal loops at a specified time interval. By means of the Gaussian function fitting to\neach time element of these images, parameters such as the spa tial oscillation amplitudes and\nthe width of the loops are extracted. Then, through analyzin g the spatial oscillation amplitudes\nof the loop, the periods and the damping time are obtained. In this study, consecutive images\nof loops on 11 October 2013 at 07:11:59 to 08:21:59 UT (Figure 1, a) and 22 January 2013\nat 02:20:00 to 03:41:00 UT (Figure 1, b) provided by http://j soc.stanford.edu are investigated.\nTo extract the oscillations for each data set, we applied the displacements correction due to the\ndifferential rotation of the Sun. To co-align the consecuti ve images to one reference, all data\nwere derotated (see Alipour, Safari (2015)).\nIn order to create the space time image, arbitrary number of p oints (based on length and\nwidth of the loop) are selected using the spline interpolati on in two directions to form a rectan-\ngular region perpendicular to the loop axis (as shown with gr een lines in Fig. 1). Averaging over\nintensities of distinct pixels on different rows within the appointed box creates an element of\nthe space time image for the desirable locations at a specifi c time. Performing the same proce-\ndure for each successive image results in the space time imag e (Fig. 2), in which the transverse\noscillating mode is extracted by averaging over the intensi ties perpendicular to the loop axis.\n3/10Figure 1. Partial image of the Sun at 171 Å provided by SDO/AIA. Left pan el: the observed\nloop on 11 October 2013. Right panel: the observed loop on 22 J anuary 2013.\n3 Results and Discussion\nIn the space-time image of Fig. 2, the kink oscillation modes of several loops observed on 22\nJanuary 2013 are shown. Considering the complexity of the si multaneous analysis of all these\noscillations, we just extract the most noticeable loop from this space-time image and address\nits parameters (Figure 3). This oscillations started at 07: 11:59 UT and ended at 08:21:59 UT. A\nGaussian function was employed to derive the oscillation am plitudes as follows:\nF(x,t)=f(t)exp/parenleftBigg\n−/parenleftbiggx−a(t)√\n2σ(t)/parenrightbigg2/parenrightBigg\n+b(t). (1)\nThis fitting is performed for each column of the space-time i mage. In this regard, the parameter\nf(t)is the index of intensity oscillation amplitudes, a(t)is the index of spatial oscillation am-\nplitudes, xis the length of the space-time rectangle created in pixel un it,σ(t)is related to the\nthe width of Gaussian function, and brepresents the background intensity. The loop width is\nalso obtained by w=2σ√\n2ln2.\n100 200 300 400 500 600\nTime × 12 (s)204060Space (pixel)\nFigure 2. The space-time image of the loop observed on 11 October 2013, obtaining from 650\nconsecutive EUV images provided by SDO/AIA.\nThe lengths of the two studied loops are obtained about 345 ±35 Mm (Figure 1, a) and\n195±20 Mm (Figure 1, b), respectively. Therefore, using the valu es of σ, the width of loops\nare 6.85 Mm and 3 .7 Mm, respectively.\n4/10100 200 300 400 500 600\nTime × 12 (s)204060Space (pixel)\nFigure 3. The extracted oscillating wave from the space-time image of Figure 1, corresponding\nto the loop observed on 11 October 2013.\nAt the next stage the period of studied loop is attainable by f itting the following func-\ntion ( A(t)) describing equilibrium position of the spatial oscillati on amplitudes (e.g., Su et al.\n(2018)):\nA(t)=a0+a1exp/parenleftbigg−(t−t0)\nτ/parenrightbigg\ncos/parenleftbigg2π(t−t0)\nP0+kt−φ/parenrightbigg\n+a2t−t0\np0, (2)\nwhere a0is the amplitude, tis time, and τrepresents damping time. The parameters P0,k, and\nφare the period of oscillation, the evolution rate, and the os cillation phase, respectively. The\nspatial oscillation amplitudes of the loop recorded on 11 Oc tober 2013 and the corresponding\nfit are represented in Fig. (4).\nAccording to the fit results, τis obtained about 70 minutes and the oscillation period is\nobtained around 19 minutes. The damping quality, the ratio o f the damping time to the period\nof oscillation (Q=τ\nP)Ruderman, Roberts (2002); Safari et al. (2006), is computed about 3.6.\nIt seems that the main mechanism for the strong damping of the loop oscillation is the resonant\nabsorption.\nWe have also studied the loop recorded on 22 January 2013 at 02 :20:00 UT (Figure 1, b),\nfollowing the same procedure described earlier. The oscill ation of this loop started at 02:20:00\nUT and ended at 03:41:00 UT. The the space-time image corresp onding to this data set is shown\nin Figure 5. The period and damping time are obtained about 6. 79 and 81 minutes, respectively.\nThe damping quality is computed about 12. Figure (6) represe nts the result of fitting Eq. (2) to\nthe spatial oscillation amplitudes of the loop over the inte rval 720 to 3182 seconds.\n0 500 1000 1500 2000 2500 3000 3500 4000 4500\nTime (s)10152025303540Amplitude (pixel)Data points\nFitted curve\nFigure 4. The spatial oscillation amplitudes of the loop recorded on 1 1 October 2013, and the\nresult of fitting Eq. (2) to the amplitudes.\n5/1050 100 150 200 250 300 350 400 450\nTime × 12 (s)2040Amplitude (pixel)\nFigure 5. The space-time image belonging to the loop recorded on 22 Jan uary 2013 obtained\nfrom 450 consecutive EUV images provided by SDO/AIA.\n0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000\nTime (s)15202530354045Amplitude (pixel)\nFitted curve.     Data points\nFigure 6. The spatial oscillation amplitudes of the loop recorded on 2 2 January 2013, and the\nresult of fitting Eq. (2) to the amplitudes over the interval 720 to 3182 seconds.\n4 Conclusions\nIn this paper, we studied the period and the damping time of tr ansverse kink oscillations using\nsuccessive EUV images for two loops recorded on 11 October 20 13 (loop 1) and 22 January\n2013 (loop 2) provided by SDO/AIA. The spatial oscillation a mplitudes for each data set are\nderived using the appropriate Gaussian function (Eq. 1). Th e lengths of the loop 1 and 2\nare calculated 345 ±35 Mm and 195 ±20 Mm, respectively. We determined that the period\nand the damping time for the loop 1 are about 19 and 70 minute, r espectively. For the loop\n2, the period and damping time are also calculated about 81 an d 6.79 minute, respectively.\nTherefore, the damping quality for the loop 1 and 2 are 12 and 3 .6, respectively. According\nto the obtained damping qualities for each studied case, whi ch can be classified as the strong\ndamping, we conclude that the resonant absorption may be the main mechanism for damping\nof these oscillations.\nSeveral damping mechanisms (e.g., non-ideal MHD effects, l ateral wave leakage, footpoints\nwave leakage, phase mixing and resonant absorption) were in vestigated for the damping of\ncoronal loop oscillations (e. g. Aschwanden (2005)) . The re sults of the present study and also\nmany other previous research show that the kink mode oscilla tion of coronal loops are damped\ndue to the resonant absorption in the strong damping regime. From the theoretical point of\nview (e. g., Ruderman, Roberts (2002), Safari et al. (2007)) , at a thin resonant layer (boundary\n6/10layer) at the lateral boundary of the coronal loops with the i nhomogeneous density along the\ncross section, the energy of the kink mode oscillations can b e transferred to the localized Alfvén\nwaves. This energy may heat up the coronal loops to several mi llion kelvins.\nAcknowledgements\nThe authors thank NASA/SDO and the AIA science team for provi ding data publicly available.\nReferences\nAbedini A. Phase speed and frequency-dependent damping of longitudin al intensity oscillations\nin coronal loop structures observed with AIA/SDO // Astroph ysics and Space Science. mar\n2016. 361, 4.\nAbedini A. 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IX 2004b . 424. 1065–1074.\n10/10"    },    {        "title": "2311.10053v1.Near_optimal_Closed_loop_Method_via_Lyapunov_Damping_for_Convex_Optimization.pdf",        "content": "Near-optimal Closed-loop Method via Lyapunov Damping\nfor Convex Optimization\nSeverin Maier Camille Castera∗\nDepartment of Mathematics\nUniversity of T ¨ubingen\nGermanyPeter Ochs\nDepartment of Mathematics and Computer Science\nSaarland University\nGermany\nABSTRACT\nWe introduce an autonomous system with closed-loop damping for first-order convex\noptimization. While, to this day, optimal rates of convergence are only achieved by\nnon-autonomous methods via open-loop damping ( e.g., Nesterov’s algorithm), we show\nthat our system is the first one featuring a closed-loop damping while exhibiting a rate\narbitrarily close to the optimal one. We do so by coupling the damping and the speed\nof convergence of the system via a well-chosen Lyapunov function. We then derive\na practical first-order algorithm called LYDIA by discretizing our system, and present\nnumerical experiments supporting our theoretical findings.\n1 Introduction\nWe consider the unconstrained minimization of a smooth convex real-valued and lower-bounded func-\ntionfover a Hilbert space H:\nmin\nx∈Hf(x).\nOne of the most famous algorithms for such optimization problems is Nesterov’s Accelerated Gradient\nmethod (NAG) [25], which is known to achieve the optimal rate of convergence for first-order methods\non convex functions with Lipschitz-continuous gradient. Among several ways to explain the efficiency\nof NAG, Su et al. [31] studied the algorithm through the lens of Ordinary Differential Equations (ODEs)\nand proposed the following model:\n¨x(t) +a\nt˙x(t) +∇f(x(t)) = 0 ,∀t≥t0, (A VD a)\nwhere a > 0,t0≥0and∇fdenotes the gradient of the smooth real-valued convex function f.\nHere, ˙xdef=dx\ndt, respectively ¨xdef=d2x\ndt2, denotes the first, resp. second, time-derivative (or velocity,\nresp. acceleration) of the solution xof the ODE. A VD stands for Asymptotically Vanishing Damping\n∗Corresponding author: camille.castera@protonmail.comarXiv:2311.10053v1  [math.OC]  16 Nov 2023and relates to the coefficient a/t. NAG is obtained by (non-straightforward) discretization of (A VD a).\nConversely (A VD a) can be seen as NAG with infinitesimal step-sizes. Following [31], many works\nstudied the system (A VD a), and notably proved that when a >3, the function values along a solution\n(or trajectory) of (A VD a) converge with the asymptotic rate o\u00001\nt2\u0001\nto the optimal value as t→+∞\n[31, 6, 24]. This matches the rate of NAG in the case of discrete (or iterative) algorithms [25]. We\nfurther discuss additional results for (A VD a) and other possible choices of alater in Section 2.\nThe interest in the connection between ODEs (or dynamical systems) and algorithms comes from\nthe abundance of theory and tools for analyzing ODEs and the insights that they provide for under-\nstanding algorithms. In particular, continuous-time analyses relying on Lyapunov functions can often\nbe adapted to the discrete setting, see e.g., [6]. In this work, we are interested in a special case of the\nInertial Damped Gradient (IDG) system:\n¨x(t) +γ(t) ˙x(t) +∇f(x(t)) = 0 ,∀t≥t0, (IDG γ)\nwhere γis a positive function and is called “damping” by analogy with mechanics. There are two\ninherently different ways of designing the damping coefficient γ: the so-called open- and closed-loop\nmanners. While obtaining fast convergence rates o\u00001\nt2\u0001\nwith open-loop damping is known, for example,\nwhen γ(t) =a\ntwitha > 3, which exactly yields (A VD a), we propose the first closed-loop damping\nthat provides (IDG γ) with near-optimal rate of convergence. Our special instance of (IDG γ) is called\n(LD) and is introduced hereafter. We first discuss the problem setup in more details.\n1.1 Problem setting\nThe dampinga\ntin (A VD a) depends explicitly on the time variable t, making the system non-\nautonomous. We say that a damping with such explicit dependence on the time tisopen-loop . In\ncontrast, a damping γthat does not explicitly depend on tis called closed-loop and makes (IDG γ) an\nautonomous ODE (since the ODE is then independent of t0): the feedback for the system, in terms of\nγ, may only depend on the state x(and its derivatives) but not explicitly on the time t. For optimization,\nautonomous ODEs are often preferable over non-autonomous ones, since the dependence on the time\nt, and hence on the initial time t0, is removed. For example, even though the asymptotic rate of conver-\ngence of (A VD a) does not depend on t0, the trajectory does depend on the choice of t0: when choosing\na large t0, the damping a/tis very small at all time and the “damping effect” is almost completely lost.\nWe illustrate this in Figure 1 where we show how the choice of t0heavily influences the solution of\n(A VD a). Additionally some important tools for analyzing ODEs only hold (or are simpler to use) for\nautonomous ODEs, see e.g., the Hartman–Grobman Theorem [20, 19], which relates the behavior of\nautonomous ODEs around their equilibrium points to that of a linear system. Another such example\nis the so-called ODE method [23, 13], which formally states that algorithms with vanishing step-sizes\nasymptotically behave like solutions of corresponding ODEs.\nIn this work we therefore tackle the following open question:\nCan one design the damping γin(IDG γ)in a closed-loop manner (so as to make the ODE\nautonomous) while still achieving the same optimal convergence rate as (A VD a)?\nThis question hides a circular argument that makes it hard to solve. Indeed, according to Cabot\net al. [16], a sufficient condition on γfor the convergence of the function values is thatR+∞\nt0γ(t)dt=\n+∞. The straightforward choice is then the constant damping γ(t) =a >0, which in (IDG γ) yields\n2t0 t0 +500 t0 +1000\ntime t1015\n1012\n109\n106\n103\n100function values f(x(t))\n0.75\n 0.00 0.75\nx10.75\n0.000.75x2\nLD (ours)\nminimizer\ninitialization x0AVD with t0=1\nAVD with t0=10\nAVD with t0=1000Figure 1: Comparison of our autonomous system (LD) and the non-autonomous one (A VD a) for differ-\nent initial times t0on the 2D function f(x1, x2) =x4\n1+ 0.1x4\n2. The left plot shows the evolution of the\nfunction values over time. The right plot shows the trajectories in the space (x1, x2). Different initial\ntimes heavily affect the solution of (A VD a), but not (LD). We approximated the solutions using NAG\nfor (A VD a) and LYDIA (see Algorithm 1) for (LD), both with very small step-sizes.\nthe Heavy Ball with Friction (HBF) algorithm [28], known to be sub-optimal for convex functions. We\ntherefore seek a closed-loop damping that converges to zero, but not “too fast”. In the open-loop setting,\nthe natural choice is γ(t) =a\nt, a > 0because this is the fastest polynomial decay that converges to\nzero while being non-integrable. Further [4] showed that any damping of the forma\ntβwithβ∈(0,1)\nyields a sub-optimal rate. Therefore we formally seek to design a closed-loop damping γthat behaves\nlike1\nt. Because it cannot depend directly on the variable t, our closed-loop damping γmust be built\nfrom other quantities of the system that may converge to zero, for example ∥˙x(t)∥.However, the rate\nat which such quantities converge to zero depend themselves on the choice of γ. To escape this loop\nof thought, we design γusing quantities that are known to behave asymptotically like1\ntin the case\nof (A VD a) and investigate whether this choice of γstill gives fast convergence rates. This idea takes\ninspiration from [9] which used ∥˙x(t)∥as damping, with a key difference that we now explain.\n1.2 Our contribution\nIn the case of (A VD a) with a >3, it is known that under standard assumptions the quantity E(t)def=\nf(x(t))−f⋆+1\n2∥˙x(t)∥2(where f⋆def= min Hf) is such that E(t) =o\u00001\nt2\u0001\n[24]. Therefore, following the\naforementioned intuition, we propose a closed-loop damping defined for all t≥t0byγ(t)def=p\nE(t).\nOur version of (IDG γ) then reads:\n¨x(t) +p\nE(t) ˙x(t) +∇f(x(t)) = 0 ,∀t≥t0, (LD)\n3and is called (LD) for Lyapunov Damping , since it will turn out later that Eis non-increasing, making\nit a so-called “Lyapunov function” for (LD). Note that (LD) assumes the availability of the optimal\nvalue f⋆, we further discuss this later in Remark 4.10. Our main result is then that our system (LD)\nwhich features a closed-loop damping yields the following rate of convergence.\nTheorem 1.1. Assume that fis a continuously differentiable convex function and that argminHf̸=∅.\nThen, for any bounded solution xof (LD) and for any δ >0,\nf(x(t))−f⋆=o\u00121\nt2−δ\u0013\n.\nThis means that the rate of our method is arbitrarily close to o\u00001\nt2\u0001\n, the optimal one achieved by\n(A VD a) with a >3. To the best of our knowledge, this is the first system with closed-loop damping\nfeaturing such rate. Additionally, note that (LD) does not make use of the hyper-parameter a, whose\nchoice is crucial in (A VD a).\n1.3 Organization\nIn Section 2 we review related work regarding ODEs for optimization, closed-loop damping, and known\nresults in the open-loop case. In Section 3 we make the setting precise and show the existence of\nsolutions to (LD). Section 4 is devoted to showing our main result. In Section 5 we derive a practical\nalgorithm from (LD) and use it to perform numerical experiments. We finish by drawing conclusions\nand further discussing our results.\n2 Related Work\nODEs for optimization. There is a long line of work in exploiting the interplay between ODEs and\noptimization algorithms, going back, at least, to the work of Polyak’s [28] Heavy Ball with Friction\n(HBF) method for acceleration. As previously stated, Su et al. [31] linked NAG [25] to the differential\nequation (A VD a), hence providing a new view on the heavily used, yet not perfectly understood algo-\nrithm. NAG is however obtained via a non-straightforward discretization of (A VD a) since the gradient\noffis evaluated at an “extrapolated point”. Recently, Alecsa et al. [2] proposed a model with “implicit\nHessian”, whose Euler explicit discretization yields NAG. Higher-order ODEs have also been intro-\nduced to better understand NAG: Attouch et al. [7] proposed a model with “Hessian damping” based\non [3], while Shi et al. [30] similarly considered higher-resolution ODEs allowing to better distinguish\nNAG from other (IDG γ) systems.\nClosed-loop damping. The closest work to ours is [9], which proposed a closed-loop damping of\nthe form γ=∥˙x∥pfor several values of p∈R. Our work builds on [9] since our damping√\nEalso\ninvolves ∥˙x∥. Yet, Attouch et al. [9] could provide counterexamples for which their systems do not\nachieve near-optimal rates, unlike ours. A key difference is that our damping√\nEis non-increasing,\nwhile ∥˙x∥may oscillate heavily. They nonetheless derived rates under additional assumptions ( e.g.\nstrong convexity or the Kurdyka-Łojasiewicz (KL) property).\nLin and Jordan [22] introduced more complex systems of closed-loop dampings relying on the\ngradient ∇fand the speed ˙x. This was followed by Attouch et al. [10] who considered a system that\n4is shown to generalize the systems in [22]. Further Attouch et al. [10] showed that their system can\nbe interpreted as a time-rescaled gradient flow combined with so-called “time averaging” techniques.\nTheir time-scaling is expressed in terms of the inverse of the norms of the speed ˙xand the gradient\n∇f. However, the improvement achieved by time rescaling cannot carry through discretization since\nit amounts to following the same trajectory but faster, which for discrete algorithms would boil down\nto increasing the step-size, yielding numerical instabilities. Nonetheless, by combining time rescaling\nand time averaging, they managed to build a practical method achieving acceleration compared to the\nO\u00001\nt\u0001\nrate of gradient flow, but still sub-optimal compared to (A VD a).\nAdly et al. [1] proposed to replace the term γ(t) ˙x(t)in (IDG γ) by a “non-smooth” potential.\nTheir non-smooth potential provides the solutions of their differential inclusion with finite length and\nconvergence to a point very close to a minimizer of f(if such a point exists). However, since the\nsolution does not converge to the minimal value, it does not have the optimal rate that we seek.\nOur damping makes use of the optimality gap f(x(t))−f⋆(see Remark 4.10 for further discus-\nsion). This idea is not new and is used, for example, in the Polyak step-sizef(xk)−f⋆\n∥∇f(xk)∥2[27]. In our\nsystem (LD) the optimality gap is rather used for designing the damping γrather than the step-size.\nOpen-loop damping and proof techniques. The convergence properties of (A VD a) (and more gen-\nerally (IDG γ) in the open-loop setting) have been intensively studied. First, for (A VD a) with a > 3,\nfollowing [31], the convergence rate o\u00001\nt2\u0001\nfor function values and the convergence of the trajectories\nwere proved in [24, 6, 17]. In the setting a < 3, [11, 8] derived the sub-optimal rate O\u0010\n1\nt2a\n3\u0011\n. For\nthe critical value a= 3only the rate O\u00001\nt2\u0001\nis known and whether this rate can be improved to o\u00001\nt2\u0001\nand whether the trajectories converge remains open. Rates have also been derived under additional\nassumptions such as the KL property [12].\nRegarding (IDG γ), Attouch and Cabot [4] developed general conditions for the convergence of the\nvalues and of the trajectories, which unify several results mentioned above. All these results have in\ncommon that they rely on the analysis of Lyapunov functions, like Epreviously introduced. We refer\nto [32, 33] for more details on Lyapunov analyses. The proof of our main result takes inspiration from\nthose in [15] where sufficient conditions to derive optimal rates are provided. They must nonetheless\nbe significantly adapted since some of the conditions in [15] do not hold generally for closed-loop\ndampings like ours. We replace them by using a specific property of our system (see Lemma 4.1\nhereafter). A more detailed comparison follows our main analysis in Remark 4.9.\n3 Preliminaries and Existence of Solutions\nThroughout the paper we fix a real Hilbert space Hwith inner product ⟨·,·⟩and induced norm ∥·∥.\nWe make the following assumptions on the function f.\nAssumption 1. The function f:H →Ris\n(i) convex and continuously differentiable with locally Lipschitz-continuous gradient ∇f;\n(ii) bounded from below by f⋆def= inf Hf.\nWe also fix an initial time t0≥0and initial conditions x(t0) =x0∈ H, and ˙x(t0) = ˙x0∈ H.\n5Definition 1. A function x: [t0,+∞[→ H , which is twice continuously differentiable on ]t0,+∞[and\ncontinuously differentiable on [t0,+∞[, is called a (global) solution ortrajectory to (IDG γ), resp. (LD),\nif it satisfies (IDG γ), resp. (LD), for all t > t 0and satisfies the initial conditions previously mentioned.\nGiven the setting above, we can ensure the existence and uniqueness of the solutions of (LD).\nTheorem 3.1. Under Assumption 1, there exists a unique solution xto (LD) with initial conditions\n(x0,˙x0)∈ H × H and initial time t0≥0.\nThe proof relies on the Picard–Lindel ¨of Theorem and is postponed to Appendix A.\nRemark 3.2. The local Lipschitz-continuity of ∇fis only required to guarantee the existence and\nuniqueness of the solutions on [t0,+∞[(see below), but is not used elsewhere in our analysis.\nWe finally recall a special case of the Landau notation for asymptotic comparison that we use\nheavily in the sequel.\nDefinition 2. For any non-negative function g:R→Rand∀α≥0,\ng(t) =o\u00121\ntα\u0013\n⇐⇒ lim\nt→+∞tαg(t) = 0 .\n4 Main Result: Convergence Rates for (LD)\nThis section is devoted to proving our main result Theorem 1.1, which states that the solution of (LD)\nachieves a convergence rate that is arbitrarily close to the optimal one for convex functions. First we\nshow that Eis indeed a Lyapunov function for the system (LD).\n4.1 The Lyapunov function of (LD)\nThroughout what follows, let xbe the solution of (LD) with the initial conditions stated in Section 3.\nRecall that the damping coefficient in (LD) isp\nE(t)where for all t≥t0,E(t) =f(x(t))−f⋆+\n1\n2∥˙x(t)∥2. We first show identities that are specific to (LD) and that play a crucial role in what follows.\nLemma 4.1. Under Assumption 1 the solution xof (LD) is such that Eis continuously differentiable\nfor all t≥t0and\ndE(t)\ndt=−p\nE(t)∥˙x(t)∥2, (1)\nso in particular Eis non-increasing. Furthermore, for all t≥t0it holds that\ndp\nE(t)\ndt=−1\n2∥˙x(t)∥2,or equivalently,Zt\nt0∥˙x(s)∥2ds= 2p\nE(t0)−2p\nE(t). (2)\nProof. We apply the chain rule and use the fact that xsolves (LD) to obtain:\ndE(t)\ndt=⟨∇f(x(t)),˙x(t)⟩+⟨¨x(t),˙x(t)⟩\n=D\n∇f(x(t))−p\nE(t) ˙x(t)− ∇f(x(t)),˙x(t)E\n=−p\nE(t)∥˙x(t)∥2,(3)\n6which proves (1). As for the second part, the chain rule and (1) imply\ndp\nE(t)\ndt=1\n2p\nE(t)dE(t)\ndt=−1\n2p\nE(t)p\nE(t)∥˙x(t)∥2=−1\n2∥˙x(t)∥2.\nFinally, by the Fundamental Theorem of Calculus and the continuity of ˙xthis is equivalent to\nZt\nt0∥˙x(s)∥2ds= 2p\nE(t0)−2p\nE(t).\nThe function Edescribes a quantity which is non-increasing along the trajectory x, and note that\nE(t) = 0 if, and only if, f(x(t)) =f⋆and˙x(t) = 0 . Such a function is called a Lyapunov function for\nthe system (LD).\n4.2 Preliminary convergence results\nWe make the following assumptions, which are consistent with those in Theorem 1.1.\nAssumption 2. We assume that\n(i)argminHf̸=∅;\n(ii) the solution xof (LD) is uniformly bounded on [t0,+∞[.\nRemark 4.2. Assumption 2-(ii) holds, for example, when fis coercive.\nWe begin our analysis by showing that the trajectory xminimizes the function f. We do so by\nshowing that E(t)tends to zero as t→+∞.\nTheorem 4.3. Under Assumptions 1 and 2, E(t)converges to zero as t→+∞. This implies in\nparticular that f(x(t))− − − − →\nt→+∞f⋆and∥˙x(t)∥ − − − − →\nt→+∞0.\nWe make use of the following classical result to prove Theorem 4.3.\nLemma 4.4. Ifgis a non-negative continuous function on [t0,+∞[andR+∞\nt0g(t)dtis finite, then either\nlim\nt→+∞g(t)does not exist or lim\nt→+∞g(t) = 0 .\nThe proof of Lemma 4.4 is postponed to Appendix B. We now prove Theorem 4.3.\nProof of Theorem 4.3. Letz∈argminHf, and for all t≥t0define the so-called “anchor function”\nhz(t)def=1\n2∥x(t)−z∥2. Then hzis twice differentiable for all t≥t0and we have:\n˙hz(t) =⟨x(t)−z,˙x(t)⟩. (4)\n¨hz(t) =∥˙x(t)∥2+⟨x(t)−z,¨x(t)⟩\n(LD)=∥˙x(t)∥2+D\nx(t)−z,−p\nE(t) ˙x(t)− ∇f(x(t))E\n=∥˙x(t)∥2+D\nx(t)−z,−p\nE(t) ˙x(t)E\n− ⟨x(t)−z,∇f(x(t))⟩\n(4)\n≤ ∥˙x(t)∥2−p\nE(t)˙hz(t) +f⋆−f(x(t)),(5)\n7where we used the first-order characterization of the convexity of fin the last step. So hz(t)fulfills the\nfollowing differential inequality:\n¨hz(t) +f(x(t))−f⋆+1\n2���˙x(t)∥2\n| {z }\n=E(t)≤ ∥˙x(t)∥2+1\n2∥˙x(t)∥2−p\nE(t)˙hz(t),\nor equivalently,\nE(t)≤3\n2∥˙x(t)∥2−¨hz(t)−p\nE(t)˙hz(t). (6)\nWe integrate (6) from t0toT > t 0:\nZT\nt0E(t)dt≤3\n2ZT\nt0∥˙x(t)∥2dt−ZT\nt0¨hz(t)dt−ZT\nt0p\nE(t)˙hz(t)dt\n= 3\u0010p\nE(t0)−p\nE(T)\u0011\n−˙hz(T) +˙hz(t0)−p\nE(t)hz(T) +p\nE(0)hz(t0)\n−1\n2ZT\nt0∥˙x(t)∥2hz(t)dt,(7)\nwhere we performed integration by parts on the last integral and used Lemma 4.1. Note that by the\nboundedness of Eand the continuity of f,f(x)−f⋆and∥˙x∥are uniformly bounded on [t0,+∞[.\nThis implies together with Assumption 2 that hzand ˙hzare uniformly bounded as well. Therefore,\n˙hz(T),p\nE(T)andp\nE(T)hz(T)are uniformly bounded from above for all T∈[t0,+∞[. Further\n−RT\nt0∥˙x(s)∥2hz(s)ds≤0and˙hz(t0)andp\nE(t0)hz(t0)are constants. So the right-hand side in (7) is\nuniformly bounded from above for all T∈[t0,+∞[. Therefore we deduce that\nZ+∞\nt0E(t)dt <+∞. (8)\nFinally, Eis non-negative and non-increasing by (1), so it converges to some value E∞∈[0, E(t0)]as\nt→ ∞ . Further since Eis continuous we can conclude by Lemma 4.4 that lim\nt→+∞E(t) = 0 .\n4.3 Rates of convergence for E\nThe proof of our main result Theorem 1.1 relies on the following lemma.\nLemma 4.5. Letα≥0: IfR+∞\nt0tαE(t)dt <+∞, then lim\nt→+∞tα+1E(t) = 0 .\nThis result follows from standard calculus arguments and is proved later in Appendix B.\nObserve that (8) already provides us with a first rate by applying Lemma 4.5 with α= 0.\nProposition 4.6. Under Assumption 1 and 2, it holds that\nE(t) =o\u00121\nt\u0013\n.\nSince Eis a sum of non-negative quantities, any convergence rate of Etranslates into a rate for\nf, hence in particular Proposition 4.6 implies f(x(t))−f⋆=o\u00001\nt\u0001\n. The proof of Theorem 1.1 follows\nsimilar steps as that of Theorem 4.3. The main idea is that by combining (8) with (2) we can improve\nthe rate from o\u00001\nt\u0001\ntoo\u00001\nt3/2\u0001\nand iteratively repeat this process. This is stated in the following theorem.\n8Theorem 4.7. Under Assumptions 1 and 2 for all 0< ε <1\n2and all n∈N≥1it holds that\nE(t) =o\u00121\ntαn−αn−1ε\u0013\n,where αndef= 2−\u00121\n2\u0013n−1\n. (9)\nBefore proving Theorem 4.7, we show that Theorem 1.1 is a direct consequence of this result.\nProof of Theorem 1.1. For any 0< δ < 1, choose 0< ε <δ\n2and observe that\nlim\nn→+∞αn= 2, hence lim\nn→+∞(αn−αn−1ε) = 2−2ε >2−δ,\nwhich shows that there exists N∈N≥1, such that αN−αN−1ε >2−δand therefore by Theorem 4.7\n0 = lim\nt→+∞tαN−αN−1εE(t)≥lim\nt→+∞t2−δE(t)≥0,\nhence\nlim\nt→+∞t2−δE(t) = 0 .\nFinally the case δ≥1is covered by Proposition 4.6 and, as previously discussed, the following impli-\ncation holds for all δ >0:\nE(t) =o\u00121\nt2−δ\u0013\n=⇒f(x(t))−f⋆=o\u00121\nt2−δ\u0013\n.\nIt now only remains to prove Theorem 4.7. We make use of the following lemma.\nLemma 4.8. Letgbe continuous and non-negative on [t0,+∞[. Then for any β >1,\nlim\nt→+∞tβg(t)exists and is finite =⇒Z+∞\nt0g(t)dt <+∞.\nThe proof of Lemma 4.8 is postponed to Appendix B\nProof of Theorem 4.7. Fix0< ε <1\n2. We show (9) by induction over n∈N>1. The case n= 1holds\nfrom Proposition 4.6. Observe that for all n≥1, we have the following:\nαn−1 = 1−\u00121\n2\u0013n−1\n=1\n2 \n2−\u00121\n2\u0013n−2!\n=αn−1\n2. (10)\nLet us now assume that there exists n≥1such that:\nE(t) =o\u00121\ntαn−αn−1ε\u0013\n⇐⇒p\nE(t) =o\u00121\ntαn\n2−αn−1\n2ε\u0013\n. (I.H.)\nTo proceed by induction we need to show that (I.H.) implies\nE(t) =o\u00121\ntαn+1−αnε\u0013\n. (11)\n9To this aim, it is actually sufficient to prove that\nZ+∞\nt0tαn\n2−αnεE(t)dt <+∞. (12)\nIndeed, according to Lemma 4.5 if (12) holds, then\n0 = lim\nt→+∞tαn\n2+1−αnεE(t)(10)= lim\nt→+∞tαn+1−αnεE(t),\nwhich is equivalent to (11).\nTo show (12) we multiply both sides of (6) by tαn\n2−αnεand integrate from t0toT≥t0:\nZT\nt0tαn\n2−αnεE(t)dt≤3\n2ZT\nt0tαn\n2−αnε∥˙x(t)∥2dt−ZT\nt0tαn\n2−αnε¨hz(t)dt\n−ZT\nt0tαn\n2−αnεp\nE(t)˙hz(t)dt.(13)\nNow it only remains to show that the limit as T→+∞of each term on the right-hand side of (13)\nis uniformly bounded from above for T∈[t0,+∞[, which will imply that the left-hand side remains\nuniformly bounded as T→+∞. This approach is similar to the proof of Theorem 4.3, but the key\nequation (2) allows deducing further integrability results for ∥˙x(t)∥2by using the induction hypothesis\n(I.H.), which is new and specific to the system (LD).\nWe first make the following observation using (10):\nt1+ε\nt1−αn\n2+αnεp\nE(t) =tαn\n2−αn−1\n2εp\nE(t)(I.H.)− − − − →\nt→+∞0.\nTherefore according to Definition 2, we have\n1\nt1−αn\n2+αnεp\nE(t) =o\u00121\nt1+ε\u0013\n, (14)\nand using Lemma 4.8 with g(t) =1\nt1−αn2+αnεp\nE(t)andβ= 1 + ε,\nZ+∞\nt01\nt1−αn\n2+αnεp\nE(t)dt <+∞. (15)\nWe now use (15) to show the integrability of tαn\n2−αnε∥˙x(t)∥2, which shows the uniform boundedness\nof the first term on the right-hand side of (13) as T→+∞. For any T≥t0:\nZT\nt0tαn\n2−αnε∥˙x(t)∥2dtI.P., (2)=h\n−2tαn\n2−αnεp\nE(t)iT\nt0−ZT\nt0\u0010αn\n2−αnε\u0011\ntαn\n2−1−αnε(−2p\nE(t))dt\n(10)=−2Tαn\n2−αnεp\nE(T) + 2tαn\n2−αnε\n0p\nE(t0) + 2\u0010αn\n2−αnε\u0011ZT\nt01\nt1−αn\n2+αnεp\nE(t)dt,\n10where the second term is constant, the last integral is finite as T→+∞by (15) and\n−2 lim\nT→+∞Tαn\n2−αnεp\nE(T)≤0.\nTherefore the first term in (13) is bounded:\nZ+∞\nt0tαn\n2−αnε∥˙x(t)∥2dt <+∞. (16)\nLooking at the second term in (13) we have:\n−ZT\nt0tαn\n2−αnε¨hz(t)dtI.P.=−h\ntαn\n2−αnε˙hz(t)iT\nt0+\u0010αn\n2−αnε\u0011ZT\nt0tαn\n2−1−αnε˙hz(t)dt\nI.P.=−Tαn\n2−αnε˙hz(T) +tαn\n2−αnε\n0˙hz(t0) +\u0010αn\n2−αnε\u0011h\ntαn\n2−1−αnεhz(t)iT\nt0\n−\u0010αn\n2−αnε\u0011\u0010αn\n2−1−αnε\u0011ZT\nt0tαn\n2−2−αnεhz(t)dt.(17)\nRecall that there exists a 0≤M < +∞such that ∀t≥t0hz(t)∈[0, M], since by Assumption 2, x\nis bounded. Due to −\u0010αn\n2−αnε\u0011\n|{z}\n>0\u0010αn\n2−1−αnε\u0011\n| {z }\n<0>0andαn\n2−2−αnε <−1, we can bound the\nlimit of the last term in (17) as T→+∞:\n−\u0010αn\n2−αnε\u0011\u0010αn\n2−1−αnε\u0011Z+∞\nt0tαn\n2−2−αnεhz(t)dt\n≤ −M\u0010αn\n2−αnε\u0011\u0010αn\n2−1−αnε\u0011Z+∞\nt0tαn\n2−2−αnεdt <+∞.\nAgain by boundedness of xand by the definition of ˙hzwe get:\n−lim\nT→+∞Tαn\n2−αnε˙hz(T)C.S.\n≤lim\nT→+∞Tαn\n2−αnε∥x(T)−z∥ ∥˙x(T)∥\n≤√\nMlim\nT→+∞Tαn\n2−αnε∥˙x(T)∥= 0.\nIndeed,\n0(I.H.)= lim\nt→+∞tαn−αn−1εE(t) = lim\nt→+∞tαn−αn−1ε\u0012\nf(x(t))−f⋆+1\n2∥˙x(t)∥2\u0013\n≥lim\nt→+∞tαn−αn−1ε∥˙x(t)∥2≥0,\nhence:\nlim\nt→+∞tαn−αn−1ε∥˙x(t)∥2= 0⇐⇒ lim\nt→+∞tαn\n2−αn−1\n2ε∥˙x(t)∥= 0\n⇒lim\nt→+∞tαn\n2−αn−1\n2ε−ε∥˙x(t)∥= 0(10)⇐⇒ lim\nt→+∞tαn\n2−αnε∥˙x(t)∥= 0.\n11To conclude that the limit T→+∞of the right-hand side of (17) is bounded from above, we observe\nthat two of the three remaining terms are constant and\nlim\nT→+∞\u0010αn\n2−αnε\u0011\nTαn\n2−1−αnεhz(T) = 0 ,\nsinceαn\n2−1−αnε <0andhzis bounded. We finish the proof by bounding the third term of (13). For\nT≥t0we have:\n−ZT\nt0tαn\n2−αnεp\nE(t)˙hz(t)dtI.P., (2)=−h\ntαn\n2−αnεp\nE(t)hz(t)iT\nt0\n+ZT\nt0\u0012\u0010αn\n2−αnε\u0011\ntαn\n2−1−αnεp\nE(t)−1\n2tαn\n2−αnε∥˙x(t)∥2\u0013\nhz(t)dt\n=−Tαn\n2−αnεp\nE(T)hz(T) +tαn\n2−αnε\n0p\nE(t0)hz(t0) +\u0010αn\n2−αnε\u0011ZT\nt0tαn\n2−1−αnεp\nE(t)hz(t)dt\n−1\n2ZT\nt0tαn\n2−αnε∥˙x(t)∥2hz(t)dt. (18)\nOnly the third term on the right-hand side of (18) is neither constant nor non-positive. Using again the\nboundedness of hz, it holds that:\nZ+∞\nt0tαn\n2−1−αnεp\nE(t)hz(t)dt≤MZ+∞\nt0tαn\n2−1−αnεp\nE(t)dt(15)<+∞.\nHence the limit as T→+∞of the third term of (13) is bounded from above as well. Therefore all\nlimits as T→+∞of the terms on the right-hand side of (13) are bounded from above, hence (12)\nholds. By induction, we conclude that (9) holds for all n∈N≥1.\nRemark 4.9. The induction step of Theorem 4.7 resembles the proof of [15, Lemma 3.10]. Yet, their\nresult is not directly applicable to (LD) because it requires the knowledge of an asymptotic lower-bound\non the damping, which we do not know a priori. Furthermore, Cabot and Frankel [15] showed how to\niteratively repeat the application of this result but their proof relies on a strong integrability statement\non∥˙x(t)∥that may not hold in general. The specific nature of our system (LD) allows us to deduce\nthis integrability statement instead of assuming it.\n4.4 Drawbacks of the approach\nNow that we presented the main benefits of using a Lyapunov function as damping in (LD), we discuss\nsome drawbacks of our approach, starting with the following remark.\nRemark 4.10. Our system (LD) makes use of the optimal value f⋆. The latter might be unknown in\ngeneral but is known in several practical cases such as over-determined regression with squared loss, or\nempirical risk minimization of some over-parameterized machine learning problems [18]. We refer to\n[14] for more examples and further discussion. All the results above do not require the knowledge of\nf⋆, only its existence. However Algorithm 1 presented in Section 5 is only practical when f⋆is known.\n12The second drawback of our approach is that we can lower bound the convergence rate of our\ndampingp\nE(t). This not surprising as the bound actually matches the sufficient condition from [16]\nmentioned in the introduction.\nProposition 4.11. The convergence rate of the damping√\nEof the dynamical system (LD) cannot be\nasymptotically faster than1\nt,i.e., there exists no α > 1such thatp\nE(t) =o\u00001\ntα\u0001\nfor a non-constant\nbounded solution of (LD).\nRemark 4.12. Proposition 4.11 expresses that Ecan never vanish faster than o(1\nt2). However, this\ndoes not mean that the convergence rate of f(x(t))−f⋆can never be faster than the worst-case o(1\nt2);\none can sometimes obtain faster convergence of function values, as we will observe in the numerical\nexperiments.\nProof of Proposition 4.11. Assume thatp\nE(t) =o\u00001\ntα\u0001\nfor some α >1, then by Lemma 4.8,\nZ∞\nt0p\nE(t)dt <+∞. (19)\nLet us make a similar computation as in [16, Prop. 3.5]:\ndE(t)\ndt+ 2p\nE(t)E(t)(1)=−p\nE(t)∥˙x(t)∥2+ 2p\nE(t)\u0012\nf(x(t))−f⋆+1\n2∥˙x(t)∥2\u0013\n= 2p\nE(t)(f(x(t))−f⋆)≥0.(20)\nMultiply (20) by e2Rt\nt0√\nE(s)ds>0to obtain\n0≤e2Rt\nt0√\nE(s)dsdE(t)\ndt+ 2p\nE(t)e2Rt\nt0√\nE(s)dsE(t) =d\ndth\ne2Rt\nt0√\nE(s)dsE(t)i\n. (21)\nNow integrate 21 from t0tot:\n0≤e2Rt\nt0√\nE(s)dsE(t)−E(t0)\n⇐⇒ E(t)≥e−2Rt\nt0√\nE(s)dsE(t0).\nThis implies that\nlim\nt→∞E(t)≥e−2R∞\nt0√\nE(s)dsE(t0)(19)>0,\nwhich contradicts Proposition 4.3, since lim\nt→∞E(t) = 0 .Therefore, we conclude that\nZ∞\nt0p\nE(t)dt=∞, (22)\nand by Lemma 4.8 the limit of tαp\nE(t)fort→+∞does not exist or is not finite for any α > 1,\nhence E(t)̸=o\u00001\ntα\u0001\nforα >1.\n135 Algorithms and Numerical Experiments\n5.1 Practical algorithms from (LD)\nWe first detail how we discretize (LD). We use an explicit discretization with fixed step-size√s >0:\nfork∈Nwe approximate the solution xof (LD) at times tk=k√sand define xkdef=x(tk). We use\nthe approximations ˙x(t)≈xk−xk−1√sand¨x(t)≈xk+1−2xk+xk−1\ns. We also define the discrete version of\nthe damping accordingly by first defining\nEkdef=f(xk)−f⋆+1\n2\r\r\r\rxk−xk−1√s\r\r\r\r2\n. (23)\nand our damping then reads γ(xk, xk−1)def=√Ek. Using this in (LD) we then propose the following\ndiscretization scheme for all k∈N:\nxk+1−2xk+xk−1\ns+γ(xk, xk−1)xk−xk−1√s+∇f(yk) = 0\n⇐⇒ xk+1=xk+\u0000\n1−√sγ(xk, xk−1)\u0001\n[xk−xk−1]−s∇f(yk),(24)\nwhere the gradient ∇fis evaluated at yk=xk+ (1−√sγ(xk, xk−1)) [xk−xk−1], in the same fashion\nas it is done in NAG. One can optionally rather evaluate the gradient at xk.\nWe call the resulting algorithm LYDIA, for LYapunov Damped Inertial Algorithm, which is sum-\nmarized in Algorithm 1.\nAlgorithm 1: LYDIA\ninput: x0, x−1∈Rn, step-size s >0,kmax∈N\n1fork= 1tokmaxdo\n2 yk←xk+\u0010\n1−√Ek√E0\u0011\n[xk−xk−1]\n3 xk+1←yk−s∇f(yk)\n4end\nNote that compared to (24), we actually scaled the damping term√s√Ekby√s√E0in Algo-\nrithm 1. This scaling is optional but improves numerical stability1as it ensures that the coefficient in\nfront of [xk−xk−1]remains non-negative. More specifically, we have the following lemma that shows\nthatEkis non-increasing under standard assumptions.\nLemma 5.1. Letfbe a continuously differentiable convex functions whose gradient is Lipschitz con-\ntinuous with constant L >0. Let (xk, yk)k∈Nbe the sequence generated by Algorithm 1, where s <1\nL.\nThen, for all k∈N,\nEk+1≤Ek−s\n2∥∇f(xk)− ∇f(yk)∥2. (25)\nIn particular (Ek)k∈Nis non-increasing, hence it is a “discrete” Lyapunov function for Algorithm 1.\nProof. Consider for any x, y∈ H ands∈(0,1/L]the following refined version of the Descent Lemma\n[5, Remark 5.2.]:\nf(y−s∇f(y))≤f(x) +⟨∇f(y), y−x⟩ −s\n2∥∇f(y)∥2−s\n2∥∇f(x)− ∇f(y)∥2. (26)\n1An alternative to scaling is to choose sso that√s√E0≤1.\n14Take y=ykandx=xkin (26) and subtract f⋆on both sides:\nf(xk+1)−f⋆≤f(xk)−f⋆+⟨∇f(yk), yk−xk⟩ −s\n2∥∇f(yk)∥2−s\n2∥∇f(xk)− ∇f(yk)∥2.(27)\nObserve that we have the following identities:\n∥xk+1−xk∥2=∥s∇f(yk)−(yk−xk)∥2\n=s2∥∇f(yk)∥2−2s⟨∇f(yk), yk−xk⟩+∥yk−xk∥2,and, (28)\n∥yk−xk∥2(1)=\r\r\r\r\u0012\n1−√Ek√E0\u0013\n(xk−xk−1)\r\r\r\r2\n. (29)\nTherefore,\n⟨∇f(yk), yk−xk⟩=s\n2∥∇f(yk)∥2−1\n2s∥xk+1−xk∥2+1\n2s∥yk−xk∥2\n=s\n2∥∇f(yk)∥2−1\n2s∥xk+1−xk∥2+1\n2s\r\r\r\r\u0012\n1−√Ek√E0\u0013\n(xk−xk−1)\r\r\r\r2\n.\nSubstitute this in (27) to get\nf(xk+1)−f⋆+1\n2s∥xk+1−xk∥2≤f(xk)−f⋆+1\n2s\r\r\r\r\u0012\n1−√Ek√E0\u0013\n(xk−xk−1)\r\r\r\r2\n−s\n2∥∇f(xk)− ∇f(yk)∥2.(30)\nObserve that for the first iteration k= 0, (30) reads\nf(x1)−f⋆+1\n2s∥x1−x0∥2\n≤f(x0)−f⋆+1\n2s\r\r\r\r\u0012\n1−√E0√E0\u0013\n(x0−x−1)\r\r\r\r2\n−s\n2∥∇f(x0)− ∇f(y0)∥2\nwhich is equivalent to E1≤f(x0)−f⋆−s\n2∥∇f(x0)− ∇f(y0)∥2and therefore\nE1≤f(x0)−f⋆+1\n2s∥x0−x−1∥2−s\n2∥∇f(x0)− ∇f(y0)∥2\n=E0−s\n2∥∇f(x0)− ∇f(y0)∥2.\nSoE1≤E0. Let k∈N>0such that ∀1≤i≤k, E i≤Ei−1, then in particular Ek≤E0and we\ndeduce from (29) that:\n\r\r\r\r\u0012\n1−√Ek√E0\u0013\n(xk��xk−1)\r\r\r\r2\n≤ ∥xk−xk−1∥2, (31)\nwhich simplifies (30) into\nf(xk+1)−f⋆+1\n2s∥xk+1−xk∥2≤f(xk)−f⋆+1\n2s∥xk−xk−1∥2−s\n2∥∇f(xk)− ∇f(yk)∥2.\nThis is exactly (25) after using (23), hence by induction, (25) holds for all k∈N.\n15100101102103104\ntime t1016\n1013\n1010\n107\n104\n101\nfunction values fflat(x(t))LD\nAVD\nHBF\nGD\nt2\nt1\n100101102103104\ntime t1012\n1010\n108\n106\n104\n102\n100102Lyapunov function values E(t)LD\nAVD\nHBF\nGD\nt2\nt1\nFigure 2: Rate of convergence of LD, A VD, HBF and GD on the test-function fflat. Left: Evolution of\nthe function values. Right: Evolution of the Lyapunov function.\n100101102103104\ntime t1015\n1013\n1011\n109\n107\n105\n103\n101\nfunction values fnonKL(x(t))\nLD\nAVD\nHBF\nGD\nt2\nt1\n100101102103104\ntime t1015\n1013\n1011\n109\n107\n105\n103\n101\nLyapunov function values E(t)LD\nAVD\nHBF\nGD\nt2\nt2\nFigure 3: Rate of convergence of LD, A VD, HBF and GD on the test-function fnon-KL . Left: Evolution\nof the function values. Right: Evolution of the Lyapunov function.\n5.2 Numerical experiments\nMethodology. We evaluate the performance of our system and empirically corroborate Theorem 1.1.\nWe consider the following 1-dimensional test functions, defined for all x∈Rby:\nfflat(x)def=x24, f non-KL (x)def=(\n0, ifx= 0;\nexp\u0000\n−1\nx2\u0001\n,else,\nfuneven(x)def=(\nx3,ifx >0;\nx2,ifx≤0,fcontmin (x)def=\n\n(x−ε)2,ifx > ε ;\n(x+ε)2,ifx < ε ;\n0, else.\n16100101102103104\ntime t1015\n1013\n1011\n109\n107\n105\n103\n101\nfunction values funeven(x(t))LD\nAVD\nHBF\nGD\nt2\nt1\n100101102103104\ntime t1015\n1013\n1011\n109\n107\n105\n103\n101\n101Lyapunov function values E(t)LD\nAVD\nHBF\nGD\nt2\nt2\nFigure 4: Rate of convergence of LD, A VD, HBF and GD on the test-function funeven . Left: Evolution\nof the function values. Right: Evolution of the Lyapunov function.\n100101102103104\ntime t1015\n1013\n1011\n109\n107\n105\n103\n101\n101function values fcontmin(x(t))LD\nAVD\nHBF\nGD\n100101102103104\ntime t1015\n1012\n109\n106\n103\n100103Lyapunov function values E(t)LD\nAVD\nHBF\nGD\nt2\nt1\nFigure 5: Rate of convergence of LD, A VD, HBF and GD on the test-function fcontmin . Left: Evolution\nof the function values. Right: Evolution of the Lyapunov function.\nSince (LD) and (A VD a) are second-order ODEs that do not have closed-form solutions in general, we\napproximate them by using the LYDIA algorithm and NAG respectively, both with small step-sizes. We\ntakea= 3.1in NAG. We also consider Gradient Descent (GD) and the HBF algorithms mentioned in\nthe introduction. For each test function2, we present the evolution of the optimality gap f(x(t))−f⋆as\na function of the time t. Since this value is not monotonous and may oscillate heavily, we also display\nthe evolution of the (discretized) Lyapunov function E.\nResults. The function fflatis a convex polynomial that is very flat around its minimizer. As discussed\nin [6, Example 2.12] such polynomials of large degree allow emphasizing the worst-case bounds on\nthe rates of convergence. Indeed, we observe on the right of Figure 2 that the rate for GD and HBF\n2Some of these functions do not have a globally Lipschitz continuous gradient which could cause numerical instabilities.\nWe overcome this by using small-enough step-sizes to ensure boundedness of the iterates of all the algorithms considered.\n17is asymptotically close to O\u00001\nt\u0001\nwhile that of LD and (A VD a) gets close to O\u00001\nt2\u0001\nwhich empirically\nvalidates our main result Theorem 1.1.\nSome convex functions are not strongly convex but possess the KL property which allows deriving\nfaster rates of convergence (as done in the closed-loop setting by [9]). Since Theorem 1.1 does not\nassume such a property, we evaluate the algorithms on a function that is known for not having the KL\nproperty on Figure 3. The results again match the theoretical ones. Similarly, properties such as f\nbeing even ( i.e.,f(−x) =f(x),∀x) orargminHfbeing a singleton can allow one to derive faster\nconvergence results than for general convex functions. Our last-two test functions in Figures 4 and 5\nillustrate that the rate of (LD) also holds without such properties.\nFinally, while Figures 2 and 3 evidence cases where the solution of (LD) yields very fast mini-\nmization of function values, Figure 4 shows a case where (LD) is faster than GD and HBF but not as\nfast as (A VD a) and Figure 5 shows a case where (LD) is behind all other methods. Nonetheless, in all\ncases observe that Eis non-increasing and that it exhibits the rate predicted by Theorem 1.1.\n6 Conclusions and Perspectives\nWe introduced a new system (LD) which, as initially intended, is independent of the choice of the\ninitial time t0, unlike (A VD a), and which does require the hyper-parameter a > 3compared to the\nlatter, but however assumes knowledge of the optimal value f⋆. We showed that it is possible to\ndesign closed-loop dampings that achieve near-optimal rates of convergence for convex optimization.\nThe key ingredient is the coupling between the damping and the speed of convergence of the system.\nThis yielded in particular the identity (1), specific to (LD) and essential to adapt the proofs from the\nopen-loop to the closed-loop setting. This coupling provides a new understanding on how to choose\nthe damping term in second-order ODEs for optimization which may prove to be useful beyond the\nspecific cases of (LD) and (A VD a). We provided numerical experiment corroborating our theoretical\nresults as well as a new practical first-order algorithm LYDIA, derived from (LD), and for which we\nshowed that Ea still a (discrete) Lyapunov function.\nAs for future work, it remains open to know whether we can improve the rate from “arbitrarily\nclose to optimal” to optimal ( i.e., can we take δ= 0 in Theorem 1.1?). We also suspect that one can\ndrop Assumption 2-(ii) as it is not required for (A VD a) [24, 21], but we could not yet do it in the closed-\nloop setting. On the algorithmic side, our experiments suggest that the rate of (LD) is transferred to the\nLYDIA algorithm, which remains to be shown. One might also consider other discretization schemes.\nFinally, it is worth investigating whether one can find systems and algorithms with the same properties\nthat do not require the optimal value f⋆.\nAcknowledgment\nC. Castera and P. Ochs are supported by the ANR-DFG joint project TRINOM-DS under number DFG-\nOC150/5-1.\n18Appendices\nA Proof of Existence and Uniqueness\nThis section is devoted to proving Theorem 3.1. We recall some result from the theory of ODEs.\nTheorem A.1. [29, Theorem 4.2.6] (Classical Version of Picard–Lindel ¨of)\nLetHbe a Hilbert space, let Ω⊆R× H be open. Let G: Ω→ H be continuous and let (t0, x0)∈Ω.\nAssume that there exists L≥0such that for all (t, x),(t, y)∈Ωwe have\n∥G(t, x)−G(t, y)∥ ≤L∥x−y∥.\nThen, there exists δ >0such that the initial value problem\n\u001au′(t) =G(t, u(t)) ( t∈(t0, t0+δ)),\nu(t0) =x0,\nadmits a unique continuously differentiable solution u: [t0, t0+δ[→ H , which satisfies (t, u(t))∈Ω\nfor all t∈[t0, t0+δ[.\nDefinition 3. [26, Definition 3.20] Let ϕ:I→Rn,n∈N, be a solution to an ODE defined on some\nopen interval I⊂R.\n(a) We say ϕadmits an extension or continuation to the right (resp. to the left) if, and only if, there\nexists a solution ψ:J→Rnto the same ODE, defined on some open interval J⊃Isuch that\nψ|I=ϕand\nsupJ >supI(resp. infJ <infI)\nAn extension or continuation of ϕis a function that is an extension to the right or an extension\nto the left (or both).\n(b) The solution ϕis called a maximal solution if, and only if, it does not admit any extension in\nthe sense defined in (a).\nTheorem A.2. [26, Theorem 3.22] Every solution ϕ0:I0→Rnto some ODE, defined on an open\ninterval I0⊂R, can be extended to a maximal solution of this ODE.\nRemark A.3. Definition 3 is easily generalized to Hilbert spaces and Theorem A.2 is valid for an ODE\nin an infinite dimensional space as well, since the proof relies on Zorn’s Lemma.\nProof of Theorem 3.1. We first reduce the system (LD) to an equivalent first-order system:\nd\ndt\u0012x(t)\n˙x(t)\u0013\n= ˙x(t)\n−q\nf(x(t))−f⋆+1\n2∥˙x(t)∥2˙x(t)− ∇f(x(t))!\ndef=G(t,(x(t),˙x(t))). (32)\nWe first show the claim in the case argminHf̸=∅and later for the case argminHf=∅.\n19First, if (x0,˙x0)∈argminHf× {0}, then (x(t),˙x(t))≡(x0,0)is the unique global solution,\nsince ¨x(t) = 0 for all t≥t0. Now assume that (x0,˙x0)̸∈argminHf× {0}and let us check that for\nallt≥t0, the function\n(u, v)7→G(t,(u, v)) = v\n−q\nf(u)−f⋆+1\n2∥v∥2v(t)− ∇f(u)!\nis Lipschitz continuous on the open set\nAdef= (H × H )\\(argmin\nHf× {0}).\nThe first coordinate of Gis clearly locally Lipschitz continuous w.r.t. (u, v)and therefore it suffices to\ncheck that the second coordinate of Gis also locally Lipschitz continuous w.r.t. (u, v)onA.\nThe square root function is locally Lipschitz continuous everywhere except at 0 and f(u)−f⋆+\n1\n2∥v∥2̸= 0 for all (u, v)∈ A. Therefore (u, v)∈ A 7→q\nf(u)−f⋆+1\n2∥v∥2vis locally Lipschitz\ncontinuous as a product of locally Lipschitz-continuous functions. Finally ∇fis locally Lipschitz\ncontinuous by Assumption 1-(i) , hence Gis locally Lipschitz continuous on A. We can now apply\nthe Picard–Lindel ¨of Theorem A.1 to any open and bounded neighborhood of the initial conditions\n(t0,(x0,˙x0))to find some δ >0and a unique solution (x,˙x)to (32) on [t0, t0+δ[.\nIt now remains to show that the solution exists on [t0,+∞[. LetT∈]t0,+∞]be the maximal time\nof existence (as stated in Theorem A.2) of the solution (x,˙x)of (32). Assume that T < +∞, then the\nlimits of xand˙x(t)exist as t→T. Indeed, first note that the uniform boundedness of Eimplies that\n˙xis bounded on [t0, T[and by (LD) ¨xis bounded on [t0, T[as well. Now consider a Cauchy sequence\n(tn)n∈Nconverging to Tand let m, n∈N. Then by the mean value Theorem we have:\n∥x(tn)−x(tm)∥ ≤ sup\ns∈]t0,T[∥˙x(s)∥|tn−tm|,and\n∥˙x(tn)−˙x(tm)∥ ≤ sup\ns∈]t0,T[∥¨x(s)∥|tn−tm|.\nTherefore (x(tn))n∈Nand( ˙x(tn))n∈Nare Cauchy sequences which implies that lim\nt→Tx(t)andlim\nt→T˙x(t)\nexist.\nThen, if (x(T),˙x(T))∈ A then we can extend the solution by applying Picard–Lindel ¨of to an open\nand bounded neighborhood of ((T,(x(T),˙x(T)))which contradicts the fact that Twas the maximal\ntime of existence. If (x(T),˙x(T))∈∂A ⊂ argminHf× {0}, where ∂Adenotes the boundary\nofA, then ¨x(T) = 0 and we can extend the solution on [T, T +δ[, for some δ > 0, by taking\n(x(t),˙x(t)) = ( x(T),0). This is again a contradiction, therefore we necessarily have T= +∞.\nFinally, in the case where argminHf=∅, we see similarly to before that G(t,(u, v))is locally\nLipschitz continuous w.r.t. (u, v)everywhere on H × H . So the Picard–Lindel ¨of Theorem implies the\nexistence of a unique local solution for any initial value, which we can again extend to a global solution\non[t0,+∞[.\n20B Missing Proofs of Lemmas\nProof of Lemma 4.4. Assume that there exists c > 0such that lim\nt→+∞g(t) = c, which implies the\nexistence of t1≥t0such that ∀t≥t1,g(t)≥c\n2. Then\nZ+∞\nt0g(t)dt≥Z+∞\nt1g(t)dt≥Z+∞\nt1c\n2dt= +∞,\nwhich contradicts the assumption thatR+∞\nt0g(t)dt <+∞.\nProof of Lemma 4.5. Denote the positive and negative parts of a function gby\n[g(·)]+def= max( g(·),0)and[g(·)]−def= max( −g(·),0)≥0,\nso that ∀x∈ H,\ng(x) = [g(x)]+−[g(x)]−.\nThen note that\nh\u0000\ntα+1E(t)\u0001′i\n+(1)=h\n(α+ 1)tαE(t)−tα+1p\nE(t)∥˙x(t)∥2i\n+= (α+ 1)tαE(t), (33)\nand that,\nZ+∞\nt0\u0000\ntα+1E(t)\u0001′dt=Z+∞\nt0h\u0000\ntα+1E(t)\u0001′i\n+−h\u0000\ntα+1E(t)\u0001′i\n−dt\n≤Z+∞\nt0h\u0000\ntα+1E(t)\u0001′i\n+dt(33)= (α+ 1)Z+∞\nt0tαE(t)dtAss.<+∞.(34)\nTherefore by definition of the improper integral, lim\nt→+∞tα+1E(t)admits a finite limit l∈R≥0.\nNow, assume that l >0, then for any ε∈]0, l[there exists t1≥t0such for all t≥t1we have\nE(t)>l−ε\ntα+1. Then,\nZ+∞\nt0tαE(t)dt≥Z+∞\nt1tαE(t)dt >Z+∞\nt1tαl−ε\ntα+1dt= (l−ε)Z+∞\nt11\ntdt= +∞,\nwhich contradicts the integrability assumption on tαE(t).\nProof of Lemma 4.8. Letldef= lim\nt→∞tβg(t)≥0. Then for any ε > 0we can find t1≥t0such that\n∀t≥t1,\ng(t)≤(l+ε)\ntβ.\nTherefore,\nZ+∞\nt0g(t)dt=Zt1\nt0g(t)dt\n|{z}\ndef=I<+∞+Z+∞\nt1g(t)dt≤I+ (l+ε)Z+∞\nt11\ntβdt <+∞,\nsince β >1, which proves the result.\n21References\n[1] Samir Adly, Hedy Attouch, and Alexandre Cabot. Finite time stabilization of nonlinear oscillators\nsubject to dry friction. In Nonsmooth Mechanics and Analysis , pages 289–304, 2006.\n[2] Cristian Daniel Alecsa, Szil ´ard Csaba L ´aszl´o, and Titus Pint ¸a. An extension of the second or-\nder dynamical system that models Nesterov’s convex gradient method. Applied Mathematics &\nOptimization , 84:1687–1716, 2021.\n[3] Felipe Alvarez, Hedy Attouch, J ´erˆome Bolte, and Patrick Redont. A second-order gradient-like\ndissipative dynamical system with Hessian-driven damping: Application to optimization and me-\nchanics. Journal de math ´ematiques pures et appliqu ´ees, 81(8):747–779, 2002.\n[4] Hedy Attouch and Alexandre Cabot. 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Journal of Machine Learning Research , 22(1):5040–5073, 2021.\n23"    },    {        "title": "1709.10365v1.Non_local_Gilbert_damping_tensor_within_the_torque_torque_correlation_model.pdf",        "content": "Non-local Gilbert damping tensor within the torque-torque correlation model\nDanny Thonig,1,\u0003Yaroslav Kvashnin,1Olle Eriksson,1, 2and Manuel Pereiro1\n1Department of Physics and Astronomy, Material Theory, Uppsala University, SE-75120 Uppsala, Sweden\n2School of Science and Technology, Orebro University, SE-701 82 Orebro, Sweden\n(Dated: July 19, 2018)\nAn essential property of magnetic devices is the relaxation rate in magnetic switching which\ndepends strongly on the damping in the magnetisation dynamics. It was recently measured that\ndamping depends on the magnetic texture and, consequently, is a non-local quantity. The damping\nenters the Landau-Lifshitz-Gilbert equation as the phenomenological Gilbert damping parameter\n\u000b, that does not, in a straight forward formulation, account for non-locality. E\u000borts were spent\nrecently to obtain Gilbert damping from \frst principles for magnons of wave vector q. However,\nto the best of our knowledge, there is no report about real space non-local Gilbert damping \u000bij.\nHere, a torque-torque correlation model based on a tight binding approach is applied to the bulk\nelemental itinerant magnets and it predicts signi\fcant o\u000b-site Gilbert damping contributions, that\ncould be also negative. Supported by atomistic magnetisation dynamics simulations we reveal the\nimportance of the non-local Gilbert damping in atomistic magnetisation dynamics. This study gives\na deeper understanding of the dynamics of the magnetic moments and dissipation processes in real\nmagnetic materials. Ways of manipulating non-local damping are explored, either by temperature,\nmaterials doping or strain.\nPACS numbers: 75.10.Hk,75.40.Mg,75.78.-n\nE\u000ecient spintronics applications call for magnetic ma-\nterials with low energy dissipation when moving magnetic\ntextures, e.g. in race track memories1, skyrmion logics2,3,\nspin logics4, spin-torque nano-oscillator for neural net-\nwork applications5or, more recently, soliton devices6. In\nparticular, the dynamics of such magnetic textures |\nmagnetic domain walls, magnetic Skyrmions, or magnetic\nsolitons | is well described in terms of precession and\ndamping of the magnetic moment mias it is formulated\nin the atomistic Landau-Lifshitz-Gilbert (LLG) equation\nfor sitei\n@mi\n@t=mi\u0002\u0012\n\u0000\rBeff\ni+\u000b\nms@mi\n@t\u0013\n; (1)\nwhere\randmsare the gyromagnetic ratio and the\nmagnetic moment length, respectively. The precession\n\feldBeff\niis of quantum mechanical origin and is ob-\ntained either from e\u000bective spin-Hamilton models7or\nfrom \frst-principles8. In turn, energy dissipation is\ndominated by the ad-hoc motivated viscous damping in\nthe equation of motion scaled by the Gilbert damping\ntensor\u000b. Commonly, the Gilbert damping is used as\na scalar parameter in magnetization dynamics simula-\ntions based on the LLG equation. Strong e\u000borts were\nspend in the last decade to put the Gilbert damping\nto a \frst-principles ground derived for collinear mag-\nnetization con\fgurations. Di\u000berent methods were pro-\nposed: e.g. the breathing Fermi surface9{11, the torque-\ntorque correlation12, spin-pumping13or a linear response\nmodel14,15. Within a certain accuracy, the theoretical\nmodels allow to interpret16and reproduce experimental\ntrends17{20.\nDepending on the model, deep insight into the fun-\ndamental electronic-structure mechanism of the Gilbertdamping\u000bis provided: Damping is a Fermi-surface ef-\nfect and depending on e.g. scattering rate, damping\noccurs due to spin-\rip but also spin-conservative tran-\nsition within a degenerated (intraband, but also inter-\nband transitions) and between non-degenerated (inter-\nband transitions) electron bands. As a consequence of\nthese considerations, the Gilbert damping is proportional\nto the density of states, but it also scales with spin-orbit\ncoupling21,22. The scattering rate \u0000 for the spin-\rip tran-\nsitions is allocated to thermal, but also correlation ef-\nfects, making the Gilbert damping strongly temperature\ndependent which must be a consideration when applying\na three-temperature model for the thermal baths, say\nphonon14, electron, and spin temperature23. In particu-\nlar, damping is often related to the dynamics of a collec-\ntive precession mode (macrospin approach) driven from\nan external perturbation \feld, as it is used in ferromag-\nnetic resonance experiments (FMR)24. It is also estab-\nlished that the Gilbert damping depends on the orien-\ntation of the macrospin25and is, in addition, frequency\ndependent26.\nMore recently, the role of non-collective modes to the\nGilbert damping has been debated. F ahnle et al.27\nsuggested to consider damping in a tensorial and non-\nisotropic form via \u000bithat di\u000bers for di\u000berent sites i\nand depends on the whole magnetic con\fguration of the\nsystem. As a result, the experimentally and theoret-\nically assumed local Gilbert equation is replaced by a\nnon-local equation via non-local Gilbert damping \u000bijac-\ncounting for the most general form of Rayleigh's dissi-\npation function28. The proof of principles was given for\nmagnetic domain walls29,30, linking explicitly the Gilbert\ndamping to the gradients in the magnetic spin texture\nrm. Such spatial non-locality, in particular, for discrete\natomistic models, allows further to motivate energy dis-arXiv:1709.10365v1  [cond-mat.mtrl-sci]  29 Sep 20172\nij\nαij\nq\nFIG. 1: Schematic illustration of non-local energy dissipation\n\u000bijbetween site iandj(red balls) represented by a power\ncord in a system with spin wave (gray arrows) propagation q.\nsipation between two magnetic moments at sites iand\nj, and is represented by \u000bij, as schematically illustrated\nin Fig. 1. An analytical expression for \u000bijwas already\nproposed by various authors14,31,32, however, not much\nwork has been done on a material speci\fc, \frst-principle\ndescription of the atomistic non-local Gilbert damping\n\u000bij. An exception is the work by Gilmore et al.32who\nstudied\u000b(q) in the reciprocal space as a function of the\nmagnon wave vector qand concluded that the non-local\ndamping is negligible. Yan et al.29and Hals et al.33, on\nthe other hand, applied scattering theory according to\nBrataas et al.34to simulate non-collinearity in Gilbert\ndamping, only in reciprocal space or continuous meso-\nscopic scale. Here we come up with a technical descrip-\ntion of non-locality of the damping parameter \u000bij, in\nreal space, and provide numerical examples for elemental,\nitinerant magnets, which might be of high importance in\nthe context of ultrafast demagnetization35.\nThe paper is organized as follows: In Section I, we\nintroduce our \frst-principles model formalism based on\nthe torque-torque correlation model to study non-local\ndamping. This is applied to bulk itinerant magnets bcc\nFe, fcc Co, and fcc Ni in both reciprocal and real space\nand it is analysed in details in Section II. Here, we will\nalso apply atomistic magnetisation dynamics to outline\nthe importance in the evolution of magnetic systems. Fi-\nnally, in the last section, we conclude the paper by giving\nan outlook of our work.\nI. METHODS\nWe consider the torque-torque correlation model in-\ntroduced by Kambersk\u0013 y10and further elaborated on by\nGilmore et al.12. Here, \fnite magnetic moment rotations\ncouple to the Bloch eigenenergies \"n;kand eigenstates\njnki, characterised by the band index nat wave vec-tork, due to spin-orbit coupling. This generates a non-\nequilibrium population state (a particle-hole pair), where\nthe excited states relax towards the equilibrium distribu-\ntion (Fermi-Dirac statistics) within the time \u001cn;k=1=\u0000,\nwhich we assume is independent of nandk. In the adi-\nabatic limit, this perturbation is described by the Kubo-\nGreenwood perturbation theory and reads12,36in a non-\nlocal formulation\n\u000b\u0016\u0017(q) =g\u0019\nmsZ\n\nX\nnmT\u0016\nnk;mk+q\u0000\nT\u0017\nnk;mk+q\u0001\u0003Wnk;mk+qdk:\n(2)\nHere the integral runs over the whole Brillouin zone\nvolume \n. A frozen magnon of wave vector qis consid-\nered that is ascribed to the non-locality of \u000b. The scat-\ntering events depend on the spectral overlap Wnk;mk+q=R\n\u0011(\")Ank(\";\u0000)Amk+q(\";\u0000) d\"between two bands \"n;k\nand\"m;k+q, where the spectral width of the electronic\nbandsAnkis approximated by a Lorentzian of width \u0000.\nNote that \u0000 is a parameter in our model and can be spin-\ndependent as proposed in Ref. [37]. In other studies, this\nparameter is allocated to the self-energy of the system\nand is obtained by introducing disorder, e.g., in an al-\nloy or alloy analogy model using the coherent potential\napproximation14(CPA) or via the inclusion of electron\ncorrelation38. Thus, a principle study of the non-local\ndamping versus \u0000 can be also seen as e.g. a temperature\ndependent study of the non-local damping. \u0011=@f=@\"is\nthe derivative of the Fermi-Dirac distribution fwith re-\nspect to the energy. T\u0016\nnk;mk+q=hnkj^T\u0016jmk+qi, where\n\u0016=x;y;z , are the matrix elements of the torque oper-\nator ^T= [\u001b;Hso] obtained from variation of the mag-\nnetic moment around certain rotation axis e.\u001band\nHsoare the Pauli matrices and the spin-orbit hamilto-\nnian, respectively. In the collinear ferromagnetic limit,\ne=ezand variations occur in xandy, only, which al-\nlows to consider just one component of the torque, i.e.\n^T\u0000=^Tx\u0000i^Ty. Using Lehmann representation39, we\nrewrite the Bloch eigenstates by Green's function G, and\nde\fne the spectral function ^A= i\u0000\nGR\u0000GA\u0001\nwith the\nretarded (R) and advanced (A) Green's function,\n\u000b\u0016\u0017(q) =g\nm\u0019Z Z\n\n\u0011(\")^T\u0016^Ak\u0010\n^T\u0017\u0011y^Ak+qdkd\":(3)\nThe Fourier transformation of the Green's function G\n\fnally is used to obtain the non-local Gilbert damping\ntensor23between site iat positionriand sitejat position\nrj,\n\u000b\u0016\u0017\nij=g\nm\u0019Z\n\u0011(\")^T\u0016\ni^Aij\u0010\n^T\u0017\nj\u0011y^Ajid\": (4)\nNote that ^Aij= i\u0000\nGR\nij\u0000GA\nji\u0001\n. This result is consis-\ntent with the formulation given in Ref. [31] and Ref. [14].\nHence, the de\fnition of non-local damping in real space3\nand reciprocal space translate into each other by a\nFourier transformation,\n\u000bij=Z\n\u000b(q) e\u0000i(rj\u0000ri)\u0001qdq: (5)\nNote the obvious advantage of using Eq. (4), since it\nallows for a direct calculation of \u000bij, as opposed to tak-\ning the inverse Fourier transform of Eq. (5). For \frst-\nprinciples studies, the Green's function is obtained from\na tight binding (TB) model based on the Slater-Koster\nparameterization40. The Hamiltonian consists of on-site\npotentials, hopping terms, Zeeman energy, and spin-orbit\ncoupling (See Appendix A). The TB parameters, includ-\ning the spin-orbit coupling strength, are obtained by \ft-\nting the TB band structures to ab initio band structures\nas reported elsewhere23.\nBeyond our model study, we simulate material spe-\nci\fc non-local damping with the help of the full-potential\nlinear mu\u000en-tin orbitals (FP-LMTO) code \\RSPt\"41,42.\nFurther numerical details are provided in Appendix A.\nWith the aim to emphasize the importance of non-\nlocal Gilbert damping in the evolution of atomistic\nmagnetic moments, we performed atomistic magnetiza-\ntion dynamics by numerical solving the Landau-Lifshitz\nGilbert (LLG) equation, explicitly incorporating non-\nlocal damping23,34,43\n@mi\n@t=mi\u00020\n@\u0000\rBeff\ni+X\nj\u000bij\nmj\ns@mj\n@t1\nA:(6)\nHere, the e\u000bective \feld Beff\ni =\u0000@^H=@miis allo-\ncated to the spin Hamiltonian entails Heisenberg-like ex-\nchange coupling\u0000P\nijJijmi\u0001mjand uniaxial magneto-\ncrystalline anisotropyP\niKi(mi\u0001ei)2with the easy axis\nalongei.JijandKiare the Heisenberg exchange cou-\npling and the magneto-crystalline anisotropy constant,\nrespectively, and were obtained from \frst principles44,45.\nFurther details are provided in Appendix A.\nII. RESULTS AND DISCUSSION\nThis section is divided in three parts. In the \frst part,\nwe discuss non-local damping in reciprocal space q. The\nsecond part deals with the real space de\fnition of the\nGilbert damping \u000bij. Atomistic magnetization dynam-\nics including non-local Gilbert damping is studied in the\nthird part.\nA. Non-local damping in reciprocal space\nThe formalism derived by Kambersk\u0013 y10and Gilmore12\nin Eq. (2) represents the non-local contributions to the\nenergy dissipation in the LLG equation by the magnonwave vector q. In particular, Gilmore et al.32con-\ncluded that for transition metals at room temperature\nthe single-mode damping rate is essentially independent\nof the magnon wave vector for qbetween 0 and 1% of\nthe Brillouin zone edge. However, for very small scat-\ntering rates \u0000, Gilmore and Stiles12observed for bcc Fe,\nhcp Co and fcc Ni a strong decay of \u000bwithq, caused by\nthe weighting function Wnm(k;k+q) without any sig-\nni\fcant changes of the torque matrix elements. Within\nour model systems, we observed the same trend for bcc\nFe, fcc Co and fcc Ni. To understand the decay of the\nGilbert damping with magnon-wave vector qin more de-\ntail, we study selected paths of both the magnon qand\nelectron momentum kin the Brillouin zone at the Fermi\nenergy\"Ffor bcc Fe (q;k2\u0000!Handq;k2H!N),\nfcc Co and fcc Ni ( q;k2\u0000!Xandq;k2X!L) (see\nFig. 2, where the integrand of Eq. (2) is plotted). For\nexample, in Fe, a usually two-fold degenerated dband\n(approximately in the middle of \u0000H, marked by ( i)) gives\na signi\fcant contribution to the intraband damping for\nsmall scattering rates. There are two other contributions\nto the damping (marked by ( ii)), that are caused purely\nby interband transitions. With increasing, but small q\nthe intensities of the peaks decrease and interband tran-\nsitions become more likely. With larger q, however, more\nand more interband transitions appear which leads to an\nincrease of the peak intensity, signi\fcantly in the peaks\nmarked with ( ii). This increase could be the same or-\nder of magnitude as the pure intraband transition peak.\nSimilar trends also occur in Co as well as Ni and are\nalso observed for Fe along the path HN. Larger spectral\nwidth \u0000 increases the interband spin-\rip transitions even\nfurther (data not shown). Note that the torque-torque\ncorrelation model might fail for large values of q, since\nthe magnetic moments change so rapidly in space that\nthe adiababtic limit is violated46and electrons are not\nstationary equilibrated. The electrons do not align ac-\ncording the magnetic moment and the non-equilibrium\nelectron distribution in Eq. (2) will not fully relax. In\nparticular, the magnetic force theorem used to derive\nEq. (3) may not be valid.\nThe integration of the contributions in electron mo-\nmentum space kover the whole Brillouin zone is pre-\nsented in Fig. 3, where both `Loretzian' method given\nby Eq. (2) and Green's function method represented\nby Eq. (3) are applied. Both methods give the same\ntrend, however, di\u000ber slightly in the intraband region,\nwhich was already observed previously by the authors\nof Ref. [23]. In the `Lorentzian' approach, Eq. (2), the\nelectronic structure itself is una\u000bected by the scattering\nrate \u0000, only the width of the Lorentian used to approx-\nimateAnkis a\u000bected. In the Green function approach,\nhowever, \u0000 enters as the imaginary part of the energy\nat which the Green functions is evaluated and, conse-\nquently, broadens and shifts maxima in the spectral func-\ntion. This o\u000bset from the real energy axis provides a more\naccurate description with respect to the ab initio results\nthan the Lorentzian approach.4\nΓHq(a−1\n0)\nΓ H\nk(a−1\n0)\nFe\nΓX\nΓ X\nk(a−1\n0)\n Co\nΓX\nΓ X\nk(a−1\n0)\n Ni\n(i) (ii) (ii)\nFIG. 2: Electronic state resolved non-local Gilbert damping obtained from the integrand of Eq. (3) along selected paths in the\nBrillouin zone for bcc Fe, fcc Co and fcc Ni. The scattering rate used is \u0000 = 0 :01 eV. The abscissa (both top and bottom in\neach panels) shows the momentum path of the electron k, where the ordinate (left and right in each panel) shows the magnon\npropagation vector q. The two `triangle' in each panel should be viewed separately where the magnon momentum changes\naccordingly (along the same path) to the electron momentum.\nWithin the limits of our simpli\fed electronic structure\ntight binding method, we obtained qualitatively similar\ntrends as observed by Gilmore et al.32: a dramatic de-\ncrease in the damping at low scattering rates \u0000 (intra-\nband region). This trend is common for all here ob-\nserved itinerant magnets typically in a narrow region\n0<jqj<0:02a\u00001\n0, but also for di\u000berent magnon propa-\ngation directions. For larger jqj>0:02a\u00001\n0the damping\ncould again increase (not shown here). The decay of \u000b\nis only observable below a certain threshold scattering\nrate \u0000, typically where intra- and interband contribu-\ntion equally contributing to the Gilbert damping. As\nalready found by Gilmore et al.32and Thonig et al.23,\nthis point is materials speci\fc. In the interband regime,\nhowever, damping is independent of the magnon propa-\ngator, caused by already allowed transition between the\nelectron bands due to band broadening. Marginal vari-\nations in the decay with respect to the direction of q\n(Inset of Fig. 3) are revealed, which was not reported be-\nfore. Such behaviour is caused by the break of the space\ngroup symmetry due to spin-orbit coupling and a selected\nglobal spin-quantization axis along z-direction, but also\ndue to the non-cubic symmetry of Gkfork6= 0. As a re-\nsult, e.g., in Ni the non-local damping decays faster along\n\u0000Kthan in \u0000X. This will be discussed more in detail in\nthe next section.\nWe also investigated the scaling of the non-local\nGilbert damping with respect to the spin-orbit coupling\nstrength\u0018dof the d-states (see Appendix B). We observe\nan e\u000bect that previously has not been discussed, namely\nthat the non-local damping has a di\u000berent exponential\nscaling with respect to the spin-orbit coupling constant\nfor di\u000berentjqj. In the case where qis close to the Bril-\nlouin zone center (in particular q= 0),\u000b/\u00183\ndwhereas\nfor wave vectors jqj>0:02a\u00001\n0,\u000b/\u00182\nd. For largeq,\ntypically interband transitions dominate the scatteringmechanism, as we show above and which is known to\nscale proportional to \u00182. Here in particular, the \u00182will\nbe caused only by the torque operator in Eq. (2). On the\nother hand, this indicates that spin-mixing transitions\nbecome less important because there is not contribution\nin\u0018from the spectral function entering to the damping\n\u000b(q).\nThe validity of the Kambserk\u0013 y model becomes ar-\nguable for\u00183scaling, as it was already proved by Costa\net al.47and Edwards48, since it causes the unphysical\nand strong diverging intraband contribution at very low\ntemperature (small \u0000). Note that there is no experi-\nmental evidence of such a trend, most likely due to that\nsample impurities also in\ruence \u0000. Furthermore, various\nother methods postulate that the Gilbert damping for\nq= 0 scales like \u00182 9,15,22. Hence, the current applied\ntheory, Eq. (3), seems to be valid only in the long-wave\nlimit, where we found \u00182-scaling. On the other hand,\nEdwards48proved that the long-wave length limit ( \u00182-\nscaling) hold also in the short-range limit if one account\nonly for transition that conserve the spin (`pure' spin\nstates), as we show for Co in Fig. 11 of Appendix C. The\ntrends\u000bversusjqjas described above changes drastically\nfor the `corrected' Kambersk\u0013 y formula: the interband re-\ngion is not a\u000bected by these corrections. In the intraband\nregion, however, the divergent behaviour of \u000bdisappears\nand the Gilbert damping monotonically increases with\nlarger magnon wave vector and over the whole Brillouin\nzone. This trend is in good agreement with Ref. [29].\nFor the case, where q= 0, we even reproduced the re-\nsults reported in Ref. [21]; in the limit of small scattering\nrates the damping is constant, which was also reported\nbefore in experiment49,50. Furthermore, the anisotropy\nof\u000b(q) with respect to the direction of q(as discussed\nfor the insets of Fig. 3) increases by accounting only for\npure-spin states (not shown here). Both agreement with5\n510−22Fe\n0.000\n0.025\n0.050\n0.075\n0.100\nq: Γ→H\n2510−2α(q)Co\nq: Γ→X\n510−225\n10−310−210−110+0\nΓ (eV)Ni\nq: Γ→X\nFIG. 3: (Color online) Non-local Gilbert damping as a func-\ntion of the spectral width \u0000 for di\u000berent reciprocal wave vector\nq(indicated by di\u000berent colors and in units a\u00001\n0). Note that q\nprovided here are in direct coordinates and only the direction\ndi\u000bers between the di\u000berent elementals, itinerant magnets.\nThe non-local damping is shown for bcc Fe (top panel) along\n\u0000!H, for fcc Co (middle panel) along \u0000 !X, and for fcc Ni\n(bottom panel) along \u0000 !X. It is obtained from `Lorentzian'\n(Eq. (2), circles) and Green's function (Eq. (3), triangles)\nmethod. The directional dependence of \u000bfor \u0000 = 0:01 eV is\nshown in the inset.\nexperiment and previous theory motivate to consider \u00182-\nscaling for all \u0000.\nB. Non-local damping in real space\nAtomistic spin-dynamics, as stated in Section I (see\nEq. (6)), that includes non-local damping requires\nGilbert damping in real-space, e.g. in the form \u000bij. This\npoint is addressed in this section. Such non-local con-\ntributions are not excluded in the Rayleigh dissipation\nfunctional, applied by Gilbert to derive the dissipation\ncontribution in the equation of motion51(see Fig. 4).\nDissipation is dominated by the on-site contribution\n-101 Fe\nαii= 3.552·10−3\n˜αii= 3.559·10−3\n-101αij·10−4Co\nαii= 3.593·10−3\n˜αii= 3.662·10−3\n-10\n1 2 3 4 5 6\nrij/a0Ni\nαii= 2.164·10−2\n˜αii= 2.319·10−2FIG. 4: (Color online) Real-space Gilbert damping \u000bijas\na function of the distance rijbetween two sites iandjfor\nbcc Fe, fcc Co, and fcc Ni. Both the `corrected' Kambersk\u0013 y\n(red circles) and the Kambersk\u0013 y (blue squares) approach is\nconsidered. The distance is normalised to the lattice constant\na0. The on-site damping \u000biiis shown in the \fgure label. The\ngrey dotted line indicates the zero line. The spectral width is\n\u0000 = 0:005 eV.\n\u000biiin the itinerant magnets investigated here. For both\nFe (\u000bii= 3:55\u000110\u00003) and Co ( \u000bii= 3:59\u000110\u00003) the\non-site damping contribution is similar, whereas for Ni\n\u000biiis one order of magnitude higher. O\u000b-site contri-\nbutionsi6=jare one-order of magnitude smaller than\nthe on-site part and can be even negative. Such neg-\native damping is discernible also in Ref. [52], however,\nit was not further addressed by the authors. Due to\nthe presence of the spin-orbit coupling and a preferred\nglobal spin-quantization axis (in z-direction), the cubic\nsymmetry of the considered itinerant magnets is broken\nand, thus, the Gilbert damping is anisotropic with re-\nspect to the sites j(see also Fig. 5 left panel). For ex-\nample, in Co, four of the in-plane nearest neighbours\n(NN) are\u000bNN\u0019\u00004:3\u000110\u00005, while the other eight are\n\u000bNN\u0019\u00002:5\u000110\u00005. However, in Ni the trend is opposite:\nthe out-of-plane damping ( \u000bNN\u0019\u00001:6\u000110\u00003) is smaller\nthan the in-plane damping ( \u000bNN\u0019 \u00001:2\u000110\u00003). In-\nvolving more neighbours, the magnitude of the non-local6\ndamping is found to decay as 1=r2and, consequently, it\nis di\u000berent than the Heisenberg exchange parameter that\nasymptotically decays in RKKY-fashion as Jij/1=r353.\nFor the Heisenberg exchange, the two Green's functions\nas well as the energy integration in the Lichtenstein-\nKatsnelson-Antropov-Gubanov formula54scales liker\u00001\nij,\nG\u001b\nij/ei(k\u001b\u0001rij+\b\u001b)\njrijj(7)\nwhereas for simplicity we consider here a single-band\nmodel but the results can be generalized also to the multi-\nband case and where \b\u001bdenotes a phase factor for spin\n\u001b=\";#. For the non-local damping the energy integra-\ntion is omitted due to the properties of \u0011in Eq. (4) and,\nthus,\n\u000bij/sin\u0002\nk\"\u0001rij+ \b\"\u0003\nsin\u0002\nk#\u0001rij+ \b#\u0003\njrijj2:(8)\nThis spatial dependency of \u000bijsuperimposed with\nRuderman-Kittel-Kasuya-Yosida (RKKY) oscillations\nwas also found in Ref. [52] for a model system.\nFor Ni, dissipation is very much short range, whereas in\nFe and Co `damping peaks' also occur at larger distances\n(e.g. for Fe at rij= 5:1a0and for Co at rij= 3:4a0).\nThe `long-rangeness' depends strongly on the parameter\n\u0000 (not shown here). As it was already observed for the\nHeisenberg exchange interaction Jij44, stronger thermal\ne\u000bects represented by \u0000 will reduce the correlation length\nbetween two magnetic moments at site iandj. The same\ntrend is observed for damping: larger \u0000 causes smaller\ndissipation correlation length and, thus, a faster decay\nof non-local damping in space rij. Di\u000berent from the\nHeisenberg exchange, the absolute value of the non-local\ndamping typically decreases with \u0000 as it is demonstrated\nin Fig. 5.\nNote that the change of the magnetic moment length\nis not considered in the results discussed so far. The\nanisotropy with respect to the sites iandjof the non-\nlocal Gilbert damping continues in the whole range of the\nscattering rate \u0000 and is controlled by it. For instance, the\nsecond nearest neighbours damping in Co and Ni become\ndegenerated at \u0000 = 0 :5 eV, where the anisotropy between\n\frst-nearest neighbour sites increase. Our results show\nalso that the sign of \u000bijis a\u000bected by \u0000 (as shown in\nFig. 5 left panel). Controlling the broadening of Bloch\nspectral functions \u0000 is in principal possible to evaluate\nfrom theory, but more importantly it is accessible from\nexperimental probes such as angular resolved photoelec-\ntron spectroscopy and two-photon electron spectroscopy.\nThe importance of non-locality in the Gilbert damping\ndepend strongly on the material (as shown in Fig. 5 right\npanel). It is important to note that the total | de\fned as\n\u000btot=P\nj\u000bijfor arbitrary i|, but also the local ( i=j)\nand the non-local ( i6=j) part of the Gilbert damping do\nnot violate the thermodynamic principles by gaining an-\ngular momentum (negative total damping). For Fe, the\n-101\n1. NN.\n2. NN.Fe\n34567αii\nαtot=/summationtext\njαijαq=0.1a−1\n0αq=0\n-10αij·10−4Co\n123456\nαij·10−3\n-15-10-50\n10−210−1\nΓ (eV)Ni\n5101520\n10−210−1\nΓ (eV)FIG. 5: (Color online) First (circles) and second nearest\nneighbour (triangles) Gilbert damping (left panel) as well as\non-site (circles) and total Gilbert (right panel) as a function of\nthe spectral width \u0000 for the itinerant magnets Fe, Co, and Ni.\nIn particular for Co, the results obtained from tight binding\nare compared with \frst-principles density functional theory\nresults (gray open circles). Solid lines (right panel) shows the\nGilbert damping obtained for the magnon wave vectors q= 0\n(blue line) and q= 0:1a\u00001\n0(red line). Dotted lines are added\nto guide the eye. Note that since cubic symmetry is broken\n(see text), there are two sets of nearest neighbor parameters\nand two sets of next nearest neighbor parameters (left panel)\nfor any choice of \u0000.\nlocal and total damping are of the same order for all\n\u0000, where in Co and Ni the local and non-local damp-\ning are equally important. The trends coming from our\ntight binding electron structure were also reproduced by\nour all-electron \frst-principles simulation, for both de-\npendency on the spectral broadening \u0000 (Fig. 5 gray open\ncircles) but also site resolved non-local damping in the\nintraband region (see Appendix A), in particular for fcc\nCo.\nWe compare also the non-local damping obtain from\nthe real and reciprocal space. For this, we used Eq. (3)\nby simulating Nq= 15\u000215\u000215 points in the \frst magnon\nBrillouin zone qand Fourier-transformed it (Fig. 6). For7\n-1.0-0.50.00.51.0αij·10−4\n5 10 15 20 25 30\nrij/a0FFT(α(q));αii= 0.003481\nFFT(G(k));αii= 0.003855\nFIG. 6: (Color online) Comparing non-local Gilbert damping\nobtained by Eq. (5) (red symbols) and Eq. (4) (blue symbols)\nin fcc Co for \u0000 = 0 :005 eV. The dotted line indicates zero\nvalue.\nboth approaches, we obtain good agreement, corroborat-\ning our methodology and possible applications in both\nspaces. The non-local damping for the \frst three nearest\nneighbour shells turn out to converge rapidly with Nq,\nwhile it does not converge so quickly for larger distances\nrij. The critical region around the \u0000-point in the Bril-\nlouin zone is suppressed in the integration over q. On\nthe other hand, the relation \u000btot=P\nj\u000bij=\u000b(q= 0)\nfor arbitrary ishould be valid, which is however violated\nin the intraband region as shown in Fig. 5 (compare tri-\nangles and blue line in Fig. 5): The real space damping\nis constant for small \u0000 and follows the long-wavelength\nlimit (compare triangles and red line in Fig. 5) rather\nthan the divergent ferromagnetic mode ( q= 0). Two\nexplanations are possible: i)convergence with respect to\nthe real space summation and ii)a di\u000berent scaling in\nboth models with respect to the spin-orbit coupling. For\ni), we carefully checked the convergence with the summa-\ntion cut-o\u000b (see Appendix D) and found even a lowering\nof the total damping for larger cut-o\u000b. However, the non-\nlocal damping is very long-range and, consequently, con-\nvergence will be achieved only at a cut-o\u000b radius >>9a0.\nForii), we checked the scaling of the real space Gilbert\ndamping with the spin-orbit coupling of the d-states\n(see Appendix B). Opposite to the `non-corrected' Kam-\nbersk\u0013 y formula in reciprocal space, which scales like\n\u00183\nd, we \fnd\u00182\ndfor the real space damping. This indi-\ncates that the spin-\rip scattering hosted in the real-space\nGreen's function is suppressed. To corroborate this state-\nment further, we applied the corrections proposed by\nEdwards48to our real space formula Eq. (4), which by\ndefault assumes \u00182(Fig. 4, red dots). Both methods, cor-\nrected and non-corrected Eq. (4), agree quite well. The\nsmall discrepancies are due to increased hybridisations\nand band inversion between p and d- states due to spin-\norbit coupling in the `non-corrected' case.\nFinally, we address other ways than temperature (here\nrepresented by \u0000), to manipulate the non-local damping.\nIt is well established in literature already for Heisenberg\nexchange and the magneto crystalline anisotropy that\n-0.40.00.40.81.2αij·10−4\n1 2 3 4 5 6 7\nrij/a0αii= 3.49·10−3αii= 3.43·10−3FIG. 7: (Color online) Non-local Gilbert damping as a func-\ntion of the normalized distancerij=a0for a tetragonal dis-\ntorted bcc Fe crystal structure. Here,c=a= 1:025 (red circles)\nandc=a= 1:05 (blue circles) is considered. \u0000 is put to 0 :01 eV.\nThe zero value is indicated by dotted lines.\ncompressive or tensial strain can be used to tune the mag-\nnetic phase stability and to design multiferroic materials.\nIn an analogous way, also non-local damping depends on\ndistortions in the crystal (see Fig. 7).\nHere, we applied non-volume conserved tetragonal\nstrain along the caxis. The local damping \u000biiis marginal\nbiased. Relative to the values of the undistorted case,\na stronger e\u000bect is observed for the non-local part, in\nparticular for the \frst few neighbours. Since we do a\nnon-volume conserved distortion, the in-plane second NN\ncomponent of the non-local damping is constant. The\ndamping is in general decreasing with increasing distor-\ntion, however, a change in the sign of the damping can\nalso occur (e.g. for the third NN). The rate of change\nin damping is not linear. In particular, the nearest-\nneighbour rate is about \u000e\u000b\u00190:4\u000110\u00005for 2:5% dis-\ntortion, and 2 :9\u000110\u00005for 5% from the undistorted case.\nFor the second nearest neighbour, the rate is even big-\nger (3:0\u000110\u00005for 2:5%, 6:9\u000110\u00005for 5%). For neigh-\nbours larger than rij= 3a0, the change is less signi\fcant\n(\u00000:6\u000110\u00005for 2:5%,\u00000:7\u000110\u00005for 5%). The strongly\nstrain dependent damping motivates even higher-order\ncoupled damping contributions obtained from Taylor ex-\npanding the damping contribution around the equilib-\nrium position \u000b0\nij:\u000bij=\u000b0\nij+@\u000bij=@uk\u0001uk+:::. Note that\nthis is in analogy to the magnetic exchange interaction55\n(exchange striction) and a natural name for it would\nbe `dissipation striction'. This opens new ways to dis-\nsipatively couple spin and lattice reservoir in combined\ndynamics55, to the best of our knowledge not considered\nin todays ab-initio modelling of atomistic magnetisation\ndynamics.\nC. Atomistic magnetisation dynamics\nThe question about the importance of non-local damp-\ning in atomistic magnetization dynamics (ASD) remains.8\n0.40.50.60.70.80.91.0M\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nt(ps)0.5\n0.1\n0.05\n0.01αtot\nαij\n0.5 1.0 1.5 2.0 2.5 3.0\nt(ps)Fe\nCo\nFIG. 8: (Color online) Evolution of the average magnetic mo-\nmentMduring remagnetization in bcc Fe (left panel) and\nfcc Co (right panel) for di\u000berent damping strength according\nto the spectral width \u0000 (di\u000berent colors) and both, full non-\nlocal\u000bij(solid line) and total, purely local \u000btot(dashed line)\nGilbert damping.\nFor this purpose, we performed zero-temperature ASD\nfor bcc Fe and fcc Co bulk and analysed changes in the\naverage magnetization during relaxation from a totally\nrandom magnetic con\fguration, for which the total mo-\nment was zero (Fig. 8)\nRelated to the spectral width, the velocity for remag-\nnetisation changes and is higher, the bigger the e\u000bective\nGilbert damping is. For comparison, we performed also\nASD simulations based on Eq. (2) with a scalar, purely\nlocal damping \u000btot(dotted lines). For Fe, it turned out\nthat accounting for the non-local damping causes a slight\ndecrease in the remagnetization time, however, is overall\nnot important for relaxation processes. This is under-\nstandable by comparing the particular damping values\nin Fig. 5, right panel, in which the non-local part ap-\npear negligible. On the other hand, for Co the e\u000bect\non the relaxation process is much more signi\fcant, since\nthe non-local Gilbert damping reduces the local contribu-\ntion drastically (see Fig. 5, right panel). This `negative'\nnon-local part ( i6=j) in\u000bijdecelerates the relaxation\nprocess and the relaxation time is drastically increased\nby a factor of 10. Note that a `positive' non-local part\nwill accelerate the relaxation, which is of high interest for\nultrafast switching processes.\nIII. CONCLUDING REMARKS\nIn conclusion, we have evaluated the non-locality of\nthe Gilbert damping parameter in both reciprocal and\nreal space for elemental, itinerant magnets bcc Fe, fcc\nCo and fcc Ni. In particular in the reciprocal space,\nour results are in good agreement with values given in\nthe literature32. The here studied real space damping\nwas considered on an atomistic level and it motivates\nto account for the full, non-local Gilbert damping in\nmagnetization dynamic, e.g. at surfaces56or for nano-\nstructures57. We revealed that non-local damping canbe negative, has a spatial anisotropy, quadratically scales\nwith spin-orbit coupling, and decays in space as r\u00002\nij.\nDetailed comparison between real and reciprocal states\nidenti\fed the importance of the corrections proposed by\nEdwards48and, consequently, overcome the limits of the\nKambersk\u0013 y formula showing an unphysical and experi-\nmental not proved divergent behaviour at low tempera-\nture. We further promote ways of manipulating non-local\nGilbert damping, either by temperature, materials dop-\ning or strain, and motivating `dissipation striction' terms,\nthat opens a fundamental new root in the coupling be-\ntween spin and lattice reservoirs.\nOur studies are the starting point for even further in-\nvestigations: Although we mimic temperature by the\nspectral broadening \u0000, a precise mapping of \u0000 to spin\nand phonon temperature is still missing, according to\nRefs. [14,23]. Even at zero temperature, we revealed a\nsigni\fcant e\u000bect of the non-local Gilbert damping to the\nmagnetization dynamics, but the in\ruence of non-local\ndamping to \fnite temperature analysis or even to low-\ndimensional structures has to be demonstrated.\nIV. ACKNOWLEDGEMENTS\nThe authors thank Lars Bergqvist, Lars Nordstr om,\nJustin Shaw, and Jonas Fransson for fruitful discus-\nsions. O.E. acknowledges the support from Swedish Re-\nsearch Council (VR), eSSENCE, and the KAW Founda-\ntion (Grants No. 2012.0031 and No. 2013.0020).\nAppendix A: Numerical details\nWe performkintegration with up to 1 :25\u0001106mesh\npoints (500\u0002500\u0002500) in the \frst Brillouin zone for bulk.\nThe energy integration is evaluated at the Fermi level\nonly. For our principles studies, we performed a Slater-\nKoster parameterised40tight binding (TB) calculations58\nof the torque-torque correlation model as well as for the\nGreen's function model. Here, the TB parameters have\nbeen obtained by \ftting the electronic structures to those\nof a \frst-principles fully relativistic multiple scattering\nKorringa-Kohn-Rostoker (KKR) method using a genetic\nalgorithm. The details of the \ftting and the tight binding\nparameters are listed elsewhere23,59. This puts our model\non a \frm, \frst-principles ground.\nThe tight binding Hamiltonian60H=H0+Hmag+\nHsoccontains on-site energies and hopping elements H0,\nthe spin-orbit coupling Hsoc=\u0010S\u0001Land the Zeeman\ntermHmag=1=2B\u0001\u001b. The Green's function is obtained\nbyG= (\"+ i\u0000\u0000H)\u00001, allows in principle to consider\ndisorder in terms of spin and phonon as well as alloys23.\nThe bulk Greenian Gijin real space between site iandj\nis obtained by Fourier transformation. Despite the fact\nthat the tight binding approach is limited in accuracy, it\nproduces good agreement with \frst principle band struc-\nture calculations for energies smaller than \"F+ 5 eV.9\n-1.5-1.0-0.50.00.51.01.5\n5 10 15 20 25 30\nrij(Bohr radii)Γ≈0.01eVTB\nTBe\nDFT\nαDFT\nii= 3.9846·10−3\nαTB\nii= 3.6018·10−3-1.5-1.0-0.50.00.51.01.5\nΓ≈0.005eV\nαDFT\nii= 3.965·10−3\nαTB\nii= 3.5469·10−3αij·10−4\nFIG. 9: (Colour online) Comparison of non-local damping ob-\ntained from the Tight Binding method (TB) (red \flled sym-\nbols), Tight Binding with Edwards correction (TBe) (blue\n\flled symbols) and the linear mu\u000en tin orbital method (DFT)\n(open symbols) for fcc Co. Two di\u000berent spectral broadenings\nare chosen.\nEquation (4) was also evaluated within the DFT and\nlinear mu\u000en-tin orbital method (LMTO) based code\nRSPt. The calculations were done for a k-point mesh\nof 1283k-points. We used three types of basis func-\ntions, characterised by di\u000berent kinetic energies with\n\u00142= 0:1;\u00000:8;\u00001:7 Ry to describe 4 s, 4pand 3dstates.\nThe damping constants were calculated between the 3 d\norbitals, obtained using using mu\u000en-tin head projection\nscheme61. Both the \frst principles and tight binding im-\nplementation of the non-local Gilbert damping agree well\n(see Fig. 9).\nNote that due to numerical reasons, the values of\n\u0000 used for the comparisons are slightly di\u000berent in\nboth electronic structure methods. Furthermore, in the\nLMTO method the orbitals are projected to d-orbitals\nonly, which lead to small discrepancies in the damping.\nThe atomistic magnetization dynamics is also per-\nformed within the Cahmd simulation package58. To\nreproduce bulk properties, periodic boundary condi-\ntions and a su\u000eciently large cluster (10 \u000210\u000210)\nare employed. The numerical time step is \u0001 t=\n0:1 fs. The exchange coupling constants Jijare\nobtained from the Liechtenstein-Kastnelson-Antropov-\nGubanovski (LKAG) formula implemented in the \frst-\nprinciples fully relativistic multiple scattering Korringa-\nKohn-Rostoker (KKR) method39. On the other hand,\nthe magneto-crystalline anisotropy is used as a \fxed pa-\nrameter with K= 50\u0016eV.\n012345678α·10−3\n0.0 0.02 0.04 0.06 0.08 0.1\nξd(eV)2.02.22.42.62.83.03.2γ\n0.0 0.1 0.2 0.3 0.4\nq(a−1\n0)-12-10-8-6-4-20αnn·10−5\n01234567\nαos·10−3 1.945\n1.797\n1.848\n1.950\n1.848\n1.797\n1.950FIG. 10: (Color online) Gilbert damping \u000bas a function of\nthe spin-orbit coupling for the d-states in fcc Co. Lower panel\nshows the Gilbert damping in reciprocal space for di\u000berent\nq=jqjvalues (di\u000berent gray colours) along the \u0000 !Xpath.\nThe upper panel exhibits the on-site \u000bos(red dotes and lines)\nand nearest-neighbour \u000bnn(gray dots and lines) damping.\nThe solid line is the exponential \ft of the data point. The\ninset shows the \ftted exponents \rwith respect wave vector\nq. The colour of the dots is adjusted to the particular branch\nin the main \fgure. The spectral width is \u0000 = 0 :005 eV.\nAppendix B: Spin-orbit coupling scaling in real and\nreciprocal space\nKambersk\u0013 y's formula is valid only for quadratic spin-\norbit coupling scaling21,47, which implies only scattering\nbetween states that preserve the spin. This mechanism\nwas explicitly accounted by Edwards48by neglecting the\nspin-orbit coupling contribution in the `host' Green's\nfunction. It is predicted for the coherent mode ( q= 0)21\nthat this overcomes the unphysical and not experimen-\ntally veri\fed divergent Gilbert damping for low tem-\nperature. Thus, the methodology requires to prove the\nfunctional dependency of the (non-local) Gilbert damp-\ning with respect to the spin-orbit coupling constant \u0018\n(Fig. 10). Since damping is a Fermi-surface e\u000bects, it\nis su\u000ecient to consider only the spin-orbit coupling of\nthe d-states. The real space Gilbert damping \u000bij/\u0018\r\nscales for both on-site and nearest-neighbour sites with\n\r\u00192. For the reciprocal space, however, the scaling is\nmore complex and \rdepends on the magnon wave vec-\ntorq(inset in Fig. 10). In the long-wavelength limit,\nthe Kambersk\u0013 y formula is valid, where for the ferromag-\nnetic magnon mode with \r\u00193 the Kambersk\u0013 y formula\nis inde\fnite according to Edwards48.10\n10−32510−2α(q)\n10−310−210−110+0\nΓ (eV)0.000\n0.025\n0.050\n0.075\n0.100\nq: Γ→XCo\nFIG. 11: (Colour online) Comparison of reciprocal non-local\ndamping with (squares) or without (circles) corrections pro-\nposed by Costa et al.47and Edwards48for Co and di\u000berent\nspectral broadening \u0000. Di\u000berent colours represent di\u000berent\nmagnon propagation vectors q.\nAppendix C: Intraband corrections\nFrom the same reason as discussed in Section B, the\nrole of the correction proposed by Edwards48for magnon\npropagations di\u000berent than zero is unclear and need to\nbe studied. Hence, we included the correction of Ed-\nward also to Eq. (3) (Fig. 11). The exclusion of the spin-\norbit coupling (SOC) in the `host' clearly makes a major\nqualitative and quantitative change: Although the in-\nterband transitions are una\u000bected, interband transitions\nare mainly suppressed, as it was already discussed by\nBarati et al.21. However, the intraband contributions are\nnot totally removed for small \u0000. For very small scat-\ntering rates, the damping is constant. Opposite to the\n`non-corrected' Kambersk\u0013 y formula, the increase of the\nmagnon wave number qgives an increase in the non-\nlocal damping which is in agreement to the observation\nmade by Yuan et al.29, but also with the analytical modelproposed in Ref. [52] for small q. This behaviour was ob-\nserved for all itinerant magnets studied here.\nAppendix D: Comparison real and reciprocal\nGilbert damping\nThe non-local damping scales like r\u00002\nijwith the dis-\ntance between the sites iandj, and is, thus, very long\nrange. In order to compare \u000btot=P\nj2Rcut\u000bijfor arbi-\ntraryiwith\u000b(q= 0), we have to specify the cut-o\u000b ra-\ndius of the summation in real space (Fig. 12). The inter-\nband transitions (\u0000 >0:05 eV) are already converged for\nsmall cut-o\u000b radii Rcut= 3a0. Intraband transitions, on\nthe other hand, converge weakly with Rcutto the recipro-\ncal space value \u000b(q= 0). Note that \u000b(q= 0) is obtained\nfrom the corrected formalism. Even with Rcut= 9a0\nwhich is proportional to \u001930000 atoms, we have not\n0.81.21.62.0αtot·10−3\n4 5 6 7 8 9\nRcut/a00.005\n0.1\nFIG. 12: Total Gilbert damping \u000btotfor fcc Co as a function\nof summation cut-o\u000b radius for two spectral width \u0000, one in\nintraband (\u0000 = 0 :005 eV, red dottes and lines) and one in the\ninterband (\u0000 = 0 :1 eV, blue dottes and lines) region. The\ndotted and solid lines indicates the reciprocal value \u000b(q= 0)\nwith and without SOC corrections, respectively.\nobtain convergence.\n\u0003Electronic address: danny.thonig@physics.uu.se\n1S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320,\n190 (2008), URL http://www.sciencemag.org/cgi/doi/\n10.1126/science.1145799 .\n2J. Iwasaki, M. Mochizuki, and N. Nagaosa, Nature\nNanotech 8, 742 (2013), URL http://www.nature.com/\ndoifinder/10.1038/nnano.2013.176 .\n3A. Fert, V. Cros, and J. 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As an ap-\nplication of our approach, we also study the asymptotic beha vior of the damped wave equation in\nEuclidean space under the geometric control condition.\nKEY WORDS: Damped wave equation, Asymptotic expansion, Hea t equation, Diffusion phenom-\nena, Geometric control condition, Variable damping, Energ y decay.\nAMS SUBJECT CLASSIFICATION: 35L05, 35K05, 35 Q99, 46N20\n1 Introduction\nWe study the following abstract damped wave equation\n(1.1) ( ∂2\nt+A+B∂t)u= 0.\nHere∂t=∂\n∂tandA,Bare self-adjoint operators on a Hilbert space H. We assume Ais a densely-defined\npositive operator i.e. A>0.Bis a strictly positive bounded operator on Hi.e.B≥c Idfor a constant\nc >0. We also assume B1/2andB−1/2are continuous operators on D(A), i.e. there exists a constant\nC >0 such that for any u∈D(A), the following bounds hold\n/ba∇dblAB1/2u/ba∇dblH≤C(/ba∇dblu/ba∇dblH+/ba∇dblAu/ba∇dblH),\n/ba∇dblAB−1/2u/ba∇dblH≤C(/ba∇dblu/ba∇dblH+/ba∇dblAu/ba∇dblH).\nAn example of this equation is the usual damped wave equation\n(1.2) ( ∂2\nt−∆+a(x)∂t)u= 0 inR×Rd\nwhere ∆ is the constant coefficient Laplace operator on Rd,a(x) is a strictly positive definite smooth\nfunction with bounded derivatives. It is known that eventually the s olution of (1 .2) are close to the\nsolution of a heat type equation. Here we see this by heuristic argum ent. The stationary equation\nis closely related to the asymptotic profile. So we take u=etλu0(x) in (1.2), we have the following\nstationary damped wave equation\n(1.3) ( λ2+a(x)λ−∆)u0= 0.\nTo the solution of (1 .2), the influence of large frequency part becomes smaller as time te nd to infinity by\ndamping. So for small λ, if we neglect λ2term in (1.3), we obtain\n(a(x)λ−∆)u0= 0.\nThis equation is a stationary equation to the following heat equation\n(a(x)∂t−∆)u0= 0.\nSo we may expect the asymptotic behavior of the solution to (1 .1) is close to the following abstract heat\nequation\n(1.4) B1/2(∂t+˜A)B1/2v= 0.\n1Here˜A=B−1/2AB−1/2. The solution of this equation can be written as follows\nB−1/2e−t˜AB1/2u0.\nThus we can expect that the solution of the damped wave equation a pproach to the above solution as\ntime tend to infinity. The following statement gives a justification of t his argument.\nTheorem 1.1. Letube the solution to the Cauchy problem of the following abstra ct damped wave equation\n(1.5)/braceleftBigg\n(∂2\nt+A+B∂t)u= 0,\nu|t=0=u0,∂tu|t=0=u1.\nThen there exists C >0such that for any u0∈D(√\nA),u1∈Handt >1, the following asymptotic\nprofile holds\n/ba∇dblu(t)−B−1/2e−t˜AB1/2u0−B−1/2e−t˜AB−1/2u1/ba∇dblH≤Ct−1(/ba∇dblu0/ba∇dblH+/ba∇dbl√\nAu0/ba∇dblH+/ba∇dblu1/ba∇dblH).\nRemark 1.2. We can get more sharp estimate, see Remark 5.5. On the other hand, this decay rate is\noptimal, see Proposition 5.6which is a generalization of the result of [6].\nIn many situation, we impose some assumptions to the initial data. In the case, Theorem 1 .1 does\nnot give sufficient information. For example, we consider the consta nt coefficient damped wave equation\nonRd\n(∂2\nt−∆+∂t)u= 0.\nThe correspondence heat propagator satisfies\net∆:L1→L2=O(t−d/4).\nFrom this decay estimate, we can not get the top term from Theore m 1.1 if the initial date u0,u1are in\nL1andd>3. So we need to impose some more assumption to the equation. For t he purpose, we specify\nH=L2(Ω,µ) where (Ω ,µ) is aσ-finite measure space and assume that {e−t˜A}t>0is also a bounded\nsemi-group on L1(Ω,µ)\n(1.6) /ba∇dble−t˜Au/ba∇dblL1=O(1)/ba∇dblu/ba∇dblL1,t>0.\nWe also assume that there exists m>0 such that the following diffusion estimate holds\n(1.7) /ba∇dble−t˜Au/ba∇dblL2=O(t−m)/ba∇dblu/ba∇dblL1, t>0.\nFurthermore B1/2is also a bounded operator on L1satisfying\n(1.8)1\nC/ba∇dblu/ba∇dblL1≤ /ba∇dblB1/2u/ba∇dblL1≤C/ba∇dblu/ba∇dblL1\nfor a constant C >0. Our model problem (1 .2), satisfies this assumption. The next result is a diffusion\ntype estimate for such operators.\nTheorem 1.3. Under the assumptions of Theorem 1.1, we also assume (1.6),(1.7)and(1.8). Then\nthere exists a constant C >0such that for any u0∈D(√\nA)∩L1,u1∈L2∩L1,t>1, the solution u(t)\nof(1.5)satisfies the following asymptotic profile\n/ba∇dblu(t)−B−1/2e−t˜AB1/2u0−B−1/2e−t˜AB−1/2u1/ba∇dblL2≤Ct−m−1(/ba∇dblu0/ba∇dblL2∩L1+/ba∇dbl√\nAu0/ba∇dblL2+/ba∇dblu1/ba∇dblL2∩L1).\nIn the final section, as an application of the above theorems, we co nsider an example of damped wave\nequation on Euclidean space and we give an asymptotic profile of the s olution for a variable coefficient\ndamped wave equation, see Theorem 7 .1. We also treat a perturbation of this case for which the strictly\npositivity of the damping term may not be satisfied but the geometric control condition holds. In this\ncase, we can give the similar asymptotic profile, see Theorem 7 .9.\nNow we remark some related result. On bounded domains or manifolds , under the geometric control\ncondition, the exponential energy decay of the solution was prove d in [1], [16]. On the other hand, for\nunbounded domains the energy does not decay exponentially. In [13 ], the fact that the decay late of the\nsolution is similar to the heat equation is proved to study the semi-linea r damped wave equation. The\n2solution of the damped wave equation tend to the solution of the hea t equation is well-known for example\n[14]. This type results may be called diffusion phenomena for the dampe d wave equation. For variable\ncoefficient damping, Y, Wakasugi ([18]) proved the diffusion phenome na for spacial decreasing damping.\nFor abstract setting, the diffusion phenomena may be proved in [17 ] but at present, the manuscript does\nnot published and their result seems to slightly different from ours.\nIf the damping term is too weak, for example short rangecase, the asymptotic behavior of the solution\nis quite different. In the case, the solution tend to the solution of th e wave equation and the local energy\ndecay is an important problem, see [3].\n2 Abstract damped wave equation\nWerecallthe settingofthe problem. Inthis section, weonlyassume thatBisabounded andnon-negative\noperator. We reduce (1 .1) to a first order system. We introduce the following operator\nA=/parenleftbigg0Id\n−A−B/parenrightbigg\n:H0→ H0, D(A) ={t(u,v)∈H1×H;Au∈H,v∈D(√\nA)}.\nHereH0=H1×His the energy space, H1is the Hilbert space completion of D(A) by the norm\n/ba∇dblu/ba∇dblH1=/ba∇dbl√\nAu/ba∇dbl2\nH.\nH−1denotes the dual space of H1. We noteD(√\nA) is naturally embedded in H1. So in what follows,\nwe consider D(√\nA) is a subspace of H1. By using this operator, we can rewrite the wave equation (1.5)\nas follows,\n(2.1) ∂t/parenleftbiggu\n∂tu/parenrightbigg\n=A/parenleftbiggu\n∂tu/parenrightbigg\n,/parenleftbiggu\n∂tu/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt=0=/parenleftbiggu0\nu1/parenrightbigg\n∈D(√\nA)×H.\nSince we are considering the damped wave equation, we have the follo wing proposition.\nProposition 2.1. Ais a m-dissipative operator.\nProof. From the definition, we easily prove Ais closed. For ( u,v)∈D(A), we compute as follows\nRe(At(u,v),t(u,v))H0= Re{(A1/2v,A1/2u)H−(Au+Bv,v)H}=−(Bv,v)H≤0.\nThusAis dissipative.\nFrom this proposition, we have the following resolvent estimate for R eλ>0\n/ba∇dbl(λ−A)−1/ba∇dblH0→H0≤1\nReλ.\nSo by Hille-Yosida Theorem, Agenerates a contraction semi-group\nU(t) =etA:H0→ H0.\nTo an initial datet(u0,u1)∈ H0, we write the solution as follows\nt(u(t),v(t)) =U(t)t(u0,u1).\nWe restrict the initial date since the energy space H0is a exotic space. Fort(u0,u1)∈D(A)×D(√\nA)⊂\nD(A), sinceU(t)t(u0,u1) isC1, we have the following identity in H1\nu(t) =u0+/integraldisplayt\n0v(s)ds.\nBy the density argument, the above identity also holds fort(u0,u1)∈D(√\nA)×H. From the right hand\nside of the above identity.,we can regard u(t)∈H. So we introduce the space H=D(√\nA)×Hby the\nnorm\n/ba∇dbl(u,v)/ba∇dblH={/ba∇dblu/ba∇dbl2\nH+/ba∇dblu/ba∇dbl2\nH1+/ba∇dblv/ba∇dbl2\nH}1/2.\nSinceU(t) is a contraction operator, from the above identity, we have the f ollowing bound for the\npropagator\n(2.2) /ba∇dblU(t)/ba∇dblH→H≤C(1+t).\nIn this paper, we call u(t)∈D(√\nA) fort(u(t),v(t)) =U(t)t(u0,u1), (u0,u1)∈ Has the solution to (1.5).\n33 Resolvent estimate\nWe prove some related estimates to the resolvent of A. Probably these estimates are well-known, c.f. [5],\nbut for completeness, we give the proof. First we study the followin g operator which is almost similar to\nthe resolvent\n(3.1) R(λ) = (λ2+Bλ+A)−1.\nFrom the positivity of B, we have the following estimate.\nLemma 3.1. There exists δ >0such that for Reλ>−δandImλ/\\e}atio\\slash= 0,R(λ) :H→D(A)exists and the\nfollowing bound holds\n(3.2) /ba∇dblR(λ)/ba∇dblH→H=O(|Imλ|−1).\nProof. For u∈D(A), we have\n((λ2+Bλ+A)u,u)H= ((|Reλ|2−|Imλ|2+BReλ+A)u,u)H+iImλ((2Reλ+B)u,u)H. (3.3)\nTaking imaginary part of the above identity and using B≥cIdwe have the following bound\n|((λ2+Bλ+A)u,u)H| ≥ |Imλ((2Reλ+B)u,u)H|\n≥ |Imλ|(c+2Reλ)(u,u)H\nfor Reλ >−c/2. So (λ2+Bλ+A) is injective if Re λis sufficiently small and Im λ/\\e}atio\\slash= 0. We can also\nprove (λ2+Bλ+A) is surjective by considering adjoint operator ( λ2+Bλ+A) and applying similar\nargument. So the inverse exists and we have the bound.\nTaking real part of the identity (3.3) and applying similar argument, w e can also prove the following\nlemma.\nLemma 3.2. IfReλ>|Imλ|, thenR(λ)exists and the following bound holds\n(3.4) /ba∇dblR(λ)/ba∇dblH→H=O(|Reλ|−1).\nRemark 3.3. From the identity (λ2+Bλ+A)∗=λ2+Bλ+A, we haveR(λ)∗=R(λ).So ifR(λ)\nexists then R(λ)exists.\nNext we estimate R(λ) as an operator between different Hilbert spaces.\nLemma 3.4. IfR(λ)exists, then the following bound holds\n(3.5) /ba∇dblR(λ)/ba∇dblH→H1=O(/ba∇dblR(λ)/ba∇dbl1/2\nH→H+(|λ|1/2+|λ|)/ba∇dblR(λ)/ba∇dblH→H).\nProof. For u∈H, we have\n/ba∇dblR(λ)u/ba∇dbl2\nH1= (√\nAR(λ)u,√\nAR(λ)u)H\n= (AR(λ)u,R(λ)u)H\n= ((A+λ2+Bλ)R(λ)u,R(λ)u)H−((λ2+Bλ)R(λ)u,R(λ)u)H\n= (u,R(λ)u)H−((λ2+Bλ)R(λ)u,R(λ)u)H\n=O(/ba∇dblR(λ)/ba∇dblH→H+(|λ|+|λ|2)/ba∇dblR(λ)/ba∇dbl2\nH→H)/ba∇dblu/ba∇dbl2\nH.\nLemma 3.5. IfR(λ)exists, then R(λ)A:D(A)→Hcan be extend to H1→Hand the following bound\nholds\n(3.6) /ba∇dblR(λ)A/ba∇dblH1→H=O(/ba∇dblR(λ))/ba∇dbl1/2\nH→H+(|λ|1/2+|λ|)/ba∇dblR(λ)/ba∇dblH→H).\n4Proof. ByR(λ)∗=R(λ)andLemma 3 .4,R(λ)isaboundedoperatorfrom HtoH1. SinceR(λ)∗=R(λ),\nwe can regard R(λ) as an operator from H−1toHby duality argument. From Lemma 3 .4, we have the\nfollowing estimate\nR(λ) :H−1→H=O(/ba∇dblR(λ)/ba∇dbl1/2\nH→H+(|λ|1/2+|λ|)/ba∇dblR(λ)/ba∇dblH→H).\nWe extend Ato the operator form H1toH−1by using the following identity\n/a\\}b∇acketle{tAu,v/a\\}b∇acket∇i}htH−1,H1= (√\nAu,√\nAv)H.\nFrom the definition, Asatisfies\nA:H1→H−1=O(1).\nThus we obtain the lemma.\nLemma 3.6. IfR(λ)exists for a λ/\\e}atio\\slash= 0thenR(λ)(λ+B) :D(A)→H1can be extend to H1→H1and\nthe following bound holds\n(3.7) /ba∇dblR(λ)(λ+B)/ba∇dblH1→H1=O((|λ|−1/2+1)/ba∇dblR(λ)/ba∇dbl1/2\nH→H+(1+|λ|)/ba∇dblR(λ)/ba∇dblH→H+|λ|−1).\nProof. We will estimate R(λ)(λ+B) using the following identity\nR(λ)(λ+B) =1\nλ−1\nλR(λ)A.\nForf∈D(A), we consider the following equation\nR(λ)Af=u.\nThis can be written as follows\nAf= (λ2+Bλ+A)u.\nSo we have\n(Au,u)H= (Af,u)H−((λ2+Bλ)u,u)H.\nTaking real part, for sufficiently small ε>0, we obtain\n/ba∇dblu/ba∇dbl2\nH1≤C((|λ|2+|λ|)/ba∇dblu/ba∇dbl2\nH+/ba∇dblf/ba∇dbl2\nH1)+ε/ba∇dblu/ba∇dbl2\nH1.\nThus\n/ba∇dblR(λ)Af/ba∇dblH1≤C((|λ|1/2\nH+|λ|)/ba∇dblR(λ)Af/ba∇dblH+/ba∇dblf/ba∇dblH1).\nApplying (3 .6), the following bound holds\n/ba∇dblR(λ)Af/ba∇dblH1≤C((|λ|1/2+|λ|)/ba∇dblR(λ)/ba∇dbl1/2\nH→H+(|λ|+|λ|2)/ba∇dblR(λ)/ba∇dblH→H)+1)/ba∇dblf/ba∇dblH1).\nBy density, we conclude the extension and the estimate.\nBy these lemmas, we can give the existence of the resolvent.\nProposition 3.7. IfR(λ)exists for a λ/\\e}atio\\slash= 0, then the resolvent (λ− A)−1:H0→ H0exists and the\nfollowing identity holds\n(3.8) ( λ−A)−1=/parenleftbigg\nR(λ)(λ+B)R(λ)\n−R(λ)A R(λ)λ/parenrightbigg\n.\nThe following estimate also holds\n(3.9) /ba∇dbl(λ−A)−1/ba∇dblH0→H0=O((1+|λ|−1/2)/ba∇dblR(λ)/ba∇dbl1/2\nH→H+(1+|λ|)/ba∇dblR(λ)/ba∇dblH→H)+|λ|−1).\nProof. From Lemma 3 .4, Lemma 3 .5 and Lemma 3 .6, the right hand side of (3 .8) can be extend as an\nbounded operator on H0. By the direct computation, ift(u,v)∈D(A)×D(A) then\n/parenleftbigg\nR(λ)(λ+B)R(λ)\n−R(λ)A R(λ)λ/parenrightbigg\n(λ−A)/parenleftbigg\nu\nv/parenrightbigg\n=/parenleftbigg\nR(λ)(λ+B)R(λ)\n−R(λ)A R(λ)λ/parenrightbigg/parenleftbigg\nλ−Id\nA λ+B/parenrightbigg/parenleftbigg\nu\nv/parenrightbigg\n=/parenleftbigg\nu\nv/parenrightbigg\n5So by density it is the left inverse. On the other hand fort(u,v)∈D(A)×D(√\nA)\n/parenleftbiggλ−Id\nA λ+B/parenrightbigg/parenleftbiggR(λ)(λ+B)R(λ)\n−R(λ)A R(λ)λ/parenrightbigg/parenleftbiggu\nv/parenrightbigg\n=/parenleftbiggId 0\nAR(λ)(λ+B)−(λ+B)R(λ)A Id/parenrightbigg/parenleftbiggu\nv/parenrightbigg\nWe use the following trick, for a λ/\\e}atio\\slash= 0\nλ(AR(λ)(λ+B)−(λ+B)R(λ)A)u= (AR(λ)(λ2+Bλ+A)−(λ2+Bλ+A)R(λ)A)u= 0.\nThus the density argument, it is also the right inverse. From Lemma 3 .4, Lemma 3 .5 and Lemma 3 .6,\nwe have the bound.\nRemark 3.8. From this proposition, we have the following identities\n((λ−A)−1t(0,u),t(0,v))H0=λ(R(λ)u,v)H,\n((λ−A)−1t(0,u),t(v,0))H0= (R(λ)u,v)H1.\nThe left hand side is complex analytic since the resolvent is analytic. So the right had side is complex\nanalytic so R(λ) :H→H,H→H1are complex analytic family of operators.\nApplying Lemma 3 .4 directly to R(λ)(λ+B) :H→H1, we also have the following estimates to the\nresolvent.\nProposition 3.9. IfR(λ)exists for a λ/\\e}atio\\slash= 0, then we have the following estimate\n(3.10) /ba∇dbl(λ−A)−1/ba∇dblH→H 0=O((|λ|+1)/ba∇dblR(λ)/ba∇dbl1/2\nH→H+(|λ|1/2+|λ|2)/ba∇dblR(λ)/ba∇dblH→H).\nThis proposition says if we restrict the domain of the resolvent to Hthen the singularity, as λtend to\n0, become weaker one. This estimate is important for energy decay . We also have the following estimate.\nProposition 3.10. IfR(λ)exists for a λ/\\e}atio\\slash= 0, then we have the following estimate\n(3.11) /ba∇dbl(λ−A)−1/ba∇dblH→H=O((1+|λ|−1/2)/ba∇dblR(λ)/ba∇dbl1/2\nH→H+(1+|λ|)/ba∇dblR(λ)/ba∇dblH→H+|λ|−1).\nBy Lemma 3 .1 and Lemma 3 .2, we conclude the following estimates of the resolvent.\nProposition 3.11. If|Reλ|<cfor sufficient small c>0andImλ/\\e}atio\\slash= 0, then the resolvent satisfies\n/ba∇dbl(λ−A)−1/ba∇dblH0→H0=O(|Imλ|−1+1),\n/ba∇dbl(λ−A)−1/ba∇dblH→H 0=O((1+|λ|)|Imλ|−1/2+(|λ|1/2+|λ|2)|Imλ|−1),\n/ba∇dbl(λ−A)−1/ba∇dblH→H=O(|Imλ|−1+1).\nIfReλ>0, the following estimate also holds\n/ba∇dbl(λ−A)−1/ba∇dblH0→H0=O(|λ|−1+1),\n/ba∇dbl(λ−A)−1/ba∇dblH→H 0=O(1+|λ|+|λ|−1/2),\n/ba∇dbl(λ−A)−1/ba∇dblH→H=O(|λ|−1+1).\n4 Abstract energy decay\nWe prove the following abstract energy decay estimate to the damp ed wave equation. The theorem may\nbe well known but we prove here for later argument.\nTheorem 4.1. There exists a constant Csuch that for anyt(u0,u1)∈ H, andt>1, the following bound\nholds\n/ba∇dblU(t)t(u0,u1)/ba∇dblH0≤C(t−1/2/ba∇dblt(u0,u1)/ba∇dblH).\nRemark 4.2. We give one example which may be useful to understand the proo f of this theorem. We\nconsider the constant coefficient damped wave equation on Rd\n(∂2\nt−∆+∂t)u= 0.\n6The Fourier transform of the solution is the following form\nu+(ξ)exp/parenleftBigg\n−1+/radicalbig\n1−4|ξ|2\n2t/parenrightBigg\n+u−(ξ)exp/parenleftBigg\n−1−/radicalbig\n1−4|ξ|2\n2t/parenrightBigg\n.\nFor simplicity, we assume u+andu−are inL2, thenu−part decays exponentially so it is no interest.\nIfξ= 0,u+part does not decay therefore uniform decay does not occur if we only assume u+(ξ)∈\nL2. However if we assume |ξ|u+(ξ)∈L2, we have the uniform decay estimate, by using the following\ndecomposition,\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble|ξ|u+(ξ)exp(−1+/radicalbig\n1−4|ξ|2\n2t)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2\n≤/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleχ{|ξ|<1/2}|ξ|u+(ξ)exp(−1+/radicalbig\n1−4|ξ|2\n2t)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleχ{|ξ|≥1/2}|ξ|u+(ξ)exp(−1+/radicalbig\n1−4|ξ|2\n2t)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2\n≤/vextenddouble/vextenddoubleχ{|ξ|<1/2}|ξ|exp(−|ξ|2t)/vextenddouble/vextenddouble\nL∞/ba∇dblu+(ξ)/ba∇dblL2+O(e−t)/ba∇dbl|ξ|u+(ξ)/ba∇dblL2\n≤ O(t−1/2)(/ba∇dblu+(ξ)/ba∇dblL2+/ba∇dbl|ξ|u+(ξ)/ba∇dblL2).\nHereχ{·}denote the characteristic functions. The high frequency pa rt decays exponentially and the low\nfrequency part becomes the main part. This type decompositi on will be used in the proof of the theorem.\nWe also remark that the above estimate is optimal without any further assumption for u+.\nWe give a proof of the theorem. First we assume the Cauchy data f=t(u0,u1) is inD(Ak) for large\nkandu0∈H. SoU(t)fis inCkastvariable function. We take a smooth cut-off function ψsatisfying\nψ(t) =/braceleftBigg\n1, t≥2,\n0, t<1.\nWe shall estimate the following cut-off propagator as in [4] ,[11], [19].\nV(t)f=ψ(t)U(t)f.\nFrom (2.2), we can define its Fourier-Laplace transform by\n/hatwiderVf(λ) =1√\n2π/integraldisplay∞\n−∞e−iλtV(t)fdt\nif Imλ=−Reiλ=−ε<0. Then by Fourier inversion formula, the following identity holds\nV(t)f=1√\n2π/integraldisplay\nImλ=−εeiλt/hatwiderVf(λ)dλ.\nBy definition, we have the identity\n(∂t−A)V(t)f=ψ′(t)U(t)f.\nTaking its Fourier-Laplace transform, we get\n(iλ−A)/hatwiderVf(λ) =/hatwiderψ′Uf(λ).\nif Imλ=−ε<0. Since Ais a dissipative operator the resolvent exists for Im λ=−ε<0 and we obtain\n/hatwiderVf(λ) = (iλ−A)−1/hatwiderψ′Uf(λ).\nWe take its inverse Fourier transform and we get\nV(t)f=1√\n2π/integraldisplay\nImλ=−ε(iλ−A)−1eiλt/hatwiderψ′Uf(λ)dλ.\nSinceψ′(t) has a compact support, /hatwiderψ′Uf(λ) is a holomorphic function and by the results of previous\nsection, (iλ−A)−1can be holomorphically extended to Re iλ=−a <0 ifa >0 is a sufficiently small\n7constant and Im iλ/\\e}atio\\slash= 0. Since U(t)fisCkinD(A)×H,/hatwiderψ′Uf(λ) decays sufficiently first as λtend to\ninfinity, so we can change the integral contour as follows\nV(t)f=1√\n2π/integraldisplay\nImλ=aχ(λ)(iλ−A)−1eiλt/hatwiderψ′Uf(λ)dλ+1√\n2π/integraldisplay\nC(iλ−A)−1eiλt/hatwiderψ′Uf(λ)dλ\n=e−atI1f(t)+I2f(t) = (High frequency part) + (low frequency part) .\nHereχis a cut-off function defined by\nχ(λ) =/braceleftBigg\n1,|Reλ| ≥δ,\n0,|Reλ|<δ,\nfor sufficient small δ >0.Cwhich we choose later, is a contour around the origin connecting two points\n±δ+ia. We prove the following bound of I1by using the method of [11] which is known as Morawetz’s\nargument\n(4.1) /ba∇dble−atI1f(t)/ba∇dblH0≤Ce(−a+ε)t/ba∇dblf/ba∇dblH0.\nFirst we estimate L2norm ofI1f. Using Plancheral’s identity, we have\n/integraldisplay∞\n−∞/ba∇dblI1f(τ)/ba∇dbl2\nH0dτ=/integraldisplay∞\n−∞/vextenddouble/vextenddouble/vextenddouble/vextenddouble1√\n2π/integraldisplay\nImλ=ae(iλ+a)τχ(λ)(iλ−A)−1/hatwiderψ′Uf(λ)dλ/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nH0dτ\n≤/integraldisplay∞\n−∞/ba∇dblχ(λ+ia)(iλ−a−A)−1/hatwiderψ′Uf(λ+ia)/ba∇dbl2\nH0dλ.\nBy proposition 3 .11 andχcut low frequency, χ(λ+ia)(iλ−a−A)−1are uniformly bounded operators\nonH0so we estimate\n/integraldisplay∞\n−∞/ba∇dblI1f(τ)/ba∇dbl2\nH0dτ≤C/integraldisplay∞\n−∞/vextenddouble/vextenddouble/vextenddouble/hatwiderψ′Uf(λ+ia)/vextenddouble/vextenddouble/vextenddouble2\nH0dλ\n=C/integraldisplay+∞\n−∞e2at/ba∇dblψ′(t)U(t)f/ba∇dbl2\nH0dt.\nFor last identity, we use Plancheral’s identity again. Since ψ′is a compact support function, we have the\nfollowing estimate\n(4.2)/integraldisplay∞\n−∞/ba∇dblI1f(τ)/ba∇dbl2\nH0dτ≤C/ba∇dblf/ba∇dbl2\nH0.\nWe give point-wise bound of I1f(t) using this estimate. From this inequality, there exist 0 <s<2 such\nthat\n/ba∇dblI1f(s)/ba∇dblH0≤C/ba∇dblf/ba∇dblH0.\nWe define\n∂tI1f(t)−AI1f(t) =1√\n2π/integraldisplay\nImλ=aχ(λ)e(iλ+a)t/hatwiderψ′Uf(λ)dλ+aI1f(t)\n=I3f(t)+aI1f(t).\nBy Duhamel’s principle, we have\n(4.3) I1f(t) =U(t−s)I1f(s)+/integraldisplayt\nsU(t−τ)(I3f(τ)+aI1f(τ))dτ.\nWe shall estimate the last part. First we obtain\n/integraldisplayt\ns/ba∇dblU(t−τ)I3f(τ)/ba∇dbl2\nH0dτ=/integraldisplayt\ns/vextenddouble/vextenddouble/vextenddouble/vextenddoubleU(t−τ)/integraldisplay\nImλ=ae(iλ+a)τχ(λ)/hatwiderψ′Uf(λ))dλ/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nH0dτ\n≤/integraldisplay+∞\n−∞/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplay\nImλ=ae(iλ+a)τχ(λ)/hatwiderψ′Uf(λ))dλ/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nH0dτ(4.4)\n8sinceU(t) is a propagator of a dissipative operator. By Plancheral’s identity w e have\n/integraldisplayt\ns/ba∇dblU(t−τ)I3f(τ)/ba∇dbl2\nH0dτ≤/integraldisplay+∞\n−∞/ba∇dblχ(λ+ia)/hatwiderψ′Uf(λ+ia)/ba∇dbl2\nH0dλ\n≤C/integraldisplay+∞\n−∞/ba∇dbl/hatwiderψ′Uf(λ+ia)/ba∇dbl2\nH0dλ.\nWe use Plancheral’s identity again and since ψ′is a compact support function, we have\n/integraldisplayt\ns/ba∇dblU(t−τ)I3f(τ)/ba∇dbl2\nH0dτ≤C/integraldisplay+∞\n−∞e2at/ba∇dblψ′(t)U(t)f/ba∇dbl2\nH0dt\n≤C/ba∇dblf/ba∇dbl2\nH0.(4.5)\nBy the Schwartz inequality, from (4 .3), (4.2) and (4.5), we have\ne−εt/ba∇dblI1f(t)/ba∇dblH0≤e−εt/ba∇dblU(t−s)I1f(s)/ba∇dblH0+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayt\nse−εtU(t−τ)(I3f(τ)+aI1f(τ))dτ/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nH0\n≤C/ba∇dblf/ba∇dblH0+/braceleftbigg/integraldisplayt\nse−2εtdτ/bracerightbigg1/2/braceleftbigg/integraldisplayt\ns/ba∇dblU(t−τ)I3f(τ)/ba∇dbl2\nH0dτ+/integraldisplayt\ns/ba∇dblU(t−τ)aI1f(τ)/ba∇dbl2\nH0dτ/bracerightbigg1/2\n≤C/ba∇dblf/ba∇dblH0+C/braceleftbigg/integraldisplayt\ns/ba∇dblU(t−τ)I3f(τ)/ba∇dbl2\nH0dτ+/integraldisplay∞\n−∞/ba∇dblI1f(τ)/ba∇dbl2\nH0dτ/bracerightbigg1/2\n≤C/ba∇dblf/ba∇dblH0.\nSoe−atI1f(t) decays exponentially.\nNext we study I2fpart. We show the following estimate.\nProposition 4.3. There exists C >0such that for any t>1andf∈ H, the following bound holds\n/ba∇dblI2f(t)/ba∇dblH0≤C(t−1/2/ba∇dblf/ba∇dblH).\nProof. We choose the contour as follows C=Co∪C+∪C−.Co={1\nteis;s∈[−π,0]},C+={(1−s)1\nt+\ns(δ+ia);s∈[0,1]},C−={−(1−s)1\nt+s(−δ+ia);s∈[0,1]}. Here we impose suitable orientation on\nthese contours. We estimate each contours in the following integra l,\nI2f(t) =1√\n2π/integraldisplay\nCo∪C+∪C−(iλ−A)−1eiλt/hatwiderψ′U(λ)fdλ.\nWe have /ba∇dbl(iλ−A)−1/ba∇dblH→H 0=O(t1/2) onCoby Proposition 3 .11 and the length of CoisO(1/t). So the\nfollowing bound holds\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplay\nCo(iλ−A)−1eiλt/hatwiderψ′Uf(λ)dλ/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nH0=O/parenleftbigg1\ntt1/2sup\nλ∈Co/ba∇dbl/hatwiderψ′Uf(λ)/ba∇dblH/parenrightbigg\n.\nFrom (2.2) andψ′is a compact support function, we have\nsup\nλ∈Co/ba∇dbl/hatwiderψ′Uf(λ)/ba∇dblH= sup\nλ∈Co/vextenddouble/vextenddouble/vextenddouble/vextenddouble1√\n2π/integraldisplay∞\n−∞e−iλsψ′(s)U(s)fds/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nH=O(1)/ba∇dblψ′/ba∇dblL1/ba∇dblf/ba∇dblH\nSo we have the estimate for Copart. Next we estimate C+part, for sufficient large t, we have |λ| ∼\n|Reλ| ∼s+1/tonC+. So by Proposition 3 .11, we obtain\n/ba∇dbl(iλ−A)−1/ba∇dblH→H 0=O(t1/2).\nThus\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplay\nC+(iλ−A)−1eiλt/hatwiderψ′Uf(λ)dλ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nH0=O/integraldisplay1\n0t1/2e−astdssup\nλ∈C+/ba∇dbl/hatwiderψ′Uf(λ)/ba∇dblH\n=O(t−1/2/ba∇dblf/ba∇dblH).\nIn the same way, we can estimate C−part and we have proved.\n9Thus we have proved the abstract energy decay of the damped wa ve equation for sufficiently regular\ninitial data. Then by density argument, we obtain the theorem.\nRemark 4.4. In the proof, we essentially use the following facts\n(i)the existence of the resolvent for Imλ/\\e}atio\\slash= 0andReλ>−c,c>0,\n(ii)uniform estimate of resolvent for large λwhich comes from |Imλ|−1order estimate of R(λ),\n(iii)|λ|−1/2order estimate of the resolvent near the origin which comes f rom|λ|−1estimate of R(λ)\nforReλ>0andImλ∼ |λ|regions.\n5 Proof of Theorem 1.1\nFrom now on we shall prove Theorem 1 .1. We recall ˜A=B−1/2AB−1/2.SinceB−1/2is bounded from\nD(A) toD(A), this operator is also a self-adjoint operator whose domain is D(A). By the spectral\ntheorem, we introduce a projection operator φ(˜A). Hereφis a cut-off function defined as follows\n(5.1) φ(x) =/braceleftBigg\n1, x≥ε,\n0, x<ε.\nBy using this function, we define a cut-off operator by\nφB(A) =B−1/2φ(˜A)B1/2.\nThen we have\nφB(A)2=φB(A), φB(A)∗=B1/2φ(˜A)B−1/2.\nSinceU(t) :H → H, we can apply φB(A) toU(t)f=t(u(t),v(t)),f=t(u0,u1)∈ Hby\nφB(A)U(t)f=t(φB(A)u(t),φB(A)v(t)).\nTo prove Theorem 1 .3, we use the following decomposition\nU(t)f=φB(A)U(t)f+(1−φB(A))U(t)f\n= (High frequency part)+(Low frequency part)\nand estimate each part. First we estimate the High frequency part .\nProposition 5.1. There exists C >0such that for any t>1andf∈ H, we have the following bound\n/ba∇dblφB(A)U(t)f/ba∇dblH≤Ct−2/ba∇dblf/ba∇dblH.\nProof. Applying similar argument to the previous section, for sufficie nt regularf, we write\nφB(A)ψ(t)U(t)f=φB(A)1√\n2π/integraldisplay\nImλ=aχ(λ)(iλ−A)−1eiλt/hatwiderψ′Uf(λ)dλ\n+φB(A)1√\n2π/integraldisplay\nC(iλ−A)−1eiλt/hatwiderψ′Uf(λ)dλ\n=φB(A)e−atI1f(t)+φB(A)I2f(t)\n= (High frequency part)+(Low frequency part)\nHereψ(t) andχ(λ) arecut-offfunctions introduced in previoussection, a>0 is a sufficient small constant\nandCis the contour around the origin which is also used in the former sectio n. Sinceψ(t) = 1 for large\nt, we have\nφB(A)U(t)f=φB(A)ψ(t)U(t)f\nfor larget. So we need the estimate of φB(A)ψ(t)U(t)f. The estimate of e−atI1f(t) is essentially similar\nto the argument to Proposition 4 .1 but we need small modification in (4 .4) since we only know the\nestimate (2 .2) andU(t−τ) may be not bounded. In this case, we can prove the following estima te by\nsimilar argument for arbitrary small ε>0\n/integraldisplayt\ns/ba∇dble−εtU(t−τ)I3f/ba∇dbl2\nHdτ≤C/ba∇dblf/ba∇dblH.\n10So we conclude the following bound\n(5.2) /ba∇dblφB(A)e−atI1f(t)/ba∇dblH=O(e−(a−2ε)t/ba∇dblf/ba∇dblH).\nNext we shall estimate φB(A)I2fby using the operator\n(Bλ+A)−1=B−1/2(λ+˜A)−1B−1/2.\nThis identity is easily seen from ( Bλ+A) =B1/2(λ+˜A)B1/2. From the following identity\n(λ2+Bλ+A)(Bλ+A)−1=Id+λ2(Bλ+A)−1,\nwe also have\n(5.3) R(λ) = (Bλ+A)−1−λ2(Bλ+A)−1R(λ).\nHereR(λ) = (λ2+Bλ+A)−1and (Bλ+A)−1are commutative operators. We recall the identity\n(λ−A)−1=R(λ)/parenleftbigg(λ+B)Id\n−A λ/parenrightbigg\n=R(λ)M(λ). (5.4)\nApplying these identities, we have\nφB(A)I2f(t) =1√\n2π/integraldisplay\nCφB(A){(Biλ+A)−1+λ2(Biλ+A)−1R(iλ)}M(iλ)eiλt/hatwiderψ′Uf(λ)dλ\n=J1f(t)+J2f(t).\nHere\nJ1f(t) =1√\n2π/integraldisplay\nCB−1/2φ(˜A)(iλ+˜A)−1B−1/2eiλtM(iλ)/hatwiderψ′Uf(λ)dλ,\n(5.5) J2f(t) =1√\n2π/integraldisplay\nCB−1/2φ(˜A)(iλ+˜A)−1B−1/2λ2(iλ−A)−1eiλt/hatwiderψ′Uf(λ)dλ.\nSinceφ(˜A) is a cut-off operator to the high frequency part. φ(˜A)(iλ+˜A)−1can be holomorphically\nextended across the origin. Since B1/2andB−1/2are continuous on D(A), by interpolation we have the\nfollowing estimate if λis sufficiently near the origin,\n/ba∇dblB−1/2φ(˜A)(iλ+˜A)−1u/ba∇dbl2\nH1=/ba∇dbl√\nAB−1/2φ(˜A)(iλ+˜A)−1u/ba∇dbl2\nH\n= (˜Aφ(˜A)(iλ+˜A)−1u,φ(˜A)(iλ+˜A)−1u)H\n=/ba∇dbl/radicalbig\n˜Aφ(˜A)(iλ+˜A)−1u/ba∇dbl2\nH\n=/ba∇dblφ(˜A)(iλ+˜A)−1/radicalbig\n˜Au/ba∇dbl2\nH\n=O(/ba∇dbl/radicalbig\n˜Au/ba∇dbl2\nH) =O(/ba∇dblu/ba∇dbl2\nH1+/ba∇dblu/ba∇dbl2\nH).\nIn the same way, using ( iλ+˜A)−1˜A=Id−iλ(iλ+˜A)−1, near the origin, we have the following bound\n/ba∇dblB−1/2φ(˜A)(iλ+˜A)−1B−1/2M(iλ)/ba∇dblH→H=O(1).\nForJ1f, we deform the contour to Re iλ<0. Then we have\n/ba∇dblJ1f(t)/ba∇dblH=1√\n2π/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplay\nImλ=a,|Reλ|<δB−1/2φ(˜A)(iλ+˜A)−1B−1/2eiλtM(iλ)/hatwiderψ′Uf(λ)dλ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nH\n≤C/integraldisplay\nImλ=a,|Reλ|<δ|eiλt|/ba∇dbl/hatwiderψ′Uf(λ)/ba∇dblHdλ\n≤Ce−at/ba∇dblf/ba∇dblH.\nThus this part decays exponentially. We apply same contour deform ation as in Proposition 4 .3 toJ2f.\nSince/ba∇dblλ2(iλ+A)−1/ba∇dblH→H=O(λ) for Reiλ>0, we obtain /ba∇dblλ2(iλ+A)−1/ba∇dblH→H=O(1/t) onCowhich is\n11defined in the proof of Proposition 4 .3. Then the length of CoisO(1/t) andφ(˜A)(iλ+˜A)−1is bounded,\nwe obtain\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplay\nCoB−1/2φ(˜A)(iλ+˜A)−1B−1/2λ2(iλ+A)−1eiλt/hatwiderψ′Uf(λ)dλ/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nH=O(t−2/ba∇dblf/ba∇dblH).\nOnC+, we have |λ| ∼ |Reλ| ∼s+1/t. So/ba∇dbl(iλ+A)−1/ba∇dblH→H=O(|λ|−1) =O(t) and\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplay\nC+B−1/2φ(˜A)(iλ+˜A)−1B−1/2λ2(iλ+A)−1eiλt/hatwiderψ′Uf(λ)dλ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nH\n=O(1)/integraldisplay1\n0t(s+1/t)2e−astds/ba∇dblf/ba∇dblH\n=O(t−2/ba∇dblf/ba∇dblH).\nWe can get the same estimate for C−. Now by density, we have proved.\nNext we estimate the low frequency part: (1 −φB(A))U(t)f=˜φB(A)U(t)f. Here we write\n˜φ= 1−φ.\nWe assume fis sufficiently regular then by the similar argument used in the previous section, for a cut-off\nfunctionψ, the following identity holds\n˜φB(A)ψ(t)U(t)f=1√\n2π/integraldisplay\nImλ=−ε˜φB(A)R(iλ)M(iλ)eiλt/hatwiderψ′Uf(λ)dλ.\nWritingU(t)f=t(u(t),v(t)),/hatwiderψ′Uf(λ) =t(/hatwiderψ′Uf(λ)0,/hatwiderψ′Uf(λ)1), we have the following identity for large\nt\n(5.6) ˜φB(A)u(t) =1√\n2π/integraldisplay\nImλ=−ε˜φB(A)R(iλ)eiλt/parenleftBig\n(iλ+B)/hatwiderψ′Uf(λ)0+/hatwiderψ′Uf(λ)1/parenrightBig\ndλ.\nWe defineψn(t) =ψ(nt) for the cut-off function ψsatisfyingψ′≧0. Thenψ′\nnbecome a Dirac sequence\ni.e.ψ′\nn(t)→δ(t), asn→ ∞whereδ(t) is the Dirac measure. We change ψin the above integral to a\nsequence of cut-off functions {ψn}. So/hatwiderψ′nUf(λ)→1√\n2πU(0)f=1√\n2πfasn→ ∞and we would like to\ntake this limit in the integral. To verify this limiting argument, we use the following lemma.\nLemma 5.2. IfR(λ)exists for a λ/\\e}atio\\slash= 0then we have the following bound\n/ba∇dbl˜φB(A)R(λ)u/ba∇dblH=O(|λ|−2/ba∇dblu/ba∇dblH)+O(|λ|−2(|λ|+1)/ba∇dblR(λ)u/ba∇dblH),\n/ba∇dbl˜φB(A)R(λ)Av/ba∇dblH=O(|λ|−2/ba∇dblv/ba∇dblH)+O(|λ|−2(|λ|+1)/ba∇dblR(λ)Av/ba∇dblH),\nforu∈Handv∈D(A).\nRemark 5.3. IfAis bounded, then λ2+λB+A=O(λ2)for largeλ. So we can expect nearly |λ|−2\nestimate of the high frequency cut-off resolvent.\nProof. Since ˜φ(˜A) :H→D(A) andB1/2:D(A)→D(A) are bounded, we have\n/ba∇dbl˜φB(A)Au/ba∇dblH=/ba∇dbl˜φB(A)A/ba∇dblH→H/ba∇dblu/ba∇dblH=/ba∇dbl(˜φB(A)A)∗/ba∇dblH→H/ba∇dblu/ba∇dblH\n=/ba∇dblAB1/2˜φ(˜A)B−1/2/ba∇dblH→H/ba∇dblu/ba∇dblH≤C/ba∇dblu/ba∇dblH.\nSo we obtain\n/ba∇dbl(˜φB(A)(λ2+Bλ+A)u/ba∇dblH≥ |λ|2/ba∇dbl˜φB(A)u/ba∇dblH−C(|λ|+1)/ba∇dblu/ba∇dblH.\nThus\n/ba∇dbl˜φB(A)u/ba∇dblH≤ |λ|−2/ba∇dbl(˜φB(A)(λ2+Bλ+A)u/ba∇dblH+C|λ|−2(|λ|+1)/ba∇dblu/ba∇dblH.\nTakingu=R(λ)vandu=R(λ)Av, we have the desired inequality.\n12SinceR(λ)(λ+B) = (1+R(λ)A)/λ, by (5.6), we obtain\n˜φB(A)u(t) =1√\n2π/integraldisplay\nImλ=−ε˜φB(A)eiλt1\niλ/hatwiderψ′Uf(λ)0dλ\n+1√\n2π/integraldisplay\nImλ=−ε˜φB(A)eiλt/parenleftbigg1\niλR(iλ)A/hatwiderψ′Uf(λ)0+R(iλ)/hatwiderψ′Uf(λ)1/parenrightbigg\ndλ\n=˜J1(t)+˜J2(t).\nIf suppψ′⊂[0,c] we have the following estimate\n/ba∇dbl/hatwiderψ′Uf(λ)/ba∇dblH=/vextenddouble/vextenddouble/vextenddouble/vextenddouble1√\n2π/integraldisplay∞\n−∞e−iλsψ′(s)U(s)fds/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nH≤CecImλ/ba∇dblψ′/ba∇dblL1/ba∇dblf/ba∇dblH=CecImλ/ba∇dblf/ba∇dblH.\nSo for sufficient large t, in the integral ˜J1, we change the contour to ˜C=˜C+∪˜C−∪ C. Here˜C±=\n{±s+is;s∈[δ,∞)}andCis a contour around the origin. Thanks to the estimate |eiλt|≦Ce−tImλon\n˜C+∪˜C−, iftis sufficiently large, we have\n˜J1(t) =1√\n2π/integraldisplay\n˜C+∪˜C−∪C˜φB(A)eiλt1\niλ/hatwiderψ′Uf(λ)0dλ\n=1√\n2π/integraldisplay\nC˜φB(A)eiλt1\niλ/hatwiderψ′Uf(λ)0dλ+O(e−δ(t−c)/ba∇dblf/ba∇dblH)\ninH. For˜J2, we change the contour to /hatwideC=/hatwideC+∪/hatwideC−∪ C. Here/hatwideC±={±s+iε;s∈[δ,∞)}andCis a\ncontour around the origin as in the proof of Proposition 4 .3. By Lemma 5 .2, Lemma 3 .5 and Lemma 3 .1,\non/hatwideC±, we have the following estimate,\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble1\niλ˜φB(A)R(iλ)A/hatwiderψ′Uf(λ)0+˜φB(A)R(iλ)/hatwiderψ′Uf(λ)1/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nH=O(|λ|−2/ba∇dbl/hatwiderψ′Uf(λ)/ba∇dblH) =O(|λ|−2/ba∇dblf/ba∇dblH).\nThis is integrable and changing the contour, we obtain\n˜J2(t) =1√\n2π/integraldisplay\nC/parenleftbigg1\niλ˜φB(A)R(iλ)A/hatwiderψ′Uf(λ)0+˜φB(A)R(iλ)/hatwiderψ′Uf(λ)1/parenrightbigg\ndλ+O(e−εt/ba∇dblf/ba∇dblH)\ninH. Thus for sufficient large t, we have\n˜φB(A)u(t) =1√\n2π/integraldisplay\nC˜φB(A)eiλt/parenleftBig\nR(iλ)(iλ+B)/hatwiderψ′Uf(λ)0+R(iλ)/hatwiderψ′Uf(λ)1/parenrightBig\ndλ+O(e−εt/ba∇dblf/ba∇dblH).\nChangingψin the above identity to the sequence {ψn}and we take nto infinity. Since /ba∇dblψ′\nn/ba∇dblL1=/ba∇dblψ′/ba∇dblL1\nand suppψn⊂[0,c], the exponential term decays uniformly with respect to nandCis compact, we can\ntake limit under the integral sign. So tending nto infinity, we conclude\n˜φB(A)u(t) =1\n2π/integraldisplay\nC˜φB(A)eiλt(R(iλ)(iλ+B)u0+R(iλ)u1)dλ+O(e−εt/ba∇dblf/ba∇dblH)\n=˜J3(t)+O(e−εt/ba∇dblf/ba∇dblH).(5.7)\nFinally we apply the identity (5 .3) to get the asymptotic profile. By (5 .3), we have\n˜J3(t) =1\n2π/integraldisplay\nCB−1/2˜φ(˜A)(iλ+˜A)−1B−1/2eiλt(Bu0+u1)dλ\n+1\n2π/integraldisplay\nCB−1/2˜φ(˜A)(iλ+˜A)−1B−1/2eiλt(iλu0)dλ\n+1\n2π/integraldisplay\nCB−1/2˜φ(˜A)(iλ+˜A)−1B−1/2λ2R(iλ)eiλt((iλ+B)u0+u1)dλ.\nIn the former integral, we can add contours ˜C±={±s+is;s∈[ε,∞)}, modulo exponential decaying\nfactor. To the latter integrals, we apply same contour deformatio n as in Proposition 4 .3. Then on the\ncontour, ˜φ(˜A)(iλ+˜A)−1andR(iλ) are|λ|−1order. So we obtain these integrals have 1 /tbound. Thus\nwe get\n˜J3(t) =1\n2π/integraldisplay\nC∪˜C+∪˜C−B−1/2˜φ(˜A)(iλ+˜A)−1B−1/2eiλt(Bu0+u1)dλ+O(1/t)(/ba∇dblu0/ba∇dblH+/ba∇dblu1/ba∇dblH)\n=˜J4(t)+O(1/t)(/ba∇dblu0/ba∇dblH+/ba∇dblu1/ba∇dblH).(5.8)\n13Since˜Ais self-adjoint operator, it is a generator of an analytic semi-group and we have\n1\n2π/integraldisplay\nCo∪˜C+∪˜C−(iλ+˜A)−1eiλtdλ=e−t˜A.\nThus we obtain\n˜J4(t) =B−1/2e−t˜AB−1/2(Bu0+u1)\n+1\n2π/integraldisplay\nCo∪C+∪C−B−1/2(1−˜φ(˜A))(iλ+˜A)−1B−1/2eiλt(Bu0+u1)dλ.(5.9)\nSince (5.1), 1−˜φ(˜A) =φ(˜A) cut low frequency, φ(˜A)(iλ+˜A)−1is analytic near the origin. Thanks to\ntheeitλfactor, by the contour deformation across the origin, we know th e latter decays exponentially.\nBy (5.7), (5.8), (5.9) and density, we conclude\nProposition 5.4. Ifφcut high frequency, then there exists C >0such that the following bound holds\nfort>1andu(t)which is the solution of the problem (1.5)with the initial detat(u0,u1)∈ H\n/ba∇dbl(1−φB(A))u(t)−B−1/2e−t˜AB−1/2(Bu0+u1)/ba∇dblH≤C(t−1(/ba∇dblu0/ba∇dblH+/ba∇dblu1/ba∇dblH)+t−2/ba∇dblu0/ba∇dblH1).\nFrom Proposition 5 .1 and 5.4, we have proved Theorem 1 .1.\nRemark 5.5. From the proof, we can get more sharp decay i.e. we can change t−1/ba∇dbl√\nAu0/ba∇dblHterm to\nt−2/ba∇dbl√\nAu0/ba∇dblH. Moreover if we apply the argument of Proposition 6.3, we can get this term is t−∞order.\nNext we show the optimality and an improvement of Theorem 1 .1. The following statement says the\noptimality of the decay rate of Theorem 1 .1.\nProposition 5.6. Assume that 0belongs to the spectrum of A. Then\nlimsup\nt→∞sup\n/bardblt(u0,u1)/bardblH≤1t/ba∇dblu(t)−B−1/2e−t˜AB1/2u0−B−1/2e−t˜AB−1/2u1/ba∇dblH>0\nwhereu(t)is the solution of the problem (1.5).\nTheorem 1.1 is an uniform estimate and for individual solution, we can obtain the f ollowing result.\nProposition 5.7. For each solution u(t)of the problem (1.5)witht(u0,u1)∈ H, we have\nlim\nt→∞t/ba∇dblu(t)−B−1/2e−t˜AB1/2u0−B−1/2e−t˜AB−1/2u1/ba∇dblH= 0.\nProof of Proposition 5.7.From the proof of Theorem 1 .1, the main part of u(t) is˜J3(t). Since (iλ+\n˜A)−1B−1/2λ2R(iλ)iλ=O(|λ|) onC, we can apply the same argument as in the proof of Proposition 4 .3\nand 5.1, this part is negligible in ˜J3. So we have\nI(t) =u(t)−B−1/2e−t˜AB1/2u0−B−1/2e−t˜AB−1/2u1\n=1\n2π/integraldisplay\nCB−1/2˜φ(˜A)(iλ+˜A)−1B−1/2eiλt(iλu0)dλ\n+1\n2π/integraldisplay\nCB−1/2˜φ(˜A)(iλ+˜A)−1B−1/2λ2R(iλ)eiλt(Bu0+u1)dλ+O(t−2/ba∇dblf/ba∇dblH).\nFor latter integral, we apply (5 .3). SinceR(iλ) and (Biλ+A)−1areO(|λ|−1) onC, we can neglect\nthe remainder term. For former integral, we apply the similar argume nt to get heat type asymptotic\nbehavior. We obtain\nI(t) =−B−1/2˜Ae−˜AtB−1/2u0+1\n2π/integraldisplay\nCB−1/2˜φ(˜A)(iλ+˜A)−1B−1λ2(iλ+˜A)−1eiλtB−1/2(Bu0+u1)dλ\n+O(t−2/ba∇dblf/ba∇dblH).\nForφsatisfying (5 .1), we define φn(˜A) =φ(n˜A). Then by the spectral theorem, we have\nφn(˜A)u→uinHasn→ ∞.\n14We decompose I(t) as follows\nI(t) =−B−1/2˜Ae−˜AtB−1/2u0\n+1\n2π/integraldisplay\nCB−1/2˜φ(˜A)(iλ+˜A)−1B−1λ2(iλ+˜A)−1eiλtφn(˜A)B−1/2(Bu0+u1)dλ\n+1\n2π/integraldisplay\nCB−1/2˜φ(˜A)(iλ+˜A)−1B−1λ2(iλ+˜A)−1eiλt(1−φn(˜A))B−1/2(Bu0+u1)dλ\n+O(t−2/ba∇dblf/ba∇dblH).\nSinceφn(˜A) cut low frequency, ( iλ+˜A)φn(˜A) is bounded near the origin. So in the first integral,\nthe integrand is |λ|order and can be estimate Cnt−2as in the proof of Proposition 4 .3 and 5.1. The\nsecond integral can be estimated as in the estimate of (5 .8) and we get uniform t−1estimate. Since\n(1−φn(˜A))B−1/2(Bu0+u1)→0 inHasn→H. By letting nsufficiently large, this part decays εt−1\norder for any ε>0. Now we recall the fact, t˜Ae−˜Atu→0 ast→ ∞, for individual u. This fact is easily\nobserved from the spectral theory. Thus we have proved.\nProof of Proposition 5.6.For simplicity, we can assume u0= 0. From the proof of Proposition 5 .7, we\nhave\nu(t)−B−1/2e−t˜AB−1/2u1\n=1\n2π/integraldisplay\nCB−1/2˜φ(˜A)(iλ+˜A)−1B−1/2λ2B−1/2(iλ+˜A)−1B−1/2eiλtu1dλ+O(t−2/ba∇dblu1/ba∇dblH).\nSo we get\n(u(t)−B−1/2e−t˜AB−1/2u1,u1)H\n=1\n2π/integraldisplay\nC(B−1λ2eiλt(iλ+˜A)−1B−1/2u1,˜φ(˜A)(iλ+˜A)−1B−1/2u1)Hdλ+O(t−2/ba∇dblu1/ba∇dbl2\nH).\nFora∈σ(˜A)∩supp˜φ, we take a sequence of functions fnsuch thatψn(˜A)fn=fnand/ba∇dblfn/ba∇dblH= 1. Here\nψn(x) =/braceleftBigg\n1,|x−a| ≤1/n,\n0,|x−a|>1/n.\nWe takeun\n1=B1/2fn, then\n1\nC=1\nC/ba∇dblB−1/2un\n1/ba∇dblH≤ /ba∇dblB−1un\n1/ba∇dblH≤C/ba∇dblB−1/2un\n1/ba∇dblH=C.\nSo if necessary, we take a subsequence and we can assume /ba∇dblB−1un\n1/ba∇dblHconverge to βa≥1/Casntends\nto infinity. We also note /ba∇dblAfn/ba∇dblHis uniformly bounded so from the continuity of B1/2and interpolation,\n/ba∇dblun\n1/ba∇dblH1is uniformly bounded. For λ∈ C, we easily see\n(iλ+˜A)−1B−1/2un\n1−(iλ+a)−1B−1/2un\n1→0\ninHasntends to infinity and the convergence is locally uniform. We have\nlim\nn→∞(un(t)−B−1/2e−t˜AB−1/2un\n1,un\n1)H=βa1\n2π/integraldisplay\nCλ2eiλt(iλ+a)−2dλ+O(t−2).\nIn the above integral, we can change Cto a simple closed contour ˜Cby adding a contour with exponential\nerror terms. So we have\nlim\nn→∞(un(t)−B−1/2e−t˜AB−1/2un\n1,un\n1)H=iβa1\n2π/integraldisplay\n˜Cλ2eiλt∂λ(iλ+a)−1dλ+O(t−2)\n=−βa1\n2πi/integraldisplay\n˜C(2iλeiλt−λ2teiλt)(iλ+a)−1d(iλ)+O(t−2)\n=βa(2ae−at−a2te−at)+O(t−2).\nTakingt= 1/aanda→0. Sinceβa≥1/C, we obtain this integral does not decay faster than t−1order.\nSo we have proved Proposition 5 .6.\n156 Proof of Theorem 1.3\nHere we study the asymptotic expansion of the propagator for pr oving Theorem 1 .3. In this section, we\nassumeH=L2. We apply the identity (5 .3) to (5.7) and get\n˜φB(A)u(t) =1\n2π/integraldisplay\nCB−1/2˜φ(˜A)(iλ+˜A)−1B−1/2eiλt(Bu0+u1)dλ\n+1\n2π/integraldisplay\nCB−1/2˜φ(˜A)(iλ+˜A)−1B−1/2eiλt(iλu0)dλ\n+1\n2π/integraldisplay\nCB−1/2˜φ(˜A)(iλ+˜A)−1B−1/2R(iλ)eiλt(Bu0+u1)dλ\n+1\n2π/integraldisplay\nCB−1/2˜φ(˜A)(iλ+˜A)−1B−1/2R(iλ)eiλt(iλu0)dλ+O(e−εt/ba∇dblf/ba∇dblH)\n=I+˜I+R1+˜R1+O(e−εt/ba∇dblf/ba∇dblH) =I+˜I+R+O(e−εt/ba∇dblf/ba∇dblH).(6.1)\nR=R1+˜R1is the reminder term and Cis a contour around the origin as in the proof of Proposition\n4.3. We have estimated Iin previous section and get heat type asymptotic behavior modulo ex ponential\ndecay.\n(6.2) I=B−1/2e−t˜AB−1/2(Bu0+u1)+O(e−εt/ba∇dblf/ba∇dblH).\nSo we shall estimate ˜IandR. Since (λ+˜A)−1(λ+˜A) =Id, we have\n˜I=1\n2π/integraldisplay\nCB−1/2˜φ(˜A)B−1/2eiλtu0dλ−1\n2π/integraldisplay\nCB−1/2˜φ(˜A)˜A(iλ+˜A)−1B−1/2eiλtu0dλ\nIn the former integral, the integrand is holomorphic. So by the cont our deformation across the origin, we\nget exponential decay from eitλterm. In the latter integral, we can apply similar argument for provin g\nheat type asymptotic behavior and obtain the following asymptotics\n(6.3) ˜I=−B−1/2˜Ae−t˜AB−1/2u0+O(e−εt/ba∇dblf/ba∇dblH).\nBy the assumption, we have the bound\ne−t/2˜A:L1→L2=O(t−m).\nSincexe−x, x≥0 is bounded, tAe−tAis bounded. So we have\n˜Ae−t/2˜A:L2→L2=O(t−1).\nB−1/2is also a bounded operator on L1so ifu0∈L1, using the identity ˜Ae−t˜A=˜Ae−t/2˜Ae−t/2˜A, we\nobtain\n(6.4) /ba∇dbl˜I/ba∇dblL2=O(t−m−1(/ba∇dblu0/ba∇dblL1+/ba∇dblf/ba∇dblH)).\nNext we study the remainder term R. Sinceλ2B−1/2(iλ+˜A)−1B−1/2:H→H=O(|λ|) on the contour,\nusing (5.3) repeatedly, we obtain, for sufficiently large N\nR=1\n2π/integraldisplay\nCeiλtB−1/2˜φ(˜A)(iλ+˜A)−1B−1λ2(iλ+˜A)−1B−1/2(Bu0+u1)dλ\n+1\n2π/integraldisplay\nCeiλtB−1/2˜φ(˜A)(iλ+˜A)−1B−1λ2(iλ+˜A)−1B−1/2iλu0dλ\n+1\n2π/integraldisplay\nCeiλtB−1/2˜φ(˜A)(iλ+˜A)−1B−1λ2(iλ+˜A)−1B−1λ2(iλ+˜A)−1B−1/2(Bu0+u1)dλ\n+···+/integraldisplay\nCeiλtO(|λ|N)/ba∇dblf/ba∇dblHdλ=1\n2πK1+1\n2π˜K1+1\n2πK2+···+1\n2πKN+1\n2π˜KN+O(t−N−1)/ba∇dblf/ba∇dblH.\nFor the latter estimate, we apply same contour deformation as in Pr oposition 4.3 by using the estimate\nO(|λ|N) on the contour. Here\nKn=/integraldisplay\nCeiλtB−1/2˜φ(˜A)(iλ+˜A)−1B−1/2(λ2B−1/2(iλ+˜A)−1B−1/2))n(Bu0+u1)dλ,\n˜Kn=/integraldisplay\nCeiλtB−1/2˜φ(˜A)(iλ+˜A)−1B−1/2(λ2B−1/2(iλ+˜A)−1B−1/2))niλu0dλ.\nFirst we estimate K1and˜K1.\n16Proposition 6.1. There exists a constant C >0such that the following estimates hold for any t >1\nandu0∈L1∩D(√\nA),u1∈L2∩L1\n/ba∇dblK1/ba∇dblL2≤C(t−m−1(/ba∇dblu0/ba∇dblL1∩L2+/ba∇dblu1)/ba∇dblL1∩L2),\n/ba∇dbl˜K1/ba∇dblL2≤C(t−m−2(/ba∇dblu0/ba∇dblL1∩L2+/ba∇dblu1/ba∇dblL1∩L2).\nProof. We use the following identity\n(6.5) ( iλ+˜A)−1= (iλ+˜A)−1e−t(iλ+˜A)/2+/integraldisplayt/2\n0e−t1(iλ+˜A)dt1.\nInserting this identity, we obtain\nK1=/integraldisplay\nCeiλt/2B−1/2˜φ(˜A)(iλ+˜A)−1B−1λ2(iλ+˜A)−1e−t˜A/2B−1/2(Bu0+u1)dλ\n+/integraldisplay\nCdλ/integraldisplayt/2\n0dt1eiλ(t−t1)B−1/2˜φ(˜A)λ2(iλ+˜A)−1B−1e−t1˜AB−1/2(Bu0+u1).\nThe first integral is easily treated since we have the ( iλ+˜A)−1=O(|λ|−1) onC. For latter integrals, we\napply the following identity\nλ(iλ+˜A)−1=−iId+i˜A(iλ+˜A)−1.\nSo we have\nK1=/integraldisplay\nCeiλt/2O(1)e−t/2˜AB−1/2(Bu0+u1)dλ\n−i/integraldisplay\nCdλ/integraldisplayt/2\n0dt1eiλ(t−t1)B−1/2˜φ(˜A)λB−1e−t1˜AB−1/2(Bu0+u1)\n+/integraldisplay\nCdλ/integraldisplayt/2\n0dt1eiλ(t−t1)B−1/2˜φ(˜A)˜AB−1e−t1˜AB−1/2(Bu0+u1)\n−/integraldisplay\nCdλ/integraldisplayt/2\n0dt1eiλ(t−t1)B−1/2˜φ(˜A)˜A2(iλ+˜A)−1B−1e−t1˜AB−1/2(Bu0+u1).\nIn second and third integrals, integrands are holomorphic with resp ect toλvariable, and t−t1≧t/2, we\ncan deform the contour across the origin and get exponential dec ay. For the fourth integral, we can get\nheat type semi-group modulo exponential error by applying same ar gument as in the previous section.\nWe obtain\nK1=/integraldisplay\nCeiλt/2O(1)dλe−t/2˜AB−1/2(Bu0+u1)\n+/integraldisplayt/2\n0B−1/2φ(˜A)˜A2e−(t−t1)˜AB−1e−t1˜AB−1/2(Bu0+u1)dt1+O(e−εt)(/ba∇dblu0/ba∇dblL2+/ba∇dblu1/ba∇dblL2)\n=˜I1+˜I2+O(e−εt)(/ba∇dblu0/ba∇dblL2+/ba∇dblu1/ba∇dblL2).\nApplying same contour deformation as in Proposition 4 .3, we get\n/ba∇dbl˜I1/ba∇dblL2=O(t−1/ba∇dble−t/2˜AB−1/2(Bu0+u1)/ba∇dblL2)\n=O(t−m−1)(/ba∇dblu0/ba∇dblL1+/ba∇dblu1/ba∇dblL1).\nSince/integraltextt/2\n0dt1=O(t) and/ba∇dblφ(˜A)˜A2e−t/4˜A/ba∇dblL2→L2=O(t−2), the following estimate holds\n/ba∇dbl˜I2/ba∇dblL2=O(t/ba∇dblφ(˜A)˜A2e−t/4˜A/ba∇dblL2→L2sup\n0≤t1≤t/2/ba∇dble−˜A(3t/4−t1)B−1e−t1˜AB−1/2(Bu0+u1)/ba∇dblL2)\n=O(t−1sup\n0≤t1≤t/2/ba∇dble−˜A(3t/4−t1)/ba∇dblL1→L2/ba∇dblB−1e−t1˜AB−1/2(Bu0+u1)/ba∇dblL1)\n=O(t−m−1(/ba∇dblu0/ba∇dblL1+/ba∇dblu1)/ba∇dblL1).\nThus we have estimated K1part. The estimate of ˜K1is almost similar.\n17We can apply similar idea for the estimate of Knand˜Knterms, as an example, we next prove the\nfollowing estimate\n/ba∇dblK2/ba∇dblL2=O(t−m−2(/ba∇dblu0/ba∇dblL1∩L2+/ba∇dblu1/ba∇dblL1∩L2)).\nBy the identity (6 .5), we have\nK2=/integraldisplay\nCeiλt/2B−1/2˜φ(˜A)(iλ+˜A)−1B−1λ2(iλ+˜A)−1B−1λ2(iλ+˜A)−1e−t/2˜AB−1/2(Bu0+u1)dλ\n+/integraldisplay\nCdλ/integraldisplayt/2\n0dt1eiλ(t−t1)B−1/2˜φ(˜A)(iλ+˜A)−1B−1λ2(iλ+˜A)−1B−1λ2e−t1˜AB−1/2(Bu0+u1).\nApplying the following identity to the second integral\n(6.6) ( iλ+˜A)−1= (iλ+˜A)−1e−t/4(iλ+˜A)+/integraldisplayt/4\n0e−t2(iλ+˜A)dt2,\nwe get\nK2=/integraldisplay\nCeiλt/2O(|λ|)dλe−t/2˜AB−1/2(Bu0+u1)\n+/integraldisplay\nCdλ/integraldisplayt/2\n0dt1eiλ(3t/4−t1)B−1/2˜φ(˜A)(iλ+˜A)−1B−1λ4(iλ+˜A)−1e−t/4˜AB−1e−t1˜AB−1/2(Bu0+u1)\n+/integraldisplay\nCdλ/integraldisplayt/2\n0dt1/integraldisplayt/4\n0dt2eiλ(t−t1−t2)B−1/2˜φ(˜A)λ4(iλ+˜A)−1B−1e−t2˜AB−1e−t1˜AB−1/2(Bu0+u1).\nWe can apply similar argument as in the proof of Proposition 6 .1 and we can get the estimate.\nIn the same way, we obtain\nProposition 6.2. There exists a constant C >0such that the following estimates holds for any t >1\nandu0∈L1∩D(A),u1∈L2∩L1\n/ba∇dblKn/ba∇dblL2≤C(t−m−n(/ba∇dblu0/ba∇dblL1∩L2+/ba∇dblu1/ba∇dblL1∩L2),\n/ba∇dbl˜Kn/ba∇dblL2≤C(t−m−n−1(/ba∇dblu0/ba∇dblL1∩L2+/ba∇dblu1/ba∇dblL1∩L2).\nBy these propositions, we have estimated ˜φB(A)u(t) part and we have\n(6.7) ˜φB(A)u(t) =B−1/2e−t˜AB−1/2(Bu0+u1)+O(t−m−1/ba∇dblu0/ba∇dblL1+/ba∇dblu1/ba∇dblL1+/ba∇dblf/ba∇dblH).\nNext we estimate (1 −˜φB(A))u(t) =φB(A)u(t) term by sharpening the estimate of Proposition 5 .1. In\nthe proof of Proposition 5 .1, we have the decomposition\nφB(A)ψ(t)U(t)f=1\n2πJ2f+O(e−εt/ba∇dblf/ba∇dblH).\nHereψ(t) = 1 fort>2 and\nJ2f=/integraldisplay\nCdλ/integraldisplay∞\n−∞ds B−1/2λ2φ(˜A)(iλ+˜A)−1eiλ(t−s)B−1/2ψ′(s)U(s)(iλ−A)−1f.\nSinceAis a generatorof U(t), we exchange U(t) to (iλ−A)−1. We apply similarargument to Proposition\n6.1. By (5.3) and (5.4), we have\nJ2f=/integraldisplay\nCdλ/integraldisplay∞\n−∞ψ′(s)ds B−1/2λ2φ(˜A)(iλ+˜A)−1eiλ(t−s)B−1/2U(s)B−1/2(iλ+˜A)−1B−1/2M(λ)f+R2.(6.8)\nHereR2is the reminder term. We recall the support of ψ′is in [1,2]. We apply identity (6 .5) andJ2f\nbecomes\n/integraldisplay\nCdλ/integraldisplay2\n1ψ′(s)ds B−1/2λ2φ(˜A)(iλ+˜A)−1eiλ(t/2−s)B−1/2U(s)B−1/2(iλ+˜A)−1e−t˜A/2B−1/2M(λ)f\n+/integraldisplay\nCdλ/integraldisplay2\n1ψ′(s)ds/integraldisplayt/2\n0dt1B−1/2λ2φ(˜A)(iλ+˜A)−1eiλ(t−t1−s)B−1/2U(s)B−1/2e−t1˜A/2B−1/2M(λ)f+R2.(6.9)\n18For the first integral, on the contour, we have\n/ba∇dbl(iλ+˜A)−1/ba∇dblL2→H1=O(|λ|−1),\nand\n/ba∇dble−t˜AB−1/2A/ba∇dblL1→L2=/ba∇dbl˜Ae−t˜AB1/2/ba∇dblL1→L2=O(t−m−1).\nFrom these estimate, we can estimate the first integral by similar co ntour deformation as in Proposition\n4.3 and gett−m−2order decay. In the second integral, we need the following estimate\n/ba∇dblB−1/2e−t1˜A/2B−1/2M(λ)f/ba∇dblH=O(t−1/2\n1+1)/ba∇dblf/ba∇dblH.\nThis estimate follows from\n/ba∇dblB−1/2e−t1˜A/2u/ba∇dblH1=O(t−1/2\n1+1)/ba∇dblu/ba∇dblH\nand\n/ba∇dblB−1/2e−t1˜A/2B−1/2Au/ba∇dblH=O(t−1/2\n1+1)(/ba∇dblu/ba∇dblH1).\nIn the above, the first estimate holds by interpolating the following e stimate and boundedness on H\n/ba∇dblAB−1/2e−t1˜A/2B−1/2u/ba∇dblH=/ba∇dblB1/2˜Ae−t1˜A/2B−1/2u/ba∇dblH=O(t−1\n1)/ba∇dblu/ba∇dblH\nThe second one is obtained from the first estimate by taking its adjo int\n/ba∇dblB−1/2e−t1˜A/2B−1/2Au/ba∇dblH=/ba∇dblB−1/2e−t1˜A/2B−1/2A1/2A1/2u/ba∇dblH\n≤ /ba∇dbl(B−1/2e−t1˜A/2B−1/2A1/2)/ba∇dblH→H/ba∇dblA1/2u/ba∇dblH\n=/ba∇dbl(B−1/2e−t1˜A/2B−1/2A1/2)∗/ba∇dblH→H/ba∇dblA1/2u/ba∇dblH\n=/ba∇dblA1/2B−1/2e−t1/2˜AB−1/2/ba∇dblH→H/ba∇dbl√\nAu/ba∇dblH\n=O(t−1/2\n1+1)/ba∇dblu/ba∇dblH1.\nIn the second integral, since the integrand is holomorphic and t−1/2\n1is integrable near t1= 0, we obtain\nexponential decay by contour deformation. Using the above estim ates, we can repeat similar argument\nfor reminder term R2. In this case, one may think the L1L1estimate of e−t1˜A/2˜Ais needed but since\ne−t˜A/2˜A=−∂te−t˜A/2and using integration by parts, we can avoid this term. Since the arg ument is\nessentially similar, we omit the detail and we get\nProposition 6.3. There exists a constant C >0such that the following estimates holds for any t >1\nandf∈ H\n/ba∇dbl(1−˜φB(A))u(t)/ba∇dblH≤Ct−m−2/ba∇dblf/ba∇dblH.\nFrom these propositions, we have proved Theorem 1.3.\n7 Applications\nAs applications of our approach, here we consider some examples of damped wave equation on Rd. We\nstudy the following divergence form damped wave equation\n(7.1)\n∂2\nt−d/summationdisplay\ni,j=1∂igij(x)∂j+a(x)∂t\nu(x,t) = (∂2\nt+P+a(x)∂t)u(x,t) = 0.\nHere we write ∂i=∂\n∂xiandP=−/summationtextd\ni,j=1∂igij(x)∂j. We assume gijare smooth functions with bounded\nderivatives and ( gij(x)) is a family of uniformly elliptic real symmetric matrices, i.e. there exis ts a\nconstantC >0 such that1\nCId≤(gij(x))≤CId\nholds for any x∈Rd. Under this assumption Pis a positive definite self-adjoint operator on H2and\nD(√\nP) =H1whereH1andH2are usual sobolev spaces on Rd. The damping term a(x) is a smooth\nnon-negative function with bounded derivatives. If a(x)>cfor a constant c>0, by Lemma 7 .12 below,\nwe can apply Theorem 1 .1 and Theorem 1 .3. Thus we obtain similar decay estimates for the constant\ncoefficient case.\n19Theorem 7.1. Letube a solution of (7.1)with the initial data u|t=0=u0and∂tu|t=0=u1. We assume\na(x)>cforc>0and define\nv(t) =a(·)−1/2e−ta−1/2Pa−1/2a(·)1/2u0−a(·)−1/2e−ta−1/2Pa−1/2a(·)−1/2u1.\nThen there exists C >0such that we have the following asymptotic profile\n/ba∇dblu(t)−v(t)/ba∇dblL2≤Ct−1(/ba∇dblu0/ba∇dblH1+/ba∇dblu1/ba∇dblL2).\nforu0∈H1,u1∈L2andt >1. Furthermore, if we assume u0∈H1∩L1andu1∈L2∩L1, then we\nalso have the following bound for t>1,\n/ba∇dblu(t)−v(t)/ba∇dblL2≤Ct−d/4−1(/ba∇dblu0/ba∇dblL2∩L1+/ba∇dblu1/ba∇dblL2∩L1).\nWe shall study the case that a(x) may be vanish on a compact set however the geometric control\ncondition holds. So we recall about the geometric control condition . For the operator P, we define its\nHamiltonian flow φt(x,ξ) = (x(t),ξ(t)) by the solution of the following the Hamiltonian equation\n\n\ndx\ndt=∂p\n∂ξ,dξ\ndt=−∂p\n∂x\n(x(0),ξ(0)) = (x,ξ).\nHere the symbol p(x,ξ) =/summationtextd\ni,j=1gij(x)ξiξjis associated with P. By the uniform ellipticity, the above\nHamilton equation has time global solutions for any initial date and we c an define the Hamiltonian flow.\nFor the damped wave equation, it is known that the mean value of the damping term along a geodesic\nis closely related to the asymptotic profile of the solution c.f. [7], [16]. S o we introduce\n/a\\}b∇acketle{ta/a\\}b∇acket∇i}htT(x,ξ) =1\nT/integraldisplayT\n0a(φt(x,ξ))dt\nwhereT >0 and we use the notion a(x,ξ) :=a(x). We study the asymptotic behavior of the damped\nwave equation under the following geometric control condition.\nGeometric control condition : There exist T >0 andα>0 such that for all ( x,ξ)∈p−1(1), we have\n/a\\}b∇acketle{ta/a\\}b∇acket∇i}htT(x,ξ)≥α.\nUnder the geometric control condition, we can expect the asympt otic profile is similar to the strictly\npositive case. We first obtain the energy decay. Let U(t) be the propagatorof the damped waveequation.\nWe have the following energy decay estimate.\nTheorem 7.2. We assume the geometric control condition and a(x)> cfor|x|> Rwherec >0and\nR >0are some constants. Let ube a solution of (7.1)with the initial data u|t=0=u0∈H1and\n∂tu|t=0=u1∈L2. Then there exists a constant Csuch that for t>1the following bound holds\n/ba∇dbl∇u/ba∇dbl2\nL2+/ba∇dbl∂tu/ba∇dbl2\nL2≤C(d/summationdisplay\ni,j=1(gij∂iu,∂ju)L2+/ba∇dbl∂tu/ba∇dbl2\nL2)≤Ct−1(/ba∇dblu0/ba∇dbl2\nH1+/ba∇dblu1)/ba∇dblL2).\nIn this case, the damping term is not uniformly positive. But we can pr ove the similar resolvent\nestimate and we can apply the argument of proving Theorem 4.1. We s tudy the resolvent by dividing\nthree range of frequencies.\nFor low frequency, we use the following estimate\nLemma 7.3. Ifλandδ>0are sufficiently small, then (P+λ2+λa(·))−1:L2→L2exists for Reλ>0\nor|Imλ|>δ|Reλ|and we have the following bound\n(P+λ2+λa(·))−1=O(|λ|−1).\nProof. First we prove the following bound, for sufficient small λ>0\n(7.2)/integraldisplay\nRd|∇u|2dx+λ/integraldisplay\nRda(x)|u(x)|2dx≥Cλ/integraldisplay\nRd|u|2dx.\nTakingasmoothcompactsupportnon-negativefunction χ(x),χ(x) = 1for |x|<1. We setχr(x) =χ(rx)\nforr>0. We have\n/ba∇dbl∇u/ba∇dblL2≥ /ba∇dblχr∇u/ba∇dblL2≥ /ba∇dbl∇(χru)/ba∇dblL2−/ba∇dbl[∇,χr]u/ba∇dblL2.\n20Sincea>cfor|x|>Rand (∇χ)r(x) =∇χ(rx) = 0 for |x|<1/r, we obtain\n/ba∇dbl[∇,χr]u/ba∇dblL2=r/ba∇dbl(∇χ)ru/ba∇dblL2≤Cr/parenleftbigg/integraldisplay\nRda(x)|u(x)|2dx/parenrightbigg1/2\nfor sufficient small rand someC >0. Sinceχru= 0 for|x|>1/r, by the Poincar´ e inequality, we have\n/ba∇dblχru/ba∇dblL2≤C1\nr/ba∇dbl∇(χru)/ba∇dblL2.\nSo we get\n/ba∇dbl∇u/ba∇dblL2≥1\nCr/ba∇dblχru/ba∇dblL2−Cr/parenleftbigg/integraldisplay\nRda(x)|u(x)|2dx/parenrightbigg1/2\n.\nTakingr=ε√\nλfor sufficient small ε, we have (7 .2) sincea>cfor|x|>R. Now we prove the lemma.\nWe use the following identity\n((P+λ2+λa(·))u,u)L2= (Pu,u)L2+λ2/ba∇dblu/ba∇dbl2\nL2+λ/integraldisplay\nRda(x)|u(x)|2dx,\nFrom this identity and uniform ellipticity of P, we have\n|((P+λ2+λa(·))u,u)L2| ≥C/ba∇dbl∇u/ba∇dblL2+C|λ|/integraldisplay\nRda(x)|u(x)|2dx−O(|λ|2/ba∇dblu/ba∇dblL2).\nfor Reλ>0 or|Imλ|>δ|Reλ|ifδis sufficient small . So for sufficient small |λ|, applying (7 .2) we get\n|((P+λ2+λa(·))u,u)L2| ≥C|λ|/ba∇dblu/ba∇dbl2\nL2.\nFromthisboundweeasilyprovetheexistenceoftheinverseandthe estimatebyapplyingsimilarargument\nfor Lemma 3.1.\nRemark 7.4. The above estimate holds for u∈C∞\n0(Ω)for exterior domain Ωand we can generalize the\ntheorem for exterior problem with Dirichlet boundary condi tion.\nBy the geometric control condition, we can control high frequenc y part and we can provethe following\nstatement.\nLemma 7.5. Under the geometric control condition, R(λ) = (P+λ2+λa(·))−1:L2→L2exists for\nλ=iτ,τ∈R\\{0},|τ|>Cfor sufficient large C >0and we have the following bound\n(P+λ2+λa(·))−1=O(|λ|−1)\nThis type lemma is well-known, e.g. [7], [19], so we omit the proof. From t his Lemma and by\nProposition 3 .7, we have\n(λ−A)−1=O(1+|λ|−1).\nThis estimate guarantees the analytic continuation of the resolven t across the imaginary axis for high\nfrequency region.\nFor middle frequency region, we can use the Fredholm theory. We fo llow the argument in [19]. We\ntake a smooth compact support function b(x) satisfying,\nc(x) =a(x)+b(x)>c\nfor a constant c>0. Then (P+λ2+λc(·))−1exists for pure imaginary λ=iτ,τ∈R\\{0}and we have\n(P+λ2+λa(·))(P+λ2+λc(·))−1=Id−λb(·)(P+λ2+λc(·))−1\nSince (P+λ2+λc(·))−1takesL2toH1andb(x) is a compact support function, by Rellich’s compact\nembedding theorem, b(·)(P+λ2+λc(·))−1is a compact operator. Thus by the Fredholm theory, the\nresolvent exists if and only if the following equation have a solution\nλb(·)(P+λ2+λc(·))−1u=u.\nTakingu= (P+λ2+λc(·))v, the equation can be written as follows\n(P+λ2+λa(x))v= 0.\n21For pure imaginary λ=iτ,τ∈R\\{0}, multiplying ¯ vto the equation, integrating on Rdand taking\nimaginary part, we have /integraldisplay\nRda(x)|v(x)|2dx= 0.\nSov= 0 on the support of a. Then by the unique continuation of second order elliptic operators , we\nconcludev= 0. Thus for pure imaginary λ/\\e}atio\\slash= 0, we have u= 0. So the resolvent exist and by continuity,\nfor middle frequency, we have the existence of the resolvent acro ss the imaginary axis. Combining the\nexistence of the resolvent for high and middle frequency, we have t he following lemma.\nLemma 7.6. For anyε>0there exists δ>0such thatR(λ) = (P+λ2+λa(·))−1:L2→L2exists for\nReλ>−δif|Imλ|>ε.\nFrom these lemmas, we can conclude the energy decay by applying sim ilar argument as in section 4\nc.f. Remark 4 .4.\nRemark 7.7. We can also prove the energy decay only assuming the geometri c control condition, by\nusing the argument in [7], see Appendix.\nRemark 7.8. We can also treat a small perturbation of the above case and we can get the energy decay\nfor the case that a(x)≥0does not hold. In the above setting, we take a smooth function b(x)>0with\nbounded derivatives and we consider the following damped wa ve equation.\n(∂2\nt+P+a(x)∂t−εb(x)∂t)u(t,x) = 0.\nFrom the above argument, we get the following bound for λ=iτ,τ∈R\\{0}\n/ba∇dbl(P+λ2+λa(·))−1u/ba∇dblL2≤C|λ|−1/ba∇dblu/ba∇dblL2.\nTakingu= (P+λ2+λa(·))v, we get\n|λ|/ba∇dblv/ba∇dblL2≤C/ba∇dbl(P+λ2+λa(·))v/ba∇dblL2.\nSo we have\n/ba∇dbl(P+λ2+λa(·)−ελb(·))v/ba∇dblL2≥ /ba∇dbl(P+λ2+λa(·))v/ba∇dblH−Cε|λ|/ba∇dblv/ba∇dblL2≥C|λ|/ba∇dblv/ba∇dblL2\nifεis sufficiently small. This inequality implies the existence of the resolvent and we can apply previous\nargument to get the energy decay though for applying the prev ious argument, we need a small modification\nsince at first we only know U(t)may be growth like eCεtorder.\nWe also give the asymptotic profile of the solution. We take a smooth c ompact support function b(x)\nsatisfying\nc(x) =a(x)+b(x)>c\nfor a constant c>0. We take\n˜P:=c(·)−1/2Pc(·)−1/2.\nThe following theorem is a generalization of Theorem 1 .1.\nTheorem 7.9. Letu,vbe the solution of the following equations\n/braceleftBigg\n(∂2\nt+P+a(·)∂t)u= 0,\nu|t=0=u0,∂tu|t=0=u1,(7.3)\n/braceleftBigg\n(∂t+˜P)v= 0,\nv|t=0=c(·)1/2u0+c(·)−1/2u1(7.4)\nonR×Rdford>2, respectively. Under the assumption of Theorem 7.2, there exists C >0such that\nfor anyu0∈H1,u1∈L2andt>1, the following asymptotic profile holds\n/ba∇dblu(t)−c(·)1/2v(t)/ba∇dblL2≤Ct−1(/ba∇dblu0/ba∇dblH1+/ba∇dblu1/ba∇dblL2).\n22To prove the theorem, we need the asymptotic expansion of the re solvent near the origin. Since c(x)\nis uniformly positive, the resolvent R(λ) = (P+λ2+λc(·))−1exists near the imaginary axis. We have\nthe following identity\n(7.5) ( P+λ2+λc(·))−1(P+λ2+λa(·)) =Id+λ(P+λ2+λc(·))−1b(·).\nWe use the following estimate for the cut-off resolvent.\nLemma 7.10. Letd >2andχis a continuous compact support function then there exists a constant\nδ>0such taht the following estimate holds for |λ|withReλ>0or|Imλ|>δ|Reλ|,\n/ba∇dblχR(λ)χ/ba∇dblL2→L2=O(1).\nProof. By uniform ellipticity, we have\n|((P+λ2+λc(·))u,u)L2| ≥C/ba∇dbl∇u/ba∇dbl2\nL2+C|λ|/integraldisplay\nRdc(x)|u(x)|2dx−C|λ|2/ba∇dblu/ba∇dbl2\nL2\nfor Reλ>0 or|Imλ|>δ|Reλ|ifδis sufficient small. Since c(x)>c, for sufficient small λ, we have\n|λ|/integraldisplay\nRdc(x)|u(x)|2dx−|λ|2/ba∇dblu/ba∇dbl2\nL2≥0.\nIfd>2, by the Hardy’s inequality, we obatain\n/ba∇dbl∇u/ba∇dbl2\nL2≥C/ba∇dbl|x|−1u/ba∇dbl2\nL2≥C/ba∇dbl/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−1u/ba∇dbl2\nL2.\nHere/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht=√\n1+x2. So we have\n/ba∇dbl(/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht(P+λ2+λc(x))u/ba∇dblL2/ba∇dbl/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−1u/ba∇dblL2≥ |(/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht(P+λ2+λc(x))u,/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−1u)L2|\n=|(P+λ2+λc(x))u,u)L2| ≥C/ba∇dbl/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−1u/ba∇dbl2\nL2.\nTakingu=R(λ)/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−1v, we obtain\n(7.6) /ba∇dbl/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−1R(λ)/a\\}b∇acketle{t·/a\\}b∇acket∇i}ht−1v/ba∇dblL2≤1\nC/ba∇dblv/ba∇dblL2.\nFrom this inequality, we have the estimate\nFord= 1,2, we can give the following results.\nLemma 7.11. Ifχis a real-valued continuous compact support function, then there exists a constant\nδ>0such that the following estimates hold for sufficient small |λ|withReλ>0or|Imλ|>δ|Reλ|,\n/ba∇dblχR(λ)χ/ba∇dblL2→L2=O(log|λ|) ( d= 2),\n/ba∇dblχR(λ)χ/ba∇dblL2→L2=O(|λ|−1/2) ( d= 1).\nProof. By uniform ellipticity, we have\n(7.7) |((P+λ2+λc(·))u,u)L2| ≥C(/ba∇dbl∇u/ba∇dbl2\nL2+|λ|/ba∇dblu/ba∇dbl2\nL2)≥C/ba∇dbl(|∇|+|λ|1/2)u/ba∇dbl2\nL2.\nWe prove the following estimate.\n(7.8) /ba∇dbl(|∇|+|λ|1/2)u/ba∇dbl2\nL2≥/braceleftBigg\nC1\nlog|λ|/ba∇dblχu/ba∇dbl2\nL2,(d= 2),\nC|λ|1/2/ba∇dblχu/ba∇dbl2\nL2,(d= 1).\nWe take a compact support cut off function ˜ χ(ξ) which satisfies ˜ χ(ξ) = 1 if|ξ|is sufficient small. Then\nby the Plancherel theorem and ( |ξ|+|λ|1/2)−1(1−˜χ(ξ)) is bounded for sufficient small λ, we have\n/ba∇dblχ(|∇|+|λ|1/2)−1u/ba∇dbl2\nL2≤ /ba∇dblχ(|∇|+|λ|1/2)−1˜χ(D)u/ba∇dbl2\nL2+/ba∇dbl(|∇|+|λ|1/2)(1−˜χ(D))u/ba∇dbl2\nL2\n≤ /ba∇dbl(|∇|+|λ|1/2)−1˜χ(D)u/ba∇dbl2\nL∞+/ba∇dbl(|ξ|+|λ|1/2)−1(1−˜χ(ξ))/hatwideu/ba∇dbl2\nL2\n≤ /ba∇dbl(|ξ|+|λ|1/2)−1˜χ(ξ)u/ba∇dbl2\nL1+C/ba∇dblu/ba∇dbl2\nL2\n≤ /ba∇dbl(|ξ|+|λ|1/2)−1˜χ(ξ)/ba∇dbl2\nL2/ba∇dblu/ba∇dbl2\nL2+C/ba∇dblu/ba∇dbl2\nL2.\n23By easy computations, we obtain\n/ba∇dbl(|ξ|+|λ|1/2)−1˜χ(ξ)/ba∇dbl2\nL2≤/braceleftBigg\nClog|λ|,(d= 2),\nC|λ|−1/2,(d= 1).\nTakingu= (|∇|+|λ|1/2)v, we have (7 .8). From (7 .7) and (7.8), we get\n/ba∇dblχu/ba∇dbl2\nL2≤/braceleftBigg\nClog|λ||((P+λ2+λc(x))u,u)L2|,(d= 2),\nC|λ|−1/2|((P+λ2+λc(x))u,u)L2|,(d= 1).\nNow we take u= (P+λ2+λc(x))−1χvthen we have\n/ba∇dblχu/ba∇dblL2/ba∇dblχ(P+λ2+λc(x))−1χv/ba∇dblL2≤/braceleftBigg\nClog|λ||((v,χu)L2≤Clog|λ|/ba∇dblv/ba∇dblL2/ba∇dblχu/ba∇dblL2,(d= 2),\nC|λ|−1/2|((v,χu)L2≤C|λ|−1/2/ba∇dblv/ba∇dblL2/ba∇dblχu/ba∇dblL2,(d= 1).\nDividing by /ba∇dblχu/ba∇dblL2, we have proved.\nWe take the inverse of the right hand side of the identity (7 .5). We define\n˜R(λ) =/radicalbig\nb(·)λ(P+λ2+λc(·))−1/radicalbig\nb(·).\nSinceb(x) is compact support function, by the above lemmas, we can assume /ba∇dbl˜R(λ)/ba∇dblL2→L2<1 for\nsufficient small λ. For suchλ, we can construct the inverse by Neumann series.\n(Id−λ(P+λ2+λc(·))−1b(·))−1\n=Id+∞/summationdisplay\nn=0(−1)nλ(P+λ2+λc(·))−1/radicalbig\nb(·){λ˜R(λ)}n/radicalbig\nb(·).\nThus we have the following identity for the resolvent R(λ) = (P+λ2+λa(·))−1andRc(λ) = (P+λ2+\nλc(·))−1\nR(λ) = (P+λ2+λc(·))−1\n+∞/summationdisplay\nn=0(−1)nλ(P+λ2+λc(·))−1/radicalbig\nb(·){λ˜R(λ)}n/radicalbig\nb(·)(P+λ2+λc(·))−1\n=Rc(λ)+λRc(λ)b(·)Rc(λ)+R1(λ).(7.9)\nHereR1(λ) is the reminder term. So we have the asymptotic expansion of the r esolvent. We shall prove\nTheorem 7.3. Ifd >2 then˜R(λ) is bounded so we have /ba∇dblλ˜R(λ)/ba∇dblL2→L2=O(|λ|) and/ba∇dblR1(λ)/ba∇dbll2→L2=\nO(1) near the origin. To the first term Rc(λ), we can apply similar argument in Theorem 1 .1 and get\nthe heat type asymptotic. R1(λ) term becomes lower order term since /ba∇dblR1(λ)/ba∇dbl=O(1) is lower than the\nmain order |λ|−1and we can get t−1order decay. The main difference is the second term and we study\nthe term. We have /ba∇dblφc(P)λRc(λ)b(x)Rc(λ)/ba∇dbl=O(1) ifφc(P) cut low frequency so, corresponding to\nProposition 5.1, we have t−1order decay to the high frequency part. For low frequency part, we follow\nthe proof of Theorem 1.1. To obtain (5 .7), we do not use the identity (5 .3) and we only use these estimate\nof the resolvent which is similar in this situation. Thus the main term is ˜J3and we have\nu(t) =1\n2π/integraldisplay\nC˜φc(P)eiλt(R(iλ)(iλ+c(·))u0+R(iλ)u1)dλ+O(t−1). (7.10)\nInserting (7 .9), sinceR1(λ) and the first term are a lower order, we get\nu(t) =1\n2π/integraldisplay\nC˜φc(P)eiλt(Rc(iλ)+iλRc(iλ)b(·)Rc(iλ))(c(·)u0+u1)dλ+O(t−1)\n=L1+1\n2πL2+O(t−1).(7.11)\nIn the above integral the first term is similar to the strictly positive c ase, so we can get the following\nasymptotic profile by applying similar argument to prove Theorem 1.1.\nL1=c(·)−1/2e−t˜Pc(·)−1/2(c(·)u0+u1)+O(t−1).\nTo estimate L2part, we recall the following properties of our heat propagator.\n24Lemma 7.12. There exists a constant C >0such that the following bounds hold for u∈L1∩L2and\nt>1\n/ba∇dble−t˜Pu/ba∇dblL1≤C/ba∇dblu/ba∇dblL1,\n/ba∇dble−t˜Pu/ba∇dblL2≤Ct−d/4/ba∇dblu/ba∇dblL1,\n/ba∇dble−t˜Pu/ba∇dblL∞≤Ct−d/4/ba∇dblu/ba∇dblL2.\nProof. Probably this result is well know c.f. [8] and we give an outline o f the proof. We use the following\nNash’s inequality\n/ba∇dblu/ba∇dbl2+4/d\nL2≤C(˜Pu,u)L2/ba∇dblc(·)1/2u/ba∇dbl2/d\nL1.\nSincePis uniformlyelliptic and c(·) isstrictlypositive boundedfunction, this inequalityiseasilyobtained\nfrom the following usual Nash’s inequality\n/ba∇dblu/ba∇dbl2+4/d\nL2≤C/ba∇dbl∇u/ba∇dbl2\nL2/ba∇dblu/ba∇dbl4/d\nL1.\nLetu(t) =u(t,·) =e−t˜Pu0, then by the integration by parts, we have\nd\ndt/integraldisplay\nRdc(x)1/2u(t,x)dx=−/integraldisplay\nRd∂i/parenleftBig\ngij(x)∂jc(x)−1/2u(t,x)/parenrightBig\ndx= 0.\nThus the propagator et˜Ppreserves this modified total heat. Under our assumption, et˜Palso preserve\npositivity see [9]. So we can only consider the positive solution and we as sumeu(t) is positive. From the\nconservation of the modified total heat, we have the L1L1bound. Next we define\nH(t) :=/ba∇dblu(t)/ba∇dbl2\nL2\n/ba∇dblc(·)1/2u(t)/ba∇dbl2\nL1.\nSince the modified total heat is preserved, by the Nash’s inequality, we have\nd\ndtH(t) =−2(˜Pu,u)\n/ba∇dblc(·)1/2u(t)/ba∇dbl2\nL1≤ −CH(t)1+2/d.\nThis inequality implies\n−H(t)−2/d≤H(0)−2/d−H(t)−2/d≤ −Ct.\nSo we obtain\n/ba∇dblu(t)/ba∇dblL2≤Ct−d/4/ba∇dblu0/ba∇dblL1.\nFrom this estimate, we can prove the L∞estimates by using duality argument. For positive vands≤t,\nwe take\nv(s,x) =v(s) :=e(s−t)˜Pv.\nThen we have\nd\nds/integraldisplay\nRdu(s,x)v(s,x)dx=/integraldisplay\nRd/parenleftBig\n−˜Pu(s,x)/parenrightBig\nv(s,x)dx+/integraldisplay\nRdu(s,x)/parenleftBig\n˜Pv(s,x)/parenrightBig\ndx= 0.\nThus the above integral does not depend sso takings=tands=t/2, we have\n/integraldisplay\nRdu(t,x)vdx=/integraldisplay\nRdu(t/2,x)v(t/2,x)dx≤ /ba∇dblu(t/2)/ba∇dblL2/ba∇dblv(t/2)/ba∇dblL2≤Ct−d/4/ba∇dblu/ba∇dblL2/ba∇dblv/ba∇dblL1.\nBy this inequality and duality, we get the L∞L2estimate.\nUsing this bound, we estimate L2in the decomposition (7 .10). From the identity (5 .3), we have\n(7.12) Rc(λ) =c(·)−1/2(λ+˜P)−1c(·)−1/2−λ2c(·)−1/2(λ+˜P)−1c(·)−1/2Rc(λ).\nInserting this identity to L2, sinceλ2c(·)−1/2(λ+˜P)−1c(·)−1/2Rc(λ) =O(|λ|), we obtain\nL2=/integraldisplay\nC˜φc(P)eiλtiλc(·)−1/2(λ+˜P)−1c(·)−1/2b(x)c(·)−1/2(λ+˜P)−1c−1/2(c(·)u0+u1)dλ+O(t−1)\n=L3+O(t−1).\n25So we estimate L3part. We use identities (6 .5) and (6.6). By these identities, we have\nL3=/integraldisplay\nC˜φc(P)iλc(·)−1/2(λ+˜P)−1e−t˜P/2c(·)−1/2b(·)c(·)−1/2e−t˜P/4(iλ+˜P)−1c(·)−1/2(c(·)u0+u1)dλ\n+/integraldisplay\nC˜φc(P)eiλtiλc−1/2/integraldisplayt/2\n0e−t1(iλ+˜P)dt1c−1/2b(·)c(·)−1/2(iλ+˜P)−1e−t(iλ+˜P)/4c−1/2(c(·)u0+u1)dλ\n+/integraldisplay\nC˜φc(P)eiλtiλc(·)−1/2(λ+˜P)−1e−t˜P/2c(·)−1/2b(·)c(·)−1/2/integraldisplayt/4\n0e−t2(iλ+˜P)dt2c(·)−1/2(c(·)u0+u1)dλ\n+/integraldisplay\nC˜φc(P)eiλtiλc(·)−1/2/integraldisplayt/2\n0e−t1(iλ+˜P)dt1c(·)−1/2b(·)c(·)−1/2/integraldisplayt/4\n0e−t2(iλ+˜P)dt2c(·)−1/2(c(·)u0+u1)dλ\n=L4+L5+L6+L7\nSinceb(x) is compact support function and d>2, from Lemma 7 .12, we have\n/ba∇dble−t˜P/2c(x)−1/2b(x)c(x)−1/2e−t˜P/4/ba∇dblL2→L2≤ /ba∇dble−t˜P/2/ba∇dblL1→L2/ba∇dblc(x)−1/2b(x)c(x)−1/2e−t˜P/4/ba∇dblL2→L1\n≤Ct−d/4/ba∇dblc(x)−1/2b(x)c(x)−1/2/ba∇dblL1/ba∇dble−t˜P/4/ba∇dblL2→L∞\n≤Ct−d/2≤Ct−1.\nFrom this estimate, we know L4is a lower order term. For L5, since\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayt/2\n0e−t1˜Pdt1/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL1→L2= max{1,t−d/4+1+ε}\nfor anyε>0, we have\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayt/2\n0eiλ(3t−4t1)/4(iλc−1/2e−t1˜Pc−1/2b(·)c(·)−1/2(iλ+˜P)−1e−t˜P/4c−1/2dt1/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2→L2\n≤/integraldisplayt/2\n0/vextenddouble/vextenddouble/vextenddoubleeiλ(3t−4t1)/4c−1/2e−t1˜P/vextenddouble/vextenddouble/vextenddouble\nL1→L2/vextenddouble/vextenddouble/vextenddoublec−1/2b(·)c(·)−1/2e−t˜P/4iλ(iλ+˜P)−1c−1/2/vextenddouble/vextenddouble/vextenddouble\nL2→L1dt1\n≤/integraldisplayt/2\n0/vextenddouble/vextenddouble/vextenddoubleeiλ(3t−4t1)/4c−1/2e−t1˜P/vextenddouble/vextenddouble/vextenddouble\nL1→L2dt1/vextenddouble/vextenddouble/vextenddoublec−1/2b(·)c(·)−1/2/ba∇dblL1/ba∇dble−t˜P/4/ba∇dblL2→L∞/ba∇dbliλ(iλ+˜P)−1c−1/2/vextenddouble/vextenddouble/vextenddouble\nL2→L2\n≤Cmax{e−3tImλ/4,e−tImλ/4}max{1,t−d/4+1+ε}t−d/4.\nFrom this estimate the integrand is O(1) asλvariable and by contour deformation as in the proof of\nProposition 4 .3, we get further t−1decay. Thus we have\n/ba∇dblL5/ba∇dblL2≤Cmax{1,t−d/4+1+ε}t−d/4−1≤Ct−1.\nWe can apply similar argumetn to estimate L6term and we get t−1order decay. For L7term, the\nintegrand is holomorphic so thanks to the e−iλ(t−t1−t2)factor, we can get the exponential decay estimate,\nby contour deformation across the origin. Thus we have proved Th eorem 7.9. Ford= 1,2, in the same\nway, we can also prove the following result.\nTheorem 7.13. Under the assumption of Theorem 7.2, for anyε>0andd= 1,2, there exists C >0\nsuch that for t>2the following estimates between the solution uof the equation (7.3)and the solution\nvof the equation (7.4)hold\n/ba∇dblu(t)−c(·)1/2v(t)/ba∇dblL2≤Ct−1logt(/ba∇dblu0/ba∇dblH1+/ba∇dblu1/ba∇dblL2), (d= 2),\n/ba∇dblu(t)−c(·)1/2v(t)/ba∇dblL2≤Ct−1/2(/ba∇dblu0/ba∇dblH1+/ba∇dblu1/ba∇dblL2), (d= 1).\nRemark 7.14. In the above theorems we have the arbitrariness to the choice ofc(·)term. Probably our\nestimates come from the arbitrariness of c(·). For simplicity, we consider the following heat equation\n((1+a(·))∂t−∆)u= 0, u|t=0= (1+a(·))−1u0\nHerea(·)is a non-negative compact support function. Then by Duhamel ’s principle we have\nu(t) =et∆u(0)−/integraldisplayt\n0e(t−s)∆a(·)∂su(s)ds.\n26By integration by parts, we obtain\nu(t) =et∆u0−/integraldisplayt\nt/2e(t−s)∆a(·)∂su(s)ds−/integraldisplayt/2\n0/parenleftBig\n∂te(t−s)∆/parenrightBig\na(·)u(s)ds−et/2∆a(·)u(t/2).\nIn the right hand side, the first term is the top term. Since a(x)is a compact support function, we have\n/ba∇dblet∆/2a(·)u(t/2)/ba∇dblL2≤ /ba∇dblet/2∆/ba∇dblL1→L2/ba∇dbla(·)u(t/2)/ba∇dblL1\n≤ /ba∇dblet/2∆/ba∇dblL1→L2/ba∇dbla(·)/ba∇dblL1/ba∇dblu(t/2)/ba∇dblL2→L∞≤Ct−d/2/ba∇dblu(0)/ba∇dblL2.\nFrom well-known estimates of heat propagators, we get the fo llowing estimate\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayt\nt/2e(t−s)∆a(·)∂su(s)ds/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2≤C/integraldisplayt\nt/2(t−s)−d/4s−d/4−1/ba∇dbla(·)/ba∇dblL1/ba∇dblu(0)/ba∇dblL2ds≤C/braceleftBigg\nt−1/ba∇dblu0/ba∇dblL2(d= 2),\nt−1/2/ba∇dblu0/ba∇dblL2(d= 1).\nWe can treat the another integral by same manner and get simil ar estimate. So the above estimate seems\nalmost optimal.\nIn the same manner, we can generalize Theorem 1 .3.\nTheorem 7.15. Letu,vare the solution of the equations (7.3),(7.4)respectively. Then under the\nassumption of Theorem 7.1, for any ε >0, there exists C >0such that for any u0∈H1∩L1and\n/a\\}b∇acketle{tx/a\\}b∇acket∇i}htu0∈L2,u1∈L2∩L1andt>2, the following asymptotic profile hold\n/ba∇dblu(t)−c(·)1/2v(t)/ba∇dblL2≤Ct−1−d/4(/ba∇dblu0/ba∇dblH1∩L1+/ba∇dblu1/ba∇dblL2∩L1)\nProof. The proof is essentially same and we give an outline of the proo f. We use the expansion (7 .9).Rc\nterm can be treated by similar manner in the proof of Theorem 1 .3 and we get the heat type asymptotic.\n8 Appendix\nIn this appendix, we prove the energy decay estimate under the ge ometric control condition. We estimate\nthe resolvent dividing three frequency parts in the same way as The orem 7.2. We can use Lemma 7 .5 to\nestimate the high frequency part. So we need to estimate middle and low frequency parts. We take\nL=P+λ2+λa(·)\nand estimate /ba∇dblLu/ba∇dblL2from below. The following argument is essentially similar to [7]. By the geo metric\ncontrol condition, we can take a sequence ( xn)⊂Rdsuch thata(xn)≥α/Tand any point of Rdis\nat bounded distance from the set ∪n{xn}. By the uniformly continuity, we can find r,β >0 such that\na(x)≥β >0 onω=∪nB(xn,r).\nWe estimate the low frequency part. First we assume |Reλ|<|Imλ|. By taking imaginary part of\n(Lu,u)L2, we easily have\n(8.1) /ba∇dblLu/ba∇dbl2\nL2+ε2|λ|2/ba∇dblu/ba∇dbl2\nL2+Cε|λ|3/ba∇dblu/ba∇dbl2\nL2≥ε|λ|2\nC/ba∇dbl√au/ba∇dbl2\nL2.\nWe can take ε>0 arbitrary small and C >0 is some constant. As in [7], we set\nQh=h2eϕ/h(P+λ2)e−ϕ/h\nhereh>0 is sufficiently small. Then we can take a uniformly bounded weight ϕsuch that the following\nsub-elliptic estimate holds\n(8.2) /ba∇dblQhu/ba∇dbl2\nL2≥Ch/ba∇dblu/ba∇dbl2\nL2\nhereC >0 andusatisfiesu|ω= 0, see [7] for the proof. We take a real-valued cut-off function χ∈C∞\n0\nsuch thatχ(s) = 0 fors≥βandχ≡1 in a neighborhood of 0. Then ac=χ◦avanishes on ωand any\nderivatives of acis controlled by a. We setv=eφ/hacuand compute\nQhv=h2eφ/hacLu−h2eφ/hλacau+h2eφ/h[P,ac]u.\n27Sincev|ω= 0, from this identity and (8 .2), we obtain\n/ba∇dblh2eφ/hacLu/ba∇dbl2\nL2+/ba∇dblh2eφ/hλacau/ba∇dbl2\nL2+/ba∇dblh2eφ/h[P,ac]u/ba∇dbl2\nL2≥ /ba∇dblQhv/ba∇dbl2\nL2≥Ch/ba∇dblv/ba∇dbl2\nL2.\nSinceϕis uniformly bounded, for fixed h,eφ/his uniformly positive and bounded . Thus we have\n/ba∇dblLu/ba∇dbl2\nL2+C|λ|2/ba∇dblau/ba∇dbl2\nL2+C/ba∇dbl[P,ac]u/ba∇dbl2\nL2≥1\nC/ba∇dblacu/ba∇dbl2\nL2.\nSince derivatives of acare controlled by a, for sufficient small |λ|, we have\n(8.3)ε3/2|λ|2/ba∇dblLu/ba∇dbl2\nL2+Cε3/2|λ|2/ba∇dbl√au/ba∇dbl2\nL2+Cε3/2|λ|2d/summationdisplay\ni,j=1/ba∇dbl(∂jac)∂iu/ba∇dbl2\nL2≥ε3/2|λ|2\nC/ba∇dblacu/ba∇dbl2\nL2.\nFork= 1,2,···,d, using integration by parts, we have\n(Pu,(∂kac)2u)L2=d/summationdisplay\ni,j=1(gij(∂kac)∂iu,(∂kac)∂ju)L2+2d/summationdisplay\ni,j=1((∂kac)∂iu,gij(∂j∂kac)u)L2.\nSince derivatives of acare controlled by a, from this identity, we have\n/ba∇dblLu/ba∇dbl2\nL2+/ba∇dbl√au/ba∇dbl2\nL2≥Cd/summationdisplay\nk,i=1/ba∇dbl(∂kac)∂iu/ba∇dbl2\nL2\nfor sufficiently small |λ|. So from (8 .3), we get\n(8.4) ε3/2|λ|2/ba∇dblLu/ba∇dbl2\nL2+Cε3/2|λ|2/ba∇dbl√au/ba∇dbl2\nL2≥ε3/2|λ|2\nC/ba∇dblacu/ba∇dbl2\nL2.\nThus from (8 .1) and (8.4), we obtain\n/ba∇dblLu/ba∇dbl2\nL2+Cε2|λ|2/ba∇dblu/ba∇dbl2\nL2+Cε3/2|λ|2/ba∇dbl√au/ba∇dbl2\nL2+Cε|λ|3/ba∇dblu/ba∇dbl2\nL2≥ε|λ|2\nC/ba∇dbl√au/ba∇dbl2\nL2+ε3/2|λ|2\nC/ba∇dblacu/ba∇dbl2\nL2.\nNow taking εsufficiently small and |λ|<ε, sinceac(x) = 1 ifa(x) = 0, we conclude\n(8.5) /ba∇dblLu/ba∇dbl2\nL2≥C|λ|2/ba∇dblu/ba∇dbl2\nL2.\nFrom this estimate, we get R(λ) =O(|λ|−1) for sufficiently small |λ|around the imaginary axis. If\nReλ≥ |Imλ|, by taking real part of ( Lu,u)L2, we have\n/ba∇dblLu/ba∇dbl2\nL2+ε2|λ|2/ba∇dblu/ba∇dbl2\nL2≥Cε|λ|2/ba∇dbl√au/ba∇dbl2\nL2.\nBy similar argument, we can also prove (8 .4) in this case. So we have (8 .5) and getR(λ) =O(|λ|−1) for\nReλ >0. Thus we have estimated the low frequency part. The estimate of the middle frequency part\nis essentially same and more easy (we only need the reversibility of P−λ2+iλaifλis real and in a\ncompact subset of (0 ,∞)) so we omit the proof. From these estimate, We conclude the follow ing energy\ndecay estimate.\nTheorem 8.1. We assume the geometric control condition. Let ube a solution of (7.1)with the initial\ndatau|t=0=u0∈H1and∂tu|t=0=u1∈L2. Then there exists a constant Csuch that for t >1the\nfollowing bound holds\n/ba∇dbl∇u/ba∇dbl2\nL2+/ba∇dbl∂tu/ba∇dbl2\nL2≤C(d/summationdisplay\ni,j=1(gij∂iu,∂ju)L2+/ba∇dbl∂tu/ba∇dbl2\nL2)≤Ct−1(/ba∇dblu0/ba∇dbl2\nH1+/ba∇dblu1)/ba∇dblL2).\nReferences\n[1] C. Bardos, G. Lebeau and J. Rauch: Un example d’urilizati on des notions de propagation pour le controle\net la stabilisation de problems hyperboliques. Rend. sem. M at. Univ. Pol. Torino Fascicolo speciale 1988\nHyperbolic equations, (1987).\n28[2] C. Bardos, G. Lebeau and J. 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Friedman, Partial differential equations of paraboli c type, Prentice-Hall, (1964)\n[10] R. Ikehata, G. Todorova, B, Yordanov, Optimal decay rat e of the energy for wave equations with critical\npotential, J. Differential Equations 254 (2013), no. 8, 3352 -3368.\n[11] M. Khenissi: ´Equation des ondes amorties dans un domaine ext´ erieur, Bul l. Soc. math. France 131 (2),\n(2003), 211–228\n[12] G. Lebeau: Equation des ondes amorties. in Algebraic an d Geometric Methods in Mathematical Physics, A.\nBoutet de Monvel and V. Marchenko eds. (1996), 73-109.\n[13] A. Matsumura, On the asymptotic behavior of solutions o f semi-linear wave equations, Publ. Res. Inst. Math.\nSci. 12 (1976) 169–189.\n[14] K. Nishihara, Lp-Lq estimates of solutions to the dampe d wave equation in 3-dimensional space and their\napplication, Math. Z., 244 (2003), 631–649.\n[15] K. Ono, Decay estimates for dissipative wave equations in exterior domains, J. Math. Anal. Appl. 286 (2003)\n540–562.\n[16] J. Rauch and M. Taylor: Exponential decay of solutions t o symmetric hyperbolic equations in bounded\ndomeins. Indiana J.Math. 24 (1974), 79–86.\n[17] P. Radu, G. Todorova, B. Yordanov: On the generalized di ffusion phenomenon of dissipative wave equations.\n[18] Y. Wakasugi: On diffusion phenomena for the linear wave e quation with space-dependent damping, J. Hyper.\nDifferential Equations, 11, 795 (2014).\n[19] M.Zworsky: semiclassical analysis. GraduateStudies inMathematics no138. AmericanMathematical Society,\nProvidence, R.I., (2012).\nDepartment of Education, Wakayama University,\n930, Sakaedani, Wakayama city , Wakayama 640-8510, Japan\nE-mail:h2480@center.wakayama-u.ac.jp\n29"    },    {        "title": "0708.3855v1.Ising_Dynamics_with_Damping.pdf",        "content": "arXiv:0708.3855v1  [cond-mat.mtrl-sci]  28 Aug 2007Ising Dynamics with Damping\nJ. M. Deutsch\nDepartment of Physics, University of California, Santa Cru z, CA 95064.\nA. Berger\nCIC Nanogune, Mikeletegi Pasealekua 56, 301 E-20009 Donost ia Spain\nWe show for the Ising model that is possible construct a discr ete time stochastic model analogous\nto the Langevin equation that incorporates an arbitrary amo unt of damping. It is shown to give\nthe correct equilibrium statistics and is then used to inves tigate nonequilibrium phenomena, in\nparticular, magnetic avalanches. The value of damping can g reatly alter the shape of hysteresis\nloops, andfor small dampingandhighdisorder, themorpholo gyoflarge avalanchescanbedrastically\neffected. Small damping also alters the size distribution of avalanches at criticality.\nPACS numbers: 75.40.Mg, 75.60.Ej, 05.45.Jn,\nI. INTRODUCTION\nIn many situations, it is useful to discretize continu-\nous degrees of freedom to better understand them, both\nfrom a theoretical standpoint and for numerical effi-\nciency. Ising models are perhaps the best example of this\nand have been the subject of numerous theoretical and\nnumerical studies. Renormalization group arguments1\nhave explained the reason why this discretization gives\nequilibrium critical properties of many experimental sys-\ntems, and these kinds of arguments have been extended\nto understanding their equilibrium dynamics2. For non-\nequilibrium situations, such as the study of avalanches,\nsuch arguments probably do also apply to large enough\nlength and time scales as well. However there are many\nsituations where it would be desirable to understand\nsmaller length scales where other factors should become\nrelevant.\nThis is particularly true with dynamics of magnetic\nsystems, where damping is often weak in comparison to\nprecessional effects. For studies of smaller scales, it has\nbeen necessary to use more time consuming micromag-\nnetic simulations utilizing continuous degrees of freedom,\nsuch as the Landau Lifshitz Gilbert equations3which is a\nkind of Langevin equation that gives the stochastic evo-\nlution of Heisenberg spins.\nds\ndt=−s×(B−γs×B), (1)\nwheresis a microscopic magnetic moment, Bis the local\neffective field, and γis a damping factor, measuring the\nrelative importance of damping to precession. In real\nmaterials it ranges4from small damping γ=.01, to 1.\nIn contrast, the dynamical rules implemented for Ising\nmodels are most often “relaxational” so that energy is\ninstantaneously dissipated when a spin flips, as with the\nMetropolis algorithm.\nHowever there is a class of “microcanonical” Ising dy-\nnamics5reviewed in section II where auxiliary degrees of\nfreedom are introduced and all moves conserve the total\nenergy. The other degrees of freedom can be taken to bevariables associated with each spin, and allowed moves\ncan change both the state of the spins and the auxiliary\nvariables. Thiscanbe thoughtofcrudely, asadiscretized\nanalogyto moleculardynamics, and is also similar to dis-\ncrete lattice gas models of fluids6,7. These models give\nthe correct equilibrium Ising statistics of large systems\nand can also be used to understand dynamics in a differ-\nent limit than the relaxational case.\nReal spin systems are intermediate between these two\nkindsofdynamicsand asmentioned above, arebetter de-\nscribed by Langevin dynamics. In the context of spins,\nthe question posed and answered here is: how does one\nformulate a discrete time version of stochastic dynamics\nthat includes damping and gives the correct equilibrium\nstatistics? In section IIA we are able to show that there\nis a fairly simple method for doing this using a combina-\ntion of microcanonical dynamics, and an elegant proce-\ndure that incorporates damping and thermal noise. This\nprocedure differs from that of the Langevin equation in\nthat it requires non-Gaussian noise. Despite this, the\nnoise has surprisingly simple but unusual statistics.\nWe will then showthat this proceduregivesthe correct\nequilibrium statistics and verify this numerical in section\nIIB with a simulation ofthe two dimensional Ising model\nwith different amounts of damping.\nBecause the value of damping is an important phys-\nical parameter in many situations it is important that\nthere is a straightforward way of incorporating its effects\nin Ising simulations. This is particularly noteworthy as\nIsing kinetics are a frequently used means of understand-\ning dynamics in many condensed matter systems.\nAfter this in section III we will turn to nonequilibrium\nproblems where, using this approach, we can study the\neffects of damping on a number of interesting properties\nof systems displaying avalanches and Barkhausen noise8.\nWe firstshowhowto modify the kinetics forthis caseand\nthen study systems in two and three dimensions. With\nmodest amounts of computer time, we can analyze prob-\nlems that are out of the reach of micromagnetic simula-\ntions and allow us to probe the effects of damping on the\nproperties of avalanches. This is related to recent work92\nby the present authors using both the Landau Lifshitz\nGilbert equation, Eqn. 1, and theoretical approaches, to\nunderstand how relaxational dynamics of avalanches10,\nare modified at small to intermediate scales by this more\nrealistic approach. With the present approach we find\nnew features and modifications of avalanche dynamics.\nWe find that the shape ofhysteresisloopscanbe strongly\ninfluenced by the amount of damping. One of the most\nstriking findings is that there exists a parameter regime\nof high disorder and small damping where single system-\nsize avalanches occur that are made up of a large number\nof disconnected pieces. We can also analyze the criti-\ncal properties of avalanches when damping is small and\ngive evidence that there is a crossoverlength scale, below\nwhich avalanches have different critical properties.\nII. NON-RELAXATIONAL DYNAMICS\nWestartbyconsideringamodelforamagnetwithcon-\ntinuous degrees of freedom, such as a Heisenberg model\nwith anisotropy. The Ising approximation simplifies the\nstate of each spin to either up or down, that is si=±1,\ni= 1,...,N. One important effect that is ignored by\nthis approximation is that of spin waves that allow the\ntransfer of energy between neighbors, and for small os-\ncillations, give an energy contribution per spin equal to\nthe temperature T(here we set kB= 1). This moti-\nvates the idea that there are extra degrees of freedom\nassociated with every spin that can carry (a positive)\nenergyei. Creutz introduced such degrees of freedom5\nand posited that they could take any number of dis-\ncrete values. He used these auxiliary variables eito\nconstruct a cellular automota to give the correct equi-\nlibrium statistics for the Ising model, in a very efficient\nway that did not require the generation of random num-\nbers. Thus we have a Hamiltonian Htotthat is the sum\nof both spin Hspinand auxiliary degrees of freedom He:\nHtot=Hspin+He.Hspincan be a general Ising spin\nHamiltonian and He=/summationtext\niei. In our model there is a\nsingle auxiliary variable eiassociated with each lattice\nsitei, that can take on any real value ≥0.\nHowever for the purposes of trying to model dynamics\nof spins, it also makes sense to allow the ei’s to interact\nand exchange energy between neighbors. For example,\none precessing spin should excite motion in its neighbors.\nThis exchange was formulated in the context of solidifi-\ncation using a Potts model instead of an Ising model by\nContiet al.11, but can equally well be used here.\nNow we can formulate a microcanonical algorithm for\nthe Ising model using a procedure very similar to their\nprescription. In each step:\n1. We choose a site iat random.\n2. We randomly pick with equal probability either a\nspin or an auxiliary degree of freedom, siofei:(a)si’s: We attempt to move spins (such as the\nflipping of a single spin). If the energy cost in\ndoing this is ≤eiwe perform the move and\ndecrease eiaccordingly. Otherwise we reject\nthe move.\n(b)ei’s: We pick a nearest neighbor j, and repar-\ntition the total energy with uniform probabil-\nity between these two variables. That is, af-\nter repartitioning, e′\ni= (ei+ej)rande′\nj=\n(ei+ej)(1−r), where 0 < r <1 is uniform\nrandom variable.\nNote that these rules preserve the total energy and the\ntransitions between any two states have the same proba-\nbility. Therefore this will give the correct microcanonical\ndistribution. For large N, this is, for most purposes12,\nequivalent to the canonical distribution ∝exp(−βHtot).\nNote that the probability distribution for each variable\nei,P(ei) =βexp(−βei), so that the ∝angbracketleftei∝angbracketright=T. That is,\nmeasurement of averageof ei’s directly gives the effective\ntemperature of the system.\nA. Extension To Damping\nThe question we asked, is how to extend this equilib-\nrium simulation method to include damping. In this case\nthe system is no longer closed and energy is exchanged\nwith an outside heat bath through interaction with the\nauxiliaryvariables. As with the Langevinequation, there\nare two effects. The first is that the energy is damped.\nCall the dissipation parameter for each step α, which will\nlie between 0 and 1. Then at each time step we lower\nthe energy with ei→αeifor all sites i. By itself, this\nclearly will not give a system at finite temperature and\nwe must also include the second effect of a heat bath,\nwhich adds energy randomly to the system. In the case\nof the Langevin equation, a Gaussian noise term n(t) is\nadded to keep the system at finite temperature. A dis-\ncretized version of this, that evolves the energy e(t) at\ntime step tis\ne(t+1) =αe(t)+n(t). (2)\nThis equation will not work if the noise n(t) is Gaussian\nas this does not give the Gibbs distribution Peq(e) =\nβexp(−βe). Therefore we need to modify the statistics\nofn(t). It is possible to do so if we choose n(t) at each\ntimetfrom a distribution\np(n) =αδ(n)+(1−α)βe−βnθ(n) (3)\nwhereθis the Heaviside step function. To show this, we\nwrite down the corresponding equation for the evolution\nof the probability distribution for e:3\nP(e′,t+1) =∝angbracketleftδ(e′−(αe+n))∝angbracketright=/integraldisplay /integraldisplay\nP(e,t)p(n)δ(e′−(αe+n))de dn (4)\nWe require that the Past→ ∞obeysP(e,t+ 1) =\nP(e,t) =Peq(e) =βexp(−βe), fore >0. It is easily ver-\nified thatby choosingthisform of P(e,t)and bychoosing\nP(n) as in Eq. 3, we satisfy Eq. 4.\nTherefore to add damping to this model, we add the\nfollowing procedure to the steps stated above:\n3. Chooseauniformrandomnumber0 < r <1. Ifr <\nα, thenei→αei. Otherwise ei→αei−Tln(r′),\nwherer′is another uniform random number be-\ntween 0 and 1.\nIf we assume that the probability distribution for the\ntotal system is of the form PGibbs∝exp(−βHtot) =\nexp(−βHspin)exp(−βHe), we will now show that the\nsteps 1, 2, and 3, of this algorithm preserve this distri-\nbution. Following the same reasoning as above for the\nmicrocanonical simulation, moves implementing steps 1\nand 2 do not change the total energy, and they preserve\nthe form of PGibbsbecause PGibbsdepends only on the\ntotal energy ( Htot), and 1 and 2 explore each state in\nan energy shell with uniform probability. Because of\nthe form of PGibbs, its dependence on the variable eiis\n∝exp(−βei). According to the above argument, after\nstep 3, it will remain unchanged. Therefore all steps in\nthis algorithm leave PGibbsunchanged. The algorithm is\nalsoergodic,and thereforethis willconvergetothe Gibbs\ndistribution13ast→ ∞.\nBecausethe stepseachpreservethe Gibbsdistribution,\nthe ordering of them is not important in preserving equi-\nlibrium statistics. For example, we could sweep through\nthe lattice sequentially instead of picking iat random.\nWe could perform step 3 after steps 1 and 2 were per-\nformedNtimes.\nB. Equilibrium Tests\nWe performed tests on this algorithm and verified that\nit did indeed work as expected. We simulated the two\ndimensional Ising model on a 1282lattice with differ-\nent values of the damping parameter, and compared it\nwith the exact results. The average magnetization per\nspinmis plotted in Fig. 1 as a function of the tempera-\ntureTand compared with the exact result14for large N\n(dashedcurve). The ×’sarethecase α= 1, whichisthen\njust an implementation of the microcanonical method11\ndescribed above. In this case, the temperature was ob-\ntained by measuring ∝angbracketleftei∝angbracketrightbecause the energy was fixed\nat the start of the simulation. The only point which is\nslightly off the exact solution is in the critical region, as\nis to be expected. The case α= 0.5 is shown with the\n+’s and lie on the same curve. Results were obtained forα= 0.9 but are so close as to be indistinguishable and\nare therefore not shown. We also checked that the distri-\nbution of auxiliary variables had the correct form. The\nprobability distribution for the energy eis shown in Fig.\n2. Fig. 2 plots the distribution P(ei) versus energy ei,\naveraged over all sites ion a linear-log scale for α= 0.5\nandT= 0.8, and 1.1. The curves are straight lines over\nfour decades and show the correct slopes, for T= 0.8,\n∝angbracketleftei∝angbracketright= 0.8002 and for T= 1.1,∝angbracketleftei∝angbracketright= 1.1003.\nFIG. 1: Plot of results obtained for the two dimensional Isin g\nmodel on a1282lattice for twodifferent values of the damping\nparameter. This is a plot of the average magnetization per\nspinmvs.T. The×’s are for no dissipation, α= 1, which is\na purely microcanonical simulation. The +’s are for α= 0.5.\nThe dashed curve is the exact solution to this model in the\nthermodynamic limit.\nIII. AVALANCHE DYNAMICS\nAvalanche dynamics of spin systems have been mainly\nstudied using models that are purely relaxational. There\nis a whole rangeof interesting phenomena that have been\nelucidated by such studies and have yielded very inter-\nesting properties. The simplest model that can be used\nin this context is the random field Ising model (RFIM)\nwith a Hamiltonian\nH=−/summationdisplay\n<ij>Jsisj−/summationdisplay\nihisi−h/summationdisplay\nsi(5)\nwhereJis the strength of the nearest neighbor coupling,\nhiis a random field, with zero mean, and his an exter-\nnally applied field. A magnet is placed in a high field h4\nFIG. 2: Plot of results obtained for the two dimensional\nIsing model on a 1282lattice for the probability distribution\nfor the auxiliary variables ei, at two different temperatures\nwith a damping parameter α= 0.5. The upper curve is for\nT= 1.1 and the lower for T= 0.8.\nandthen this isveryslowlylowered. As thishappens, the\nspins will adjust to the new field byflipping to lowertheir\nenergy. In the usual situation, the system is taken to be\natT= 0, so that only moves that lower the energy are\naccepted. The flipping of one spin can cause a cascade of\nadditional spins to flip, causing the total magnetization\nMto further decrease. The occurrence of these cascades\nis called an “avalanche”. At zero temperature there is\none parameter jthat characterizes the system, the ratio\nof nearest neighbor coupling to the distribution width of\nthe random field. One considers the behavior of a sys-\ntem when its starts in a high field and is slowly lowered.\nWhenjis small the system is strongly pinned and the\nsystem will have a number of small avalanches generat-\ning a smooth hysteresis loop. For large j, the system\nwill have a system-size avalanche involving most of the\nspins in the system, leading to a precipitous drop in the\nhysteresis loop. There is a critical value of jwhere the\ndistribution of avalanche sizes is a power law and self-\nsimilar scaling behavior is observed.\nHere we investigate how this is modified by adding\ndamping to these zero temperature dynamics according\nto the following rules:\n1. The field is slowly lowered by finding the next field\nwhere a spin can flip.\n2. The spins then flip, exchanging energy with auxil-\niaryvariables eias describedabove. The numberof\ntimes this is attempted is nmtimes the total num-\nber of spins in the system. Here we set nm= 16.\nIn more detail:\n(i)Spin moves: An attempt to move each spin\non the lattice is performed by attempting to\nflip sequentially every third spin, in order tominimize artifacts in the dynamics due to up-\ndating contiguous spins. (The lattice sites are\nlinearly ordered using “skew” boundary con-\nditions). Then all three sublattices are cycled\nover.\n(ii)Energy moves: Exchange of energy\nwith nearest neighbors is performed cycling\nthrough all directions of nearest neighbors.\nUsing the same sequence of updates, the ei’s\nexchange energy with their nearest neighbors\nin one particular direction.\n(iii)Dissipation: The energyofeach eiis lowered\ntoαei.\n3. We check for when the spins have settled down as\nfollows: if the ei’s are not all below some energy\nthreshold ethresh, set below to be 10−4, or the spin\nconfiguration has changed, step 2 is repeated until\nthese conditions are both met.\n4. When the spins have settled down, we go to step 1.\nThe parameters nmandethreshwere varied to check that\nthe correct dynamics were obtained. The larger α, the\nsmaller the dissipation and the larger the number of iter-\nations necessary to achieve the final static configuration.\nFIG. 3: The major branch of the descending hysteresis loop\nfor 642systems using different values of the damping parame-\nter and the spin coupling. Strong damping, α= 0.5 is shown\nin the left most curve (as judged from the top of the plot)\nfor coupling j= 0.3 which starts decreasing from M= 1 at\nh=−0.2, and does not have large abrupt changes. All the\nother curves are for weak damping, α= 0.99. In this case but\nalso forj= 0.3, we see that although Mstarts to decrease at\nthe same location as for strong damping, it drops abruptly as\nthe field is lowered. As the coupling jis decreased, smooth\ncurvesare eventuallyseenagain. Goinglefttoright, as jud ged\nfrom the top, are j= 0.3, 0.25, 0.2, and 0.15.5\n(a)\n (b)\nFIG. 4: (a) The spin configuration for a 2562system with j= 0.35,α= 0.9 during a system size avalanche at the field\nh=−0.400007. (b) A gray-scale plot of the auxiliary variables at t he same time.\n(a)\n (b)\n (c)\nFIG. 5: Spin configurations for a 2562system with j= 0.25,α= 0.99 during an avalanche at the field h=−7×10−5. (a)\nThe beginning of the avalanche. (b) When the avalanche is of o rder of half the system size. (c) The final configuration of the\navalanche.\nA. Two Dimensional Patterns\nWe first investigate the case of two dimensions where\nit is simpler to visualize the avalanches in various condi-\ntions than in three dimensions. Much experimental work\nand theoretical work on avalanches has been done on two\ndimensionalmagneticfilmsandthiscaseshouldbehighly\nrelevant10.\nWe first examine how the hysteresis loops change as\na function of the coupling jand the damping parame-\nterαfor a 642system. The major downwards hystere-\nsis loops are shown in Fig. 3 for a variety of param-\neters described below. We first examine strong damp-\ningα= 0.5. Forj= 0.3 the hysteresis curve is quite\nsmooth with all avalanches much less than the systemsize (left most curve). Now consider the same value of j\nbut with with small damping, α= 0.99. The curve now\nis a single downwards step with a small tail at negative\nh. The lower damping has allowed that system to form\na system size avalanche. The difference is due to the\nfact that with small damping, the energy of avalanched\nspinsis not immediately dissipatedand asaconsequence,\nheats up neighboring spins, allowing them to more easily\navalanche as well. Therefore a system size avalanche is\nseenin the smalldampingcase, leadingtothe precipitous\ndrop in the hysteresis loop.\nWhen the value of the coupling jis lowered to 0 .15 for\nα= 0.99, smooth loops are obtained. The Fig. 3 shows\nintermediate values of the coupling parameter as well.\nTo better understand the reason why the energy of6\nthe auxiliary variables can trigger further spins to flip, in\nFig. 4 we show the state of a system during a system size\navalanche for j= 0.35 and a moderately small damping\nvalue,α= 0.9,withh=−0.400007. Fig. 4(a)showsthat\nthe flipped spins form a fairly compact cluster and Fig.\n4(b) shows the corresponding values of the ei’s in a gray\nscale plot, suitably normalized. It has the appearance of\na halo around the growth front of the avalanche. The\nspins in the growth front have just flipped and so energy\nthere has not had a chance to diffuse or dissipate and\nso has a higher spin temperature. The interior is cold\nbecause damping has removed energy from the auxiliary\ndegreesoffreedom. This highertemperature diffuses into\nthe the unflipped region allowing spins to flip by thermal\nactivation.\nBecause large avalanches are possible for small damp-\ning in a parameter range where the relative effect of the\nrandom field is much larger, it is of interest to see if\navalancheshaveadifferentmorphologythan typicallarge\navalanches for high damping systems. Fig. 5 shows such\nspin configurationsfirst at the beginning of the avalanche\nandfurtheralongduringpropagationwhenithasreached\nroughly half the system size, and finally when it has\nreached its final configuration and the maximum aux-\niliary variable value is <4×10−4. The morphology of\nthisisverydifferentthanwhatisseenforlargeavalanches\nwith strongercoupling, for exampleFig. 4. At verysmall\nfields, in this figure h=−7×10−5, surface tension pre-\ncludes the formation of minority domains, but because\ndisorder is large, there will be many small regions where\nthe local field is much stronger and these will want to\nform downward oriented (black) domains. There is a fi-\nnite activation barrier to forming these that can only be\novercome at finite temperature. However the majority\nof the spins still strongly disfavor flipping. But because\ndamping is small, heat has a chance to diffuse through\nthese regions into the favorable regions, allowing discon-\nnected regions to change orientation by thermal activa-\ntion. Note that we have checked numerically that small\ndamping with strong coupling also leads to compact con-\nfigurations, so disorder is an essential ingredient in this\nnew morphology.\nB. Three Dimensions\nWe first check that as with two dimensions, the value\nof the damping parameter can have a large effect on the\nshape of a hysteresis loop. Fig. 6 shows the downward\nbranches of the major hysteresis loop when the only pa-\nrameter that is changed is the damping, α. The system\nis a 323lattice with j= 0.19. A value for high damping,\nα= 0.5, is the upper line. The lower line is for small\ndamping with α= 0.99.\nA more subtle effect, is that of damping on what hap-\npens near criticality. In this case the value of the criti-\ncaljwill depend on the value of αas is apparent from\nthe results of Fig. 6. At this point, the distribution ofFIG. 6: The major branch of the descending hysteresis loops\nin two 323systems with j= 0.19, for two different values of\nthe dissipation, upper curve: α= 0.5, lower curve: α= 0.99.\navalanchesizesisexpected tofollowapowerlawdistribu-\ntion for large sizes. We located this point and examined\nsystem properties in this vicinity. Fig. 7 shows examples\nof such runs for 323systems. Fig. 7(a) shows a plot of\nthe magnetization per spin M, versus the applied field h\nforj= 0.165 and j= 0.167. For larger values of j, the\navalanches rapidly become much larger as is seen in Fig.\n6, and for smaller values, avalanches all become small.\nFig. 7(b) shows a plot of the same quantity with relax-\national dynamics near criticality. The avalanches take\nplace over a much smaller range in applied field.\nTo quantify this difference, we studied the avalanche\nsize distribution exponent that is obtained by calculating\nthe distribution of avalanche sizes over the entire hys-\nteresis loop. This was studied by averaging avalanches of\nmanyruns, (200for α= 0.99)for 323systemsand fordif-\nferentvaluesofparameters. Weshowacomparisonofthe\navalanche size distribution for α= 0.99, shown with +’s\nand forα= 0.9, shown with ×’s in Fig. 8. For α= 0.99\nthe curve fits quite well to a power law with an exponent\nof−1.4±.1 as shown in the figure. For purely relax-\national dynamics, the same exponent has been carefully\nmeasured15to be 2.03±.03 (which is consistent with our\nresults for relaxational dynamics on much smaller sys-\ntems than theirs). With smaller damping we expect to\nhave a crossover length corresponding to the length scale\nassociated with the damping time, above which the dy-\nnamics should appear relaxational. α= 0.9 appears to\nshowsuchacrossoverfromaslope ofapproximately −1.4\nfor small avalanches, to a higher slope for large ones. A\nlinewithslopeof −2isshownforcomparisonandappears\nto be consistent with this interpretation.7\n(a) (b)\nFIG. 7: (a) Magnetization versus field for the Ising model wit h damping described in the text. The system size is 323and\nthe two lines represent two runs close to criticality, one wi th a coupling of j= 0.165 and the other of 0 .167. (b) The plot for\nrelaxational dynamics (large damping) with couplings of .21 and.212.\nFIG. 8: The avalanche size distribution, measured of the\nentirehysteresisloopfor α= 0.99(+symbols)and α= 0.9(×\nsymbols). The x-axis is the number of avalanches normalized\nby it’s mean size. The y-axis is the normalized distribution\nof sizes. The less negative sloped straight line is a fit of the\nα= 0.99 curve and has a slope of −1.4. The more strongly\nsloped one has a slope of −2.\nIV. DISCUSSION\nThis paper has introduced a new set of dynamics for\nIsing models that incorporates damping in a way that\nhas not before been achieved. The dynamics that have\nbeen devised have a lot in common with Langevin dy-\nnamics, except they are for discrete rather than contin-\nuous systems. In Langevin equations, a continuous set\nof stochastic differential equations are used to model a\nsystem. It differs from molecular dynamics in that ther-\nmal noise and damping are both added so that the sys-\ntem obeys the correct equilibrium statistics. In the case\nstudied here, we start by considering microcanonical dy-namics5,11whichintroduces auxiliarydegreesoffreedom.\nWe then add damping and thermal noise. Whereas the\nthermal noise is typically Gaussian in the case of the\nLangevin equation, here it must be taken to be of a spe-\ncial exponential form, Eq. 3, in order for it to satisfy the\ncorrect equilibrium statistics.\nThe form of this noise, although quite unusual, can\nbe understood, to some extent qualitatively. For large\ndamping, or small α, the strength of the δfunction be-\ncomes small, and the effect is dominated by the second\ntermwhichis ∝exp(−βn)(forpositive n). Althoughthis\nisnon-Gaussian, ncanbethoughtofasarandomamount\nof positive energy. In the Langevin equation, noise is of-\nten added to a velocity degree of freedom. In terms of\na velocity, the exponential form that we have obtained\nwould correspond to a Gaussian if this was expressed in\nterms of a velocity instead. In the limit of small damp-\ning, where αis close to 1, the effect of the noise becomes\nsmallbecausethefirstterm, whichisto addnonoise, will\ndominate the distribution. This is in accord with what\nhappens in the Langevin equation where if dissipation is\nsmall, little thermal noise is needed to keep the system\nat a given temperature.\nThe fact that it is possible to model damped systems\nin this discrete manner should have many useful applica-\ntions, and is easily extended to other kinds of systems,\naside from Ising models, especially in applications where\ncomputational efficiency is an important criterion.\nThe case of avalanches in magnetic systems is an in-\nteresting nonequilibrium use of these dynamics. Al-\nthoughonemightexpectthatinmostsituations,forlarge\nenough distance and time scales, finite damping will be\nunimportant, physics at smaller scales is still of great in-\nterest and effects at those scales can propagate to larger\nscales. Because damping in real materials can be quite\nsmall, their effects are readily observable experimentally.\nThis work is expected to be important at intermediate\nscales. We haveinvestigatedthephenomenonseenin this\nmodel with varying degrees of damping and found that8\nit makes a qualitative difference to many of the features\nseen on small and intermediate scales. This workis by no\nmeans exhaustive and there are many other effects that\ncan be investigated by straightforward extensions. The\neffect of dipolar interactions in conjunction with damp-\ning could alsobe explored. We havechosento update the\nspinandauxiliaryvariablesatequalfrequencies. Varying\nthis should lead to a different value for the heat diffusion\ncoefficient which should change the quantitative values\nfor length and time scales.\nThe phenomena we have found was in qualitative\nagreement with earlier work using the Landau Lifshitz\nGibbs equations9. As avalanches progress, the effective\ntemperature, which we have seen can be quantified by\n∝angbracketleftei∝angbracketrightat sitei, will increase as energy is released. Thisenergy then diffuses to the surrounding regions, giving\nthosespins the opportunity tolowertheir energyby ther-\nmal activation. This allows avalanches to more easily\nprogress when the damping is small in contrast to relax-\national dynamics, which has effectively infinite damping,\nα= 0. This can lead to some substantial differences in\navalanche morphology, particularly as for small damp-\ning, highly disordered systems can avalanche. At low\nfields this leads to a single avalanche being composed of\nmany disconnected pieces. Experiments have been de-\nvised16that are close experimental realization of the two\ndimensional random field Ising model, and it would be\ninteresting to determine if systems such as this one, or\nsimilar to it, show avalanches with this morphology.\n1S.K.Ma, “Modern Theory of Critical Phenomena”, Fron-\ntiers in Physics, No. 46, Perseus Books (1976).\n2B.I. Halperin, P.C. Hohenberg, and S.K. Ma, Phys. Rev.\nB10, 139 (1974).\n3F.H. de Leeuw, R. van den Doel and U. Enz, Rep. Prog.\nPhys.43, 689 (1980).\n4Q.PengandH.N.Bertram, J. Appl.Phys. 81, 4384 (1997);\nA. Lyberatos, G. Ju, R.J.M. van de Veerdonk, and D.\nWeller, J. Appl. Phys. 91, 2236 (2002).\n5M. Creutz, Ann. Phys 16762 (1986).\n6U. Frisch, B. Hasslacher, and Y. Pomeau, Phys. Rev. Lett.\n56, 1505 (1986).\n7G. Zanetti, Phys. Rev. A 40, 1539 (1989).\n8H.Barkhausen, Z. Phys. 20, 401 (1919); P.J. Cote and\nL.V. Meisel, Phys. Rev. Lett. 671334 (1991); J.P. Sethna,\nK.A. Dahmen and C.R. Myers Nature 410242(2001); B.\nAlessandro, C. Beatrice, G. Bertotti, and A. Montorsi, J.\nAppl. Phys. 682901 (1990); ibid.682908 (1990); J.S. Ur-\nbach, R.C.Madison, andJ.T. Markert, Phys.Rev.Lett. 75\n276 (1995); O. Narayan, Phys. Rev. Lett. 773855 (1996);S. Zapperi, P. Cizeau, G. Durin, and H.E. Stanley, Phys.\nRev. B586353 (1998);\n9J.M. Deutsch and A. Berger, Phys. Rev. Lett. 99, 027207\n(2007)\n10J.P. Sethna, K.A. Dahmen and O. Perkovic, “The Science\nof Hysteresis II” , edited by G. Bertotti and I. Mayergoyz,\nAcademic Press, Amsterdam, 2006, p. 107-179.\n11M. Conti, U. M. B. Marconi and A. Crisanti, Europhys.\nLett.47338 (1999).\n12M. Lax, Phys. Rev. 971419, (1955).\n13J.P. Sethna “Statistical Mechanics Entropy, Order Param-\neters and Complexity” Page 170. Oxford University Press\n(2006),\n14B.M. McCoy and T.T. Wu, “The Two-dimensional Ising\nModel” Harvard University Press, Cambridge, MA, 1973.\n15O.Perkovi´ c., K.Dahmen, J. Sethna.Phys.Rev.B 59, 6106\n(1999).\n16A. Berger, A. Inomata, J. S. Jiang, J. E. Pearson, and S.\nD. Bader, Phys. Rev. Lett. 854176 (2000);"    },    {        "title": "1402.7194v2.Escape_rate_for_the_power_law_distribution_in_low_to_intermediate_damping.pdf",        "content": "arXiv: 1402.7194 \n \nEscape rate for the power-law distribution in low-to-intermediate damping  \n \nZhou Yanjun, Du Jiulin \nDepartment of Physics, School of Science, Tianjin University, Tianjin 300072, China \n \nAbstract ：Escape rate in the low-to-interme diate damping connecting the low \ndamping with the intermediate damping is established for the pow er-law distribution \non the basis of flux over population theory. We extend the escape rate in the low \ndamping to the low-to-intermediate dampi ng, and get an expression for the power-law \ndistribution. Then we apply the escape ra te for the power-law distribution to the \nexperimental study of the excited-state isomerization, and show a good agreement \nwith the experimental value. The extra cu rrent and the improvement of the absorbing \nboundary condition are discussed. \n \nKeywords ：Escape rate; low-to-intermediate da mping; power-law distributions \n \n1. Introduction \nIn 1940, Kramers proposed a thermal escape of a Brownian particle out of a \nmetastable well [1], and according to the very low and intermediate to high dissipative \ncoupling to the bath, he yielded three explic it formulas of the escape rates in the low \ndamping, intermediate-to-high damping (IHD)  and very high damping respectively, \nall of which has been received great attenti ons and interests in physics, chemistry, and \nbiology etc [2-3]. In the IHD region, he got an  expression of escape rate in the infinite \nbarrier (i.e. the barrier height ) and successfully extende d it to high damping \nregion; in the low damping region, he derived a rate in energy diffusion regime; as for \nthe intermediate region, he had not gi ven an expression, which was known as \nKramers turnover problem. Later, plenty of researches had been continued. Carmeli et al derived an expression for the escape rate  in the Kramers model valid for the entire \nfriction coefficient by assuming that the stationary solutions of the low damping and moderate-to-high damping overlap in some  region of phase space and are equal to \neach other (see Eq. (17) in [4] ); B üttiker  et al extended the low damping result to the CBEk\u0015 T\n 1larger range of damping by reconsidering absorbing boundary cond ition at the barrier \nand introducing an extra flux (see Eq.(3.11) in [5] ); Pollak et al got a general \nexpression in non-Markov processes (see Eq.(3.33) in [6]); Hänggi et al introduced a \nsimple interpolation formula (see Eq.(6.1) in [7]) for the arbitrary friction coefficient. \nHowever, it has been noticed that the a bove bridging expressions yield results that \nagree roughly to within  with the numerically precise answers inside the \nturnover region; in higher dimensions a nd for the case of memory friction, these \ninterpolation formulas may eventually fail seriously [7]. At the same time, more attentions need to be paid that the systems studied in above theories are all in thermal \nequilibrium and the distributions all follows  a Maxwell-Boltzmann (MB) distribution, C EEJ>\n20%≤\n()0BEkT\neqEeρρ−= , where E  is the energy, 0ρis the normalization  constant , kB is the \nBoltzmann  constant , and T is the temperature. It should be considered that a complex \nsystem far away from equilibrium has not to  relax to a thermal equilibrium state with \nMB distribution, but often asymptotic ally approaches to a nonequilibrium \nstationary-state with power- law distributions. In these situations, the Kramers escape \nrate should be restudied. B\nIn fact, plenty of the th eoretical and experimental studies have shown that \nnon-MB distributions or power-law dist ributions are quite common in some \nnonequilibrium complex systems, such as in  glasses [8,9], disordered media [10-12], \nfolding of proteins [13], single-molecule  conformational dynamics [14,15], trapped \nion reactions [16], chemical kinetics,  and biological and ecological population \ndynamics [17, 18], reaction–diffusion proces ses [19], chemical reactions [20], \ncombustion processes [21], gene expression  [22], cell reproduc tions [23], complex \ncellular networks [24], sma ll organic molecules [25], and astrophysical and space \nplasmas [26]. The typical forms of such pow er-law distributions include the noted  \nκ-distributions in the solar wind and space plasmas [26,27], the q-distributions in \ncomplex systems within nonextensi ve statistics [28], and the α-distributions noted in \nphysics, chemistry and elsewhere like P(E)~E−α with an index α >0 [16,19,20,25,29]. \nThese power-law distributions may lead to pr ocesses different from those in the realm \n 2governed by Boltzmann-Gibbs statistics w ith MB distributions. Simultaneously, a \nclass of statistical mechanical theories studying the power-law distributions in \ncomplex systems has been constructed, for instance, by ge neralizing Boltzmann \nentropy to Tsallis entropy [28] , by generalizing Gibbsian theory  [30] to a system away \nfrom thermal equilibrium, and so forth. Re cently, a stochastic dynamical theory of \npower-law distributions has been developed by means of studying the Brownian motion in a complex system [31,32], whic h lead the new fluctuation-di ssipation \nrelations (FDR) for power-law distributi ons, a generalized Klein-Kramers equation \nand a generalized Smoluchowski equation. Ba sed on the statistica l theory, one can \ngeneralize the transition state theory (TST) to the nonequilibrium systems with \npower-law distributions [33]; one can study the power-law re action rate coefficient for \nan elementary bimolecular r eaction [34], the mean first passage time for power-law \ndistributions [35], and the es cape rate for power-law dist ributions in the overdamped \nsystems [36].  \nIn this work, the Kramers escape rate  for power-law distributions in the \nlow-to-intermediate damping (LID) will be studied. The paper is organized as follows. In section 2, a generalized escape rate in th e LID region is obtained for the power-law \ndistribution and compared with the results of the low damping Kramers’ escape rate, \nand then we apply our theory to the excited-state isomerization of 2-alkenylanthracene in alkane. Further discussi on of extra current is  given in section 3, \nand finally the conclusion is made in section 4. \n \n2. Escape rate for the power-law distribution in the LID  \n    We have mentioned in the introducti on that Büttiker et al. got a Kramer’s escape \nrate in a wider frictional range on the assumption that the system follows the thermal equilibrium distribution. However, for th e low damping systems, it is always \nnonequilibrium. Because the coupling to the ba th is very weak and the time to reach \nthermal equilibrium is very long in low da mping systems, the escape of particles may \nbe established before thermal equilibrium, and thus nonequilibrium effects dominate \nthe process [37]. Thereby, the nonequilibrium distribution, such as \nκ-distribution, may \n 3be used here. \nLow damping or small viscosity means th at the Brownian forces cause only a \ntiny perturbation in the undamped energy, so it is helpful to replace the momentum by \nthe energy. In the energy region, the Klei n-Kramers equation can be written [3] as \n       ()()() ()=+22 2IIIDtE Eωω ω I\nEρ ργρππ⎛⎞ ∂∂ ∂\n⎜∂∂ ∂ ∂ ⎝⎠ π∂\n⎟,             ( 1 )  \nwhere ωis the angular frequency of osc illation frequency and it satisfies \n()=2 /I dE dI ωπ , D is the diffusion coefficient, γ is the friction coefficient, I is the \naction defined as ()\nEC o n s tIE pdx\n==∫v. In energy space, the continuity equation [3] is \n                       ()=2IJ\ntEω ρ\nπ∂∂−∂∂.                              ( 2 )  \nTake Eq. (2) into E q. (1) and the current J becomes \n()\n() ()22=e x p e x p2DI IIJd EDI E DIω πγ πγρπω ω⎛⎞ ⎛ ∂−− ⎜⎟ ⎜∂⎝⎠ ⎝∫∫dE⎞\n⎟\n⎠         \n() ( )1\n2s\nsDI\nEρρ ωρπ−∂\n=−∂.                                 ( 3 )  \nwheresρis the stationary-state distribution1 2expsIZ dEDπγρω− ⎛=− ⎜⎝⎠∫⎞\n⎟, and Z is the \nnormalization constant. In the previous wor k, we derived the Kramers’ escape rate for \nthe power-law distribution in the low dampi ng, and showed that the stationary-state \ndistribution is the power-law κ-distribution, \n() ( )1 1\n+=1sEZ Eκρκ−−β ,                            ( 4 )  \nif the FDR, (1 2=1DI )Eπγβκ βω−− , is satisfied (see the Appendix in [35]). Thus this \nFDR is a condition under which the κ-distribution can be created from the stochastic \ndynamics of the Langevin equations. When  we take the limit of the power-law \nparameter, , the power-law κ-distribution becomes MB  distribution, and the \nFDR becomes the standard one in the traditional statistics. 0 κ→\nSupposing that the distribution function ()Eρ can be written as the following \n 4form [5], i.e. , and Eq. (3) becomes [35], () () ()s EEρξ ρ = E\n()()()\n2sDI EJEEωξρπ∂=−∂;                        ( 5 )  \nintegrate over E  in both sides of Eq. (5) ( J is treated as a c onstant), we have, \n         () ( )()1\n1 2()C\nBE\nCs\nkTJE E E d EDIπξξ ρω−\n−⎡ ⎤\n=−⎡⎤ ⎢ ⎥ ⎣⎦⎢ ⎥ ⎣ ⎦∫.              ( 6 )  \nIn the low damping systems, the absorbing boundary conditions are  and \n in the bottom and in the well, respectively [35]. Now, in order to extend the \nlow damping regime to the LID regime, th e absorbing boundary cond ition needs to be \nimproved, and an extra current for which the energy E is larger than the barrier \nenergy E() 0CEξ =\n() 1 Eξ =\nC EEJ>\nC is considered [5], i.e. ()0CEξ ≠. Therefore, the continuity equation is \nrewritten as \n                    ()( )=2C EE JJ I\ntEω ρ\nπ> ∂+ ∂−∂∂.                        ( 7 )  \nWhen the system reaches the steady state, and J keep balance, Eq.(7) becomes, \nC EEJ>\nC EEJ J\nEE>∂ ∂=−∂∂,                             ( 8 )  \nand the left side of Eq.(8) is the currentC EE< induced, so it is still Eq.(6); the right of \nEq.(8) comes from the complete integral of the phase-space density multiply velocity \nat the location of the barrie r in the momentum space [5], \n() ()\n() (),,\n        ,      ,C EE C s C\ns CJx v x v v d p\nE Ed E E Eξρ\nαξ ρ∞\n>\n−∞\n∞\n−∞=\n=≥∫\n∫                 ( 9 )  \nwhere the factor αis a constant of order unit.  The reason for introducing αis that in \nthe low damping, the particles move along the orbit of constant energy and the \nphase-space density increases as one moves them away from the barrier and into the \nwell, due to particles boiling up into this energy range. Thus, the actual distribution \n 5function ρ(x,v), at the barrier peak, differs from  the average distribution function  ρ(E). \nThis is taken into account in Eq. (9) by the factor α[3]. \nTake the derivative of the energy for Eq. (5), we have \n() ()()\n2s IDE E J\nEE Z Eωρ ξ\nπ∂ ⎡ ⎤ ∂∂=− ⎢ ⎥∂∂ ∂ ⎣ ⎦.                    ( 1 0 )  \nEq.(9) and Eq.(10) are then brought together into Eq.(8), \n() ()() ()2s\nsID EE EEEωρ ξαξ ρπ∂ ⎡⎤∂= ⎢⎥∂∂⎣⎦.                ( 1 1 )  \nWe use the generalized FDR to go on the further simplification. Because a relatively \nnarrow energy range above E=E C is concerned, I can be taken as sensibly constant, \ni.e.I=IC [5], the frequency, i.e. ω(I)/2π ≈ ω0/2π. In the low damping, the friction \ncoefficient has less effect on the energy, so  it also can be taken as a constant, i.e. γ = γC. \nTherefore, Eq. (11) becomes, \n() ()0 0\n02'' ' 022CCDE DE I\nEω ωπ γξππ ω∂⎛⎞+− − ⎜⎟∂⎝⎠ξαξ=,             ( 1 2 )  \nwithin a small energy range above EC one can assume essentially a constant diffusion \ncoefficient [38], we might as well take \n ()1\n02=1\nC EE CC C DI E() π0\nC EEDE\nE≈∂γβ κ βω−\n≈ − , =∂\n0,            ( 1 3 )  \nhence Eq. (12) turns into  a conventional ordinary differential equation for E, \n()11' ' 'CC C CC IE Iγβ κ β ξ γξ α ξ−− −− =,                ( 1 4 )  \nwhich has a solution, \n()()()\n()()1\n241exp 1 121\n41           exp 1 121C\nCC C\nC\nCC CE EECEI\nE ECEIακ β βξκβ γ β\nακ β β\nκβ γ β⎧⎫ ⎡ ⎤ − ⎪⎪=+ + ⎢ ⎥ ⎨⎬− ⎢ ⎥ ⎪⎪ ⎣ ⎦ ⎩⎭\n⎧⎫ ⎡ ⎤ − ⎪⎪+− + ⎢ ⎥ ⎨⎬− ⎢ ⎥ ⎪⎪ ⎣ ⎦ ⎩⎭,             ( 1 5 )  \nwhere C1and C2 are two integral constants respec tively. The density should be the \ndefinite value when the energy increases, thus the first term of the right side of Eq.(15) \nis abandoned and the solution is written as, \n 6()()( )\n241exp 1 121C\nCC CE EECEIακ β βξκβ γ β⎧ ⎫ ⎡ ⎤ − ⎪ ⎪=− + ⎢ ⎥ ⎨ ⎬− ⎢ ⎥ ⎪ ⎪ ⎣ ⎦ ⎩⎭.          ( 1 6 )  \nMake a substitution, \n()() 1 41 11211C\nC\nCCsEEIακ βκβγβ−⎡ ⎤ −−− + ⎢ ⎥\n⎢=\n⎥ ⎣ ⎦,             ( 1 7 )  \nand Eq.(16) becomes a more convenient form, \n()2sEE Ceβξ = .                               ( 1 8 )  \nIn order to keep ()Eξ continuous at E=E C, assuming Eq.(18) has the following form \n[5], \n() ()( )C sEE\nC EE eβξξ−= .                         ( 1 9 )  \nNext we derive the expression of ()CEξ according to equating the current of E< EC \nand the current of E >EC at E=EC. The current of E >EC is \n()()\n()() ( )()0\n11,2\n 1 C sE E CCs\nC\nCEIEEseDE EJE\nZκ β ξγκβωρ\nκβξ\nπ\n−∂=−∂\n=− − −    ( 2 0 )  \nand the current of E<EC (Ref. [35] Appendix (A.22) where Eb is replaced by EC) is \n()\n()\n()() ( )1\n11\n2\n1  11C\nBC\nE\ns\nkT\nCC\nCCEJ\ndEDI\nIEE .Zκξ\nπρω\nγκξκ β−−=\n−=− − ⎡⎤⎣⎦∫             ( 2 1 )  \nEquating the above two expressions at  E=EC gives \n()1( 1\n1( )C\nCsEEs) κ βξκ− +=−+.                           ( 2 2 )  \nOne of the most significant escape rate th eories is the flux over population theory \n[2,3,7]. If the steady-state current J and the (nonequilibrium) population inside the \ninitial domain n are got, the rate of escape is then given by the ratio, k =Jn. Hence, \naccording to the appendix Eq.(A.9) in Ref.[35], the escape rate for the power-law \ndistribution can be finally obtained as \n 7() ()()\n()() ( )TST11\n+2 14111\n4111 1C\nCC\nC CCC\nCCII\nkk\nIE\nEEκ κγακ β\nγβ\nακβκ\nβκκ βγβκ\n−⎡⎤\n−+ ⎢⎥\n⎢⎣−+−\n−+− −=\n−⎥⎦.         ( 2 3 )  \nwhere is the TST rate for the power-law distribution if we choose VTSTkκ− a=0 and \nκ =ν−1 (see Eq. (63) and Eq. (64) in [33] ). In the limit κ→0, Eq. (23) can return to \nthe traditional result of the escape rate fo r MB distribution (see Eq.(3.11) in [5]), \n 0\nBHL2+1411\n41C C C CC\nCCE I Ik\nIβ ωγ β\nπα\nγβ\nα\nγβ−+−\n+= e .                 ( 2 4 )  \nAs the friction coefficient tends to zero γC→0, Eq.(23) reduces to \n( )( )1\n0112CC\nCIk E κκκγω β κχκ βπ−= − ,                  ( 2 5 )  \nw i t h            \n()\n()22\n2211 1,  2< <02\n13 1+ +1 ,     >02κκκκκχ\nκκκκ⎧ ⎛⎞ ⎛ ⎞−Γ − Γ − − − ⎜⎟ ⎜ ⎟ ⎪⎪⎝ ⎠ ⎝ ⎠=⎨⎛⎞ ⎛ ⎞ ⎪ΓΓ⎜⎟ ⎜ ⎟⎪ ⎝⎠ ⎝ ⎠ ⎩      \nwhich coincides with the Kramers escape ra te in the low damping for the power-law \ndistribution (see Appendix (A.14) in [35]);  therefore Eq.(23) contains the low \ndamping region. When friction co efficient tends to infinity  γC→∞, Eq.(23) becomes \n() ()1\n0\nTST 12C kE κκκ κ1kαωχκ β α κπ− =− = + .                ( 2 6 )  \nIn Fig.1, we plot the low damping, LI D and IHD rates for the power-law \ndistribution, which are all normalized by th e TST rate for MB distribution. The IHD \nrate for the power-law distribution is derived in the Appendix  (see Eq.(A.15)). We see \nfrom Fig.1 that two solid curves of the LID with MB distribution and power-law \ndistribution almost overlap with the one s of the low damping, which has been \nexplained in the above deduction. As the damping γ  increases, the traditional curves \nof the LID and IHD intersect at about γ =0.8 and the ratio of the rate normalized by \nthe TST result is about BHL TST 0.93 kk = ; whereas they in tersect at about  γ =5 for the \n 8power-law distribution a nd the ratio is about kκ /kTST=0.64. When the larger \npower-law parameter κ is taken, the intersection point will move to the higher \ndamping. With much higher damping  γ , two solid curves both approach to the TST \nrate deviating from each curv e of IHD. Though the result of  the LID overestimates the \nrate in the very higher damping, the transition from the low damping to LID is \nreasonably achieved and predicts a lower escape rate than the Kramers results.  \n \n \nFig.1. Theoretical estimation of the escape rate normalized by the TST result in three \ndamping ranges for power-law parameter κ =−0.28 , the barrier height EC = 5.77 kBT, the \nfrequency of the barrier ωB\nC =5.45, the mass  of particles m=1, β=1 and α=1 [5, 38]. kL \nand kM are the Kramers’ low damping rate and IHD rate respectively. The inset showed \nthe enlarged parts for the rate in the low damping. \n \nThen we apply our result to the experiment and check the prediction. Hara et al \nstudied the Kramers turnover behavior fo r the excited-state isomerization of \n2-alkenylanthracene in alkane at the high pr essure [39]. The experimental material \nwas 2-(2-propenyl) anthracene (22PA), synthesized using the method of Stolka et al. \n[40] and purified by TLC. Steady-state a nd time-resolved fluorescence spectra in \nsupercritical (SC) ethane  (99.95%) and SC CO2 (99.999%) were measured at 323 K \nand at pressures up to 15.1 and 17.4 MPa respec tively. FIG.3 of [39] indicated a clear \ndemonstration of the Kramers turnover behavior with increasing the viscosity. At the same time, the interaction (i.e. dynamic solv ent effect) between th e solute and solvent \n 9was also studied (see FIG.5 in [39]), and the consequence can be well explained by \nour LID result. These parameters we adopt in Fig.1 keep the same with the \nexperimental data in [39] , i.e. activation energy, E0=5.77k BT, the mass of particle, \n3.223×10-25kg, and the barrier top frequency, 5.45bmC ω ω = = . At the turning point, \nour result kκ /kTST=0.64 with the power-law parameter κ=−0.28  agrees with the \nexperimental value  (see Table ΙΙ in [39]). It is therefore \nconcluded that our theory represents excelle ntly the experimental result as compared \nto the traditional theory. max TST (/ ) 0 . 6fkk κ == 4\n \n3. Further discussion of extra current \nIn Section 2, an extra current  is introduced. The problem naturally arises \nwhether it exists or not a nd how the friction coefficien t affects it. Now we make \nfurther discussion. First we calculate the probability P(E)dE that an escaping particle \nhas an energy E between E and E+dE. The current J that E<EC EEJ>\nC produces is zero due to \nthe steady-state distribution; the excess energy ∆E=E-E C is introduced and the \nprobability is \n()()\n()\n()()\n()1\n1\n1\n0           1\n11.\n1C\nCC\nC\nsE EEE\nE\nE\nCE\ns\nCE\nedJ Ed EPEd EJJEd E\ndE\ndEE\nEEeEβ κ\nβκ\nκκβ\nκβκβαρ\nαρ\nκβ−>\n∞\n>\n∞\nΔ==+\n=\n⎛−\nΔ−−−⎞Δ ⎜⎟⎝⎠∫\n∫         ( 2 7 )  \nThe average of the excess energy is then given by \n  () ()\nCEEE P E d E∞\nΔ= Δ ∫1\n1\n0\n011.\n() 11 \n()E s\nC\nsE\nCE\ndE\ndEEeE\nEeEκ\nβ\nβκκβ\nκβ\nκβ\nκβΔ\n∞\n∞\nΔ⎛⎞Δ ⎜⎟⎝⎠=Δ\n⎛⎞ΔΔ−−\nΔ−−⎜⎟⎝⎠∫\n∫        ( 2 8 )  \nWe do numerical integral about Eq. (28) a nd plot the average energy in the extremely \nlow damping and the extremely high damping, respectively, for different power-law \nparameters; other parameters are taken as α =1 [5, 38] and βEC=10. \n 10 \n \n(a) Extremely low damping, 0Cγ→            (b) Extremely high damping, Cγ→∞ \nFig.2 The influence of the friction coefficient on the average energy for different power-law \nparameters \n \nIn Fig.2, with decreasing damping the average energy of escaping particles \ndecreases to zero both for the MB  distribution and the power-law κ-distribution. So \nthere does not exist the extra current in extremely low damping, i.e. ρ(E\nC EEJ> C)=0 \nwhich Kramers had ever assumed, and the Kramers low damping rate corresponds to \nthe underdamped case. For extremely high da mping, the average energy approaches \nthe constant independent of the friction coefficient in both cases, and thus there \ndefinitely exists the extra current . Thereby, we get a conclusion that when the \ndamping is extremely low, the absorbing boundary condition at the barrier which was \nalways used in the past is right; once the damping is not very low, the absorbing \nboundary condition becomes an approximation and then needs to be improved by \ntaking the extra current into account. C EEJ>\nC EEJ>\n4. Conclusion \nMany physical, chemical and biological systems are complex, and usually open \nand nonequilibrium. In fact, a complex syst em far away from equilibrium does not \nhave to relax to thermal equilibrium with a MB distribution, but often asymptotically \napproaches a stationary nonequi librium with a power-law distribution. Therefore, the \nescape rate theory should be reestablished under the framework of  the statistics of \npower-law distributions. According to the flux over population theory, we have \n 11extended the result in the low damping to a wider range of the friction coefficient by \nimproving the absorbing boundary condition, and get the expression of escape rate in \nthe low-to-intermediate damping (LID) for the power-law κ-distribution. When the \ndamping is extremely low, it returns to the Kramers escape rate in the low damping; when the damping is extremely high, it reduces to the TST rate.  \nWe have applied our theory  to the experimental st udy of the excited-state \nisomerization of 2-alkenylanthracene in alkane, checked the prediction and concluded that the result was a good agreement with th e experimental value. Furthermore, we \nhave made the numerical analyses and further discussions about the extra current. \n \nAppendix \nParticles move in the IHD systems and th e process is governed by the Klein- \nKramers equation [3], \n       ()() () ,,dV x pxp Dxptm x p d x p pρ ρργρ⎛⎞ ⎛ ∂∂ ∂ ∂ ∂=− + + + ⎜⎟ ⎜⎞\n⎟∂ ∂∂ ∂ ∂⎝⎠ ⎝⎠.         ( A 1 )  \nIn Eq.(A1), if the coefficients (),Dxp and (),xp γ satisfy the generalized fluctuation- \ndissipation relation given [31] by \n                        ( )1= 1Dm E γβκ β−− ,                         ( A 2 )  \nthen the stationary-state solution is the power-law κ-distribution, \n                     () ( )1 1\n+=1sEZ Eκρκ−−β ,                       ( A 3 )  \nfor the energy E. In the limit , the distribution returns to the MB distribution. 0 κ→\nSupposing the barrier is located atCxand the potential can be expanded as a \nTaylor series about xC, one can write taking the barrier top as the zero of the potential \n[3], \n                      (2 2\n11=2CC Vx ω−− )x.                          ( A 4 )  \nNear the bottom of the well, Ax(0Ax≈), the potential is approximated by \n                       22\n2=+A VV x ω −Δ 2,                          ( A 5 )  \n 12where . Take Eq.(A4) into Eq.(A1), and Eq.(A1) becomes () (=C VVx VxΔ− )A\n         2'0'Cxp p Dpx p pρωρ γ ρ⎛ ∂∂ ∂ ∂+− + ⎜∂∂ ∂ ∂ ⎝⎠ρ⎞=⎟ ,                ( A 6 )  \nwhere 'C xxx≡− . Now make the substitution [3] and take the power-law steady-state \nsolution at the barrier, \n            () () ( )122 2', ', 1 ' 2s xp xp p xCκ\nρξ ρ ξ κ β ω ⎡ ⎤ ≡= − −⎣ ⎦,          ( A 7 )  \nsubstituted into Eq.(A6), combining Eq.(A2) to simplify, we have \n           2\n2\n2'( )'CDxp p Dpx p ppξξ ξ ξωγ∂∂ ∂ ∂∂++ − − =∂∂ ∂ ∂∂0.              ( A 8 )  \nHere we adopted a special case, i.e. assu me the friction coefficient is a constant, γ =γC, \nbut the diffusion coefficient is a function of the energy, /C Dp p κγ ∂∂= − , then Eq.(A8) \nbecomes \n             ( ) { }2'[ 1 ] ' ' ' 0CCxa p Dωκ γ ξ ξ−+ − + + = .                ( A 9 )  \nTo solve this equation, one wish es to write the coefficients 'ξand ''ξ in terms of \nthe single variable, ', rather than x' and p [3], where a is an undetermined \nconstant. It can be achieved in a very neat way if one writes up a x≡−\n( ) ( ) ( )2'1 1CC  C xap a ωκ γ κ γ−+ − + = − + u ⎡ ⎤⎣ ⎦,               ( A 1 0 )  \nwhich imposes on a the condition: \n              () ()2 2 111122CC a κγ κ γ ω±=+± + +24C.                 ( A 1 1 )  \nEq.(A9) then takes the form of a conven tional ordinary diff erential equation in u, \n            ()\n()+1 ''' = 'C auduDu\nuCe d uκγ\nξ−⎡⎤⎣⎦−∫∫,                         ( A 1 2 )  \nwhere C is an integral constant and a takes a+so as to make the distribution finite.  \nThe probability current J crossing the barrier can be  obtained by integrating for \npρ over p from minus infinity to infinity, \n 13()\n()12'1121\n--1'2C\npCaxp dx\nx pJ p dp C p dp dp eκγκ\nβγ κ βρκ β−−+⎡⎤⎣⎦− ∞∞\n−\n∞∞ −∞∫ ⎛⎞==− ⎜⎟⎝⎠∫∫ ∫.        ( A 1 3 )  \nWhile the number of particles n trapped near the minimum A is \n  ()()\n()11''' 21C\npapdpDp\nAVnC d p eκγ\nκκπχ κβ\nκβω−+⎡ ⎤⎣ ⎦ ∞−\n−∞+Δ ∫= ∫.              ( A 1 4 )  \nThe probability of the escape is therefore th e number crossing the saddle line in unit \ntime divided by the number in the well, \n()()\n()\n()\n()12 '11212\nTST 1'''11 '2C\npC\nC\npaxp dx\nx\napdpDpppd p e d p\nJkkn\ned pκγκ\nβγ κ β\nκ κγβκ κ β−−+⎡⎤⎣⎦− ∞\n−\n−∞ −∞\n− −+⎡⎤⎣⎦ ∞−\n−∞∫ ⎛⎞+− ⎜⎟⎝⎠==\n∫∫∫\n∫.   (A15) \n \nAcknowledgments \nThis work is supported by the Nationa l Natural Science Foundation of China \nunder Grant No 11175128 and by the Higher School Specialized Research Fund for \nDoctoral Program under Grant No 20110032110058. \n \nReferences \n[1] H. A. Kramers, Physica (Utrecht) 7 (1940) 284. \n[2] C W Gardiner, handbook of stochastic methods for physics, chemistry and the \nNatural Sciences,  Springer, 2004.  \n[3] W. T. Coffey, Y . P. Kalmykov and J. T. Waldron, The Langevin Equation: With \nApplications to Stochastic Problems in Physics Chemistry and Electrical \nEngineering, Singapore: World Scientific, 2004. \n[4] B. Carmeli and A. Nitzan, Phys. Rev. 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Pearson, Macromolecules 9 (1976) 710. \n 15"    },    {        "title": "0802.0581v1.Perspectives_on_Lorentz_and_CPT_Violation.pdf",        "content": "arXiv:0802.0581v1  [gr-qc]  5 Feb 2008IUHET 513, September 2007\nPERSPECTIVES ON LORENTZ AND CPT VIOLATION\nV. ALAN KOSTELECK ´Y\nPhysics Department, Indiana University\nBloomington, IN 47405, U.S.A.\nThis talk offers some comments and perspectives on Lorentz an d CPT violation.\n1. Introduction\nLorentz symmetry is the invariance of physical laws under rotation s and\nboosts. As a global symmetry over Minkowski spacetime, it underlie s the\ntheory of Special Relativity and the Standard Model of particle phy sics,\nwhere it is intimately tied to CPT invariance. As a local symmetry of\nfreely falling frames, it is an essential component of General Relativ ity.\nNonetheless, the possibility exists that nature may exhibit tiny violat ions\nof Lorentz symmetry. This talk presents some perspectives on th e basic\nmotivations and reasoning in this subject.\nSince Lorentz symmetry has been verified in many experiments, as h as\nCPT invariance, it is reasonable to ask why relativity violations are wor th\nconsidering. A sufficient theoretical motivation is the need for a con sistent\ndescription of Lorentz and CPT violation to offer guidance for exper imen-\ntal tests. However, a stronger motivation is the prospect that L orentz and\nCPT violation can serve as a sensitive potential signal for physics at the\nPlanck scale. In fact, the present interest in the subject was trig gered by\nthe realization that natural mechanisms for Lorentz and CPT violat ion\nexist in unified theories at the Planck scale.1The large range of existing\nphenomenological and experimental activities stems from the applic ation\nof effective field theory2and the construction of the Standard-Model Ex-\ntension (SME)3,4to catalogue and predict observable effects.\n2. Approaches and origins\nThe study of Lorentz and CPT violation can be approached on three dis-\ntinct levels. First, at the level of fundamental theory at the unific ation\n12\nscale, one can investigate possible mechanisms and determine their f ea-\ntures and implications. Second, at the level of theory at accessible scales,\none can seek a description of the resulting effects that is quantitat ive, gen-\neral, and compatible with the established physics of the Standard Mo del\nand General Relativity. Finally, at the level of observation and expe riment,\none can study and perform both high sensitivity tests and broad se arches,\npreferably ones that are feasible with existing or near-future tec hnology.\nThefirstpointtoestablishiswhetherLorentzviolationcanindeedoc cur\nin a fundamental theory. Without at least one viable mechanism, the inter-\nest of the idea would be much reduced. A plausible origin for Lorentz v iola-\ntion has been identified in string field theory,1which has interactions with\na generic structure that could in principle trigger spontaneous viola tion of\nLorentzsymmetryand generatevacuum expectationvalues forL orentzten-\nsor fields. More recently, numerous other mechanisms for Lorent z violation\natthefundamentallevelhavebeenproposedincluding, forexamp le,onesin-\nvolvingnoncommutativefield theories,5spacetime-varyingfields,6quantum\ngravity,7random-dynamics models,8multiverses,9brane-world scenarios,10\nsupersymmetry,11and massive gravity.12\nGiven that mechanisms exist for Lorentz and CPT violation in an un-\nderlying theory, it is natural to ask about the consequences for p hysics at\nobservable scales. In particular, the question arises as to the bes t method\nto develop a description of the possible effects.\n3. Describing Lorentz violation\nSome key criteria offer valuable guidance in the search for a suitable t heo-\nretical framework for describing Lorentz violation at attainable sc ales. The\nfirst iscoordinate independence. It hasbeen accepted since long before 1905\nthat the physics of a system should be independent of a change of o bserver\ncoordinates. This holds whether a coordinate change is implemented via a\nLorentz transformation or in any other way.\nThe second is realism. Since 1905, when virtually no fundamental parti-\ncles were known and quantum physics was at its dawn, thousands of people\nhave invested millions of person-hours and billions of dollars in establish -\ning the Standard Model of particle physics and General Relativity as an\naccurate description of nature. To be of real interest nowadays , any pro-\nposed theoretical framework for Lorentz violation must incorpor ate this\nwell-established physics.\nThe third is generality. No compelling evidence for Lorentz violation3\nexists at present. Physics is therefore currently in the position of searching\nfor a violation, as opposed to attempting to understand an observ ed effect.\nIn the searching phase, it is desirable to have the most general pos sible\nformulationsothatnoregionisleftunexplored. Thisisinstrongcont rastto\nthe modeling phase, where considerations such as simplicity are impor tant\nin attempts to understand a known effect.\nArmed with these criteria, we can follow the basic reasoning that lead s\nto the application of effective field theory and the construction of t he SME.\n3.1.Modified Lorentz transformations\nSince the essential content of Special Relativity is the idea that phy sics\nis invariant under Lorentz transformations, the most obvious app roach to\ndescribing relativity violations is to investigate modifications of the Lo rentz\ntransformations. In fact, the literature since 1905 abounds with variousad\nhocproposals of this type. However, independent of any specific prop osal,\nthis approach has some serious disadvantages.\nOne is that a textbook Lorentz transformation acts on the obser ver and\ntherefore corresponds merely to a change of coordinates, i.e., a c hange of\nreference frame. However, according to the above criterion of c oordinate\nindependence, a frame change cannot have physical implications by itself.\nThe key feature of Special Relativity is really the requirement that t he\nequations for the system being observed must be covariant under a Lorentz\ntransformation,whichintrinsicallyassumesthatLorentzsymmetr yisexact.\nThis approach is therefore problematic for investigating violations.\nIt is of coursepossible to construct special models imposing form co vari-\nance of the system under some ad hocalternative transformation. However,\nany specific such proposal runs counter to the criterion of gener ality. More-\nover, some kinds of violations are difficult and perhaps even impossible to\ncountenance via this approach. For example, Lorentz violation in na ture\nmight well be particle-species dependent, but it is very challenging to for-\nmulate a description of this flavor dependence based on modified Lor entz\ntransformations of the observer. The criterion of realism presen ts a further\nsubstantial obstacle, since it is awkward at best to implement such m odels\nin the context of the Standard Model and General Relativity.\n3.2.Modified dispersion laws\nThe above discussion suggests that a general and realistic investig ation\nof Lorentz violation is most naturally performed directly in terms of t he4\nproperties of a system rather than via modifications of the Lorent z trans-\nformations. A simple implementation of this is to study modifications of\nparticle dispersion laws. However, this also suffers from serious dra wbacks.\nOneissueinvolvesthecriterionofgenerality. Modificationsofdisper sion\nlaws can only describe changes in the free propagation of particles a nd\nperhaps also partially account for interaction kinematics. However , physics\nis far more than free propagation, and this approach therefore d isregards a\nlarge range of interesting Lorentz-violating effects involving intera ctions.\nThere are also various issues associated with the choice of modifica-\ntions to the dispersion law. Not all choices are compatible with desirab le\nfeatures such as originating from an action. Also, meaningful phys ical mea-\nsurements must necessarily compare two quantities, so some choic es may\nbe unphysical. In particular, calculations with a modified dispersion law\nthat yield apparent changes of properties are insufficient by thems elves to\ndemonstrate physical Lorentz violation. A simple example of a modifie d\ndispersion law with no observable consequences in Minkowski spacet ime\nis3pµpµ=m2+aµpµ, whereaµis a prescribed set of four numbers in a\ngiven frame. Direct calculations with this dispersion law appear to give\nLorentz-violating properties that depend on the preferred vect oraµ, but in\nfact they are unobservable because aµcan be eliminated via a physically\nirrelevant redefinition of the energy and momentum. The observab ility of\nmodifications to a dispersion law can be challenging to demonstrate.\n3.3.Effective field theory and the SME\nWe seethat the desiderataforasatisfactorydescriptionofLore ntzviolation\ninclude a comprehensive treatment of free and interacting effects in all\nparticle species. Remarkably, a model-independent and general ap proach\nof this type exists.\nThekey isto takeadvantageofthe ideathat Lorentzviolationat at tain-\nable energies can be described using effective field theory, independ ent of\nthe underlying mechanism.2,13Starting from the Standard Model coupled\nto General Relativity, we can add to the action all possible scalar ter ms\nformed by contracting operators for Lorentz violation with coeffic ients that\ncontrol the size of the effects. The operators are naturally orde red accord-\ning to their mass dimension. The resulting realistic effective field theor y\nis the SME.3,4Since CPT violation in realistic field theories comes with\nLorentz violation,14the SME also incorporates general CPT violation.\nBy virtue of its construction, the SME satisfies the three guiding cr ite-5\nria of coordinate invariance, realism, and generality. Moreover, it h andles\nsimultaneously all particle species, including both propagation and int er-\naction properties, so its equations of motion contain all action-com patible\nmodifications ofrealistic dispersionlaws. The coordinateinvarianceim plies\nthat physics is unaffected by observerframe changes, including Lo rentz and\nother transformations, while particle transformations can produ ce observ-\nable effects of Lorentz violation.\nThe primary disadvantage of the SME approach is its comparative co m-\nplexity and the investment required to become proficient with its use . How-\never, this is outweighed by its advantages as a realistic, general, an d cal-\nculable framework for describing Lorentz violation. Judicious choice s es-\ntablishing relations among the SME coefficients for Lorentz violation y ield\nelegant and simple models that can serve as a theorist’s playground, while\nthe general case offers guidance for broad-based experimental searches.\n4. Gravity and Lorentz violation\nThe SME allows for both global3and local4Lorentz violation, and inter-\nesting effects arise from local Lorentz violation in the gravitational context.\nIn general, local Lorentz violation can be understood as arising whe n a\nnonzero coefficient tabc...for Lorentz violation exists in local freely falling\nframes.4The cofficient tabc...can be converted to a coefficient tλµν...on the\nspacetime manifold using the vierbein ea\nµ.\nOne result is that spontaneous violation of local Lorentz symmetry is\nalways accompanied by spontaneous diffeomorphism violation, and vic e\nversa.15A nonzero coefficient tabc...is the vacuum value of a local Lorentz\ntensor field, and it implies Lorentz violation because it is invariant inste ad\nof transforming like a tensor under particle transformations. The vierbein\nensures that there is a corresponding spacetime tensor field with v acuum\nvaluetλµν...on the spacetime manifold, which in turn implies spontaneous\ndiffeomorphism breaking because it is invariant instead of transform ing like\na tensor under particle diffeomorphisms.\nA more surprising result is that explicit Lorentz violation is generically\nincompatible with Riemann geometry.4Explicit violation occurs when the\nSME coefficients are externally prescribed, but the ensuing equatio ns of\nmotion turn out to be inconsistent with the Bianchi identities. This re sult\nalso holds in Riemann-Cartan spacetime. However, spontaneous vio lation\nevades the difficulty because it generates the SME coefficients dyna mically,\nthereby ensuring compatibility with the underlying spacetime geomet ry.6\nSpontaneouslocalLorentzviolationis accompaniedby up to10Namb u-\nGoldstone (NG) modes.15With a suitable choice of gauge, these modes can\nbeidentifiedwith componentsofthevierbeinnormallyassociatedwith local\nLorentz and diffeomorphism gauge freedoms. The physical role of t he NG\nmodes varies, but in general they represent long-range forces t hat can be\nproblematic for phenomenology. However, in certain models the NG m odes\ncan be interpreted as photons, thus offeringthe intriguing prospe ct that the\nexistenceoflightcouldbeaconsequenceofLorentzviolationinstea doflocal\nU(1) gauge invariance, with concomitant observable signals.15,16A similar\ninterpretation is possible for the graviton.17Other potential experimental\nsignals arise from NG modes in the gravity18and matter19sectors, and\nfrom torsion.20The spectrum of vacuum excitations typically also includes\nmassive modes that may lead to additional observable effects.21\n5. The search for signals\nThe SME predicts some unique signals, such as rotational, sidereal, a nd\nannual variations. The effects are likely to be heavily suppressed, p erhaps\nas some power of the ratio of an accessible scale to the underlying sc ale,\nbut they could be detected using sensitive tools such as interferom etry.\nFor example, meson interferometry offers the potential to identif y flavor-\nand direction-dependent energy shifts of mesons relative to antim esons,22\nwhile exquisite interferometric sensitivity to polarization-dependen t effects\nofphotonsisattainedusingcosmologicalbirefringence.23Conceivably,SME\neffects might even be reflected in existing data, such as those for fl avor\noscillationsofneutrinos.24Overall,animpressiverangeofsensitivitiesinthe\nmatter, gauge, and gravitational sectors of the SME has been ac hieved.25\nDespite a decade of intense activity, most of the SME coefficient spa ce\nis still unexplored by experiments, and many basic theoretical issue s are\nunaddressed. The study of relativity violations remains fascinating , with\nthe enticing prospect of identifying a signal from the Planck scale.\nAcknowledgments\nThis work was supported in part by DoE grant DE-FG02-91ER40661 and\nNASA grant NAG3-2914.\nReferences\n1. V.A. Kosteleck´ y and S. Samuel, Phys. Rev. D 39, 683 (1989); V.A. Kost-\neleck´ y and R. Potting, Nucl. Phys. B 359, 545 (1991).7\n2. V.A. Kosteleck´ y and R. Potting, Phys. Rev. D 51, 3923 (1995).\n3. D. Colladay and V.A. Kosteleck´ y, Phys. Rev. D 55, 6760 (1997); Phys. Rev.\nD58, 116002 (1998).\n4. V.A. Kosteleck´ y, Phys. Rev. D 69, 105009 (2004).\n5. See, for example, I. Mocioiu et al., Phys. Lett. B 489, 390 (2000); S.M.\nCarrollet al., Phys. Rev. Lett. 87, 141601 (2001).\n6. V.A. Kosteleck´ y et al., Phys. Rev. D 68, 123511 (2003).\n7. See, for example, G. Amelino-Camelia et al., AIP Conf. Proc. 758, 30 (2005);\nN.E. Mavromatos, Lect. Notes Phys. 669, 245 (2005); Y. Bonder and D.\nSudarsky, arXiv:0709.0551.\n8. C.D. Froggatt and H.B. Nielsen, hep-ph/0211106.\n9. J.D. Bjorken, Phys. Rev. D 67, 043508 (2003).\n10. See, for example, C.P. Burgess et al., JHEP0203, 043 (2002).\n11. M. Berger and V.A. Kosteleck´ y, Phys. Rev. D 65, 091701(R) (2002); P.A.\nBolokhov et al., Phys. Rev. D 72, 015013 (2005).\n12. See, for example, G. Dvali et al., Phys. Rev. D 76, 044028 (2007); D.S.\nGorbunov and S.M. Sibiryakov, JHEP 0509, 082 (2005); M.V. Libanov and\nV.A. Rubakov, JHEP 0508, 001 (2005); N. Arkani-Hamed et al., JHEP\n0507, 029 (2005); V.A. Kosteleck´ y and S. Samuel, Phys. Rev. D 42, 1289\n(1990); Phys. Rev. Lett. 66, 1811 (1991).\n13. V.A. Kosteleck´ y and R. Lehnert, Phys. Rev. D 63, 065008 (2001).\n14. O.W. Greenberg, Phys. Rev. Lett. 89, 231602 (2002).\n15. R. Bluhm and V.A. Kosteleck´ y, Phys. Rev. D 71, 065008 (2005).\n16. B. Altschul and V.A. Kosteleck´ y, Phys. Lett. B 628, 106 (2005).\n17. V.A. Kosteleck´ y and R. Potting, Gen. Rel. Grav. 37, 1675 (2005).\n18. Q.G. Bailey and V.A. Kosteleck´ y, Phys. Rev. D 74, 045001 (2006).\n19. V.A. Kosteleck´ y and J.D. Tasson, in preparation.\n20. V.A. Kosteleck´ y, N. Russell, and J.D. Tasson, arXiv:07 12.4393.\n21. R. Bluhm et al., arxiv:0712.4119; V.A. Kosteleck´ y and S. Samuel, Phys. Re v.\nD40, 1886 (1989); Phys. Rev. Lett. 63, 224 (1989).\n22. H. Nguyen (KTeV), hep-ex/0112046; A. Di Domenico et al.(KLOE), these\nproceedings; B. Aubert et al.(BaBar), hep-ex/0607103; arXiv:0711.2713;\nD.P. Stoker (BaBar), these proceedings; J.M. Link et al.(FOCUS), Phys.\nLett. B556, 7 (2003); V.A. Kosteleck´ y, Phys. Rev. Lett. 80, 1818 (1998);\nPhys. Rev. D 61, 016002 (2000); Phys. Rev. D 64, 076001 (2001).\n23. V.A. Kosteleck´ y and M. Mewes, Phys. Rev. Lett. 87, 251304 (2001); Phys.\nRev. D66, 056005 (2002); Phys. Rev. Lett. 97, 140401 (2006); Phys. Rev.\nLett.99, 011601 (2007).\n24. L.B. Auerbach et al., Phys. Rev. D 72, 076004 (2005); B.J. Rebel and S.F.\nMufson, these proceedings; V.A. Kosteleck´ y and M. Mewes, P hys. Rev. D\n69, 016005 (2004); Phys. Rev. D 70, 031902(R) (2004); Phys. Rev. D 70,\n076002 (2004); T. Katori et al., Phys. Rev. D 74, 105009 (2006); V. Barger\net al., Phys. Lett. B 653, 267 (2007); K. Whisnant, these proceedings.\n25. Results are tabulated in V.A. Kosteleck´ y and N. Russell , arXiv:0801.0287."    },    {        "title": "2009.07620v1.Fast_convex_optimization_via_inertial_dynamics_combining_viscous_and_Hessian_driven_damping_with_time_rescaling.pdf",        "content": "Noname manuscript No.\n(will be inserted by the editor)\nFast convex optimization via inertial dynamics combining\nviscous and Hessian-driven damping with time rescaling\nHedy ATTOUCH \u0001A \u0010cha BALHAG \u0001Zaki\nCHBANI \u0001Hassan RIAHI\nthe date of receipt and acceptance should be inserted later\nAbstract In a Hilbert setting, we develop fast methods for convex unconstrained\noptimization. We rely on the asymptotic behavior of an inertial system combining\ngeometric damping with temporal scaling. The convex function to minimize enters\nthe dynamic via its gradient. The dynamic includes three coe\u000ecients varying with\ntime, one is a viscous damping coe\u000ecient, the second is attached to the Hessian-\ndriven damping, the third is a time scaling coe\u000ecient. We study the convergence\nrate of the values under general conditions involving the damping and the time\nscale coe\u000ecients. The obtained results are based on a new Lyapunov analysis and\nthey encompass known results on the subject. We pay particular attention to the\ncase of an asymptotically vanishing viscous damping, which is directly related to\nthe accelerated gradient method of Nesterov. The Hessian-driven damping signi\f-\ncantly reduces the oscillatory aspects. As a main result, we obtain an exponential\nrate of convergence of values without assuming the strong convexity of the objec-\ntive function. The temporal discretization of these dynamics opens the gate to a\nlarge class of inertial optimization algorithms.\nKeywords damped inertial gradient dynamics; fast convex optimization;\nHessian-driven damping; Nesterov accelerated gradient method; time rescaling\nMathematics Subject Classi\fcation (2010) 37N40, 46N10, 49M30, 65K05,\n65K10, 90B50, 90C25.\nHedy ATTOUCH\nIMAG, Univ. Montpellier, CNRS, Montpellier, France\nhedy.attouch@umontpellier.fr,\nSupported by COST Action: CA16228\nA \u0010cha BALHAG \u0001Zaki CHBANI\u0001Hassan RIAHI\nCadi Ayyad University\nS\u0013 emlalia Faculty of Sciences 40000 Marrakech, Morroco\naichabalhag@gmail.com \u0001chbaniz@uca.ac.ma \u0001h-riahi@uca.ac.maarXiv:2009.07620v1  [math.OC]  16 Sep 20202 Hedy ATTOUCH et al.\n1 Introduction\nThroughout the paper, His a real Hilbert space with inner product h\u0001;\u0001iand\ninduced normk\u0001k;andf:H!Ris a convex and di\u000berentiable function. We aim\nat developping fast numerical methods for solving the optimization problem\n(P) min\nx2Hf(x):\nWe denote by argminHfthe set of minimizers of the optimization problem ( P),\nwhich is assumed to be non-empty. Our work is part of the active research stream\nthat studies the close link between continuous dissipative dynamical systems and\noptimization algorithms. In general, the implicit temporal discretization of con-\ntinuous gradient-based dynamics provides proximal algorithms that bene\ft from\nsimilar asymptotic convergence properties, see [28] for a systematic study in the\ncase of \frst-order evolution systems, and [5,6,8,11,12,19,20,21] for some recent\nresults concerning second-order evolution equations. The main object of our study\nis the second-order in time di\u000berential equation\n(IGS)\r;\f;bx(t) +\r(t) _x(t) +\f(t)r2f(x(t)) _x(t) +b(t)rf(x(t)) = 0;\nwhere the coe\u000ecients \r;\f: [t0;+1[!R+take account of the viscous and Hessian-\ndriven damping, respectively, and b:R+!R+is a time scale parameter. We take\nfor granted the existence and uniqueness of the solution of the corresponding\nCauchy problem with initial conditions x(t0) =x02H, _x(t0) =v02H. Assuming\nthatrfis Lipschitz continuous on the bounded sets, and that the coe\u000ecients are\ncontinuously di\u000berentiable, the local existence follows from the nonautonomous\nversion of the Cauchy-Lipschitz theorem, see [24, Prop. 6.2.1]. The global existence\nthen follows from the energy estimates that will be established in the next section.\nEach of these damping and rescaling terms properly tuned, improves the rate of\nconvergence of the associated dynamics and algorithms. An original aspect of our\nwork is to combine them in the same dynamic. Let us recall some classical facts.\n1.1 Damped inertial dynamics and optimization\nThe continuous-time perspective gives a mechanical intuition of the behavior of the\ntrajectories, and a valuable tool to develop a Lyapunov analysis. A \frst important\nwork in this perspective is the heavy ball with friction method of B. Polyak [29]\n(HBF) x(t) +\r_x(t) +rf(x(t)) = 0:\nIt is a simpli\fed model for a heavy ball (whose mass has been normalized to one)\nsliding on the graph of the function fto be minimized, and which asymptoti-\ncally stops under the action of viscous friction, see [14] for further details. In this\nmodel, the viscous friction parameter \ris a \fxed positive parameter. Due to too\nmuch friction (at least asymptotically) involved in this process, replacing the \fxed\nviscous coe\u000ecient with a vanishing viscous coe\u000ecient ( i.e.which tends to zero as\nt!+1) gives Nesterov's famous accelerated gradient method [26] [27]. The other\ntwo basic ingredients that we will use, namely time rescaling, and Hessian-driven\ndamping have a natural interpretation (cinematic and geometric, respectively) inInertial dynamics with Hessian damping and time rescaling 3\nthis context. We will come back to these points later. Precisely, we seek to develop\nfast \frst-order methods based on the temporal discretization of damped inertial\ndynamics. By fast we mean that, for a general convex function f, and for each\ntrajectory of the system, the convergence rate of the values f(x(t))\u0000infHfwhich\nis obtained is optimal ( i.e.is achieved of nearly achieved in the worst case). The\nimportance of simple \frst-order methods, and in particular gradient-based and\nproximal algorithms, comes from the applicability of these algorithms to a wide\nrange of large-scale problems arising from machine learning and/or engineering.\n1.1.1 The viscous damping parameter \r(t).\nA signi\fcant number of recent studies have focused on the case \r(t) =\u000b\nt,\f= 0\n(without Hessian-driven damping), and b= 1 (without time rescaling), that is\n(AVD)\u000bx(t) +\u000b\nt_x(t) +rf(x(t)) = 0:\nThis dynamic involves an Asymptotically Vanishing Damping coe\u000ecient (hence\nthe terminology), a key property to obtain fast convergence for a general convex\nfunctionf. In [32], Su, Boyd and Cand\u0012 es showed that for \u000b= 3 the above system\ncan be seen as a continuous version of the accelerated gradient method of Nesterov\n[26,27] with f(x(t))\u0000minHf=O(1\nt2) ast!+1. The importance of the parame-\nter\u000bwas put to the fore by Attouch, Chbani, Peypouquet and Redont [9] and May\n[25]. They showed that, for \u000b>3, one can pass from capital Oestimates to small\no. Moreover, when \u000b>3, each trajectory converges weakly, and its limit belongs to\nargminf1. Recent research considered the case of a general damping coe\u000ecient \r(\u0001)\n(see [4,7]), thus providing a complete picture of the convergence rates for (AVD)\u000b:\nf(x(t))\u0000minHf=O(1=t2) when\u000b\u00153, andf(x(t))\u0000minHf=O\u0010\n1=t2\u000b\n3\u0011\nwhen\n\u000b\u00143, see [7,10] and Apidopoulos, Aujol and Dossal [3].\n1.1.2 The Hessian-driven damping parameter \f(t).\nThe inertial system\n(DIN)\r;\fx(t) +\r_x(t) +\fr2f(x(t)) _x(t) +rf(x(t)) = 0;\nwas introduced by Alvarez, Attouch, Bolte, and Redont in [2]. In line with (HBF),\nit contains a \fxed positive friction coe\u000ecient \r. As a main property, the introduc-\ntion of the Hessian-driven damping makes it possible to neutralize the transversal\noscillations likely to occur with (HBF), as observed in [2]. The need to take a\ngeometric damping adapted to fhad already been observed by Alvarez [1] who\nconsidered the inertial system\nx(t) +D_x(t) +rf(x(t)) = 0;\nwhereD:H!H is a linear positive de\fnite anisotropic operator. But still this\ndamping operator is \fxed. For a general convex function, the Hessian-driven damp-\ning in (DIN)\r;\fperforms a similar operation in a closed-loop adaptive way. (DIN)\nstands shortly for Dynamical Inertial Newton, and refers to the link with the\n1Recall that for \u000b= 3 the convergence of the trajectories is an open question4 Hedy ATTOUCH et al.\nLevenberg-Marquardt regularization of the continuous Newton method. Recent\nstudies have been devoted to the study of the inertial dynamic\nx(t) +\u000b\nt_x(t) +\fr2f(x(t)) _x(t) +rf(x(t)) = 0;\nwhich combines asymptotic vanishing damping with Hessian-driven damping [17].\n1.1.3 The time rescaling parameter b(t).\nIn the context of non-autonomous dissipative dynamic systems, reparameteriza-\ntion in time is a simple and universal means to accelerate the convergence of\ntrajectories. This is where the coe\u000ecient b(t) comes in as a factor of rf(x(t)).\nIn [11] [12], in the case of general coe\u000ecients \r(\u0001) andb(\u0001) without the Hessian\ndamping, the authors made in-depth study. In the case \r(t) =\u000b\nt, they proved that\nunder appropriate conditions on \u000bandb(\u0001),f(x(t))\u0000minHf=O(1\nt2b(t)). Hence a\nclear improvement of the convergence rate by taking b(t)!+1ast!+1.\n1.2 From damped inertial dynamics to proximal-gradient inertial algorithms\nLet's review some classical facts concerning the close link between continuous\ndissipative inertial dynamic systems and the corresponding algorithms obtained\nby temporal discretization. Let us insist on the fact that, when the temporal\nscalingb(t)!+1ast!+1, the transposition of the results to the discrete\ncase naturally leads to consider an implicit temporal discretization, i.e.inertial\nproximal algorithms. The reason is that, since b(t) is in front of the gradient, the\napplication of the gradient descent lemma would require taking a step size that\ntends to zero. On the other hand, the corresponding proximal algorithms involve\na proximal coe\u000ecient which tends to in\fnity (large step proximal algorithms).\n1.2.1 The case without the Hessian-driven damping\nThe implicit discretization of (IGS)\r;0;bgives the Inertial Proximal algorithm\n(IP)\u000bk;\u0015k(\nyk=xk+\u000bk(xk\u0000xk\u00001)\nxk+1= prox\u0015kf(yk)\nwhere\u000bkis non-negative and \u0015kis positive. Recall that for any \u0015>0, the proximity\noperator prox\u0015f:H!H is de\fned by the following formula: for every x2H\nprox\u0015f(x) := argmin\u00182H\u001a\nf(\u0018) +1\n2\u0015kx\u0000\u0018k2\u001b\n:\nEquivalently, prox\u0015fis the resolvent of index \u0015of the maximally monotone opera-\ntor@f. When passing to the implicit discrete case, we can take f:H!R[f+1g\na convex lower semicontinuous and proper function. Let us list some of the main\nresults concerning the convergence properties of the algorithm (IP)\u000bk;\u0015k:\n\u000f1. Case\u0015k\u0011\u0015>0 and\u000bk= 1\u0000\u000b\nk. When\u000b= 3, the (IP)1\u00003=k;\u0015algorithm\nhas a similar structure to the original Nesterov accelerated gradient algorithm[26],Inertial dynamics with Hessian damping and time rescaling 5\njust replace the gradient step with a proximal step. Passing from the gradient to the\nproximal step was carried out by G uler [22,23], then by Beck and Teboulle [18] for\nstructured optimization. A decisive step was taken by Attouch and Peypouquet in\n[16] proving that, when \u000b>3,f(xk)\u0000minHf=o\u00001\nk2\u0001. The subcritical case \u000b<3\nwas examined by Apidopoulos, Aujol, and Dossal [3] and Attouch, Chbani, and\nRiahi [10] with the rate of convergence rate of values f(xk)\u0000minHf=O\u0010\n1\nk2\u000b\n3\u0011\n.\n\u000f2. For a general \u000bk, the convergence properties of (IP)\u000bk;\u0015were analyzed\nby Attouch and Cabot [5], then by Attouch, Cabot, Chbani, and Riahi [6], in\nthe presence of perturbations. The convergence rates are then expressed using the\nsequence ( tk) which is linked to ( \u000bk) by the formula tk:= 1 +P+1\ni=kQi\nj=k\u000bj.\nUnder growth conditions on tk, it is proved that f(xk)\u0000minHf=O(1\nt2\nk). This\nlast results covers the special case \u000bk= 1\u0000\u000b\nkwhen\u000b\u00153.\n\u000f3. For a general \u0015k, Attouch, Chbani, and Riahi \frst considered in [11] the\ncase\u000bk= 1\u0000\u000b\nk. They proved that under a growth condition on \u0015k, we have\nthe estimate f(xk)\u0000minHf=O(1\nk2\u0015k). This result is an improvement of the one\ndiscussed previously in [16], because when \u0015k=k\u000ewith 0<\u000e<\u000b\u00003, we pass from\nO(1\nk2) toO(1\nk2+\u000e). Recently, in [13] the authors analyzed the algorithm (IP)\u000bk;\u0015kfor general \u000bkand\u0015k. By including the expression of tkpreviously used in [5,6],\nthey proved that f(xk)\u0000minHf=O\u00001=t2\nk\u0015k\u00001\u0001under certain conditions on \u0015k\nand\u000bk. They obtained f(xk)\u0000minHf=o\u00001=t2\nk\u0015k\u0001, which gives a global view of\nof the convergence rate with small o, encompassing [5,13].\n1.2.2 The case with the Hessian-driven damping\nRecent studies have been devoted to the inertial dynamic\nx(t) +\u000b\nt_x(t) +\fr2f(x(t)) _x(t) +rf(x(t)) = 0;\nwhich combines asymptotic vanishing viscous damping with Hessian-driven damp-\ning. The corresponding algorithms involve a correcting term in the Nesterov ac-\ncelerated gradient method which reduces the oscillatory aspects, see Attouch-\nPeypouquet-Redont [17], Attouch-Chbani-Fadili-Riahi [8], Shi-Du-Jordan-Su [30].\nThe case of monotone inclusions has been considered by Attouch and L\u0013 aszl\u0013 o [15].\n1.3 Contents\nThe paper is organized as follows. In section 2, we develop a new Lyapunov anal-\nysis for the continuous dynamic (IGS)\r;\f;b. In Theorem 1, we provide a system of\nconditions on the damping parameters \r(\u0001) and\f(\u0001), and on the temporal scaling\nparameter b(\u0001) giving fast convergence of the values. Then, in sections 3 and 4,\nwe present two di\u000berent types of growth conditions for the damping and tempo-\nral scaling parameters, respectively based on the functions \u0000 \randp\r, and which\nsatisfy the conditions of Theorem 1. In doing so, we encompass most existing re-\nsults and provide new results, including linear convergence rates without assuming\nstrong convexity. This will also allow us to explain the choice of certain coe\u000ecients\nin the associated algorithms, questions which have remained mysterious and only6 Hedy ATTOUCH et al.\njusti\fed by the simpli\fcation of often complicated calculations. In section 5, we\nspecialize our results to certain model situations and give numerical illustrations.\nFinally, we conclude the paper by highlighting its original aspects.\n2 Convergence rate of the values. General abstract result\nWe will establish a general result concerning the convergence rate of the values\nveri\fed by the solution trajectories x(\u0001) of the second-order evolution equation\n(IGS)\r;\f;b x(t) +\r(t) _x(t) +\f(t)r2f(x(t)) _x(t) +b(t)rf(x(t)) = 0:\nThe variable parameters \r(\u0001),\f(\u0001) andb(\u0001) take into account the damping, and\ntemporal rescaling e\u000bects. They are assumed to be continuously di\u000berentiable.\nTo analyze the asymptotic behavior of the solutions trajectories of the evolution\nsystem (IGS) \r;\f;b, we will use Lyapunov's analysis. It is a classic and powerful tool\nwhich consists in building an associated energy-like function which decreases along\nthe trajectories. The determination of such a Lyapunov function is in general a\ndelicate problem. Based on previous works, we know the global structure of such\na Lyapunov function. It is a weighted sum of the potential, kinetic and anchor\nfunctions. We will introduce coe\u000ecients in this function that are a priori unknown,\nand which will be identi\fed during the calculation to verify the property of decay.\nOur approach takes advantage of the technics recently developed in [4], [17], [12].\n2.1 The general case\nLetx(\u0001) be a solution trajectory of (IGS) \r;\f;b. Givenz2argminHf, we introduce\nthe Lyapunov function t7!E(t) de\fned by\nE(t) :=c(t)2b(t)(f(x(t))\u0000f(z))+\u0012(t)\u001b(t)2\n2kv(t)k2+\u0018(t)\n2kx(t)\u0000zk2;(1)\nwherev(t) :=x(t)\u0000z+1\n\u001b(t)(_x(t) +\f(t)rf(x(t))):\nThe four variable coe\u000ecients c(t);\u0012(t);\u001b(t);\u0018(t) will be adjusted during the calcu-\nlation. According to the classical derivation chain rule, we obtain\nd\ndtE(t) =d\ndt\u0010\nc2(t)b(t)\u0011\n(f(x(t))\u0000f(z))+c(t)2b(t)hrf(x(t));_x(t)i\n+1\n2d\ndt(\u0012(t)\u001b2(t))kv(t)k2+\u0012(t)\u001b2(t)h_v(t);v(t)i\n+1\n2_\u0018(t)kx(t)\u0000zk2+\u0018(t)h_x(t);x(t)\u0000zi:Inertial dynamics with Hessian damping and time rescaling 7\nFrom now, without ambiguity, to shorten formulas, we omit the variable t.\nAccording to the de\fnition of v, and the equation (IGS) \r;\f;b, we have\n_v= _x\u0000_\u001b\n\u001b2(_x+\frf(x))+1\n\u001bd\ndt(_x+\frf(x))\n= _x\u0000_\u001b\n\u001b2(_x+\frf(x))+1\n\u001b\u0010\nx+\fr2f(x) _x+_\frf(x)\u0011\n= _x\u0000_\u001b\n\u001b2(_x+\frf(x))+1\n\u001b\u0000\n\u0000\r_x\u0000brf(x) +_\frf(x)\u0001\n=\u0012\n1\u0000_\u001b\n\u001b2\u0000\r\n\u001b\u0013\n_x+\u0012_\f\n\u001b\u0000_\u001b\f\n\u001b2\u0000b\n\u001b\u0013\nrf(x):\nTherefore,\nh_v;vi=\u001c\u0012\n1\u0000_\u001b\n\u001b2\u0000\r\n\u001b\u0013\n_x+\u0012_\f\n\u001b\u0000_\u001b\f\n\u001b2\u0000b\n\u001b\u0013\nrf(x); x\u0000z+1\n\u001b(_x+\frf(x))\u001d\n=\u0012\n1\u0000_\u001b\n\u001b2\u0000\r\n\u001b\u0013\nh_x; x\u0000zi+\u0012\n1\u0000_\u001b\n\u001b2\u0000\r\n\u001b\u00131\n\u001bk_xk2\n+\u0012\u0012\n1\u0000_\u001b\n\u001b2\u0000\r\n\u001b\u0013\n\f\n\u001b+\u0012_\f\n\u001b\u0000_\u001b\f\n\u001b2\u0000b\n\u001b\u00131\n\u001b\u0013\nhrf(x);_xi\n+\u0012_\f\n\u001b\u0000_\u001b\f\n\u001b2\u0000b\n\u001b\u0013\nhrf(x); x\u0000zi+\u0012_\f\n\u001b\u0000_\u001b\f\n\u001b2\u0000b\n\u001b\u0013\n\f\n\u001bkrf(x)k2:\nAccording to the de\fnition of v(t), after developing kv(t)k2, we get\nkvk2=kx\u0000zk2+1\n\u001b2\u0010\nk_xk2+\f2krf(x)k2\u0011\n+2\n\u001bh_x; x\u0000zi\n+2\f\n\u001bhrf(x); x\u0000zi+2\f\n\u001b2hrf(x);_xi:\nCollecting the above results, we obtain\nd\ndtE(t) =d\ndt\u0010\nc2b\u0011\n(f(x)\u0000f(z))+c2bhrf(x);_xi+1\n2_\u0018kx\u0000zk2+\u0018h_x;x\u0000zi\n+1\n2d\ndt(\u0012\u001b2)\u0010\nkx\u0000zk2+1\n\u001b2\u0010\nk_xk2+\f2krf(x)k2\u0011\n+2\n\u001bh_x; x\u0000zi\u0011\n+1\n2d\ndt(\u0012\u001b2)\u00102\f\n\u001bhrf(x); x\u0000zi+2\f\n\u001b2hrf(x);_xi\u0011\n+\u0012\u001b2\u0010\u0012\n1\u0000_\u001b\n\u001b2\u0000\r\n\u001b\u0013\nh_x; x\u0000zi+\u0012\n1\u0000_\u001b\n\u001b2\u0000\r\n\u001b\u00131\n\u001bk_xk2\u0011\n+\u0012\u001b2\u0012\u0012\n1\u0000_\u001b\n\u001b2\u0000\r\n\u001b\u0013\n\f\n\u001b+\u0012_\f\n\u001b\u0000_\u001b\f\n\u001b2\u0000b\n\u001b\u00131\n\u001b\u0013\nhrf(x);_xi\n+\u0012\u001b2\u0010\u0012_\f\n\u001b\u0000_\u001b\f\n\u001b2\u0000b\n\u001b\u0013\nhrf(x); x\u0000zi+\u0012_\f\n\u001b\u0000_\u001b\f\n\u001b2\u0000b\n\u001b\u0013\n\f\n\u001bkrf(x)k2\u0011\n:\nIn the second member of the above formula, let us examine the terms that contain\nhrf(x); x\u0000zi. By grouping these terms, we obtain the following expression\n\u0010\f\n\u001bd\ndt(\u0012\u001b2) +\u0012\u001b2\u0012_\f\n\u001b\u0000_\u001b\f\n\u001b2\u0000b\n\u001b\u0013\u0011\nhrf(x); x\u0000zi:8 Hedy ATTOUCH et al.\nTo majorize it, we use the convex subgradient inequality hrf(x);x\u0000zi\u0015f(x)\u0000\nf(z);and we make a \frst hypothesis\f\n\u001bd\ndt(\u0012\u001b2)+\u0012\u001b2\u0010_\f\n\u001b\u0000_\u001b\f\n\u001b2\u0000b\n\u001b\u0011\n\u00140:Therefore,\nd\ndtE(t)\u0014\u0014\nd\ndt\u0010\nc2b\u0011\n+\f\n\u001bd\ndt(\u0012\u001b2) +\u0012\u001b2\u0012_\f\n\u001b\u0000_\u001b\f\n\u001b2\u0000b\n\u001b\u0013\u0015\n(f(x)\u0000f(z))\n+\u0014\nc2b+\f\n\u001b2d\ndt(\u0012\u001b2) +\u0012\u001b\u0012\u0012\n1\u0000_\u001b\n\u001b2\u0000\r\n\u001b\u0013\n\f+\u0012_\f\n\u001b\u0000_\u001b\f\n\u001b2\u0000b\n\u001b\u0013\u0013\u0015\nhrf(x);_xi\n+\u00141\n\u001bd\ndt(\u0012\u001b2) +\u0012\u001b2\u0012\n1\u0000_\u001b\n\u001b2\u0000\r\n\u001b\u0013\n+\u0018\u0015\nh_x; x\u0000zi\n+1\n2\u0014\nd\ndt(\u0012\u001b2) +_\u0018\u0015\nkx\u0000zk2+\u00141\n2\u001b2d\ndt(\u0012\u001b2) +\u0012\u001b\u0012\n1\u0000_\u001b\n\u001b2\u0000\r\n\u001b\u0013\u0015\nk_xk2\n+\u0014\n\f2\n2\u001b2d\ndt(\u0012\u001b2) +\u0012\u001b\f\u0012_\f\n\u001b\u0000_\u001b\f\n\u001b2\u0000b\n\u001b\u0013\u0015\nkrf(x)k2: (2)\nTo getd\ndtE(t)\u00140, we are led to make the following assumptions:\n(i)\f\n\u001bd\ndt(\u0012\u001b2) +\u0012\u001b2\u0012_\f\n\u001b\u0000_\u001b\f\n\u001b2\u0000b\n\u001b\u0013\n\u00140\n(ii)d\ndt\u0010\nc2b\u0011\n+\f\n\u001bd\ndt(\u0012\u001b2) +\u0012\u001b2\u0012_\f\n\u001b\u0000_\u001b\f\n\u001b2\u0000b\n\u001b\u0013\n\u00140;\n(iii)c2b+\f\n\u001b2d\ndt(\u0012\u001b2) +\u0012\u001b\u0012\u0012\n1\u0000_\u001b\n\u001b2\u0000\r\n\u001b\u0013\n\f+\u0012_\f\n\u001b\u0000_\u001b\f\n\u001b2\u0000b\n\u001b\u0013\u0013\n= 0;\n(iv)1\n\u001bd\ndt(\u0012\u001b2) +\u0012\u001b2\u0012\n1\u0000_\u001b\n\u001b2\u0000\r\n\u001b\u0013\n+\u0018= 0;\n(v)d\ndt(\u0012\u001b2) +_\u0018\u00140;\n(vi)1\n2\u001b2d\ndt(\u0012\u001b2) +\u0012\u001b\u0012\n1\u0000_\u001b\n\u001b2\u0000\r\n\u001b\u0013\n\u00140;\n(vii)\f2\n2\u001b2d\ndt(\u0012\u001b2) +\u0012\u001b\f\u0012_\f\n\u001b\u0000_\u001b\f\n\u001b2\u0000b\n\u001b\u0013\n\u00140:\nAfter simpli\fcation, we get the following equivalent system of conditions:\nA: Lyapunov system of inequalities involving c(t);\u0012(t);\u001b(t);\u0018(t):\n(i)d\ndt(\f\u0012\u001b)\u0000\u0012b\u001b\u00140\n(ii)d\ndt\u0000\nc2b+\f\u0012\u001b\u0001\n\u0000\u0012b\u001b\u00140;\n(iii)b(c2\u0000\u0012) +\f\u0012(\u001b\u0000\r) +d\ndt(\f\u0012) = 0;\n(iv)d\ndt(\u0012\u001b) +\u0012\u001b(\u001b\u0000\r)+\u0018= 0;\n(v)d\ndt(\u0012\u001b2+\u0018)\u00140;\n(vi)_\u0012+ 2(\u001b\u0000\r)\u0012\u00140;\n(vii)\f\u0010\n\f_\u0012+ 2\u0000_\f\u0000b\u0001\n\u0012\u0011\n\u00140:Inertial dynamics with Hessian damping and time rescaling 9\nLet's simplify this system by eliminating the variable \u0018. From (iv) we get\u0018=\n\u0000d\ndt(\u0012\u001b)\u0000\u0012\u001b(\u001b\u0000\r), that we replace in ( v), and recall that \u0018is prescribed to\nbe nonnegative. Now observe that the unkown function ccan also be eliminated.\nIndeed, it enters the above system via the variable bc2, which according to ( iii)\nis equal to bc2=b\u0012\u0000\f\u0012(\u001b\u0000\r)\u0000d\ndt(\f\u0012):Replacing in ( ii), which is the only\nother equation involving bc2, we obtain the equivalent system involving only the\nvariables\u0012(t);\u001b(t).\nB: Lyapunov system of inequalities involving the variables: \u0012(t);\u001b(t)\n(i)d\ndt(\f\u0012\u001b)\u0000\u0012b\u001b\u00140;\n(ii)d\ndt(b\u0012+\f\u0012\r)\u0000d2\ndt2(\f\u0012)\u0000\u0012b\u001b\u00140;\n(iii)b\u0012\u0000\f\u0012(\u001b\u0000\r)\u0000d\ndt(\f\u0012)\u00150;\n(iv)d\ndt(\u0012\u001b) +\u0012\u001b(\u001b\u0000\r)\u00140;\n(v)d\ndt\u0000\n\u0000d\ndt(\u0012\u001b) +\u0012\u001b\r\u0001\n\u00140;\n(vi)_\u0012+ 2(\u001b\u0000\r)\u0012\u00140;\n(vii)\f\u0010\n\f_\u0012+ 2\u0000_\f\u0000b\u0001\n\u0012\u0011\n\u00140:\nThen, the variables \u0018andcare obtained by using the formulas\n\u0018=\u0000d\ndt(\u0012\u001b)\u0000\u0012\u001b(\u001b\u0000\r)\nbc2=b\u0012\u0000\f\u0012(\u001b\u0000\r)\u0000d\ndt(\f\u0012):\nThus, under the above conditions, the function E(\u0001) is nonnegative and nonincreas-\ning. Therefore, for every t\u0015t0,E(t)\u0014E(t0);which implies that\nc2(t)b(t)f(x(t))\u0000min\nHf)\u0014E(t0):\nTherefore, as t!+1\nf(x(t))\u0000min\nHf=O\u00121\nc2(t)b(t)\u0013\n:\nMoreover, by integrating (2) we obtain the following integral estimates:\na) On the values:\nZ+1\nt0\u0010\n\u0012(t)b(t)\u001b(t)\u0000d\ndt\u0010\nc2(t)b(t) +\f(t)\u0012(t)\u001b(t)\u0011\u0011\u0012\nf(x(t))\u0000inf\nHf\u0013\ndt< +1;\nwhere we use the equality:\n\u0000\u0010\nd\ndt\u0000\nc2b\u0001+\f\n\u001bd\ndt(\u0012\u001b2) +\u0012\u001b2\u0010_\f\n\u001b\u0000_\u001b\f\n\u001b2\u0000b\n\u001b\u0011\u0011\n=\u0012b\u001b\u0000d\ndt\u0000\nc2b+\f\u0012\u001b\u0001\nand the fact that, according to ( ii), this quantity is nonnegative.10 Hedy ATTOUCH et al.\nb) On the norm of the gradients:\nZ+1\nt0q(t)krf(x(t))k2dt< +1:\nwhereqis the nonnegative weight function de\fned by\nq(t) :=\u0012(t)\f(t)\u0012_\u001b(t)\f(t)\n\u001b(t)+b(t)\u0000_\f(t)\u0013\n\u0000\f2(t)\n2\u001b2(t)d\ndt(\u0012\u001b2)(t)\n=b(t)\u0012(t)\f(t)\u00001\n2d\ndt(\u0012\f2)(t): (3)\nWe can now state the following Theorem, which summarizes the above results.\nTheorem 1 Letf:H!Rbe a convex di\u000berentiable function with argminHf6=;.\nLetx(\u0001)be a solution trajectory of\n(IGS)\r;\f;b x(t) +\r(t) _x(t) +\f(t)r2f(x(t)) _x(t) +b(t)rf(x(t)) = 0:\nSuppose that \r(\u0001),\f(\u0001), andb(\u0001), areC1functions on [t0;+1[such that there exists\nauxiliary functions c(t);\u0012(t);\u001b(t);\u0018(t)that satisfy the conditions (i)\u0000(vii)above. Set\nE(t) :=c(t)2b(t)(f(x(t))\u0000f(z))+\u0012(t)\u001b(t)2\n2kv(t)k2+\u0018(t)\n2kx(t)\u0000zk2;(4)\nwithz2argminHfandv(t) =x(t)\u0000z+1\n\u001b(t)(_x(t) +\f(t)rf(x(t))).\nThen,t7!E(t)is a nonincreasing function. As a consequence, for all t\u0015t0,\n(i)f(x(t))\u0000min\nHf\u0014E(t0)\nc2(t)b(t); (5)\n(ii)Z+1\nt0\u0010\n\u0012(t)b(t)\u001b(t)\u0000d\ndt\u0000\nc2b+\f\u0012\u001b\u0001(t)\u0011\u0012\nf(x(t))\u0000inf\nHf\u0013\ndt< +1; (6)\n(iii)Z+1\nt0\u0012\nb(t)\u0012(t)\f(t)\u00001\n2d\ndt\u0000\n\u0012\f2\u0001(t)\u0013\nkrf(x(t))k2dt< +1: (7)\n2.2 Solving system ( i)\u0000(vii)\nThe system of inequalities ( i)\u0000(vii) of Theorem 1 may seem complicated at \frst\nglance. Indeed, we will see that it simpli\fes notably in the classical situations.\nMoreover, it makes it possible to unify the existing results, and discover new\ninteresting cases. We will present two di\u000berent types of solutions to this system,\nrespectively based on the following functions:\np\r(t) = exp\u0012Zt\nt0\r(u)du\u0013\n; (8)\nand\n\u0000\r(t) =p\r(t)Z+1\ntdu\np\r(u): (9)\nThe use of \u0000 \rhas been considered in a series of articles that we will retrieve as a\nspecial case of our approach, see [4], [5], [7], [12]. Using p\rwill lead to new results,\nsee section 4.Inertial dynamics with Hessian damping and time rescaling 11\n3 Results based on the function \u0000 \r\nIn this section, we will systematically assume that condition ( H0) is satis\fed.\n(H0)Z+1\nt0ds\np(s)<+1:\nUnder (H0), the function \u0000 \r(\u0001) is well de\fned. It can be equally de\fned as the\nsolution of the linear non autonomous di\u000berential equation\n_\u0000\r(t)\u0000\r(t)\u0000\r(t) + 1 = 0; (10)\nwhich satis\fes the limit condition lim t!+1\u0000\r(t)\np\r(t)= 0.\n3.1 The case without the Hessian, i.e.\f\u00110\nThe dynamic writes\n(IGS)\r;0;b x(t) +\r(t) _x(t) +b(t)rf(x(t)) = 0:\nTo solve the system ( i)\u0000(vii) of Theorem 1, we choose\n\u0018\u00110; c(t) = \u0000\r(t); \u001b(t) =1\n\u0000\r(t); \u0012(t) =\u0000\r(t)2:\nAccording to (10), we can easily verify that conditions ( i);(iii)\u0000(vii) are satis\fed,\nand (ii) becomes\nd\ndt\u0010\n\u0000\r(t)2b(t)\u0011\n\u0000\u0000\r(t)b(t)\u00140:\nAfter dividing by \u0000 \r(t), and using (10), we obtain the condition\n0\u0015\u0000\r(t)_b(t)\u0000(3\u00002\r(t)\u0000\r(t))b(t):\nThis leads to the following result obtained by Attouch, Chbani and Riahi in [12].\nTheorem 2 [12, Theorem 2.1] Suppose that for all t\u0015t0\n\u0000\r(t)_b(t)\u0014b(t)(3\u00002\r(t)\u0000\r(t)); (11)\nwhere \u0000\ris de\fned from \rby(9). Letx: [t0;+1[!H be a solution trajectory of\n(IGS)\r;0;b. Givenz2argminHf, set\nE(t) := \u00002\n\r(t)b(t)(f(x(t))\u0000f(z))+1\n2kx(t)\u0000z+ \u0000\r(t) _x(t)k2: (12)\nThen,t7!E(t)is a nonincreasing function. As a consequence, as t!+1\nf(x(t))\u0000min\nHf=O\u00121\n\u0000\r(t)2b(t)\u0013\n: (13)\nPrecisely, for all t\u0015t0\nf(x(t))\u0000min\nHf\u0014C\n\u0000\r(t)2b(t); (14)\nwithC= \u0000\r(t0)2b(t0)(f(x(t0))\u0000minHf)+d(x(t0);argminf)2+\u0000\r(t0)2k_x(t0)k2:\nMoreover,\nZ+1\nt0\u0000\r(t)\u0010\nb(t)(3\u00002\r(t)\u0000\r(t))\u0000\u0000\r(t)_b(t)\u0011\n(f(x(t))\u0000min\nHf)dt< +1:\nRemark 1 Whenb\u00111, condition (11) reduces to \r(t)\u0000\r(t)\u00143\n2, introduced in [4].12 Hedy ATTOUCH et al.\n3.2 Combining Nesterov acceleration with Hessian damping\nLet us specialize our results in the case \f(t)>0, and\r(t) =\u000b\nt. We are in the case\nof a vanishing damping coe\u000ecient ( i.e.\r(t)!0 ast!+1). According to Su,\nBoyd and Cand\u0012 es [32], the case \u000b= 3 corresponds to a continuous version of the\naccelerated gradient method of Nesterov. Taking \u000b > 3 improves in many ways\nthe convergence properties of this dynamic, see section 1.1.1. Here, it is combined\nwith the Hessian-driven damping and temporal rescaling. This situation was \frst\nconsidered by Attouch, Chbani, Fadili and Riahi in [8]. Then the dynamic writes\n(IGS)\u000b=t;\f;b x(t) +\u000b\nt_x(t) +\f(t)r2f(x(t)) _x(t) +b(t)rf(x(t)) = 0:\nElementary calculus gives that ( H0) is satis\fed as soon as \u000b>1. In this case,\n\u0000\r(t) =t\n\u000b\u00001:\nAfter [8], let us introduce the following quantity which will simplify the formulas:\nw(t) :=b(t)\u0000_\f(t)\u0000\f(t)\nt: (15)\nThe following result will be obtained as a consequence of our general abstract The-\norem 1. Precisely, we will show that under an appropriate choice of the functions\nc(t);\u0012(t);\u001b(t);\u0018(t), the conditions ( i)\u0000(vii) of Theorem 1 are satis\fed.\nTheorem 3 [8, Theorem 1] Letx: [t0;+1[!H be a solution trajectory of\n(IGS)\u000b=t;\f;b x(t) +\u000b\nt_x(t) +\f(t)r2f(x(t)) _x(t) +b(t)rf(x(t)) = 0:\nSuppose that \u000b>1, and that the following growth conditions are satis\fed: for t\u0015t0\n(G2)b(t)>_\f(t) +\f(t)\nt;\n(G3)t_w(t)\u0014(\u000b\u00003)w(t):\nThen,w(t) :=b(t)\u0000_\f(t)\u0000\f(t)\ntis positive and\n(i)f(x(t))\u0000min\nHf=O\u00121\nt2w(t)\u0013\nast!+1;\n(ii)Z+1\nt0t\u0010\n(\u000b\u00003)w(t)\u0000t_w(t)\u0011\n(f(x(t))\u0000min\nHf)dt< +1;\n(iii)Z+1\nt0t2\f(t)w(t)krf(x(t))k2dt< +1:\nProof Take\u0012(t) = \u0000\r(t)2; \u001b(t) =1\n\u0000\r(t); \u0018(t)\u00110;and\nc(t)2=1\n(\u000b\u00001)2t\nb(t)\u0000\ntb(t)\u0000\f(t)\u0000t_\f(t)\u0001\n: (16)\nThis formula for c(t) will appear naturally during the calculation. Note that the\ncondition (G2) ensures that the second member of the above expression is positive,\nwhich makes sense to think of it as a square. Let us verify that the conditions ( i)\nand (iv);(v);(vi);(vii) are satis\fed. This is a direct consequence of the formula\n(10) and the condition ( G2):Inertial dynamics with Hessian damping and time rescaling 13\n(i)d\ndt(\f\u0012\u001b)\u0000\u0012b\u001b=d\ndt\f\u0000\u0000\u0000b=1\n\u000b\u00001\u0000d\ndt(t\f)\u0000tb\u0001=t\n\u000b\u00001\u0010\n_\f+\f\nt\u0000b\u0011\n\u00140.\n(iv)d\ndt(\u0012\u001b) +\u0012\u001b(\u001b\u0000\r)+\u0018=_\u0000 + \u0000\u00001\n\u0000\u0000\r\u0001=_\u0000 + 1\u0000\r\u0000 = 0.\n(v) Since\u0012\u001b2\u00111 and\u0018\u00111, we haved\ndt(\u0012\u001b2+\u0018) = 0.\n(vi)_\u0012+ 2(\u001b\u0000\r)\u0012= 2\u0000 _\u0000 + 2(\u0000\u0000\r\u00002) = 2\u0000( _\u0000 + 1\u0000\r\u0000) = 0.\n(vii)\f_\u0012+ 2\u0000_\f\u0000b\u0001\n\u0012= 2\u0000(\f_\u0000 + ( _\f\u0000b)\u0000) = 2\u00002(_\f\u0000b+\f\nt)\u00140:\nLet's go to the conditions ( ii) and (iii). The condition ( iii) gives the formula (16)\nforc(t). Then replacing c(t)2by this value in ( ii) gives the condition ( G3). Note\nthen thatb(t)c(t)2=1\n(\u000b\u00001)2t2!(t), which gives the convergence rate of the values\nf(x(t))\u0000min\nHf=O\u00121\nt2w(t)\u0013\n:\nLet us consider the integral estimate for the values. According to the de\fnition\n(16) forc2band the de\fnition of w, we have\n\u0012b\u001b\u0000d\ndt\u0010\nc2b+\f\u0012\u001b\u0011\n=1\n\u000b\u00001tb\u0000d\ndt\u00101\n\u000b\u00001t2w(t) +1\n\u000b\u00001t\f\u0011\n=t\n(\u000b\u00001)2\u0010\n(\u000b\u00001)b\u00002w\u0000t_w\u0000(\u000b\u00001)(_\f+\f\nt)\u0011\n=t\n(\u000b\u00001)2\u0010\n(\u000b\u00003)w\u0000t_w\u0011\n:\nAccording to Theorem 1 ( ii)\nZ+1\nt0t\u0010\n(\u000b\u00003)w(t)\u0000t_w(t)\u0011\n(f(x(t))\u0000min\nHf)dt< +1:\nMoreover, since \u0012\u001b2= 1, the formula giving the weighting coe\u000ecient q(t) in the\nintegral formula simpli\fes, and we get\nq(t) =\u0012(t)\u001b(t)\f(t) _\u001b(t)\f(t)\n\u001b2(t)+b(t)\n\u001b(t)\u0000_\f(t)\n\u001b(t)!\n=\f(t)\u0000\r(t)\u0010\n\u0000\f(t)_\u0000\r(t) +b(t)\u0000\r(t)\u0000_\f(t)\u0000\r(t)\u0011\n=\f(t)\u0000\r(t)2!(t):\nAccording to Theorem 1 ( iii)\nZ+1\nt0t2\f(t)w(t)krf(x(t))k2dt< +1\nwhich gives the announced convergence rates. u t\nRemark 2 Take\f= 0. Then, according to the de\fnition (15) of w, we havew=b,\nand the conditions of Theorem 3 reduce to\nt_b(t)\u0000(3\u0000\u000b)b(t)\u00140 fort2[t0;+1[:14 Hedy ATTOUCH et al.\nWe recover the condition introduced in [12, Corollary 3.4]. Under this condition,\neach solution trajectory xof\n(IGS)\u000b=t;0;b x(t) +\u000b\nt_x(t) +b(t)rf(x(t)) = 0;\nsatis\fes\nf(x(t))\u0000min\nHf=O\u00121\nt2b(t)\u0013\nast!+1:\n3.3 The case \r(t) =\u000b\nt,\fconstant\nDue to its practical importance, consider the case \r(t) =\u000b\nt,\f(t)\u0011\fwhere\fis a\n\fxed positive constant. In this case, the dynamic (IGS)\r;\f;bis written as follows\nx(t) +\u000b\nt_x(t) +\fr2f(x(t)) _x(t) +b(t)rf(x(t)) = 0: (17)\nThe set of conditions ( G2), (G3) boils down to: for t\u0015t0\n(G2)b(t)>\f\nt;\n(G3)t_w(t)\u0014(\u000b\u00003)w(t);\nwherew(t) =b(t)\u0000\f\nt. Therefore, b(\u0001) must satisfy the di\u000berential inequality\ntd\ndt\u0010\nb(t)\u0000\f\nt\u0011\n\u0014(\u000b\u00003)\u0012\nb(t)\u0000\f\nt\u0013\n:\nEquivalently\ntd\ndtb(t)\u0000(\u000b\u00003)b(t) +\f(\u000b\u00002)1\nt\u00140:\nLet us integrate this linear di\u000berential equation. Set b(t) =k(t)t\u000b\u00003wherek(\u0001) is\nan auxiliary function to determine. We obtain\nd\ndt\u0010\nk(t)\u0000\f\nt\u000b\u00002\u0011\n\u00140;\nwhich gives k(t) =\f\nt\u000b\u00002+d(t) withd(\u0001) nonincreasing. Finally, b(t) =\f\nt+d(t)t\u000b\u00003;\nwithd(\u0001) a nonincreasing function to be chosen arbitrarily. In summary, we get\nthe following result:\nProposition 1 Letx: [t0;+1[!H be a solution trajectory of\nx(t) +\u000b\nt_x(t) +\fr2f(x(t)) _x(t) +\u0010\f\nt+d(t)t\u000b\u00003\u0011\nrf(x(t)) = 0 (18)\nwhered(\u0001)is a nonincreasing positive function. Then, the following properties are sat-\nis\fed:\n(i)f(x(t))\u0000min\nHf=O\u00121\nt\u000b\u00001d(t)\u0013\nast!+1;\n(ii)Z+1\nt0\u0000_d(t)t\u000b\u00001(f(x(t))\u0000inf\nHf)dt< +1:\n(iii)Z+1\nt0t\u000b\u00001d(t)krf(x(t))k2dt< +1:\nProof According to the de\fnition of w(t) andb(t), we have the equalities\nt2w(t) =t2\u0010\nb(t)\u0000\f\nt\u0011\n=t2d(t)t\u000b\u00003=t\u000b\u00001d(t). Then apply Theorem 3. u tInertial dynamics with Hessian damping and time rescaling 15\n3.4 Particular cases\nAccording to Theorem 3 and Proposition 1, let us discuss the role and the impor-\ntance of the scaling coe\u000ecient b(t) in front of the gradient term.\na)The \frst inertial dynamic system based on the Nesterov method, and which\nincludes a damping term driven by the Hessian, was considered by Attouch, Pey-\npouquet, and Redont in [17]. This corresponds to b(t)\u00111, which gives:\nx(t) +\u000b\nt_x(t) +\fr2f(x(t)) _x(t) +rf(x(t)) = 0:\nIn this case, we have w(t) = 1\u0000\f\nt, and we immediately get that ( G2), (G3) are\nsatis\fed by taking \u000b>3 andt>\f . This corresponds to take d(t) =1\nt\u000b\u00003\u0000\f\nt\u000b\u00002,\nwhich is nonincreasing when t\u0015\u000b\u00002\n\u000b\u00003.\nCorollary 1 [17, Theorem 1.10, Proposition 1.11] Suppose that \u000b >3and\f >0.\nLetx: [t0;+1[!H be a solution trajectory of\nx(t) +\u000b\nt_x(t) +\fr2f(x(t)) _x(t) +rf(x(t)) = 0: (19)\nThen,\n(i)f(x(t))\u0000min\nHf=O\u00121\nt2\u0013\nast!+1;\n(ii)Z+1\nt0t(f(x(t))\u0000inf\nHf)dt< +1;\n(iii)Z+1\nt0t2krf(x(t))k2dt< +1:\nb)Another important situation is obtained by taking d(t) =1\nt\u000b\u00003. This is the\nlimiting case where the following two properties are satis\fed: d(\u0001) is nonincreasing,\nand the coe\u000ecient of rf(x(t)) is bounded. This o\u000bers the possibility of obtaining\nsimilar results for the explicit temporal discretized dynamics, that is to say the\ngradient algorithms. Precisely, we obtain the dynamic system considered by Shi,\nDu, Jordan, and Su in [30], and Attouch, Chbani, Fadili, and Riahi in [8].\nCorollary 2 [8, Theorem 3], [30, Theorem 5]\nSuppose that \u000b\u00153. Letx: [t0;+1[!H be a solution trajectory of\nx(t) +\u000b\nt_x(t) +\fr2f(x(t)) _x(t) +\u0010\n1 +\f\nt\u0011\nrf(x(t)) = 0 (20)\nThen, the conclusions of Theorem 3are satis\fed:\n(i)f(x(t))\u0000min\nHf=O\u00121\nt2\u0013\nast!+1;\n(ii)When\u000b>3;Z+1\nt0t(f(x(t))\u0000inf\nHf)dt< +1:\n(iii)Z+1\nt0t2krf(x(t))k2dt< +1:16 Hedy ATTOUCH et al.\nNote that (20) has a slight advantage over (19): the growth conditions are valid\nfort>0, while for (19) one has to take t>\f . Accordingly, the estimates involve\nthe quantity1\nt2instead of1\nt2(1\u0000\f\nt).\nc)Taked(t) =1\ntswiths >0. According to Proposition 1, for any solution\ntrajectory x: [t0;+1[!H of\nx(t) +\u000b\nt_x(t) +\fr2f(x(t)) _x(t) +\u0010\f\nt+t\u000b\u00003\u0000s\u0011\nrf(x(t)) = 0 (21)\nwe have:\n(i)f(x(t))\u0000min\nHf=O\u00121\nt\u000b\u00001\u0000s\u0013\nast!+1;\n(ii)Z+1\nt0t\u000b\u0000s\u00002(f(x(t))\u0000inf\nHf)dt< +1;Z+1\nt0t\u000b\u0000s\u00001krf(x(t))k2dt< +1:\n4 Results based on the function p\r\nIn this section, we examine another set of growth conditions for the damping\nand rescaling parameters that guarantee the existence of solutions to the system\n(i)\u0000(vii) of Theorem 1. In the following theorems, the Lyapunov analysis and the\nconvergence rates are formulated using the function p\r: [t0;+1[!R+de\fned by\np\r(t) := exp\u0012Zt\nt0\r(s)ds\u0013\n:\nIn Theorems 2 and 3, in line with the previous articles devoted to these questions\n(see [4], [7], [12]), the convergence rate of the values was formulated using the\nfunction\u0000\r(t) =p\r(t)R+1\nt1\np\r(s)ds. In fact, each of the two functions p\rand\u0000\r\ncaptures the properties of the viscous damping coe\u000ecient \r(\u0001), but their growths\nare signi\fcantly di\u000berent. To illustrate this, in the model case \r(t) =\u000b\nt,\u000b > 1,\nwe havep\r(t) =\u0000t\nt0\u0001\u000b, while\u0000\r(t) =t\n\u000b\u00001. Therefore, p\rgrows faster than \u0000\r\nast!+1, and we can expect to get better convergence rates when formulating\nthem using p\r. Moreover, p\rmakes sense and allows to analyze the case \u000b\u00141,\nwhile\u0000\rdoes not. Thus, we will see that the approach based on p\rprovides results\nthat cannot be captured by the approach based on \u0000\r. To illustrate this, we start\nwith a simple situation, then we consider the general case.\n4.1 A model situation\nConsider the system\n(IGS)\r;0;b x(t) +\r(t) _x(t) +b(t)rf(x(t)) = 0\nwith\r(t) =\r0(t) +1\np0(t)and p 0(t) = exp\u0012Zt\nt0\r0(s)ds\u0013\n:\nChoose\n\u0018\u00110; c(t) = p 0(t); \u001b(t) =1\np0(t); \u0012(t) = p 0(t)2:Inertial dynamics with Hessian damping and time rescaling 17\nAccording to _ p0(t) =\r0(t)p0(t), we can easily verify that the conditions ( i);(iii)\u0000\n(vii) of Theorem 1 are satis\fed, and ( ii) becomesd\ndt\u0000p0(t)2b(t)\u0001\n\u0000p0(t)b(t)\u00140:\nThen, a direct application of Theorem 1 gives the following result.\nTheorem 4 Suppose that for all t\u0015t0\np0(t)_b(t) +\u0010\n2\r0(t)p0(t)\u00001\u0011\nb(t)\u00140: (22)\nLetx: [t0;+1[!H be a solution trajectory of (IGS)\r;0;b. Then, ast!+1\nf(x(t))\u0000min\nHf=O\u00121\np0(t)2b(t)\u0013\n: (23)\nMoreover,R+1\nt0p0(t)\u0010\n1\u0000(2\r0(t)p0(t))\u0000p0(t)_b(t)\u0011\n(f(x(t))\u0000minHf)dt< +1:\nRemark 3 Let us rewrite the linear di\u000berential inequality (22) as follows:\n_b(t)\nb(t)\u00141\np0(t)\u00002_ p0(t)\np0(t):\nA solution corresponding to equality is b(t) = p 0(t)\u00002exphZt\nt0\u00101\np0(s)\u0011\ndsi\n.\nIn the case \r0(t)(t) =\u000b\nt, 0<\u000b< 1,t0= 1, we have p 0(t) =t\u000b;which gives\nb(t) =t\u00002\u000bexpht1\u0000\u000b\u00001\n1\u0000\u000bi\n:\nTherefore, for 0 <\u000b< 1;and for this choice of b, (23) gives\nf(x(t))\u0000min\nHf=O\u00101\nexph\nt1\u0000\u000b\n1\u0000\u000bi\u0011\n: (24)\nThus, we obtain an exponential convergence rate in a situation that cannot be\ncovered by the \u0000\rapproach.\n4.2 The general case, with the Hessian-driven damping\nTheorem 5 Letf:H!Rbe a convex function of class C1such that argminHf6=;.\nSuppose that \r(\u0001);\f(\u0001)areC1functions and b(\u0001)is aC2function which is nondecreasing.\nSuppose that randmare positive parameters which satisfy 0<r\u00141\n3and2r\u0014m\u0014\n1\u0000r. Suppose that the following growth conditions are satis\fed: for t\u0015t0\n(H1)\u00180(t)\u00150;\n(H2)\u00180(t)\u00002\u001b(t)\u0000(m+r)\r(t)\u0001\n\u00001\n2_\u00180(t) +\u0000\nm\u0000(1\u0000r)\u0001\n\r(t)\u001b2(t)\u00150;\n(H3)b(t)\u0000_\f(t) +\f(t)\u0000\n\u001b(t)\u0000\r(t)\u0001\n\u00150;\n(H4)d\ndt\u0010\n\u0012(w+\f\u001b)\u0011\n(t)\u0000\u0012(t)b(t)\u001b(t)\u00140:18 Hedy ATTOUCH et al.\nwhere\n\u00180(t) :=\u0000(1\u00002(r+m))\r(t) +\u001b(t)\u0001\n\u001b(t)\u0000_\u001b(t); (25)\n\u001b(t) :=m\r(t) +1\n3_b(t)\nb(t): (26)\nw(t) =b(t)\u0000_\f(t) +\f(t)\u001b(t) + (1\u00002r\u00002m)\r(t)\f(t): (27)\nThen, for each solution trajectory of x: [t0;+1[!H of(IGS)\r;\f;b, we have,\n(i)f(x(t))\u0000min\nHf=O \n1\np\r(t)2rw(t)b(t)\u00002\n3!\nast!+1 (28)\n(ii)Z+1\nt0p2r\n\r(t)\u0007(t)\u0012\nf(x(t))\u0000inf\nHf\u0013\ndt< +1; (29)\n(iii)Z+1\nt0\u0012\np2r\n\r(t)b1\n3(t)\f(t)\u0000d\ndt\u0000\np2r\n\rb\u00002\n3\f2\u0001(t)\u0013\nkrf(x(t))k2dt< +1:(30)\nHere\u0007(t) :=\u0010\n3\u001b(t)\u00002(r+m)\r(t)\u0011\nw(t)\u0000_w(t)\u00002(1\u0000r\u0000m)\r(t).\nProof According to Theorem 1, it su\u000eces to show that, under the hypothesis\n(H1)\u0000(H4);there exists c;\u0012;\u001b;\u0018 which satisfy the conditions ( i)\u0000(vii) of Theorem\n1. To perform the corresponding derivative calculation, let's start by establishing\nsome preliminary results.\n\u000flnp\r(t) =Rt\nt0\r(s)ds, which by derivation gives_p\r\np\r=\r, that is to say _ p\r=\rp\r:\n\u000fAccording to the de\fnition of \u001b,\nd\ndt\u0010\np2r\n\rb\u00002\n3\u0011\n= 2p2r\n\rb\u00002\n3\u0012\nr\r\u00001\n3_b\nb\u0013\n(31)\n= 2\u0012((r+m)\r\u0000\u001b): (32)\nLet us show that the following choice of the unknown parameters c;\u0012;\u001b;\u0018 satis\fes\nthe conditions ( i)\u0000(vii) of Theorem 1:\n\u0012:=p2r\n\rb\u00002\n3; \u001b :=m\r+1\n3_b\nb; \u0018 :=\u0012\u00180;\nand\nc2b:=\u0012w:=\u0012\u0010\nb\u0000_\f+\f\u001b+ (1\u00002r\u00002m)\r\f\u0011\n; (33)\nwhere\u00180has been de\fned in (25). We underline that under condition ( H3);\nc2b=\u0012\u0010\nb\u0000_\f+\f\u001b\u0000\r\f|{z}\n\u00150+2 (1\u0000r\u0000m)|{z}\n\u00150\r\f\u0011\n\u00150:\nAlso, according to (32), we have _\u0012= 2\u0012\u0000(r+m)\r\u0000\u001b\u0001\n:Inertial dynamics with Hessian damping and time rescaling 19\n(i)d\ndt(\f\u0012\u001b)\u0000\u0012b\u001b=_\f\u0012\u001b+\f(_\u0012\u001b+\u0012_\u001b)\u0000b\u0012\u001b\n=\u0012h\n_\f\u001b+ 2\u0010\n(r+m)\r\u0000\u001b\u0011\n\f\u001b+\f_\u001b\u0000b\u001bi\nbecause _\u0012= 2((r+m)\r\u0000\u001b)\n=\u0012\u0010\n\u0000\f\u00180\u0000\u001b(b\u0000_\f+\f\u001b\u0000\r\f)\u0011\n:\n(34)\nSincebis nondecreasing, then \u001b\u00150;so by (H1) and (H3);we get\nd\ndt(\f\u0012\u001b)\u0000\u0012b\u001b=\u0012\u0010\n\u0000\f\u00180\u0000\u001b(b\u0000_\f+\f\u001b\u0000\r\f)|{z}\n\u00150\u0011\n\u00140\n(ii) According to the derivation chain rule and ( H4), we conclude that\nd\ndt\u0010\nc2b\u0011\n+d\ndt(\f\u0012\u001b)\u0000\u0012b\u001b=d\ndt(\u0012w)+d\ndt(\f\u0012\u001b)\u0000\u0012b\u001b\n=d\ndt(\u0012w+\f\u0012\u001b)\u0000\u0012b\u001b\u00140:\n(iii)b(c2\u0000\u0012) +\f\u0012(\u001b\u0000\r) +d\ndt(\f\u0012) = 0 results from (33).\n(iv) According to the derivation chain rule, (31), and the de\fnition of \u001b\nd\ndt(\u0012\u001b) +\u0012\u001b(\u001b\u0000\r)+\u0018=_\u0012\u001b+\u0012_\u001b+\u0012\u001b(\u001b\u0000\r)+\u0018\n= 2\u0012\u001b\u0000(r+m)\r\u0000\u001b\u0001+\u0012_\u001b+\u0012\u001b(\u001b\u0000\r)+\u0018\n=\u0012\u0010\n_\u001b\u0000\u001b\u0000(1\u00002r\u00002m)\r+\u001b\u0001\u0011\n+\u0018:\nFor this quantity to be equal to zero, we therefore take \u0018=\u0012\u00180, where\u00180is\nde\fned in (26).\n(v) According to our choice \u0018=\u0012\u00180, we have\n(v)()d\ndt\u0010\n\u0012(\u001b2+\u00180)\u0011\n\u00140:\nLet's compute this quantity. According to the derivation chain rule and (32)\nd\ndt\u0010\n\u0012(\u001b2+\u00180)\u0011\n=\u0010\n\u0012_\u00180+_\u0012(\u001b2+\u00180) + 2\u0012_\u001b\u001b\u0011\n= 2\u0012\u00101\n2_\u00180+ (\u001b2+\u00180)((r+m)\r\u0000\u001b)+ _\u001b\u001b\u0011\n= 2\u0012\u00101\n2_\u00180+\u00180\u0000(r+m)\r\u00002\u001b\u0001+\u001b\u0000\n\u00180+\u001b\u0000(r+m)\r\u0000\u001b\u0001+ _\u001b\u0001\u0011\n:\nBy de\fnition of \u00180, we have\u00180+ _\u001b=\u0010\n(1\u00002(r+m))\r+\u001b\u0011\n\u001b. Therefore\nd\ndt\u0010\n\u0012(\u001b2+\u00180)\u0011\n= 2\u0012\u00101\n2_\u00180+\u00180((r+m)\r\u00002\u001b)+\r\u001b2(1\u0000(r+m))\u0011\n:\nSo, (v) is satis\fed under the condition\n1\n2_\u00180+\u00180\u0000(r+m)\r\u00002\u001b\u0001+\r\u001b2\u00001\u0000(r+m)\u0001\n\u00140;\nwhich is precisely ( H2).20 Hedy ATTOUCH et al.\n(vi) Let's compute\n_\u0012+ 2(\u001b\u0000\r)\u0012= 2\u0012\u0010\nr\r\u00001\n3_b\nb+ (m\u00001)\r+1\n3_b\nb\u0011\n= 2(r+m\u00001)\u0012\r:\nAccording to the assumption m\u00141\u0000r, this quantity is less or equal than zero.\nWe have (vii)()\f(\f_\u0012+ 2( _\f\u0000b)\u0012)\u00140:\nAccording to condition H3and the assumption m\u00141\u0000r;we conclude\n\f_\u0012+ 2( _\f\u0000b)\u0012= 2\u0012\u0000\n\f(r+m)\r\u0000\f\u001b\u0000b\u0001\n= 2\u0012h\n\u0000(b\u0000_\f+\f\u001b\u0000\r\u001b)|{z}\n\u00150\u0000\f\r(1\u0000r\u0000m)|{z}\n\u00150i\n\u00140:\nSo, (vii) is satis\fed\nAccording to Theorem 1, we obtain (28)-(29)-(30) which completes the proof. u t\n4.3 The case without the Hessian\nLet us specialize the previous results in the case \f= 0, i.e.without the Hessian:\n(IGS)\r;0;b x(t) +\r(t) _x(t) +b(t)rf(x(t)) = 0:\nTheorem 6 Suppose that the conditions (H1)and(H2)of Theorem 5are satis\fed.\nThen, for each solution trajectory x: [t0;+1[!H of(IGS)\r;0;b, we have, as t!+1\nf(x(t))\u0000min\nHf=O \n1\np\r(t)2rb(t)1\n3!\n: (35)\nMoreover, when m> 2r\nZ+1\nt0p\r(t)2rb(t)1\n3\r(t)\u0012\nf(x(t))\u0000inf\nHf\u0013\ndt< +1: (36)\nProof Conditions (H1) and (H2) in Theorem 5 remain unchanged since they are\nindependent of \f. We just need to verify ( H4), because (H3) is written b(t)\u00150\nand becomes obvious. Since \f= 0, we have (H4)()d\ndt\u0010\n\u0012b\u0011\n(t)\u0000\u0012(t)b(t)\u001b(t)\u00140:\nAccording to\nd\ndt\u0010\n\u0012b\u0011\n(t)\u0000\u0012(t)b(t)\u001b(t) =_\u0012(t)b(t) +\u0012(t)_b(t)\u0000\u0012(t)b(t)\u0012\nm\r+_b(t)\n3b(t)\u0013\n=b(t)1=3\u0014\nd\ndt\u0010\n\u0012(t)b(t)2=3\u0011\n\u0000m\r(t)\u0010\n\u0012(t)b(t)2=3\u0011\u0015\n=b(t)1=3\u0014\nd\ndt\u0010\np\r(t)2r\u0011\n\u0000m\r(t)\u0010\np\r(t)2r\u0011\u0015\n= (2r\u0000m)\r(t)b(t)1=3p\r(t)2r\u00140 since 2r\u0014m;\nwe conclude that ( H4) holds, which completes the proof. u t\nNext, we show that the condition ( H2) on the coe\u000ecients \r(\u0001) andb(\u0001) can be\nformulated in simpler form which is useful in practice.Inertial dynamics with Hessian damping and time rescaling 21\nTheorem 7 The conclusions of Theorem 6remain true when we replace (H2)by\n(H+\n2)\u001b(t)\u0000\n\u001b(t)\u0000(r+m)\r(t)\u0001\u00002\u001b(t) +\u00001\u00002(r+m)\u0001\n\r(t)\u0001+1\n2\u001b(t)\u00150,\nand assume moreover that b(\u0001)is log-concave, i.e.,d2\ndt2(ln(b(t)))\u00140.\nProof According to Theorem 6, it su\u000eces to show that ( H2) is satis\fed under the\nhypothesis (H+\n2). By de\fnition of \u001b, we have\n\u00002\u001b(t)\u0000(m+r)\r(t)\u0001=\u0010\n(m\u0000r)\r(t) +2\n3_b(t)\nb(t)\u0011\n:\nSo (H2) can be written equivalently as A\u00150, where\nA=:\u00180(t)\u0010\n(m\u0000r)\r(t) +2\n3_b(t)\nb(t)\u0011\n\u00001\n2_\u00180(t) +\u0010\nm+r\u00001\u0011\n\r(t)\u001b2(t): (37)\nA calculation similar to the one above gives\n\u00180(t) =\u0010\n(1\u00002r\u0000m)\r(t)\u0000m\r(t) +\u001b(t)\u0011\n\u001b(t)\u0000_\u001b(t);\n=\u0010\n(1\u00002r\u0000m)\r(t) +1\n3_b(t)\nb(t)\u0011\n\u001b(t)\u0000_\u001b(t): (38)\nIn (37), let's replace \u00180(\u0001) by its formulation (38), we obtain\nA=1\n2d2\ndt2\u001b(t)\u00001\n2d\ndt\u0014\n\u001b(t)\u0012\n(1\u00002r\u0000m)\r(t) +1\n3_b(t)\nb(t)\u0013\u0015\n\u0000_\u001b(t)\u0012\n(m\u0000r)\r(t) +2\n3_b(t)\nb(t)\u0013\n+\u0010\nm+r\u00001\u0011\n\r(t)\u001b2(t)\n+\u0010\n(m\u0000r)\r(t) +2\n3_b(t)\nb(t)\u0011\u0010\n(1\u00002r\u0000m)\r(t) +1\n3_b(t)\nb(t)\u0011\n\u001b(t):\nSet\nB:=\u0010\nm+r\u00001\u0011\n\r(t)\u001b2(t) +\u0010\n(m\u0000r)\r(t) +2\n3_b(t)\nb(t)\u0011\u0010\n(1\u00002r\u0000m)\r(t) +1\n3_b(t)\nb(t)\u0011\n\u001b(t);\nthen we have (by omitting the variable tto shorten the formulas)\nB=\u001bh\n(m+r\u00001)\r\u001b+\u0012\n(m\u0000r)\r(t) +2\n3_b\nb\u0013\u0012\n(1\u00002r\u0000m)\r+1\n3_b\nb\u0013i\n=\u001bh\n(m+r\u00001)\r\u001b+\u0012\n\u0000r\r+1\n3_b\nb+\u001b\u0013\u0012\n(1\u00002r)\r+2\n3_b\nb\u0000\u001b\u0013i\n=\u001bh\n(m+r\u00001)\r\u001b\u0000\u001b2+\r\u001b(\u0000m+ 1\u0000r)+\u001b2+\u0012\n\u0000r\r+1\n3_b\nb\u0013\u0012\n(1\u00002r)\r+2\n3_b\nb\u0013i\n=\u001b\u0012\n\u0000r\r+1\n3_b\nb\u0013\u0012\n(1\u00002r)\r+2\n3_b\nb\u0013\n:\nReplacingBinA, we obtain\nA=\u001b(t)\u0010\n\u001b(t)\u0000(m+r)\r(t)\u0011\u0010\n2\u001b(t) + (1\u00002(m+r))\r(t)\u0011\n+1\n2d2\ndt2\u001b(t) +C(t) (39)22 Hedy ATTOUCH et al.\nwhere\nC(t) :=\u0000_\u001b(t)\u0012\n(m\u0000r)\r(t) +2\n3_b(t)\nb(t)\u0013\n\u00001\n2d\ndt\u0014\n\u001b(t)\u0012\n(1\u00002r\u0000m)\r(t) +1\n3_b(t)\nb(t)\u0013\u0015\n:\nLet us show that C(t) is nonnegative. After replacing \u001b(t) by its value m\r(t)+1\n3_b(t)\nb(t),\nand developing, we get\nC(t) =\u0000m_\r(t)\r(t)(1\u00003r)\u00001\n6(4m\u00002r+ 1)_\r(t)_b(t)\nb(t)\n\u00001\n6d2\ndt2(ln(b(t)))\u0012\n(1 + 2(m\u00002r))\r(t) + 2_b(t)\nb(t)\u0013\n:\nBy assumption, m\u00002r\u00150, 1\u00003r\u00150,\r(\u0001) is nonincreasing, b(\u0001) is nondecreasing,\nandd2\ndt2(ln(b(t)))\u00140. We conclude that C(t)\u00150. According to (39), we obtain\nA\u0015\u001b(t)\u0010\n\u001b(t)\u0000(m+r)\r(t)\u0011\u0010\n2\u001b(t) + (1\u00002(m+r))\r(t)\u0011\n+1\n2d2\ndt2\u001b(t):\nThe condition (H+\n2) expresses that the second member of the above inequality is\nnonnegative. Therefore ( H+\n2) implies (H2), which gives the claim. u t\n4.4 Comparing the two approaches\nAs we have already underlined, Theorems 2 and 7 are based on the Lyapunov\nanalysis of the dynamic (IGS)\r;0;busing the functions \u0000\randp\r, respectively. As\nsuch, they lead to signi\fcantly di\u000berent growth conditions on the coe\u000ecients of\nthe dynamic. Precisely, using the following example, we will show that Theorem\n7 better captures the case where bhas an exponential growth. Take\nb(t) =e\u0016tqand\r(t) =\u000b\nt1\u0000qwith\u000b=\u0016q> 0; q2(0;1):\na)First, let us show that the condition ( H+\n2) of Theorem 7 is satis\fed. We have\n1\n2\u001b(t) +\u001b(t)\u0010\n\u001b(t)\u0000(m+r)\r(t)\u0011\u0010\n2\u001b(t) + (1\u00002(m+r))\r(t)\u0011\n= (\u0016q)3\u0012\nm+1\n3\u0013\u00121\n3\u0000r\u0013\u00125\n3\u00002r\u00131\nt3\u00003q+1\n2\u0016q\u0012\nm+1\n3\u0013\n(1\u0000q)(2\u0000q)1\nt3\u0000q\nwhich is nonnegative because of the hypothesis r\u00141\n3andq<1.\nb)Let us now examine the growth condition used in Theorem 2:\n\u0000(t)_b(t)\u0014b(t)\u0010\n3\u00002\r(t)\u0000(t)\u0011\nwhere\u0000(t) :=p(t)Z+1\ntds\np(s): (40)\nHerept) =e\u0016(tq\u0000tq\n0). Therefore \u0000(t) =e\u0016tqZ+1\nte\u0000\u0016sq\nds, which gives\n\u0000(t)_b(t)\u0000b(t)\u0010\n3\u00002\r(t)\u0000(t)\u0011\n= 3e\u0016tq\u0012\n\u0016qtq\u00001e\u0016tqZ+1\nte\u0000\u0016sq\nds\u00001\u0013\n:Inertial dynamics with Hessian damping and time rescaling 23\nLet us analyze the sign of the above quantity, which is the same as\nD(t) :=\u0016qtq\u00001e\u0016tqZ+1\nte\u0000\u0016sq\nds\u00001\n=\u0000\u0016qtq\u00001e\u0016tqZ+1\ntd\nds\u0010\ne\u0000\u0016sq\u00111\n\u0016qs1\u0000qds\u00001\nAfter integration by parts, we get\nD(t) :=\u00121\nq\u00001\u0013\n+1\u0000q\nqtq\u00001e\u0016tqZ+1\nte\u0000\u0016sq1\nsqds>\u00121\nq\u00001\u0013\n>0:\nTherefore, the condition (40) is not satis\fed.\n5 Illustration of the results\nLet us particularize our results in some important special cases, and compare them\nwith the existing litterature. We do not detail the proofs which result from the\ndirect applications of the previous theorems and the classical di\u000berential calculus.\n5.1 The case b(t) =p(t)3p0.\nRecall that p(t) = exp\u0012Zt\nt0\r(s)ds\u0013\n. We start with results in [7] concerning the\nrate of convergence of values in the case b(t) =c0p(t)3p0withp0\u00150 andc0\u00150.\nIn this case, the system (IGS) \r;0;bbecomes:\nx(t) +\r(t) _x(t) +c0exp\u0012\n3p0Zt\nt0\r(s)ds\u0013\nrf(x(t) = 0: (41)\nObserve that_b(t)\n3b(t)=p0\r(t) and\u00180(t) = (m+p0)\u0000(1\u00002r\u0000m+p0)\r2(t)\u0000_\r(t)\u0001.\nTherefore, conditions ( H1) and (H2) of Theorem 6 become after simpli\fcation:\n(H1) [(p0\u0000r) + (1\u0000r\u0000m)]\r2(t)\u0000_\r(t)\u00150;\n(H2) 2(p0\u0000r)(1 + 2(p0\u0000r))\r3(t)\u00002(1 + 3(p0\u0000r))\r(t)_\r(t) + \r(t)\u00150.\nSincem\u00141\u0000r, instead of (H1), it su\u000eces to verify\n(H+\n1)\u0000\np0\u0000r\u0001\n\r2(t)\u0000_\r(t)\u00150.\nTheorem 8 Let\r: [t0;+1)!R+be a nonincreasing and twice continuously di\u000ber-\nentiable function. Suppose that there exists r2(0;1\n3\u0003\nsuch that\n\r(t)\u00152\u0002min(0;p0\u0000r)\u00032\r3(t)on[t0;+1): (42)\nThen, for each solution trajectory x(\u0001)of(41), we have as t!+1\nf(x(t))\u0000min\nHf=O\u00121\np(t)2r+p0\u0013\n: (43)24 Hedy ATTOUCH et al.\nProof To prove the claim, we use Theorem 5 and distinguish two cases:\n?Supposep\u0000r\u00150, then (42) implies  \r(t)\u00150, and since \ris a nonincreasing,\nwe also have _ \r(t)\u00140; thus both conditions ( H+\n1) and (H2) are satis\fed.\n?Supposep\u0000r<0, then (42) becomes\n\r(t)\u0015(2p\u0000r)2\r3(t) on [t0;+1): (44)\nSince\r(\u0001) is a positive and nonincreasing, lim t!+1\r(t) =`exists and is equal to\nzero. Otherwise, by integrating (44) on [ t0;t] fort>t 0, we would have\n_\r(t)\u0000_\r(t0)\u00152(p\u0000r)2Zt\nt0\r(s)3ds\u00152(p\u0000r)2`3(t\u0000t0):\nThis in turn gives lim t!+1_\r(t) = +1, which implies lim t!+1\r(t) = +1, that is\na contradiction. Then, multiply (44) by _ \r(t). Since\r(\u0001) is nonincreasing, we obtain\n\r(t)_\r(t)\u00142(p\u0000r)2\r3(t)_\r(t)()1\n2d\ndt(_\r(t)2)\u0014(p\u0000r)2\n2d\ndt(\r4(t)):\nBy integrating this inequality from ttoT >t , we get\n_\r(T)2\u0000_\r(t)2\u0014(p\u0000r)2(\r4(T)\u0000\r4(t));\nLettingT!+1;and using lim T!+1\r(T) = 0, we obtain _ \r2(t)\u0015(p\u0000r)2\r4(t);\nwhich is equivalent to j_\r(t)j\u0015jp\u0000lj\r2(t):Since _\r(t)\u00140 andp < r , this gives\n\u0000_\r(t)\u0015(r\u0000p)\r2(t);8t>t 0, that is (H+\n1):We have\n[(p\u0000r) + (1\u0000r\u0000m)]\r2(t)\u0000_\r(t)\n=\u00002(p\u0000r)2\r3(t) + \r(t)|{z}\n\u00150 by (44)+2 (1\u00003r+ 3p)|{z}\n\u00150 sincep<r\r(t)\u0000(p\u0000r)\r2(t)\u0000_\r(t)|{z}\n\u00150 by (H+\n1)\u0001\n\u00150:\nTherefore, (H+\n1) and (H2) are satis\fed. Applying Theorem 5, we conclude. u t\nAs a particular case of Theorem 8, with p0= 0, we obtain the following result.\nTheorem 9 [7, Theorem 2.1] Let\r(\u0001)be a nonncreasing function of class C2, and\nx(\u0001)a solution trajectory of\nx(t) +\r(t) _x(t) +c0rf(x(t) = 0: (45)\nSuppose that\n(Hr;\r)9r>0such that\u00002r2\r3(t) + \r(t)\u00150fortlarge enough.\nThen,f(x(t))\u0000minHf=O\u0010\ne\u00002 min(r;1\n3)Rt\nt0\r(s)ds\u0011\nast!+1.\nRemark 4 The case\r(t) =1\nt(lnt)\u001a, for 0\u0014\u001a\u00141, was developed in [7]. In that case\ncondition (H3;\r) writes as\n2(lnt)2+ 3\u001alnt+\u001a(\u001a+ 1)\u00152r2(lnt)2(1\u0000\u001a);\nwhich is satis\fed for any r\u00141 and any t\u0015e.Inertial dynamics with Hessian damping and time rescaling 25\n{If\u001a= 1, thenp(t) = exp\u0012Zt\nt01\ns(lns)\u001ads\u0013\n= exp Zlnt\nlnt0du\nx!\n=lnt\nlnt0;\nand forr=1\n3, we getf(x(t))\u0000minHf=O\u0012\n1\n(lnt)2\n3\u0013\n:\n{If 0\u0014\u001a<1, thenp(t) = exp Zlnt\nlnt01\nu\u001adu!\n= exp\u0010\n1\n1\u0000\u001a\u0000(lnt)1\u0000\u001a\u0000(lnt0)1\u0000\u001a\u0001\u0011\n;\nand, forr=1\n3, we also get f(x(t))\u0000minHf=O\u0012\n1\nexp\u0010\n2\n3(1\u0000\u001a)(lnt)1\u0000\u001a\u0011\u0013\n:\n5.2 The case b(t) =c0tqand\r(t) =\u000b\nt.\nWhenb(t) =c0tqand\r(t) =\u000b\ntwhere\u000b > 0 andq\u00150, we \frst observe that\np(t) = exp\u0010Rt\nt0\r(s)ds\u0011\n=\u0000t\nt0\u0001\u000b:The second-order continuous system becomes:\nx(t) +\u000b\nt_x(t) +c0tqrf(x(t)) = 0: (46)\nApplying Theorem 8, we obtain the following new result.\nTheorem 10 Letx(\u0001)be a solution trajectory of (46) with\u000b>1andq\u00150. Suppose\nthat1<\u000b\u00143 +q. Then,\nf(x(t))\u0000min\nHf=O\u00121\nt2\u000b+q\n3\u0013\n;ast!+1: (47)\nRemark 5 Takingq= 0, a direct application of the above result covers the results\nobtained in [9,32] (case \u000b\u00153), and in [3,10], (case \u000b\u00143). It su\u000eces to take\n\r(t) =\u000b\ntandr=1\n\u000b. More precisely, we get :\n{if 0<\u000b\u00143 thenf(x(t))\u0000minHf=O(t\u00002\u000b\n3),\n{if\u000b>3 thenf(x(t))\u0000minHf=O(1\nt2).\n5.3 The case b(t) =e\u0016tqand\r(t) =\u000b\nt1\u0000q.\nSuppose that \u0016\u00150;0\u0014q\u00141 and\u000b>0. This will allow us to obtain the following\nexponential convergence rate of the values.\nTheorem 11 Letx: [t0;+1[\u0000!H be a solution trajectory of\nx(t) +\u000b\nt1\u0000q_x(t) +e\u0016tq\nrf(x(t)) = 0: (48)\nSuppose that \u000b\u0014\u0016q, then, ast!+1\nf(x(t))\u0000min\nHf=O\u0010\ne\u00002\u000b+\u0016q\n3tq\u0011\n:26 Hedy ATTOUCH et al.\nRemark 6 a) Forq=\u0016= 0, (48) reduces to the system initiated in [32], i.e.\nx(t) +\u000b\nt_x(t) +rf(x(t)) = 0:\nJust assuming \u000b>0, we obtain lim\nt!+1\u0012\nf(x(t))\u0000min\nHf\u0013\n= 0.\nb) Forq=1\n2we get\n{If\u000b\u0014\u0016, thenf(x(t))\u0000minHf=O\u0010\ne\u00002(2\u000b+\u0016)\n3)p\nt\u0011\n:\n{If\u000b\u0015\u0016, thenf(x(t))\u0000minHf=O\u0010\ne\u00002\u0016p\nt\u0011\n:\nc) Forq= 1, direct application of Theorem 11 gives:\nCorollary 3 (Linear convergence) Letx: [t0;+1[!H be a solution trajectory of\nx(t) +\u000b_x(t) +e\u0016trf(x(t)) = 0: (49)\nIf\u000b\u0014\u0016;thenf(x(t))\u0000minHf=O\u0010\ne\u00002\u000b+\u0016\n3t\u0011\n:\nLet us illustrate these results. Take f(x1;x2) :=1\n2\u0000\nx2\n1+x2\n2\u0001\n\u0000ln(x1x2), which is a\nstrongly convex function. Trajectories of\nx(t) +\u000b_x(t) +e\u0016trf(x(t)) +ce\u0017tr2f(x(t)) _x(t) = 0;\ncorresponding to di\u000berent values of the parameters \u000b,\u0016,\u0017, andc, are plotted in\nFigure 12. The parameter cshows the importance of the Hessian-damping.\nFig. 1 Evolution of f(x(t))\u0000minffor solutions of (49), (50), and f(x1;x2) =1\n2\u0000\nx2\n1+x2\n2\u0001\n\u0000\nln(x1x2).\n2From Scilab version 6.1.0 http://www.scilab.org as an open source softwareInertial dynamics with Hessian damping and time rescaling 27\n\r(t)\f(t)b(t) f(x(t))\u0000minf Reference\nCte 0 1 O\u0000\nt\u00001\u0001\n(1964) [29]\nCte Cte 1 O\u0000\nt\u00001\u0001\n(2002) [2]\n\u000b=t 0 1O\u0010\nt\u00002\n3\u000b\u0011\nif 0<\u000b\u00143\nO\u0000\nt\u00002\u0001\nif\u000b\u00153(2019) [10]\n(2014) [32]\n\u000b=t Cte 1O\u0000\nt\u00002\u0001\nif\u000b\u00153;\f> 0 (2016) [17]\n\r(t) 0b(t)O\u0012\u0010\np(t)R+1\nt(p(s))\u00001ds\u0011\u00002\n(b(t))\u00001\u0013\nwherep(t) := exp\u0010Rt\nt0\r(s)ds\u0011 (2019) [10]\n\u000b=t\f(t)b(t)O \u0012\nt2b(t)\u0000_\f(t)\u0000\f(t)\nt\u0013\u00001!\n(2020) [8]\nFig. 2 Convergence rate of f(x(t))\u0000minffor instances of Theorem 1 and general f.\n5.4 Numerical comparison\nFigure 2 summarizes our convergence results, according to the behavior of the\nparameters \r(t),\f(t),b(t). Let's comment on them and compare them, separately\nconsidering fto be strongly convex or not.\n5.4.1 Strongly convex case\nSuppose that fiss-strongly convex. Following Polyak's [29], the system\nx(t) + 2ps_x(t) +rf(x(t)) = 0 (50)\nprovides the linear convergence rate f(x(t))\u0000infHf\u0014Ce\u0000pst, see also [31, The-\norem 2.2]. In the presence of an additional Hessian-driven damping term\nx(t) + 2ps_x(t) +\fr2f(x(t)) _x(t) +rf(x(t)) = 0 (\f\u00150) (51)\na related linear rate of convergence can be found in [8, Theorem 7]. Let us insist\non the fact that, in Corollary 3, we obtain a linear convergence rate for a general\nconvex di\u000berentiable function f. In Figure 1, for the strongly convex function\nf(x1;x2) =1\n2\u0000\nx2\n1+x2\n2\u0001\n\u0000ln(x1x2);we can observe that some values of \u0016give a\nbetter speed of convergence of f(x(t))\u0000minf. We can also note that for \u0016correctly\nset, the system (49) provides a better linear convergence rate than the system (50).\n5.4.2 Non-strongly convex case\nWe illustrate our results on the following simple example of a non strongly convex\nminimization problem, with non unique solutions.\nmin\nR2f(x1;x2) =1\n2(x1+ 103x2)2: (52)\nFrom Figure 3 we get the following properties:\na) The convergence rate of the values is in accordance with Figure 2.\nb) The system (49) is best for its linear convergence of values.\nc) The Hessian-driven damping reduces the oscillations of the trajectories.28 Hedy ATTOUCH et al.\nFig. 3 Evolution of f(x(t))\u0000minffor systems in Figure 2, and f(x1;x2) =1\n2\u0000\nx2\n1+ 103x2\n2\u0001\n.\n6 Conclusion, perspectives\nOur study is one of the \frst works to simultaneously consider the combination of\nthree basic techniques for the design of fast converging inertial dynamics in con-\nvex optimization: general viscous damping (and especially asymptotic vanishing\ndamping in relation to the Nesterov accelerated gradient method), Hessian-driven\ndamping which has a spectacular e\u000bect on the reduction of the oscillatory aspects\n(especially for ill-conditionned minimization problems), and temporal rescaling.\nWe have introduced a system of equations-inequations whose solutions provide\nthe coe\u000ecients of a general Lyapunov functions for these dynamics. We have been\nable to encompass most of the existing results and \fnd new solutions for this sys-\ntem, thus providing new Lyapunov functions. Also, we have been able to explain\nthe mysterious coe\u000ecients which have been used in recent algorithmic develope-\nments, and which were just justi\fed until now by the simpli\fcation of complicated\ncalculations. Finally, by playing on fast rescaling methods, we have obtained linear\nconvergence results for general convex functions. This work provides a basis for\nthe development of corresponding algorithmic results.\nReferences\n1. F. \u0013Alvarez, On the minimizing property of a second-order dissipative system in Hilbert\nspaces, SIAM J. Control Optim., 38 (4) (2000), 1102-1119.\n2. F. \u0013Alvarez, H. Attouch, J. Bolte, P. Redont, A second-order gradient-like dissipative dy-\nnamical system with Hessian-driven damping, J. Math. Pures Appl., 81 (2002), 747{779.\n3. V. Apidopoulos, J.-F. Aujol, Ch. 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Control Optim., 54 (2016), 1423-1443.\n20. R. I. Bot \u0018, E. R. Csetnek, S.C. L\u0013 aszl\u0013 o, Approaching nonsmooth nonconvex minimization\nthrough second order proximal-gradient dynamical systems, J. Evol. Equ., 18(3) (2018),\n1291{1318.\n21. R. I. Bot \u0018, E. R. Csetnek, S.C. L\u0013 aszl\u0013 o, Tikhonov regularization of a second order dynamical\nsystem with Hessian damping, Math. Program., DOI:10.1007/s10107-020-01528-8.\n22. O. G uler, On the convergence of the proximal point algorithm for convex optimization,\nSIAM J. Control Optim., 29 (1991), 403{419.\n23. O. G uler, New proximal point algorithms for convex minimization, SIAM Journal on\nOptimization, 2 (4) (1992), 649{664.\n24. A. Haraux, Syst\u0012 emes Dynamiques Dissipatifs et Applications, Recherches en\nMath\u0013 ematiques Appliqu\u0013 ees 17, Masson, Paris, 1991.\n25. R. May, Asymptotic for a second order evolution equation with convex potential and\nvanishing damping term, Turkish Journal of Mathematics, 41 (3) (2016), 681{685.\n26. Y. Nesterov, A method of solving a convex programming problem with convergence rate\nO(1=k2), Soviet Mathematics Doklady, 27 (1983), 372{376.\n27. Y. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course. Springer\nScience+Business Media New York (2004).\n28. J. Peypouquet, S. Sorin, Evolution equations for maximal monotone operators: asymptotic\nanalysis in continuous and discrete time, J. Convex Anal, 17 (3-4) (2010), 1113{1163.\n29. B.T. Polyak, Some methods of speeding up the convergence of iteration methods, U.S.S.R.\nComput. Math. Math. Phys., 4 (1964), 1{17.\n30. B. Shi, S.S Du, M.I. Jordan, W.J. Su, Understanding the acceleration phenomenon via\nhigh-resolution di\u000berential equations, arXiv:submit/2440124[cs.LG] 21 Oct 2018.\n31. W. Siegel, Accelerated \frst-order methods: Di\u000berential equations and Lyapunov functions.\narXiv:1903.05671v1 [math.OC] (2019).\n32. W.J. Su, S. Boyd, E.J Cand\u0012 es, A di\u000berential equation for modeling Nesterov's accelerated\ngradient method: theory and insights,Neural Information Processing Systems, 27 (2014),\n2510{2518."    },    {        "title": "1905.09915v1.Escaping_Locally_Optimal_Decentralized_Control_Polices_via_Damping.pdf",        "content": "Escaping Locally Optimal Decentralized Control\nPolices via Damping\u0003\nHan Feng and Javad Lavaeiy\nApril 20, 2022\nAbstract\nWe study the evolution of locally optimal decentralized controllers with\nthe damping of the control system. Empirically it is shown that even for\ninstances with an exponential number of connected components, damping\nmerges all local solutions to the one global solution. We characterize the\nevolution of locally optimal solutions with the notion of hemi-continuity\nand further derive asymptotic properties of the objective function and of\nthe locally optimal controllers as the damping becomes large. Especially,\nwe prove that with enough damping, there is no spurious locally optimal\ncontroller with favorable control structures. The convoluted behavior of\nthe locally optimal trajectory is illustrated with numerical examples.\n1 Introduction\nThe optimal decentralized control problem (ODC) adds controller constraints to\nthe classical centralized optimal control problem. This addition breaks down\nthe separation principle and the classical solution formulas culminated in [ 4].\nAlthough ODC has been proved intractable in general [ 23,1], the problem\nhas convex formulations under assumptions such as partially nestedness [ 19],\npositiveness [ 17], and quadratic invariance [ 10]. A recently proposed System\nLevel Approach [ 21] convexified the problem in the space of system response\nmatrix. Convex relaxation techniques have been extensively documented in [ 2],\nthough it is considered challenging to solve large scale optimization problems\nwith linear matrix inequalities.\nThe line of research on convexification is in contrast with the success of\nstochastic gradient descent well-documented in machine learning practice [ 9,\n8]. Admittedly, the problem of generalizability, training speed, and fairness\nin machine learning departs from the traditional control focus on stability,\nrobustness, and safety. Nevertheless, the interplay of the two has inspired fruitful\nresults. As an example, to solve the linear-quadratic optimal control problem, the\ntraditional nonlinear programming methods include Gauss-Newton, augmented\n\u0003Email: han_feng@berkeley.edu, lavaei@berkeley.edu\nyThis work was supported by grants from ARO, ONR, AFOSR, and NSF.\n1arXiv:1905.09915v1  [math.OC]  23 May 2019Lagrangian, and Newton’s methods [ 11,22,12,13]. Only in the last few years\ndo researchers started to look at the classical problem with the newly developed\noptimization techniques and proved the efficiency of policy gradient methods\nin model-based and model-free optimal control problems [ 6]. This efficiency\nstatement of local search, however, is unlikely to carry over trivially to ODC, due\nto the NP-hardness of the problem and the recent investigation of the topological\nproperties of ODC in [ 7]. Nevertheless, questions can be answered without\ncontradicting the general complexity statement. For example, it is known that\ndamping of the system reduces the number of connected components of the set\nof stabilizing decentralized controllers. Does damping reduce the number of\nlocally optimal decentralized controllers? This paper attempts an answer with\n(1) a study of the continuity properties of the trajectories of the locally optimal\nsolutions formed by varying damping, and (2) an asymptotic analysis of the\ntrajectories as the damping becomes large. The observation of our study shall\nshed light on the properties of local minima in reinforcement learning, whose\naim is to design optimal control policies and different local minima have different\npractical behaviors.\nThis work is closely related to continuation methods such as homotopy. They\nare known to be appealing yet theoretically poorly understood [15]. Homotopy\nhas been used as an initialization strategy in optimal control: in [ 3], the author\nmentioned the idea of gradually moving from a stable system to the original\nsystem to obtain a stabilizing controller. The paper [ 24] considered H2-reduced\norder problem and proposed several homotopy maps and initialization strategies;\nin its numerical experiments, initialization with a large multiple of \u0000Iwas\nfound appealing. [ 5] compared descent and continuation algorithms for H2\noptimal reduced-order control problem and concluded that homotopy methods\nare empirically superior to descent methods. The difficulty of obtaining a\nconvergence theory for general constrained optimal control problem can be\nappreciated from the examples in [ 14]. Compared with those earlier works, we\nconsider a special kind of continuation, that is, damping, to improve the locally\noptimal solutions in optimal decentralized control. Our focus is not so much on\nfollowing a specific path but on the evolution of several paths and the movement\nof locally optimal solutions from one path to another.\nThe remainder of this paper is organized as follows. Notations and problem\nformulations are given in Section 2. Continuity and asymptotic properties of\nour damping strategies are outlined in Section 3 and Section 4, respectively.\nNumerical experiments are detailed in Section 5. Concluding remarks are drawn\nin Section 6.\n2 Problem Formulation\nConsider the linear time-invariant system\n_x(t) =Ax(t) +Bu(t);\n2whereA2Rn\u0002nandB2Rn\u0002mare real matrices of compatible sizes. The\nvectorx(t)is the state of the system with an unknown initialization x(0) =x0,\nwherex0is modeled as a random variable with zero mean and a positive definite\ncovariance E[x(0)x(0)>] =D0. The control input u(t)is to be determined via\na static state-feedback law u(t) =Kx(t)with the gain K2Rm\u0002nsuch that\nsome quadratic performance measure is maximized. Given a controller K, the\nclosed-loop system is\n_x(t) = (A+BK)x(t):\nA matrix is said to be stable if all its eigenvalues lie in the open left half plane.\nThe controller Kis said to stabilize the system if A+BKis stable. ODC\noptimizes over the set of structured stabilizing controllers\nfK:A+BKis stable;K2Sg;\nwhereS \u0012Rm\u0002nis a linear subspace of matrices, often specified by fixing\ncertain entries of the matrix to zero. In that case, the sparsity pattern can be\nequivalently described with the indicator matrix IS, whose (i;j)-entry is defined\nto be\n[IS]ij=(\n1;ifKijis free\n0;ifKij= 0.\nThe structural constraint K2Sis then equivalent to K\u000eIS=K, where\n\u000edenotes entry-wise multiplication. In the following, we will consider the\ndiscounted, or damped cost, which is defined as\nJ(K;\u000b) =EZ1\n0\u0002\ne\u00002\u000bt\u0000\n^x>(t)Q^x(t) + ^u>(t)R^u(t)\u0001\u0003\ndt\ns:t: ^_x(t) =A^x(t) +B^u(t)\n^u(t) =K^x(t):(1)\nwhereQ\u00170is positive semi-definite and R\u001f0is positive definite. The\nexpectation is taken over x0. Settingx(t) =e\u0000\u000bt^x(t);u(t) =e\u0000\u000bt^u(t), the cost\nJ(K;\u000b)can be equivalently written as\nJ(K;\u000b) =EZ1\n0\u0002\nx>(t)Qx(t) +u>(t)Ru(t)\u0003\ndt\ns:t: _x(t) = (A\u0000\u000bI)x(t) +Bu(t)\nu(t) =Kx(t);(2)\nThe two equivalent formulations above motivate the notion of “damping\nproperty”. We make a formal statement below.\nLemma 1. The function J(K;\u000b)defined in (1)and(2)satisfies the following\n“damping property”: suppose that Kstabilizes the system (A\u0000\u000bI;B ), then for\nall\f >\u000b,Kstabilizes the system (A\u0000\fI;B )withJ(K;\f)<J(K;\u000b).\n3Proof.From the formulation (3), whenA\u0000\u000bI+BKis stable and \f > \u000b, it\nholds that A\u0000\fI+BK = (A\u0000\u000bI+BK)\u0000(\f\u0000\u000b)Iis stable. Therefore,\nJ(K;\f)is well-defined. From formulation (1), J(K;\f)<J(K;\u000b).\nThe ODC problem can be succinctly written as\nminJ(K;\u000b)\ns:t: K2S\nA\u0000\u000bI+BKis stable:(3)\nWe denote its set of globally optimal controllers by K\u0003(\u000b), and its set of locally\noptimal controllers by Ky(\u000b). The paper studies the properties of K\u0003(\u000b),Ky(\u000b),\nandJ(K;\u000b)forK2K\u0003(\u000b)orKy(\u000b).\nTo motivate the study of Ky(\u000b), consider Figure 1 below. The set-up of\nthe experiments will be detailed in Section 5. It is known that systems of this\ntype have a large number of locally optimal controllers [ 7]. The left figure plots\nselected trajectories of J(K;\u000b)against\u000b, whereK2Ky(\u000b). The selected\ntrajectories are connected to a stabilizing controller in Ky(0). The lowest curve\ncorresponds to J(K\u0003(\u000b);\u000b). The right figure plots the distance of the selected\nK2Ky(\u000b)to the oneK2K\u0003(\u000b).\nFigure 1: Trajectory of system in equation (11)\nThe fact that modest damping causes the locally optimal trajectories to\n“collapse” to each other is a very attractive phenomenon. Especially, they suggest\ntwo improving heuristics.\n\u000fSolve (3) from a large \u000band then gradually decrease \u000bto0.\n\u000fStart from a locally optimal K2Ky(\u000b), solve(3)while gradually increase\n\u000bto a positive value and then decrease \u000bto0.\nThe first idea shall avoid many unnecessary local optimum and its empirical\nbehavior has been documented in [ 24]. The second idea has the potential to\nimprove the locally optimal controllers obtained from many other methods. Due\nto the NP-hardness of general ODC, we expect no guarantee of producing a\n4globally optimal, or even a stabilizing, decentralized controller. The breakdown\nof these heuristics will be discussed in Section 5.\n3 Continuity\nThis section studies the continuity properties of K\u0003(\u000b)andKy(\u000b). The key\nnotion of hemi-continuity captures the evolution of parametrized optimization\nproblems.\nDefinition1. The set valued map \u0000 :A!Bis said to be upper hemi-continuous\n(uhc) at a point aif for any open neighborhood Vof\u0000(a)there exists a neigh-\nborhoodUofasuch that \u0000(U)\u0012V.\nA related notion of lower hemi-continuity is provided in the supplement.\nA set-valued map is said to be continuous if it is both upper and lower hemi-\ncontinuous. A single-valued function is continuous if and only if it is uhc. We\nrestate a version of Berge Maximum Theorem with a compactness assumption\nfrom [16].\nLemma 2 (Berge Maximum Theorem) .LetA\u0012RandS\u0012Rm\u0002n, assume\nthatJ:S\u0002A! Ris jointly continuous and \u0000 :A!Sis a compact-valued\ncorrespondence. Define\nK\u0003(\u000b) = arg minfJ(K;\u000b)jK2\u0000(\u000b)g;for all\u000b2A; (4)\nand\nJ(K\u0003(\u000b);\u000b) = minfJ(K;\u000b)jK2\u0000(\u000b)g;for all\u000b2A:\nIf\u0000is continuous at some \u000b2A, thenJ(K\u0003(\u000b);\u000b)is continuous at \u000b. Fur-\nthermore,K\u0003is non-empty, compact-valued, closed, and upper hemi-continuous.\nBerge Maximum Theorem does not trivially apply to ODC: the set of stabi-\nlizing controllers is open and often unbounded. However, a lower-level set trick\napplies.\nTheorem 1. Assume that K\u0003(0)is non-empty, then the set K\u0003(\u000b)is non-empty\nfor all\u000b>0.K\u0003(\u000b)is upper hemi-continuous and the optimal cost J(K\u0003(\u000b);\u000b)\nis continuous and strictly decreasing in \u000b.\nProof.WhenK\u0003(0)is non-empty, there is an optimal decentralized controller\nfor the undamped system. With the set of stabilizing controller non-empty, we\nincur the “damping property” in Lemma 1 and conclude\nJ(K\u0003(\u000b);\u000b)\u0014J(K\u0003(0);\u000b)<J(K\u0003(0);0):\nThe inequality above assumed existence of the globally controller for all values\nof damping parameter \u000b. This is true because the lower-level set of J(K;\u000b)is\ncompact [20]. Precisely, define \u0000M(\u000b)to be\n\u0000M(\u000b) =fK2S:A\u0000\u000bI+BKis stable and J(K;\u000b)\u0014Mg:\n5The set-valued function \u0000Mis compact-valued for all fixed \u000bgiven a fixed\nM. From the damping property, we can select any M > J (K\u0003(0);0)and\noptimize instead over \u0000M(\u000b)without losing any globally optimal controller. The\ncontinuity of \u0000M(\u000b)at\u000bfor almost all Mis proved in the supplement. Berge\nmaximum theorem then applies and yields the desired continuity of K\u0003(\u000b)and\nJ(K\u0003(\u000b);\u000b).\nThe argument above can be extended to characterize all locally optimal\ncontrollers. A caveat is the possible existence of locally optimal controllers with\nunbounded cost. Their existence does not contradict the damping property —\ndamping can introduce locally optimal controllers that are not stabilizing without\nthe damping.\nTheorem 2. Assume that Ky(0)is non-empty, then the set Ky(\u000b)is nonempty\nfor all\u000b>0. Suppose furthermore that at an \u000b0>0\nlim\n\u000f!0+sup\n\u000b2[\u000b0\u0000\u000f;\u000b0+\u000f]sup\nK2Ky(\u000b)J(K;\u000b)<1;\nthenKy(\u000b)is upper hemi-continuous at \u000b0and the optimal cost J(Ky(\u000b);\u000b)is\nupper hemi-continuous at \u000b0.\nProof.ThatKy(\u000b)is non-empty follows from the existence of globally optimal\ncontrollers in Theorem 1. Consider the parametrized optimization problem\nminkrJ(K;\u000b)k\ns:t: K2\u0000M(\u000b): (5)\nThe assumption ensures the existence of an Mand an\u000f >0such thatM >\nJ(K;\u000b)forK2Ky(\u000b)where\u000b2[\u000b0\u0000\u000f;\u000b0+\u000f]. This choice of Mguarantees\nthat the formulation (5)does not cut off any locally optimal controllers. As\nproved in the supplement, \u0000M(\u000b)is continuous at \u000b0for almost any M, and\na largeMcan be selected to make \u0000M(\u000b)continuous at \u000b0. Berge Maximum\nTheorem applies to conclude that Ky(\u000b)is upper hemi-continuous. Since J(K;\u000b)\nis jointly continuous in (K;\u000b),J(Ky(\u000b);\u000b)is upper hemi-continuous.\n4 Asymptotic Properties\nIn this section, we state asymptotic properties of the local solutions Ky(\u000b). The\ncontrollersK2Ky(\u000b)satisfy the first order necessary conditions in the following\nequations (6)-(9); their derivation can be found in [18].\n(A\u0000\u000bI+BK)>P\u000b(K) +P\u000b(K)(A\u0000\u000bI+BK) +K>RK+Q= 0(6)\nL\u000b(K)(A\u0000\u000bI+BK)>+ (A\u0000\u000bI+BK)L\u000b(K) +D0= 0 (7)\n((B>P\u000b(K) +RK)L\u000b(K))\u000eIS= 0 (8)\nK\u000eIS=K: (9)\n6The above conditions provide a closed-form expression of the cost\nJ(K;\u000b) = tr(D0P\u000b(K)): (10)\nIt is worth pointing out that equations (6)-(10)are algebraic, involving only\npolynomial functions of the unknown matrices K;P\u000bandL\u000b. The matrices P\u000b\nandL\u000bare written as a function of Kbecause they are uniquely determined\nfrom(6)and(7)given a stabilizing controller K. The following theorem charac-\nterizes the evolution of locally optimal controllers for a specific sparsity pattern.\nThe theorem justifies the practice of random initialization around zero.\nTheorem 3. Suppose that the sparsity pattern ISis block-diagonal with square\nblocks and that Rhas the same sparsity pattern as IS. Then, all points in Ky\nconverge to the zero matrix as \u000b!1. Furthermore, J(K;\u000b)!0as\u000b!1\nfor allK2Ky(\u000b).\nNot only do all locally optimal controllers approach zero, the problem is in\nfact convex over bounded regions with enough damping.\nTheorem 4. For any given r >0, the Hessian matrix r2J(K;\u000b)is positive\ndefinite overkKk\u0014rfor all large \u000b.\nThe proof of the two theorems above is given in the supplement.\nCorollary 1. With the assumption of Theorem 3, there is no spurious locally\noptimal controller for large \u000b. That is,Ky(\u000b) =K\u0003(\u000b)for all large \u000b.\nProof.For any given r >0, all controllers in the ball B=fK:kKk\u0014rgare\nstabilizing when \u000bis large. As a result, stability constraints can be relaxed\noverB. Furthermore, from Theorem 3, when \u000bis large, all locally optimal\ncontrollers will be inside B. From Theorem 4, the objective function become\nconvex overBfor large enough \u000b. The observations imply local and global\nsolutions coincide.\nThe theorems above rely on the “damping property” in Lemma 1. It is worth\ncommenting that damping the system with \u0000Iis almost the only continuation\nmethod for general system matrices Athat achieves the monotonic increasing of\nstable sets. Formally,\nTheorem 5. Whenn\u00153, for anyn-by-nreal matrix Hthat is not a multiple\nof\u0000I, there exists a stable matrix Afor whichA+His unstable.\nThe proof is given in the supplement. This theorem justifies the use of \u0000\u000bIas\nthe continuation parameter. However, in a given system with structure, matrices\nother than\u0000Imay be appropriate.\n75 Numerical Experiments\nIn this section, we document various homotopy behaviors as the damping param-\neter\u000bvaries. The focus is on the evolution of locally optimal trajectories, which\ncan be tracked by any local search methods. The experiments are performed on\nsmall-sized systems so the random initialization can find a reasonable number\nof distinct locally optimal solutions. Despite the small system dimension, the\nexistence of many locally optimal solutions and their convoluted trajectories\ndemonstrates what is possible in a theory of homotopy.\nThe local search methods we used is the simplest projected gradient descent.\nAt a controller Ki, we perform line search along the direction ~Ki=\u0000rJ(K)\u000eIS.\nThe step size is determined with backtracking and Armijo rule, that is, we select\nsias the largest number in f\u0016s;\u0016s\f;\u0016s\f2;:::gsuch thatKi+si~Kiis stabilizing\nwhile\nJ(Ki+si~Ki)<J(Ki) +\u000bsihrJ(Ki);~Kii:\nOur choice of parameters are \u000b= 0:001,\f= 0:5, and \u0016s= 1. We terminate the\niteration when the norm of the gradient is less than 10\u00003.\n5.1 Systems with a large number of local minima\nWe first consider the examples from [ 7], where the feasible set is reasonably\ndisconnected and admits many local minima. The system matrices are given by\nA=2\n664\u00001 2 0 0\n\u00002 0 1 0\n0\u00001 0 2\n0 0\u00002 03\n775;B=2\n6640 1 0 0\n\u00001 0 1 0\n0\u00001 0 1\n0 0\u00001 03\n775;D0=I; IS=I:(11)\nWhen the dimension nis4, it is known that the set of stabilizing decentralized\ncontrollers has at least 5connected components. We sample the initial controllers\nfromN(0;1)and, after 1000 samples, obtain 5initial optimal solutions. We\ngradually increase the damping parameter from 0to0:6with 0:002increment,\nand track the trajectories of locally optimal solutions by solving the newly\ndamped system with the previous local optimal solution as the initialization.\nThe evolution of the optimal cost and the distance from the best known optimal\ncontroller is plotted Figure 1. Notice that all sub-optimal local trajectories\nterminate after a modest damping \u000b\u00190:2. After that, the minimization\nalgorithm always tracks a single trajectory. This illustrates the prediction of\nCorollary 1. Especially, if we start tracking a sub-optimal controller trajectory\nfrom\u000b= 0, we will be on the better trajectory when \u000b\u00190:2. At that time, if\nwe gradually decrease \u000bto zero, we obtain a stabilizing controller with a lower\ncost.\n5.2 Experiments on Random Systems\nWith the same initialization and optimization procedure, we perform the ex-\nperiments with 3-by-3system matrices AandBrandomly generated from the\n8distribution N(0;1). For 92 out of 100 samples we are not able to find more than\none locally optimal trajectory. Examples with more than one local trajectories\nare listed below. All figures to the left plot the cost of locally optimal controllers.\nAll figures to the right plot the distance of the locally optimal controllers to the\ncontroller with the lowest cost. Note that the order of the cost of the trajectories\nmay be preserved during the damping (Figure 2) and may also be disrupted\n(Figure 3). More than one trajectory may have the lowest cost during the\ndamping (Figure 4).\nFigure 2: Trajectory of a randomly generated system where the order of locally\noptimal controller is preserved.\nFigure 3: Trajectories of a randomly generated system where the order of locally\noptimal controller is disrupted.\nFigure 5 shows a hysteresis-like loop as the damping coefficient is first\ndecreased and then increased. The trajectory of the controller first leads up to\nlarge cost and, the local search method escapes this local minimum to another\none with a smaller cost. As the damping decreases, it returns where it starts\nalong a different route.\n6 Conclusion\nThis paper studied the trajectory of locally and globally optimal solution to\nthe optimal decentralized control problem as the damping of the decentralized\n9Figure4: Trajectoryofarandomlygeneratedsystemwithacomplicatedbehavior.\nFigure 5: First decrease damping and then increase damping.\n10control system varies. Asymptotic and continuity properties of trajectories\nare proved. The complicated phenomenon of continuation is illustrated with\nnumerical examples. The fact that damping merges all locally optimal solutions\nis strong evidence that the idea of homotopy can be fruitfully used to improve\nlocally optimal solutions.\nAcknowledgments\nThe authors are grateful to Salar Fattahi and Cédric Josz for their construc-\ntive comments and feedback. The author thanks Yuhao Ding for sharing the\nimplementation of local search algorithms.\nReferences\n[1]Vincent D. Blondel and John N. Tsitsiklis. A survey of computational complexity\nresults in systems and control. Automatica , 36(9):1249–1274, 2000.\n[2]Stephen P. Boyd, L El Ghaoui, E Feron, and V Balakrishnan. Linear Matrix\nInequalities in System and Control Theory , volume 15. 1994.\n[3]J. R. Broussard and N. Halyo. Active Flutter Control using Discrete Optimal\nConstrained Dynamic Compensators. 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Homotopy Approaches to the H2 Reduced Order Model Problem.\nhttp://eprints.cs.vt.edu/archive/00000269/, January 1991.\nA Notions of continuity\nWe recount the notion of upper and lower hemi-continuity and prove the continuity\nproperties of the lower level-set map. The reader is referred to [ 16] for an accessible\ntreatment.\nDefinition 2. The set valued map \u0000 :A!Bis said to be upper hemi-continuous\n(uhc) at a point aif for any open neighborhood Vof\u0000(a)there exists a neighborhood\nUofasuch that \u0000(U)\u0012V.\nIfBis compact, uhc is equivalent to the graph of \u0000being closed, that is, if an!a\u0003\nandbn2\u0000(an)!b\u0003, thenb\u00032\u0000(a\u0003).\n12Definition 3. The set valued map \u0000 :A!Bis said to be lower hemi-continuous (lhs)\nat a pointaif for any open neighborhood Vintersecting \u0000(a)there exists a neighborhood\nUofasuch that \u0000(x)intersectsVfor allx2U.\nEquivalently, for all am!a2Aandb2\u0000(a), there exists amksubsequence of am\nand a corresponding bk2\u0000(amk), such that bk!b.\nWe prove the upper hemi-continuity of the lower level set map in Lemma 3 below.\nLemma 3. Given matrices A;Band the objective cost J(K;\u000b)that satisfies the\ndamping property. Define\n\u0000M(\u000b) =fK2S:A\u0000\u000bI+BKis stable and J(K;\u000b)\u0014Mg:\nAssume that \u0000M(\u000b)is not empty for all \u000b\u00150and a given M > 0, then \u0000M(\u000b)is an\nupper hemi-continuous set-valued map.\nProof.From [20],\u0000M(\u000b)is compact for all \u000b. From the damping property, for any\n\u000b<\f, we have \u0000M(\u000b)\u0012\u0000M(\f). Therefore, to characterize the continuity of \u0000at a\n\u000b\u0003\u00150, it suffices to consider the restricted map \u0000M: [\u000b\u0003\u0000\u000f;\u000b\u0003+\u000f]!\u0000M(\u000b\u0003+\u000f)for\nsome\u000f>0, that is, to consider the range of \u0000Mto be compact. Therefore, the sequence\ncharacterization of uhc applies. Suppose \u000bi!\u000b\u0003, pick a sequence of Ki2\u0000M(\u000bi)\nthat converges to K\u0003. The continuity of J(K;\u000b)impliesJ(K\u0003;\u000b\u0003)\u0014M. The fact\nthat the cost is bounded implies A\u0000\u000b\u0003I+BKis stable. Since subspaces of matrices\nare closed, K\u00032S. We have verified all conditions for K\u00032\u0000(\u000b\u0003), so\u0000Mis upper\nhemi-continuous.\nThe lower hemi-continuity of \u0000Mis more subtle.\nLemma 4. At any given \u000b\u0003\u00150,\u0000M(\u000b)is lower hemi-continuous at \u000b\u0003except when\nM2fJ(K;\u000b\u0003) :K2Ky(\u000b\u0003)g, which is a finite set of locally optimal costs.\nProof.Prove by contradiction, consider a sequence \u000bi!\u000b\u0003and aK\u00032\u0000(\u000b\u0003), but\nthere exists no subsequence of \u000biandKi2\u0000(\u000bi)such thatKi!K\u0003. We must\nhaveJ(K\u0003;\u000b\u0003) =M— otherwise J(K\u0003;\u000bi)< Mfor largeiand, since the set of\nstabilizing controllers is open, K\u00032\u0000M(\u000bi)for largei. Furthermore, K\u0003must be\na local minimum of J(K;\u000b\u0003)— otherwise there exists a sequence Kj!K\u0003with\nJ(Kj;\u000b\u0003)<Mand, by the continuity of J, there exists as sequence of large enough\nindicesnjsuch thatJ(Kj;\u000bnj)<M; the sequence Kj2\u0000M(\u000bnj)converges to K\u0003.\nThe argument above suggests that Mbelongs to the cost locally optimal controllers at\n\u000b\u0003. BecauseJ(K;\u000b\u0003)as a function over Kcan be described as a linear function over\nan algebraic set, the value of local minimum is finite.\nB Convergence of locally optimal controllers\nWe prove the asymptotic properties of the locally optimal controllers in Section 4 of\nthe main paper.\nTheorem. Suppose the sparsity pattern ISis block-diagonal with square blocks, and R\nhas the same sparsity pattern as IS. Then all points in Ky(\u000b)converges to the zero\nmatrix as\u000b!1. Furthermore, J(K;\u000b)!0as\u000b!1for allK2Ky(\u000b).\n13Proof.Recall the expression of the objective function\nJ(K;\u000b) =EZ1\n0h\ne\u00002\u000bt\u0010\n^x>(t)Q^x(t) + ^u>(t)R^u(t)\u0011i\ndt\ns:t: ^_x(t) =A^x(t) +B^u(t)\n^u(t) =K^x(t);(12)\nand the first order necessary conditions\n(A\u0000\u000bI+BK)>P\u000b(K) +P\u000b(K)(A\u0000\u000bI+BK) +K>RK+Q= 0(13)\nL\u000b(K)(A\u0000\u000bI+BK)>+ (A\u0000\u000bI+BK)L\u000b(K) +D0= 0 (14)\n((B>P\u000b(K) +RK)L\u000b(K))\u000eIS= 0 (15)\nK\u000eIS=K: (16)\nThose first order conditions can be used to characterize the objective function\nJ(K;\u000b) = tr(D0P\u000b(K)): (17)\nAs\u000bincreases, some local solution may disappear, some new local solution may appear.\nThe appearance cannot happen infinitely often because the equations (13)-(16)are\nalgebraic. Suppose when \u000b\u0015\u000b0, the number of local solutions does not change. The\ndamping property ensures for \f >\u000b>\u000b 0,\nmax\nK2Ky(\f)J(K;\f)\u0014max\nK2Ky(\u000b)J(K;\f)\nThe right hand side optimizes over a fixed, finite set of controllers and goes to zero as\n\f!1from the formulation (12)and the dominated convergence theorem. The left\nhand side, therefore, also converges to zero as \f!1. From(17)and the assumption\nthatD0is positive definite, kP\f(K)k!0for allK2Ky(\f)as\f!1.\nThe assumption on sparsity allows the expression of the locally optimal controllers\nin (15) as\nK=\u0000R\u00001((B>P\u000b(K)L\u000b(K))\u000eIS)(L\u000b(K)\u000eIS)\u00001:\nEspecially we can bound\nkBKk\u0014kBR\u00001B>P\u000b(K)L\u000b(K)k\u0015min(L\u000b(K))\u00001:\nPre- and post- multiply (14) by L\u000b(K)’s unit minimum eigenvector v,\n\u0015min(L\u000b(K))(2a\u00002v>(A+BK)v) =v>D0v: (18)\nTherefore\n\u0015min(L\u000b(K))\u0015\u0015min(D0)\n2\u000b+ 2kA+BKk(19)\n\u0015\u0015min(D0)\n2\u000b+ 2kAk+ 2kBKk(20)\n\u0015\u0015min(D0)\n2\u000b+ 2kAk+ 2kBR\u00001B>P\u000b(K)L\u000b(K)k\u0015min(L\u000b(K))\u00001:(21)\nThis simplifies to\n\u0015min(L\u000b(K))\u0015\u0015min(D0)\u00002kBR\u00001B>P\u000b(K)L\u000b(K)k\n(2\u000b+ 2kAk)(22)\n14Take the trace of (14) and consider the estimate\n2nkAkkL\u000bk+ tr(D0)\u00152kAktr(L\u000b) + tr(D0)\n\u00152\u000btr(L\u000b) + 2 tr(BR\u00001((B>P\u000bL\u000b)\u000eIS)(L\u000b\u000eIS)\u00001L\u000b)\n\u00152\u000btr(L\u000b)\u00002kBR\u00001((B>P\u000bL\u000b)\u000eIS)ktr((L\u000b\u000eIS)\u00001L\u000b)\n= 2\u000btr(L\u000b)\u00002kBR\u00001((B>P\u000bL\u000b)\u000eIS)kn\n\u00152\u000bkLk\u00002nkBR\u00001kkB>kkkP\u000bkkL\u000bk;\nwhere for clarity L\u000bdenotesL\u000b(K)andP\u000bdenotesP\u000b(K). The second and the third\ninequalities use the fact that jtr(AL)j\u0014kAktr(L)for a positive definite matrix L\nand any matrix A. This estimate, combined with previous argument that kP\u000bk!0,\nconcludeskL\u000bk!0. We also obtain from the inequality that\nkL\u000bk\u0014tr(D0)\n2a\u00002nkAk\u00002nkBR\u00001kkB>kkkP\u000bk; (23)\nfor small enough P\u000b. Combining (22) and (23)\nkKk\u0014kR\u00001k\u0001k(B>P\u000bL\u000b)\u000eISk\u0001k(L\u000b\u000eIS)\u00001k\n\u0014kR\u00001k\u0001kB>k\u0001kP\u000bk\u0001kL\u000bk\u0001k\u0015min(P\u000b)\u00001\n\u0014kR\u00001k\u0001kB>k\u0001kP\u000bktr(D0)\n2\u000b\u00002nkAk\u00002nkBR\u00001kkB>kkkP\u000bk(2\u000b+ 2kAk)\n\u0015min(D0)\u00002kBR\u00001B>P\u000bL\u000bk;\nwhich converges to 0as\u000b!1.\nC The Positive Definiteness of Hessian\nTheorem. For any given r >0, the Hessian matrix r2J(K;\u000b)is positive definite\noverkKk\u0014rfor all large \u000b.\nProof.The proof requires the vectorized Hessian formula given in Lemma 3.7 of [ 18],\nrestated below.\nLemma 5 ([18]).Definej\u000b:Rm\u0001n!Rbyj\u000b(vec(K)) =J(K;\u000b). The Hessian of j\u000b\nis given by the formula\nH\u000b(K) = 2n\n(L\u000b(K)\nR) +G\u000b(K)>+G\u000b(K)o\n;\nwhere\nG\u000b(K) = [I\n(BTP\u000b(K) +RK)] [I\n(A\u0000\u000bI+BK) + (A\u0000\u000bI+BK)\nI]\u00001(In;n+P(n;n))[L\u000b(K)\nB]\nandP(n;n)is ann2\u0002n2permutation matrix.\nWe show that H\u000b(K)in the lemma is positive definite for any fixed Kwhen\u000bis\nlarge. Recall the definition of L\u000bandK\u000b.\nL\u000b(K)(A\u0000\u000bI+BK)T+ (A\u0000\u000bI+BK)L\u000b(K) +D0= 0;(24)\nP\u000b(K)(A\u0000\u000bI+BK) + (A\u0000\u000bI+BK)TP\u000b(K) +KTRK+Q= 0:(25)\n15With triangle inequality\n2\u000bkL\u000b(K)k\u0014kD0k+ 2kA+BKkkL\u000b(K)k\n2\u000bkP\u000b(K)k\u0014kQk+ 2kA+BKkkP\u000b(K)k+kRkkKk2\nwhich meanskP\u000b(K)k!0andkL\u000b(K)k!0as\u000b!1. The minimum eigenvalue of\nL\u000b(K)can be bounded similarly: let vbe the unit eigenvector of L\u000b(K)corresponding\nto\u0015min(L\u000b(K)), pre- and post- multiply (24) by v, we obtain\n\u0015min(L\u000b(K))\u0015vTD0v\n2\u000b\u00002vT(A+BK)v\u0015\u0015min(D0)\n2\u000b+ 2kA+BKk: (26)\nThe first Hessian term L\u000b(K)\nRcan bounded from below with (26)\n\u0015min(L\u000b(K)\nR) =\u0015min(L\u000b(K))\u0015min(R)\u0015\u0015min(D0)\u0015min(R)\n2\u000b+ 2kA+BKk:\nWe bound the norm of the second and the third Hessian term kG\u000b(K)kas follows,\nwhere .hides constants that do not depend on \u000b.\nkG\u000b(K)k\u0014kI\n(BTP\u000b(K) +RK)k\n\u0001k[I\n(A\u0000\u000bI+BK) + (A\u0000\u000bI+BK)\nI]\u00001k\u0001k(In;n+P(n;n))[L\u000b(K)\nB]k\n.(\u0000\u0015max(I\n(A\u0000\u000bI+BK) + (A\u0000\u000bI+BK)\nI))\u00001kL\u000b(K)k\n.(2\u000b)\u00001kL\u000b(K)k:\nComparing the two estimates above, we find the first term dominates the two following\nterms with large \u000b, uniformly over bounded K. The Hessian H\u000b(K)is therefore\npositive definite over bounded Kwhen\u000bis large. The conclusion carries over to the\nHessian of the decentralized controller, which is a principal sub-matrix of the Hessian\nof the centralized controller.\nD The uniqueness of the continuation direction\nThis section aims to prove the following result\nTheorem. Whenn\u00153, for anyn-by-nreal matrix Hthat is not a multiple of \u0000I,\nthere exists a stable matrix Afor whichA+His unstable.\nDefine the set of stable directions\nH=fH:A+tHis stable whenever Ais stable and t\u00150g; (27)\nwhereAandHaren-by-nreal matrices.\nLemma 6. All matrices inHis similar to a diagonal matrix with non-positive diagonal\nentries. Especially, they cannot have complex eigenvalues.\nProof.Whentis large,A+tHis a small perturbation of tH, hence the eigenvalues of\nHhas to be in the closed left half plane. With a suitable similar transform assume H\nis in real Jordan form. First consider the case of two by two matrices, and we denote\nthe matrices by H2andA2. Assume for contradiction that H2is not diagonalizable.\nThe non-diagonal real Jordan form of H2has the following possibilities:\n16\u000fH2=\u0014h1\n0h\u0015\n, whereH2has real eigenvalues h<0. PickA2=\u00144h\u00002\n10h2\u00003h\u0015\n,\nwhich is stable because tr(A2) =h < 0anddet(A2) = 8h2>0. We have\nA2+tH2=\u0014ht+ 4hby t\u00002\n10h2ht\u00003h\u0015\n, whose stability criterion tr(A2+tH2)<0\nanddet(A2+tH2)>0amounts to\n2ht+h<0\nh2(t2\u00009t+ 8)>0;\nor equivalently t2(\u00001=2;1)[(8;+1). Especially when t= 2,A2+tH2is not\nstable.\n\u000fH2=\u00140 1\n0 0\u0015\n. Pick a stable matrix A=\u0014\u00001 0\n1\u00001\u0015\n.A+tHis not stable when\nt= 2.\n\u000fH2=\u00140f\n\u0000f0\u0015\n, wheref >0, PickA=\u0014\u00001\u00004\n1\u00001\u0015\n,A+2\nfH2=\u0014\u00001\u00002\n\u00001\u00001\u0015\nis\nnot stable.\n\u000fH2=\u0014h f\n\u0000f h\u0015\n, whereh<0andf >0. By rescaling assume f= 1. Consider\nthe following matrix function\nG(t) =\u001401\n2+ (u+w)h\n\u00001\n2+ (u\u0000w)h h\u0015\n+t\u0014h1\n\u00001h\u0015\n(28)\nWe have\ntr(G(t)) =h+ 2ht\ndet(G(t)) = (1 +h2)t2+ (1 +h2+ 2hw)t+h2(w2\u0000u2) +hw+1\n4:\nEspeically,\ntr(G(\u00001\n2)) = 0\nd\ndttrG(t) = 2h\ndet(G(\u00001\n2)) =h2(\u00001\n4\u0000u2+w2)\nd\ndtdetG(t)\f\f\f\f\nt=\u00001\n2= 2hw\nHence as long as\nw>0and\u00001\n4\u0000u2+w2>0 (29)\nfor small enough \u000f >0,A2=G(\u00001\n2+\u000f)is a stable matrix and there will be\nmatricesG(t)witht >\u00001\n2whose trace is negative and whose determinant is\nsmaller. Consider the minimal value the determinant can take\ndetG\u0012\n\u00001\n2\u0000hw\n1 +h2\u0013\n=h2\u0012\n\u00001\n4\u0000u2+h2\n1 +h2w2\u0013\n17which means when\n\u00001\n4\u0000u2+h2\n1 +h2w2<0 (30)\nThe matrix G(t)witht=\u00001\n2\u0000hw\n1+h2is unstable. There certainly exist uandw\nthat satisfies (29) and (30).\nFor general n,H’s real Jordan form is an block upper-triangular matrix\nH=\u0014H2\u0003\n0\u0003\u0015\nwhereH2can take the four possibilities mentioned above. We take the corresponding\nstableA2constructed above, which has the property that A2+t0H2is not stable for\nsomet0>0. Form the block diagonal matrix\nA=\u0014A20\n0\u0000I\u0015\nThenAis stable, while A+t0H=\u0014A2+t0H2\u0003\n0\u0003\u0015\nis not stable.\nWe can strengthen the argument above and further characterize Hin the case\nn\u00153.\nLemma 7. Whenn\u00153, the set of stable directions Hdoes not contain any matrices\nof rank 1,2, ...,n\u00002.\nProof.From lemma 6, we only need to consider the case where His diagonal with\nnegative diagonal entries. Assume there is a rank one matrix H2H, write\nH=\u0014H30\n0\u0003\u0015\n;\nwhereH3=diag(\u00001;0;0). This is possible with the rank assumption. We will construct\na stable 3-by-3matrixA3, such that there is some t0>0that makes A3+t0H3unstable,\nand then carry the instability to A+t0Hwith the extended matrix\nA=\u0014A30\n0\u0000I\u0015\n:\nFrom [7], the set\nT=8\n<\n:t:2\n40 1 0\n0 0 1\n5 1\u000013\n5+t2\n40\n0\n\u000013\n5\u0002\n0:85 0:2 0:2\u0003\nis stable9\n=\n;\nhas two disconnected components. Consider the Jordan decomposition of the matrix\n2\n40\n0\n\u000013\n5\u0002\n0:85 0:2 0:2\u0003\n=Pdiag(\u00000:2;0;0)P\u00001;\n18wherePis some invertible matrix. Write\nG(t) = 5P\u000012\n40 1 0\n0 0 1\n5 1\u000013\n5P+t\u0002diag(\u00001;0;0):\nAfter this similar transform, the set Tcan be written with G(t;0).\nT=ft:G(t)is stableg\nSinceTis disconnected there exists some t1<t2such thatG(t1)is stable, while G(t2)\nis unstable with some eigenvalue in the right half plane. Setting A3=G(t1)and\nt0=t2\u0000t1completes the proof.\nSince we can perturb the direction and make Hfull-rank, the fact that Hhas rank\none is not the substantial property. This is indeed the case.\nLemma 8. Whenn\u00153,H=f\u0000\u0015I;\u0015\u00150g.\nProof.From lemma 6, we only need to consider the case where His diagonal with\nnegative diagonal entries. Write\nH=\u0014H30\n0\u0003\u0015\n;\nwhereH3=diag(h1;h2;h3). The diagonal entries hi;i= 1;2;3are non-positive and\nnot all equal. We will construct an A3and a corresponding t0such thatA3is stable\nwhileA3+t0H3is not stable, and extend to the general Aas in Lemma 7. The case\nwhereH3has rank 1has been considered in Lemma 7. We show the remaining rank is\nimpossible. Without loss of generality we rescale H3and assume h1=\u00001.\n\u000fH3= diag(\u00001;h2;0), whereh2<0. Consider the matrix function\nG(t) =2\n40\u00001 0\n0 0\u0000h2\n2 1 03\n5+tH3=2\n4\u0000t\u00001 0\n0th2\u0000h2\n2 1 03\n5:\nThe characteristic polynomial of G(t)is\n\u001eG(t)(x) =x3+ (t\u0000th2)x2+ (h2\u0000t2h2)x+ (t\u00002)h2:\nThe Routh-Hurwitz Criterion insists\nt(1\u0000h2)>0\n(t\u00002)h2>0\nt(1\u0000h2)h2(1\u0000t2)>(t\u00002)h2:\nwhich is simplified with h2<0to\n0<t< 2 (31)\n(1\u0000h2)t3+th2\u00002>0: (32)\nEspecially, when t=3\n2,(32)simplifies to the obvious expression1\n8(11\u000015h2)>0.\nwhent= 3,(31)impliesG(t)is not stable. Setting A3=G(3\n2)andt0=3\n2\nconcludes the proof.\n19\u000fH3= diag(\u00001;h2;h3), where without loss of generally we assume\n\u00001\u0014h2;h3<0;and one of them is not \u00001: (33)\nConsider the matrix\nG(t) =2\n40\u00001 0\n0 0h2\nah3h303\n5+tH3=2\n4\u0000t\u00001 0\n0th2h2\nah3h3th33\n5\nIts Routh-Hurwitz Criterion insists\nt>0\nf1(t) =a\u0000t+t3>0(34)\nf2(t) =\u0000ah2h3+th2h3(h2+h3) +t3(1\u0000h2)(1\u0000h3)(\u0000h2\u0000h3)>0(35)\nWe claim that when\ns\nh2h3(h2+h3)2\n(\u0000h2\u0000h3+h2h3)3<a<r\n4\n27(36)\nthe set oftthat satisfy Routh-Hurwitz Criterion is disconnected. To see this,\nwrite the positive local minimum of f1(t)in(34)ast1=q\n1\n3, and write the\npositive local minimum of f2(t)in(35)ast2=q\nh2h3\n3(1\u0000h1)(1\u0000h2). The condition\n(33)ensures that t1<t2and the condition (36)ensures that f1(t1)andf2(t2)\nare negative. Furthermore, consider t0=ah2+h3\u0000h2h3\nh2+h3, which is the root of\n(1\u0000h2)(1\u0000h3)(\u0000h2\u0000h3)f1(t)\u0000f2(t). It holds that t1< t0< t2and both\nf1(t0)andf2(t0)are positive, which implies that the positive intersection f1(t)\nandf2(t)are positive. We conclude that when t=t0, the matrix G(t0)is stable,\nand whentis large,G(t)is again stable. Yet when t=t22(t0;1), the matrix\nG(t2)is not stable.\n20"    },    {        "title": "1503.07122v1.On_damping_created_by_heterogeneous_yielding_in_the_numerical_analysis_of_nonlinear_reinforced_concrete_frame_elements.pdf",        "content": "ON DAMPING CREATED BY HETEROGENEOUS YIELDING IN\nTHE NUMERICAL ANALYSIS OF NONLINEAR REINFORCED\nCONCRETE FRAME ELEMENTS\nPierre Jehel1,2\u0003and R\u0013 egis Cottereau1\n1Laboratoire MSSMat / CNRS-UMR 8579, \u0013Ecole Centrale Paris, Grande voie\ndes Vignes, 92295 Ch^ atenay-Malabry Cedex, France\n2Department of Civil Engineering and Engineering Mechanics, Columbia\nUniversity, 630 SW Mudd, 500 West 120th Street, New York, NY, 10027, USA\nMarch 2nd, 2015: Paper accepted for publication in Computers and Structures\nDOI: 10.1016/j.compstruc.2015.03.001\nAbstract. In the dynamic analysis of structural engineering systems, it is\ncommon practice to introduce damping models to reproduce experimentally\nobserved features. These models, for instance Rayleigh damping, account for\nthe damping sources in the system altogether and often lack physical basis. We\nreport on an alternative path for reproducing damping coming from material\nnonlinear response through the consideration of the heterogeneous character\nof material mechanical properties. The parameterization of that heterogeneity\nis performed through a stochastic model. It is shown that such a variability\ncreates the patterns in the concrete cyclic response that are classically regarded\nas source of damping.\nKeywords: damping ; concrete ; nonlinear constitutive relation ; material hetero-\ngeneity ; stochastic \feld\n1.Introduction\nIn the last few decades, a great deal of attention was paid to the comprehension\nand modeling of damping mechanisms in inelastic time-history analyses (ITHA) of\nconcrete and reinforced concrete (RC) structures [2, section 2.4]. Figure 1, adapted\nfrom [33], shows the uniaxial cyclic compressive strain-stress ( E-\u0006) response mea-\nsured on a concrete test specimen. Throughout this paper, the term \\uniaxial\"\nimplies that there is only one loading direction and that the stress, respectively\nstrain, of interest is the normal component of the stress, respectively strain, vector\nin the loading direction. In other words, when it comes to constitutive relation be-\ntween stress and strain, the work presented thereafter is developed in a 1D setting.\nIn \fgure 1, the so-called backbone curve, which is the envelope of the response\n(dashed line), shows the following phases: (i) an inelastic phase with positive slope\n(E\u00142:7\u000210\u00003for that particular example, where Eis the measured strain),\nand (ii) an inelastic phase with negative slope before the specimen collapses. For\nconcrete, no elastic phase can really be identi\fed, and hysteresis loops appear in\n\u0003Corresponding author: pierre.jehel[at]centralesupelec.fr\n1arXiv:1503.07122v1  [cs.CE]  21 Mar 20152\nunloading-reloading cycles even for limited strain amplitudes. Other salient fea-\ntures include: (i) a residual deformation after unloading, and (ii) a progressive\ndegradation of the sti\u000bness (slope of the unloading-loading segments). The hys-\nteresis loops are one of the sources of the damping that is observed in free vibration\nrecordings of concrete beams. Other sources include friction at joints [27] or at the\nconcrete-steel interface in reinforced concrete [14]. These other sources of damping\nwill not be discussed in this paper, where we will concentrate on material damping.\n2\nunloading-reloading cycles even for limited strain amplitudes. Other s alient fea-\ntures include: (i) a residual deformation after unloading, and (ii) a p rogressive\ndegradation of the sti ffness (slope of the unloading-loading segmen ts). The hys-\nteresis loops are one of the sources of the damping that is observe d in free vibration\nrecordings of concrete beams. Other sources include friction at j oints [27] or at the\nconcrete-steel interface in reinforced concrete [14]. These oth er sources of damping\nwill not be discussed in this paper, where we will concentrate on mate rial damping.\nFigure 1. Strain-stress concrete experimental response in pseudo-\nstatic cyclic uniaxial compressive loading (adapted from [33]). Σ\nand Eare the homogeneous compression stress and strain in the\nconcrete test specimen that are measured in the loading direction.\nΣand Eare spatial mean quantities in the sense that Σis com-\nputed as the load in the hydraulic cylinder of the testing machine\ndivided by the area of the specimen cross section, and Eis com-\nputed as the displacement of the cylinder divided by the length of\nthe concrete specimen.\nMost classical uniaxial constitutive models of concrete for numeric al simulation\ndo not dissipate any energy in unloading-reloading cycles (see e.g. fig ure 2 [top left]).\nIt is then common practice to add a viscous damping model (Rayleigh d amping)\nto the inelastic structural model, to reproduce phenomena that a re experimentally\nobserved at the structural level (decreasing amplitude of displac ements in free vi-\nbration). However, Rayleigh damping is well known to lack physical ju stification,\neven when care is taken to avoid generating spurious damping force s [7, 18].\nAnother class of approach aims at reproducing more precisely the f eatures of\nFigure 1 through elaborate inelastic constitutive relations. Figure 2 shows typi-\ncal examples of uniaxial relations found in the literature. The relatio n described\nin [9] [top left] defines di fferent response phases for di fferent stra in intensities, with\nan additional coe fficient to control the loss of sti ffness. The const itutive relation\ndescribed in [23] [bottom right] comes from a formulation developed in the frame-\nwork of thermodynamics with internal variables. It reproduces da mping features\nreasonably well, in particular for higher amplitudes, but requires the identifica-\ntion of a rather large number of parameters. These first two type s of relations\nFigure 1. Strain-stress concrete experimental response in pseudo-\nstatic cyclic uniaxial compressive loading (adapted from [33]). \u0006\nandEare the homogeneous compression stress and strain in the\nconcrete test specimen that are measured in the loading direction.\n\u0006 andEare spatial mean quantities in the sense that \u0006 is com-\nputed as the load in the hydraulic cylinder of the testing machine\ndivided by the area of the specimen cross section, and Eis com-\nputed as the displacement of the cylinder divided by the length of\nthe concrete specimen.\nMost classical uniaxial constitutive models of concrete for numerical simulation\ndo not dissipate any energy in unloading-reloading cycles (see e.g. \fgure 2 [top left]).\nIt is then common practice to add a viscous damping model (Rayleigh damping)\nto the inelastic structural model, to reproduce phenomena that are experimentally\nobserved at the structural level (decreasing amplitude of displacements in free vi-\nbration). However, Rayleigh damping is well known to lack physical justi\fcation,\neven when care is taken to avoid generating spurious damping forces [7, 18].\nAnother class of approach aims at reproducing more precisely the features of\nFigure 1 through elaborate inelastic constitutive relations. Figure 2 shows typi-\ncal examples of uniaxial relations found in the literature. The relation described\nin [9] [top left] de\fnes di\u000berent response phases for di\u000berent strain intensities, with\nan additional coe\u000ecient to control the loss of sti\u000bness. The constitutive relation\ndescribed in [23] [bottom right] comes from a formulation developed in the frame-\nwork of thermodynamics with internal variables. It reproduces damping features\nreasonably well, in particular for higher amplitudes, but requires the identi\fca-\ntion of a rather large number of parameters. These \frst two types of relations\nare somehow de\fned by parts for di\u000berent loading regimes. They hence require a3\nwider set of parameters and seem to contradict the seemingly smooth transition be-\ntween regimes observed experimentally. The constitutive relation described in [48]\n[top right] is heuristically de\fned from a database of experiments. It reproduces\nunloading-reloading hysteresis mechanisms, but lacks a theoretical basis. Finally,\nthe relation described in [32] [bottom left] is based on a physical model of damage\nand friction. It manages to dissipate energy in unloading-reloading cycles, but the\nlack of obvious physical meaning for some parameters can render their identi\fcation\ndi\u000ecult.\n3\nare somehow defined by parts for di fferent loading regimes. They he nce require a\nwider set of parameters and seem to contradict the seemingly smoo th transition be-\ntween regimes observed experimentally. The constitutive relation d escribed in [48]\n[top right] is heuristically defined from a database of experiments. I tr e p r o d u c e s\nunloading-reloading hysteresis mechanisms, but lacks a theoretica l basis. Finally,\nthe relation described in [32] [bottom left] is based on a physical mode lo fd a m a g e\nand friction. It manages to dissipate energy in unloading-reloading c ycles, but the\nlack of obvious physical meaning for some parameters can render t heir identification\ndifficult.\nFigure 2. Typical strain-stress relations in pseudo-static cyclic\ncompressive loading for di fferent models: [9] [top left], [48] [top\nright], [32] [bottom left], and [23] [bottom right].\nThe main purpose of this paper is to present a multi-scale stochastic nonlinear\nconcrete model that can be accommodated in an e fficient structur al frame element\n(fiber element), and that participates to the overall structural damping in dynamic\nloading. In particular, this implies the developed concrete model be c apable of rep-\nresenting hysteresis loops in unloading-reloading cycles at macro-s cale (the scale\nwhere such behavior as in Figure 1 can be observed). In this work, t his is achieved\nby the introduction, at an underlying meso-scale, of spatial variab ility in the pa-\nrameters. At meso-scale, an elasto-plastic response with linear kin ematic hardening\nand heterogeneous yield stress is considered. This choice is mainly dr iven by its\nsimplicity and its relevance is illustrated in the numerical applications.\nThe main issue with modeling the heterogeneity of the yield stress lies in the pa-\nrameterization. On the one hand, local information on the heterog eneity of concrete\nis available at a scale that we wish to avoid (because of the associated computa-\ntional costs). On the other hand, identification becomes extreme ly di fficult when\nvery fine models are considered. We therefore choose to model th e heterogeneity of\nthe yield stress through a stochastic model. Hence, only three par ameters control\nthat heterogeneity: a mean value, a variance, and a correlation len gth. The choice of\nparameterizing the fluctuating field of constitutive parameters by statistical quan-\ntities means that there might be fluctuations in the quantities of inte rest measured\nfor di fferent realizations of the random model. However, as will beco me apparent\nin the examples, some sort of homogenization comes in and these fluc tuations can\nrightfully be ignored.\nFigure 2. Typical strain-stress relations in pseudo-static cyclic\ncompressive loading for di\u000berent models: [9] [top left], [48] [top\nright], [32] [bottom left], and [23] [bottom right].\nThe main purpose of this paper is to present a multi-scale stochastic nonlinear\nconcrete model that can be accommodated in an e\u000ecient structural frame element\n(\fber element), and that participates to the overall structural damping in dynamic\nloading. In particular, this implies the developed concrete model be capable of rep-\nresenting hysteresis loops in unloading-reloading cycles at macro-scale (the scale\nwhere such behavior as in Figure 1 can be observed). In this work, this is achieved\nby the introduction, at an underlying meso-scale, of spatial variability in the pa-\nrameters. At meso-scale, an elasto-plastic response with linear kinematic hardening\nand heterogeneous yield stress is considered. This choice is mainly driven by its\nsimplicity and its relevance is illustrated in the numerical applications.\nThe main issue with modeling the heterogeneity of the yield stress lies in the pa-\nrameterization. On the one hand, local information on the heterogeneity of concrete\nis available at a scale that we wish to avoid (because of the associated computa-\ntional costs). On the other hand, identi\fcation becomes extremely di\u000ecult when\nvery \fne models are considered. We therefore choose to model the heterogeneity of\nthe yield stress through a stochastic model. Hence, only three parameters control\nthat heterogeneity: a mean value, a variance, and a correlation length. The choice of4\nparameterizing the \ructuating \feld of constitutive parameters by statistical quan-\ntities means that there might be \ructuations in the quantities of interest measured\nfor di\u000berent realizations of the random model. However, as will become apparent\nin the examples, some sort of homogenization comes in and these \ructuations can\nrightfully be ignored.\nSeveral authors in the literature have considered random models of \ructuating\nnonlinear materials [16, 20, 1, 35], in particular for concrete [21, 34, 26, 49, 45,\n50, 30] or in the context of dynamic analysis [36, 28, 44]. We consider here a\nmodeling framework that is a combination of ingredients found in several previous\npapers [29, 5, 21, 30], with a \ructuating yield stress modeled as a random \feld with\nnon-zero correlation length. However, the objective in these papers was to assess\nthe in\ruence of parameter uncertainty on some quantity of interest. An objective\nwith the current paper is to observe the e\u000bect of randomness at a meso-scale on the\nnonlinear stress-strain relation at macro-scale. The work herein presented should\ntherefore be seen as an innovative proposal for parameterization of a nonlinear\nstress-strain relation.\nIn Section 2, we recall the theoretical formulation of the inelastic beam model\nthat will be used throughout this paper. The stochastic multi-scale constitutive re-\nlation developed to represent concrete cyclic behavior in reinforced concrete frame\nelements is introduced in section 3. Concrete behavior at macro-scale is retrieved\nfrom the description of a meso-scale where elasto-plastic response with linear kine-\nmatic hardening and spatially variable yield stress is assumed. In particular, we\nemphasize in sections 3.3 and 3.4 the heterogeneity of the yield stress and the pa-\nrameterization of that heterogeneity through a random model. In section 4, we\nreport on the limiting case of vanishing correlation length and monotonic loading,\nfor which several results can be derived analytically. Section 5 presents numerical\napplications of the model in the context of dynamic structural analysis of reinforced\nconcrete frame elements.\n2.2D continuum Euler-Bernoulli inelastic beam\nClassical displacement-based formulation has been retained here although other\nmixed formulations can in certain cases show better performances [46]. For the\nsake of conciseness, we present the beam element in the 2D case, extension to 3D\nis straightforward.\n2.1.Euler-Bernoulli kinematics. We de\fne the continuum beam B=fx2\nR3jx12[0;L];x22[\u0000h=2;h=2];x32[\u0000w=2;w=2]g. Such a beam has length L\nand uniform rectangular cross-section Sof sizew\u0002h. We consider an orthonormal\nbasis ( i1;i2;i3) ofR3, so that any material point in space x=P3\ni=1xiii. In the 2D\nsetting adopted here, Euler-Bernoulli kinematics can be written at any point x2B\nand at any time t2[0;T] as\n(1) u(x;t) =\u0012u1(x;t) =uS\n1(x1;t)\u0000x2\u0012S\n3(x1;t)\nu2(x;t) =uS\n2(x1;t)\u0013\nwith\n\u0012S\n3(x1;t) =@uS\n2(x1;t)\n@x1:\nu1andu2are the longitudinal and transversal components of the displacement\nvector uat any point in the beam. uS\n1anduS\n2are rigid body translations and \u0012S\n35\nis rigid body rotation of section Sat position x1along the beam axis. Thus, uS\n1,\nuS\n2and\u0012S\n3only depend on x1.\nFor small transformations, strain tensor reads E=1\n2\u0000\nD(u) +DT(u)\u0001\n, where\nD(\u0001) =P3\ni=1@\u0001\n@xi\nii, with\nthe tensor product and \u0001Tthe transpose operation.\nThen, de\fning the axial strain \u000fS=@uS\n1=@x 1and the curvature \u001fS=@2uS\n2=@x2\n1,\nit comes:\n(2) E(x;t) =E(x;t)i1\ni1\nwith\n(3) E(x;t) =\u000fS(x1;t)\u0000x2\u001fS(x1;t):\n2.2.Variational formulation. SupposeBis loaded with forces per unit length\nof beam band concentrated forces applied at beam ends. A variational form of the\nproblem of \fnding usuch that beam equilibrium is satis\fed is: \fnd usuch that\n(4) 0 =Z\nL\u001aZ\nS\u0000\n\u000e\u000fS\u0000x2\u000e\u001fS\u0001\n\u0006dS\u001b\ndx1\u0000Z\nL\u000eu\u0001bdx1\u0000\u000e\u0005bc;\nwhere\u000euis any kinematically admissible displacement \feld \u0006 = \u0006\u0001i1\ni1with\u0006\nthe stress tensor, \u0001a matrix product here, and \u000e\u0005bcthe potential for the forces at\nbeam ends.\nIntroducing the normal and bending forces in the beam cross-sections as\n(5) N=Z\nS\u0006dSandM=\u0000Z\nSx2\u0006dS;\nequation (4) can then be rewritten as\n(6) 0 =Z\nL\u000eeS\u0001qdx1\u0000Z\nL\u000eu\u0001bdx1\u0000\u000e\u0005bc;\nwhere q= (N;M )TandeS=\u0000\n\u000fS;\u001fS\u0001T.\n2.3.Inelastic constitutive behavior. Cross-section inelastic constitutive response\nq(eS;t) is thereafter represented using uniaxial material constitutive response \u0006( E;x;t)\nintegrated over the cross-section, rather than using a direct relation between section\ndisplacements and forces. This approach leads to what is often referred to as \fber\nbeam element.\nWith \u0001 denoting an increment of some quantity, we introduce the tangent mod-\nulusDas \u0001\u0006 =D\u0002\u0001E, that is, from equation (3),\n(7) \u0001\u0006( x;t) =D(x;t)\u0000\n\u0001\u000fS(x1;t)\u0000x2\u0001\u001fS(x1;t)\u0001\n:\nIntroducing relation (7) in (5), beam section inelastic constitutive equation reads\n\u0001q=KS\u0001eSwith tangent sti\u000bness matrix\n(8) KS(x1;t) =\u0014Z\nS\u00121\n\u0000x2\u0013\nD(x;t)\u00001\u0000x2\u0001\ndS\u0015\n:6\n2.4.Numerical implementation (structural level). The \fnite element method\nis used to approximate the displacement \felds. Classically, we have for each ele-\nment u(x;t) =N(x)d(t) and eS(x;t) =B(x)d(t), where the vector dgathers the\ndisplacements uS\n1;uS\n2;\u0012S\n3at the element nodes, and matrices NandBgather the\nclassical shape functions for Euler-Bernoulli kinematics. Choosing, in equation (6),\n\u000eu=N\u000edand\u000eeS=B\u000ed, and then linearizing the resulting relation, we have\n(9) K(k)\nn\u0001d(k)\nn=r(k)\nn;\nwhere K=R\nLBTKSBdx1andr=f\u0000R\nLBTqdx1.fis the vector of nodal\nforces calculated from band from any concentrated force applied at a beam ends.\nSubscriptnrefers to any time step in the loading history; superscript krefers to\nany Newton-Raphson iteration.\nFor any integrable function g, line integrals are numerically approximated asR\nLg(x)dx\u0019PNl\nl=1g(xl)Wlwhere subscript lrefers to a quadrature point and\nWldenotes quadrature weight and length. Section integrals are estimated asR\nSlg(xl)dSl\u0019PNF\nF=1AFg(xF\nl), whereAFis the section area of the so-called \fber\nFandxF\nlis the position of the \fber centroid in the control section Slat quadrature\npointl.\n3.Multi-scale uniaxial cyclic model for concrete\nIn this section, we present the core objective of the paper, which is a nonlin-\near uniaxial constitutive model capable of representing salient features of concrete\ncompressive response in cyclic loading. It is based on a simple local (meso-scale)\nelasto-plastic constitutive relation, for which the yield stress is modeled as a random\n\feld. The spatial \ructuations of the yield stress induce at macro-scale constitutive\nrelation \u0006(E) which resembles that encountered experimentally. Again, as already\nstated in the introduction, the idea of considering an elasto-plastic constitutive re-\nlation with \ructuating yield stress is not novel per se [29, 5, 21, 30]. It has been\nproposed to assess e\u000bects of uncertain parameters on model outputs of interest for\nengineering practice while our aim here is to stress on the fact that this can be seen\nas a way of parameterizing material nonlinear constitutive relations. In particu-\nlar, it will be shown that, in some circumstances, even if randomness is present at\nmeso-scale, our model can predict non-random outputs.\n3.1.Meso- and macro-scale modeling of concrete. Concrete is a heteroge-\nneous material (see \fgure 3). Two scales are classically considered for its modeling:\n(i) a micro-scale at which each phase (aggregate, concrete, cement paste) is clearly\nidenti\fed and modeled with its own constitutive relation; and (ii) a macro-scale at\nwhich concrete is considered as homogeneous. The macro-scale is the relevant scale\nfor structural engineering applications but the behavior at that scale is strongly\nin\ruenced by phenomena occurring at the micro-scale. In particular, the geome-\ntry of the phases is important as it controls in a large part local concentrations\nof stresses. As pointed out in the introduction to this paper, formulating and im-\nplementing concrete constitutive laws at the macro-scale can then turn out to be\nchallenging, even in the uniaxial case, and especially when it comes to accounting\nfor material energy dissipation sources. We follow here another path, considering\na meso-scale at which the parameters of the constitutive relation are assumed to\nvary continuously. This scale is intermediary between the macro-scale, at which7\nthe parameters are homogeneous, and the micro-scale, at which the parameters are\ndiscontinuous.\n7\nFigure 3. Polished concrete section where the two phases human\neyes can see are represented: aggregates (crushed gravel an ds a n d )\nand cement paste in-between.\nand stress at macro-scale Σis computed as the spatial mean stres so v e r R:\n(11) Σ(E;θ)=1\n|R|/integraldisplay\nRσ(ϵ;θ)dR,\nwhere |R|denotes the area of the fiber section R,σand ϵthe stress and strain at\nmeso-scale in the uniaxial case. That is, analogously to Eand Σ,ϵ=ϵ·i1⊗i1and\nσ=σ·i1⊗i1,w h e r e ϵand σare the strain and stress tensors at meso-scale such\nthat the constitutive model is considered in a 1D setting. Besides, w ee n h a n c et h e\nfact that Σis computed as the spatial mean of σ(x) and not as the sample mean of\nσ(θ). Local constitutive relation presented in section 3.2 below govern s the relation\nbetween heterogeneous stress field σ(x,t) and strain field ϵ(x,t).\nThen, we introduce the tangent modulus at meso-scale Das∆σ=D×∆ϵ.\nAccording to equations (7), (10) and (11), we have the tangent m odulus at macro-\nscale\n(12) D(x,t;θ)=1\n|R|/integraldisplay\nRD(x,t;θ)dR.\nWe will see in the examples below that, for a wide range of relative corr elation\nlength and size of the section R,Σand Ddo not depend on θ.I n t h a t c a s e ,\neven though the meso-scale model of concrete is stochastic, the resulting macro-\nscale model is deterministic, and independent of the actual realizat ion of the local\nparameters that is being considered.\nFinally, we recall that concrete specimens exhibit quasi-brittle beha vior in ten-\nsion with tensile strength generally 10 times smaller than compressive strength.\nBesides, we point out here that it is at macro-scale that these notio ns of compres-\nsion ( Σ≤0) and tension ( Σ>0) are relevant for the modeling presented in this\nwork.\n3.2. Model of the uniaxial cyclic behavior at meso-scale. We now concen-\ntrate on the local uniaxial cyclic inelastic constitutive relation that w ill be consid-\nered in this paper at meso-scale. We define it to be a simple elasto-plas tic model\nwith linear kinematic hardening, as illustrated in Figure 4. We provide he re, in\nthe setting of computational inelasticity [41, 22], the assumptions a nd resulting\nequations corresponding to this relation:\nFigure 3. Polished concrete section where the two phases human\neyes can see are represented: aggregates (crushed gravel and sand)\nand cement paste in-between.\nFor practical implementation, the heterogeneity will be conveyed in our model\nby the \ructuations of a random \feld p(x;\u0012), where\u0012represents randomness. Con-\nsistently with the \fber beam formulation presented in the previous section, we\nconsider a mesh of \fbers Fspanning beam cross-sections S. These \fbers have a\ncentroid located at position xF\nland a cross-section denoted by R. In the spirit of\nstrain-controlled tests on concrete specimens (see \fgure 1 along with its caption),\nstrainEis assumed homogeneous over R:\n(10) \u000f(x;t) =E(xF\nl;t)8x2R;\nand stress at macro-scale \u0006 is computed as the spatial mean stress over R:\n(11) \u0006( E;\u0012) =1\njRjZ\nR\u001b(\u000f;\u0012)dR;\nwherejRjdenotes the area of the \fber section R,\u001band\u000fthe stress and strain at\nmeso-scale in the uniaxial case. That is, analogously to Eand \u0006,\u000f=\u000f\u0001i1\ni1and\n\u001b=\u001b\u0001i1\ni1, where \u000fand\u001bare the strain and stress tensors at meso-scale such\nthat the constitutive model is considered in a 1D setting. Besides, we enhance the\nfact that \u0006 is computed as the spatial mean of \u001b(x) and not as the sample mean of\n\u001b(\u0012). Local constitutive relation presented in section 3.2 below governs the relation\nbetween heterogeneous stress \feld \u001b(x;t) and strain \feld \u000f(x;t).\nThen, we introduce the tangent modulus at meso-scale Das \u0001\u001b=D\u0002\u0001\u000f.\nAccording to equations (7), (10) and (11), we have the tangent modulus at macro-\nscale\n(12) D(x;t;\u0012) =1\njRjZ\nRD(x;t;\u0012)dR:\nWe will see in the examples below that, for a wide range of relative correlation\nlength and size of the section R, \u0006 andDdo not depend on \u0012. In that case,8\neven though the meso-scale model of concrete is stochastic, the resulting macro-\nscale model is deterministic, and independent of the actual realization of the local\nparameters that is being considered.\nFinally, we recall that concrete specimens exhibit quasi-brittle behavior in ten-\nsion with tensile strength generally 10 times smaller than compressive strength.\nBesides, we point out here that it is at macro-scale that these notions of compres-\nsion (\u0006\u00140) and tension (\u0006 >0) are relevant for the modeling presented in this\nwork.\n3.2.Model of the uniaxial cyclic behavior at meso-scale. We now concen-\ntrate on the local uniaxial cyclic inelastic constitutive relation that will be consid-\nered in this paper at meso-scale. We de\fne it to be a simple elasto-plastic model\nwith linear kinematic hardening, as illustrated in Figure 4. We provide here, in\nthe setting of computational inelasticity [41, 22], the assumptions and resulting\nequations corresponding to this relation:\n(i) The total deformation \u000fis split into elastic ( \u000fe) and plastic ( \u000fp) parts:\n(13) \u000f=\u000fe+\u000fp:\n(ii) The following state equation holds (upper dot denotes derivative with re-\nspect to time):\n(14) _ \u001b=C_\u000fe;\nwhereCis the elastic modulus.\n(iii) We impose that the stress \u001bcorrected by \u000b, the so-called back stress due\nto kinematic hardening, satis\fes yielding criterion\n(15) \u001ep=j\u001b+\u000bj\u0000\u001by\u00140;\nwhere\u001by\u00150 is the yield stress. As yielding function \u001ep(\u001b;\u000b) is negative,\nthe material is elastic; otherwise, plasticity is activated and the material\nstate evolves such that the condition \u001ep(\u001b;\u000b) = 0 is satis\fed.\n(iv) A change in \u000fpcan only take place if \u001ep= 0 and yielding occurs in the\ndirection of \u001b+\u000b, with a constant rate _ \rp\u00150:\n(16) _ \u000fp=\u001a_\rpsign(\u001b+\u000b) if\u001ep(\u001b;\u000b) = 0\n0 otherwise:\n_\rpis the so-called plastic multiplier.\n(v) WithHthe kinematic hardening modulus, the evolution of \u000bis de\fned as:\n(17) _ \u000b=\u0000H_\u000fp=\u0000_\rpHsign(\u001b+\u000b):\nAccordingly, admissible stresses \u001band\u000bremain in the set Ke=f(\u001b;\u000b)j\u001ep\u00140g\nand two kinds of evolutions are possible:\n(i) If (\u001b;\u000b)2\u0016Ke=f(\u001b;\u000b)j\u001ep<0g, the response is elastic:\n(18) _ \u000fp= 0) _\u001b=C_\u000f :\n(ii) If (\u001b;\u000b)2@Ke=f(\u001b;\u000b)j\u001ep= 0g, any evolution is possible only if _\u001ep= 0:\n(19)@\u001ep\n@\u001b_\u001b+@\u001ep\n@\u000b_\u000b= 0)_\rp=Csign(\u001b+\u000b)_\u000f\nC+H)_\u001b=CH\nC+H_\u000f :9\n8\n(i) The total deformation ϵis split into elastic ( ϵe) and plastic ( ϵp)p a r t s :\n(13) ϵ=ϵe+ϵp.\n(ii) The following state equation holds (upper dot denotes derivative with re-\nspect to time):\n(14) ˙ σ=C˙ϵe,\nwhere Cis the elastic modulus.\n(iii) We impose that the stress σcorrected by α, the so-called back stress due\nto kinematic hardening, satisfies yielding criterion\n(15) φp=|σ+α|−σy≤0,\nwhere σy≥0 is the yield stress. As yielding function φp(σ,α) is negative,\nthe material is elastic; otherwise, plasticity is activated and the mat erial\nstate evolves such that the condition φp(σ,α) = 0 is satisfied.\n(iv) A change in ϵpcan only take place if φp= 0 and yielding occurs in the\ndirection of σ+α, with a constant rate ˙ γp≥0:\n(16) ˙ ϵp=/braceleftbigg˙γpsign( σ+α) if φp(σ,α)=0\n0 otherwise.\n˙γpis the so-called plastic multiplier.\n(v) With Hthe kinematic hardening modulus, the evolution of αis defined as:\n(17) ˙ α=−H˙ϵp=−˙γpHsign( σ+α).\nFigure 4. Compressive cyclic behavior at meso-scale. The yield\nstress σy(x) fluctuates over the concrete section R: two local re-\nsponses at two distinct material points x1[left] and x2[right] are\nrepresented in the figure. During elastic loading/unloading, the\nslope is C; in yielding phases, the slope isCH\nC+H<C. With this\ntype of model first yielding occurs once σ=σyand the elastic do-\nmain keeps constant amplitude 2 σy. For low values of σy, yielding\ncan be observed in compression both during loading and unloading\n[right].\nAccordingly, admissible stresses σandαremain in the set Ke={(σ,α)|φp≤0}\nand two kinds of evolutions are possible:\n(i) If ( σ,α)∈¯Ke={(σ,α)|φp<0}, the response is elastic:\n(18) ˙ ϵp=0 ⇒ ˙σ=C˙ϵ.\n(ii) If ( σ,α)∈∂Ke={(σ,α)|φp=0}, any evolution is possible only if ˙φp=0 :\n(19)∂φp\n∂σ˙σ+∂φp\n∂α˙α=0 ⇒˙γp=Csign( σ+α)˙ϵ\nC+H⇒˙σ=CH\nC+H˙ϵ.\nFigure 4. Compressive cyclic behavior at meso-scale. The yield\nstress\u001by(x) \ructuates over the concrete section R: two local re-\nsponses at two distinct material points x1[left] and x2[right] are\nrepresented in the \fgure. During elastic loading/unloading, the\nslope isC; in yielding phases, the slope isCH\nC+H< C. With this\ntype of model \frst yielding occurs once \u001b=\u001byand the elastic do-\nmain keeps constant amplitude 2 \u001by. For low values of \u001by, yielding\ncan be observed in compression both during loading and unloading\n[right].\nIt is then possible, from equations (18) and (19), to give the expression of the\ntangent modulus D:\n(20) _ \u001b=D_\u000fwith D=\u001aC if (\u001b;\u000b)2\u0016Ke\nCH\nC+Hif (\u001b;\u000b)2@Ke:\nIt should be reminded at this point that, as mentioned in the introduction and\nillustrated in Figure 4, the yield stress is assumed heterogeneous. The relation\npresented here is therefore de\fned between stress and strain in each point in space\nwith a di\u000berent yield stress.\n3.3.Description of the yield stress random \feld. In this section, we describe\nthe choice that is made for the modeling of the heterogeneous yield stress: the yield\nstress is represented by a 2D log-normal homogeneous random \feld over the con-\ncrete areaR. We note here that, to the best of our knowledge, there currently exists\nno experimental dataset of local stress-strain uniaxial concrete responses recorded\nat many points over a concrete area. Here, the choice of using random \felds to con-\nvey heterogeneity of the yield stress is mainly motivated by the e\u000bectiveness of the\nmethod. We hope that this proposed interpretation of concrete meso-structure will\nfoster interaction between numerical and material scientists and help designing ex-\nperimental investigations that would eventually support or invalidate the numerical\nmodel we propose in this paper.\nLet us then consider a probability space (\u0002 ;\n;Pr), where \n is a \u001b-algebra of\nelements of \u0002 and Pr is a probability measure. The 2D random \feld of yield stress\nis constructed as a nonlinear point-wise transformation [17] Sy(x;\u0012) =f(G(x;\u0012))\nof a homogeneous unit centered Gaussian random \feld G(x;\u0012) with given power\nspectral density (PSD) SGG(\u0014). The PSD is chosen here as the product of triangle\nfunctions with identical properties in the two orthogonal directions of the 2D plane,10\ndenoted by the subscript 1 and 2 throughout sections 3.3 and 3.4:\n(21) SGG(\u0014) =1\n\u00142u\u0003\u0012\u00141\n\u0014u\u0013\n\u0003\u0012\u00142\n\u0014u\u0013\n;\nwhere \u0003(\u0014) = 1\u0000j\u0014jifj\u0014j\u00141 and cut-o\u000b wave numbers \u0014u;1=\u0014u;2=\u0014u. For\nwave numbers above the cut-o\u000b \u0014u, the spectral density vanishes. In the spatial\ndomain, this PSD corresponds to the following autocorrelation function (the Fourier\ntransform of SGG(\u0014)):\n(22) RGG(\u0010) = sinc2\u0010\u0014u\n2\u0019\u00101\u0011\nsinc2\u0010\u0014u\n2\u0019\u00102\u0011\n;\nwhere sinc( x) = sin(\u0019x)=(\u0019x). The random \feld G(x;\u0012) therefore \ructuates over\ntypical lengths `c;1=`c;2=`c= 2\u0019=\u0014u, the so-called correlation length.\nThe nonlinear point-wise transformation fcontrols the \frst-order marginal dis-\ntribution of \u001by. In particular, it controls the desired expectation mand variance s2\nof the yield stress homogeneous random \feld. In this paper, we choose to consider\na log-normal \frst-order marginal density, to ensure that the realizations are almost-\nsurely and almost everywhere positive, as expected. The nonlinear transformation\nis then given by:\n(23) Sy(x;\u0012) = exp(mG+sG\u0002G(x;\u0012))>0;\nwhere\n(24)mG=\u0000ln \n1\nmr\n1 +s2\nm2!\nandsG=s\nln\u0012\n1 +s2\nm2\u0013\n:\nOther \frst-order marginal densities could be considered, for example using the\nmaximum entropy principle [37, 47, 42, 10, 11] or Bayesian identi\fcation [4, 19].\nAlso, the PSD function is translated by the nonlinear transformation fso that the\nPSD of the yield stress and of the underlying Gaussian \feld are di\u000berent, with\npossible incompatibilities with the chosen \frst-order marginal density [17, 31, 38].\nThese important but technical issues go beyond the scope of this paper and will\nnot be further discussed here.\n10\nwhere\n(24) mG=−ln/parenleftBigg\n1\nm/radicalbigg\n1+s2\nm2/parenrightBigg\nand sG=/radicalBigg\nln/parenleftbigg\n1+s2\nm2/parenrightbigg\n.\nOther first-order marginal densities could be considered, for exa mple using the\nmaximum entropy principle [37, 47, 42, 10, 11] or Bayesian identificat ion [4, 19].\nAlso, the PSD function is translated by the nonlinear transformatio nfso that the\nPSD of the yield stress and of the underlying Gaussian field are di fferent, with\npossible incompatibilities with the chosen first-order marginal densit y [17, 31, 38].\nThese important but technical issues go beyond the scope of this p aper and will\nnot be further discussed here.\n00.20.40.60.81\n00.20.40.60.8100.511.522.5\nx2/dx3/dσy/m\nFigure 5. Realizations of log-normal random fields over a square\nof size dfor di fferent correlation lengths: [left] ℓc/d= 1, [center]\nℓc/d=0.1, [right] ℓc/d= 0 (white noise). Coordinates x2and\nx3, previously introduced in the description of the beam element\nin section 2, are reused here to recall that the random fields are\ngenerated to parameter heterogeneous yield stress over beam c ross-\nsection areas.\n3.4. Numerical implementation (at each quadrature point in each beam\nfiber). At each material point xF\nl(quadrature point l,b e a mfi b e r F), relations (10),\n(11) and (12) have to be calculated.\nOn the one hand, Gaussian random field Gis digitized using the spectral rep-\nresentation method, in its FFT implementation [39, 13, 40]. As an illustr ation,\nconsidering a 2D random field with identical properties in orthogonal directions 1\nand 2 (see [40] for more details):\n(25) G(p1∆x, p 2∆x;θ)=R eM−1/summationdisplay\nn1=0M−1/summationdisplay\nn2=0/parenleftBig\nBn1n2(θ)e x p/parenleftBig\n2iπ/parenleftBign1p1\nM+n2p2\nM/parenrightBig/parenrightBig\n+˜Bn1n2(θ)e x p/parenleftBig\n2iπ/parenleftBign1p1\nM−n2p2\nM/parenrightBig/parenrightBig/parenrightBig\nwhere i2=−1, (p1,p2)∈[0,...,M −1]2,\nBn1n2=2∆κ/radicalbig\nSGG(n1∆κ,n2∆κ)e x p ( iφn1n2(θ))\n˜Bn1n2=2∆κ/radicalbig\nSGG(n1∆κ,−n2∆κ)e x p ( iψn1n2(θ)) (26)\nand∆κ=κu/N(N∈N⋆→∞),M≥2N,φn1n2(θ)a n d ψn1n2(θ) are independent\nrandom phase angles uniformly distributed in [0 ,2π]. The resulting 2D random\nFigure 5. Realizations of log-normal random \felds over a square\nof sizedfor di\u000berent correlation lengths: [left] `c=d= 1, [center]\n`c=d= 0:1, [right]`c=d= 0 (white noise). Coordinates x2and\nx3, previously introduced in the description of the beam element\nin section 2, are reused here to recall that the random \felds are\ngenerated to parameter heterogeneous yield stress over beam cross-\nsection areas.11\n3.4.Numerical implementation (at each quadrature point in each beam\n\fber). At each material point xF\nl(quadrature point l, beam \fber F), relations (10),\n(11) and (12) have to be calculated.\nOn the one hand, Gaussian random \feld Gis digitized using the spectral rep-\nresentation method, in its FFT implementation [39, 13, 40]. As an illustration,\nconsidering a 2D random \feld with identical properties in orthogonal directions 1\nand 2 (see [40] for more details):\n(25)G(p1\u0001x;p 2\u0001x;\u0012) = ReM\u00001X\nn1=0M\u00001X\nn2=0\u0010\nBn1n2(\u0012) exp\u0010\n2i\u0019\u0010n1p1\nM+n2p2\nM\u0011\u0011\n+~Bn1n2(\u0012) exp\u0010\n2i\u0019\u0010n1p1\nM\u0000n2p2\nM\u0011\u0011\u0011\nwherei2=\u00001, (p1;p2)2[0;:::;M\u00001]2,\nBn1n2= 2\u0001\u0014p\nSGG(n1\u0001\u0014;n 2\u0001\u0014) exp(i\u001en1n2(\u0012))\n~Bn1n2= 2\u0001\u0014p\nSGG(n1\u0001\u0014;\u0000n2\u0001\u0014) exp(i n1n2(\u0012)) (26)\nand \u0001\u0014=\u0014u=N(N2N?!1 ),M\u00152N,\u001en1n2(\u0012) and n1n2(\u0012) are independent\nrandom phase angles uniformly distributed in [0 ;2\u0019]. The resulting 2D random\n\feld is periodic with a two-dimensional period L0\u0002L0withL0=M\u0001x= 2\u0019=\n\u0001\u0014. Realizations of log-normal random \felds with di\u000berent correlation lengths are\nshown in \fgure 5.\nOn the other hand, a generic concrete section Ris built as a square with edge\nof lengthdandRis meshed by a square grid of N2\nfidentical squares. Then, the\nmesh size is d=Nfand, for any integrable function g,R\nRg(x)dR\u0019d2\nN2\nfPN2\nf\nf=1g(xf),\nwhere xfis the position of the centroid of the f-th mesh overR.\nThen, digitized random \feld Syis mapped onto the xf's overR. To this purpose,\nwe imposeL0\u0015d, that isjRjis smaller or equal to a period of the random \feld, and\nmapping is performed according to the following method. First, Nfis calculated\nas:\n(27)d= Int\u0012d\n\u0001x\u0013\n\u0001x+ Res)Nf=\u001aInt(d=\u0001x) if Res = 0\nInt(d=\u0001x) + 1 otherwise:\nThen, at the N2\nfpoints xf2R,Sy(xf;\u0012) is calculated as the linear interpolation\nof the four digitized values of Sy(x;\u0012) in ]xf\u0000\u0001x;xf+ \u0001x]2, as illustrated in\n\fgure 6.\nWith the spatially variable yield stress now known at each point xf,f2[1;::;N2\nf]\ninR, the equations presented in section 3.2 can be solved to update the variables at\nmeso-scale. This is done numerically at each of the N2\nfpositions following classical\nreturn-mapping computational procedure [41, 22].\nFinally, as a transition from compression to tension is detected during global\nNewton-Raphson iterative process to solve structural equilibrium equations, that is\n\u0006(k+1)\nn>0 while \u0006(k)\nn\u00140, a local Newton-Raphson precess is implemented to \fnd\nthe strainEcfor which \u0006(k+1)\nn (Ec) = 0, to update the meso-structure accordingly,\nand to set \u0006(k+1)\nn = 0 andD(k+1)\nn = 0.\nBefore observing on numerical tests the shape of the stress-strain curves obtained\nwith this model, we turn to the simple case of vanishing correlation length ( `c!\n0). The interest of this particular case is that some analytical expressions can be12\n11\nfield is periodic with a two-dimensional period L0×L0with L0=M∆x=2π/\n∆κ. Realizations of log-normal random fields with di fferent correlation le ngths are\nshown in figure 5.\nOn the other hand, a generic concrete section Ris built as a square with edge\nof length dand Ris meshed by a square grid of N2\nfidentical squares. Then, the\nmesh size is d/N fand, for any integrable function g,/integraltext\nRg(x)dR≈d2\nN2\nf/summationtextN2\nf\nf=1g(xf),\nwhere xfis the position of the centroid of the f-th mesh over R.\nThen, digitized random field Syis mapped onto the xf’s over R. To this purpose,\nwe impose L0≥d, that is |R|is smaller or equal to a period of the random field, and\nmapping is performed according to the following method. First, Nfis calculated\nas:\n(27) d= Int/parenleftbiggd\n∆x/parenrightbigg\n∆x+R e s ⇒Nf=/braceleftbiggInt(d/∆x) if Res = 0\nInt(d/∆x) + 1 otherwise.\nThen, at the N2\nfpoints xf∈R,Sy(xf;θ) is calculated as the linear interpolation\nof the four digitized values of Sy(x;θ) in ] xf−∆x,xf+∆x]2, as illustrated in\nfigure 6.\nFigure 6. On the one hand, the random field is digitized on a\nsquare grid (dashed lines) of ( M+1 )2points — here M=6—\nspanning an area of size L0×L0, with L0=M×∆x.O n t h e o t h e r\nhand, generic concrete square section Rhas an area of size d×d\n(d≤L0) that is divided into N2\nfidentical meshes (plain lines) —\nhere Nf= 5. The values of the random field at the positions xf\noccupied by the centroids of the N2\nfmeshes ( ×) is calculated as\nthe linear interpolation of the four surrounding digitized values of\nthe random field ( ◦).\nWith the spatially variable yield stress now known at each point xf,f∈[1, .., N2\nf]\ninR, the equations presented in section 3.2 can be solved to update the variables at\nmeso-scale. This is done numerically at each of the N2\nfpositions following classical\nreturn-mapping computational procedure [41, 22].\nFinally, as a transition from compression to tension is detected durin g global\nNewton-Raphson iterative process to solve structural equilibrium equations, that is\nΣ(k+1)\nn >0 while Σ(k)\nn≤0, a local Newton-Raphson precess is implemented to find\nthe strain Ecfor which Σ(k+1)\nn(Ec) = 0, to update the meso-structure accordingly,\nand to set Σ(k+1)\nn =0a n d D(k+1)\nn =0 .\nFigure 6. On the one hand, the random \feld is digitized on a\nsquare grid (dashed lines) of ( M+ 1)2points | here M= 6 |\nspanning an area of size L0\u0002L0, withL0=M\u0002\u0001x. On the other\nhand, generic concrete square section Rhas an area of size d\u0002d\n(d\u0014L0) that is divided into N2\nfidentical meshes (plain lines) |\nhereNf= 5. The values of the random \feld at the positions xf\noccupied by the centroids of the N2\nfmeshes (\u0002) is calculated as\nthe linear interpolation of the four surrounding digitized values of\nthe random \feld ( \u000e).\nderived, so that discussion is more straightforward. The more general case with\n\fnite correlation length will be considered later in Section 5.1.\n4.A particular case: vanishing correlation length and monotonic\nloading\n4.1.Preliminaries. The case of vanishing correlation length along with uniaxial\ncyclic loading has been treated in a general setting. Indeed in [24], the stress-\nstrain uniaxial response is given as a probability density function ( pdf) of stress\nwith respect to the time-dependent strain and a second-order exact expression of\nthepdfevolution is computed solving the Fokker-Planck-Kolmogorov equation that\ngoverns the problem. This latter method is valid for monotonic as well as cyclic\nloading. Hereafter, the validity of the results is limited to monotonic loading, but\nthe problem is cast in a di\u000berent and simpler mathematical setting that can be\nsolved analytically. These analytical developments shed light on some capabilities\nof the model introduced in the previous section and that will be retrieved in the\nmore general case of non-zero correlation in Section 5.1.\n4.2.Constitutive response at macro-scale. Let respectively denote Reand\nRpthe shares of a \fber cross-section that remain elastic and yield. According\nto the developments in section 3.2: Re(t;\u0012) =fx2 R j D(x;t;\u0012) =Cgand\nRp(t;\u0012) =fx2Rj D(x;t;\u0012) =CH= (C+H)g. Also,Re\\Rp=;andR=Re[Rp.\nNote thatReandRpare time-dependent because Ddepends on the loading history.\nWe denote byj\u000fjthe area of\u000f. Then, considering a subset AofR, we have,8x2R,\nthe probability measure Pr[ x2A] =jAj=jRj.13\nUsing the fact that jRj=jRej+jRpj, we \frst rewrite the tangent modulus at\nmacro-scale in equation (12) as:\n(28)D=1\njRj\u0012\njRejC+jRpjCH\nC+H\u0013\n=C\nC+H\u0012jRej\njRjC+H\u0013\n:\nWe now seek an explicit expression for jRej=jRj.\nFirst, suppose the state of the material is known at time t0, then we de\fne the\ntrial stresses\n(29) \u001btr(x;t) =\u001b0(x) +C(\u000f(t)\u0000\u000f0) and\u000btr(x;t) =\u000b0(x);\nwhere subscript 0 refers to time t0. In the particular case of monotonic loading, a\nnecessary and su\u000ecient condition for xto be inRpat timet>t 0is\u001ep;tr(x;t)\u00150,\nthat is\u001by(x)\u0014j\u001btr(x;t) +\u000b0(x)j(see equation 15). We then have:\n(30)jRej=jRj= Pr[ x2Re] = 1\u0000Pr[Sy(x)\u0014j\u001btr(x;t) +\u000b0(x)j]:\nThen, in the particular case of vanishing correlation length, the random variables\nSy(x) are independent and identically distributed over R. For the log-normal\ndistribution assumption made throughout this work, it means that the cumulative\ndensity function of Sy(x) is,8x2R:\n(31)FSy(x)(\u001by) = Pr[ Sy(x)\u0014\u001by] =1\n2\u0012\n1 + erf\u0012ln\u001by\u0000mGp\n2sG\u0013\u0013\n;\nwhere erf is the so-called error function.\nFinally, for the sake of simplicity and without any loss of generality, we assume\n\u001b0=\u000b0=\u000f0= 0. Accordingly, and using equations (29), along with (24) to\nreplacemGandsGby the mean mand standard deviation sof the homogeneous\nlog-normal random \feld Sy, it comes:\n(32)jRej\njRj= 1\u0000FSy(x)(jC\u000f(t)j) =1\n20\nBB@1\u0000erf0\nBB@ln\u0012\njC\u000f(t)j\nmq\n1 +s2\nm2\u0013\nq\n2 ln\u0000\n1 +s2\nm2\u00011\nCCA1\nCCA:\nEquations (28) and (32) are used to plot \fgure 7 where the response of the model\nat macro-scale is shown for di\u000berent sets of mean and variance parameters for the\nlog-normal random yield stress \feld Sy.\n4.3.Asymptotic response of the model at macro-scale. The following as-\nymptotic behaviors can be observed at macro-scale:\n(i) Suppose s2=m2approaches 0. Then, according to equation (32), jRej=jRj\napproaches the Heaviside's function H(m\u0000jCE(t)j), that isjRej=jRj= 0 if\njCE(t)j>m andjRej=jRj= 1 ifjCE(t)j\u0014m. According to equation (28),\nthe model response at macro-scale is then as follows:\n(33) _\u0006(t) =D(t)_E(t) whereD=\u001aC ifjCE(t)j\u0014m\nCH\nC+HifjCE(t)j>m:\n(ii) Ifs2=m2!1 , thenjRej=jRj! 0 and consequently _\u0006(t)!CH= (C+\nH)_E(t).14\n13\nFinally, for the sake of simplicity and without any loss of generality, we assume\nσ0=α0=ϵ0= 0. Accordingly, and using equations (29), along with (24) to\nreplace mGand sGby the mean mand standard deviation sof the homogeneous\nlog-normal random field Sy, it comes:\n(32)|Re|\n|R|=1−FSy(x)(|Cϵ(t)|)=1\n2⎛\n⎜⎜⎝1−erf⎛\n⎜⎜⎝ln/parenleftbigg\n|Cϵ(t)|\nm/radicalBig\n1+s2\nm2/parenrightbigg\n/radicalBig\n2 ln/parenleftbig\n1+s2\nm2/parenrightbig⎞\n⎟⎟⎠⎞\n⎟⎟⎠.\nEquations (28) and (32) are used to plot figure 7 where the respon se of the model\nat macro-scale is shown for di fferent sets of mean and variance par ameters for the\nlog-normal random yield stress field Sy.\n−3.5 −3 −2.5 −2 −1.5 −1 −0.5 0\nx 10−3−50−45−40−35−30−25−20−15−10−50\nEΣ[MPa]increasing s\n−3.5 −3 −2.5 −2 −1.5 −1 −0.5 0\nx 10−3−110−100−90−80−70−60−50−40−30−20−100\nEΣ[MPa]\nincreasing mFigure 7. Monotonic macro-scale response of the model with\nvanishing correlation length. Parameters C=3 0 G P a a n d\nH= 10 GPa are used. [top] m=3 0 M P a a n d s/\nm=1 0−4,0.4,2,10,109; [bottom] s=3 0M P aa n d s/m =\n10−3,0.5,1,2.5,103. Plain curves show asymptotic behaviors as\ns/m →0o r+ ∞.\n4.3. Asymptotic response of the model at macro-scale. The following as-\nymptotic behaviors can be observed at macro-scale:\n(i) Suppose s2/m2approaches 0. Then, according to equation (32), |Re|/|R|\napproaches the Heaviside’s function H(m−|CE(t)|), that is |Re|/|R|= 0 if\n13\nFinally, for the sake of simplicity and without any loss of generality, we assume\nσ0=α0=ϵ0= 0. Accordingly, and using equations (29), along with (24) to\nreplace mGand sGby the mean mand standard deviation sof the homogeneous\nlog-normal random field Sy, it comes:\n(32)|Re|\n|R|=1−FSy(x)(|Cϵ(t)|)=1\n2⎛\n⎜⎜⎝1−erf⎛\n⎜⎜⎝ln/parenleftbigg\n|Cϵ(t)|\nm/radicalBig\n1+s2\nm2/parenrightbigg\n/radicalBig\n2 ln/parenleftbig\n1+s2\nm2/parenrightbig⎞\n⎟⎟⎠⎞\n⎟⎟⎠.\nEquations (28) and (32) are used to plot figure 7 where the respon se of the model\nat macro-scale is shown for di fferent sets of mean and variance par ameters for the\nlog-normal random yield stress field Sy.\n−3.5 −3 −2.5 −2 −1.5 −1 −0.5 0\nx 10−3−50−45−40−35−30−25−20−15−10−50\nEΣ[MPa]increasing s\n−3.5 −3 −2.5 −2 −1.5 −1 −0.5 0\nx 10−3−110−100−90−80−70−60−50−40−30−20−100\nEΣ[MPa]\nincreasing mFigure 7. Monotonic macro-scale response of the model with\nvanishing correlation length. Parameters C=3 0 G P a a n d\nH= 10 GPa are used. [top] m=3 0 M P a a n d s/\nm=1 0−4,0.4,2,10,109; [bottom] s=3 0M P aa n d s/m =\n10−3,0.5,1,2.5,103. Plain curves show asymptotic behaviors as\ns/m →0o r+ ∞.\n4.3. Asymptotic response of the model at macro-scale. The following as-\nymptotic behaviors can be observed at macro-scale:\n(i) Suppose s2/m2approaches 0. Then, according to equation (32), |Re|/|R|\napproaches the Heaviside’s function H(m−|CE(t)|), that is |Re|/|R|= 0 if\nFigure 7. Monotonic macro-scale response of the model with\nvanishing correlation length. Parameters C= 30 GPa and\nH= 10 GPa are used. [top] m= 30 MPa and s=\nm= 10\u00004;0:4;2;10;109; [bottom] s= 30 MPa and s=m =\n10\u00003;0:5;1;2:5;103. Plain curves show asymptotic behaviors as\ns=m!0 or +1.\n(iii) Now with \fnite and non-zero s2=m2:\n(34) _\u0006(t) =D(t)_E(t) whereD!\u001aC ifE(t)!0\nCH\nC+HifE(t)!1:\nThese asymptotic responses at macro-scale are illustrated in \fgure 7 (plain lines).\n5.Numerical applications\n5.1.Concrete uniaxial compressive cyclic response at macro-scale. First\nnumerical applications aim at demonstrating the capability of the model introduced\nabove in section 3 to represent the response of concrete in uniaxial compressive\ncyclic loading. Five model parameters need to be considered: elastic and harden-\ning moduli CandH, along with mean m, standard deviation sand correlation\nlength`cused to build realizations of a homogeneous log-normal random \feld that\nparameterizes the \ructuations of the yield stress \u001byover beam sections.\nThe e\u000bects of m,sand`con the material response at macro-scale will be further\ninvestigated below. Right now however, we set:\n\u000fC= 27:5 GPa, which corresponds to the elastic modulus measured on\nspecimens made of the concrete actually cast to build the frame element\nused in the next numerical application (Section 5.2).\n\u000fH= 0 according to both (i) the fact that Hcontrols tangent modulus at\nmacro scale as strain becomes large (see equation (34)), and (ii) that we\nseek a numerical response that ultimately exhibits null tangent modulus\nin monotonic loading at macro-scale. We anticipate here stressing that\nthe model developed in previous sections is not capable of representing\nthe softening phase as strain increases while stress decreases (non-positive\ntangent modulus).15\n15\nFigure 8. Sample mean (thick plain line) plus/minus standard\ndeviation (boundaries of the shaded areas) monotonic response a t\nmacro-scale computed from a sample of 100 realizations of the\nmaterial structure at meso-scale with m=3 0M P aa n d s/m=1\nfor the log-normal marginal law. Meso-structures are generate d\nwith di fferent correlation lengths: [left] ℓc/d=0.1, [center] ℓc/\nd=0.2, [right] ℓc/d=0.4. Cyclic response for one particular\nrealization of the meso-structure is also shown (thin plain line).\nincreases. The area Rdefined with ℓc/d=0.1,N=1 0a n d M= 64, is statistically\nrepresentative in the sense that there is almost independence bet ween the random\nrealization of the meso-structure and the response at macro-sc ale. We remark here\nthat the model is capable of representing variability from one concr ete sample to\nanother and that this variability can bring information on the correla tions in the\nmeso-structure. Suppose indeed that we had 100 concrete samp les and a variability\nof the responses at macro-scale close to that shown by the grey a rea in figure 8\n[center] for instance. Then, the meso-structure of the tested concrete would be best\nrepresented by the ratio ℓc/d=0.2. Consequently, the sample standard deviation\ncan bring information on the actual correlation length.\nFinally, we can notice that the shape of the cyclic response (thin line) represents\nmost of the salient features exhibited experimentally in uniaxial comp ression test\nfor concrete (remember figure 1). We point out here that streng th degradation\n(softening) along with sti ffness degradation (observed experimen tally one cycle after\nanother) are not represented by this model. However, a key point for representing\nmaterial damping is the capability of the model to generate local hys teresis loops\nin unloading-loading cycles.\n5.1.2. Influence of mand son the macroscopic response. We use here, beside N=\n10 and M=6 4( Nf=6 4 ) , ℓc/d=0.1 so that little variability is expected to be\nobserved in the model response at macro-scale from one realizatio no ft h em e s o -\nstructure to another (see figure 8 [left]). Figure 9 shows material response at macro-\nscale for di fferent sets of mean mand standard deviation sof the log-normal random\nfield that conveys spatial variability in the material structure at me so-scale.\nIt can be observed that for a small value of s, response approaches bi-linear elasto-\nplastic behavior (actually perfectly plastic because His set to zero here) where there\nis almost no hysteresis observed during unloading-loading cycle (figu re 9 [left] and\n[right]). This comes from the fact that, if sapproaches 0, there is almost no spatial\nvariability of the yield stress because it is almost homogeneous over Rand takes\nFigure 8. Sample mean (thick plain line) plus/minus standard\ndeviation (boundaries of the shaded areas) monotonic response at\nmacro-scale computed from a sample of 100 realizations of the\nmaterial structure at meso-scale with m= 30 MPa and s=m = 1\nfor the log-normal marginal law. Meso-structures are generated\nwith di\u000berent correlation lengths: [left] `c=d= 0:1, [center] `c=\nd= 0:2, [right]`c=d= 0:4. Cyclic response for one particular\nrealization of the meso-structure is also shown (thin plain line).\n5.1.1. In\ruence of `con the macroscopic response. We \frst illustrate how `cin-\n\ruences the macroscopic response by considering the three following cases: (i) `c=\nd= 0:1, (ii)`c=d= 0:2 and (iii) `c=d= 0:4. For each of these three cases, we\ntakeN= 10 andM= 64, that is \u0001 x=d= 0:016 andNf= 64 (see equation (27)).\nWe recall that the random \feld characteristics are taken as identical in both space\ndirections ( `c=`c;1=`c;2,N=N1=N2, . . . ). Besides, a sample of 100 indepen-\ndent homogeneous log-normal random \felds with targeted mean m= 30 MPa and\ncoe\u000ecient of variation s=m = 1 for the marginal log-normal law is generated for\neach case.\nResulting material responses at macro-scale are shown in \fgure 8. A \frst ob-\nvious observation is that model response at macro-scale is much richer than at\nmeso-scale (see \fgure 4). We can then observe that sample mean response (thick\nline) is not sensitive to the correlation length. However the variability of the macro-\nscopic response from one realization of the meso-structure to another depends on\nthe correlation length: it is almost null for `c=d= 0:1 while it is enhanced as `c\nincreases. The area Rde\fned with `c=d= 0:1,N= 10 andM= 64, is statistically\nrepresentative in the sense that there is almost independence between the random\nrealization of the meso-structure and the response at macro-scale. We remark here\nthat the model is capable of representing variability from one concrete sample to\nanother and that this variability can bring information on the correlations in the\nmeso-structure. Suppose indeed that we had 100 concrete samples and a variability\nof the responses at macro-scale close to that shown by the grey area in \fgure 8\n[center] for instance. Then, the meso-structure of the tested concrete would be best\nrepresented by the ratio `c=d= 0:2. Consequently, the sample standard deviation\ncan bring information on the actual correlation length.\nFinally, we can notice that the shape of the cyclic response (thin line) represents\nmost of the salient features exhibited experimentally in uniaxial compression test\nfor concrete (remember \fgure 1). We point out here that strength degradation\n(softening) along with sti\u000bness degradation (observed experimentally one cycle after16\nanother) are not represented by this model. However, a key point for representing\nmaterial damping is the capability of the model to generate local hysteresis loops\nin unloading-loading cycles.\n16\n−3.5 −3 −2.5 −2 −1.5 −1 −0.5 0\nx 10−3−1−0.9−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.10\nEΣ/mincreasing s\n−3.5 −3 −2.5 −2 −1.5 −1 −0.5 0\nx 10−3−1−0.9−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.10\nEΣ/mincreasing m\n−3.5 −3 −2.5 −2 −1.5 −1 −0.5 0\nx 10−3−1−0.9−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.10\nEΣ/mincreasing m=sFigure 9. Sample mean (plain line) plus/minus standard devia-\ntion (dashed lines) response at macro-scale computed from a sam-\nple of 100 di fferent realizations of the material structure at meso-\nscale. Meso-structures are generated with di fferent targeted m ean\nmand coe fficients of variation s/mfor the log-normal marginal\nlaw: [left] m=3 0M P aa n d s/m=0.1,1,3; [center] s=3 0M P a\nand s/m=0.6,1,3; [right] s/m=1a n d m=1 0 ,30,50 MPa.\nvalues close to its mean m; then the response at macro-scale coincides with that\nat meso-scale (elasto-plasticity with H= 0 here). This is also in accordance with\nwhat was shown already in figure 7.\nResponses shown in figure 9 [left] lie in-between this latter extreme c ase and\nthe other extreme case of sapproaching infinity. In this situation, log-normal\ndistribution approaches 0 all over the positive real semi-line and, co nsequently,\nplasticity is activated almost everywhere over Rresulting in a macro-scale response\nthat is perfectly plastic without elastic phase (that is here Σ=0f o ra nyEbecause\nH=0 ) .\nAlso, it is shown in figure 9 that the value of the strain Eat which stress Σ\nreaches zero when unloading (residual plastic deformation) is much more sensitive\nto parameter mthan s. Furthermore, the thickness of the hysteresis loops obviously\ndepends on the so-called coe fficient of variation s/mbut it is not clear whether it\nis more sensitive to either of the two parameters. Finally, the variab ility in the\nsample of responses at macro-scale increases with s/mand is more sensitive to m\nthan s, at least as far as the range of values chosen here for both param eters is\nconcerned.\n5.2. Damping in a reinforced concrete column in free vibration. We now\nshow how the material model developed in the previous sections can be used to\nrepresent the experimental backbone curve of a concrete test specimen in uniaxial\nloading. Then, we implement this material law in the fiber beam element p resented\nin section 2 and show how damping is generated in a reinforced concre te (RC)\ncolumn in free vibration. The observed damping does not result from the addition\nof damping forces in the balance equation of the RC column but from t he hysteresis\nloops in the concrete material law at macro-scale.\n5.2.1. Geometry of the column and loading. The column considered here corre-\nsponds to the 1st-floor external column of the ductile ( R=4 )R Cf r a m et e s t e d\nin [15, 25]. The loading is however here di fferent: it consists of a mass M=5 0 0k g\nimposed step by step and kept constant while the column oscillates in f ree vibration\nFigure 9. Sample mean (plain line) plus/minus standard devia-\ntion (dashed lines) response at macro-scale computed from a sam-\nple of 100 di\u000berent realizations of the material structure at meso-\nscale. Meso-structures are generated with di\u000berent targeted mean\nmand coe\u000ecients of variation s=m for the log-normal marginal\nlaw: [left]m= 30 MPa and s=m = 0:1;1;3; [center]s= 30 MPa\nands=m = 0:6;1;3; [right]s=m = 1 andm= 10;30;50 MPa.\n5.1.2. In\ruence of mandson the macroscopic response. We use here, beside N=\n10 andM= 64 (Nf= 64),`c=d= 0:1 so that little variability is expected to be\nobserved in the model response at macro-scale from one realization of the meso-\nstructure to another (see \fgure 8 [left]). Figure 9 shows material response at macro-\nscale for di\u000berent sets of mean mand standard deviation sof the log-normal random\n\feld that conveys spatial variability in the material structure at meso-scale.\nIt can be observed that for a small value of s, response approaches bi-linear elasto-\nplastic behavior (actually perfectly plastic because His set to zero here) where there\nis almost no hysteresis observed during unloading-loading cycle (\fgure 9 [left] and\n[right]). This comes from the fact that, if sapproaches 0, there is almost no spatial\nvariability of the yield stress because it is almost homogeneous over Rand takes\nvalues close to its mean m; then the response at macro-scale coincides with that\nat meso-scale (elasto-plasticity with H= 0 here). This is also in accordance with\nwhat was shown already in \fgure 7.\nResponses shown in \fgure 9 [left] lie in-between this latter extreme case and\nthe other extreme case of sapproaching in\fnity. In this situation, log-normal\ndistribution approaches 0 all over the positive real semi-line and, consequently,\nplasticity is activated almost everywhere over Rresulting in a macro-scale response\nthat is perfectly plastic without elastic phase (that is here \u0006 = 0 for any Ebecause\nH= 0).\nAlso, it is shown in \fgure 9 that the value of the strain Eat which stress \u0006\nreaches zero when unloading (residual plastic deformation) is much more sensitive\nto parameter mthans. Furthermore, the thickness of the hysteresis loops obviously\ndepends on the so-called coe\u000ecient of variation s=m but it is not clear whether it\nis more sensitive to either of the two parameters. Finally, the variability in the\nsample of responses at macro-scale increases with s=m and is more sensitive to m17\nthans, at least as far as the range of values chosen here for both parameters is\nconcerned.\n5.2.Damping in a reinforced concrete column in free vibration. We now\nshow how the material model developed in the previous sections can be used to\nrepresent the experimental backbone curve of a concrete test specimen in uniaxial\nloading. Then, we implement this material law in the \fber beam element presented\nin section 2 and show how damping is generated in a reinforced concrete (RC)\ncolumn in free vibration. The observed damping does not result from the addition\nof damping forces in the balance equation of the RC column but from the hysteresis\nloops in the concrete material law at macro-scale.\n5.2.1. Geometry of the column and loading. The column considered here corre-\nsponds to the 1st-\roor external column of the ductile ( R= 4) RC frame tested\nin [15, 25]. The loading is however here di\u000berent: it consists of a mass M= 500 kg\nimposed step by step and kept constant while the column oscillates in free vibration\nconsequently to a horizontal force F(t). The geometrical and loading characteristics\nof the column are depicted in \fgure 10.\n17\nconsequently to a horizontal force F(t). The geometrical and loading characteristics\nof the column are depicted in figure 10.\nFigure 10. Geometry and loading of the column.\n5.2.2. Concrete constitutive model. In [25], the monotonic uniaxial response (back-\nbone curve) of the concrete cast to build the RC column is detailed. I t is used\nhere as the baseline to identify the parameters of the concrete mo del. According\nto this report, we set C=2 7 .5G P a , H=0 ;t h e nw eu s e N=1 0 , M=6 4\n(Nf=6 4 )a n d ℓc/d=0.1 to ensure a response at macro-scale that is almost in-\ndependent of the realization of the meso-structure; finally, mand sare identified.\nFigure 11 shows model response macro-scale (plain line) with m=3 0 .5M P aa n d\ns/m=0.943. Comparing the numerical backbone curve (solid line) and the ex peri-\nmental response (dashed line), this figure illustrates the capability of the developed\nnumerical model to represent actual experimental concrete mo notonic response, at\nleast as far as the monotonic behavior is concerned. Note that no e xperimental\ndata was available for the cyclic behavior.\n−3.5 −3 −2.5 −2 −1.5 −1 −0.5 0\nx 10−3−30−25−20−15−10−50\nEΣ[MPa]Figure 11. Sample mean (—) response at macro-scale obtained\nnumerically from a sample of 2,000 meso-structures, along with\nbackbone curve (- -) recorded during uniaxial test on a specimen o f\nthe concrete used to build the RC column of interest here. Targete d\nmean and standard deviation of the marginal log-normal law are\nm=3 0 .5M P aa n d s/m=0.943. No experimental data is available\nfor the cyclic behavior.\nFigure 10. Geometry and loading of the column.\n5.2.2. Concrete constitutive model. In [25], the monotonic uniaxial response (back-\nbone curve) of the concrete cast to build the RC column is detailed. It is used\nhere as the baseline to identify the parameters of the concrete model. According\nto this report, we set C= 27:5 GPa,H= 0; then we use N= 10,M= 64\n(Nf= 64) and `c=d= 0:1 to ensure a response at macro-scale that is almost in-\ndependent of the realization of the meso-structure; \fnally, mandsare identi\fed.\nFigure 11 shows model response macro-scale (plain line) with m= 30:5 MPa and\ns=m = 0:943. Comparing the numerical backbone curve (solid line) and the experi-\nmental response (dashed line), this \fgure illustrates the capability of the developed\nnumerical model to represent actual experimental concrete monotonic response, at\nleast as far as the monotonic behavior is concerned. Note that no experimental\ndata was available for the cyclic behavior.\n5.2.3. Steel cyclic model. Young modulus Cs= 224:6 GPa, yield stress \u0006 y= 438\nMPa and ultimate stress \u0006 u= 601 MPa have been experimentally measured during18\n17\nconsequently to a horizontal force F(t). The geometrical and loading characteristics\nof the column are depicted in figure 10.\nFigure 10. Geometry and loading of the column.\n5.2.2. Concrete constitutive model. In [25], the monotonic uniaxial response (back-\nbone curve) of the concrete cast to build the RC column is detailed. I t is used\nhere as the baseline to identify the parameters of the concrete mo del. According\nto this report, we set C=2 7 .5G P a , H=0 ;t h e nw eu s e N=1 0 , M=6 4\n(Nf=6 4 )a n d ℓc/d=0.1 to ensure a response at macro-scale that is almost in-\ndependent of the realization of the meso-structure; finally, mand sare identified.\nFigure 11 shows model response macro-scale (plain line) with m=3 0 .5M P aa n d\ns/m=0.943. Comparing the numerical backbone curve (solid line) and the ex peri-\nmental response (dashed line), this figure illustrates the capability of the developed\nnumerical model to represent actual experimental concrete mo notonic response, at\nleast as far as the monotonic behavior is concerned. Note that no e xperimental\ndata was available for the cyclic behavior.\n−3.5 −3 −2.5 −2 −1.5 −1 −0.5 0\nx 10−3−30−25−20−15−10−50\nEΣ[MPa]Figure 11. Sample mean (—) response at macro-scale obtained\nnumerically from a sample of 2,000 meso-structures, along with\nbackbone curve (- -) recorded during uniaxial test on a specimen o f\nthe concrete used to build the RC column of interest here. Targete d\nmean and standard deviation of the marginal log-normal law are\nm=3 0 .5M P aa n d s/m=0.943. No experimental data is available\nfor the cyclic behavior.\nFigure 11. Sample mean (|) response at macro-scale obtained\nnumerically from a sample of 2,000 meso-structures, along with\nbackbone curve (- -) recorded during uniaxial test on a specimen of\nthe concrete used to build the RC column of interest here. Targeted\nmean and standard deviation of the marginal log-normal law are\nm= 30:5 MPa and s=m = 0:943. No experimental data is available\nfor the cyclic behavior.\nuniaxial tests on longitudinal steel rebars [25]. An elasto-plastic model with kine-\nmatic hardening is used to represent steel response in cyclic loading. The model\nimplemented with these latter measured parameters is shown in \fgure 12.\n18\n5.2.3. Steel cyclic model. Young modulus Cs=2 2 4 .6 GPa, yield stress Σy=4 3 8\nMPa and ultimate stress Σu= 601 MPa have been experimentally measured during\nuniaxial tests on longitudinal steel rebars [25]. An elasto-plastic mo del with kine-\nmatic hardening is used to represent steel response in cyclic loading .T h e m o d e l\nimplemented with these latter measured parameters is shown in figur e1 2 .\n−4 −3 −2 −1 0 1 2 3 4 5\nx 10−3−500−400−300−200−1000100200300400500\nEΣ[MPa]Figure 12. Numerical cyclic response of a steel longitudinal rebar\nused to build the frame. Cyclic behavior has not been observed\nexperimentally.\n5.2.4. Free vibration – Structural damping. Those concrete and steel uniaxial con-\nstitutive models are implemented in the fiber frame element presente d in section 2.\nThe column is modeled with one frame element with Nl= 2 control sections and\nNF= 6 fibers (actually layers here in the case of a 2D problem). As alread ym e n -\ntioned in section 5.2.1, the mass M= 500 kg is imposed step by step and kept\nconstant while the column oscillates in free vibration consequently to the horizon-\ntal force F(t). The column possibly exhibits nonlinear response while the mass M\nis applied and while F(t) increases from 0 to F0. Figure 13 shows typical column\ntop-displacement time histories for two di fferent values of F0.I t c a n b e o b s e r v e d\nthat damping depends on the amplitude of the oscillations: the column is clearly\ndamped for F0= 15 kN (grey curve) while damping is much lower for F0=5k N\n(black curve). One can also notice the di fferent vibration periods fo r both hori-\nzontal forces; this is due to the fact that the larger force activa tes some nonlinear\nmechanisms in the structure, which leads to an elongation of the str uctural vibra-\ntion period. We finally stress again here that there is no damping forc ea d d e d\nin the dynamic balance equations, such as for instance Rayleigh damp ing forces:\nthe damping e ffect shown in figure 13 only comes from the hysteresis loops in the\nconcrete response during unloading-reloading cycles.\nWe now define what we will refer to as “viscous-like damping ratio” and hereafter\ndenote by ξv. Considering the column top-displacement time history Xtop(t) in free\nvibration, we appeal to the so-called log-decrement method to eva luate the modal\ndamping ratio ξv(see e.g. [8, §4.6]):\n(35) ξv=1\n2πNclnXpeak\ntop(tN1)\nXpeak\ntop(tN2)\nXpeak\ntop(tN1)a n d Xpeak\ntop(tN2) are the amplitudes of any two peaks separated by Nc=\nN2−N1cycles. It is worth recalling here that this is only valid in case damping is\nFigure 12. Numerical cyclic response of a steel longitudinal rebar\nused to build the frame. Cyclic behavior has not been observed\nexperimentally.\n5.2.4. Free vibration { Structural damping. Those concrete and steel uniaxial con-\nstitutive models are implemented in the \fber frame element presented in section 2.\nThe column is modeled with one frame element with Nl= 2 control sections and\nNF= 6 \fbers (actually layers here in the case of a 2D problem). As already men-\ntioned in section 5.2.1, the mass M= 500 kg is imposed step by step and kept\nconstant while the column oscillates in free vibration consequently to the horizon-\ntal forceF(t). The column possibly exhibits nonlinear response while the mass M\nis applied and while F(t) increases from 0 to F0. Figure 13 shows typical column19\ntop-displacement time histories for two di\u000berent values of F0. It can be observed\nthat damping depends on the amplitude of the oscillations: the column is clearly\ndamped for F0= 15 kN (grey curve) while damping is much lower for F0= 5 kN\n(black curve). One can also notice the di\u000berent vibration periods for both hori-\nzontal forces; this is due to the fact that the larger force activates some nonlinear\nmechanisms in the structure, which leads to an elongation of the structural vibra-\ntion period. We \fnally stress again here that there is no damping force added\nin the dynamic balance equations, such as for instance Rayleigh damping forces:\nthe damping e\u000bect shown in \fgure 13 only comes from the hysteresis loops in the\nconcrete response during unloading-reloading cycles.\n19\n0 2 4 6 8 10 12−0.03−0.02−0.0100.010.020.03\ntime [s]top-displacement [m]Figure 13. Top displacement time history in free vibration for\nF0= 5 kN (black) and F0=1 5k N( g r e y ) . M a s s Mand horizon-\ntal forces F0are applied step-by-step during the first and second\nseconds, then horizontal force Fabruptly drops to zero and the\ncolumn oscillates in free vibration.\nlinear viscous, which in our case is not necessarily the case. Indeed, equations (35)\ncomes from the assumption that the envelope of the decaying top- displacement is\ndescribed as Xtop(t)= X0e−2ξπftwith fthe modal frequency. Hence the terms\n“viscous-like” to characterize the calculated damping ratios.\n0 10 20 30 4000.050.10.150.20.250.30.35\nNumber of cycles N1in free vibrationViscous-like damping ratio ξv[%]\nFigure 14. Viscous-like damping ratio time history ξvforF0=\n5 kN (black) and F0=1 5k N( g r e y ) .\nBased on the top-displacement time histories in figure 13, figure 14 s hows how ξv\ndecreases throughout free vibration time history for both values ofF0. Viscous-like\ndamping ratios ξv(tN1), are computed according to equations (35), with Nc=5 .\nNote that such damping ratios depend on the parameters mands/mof the random\nfield along with the hardening parameter Hat meso-scale. For the sake of illus-\ntration, figure 15 shows other results for another set of materia l parameters that is\nnot optimal for representing the monotonic response in compress ion of the concrete\nused to build the tested column. The capability of the proposed mate rial model\nfor generating structural damping has been demonstrated and t he development of\nan automatic procedure for identifying the full set of parameters targeting accurate\nrepresentation of both cyclic concrete response and damping is lef tf o rf u t u r ew o r k .\n6.Conclusions\nIn this paper, a multi-scale stochastic uniaxial cyclic model suitable f or rep-\nresenting most of the salient features of concrete nonlinear resp onse observed in\nFigure 13. Top displacement time history in free vibration for\nF0= 5 kN (black) and F0= 15 kN (grey). Mass Mand horizon-\ntal forcesF0are applied step-by-step during the \frst and second\nseconds, then horizontal force Fabruptly drops to zero and the\ncolumn oscillates in free vibration.\nWe now de\fne what we will refer to as \\viscous-like damping ratio\" and hereafter\ndenote by\u0018v. Considering the column top-displacement time history Xtop(t) in free\nvibration, we appeal to the so-called log-decrement method to evaluate the modal\ndamping ratio \u0018v(see e.g. [8,x4.6]):\n(35) \u0018v=1\n2\u0019NclnXpeak\ntop(tN1)\nXpeak\ntop(tN2)\nXpeak\ntop(tN1) andXpeak\ntop(tN2) are the amplitudes of any two peaks separated by Nc=\nN2\u0000N1cycles. It is worth recalling here that this is only valid in case damping is\nlinear viscous, which in our case is not necessarily the case. Indeed, equations (35)\ncomes from the assumption that the envelope of the decaying top-displacement is\ndescribed as Xtop(t) =X0e\u00002\u0018\u0019ftwithfthe modal frequency. Hence the terms\n\\viscous-like\" to characterize the calculated damping ratios.\nBased on the top-displacement time histories in \fgure 13, \fgure 14 shows how \u0018v\ndecreases throughout free vibration time history for both values of F0. Viscous-like\ndamping ratios \u0018v(tN1), are computed according to equations (35), with Nc= 5.\nNote that such damping ratios depend on the parameters mands=m of the random\n\feld along with the hardening parameter Hat meso-scale. For the sake of illus-\ntration, \fgure 15 shows other results for another set of material parameters that is20\n19\n0 2 4 6 8 10 12−0.03−0.02−0.0100.010.020.03\ntime [s]top-displacement [m]Figure 13. Top displacement time history in free vibration for\nF0= 5 kN (black) and F0=1 5k N( g r e y ) . M a s s Mand horizon-\ntal forces F0are applied step-by-step during the first and second\nseconds, then horizontal force Fabruptly drops to zero and the\ncolumn oscillates in free vibration.\nlinear viscous, which in our case is not necessarily the case. Indeed, equations (35)\ncomes from the assumption that the envelope of the decaying top- displacement is\ndescribed as Xtop(t)= X0e−2ξπftwith fthe modal frequency. Hence the terms\n“viscous-like” to characterize the calculated damping ratios.\n0 10 20 30 4000.050.10.150.20.250.30.35\nNumber of cycles N1in free vibrationViscous-like damping ratio ξv[%]\nFigure 14. Viscous-like damping ratio time history ξvforF0=\n5 kN (black) and F0=1 5k N( g r e y ) .\nBased on the top-displacement time histories in figure 13, figure 14 s hows how ξv\ndecreases throughout free vibration time history for both values ofF0. Viscous-like\ndamping ratios ξv(tN1), are computed according to equations (35), with Nc=5 .\nNote that such damping ratios depend on the parameters mands/mof the random\nfield along with the hardening parameter Hat meso-scale. For the sake of illus-\ntration, figure 15 shows other results for another set of materia l parameters that is\nnot optimal for representing the monotonic response in compress ion of the concrete\nused to build the tested column. The capability of the proposed mate rial model\nfor generating structural damping has been demonstrated and t he development of\nan automatic procedure for identifying the full set of parameters targeting accurate\nrepresentation of both cyclic concrete response and damping is lef tf o rf u t u r ew o r k .\n6.Conclusions\nIn this paper, a multi-scale stochastic uniaxial cyclic model suitable f or rep-\nresenting most of the salient features of concrete nonlinear resp onse observed in\nFigure 14. Viscous-like damping ratio time history \u0018vforF0=\n5 kN (black) and F0= 15 kN (grey).\nnot optimal for representing the monotonic response in compression of the concrete\nused to build the tested column. The capability of the proposed material model\nfor generating structural damping has been demonstrated and the development of\nan automatic procedure for identifying the full set of parameters targeting accurate\nrepresentation of both cyclic concrete response and damping is left for future work.\n20\n0 2 4 6 8 10 12−0.03−0.02−0.0100.010.020.03\ntime [s]top-displacement [m]\n0 5 10 15 20 25 30 350.10.20.30.40.50.60.70.8\nNumber of cycles N1in free vibrationViscous-like damping ratio ξv[%]\nFigure 15. [top] Top-displacement time history in free vibration\nand [bottom] viscous-like damping ratio time history ξvfor a set of\nmaterial parameters that is not optimal for the column considered\nabove: m=2 0M P a , s/m=1 0 , C=2 7 .5G P aa n d H=1 0G P a .\ncompressive experimental tests has been developed. It is based o n the construction\nof a meso-scale where the response at each material point is elasto -plastic with\nkinematic hardening and heterogeneous yield stress. This implies tha t the transi-\ntion from elastic to plastic regime occurs at a loading level that is di fferent in each\nmaterial point. Heterogeneity is parameterized by a 2D homogeneo us log-normal\nrandom field. As a first illustration of the capabilities of the model, som e analyt-\nical results are derived in the particular case of monotonic loading an d vanishing\ncorrelation length for the random field. Then, numerical simulations are performed\nand the e ffects of the parameters of the random field – that is the m eanm,c o e f -\nficient of variation s/mand correlation length ℓc– are investigated. It is shown\nthat for small values of the correlation length, material response at macro-scale\ndoes not depend on the realization of the random field, showing that the devel-\noped model is suitable for an objective representation of the mate rial behavior.\nBesides, it is shown that the mean mand standard deviation scan be identified\nso that the monotonic compressive response of an actual concre te test specimen\ncan be accurately represented by the developed model. The develo ped model how-\never lacks the ingredients for representing both strength and st iffness degradation\nmechanisms. Finally, the developed material model is implemented in a f rame ele-\nment in the purpose of representing the dynamic response of an ac tual reinforced\nconcrete column. The numerical analysis of the column in free vibrat ion shows the\ncapability of the developed material model to create patterns clas sically associated\n20\n0 2 4 6 8 10 12−0.03−0.02−0.0100.010.020.03\ntime [s]top-displacement [m]\n0 5 10 15 20 25 30 350.10.20.30.40.50.60.70.8\nNumber of cycles N1in free vibrationViscous-like damping ratio ξv[%]\nFigure 15. [top] Top-displacement time history in free vibration\nand [bottom] viscous-like damping ratio time history ξvfor a set of\nmaterial parameters that is not optimal for the column considered\nabove: m=2 0M P a , s/m=1 0 , C=2 7 .5G P aa n d H=1 0G P a .\ncompressive experimental tests has been developed. It is based o n the construction\nof a meso-scale where the response at each material point is elasto -plastic with\nkinematic hardening and heterogeneous yield stress. This implies tha t the transi-\ntion from elastic to plastic regime occurs at a loading level that is di fferent in each\nmaterial point. Heterogeneity is parameterized by a 2D homogeneo us log-normal\nrandom field. As a first illustration of the capabilities of the model, som e analyt-\nical results are derived in the particular case of monotonic loading an d vanishing\ncorrelation length for the random field. Then, numerical simulations are performed\nand the e ffects of the parameters of the random field – that is the m eanm,c o e f -\nficient of variation s/mand correlation length ℓc– are investigated. It is shown\nthat for small values of the correlation length, material response at macro-scale\ndoes not depend on the realization of the random field, showing that the devel-\noped model is suitable for an objective representation of the mate rial behavior.\nBesides, it is shown that the mean mand standard deviation scan be identified\nso that the monotonic compressive response of an actual concre te test specimen\ncan be accurately represented by the developed model. The develo ped model how-\never lacks the ingredients for representing both strength and st iffness degradation\nmechanisms. Finally, the developed material model is implemented in a f rame ele-\nment in the purpose of representing the dynamic response of an ac tual reinforced\nconcrete column. The numerical analysis of the column in free vibrat ion shows the\ncapability of the developed material model to create patterns clas sically associated\nFigure 15. [top] Top-displacement time history in free vibration\nand [bottom] viscous-like damping ratio time history \u0018vfor a set of\nmaterial parameters that is not optimal for the column considered\nabove:m= 20 MPa, s=m = 10,C= 27:5 GPa and H= 10 GPa.\n6.Conclusions\nIn this paper, a multi-scale stochastic uniaxial cyclic model suitable for rep-\nresenting most of the salient features of concrete nonlinear response observed in\ncompressive experimental tests has been developed. It is based on the construction\nof a meso-scale where the response at each material point is elasto-plastic with\nkinematic hardening and heterogeneous yield stress. This implies that the transi-\ntion from elastic to plastic regime occurs at a loading level that is di\u000berent in each\nmaterial point. Heterogeneity is parameterized by a 2D homogeneous log-normal\nrandom \feld. As a \frst illustration of the capabilities of the model, some analyt-\nical results are derived in the particular case of monotonic loading and vanishing21\ncorrelation length for the random \feld. Then, numerical simulations are performed\nand the e\u000bects of the parameters of the random \feld { that is the mean m, coef-\n\fcient of variation s=m and correlation length `c{ are investigated. It is shown\nthat for small values of the correlation length, material response at macro-scale\ndoes not depend on the realization of the random \feld, showing that the devel-\noped model is suitable for an objective representation of the material behavior.\nBesides, it is shown that the mean mand standard deviation scan be identi\fed\nso that the monotonic compressive response of an actual concrete test specimen\ncan be accurately represented by the developed model. The developed model how-\never lacks the ingredients for representing both strength and sti\u000bness degradation\nmechanisms. Finally, the developed material model is implemented in a frame ele-\nment in the purpose of representing the dynamic response of an actual reinforced\nconcrete column. The numerical analysis of the column in free vibration shows the\ncapability of the developed material model to create patterns classically associated\nto damping e\u000bects. In this simulation, damping does no come from some damping\nforces added in the dynamic balance equation (e.g. Rayleigh damping) but from\nthe multi-scale stochastic nonlinear model. Although the underlying model is sto-\nchastic, the simulations and results shown are the same for any realization of the\nstochastic model.\nThe main research prospects lie (i) in the enhancement of the model at meso-\nscale so that it can represent sti\u000bness and strength degradation mechanisms at\nmacro-scale; (ii) in the precise characterization of the stochastic model based on\ninformation from lower scales. This will consist in choosing, based on rational\narguments, the type of \frst-order marginal law and correlation model, as well as\nthe value of the corresponding parameters (mean, variance and correlation length).\nAlthough in another context, such an interaction between structural and material\nscientists has already been appealed for in [6]. Also, these issues could be considered\nin the context of stochastic micro-meso scale transition [43, 3, 12].\nAcknowledgement\nThe \frst author is supported by a Marie Curie International Outgoing Fellow-\nship within the 7th European Community Framework Programme (proposal No.\n275928). The second author, working within the SINAPS@ project, bene\fted from\nFrench state funding managed by the National Research Agency under program\nRNSR Future Investments bearing reference No. ANR-11-RSNR-0022-04.\nReferences\n[1] M. Anders and M. Hori. Three-dimensional stochastic \fnite element method for elasto-plastic\nbodies. Int. J. Numer. Meth. Engr. , 51(4):449{478, 2001.\n[2] Applied Technology Council. Modeling and acceptance criteria for seismic design and anal-\nysis of tall buildings. Technical Report PEER/ATC-72-1, Paci\fc Earthquake Engineering\nResearch Center, Richmond (CA), October 2010.\n[3] M Arnst and R Ghanem. 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Stochastic heterogeneity as fundamental basis for the design\nand evaluation of experiments. Cement Concrete Composites , 30(6):506{514, 2008.\n[46] R L Taylor, F C Filippou, A Saritas, and F Auricchio. A mixed \fnite element method for\nbeam and frame problems. Computational Mechanics , 31:192{203, 2003.\n[47] F. E. Udwadia. Some results on maximum entropy distributions for parameters known to lie\nin \fnite intervals. SIAM Rev. , 31(1):103{109, 1989.\n[48] P S Wong and F J Vecchio. VecTor2 & Formworks User's Manuals . University of Toronto,\nDepartment of Civil Engineering, Toronto, ON, Canada, 2002.\n[49] P. Wriggers and S. O. Moftah. Mesoscale models for concrete: homogenisation and damage.\nFinite Element Anal. Design , 42(7):623{636, 2006.\n[50] Z. J. Yang, X. T. Su, J. F. Chen, and G. H. Liu. Monte Carlo simulation of complex cohesive\nfracture in random heterogeneous quasi-brittle materials. Int. J. Solids Struct. , 46(17):3222{\n3234, 2009."    },    {        "title": "2206.11002v2.Lorentz_symmetry_breaking_and_supersymmetry.pdf",        "content": "arXiv:2206.11002v2  [hep-th]  23 Jun 2022Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022\n1\nLorentz symmetry breaking and supersymmetry\nJ. R. Nascimento, A. Yu. Petrov\n1Departamento de F´ ısica, Universidade Federal da Para´ ıba ,\nCaixa Postal 5008, 58051-970, Jo˜ ao Pessoa - PB, Brazil\nWe discuss three manners to implement Lorentz symmetry brea king in a super-\nfield theory formulated within the superfield formalism, tha t is, deformation of\nthe supersymmetry algebra, introducing of an extra superfie ld whose compo-\nnents can depend on Lorentz-violating (LV) vectors (tensor s), and adding of\nnew terms proportional to LV vectors (tensors) to the superfi eld action. We\nillustrate these methodologies with examples of quantum ca lculations.\n1. Introduction\nThesupersymmetry(SUSY) istreatednowasanimportantingredie ntofan\nexpected unified model of fundamental interactions. Therefore , the natural\nquestion consists in a possibility of constructing supersymmetric ex tensions\nfor known LV models. To do this, it is very convenient to use the supe rfield\nformalism known to be very advantageous (see f.e. Ref. 1). To con struct a\nLV superfield theory, one can introduce the Lorentz symmetry br eaking at\nthree possible levels – (i) at the level of the SUSY algebra, (ii) at the le vel\nof the superfield, (iii) at the level of the superfield Lagrangian.\n2. Deformation of the SUSY algebra\nThe most convenient method to construct superfield LV models is ba sed on\nusing the Kostelecky-Berger construction2. Within it, the deformed SUSY\ngenerators, in the four-dimensional spacetime, are defined as\nQα=i(∂α−i¯θ˙βσm\n˙βα∇m);¯Q˙α=i(∂˙α−iθβ¯σm\nβ˙α∇m),(1)\nwhere∇m=∂m+kmn∂nis the ”twisted” derivative, with kmnis a con-\nstant tensor implementing the Lorentz symmetry breaking. The an alogous\nmanner of modification of SUSY generators, i.e. replacement of ∂mby∇m,\nis employed also within other representations of the SUSY algebra, a s well\nas in the three-dimensional space-time. The Qα,¯Q˙αsatisfy the deformedProceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022\n2\nanticommutation relation: {Qα,¯Q˙α}=−2iσm\n˙αα∇m, while other anticom-\nmutators are not changed. The same deformation must be perfor med in\nspinor supercovariant dervatives Dα,¯D˙αas well, to ensure their anticom-\nmuting with the SUSY generators. The Dα,¯D˙αsatisfy the relations:\n{Dα,¯D˙α}= 2iσm\n˙αα∇m;D2¯D2D2= 16˜/squareD2;DαDβDγ= 0.(2)\nThe superfield actions defined in a superspace with the deformed SU SY\nalgebra formally reproduce the same expressions as in usual super field the-\nories, but being rewritten in components, they involve additional LV terms,\nf.e. the LV Wess-Zumino (WZ) model in components looks like\nS=/integraldisplay\nd4x/bracketleftBig\nφ˜/square¯φ+ψαiσm\nα˙α∇m¯ψ˙α+F¯F+\n+/parenleftBig\nm(ψαψα+φF)+λ(φψαψα+1\n2φ2F)+h.c./parenrightBig/bracketrightBig\n, (3)\ni.e. it involves CPT-even aether-like terms3for scalar and spinor fields.\nHere,˜/square=∇m∇mis the deformed d’Alembertian operator.\nThe superfield propagators in this theory look like\n<Φ(z1)¯Φ(z2)>=1\n˜/square−m2δ12;<Φ(z1)Φ(z2)>=mD2\n4˜/square(˜/square−m2)δ12.(4)\nTheir difference from the usual case consists in the presence of ˜/square.\nIn this theory, one can calculate the one-loop low-energyeffective action\ndescribedbytheK¨ ahlerianeffectivepotential. Itiscontributedby theseries\nof supergraphs depicted at Fig. 1, with external legs are for alter nating\nΨ =m+λΦ and¯Ψ =m+λ¯Φ.\n✧✦★✥\n✧✦★✥\n✧✦★✥\n\u0000\u0000 ❅❅❅❅ \u0000\u0000...\nFig. 1. One-loop contributions to the K¨ ahlerian effetive po tential.\nThe sum of these supergraphs is easily found to be4\nK(1)=−1\n2/integraldisplay\nd8z/integraldisplayd4q\n(2π)41\n(qm+kmnqn)2ln(1−¯ΨΨ\n(qm+kmnqn)2).(5)\nWe change qm+kmnqn→˜qm:\nK(1)=−1\n2∆/integraldisplay\nd8z/integraldisplayd4˜q\n(2π)41\n˜q2ln(1−¯ΨΨ\n˜q2), (6)Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022\n3\nwhere ∆ = det(∂qm\n∂˜qn) = det−1(δm\nn+km\nn) is a Jacobian ofchange of variables.\nAfter integration and renormalization we arrive at\nK(1)=−1\n32π2∆/integraldisplay\nd8zΨ¯ΨlnΨ¯Ψ\nµ2. (7)\nWe conclude that in this case quantum calculations are no more difficult\nthat intheusualLorentz-invariantcase. Thisapproachwasalsog eneralized\nfor supergauge theories5.\n3. Introduction of the additional superfield\nWithin this approach we introduce a new superfield whose some compo -\nnents depend on LV vectors (tensors). The most interesting exa mple of its\napplication allowed to construct the SUSY extension of the Carroll-F ield-\nJackiw (CFJ) term6: we introduce the new superfield S=s+..., wheres\nis its lower component, and define the CPT-odd term\nSodd=/integraldisplay\nd8zSWαDαV+h.c., (8)\nwhereVis the gauge scalar superfield, and Wα=−1\n4¯D2DαVis the corre-\nsponding superfield strength. In components this term looks like\nSodd=i\n2/integraldisplay\nd4x∂a(s−s∗)ǫabcdFbcAd+..., (9)\nthus, ifs(x) =−ikaxa, we reproduce the CFJ term. Actually, this is the\nonly known manner to obtain the CFJ term within the superfield frame -\nwork. This methodology can be used to obtain a superfield extension for\nthe CPT-even aether-like term as well7.\n4. Direct modification of the superfield Lagrangian\nIn this case, we add Lorentz-breaking term to a Lagrangian. The s implest\nexample is the following Horava-Lifshitz-like action of the chiral supe rfield,\nwithzis the critical exponent measuring the space-time anisotropy:\nS=/integraldisplay\nd8zΦ(1+ρ∆z−1)¯Φ+(/integraldisplay\nd6zW(Φ)+h.c.), (10)\nwithW(Φ) is a some potential. For this theory, we can calculate the one-\nloop low-energy effective action, using the same scheme as in the sec tionProceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022\n4\n2, with Ψ = W′′, the supergraphs are again given by Fig. 1, and the\npropagators are correspondingly modified. As a result, we arrive a t\nK(1)=−i\n2/integraldisplay\nd4θd4k\n(2π)41\nk2ln/bracketleftBigg\n1+Ψ¯Ψ\nk2(1+ρ(/vectork2)z−1)2/bracketrightBigg\n.(11)\nThis integral can be calculated only approximately. Under the schem e\ndeveloped in8, where subleading orders of /vectorkare disregarded, we find\nK(1)=1\n12πcsc(π\nz)(4ρ\n3)−1/z(Ψ¯Ψ)1/z. (12)\nThis expression is singular at z= 1, as it must be, since this choice of zis\nnecessary to achieve the Lorentz invariance.\n5. Summary\nWepresentedthreeschemesforconstructingLVsuperfieldtheo ries. Itturns\nout to be that, first, in principle, perturbative calculations in LV sup erfield\ntheories are no more difficult as in usual ones, second, the CPT-eve n sce-\nnario is easier to reproduce in a superfield form. So, the main problem s\nto study are, first, search for other manners to construct LV s uperfield\ntheories, second, generalization of these schemes to a curved sp ace-time.\nReferences\n1. I. L. Buchbinder, S. M. Kuzenko, Ideas and methods of super symmetry and\nsupergravity, or A Walk through superspace, IOP Publishing , Bristol and\nPhiladelphia, 1998.\n2. M.S. Berger and V.A. Kostelecky, Phys. Rev. D 65, 091701 (2002).\n3. S.M. Carroll and H. Tam, Phys. Rev. D 78, 044047 (2008).\n4. C.F. Farias, A.C. Lehum, J.R. Nascimento and A.Y. Petrov, Phys. Rev. D\n86, 065035 (2012).\n5. A.C. Lehum, J.R. Nascimento, A.Y. Petrov and A.J. da Silva , Phys. Rev.\nD88, 045022 (2013).\n6. H. Belich, J. L. Boldo, L. P. Colatto, J. A. Helayel-Neto an d\nA. L. M. A. Nogueira, Phys. Rev. D 68, 065030 (2003).\n7. H. Belich, L. D. Bernald, P. Gaete, J. A. Helay¨ el-Neto and F. J. L. Leal, Eur.\nPhys. J. C 75, 291 (2015).\n8. M. Gomes, J. R. Nascimento, A. Y. Petrov and A. J. da Silva, P hys. Rev. D\n90, 125022 (2014).\nAcknowledgments\nThis study was partially supported by Conselho Nacional de Desenvo lvi-\nmento Cient´ ıfico e Tecnol´ ogico (CNPq) via the grant 301562/2019 -9."    },    {        "title": "2306.12254v1.The_effect_of_singularities_and_damping_on_the_spectra_of_photonic_crystals.pdf",        "content": "The effect of singularities and damping on the spectra of photonic\ncrystals\nKonstantinos Alexopoulos∗Bryn Davies†\nAbstract\nUnderstanding the dispersive properties of photonic crystals is a fundamental and well-studied\nproblem. However, the introduction of singular permittivities and damping complicates the other-\nwise straightforward theory. In this paper, we study photonic crystals with a Drude-Lorentz model\nfor the permittivity, motivated by halide perovskites. We demonstrate how the introduction of\nsingularities and damping affects the spectral band structure and show how to interpret the notion\nof a “band gap” in this setting. We present explicit solutions for a one-dimensional model and show\nhow integral operators can be used to handle multi-dimensional systems.\nContents\n1 Introduction 2\n2 Problem setting 3\n2.1 Initial Helmholtz formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3\n2.2 Periodic formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3\n2.3 Floquet-Bloch theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4\n3 One dimension 5\n3.1 Dispersion relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6\n3.2 Properties of the dispersion relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7\n3.2.1 Real and imaginary parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8\n3.2.2 Imaginary part decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10\n3.3 The effect of singularities and damping . . . . . . . . . . . . . . . . . . . . . . . . . . . 12\n3.3.1 Constant permittivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13\n3.3.2 Singular permittivity with no damping . . . . . . . . . . . . . . . . . . . . . . . 14\n3.3.3 Complex permittivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17\n3.3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20\n4 Multiple dimensions 20\n4.1 Integral formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21\n4.2 Dispersion relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21\n4.2.1 Matrix representaion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21\n4.2.2 Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22\n4.3 The two-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24\n4.3.1 Integral operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24\n4.3.2 Spectral results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24\n4.3.3 Resonant frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25\n5 Conclusion 26\n∗Department of Mathematics, ETH Zurich, R¨ amistrasse 101, CH-8092 Zurich, Switzerland.\n†Department of Mathematicsbb, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, United Kingdom.\n1arXiv:2306.12254v1  [math.AP]  21 Jun 20231 Introduction\nPhotonic crystals present interesting and useful wave properties. Even very simple photonic crystals,\nsuch as those composed of periodically alternating layers of non-dispersive materials, can display exotic\ndispersive properties. As a result, they are able to support band gaps: ranges of frequencies that are\nunable to propagate through the material [22]. These band gaps are the fundamental building blocks of\nthe many different wave guides and wave control devices that have been conceived. Notable examples\ninclude flat lenses [17], invisibility cloaks [14], rainbow trapping filters [24] and topological waveguides\n[8].\nWhen working at certain electromagnetic frequencies (which often includes the visible spectrum),\nit is important to take into account the oscillatory behaviour of the free electrons in a metal. This be-\nhaviour leads to resonances at characteristic frequencies and gives metals a highly dispersive character\n(even before the introduction of macroscopic structure, as in a photonic crystal). Several different\nmodels exist to describe this behaviour. Most models are variants of the Lorentz oscillator model,\nwhereby electrons are modelled as damped harmonic oscillators due to electrostatic attractions with\nnuclei [12]. A popular special case of this is the Drude model, in which case the restoring force is ne-\nglected (to reflect the fact that most electrons in metals are not bound to any specific nucleus, so lack\na natural frequency of oscillation). Many other variants of these models exist, for instance by adding\nor removing damping from the various models, cf.[18] or [11], and by taking linear combinations of\nthe different models, as in [19].\nA key feature that unites dispersive permittivity models is the existence of singularities in the\npermittivity. The position of these poles in the complex plane, which correspond to resonances, are\none of the crucial properties that determines how a metal interacts with an electromagnetic wave.\nIn conventional Lorentz models the poles appear in the lower complex plane [12]. The imaginary\npart of the singular frequency is determined by the magnitude of the damping, and the singularities\naccordingly fall on the real line if the damping is set to zero. In the Drude model, the removal of the\nrestorative force causes the singularities to fall at the origin and on the negative imaginary axis.\nA particularly important example of dispersive materials, that are central to the motivation for\nthis study, are halide perovskites. They have excellent optical and electronic properties and are cheap\nand easy to manufacture at scale [13]. As a result, they are being used in many applications, including\noptical sensors [7], solar cells [21] and light-emitting diodes [25]. The dielectric permittivity of halide\nperovskites has been shown to depend heavily on excitonic transitions, leading to a permittivity that\nhas symmetric poles in the lower complex plane [13].\nThere are a range of methods that can be used to capture the spectra of photonic crystals. For\none-dimensional systems, explicit solutions typically exist and transfer matrices are particularly con-\nvenient. These were used for Drude materials in [20] and for undamped Lorentz materials in [11],\nfor example. In multiple dimensions, studies often resort to numerical simulation (for instance with\nfinite elements). A valuable approximation strategy is a multi-scale asymptotic method known as\nhigh-frequency homogenisation [6], which can be extended to approximate the dispersion curves in\ndispersive media [23].\nIn this work, we will study photonic crystals composed of metals with permittivity inspired by\nthat of halide perovskites, in the sense that it has symmetric poles in the lower complex plane. After\nsetting out the Floquet-Bloch formulation of the periodic problem in section 2, we will study the\none-dimensional periodic Helmholtz problem in section 3. We retrieve the dispersion relation which\ncharacterizes the halide perovskite system and show how its properties depend on the characteristics\nof the dispersive permittivity (namely, being real or being complex and having poles either on or\nbelow the real axis). Finally, in section 4, we use integral methods and asymptotic analysis in order\nto obtain the dispersion relation for the two- and three-dimensional cases, showing how to extend our\nanalysis to multi-dimensional photonic crystals.\n22 Problem setting\n2.1 Initial Helmholtz formulation\nLet us consider N∈Nparticles D1, D2, . . . , D Nwhich together occupy a bounded domain Ω ⊂Rd, for\nd∈ {1,2,3}. The collection of particles Ω will be the repeating unit of the periodic photonic crystal.\nWe suppose that permittivity of the particles is given by a Drude–Lorentz-type model, given by\nε(ω) =ε0+α\n1−βω2−iγω, (2.1)\nwhere ε0denotes the background dielectric constant and α, β, γ are positive constants. αdescribes\nthe strength of the interactions, βdetermines the natural resonant frequency and γis the damping\nfactor. This is motivated by the measured permittivity of halide perovskites, as reported in [13]. We\nchoose to use this expression as a canonical model for dispersive materials whose permittivities have\nsingularities in the complex frequency space. Notice that (2.1) is singular at two complex values of ω.\nThese are given by\nω∗\n±=1\n2β\u0010\n−iγ±p\n4β−γ2\u0011\n. (2.2)\nBy varying the parameters α,βandγwe can force these singularities to lie in the lower half of the\ncomplex plane ( γ >0), on the real line ( γ= 0) or to vanish completely ( β=γ= 0). We will make use\nof this property when trying to interpret the dispersion diagrams we obtain in the following analysis.\nWe suppose that the particles are surrounded by a non-dispersive medium with permittivity ε0. We\nassume that the particles are non-magnetic, meaning the magnetic permeability µ0is constant on all\nofRd.\nWe consider the Helmholtz equation as a model for the propagation of time-harmonic waves with\nfrequency ω. This is a reasonable model for the scattering of transverse magnetic polarised light\n(seee.g.[15, Remark 2.1] for a discussion). The wavenumber in the background Rd\\Ω is given by\nk0:=ωε0µ0and we will use kto denote the wavenumber within Ω. Let us note here that, from now\non, we will suppress the dependence of k0andkonωfor brevity. We, then, consider the system of\nequations\n\n\n∆u+ω2ε(ω)µ0u= 0 in Ω ,\n∆u+k2\n0u= 0 in Rd\\Ω,\nu|+−u|−= 0 on ∂Ω,\n∂u\n∂ν|+−∂u\n∂ν|−= 0 on ∂Ω,\nu(x)−uin(x) satisfies the outgoing radiation condition as |x| → ∞ ,(2.3)\nwhere uinis the incident wave, assumed to satisfy (∆ + k2\n0)uin= 0, and the appropriate outgoing\nradiation condition depends on the dimension of the problem and of the periodic lattice.\n2.2 Periodic formulation\nWe will assume that the collection of Nparticles is repeated in a periodic lattice Λ. We suppose that\nthe lattice has dimension dl, in the sense that there are lattice vectors l1, . . . , l dl∈Rdwhich generate\nΛ according to\nΛ :={m1l1+···+mdlldl|mi∈Z}. (2.4)\nThe fundamental domain of the lattice Λ is the set Y∈Rdgiven by\nY:={c1l1+···+cdlldl|0≤c1, . . . , c dl≤1}. (2.5)\nThe dual lattice of Λ, denoted by Λ∗, is generated by the vectors α1, . . . , α dlsatisfying αi·lj= 2πδij\nfori, j= 1, . . . , d l. Finally, the Brillouin zone Y∗is defined by\nY∗:= (Rdl× {0})/Λ∗, (2.6)\n3· · · · · ·\nD1D2 D3\nD4D5D6\nD1D2 D3\nD4D5D6\nD1D2 D3\nD4D5D6ΩFigure 1: A periodic array of halide perovskite particles. Here, we have six particles D1, . . . , D 6repeated\nperiodically in one dimension. Each of them has the halide perovskite permittivity ε(ω)defined by (2.1).\nwhere 0is the zero vector in Rd−dl. The Brillouin zone Y∗is the space that the reduced unit cell of\nreciprocal space.\nThe periodic structure, denoted by D, is given by\nD=N[\ni=1 [\nm∈ΛDi+m!\n.\nHence, the problem we wish to study is the following:\n\n\n∆u+ω2ε(ω)µ0u= 0 in D,\n∆u+k2\n0u= 0 in Rd\\D,\nu|+−u|−= 0 on ∂D,\n∂u\n∂ν|+−∂u\n∂ν|−= 0 on ∂D,\nu(xl, x0) satisfies the outgoing radiation condition as |x0| → ∞ .(2.7)\n2.3 Floquet-Bloch theory\nIn order to study the problem (2.7), we will make use of Floquet-Bloch theory [10]. Let us first give\ncertain definitions which will help with the analysis of the problem.\nDefinition 2.1. A function f(x)∈L2(Rd)is said to be κ-quasiperiodic, with quasiperiodicity κ∈Y∗,\nife−iκ·xf(x)isΛ-periodic.\nDefinition 2.2 (Floquet transform) .Letf∈L2(Rd). The Floquet transform of fis defined as\nF[f](x, κ) :=X\nm∈Λf(x−m)e−iκ·x, x∈Rd, κ∈Y∗.\nWe have that F[f] isκ-quasiperiodic in xand periodic in κ. The Floquet transform is an invertible\nmapF:L2(Rd)→L2(Y×Y∗), with inverse given by\nF−1[g](x) =1\n|Y∗\nl|Z\nY∗g(x, κ)dκ, x ∈Rd,\nwhere g(x, κ) is extended quasiperiodically for xoutside of the unit cell Y.\nLet us define uκ(x) :=F[u](x, κ). Then, applying the Floquet transform to (2.7), we obtain the\n4following system:\n\n\n∆uκ+ω2ε(ω)µ0uκ= 0 in D,\n∆uκ+k2\n0uκ= 0 in Rd\\D,\nuκ|+−uκ|−= 0 on ∂D,\n∂uκ\n∂ν|+−∂uκ\n∂ν|−= 0 on ∂D,\nuκ(xd, x0) is κ-quasiperiodic in xd,\nuκ(xd, x0) satisfies the κ-quasiperiodic radiation condition as |x0| → ∞ .(2.8)\nThe solutions to (2.8) typically take the form of a countable collection of spectral bands, each of\nwhich depends continuously on the Bloch parameter κ. The goal of our analysis is identifying and\nexplaining the gaps between the spectral band. At frequencies within these band gaps, waves do not\npropagate in the material and their amplitude decays exponentially. As a result, they are the starting\npoint for building waveguides and other wave control devices.\nFor real-valued permittivities, it is straightforward to define band gaps as the intervals between\nthe real-valued bands:\nDefinition 2.3 (Band gap for real permittivities) .A frequency ω∈Ris said to be in a band gap of\nthe periodic structure Dif it is such that (2.8) does not admit a non-trivial solution for any κ∈R.\nWe are interested in materials for which the permittivity takes complex values, corresponding to\nthe introduction of damping to the model. In which case, we elect to keep the frequency ω∈Ras a\nreal number but allow the Bloch parameter κto take complex values. In which case, the imaginary\npart of κdescribes the rate at which the waves amplitude decays. It should be noted that it is also\nquite common to do the opposite by forcing κto be real and allowing ωto be complex valued, as in\n[4, 23] for example.\nIn the real-valued case, it is clear that κbelongs to the Brillouin zone Y∗(which has the topology\nof a torus, due to the periodicity in κ). When κis complex valued, its real part still lives in Y∗but\nits imaginary part can take arbitrary values. Thus, κlives in a space that is isomorphic to Y∗×R.\nIn our setting, which is a damped model that is characterized by a complex permittivity, we have\nthatκ∈Cand it is less clear how to define a band gap. Intuitively, a band gap is a range of frequencies\nat which the damping is particularly large. Hence, we provide a modified definition for the notion of\na band gap for complex permittivities in terms of local maxima of the amplitude decay:\nDefinition 2.4 (Band gap for complex permittivities) .We define a band gap for complex permittivities\nto be the set of frequencies ω∈Rfor which (2.8) admits a non-trivial solution with quasiperiodicity\nκ∈Cand|ℑ(κ)|is at a local maximum.\nWe will first study the problem in the one-dimensional setting. In one dimension, the problem\nis easier to manipulate and we are able to retrieve explicit expressions. Hence, we can get a variety\nof results concerning the characteristics of the quasiperiodic system. In particular, our main goal is\nto obtain the dispersion relation, an expression which relates the quasiperiodicities κ∈Cwith the\nfrequencies ω∈R, and study its properties. Then, in section 4 we will provide an analysis for higher\ndimensional systems and we will give the equivalent relation.\n3 One dimension\nLet us first treat the Helmholtz problem (2.8) in the one-dimensional case. We will work on the\ninterval [ −1,1], with [ −1,0) denoting the background and [0 ,1) the particle. A schematic depiction\nof this is given in Figure 2. Hence, the problem reads as follows:\nd\ndx\u00121\nε(x, ω)du\ndx\u0013\n+ω2µ0u(x) = 0 , (3.1)\n5−1 0 1\nDε0 ε(ω)Figure 2: The one-dimensional setting. The periodically repeated cell is of length 2. Here the interval\n[−1,0)is the background and the interval [0,1)is the particle.\non the domain [ −1,1], where\nε(x, ω) :=(\nε0, x ∈[−1,0) (background) ,\nε(ω), x ∈[0,1] (particle) ,(3.2)\nwith the boundary conditions\nu(1) = e2iκu(−1) anddu\ndx(1) = e2iκdu\ndx(−1). (3.3)\n3.1 Dispersion relation\nWe will now retrieve an expression for the solution to (3.1). Let us define the quantities\nσ0:=ω√ε0µ0,and σc:=ωp\nε(ω)µ0. (3.4)\nFor many of the results that follow, the crucial quantity will be the contrast between the material inside\nthe particles and the background medium. With this in mind, we introduce the frequency-dependent\ncontrast ρas\nρ(ω) :=σc\nσ0=s\nε(ω)\nε0. (3.5)\nThen, the following expression holds for the solution to (3.1).\nLemma 3.1. Letudenote a solution to (3.1). Then, uis given by\nu(x) =\n\nAρsin\u0010\nσ0x\u0011\n+Bcos\u0010\nσ0x\u0011\n, x∈[−1,0)\nAsin\u0010\nσcx\u0011\n+Bcos\u0010\nσcx\u0011\n, x ∈[0,1],(3.6)\nwhere A,B ∈Care two constants.\nProof. We know that a solution to (3.1) must be given by\nu(x) =\n\nA1sin\u0010\nω√ε0µ0x\u0011\n+B1cos\u0010\nω√ε0µ0x\u0011\n, x ∈[−1,0),\nA2sin\u0010\nωp\nε(ω)µ0x\u0011\n+B2cos\u0010\nωp\nε(ω)µ0x\u0011\n, x ∈[0,1],(3.7)\nwhere A1,A2,B1,B2∈Care constants to be defined. This, also, gives\ndu\ndx(x) =\n\nA1ω√ε0µ0cos\u0010\nω√ε0µ0x\u0011\n− B 1ω√ε0µ0sin\u0010\nω√ε0µ0x\u0011\n, x ∈[−1,0),\nA2ωp\nε(ω)µ0cos\u0010\nωp\nε(ω)µ0x\u0011\n− B 2ωp\nε(ω)µ0sin\u0010\nωp\nε(ω)µ0x\u0011\n, x ∈[0,1].\nNow, from the boundary transmission conditions in (2.7), we require\nlim\nx→0−u(x) = lim\nx→0+u(x) and lim\nx→0−du\ndx(x) = lim\nx→0+du\ndx(x).\nThese conditions mean we must have that B1=B2andA1=p\nε(ω)/ε0A2=ρ(ω)A, which gives the\ndesired result.\n6Using the boundary conditions (3.3), we can obtain the dispersion relation for the one-dimensional\nproblem. This is a well-known result, that first appeared in a quantum-mechanical setting [9] and has\nsince been shown to describe a range of periodic classical wave systems also [1, 16]. We include a brief\nproof, for completeness.\nTheorem 3.2 (Dispersion relation) .Letudenote the solution to (3.1) along with the boundary condi-\ntions (3.3). Then, for uto be non-trivial, the quasiperiodicities κ∈Csatisfies the dispersion relation\ncos(2 κ) = cos( σ0) cos( ρσ0)−1 +ρ2\n2ρsin(σ0) sin(ρσ0). (3.8)\nProof. From Lemma 3.1, we have that uis given by (3.6). Then, using (3.3), we have\n\n\nh\nsin(σc) +e2iκρsin(σ0)i\nA+h\ncos(σc)−e2iκcos(σ0)i\nB= 0,h\nσccos(σc)−e2iκρσ0cos(σ0)i\nA −h\nσcsin(σc) +e2iκσ0sin(σ0)i\nB= 0.(3.9)\nWe observe that for (3.1) to have a non-zero solution, it should hold\nh\nsin(σc) +e2iκρsin(σ0)i\n·h\nσcsin(σc) +e2iκσ0sin(σ0)i\n+\n+h\ncos(σc)−e2iκcos(σ0)i\n·h\nσccos(σc)−e2iκρσ0cos(σ0)i\n= 0,\nwhich gives\np\nε(ω)e4iκ+\u0014ε0+ε(ω)√ε0sin(σ0) sin(σc)−2p\nε(ω) cos( σ0) cos( σc)\u0015\ne2iκ+p\nε(ω) = 0 . (3.10)\nMaking some algebraic rearrangements, we observe that\n2s\nε(ω)\nε0h\ncos(σ0) cos( σc)−cos(2 κ)i\n−ε0+ε(ω)\nε0sin(σ0) sin(σc) = 0 . (3.11)\nFinally, making the substitutions σc=ρσ0andp\nε(ω) =ρ√ε0, we obtain the desired result.\nThe dispersion relation (3.8) can be used to plot the dispersion curves. For a given frequency ω,\nρ(ω) can be calculated to yield the right hand side of (3.8), which can subsequently be solved to find\nκ. This is shown in Figure 3. Since ϵ(ω) is complex valued, κwill generally take complex values.\nWe plot only the absolute values of both the real and imaginary parts; as we will see below, this is\nsufficient to characterise the full dispersion relation. Notice also that ℜ(κ)∈Y∗= [−π/2, π/2).\n3.2 Properties of the dispersion relation\nThe dispersion relation (3.8) describes the behaviour of the periodic system and reveals the relationship\nbetween the quasiperiodicities κ∈C, the frequencies ω∈Rand the permittivity ε(ω) of the material.\nWe can use it to derive some simple results about the dispersion curves. The first thing to understand\nis the symmetries of the dispersion curves.\nLemma 3.3 (Opposite quasiperiodicities) .Letκ∈Cbe a complex quasiperiodicity satisfying the\ndispersion relation (3.8) for a given frequency ω∈R. Then, the opposite quasiperiodicity, i.e. −κ,\nsatisfies the same dispersion relation.\nProof. We just have to use that cos( ·) is an even function. Then, if κ∈Cis such that (3.8) holds,\nfrom the fact that cos( −2κ) = cos(2 κ), we get that −κ∈Calso satisfies (3.8). This concludes the\nproof.\nIt is with Lemma 3.3 in mind that we are able to plot only the absolute values of the imaginary\nparts in Figure 3 and the subsequent figures.\n7Singularities:\nRe(ω)Im(ω)\n×ω∗\n+ ω∗\n−×0.97 -0.97\n-0.25\n.\nFigure 3: The dispersion relation of the halide perovskite photonic crystal. We model a material with\npermittivity given by (2.1) with α= 1,β= 1 andγ= 0.5. The frequency ωis chosen to be real\nand the Bloch parameter κallowed to take complex values. The permittivity is singular at two points,\nwhich are in the lower complex plane and are symmetric about the imaginary axis, as indicated in the\nsketch on the right and (the real parts) by the crosses on the frequency axes of the plots.\n3.2.1 Real and imaginary parts\nIn the analysis that will follow, it will be useful to be able to describe the behaviour of the real\nand imaginary part of the quasiaperiodicity with respect to the permittivity. In particular, we will\ndecompose both the quasiperiodicity κandρinto real and imaginary parts, and we will derive this\ndependence from the dispersion relation. Since κ∈Candρ∈C, let us define:\nκ=κ1+iκ2and ρ=ρ1+iρ2, (3.12)\nwith κ1, κ2, ρ1, ρ2∈R. We will also define L1andL2, which depend on ω∈R, as follows:\nL1(ω) := cos( σ0) cos( σ0ρ1) cosh( σ0ρ2)−sin(σ0)\n2(ρ2\n1+ρ2\n2)h\nρ1(1 +ρ2\n1+ρ2\n2) sin(σ0ρ1) cosh( σ0ρ2)−\n−ρ2(ρ2\n2−1 +ρ2\n1) cos( σ0ρ1) sinh( σ0ρ2)i\n,(3.13)\nand\nL2(ω) := cos( σ0) sin(σ0ρ1) sinh( σ0ρ2) +sin(σ0)\n2(ρ2\n1+ρ2\n2)h\nρ2(ρ2\n2−1 +ρ2\n1) sin(σ0ρ1) cosh( σ0ρ2)+\n+ρ1(1 +ρ2\n1+ρ2\n2) cos( σ0ρ1) sinh( σ0ρ2)i\n,(3.14)\nwhere we note that ρ1,ρ2andσ0all depend on the frequency ω, as specified in (3.4) and (3.5). Then,\nwe have the following result.\nProposition 3.4. Letκ∈C, given by (3.12) , satisfying the dispersion relation (3.8) for a given\nfrequency ω∈R. Then, its real and imaginary parts are given by\nℜ(κ) =±1\n2arccos\u0012L1\ncosh(2 ℑ(κ))\u0013\n, (3.15)\nand\nℑ(k) =1\n2arcsinh \n±r\n1\n2h\nL2\n1+L2\n2−1 +q\n(1−L2\n1−L2\n2)2+ 4L2\n2i!\n, (3.16)\nwhereL1andL2are given by (3.13) and(3.14) , respectively. We, also, note that the choice of +or\n−should be the same in (3.15) and(3.16) .\n8Proof. From (3.12), the dispersion relation (3.8) becomes\ncos(2 κ1+i2κ2) = cos( σ0) cos( σ0ρ1+iσ0ρ2)−1 + (ρ1+iρ2)2\n2(ρ1+iρ2)sin(σ0) sin(σ0ρ1+iσ0ρ2),\nwhich is,\ncos(2 κ1) cosh(2 κ2)−isin(2κ1) sinh(2 κ2) =\n= cos( σ0)h\ncos(σ0ρ1) cosh( σ0ρ2)−isin(σ0ρ1) sinh( σ0ρ2)i\n−\n−1 +ρ2\n1+ 2iρ1ρ2−ρ2\n2\n2(ρ2\n1+ρ2\n2)(ρ1−iρ2) sin(σ0)h\nsin(σ0ρ1) cosh( σ0ρ2) +icos(σ0ρ1) sinh( σ0ρ2)i\n.\nTaking real and imaginary parts, we obtain, for the real part,\ncos(2 κ1) cosh(2 κ2) = cos( σ0) cos( σ0ρ1) cosh( σ0ρ2)−sin(σ0)\n2(ρ2\n1+ρ2\n2)h\nρ1(1 +ρ2\n1+ρ2\n2) sin(σ0ρ1) cosh( σ0ρ2)+\n−ρ2(ρ2\n2−1 +ρ2\n1) cos( σ0ρ1) sinh( σ0ρ2)i\n, (3.17)\nand, for the imaginary part,\nsin(2κ1) sinh(2 κ2) = cos( σ0) sin(σ0ρ1) sinh( σ0ρ2) +sin(σ0)\n2(ρ2\n1+ρ2\n2)h\nρ2(ρ2\n2−1 +ρ2\n1) sin(σ0ρ1) cosh( σ0ρ2)+\n+ρ1(1 +ρ2\n1+ρ2\n2) cos( σ0ρ1) sinh( σ0ρ2)i\n. (3.18)\nSo, from (3.13) and (3.14), we obtain the system\n(\ncos(2 κ1) cosh(2 κ2) =L1,\nsin(2κ1) sinh(2 κ2) =L2.(3.19)\nFrom the first equation, we immediately see that\nκ1=±1\n2arccos\u0012L1\ncosh(2 κ2)\u0013\n,\nthen, substituting into the second equation gives\nsin\u0014\narccos\u0012L1\ncosh(2 κ2)\u0013\u0015\nsinh(2 κ2) =±L2.\nWe know that for x∈[−1,1], we have the identity sin[arccos( x)] =√\n1−x2. Hence, from the above,\nwe get\ns\n1−L2\n1\ncosh2(2κ2)sinh(2 κ2) =±L2. (3.20)\nSimilarly, using the fact that cosh2(x)−sinh2(x) = 1 for x∈R, we find that\ns\n1−L2\n1\n1 + sinh2(2κ2)sinh(2 κ2) =±L2. (3.21)\nHence, we have\nsinh4(2κ2) + (1 −L2\n1−L2\n2) sinh2(2κ2)−L2\n2= 0.\n9Using the quadratic formula, this gives\nsinh2(2κ2) =1\n2h\nL2\n1+L2\n2−1 +q\n(1−L2\n1−L2\n2)2+ 4L2\n2i\n,\nand so, we get\nκ2=1\n2arcsinh \n±r\n1\n2h\nL2\n1+L2\n2−1 +q\n(1−L2\n1−L2\n2)2+ 4L2\n2i!\n.\nThis gives the desired result.\nRemark. Another way of viewing that the choice of +or−in(3.15) is the same as the one in (3.16)\nis from the fact that we have shown that if κ∈Csatisfies (3.8), then −κdoes as well, but κdoes not.\n3.2.2 Imaginary part decay\nFrom (3.16), we obtain a result on the decay of the imaginary part of the quasiperiodicity κasω→ ∞ .\nWe will first state some preliminary results, before proving the main theorem.\nLemma 3.5. Let the frequency-dependent contrast ρ∈Cbe given by (3.5). Then, it holds\nlim\nω→∞|ρ|= 1, (3.22)\nand\nlim\nω→∞|ℜ(ρ)|= 1 and lim\nω→∞|ℑ(ρ)|= 0. (3.23)\nProof. From (3.4), we have\nρ=r\n1 +α\nε0(1−βω2−iγω)\nwhich gives directly lim ω→∞|ρ|= 1. This can be rewritten as\nρ=s\n1 +α\nε01−βω2\n(1−βω2)2+γ2ω2+iα\nε0γω\n(1−βω2)2+γ2ω2\nTo ease the notation, let us write\na(ω) := 1 +α\nε01−βω2\n(1−βω2)2+γ2ω2and b(ω) :=α\nε0γω\n(1−βω2)2+γ2ω2. (3.24)\nThen, we have\nρ=±\nsp\na(ω)2+b(ω)2+a(ω)\n2+ib(ω)\n|b(ω)|sp\na(ω)2+b(ω)2−a(ω)\n2\n (3.25)\nWe observe, from (3.24), that, as ω→ ∞ ,\na(ω) = 1 + O\u00121\nω2\u0013\nand b(ω) =O\u00121\nω3\u0013\n, (3.26)\nwhich gives\nlim\nω→∞a(ω) = 1 and lim\nω→∞b(ω) = 0 .\n10Also, since α, γ, ε 0>0, it holds that\nlim\nω→∞b(ω)\n|b(ω)|= 1.\nHence, combining these results, we get\nlim\nω→∞|ρ1|= lim\nω→∞\f\f\f\f\f\fsp\na(ω)2+b(ω)2+a(ω)\n2\f\f\f\f\f\f= 1\nand\nlim\nω→∞|ρ2|= lim\nω→∞\f\f\f\f\f\fb(ω)\n|b(ω)|sp\na(ω)2+b(ω)2−a(ω)\n2\f\f\f\f\f\f= 0.\nThis concludes the proof.\nLemma 3.6. Asω→ ∞ , we have that\n|L1| ≤1andL2→0, (3.27)\nwhereL1=L1(ω)andL2=L2(ω)were defined in (3.13) and(3.14) .\nProof. From Lemma 3.5, we have that, as ω→ ∞ ,\n|ρ1| →1,|ρ2| →0\nand from (3.4), we have that\nσ0→ ∞ .\nSo, it is essential to understand the behaviour of σ0ρ2asω→ ∞ . Using the same notations as in the\nproof of Lemma 3.5, we have, without loss of generality on the ±of (3.25),\nρ2=b(ω)\n|b(ω)|sp\na(ω)2+b(ω)2−a(ω)\n2,\nand hence, from (3.4), we have\nσ0ρ2=√µ0ε0b(ω)\n|b(ω)|vuuutr\nω4\u0010\na(ω)2+b(ω)2\u0011\n−ω2a(ω)\n2.\nFrom (3.26), we see that, as ω→ ∞ ,\nω4a(ω)2=ω4+O(1), ω2a(ω) =ω2+O(1) and ω4b(ω)2=O\u00121\nω2\u0013\n.\nHence, as ω→ ∞ ,\nvuuutr\nω4\u0010\na(ω)2+b(ω)2\u0011\n−ω2a(ω)\n2→0,\nwhich gives,\nlim\nω→∞σ0ρ2= 0,\n11and so\nlim\nω→∞|cosh( σ0ρ2)|= 1 and lim\nω→∞|sinh(σ0ρ2)|= 0. (3.28)\nThus, (3.13) gives\nlim\nω→∞|L1|= lim\nω→∞\f\f\fcos(σ0) cos( σ0ρ1)−sin(σ0) sin(σ0ρ1)\f\f\f\n= lim\nω→∞\f\f\fcos\u0010\nσ0(1 +ρ1)\u0011\f\f\f≤1,\nwhich is the desired bound for L1. Similarly, from the triangle inequality applied on (3.14), we have\n|L2| ≤ |sinh(σ0ρ2)|+1\n2(ρ2\n1+ρ2\n2)h\n|ρ2(ρ2\n2−1 +ρ2\n1)||cosh( σ0ρ2)|+\n+|ρ1|(1 +ρ2\n1+ρ2\n2)|sinh(σ0ρ2)|i\n.(3.29)\nUsing Lemma 3.5 and (3.28), we obtain\nlim\nω→∞L2= 0.\nThis concludes the proof.\nUsing these results, we will describe the behaviour of the imaginary part κ2of the quasiperiodicity\nκ∈Cas the frequency tends to infinity, i.e. ω→ ∞ .\nProposition 3.7. Let us consider a complex quasiperiodicity κ∈Csatisfying the dispersion relation\n(3.8) withα, β, γ ∈R>0. Then, it holds that\nlim\nω→∞ℑ(κ) = 0 . (3.30)\nProof. Indeed, since κ∈C, let us define κ1:=ℜ(κ) and κ2:=ℑ(κ). Then, from (3.16), we have that\nκ2is given by\nκ2=1\n2arcsinh \n±r\n1\n2h\nL2\n1+L2\n2−1 +q\n(1−L2\n1−L2\n2)2+ 4L2\n2i!\nFrom Lemma 3.6, we have that, as ω→ ∞ ,L1remains bounded, whereas L2→0. Thus, the\nfollowing holds\nlim\nω→∞h\nL2\n1+L2\n2−1 +q\n(1−L2\n1−L2\n2)2+ 4L2\n2i\n=h\nL2\n1−1 +|1−L2\n1|i\n= 0,\nsince we have that |L1| ≤1 also from Lemma 3.6. Then, from the continuity of the arcsinh( ·) function,\nthe desired result follows.\nThe decay predicted by Proposition 3.7 is shown in Figure 3. Due to the damping in the model,\nthe imaginary part has discernible peaks at the first few gaps, but then decays steadily to zero at\nhigher frequencies.\n3.3 The effect of singularities and damping\nAs mentioned before, the dispersion relation of the halide perovskite particles leads to dispersion\ncurves which are not trivial to understand in terms of the traditional viewpoint of band gaps. In order\nto understand the behaviour, we will examine each distint feature of the halide perovskite permittivity,\nto understand the effect it has on the spectrum of the photonic crystal.\nIn particular, we will begin with the simplest case when the permittivity is real and constant\nwith respect to the frequency ω. Then, we will introduce a dispersive behaviour to the permittivity\nby adding singularities at non-zero frequencies. We will initially suppose that these poles lie on the\nreal axis and will study the behaviour close to these regions. Finally, we will study the effect of\nintroducing a complex permittivity, corresponding to damping. Taken together, these results will\nallow us to explain the spectra the halide perovskite photonic crystal.\n12Figure 4: The dispersion relation of a photonic crystal with frequency-independent material parame-\nters. We model a material with permittivity given by (2.1) withα= 1,β= 0andγ= 0. The frequency\nωis chosen to be real and the Bloch parameter κallowed to take complex values. The permittivity is\nnever singular in this case.\n3.3.1 Constant permittivity\nThe first case we will consider is the one of a real-valued, non-dispersive permittivity, constant with\nrespect to the frequency ω. In our setting this translates into having (2.1) with β=γ= 0 and α >0,\ni.e.\nε(ω) =ε0+α∈R>0. (3.31)\nThis setting has been studied quite extensively. We refer to [5], as a classical reference for studying the\ndispersive nature of waves in periodic systems. In Figure 4, we provide an example for the dispersion\ncurves when the permittivity is constant and real. It is worth noting the following result.\nLemma 3.8. Letε(ω)be the real-valued, non-dispersive permittivity given by (3.31) . Then, if κ∈C\nis a quasiperiodicity satisfying (3.8) for a given frequency ω∈R, then so does κ∈C.\nProof. Indeed, since ε(ω)∈R>0, then ρ∈R>0. Let us take κ∈Csatisfying (3.8). Then, we can\nwrite κ=κ1+iκ2, with κ1, κ2∈R. Now, since ρ >0, we can write\ncos(σ0) cos( ρσ0)−1 +ρ2\n2ρsin(σ0) sin(ρσ0) =:A >0.\nThus, (3.8) gives us\ncos(2 κ1+ 2iκ2) =A,\nwhich becomes the following system\n(\ncos(2 κ1)·cosh(2 κ2) =A,\nsin(2κ1)·sinh(2 κ2) = 0 .\nThis implies that\nκ2= 0 or κ1=m\n2π, m∈Z.\nIfκ2= 0, then κ=κ1∈R. Thus, κ=κ, which gives the desired result. If κ1=m\n2π, form∈Z, then\nκsatisfies (3.8) if and only if\ncosh(2 κ2) =±A. (3.32)\nSince cosh( ·) is an even function, we have that −κ2satisfies (3.32) for the same frequency ω. Hence,\nκ=κ1−iκ2satisfies (3.8) and this concludes the proof.\n13Crucially, the dispersion curves shown in Figure 4 consist of a countable sequence of disjoint bands\nin which κis real valued. Between each band there is a band gap, defined in the sense of Definition 2.3,\nin which κis purely imaginary, corresponding to the decay of the wave. The occurence of κbeing\neither purely real or purely imaginary is the mechanism behind Lemma 3.8. As we will see below,\nwhen we add singularities or damping to the model, the band gap structure is less straightforward to\ninterpret.\n3.3.2 Singular permittivity with no damping\nLet us now study the case where the permittivity has a dispersive (and singular) character with respect\nto the frequency ω, but there is no damping, i.e. we consider α, β > 0 and γ= 0. This implies that\nε(ω) =ε0+α\n1−βω2. (3.33)\nThe interesting aspect in this setting is the existence of real poles for the permittivity. They are given\nby\nω∗\n±=±1√β.\nIn Figure 5, we observe that near the pole of the permittivity there are infinitely many band-gaps.\nThis was similarly observed recently by [23]. Noting that a band gap occurs when the magnitude of\nthe right-hand side of (3.8) is greater than one. We define the function\nf(ω) := cos\u0010\nσ0(ω)\u0011\ncos\u0010\nρ(ω)σ0(ω)\u0011\n−1 +ρ(ω)2\n2ρ(ω)sin\u0010\nσ0(ω)\u0011\nsin\u0010\nρ(ω)σ0(ω)\u0011\n, (3.34)\nwhich is the right-hand side of (3.8). We will prove that this takes values greater than one on a\ncountably infinite number of disjoint intervals within any neighbourhood of the singularity. To do so,\nwe will introduce the following notation, which will be used in our analysis.\nNotation. Letx, y∈R. Then, we use x↓ywhen x→yandx > y . Similarly, we use x↑ywhen\nx→yandx < y .\nThen, close to a permittivity pole, the following holds.\nTheorem 3.9. Letω∗denote a pole of the permittivity ε(ω)given by (3.33) , i.e. ω∗∈n\n±1√βo\n. Then,\nforδ >0, the intervals [ω∗−δ, ω∗)and(ω∗, ω∗+δ]contain infinitely many disjoint sub-intervals,\ndenoted by IiandJi,i= 1,2, . . ., respectively, that are band gaps.\nProof. We will first prove this result for the interval [ ω∗−δ, ω), for δ >0. It suffices to show that\nthere are infinitely many points ω†\ni∈[ω∗−δ, ω∗),i= 1,2, . . . , forδ >0 for which f(ω†\ni)>1 or\nf(ω†\ni)<−1. Then, the continuity of faround these points gives us the existence of intervals of the\nformIi:= [ω†\ni−s, ω†\ni+s], for i= 1,2, . . ., for small s >0, such that,\nf(ω)>1 or f(ω)<−1,∀ω∈ Ii, i= 1,2, . . . .\nFrom (3.8) and (3.34), this gives\ncos(2 κ)>1 or cos(2 κ)<−1,∀ω∈ Ii, i= 1,2, . . . ,\nThis is equivalent to the Ii’s,i= 1,2, . . .being band gaps, since κbecomes complex in these intervals,\ni.e.|ℑ(κ)| ̸= 0. Hence, since |ℑ(κ)|is continuous with respect to ω, we get that it has a local maximum\nin each of the Ii’s, for i= 1,2, . . ..\nWe observe that lim ω↑ω∗ε(ω) = +∞. Then, this implies that lim ω↑ω∗ρ(ω) = +∞, and so, we get\nlim\nω↑ω∗1 +ρ(ω)2\n2ρ(ω)= +∞.\n14Singularities:\nRe(ω)Im(ω)\n×ω∗\n+×ω∗\n−\n1 -1\n.\nFigure 5: The dispersion relation of a photonic crystal with frequency-independent material parame-\nters. We model a material with permittivity given by (2.1) withα= 1,β= 1andγ= 0. The frequency\nωis chosen to be real and the Bloch parameter κallowed to take complex values. The permittivity is\nsingular when ω= 1. The lower two plots are display the same dispersion curves, zoomed into the\nregion around the singularity.\n15Also, as ω↑ω∗, we have that σ0is constant and so, without loss of generality, we can assume that\nsin(σ0)>0 (the same argument holds for taking sin( σ0)<0). Hence, there exists δ1>0 such that\nfor all ω∈[ω∗−δ1, ω∗), we have that\n1 +ρ(ω)2\n2ρ(ω)>1\nsin\u0010\nσ0(ω)\u0011≥1. (3.35)\nNow, since lim ω↑ω∗ρ(ω) = +∞, we have that\nlim\nω↑ω∗ρ(ω)σ0(ω) = +∞. (3.36)\nThis implies that, for all K > 0, there exists δ2>0 such that for all ω∈[ω∗−δ2, ω∗) it holds that\n|ρ(ω)σ0(ω)|> K.\nNow, let δ:= max {δ1, δ2}and let I(−)\nδ:= [ω∗−δ, ω∗). Then, (3.36) implies that the exist two families\nof infinitely many points in I(−)\nδ, denoted by {ω(+)\ni}i=1,...,∞and{ω(−)\ni}i=1,...,∞, such that, for all\ni= 1, . . . ,∞, we have\nsin\u0010\nρ(ω(+)\ni)ω(+)\ni\u0011\n= 1 and sin\u0010\nρ(ω(−)\ni)ω(−)\ni\u0011\n=−1. (3.37)\nWe also note that this implies, for all i= 1, . . . ,∞, that\ncos\u0010\nρ(ω(+)\ni)ω(+)\ni\u0011\n= cos\u0010\nρ(ω(−)\ni)ω(−)\ni\u0011\n= 0. (3.38)\nThus, for all i= 1, . . . ,∞, we have\nf\u0010\nω(+)\ni\u0011\n=−1 +ρ(ω(+)\ni)2\n2ρ(ω(+)\ni)sin\u0010\nσ0(ω(+)\ni)\u0011\n<−1 (3.39)\nand\nf\u0010\nω(−)\ni\u0011\n=1 +ρ(ω(−)\ni)2\n2ρ(ω(−)\ni)sin\u0010\nσ0(ω(−)\ni)\u0011\n>1 (3.40)\nIn particular, without loss of generality, let us assume that ω(+)\n0is the smallest of the elements in both\nfamilies {ω(+)\ni}i=1,...,∞and{ω(−)\ni}i=1,...,∞. Then, the periodicity of sin( ·) shows that the elements of\nthese families respect the following ordering:\nω(+)\n0< ω(−)\n0< ω(+)\n1< ω(−)\n1< . . . . (3.41)\nNow, the continuity of faround these points allows us to take s >0 such that\nf(ω)<−1,∀ω∈h\nω(+)\ni−s, ω(+)\ni+si\n, i= 1,2, . . . ,\nf(ω)>1,∀ω∈h\nω(−)\ni−s, ω(−)\ni+si\n, i= 1,2, . . . ,\nand\nh\nω(+)\ni−s, ω(+)\ni+si\\h\nω(−)\ni−s, ω(−)\ni+si\n=∅, i= 1,2, . . . .\nFinally, the infinity of elements in the families {ω(+)\ni}i=1,...,∞and{ω(−)\ni}i=1,...,∞gives us the desired\nresult.\nWe note that for the neighborhood of the form ω∈(ω∗, ω∗+δ] for δ >0, the proof remains the\nsame with the slight change of taking the limits as ω↓ω∗.\n16In addition to the occurrence of a countable number of band gaps close to the pole, in Figure (5),\nwe observe that there is an interesting behaviour of the imaginary part ℑ(κ) of the quasiperiodicity\nκas the frequency ω∈Rapproaches a permittivity pole. In fact, we see that close to a pole, |ℑ(κ)|\nbecomes arbitrarily big. This due to the resonance occurring here and is strongly related to the\nexistence of infinitely many band gaps close to the pole. Actually, it is a corollary of Theorem 3.9.\nCorollary 3.9.1. Letω∈Randκ∈Cbe the associated quasiperiodicity satisfying the dispersion\nrelation (3.8) and let f(ω)be the function defined in (3.34) . Let ω∗∈Rdenote a pole of the undamped\npermittivity ε(ω), given by (3.33) . Then, for all K > 0, there exists δ >0, such that for all p∈\n[−K, K ], there exists ωp∈[ω∗−δ, ω∗)such that ℑ(κ(ωp)) = p. That is, |ℑ(κ(ω))|takes arbitrarily\nlarge values as ω↑ω∗. The same result holds as ω↓ω∗.\nProof. From Theorem 3.9, we have that for all K > 0, there exists δ > 0, such that, for all ω∈\n[ω∗−δ, ω),\n1 +ρ(ω)2\n2ρ(ω)> K.\nWe have that cos( σ0(ω)) cos( ρ(ω)σ0(ω)) remains bounded close to ω∗and because of the continuity of\nsin(·), we can take sin( σ0(ω))>0 in [ ω∗−δ, ω). Then, it follows that, for all K > 0, in [ ω∗−δ, ω),\n1 +ρ(ω)2\n2ρ(ω)sin\u0010\nσ0(ω)\u0011\n> K.\nBut, from Theorem (3.9), we have the existence of infinitely many points ω(+)\n0< ω(−)\n0< ω(+)\n1< ω(−)\n1<\n. . .in [ω∗−δ, ω), for which, sin\u0010\nρ(ω)σ0(ω)\u0011\noscillates between 1 and -1 in each of the intervals of\nthe form [ ω(+)\n0, ω(−)\n0], [ω(−)\n0, ω(+)\n1],. . ., denoted by Ii,i= 1,2, . . . . This implies that, for all K > 0,\nfor all p∈[−K, K ], there exists ωp∈ Ii, fori= 1,2, . . . , such that f(ωp) =p. Since this holds for all\nK > 0, it translates to foscillating and taking all values between + ∞and−∞in [ω∗−δ, ω∗) as we\nget closer to ω∗.\nNow, from (3.8) and (3.34), we get that\nκ=−i\n2ln\u0010\nf(ω)±p\nf(ω)2−1\u0011\n, (3.42)\nwhere ln( ·) denotes the complex logarithm. We see that\nℑ(κ) =1\n2ℜ\u0010\nln\u0010\nf(ω)±p\nf(ω)2−1\u0011\u0011\n.\nAlthough, since we are using the complex logarithm, we have that\nℜ\u0010\nln\u0010\nf(ω)±p\nf(ω)2−1\u0011\u0011\n= ln\f\f\ff(ω)±p\nf(ω)2−1\f\f\f.\nHence, since we have shown that in [ ω∗−δ, ω∗), the function f(ω) oscillates between + ∞and−∞, we\nhave that\f\f\ff(ω)±p\nf(ω)2−1\f\f\fhas the same behaviour in [ ω∗−δ, ω∗), but the oscillation takes place\nbetween 0 and + ∞. Finally, since ln( ·) is an increasing function, we obtain the desired result. Let\nus note that the proof is the same when we consider ω↓ω∗, with the slight change that we consider\nneighborhoods of the form ( ω∗, ω∗+δ].\n3.3.3 Complex permittivity\nWe will now study the effect that introducing damping though allowing the permitivitty to be complex\nhas on our one-dimensional system. Starting from the straightforward real-valued, non-dispersive\n17model considered in Section 3.3.1, we subsequently add damping. For this, we take α∈Cand\nβ=γ= 0, i.e.\nε(ω) =ε0+α∈C. (3.43)\nThe dispersion curves for this setting are shown in Figure 6. They are plotted for αwith successively\nlarger imaginary parts, to show the effect of gradually increasing the damping. We see that the\nclear structure of successive bands and gaps is gradually blurred out, eventually to the point that the\nspectrum bears no clear relation to the original undamped spectrum.\nIn many ways, the spectrum we obtain in this setting appears to be similar to the actual halide\nperovskite particles, as plotted in Figure 3. Indeed, all of the results proved in subsection 3.2 hold,\nwith the exception of the imaginary part decay. In fact, the converse is true, as made precise by the\nfollowing result.\nLemma 3.10. Letκ∈Candω∈Rbe a quasiperiodicity and a frequency, respectively, satisfying the\ndispersion relation (3.8) with complex-valued, non-dispersive permittivity given by (3.43) . Then,\nlim\nω→+∞|ℑ(κ)|= +∞. (3.44)\nProof. Let us recall that κ2, denoting the imaginary part of κ∈C, is given by (3.16), where L1and\nL2are given by (3.13) and (3.14), respectively. We have that\nlim\nω→∞ρ(ω)σ0(ω) = +∞. (3.45)\nsince σ0is linear with respect to ωandρdoes not depend on ωin this setting. This implies\nlim\nω→∞sinh\u0010\nρ(ω)σ0(ω)\u0011\n= +∞and lim\nω→∞cosh\u0010\nρ(ω)σ0(ω)\u0011\n= +∞.\nWe note that it is enough to show that\nlim\nω→∞|L1|= lim\nω→∞|L2|= +∞,\nsince applying this on (3.16) gives that |κ2| →+∞asω→+∞. Indeed, (3.13) gives that\nL1=C1cosh( σ0ρ2)−C2sinh(σ0ρ2),\nwhere\nC1:= cos( σ0) cos( σ0ρ1)−ρ1(1 +ρ2\n1+ρ2\n2)\n2(ρ2\n1+ρ2\n2)sin(σ0) sin(σ0ρ1)\nand\nC2:=ρ2(ρ2\n1−1 +ρ2\n2)\n2(ρ2\n1+ρ2\n2)sin(σ0) cos( σ0ρ1).\nUsing the exponential formulation of the hyperbolic trigonometric functions, we get\nL1=C1−C2\n2eσ0ρ2+C1+C2\n2e−σ0ρ2.\nNow, we observe that, as ω→+∞,C1andC2are both bounded and C1−C2̸= 0. Then, from (3.45),\nwe get that\nlim\nω→∞|L1|= +∞.\nSimilarly, (3.14) gives that\nL2=˜C1sinh(σ0ρ2)−˜C2cosh( σ0ρ2),\n18(a)α= 1 + 0 .001i\n. .\n(b)α= 1 + 0 .01i\n. .\n(c)α= 1 + 0 .1i\n. .\n(d)α= 1 + i\n. .\nFigure 6: The dispersion relation of a photonic crystal with frequency-independent complex-valued\nmaterial parameters. We model a material with permittivity given by (2.1) with α∈C,β= 0and\nγ= 0. The frequency ωis chosen to be real and the Bloch parameter κallowed to take complex values.\nThe permittivity is never singular in this case.\n19where\n˜C1:= cos( σ0) sin(σ0ρ1) +ρ1(1 +ρ2\n1+ρ2\n2)\n2(ρ2\n1+ρ2\n2)sin(σ0) cos( σ0ρ1)\nand\n˜C2:=ρ2(ρ2\n2−1 +ρ2\n1)\n2(ρ2\n1+ρ2\n2)sin(σ0) sin(σ0ρ1)\nThen, we can write\nL2=˜C1+˜C2\n2eσ0ρ2+˜C2−˜C1\n2e−σ0ρ2\nAs before, we observe that, as ω→+∞,˜C1and ˜C2are both bounded and ˜C1−˜C2̸= 0. Then, from\n(3.45), we get that\nlim\nω→∞|L2|= +∞.\nThis concludes the proof\n3.3.4 Discussion\nThe analysis in this section can be used to understand the dispersion diagram for the halide perovskite\nphotonic crystal that was presented in Figure 3. There are two crucial observations. First, we saw in\nSection 3.3.2 that the introduction of singularities in the permittivity led to the creation of countably\ninfinitely many band gaps in a neigbourhood of the pole, when the pole falls on the real axis. However,\nthis exotic behaviour is not seen in Figure 3, due in part to the introduction of damping causing the\npoles to fall below the real axis. This effect can also be exaplined in terms of the results in Section 3.3.3,\nwhere we saw that the introduction of damping to a simple non-dispersive model smoothed out the\nband gaps. The behaviour shown in Figure 3 is a combination of these phenomena.\n4 Multiple dimensions\nLet us now treat the periodic structures in two- and three-dimensions. In this case, to be able to handle\nthe problem concisely using asymptotic methods, we are interested in the case of small resonators.\nWe will assume that there exists some fixed domain D, which the the union of the Ndisjoint subsets\nD=D1∪D2∪ ··· ∪ DN, such that Ω is given by\nΩ =δD+z, (4.1)\nfor some position z∈Rd,d= 2,3, and characteristic size 0 < δ≪1. Then, making a change of\nvariables, the quasiperiodic Helmholtz problem (2.8) becomes\n\n\n∆uκ+δ2ω2ε(ω)µ0uκ= 0 in D,\n∆uκ+δ2k2\n0uκ= 0 in Rd\\D,\nuκ|+−uκ|−= 0 on ∂D,\n∂uκ\n∂ν|+−∂uκ\n∂ν|−= 0 on ∂D,\nuκ(xd, x0) is κ-quasiperiodic in xd,\nuκ(xd, x0) satisfies the κ-quasiperiodic radiation condition as |x0| → ∞ .(4.2)\nWe will also make an additional assumption on the dimensions of the nano-particles. This will allow\nus to prove an approximation for the values of the modes u|Di,i= 1, . . . , N, on each particle. The\nassumption is one of diluteness , in the sense that the particles are small relative to the separation\n20distances between them. To capture this, we introduce the parameter ρito capture the size of the\nreference particles D1, . . . , D N. We define ρi:=1\n2(diam( Di)) where diam( Di) is given by\ndiam( Di) = sup {|x−y|:x, y∈Di}. (4.3)\nWe will assume that each ρi→0 independently of δ. This regime means that the system is dilute in\nthe sense that the particles are small relative to the distances between them.\nWe will first present certain general results for this system. Then we will give a more qualitative\ndescription of the two-dimensional setting.\n4.1 Integral formulation\nLetGk(x) be the outgoing Helmholtz Green’s function in Rd, defined as the unique solution to\n(∆ + k2)Gk(x) =δ0(x) inRd,\nalong with the outgoing radiation condition, where δ0is the Dirac delta. It is well known that Gkis\ngiven by\nGk(x) =(\n−i\n4H(1)\n0(k|x|), d = 2,\n−eik|x|\n4π|x|, d = 3,(4.4)\nwhere H(1)\n0is the Hankel function of first kind and order zero. We define the quasiperiodic Green’s\nfunction Gκ,k(x) as the Floquet transform of Gk(x) in the first dlcoordinate dimensions, i.e.\nGκ,k(x) =\n\n−i\n4P\nm∈ΛH(1)\n0(k|x−m|)eim·κ, d = 2,\n−1\n4πP\nm∈Λeik|x−m|\n|x−m|eiκ·m, d = 3.(4.5)\nThen, from [2, 3], we know that (2.8) has the following integral representation expression.\nTheorem 4.1 (Lippmann-Schwinger integral representation formula) .A function uκsatisfies the\ndifferential system (2.8) if and only if it satisfies the following equation\nuκ(x)−uκ\nin(x) =−δ2ω2ξ(ω)Z\nDGκ,δk 0(x−y)u(y)dy, x∈Rd, (4.6)\nwhere the function ξ:C→Cdescribes the permittivity contrast between Dand the background and is\ngiven by\nξ(ω) =µ0(ε(ω)−ε0).\n4.2 Dispersion relation\nWe will now retrieve an expression which relates the subwavelength resonances of the system and the\nquasiperiodicities. The method used is similar to the one developed in [3] for systems of finitely many\nparticles.\n4.2.1 Matrix representaion\nLet us define the following κ-quasiperiodic integral operators\nKκ,r\nDi:u|Di∈L2(Di)→ −Z\nDiGκ,r(x−y)u(y)dy\f\f\f\nDi∈L2(Di) (4.7)\nand\nRκ,r\nDiDj:u|Di∈L2(Di)→ −Z\nDiGκ,r(x−y)u(y)dy\f\f\f\nDj∈L2(Dj). (4.8)\n21Then, as in [3], the scattering problem has the following matrix representation:\n\n1−δ2ω2ξ(ω)Kκ,δk 0\nD1−δ2ω2ξ(ω)Rκ,δk 0\nD2D1. . .−δ2ω2ξ(ω)Rκ,δk 0\nDND1\n−δ2ω2ξ(ω)Rκ,δk 0\nD1D21−δ2ω2ξ(ω)Kκ,δk 0\nD2. . .−δ2ω2ξ(ω)Rκ,δk 0\nDND2............\n−δ2ω2ξ(ω)Rκ,δk 0\nD1DN−δ2ω2ξ(ω)Rκ,δk 0\nD2DN. . . 1−δ2ω2ξ(ω)Kκ,δk 0\nDN\n\nuκ|D1\nuκ|D2...\nuκ|DN\n=\nuκ\nin|D1\nuκ\nin|D2...\nuκ\nin|DN\n\nSince the scattered field is fully determined by the value within each resonator, we will introduce the\nnotation\nuκ\ni:=uκ|Di, i= 1, . . . , N. (4.9)\nThen, the resonance problem is to find ω∈C, such that there exists ( uκ\n1, uκ\n2, . . . , uκ\nN)∈L2(D1)×\nL2(D2)× ··· × L2(DN),uκ\ni̸= 0, for i= 1, . . . , N, such that\n\n1−δ2ω2ξ(ω)Kκ,δk 0\nD1−δ2ω2ξ(ω)Rκ,δk 0\nD2D1. . .−δ2ω2ξ(ω)Rκ,δk 0\nDND1\n−δ2ω2ξ(ω)Rκ,δk 0\nD1D21−δ2ω2ξ(ω)Kκ,δk 0\nD2. . .−δ2ω2ξ(ω)Rκ,δk 0\nDND2............\n−δ2ω2ξ(ω)Rκ,δk 0\nD1DN−δ2ω2ξ(ω)Rκ,δk 0\nD2DN. . . 1−δ2ω2ξ(ω)Kκ,δk 0\nDN\n\nuκ\n1\nuκ\n2...\nuκ\nN\n=\n0\n0\n...\n0\n(4.10)\n4.2.2 Resonances\nLet us now retrieve the relation between the subwavelength resonant frequencies and the quasiperi-\nodicities, obtained by studying the solutions to (4.10). We will first recall a definition and a lemma\nwhich will help in the analysis of the problem.\nDefinition 4.2. Given N∈N, we denote by ⌊N⌋:N→ {1,2, . . . , N }a modified version of the\nmodulo function, i.e.the remainder of euclidean division by N. In particular, for all M∈N, there\nexists unique τ∈Z≥0andr∈Nwith 0< r≤N, such that\nM=τ·N+r.\nThen, we define M⌊N⌋to be\nM⌊N⌋:=r.\nWe recall the diluteness assumption that we have made on our system, which is captured by\nconsidering small particle size ρ. We define ρ:=1\n2max i(diam( Di)) where diam( Di) is given by\ndiam( Di) = sup {|x−y|:x, y∈Di}. (4.11)\nThe next lemma is a variation of Lemma 2.6 in [3].\nLemma 4.3. For all i= 1, . . . , N , we denote uκ\ni=uκ|Di, where uκis a resonant mode, in the sense\nthat it is a solution to (4.6) with no incoming wave. Then, for characteristic size δof the same order\nasρ, we can write that\nuκ\ni=⟨uκ, ϕ(i)\nκ⟩ϕ(i)\nκ+O(ρ2), i= 1, . . . , N, (4.12)\nasρ→0, where ϕ(i)\nκdenotes the eigenvector associated to the particle Diof the potential Kκ,δk 0\nDiand\nρ >0denotes the particle size parameter of D1, . . . , D N. Here, δandρare of the same order in the\nsense that δ=O(ρ)andρ=O(δ). In this case, the error term holds uniformly for any small δandρ\nin a neighbourhood of 0.\nLet us now state the main result of this section.\n22Theorem 4.4. The resonance problem, as δ→0andρ→0, with δ=O(ρ)andρ=O(δ),(4.10) in\ndimensions d= 2,3, becomes finding ω∈Csuch that\ndet\u0010\nKκ(ω)\u0011\n= 0,\nwhere\nKκ(ω)ij:=\n\n⟨Rκ,δk 0\nDiDi+1⌊N⌋ϕ(i)\nκ, ϕ(i+1⌊N⌋)\nκ ⟩, ifi=j,\n−Aκ\ni(ω, δ)⟨Rκ,δk 0\nDjDiϕ(j)\nκ, ϕ(i)\nκ⟩⟨Rκ,δk 0\nDiDi+1⌊N⌋ϕ(i)\nκ, ϕ(i+1⌊N⌋)\nκ ⟩,ifi̸=j,(4.13)\nHere, k0=ω√µ0ε0and\nAκ\ni(ω, δ) :=δ2ω2ξ(ω)\n1−δ2ω2ξ(ω)λ(i)\nκ, i= 1, ..., N. (4.14)\nwithλ(i)\nκandϕ(i)\nκbeing the eigenvalues and the respective eigenvectors associated to the particle Diof\nthe potential Kκ,δk 0\nDi, for i= 1,2, . . . , N .\nProof. We will provide an outline of the proof of this result, since it follows the exact same reasoning\nas, for example, the proof of Theorem 2.8 in [3]. Using the following pole pencil decomposition,\n\u0010\n1−δ2ω2ξ(ω)Kκ,δk 0\nDi\u0011−1\n(·) =⟨·, ϕ(i)\nκ⟩ϕ(i)\nκ\n1−δ2ω2ξ(ω)λ(i)\nκ+Ri[ω](·), i= 1, . . . , N, (4.15)\nwe get that (4.10) is equivalent to the system of equations\nuκ\ni−δ2ω2ξ(ω)\n1−δ2ω2ξ(ω)λ(i)\nκNX\nj=1,j̸=1⟨Rκ,δk 0\nDjDiuκ\nj, ϕ(i)\nκ⟩ϕ(i)\nκ= 0,for each i= 1, . . . , N.\nThe above system is equivalent to\n⟨Rκ,δk 0\nDiDi+1⌊N⌋uκ\ni, ϕ(i+1⌊N⌋)\nκ ⟩ −δ2ω2ξ(ω)\n1−δ2ω2ξ(ω)λ(i)\nκNX\nj=1,j̸=i⟨Rκ,δk 0\nDjDiuκ\nj, ϕ(i)\nκ⟩⟨Rκ,δk 0\nDiDi+1⌊N⌋ϕ(i)\nκ, ϕ(i+1⌊N⌋)\nκ ⟩= 0.\nFrom Lemma 4.3, we have\nuκ\ni≃ ⟨uκ, ϕ(i)\nκ⟩ϕ(i)\nκ,\nwhich gives\nKκ(ω)\n⟨uκ, ϕ(1)\nκ⟩\n⟨uκ, ϕ(2)\nκ⟩\n...\n⟨uκ, ϕ(N)\nκ⟩\n=\n0\n0\n...\n0\n, (4.16)\nwhere\nKκ(ω)ij:=\n\n⟨Rκ,δk 0\nDiDi+1⌊N⌋ϕ(i)\nκ, ϕ(i+1⌊N⌋)\nκ ⟩, ifi=j,\n−Aκ\ni(ω, δ)⟨Rκ,δk 0\nDjDiϕ(j)\nκ, ϕ(i)\nκ⟩⟨Rκ,δk 0\nDiDi+1⌊N⌋ϕ(i)\nκ, ϕ(i+1⌊N⌋)\nκ ⟩,ifi̸=j.(4.17)\nThen, for the system to have a non-trivial solution, we require\ndet\u0010\nKκ(ω)\u0011\n= 0,\nwhich gives the desired result.\n234.3 The two-dimensional case\nIn the particular case of a two-dimensional system, it is possible to provide a more detailed and\nsimplified version of the result in Theorem 4.4. In particular, we will consider the setting where the\nperiodic structure Dis composed of N∈Nresonators, denoted by Di,i= 1, . . . , N , which are repeated\nperiodically in the lattice Λ. Let us recall that in the dimension d= 2, the κ-quasiperiodic Green’s\nfunction Gκ,r(x) is given by\nGκ,k(x) =−i\n4X\nm∈ΛH(1)\n0\u0010\nk|x−m|\u0011\neim·κ, (4.18)\nwhere H(1)\n0denotes the Hankel function of the first kind of order zero and has the following asymptotic\nexpansion in terms of a small argument:\nH(1)\n0(s) =2i\nπ∞X\nm=0(−1)ms2m\n22m(m!)2\nlog(ˆγs)−mX\nj=11\nj\n. (4.19)\n4.3.1 Integral operators\nWe define the integral operators Kκ,(−1)\nDi:L2(Di)→L2(Di) and Kκ,(0)\nDi:L2(Di)→L2(Di) by\nKκ,(−1)\nDi[u](x) :=−1\n2πlog(ˆγδk0)Z\nDiX\nm∈Λeim·κu(y)dy\f\f\f\nDi,\nKκ,(0)\nDi[u](x) :=−1\n2πZ\nDiX\nm∈Λlog\u0010\n|x−y−m|\u0011\neim·κu(y)dy\f\f\f\nDi,\nand the integral operators Rκ,(−1)\nDiDj:L2(Di)→L2(Dj) and Rκ,(0)\nDiDj:L2(Di)→L2(Dj) by\nRκ,(−1)\nDiDj[u](x) :=−1\n2πlog(ˆγδk0)Z\nDiX\nm∈Λeim·κu(y)dy\f\f\f\nDj,\nRκ,(0)\nDiDj[u](x) :=−1\n2πZ\nDiX\nm∈Λlog\u0010\n|x−y−m|\u0011\neim·κu(y)dy\f\f\f\nDj,\nfori= 1, . . . , N . We will provide some results which will help us in the analysis of the problem.\nDefinition 4.5. We define the integral operators Mδk0\nDiandNδk0\nDiDjfori, j= 1,2by\nMδk0\nDi:=Kκ,(−1)\nDi+Kκ,(0)\nDiand Nδk0\nDiDj:=Rκ,(−1)\nDiDj+Rκ,(0)\nDiDj. (4.20)\nFrom the asymptotic expansion of the Hankel function in (4.19), the following holds.\nProposition 4.6. For the integral operators Kκ,δk 0\nDiandRκ,δk 0\nDiDj, defined in (4.7) and(4.8) respectively,\nwe can write\nKκ,δk 0\nDi=Mκ,δk 0\nDi+O\u0010\nδ2log(δ)\u0011\nand Rκ,δk 0\nDiDj=Nκ,δk 0\nDiDj+O\u0010\nδ2log(δ)\u0011\n, (4.21)\nasδ→0and with k0fixed.\n4.3.2 Spectral results\nWe have the following spectral results for the operators Kκ,(−1)\nDiandKκ,δk 0\nDi, fori= 1, . . . , N .\n24Lemma 4.7. Letν(κ)\n−1,iandΨ(κ)\n−1,idenote a non-zero eigenvalue and the associated eigenvector of the\noperator Kκ,(−1)\nDi, for i= 1, . . . , N . Then,\nν(κ)\n−1,i=−|Di|\n2πX\nm∈Λeim·κand Ψ(κ)\n−1,i=ˆ1Di, (4.22)\nfori= 1, . . . , N , where ˆ1Di:=1Di√\n|Di|and|Di|denotes the volume of Di.\nProof. From the definition of Kκ,(−1)\nDi, for i= 1, . . . , N , we observe that Kκ,(−1)\nDiis independent of\nx∈Di, and so, normalizing on L2(Di), we get\nΨ(κ)\n−1,i=ˆ1Di.\nThen, the following must hold:\nν(κ)\n−1,iˆ1Di=Kκ,(−1)\nDi[ˆ1Di]⇒ν(κ)\n−1,iˆ1Di=−|Di|\n2πˆ1DiX\nm∈Λeim·κ\n⇒ν(κ)\n−1,i=−|Di|\n2πX\nm∈Λeim·κ.\nThis concludes the proof.\nLemma 4.8. Letν(κ)\nidenote a non-zero eigenvalue of the operator Mκ,δk 0\nDi, for i= 1, . . . , N , in\ndimension 2. Then, for small δ, it is approximately given by:\nν(κ)\ni= log( δk0ˆγ)ν(κ)\n−1,i+⟨Kκ,(0)\nDiΨ(κ)\n−1,i,Ψ(κ)\n−1,i⟩+O(δ2log(δ)), (4.23)\nwhere ν(κ)\n−1,iandΨ(κ)\n−1,idenote an eigenvalue and the associated eigenvector of the potential Kκ,(−1)\nDi, for\ni= 1, . . . , N , respectively.\nProof. This was proved in Lemma 2.6 of [2].\nSince we have considered identical resonators, the symmetry of the system leads to the following\nsimple result.\nLemma 4.9. Letν(κ)\n−1,idenote a non-zero eigenvalue of the operator Kκ,(−1))\nDi, for i= 1, . . . , N . Then,\nit holds that\nν(κ)\n−1,1=ν(κ)\n−1,2=···=ν(κ)\n−1,N=:ν(κ)\n−1.\n4.3.3 Resonant frequencies\nWe will now state a more explicit version of Theorem 4.4. This is the main result of our analysis of\nthe two-dimensional system, which fully characterises the resonant frequencies of the periodic system.\nProposition 4.10. The resonance problem, as δ→0andρ→0, with δ=O(ρ)andρ=O(δ),(4.10)\nin dimensions d= 2,3, becomes finding ω∈Csuch that\ndet\u0010\nKκ(ω)\u0011\n= 0,\nwhere\nKκ(ω)ij=(\n⟨Nκ,δk 0\nDiDi+1⌊N⌋ˆ1Di,ˆ1D(i+1⌊N⌋)⟩, ifi=j,\n−Aκ\ni(ω, δ)⟨Nκ,δk 0\nDjDiˆ1Dj,ˆ1Di⟩⟨Nκ,δk 0\nDiDi+1⌊N⌋ˆ1Di,ˆ1D(i+1⌊N⌋)⟩,ifi̸=j.(4.24)\n25Here, k0=ω√µ0ε0and\nAκ\ni(ω, δ) :=δ2ω2ξ(ω)\n1−δ2ω2ξ(ω)ν(κ), i= 1, ..., N. (4.25)\nwithν(κ)denoting a non-zero eigenvalue of the potential Mκ,δk 0\nDi, for i= 1,2, . . . , N .\nProof. This is a direct consequence of Theorem 4.4. We just have to apply (4.21) and (4.22) to (4.13)\nand get\nKκ(ω)ij=(\n⟨Nκ,δk 0\nDiDi+1⌊N⌋ˆ1Di,ˆ1D(i+1⌊N⌋)⟩, ifi=j,\n−Aκ\ni(ω, δ)⟨Nκ,δk 0\nDjDiˆ1Dj,ˆ1Di⟩⟨Nκ,δk 0\nDiDi+1⌊N⌋ˆ1Di,ˆ1D(i+1⌊N⌋)⟩,ifi̸=j,\nwhich gives the desired result.\n5 Conclusion\nWe have used analytic methods to understand the dispersive nature of photonic crystals fabricated\nfrom metals with singular permittivities. In particular, we considered a Drude–Lorentz model inspired\nby halide perovskites that has poles in the lower complex plane. For a one-dimensional system, we\ncharacterised the effect that each feature of this model has on the dispersion relation. In particular,\nwe showed that the introduction of singularities leads to the creation of countably many band gaps\nnear the poles, whereas the introduction of damping smooths out the band gap structure. Finally,\nwe showed how the integral methods developed in [2, 3] can be used to extend this theory to multi-\ndimensional systems.\nConflicts of interest\nThe authors have no conflicts of interest to disclose.\nData availability\nThe code used to perform the numerical simulations presented in this work can be found at https:\n//doi.org/10.5281/zenodo.8055547 . No other data were generated in this project.\nAcknowledgements\nThe work of KA was supported by ETH Z¨ urich under the project ETH-34 20-2. The work of BD was\nfunded by the Engineering and Physical Sciences Research Council through a fellowship with grant\nnumber EP/X027422/1.\nReferences\n[1] S. D. M. Adams, R. V. Craster, and S. Guenneau. Bloch waves in periodic multi-layered acoustic\nwaveguides. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences ,\n464(2098):2669–2692, 2008.\n[2] K. Alexopoulos and B. Davies. 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Perovskite-molecule composite thin films for efficient and stable light-emitting diodes.\nNature Communications , 11(1):1–9, 2020.\n28"    },    {        "title": "2205.10525v1.Noether_symmetries_and_first_integrals_of_damped_harmonic_oscillator.pdf",        "content": "arXiv:2205.10525v1  [math-ph]  21 May 2022Noether symmetries and first integrals of damped harmonic\noscillator\nM. Umar Farooq1,∗and M. Safdar2,†\n1Department of Basic Sciences &Humanities, College of E &ME,\nNational University of Sciences and Technology (NUST), H-12 , Islamabad Pakistan\n2School of Mechanical &Manufacturing Engineering,\nNational University of Sciences and Technology (NUST), H-12 , Islamabad Pakistan\nAbstract\nNoether’s theorem establishes aninteresting connection b etween symmetriesof the\naction integral and conservation laws (first integrals) of a dynamical system. The\naim of the present work is to classify the damped harmonic osc illator problem with\nrespect to Noether symmetries and to construct correspondi ng conservation laws\nfor all over-damped, under-damped and critical damped case s. For each case we\nobtain maximum five linearly independent group generators w hich provide related\nfive conserved quantities. Remarkably, after obtaining com plete set of invariant\nquantities we obtain analytical solutions for each case. In the current work, we\nalso introduce a new Lagrangian for the damped harmonic osci llator. Though the\nform of this new Lagrangian and presented by Bateman are comp letely different,\nyet it generates same set of Noether symmetries and conserve d quantities. So, this\nnew form of Lagrangian we are presenting here may be seriousl y interesting for\nthe physicists. Moreover, we also find the Lie algebras of Noe ther symmetries and\npoint out some interesting aspects of results related to Noe ther symmetries and first\nintegrals of damped harmonic oscillator which perhaps not r eported in the earlier\nstudies.\nKey words: Damped harmonic oscillator, Noether symmetries , conservation laws.2\nI. INTRODUCTION\nThe study of Noether symmetries and corresponding first integra ls of underlying dynam-\nical system has important mathematical significance as well as it has profound physical\nbackground. Discussion of Lie point transformations which leave th e action integral of the\ndamped harmonic oscillator (DHO) invariant has widened in recent yea rs. Among the enor-\nmous and useful results concerning DHO, the discussion of Noethe r symmetries and the\nrelevant conserved quantities have attracted attention of many researchers. If a Lie point\ntransformation leaves the action integral invariant, the classical Noether’s theorem [ 1] pro-\nvides a corresponding first integral. Another significance of these conserved quantities is\nthat they help us in establishing solutions of underlying systems. Mor eover, Lagrangian\nand Hamiltonian functions play essential role in deriving equations of m otion. Therefore,\nin order to study relation of symmetries and conservation laws using Noether’s theorem,\none needs the corresponding equation of motion that arise from th e action principle [ 2]. A\nnumber of studies have been reported to construct integral of m otions of DHO [ 3–5]. In the\ncurrent study, we intend to investigate the symmetry properties of DHO equation that is\nderivable from a variational principle. We shall see that the full eight parameter group that\nleaves the equation of motion invariant, only five of them leave the ac tion integral invariant\nand become Noether symmetries. Consequently, we obtain only five conserved quantities for\nDHO. It is commonly known that maximum number of conserved quant ities of a dynamical\nsystem often leads to find the solution of underlying equation. In th is paper, we find many\nresults on Noether symmetries and conservation laws of DHO and po int out some interest-\ning results related to Noether symmetries and first integrals to tha t already available in the\nliterature. We also present and discuss a new form of Lagragian (ma y be taken as standard\nLagragian) for DHO system which has not been reported so far. Th ough the new form of\nLagrangian is completely different yet it yields same results as derived from the Bateman\nLagrangian. So this new form of optimal Lagragian function may con ceal something inter-\nesting for the physicists.\nA brief outline of the present work is as follows: In Section II, we out line our treatment\non Noether symmetries and Noether’s theorem. In Section III, we give a classification of\n∗Electronic address: mufarooq@yahoo.com\n†Electronic address: safdar.camp@gmail.com3\nDHO with respect to Noether symmetries and construct correspo ndent conserved quantities\nfor three cases. We provide Lie algebras of Noether operators an d also discuss new form of\nLagrangian for DHO. Finally, we present a brief summary of our work in Section IV.\nII. PRELIMINARIES ON NOETHER SYMMETRIES AND FIRST INTEGRALS\nIn order to present results of the current study in a straightfor ward manner, we start with\nreviewing expressions on finding Noether symmetry generators an d associated conservation\nlaws for DHO. We shall use the salient features of this section to der ive important results\nin the next section.\nConsider the point type vector field\nZ=ξ(t,u)∂t+η(t,u)∂u, (1)\nthen its first prolongation can be expressed as\nZ[1]=Z+(ηt+ηuu′−ξtu′−ξuu′2)∂u′. (2)\nFurther assume that we have a first-order Lagrangian function L(t,u,u′) whose insertion in\nthe Euler-Lagrange equation\nd\ndt(∂L\n∂u′)−∂L\n∂u= 0, (3)\nyields the following equation of motion\nu′′=w(t,u,u′). (4)\nWe recall that the vector field Zis called a Noether point symmetry generator corresponding\nto the Lagrangian L(t,u,u′) if a gauge function B(t,u) exists such that\nZ[1](L)+D(ξ)L=D(B), (5)\nholds, where\nD=∂t+u′∂u+u′′∂u′+.... (6)\nThe significance of a Noether symmetry is that it provides the first in tegral as well as double\nreduction of order of underlying equation of motion.4\nNoether’s Theorem: Suppose that a Lie point symmetry generator Zsatisfies the the\ncondition given in Eq. (5), then the following expression leads to the c onserved quantity or\nconstant of motion of (4) related to Z.\nI=ξL+(η−u′ξ)∂L\n∂u′−B. (7)\nIII. NOETHER SYMMETRIES AND FIRST INTEGRALS OF DAMPED\nHARMONIC OSCILLATOR\nIn this section we wish to find Noether symmetries and then utilizing th em in the\nNoether’s theorem to determine first integrals of a famous dynamic al equation depicting\nmotion of DHO. We shall classify the problem with respect to Noether ’s symmetries and\nconstruct the associated conserved quantities for over-dampe d, under-damped and critically\ndamped cases and also determine their Lie algebras. So for this we pr oceed as:\nConsider the physical system of DHO whose dynamics is governed by the following second-\norder ordinary differential equation\nu′′+c\nmu′+k\nmu= 0. (8)\nThe well known time dependent Lagrangian of above Eq. (8) is given b y the so called\nBateman Lagrangian [ 6–9]\nL=1\n2exp/parenleftbiggct\nm/parenrightbigg\n(mu′2−ku2),. (9)\nFrom Eq. (9), it is evident that the time dependency of Lagrangian d oes not reveal the\nenergy conservation of the system as one observes in the usual h armonic oscillator case.\nAfter inserting the above Lagrangian in eq. (5) and solving the resu lting set of partial\ndifferential equations, we obtain the following Noether symmetries\nX1=∂\n∂t−cu\n2m∂\n∂u,\nX2= sin/parenleftBig√\n4km−c2t\nm/parenrightBig∂\n∂t+/parenleftBigu√\n4km−c2\n2mcos/parenleftBig√\n4km−c2t\nm/parenrightBig\n−cu\n2msin/parenleftBig√\n4km−c2t\nm/parenrightBig/parenrightBig∂\n∂u,\nX3= cos/parenleftBig√\n4km−c2t\nm/parenrightBig∂\n∂t−/parenleftBigu√\n4km−c2\n2msin/parenleftBig√\n4km−c2t\nm/parenrightBig\n+cu\n2mcos/parenleftBig√\n4km−c2t\nm/parenrightBig/parenrightBig∂\n∂u,\nX4= exp/parenleftbigg(−c−√\nc2−4km)t\n2m/parenrightbigg∂\n∂u,\nX5= exp/parenleftBig(−c+√\nc2−4km)t\n2m/parenrightBig∂\n∂u.\n(10)5\nNow by employing these operators in the Noether’s theorem gives ris e respectively the\nfollowing 5-first integrals\nI1=−exp/parenleftBigct\nm/parenrightBig\n(mu′2+cuu′+ku2),\nI2=−exp/parenleftBig\nct\nm/parenrightBig\n4m(2m2u′2+2mu(cu′−ku)+c2u2)sin/parenleftBig√\n4km−c2t\nm/parenrightBig\n+exp/parenleftBig\nct\nm/parenrightBig\n4mu√\n4km−c2(cu+2mu′)cos/parenleftBig√\n4km−c2t\nm/parenrightBig\n,\nI3=−exp/parenleftBig\nct\nm/parenrightBig\n4m(2m2u′2+2mu(cu′−ku)+c2u2)cos/parenleftBig√\n4km−c2t\nm/parenrightBig\n−exp/parenleftBig\nct\nm/parenrightBig\n4mu√\n4km−c2(cu+2mu′)sin/parenleftBig√\n4km−c2t\nm/parenrightBig\n,\nI4= exp/parenleftBigct\nm/parenrightBig\nexp/parenleftBig−(c+√\nc2−4km)t\n2m/parenrightBig\nmu′+1\n2(c+√\nc2−4km)uexp/parenleftBig(c−√\nc2−4km)t\n2m/parenrightBig\n,\nI5= exp/parenleftBigct\nm/parenrightBig\nexp/parenleftBig(−c+√\nc2−4km)t\n2m/parenrightBig\nmu′+1\n2(c−√\nc2−4km)uexp/parenleftBig(c+√\nc2−4km)t\n2m/parenrightBig\n.\n(11)\nHenceforth, we shall discuss 3-forms of DHO which include over-da mped, under-damped\nand critical damped. We shall use conditions for each form and retr ieve some known as well\nas some new results.\nCase 1. Over-damped oscillator:\nIf the damping force is much stronger than the restoring force of the system, then it is called\nthe over-damped harmonic oscillator. In this case 4 km < c2, and the set of generators given\nin Eq. (10) takes the form\nX1=∂\n∂t−cu\n2m∂\n∂u,\nX2=ιsinh/parenleftBig√\nc2−4kmt\nm/parenrightBig∂\n∂t+ι/parenleftBigu√\nc2−4km\n2mcosh/parenleftBig√\nc2−4kmt\nm/parenrightBig\n−cu\n2msinh/parenleftBig√\nc2−4kmt\nm/parenrightBig/parenrightBig∂\n∂u,\nX3= cosh/parenleftBig√\nc2−4kmt\nm/parenrightBig∂\n∂t−/parenleftBigu√\nc2−4km\n2msinh/parenleftBig√\nc2−4kmt\nm/parenrightBig\n+cu\n2mcosh/parenleftBig√\nc2−4kmt\nm/parenrightBig/parenrightBig∂\n∂u,\nX4= exp/parenleftBig(−c−√\nc2−4km)t\n2m/parenrightBig∂\n∂u,\nX5= exp/parenleftBig(−c+√\nc2−4km)t\n2m/parenrightBig∂\n∂u.\n(12)\nOn closely viewing these operators, we note that the Noether oper atorsX2andX3attain\ndifferent forms to those given in Eq. (10) while the other three gene ratorsX1,X4,X56\nand the associated conserved quantities I1,I4,I5remained unchanged. Hence, from the\nconstants of the motion I4andI5we obtain the following solution for the over-damped\noscillator\nu(x) =2m\nc2−4km/bracketleftBig\nI4e−c+√\nc2−4km\n2m−I5e−c−√\nc2−4km\n2m/bracketrightBig\n. (13)\nMoreover, by utilizing the Noether symmetries X2andX3along with the Lagrangian (9) in\nNoether’stheorem, wedeterminerespectively thefollowingtwonew correspondentconserved\nquantities\nI2=1\n16km2/parenleftBig\n−4kmexp/parenleftBigct\nm/parenrightBig\n(2m2u′2+2m(cuu′−ku2)+c2u2)sinh/parenleftBig√\nc2−4kmt\nm/parenrightBig\n+\nexp/parenleftBigct\nm/parenrightBig\n(cu2+2muu′)(c2√\nc2−4km−(c2−4km)3/2)cosh/parenleftBig√\nc2−4kmt\nm/parenrightBig/parenrightBig\n,\nI3=1\n16km2/parenleftBig\n−4kmexp/parenleftBigct\nm/parenrightBig\n(2m2u′2+2m(cuu′−ku2)+c2u2)cosh/parenleftBig√\nc2−4kmt\nm/parenrightBig\n+\nexp/parenleftBigct\nm/parenrightBig\n(cu2+2muu′)(c2√\nc2−4km−(c2−4km)3/2)sinh/parenleftBig√\nc2−4kmt\nm/parenrightBig/parenrightBig\n.\n(14)\nFurthermore, the Lie algebra of these operators is shown in the fo llowing table\nX1 X2 X3 X4 X5\nX10√\nc2−4km\nmX3√\nc2−4km\nmX2−√\nc2−4km\n2mX4√\nc2−4km\n2mX5\nX2−√\nc2−4km\nmX30−√\nc2−4km\nmX1−√\nc2−4km\n2mX5−√\nc2−4km\n2mX4\nX3−√\nc2−4km\nmX2√\nc2−4km\nmX10−√\nc2−4km\n2mX5√\nc2−4km\n2mX4\nX4√\nc2−4km\n2mX4√\nc2−4km\n2mX5√\nc2−4km\n2mX5 0 0\nX5−√\nc2−4km\n2mX5√\nc2−4km\n2mX4−√\nc2−4km\n2mX40 0\nTABLE I: Commutators Table\nThe list of generatorsprovided inEq. (12)except X1isentirely different fromthatpresented\nin [5]. Consequently, the associated first integrals are also non-identic al to those mentioned\nin [5]. Moreover, from above we see that if we put c= 0, the operator X1becomes time\ntranslational symmetry which yields integral I1asenergy conservation of the usual harmonic\noscillator. Furthermore, under the assumption that the system is free of damping term, all\nthe above symmetry generators and corresponding first integra ls become identical to what\ndetermine by Lutzky [ 10].\nCase 2. Under-damped Oscillator:7\nIf the restoring force is large compared to the damping force term , then the system is termed\nas the under-damped. This situation occurs when 4 km > c2, so the resulting Noether\nsymmetry generators are expressed as\nX1=∂\n∂t−cu\n2m∂\n∂u,\nX2= sin/parenleftBig√\n4km−c2t\nm/parenrightBig∂\n∂t+/parenleftBigu√\n4km−c2\n2mcos/parenleftBig√\n4km−c2t\nm/parenrightBig\n−cu\n2msin/parenleftBig√\n4km−c2t\nm/parenrightBig/parenrightBig∂\n∂u,\nX3= cos/parenleftBig√\n4km−c2t\nm/parenrightBig∂\n∂t−/parenleftBigu√\n4km−c2\n2msin/parenleftBig√\n4km−c2t\nm/parenrightBig\n+cu\n2mcos/parenleftBig√\n4km−c2t\nm/parenrightBig/parenrightBig∂\n∂u,\nX4= exp/parenleftBig−ct\n2m/parenrightBig/parenleftBig\ncos/parenleftBig√\n4km−c2t\n2m/parenrightBig\n−ιsin/parenleftBig√\n4km−c2t\n2m/parenrightBig/parenrightBig∂\n∂u,\nX5= exp/parenleftBig−ct\n2m/parenrightBig/parenleftBig\ncos/parenleftBig√\n4km−c2t\n2m/parenrightBig\n+ιsin/parenleftBig√\n4km−c2t\n2m/parenrightBig/parenrightBig∂\n∂u.\n(15)\nIn this case the operators X4andX5, are different from that shown in (10). Interestingly,\nthe splitting one of these two symmetries X4andX5into their real and imaginary parts\nresults in two new Noether symmetries as shown\nG4= exp/parenleftBig−ct\n2m/parenrightBig\ncos/parenleftBig√\n4km−c2t\n2m/parenrightBig∂\n∂u,\nG5= exp/parenleftBig−ct\n2m/parenrightBig\nsin/parenleftBig√\n4km−c2t\n2m/parenrightBig∂\n∂u.(16)\nNow insertion of the Lagrangian (9) along with Noether symmetries m entioned in Eq. (16),\nin Noether’s theorem (7) leads towards following pair of conserved q uantities\nI4= exp/parenleftBigct\n2m/parenrightBig/bracketleftBig\n(mu′+cu\n2)cos/parenleftBig√\n4km−c2t\n2m/parenrightBig\n+1\n2u√\n4km−c2sin/parenleftBig√\n4km−c2t\n2m/parenrightBig/bracketrightBig\n,\nI5= exp/parenleftBigct\n2m/parenrightBig/bracketleftBig\n(mu′+cu\n2)sin/parenleftBig√\n4km−c2t\n2m/parenrightBig\n−1\n2u√\n4km−c2cos/parenleftBig√\n4km−c2t\n2m/parenrightBig/bracketrightBig\n.(17)\nInterestingly, though forms of both pairs of generators X4,X5given in Eq. (15) and G4,G5\nin (16) are different, even then they provide the same first integra ls. Moreover, the Lie\nalgebra of the operators G4andG5is Abelian, i.e., [ G4,G5] = 0 and the elimination of u′\nfrom the Eq. (17) yields an analytical solution for the equation of un der-damped oscillator\nas\nu(t) =1√\n4km−c2exp/parenleftBig\n−ct\n2m/parenrightBig/bracketleftBig\nC1sin/parenleftBig√\n4km−c2t\n2m/parenrightBig\n−C2cos/parenleftBig√\n4km−c2t\n2m/parenrightBig/bracketrightBig\n.(18)\nThe commutation relations in view of operators (14) are displayed in T ABLE II8\nX1 X2 X3 X4 X5\nX10√\n4km−c2\nmX3−√\n4km−c2\nmX2−√\n4km−c2\n2mX5√\n4km−c2\n2mX4\nX2−√\n4km−c2\nmX30−√\n4km−c2\nmX1−√\n4km−c2\n2mX4√\n4km−c2\n2mX5\nX3√\n4km−c2\nmX2√\n4km−c2\nmX1 0√\n4km−c2\n2mX5√\n4km−c2\n2mX4\nX4√\n4km−c2\n2mX5√\n4km−c2\n2mX4−√\n4km−c2\n2mX50 0\nX5−√\n4km−c2\n2mX4−√\n4km−c2\n2mX5−√\n4km−c2\n2mX40 0\nTABLE II: Commutators Table\nCase 3. Critical-damped oscillator:\nIf the restoring force and damping force are comparable in effect t hen the dynamical system\nbehaves as critical damping oscillator. In this situation, we have 4 km−c2= 0 and the Eq.\n(5) provides the following 5-Noether symmetries\nX1= exp/parenleftBigg\n−/radicalbigg\nk\nmt/parenrightBigg\n∂\n∂u,\nX2=texp/parenleftBigg\n−/radicalbigg\nk\nmt/parenrightBigg\n∂\n∂u,\nX3=∂\n∂t−/radicalbigg\nk\nmu∂\n∂u,\nX4=t∂\n∂t−1\n2/parenleftBig\n2t/radicalbigg\nk\nm−1/parenrightBig\nu∂\n∂u,\nX5=1\n2t2∂\n∂t−1\n2/parenleftBigg\nt2/radicalbigg\nk\nm−t/parenrightBigg\nu∂\n∂u.(19)\nWe mention here that the forms of Noether symmetry generators in the critical damped\nharmonic case are entirely different from what we have seen for ove r-damped and under-\ndamped cases.\nThe Lie algebra of these operators is shown below\nNow considering these Noether symmetries and employing the classic al Noether’s theorem9\nX1X2X3X4X5\nX10 0 01\n2X11\n2X2\nX20 0 −X1−1\n2X20\nX30X10X3X4\nX4−1\n2X11\n2X2−X30X5\nX5−1\n2X20−X4−X50\nTABLE III: Commutators Table\nprovide respectively, the five first integrals\nI1= exp/parenleftBig/radicalbigg\nk\nmt/parenrightBig/parenleftBig/radicalbigg\nk\nmu+u′/parenrightBig\n,\nI2= exp/parenleftBig/radicalbigg\nk\nmt/parenrightBig/parenleftBig/radicalbigg\nk\nmtu+tu′−u/parenrightBig\n,\nI3=/parenleftBig\nexp/parenleftBig/radicalbigg\nk\nmt/parenrightBig/parenrightBig2/parenleftBigku2\n2m+/radicalbigg\nk\nmuu′+1\n2u′2/parenrightBig\n,\nI4=/parenleftBig\nexp/parenleftBig/radicalbigg\nk\nmt/parenrightBig/parenrightBig2/parenleftBigt\n2/parenleftBig/radicalbigg\nk\nmu+u′/parenrightBig2\n−/parenleftBig/radicalbigg\nk\nmtu−u\n2+tu′/parenrightBig/parenleftBig/radicalbigg\nk\nmu+u′/parenrightBig/parenrightBig\n,\nI5=u2\n4exp/parenleftBig\n2t/radicalbigg\nk\nm/parenrightBig\n+t\n4m(2mu(tu′−u)/radicalbigg\nk\nm+tmu′2+tku2−2muu′)/parenleftBig\nexp/parenleftBig/radicalbigg\nk\nmt/parenrightBig/parenrightBig2\n.\n(20)\nWe stress here that the forms of Noether symmetry generators X1,X2,X3,X4given in Eq.\n(20) are similar to that found in [ 5] and the first integrals as well. However, the form of\noperator X5is different and the resulting first integral is also new and fulfill the co ndition\nto becomes first integral of the critical damped harmonic oscillator . Furthermore, from the\ninvariant quantities I1,I2given in Eq. (20), we obtain following solution for the critical\ndamped harmonic oscillator\nu(x) =e−√\nk/mt/bracketleftBig\nI1t−I2/bracketrightBig\n. (21)\nInterestingly, we point out here that the insertion of the following L agragian in Euler-\nLagrange equation (3) also yields the same damped harmonic oscillato r equation (8)\nL=ec\nmt\n4m2/bracketleftBig\n2m2u′2+2cmuu′+(c2−2km)u2/bracketrightBig\n. (22)\nRemarkably, if we employ this new form of Lagragian for the DHO in Noe ther symmetry\ncondition and classical Noether’s theorem, we obtain same set of No ether symmetries and10\nconstants of motion as we have determined for over-, under- and critical-damped oscillators.\nSo this Lgargian can also be considered as the standard Lagragian. Though the new form\nof Lagrangian is completely different yet it yields same results as we de rived using the\nBateman Lagrangian. So this new form of optimal Lagragian functio n given in (22) may\nconceal something interesting for the physicists. We conclude her e that this new form of\nLagrangian may lead to entirely different quantum mechanical prope rties related to damped\nharmonic oscillator. Moreover, some interesting results related to DHO employing Eq. (22)\nwill be discussed in future work.\nIV. CONCLUSION\nIn this paper we have classified the harmonic oscillator problem with re spect to Noether\nsymmetries by constructing related conserved quantities. We hav e discussed three cases of\ndamped harmonic oscillator namely, over-damped, under-damped a nd critical-damped. We\nobserve that when we impose over-damped, under-damped and cr itical-damped conditions,\nthe forms of few generators given in Eq. (10) get changed and pro vide new first integrals\ncompared to that mentioned in Eq (11). Being an old and classical pro blem, several authors\nhave attempted to study the group theoretic properties of varia tional DHO equation. In the\ncurrent study we have also investigated the DHO equation of motion from group theoretic\nview point. In this paper we have determined some new results along w ith old one which\nhave already been reported in the literature. For instance, in the o ver-damped harmonic\noscillator case, the forms of all generators except X1are entirely different to what mentioned\nin [5] and consequently the corresponding conserved quantities are a lso different. Similarly,\nfor the under-damped harmonic oscillator case the real and imagina ry parts of operator X4\nprovide two Noether symmetry generators. These resulting oper ators form an Abelian alge-\nbra and yield two functionally independent first integrals. Furtherm ore, these two integrals\nsuffice to provide the exact solution of under-damped oscillator. In the critical case the four\nNoether symmetry generators are similar to that mentioned in [ 5], while the operator X5is\nnew and distinct from the list provided in [ 5] and the resulting first integral also fulfills the\ncondition to become the first integral of critical damped harmonic o scillator. In all three\ncases of damped harmonic oscillator the algebra of resulting Noethe r symmetry generators\nis closed.11\nWehavealsopresentedanewformofaLagrangianfunctionforthe DHOsystem. Weobserve\nthat forDHO onusing new formofLagrangianprovides same set of N oether symmetries and\nassociated conservation Laws as previously discussed for over-, under- and critical-damped\noscillators. So this Lagrangian can also be considered as the standa rd Lagrangian. Though\nthe new form of Lagrangian is completely different to that Bateman L agrangian but it\nyields same results as we derive employing Bateman Lagrangian. So th is new form of opti-\nmal Lagrangian function given in (22) may be usefully applied to explor e some mechanical\nproperties related to DHO.\n[1] E Noether, Invariante Variationsprobleme, Nachr. d. Ko nig. Gessellsch. d. Wiss. zu Gottingen,\nMath-phys. Klasse 235?257 (1918); English translation: M A Travel,Transport Theory and\nStatistical Physics ,1(3), 183-207 (1971)\n[2] H Goldstein, Classical mechanics (Narosa Publishing Ho use, New Delhi, India, 1998)\n[3] A. Choudhuri, S. Ghosh, B. Talukdar, Symmetries and cons ervation laws of the damped\nharmonic oscillator, Pramana J. Phys. ,4(70), 657-667 (2008)\n[4] A.J.Sinclair, J.E.Hurtado, C.B., P.Betsch, OnNoether ’sTheoremandtheVariousIntegrals\nof the Damped Linear Oscillator, J. Astronaut. Sci. ,60, 396-407 (2013)\n[5] J. M. Cervero, J. Villarroel, SL(3,R) realisations and t hedampedharmonic oscillator, J. Phys.\nA: Math. Gen. ,17, 1777-1786 (1984)\n[6] H. Bateman, On dissipative systems and related variatio nal principles, Phys. Rev. ,38(4),\n815-819 (1931)\n[7] C. C. Yan, Construction of Lagrangians and Hamiltonians from the equation of motion Am.\nJ. Phys.,46, 671 (1978)\n[8] I. K. Edwards, Quantization of inequivalent classical H amiltonians Am. J. Phys. ,47, 153\n(1979)\n[9] J. R. Ray, Lagrangians and systems they describe?how not to treat dissipation in quantum\nmechanics, Am. J. Phys. ,47, 626 (1979)\n[10] M Lutzky, Symmetry group and conserved quantities for t he harmonic oscillator, J. Phys. A:\nMath. Gen. ,2(11), 249-258 (1978)"    },    {        "title": "1104.0565v1.Plasmonic_abilities_of_gold_and_silver_spherical_nanoantennas_in_terms_of_size_dependent_multipolar_resonance_frequencies_and_plasmon_damping_rates.pdf",        "content": "1\nPlasmonic abilitiesofgoldandsilversphericalnanoantennas \nintermsofsizedependent multipolar resonance frequencies and\nplasmondampingrates.\n\nK.KolwasandA.Derkachova \nInstituteofPhysics,PolishAcademyofSciences,\nAl.Lotników32/46,02‐668Warsaw,Poland\nE‐mail:Krystyna.Kolwas@ifpan.edu.pl \n \n \nAbstract\n\nAbsorbing andemittingopticalproperties ofasphericalplasmonic nanoantenna aredescribed \nintermsofthesizedependent resonance frequencies anddampingratesofthemultipolar surface\nplasmons (SP).Weprovidetheplasmonsizecharacteristics forgoldandsilversphericalparticlesupto\nthelargesizeretardation regimewheretheplasmonradiativedampingissignificant. Weunderline the\nroleoftheradiationdampingincomparison withtheenergydissipation dampinginformation of\nreceivingandtransmitting properties ofaplasmonic particle.Thesizedependence ofboth:the\nmultipolar SPresonance frequencies andcorresponding dampingratescanbeaconvenient toolin\ntailoringthecharacteristics ofplasmonic nanoantennas forgivenapplication. Suchcharacteristics \nenabletocontrolanoperation frequency ofaplasmonic nanoantenna andtochangetheoperation \nrangefromthespectrally broadtospectrally narrowandviceversa.Itisalsopossibletoswitch\nbetweenparticlereceiving(enhanced absorption) andemitting(enhanced scattering) abilities.\nChanging thepolarization geometry ofobservation itispossibletoeffectively separatethedipoleand\nthequadrupole plasmonradiationfromallthenon‐plasmonic contributions tothescattered light.\n \nKeywords :surfaceplasmon(SP)resonance, plasmondampingrates,multipolar plasmon\nmodes,Mietheory,opticalproperties ofgoldandsilvernanospheres, noblemetalnanoparticles, \nreceivingandemittingnanoantennas, dispersion relation,nanophotonics, SERStechnique \n \n \n1.Introduction  \n \nTheuniqueproperties ofmetalnanostructures [1],[2],[3],[4],[5],[6]areknowntobedueto\nexcitations ofcollective oscillations ofelectronsurfacedensities: surfaceplasmons (SPs).SPsproperties \nformanattractive startingpointforemerging researchfieldsandpracticalapplications (Ref.[7],[8]and\nreferences therein).Excitation ofSPsatopticalfrequencies, guidingthemalongthemetal‐dielectric\ninterfaces, andtransferring thembackintofreelypropagating light,areprocesses ofgreatimportance \nformanipulation andtransmission oflightatthenanoscale. Anon‐diffraction limitedtransferoflight\nviaalinearchainofgoldnanospheres [7],[9],[10],[11],[12],[13]andnanowires [14]canbean\nexampleforsuchcontrol.Thegeometry ‐andsize‐dependent properties ofnanoparticles [1],[2],[3],\n[4],[5],[6]havepotentialapplications innanophotonics, biophotonics [15],microscopy, datastorage,2\nsensing[16],[17]biochemistry [18],[19],medicine[20]andspectroscopic measurements, e.g.inthe\nSurfaceEnhanced RamanScattering (SERS)technique [21],[22],[23],[24].  \nSizedependence oftheSPresonance frequency ofmetallicnanoparticles isessentialin\napplications. Thespectralresponseofmetallicnanoparticles canbecontrolled bychangingtheirsize\nandenvironment. However, forsomeapplications notonlythevalueoftheplasmonresonance \nfrequency isimportant, butalsodifferences inreceivingandtransmitting abilitiesofaplasmonic \nparticleofagivensizearecrucial.Plasmonic particleofpropersizecanbeusedasaneffective\ntransmitting nanoantenna andcanserveasatransitional structureabletoreceiveor/andtransmit\nelectromagnetic radiationatopticalfrequencies. \nTheexperimental dataconcerning resonance frequencies ofthedipoleSPforsphericalparticles\nofsomechosensizes[1],[25],[26],[27],[28],[29],comesfrommeasuring positionsofmaximain\nspectraoflightscattered or/andabsorbed bytheseparticles.Usually,experimental dataareinquite\ngoodagreement withthepredictions ofMietheory[30],[31],[32],[33].However, Mietheorydoesnot\ndealwiththenotionofSPs[29].Asfarasweknow,thereareonlyfewstudiesthattreatSPasan\nintrinsicpropertyofametallicsphereandthatproviderigorousdescription oftheSPparameters asa\ndirect,continuous functionofparticlesize[28],[29],[34].Itissometimes supposed [35],thatthe\nexistingtheoriesratherdonotallowdirectandaccuratecalculations ofthefrequencies oftheSP\nresonantmodesasafunctionofparticlesize. \nInthispaper,wedescribemultipolar resonance frequencies anddampingratesofthe\nmultipolar SPsforgoldandsilversphericalparticlesasafunctionoftheirradiusuptothelargesize\nretardation regimewheretheplasmonradiativedampingissignificant. Weusetheself‐consistent \nelectromagnetic approach basedontheanalysisoftheSPdispersion relations[28],[34],[36],[37].We\ntreatapossibility ofexcitation ofSPoscillations andtheirdampingasabasic,intrinsicpropertyofa\nconducting sphere.Weprovidesomeready‐to‐usesurfacemultipolar plasmonsizecharacteristics of\nthedipoleandhigherpolarityplasmonresonances ingoldandsilvernanospheres uptotheradiusof\nnm150coveringthemultipolar plasmonresonance frequencies intherangeofabout eV41forsilver\nand eV2.7 1.2forgoldnanoparticles. Opticalproperties ofthesemetalsaredescribed byrealistic,\nfrequency dependent refractive indicesofcorresponding puremetals[38].Themultipolar SPs\nresonance frequencies andtheSPsdampingratesaretreatedconsistently.  \nImplications ofdampingprocesses inSPapplications areextremely important ([4],[39],[40]\nandreferences therein).Inthispaper,weunderline theroleoftheplasmonradiationdampingin\ncomparison withtheinternalenergydissipation dampinginformation ofreceivingandtransmitting \nproperties ofaplasmonic particle.Inparticular wedemonstrate thattheenhancement factorof\nabsorbing andscattering particleabilitiesdependsontherelativecontribution oftheenergydissipative \nandradiativeprocesses tothetotaldampingofSPoscillations. Theseprocesses aresizedependent. \nWeshowthatonlyknowingthesizedependence ofboth:themultipolar SPresonance \nfrequency andcorresponding dampingrate,aneffectivedescription ofplasmonic properties is\npossible.SPsizecharacteristics westudyhereallownotonlytopredictafrequency ofSPresonance in\ngivenpolarityorder,butalsotoexploitaplasmonic particleasaneffectivereceiving(enhanced \nabsorption) oremitting(enhanced scattering) nanoantenna. ThesefeaturesarecrucialforSPs\napplications. \n\n\n3\n2.Modelling ofSPsinherentsizecharacteristics  \n \nCollective motionofsurfacefree‐electrons inametallicparticlecanbeexcitedbytheEMfield\nundertheresonance conditions whichisdefinedbyintrinsicproperties ofaplasmonic sphereduetoits\nsize,optical(conductive) properties ofametalanddielectricproperties ofthesphere’senvironment. \nTheSPresonance takesplacewhenthefrequency oftheincominglightapproaches atleastoneof\nthecharacteristic eigenfrequencies  )(Rl ,1,2,3,...=l ofasphereofradiusR.Thedynamicplasmon\nchargedensitydistribution, inducedbyEMfield,canbequitecomplicated. Withtheincreasing ofthe\nspheresurface,thesurfacechargedensitydistributions ofpolarityhigherthanthedipoleone(1=l )\ncomeintoplay[34].  \nThecollectively oscillating electrons ofSPsatcurvedsurfacemustemitEMenergythrough\nradiation. Radiative dampingis,then,theinherentpropertyofSPs.Itenablestheenhancement ofthe\nEMfieldscattered byasphereandisinseparably associated withSPoscillations atcharacteristic \nfrequencies ofmultipolar plasmonmodes.  \nTofindtheSPintrinsicsizecharacteristics weuseaself‐consistent rigorousEMapproach based\non[36],anddescribed inmoredetailse.g.in[28],[29],[34].Weconsidercontinuity relationsatthe\nsphericalboundary forthetangentcomponent ofthetransverse magnetic (TM)solutionofthe\nHelmholtz equationinsphericalcoordinates, whileTMsolutionsonlypossessnonzeronormaltothe\nsurfacecomponent oftheelectricfieldrE.Thiscomponent isabletocouplewiththechargedensities\nattheboundary. Resulting conditions definethedispersion relationsinsphericalcoordinates [28],[34]:\n  0,=)( )( )( )( )( )( R k R k R k R kin'\nl out l out in l out'\nl in    (1) \nwhicharefulfilledforthecomplexeigenfrequencies ofthefieldsl,1=l ,2,3,...atthe Rr=\ndistancefromaspherecentre.zlandzlareRiccati‐Besselsphericalfunctions, theprimemarker\n(')indicatesdifferentiation withrespecttothefunction’s argument and )( =\nin inck and\nout outck= . \nTherealisticmodelling oftheSPcharacteristics ispossibleifthefunctional dependences ofthe\ndielectricfunctiononfrequency inanalytical formforboththesphere )(inandsurrounding medium\n)(outareknown.Itassuresthecorrectcouplingofthemetaldispersion totheoverallfrequency \ndependence oftheplasmondispersion relationEq.(1).Weusedthemodifieddielectricfunctionofthe\nDrudeelectrongasmodelbasedonthekineticgastheory: ) /( =)(=)(2 2 ip D in  which\nquitewelldescribes theopticalproperties ofmanymetalswithinrelativelywidefrequency range[38].\nisthephenomenological parameter describing thecontribution oftheboundelectrons tothe\npolarizability [41]whichequalsto1onlyiftheconduction bandelectrons areresponsible forthe\nopticalproperties ofametal(e.g.sodium)[28].Forgoldandsilver,theinterband transitions are\nimportant fordefiningtheiropticalproperties. pisthebulkplasmonfrequency, isthe\nphenomenological relaxation constantofthebulkmaterial.Foraperfectfree‐electronbulkmetalwith\ninfiniteboundaries, electronrelaxation isduetoelectron‐electron,electron‐phonon,andelectron‐\ndefect(grainboundaries, impurities, anddislocations) scattering processes. resultsfromtheaverage\noftherespective collisionfrequencies ofelectrons andthusiscloselyrelatedtotheelectricalresistivity \nofthemetal[24].Thefunctions  )(inwitheffectiveparameters:  9.84= , eVp9.096= ,4\neV 0.072= forgold,and eV3.7= , eVp8.9= , eV 0.021= forsilversatisfactorily reproduce the\nexperimental valuesof )]([2nRe ( )]([2nIm )inthefrequency ranges: eV eV 5.0 0.8forgoldand\n4.2 0.8eV (4.0)eVforsilver(Figure1).Asillustrated inFigure1,theagreement ofthemodeland\nmeasured frequency dependence ofthedielectricfunctionisverygood,withexception ofImin()\ndependence forsilverathigherfrequencies ofthestudiedfrequency range.(above 4eV,itisforsome\nwavelengths outsidethevisiblerange(<310nm)). \n\n\n\nFigure1:Realandimaginary partsofthedielectricfunctionwitheffectiveparameters:  9.84= ,\neVp9.096= , eV 0.072= forgold,and eV3.7= , eVp8.9= , eV 0.021= forsilver.Squaresand\ncirclesarethecorresponding valuesofRe(n2)andofIm(n2)according to[38].\n\nInthecalculations theopticalproperties ofthedielectricmediumoutsidethesphereare\ndescribed by2=out out n ,withoutnischosenheretobe1or1.5.However, theformalism, ingeneral,\nalsoincludesthefrequency dependence oftheopticalproperties ofthesurroundings. Insuchacase,\nthefrequency dependence ofboththemetallicsphereandthesurrounding shouldbeconvoluted into\nthesizedependence ofmultipolar plasmoneigenfrequencies andthecorresponding plasmondamping\nrates.\nThedispersion relationEq.(1)issolvednumerically withrespecttocomplexvaluesof\n)( = Rl foreachl.Theydefinethesizedependent multipolar plasmonoscillation frequencies \n)( =)( R Re Rl'\nl  andthedampingrates )( =)( R Im Rl''\nl  ofplasmonic oscillations. Thesphereradius\nRistreatedasanoutside,independent parameter. Theresultsarepresented inFigure2forsilverand\ninFigure3forgoldspheres.Figuresaandcshowtheplasmonsizecharacteristics forfreespheresin\nvacuum( 1=outn ),whileFiguresbanddshowthesamesizedependencies forspheressuspended ina\ndielectricmediumof 1.5=outn .Thefirstsevenmultipolar plasmoncharacteristics arepresented (with\n1=lforadipoleplasmonand2=lforaquadrupole plasmon) uptotheradius nm150 .Themultipolar \nplasmonresonance frequencies arelowerforgoldthanforsilvernanospheres ofthesamesizeand\nwiththesamesurroundings. 5\n \n\nFigure2:(a),(b)Multipolar plasmonresonance \nfrequencies  )(R'\nland(c),(d)plasmondampingrates\n)(R''\nl(1,2,..7=l )forsilvernanoparticles infree\nspace( 1=outn )andinasuspension ( 1.5=outn )asa\nfunctionofsphereradius.Figure3:(a),(b)Multipolar plasmonresonance \nfrequencies  )(R'\nland(c),(d)plasmondampingrates\n)(R''\nl(1,2,..7=l )forgoldnanoparticles infree\nspace( 1=outn )andinasuspension ( 1.5=outn )asa\nfunctionofsphereradius.  \n\nAsournumerical resultsshow(Figures2and3forsilverandgoldand,[28],[34]foralkalies),the\nplasmonmultipolar resonance frequencies '\nl0,andthecorresponding dampingrates | |0,''\nl(Eq.(6))\ncanbeattributed onlytoasphereofaradiuslargerthanminimalradiusl Rmin,ineachplasmonmode.\nl R,ministhefastincreasing functionofl.\nTheextremevaluesof )(R'\nland )(R''\nlcanbefoundfromanapproximated andveryrough\nconsideration usingthepowerseriesexpansion ofthesphericalBasselandHankelfunctions: \n,...)52)(32(!25.0\n)32(!15.01!)!12()(22 2\n\n\n\n\nl lz\nlz\nlzzjl\nl  (2)\n\n\n\n\n\n...)23)(21(!25.0\n)21(!15.01!)!12()(22 2\n1l lz\nlz\nzli zhl l , (3)\nwhere(2l±1)!!≡1×3×5×…× (2l±1). Applyingthelimitofsmallarguments z (socalled\"quasistatic \napproximation\"), andkeepingonlythefirsttermsofthepowerseriesEqs.(2)and(3)(whatisnot\njustifiedforlargerl values),thedispersion relationEq.(1)isfulfilledunderthecondition below:6\n11\noutin\nll\n,   (4)\nForequationin()intheformgivenbyEq.(1),onegets:\n,2 1=1/2\n2 2\n0,\n\n\n\n\n\n\n\n\noutp '\nl\nll (5) \n.2=0,''\nl (6) \nNeglecting relaxation ( 0= )foraperfectfree‐electronmetal( 1= )and 1=outEq.(5)leads\ntothewell‐knownmultipolar plasmonfrequency valuesofametalspherewithinthe''quasistatic \napproximation'' [37],[42],[43]: l lout p'\nl 1)/( 1/ =0,  ,and,inparticular, tothedipole(1=l )\nplasmonfrequency  3/ =1=0, p'\nl ,thevalueknownasthegiantMieresonance frequency. \nPlasmonoscillations arealwaysdamped(seedrawingscanddinFigures2and3dueto\nradiationlossesandalltherelaxation processes includedintherelaxation rate.Withincreasing size,\nthedampingrates )(R''\nlinitiallyincreasestartingfromthevalues /2= ||0,''\nl''\nl ineachplasmon\nmode.Ifaccountsforelectronic relaxation processes leadingtodissipation (absorption) ofenergyin\nthemetal,''\nl0,accountsforthedecayoftheSPoscillations duetodephasing ofthecollective electron\nmotionwhichisoftenassumedtobestatistically ''memory destroying''. Thisquantityhasbeen\ninvestigated byvariousexperimental methods([4],[39],[40]andreferences therein).Forparticlesof\nradiilargerthanl R,min,thesizedependence ofthetotaldampingrates )(R''\nlisduetothesize\ndependence oftheradiativedampingrates )(Rrad\nl .Whereasradiativedumpingandenergy\ndissipation areuncorrelated processes, thetotaldampingrate )(R''\nlcanbewrittenasasum:  \n , 1,2,3...=, )( =)( l R Rdiss rad\nl''\nl    (7) \nwhere /2=dissforanymultipolar model.Sizedependence oftherate )(R''\nlisresultsfrom\n)(Rrad\nlsizedependence. Theinitialincreaseof )(R''\nlwithparticlesizeisfollowedbythe\nsubsequent decreaseforsufficiently largespheresasisillustrated inFigures2c,dand3c,dforthe\ndipolemodedamping)(1=R''\nlforsilverandgoldnanospheres, respectively.  \nTheopticalproperties ofthesurrounding mediumcanintroduce asignificant alteration inthe\nmultipolar SPresonance frequencies anddampingratesforaparticleofgivenradius.Inparticular, an\nincreaseoftheopticaldensityofthesurroundings introduces animportant ''redshift''oftheplasmon\nresonance frequencies  )(R'\nl ,asisillustrated inFigures2,3aandb. \n \n3.Properties ofplasmonic nanoantennas intermsofmultipolar SP\nresonance frequencies andplasmonoscillation dampingrates \n \nThederivedSPsizecharacteristics canbeusedtotunetheabsorbing oremittingproperties of\ntheplasmonic sphericalnanoantennas. Knowingboththemultipolar SPresonance frequencies  )(R'\nl7\nandcorresponding dampingrates |)(| R''\nl ,onecanpredictproperties ofplasmonic particlein\nresponsetotheexternalEMfieldandfindtheoptimalsizeofananosphere forachosenapplication. It\ncanbe,forexample,aneffectivecoherentcouplingwithsomeneighbouring particlesinametal\nnanoparticle array,anenhancement oftheEMfieldnearthespheresurface(usede.g.forSERS\nspectroscopic technique) oramodification ofthefarfieldscattering properties.  \nResonant excitation ofSPoscillations ispossiblewhenthefrequency oftheincominglightin\napproaches aneigenfrequency ofaplasmonic nanoantenna ofgivenradius R: )(= R'\nl in ,1,2,3,...=l .\n(Figures2,3aandb).Ifexcited,plasmonoscillations aredampedatthecorresponding rates |)(| R''\nl .\nThesphereacts,then,asareceivingor/andemittingmultipolar EMnanoantenna inmode(s)l.The\nnanoantennas performance canbeadjustedbychangingthespheresize.Notonlytheresonance \nfrequencies  )(R'\nl(with1=lforthedipoleantennamode),butalsoreceiving(absorbing) and\nscattering (transmitting) abilitiesofplasmonic nanoantennas aresizedependent asfollowsfromtheSP\ndampingrates |)(| R''\nlsizedependence (Figures2,3c,d).Itisconvenient tokeepexpressions for ,'\nl\nand''\nlin[eV]unitsinthediscussion, butforcompleteness wegivethecorresponding magnitudes in\n[1/s]onesontheleftverticalaxisofFigures2,3c,d. \n \n3.1.Receiving andtransmitting abilitiesofplasmonic nanoantenna asan\nintrinsicpropertyofaplasmonic particle  \n \nItiswidelyknown,thatopticalproperties ofaperfect‐metalsphericalparticle,whichismuch\nsmallerthanthewavelength ofincominglight,aremainlyduetothegiantabsorption ataresonance \nfrequency 3/p .Thatvaluecorresponds totheSPdipoleresonance frequency  )(,0R'\nlgivenbyEq.(5)\nfor1=l .However, aplasmonic particlecanactasanefficientabsorbing antennanotonlyinthe\ndipole,butalsoinlargerpolaritymodesforparticlesoflargersizes.Moreover, insomerangesof\nparameters Randl,thereceivingandradiativeabilitiesofaplasmonic nanoantenna maybe\ncomparable. Insuchacasetheplasmonic fractioninthetotalextinction spectrum wouldbemanifested \nbycomparable contributions oftheabsorption andscattering. \nScattering andabsorbing abilitiesofsphericalnanoantennas canbeconveniently described by\ntherelativecontribution oftheradiativeandnonradiative dampingratestothetotaldamping\nrate |)(| R''\nl .(Figures2,3,a,b).Plasmonic nanoparticles canactastheabsorbing antennas whenthe\ncontribution ofthenonradiative dampingrateinthetotaloneislarge.Thatispossibleforparticlesof\nsomeradiiR,suchthatdiss ''\nl l''\nlR   = |) (|0, ,min .Insuchcase,theSPresonance manifestation inthe\nabsorption spectrum isexpectedtobewellpronounced, withmaximaattheresonance \nfrequencies ),(R'\nl1,2,...=l .Italsocontributes considerably totheextinction spectrum regardless of\nhowlargetheparticleis.Ontheotherhand,ifcontribution oftheradiativedamping )(Rrad\nltothe\ntotalSPdampingrate |)(| R''\nlislargeincomparison withthenonradiative dampingdiss:\n),( |)(| R Rrad\nl''\nl aparticleofradiusRisabletoactasanefficientscattering antenna.Therangeof\nparticlesizeswhichfulfiltheseconditions resultsfromthesizecharacteristics of |)(| R''\nl presented in\nFigure2andFigure3c,dforsilverandgoldplasmonic nanospheres, respectively. Thebasic,intrinsic\nproperties ofaparticledescribed by )(R'\nlandthesizecharacteristics of |)(| R''\nlarereflectedina\nwaytheparticlerespondstothelightfield.8\n\n3.2.SPresonance manifestation inopticalsignals  \n \nLetusdiscusssomeexamples oflightelastically scattered bygoldnanoparticles ofchosensizes\ninenvironment with 1,5=outn .According toMietheory[30],[31],[32],[33],[44],theopticalresponse\ntotheincomingEMwavecanbedescribed insphericalcoordinates assquareofsumofthepartial\nwavesoftheTM(''electric'') andtheTE(''magnetic'') EMcontributions. Therefore, thefields\ncontributing tothelightintensityareinevitably composed ofbothTMandTEcomponents ofdifferent\npolarities l,whiletheelectromagnetic fieldsofSPdispersion relation(Eq.(1))aretransverse magnetic. \nConstructive anddestructive interference ofTMandTEcomponents influences themannerof\nplasmons manifestation inthescattered lightintensity(irradiance). Forgivenparticlesize,the\nmanifestation oftheSPresonance dependsontheobservable quantity,observation angleand\npolarization geometry. Figure4illustrates thespectraofthetotalscattering  )(scat ,extinction \n)(extandabsorption  )(abscross‐sectionsforgoldsphereswithradius nm nm,75 10 and nm130\n(Figure4a,bandc,respectively) calculated withinMietheoryforasuspension ( 1.5=outn )ofmono‐\nsizedgoldspheresasanexample.\n \n \n \nFigure4:Totalextinction, scattering andabsorption cross‐sectionsspectraforgoldnanospheres of\nradii:a)nm10 ,b)nm75andc) nm130insuspension ( 1.5=outn ).DottedverticallinesindicatetheSP\nmultipolar resonance frequencies  )(R'\nl ,1,2,3,..=l fortheseradii,according tothedatainFigure3b. \n 9\nFor nm R10=(Figure4a),thedipoleplasmonresonance at ) 10=(1= nm R'\nl becomesapparent\nmainlyintheenhanced absorptive properties ofananosphere (themaximum in )(absdependence, \ngraylineinFigure4a),duetothelargecontribution ofthenonradiative dampingtothetotaldamping\nrate: eV nm R''\nl 0,036=/2|) 10=(|    .Theon‐resonance enhancement factorisofthreeordersin\nmagnitude. Thepeakinthetotalscattering cross‐section )(scatisbyafactoroftensmallerdueto\nthepoorradiationdamping:  ) 10=(1= nm R''\nl  )(Rrad\nl (Figure3d).Therefore, theopticalproperties \n(including colours)ofsuchmetallicnanoparticles aredominated bytheresonantnonradiative plasmon\nenergydissipation atthedipoleSPeigenfrequency  ) 10=(1= nm R'\nl (seeFigure3b).Thementioned \nabovealsoappliestotheabsorptive properties ofatomsormolecules associated withtheresonant\n(dipole)transition totheshort‐livingexcitedstatesoflifetimesoftheorderofpicoseconds. Thefull\nwidthathalfmaximum (FWHM)ofanarrow,Lorentzian ‐likepeakinthetotalabsorption cross‐\nsection )(abs(Figure4a)iswelldescribed bytheplasmondampingrate ) 10=(1= nm R''\nl (seeFigure\n3d): \n . ) 10=( 2=) 10=(1=     nm R nm R''\nl  (8) \nThecorresponding plasmondampingtime sec 18,3= p . \nScattering effectsbecomeimportant fornanoparticles oflargersizes,asknownfromMiework\n[30].Figure4billustrates thiseffectforspherewithradius nm R75= .Themagnitude ofthetotal\nscattering cross‐section )(scatisbiggerthanthatofthetotalabsorption cross‐section )(absinthe\nfullopticalrange.Thescattering spectrum isbroadened andcomposed ofsomepartiallyoverlapping \nmaximawhichareblueshiftedinrespecttotheSPmultipolar resonance frequencies  )(R'\nl .Using\nagaintheSPdampingratecharacteristics, onecanpredictthemultipolar plasmonic contribution tothe\nscattering.  \nThecollectively oscillating electrons attheresonance frequency  eV nm R'\nl 1,6=) 75=( lose\ntheirenergymainlyduetoradiation, theeffectaccounted inthesizeaugmented radiationrate\ncontribution tothetotaldampingrate: ) 75=( 0,49=) 75=( nm R eV nm Rrad\nl''\nl    (Figure3d).Thatis\nwhythefractionofthedipoleplasmonic absorption isnegligibly small.Corresponding dipoledamping\nrate eV nm R''\nl 0,49=) 75=(1= (Figure3d)isdominated bytheradiationdamping: \neV nm Rdiss rad\nl 0,036= ) 75=(    .Therefore, iftheplasmonoscillations arecontinuously excitedby\nlightattheresonance frequency ofadipoleplasmon ) 75=( =1= nm R'\nl ,asphereisabletoefficiently \nscatterlightthroughtheplasmonic mechanism andtoworkasanexcellentemittingantennaatthat\nfrequency. Enhancement ofthescattered nearandfarfieldinthespacearoundtheparticleisthan\npossible.  \nThelargeristherate )(Rrad\nlincomparison withthenonradiative dampingrateforsuccessive \n1,2,3...=l ,thehigheristhepeakinthescattering spectrum  )(scatduetotheplasmonic \ncontribution, asdemonstrated inFigure4bandc.Butalso,thehigheristherate )(Rrad\nl ,thelargeris\nthespectralbandwidth oftheSPparticipation inthescattering spectrum aroundtheSPresonance \nfrequency  )(R'\nl .\nAsillustrated inFigure4andFigure6b,thelargeristhespectralbandwidth oftheSP\ncontribution ofpolarityltothespectrum (definedbytherate )(Rrad\nl ),thestrongeristheblueshiftof10\nthemaximum inrespecttotheSPresonance frequency  )(R'\nl .Asillustrated inFigure4b,c,allthe\nlarge‐bandwidth maximainthescattering spectrum sufferfromthiseffectandareblue‐shiftedin\nrespecttotheSPresonance frequencies  )(R'\nl .Thiseffectisduetotheimportant changeswith\nfrequency intheimaginary partoftheindexofrefraction  )(inn(Figure6c),thataffectsthefrequency \ndependence ofthenonplasmonic contribution (specularreflection) tothescattering. Butalso,the\nlargeristhespectralbandwidth ofthemaximum, themoreimportant isoftheinterference effects\n(constructive anddestructive) ofthepartialTMandTEwaves.\nEvenlargeplasmonic particles(suchasthoseofradius nm R75= orlarger(seeFigure4b,c))can\nactasreceivingantennas producing narrow,wellpronounced peaksintheabsorption (andextinction) \nspectra,ifonlytheparticipation oftheenergydissipation ratedissinthetotalplasmondampingrates\n)(R''\nlinmodeliscomparable withtheradiativerate )(Rrad\nl :diss rad\nlR)( (seeEq.(7)).Thewell\npronounced maximainthetotalabsorption cross‐sections )(abs(blacklinesinFigure4)nearthe\nplasmonfrequencies  ) 75=(2,3= nm R'\nl (Figure4b)and ) 130=(4= nm R'\nl (Figure4c)aresome\nexamples. Thewellpronounced absorption peaksduetoSPsarenotaffectedbyshiftingandsmearing\nouteffectssufferedbymaximainthescattered spectra,whiletheSPabsorption isduetothe\nplasmonic dependent energydissipation only.Therefore, interference ofEMwavesdoesnotaffectthe\npositionandwidthofmaximain )(abs(blacklinesinFigure4).Thecontribution ofthedipoleSP\nresonance tothemaximum inthescattering spectrum ofaspherewith nm R130=issmallerthanfora\nspherewith nm R75= ,asonecanexpectknowingthat ) 130=(1= nm R''\nl < ) 75=(1= nm R''\nl (see\nFigure3d).Suchrelationresultsfromthesmallercontribution oftheradiativedampinginthetotalSP\ndampingrate: ) 130=(1= nm Rrad\nl < ) 75=(1= nm Rrad\nl . \nAswasjustmentioned, thepartialsmearingoutandblueshiftofthemaximainthescattering \n(extinction) spectraisduetotheinterference ofEMfieldsscattered byasphere.Suchfieldsare\ninevitably composed ofboththeTMandTEcomponents ofdifferentpolarityl,whiletheresonantSP's\ncontributions areduetothenormaltothespheresurfacecomponent oftheelectricfieldrE,thatis\npresentintheTMpolarization modeonly.TheTEfractionofeddycurrentsisalsosizedependent and\nincreases monotonically withR,contributing alsoatsomeamounttotheblueshiftofthemaximain\nthetotalscattering cross‐section[29].\nOnecanconclude, thatiftheSPdampingrates )(1=R''\nlfor1,2,...=lofaplasmonactive\nsphereofradiusRaremainlyduetothenonradiative damping )(R''\nldiss(Figure3d),suchsphere\nisabletoefficiently absorblightthroughtheplasmonic mechanism nearthecorresponding SP\nresonance frequency  )(R'\nlinspectrally narrowbandwidth dissdefinedbytherelaxation rate.\nHowever, suchreceivingantennas areratherunabletocoupleelectromagnetically withanother\nparticlebytheplasmonic mechanism. \nIftheSPdampingrate )(1=R''\nlaremuchlargerthandiss,plasmonic particleofthesizeRisa\ngoodradiatingantennaatresonance frequency  )(R'\nlofgivenpolarityl.Radiative damping, whichis\ninherently coupledtoplasmonoscillations, affectsnotonlythespectralwidthoftheplasmonrelated\nmaximainthescattered lightintensity, butalsoenhances theplasmonrelatedcontributions tothe\nscattered spectrum. Forlargeradiationdampingrates )(1=R''\nl ,theenergyoftheincidentEMwave11\ncanbeeffectively redistributed intothescattered fieldenergyaroundthesphereduetotheplasmonic \nmechanism (asphereactsasaradiatingantenna).  \n \n3.3.Plasmonic scattering inorthogonal polarization geometries  \n \nAdistinctdipoleandquadrupole scattering abilitiesoftheplasmonic nanoantenna canbe\nconveniently demonstrated bylinearlypolarized lightilluminating theplasmonic sphereattheright\nangletothedirectionoftheincominglightbeam(Figure5). \n \n \n \nFigure5:Orthogonal observation geometry  \n \nSuch''clinical''orthogonal scattering geometry enablestoexposeacontribution ofboth:a\nsingledipole(1=l )andasinglequadrupole (2)=lSPtothescattering signalsandtoseparatethese\ncontributions spatiallybyobserving  ),(R Iand ),(|| R Iintensities (Figure4)[29],[45].WhereasI\nand||Iaretheintensities ofthepurelyscattered lightwhilethereisnoinfluence oftheinterference \neffectwiththeEMfieldoftheincominglightwavefromtheperpendicular direction. Therefore, such\nobservation enablestostudypurescattering abilitiesofaplasmonic sphere.Usingsuchan\nexperimental geometry andobserving thechangesin2)/2( R RI and2\n||)/2( R RIwithsize[46]we\nhavestudiedthesizedependence ofthedipoleandquadrupole plasmonresonances inourexperiment \nonspontaneously growingsodiumdroplets(uptothedropletradius nm R150= )inducedbylaserlight\n[28],[29].Studyingtheintensityscattered bytheparticleunitareaallowedustodiminisha\ncontribution ofthebackground duetoeddycurrentsandtoemphasize SPmanifestation. \nThechangesinthespectraof )(Iand )(||Iwithparticleradius,discussed intermsofthe\nintrinsicSPmultipolar sizecharacteristics (Section2),candirectlyillustratetheroleoftheSPradiative\ndampingrates )(R''\nlinformation oftheradiativeproperties oftheplasmonic nanoantennas.  \nTheFWHMscat\nl1=ofthe )(Ispectrum (Figure6a,b,solidline),andscatt\nl2=ofthe )(||I\nspectrum (Figure6a,b,dashedline)canbedescribed bytheLorentzian functionofFWHM\ncorresponding valuesoftheplasmondampingrates )(1=R''\nl and )(2=R''\nl : \n 1,2.=,)(2)( l R R''\nlscatt\nl   (9) \nThepeaksinthespectraforspheresoflargersizesbecomeblueshiftedinrespecttothe\nintrinsicvalue )(1,2=R'\nloftheSPdipoleandquadrupole SPresonances, asillustrated inFigure6aand\nc.Thelargerthespectralwidthscatt\nl ,thestrongertheblueshiftofthemaximum positionmax\nlin\nrespecttotheSPplasmonresonance position )(l'R ,asillustrated inFigure6b.ThemaximaduetoSP\nresonances inthescattering spectraaregoverned bytheradiativedampingrates , )( =diss\nl''\nlrad\nl R \nfordipole(1=l )andquadrupole (2=l )plasmonmode,correspondingly.  12\n \n \nFigure6:a)andb) )(Iand )(||Ispectrafor nm R30=and nm R75= ,respectively, withtheSP\ndipoleresonance at )( =1=R'\nl andthequadrupole resonance at )( =2=R'\nl ,correspondingly. Dashedlines\nrepresent LorentzprofilesofFWHMscat\nl1=Eq.(9).c)Thedependence oftherefraction coefficient (ReandIm\npart)ofthebulkgoldoninthestudiedfrequency range. 1.5=outn . \n \nOnecanconcludethatchangingthepolarization geometry itispossibletoseparatethe\nquadrupole plasmonradiationfromthedipoleandfromthelargermultipolarity plasmoncontribution \ntothescattered light.Knowingthemannerofdipoleandquadrupole plasmonmanifestation in )(I\nand )(||Ispectracorrespondingly, and )(1,2=R'\nland )(1,2=R''\nl sizedependencies, itispossibleto\nswitchbetweenthereceiving(enhanced absorption) andemitting(enhanced scattering) abilitiesofa\nplasmonic nanoantenna bychangingthedirectionofobservation (Figure5)ordirectionofpolarization \ninrespecttotheobservation plane.Itisalsopossibletocontrolabandwidth ofparticleextinction or\nscattering spectrachangingitfromspectrally broadtospectrally narrowandviceversa,bychanging\nthewavelength (seeFigure4borFigure6basanexample) oftheofilluminating light,ifaparticleis\nsufficiently large . \n \n\n13\n4.Conclusions  \n \nReceiving or/andemittingproperties ofthesphericalplasmonic nanoantenna aregoverned by\ntheinherentparameters: thesizedependent plasmonresonance frequencies  )(R'\nlanddampingrates\ndiss rad\nl''\nl R R  )( =)( ,whichprovideacomplete description ofparticleplasmonic properties. \nKnowledge ofthesizedependence ofboth:themultipolar SPresonance frequencies andcorresponding \ndampingratesmakespossibletoeffectively describeparticleplasmonic properties inresponsetothe\nEMfield.Takingadvantage oftheparticleintrinsicsizecharacteristics itispossibletopredictnotonly\ntheSPresonance frequencies  )(R'\nlbutalsothestrengthofSPresonances ingivenpolarityorder\n1,2,3,...=l afterproperadjustment oftherelativemagnitude oftheradiative )(Rrad\nlandenergy\ndissipative dissdampingrates.\nIftheSPmultipolar dampingrate )(1=R''\nlofaplasmonactivesphereofradiusRismainlydue\ntothenonradiative rate: )(R''\nl 2/diss(Figure3d),suchsphereisabletoefficiently absorb\nlightthroughtheplasmonic mechanism nearthecorresponding SPresonance frequency  )(R'\nl ,in\nspectrally narrowbandwidth diss,evenforlargersizes(andlargerl).However, suchreceivingantenna\nisunabletocoupleelectromagnetically withanotherparticlebytheplasmonic mechanism. IftheSP\ndampingrate )(1=R''\nlismuchlargerthandiss,aplasmonic particleofthecorresponding sizeRisa\ngoodradiatingantennaattheresonance frequency  )(R'\nlandisabletoenhancetheEMfieldinfar\nandnearfieldregion. \nUsingintrinsicplasmonsizecharacteristics  )(R'\nland )(R''\nlitispossibletocontrolan\noperation resonance frequency ofplasmonic nanoantenna andchangetheoperation rangefrom\nspectrally broadtospectrally narrowandviceversa(largeorsmall )(R''\nl ).Itisalsopossibletoswitch\nbetweentheparticlereceiving(enhanced absorption) andemitting(enhanced scattering) abilitiesin\nthequalitatively controlled manner.Changing thepolarization geometry (Section3.2)itispossibleto\neffectively separatethedipoleorquadrupole plasmonradiationfromallthenon‐plasmonic \ncontributions tothescattered light.\n \nAcknowledgment. 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,7,1935(1977).\n44.M.I.Mishchenko, L.D.Travis,andA.A.Lacis,Scattering, Absorption andEmissionofLightbySmall\nParticles ,Cambridge University Press,Cambridge, 2002.\n45.K.Kolwas,S.Demianiuk, andM.Kolwas,\"Dipoleandquadrupole plasmonresonances inlarge\nsodiumclustersobserved inscattered light,\"J.Chem.Phys. ,106,8436‐8441(1997).\n46.S.Demianiuk andK.Kolwas,\"Dynamics ofspontaneous growthoflight‐inducedsodiumdroplets\nfromthevapourphase,\"J.Phys.B,34,1651‐1671(2001).\n "    },    {        "title": "2108.04150v1.Effect_of_stepwise_adjustment_of_Damping_factor_upon_PageRank.pdf",        "content": "1\nE f f e c t  o f  s t e p w i s e  a d j u s t m e n t  o f\nD a m p i n g  f a c t o r  u p o n  P a g e R a n k\nS u b h a j i t  S a h u\n1\n,\nK i s h o r e  K o t h a p a l l i\n1\n,\nD i p  S a n k a r  B a n e r j e e\n2\n1\nI n t e r n a t i o n a l  I n s t i t u t e  o f  I n f o r m a t i o n  T e c h n o l o g y ,\nH y d e r a b a d\n2\nI n d i a n  I n s t i t u t e  o f  T e c h n o l o g y ,  J o d h p u r\nA b s t r a c t\n—\nT h e\ne f f e c t\no f\na d j u s t i n g\nd a m p i n g\nf a c t o r\nα\n,\nf r o m\na\ns m a l l\ni n i t i a l\nv a l u e\nα\n0\nt o\nt h e\nfi n a l\nd e s i r e d\nα\nf\nv a l u e\n,\nu p o n\nt h e n\ni t e r a t i o n s\nn e e d e d\nf o r\nP a g e R a n k\nc o m p u t a t i o n\ni s\no b s e r v e d .\nA d j u s t m e n t\no f\nt h e\nd a m p i n g\nf a c t o r\ni s\nd o n e\ni n\no n e\no r\nm o r e\ns t e p s\n.\nR e s u l t s\ns h o w\nn o  i m p r o v e m e n t\ni n  p e r f o r m a n c e  o v e r  a\nfi x e d  d a m p i n g\nf a c t o r\nb a s e d  P a g e R a n k .\nI n d e x  t e r m s\n—  P a g e R a n k  a l g o r i t h m ,  S t e p - w i s e  a d j u s t m e n t ,\nD a m p i n g  f a c t o r .\n1 .\nI n t r o d u c t i o n\nW e b\ng r a p h s\na r e\nb y\nd e f a u l t\nr e d u c i b l e ,\na n d\nt h u s\nt h e\nc o n v e r g e n c e\nr a t e\no f\nt h e\np o w e r - i t e r a t i o n\nm e t h o d\ni s\nt h e\nr a t e\na t\nw h i c h\nα\nk\n→\n0\n,\nw h e r e\nα\ni s\nt h e\nd a m p i n g\nf a c t o r\n,\na n d\nk\ni s\nt h e\ni t e r a t i o n\nc o u n t\n.\nA n\ne s t i m a t e\no f\nt h e\nn u m b e r\no f\ni t e r a t i o n s\nn e e d e d\nt o\nc o n v e r g e\nt o\na\nt o l e r a n c e\nτ\ni s\nl o g\n1 0\nτ\n/\nl o g\n1 0\nα\n[ 1 ]\n.\nF o r\nτ\n=\n1 0\n- 6\na n d\nα\n=\n0 . 8 5\n,\ni t\nc a n\nt a k e\nr o u g h l y\n8 5\ni t e r a t i o n s\nt o\nc o n v e r g e .\nF o r\nα\n=\n0 . 9 5\n,\na n d\nα\n=\n0 . 7 5\n,\nw i t h\nt h e\ns a m e\nt o l e r a n c e\nτ\n=\n1 0\n- 6\n,\ni t\nt a k e s\nr o u g h l y\n2 6 9\na n d\n4 8\ni t e r a t i o n s\nr e s p e c t i v e l y .\nF o r\nτ\n=\n1 0\n- 9\n,\na n d\nτ\n=\n1 0\n- 3\n,\nw i t h\nt h e\ns a m e\nd a m p i n g\nf a c t o r\nα\n=\n0 . 8 5\n,\ni t\nt a k e s\nr o u g h l y\n1 2 8\na n d\n4 3\ni t e r a t i o n s\nr e s p e c t i v e l y .\nT h u s ,\na d j u s t i n g\nt h e\nd a m p i n g\nf a c t o r\no r\nt h e\nt o l e r a n c e\np a r a m e t e r s\no f\nt h e\nP a g e R a n k\na l g o r i t h m\nc a n\nh a v e  a  s i g n i fi c a n t  e f f e c t  o n  t h e\nc o n v e r g e n c e  r a t e\n.\n2 .\nM e t h o d\nT h e\ni d e a\nb e h i n d\nt h i s\ne x p e r i m e n t\nw a s\nt o\na d j u s t\nt h e\nd a m p i n g\nf a c t o r\nα\ni n\ns t e p s\n,\nt o\ns e e\ni f\ni t\nm i g h t\nh e l p\nr e d u c e\nP a g e R a n k\nc o m p u t a t i o n\nt i m e .\nT h e\nP a g e R a n k\nc o m p u t a t i o n\nfi r s t\ns t a r t s\nw i t h\na\ns m a l l\ni n i t i a l\nd a m p i n g\nf a c t o r\nα\n=\nα\n0\n.\nA f t e r\nt h e2\nr a n k s\nh a v e\nc o n v e r g e d\n,\nt h e\nd a m p i n g\nf a c t o r\nα\ni s\nu p d a t e d\nt o\nt h e\nn e x t\nd a m p i n g\nf a c t o r\ns t e p ,\ns a y\nα\n1\na n d\nP a g e R a n k\nc o m p u t a t i o n\ni s\nc o n t i n u e d\na g a i n .\nT h i s\ni s\nd o n e\nu n t i l\nt h e\nfi n a l\nd e s i r e d\nv a l u e\no f\nα\nf\ni s\nr e a c h e d .\nF o r\ne x a m p l e ,\nt h e\nc o m p u t a t i o n\ns t a r t s\ni n i t i a l l y\nw i t h\nα\n=\nα\n0\n=\n0 . 5\n,\nl e t s\nr a n k s\nc o n v e r g e\nq u i c k l y ,\na n d\nt h e n\ns w i t c h e s\nt o\nα\n=\nα\nf\n=\n0 . 8 5\na n d\nc o n t i n u e s\nP a g e R a n k\nc o m p u t a t i o n\nu n t i l\ni t\nc o n v e r g e s .\nT h i s\ns i n g l e - s t e p\nc h a n g e\ni s\na t t e m p t e d\nw i t h\nt h e\ni n i t i a l\n( f a s t\nc o n v e r g e )\nd a m p i n g\nf a c t o r\nα\n0\nf r o m\n0 . 1\nt o\n0 . 8\n.\nS i m i l a r\nt o\nt h i s ,\nt w o - s t e p ,\nt h r e e - s t e p ,\na n d\nf o u r - s t e p\nc h a n g e s\na r e\na l s o\na t t e m p t e d .\nW i t h\na\nt w o - s t e p\na p p r o a c h ,\na\nm i d p o i n t\nb e t w e e n\nt h e\ni n i t i a l\nd a m p i n g\nv a l u e\nα\n0\na n d\nα\nf\n=\n0 . 8 5\ni s\ns e l e c t e d\na s\nw e l l\nf o r\nt h e\ns e c o n d\ns e t\no f\ni t e r a t i o n s .\nS i m i l a r l y ,\nt h r e e - s t e p\na n d\nf o u r - s t e p\na p p r o a c h e s  u s e\nt w o\na n d\nt h r e e\nm i d p o i n t s  r e s p e c t i v e l y .\n3 .\nE x p e r i m e n t a l  s e t u p\nA\ns m a l l\ns a m p l e\ng r a p h\ni s\nu s e d\ni n\nt h i s\ne x p e r i m e n t ,\nw h i c h\ni s\ns t o r e d\ni n\nt h e\nM a t r i x M a r k e t\n( . m t x )\nfi l e\nf o r m a t .\nT h e\ne x p e r i m e n t\ni s\ni m p l e m e n t e d\ni n\nN o d e . j s ,\na n d\ne x e c u t e d\no n\na\np e r s o n a l\nl a p t o p .\nO n l y\nt h e\ni t e r a t i o n\nc o u n t\no f\ne a c h\nt e s t\nc a s e\ni s\nm e a s u r e d .\nT h e\nt o l e r a n c e\nτ\n=\n1 0\n- 5\ni s\nu s e d\nf o r\na l l\nt e s t\nc a s e s .\nS t a t i s t i c s\no f\ne a c h\nt e s t\nc a s e\ni s\np r i n t e d\nt o\ns t a n d a r d\no u t p u t\n( s t d o u t )\n,\na n d\nr e d i r e c t e d\nt o\na\nl o g\nfi l e\n,\nw h i c h\ni s\nt h e n\np r o c e s s e d\nw i t h\na\ns c r i p t\nt o\ng e n e r a t e\na\nC S V\nfi l e\n,\nw i t h\ne a c h\nr o w\nr e p r e s e n t i n g\nt h e\nd e t a i l s\no f\na\ns i n g l e\nt e s t\nc a s e\n.\nT h i s\nC S V\nfi l e\ni s\ni m p o r t e d\ni n t o\nG o o g l e\nS h e e t s\n,\na n d\nn e c e s s a r y\nt a b l e s\na r e\ns e t\nu p\nw i t h\nt h e\nh e l p\no f\nt h e\nF I L T E R\nf u n c t i o n  t o  c r e a t e  t h e\nc h a r t s\n.\n4 .\nR e s u l t s\nF r o m\nt h e\nr e s u l t s ,\na s\ns h o w n\ni n\nfi g u r e\n4 . 1 ,\ni t\ni s\nc l e a r\nt h a t\nm o d i f y i n g\nt h e\nd a m p i n g\nf a c t o r\nα\ni n\ns t e p s\ni s\nn o t\na\ng o o d\ni d e a\n.\nT h e\ns t a n d a r d\nfi x e d\nd a m p i n g\nf a c t o r\nP a g e R a n k\n,\nw i t h\nα\n=\n0 . 8 5\n,\nc o n v e r g e s\ni n\n3 5\ni t e r a t i o n s\n.\nU s i n g\na\ns i n g l e - s t e p\na p p r o a c h\ni n c r e a s e s\nt h e\nt o t a l\nn u m b e r\no f\ni t e r a t i o n s\nr e q u i r e d ,\nb y\na t\nl e a s t\n4\ni t e r a t i o n s\n( w i t h\na n\ni n i t i a l\nd a m p i n g\nf a c t o r\nα\n0\n=\n0 . 1\n) .\nI n c r e a s i n g\nα\n0\nf u r t h e r\ni n c r e a s e s\nt h e\nt o t a l\nn u m b e r\no f\ni t e r a t i o n s\nn e e d e d\nf o r\nc o m p u t a t i o n .\nS w i t c h i n g\nt o\na\nm u l t i - s t e p\na p p r o a c h\na l s o\ni n c r e a s e s\nt h e\nn u m b e r\no f\ni t e r a t i o n s\nn e e d e d\nf o r\nc o n v e r g e n c e .\nT h e\nt w o - s t e p ,\nt h r e e - s t e p ,\na n d\nf o u r - s t e p\na p p r o a c h e s3\nr e q u i r e\na\nt o t a l\no f\na t l e a s t\n4 9\n,\n6 0\n,\na n d\n7 1\ni t e r a t i o n s\nr e s p e c t i v e l y .\nA g a i n ,\ni n c r e a s i n g\nα\n0\nc o n t i n u e s\nt o\ni n c r e a s e\nt h e\nt o t a l\nn u m b e r\no f\ni t e r a t i o n s\nn e e d e d\nf o r\nc o m p u t a t i o n .\nA\np o s s i b l e\ne x p l a n a t i o n\nf o r\nt h i s\ne f f e c t\ni s\nt h a t\nt h e\nr a n k s\nf o r\nd i f f e r e n t\nv a l u e s\no f\nt h e\nd a m p i n g\nf a c t o r\nα\na r e\ns i g n i fi c a n t l y\nd i f f e r e n t\n,\na n d\ns w i t c h i n g\nt o\na\nd i f f e r e n t\nd a m p i n g\nf a c t o r\nα\na f t e r\ne a c h\ns t e p\nm o s t l y\nl e a d s\nt o\nr e c o m p u t a t i o n\n.\nF i g u r e\n4 . 1 :\nI t e r a t i o n s\nr e q u i r e d\nf o r\nP a g e R a n k\nc o m p u t a t i o n ,\nw h e n\nd a m p i n g\nf a c t o r\nα\ni s\na d j u s t e d\ni n\n1 - 4\ns t e p s\n,\ns t a r t i n g\nw i t h\na n\ni n i t i a l\ns m a l l\nd a m p i n g\nf a c t o r\nα\n0\n( d a m p i n g _ s t a r t )\n.\n0 - s t e p\ni s  t h e\nfi x e d  d a m p i n g  f a c t o r\nP a g e R a n k\n,  w i t h\nα\n =  0 . 8 5\n.\n5 .\nC o n c l u s i o n\nA d j u s t i n g\nt h e\nd a m p i n g\nf a c t o r\nα\ni n\ns t e p s\ni s\nn o t\na\ng o o d\ni d e a ,\nm o s t\nl i k e l y\nb e c a u s e\nr a n k s\no b t a i n e d\nf o r\ne v e n\nn e a r b y\nd a m p i n g\nf a c t o r s\na r e\ns i g n i fi c a n t l y\nd i f f e r e n t .\nF i x e d\nd a m p i n g\nf a c t o r\nP a g e R a n k\nc o n t i n u e s\nt o\nb e\nt h e\nb e s t\na p p r o a c h .\nT h e\nl i n k s\nt o\ns o u r c e\nc o d e ,\na l o n g\nw i t h\nd a t a\ns h e e t s\na n d\nc h a r t s ,\nf o r\na d j u s t i n g\nd a m p i n g  f a c t o r  i n  s t e p s\n[ 2 ]\ni s  i n c l u d e d  i n  r e f e r e n c e s .\n4\nR e f e r e n c e s\n[ 1 ]\nA .  L a n g v i l l e  a n d  C .  M e y e r ,  “ D e e p e r  I n s i d e  P a g e R a n k , ”\nI n t e r n e t  M a t h .\n,\nv o l .  1 ,  n o .  3 ,  p p .  3 3 5 – 3 8 0 ,  J a n .  2 0 0 4 ,  d o i :\n1 0 . 1 0 8 0 / 1 5 4 2 7 9 5 1 . 2 0 0 4 . 1 0 1 2 9 0 9 1 .\n[ 2 ]\nS .  S a h u ,  “ p u z z l e f / p a g e r a n k - a d j u s t - d a m p i n g - f a c t o r - s t e p w i s e . j s :\nE x p e r i m e n t i n g  P a g e R a n k  i m p r o v e m e n t  b y  a d j u s t i n g  d a m p i n g\nf a c t o r  (\nα\n)\nb e t w e e n  i t e r a t i o n s . ”\nh t t p s : / / g i t h u b . c o m / p u z z l e f / p a g e r a n k - a d j u s t - d a m p i n g - f a c t o r - s t e p w i s e . j s\n( a c c e s s e d  A u g .  0 6 ,  2 0 2 1 ) ."    },    {        "title": "1907.02176v1.Testing_Lorentz_violation_by_the_comparison_of_atomic_clocks.pdf",        "content": "arXiv:1907.02176v1  [gr-qc]  4 Jul 2019Proceedings of the Eighth Meeting on CPT and Lorentz Symmetr y (CPT’19), Indiana University, Bloomington, May 12–16, 2019\n1\nTesting Lorentz violation by the comparison of atomic clock s\nXiao-yu Lu, Yu-Jie Tan, and Cheng-Gang Shao\nMOE Key Laboratory of Fundamental Physical Quantities Meas urements &Hubei Key\nLaboratory of Gravitation and Quantum Physics,\nPGMF and School of Physics, Huazhong University of Science a nd Technology,\nWuhan 430074, Peoples Republic of China\nA more complete theoretical model of testing Lorentz violat ion by the compar-\nison of atomic clocks is developed in the Robertson-Mansour i-Sexl kinematic\nframework. As this frame postulates the deviation of the coo rdinate transfor-\nmation from the Lorentz transformation, from the viewpoint of the transforma-\ntion violations on time and space, the frequency shift effect in the atomic clock\ncomparison can be explained as two parts: time-delay effect αv2\nc2and structure\neffect−β+2δ\n3v2\nc2. Standard model extension is a widely used dynamic frame\nto characterize the Lorentz violation, in which a space-ori entation dependence\nviolating background field is added as the essential reason f or the Lorentz viola-\ntion effect. Compared with the RMS frame which only indicates the kinematic\nproperties with the coordinate transformation, this dynam ic frame provides a\nmore complete and clear description for the possible Lorent z violation effect.\n1. Introduction\nLorentz invariance (LI) is the fundamental symmetry of spacetim e, which\npostulates the experimental result independent on the motion sta te of the\napparatus1. As LI is at the foundation of both the Standard model of\nparticles physics and general relativity, its related research is an im portant\nsubject in the physics science. Here, we studied the LI effect in Rob ertson-\nMansouri-Sexl (RMS) framework,2,3and made a simple comparison with\nthat in Standard-Model Extension (SME) framework.4,5RMS framework\nconsiders the speed of light anisotropic, and also postulates there is a pre-\nferred universalframe in which light propagatesconventionallyasm easured\nusing a set of rods and clocks. For these RMS rods and clocks, they are\nisotropic and the photon is anisotropic, while for the SME rods and clo cks,\nthe case is opposite. Since LI violation results in the difference of tra nsition\nfrequency, the atomic clock comparison is a good means to test this violat-\ning effect, and we focus on analyzing this violation between compariso ns.Proceedings of the Eighth Meeting on CPT and Lorentz Symmetr y (CPT’19), Indiana University, Bloomington, May 12–16, 2019\n2\n2. Lorentz violation of atomic clock comparison\nThe violation of LI is described in RMS kinematic framework as the defo r-\nmation of Lorentz transformation, and it postulates the existenc e of a pre-\nferred frame Σ, that is, cosmological microwave background (CMB ) frame.\nIf the laboratory reference frame Shas the velocity vwith respect to Σ\nframe, the transformation between these two frames can be writ ten as2,3\nt=aT+/vector ε·/vector x, x=b(X−vT), y=dY, z=dZ (1)\nwitha(v) = 1 + ( α−1\n2)v2\nc2,b(v) = 1 + ( β+1\n2)v2\nc2andd(v) = 1 + δv2\nc2,6\nwhich returns to the Lorentz transformation with α=β=δ= 0. For the\ncomparison of clock frequencies, the violation of LI in the Sframe can be\ndetected through measuring the anisotropic of light speed. Analyz ing light-\nclock and atom-clock comparisons in Fig.1(a) and 1(b),7,8the frequency\nshift signal of clock-comparison experiment contains the time-dela y and\nstructure effects9\n∆LV=αv2\nc2−β+2δ\n3v2\nc2, (2)\nwhere the first part means time delay and the other is structure eff ect.\nABC\n0L\nLaser\nD12\n3\n69\nLightClockIIRotation\n12\n3\n69\nLightClockI- -\nr r\nAtomicclockIAtomicclockII\n(a) (b)0L\nFig. 1. Two kinds of clock-comparison experiments. (a) The c omparison of light clocks:\nit is similar to a Michelson interferometer with each arm len gthL0, where each interfer-\nence arm can be considered as a light clock. (b) The compariso n of atomic clocks: two\natomic clocks are located in different places.\nTheSMEframeworkprovidesageneraltheoreticalframeworkfo rstudy-\ning the violation of LI, such as the violation in photon, matter, gravit y sec-\ntors and so on. Compared with RMS framework, SME framework pro videsProceedings of the Eighth Meeting on CPT and Lorentz Symmetr y (CPT’19), Indiana University, Bloomington, May 12–16, 2019\n3\na vast parameter space. In this framework, a LI violating backgro und field\nis postulated, and different coordinate systems are linked by the Lo rentz\ntransformation. Therefore, for the atomic clock comparison, th ere are no\nLI violation in photon sector, and the main violating effect is embodied in\nthe matter sector.10RMS formalism can be regarded as a special limit of\nSME LI violation formalism.\n3. conclusion\nBased on the theoretical analysis of testing LI violation by light-cloc k com-\nparison in RMS frame, we studied the test by atomic clock comparison , and\nthe result indicates LI violation effect includes the time-delay and str ucture\neffects. In addition, we also make a simple explanation for the differen t in-\ndications of LI violation for atomic clock comparison in the RMS and SME\nframes. For RMS framework, the violating effect arises from the de forma-\ntion of Lorentz transformation, while for the SME frame, it results from\nthe existed violating background field.\n4. Acknowledgments\nThis work is supported by the Post-doctoral Science Foundation o f China\n(Grant Nos. 2017M620308 and 2018T110750).\nReferences\n1. O.Bertolami andJ. Paramos, The experimental status of Special and General\nRelativity , to appear in Handbook of Spacetime, Springer, Berlin (2013 );\narXiv:1212.2177 [gr-qc].\n2. H.P. Robertson, Rev. Mod. Phys. 21, 378 (1949).\n3. R. Mansouri and R.U. Sexl, Gen. Relativ. Gravit. 8, 497 (1977).\n4. D. Colladay and V.A. Kosteleck´ y, Phys. Rev. D 55, 6760 (1997); 58, 116002\n(1998).\n5. V.A. Kosteleck´ y, Phys. Rev. D 69, 105009 (2004).\n6. Y.Z. Zhang, Gen. Relativ. Gravit. 27, 5 (1995).\n7. C. Eisele, A.Y.Nevsky, andS.Schiller, Phys. Rev. Lett. 103, 090401 (2009).\n8. P. Delva et al., Phys. Rev. Lett. 118, 221102 (2017).\n9. X.Y. Lu, Y.J. Wang, Y.J. Tan, and C.G. Shao, Phys. Rev. D 98, 096022\n(2018).\n10. V.A. Kosteleck´ y and M. Mewes, Phys. Rev. D 80, 015020 (2009)."    },    {        "title": "2303.04920v1.Material_Geometry_Interplay_in_Damping_of_Biomimetic_Scale_Beams.pdf",        "content": "Material-Geometry Interplay in Damping of Biomimetic Scale Beams\nH. Ebrahimi,1M. Krsmanovic,1H. Ali,2and R. Ghosh1\n1)Department of Mechanical and Aerospace Engineering, University of Central Florida, 4000 Central Florida Blvd, Orlando,\nFL 32816\n2)Stress Engineering Services, Inc., 7030 Stress Engineering Way, Mason, OH 45040\n(*Electronic mail: ranajay.ghosh@ucf.edu)\n(Dated: 10 March 2023)\nBiomimetic scale-covered substrates are architected meta-structures exhibiting fascinating emergent nonlinearities via\nthe geometry of collective scales contacts. In spite of much progress in understanding their elastic nonlinearity, their\ndissipative behavior arising from scales sliding is relatively uninvestigated in the dynamic regime. Recently discov-\nered is the phenomena of viscous emergence, where dry Coulomb friction between scales can lead to apparent viscous\ndamping behavior of the overall multi-material substrate. In contrast to this structural dissipation, material dissipation\ncommon in many polymers has never been considered, especially synergestically with geometrical factors. This is ad-\ndressed here for the first time, where material visco-elasticity is introduced via a simple Kelvin-V oigt model for brevity\nand clarity. The results contrast the two damping sources in these architectured systems: material viscoelasticity, and\ngeometrical frictional scales contact. It is discovered that although topically similar in effective damping, viscoel-\nsatic damping follows a different damping envelope than dry friction, including starkly different effects on damping\nsymmetry and specific damping capacity.\nBiologically inspired scale-covered substrates are under\nsustained scrutiny as a structural platform with unique prop-\nerty combinations akin to metamaterials using the geometry\nand kinematics of scales sliding Fig. 11–5. Physically, such\nmulti-material systems comprise of a soft deformable sub-\nstrate with protruding stiff plates acting as scales, Fig. 1 (a)6,7.\nWhen the substrate deforms, the stiff scales eventually contact\nas a collective giving rise to a fascinating spectrum of me-\nchanical and optical properties8–10. Many of these behaviors\nemerge from the collective sliding motion of the scales on the\nsubstrate. Such kinematic origins of nonlinear behavior mean\nthat their fundamental source of nonlinearity lie in the distri-\nbution and orientation of the stiff scales11–13. This results in a\ngeometry-dictated landscape of nonlinear elasticity and frac-\nture.\nIn spite of deep scrutiny of the elastic and fracture charac-\nteristics, interest in the dissipative behavior of these substrates\nhas been more recent14–17. In the static regime where dry fric-\ntion was postulated between scales, friction was found to play\na dual role in adding stiffness to the substrate as well as limit-\ning the range of motion by introducing an additional locking\nphase14,16,18. Such conflicting roles prompted further exten-\nsion of friction onto the dynamic regime. Here, a new type of\ndissipative behavior was discovered - emergent viscosity. In\nother words, even when dry friction was assumed between the\nscales, the overall damped oscillation of the substrates indi-\ncated viscous-like exponential damping15.\nIn these works, the role of substrate polymer viscoelastic-\nity was not investigated15. Thus, the only source of damping\nwas from the scales sliding. However, many polymers ex-\nhibit viscoelastic behavior19,20, and its interplay with scales\nsliding dissipation remains unknown. Specifically, the effect\nof damping that emerges from the synergistic combination\nof the viscoelasticity of the substrate and the dry interfacial\nscale friction has not been revealed. In this letter, we include\nthe viscoelasticity of the substrate for the first time to under-\nstand its role of damping during the oscillation of an Euler-Bernoulli substrate. For this study, we chose a simple Kelvin-\nV oigt model to represent the viscoelasticity21,22, and assume\nCoulomb friction between the scales. We investigate both free\nvibration and forced vibration.\nViscoelasticity of real polymers is a highly complex phe-\nnomenon encompassing both linear and nonlinear deforma-\ntions and multiple intrinsic time scales. In this letter, a simple\nKelvin-V oigt model is chosen for this study for brevity and\nfundamental understanding of the complex geometry-material\ninterplay with the aim for further detailed numerical stud-\nies in later publications. We also note that the major source\nof nonlinearity in these slender substrates come from geo-\nmetrical and contact sources and not the material due to the\ngeometrical locking phenomena8. The Kelvin-V oigt model\nfor viscoelastic behavior can be represented as a purely vis-\ncous damping element (damper) and a purely elastic element\n(spring) connected parallel together. The relationship between\nstress s, strain e, and strain rate d e=dtis governed by21,22:\ns=EBe+˜xde\ndt; (1)\nwhere EBand˜xare the Young’s modulus and viscosity of\nthe substrate, respectively. The scales are considered rigid to\nisolate the purely geometric effect of scales.\nThe length of substrate is considered as LB, and the height\nhB. We match the material properties of a typical silicone\nrubber, which can be used to fabricating the soft substrate.\nThese material properties are as follows: the Young’s modu-\nlusEB=1:5 MPa, Poisson’s ratio n=0:4210,23, and density\nrB=854 kg/m315. The viscosity of silicone rubbers is in the\nrange of 1 to 108mPa.s24,25, where the lower range is related\nto the liquid form of silicone polymer and the higher range\nis related to the solid form of silicone polymer. It should be\nnoted that these material properties are just an example of a\nsoft viscoelastic silicone polymer, and their exact values are\nnot critical to the central discoveries of this research work.\nThe rectangular scales with thickness Dare partially em-arXiv:2303.04920v1  [cond-mat.soft]  8 Mar 20232\ny~\nx~\nApplied ForceRepresentative Volume Element (RVE)\nf0 (x,t) = f 0 sin(πx)  cos Ωt   ̃   ̃   ̃   ̃  ̃   ̃   ̃\n(a) (b)\nr \nl N l l  ffr \n ffr \nd ~ N \n(c) (d)\nFIG. 1. (a) The overlapped arrangement of natural fish scales, and their artificial biomimetic reproduction. The picture is adapted under CC BY\n2.07. (b) Schematic diagram of a simply supported biomimetic scaled beam and the representative volume element (RVE) selected from the\nmiddle of the beam marked with the blue rectangle. (c) Schematic diagram of scales with characteristic dimensions and angles. (d) Schematic\ngeometry of the representative volume element (RVE) with detailed geometric parameters.\nbedded into the top surface of the substrate with initial angle\nq0, Fig. 1 (c). The total length of scales lsis including the\nexposed length l, and embedded length L, (ls=l+L). The\nexposed length of scales lis non-dimensionalized by the spac-\ning between the neighboring scales ˜d, , Fig. 1 (c), as h=l=˜d\ncalled overlap ratio8,15. We assume that the scale’s thickness\nDis negligible in comparison with the length of the scales,\nls(D\u001cls), and the scale’s embedded length is negligible\nin comparison with the substrate’s height ( L\u001chB)8–10,14–16.\nThese assumptions allow us to consider each scales as a linear\ntorsional spring with constant ˜Ks, due to resistance of sub-\nstrate against rotation of embedded part of scales8,10,26. For\nsuch a system, the constant was obtained as scaling expres-\nsion ˜Ks=EBD2CB(L=D)n, where CBandnare constants with\ncorresponding values 0 :66 and 1 :75, respectively8,15.\nThe dynamic equation of motion of a viscoelastic plain\nbeam (without any scale) is derived using Hamilton’s prin-\nciple, dR˜t2\n˜t1(ˆT\u0000ˆV+W)d˜t=0 for a viscoelastic beam. In\nthis relationship, ˆT,ˆV, and Ware the kinetic energy per unit\nlength, the strain energy per unit length, and the work done by\nthe applied traction, respectively,15,27–which leads to the fol-\nlowing differential equation for a plain viscoelastic beam21,28:\nrBAB¶2˜y\n¶˜t2+EBIB¶4˜y\n¶˜x4+˜xIB¶\n¶˜t\u0010¶4˜y\n¶˜x4\u0011\n=˜f(˜x;˜t);(2)\nHere, ABandIBare the area of the substrate’s cross-section\nand the second moment of area, respectively. The quanti-ties: ˜t, and ˜ xand ˜yare time, and the two spatial coordi-\nnates shown on Fig. 1 (b), respectively. The ˜f(˜x;˜t)is the\napplied force function, which is shown in Fig. 1 (b) schemat-\nically, for a scale-covered beam. For the free vibration case,\nthe function ˜f(˜x;˜t)is equal to zero and the equation of mo-\ntion is a homogeneous equation, whereas for the forced vi-\nbration case, applied force can be considered as the first mode\n˜f(˜x;˜t) =˜f0f(˜x)cos˜W˜t15,29, where ˜f0,f(˜x), and ˜Ware the load\namplitude, the first mode shape function for the simply sup-\nported beam, and the load frequency15,30. First mode shape\nfunction for a simply supported beam with length LBis known\nasf(˜x) =sin\u0000p˜x\nLB\u0001\n.\nUnder pure bending, the relationship between scales incli-\nnation angle q, and the substrate bending angle y, is given as\nq=sin\u00001(hycosy=2)\u0000y=28,14,26. Note that although this\nrelationship is not satisfied globally except pure bending, local\nperiodicity could be assumed (dense scales assumption)10,15.\nWith these considerations, the global deformation of the\nscaly beam can be envisioned as a combination of the two de-\nformation modes comprising of the substrate bending and the\nlocal scales rotating in all RVEs8,10. This kinematics allows\nthe inclusion of the work of the friction between scales as they\nslide. The frictional work can be included in the Hamilton’s\nprinciple now. The friction is modeled based on the Coulomb\ndry friction by considering different coefficients of friction m,\nand the effect of scales’ mass on the kinetic energy of the sys-\ntem is neglected15. These considerations lead to the follow-\ning partial differential equation for a scale-covered viscoelas-\ntic beam15(see Supplementary Material):3\n𝑡/𝜏𝑛𝑦𝑚𝑖𝑑𝜇=0,ሚ𝜉=0\n𝜇=2,ሚ𝜉=0.005𝜇=2,ሚ𝜉=0\n𝜇=2,ሚ𝜉=0.010\n𝜇=2,ሚ𝜉=0.015𝑨𝒊𝑨𝒊+𝟏\n𝑻𝒊\nFIG. 2. The response of middle point of the scaly viscoelastic beam\nwithh=5 under velocity initial condition for various viscosity co-\nefficients˜x(Different˜xare described with the unit MPa.s). For the\ngray plot m=0, and for other plots m=2.\nrBAB¶2˜y\n¶˜t2+EBIB¶4˜y\n¶˜x4+˜xIB¶\n¶˜t\u0010¶4˜y\n¶˜x4\u0011\n+¶2\n¶˜x21\nNå\u0014\n˜Ks(q\u0000q0)\n¶q\n¶y+sin(b)˜Ks(q\u0000q0)sgn(˙˜y)\ncos(y+b)\u0000¯rcos(b)¶¯r\n¶y\u0015\nH(˜k\u0000˜ke) = ˜f(˜x;˜t):(3)\nHere, b=tan\u00001m, and ¯ ris the non-dimensionalized form\nofr, which is the distance between the scale’s base to the in-\nteraction point with the left neighboring scale, with respect to\nthe exposed length of scale l, as ¯r=r=l, Fig. 1 (d) for a par-\nticular RVE. By considering the geometrical arrangements in\neach RVE shown in Fig. 1 (d) ¯ ris derived as ¯ r=sin\u0000\nq\u0000y=2\u0001\nsin\u0000\nq+y=2\u0001.\nIn Eq. (3), the Heaviside step function ensures that the terms\nregarding the strain energy due to the scales rotation, and the\ndissipation energy due to the friction between the scales, are\nonly contributed after scales engagement at each RVE level.\nThat is, only in the case of downward deflection of the beam,\nand when q>q0or, in another word, when ˜k>˜ke. The num-\nber of RVEs utilized in the solution of the system has been\nshown as N.\nWe first verify our model by comparing the midpoint de-\nflection during free vibration of the beam with finite element\nsimulation of an equivalent system, Fig. 2. In this figure,\nwhich is a displacement-time plot of the midpoint of the beam,\nwe fix h=5 ,q0=5\u000eand vary the coefficient of friction.\nThe black dots indicate FE simulations, which are in excel-\nlent agreement with our model results. Overall, comparing\nthis plot with purely Coulombic friction case15, it looks as if\nthe material viscosity effects are very similar to dry friction\neffect. They both lead to viscous damping, and increase with\ntime. Thus, it would seem that dry friction and material vis-\ncosity effects reinforce in tandem, the viscous damping of the\nbeam.\nHowever, this is where their similarities end. The material\nviscosity is essentially an symmetric source of dissipation -acting on both sides of the bending whereas the friction is\nasymmetric, acting only on scales side. In addition, the scales\nthemselves add asymmetry to the overall vibration by dent of\nbeing on only one side.\nThe asymmetry brought about by scales have a pronounced\ngeometrical component. On the one hand, if the scales are\ndense, the additional stiffness would be higher on the scales\nside. Similarly, frictional effects would also be higher. Thus\nit would seem like denser scales add to asymmetry of the\nmedium. On the other hand, if the scales initial inclination\nis higher, then they engage at a greater curvature and hence\ntheir impact on symmetry would be lesser. The effect of ma-\nterial viscoelasticity seems to be symmetric in nature because\nit acts on both sides. However, the scales on one side also\ninhibit displacement on the other side, and hence the symmet-\nric effect of material viscosity can also be broken. A suitable\nmeasure of asymmetry would be the logarithmic decrement\nfactor ( d=1\nDlogAAn+1\nAAn) that measures the relative decline of\namplitudes in successive cycles. This parameter is related to\nthe overall damping coefficient and the Q-factor of the vibra-\ntion. We could expect that these vibration asymmetries would\nlead to a split in the dvalues between the concave and con-\nvex side. We define the ratio of the two ds by an asymmetry\nratio a=dconvex =dconcave , and take them as a measure of bi-\ndirectional asymmetry.\nTopically, it would seem that increasing friction of scales\nwould cause greater asymmetry as it acts selectively on only\none direction whereas material dissipation would cause dis-\nsipation symmetrically in both direction. Hence, increasing\nCoulomb friction should accentuate asymmetry, whereas vis-\ncoelasticity should leave it unaffected. In order to investi-\ngate these effects, we develop phase maps of amapped by\nmand¯xfor various values of handq0, Fig. 3. The first row\nof this asymmetry map Fig. 3 (a-b) shows the effect of in-\ncreasing q0while his kept constant. Fig. 3 (a) shows that\nincreasing Coulomb friction does not lead to an increase in\nanisotropy of logarithmic damping, even though its effect on\ndisplacement asymmetry is pronounced. A rather surprising\nand counter-intuitive result. It seems like symmetry is bro-\nken only when the initial inclination angle changes. Once that\noccurs, increasing inclination angle causes the asymmetric re-\ngion to flatten and spread to lower values of Coulomb friction.\nWe also investigate the effect of scale density htowards the\nasymmetry, Fig. 3 (c-d). As expected the effect of higher den-\nsity is also to further anisotropy. For the same combination of\nmand¯x, the anisotropy is greatly pronounced with higher h,\nFig. 3 (c-d).\nIn order to gain better physical insight into the system\nwe probe the fundamentals using an asymmetric spring mass\ndamper system (SMD) that is damped more on one side,\nFig. 4. This system can be integrated to obtain analytical\nclosed-form expressions for logarithmic damping ratio (See\nSupplementary Material).\nThe symmetry breaking of this system can result from two\ndifferent sources - scales sliding on one side, and delayed en-\ngagement of scales on the other side only due to initial scale\nangle. Damping phase maps show that although introduction\nof asymmetry (in one direction) is sufficient to cause overall4\nത𝛼\n𝜇ሚ𝜉(MPa.s)\nത𝛼\n𝜇ሚ𝜉(MPa.s)\n(a) (b)\nത𝛼\n𝜇ሚ𝜉(MPa.s)\nത𝛼\n𝜇ሚ𝜉(MPa.s)\n(c) (d)\nFIG. 3. Phase map of the ratio between the convex and concave damping coefficient ( ¯a), known as asymmetry coefficient, spanned by˜xand\nm, for different cases: (a) h=5, and q0=0\u000e. (b)h=5, and q0=5\u000e. (c)q0=5\u000e, and h=3. (d) q0=5\u000e, and h=7.\nasymmetry of the system, the logarithmic decrement is still\nthe same for both sides of damped oscillation. Thus damp-\ning on just one side is not sufficient to cause a major change\nin symmetry in logarithmic damping ratio. The symmetry of\nthe SMD system can be further broken if we introduce an en-\ngagement asymmetry to mimic initial angle q0. The asym-\nmetry in stiffness in our SMD model is addressed by letting\nthat damping start at y=0 in the domain y<0, and at an\narbitrary ye>0 in the domain y>0, Fig. 4 (a). We vary the\nnatural frequencies wjy>yeandwjy<0while keeping the damp-\ning coefficients cjy>0andcjy<0constant, so that damping ra-\ntioszjy>0andzjy<0change together with the respective natu-\nral frequencies. These parameters are defined to approximate\nour architectured system – high natural frequencies indicate\nhigher stiffness, where higher damping coefficient is meant\nto simulate higher stiffness, whereas ye—the offset–simulates\nthe initial angle. In Fig. 4 (b) we plot the effect of scales with\nno offset. We clearly see that the lack of offset results in neg-\nligible difference in damping between the scales and the plain\nside. However, as soon as offset is added we see that a visible\nasymmetry in damping emerges. The asymmetry in dampingcoefficient increases as the contrast between the two sides in-\ncreases. In Fig. 4 (c), we plot the effect of different damping\ncoefficients with a given offset, and in Fig 4 (d), we increase\nthe ratio of the stiffnesses while keeping the damping coeffi-\ncients the same. This would be a comparison between systems\nwith a different overlap ratio. We find that higher overlap ra-\ntio clearly accentuates the asymmetry in logarithmic damping,\nceteris paribus. This confirms the trends from phase map Fig.\n3 (c), (d). Here, we compare two asymmetric SMD systems,\none with no offset and another with offset and find that addi-\ntion of offset, accentuates asymmetry, confirming our findings\nin Fig 3 (a)-(b). In addition to free vibration, we also con-\nsider dissipative effects in forced oscillations. We quantify\ndissipation using \"Specific Damping Capacity ( SDC )\", which\nmeasures a material’s ability to dissipate elastic strain energy\nthrough a mechanical vibration motion30,31. The SDC can be\ndefined as follows:5\n(a) (b)\n(c) (d)\nFIG. 4. Logarithmic decrement asymmetry exhibited when partial engagement at ye>0 is present in a simplified system of two coupled\ndamped oscillators. (a) A schematic of asymmetric spring mass damper system simulating the biomimetic metastructure. The offset mimics\ninitial scale angle after which engagement occurs. The engagement adds additional stiffness Dwand additional damping Dc(b) Damped\noscillations with no offset, (c) with offset, (d) with offset and higher stiffness on scales side than (c).\nSDC =Dissipated Energy per Steady State Cycle (DU)\nMaximum Stored Energy (U):\n(4)\nThese dissipated and stored energies can be calculated nu-\nmerically through a computational model.\nHere, in Fig. 5 (a), we plot the specific damping capacity\nfrom material sources¯xand find that it increases (barring a\nfew peaks at sub-harmonic frequencies due to complex nature\nof oscillations) as the frequency increases. This is a traditional\nviscous damping response. In contrast, the effect of inter-scale\nfriction is dramatically different. The specific damping ca-\npacity, Fig. 5 (b) shows a pronounced and sharp increase nearresonance with higher peaks corresponding to higher friction.\nAfter the resonance, the damping begins to decrease sharply.\nThe overall reason for this behavior is due to lack of rate-\ndependence of the frictional component of the force. As the\namplitude of the vibration decreases post resonance, so does\nthe work done by friction.\nIn conclusion, we find that although the viscoelastic and\nfrictional sources of dissipation are two apparently similar\nsources of damping in a biomimetic scale architectured sub-\nstrate, on closer scrutiny they are quite different. Their effects\non displacement, damping asymmetry, and specific damping\nmarkedly diverge. The geometry-material interplay is investi-\ngated for the first time. Real world polymers exhibit far more\ncomplexity in their material behavior. In linear regime they6\nΩ/𝜔𝑛𝑆𝐷𝐶\n~\n~\n~\n~\n~\nΩ/𝜔𝑛𝑆𝐷𝐶\n(a) (b)\nFIG. 5. (a) Variation of Specific damping coefficient (SDC) with frequency for forced vibration for various material viscosity parameters,\nfriction is absent. (b) Variation of Specific damping coefficient (SDC) with frequency for forced vibration for various coefficients of friction.\nMaterial viscosity is negligible (<0.005)\nare often represented by a combination of Kelvin-V oigt ele-\nments. We aim to study such complexities in later iterations\nof this study. In spite of this limitation, the current findings\nhave wide implications in the design of fish scale like smart\nskins and appendages for soft robotics, tailored prosthetic ap-\nplications.\nACKNOWLEDGMENTS\nThis work was supported by the United States National Sci-\nence Foundation’s Civil, Mechanical, and Manufacturing In-\nnovation, Grant No. 2028338.\n1G. N. S and B. R. R. J, “Performance enhancement of futuristic air-\nplanes by nature inspired biomimetic fish scale arrays—a design approach,”\nBiomimetic Intelligence and Robotics 2, 100045 (2022).\n2M. Muthuramalingam, D. K. Puckert, U. Rist, and C. Bruecker, “Transi-\ntion delay using biomimetic fish scale arrays,” Scientific Reports 10, 1–13\n(2020).\n3H. Jiakun, H. Zhe, T. Fangbao, and C. Gang, “Review on bio-inspired\nflight systems and bionic aerodynamics,” Chinese Journal of Aeronautics\n34, 170–186 (2021).\n4J. B. R. Rose, S. G. Natarajan, and V . Gopinathan, “Biomimetic flow con-\ntrol techniques for aerospace applications: a comprehensive review,” Re-\nviews in Environmental Science and Bio/Technology , 1–33 (2021).\n5B. Jenett, S. Calisch, D. Cellucci, N. Cramer, N. Gershenfeld, S. Swei, and\nK. C. Cheung, “Digital morphing wing: active wing shaping concept using\ncomposite lattice-based cellular structures,” Soft robotics 4, 33–48 (2017).\n6J. Long, M. Hale, M. Mchenry, and M. Westneat, “Functions of fish skin:\n��exural stiffness and steady swimming of longnose gar, lepisosteus osseus,”\nThe Journal of experimental biology 199, 2139–2151 (1996).\n7A. Aini, “Sting & aro,” (2008), (accessed June 23, 2022).\n8R. Ghosh, H. Ebrahimi, and A. Vaziri, “Contact kinematics of biomimetic\nscales,” Applied Physics Letters 105, 233701 (2014).\n9H. Ebrahimi, H. Ali, R. A. Horton, J. Galvez, A. P. Gordon, and R. Ghosh,“Tailorable twisting of biomimetic scale-covered substrate,” EPL (Euro-\nphysics Letters) 127, 24002 (2019).\n10H. Ali, H. Ebrahimi, and R. Ghosh, “Bending of biomimetic scale covered\nbeams under discrete non-periodic engagement,” International Journal of\nSolids and Structures 166, 22–31 (2019).\n11B. Wang, W. Yang, V . R. Sherman, and M. A. Meyers, “Pangolin armor:\noverlapping, structure, and mechanical properties of the keratinous scales,”\nActa biomaterialia 41, 60–74 (2016).\n12W. Yang, I. H. Chen, B. Gludovatz, E. A. Zimmermann, R. O. Ritchie, and\nM. A. Meyers, “Natural flexible dermal armor,” Advanced Materials 25,\n31–48 (2013).\n13H. Ehrlich, “Materials design principles of fish scales and armor,” in Bio-\nlogical Materials of Marine Origin (Springer, 2015) pp. 237–262.\n14R. Ghosh, H. Ebrahimi, and A. Vaziri, “Frictional effects in biomimetic\nscales engagement,” EPL (Europhysics Letters) 113, 34003 (2016).\n15H. Ali, H. Ebrahimi, and R. Ghosh, “Frictional damping from biomimetic\nscales,” Scientific reports 9, 1–7 (2019).\n16H. Ebrahimi, H. Ali, and R. Ghosh, “Coulomb friction in twisting of\nbiomimetic scale-covered substrate,” Bioinspiration & Biomimetics 15,\n056013 (2020).\n17S. Dharmavaram, H. Ebrahimi, and R. Ghosh, “Coupled bend–twist me-\nchanics of biomimetic scale substrate,” Journal of the Mechanics and\nPhysics of Solids 159, 104711 (2022).\n18H. Ali, H. Ebrahimi, J. Stephen, P. Warren, and R. Ghosh, “Tailorable\nstiffness lightweight soft robotic materials with architectured exoskeleton,”\ninAIAA Scitech 2020 Forum (2020) p. 1551.\n19J. D. Ferry, Viscoelastic properties of polymers (John Wiley & Sons, 1980).\n20G. Polacco, J. Stastna, D. Biondi, and L. Zanzotto, “Relation between poly-\nmer architecture and nonlinear viscoelastic behavior of modified asphalts,”\nCurrent opinion in colloid & interface science 11, 230–245 (2006).\n21A. Craifaleanu, N. OR ˘a¸ SANU, and C. Dragomirescu, “Bending vibrations\nof a viscoelastic euler-bernoulli beam–two methods and comparison,” in\nApplied Mechanics and Materials , V ol. 762 (Trans Tech Publ, 2015) pp.\n47–54.\n22J. Freundlich, “Transient vibrations of a fractional kelvin-voigt viscoelastic\ncantilever beam with a tip mass and subjected to a base excitation,” Journal\nof Sound and Vibration 438, 99–115 (2019).\n23H. Ali, H. Ebrahimi, and R. Ghosh, “Tailorable elasticity of cantilever\nusing spatio-angular functionally graded biomimetic scales,” Mechanics of7\nSoft Materials 1, 1–12 (2019).\n24MatWeb, “Overview of materials for silicone rubber,” (2022), (accessed\nFebruary 3, 2022).\n25R. Suriano, O. Boumezgane, C. Tonelli, and S. Turri, “Viscoelastic prop-\nerties and self-healing behavior in a family of supramolecular ionic blends\nfrom silicone functional oligomers,” Polymers for Advanced Technologies\n31, 3247–3257 (2020).\n26F. J. Vernerey and F. Barthelat, “On the mechanics of fishscale structures,”\nInternational Journal of Solids and Structures 47, 2268–2275 (2010).\n27M. Javadi and M. Rahmanian, “Nonlinear vibration of fractional kelvin–voigt viscoelastic beam on nonlinear elastic foundation,” Communications\nin Nonlinear Science and Numerical Simulation 98, 105784 (2021).\n28J. Diani, “Free vibrations of linear viscoelastic polymer cantilever beams,”\nComptes Rendus. Mécanique 348, 797–807 (2020).\n29N. Abhyankar, E. Hall, and S. Hanagud, “Chaotic vibrations of beams: nu-\nmerical solution of partial differential equations,” Journal of Applied Me-\nchanics 60, 167–174 (1993).\n30S. S. Rao, Vibration of continuous systems (Wiley, 2019).\n31J. Zhang, R. Perez, and E. Lavernia, “Documentation of damping capac-\nity of metallic, ceramic and metal-matrix composite materials,” Journal of\nmaterials science 28, 2395–2404 (1993)."    },    {        "title": "2012.13859v2.Quantum_speed_limit_time_in_relativistic_frame.pdf",        "content": "Quantum speed limit time in relativistic frame\nN. A. Khan\u0003\nCentro de F\u0013 \u0010sica das Universidades do Minho e Porto\nDepartamento de F\u0013 \u0010sica e Astronomia, Faculdade de Ci^ encias,\nUniversidade do Porto, 4169-007 Porto, Portugal\nMunsif Jany\nKey Laboratory of Quantum Information of Chinese Academy of Sciences (CAS),\nUniversity of Science and Technology of China, Hefei 230026, P. R. China\nWe investigate the roles of relativistic e\u000bect on the speed of evolution of a quantum system\ncoupled with amplitude damping channels. We \fnd that the relativistic e\u000bect speed-up the quantum\nevolution to a uniform evolution speed of an open quantum systems for the damping parameter\np\u001c.p\u001cc0:Moreover, we point out a non-monotonic behavior of the quantum speed limit time\n(QSLT) with acceleration in the damping limit p\u001cc0.p\u001c.p\u001cc1;where the relativistic e\u000bect \frst\nspeed-up and then slow down the quantum evolution process of the damped system. For the damping\nstrengthp\u001cc1.p\u001c, we observe a monotonic increasing behavior of QSLT, leads to slow down the\nquantum evolution of the damped system. In addition, we examine the roles of the relativistic e\u000bect\non the speed limit time for a system coupled with the phase damping channels.\nI. INTRODUCTION\nThe \feld of quantum information under relativistic\nconstraints leads to the emergence of a new \feld of high\nresearch intensity, known as Relativistic Quantum Infor-\nmation (RQI). The most spectacular research in this \feld\nhave been devoted to the study of: entanglement between\nquantum \feld modes in the accelerated frames1{14, en-\ntanglement in black-hole space-times15,16,entanglement\nof Dirac \feld in an expanding space-time17, relativistic\nquantum metrology18,19and teleportation with a uni-\nformly accelerated partner20. The \feld aims to under-\nstand the uni\fcation of the theory of relativity and quan-\ntum information. One of the most fundamental man-\nifestations of RQI is the Unruh entanglement degrada-\ntion between quantum \feld modes in the accelerated\nframes1,2,6under single mode approximation. More-\nover, the observer-dependent property of entanglement\nhas been successfully examined beyond the single-mode\napproximation3.\nThe tool of quantum information theory plays a promi-\nnent role in the understanding of entanglement witness\nof polarization-entangled photon pairs21. It has been ex-\nperimentally tested that the photonic quantum entangle-\nment persist in the accelerated frames. An other promis-\ning experimental realization of relativistic scenario is\\en-\ntanglement swapping protocol22\", where a maximum\nBell violation occurred in a suitable reference frame.\nThe minimal time required for the evolution of a quan-\ntum system is known as \\Quantum Speed Limit Time\"\n(QSLT)23. There exists a considerable amount of work\ndedicated to estimate the minimal evolution time of a\nquantum system23{26. For instance, QSLT was success-\nfully investigated for the damped Jaynes-Cummings and\nthe Ohmic-like dephasing model23. Moreover, it was\nfound that the relativistic e\u000bect slow down the quantum\nevolution of the qubit in the damped Jaynes-Cummingsmodel. There have also been studies involving the speed\nof quantum evolution of a single free spin-1 =2 parti-\ncle coupled with phase damping channels in the rela-\ntivistic framework27. In addition, the nature of QSLT\nin Schwarzschild space-time for the damped Jaynes-\nCummings and Ohmic-like dephasing models have been\nexamined28. Their results show that the QSLT decreased\nand increased by increasing relative distance of quantum\nsystem to event horizon for damped Jaynes-Cummings\nand Ohmic-like dephasing model, respectively.\nRecently, the relativistic e\u000bects on the speed of quan-\ntum evolution have been reported for a free Dirac \feld in\nnon-inertial frames29. It is pointed out that the relativis-\ntic e\u000bects speed-up the evolution of the quantum system\ncoupled with the amplitude damping channels. However,\nno relativistic e\u000bects have been encountered for the speed\nof quantum evolution of the phase dapmed-system in a\nnon-inertial frame.\nThe aim of this article is to explore the role of rela-\ntivistic e\u000bects on the speed of evolution of a quantum\nsystem for a free scalar \feld which manifests itself in the\nquantum noise. We point out that the QSLT initially\nreduces to a minimum with increasing acceleration and\nthen trapped to a uniform \fxed value for damping pa-\nrameterp\u001c.p\u001cc0. This phenomenon leads to a speed-up\nof the quantum evolution initially, and then reaches to\na uniform evolution speed of an open quantum system.\nIn the region p\u001cc0.p\u001c.p\u001cc1, the QSLT \frst decreases\nto a minimum value, and then gradually increases to a\nmaximum uniform value as depicted in Fig. 1.\nThis shows that the relativistic e\u000bect speed-up the\nquantum evolution in the beginning and then slow down\nthe speed of evolution of the system. However, the quan-\ntum evolution of the system exhibits a slow down behav-\nior with increasing acceleration for p\u001cc1.p\u001c, leads to\na larger QSLT in non-inertial frame. For each case, we\nnotice a uniform speed of evolution of the system in the\nlarge acceleration limit, where the QSLT trapped to aarXiv:2012.13859v2  [quant-ph]  7 Jan 20212\n\fxed value. In addition, for the phase damped-system,\nwe obtain an acceleration independent speed limit time\nin relativistic frame.\nDamping parameter (𝑝𝜏)Acceleration parameter (𝑟)\nuniform evolution\nslow -down evolutionspeed -up evolution\nFigure 1. (Color online) The roles of relativistic e\u000bect on\nquantum speed limit time of an open system coupled with\nthe amplitude damping channels.\nThe structure of the paper is as follows. Sec. II, is\ndevoted to the theoretical background of the scalar \feld\nas observed by uniformly accelerated observer. In par-\nticular, we review the mathematical transformations be-\ntween Minkowski and Rindler modes under single mode\napproximation. In Sec. III, we present the physical sce-\nnario and the mathematical procedure for calculating the\nQSLT of a quantum system coupled with the amplitude\ndamping channels in non-inertial frames. Moreover, we\nanalyze the relativistic e\u000bects on QSLT for the amplitude\ndamped open quantum system, when one observer move\nwith a uniform acceleration. In the last section, we sum\nup our conclusions.\nII. SCALAR FIELD\nA real scalar \feld \u001ein two dimensional Minkowski\nspace time can be described by the massless Klein-\nGordon equation, @\u001e= 0. This \feld can be expressed\nin terms of the positive and negative energy solutions of\nthe Klein-Gordon equation, given by3,30\n\b!;M=Z1\n0(a!;M'!;M+ay\n!;M'\u0003\n!;M)d! (1)\nwherea!;Manday\n!;Mare the Minkowski annihilation and\ncreation operators, obeying the bosonic commutation re-\nlations. The positive-energy mode solution '!;M, with\nrespect to the timelike Killing vector \feld @t, for an iner-\ntial observer in Minkowski coordinates ( t; x);with posi-\ntive Minkowski frequency !is given by\n'!;M(t;x) =1p\n4\u0019!exp [\u0000i!(t\u0000\"x)]; (2)where\"can takes the value 1 and \u00001 for modes with\npositive (right movers) and negative (left movers) mo-\nmentum, respectively. The mode solutions satisfy the\nfollowing relations\n('!;M;'!0;M) =\u0000('\u0003\n!;M;'\u0003\n!0;M) =\u000e!!0;\n('\u0003\n!;M;'!0;M) = 0: (3)\nThe Klein-Gordon equation for a uniformly accelerated\nobserver can be more appropriately described by Rindler\nspace-time. The Rindler coordinates and Minkowski co-\nordinates are related by3,30\n\u0011= arctan\u0012t\nx\u0013\n; \u001f =p\nx2\u0000t2; (4)\nwhere\u001f=a\u00001, is the position and \u0011=ais the proper\ntime of the accelerated observer in region I. Here ais\na positive constant, referred as the acceleration of the\nuniformly accelerated observer. The Rindler coordinates\nin region II can simply be obtained by replacing \u0011=\u0000\u0011.\nThe Rindler coordinates have ranges 0 < \u001f <1and\n\u00001<\u0011<1.\nThe \feld can be expanded in terms of the energy solu-\ntions of the Klein-Gordon equation in region I and II in\nthe Rindler coordinates is3,30\n\b\n;R=Z1\n0(a\n;I'\n;I+ay\n\n;I'\u0003\n\n;I+a\n;II'\n;II+ay\n\n;II'\u0003\n\n;II)d\n;\n(5)\nwherea\n;\u001banday\n\n;\u001bare the Rindler annihilation and\ncreation operators for \u001b2fI, IIg, respectively. It obeys\nthe bosonic commutation relations. The '\n;\u001b(t;x) is the\npositive frequency mode functions with respect to the\ntimelike Killing vector \feld \u0006@\u0011, for the accelerated ob-\nserver in region \u001b, as given by\n'\n;I(t;x) =1p\n4\u0019\n\u0012x\u0000\"t\nl\n\u0013i\"\n;\n'\n;II(t;x) =1p\n4\u0019\n\u0012\"t\u0000x\nl\n\u0013\u0000i\"\n; (6)\nwhere \n is a dimensionless Rindler frequency, l\nis a pos-\nitive constant of dimension length.\nThe \feld solution can also be expressed in the Unruh\nbases3,30\n\b\n;U=Z1\n0(a\n;R'\n;R+ay\n\n;R'\u0003\n\n;R+a\n;L'\n;L+ay\n\n;L'\u0003\n\n;L)d\n;\n(7)\nwherea\n;\u0017anday\n\n;\u0017are the Unruh annihilation and cre-\nation operators for \u00172 fR, Lg, respectively, obey the\nbosonic commutation relations. The solution of the \feld\n'\n;\u0017(t;x) in the Unruh bases are related by\n'\n;R= coshr\n'\n;I+ sinhr\n'\u0003\n\n;II;\n'\n;L= coshr\n'\n;II+ sinhr\n'\u0003\n\n;I; (8)3\nThe expression Eq. 8 shows the transformation between\nthe Unruh and the Rindler bases. Similarly, the trans-\nformation between the Minkowski and the Unruh modes\nare related by3,30\n'!;M(t;x) =Z1\n0\u0000\n\u000bR\n!\n'\n;R+\u000bL\n!\n'\n;L\u0001\nd\n;(9)\nwhere\n'\n;\u0017=Z1\n0(\u000b\u0017\n!\n)\u0003'!;Md!; \u00172fR, Lg; (10)\n\u000bR\n!\n=1p\n2\u0019!(!l)i\"\n; \u000bL\n!\n=\u0000\n\u000bR\n!\n\u0001\u0003; (11)\nwherelis constant of dimension length.\nA similar transformation between the Minkowski\nand the Unruh operators may yield the following\nexpression3,30\na!;M=Z1\n0\u0010\u0000\n\u000bR\n!\n\u0001\u0003a\n;R+\u0000\n\u000bL\n!\n\u0001\u0003a\n;L\u0011\nd\n;(12)\nwhere\na\n;\u0017=Z1\n0\u000b\u0017\n!\na!;Md!; \u00172fR, Lg; (13)\nOne can obtain a relationship between the Minkowski\nand the Unruh operators\na\n;I= coshr\na\n;R+ sinhr\nay\n\n;L;\na\n;II= coshr\na\n;L+ sinhr\nay\n\n;R: (14)\nNow we are in the position to relate the vacua and excited\nstates of the Minkowski, Rindler and Unruh modes. It is\nimportant to mention that the Minkowski and the Unruh\nmodes have common vacuum state, i.e., j0\niM=j0\niU.\nThe Minkowski vacuum state for a free scalar \feld can\nbe expressed as a two-mode squeezed state of the Rindler\nvacuum\nj0\niM=1\ncoshr1X\nn=0tanhnrjn\niIjn\niII; (15)\nwherejn\niIandjn\niIIindicate the Rindler particle mode\nin region I and II ;respectively. Using single mode ap-\nproximations, the Minkowski excited state can be ex-\npressed as\nj1\niM=1\ncosh2r1X\nn=0p\nn+ 1 tanhnrjn+ 1\niIjn\niII;\n(16)\nwhereris the acceleration parameter, de\fned as tanh r=\nexp(\u0000\u0019\n), with \n =jkjc=a, such that 0\u0014r\u00141 for\n0\u0014a\u00141. Here,kis a wave vector denotes the modes\nof the scalar \feld.III. RELATIVISTIC EFFECTS ON THE QSLT\nIn the following, we discuss the notion of a QSLT,\nwhich sets the ultimate maximal speed of evolution of\nan open quantum system. It basically determines the\nminimal time (lower bound) of evolution from a mixed\nstate%0to its \fnal mixed state %\u001c. A general expression\nfor the QSLT of an open systems can be written as29,31,\n\u001cQSL=k%0\u0000%\u001ckhs\nk_%tkhs; (17)\nwith\nk_%tkhs=1\n\u001cZ\u001c\n0dtk_%tkhs; (18)\nwherekOkhsis the the Hilbert-Schmidt norm of density\noperatorO. ThekOkhs=pP\nie2\ni, witheibeing the\nsingular values of O. The term in the numerator coined\nas the Euclidean distance D(%0; %\u001c) =k%0\u0000%\u001ckhs. It is\nimportant to mention that the QSLT turns out to be the\nactual evolution time in the limit D(%0; %\u001c) =k_%tkhs.\nWe assume that the initial maximally entangled state\nbetween AliceAand BobBof single Minkowski mode k\nreads;\nj i=1p\n2(j0kiAj0kiB+j1kiAj1kiB); (19)\nGiven that the observer Bundergoes a uniform accelera-\ntion with respect to the inertial observer A, the scalar\nparticle vacuum and excited state of Bin Minkowski\nspace are transformed into two causally disconnected\nRindler regions I and II for particles and anti-particles,\nrespectively. Using single mode approximation, the state\n(Eq. 19) can be expressed in terms of Rindler states of\nthe free scalar \feld, as given by\nj\ti=1p\n2 coshr1X\nn=0tanhnr(j0nni+p\nn+ 1\ncoshrj1n+ 1ni);\n(20)\nwherejxyzi=jxkiAjykiBIjz\u0000kiBIIforx2(0;1) and\ny;z2(n; n+ 1).\nIn what follows, we study the evolution of the\namplitude-damped quantum system in the relativistic\nframework (especially Rindler basis in region I). Taking\nthe trace over the inaccessible state of region II, we obtain\na reduced density operator , \u001aABI, of Alice and physically\naccessible region of Bob, as given by\n\u001aABI=j0ih0j\nMnn+j1ih1j\nMn+1n+1\n+j1ih0j\nMn+1n+j0ih1j\nMnn+1; (21)4\nwhere\nMnn=1\n2 cosh2r1X\nn=0tanh2nrjnihnj;\nMn+1n=1\n2 cosh3r1X\nn=0p\nn+ 1 tanh2nrjn+ 1ihnj;\nMn+1n+1=1\n2 cosh4r1X\nn=0(n+ 1) tanh2nrjn+ 1ihn+ 1j;\nMnn+1=M\u0003\nn+1n: (22)\nLet consider the inertial observer Aof the system is under\nthe action of an amplitude damping channel. In this case,\nnothing happens, if the system is in the ground state.\nHowever, it change the dynamic of the excited state of\nthe system. The Krauss operators Kifori2(0;1) of this\nmodel is represented by the following positive quantum\nmap32\nK0=\u0014\n1 0\n0ppt\u0015\n;K1=\u0014\n0p1\u0000pt\n0 0\u0015\n; (23)\nwherept= exp (\u0000\u0000t);is a damping parameter, an expo-\nnential decay of the excited population with decay rate \u0000.\nThe system-environment interaction can be represented\nby the Krauss decomposition of the quantum channel,\nE(\u001a0) =P\niKy\ni\u001a0Ki, satisfyingP\niKy\niKi= I. Thus, the\nreduced density operator Eq. 21, after inertial observer\ninteracting with amplitude damping channel is given by\n\u001aABI(pt) =j0ih0j\nMnn+ppt(j1ih0j\nMn+1n+ h.c)\n+ (ptj1ih1j+ (1\u0000pt)j0ih0j)\nMn+1n+1;\n(24)\nIn order to characterize the speed of evolution for the\namplitude damped quantum systems, we \frst calculate\nthe Euclidean distance D(p\u001c; r);between the noise-free\ninitial state \u001aABIto its amplitude decoherence state\n\u001aABI(p\u001c) in the accelerated frame. Calculating the sin-\ngular values of the reduced system \u001aABI\u0000\u001aABI(p\u001c) and\ntaking the square root of the sum of square of them, one\n\fnds that the Euclidean distance,\nD(p\u001c; r) =1\u0000pp\u001cp\n2q\n(1 +pp\u001c)2+a2(r); (25)\nwherea(r) is the acceleration dependent parameter,\ngiven by\na(r) =1\ncosh3r1X\nn=0p\nn+ 1 tanh2nr;\n=1\ncoshrsinh2rLi\u00001\n2\u0000\ntanh2r\u0001\n: (26)\nThe trace norm of the _ \u001aABI(p\u001c) can be obtained as\nk_\u001aABI(p\u001c)khs=\u00001\n2p\n2\u001cZp\u001c\n1dpts\n4pt+a2(r)\npt;(27)\n0 2 4 6 8r-8e-04-4e-040e+004e-048e-04∆τ(pτ, r)/τpτ = 0.3\npτ = 0.5\npτ = 5.0×10-4pτ = 1.5×10-3\npτ = 1.0×10-4pτ = 0.2\npτ = 1.0×10-3pτ = 0.1Figure 2. (Color online) The \u0001 \u001c(p\u001c; r) as a function of ac-\nceleration parameter rof the accelerated bosonic observer\n(scalar \feld) for di\u000berent values of damping strength p\u001cfor\np\u001c\u00150:1 &p\u001c\u00141:5\u000210\u00003of the amplitude damping chan-\nnels.\n0 2 4 6 8r-4e-04-2e-040e+002e-044e-04∆τ(pτ, r)/τpτ = 8.0×10-3\npτ = 5.0×10-3\npτ = 3.0×10-3pτ = 4.0×10-3\npτ = 2.0×10-3pτ = 1.0×10-2\npτ = 6.0×10-3\npτ = 1.5×10-3pτ = 1.5×10-2\nFigure 3. (Color online) The \u0001 \u001c(r; p\u001c) as a function of ac-\nceleration parameter rof the accelerated bosonic observer\n(scalar \feld) for di\u000berent values of damping strength p\u001cfor\n1:5\u000210\u00002\u0015p\u001c\u00151:5\u000210\u00003of the amplitude damping chan-\nnels.\nThe expression for quantum speed limit time \u001c(p\u001c; r),\nof the amplitude-damped state in the relativistic frame\nhas the form\n\u001c(p\u001c; r) =2\u001c(1\u0000pp\u001c)q\u0000\n1 +pp\u001c\u00012+a2(r)\n\u0000Rp\u001c\n1dptq\n4pt+a2(r)\npt;(28)\nThe integral in the denominator is given by33. It is\nstraightforward to show that the QSLT \u001c(p\u001c; r);reduce5\nCritical damping parameters Numerical values\np\u001cc0 1:5\u000210\u00003\np\u001cc1 1:5\u000210\u00002\nTable I. Numerical values of the critical damping parameter,\nwhere the QSLT shows di\u000berent behavior with respect to ac-\nceleration.\nto pure Unruh decoherence form of the scalar \feld in\nthe absence of quantum noise ( p\u001c= 0). In this limit,\nthe\u001c(p\u001c= 0; r);only depends on the acceleration pa-\nrameter of the accelerated observer. On the other hand,\nthe\u001c(p\u001c; r= 0), represents the QSLT of the purely am-\nplitude damped quantum system in the non-relativistic\nframe.\nOur analysis of the quantum system coupled with the\namplitude damping channel indicates that the relativis-\ntic e\u000bect may speed-up or slow down the quantum evo-\nlution of the damped-system, leads to a smaller or larger\n\u001c(p\u001c; r), respectively, in non-inertial frame as illustrated\nin Fig. 2. It is shown that the QSLT \u0001 \u001c(p\u001c; r) =\n\u001c(p\u001c; r)\u0000\u001c(p\u001c; r= 0), is monotonically an increasing\nand a decreasing function of acceleration parameter rfor\np\u001c\u00150:1 andp\u001c\u0014p\u001cc0, respectively. Hence, the relativis-\ntic e\u000bect slow down the speed of evolution in the former\ncase, whereas in the later case speed-up the evolution of\nthe quantum system. In the limit p\u001c\u00181;there exist no\nrelativistic e\u000bects on the speed of evolution of strongly\ndamped system i.e., \u001c(p\u001c; r)\u0019\u001c(p\u001c; r= 0), where the\n\u0001\u001c(p\u001c; r) approaches to a vanishingly small value. On\nthe other hand, the relativistic e\u000bect speed-up the quan-\ntum evolution of the weakly damped system ( p\u001c\u0014p\u001cc1),\nresults a smaller \u0001 \u001c(p\u001c; r) in the non-inertial frame. It\nis important to mention that the \u0001 \u001c(p\u001c; r) for various p\u001c\ngradually approach to a \fxed value, leads to a uniform\nevolution speed of the system.\nThe anomalous behavior of QSLT under Unruh deco-\nherence as a function of acceleration parameter for var-\nious damping strength, on the interval p\u001c2(p\u001cc1;p\u001cc0)\nare depicted in Fig. 3. One can clearly see that the QSLT\ndecreases to a minimum value in the beginning, then in-\ncreases to a \fxed value with increasing acceleration. This\nshows that the relativistic e\u000bect \frst speed-up the evolu-\ntion process, then exhibit a gradual deceleration process\nto a uniform evolution of the system. Furthermore, we\nnotice an acceleration independent behavior of QSLT for\np\u001c\u00195\u000210\u00003;in the large acceleration limit.\nThe anomalous |monotonic and non-monotonic| be-\nhavior of QSLT is due to the competition between accel-\nerated parameter rand damping parameter p\u001c. It is easy\nto see an enhancement of the QSLT with acceleration for\np\u001c\u00150:1, trapped to a \fxed value for the large acceler-\nation limit. However, this behavior ceased out for the\nhighly damped limit p\u001c\u00181, where the QSLT turns out\nto be acceleration independent. In this limit, we may\nsay that the system is completely evolved. For a weakly\ncoupled system ( p\u001c\u001c5\u000210\u00003), the acceleration pa-\nrameter dominates, leads to the degradation of QSLT in\nnon-inertial frames. In particular, the QSLT of the sys-tem in the absence of noise turns out:\n\u001c(p\u001c= 0; r) =2p\na2(r) + 1p\na2(r) + 4 +1\n2a2(r) sinh\u000012\na(r):(29)\nIn addition, on the interval p\u001c2(p\u001cc1;p\u001cc0), \frst the\nacceleration dominates the damping parameter, then\ndamping parameter dominates the acceleration, results\nin decreasing and increasing of QSLT of the open quan-\ntum system.\nFurthermore, the in\ruence of the relativistic e\u000bects on\nthe QSLT for the quantum systems coupled with the\nphase damping channels in non-inertial frames has also\nbeen examined. The relativistic QSLT can be calculated\nas\n\u001c(q\u001c) =1\u0000p1\u0000q\u001c\n1\u0000pq\u001c: (30)\nwhereq\u001cis a damping parameter of the phase damp-\ning channels. Eq. 30 suggest that the QSLT is inde-\npendent of acceleration parameter for the quantum sys-\ntems coupled with the phase damping channels in non-\ninertial frames. This acceleration-independent behavior\nof QSLT is in consistent with the result obtained for\nthe phased-damped system of the fermionic \feld in non-\ninertial frame29\nIV. CONCLUSIONS\nWe have investigated the speed of evolution of the am-\nplitude damped quantum system under Unruh decoher-\nence from the perspective of QSLT. For the scalar \feld,\nwe have observed a speed-up of quantum evolution of\nthe system due to Unruh decoherence for the damping\nparameterp\u001c.p\u001cc0. The QSLT in this region turned\nout to be a decreasing function of the acceleration pa-\nrameter for a given damping. Moreover, on the interval\np\u001c2(p\u001cc1;p\u001cc0), the relativistic e\u000bect \frst speeded-up,\nand then decelerated the quantum evolution process of\nthe system. On the other hand, we have noticed a decel-\neration quantum evolution process for p\u001cc1.p\u001cin the\nrelativistic frame. 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Press, 2000).\n33The integral has the simpli\fed form\n\u0000Zp\u001c\n1dpts\n4pt+a2(r)\npt=p\n4 +a2(r)\u0000p\np\u001c(4p\u001c+a2(r))\n\u00001\n2a2(r)(sinh\u000012pp\u001c\na(r)\u0000sinh\u000012\na(r)):\n."    },    {        "title": "1801.05630v1.On_Global_Existence_and_Blow_up_for_Damped_Stochastic_Nonlinear_Schrödinger_Equation.pdf",        "content": "arXiv:1801.05630v1  [math.PR]  17 Jan 2018On Global Existence and Blow-up for Damped Stochastic\nNonlinear Schr¨ odinger Equation\nJianbo Cui, Jialin Hong, and Liying Sun\nAbstract. In thispaper, weconsiderthewell-posednessofthe weaklyd amped\nstochastic nonlinear Schr¨ odinger(NLS) equation driven b y multiplicative noise.\nFirst, we show the global existence of the unique solution fo r the damped sto-\nchastic NLS equation in critical case. Meanwhile, the expon ential integrability\nof the solution is proved, which implies the continuous depe ndence on the\ninitial data. Then, we analyze the effect of the damped term an d noise on\nthe blow-up phenomenon. By modifying the associated energy , momentum\nand variance identity, we deduce a sharp blow-up condition f or damped sto-\nchastic NLS equation in supercritical case. Moreover, we sh ow that when the\ndamped effect is large enough, the damped effect can prevent th e blow-up of\nthe solution with high probability.\n1. Introduction\nThe nonlinear Schr¨ odinger equation, as one of the basic models for nonlinear\nwaves, has many physical applications to, e.g. nonlinear optics, plas ma physics and\nquantum field theory and so on (see e.g. [ 3, 5, 10, 12, 17 ]).\nIn this paper, we consider the weakly damped stochastic NLS equat ion driven\nby a linear multiplicative noise in focusing mass-(super)critical range ,\ndu=i(∆u+λ|u|2σu)dt−audt+iu◦dW(t), (1)\nu(0) =u0,\nwhere2\nd≤σ <2\n(d−2)+,λ= 1,a≥0,x∈Rd,t≥0 and “ ◦” stands for a\nStratonovich product. Here W={W(t) :t∈[0,T]}is anL2(Rd)-valued Q-\nWiener process on a stochastic basis (Ω ,F,Ft,P), i.e., there exists an orthonormal\nbasis{ek}k∈N+ofL2(Rd) and a sequence of mutually independent, real-valued\nBrownian motions {βk}k∈N+such that W(t) =/summationtext\nk∈N+Q1\n2ekβk(t),t∈[0,T]. We will\nuse frequently the equivalent Itˆ o form of Eq. (1)\ndu=i(∆u+|u|2σu)dt−(a+1\n2FQ)udt+iudW(t),\nu(0) =u0\n2010Mathematics Subject Classification. Primary 60H35; Secondary 60H15, 60G05.\nKey words and phrases. stochastic Schr¨ odinger equation, global existence, blow -up, expo-\nnential integrability .\n12 JIANBO CUI, JIALIN HONG, AND LIYING SUN\nwithFQ:=/summationtext\nk∈N+(Q1\n2ek)2.\nIn recent twenty years, much effort has been devoted to studyin g the well-\nposedness of stochastic NLS equation, see [ 1, 4, 10, 12, 14 ] and the references\ntherein. In [ 10] and [12], the local and global existence of the mild solution of sto-\nchastic NLS equation in L2and inH1are investigated, respectively. For stochastic\nNLS equation on a manifold, [ 4] considers the existence and uniqueness of the solu-\ntion based on the stochastic Strichartz estimate in UMD Banach spa ce. [14] shows\nthe local well-posedness in L2via stochastic Strichartz estimates and the global\nwell-posedness in subcritical case. With the help of the rescaling tra nsformation,\n[1] obtains that the local well-posedness in H1forσ <2\n(d−2)+with some decay\nconditions on the noise, and that the global existence when λ=−1,σ <2\n(d−2)+\norλ= 1,σ <2\nd.The global existence of the solution in H2is presented in [ 8]\nfor one dimensional stochastic NLS equation driven by linear multiplica tive noises.\nFor the global existence of the solution of the NLS equation in critica l case, there\nexist much more results in the deterministic case than those in stoch astic case. For\ninstance, in the deterministic case, [ 19] finds a threshold R by the optimal constant\nof Gagliardo–Nirenberg’s inequality, and proves the global existenc e of the solution\nwhenσ=2\ndand/ba∇dblu0/ba∇dbl</ba∇dblR/ba∇dbl. After adding the noise and damped effect, we wonder\nwhether these effect will influence the existence of a threshold in ma ss critical case\nfor the damped stochastic NLS equation, which is one of the main inte rests in this\npaper.\nThe another main interest is to study the damped effect and the nois e effect on\nthe blow-up in the focusing mass-(super)critical case. It’s well kn own that the solu-\ntion of deterministic NLS equation with a= 0 in the focusing mass-(super)critical\ncase will blow up in some finite time when u0possesses some negative Hamilton-\nian, see [ 3, 5, 17 ] and references therein. When a >0, the damped term has the\neffect to delay the blow-up, see [ 16, 17, 18 ] and references therein. For instance,\nforσ >2\nd, the blow-up may occur for small values of a(see e.g. [ 18]) and large\nenough values of acan ensure the global existence of the solution for σ≥2\nd(see e.g.\n[16]). In stochastic case, the noise also has an impact on blow-up solutio ns. [13]\nshows that the noise effect can accelerate the formation of singula rity, and that the\nsolution of Eq. (1) with a= 0 in focusing supercritical case will blow up in a finite\ntime with a positive probability when the variance of the initial datum is fi nite.\nThe blow-up solution for the stochastic NLS equation driven by addit ive noises is\nconsidered in [ 11]. When the noise of stochastic NLS equation is non-conservative,\n[2] shows that adding a large multiplicative Gaussian noise can prevent t he blow-up\nin any finite time with high probability.\nThroughout this paper, we assume that the local-wellposedness of the solu-\ntion of Eq. (1) holds. The local solution u(·) is defined on a random interval\n[0,τ∗(u0,ω)), where τ∗(u0,ω) is a stopping time such that\nτ∗(u0,ω) = +∞,or lim\nt→τ∗(u0,ω)/ba∇dblu(t)/ba∇dblH1= +∞.\nFirst, the evolution of charge and energy of the local solution are in troduced. By\nusing the optimal constant of Gagliardo–Nirenberg’s inequality, we s how the a\npriori estimation in H1-norm, and prove that the threshold R is unchanged when\na≥0,σ=d\n2and initial datum is deterministic. Moreover, based on the provedON GLOBAL EXISTENCE AND BLOW-UP FOR DAMPED STOCHASTIC NLS EQ UATION 3\nexponential integrability of the solution u, i.e.,\nsup\nt∈[0,∞)E/bracketleftBig\nexp/parenleftBig/ba∇dblu(t)/ba∇dbl2\nH1\neαt/parenrightBig/bracketrightBig\n≤C(u0,a,Q)\nwithαdepending on u0,aandQ,we obtain the strong continuous dependence\non the initial data in one dimensional case, which is not a trivial proper ty for\nstochastic partial differential equation with non-global coefficient s, see [6, 8] and\nreferences therein. We would like to mention that this exponential in tegrability is\nuseful for studying the continuous dependence on noises, expon ential tail estimate\nof the solution, strong and weak convergence rates of numerical approximations,\nsee [6, 7, 8, 9, 15 ] and references therein.\nNext we consider the influence of damped term and noise on the blow- up. For\nthe damped stochastic NLS equation, that is, a >0, the method used in [ 13] to\nget the blow-up condition is not available since the variance identity of Eq. (1)\ndo not have a polynomial expansion. To overcome this difficulty, we mo dify the\nenergy, momentum and variance identity which is similar to [ 18], and deduce a\nsharp blow-up condition. Indeed, we show that under some mild assu mptions on\nu0andQ, if there exist z≥4aσ\nσd−2and¯tsuch that\nE/bracketleftBig\nV(u0)/bracketrightBig\n+4¯tE/bracketleftBig\nG(u0)/bracketrightBig\n+(8¯t2+8\n3z¯t3)E/bracketleftBig\nH(u0)/bracketrightBig\n+/parenleftBig4\n3¯t3+4\n3z¯t4/parenrightBig\nE/bracketleftBig\n/ba∇dblu0/ba∇dbl2/bracketrightBig\n/ba∇dblfQ/ba∇dblL∞≤0,\nwherefQ=/summationtext\nk∈N+|∇Q1\n2ek|2, then\nP(τ∗(u0)≤¯t)>0.\nThis implies that no matter how large the damped effect is, the blow-up phenom-\nenon will not disappear. We remark that the above blow-up condition can be\ndegenerated to the blow-up condition in conservative stochastic c ase and in the\ndeterministic case. On the other hand, if the noise satisfies more co nditions, using\nthe rescaling transform idea in [ 1], we prove that when a→ ∞andσ≥2\nd, for any\nfixed time T, the blow-up of the solution does not happen with probab ility 1.\nThis paper is organized as follows. In Section 2, we study the evolutio n of\ncharge and energy, and show the global existence of the unique so lution. In Section\n3, the modified variance identity is given. Based on it, we obtain a shar p blow-up\ncondition. Furthermore, we prove that when the value of the damp ed coefficient a\nbecomes large enough, the solution does not blow up at any finite time with high\nprobability. At last, We give a short conclusion in Section 4.\n2. Global existence of solutions for critical stochastic NL S equations\nIn this section, we focus on the global existence and some propert ies of the so-\nlution for Eq. (1). Throughout this paper, we assume that the loca l well-posedness\nfor Eq. (1) holds. For the local well-posedness for Eq. (1), we ref er to [1, 12, 14 ]\nand references therein. When consider the focusing mass-(supe r)critical case, [ 11]\nproves that the solution of Eq. (1) blows up with any initial data for t he additive\ncase. For the stochastic NLS driven by the multiplicative noise, similar situation\nhappens with any initial datum in the super-critical case (see e.g. [ 2, 13]). This\nphenomenon is different from the deterministic case, where the solu tion will blow4 JIANBO CUI, JIALIN HONG, AND LIYING SUN\nup in some finite time when u0possesses some negative Hamiltonian in the focus-\ning mass-(super)critical case (see e.g. [ 3, 5, 17 ]). However, it is still not clear\non whether or not the solution of Eq. (1) equation globally exists in cr itical case.\nNotice that when the noise is independent of space and a= 0, i.e.,\ndu=i(∆u+|u|2σu)dt−1\n2udt+iudβ(t),\nu(0) =u0,\ntheglobalexistenceandblow-upresultsbecomemoreclear. Inthis case,onecanuse\nthe infinite dimensional Doss–Sussman type transformation u(t) = exp( iβ(t))y(t)\nto get the well-posedness and blow-up results, where y(t) satisfies\ndy=i∆ydt+i|y|2σydt,\ny(0) =u0.\nWe first study the global exsitence of the solution of Eq. (1) in the f ocusing\ncritical case. This suggests that the critical nonlinearity in multiplica tive cases\nis different from the supercritical nonlinearity, and that the critica l nonlinear-\nity combined with the dispersion term dominates the behavior of the s olution.\nFor convenience, we assume that u0∈H1is a deterministic function and that/summationtext\nk/ba∇dblQ1\n2ek/ba∇dbl2\nH1+/ba∇dblfQ/ba∇dblL∞<∞withfQ=/summationtext\nk|∇Q1\n2ek|2. The solution u(·) of Eq.\n(1) is defined on a random interval [0 ,τ∗(u0,ω)), where τ∗(u0,ω) is a stopping time\nsuch that\nτ∗(u0,ω) = +∞,or lim\nt→τ∗(u0,ω)/ba∇dblu(t)/ba∇dblH1= +∞.\nTo get a priori estimate of u, we first study the evolution of charge M(u(t)) :=\n/ba∇dblu(t)/ba∇dbl2and energy H(u) :=1\n2/ba∇dbl∇u/ba∇dbl2−λ\n2σ+2/ba∇dblu/ba∇dbl2σ+2\nL2σ+2in the following lemma.\nLemma2.1.Assume that u0∈H1and/summationtext\nk/ba∇dblQ1\n2ek/ba∇dbl2\nH1+/ba∇dblfQ/ba∇dblL∞<∞. For any\nτ < τ∗(u0), we have\nM(u(τ)) =e−2aτM(u0),a.s., (2)\nand\nH(u(τ)) =H(u0)−a/integraldisplayτ\n0(/ba∇dbl∇u(s)/ba∇dbl2−/ba∇dblu(s)/ba∇dbl2σ+2\nL2σ+2)ds (3)\n−Im/integraldisplay\nRd/integraldisplayτ\n0¯u(s)∇u(s)∇dW(s)dx\n+1\n2/summationdisplay\nk∈N+/integraldisplayτ\n0/ba∇dblu(s)∇Q1\n2ek/ba∇dbl2ds,a.s.\nProofFor purpose of obtaining the charge and energy evolution of u, the\ntruncated argument in [ 12] is applied. In detail, let N∈N+andK >0 and define\nthe operators Θ N,N∈Nby\nΘNv:=F−1/parenleftBig\nθ(|·|\nj)∗N/parenrightBig\n,\nwhereFis the Fourier transform and θ∈C∞\ncis a real-valued and nonnegative\nfunction satisfying θ(x) = 1 for |x| ≤1,θ(x) = 0 for |x|>2. Using the aboveON GLOBAL EXISTENCE AND BLOW-UP FOR DAMPED STOCHASTIC NLS EQ UATION 5\nnotation, we have the truncated approximation, for m= (m1,m2)∈N2,\ndum\nK=i/parenleftbigg\nΘm1∆um\nK+θ/parenleftBig/ba∇dblum\nK/ba∇dblH1\nK/parenrightBig\nΘm2(|um\nK|2σum\nK)−aum\nK−FQm2\n2um\nK/parenrightbigg\ndt\n+ium\nKΘm2dW(t),(4)\nwhereFQm2:=/summationtext\nk∈N+(Θm2(Q1\n2)ek)2. Combining with Itˆ o formula in [0 ,τ] and\ntaking limits as m→ ∞, the evolution of the charge (2) is obtained by choosing\na large enough K. Similarly, using the above arguments, the energy evolution law\n(3) can be proved. /square\nRemark 2.1.The truncatedargument is also available for stochastic NLS equa-\ntion with the homogenous Dirichlet boundary condition. In t his case, replacing ΘN\nby the projection operator PN, then the truncated Galerkin approximated equation\nbecomes\nduN\nK=i/parenleftbigg\n∆uN\nK+θ/parenleftBig/ba∇dbluN\nK/ba∇dblH1\nK/parenrightBig\nPN(|uN\nK|2σuN\nK)−aPNuN\nK−PN/parenleftBigFQ\n2uN\nK/parenrightBig/parenrightbigg\ndt\n+iPN(uN\nKdW(t)),(5)\nwhereK >0,N∈N+. The inverse inequality, for s≥1,\n/ba∇dbluN\nK/ba∇dblHs≤C(N)/ba∇dbluN\nK/ba∇dblH1\nimplies the coefficients of Eq. (5)are globally Lipschitz. Therefore, by the arguments\nin[12], the result of Lemma 2.1 holds.\nIn order to illustrate the global well-posedness result, we introduc e the optimal\nconstant for Gagliardo–Nirenberg inequality and its corresponding ground state\nsolution (see e.g. [ 19]).\nLemma2.2.The best constant Cσ,dfor Gagliardo–Nirenberg inequality\n/ba∇dblf/ba∇dbl2σ+2\nL2σ+2≤Cσ,d/ba∇dbl∇f/ba∇dblσd/ba∇dblf/ba∇dbl2+σ(2−d)(6)\nwithf∈H1(Rd),0< σ <2\nd−2andd≥2is given by\nCσ,d= (σ+1)2(2+2σ−σd)−1+σd\n2\n(σd)σd\n21\n/ba∇dblR/ba∇dbl2σ,\nwhereRis the positive solution (ground state solution) of ∆R−R+R2σ+1= 0.\nBased on Lemma 2.1 and Lemma 2.2, we are in position to show the global\nexistence of u. For the sake of simplicity, the procedure about truncated argum ents\nand taking limits is omitted in the rest of this paper.\nTheorem 2.1.Assume that u0∈H1with/ba∇dblu0/ba∇dbl</ba∇dblR/ba∇dbl,/summationtext\nk/ba∇dblQ1\n2ek/ba∇dbl2\nH1+/ba∇dblfQ/ba∇dblL∞<\n∞. Then there exists a unique global solution of Eq. (1)inH1, i.e.,τ∗(u0) =∞.\nProofSince the local well-posedness of Eq. (1) is shown by [ 12, 4, 1 ], we\nonly need to get the uniform boundedness of /ba∇dblu/ba∇dblH1to ensure the global existence\nof the solution. By the charge conservation law, Gagliardo–Nirenbe rg inequality\nandσd= 2,\n/ba∇dbl∇u/ba∇dbl2≤2H(u)+1\nσ+1Cσ,d/ba∇dblu/ba∇dbl2σ/ba∇dbl∇u/ba∇dbl2,6 JIANBO CUI, JIALIN HONG, AND LIYING SUN\nwhereCσ,d=σ+1\n/bardblR/bardbl2σandRis the ground state solution of ∆ R−R+R2σ+1= 0.\nThen the energy evolution of uimplies that for any T0>0, any stopping time\nτ <inf(T0,τ∗(u0)) and any time t≤τ,\n/parenleftbigg\n1−/ba∇dblu(t)/ba∇dbl2σ\n/ba∇dblR/ba∇dbl2σ/parenrightbigg\n/ba∇dbl∇u(t)/ba∇dbl2≤2H(u0)−2a/integraldisplayt\n0(/ba∇dbl∇u(s)/ba∇dbl2−/ba∇dblu(s)/ba∇dbl2σ+2\nL2σ+2)ds\n−2Im/integraldisplay\nO/integraldisplayt\n0¯u(s)∇u(s)∇dW(s)dx\n+/summationdisplay\nk∈N+/integraldisplayt\n0/integraldisplay\nO|u(s)|2|∇Q1\n2ek|2dxds,\nwhereH(u0)>0 since/ba∇dblu0/ba∇dbl</ba∇dblR/ba∇dbl. After taking expectation, we have\n/parenleftbigg\n1−/ba∇dblu0/ba∇dbl2σ\n/ba∇dblR/ba∇dbl2σ/parenrightbigg\nE/bracketleftBig\n/ba∇dbl∇u(t)/ba∇dbl2/bracketrightBig\n≤2H(u0)−4a/integraldisplayt\n0E/bracketleftBig\nH(u(s))/bracketrightBig\nds\n+/integraldisplayt\n02aσ\nσ+1E/bracketleftBig\n/ba∇dblu(s)/ba∇dbl2σ+2\nL2σ+2/bracketrightBig\nds\n+E/parenleftBigg/summationdisplay\nk∈N+/integraldisplayt\n0/integraldisplay\nO|u(s)|2|∇Q1\n2ek|2dxds/parenrightBigg\n.\nThen H¨ older inequality and Sobolev embedding theorem yield that\n/parenleftbigg\n1−/ba∇dblu0/ba∇dbl2σ\n/ba∇dblR/ba∇dbl2σ/parenrightbigg\nE/bracketleftBig\n/ba∇dbl∇u(t)/ba∇dbl2/bracketrightBig\n≤2H(u0)−4a/integraldisplayt\n0E/bracketleftBig\n/ba∇dbl∇u(s)/ba∇dbl2/bracketrightBig\nds+2aσ\nσ+1Cσ,d/integraldisplayt\n0E/bracketleftBig\n/ba∇dbl∇u(s)/ba∇dbl2/ba∇dblu(s)/ba∇dbl2σ/bracketrightBig\nds\n+/integraldisplayt\n0/summationdisplay\nk∈N+/ba∇dbl∇Q1\n2ek/ba∇dbl2\nL∞E/bracketleftBig\n/ba∇dblu(s)/ba∇dbl2/bracketrightBig\nds\n≤2H(u0)+2aσ\nσ+1Cσ,d/integraldisplayt\n0e−2asE/bracketleftBig\n/ba∇dbl∇u(s)/ba∇dbl2/bracketrightBig\n/ba∇dblu0/ba∇dbl2σds\n+/integraldisplayt\n0/summationdisplay\nk∈N+/ba∇dbl∇Q1\n2ek/ba∇dbl2\nL∞e−2as/ba∇dblu0/ba∇dbl2ds.\nGronwall inequality implies that\nsup\nt≤τE/bracketleftBig\n/ba∇dbl∇u(t)/ba∇dbl2/bracketrightBig\n≤C(u0,R,a,Q,T 0).ON GLOBAL EXISTENCE AND BLOW-UP FOR DAMPED STOCHASTIC NLS EQ UATION 7\nMoreover, using Burkholder–Davis–Gundy inequality and Young ineq uality,\nE/bracketleftBig\nsup\nt≤τ/ba∇dbl∇u(t)/ba∇dbl2/bracketrightBig\n≤C(u0,R,a,Q,T 0)+CE/bracketleftBig\nsup\nt≤τ/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0/integraldisplay\nRd¯u(s)∇u(s)∇dW(t)dx/vextendsingle/vextendsingle/vextendsingle/bracketrightBig\n≤C(u0,R,a,Q,T 0)+CE/bracketleftBig/vextendsingle/vextendsingle/vextendsingle/integraldisplayτ\n0/summationdisplay\nk∈N+/ba∇dblu(s)∇u(s)∇Q1\n2ek/ba∇dbl2dt/vextendsingle/vextendsingle/vextendsingle1\n2/bracketrightBig\n≤C(u0,R,a,Q,T 0)+ǫE/bracketleftBig\nsup\nt≤τ/ba∇dbl∇u(t)/ba∇dbl2/bracketrightBig\n+C(ǫ)E/bracketleftBig/integraldisplayτ\n0/summationdisplay\nk∈N+/ba∇dbl∇Q1\n2ek/ba∇dbl2\nL∞/ba∇dblu(s)/ba∇dbl2dt/bracketrightBig\n,\nwhereǫ <1\n2. The last term of the above inequality is bounded by H¨ older inequality\nand the charge evolution law (2), which in turns implies that\nE/bracketleftBig\nsup\nt≤τ/ba∇dbl∇u(t)/ba∇dbl2/bracketrightBig\n≤C(u0,R,a,Q,T 0).\nThese a priori estimations combined with local well-posedness imply th e global ex-\nistence of the unique solution. /square\nThe condition /ba∇dblu0/ba∇dbl</ba∇dblR/ba∇dblin Theorem2.1is asufficient conditionfor the global\nexistence of the solution. Based on it, we get a upper bound of the p robability of\nthe blow-up with a random initial datum at any finite time.\nCorollary 2.1.Assume that u0is random initial datum. Under the condition\nof Theorem 2.1, we have\nP(τ∗(u0)<∞)≤P(/ba∇dblu0/ba∇dbl ≥ /ba∇dblR/ba∇dbl).\nMoreover, we can get the following exponential integrability of u, which is\nuseful for studying the continuous dependence on initial data and noise, exponen-\ntial tail estimate of the solution, strong and weak convergence ra tes of numerical\napproximations (see e.g. [ 7, 8, 9]).\nProposition 2.1.Assume that the condition of Theorem 2.1 holds, then there\nexist constants C=C(u0,R,a,Q)andα=α(u0,R,a,Q)such that\nsup\nt∈[0,T]E/bracketleftBig\nexp/parenleftbigg/ba∇dblu/ba∇dbl2\nH1\neαt/parenrightbigg/bracketrightBig\n≤C.\nProofDenote by a positive number c:= 1−/bardblu0/bardbl2σ\n/bardblR/bardbl2σ<1,µ(u) =i∆u+i|u|2u−\n1\n2FQu−aandσ(u) =iuQ1\n2. Based on Gagliardo–Nirenberg inequality (6), we only\nneed to show the boundedness of sup\nt∈[0,T]E/bracketleftBig\nexp/parenleftBig\n2H(u(t))\nceαt/parenrightBig/bracketrightBig\n. Using the truncated8 JIANBO CUI, JIALIN HONG, AND LIYING SUN\narguments and taking limits, we get\n2DH(u(s))µ(u(s))\nc+tr[D2H(u(s))σ(u(s))σ∗(u(s))]\nc+4/ba∇dblσ∗(u(s))DH(u(s))/ba∇dbl2\nc2eαt\n=1\nc/summationdisplay\nk/ba∇dblu(s)∇Q1\n2ek/ba∇dbl2−2a\nc/ba∇dbl∇u(s)/ba∇dbl2+2a\nc/ba∇dblu(s)/ba∇dbl2σ+2\nL2σ+2\n+4\nc2eαt/summationdisplay\nk/ba∇dbl∇u(s)/ba∇dbl2/ba∇dblu(s)∇Q1\n2ek/ba∇dbl2.\nAgain by the Gagliardo–Nirenberg inequality (6) and the Sobolev embe dding the-\norem, we obtain\n2DH(u(s))µ(u(s))\nc+tr[D2H(u(s))σ(u(s))σ∗(u(s))]\nc+4/ba∇dblσ∗(u(s))DH(u(s))/ba∇dbl2\nc2eαt\n≤1\nc/ba∇dblu(s)/ba∇dbl2/ba∇dblfQ/ba∇dblL∞−4a\ncH(u(s))+2aσ\nc(σ+1)Cσ,d/ba∇dbl∇u(s)/ba∇dbl2/ba∇dblu(s)/ba∇dbl2σ\n+4\nc2eαt/ba∇dbl∇u(s)/ba∇dbl2/ba∇dblu(s)/ba∇dbl2/ba∇dblfQ/ba∇dblL∞\n≤1\nc/ba∇dblu(s)/ba∇dbl2/ba∇dblfQ/ba∇dblL∞−4a\ncH(u(s))+4aσ\nc2(σ+1)/ba∇dblu(s)/ba∇dbl2σCσ,dH(u(s))\n+8\nc3eαt/ba∇dblu(s)/ba∇dbl2/ba∇dblfQ/ba∇dblL∞H(u(s)).\nCharge evolution law in Lemma 2.1 leads that\n2DH(u(s))µ(u(s))\nc+tr[D2H(u(s))σ(u)σ∗(u)]\nc+4/ba∇dblσ∗(u)DH(u(s))/ba∇dbl2\nc2eαt\n≤e−2at\nc/ba∇dblu0/ba∇dbl2/ba∇dblfQ/ba∇dblL∞\n+/parenleftBigg\n−2a+2aσ\nc(σ+1)/ba∇dblu0/ba∇dbl2σCσ,d+4\nc2e(α+2a)t/ba∇dblu0/ba∇dbl2/ba∇dblfQ/ba∇dblL∞/parenrightBigg\n2\ncH(u(s)).\nThen Lemma 3.1 in [ 8] implies that\nsup\nt∈[0,T]E/bracketleftBigg\nexp/parenleftBigg\n2\ncH(u(t))\neαt/parenrightBigg/bracketrightBigg\n≤E/bracketleftBig\nexp/parenleftBig2\ncH(u0)/parenrightBig/bracketrightBig\nexp/parenleftBig/integraldisplayT\n0/ba∇dblu0/ba∇dbl2/ba∇dblfQ/ba∇dblL∞\ne(α+2a)tdt/parenrightBig\n,\nwhereα≥ −2a+2aσ\nc(σ+1)/ba∇dblu0/ba∇dbl2σCσ,d+4\nc2/ba∇dblu0/ba∇dbl2/ba∇dblfQ/ba∇dbl2\nL∞. /square\nApplying the above exponential integrability of exact solution, we de duce the\nfollowing strongly continuous dependence on the initial data.\nCorollary 2.2.Assumethat d= 1,σ= 2,a >0,u0,v0∈H1andmax(/ba∇dblu0/ba∇dbl,/ba∇dblv0/ba∇dbl)<\nmin/parenleftBig\n4ac(2c−1)\n4c2+/bardblfQ/bardblL∞,ac(2c−1)\n/bardblfQ/bardbl∞/parenrightBig\nwith1\n2< c <1. Letu={u(t) :t∈[0,T]}and\nv={v(t) :t∈[0,T]}be the solutions of Eq. (1)with initial data u0andv0, re-\nspectively. Under the condition of Proposition 2.1, then th ere exists a constant\nC=C(p,u0,v0,Q,a)such that\nsup\nt∈[0,T]E/bracketleftBig\n/ba∇dblu(t)−v(t)/ba∇dbl2/bracketrightBig\n≤CE/bracketleftBig\n/ba∇dblu0−v0/ba∇dbl2/bracketrightBig\n.ON GLOBAL EXISTENCE AND BLOW-UP FOR DAMPED STOCHASTIC NLS EQ UATION 9\nProofApplying the truncated arguments, Itˆ o fromula and taking limits yie ld\nthat\n/ba∇dblu(t)−v(t)/ba∇dbl2=/ba∇dblu0−v0/ba∇dbl2−2a/integraldisplayt\n0/ba∇dblu(s)−v(s)/ba∇dbl2ds\n+2/integraldisplayt\n0/an}b∇acketle{tu(s)−v(s),i∆(u(s)−v(s))/an}b∇acket∇i}htds\n+2/integraldisplayt\n0/an}b∇acketle{tu(s)−v(s),i(|u(s)|4u(s)−|v(s)|4v(s))/an}b∇acket∇i}htds\n=/ba∇dblu0−v0/ba∇dbl2−2a/integraldisplayt\n0/ba∇dblu(s)−v(s)/ba∇dbl2ds\n+2/integraldisplayt\n0/an}b∇acketle{tu(s)−v(s),i(|u(s)|4u(s)−|v(s)|4v(s))/an}b∇acket∇i}htds,\nSince for a,b∈C,|a|4a−|b|4b= (|a|4+|b|4)(a−b)+ab(|a|2+|b|2)(¯a−¯b)+|a|2|b|2(a−\nb), combining with Young inequality and Gagliardo–Nirenberg inequality, we get\n/ba∇dblu(t)−v(t)/ba∇dbl2=/ba∇dblu0−v0/ba∇dbl2−2a/integraldisplayt\n0/ba∇dblu(s)−v(s)/ba∇dbl2ds\n+2/integraldisplayt\n0/an}b∇acketle{tu(s)−v(s),iu(s)v(s)(|u(s)|2+|v(s)|2)(¯u(s)−¯v(s))/an}b∇acket∇i}htds\n≤ /ba∇dblu0−v0/ba∇dbl2−2a/integraldisplayt\n0/ba∇dblu(s)−v(s)/ba∇dbl2ds\n+2/integraldisplayt\n0/parenleftBig\n/ba∇dblu(s)/ba∇dbl4\nL∞+/ba∇dblv(s)/ba∇dbl4\nL∞/parenrightBig\n/ba∇dblu(s)−v(s)/ba∇dbl2ds.\nGronwall inequality and Gagliardo–Nirenberg inequality lead that\n/ba∇dblu(t)−v(t)/ba∇dbl2≤exp/parenleftBig/integraldisplayt\n02(/ba∇dblu(s)/ba∇dbl4\nL∞+/ba∇dblv(s)/ba∇dbl4\nL∞)ds/parenrightBig\n/ba∇dblu0−v0/ba∇dbl2\n≤exp/parenleftBig/integraldisplayt\n08e−2as/ba∇dblu0/ba∇dbl2/ba∇dbl∇u(s)/ba∇dbl2+8e−2as/ba∇dblv0/ba∇dbl2/ba∇dbl∇v(s)/ba∇dbl2ds/parenrightBig\n/ba∇dblu0−v0/ba∇dbl2.\nAfter taking expectation, by Young inequality, we obtain\nE/bracketleftBig\nsup\nt∈[0,T]/ba∇dblu(t)−v(t)/ba∇dbl2/bracketrightBig\n≤/radicalBigg\nE/bracketleftBig\nexp/parenleftBig/integraldisplayT\n016e−2as/ba∇dblu0/ba∇dbl2/ba∇dbl∇u(s)/ba∇dbl2ds/parenrightBig/bracketrightBig(7)\n×/radicalBigg\nE/bracketleftBig\nexp/parenleftBig/integraldisplayT\n016e−2as/ba∇dblv0/ba∇dbl2/ba∇dbl∇v(s)/ba∇dbl2ds/parenrightBig/bracketrightBig\n/ba∇dblu0−v0/ba∇dbl2\nIt is obvious that if these two exponential moments in the above ineq uality are\nbounded,thetheoremisproved. Forsimplicity, wetake E/bracketleftBig\nexp/parenleftBig/integraltextT\n016e−2as/ba∇dblu0/ba∇dbl2/ba∇dbl∇u(s)/ba∇dbl2ds/parenrightBig/bracketrightBig10 JIANBO CUI, JIALIN HONG, AND LIYING SUN\nas example. By Jensen inequality, for α <2a,\nE/bracketleftBig\nexp/parenleftBig/integraldisplayT\n016e−2as/ba∇dblu0/ba∇dbl2/ba∇dbl∇u(s)/ba∇dbl2ds/parenrightBig/bracketrightBig\n=E/bracketleftBig\nexp/parenleftBig/integraldisplayT\n0e−(2a−α)s16/ba∇dblu0/ba∇dbl2e−αs/ba∇dbl∇u(s)/ba∇dbl2ds/parenrightBig/bracketrightBig\n≤E/bracketleftBig\nexp/parenleftBig16\n2a−α/ba∇dblu0/ba∇dbl2e−αs/ba∇dbl∇u(s)/ba∇dbl2/parenrightBig/bracketrightBig\n.\nThenwetake α=−2a+2aσ\nc(σ+1)/ba∇dblu0/ba∇dbl2σCσ,d+4\nc2/ba∇dblu0/ba∇dbl2/ba∇dblfQ/ba∇dblL∞suchthat16\n2a−α/ba∇dblu0/ba∇dbl2≤\n1 andα <2a. Indeed, the assumption on u0andv0, together with Gagliardo–\nNirenberg inequality Eq. (6), implies\n/ba∇dblu0/ba∇dbl2≤a\n2−a\n4c−1\n4c2/ba∇dblu0/ba∇dbl2/ba∇dblfQ/ba∇dbl∞\n=a(2c−1)\n4c−1\n4c2/ba∇dblu0/ba∇dbl2/ba∇dblfQ/ba∇dbl∞.\nand\n1\nc+1\nac2/ba∇dblu0/ba∇dbl2/ba∇dblfQ/ba∇dbl∞<2,\nwhich ensure that16\n2a−α/ba∇dblu0/ba∇dbl2≤1 andα <2a. Proposition 2.1, combining the\nabove estimations, implies the uniform boundedness of the exponen tial moments of\nEq. (7), which completes the proof. /square\nRemark 2.2.Due to the fact that the ground state solution R(x) in one dime n-\nsion is31\n4√\ncosh(2x)with/ba∇dblR/ba∇dbl2=π√\n3, the above strong continuous dependence result\non initial data holds with max(/ba∇dblu0/ba∇dbl,/ba∇dblv0/ba∇dbl)<4/radicalBig\n3π2\n2whenabecomes large enough.\nFor/ba∇dblu0/ba∇dbl</ba∇dblR/ba∇dbl,a≥0, by the above arguments, we can get the continuous depen-\ndence on initial data in pathwise sense,\nsup\nt∈[0,T]/ba∇dblu(t)−v(t)/ba∇dbl ≤C(ω)/ba∇dblu0−v0/ba∇dbl.\n3. Blow-up of solutions in focusing mass-(super)critical c ase\nAs shown in Section 2, the result about well-posedness for Eq. (1) in the\ncritical case is similar to that in deterministic case. Notice that this ph enomenon is\ndifferent from the additive case with σ≥d\n2and the multiplicative case with σ >d\n2\n(see e.g. [ 11, 13]), where the singularity happens in any finite time with a positive\nprobability for any initial datum.\nInfact,theauthorsin[ 13]showthatfor σ≥2\nd, ifu0∈L2(Ω;Σ)∩L2σ+2(Ω;L2σ+2(Rd)),\nfQ=/summationtext\nk∈N+|∇Q1\n2ek|2and for some ¯t >0,\nE/bracketleftBig\nV(u0)/bracketrightBig\n+4E/bracketleftBig\nG(u0)/bracketrightBig\n¯t+8E/bracketleftBig\nH(u0)/bracketrightBig\n¯t2+4\n3¯t3/ba∇dblfQ/ba∇dblL∞E/bracketleftBig\nM(u0)/bracketrightBig\n<0,\nthenP(τ∗(u0)≤¯t)>0. The above result implies that if the energy of u0is a.s.\nnegative, then P(τ∗(u0)≤t)>0 for some t >0 provided the noise is not too\nstrong, i.e., /ba∇dblfQ/ba∇dblL∞is small enough. The natural question is whether the damped\neffect can prevent the blow-up phenomenon or not in stochastic ca se.ON GLOBAL EXISTENCE AND BLOW-UP FOR DAMPED STOCHASTIC NLS EQ UATION11\nTo study the blow-up phenomenon, we introduce the finite variance space\nΣ ={v∈H1:|x|v∈H}\nendowed with the norm /ba∇dbl·/ba∇dblΣ:\n/ba∇dblv/ba∇dbl2\nΣ=/ba∇dbl|x|v/ba∇dbl2+/ba∇dblv/ba∇dbl2\nH1,\nthe variance\nV(v) =/integraldisplay\nRd|x|2|v(x)|2dx, v∈Σ\nand the momentum\nG(v) = Im/integraldisplay\nRd¯v(x)x·∇v(x)dx, v∈Σ.\nWith the help of a smoothing procedure and truncated arguments ( see e.g. [ 13]),\nwe can prove rigorously the evolution laws of VandGfor the damped stochastic\nNLS equation.\nProposition 3.1.Assume that u0∈Σ. Under the conditions of Lemma 2.1,\nfor any stopping time τ < τ∗(u0)a.s., we have\nV(u(τ)) =V(u0)+4/integraldisplayτ\n0G(u(s))ds−2a/integraldisplayτ\n0V(u(s))ds,\nand\nG(u(τ)) =G(u0)+4/integraldisplayτ\n0H(u(s))ds−2a/integraldisplayτ\n0G(u(s))ds+2−σd\nσ+1/integraldisplayτ\n0/ba∇dblu(s)/ba∇dbl2σ+2\nL2σ+2ds\n+/summationdisplay\nk∈N/integraldisplayτ\n0/integraldisplay\nRd|u(s,x)|2x·∇(Q1\n2ek)(x)dxdβk(s).\nProofApplying Itˆ o formula to VandG, integration by parts and taking the\nimaginary part of the integration, we obtain\nV(u(τ)) =V(u0)+4/integraldisplayτ\n0G(u(s))ds−2a/integraldisplayτ\n0V(u(s))ds,12 JIANBO CUI, JIALIN HONG, AND LIYING SUN\nand\nG(u(τ)) =G(u0)+2/integraldisplayτ\n0Im/integraldisplay\nRdx·∇u/parenleftBig\n−i∆¯u−i|u|2σ¯u−a¯u−1\n2FQ¯u/parenrightBig\ndxds\n−d/integraldisplayτ\n0Im/integraldisplay\nRd/parenleftBig\ni∆u+i|u|2σu−au−1\n2FQu/parenrightBig\n¯udxds\n+2/integraldisplayτ\n0Im/integraldisplay\nRd−ix∇u¯udxdW(s)−d/integraldisplayτ\n0Im/integraldisplay\nRdi|u|2dxdW(s)\n+/integraldisplayτ\n0Im/integraldisplay\nRdx·∇(uQ1\n2ek)Q1\n2ek¯udxds\n=G(u0)+d/integraldisplayτ\n0/parenleftBig\n/ba∇dbl∇u(s)/ba∇dbl2−/ba∇dblu(s)/ba∇dbl2σ+2\nL2σ+2/parenrightBig\nds\n−2/integraldisplayτ\n0Im/integraldisplay\nRdx·∇ui∆¯udxds\n−2/integraldisplayτ\n0Im/integraldisplay\nRdx·∇u(i|u|2σ+2¯u)dxds−2a/integraldisplayτ\n0G(u(s))ds\n+/summationdisplay\nk∈N/integraldisplayτ\n0/integraldisplay\nRd|u(s,x)|2x·∇(Q1\n2ek)(x)dxdβk(s).\nBy the definition of Handσd= 2, we get\nG(uτ) =G(u0)+4/integraldisplayτ\n0H(u(s))ds+2−σd\nσ+1/integraldisplayτ\n0/ba∇dblu(s)/ba∇dbl2σ+2\nL2σ+2ds−2a/integraldisplayτ\n0G(u(s))ds\n+/summationdisplay\nk∈N/integraldisplayτ\n0/integraldisplay\nRd|u(s,x)|2x·∇(Q1\n2ek)(x)dxdβk(s).\n/square\nFor the damped stochastic NLS equation, the method in [ 13] is not available\nsince the damped effect will lead that the expansion of Vproduces many addition\nterms which can not be estimated directly. We introduce the modified energy,\ninvariance and momentum as in [ 18] and study the evolution of these modified\nquantities to investigate the blow-up condition for supercritical ca seσd >2,a >0.\nLemma3.1.Letb∈R. Under the same condition of Proposition 3.1, for any\nstopping time τ < τ∗(u0), we have\nebτH(u(τ)) =H(u0)+b/integraldisplayτ\n0ebsH(u(s))ds−a/integraldisplayτ\n0ebs/parenleftBig\n/ba∇dbl∇u(s)/ba∇dbl2−/ba∇dblu(s)/ba∇dbl2σ+2\nL2σ+2/parenrightBig\nds\n−Im/integraldisplay\nO/integraldisplayτ\n0ebs¯u∇u∇dWdx+1\n2/summationdisplay\nk∈N+/integraldisplayτ\n0ebs/ba∇dblu∇Q1\n2ek/ba∇dbl2ds, (8)\nebτG(u(τ)) =G(u0)−(2a−b)/integraldisplayτ\n0ebsG(u(s))ds\n+2/integraldisplayτ\n0ebs/parenleftBig\n2H(u(s))+2−σd\n2σ+2/ba∇dblu(s)/ba∇dbl2σ+2\nL2σ+2/parenrightBig\nds (9)\n+/summationdisplay\nk∈N/integraldisplayτ\n0/integraldisplay\nRdebs|u(s,x)|2x·∇(Q1\n2ek)(x)dxdβk(s),ON GLOBAL EXISTENCE AND BLOW-UP FOR DAMPED STOCHASTIC NLS EQ UATION13\nand\nebτV(u(τ)) =V(u0)+4/integraldisplayτ\n0ebsG(u(s))ds−(2a−b)/integraldisplayτ\n0ebsV(u(s))ds. (10)\nProofThe proof is similar to the proof of Lemma 2.1 and Proposition 3.1\nby using smoothing procedures, truncated arguments, integrat ion by parts and Itˆ o\nformula. More details, we refer to [ 13]. /square\nBased on Lemma 3.1, we prove a preliminary result on the blow-up cond ition\nfor Eq. (1) in the supercritical case.\nProposition 3.2.Letσd >2, a >0,b <2asatisfy4aσ\nσd−2≤2a−b, u0∈\nL2(Ω;Σ)∩L2σ+2(Ω;L2σ+2)and/summationtext\nk∈N+/ba∇dblQ1\n2ek/ba∇dbl2\nH1+/ba∇dblfQ/ba∇dblL∞<∞. Assume in\naddition that for some 0< y≤1\n2a−bsuch that\nE/bracketleftBig\nV(u0)+4yG(u0)+16y2H(u0)+8y3/ba∇dblu0/ba∇dbl2/ba∇dblfQ/ba∇dbl∞/bracketrightBig\n<0, (11)\nthen for some ¯t, we have\nP(τ∗(u0)≤¯t)>0.\nProofWe prove the assertion by contradiction. Assume that the solution u\nexists globally. Then for any t >0,τ∗(u0)> ta.s. Then we take τ=t. The\nevolution law of modified energy Eq. (8) , charge evolution law Eq. (2) and taking\nexpectation leads that\nE/bracketleftBig\nebtH(u(t))/bracketrightBig\n=H(u0)+b/integraldisplayt\n0E/bracketleftBig\nebsH(u(s))/bracketrightBig\nds−a/integraldisplayt\n0E/bracketleftBig\nebs/parenleftBig\n/ba∇dbl∇u(s)/ba∇dbl2−/ba∇dblu(s)/ba∇dbl2σ+2\nL2σ+2/parenrightBig/bracketrightBig\nds\n+1\n2/summationdisplay\nk∈N+/integraldisplayt\n0E/bracketleftBig\nebs/ba∇dblu∇Q1\n2ek/ba∇dbl2/bracketrightBig\nds\n≤H(u0)+(b\n2−a)/integraldisplayt\n0ebsE/bracketleftBig\n/ba∇dbl∇u/ba∇dbl2+2\nb−2a(a−b\n2σ+2)/ba∇dblu/ba∇dbl2σ+2\nL2σ+2/bracketrightBig\nds\n+1\n2/integraldisplayt\n0E/bracketleftBig\ne(b−2a)s/ba∇dblu0/ba∇dbl2/ba∇dblfQ/ba∇dblL∞/bracketrightBig\nds.\nNext, we aim to show a priori estimate on ebtG(u(t)). For simplicity, we denote\n˜H(u) :=/ba∇dbl∇u/ba∇dbl2−σd\n2σ+2/ba∇dblu/ba∇dbl2σ+2\n2σ+2. Applying the evolution of modified momentum\nEq. (9) and taking expectation, we obtain\nE/bracketleftBig\nebtG(u(t))/bracketrightBig\n=E/bracketleftBig\nG(u0)/bracketrightBig\n+2/integraldisplayt\n0ebsE/bracketleftBig\n˜H(u(s))/bracketrightBig\nds−(2a−b)/integraldisplayt\n0E/bracketleftBig\nebsG(u(s))/bracketrightBig\nds(12)\nTo control the second term E/bracketleftBig\n˜H(u(s))/bracketrightBig\nuniformly, we take b≤a[2−4σ\nσd−2] such\nthat\n/ba∇dbl∇u/ba∇dbl2+2\nb−2a(a−b\n2σ+2)/ba∇dblu/ba∇dbl2σ+2\n2σ+2≥2H(u)+2−σd\n2σ+2/ba∇dblu/ba∇dbl2σ+2\n2σ+2\n=˜H(u(s)).14 JIANBO CUI, JIALIN HONG, AND LIYING SUN\nThen the fact that ˜H(u)≤2H(u) leads that\nE/bracketleftBig1\n2ebt˜H(u(t))/bracketrightBig\n+(2a−b)/integraldisplayt\n0E/bracketleftBig1\n2ebs˜H(u(s))/bracketrightBig\nds\n≤H(u0)+1−e(b−2a)t\n4a−2b/ba∇dblu0/ba∇dbl2/ba∇dblfQ/ba∇dblL∞.\nUsing Gronwall inequality, we have\n/integraldisplayt\n0E/bracketleftBig1\n2ebs˜H(u(s))/bracketrightBig\nds≤1−e(b−2a)t\n2a−bH(u0)+1−e(b−2a)t\n2(2a−b)2/ba∇dblu0/ba∇dbl2/ba∇dblfQ/ba∇dblL∞\n−te(b−2a)t\n4a−2b/ba∇dblu0/ba∇dbl2/ba∇dblfQ/ba∇dblL∞,\nwhich derives that\n/integraldisplayt\n0/bracketleftBig\nebs˜H(u(s))/bracketrightBig\nds≤1\n2a−b/parenleftBig\n2H(u0)+1\n2a−b/ba∇dblu0/ba∇dbl2/ba∇dblfQ/ba∇dblL∞/parenrightBig\n.\nThe above estimation and Eq. (12) yield that\nE/bracketleftBig\nebtG(u(t))/bracketrightBig\n≤E/bracketleftBig\nG(u0)/bracketrightBig\n−(2a−b)/integraldisplayt\n0ebsG(u(s))ds\n+2\n2a−b/parenleftBig\n2H(u0)+1\n2a−b/ba∇dblu0/ba∇dbl2/ba∇dblfQ/ba∇dblL∞/parenrightBig\n.\nAgain by Gronwall inequality,\nE/bracketleftBig/integraldisplayt\n0ebsG(u(s))ds/bracketrightBig\n≤1−e(b−2a)t\n2a−bE/bracketleftBig\nG(u0)+2\n2a−b/parenleftBig\n2H(u0)+1\n2a−b/ba∇dblu0/ba∇dbl2/ba∇dblfQ/ba∇dblL∞/parenrightBig/bracketrightBig\n.\nThe above inequality, Eq. (10) and the non-negativity of Vyield that\nE/bracketleftBig\nebtV(u(t))/bracketrightBig\n=E/bracketleftBig\nV(u0)/bracketrightBig\n+4/integraldisplayt\n0ebsE/bracketleftBig\nG(u(s))/bracketrightBig\nds−(2a−b)/integraldisplayτ\n0ebsE/bracketleftBig\nV(u(s))/bracketrightBig\nds\n≤E/bracketleftBig\nV(u0)/bracketrightBig\n+4\n2a−b(1−e(b−2a)t)E/bracketleftBig\nG(u0)+2\n2a−b/parenleftBig\n2H(u0)\n+1\n2a−b/ba∇dblu0/ba∇dbl2/ba∇dblfQ/ba∇dblL∞/parenrightBig/bracketrightBig\n.\nSince the assumption means that E/bracketleftBig\n2a−b\n4V(u0)+G(u0)+2\n2a−b/parenleftBig\n2H(u0)+1\n2a−b/ba∇dblu0/ba∇dbl2\n/ba∇dblfQ/ba∇dblL∞/parenrightBig/bracketrightBig\n<0, we only need to make\n¯t≥ −1\n2a−bln/parenleftBigg2a−b\n4E/bracketleftBig\nV(u0)/bracketrightBig\n+E/bracketleftBig\nG(u0)+2\n2a−b/parenleftBig\n2H(u0)+1\n2a−b/ba∇dblu0/ba∇dbl2/ba∇dblfQ/ba∇dblL∞/parenrightBig/bracketrightBig\nE/bracketleftBig\nG(u0)+2\n2a−b/parenleftBig\n2H(u0)+1\n2a−b/ba∇dblu0/ba∇dbl2/ba∇dblfQ/ba∇dblL∞/parenrightBig/bracketrightBig/parenrightBigg\n.ON GLOBAL EXISTENCE AND BLOW-UP FOR DAMPED STOCHASTIC NLS EQ UATION15\nBy the above inequality and the positivity of ebt, there exists some ¯tsuch that\nlim\nt→¯tE/bracketleftBig\nV(u(t))/bracketrightBig\n= 0. By the uncertainty principle /ba∇dblu/ba∇dbl2≤2\nd/ba∇dbl∇u/ba∇dbl/vextenddouble/vextenddouble|x|u/vextenddouble/vextenddouble, we get\n/radicalbigg\nE/bracketleftBig\n/ba∇dbl∇u(t)/ba∇dbl2/bracketrightBig\n≥d/ba∇dblu0/ba∇dbl2\n2e2at/radicalbigg\nE/bracketleftBig/vextenddouble/vextenddouble|x|u/vextenddouble/vextenddouble2/bracketrightBig\n=d/ba∇dblu0/ba∇dbl2\n2e(2a−b)t/radicalbigg\nE/bracketleftBig\nV(u(t))/bracketrightBig.\nThe above estimation yields that/radicalbigg\nE/bracketleftBig\n/ba∇dbl∇u(t)/ba∇dbl2/bracketrightBig\ngoes into ∞whenttends to ¯t,\nwhich leads to a contradiction and finishes the proof. /square\nRemark 3.1.The above proposition implies that when\nE/bracketleftBig\nV(u0)+σd−2\naσG(u0)+(σd−2\naσ)2H(u0)+1\n8(σd−2\naσ)3/ba∇dblu0/ba∇dbl2/ba∇dblfQ/ba∇dbl∞/bracketrightBig\n<0,\nthe solution of Eq. (1)will blow up in a finite time with a positive probability.\nClearly, if the energy of u0is negative a.s., damped effect is not strong and the\nnoise is small enough, the blow up phenomenon of the solution always happens.\nThe blow-up condition Eq. (11) presents the effect of the damped t erm. The\nfollowing result gives a sharp time-dependent blow-up condition.\nTheorem 3.1.Letσd >2,a≥0,u0∈L2(Ω;Σ)∩L2σ+2(Ω;L2σ+2), and/summationtext\nk∈N+/ba∇dblQ1\n2ek/ba∇dbl2+/ba∇dblfQ/ba∇dblL∞<∞. Assume also that for some z≥4aσ\nσd−2and¯tsuch\nthat\nE/bracketleftBig\nV(u0)/bracketrightBig\n+4¯tE/bracketleftBig\nG(u0)/bracketrightBig\n+(8¯t2+8\n3z¯t3)E/bracketleftBig\nH(u0)/bracketrightBig\n+/parenleftBig4\n3¯t3+4\n3z¯t4/parenrightBig\nE/bracketleftBig\n/ba∇dblu0/ba∇dbl2/bracketrightBig\n/ba∇dblfQ/ba∇dblL∞≤0,(13)\nthen we have\nP(τ∗(u0)≤¯t)>0.\nProofThe proofis similarto the proofofProposition3.2. Usingthe evolution s\nof the modified energy Eq. (8), the new energy ˜Hin Proposition 3.2 leads that for\nz≥4aσ\nσd−2,\n/integraldisplayt\n0E/bracketleftBig\nebs˜H(u(s))/bracketrightBig\nds\n≤1−e−zt\nzE/bracketleftBig\n2H(u0)/bracketrightBig\n+1−(1+zt)e−zt\nz2E/bracketleftBig\n/ba∇dblu0/ba∇dbl2/bracketrightBig\n/ba∇dblfQ/ba∇dblL∞.16 JIANBO CUI, JIALIN HONG, AND LIYING SUN\nApplying the the evolution ofthe modified momentum Eq. (9), invarian ceEq. (10),\nand combining with the Taylor expansion of ez,z∈R, we get\nE/bracketleftBig\nebtV(u(t))/bracketrightBig\n≤E/bracketleftBig\nV(u0)/bracketrightBig\n+1−e−zt\nzE/bracketleftBig\n4G(u0)/bracketrightBig\n+1−(1+zt)e−zt\nz2E/bracketleftBig\n16H(u0)/bracketrightBig\n+1−(1+zt+1\n2(zt)2)e−zt\nz38E/bracketleftBig\n/ba∇dblu0/ba∇dbl2/bracketrightBig\n/ba∇dblfQ/ba∇dblL∞\n≤E/bracketleftBig\nV(u0)/bracketrightBig\n+4tE/bracketleftBig\nG(u0)/bracketrightBig\n+(8t2+8\n3zt3)E/bracketleftBig\nH(u0)/bracketrightBig\n+/parenleftBig4\n3t3+4\n3zt4/parenrightBig\nE/bracketleftBig\n/ba∇dblu0/ba∇dbl2/bracketrightBig\n/ba∇dblfQ/ba∇dblL∞\nSimilar arguments in Proposition 3.2 yield that Eq. (13) is the blow-up co ndition.\n/square\nRemark 3.2.The above blow-up condition is sharp in the sense that if z→0,\nthe above condition can be degenerated to the blow-up condit ion in the conservative\ncase, i.e. for some ¯t\nE/bracketleftBig\nV(u0)/bracketrightBig\n+4E/bracketleftBig\nG(u0)/bracketrightBig\n¯t+8E/bracketleftBig\nH(u0)/bracketrightBig\n¯t2+4\n3¯t3/ba∇dblfQ/ba∇dblL∞E/bracketleftBig\nM(u0)/bracketrightBig\n<0.\nSince the blow-up condition for Eq. (1)in supercritical case is similar tothe condition\nof Theorem 4.1 in [13], it is possible to apply the skills and arguments in [13]to get\na stronger result that P(τ∗(u0,a)> t)>0, for any tandu0,u0/ne}ationslash= 0under some\nassumptions on u0,dandQ1\n2.\nRemark 3.3.The blow-up condition in critical case can not be obtained by the\nmethod in Proposition 3.2, we only get some necessary condit ion in Remark 2.1.\nIt is our future work to study the blow-up phenomena of the sol ution and show the\nsufficient blow-up condition for damped stochastic NLS equat ions in critical case.\nIt seems that when abecomes larger,the blow-up time becomes longerand that\nwhenagoes to∞, the blow-up condition is not satisfied. Indeed, we can show that\nwhen the damped effect is large enough, the damped effect can prev ent the blow-\nup of the solution with high probability. The key of the proof is using th e infinite\ndimensional Doss–Sussman type transformationin [ 1, 2]. In the following theorem,\nwe assume that the noise satisfies/summationtext\nk∈N+/ba∇dblQ1\n2ek/ba∇dblC2\nb<∞and the following decay\ncondition\nlim\nx→∞η(x)(|Q1\n2em(x)|+|∇Q1\n2em(x)|+|∆Q1\n2em(x)|) = 0,\nwhereη(x) = 1+|x|2ifd/ne}ationslash= 2, and η(x) = (1+ |x|2)(ln(2+|x|2))2ifd= 2. Under\nthese assumptions, the local well-posednessis obtained in [ 1]. We alsoremarkwhen\n{ek}k∈N+is an orthonormal basis of H, the decay condition natural holds. In this\ncase,ekcan be chosen as the k-th Hermite function in H, and meanwhile as the\nk-th eigenvector of the operator Q1\n2.\nTheorem 3.2.Assume that a >0,σd≥2and/summationtext\nk/ba∇dbl∆Q1\n2ek/ba∇dbl2\nL∞<∞. Then\nfor anyu0∈H1and0< T <∞, we have\nlim\na→∞P(τ∗(u0,a)> T) = 1.ON GLOBAL EXISTENCE AND BLOW-UP FOR DAMPED STOCHASTIC NLS EQ UATION17\nProofWe apply the rescaling transformation v(t) =eat−iW(t)u(t) to Eq. (1)\nand get the following random partial di���erential equation\ndv=iexp/parenleftBig\nat−iW(t)/parenrightBig\n∆u(t)dt+iexp/parenleftBig\nat−iW(t)/parenrightBig\n|u|2σudt (14)\n=i/parenleftBig\n∆+2i∇(W(t))·∇+|∇W(t)|2+i∆W(t)/parenrightBig\nv\n+iexp/parenleftBig\n−2aσt/parenrightBig\n|v|2σvdt\n:=A(t)vdt+iexp/parenleftBig\n−2aσt/parenrightBig\n|v|2σvdt.\nBy Lemma 2.4 in [ 1], the solutions of Eq. (14) and Eq. (1) are equivalent. Let\nr=4σ+4\nσdsuch that (2 σ+2,r) is a Strichartz pair. Next we recall the proof of the\nlocal well-posedness and set\nXτ\nR:=/braceleftBig\nv∈C(0,τ;H1)∩Lr(0,τ;W1,2σ+2)/vextendsingle/vextendsingle/vextendsingle/ba∇dblv/ba∇dblC(0,τ;H1)+/ba∇dblv/ba∇dblLr(0,τ;W1,2σ+2)≤R/bracerightBig\n.\nConsidering the solution map Gof Eq. (14), by the random Strichartz estimate\nand similar arguments in [ 1], we obtain\n/ba∇dblG(v)/ba∇dblC(0,τ;H1)+/ba∇dblG(v)/ba∇dblLr(0,τ;W1,2σ+2)\n≤2Cτ/ba∇dblu0/ba∇dblH1+2R2σ+1CSCτ/ba∇dblexp(−2aσt)/ba∇dblLq(0,τ),\n/ba∇dblG(v)−G(w)/ba∇dblC(0,τ;H)+/ba∇dblG(v)−G(w)/ba∇dblLr(0,τ;L2σ+2)\n≤4CSCτR2σ/ba∇dblexp(−2aσt)/ba∇dblLq(0,τ)/ba∇dblv−w/ba∇dblLr(0,τ;L2σ+2),\nwherev,w∈ Xτ\nR,CS= (2σ+ 1)D2σ,Dis the Sobolev embedding coefficient\nformL2σ+2toH1,Cτis the random Strichartz estimate coefficient, q >1 and\n1\nq= 1−2\nr>0. Now we take R= 4Cτ/ba∇dblu0/ba∇dblH1and\nτ(a) = inf/braceleftBig\nt >0/vextendsingle/vextendsingle/vextendsingle2CSCt\naσR2σ>1/bracerightBig\nsuch that the mapping G:Xτ\nR−→Xτ\nRhas a fixed point in the Banach space/parenleftBig\nXτ\nR,/ba∇dbl·/ba∇dblC(0,τ;H)+/ba∇dbl·/ba∇dblLr(0,τ;L2σ+2)/parenrightBig\n, which implies the local well-posedness of Eq.\n(14).\nNow, we aim to show that lim a→∞P(τ∗(u0,a)> T) = 1. Based on the result\nin [1, Lemma 2.7] that Ct,t >0 isFtmeasurable, increasing and continuous, the\ndefinition of τ∗(u0,a) and the a.s. boundedness of Ctyield that\nlim\na→∞P(τ∗(u0,a)> T)≥lim\na→∞P(τ(a)> T)\n= lim\na→∞P(2CSCt\naσR2σ≤1, t∈[0,T])\n≥lim\na→∞P(2CSCT\naσR2σ≤1)\n≥lim\na→∞P(2CSCT\nσR2σ≤a) = 1.\n/square\nRemark 3.4.If the noise is space-independent or disappears, applying t he ar-\nguments in Theorem 2.1 and Theorem 3.2, one can get global exi stence and blow-up\nresults of the solutions for the damped stochastic NLS equat ion.18 JIANBO CUI, JIALIN HONG, AND LIYING SUN\n4. Conclusions\nIn this paper, we consider the influence of both damped term and no ise on\nthe stochastic nonlinear Schr¨ odinger equation driven by multiplicat ive noise. We\nfirst show the global existence of the unique solution for damped st ochastic NLS\nequation in critical case and study the exponential integrability and the continuous\ndependence on the initial data of the solution. Then based on the mo dified variance\nidentity, we deduce a sharp blow-up condition for damped stochast ic NLS equation\nin supercritical case. Moreover, we prove that the large damped e ffect can prevent\nthe blow-up with high probability.\nReferences\n[1] V. Barbu, M. R¨ ockner, and D. Zhang. Stochastic nonlinea r Schr¨ odinger equations. Nonlinear\nAnal., 136:168–194, 2016.\n[2] V.Barbu, M.R¨ ockner, and D.Zhang. Stochastic nonlinea rSchr¨ odinger equations: no blow-up\nin the non-conservative case. J.Differential Equations, 263(11):7919–7940, 2017.\n[3] J. Bourgain. Globalsolutions ofnonlinear Schr¨ odinger equations, volume 46 of American\nMathematical SocietyColloquium Publications.American Mathematical Society, Providence ,\nRI, 1999.\n[4] Z. Brze´ zniak and A. Millet. On the stochastic Strichart z estimates and the stochastic nonlin-\near Schr¨ odinger equation on a compact Riemannian manifold .Potential Anal., 41(2):269–315,\n2014.\n[5] T. Cazenave. Semilinear Schr¨ odinger equations, volume 10 of Courant Lecture Notesin\nMathematics. New York University, Courant Institute of Mat hematical Sciences, New York;\nAmerican Mathematical Society, Providence, RI, 2003.\n[6] S. Cox, M. Hutzenthaler, and A. Jentzen. Local lipschitz continuity in the initial value and\nstrong completeness for nonlinear stochastic differential equations. arXiv:1309.5595.\n[7] J.Cuiand J.Hong. Analysisofa splitting scheme fordamp ed stochastic nonlinear schrodinger\nequation with multiplicative noise. arXiv:1711.00516.\n[8] J. Cui, J. Hong, and Z. Liu. Strong convergence rate of fini te difference approximations for\nstochastic cubic Schr¨ odinger equations. J.Differential Equations, 263(7):3687–3713, 2017.\n[9] J. Cui, J. Hong, Z. Liu, and W. Zhou. Strong convergence ra te of splitting schemes for\nstochastic nonlinear Schr¨ odinger equations. arXiv:1701.05680.\n[10] A. de Bouard and A. Debussche. A stochastic nonlinear Sc hr¨ odinger equation with multi-\nplicative noise. Comm.Math.Phys., 205(1):161–181, 1999.\n[11] A. de Bouard and A. Debussche. On the effect of a noise on th e solutions of the focusing\nsupercritical nonlinear Schr¨ odinger equation. Probab. TheoryRelated Fields, 123(1):76–96,\n2002.\n[12] A. de Bouard and A. Debussche. The stochastic nonlinear Schr¨ odinger equation in H1.\nStochastic Anal.Appl., 21(1):97–126, 2003.\n[13] A. de Bouard and A. Debussche. Blow-up for the stochasti c nonlinear Schr¨ odinger equation\nwith multiplicative noise. Ann.Probab., 33(3):1078–1110, 2005.\n[14] F.Hornung. The nonlinear stochastic schr¨ odinger equ ation via stochastic strichartz estimates.\narXiv:1611.07325.\n[15] M.Hutzenthaler and A.Jentzen. Ona perturbation theor yand onstrongconvergence ratesfor\nstochastic ordinary and partial differential equations wit h non-globally monotone coefficients.\narXiv:1401.0295.\n[16] M. Ohta and G. Todorova. Remarks on global existence and blowup for damped nonlinear\nSchr¨ odinger equations. Discrete Contin. Dyn.Syst., 23(4):1313–1325, 2009.\n[17] C. Sulem and P. Sulem. Thenonlinear Schr¨ odinger equation, volume 139 of Applied\nMathematical Sciences. Springer-Verlag, New York, 1999. Self-focusing and wave collapse.\n[18] M. Tsutsumi. Nonexistence of global solutions to the Ca uchy problem for the damped non-\nlinear Schr¨ odinger equations. SIAMJ.Math.Anal., 15(2):357–366, 1984.\n[19] Michael I. Weinstein. Nonlinear Schr¨ odinger equatio ns and sharp interpolation estimates.\nComm.Math.Phys., 87(4):567–576, 1982/83.ON GLOBAL EXISTENCE AND BLOW-UP FOR DAMPED STOCHASTIC NLS EQ UATION19\n1. LSEC, ICMSEC, Academy of Mathematics and Systems Science , Chinese Academy\nof Sciences, Beijing, 100190, China 2. School of Mathematic al Science, University\nof Chinese Academy of Sciences, Beijing, 100049, China\nE-mail address :jianbocui@lsec.cc.ac.cn\n1. LSEC, ICMSEC, Academy of Mathematics and Systems Science , Chinese Academy\nof Sciences, Beijing, 100190, China 2. School of Mathematic al Science, University\nof Chinese Academy of Sciences, Beijing, 100049, China\nE-mail address :hjl@lsec.cc.ac.cn\n1. LSEC, ICMSEC, Academy of Mathematics and Systems Science , Chinese Academy\nof Sciences, Beijing, 100190, China 2. School of Mathematic al Science, University\nof Chinese Academy of Sciences, Beijing, 100049, China\nE-mail address :liyingsun@lsec.cc.ac.cn"    },    {        "title": "0807.1998v1.Atomic_and_optical_tests_of_Lorentz_symmetry.pdf",        "content": "arXiv:0807.1998v1  [hep-ph]  12 Jul 2008Atomic and optical tests of Lorentz symmetry\nNeil Russell\nPhysics Department, Northern Michigan University, Marquette, M I, USA\nE-mail:nrussell@nmu.edu\nAbstract. This article ‡reports on the Fourth Meeting on Lorentz and CPT\nSymmetry, CPT ’07, held in August 2007 in Bloomington, Indiana, USA. The focus\nis on recent tests of Lorentz symmetry using atomic and optical ph ysics. Results\npresented at the meeting include improved bounds on Lorentz violat ion in the photon\nsector, and the first bounds on several coefficients in the gravity sector.\n1. Introduction\nThe AMO community has played a major role in testing Lorentz symmet ry over the last\ndecade. Much of this is due the innovative design work of experiment alists, who have\nsteadily improved the attainable levels of precision in various experime nts. Exquisite\ntests of Lorentz symmetry in AMO and other areas have been perf ormed with optical\nand microwave cavities [1, 2], with atomic clocks and masers [3], with to rsion pendula\n[4, 5], and with Penning traps confining electrons or protons [6]. The se endeavors have\nvigorously sought to test whether nature is exactly Lorentz symm etric.\nThe creation in the 1990s of a broad theoretical framework for Lo rentz violation is\na major reason for the surge of interest in studies of Lorentz sym metry. This framework,\nwhich spans the spectrum of quantum and gravitational physics, is called the Standard-\nModel Extension, or SME [7]. It has opened numerous avenues to pr obe Planck-scale\nphysics, where Lorentz violations may occur, without the need to a ttain the 1019GeV\nenergies at which the theories of particle physics and gravitation ar e expected to merge.\nLorentz violations are possible, for example, in string theory with sp ontaneous\nsymmetry breaking. This idea, of using a potential to spontaneous ly break Lorentz\nsymmetry, thus enforcing a nonzero vacuum value for a tensor fie ld, was introduced\nby Kosteleck´ y and Samuel [8]. Several models for such fields have b een created as\nuseful test cases, and include the ones known as the bumblebee fie ld and the cardinal\nfield [9]. It is remarkable that AMO experiments, such as the ones men tioned above,\nare able in principle to achieve sensitivity to the vacuum expectation v alues of these\nfields. Alternative approaches to Lorentz violation include ones invo lving spacetime-\nvarying fields [10], noncommutative field theories [11], quantum-grav ity [12], branes\n[13], supersymmetry [14], and a variety of others [15].\n‡Invited Comment for CAMOP section of Physica ScriptaAtomic and optical tests of Lorentz symmetry 2\nThe goal of this article is to provide background and details of exper imental and\ntheoreticalworkonLorentzviolationintheareaofAMOphysicspre sented attheFourth\nMeeting on CPT and Lorentz Violation, held in Bloomington, Indiana, in A ugust 2007\n[16]. A number of new bounds on SME coefficients were presented at t he meeting. A\nfull listing of the experimental measurements of the SME coefficient s in all sectors can\nbe found in Ref. [17].\n2. The Standard-Model Extension\nThe SME is defined at the level of the effective field-theory action SSME, with a variety\nof Lorentz-violating terms appearing in the lagrangian density LSME:\nSSME=/integraldisplay\nLSMEd4x. (1)\nOne way to evaluate the content of the lagrangian density LSMEis to separate the\ngravity and matter sectors:\nLSME=Lmatter+Lgravity. (2)\nTerms in the gravitational piece Lgravityare constructed only from the basic\ngravitational fields. The choice of these is guided by the need for a r ealistic description\nof nature, which must include particles with spin. Since Riemann-Cart an spacetimes\nincorporate spinors in curved spacetimes and can be constructed using the vierbein and\nthe spin connection [18], these are the chosen basic fields. The familia r gravitational\nfields, such as the curvature and the torsion, can be expressed in terms of the vierbein\nandthespinconnection. Thematterpiece Lmatterconsistsofallotherterms. Itincludes\nones constructed from the spinors ψdescribing ordinary matter (protons, neutrons, and\nelectrons), gauge fields such as Aµfor the photon, andthe fields describing particles that\nare not ‘ordinary,’ like muons, mesons, neutrinos and so on. The ter ms inLmattercan\ninclude the basic gravitational fields together with these matter fie lds, whereas Lgravity\nis ‘pure,’ containing only gravitational fields.\nLorentz violation in the flat-spacetime (Minkowski) limit with no torsion has been\nstudied extensively since the basic theoretical framework was intr oduced [7]. In this\nlimit, the metric gµνhas nonzero constant values on the diagonal only, there are no\ngravitational fields to consider, and the only lagrangian density of r elevance is Lmatter.\nThe Lorentz-preserving part of this lagrangian density is the stan dard model of particle\nphysics, while the Lorentz-violating part contains terms with coeffic ients that can be\nexperimentally probed.\nFor more than a decade, experimental limits have been placed on coe fficients\nfor Lorentz violation in the torsion-free Minkowski limit of the SME. I n the case of\nordinary matter and radiation, relevant studies include the ones me ntioned above, as\nwell as others involving high-speed ions [19], cosmological birefringen ce [20, 21], and\nsatellite-mounted oscillators [22]. For other particles and fields in the Minkowski limit\nof the matter sector, theoretical and experimental studies inclu de ones looking at muons\n[23, 24], neutral mesons [25], neutrinos [26], the Higgs [27], and baryo genesis [28]. AAtomic and optical tests of Lorentz symmetry 3\nvariety of astrophysical processes involving both ordinary and ot her matter place limits\non Lorentz violation [29].\nIn the case of nonzero torsion, the Minkowski limit of the matter se ctor has recently\nbeen studied. Several new bounds on components of the torsion t ensor have been found\n[30] based on experiments in the AMO field.\nIn the pure-gravity sector, the Lorentz-preserving part of Lgravitycontains the\nconventional Einstein-Hilbert lagrangian from which the Einstein field equations follow\nwhen torsion is zero and no other terms are present. The Lorentz -violating terms in\nLgravityprovide a framework for a variety of experimental tests of Loren tz symmetry in\nthe context of pure gravity [31]. The first experimental measure ments of coefficients for\nLorentz-violating terms in this sector were presented at CPT ’07 [3 2, 33].\nThe following sections provide anoverview of recent experiments an dtheory ineach\nof these sectors, with emphasis on results relating to AMO physics t hat were presented\nat CPT ’07.\n3. AMO Lorentz tests of the Minkowski limit of the SME\n3.1. Couplings of fermions to the SME background\nSince Lorentz-violating background fields are known to be small, the analysis of effects\nthat might occur is readily handled using perturbation theory. For m ost applications\nwith ordinary fermionic matter, the unperturbed system is obtaine d from the Dirac\nequation with solutions being the spinors ψ. Many of the principles encapsulated in the\nDirac equation have recently been studied with an eye towards atom ic-interferometry\nbased tests of basic principles including Lorentz symmetry, the univ ersality of free fall,\nlocality, and the superposition principle [34]. Other issues such asst ability and causality\nhavebeenresearched inthiscontext, aswell asinfieldtheory[35]. A varietyofcouplings\nof fermions to Lorentz-violating background fields have been stud ied. For example, one\nterm appearing in the lagrangian density is [7]:\nLmatter⊃bµψγ5γµψ. (3)\nDistinct coefficients bµare used to quantify Lorentz violation for each fermion.\nPerturbative analysis of this term can be used to find the shifts in th e spectra of\nelectrons in Penning traps [6], hydrogen and antihydrogen [36], atom ic clocks and\nmasers[3], torsion pendula [5], and other systems. Many of these ex periments involve\nthe comparison of highly stable frequencies with each other [37].\nIn the neutron sector of the SME, the most stringent bounds on L orentz violation\nhave been obtained using a He-Xe dual maser at the Harvard-Smith sonian Center for\nAstrophysics. Limits on symmetry breaking among the rotational c omponents of the\nLorentz group are at the level of 10−31GeV [38] and, on the boost components, at the\nlevel of 10−27GeV [39]. An improvement in precision of about an order of magnitude\nis expected after current upgrades are completed. These include upgraded temperature\ncontrols, optimized noblegaspressures andcell geometries, incre ased Zeeman frequency,Atomic and optical tests of Lorentz symmetry 4\nproperspatialdefinitionofmasingensembles, andimprovedstability ofthedouble-tuned\nresonator [40].\nA group at Princeton University has designed, built, and operated a potassium-\nhelium co-magnetometer with sensitivity to electron, proton, and n eutron coefficients\nfor Lorentz violation. The potassium and helium atoms are confined w ithin a glass\ncell and controlled using optical pumping techniques. Using data tak en over a period\nof 15 months, this magnetometer, dubbed CPT-I, has achieved ex cellent sensitivity\nto a variety of effects including sidereal signals that would be expect ed from a fixed\nLorentz-violating background. Preliminary results include a bound a t the level of about\n10−30GeV on the equatorial components of the proton bµcoefficient [41]. A second-\ngeneration co-magnetometer, CPT-II, is currently being implemen ted to achieve yet\nhigher sensitivities. This device is mounted on a turntable, making pos sible cycle times\nof much less than a day. This is expected to much improve the sensitiv ity to sidereal\neffects. Other innovations have been introduced to improve sensit ivities in various ways.\nThese include shorter optical path lengths, reduction of convect ion noise in the oven\narea, evacuation of air from the optical path, and improved magne tic shielding. CPT-II\nis expected to surpass the sensitivity of CPT-I by several orders of magnitude [42].\nExperiments with antihydrogen have the potential to find signals of Lorentz\nviolation that are not accessible with other systems. There are thr ee groups working on\nantihydrogen physics at CERN: the ‘Antihydrogen Laser Physics Ap paratus’ (ALPHA)\ncollaboration, the ‘Atomic Spectroscopy and Collisions using Slow Antip rotons’\n(ASACUSA) collaboration , and the ‘Antihydrogen Trap’ (ATRAP) co llaboration.\nALPHA and ASACUSA were represented at CPT ’07.\nThe ASACUSA collaboration has conducted several precision exper iments using\nthe laser spectroscopy of antiprotonic helium. The group has meas ured the antiproton-\nto-electron mass ratio to a precision of 2 parts per billion [43], which is w ithin an\norder of magnitude of the proton-to-electron mass ratio found u sing a Penning-trap\ncomparison of a proton and an electron. Theoretical studies of Lo rentz violation\nin antihydrogen have shown that unsuppressed signals could poten tially occur in the\ncomparison of the hyperfine spectral lines of hydrogen and antihy drogen [36]. The\nASACUSAcollaborationplanstomeasurethehyperfinelinesofantihy drogeninaStern-\nGerlach beam arrangement [44]. The expected resolution is at the lev el of 10−21GeV.\nThe ALPHA collaborationaimsto produce trappedantihydrogen with theeventual\ngoalofconductingprecisecomparisonsofthespectraofantihyd rogenandhydrogen. The\ngroup demonstrated the trapping of antiprotons from the CERN a ntiproton decelerator\nin 2006. The design involves a Penning-Malmberg trap featuring a mag netic octopole\nconfiguration [45] to confine positrons and antiprotons in the same region. Methods of\ncooling and compressing the plasmas to enhance the rate of antihyd rogen formation are\nbeing investigated.\nThe E¨ ot-Wash group at the University of Washington in Seattle has investigated\ncouplingsofspintoLorentz-violatingSMEbackgroundfields[4]. Thea pparatususedfor\nthis consists of a spin-polarized torsion pendulum suspended by a 75 -cm tungsten fiber.Atomic and optical tests of Lorentz symmetry 5\nIt has minimal magnetic and gravitational moments, and a net numbe r of polarized\nspins on the order of 1023, making it highly sensitive to the coupling of these spins\nto the Lorentz-violating background field bµfor the electron. The component of this\nbackground that is parallel to the rotation axis of the Earth has be en bounded at the\nlevel of a few parts in 10−30GeV by this experiment. The limits it places on the two\ncomponents in the equatorial plane are an additional order of magn itude tighter.\nAnothersystemwherelargenumbersofspin-polarizedatomsmayb eabletoamplify\nLorentz-violating effects is the Bose-Einstein condensate. Since t his involves atoms that\narebosonic, thestatistical properties canbeexpected to bever y different fromfermionic\nsystems. Under suitable conditions, spin-polarized Bose-Einstein c ondensates may be\nsensitive to Lorentz-violating background fields at a level compara ble to other existing\ntests [46].\nSpace-based experimental tests of fundamental physics are mo tivated by their\npotential to reach higher precisions than earth-based ones and t o probe otherwise\ninaccessible observables. A number of proposals and projects at v arious stages\nof development exist in the European Space Agency and NASA commu nities.\nThese include the Laser Interferometer Space Antenna (LISA), its precursor LISA\nPathfinder (LISAPF), theGrandUnificationandGravity Explorer ( GAUGE), theLaser\nAstrometric Test of Relativity (LATOR), the Astronomical Space T est of Relativity\nusingOpticalDevices(ASTROD),theOdysseyMissionaimedatexplor inggravityinthe\nSolar System, and the Matter-Wave Explorer of Gravity (MWXG) [47 ]. Technological\nadvances making such missions attractive for physics experiments include the ability to\ncreate drag-free platforms using systems such as micronewton t hrusters, and precision\ncapacitive, magnetic, and optical sensing of proof-mass behavior . Other proposals for\nhigh-precision space tests include ones based on atomic-clock comp arisons [22].\nA recent muon experiment at the Brookhaven National Laborator y, while not\ndirectly in the field of AMO physics, may be of interest since it has many similarities\nwith the Penning-trap system. The E821 experiment, run by the g−2 collaboration,\nmeasured the anomaly frequency of positive and negative muons st ored in the AGS\nring. Analysis of sidereal variations in these frequencies limited the e quatorial-plane\n˜bµcoefficients for Lorentz violation at the level of 10−24GeV [23]. Further analysis is\nexpected tobeableto placeconstraints ona variety ofcombination s ofmuon coefficients\nfor Lorentz violation.\n3.2. Couplings of photons to the SME background\nIn the Minkowski limit of the SME without torsion, the following photon -sector term\nhas been the primary focus of a number of experiments:\nLmatter⊃ −1\n4(kF)κλµνFκλFµν. (4)\nTo date, most of the tests in this sector have focussed on propag ating electromagnetic\nfields, although practical tests are possible in statics [48]. After a ccounting forAtomic and optical tests of Lorentz symmetry 6\nTable 1. Photon-sector results reported by the Humboldt University grou p [50]. The\nvalue ofβis 10−4\nCombination Result\n(˜κe−)XY(−0.1±0.6)×10−17\n(˜κe−)XZ(−2.0±0.9)×10−17\n(˜κe−)YZ(−0.3±1.4)×10−17\n(˜κe−)XX−(˜κe−)YY(−2.0±1.7)×10−17\n(˜κe−)ZZ(−0.2±3.1)×10−17\nβ(˜κo+)XY(−2.5±2.5)×10−17\nβ(˜κo+)XZ(1.5±1.7)×10−17\nβ(˜κo+)YZ(−1.0±1.5)×10−17\nsymmetries there are nineteen independent components for the c oefficients ( kF)κλµν.\nThere are ten linear combinations of these that imply birefringence, and these have\nbeen constrained tightly using observations of distant cosmologica l sources [20]. The\nremaining nine have been studied extensively in laboratory experimen ts with microwave\nand optical cavity oscillators. Experiments have placed bounds on lin ear combinations\nof these nine coefficients, denoted by ˜ κe−and ˜κo+. Recent results from two such\nexperiments were presented at CPT ’07.\nAn experiment at the University of Western Australia involves two cr yogenic\nsapphire oscillators, rotated about the vertical axis with a period o f 18 seconds. By\ntaking data over a time scale of about one year, measurements hav e been made of all\neight independent ˜ κe−and ˜κo+components without any non-cancelation assumptions\n[49]. A second experiment by this group at the University of Western Australia consists\nof a Mach-Zehnder microwave interferometer mounted on a rotat ing platform. After\ncompletion of the development stages, it is expected to measure ˜ κtrat competitive\nlevels [49].\nAn order of magnitude improvement in sensitivity is expected in an exp eriment\nat the Humboldt University in Berlin, Germany. It compares the optic al frequencies\nin two orthogonal cavities created in a single block of fused silica [50]. T he system is\nmaintained in a thermally insulated and vibration isolated vacuum chamb er, which is\nmounted on a turntable with a period of 45 seconds. This and other e xperiments have\nutilized such rotatingturntables to improve precisions over earlier v ersions that relied on\nthe rotation of the earth to seek anisotropies. The preliminary res ults of this experiment\nplace some of the tightest constraints on a variety of the ˜ κJK\ne+and ˜κJK\no−coefficients for\nLorentz violation [17] and are listed in Table 1.Atomic and optical tests of Lorentz symmetry 7\nThe results from geographically distant experiments, such as the o nes discussed\nabove, can be combined to obtain additional information. Furtherm ore, coordinate and\nfield redefinitions can be used to establish various links between resu lts in different\nsectors of the SME. Recent work along these lines has led to severa l results [2].\nTheoretical considerations of precision Doppler-shift experiment s show that\nsensitivity to some coefficients for Lorentz violation, such as the cµνfor protons and\nelectrons, is possible in principle [51]. An experiment at the Max Planck I nstitute for\nNuclear Physics in Heidelberg, Germany, has conducted a test of sp ecial relativity by\nmeasuring the Doppler shifts of beams of lithium atoms traveling at sp eeds of about 3\nto 6 % of the speed of light [19]. One of the results of this experiment is\n|˜κtr|<8.4±10−8, (5)\naboundonLorentzviolationinthephotonsector. Additionalsensit ivity, tocomponents\nin the fermion sector, may be possible through the use of circularly p olarized lasers.\nHigher-order couplings constructed from the electromagnetic fie ldsAµorFµν,and\nderivatives, have been studied recently [21]. They include, for exam ple, the term\nLmatter⊃ −1\n2ǫκλµν(k(5)\nAF)γτ\nκAλ∂γ∂τFµν. (6)\nThe constant coefficient ( k(5)\nAF)γτ\nκhas the dimension of inverse mass, which ensures that\nthe full term is of dimension four in the mass. In general, there is an in finite number\nof terms constructed in this way by the inclusion of further derivat ives. Effects of such\ntermsinclude vacuumbirefringence, andrecent workhasplacedlimit sonthecoefficients\n(k(5)\nAF)γτ\nκand higher-order coefficients by studying polarization data from ob servations\nof the cosmic microwave background [21].\n4. AMO Lorentz tests in the gravitational sector\n4.1. Pure gravity sector\nThe first constraints on pure-gravity sector SME coefficients wer e presented at CPT ’07\nby two experimental groups, one working with lunar-laser ranging, and the other with\natomic interferometry. The SME terms of interest in these experim ents appear in the\nlagrangian density in the form [31]\nLgravity⊃1\n16πGsµνRT\nµν, (7)\nwhereGis the universal gravitational constant. This term couples the tra celess Ricci\ntensorRT\nµν, obtainedbycontractionofthecurvaturetensor Rµναβ, toaLorentz-violating\nbackground expressed as sµν. The coefficients sµν(x) have vacuum expectation values\nsµνinduced by spontaneous violation of local Lorentz symmetry. The fl uctuations of\nfields likesµν(x) about the vacuum expectation values sµνhave fascinating implications\nfor physics [9]. The coefficients of interest at present are sµν, which are traceless and\nantisymmetric and so have 9 independent values.Atomic and optical tests of Lorentz symmetry 8\nTable 2. Pure-gravity sector results reported by the Stanford Universit y group, using\na Cesium-based atomic gravimeter [33]. The σcoefficients are a combination of pure-\ngravity sector scoefficients and photon-sector coefficients.\nCombination Result\nσXX−σYY(−5.6±2.1)×10−9\nσXY(−0.09±79)×10−9\nσXZ(−13±37)×10−9\nσYZ(−61±38)×10−9\nσTY(−2.0±4.4)×10−5\nσTX(5.4±4.5)×10−5\nσTZ(1.1±26)×10−5\nA group at Stanford University has used a highly sensitive atomic gra vimeter to\nplace bounds on combinations involving the scoefficients and photon-sector coefficients.\nThe outstanding precision of gravimeters based on atom interfero metry stems from the\nability of neutral atoms to approach a freely falling reference fram e with high accuracy,\nand the ability of lasers to interrogate the motion with fantastic pre cision. The Stanford\ngroup controls and measures the behavior of matter waves forme d using clouds of Cs\natoms. The device has resolved the acceleration of gravity more th an three times better\nthan the best previously reported value [33]. Their results are given in Table 2.\nA Harvard group presented results of an analysis of more than 30 y ears of lunar\nlaser-ranging data, constraining six independent combinations of scoefficients at the\nlevel of 10−6and 10−11. These data were collected primarily at the McDonald Laser-\nRanging Station in Texas, USA, and the Cˆ ote d’Azur station in Grass e, France, during\nthe period spanning September 1969 and December 2003. Their res ults are reported in\nRef. [32], and are summarized in Table 3. The Apache Point Observato ry Lunar Laser-\nranging Operation (APOLLO) [52] is expected to improve on these re sults by about an\norder of magnitude. The telescope at Apache Point in New Mexico, US A, can detect\nreflections from the lunar retroreflector arrays even in daylight c onditions, and ranging\ncan be achieved at the millimeter level.\n4.2. Couplings of matter with gravitational fields\nTerms in Lmatterthat couple matter to gravitational fields are currently being stud ied\n[53] since they offer the possibility of obtaining new sensitivities to Lor entz violation in,\nfor example, the fermion sector. One approach is to start from th e relativistic theory\nusing the spin connection and the vierbein [18] as the basic gravitatio nal objects, andAtomic and optical tests of Lorentz symmetry 9\nTable 3. Pure-gravity sector results reported by the Harvard-Smithson ian group,\nbased on archival lunar laser-ranging data [32].\nCombination Result\ns11−s22(1.3±0.9)×10−10\ns12(6.9±4.5)×10−11\ns02(−5.2±4.8)×10−7\ns01(−0.8±1.1)×10−6\nsΩ⊕c (0.2±3.9)×10−7\nsΩ⊕s(−1.3±4.1)×10−7\nthe Dirac fermion ψand the photon field Aµas the basic non-gravitational objects.\nOne can then extract the nonrelativistic limit using, for example, a Fo uldy-Wouthuysen\ntransformation, to obtain a formalism appropriate for direct expe rimental analysis.\nAnother approach of interest is the classical theory involving point -particles rather than\nwave functions. Results are expected to provide the first direct s ensitivities to the aµ\ncoefficients for the proton, neutron, and electron [53].\nTorsionisabasicfieldinRiemann-Cartantheoriesofgravity, givingtw istingdegrees\nof freedom that are distinct from curvature. This field Tµ\nαβ, which has 24 independent\ncomponents, can be nonzero even in the Minkowski flat-spacetime limit. Couplings of\nfermions and other particles to this field have similarities with couplings of fermions to\nLorentz-violating background fields in the SME. This fact has been e xploited recently\nto deduce new bounds on 15 of the torsion components and the mos t stringent bounds\non the four minimally-coupled torsion components [30]. The latter fou r bounds, on the\naxial components Aµ≡ǫαβγµTαβγ/6, are:\n|AT|<2.9×10−27GeV,|AX|<2.1×10−31GeV,\n|AY|<2.5×10−31GeV,|AZ|<1.0×10−29GeV.\nThese results are based on experiments with a spin-polarized torsio n pendulum [4], and\nwith a helium-xenon dual maser [39].\n5. Closing\nThe Standard-Model Extension is an umbrella framework for tests of Lorentz symmetry\nin nature. 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Tasson, in preparation."    },    {        "title": "2311.08215v1.Electronic_plasma_diffusion_with_radiation_reaction_force_and_time_dependent_electric_field.pdf",        "content": "arXiv:2311.08215v1  [cond-mat.stat-mech]  14 Nov 2023Electronic plasma diffusion with radiation reaction force a nd\ntime-dependent electric field\nJ. F. Garc´ ıa-Camacho1,3,∗O. Contreras-Vergara2, N.\nS´ anchez-Salas2,†G. Ares de Parga2, and J. I. Jim´ enez-Aquino3\n1Departamento de Matem´ aticas, Unidad Profesional Interdi sciplinaria\nde Energ´ ıa y Movilidad, Instituto Polit´ ecnico Nacional,\nUP Zacatenco, CP 07738, CDMX, M´ exico.\n2Departamento de F´ ısica, Escuela Superior de F´ ısica y Mate m´ aticas,\nInstituto Polit´ ecnico Nacional, Edif. 9 UP Zacatenco, CP 0 7738, CDMX, M´ exico. and\n3Departamento de F´ ısica, Universidad Aut´ onoma\nMetropolitana-Iztapalapa, C.P. 09340, CDMX, M´ exico.\n(Dated: November 15, 2023)\nAbstract\nIn this work the explicit solution of the electronic plasma d iffusion with radiation reaction force,\nunder the action of an exponential decay external electric fi eld is given. The electron dynamics\nis described by a classical generalized Langevin equation c haracterized by an Ornstein-Uhlenbeck-\ntypefriction memorykernel, withaneffective memorytimewhi chaccounts fortheeffective thermal\ninteraction between the electron and its surroundings (the rmal collisions between electrons + ra-\ndiation reaction force). The incident electric field exerts an electric force on the electron, which in\nturn can induce an additional damping to the braking radiati on force, allowing a delay in the elec-\ntron characteristic time. This fact allows that the effective memory time be finite and positive, and\nas a consequence, obtaining physically admissible solutio ns of the stochastic Abraham-Lorentz-like\nequation. It is shown that the diffusion process is quasi-Mark ovian which includes the radiation\neffects.\nPACS numbers: 05.40.-a; 52.20.Fs\n∗Electronic address: jfgarciac@ipn.mx\n†Electronic address: nsanchezs@ipn.mx\n1I. INTRODUCTION\nThe Abraham-Lorentz equation [1, 2] is basically related to the stud y of an electron\ndynamics with radiation reaction force. It was derived from a Newto n second law and\ndue to an additional term proportional to the electron’s accelerat ion rate of change, it\nrepresents a classical non-relativistic third-order time derivative equation. The equation\nleads to paradoxical solutions such as, runaway solutions and violat ion to causality [3]. The\nsolution of such inconsistencies goes back to the works reported in the context of classical\n[4–7] and quantum [8–11] electrodynamics. It was precisely in the pa pers [8, 9, 11] in which\nthe electronic plasma diffusion was considered as a fluctuation-dissip ation phenomenon,\ndescribed by quantum GLE associated with an electron embedded in a heat bath.\nIn a recent paper [12], it has been shown, using the classical GLE tha t, the effects due\nto the radiation reaction force can be neglected. According to the data reported in [12]\nit is shown that, in a classical non relativistic regime, the collision time τis greater than\nthe electron’s characteristic time τeand thus the effective memory time ˆ τe=τ−τe≃τ.\nHereτis of the order of magnitude of the collision time between electrons, c oming from a\nBrownian motion-like manner, and τe= 6.26×10−24s, arises due to the radiation reaction\nforce. So that, in this classical description, the effective friction f orcem(τ−τe)...x, appearing\nin the Stochastic Abraham-Lorentz-like Equation (SALE), must be m(τ−τe)...x≈mτ...x,\nand therefore, the radiation reaction force mτe...x, does not have any effect on the electronic\nplasma diffusion. Due to this fact, it could be assumed that a classical description of the\nproblem would not be possible. However, it will be shown in the present contribution that,\nit is possible to achieve the goal if the following proposal is considered .\nThe idea is to make the amplitude of the effective friction force finite a nd positive, to take\ninto account the influence of the radiation reaction force. This can be done by considering\nthe incidence on the system of a time-dependent external electric field of the form Eext(t) =\nE0e−ωt, beingE0itsamplitude and ωitscharacteristic frequency whichisproposedtosatisfy\nω=1\nkτe, and the constant kmust be such that k≫1 (kτeis the electric field time decay).\nThe applied electric field exerts a damping electric force on each elect ron (with charge −e)\ngiven by Fext(t) =−eE0e−ωt, which in turn is added to the radiationreaction force, allowing\na delay in the characteristic time by kτe. Thek≫1 condition allows a control on the value\nofkτe, to be of the same order of magnitude than τ, in order to have a finite and positive\n2effective memory time, defined by ˆ τk=τ−kτe>0. This approach enables the derivation\nof physically consistent solutions for the SALE and avoids violation of causality. It is worth\nhighlighting that, even with the presence of the external electric f orce, the fluctuation-\ndissipation relation of the second kind is valid, an therefore the GLE f or the velocity reaches\nits equilibrium stationary state. Our present contribution adds to t he list of works reported\nintheliterature relatedto thestudy of Brownianmotionwith radiatio nreactionforcewithin\nClassical andQuantumMechanics, andconsidering gravitationaleff ects, including therecent\ncontribution [13–18].\nThis work is organized as follow. In Sec. II the classical GLE and its as sociated third\norder time-derivative Langevin equation are studied. The solution o f the dimensionless\nthree-order time derivative Langevin equation, is explicitly given for three specific cases of\na dimensionless parameter J, as shown in Sec. III. In Sec. IV, the numerical simulations\nare compared with theory and finally, the concluding remarks are giv en in Sec. V.\nII. THEORETICAL APPROACH\nConsider the GLE associated with a free particle Brownian motion\nm˙v=−/integraldisplayt\n0γ(t−t′)v(t′)dt′+f(t), (1)\nγ(t) being the generalized friction memory kernel and f(t) a Gaussian noise with zero mean\nvalue,/an}bracketle{tf(t)/an}bracketri}ht= 0, and a correlation function which satisfies the fluctuation-dissip ation rela-\ntion of the second kind [19]\n/an}bracketle{tf(t)f(t′)/an}bracketri}ht=kBTγ(t−t′), (2)\nwithkBthe Boltzmann’s constant and Tthe bath temperature. This fluctuation-dissipation\nrelationguarantees thenon-Markovian process (1) becomes sta tionaryinthe longtime limit.\nIn a recent publication [12], the classical SALE for an electron gas wit h radiation reaction\nforce, was derived by means of a classical GLE characterized by an Ornstein-Uhlenbeck-type\nfriction memory kernel. In this case, the fluctuation-dissipation re lation of the second kind\nsatisfies /an}bracketle{tf(t)f(t′)/an}bracketri}ht=γ0kBT\nˆτee−|t−t′|/ˆτe, beingγ0the friction coefficient and ˆ τe=τ−τe,\nan effective memory time. However, it has been demonstrated that within this classical\nnon-relativistic regime, where τ≫τe, the effects induced by the radiation reaction force\nare imperceptible. To incorporate these effects into a classical des cription of the electronic\n3plasma diffusion, we propose considering the influence of an externa l time-dependent electric\nfield, the purpose of which is to attenuate the electron’s characte ristic time, as explicitly\ndiscussed in the introduction of this work. The classical GLE is thus w ritten as\nm˙v=−/integraldisplayt\n0γ(t−t′)v(t′)dt′−F0e−ωt+f(t), (3)\nbeingF0=eE0. Ontheotherside, duetotheexponential decayoftheexternal electricforce,\nthe fluctuation-dissipation relation of the second kind is also valid, an d thus the stochastic\nprocess (3) is stationary in the long time limit. The electric force must induce a delay in\nthe electron’s characteristic time by a quantity kτe, which allows to get a finite memory\ntime ˆτk=τ−kτe>0, for an appropriate value of k≫1. As a consequence of this fact,\nthe friction memory kernel is assumed to satisfy γ(t−t′) =γ0\nˆτke−|t−t′|/ˆτk, and therefore the\nsecond kind fluctuation-dissipation relation reads\n/an}bracketle{tf(t)f(t′)/an}bracketri}ht=γ0kBT\nˆτke−|t−t′|/ˆτk. (4)\nSo, the classical GLEassociated with an electron into anelectron ga s with radiationreaction\nforce becomes\nm˙v=−γ0\nˆτk/integraldisplayt\n0e−(t−t′)/ˆτkv(t′)dt′−F0e−ωt+f(t). (5)\nUsing the provided expressions in [12]\nη(t) =−γ0\nˆτk/integraldisplayt\n0e−(t−t′)/ˆτkv(t′)dt′+f(t), (6)\nf(t) =√\nλ\nˆτk/integraldisplayt\n0e−(t−t′)/ˆτkξ(t′)dt′, (7)\nwhereλ=γ0kBT, andξ(t) is a Gaussian white noise with zero mean and a correlation\nfunction /an}bracketle{tξ(t)ξ(t′)/an}bracketri}ht= 2δ(t−t′), it becomes straightforward to derive\nmˆτk...x+m¨x=−γ0v+c0F0e−ωt+√\nλ ξ(t), (8)\nwithc0=ωτ−2. This is a classical SALE which describes the electronic plasma diffusio n\ntaking into account the braking to radiation force, when the syste m is under the action of\nan exponential decay electric field. In terms of dimensionless variab lesy=c1xands=c2t,\nwithc1=m\nˆτ2\nkc0F0andc2=1\nˆτk, Eq. (8) is further expressed as\nd3y\nds3+d2y\nds2+Jdy\nds=1\nc0e−(ωτ−1)s+γ0\nc0F0√\nD ξ(ˆτks), (9)\n4beingJ=ˆτk\nτr, withτr=m\nγ0,therelaxationtime, and D=kBT\nγ0, Einstein’sdiffusioncoefficient.\nThe product ωτ=τ\nkτe>2, accounts for the coupling effect between both the memory time τ\nandscaled electron characteristic time kτe. Inthe next section we will calculate the statistics\nof the electron gas Brownian motion by means of the solution of Eq. ( 9).\nIII. EXPLICIT SOLUTIONS\nThe explicit solution of Eq. (9) has three roots given by\nλ1= 0, λ 2=−1\n2+1\n2√\n1−4J, λ 3=−1\n2−1\n2√\n1−4J, (10)\nfrom which also three cases can be analyzed, namely: the cases of r eal, complex, and critical\nroots. Therefore, this highlights the essential role played by the J parameter.\nA. The case of real roots 1−4J >0\nThis case implies that J <1\n4, which means that ˆ τk<1\n4τr. Taking into account that\nˆτk=τ−kτe=1\nω(ωτ−1) andωτ >2, thus 2 < ωτ <1+1\n4ωτr. For zero initial conditions,\nthe solution for the average of the dimensionless quantities y,vy, anday, beingvy=dy\nds, and\nay=dvy\nds, are shown to be\n/an}bracketle{ty/an}bracketri}ht=1−e−(ωτ−1)s\nλ2λ3(ωτ−1)−e−λ2s−e−(ωτ−1)s\nλ2(λ3−λ2)(λ2+ωτ−1)\n+e−λ3s−e−(ωτ−1)s\nλ3(λ3−λ2)(λ3+ωτ−1), (11)\n/an}bracketle{tvy/an}bracketri}ht=−e−λ2s−e−(ωτ−1)s\n(λ3−λ2)(λ2+ωτ−1)+e−λ3s−e−(ωτ−1)s\n(λ3−λ2)(λ3+ωτ−1), (12)\n/an}bracketle{tay/an}bracketri}ht=−λ2(e−λ2s−e−(ωτ−1)s)\n(λ3−λ2)(λ2+ωτ−1)+λ3(e−λ3s−e−(ωτ−1)s)\n(λ3−λ2)(λ3+ωτ−1). (13)\nAlso, upon the definition of dimensionless fluctuating variables Y=y−/an}bracketle{ty/an}bracketri}ht,Vy=vy−/an}bracketle{tvy/an}bracketri}ht,\nAy=ay−/an}bracketle{tay/an}bracketri}ht, it can be shown that the variances σ2\nY=/an}bracketle{tY2/an}bracketri}ht,σ2\nVy=/an}bracketle{tV2\ny/an}bracketri}ht, andσ2\nAy=/an}bracketle{tA2\ny/an}bracketri}ht,\nare shown to satisfy\n/an}bracketle{tY2/an}bracketri}ht=γ2\n0D\nˆτkc2\n0F2\n0/braceleftbigg2s\n(λ2λ3)2+e2λ2s−1\nλ3\n2(λ3−λ2)2+e2λ3s−1\nλ2\n3(λ3−λ2)2\n5+ 4/bracketleftbiggeλ3s−1\nλ2λ3\n3(λ3−λ2)−eλ2s−1\nλ3λ3\n2(λ3−λ2)−e(λ3+λ2)s−1\nλ3λ2(λ3−λ2)2(λ3+λ2)/bracketrightbigg/bracerightbigg\n, (14)\n/an}bracketle{tV2\ny/an}bracketri}ht=γ2\n0D\nˆτkc2\n0F2\n01\n(λ3−λ2)2/bracketleftbigge2λ2s−1\nλ2−4e(λ3+λ2)s−1\nλ3+λ2+e2λ3s−1\nλ3/bracketrightbigg\n, (15)\n/an}bracketle{tA2\ny/an}bracketri}ht=γ2\n0D\nˆτkc2\n0F2\n01\n(λ3−λ2)2/bracketleftbigg\nλ2(e2λ2s−1)−4λ2λ3(e(λ3+λ2)s−1)\nλ2+λ3+λ3(e2λ3s−1)/bracketrightbigg\n.(16)\nWe can return to the original variables using again the transformat ionsy=c1x,s=c2t,\nand also the fluctuating variables X=x−/an}bracketle{tx/an}bracketri}ht,V=v−/an}bracketle{tv/an}bracketri}ht, andA=a−/an}bracketle{ta/an}bracketri}ht. In particular,\nin the limit of long time the variances in the original variables become\n/an}bracketle{tX2/an}bracketri}ht= 2D/parenleftbigg\nt+J−3\n2τr/parenrightbigg\n,/an}bracketle{tV2/an}bracketri}hteq=kBT\nm,/an}bracketle{tA2/an}bracketri}hteq=1\nJ τ2r/an}bracketle{tV2/an}bracketri}hteq.(17)\nThe expression /an}bracketle{tV2/an}bracketri}hteq=kBT\nm, is the equilibrium expected result, and the acceleration vari-\nance is proportional to /an}bracketle{tV2/an}bracketri}hteq, andJ-dependent. However, due to the fact that 0 < J <1\n4,\nit can be neglected respect to number 3 and thus the variance /an}bracketle{tX2/an}bracketri}htcan be approximated\nby\n/an}bracketle{tX2/an}bracketri}ht= 2D/parenleftbigg\nt−3\n2τr/parenrightbigg\n, (18)\nwhich is independent of the Jparameter. The comparison of these theoretical results with\nthe numerical simulation of Eq. (9), is given in detail in Sec. IV.\nB. The case of complex roots 1−4J <0\nIn this case J >1\n4or ˆτk>1\n4τr, which means that ωτ >1+1\n4ωτr. The two complex roots\ncan be defined as λ2=a+ib, andλ3=a−ib, beinga=−1\n2andb=1\n2√\n4J−1. So, for\nzero initial conditions, the mean values /an}bracketle{ty/an}bracketri}ht,/an}bracketle{tvy/an}bracketri}ht, and/an}bracketle{tay/an}bracketri}ht, can be written as\n/an}bracketle{ty/an}bracketri}ht=1\nJ/bracketleftbig\nI0+K1I1+K2I2/bracketrightbig\n, (19)\n/an}bracketle{tvy/an}bracketri}ht=1\nJ/bracketleftbig\n−K′\n1I1+K′\n2I2/bracketrightbig\n, (20)\n/an}bracketle{tay/an}bracketri}ht=1\nJ/bracketleftbig\n−K′′\n1I1+K′′\n2I2/bracketrightbig\n, (21)\n6whereK′\n1andK′\n2are the derivative respect to svariable, and\nI0=1\nωτ/bracketleftbig\n1−e−ωτs/bracketrightbig\n, (22)\nI1=1\n(a+ωτ)2+b2/braceleftbig\nb−e−s(ωτ+a)/bracketleftbig\nbcosbs+(a+ωτ)sinbs/bracketrightbig/bracerightbig\n, (23)\nI2=1\n(a+ωτ)2+b2/braceleftbig\na+ωτ+e−s(ωτ+a)/bracketleftbig\nbsinbs−(a+ωτ)cosbs/bracketrightbig/bracerightbig\n,(24)\nK1=eas/bracketleftbiga\nbcosbs+sinbs/bracketrightbig\n, (25)\nK2=eas/bracketleftbiga\nbsinbs−cosbs/bracketrightbig\n. (26)\nThe variances have respectively the following expressions\n/an}bracketle{tY2/an}bracketri}ht=γ2\n0D\nˆτkc2\n0F2\n01\nJ3/bracketleftbig\n2Js+K2\n1B+K2\n2C −2/parenleftbig\nK1D −K2E+K1K2F/parenrightbig/bracketrightbig\n,(27)\n/an}bracketle{tV2\ny/an}bracketri}ht=γ2\n0D\nˆτkc2\n0F2\n01\nJ3/bracketleftbig\nK′2\n1B −2K′\n1K′\n2F+K′2\n2C/bracketrightbig\n, (28)\n/an}bracketle{tA2\ny/an}bracketri}ht=γ2\n0D\nˆτkc2\n0F2\n01\nJ3/bracketleftbig\nK′′2\n1B −2K′′\n1K′′\n2F+K′′2\n2C/bracketrightbig\n, (29)\nwith\nB=−/bracketleftbig\nb2+e−2as/parenleftbig\na2cos2bs−absin2bs−J/parenrightbig/bracketrightbig\n, (30)\nC=−/bracketleftbig\na2+J−e−2as/parenleftbig\na2cos2bs−absin2bs+J/parenrightbig/bracketrightbig\n, (31)\nD=/bracketleftbig\n2b−2e−as/parenleftbig\nasinbs+bcosbs/parenrightbig/bracketrightbig\n, (32)\nE=/bracketleftbig\n2a−2e−as/parenleftbig\nacosbs−bsinbs/parenrightbig/bracketrightbig\n, (33)\nF=1\n2/bracketleftbig\nb−e−2as/parenleftbig\nasin2bs+bcos2bs/parenrightbig/bracketrightbig\n. (34)\nAlso in the long time limit, the variances /an}bracketle{tX2/an}bracketri}ht,/an}bracketle{tV2/an}bracketri}htand/an}bracketle{tA2/an}bracketri}htare the same as those given\nin Eq. (17). In this case, the variance /an}bracketle{tX2/an}bracketri}htis valid for all J >0.25, and in particular for\nJ= 3, the variance reduces to /an}bracketle{tX2/an}bracketri}ht= 2Dt, which is the same as the Markovian result. The\ndetails are also given in Sec. IV.\nC. The critical case 1−4J= 0\nIt is clear in this case that J=1\n4or ˆτk=1\n4τr. So, the dimensionless variables now read\ny=16m\nτ2rc0F0x,s=4\nτrt, andc0=1\n4(ωτ−4), and therefore, the solution of the corresponding\ndimensionless stochastic differential equation leads to the following s tatistics\n/an}bracketle{ty/an}bracketri}ht=4[1−e−(ωτ−1)s]\nωτ−1−2[s+2][1−e−(ωτ−3\n2)s]e−s/2\nωτ−3\n2\n7+2{1+[(ωτ−1\n2)s−1]e−(ωτ−3\n2)s}e−s/2\n(ωτ−3\n2)2, (35)\n/an}bracketle{tvy/an}bracketri}ht=s[1−e−(ωτ−3\n2)s]e−s/2\nωτ−3\n2−{1+[(ωτ−1\n2)s−1]e−(ωτ−3\n2)s}e−s/2\n(ωτ−3\n2)2, (36)\n/an}bracketle{tay/an}bracketri}ht=(1−s\n2)[1−e−(ωτ−3\n2)s]e−s/2\nωτ−3\n2−{1+[(ωτ−1\n2)s−1]e−(ωτ−3\n2)s}e−s/2\n2(ωτ−3\n2)2.(37)\nThe variances are shown to be\n/an}bracketle{tY2/an}bracketri}ht=γ2\n0D\nˆτkc2\n0F2\n0/bracketleftbig\n32s−176+64( s+4)e−s/2−8(s2+6s+10)e−s/bracketrightbig\n, (38)\n/an}bracketle{tV2\ny/an}bracketri}ht=γ2\n0D\nˆτkc2\n0F2\n0/bracketleftbig\n4−2(s2+2s+2)e−s/bracketrightbig\n, (39)\n/an}bracketle{tA2\ny/an}bracketri}ht=γ2\n0D\nˆτkc2\n0F2\n0/bracketleftbig\n1−1\n2(s2−2s+2)e−s/bracketrightbig\n. (40)\nIn the long time limit and in the original variables, evolve as follows:\n/an}bracketle{tX2/an}bracketri}ht= 2D/parenleftbigg\nt−11\n8τr/parenrightbigg\n,/an}bracketle{tV2/an}bracketri}ht=kBT\nm,/an}bracketle{tA2/an}bracketri}ht=1\nJ τ2r/an}bracketle{tV2/an}bracketri}hteq (41)\nHere, the variance /an}bracketle{tX2/an}bracketri}htis consistent with the one given in Eq. (17) if J=1\n4. In this case,\nthe numerical simulation of Eq. (9) are carried out taking into accou nt the corresponding\nexpression of each parameter J, ˆτk,y, ands. See the details in Sec IV.\nIV. COMPARISON WITH THE NUMERICAL SIMULATION\nIn this section we compare the explicit solutions reported in Sec. III with those derived\nfrom the numerical simulations of Eq. (9)\n•In Fig. 1, we show the comparison between the variance or mean squ are displacement\n(MSD)/an}bracketle{tX2/an}bracketri}htand both the theoretical result (18) and numerical simulation of Eq . (9), for\ntwo specific values of Jin the interval 0 < J <0.25. As can be seen, in this interval the\nvariance (14) for the original variable Xis practically the same as Eq. (18), and therefore\nthe diffusion process is J-independent. In this case the variance (18) is the same as the\nMarkovian one except for a time delay t−3\n2τr.\n•The plot given in Fig. 2 corresponds to the comparison between the t heoretical velocity\nvariance /an}bracketle{tV2/an}bracketri}htand the numerical solution of Eq. (9). Both results show that the v elocity\n82.0 2.5 3.0 3.5 4.0 4.5 5.0\nt405060708090100< x2\n>< x2\n> = 2 Dt\nωτ = 20, J= 0. 20\nωτ = 20, J= 0.05\nωτ = 20, J= 0.05\nωτ = 20, J= 0. 20\n< x2\n> = 2 Dt − 3 Dτ\nr\nFIG. 1: Mean Square Displacement (MSD), compared with both t he Markovian one and numerical\nsimulation. Black line is the Markovian result, Orange line and blue stars the theoretical results\n(14) in the original variable X, for one value of ωτ= 20 and two different values of J= 0.2,0.05,\nrespectively. Red diamonds and green triangles are numeric al simulation of Eq. (9), for ωτ= 20\nandJ= 0.05,0.2, respectively. Black dots represent the theoretical appr oximation (18)\n0 1 2 3 4 5\nt01020304050< v2\n>\nωτ = 20, J = 0. 20\nωτ = 20, J = 0.05\nωτ = 20, J = 0. 20\nωτ = 20, J = 0.05\nFIG. 2: Velocity variance or Mean Square Velocity /an}bracketle{tV2/an}bracketri}htcompared with the numerical simulation.\nOrange line and blue dashed line are the theoretical results (15), in the original variable V, for a\nspecific value of ωτ= 20 and two values of J= 0.2,0.05, respectively. The numerical simulation\nis plotted for the same values of ωτandJ. As the time gets large, all the curves tend toward the\nequilibrium valuekBT\nm.\nreaches its equilibrium expected result /an}bracketle{tV2/an}bracketri}hteq=kBT\nm, as expected. The variance /an}bracketle{tA2/an}bracketri}htin the\noriginal variable has a similar dynamic behavior and plotted in Fig. 8, for J= 0.05 and\n90 10 20 30 40 50 60\nt020040060080010001200< x2\n>2 Dt\nωτ = 20, J= 40\nωτ = 20, J= 10\nωτ = 20, J= 40\nωτ = 20, J= 10\nFIG. 3: MSD /an}bracketle{tX2/an}bracketri}ht, compared with both theMarkovian result andnumerical simu lation. Black line\nis the Markovian MSD. The orange and blue colors correspond t o oscillatory behavior of Eq. (27)\nin the original variable X, for a specific value of ωτ= 20 and two values of J= 40,10,respectively.\nBoth curves are attenuated taking the shape straight lines, and are above of the Markovian result.\nThe green triangles and red diamonds are the numerical simul ation, for the same values of ωτand\nJ.\n0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5\nt020406080100< x2\n>ωτ = 20, J = 3\nωτ = 20, J = 3\n2 Dt\nFIG. 4: MSD /an}bracketle{tX2/an}bracketri}ht, compared with both the Markovian result and numerical simu lation. Black\nline is the Markovian MSD. The blue color correspond to oscil latory behavior of Eq. (27) in the\noriginal variable X, for specific values of ωτ= 20 and J= 3. It is shown that, in the long time\nlimit, the oscillatory behavior is attenuated as a straight line towards the Makovian result. The\nred diamonds are the numerical simulation.\ncompared with the numerical simulation of Eq. (9). Its equilibrium valu e is consistent with\n10the result /an}bracketle{tV2/an}bracketri}hteq/Jτ2\nr.\n•In Fig. 3, we plot the theoretical variance /an}bracketle{tX2/an}bracketri}htcoming from Eq. (27), for two values\nofJ= 10,20 ( recall that J >0.25). In this case the oscillatory behavior is due to complex\nnature of the roots λ2andλ3. As can be seen, as time is longer, the oscillatory behavior\nbecomes attenuated taking the shape of a straight line. This can be corroborated because\nin the long time limit the variance /an}bracketle{tX2/an}bracketri}htis the straight line given in Eq. (17). In this\nlimit case, a remarkable result is obtained when J= 3, because /an}bracketle{tX2/an}bracketri}ht= 2Dt, which is the\nMarkovian result. This fact is clearly seen in Fig. 4, in which the oscillato ry variance follows\nthe Markovian straight line, as well as the numerical simulation result s.\n•In the case of complex roots, the theoretical velocity variance /an}bracketle{tV2/an}bracketri}htcoming from Eq.\n(28), is plotted in Fig. 5, where it is compared with the numerical simula tion. Also its\noscillatory behavior at early times, as well as its equilibrium valuekBT\nmas the time goes out,\nare shown. The variance /an}bracketle{tA2/an}bracketri}htis plotted in Fig. 8 for the value of J= 3.0 and also compared\nwith the numerical simulation. With values close to zero.\n0 10 20 30 40 50 60\nt01020304050< v2\n>\nωτ = 20, J= 40\nωτ = 20, J= 10\nωτ = 20, J= 40\nωτ = 20, J= 10\nFIG. 5: Velocity variance /an}bracketle{tV2/an}bracketri}htcompared with the numerical simulation. Blue and black line s are\nthe theoretical results (28), in the original variable V, for a specific value of ωτ= 20 and two\nvalues of J= 40,10, respectively. The numerical simulation is plotted for t he same values of ωτ\nandJ. In the long time limit, all the curves tend toward the equili brium valuekBT\nm, as expected.\n•In the critical case J=1\n4and thus ˆ τk=1\n4τr, the theoretical variance /an}bracketle{tX2/an}bracketri}htcoming from\nEq. (38), is plotted in Fig. 6. In this case, it is shown that no matter w hat the value of\nωτis, both the theoretical and numerical simulation results coincide. M oreover, they also\n11coincide with the Markovian result except for a time delay t−11\n8τr.\n0 1 2 3 4 5\nt020406080100< x2\n>< x2\n> = 2 Dt\nωτ = 20\nωτ = 10\nωτ = 20\nωτ = 10\n< x2\n> = 2 D (t −11\n8τ\nr)\nFIG. 6: MSD /an}bracketle{tX2/an}bracketri}htcompared with both the Markovian result and numerical simul ation. The blue\nstraight line is the Markovian MSD. The red line and green tri angles are the theoretical variance\ngiven by Eq. (38) in the original variable X, for fixed value of J= 0.25, and two values of\nωτ= 20,10. Red diamonds and circles correspond to the numerical sim ulation, and black dots\nthe MSD given in Eq. (41)\n.\n•In Fig. 7, the variance /an}bracketle{tV2/an}bracketri}htcoming from Eq. (39) is compared with the numerical\nsimulation results. Both results are consistent and tend towards t he equilibrium valuekBT\nm\nas it should be. The variance /an}bracketle{tA2/an}bracketri}htis also plotted in Fig. 8 for the value of J= 0.25 and\ncompared with the numerical simulation. It is clearly shown that, as Jis greater the value\nof acceleration variance is lesser.\n120 1 2 3 4 5\nt01020304050< v2\n>\nωτ = 10\nωτ = 20\nωτ = 10\nωτ = 20\nFIG. 7: Mean square velocity /an}bracketle{tV2/an}bracketri}ht, given by Eq. (39) in the original variable X, for fixed value\nofJ= 0.25 and two values of ωτ= 10,20. The orange and blue lines are the theoretical results;\nred diamonds and green triangles are the numerical simulati ons. All the results coincide no matter\nwhat the value of ωτis.\n0.0 0.2 0.4 0.6 0.8 1.0\nt0500010000150002000025000< A2\n>ωτ = 20, J = 0. 25\nωτ = 20, J = 0.05\nωτ = 20, J = 3.00\nωτ = 20, J = 0. 25\nωτ = 20, J = 0.05\nωτ = 20, J = 3.00\nFIG. 8: Mean square acceleration /an}bracketle{tA2/an}bracketri}htfor fixed value of J= 0.25 and a value of ωτ= 20. Green\nline corresponds to Eq. (16), black line to eq. (29), and oran ge line to (40), all in the original\nvariable A. The circles are the corresponding simulation results.\nV. CONCLUDING REMARKS\nIn this work we have been able to describe the statistic of the electr onic plasma diffusion\nin a classic way, taking into account the radiation reaction force. Th is has successfully\nbeen achieved by means of a GLE under the action of a time-depende nt electric field, in\n13the form of an exponential decay. The proposed electric field induc es an electric force on\nthe electron, which is capable of producing a delay in the electron’s ch aracteristic time.\nThe theoretical approach allows to establish a quasi-Markovian thir d-order time derivative\nstochastic differential equation, containing a frictionforce m(τ−kτe)...xinwhich theeffective\nmemory time ˆ τk=τ−kτeis required to be positive and finite in order to avoid runaway\nsolutions.\nThe electron gas statistic is better calculated using the dimensionles s stochastic differ-\nential equation, in which the Jparameter plays an important role. For the real and critical\nroots, the effects of the radiation reaction force through the Jparameter, appear only\nin the acceleration variance, as shown in Fig. 8. However, in the case of complex roots,\n(J >1\n4, or ˆτk>1\n4τr), the influence of the radiation reaction force is more noticeable in\nthe variance /an}bracketle{tX2/an}bracketri}htthan the acceleration one. In this case, the MSD exhibits an oscillato ry\nbehavior whose amplitude of oscillation decreases as time progresse s, taking the shape of\na straight line. The influence of the radiation reaction force still rem ains in the long time\nlimit for which the /an}bracketle{tX2/an}bracketri}htis indeed a straight line, given by /an}bracketle{tX2/an}bracketri}ht= 2D(t−J−3\n2τr), (see\nFigs. 3 and 4 ). In this limiting case, the mean square displacement MSD is the same as\nthe Markovian one except for a time delay t−J−3\n2τr. In the case of real roots, the variance\n/an}bracketle{tX2/an}bracketri}htdoes not depend on Jand it is given by Eq. (18). In the critical case, for a given\nvalue ofJ= 0.25, the mean square displacement is also independent of the value of ωτand\nall coincide with one given in Eq. (41). Lastly, according to the result s reported in Fig. 8,\nthe acceleration variance decreases as Jincreases, and reaches its equilibrium value faster\nthan the velocity variance.\nAcknowledgments\nJ.F.G.C. thanks to CONAHCyT for a grant in a Postdoc position at Metr opolitan Au-\ntonomus University, campus Iztapalapa. Also, O.C.V thanks to CONA HCyT for a grant.\nGAP and NSS thank COFAA, EDI IPN and CONAHCyT.\n[1] H.A.Lorentz. La th´ eorie ´ electromagn´ etique de Maxwell et son applicat ion aux corps mouvants ,\nvolume 25. EJ Brill, 1892.\n14[2] M. Abraham. Theorie der elektrizit¨ at . BG Teubner, Leiptzig, 1905.\n[3] F. Rohrlich. Classical charged particles . Addison-Wesley, Reading, 1965.\n[4] P. A. M. Dirac. Classical theory of radiating electrons. Proc. Roy. Soc. London. A , 167(929):\n148–169, 1938.\n[5] B. S. DeWitt and R. W. Brehme. Radiation damping in a gravi tational field. Ann. Phys. , 9\n(2):220–259, 1960.\n[6] L. D. Landau and E. M. Lifshitz. The Classical Theory of Fields: Volume 2 . Butterworth-\nHeinemann, Oxford, 1975.\n[7] F. Rohrlich. The self-force and radiation reaction. Am. J. Phys. , 68(12):1109–1112, 2000.\n[8] G. W. Ford, J. T. Lewis, and R.F. O’Connell. Quantum lange vin equation. Phys. Rev. A , 37\n(11):4419, 1988.\n[9] G. W. Ford and R. F. O’Connell. Radiation reaction in elec trodynamics and the elimination\nof runaway solutions. Phys. Lett. A , 157(4-5):217–220, 1991.\n[10] V.S. Krivitskii and V.N. Tsytovich. Average radiation -reaction force in quantum electrody-\nnamics.Sov. Phys. Usp. , 34(3):250, 1991.\n[11] R. F. O’Connell. Radiation reaction: general approach and applications, especially to electro-\ndynamics. Contemporary Physics , 53(4):301–313, 2012.\n[12] G. Ares de Parga, N. S´ anchez-Salas, and J. I. Jim´ enez- Aquino. Electronic plasma brownian\nmotion with radiation reaction force. Physica A: Statistical Mechanics and its Applications ,\n600:127556, 2022. doi: https://doi.org/10.1016/j.physa.20 22.127556.\n[13] J. T. Hsiangand B. L. Hu. Non-markovian abraham-lorenz -dirac equation: Radiation reaction\nwithout pathology. Physical Review D ,106:125018, 2022. doi: https://link.aps.org/doi/10.\n1103/PhysRevD.106.125018.\n[14] J. T. Hsiang and B. L. Hu. Atom-field interaction: From va cuum fluctuations to quantum\nradiation and quantum dissipation or radiation reaction. Physics,1:430, 2019. doi: https:\n//doi.org/10.3390/physics1030031.\n[15] M. Bravo, J. T. Hsiang, and B. L. Hu. Quantum radiation an d dissipation from an\natom in a squeezed quantum field. Physics,5(2):554, 2023. doi: https://doi.org/10.3390/\nphysics5020040.\n[16] C. R. Garley and B. L. Hu. Self-force with a stochastic co mponent from radiation reaction\nof a scalar charge moving in curved spacetime. Physical Review D ,72:084023, 2005. doi:\n15https://doi.org/10.1103/PhysRevD.72.084023.\n[17] C. R. Garley, B. L. Hu, and S. Y. Lin. Electromagnetic and gravitational radiation reaction\nin curved spacetime: selfforce derivation from stochastic fie ld theory. Physical Review D ,74:\n024017, 2006. doi: https://doi.org/10.1103/PhysRevD.74 .024017.\n[18] P. R. Johnson, B. L. Hu, and S. Y. Lin. Stochastic theory o f relativistic particles moving\nin a quantum field: Scalar abraham-lorentz-dirac-langevin equation, radiation reaction, and\nvacuum fluctuations. Physical Review D ,65:065015, 2002. doi: https://doi.org/10.1103/\nPhysRevD.74.024017.\n[19] R. Kubo. The fluctuation-dissipation theorem. Rep. Prog. Phys. , 29(1):255, 1966.\n16"    },    {        "title": "2311.13114v2.Narrow_spectra_of_repeating_fast_radio_bursts__A_magnetospheric_origin.pdf",        "content": "Astronomy &Astrophysics manuscript no. AAmain_clean ©ESO 2024\nFebruary 23, 2024\nNarrow spectra of repeating fast radio bursts: A magnetospheric\norigin\nWei-Yang Wang ( 王维扬)1,2,3, Yuan-Pei Yang ( 杨元培)4,5, Hong-Bo Li ( 李洪波)3,6, Jifeng Liu ( 刘继峰)1,7,8, and\nRenxin Xu ( 徐仁新)2,3,6\n1School of Astronomy and Space Science, University of Chinese Academy of Sciences, Beijing 100049, PR China\ne-mail: wywang@ucas.ac.cn\n2State Key Laboratory of Nuclear Physics and Technology, School of Physics, Peking University, Beijing 100871, PR China\n3Department of Astronomy, School of Physics, Peking University, Beijing 100871, PR China\n4South-Western Institute for Astronomy Research, Yunnan University, Kunming, Yunnan 650504, PR China\n5Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210023, PR China\n6Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, PR China\n7New Cornerstone Science Laboratory, National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, PR\nChina\n8Institute for Frontiers in Astronomy and Astrophysics, Beijing Normal University, Beijing 102206, PR China\nReceived XXX /Accepted XXX\nABSTRACT\nFast radio bursts (FRBs) can present a variety of polarization properties, and some of them have narrow spectra. We study spectral\nproperties from perspectives of intrinsic radiation mechanisms and absorption during the waves propagating in the magnetosphere. The\nintrinsic radiation mechanisms are considered by invoking quasi-periodic bunch distribution and perturbations on charged bunches\nmoving on curved trajectories. The narrow-band emission likely reflects some quasi-periodic structure on the bulk of bunches, which\nmay be due to quasi-periodically sparking in a “gap” or quasi-monochromatic Langmuir waves. A sharp spike would appear at the\nspectrum if the perturbations can induce a monochromatic oscillation of bunches, however, it is hard to create a narrow spectrum\nbecause the Lorentz factor has large fluctuations so that the spike disappears. Both the bunching mechanism and perturbations scenar-\nios share the same polarization properties with a uniformly distributed bulk of bunches. We investigate absorption e ffects including\nLandau damping and curvature self-absorption in the magnetosphere, which are significant at low frequencies. Subluminous O-mode\nphotons can not escape from the magnetosphere due to the Landau damping, leading to a height-dependent lower frequency cut-o ff.\nSpectra can be narrow when the frequency cut-o ffis close to the characteristic frequency of curvature radiation, while such conditions\ncan only be met sometimes. The spectral index is 5 /3 at low-frequency bands due to the curvature self-absorption but not as steep as\nthe observations. The intrinsic radiation mechanisms are more likely to generate the observed narrow spectra of FRBs rather than the\nabsorption e ffects.\nKey words. polarization – radiation mechanisms: non-thermal – stars: magnetars – stars: neutron\n1. Introduction\nFast radio bursts (FRBs) are millisecond-duration and ener-\ngetic radio pulses (see Cordes & Chatterjee 2019; Petro ffet al.\n2019, for reviews). They are similar in some respects to single\npulses from radio pulsars, however, the bright temperature of\nFRBs is many orders of magnitude higher than that of normal\npulses from Galactic pulsars. The physical origin(s) of FRB are\nstill unknown but the only certainty is that their radiation mech-\nanisms must be coherent.\nMagnetars, as the most likely candidate, have been invoked\nto interpret the emission of FRBs, especially repeating ones. The\nmodels can be generally divided into two categories by consider-\ning the distance of the emission region from the magnetar (Zhang\n2020): emission within the magnetosphere (pulsar-like model,\ne.g., Katz 2014; Kumar et al. 2017; Yang & Zhang 2018; Wa-\ndiasingh & Timokhin 2019; Lu et al. 2020; Cooper & Wijers\n2021; Wang et al. 2022c; Zhang 2022; Liu et al. 2023a; Qu &\nZhang 2023), and emission from relativistic shocks region far\noutside the magnetosphere (GRB-like model, e.g., Metzger et al.\n2019; Beloborodov 2020; Margalit et al. 2020; Chen et al. 2022;Khangulyan et al. 2022). The magnetar origin for at least some\nFRBs was established after the detection of FRB 20200428D,\n1an FRB-like burst from SGR J1935 +2154, which is a Galac-\ntic magnetar (Bochenek et al. 2020; CHIME /FRB Collaboration\net al. 2020). Nevertheless, the localization of FRB 20200120E\nto an old globular cluster in M81 (Bhardwaj et al. 2021; Majid\net al. 2021; Kirsten et al. 2022) challenges that a young mag-\nnetar born via a massive stellar collapse, which means that the\nmagnetar formed in accretion-induced collapse or a system as-\nsociated with a compact binary merger (Kremer et al. 2021; Lu\net al. 2022).\nSpectral time characteristics of FRBs are significant tools\nto gain a deeper understanding of emission mechanisms. The\nburst morphologies of FRBs are varied. Some FRBs have narrow\nbandwidth, for instance, the sub-bursts of FRB 20220912A show\nthat the relative spectral bandwidth of the radio bursts was dis-\ntributed near at ∆ν/ν≃0.2 (Zhang et al. 2023). Similar phenom-\nena of narrow-band can appear at FRB 20121102A, in which\n1FRB 20200428D was regarded as FRB 20200428A previously until\nother three burst events detected (Giri et al. 2023).\nArticle number, page 1 of 19arXiv:2311.13114v2  [astro-ph.HE]  22 Feb 2024A&A proofs: manuscript no. AAmain_clean\nspectra are modeled by simple power laws with indices ranging\nfrom−10 to +14 (Spitler et al. 2016). Simple narrow-band and\nother three features (simple broadband, temporally complex, and\ndownward drifting) are summarized as four observed archetypes\nof burst morphology among the first CHIME /FRB catalog (Pleu-\nnis et al. 2021). The narrow spectra may result from small deflec-\ntion angle radiation, coherent processes or modulated by scintil-\nlation and plasma lensing in the FRB source environment (Yang\n2023).\nDownward drift in the central frequency of consecutive sub-\nbursts with later-arriving time, has been discovered in at least\nsome FRBs (CHIME /FRB Collaboration et al. 2019b; Hessels\net al. 2019; Fonseca et al. 2020). Besides, a variety of drifting\npatterns including upward drifting, complex structure, and no\ndrifting subpulse, was found in a sample of more than 600 bursts\ndetected from the repeater source FRB 20201124A (Zhou et al.\n2022). Such a downward drifting structure was unprecedentedly\nseen in pulsar PSR B0950 +08 (Bilous et al. 2022), suggesting\nthat FRBs originate from the magnetosphere of a pulsar. Down-\nward drifting pattern can be well understood by invoking mag-\nnetospheric curvature radiation (Wang et al. 2019), however, it\nis unclear why the spectral bandwidth can be extremely narrow,\ni.e.,∆ν/ν≃0.1.\nPolarization properties carry significant information about\nradiation mechanisms to shed light on the possible origin of\nFRBs. High levels of linear polarization are dominant for most\nsources and most bursts for individual repeating sources (e.g.,\nMichilli et al. 2018; Day et al. 2020; Luo et al. 2020; Sherman\net al. 2023), while some bursts have significant circular polariza-\ntion fractions, which can be even up to 90% (e.g., Masui et al.\n2015; Xu et al. 2022; Jiang et al. 2022; Kumar et al. 2022). The\nproperties are reminiscent of pulsars, which show a wide variety\nof polarization fractions between sources. The emission is highly\nlinearly polarized when the line of sight (LOS) is confined into\na beam angle, while highly circular polarization presents when\nLOS is outside, by considering bunched charges moving at ultra-\nrelativistic speed in magnetosphere (Wang et al. 2022a; Liu et al.\n2023b).\nIn this work, we attempt to understand the origin(s) of the ob-\nserved narrow spectra of FRBs from the views of intrinsic radi-\nation mechanisms and absorption e ffects in the magnetosphere.\nThe paper is organized as follows. The intrinsic radiation mech-\nanisms by charged bunches in the magnetosphere are discussed\nin Section 2. We investigate two scenarios of the spectrum, in-\ncluding the bunch mechanism (Section 2.1) and perturbations\non the bunch moving at curved trajectories (Section 2.2). The\npolarization properties of the intrinsic radiation mechanisms are\nsummarized in Section 2.3. In Section 3, we discuss several pos-\nsible mechanisms as triggers that may generate emitting charged\nbunches (Section 3.1), and oscillation and Alfvén wave. Some\nabsorption e ffects during the wave propagating in the magne-\ntosphere are investigated in Section 4. We discuss the bunch’s\nevolution in the magnetosphere in Section 5. The results are\ndiscussed and summarized in Section 6 and 7. The convention\nQx=Q/10xin cgs units and spherical coordinates ( r, θ, φ ) con-\ncerning the magnetic axis are used throughout the paper.\n2. Intrinsic Radiation Mechanism\nA sudden trigger may happen on the stellar surface and sustain\nat least a few milliseconds to create free charges. Such a trigger\nevent is a sudden and violent process in contrast to a consecu-\ntive “sparking” process in the polar cap region of a pulsar (Rud-\nerman & Sutherland 1975). The charges can form bunches, and\nthe total emission of charges is coherently enhanced significantlyif the bunch size is smaller than the half wavelength. Note that\nelectrons as the emitting charges are the point of interest in the\nfollowing discussion, leading to negatively net-charged bunches.\nWe attempt to study how to trigger the emitting charges and how\nto form bunches in Section 3.\nWithin the magnetosphere, the charged particles are sud-\ndenly accelerated to ultra-relativistic velocities and stream out-\nward along the magnetic field lines. The charges hardly move\nacross the magnetic field lines because the cyclotron cooling is\nvery fast in a strong magnetic field and charges stay in the low-\nest Landau level so that charges’ trajectories track with the field\nlines essentially.\nCurvature radiation can be produced via the perpendicular\nacceleration for a charge moving at a curved trajectory. It is a nat-\nural consequence that charges moving at curved trajectories, that\nhas been widely discussed in pulsars (Ruderman & Sutherland\n1975; Cheng & Ruderman 1977; Melikidze et al. 2000; Gil et al.\n2004; Gangadhara et al. 2021) and FRBs (Kumar et al. 2017;\nKatz 2018; Lu & Kumar 2018; Ghisellini & Locatelli 2018;\nYang & Zhang 2018; Wang et al. 2022c; Cui et al. 2023). How-\never, the noticeable di fference is that there should be an electric\nfield parallel ( E∥) to the local magnetic field Bin the FRB emis-\nsion region because the enormous emission power of FRB lets\nthe bunches cooling extremely fast. The E∥sustains the energy\nof bunches so that they can radiate for a long enough time to\npower FRBs. The equation for motion in the FRB emission re-\ngion in the lab frame can be written as\nNeE∥eds−L bdt=Nemec2dγ, (1)\nwhere Neis the number of net charges in one bunch that are\nradiating coherently, cis the speed of light, eis the elementary\ncharge, meis the positron mass, γis the Lorentz factor of the\nbunch, andLbis the luminosity of a bunch. The timescale for\ntheE∥that can be balanced by radiation damping is essentially\nthe cooling timescale of the bunches, which is much shorter than\nthe FRB duration. As a result, bunches stay balanced throughout\nthe FRB emission process.\nBunches in the magnetosphere are accelerated to ultra-\nrelativistic velocity. A stable E∥(∂E/∂t=0) may sustain the\nLorentz factor of bunches to be a constant. The emission of\nbunches is confined mainly in a narrow conal region due to the\nrelativistic beaming e ffect, so the line of sight (LOS) sweeps\nacross the cone within a very short time. According to the prop-\nerties of the Fourier transform, the corresponding spectrum of\nthe beaming emission within a short time is supposed to be wide-\nband. This is a common problem among the emission from rela-\ntivistic charges. In the following discussion, we attempt to under-\nstand the narrow-band FRB emission by considering the bunch-\ning mechanism (Section 2.1) and perturbation on the curved tra-\njectories (Section 2.2).\n2.1. Bunching Mechanism\nThe waves are coherently enhanced significantly when the emit-\nting electrons form charged bunches, which is the fundamen-\ntal unit of coherent emission. Di fferent photon arrival delays\nare attributed to charges moving in di fferent trajectories, so that\ncharges in the horizontal plane (a light solid blue line denotes\na bunch shown in Figure 1) have roughly the same phases. The\nbunch requires a longitude size smaller than the halfwavelength,\nwhich is∼10 cm for GHz wave.\nInside a bunch, the total emission intensity is the summa-\ntion of that of single charges with coherently adding. Consider\nArticle number, page 2 of 19Wang, W.-Y ., et al.: Narrow spectra of repeating fast radio bursts: A magnetospheric origin\nΩμ\nn\n\tn\nkφ\nijχz0\nᵝ.\nijk, ]]ᵝ.\nijk, \nᵝ0, ||ϵ2\n1\nNeutron starFRB\nϵ2 ϵ1ϵ\nFig. 1. Left: Schematic diagram of a bulk of bunches moving in the magnetosphere. The light solid blue line shows the slice in which electrons\nemit roughly the same phase. The unit vector of the LOS is denoted by n, andϵ1andϵ2denote the two polarization components. Right: Geometry\nfor instantaneous circular motion for a particle identified by i jkin a bunch. The trajectory lies in the β0,∥-ϵ1plane. At the retarded time t=0, the\nelectron is at the origin. The dotted trajectory shows an electron at the origin. The angle between the solid and dotted trajectory is χi jatt=0. The\nangle between LOS ( z0) and the trajectory plane is φk.\na single charge that moves along a trajectory rs(t). The energy\nradiated per unit solid angle per unit frequency interval is in-\nterested in considering spectrum, which is given by (Rybicki &\nLightman 1979; Jackson 1998)\nd2W\ndωdΩ=e2ω2\n4π2c\n×\f\f\f\f\f\f\fZ+∞\n−∞NsX\ns−βs⊥exp [iω(t−n·rs/c)]dt\f\f\f\f\f\f\f2\n,(2)\nwhereβs⊥is the component of βsin the plane that is perpendic-\nular to the LOS:\nβs⊥=−n×(n×βs). (3)\nϵ1is the unit vector pointing to the center of the instantaneous\ncircle,ϵ2=n×ϵ1is defined. If there is more than one charged\nparticle, one can use subscripts ( i,j,k) to describe any charge\nin the three-dimensional bunch. The subscripts contain all the\ninformation about the location. We assume that bunches are uni-\nformly distributed in all directions of ( ˆr,ˆθ,ˆφ). Equation (3) can\nbe written as\nd2W\ndωdΩ=e2ω2\n4π2c\n×\f\f\f\f\f\f\f\fZ+∞\n−∞NiX\niNjX\njNkX\nk−βi jk,⊥exph\niω\u0010\nt−n·ri jk(t)/c\u0011i\ndt\f\f\f\f\f\f\f\f2\n,(4)\nwhere Ne=NiNjNk. The emission power from charges with the\nsame phase is N2\neenhanced. Detailed calculations are referred to\nAppendix A.\nThe bunches thus act like single macro charges in some re-\nspects. Define a critical frequency of curvature radiation ωc=\n3cγ3/(2ρ), whereρis the curvature radius. Spectra of curva-\nture radiation from a single charge can be described as a power\nlaw forω≪ωcand exponentially drop when ω≫ωc. The\nspectrum and polarization properties of a bunch with a longi-\ntude size smaller than the half-wavelength and half-opening an-\ngleφt≲1/γ, are similar to the case of a single charge. In gen-\neral, the spectra can be characterized as multisegmented broken\npower laws by considering a variety of geometric conditions of\nbunch (Yang & Zhang 2018; Wang et al. 2022c).\n2.1.1. Bunch Structure\nThere could be a lot of bunches contributing to instantaneous\nradiation in a moving bulk and their emissions are added inco-\nherently. The spread angle of relativistic charge is\nθc(ω)≃1\nγ 2ωc\nω!1/3\n= 3c\nωρ!1/3\n, ω≪ωc\n1\nγ 2ωc\n3ω!1/2\n, ω≫ωc. (5)\nArticle number, page 3 of 19A&A proofs: manuscript no. AAmain_clean\nForω∼ωc, bunches within a layer of ρ/γcan contribute to the\nobserved luminosity at an epoch. The number of that contribut-\ning charges is estimated as Nb∼2ρν/(cγ)≃104ρ7ν9γ−1\n2.\nWe assume that the number density in a bunch has ∂ne/∂t=\n0. The case of fluctuating bunch density can give rise to a sup-\npressed spectrum, and there is a quasi-white noise in a wider\nband in the frequency domain (Yang & Zhang 2023). The fluc-\ntuating density may cause the integral of Equation (A.5) in Ap-\npendix A to not converge.\nDifferent bunches may have di fferent bulk Lorentz factors.\nNote that emissions from relativistic charges may have a char-\nacteristic frequency determined by the Lorentz factor. The scat-\ntering of the Lorentz factor from di fferent emitting units likely\nenables the characteristic frequency to be polychromatic. There-\nfore, non-monoenergetic distributed charges give rise to a wide\nspectrum. However, for the bunch case discussed in Section 2,\nthe layer contributing to the observed luminosity simultaneously\nis∼ρ/γmuch smaller than the curvature radius. Emission power\nin the layers is thus roughly the same. The Lorentz factors be-\ntween di fferent bunch layers can be regarded as a constant.\nThe frequency structure of a burst is essentially the Fourier\ntransform of the spatial structure of the radiating charge den-\nsity (Katz 2018). Therefore, a relatively narrow spectrum may\nbe produced via the emission from a quasi-periodically struc-\ntured bulk of bunches. The possible mechanisms that generate\nsuch quasi-periodically structure are discussed in Section 3.\n2.1.2. Spectrum of the Bunch Structure\nWe focus on the radiation properties due to the quasi-periodical\nstructure. The charged bunch distribution is thought to be quasi-\nperiod in the time domain, which may be attributed to some\nquasi-periodic distribution in space between bunches. Each emit-\nting bunch has the same E(t) but with di fferent arrival times. The\ntotal received electric field from the multiple bunches simultane-\nously is given by\nE(t)=NbX\nnEn(t−tn), (6)\nand its time-shifting property of the Fourier transform is\nE(ω)=NbX\nnEn(ω) exp( iωtn). (7)\nA quasi-periodically distributed bulk of bunches means that\nωtn=nω/ω M+δϕn, where 1/ωMdenotes the period of the bunch\ndistribution, and ϕnis the relative random phase.\nA strictly periodically distributed bulk means that the relative\nrandom phase is zero strictly. The energy per unit time per unit\nsolid angle is (Yang 2023)\nd2W\ndωdΩ=c2R2|En(ω)|2sin2[Nbω/(2ωM)]\nsin2[ω/(2ωM)], (8)\nwhereRis the distance from the emitting source to the observer.\nThe coherence properties of the radiation by the multiple bulks\nof bunches are determined by ωM. The radiation is coherently\nenhanced when ωis 2πωMof integer multiples so that the en-\nergy is radiated into multiples of 2 πωMwith narrow bandwidths.\nEvenϕnis not zero, the Monte Carlo simulation shows that the\ncoherent radiation energy is still radiated into multiples of 2 πωM\nwith narrow bandwidths (Yang 2023).Figure 2 shows the simulation of radiation spectra from\nquasi-periodically distributed bulk of bunches with ωM=108\nHz andωM=5×108Hz. We assume that Nb=104,γ=102,\nδϕn=0 andρ=107cm. Since the flux drops to a small number\nrapidly when both 1 /γ < φ′and 1/γ < χ′, we fixχ′=10−3\nin the simulations for simplicity. Spectra from uniformly dis-\ntributed bulk of bunches are plotted as comparisons. By assum-\ning there is an observation threshold exceeding the peak flux of\nthe pure curvature radiation, the spectrum is characterized by\nmultiple narrow band emissions with a bandwidth smaller than\n2πωM. AsωMbecomes larger, the number of spikes is smaller.\nEven though there are multi-spikes in the spectra, there is a main\nmaximum at near ν=νcordered larger than other spikes, so\nthat multi-narrow-band emissions are di fficult to detect from an\nultra-wide-band receiver.\n2.2. Perturbation on Curved Trajectories\nWe consider two general cases that the perturbations exist both\nparallel ( E1,∥) and perpendicular ( E1,⊥) to the local B-field inde-\npendently. The perturbations can produce extra parallel c˙β∥and\nperpendicular accelerations c˙β⊥to the local B-field, as shown in\nFigure 1. The perturbation electric fields are much smaller than\ntheE∥, which is∼107esu, sustaining a constant Lorentz factor\n(Wang et al. 2019). A ffected by these accelerations, electrons not\nonly emit curvature radiation, but also emissions caused by the\nperturbations on the curved trajectory. As a general discussion,\nwe do not care about what is the perturbation in this Section but\nrequire that the velocity change slightly, i.e., δβ≪β.\n2.2.1. Parallel Acceleration\nThe emission power by the acceleration parallel to the velocity,\ni.e., longitudinal acceleration, is given by (Rybicki & Lightman\n1979)\nP=2e2\n3m2ec3 dp\ndt!2\n. (9)\nAccording to Equation (9), if the ratio of two longitudinal forces\nisN, the ratio of their emission power to N2, so that we can\nignore the radiation damping caused by E1,∥. The energy loss\nby radiation is mainly determined by the coherent curvature ra-\ndiation. The balance between the E∥and radiation damping is\nstill satisfied approximately, leading to a constant Lorentz fac-\ntor. Thus, the velocity change caused by E1,∥of a single charge\ncan be described as\nE1,∥edt≃meγ3\nsdvs. (10)\nThe velocity change is so slight that the partial integration of\nEquation (A.5) converges, thus Equation (A.6) can be obtained\nfrom Equation (A.5).\n2.2.2. Spectrum of Parallel Acceleration\nThe dimensionless velocity of a single charge reads βs=βs,0+\nβ1,∥, whereβs,0is the mean dimensionless velocity and β1,∥is\nits perturbation parallel to the field lines, so that the emitting\nelectric field can be calculated by the curvature radiation and\nperturbation-induced emission independently. Let us consider a\ncharge with a dentifier of i,j,kin a bunch. The polarization com-\nponents of the parallel perturbation-induced wave are calculated\nArticle number, page 4 of 19Wang, W.-Y ., et al.: Narrow spectra of repeating fast radio bursts: A magnetospheric origin\n10 810 910 10 \n (Hz)10 010 5Normalized F \n(a) (b) 10 810 910 10 \n (Hz)10 010 5Normalized F \nFig. 2. Normalized spectra of the quasi-periodically structured bulk of bunches: (a) ωM=108Hz; (b)ωM=5×108Hz. The black solid line is the\nspectrum due to the quasi-periodic structure. The blue dashed line is the spectrum of a uniformly distributed bulk of bunches.\n10 910 10 \n (Hz)10 -3 10 -2 10 -1 10 0Normalized F 1.5 2 2.510 90.40.60.81\n10 8\nFig. 3. Normalized spectra of the parallel perturbation cases at φ=0:\nφt=0.1/γ,β1=10−6(blue dashed line); φt=0.1/γ,β1=10−5(red\ndotted line); φt=1/γ,β1=10−6(black solid line); and φt=1/γ,\nβ1=10−5(green dashed-dotted line).\nas\n˜Ai jk,∥≃Z+∞\n−∞β1,∥(t) sin ct\nρ+χi j!\n×exp \niω\n2\" 1\nγ2+φ2\nk+χ2\ni j!\nt+c2t3\n3ρ2+ct2\nρχi j#!\ndt,\n˜Ai jk,⊥≃Z+∞\n−∞β1,∥(t) sinφkcos ct\nρ+χi j!\n×exp \niω\n2\" 1\nγ2+φ2\nk+χ2\ni j!\nt+c2t3\n3ρ2+ct2\nρχi j#!\ndt.(11)\nA complex spectrum can be caused by a complexly varying\nβ1,∥(t). A simple case is that the integrals of Equation (11) can\nbe described by the modified Bessel function Kν(ξ) ifβ1is a\nconstant. The function is approximated to be power-law-like for\nξ≪1 and exponential-like for ξ≫1 whenν,0. Here, it isworth noting that Kν(ξ=0)→+∞whenξ=0, exhibiting a\nvery narrow spectrum.\nIn particular, if a waveform E(t) is quasi-sinusoid periodic,\nthe spectrum would be quasi-monochromatic, equivalently to\nnarrow-band. The quasi-sinusoid periodic waveform can be pro-\nduced by the intrinsic charge radiation if the charge is under a pe-\nriodic acceleration during its radiation beam pointing to the ob-\nserver. For a perturbation E1.∥which is monochromatic oscillat-\ning with time of interest, according to Equation (10) in Appendix\nA, the velocity is also monochromatic oscillating. Therefore, we\nintroduce a velocity form with monochromatic oscillating in the\nlab frame, i.e., β1,∥(t)=β1exp(−iωat). The two parameters intro-\nduced in the curvature radiation in Equation (A.11) are replaced\nby\nua=ct\nρ 1\nγ2+φ2\nk+χ2\ni j−2ωa\nω!−1/2\n,\nξa=ωρ\n3c 1\nγ2+φ2\nk+χ2\ni j−2ωa\nω!3/2\n.(12)\nThe radiation can be narrow-band when γis a constant and the\ncenter frequency is\nω=2ωaγ2\n1+γ2(χ2\ni j+φ2\nk). (13)\nFor the bunching emission, charges are located with di fferent\nχandφwhich results in the bandwidth getting wider. Now let’s\nconsider the change of the E∥is much smaller than l∂E∥/∂r≪\nE∥, where lis the longitude bunch size. The charges in the bunch\nare then monoenergic. However, for an ultra-relativistic particle,\na small change of velocity can create a violent change of Lorentz\nfactor, i.e.,δγ≃γ3βδβ∥. To keep electrons being monoenergic,\nit is required that the amplitude β1≪1/γ2.\nFollowing the Appendix A, one can obtain the spectrum of\nbunching charges. Adopting γ=100,ρ=107cm,φ=0 and\nωa=2π×105Hz, we simulate the spectra in four cases: φt=\n0.1/γ,β1=10−6;φt=0.1/γ,β1=10−5;φt=1/γ,β1=10−6;\nandφt=1/γ,β1=10−5, shown as Figure 3. The spectra are\nnormalized to the Fνof curvature radiation with φt=10−3at\nArticle number, page 5 of 19A&A proofs: manuscript no. AAmain_clean\n1081091010\n (Hz)10-310-210-1100Normalized F\nFig. 4. Normalized spectra of the perpendicular perturbation cases at\nφ=0:φt=0.1/γ,β1=10−4(blue dashed line); φt=0.1/γ,β1=10−3\n(red dotted line); φt=1/γ,β1=10−4(black solid line); and φt=1/γ,\nβ1=10−3(green dashed-dotted line).\nω=ωc. Forφt=0.1/γ, the spectra show spikes at 2 νaγ2. The\namplitude of spikes decreases as β1decreases. For φt=1/γ, the\nscattering of φ′leads the spikes to become wider and shorter. The\nsmall amplitude β1lets the spike disappear, and the spectrum is\nthen determined by the curvature radiation. As φbecomes larger,\nmore di fferent trajectories may have more di fferent azimuths so\nthat the coherence of emission decreases. The spectrum has a\nflat component at ω<ω cwhen 3 c/ρ<ω c(χ′2+φ′2)3/2(Yang &\nZhang 2018).\n2.2.3. Perpendicular Acceleration\nFor comparable parallel and perpendicular forces d p/dt, the ra-\ndiation power from the parallel component is of order 1 /γ2com-\npared to that from the perpendicular component. The energy loss\nby radiation is also mainly determined by the coherent curvature\nradiation. Similar to the parallel scenario, the velocity change\ncaused by E1,⊥of a single charge can be described as\nE1,⊥edt≃meγscdβ1,⊥, (14)\nwhenβ1,⊥≪β∥. The velocity change is also slight, therefore,\none can use Equation (A.6) to calculate the spectrum (see Ap-\npendix A).\n2.2.4. Spectrum of Perpendicular Acceleration\nThe polarization components of the perpendicular perturbation-\ninduced wave of a charge with an identifier of i jkare calculated\nas\n˜Ai jk,∥≃Z+∞\n−∞1\n2β1,⊥(t)h\nφ2\nk−(χi j+ct/ρ)2i\n×exp \niω\n2\" 1\nγ2+φ2\nk+χ2\ni j!\nt+c2t3\n3ρ2+ct2\nρχi j#!\ndt,\n˜Ai jk,⊥≃Z+∞\n−∞β1,⊥(t)φk(χi j+ct/ρ)\n×exp \niω\n2\" 1\nγ2+φ2\nk+χ2\ni j!\nt+c2t3\n3ρ2+ct2\nρχi j#!\ndt.(15)\n-0.04 -0.02 0 0.02 0.04 -0.5 00.5 1Normalized Intensity -100 0100 Fraction (%) -90090 PA ( °)Fig. 5. Simulated polarization profiles for the bunch with γ=102,\nφt=10−2,ρ=107cm. Top panel: the polarization position angle\nenvelope. Middle panel: total polarization fraction (black solid line),\nlinear polarization fraction (red dashed line), and circular polarization\nfraction (blue dashed-dotted line). Bottom: total intensity I(black solid\nline), linear polarization L(red dashed line), and circular polarization\nV(blue dashed-dotted line) as functions of φ. All three functions are\nnormalized to the value of Iatφ=0.\nwhereχi jandφkare small angles. The amplitudes are much\nsmaller than those of parallel perturbations with comparable β1.\nSimilar to the parallel scenario, the total amplitudes of a bunch\nfor the perpendicular perturbation case can be calculated by sum-\nmation of Equation (A.17).\nA narrow spectrum is expected when γis a constant with\nthe scattering of χandφis small. The spike is at ω=2γ2ωa\nresembling that of the parallel perturbation case. A small change\nin velocity can also make a violent change in the Lorentz factor\nbut with a noticeable di fference from that of longitude motion.\nUnder the energy supply from the E∥, charges have β∥≈1. The\nchange of Lorentz factor is δγ≃γ3β⊥δβ⊥whenβ1,⊥≪1. To\nkeep electrons being monoenergic, the condition of β1≪1/γ\nshould be satisfied at least.\nFollowing the calculation steps in the Appendix A, we simu-\nlate the spectra in four cases: φt=0.1/γ,β1=10−4;φt=0.1/γ,\nβ1=10−3;φt=1/γ,β1=10−4; andφt=1/γ,β1=10−3,\nshown as Figure 4. Here, we take γ=100,ρ=107cm,φ=0\nandωa=2π×105Hz. The spectra are normalized to the Fνof\ncurvature radiation with φt=10−3atω=ωc. The spikes for\nthe four cases are very small so the spectra are dominated by\ncurvature radiations.\n2.3. Polarization of the Intrinsic Radiation Mechanisms\nFour Stokes parameters can be used to describe the polarization\nproperties of a quasi-monochromatic electromagnetic wave (see\nArticle number, page 6 of 19Wang, W.-Y ., et al.: Narrow spectra of repeating fast radio bursts: A magnetospheric origin\nAppendix B). Based on Equation (A.6), the electric vectors of\nthe emitting wave reflect the motion of charge in a plane normal\nto the LOS. For instance, if the LOS is located within the trajec-\ntory plane of the single charge, the observer will see electrons\nmoving in a straight line so that the emission is 100% linearly\npolarized. Elliptical polarization would be seen if the LOS is\nnot confined to the trajectory plane. We define that φ < θ cas\non-beam and φ > θ cas off-beam. For the bunch case, similar\non-beam and o ff-beam cases can also be defined in which θcis\nreplaced by φt+θc(Wang et al. 2022a). The general conclusion\nof the bunching curvature radiation is that high linear polariza-\ntion appears for the on-beam geometry, whereas high circular\npolarization is present for the o ff-beam geometry.\nWe consider the polarization properties of the bunch struc-\nture case (Section 2.1.2), and the perturbation scenarios includ-\ning the parallel and the perpendicular cases (Section 2.2). For the\nquasi-periodically distributed bulk of bunches scenario, both A∥\nandA⊥multiply the same factor, so that the polarization proper-\nties are similar with a bulk of bunches, which bunches are uni-\nformly distributed (e.g., Wang et al. 2022c). For the perturbation\nscenario, the perturbation on the curved trajectory can bring new\npolarization components, which generally obey the on(o ff)-beam\ngeometry. However, the intensity of the perturbation-induced\ncomponents is too small then the polarization profiles are mainly\ndetermined by the bunching curvature radiation. Consequently,\nthe polarization properties for both the bunching mechanism and\nthe perturbation scenarios can be described by the uniformly dis-\ntributed bulk of bunches.\nWe takeγ=102,φt=10−2andρ=107cm. The beam angle\natω=ωcis 0.02. According to the calculations in Appendix\nB, we simulate the I,L,V, and PA profiles, shown in Figure\n5. These four parameters can completely describe all the Stokes\nparameters. The total polarization fraction is 100% because the\nelectric vectors are coherently added. The polarization position\nangle (PA) for the linear polarization exhibits some variations\nbut within 30◦. The di fferential of Vis the largest at φ=0 and\nbecomes smaller when |φ|gets larger. High circular polarization\nwith flat fractions envelope would appear at o ff-beam geome-\ntry. In this case, the flux is smaller but may still be up to the\nsame magnitude as the peak of the profile (e.g., Liu et al. 2023b).\nThe spectrum at the side of the beam becomes flatter due to the\nDoppler e ffect (Zhang 2021). The sign change of Vwould be\nseen whenφ=0 is confined into the observation window (Wang\net al. 2022a).\n3. Trigger of Bunches\nThe trigger mechanism plays an important role in producing\nbunching charges and creating the E∥. Intriguingly, a giant glitch\nwas measured 3 .1±2.5 days before FRB 20200428A associated\nwith a quasi-periodic oscillations (QPOs) of ∼40 Hz in the X-\nray burst from the magnetar SGR J1935 +2154 (Mereghetti et al.\n2020; Li et al. 2021; Ridnaia et al. 2021; Tavani et al. 2021; Ge\net al. 2022; Li et al. 2022). Another glitch event was found on\n5 October 2020 and the magnetar emitted three FRB-like radio\nbursts in subsequent days (Younes et al. 2023; Zhu et al. 2023).\nBoth the glitches and QPOs are most likely the smoking guns of\nstarquakes. Therefore, magnetar quakes are considered a promis-\ning trigger model of FRB in the following.\nAs discussed in Section 2.1.2, we are particularly inter-\nested in the mechanisms that can generate quasi-periodically dis-\ntributed bunches. Two possible scenarios are discussed indepen-\ndently in Section 3.1.1 and Section 3.1.2. Motivated by the fact\nthat Alfvén waves associated with the quake process have beenproposed by lots of models (e.g., Wang et al. 2018; Kumar &\nBošnjak 2020; Yang & Zhang 2021; Yuan et al. 2022), we dis-\ncuss the Alfvén waves as the perturbations in Section 2.2.\n3.1. Structured Bunch Generation\n3.1.1. Pair Cascades from “Gap”\nA magnetized neutron star’s rotation can create a strong electric\nfield that extracts electrons from the stellar surface and acceler-\nates them to high speeds (Ruderman & Sutherland 1975). The\ninduced electric field region lacks adequate plasma, i.e., charge-\nstarvation, referred to as a “gap”. The extracted electrons move\nalong the curved magnetic field lines near the surface, emitting\ngamma rays absorbed by the pulsar’s magnetic field, resulting\nin electron-positron pairs. Such charged particles then emit pho-\ntons that exceed the threshold for e ffective pair creation, result-\ning in pair cascades that populate the magnetosphere with sec-\nondary pairs. The electric field in the charge-starvation region is\nscreened once the pair plasma is much enough and the plasma\ngeneration stops (Levinson et al. 2005). When the plasma leaves\nthe region, a gap with almost no particles inside is formed forms\nagain, and the vacuum electric field is no longer screened.\nThe numerical simulations show that such pair creation is\nquasi-periodic (Timokhin 2010; Tolman et al. 2022), leading to\nquasi-periodically distributed bunches. The periodicity of cas-\ncades is∼3hgap/c, where hgapis the height of the gap. The gap’s\nheight is written as (Timokhin & Harding 2015)\nhgap=1.1×104ρ2/7\n7P3/7\n0B−4/7\n12|cosα|−3/7cm, (16)\nwhere Pis the spin period and αis the inclination angle of mag-\nnetic axis. For a typical radio pulsar, we take B=1012G and\nP=1 s. The corresponding ωMis 107Hz. According to Equa-\ntion (8), radiation energy is emitted into multiples of ∼10 MHz\nnarrow bandwidths, which is consistent with the narrow-band\ndrifting structure of PSR B0950 +08 (Bilous et al. 2022).\nIn contrast to the continuous sparking in the polar cap region\nof normal pulsars, FRB models prefer a sudden, violent spark-\ning process. Starquake as a promising trigger mechanism can\noccur when the pressure induced by the internal magnetic field\nexceeds a threshold stress. The elastic and magnetic energy re-\nleased from the crust into the magnetosphere are much higher\nthan the spin-down energy for magnetars. Crust shear and mo-\ntion may create hills on the surface. The height of the gap be-\ncomes smaller and the electric field is enhanced due to the so-\ncalled point-discharging. Quasi-periodically distributed bunches\nwithωM∼108may be formed from such shorter gaps. Observa-\ntional consequences of small hills on stellar surfaces could also\nbe known by studying regular pulsars (e.g., Wang et al. 2023).\nNote that the sparking process to generate FRB-emitting\nbunches is powered by magnetic energy rather than spin-down\nenergy. The stellar oscillations during starquakes can provide ad-\nditional voltage, making the star active via sparking (Lin et al.\n2015). The oscillation-driven sparking process may generate\nmuch more charges than the spin-down powered sparking of a\nnormal rotation-powered pulsar.\n3.1.2. Two-stream Instability\nAlternatively, another possibility to form charged bunches is\nthe two-stream instability between electron-positron pairs. The\ngrowth rate of two-stream instability is discussed by invoking\nlinear waves. Following some previous works (e.g., Usov 1987;\nGedalin et al. 2002; Yang & Zhang 2023), we consider that there\nArticle number, page 7 of 19A&A proofs: manuscript no. AAmain_clean\nare two plasma components denoted by “1” and “2” with a rel-\native motion along the magnetic field line. The Lorentz factor\nand the particle number density are γjandnjwith j=1,2. Each\nplasma component is assumed to be cold in the rest frame of each\ncomponent. Then the dispersion relation of the plasma could be\nwritten as (Asseo & Melikidze 1998)\n1−ω2\np,j=1\nγ3\nj=1(ω−βj=1kc)2−ω2\np,j=2\nγ3\nj=2(ω−βj=2kc)2=0, (17)\nwhereωp,jis the plasma frequency of the component j. For\nthe resonant reactive instability as the bunching mechanism, the\ngrowth rate and the characteristic frequency of the Langmuir\nwave are given by (e.g., Usov 1987; Gedalin et al. 2002; Yang &\nZhang 2023)\nΓ∼γ−1\nj=1γ−1\nj=2 nj=1\nnj=2!1/3\nωL, (18)\nωL∼2γ1/2\nj=2ωp,j=2. (19)\nThe linear amplitude of the Langmuir wave depends on the gain\nG∼Γr/c. According to Rahaman et al. (2020), a threshold gain\nGthindicating the breakdown of the linear regime is about Gth∼\na few. The su fficient amplification would drive the plasma be-\nyond the linear regime once G∼Gth, and the bunch forma-\ntion rate is Γ(G∼Gth). The bunch separation corresponds to\nthe wavelength of the Langmuir wave for G∼Gth, i.e.,\nlsep∼πc\nγ1/2\nj=2ωp,j=2. (20)\nA narrow-band spectrum with ∆ν∼100 MHz would be created\nifγ1/2\nj=2ωp,j=2∼108Hz.\nGenerally, the charged bunches would form and disperse dy-\nnamically due to the instability, charge repulsion, velocity dis-\npersion, etc (e.g., Lyutikov 2021; Yang & Zhang 2023). The\nspectrum of the fluctuating bunches depends on the bunch for-\nmation rate λB, lifetimeτB, and bunch Lorentz factor γ, as pro-\nposed by Yang & Zhang (2023). The coherent spectrum by a\nfluctuating bunch is suppressed by a factor of ( λBτB)2compared\nwith that of a persistent bunch, and there is a quasi-white noise\nin a wider band in the frequency domain. The observed narrow\nspectrum implies that, at the minimum, the quasi-white noise\nshould not be dominated in the spectrum, in which case, the con-\ndition of 2γ2λB≳min(ωpeak,2γ2/τB) would be required, where\nωpeakis the peak frequency of curvature radiation, see Yang &\nZhang (2023) for a detailed discussion.\n3.2. Quake-induced Oscillation and Alfvén Wave\nAn Alfvén wave packet can be launched from the stellar surface\nduring a sudden crustal motion and starquake. The wave vector\nis not exactly parallel to the field line so the Alfvén waves have\na non-zero electric current along the magnetic field lines. Star-\nquakes as triggers of FRBs are always accompanied by Alfvén\nwave launching. We investigate whether Alfvén waves as the\nperturbations discussed in Section 2.2 can a ffect the spectrum.\nThe angular frequencies of the torsional modes Ωcaused by\na quake are calculated as follows (also see Appendix C). We\nchose the equation of state (EOS A) (Pandharipande 1971) and\nEOS WFF3 (Wiringa et al. 1988), which describe the core of\nneutron stars, as well as consider two di fferent proposed EOSs\nfor the crusts, including EOSs for NV (Negele & Vautherin\n(Hz)ΩFig. 6. The frequencies of the second overtone n=2 andℓ=2 torsional\nmodes as a function of the normalized magnetic field ( B/Bµ). The NS\nmass is M=1.4M⊙. The dashed lines correspond to our fits using\nthe empirical formula (21) with di fferent coe fficient values. The fitting\nvalues are 2 .4, 0.78, 2.2, and 0.69 for WFF3 +DH model, WFF3 +NV\nmodel, A +DH model, and A +NV model, respectively.\n1973) and DH (Douchin & Haensel 2001) models. We match\nthe various core EOS to two di fferent EOSs for the crust. The\ncrust-core boundary for the NV and DH EOSs is defined at\nρm≈2.4×1014g cm−3, andρm≈1.28×1014g cm−3, respec-\ntively.\nThe frequencies of torsional modes are various with a strong\nmagnetic field (Messios et al. 2001). As already emphasized by\nSotani et al. (2007), the shift in the frequencies would be signif-\nicant when the magnetic field exceeds ∼1015G. In the presence\nof magnetic fields, frequencies are shifted as,\nℓfn=ℓf(0)\nn1+ℓαn B\nBµ!21/2\n, (21)\nwhereℓαnis a coe fficient depending on the structure of the star,\nℓf(0)\nnis the frequency of the non-magnetized star, and Bµ≡\n(4πµ)1/2is the typical magnetic field strength.\nIn Fig. 6, we show the e ffects of the magnetic field on the fre-\nquencies of the torsional modes. The magnetic field strength is\nnormalized by Bµ=4×1015G. Di fferent dashed lines in Fig. 6\nare our fits to the calculated numerical data with a high accu-\nracy. For B>Bµ, we find that the frequencies follow a quadratic\nincrease against the magnetic field, and tend to become less sen-\nsitive to the NS parameters. The oscillation frequencies exceed\n103Hz for n=2 and Ω∼104Hz for typical magnetars.\nAlfvén waves are launched from the surface at a frequency\nofΩ, which is much faster than the angular frequency of the\nspin, and propagate along the magnetic field lines. Modifying the\nGoldreich-Julian density can produce electric fields to accelerate\nmagnetospheric charged particles. The charged particles oscil-\nlate at the same frequency as the Alfvén waves in the comoving\nframe. In the lab frame, the perturbation has ωa= Ωγ2/γ2\nAdue to\nthe Doppler e ffect, whereγA=p\n1+B2/(4πρmc2).ωais much\nsmaller than 108Ω4γ2\n2Hz because of the strong magnetic field\nand low mass density in the magnetosphere.\nTherefore, by considering that Alfvén waves play a role in\nthe electric perturbations discussed in Section 2.2, the spectra\nhave no spike from 100 MHz to 10 GHz, which is the observa-\ntion bands for FRBs so far. The spectral spike would appear at\nArticle number, page 8 of 19Wang, W.-Y ., et al.: Narrow spectra of repeating fast radio bursts: A magnetospheric origin\nthe observation bands when the perturbations are global oscilla-\ntions, meaning that all positions in the magnetosphere oscillate\nsimultaneously with Ωrather than a local Alfvén wave packet.\n4. Absorption Effects in Magnetosphere\nIn this Section, we investigate absorption e ffects during the\nwave propagating the magnetosphere. The absorption e ffects dis-\ncussed as possible origins of narrow-band emission are indepen-\ndent of the intrinsic mechanisms in Section 2. To create narrow-\nband emission, there has to be a steep frequency cut-o ffon the\nspectrum due to the absorption conditions. Absorption condi-\ntions should also change cooperatively with the emission region,\nthat is, the cut-o ffappears at a lower frequency as the flux peak\noccurs at a lower frequency. Curvature radiation spectra can be\ncharacterized as a power law for ω < ω cand exponential func-\ntion atω > ω c(Jackson 1998). In the following discussion, we\nmainly consider two absorption mechanisms including Landau\ndamping and self-absorption of curvature radiation.\n4.1. Wave Modes\nWe consider the propagation of electromagnetic waves in mag-\nnetospheric relativistic plasma. The magnetosphere mainly con-\nsists of relativistic electron-positron pair plasma, which can af-\nfect the radiation created in the inner region. We assume that\nthe wavelength is much smaller than the scale lengths of the\nplasma and of the magnetic field in which the plasma is im-\nmersed. The scale lengths of the magnetic field can be estimated\nasB/(∂B/∂r)∼r. The plasma is consecutive and its number\ndensity is thought to be multiples of the Goldreich-Julian den-\nsity (Goldreich & Julian 1969), leading to the same scale lengths\nas that of the magnetic field. Both the scale lengths of magnetic\nfield and plasma are much larger than the wavelength λ=30ν−1\n9\ncm, even for outer magnetosphere r∼4.8×109P0cm.\nWhatever a neutron star or a magnetar, the magnetospheric\nplasma is highly magnetized. It is useful to define the (non-\nrelativistic) cyclotron frequency and plasma frequency:\nωB=eB\nmec=1.78×1019B12Hz, (22)\nωp= 4πnpe2\nme!1/2\n=4.73×1011M1/2\n3B1/2\n12P1/2\n0Hz. (23)\nwhere np=MnGJ=MB/(ceP), in which nGJis the Goldreich-\nJulian density, and Mis a multiplicity factor. In the polar-cap\nregion of a pulsar, a field-aligned electric field may accelerate\nprimary particles to ultra-relativistic speed. The pair cascading\nmay generate a lot of plasma with lower energy and a multiplic-\nity factor ofM∼ 102−105(Daugherty & Harding 1982). The\nrelationships that ω≪ωBandωp≪ωBare satisfied at most\nregions inside the magnetosphere.\nThe vacuum polarization may occur at the magnetosphere\nfor high energy photons and radio waves by the plasma e ffect in\na strong magnetic field (Adler 1971). At the region of r≳10R\nwhere FRBs can be generated, in which Ris the stellar radius, the\nmagnetic field strength B<BQ=m2\nec3/(eℏ) with assumption of\nBs=1015at stellar surface. In the low-frequency limit ℏω≪\nmec2, the vacuum polarization coe fficient can be calculated as\nδV=1.6×10−4(B/BQ)2=2.65×10−8B2\n12(Heyl & Hernquist\n1997; Wang & Lai 2007). Therefore, the vacuum polarization\n10 810 910 10 \n (Hz)10 010 210 410 610 8Optical Depth \nGaussian \nJüttner \nPower-law, p = 3 \nPower-law, p = 4 Fig. 7. Optical depths for four distributions: Jüttner with ρT=0.1 (black\nsolid line), Gaussian with um=10/√\n2π(blue solid line), power-law\nwith p=3 (red dashed line) and power-law with p=4 (red dotted-\ndashed line).\ncan be neglected in the following discussion about the dispersion\nrelationship in the magnetosphere.\nWaves have two propagation modes: O-mode and X-mode.\nIn the highly magnetized relativistic case, the dispersion of X-\nmode photon is n≈1. O-mode photons have two branches in the\ndispersion relationship, which are called subluminous O-mode\n(n>1) and superluminous O-mode ( n<1), shown as Figure\nD.1. The subluminous O-mode cuts o ffat 1/cosθB, whereθBis\nthe angle between angle between ˆkand ˆB. Following the calcu-\nlation shown in Appendix D, the dispersion relation of magneto-\nspheric plasma for subluminous O-mode photons can be found\nas (Arons & Barnard 1986; Beskin et al. 1988)\n\u0010\nω2−c2k2\n∥\u00111−X\nsω2\np,s\nω2gs−c2k2\n⊥=0, (24)\nwith\ngs=PZ∞\n−∞f(u)du\nγ3\u00001−n∥β\u00012+iπβ2\nnγ3\nn\"df\ndu#\nβ=βn, (25)\nwhereβn=n−1\n∥=ω/(k∥c),γn=(1−β2\nn)−1/2and f(u) is the\ndistribution function of plasma.\n4.2. Landau Damping\nWave forces can exist many collective waves in a beam and ex-\nchange energy between waves and particles. If the distribution\nfunction is d f/dv <0 close to the phase speed of the wave, the\nnumber of particles with speeds v<ω/ kwill be larger than that\nof the particles having v>ω/ k. Then the transmission of energy\nto the wave is then smaller than that absorbed from the wave,\nleading to a Landau damping of the wave. According to the dis-\npersion relationship given in Section 4.1, there is no linear damp-\ning of the X-mode since wave-particle resonance is absent. The\nsuperluminous branch of the O-mode also has no Landau damp-\ning because the phase speed exceeds the speed of light. Landau\ndamping only occurs at the subluminous O-mode photons.\nArticle number, page 9 of 19A&A proofs: manuscript no. AAmain_clean\nThe Landau damping decrement can be calculated by the\nimaginary part of the dispersion relationship (Boyd & Sander-\nson 2003). We let k=kr+iki, where ki≪kr≈k. If a wave has a\nform of E∝exp(−iωt+ikr), the imaginary kican make the am-\nplitude drop exponentially. The imaginary part of Equation (24)\nby considering a complex waveform of kis given by\nki=n2ω2\npcos2θB−ω2\np\n2kc2(1−grω2pcos2θB/ω2)gi, (26)\nwheregrandgidenote the real and imaginary parts of Equa-\ntion (25). Consider emission at radius realong a field line whose\nfootprint intersects the crust at magnetic colatitude Θe(R/re)1/2\nforΘe≪1, one can obtain the angle between kandBat each\npoint:\nθB=3\n8Θe r\nre!1/2\u0012\n1−re\nr\u00132\n. (27)\nConsequently, the optical depth for waves emitted at reto reach\nris\nτLD=Zr\nre2kidr. (28)\nEven the distribution function for the magnetospheric elec-\ntrons and positrons is not known, a relativistic thermal distribu-\ntion is one choice. Since the vertical momentum drops to zero\nrapidly, for instance, a 1-D Jüttner distribution is proposed here\n(e.g., Melrose et al. 1999; Rafat et al. 2019):\nf(u)=1\n2K1(ρT)exp(−ρTγp), (29)\nwhereρT=mec2/(kBT), in which Tis the temperature of plasma\nandkBis the Boltzmann constant. Alternatively, another widely\nused distribution is a Gaussian distribution (e.g., Egorenkov et al.\n1983; Asseo & Melikidze 1998)\nf(u)=1√\n2πumexp \n−u2\n2u2m!\n, (30)\nwhere umcan be interpreted as the average uover the distribution\nfunction. Besides a thermal distribution, power law distributions\nf(γp)=(p−1)γ−p\np, where pis the power law index ( p>2),\nare representative choice to describe the plasma in the magneto-\nsphere (e.g., Kaplan & Tsytovich 1973).\nThe optical depths as a function of wave frequency with four\ndifferent distributions are shown in Figure 7. Here, we assume\nthatre=107cm,M=103,γp=10 and Θe=0.05. For all\nconsidered distributions, optical depths decrease with frequen-\ncies which means that the Landau damping is more significant\nfor lower-frequency subluminous O-mode photons. The optical\ndepths are much larger than 1 for the FRB emission band so that\nthe subluminous O-mode photon can not escape from the mag-\nnetosphere. Superluminous O-mode photons which frequency\nhigher than ωcut=ωpγ−3/2\npcan propagate. The bunching curva-\nture radiation is exponentially dropped at ω > ω c. If the super-\nluminous O-mode cut-o fffrequency is close to ωc, the spectrum\nof the emission would be narrow. A simple condition could be\ngiven when a dipole field is assumed:\nr≃13.8RM3Bs,15P−1\n0γ−6\n2γ−3\np,1.3, (31)\nwhere the emission region is considered at closed field lines in\nwhichρ∼r. The cut-o fffrequencyωcut∝r−3/2which evolves\n1081091010\n (Hz)10-310-210-1100Normalized FFig. 8. Normalized spectra of self-absorbed curvature radiation. Black\nlines denote purely bunching curvature radiations for φt=0.1/γ(solid\nline) andφt=1/γ(dashed-dotted line). Blue lines denote the radiation\nafter curvature self-absorption for φt=0.1/γ(solid line) and φt=1/γ\n(dashed-dotted line).\nsteeper than ωc∝r−1. The bandwidth is suggested to become\nwider as the burst frequency gets lower.\nCoherent curvature radiation can propagate out of the mag-\nnetosphere via superluminous O-mode with a frequency cut-o ff,\nleading to narrow spectra likely. However, the condition that the\nfrequency cut-o ffis close to the characteristic frequency of cur-\nvature radiation could not be satisfied at all heights, leading to\nnarrow spectra sometimes. The waves can also propagate via X-\nmode photon which has no frequency cut-o ff. The spectrum of\nX-mode photons ought to be broad. The waves propagate via\nwhether X-mode or O-mode is hard to know because the real\nmagnetic configuration is complex. Narrow spectra have been\nfound in some FRBs, while some FRBs seem to have broad\nspectra, and there are also some not sure due to the limitation\nof bandwidth of the receiver (Zhang et al. 2023).\nIn principle, the spectrum of a radio magnetar would be nar-\nrow, if the emission is generated from r>10Rpropagating via\nO-mode. However, radio pulses are rarely detected among mag-\nnetars. For normal pulsars, the lower magnetic fields may in-\nduce lower number density so that the Landau damping is not as\nstrong as in magnetars.\n4.3. Self-absorption by Curvature Radiation\nThe self-absorption is also associated with the emission. We as-\nsume that the momentum of the particle is still along the mag-\nnetic field line even after having absorbed the photon. The cross-\nsection can be obtained through the Einstein coe fficients when\nhν≪γsmec2(e.g., Locatelli & Ghisellini 2018):\ndσc\ndΩ=1\n2meν2∂\n∂γs\u0002js(ν,γ s,φs)\u0003, (32)\nwhereσcis the curvature cross section and jsis the emissivity.\nThere are special cases which the stimulated emission be-\ncomes larger than the true absorption. As a result, the total\ncross section becomes negative, and there is a possible maser\nmechanism. However, such maser action is ine ffective when the\nmagnetic energy density is much higher than that for the ki-\nnetic energy of charges (Zhelezniakov & Shaposhnikov 1979).\nArticle number, page 10 of 19Wang, W.-Y ., et al.: Narrow spectra of repeating fast radio bursts: A magnetospheric origin\nFitting for FRB 121102 \nFitting for FRB 180916 \nFitting for all FRBs \nFRB 121102 Gajjar+2018 \nFRB 121102 Michilli+2018 \nFRB 121102 Hessels+2019\nFRB 121102 Josephy+2019\nFRB 121102 Caleb+2020 &Platts+2021 FRB 201124 Hilmarsson+2021 \nFRB 201124 Zhou+2022 FRB 180301 Kumar+2023FRB 180814 CHIME/FRB+2019 \nFRB 180916 CHIME/FRB+2020 & Chamma+2021 \nFRB 180916 Chawla+2020 \nFRB 180916 Pastor-Marazuela+2021 \nFRB 180916 Sand+2022 \nFRB 180916 Gopinath+2023 \nFRB 190711 Day+2020 10 210 310 4\nFrequency (MHz) -10 6-10 4-10 2-10 0-10 -2 Dirft Rate (MHz ms -1)\nFig. 9. Comparison of drift rates at di fferent frequencies. Di fferent re-\ngions show the 1 σregion of the best fitting results: FRB 20121102A\n(blue), FRB 180916B(red) and all FRBs (grey).\nThe isotropic equivalent luminosity for the curvature radiation is\n∼1042erg s−1ifNe=1022atr=107cm forγ=102(Kumar\net al. 2017). The number density in a bunch is estimated as ne∼\n1011cm−3by Wang et al. (2022c), so that neγmec2≪B2/(8π).\nThus, the maser process is ignored in the following discussion.\nBy inserting the curvature power per unit frequency (see e.g.,\nJackson 1998), the total cross section integrated over angles is\ngiven by (Ghisellini & Locatelli 2018)\nσc≈25/3√\n3Γ(5/3)πe2\nγ3smecν, ν≪νc, (33)\nwhere Γ(ν) is the Gamma function. As discussed in Section 2,\nthe charges are approximately monoenergetic due to the rapid\nbalance between E∥and the curvature damping. The absorption\noptical depth is\nτCR≃σcneρ\nγs. (34)\nThe absorption is more significant for lower frequencies.\nFigure 8 exhibits the comparison between purely bunching\ncurvature radiation and it after the absorption. We adopt that γs=\n102,ne=1011cm−3andρ=107cm. The plotted spectra are\nnormalized to the Fνof curvature radiation with φt=10−3at\nω=ωc. The spectral index is 5 /3 at lower frequencies due to the\nself-absorption, which is steeper than pure curvature radiation,\nbut it is still wide-band emission. Alternatively, the E∥can let\nelectrons and positrons decouple, which leads to a spectral index\nof 8/3 at lower frequencies (Yang et al. 2020). The total results\nof these independent processes can make a steeper spectrum with\nan index of 13 /3.\n5. Bunch Expansion\nAs the bunch moves to a higher altitude, the thickness of the\nbunch (the size parallel to the field lines) becomes larger due to\n10 010 210 410 6\nFrequency/Width (MHz ms -1 )-10 10 -10 5-10 0-10 -5 Dirft Rate (MHz ms -1 )Fig. 10. Same as Figure 9 but for comparison of drift rates at di fferent\nfrequency /width.\nthe repulsive force of the same sign charge. If the bunch travels\nthrough the emission region faster than the LOS sweeps across\nthe whole emission beam, the thickness can be estimated as cw,\nwherewis the burst width (Wang et al. 2022a). For simplicity,\nwe consider the motion of electrons at the boundary due to the\nCoulomb force from bunches in the co-moving rest frame. The\nacceleration of an electron at the boundary is estimated as\na′\nb=Ze2nel′\n(R′2\nb+l′2)3/2γmedV′, (35)\nwhere R′\nb=Rbis the transverse size of bunch. Note that the\ntransverse size of a bunch is much larger than the longitude size\neven in the co-moving rest frame. Equation (35) is then calcu-\nlated as a′\nb≃πe2nes′/(γme). Charged particles in the bunch can\nnot move across the B-field due to the fast cyclotron cooling.\nThe transverse size is essentially Rb∝rdue to the magnetic\nfreezing, thus, we have a′\nb∝r−2. The distance that the electron\ntravels in the co-moving frame is s′∝rby assuming r≫r0(r0\nis an initial distance). As a result, the thickness of the bunch is\ncw∝rin the lab frame.\nBurst width wwould be larger if the emission is seen from\na higher altitude region. If there is more than one bulk of\nbunches being observed coincidentally but with di fferent tra-\njectory planes, one would see the drifting pattern (Wang et al.\n2019). The geometric time delay always lets a burst from a lower\nregion be seen earlier, that is, a downward drifting pattern is ob-\nserved. Subpulse structure with such a drifting pattern is a nat-\nural consequence caused by the geometric e ffect when the char-\nacteristic frequency of the emission is height-dependent.\nFollowing the previous work (Wang et al. 2022c), we con-\ntinuously investigate the drifting rate as a function of the burst\ncentral frequency and width. The drift pattern is a consequence\nof narrow-band emission. Compared with our previous work, we\nadd the fitting results of FRB 20180916B. Figure 9 and 10 show\nthe best fitting results for FRB 20121102A (blue dashed-dotted\nlines), FRB 20180916B (red dashed lines) and all FRBs (black\nsolid lines) and their corresponding 1 σregions. The best fit-\nting of ˙ν−νrelationship for FRB 20121102A, FRB 20180916B\nand all FRBs are (˙ ν/MHz ms−1)=−10−5.30×(ν/MHz)2.29,\n(˙ν/MHz ms−1)=−10−4.52×(ν/MHz)1.93, and (˙ν/MHz ms−1)=\n−10−6.25×(ν/MHz)2.52. The best fitting of ˙ νas a function\nArticle number, page 11 of 19A&A proofs: manuscript no. AAmain_clean\nofν/width for FRB 20121102A, FRB 20180916B and all\nFRBs are (˙ν/MHz ms−1)=−10−1.75×(ν/MHz)1.10(w/ms)−1.10,\n(˙ν/MHz ms−1)=−10−1.11×(ν/MHz)0.88(w/ms)−0.88, and\n(˙ν/MHz ms−1)=−10−0.99×(ν/MHz)0.94(w/ms)−0.94. The fitting\nresults generally match the case of curvature radiation, which has\n˙ν∝ν2. The relationship ˙ ν∝νw−1is attributed to w∝r. Data are\nquoted from references as follows: Michilli et al. (2018); Gaj-\njar et al. (2018); Hessels et al. (2019); Josephy et al. (2019);\nCaleb et al. (2020); Platts et al. (2021); Kumar et al. (2023);\nCHIME /FRB Collaboration et al. (2019a); Chime /Frb Collabo-\nration et al. (2020); Chamma et al. (2021); Chawla et al. (2020);\nPastor-Marazuela et al. (2021); Sand et al. (2022); Day et al.\n(2020); Hilmarsson et al. (2021); Zhou et al. (2022); Gopinath\net al. (2024).\n6. Discussion\n6.1. Curvature Radiation\nCurvature radiation is formed since the perpendicular accelera-\ntion on the curved trajectory and its spectrum is naturally wide-\nband. Spectra of bunching curvature radiation are more complex\ndue to the geometric conditions and energy distribution (e.g.,\nYang & Zhang 2018; Wang et al. 2022c). Despite considering\nthe complex electron energy distributions, the spectrum is still\nhard to be narrow-band.\nA quasi-periodically distributed bunches can create such\nnarrow-band emission. The radiation from the bulk of bunches is\nquasi-periodically enhanced and the bandwidth depends on ωM\nand the threshold of the telescope. As shown in Figure 2, narrow-\nband emissions with roughly the same bandwidth are expected\nto be seen from multi-band observations simultaneously. The\nenergy is radiated into several multiples of 2 πωMwith narrow\nbandwidths and the required number of particles in the bunch is\nsmaller than that of pure curvature radiation.\n6.2. Inverse Compton scattering\nInverse Compton scattering (ICS) occurs when low-frequency\nelectromagnetic waves enter bunches of relativistic particles\n(Zhang 2022). The bunched particles are induced to oscillate at\nthe same frequency in the comoving frame. The outgoing ICS\nfrequency in the lab-frame is calculated as\nω≃ωinγ2(1−cosθin), (36)\nwhereωinis the angular frequency of the incoming wave and\nθinis the angle between the incident photon momentum and the\nelectron momentum. This ICS process could produce a narrow\nspectrum for a single electron by giving a low-frequency wave\nof narrow spectrum (Zhang 2023). In order to generate GHz\nwaves, dozens of kHz incoming wave is required for bunches\nwithγ=102andθinshould be larger than 0 .5. This requires\nthe low-frequency waves to be triggered from a location very\nfar from the emission region. The X-mode of these waves could\npropagate freely in the magnetosphere, but O-mode could not\nunless the waves’ propagation path approaches a vacuum-like\nenvironment.\nAs discussed in Section 2, the Lorentz factor is constant\ndue to the balance between E∥and radiation damping. θinvaries\nslightly as the LOS sweeps across di fferent field lines. If the in-\ncoming wave is monochromatic, the outgoing ICS wave would\nhave a narrow value range matching the observed narrow-band\nFRB emissions. The polarization of ICS from a single charge isstrongly linearly polarized, however, the electric complex vec-\ntors of the scattered waves in the same direction are added for all\nthe particles in a bunch. Therefore, circular polarization would\nbe seen for o ff-beam geometry (Qu & Zhang 2023), similar to\nthe case of bunching curvature radiation but they are essentially\ndifferent mechanisms(Wang et al. 2022b). Doppler e ffects may\nbe significant at o ff-beam geometry so that the burst duration\nand spectral bandwidth would get larger. The model predicts\nthat a burst with higher circular polarization components tends to\nhave wider spectral bandwidth. This may di ffer from the quasi-\nperiodically distributed bunch case, in which the mechanism to\ncreate narrow spectra is independent of polarization.\nThe perturbation discussed in Section 2.2 can let charges\noscillate at the same frequency in the comoving frame. This\nprocess is essentially the same as the ICS emission. The ac-\nceleration caused by the perturbation is proposed to be pro-\nportional to exp(−iωat), which leads to the wave frequency of\nωa(1−βcosθin) due to the Doppler e ffect in the lab frame. Tak-\ning the wave frequency into Equation (13), the result is the same\nas Equation (36). We let F(t)=R\nE(t)em−1\nedtand ignore the ra-\ndiation damping as a simple case. The dimensionless velocity\nis then given by β=F(t)/p\n1+F(t)2. By assuming F(t) has\na monochrome oscillating form, according to Equation (A.6) if\nF(t)≪1, the result is reminiscent of the perturbation cases and\nICS. If F(t)≫1, we have β∼1 which implies a wide-band\nemission.\n6.3. Interference Processes\nSome large-scale interference processes, e.g., scintillation, grav-\nitational lensing, and plasma lensing, could change the radiation\nspectra via wave interference (Yang 2023). The observed spec-\ntra could be coherently enhanced at some frequencies, leading\nto the modulated narrow spectra. We consider that the multipath\npropagation e ffect of a given interference process (scintillation,\ngravitational lensing, and plasma lensing) has a delay time of\nδt, then the phase di fference between the rays from the multi-\nple images is δϕ∼2πνδt, whereνis the wave frequency. Thus,\nfor GHz waves, a significant spectral modulation requires a de-\nlay time of δt∼10−10ν−1\n9s, which could give some constraints\non scintillation, gravitational lensing, or plasma lensing, if the\nobserved narrow-band is due to these processes.\nFor scintillation, according to Yang (2023), the plasma\nscreen should be at a distance of ∼1015cm from the FRB source,\nand the medium in the screen should be intermediately dense and\nturbulent. For plasma lensing, the lensing object needs to have an\naverage electron number density of ∼10 cm−3and a typical scale\nof∼10−3AU at a distance of ∼1 pc from the FRB source. The\npossibility of gravitational lensing could be ruled out because the\ngravitational lensing event is most likely a one-time occurrence,\nmeanwhile, it cannot explain the remarkable burst-to-burst vari-\nation of the FRB spectra.\n7. Summary\nIn this work, we discuss the spectral properties from the per-\nspectives of radiation mechanisms and absorption e ffects in the\nmagnetosphere, and the following conclusions are drawn:\n1. A narrow spectrum may reflect the periodicity of the\ndistribution of bunches, which have roughly the same Lorentz\nfactor. The energy is radiated into multiples narrow bands of\n2πωMwhen the bulk of bunches is quasi-periodic distributed\n(Yang 2023). Such quasi-periodically distributed bunches may\nArticle number, page 12 of 19Wang, W.-Y ., et al.: Narrow spectra of repeating fast radio bursts: A magnetospheric origin\nbe produced due to quasi-monochromatic Langmuir waves or\nfrom a “gap”, which experiences quasi-periodically sparking.\nThe model predicts that narrow-band emissions may be seen in\nother higher or lower frequency bands.\n2. The quasi-periodically distributed bunches and the pertur-\nbation scenarios share the same polarization properties with the\nuniformly distributed bunches’ curvature radiation (Wang et al.\n2022c). The emission is generally dominated by linear polariza-\ntion for the on-beam case whereas it shows highly circular polar-\nization for the o ff-beam case. The di fferential of Vis the largest\nand sign change of circular polarization appears at φ=0. The\nPA across the burst envelope may show slight variations due to\nthe existence of circular polarization.\n3. Spectra of bunching curvature radiation with normal dis-\ntribution are wide-band. We investigate the spectra from charged\nbunches with force perturbations on curved trajectories. If the\nperturbations are sinusoid with frequency of ωa, there would be\na spike at 2 γ2ωaon the spectra. However, the perturbations are\nconstrained by the monoenergy of bunches unless the scatter-\ning of the Lorentz factor can make the spike wider leading to a\nwide-band spectrum. Monochromatic Alfvén waves as the per-\nturbations would not give rise to a narrow-band spectrum from\n100 MHz to 10 GHz.\n4. Landau damping as an absorption mechanism may gener-\nate a narrow-band spectrum. We consider three di fferent forms of\ncharge distribution in the magnetosphere. Subluminous O-mode\nphotons are optically thick, but there is no Landau damping for\nX-mode and superluminous O-mode photons. The superlumi-\nnous O-mode photons have a low-frequency cut-o ffofωpγ−3/2\np\nwhich depends on the height in the magnetosphere. If so, the\nspectrum would be narrow when ωpγ−3/2\npis close toωcand the\nbandwidth is getting wider as the frequency becomes lower. The\ncondition of ωpγ−3/2\np≈ωccan only be satisfied sometimes. For\nsome FRBs, the spectrum of some FRBs can be broader than sev-\neral hundred MHz. This may be due to the frequency cut-o ffof\nthe superluminous O-mode being much lower than the character-\nistic frequency of curvature radiation, or the waves propagating\nvia X-mode.\n5. Self-absorption by curvature radiation is significant at\nlower frequencies. By considering the Einstein coe fficients, the\ncross section is larger for lower frequencies. The spectral index\nis 5/3 steeper than that of pure curvature radiation at ω≪ωc,\nhowever, the spectrum is not as narrow as ∆ν/ν=0.1.\n6. The two intrinsic mechanisms by invoking quasi-periodic\nbunch distribution and the perturbations are more likely to gen-\nerate the observed narrow spectra of FRBs rather than the ab-\nsorption e ffects.\n7. As a bulk of bunches moves to a higher region, the lon-\ngitude size of the bulk becomes larger, suggesting that burst du-\nration is larger for bulks emitting at higher altitudes. By inves-\ntigating drifting rates as functions of the central frequency and\nν/w, we find that drifting rates can be characterized by power\nlaw forms in terms of the central frequency and ν/w. The fitting\nresults are generally consistent with curvature radiation.\nAcknowledgements. We are grateful to the referee for constructive comment\nand V . S. Beskin, He Gao, Biping Gong, Yongfeng Huang, Kejia Lee, Ze-Nan\nLiu, Jiguang Lu, Rui Luo, Chen-Hui Niu, Jiarui Niu, Yuanhong Qu, Hao Tong,\nShuangqiang Wang, CHengjun Xia, Zhenhui Zhang, Xiaoping Zheng, Enping\nZhou, Dejiang Zhou, and Yuanchuan Zou for helpful discussion. W.Y .W. is grate-\nful to Bing Zhang for the discussion of the ICS mechanism and his encourage-\nment when I feel confused about the future. This work is supported by the Na-\ntional SKA Program of China (No. 2020SKA0120100) and the Strategic Prior-\nity Research Program of the CAS (No. XDB0550300). Y .P.Y . is supported by\nthe National Natural Science Foundation of China (No.12003028) and the Na-\ntional SKA Program of China (2022SKA0130100). J.F.L. acknowledges supportfrom the NSFC (Nos.11988101 and 11933004) and from the New Cornerstone\nScience Foundation through the New Cornerstone Investigator Program and the\nXPLORER PRIZE.\nArticle number, page 13 of 19A&A proofs: manuscript no. AAmain_clean\nAppendix A: Bunch Radiation\nIn order to obtain the spectrum from bunches, we briefly sum-\nmarize the physics of radiation from a single charge. As shown\nin Figure 1, the angle between the electron velocity direction and\nx-axis at t=0 is defined as χi j, and that between the LOS and the\ntrajectory plane is φk. The observation point is far enough away\nfrom the region of space where the acceleration occurs. Based\non the classical electrodynamics, the radiation electric vector is\ngiven by (Jackson 1998)\nE(t)=e\nc(n×[(n−βs)×˙βs]\n(1−βs·n)3R)\n, (A.1)\nwhere the expression in the brackets is evaluated at the retarded\ntime tret=t−R/c. By inserting the two accelerations ˙β∥and\n˙β⊥, the vector component of Equation (A.1) has (the subscript\n‘i,j,k’ has been replaced by ‘ s’)\nn×[(n−βs)×˙βs]\n=−˙βs∥sin(χs+vst/ρ)ϵ1+˙βs⊥[cosφs−cos(χs+vst/ρ)]ϵ1\n+˙βs∥sinφscos(χs+vst/ρ)ϵ2−˙βs⊥sinφssin(χs+vst/ρ)ϵ2.\n(A.2)\nConsider the small value of φs,χsandvst/ρ, if˙βs∥≃˙βs⊥, the\nradiation power caused by ˙βs⊥is much smaller than that for ˙βs∥.\nThe energy per unit frequency per unit solid angle of a single\nparticle is given by\nd2W\ndωdΩ=c\n4π2\f\f\fREs(t) exp( iωt)dt\f\f\f2\n=e2\n4π2c\f\f\f\f\f\fZ∞\n−∞n×[(n−βs)×˙βs]\n(1−βs·n)2\n×exp[iω(t′−n·rs(t′)/c)]dt′\f\f\f2.(A.3)\nUsing the identity:\nn×[(n−βs)×˙βs]\n(1−βs·n)2=d\ndt′\"n×(n×βs)\n1−βs·n#\n, (A.4)\nEquation (A.3) becomes\nd2W\ndωdΩ=e2\n4π2c\f\f\f\f\f\fZ∞\n−∞exp[iω(t′−n·rs(t′)/c)]d\"n×(n×βs)\n1−βs·n#\f\f\f\f\f\f2\n.\n(A.5)\nIt can be proven that by adding or subtracting the integrals over\nthe times while maintaining a constant velocity from Equation\n(A.5), one can use partial integration to obtain\nd2W\ndωdΩ=e2ω2\n4π2c\f\f\f\f\fZ∞\n−∞n×(n×βs) exp[ iω(t′−n·rs(t′)/c)]dt′\f\f\f\f\f2\n,\n(A.6)\nas long as a convergence factor (e.g., exp( −ϵ|t|), where taking\nthe limitϵ→0) is inserted to the integrand function of Equation\n(A.5) to eliminate instability at t=±∞.\nThe energy radiated per unit solid angle per unit frequency\ninterval in terms of the two polarization components can be writ-\nten as\nd2W\ndωdΩ=e2ω2\n4π2c\f\f\f−ϵ∥A∥+ϵ⊥A⊥\f\f\f2. (A.7)As shown in Figure 1, we consider a charge of identifier of i jk.\nThe vector part of Equation (A.6) can be written as\nn×\u0010\nn×βi jk,∥\u0011\n=β∥\"\n−ϵ∥sin vt\nρ+χi j!\n+ϵ⊥cos vt\nρ+χi j!\nsinφk#\n,(A.8)\nand the argument of the exponential is\nω \nt−n·ri jk(t)\nc!\n=ω\"\nt−2ρ\ncsin vt\n2ρ!\ncos vt\n2ρ+χi j!\ncosφk#\n≃ω\n2\" 1\nγ2+φ2\nk+χ2\ni j!\nt+c2t3\n3ρ2+ct2\nρχi j#\n,\n(A.9)\nwhere tis adopted to replace t′in the following calculations.\nTherefore, the two amplitudes for Equation (A.7) are given by\nAi jk,∥≃Z∞\n−∞ ct\nρ+χi j!\n×exp \niω\n2\" 1\nγ2+φ2\nk+χ2\ni j!\nt+c2t3\n3ρ2+ct2\nρχi j#!\ndt,\nAi jk,⊥≃φkZ∞\n−∞exp \niω\n2\" 1\nγ2+φ2\nk+χ2\ni j!\nt+c2t3\n3ρ2+ct2\nρχi j#!\ndt.\n(A.10)\nMaking the changes of variables\nu=ct\nρ 1\nγ2+φ2\nk+χ2\ni j!−1/2\n,\nξ=ωρ\n3c 1\nγ2+φ2\nk+χ2\ni j!3/2\n,(A.11)\nEquation (A.10) becomes\nAi jk,∥≃ρ\nc 1\nγ2+φ2\nk+χ2\ni j!\n×Z∞\n−∞u+χi jq\n1/γ2+χ2\ni j+φ2\nk\n×expi3\n2ξu+1\n3u3+χi jq\n1/γ2+φ2\nk+χ2\ni ju2du,\nAi jk,⊥≃ρ\ncφk 1\nγ2+φ2\nk+χ2\ni j!1/2\n×Z∞\n−∞expi3\n2ξu+1\n3u3+χi jq\n1/γ2+φ2\nk+χ2\ni ju2du.\n(A.12)\nNote that\nlim\nu→+0u+1\n3u3+χi jq\n1/γ2+φ2\nk+χ2\ni ju2=u,\nlim\nu→+∞u+1\n3u3+χi jq\n1/γ2+φ2\nk+χ2\ni ju2=1\n3u3.(A.13)\nArticle number, page 14 of 19Wang, W.-Y ., et al.: Narrow spectra of repeating fast radio bursts: A magnetospheric origin\nThe integrals in Equation (A.12) are identifiable as Airy inte-\ngrals. Consequently, the two amplitudes are\nAi jk,∥≃i2√\n3ρ\nc 1\nγ2+φ2\nk+χ2\ni j!\nK2\n3(ξ)\n+2√\n3ρ\ncχi j 1\nγ2+φ2\nk+χ2\ni j!1/2\nK1\n3(ξ),\nAi jk,⊥≃2√\n3ρ\ncφk 1\nγ2+φ2\nk+χ2\ni j!1/2\nK1\n3(ξ),(A.14)\nwhere Kν(ξ) is the modified Bessel function, in which Kν(ξ)→\n(Γ(ν)/2)(ξ/2)−νforξ≪1 andν,0, and Kν(ξ)→p\nπ/(2ξ) exp(−ξ) forξ≫1, leading to a wide-band spectrum.\nThe curvature radius can be regarded as a constant since the\nbunch’s size is much smaller than its altitude. We assume that\neach bunch has roughly the same χuandχdeven with di fferent k\nso that an average height can be derived to replace the complex\nboundary condition of the bulk (Wang et al. 2022c). Consider\nthat the bunch longitude size is l<4ρ/(3γ3). The emission with\nω<ω cis added coherently. For the emission band with ω>ω l,\nthe emission is added incoherently and the fluxes drop with ω\nvery quickly. The spectrum becomes much deeper than an expo-\nnential function. The total amplitudes of a bunch are then given\nby\nA∥≃2√\n3ρ\ncNj\n∆χNk\n2φtNi\n×Zχu,i\nχd,idχ′Zφu\nφd\"\ni 1\nγ2+φ′2+χ′2!\nK2\n3(ξ)\n+χ′ 1\nγ2+φ′2+χ′2!1/2\nK1\n3(ξ)cosφ′dφ′,\nA⊥≃2√\n3ρ\ncNj\n∆χNk\n2φtNi\n×Zχu,i\nχd,idχ′Zφu\nφd 1\nγ2+φ′2+χ′2!1/2\nK1\n3(ξ)φ′cosφ′dφ′,\n(A.15)\nwhereχuis the upper boundary χ′,χdis the lower boundary χ′,\nand the boundaries of φ′readφu=φt+φandφd=−φt+φ.\nWith the particular interest of β1,∥(t)=β1exp(−iωat), the\ntotal amplitudes of a bunch for the parallel perturbation are given\nby\n˜A∥≃2√\n3ρ\ncNj\n∆χNk\n2φtNiβ1\n×Zχu,i\nχd,idχ′Zφu\nφd\"\ni 1\nγ2+φ′2+χ′2−2ωa\nω!\nK2\n3(ξa)\n+χ′ 1\nγ2+φ′2+χ′2−2ωa\nω!1/2\nK1\n3(ξa)cosφ′dφ′,\n˜A⊥≃2√\n3ρ\ncNj\n∆χNk\n2φtNiβ1\n×Zχu,i\nχd,idχ′Zφu\nφd 1\nγ2+φ′2+χ′2−2ωa\nω!1/2\n×K1\n3(ξa)φ′cosφ′dφ′.(A.16)\nThe monochromatic perturbation frequency is Doppler boosted\nto 2γ2ωaso that there is a spike in the spectrum at ω=2γ2ωa,and the width of the spike is determined by the boundaries of χ′\nandφ′.\nWe consider the case with a similar formula but for the per-\npendicular perturbation, i.e., β1,⊥(t)=β1exp(−iωat). Follow-\ning the discussion in Section 2.2.4, we insert Equation (12) into\nEquation (15) and the amplitudes are calculated as\n˜Ai jk,∥≃1√\n3ρ\ncβ1(φ2\nk−χ2\ni j) 1\nγ2+φ′2+χ′2−2ωa\nω!1/2\nK1\n3(ξa)\n−i2√\n3ρ\ncβ1χi j 1\nγ2+φ′2+χ′2−2ωa\nω!\nK2\n3(ξa)\n+ρ\ncβ1 1\nγ2+φ′2+χ′2−2ωa\nω!3/2\n×\"1√\n3K1\n3(ξa)−πδ(3ξa/2)#\n,\n˜Ai jk,⊥≃−i2√\n3ρ\ncφkβ1 1\nγ2+φ2\nk+χ2\ni j!\nK2\n3(ξa)\n−2√\n3ρ\ncφkβ1χi j 1\nγ2+φ2\nk+χ2\ni j!1/2\nK1\n3(ξa),\n(A.17)\nwhereδ(x) is the Dirac function. The total amplitudes of a bunch\nfor the perpendicular perturbation case can be given by the inte-\ngrals of Equation (A.17).\nAppendix B: Stokes Parameters\nFour Stokes parameters are significant tools to describe the po-\nlarization properties of a quasi-monochromatic electromagnetic\nwave. In this section, we consider the monochrome Stokes pa-\nrameters which can be derived by the Fourier-transformed elec-\ntric vector. The Fourier-transformed polarized components of\nelectric waves can be written as\nE∥(ω)=eωA∥\n2πcR,E⊥(ω)=eωA⊥\n2πcR. (B.1)\nBy inserting Equation (B.1), four Stokes parameters of observed\nwaves can be calculated as\nI=µ\u0010\nA∥A∗\n∥+A⊥A∗\n⊥\u0011\n,\nQ=µ\u0010\nA∥A∗\n∥−A⊥A∗\n⊥\u0011\n,\nU=µ\u0010\nA∥A∗\n⊥+A⊥A∗\n∥\u0011\n,\nV=−iµ\u0010\nA∥A∗\n⊥−A⊥A∗\n∥\u0011\n,(B.2)\nwhereµ=ω2e2/(4π2R2cT) is the proportionality factor, Ide-\nfines the total intensity, QandUdefine linear polarization and\nits position angle, and Vdescribes circular polarization. The\ntimescaleTis chosen as the average timescale the flux density I\nrepeated. The corresponding linear polarization and PA are then\ngiven by\nL=p\nU2+Q2,\nΨ =1\n2tan−1 Us\nQs!\n,(B.3)\nwhere \nUs\nQs!\n= \ncos 2ψ sin 2ψ\n−sin 2ψcos 2ψ! \nU\nQ!\n, (B.4)\nin whichψis given by the rotation vector model (RVM; Rad-\nhakrishnan & Cooke 1969).\nArticle number, page 15 of 19A&A proofs: manuscript no. AAmain_clean\nAppendix C: Quake-induced Oscillation\nWe calculate the frequencies of the torsional modes for the rela-\ntivistic star caused by a quake and discuss the e ffect of the mag-\nnetic field on the frequencies of the various torsional modes. Os-\ncillations after the quake have two types: spherical and toroidal,\nin which only the toroidal oscillations can e ffectively modify\nthe spin of the star and the Goldreich-Julian density. The ax-\nial perturbation equations for the elastic solid star in the Cowl-\ning approximation is written as (Samuelsson & Andersson 2007;\nSotani et al. 2012)\nY\"ρm+P\nµΩ2exp(−2Φ)−(ℓ+2)(ℓ−1)\nr2#\nexp(2 Λ)+Y′′\n+ 4\nr+ Φ′−Λ′+µ′\nµ!\nY′=0,(C.1)\nwhereµis the shear modulus, ΦandΛare functions of r,ρmis\nthe mass-energy density, Y(r) describes the radial part of the an-\ngular oscillation amplitude, and the integer ℓis the angular sepa-\nration constant which enters when Y(r) is expanded in spherical\nharmonics Yℓm(θ,ϕ).\nTo discuss the possible e ffects of the magnetic field, we ex-\ntend our studies to the torsional oscillation of a magnetized rel-\nativistic star (e.g., Li et al. 2024). Sotani et al. (2007) derived\nthe perturbation equations of the magnetized relativistic star us-\ning the relativistic Cowling approximation. The final perturba-\ntion equation is\nAℓ(r)Y′′+Bℓ(r)Y′+Cℓ(r)Y=0, (C.2)\nwhere the coe fficients are given in terms of the functions de-\nscribing the equilibrium metric, fluid, and the magnetic field of\nthe star,\nAℓ(r)=µ+(1+2λ1)a12\nπr4, (C.3)\nBℓ(r)= 4\nr+ Φ′−Λ′!\nµ+µ′\n+(1+2λ1)a1\nπr4\u0002\u0000Φ′−Λ′\u0001a1+2a1′\u0003, (C.4)\nCℓ(r)=(2+5λ1)a1\n2πr4\b\u0000Φ′−Λ′\u0001a1′+a1′′\t\n−(λ−2) µexp(2 Λ)\nr2−λ1a1′2\n2πr4!\n+\" \nρm+P+(1+2λ1)a12\nπr4!\nexp(2 Λ)−λ1a1′2\n2πr2#\n(C.5)\n×Ω2exp(−2Φ),\nwhereλ=ℓ(ℓ+1), andλ1=−ℓ(ℓ+1)/[(2ℓ−1)(2ℓ+3)]. To solve\nEqs. (C.1) and (C.2) and determine the oscillation frequencies,\nthe boundary conditions require that the traction vanishes at the\ntop and the bottom of the crust.\nThe toroidal oscillations can e ffectively modify the spin and\nthe electric field in the gap. The velocity components of toroidal\noscillations can be written as\nδvˆi=\"\n0,1\nsinθ∂ϕYℓm(θ,ϕ),−∂θYℓm(θ,ϕ)#\n˜η(r)e−iΩt, (C.6)\nwhere ˜η(r) is a parameter denoting the amplitude of the oscil-\nlations. Owing to unipolar induction, we have (Lin et al. 2015)\nEθ=−1\nNc\"2πr\nPsinθ \n1−R3κ\nr3!\n−∂θYℓm˜η(r)#\nBsf(r)\nf(R)R3cosθ\nr3,\n0 2 4 6 8 10 \n 3/2/p00.5 11.5 22.5  n=ck/ \nX-mode\nSuperluminous  O-mode Subluminous  O-mode Fig. D.1. The refractive indices of wave modes as a function of ωγ3/ωp\nwithθB=0.2 for two cases: γ=2 (dashed lines) and γ=100 (solid\nlines). The black dotted-dashed line shows the X-mode ( n=1). The\nO-modes have two branches: subluminous O-mode (cyan dashed line\nforγ=2 and blue solid line for γ=10) and superluminous O-mode\n(magenta dashed line for γ=2 and red solid line for γ=10).\n(C.7)\nwhere N=p\n1−2GM/(rc2),κ=2GIm/(R3c2),Imis the mo-\nment of inertia, and f(r) is\nf(r)=−3 rc2\n2GM!3\"\nln \n1−2GM\nrc2!\n+2GM\nrc2\u0012\n1+GM\nrc2\u0013#\n.(C.8)\nAppendix D: Dispersion Relationship of\nUltra-relativistic Plasma\nIn this section, we briefly summarize the physics of waves in\nultra-relativistic plasma. Basically, the equation of motion of a\ngiven charge species reads\nd\ndt(γsmeus)=qsE+qs\ncus×B, (D.1)\nwhereus=us,0+δus,E=δE,B=B0+δB, in whichδus,δE,δB\nare associated with the disturbance in the plasma and have a form\nof exp [−i(ωt−k·r)]. Combined with\ndδus/dt=−iωδus+ic\u0000k·us,0\u0001δus, (D.2)\nand\nδB=(ck/ω)×δE, (D.3)\nδuscan be then described in terms of δE. The continuity equation\nreads\n∂ns\n∂t+∇·(nsus,0)=0, (D.4)\nso that the density perturbation is given by\nδns=ns(k·δus)\nω−k·us,0. (D.5)\nThe conductivity tensor σis defined by\nδJ=σ·δE= Σ s(nsqsδus+δnsqsus,0), (D.6)\nArticle number, page 16 of 19Wang, W.-Y ., et al.: Narrow spectra of repeating fast radio bursts: A magnetospheric origin\nand the dielectric tensor is given by\nϵ=I+i4π\nωσ, (D.7)\nwhere Iis the unit tensor. Since the electric displacement vector\nis defined as D=ϵ·E, we can obtain the equation for plane\nwaves in a coordinate system ( x′, y′,z′) where B0along z′\nϵ=S iD A\n−iD S−iC\nA iC P, (D.8)\nwith\nS=1+X\nsfs,11,\nD=X\nsfs,12,\nA=X\nsζsfs,11,\nC=X\nsζsfs,12,\nP=1+X\ns\u0010\nfs,η+ζ2\nsfs,11\u0011\n,(D.9)\nwhere\nfs,11=−ω2\npγ−1\ns\nω2−ω2\nBγ−2s(1−nβscosθB)−2,\nfs,12=−sign(qs)ωBω2\npω−3γ−2\ns(1−nβscosθB)−1\n1−ω2\nBω−2γ−2s(1−nβscosθB)−2,\nfs,η=−ω2\np\nω2γ3s(1−nβscosθB)2,\nζs=nβssinθB\n1−nβscosθB,(D.10)\nhere the subscript ‘0’ has been suppressed. In the coordinate sys-\ntemx0y0z0with kalong the z0-axis and Bin the x0−z0plane,\nthe components of dielectric tensor are given by\nϵxx=Scos2θB−2AsinθBcosθB+Psin2θB,\nϵyy=S,\nϵzz=Ssin2θB+2AsinθBcosθB+Pcos2θB,\nϵxy=−ϵyx=i(DcosθB−CsinθB),\nϵxz=ϵzx=Acos 2θB+(S−P)sinθBcosθB,\nϵyz=−ϵzy=−i(DsinθB+CcosθB).(D.11)\nThe three electric components have\nEz=−ϵ−1\nzz\u0010\nϵzxEx+ϵzyEy\u0011\n. (D.12)\nReinserting this back into the definition of electric displacement\nvector yields\n \nηxx−n2ηxy\nηyxηyy−n2! \nEx\nEy!\n=0 (D.13)where\nηxx=1\nϵzz(ϵzzϵxx−ϵxzϵzx)=1\nϵzz\u0010\nS P−A2\u0011\nηyy=1\nϵzz\u0010\nϵzzϵyy−ϵyzϵzy\u0011\n=1\nϵzzh\u0010\nS2−D2−S P+C2\u0011\nsin2θBi\n+1\nϵzzh\n2(AS−CD)sinθBcosθB+S P−C2i\nηyx=−ηxy=1\nϵzz\u0010\nϵzzϵyx−ϵyzϵzx\u0011\n=−i\nϵzz[PDcosθB−S CsinθB+A(DsinθB−CcosθB)]\n(D.14)\nFor a normal pulsar or magnetar, one can obtain ωB≫ω\nandωB≫ωpinside the magnetosphere. Under these conditions,\nsome parameters can be reduced as\nf11∼ω2\np,sγs/ω2\nB∼0,f12∼0,\nfη≃−ω2\np,sω−2γ−3\ns(1−nβscosθB)−2,\nS≃1,P≃1+fη,D≃0,A≃0,\nC≃0, ϵ zz≃1+fηcos2θB,\nηxx≃1+fη\n1+fηcos2θB, ηyy≃1, ηyx≃−ηxy≃0.(D.15)\nReinserting this back into Equation (D.13) and the solution of\nrefractive index is given by (Arons & Barnard 1986)\nn2=1,\nn2=ηxx,\n\u0010\nω2−c2k2\n∥\u00111−ω2\np,s\nγ3sω2(1−βsck∥/ω)−c2k2\n⊥=0,(D.16)\nwhere k∥=kcosθBandk⊥=ksinθB. For the mode with n2=1,\nthe polarization is given by\n\f\f\f\f\f\fEx\nEy\f\f\f\f\f\f=0,\f\f\f\f\f\fEz\nEy\f\f\f\f\f\f=0. (D.17)\nAs a result, the mode is a transverse wave with the electric field\nvector in the k×Bdirection, that is, the X-mode. For the other\ntwo refractive index solutions, one can obtain\n\f\f\f\f\fEy\nEx\f\f\f\f\f=0,\f\f\f\f\fEz\nEx\f\f\f\f\f=\f\f\f\f\f\fn2−1\ntanθB\f\f\f\f\f\f. (D.18)\nThe mode is called O-mode which the electric field oscillates at\nk−Bplane.\nWe plot the dispersion relationship of the X-mode and the O-\nmode for two cases ( γ=2 andγ=10) in Figure D.1. For both\ncases,θB=0.2 is adopted. The O-modes have two branches:\nthe superluminous branch ( n<1) and the subluminous branch\n(n>1). The superluminous O-mode has a frequency cut-o ffat\nωγ3=ωp. The latter one has a refractive index cut-o ffatn=\n1/cosθB. This mode looks to have two branches as shown in\nFigure D.1, but the branches can get crossed at high frequencies.\nArticle number, page 17 of 19A&A proofs: manuscript no. AAmain_clean\nAppendix E: Notation List\nSubscript i,j,k: The identifier of each charged particle\nSubscript s: Species of a single particle c: Speed of light\nd2W/dωdΩ: Energy radiated per unit solid angle per unit\nfrequency interval\ne: Charge (absolute value) of electron\nf(u): Distribution function of plasma\nhgap: Height of the gap\nk: Wave vector\nkB: Boltzmann constant\nl: Longitude bunch size\nℓ: Angular separation constant\nme: Mass of the electron\nn: Unit vector of the line of sight\nn: Refractive index\nne: Number density of charges\nnGJ: Goldreich-Julian density\nnp: Number density of magnetospheric plasma\np: Power law index\nr: Distance to the neutron star center\ns: Distance that charges traveled along the field lines\nu: Space-like part of the four-velocity\num: Average uover the Gaussian distribution\nv: Velocity of electron\nw: Burst width\nA∥: Parallel component of amplitude\nA⊥: Perpendicular component of amplitude\nB: Magnetic field strength\nE: Wave electric field\nE1,∥: Perturbation electric field parallel to the B-field\nE1,⊥: Perturbation electric field perpendicular to the B-field\nE∥: Electric field parallel to the B-field\nI,Q,U,V: Stokes parameters\nJ: Current density\nj: Emissivity\nKν: Modified Bessel function\nLb: Luminosity of a bunch\nM: Stellar mass\nM: Multiplicity factor\nNb: Number of bunches that contribute to the observed power at\nan epoch\nNe: Number of electrons in one bunch\nNi: Maximum number of the subscript of i\nNj: Maximum number of the subscript of j\nNk: Maximum number of the subscript of k\nP: Spin period\nR: Stellar radius\nRb: Transverse size of bunch\nR: Distance from the emitting source to the observer\nT: Temperature\nT: The average timescale of the flux density Irepeated\nYℓm(θ,ϕ): Spherical harmonics\nα: Angle between spin axis and magnetic axis\nβ: Dimensionless velocity of one charge\n˙β∥: Dimensionless perturbation acceleration parallel to the\nB-field\n˙β⊥: Dimensionless perturbation acceleration perpendicular to\ntheB-field\nδV: Vacuum polarization coe fficient\nγ: Lorentz factor\nµ: Shear modulus ν: Frequency, index of the modified Bessel\nfunction\n˙ν: Drift rateχ: Angle between ˆβfor di fferent trajectories\nχd: Lower boundary of angle between ˆβfor di fferent trajectories\nχu: Upper boundary of angle between ˆβfor di fferent trajectories\nϵ: Dielectric tensor\nλ: Wavelength\nψ: Polarization angle given by the RVM model\nρ: Curvature radius\nσ: Conductivity tensor\nσc: Curvature cross section\nτ: Optical depth\nθ: Poloidal angle with respect to the magnetic axis\nθc: Spread angle of the curvature radiation\nθin: Angle between the incident photon momentum and the\nelectron momentum\nθs: Angle of the footpoint for field line at the surface\nθB: Angle between angle between ˆkand ˆB\nω: Angular frequency\nωa: Frequency of the perturbation-induced acceleration\nωB: Cyclotron frequency\nωc: Critical frequency of curvature radiation\nωcut: Angular frequency cut-o ffof the superluminous O-mode\nωin: Angular frequency of incoming wave of ICS\nωM: Frequency of bulk of bunches distribution\nωp: Plasma frequency\nφ: Azimuth angle to the magnetic axis, the angle between LOS\nand the trajectory plane\nφd: Lower boundary of angle between LOS and the trajectory\nplane\nφt: Half opening angle of a bulk of bunches\nφu: Upper boundary of angle between LOS and the trajectory\nplane\nΓ(ν): Gamma function\nΨ: Polarization angle\nΘe: Magnetic colatitude of the emission point\nΩ: Solid angle of radiation, angular frequency of the oscillations.\nReferences\nAdler, S. 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L., Zhang, B., et al. 2022, Research in Astronomy and As-\ntrophysics, 22, 124001\nZhu, W., Xu, H., Zhou, D., et al. 2023, Science Advances, 9, eadf6198\nArticle number, page 19 of 19"    },    {        "title": "1706.00607v1.Vanishing_viscosity_limit_for_global_attractors_for_the_damped_Navier__Stokes_system_with_stress_free_boundary_conditions.pdf",        "content": "arXiv:1706.00607v1  [math.AP]  2 Jun 2017VANISHING VISCOSITY LIMIT FOR GLOBAL\nATTRACTORS FOR THE DAMPED NAVIER–STOKES\nSYSTEM WITH STRESS FREE BOUNDARY\nCONDITIONS\nVLADIMIR CHEPYZHOV1,3, ALEXEI ILYIN1,2AND SERGEY ZELIK2,4\nTo Edriss Titi on the occasion of his 60-th birthday with warm est regards\nAbstract. We consider the damped and driven Navier–Stokes\nsystem with stress free boundary conditions and the damped Euler\nsystem in a bounded domain Ω ⊂R2. We show that the damped\nEuler system has a (strong) global attractor in H1(Ω). We also\nshow that in the vanishing viscosity limit the global attractors of\ntheNavier–Stokessystemconvergeinthenon-symmetricHausdo rff\ndistance in H1(Ω) to the the strong global attractor of the limiting\ndamped Euler system (whose solutions are not necessarilyunique).\n1.Introduction\nIn this paper, we study from the point of view of global attractors\nthe 2D damped and driven Navier–Stokes system\n/braceleftbigg\n∂tu+(u,∇)u+∇p+ru=ν∆u+g(x),\ndivu= 0, u(0) =u0,(1.1)\nand the corresponding limiting ( ν= 0) damped/driven Euler system\n/braceleftbigg\n∂tu+(u,∇)u+∇p+ru=g(x),\ndivu= 0, u(0) =u0.(1.2)\nBoth systems are considered in a bounded multiply connected smoot h\ndomain Ω ⊂R2with standard non-penetration boundary condition\nu·n|∂Ω= 0, (1.3)\n2000Mathematics Subject Classification. 35B40, 35B41, 35Q35.\nKey words and phrases. Damped Euler equations, global attractors, vanishing\nviscosity limit.\nThe research of V. Chepyzhov and A. Ilyin was carried out in the Ins titute for\nInformation Transmission Problems, Russian Academy of Sciences, at the expense\nof the Russian Science Foundation (project 14-50-00150). The w ork of S. Zelik was\nsupported in part by the RFBR grant 15-01-03587.\n12 V.V. CHEPYZHOV, A.A. ILYIN, AND S.V ZELIK\nwhile the system (1.1) is supplemented with the so-called stress free or\nslip boundary conditions\nu·n|∂Ω= 0,curlu|∂Ω= 0. (1.4)\nTheLaplaceoperatorwith(1.4)commutes theLerayprojection. T hese\nboundary conditions guarantee the absence of the boundary laye r and\nyield the conservation of enstrophy in the unforced and undamped case\nof (1.2). They are also convenient for studying the limit as ν→0+of\nthe individual solutions of the 2D Navier–Stokes system [4, 25].\nSystems (1.1) and (1.2) are relevant in geophysical hydrodynam-\nics and the damping term −rudescribes the Rayleigh or Ekman fric-\ntion and parameterizes the main dissipation occurring in the planetar y\nboundarylayer (see, for example, [27]). The viscous term −ν∆uinsys-\ntem (1.1) is responsible for the small scale dissipation. We also observ e\nthat in physically relevant cases we have ν≪r|Ω|.\nThe damped and driven 2D Euler and Navier-Stokes systems at-\ntracted considerable attention over the last years and were stud ied\nfrom different points of view. The regularity, uniqueness, and stab ility\nof the stationary solutions for (1.2) were studied in [5, 29, 33]. The\nvinishing viscosity limit for system (1.1) was studied for steady-stat e\nstatistical solutions in [14].\nIn the presence of the damping term the weak attractor for the s ys-\ntem (1.2) was constructed in [17] in the phase space H1. In the trajec-\ntory phase space the weak attractor was constructed in [6, 7].\nThe dynamical effects of the damping term −ruin the case of the\nNavier–Stokes system (1.1) were studied in [21] on the torus, on t he 2D\nsphere, and in bounded (simply connected) domain Ω with boundary\nconditions (1.4). Specifically, it was shown that the fractal dimensio n\nof the global attractor Aνsatisfies the estimate\ndimfAν≤min/parenleftbigg\nc1(Ω)/ba∇dblcurlg/ba∇dbl|Ω|1/2\nνr, c2(Ω)/ba∇dblcurlg/ba∇dbl2\nνr3/parenrightbigg\n,(1.5)\nwhere|Ω|is the area of the spatial domain. This estimate is sharp in\nthe limitν→0+and the lower bound is provided by the corresponding\nfamily of Kolmogorov flows. Furthermore, the constants c1andc2are\ngiven explicitly for the torus Ω = T2and for the sphere Ω = S2. The\ncase of an elongated torus T2\nαwith periods LandL/α, whereα→0+\nwas studied in [24], where it was shown that (1.5) still holds for T2\nαand\nis sharp as both α→0 andν→0.\nThe essential analytical tool used in the proof of (1.5), especially in\nfindingexplicit valuesof c1andc2, istheLieb–Thirring inequality. NewDAMPED NAVIER–STOKES SYSTEM 3\nbounds for the Lieb–Thirring constants for the anisotropic torus were\nrecently obtained in [20] with applications to the system (1.1) on T2\nα.\nOne might expect that in the case of the damped Navier–Stokes\nsystem (1.1) in R2in the space of finite energy solutions the attractor\nAνexists and its fractal dimension is bounded by the second number\non the right-hand side in (1.5). It was recently shown in [22] that it is\nindeed the case:\ndimfAν≤1\n16√\n3/ba∇dblcurlg/ba∇dbl2\nνr3. (1.6)\nMoreover, due to convenient scaling available for R2this estimate of\nthe dimension is included in [22] in the family of estimates depending\non the norm of gin the scale of homogeneous Sobolev spaces ˙Hs(R2),\n−1≤s≤1; the case s= 1 being precisely (1.6).\nEstimates for the degrees of freedom for the damped Navier–Sto kes\nsystem(1.1)expressedintermsofvariousfinitedimensionalproje ctions\nwere obtained in [23]. They are also of the order (1.5).\nWepoint out two important differences between thedamped Navier–\nStokes system (1.1)andthedampedEuler system (1.2)which make t he\nconstruction of the global attractor for (1.2) less straightforw ard. The\nfirst is the absence for (1.2) of the instantaneous smoothing prop erty\nof solutions and explains why the existence of only a weak attractor\nwas first established [17]. The second is that the uniqueness is only\nknown for the solutions with bounded vorticity [34] and is not known\nin the natural Sobolev space H1, which makes the trajectory attractors\nvery convenient for (1.2), see [6, 7, 8, 9, 32]. The trajectory att ractors\nfor (1.2) in the weak topology of H1were constructed in [7] (see also\n[6]) and in [12] for the non-autonomous case. In addition, the upper\nsemi-continuous dependence as ν→0+ of the trajectory attractors of\nthesystem (1.1)onthetoruswasestablishedin[7]intheweaktopo logy\nofH1(T2).\nThe existence of the strong H1trajectory attractors for the dissi-\npative Euler system (1.2) on the 2D torus was proved in [10] under\nthe assumption that curl g∈L∞which was used to prove the enstro-\nphy equality. The strong attraction and compactness for the tra jec-\ntory attractor were established using the energy method develop ed in\n[3, 16, 26, 28] for the equations in unbounded, non-smooth domain s or\nfor equations without uniqueness. This method is based on the corr e-\nsponding energy balance for the solutions and leads to the asympto tic\ncompactness of the solution semigroups or collections the traject ories.\nMost closely related to the present work is the paper [13] where the\nstrong global and trajectory W1,p-attractors were constructed for the4 V.V. CHEPYZHOV, A.A. ILYIN, AND S.V ZELIK\nsystem (1.2) in R2. The crucial equation of the enstrophy balance is\nproved there in the Sobolev spaces W1,p, 2≤p <∞without the\nassumption on gthat guarantees the uniqueness of solutions on the\nattractor. Instead the authors used the fact that in the 2D cas e the\nvorticity satisfies a scalar transport equation, and the required e nstro-\nphy equality directly follows from the results of [15].\nInunboundeddomainsthedampedNavier–StokesandEulersystem s\ncan be studied from the point of view of uniformly local spaces (wher e\nthe energy is infinite) and one of the main issues is the proof of the\ndissipative estimate, which is achieved by means of delicate weighted\nestimates. In the uniformly local spaces in the viscous case ν >0\nthe global attractors for (1.1) in the strong topology were const ructed\nin [37], see also [35, 36] for similar results in channel-like domains. In\nthe inviscid case the strong attractor for (1.2) in the uniformly loca l\nH1space was recently constructed in [11].\nInthepresentpaperwestudytheconvergenceoftheglobalatt ractors\nAνof the system (1.1), (1.4) in the vanishing viscosity limit ν→0+,\nand our main result is as follows. The system (1.2), (1.3) has a global\nattractor A0⋐H1(Ω). For every δ-neighbourhood OδofA0inH1(Ω)\nthere exits ν(δ)>0 such that\nAν⊂Oδ(A0) for all ν≤ν(δ), (1.7)\nwhere Aνforν >0 are the attractors of the damped Navier–Stokes\nsystem (1.1), (1.4).\nWe point out that despite the fact that the dimension of Aνcan be\nof order 1/νasν→0+(at least in the periodic case and the special\nfamily Kolmogorov-type forcing terms) the limiting attractor A0is,\nnonetheless, a compact set in H1(Ω).\nThis paper has the following structure. In Section 2 we define the\nfunction spaces, paying attention the case when the domain Ω is mul-\ntiply connected, and construct the global attractors Aνfor (1.1), (1.4).\nIn Section 3 we prove the existence of weak solutions of the damped\nEuler system (1.2). We adapt the theory of renormalized solutions\nfrom [15] to the vorticity equation in a bounded domain which gives\nus the crucial equation of the enstrophy balance for an arbitrary weak\nsolution of (1.2). In Section 4 we consider the generalized solution\nsemigroup for the system (1.2) and define weak and strong global a t-\ntractorsforthegeneralized semigroup. Wefirst construct awea k global\nattractor A0inH1for (1.2) and then we prove the asymptotic com-\npactness of the generalized semigroup which gives that the weak glo bal\nattractor A0is, in fact, the H1strong global attractor. In Section 5\nwe prove (1.7).DAMPED NAVIER–STOKES SYSTEM 5\n2.Equations and function spaces\nWe shall be dealing with the damped and driven Navier–Stokes sys-\ntem (1.1) with boundary conditions (1.4) and the corresponding limit-\ning (ν= 0) damped Euler system (1.2) with standard non-penetration\ncondition (1.3).\nBoth systems are studied in a bounded domain Ω ⊂R2. We consider\nthe general case when Ω can be multiply connected with boundary\n∂Ω = Γ = Γ 0∪Γ1∪···∪Γk.\nIn other words, Γ 0is the outer boundary, and the Γ i’s are the bound-\naries ofkislands inside Γ 0. We assume that ∂Ω is smooth ( C2will be\nenough) so that there exists a well-defined outward unit normal nand\nalso an extension operator E:\nE:H2(Ω)→H2(R2),/ba∇dblEu/ba∇dblH2(R2)≤const/ba∇dblu/ba∇dblH2(Ω).\nWe now introduce the required function spaces and their orthogon al\ndecompositions. We set\nH={u∈L2(Ω),divu= 0, u·n|∂Ω= 0}.\nThe following orthogonal decomposition holds [31, Appendix 1]:\nH=H0⊕Hc, (2.1)\nwhere\nH0={u∈L2(Ω),divu= 0, u·n|∂Ω= 0, u=∇⊥ϕ, ϕ∈H1\n0(Ω)},\nthat is, the vector functions in H0have a unique single valued stream\nfunctionϕvanishing at all components of the boundary ∂Ω. Hereϕis\na scalar function, and\n∇⊥ϕ:={−∂2ϕ,∂1ϕ}=−curlϕ, u⊥:={−u2, u1}.\nAccordingly, theorthogonalcomplementto H0inHisthek-dimensional\nspace of harmonic (and hence infinitely smooth) vector functions:\nHc={u∈L2(Ω),divu= 0,curlu= 0, u·n|∂Ω= 0},\nIn the similar way, for smoothness of order one we have\nH1:={u∈H1(Ω),divu= 0, u·n|∂Ω= 0}=H1⊕Hc,\nwhereHcis as before and\nH1={u=∇⊥ϕ, ϕ∈H2(Ω)∩H1\n0(Ω)},/ba∇dblu/ba∇dblH1=/ba∇dbl∆ϕ/ba∇dbl.\nFor smoothness of order two\nH2:={u∈H2(Ω),divu= 0, u·n|∂Ω= 0}=H2⊕Hc,6 V.V. CHEPYZHOV, A.A. ILYIN, AND S.V ZELIK\nwhere\nH2={u=∇⊥ϕ, ϕ∈H3(Ω)∩H1\n0(Ω)},/ba∇dblu/ba∇dblH2=/ba∇dbl∇∆ϕ/ba∇dbl.\nCorresponding to the second boundary condition in (1.4) is the fol-\nlowing closed subspace in H2:\nH0\n2={u=∇⊥ϕ, ϕ∈H3(Ω)∩H1\n0(Ω)∩{∆ϕ|∂Ω= 0}}.\nThe space of all divergence free vector functions of class H2(Ω) satis-\nfying the boundary conditions (1.4) is denoted by H2\n0:\nH2\n0=H0\n2⊕Hc. (2.2)\nThe orthonormal basis in H0is made up of vector functions\nuj=λ−1/2\nj∇⊥ϕj,\nwhereλjandϕjare the eigenvalues and eigenfunctions of the scalar\nDirichlet Laplacian [18]\n−∆ϕj=λjϕj, ϕj|∂Ω= 0,0<λ1<λ2≤λ3··· →+∞.\nIn fact,\n/ba∇dbluj/ba∇dbl2=λ−1\nj(∇⊥ϕj,∇⊥ϕj) =λ−1\nj/ba∇dbl∇ϕj/ba∇dbl2= 1.\nFurthermore, since on scalars\ncurl∇⊥=−curlcurl = ∆ ,curl =−∇⊥,(2.3)\ntheuj’s satisfy (1.4), and the system {uj}∞\nj=1is the complete orthonor-\nmal basis of eigen vector functions with eigenvalues {λj}∞\nj=1of the\nvector Laplacian\n∆ =∇div−curlcurl (2.4)\nwith boundary conditions (1.4):\n−∆uj= curlcurluj=λjuj.\nWe can express the fact that a vector function ubelongs toH0,H1,\norH0\n2in terms of its Fourier coefficients as follows. Let\nu=∞/summationdisplay\nj=1cjuj, c j= (u,uj) =λ−1/2\nj(u,∇⊥ϕj),(2.5)\nwhere (setting ω:= curlu)\n(u,∇⊥ϕj) = (u⊥,∇ϕj) =−(divu⊥,ϕj) = (curlu,ϕj) = (ω,ϕj).DAMPED NAVIER–STOKES SYSTEM 7\nThis gives that\nu∈H0⇔∞/summationdisplay\nj=1c2\nj=/ba∇dblu/ba∇dbl2=/ba∇dblω/ba∇dbl2\nH−1(Ω)<∞,\nu∈H1⇔∞/summationdisplay\nj=1λjc2\nj=/ba∇dblω/ba∇dbl2<∞,\nu∈H0\n2⇔∞/summationdisplay\nj=1λ2\njc2\nj=/ba∇dblcurlcurlu/ba∇dbl2=/ba∇dblω/ba∇dbl2\nH1\n0(Ω)<∞.\nThe basis in the k-dimensional space of harmonic vector functions\nHcis given in [31, Appendix 1, Lemma 1.2] in terms of the gradients of\nharmonic multi valued functions. In our 2D case it is more convenient\nto construct a basis in Hcin terms of single valued stream functions.\nLemma 2.1. The system {∇⊥ψj}k\nj=1is a basis in Hc. Hereψjis\nthe solution in Ωof the equation ∆ψj= 0, whereψj= 0at all the\ncomponents of the boundary Γexcept for Γj, whereψj= 1.\nProof.The vector functions ∇⊥ψj∈Hcand are linearly independent.\n/square\nNext, we consider the Leray projection PfromL2(Ω) onto H. In\naccordance with (2.1) we have P=P0⊕Pc. For the projection P0onto\nH0we have\nP0u=∇⊥(∆D\nΩ)−1curlu, (2.6)\nwhere ∆D\nΩis the (scalar) Dirichlet Laplacian, which is an isomorphism\nfromH1\n0(Ω) ontoH−1(Ω).\nLemma 2.2. OnH2\n0the projection Pcommutes with the Laplacian ∆\nwith boundary conditions (1.4).\nProof.SincePc∆ = ∆Pc= 0 onHc, it suffices to consider P0. Let\nu∈H0\n2, see (2.2), so that P0u=u. Then interpreting curl uas a scalar\nand using (2.3) we obtain\nP0∆u=−∇⊥(∆D\nΩ)−1curlcurlcurl u=∇⊥(∆D\nΩ)−1∆curlu=\n=∇⊥curlu=−curlcurlu= ∆u= ∆P0u.\n/square\nThis lemma makes the subsequent analysis very similar to the 2D\nperiodic case or the case of a manifold without boundary.\nWe also recall the familiar formulas\n(∇ϕ,v) =−(ϕ,divv), v ·n|∂Ω= 0,\n(curlϕ,v) = (ϕ,curlv),curlv|∂Ω= 0.(2.7)8 V.V. CHEPYZHOV, A.A. ILYIN, AND S.V ZELIK\nLemma 2.3. [18]Letu∈ H0\n2(see(2.2)). Then\n((u,∇)u,∆u) = 0. (2.8)\nProof.We use the invariant expression for the convective term\n(u,∇)u= curlu×u+1\n2∇u2.\nLetu=u0+uc, whereu0∈H0\n2,uc∈Hc. Then, taking into ac-\ncount (2.4), for the second term in the above expression we have\n(∇u2,curlcurlu0) = (curl ∇u2,curlu0) = 0,\nsince curl ∇= 0 algebraically, and the first equality follows from (2.7)\nwith boundary condition curl u0|∂Ω= 0\nFor the first term we have setting ω= curlu0and using (2.3)\n(curlu0×u,curlcurlu0) =−(ωu⊥,∇⊥ω) =\n−(ωu,∇ω) =−1\n2(u,∇ω2) =1\n2(divu,ω2) = 0,\nwhere we used u·n|∂Ω= 0 for the integration by parts. /square\nWe also recall the familiar orthogonality relation\nb(u,v,v) = 0, (2.9)\nwhere the trilinear form b\nb(u,v,w) =/integraldisplay\nΩ2/summationdisplay\ni,j=1ui∂ivjwjdx\nis continuous on H1.\nThe spaceHcof (infinitely smooth) harmonic vector functions is k-\ndimensional, andevery Sobolevnorm Hk(Ω)isequivalent tothe L2(Ω)-\nnorm. Therefore the H1(Ω)-norm on H1foru=u0+uc∈H1⊕Hc\ncan be given by\n/ba∇dblu/ba∇dbl2\n1:=/ba∇dblu/ba∇dbl2+/ba∇dblcurlu/ba∇dbl2=/ba∇dblu/ba∇dbl2+/ba∇dblcurlu0/ba∇dbl2.\nAccordingly, the H2(Ω)-norm on H2\n0is given by\n/ba∇dblu/ba∇dbl2\n2:=/ba∇dblu/ba∇dbl2+/ba∇dblcurlcurlu/ba∇dbl2=/ba∇dblu/ba∇dbl2+/ba∇dblcurlcurlu0/ba∇dbl2.\nTheorem 2.4. Let the initial data u0and the right-hand side gin the\ndamped Navier–Stokes system (1.1),(1.4)satisfy\nu0∈ H1, g∈ H1.\nThenthere existsa unique strongsolution u∈C([0,T];H1)∩L2(0,T;H2\n0)\nof(1.1),(1.4). Thus, a semigroup of solution operators\nu(t) =S(t)u(0),DAMPED NAVIER–STOKES SYSTEM 9\ncorresponding to (1.1),(1.4)is well defined.\nThe solution satisfies the equation of balance of energyand e nstrophy:\n1\n2d\ndt/ba∇dblu/ba∇dbl2\n1+ν/ba∇dbl∆u/ba∇dbl2+r/ba∇dblu/ba∇dbl2\n1= (g,u)1, (2.10)\nwhere\n(g,u)1:= (g,u)+(curlg,curlu).\nProof.TheproofisstandardandusestheGalerkinmethod. Weusethe\nspecial basis (2.5) in H2\n0⊂ H1and supplement it with a k-dimensional\nbasis inHc, for example, with the one from Lemma 2.1 starting the\nenumeration from the basis in Hc.\nThen for every approximate Galerkin solution\nu=u(n)=n/summationdisplay\nk=1ckuk∈ H2\n0\nwe have the orthogonality relations (2.8), (2.9). We take the scalar\nproduct of (1.1) with u, and also with ∆ u, integrate by parts us-\ning (2.7), drop the ν-terms and use Growwall’s inequality to obtain\nin the standard way the estimates\n/ba∇dblu(t)/ba∇dbl2≤ /ba∇dblu(0)/ba∇dble−rt+r−2/ba∇dblg/ba∇dbl2,\n/ba∇dblcurlu(t)/ba∇dbl2≤ /ba∇dblcurlu(0)/ba∇dble−rt+r−2/ba∇dblcurlg/ba∇dbl2,\nwhich gives\n/ba∇dblu(t)/ba∇dbl2\n1≤ /ba∇dblu(0)/ba∇dbl1e−rt+r−2/ba∇dblg/ba∇dbl2\n1 (2.11)\nforu=u(n), uniformlyfor nandν >0. Theremainingassertionsofthe\ntheoremareprovedverysimilarlytotheclassicalcaseofthe2DNav ier–\nStokes system with Dirichlet boundary conditions (even simpler, sinc e\nwe now have more regularity, see, for instance, [2],[31]). /square\nWe recall the following definition of the (strong) global attractor\n(see, for instance [2],[30]).\nDefinition 2.5. LetS(t),t≥0, be a semigroup acting in a Banach\nspaceB. Then the set A⊂ Bis a global attractor of S(t) if\n1)Ais compact in B:A⋐B.\n2)Ais strictly invariant: S(t)A=A.\n3)Ais globally attracting, that is,\nlim\nt→∞dist(S(t)B,A) = 0,for every bounded set B⊂ B.\nTheorem 2.6. The semigroup S(t)corresponding to (1.1),(1.4)has\na global attractor A⋐H1.10 V.V. CHEPYZHOV, A.A. ILYIN, AND S.V ZELIK\nProof.It follows from (2.11) that the ball\nB0={u∈ H1,/ba∇dblu/ba∇dbl2\n1≤2r−2/ba∇dblg/ba∇dbl2\n1} (2.12)\nis the absorbing ball for S(t). The semigroup S(t) is continuous in H1\nand has the smoothing property (which can be shown similarly to the\nclassical 2D Navier–Stokes system [2], [30]). Therefore the set\nB1=S(1)B0\nis a compact absorbing set, which gives the existence of the attrac tor\nA⋐H1. We finally point out that for u(t)∈Awe have for all t∈R\n/ba∇dblu(t)/ba∇dbl1≤r−1/ba∇dblg/ba∇dbl1 (2.13)\nuniformly with respect to ν >0. /square\n3.Weak solutions for the Euler system and\nenergy-enstrophy balance\nWe now turn to the damped and driven Euler system (1.2), (1.3).\nDefinition 3.1. Letu(0),g∈ H1. A vector function u=u(t,x) is\ncalled a weak solution of (1.2), (1.3) if u∈L∞(0,T;H1) and satisfies\nthe integral identity\n−/integraldisplayT\n0(u,vη′(t))dt+/integraldisplayT\n0b(u,u,vη(t))dt+\n+r/integraldisplayT\n0(u,vη(t))dt=/integraldisplayT\n0(g,vη(t))dt(3.1)\nfor allη∈C∞\n0(0,T) and allv∈ H1.\nTheorem 3.2. There exists at least one solution of the damped Euler\nsystem(1.2),(1.3). Moreover, every weak solution in the sense of Def-\ninition 3.1 is of class C([0,T];H)and satisfies the equation of balance\nof energy\n1\n2d\ndt/ba∇dblu(t)/ba∇dbl2+r/ba∇dblu(t)/ba∇dbl2= (g,u(t)). (3.2)\nProof.As before we use the special basis and see that approximate\nGalerkin solutions unsatisfy (2.11) and therefore we obtain that uni-\nformly with respect to n\nun∈L∞(0,T;H1).\nNext, we see fromequation (1.2) that ∂tunis bounded in L2(0,T;H−1).\nTherefore we can extract a subsequence (still denoted by un) such that\nun→u∗-weakly inL∞(0,T;H1) and strongly in L2(0,T;H).DAMPED NAVIER–STOKES SYSTEM 11\nThis is enough to pass to the limit in the non-linear term in (3.1) and\ntherefore to verify that usatisfies (3.1). Since ∂tu∈L2(0,T;H−1), it\nfollows that we can take the scalar product of (1.2) with the solution\nuto obtain (3.2), see [31]. /square\nWe now derive the scalar equation for ω= curlu. We set in (3.1)\nv= curlϕ, ϕ ∈C∞\n0(Ω)\nand integrate by parts the linear terms in (3.1) by using the second\nformula in (2.7). For the non-linear term we have\nb(u,u,v) =/integraldisplay\nΩ(u,∇)u·curlϕdx=/integraldisplay\nΩ(curlu×u)·curlϕdx=\n/integraldisplay\nΩ(ωu⊥)·curlϕdx=/integraldisplay\nΩcurl(ωu⊥)ϕdx=/integraldisplay\nΩu∇ωϕdx,(3.3)\nsince algebraically curl( ωu⊥) =ωdivu+u∇ω.\nThus, we have shown that ω= curlusatisfies in Ω the following\nequation (in the sense of distributions)\n∂tω+u∇ω+rω=G,\nω(0) =ω0.(3.4)\nwhereG= curlg,ω0= curlu(0).\nWe observe that we can integrate by parts the last term in (3.3)\nanother time using the boundary condition for uonly:u·n|∂Ω= 0.\nNamely, for every ϕ∈C∞(¯Ω) it holds\n/integraldisplay\nΩu∇ωϕdx=−/integraldisplay\nΩωdiv(uϕ)dx,\nwhereϕdoes not necessarily vanish at ∂Ω, we useu·n|∂Ω= 0 instead.\nThe above argument shows that if uis a weak solution of the Euler\nsystem (1.2), then ω= curlusatisfies the following integral identity:\n−/integraldisplayT\n0/integraldisplay\nΩωϕη′(t)dxdt−/integraldisplayT\n0/integraldisplay\nΩωdiv(uϕ)η(t)dxdt+\n+r/integraldisplayT\n0/integraldisplay\nΩωϕη(t)dxdt=/integraldisplayT\n0/integraldisplay\nΩGϕη(t)dxdt,\n(3.5)\nholding for all ϕ∈C∞(¯Ω).\nWe now extend ωby zero outside Ω setting for all t\n/tildewideω=/braceleftbigg\nω,in Ω;\n0,in Ωc=R2\\Ω.12 V.V. CHEPYZHOV, A.A. ILYIN, AND S.V ZELIK\nIn the similar way by define /tildewideG. The vector function uis extended to\na/tildewideu∈H1(R2) in a certain way that will be specified later. Since ϕ\nin (3.5) is an arbitrary smooth function in C∞(¯Ω), it follows that the\nfollowing integral identity holds in the whole R2\n−/integraldisplayT\n0/integraldisplay\nR2/tildewideωϕη′(t)dxdt−/integraldisplayT\n0/integraldisplay\nR2/tildewideωdiv(/tildewideuϕ)η(t)dxdt+\n+r/integraldisplayT\n0/integraldisplay\nR2/tildewideωϕη(t)dxdt=/integraldisplayT\n0/integraldisplay\nR2/tildewideGϕη(t)dxdt,\n(3.6)\nholding for all ϕ∈C∞\n0(R2),η∈C∞\n0(0,T).\nIn other words, we have shown that /tildewideωis a weak solution in the whole\nR2of the equation\n∂t/tildewideω+/tildewideu∇/tildewideω+r/tildewideω=/tildewideG,\n/tildewideω(0) =/tildewideω0.(3.7)\nWe shall now specify the construction of /tildewideu. Recall that\nu=u0⊕uc, u0∈H1, uc∈Hc,\nwhereu∈L∞(0,T;H1), and where u0has a single valued stream\nfunctionψ0:u0=∇⊥ψ0,ψ0∈H2(Ω) (we do not use the additional\ninformation that ψ0= 0 at Γ). In view of Lemma 2.1, so does uc:\nuc=∇⊥ψc, whereψc∈H2(Ω) (at least). We set ψ=ψ0+ψcand\napply the extension operator E:/tildewideψ=Eψ,\n/tildewideψ∈H2(R2),/ba∇dbl/tildewideψ/ba∇dblH2(R2)≤c(Ω)/ba∇dblψ/ba∇dblH2(Ω).\nThen/tildewideu:=∇⊥/tildewideψis the required extension of the vector function uwith\n/ba��dbl/tildewideu/ba∇dblH1(R2)≤c(Ω)/ba∇dblu/ba∇dblH1,div/tildewideu= 0 in the whole R2.\nWe are now in a position to apply the theory developed in [15]. In\nparticular, it follows from [15, Theorem II.3] that the weak solution /tildewideω\nof (3.7) in the sense (3.6) is a renormalized solution, that is, satisfies\n∂tβ(/tildewideω)+/tildewideu∇β(/tildewideω)+r/tildewideωβ′(/tildewideω) =β′(/tildewideω)/tildewideG\nfor allβ∈C1\nb(R) withβ(0) = 0. This gives that\nd\ndt/integraldisplay\nR2β(/tildewideω)dx+r/integraldisplay\nR2/tildewideωβ′(/tildewideω)dx=/integraldisplay\nR2/tildewideGβ′(/tildewideω)dx.\nSinceβ(0) = 0 and /tildewideω= 0 outside Ω, the last equation goes over to\nd\ndt/integraldisplay\nΩβ(ω)dx+r/integraldisplay\nΩωβ′(ω)dx=/integraldisplay\nΩGβ′(ω)dx.DAMPED NAVIER–STOKES SYSTEM 13\nChoosing now for βappropriate approximations of the function s→s2\nwe finally obtain\n1\n2d\ndt/ba∇dblω(t)/ba∇dbl2+r/ba∇dblω(t)/ba∇dbl2= (ω(t),G). (3.8)\nThus, we have proved the following result.\nTheorem 3.3. Every weak solution of the damped and driven Euler\nequation is of class C([0,T];H1)and satisfies the equation of balance\nof energy and enstrophy\n1\n2d\ndt/ba∇dblu/ba∇dbl2\n1+r/ba∇dblu/ba∇dbl2\n1= (u,g)1. (3.9)\nProof.The equation of balance (3.9) follows from (3.2) and (3.8). The\ncontinuity in H1follows from the continuity in H(and, hence, weak\ncontinuity in H1) and the continuity of the norm t→ /ba∇dblω(t)/ba∇dbl2, which\nfollows from (3.8), see [31]. /square\n4.Global attractor for the damped Euler system\nFor every solution of the damped Euler system we obtain from (3.9)\nthat\nd\ndt/ba∇dblu/ba∇dbl2\n1+2r/ba∇dblu/ba∇dbl2\n1= 2(g,u)≤2/ba∇dblg/ba∇dbl1/ba∇dblu/ba∇dbl1≤r/ba∇dblu/ba∇dbl2\n1+r−1/ba∇dblg/ba∇dbl2\n1,\nso that by the Grownwall inequality\n/ba∇dblu/ba∇dbl2\n1≤ /ba∇dblu(0)/ba∇dbl2\n1+r−2/ba∇dblg/ba∇dbl2\n1(1−e−rt)\nthe ball (2.12) is also the absorbing ball for the generalized semigrou p\nof solution operators\nS(t)u0={u(t)}\nfor the damped Euler system, where {u(t)}is the section at time tof\nall weak solutions with u(0) =u0.\nOur goal is to show that the generalized semigroup S(t) has a weak\n(H1,H1\nw) attractor in the sense of the following definition (see [1], [2]).\nDefinition 4.1. A set A⊂ H1is called an ( H1,H1\nw) attractor of the\ngeneralized semigroup S(t) if\n1)Ais compact in the weak topology H1\nw.\n2)Ais strictly invariant: S(t)A=A.\n3)Aattracts in the weak topology H1\nwbounded sets in H1.\nWe first show that S(t) a semigroup in the generalized sense.14 V.V. CHEPYZHOV, A.A. ILYIN, AND S.V ZELIK\nLemma 4.2. The family S(t)has the semigroup property\nS(t+τ)u0=S(t)S(τ)u0(4.1)\nin the sense of the equality of sets.\nProof.Theinclusion S(t+τ)u0⊂S(t)S(τ)u0holdssince every solution\nin the sense of Definition 3.1 on the interval [0 ,T] is also a solution on\nevery smaller interval [ τ,T]. Let us prove the converse inclusion:\nS(t)S(τ)u0⊂S(t+τ)u0(4.2)\nAny solution u(t) satisfies on [0 ,T] the integral identity\n−/integraldisplayT\n0(u,vη′(t))dt+/integraldisplayT\n0b(u,u,vη(t))dt+r/integraldisplayT\n0(u,vη(t))dt−\n−/integraldisplayT\n0(g,vη(t))dt= (u(0),vη(0))−(u(T),vη(T))(4.3)\nfor everyv∈ H1andη∈C∞[0,T]. If this identity holds on the\nintervals [0,τ] and [τ,t+τ, then adding them we see that it holds on\n[0,t+τ] for everyη∈C∞[0,t+τ]. This proves (4.2). /square\nThegeneralizedsemigroupisnotknowntobecontinuous(theunique -\nness is not proved), however, the following two properties of it are , in\na sense, a substitution for the continuity and make it possible to con -\nstruct a weak attractor [1], [2].\nLemma 4.3. The generalized semigroup S(t)satisfies the following:\n1) [S(t)X]w⊂S(t)[X]wfor anyX⊂B0,\n2)for everyy∈ H1the setS(t)−1y∩B0is compact in H1\nw.\nHereB0is the absorbing ball (2.12), and[ ]wis the closure in H1\nw.\nProof.1)Letu=uT∈[S(T)X]w. Thenthereexistsasequence xn∈X\nsuch thatS(T)xn→uTweakly in H1\nw. The sequence {xn}is bounded\ninH1and contains a subsequence weakly converging to x0∈[X]w.\nThe set of all solutions un(t) =S(t)xnis bounded in C([0,T];H1),\nwhere the set ∂tunis bounded in L∞(0,T;L2−ε(Ω)). Therefore we can\nextract a subsequence unsuch that\nun→u∗-weakly inL∞(0,T;H1) and strongly in L2(0,T;H).(4.4)\nEachunsatisfies (4.3):\n−/integraldisplayT\n0(un,vη′(t))dt+/integraldisplayT\n0b(un,un,vη(t))dt+r/integraldisplayT\n0(un,vη(t))dt−\n−/integraldisplayT\n0(g,vη(t))dt= (xn,vη(0))−(S(T)xn,vη(T)).DAMPED NAVIER–STOKES SYSTEM 15\nThe convergence (4.4) makes it possible to pass to the limit in the\nintegral terms, while by hypotheses we have\n(S(T)xn,vη(T))→(uT,vϕ(T)),(xn,vη(T))→(uT,vη(T)).\nThis proves 1), since uis a solution with u(0) =x0andu(T) =uT,\nwherex0∈[X]w.\n2) The second property is proved similarly. Let\nun(0) =xn, un(t) =y, xn∈B0, xn→x∈B0weakly in H1.\nPassing to the limit as inin part 1) we obtainthat the limiting function\nuis a solution with u(0) =x,u(t) =y,x∈B0. /square\nThis lemma shows that the hypotheses of [1, Theorem 6.1] or [2,\nTheorem II.1.1] are satisfied for the generalized semigroup S(t). As a\nresult we have proved the existence of the weak attractor.\nTheorem 4.4. The generalized semigroup S(t)corresponding to the\ndamped Euler system has a weak (H1,H1\nw)-attractor A.\nOur next goal is to show that the attractor Ais in fact a (strong)\nglobal attractor in the sense of Definition 2.5, the only difference be ing\nthat the semigroup S(t) now is a generalized (multi-valued) semigroup.\nThe key role below is played by the equation of balance of energy and\nenstrophy (3.9).\nTheorem 4.5. The attractor Ais the (strong) global attractor.\nProof.We have to prove the asymptotic compactness of S(t), that is,\nfor every sequence {u0\nn}bounded in H1and every sequence tn→+∞\nthe sequence (of sets) S(tn)u0\nnis precompact in H1.\nLetun(t),t≥ −tnbe a sequence of solutions of the damped Euler\nsystem:/braceleftbigg\n∂tun+(un,∇)un+∇pn+run=g(x),\ndivun= 0, un|t=−tn=u0\nn.\nThenun(0)∈S(tn)u0\nnand we have to verify that {un(0)}∞\nn=0is pre-\ncompact in H1.\nThe solutions un(t),t≥ −tn, are bounded in Cb([−T,∞),H1) for\nT≤tnand we can extract a subsequence\nun(0)→¯u∈ H1weakly in H1.\nAlong a further subsequence we have\nun→u∗-weakly inL∞(−T,T;H1) and strongly in L2(−T,T;H).16 V.V. CHEPYZHOV, A.A. ILYIN, AND S.V ZELIK\nThis is enough to pass to the limit in the integral identities satisfied by\nunto obtain that the following integral identity holds for u:\n−/integraldisplay\nR(u,vη′(t))dt+/integraldisplay\nRb(u,u,vη(t))dt+\nr/integraldisplay\nR(u,vη(t))dt−/integraldisplay\nR(g,vη(t))dt= 0, η∈C∞\n0(R),\nwhich gives that uis a solution of the damped Euler system bounded\nont∈R. Next, we have\nu(0) = ¯u. (4.5)\nThis is standard [31]. On one hand, for η(0)/\\e}atio\\slash= 0 we have\n−/integraldisplay0\n−∞(u,vη′(t))dt+/integraldisplay0\n−∞b(u,u,vη(t))dt+\nr/integraldisplay0\n−∞(u,vη(t))dt−/integraldisplay0\n−∞(g,vη(t))dt=−(¯u,v)η(0),\n(4.6)\nOn the other hand, multiplying the equation\nd\ndt(u,v)+b(u,u,v)+r(u,v) = (g,v)\nby the same ηand integrating from −∞to 0 we obtain equality (4.6)\nwith the right-hand side equal to −(u(0),v)η(0). This gives (4.5).\nThus, we have that un(0)→u(0) weakly in H1, we now show that\nun(0)→u(0) strongly in H1. We multiply the balance equation (3.9)\nforunbye2rtand integrate from −tnto 0. We obtain\n/ba∇dblun(0)/ba∇dbl2\n1=/ba∇dblun(−tn)/ba∇dbl2\n1e−2rtn+2/integraldisplay0\n−tn(un(t),g)1e2rtdt.\nSinceun(−tn) are uniformly bounded in H1and\nun→u∗-weakly in L∞\nloc(R;H1)\nwe can pass to the limit as n→ ∞to obtain\nlim\nn→∞/ba∇dblun(0)/ba∇dbl2\n1= 2/integraldisplay0\n−∞(u(t),g)1e2rtdt.\nThe complete trajectory u(t) also satisfies the balance equation, and\nacting similarly we obtain\n/ba∇dblu(0)/ba∇dbl2\n1= 2/integraldisplay0\n−∞(u(t),g)1e2rtdt.DAMPED NAVIER–STOKES SYSTEM 17\nThus, we have shown that\nlim\nn→∞/ba∇dblun(0)/ba∇dbl2\n1=/ba∇dblu(0)/ba∇dbl2\n1,\nwhich along with the established weak convergence gives that\nun(0)→u(0) strongly in H1,\nand completes the proof. /square\n5.Upper semi-continuity of the attractors in the limit\nof vanishing viscosity\nIn this concluding section we study the dependence of the attract ors\nAνof the damped Navier–Stokes system on the viscosity coefficient ν\nasν→0+. In the previous section we have shown that the damped\nEuler system (with ν= 0) has the global attractor\nAν=0=:A0.\nFurthermore, uniformly for ν≥0 the following estimate holds:\nsup\nu∈Aν/ba∇dblu/ba∇dbl1≤/ba∇dblg/ba∇dbl1\nr.\nTheorem 5.1. The attractors Aνdepend upper semi-continuously on\nνasν→0+. In other words\nlim\nν→0+distH1(Aν,A0) = 0, (5.1)\nwhere\ndistH1(X,Y) := sup\nx∈Xinf\ny∈Y/ba∇dblx−y/ba∇dblH1. (5.2)\nProof.We take an arbitrary sequence νn→0+, and for every νnchoose\na point on the attractor Aνnof equation (1.1) with ν=νn. Specifically,\nwe choose the point on Aνn, whose distance from A0is equal to the\ndistance from AνntoA0. In view of the compactness of AνnandA0\nsuch a point exists. These points lie on Aνnand therefore there are\ncomplete trajectories passing through them, and we can denote t hese\npoints byun(0), so that\nun(0)∈Aνn, un∈Cb(R,H1),/ba∇dblun/ba∇dblCb(R,H1)≤r−1/ba∇dblg/ba∇dbl1,(5.3)\nand in view of our choice\ndistH1(un(0),A0) = dist H1(Aνn,A0). (5.4)\nInviewof (5.3)wecanextractasubsequence uνnforwhichfora ¯ u∈ H1\nuνn(0)→¯uweakly in H1asn→ ∞,18 V.V. CHEPYZHOV, A.A. ILYIN, AND S.V ZELIK\nand along a further subsequence we have\nuνn(0)→u0∗-weakly inL∞\nloc(R,H1) and strongly in L2\nloc(R,H).\nThe solutions uνn, by definition, satisfy the integral identity\n−/integraldisplay\nR(uνn,vη′(t))dt+/integraldisplay\nRb(uνn,uνn,vη(t))dt+\n+/integraldisplay\nRνn(curluνn,curlvη(t))dt+r/integraldisplay\nR(uνn,vη(t))dt−/integraldisplay\nR(g,vη(t))dt= 0.\nWe now pass to the limit in this identity taking into account that\nνn(curluνn,curlv)→0 asνn→0,\nand obtain that u0is a solution (a complete trajectory) of the damped\nEuler system and therefore satisfies the balance equation (3.9). I n\naddition, as in Theorem 4.5, we can show that u(0) = ¯u, so that\nuνn(0)→u(0) weakly in H1.\nThe complete trajectories un=un(t) of the damped Navier–Stokes\nsystem (1.1) satisfy the balance equation (2.10). We drop there th e\nsecond (non-negative) term multiply the resulting inequality by e2rt\nand integrate from −tnto 0, where tn→+∞. We obtain\n/ba∇dbluνn(0)/ba∇dbl2\n1≤ /ba∇dbluνn(−tn)/ba∇dbl2\n1e−2rtn+2/integraldisplay0\n−tn(uνn(t),g)1e2rtdt.\nIn the limit as n→ ∞this gives that\nlimsup\nn→∞/ba∇dbluνn(0)/ba∇dbl2\n1≤2/integraldisplay0\n−∞(u0(t),g)1e2rtdt.\nFor the solution u0as in Theorem 4.5 we have\n/ba∇dblu0(0)/ba∇dbl2\n1= 2/integraldisplay0\n−∞(u0(t),g)1e2rtdt,\nand together with the previous inequality this gives that\nlimsup\nn→∞/ba∇dbluνn(0)/ba∇dbl2\n1≤ /ba∇dblu0(0)/ba∇dbl2\n1. (5.5)\nSince by the weak convergence we always have\n/ba∇dblu0(0)/ba∇dbl1≤liminf\nn→∞/ba∇dblun(0)/ba∇dbl1,\nit follows from (5.5) that\nlim\nn→∞/ba∇dbluνn(0)/ba∇dbl1=/ba∇dblu0(0)/ba∇dbl1,\nand, finally, that\nlim\nn→∞/ba∇dbluνn(0)−u0(0)/ba∇dbl1= 0.DAMPED NAVIER–STOKES SYSTEM 19\nTaking into account (5.4) we obtain that\nlim\nn→∞distH1(Aνn,A0) = 0. 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Temam, Infinite-Dimensional Dynamical Systems in Mechanics and\nPhysics, Springer, New York, 1997.\n[31] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis , North-\nHolland, Amsterdam–New York-Oxford, 1977.\n[32] M.I. Vishik and V.V. Chepyzhov, Trajectory attractors of equ ations of math-\nematical physics, Uspekhi Mat. Nauk 66:4 (2011), 3–102; English tarnsl. in\nRussian Math. Surveys. 66:4 (2011).\n[33] G. Wolansky, Existence, uniqueness, and stability of stationar ybarotropicflow\nwith forcing and dissipation, Comm. Pure Appl. Math. 41(1988), 19–46.\n[34] V.I. Yudovich, Non-Stationary flow of an ideal incompressible flu id,Zh. Vy-\nchisl. Mat. Mat. Fiz. 3(1963), 1032–1066.\n[35] S. Zelik, Spatially nondecaying solutions of the 2D Navier-Stokes equation in\na strip,Glasg. Math. J. 49(2007), 525–588.DAMPED NAVIER–STOKES SYSTEM 21\n[36] S. Zelik, Weak spatially nondecaying solutions of 3D Navier-Stoke s equations\nin cylindrical domains. Instability in models connected with fluid flows. II , 255–\n327, Int. Math. Ser. (N. Y.), 7, Springer, New York, 2008.\n[37] S. Zelik, Infinite energy solutions for damped Navier-Stokes eq uations in R2,\nJ. Math. Fluid Mech. 15(2013), 717–745.\n1Institute for Information Transmission Problems, Moscow 1 27994,\nRussia,\n2Keldysh Institute of Applied Mathematics, Moscow 125047, R ussia,\n3National Research University Higher School of Economics, M oscow\n101000, Russia,\n4University of Surrey, Department of Mathematics, Guildfor d, GU2\n7XH, UK.\nE-mail address :chep@iitp.ru\nE-mail address :ilyin@keldysh.ru\nE-mail address :s.zelik@surrey.ac.uk"    },    {        "title": "1910.09762v1.Modifications_to_Plane_Gravitational_Waves_from_Minimal_Lorentz_Violation.pdf",        "content": "arXiv:1910.09762v1  [gr-qc]  22 Oct 2019Modifications to Plane Gravitational Waves from Minimal\nLorentz Violation\nRui Xu\nKavli Institute for Astronomy and Astrophysics,\nPeking University, Beijing 100871, China\n(Dated: October 2019)\nAbstract\nGeneral Relativity predicts two modes for plane gravitatio nal waves. When a tiny violation of\nLorentz invariance occurs, the two gravitational wave mode s are modified. We use perturbation\ntheory to study the detailed form of the modifications to the t wo gravitational wave modes from\nthe minimal Lorentz-violation coupling. The perturbation solution for the metric fluctuation up to\nthe first order in Lorentz violation is discussed. Then, we in vestigate the motions of test particles\nunder the influence of the plane gravitational waves with Lor entz violation. First-order deviations\nfrom the usual motions are found.\n1I. INTRODUCTION\nGeneral Relativity (GR), as the standard classical gravitational t heory, has been making\npredictions consistent with all the terrestrial experiments and mo st of the astrophysical\nobservations [1, 2]. However, the fact that it is incompatible with qua ntum theory motivates\nceaseless newtestsandalargeamountofalternativetheories[3,4 ]. Lorentzinvariance, being\none of the fundamental principles in GR, has been suffering constan t tests in various high-\nprecision experiments and observations [5–9]. Especially, gravitat ional wave observations,\nproviding us the unique access to strong-field environments, have recently put new stringent\nconstraints on Lorentz violation based on the analysis of the modifie d dispersion relation of\ngravitational waves in the Standard-Model Extension (SME) fram ework [10, 11].\nThe SME framework is a tool to study Lorentz violation in a model-inde pendent way [12–\n18]. It incorporates all possible Lorentz-violation couplings into the Lagrangian density of\nGR and the Standard Model by employing the so called Lorentz-violat ion coefficients which\ncan be measured or constrained with experimental data. The sect or that describes gravity\nwith Lorentz violation in vacuum is called the pure gravity sector of th e SME [19, 20], and\nit is the theoretical basis from which the modified dispersion relation o f gravitational waves\nis derived [10, 21].\nUsing the modified dispersion relation to constrain Lorentz violation m arks the beginning\nof testing Lorentz invariance with gravitational wave observation s [22]. As the number and\nsensitivity of gravitational wave observatories increase [23, 24], we can extract more informa-\ntion about the incoming waves from the observed signals, including th e polarization status\nof them. Recently, a detailed investigation on plane-wave solutions f or arbitrary Lorentz\nviolation in the pure gravitationa SME is carried out, and the modificat ions to the two po-\nlarization modes of the gravitational waves from coalescing compac t binaries are considered\n[25]. Here we study a similar question but only with the simplest Lorentz -violation coupling\nin the pure gravity sector of the SME so that the calculations are mo re transparent. We\nhave to point out that there are much bigger indicators of Lorentz violation [26] than what\nis described here. Therefore our result is mostly pedagogical. In ca se it is to be used to con-\nstrain Lorentz violation in gravitational wave observations, a more comprehensive treatment\nto strain signals in gravitational-wave detectors is required.\nWe start with describing the basics of the minimal Lorentz-violation c oupling [13, 19]\n2and show that a plane wave ansatz gives a naive modification to the us ual plane wave\nsolution in Section II. In Section III, we generalize the naive modifica tion to serve as a\nrigorous perturbation solution to the Lorentz-violation field equat ions. In Section IV, the\nperturbation solution is used to find the geodesic deviation of test p articles on a ring under\nthe effect of gravitational waves with Lorentz violation.\nII. PLANE WAVES WITH MINIMAL LORENTZ VIOLATION\nThe Lorentz-violation couplings in the SME framework are construc ted as coordinate\nscalars of the Lorentz-violation coefficients and conventional field operators. The simplest\nterm in the pure gravity sector is [13, 19]\nL(4)=1\n16πG(−uR+sµνRT\nµν+tαβγδCαβγδ), (1)\nwhereu,sµν, andtαβγδare called the minimal Lorentz-violation coefficients as the coupling\ninvolves no derivatives of the Riemann tensor. sµνandtαβγδinherit the symetries and trace-\nless property of the trace-free Ricci tensor, RT\nµν, and the Weyl conformal tensor, Cαβγδ,\nseparately. Note that the superscript 4 on Lrepresents the mass dimension of the gravi-\ntational operators (including the gravitational constant factor G). Therefore, the minimal\nLorentz-violation coefficients u,sµν, andtαβγδ, are also called the Lorentz-violation coeffi-\ncients with mass dimension d= 4.\nAdding to the Einstein–Hilbert term, the Lagrangian density (1) give s modifications\nfrom minimal Lorentz-violation to the Einstein field equations. The de tails on linearizing\nthe modified field equations and expressing them in terms of the back ground values of u,sµν,\nandtαβγδare demonstrated in Ref. [19]. Here we just show the result which is t he starting\npoint of our calculation, namely the linearized vacuum field equations w ith minimal Lorentz-\nviolation. They are\nRµν= ¯sαβRαµνβ, (2)\nwith ¯sαβbeing the background value of sµν. Note that the background values of uandtαβγδ\ndo not appear [19]. We also point out that the word ”linearized” has tw o meanings here.\nOne is the same as usual, namely the gravitational field is linearized. Th e second is that\nEquation (2) holds up to the first order in ¯ sαβ. There is no need to keep terms at higher\n3orders in ¯ sαβbecause Lorentz violationshould be tiny to beconsistent with theex perimental\nsupport for Lorentz invariance.\nThe dispersion relation implied by a generalized form of Equation (2) is s tudied in Ref.\n[27] to predict gravitational ˇCerenkov radiation from Lorentz violation. They proposed the\nmodified harmonic gauge condition,\n(ηλκ+ ¯sλκ)∂λhκµ=1\n2(ηλκ+ ¯sλκ)∂µhλκ, (3)\nthat simplifies the field equations (2) to\n(ηαβ+ ¯sαβ)∂α∂βhµν= 0, (4)\nwherehµν=gµν−ηµνis the fluctuation of the metric. Using the plane wave ansatz\nhµν(x) =Aµνeikx, (5)\nthe modified dispersion relation up to the first order in ¯ sαβis found to be\nk0=|/vectork|+1\n2¯sαβkαkβ\n|/vectork|. (6)\nNamely the wave vector can be written as kµ= (ω+δω,/vectork) withω=|/vectork|andδω=1\n2¯sαβkαkβ\n|/vectork|.\nThus, the plane wave solution can be written as\nhµν(x) =Aµνe−i(ωt−/vectork·/vector x)−iδωtA µνe−i(ωt−/vectork·/vector x)+.... (7)\nThe first term, Aµνe−i(ωt−/vectork·x), is apparently the plane wave solution in GR, and the rest\nconsists of corrections from Lorentz violation. Up to the first ord er in ¯sαβ, the correction is\nh(1)\n��ν=−iδωtA µνe−i(ωt−/vectork·/vector x). (8)\nAs there is a factor of tin the amplitude of h(1)\nµν, the correction is only valid during a finite\ntime period. The plane wave solution (7) is insufficient to describe the e ntire content of\nthe Lorentz-violation modification to gravitational waves. Howeve r, Equation (8) provides\nus an insight into how the modification might look. In the next section, we will take the\ngeneralized form of Equation (8), which is\nh(1)\nµν=Cµναxαe−i(ωt−/vectork·/vector x), (9)\nas an ansatz to solve the field equations (4) up to the first order in ¯ sαβ. The constants Cµνα\nare going to be determined by the gauge condition (3) and the field eq uations (4). Note that\nthe Lorentz-violation modification shown in (9) applies only to a finite s pacetime region as\nthe coordinates xαappear in the amplitude.\n4III. THE PERTURBATION SOLUTION\nWe seek a perturbation solution up to the first order in ¯ sαβfor the field equations (4).\nTo proceed, we assume that the zeroth-order plane wave travels along the zdirection with\nthe conventional wave vector\nk(0)µ= (ω,0,0, ω), (10)\nand that its amplitude Aµνtakes the usual form\nAµν=\n0 0 0 0\n0A11A120\n0A12−A110\n0 0 0 0\n, (11)\nwhereA11is the amplitude of the plus wave and A12is the amplitude of the cross wave. By\nsubstituting\nhµν(x) =Aµνe−i(ωt−kz)+Cµναxαe−i(ωt−kz), (12)\ninto the field equations (4) and keeping only the first-order terms, we have\n2Cµναik(0)α= ¯sαβk(0)\nαk(0)\nβAµν. (13)\nWritting the above equations explicitly, they are\nCµν0+Cµν3=−iω\n2(¯s00−2¯s03+ ¯s33)Aµν. (14)\nIn addition, using Equation (12) in the gauge condition (3), up to the first order we have\nηλκCκµαik(0)\nλxα+ηλκCκµλ+ ¯sλκik(0)\nλAκµ=1\n2/parenleftbig\nCαik(0)\nµxα+Cµ+ ¯sλκik(0)\nµAλκ/parenrightbig\n,(15)\nwhereCα=ηµνCµνα. The relations (15) imply two sets of equations:\nηλκCκµαk(0)\nλ−1\n2Cαk(0)\nµ= 0, (16)\nand\nηλκCκµλ−1\n2Cµ=i\n2¯sλκk(0)\nµAλκ−i¯sλκk(0)\nλAκµ. (17)\n5Using the expressions (10) and (11), we find that Equation (16) ca n be simplified to\nC00α+2C03α+C33α= 0,\nC11α+C22α= 0, (18)\nC01α+C31α= 0,\nC02α+C32α= 0,\nand that Equation (17) can be simplified to\nC011+C022=−1\n2iω/parenleftbig\n(¯s11−¯s22)A11+2¯s12A12/parenrightbig\n,\nC111+C122+1\n2(C001−C331) =iωA11(¯s01−¯s31)+iωA12(¯s02−¯s32),(19)\nC121−C112+1\n2(C002−C332) =iωA12(¯s01−¯s31)−iωA11(¯s02−¯s32).\nNote that Equation (17) turns out to have only 3 independent equa tions.\nEquation (19) shows that there are 6 independent components in t he first-order solution\nh(1)\nµν, which can be written as\nh(1)\nµν=\nh(1)\n00 h(1)\n01h(1)\n02−1\n2(h(1)\n00+h(1)\n33)\nh(1)\n01 h(1)\n11h(1)\n12 −h(1)\n01\nh(1)\n02 h(1)\n12−h(1)\n11−h(1)\n02\n−1\n2(h(1)\n00+h(1)\n33)−h(1)\n01−h(1)\n02 h(1)\n33\n. (20)\nThe 6 independent components are easily divided into 3 groups, {h(1)\n11, h(1)\n12},{h(1)\n00, h(1)\n33}, and\n{h(1)\n01, h(1)\n02}. The remaining equations in (14) and (20) are insufficient to determin e any of\nthem. This indicates that the ansatz (9) does not lead to a unique fir st-order solution. We\nneed extra information to fix h(1)\nµν. Next, we discuss the solutions for {h(1)\n11, h(1)\n12},{h(1)\n00, h(1)\n33}\nand{h(1)\n01, h(1)\n02}separately.\nA.{h(1)\n11, h(1)\n12}\nWe expect these two components recover the correction (8). Th is is indeed the case if we\ntake all the components of C11αandC12αto be zero except for\nC110=−iω\n2(¯s00−2¯s03+ ¯s33)A11,\nC120=−iω\n2(¯s00−2¯s03+ ¯s33)A12. (21)\n6In this way, h(1)\n11andh(1)\n12are fixed, and the dispersion relation (6) can be recovered in the\nperturbation solution.\nB.{h(1)\n00, h(1)\n33}\nWithC111=C112=C121=C122= 0, we have\nC001−C331= 2iωA11(¯s01−¯s31)+2iωA12(¯s02−¯s32),\nC002−C332= 2iωA12(¯s01−¯s31)−2iωA11(¯s02−¯s32). (22)\nIt turns out the combinations C001−C331andC002−C332are the only terms involving\nC00αandC33αin the first-order Riemann tensor (see the Appendix A). Therefor e, without\nany ambiguity in observables, we can safely assume all the componen ts ofC00αandC33α\nvanishing except for C001andC002, which are given by Equation (22).\nC.{h(1)\n01, h(1)\n02}\nIn the Appendix A, we can see that C010, ,C013, C020, andC023do not appear in the\nfirst-order Riemann tensor. Therefore, they can be taken as ze ro. However, C011andC022\nappear, and they do not appear as the combination C011+C022as shown in Equation (20).\nIn addition, C012andC021also show up in the first-order Riemann tensor. Namely, we have\none equation in (20) to use but 4 unknowns, C011, C012, C021, andC022, to determine. The\ninadequacy is likely from the fact that we are missing certain informat ion about the specific\ndynamic model of the Lorentz-violation coefficient sαβ. In other words, we expect sαβto\nhave itsown field equations with the metric involved. Then, when sαβis approximated by its\nbackground value ¯ sαβ, some of these field equations degenerate to constraints on the m etric\nthough most of them vanish trivially.\nBuilding a specific dynamic model for sαβsimply lies beyond the scope of the present\nwork. For the calculation in the next section, we decide to choose th e simplest solution for\nh(1)\n01andh(1)\n02, by which we mean that all the components of C01αandC02αvanish except for\nC011=−1\n2iω/parenleftbig\n(¯s11−¯s22)A11+2¯s12A12/parenrightbig\n. (23)\n7IV. GEODESIC DEVIATION\nNow we use the above first-order solution to calculate the effects o f Lorentz violation on\nthe motions of test particles when plane gravitational waves pass t hrough. Similarly to the\nusual case, it is illustrative to consider a ring of test particles whose initial positions form a\ncircle\n(X(0))2+(Y(0))2=d2, (24)\nin a local inertial frame with local coordinates {X, Y, Z}. Assuming the local coordinates\narealignedwiththegeneralcoordinates {x, y, z}, thenthenonrelativistic geodesicdeviation\nequations that determine the motions of the test particles in the loc al frame are [28]\nd2X\ndt2=−R0101X(0)−R0102Y(0)−R0103Z(0),\nd2Y\ndt2=−R0201X(0)−R0202Y(0)−R0203Z(0), (25)\nd2Z\ndt2=−R0301X(0)−R0302Y(0)−R0303Z(0).\nThe zeroth-order solution for X(t), Y(t) andZ(t) is the usual deformation\nX(0)(t)−X(0) =1\n2/parenleftbig\nA11X(0)+A12Y(0)/parenrightbig\n(e−iωt−1),\nY(0)(t)−Y(0) =1\n2/parenleftbig\nA12X(0)−A11Y(0)/parenrightbig\n(e−iωt−1), (26)\nZ(0)(t)−Z(0) = 0.\nNote that we have assumed that the local frame is moving along the g eodesicx(t) =y(t) =\nz(t) = 0. The first-order solution turns out to be\nX(1)(t) =−1\nω2/parenleftbig\nαX+βX(ωt−2i)/parenrightbig\ne−iωt+1\nω2(αX−2iβX),\nY(1)(t) =−1\nω2/parenleftbig\nαY+βY(ωt−2i)/parenrightbig\ne−iωt+1\nω2(αY−2iβY), (27)\nZ(1)(t) =−1\nω2αZe−iωt+1\nω2αZ,\nwhere\nαX=−iω(C110−C011)X(0)−iωC120Y(0)+1\n4iωC001Z(0),\nβX=−1\n2ωC110X(0)−1\n2ωC120Y(0),\nαY=−iωC120X(0)+iωC110Y(0)+1\n4iωC002Z(0), (28)\nβY=−1\n2ωC120X(0)+1\n2ωC110Y(0),\nαZ=1\n4iωC001X(0)+1\n4iωC002Y(0).\n8The solution (27) as well as Equation (28) is written with the underst anding that only the\nreal parts are taken.\nThe most notable correction is that Lorentz violation causes an osc illation along the z\ndirection in general, which does not happen in the case of the usual p lane gravitational\nwaves. Then, for the ring of the test particles in the XYplane, we find that the shape is\nstill deformed into ellipses. Butthe semi axes arecorrected byLor entzviolation. Specifically\nspeaking, when A11is real and A12= 0, the semi axes of the ellipse at time tare\na=d/parenleftbig\n1+1\n2A11(cosωt−1)−1\n2A11(¯s11−¯s22)(cosωt−1)−1\n4A11(¯s00−2¯s03+ ¯s33)ωtsinωt/parenrightbig\n,\nb=d/parenleftbig\n1−1\n2A11(cosωt−1)+1\n4A11(¯s00−2¯s03+ ¯s33)ωtsinωt/parenrightbig\n; (29)\nwhenA12is real and A11= 0, the semi axes of the ellipse at time tare\na=d/parenleftbig\n1+A12(cosωt−1)−A12¯s12(cosωt−1)−1\n2A12(¯s00−2¯s03+ ¯s33)ωtsinωt/parenrightbig\n,\nb=d/parenleftbig\n1−A12(cosωt−1)−A12¯s12(cosωt−1)+1\n2A12(¯s00−2¯s03+ ¯s33)ωtsinωt/parenrightbig\n.(30)\nLast but not least, we point out that when A12is real and A11= 0, the rotation angle of\nthe ellipses from the standard position\nX2\na2+Y2\nb2= 1, (31)\nis not±π\n4any more. A time-independent deviation of1\n2¯s12occurs in the presence of Lorentz\nviolation.\nV. CONCLUSIONS\nWe used the ansatz (9) to find the correction to plane gravitationa l waves from minimal\nLorentzviolation. ItwasshownthatuptothefirstorderinLorent zviolation, the correction,\nh(1)\nµν, has 6 independent components, with 4 of them fixed in the SME fram ework. To\ndetermine the remaining two components, extra information about the dynamics of the\nLorentz-violation coefficient sαβis necessary. This requires treating sαβas a dynamic field\nand assigning it a kinetic term in the Lagrangian density. This lies beyon d the scope of the\npresent work.\nThen, to demonstrate the effects of Lorentz violation on the motio ns of test particles\nunder the influence of plane gravitational waves, we artificially fixed the two undetermined\n9components of h(1)\nµν. Together with the other 4 determined components, two notable e ffects\nwere found. One is the oscillation of a test particle along the propaga ting direction of the\ngravitational waves, and the other is the deviation from ±π\n4for the rotation angle of the\ndeformed ellipse in the presence of the cross wave. Note that the a mplitude of the oscillation\nalong the Z-direction is proportional to the amplitude of the zeroth-order gr avitational\nwave but suppressed by the components of the Lorentz-violation coefficient ¯ sαβ. Taking the\ncurrent best bound of 10−15[11] on ¯sαβinto consideration, it is unlikely that this oscillation\nprovides a viable test of Lorentz violation even in the near future. O n the other hand, as we\nare getting access to the polarization information of incoming gravit ational waves with more\nand more detectors in construction, the deviation of the rotation angle suggests a Lorentz-\nviolation phase difference between the two polarization modes to tes t in future observations\nof polarized gravitational waves. To conduct such tests, a more c omprehensive treatment in\nthe context of existing and future gravitational-wave detectors is required, which deserves\nanother paper for investigation.\nAcknowledgments\nR.X. is thankful to Alan Kosteleck´ y, Lijing Shao, and Jay Tasson fo r valuable comments.\nAppendix A: The First-Order Riemann Tensor\nThe first-order Riemann tensor is calculated by\nR(1)\nαβγδ=1\n2(∂γ∂βh(1)\nαδ+∂α∂δh(1)\nβγ−∂γ∂αh(1)\nβδ−∂δ∂βh(1)\nαγ). (A1)\nPlugging Equation (9) into it, and using /vectork= (0,0, ω), we find\nR(1)\n0101=1\n2/parenleftbig\n2iω(C110−C011)+ω2C11αxα/parenrightbig\ne−i(ωt−/vectork·/vector x),\nR(1)\n0102=1\n2/parenleftbig\niω(2C120−C021−C012)+ω2C12αxα/parenrightbig\ne−i(ωt−/vectork·/vector x),\nR(1)\n0103=−1\n4iω(C001−C331)e−i(ωt−/vectork·/vector x), (A2)\nR(1)\n0202=−1\n2/parenleftbig\n2iω(C110+C022)+ω2C11αxα/parenrightbig\ne−i(ωt−/vectork·/vector x),\nR(1)\n0203=−1\n4iω(C002−C332)e−i(ωt−/vectork·/vector x),\nR(1)\n0303= 0,\n10R(1)\n0112=−1\n2iω(C112−C121)e−i(ωt−/vectork·/vector x),\nR(1)\n0113=1\n2/parenleftbig\niω(C110−C113−2C011)+ω2C11αxα/parenrightbig\ne−i(ωt−/vectork·/vector x),\nR(1)\n0123=−1\n2iωC012e−i(ωt−/vectork·/vector x),\nR(1)\n0212=−1\n2iω(C111+C122)e−i(ωt−/vectork·/vector x),\nR(1)\n0213=1\n2/parenleftbig\niω(C120−C123−C021−C012)+ω2C12αxα/parenrightbig\ne−i(ωt−/vectork·/vector x), (A3)\nR(1)\n0223=1\n2/parenleftbig\n−iω(C110−C113+2C022)−ω2C11αxα)e−i(ωt−/vectork·/vector x),\nR(1)\n0313=−1\n4iω(C001−C331)e−i(ωt−/vectork·/vector x),\nR(1)\n0323=−1\n4iω(C002−C332)e−i(ωt−/vectork·/vector x),\nand\nR(1)\n1212= 0,\nR(1)\n1213=−1\n2iω(C112−C121)e−i(ωt−/vectork·/vector x),\nR(1)\n1223=−1\n2iω(C111+C122)e−i(ωt−/vectork·/vector x),\nR(1)\n1313=1\n2/parenleftbig\n−2iω(C113+C011)+ω2C11αxα/parenrightbig\ne−i(ωt−/vectork·/vector x), (A4)\nR(1)\n1323=1\n2/parenleftbig\n−iω(C012+C021+2C123)+ω2C12αxα/parenrightbig\ne−i(ωt−/vectork·/vector x),\nR(1)\n2323=1\n2/parenleftbig\n2iω(C113−C022)−ω2C11αxα/parenrightbig\ne−i(ωt−/vectork·/vector x).\n[1] Debono, I.; Smoot, G.F. 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Detecting Lorentz Violations with Grav itational Waves from Black Hole Bina-\nries.Phys. Rev. Lett. 2018,120, 041104.\n[27] Kosteleck´ y, V.A.; Tasson, J.D. Constraints on Lorent z violation from gravitational ˇCerenkov\nradiation. Phys. Lett. B 2015,749, 551.\n[28] Poisson E.; Will, C.M. Gravity; Cambridge University Press: Cambridge, UK, 2014.\n13"    },    {        "title": "1107.5539v2.Constraint_damping_for_the_Z4c_formulation_of_general_relativity.pdf",        "content": "arXiv:1107.5539v2  [gr-qc]  20 Feb 2012Constraint damping for the Z4c formulation of general relat ivity\nAndreas Weyhausen, Sebastiano Bernuzzi and David Hilditch\nTheoretical Physics Institute, University of Jena, 07743 J ena, Germany\n(Dated: November 7, 2018)\nOne possibility for avoiding constraint violation in numer ical relativity simulations adopting free-\nevolution schemes is to modify the continuum evolution equa tions so that constraint violations\nare damped away. Gundlach et. al. demonstrated that such a sc heme damps low-amplitude, high-\nfrequency constraint-violating modes exponentially for t he Z4 formulation of general relativity. Here\nwe analyze the effect of the damping scheme in numerical appli cations on a conformal decomposition\nof Z4. After reproducing the theoretically predicted dampi ng rates of constraint violations in the\nlinear regime, we explore numerical solutions not covered b y the theoretical analysis. In particular\nwe examine the effect of the damping scheme on low-frequency a nd on high-amplitude perturbations\nof flat spacetime as well and on the long-term dynamics of punc ture and compact star initial data\nin the context of spherical symmetry. We find that the damping scheme is effective provided that\nthe constraint violation is resolved on the numerical grid. On grid noise the combination of artificial\ndissipation and damping helps to suppress constraint viola tions. We find that care must be taken\nin choosing the damping parameter in simulations of punctur e black holes. Otherwise the damping\nscheme can cause undesirable growth of the constraints, and even qualitatively incorrect evolutions.\nIn the numerical evolution of a compact static star we find tha t the choice of the damping parameter\nis even more delicate, but may lead to a small decrease of cons traint violation. For a large range of\nvalues it results in unphysical behavior.\nI. INTRODUCTION\nThe most common way to construct numerical solu-\ntions to the field equations of general relativity is to take\na free-evolution approach. The Hamiltonian and mo-\nmentum constraints of the theory are explicitly solved\nonly for initial data. Then the remaining field equa-\ntions are rewritten in a suitable hyperbolic form, and\nthe initial data can be evolved using the desired numer-\nical method with this hyperbolic formulation . In the ab-\nsence of boundaries the contracted Bianchi identities can\nbe used to show that if the constraints are satisfied on\none spacelike slice of a foliation, then they will be sat-\nisfied everywhere. However, numerical solutions violate\nthe constraints. This violation can be considered under\ncontrol, if when one applies more resolution to the prob-\nlem, the constraint violationconvergesawayat an appro-\npriate rate. Nonetheless, even if the constraint violation\nconverges away, at finite resolution constraint violation\nis undesirable. A number of strategies to minimize the\nviolation have been considered. One is to choose the\nformulation such that every constraint propagates. In\ncombination with suitable boundary conditions, the con-\nstraint violation on the numerical grid should then be\npropagated away. On the other hand, if the constraints\ndo not propagate then any violation may sit on the grid\nand grow. Another strategy is to use a constraint damp-\ning scheme , namely to modify the evolution equations\nso that the constraint-satisfying hypersurface becomes\nan attractor in phase space; such an evolution scheme\nis sometimes called a λ-system [1]. The constraints are\nthen referred to as the λ-variables.\nThe Z4 formulation [2–7] has both propagating con-\nstraintsandadmitsaconstraintdampingscheme[8]. The\nZ4 formulation has a close relationship with the gener-alized harmonic formulation, and the damping scheme\nis essentially the same for both systems. The damping\nschemewasacrucialingredientinthe firstsuccessful evo-\nlution of orbiting binary black holes through merger [9].\nAnalytic calculations demonstrating that constraint vio-\nlations will be damped away are performed in the frozen\ncoefficient approximation. On the basis of these calcula-\ntions one expects that the damping scheme will be effec-\ntive, in numerical applications, on constraint violations\nthat are of low amplitude and high frequency in space-\ntimes that are close to stationarity. Since the constraint\ndampingschemeisamodification ofthe continuumequa-\ntions, this high-frequency should be resolved on the nu-\nmerical mesh. It is not clear what effect the damping\nscheme will have on ill-resolved numerical noise.\nA conformal decomposition of Z4, called Z4c, was pro-\nposed [10, 11] with the hope of bringing the advantages\nof propagating constraints and the constraint damping\nscheme to the puncture method [12, 13] for the evolu-\ntion of binary black holes. Here we continue that in-\nvestigation, considering more carefully the effect of the\nconstraint damping scheme on numerical evolutions. We\naddress the following questions: (i). Under what condi-\ntions can the theoretically predicted damping rates be\nrecovered in the numerical approximation? (ii). How ef-\nfective is the damping scheme in astrophysically relevant\nspacetimes? (iii). In practical applications what are rea-\nsonable values for the constraint damping coefficients?\nA variation of the conformal decomposition has been\nrecently presented in [14]. There it was found that the\nconstraint damping terms are essential for stable long-\nterm 3D evolutions of binary black holes and the gauge\nwave test. Since the Z4c conformal decomposition differs\nfrom that in [14] by nonprincipal terms and implementa-\ntion details (e.g. constraints projection and summation-2\nby-part operators), our study does not necessarily apply\nin that case. More work is required to carefully evaluate\nthe role of the constraint damping scheme in that case.\nBecause of the obvious computational overhead of\nworking in three dimensions, we once again present nu-\nmerical results in spherical symmetry. Note that since\nthe constraint damping scheme is a modification to the\ncontinuum Z4c formulation, our results are expected to\nreflect the behavior of the full system. Working in spher-\nical symmetry furthermore affords us the possibility of\nperforming a thorough study in the parameter space of\nconstraint damping coefficients.\nIn Sec. II we summarize the equations of motion for\nthe Z4c formulation and describe the constraint damp-\ning scheme we employ. We also present the expected\ntheoreticalratesof dampingin the high-frequency, frozen\ncoefficient approximation. In Sec. III we present our nu-\nmerical study. Finally we conclude in Sec. IV.\nNotation. Geometric units are employed. Standard\nnotation for the 3+1 general relativity is used, e.g.\n∂i, partial derivative with respect to coordinates xi,\ni= 1,2,3,Di, 3-covariant derivative, Lβ, Lie derivative\nalong the vector βi,γ, determinant of the 3-metric γij,\nKij, α, βi, extrinsic curvature, lapse function and shift\nvector. In the perturbed flat-space simulations there is\nno natural scale of time; there we give the time in abi-\ntrary units.\nII. THE Z4C CONSTRAINT DAMPING\nSCHEME\nA. The Z4c formulation\nIn 3+1 form the field equations of the Z4c formulation\nof general relativity for the three-metric and extrinsic\ncurvature read\n∂tγij=−2αKij+Lβγij, (1)\n∂tKij=−DiDjα+α[Rij−2KikKkj+KKij\n+2ˆD(iZj)−κ1(1+κ2)γijΘ]\n+4πα[γij(S−ρADM)−2Sij]+LβKij,(2)\nwhere we use the notation\nˆDiZj≡γ−1\n3γkj∂i[γ1\n3Zk]. (3)\nThe constraints Θ ,Zievolve according to\n∂tΘ =α[1\n2H+ˆDiZi−κ1(2+κ2)Θ]+LβΘ, (4)\n∂tZi=α[Mi+DiΘ−κ1Zi]+γ1\n3Zj∂t[γ−1\n3γij]+βjˆDjZi,\n(5)\nwhere the Hamiltonian and momentum constraints\nH, Miare given by\nH=R−KijKij+K2−16πρADM= 0,(6)\nMi=Dj[Kij−γijK]−8πSi= 0. (7)Their time dependence can be computed as (neglecting\nmatter terms),\n∂tH=−2αDiMi−4MiDiα+2αKH\n+2α/parenleftbig\n2Kγij−Kij/parenrightbigˆD(iZj)\n−4κ1(1+κ2)αKΘ+LβH, (8)\n∂tMi=−1\n2αDiH+αKMi−(Diα)H\n+Dj/parenleftBig\n2αˆD(iZj)/parenrightBig\n−Di/parenleftBig\n2αγklˆD(kZl)/parenrightBig\n+2κ1(1+κ2)Di(αΘ)+LβMi. (9)\nFrom Eq. (4-5) one sees that Θ , Zibehave asλvariables\nif the free parameters κ1,2are properly chosen. Conse-\nquently the Einstein constraint are damped for κ1>0\nandκ2>−1 [8]. The constraint subsystem is closed.\nIf the constraints Θ ,ZiandH,Miare satisfied in one\nhypersurface they will remain satisfied at all times. In-\ntroducing the following variables and definitions,\n˜γij=γ−1\n3γij, χ=γ−1\n3, (10)\nˆK=γijKij−2Θ,˜Aij=γ−1\n3(Kij−1\n3γijK),(11)\n˜Γi= 2˜γijZj+ ˜γij˜γkl˜γjk,l,˜Γdi= ˜γjk˜Γijk,(12)\nthe conformal evolutions equations, Z4c, read,\n∂tχ=2\n3χ[α(ˆK+2Θ)−Diβi], (13)\n∂t˜γij=−2α˜Aij+βk˜γij,k+2˜γk(iβk\n,j)−2\n3˜γijβk\n,k,(14)\n∂tˆK=−DiDiα+α[˜Aij˜Aij+1\n3(ˆK+2Θ)2]\n+4πα[S+ρADM]+ακ1(1−κ2)Θ+βiˆK,i(15)\n∂t˜Aij=χ[−DiDjα+α(Rij−8πSij)]tf\n+α[(ˆK+2Θ)˜Aij−2˜Aki˜Akj]\n+βk˜Aij,k+2˜Ak(iβk\n,j)−2\n3˜Aijβk,k (16)\n∂t˜Γi=−2˜Aijα,j+2α[˜Γi\njk˜Ajk−3\n2˜Aijln(χ),j\n−1\n3˜γij(2ˆK+Θ),j−8π˜γijSj]+ ˜γjkβi\n,jk\n+1\n3˜γijβk\n,kj+βj˜Γi\n,j−˜Γdjβi\n,j+2\n3˜Γdiβj\n,j\n−2ακ1(˜Γi−˜Γdi), (17)\n∂tΘ =α[1\n2R−1\n2˜Aij˜Aij+1\n3(ˆK+2Θ)2\n−8πρADM−κ1(2+κ2)Θ]+LβΘ. (18)3\nHere the intrinsic curvature is written as\nRij=Rχij+˜Rij, (19)\n˜Rχ\nij=1\n2χ˜Di˜Djχ+1\n2χ˜γij˜Dl˜Dlχ\n−1\n4χ2˜Diχ˜Djχ−3\n4χ2˜γij˜Dlχ˜Dlχ,(20)\n˜Rij=−1\n2˜γlm˜γij,lm+ ˜γk(i|˜Γk\n|,j)+˜Γdk˜Γ(ij)k\n+ ˜γlm/parenleftBig\n2˜Γk\nl(i˜Γj)km+˜Γk\nim˜Γklj/parenrightBig\n.(21)\nThe equations above are constrained by two algebraic\nexpressions, ln(det˜ γ) = 0 and ˜ γij˜Aij= 0, which are\nexplicitly imposed during the numerical evolution. A so-\nlution of the evolution system is a solution of the Ein-\nstein system provided that Θ, Zi,HandMivanish. In\nour numerical applications we close the system with the\npuncture gauge choice [15, 16],\n∂tα=−µLα2ˆK+Lβα, (22)\n∂tβi=µSα2˜Γi−ηβi+βj∂jβi, (23)\nwithµL= 2/α,µS= 1/α2andη= 2/M, whereMis\ntheADMmassofthespacetime. Unlessstatedotherwise,\nwe employ the constraint-preserving boundary condition\nof [11].\nFor a more thorough introduction to the Z4c formula-\ntion we refer the reader to [10, 11]. The full formulation\ngenerically forms a strongly hyperbolic system of partial\ndifferential equations, except in special cases which we\ndo not discuss here.\nB. Theoretical damping rates\nHere webrieflyreviewthe resultsof[8], seealso[17] for\na discussion of stability of the undamped nonlinear con-\nstraint system. Consider the subsystem (4-5, 8-9) when\nthe initial data is constraint violating. We consider a\nsmall constraint-violating perturbation on a background\nwhich satisfies the Z4c equations of motion. Working in\nthe frozen coefficient approximation we can choose coor-\ndinates at a point so that the spatial metric is just that\nof flat-space, and the lapse is unity. The only nontriv-\nial component of the metric is the shift, which can only\nbe made to vanish if we allow ourselves the freedom to\nchange the spatial slice, which we take as given [18]. We\ndiscard nonprincipal terms involvingproducts of the per-\nturbation and the backgroundcurvature, extrinsic curva-\nture, and matter sources, which is inconsistent with our\nfirst-order perturbation approach. The inclusion of mat-\nter sources without background curvature terms should\ngive the correct damping rates for test matter fields on\nflat space. Such an analysis is not expected to give the\nrates for compact stars as evolved in Sec. III; nontriv-\nial background curvature, extrinsic curvature and mattersource terms will affect the damping rates, but for a con-\nsistent analysis all of these effects should be considered\ntogether. Already in the variable coefficient approxima-\ntion such an analysis will be challenging. We will use the\nshift only to illustrate that it does not play any role on\nthe damping rates. Besides the inclusion of the shift our\ncalculations are exactly the same as those of [8]. To start\nwith, we make a plane wave ansatz\nΘ =est+iωixiˆΘ, Z i=est+iωixiˆZi,(24)\nfor the solution of the constraint subsystem, with com-\nplexsand realωi. We restrict to the special case κ2= 0,\nand writeκ1=k. Rewriting the constraint subsystem\nin fully second order in time form, we find that following\neigenvalue problem\n/parenleftbigg\nλ+2sk+2iωβk iωk\n0λ+sk−iωβk/parenrightbigg/parenleftbiggˆΘ\nˆZˆω/parenrightbigg\n= 0,(25)\n/parenleftbigλ+sk−iωkβ/parenrightbig/parenleftbigˆZA/parenrightbig\n= 0,(26)\nmust be satisfied, where we write\nλ=s2+ω2−2iωsβ−ω2β2, (27)\nandˆZˆωstands for the component of ˆZiin the ˆωidirec-\ntion, while ˆZAare the components transverse to the unit\nwave vector ˆ ωiandβ=βiˆωi. The symbol of this equa-\ntion generically has a complete set of eigenvectors, so the\nsystem can be rotated to diagonal form,\n/parenleftbigg\nλΘ0\n0λZˆω/parenrightbigg/parenleftbigg\nsˆΘ+iω(ˆZˆω−βˆΘ)\nˆZˆω/parenrightbigg\n= 0,(28)\n/parenleftbigλZT/parenrightbig/parenleftbigˆZA/parenrightbig\n= 0.(29)\nIn one dimension, identifying the direction r, one expects\nthat the behavior of the combinations of primitive vari-\nables,\nuΘ=∂tΘ+∂rZr, (30)\nuω=Zr (31)\nwill be determined, in the linear regime, by the eigenval-\nues of the symbol provided that the shift is small. The\neigenvalues are\nλΘ=λ+2sk−2ikωβ,\ns=−k+iωβ±/radicalbig\nk2−ω2, (32)\nand\nλZs=λ+sk−iωkβ,\ns=−k\n2+iωβ±/radicalBigg/parenleftbiggk\n2/parenrightbigg2\n−ω2,(33)\nand finally\nλZT=λ+sk−iωkβ,\ns=−k\n2+iωβ±/radicalBigg/parenleftbiggk\n2/parenrightbigg2\n−ω2.(34)4\nTABLE I: Setting used for the numerical simulations pre-\nsented in this paper. The resolution is given in grid points,\nroutis the coordinate location of the computational bound-\nary, and αCFLis the Courant-Friedrichs-Levy factor used in\nthe time-stepping.\nInitial data Resolution routαCFL\nFlat 4000 100 a.u. 0.5\nPuncture 1000 50 M 0.5\nStar 2000 50 M 0.4\nIn the low-frequency limit ω≪kwe find that\ns≃ −2k+iωβ, s ≃iωβ−ω2\n2k,\ns≃ −k+iωβ+ω2\nk, s ≃iωβ−ω2\nk,(35)\nwhereas in the high-frequency limit k≪ω, we have\ns≃ −k+iω(β∓1), s≃ −k\n2+iω(β±1),(36)\nso at lowerfrequencies halfofthe modes aredamped less.\nIn the high-frequency limit, the damping scheme causes\na exponential decay of the constraints with a decay rate\nof−kand−k/2 respectively.\nIII. NUMERICAL RESULTS\nInthis sectionwepresentournumericalresults. Spher-\nical symmetry is assumed. We perform a detailed anal-\nysis of the damping scheme applied to the evolution of\nflat spacetime with different constraint-violating pertur-\nbations in order to reproduce the analytic results and ex-\nplore the regime not accessible to a pen-and-paper analy-\nsis. We consider then nontrivial initial data composed of\neither punctures or a compact star and evolve them us-\ning different values for the damping parameters. In this\ncase the performance of the damping scheme in a strong-\nfield regime on constraint violations related, essentially,\nto different truncation errors are investigated. The code\nemployed is described in detail in [10].\nNumerical setup. In all numerical simluations we use\nfourth-order finite differences for the discretization of the\nspatial derivatives of the metric fields. For the time inte-\ngration we use Runge-Kutta fourth order in the vacuum\nand puncture tests. In the simulation of the compact\nstar we employ the Runge-Kutta third order in combi-\nnation with a high-resolution shock capturing based on\nthe local Lax-Friedrichs flux and the convex-essentially-\nnon-oscillatoryinterpolationfor the reconstructionofthe\nmatter fields. Table I summarizes the numerical set-\ntings employed for the results presented; convergence\ntests were run on some cases (see also discussions in the\nfollowing paragraphs).A. Perturbed flat space-time experiments\nInitial data and parameter space. Perturbations of\nflat space-time are constructed, depending on which of\nthe twoeigenmodes, uΘ,uω, we wantto analyze, bymod-\nifying theχvariable,\nχ(0,r) = 1+Aexp/parenleftbigg\n−r2\n2b2/parenrightbigg\ncos/parenleftbigg2πν\nbr/parenrightbigg\n,(37)\nor the˜Γrvariable,\n˜Γr(0,r) =Arexp/parenleftbigg\n−r2\n2b2/parenrightbigg\ncos/parenleftbigg2πν\nbr/parenrightbigg\n.(38)\nSimulations employing the first kind of initial data are\nanalyzed by looking at the eigenmode uΘ, while those\nemploying the second kind by looking at uω. The eigen-\nmodes are Fourier-transformed in space for every time\nstep, and their decay is studied by means of the power\nspectral density (PSD) at the frequency ˜ ν=ν\nb.\nThe parameter space, depicted in Fig. 1 (left panel),\nis spanned by the amplitude Aand the frequency νof\nthe initial constraint violation. We vary also κ1=k∈\n[0,1] but keep for simplicity κ2= 0. The parameter\nb= 10 (fixed) introduces a length scale to the problem,\nwhich is useful for tuning the frequency ν(number of\ncycles in the period b, see right panel of Fig. 1). To\nevaluate the strength of the perturbation, the value of\nthe perturbation’s amplitude Ashould be compared with\nunity, see Eq. (37).\nHigh-frequency, low-amplitude corner. In this region\noftheparameterspacetheanalyticalresultshold. Weset\nν= 10 (˜ν= 1) andA= 10−4, Fig. 2 display the results\nobtained. The numerical datashow an exponentialdecay\nof the PSD at the induced frequency ˜ νfor both eigen-\nmodes and for different values of the damping parameter\nk. The rate is quantified by linear fitting as displayed\nin Fig. 3. Table II reports the decay rates for different\nvalues ofkand different grid resolution ( nis the num-\nber of grid points), together with the computed fitting\nerror. Almost for every case the analytically predicted\nvalues, i.e. s≈k(foruΘ) ands=k\n2(foruω), lie within\nthe error range of the numerically found number. For\nincreasing resolution the error gets smaller. Note that\nthe large error is caused by the oscillations of the modes.\nThe decay rates of the modes agree much better with the\nanalytically predicted values than our conservative error\nestimate suggests (see Fig. 3)\nFrom high to low frequency. To explore the low-\nfrequency regime, we keep the low amplitude A= 10−4\nand vary the frequency in the range ν= [0,10]. Analytic\nresults indicates that there is no damping for constant in\nspace modes (zero-frequency modes). The numerical ex-\nperiments show that, for decreasing values of the initial\nperturbation frequency in the range [2 ,10], the damping\nscheme remains effective with the analytic exponential\ndecay rates. The behavior is displayed in Fig. 4.5\nFrequency νAmplitude A\nk\nNo damping All modes are damped ex-\nponentially 00.40.81.21.62.0\n0 10 20 30 40 50\nr[a.u.]χ(0)\nFIG. 1: (Left) Parameter space for the constraint violation of flat space-time. The violation can be tuned by its amplitud eA\nand frequency ν. Analytically known are the low-frequency corner, where th e violation is not damped and the high-frequency,\nlow amplitude-corner where all modes are damped exponentia lly. (Right) Shape of the constraint-violating initial dat a. A\nGaussian curve with amplitude Aand full-width half-maximum bis modulated with the frequency ν/b. The parameter νtells\nhow many oscillations are within b. The figure shows the constraint violation in χatt= 0 with A= 1,b= 10 and ν= 5.\n−24−20−16−12−8\n0 25 50 75ln(PSD[ ruΘ])\nTime [a.u.]/Bulletk= 0.00\n/Bulletk= 0.25\n/Bulletk= 0.50\n/Bulletk= 0.75\n/Bulletk= 1.00\n−24−20−16−12−8\n0 25 50 75ln(PSD[ ruω])\nTime [a.u.]/Bulletk= 0.00\n/Bulletk= 0.25\n/Bulletk= 0.50\n/Bulletk= 0.75\n/Bulletk= 1.00\nFIG. 2: Behavior of the eigenmodes uΘ(left) and uω(right) for high-frequency ν= 10, low-amplitude A= 10−4constraint\nviolation. The modes are extracted in the Fourier space at ¯ ν= 1. For no damping ( k= 0) the modes are stay constant. For a\nk >0 the modes are damped exponentially with different damping r ates.\nThe transition from exponential damping to no damp-\ning happens after the second octave ν∈[0,2], Fig. 5.\nAs demonstrated by the plot the transition is smooth\nand quite rapid. The experimental fact, observed here,\nthat constraint -violating modes of “almost all” nonzero\nfrequencies are killed by the damping scheme can be im-\nportant in numerical relativity simulation.\nFrom low to high amplitude. Increasing the ampli-\ntude of the perturbation, i.e. moving from a perturbative\nregime to a fully nonlinear situation is a delicate proce-\ndure. Our results can be summarized as follows.\nHigh-amplitude perturbations, up to A≃0.1, aredamped and the damping rates unaffected. The use of\nprogressively higher amplitudes first modifies the damp-\ning rates (constraint-violating modes are less damped)\nand secondly leads to unphysical results and code fail-\nures. The maximum amplitude which can be reached\nwithout changing the damping parameter kdepends\non the frequency of the initial perturbation. For low-\nfrequency initial perturbations higher amplitudes are ef-\nfectively damped by the damping scheme than for high\nfrequencies. Furthermore, it is not true that increasing\nkgenerically allows for higher amplitudes. The depen-\ndence on the damping parameter is not monotonic, the6\n−22−20−18−16−14−12−10−8−6\n0 10 20 30 40 50/BoldMul/BoldMul\n/BoldMul\n/BoldMul/BoldMul/BoldMul\n/BoldMul\n/BoldMul/BoldMul/BoldMul\n/BoldMul\n/BoldMul/BoldMul/BoldMul\n/BoldMul\n/BoldMul/BoldMul/BoldMul\n/BoldMul\n/BoldMul/BoldMul /BoldMul\n/BoldMul\n/BoldMul/BoldMul/BoldMul\n/BoldMul/BoldMul/BoldMul\n/BoldMul\n/BoldMul /BoldMul/BoldMul\n/BoldMul\n/BoldMul/BoldMul /BoldMul\n/BoldMul\n/BoldMul/BoldMul/BoldMul\n/BoldMul\n/BoldMul/BoldMul/BoldMul\n/BoldMul\n/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul\n/BoldMul/BoldMul/BoldMul\n/BoldMul\n/BoldMul/BoldMul/BoldMul\n/BoldMul\n/BoldMul/BoldMul/BoldMul\n/BoldMul\n/BoldMul /BoldMul/BoldMul\n/BoldMul/BoldMul/BoldMul\n/BoldMul\n/BoldMul /BoldMul/BoldAdd/BoldAdd\n/BoldAdd\n/BoldAdd\n/BoldAdd/BoldAdd\n/BoldAdd\n/BoldAdd/BoldAdd/BoldAdd\n/BoldAdd\n/BoldAdd/BoldAdd/BoldAdd\n/BoldAdd\n/BoldAdd/BoldAdd/BoldAdd\n/BoldAdd\n/BoldAdd/BoldAdd /BoldAdd\n/BoldAdd\n/BoldAdd/BoldAdd/BoldAdd\n/BoldAdd\n/BoldAdd/BoldAdd/BoldAdd\n/BoldAdd\n/BoldAdd\n/BoldAdd /BoldAdd\n/BoldAdd\n/BoldAdd\n/BoldAdd/BoldAdd\n/BoldAdd\n/BoldAdd/Bullet/Bullet\n/Bullet\n/Bullet\n/Bullet/Bullet\n/Bullet\n/Bullet\n/Bullet/Bullet\n/Bullet\n/Bullet\n/Bullet/Bullet\n/Bullet\n/Bullet\n/Bullet/Bullet\n/Bullet\n/Bullet\n/Bullet/Bullet\n/Bullet\n/Bullet\n/Bullet/Bullet/Bullet\n/Bullet/SolidTriangle/SolidTriangle\n/SolidTriangle\n/SolidTriangle\n/SolidTriangle/SolidTriangle\n/SolidTriangle\n/SolidTriangle\n/SolidTriangle/SolidTriangle\n/SolidTriangle\n/SolidTriangle\n/SolidTriangle/SolidTriangle\n/SolidTriangle\n/SolidTriangle\n/SolidTriangle/SolidTriangle/SolidTriangle\n/SolidTriangle\nTime [a.u.]ln(PSD[ ruΘ])/BoldMulk= 0.25\n/BoldAddk= 0.50\n/Bulletk= 0.75\n/SolidTrianglek= 1.00\n−22−20−18−16−14−12−10−8−6\n0 20 40 60 80 100/BoldMul\n/BoldMul\n/BoldMul/BoldMul/BoldMul\n/BoldMul\n/BoldMul/BoldMul/BoldMul\n/BoldMul\n/BoldMul/BoldMul /BoldMul\n/BoldMul\n/BoldMul/BoldMul /BoldMul\n/BoldMul\n/BoldMul/BoldMul /BoldMul\n/BoldMul\n/BoldMul/BoldMul /BoldMul\n/BoldMul/BoldMul/BoldMul /BoldMul\n/BoldMul/BoldMul/BoldMul/BoldMul\n/BoldMul /BoldMul/BoldMul/BoldMul\n/BoldMul /BoldMul /BoldMul/BoldMul/BoldMul /BoldMul /BoldMul/BoldMul/BoldMul/BoldMul /BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldMul/BoldAdd\n/BoldAdd\n/BoldAdd/BoldAdd/BoldAdd\n/BoldAdd\n/BoldAdd/BoldAdd /BoldAdd\n/BoldAdd\n/BoldAdd/BoldAdd /BoldAdd\n/BoldAdd\n/BoldAdd/BoldAdd /BoldAdd\n/BoldAdd\n/BoldAdd/BoldAdd/BoldAdd\n/BoldAdd\n/BoldAdd/BoldAdd/BoldAdd\n/BoldAdd\n/BoldAdd/BoldAdd/BoldAdd\n/BoldAdd\n/BoldAdd/BoldAdd/BoldAdd\n/BoldAdd/BoldAdd /BoldAdd/BoldAdd\n/BoldAdd/BoldAdd /BoldAdd/BoldAdd\n/BoldAdd/BoldAdd/BoldAdd/BoldAdd\n/BoldAdd/BoldAdd/BoldAdd/BoldAdd\n/BoldAdd/BoldAdd/BoldAdd/BoldAdd\n/BoldAdd/BoldAdd/BoldAdd/BoldAdd\n/BoldAdd/BoldAdd/BoldAdd/BoldAdd\n/BoldAdd/BoldAdd/BoldAdd\n/BoldAdd\n/BoldAdd/BoldAdd/BoldAdd\n/BoldAdd\n/BoldAdd/BoldAdd/BoldAdd\n/BoldAdd\n/BoldAdd /BoldAdd/BoldAdd\n/BoldAdd\n/BoldAdd/BoldAdd/BoldAdd\n/BoldAdd/Bullet\n/Bullet\n/Bullet/Bullet /Bullet\n/Bullet\n/Bullet/Bullet /Bullet\n/Bullet\n/Bullet/Bullet/Bullet\n/Bullet\n/Bullet/Bullet/Bullet\n/Bullet\n/Bullet/Bullet/Bullet\n/Bullet\n/Bullet/Bullet/Bullet\n/Bullet\n/Bullet/Bullet/Bullet\n/Bullet\n/Bullet/Bullet\n/Bullet\n/Bullet\n/Bullet /Bullet\n/Bullet\n/Bullet\n/Bullet/Bullet\n/Bullet\n/Bullet\n/Bullet/Bullet\n/Bullet\n/Bullet\n/Bullet/Bullet\n/Bullet\n/Bullet\n/Bullet/Bullet\n/Bullet\n/Bullet/Bullet/SolidTriangle\n/SolidTriangle\n/SolidTriangle/SolidTriangle/SolidTriangle\n/SolidTriangle\n/SolidTriangle/SolidTriangle/SolidTriangle\n/SolidTriangle\n/SolidTriangle/SolidTriangle/SolidTriangle\n/SolidTriangle\n/SolidTriangle/SolidTriangle/SolidTriangle\n/SolidTriangle\n/SolidTriangle/SolidTriangle/SolidTriangle\n/SolidTriangle\n/SolidTriangle/SolidTriangle/SolidTriangle\n/SolidTriangle\n/SolidTriangle/SolidTriangle/SolidTriangle\n/SolidTriangle\n/SolidTriangle /SolidTriangle/SolidTriangle\n/SolidTriangle\n/SolidTriangle/SolidTriangle /SolidTriangle\n/SolidTriangle\n/SolidTriangle/SolidTriangle /SolidTriangle\n/SolidTriangle\n/SolidTriangle\nTime [a.u.]ln(PSD[ ruω])/BoldMulk= 0.25\n/BoldAddk= 0.50\n/Bulletk= 0.75\n/SolidTrianglek= 1.00\nFIG. 3: Fit of the exponential decay of the eigenmodes uΘ(left) and uω(right) for high-frequency ν= 10 and low-amplitude\nA= 10−4constraint violation. The analytically predicted decay ra tes (solid lines) lie in every case within the 68% confidence\ninterval of the fits which are represented in the figure by the s haded regions.\n−28−24−20−16−12−8\n0 25 50 75ln(PSD[ ruΘ])\nTime [a.u.]/Bulletν= 2.0\n/Bulletν= 4.0\n/Bulletν= 6.0\n/Bulletν= 8.0\n/Bulletν= 10.0\n−28−24−20−16−12−8\n0 25 50 75ln(PSD[ ruω])\nTime [a.u.]/Bulletν= 2.0\n/Bulletν= 4.0\n/Bulletν= 6.0\n/Bulletν= 8.0\n/Bulletν= 10.0\nFIG. 4: The rate of the exponential decay of the eigenmodes uΘ(left) and uω(right) for low-amplitude A= 10−4constraint\nviolations stays constant in the frequency range ν∈[2,10].\noptimal value for this problem has been experimentally\nfound to be k= 0.5.\nFigure6 showsthat, for k= 0.5andν= 10, the damp-\ning rates stay the same as in the low-amplitude case for\nan amplitude range A∈[0.0001,0.01]. Setting the am-\nplitude toA= 0.1, the damping is no longer exponential\nand, for higher values, the code gives no reasonable re-sult.\nResolution dependency. Convergence of the results\nhas been already reported in Table II and briefly dis-\ncussed; we further comment here focusing on represen-\ntative simulations with k= 0.5,ν= 10,A= 10−4\nwith varying resolutions. As shown in Fig. 7 the damp-\ning effect holds longer for higher resolutions, i.e. if the7\n−28−24−20−16−12−8\n0 25 50 75ln(PSD[ ruΘ])\nTime [a.u.]ν= 0.0. . .2.0\n/Bullet /Bullet /Bullet /Bullet /Bullet /Bullet /Bullet /Bullet /Bullet /Bullet /Bullet /Bullet /Bullet /Bullet /Bullet /Bullet /Bullet /Bullet /Bullet /Bullet\n−28−24−20−16−12−8\n0 25 50 75ln(PSD[ ruω])\nTime [a.u.]ν= 0.0. . .2.0\n/Bullet /Bullet /Bullet /Bullet /Bullet /Bullet /Bullet /Bullet /Bullet /Bullet /Bullet /Bullet /Bullet /Bullet /Bullet /Bullet /Bullet /Bullet /Bullet /Bullet\nFIG. 5: In the small frequency-range ν∈[0,2] the transition between exponential damping and no dampin g happens. The\nfigure shows for low-amplitude A= 10−4constraint violations the decay of the eigenmodes uΘ(left) and uω(right).\nTABLE II: Fits results for the decay rates for different resol u-\ntions. The parameters of the initial perturbation are ν= 10\nandA= 10−4.\nks(n= 2000) s(n= 4000) s(n= 8000) sanalytic\n0.25−0.21±0.02−0.24±0.02−0.25±0.01−0.25\n0.50−0.49±0.06−0.51±0.04−0.50±0.04−0.50\n0.75−0.74±0.09−0.78±0.08−0.77±0.05−0.75\n1.00−0.93±0.16−1.02±0.20−1.02±0.14−1.00\n0.25−0.12±0.01−0.12±0.01−0.12±0.01−0.125\n0.50−0.24±0.02−0.24±0.02−0.24±0.01−0.250\n0.75−0.35±0.04−0.35±0.04−0.36±0.02−0.375\n1.00−0.47±0.07−0.48±0.07−0.48±0.03−0.500\nfrequency of the constraint violation is better resolved,\nthen the damping scheme works more effectively. This\nsuggests that the damping scheme works as long as the\nfrequency of the perturbation is well-resolved and it is\npartially expected since it acts at the continuum level.\nThe sameeffect could havebeen alreadyanticipatedfrom\nFig.4, which referstoa singleresolutionbut different fre-\nquencies which are resolved differently on the grid.\nVery high frequencies, grid modes and dependency on\nartificial dissipation. The effect of the damping scheme\non frequencies comparable to or of the order of the nu-\nmerical grid (grid modes) is finally studied [31] . For\nthese tests we use k= 0.5 and vary the frequency\nν∈[10,30] (Note the grid spacing is h= 0.025.). As\ndemonstrated in Fig. 8, if the frequency νof the pertur-\nbation is increased further to a regime where the signal\nis not well-resolved, the damping becomes progressively\nless effective and deviates from the analytic expectation.\nMoreimportantly,theamountofartificialdissipation[19]−24−20−16−12−8−40\n0 25 50 75ln(PSD[ ruω])\nTime [a.u.]/BulletA= 0.0001\n/BulletA= 0.001\n/BulletA= 0.01\n/BulletA= 0.1\nFIG. 6: Decay of the eigenmode uωfor high-frequency con-\nstraint violation with increasing amplitude A. Between A=\n10−4andA= 10−2the damping rate does not change. For\nvery high-amplitude A= 10−1the damping of the eigenmode\nis not exponential anymore and the code does not give phys-\nically reasonable results.\nplays a significant role. The use of the artificial dissipa-\ntionfiltersoutgridmodesandgenericallyattenuateshigh\nfrequencies which are aliasedto lowerones. Figure 9 (left\npanel) shows that the use of different amounts of dissi-\npation,σ, quantitatively changes the decay rate of the\neigenmodes. The higher σis, the higher the damping\nrate. However one cannot expect to use arbitrary large\nvalues ofσ, so a balance between kandσand the prop-\nerty of the solution have to be studied case by case by8\n−24−20−16−12−8\n0 25 50 75ln(PSD[ ruΘ])\nTime [a.u.]/Bulletn= 2000\n/Bulletn= 4000\n/Bulletn= 8000\nFIG. 7: The damping effect depends on the resolution of\nthe initial constraint violation. The figure shows for high-\nfrequency ν= 10, low-amplitude A= 10−4the damping of\nthe eigenmode uΘfor different resolutions. For high resolu-\ntion the frequency is well-resolved and therefore the dampi ng\neffects lasts longer than for less well-resolved case.\n−24−20−16−12−8\n0 25 50 75ln(PSD[ ruΘ])\nTime [a.u.]/Bulletν= 10\n/Bulletν= 20\n/Bulletν= 30\nFIG. 8: Increasing the frequency of the initial constraint\nviolation with low-amplitude A= 10−4even further to very\nhigh frequencies. At these high frequencies the constraint\nviolation get less resolved which weakens the effect of the\ndamping.\nperforming convergence tests. In our test case we found\nthatσ= 0.05 roughly reproduce the analytic damping\nrates.\nAs a finaltest weevolveinitial datacontainingrandomnoise,\nχ(0,r) = 1+Aexp/parenleftbigg\n−r2\n2b2/parenrightbigg\nrand([−1,1]).(39)\nNote that, in principle, random noise differs from the\nhigh-frequency perturbation used before because it has a\nflatspectrum. Figure9(rightpanel)showsthattheeffec-\ntiveness of the damping scheme depends again strongly\non the value of σin the artificial dissipation operator\nused. However, differently from the previous case, in this\ncase we were not able to recover the analytic damping\nrates\nB. Punctures and compact star experiments\nSingle puncture. In order to test the damping ef-\nfect on strong-field evolution, we evolve puncture ini-\ntial data [20]. While the initial data are constraint-\nsatisfying, a constraint-violating wave leaving the hole\nand propagating outside is observed in numerical simula-\ntions. This feature is generic and not related to the use\nof Z4, but observed also in BSSNOK evolutions, e.g. [21].\nNote however that the constraint violation is converging\naway with resolution, thus not a continuum feature (the\nconstraint subsystem in Z4c does not have superlumi-\nnal speeds). Figure 10 (left panel) shows a snapshot of\nthe constraint violation leaving the horizon. During the\nevolution the biggest violation is instead found at the\npuncture, where the solution is not smooth. A priori,\nthe frequency of the initial constraint-violating wave, as\nwell asthe later violationat the puncture, cannotbe esti-\nmated, whereastheiramplitude is expectedto be “small”\n(in the sense that it is converging away). From Fig. 10 it\nis evident that the frequency of the constraint-violating\nwave spans a certain range of frequencies; in terms of\nthe length scale given by the mass, M= 1, we mainly\nobserved violation at a peak frequency ¯ ν∼0.5. It can\nbe considered as “high” and we expect it to be damped\nsince it is within the first octave.\nThe numerical evolutions performed with different val-\nues ofkshow that the use of the damping scheme gener-\nically introduces a certain dynamics in the constraints,\nwhose values in space oscillates in time around a small\nvalue close to zero. The evolution of the L2 norm of the\nconstraint monitor\nC=/radicalbig\nH2+MiMi+Θ2+ZiZi (40)\nisreportedinFig.11(left panels)fordifferentvaluesof k.\nIn these tests artificial dissipation is used with σ= 0.007.\nIn all the cases the norm at early times is dominated by\na violation inside the horizon during the gauge adjust-\nment which leads to the trumpet solution [22–27]. The\ninitial constraint violation wave is also propagated out\nduring this phase, and eventually damped depending on\nthe value of k. The bottom left panel shows the norm\noutside the horizon at early times and highlights the ef-\nfect of the damping scheme for several values of k. At9\n−24−20−16−12−8\n0 25 50 75ln(PSD[ ruΘ])\nTime [a.u.]/Bulletσ= 0\n/Bulletσ= 0.0007\n/Bulletσ= 0.005\n−24−20−16−12−8\n0 25 50 75ln(PSD[ ruω])\nTime [a.u.]/Bulletσ= 0\n/Bulletσ= 0.007\n/Bulletσ= 0.05\nFIG. 9: For initial constraint violations with low amplitud e and very high frequency, the artificial dissipation starts to have\nan effect. The high-frequency modes are shifted by the artific ial dissipation to lower frequencies which are then damped. The\nfigure shows the dependence of the decay of the eigenmode uΘon the artificial dissipation parameter σusing high-frequency\nν= 30 (left) and random noise constraint violation initial da ta (right).\n−0.10−0.08−0.06−0.04−0.0200.020.04\n0 10 20 30\nr[M]H×1000/Bullett= 30/Bullett= 15\n−1.0−0.8−0.6−0.4−0.200.20.4\n0 10 20 30\nr[M]H×1000/Bullett= 30/Bullett= 15\nFIG. 10: The puncture and the stuffed puncture initial data vi olate the constraints. This constraint violations leave th e black\nhole horizon. The figure shows the Hamiltonian constraint Hat timet= 15Mandt= 30M. (Left) Puncture initial data,\n(right) Stuffed puncture initial data.\nlater times the amplitude of the oscillations in the con-\nstraints amplifies around t∼1.25×103Mas shown in\nthe top left panel. Depending on the value of k, the am-\nplification is observed to saturate and damp ( k <0.2)\nor to keep on growing, contaminating the numerical so-\nlution (k >0.2). In the latter case the code eventually\nfails because the boundary conditions implemented [11]\ncan not sustain such a large violation.\nTo assess the relative importance of boundary con-\nditions and the constraint damping scheme in our nu-\nmerical experiments, we performed long evolutions of\na single puncture with and without both the damping\nscheme, and either constraint-preserving [11] or Som-\nmerfeld boundary conditions. The outer boundary wasplaced at 50 M. The results are presented in Fig. 12.\nWe find that the use of constraint-preserving boundary\nconditions is more important than that of the damping\nscheme in avoiding violations. Although the use of the\ndamping scheme with k= 0.02 reduces the violation by\na factor of 9 when Sommerfeld conditions are used. This\ntest is not expected to be representative of more general\nscenarios in which the outer boundary is placed farther\nout with the same resolution, since then, experimentally\nwe find that a smaller constraint violation interacting\nwith the Sommerfeld condition results in smaller reflec-\ntions. On the other hand, since the constraint-preserving\nconditions are found to converge numerically, and the\nSommerfeld conditions do not, it is expected that at10\n00.10.20.3\n0 0 .5 1 .0 1 .5 2 .0 2 .5/Bullet\n/Bullet\n/Bullet\n/Bullet\n/Bullet\n/Bulletk= 0.00\nk= 0.04\nk= 0.08\nk= 0.10\nk= 0.20\nk= 0.30/bardblC/bardbl2×10−2\nTime/1000 [M]00.10.20.3\n0 0 .5 1 .0 1 .5 2 .0 2 .5/Bullet\n/Bullet\n/Bullet\n/Bullet\n/Bullet\n/Bulletk= 0.00\nk= 0.04\nk= 0.08\nk= 0.10\nk= 0.20\nk= 0.30/bardblC/bardbl2×10−2\nTime/1000 [M]\n00.10.20.3\n0.0 0.02 0.04 0.06 0.08/Bullet\n/Bullet\n/Bullet\n/Bullet\n/Bullet\n/Bulletk= 0.00\nk= 0.04\nk= 0.08\nk= 0.10\nk= 0.20\nk= 0.30/bardblC/bardbl2×10−3\nTime/1000 [M]00.10.20.3\n0.0 0 .02 0 .04 0 .06 0 .08/Bullet\n/Bullet\n/Bullet\n/Bullet\n/Bullet\n/Bulletk= 0.00\nk= 0.04\nk= 0.08\nk= 0.10\nk= 0.20\nk= 0.30/bardblC/bardbl2×10−2\nTime/1000 [M]\nFIG. 11: Time evolution of the norm of the constraint monitor . For the figures on the left puncture initial data was used, fo r\nthe figures on the right stuffed puncture initial data. The lon g time behavior of both cases is the same which can be seen in th e\npictures on the top. The difference between the two initial da ta can be seen in the two figures on the bottom which show the\nnorm of the constraint monitor outside the horizon for early time. The stuffing introduces a big constraint violation comp ared\nto the normal puncture data. This violation is damped away. T he longtime behavior depends on the value of the damping\nparameter while for damping parameter k <0.3 the damping scheme does not cause problems in the evolution , damping\nparameter k≥0.3 introduce dynamics to the system which leads to increasing constraint violation and a crash of the code.\nsome resolution the constraint violation induced by the\nSommerfeld conditions will become dominant, even if the\nouter boundary is placed far out. This has been recently\npointed out for the case of 3d matter simulations and of\nBSSNOK in [28].\nStuffed puncture. A second series of tests performed\nis the evolution of puncture initial data stuffed in the\nblack-hole interior [21, 29, 30]. The hole has been stuffed\ninside the horizon rh=M/2 atrex= 0.475 using a\nfourth-order polynomial in the conformal factor\nψ(0,r) = 2.97368−5.83175/parenleftBigr\nM/parenrightBig2\n+7.75413/parenleftBigr\nM/parenrightBig4\n.\n(41)\nThe polynomial matches the puncture data at rex\nup to the second derivatives. The initial data are\nclearly constraint-violating and also show the outgoing\nconstraint-violation wave Fig. 10 (right panel).The evo-\nlution ofthe constraintmonitorfor different kis reportedin the right panels of Fig. 11. Only quantitative differ-\nences with respect to the puncture case are observed. As\ndemonstrated in the right bottom panel of Fig. 11 (note\nthe difference in the scalewith respect the left panel), the\ndamping scheme is again effective in reducing the outgo-\ning constraint violation during the initial adjustment. At\nlate times the situation is completely analogous to the\npuncture evolution and we do not repeat the description,\nsee top right panel.\nStable star. As a final test, we present a study of\nthe influence of the damping scheme on the evolution\nof a stable equilibrium model of a compact star of mass\nM= 1.4M⊙described by the ideal gas equation of state.\nInitial data are the same as those employed in previous\nworks [10, 11] and constraint-satisfying. Furthermore\nthey provide the exact solution for the evolution prob-\nlem since the system is static. At the typical resolutions\nemployed (and without damping, k= 0) stable evolu-11\n0246810121416\n0 2 4 6 8 10 12/Bullet/Bullet/Bullet/Bullet\nSFBC k= 0.00SFBC k= 0.02CPBC k= 0.00CPBC k= 0.02/bardblC/bardbl2×10−2\nTime/1000 [M]\nFIG. 12: Long-term evolution of a puncture with the Z4c\nformulation. Different boundary conditions were tested wit h\nand without damping (with k= 0.02). Sommerfeld boundary\nconditions lead to a high constraint violation, which is effe c-\ntively suppressed by the damping scheme. Using Sommerfeld,\nthe constraint violation is roughly a factor of 9 lower with a nd\nwithout damping at late times. Using constraint-preservin g\nboundary conditions, without damping, leads to a late-term\nconstraint violation which is 20 times lower than that of un-\ndamped Sommerfeld. The damping is less important in the\ncase of constraint-preserving boundary conditions. Using the\nstandard damping value k= 0.02, the improvement is by only\na factor of 2.\ntions are obtained for about 10 ms(t∼1400M). Trun-\ncation errors trigger radial oscillation in the star, which\nare small amplitude, low frequency in space constraint-\nviolating modes, ν∼1/(2R) whereRis the star coordi-\nnate radius.\nIn previous investigations on stable stars, which con-\nsidered only one value of the damping parameter, we\nfound that the constraint damping terms have a negligi-\nble effect on the dynamics, while constraint propagation\nmadealargedifferencewiththeequivalentBSSNOKsim-\nulations [10]. The new results discussed here confirm the\nprevious ones for the specific kconsidered, but they also\nshow that for certain values of the damping parameter\nthe constraint damping terms are not beneficial in the\nlong-term evolutions.\nFigure 13 shows the evolution of the central rest-mass\ndensity (left) and of the L2 norm of the constraint moni-\ntor (right) fordifferent values of kemployed in the damp-\ning scheme. For k≥0.3 the constraint damping am-\nplifies the radial oscillations and drives the star to col-\nlapse. Large constraint violations indicate the departure\nfrom the constraint-satisfying solution space (for clarity\nthey are not shown in the right panel). Smaller values of\nk≤0.06produce instead an effective constraint damping\nat early times (see right panel Fig. 13). In the long-term\nhowever, the evolution without damping scheme ( k= 0)\nis always preferable to the evolutions with k≥0.03. In\nthe latter cases a long-term growth is observed, simi-lar to that seen in the puncture simulations. The value\nk= 0.02 leads to a constraint damped evolution, but its\neffect is almost negligible. More importantly, this choice\nis not robust in other tests performed. In both simula-\ntions employing a different equation of state, specifically\na polytrope which produces different truncation errorsat\nthe surface of the star, and in the migration test of [10]\nwe werenot able to identify a value of kwhich leads toan\nefficient constraint damping. We presented here in detail\nonly the test with the most varied outcome.\nWhile the static stable star is a delicate test (every\nsmall perturbation causes a departure from the Einstein\nsolution), it providesa specific relevant example in which\nthe constraintdampingfails for alargechoiceofdamping\nparameters. This indicates that the use of the constraint\ndamping scheme without specific investigations is poten-\ntially dangerous.\nIV. CONCLUSION\nIn order to expand the body of evidence that a con-\nformal decomposition of the Z4 formulation of general\nrelativity [2] may be a useful tool for numerical relativity\nwe have presented a detailed study of the effect of the\nconstraint damping scheme of Gundlach et al. [8].\nWe have attempted to answer three questions, which\nwe address here specifically:\n(i).Under what conditions can the theoretically pre-\ndicted damping rates be recovered in the numerical ap-\nproximation? By studying the evolution of parametrized\nconstraint violating-perturbations on top of flat space,\nwe first found that the predicted damping rates of [8]\nare recovered for well-resolved high-frequency constraint\nviolations. Varying the frequency of the constraint vi-\nolation, we found that the analytically predicted expo-\nnential decay is maintained over a large, three-octave,\nrange. The cut-off in the effectiveness of the scheme oc-\ncurs over a small range at low frequencies. On grid noise,\nunsurprisingly, we find that the predicted damping rates\nare not recovered, although the combination of damping\nand artificial dissipation does help to suppress constraint\nviolations. The intuitive explanation for this is that ar-\ntificial dissipation aliases the grid noise to lower frequen-\ncies which are well-resolved, and on which the damping\nscheme is effective. Finally we increased the amplitude\nof the constraint violation. At amplitudes above A≃0.1\nthe damping scheme becomes increasingly less effective,\nafter which numerical integration is often not possible,\neither with or without constraint damping.\n(ii).How effective is the damping scheme in astrophys-\nically relevant spacetimes? For this part of the inves-\ntigation we began by evolving a single puncture black\nhole. We find that the constraint damping scheme sup-\npresses constraint-violating numerical error leaving the\nblack hole horizon, but that it generically introduces a\ndynamical behavior to the constraints. The suppression\nof the violation leaving the horizon is furthermore not12\n−0.200.20.40.6\n0 0 .5 1 .0(ρc(t)/ρc(0)−1)×1000\nTime/1000 [M]k= 0.0...1.0\n/Bullet /Bullet /Bullet /Bullet /Bullet /Bullet /Bullet /Bullet /Bullet /Bullet\n−7.2−7.0−6.8−6.6−6.4\n0 0 .5 1 .0log/bardblC/bardbl2\nTime/1000 [M]k= 0.00\nk= 0.02\nk= 0.05\nk= 0.06/Bullet\n/Bullet\n/Bullet\n/Bullet\nFIG. 13: (Left) Time evolution of central rest-mass density for a stable star obtained using different values of k. (Right) Time\nevolution of the constraint monitor corresponding to a subs et of the evolutions of the right panel.\ndramatic. For reasonable values of the damping param-\neter a factor of about 2 or 3 is gained in the norm of\nthe constraint violation. If the damping parameters are\nchosen too large, the dynamical behavior induced by the\nschemecausesalargeconstraintviolationtohit the outer\nboundary, which in our tests was placed at 50 M, even-\ntually causing a code failure, which we think it may be\npossible to avoid by including constraint damping terms\nin the constraint-preservingboundary conditions [11] ap-\npropriately. Since mixed puncture-black-hole neutron-\nstar initial data are not readily available, “black-hole\nstuffing” has been proposed [21, 29, 30]. Therefore to\ninvestigate the likely effect of the constraint damping\nscheme in mixed binary evolutions, we evolved a sin-\ngle puncture with a constraint-violating interior. Here\nwe find qualitatively the same behavior as in the single\npuncture evolutions, the only difference being that the\nsize of the constraint-violating numerical error leaving\nthe black-hole is larger. Finally on the question of astro-\nphysically relevant spacetimes evolutions of a static star\nwere performed. Here we find that using the damping\nscheme is generically of minor benefit and can cause an\nunphysical collapse.\n(iii).In practical applications what are reasonable val-\nues for the constraint damping coefficients? Our flat-\nspace tests demonstrate that higher values of the damp-\ning parameters are preferable, because then faster rates\nofexponential damping are achieved. On the otherhand,\nsince our evolutionsofcompact objects suffer from severe\nproblems when the damping parameters are chosen too\nlarge, we suggest that the damping parameter are chosen\nin the range k∈[0,0.1] for puncture evolutions while for\nmatter evolution the safest option to use k= 0 unless\nspecific damping tests are performed.In summary, at least for the spherical symmetric sys-\ntems studied within this work, the following statements\nabout the constraintdamping scheme can be made: Con-\nsidering vacuum spacetimes, the damping scheme may\nbe, forcarefullychosendampingparameters,ausefultool\nfor suppressing constraint violations. This is certainly\ntrue if there are features in the numerical setup which\ncause large constraint violations, for example, Sommer-\nfeld boundary conditions or constraint- violating initial\ndata. Ifnosuchfeaturesarepresent,thedampingscheme\nis not essential and can furthermore affect the physics of\nthesystemifthedampingparametersaretakentoolarge.\nIn the evolution of a static compact star our numeri-\ncal evidence indicates that the damping scheme some-\ntimes leads to a slight decrease of constraint violation.\nOn the other hand the damping scheme, in combina-\ntion with some numerical setups, causes growth of the\nconstraints; in the special cases we have considered the\ndamping scheme is of marginal use.\nAcknowledgments\nThe authors would like to thank Bernd Br¨ ugmann and\nMilton Ruiz for helpful discussions. We also thank the\nauthors of [14] for valuable comments on the manuscript,\nand, in particular, Carlos Palenzuela for his query on our\nneutron star results. 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Phys.Rev., D76:081503, 2007,\n0707.3101.\n[31] We use now the term “high” not related to the scale\n¯ν/greaterorsimilar1 of the continuum problem, i.e. not for ν∼b, but\nrelated to the characteristic frequency introduced by the\nnumerical grid, i.e. ν∼1/h."    },    {        "title": "2210.15697v3.Sharp_polynomial_decay_for_polynomially_singular_damping_on_the_torus.pdf",        "content": "SHARP POLYNOMIAL DECAY FOR POLYNOMIALLY SINGULAR\nDAMPING ON THE TORUS\nPERRY KLEINHENZ AND RUOYU P. T. WANG\nAbstract. We study energy decay rates for the damped wave equation with un-\nbounded damping, without the geometric control condition. Our main decay result\nis sharp polynomial energy decay for polynomially controlled singular damping on\nthe torus. We also prove that for normally Lp-damping on compact manifolds, the\nSchr odinger observability gives p-dependent polynomial decay, and \fnite time extinc-\ntion cannot occur. We show that polynomially controlled singular damping on the\ncircle gives exponential decay.\n1.Introduction\n1.1.Introduction. In this paper we study the damped wave equation. Let ( M;g)\nbe a compact Riemannian manifold without boundary and let Wbe a non-negative\nmeasurable function on M. Then the viscous damped wave equation is\n(\n(@2\nt\u0000\u0001 +W@t)u= 0;inM\n(u;@tu)jt=0= (u0;u1)2D\u001aH1(M)\u0002L2(M);(1.1)\nwhere the set of admissible initial data Dis speci\fed later in De\fnition 1.2. The\nprimary object of study in this paper is the energy\nE(u;t) =1\n2Z\njruj2+j@tuj2dx:\nWhenWis continuous, it is classical that uniform stabilization is equivalent to geo-\nmetric control by the positive set of the damping. That is, there exists r(t)!0 as\nt!1 such that\nE(u;t)\u0014Cr(t)E(u;0);\nif and only if there exists L, such that all geodesics of length at least Lintersect\nfW > 0g. In this case, due to the semigroup property of solutions, one can take\nr(t) = exp(\u0000ct), for some c>0.\nWhen the geometric control condition is not satis\fed, E(u;0) must be replaced on\nthe right hand side. Decay rates are of the form\nE(u;t)1=2\u0014Cr(t) (ku0kH2+ku1kH1): (1.2)\nKeywords. damped waves, singular damping, backward uniqueness, Schr odinger observability.\n1arXiv:2210.15697v3  [math.AP]  16 Apr 20232 PERRY KLEINHENZ AND RUOYU P. T. WANG\nFurthermore, the optimal r(t) depends not only on the geometry of MandfW > 0g,\nbut also on properties of Wnear@fW > 0g. In general, for bounded Wthe more\nsingularWis near@fW > 0g, the slower the sharp energy decay rate. Since unbounded\ndamping allows for even more singular behavior it is natural to see if this relationship\ncontinues.\nFor much of this paper we focus on M=T2, which has polynomial rates r(t) =hti\u0000\u000b\nwith\u000b > 0. In particular, when W2L1(M), is positive on a positive measure set,\nbut suppWdoes not satisfy the geometric control condition, [ALN14] showed that\n(1.2) holds with r(t) =hti\u00001\n2and cannot hold with r(t) =hti\u00001\u0000\". Although there are\ndamping functions which saturate the hti\u00001rate, there are no examples of damping for\nthat the waves must decay at hti\u00001\n2. The closest rate was proved by Nonnenmacher in\nan appendix to [ALN14]: for damping equal to the characteristic function of a strip,\nthere are solutions decaying no faster than hti\u00002\n3. This leaves a mysterious gap between\nhti\u00001\n2andhti\u00002\n3where no known bounded damping gives such a rate.\nIn this paper, we aim to address two open problems formulated in [ALN14]:\nQuestion 1. Is the a priori upper bound hti\u00001\n2for the rates optimal?\nQuestion 2. How is the vanishing rate of Wrelated to the energy decay rate?\nIn order to do this, we no longer assume W2L1and consider unbounded damping.\n1.2.Main results. Now we list our main results in this paper. First we show that on\nM=T2, for\f2(\u00001;1);y-invariant damping functions vanishing like d(\u0001;fW= 0g)\f\nnear@fW= 0gproduce sharphti\u0000\f+2\n\f+3-decay. This interpolates the decay rates between\nhti\u00001\n2andhti\u00001. The possibly unbounded polynomially controlled damping functions\ninX\f\n\u0012and the spaceDof admissible initial data are speci\fed in De\fnitions 1.1 and\n1.2 below.\nTheorem 1 (Polynomial decay on the torus) .LetM=T2and \fxn2Nand\f2\n(\u00001;1). SupposeW(x;y) =Pn\nj=1Wj(x), withWj2X\f\n\u0012j. Then there exists C > 0\nsuch that for all t>0,\nE(u;t)1\n2\u0014Chti\u0000\f+2\n\f+3(kru0kL2+ku1kH1+k\u0000\u0001u0+Wu 1kL2);\nfor the solution uof(1.1) with respect to any initial data (u0;u1)2D.\nTheorem 2 (Sharpness of the decay rates) .The rates obtained in Theorem 1 are sharp\nfor\f2(\u00001;1). Indeed, let W(x;y) = 1jxj\u0015\u0019\n2\u0000\n(jxj\u0000\u0019\n2)\f+ 1\u0001\nfor\f2(\u00001;0). Then\nfor allC;\u000f> 0, there exists t0>0and initial data (u0;u1)2D such that\nE(u;t0)1\n2>Cht0i\u0000\f+2\n\f+3\u0000\u000f(kru0kL2+ku1kH1+k\u0000\u0001u0+Wu 1kL2);\nfor the solution uof(1.1) with respect to the initial data (u0;u1).SHARP POLYNOMIAL DECAY FOR SINGULAR DAMPING 3\n−−\n−−σ −σ π −πxVβ\n0\n(a)V\fonS1\n(b)W(x)2X\f\n\u0019onT2\nFigure 1. Polynomial control functions V\fand typical damping W.\nTheorems 1 and 2 address Question 2: we show that damping of more/less singular\nbehaviour than a characteristic function gives slower/faster decay than hti\u00002\n3, and\ngive an explicit relation between the decay rates and the vanishing rates of W: when\n\f!\u0000 1+, the decay rate approaches hti\u00001\n2\u0000, the upper bound for the energy decay\nobtained in [ALN14]; when \f!1 , the decay rate approaches hti\u00001+, the lower bound.\nThis also \flls up the mysterious gap between the upper bound hti\u00001\n2they found and\nthe upper boundhti\u00002\n3they could produce with bounded damping. However, there are\nsome complications, as we discuss in Theorem 3. Note for \f2[0;1), the decay rates\nin Theorem 1 were already shown in [DK20, Sta17] and their corresponding sharpness\nresults in Theorem 2 were shown in [Kle19, ALN14], so we here only need to address\nthe case\f2(\u00001;0). Now we de\fne the terminologies:\nDe\fnition 1.1 (Polynomially controlled damping) .Fix\u00122S1. Parametrize S1by\n[\u0000\u0019;\u0019)with ends identi\fed periodically and \u0012= 0. We de\fne, X\f\n\u0012, the space of poly-\nnomially controlled functions on S1as the space of measurable functions fonS1such\nthat there are C0;\u001b> 0such thatC\u00001\n0V\f(x)\u0014f(x)\u0014C0V\f(x), where\nV\f(x) =(\n0; x 2[\u0000\u001b;\u001b];\n(jxj\u0000\u001b)\f;jxj2(\u001b;\u0019):(1.3)\nWe will mainly consider damping W(x;y) that is a sum of V\f(x) with\f2(\u00001;0)\nin this paper, so Wlocally blows up like the polynomial ( jxj\u0000\u001b)\fnear\u0006\u001b: see Figure\n1. Because of the unboundedness of the damping Wwe need to adjust the space our\ninitial data is in.\nDe\fnition 1.2 (Admissible initial data) .LetD=f(u0;u1)2H1(M)\u0002H1(M) :\n\u0000\u0001u0+Wu 12L2(M)gequipped with the norm\nk(u0;u1)k2\nD=ku0k2\nH1+ku1k2\nH1+k\u0000\u0001u0+Wu 1k2\nL2:4 PERRY KLEINHENZ AND RUOYU P. T. WANG\nRemark 1.3 (Relation between DandH2\u0002H1).We remark that when \f2(\u00001\n2;1),\nD=H2(M)\u0002H1(M) and theD-norm is equivalent to that of H2\u0002H1. When\n\f2(\u00001;\u00001\n2],\nH3\n2\u0000\u0002H1\u001bD)H2\u0002fu12H1:u1= 0 onW\u00001(1)g:\nFurthermore, in this case, DandH2\u0002H1are mutually not a subset to each other.\nWe now move on to results in the setting of normally Lp-damping on compact mani-\nfolds. Normally Lp-damping are those damping functions that look like an Lp-function\nalong the normal direction near a hypersurface, given in De\fnition 2.1. Note that for\n\f2(\u00001;0), polynomially controlled damping in X\f\n\u0012, are normally L\u00001\n\f\u0000and for\f\u00150\nthey are (normally) L1.\nThe natural question to ask, is if the a priori upper bound hti\u00001\n2, obtained via\nSchr odinger observability, still holds when W =2L1? The answer is negative: although\nwe are able to obtain an a priori decay rate for such Wvia Schr odinger observability,\nunfortunately it is slower.\nTheorem 3 (Schr odinger observability gives polynomial decay) .LetWbe normally\nLpforp2(1;1). Assume the Schr odinger equation is exactly observable from an\nopen set \n, and there exists \u000f >0;such thatW\u0015\u000falmost everywhere on an open\nneighbourhood of \n. Then\nE(u;t)1\n2\u0014Chti\u00001\n2+1\np(kru0kL2+ku1kH1+k\u0000\u0001u0+Wu 1kL2);\nfor the solution uof(1.1) with respect to any initial data (u0;u1)2D. Whenp= 1,\nE1\n2\u0014Chti\u00001=(3+). Whenp=1,E1\n2\u0014Chti\u00001=2.\nWhenp=1, we can indeed have the hti\u00001=2-decay: this bounded case was known\nin [ALN14]. Together with the Schr odinger observability results in [BZ12, AM14], we\nimmediately have:\nCorollary 1.4. SupposeM=Tdford\u00151. LetWbe normally Lpforp2(1;1), and\nassume there exists \u000f>0, such that W\u0015\u000falmost everywhere on an open set. Then\nE(u;t)1\n2\u0014Chti\u00001\n2+1\np(kru0kL2+ku1kH1+k\u0000\u0001u0+Wu 1kL2);\nfor the solution uof(1.1) with respect to any initial data (u0;u1)2D. Whenp= 1,\nE1\n2\u0014Chti\u00001=(3+). Whenp=1,E1\n2\u0014Chti\u00001=2.\nTheorems 1, 2 and 3 give both positive and negative answers to Question 1: on\none hand, in Theorem 2 for any polynomial rates between hti\u00001\n2andhti\u00002\n3, we found\nsingular damping functions that are sharply stabilized at that rate. On the other hand,\nTheorem 3 implies that for normally Lp-damping, the upper bound obtained via the\nSchr odinger observability can be as weak as hti\u00001\n3+asp!1+, instead ofhti\u00001\n2. ThusSHARP POLYNOMIAL DECAY FOR SINGULAR DAMPING 5\nwhile we \flled the gap between hti\u00001\n2andhti\u00002\n3by making Wmore singular, we also\ncreate a new gap between hti\u00001\n3andhti\u00001\n2.\nA peculiar feature to note is that observability of the Schr odinger equation does not\ndepend on the singularity of W, but the decay rate produced does. Such dependency\nis not observed in the case of bounded damping. We point out that the rates we obtain\nin Theorem 3 are better than the rates obtainable with [CPS+19]: see Remark 2.29.\nWe now move on to address some open problems concerning the \fnite time extinction\nphenomenon concerning singular damping.\nTheorem 4 (Backward uniqueness) .LetWbe normally Lpforp2(1;1]. Then the\ndamped wave semigroup etAis backward uniqueness. That is:\n(1) Ifu(t) = 0;@tu(t) = 0 at somet>0, thenu0=u1= 0.\n(2) IfE(u;t) = 0 at somet>0, thenE(u;0) = 0 .\nThis theorem states that there cannot be \fnite-time extinction of solutions or energy\nwhen the damping vanishes like x\ffor\f2(\u00001;1]. This is in contrast to the limit case\n\f=\u00001, studied in [FHS20, CC01] where particular setups with damping of the form\n2x\u00001were found and all solutions go extinct in \fnite time. Such \fnite-time extinction\nphenomenons are of note as they are rarely observed for linear equations.\nOnM=S1, our last theorem states that exponential decay occurs when Wis a\n\fnite sum of polynomially controlled functions and bounded functions.\nTheorem 5 (Exponential decay on the circle) .SupposeM=S1and for 1\u0014j\u0014n;\nWj2L1orWj2X\fj\n\u0012jwith\fj2(\u00001;0). IfW=Pn\nj=1Wj, then there exist C;c> 0\nsuch that\nE(u;t)\u0014Ce\u0000ctE(u;0);\nfor the solution uof(1.1) with respect to any initial data (u0;u1)2H1\u0002L2.\nThis theorem shows that, in one dimension, the geometric control implies exponential\ndecay even if there are some singularities, as long as the singularities are not too large.\nRemark 1.5. We heuristically interpret that the singularity of Wnear@fW= 0g\nprevents the propagation of low-frequency modes into fW > 0g. The singularity\nre\rects energy back into fW= 0gas well as transmitting some into fW > 0g;and\nthe greater the singularity the more energy it re\rects. In the setting of Theorems\n1 and 2, the low-frequency sideways propagation of vertically concentrated modes in\nfW= 0ghas a\f-dependent transmission rate into fW > 0g. This is why the energy\ndecay rate depends on \f. On the other hand, the singularity does not a\u000bect high-\nfrequency propagation much. This is best seen in the setting of Theorem 5, where all\nhigh-frequency modes penetrate into fW > 0gand are damped exponentially.6 PERRY KLEINHENZ AND RUOYU P. T. WANG\nRemark 1.6. (1) Theorems 1 and 5 hold for the limit case \f=\u00001, when the\nevolutional semigroup is restricted to f(u;v)2H1\u0002L2:ujW\u00001(1)= 0gwith its\ngenerator de\fned on f(u;v)2H2\u0002H1:ujW\u00001(1)=vjW\u00001(1)= 0g, without any\nmajor changes to the proofs. To keep this paper concise, we choose not to pursue this\ndirection.\n(2) Note, the proof of Theorem 5 su\u000eces for Wj= 1 [\u0000\u0019;\u0000\u001b](x)Vjor 1 [\u001b;\u0019)(x)Vjfor\nVj2X\fj\n\u0012j. However for ease of notation we only work with the symmetric de\fnition of\npolynomial control. See Remark 3.7 for further details.\n1.3.Literature review. The equivalence of uniform stabilization and geometric con-\ntrol for continuous damping functions was proved by Ralston [Ral69], and Rauch and\nTaylor [RT74] (see also [BLR88], [BLR92] and [BG97], where Mis also allowed to have\na boundary). For \fner results concerning discontinuous damping functions, see Burq\nand G\u0013 erard [BG20].\nDecay rates of the form (1.2) go back to Lebeau [Leb93]. If we assume only that\nW2C(M) is non-negative and not identically 0, then the best general result is\nthatr(t) = 1=log(2 +t) in (1.2) [Bur98],[Leb93]. Furthermore, this is optimal on\nspheres and some other surfaces of revolution [Leb93]. For recent work on logarithmic\ndecay see [BM23]. At the other extreme, if Mis a negatively curved (or Anosov)\nsurface,W2C1(M), andW6\u00110, thenr(t) may be chosen exponentially decaying in\n[DJ18, Jin20, DJN22].\nWhenMis a torus, these extremes are avoided and the best bounds are polynomially\ndecaying in (1.2). Anantharaman and L\u0013 eautaud [ALN14] show (1.2) holds with hti\u00001=2\nwhenW2L1, andW\u0015\u000fon some open set for some \u000f >0, as a consequence of\nSchr odinger observability/control [Jaf90, Mac10, BZ12, AM14]. The more recent result\nof Burq and Zworski on Schr odinger observability and control [BZ19] weakens the \fnal\nrequirement to merely W6\u00110. Anantharaman and L\u0013 eautaud [ALN14] further show\nthat if supp Wdoes not satisfy the geometric control condition then (1.2) cannot hold\nforhti\u00001\u0000\". They also show if there exists C > 0, such that Wsatis\fesjrWj\u0014CW1\u0000\"\nfor\"<1=29, andW2Wk0;1fork0\u00158, then (1.2) holds with hti\u00001\n1+4\". See also [BH07].\nSharp decay results have been obtained on the torus when the damping is taken to\nbe polynomially controlled, bounded and y-invariant. In particular [Kle19] and [DK20]\ntogether show that for such damping with \f > 0 (1.2) holds with hti\u0000\f+2\n\f+3and there\nare some solutions decaying no faster than this rate. See also [ALN14] and [Sta17]\nfor the original proof of the case \f= 0. For improved decay rates under di\u000berent\ngeometric assumptions on the support of the damping see [LL17] and [Sun22]. In\n[Wan21a, Wan21b], the second author showed that (1.2) holds with hti\u00001\n2when there is\nboundary damping. The boundary damping has a singularity structure similar to \u000e(x),\nwhich is in the Besov space B\u00001\n1;1, hinting that the decay rate hti\u0000\f+2\n\f+3still holds whenSHARP POLYNOMIAL DECAY FOR SINGULAR DAMPING 7\n\f=\u00001. We then conjectured that hti\u0000\f+2\n\f+3holds and is optimal for all \f2(\u00001;1).\nIn this paper, we prove that our conjecture is correct.\nAs mentioned above, using observability of an associated Schr odinger equation to\nprove energy decay for the damped wave equation was used in [ALN14] to prove hti\u00001\n2\nenergy decay. This relied on a characterization of observability due to [Mil05]. This\napproach was also applied to prove energy decay for a semilinear damped wave equation\nin [JL20]. A more abstract, semigroup focused treatment for singular damping is\nprovided in [CPS+19].\nAnother motivation for the study of unbounded damping to understand their over-\ndamping behavior. Overdamping here heuristically means \\more\" damping leads to\nslower decay. At a basic level this can be observed on M= [0;1] withW=C, a con-\nstant. Such a damping satis\fes the GCC and so experiences exponential decay, but as\nshown in [CZ94] the exponential rate is not monotone in C. The decay becomes faster\nasCincreases from 0, up to a point, and then becomes slower as Cgoes to in\fnity.\nBecause of this, one might expect that unbounded damping would exhibit slower decay\nthan bounded analogs. However, when M= [0;1] andW=2\nx;[CC01] show that all\nsolutions are identically 0 for t>2. The case where W=2\u000b\nxon [0;1] for\u000ba constant\nwas studied in [FHS20]. The authors showed that although the \fnite extinction time\nbehavior is unique to \u000b= 1, in general all solutions decay exponentially and for \u000b2N\nmost solutions have a \fnite extinction time. See also [Maj74]. Unbounded damping\nhas also been studied on non-compact manifolds in [FST18], [Ger22] and [Arn22].\n1.4.Paper outline. In Section 2, we rigorously formulate the strongly continuous\ndamped wave semigroup etA:H!H onH=H1\u0002L2. The stability of etAis related\nto theL2-resolvent estimates for P\u0015=\u0000\u0001\u0000i\u0015W\u0000\u00152, the family of stationary damped\nwave operators in \u00152C. This is characterised by the next proposition:\nProposition 1.7 (Equivalence between resolvent estimates and decay) .The following\nare true:\n(1) Fix\u000b>0. There exists C > 0such that\nE(u;t)1=2\u0014Chti\u0000\u000b(kru0kL2+ku1kH1+k\u0000\u0001u0+Wu 1kL2);\nfor all solutions uwith initial data (u0;u1)2D, if and only if, there exists C > 0such\nthat for all \u00152Rand\u00156= 0 we havekP\u00001\n\u0015kL2!L2\u0014Cj\u0015j1=\u000b\u00001.\n(2) There exist C;c > 0such thatE(u;t)\u0014Ce\u0000ctE(u;0), for all solutions uwith\ninitial data (u0;u1)2H1\u0002L2, if and only if there exist C > 0such that for all \u00152R\nand\u00156= 0 we havekP\u00001\n\u0015kL2!L2\u0014C=j\u0015j.\nHere, the exponential decay resolvent estimate result can be thought of as a re\fne-\nment of the polynomial result when \u000b!1 . We show there are pole-free regions of P\u0015\nin Propositions 2.5 and 2.7, and use them to prove Theorem 4, that etAis backward8 PERRY KLEINHENZ AND RUOYU P. T. WANG\nM\nNρ\nq\n(a)Nrelative to M.\nNρ\n−0\nU+U− (b)Nrelative to U.\nFigure 2. The normal structure near the hypersurface NinsideM.\nunique. We also prove Theorem 3 that the Schr odinger observability gives polynomial\ndecay.\nIn Section 3, we prove the necessary resolvent estimates for P\u0015using a Morawetz\nmultiplier method. We then use Proposition 1.7 to prove Theorem 1 giving polynomial\ndecay on T2, and Theorem 5 giving exponential decay on S1. In Section 4, we construct\neigenfunctions for the one-dimensional Schr odinger operator with a complex Coulomb\npotential, and quasimodes for the damped wave operator to prove Theorem 2, the\nsharpness result.\n1.5.Acknowledgement. The authors are grateful to Jared Wunsch for many dis-\ncussions around these results, and to Je\u000b Galkowski for pointing out an improvement\nto the Sobolev multiplier estimates in Section 2. The authors are grateful to Romain\nJoly, Irena Lasiecka, Je\u000brey Rauch, Cyril Letrouit and Pedro Freitas for insightful\ncomments. RPTW is partially supported by NSF grant DMS-2054424.\n2.Semigroups generated by normally Lp-damping\nIn this section, we provide a general framework for the damped wave semigroup,\nwith damping W, unbounded near a closed hypersurface. We then use this semigroup\nto prove Proposition 1.7.\n2.1.Normally Lp-damping and resolvent estimates. LetMbe a compact smooth\nmanifold without boundary. Let Nbe a closed and orientable hypersurface in M, with\n\fnitely many components and an orientable normal bundle. Take a normal neighbour-\nhoodU=U\u0000tNtU+divided into two components U\u0006byN. Denote\u001a2C1(U)\nby\n\u001a(z) =(\n\u0006dist(z;N); z2U\u0006\n0; z2N:SHARP POLYNOMIAL DECAY FOR SINGULAR DAMPING 9\n(a) Wrepresented by the\nheight of the plot on T2.\n0110102103104(b)Wrepresented by how\ndark the color is on S2.\nFigure 3. Examples of normally Lp-dampingWon manifolds.\nThere exists some small \u000e >0 such that N\u000e=\u001a\u00001((\u0000\u000e;\u000e)) is compactly embedded in\nUand\u001a\u00001(s) is di\u000beomorphic to Nfor alls2(\u0000\u000e;\u000e). We can then identify N\u000eby\nN\u0002(\u0000\u000e;\u000e)\u001a: see Figure 2 for illustration.\nDe\fnition 2.1 (NormallyLp-damping) .Assume the damping function W(z)\u00150on\nMand isL1on any compact subset of MnN. Forp2[1;1], we sayW(z)is normally\nLp(with respect to N) if\nw(\u001a) = esssupfW(q;\u001a) :q2Ng2Lp(\u0000\u000e;\u000e):\nThe name comes from the fact that Wblows up near N=\u001a\u00001(0) likew(\u001a), a\nfunctionLp-integrable along \u001a, the \fber variable of the normal bundle to N. For\n1\u0014p<q\u00141 , any normally Lq-damping is also normally Lp. The class of normally\nL1-damping coincide with L1(M). Throughout this section, being normally Lpis\nthe only assumption we impose on W(z): we frequently draw a distinction between\nfunctions which are L1andLpforp>1. We give some important examples of normally\nL1-damping that we use in other sections:\nExample 2.2. (1) LetM=T2and\f2(\u00001;0). ThenV\f(x) de\fned in (1.3) are\nnormallyL\u00001=\f\u0000with respect tof(x;y)2T2:x=\u0006\u001bg. They are also normally L1.\nWhen\f2[0;1), they are normally L1: see Figure 1(B). The same is true for X\f\n\u0012.\n(2) LetM=T2,\f2(\u00001;0) and\u000f>0 small. Then W= 1fx2+y2\u0014\u000f2g(\u000f\u0000(x2+y2)1\n2)\f\nis normally L\u00001=\f\u0000with respect tofx2+y2=\u000f2g: see Figure 3(A).\n(3) LetM=S2be equipped with the spherical coordinates ( \u0012;\u001e)2[0;2\u0019]\u0002[0;\u0019].\nLet\f2(\u00001;0), thenW= 1f\u001e\u0014\u0019=2g(\u0019=2\u0000\u001e)\fis normally L\u00001=\f\u0000with respect to\nf\u001e=\u0019=2g, the equator: see Figure 3(B).\nLemma 2.3 (Sobolev multiplier) .LetWbe normally Lpforp2(1;1). Then the\nmultiplierp\nWis a bounded map from H1\n2p(M)toL2(M), and extends to a bounded10 PERRY KLEINHENZ AND RUOYU P. T. WANG\nmap fromL2(M)toH\u00001\n2p(M). Whenp= 1,p\nWmaps from H1\n2+toL2(M)and\nextends to a bounded map from L2(M)toH\u00001\n2\u0000. Whenp=1,p\nWis bounded on\nL2(M).\nProof. 1. Letp2(1;1). We show the multiplierp\nWis bounded from H1\n2ptoL2.\nSinceW2L1(MnN\u000e), it su\u000eces to show\nkp\nWukL2(N\u000e)\u0014Ckuk\nH1\n2p(N\u000e):\nBy the Sobolev embedding we have\nu2H1\n2p(N\u000e),!L2p\np\u00001\u0000\n(\u0000\u000e;\u000e)\u001a;L2(N)\u0001\n:\nTherefore\nkp\nWuk2\nL2(N\u000e)\u0014CZ\u000e\n\u0000\u000ew(\u001a)Z\nNju(q;\u001a)j2dqd\u001a\u0014CkwkLp\n\u001akuk\nL2p\np\u00001((\u0000\u000e;\u000e);L2(N))\nis bounded bykuk\nH1\n2p(N\u000e).\n2. Letp= 1 ands>1\n2. By the Sobolev embedding we have\nu2Hs(N\u000e),!C0;0\u0000\n(\u0000\u000e;\u000e)\u001a;L2(N)\u0001\n:\nTherefore\nkp\nWuk2\nL2(N\u000e)\u0014CZ\u000e\n\u0000\u000ew(\u001a)Z\nNju(q;\u001a)j2dqd\u001a\u0014CkwkL1\u001akukL1((\u0000\u000e;\u000e);L2(N))\nis bounded bykukHs(N\u000e).\n3. Letp=1. ThenW2L1(M) andkp\nWukL2(M)\u0014CkukL2(M). Thus for all\np2[1;1] we have the desired conclusion.\n4. It su\u000eces to observe that for boundedp\nW:Hs!L2, its adjointp\nW\u0003:L2!\nH\u0000sis bounded as well. \u0003\nLemma 2.4. LetWbe normally L1. Then for any u2L2(M)with suppu\u001aMnN,\nwe haveWu2L2(M).\nProof. It su\u000eces to observe that Wis essentially bounded on supp uas a compact\nsubset ofMnN. Note that we cannot give an uniform multiplier estimate, unless\nw(\u001a)2j\u001aj\u00001L1(\u0000\u000e;\u000e). \u0003\nWe now do some spectral analysis of P\u0015=\u0000\u0001\u0000i\u0015W\u0000\u00152. We sayP\u0015has a pole\nat\u00152CifP\u0015:H1!H\u00001fails to be invertible. We show in Proposition 2.5, when\np2[1;1], we have no poles in the upper half plane. We show in Proposition 2.7,\nwhenp2(1;1), we have no poles in some regions in the lower half plane that shrink\nasp!1+. The pole-free region of normally Lp-damping in the lower half plane is\nasymptotically smaller than that of L1-damping: see Figure 4.SHARP POLYNOMIAL DECAY FOR SINGULAR DAMPING 11\n0Proposition 2.5\nProposition 2.7jRe\u0015j\u0018j Im\u0015jp\u0000\n(a)Normally Lp-damping.\n0 (b)L1-damping.\nFigure 4. Pole-free regions of P\u0015for normally Lp-damping in \u00152C.\nWe begin with the upper half plane:\nProposition 2.5 (Pole-free region in the upper half plane) .LetWbe normally L1.\nFor\u00152C, considerP\u0015=\u0000\u0001\u0000i\u0015W\u0000\u00152as a bounded operator from H1toH\u00001.\nThen the following are true:\n(1)P\u0015:H1!H\u00001is bijective on \u00152fIm\u0015\u00150;\u00156= 0g.\n(2) There is C > 0such that for all \u00152fIm\u0015>0g, we have\nkP\u00001\n\u0015kL2!H1\u0014C\nIm\u0015;kP\u00001\n\u0015kL2!L2\u0014C\njIm\u0015j2: (2.1)\n(3) There is C > 0such that for for any u2H1and any\u00152fIm\u0015\u00150g, we have\nkuk2\nH1\u0014ChRe\u0015i2kuk2\nL2+CkP\u0015uk2\nH\u00001:\n(4) For any \u00152C,P\u0015:H1!H\u00001is bijective if and only if P\u0000\u0016\u0015:H1!H\u00001is\nbijective: SpecP\u0015is symmetric about the imaginary axis.\n(5) At\u0015= 0,P0=\u0000\u0001has a simple pole. P0is surjective from H1toff2H\u00001:\nhf;1i= 0g, and KerP0= spanf1g.\nProof. 1. First we will show that P\u0015is Fredholm with index 0 for all \u00152C. Note\nhPiu;vi=h(\u0000\u0001 + 1 +W)u;vi;\nis a coercive form on H1. Indeed we have\njhPiu;vij\u0014\f\f\fhru;rvi+hu;vi+hp\nWu;p\nWvi\f\f\f\u0014CkukH1kvkH1:\nand\nhPiu;ui=kuk2\nH1+kp\nWuk2\u0015kuk2\nH1:\nBy the Lax-Milgram theorem, we have\nP\u00001\ni:H\u00001!H1;12 PERRY KLEINHENZ AND RUOYU P. T. WANG\nis bounded. Then Piis Fredholm with index 0. Now note\nP\u0015=Pi\u0000\u0000\n1 +\u00152\u0001\n\u0000(1 +i\u0015)W;\nand that (1 + \u00152) + (1 +i\u0015)W:H1!H\u00001\n2\u0000,!H\u00001compactly from Lemma 2.3.\nThusP\u0015is Fredholm with index 0 for all \u00152C. This also implies that P\u0015is bijective\nif and only if P\u0003\n\u0015=P\u0000\u0016\u0015is bijective from H1toH\u00001.\n2. We show that P\u0003\n\u0015:H1!H\u00001has trivial kernel in the upper half plane. For\nIm\u0015\u00150;\u00156= 0 let\u0015=\u000b+i\f. Consider\nP\u0003\n\u0015u=P\u0000\u0016\u0015u=\u0000\n\u0000\u0001 +\u0000\n\f2+\fW\u0000\u000b2\u0001\n+i\u000b(W+ 2\f)\u0001\nu= 0:\nPair it with uto see\nhP\u0003\n\u0015u;ui=kruk2+\u0000\n\f2\u0000\u000b2\u0001\nkuk2+\fkp\nWuk2+i\u0010\n\u000bkp\nWuk2+ 2\u000b\fkuk2\u0011\n= 0:\n(2.2)\nSuppose\f > 0. When\u000b6= 0, the imaginary part of (2.2) implies kuk= 0. When\n\u000b= 0, the real part of (2.2) implies kuk= 0. When \f= 0 and\u000b6= 0, (2.2) is reduced\nto\nkruk2\u0000\u000b2kuk2\u0000i\u000bkp\nWuk2= 0;\nandkp\nWuk= 0. From the unique continuation we know u\u00110 almost everywhere.\nNow since Ker P\u0003\n\u0015is trivial and P\u0015has index 0, we know CoKer P\u0015is trivial and P\u0015is\ninvertible.\n3. Let\u0015=\u000b+i\fwith\f >0. Consider\nP\u0015u= (\u0000\u0001 + (\f2\u0000\u000b2+\fW)\u0000i\u000b(W+ 2\f))u=f:\nPair it with uto observe\nhP\u0015u;ui=kruk2+ (\f2\u0000\u000b2)kuk2+\fkp\nWuk2\u0000i\u000b(2\fkuk2+kp\nWuk2):(2.3)\nWhen\u000b= 0, we have\nkruk2+\f2kuk2\u00141\n2\f\u00002kfk+1\n2\f2kuk2;\nthe absorption of the last term gives (2.1). When \u000b6= 0, take the imaginary part to\nsee\n2j\u000bj\fkuk2+j\u000bjkp\nWuk2\u0014j\u000bj\fkuk2+C\f\u00001j\u000bj\u00001kfk2;\nthe absorption of the \frst term on the right gives\nkuk2\u0014C\f\u00002j\u000bj\u00002kfk2: (2.4)\nThe real part of (2.3) reads\nkruk2+\f2kuk2+\fkp\nWuk2\u0014C\u000f\u00001\f\u00002kfk2+\u000b2kuk2+\u000f\f2kuk2:\nAbsorb the last term on the right and bring in (2.4) to see\nkruk2+\f2kuk2\u0014C\f\u00002kfk2;SHARP POLYNOMIAL DECAY FOR SINGULAR DAMPING 13\nwhich is (2.1).\n4. Since\f\u00150, the real part of (2.3) gives\nkruk2+ (\f2\u0000\u000b2)kuk2\u0014C\u000f\u00001kfk2\nH\u00001+\u000fkuk2\nH1;\nand the absorption of the last term gives\nkuk2\nH1\u0014Ch\u000bi2kuk2\nL2+Ckfk2\nH\u00001;\nwhereCdoes not depend on \f.\n5. Let\u0015= 0. Then Ker P\u0003\n0= KerP0= Ker(\u0000\u0001) = spanf1g, andP0is surjective\nfromH1to (KerP\u0003\n0)?=ff2H\u00001:hf;1i= 0g. \u0003\nWe now look at the lower half plane:\nLemma 2.6 (Interpolation inequality) .LetWbe normally Lpforp2[1;1), then for\nallr2(1\n2p;1)there exists C > 0such that for all \r >0,\nkp\nWukL2\u0014\rkukL2+C\r1\u0000r\n\u0000rkukH1:\nNote thatCdoes not depend on \r;u.\nProof. Use the weight ( C\u00001\r)s2\u0000r\ns1\u0000rin [DZ19, Proposition E.21] with s1= 0;s2= 1;h= 1\nto observe that for all r2(0;1), there is C > 0 such that for all \r >0 we have\nkukHr\u0014\rkukL2+C\r1\u0000r\n\u0000rkukH1: (2.5)\nFor anyr2(1\n2p;1), Lemma 2.3 implies kp\nWukL2\u0014kukHrand concludes the proof. \u0003\nProposition 2.7 (Pole-free region in the lower half plane) .LetWbe normally Lpfor\np2(1;1). Then the following are true:\n(1) For any M;\u000e > 0, there exists KM;\u000e>0such thatP\u0015:H1!H\u00001is bijective\nonf\u00152C:jRe\u0015j\u0014\u0000MjIm\u0015jp\u0000\u000e;Im\u0015\u0014\u0000KM;\u000eg.\n(2) Along any ray \u0015=j\u0015jei\u0012with\u00122(\u0019;2\u0019), there are\u00150;C > 0such that for any\nj\u0015j>\u0015 0we have\nkP\u00001\n\u0015kL2!H1\u0014Cj\u0015j\u00001;kP\u00001\n\u0015kL2!L2\u0014Cj\u0015j\u00002: (2.6)\n(3) There are C;K > 0such that for any u2H1and\u00152fIm\u0015\u0014\u0000Kgwe have\nkuk2\nH1\u0014ChRe\u0015i2kuk2\nL2+CkP\u0015uk2\nH\u00001:\nProof. 1. Let\u0015=\u000b+i\f. We show P\u0003\n\u0015has trivial kernel on\nD\u000e=f0<j\u000bj\u0014\u0000M\f1+\u000e; \f\u0014\u0000K\u000eg:\nLetP\u0003\n\u0015u= 0. Fix some r2(1\n2p\u00002\u000e;1\n2)\u001a(1\n2p;1\n2), thens=1\u0000r\nr>2p\u00002\u000e\u00001. Lemma\n2.6 implies\nkp\nWukL2\u0014\u000fj\fj1\n2kukL2+Cr;\u000fj\fj\u0000s\n2kukH1: (2.7)14 PERRY KLEINHENZ AND RUOYU P. T. WANG\n1a. When\u000b6= 0, the imaginary part of (2.2) reads\n2j\fjkuk2\u0014kp\nWuk2\u0014j\fjkuk2\nL2+Crj\fj\u0000skuk2\nH1; (2.8)\nwhich implies that for j\fj\u0015Crwe have\nkuk2\u0014Crj\fj\u0000s\u00001kruk2;\nsince\u0000s<\u00001. Substituting this back into (2.8) to observe\nkp\nWuk2\u0014Crj\fj\u0000skruk2:\nThe real part of (2.2) now reads\nkruk2+\f2kuk2=\u000b2kuk2+j\fjkp\nWuk2\u0014Cr(\u000b2j\fj\u0000s\u00001+j\fj\u0000s+1)kruk2:\nNote that\u000b\u0014Mj\fjp\u0000\u000e, and we have\nkruk2+\f2kuk2\u0014Crj\fj\u0000s+2p\u00002\u000e\u00001kruk2:\nSinces>2p\u00002\u000e\u00001, there exists K\u000e>0 large such that for j\fj\u0015K\u000e, the right hand\nside of the last equation can be absorbed by the left. We thus obtained kukH1= 0.\n1b. When\u000b= 0, consider the real part of (2.2):\nkruk2+\f2kuk2=j\fjkp\nWuk2\u0014\u000f2\f2kuk2+Cr;\u000fj\fj\u0000s+1kukH1:\nSinces>1, both terms on the right can be absorbed by the left when j\fjis su\u000eciently\nlarge. In both cases, u= 0 inH1. ThusP\u0003\n\u0015has trivial kernel in D\u000eandP\u0015:H1!H\u00001\nis bijective there.\n2. Fix\u00122(\u0019;2\u0019). Then\u0015=j\u0015jei\u0012can be parametrized by \u0015=h\u00001(\u000e\u0000i), where\nh=h\u000eij\u0015j\u00001!0 and\u000e= cot\u0012\fxed. From Step 1 we know that P\u0015is bijective forj\u0015j\nlarge. LetP\u0015u=fwhere\nP\u0015=\u0000\u0001 +h\u00002(1\u0000\u000e2)\u0000h\u00001W+ih\u00001(2h\u00001\u0000\u000eW):\nSemiclassicalize by Ph=h2P\u0015,g=h2fand\nPhu=\u0000h2\u0001 + (1\u0000\u000e2)\u0000hW+i(2\u0000h\u000eW )u=g:\nPair it with uinL2to observe\nkhruk2+ (1\u0000\u000e2)kuk2\u0000hkp\nWuk2+i\u0010\n2kuk2\u0000h\u000ekp\nWuk2\u0011\n=hg;ui: (2.9)\nFix somer2(1\n2p;1\n2), thens=1\u0000r\nr>1. Lemma 2.6 implies\nkp\nWukL2\u0014\u000fh\u000ei\u00001\n2h\u00001\n2kukL2+C\u000fh\u000eis\n2hs\n2kukH1:\nThe imaginary part of (2.9) implies\n2kuk2\u0014h\u000ekp\nWuk2+\u000fkuk2+C\u000f\u00001kgk2\u0014\u000f(1+\u000eh\u000ei\u00001)kuk2\nL2+C\u000f(\u000eh\u000eishs+1kuk2\nH1+kgk2)\n\u00142\u000fkuk2\nL2+C\u000f(\u000eh\u000eishs+1kuk2\nH1+kgk2):SHARP POLYNOMIAL DECAY FOR SINGULAR DAMPING 15\nWhenhis smaller than some \u000e-dependent bounds, the absorption of the \frst two terms\non the right gives\nkuk2\u0014Chs\u00001khruk2+Ckgk2:\nFrom now on our constants also depend on \u000e. The imaginary part of (2.9) then gives\nhkp\nWuk2\nL2\u0014Chs\u00001khruk2+Ckgk2: (2.10)\nSubstitute those two estimates back to the real part of (2.9) to see\nkhruk2+kuk2= Rehg;ui+hkp\nWuk2+\u000e2kuk2\u0014Chs\u00001khruk2+Ckgk2:\nNotes>1 and forhsmall,Chs\u00001<1. Thus by absorbing the \frst term on the right\nthere ish0>0 and for all h<h 0we havekukH1\nh\u0014CkgkL2, whereCdoes not depend\nonh. This implies\nkuk2\nL2+h\u000ei2j\u0015j\u00002kruk2\nL2\u0014Ch\u000ei4j\u0015j\u00004kfk2\nL2;\nfor allj\u0015j\u0015\u00150for some\u00150>0. This gives us the desired estimate (2.6) on the lines.\n3. Consider the pairing hP\u0015u;uiin (2.3). Its real part is\nkruk2+ (\f2\u0000\u000b2)kuk2= Rehf;ui\u0000\fkp\nWuk2:\nApply (2.7) to observe\nkruk2+ (\f2\u0000\u000b2)kuk2\u0014C\u000f\u00001kfk2\nH\u00001+\u000fkuk2\nH1+\u000fj\fj2kuk2+C\u000fj\fj\u0000s+1kuk2\nH1:\nAss >1, there exists K > 0 such that when \f\u0014\u0000K, the last three terms on the\nright can be absorbed. This gives\nkuk2\nH1\u0000Ch\u000bi2kuk2\u0014Ckfk2\nH\u00001;\nas desired. \u0003\nRemark 2.8. The proof only used the property ofp\nWbeing a bounded map from\nHrtoL2for somer<1\n2. Note that we need some elbow room for r2(1\n2p;1\n2) to show\nh1\n2kp\nWuk\u0014Ch1\n2\u0000rkukHr\nhis semiclassically small compared to kukH1\nhin (2.10). When\np= 1, (1\n2p;1\n2) is empty and there is no rfor the proof to work. This is consistent with\nthe observation thatp\nWis not a bounded map from H1\n2toL2whenp= 1. In other\nwords,h1\n2p\nWubecomes too large compared to all the other terms in (2.9). This may\nexplain why there can be \fnite time extinction in the case \f=\u00001 in [FHS20, CC01].\n2.2.Semigroup generated by normally Lp-damping. Now we consider semi-\ngroups for normally L1-damping.\nDe\fnition 2.9 (Semigroup for normally L1-damping) .LetH=f(u;v)2H1(M)\u0002\nL2(M)gwith normk(u;v)kH=kuk2\nH1+kvk2\nL2. De\fne\nA=\u00120 Id\n\u0001\u0000W\u0013\n:D(A)!H;16 PERRY KLEINHENZ AND RUOYU P. T. WANG\nwithD(A) =D=f(u;v)2H :A(u;v)2Hg . Note that the equivalent de\fnition for\nDis\nD=f(u;v)2H1\u0002H1:\u0000\u0001u+Wv2L2g;\nequipped withk(u;v)k2\nD=k(u;v)k2\nH1\u0002H1+k\u0000\u0001u+Wvk2\nL2, equivalent to the graph\nnorm ofA. Solutions of the damped wave equation (1.1) are equivalent to solutions of\n(\n@tU(t) =AU(t)\nU(0) = (u0;u1)t; U (t) =\u0012u(t;x)\n@tu(t;x)\u0013\n;\nwhereU(t) =etAU(0), whose semigroup nature will be shown soon in Propositions 2.13\nand 2.23.\nRemark 2.10. Note that the our semigroup construction applies to damping not\ncovered by [FST18]. In particular, damping functions in X\f\n\u0012are normally L1when\n\u00001<\f, but are not L2\nlocfor\u00001<\f <\u00001\n2, and cannot satisfy the relative boundedness\npiece of [FST18, Assumption 1]. The semigroup from [FST18] is used in [FHS20] on\ndamping of the form 2 =xon (0;1) with Dirichlet boundary conditions. This is possible\nbecause solutions are 0 exactly where the damping is singular, which allows use of the\nHardy inequality. We allow solutions to be non-zero where the damping is singular,\nand indeed, this is an essential feature of our quasimode construction in Section 4. One\ncan check that our semigroup is the same as in [CPS+19, Section 2.2].\nWe remind the reader that P\u0015=\u0000\u0001\u0000i\u0015W\u0000\u00152is bounded from H1!H\u00001. We\nnow draw the connection between P\u0015andA+i\u0015, which will be useful to show that\netAis a strongly continuos semigroup. Let k\u00152 andD1=D, de\fneDk=f(u;v)2\nDk\u00001:A(u;v)2Dk\u00001g, and thenAk:Dk!H is a bounded map.\nLemma 2.11 (Spectral equivalence) .Let\u00152C. Then the following are true:\n(1)A+i\u0015:D!H is bijective if and only if P\u0015:H1!H\u00001is bijective.\n(2)A+i\u0015is bijective i\u000bA\u0000i\u0016\u0015is so: SpecAis symmetric about the real axis.\n(3)A:D!f (u;v)2H :hWu+v;1i= 0gis surjective, and KerA= spanf(1;0)g.\n(4) IfW6= 0, then Ker(Ak) = Ker(A)for allk\u00152:Ahas a simple pole at 0.\nProof. 1. Assume P\u0015:H1!H\u00001is injective at some \u0015. Then for any ( f;g)2H =\nH1\u0002L2, consider\n\u0012u\nv\u0013\n= (A+i\u0015)\u00001\u0012f\ng\u0013\n=\u0012\u0000i\u0015\u00001(Id +P\u00001\n\u0015\u0001)\u0000P\u00001\n\u0015\n\u0000P\u00001\n\u0015\u0001i\u0015P\u00001\n\u0015\u0013\u0012f\ng\u0013\n: (2.11)\nas an element in H1\u0002H1that satis\fes\n(A+i\u0015)\u0012u\nv\u0013\n=\u0012i\u0015u+v\n\u0001u\u0000(W\u0000i\u0015)v\u0013\n=\u0012f\ng\u0013\n:SHARP POLYNOMIAL DECAY FOR SINGULAR DAMPING 17\nNote that \u0001 u\u0000Wv=g+i\u0015v2L2. Thus (u;v)2D and (A+i\u0015)(u;v) = (f;g).\nMoreover, (A+i\u0015)\u00001(0;0) = (0;0) sinceP\u0015is injective. Therefore A+i\u0015is bijective.\n2. AssumeA+i\u0015:D!H is bijective. For any g2L2, there exists a unique\n(u;v)2D such that\n(A+i\u0015)\u0012u\nv\u0013\n=\u0012i\u0015u+v\n\u0001u\u0000(W\u0000i\u0015)v\u0013\n=\u00120\n\u0000g\u0013\n:\nThusv=\u0000i\u0015uandP\u0015u=g. Moreover, (\u0000u;\u0000i\u0015u)2D impliesu2fu2H1:\n(\u0000\u0001\u0000i\u0015W)u2L2g\u001aH1. ThusP\u0015:L2!fu2H1: (\u0000\u0001\u0000i\u0015W)u2L2gis\nbijective. Its dual P\u0003\n\u0015=P\u0000\u0016\u0015:fu2H1: (\u0000\u0001\u0000i\u0015W)u2L2g\u0003!L2is bijective.\nLetf2H\u00001\u001a(\u0000\u0001\u0000i\u0015W)u2L2g\u0003, then there exists a unique w2L2such that\nP\u0000\u0016\u0015w= (\u0000\u0001 +i\u0016\u0015W\u0000\u0016\u00152)w=f. PairP\u0000\u0016\u0015wwithwand take the real part to see\nkrwk2\u0014jhf;wij+\u0016\u00152kwk2<1:\nThusw2H1andP\u0000\u0016\u0015:H\u00001!H1is bijective. Apply Proposition 2.5 to see P\u0015:\nH\u00001!H1is also bijective. This also implies A+i\u0015is bijective i\u000bA\u0000i\u0016\u0015is so.\n3. Let (u;v)2H andA(u;v) = (v;\u0001u\u0000Wv) = (f;g)2H. Thusv=f2H1and\n\u0000P0u= \u0001u=Wf+g2H\u00001sinceWf2H\u00001. Due to Proposition 2.5, there exists\nu2H1if and only ifhWf+g;1i= 0, and Ker \u0001 = span f1g. ThusAis surjective\nontof(f;g)2H :hWf+g;1i= 0gand KerA= spanf(1;0)g.\n4. To show Ker(Ak) = Ker(A), it su\u000eces to show that Ker( A2) = Ker(A), that\nis, for any ( u;v)2D2, ifA(u;v)2KerAthen (u;v)2KerA. Note thatD2=\nf(u;v)2H1\u0002H1: \u0001u\u0000Wv2H1;\u0000\u0001v+W(\u0001u\u0000Wv)2L2g. AssumeA(u;v) =\n(v;\u0001u\u0000Wv) =c(1;0) for some constant c. Thenv=c, and \u0001u=cW2H\u00001. By\nProposition 2.5, there exists u2H1such that \u0001 u=cW2H\u00001only ifhcW;1i= 0.\nSinceW\u00150 is not identically 0, v=c= 0, andu=c0for some constant c0. Thus\n(u;v)2KerA. Note that when W\u00110, Ker(A2) =f(u;v)2D2:hu;1i=hv;1i= 0g\nis a 2-dimensional subspace that contains Ker Aas a proper subspace. \u0003\nCorollary 2.12 (Pole-free region of A).The following are true:\n(1) LetWbe normally L1. ThenA\u0000\u0016:D!H is bijective onfRe\u0016\u00150;\u00166= 0g.\n(2) If we further assume Wis normally Lpforp2(1;1], then for any M;\u000e > 0,\nthere exists KM;\u000e>0such thatA\u0000\u0016:D!H is bijective onf\u00162C:jIm\u0016j\u0014\n\u0000MjRe\u0016jp\u0000\u000e;Re\u0016\u0014\u0000KM;\u000eg.\nProof. Let\u0015=i\u0016and apply Lemma 2.11 to the part 1 of Propositions 2.5 and 2.7. \u0003\nLetw0= ess inffW(z) :z2Mg\u00150 be the essential minimum of the damping W.\nWe now show etAis a strongly continuous semigroup on H.18 PERRY KLEINHENZ AND RUOYU P. T. WANG\nProposition 2.13 (Quasi-contraction semigroup) .LetWbe normally L1. Then the\nfollowing are true:\n(1) The generator Ais closed andDis dense inH.\n(2) The generator Agenerates a strongly continuous semigroup etA:H!H .\n(3)etAis quasi-contractive: for all t\u00150,ketAkL(H)\u0014exp(1\n2(hw0i\u0000w0)t).\n(4) The generator Ahas compact resolvent, and the spectrum of Acontains only\nisolated eigenvalues.\nProof. 1. Consider the core D0=C1(M)\u0002fv2C1(M) : suppv\u001aMnNg\u001aD .\nIndeed, for such ( u;v)2D 0, we haveWv2L2from Lemma 2.4, and thus ( u;v)2D.\nFurthermoreD0is dense inH=H1\u0002L2.\n2. Consider the Hilbert space H1\nW=f\u001e2H1:W\u001e2L2gequipped with inner\nproduct\nh\u001e; iH1\nW=h\u001e; iH1+hW\u001e;W i:\nConsider its dual space H\u00001\nW= (H1\nW)\u0003, the set of complex-valued continuous linear\nfunctionals on H1\nW. Note that H\u00001\u001aH\u00001\nW. The map G:H\u0002H!L2\u0002H\u00001\nWgiven by\nG((u;v);(f;g)) = (A(u;v)\u0000(f;g)) = (v\u0000f;\u0001u\u0000Wv\u0000g);\nis then bounded. The graph of A:D!H is the zero set of this continuous map G,\nand is thus closed. Indeed, A:D!H is the maximal closed extension of A:D0!\nC1\u0002L2.\n3. We now showAgenerates a strongly continuous semigroup. Firstly Consider that\nfor any (u;v)2D, we havev2H1andp\nWv2L2, and\nhA(u;v);(u;v)iH=h(v;\u0001u\u0000Wv);(u;v)iH=\u0000kp\nWvk2+hv;ui+ 2iImhrv;rui;\nthe real part of which is\nRehA(u;v);(u;v)iH=\u0000kp\nWvk2+ Rehv;ui\u0014\u0000w0kvk2+ Rehv;ui:\nNote1\n2(p\nw2\n0+ 1\u0000w0)\u00001\u0000w0=1\n2(p\nw2\n0+ 1\u0000w0) and\njRehv;uij\u00141\n2(q\nw2\n0+ 1\u0000w0)\u00001kvk2+1\n2(q\nw2\n0+ 1\u0000w0)kuk2;\nimplies RehA(u;v);(u;v)iH\u00141\n2(hw0i\u0000w0)k(u;v)k2\nH. Now for any real \u0016 >1\n2(hw0i\u0000\nw0)>0, Corollary 2.12 implies A\u0000\u0016:D!H is bijective. Moreover,\nk(A\u0000\u0016)(u;v)kHk(u;v)kH\u0015Reh(\u0016\u0000A)(u;v);(u;v)iH\u0015(\u0016\u00001\n2(hw0i\u0000w0))k(u;v)k2\nH:\nThis impliesk(A\u0000\u0016)\u00001kL(H)\u0014\u0000\n\u0016\u00001\n2(hw0i\u0000w0)\u0001\u00001. Apply the Hille-Yoshida the-\norem as in [EN00, Corollary II.3.6, p. 76] to conclude that A:D!H generates a\nstrongly continuous semigroup etA:H!H , andketAkL(H)\u0014exp(1\n2(hw0i\u0000w0)t) for\nallt\u00150.SHARP POLYNOMIAL DECAY FOR SINGULAR DAMPING 19\n4. Note that the range of any resolvent of Awill be a subset of D. For any (u;v)2D,\nwe haveWu2H\u00001\n2\u0000from Lemma 2.3, and \u0000\u0001u2H\u00001\n2\u0000. From the classical elliptic\nregularity, we have u2H3\n2\u0000. ThusD\u001aH3\n2\u0000\u0002H1embeds compactly into H=H1\u0002L2.\nThen any resolvent of Ais a compact operator from HtoH. This implies that the\nspectrum ofAcontains only isolated eigenvalues. \u0003\nRemark 2.14. Note that when W\u00110, theL2-mass ofumay grow linearly in time.\nWe later show that if W6= 0 thenetAis bounded in Proposition 2.23.\nWe need a more quantitative connection between the resolvent estimates for P\u0015and\nA+i\u0015to prove further results about etA:\nLemma 2.15 (Resolvent equivalence) .The following are true:\n(1) At any \u00152Cnf0gsuch thatA+i\u0015:D!H is bijective, we have\nkP\u00001\n\u0015kL2!H1\u0014k(A+i\u0015)\u00001kL(H);kP\u00001\n\u0015kL2!L2\u0014j\u0015j\u00001k(A+i\u0015)\u00001kL(H):\n(2) LetWbe normally L1. Then there is C > 0such that\nk(A+i\u0015)\u00001kL(H)\u0014j\u0015jkP\u00001\n\u0015kL2!L2+kP\u00001\n\u0015kL2!H1+\u0000\n1 +Cj\u0015j\u00001h\u0015i\u0001\nkP\u00001\n\u0000\u0016\u0015kL2!H1+Cj\u0015j\u00001;\n(2.12)\nuniformly for any \u00152fIm\u0015\u00150;\u00156= 0g.\n(3) If we further assume Wis normally Lpforp2(1;1], then there exists K > 0\nsuch that (2.12) is also uniformly true for any \u00152fIm\u0015 <\u0000Kgat which both P\u00001\n\u0015\nandP\u00001\n\u0000\u0016\u0015are bounded from L2toH1.\nProof. 1. Note that (A+i\u0015)(\u0000u;i\u0015u ) = (0;P\u0015u). This implies\nkukH1\u0014k(A+i\u0015)\u00001kL(H)kP\u0015ukL2;kukL2\u0014j\u0015j\u00001k(A+i\u0015)\u00001kL(H)kP\u0015ukL2:\n2. The normally L1andLpassumptions will allow us to use the part 3 of Propositions\n2.5 and 2.7 respectively on each domain. Firstly note\n\r\rP\u00001\n\u0015\r\r\nH\u00001!L2=\r\r\u0000\nP\u00001\n\u0015\u0001\u0003\r\r\nL2!H1=\r\r(P\u0003\n\u0015)\u00001\r\r\nL2!H1=\r\r\rP\u00001\n\u0000\u0016\u0015\r\r\r\nL2!H1:\nNow let (A+i\u0015)(u;v) = (f;g) and (2.11) implies\nu=\u0000i\u0015\u00001f\u0000i\u0015\u00001P\u00001\n\u0015\u0001f\u0000P\u00001\n\u0015g; v =\u0000P\u00001\n\u0015\u0001f+i\u0015P\u00001\n\u0015g:\nNote\nkP\u00001\n\u0015\u0001fkL2\u0014kP\u00001\n\u0015kH\u00001!L2kfkH1=kP\u00001\n\u0000\u0016\u0015kL2!H1kfkH1: (2.13)\nThus\nkvkL2\u0014\u0010\nkP\u00001\n\u0000\u0016\u0015kL2!H1+j\u0015jkP\u00001\n\u0015kL2!L2\u0011\nk(f;g)kH:\nOn another hand, apply the part 3 of Propositions 2.5 and 2.7 to (2.13) to see\nkP\u00001\n\u0015\u0001fkH1\u0014Ch\u0015ikP\u00001\n\u0015\u0001fkL2+Ck\u0001fkH\u00001\u0014C\u0010\nh\u0015ikP\u00001\n\u0000\u0016\u0015kL2!H1+ 1\u0011\nkfkH1:20 PERRY KLEINHENZ AND RUOYU P. T. WANG\nThen\nkukH1\u0014\u0010\nkP\u00001\n\u0015kL2!H1+Cj\u0015j\u00001h\u0015ikP\u00001\n\u0000\u0016\u0015kL2!H1+Cj\u0015j\u00001\u0011\nk(f;g)kH:\nBring together the estimates for uandvto conclude. \u0003\nWe cite a backward uniqueness result for semigroups and use it to prove our semi-\ngroupetAis backward unique when Wis normally Lp:\nProposition 2.16 (Lasiecka-Renardy-Triggiani, Theorem 3.1 of [LRT01]) .LetAgen-\nerates a strongly continuous semigroup etAon a Banach space X. Assume there exist\n\u00122(\u0019=2;\u0019);R;C > 0such that uniformly for all r\u0015Rwe have\nk(A\u0000re\u0006i\u0012)\u00001kL(X)\u0014C:\nThenetAis backward unique, that is, if etAx= 0 at somet>0, thenx= 0.\nProposition 2.17 (Backward uniqueness for etA).LetWbe normally Lpforp2\n(1;1]. ThenetAis backward unique, that is, if etA(u;v) = 0 at somet >0, then\n(u;v) = 0 .\nProof. 1. We claim for any \u0012such that\u00122(\u0019=2;3\u0019=2), along the ray \u0000i\u0015=j\u0015jei\u0012,\nthere exists C;\u0015 0>0 such that for all j\u0015j>\u0015 0, we havek(A+i\u0015)\u00001kL(H)\u0014Cj\u0015j\u00001.\nNote that arg( \u0015)2(\u0019;2\u0019) and so is arg(\u0000\u0016\u0015). From Proposition 2.7, we have\nkP\u00001\n\u0015kL2!H1\u0014Cj\u0015j\u00001;kP\u00001\n\u0015kL2!L2\u0014Cj\u0015j\u00002;\nand\nkP\u00001\n\u0000\u0016\u0015kL2!H1\u0014Cj\u0015j\u00001;\nforj\u0015j\u0015\u00150. Invoke Lemma 2.15 to see k(A+i\u0015)\u00001kL(H)\u0014Cj\u0015j\u00001.\n2. Now invoke Proposition 2.16 to conclude the backward uniqueness. \u0003\n2.3.Semigroup decomposition. In this subsection, we always assume W6= 0. We\nwould like to apply the following results to etA, if possible.\nProposition 2.18 (Borichev-Tomilov, Theorem 2.4 of [BT10]) .Letet_Abe a strongly\ncontinuous semigroup on a Hilbert space _H, generated by _A. IfiR\\Spec( _A) =;, then\nthe following conditions are equivalent:\nket_A_A\u00001kL(_H)=O(hti\u0000\u000b) ast!1;\nk(i\u0015Id\u0000_A)\u00001kL(_H)=O(j\u0015j1=\u000b) as\u0015!1:\nProposition 2.19 (Gearhart-Pr uss-Huang, [Gea78, Pr u84, Hua85]) .Letet_Abe a\nstrongly continuous semigroup on a Hilbert space _Hand assume that there exists aSHARP POLYNOMIAL DECAY FOR SINGULAR DAMPING 21\npositive constant M > 0such thatket_Ak\u0014Mfor allt\u00150. Then there exist C;c> 0\nsuch that for all t>0\nket_AkL(_H)\u0014Ce\u0000ct;\nif and only if iR\\Spec( _A) =;and\nsup\n\u00152Rk(_A\u0000i\u0015Id)\u00001kL(X)<1:\nAn issue with applying these is that Acan have spectrum at 0. The outline for\nSection 2.3 is to de\fne a semigroup generator _Awith no spectrum at 0 and which\nprovides energy decay information for etA. We then establish an equivalence of resolvent\nestimates for _AandP\u0015, which we use to prove Propositions 1.7 using Propositions 2.18\nand 2.19. We follow the strategy of [ALN14] to separate the zero-frequency modes\nfrom others.\nDe\fnition 2.20 (Spectral decomposition) .AssumeW6= 0, thenAhas a simple pole\nat0. Let \u00050be the Riesz projector of Athat projectsHonto KerA, given by the\nspectral resolution\n\u00050=1\n2\u0019iZ\n\r(zId\u0000A)\u00001dz;\nwhere\ris a small circle around 0inCcontaining 0as the only eigenvalue of Ain its\ninterior. Consider the range of A,\n_H=A(D) =f(u;v)2H : Av(Wu+v) = 0g;\na codimension-1 subspace of H, equipped with the norm k(u;v)k2\n_H:=kruk2\nL2+kvk2\nL2.\nNote \u0005\u000f= Id\u0000\u00050projectsHonto _H. The Riesz projectors non-orthogonally decom-\nposesHas\u00050H\b _H. Concretely, letjMjbe the volume of Mand for any (u;v)2H,\nthe Riesz projectors are\n\u00050(u;v) = (Av(W)\u00001Av(Wu+v);0);\u0005\u000f(u;v) = (u\u0000Av(W)\u00001Av(Wu+v);v);\nwhere Av(u) =1\njMjR\nMu dM , the average of uoverM, and Av(W)>0sinceW6= 0.\nNote \u00050\u0005\u000f= \u0005\u000f\u00050= 0, and thus _H= Ker \u0005 0,\u00050H= KerA= Ker \u0005\u000f= Ker \u0001\u0002\nf0g= spanf(1;0)g, andk\u0005\u000f(u;v)k_H=k(u;v)k_Hfor any (u;v)2H. Let\n_A=AjD(_A)=\u00120 Id\n\u0001\u0000W\u0013\n:D(_A)!_H;\nwhere _D=D(_A) = \u0005\u000fD(A) =D\\ _H, equipped with the norm\nk(u;v)k2\nD(_A)=kruk2\nL2+kvk2\nH1+k\u0000\u0001u+Wvk2\nL2:\nThenD=_D\b \u00050Dand\u00050D= \u0005 0H= KerA. Note that the nature of spectral\nresolution implies \u0005\u000fA=A\u0005\u000f=_A\u0005\u000f,\u00050A=A\u00050= 0 and thusA=_A\u0005\u000f.22 PERRY KLEINHENZ AND RUOYU P. T. WANG\nRemark 2.21. Without assuming W6= 0, we can always orthogonally decompose H=\nKerA\b(KerA)?. Doing this will not invalidate most of the theorems in this section\nnor the main results, but we point out that it is natural to use the non-orthogonal\ndecomposition Ker A\b _H. WhenAis not normal, we do not expect its eigenspaces\nwith distinct eigenvalues to be orthogonal to each other. In such a case, the lack of\northogonality of \u0005 0is a known phenomenon: see further in [HS96, Proposition 6.3],\n[DZ19, Remarks(1), p. 84], [Heu82, Proposition 50.2].\nLemma 2.22 (Spectral and resolvent equivalence between _AandA).Let\u00162Cnf0g,\nW6= 0. Then there exists C > 0independent of \u0016such that the following are true:\n(1) _A:_D! _His bijective.\n(2) _A\u0000\u0016:_D! _His bijective if and only if A\u0000\u0016:D!H is bijective.\n(3) On _H, the _H-norm andH-norm are equivalent.\n(4) If both _A\u0000\u0016andA\u0000\u0016are bijective, then\nC\u00001k(_A\u0000\u0016)\u00001kL(_H)\u0014k(A\u0000\u0016)\u00001kL(H)\u0014C\u0010\nk(_A\u0000\u0016)\u00001kL(_H)+j\u0016j\u00001\u0011\n:\nProof. 1. Note that from Lemma 2.11 we know A:D=_D\bKerA! _His surjective\nand thus _A:_D! _His bijective. Note that k(u;v)k_H\u0014k(u;v)kHand ( _H;k\u0001kH) is\ncontinuously embedded in ( _H;k\u0001k _H). Since the embedding is a bijective continuous\nmap, it is further an open map and admits a continuous inverse. This implies that the\nnorms are equivalent and we write k\u0001k _H\u0014k\u0001kH\u0014Ck\u0001k _H.\n2. Let\u00162Cnf0g. Assume _A\u0000\u0016:_D! _His bijective. Note that \u0005 \u000fA=_A\u0005\u000f.\nThen consider the decomposition\nA\u0000\u0016= (\u0005\u000f+ \u0005 0)A\u0000\u0016= (_A\u0000\u0016)\u0005\u000f+ \u0005 0(A\u0000\u0016):\nFor anyV2H, since _A\u0000\u0016is surjective, there exists _U2_Dsuch that ( _A\u0000\u0016)_U= \u0005\u000fV.\nThen\n(A\u0000\u0016)\u0010\n_U\u0000\u0016\u00001\u00050V\u0011\n= (_A\u0000\u0016)_U\u0000\u00050(A\u0000\u0016)\u0016\u00001\u00050V= \u0005\u000fV+ \u0005 0V=V;\nwhere we usedA\u00050= 0. Thus _A\u0000\u0016is surjective. To show it is injective, assume\n(A\u0000\u0016)U= 0 for some U2D. Then\n0 = \u0005\u000f(A\u0000\u0016)U= (_A\u0000\u0016)\u0005\u000fU:\nAs ( _A\u0000\u0016) is injective, \u0005 \u000fU= 0. ThusU= \u0005 0U. Then\n0 = \u0005 0(A\u0000\u0016)U=\u0000\u0016U:\nAs\u00166= 0,U= 0 andA\u0000\u0016is also injective.\n3. AssumeA\u0000\u0016:D!H is bijective fromD=_D\b\u00050DtoH=_H\b \u00050H. Note\nthatA= 0 on \u0005 0DandA\u0000\u0016maps \u0005 0Dto \u0005 0D= \u0005 0Hbijectively. Thus A\u0000\u0016isSHARP POLYNOMIAL DECAY FOR SINGULAR DAMPING 23\nbijective from _Dto_H. Eventually observe that _A\u0000\u0016= \u0005\u000f(A\u0000\u0016) on _Hto conclude\nit is bijective.\n4. Now assume both _A\u0000\u0016:_D! _HandA\u0000\u0016:D!H are bijective. Fix _V2_D,\nand let _Ube the unique element in _Hsuch that ( _A\u0000\u0016)_U=_V. Since _V2D, there\nexists a unique U2H such that (A\u0000\u0016)U=_V. Moreover,\n(_A\u0000\u0016)\u0005\u000fU= \u0005\u000f(A\u0000\u0016)U= \u0005\u000f_V=_V;\nimplies _U= \u0005\u000fU. Thus\nk_Uk_H\u0014k_UkH\u0014CkUkH\u0014Ck(A\u0000\u0016)\u00001kL(H)k_VkH\u0014Ck(A\u0000\u0016)\u00001kL(H)k_Vk_H\nandk(_A\u0000\u0016)\u00001kL(_H)\u0014Ck(A\u0000\u0016)\u00001kL(H). On another hand, let ( A\u0000\u0016)U=Vfor\nU2D,V2H. Then\n(_A\u0000\u0016)\u0005\u000fU= \u0005\u000f(A\u0000\u0016)U= \u0005\u000fV;\nimpliesk\u0005\u000fUkH\u0014k(_A\u0000\u0016)\u00001kL(_H)k\u0005\u000fVkH. Meanwhile,\n\u00050V= \u0005 0(A\u0000\u0016)U=\u0000\u0016\u00050U;\nwhere we used \u0005 0A= 0. Thusj\u0016jk\u00050UkH\u0014k\u00050VkH, and\nkUkH\u0014C\u0010\nk(_A\u0000\u0016)\u00001kL(_H)+j\u0016j\u00001\u0011\nkVkH;\nas we want. \u0003\nWe now show that this _Aindeed generates a contraction semigroup that provides\nenergy decay information about etA.\nProposition 2.23 (Semigroup decomposition) .LetWbe normally L1andW6= 0.\nThen the following are true:\n(1) The generator _A:_D! _His maximally dissipative.\n(2) The generator _Agenerates a contraction semigroup et_Aon_H.\n(3) The strongly continuous semigroup generated by AonHcan be decomposed as\netA=et_A\u0005\u000f+ \u0005 0; (2.14)\n(4) We have \u0005\u000fetA=et_A\u0005\u000f.\n(5) There exists C > 0such thatketAkL(H)\u0014Cfor allt\u00150.\nProof. 1. We show _A:_D! _His maximally dissipative. Firstly note that Corollary\n2.12 impliesA\u0000 1 is bijective from D=_D\b \u00050DtoH=_H\b \u00050H. Apply Lemma\n2.22 to conclude that _A\u0000 1 is bijective from _Dto_H. Then it is straightforward to\ncompute for any ( u;v)2_D,\nReD\n_A(u;v);(u;v)E\n_H=\u0000Z\nMWjvj2\u00140;24 PERRY KLEINHENZ AND RUOYU P. T. WANG\nwhich demonstrates the dissipative nature of _A, and so by the Lumer-Phillips theorem\nas in [EN00, Theorem II.3.15, p.83], _Agenerates a contraction semigroup et_Aon_H,\nandket_Ak_H\u00141 for allt\u00150.\n2. We begin with the initial decomposition\netA=etA\u0005\u000f+etA\u00050;\nand simplify both terms. Consider for the \frst term that on _H,\n@tetA=A=_A; etAjt=0= Id;\nThus _AgeneratesetA: by the uniqueness of the generator we know etA=et_Aon_H.\nOn another hand, for the third term, on \u0005 0H= KerA, we have\n@tetA=A= 0; etAjt=0= Id;\nand thusetA\u00050= \u0005 0. The above observations give the desired decomposition (2.14).\nApply \u0005\u000fto (2.14) and note \u0005 \u000f\u00050= 0 to obtain \u0005 \u000fetA=et_A\u0005\u000f.\n3. For the boundedness of etAonH, note for any U2H,\nketAUkH\u0014Cket_A\u0005\u000fUk_H+Ck\u00050UkH\u0014CkUkH;\nwhere we used _H-norm andH-norm are equivalent on _H, andket_AkL(_H)\u00141.\u0003\nProposition 2.24 (Backward uniqueness for et_A).LetWbe normally Lpforp2\n(1;1]. Thenet_Ais backward unique, that is, if et_A(u;v) = 0 at somet >0, then\n(u;v) = 0 .\nProof. Similar to Proposition 2.17: Lemma 2.22 implies on the spectral region of inter-\nest, the resolvents k(A+i\u0015)\u00001kL(H)andk(_A+i\u0015)\u00001kL(_H)obey the same bounds. \u0003\nProof of Theorem 4. Part 1 immediately follows from Proposition 2.17. To prove the\nbackward uniqueness for the energy, assume E(u;t) = 0 at some t>0. This implies\n0 =ketA(u0;u1)k_H=k\u0005\u000fetA(u0;u1)k_H=ket_A\u0005\u000f(u0;u1)k_H;\nwhere we used that k\u0005\u000f(u;v)k_H=k(u;v)k_Hfor any (u;v)2H and Proposition 2.23.\nProposition 2.24 implies \u0005 \u000f(u0;u1) = 0, and thus\nE(u;0)1\n2=1\n2k(u0;u1)k_H=1\n2k\u0005\u000f(u0;u1)k_H= 0;\nas we want in Part 2. \u0003\nLemma 2.25 (Resolvent equivalence on the real line) .Let\u000b\u0015\u00001and\u00150>0. The\nfollowing are equivalent:\n(1) There exists C > 0such that for all \u00152Rwithj\u0015j\u0015\u00150we have\nkP\u00001\n\u0015kL2!L2\u0014Ch\u0015i\u000b: (2.15)SHARP POLYNOMIAL DECAY FOR SINGULAR DAMPING 25\n(2) There exists C > 0such that for all \u00152Rwithj\u0015j\u0015\u00150we have\nk(_A+i\u0015)\u00001kL(_H)\u0014Ch\u0015i\u000b+1: (2.16)\nProof. 1. Assume (2.16) holds. Lemma 2.22 implies k(A+i\u0015)\u00001kL(H)\u0014Ch\u0015i\u000b+1and\nLemma 2.15(1) implies (2.15).\n2. Assume (2.15) holds. Proposition 2.5(3) implies\nkP\u00001\n\u0015kL2!H1\u0014Ch\u0015i\u000b+1+C\u0014Ch\u0015i\u000b+1;\nuniformly for all \u00152f\u00152R:j\u0015j\u0015\u00150g. For\u00152f\u00152R:j\u0015j\u0015\u00150g,\u0000\u0016\u0015=\u0000\u0015is also\nin this set. Lemma 2.15 implies\nk(A+i\u0015)\u00001kL(H)\u0014j\u0015jh\u0015i\u000b+ (2 +Cj\u0015j\u00001h\u0015i)(Ch\u0015i\u000b+1+C) +Cj\u0015j\u00001\u0014Ch\u0015i\u000b+1:\nInvoke Lemma 2.22 to see (2.16). \u0003\nWe now give full proof to Proposition 1.7, that resolvent estimates of P\u0015are equiv-\nalent to energy decay.\nProof of Proposition 1.7. 1. We assume P\u00001\n\u0015, forj\u0015j\u0015\u00150\nkP\u00001\n\u0015kL2!L2\u0014Cj\u0015j1=\u000b\u00001;\nby Lemma 2.25 this is equivalent to\nk(_A+i\u0015)\u00001kL(_H)\u0014Ch\u0015i1=\u000b;\nforj\u0015j\u0015\u00150. Note that Corollary 2.12(1) with Lemma 2.22 implies iR\\Spec _A=;.\nBy Proposition 2.18 of Borichev-Tomilov, this is equivlent to\nket_A_A\u00001kL(_H)=O(hti\u0000\u000b):\nThis is equivalent to that, the energy of solution uto the damped wave equation (1.1)\nis bounded by\nE(u;t)1\n2\u0014C\r\retA(u0;u1)\r\r_H=C\r\r\ret_A\u0005\u000f(u0;u1)\r\r\r_H\u0014\r\r\ret_A_A\u00001\r\r\r\nL(_H)k(u0;u1)k_D\n\u0014Chti\u0000\u000b(kru0kL2+ku1kH1+k\u0000\u0001u0+Wu 1kL2);\nas desired.\n2. WhenkP\u00001\n\u0015kL2!L2\u0014C, we apply Proposition 2.19 of Gearhart-Pr uss-Huang. \u000326 PERRY KLEINHENZ AND RUOYU P. T. WANG\n2.4.Schr odinger observability gives polynomial decay. Lete\u0000it\u0001be the unitary\nSchr odinger operator group on L2generated by the anti-self-adjoint operator \u0000i\u0001 :\nH2!L2. The Schr odinger equation\n(i@t\u0000\u0001)u= 0; ujt=0=u02L2;\nis unique solved by u=e\u0000it\u0001u0.\nDe\fnition 2.26 (Schr odinger observability) .We say that the Schr odinger equation is\nexactly observable from an open set \n\u001aMif there exists T >0;CT>0such that for\nanyu02L2,\nku0kL2\u0014CTZT\n0k 1\ne\u0000it\u0001u0kL2dt:\nWe now consider the spectral theory of \u0000\u0001. Since\u0000\u0001 is essentially self-adjoint and\npositive-de\fnite on L2, we have a spectral resolution\n\u0000\u0001u=Z1\n0\u001a2dE\u001a(u);\nwhereE\u001ais a projection-valued measure on L2and suppE\u001a\u001a[0;1). De\fne the\nscaling operators\n\u0003\u0000s=Z1\n0(1 +\u001a2)\u0000sdE\u001a(u):\nThose operators \u0003\u0000s:H\u0000s!L2are elliptic and bounded from above and below, and\nthey commute with \u0000\u0001.\nLemma 2.27. Fixs;N > 0. Assume the Schr odinger equation is exactly observable\nfrom \n. Let\u001f2C1where\u001f= 1on a open neighbourhood of \n. Then there is C > 0\nsuch that\nkukH\u0000s\u0014Ck(\u0000\u0001\u0000\u00152)ukH\u0000s+Ck\u001fuk2\nL2+Ckuk2\nH\u0000N:\nfor all\u00152R.\nProof. When the Schr odinger equation is exactly observable from \n, [Mil05, Theorem\n5.1] implies for any v2L2\nkvkL2\u0014Ck(\u0000\u0001\u0000\u00152)vkL2+Ck 1\nvkL2: (2.17)\nNow letu2H\u0000sandv= \u0003\u0000su2L2. Apply (2.17) to see\nk\u0003\u0000sukL2\u0014Ck(\u0000\u0001\u0000\u00152)\u0003\u0000sukL2+Ck 1\n\u0003\u0000sukL2=Ck\u0003\u0000s(\u0000\u0001\u0000\u00152)ukL2+Ck 1\n\u0003\u0000sukL2;\nwhich implies\nkukH\u0000s\u0014Ck(\u0000\u0001\u0000\u00152)ukH\u0000s+Ck 1\n\u0003\u0000sukL2:SHARP POLYNOMIAL DECAY FOR SINGULAR DAMPING 27\nNow \fx a cuto\u000b \u001f0such that\u001f0\u00111 on \n, supp \u001f0\u001asupp\u001fcompactly, and \u001f\u001f0=\u001f0.\nNote that since WF ( \u001f0\u0003\u0000s)\u001aell\u001f, we have\nk 1\n\u0003\u0000suk2\nL2\u0014k\u001f0\u0003\u0000suk2\nL2\u0014Ck\u001fuk2\nH\u0000s+Ckuk2\nH\u0000N;\nfor anyN > 0 from the elliptic estimate [DZ19, Theorem E.33]. \u0003\nProposition 2.28. LetWbe normally Lpforp2(1;1). Assume the Schr odinger\nequation is exactly observable from \n, and there exists \u000f >0such thatW\u0015\u000falmost\neverywhere on an open neighbourhood of \n. Then there exists \u00150;C > 0such that for\nallj\u0015j\u0015\u00150\nkP\u00001\n\u0015kL2!L2\u0014Ch\u0015i1+1\np:\nWhenp= 1,kP\u00001\n\u0015kL2!L2\u0014Ch\u0015i2+. Whenp=1,kP\u00001\n\u0015kL2!L2\u0014Ch\u0015iuniformly for\nallj\u0015j\u0015\u00150.\nProof. 1. Letp2(1;1) ands=1\n2p. LetP\u0015u= (\u0000\u0001\u0000i\u0015W\u0000\u00152)u=f. There exists\n\u001f2C1such that\u001f= 1 on \n, while supp \u001fis compactly supported in fW\u0015\u000fg.\nSince on supp \u001f,Wis bounded from below, we have k\u001fukL2\u0014k\u001fp\nWukL2. Lemma\n2.27 then implies\nkuk2\nH\u0000s\u0014Ck(\u0000\u0001\u0000i\u0015W\u0000\u00152)uk2\nH\u0000s+\u00152kWuk2\nH\u0000s+Ck\u001fp\nWuk2\nL2+Ckuk2\nH\u0000N:\nSincep\nW:L2!H\u0000sis bounded, we have\nkuk2\nH\u0000s\u0014Ckfk2\nH\u0000s+C\u00152kp\nWuk2\nL2+Ckuk2\nH\u0000N: (2.18)\nWe now get rid of the last term on the right. Pair P\u0015uwithuinH\u0000Nto observe\n\u00152kuk2\nH\u0000N=kruk2\nH\u0000N\u0000i\u0015hWu;uiH\u0000N\u0000hf;uiH\u0000N:\nThis implies\nkuk2\nH\u0000N\u0014C\u0015\u00002kuk2\nH1\u0000N+Cj\u0015j\u00001jhWu;uiH\u0000Nj+C\u0015\u00002kfk2\nH\u0000N: (2.19)\nNote thathWu;uiH\u0000N=h\u0003\u0000Np\nWp\nWu; \u0003\u0000Nui. WhenN\u0015s, \u0003\u0000Np\nWis a bounded\nmap onL2. Thus we have\njhWu;uiH\u0000Nj\u0014C\u000f\u00001kp\nWuk2\nL2+\u000fkuk2\nH\u0000N;\nfor anyN\u0015s. Now \fxN >s + 1, (2.19) implies\nkuk2\nH\u0000N\u0014C\u0015\u00002kukH\u0000s+Cj\u0015j\u00001kp\nWuk2\nL2+C\u0015\u00002kfkH\u0000s\u00001:\nApplying this to (2.18) implies that for large j\u0015j\u0015\u00150,\nkuk2\nH\u0000s\u0014C\u0010\nkfk2\nH\u0000s+\u00152kp\nWuk2\nL2\u0011\n:\nNow pairP\u0015uwithuinH\u0000sto observe\nkruk2\nH\u0000s\u0000\u00152kuk2\nH\u0000s\u0000i\u0015hWu;uiH\u0000s=hu;fiH\u0000s:28 PERRY KLEINHENZ AND RUOYU P. T. WANG\nNotejhWu;uiH\u0000sj\u0014C\u000f\u00001kp\nWuk2\nL2+\u000fkuk2\nH\u0000sand thus\nkuk2\nH1\u0000s\u0014Ch\u0015i2kuk2\nH\u0000s+Ch\u0015i\u00002kfk2\nH\u0000s+j\u0015jjhWu;uiH\u0000sj\u0014Ch\u0015i2\u0010\nkfk2\nH\u0000s+\u00152kp\nWuk2\nL2\u0011\n:\nApply the interpolation inequality (2.5) to \u0003\u0000suwith\r=\u0015sandr=sto see\nkuk2\nL2\u0014Ch\u0015i2s\u0010\nkfk2\nH\u0000s+\u00152kp\nWuk2\nL2\u0011\n:\nPairP\u0015uwithuinL2to observekruk2\u0000\u00152kuk2\u0000i\u0015kp\nWuk2=hf;uiand\nj\u0015jkp\nWuk2\u0014C\u000f\u00001h\u0015i2s+1kfk2+\u000fh\u0015i\u00002s\u00001kuk2:\nThenkuk2\u0014Ch\u0015i2+4skfk2andkP\u00001\n\u0015kL2!L2\u0014Ch\u0015i1+2s.\n2. Whenp= 1, for any s >1\n2, the proof above still works. When p=1,p\nWis\nbounded on L2, and the above proof works with s= 0. \u0003\nProof of Theorem 3. Proposition 1.7 gives E1\n2\u0014Chti\u00001=(2+1\np)forp2(1;1), and the\ncorresponding rates for p= 1 orp=1. \u0003\nRemark 2.29. Assumep2(1;1). By a argument similar to the proof of Proposition\n2.28, observe that k(\u0000\u0001 + (1 +i\u0015)2)\u00001kH\u0000s!Hs\u0014Ch\u0015i2s\u00001. Fors=1\n2p, we have\nkp\nW(\u0000\u0001 + (1 +i\u0015)2)\u00001p\nWkL(L2)\u0014Ch\u0015i2s\u00001. Apply [CPS+19, Proposition 3.10] to\nobtaink(_A+i\u0015)\u00001k_H\u0014Ch\u0015i2+4swhich gives E1\n2\u0014Chti\u00001=(2+2\np), which is slower than\nour rates given in Theorem 3.\n3.Resolvent estimates in one dimension\nConsider the equation\nP\u0015u= (\u0000\u0001\u0000i\u0015W\u0000\u00152)u=f: (3.1)\nTo showkP\u00001\n\u0015kL2!L2\u0014C\ng(\u0015);so that Propositions 1.7 can be applied, it is enough to\nshow that there exist C;\u0015 0\u00150 such that for any f2L2(M) and anyj\u0015j\u0015\u00150, if\nu2H2(M) solves (3.1), then\nkuk2\nL2\u0014Cg(\u0015)2kfk2\nL2: (3.2)\nTo show Theorem 1 we must show (3.2) holds with g(\u0015)2=j\u0015j2\n\f+2. To show Theorem\n5 we must show (3.2) holds with g(\u0015)2=1\nj\u0015j2.\nThe main estimate for this section is the following one-dimensional resolvent esti-\nmate:\nProposition 3.1 (1D Resolvent Estimate) .Consideru2H2(S1)that satis\fes\n\u0000\n\u0000@2\nx\u0000i\u0015W\u0000\u00162\u0001\nu(x) =f(x); (3.3)SHARP POLYNOMIAL DECAY FOR SINGULAR DAMPING 29\nthen there is C;\u0015 0>0such that for \u00162\u0014\u00152andj\u0015j\u0015\u00150we have\nkuk2\nL2+h\u00162i\u00001k@xuk2\nL2\u0014Ch\u00162i\u00001\u0010\n1 +h\u00162i\u00001j\u0015j2\n2+\f\u0011\nkfk2\nL2:\nThe proof of Proposition 3.1 will be delayed to the second part of this section. Note\nthat Proposition 3.1 with \u00162=\u00152and Proposition 1.7 together imply Theorem 5.\nProposition 3.1 can also be used to show the following proposition, which along with\nProposition 1.7 implies Theorem 1.\nProposition 3.2 (Resolvent Estimate on Tori) .Letu2H2(T2)be the solution to\nP\u0015u(x;y) =\u0000\n\u0000\u0001\u0000i\u0015W\u0000\u00152\u0001\nu(x;y) =f(x;y): (3.4)\nThen there exists C;\u0015 0>0such that for \u00152Rwithj\u0015j>\u0015 0we have\nkuk2\nL2+\u0015\u00002kruk2\nL2\u0014C\u0010\n1 +j\u0015j2\n2+\f\u0011\nkfk2\nL2:\nProof. Consider the eigenfunctions en(y) with\n\u0000@2\nyen(y) =\u00152\nnen(y); \u00152\nn!1:\nDecompose\nu(x;y) =X\nun(x)en(y); f(x;y) =X\nfn(x)en(y);\nand the equation (3.4) is reduced to\n\u0000\n\u0000@2\nx\u0000i\u0015W\u0000\u00162\u0001\nun(x) =fn(x); \u00162=\u0000\n\u00152\u0000\u00152\nn\u0001\n:\nApply Proposition 3.1 to see uniformly in n;\u0015that forj\u0015j>\u0015 0we have\nkunk2\nL2+h\u00162i\u00001krunk2\nL2\u0014Ch\u00162i\u00001\u0010\n1 +h\u00162i\u00001j\u0015j2\n2+\f\u0011\nkfnk2\nL2:\nIn particular uniformly in n;\u0015\nkunk2\u0014C\u0010\n1 +j\u0015j2\n2+\f\u0011\nkfnk2:\nApply the Parseval theorem to obtain\nkuk2\u0014C\u0010\n1 +j\u0015j2\n2+\f\u0011\nkfk2: (3.5)\nPair (3.4) with uand take the real part to see\n\f\fkruk2\u0000\u00152kuk2\f\f\u0014C\u0015\u00002kfk2+1\n2\u00152kuk2:\nThis along with (3.5) produces the desired estimate. \u000330 PERRY KLEINHENZ AND RUOYU P. T. WANG\nThe rest of this section is devoted to the proof of Proposition 3.1. In particular, we\nwill show that there exists \u00150;\u00162\n0>0 such that for any real \u0015\u0015\u00150and\u00162\u0014\u00152and\nu2H2(S1);f2L2(S1) solving (3.3) then\nZ\njuj2+ju0j2\u0014CZ\njfj2when\u00162<\u00162\n0; (3.6)\nZ\njuj2+1\n\u00162ju0j2\u0014C\n\u00162 \n\u00152\n2+\f\n\u00162+ 1!Z\njfj2when\u00162\u0015\u00162\n0: (3.7)\nHere and below integrals are taken over S1. The general case of j\u0015j\u0015\u00150follows by an\nidentical argument, but we focus on \u0015>0 for ease of notation. Note that in the case\n\u00162< \u00162\n0we can actually have \u00162<0. However, the bulk of our argument is devoted\nto the proof of (3.7), where \u00162is indeed a positive real number.\nIn our proof we use a version of the Morawetz multiplier method, which is arranged\nvia the energy functional\nF(x) =ju0(x)j2+\u00162ju(x)j2:\nThis method was introduced by [Mor61]. It has been used in [CV02] and [CD21], we\nwill follow its use in [DK20].\nFollowing the proof of Lemma 1 from [DK20], with the modi\fcation that bmust\nbe chosen such that b00<0 on a neighborhood of each zero interval for W, we obtain\nbasic estimates on the size of uandu0on the damped region, and (3.6). The proof is\notherwise identical, so we do not include the details.\nLemma 3.3. If\u0015>0;\u001622Randu;f solve (3.3) then\nZ\nWjuj2\u00141\n\u0015Z\njfuj: (3.8)\nFurthermore if for some c >0; 2C1\n0is supported infW > cg, then there exists\nC > 0such thatZ\n ju0j2\u0014C\u0012\n1 +\u00162\n\u0015\u0013Z\njfuj: (3.9)\nFinally there are positive constants \u00162\n0andCsuch that for any \u0015>0;\u00162\u0014\u00162\n0andu;f\nsolving (3.3) we have (3.6)\nWe now set up a multiplier, which we call b. The multiplier method then provides\nthe following estimate, which must be re\fned to obtain our desired resolvent estimates.\nLemma 3.4. Let\u000ej>0;j= 1;:::;n be \fxed constants and let\n\u001e=\u001e(x) =(\n\u0015\u000ejjxj2[\u001bj\u0000\u0015\u0000\u000ej;\u001bj+\u0015\u0000\u000ej]\n1 otherwise:SHARP POLYNOMIAL DECAY FOR SINGULAR DAMPING 31\nIf\u0015;\u00162>0andu;f solve (3.3) then\nZ\n\u001eju0j2+\u00162\u001ejuj2.Z\njfj2+X\nj\u0015Z\nWjjuu0j:\nProof. Choosec>0 small enough so that fW >cgintersects the support of each Wj\nand letbbe piecewise linear on S1with\nb0(x) =8\n>><\n>>:\u0015\u000ejjxj2[\u001bj\u0000\u0015\u0000\u000ej;\u001bj+\u0015\u0000\u000ej]\n\u0000MfW >cgandjxj62[\u001bj\u0000\u0015\u0000\u000ej;\u001bj+\u0015\u0000\u000ej]\n1 elsewhere :\nIt is possible to choose Mbig enough so that bis indeed periodic on S1. So then let\nF=ju0j2+\u00162juj2and compute\n0 =Z2\u0019\n0(bF)0=Z\nb0ju0j2+\u00162b0juj2+ 2Reu00\u0016u0+\u001622Reu0\u0016u0\n=Z\nb0ju0j2+\u00162b0juj2\u00002bRef\u0016u0\u00002\u0015bReiWu\u0016u0:\nThereforeZ\nb0ju0j2+\u00162b0juj2\u00142Z\nbjfu0j+ 2\u0015Z\nbWjuu0j:\nNow adding a multiple of (3.8) and (3.9) to both sides gives\nZ\n\u001eju0j2+\u001e\u00162juj2\u00142Z\njfu0j+ 2\u0015Z\nWjuu0j+\u0015\u00001Z\njfuj+C\u0012\n1 +\u00162\n\u0015\u0013Z\njfuj:\nApplying Young's inequality for products to the jfujterms on the right hand side,\nabsorbing the resultant juj2terms back into the left hand side and recalling W=PWj\nand\u00162\u0014\u00152gives the desired inequality. \u0003\nIt now remains to estimate the Wjterms. We begin with an estimate in the case\nM=S1, so\u00162=\u00152and consider Wj2L1. BecauseWjsatis\fes hypotheses of the\nclassical geometric control argument, one expects this argument to be straightforward\nand it is.\nLemma 3.5. WhenWj2L1andM=S1for any\">0there exists C > 0such that\nif\u0015>0;\u00162=\u00152andu;f solve (3.3)\n\u0015Z\nWjjuu0j\u0014CZ\njfj2+\"\u00162\n2Z\njuj2+\"\n2Z\nju0j2:32 PERRY KLEINHENZ AND RUOYU P. T. WANG\nProof. Well, using that Wjis bounded and (3.8)\n\u0015Z\nWjjuu0j\u0014\u00152\n2\"Z\nW2\njjuj2+\"\n2Z\nju0j2\u0014C\u00152\n2\"Z\nWjjuj2+\"\n2Z\nju0j2\n\u0014C\u0015\n2\"Z\njfuj+\"\n2Z\nju0j2\n\u0014C\u00152\n\u00162Z\njfj2+\"\u00162\n2Z\njuj2+\"\n2Z\nju0j2:\nFinally, since \u00162=\u00152this gives the desired inequality. \u0003\nWe now turn to Wjwith polynomial type singularities. Following the structure of\n[DK20] we prove an intermediate result and reduce the problem to estimating Vj\u001fjjfuj.\nIn this proof we specify \u000ej=1\n2+\fjin order to control the growth of a term. The proof of\nthis lemma requires a change in technique from the proof in [DK20] in order to account\nfor the singularity and the fact that W0\njis larger than Wjnear its singularity.\nLet\u001f2C1\n0be supported onjxj>1=2 and be identically 1 on jxj>1.\nLemma 3.6. IfWj2X\fj\n\u0012j, then for any e\" >0, there exists C > 0and\" >0, such\nthat if\u001fj(x) =\u001f\u0010\nx\u0000\u001bj\n\"\u0015\u0000\u000ej\u0011\nand if\u0015;\u00162>0andu;f solve (3.3) , then\n\u0015Z\nWjjuu0j\u0014C(\u0015\u000ej+\u0016)Z\njfuj+C\u00151=2\u0012Z\njfuj\u00131=2\u0012Z\nVj\u001fjjfuj\u00131=2\n+e\"Z\n\u001eju0j2:\nProof. Recall throughout that \fj2[\u00001;0). FromWj2X\fj\n\u0012jwe have that there exists\nCj>0 andVj= (jxj\u0000\u001bj)\fj\n+such that1\nCjVj(x)\u0014W(x)\u0014CjVj(x).\nTo begin make a change of variables so that \u001bj= 0 thenVj=jxj\fjandR\nWjjuu0j\u0014\nCR\nVjjuu0j. The strategy is to split this integral into jxj>\"\u0015\u0000\u000ejandjxj<\"\u0015\u0000\u000ejwhere\n\">0 is to be chosen later.\nCase 1:jxj> \"\u0015\u0000\u000ej. Note\u001fjas de\fned above is supported on jxj>\"\u0015\u0000\u000ej\n2and is\nidentically 1 onjxj>\"\u0015\u0000\u000ej. So applying Cauchy-Schwarz and (3.8)\nZ\njxj>\"\u0015\u0000\u000ejVjjuu0j\u0014C\u0012Z\nVjjuj2\u00131=2\u0012Z\nVj\u001fjju0j\u00131=2\n\u0014\u0015\u00001=2\u0012Z\njfuj\u00131=2\u0012Z\nVj\u001fjju0j\u00131=2\n:\nUsing integration by parts\nZ\nVj\u001fjju0j2=\u0000ReZ\n(Vj\u001fj)0u0\u0016u\u0000ReZ\nVj\u001fju00\u0016u: (3.10)SHARP POLYNOMIAL DECAY FOR SINGULAR DAMPING 33\nTo control the \frst term note jxj\u0015\"\u0015\u0000\u000ej\n2on supp\u001fjso1\njxj\u00142\u0015\u000ej\n\"there, alsoj\u001f0\njj.\u0015\u000ej\n\"\nand soj(Vj\u001fj)0j\u0014C\u0015\u000ejVj\n\". Therefore\nReZ\n(Vj)0u0\u0016u\u0014C\u0015\u000ej\n\"Z\nVjjuu0j: (3.11)\nFor the second term apply (3.3) and (3.8) to get\nReZ\nVj\u001fu00\u0016u= ReZ\nVj\u001f(\u0000i\u0015Wu\u0000\u00162u\u0000f)\u0016u\u0014ReZ\nVj\u001f\u00162juj2+Vj\u001fjfuj\n\u0014ReZ\u00162\n\u0015jfuj+Vj\u001fjfuj: (3.12)\nCombining (3.11) and (3.12) with (3.10) gives\n\u0015Z\njxj>\"\u0015\u0000\u000ejVjjuu0j\u0014C\u00151=2\u0012Z\njfuj\u00131=2\u0012\u0015\u000ej\n\"Z\nVjjuu0j+\u00162\n\u0015jfuj+Vj\u001fjfuj\u00131=2\n:\n(3.13)\nCase 2:jxj<\"\u0015\u0000\u000ej. To begin note that\n\u001e\u0015\u0000\u000ej= 1 onjxj<\"\u0015\u0000\u000ej<\u0015\u0000\u000ej: (3.14)\nSo applying Cauchy-Schwarz\n\u0015Z\njxj<\"\u0015\u0000\u000ejVjjuu0j\u0014\u00151\u0000\u000ej\n2\u0012Z\njxj<\"\u0015\u0000\u000ejV2\njjuj2\u00131=2\u0012Z\n\u001eju0j2\u00131=2\n: (3.15)\nNow let be a cuto\u000b supported in jxj<2\u0015\u0000\u000ejand identically 1 on jxj< \u0015\u0000\u000ejthen\nlet \"(x) = (x=\") and insert it into the below integral. Then rewriting jxj2\fj=\n\u0000C@xjxj2\fj+1and integrating by parts\nZ\njxj<\"\u0015\u0000\u000ejV2\njjuj2\u0014Z\njxj2\fj \"juj2=\u0000CZ\n@x\u0000\njxj2\fj+1\u0001\n \"juj2\n=CZ\njxj2\fj+1@x( \"juj2)\n\u0014CZ\njxj2\fj+1 0\n\"juj2+CZ\njxj2\fj+1 \"juu0j: (3.16)\nTo control the \frst term of (3.16) note\nZ\njxj<2\"\u0015\u0000\u000ejjxj2\fj+1 0\n\"juj2=Z\njxj<2\"\u0015\u0000\u000ejjxj1+\fjjxj\fj\"\u00001 0juj2\n\u0014C\"\fj\u0015\u0000\fj\u000ejZ\njxj<\"\u0015\u0000\u000ejVjjuj2\n\u0014C\"\fj\u0015\u0000\fj\u000ejZ\nWjuj2\u0014C\"\fj\u0015\u0000\fj\u000ej\u00001Z\njfuj:(3.17)34 PERRY KLEINHENZ AND RUOYU P. T. WANG\nNow let\u0011>0 be a constant to be speci\fed and applying Young's inequality for products\nto the second term of (3.16), then using that  \"is supported onfjxj<2\"\u0015\u0000\u000ejgand\n(3.14)\nZ\njxj2\fj+1 \"juu0j\u0014\u0011\n2Z\njxj2\fj \"juj2+1\n2\u0011Z\njxj2+2\fj \"ju0j2\n\u0014\u0011\n2Z\njxj2\fj \"juj2+C\n\u0011\"2+2\fj\u0015\u00002\fj\u000ej\u00003\u000ejZ\n\u001eju0j2:(3.18)\nSo then combining (3.16), (3.17) and (3.18)\nZ\n \"V2\njjuj2\u0014C\"\fj\u0015\u0000\fj\u000ej\u00001Z\njfuj+\u0011CZ\njxj2\fj \"juj2+C\n\u0011\"2+2\fj\u0015\u00002\fj\u000ej\u00003\u000ejZ\n\u001eju0j2\nZ\n \"V2\njjuj2\u0014C\"\fj\u0015\u0000\fj\u000ej\u00001Z\njfuj+C\"2+2\fj\u0015\u00002\fj\u000ej\u00003\u000ejZ\n\u001eju0j2:\nWhere the second integral in the \frst inequality was absorbed back into the left hand\nside by choosing \u0011small enough. Combining this last inequality with (3.15), letting\n\"0>0 be a constant to be speci\fed and applying Young's inequality for products\n\u0015Z\njxj<\u0015\u0000\u000ej\"Vjjuu0j\u0014\u00151\u0000\u000ej=2\u0012\nC\"\fj\u0015\u0000\fj\u000ej\u00001Z\njfuj+C\"2+2\fj\u0015\u00002\fj\u000ej\u00003\u000ejZ\n\u001eju0j2\u00131=2\u0012Z\n\u001eju0j2\u00131=2\n\u0014\u00152\u0000\u000ej\n2\"0\u0012\nC\"\fj\u0015\u0000\fj\u000ej\u00001Z\njfuj+C\"2+2\fj\u0015\u00002\fj\u000ej\u00003\u000ejZ\n\u001eju0j2\u0013\n+\"0\n2Z\n\u001eju0j2\n=C\"\fj\u00151\u0000(\fj+1)\u000ejZ\njfuj+C\"2+2\fj\n\"0\u0015\u0000(2\fj+4)\u000ej+2Z\n\u001eju0j2+\"0\n2Z\n\u001eju0j2:\n(3.19)\nNow combining (3.13) and (3.19) then applying Young's inequality for products to\nabsorb the Vjjuu0jterm and a \u00151=2from the right hand side back into the left hand\nside.\n\u0015Z\nVjjuu0j\u0014C\u00151=2\u0012Z\njfuj\u00131=2\u0012\u0015\u000ej\n\"Z\nVjjuu0j+\u00162\n\u0015jfuj+Vj\u001fjfuj\u00131=2\n+C\"\fj\u00151\u0000(\fj+1)\u000ejZ\njfuj+C\"2+2\fj\n\"0\u0015\u0000(2\fj+4)\u000ej+2Z\n\u001eju0j2+\"0\n2Z\n\u001eju0j2\n\u0014C(\u0015\u000ej+\u0016)Z\njfuj+C\u00151=2\u0012Z\njfuj\u00131=2\u0012Z\nVj\u001fjfuj\u0013\n+\u0012C\"2+2\fj\n\"0+\"0\n2\u0013Z\n\u001eju0j2:\nWhere the \u0015dependence on the second to last \u001eju0j2term is eliminated by setting\n\u000ej=1\n\fj+2, as then\u0000(2\fj+ 4)\u000ej+ 2 = 0. This also ensures that 1 \u0000(\fj+ 1)\u000ej=\u000ej.\nChoosing\"0small enough and then \"small enough gives the desired inequality. \u0003SHARP POLYNOMIAL DECAY FOR SINGULAR DAMPING 35\nRemark 3.7. Note that when M=S1, the estimate ofR\nWjjuu0jcan be modi\fed to\nhave a third case without changing the result. That is for some small c>0, consider\nc>jxj>\"\u0015\u0000\u000ej;jxj<\"\u0015\u0000\u000ejandjxj>c. The \frst two cases are proved as normal and\nthe casec >jxjcan be controlled by Lemma 3.5. This is what makes it possible to\naddressWj= 1 [\u0019;\u0000\u001b](x)Vjor 1 [\u001b;\u0019)(x)VjforVj2X\fj\n\u0012jwhenM=S1.\nTo obtain the desired resolvent estimate it remains to control the Vj\u001fjjfujterm.\nLemma 3.8. If\u0015>0;\u001622Randu;f solve (3.3) then\n\u00151=2\u0012Z\njfuj\u00131=2\u0012Z\nVj\u001fjjfuj\u00131=2\n\u0014CZ\njfj2:\nProof. By linearity there are two cases\n(1) suppf\u001a(suppVj)c\n(2) suppf\u001asuppVj\nIn case 1 the termR\nVj\u001fjjfujvanishes.\nIn case 2 note that Vj\u0015Con suppVjso\nZ\njfuj\u0014cZ\nV1=2\njjfuj:\nThen using Cauchy-Schwarz and (3.8)\nZ\njfuj\u0014c\u0012Z\njfj2\u00131=2\u0012Z\nVjjuj2\u00131=2\n\u0014C\n\u00151=2\u0012Z\njfj2\u00131=2\u0012Z\njfuj\u00131=2\n:\nThereforeR\njfuj\u0014C\n\u0015R\njfj2. From this and Cauchy-Schwarz\n\u00151=2\u0012Z\njfuj\u00131=2\u0012Z\n\u001fjVjjfuj\u00131=2\n\u0014\u0012Z\njfj2\u00131=2\u0012Z\njfj2\u00131=4\u0012Z\n\u001fjV2\njjuj2\u00131=4\n:\nIt remains to control the \fnal term on the right hand side. Because \u001fjis supported on\nx>\"\u0015\u0000\u000ejandVj=jxj\fj.\nZ\n\u001fV2\njjuj2\u0014C\u0015\u0000\u000ej\fjZ\nVjjuj2\u0014C\u0015\u0000\u000ej\fj\u00001Z\njfuj\u0014C\u0015\u0000\u000ej\fj\u00002Z\njfj2\u0014CZ\njfj2:\nWhere the \fnal inequality holds because \f\u0015\u00001>\u00004=3 and\u000e\f=\f\n\f+2\u00142 when\n\f\u00142\f+ 4 so\u0000\u000e\f\u00002\u00140 . Therefore in case 2\n\u00151=2\u0012Z\njfuj\u00131=2\u0012Z\nVj\u001fjfuj\u00131=2\n\u0014CZ\njfj2:\n\u000336 PERRY KLEINHENZ AND RUOYU P. T. WANG\nCombining Lemmas 3.6 and 3.8 and, when M=S1, Lemma 3.5\n\u0015Z\nWjjuu0j\u0014C\u0000\n\u0015\u000ej+\u0016\u0001Z\njfuj+CZ\njfj2+ ~\"Z\n\u001eju0j2:\nThis along with Lemma 3.4\nZ\n\u001eju0j2+\u00162\u001ejuj2\u0014CX\nj\u0000\n\u0015\u000ej+\u0016\u0001Z\njfuj+CZ\njfj2+ ~\"Z\n\u001eju0j2:\nThen sincee\"can be taken small enough to allow the \u001eju0j2on the right hand side to\nbe absorbed into the left hand side\nZ\n\u001eju0j2+\u00162\u001ejuj2\u0014C\u0012\u0015\u000ej\n\u00162+1\nj\u0016j\u0013Z\njfj2:\nAnd soZ1\n\u00162ju0j2+Z\njuj2\u0014C\n\u00162\u0012\u00152\u000e\n\u00162+ 1\u0013Z\njfj2;\nwhich is exactly (3.7).\n4.Sharpness of energy decay\nThroughout this section we assume \f2(\u00001;0). We will follow the strategy of\n[Kle19] with some changes. We begin by constructing a sequence of solutions to a\nclassical equation with a semiclassically small perturbation term.\nLemma 4.1. Fix\f2(\u00001;0). There exists h0>0small such that for each h2(0;h0)\nandz2Cwithjzj\u00141\n2h\u00002, there exists vh;z2H1(0;1)such that\nQh;zvh;z=\u0010\n\u0000@2\ny\u0000i(y\f+h\u00002\f\n2+\f)\u0000h4\n2+\fz\u0011\nvh;z= 0;\nwith@yvh;z(0) = 1 and\nC\u00001\u0014kvh;zkH1(0;1)\u0014Ch\f\n2+\f:\nMoreovervh;z(0)is bounded and analytic in fz2C:jzj \u00141\n2h\u00002gfor each \fxed\nh2(0;h0), and there exists K > 0such that\nC\u00001h\u00002\f\n2+\f\u0014jvh;z(0)j\u0014K:\nThe constants are independent of hand\u0011.\nProof. 1. We claim for those handz,Qh;zis an invertible operator from H1(0;1) to\nH\u00001\n0(0;1), the dual space of H1(0;1). Consider the bounded bilinear form\nB(u;v) =h@yu;@yvi\u0000h4\n2+\fzhu;vi\u0000i\u0010\nhy\fu;vi+h\u00002\f\n2+\fhu;vi\u0011\n;SHARP POLYNOMIAL DECAY FOR SINGULAR DAMPING 37\nonH1(0;1), as the variational formulation for Qh;zwith Neumann conditions at 0.\nFor \fxedh>0, the bilinear form is strongly coercive:\njB(u;u)j2=\u0010\nk@yuk2\u0000h4\n2+\fRezkuk2\u00112\n+\u0010\nky\f\n2uk2+h\u00002\f\n2+\fkuk2+h4\n2+\fImzkuk2\u00112\n\u00151\n2k@yuk4\u0000h8\n2+\fjzj2kuk2+1\n2ky\f\n2uk4+1\n2h\u00004\f\n2+\fkuk2;\nwhere we used the algebraic inequality ( a\u0000b)2\u00151\n2a2\u0000b2for non-negative a;b. Note\nh8\n2+\fjzj2\u00141\n4h\u00004\f\n2+\fand we have\njB(u;u)j\u00151p\n2\u00121p\n2k@yuk4+1p\n2ky\f\n2uk4+1\n2h\u00002\f\n2+\fkuk2\u0013\n\u00151p\n8h\u00002\f\n2+\fkuk2\nH1(0;1);\nthe coercivity we need. By the Lax-Milgram theorem we know for any f2H\u00001\n0(0;1),\nthere is a unique v2H1(0;1) denoted by v=Rh;zfsuch thatB(v;\u0001) =hf;\u0001ias a\nfunctional on H1(0;1), that is,Qh;zv=fwith@yv(0) = 0. We further note Rh;zis a\nright inverse to Qh;z:H1!H\u00001\n0and\nC\u00001\n1\u0014kRh;zkH\u00001\n0!H1\u0014Ch\f\n2+\f; (4.1)\nthe upper and lower bounds are respectively from the Lax-Milgram theorem and the\nboundedness of Qh;z.\n2. We claim that there exists C > 0, such that for any f2y\f\n2L2(0;1) we have\njRh;zf(0)j\u0014Cky\u0000\f\n2fk:\nNote that Lemma 2.3 implies that f2H\u00001. Denotev=Rh;zf. Observe\nhf;vi=B(v;v) =k@yvk2\u0000h4\n2+\fRezkvk2\u0000i\u0010\nky\f\n2vk2+h\u00002\f\n2+\fkvk2+h4\n2+\fImzkvk2\u0011\n;\n(4.2)\nthe imaginary part of which implies\nky\f\n2vk2+h\u00002\f\n2+\fkvk2+h4\n2+\fImzkvk2\u0014jhf;vij\u0014C\u000f\u00001ky\u0000\f\n2fk2+\u000fky\f\n2vk2:\nNote thatjzj\u00141\n2h\u00002and absorb the last term on the right by the left, to obtain\nky\f\n2vk2+h\u00002\f\n2+\fkvk2\u0014C\u000f\u00001ky\u0000\f\n2fk2: (4.3)\nFurthermore, since y\f\n2\u00151 on (0;1], we havekvk2\nL2(0;1)\u0014Cky\u0000\f\n2fk. The real part of\n(4.2) and (4.3) implies\nk@yvk2\u0014Cky\u0000\f\n2fk2+Cky\f\n2vk2+h4\n2+\fRezkvk2\u0014Cky\u0000\f\n2fk2:\nThus, by the trace theorem, jv(0)j\u0014kvkH1(0;1)\u0014Cky\u0000\f\n2fkas we need.38 PERRY KLEINHENZ AND RUOYU P. T. WANG\n3. We claim Rh;zf(0) is analytic in z2fz2C:jzj\u00141\n2h\u00002g, at each \fxed h.\nFixz02fjzj\u00141\n2h\u00002g. For anyz2fjz\u0000z0j\u0014kRh;z0k\u00001\nH\u00001\n0!H1g, we have from the\nresolvent identity that\nRh;z=Rh;z0(1\u0000(z\u0000z0)Rh;z0)\u00001=1X\nk=0(z\u0000z0)kRk+1\nh;z0;\nwhich converges in norm. Thus Rh;z:H\u00001\n0!H1is analytic infjzj\u00141\n2h\u00002g, and\nbecause the trace at 0 is bounded by the H1-norm, we have the desired analyticity.\n4. We now construct vh;\u0011with speci\fc Neumann data at 0. Fix a real \u001f2C1\nc([0;1))\nwith\u001f(0) = 0,@y\u001f(0) = 1. We pick \u001fsuch thatky\f\u001fk\u00152(C1+ 1)k\u001fk, whereC1is\nthe bound in (4.1). Then Qh;z\u001fis smooth and compactly supported and\n2C1k\u001fk\u0014ky\f\u001fk\u0000h\u00002\f\n2+\fk\u001fk\u0014kQh;z\u001fkL2\u0014Ck\u001fkH2\u0014C\u001f:\nNow letvh;z=\u001f\u0000Rh;zQh;z\u001f. Immediately, we see Qh;zvh;z= 0 and by the construction\nofRh;zin Step 1, @yRh;zQh;z\u001f= 0 so@yvh;z(0) = 1. Now, note that since Rh;zis\nbounded below as an operator\nkRh;zQh;z\u001fkH1\u00151\nC1kQh;z\u001fkL2\u00152k\u001fk:\nAnd sinceRh;zis bounded above as an operator\nkRh;zQh;z\u001fkH1\u0014Ch\f\n2+\f:\nTherefore\nC\u0014kvh;zkH1\u0014Ch\f\n2+\f:\nFurthermore, note that\r\r\ry\u0000\f\n2Qh;z\u001f\r\r\r\u0014C\u001f, so by Step 2 and \u001f(0) = 0;jvh;z(0)j\u0014K.\n5. It remains to show jvh;z(0)jis bounded from below. Observe\n0 =hQh;zvh;z;vh;zi=vh;z(0)@yvh;z(0) +k@yvh;zk2\u0000h4\n2+\fRezkvh;zk2\n\u0000i\u0010\nky\f\n2vh;zk2+h\u00002\f\n2+\fkvh;zk2+h4\n2+\fImzkvh;zk2\u0011\n;\nwhich after simpli\fcation becomes\njvh;z(0)j\u0015C\u00001\u0010\nk@yvh;zk2+ky\f\n2vh;zk2+h\u00002\f\n2+\fkvh;zk2\u0011\n\u0015C\u00001h\u00002\f\n2+\fkvh;zk2\nH1\u0015C\u00001h\u00002\f\n2+\f;\nwhere we used @yvh;z(0) = 1. \u0003\nRemark 4.2. Note that the resolvent Rh;zis not de\fned at h= 0, which makes it\ndi\u000berent from the case of [Kle19, Lemma 4.3]. We use a di\u000berent strategy below, rather\nthan invoking the implicit function theorem at h= 0;z= 0, ash!0.\nWe now move to construct the solutions to a complex absorbing potential problem\nwith a complex Coulomb potential.SHARP POLYNOMIAL DECAY FOR SINGULAR DAMPING 39\nProposition 4.3 (Eigenmodes on the half-line) .Fix\f2(\u00001;0)and letV(x) =\n1x\u00150(x\f+ 1) . There exist h0;C > 0such that for all h2(0;h0), there are \u0011h2C,\nvh2H1(\u0000\u0019\n2;1)withvh(\u0000\u0019\n2) = 0 andkvhkH1(\u0000\u0019\n2;1)\u0014Csuch that\nQhvh(x) =\u0000\n\u0000h2@2\nx\u0000iV(x)\u0000h2\u00112\nh\u0001\nvh(x) = 0: (4.4)\nFurthermore\nj\u0011h\u00002j\u00149K\n\u0019h2\n2+\f;\nwhereKis the distinguished bound in Lemma 4.1 and is independent of h.\nProof. 1. We \frst solve the equation (4.4) on (0 ;1). By rescaling via y=h\u00002\n2+\fx, the\nequation on (0 ;1) is reduced to\nQhvh=h2\f\n2+\fQh;\u0011vh(y) =h2\f\n2+\f\u0010\n\u0000@2\ny\u0000i(y\f+h\u00002\f\n2+\f)\u0000h4\n2+\f\u00112\u0011\nvh(y) = 0;\nony2(0;1), whereQh;\u0011isQh;zin Lemma 4.1 with z=\u00112\nh. Apply Lemma 4.1 to\nsee that for any h2(0;h0),\u00112f\u00112C:j\u0011j\u00141\n2h\u00002g, there exists a sequence of\nvh;\u00112H1(0;1) thatQh;\u0011vh;\u0011(y) = 0 withvh;\u0011(0) is analytic in \u0011and\njvh;\u0011(0)j\u0014K; @yvh;\u0011(0) = 1:\nLetv+\nh;\u0011(x) =vh;\u0011(h\u00002\n2+\fx)2H1(0;1) denote such solutions in x2(0;1). Note that\nQhv+\nh;\u0011= 0 on (0;1),kv+\nh;\u0011kH1\u0014Ch\u00001+\f\n2+\fandv+\nh;\u0011(0) is analytic in \u0011and\n\f\fv+\nh;\u0011(0)\f\f\u0014K; @xv+\nh;\u0011(0) =h\u00002\n2+\f:\nThis is our solution to (4.4) corresponding to parameters h;\u0011on the positive half real\nline (0;1).\n2. We then solve the equation (4.4) on ( \u0000\u0019\n2;0). On (\u0000\u0019\n2;0) the potential W(x)\u00110\nand the equation (4.4) is solved by v\u0000\nh;\u0011= sin(\u0011(x+\u0019\n2)). We take note of the Cauchy\ndata atx= 0:\nv\u0000\nh;\u0011(0) = sin(\u0011\u0019\n2); @xv\u0000\nh;\u0011(0) =\u0011cos(\u0011\u0019\n2):\n3. We now match v\u0006\nh;\u0011at 0. The transition condition is now\nv\u0000\nh;\u0011(0)\n@xv\u0000\nh;\u0011(0)=v+\nh;\u0011(0)\n@xv+\nh;\u0011(0);\nwhich is\ntan(\u0011\u0019\n2) =h2\n2+\f\u0011v+\nh;\u0011(0): (4.5)\nWe will be looking for solutions \u0011= 2+h2\n2+\f\u0016with parameter \u0016inf\u00162C:j\u0016j\u00149K\n\u0019g.\nLetv+\nh;\u0016=v+\nh;\u0011(\u0016)be parametrised in \u0016. Consider the objective function\nFh(\u0016) = (2 +h2\n2+\f\u0016)v+\nh;\u0016(0)\u0000h\u00002\n2+\ftan(\u0019(2 +h2\n2+\f\u0016)=2);40 PERRY KLEINHENZ AND RUOYU P. T. WANG\nthe zeros of which solve (4.5). We now assume h0is small such that v+\nh;\u0016(0) andFh(\u0016)\nare analytic infj\u0016j\u00149K\n\u0019g, at eachh2(0;h0). Note\ntan(\u0019(2 +h2\n2+\f\u0016)=2) =1\n2\u0019h2\n2+\f\u0016+O(h6\n2+\f\u00163);\nand sincev+\nh;\u0016(0) and\u0016are bounded by h-independent constants, we have\nFh(\u0016) = 2v+\nh;\u0016(0)\u00001\n2\u0019\u0016+O(h2\n2+\f\u0016):\nAssume against contradiction that Fh(\u0016) does not have a zero in fj\u0016j\u00149K\n\u0019g. Since\nFh(\u0016) is analytic in the disk, by the minimum modulus principle, jFh(\u0016)jachieves its\nminimum overfj\u0016j\u00149K\n\u0019gon its boundary. Consider any j\u0016j=9K\n\u0019, we have when his\nsmall that\njFh(\u0016)j\u00151\n2\u0019j\u0016j\u00002\f\fv+\nh;\u0016(0)\f\f+O(h2\n2+\fj\u0016j)\u00159\n4K:\nHowever,Fh(0) =v+\nh;0(0) andjFh(0)j\u00142K\u00149\n4K, which is a contradiction. Thus for\nanyh2(0;h0), there exists \u0016hinf\u00162C:j\u0016j\u00149K\n\u0019gsuch thatFh(\u0016h) = 0. Then\n\u0011h= 2 +h2\n2+\f\u0016hsatis\fes the transition condition (4.5) at 0, and Qhvh= 0 on (\u0000\u0019;1)\nwith\nvh=v\u0000\nh;\u0011h\u0000\fhv+\nh;\u0011h; \fh= 2h2\n2+\f+\u0016hh4\n2+\f+O(h6\n2+\f);\nandkvhkH1\u0018Cinh2(0;h0). \u0003\nRemark 4.4. Those eigenfunctions have most of its H1-mass infV= 0g, and pene-\ntrates intofV > 0gwith the Dirichlet data vh(0) =O(h2\n2+\f) at the boundary between\nthose two regimes. The eigenvalues are h2\n2+\f-sized perturbation from the Dirichlet\neigenvalues 2 corresponding to those eigenfunctions that vanish at \u0000\u0019\n2and 0. When\n\f!0\u0000, this is consistent with the observation made by Nonnenmacher in [ALN14] in\nthe limit case.\nProposition 4.5 (Low-frequency quasimodes on S1).Parametrize S1by(\u0000\u0019;\u0019)and\nletW(x) = 1jxj\u0015\u0019\n2\u0000\n(jxj\u0000\u0019\n2)\f+ 1\u0001\n. Then for any sequence f\u0015kg\u001aR,\u0015k!1 , there\nexists a sequence of uk2H1(S1),kukkL2\u00111such that\n\u0000\n\u0000@2\nx\u0000i\u0015kW(x)\u00002\u0001\nuk=OL2(\u0015\u00001\n2+\f\nk):\nProof. Consider a cuto\u000b \u001f2C1[\u0000\u0019\n2;\u0019\n2] with\u001f\u00111 on [\u0000\u0019\n2;\u0019\n4], and\u001f\u00110 on [3\u0019\n8;\u0019\n2].\nNote that [Qh;\u001f]2\t1\nhhas semiclassical microsupport inside T\u0003((\u0019\n4;3\u0019\n8)), on which Qh\nis semiclassically elliptic. For ( x;\u0018)2T\u0003((\u0019\n4;3\u0019\n8))\nj\u001bh(Qh)j=\f\f\u00182\u0000iV\f\f\u0015j\u0018j2+ 1\u00151:\nTherefore, by the elliptic estimate in [DZ19, Theorem E.33], for any N2N, there\nexistsC > 0, dependent of N, such that for any h2(0;h0)\nk[Qh;\u001f]vhkH1\nh\u0014CkQhvhkL2+ChNkvhkH\u0000N\nh\u0014ChNkvhkL2\u0014ChN;SHARP POLYNOMIAL DECAY FOR SINGULAR DAMPING 41\nand herebykQh\u001fvhkL2=O(hN). Lethk=\u0015\u00001\n2\nk!0 and\nuk(x) = sgn(x)(\u001fvhk)(jxj\u0000\u0019\n2)=\r\r\rsgn(x)(\u001fvhk)(jxj\u0000\u0019\n2)\r\r\r\nL2(\u0000\u0019;\u0019):\nThenuk2H1(S1) areL2-normalized, vanish near \u0006\u0019and\n\u0000\n\u0000@2\nx\u0000i\u0015kW(x)\u0000(2 +\u0016k)2\u0001\nuk=OL2(\u0015\u0000N\n2+1\nk );\nwhere\u0016k=\u0016hk=O(\u0015\u00001\n2+\f\nk). PickNlarge such that\n\u0000\n\u0000@2\nx\u0000i\u0015kW(x)\u00002\u0001\nuk=OL2(\u0015\u00001\n2+\f\nk):\nSo theseukare appropriate quasimodes. \u0003\nProof of Theorem 2. In Proposition 4.5, we can pick a sequence \u0015k= (2 +k2)1\n2!1 ,\nwk(x;y) =uk(x) sin(ky):\nThen\nP\u0015kwk=\u0000\n\u0000@2\nx\u0000@2\ny\u0000i\u0015kW(x)\u0000\u00152\nk\u0001\nwk=OL2x(\u0015\u00001\n2+\f\nk) sin(ky):=OL2(T2)(\u0015\u00001\n2+\f\nk):\nand this means that for any \u000f>0,\nkP\u0015kwkkL2(T2)=o(\u0015\u00001\n2+\f+\u000f\nk )kwkkL2(T2):\nBy Proposition 1.7, the solution to the damped wave equation (1.1) cannot be stable\nat the rate ofhti\u0000\f+2\n\f+3\u0000\u000ffor any\u000f>0. \u0003\nReferences\n[ALN14] N. Anantharaman, M. L\u0013 eautaud, and S. Nonnenmacher. Sharp polynomial decay rates for\nthe damped wave equation on the torus. Anal. PDE , 7(1):159{214, 2014.\n[AM14] N. Anantharaman and F. Maci\u0012 a. Semiclassical measures for the Schr odinger equation on\nthe torus. J. Eur. Math. Soc. (JEMS) , 16(6):1253{1288, 2014.\n[Arn22] A. Arnal. 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Anal. , 218(2):425{444, 2005.\n[Mor61] C. S. Morawetz. The decay of solutions of the exterior initial-boundary value problem for\nthe wave equation. Comm. Pure Appl. Math. , 14:561{568, 1961.\n[Pr u84] J. Pr uss. On the spectrum of C0-semigroups. Trans. Amer. Math. Soc. , 284(2):847{857,\n1984.\n[Ral69] J. V. Ralston. Solutions of the wave equation with localized energy. Comm. Pure Appl.\nMath. , 22:807{823, 1969.\n[RT74] J. Rauch and M. Taylor. Exponential decay of solutions to hyperbolic equations in bounded\ndomains. Indiana Univ. Math. J. , 24:79{86, 1974.\n[Sta17] R. Stahn. Optimal decay rate for the wave equation on a square with constant damping on\na strip. Z. Angew. Math. Phys. , 68(2), 2017.\n[Sun22] C. Sun. Sharp decay rate for the damped wave equations with convex-shaped damping. Int.\nMath. Res. Not. IMRN , to appear, 2022.\n[Wan21a] R. P. T. Wang. Sharp polynomial decay for waves damped from the boundary in cylindrical\nwaveguides. Math. Res. Lett. , to appear, 2021.\n[Wan21b] R. P. T. Wang. Stabilisation of waves on product manifolds by boundary strips. Proc.\nAmer. Math. Soc. , to appear, 2021.\nDepartment of Mathematics, Michigan State University, East Lansing, MI 48824,\nUSA\nE-mail address , P. Kleinhenz: kleinh29@msu.edu\nDepartment of Mathematics, Northwestern University, Evanston, IL 60208, USA\nE-mail address , R. P. T. Wang: rptwang@math.northwestern.edu"    },    {        "title": "0908.1683v1.Linear_Fractionally_Damped_Oscillator.pdf",        "content": " 1 Linear Fractionally Damped Oscillator \nMark Nabera) \nDepartment of Mathematics \nMonroe County Community College \nMonroe, Michigan, 48161-9746  \n In this paper the linearly damped oscillator equation is considered with the damping term generalized to a Caputo fractional derivative.  The order of the derivative being considered is \n0\u0002\u0001\u00021.  At the lower end ( \u0001=0) the equation represents an un-damped \noscillator and at the upper end ( \u0001=1) the ordinary linearly damped oscillator equation is \nrecovered.  A solution is found analytically and a comparison with the ordinary linearly \ndamped oscillator is made.  It is found that there are nine distinct cases as opposed to the usual three for the ordinary equation (damped, over-damped, and critically damped).  For three of these cases it is shown that the frequency of oscillation actually increases with increasing damping order before eventually falling to the limiting value given by the \nordinary damped oscillator equation.  For the other six cases the behavior is as expected, \nthe frequency of oscillation decreases with increasing order of the derivative (damping term).     \na)Electronic mail: mnaber@monroeccc.edu  21.  INTRODUCTION  \n \n In this paper the linearly damped oscillator equation is considered with the \ndamping term replaced by a fractional derivative [1] whose order, \u0001, will be restricted to, \n0\u0002\u0001\u00021, \n \n  Dt2x+\u00010Dt\u0002x+\u00032x=0. (1) \n \nBurov and Barki [2, 3] examined such an equation in connection with critical behavior.  \nThey were able to determine a solution in terms of generalized Mittag-Leffler functions.  \nNon-linear fractional oscillators have been studied numerically by Zaslavsky [4].  He was \nprimarily interested in chaotic behavior.  It is hoped that a careful study of the analytic \nsolution to the linear fractionally damped equation will help shed light on properties of \nthe nonlinear equation and be of use for direct applications of fractionally damped \noscillations (see for example [5, 6]). \n \n In this paper the Caputo formulation of the fractional derivative will be used.  The \nCaputo derivative is preferred over the Riemann-Liouville derivative for physical \nreasons.  Consider the Laplace transform of the two formulations of the fractional \nderivative for 0<\u0001<1 \n \n   L0RLDt\u0002f(t) () =s\u0002F(s)\u00010RLDt\u0002\u00011f(t)0, (2) \n \n   L0CDt\u0002f(t) () =s\u0002F(s)\u0001f(t)0. (3) \n \nThe constant term arising from the Laplace transform of the Caputo derivative is merely \nthe initial value of the function.  For the Riemann-Liouville derivative this is not the case.  \nThe constant term arising from the Laplace transform currently has no simple physical \ninterpretation.  Hence the Caputo fractional derivative seems to be more useful for \nmodeling physical systems. \n \n If the order of the fractional damping term is allowed to become 3/2 (outside the \nrange of values considered in this paper) the equation is usually referred to as the Bagley-\nTorvik equation (see for example [1, 7, 8]).  The solution of this equation exhibits \ndamped oscillatory behavior similar to what we expect to find for the equation studied in \nthis paper.  The Bagley-Torvik equation was originally derived to study the motion of a \nrigid plate in a Newtonian fluid [7]. \n \n The analytic solution to the fractionally damped equation is found by means of \nLaplace transform.  For the sake of clarity, and for pointing out some unique difficulties \nwith the factional equation, a comparison with the Laplace transform method as applied \nto the non-fractional case is made.  It is found that there are nine distinct cases for the \nfractionally damped equation as opposed to the usual three cases for the non-fractional \nequation.  In six of the nine cases the results are as expected; increasing the order of the \nfactional derivative increases the effects of the damping (i.e. the frequency of the  3damping slows as the order of the derivative increases).  However, in three cases, the \nfrequency of the damping actually increases as the order of the fractional derivative \nincreases until a peak value is reached after which the frequency falls to its non-fractional \nlimit.  The physical reason for this increase in the oscillation frequency is not yet clear. \n \n \n2.  THE NON-FRACTIONAL CASE  \n \n Before a solution to the linear fractionally damped oscillator equation is \nconstructed it will be useful to review the Laplace transform method of solution for the \nlinearly damped oscillator equation, \n \n  Dt2x+\u0001Dtx+\u00022x=0. (4) \n \nThe constants \u0001 and \u0001 are taken to be real and positive. \u0001 is the damping force per unit \nmass. \u00012 is the restoring force per unit mass.  In both cases (fractional and non-\nfractional) the following initial conditions will be used \n \n  x(0)=x0,\nDtx(0)=x1. (5) \n \nTransforming Eq. (4) together with the initial conditions Eq. (5) gives the following \n \n  s2X(s)\u0001sx0\u0001x1+\u0002sX(s)\u0001x0 () +\u00032X(s)=0, (6) \n \nor, \n \n  X(s)=sx0\ns2+\u0001s+\u00022+x1+\u0001x0\ns2+\u0001s+\u00022. (7) \n \nEq. (7) can be inverted using tables, however, to shed light on a problem that will happen \nlater, Eq. (7) will be inverted via the complex inversion integral.  The exponents on the s \nvariable in both terms are whole numbers, hence there will not be a branch cut in the \ncontour integral and the Bromwich contour can be used, \n \n  x(t)=Residue \u00011\n2\u0002iestX(s)ds\u0003. (8) \n \nRecall that the Bromwich contour begins at \u0002\u0001i\u0003 goes vertically up to \u0001+i\u0002  (where \n\u0001 is chosen so that all poles will lie to the left of the vertical contour line and thus all \npoles will be captured within the contour) and then travels in a half circle (to the left, \ncounter clockwise) back to \u0002\u0001i\u0003.  For this problem there is no contribution from the \ncontour integral.  The only contribution comes from the residue.  The residue is generated \nfrom the roots of the following quadratic equation,  4\n  s2+\u0001s+\u00022=0. (9) \n \nThere are three different cases. \n \n1) \u0001>2\u0002; 2 unequal real roots that are negative. \n \n  s1, 2 =\u0001\u0002±\u00022\u00014\u00032\n2 (10) \n \n2) \u0001=2\u0002; 2 repeated real roots that are negative. \n \n  s3=\u0001\u0002\n2=\u0001\u0003 (11) \n \n3) \u0001<2\u0002; 2 complex roots whose real parts are negative. \n \n  s4,5 =\u0001\u0002±i4\u00032\u0001\u00022\n2 (12) \n \nSee fig. 1 below for a graphical representation of the location of the roots. \n \n \n \n \n \n \n \n \n \n \n \n           Fig. 1 \n  Note that in cases one and three the poles will be of order one and in case two the \npole will be of order two.  Note also that if there were no damping the poles would be on \nthe imaginary axis at \n±i\u0001. \n \n Computing the residue for case one gives \n \n  Residue =lim\ns\u0004s1,2s\u0001s1,2 () est sx0\ns2+\u0002s+\u00032+x1+\u0002x0\ns2+\u0002s+\u00032\u0005\n\u0007\u0006\b\n\n\t, (13) \n \nor, \n  x(t)=es1t\n2s1+\u0001s1x0+x1+\u0001x0 () +es2t\n2s2+\u0001s2x0+x1+\u0001x0 () . (14)  5As s1 and s2 are both negative this solution will decay exponentially.  This is usually \nreferred to as the over-damped case. \n \n Case three is computed the same way as case one.  Now the poles are complex so \nthe exponential function can be expressed using sine and cosine with an over-all \nexponential damping factor \n \n  x(t)=e\u0001\u0002tx0cos(\u0004t)+(2x1+\u0003x0)\n2\u0004sin(\u0004t)\u0005\n\u0007\u0006\b\n\n\t, (15) \n \nwhere\u0001\u0004=\u00052\u0001\u00032/4\u0001and\u0001\u0002=\u0003\n2.  Notice that the presence of damping causes the \neffective angular frequency, \u0001, to be smaller than the un-damped angular frequency, i.e. \nthe oscillations go slower, as one might expect if there were damping to impede the \nmotion.  This is usually referred to as the under-damped case.  By comparison, cases one \nand two could be viewed as having a zero frequency or an infinite period. \n \n For case two the pole is of order 2, and the residue is given by \n \n  lim\ns\u0004\u0001\u0003d\ndss+\u0003 ()2estsx0+x1+\u0002x0\ns2+\u0002s+\u00032\u0005 \n\u0007 \u0006 \b \n\n \t \u0005 \n\u0007 \u0006 \b\n\n\t. (16) \n \nRecall that \u0001=2\u0002, this allows the denominator to be factored.  The limit then becomes, \n \n  lim\ns\u0003\u0001\u0002d\ndsestsx0+x1+2\u0002x0 () () , (17) \n \nor, \n \n  x(t)=e\u0001\u0002tt\u0002x0+x1 () +x0 () . (18) \n \nThis is usually called the critically-damped case.  Graphs of sample solutions to these \nthree cases can be found in any introductory book on differential equations. \n \n \n3.  THE FRACTIONAL CASE  \n \n Now consider Eq. (1).  In this case, \u0001 has units of time raised to the power \u0002\u00012.  \nHence the over all units of the second term remain the same as in Eq. (4).  There are two \ncases to consider, 0<\u0001<1 and 1<\u0001<2.  We will consider the former in this paper.  \nThe Laplace transform of Eq. (1) is \n \n  s2X\u0001sx0\u0001x1+\u0002s\u0003X\u0001s\u0003\u00011x0 () +\u00042X=0, (19)  6or, \n \n  X=sx0+x1+\u0002s\u0003\u00011x0\ns2+\u0002s\u0003+\u00042. (20) \n \nIf Eq. (20) is inverted using a contour integral a branch cut is needed on the negative real \naxis due to the fractional exponents on the complex variable s.  Hence, a Hankel contour \nwill be used.  This contour starts at \u0002\u0001i\u0003 goes vertically up to \u0001+i\u0002 (where \u0001 is again \nchosen so that all poles will lie to the left of the vertical contour line) and then travels in a \nquarter circle arc (to the left) to just above the negative real axis (i.e. \u0001\u0002).  The contour \nthen has a cut that goes into the origin (following the negative real axis), around the \norigin in a clockwise sense (to just below the negative real axis) and then back out to \u0001\u0002\n.  The contour then has another quarter circle arc to \u0002\u0001i\u0003. \n \n Now the question is, where are the poles?  This is a somewhat more involved \nquestion than in the standard linearly damped model.  To find the poles the following \nequation needs to be solved \n \n  s2+\u0001s\u0002+\u00032=0. (21) \n \nWhich, for an arbitrary \u0001, is not a trivial problem.  To determine if there are solutions, \nand if so how many, let s=rei\u0001 then Eq. (21) breaks into 2 equations, a real and an \nimaginary part \n \n  r2cos(2\u0003)+\u0001r\u0002cos(\u0002\u0003)+\u00042=0,\n\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001 r2sin(2\u0003)+\u0001r\u0002sin(\u0002\u0003)=0. (22) \n \nCould there be a solution on the positive real axis?  No, in this case \u0001=0 and the first \nequation of Eq. (22) would be the sum of three positive non-zero terms, which would \nnever be zero.  Could there be a solution on the negative real axis?  No, in this case \n\u0002=\u0001 and the second term of the second equation of Eq. (22) would never be zero.  \nUsing similar arguments we can show that there are no solutions on the positive or \nnegative imaginary axes, recall 0<\u0001<1.  It can also be shown that no solutions are in \nthe right half plane (both terms of the second equation would always be positive).  If \nthere are solutions they should be in pairs, complex conjugates, with \u0001/2<\u0002<\u0001 and \n\u0001\u0002/2>\u0003>\u0001\u0002.  To attempt to find a solution first solve the second equation of Eq. (22) \nfor r and substitute this into the first equation (only look for positive \u0001 values first), \n \n  \u0001\u0002sin(\u0003\u0004)\nsin(2\u0004)\u0006\n\b\u0007\t\n\u000b\n2/(2\u0001\u0003)\ncos(2\u0004)+\u0002\u0001\u0002sin(\u0003\u0004)\nsin(2\u0004)\u0006\n\b\u0007\t\n\u000b\n\u0003/(2\u0001\u0003)\ncos(\u0003\u0004)+\u00052=0. (23) \n \nThe reader may be worried about the negative sign and the fractional exponent in Eq. \n(23), however, for the restricted angular range being considered, \u0001/2<\u0002<\u0001, sin(2\u0001) is \nalways negative.  So, the argument of the root will always be positive.  7 Given values for \u0002,\u0001\u0001,\u0001and\u0001\u0003 it would appear to be impossible to solve Eq. (23) \nfor \u0001.  Equation (23) can be simplified to a more aesthetically pleasing form, \n \n  sin(\u0003\u0004) ()\u0003\nsin(2\u0004) ()2\u0006\n\b\u0007\t\n\u000b\n1/(2\u0001\u0003)\nsin((2\u0001\u0003)\u0004)=\u0005\n\u00021/(2\u0001\u0003)\u0006\n\b\u0007\t\n\u000b\n2\n. (24) \n \n Now it needs to be seen if there is a \u0001 value that will satisfy Eq. (24).  For this \nequation to be true sin((2\u0001\u0002)\u0003) needs to be positive.  This will only happen on the \nrestricted domain \u0003\n2<\u0004<\u0003\n2\u0001\u0002.  Now the question becomes, on this restricted domain \ncan we pick a \u0001 value that will make the left hand side of Eq. (24) as large or as small as \nwe wish?  Thus ensuring that no matter what the values of \u0001, \u0001, and \u0001 we are given we \ncan always find a \u0001 value that will satisfy Eq. (24).  Consider the two limits \n \n  lim\n\u0004\u0006\u0003/2+sin(\u0002\u0004) ()\u0002\nsin(2\u0004) ()2\u0007\n\t\b\n\f\u000b1/(2\u0001\u0002)\nsin((2\u0001\u0002)\u0004)=\u0005, (25) \n \n  lim\n\u0004\u0005\u0003/(2\u0001\u0002)\u0001sin(\u0002\u0004) ()\u0002\nsin(2\u0004) ()2\u0006\n\b\u0007\t\n\u000b\n1/(2\u0001\u0002)\nsin((2\u0001\u0002)\u0004)=0. (26) \n \nSince the left hand side of Eq. (24) is continuous in \u0001 and we have the two limits above, \nEqs. (25) and (26), it is guaranteed that there will be at least one solution to Eq. (24) and \nhence there will be at least two poles for the residue calculation.  If we can show that the \nleft hand side of Eq. (24) decreases monotonically in \u0001 over the restricted domain then \nwe know that there will be only one solution to Eq. (24), and thus only two poles in the \nresidue calculation.  To show that the left hand side of Eq. (24) decreases monotonically \nin \u0001 we need to show that the derivative of the left hand side of Eq. (24) with respect to \n\u0001 is always negative, i.e. \n \n  \u0004\n\u0004\u0003sin(\u0002\u0003) ()\u0002\nsin(2\u0003) ()2\u0005\n\u0007\u0006\f\n\u000e\r1/(2\u0001\u0002)\nsin((2\u0001\u0002)\u0003)\b\n\t\u000b\n\n\u000b\u000f\n\u0010\u000b\n\u0011\u000b<0. (27) \n \nComputing the derivative, doing some algebra, and throwing away over all factors that \nare always positive we have, \n \n  \u00022sin2(2\u0003)\u00014\u0002sin(2\u0003)sin(\u0002\u0003)cos((2\u0001\u0002)\u0003)+4s i n2(\u0002\u0003)>0. (28) \n \nOn the restricted domain sin(\u0001\u0002)>0, sin(2\u0001)<0, and cos((2\u0001\u0002)\u0003)\u00041.  This reduces \nEq. (28) to \n  8\n  \u00022sin2(2\u0003)\u00014\u0002sin(2\u0003)sin(\u0002\u0003)+4s i n2(\u0002\u0003)>0. (29) \n \nEq. (29) can now be factored into a perfect square and prove the assertion made in Eq. \n(27), \n \n  \u0002sin(2\u0003)\u00012s i n (\u0002\u0003) ()2>0. (30) \n \nHence the left hand side of Eq. (24) will decrease monotonically on the restricted domain \nwith the upper bound being \u0001 and the lower bound being 0.  To summarize, it has just \nbeen shown that there is always one solution, with a positive angle, to Eq. (24) and this \nsolution must be such that \u0003\n2<\u0004<\u0003\n2\u0001\u0002.  Consequently there will be two poles for the \nresidue calculation and they will be complex conjugates of each other.  Notice that for the \nfractionally damped equation repeated roots are not possible.  Repeated roots can only \nhappen when the order of the derivative becomes one.  See fig. 2 below for a graphical \nrepresentation of the location of the roots. \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 2 \n \n Now that the question of the poles has been settled the solution to Eq. (1) can be \ngenerated.  Denote the two poles as \n \n  s6,7=\u0001±i\u0003=re±i\u0002, (31) \n \nwhere \u0001 and \u0001 are determined from the r and \u0001 values that satisfy Eq. (24) in the usual \nway, r=\u00012+\u00022 and tan(\u0002)=\u0003/\u0001.  Note that \u0001 is negative, the two solutions are in \nthe second and third quadrants and s7 is the complex conjugate of s6.  Note also that \u0001 \nplays the role of \u0001\u0002/2 from the non-fractional case.  The poles are of order one and the \nresidue is given by, \n \nResidue =lim\ns\u0005s6s\u0001s6 () estsx0+x1+\u0002s\u0003\u00011x0\ns2+\u0002s\u0003+\u00042\u0006\n\b\u0007\t\n\u000b\n+lim\ns\u0005s7s\u0001s7 () estsx0+x1+\u0002s\u0003\u00011x0\ns2+\u0002s\u0003+\u00042\u0006\n\b\u0007\t\n\u000b\n, (32) \n  9  =es6ts6x0+x1+x0\u0002s6\u0003\u00011\n2s6+\u0003\u0002s6\u0003\u00011\u0004\n\u0006\u0005\u0007\n\t\b+es6ts6x0+x1+x0\u0002s6\u0003\u00011\n2s6+\u0003\u0002s6\u0003\u00011\u0004\n\u0006\u0005\u0007\n\t\b, (33) \n \nwhere s7 has been replaced by s6.  After some algebra this can be reduced to, \n \n2e\u0002tcos(\u0006t)x02r2+\u0004\u00032r2\u0004\u00012+\u0003r\u0004(\u0004+2)cos(\u0005(\u0004\u00012)) () +x12rcos(\u0005)+\u0004\u0003r\u0004\u00011cos(\u0005(\u0004\u00011)) ()\n4r2+4\u0004\u0003\u0004r\u0004cos( (2\u0001\u0004)\u0005)+\u00042\u00032\u0004r2\u0004\u00012\n+2e\u0002tsin(\u0006t)x0\u0003r\u0004(\u0004\u00012)sin((\u0004\u00012)\u0005) () +x12rsin(\u0005)+\u0004\u0003r\u0004\u00011sin(\u0005(\u0004\u00011)) ()\n4r2+4\u0004\u0003r\u0004cos( (2\u0001\u0004)\u0005)+\u00042\u00032r2\u0004\u00012\n   (34) \n  For the contour integral the only contributions come from the paths along the \nnegative real axis \n \n  \u0002\n\u0004(Rx0\u0001x1)sin(\u0003\u0004)+x0\nR(R2+\u00052)sin(\u0004(\u0003\u00011))\n(R2+\u00052)2+2\u0002R\u0003(R2+\u00052)cos(\u0003\u0004)+(\u0002R\u0003)2e\u0001RtR\u0003dR\n0\u0006\u0007. (35) \n \nThe solution to Eq. (1) is then, Eq. (35) subtracted from Eq. (34).  This may look overly \ncomplicated but the solution does have the general form of \n \n  x(t)=Ae\u0002tcos(\u0003t)+Be\u0002tsin(\u0003t)\u0001Decay\u0001function . (36) \n \nNotice that the decay function, Eq. (35), goes to zero if \u0001 goes to zero or one, i.e. if Eq. \n(1) goes to its non-fractional limits the decay function goes away, as expected.  The \ndamping factor e\u0001t is similar to the damping factor for the non-fractional case, e\u0001\u0002t/2.  \nNotice that since the poles, for the residue calculation, have non-zero imaginary and non-\nzero real parts we will not have the same three distinct cases as we did for the non-\nfractional case (critically-damped, over-damped, and under-damped). \n \n \n4.  THE OSCILLATION FREQUENCY  \n \n Consider the frequency of the oscillation component of the solution, \u0001=Im(s6).  \nOne question we might ask is: how does the frequency change as we change the order of \nthe fractional damping?  When \u0001 is set to zero we have an un-damped oscillator with \nfrequency \n \n  \u0002=\u0001+\u00032. (37) \n  10When \u0001 is set to one we have the three cases given at the beginning of the chapter; over-\ndamped, critically-damped, and under-damped.  So, the frequency may be zero or non-\nzero, i.e., \n \n  \u0003=0\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001 \u0002\u00052\u0004,\n\u0003=4\u00042\u0001\u00022\n2\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0002<2\u0004. (38) \n \nFor 0<\u0001<1 there will always be a non-zero frequency.  Note that \n0\u00044\u00032\u0001\u00022/2 <\u0002+\u00032. \n \n In the non-fractional case increasing \u0001 causes the frequency of oscillation to \nbecome smaller, monotonically, until the critical cases are reached and the oscillation \nperiod becomes infinite (these are the critical and over-damped cases).  In the fractional \ncase the frequency of oscillation, \u0001=Im(s6), now depends on the order of the derivative, \n\u0001, as well as \u0001 and \u0001.  To try to determine how \u0001 depends on these three parameters \nconsider s to be a function of \u0001, on 0\u0002\u0001\u00021, implicitly defined by \n \n  s2+\u0001s\u0002+\u00032=0, (39) \n \nfor fixed values of \u0001 and \u0001 (both being positive).  Let us restrict our attention to the \nupper half plane for s.  As such s will be one-to-one on 0\u0002\u0001<1.  Due to the fractional \nexponent causing a branch cut on the negative real axis s will not be one-to-one at \u0001=1. \n \nNow the question arises, does \u0001 fall monotonically with respect to \u0001?  To get at the \nanswer to this consider the derivative of Eq. (39) with respect to \u0001 and isolate ds /d\u0001 \n(remember, \u0001 and \u0001 are being held fixed) \n \n  ds\nd\u0003=\u0001\u0002s\u0003ln(s)s\n2s2+\u0002s\u0003\u0003. (40) \n \nThe imaginary part of this equation is, \n \n  d\u0002\nd\u0001=Imds\nd\u0001\u0003\n\u0005\u0004\u0006\n\b\u0007. (41) \n \nSpecifically, consider this equation at \u0001=0 \n \n  d\u0004\nd\u0003\u0003=0=Im(s2+\u00052)ln(s)s\n2s2\u0001\u0003(s2+\u00052)\u0006\n\b\u0007\t\n\u000b\n\u0003=0=\u0002ln(\u0002+\u00052) ()\n4\u0002+\u00052. (42) \n \nThis gives three initial slopes for the rate of change of \u0001 with respect to \u0001. \n  11\u0001+\u00022>1\u0001\u0001\u0001\u0003 The frequency initially increases with increasing damping order. \n \n\u0001+\u00022=1\u0001\u0001\u0001\u0003The frequency initially is not changing with increasing damping order. \n \n\u0001+\u00022<1\u0001\u0001\u0001\u0003 The frequency initially decreases with increasing damping order. \n \n This is not entirely what might have been expected.  In the first case the \noscillation frequency actually increases before falling.  Hence there will be some values \nof \u0001 for which the fractional damping will actually cause the oscillations to go faster \nthan the un-damped oscillator (the damping will still cause the amplitude to decrease).  \nEach of the above three cases can become any of the three non-fractionally damped cases \nby letting \u0001\u00021 (Eqs. (10), (11), and (12)).  Hence, there are nine cases for the linear \nfractionally damped oscillator. \n \n Below are some graphs of solutions to the imaginary part of Eq. (39) (the \noscillation frequency) for various values of \u0001, \u0001, and \u0001.  In all three graphs the \noscillation frequency is on the vertical axis and the order of the derivative is on the \nhorizontal axis.  The three graphs for each case correspond to what would be under-\ndamped, critically-damped, and over-damped for a damped oscillator with whole order \nderivatives. \n \nFig. 3 \n \nFig. 3 is a representative graph of case one, \u0001+\u00022>1.  The green graph is for \u0001=\u0002 =1, \nthe black graph is for \u0001=2 an d \u0002 =1, and the red graph is for \u0001=3 and \u0002 =1. \n 12 \nFig. 4 \n \nFig. 4 is a representative graph from case two, \u0001+\u00022=1, a flat start. The red graph is for \n\u0002=22\u00011 ()  and  \u0003 =\u0002/2, the green graph is for \u0001=1/2 and \u0002 =1/ 2 , and the black \ngraph is for \u0001=15/16 and \u0002 =1/4. \n 13 \nFig. 5 \n \nFig. 5 is a representative graph for case three a decreasing start.  The green graph is for \n\u0001=1/2 and \u0002 =1/8, the black graph is for \u0001=1/2 and \u0002 =1/4, and the red graph is \nfor \u0001=\u0002 =1/2. \n \n \n5.  CONCLUSION  \n \n In this paper the linear fractionally damped oscillator equation was solved \nanalytically.  It was found that the solution is very similar to the non-fractional case \n(decayed oscillations but with the inclusion of an additional decay function).  It was \nfound that there are nine distinct cases, as opposed to the usual three for the ordinary \ndamped oscillator.  An unexpected result was that for three of the cases the oscillation \nfrequency actually increases with increasing order of derivative of the damping term (till \na peak value is reached, then the frequency decreases as expected).  The physical reason \nfor this increase in oscillation frequency is not yet clear. \n \nREFERENCES  \n[1]  Igor Podlubny, Fractional Differential Equations , Academic Press, 1999. \n \n[2] S. Burov and E. Barkai, Fractional Langevin Equation: Over-Damped, Under-\nDamped, and Critical behaviors, arXiv:0802.3777v1 [cond-mat.stat-mech] 26 Feb 2008. \n 14 \n[3] S. Burov and E. Barkai, The Critical Exponent of the Fractional Langevin Equation is \n\u0001c\u00020.402 , arXiv:0712.3407v1 [cond-mat.stat-mech] 20 Dec 2007. \n \n[4] G. M. Zaslavsky, A.A. Stanislavsky, and M. Edelman, Chaotic and Pseudochaotic \nAttractors of Perturbed Fractional Oscillator , arXiv:nlin.CD/0508018 v1 10 Aug 2005. \n \n[5] Ana Cristina Galucio, Jean-François, and François Dubois, On the use of fractional \nderivative operators to describe viscoelastic damping in structural dynamics- FE \nformulation of sandwich beams and approximation of fractional derivatives by using the \nG\u0001 scheme , Derivation fractionaire en mecanique – Etat-de-l’art et applications, CNAM \nParis – 17th November, 2006. \nhttp://www.cnam.fr/lmssc/seminaires/derivfrac/galucio/17NOV.PDF \n \n[6] B. N. Narahari Achar, J.W. Hanneken, and T. Clarke, Damping characteristics of a \nfractional oscillator , Physica A: Statistical Mechanics and its Applications, Vol. 339, \nissues 3-4, 15 Aug 2004, pages 311-319. \n \n[7]  R. L. Bagley and P. J. Torvik, On the appearance of the fractional derivative in the \nbehavior of real materials, J. Appl. Mech., 51 (1984), pp. 294-298. \n \n[8]  S. Saha Ray and R. K. Bera, Analytical solutions of the Bagley Torvik equation by \nAdomian decomposition method, Appl. Math. and Comp., vol 168, issue 1, Sept. 2005, \npp. 398-410. "    },    {        "title": "2108.10929v2.Jet_Parameters_in_the_Black_Hole_X_Ray_Binary_MAXI_J1820_070.pdf",        "content": "DRAFT VERSION NOVEMBER 11, 2021\nTypeset using L ATEXtwocolumn style in AASTeX631\nJet Parameters in the Black-Hole X-Ray Binary MAXI J1820+070\nANDRZEJ A. Z DZIARSKI ,1ALEXANDRA J. T ETARENKO ,2, 3 ,\u0003AND MAREK SIKORA1\n1Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences, Bartycka 18, PL-00-716 Warszawa, Poland; aaz@camk.edu.pl\n2East Asian Observatory, 660 N. A’oh ¯ok¯u Place, University Park, Hilo, Hawaii 96720, USA\n3Department of Physics and Astronomy, Texas Tech University, Lubbock, Texas 79409-1051, USA\nABSTRACT\nWe study the jet in the hard state of the accreting black-hole binary MAXI J1820+070. From the available\nradio-to-optical spectral and variability data, we put strong constraints on the jet parameters. We find while it\nis not possible to uniquely determine the jet Lorentz factor from the spectral and variability properties alone,\nwe can estimate the jet opening angle ( \u00191:5\u00061\u000e), the distance at which the jet starts emitting synchrotron\nradiation (\u00183\u00021010cm), and the magnetic field strength there ( \u0018104G), with relatively low uncertainty, as\nthey depend weakly on the bulk Lorentz factor. We find the breaks in the variability power spectra from radio\nto sub-mm are consistent with variability damping over the time scale equal to the travel time along the jet at\nany Lorentz factor. This factor can still be constrained by the electron-positron pair production rate within the\njet base, which we calculate based on the observed X-ray/soft gamma-ray spectrum, and the jet power, required\nto be less than the accretion power. The minimum ( \u00181.5) and maximum ( \u00184.5) Lorentz factors correspond\nto the dominance of pairs and ions, and the minimum and maximum jet power, respectively. We estimate the\nmagnetic flux threading the black hole and find the jet can be powered by the Blandford-Znajek mechanism in\na magnetically-arrested flow accretion flow. We point out the similarity of our derived formalism to that of core\nshifts, observed in extragalactic radio sources.\n1.INTRODUCTION\nOur knowledge of the structure of extragalactic radio jets is\nalready quite detailed, see, e.g., the review of Blandford et al.\n(2019). While a number of aspects remains to be determined,\ne.g., the jet lateral structure (Perlman et al. 2019), radio maps\nprovide us the projected structures of the jets, in particular\ntheir opening angles. Magnetic fields can be determined via\ncore shifts, which are angular displacements of the position\nof the radio core between two frequencies (e.g., Lobanov\n1998; Zamaninasab et al. 2014; Zdziarski et al. 2015, here-\nafter Z15). Superluminal motion allows us to estimate the jet\nbulk Lorentz factors, \u0000(e.g., Jorstad et al. 2001; Kellermann\net al. 2004; Lister et al. 2019). They can also be indepen-\ndently estimated from radiative models of blazars (Ghisellini\n& Tavecchio 2015) and from the radio core luminosity func-\ntion (Yuan et al. 2018). The jet power can be estimated from\ncalorimetry of radio lobes (Willott et al. 1999; Shabala &\nGodfrey 2013) and core shifts (e.g., Pjanka et al. 2017). Fi-\nnally, the e\u0006pair content can be obtained from comparison of\nthe observed jet powers with theoretical predictions (Sikora\net al. 2020 and references therein).\nOn the other hand, our knowledge of jets in accreting\nblack-hole (BH) binaries, which are the main class of mi-\ncroquasars, is much more rudimentary. From available radio\nmaps, we can only set upper limits on the jet opening an-\ngles (e.g., Stirling et al. 2001; Miller-Jones et al. 2006). We\n\u0003NASA Einstein Fellowcan estimate the Lorentz factors of ejected transient blobs,\nwhich phenomenon is associated with transitions from the\nhard spectral state to the soft one, e.g., Atri et al. (2020);\nWood et al. (2021), but this determination strongly depends\non the distance to the source. The Lorentz factors of steady\ncompact jets commonly present in the hard spectral state are\neven more difficult to constrain, with only rough estimates\nof\u0000&1:5–2 (e.g., Stirling et al. 2001; Casella et al. 2010;\nTetarenko et al. 2019). Therefore, an accurate determination\nof the jet parameters even for a single source would be very\nimportant.\nHere we study the jet in the transient BH X-ray binary\nMAXI J1820+070 during its outburst in 2018. We use the\nobservational data of outstanding quality for that source pre-\nsented by Tetarenko et al. (2021) (hereafter T21), which gives\nus an opportunity of such an accurate parameter determina-\ntion. We interpret these data in terms of the classical model\nof Blandford & K ¨onigl (1979) and K ¨onigl (1981). Here flat\nradio spectra are interpreted in terms of a superposition of\nsynchrotron self-absorbed and optically-thin spectra, spec-\ntra above the break frequency are optically-thin synchrotron,\nand the electron distribution and the magnetic field strength\nare parametrized by power laws. The jet in the synchrotron-\nemitting part is assumed to be conical and of a constant bulk-\nmotion velocity. We provide an updated analysis of those\ndata, making corrections to the similar model used in T21.\nIn particular, T21 followed the formulation of the model that\nsuffered from some errors related to the transformation from\nthe comoving frame to that of the observer. Also, we properly\nconnect the break frequencies in the radio/mm spectra witharXiv:2108.10929v2  [astro-ph.HE]  10 Nov 20212 Z DZIARSKI ET AL .\nthe propagation time along the jet, and we correct the expres-\nsions for jet power. Furthermore, we link the dependencies\nof energy densities on the distance to the observed hard in-\nverted spectral index, as well as we use additional data from\nRodi et al. (2021). This allows us to obtain constraints based\non the full radio-through-optical spectrum.\nMAXI J1820+070 was discovered during its outburst in\n2018 (Tucker et al. 2018; Kawamuro et al. 2018). The source\nis relatively nearby, with a distance of D\u00192:96\u00060:33kpc\nmeasured based on a radio parallax (Atri et al. 2020). Then,\nWood et al. (2021) determined D\u00143:11\u00060:06kpc based\non the proper motion of the moving ejecta during the hard-\nto-soft state transition. The inclination of the radio jet is i\u0019\n64\u000e\u00065\u000e(Wood et al. 2021), while the inclination of the\nbinary is constrained to ib\u001966\u000e–81\u000e(Torres et al. 2020).\nThe BH mass is given by M\u0019(5:95\u00060:22)M\f=sin3ib\n(Torres et al. 2020).\nThe data presented in T21 were obtained during a mul-\ntiwavelength observational campaign from radio to X-rays\nperformed during a 7-h period during the hard state on 2018\nApril 12 (MJD 58220). We also use the simultaneous IR\nand optical data obtained by Rodi et al. (2021). During the\ncampaign, the source was in a part of the initial hard state\nwhich formed a plateau in the X-ray hardness vs. flux dia-\ngram (Buisson et al. 2019).\nOur theoretical model is presented in Section 2 and Ap-\npendix A. In Section 3, we fit it to the data. In Section 4,\nwe discuss various aspects of our results, and show that our\nformalism based on time lags between different frequencies\nof the flat spectrum is equivalent to the formalism based on\ncore shifts. We give our conclusions in Section 5.\n2.STEADY-STATE JETS\n2.1. Power-law dependencies\nFollowing the theoretical interpretation in T21, we con-\nsider a continuous, steady-state, jet in the range of its distance\nfrom the BH where it has constant both the bulk Lorentz\nfactor and the opening angle, i.e., it is conical. We con-\nsider its synchrotron emission and self-absorption, and as-\nsume that the hard, partially self-absorbed, part of the total\nspectrum results from superposition of spectra from different\ndistances with breaks corresponding to unit self-absorption\noptical depth (Blandford & K ¨onigl 1979). We use the formu-\nlation of the model of K ¨onigl (1981) (which is an extension\nof the model of Blandford & K ¨onigl 1979 for cases with the\nself-absorbed radio index different from zero) as developed\nin Zdziarski et al. (2019), hereafter Z19. In this model, the\njet is assumed to be laterally uniform, which is a good ap-\nproximation for i\u001d\u0002, where \u0002is the jet (half) opening\nangle. We denote the observed and comoving-frame photon\nfrequencies as \u0017and\u00170, respectively, and introduce the di-\nmensionless distance,\n\u00170=\u0017(1 +zr)\n\u000e; \u000e\u00111\n\u0000(1\u0000\fcosi); \u0018\u0011z\nz0;(1)\nwherezris the cosmological redshift (equal to null in our\ncase), \u0000and\fare the jet bulk Lorentz factor and the velocityin units of the light speed, respectively, zis the distance from\nthe BH, and z0is the distance at which the jet becomes opti-\ncally thin to self-absorption at all considered frequencies. As\nin K¨onigl (1981), we assume the electron differential density\ndistribution, n(\r;\u0018), and the magnetic field strength, B, are\nparameterized by power-law dependencies,\nR(\u0018) =z0\u0018tan \u0002; n(\r;\u0018) =n0\u0018\u0000a\r\u0000p; B(\u0018) =B0\u0018\u0000b;\n(2)\nwhereRis the jet radius and \ris the Lorentz factor of the\nemitting electrons in the jet comoving frame, with \rmin\u0014\n\r\u0014\rmax. The quantities Randzare measured in the local\nobserver’s frame, while nandBare given in the comoving\nframe (for notational simplicity, we skip the primes). For a\nconserved electron number along the jet and conserved mag-\nnetic energy flux dominated by toroidal fields, we have a= 2\nandb= 1, corresponding to the spectral index of \u000b= 0,\nindependent of the value of p(Blandford & K ¨onigl 1979).\nHere, we define \u000bby the energy flux density of F\u0017/\u0017\u000b. If\neither the electron or magnetic energy is dissipated, a > 2,\nb > 1, respectively. Then, the emission weakens with the\ndistance and the synchrotron spectrum in the partially self-\nabsorbed frequency range becomes harder than in the con-\nserved case, \u000b > 0. The spectral indices of partially self-\nabsorbed and optically thin synchrotron emission are\n\u000b=5a+ 3b+ 2(b\u00001)p\u000013\n2a\u00002 +b(p+ 2); \u000b thin=1\u0000p\n2;(3)\nrespectively (K ¨onigl 1981; Z19). Using a delta-function ap-\nproximation to the single-electron synchrotron spectrum at\n\r2\u001d1(assumed hereafter), the synchrotron frequency for a\ngiven\rand\u0018, and its range emitted by the jet are\nh\u00170\nmec2=B0\u0018\u0000b\nBcr\r2; (4)\nh\u00170\nmin\nmec2=B0\u0018\u0000b\nM\nBcr\r2\nmin;h\u00170\nmax\nmec2=B0\nBcr\r2\nmax; (5)\nrespectively. Here z0\u0018Mis the distance at which the jet ter-\nminates,his the Planck constant, Bcr= 2\u0019m2\nec3=(eh)\u0019\n4:414\u00021013G is the critical magnetic field strength, and\nmeandeis the electron mass and charge, respectively. The\nspectral density of the synchrotron emission for a single jet\nparameterized by Equation (2) and for \u0017min\u0014\u0017\u0014\u0017maxis\nthen given by (see equation A5 of Z19),\nF\u0017'F0\u0012\u0017\n\u00170\u00135\n2Z\u0018max\n\u0018mind\u0018\u00181+b=2\u001a\n1\u0000exp[\u0000\u001c(\u0017\n\u00170;\u0018)]\u001b\n;\n(6)\nF0\u0011(1 +zr)7\n2(meh\u000e)1\n2\u0019C1(p)z2\n0\u00175\n2\n0tan \u0002 sini\n6cC2(p)(B0=Bcr)1\n2D2: (7)\nHereF0is a constant proportional to the bolometric flux,\n\u001c(\u0017=\u00170;\u0018)is the synchrotron self-absorption optical depth,\n\u00170is the break frequency, see Equation (13) below, andJET PARAMETERS 3\n11010010001040.51.05.010.050.0100.0\nxdFndlnx@mJyD\nFigure 1. An example of the spatial structure of the jet emission\nat different frequencies for \u00170= 2\u0002104GHz,F0= 300 mJy,\nb= 1:1,a= 2b,p= 2. The red dots, blue dashes, magenta dots,\ncyan dashes, and green dots correspond to \u0017= 5:25, 25.9, 343.5,\n1:4\u0002105GHz, and 5\u0002106GHz, respectively. The black solid\ncurve corresponds to \u0017=\u00170. The three lowest and two highest\nfrequency curves end at \u0018min>1and at\u0018max, respectively, beyond\nwhich there is no emission at those \u0017. These values of \u0018min,\u0018max\nwere calculated for \rmin= 10 ,\u0017max= 107GHz,i= 64\u000e,\u0000 = 3\nandB0= 104G (which correspond to \rmax\u0019793).\nC1(p),C2(p)follow from averaging the synchrotron emis-\nsion and absorption coefficients over the pitch angle,\nC1(p) =3p+4\n2\u0000E\u00003p\u00001\n12\u0001\n\u0000E\u00003p+19\n12\u0001\n\u0000E\u0000p+1\n4\u0001\n25\u00191\n2\u0000\u0000p+7\n4\u0001 ; (8)\nC2(p) =3p+3\n2\u0000E\u00003p+2\n12\u0001\n\u0000E\u00003p+22\n12\u0001\n\u0000E\u0000p+6\n4\u0001\n24\u00191\n2\u0000E\u0000p+8\n4\u0001 (9)\n(cf. Jones et al. 1974; Zdziarski et al. 2012), where \u0000Eis\nthe Euler Gamma function. In the extragalactic case, Dis\nthe luminosity distance. The lower and upper limits of the\nintegral (6) are,\n\u0018min(\u0017) = max\"\n1;\u0012B0mec2\r2\nmin\nh\u00170Bcr\u00131\nb#\n; (10)\n\u0018max(\u0017) = min\u0014\u0010\u0017max\n\u0017\u00111\nb;\u0018M\u0015\n; (11)\nrespectively. Figure 1 shows an example of the spatial de-\npendencies of the emission along the jet at different frequen-\ncies for\rmin= 10 and\u0017max= 107GHz. For\rmin&30,\nthe emission at all frequencies in this case would be in the\noptically-thin regime only, cf. Equation (16) below. We note\nthat above we have assumed the single-electron synchrotron\nemission is isotropic in the plasma frame, which is strictly\nvalid for a tangled magnetic field.\nThen, power-law dependencies assuming \u0018min = 1 ,\n\u0018max=1in the optically-thick and optically thin cases are\n100100010410510610710-1410-1310-1210-1110-1010-9\nn@GHzDnFn@erg cm-2s-1DFigure 2. An example of the jet synchrotron spectrum for \u00170=\n2\u0002104GHz,F0= 300 mJy,b= 1:1,a= 2b,p= 2 ,\n\u0017max= 107GHz,i= 64\u000e. This spectrum is virtually independent\nof\u0018min(\u0017)in the shown range of \u0017, as long as\u0018min(\u0017)\u001c\u0018\u0017. The\nblue curve shows the accurate spectrum of Equation (6), and the red\ndashes show the approximation of Equation (12). The gradual high-\nenergy cutoff of the accurate spectrum is due to \u0018max decreasing\nwith increasing \u0017and reaching unity for \u0017max.\n(cf. equation A11 in Z19)\nF\u0017'2F08\n>><\n>>:\u0000E\u00142a\u00006 +b(p+ 1)\n2a\u00002 +b(p+ 2)\u0015(\u0017=\u00170)\u000b\n4 +b; \u0017\u001c\u00170;\n(\u0017=\u00170)\u000bthin\n2a\u00006 +b(p+ 1); \u0017 0\u001c\u0017\u001c\u0017max:\n(12)\nFigure 2 shows an example comparison of the accurate spec-\ntrum of Equation (6) with these power-law approximations.\nWe see they are inaccurate around \u00170as well as close to \u0017max,\nwhere they fail to reproduce the gradual cutoff of the accu-\nrate spectrum. While the power-law asymptotic solutions in-\ntersect at a\u0017slightly different from \u00170, that frequency has\nno physical meaning since the actual spectrum in that range\ndoes not follow the broken-power law form, see Figure 2. We\ncan define a broken-power law approximation by taking the\nminimum of the two branches in Equation (12).\nThe optical depth along a line of sight crossing the jet spine\ncan be written as\n\u001c(\u0017=\u00170;\u0018) = (\u0017=\u00170)\u0000(p+4)=2\u00181\u0000a\u0000b(p+2)=2; (13)\nwhere\u00170is defined by \u001c(\u0017=\u00170;\u0018= 1) = 1 . The place\n\u0018= 1, orz=z0, corresponds to the jet being optically thin\nfor all\u0017\u0015\u00170. There is no synchrotron emission1atz <z 0,\n1We note that since the partially optically-thick emission of a jet at z >z 0\nwould remain almost unaffected if there were still emission following the\nscaling of Equation (2) at z < z 0(which would, however, decrease the\nactual value of z0and increase \u00170), it is also possible to formulate the\nstructure of the partially optically-thick part without invoking z0and\u00170.\nSuch a formulation is presented in Equations (A1–A4) in Appendix A.4 Z DZIARSKI ET AL .\nand thusz0corresponds to the onset of the jet emission. The\nrelationship of \u00170to the jet parameters is given by equation\n(A8) of Z19. We express it here as a formula for the normal-\nization of the electron distribution,\nn0=\u0012Bcr\nB0\u000e\u00131+p\n2\u0014h\u00170(1 +zr)\nmec2\u00152+p\n2\u000bfsini\nC2(p)\u0019\u001bTz0tan \u0002;\n(14)\nwhere\u000bfis the fine-structure constant and \u001bTis the Thomson\ncross section. From Equation (13), the distance along the jet\nat which\u001c(\u0017;\u0018\u0017) = 1 at\u0017.\u00170is\n\u0018\u0017=\u0012\u0017\n\u00170\u0013\u0000q\n; q\u0011p+ 4\n2a+bp+ 2b\u00002; z\u0017=z0\u0018\u0017:(15)\nFora= 2 andb= 1, we haveq= 1 at anyp. This dis-\ntance is very close to that at which most of the flux at a\ngiven\u0017is emitted, which can be defined by the maximum\nofdF\u0017(\u0018)=d ln\u0018, see Figure 1, and can be calculated using\nEquation (6). For example, at a= 2:2,b= 1:1, andp= 2,\nthat maximum is at \u0018\u00191:19\u0018\u0017. The emission around the\npeak has a broad spatial distribution; the 50% values of the\nmaximum flux are reached at \u0018= 0:65\u0018\u0017and3:64\u0018\u0017.\nThen, the Lorentz factor responsible for the bulk of emis-\nsion at\u0018\u0017is\n\r\u0017=\u0012Bcr\nB0h\u00170\n\u000emec2\u00131=2\u0012\u0017\n\u00170\u0013(1\u0000bq)=2\n; (16)\nwhich is usually weakly dependent on \u0017. While the integral\nspectrum of Equation (6) is valid for any \rmin, the asymptotic\npower-laws of Equation (12) require \rminto be by a factor of\nat least a few lower than \r\u0017for values of \u0017of interest (in the\nrange<\u00170) and\rmaxis required to be a factor of a few larger\nthan\r\u0017. If a high-energy cutoff is observed, an additional\nconstraint follows from it, see Equation (4).\nIf we know\u000band\u000bthin, we still cannot determine the val-\nues ofaandbseparately. However, a likely possibility is\nthat the ratio between the electron and magnetic energy den-\nsities remains constant, i.e., maintaining the same degree of\nequipartition along the jet, in which case a= 2b. We define\nan equipartition parameter as the ratio of the energy densities,\n\feq\u0011up\nB2=8\u0019=n0mec2(1 +ki)(fE\u0000fN)\nB2\n0=8\u0019; (17)\nwhere\nfE\u00118\n<\n:\r2\u0000p\nmax\u0000\r2\u0000p\nmin\n2\u0000p; p6= 2;\nln\rmax\n\rmin; p = 2;fN\u0011\r1\u0000p\nmin\u0000\r1\u0000p\nmax\np\u00001;(18)\nthe second equality in Equation (17) is at z0,upis the particle\nenergy density, kiaccounts for the energy density in particles\nother than the power-law electrons, in particular in ions (ex-\ncluding the rest energy), and p>1has been assumed in theexpression for fN. Fora= 2b,\feqis constant along the jet\n(providedkiis also constant) at z\u0015z0, which yields\n\u000b=(b\u00001)(13 + 2p)\nb(p+ 6)\u00002; q =p+ 4\nb(p+ 6)\u00002: (19)\nBelow, we use \feqanda= 2bto constrain the jet parameters.\nWe note that the case of a > 2requires that either \rmin\ndecreases or the electrons removed from their power-law dis-\ntribution move to some low energies below \rmin(with negli-\ngible emission). Since we assume \rminto be constant along\nthe jet, the latter has to be the case.\nWe next consider the difference between the arrival times\nof two photons. The first photon, at \u00171, is emitted toward the\nobserver at\u0018\u00171. The second photon, with \u00172<\u00171, is emitted\nat\u0018\u00172by the same comoving point of the jet after the time\n\u0001te, which is further downstream in the jet by \fc\u0001te. Since\nthe jet moves at an angle iwith respect to the line of sight,\nthe distance of the emitting point to the observer will become\nshorter during this time by \fc\u0001tecosi. For an observed dif-\nference in the arrival times of \u0001ta, the intrinsic separation be-\ntween the emission points (measured in the local observer’s\nframe) will be\nz\u00172\u0000z\u00171=z0(\u0018\u00172\u0000\u0018\u00171) =\u0001ta\fc\n(1\u0000\fcosi)(1 +zr):(20)\nAt frequencies\u0014\u00170,\u0018\u0017follows from Equation (15). Here,\nwe have also taken into account the redshift, making this ex-\npression correct also for an extragalactic source. Given an\nobserved \u0001ta, Equations (15) and (20) imply\nz0=\u0001ta\u0017\u0000q\n0\fc\n(1\u0000\fcosi)\u0000\n\u0017\u0000q\n2\u0000\u0017\u0000q\n1\u0001\n(1 +zr)=t0c\f\u0000\u000e\n1 +zr;(21)\nt0\u0011\u0001ta\n\u0001\u0018=\u0001ta\u0017\u0000q\n0\n\u0017\u0000q\n2\u0000\u0017\u0000q\n1; (22)\nwheret0is the implied lag time between the BH and z0,\nwhich can be obtained from time lag data if \u00170andqare\nknown (from the spectrum).\nAppendix A provides general solutions for z\u0017,B,\u0002and\nnto the equations in this section assuming the validity of\nEquation (12) for \u0017 < \u0017 0as functions of b(assuminga=\n2b),p,\u00170,F0,t0,iandD, as well as\feq,\rminand\rmax.\n2.2. The jet power\nThe jet power can be calculated using standard expres-\nsions. Note that it is defined in terms of the proper en-\nthalpy rather than the energy density, e.g., Levinson (2006),\nZdziarski (2014). Then, the component of the jet power due\nto both the relativistic electrons and magnetic fields (assum-\ning they are predominantly toroidal at z0) including both the\njet and counterjet for a= 2bandki= 0atz\u0015z0is\nPB+Pe=\u00121\n2+\feq\n3\u0013\nc\f(B0z0\u0000 tan \u0002)2\u00182\u00002b:(23)JET PARAMETERS 5\nThe usable power in cold ions at any z(calculated at z0), is\nPi= 2\u0019\u0016en0fN\u0012\n1\u00002n+\nne\u0013\nmpc3\f\u0000(\u0000\u00001)(z0tan \u0002)2;\n(24)\nwhereneandn+is the density of both electrons and\npositrons (whose ratio is assumed to be constant along the\njet), and positrons only, respectively, \u0016e= 2=(1 +X)is the\nelectron mean molecular weight, X(\u00190:7for the cosmic\ncomposition) is the H mass fraction, mpis the proton mass,\nandfNis given by Equation (18). This is the power in ions\nthat has to be supplied to the jet, and then it can be dissipated,\nhence the factor (\u0000\u00001). Equation (24) neglects the possible\npresence of background electrons being piled up at \r <\r min\nalready atz0. On the other hand, a>2at a constant \rminre-\nquires that leptons removed from n(\r;\u0018)atz>z 0do appear\nat some low energies below \rmin(with negligible emission).\nWe note that in a steady state, 2n+=ne= 2_N+=_Ne, where\n2_N+is the total rate of advection upstream of e\u0006pairs pro-\nduced at the jet base, and _Neis the total lepton flow rate,\n_Ne\u00192\u0019n0fNc\f\u0000(z0 tan \u0002)2: (25)\nThis implies\nPi=\u0016empc2(\u0000\u00001)\u0010\n_Ne\u00002_N+\u0011\n\u00150: (26)\nSince the pair production rate at the jet base, _N+(Section\n2.3), is independent of \u0000, the condition of Pi\u00150can give a\nlower limit on \u0000.\nIn any jet model, the total usable jet power is approxi-\nmately constrained by the accretion power,\nPj=PB+Pe+Pi._Mc2=L\n\u000fe\u000b(27)\nwhere _Mis the mass accretion rate, Lis the bolometric lu-\nminosity and \u000fe\u000b\u00180:1is the accretion efficiency. This then\ngives an upper limit on \u0000. The limit _Mc2can be exceeded\nif the rotation of the BH is tapped, but only by a factor .1.3\nand for a maximally rotating BH, see the references in Sec-\ntion 2.4 below.\nFinally, we consider the power lost in synchrotron emis-\nsion. It equals the synchrotron luminosity emitted by both\njets in all directions (which is Lorentz invariant). Since \u00170\ndepends on the direction and the partially self-absorbed emis-\nsion is not isotropic in the comoving frame, we neglect its\neffect and assume the entire emission is optically thin and\nisotropic in that frame, which is a good approximation for\nhard electron distributions with p.2:5or so. This gives\nPS\u00191\n3(B0tan \u0002)2\u001bTcz3\n0n0\u0000fE2f\u0018;\nfE2\u00118\n<\n:\r3\u0000p\nmax\u0000\r3\u0000p\nmin\n3\u0000p; p6= 3;\nln\rmax\n\rmin; p = 3;(28)\nf\u0018\u0011Z1\n1d\u0018\u00182\u00002b\u0000a=1\n2b+a\u00003;\nFigure 3. A sketch of of the pair-producing geometry based on\nfig. 3.9 of Tchekhovskoy (2015), which shows the result of his 3D\nGRMHD simulation for magnetically-arrested accretion on a BH\nwith the spin parameter of a\u0003= 0:99. In our case the disk is hot in\nits inner part (up to the radius Rhot) and surrounded by a cool outer\ndisk. We consider e\u0006pair production within the jet base (shown in\ngreen), which is devoid of matter, with the wavy arrows representing\npair-producing photons. We denote the characteristic jet radius of\nthe pair-producing region as Rjet. In addition, pairs are produced\nwithin the hot disk, but it is magnetically shielded from the jet base.\nThe black solid curves show the poloidal magnetic field.\nwhere 2b+a>3is assumed. This PSapproximately equals\nthe intrinsic luminosity of both jets,\nLjet\u00198\u0019D2\u000e\u00003\u0000Z\u0017max\n0d\u0017F\u0017; (29)\nwhereF\u0017is for the approaching jet, \u0017maxis given by Equa-\ntion (4) and the transformation law is for a stationary jet emit-\nting isotropically in its comoving frame (Sikora et al. 1997).\nFor self-consistency of our equations, PS\u001cPjis required.\n2.3. Pair production\nAs we see from Equation (26), the jet power in ions (given\nan observed synchrotron spectrum) strongly depends on the\nabundance of e\u0006pairs. In the case of extragalactic jets, there\nare strong indications that they dominate by number, though\nmost of the rest mass is usually still in ions (Sikora et al.\n2020). In the case of jets in microquasars, this is uncertain.\nAn important issue for that is the origin of pairs. A likely\nmechanism is pair production in photon-photon collisions by\nphotons produced close to the BH.\nPairs can be produced within the hot flow, e.g, Svens-\nson (1987). Since the Larmor radius of either a proton or\nan electron is orders of magnitude lower than Rg(where\nRg=GM=c2is the gravitational radius), the magnetic base6 Z DZIARSKI ET AL .\nof the jet is shielded from the hot plasma, and pairs produced\nin the accretion flow cannot enter the jet. On the other hand,\npairs can also be produced within the magnetic base of the jet,\noutside the hot plasma (Sikora et al. 2020). There, photon-\nphoton collisions will create e\u0006pairs in an environment de-\nvoid of background matter, thus strongly reducing the rate\nof pair annihilation (Beloborodov 1999). A possible geome-\ntry (see also Henri & Pelletier 1991; Ferreira et al. 2006) is\nshown in Figure 3. From an observed hard X-ray spectrum\nand a radius, Rhot, of the emitting hot plasma (inferred, e.g.,\nfrom X-ray spectroscopy; Bambi et al. 2021), we can esti-\nmate the average photon density within the jet base, which\nthen gives us the rate of pair production per unit volume,\n/R\u00004\nhot. We approximate the pair-producing volume as two\ncylinders with the height Rhotand the characteristic radius\nof the jet,Rjet, i.e.,V= 2\u0019R2\njetRhot. We can then write\n(assumingRjet\u0014Rhot) the total lepton production rate as\n2_N+=A\r\rR\u00003\nhotR2\njet; (30)\nwhere the factor A\r\rwould follow from detailed calcula-\ntions. Depending on the equilibrium density of the pairs,\nsome of them would annihilate, and some would be advected\nto the BH, reducing the effective 2_N+. We address this issue\nfor the case of MAXI J1820+070 in Section 3.1.\n2.4. The Blandford-Znajek mechanism\nWe can also estimate the jet power in the framework of\nthe model with extraction of the rotational power of the BH\n(Blandford & Znajek 1977). The jet power in this case de-\npends on the magnetic flux, \bBH, threading the BH (on one\nhemisphere), which can be written as\n\bBH=\u001eBH(_Mc)1=2Rg; (31)\nwhere\u001eBHis a dimensionless magnetic flux. Its maxi-\nmum value of\u001950 is obtained in magnetically arrested disks\n(MAD; Narayan et al. 2003; Bisnovatyi-Kogan & Ruzmaikin\n1974), as it was found in GRMHD simulations of MAD ac-\ncretion (Tchekhovskoy et al. 2011; McKinney et al. 2012;\nsee its more accurate value in Davis & Tchekhovskoy 2020).\nThen it has been found that\nPj\u00191:3\u0012\u001eBH\n50\u00132\nh0:3a2\n\u0003_Mc2; (32)\nwherea\u0003is the BH spin parameter and h0:3is defined by the\nhalf-thickness of the disk being Hdisk=Rdisk0:3h0:3(Davis\n& Tchekhovskoy 2020). This maximum differs from that of\nEquation (27) by the factor 1:3h0:3a2\n\u0003. In the spectrally hard\nstate, the disk is most likely hot, in which case h0:3\u00181.\nWe can estimate \u001eBHusing the magnetic field strength\nmeasured far from the BH by using the conservation of\nthe magnetic flux. Specifically, we use the expected equal-\nity between the poloidal and toroidal field components at\nthe Alfv ´en surface (in the observer’s frame), which radius,\nfor strongly magnetized jets, approaches the light cylinder\nradius,RLC(Lyubarsky 2010). This implies \u0000hB0\n\u001ei \u0019(R=R LC)Bp, whereBpis the poloidal field (which has the\nsame value in the comoving and BH frames) and hB0\n\u001eiis the\naverage toroidal field strength in the comoving frame, de-\nnoted byBin the remainder of this paper. Then, the toroidal\nfield dominates at zsatisfyingR(z)\u001d\u0000RLC, and, presum-\nably, atz\u0015z0. The magnetic flux using this method was\ndetermined for a sample of radio loud active galactic nuclei\nin Zamaninasab et al. (2014) and Z15. We use the resulting\nformula as derived in Z15,\n\bj=23=2\u0019RHsz0B0(1 +\u001b)1=2\n`a\u0003; (33)\nwhich allows us to estimate \u001eBHfor a givena\u0003by setting\n\bj= \b BH. HereRH= [1 + (1\u0000a2\n\u0003)1=2]Rgis the BH hori-\nzon radius,`.0:5is the ratio of the field and BH angular\nfrequencies, and sis the scaling factor relation between the\njet opening angle and the bulk Lorentz factor (Komissarov\net al. 2009; Tchekhovskoy et al. 2009), limited by causality\nto.1,\n\u0002\u0019s\u001b1=2=\u0000: (34)\nHere,\u001bis the magnetization parameter, which is defined as\nthe ratio of the proper magnetic enthalpy to that for particles\nincluding the rest energy,\n\u001b\u0011B2=4\u0019\n\u0011up+\u001ac2=2\n\feq\u0014\n\u0011+\u0016emp(1\u00002n+=ne)fN\nme(fE\u0000fN)(1 +ki)\u0015\u00001\n;\n(35)\nwhere\u001ais the rest-mass density, and 4=3< \u0011 < 5=3is\nthe particle adiabatic index. The second equality relates \u001b\nto\feqassuming that the only ions are those associated with\nthe power law electrons (i.e., neglecting the possible pres-\nence of ions associated with electrons with \r < \r min, e.g.,\nwith a quasi-Maxwellian distribution). For p>2and a large\nenough\rmax,fE=fN\u0019\rmin(p\u00001)=(p\u00002). Then, for\n\feq(1\u00002n+=ne)=\rmin\u001dme=mp, we have\u001b\u001c1.\n3.APPLICATION TO MAXI J1820+070\nHere, we apply the model of Section 2 to the source. We\nuse the VLA fluxes at 5.25, 7.45, 8.5, 11.0, 20.9, 25.9 and\nthe ALMA flux at 343.5 (from table 1 in T21). We also use\nthe IR flux at 1:4\u0002105from VLT/HAWK-I, and the optical\nflux at 3:9\u0002105GHz from NTT/ULTRACAM (T21), and the\n13 fluxes between 1.37 and 7:00\u0002105GHz from the VLT\nX-shooter and the INTEGRAL /OMC flux at 5:66\u0002105GHz\n(Rodi et al. 2021). All of the IR/optical fluxes have been\nde-reddened with E(B\u0000V) = 0:18(Tucker et al. 2018), as\nassumed in T21. We use the time lags between 25.9 GHz and\nlower frequencies, 11.0 GHz and lower frequencies, and be-\ntween 343.5 GHz and lower frequencies. The lags are given\nin tables 3 and 4 of T21. This gives us 23 spectral measure-\nments and 14 time lags. We present analytical and numerical\nestimates in Sections 3.1 and 3.2, respectively. We assume\nthe ratio between the electron and magnetic energy densities\nto be constant along the jet, i.e., a= 2b.\nIn our fits below, we use the Markov-Chain Monte Carlo\n(hereafter MCMC) technique with wide uniform priors, seeJET PARAMETERS 7\nb= 1.10+0.01\n−0.01\n1.62.02.42.83.2pp= 2.21+0.22\n−0.19\n160240320400F0(mJy)\nF0(mJy) = 297 .61+31.76\n−37.87\n1.62.43.24.0ν0(×104GHz)\nν0(×104GHz) = 2.32+0.65\n−0.60\n0.60.91.21.51.8t0(s)\nt0(s) = 0.79+0.30\n−0.19\n2030405060νmax(×104GHz)\nνmax(×104GHz) = 32.51+5.02\n−4.37\n1.0881.0961.1041.1121.120\nb0.40.60.8αdisc\n1.62.02.42.83.2\np\n160 240 320 400\nF0(mJy)\n1.62.43.24.0\nν0(×104GHz)\n0.60.91.21.51.8\nt0(s)\n20 30 40 50 60\nνmax(×104GHz)0.40.60.8\nαdiscαdisc= 0.54+0.13\n−0.12\nFigure 4. MCMC fit results for the seven model-independent quantities, which require only the assumptions of a= 2b, see Section 3.1. Here\nand in Figure 8 below, the panels show the histograms of the one-dimensional posterior distributions for the model parameters, and the two-\nparameter correlations, with the best-fitting values of the parameters indicated by green lines/squares. The best-fit results for fitted quantities\nare taken as the medians of the resulting posterior distributions, and are shown by the middle vertical dashed lines in the distribution panels.\nThe surrounding vertical dashed lines correspond approximately to a 1\u001buncertainty.8 Z DZIARSKI ET AL .\nFigure 5. The radio-to-optical spectrum from T21 (VLA, ALMA,\nVLT/HAWK-I, NTT/ULTRACAM; red error bars), the 339 MHz\nmeasurement (VLITE, magenta error bar; Polisensky et al. 2018),\nand from the VLT/X-shooter (blue error bars) and the INTE-\nGRAL /OMC (cyan error bar) as obtained by Rodi et al. (2021), but\nwith the de-reddening correction for E(B\u0000V) = 0:18(Tucker et al.\n2018). The error bars for the radio and sub-mm measurements of\nT21 are the square roots of the squares of their statistical and sys-\ntematic errors, and 10% systematic errors are assumed for the IR\nand optical measurements. The spectrum above 5 GHz is fitted by\nthe jet model of Equation (6) using the best-fit parameters shown\nin Figure 4 (cyan dashed curve) and a phenomenological power\nlaw approximating the disk component with \u000bdisk= 0:53(green\nsolid curve). The former is virtually independent of \u0018minwithin\nthe ranges obtained in the full fits (Equation 10; Section 3.2), so it\ncan be assumed to be unity. The sum is shown by the solid black\ncurve. The corresponding asymptotic optically thick and optically\nthin spectra of Equation (12) are shown by the magenta dotted lines.\nT21 for details. We assume i= 64\u000e\u00065\u000e(Wood et al. 2021)\nwith a Gaussian prior, and D= 2:96\u00060:33kpc (Atri et al.\n2020) with a Gaussian prior, but truncated at the upper limit\nofDmax= 3:11kpc found by Wood et al. (2021).\nWe assume the observed spectrum is from the approaching\njet only. Ati\u001964\u000e, this assumption is satisfied only roughly.\nThe ratio of the jet-to-counterjet fluxes in the optically-\nthick part of the spectrum is given by [(1 +\fcosi)=(1\u0000\n\fcosi)](7+3p)=(4+p)(which follows from Zdziarski et al.\n2012), which is\u00197 at the fitted p\u00192:21(see below).\n3.1. The initial fit and analytical estimates\nWe can solve for b,p,\u00170,F0,\u0017maxandt0by only assum-\ninga= 2b. From the measured fluxes, we obtain b,p,\u00170,F0\nand\u0017maxusing Equations (6–7). We find the fitted spectrum\nis very insensitive to \u0018min(\u0017), Equation (10), as long as it is\nlow enough. We can just use any low value of it, or just as-\nFigure 6. The time lags measured by T21 vs. the theoretically\nexpected distance in units of z0between the emission points for\nthe partially-self-absorbed part of the spectrum. The blue and red\nsymbols correspond to the lags between 343.5 GHz and radio fre-\nquencies (5.25–25.9 GHz), and within the radio frequencies, re-\nspectively, where we assumed a constant ratio between the elec-\ntron and magnetic energy densities, \u0018\u0017= (\u0017=\u00170)\u0000q,q\u00190:88,\nand\u00170= 2:32\u0002104GHz. The diagonal line gives the best-fit\ntheoretical relationship between the two quantities corresponding to\nt0= 0:79s, see text.\nsume\u0018min= 1, and check a posteriori the consistency of the\nchoice. Similar to Rodi et al. (2021), we find the presence of\nan additional hard component beyond \u0017max, apparently due\nto the emission of an accretion disk. Given the limited range\nof the fitted frequencies, we fit it phenomenologically as a\npower law,F\u0017;disk=Fdisk(\u0017=105GHz)\u000bdisk. Then, using\nthe obtained values of b,pand\u00170, we can fit the time lags\nusing Equations (19) and (22). However, with the MCMC\ntechnique, we fit the flux and time-lag measurements simul-\ntaneously. The fitted parameters and the correlations between\nthem are shown in Figure 4, and the parameters are listed in\nTable 1. The best-fitting values are given as the median of the\nresulting posterior distributions, and the lower and upper un-\ncertainties are reported as the range between the median and\nthe 15th percentile, and the 85th percentile and the median,\nrespectively. These uncertainties correspond approximately\nto1\u001berrors. We use these best-fit values as well as the best-\nfit values of Dandiin our estimates in this subsection.\nFigure 5 shows the observed average radio-to-optical spec-\ntrum fitted by the above model. The best-fitting spectral in-\ndices in the optically thick and optically thin regimes are then\n\u000b\u00190:25and\u000bthin\u0019\u00000:61, respectively. We show the the-\noretical spectrum calculated by integrating Equation (6) and\nthe asymptotic optically thick and thin spectra of Equation\n(12) for this fit.\nWe then show the time lags in Figure 6, where we plot\nthe values of the measured \u0001taagainst the separation in the\ndimensionless units, \u0018, using Equations (15) and (22). AtJET PARAMETERS 9\nTable 1. The basic parameters of the jet in MAXI J1820+070.\nb p \u0017 0F0\u0017max\u000bdiskFdiskt0\n104GHz mJy 104GHz mJy s\n1:10+0:01\n\u00000:012:21+0:22\n\u00000:192:32+0:65\n\u00000:60298+31\n\u00003832:5+5:0\n\u00004:40:54+0:13\n\u00000:1230:4+6:0\n\u00005:90:79+0:30\n\u00000:19\nNOTE—The fits are done with the MCMC method assuming a= 2b.Fdiskgives the flux density at 105GHz. 1 mJy\n\u001110\u000026erg cm\u00002s\u00001Hz\u00001.\nthe best-fit values of bandp,q\u00190:883, see Equation (19).\nThe actual lags have to follow a single dependence relating\nthe physical separation between the emission points to \u0001ta,\nwhich is shown by the diagonal line showing the linear re-\nlationship between taand\u0001\u0018\u0017corresponding to the best-fit\nvalue oft0, see Equation 22. We see a certain offset between\nthe points corresponding to the lags between the sub-mm\nfrequency of 343.5 GHz and 6 radio frequencies (blue error\nbars), and the lags measured between the radio frequencies\n(red error bars). This may be related to the different methods\nused in T21 to determine those. On the other hand, the offset\nis significantly reduced for q= 0:8, which value of q, how-\never, is not compatible with \u000b\u00190:25. This may indicate that\nthe jet is more complex than we assume, e.g., either \u0000,\u0002or\nbare not constant at z\u0015z0.\nOur formalism assumes the lags correspond to propaga-\ntion of perturbations between different values of z\u0017at the jet\nspeed,\f. With this assumption, we obtain z0as a function of\n\u0000, see Equation (21),\nz0\u0019(2:37\u00021010cm)(t0=0:79 s)\f\u0000\u000e: (36)\nWe then use the solutions obtained in Appendix A assum-\ning\rmin= 3,ki= 0. However, we only know \u0017maxrather\nthan\rmax, see Equation (5). Since the solutions depend on\n\rmax relatively weakly, we assume here the best-fit values\nofB0= 104G and \u0000 = 2:2obtained in Section 3.2 for\n\rmin= 3, which yield \rmax= 125 , which we will use here-\nafter in this subsection. Using Equation (A5), we obtain at\nthe best fit\n\u0002\u00192:32\u000e\n(\f\u0000)1:89\u000e2:57\f0:11eq: (37)\nAt\f\u00191and\feq= 1,\u0002\u00190:53\u000e\u00000:67. Next, Equations\n(A6–A7) give at the best fit,\nB0\u00198:2\u0002103G(\f\u0000)0:22\n\u000e0:13\f0:22eq; (38)\nn0\u00191:8\u00021012cm\u00003(\f\u0000)0:43\f0:57\neq\n\u000e0:26; (39)\nwithB0/\u00000:35andn0/\u00000:70at\f\u00191. Equation (37)\nshows that we cannot determine both \u0002and\u000eeven assuming\na value of\feq(on which \u0002depends very weakly). We can\nalso calculate the Thomson scattering optical depth along the\njet radius at z\u0015z0, which equals\n\u001cT(\u0018) =\u001bTn0fNz0tan \u0002\u00181\u00002b\u00192:5\u000210\u00004\f0:46\neq\n(\f\u0000)0:46\u000e1:82\u00181:2:\n(40)\nFigure 7. The locations of the emission at the observed frequencies\ninferred from the break in the power spectra with the assumption of\nzb=\fc=f break , shown as their ratio to the locations based on time\nlags and the slope of the partially self-absorbed spectrum, z\u0017\u0019\nz0(\u0017=\u00170)\u00000:88(withz0for\u0000 = 3 andi= 64\u000e).\nAti= 64\u000eand\u0000 = 2 , 3, 4, we have \u000e\u00190:81,\n0.57, 0.43, and, at \feq= 1,\u0002\u00191:4\u000e,1:4\u000e,1:5\u000e,B0\u0019\n1:0;1:1;1:2\u0002104G,z0\u00193:3;3:8;4:0\u00021010cm, and\n\u001cT(\u0018= 1)\u00192:9;4:3;6:1\u000210\u00004, respectively. The val-\nues ofz0correspond to\u0019(2:8–3:3)\u0002104Rgat an assumed\nM= 8M\f. We find that \u0002,B0andz0depend relatively\nweakly on \u0000for1:5.\u0000.5.\nWe determine the typical Lorentz factors, \r\u0017, of relativis-\ntic electrons giving rise to the emission at \u0017, which in the\npartially self-absorbed regime originates mostly from z\u0017, see\nEquation (16). We obtain\n\r\u0017\u001932\f0:11\neq(\f\u0000)\u00000:11\u000e\u00000:43(\u0017=\u00170)0:014: (41)\nIn order to obtain a power-law emission in that regime, we\nneed\rminto be a factor of a few smaller. Thus, we require\n\rmin.10for the validity of the model. The maximum \r\ncorresponds to the fitted \u0017max, Equation (5). From that, we\nobtain\rmaxranging from\u0019123 to 147 for \u0000increasing from\n2 to 4. Combining this with the values of \u001cTfrom Equation\n(40), we find that the power in the synchrotron self-Compton\ncomponent is relatively similar to that in the synchrotron one,\nPSSC.\u001cT\r2\nmaxPS.\nWe can then consider implications of the break frequen-\ncies,fb, in the power spectra for different frequencies mea-10 Z DZIARSKI ET AL .\nsured by T21. For those power spectra, most of the variabil-\nity power per lnfoccurs atf\u0014fb, with the variability at\nhigher frequencies strongly damped, see figs. 3 and 5 in T21.\nWe define the distance, zb, as that covered by a jet element\nmoving with the jet velocity during the time21=fb,\nzb(\u0017)\u0011\fc=f b(\u0017): (42)\nWe can compare it to the distance along the jet from the\nBH up to the location of the peak emission at \u0017, i.e.,z\u0017\n(Equations 15, 21). In our model, z\u0017\u0019z0(\u0017=\u00170)\u00000:88with\nz0/\f=(1\u0000\fcosi), givingzb=z\u0017/1\u0000\fcosi. Then, this\nratio depends only weakly on \f(or\u0000); ati= 64\u000e,1\u0000\fcosi\nchanges only from 1 at \f\u001c1to 0.56 at\f\u00191. This implies\nthat this correlation cannot be used to determine the actual\nbulk Lorentz factor of the jet.\nFigure 7 shows zb=z\u0017vs.z\u0017for\u0000 = 3 . We see an ap-\nproximately constant ratio of zb=z\u0017\u00191:5–2. Therefore, zb\nis proportional and close to the travel time along z\u0017in all of\nthe cases. A possible explanation of the damping of the vari-\nability at frequencies >c=z\u0017appears to be a superposition of\nthe contributions to the emission from different parts of the\nregion dominating at a given \u0017, which is/z\u0017, as shown in\nFigure 1. The peak of dF\u0017=d lnzforp= 2:21is at\u00191:15z\u0017\nand its width defined by dF\u0017=d lnzdecreasing to the 50%\nof the peak is (0:65–3:16)z\u0017. Thus, if different parts vary in-\ndependently, the variability will be damped at f&c=(2z\u0017),\nas observed.\nAlternatively, the observed radio/IR variability can be\ndriven by the variable power supplied from the vicinity of the\nBH with a wide range of frequencies (Malzac 2013, 2014)\nand then transferred upstream. Then, the travel time can act\nas a low-pass filter, removing most of the variability at fre-\nquenciesf >\fc=z\u0017. This can happen due to damping of per-\nturbations along the jet due to some kind of internal viscosity,\ne.g., collisions between shells within the jet moving with a\nrange of velocities (Jamil et al. 2010). The process would be\nthen analogous to viscous damping in accretion disks, where\nmodulations with a period shorter than the signal travel time\nacross the disk are strongly damped (Zdziarski et al. 2009).\nThis picture is also compatible with the integrated fractional\nvariability of the power spectra (RMS) decreasing with the\ndecreasing\u0017(as shown in fig. 5 of T21). This means increas-\ning the distance travelled along the jet leads to the increasing\ndamping.\nWe note that the break frequencies in the power spectra\nof T21 have been defined by choosing a specific, and not\n2T21 assumed z\u0017=zb\u0011\fc\u000e=f b(\u0017), which they used as the final condi-\ntion determining the jet parameters. Thus, they transformed the observed\nvariability frequency to the jet frame ,fb=\u000e, and multiplied the resulting\ntime scale,\u000e=fb, by the jet velocity in the observer’s frame ,\fc, which\ndoes not appear to be correct. We note that in the present case we consider\nthe light curve originating from a fixed region of the jet around z\u0017. While\nthe plasma in that region is moving, two adjacent maxima in the observed\nlight curve are emitted from the same region in the frame connected to the\nBH, which is the same frame as the observer’s one (in the absence of a\nredshift). Thus, a frequency inferred from the variability power spectrum\nshould not be transformed.unique, algorithm, as well as the obtained values of fbare\nclose to the minimum frequency at which the power spec-\ntrum is measured for f < 10GHz, which limits the accu-\nracy of the determination of those fb. Also, while the damp-\ning of variability above \fc=z\u0017clearly occurs, details of the\nphysics behind it remain uncertain, and the damping could\nstart atf\u0018\fc=(2z\u0017)instead of exactly \fc=z\u0017. Summa-\nrizing, our results are completely compatible with the vari-\nability damping at time scales shorter than the light/jet travel\ntime acrossz\u0017. However, unlike our previous estimates from\nthe observed spectrum and time lags, which are based on a\nrelatively rigorous and well-understood model, the detailed\ncause of the connection between the break frequencies and\nthe distance along the jet remains uncertain.\nWe can also consider the prediction of the location of\nthe bulk of the 15 GHz emission, z\u0017\u0019z0(\u0017=\u00170)\u00000:88\u0019\n2:5\u00021013cm (at \u0000 = 3 , but weakly dependent on it),\nwith the jet angular size at this frequency from the VLBA\nobservation on 2018 March 16 (MJD 58193), reported in\nT21 as 0:52\u00060:02mas. the deprojected size is (2:60\u0006\n0:10)\u00021013cm. The total flux density at 15 GHz was mea-\nsured asF\u0017\u001920:0\u00060:1mJy. However, the VLBA ob-\nservation was 27 d before the radio/sub-mm ones. On MJD\n58220, our best-fit spectral model yields F\u0017\u001956\u00061mJy.\nWithin the framework of the continuous conical jet model,\nwe havez\u0017/F(p+6)=(2p+13)\n\u0017 (Equation A2; Zdziarski et al.\n2012). Thus, for p= 2:2we predict the size at 15 GHz\non MJD 58220 being (56=20)0:47\u00191:6times larger than\nthat on MJD 58193, namely \u00184\u00021013cm. While some-\nwhat larger than the above z\u0017, this size appears consistent\nwith it since the peak of dF\u0017=d lnzforp= 2:21is at\n\u00191:15z\u0017\u00193:0\u00021013cm, and that spatial distribution is\nbroad and skewed toward higher distances, see Figure 1 and\nthe discussion of it above.\nWe then estimate the rate of pair production. For MAXI\nJ1820+070, pair production within the hot plasma was cal-\nculated by Zdziarski et al. (2021) based on the spectrum\nobserved by INTEGRAL in the hard state. That spectrum\nwas measured up to \u00182 MeV , well above the pair produc-\ntion threshold of 511 keV , and modelled by Comptonization.\nIt was found that an appreciable pair abundance can be ob-\ntained only provided the hard X-ray source size is as small as\nseveralRg, while the spectroscopy based on the relativistic\nbroadening of the fluorescent Fe K \u000bline indicates a size of\n&20Rg. Then, the pair abundance within the Comptonizing\nplasma is very low.\nHowever, as discussed in Section 2.3, pair production\nwithin the jet base can be much more efficient. To calculate\nit, we adapt the results of Zdziarski et al. (2021). We mod-\nify their equation (1) to calculate the photon density above\nthe hot disk, dividing the total rate of the photon emission\nby2\u0019R2\nhot(including both sides). We then use this photon\ndensity in equation (3) of that paper for the spectral param-\neters of average spectrum (table 2 in Zdziarski et al. 2021).\nThis gives the pair production rate per unit volume. With theJET PARAMETERS 11\nassumptions as in Section 2.3, we have\n2_N+\u00194:65\u00021040s\u00001\u0012Rhot\n20Rg\u0013\u00003\u0012Rjet\n10Rg\u00132\n:(43)\nThis is then balanced by the sum of the rates of pair annihi-\nlation and pair advection. Using formulae in Zdziarski et al.\n(2021), we have found that pair annihilation can be neglected\nfor the advection velocity of \f\u0006&0:1. It appears that such\na velocity can be achieved due to the net momentum com-\nponent of the pair-producing photons along the zaxis, see\nFigure 3, and due to pair acceleration by radiation pressure\nof the disk photons (Beloborodov 1999). Thus, while some\nof the produced pairs will annihilate (and a small fraction will\nbe advected to the BH), a major fraction of the produced pairs\nwill have a sufficient net bulk velocity to escape upstream.\nThen, the lepton flow rate through the jet, Equation (25),\nfor\rmin= 3is\n_Ne\u00196:7\u00021040s\u00001\f0:35\neq\n(\f\u0000)0:35\u000e3:39/\u00003:05; (44)\nwhere Equations (36), (37), (39) have been used and the pro-\nportionality assumes \f\u00191. Comparing with Equation (43),\nwe find _Ne>2_N+at any \u0000forRhot= 20Rg,Rjet= 10Rg\nand\rmin= 3. Thus, at these parameters the synchrotron-\nemitting plasma is never composed of pure pairs. If we as-\nsume either Rjet= 15Rgor\rmin= 10 , we find _Ne= 2_N+\nat\u0000\u00192, which thus represent the minimum possible \u0000for\nthese parameters. While the hot disk and jet radii and \rmin\nare poorly constrained, we consider the fact that the numbers\nin Equations (43) and (44), obtained with completely differ-\nent physical considerations, are of the same order of magni-\ntude, to be highly remarkable and indicating that indeed the\ntwo rates may be similar in this source. Then, the jet can\ncontain a large fractional abundance of pairs, and they can\ndominate by number over the ions.\nThe pairs produced in the jet base and then advected to\nlarge distances will eventually leave the jet and enter the ISM.\nOur estimated rate, \u00181040–1041s\u00001, is lower than the ap-\nproximate estimates of the positron production rates in mi-\ncroquasars of Guessoum et al. (2006). It is also a few or-\nders of magnitude below the total rate of pair annihilation\nin the Galaxy, which has been estimated by Siegert et al.\n(2016) as\u0019(3–6)\u00021043s\u00001. With the past and present\nX-ray monitors (ASM, Levine et al. 1996; MAXI, Matsuoka\net al. 2009; BAT, Barthelmy et al. 2005), we can detect all\nof the outbursting accreting BH binaries with luminosities of\nmore than a few percent of the Eddington luminosity. Based\non the MAXI data, there is \u00180.5 source in outburst at given\ntime (the MAXI team, private communication). Thus, the av-\nerage contribution of such sources to the Galactic positrons\nappears to be negligible, contrary to earlier estimates (Gues-\nsoum et al. 2006; Weidenspointner et al. 2008).\nNext, we calculate the jet power. The power in the rela-\ntivistic electrons and magnetic fields, Equation (23), becomes\natz0\nPB+Pe\u00191:9\u00021036erg s\u000013 + 2\feq\n6\f0:65eq\u00000:65\n\f0:35\u000e3:39:(45)which increases very fast with \u0000, approximately as /\u00004at\n\f\u00191. At\feq= 1,\u0000 = 3 , this power is\u00192:2\u00021037erg s\u00001.\nThe power associated with the bulk motion of cold matter,\nEquations (24), (26), is\nPi\u00191:2\u00021038erg s\u00001(\u0000\u00001)\u0002 (46)\"\n\f0:35\neq\n(\f\u0000)0:35\u000e3:39\u00000:7\u0012Rhot\n20Rg\u0013\u00003\u0012Rjet\n10Rg\u00132#\n:\nThe first term is approximately /\u00003(\u0000\u00001).\nTo constrain Pjby the accretion power, we use the es-\ntimate of the hard-state bolometric flux of Fbol\u00191:4\u0002\n10\u00007erg cm\u00002s\u00001(Shidatsu et al. 2019). This yields L\u0019\n1:5(D=2:96 kpc)21038erg s\u00001and\n_Mc2\u00191:5\u00021039\u0012D\n2:96 kpc\u00132\u0010\u000fe\u000b\n0:1\u0011\u00001\nerg s\u00001:(47)\nFor the default parameter values, Pj._Mc2implies \u0000.3:3.\nIf pair production is efficient enough, we also have a lower\nlimit on \u0000from the requirement of Pi>0. The allowed\nrange depends significantly on the assumed parameters, in\nparticular\rmin,RhotandRjet. E.g., at\rmin= 10 ,Rhot=\n20RgandRjet= 10Rg,\u0000&2:4is required.\nWe can then compare the total jet power, Pj, with the syn-\nchrotron power. At the low \rmax implied by the \u0017max fit-\nted to the spectrum, we find PS\u001cPjalways. For ex-\nample,PS\u00190:009Pjat the maximum allowed \u0000\u00193:3,\nandPS\u00190:02Pjat\u0000 = 2 . On the other hand, we have\nfoundPS\u00180:5(PB+Pe)(z0), weakly depending on ei-\nther\u0000or\rmin. Thus, the synchrotron emission can be\nentirely accounted for by the power in electrons and mag-\nnetic fields at z0, and most of the decline of PB+Pewith\nthe distance can be due to the synchrotron losses. How-\never, we may see that the decline of (PB+Pe)with\u0018is\nslower than that of the synchrotron power. If the former\nwould be just to the synchrotron emission, we would have\nd(PB+Pe)=d\u0018+ dPS=d\u0018= 0, while the former and the\nlatter terms are/\u0000\u00181\u00002band/\u00182\u00004b. This implies either\nsome electron re-acceleration at z >z 0at the expense of Pi,\nor more complexity of the actual physical situation, with the\ninitial energy loss in the flow being faster and followed by a\nslower one.\nIn the framework of models with the jet dissipation mech-\nanism being the differential collimation of poloidal magnetic\nsurfaces, the obtained \u0002\u0000\u001c1indicate the jet magnetization\natz&z0is low. Using Equation (34), we have (at \f\u00191)\n\u001b= (\u0002\u0000=s)2\u00198:4\u000210\u00005\u00003:35\n\f0:22eqs2: (48)\nAt\feq= 1 and assuming s= 0:6(as found as the average\nvalue for a large sample of radio-loud AGNs by Pjanka et al.\n2017), we obtain \u001b\u00190:0093(\u0000=3)3:35. This can be com-\npared to\u001bfrom its definition, Equation (35), which equals,\n\u001b\u0019\f\u00001\neq[2=3 + 130(1\u00002n+=ne)]\u00001: (49)12 Z DZIARSKI ET AL .\nΓ = 2.20+0.69\n−0.46\n0.61.21.82.4ΘΘ = 1.04+0.48\n−0.35\n10.410.610.811.0log[z0] (cm)\nlog[z0] (cm) = 10 .63+0.09\n−0.08\n0.81.21.62.0B0(×104G)\nB0(×104G) = 0.99+0.22\n−0.18\n36.637.237.838.4log[Pj] (erg s−1)\nlog[Pj] (erg s−1) = 38.31+0.32\n−0.60\n0.20.40.60.82n+/ne\n2n+/ne= 0.35+0.28\n−0.15\n1.53.04.56.0\nΓ90120150180γmax\n0.61.21.82.4\nΘ\n10.4\n10.6\n10.8\n11.0\nlog[z0] (cm)\n0.81.21.62.0\nB0(×104G)\n36.6\n37.2\n37.8\n38.4\nlog[Pj] (erg s−1)\n0.20.40.60.8\n2n+/ne90120 150 180\nγmaxγmax= 119.51+7.54\n−10.84\nFigure 8. (a) The MCMC fit results for \u0000,\u0002,z0,B0,Pj,2n+=neand\rmaxassuming\rmin= 3 and\u000fe\u000b= 0:3. The meaning of the panels\nand lines is the same as in Figure 4. See Section 3.2 for details.JET PARAMETERS 13\nΓ = 3.10+1.03\n−0.85\n1234ΘΘ = 1.41+0.56\n−0.47\n10.210.410.610.8log[z0] (cm)\nlog[z0] (cm) = 10 .57+0.10\n−0.13\n0.51.01.52.02.5B0(×104G)\nB0(×104G) = 1.21+0.29\n−0.22\n37.237.838.439.0log[Pj] (erg s−1)\nlog[Pj] (erg s−1) = 38.66+0.37\n−0.59\n0.20.40.60.82n+/ne\n2n+/ne= 0.29+0.26\n−0.12\n1.53.04.56.07.5\nΓ80160240320400γmax\n1 2 3 4\nΘ\n10.2\n10.4\n10.6\n10.8\nlog[z0] (cm)\n0.51.01.52.02.5\nB0(×104G)\n37.2\n37.8\n38.4\n39.0\nlog[Pj] (erg s−1)\n0.20.40.60.8\n2n+/ne80160 240 320 400\nγmaxγmax= 123.96+21.14\n−13.66\nFigure 8. (b) The MCMC fit results for \rmin= 10 and\u000fe\u000b= 0:1.14 Z DZIARSKI ET AL .\nTable 2. The parameters of the jet in MAXI J1820+070 other than those given in Table 1.\n\rmin\u000fe\u000b \u0000 \u0002 log10z0B0 log10Pj\rmax\n\u000ecm 104G erg s\u00001\n3f 0.3f 2:20+0:69\n\u00000:461:04+0:48\n\u00000:3510:63+0:09\n\u00000:080:99+0:22\n\u00000:1838:31+0:32\n\u00000:60120+8\n\u000011\n10f 0.1f 3:10+1:03\n\u00000:851:41+0:47\n\u00000:5610:57+0:10\n\u00000:131:21+0:29\n\u00000:2238:66+0:37\n\u00000:59124+21\n\u000014\nNOTE—The fits are done with the MCMC method for the assumed the fixed (marked by ’f’) minimum electron Lorentz factor, \rmin, and the\naccretion efficiency, \u000fe\u000b.\nIn the absence of pairs, \u001b\u00190:0078=\feq. Comparing the\ntwo estimates of \u001b, we see it requires \u0000&3at\feq= 1.\nHowever, the actual value of sis uncertain, there could be\nions associated with background electrons piled up at \r <\n\rmin, and, importantly, \feqcould be\u001d1. Still, the low\nmagnetization implied by Equation (48) disfavors the case of\nstrong pair dominance, (1\u00002n+=ne)\u001c1.\nUsing\u001b\u001c1, we can calculate the magnetic fluxes in the\nmodel with extraction of the BH rotational power. The jet\nmagnetic flux from Equation (33) with z0B0from Equations\n(36) and (38) is then\n\bj\u0019(4:1\u00021021G cm2)s[1 + (1\u0000a2\n\u0003)1\n2](\f\u0000)1:22\u000e0:87\n(`=0:5)a\u0003\f0:22eq;\n(50)\nwhich is\u00195:3\u00021021G cm2fora\u0003= 1,\u0000 = 3 ,`= 0:5,\ns= 0:6,\feq= 1. The flux threading the BH, Equation (31)\nwith _Mestimated as above from L, is\n\bBH\u0019(1:3\u00021022G cm2)\u001eBH\n50D\n3 kpc\u0010\u000fe\u000b\n0:1\u0011\u00001=2\n;(51)\nwhereM= 8M\fwas assumed for both ( \b/M). At\n\u001eBH= 50 and the assumed parameters, the two fluxes are\napproximately equal for a\u0003\u00190:7. We consider the close\nagreement of the above two estimates to be very remarkable.\nThey are based on completely different physical considera-\ntions. Thus, our results are consistent with the jet being pow-\nered by the BH rotation and the accretion flow being mag-\nnetically arrested. In this case, the jet power is maximal and\ngiven by Equation (32). However, we have found that if pairs\ndominate in the jet, Pj\u001c_Mc2. This requires either a\u0003\u001c1,\nthe magnetic field in the flow is weaker than that in a MAD,\nor that some assumption in the model with extraction of the\nBH rotation power, e.g., the ideal MHD, are not satisfied.\n3.2. Numerical estimates\nIn order to solve directly for the physical jet parameters\nand their uncertainties, we use again the MCMC method. In\nthe fits shown in Figure 4, we fitted b,p,\u00170,\u0017max,F0,t0,\nFdiskand\u000bdiskwith the minimum assumption of a= 2b,\nand, in particular, without the need to specify the value of\n\u0000. Now we fit for all of the parameters. However, since the\nsolution given in Appendix A is given in terms of \rmaxrather\nthan\u0017max, we fit for the former (which yields \u0017maxgiven the\nvalues of \u0000,iandB0, see Equation 5).\nIn particular, we determine \u0002from Equation (A5), z0from\nEquation (21) and B0from Equation (A6). That requiresspecifying \u0000(which is then a free parameter) and \rmin. We\nfix\feq= 1 andki= 0. However, in order to be able to\nconstrain \u0000rather than have it entirely free, we include fur-\nther constraints, using the pair production rate of Equation\n(43) and requiring 2_N+=_Ne\u00141in Equation (26) and from\nthe maximum possible jet power, Pj\u0014_Mc2, Equations (23–\n27). These constraints require specifying RhotandRjet, the\nbolometric luminosity, L, and the accretion efficiency, \u000fe\u000b.\nWe then solve simultaneously for all of the parameters, in-\ncludingb,p,\u00170,F0,t0,Fdiskand\u000bdisk. Those parameters\nhave now values similar to those shown in Figure 4, and we\nthus do not show them again.\nIn the solution, we sample Dandias described at the be-\nginning of Section 3. We assume L= 1:5\u00021038erg s\u00001,\nX= 0:7,Rhot= 20Rg,Rjet= 10Rg(forM= 8M\f). We\nshow the resulting posterior distributions for two cases with\n(\rmin= 3,\u000fe\u000b= 0:3), and with ( \rmin= 10 ,\u000fe\u000b= 0:1), in\nFigures 8(a), (b), respectively, and list the fitted parameters in\nTable 2. We see that the obtained ranges of \u0000and\u0002depend\non those two sets of assumptions, being larger for for the lat-\nter case. The allowed maximum jet power is /\u000f\u00001\ne\u000b, and then\nit is higher in case (b). On the other hand, the obtained values\nofz0\u00192–4\u00021010cm andB0\u0019104G depend relatively\nweakly on those assumptions. For the sake of brevity, we\nhave not shown the effect of changing the values of Rhotand\nRjet. For example, for Rhot>20Rg, pair production will be\nless efficient, which would in turn allow fewer leptons in the\nflow and lower values of \u0000, see Equations (43–44). Thus, we\ncannot conclusively rule out values of \u0000.1:5. Then, values\nof\u0000higher than those obtained above would be possible for\n\u000fe\u000b<0:1.\nFigures 8(a–b) also show \rmax and the pair abundance,\n2n+=ne. The former ir relatively tightly constrained in the\n\u0019110–150 range. The latter is strongly anticorrelated with\nthe jet power, being low at the maximum Pjand close to unity\nat the minimum Pj, in agreement with our considerations in\nSection 3.1. We find the synchrotron power, Equation (28), is\ntypicallyPS\u00180:01Pj, as in Section 3.1, and thus the jet ra-\ndiative efficiency, PS=Pj, is low. In our fits, we have not used\nconstraints from the break frequencies in the power spectra\nand from the jet spatial extent measurement, following our\ndiscussion in Section 3.1.\n4.DISCUSSION\n4.1. The location of the dissipation zone\nOur results indicate that the jet synchrotron emission, and\nthus electron acceleration, starts at the distance of z0\u0018JET PARAMETERS 15\n3\u0002104Rgaway from the BH. This is similar to the situa-\ntion in blazars, in which the so-called blazar zones are found\nto be at distances between that of the broad-line regions and\nof the molecular torii, i.e., z0\u0018103–105Rg(Madejski &\nSikora 2016 and references therein). Then, a radio-optical\nlag measured in the blazar BL Lac yields z0of several times\n104Rg(Marscher et al. 2008). The region upstream to the\nonset of the emission is called an acceleration and collima-\ntion zone (ACZ). This similarity of z0=Rgin jets in blazars\nand in binaries accreting onto BHs is consistent with the scale\ninvariance of relativistic jets (Heinz & Sunyaev 2003). The\nabsence/weakness of emission closer to the BH is also con-\nsistent with the jets formed by magnetic processes and be-\ning initially Poynting-flux dominated (Blandford & Znajek\n1977; Blandford & Payne 1982; Blandford et al. 2019). The\nonset of the electron acceleration is often associated with the\npresence of a standing shock. For example, Ceccobello et al.\n(2018) invoke recollimation shocks at a fast magnetosonic\npoint. A requirement for a formation of a shock is \u001b < 1,\nwhich agrees with our estimates of \u001b\u001c1in the synchrotron\nemission region, see Equations (48–49). An alternative ex-\nplanation for the presence of an ACZ in BH X-ray binaries\ninvokes the colliding shell model, in which the dissipation re-\ngion is associated with the distance at which the shells begin\nto collide (Malzac 2013, 2014).\nOur determination of z0is within that found for the jet of\nthe BH X-ray binary MAXI J1836–194 of 2\u0002103–106Rgin\nthe model of Lucchini et al. (2021). On the other hand, our\nvalues ofz0are significantly larger than that of z0\u0018103Rg\ninferred from lags of the IR/optical emission with respect to\nX-rays measured in the BH X-ray binaries GX 339–4 and\nV404 Cyg (Gandhi et al. 2008, 2017; Casella et al. 2010). We\npoint out that the analysis of T21 found the cross-correlation\nbetween the optical light curve ( 3:9\u0002105GHz) and that of\nX-rays (shown in their fig. 8) to be rather complex in MAXI\nJ1820+070. The dominant feature was an anticorrelation\ncentered on the zero lag, on top of which there was a much\nweaker positive correlation with the peak at the optical vs. X-\nrays of 150+500\n\u0000700ms, and a rather low relative amplitude of the\ncorrelation peak of \u00190.03, in contrast wit the IR/optical rms\nvariability of a few tens of per cent (T21). Given that weak-\nness and the very large error bar on the lag, we have not used\nthat constraint in our modelling. The optical/X-ray cross cor-\nrelation in MAXI J1820+070 was later studied in more detail\nby Paice et al. (2021). They found that the cross-correlation\naveraged over 2-s segments of the light curves shows the pos-\nitive correlation to dominate, with the optical lag of \u0019150–\n200 ms. Curiously, this lag was almost the same for all six\nepochs they studied, in spite of a spread of the X-ray flux up\nto a factor of three (see fig. 1 in Paice et al. 2021).\nAn interpretation of the positive lag of the optical emis-\nsion with respect to X-rays as due to the signal propation\nfrom the BH vicinity to the region of the onset of the jet dis-\nsipation implies t0\u0019150–200 ms. Thus, this region would\nlie at a significantly lower distance than that found in our\nstudy, where t0\u0019790+300\n\u0000190ms, see Figure 4. We have found\nthat including this constraint would worsen our fit to the re-maining observables, especially to the 343.5 GHz flux. Given\nthe weakness of the correlation, the complexity of the cross-\ncorrelation shape and the unexplained constancy of the pos-\nitive lag component, we consider that alternative interpreta-\ntion to be uncertain. A model including that lag would have\nto also account for the dominant anticorrelation, possibly due\nto the effect of synchrotron cooling on the X-ray spectrum\nemitted by the accretion flow (Veledina et al. 2013), which is\nbeyond the scope of the present paper.\n4.2. Electron energy losses and re-acceleration\nWe have parametrized the electron distribution as a power-\nlaw function of the distance, and assume that distribution\nkeeps a constant shape. Such a situation requires the elec-\ntron energy losses are moderate and satisfying _\r/\r. We\ncompare here the time scale for synchrotron energy losses,\ntsyn=6\u0019mec\u00182b\n\u001bTB2\n0\r; (52)\nwith the adiabatic/advection time scale,\ntad=3z0\u0018\n2\f\u0000c(53)\n(e.g., Z19). We consider the solution in Section 3.1 for\n\u0000 = 3 . At\r\u001930, which corresponds to the bulk of the\npartially self-absorbed emission, tsynis shorter than tadfor\n\u0018.3, and it is\u00193 times shorter at z0. This implies that\nelectrons responsible for the optically-thin part of the syn-\nchrotron emission have to be re-accelerated above z0.\nCalculating the electron distribution self-consistently as a\nfunction of the distance as well as accounting for the slope\nof the spectrum at \u0017 < \u0017 0is relatively complex, involving\nsolving a kinetic equation with both losses and spatial advec-\ntion (e.g., Z19). This also requires taking into account losses\nfrom Compton scattering of synchrotron photons as well as\nthe reduction of the electron energy loss rate due to self-\nabsorption (Ghisellini et al. 1988; Katarzy ´nski et al. 2006).\nSuch a model is beyond the scope of the present work.\n4.3. Comparison with other jet models of accreting black\nholes\nThe main independent study of the hard-state jet of MAXI\nJ1820+070 is that by Rodi et al. (2021). They had at their\ndisposal only the spectral data. They assumed R=z = 0:1,\ncorresponding to \u0002 = 5:7\u000e, which is much larger than that\nfound by us. They assumed \u0000 = 2:2following the result\nof Bright et al. (2020) for the ejection during the hard-to-\nsoft transition, but we note that \u0000of the hard-state jet is\nlikely to be different. The jet model of Rodi et al. (2021)\nis also different from ours, and considers an initial accel-\neration event followed by synchrotron cooling assuming no\nadiabatic losses (following Pe’er & Casella 2009). They do\nnot show the spatial structure of their jet model, and thus\nwe are not able to check whether that model would agree\nwith our time-lag data. Still, they obtain relatively similar16 Z DZIARSKI ET AL .\nvalues of the distance of the onset of electron acceleration,\nz0\u00192:8\u00021010cm, and the magnetic field strength at that\ndistance,B0\u00191:8\u0002104G.\nThe very long time lags found in T21 unambiguously show\nthat the radio/sub-mm emission originates at size scales sev-\neral orders of magnitude higher than Rg. The time lags be-\ntween\u00171and\u00172are found to be approximately proportional\nto\u0017\u00001\n2\u0000\u0017\u00001\n1. Knowing the break frequency, \u00170, above which\nthe entire synchrotron emission is optically thin, we can ex-\ntrapolate this correlation and find the location corresponding\nto\u00170. This is found to be z0\u00183\u0002104Rg, with the uncer-\ntainty of a factor of at most a few. This rules out jet mod-\nels predicting the onset of the synchrotron emission to in an\nimmediate vicinity of the BH, for example that described in\nGiannios (2005) (based on the model of Reig et al. 2003).\nThe model of Reig et al. (2003) was developed in order\nto explain time lags of harder X-rays with respect to softer\nones by Compton scattering. For that reason, the authors in-\nvoke a rather massive and extended jet, where multiple scat-\ntering of disk photons place. Consequently, this model (fur-\nther developed in a number of subsequent papers of those\nauthors) requires a rather large rate of the electron flow. For\nthe parameters of Giannios (2005) (similar to those in Reig\net al. 2003), the base of the jet has a radius of R0= 100Rg\nwith the electron density of ne;0\u00193:43\u00021016cm\u00003(for\nM= 8M\f), which electrons flow upward with \f= 0:8. For\nmatter with cosmic composition, this corresponds to the mass\nflow in the both jets of 2:4\u00021020g/s. On the other hand, the\nbolometricLestimated for MAXI J1820+070 corresponds\nto_M\u00191:7(D=2:96 kpc)2(\u000fe\u000b=0:1)\u000011018g/s, i.e., two or-\nders of magnitude less, which rules out this case. We can also\nconsider the case in which the leptons in the flow are pairs.\nIn that case, 2_N+= 2\u0019R2\n0n2\ne;0\f\u0000c\u00191:2\u00021044s\u00001. This\nis 3–4 orders of magnitude higher than the pair production\nrate in this source calculated in Equation (43), which rules\nout this case too. Since the hard state of MAXI J1820+070\nis rather similar to that of other X-ray binaries with accreting\nBHs, we find we can rule out this model in general.\n4.4. Other constraints and caveats\nOur model is based on that of Blandford & K ¨onigl (1979)\nand K ¨onigl (1981), and it assumes uniform scaling of the\nemission regions, through the coefficients aandb. As we\nsee in Figure 5, this model does not account for the observed\nflux at 339 MHz, measured by Polisensky et al. (2018). This\nhints for the decline of the energy content in the relativistic\nelectrons and magnetic field being initially faster (responsi-\nble for the emission closer to z0) and then slower (responsible\nfor the emission farther away from z0). This would introduce\nmore complexity in the modelling, and is beyond the scope\nof this work. On the other hand, the flux at 339 MHz could be\ndue to another component, in particular a pair of radio lobes\nat the jet ends. An assumption of our model is that the bulk of\nthe emission at a given distance in the partially self-absorbed\npart of the spectrum occurs at a \u0017corresponding to \u001c\u00191.\nAs we have found out, this corresponds to the synchrotron\nemission by electrons with \r\u001830. If the minimum Lorentzfactor of the electron distribution were higher, \rmin>30,\nthen the emission at a given distance in that part of the spec-\ntrum would be dominated by the electrons at \rmininstead,\nwith no contribution from self-absorption.\nWe assumed the jet is already fully accelerated at z0and\nthen does not decelerate. This may be not the case, and the\navailable data do not exclude that. The jet model of Z19 al-\nlows for a variable \u0000, and we could use some parametriza-\ntion of \u0000(z)and refit our data (as done in Zdziarski 2019 for\nanother source). This would, however, introduce more free\nparameters, and make the resulting fits less constrained than\nin the present case. We have also considered the steady state,\nwhile variability has been observed. However, the fractional\nvariability was\u00180:3at the sub-mm range and much less\nthan that in the radio regime. Thus, the variability can be\nconsidered as a small perturbation of the steady state.\nWe also use a \u000e-function approximation to the synchrotron\nprocess, which is a good approximation for power-law\nparts of the spectra, but becomes less accurate at cutoffs,\ngiven the single-electron synchrotron spectrum is quite broad\n(Ginzburg & Syrovatskii 1965). We assume the synchrotron\nemission of a single electron is isotropic in the plasma frame,\nwhich is valid for a tangled magnetic field, while we assume\na toroidal field in some of our equations. Furthermore, we\nassume a sharp cutoff in the electron distribution at \rmax.\nWhile this is not realistic, the actual form of the cutoff de-\npends on details of the acceleration process and is poorly\nconstrained. Thus, our determination of \rmaxbased on the\nobserved cutoff in the optical range is only approximate.\nThen, we have used our self-consistent set of equations, in\nwhich the slope of the partially self-absorbed part of the syn-\nchrotron spectrum is connected to the rate of decline of the\nenergy density along the jet. The latter determines the rela-\ntionship between the characteristic emitted frequency and the\ndistance (Equation 15), and thus the time-lag vs. frequency\nrelation. A significant discrepancy between the spectral slope\nand time lags vs. frequency was found in Cyg X-1 (Tetarenko\net al. 2019). In our case, the two are in an approximate mu-\ntual agreement.\nWe have found that the break frequencies in the power\nspectra,fb(\u0017), are compatible with the origin of the emis-\nsion atz\u0017, which are roughly equal to \fc=f b(\u0017)for\u0017 <\u0017 0.\nHowever, an increasing fbwith increasing \u0017is also observed\nfor the IR and optical data (see fig. 5 of T21), for which\n\u0017 > \u0017 0. In our jet model, the emission at \u0017 > \u0017 0is the\noptically-thin synchrotron from the entire part of the jet at\nz >z 0, which implies z\u0017>\u0017 0=z0. Thus, we expect that the\nabove scaling of fb/z\u00001\n\u0017no longer holds at \u0017 >\u0017 0. Then,\nthe IR/optical variability at high Fourier frequencies may be\nmostly due to electron energy losses and the re-acceleration\n(see Section 4.2) rather than due to propagation of some dis-\nturbances from z<z 0.\nAs shown in fig. 8 of T21, the optical and IR light curves\nare tightly correlated, with no measurable lag ( \u000018+30\n\u000050ms),\nin spite of a relatively large disk contribution in the optical\nrange ( 3:7\u0002105GHz), as shown in Figure 5. This shows the\nthe disk contribution is constant on the studied time scales,JET PARAMETERS 17\nwhich is consistent with the rms variability in the optical\nrange reduced with respect to the IR one, see fig. 5 in T21.\nAs shown in T21, the upper limit on the lag is consistent with\nthe synchrotron energy losses at the magnetic field strength\nof\u0018104G, which agrees with our determination of B0.\n4.5. Relationship to core shifts\nTime lags are closely related to core shifts, \u0001\u0012, which are\nangular displacements of the radio cores, observed at fre-\nquencies where the synchrotron emission is partially self-\nabsorbed. They are commonly found in radio-loud active\ngalactic nuclei (e.g., Pushkarev et al. 2012). The physical\ncause of the physical displacement along the jet, z\u00172\u0000z\u00171,\nis the same for both the core shifts and time lags; only the\nmethods to determine it are different. Using equation (4) in\nZ15 and Equation (20), the relationship of \u0001\u0012to\u0001tais\n\u0001\u0012=\u0001ta\fc(1 +zr) sini\nD(1\u0000\fcosi): (54)\nWe can then relate \u0001tatoz\u0017,z0and\u00170using Equations (20–\n21) and (A2).\nWe can estimate B0using the core-shift method, but only\nassuminga= 2,b= 1, which parameters have been as-\nsumed in published core-shift studies, including Z15. Us-\ning equation (7) of Z15 and Equation (54), we obtain B0\u0019\n1:0\u0002104G atp= 2,\u000e= 1 and\feq= 1, in a good agree-\nment with our estimate of Equation (38). We can also obtain\nB0from equation (8) in Z15 without specifying \feq.\n5.CONCLUSIONS\nWe have based our study on the results of a multiwave-\nlength campaign observing MAXI J1820+070 when it was\nclose to the peak of its luminous hard spectral state, at \u001815%\nof the Eddington luminosity. We have used mostly the data\npublished in T21 as well as the IR/optical spectrum from\nRodi et al. (2021). Our main conclusions are as follows.\nA major model-independent result of our study is the es-\ntimate of the distances at which the jet emits below the ob-\nserved break frequency, based on the time lags between vari-\nous frequencies. These distances are definitely several orders\nof magnitude higher than Rg. By extrapolating the observed\napproximate correlation of the time lags with the differences\nbetween the photon wavelengths, the place where that emis-\nsion begins can be estimated to be at the distance of several\ntens of thousands of Rgfrom the BH. This value of that dis-\ntance agrees with the corresponding finding of Rodi et al.\n(2021), based on spectral modelling alone.\nWe then use the classical model of Blandford & K ¨onigl\n(1979), as formulated in detail in later works, to determine\nthe parameters of the jet emitting in the radio-to-optical\nrange. The model assumes the hard inverted spectrum is due\nto the superposition of locally-emitted spectra that are self-\nabsorbed up to some frequency and then are optically thin.\nApart from some details, this is the same model as that used\nby T21. The values of the jet parameters obtained by us up-\ndate those of T21, which suffered from some errors (see Sec-\ntion 1 and Appendix A). Our analysis is also broader thanthat of T21, utilizing constraints from the break frequency\nand the optically thin part of the spectrum.\nBy applying the model to the data, we find we cannot\nuniquely determine the jet bulk Lorentz factor, \u0000, which then\nneeds to be specified as a free parameter. However, it can be\nconstrained from above by the requirement of the jet power\nbeing less than the accretion power. It can also be constrained\nfrom below by estimating the e\u0006pair production rate in the\nbase of the jet and comparing it to the flux of e\u0006required to\naccount for the observed synchrotron emission. We then use\na Bayesian MCMC method to determine all of the jet param-\neters, and find the most likely range of 1:7.\u0000.4:1. We\nfind the jet half-opening angle, \u0002constrained to\u00190.6–2\u000e.\nThe onset of the emission is at z0\u00193–4\u00021010cm, where\nthe magnetic field strength is B0\u00190:8–1:4\u0002104G. The to-\ntal jet power is between Pj\u00181037and\u00181039erg s\u00001. The\njet composition is strongly correlated with Pj, being mostly\npairs at the lower limit and mostly e\u0000and ions at the upper\nlimit. The optical spectral data imply a rather low value of\nthe maximum electron Lorentz factor, of \rmin\u0019110–150.\nIn order to explain the possible presence of e\u0006pairs in the\njet, we calculate the rate of pair production in the jet base\nin immediate vicinity of the hot accretion flow. We use the\nmeasurement of a power-law spectral component extending\nat least to 2 MeV in the same state of MAXI J1820+070.\nThis rate depends on the geometry, see Figure 3, but we find\nit to be of the same order as the rate of the electron flow\nthrough the synchrotron-emitting part of the jet, both being\n\u00181040–1041s\u00001. We find this coincidence to be a strong\nargument for the presence of pairs in the hard-state jet of\nMAXI J1820+070. We have also estimated the contribution\nof transient accreting BH binaries to the total rate of positron\nannihilation in the Galaxy and found it to be very small.\nWe also consider the possibility of the jet power being lim-\nited by the power from the rotation of the BH in the pres-\nence of magnetically arrested accretion flow. To test it, we\ncalculate the magnetic flux of the jet in the emitting region\nand that threading the BH. We find them to be very similar,\n\u00181021G cm2, which remarkable coincidence argues for that\nscenario. Then, the jet is initially magnetic, Poynting-flux\ndominated, slow, and not emitting, and accelerates and dis-\nsipates its energy only at large distances, in agreement with\nour finding of the emission being far from the BH.\nWe find the synchrotron power to be only a small fraction,\n\u001810\u00002, of the total jet power. On the other hand, the syn-\nchrotron power is very similar to either the electron or mag-\nnetic powers at the onset of the dissipation, showing that de-\ncline of those powers with the distance necessary to explain\nthe observations can be due to the synchrotron emission.\nFinally, we show the correspondence between the methods\nto determine the jet parameters based on time lags and ra-\ndio core shifts. We give a formula relating the lags and the\nangular displacements of the radio cores.\nACKNOWLEDGMENTS\nWe thank M. B ¨ottcher, B. De Marco, P. Gandhi, N. Ky-\nlafis, P.-O. Petrucci, Th. Siegert and A. Veledina for valu-18 Z DZIARSKI ET AL .\nable comments and discussions, A. Tchekhovskoy for per-\nmission to use his plot of GRMHD simulations, and the\nreferee for valuable comments. We acknowledge support\nfrom the Polish National Science Centre under the grants\n2015/18/A/ST9/00746 and 2019/35/B/ST9/03944, and from\nthe International Space Science Institute (Bern). Support\nfor this work was provided by NASA through the NASA\nHubble Fellowship grant #HST-HF2-51494.001 awarded\nby the Space Telescope Science Institute, which is op-\nerated by the Association of Universities for Research\nin Astronomy, Inc., for NASA, under contract NAS5-26555. This paper makes use of the following ALMA data:\nADS/JAO.ALMA#2017.1.01103.T. ALMA is a partnership\nof ESO (representing its member states), NSF (USA) and\nNINS (Japan), together with NRC (Canada), MOST and\nASIAA (Taiwan), and KASI (Republic of Korea), in coop-\neration with the Republic of Chile. The Joint ALMA Obser-\nvatory is operated by ESO, AUI/NRAO and NAOJ. The Na-\ntional Radio Astronomy Observatory is a facility of the Na-\ntional Science Foundation operated under cooperative agree-\nment by Associated Universities, Inc.\nAPPENDIX\nA.THE GENERAL SOLUTION\nWe provide here the general solution to the equations pro-\nviding the jet structure, given in Section 2.1. We first give\nthe solution parametrized by the equipartition parameter, \feq,\nbut without utilizing the time-lag constraint. This solution is\nanalogous to that given by equations (28–29) in Zdziarski\net al. (2012), which is valid for a= 2,b= 1 only. Here,\nwe assume that equipartition holds along the entire emitting\njet, i.e.,a= 2b; otherwise, it would be artificial to impose\nit only atz0. We note that the solutions below are functions\nof\rmax(throughfEandfN), while observationally we de-\ntermine\u0017max. The relation between the two involves B0,\u0000\nandi, see Equations (1) and (5). As a consequence, explicit\nsolutions in terms of \u0017maxwould be rather complicated, and\nwe do not provide them.\nWe determine B0by settingn0from Equation (14) equal\nto that following from Equation (17), and then use z\u0017=\nz0(\u00170=\u0017)qfrom Equation (15), finding\nB(z\u0017) =mec\ne\u0010\u0019\n\u000e\u00112+p\n6+p\u00143c(1 +ki)(fE\u0000fN) sini\n\feqC2(p)z\u0017tan \u0002\u00152\n6+p\n\u0002[2\u0017(1 +zr)]4+p\n6+p (A1)\nforz\u0017\u0015z0,\u0017\u0014\u00170. Then,B0=B(z0). We then sub-\nstituteB(z\u0017)of Equation (A1) in the formula for F\u0017in the\noptically-thick case, Equation (12), which yields for z\u0017\u0015z0\nz\u0017=1\n\u000e4+p\n13+2 p\u0017\u0014c(1 +ki)(fE\u0000fN)\n2\u0019\feq\u0015 1\n13+2 p\n\u0002\n\u0014C2(p)\nsini\u00155+p\n13+2 p\u0014F\u0017D2\nmeC1(p)gbp\u00156+p\n13+2 p\n\u0002 (A2)\n\u00123\n\u0019tan \u0002\u00137+p\n13+2 p\n(1 +zr)\u000019+3 p\n13+2 p;\nwheregbpfollows from Equation (12) for a= 2b,\ngbp\u0011\u0000E\u0014b(p+ 5)\u00006\nb(p+ 6)\u00002\u0015\n=(4 +b): (A3)Then,z0=z\u00170, at whichF\u00170= 2F0gbp, see Equation (12)3.\nWe can then substitute the above z\u0017(\u0015z0)into Equation\n(A1),\nB(\u0017) =\u0017\u00143C1(p)gbp(1 +ki)2(fE\u0000fN)2sin3i\nC2(p)3\f2eqD2F\u0017tan \u0002\u00152\n13+2 p\n\u0002\u00197+2p\n13+2 p29+2p\n13+2 pc17+2 p\n13+2 pm15+2 p\n13+2 pe (1 +zr)15+2 p\n13+2 p\ne\u000e3+2p\n13+2 p; (A4)\nandB0=B(\u00170). We see that the above solutions are ob-\ntained without specifying either \u00170orz0. Also, the spatial\nindexbenters only in the factor gbp, and does not modify the\nfunctional dependencies. Equations (A2) and (A4) are equiv-\nalent to equations (28–29) in Zdziarski et al. (2012), which\nare fora= 2,b= 1, and differ only in the definition of \feq\nand by factors of the order of unity due to a slightly different\nway of integrating the emission along the jet.\nNext, we can use the independent determination of z\u0017from\nthe time lags, \u0001ta. A single measured lag between the fre-\nquencies\u00172and\u00171determines, via Equation (20), z\u00172\u0000z\u00171.\nThis can be compared to the prediction using z\u0017of Equation\n(A2), which yields a constraint between \u0002and\u0000. However,\na single measurement of \u0001tahas typically a large error. We\ncan combine them by fitting the relationship between \u0001tavs.\nz\u00172\u0000z\u00171. This can be done even when the break frequency,\n\u00170, is unknown. However, here it is known, and we find it\nconvenient to define t0by\u0001ta=t0(z\u00172\u0000z\u00171)=z0, fitted to\na number of measured lags. This then implies z0=ct0\f\u0000\u000e.\nWe can set it equal to that implied by Equation (A2), and\n3Equation (A2) also provides the correct form of equation (5) in Heinz\n(2006) forp= 2 ,b= 1 . His equation should be multiplied by \u000e1=2\nfactor, which is due to that factor missing in his equation (1), which should\nhave accounted for the frame transformation from \u00170to\u0017. That incorrect\nmodel formulation was used in T21.JET PARAMETERS 19\nsolve for tan \u0002 as a function of \u0000,\ntan \u0002 =3 (\f\u0000\u00170t0)\u000013+2 p\n7+p\n\u00198+p\n7+p\u000e17+3 p\n7+p\u0014(1 +ki)(fE\u0000fN)\n\feq\u00151\n7+p\n\u0002\n\u00142C2(p)\nsini\u00155+p\n7+p\u0014F0D2\nmec2C1(p)\u00156+p\n7+p\n(1 +zr)\u000019+3 p\n7+p: (A5)Note a relatively strong dependence of \u0002ont0,\u0002/\u0018t\u00002\n0.\nWe can then insert this tan \u0002 into Equation (A4) to obtain\nB0=23+p\n7+p\u00195+p\n7+p(me\u00170)9+p\n7+pc11+p\n7+p\ne\u000ep\u00001\n7+p\u0002 (A6)\n\u0014\f\u0000t0C1(p)(1 +ki)(fE\u0000fN) sin2i\nF0\feqC2(p)2D2\u00152\n7+p\n(1 +zr)11+p\n7+p:\nWe determine n0using the above B0in Equation (17),\nn0=\u001718+2 p\n7+p\n0m11+p\n7+pe\n215+p\n7+p\u000e2p\u00002\n7+pe2\u0014c2\f\u0000t0C1(p) sin2i\nF0C2(p)2D2\u00154\n7+p\n\u0002\u0014\u0019\feq\n(1 +ki)(fE\u0000fN)\u00153+p\n7+p\n(1 +zr)22+2 p\n7+p: (A7)\nREFERENCES\nAtri, P., Miller-Jones, J. C. 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In this article some features of a Lorentz-violating (but Lorentz-\ncovariant) Lagrangian of a scalar tachyon field are consider ed. It is shown that the equation\nof motion and Feynman propagator resulting from it are Loren tz-invariant, while the Lorentz\nsymmetry of the suggested tachyon field model can be defined as covariantly broken.\nKeywords: tachyons, causality, tachyon vacuum, Lorentz-non-invari ance, Feynman propagator\n1. Introduction\nThegenerallyaccepted opinionexistinginsidethephysics communityisthatfaster-than-light\nsignals and particles (tachyons, [1, 2]) would lead inevita bly to causality violations1. Another\nserious problem related to the tachyons is the presumable in stability of the tachyon vacuum.\nSometimes also the violation of the unitarity by tachyons in teracting with ordinary particles is\ndeclared [4, 5].\nMeanwhileit isknownsincethe1970’s that tachyons donot vi olate causality ifonepostulates\nthe existence of a preferred reference frame in which the pro pagation of free tachyons is ordered\nby retarded causality, see e.g.[6, 7].\nWith similar ideas it has been shown recently [3] that the tac hyon hypothesis, if one treats\ntachyons properly, does not lead to the appearance of causal paradoxes, while the causality\nprinciplehastoberedefinedas arequirement oftheabsenceo fcausal loops, i.e. theimpossibility\nof a transfer of information to the past of an observer. There solution of causal paradoxes comes,\nindeed, fromatrivial observation that fasttachyons canpr obecosmological distances. Thisleads\nto a conclusion that the completely correct description of t achyon behaviour can be made within\ngeneral relativity only, while the Lorentz symmetry has to b e considered as an approximate one\nwhenappliedtotachyons. Thismakes theuseof tachyons fort heconstructionof thecausal loops\nimpossible (the reasons for this are formulated in the next p aragraph). It has been shown in [3]\nthat the correct approach to the tachyon theory can be achiev ed only within the postulate of a\ntight association of the tachyon preferred reference frame with the comoving frame of relativistic\ncosmology (see the definition of the latter, for example, in [ 8]). This is the absolute rest frame\n1Often causality violation is related intrinsically to the m ere possibility of existence of tachyons by pointing\nout that the positive time interval between events connecte d by a space-like world line can be converted to a\nnegative time interval by a suitable Lorentz transformatio n. Such a change of the event time order can indeed\ntake place in the case of tachyon signals, but it does not mean yet causality violation . The causality violation\nappears only when a causal loop can be constructed, i.e. a sen ding by an observer a signal to its own past, see\n[3].\nPreprint submitted to Elsevier June 17, 2019[9] in which our universe is embedded. In particular, the dis tribution of matter in the universe\nis isotropic in this frame only, the same is true for the relic black body radiation. The causality\nprotection formula, valid in all inertial frames, was formu lated in [3] as follows:\nPu≥0, (1.1)\nwherePis a 4-momentum of particles transferring a signal and uis a 4-velocity of the preferred\nreference frame with respect to (any particular) inertial o bserver. It is a boundary condition\nwhich should be imposed on solutions of any tachyon equation of motion.\nThe next step, the introduction of the concept of the preferr ed reference frame into the\nMinkowski space, can be considered as an action approximati ng that space to the real world\nspace-time. It does not destroy the mathematical perfectne ss of the Lorentz group since the\nderivation of the Lorentz transformations is based on the re quirement of the invariance of the\ninterval between world points when passing from one inertia l frame to another, and the presence\nor the absence of the preferred reference frame among the fra mes under consideration does not\naffect the derivation to any extent. In view of this one can reta in the Lorentz group (as well as\nthePoincar´ e group)whentreating tachyons. Together with this theintroduction of thepreferred\nreference frame into the tachyon theory removes the problem of the instability of the tachyon\nvacuum (see [3] and sect. 2 in this article). The space of the p referred frame turns out to be\nspanned by the continuous background of free, zero-energy o n-mass-shell tachyons propagating\nisotropically, i.e. the eigenvalues of the tachyon Hamilto nian are restricted from below, in this\nframe, by zero value. This excludes the possibility of the co nstruction of the causal loops using\ntachyons since for such a construction negative energy tach yons (propagating backward in time)\nare necessary. In the frames moving with respect to the prefe rred one negative energy tachyons\ncan appear due to Lorentz boosts, but causal loops cannot, si nce the presence or the absence\nof the causal loops is an invariant property of the relativis tic theory describing tachyons. For\nexample, the energy boundaries of the tachyon vacuum, which has to be defined in tachyon\nquantum field theories by the tachyon vacuum gauge\nPu= 0, (1.2)\nin the frames moving with respect to the preferred one are giv en by expressions\nE+\n0=µ|u|√\n1−u2(1.3)\nfor the direction coinciding with the preferred frame veloc ityuand by\nE−\n0=−µ|u|√\n1−u2. (1.4)\nfor the opposite direction.\nSimultaneously it turns out that in any reaction in which tac hyons participate asymptotic\n“in”and“out”tachyonicFockspacesareunitarilyequivale nt, whichsolvestheunitarityproblem.\nIt was argued in [3] that a realistic model of a tachyon theory should be built upon the\ninfinite-dimensional unitary irreducible representation s of the Poincar´ e group (so called “infinite\nspin” tachyons). Within the conjecture that elementary par ticles are realizations of the unitary\nirreducible representations of the Poincar´ e group the onl y alternative to the infinite spin tachyon\nmodels would be a scalar tachyon model. However this model ca nnot represent tachyons at a\nfundamental level since it possesses several diseases; in p articular, such a model would lead to\nthe instability of photons via their decay to tachyon-antit achyon pairs [3] (note that decays of\nphotons to the infinite spin tachyon-antitachyon pairs are f orbidden by the angular momentum\nconservation combined with kinematic restrictions impose d on this process).\n2Nevertheless, a scalar tachyon field model has to be consider ed first for the following reasons.\nBeing simpler than a theory based on the infinite-dimensiona l wave equation, it can serve as a\ntoy model in which several new theoretical ideas can be teste d, namely, the realization of the\npostulate of a preferred reference frame as applied to quant um field theory in a covariant way,\nthe construction of a stable tachyon vacuum with a confinemen t of acausal tachyon modes, the\nLorentz-invariance of the tachyon Feynman propagator, nec essary to restrict Lorentz-violating\neffects to the free-tachyon sector only, etc. Moreover, all in dividual components of the infinite-\ndimensional wave equation must satisfy the Klein-Gordon eq uation.\nTherefore the scalar tachyon models were considered in ref. [3]. They are based on Lorentz-\ncovariant scalar tachyon Lagrangians with a covariant brea king of the Lorentz symmetry, so the\nLorentz invariance violation appears to be affecting the asym ptotic tachyon states only, while\nleaving the sector of ordinary particles within the Standar d Model untouched, at least up to\npossible small radiative corrections. For example, the Her mitian tachyon field operator with the\ncausal Θ-function accounting for the boundary condition (1 .1), Θ(ku), reads as follows:\nΦ(x) =1/radicalbig\n(2π)3/integraldisplay\nd4k/bracketleftBig\na(k)exp(−ikx)+a+(k)exp(ikx)/bracketrightBig\nδ(k2+µ2) Θ(ku),(1.5)\nwherekis a tachyon four-momentum, a(k),a+(k) are annihilation and creation operators with\nbosonic commutation rules, annihilating or creating tachy onic states with 4-momentum k, andµ\nis a tachyon mass parameter. As can beseen, the expression (1 .5) is explicitly Lorentz-covariant.\nThiscovariance includes theinvariant meaningofthecreat ion andannihilation operators defined\nin the preferred frame; thus, for example, an annihilation o peratora(k) remains an annihilation\noperatora(k′) in the boosted frame, even if the zero component of k′may become negative.\nThis is because the one-sheeted tachyon mass-shell hyperbo loid is divided by the covariant\nboundary ( ku) = 0 into two parts separated in an invariant way. This result s, in particular,\nin a possibility of the standard operator definition of the in variant vacuum state |0>via the\nannihilation operators a(k),a(k)|0>= 0 for all ksuch that |k|>µ, because the vacuum state\nenergy and, as a consequence, the field Hamiltonian turn out t o be bounded from below in any\nreference frame, see [3] and formulae (2.12), (2.13) below.\nInpaper[10]aformalwayofintroducingthecausalΘ-functi onintothetachyon fieldoperator\n(1.5) has been suggested. In this article we continue the con sideration of Lorentz-invariant and\nLorentz-non-invariant properties of the tachyon field mode ls mentioned above, including an\nimportant element of the models such as the Lorentz-invaria nce of the Feynman propagator of\nthe tachyon scalar fields.\nThis note is organized as follows. In Section 2 we suggest a Lo rentz-non-invariant, but\nLorentz-covariant modification of the scalar tachyon Lagra ngian which leads to a tachyon Hamil-\ntonian possessing Lorentz-non-invariant boundaries of th e tachyon vacuum. The modified La-\ngrangian leaves however the tachyon equation of motion unch anged which leads to the Lorentz\ninvariance of the tachyon Feynman propagator considered in Section 3. Representations of this\npropagator in the configuration space are given in Section 4. In Section 5 some Lorentz-invariant\nand Lorentz-non-invariant two-point functions of scalar t achyon fields are presented. Sections 6\nand 7 contain the note summary and conclusion.\nIn formulae used in this article the velocity of light cand the Planck constant ¯ hare taken\nto be equal to 1.\n2. Proposed Lorentz-non-invariance of the tachyon Lagrangian\nWe start with a Lorentz-invariant Lagrangian of a free scala r tachyon field\nL=1\n2/integraldisplay\nd3x/bracketleftBig˙Φ2(x)−/parenleftBig\n∇Φ(x)/parenrightBig2+µ2Φ2(x)/bracketrightBig\n(2.1)\n3resulting in the (Klein-Gordon) equation of motion\n/parenleftBig∂2\n∂t2−∂i∂i−µ2/parenrightBig\nΦ(x) = 0, i= 1,2,3 (2.2)\nand in the Hamiltonian\nH=1\n2/integraldisplay\nd3x/bracketleftBig˙Φ2(x)+/parenleftBig\n∇Φ(x)/parenrightBig2−µ2Φ2(x)/bracketrightBig\n. (2.3)\nA standard approach to finding the minimum of the Hamiltonian (2.3), which could present\nthe field ground state Φ 0, i.e. the vacuum (requiring δH= 0), is reduced to an analysis of its\npotential term. This assumes, implicitly, that the search f or the ground state of the Hamiltonian\nis replaced by looking for its minimum under restrictions co nditioned by the Lorentz-invariant\npair of the vacuum “initial” conditions:\nΦ0=const in time, (2.4)\nΦ0=uniform (const)in space. (2.5)\nIn the case of ordinary particles, with a positive m2in the corresponding Hamiltonian, the\nexercise of ground state finding under these conditions succ eeds at Φ = 0, but the potential\nterm of the Hamiltonian (2.3) has a maximum at Φ = 0 instead of t he necessary minimum. This\nis interpreted as an impossibility (instability) of the tac hyon vacuum.\nHowever, thehypothesisofthefaster-than-lightparticle srequirestheconsiderationoftachyons\nunder a postulate of a preferred reference frame which is nec essary for the causal ordering of\nthe signals propagating over the spacelike intervals. This requirement must be respected by the\nprocedure of the tachyon ground state finding also. Therefor e the Lorentz-invariant pair of the\ninitial conditions (2.4), (2.5) must be replaced by a single , Lorentz-non-invariant one:\nΦ0=const in time (2.6)\nwhich separates, obviously, the preferred reference frame , while the condition (2.5) should be\navoided, thus excluding spatially uniform fields from the co nsideration . Then the Hamiltonian\nwhich has to be analysed in the search for the ground state wil l contain, together with the\npotential term, a gradient energy term which compensates th e former. The equation δH= 0\nbecomes equivalent to the equation δL= 0, i.e. to the equation of motion. In our case it is\nthe Klein-Gordon equation (2.2) which has, in general, solu tions in the form of plane waves\nexp±i(Et−kx) with the dispersion relation\nE≡k0=/radicalBig\nk2−µ2, (2.7)\nunder the prescription [1, 2]\n|k| ≥µ (2.8)\n(note, this prescription is Lorentz-invariant2).\n2To prove this statement let us consider the inequality |k| ≥µ, valid in some inertial frame, in a boosted frame.\nIn order to simplify the proof we take the boost direction opp osite to the tachyon momentum k: this is a critical\ncase for the validity of the inequality; then it transforms t o (|k|−E|u|)/√\n1−u2≥µ. This can be rewritten as\n|v|−|u|√\nv2−1√\n1−u2≥1, (2.9)\nwherevis a tachyon velocity, v=k/E. Consider now the function of modules of 3-dimensional vect orsuandv,\nf(u,v) = (v−u)/(√\nv2−1√\n1−u2). This function reaches its minimal value of 1 at uv= 1. This means that the\nexpression (2.9) holds always, i.e. the condition |k| ≥µis Lorentz-invariant.\n4Now we have to require the condition (2.6) to be fulfilled, and this can be easily satisfied by\nputtingE= 0 in the obtained solutions, which evidently minimizes the Hamiltonian density in\n(2.3) to zero value. If, for example, one “pumps” in some way ( via interactions) the energy into\nparticular vacuummodes promotingtheirconversion toreal tachyons (field excitations satisfying\nrelations (2.7), (2.8) at E >0), the field energy H will be increased due to an appearance of the\nkinetic energy term.\nLet us consider now a possible modification of the Lagrangian (2.1) by adding to it an\napparently Lorentz-non-invariant, but Lorentz-covarian t term proportional to the 4-velocity u\nof the preferred reference frame:\nL=1\n2/integraldisplay\nd3x/bracketleftBig˙Φ2(x)−/parenleftBig\n∇Φ(x)/parenrightBig2+µ2Φ2(x)+λuµ∂µΦ(x)/bracketrightBig\n, (2.10)\nwhereλhas the dimensionality of the mass squared. For the suggeste d tachyon field model\nviability it is important that the additional term does not c hange the equation of motion (2.2).\nChoosingλ=µ2one gets the corresponding Hamiltonian\nH=1\n2/integraldisplay\nd3x/bracketleftBig˙Φ2(x)+/parenleftBig\n∇Φ(x)/parenrightBig2−µ2Φ2+µ2u√\n1−u2∇Φ(x)/bracketrightBig\n. (2.11)\nThus, theadditional term intheintegrand of (2.11)shiftst hetachyon vacuumenergy boundaries\ndepending on the value and direction of the 3-velocity of the preferred reference frame uwith\nrespect to the tachyon source (illustrated by formulae (1.3 ), (1.4) and by fig. 1), which was just\nthe aim of the introduction of this term.\na) b)\nEtA EtB\nuA = 0\n0uB\n0E0\nE0forward region backward region\nFig. 1. The tachyon vacuum energy boundaries as seen a) from the preferred reference frame A\n5andb) from a frame B moving with respect to the preferred one with 3 -velocity u. The direction\nof the preferred frame motion as seen from the frame B is indic ated by an arrow in the top part\nofb).E+\n0andE−\n0mark the “forward” and the “backward” tachyon vacuum energy boundaries\nin the moving frame given by (1.3) and (1.4). The vertical axe s on both figures are for tachyon\nenergies, with the hatched regions to be excluded domains fo r free tachyons.\nAfter second quantization procedure the Hamiltonian reads (see [3]):\nH=/integraldisplay\n|k|>µ,ω>kud3k\n(2π)3ω−ku√\n1−u2a+\nkak. (2.12)\nThus the Hamiltonian is bounded from below and is Hermitian. In the preferred reference frame\nH=/integraldisplay\n|k|>µ,ω>0d3k\n(2π)3ω a+\nkak (2.13)\nhaving non-negative eigenvalues.\nTo conclude this section we formulate its result: the Lagran gian (2.10) differs from the\nLagrangian (2.1) by a Lorentz-non-invariant term presente d in the former; and since this ad-\nditional term, written down as λ∂µFµ(x), whereFµ(x)≡uµΦ(x), is proportional to the total\ndivergence of the 4-vector Fµ(x), the two Lagrangians, with and without the additional term ,\nare physically equivalent since the term with ∂µFµ(x) does not contribute to physical quantities,\nexcepting those related to the tachyon vacuum.\nFurthermore, since the additional term does not change the a ction we have, by the principle\nof least action, the same tachyon equation of motion (2.2), a s mentioned above3. Therefore\nwithinour approach theLorentz invariance can bedefinedas c ovariantly broken andits violation\nappears to be restricted to the asymptotic-tachyon-states sector only, as will become clear from\nthe further considerations.\n3. Lorentz invariance of tachyon Feynman propagator\nConsidering a tachyon propagator in momentum space as an inv erse of a Fourier transform\nof the wave equation (2.2) with the condition (2.8) imposed, we can write down, for example,\nthe Feynman propagator as\n˜DF(k) =i θ(|k|−µ)\nk2+µ2+iǫ(3.1)\nto be used in Feynman diagrams describing tachyon interacti ons, of course, only within our toy\nmodel of scalar tachyons. In the configuration space\nDF(x−y) =/integraldisplay\n|k|≥µd4k\n(2π)4iexp[−ik(x−y)]\nk2+µ2+iǫ. (3.2)\nOne can see that the tachyon Feynman propagator defined by for mula (3.2) is explicitly Lorentz-\ninvariant since all ingredients in this formula, including integration limits, are Lorentz-invariant\n3Generally speaking, when passing from a Lagrangian to a Hami ltonian one should take into account that there\nis a term in the Hamiltonian density proportional to a (parti al) derivative of the Lagrangian density over ˙Φ (the\nmomentum density conjugate to Φ). The term with this derivat ive, resulting in a transition from the Lagrangian\nto the Hamiltonian, destroys the immunity of the latter from the adding of an extra term (the gradient term in\nthis case) to it.\nThe different influence of given extra terms on the Lagrangian and on the Hamiltonian should not surprise us:\nwhile the Lagrangian is a Lorentz scalar (the same is true for the action), the Hamiltonian is the 0-component of\nthe field 4-momentum.\n6(in particular, if the integration limit |k| ≥µholds in some inertial frame it holds in any such\nframe4).\nLet us note that, as a consequence, the same invariance holds also for virtual tachyons and\ntachyon loops appearing in Feynman diagrams of reactions co ntaining only ordinary particles\nin the initial and final states; thus no Lorentz-violating effe cts induced by virtual tachyons\ncan appear in such reactions. Representations of the tachyo n Feynman propagator (3.2) in\nconfiguration space are given in the next section.\nIt is not a problem (see Section 5) to obtain (3.2) as a time-or dered product of the tachyon\nfield operators (1.5), taken at points xandyand averaged over the tachyonic vacuum, i.e. as\nan amplitude of the tachyon transition from xtoyorvice versa :\nDF(x−y)≡ /angb∇acketleft0|TΦ(x)Φ(y)|0/angb∇acket∇ight=/braceleftbiggD(x−y) ifx0>y0\nD(y−x) ifx0<y0, (3.3)\nwhereD(x−y) is a correlation function of the tachyon field operators Φ( x),Φ(y):\nD(x−y)≡ /angb∇acketleft0|Φ(x)Φ(y)|0/angb∇acket∇ight=/integraldisplay\n|k|≥µd3k\n(2π)3exp[−ik(x−y)]\n2ω(3.4)\nAccording to the Lehman-Symanzik-Zimmerman reduction for mula, in the computation of\ntheS-matrixelementsusingtheFeynmandiagramstheFeynmanpro pagatorsshouldbeattached\nto the internal lines of the diagrams only. Therefore the Fey nman rules for the construction of\nFeynman diagrams which include tachyons appear to remain st andard, with a minor exception:\nto each external tachyon leg (asymptotic tachyon state) the causal Θ function Θ( ku) should be\nattached to ensure a cut of the phase space of reactions with s uch states as described in [7].\nWe observe thus that the operation of the time ordering of fiel d operators in (3.3) realises,\nin general, the same function as causal Θ-terms in the field op erators (1.5). Therefore we can\ninterpret heuristically the obtained results as follows.\nThe standard iǫprescription, aimed at the definition of the integration con tour on the com-\nplex plane of k0, allows virtual tachyons (as well as virtual ordinary parti cles) to avoid causal\nrestrictions, which are otherwise (i.e. in the case of real f aster-than-light particles) imposed by\nthe causal Θ function on the propagation of free tachyons. In other words, the iǫprescription,\nensuring the time ordering of the tachyon field operators whi ch contain the Θ functions appar-\nently breaking the Lorentz invariance, makes the virtual ta chyons insensitive to the existence\nof the preferred reference frame (which indeed is intuitive ly obvious: virtual tachyons cannot\nbe used as signal carriers), and this results in the Lorentz i nvariance of the tachyon Feynman\npropagator, similarly to the Feynman propagators of ordina ry particles.\nThe Lorentz invariance of the tachyon Feynman propagator, o btained in our model with the\ncovariantly broken Lorentz invariance, means, together wi th the assumed Lorentz invariance\nof the Feynman propagators of all other particles, that the s peed of light remains a unique,\nuniversal velocity constant which limits particle velocit ies on both sides of the light barrier,\nbounding the maximum attainable velocities of ordinary par ticles and restricting the tachyon\nvelocities from below. In particular, an explicit breaking of the Lorentz symmetry by adding to\nthe Lagrangian the Lorentz-violating terms which affect the p article propagators, suggested by\nauthors of the Standard Model Extension (SME) [11] (see also [12, 13]), which lead to individual\nmaximum attainable velocity for each fundamental field, diffe ring from the velocity of light, is\nnot relevant to our approach. For the same reason the strong r estrictions on multiple Lorentz-\nviolating coefficients compiled in the “Data Tables for Loren tz and CPT violation” [14] (see also\n[15]) are not applicable to our considerations.\n4The proof of this statement is simple. If we consider the cond ition|k| ≥µin combination with the tachyon\nmass-shell hyperboloid k2+µ2= 0 at the tachyon energy k0= 0, i.e. acting along the |k|axis, its validity extends\nover the range from |k|=µto|k|=∞. Since the k2is a Lorentz invariant this means that the above condition\nis valid in the whole external volume outside the tachyon mas s shell.\n74. The Feynman propagator for scalar tachyons in the configuration space\nLet us obtain DF(x−y) explicitly:\nDF(x−y) =/integraldisplay\n|k|≥µd4k\n(2π)4iexp[−ik(x−y)]\nk2+µ2+iǫ\n=/integraldisplay\n|k|≥µd3kdk0\n(2π)4iexp[−ik0∆t+ik(x−y)]\nk2+µ2+iǫ\n=/integraldisplay\n|k|≥µd3k\n(2π)32ωexp[−iω|∆t|+ik(x−y)], (4.1)\nwhere ∆t=x0−y0, and we have used the integral representation\nexp(−iω|∆t|) =iω\nπ/integraldisplay+∞\n−∞dk0exp(−ik0∆t)\nk2\n0−ω2+iǫ, ǫ →0+. (4.2)\nIntegration of (4.1) over the angles of kgives\nDF(x−y) =1\n4π2/integraldisplay∞\nµdkk2\nωsin(|k||x−y|)\n|k||x−y|exp(−iω|∆t|). (4.3)\nWith the definition of the interval\ns≡∆t2−(x−y)2= ∆t2−r2, (4.4)\nwherer≡ |(x−y)|, we can investigate the behaviour of the DFoutside and inside the light\ncone. For spacelike intervals, s<0, we can put ∆ t= 0 to obtain\nDF(r) =1\n4π2r/integraldisplay∞\nµ|k|dk/radicalbig\nk2−µ2sin|k|r=−µ\n8πrY1(µr), (4.5)\nwhereY1is the Bessel function of the second kind; it represents an ou tgoing wave for large\nrindicating the allowance of tachyon asymptotic states. To c ompare: for an ordinary scalar\nparticle with the mass mthe corresponding Feynman propagator\nDord\nF(r) =m\n4π2rK1(mr), (4.6)\nwhereK1is the modified Bessel function of the second kind [16]; it dam ps exponentially for\nlarger, the characteristic damping length being the particle Comp ton lengthλ= 1/m.\nFor timelike intervals, s>0, we can put |x−y| →0 in (4.3), i.e. r= 0 in (4.4):\nDF(|∆t|) =1\n4π2/integraldisplay∞\nµk2dk/radicalbig\nk2−µ2exp(−i/radicalBig\nk2−µ2|∆t|) =µ\n4π2|∆t|K1(µ|∆t|),(4.7)\ni.e. it dampsexponentially for large |∆t|, withthecharacteristic dampingtimebeingthetachyon\nCompton length λ= 1/µ. For an ordinary scalar particle\nDord\nF(|∆t|) =im\n8π|∆t|H(1)\n1(m|∆t|), (4.8)\nwhereH(1)\n1is the Hankel function of the first kind which represents an ou tgoing wave for large\n|∆t|[16].\n85. Lorentz-invariant and Lorentz-non-invariant two-point functions of scala r tachyon\nfields\nThe correlation function D(x−y) can be represented as\nD(x−y)≡ /angb∇acketleft0|Φ(x)Φ(y)|0/angb∇acket∇ight=/angbracketleftBig/integraldisplayd4k\n(2π)3exp[−ik(x−y)]δ(k2+µ2) Θ(ku)/angbracketrightBig\nEvac\n=/angbracketleftBig/integraldisplay\n|k|≥µ,ω≥kud3k\n(2π)3exp[−iω∆t+ik(x−y)]\n2ω/angbracketrightBig\nEvac(5.1)\nwhere ∆t=x0−y0and the angular brackets /angb∇acketleft /angb∇acket∇ightEvac, surrounding the integrals in (5.1), denote\nthe averaging over the tachyon vacuum energy boundaries. Su ch an averaging is a necessary\naction since the tachyon vacuum energy boundaries are not, i n general (in the frames moving\nwith respect to the preferred one), rotationally invariant .\nGenerally speaking, the calculation of tachyon vacuum expe ctation values of any combina-\ntion of tachyon operators requires such an averaging as dist inct to the calculation of analogous\nvacuum expectation values in the case of ordinary particles , when such calculations result in\nLorentz-invariant c-number functions due to Lorentz-inva riance of the ordinary particle vac-\nuum. In our case (with tachyons) the expression inside the va cuum brackets /angb∇acketleft /angb∇acket∇ightEvacin (5.1) is\na Lorentz-non-invariant c-number function due to the Loren tz-non-invariant (though Lorentz-\ncovariant) integration limits, ω≥kuin (5.1), coming from the causal Θ-function (these limits\nare illustrated, in particular, by formulae (1.3), (1.4) an d by fig. 1). Note, an observer in a\nframe moving with respect to the preferred one also can detec t that the energy boundaries of\nthe tachyon vacuum are different in the forward and backward he mispheres of his motion5.\nFortunately, the averaging of the expression inside the vac uum brackets in (5.1) over the\ntachyonvacuumenergyboundariescontractstheaboveLoren tz-non-invariancesincethesebound-\naries are governed by formula (1.2), the same one which impos es those integration limits. This\noccurs owing to the fact that the boundaries are symmetric wi th respect to the zero energy\nlevel in the preferred reference frame, depending on the dir ection of the observer’s motion. The\nstatement about Lorentz-invariance of thecorrelation fun ctionD(x−y) can beproved as follows.\nFirst of all, we note that the averaging over the tachyonic va cuum energy boundaries can be\ndone for each individual direction of k, i.e. for fixed values of the angles θandφ, the polar and\nazimuthal angles with respect to the directions of kandx−y. Changing the first integration\noverd4kin (5.1) from dωtod|k|we can rewrite it as\nD(x−y) =/integraldisplay/bracketleftBigg\ndcosθdφ\n2(2π)3\n×/angbracketleftBig/integraldisplay\nω≥√\nω2+µ2|u|cosψdω/radicalBig\nω2+µ2exp(−iω∆t+i/radicalBig\nω2+µ2|x−y|cosθ)/angbracketrightBig\nEvac/bracketrightBigg\n(5.2)\nHereψis the angle between the directions of kandu. Obviously, the expression enclosed by\nthe vacuum brackets in (5.2) can be written as a sum\n1\n2/parenleftBig\nIFW(x−y,u)+IBW(x−y,u)/parenrightBig\n, (5.3)\nwhere\nI(x−y,u) =/integraldisplay\nω≥√\nω2+µ2|u|cosψdω/radicalBig\nω2+µ2exp(−iω∆t+i/radicalBig\nω2+µ2|x−y|cosθ),(5.4)\n5An attempt to detect these boundaries was undertaken in [7].\n9and the subscripts FWandBWdefine the corresponding integrals in the forward and backwa rd\nhemispheres of the vector u, where cos ψ>0 and cosψ<0, respectively.\nEach of the hemisphere integrals I(x−y,u) can be written down as a combination of two\nterms, one of which does not depend on the vector u, and another which does. So,\nIFW(x−y,u) =/integraldisplay∞\n0dω/radicalBig\nω2+µ2exp(−iω∆t+i/radicalBig\nω2+µ2|x−y|cosθ)\n−/integraldisplayE+\n0(|u|,ψ)\n0dω/radicalBig\nω2+µ2exp(−iω∆t+i/radicalBig\nω2+µ2|x−y|cosθ), (5.5)\nwhere\nE+\n0(|u|,ψ) =µ|u|cosψ√\n1−u2,cosψ>0. (5.6)\nAnalogously,\nIBW(x−y,u) =/integraldisplay∞\n0dω/radicalBig\nω2+µ2exp(−iω∆t+i/radicalBig\nω2+µ2|x−y|cosθ)\n+/integraldisplay0\nE−\n0(|u|,ψ)dω/radicalBig\nω2+µ2exp(−iω∆t+i/radicalBig\nω2+µ2|x−y|cosθ), (5.7)\nwhere\nE−\n0(|u|,ψ) =µ|u|cosψ√\n1−u2,cosψ<0, (5.8)\nAs follows from (5.6), (5.8), |E−\n0(|u|,ψ)|=E+\n0(|u|,ψ). Therefore6\n/integraldisplayE+\n0(|u|,ψ)\n0dω/radicalBig\nω2+µ2exp(−iω∆t+i/radicalBig\nω2+µ2|x−y|cosθ)\n=/integraldisplay0\nE−\n0(|u|,ψ)dω/radicalBig\nω2+µ2exp(−iω∆t+i/radicalBig\nω2+µ2|x−y|cosθ). (5.9)\nAs a result\n1\n2/parenleftBig\nIFW(x−y,u)+IBW(x−y,u)/parenrightBig\n=/integraldisplay∞\n0dω/radicalBig\nω2+µ2exp(−iω∆t+i/radicalBig\nω2+µ2|x−y|cosθ). (5.10)\nCollectingalltheingredientsof(5.2)andrevertingbackf romtheintegration over dcosθdφωdω\nto the integration over d3kwe obtain finally\nD(x−y) =/integraldisplay\n|k|≥µd3k\n(2π)3exp[−iω∆t+ik(x−y)]\n2ω, (5.11)\nwhich does not depend on u, i.e. it is manifestly Lorentz-invariant, as has been state d above.\nThis function can be used to construct the Feynman propagato r\nDF(x−y) =/braceleftbiggD(x−y) ifx0>y0\nD(y−x) ifx0<y0=/integraldisplay\n|k|≥µd3k\n(2π)3exp[−iω|∆t|+ik(x−y)]\n2ω(5.12)\n6Probably, it is worth notingthat the exponential terms in (5 .9) below are identical eventhough the integration\ndomains over the variable ωin them are quite different: this is due to the fact that expone ntial indices, containing\ntheω, are scalar products of the 4-vectors kandx−y(multiplied by i), i.e. they are Lorentz scalars by definition.\n10in which we have changed the integration variable from kto−kwhen replacing ∆ tby|∆t|in\nthe case ofx0<y0. The resulting expression in (5.12) coincides with the last line of (4.1) which\nproves (3.3).\nThe commutator of scalar tachyon fields reads [3]\n∆(x−y)≡[Φ(x),Φ(y)] =/integraldisplayd4k\n(2π)3/braceleftBig\nexp[−ik(x−y)]−exp[(ik(x−y)]/bracerightBig\nδ(k2+µ2) Θ(ku)\n(5.13)\nwhich is not automatically zero at ( x−y)2<0 as distinct to the field commutators of ordinary\nparticles. Let us consider it in the preferred reference fra me:\n[Φ(x),Φ(y)] =1\n(2π)3/integraldisplay\n|k|≥µ,ω≥0d3k\n2ω/braceleftBig\nexp[−iω∆t+ik(x−y)]\n−exp[iω∆t−ik(x−y)]/bracerightBig\n(5.14)\nIf ∆t/negationslash= 0 the commutator does not vanish (excepting the case of ω= 0 corresponding to the\nexchange of vacuum tachyons). However, the commutator ∆( x−y) vanishes at ∆ t= 0 in the\npreferred reference frame, which is obvious from (5.14); th e same is true for the correspondingly\nLorentz-shifted x−yin boosted frames (due to the covariance of the expression (5 .13)).\nThough being Lorentz-covariant, the commutator (5.13) is n ot Lorentz-invariant since it\ndepends on the value of |u|and on the angle between the directions of uandx−yin the frames\nmoving with respect to the preferred one. In other words, tho ugh all terms in the integrand of\n(5.13), except Θ( ku), are Lorentz invariant, the integration limits in this exp ression, imposed\nby the causal Θ-function, are rotationally invariant in the preferred reference frame only, as\ncan be seen from (5.14). Just this general rotational non-in variance of the integration limits\nresults in the Lorentz-non-invariance of the commutator (5 .13). This means that the amplitude\nof propagation of a tachyon from xtoyis not equal to the amplitude of propagation of the same\ntachyon from ytox7, the latter being the complex conjugate of the former.\nHowever, being averaged over the tachyonic vacuum, the comm utator\n/angb∇acketleft0|[Φ(x),Φ(y)]|0/angb∇acket∇ight (5.15)\nbecomes Lorentz-invariant since the rotational non-invar iance mentioned in the previous para-\ngraph is cancelled by the averaging. Simultaneously, this l eads to the vanishing of the commu-\ntator (5.15) at spacelike separations of the xandypositions:\n/angb∇acketleft0|∆(x−y)|0/angb∇acket∇ight ≡ /angb∇acketleft0|[Φ(x),Φ(y)]|0/angb∇acket∇ight=D(x−y)−D(y��x) = 0/parenleftBig\n(x−y)2<0/parenrightBig\n.(5.16)\nExpression (5.16) can be considered as a limiting case of a ca usal loop construction with the use\nof tachyons. Its vanishing means a principal impossibility of such a construction expressed in\nterms of quantum field theory, i.e. (5.16) is the micro-causality condition of a tachyon theory.\nFinally, the equal-time commutator of the scalar tachyon fie ld with its canonical conjugate\n˙Φ does not vanish [3]:\n[Φ(x),˙Φ(y)]x0=y0=i¯δ3(x−y), (5.17)\nwhere¯δ3(x−y) is atruncated delta function which acts like the standard delta function w ith\nrespecttothosefunctionswhoseFourier transformsvanish at|k|<µ, ω< (ku)(acting similarly\nto the truncated delta functions introduced in [2, 17]); wit h this specification (5.17) corresponds\nto the analogous commutator for ordinary scalar particle fie lds containing the standard delta\nfunction.\n7In the case of a complex tachyon field the propagation of the ta chyon from ytoxwould be replaced by the\npropagation of an antitachyon in that direction.\n116. Summary\nThe overall approach to tachyon field models emerging from th e consideration presented in\nthis note possesses the following attractive features:\n1. Its main concept is based on the experimentally proved phe nomenon [9]: the existence of\na preferred reference frame which is the comoving frame of th e relativistic cosmology [8].\n2. The introduction of this concept into the tachyon theory c an be done in a covariant way\nwhich leads to the result that the Lorentz symmetry of the tac hyon theory appears to be\ncovariantly broken.\n3. The covariant breaking of the Lorentz symmetry allows one to avoid introducing “by\nhand” Lorentz-violating terms into Lagrangians as it is sug gested by authors of the Stan-\ndard Model Extension (SME) [11, 12, 13], which would lead to t he Lorentz-non-invariant\ntachyon propagators making tachyons subject to strict rest rictions imposed on the Lorentz\nviolation by a variety of experiments (see [14, 15]).\n4. Faster-than-light velocities of particles and signals a ppear to be allowed in a wide range\nofc <v < ∞as distinct to the standard models of the Lorentz invariance violation (i.e.\nSME) in which only tiny positive deviations from the velocit y of light are permitted for\nparticles having energies below the Plank mass scale.\n5. So formulated the tachyon hypothesis results in the possi bility of the existence of a new\nworld of elementary particles residing beyond the light bar rier as distinct to the SME\nin which the Lorentz violating particles are assumed to be (s ome of the) already known\nparticles, e.g.neutrinos, electrons, muons, pions, etc.\n6. The velocity of light remains an invariant quantity to be a barrier between the tachyon\nworld of particles and that of subluminal particles which ca n be considered as a proof of\na fundamental character of this velocity.\n7. Conclusion\nA modification of a scalar tachyon field Lagrangian by adding t o it a Lorentz-non-invariant,\nbut Lorentz-covariant term is suggested, aimed at the conse rvation of causality, with the Lorentz\ninvariance of the tachyon Feynman propagator being conserv ed. It is shown that the standard iǫ\nprescription in the Feynman propagator realises the same fu nction over virtual particles (having\nk2≥0 as well as k2<0) as the causal Θ function in the quantum field operators of fr ee\ntachyons (asymptotic tachyon states). The result is a possi bility to have a tachyon model free of\ncausal paradoxes, tachyon vacuuminstability, andLorentz -violating radiative corrections coming\nfrom virtual tachyons to processes involving only ordinary particles, restricting Lorentz-violating\neffects to the free-tachyon sector only.\nAcknowledgements\nI express my gratitude to Profs. Boreskov K. G., Dzheparov F. S., Kancheli O. V. and\nSibiryakov S. M. for stimulating discussions, and to Dr. Fre nch B. R. for the critical reading\nof the manuscript. After this article has been completed I ha ve obtained the information from\nDr. Giudici G. F., to whom I am indebted very much, about the pa per [18], in which a stable,\nrenormalizable, scalar tachyonic quantum field theory with chronology protection is suggested,\ndeveloped with an approach quite different to that presented h ere and in [3].\n12References\n[1] O.M.P. Bilaniuk, V.K. Deshpande, E.C.G. Sudarshan, “Meta”-relativity , Amer. J. Phys. 30(1962)\n718-723.\n[2] G. Feinberg, Possibility of faster-than-light particles , Phys.Rev. 159(1967) 1089-1105.\n[3] V.F. Perepelitsa, Looking for a theory of faster-than-light particles , arXiv:gen-ph/1407.3245.\n[4] D.G. Boulware, Unitarity and interacting tachyons , Phys. Rev. D 1(1970) 2426-2427.\n[5] T. Jacobson, N.C. Tsamis, R.P. Woodard, Tachyons and perturbative unitarity , Phys. Rev. D 38\n(1988) 1823-1834.\n[6] R. Sigal, A. Shamaly, Tachyon behavior in general relativity , Phys. Rev. D 10(1974) 2358-2361.\n[7] V.P. Perepelitsa, Tachyon Michelson experiment , Phys. Lett. B 67(1977) 471-473.\n[8] L.D. Landau, E.M. Lifshitz, Theoretical Physics , vol.2, Classical Theory of Fields , Sect. 112\n(Addison-Wesley, New York, 1985).\n[9] M. Tanabashi et al.(Particle Data Group), Phys. Rev. D 98, 030001 (2018), Sect. 28.3.2.\n[10] V.F. Perepelitsa, How to implant a causal Θ-function into a tachyon field operator , arXiv:gen-\nph/1512.07921.\n[11] D. Colladay, V.A. Kosteleck´ y, Lorentz-violating extension of the standard model , Phys. Rev. D 58\n(1998) 116002, and references therein.\n[12] S. Coleman, S.L. Glashow, Cosmic ray and neutrino tests of special relativity , Phys. Lett. B 405\n(1997) 249-252.\n[13] S. Coleman, S.L. Glashow, High-energy tests of Lorentz invariance , Phys. Rev. D 59(1999) 116008-\n1−116008-14.\n[14] V.A. Kosteleck´ y, N. Russel, Data tables for Lorentz and CPT violation , Rev. Mod. Phys., 83(2011)\n11-31 [arXiv:hep-ph/0801.0287].\n[15] D. Mattingly, Modern tests of Lorentz invariance , Living Rev. Rel., 5(2005) 5-84 [arXiv:gr-\ngc/0502.097].\n[16] K. Huang, Quantum Field Theory: from Operators to Path Integrals , Sect. 2.9 (John Wiley and\nSons, New York, Chichester, Weinheim, Brisbane, Singapore , Toronto, 1998).\n[17] Arons M. E., Sudarshan E. C. G., Lorentz-invariance, local field theory and faster-than-li ght parti-\ncles, Phys. Rev. 173, 1622-1628 (1968)\n[18] M. J. Radzikowski, Stable, renormalizable, scalar tachyonic quantum field the ory with chronology\nprotection , arXiv:math-ph/0804.4534.\n13"    },    {        "title": "1804.08695v1.Lorentz_Dynamics_on_Closed_3_Manifolds.pdf",        "content": "arXiv:1804.08695v1  [math.DG]  23 Apr 2018LORENTZ DYNAMICS ON CLOSED 3-MANIFOLDS\nby\nCharles Frances\nAbstract . —In this paper, we give a complete topological, as well as geom etri-\ncal classification of closed 3-dimensional Lorentz manifol ds admitting a noncompact\nisometry group.\n1. Introduction\nA celebrated theorem of Myers and Steenrod [ MS], says that the isometry group\nof a closed Riemannian manifold is always a compact Lie transformation group. It is\nvery well known that this compactness property is specific to the R iemannian world,\nand fails for general closed pseudo-Riemannian manifolds. For insta nce the Lorentz\ntorusRn/Zn, endowedwiththe metricinduced by −dx2\n1+dx2\n2+...+dx2\nn, hasisometry\ngroup O(1,n−1)Z⋉Tn. This group is noncompact, since O(1 ,n−1)Zis a lattice in\nO(1,n−1).\nThegeometricalimplicationsofthosetwoantagonisticphenomena– noncompactness\nof the isometry group on the one hand, and compactness of the ma nifold on the other\nhand– were much studied in the Lorentzian case, on which we will focu s here. A\nsample of significant results can be found in [ Gr], [Z], [AS], [Z1], [Z3], [Z4], among\na lot of other works.\nOne remarkable point is that the noncompactness of the isometry g roup is also ex-\npected to have strong topological consequences. This was first n oticed by M. Gromov\nin [Gr] when the isometry group is “large”, for instance when it contains a noncom-\npact simple Lie group (see also further developments in [ FZ]). Without any extra\nasumption on the acting group, let us mention the following striking re sult:\nTheorem 1.1 . — [DA]Let(M,g)be a closed Lorentz manifold. We assume that\nMandgare real analytic, and Mis simply connected. Then Iso(M,g)is a compact\ngroup.2 CHARLES FRANCES\nThe analyticity condition is crucial in the proof of Theorem 1.1, and wh en the\ndimension of the manifold is >3, we actually don’t know if the result holds in the\nsmooth category.\nThe aim of this paper is to focus on the 3-dimensional situation, and t o provide a\nthorough study of all closed 3-dimensional manifolds, which can be e ndowed with a\nLorentz metric admitting a noncompact group of isometries.\n1.1. Statement of results. — Let us recall a class of closed 3-manifolds which\nwill play a prominent role in the sequel, namely the torus bundles over t he circle\n(torus bundles for short). Let T2be a 2-torus R2/Z2, and let us consider the product\n[0,1]×T2. We then make the identification (0 ,x)≃(1,Ax), whereAis a given\nelement of SL(2 ,Z). The resulting 3-manifold is denoted T3\nA. WhenA=id, we just\nget the 3-torus T3. IfA∈SL(2,Z) ishyperbolic , namely is R-split with eigenvalues of\nmodulus /\\e}atio\\slash= 1, we say that T3\nAis ahyperbolic torus bundle . IfA∈SL(2,Z) isparabolic ,\nnamely conjugated to a unipotent matrix ( A/\\e}atio\\slash=id), we saythat T3\nAis aparabolic torus\nbundle. Finally, elliptic torus bundles are those for which Ahas finite order.\n1.1.1. A topological classification . — Our first result is a topological classification of\nclosed Lorentz 3-manifolds admitting a noncompact isometry group .\nTheorem A . —Let(M,g)be a smooth, closed 3-dimensional Lorentz manifold. As-\nsume that (M,g)is orientable and time-orientable, and that Iso(M,g)is noncompact.\nThenMis homeomorphic to one of the following spaces:\n1. A quotient Γ\\/tildewidestPSL(2,R), whereΓ⊂/tildewidePSL(2,R)is any uniform lattice.\n2. A3-torusT3, or a torus bundle T3\nA, whereA∈SL(2,Z)can be any hyperbolic\nor parabolic element.\nConversely, any smooth compact 3-manifold homeomorphic to one of the examples\nabove can be endowed with a smooth Lorentz metric with a nonco mpact isometry\ngroup.\nThe assumption about orientability and time-orientability of the manif old is not\nreally relevant, and one could drop it (adding a few allowed topological types) with\nextra case-by-case arguments in our proofs. Notice that any clo sed 3-manifold will\nhave a covering of order at most four satisfying the assumptions o f Theorem A. We\nthus see that a lot of 3-manifolds do not admit coverings appearing in the list of\nthe theorem, where only four among the eight Thurston’s geometr ies are represented.\nHyperbolic manifolds are notably missing, and we can state:\nCorollary 1.2 . —LetMbe a smooth closed 3-dimensional manifold, which is home-\nomorphic to a complete hyperbolic manifold Γ\\H3. Then for every smooth Lorentz\nmetricgonM, the group Iso(M,g)is compact.\n1.1.2. Continuous versus discrete isometries . — It is interesting to compare the con-\nclusionsofTheoremAtocloselyrelatedresults, andespeciallytothe work[Z2], which\nwas a great source of motivation for the present paper. In [ Z2], A. Zeghib studies\n3-dimensional closed manifolds admitting a non equicontinuous isomet ric flow. ThisLORENTZ DYNAMICS ON CLOSED 3-MANIFOLDS 3\nhypothesis is actually equivalent to the noncompactness of the identity component\nIsoo(M,g). The classification can be briefly summarized as follows:\nTheorem 1.3 . — [Z2, Theorems 1 and 2] Let(M,g)be a smooth, closed 3-\ndimensional Lorentz manifold. If the identity component Isoo(M,g)is not compact,\nthen:\n1. Up to a finite cover, the manifold Mis homeomorphic either to a torus bundle\nT3\nA, withA∈SL(2,Z)hyperbolic, or to a quotient Γ\\/tildewidestPSL(2,R), for a uniform\nlatticeΓ⊂/tildewidestPSL(2,R).\n2. The manifold (M,g)is locally homogeneous. It is flat when Mis a hyper-\nbolic torus bundle, and locally modelled on a Lorentzian, no n-Riemannian, left-\ninvariant metric on /tildewidestPSL(2,R)otherwise.\nThe definition of Lorentzian, non-Riemannian, left-invariant metric s on/tildewidestPSL(2,R)\nwill be made precise in Section 2.1.\nDoes it make a big difference, putting the noncompactness assumpt ion on\nIsoo(M,g) instead of Iso( M,g)? At the topological level, notice that 3-tori and\nparabolic torus bundles do not show up in Theorem 1.3. For Lorentz m etrics on those\nmanifolds, Isoo(M,g) is always compact, but we will see that there exist suitable\nmetricsg, for which the full group Iso( M,g) is noncompact. It means that for those\nexamples, the noncompactness comes from the discrete part Iso(M,g)/Isoo(M,g).\nActually, there are instances of 3-manifolds (see Section 2), wher e the isometry group\nis discrete, isomorphic to Z.\nTo put more emphasis on how the general case may differ from the co nclusions of\n[Z2], let us state the following existence result:\nTheorem B . —LetMbe a closed 3-dimensional manifold which is homeomorphic\nto a3-torusT3, or a torus bundle T3\nA, withA∈SL(2,Z)hyperbolic or parabolic. Then\nit is possible to endow Mwith time-orientable Lorentz metrics ghaving the following\nproperties:\n1. The isometry group Iso(M,g)is noncompact, but the identity component\nIsoo(M,g)is compact.\n2. There is no open subset of (M,g)which is locally homogeneous.\nObserve that for any closed Lorentz manifold ( M,g) which is not locally homoge-\nneous, Isoo(M,g) is automatically compact by Theorem 1.3 above.\nThe constructions leading to theorem B are rather flexible. In part icular, on T3,\nor on any hyperbolic or pabolic torus bundle T3\nA, the moduli space of Lorentz metrics\nadmitting a noncompact isometry group is by no mean finite dimensiona l. This is\nagain in sharp constrast with the second point of Theorem 1.3.\n1.1.3. Geometrical results . — The topological classification given by Theorem A\ncomes as a byproduct of a finer, geometrical understanding of clo sed Lorentz 3-\nmanifolds with noncompact isometry group. We actually get a quite co mplete ge-\nometrical description:4 CHARLES FRANCES\nTheorem C . —Let(M,g)be a smooth, closed 3-dimensional Lorentz manifold. As-\nsume that (M,g)is orientable and time-orientable, and that Iso(M,g)is noncompact.\n1. IfMis homeomorphic to Γ\\/tildewidestPSL(2,R), then(M,g)is locally homogeneous, mod-\nelled on a Lorentzian, non-Riemanniann, left-invariant me tric on/tildewidestPSL(2,R).\n2. IfMis homeomorphic to T3\nA, withA∈SL(2,Z)hyperbolic, then there exists a\nsmooth, positive, periodic function a:R→(0,∞)such that the universal cover\n(˜M3,˜g)is isometric to R3endowed with the metric\n˜g=dt2+2a(t)dudv.\nIfgis locally homogeneous, it is flat.\n3. IfMis homeomorphic to T3\nA, withA∈SL(2,Z)parabolic, then there exists a\nsmooth, positive, periodic function a:R→(0,∞)such that the universal cover\n(˜M3,˜g)is isometric to R3endowed with the metric\n˜g=a(v)(dt2+2dudv).\nIfgis locally homogeneous, it is either flat or modelled on the Lo rentz-Heisenberg\ngeometry.\n4. IfMis homeomorphic to a 3-torusT3, then the universal cover (˜M3,˜g)is\nisometric to R3with a metric of type 2)or3)above. If the metric gis locally\nhomogeneous, it is flat.\nThe Lorentz-Heisenberg geometry will be described in Section 2.3.2.\nWe already emphasized that in some examples, the isometry group Is o(M,g) could\nbeinfinite discrete. However,itisworthmentioningthat noncompac tnessofIso( M,g)\nalwaysproduces somehow local continuous symmetries. It is indeed easy to infer from\nTheorem C the following result.\nCorollary D . —Let(M,g)be a closed 3-dimensional Lorentz manifold. If Iso(M,g)\nis noncompact, then Isoo(˜M,˜g)is noncompact. Actually (˜M,˜g)admits an isometric\naction of the group /tildewidestPSL(2,R),HeisorSOL.\n1.2. General strategy of the proof, and organization of the p aper. — One\naspect of the present work consists of existence results. This is t he topic of Section 2,\nwherewerecollectwell-known,andprobablylessknown, examplesof closedLorentz3-\nmanifolds having a noncompact isometry group. Examples are given, where Iso(M,g)\nis infinite discrete, or semi-discrete. This yields the existence part in Theorem A, and\na proof of Theorem B.\nThe remaining of the paper is then devoted to our classification resu lts, namely\nTheorems A and C. The point of view we adopted, is that of Gromov’s t heory of rigid\ngeometric structures [ Gr].\nSection 3 recall the main aspects of the theory, recast in the fram ework of Cartan\ngeometryas in [ M], [P]. The key result is the existence of a dense open subset Mint⊂\nM, called the integrability locus , where Killing generators of finite order do integrate\nintogenuinelocalKillingfields. Usingtherecurrencepropertiesofth eisometrygroup,\nthis implies the crucial fact that the noncompactness of Iso( M,g) must produce a lotLORENTZ DYNAMICS ON CLOSED 3-MANIFOLDS 5\noflocalKilling fields (Proposition3.3). Thosecontinuouslocal symetr ies, arisingfrom\napotentiallydiscreteIso( M,g), willbeofgreathelptounderstandthegeometryofthe\nconnected components of Mint, which can be roughly classified into three categories:\nconstant curvature, hyperbolic, and parabolic (see Section 3.5). To unravel the global\nstructure of M, we must understand how all the components of Mintare patched\ntogether (notice that there can be infinitely many such component s).\nThe first, and easiest case to study, is when all the components of Mintare locally\nhomogeneous. Results of [ F2] show that ( M,g) itself is then locally homogeneous,\nallowing to understand ( M,g) completely. This is done in Section 4.\nSection 5 studies the case where one component of Mintis not locally homogeneous\nand hyperbolic. One then shows that Mis a 3-torus or a hyperbolic torus bundle,\nand the geometry is that of examples 2 .and 4.of Theorem C. This is summarized in\nTheorem 5.2. The key feature in this case is to show that ( M,g) contains a Lorentz 2-\ntorus, on which an element h∈Iso(M,g) acts as an Anosov diffeomorphism (Lemma\n5.4). We then show that it is possible to push this Anosov torus by a kin d of normal\nflow, to recover the topological, as well as geometrical structure of (M,g).\nThe most tedious case to study is when ( M,g) is not locally homogeneous, and\nthere are no hyperbolic components at all. This is the purpose of Sec tions 6, 7 and 8.\nWe show there that Mis a 3-torus or a parabolic torus bundle, and the geometry is\nthe onedescribed in cases3 .and4.ofTheoremC. Thisis summarizedin Theorem8.1.\nThe main observation here is that the manifold ( M,g) is conformally flat (Section 6).\nWe then get a developing map δ:˜M3→Ein3, which is a conformal immersion from\nthe universal cover ˜M3to a Lorentz model space Ein3, called Einstein’s universe.\nAfter introducing relevant geometric aspects of Ein3in Section 7, we are in position\nto study in details the map δ:˜M3→Ein3in Section 8. We show that δmaps˜M3\nin a one-to-one way onto an open subset of Ein3, which is conformally equivalent to\nMinkowskispace. Wearethenreducedtothestudyofclosed,flat , Lorentz3-manifolds\nwith noncompact isometry groups, which was already done in Section 4.\nAll those partial results are recollected in Section 9, where we see h ow they yield\nTheorem C and Corollary D.\n2. A panorama of examples\nThe aim of this section is to construct a wide range of closed 3-dimens ional Lorentz\nmanifolds ( M,g), with noncompact isometry group. Those examples will show that\nall topologies appearing in Theorem A do really occur. Moreover, Sec tions 2.4, 2.2\nand 2.3.1 prove our Theorem B. Part of the examples presented her e are well known,\nothers like those described in Section 2.4.2, 2.2, 2.3.1 seem less classica l, though\nelementary.\n2.1. Examples on quotients Γ\\/tildewidestPSL(2,R). —The Lie group /tildewidestPSL(2,R), universal\ncover of PSL(2 ,R), admits a lot of interesting left-invariant Lorentzian metric. The\nmostsymmetric oneis the anti-de Sitter metricgAdS. It is obtainedby left-translating\nthe Killing form of the Lie algebra sl(2,R). The space ( /tildewidestPSL(2,R),gAdS) is a complete\nLorentz manifold with constant sectional curvature −1, called anti-de Sitter space6 CHARLES FRANCES\n/tildewideAdS3. Because the Killing form is Ad-invariant, the metric gAdSis invariant by\nleft and right multiplications of /tildewidestPSL(2,R) on itself. It follows that for any uniform\nlattice Γ ⊂/tildewidestPSL(2,R), the metric gAdSinduces a Lorentz metric gAdSon the quotient\nmanifold Γ \\/tildewidestPSL(2,R), with a noncompact isometry group coming from the right\naction of /tildewidestPSL(2,R) on Γ\\/tildewidestPSL(2,R).\nThere are other metrics than gAdSon/tildewidestPSL(2,R), which allow the same kind of\nconstructions. They are obtained as follows. Exponentiating the lin ear space spanned\nby the matrix/parenleftbigg\n0 1\n0 0/parenrightbigg\n(resp./parenleftbigg\n1 0\n0−1/parenrightbigg\n), one gets a unipotent (resp. R-split)\n1-parameter group {˜ut}(resp.{˜ht}) in/tildewidestPSL(2,R). The adjoint action of each of\nthose flows, admits invariant Lorentz scalar products on sl(2,R), which are not equal\nto a multiple of the Killing form. One can left-translate those scalar pr oducts and\nget metrics guandghon/tildewidestPSL(2,R) which are respectively /tildewidestPSL(2,R)× {˜ut}and\n/tildewidestPSL(2,R)× {˜ht}-invariant. Actually there are families of such metrics guandgh\nwhich are not pairwise isometric. Now, for each uniform lattice Γ ⊂/tildewidestPSL(2,R), the\nquotient Γ \\/tildewidestPSL(2,R) can be endowed with induced metrics guorghcarrying an\nisometric, noncompact action of R, coming from the right actions of, respectively,\n{˜ut}and{˜ht}on/tildewidestPSL(2,R).\nIn the sequel, the metric gAdSand metrics of the form guorgh, will be refered\nto asLorentzian, non-Riemannian, left-invariant metrics on /tildewidestPSL(2,R). Those are\nthe only left-invariant metrics on /tildewidestPSL(2,R), the isometry group of which does not\npreserve a Riemannian metric.\n2.2. Examples on hyperbolic torus bundles. — Let us start with the space\nR3endowed with coordinates ( x1,x2,t) associated to a basis ( e1,e2,et). We consider\na hyperbolic matrix Ain SL(2,Z). Hyperbolic means that Ahas two distinct real\neigenvalues λandλ−1different from ±1.\nLet us consider the group Γ generated by γ1=Te1(the translation of vector e1),\nγ2=Te2and the affine transformation γ3=/parenleftbiggA0\n0 1/parenrightbigg\n+\n0\n0\n1\n. It is clear that Γ\nis discrete, acts freely properly and discontinuously on R3, giving a quotient manifold\nΓ\\R3diffeomorphic to the hyperbolic torus bundle T3\nA.\nWe seeAas a linear transformation of Span( e1,e2). This transformation is of\nthe form (u,v)/ma√sto→(λu,λ−1v) in suitable coordinates ( u,v). For any smooth function\na:R→(0,∞), which is 1-periodic, the group Γ acts isometrically for the Lorentz\nmetric\nga=dt2+2a(t)dudv\nonR3. Hence the metric gainduces a Lorentz metric gaonM=T3\nA.\nWhenais a constant, we get for gaa flat metric on T3\nA, and the flow of translations\nTt\ne3acts on ( T3\nA,ga) as an Anosov flow. Up to finite index, the isometry group\ncoincides with this flow. It is of course noncompact.LORENTZ DYNAMICS ON CLOSED 3-MANIFOLDS 7\nMore interesting examples arise if one takes a function a:R→(0,∞) which is\n1-periodic, but is constant on no sub-interval of R. Then all Killing fields of gamust\nbe tangent to the hyperplanes t=t0. There is no nonempty open subset where the\nmetricgais locally homogeneous. The same is true for ga. The linear transformation/parenleftbigg\nA0\n0 1/parenrightbigg\ninduces an isometry fof (T3\nA,ga) which preserves individualy the Lorentz\ntorit=t0onT3\nA,andactsonthembyanAnosovdiffeomorphism. Theisometrygroup\nIso(T3\nA,ga) is thus noncompact. This group virtually coincides with the subgrou p\n<f >≃Zgenerated by f. It is thus discrete.\nThese examples prove Theorem B for hyperbolic torus bundles.\n2.3. Examples on parabolic torus bundles. —\n2.3.1. Flat, or non locally homogeneous examples . — We consider now R3with co-\nordinates (u,t,v). Let us call Hthe 3-dimensional Lie group given by the affine\ntransformations\n1z−z2\n2\n0 1−z\n0 0 1\n+\nr\ns\nz\n\nwherer,s,zdescribe R. Observethat Hisasubgroupisomorphictothe3-dimensional\nHeisenberggroupHeis. Theactionof HonR3isfreeandtransitive. Observealsothat\nHacts isometrically for the flat Lorentz metric h0=dt2+2dudv. Leta:R→(0,∞)\nbe a smooth function, which is 1-periodic, and let us consider the met ric\nha=a(v)(dt2+2dudv).\nWhenais not constant, it is no longer true that haisH-invariant. But it remains\ntrue thathais invariant under the action of the discrete subgroup Γ ⊂H, comprising\ntransformations of the form\n\n1m−m2\n2\n0 1−m\n0 0 1\n+\nn\n2l\n2\nm\n\nwherem,n,ldescribe Z. The gluing map between planes v= 0 andv= 1 is made\nby the matrix A=/parenleftbigg\n1 1\n0 1/parenrightbigg\n. Thus the quotient Γ \\R3is diffeomorphic to T3\nA,\nwithAthe unipotent matrix above. All parabolic torus bundles can be obta ined by\nconsidering finite index subgroups of Γ.\nThe metric hainduces a Lorentz metric haon the parabolic torus bundle T3\nA, and\nthe linear maps B=\n1m−m2\n2\n0 1−m\n0 0 1\n,m∈Z, normalize Γ, hence induce a group\nof isometries in ( T3\nA,ha). It is readily checked that this group does not have compact\nclosure in Iso( T3\nA,ha).\nWe now make the following observation. Let X=X1∂\n∂u+X2∂\n∂t+X3∂\n∂vbe a local\nconformalvector field for the flat metric h0, thenLXh0=αXh0for a smooth function8 CHARLES FRANCES\nαX. Assume that X3is nonzero on a small open set, the Xwill be a Killing field for\nhaif and only if\n(1)a′(v)\na(v)=−αX(x)\nX3(x)\nThe set of local conformal Killing fields for h0is finite dimensional, hence for a\ngeneric choice of smooth, 1-periodic a, the relation (1) won’t be satisfied, whatever\nthe conformal vector field Xwe are considering. It follows that for such a generic set\nof function, there won’t be any open subset of T3\nA(resp. of T3) where the metric ha\nwill be locally homogeneous.\nThese examples prove Theorem B for parabolic torus bundles.\n2.3.2. Examples modelled on Lorentz-Heisenberg geometry . — We call heisthe 3-\ndimensional Heisenberg Lie algebra, and Heis the connected, simply c onnected, asso-\nciated Lie group. Recall that heisadmits a basis X,Y,Z, for which the only nontrivial\nbracket relation is [ X,Y] =Z. LetA∈SL(2,Z) be a hyperbolic matrix, and consider\nthe automorphism ϕofheis, which in the basis X,Y,Zwrites/parenleftbiggA0\n0 1/parenrightbigg\n. It defines\nan automorphism Φ of the Lie group Heis.\nThe matrix Ais diagonal in some basis X′,Y′of Span(X,Y), with eigenvalues\nλ,λ−1. The Lorentz scalar product defined by < X′,Y′>= 1,< Z,Z > = 1, and\nall other products are zero, can be left-translated on Heis to give an homogeneous\nLorentz metric gLHcalledthe Lorentz-Heisenberg metric on Heis. By construction, Φ\nacts isometrically on (Heis ,gLH). We now consider the following lattice in Heis:\nHZ:={exp(aX+bY+cZ)|(a,b,c)∈Z3}.\nThe quotient HZ\\Heis is homeomorphic to a parabolic torus bundle, on which the\nLorentz-Heisenberg metric induces a metric gLH. The automorphism Φ preserves\nHZ, hence induces an isometry Φ on (HZ\\Heis,gLH), andΦ generates a noncompact\ngroup.\n2.4. Some examples on the 3-torusT3. —\n2.4.1. A flat example . — The most classical example, already mentioned in the in-\ntroduction, comes from the flat metric\ng0=−du2+dv2+dw2.\nWe call O(1 ,2) the group of linear transformations preserving g0, and we introduce\nΓthediscretesubgroupgeneratedbythetranslations Tu,Tv,Twofvectorsu,v,w. The\nquotient Γ \\R3inherits an induced (flat) metric g0fromg0, and the isometry group of\n(T3,g0) is O(1,2)Z⋉Z3. It is noncompact, because O(1 ,2)Zis a lattice in O(1 ,2)Z.\nThe identity component is however compact in this case.\n2.4.2. Non locally homogeneous examples . — These examples are built in the same\nway as those of Sections 2.2 and 2.3.1, so that we will be rather sketc hy in our\ndescription.\nWe consider the metric ga, introduced in Section 2.2, for a:R→(0,∞) a smooth\n1-periodic function.LORENTZ DYNAMICS ON CLOSED 3-MANIFOLDS 9\nThe metric gais invariant by the discrete group Γ generated by the translations\nof vectorse1,e2andet. Hencegainduces a metric gaonT3. As in Section 2.2, for\ngeneric choices of the function a:R→(0,∞), there is no open set on which gais\nlocally homogeneous. The isometry group is then Z⋉ T2(theZ-factor comes from\nthe transformation/parenleftbiggA0\n0 1/parenrightbigg\n, as in 2.2).\nWe can also consider the metric haintroduced in Section 2.3.1, and take for Γ the\ndiscrete subgroup generated by the translations of vectors ( eu,et,ev). This yields a\nmetrichaonT3= Γ\\R3. Forgenericchoicesofthe1-periodicfunction a:R→(0,∞),\nthere is no open set on which hais locally homogeneous, and the isometry group is\nnoncompact, isomorphic to Z⋉T2.\nThese examples prove Theorem B for 3-dimensional tori.\n3. Curvature, recurrence, and local Killing fields\n3.1. Generalized curvature map and integrability locus. — Let us consider\n(M,g), a smooth Lorentz manifold of dimension n≥2. All the material presented\nbelow holds actually in the much wider frameworkofCartan geometrie s, but we won’t\nneed such a generality.\n3.1.1. Cartan connection associated to the metric . — Letπ:ˆM→Mdenote the\nbundle of orthonormal frames on ˆM. This is a principal O(1 ,n−1)-bundle over M,\nand it is classical (see [ KN][Chap. IV.2 ]) that the Levi-Civita connection associated\ntogcan be interpreted as an Ehresmann connection αonˆM, with values in the Lie\nalgebrao(1,n−1). Letθbe the solderingform on ˆM, namely the Rn-valued 1-formon\nˆM, which to every ξ∈TˆxˆMassociates the coordinates of the vector π∗(ξ)∈TxMin\nthe frame ˆx. The sumα+θis a 1-formω:TˆM→o(1,n−1)⋉Rncalled the canonical\nCartan connection associated to ( M,g) (see [Sh, Chap. 6] for a nice introduction to\nCartan geometries).\nIn the following, we will denote by gthe Lie algebra o(1,n−1)⋉Rn. The Cartan\nconnection ωsatisfies the two crucial properties:\n- For every ˆ x∈ˆM,ωˆx:TˆxˆM→gis an isomorphism of vector spaces.\n- The form ωis O(1,n−1)-equivariant (where O(1 ,n−1) acts on gvia the adjoint\naction).\n3.1.2. Generalized curvature map . — The curvature of the Cartan connection ωis a\n2-formKonˆM, with values in g. IfXandYare two vector fields on ˆM, it is given\nby the relation:\nK(X,Y) =dω(X,Y)+[ω(X),ω(Y)].\nBecauseωˆxestablishes an isomorphism between TˆxˆMandgat each point ˆ xofˆM, any\nk-differential form on ˆM, with values in some vector space W, can be seen as a map\nfromˆMto Hom(⊗kg,W). This remark applies in particular for the curvature form,\nyielding a curvature map κ:ˆM→ W0, where the vector space W0is Hom(∧2(g/p);g)\n(the curvature is antisymmetric and vanishes when one argument is tangent to the\nfibers of ˆM).10 CHARLES FRANCES\nWe can now differentiate κ, getting a map Dκ:TˆM→ W0. Our previous remark\nallows to see Dκas a mapDκ:ˆM→ W1, withW1= Hom(g,W0). Applying this\nprocedurertimes, we define inductively the r-derivative of the curvature Drκ:ˆM→\nHom(⊗rg,Wr) (withWrdefined inductively by Wr= Hom(g,Wr−1)).\nLet us now set m= dimG. Thegeneralized curvature map of the Cartan geometry\n(M,C) is the map Dκ= (Dκ,...,Dm+1κ). The O(1 ,n−1)-module Wm+1will be\nsimply denoted Win the following.\n3.1.3. Integrability locus . — Letx∈Mand ˆx∈π−1(x). Because Dκ:ˆM→ Wis a\nO(1,n−1)-equivariant map, the rank of Dκat ˆx(namely the rank of the linear map\nDˆx(Dκ) :TˆxˆM→ W) does not depend on ˆ xin a same fiber π−1(x). Hence, it makes\nsense to speak about the rank of Dκat a pointx∈M. One defines the integrability\nlocus ofM, denotedMint, as the set of points x∈Mat which the rank of Dκis\nlocally constant. Notice that Mintis a dense open subset of M. We define in the\nsame way ˆMint⊂ˆMas the inverse image π−1(Mint) (which is thus the set of points\nwhere the rank of Dκ:ˆM→ Wis locally constant).\n3.2. Integrability theorem and structure of Isloc-orbits. — Forx∈M, the\nIsloc-orbit ofxis the set of points y∈Msuch thaty=f(x) for some local isometry\nf:U⊂M→V⊂M. Thekillloc-orbit ofxis the set of points y∈Mthat can be\nreached by flowing along (finitely many) successive local Killing fields.\nNotice that any local Killing field XonU⊂Mlifts to a Killing field (still denoted\nX) onˆM, satisfying LXω= 0. Indeed, local flows of isometries clearly induce local\nflows on the bundle of orthonormal frames. Conversely, local vec tor fields of ˆMsuch\nthatLXω= 0 will project on local Killing fields on M. The same remark holds for\nlocal isometries.\nObserve finally that if Xis a Killing field on ˆM(namelyLXω= 0), then the\nlocal flow of Xpreserves Dκ, henceXbelongs to Ker( DˆxDκ) at each point. The\nintegrability theorem below says that the converse is true on Mint.\nTheorem 3.1 (Integrability theorem) . —Let(Mn,g)be a Lorentz manifold,\nandˆMint⊂ˆMthe integrability locus.\n1. For every ˆx∈ˆMint, and every ξ∈Ker(DˆxDκ), there exists a local Killing field\nXaroundˆxsuch thatX(ˆx) =ξ.\n2. TheIsloc-orbits inMintare submanifolds of Mint, the connected components of\nwhich are killloc-orbits.\nThe deepest, and most difficult part, of the theorem is the first poin t. Such an\nintegrability result as well as the structure of Isloc-orbits first appeared in [ Gr]. The\nresults were recastin the frameworkof Cartan geometryby K. Me lnick in the analytic\ncase (see [ M]). The reference [ P] gives an alterative approach for smooth Cartan\ngeometries, leading to the statement of Theorem 3.1. A proof that the integrability\nproperty actually holds on the set where the rank of Dκis locally constant (first point\nof the theorem) can be found in Annex A of [ F2].LORENTZ DYNAMICS ON CLOSED 3-MANIFOLDS 11\nLet us recall how the second point of Theorem 3.1 easily follows from t he first one\n(see also [ P, Sec. 4.3.2]). The generalized curvature map Dκ:ˆM→ Wis invariant\nunder all local isometries. It follows that ˆMintis invariant as well. Given ˆ x∈ˆMint,\nandw=Dκ(ˆx), theIsloc-orbitIsloc(ˆx) is contained in Dκ−1(w)∩ˆMint. Now since\nDκhas locally constant rank on ˆMint,Dκ−1(w)∩ˆMintis a submanifold of ˆMint, and\nthe first point of Theorem 3.1 exactly means that the killloc-orbit of ˆxcoincides with\nthe connected component of Dκ−1(w)∩ˆMintcontaining ˆx, hence is a submanifold on\nˆMint. The setIsloc(ˆx) is a union of such connected components, hence a submanifold\ntoo. The point we have to check is that this property remains true w hen one projects\neverything on M. Observe first that the projection of Dκ−1(w)∩ˆMintonMcoincides\nwith that of Dκ−1(O.w)∩ˆMint, whereO.wstands for the O(1 ,n−1)-orbit ofwin\nW. Now, using the constancy of rank( Dκ) onDκ−1(O.w)∩ˆMint, the O(1,n−1)-\nequivariance of Dκ, and the fact that O(1 ,n−1)-orbits in Ware locally closed, one\nshows that Dκ−1(O.w)∩ˆMintis a submanifold of ˆMint. By O(1,n−1)-invariance of\nthis set, its projection on Mintis a submanifold too.\n3.3. Components of the integrability locus and killloc-algebra. — For each\nx∈M, there exists a neighborhood Uofx, so that any local Killing field defined in\na neighborhood of xis defined on U. There is thus a good notion of Lie algebra of\nlocal Killing fields that we denote by killloc(x).\nThe integrability locaus Mintsplits into a union of connected components/uniontextMi.\nTheM′\niswill be just called components in the sequel. The dimension of killloc(x)\n(which is finite) can notdecreaselocally, while the rankof Dκcannot decreaselocally.\nIt follows from Theorem 3.1 that on each component M, the dimension of the Lie\nalgebrakillloc(x) is locally constant. Hence the isomorphism class of killloc(x) does\nnot depend on x∈ M, and we will sometimes write killloc(M) instead of killloc(x).\nActually, on each component M, everything behaves as if the structure was analytic.\nForx∈M, we consider Is(x), the isotropy algebra at x, namely the Lie algebra\nof local Killing fields defined in a neighborhood of xand vanishing at x.\nFact 3.2 . —Ifx∈Mint, then the isotropy algebra Is(x)is isomorphic to the Lie\nalgebra of the stabilizer of Dκ(x)inO(1,n−1).\nProof: Let us consider ˆ x∈ˆMin the fiber of x. Every local Killing field Xaround\nxwhich vanishes at x, lifts to a local Killing field around ˆ x, still denoted X, which is\nvertical at ˆx. We callevˆxthe mapX/ma√sto→ω(X(ˆx)). The relation ϕt\nX.ˆx= ˆx.etevˆx(X),\navailablefor tin a neighborhoodof0, togetherwith the invarianceof Dκunder Killing\nflows, shows that evˆxa linear embedding from Ixto the Lie algebra of the stabilizer\nofDκ(x) in O(1,n−1) (this map is one-to-one because a local Killing field on ˆM\nvanishing at a point must be identically zero). Cartan’s formula LX=ιX◦d+d◦ιX\nshowsthatwhenever XandYaretwoKillingfieldsaround ˆ x, the relation ω([X,Y]) =\nK(X,Y)−[ω(X),ω(Y)] holds. When XorYis vertical,K(X,Y) = 0, proving that\nevˆxis an anti-morphism of Lie algebras. To see that evˆxis onto, let us consider\n{etξ}t∈R, a 1-parameter group of O(1 ,n−1) fixing Dκ(ˆx). Clearly, ξbelongs to12 CHARLES FRANCES\nKillDκ(ˆx), so that by Theorem 3.1, ω−1(ξ) is the evaluation at ˆ xof a local Killing\nfield. This Killing field being vertical at ˆ x, it corresponds to a local Killing field of\nIs(x).♦\n3.4. Nontrivial recurrence provides nontrivial Killing fie lds. — We still deal\nhere with ( Mn,g) a closedn-dimensional Lorentz manifold ( n≥2). There is, as\nin Riemannian geometry, a notion of Lorentzian volume, which provide s a smooth,\nIso(Mn,g)-invariant measure on M. This measure is finite under our asumption\nthatMis closed. When the group Iso( Mn,g) is noncompact, Poincar´ e’s recurrence\ntheorem applies and almost every point of Mis recurrent for the action of Iso( Mn,g).\nWe aregoingto seethat sucharecurrencephenomenon isrespons ibleforthe existence\nof nontrivial continuous local symetries. The precise statement is :\nProposition 3.3 . —Let(Mn,g)be a closed, n-dimensional Lorentz manifold, and\nassume that Iso(Mn,g)is noncompact. Then\n1. For every x∈Mint, the isotropy algebra Is(x)generates a noncompact subgroup\nofO(TxM).\n2. For every component M ⊂Mint, the Lie algebra kill(M)is at least 3-\ndimensional.\nProof: The proof of the first point is already contained in [ F2, Proposition 5.1].\nWe summarize here the main arguments for the reader’s convenienc e. Letxbe a\nrecurrent point for Iso( Mn,g) and choose ˆ x∈ˆMin the fiber of x. The recurrence\nhypothesis means that there exists ( fk) tending to infinity in Iso( Mn,g), and (pk)\na sequence of O(1 ,n−1) such that fk(ˆx).p−1\nktends to ˆx. By equivariance of the\ngeneralized curvature map Dκ:ˆM→ W, we also have\npk.Dκ(ˆx)→ Dκ(ˆx).\nObserve that ( pk) tends to infinity in O(1 ,n−1), because Iso( Mn,g) acts properly\nonˆM.\nThe O(1,n−1)-orbits on Ware locally closed, because the action of O(1 ,n−1) is\nlinear, hence algebraic. As a consequence, there exists a sequenc e (ǫk) in O(1,n−1)\nwithǫk→Idandǫk.pk.Dκ(ˆx) =Dκ(ˆx). Since (pk) tends to infiny by properness of\nthe action of Iso( Mn,g) onˆM, so does (ǫk.pk), proving that the stabilizer IˆxofDκ(ˆx)\nin O(1,n−1) is noncompact. This group is algebraic, hence the identity compon ent\nIo\nˆxis noncompact too. Fact 3.2 then ensures that Is(x) generates a noncompact\nsubgroup of O( TxM) (under the identification of Is(x) with a subalgebra of o(TxM)\nunder the isotropy representation), for every recurrent point x∈Mint. The property\nis thus true everywhere on Mint, by density of recurrent points on M.\nToprovethe secondpoint, westartwith x∈Mint, and considerasimply connected\nneighborhood U⊂Mintofx. Then, every algebra Is(y),y∈U, is realized as a Lie\nalgebra of Killing fields defined on U. The first point of proposition 3.3 says that\nthere exists Xa nontrivial Killing field on U, such that X(x) = 0. The zero locus\nofXis a nowhere dense set in U. We can thus pick y∈UsatisfyingX(y)/\\e}atio\\slash= 0, and\napply again the first point of the proof at y. We get a second nontrivial Killing ��eldLORENTZ DYNAMICS ON CLOSED 3-MANIFOLDS 13\nYdefined on U, and vanishing at y. Let us now pick a lightlike direction u∈TyM\nsuch thatgy(u,X(y))/\\e}atio\\slash= 0, anduis transverse to the zero locus of Y. Lett/ma√sto→γu(t)\nbe the geodesic passing through yatt= 0 and such that ˙ γu(0) =u. Then Clairault’s\nequation ensures that g(˙γu,X) is constant on γu, hence does not vanish on γuby our\nchoice ofu.\nOn the other hand, by the same Clairault’s equation, one has gγu(˙γu,Y) = 0, and\nY(γu(t))/\\e}atio\\slash= 0 for small, nonzero, values of t. Hence, on some open subset of U, the\norbits of the local Killing algebra have dimension ≥2, while for every point z∈U,\nthe dimension of Is(z) is≥1. It follows that the dimension of killloc(U) is at least 3.\n♦\nRemark 3.4 . —The proof above does not use the fact that the metric gis\nLorentzian, and Proposition 3.3 actually holds for any pseu do-Riemannian (non\nRiemannian) metric.\n3.5. Components of the integrability locus, and their class ification. — We\nnow stick to dimension 3, and we consider a closed Lorentz manifold ( M,g). We\nassume that the isometry group Iso( M,g) is noncompact.\nOn each component M ⊂Mint, the discussion of Section 3.2 shows that there is\na well-defined Lie algebra killloc(M) of local Killing fields (beware that some mon-\nodromy phenomena may occur), which by Proposition 3.3 is at least 3 d imensional.\nBecause in a finite dimensional linear representation of O(1 ,2), no point can have a\nstabilizer of dimension exactly 2, and because of Fact 3.2, the dimens ion ofIs(x) is 1\nor 3 for every x∈ M, and the dimension of killloc(M) is 6, 4 or 3.\n- When this dimension is 6, the component has constant sectional cu rvature.\n- When the dimension of killloc(M) is 4, it is not hard to check that Mis locally\nhomogeneous (see for instance [ DM, Lemma 4]). The dimension of Is(x) is then 1\nat each point x∈ M.\n- When the dimension of killloc(M) is 3, then the dimension of Is(x) is 1 or 3 at\neach point x∈ M. Thekillloc-orbits have dimension 0 or 2, and the component is\nnowhere locally homogeneous.\nProposition3.3ensuresthat wheneverthe dimension of Is(x) is 1, then this algebra\ngenerates a hyperbolic or a parabolic flow in O( TxM)≃O(1,2). In the first case, we\nsay thatxisa hyperbolic point , and in the second one we call xa parabolic point .\nDefinition 3.5 (Hyperbolic and parabolic components)\nA component MofMintwhich is not of constant curvature is said to be hyperbolic\nwhen it contains a hyperbolic point. Otherwise, it is called parabolic.\nObserve that this defintion allows a priori a hyperbolic component to contain\nparabolic points (it will turn out later that it does not occur).\nTo summarize, components of Mintsplit into three (rough) categories.\na) The first category comprises all components having constant sectional curvature .\nb) The second category comprises hyperbolic components . Those in turn split into\ntwo subgategories:\ni) The locally homogeneous ones, for which the dimension of killloc(M) is 4.14 CHARLES FRANCES\nii) The non locally homogeneous ones, for which the dimension of killloc(M) is 3.\nc) The remaining components are parabolic . They can also be splitted into:\ni) The locally homogeneous ones for which dim(killloc(M)) = 4.\nii) The non locally homogeneous ones for which dim(killloc(M)) = 3.\nLet us notice that points belonging to a component with constant cu rvature are\nthose for which the rank of Dκis locally equal to 0. In the same way, points belonging\nto locally homogeneouscomponents arethose for which the rank of Dκis locallyequal\nto 2. Finally, we prove:\nLemma 3.6 . —Pointsx∈Mbelonging to a component of Mintwhich is not locally\nhomogeneous are exactly those at which the rank of Dκis3.\nProof: Recall the generalized curvature map Dκ:ˆM→ Wintroduced in Section\n3.2. We saw in Proposition 3.3 that for every x∈Mint,killloc(x) has dimension ≥3.\nBecause Dκis invariant along killloc-orbits in ˆM, the corank of Dκis a least 3 on\nˆMint, and because ˆMhas dimension 6, the rank of ˆMis at most 3 on the dense set\nˆMint, hence on ˆM. The rank can only increase locally, hence points where the rank\nofDκis 3 actually stay in Mint.\n♦\n4. Locally homogeneous Lorentz manifolds with noncompact i sometry\ngroup\nIn this section, we prove Theorem A in the case where all the compon ents ofMint\narelocallyhomogeneous,implying that ( M,g) is locallyhomogeneouson adense open\nset. We observed in the previous section that in this case, and unde r our standing\nassumption that Iso( M,g) is noncompact, the Lie algebra of local Killing vector fields\nhas dimension ≥4 on each components. We can apply the results of [ F2], saying that\nwe must then have Mint=M, and the manifold ( M,g) is locally homogeneous.\nTheorem 4.1 ([F2], Theorem B) . —Let(M,g)be a smooth 3-dimensional\nLorentz manifold. Assume that on a dense open subset, the Lie algebra of local\nKilling fields is at least 4-dimensional. Then (M,g)is locally homogeneous.\nThere are a lot of homogeneous 3-dimensional models for Lorentz m anifolds. For-\ntunately, very few of them can appear as the local geometry of a c losed manifold with\na noncompact isometry group. We have indeed:\nTheorem 4.2 . — [DZ, Theorem 2.1] Let(M,g)be a closed locally homogeneous\nLorentz manifold. Assume that at each point x∈Mthe isotropy algebra Is(x)gen-\nerates a noncompact subgroup of O(TxM). Then the metric gis locally isometric\nto:\n1. A flat metric\n2. A Lorentzian, non-Riemannian, left-invariant metric on /tildewidestPSL(2,R).\n3. The Lorentz-Heisenberg metric gLHon the group Heis.\n4. The Lorentz-Sol metric gsolon the group SOL.LORENTZ DYNAMICS ON CLOSED 3-MANIFOLDS 15\nThe Theorem applies in our situation since Proposition 3.3 ensures tha tIs(x)\ngenerates a noncompact subgroup of O( TxM) for almost every (hence every) point x.\nLorentzian, non-Riemannian, left-invariant metrics gAdS,guandghon/tildewidestPSL(2,R)\nwere introduced in Section 2.1, while Lorentz-Heisenberg geometry was described in\nSection 2.3.2. It remains to explain what is the Lorentz-Sol geometr y.\nThe Lie algebra solis the 3-dimensional Lie algebra with basis T,X,Yand non-\ntrivial bracket relations [ T,Z] =Z, [T,X] =−X. The corresponding connected,\nsimply connected Lie group is denoted SOL. On sol, we can consider the Lorentz\nscalar product such that <T,Z > =<X,X > = 1, and all other products are 0. Af-\nter left-translating this scalar product on SOL, we get a Lorentz m etricgsolon SOL\nwhich is called the Lorentz-Sol metric. The isometry group of (SOL ,gsol) contains\nSOL (acting by left-translations), but it is actually 4-dimensional. Th e Lie algebra of\nKilling fields is obtained by adding YtoT,X,Z, with bracket relations [ T,Y] = 2Y\nand [X,Y] =Z(see [DZ, Section 4.2] for further details).\n4.1. Ruling out Lorentz- SOLgeometry. — We first establish:\nProposition 4.3 . —Let(M,g)be a closed, 3-dimensional Lorentz manifold locally\nmodelled on (SOL,gsol). ThenIso(M,g)is a compact group.\nThe key property for proving Proposition 4.3 is a Bieberbach rigidity t heorem for\nclosed manifolds modelled on Lorentz-Sol geometry.\nTheorem 4.4 . — [DZ, Theorem1.2(iv), andProofofProposition7.1] Let(M,g)be\na closed, 3-dimensional Lorentz manifold locally modelled on (SOL,gsol). Then(M,g)\nis isometric to the quotient of (SOL,gsol)by a discrete subgroup Γ⊂Iso(SOL,gsol).\nMoreover, the intersection of ΓwithIsoo(SOL,gsol)is a lattice Γ0⊂SOLacting by\nleft translations.\nWe know that ( M,g) is isometric to some quotient Γ \\SOL, by Theorem 4.4. Let\nus denote by LSOLthe subgroup of Iso(SOL ,gsol) comprising all left-translations by\nelements of SOL. The group Γ 0= Γ∩Isoo(SOL,gsol) is Zariski-dense in LSOLby\nTheorem 4.4. It follows that Nor(Γ), the normalizer of Γ in Iso(SOL ,gsol), must\nnormalizeLSOL. But the description of Iso(SOL,gsol) made above shows that LSOL\nhas finite index in its normalizer. We infer that Nor(Γ) /Γ is compact, what proves\nProposition 4.3.\n4.2. Minkowski and Lorentz-Heisenberg geometries with non compact\nisometry group. — We now focus on closed Lorentz manifolds locally modelled\non Minkowski space, or on Lorentz-Heisenberg geometry. In bot h cases, one has a\nBieberbach type theorem. This is very well known in the flat case, th anks to the\nworks [FG] and [GK], and the completeness result of Carri` ere for closed flat Lorent z\nmanifolds [ Ca]. For manifolds modelled on Lorentz-Heisenberg geometry, this is\nproved in [ DZ, Proposition 8.1]. The precise statement is the following:16 CHARLES FRANCES\nTheorem 4.5 (Bieberbach’s theorem for flat and Lorentz-Heisenberg man-\nifolds)\nLet(M,g)be a closed, 3-dimensional, Lorentz manifold.\n1. If(M,g)is flat, there exists a discrete subgroup Γ⊂Iso(R1,2)such that (M,g)\nis isometric to the quotient Γ\\R1,2. Moreover, there exists a connected 3-\ndimensional Lie group G⊂Iso(R1,2), which is isometric to R3,HeisorSOL,\nand which acts simply transitively on R1,2, satisfying that Γ0=G∩Γhas finite\nindex in Γand is a uniform lattice in G.\n2. If(M,g)is locally modelled on Lorentz-Heisenberg geometry, then i t is isometric\nto the quotient of (Heis,gHeis)by a discrete subgroup Γ⊂Iso(Heis,gHeis). More-\nover, there exists a finite index subgroup Γ0⊂Γwhich is a lattice Γ0⊂Heis\nacting by left translations.\nThistheoremsaysthat up to finite cover , aclosedLorentzmanifold( M,g)modelled\non Minkowski, or Lorentz-Heisenberg geometry, is homeomorphic t oT3or toT3\nAfor\nA∈SL(2,Z) hyperbolic or parabolic. Noncompactness of the isometry group a llows\nto be more precise.\nProposition 4.6 . —Let(M,g)be a closed, 3-dimensional Lorentz manifold, such\nthatIso(M,g)is noncompact. We assume that (M,g)is orientable and time-\norientable.\ni) If(M,g)is flat, then Mis diffeomorphic either to a torus T3, or to a torus\nbundleT3\nAwithA⊂SL(2,Z)hyperbolic, or parabolic.\nii) If(M,g)is modelled on Lorentz-Heisenberg geometry, then Mis diffeomorphic\nto a torus bundle T3\nAwithA∈SL(2,Z)parabolic (A/\\e}atio\\slash=id).\nProof: The situation provided by Theorem 4.5 is the following (both in the flat\nand Lorentz-Heisenberg case). We have a 3-dimensional Lie group G, which is either\nR3, Heis or SOL, as well as a left-invariant metric µonG, and the manifold ( M,g)\nis isometric to a quotient of ( G,µ) by a discrete subgroup Γ ⊂Iso(G,µ). Moreover,\nif we denote by LGthe group of left-translations by elements of G, the intersection\nΓ0= Γ∩LGhas finite index in Γ, and is a uniform lattice in LG.\nAn important remark is that if Nor(Γ) denotes the normalizer of Γ in I so(G,µ),\nthen Nor(Γ) normalizes LG. It is obvious in the case of Lorentz-Heisenberg geometry\n(G,µ) = (Heis,gHeis). In this case, the identity component Isoo(G,µ) is of the form\nR⋉LG, andLGis thus normalized by the full isometry group Iso( G,µ).\nIn the case of Minkowski geometry, one has to remember that the groupG(more\naccuratelyLG) is the identity component of the crystallographic hull of Γ (see [ FG,\nSection1.4]). Thelastpartof[ FG, Theorem1.4]ensuresthatinthecaseofMinkowski\ngeometry, the crystallographic hull is unique. It follows that Nor(Γ ) must normalize\nthis crystallographic hull, as well as its identity component LG.\nAs a consequence, elements of Nor(Γ) (in particular elements of Γ) belong to the\ngroup Aut(G)µ⋉LG, where Aut( G)µdenotes the automorphisms of Gpreserving the\nmetricµ. Let us denote by Γ lthe projection of Γ on Aut( G)µ. If this projection isLORENTZ DYNAMICS ON CLOSED 3-MANIFOLDS 17\ntrivial, we get that Γ ⊂LG. The manifold Mis obtained as a quotient of R3, SOL\nor Heis by a uniform lattice, and we are done.\nIf this projection is nontrivial, we are going to get a contradiction. I ndeed, Γ l\nmust be a finite subgroup of Aut( G)µ, because Γ 0⊂LGhas finite index in Γ. The\nisotropy representation ρof Aut(G)µateidentifies Γ lwith a nontrivial finite sub-\ngroup of SO( µ), which is actually in SOo(µ) because ( M,g) is orientable and time-\norientable. Let Noro(Γ) be the intersection of Nor(Γ) with Auto(G)µ⋉LG, and letNl\nthe projection of Noro(Γ) on Auto(G)µ. We get that ρ(Nl) normalizes ρ(Γl). Now,\nnontrivial finite groups in SOo(µ) have a unique fixed point in H2, so their normalizer\nin SOo(µ) are contained in a compact subgroup. It follows that Noro(Γ) is contained\nin a subgroup K⋉LG, withKcompact in Auto(G)µ. We thus see that Noro(Γ)/Γ0\nis compact, implying the compactness of Nor(Γ) /Γ. This in turns implies Iso( M,g)\ncompact: Contradiction.\n♦\n4.3. Anti-de Sitter structures with noncompact isometry gr oup. —\nIt remains to study closed Lorentz manifolds ( M,g) modelled on a Lorentzian,\nnon-Riemannian, left-invariant metric on /tildewidestPSL(2,R). We observe that the identity\ncomponents Isoo(/tildewidestPSL(2,R),gu) and Isoo(/tildewidestPSL(2,R),gh) are actually included in\nIsoo(/tildewidestPSL(2,R),gAdS). Thus, if ( M,g) is a closed, orientable and time-orientable,\nLorentz manifold modelled on ( /tildewidestPSL(2,R),gu) or (/tildewidestPSL(2,R),gh), there exists an\nanti-de Sitter metric g′onMwhich is preserved by a finite index subgroup of\nIso(M,g). Hence, it will be enough for us to focus on the topology of closed a nti-de\nSitter manifolds with noncompact isometry group. In the sequel, we will denote\n/tildewideAdS3the space ( /tildewidestPSL(2,R),gAdS)\nProposition 4.7 . —Let(M,g)be a closed, orientable and time-orientable, anti-de\nSitter manifold. If Iso(M,g)is noncompact, then Mis homeomorphic to a quotient\nΓ\\/tildewidestPSL(2,R), for a uniform lattice Γ⊂/tildewidestPSL(2,R).\nIt is worth noticing that all closed (orientable and time-orientable) 3 -dimensional\nanti-de Sitter manifolds are Seifert fiber bundles over hyperbolic or bifolds (a short\nproof of this fact can be found in [ Tho, Corollary 4.3.6]). Conversely, any Seifert\nfiber bundle over an hyperbolic orbifold, with nonzero Euler number, can be endowed\nwith an anti-de Sitter metric (see [ Sco]). The assumption that Iso( M,g) is non-\ncompact reduces the possibilities for the allowed Seifert bundles. Fo r instance, all\nnontrivial circle bundles over a closed orientable surface of genus g≥2 admit anti-de\nSitter metrics, but only those for which the Euler number divides 2 g−2 do occur in\nProposition 4.7.\nIt was shown in [ Kl] that closed anti-de Sitter manifolds are complete. It fol-\nlows that (M,g) as in Proposition 4.7 is a quotient of /tildewideAdS3by a discrete subgroup\n˜Γ⊂Iso(/tildewideAdS3). Actually ˜Γ⊂Isoo(/tildewideAdS3) because ( M,g) is orientable and time-\norientable. The center of /tildewidestPSL(2,R) is infinite cyclic, generated by an element ξ. The\ngroup/tildewidestPSL(2,R)×/tildewidestPSL(2,R) acts on /tildewideAdS3by left and right translations: ( h1,h2).g=18 CHARLES FRANCES\nh1gh−1\n2. This yields an epimorphism /tildewidestPSL(2,R)×/tildewidestPSL(2,R)→Isoo(/tildewideAdS3), with\ninfinite cyclic kernel generated by ( ξ,ξ). The group Isoo(/tildewideAdS3) has a center Z\nwhich is generated by the left action of ξ. Doing the quotient of Isoo(/tildewideAdS3) by\nZyields an epimorphism π: Isoo(/tildewideAdS3)→PSL(2,R)×PSL(2,R). Notice that\nPSL(2,R)×PSL(2,R) coincides with the identity component of the isometries of\nPSL(2,R) endowed with its anti-de Sitter metric.\nAn important result, known as finiteness of level , says that ˜Γ∩Z/\\e}atio\\slash=id. This\nwas first stated in [ KR]. A detailed proof can be found in [ Sa2, Theorem 3.3.2.3].\nGeometrically, this theorem ensures that there exists a finite grou p of isometries\nΛ⊂Iso(M,g), which acts freely and centralizes a finite index subgroup of Iso( M,g),\nsuch that the quotient manifold of ( M,g) by Λ is a quotient of PSL(2 ,R) by a discrete\ngroupΓ⊂PSL(2,R)×PSL(2,R). Letusdenoteby( M3,g)thisnewLorentzmanifold,\nand observethat ( M3,g) is still noncompact. Observealsothat the projection πmaps\n˜Γ ontoΓ.\nThe structure of the group Γ is well understood. Up to conjugacy, there exists Γ 0\na uniform lattice in PSL(2 ,R), and a representation ρ: Γ0→PSL(2,R) such that\nΓ ={(γ,ρ(γ))∈PSL(2,R)×PSL(2,R)|γ∈Γ0}.\nThis was established in [ KR, Theorem 5.2] when Γ is torsion-free. For a group with\ntorsion, the adapted proof can be found in [ Tho, Lemma 4.3.1].\nBecause the group Iso( M3,g) is noncompact, a result of Zeghib ([ Z3, Theorem\n1.2]) ensures that ( M3,g) must admit a codimension one, lightlike, totally geodesic\nfoliation. Such foliations Fin PSL(2,R) endowed with the anti-de Sitter metric\nare very well known. Let AG⊂PSL(2,R) be the connected 2-dimensional group\ncorresponding to the upper-triangular matrices (it is isomorphic to the affine group of\nthe line). Then up to conjugacy in PSL(2 ,R)×PSL(2,R), the leavesof Fare given by\n{gAG|g∈PSL(2,R)}or{AGg|g∈PSL(2,R)}. IfΓ preserves such a foliation F,\nwe infer that the leaves are of the form {gAG|g∈PSL(2,R)}, andρ(Γ0) normalizes\nAG, namelyρ(Γ0)⊂AG.\nWe now consider the normalizer HofΓ in PSL(2 ,R)×PSL(2,R). The first pro-\njectionπ1(H) must normalize Γ 0. Since uniform lattices in PSL(2 ,R) are of finite\nindex in their normalizer, we can replace Hby a finite index subgroup and assume\nπ1(H) = Γ0. Let us consider h= (h1,h2) inH, andγ∈Γ0. BecauseHnormalizes\nΓ, (h1γh−1\n1,h2ρ(γ)h−1\n2)∈Γ, which implies h2ρ(γ)h−1\n2=ρ(h1)ρ(γ)ρ(h1)−1. In other\nwords,h−1\n2ρ(h1) centralizes ρ(Γ). As a consequence, ρ(Γ) can not be Zariski dense\ninAG. Otherwise, h−1\n2ρ(h1) should be trivial implying that h= (h1,h2) actually\nbelongs to Γ. We thus would get that Iso( M3,g) is finite, a contradiction.\nAs a result, ρ(Γ) is included in a 1-parameter subgroup of AG. This implies that\nthe group ˜Γ is included in a product /tildewidestPSL(2,R)×R, where /tildewidestPSL(2,R) acts by left\ntranslations, and R⊂/tildewidestPSL(2,R) is aR-split or unipotent parameter group acting\non the right. We consider the projection π1:˜Γ→/tildewidestPSL(2,R) on the left-factor.\nThe group Γ := π1(˜Γ) projects surjectively on Γ byπ, hence is a uniform latticeLORENTZ DYNAMICS ON CLOSED 3-MANIFOLDS 19\nin/tildewidestPSL(2,R). Moreover, the kernel of π1must be trivial, otherwise some nontrivial\nelement of ˜Γ wouldbelongto {id}×R, and Nor( ˜Γ)/˜Γwould be compact, contradicting\nthe hypothesis Iso( M,g) noncompact. It follows that ˜Γ is isomorphic to Γ.\nIn conclusion, the two manifolds Γ \\/tildewidestPSL(2,R) and˜Γ\\/tildewidestPSL(2,R) are two Seifert\nbundles over the hyperbolic orbifold Γ 0\\H2. Their Euler number is nonzero, so that\nthey are both large Seifert manifolds (see [ Or, p. 92]). Their fundamental groups are\nisomorphic (to Γ), hence by [ Or, Theorem 6, p 97] they are homeomorphic.\n4.4. Conclusions. — The previous results show that closed, orientable and time-\norientable, Lorentz3-dimensional manifolds which are locally homoge neousare home-\nomorphic to T3, a torus bundle T3\nAforAhyperbolic or parabolic, or a quotient\nΓ\\/tildewidestPSL(2,R). This proves Theorem A for manifolds such that all components of Mint\narelocallyhomogeneous. Ouranalysisshowsmoreoverthatwhen Mishomeomorphic\nto an hyperbolic torus bundle or to a 3-torus, the metric must be fla t. WhenMis\nhomeomorphictoaparabolictorusbundle, themetriciseitherflat, o rlocallyisometric\nto Lorentz-Heisenberg geometry. When Mis homeomorphic to Γ \\/tildewidestPSL(2,R), the ge-\nometry is locally anti-de Sitter or locally modelled on a Lorentzian, non- Riemannian,\nleft-invariant metric on /tildewidestPSL(2,R). This is in accordance to points 2 ,3,4 of Theorem\nC.\n5. Manifolds admitting a hyperbolic component\nOur aim in this section is to prove Theorem A under the assumption tha t our\nLorentz manifold ( M,g) admits at least one hyperbolic component M. Actually\nTheorem A will be implied by a more precise description provided by Theo rem 5.2 to\nbe stated below.\n5.1. A first reduction. —\nProposition 5.1 . —Assume that the intergrability locus Mintcontains a hyperbolic\ncomponent. Then either (M,g)is locally homogeneous, or there exists a hyperbolic\ncomponent which is not locally homogeneous.\nProof: Assume that all hyperbolic components in Mintare locally homogeneous,\nand consider M ⊂Mintsuch a component. It is open by definition, and we are going\nto show that the boundary ∂Mis empty, which will yield M=Mby connectedness\nofM. Local homogeneity of Mwill follow.\nLet us assume that there exists x∈∂M. We are going to see that x∈Mint,\nwhich will yield a contradiction. We pick x0∈ M. In the sequel, we use the notation\nDκ(z) for the O(1 ,2)-orbit of Dκ(ˆz) inW(where ˆzis any point in the fiber of z). By\nassumption, the rank of Dκis constant equal to 2 on M. Moreover, by local homo-\ngeneity,Dκ(M) is a single O(1 ,2)-orbit in W. This orbit is 2-dimensional because\nMis locally homogeneous, but does not have constant curvature. Th e stabilizer of\npoints in Dκ(x0) are hyperbolic 1-parameter subgroups of O(1 ,2) by assumption that20 CHARLES FRANCES\nWis hyperbolic. In every finite-dimensional representation of O(1 ,2), such hyper-\nbolic orbits are closed. This is a standard fact, the proof of which is, for instance,\ndetailed in [ F2, Annex B]. It follows that Dκ(x) =Dκ(x0). Hyperbolic 1-parameter\ngroups are open in the set of 1-parameter groups of O(1 ,2). It follows that there is a\nsufficiently small neighborhood UofxinMsuch that the rank of DκonUis≥2,\nand for every y∈U, the orbit Dκ(y) is 2-dimensional with hyperbolic 1-parameter\ngroups as stabilizers of points (notice that Dκ(y) can not be 3-dimensional because of\nthe first point of 3.3). If at some point y∈U, the rank of Dκis 3, thenybelongs to a\ncomponentof Mintwhich is not locallyhomogeneous,by Lemma 3.6. This component\nmust be hyperbolic because stabilizers in Dκ(y) are hyperbolic. Since we assumed\nthat there are no such components, it follows that the rank of Dκis constant equal\nto 2 onU. But then x∈Mintleading to the desired contradiction.\n♦\nThe case of locally homogeneousmanifolds was already settled in Sect ion 4, so that\nwe will assume in all the remaining of this section thatMintcontains a component\nwhich is hyperbolicbut not locally homogeneous. We will call Mthis component. We\nare going to show that under these circumstances, Mis diffeomorphic to a hyperbolic\ntorus bundle. More precisely, the geometry of ( M,g) can be described as follows:\nTheorem 5.2 . —Assume that (M,g)is a closed, orientable and time-orientable 3-\ndimensional Lorentz manifold, such that Iso(M,g)is noncompact. Assume that (M,g)\nadmits a hyperbolic component which is not locally homogene ous. Then\n1. The manifold Mis diffeomorphic to a 3torusT3, or a torus bundle T3\nAwhere\nA∈SL(2,Z)is a hyperbolic matrix.\n2. The universal cover (˜M,˜g)is isometric to R3endowed with the metric dt2+\n2a(t)dudvfor some positive nonvanishing, periodic, smooth function a:R→\n(0,+∞).\n3. There is an isometric action of the Lie group SOLon(˜M,˜g).\nThe proof of Theorem 5.2, will be the aim of Sections 5.2 to 5.6 below.\n5.2. Existence of Anosov tori. — We first prove that under the hypotheses of\nTheorem 5.2, one can find an element hin Iso(M,g) which acts by an Anosov trans-\nformation of a flat Lorentz torus on M(see Lemma 5.4).\n5.2.1. Facts about flat Lorentz surfaces . — We begin by recalling elementary and\nwell known facts about closed flat Lorentz surfaces. Since Loren tz manifolds must\nhave zero Euler characteristic, such a surfaces are tori or Klein b ottles.\nLemma 5.3 . —A closed, flat Lorentz surface (Σ,g)admitting a noncompact isome-\ntry group is a torus. Moreover, any group H⊂Iso(Σ,g)which does not have compact\nclosure, contains an element hacting on Σby a hyperbolic linear transformation.\nProof: Closed Lorentz manifolds with constant curvature are geodesically complete\n([Ca], [Kl]). It follows that a closed, flat Lorentz surface (Σ ,g) is a quotient of the\nMinkowski plane R1,1, by a discrete subgroup Γ ⊂O(1,1)⋉R2, which acts freely and\nproperly on R1,1. Observe that nontrivial elements of O(1 ,1) are of two kinds. EitherLORENTZ DYNAMICS ON CLOSED 3-MANIFOLDS 21\nthey are hyperbolic (namely have two real eigenvalues of modulus /\\e}atio\\slash= 1), or they have\norder 2 (an orthogonal symmetry with respect to a spacelike (res p. timelike) line).\nIt is readily checked that Γ is either a lattice in R2, or admits a subgroup of index\n2 which is such a lattice. Assume we are in the first case, and let H⊂Iso(Σ,g) be\na subgroup which does not have compact closure. We lift Hto˜H⊂O(1,1)⋉R2.\nThe group ˜Hhas a nontrivial projection on SO(1 ,1), otherwise Hwould be compact.\nThus˜Hcontains a conjugate of a hyperbolic element of O(1 ,1), which acts as an\nAnosov diffeomorphism on Σ.\nIn the second case, where the projection of Γ on O(1 ,1) is an order 2 subgroup,\nthe normalizer Nor(Γ) must have trivial projection on SO(1 ,1), which implies that\nΓ is cocompact in Nor(Γ). This shows that flat Klein bottles have compact isometry\ngroup.♦\n5.2.2. Closed killloc-orbits. — Our next aim is to exhibit some killloc-orbits which are\nclosed surfaces.\nLemma 5.4 . —In every hyperbolic component M, there exists a killloc-orbitΣ0\nwhich is a flat Lorentz 2-torus, and such that there exists h∈Iso(M,g)leavingΣ0\ninvariant, and acting on Σ0as a linear hyperbolic automorphism.\nWe consider our distinguished component Mthat, we recall, is not locally homo-\ngeneous, and hyperbolic. We pick x∈ M, and ˆx∈ˆMin the fiber of x. We already\nobserved in the proof of Lemma 3.6 that the rank of Dκis at most 3 on M. It is\nexactly 3 at ˆ x, still by Lemma 3.6, hence remains constant equal to 3 in a neighbor-\nhood of ˆx. Hence, if U⊂ˆMis a small open set around ˆ x,Dκ(U) is a 3-dimensional\nsubmanifold of W. IfUis chosen small enough, the O(1 ,2)-orbit of every point in\nDκ(U) will be 2-dimensional and will have hyperbolic 1-parameter groups a s stabiliz-\ners of points. Let us now call ˆΛ the closed subset of ˆMwhere the rank of Dκis≤2.\nBy Sard’s theorem, the 3-dimensional Hausdorff measure of Dκ(ˆΛ) is zero. We infer\nthe existence of w∈ Dκ(U)\\Dκ(ˆΛ). Moving ˆ xinsideU, we assume that w=Dκ(ˆx),\nand we denote by O(w) the O(1,2)-orbit ofwinW. By O(1,2)-equivariance of Dκ,\nthe inverse image Dκ−1(O(w)) avoids ˆΛ, hence the rank of Dκis constant equal to\n3 onDκ−1(O(w)). Lemma 3.6 then leads to the inclusion Dκ−1(O(w))⊂ˆMint. By\nthe discussion right after Theorem 3.1, the projection of Dκ−1(O(w)) onMis a sub-\nmanifoldNofM. The stabilizer of win O(1,2) is hyperbolic, thus as mentioned in\nthe proof of Lemma 3.6, the orbit O(w) is closed in W. It follows that Nis closed in\nM, hence compact. By (the proof of ) Theorem 3.1, the Isloc-orbit ofxis a union of\nconnected components of N, and the connected component of xinN, denoted Σ 0, co-\nincides with the killloc-orbit ofx. It is a connected compact surface in M. Let us show\nthat this surface has Lorentz signature. The Lie algebra Is(x) is generated by a local\nKilling field Xaroundx, vanishing at x, and such that the flow {Dxφt\nX} ⊂O(TxM)\nis a hyperbolic flow. Linearizing Xaroundxthanks to the exponential map, we see\nthere are two distinct lightlike directions uandvinTxMsuch that the two geodesics\nγu:s/ma√sto→exp(x,su) andγv:s/ma√sto→exp(x,sv) are left invariant by φt\nX. In particular,\nfors/\\e}atio\\slash= 0 close to 0, ˙ γu(s) and ˙γv(s) are colinear to X, hence tangent to O(γu(s)) and22 CHARLES FRANCES\nO(γv(s)) respectively. By continuity, this property must still hold for s= 0. We infer\nthatTx(O(x)) containsthe two distinct lightlike directions uandv, hence has Lorentz\nsignature. By local homogeneity of the killloc-orbit Σ 0, we get that Σ 0is Lorentz, and\nmoreover has constant Gauss curvature. The only closed Lorent z surfaces of constant\ncurvature are flat tori or Klein bottles.\nNow Iso(M,g) sends Σ 0to components of the Isloc-orbit ofx, and there arefinitely\nmany such components by compactness of N. As a consequence the subgroup H⊂\nIso(M,g) leavingΣ 0invariantis noncompact. Observethat if g0is the metric induced\nbygon Σ0, then the injection H→Iso(Σ0,g0) is proper (see for instance [ Z2, Prop.\n3.6]). It follows that Iso(Σ 0,g0) is a noncompact group. Lemma 5.3 ensures that\n(Σ0,g0) is a flat Lorentz torus, and there exists h∈Iso(M,g) acting on Σ 0by a\nhyperbolic linear automorphism.\n5.3. Pushing Anosov tori along the normal flow. — From the 2-torus Σ 0and\nthe diffeomorphism h∈Iso(M,g) given by Lemma 5.4, we are going to recover the\ntopology of the whole manifold M, as well as its geometry.\n5.3.1. Preliminary definitions . — On the torus Σ 0, we choose a frame field ( E−,E+)\nwith the property that E−andE+are future lightlike, satisfy g(E−,E+) = 1, and\ngenerate the strong stable and unstable bundles of the Anosov diff eomorphism h.\nBecauseMis assumed to be orientable, this defines a smooth normal field ν: Σ0→\nTΣ⊥\n0with the property that ( E−,E+,ν) is a direct frame of TzMat each point\nz∈Σ0, andg(ν,ν) = +1.\nIn all the rest of the section, we pick once for all z0∈Σ0a periodic point of h\n(recall that the set of periodic points is dense in Σ 0). This point has period m0,\nand replacing hbyh2m0if necessary, we will assume henceforth that h(z0) =z0and\nh∗ν=ν.\nFor everyz∈Σ0,we will call γzthe oriented geodesic arc through z, with tangent\nν(z) atz. Observe that γz0is a closed spacelike geodesic. Indeed, since his a Lorentz\nisometry, the fixed points set Fix( h) is a closed, totally geodesic submanifold of M.\nThe matrix of the differential Dz0h, expressed in the basis ( E−(z0),E+(z0),ν(z0)),\nis of the form\n1\nλ00 0\n0λ00\n0 0 1\nwith|λ0|>1. Linearizing haroundz0thanks to the\nexponential map, we see that the component of Fix( h) containing z0, is precisely γz0.\n5.3.2. The normal flow, and an auxiliary pseudo-Riemannian m anifold. — It will be\nusefull in the sequel to consider the manifold N=R×Σ0. On this manifold, we have\nthe vector field∂\n∂t. Pushing the vector fields E−,E+on{0}×Σ0by the flow of∂\n∂t,\nwe get two more vector fields ˜E−,˜E+onN. The frame ��eld ( ˜E−,˜E+,∂\n∂t) provides\nNwith an orientation.\nLet us consider the map f: (t,z)/ma√sto→exp(z,tν(z)). It is well-defined and smooth on\nsome maximal open subset Umax⊂N. An easy application of the inverse mapping\ntheorem shows that f: (−ǫ,ǫ)×Σ0→Mis a one-to-one immersion for small ǫ>0.LORENTZ DYNAMICS ON CLOSED 3-MANIFOLDS 23\nA key property of the map fis its equivariance with respect to the action of h,\nnamely :\n(2) h◦f(t,z) =f(t,h(z)),\nwhich is available for ( t,z)∈Umax(observe that f(Umax) is left invariant by h).\nRelation (2) just follows from the fact that his an isometry preserving the normal\nfieldν.\nIn the following, we are going to introduce\nτm:= sup{s∈(0,∞)|f: (0,s)×Σ0⊂Umax→Mis an injective immersion }.\nIt will be sometimes moresuggestiveto restrict fto{0}×Σ0, and consider the normal\nflow ofΣ0,φt: Σ0→Mdefined by φt(z) := exp(z,tν(z)). By what we said before,\nφtis at least defined on (0 ,τm), and for all t∈(0,τm),φt: Σ0→Mis a proper\nembedding, with image Σ t⊂M. Equivariance relation (2) shows that hpreserves Σ t\nand acts on it as an Anosov diffeomorphism. In particular, the stable and unstable\nbundles must be lightlike for the metric g, showing that Σ tis a Lorentz torus. We\ndenote bygtthe restriction of gto Σt.\nThe mapt/ma√sto→φt(z0) provides a (cyclic) parametrization of the closed geodesic γz0\nat speed +1. Hence for every t∈R, the mapz/ma√sto→φt(z) is defined and smooth on\nsome small neighborhood Ut⊂Σ0containing z0. We call E±(t) :=Dz0φt(E±(z0)),\nand the formula a(t) :=gγz0(t)(E−(t),E+(t)) defines a smooth function a:R→R.\nThis in turns defines on the manifold Na symmetric (2 ,0)-tensor ˜g=dt2+a(t)g0\n(hereg0is the metric induced by gon Σ0). Observe that ˜ gis not a metric on N\nbecausea(t) might vanish for some values of t.\n5.3.3. First return time . — For any point z∈Σ0, the geodesic γzis defined on\nsome maximal interval [0 ,τ∗\nz). Ifγz((0,τ∗\nz))∩Σ0/\\e}atio\\slash=∅, then there exists a smallest\nτ(z)∈(0,τ∗\nz) such that γz(τ(z))∈Σ0. Whenγz((0,τ∗\nz))∩Σ0=∅, we just put\nτ(z) = +∞. We introduce the first return set:\nΩr={z∈Σ0|τ(z)<+∞andγ′\nz(τ(z)) is transverse to TΣ0}\nIt is clear that Ω ris an open set, and it is nonempty because any periodic point\nofhbelongs to Ω r(for such points, γ′\nz(τ(z)) is actually orthogonal to TΣ0). Now\nthe mapϕ:z/ma√sto→(τ(z),γ′\nz(τ(z))) is continuous on Ω r. Whenzis periodic for h,\nγ′\nz(τ(z)) =ǫ(z)ν(γz(τ(z))), whereǫ(z) :=±1. By density of such periodic points,\nwe get that ϕmaps continuously Ω rtoR+×{−1,+1}. Observe that Ω r, as well as\nϕareh-invariant. Because his topologically transitive on Ω r, this implies that ϕis\nactually a constant map z/ma√sto→(τr,ǫ). We call τrthefirst return time of the normal\nflow, andφτr: Ωr→Σ0the first return map .\n5.4. First geometric properties of the normal flow. — The previous section\nshows that the normal flow φtis defined on (0 ,τm). We detail here its main geometric\nproperties.\nProposition 5.5 . —1. For each t∈(0,τm),φtis an homothetic transformation\nfrom(Σ0,g0)to(Σt,gt). More precisely a(t)>0and(φt)∗gt=a(t)g0.24 CHARLES FRANCES\n2. The tensor ˜gis a Lorentz metric on (0,τm)×Σ0, andf: ((0,τm)×Σ0,˜g)→\n(M,g)is a one-to-one, orientation preserving, isometric immers ion.\nProof: Equivariance relation (2) implies that hacts as an Anosov diffeomorphism\non Σt, andE−(t) (resp. E+(t)) generates the stable (resp. unstable) bundle of hat\nγz0(t). It follows that E±(t) are lightlike, and linearly independant since Dz0φtis one-\nto-one. This implies a(t) =g(E−(t),E+(t))/\\e}atio\\slash= 0. Because a(0) = 1, we get a(t)>0\nfort∈(0,τm). Relation (2) shows that ϕtmaps the stable (resp. unstable) foliation\nofhon Σ0to the stable (resp. unstable) foliation of hon Σt. Hence the differential\nDzϕtmaps at each point zof Σ0the lightcone of TzΣ0to the lightcone of Tϕt(z)Σt,\nwhich means that ( ϕt)∗gt=σtg0, for some smooth function σt: Σ0→R∗\n+. Now,\nbecause of (2), the function σtish-invariant, hence constant since hadmits dense\norbits on Σ 0. This constant is given by g(Dz0ϕt(E−(z0)),Dz0ϕt(E+(z0))), namely\na(t).\nThe second point follows easily. Indeed, we already noticed that a(t)>0 for\nt∈(0,τm), whichensuresthat ˜ gisLorentzianon(0 ,τm)×Σ0. Bythefirstpoint, fwill\nbe isometric if we prove that Tγz(t)Σtis orthogonal to γ′\nz(t) for allt∈(0,τm). Now,\nobserve that for a linear Lorentz transformation L=\nλ0 0\n01\nλ0\n0 0 1\nwith|λ|>1,\nthe only Lorentz plane invariant by Lis the one generated by the two first basis\nvectors, namely the orthogonal to the line of fixed point of L. This remark shows\nthat ifz∈Σ0is a periodic point for h,Tγz(t)Σt⊥γ′\nz(t) holds. By density of periodic\npoints ofhon Σ0, the property actually holds for all z∈Σ0. Finallyfis orientation\npreserving because ( E−,E+,ν) is positively oriented, and we defined the orientation\nofNto make ( ˜E−,˜E+,∂\n∂t) positively oriented. ♦\nLetusalsosayafew wordsaboutthe surfacesΣ t. Weobservethat generally,Σ tare\nnot totally geodesic submanifolds of M. However, they enjoy the weaker condition:\nFact 5.6 . —For everyt∈(0,τm), the parametrized lightlike geodesics of Σtfor the\nmetricgtare parametrized geodesics for the metric g.\nThis can be checked directly by computation for the metric ˜ gon (0,τm)×Σ0.\nFrom Fact 5.6, we infer the following relation, available for all z∈Σ0,t∈(0,τm) and\ns∈R:\n(3) φt(exp(z,sE±(z))) = exp(φt(z),sDzφt(E±(z)))\n5.5. Completeness of the normal flow. — Thanks to the previous section, we\nunderstand pretty well the behaviour of the normal flow for t∈(0,τm). Our next\nstep is to show that the flow can be extended for t≥τm.\n5.5.1. Extension of the normal flow at t=τm. — An important step toward extend-\ning the normal flow for t=τmis to show that the geometry of the Lorentz manifold\n((0,τm)×Σ0,˜g) does not degenerate at the boundary. Precisely, we prove :\nLemma 5.7 . —There exists ǫ >0such that ˜gis a Lorentz metric on the open set\n(−ǫ,τm+ǫ)×Σ0⊂N.LORENTZ DYNAMICS ON CLOSED 3-MANIFOLDS 25\nProof: We just have to show that a(τm)/\\e}atio\\slash= 0.\nRecall the point z0∈Σ0we introduced at the begining of Section 5.3.1. This point\nis fixed by h, and we already observed that f(t,z0) exists for all t∈R. Hence, on\nU ⊂Σ0, a small convex neighborhood of z0(convex relatively to the metric g0), we\nhave an extended flow φt:U →Mdefined for t∈(0,τm+δ),δ >0. Saturating\nUby the action of ( hm)m∈Z, we get a dense open set Von whichφtis defined for\nt∈(0,τm+δ). Observe that Vcontains de stable and unstable manifolds of hatz0,\nnamelyW±:={exp(z0,sE±)|s∈R}.\nAlongthe geodesic t/ma√sto→γz0(t), wedefine twovectorfields( E−(t),E+(t)) byparallel\ntransporting ( E−(z0),E+(z0)). For each t∈(0,τm), relation (2) yields the existence\nof two nonzero reals λ±\ntsuch that E±(t) =Dz0ϕt(E−(z0)) =λ±\ntE±(t)). Observe that\na(t) =λ+\ntλ−\nt. Provinga(τm)/\\e}atio\\slash= 0 amounts to show that λ±\ntare bounded away from 0\nin (0,τm).\nAssume for a contradiction that there exists some sequence ( tk) in (0,τm), such\nthattk→τmandλ−\ntk→0. Relation (3) says that for s∈R, andk∈N\nφtk(exp(z0,sE−(z0))) = exp(φtk(z0),sλ−\ntkE−(tk)).\nThisimplies φτm(exp(z0,sE−(z0))) =φτm(z0) foralls∈R. Inparticular,becausethe\nunstable manifold W−is dense in V, we get that φτm(z) =φτm(z0) for every z∈ V.\nLet us choose a h-periodic point z1∈ V,z1/\\e}atio\\slash=z0, of periodq∈N∗(such a point exists\nby density of h-periodic points in Σ 0). By what we just said, γz0(τm) =γz1(τm).\nWe observe that γ′\nz0(τm) andγ′\nz1(τm) can not be linearly independent, otherwise\nDγz0(τm)hqwould fix pointwise a 2-dimensional space in Tγz0(τm)M, implying that\natγz0(τm),Dhqis trivial or has order 2. A Lorentz isometry being completely de-\ntermined by its first jet at a given point, this situation would lead to h2q=id, a\ncontradiction. We infer that γ′\nz0(τm) =−γ′\nz1(τm), andz1=γz0(2τm). Applying the\nsame argument to a periodic point z2∈ Vdifferent from z0andz1, we get a contra-\ndiction. Interverting the role of W−andW+, the same argument holds if λ+\ntk→0\nfor some sequence tk→τm, and the lemma follows.\n♦\nWe have shown that the Lorentz metric ˜ gon (0,τm)×Σ0extends to a Lorentz\nmetric on ( −ǫ,τm+ǫ)×Σ0. Our next goal is to extend our isometric embedding f\nto a mapf: [0,τm]×Σ0. This will be done thanks to the following general extension\nresult, which is of independent interest.\nProposition 5.8 . —Let(L,˜g)be a Lorentz manifold, and Ω⊂Lan open subset\nsuch that the closure Ωis a manifold with boundary. Assume that the boundary ∂Ω\nis a smooth Lorentz hypersurface of L. If(M,g)is another Lorentz manifold having\nsame dimension as N, and iff: (Ω,˜g)→(M,g)is a one-to-one isometric immersion,\nthenfextends to a smooth isometric immersion f:Ω→M.\nIn the previous proposition, smooth isometric immersion means that f:Ω→M\nadmits a well defined differential Dzf:TzN→Tf(z)Mfor everyz∈Ω, which is\nisometric with respect to ˜ gandg, and varies smoothly with z.\nProof: The main part of the proof is to show the following:26 CHARLES FRANCES\nLemma 5.9 . —Each point x∈∂Ωadmits an open neighborhood Ux⊂Nsuch that\ni) The sets Ux∩ΩandUx:=Ux∩∂Ωare connected.\nii) There exists a smooth injective immersion ˜fx:Ux→Msuch that ˜fxandf\ncoincide on Ux∩Ω.\nProof: We consider, at x, the vector νwhich is normal to Tx(∂Ω), and points\ntoward Ω. We consider γa small geodesic segment starting from xand satisfying\nγ′(0) =ν, as well as a sequence ( xk) of points of γ∩Ω converging to x. Since\nMis compact, we may assume that f(xk) converges to a point y∈M. In small\nneighborhoodsof xandy, we choosetwoorthonormalframe ��elds, whichyield at each\npointsz,z′of those neighborhoods, isometric identifications iz:R1,n−1→(TzL,˜g),\niz′:R1,n−1→(Tz′M,g) (here,R1,n−1stands forn-dimensional Minkowski space).\nObviously, one can choose our orthonormal frame fields such that i−1\nγ(t)(γ′(t)) is a\nconstant vector ξ∈ U. Also, there are U,Vneighborhoods of the origin in R1,n−1\nsuch thatu/ma√sto→exp(z,iz(u),u∈ U, andv/ma√sto→exp(z′,iz′(v)),v∈ V, make sense and are\ndiffeomorphisms on their images. In the trivialization given by the fram e fields, the\nsequence of differentials ( Dxkf) becomes a sequence of matrices ( Ak) in O(1,n−1).\nSincefis an isometry, we have the relation\n(4) f(exp(xk,u)) = exp(f(xk),Ak(u))\nfor everyu∈ U.\nWe can prove the lemma if we show that the sequence ( Ak) is contained in a\ncompact set of O(1 ,n−1). For if it is the case, we may assume Ak→A∞, and\nshrinkingmaybe U, wewillhave Ak(U)⊂ Vforallk∈N. Then, wechoose C ⊂R1,n−1\nan open cone with vertex 0, containing −ξand contained in U. Fork0large enough,\nUx= exp(xk0,ixk0(C)) contains x, and ifCis chosen connected and narrow enough\naround−ξ,Ux∩Ω andUx∩∂Ω are connected. The map fx:Ux→Mgiven\nbyfx(exp(xk0,ixk0(u))) = exp(yk0,iyk0(u)),u∈ Cis a one-to-one immersion which\ncoincides with fonUx∩Ω.\nIt remains to explain why the sequence ( Ak) must be bounded. If not, we apply\ntheKAKdecomposition of O(1 ,n−1) to the sequence ( Ak), and after considering a\nsubsequence we can write Akas a product MkDkNkwithMk→M∞(resp.Nk→\nN∞)inO(1,n−1), andDk=\nλk\n...\nλ−1\nk\n,|λk| → ∞. Weseethatthereexists\na lightlike hyperplane H ⊂R1,n−1(namely the image by N−1\n∞of Span(e2,...,e n))\nwith the following dynamical property: For every u∈ H, there exists uk→usuch\nthat after extracting a subsequence, Ak(uk)→u∞. Moreover, we see that if v/\\e}atio\\slash∈ H,\none can find a sequence of reals sk→0 such that Ak(skv)→v∞/\\e}atio\\slash= 0.\nBecause His lightlike while Tx(∂Ω) has Lorentz signature, one can find a nonzero\nu∈ H∩Usuch thatix(u)/\\e}atio\\slash∈Tx(∂Ω) andix(u) points toward Ω. We choose a sequence\n(uk)inUconvergingto usuchthatAk(uk)tendstou∞(afterextraction). Wecanalso\npicksomev∈ U\\Hsuchthatix(v) pointstowardΩ. Then wecanfind ( sk) asequence\nof reals tending to 0 such that Ak(skv) converges to v∞/\\e}atio\\slash= 0, and (uk). Observe thatLORENTZ DYNAMICS ON CLOSED 3-MANIFOLDS 27\nexp(xk,uk) and exp(xk,uk+skv) belong to Ω for klarge. Now, f(exp(xk,uk)) tends\ntof(exp(x,u)), and relation (4) shows that f(exp(x,u)) = exp(y,u∞). On the other\nhand,f(exp(xk,uk+skv)) should also converge to f(exp(x,u)), because sk→0.\nBut relation (4) says that this sequence actually converges to exp (y,u∞+v∞). Since\nv∞/\\e}atio\\slash= 0, and because we can rescale uandvso thatu∞andu∞+v∞belong to V,\nwe have exp( y,u∞+v∞)/\\e}atio\\slash= exp(y,u∞), and we get a contradiction. ♦\nLemma 5.9 easily provides a smooth extension of f,f:Ω→M, puttingf(x) =\nf(x) ifx∈Ω, andf(x) :=˜fx(x) for every x∈∂Ω. This extension map fis well\ndefined because if Ux∩Ux′/\\e}atio\\slash=∅, then˜fxand˜fx′are equal to fonUx∩Ux′∩Ω, hence\nonUx∩Ux′∩∂Ω. Observe that the relation f∗g= ˜gwhich is available on Ux∩Ω\nmust still hold on Ux∩Ω. This proves that fis an isometric immersion. ♦\n5.5.2. The normal flow at τmrealizes the first return map . — We apply Proposition\n5.8 choosing for Lthe Lorentz manifold (( −ǫ,τm+ǫ)×Σ0,˜g) and for Ω the product\n(0,τm)×Σ0. We geta smooth, orientationpreserving, extension f: ([0,τm]×Σ0,˜g)→\n(M,g) which is an isometric immersion coinciding with fon (0,τm)×Σ0.\nProposition 5.10 . —The extension fmaps{τm}×Σ0diffeomorphically and iso-\nmetrically onto Σ0. In other words, the first return time τrcoincides with τm, and\nφτm: Σ0→Σ0realizes the first return map.\nProof: In the proof, we aregoingto write ˜Στm(resp.˜Σ0) instead of {τm}×Σ0(resp.\n{0}×Σ0). We recall the notations∂\n∂t,˜E−,˜E+from Section 5.3.2. We first show that\nthe restriction of fto˜Στmis one-to-one. Assume for a contradiction that it is not the\ncase. We get two points ˜ z1= (τm,z1) and ˜z2= (τm,z2) on˜Στmsuch thatf(˜z1) =\nf(˜z2). Observethat D˜z1f(T˜z1˜Στm) =D˜z2f(T˜z2˜Στm) because atransverseintersection\nof those two subspaces would not be compatible with injectivity of fon (0,τm)×Σ0.\nLooking at orthogonal subspaces, and because fis an isometric immersion, we get\nD˜z1f(∂\n∂t) =±D˜z2f(∂\n∂t). Again,D˜z1f(∂\n∂t) =D˜z2f(∂\n∂t) would violate the injectivity of\nfon (0,τm)×Σ0. We can reformulate equality D˜z1f(∂\n∂t) =−D˜z2f(∂\n∂t), saying that\nγz1(τm) =γz2(τm), andγ′\nz1(τm) =−γ′\nz2(τm). It follows that z1andz2are in the first\nreturn set Ω r,z2=θr(z1), and the first return time τrequals 2τm. As a consequence,\nwe get for every z∈Ωr, the identity:\nf(τm,z) =f(τm,θr(z)).\nDefining ˜θr(t,z) := (t,θr(z)), this identity yields:\nD(τm,z)f(˜E−) =D˜θr(τm,z))f(D(τm,z)˜θr(˜E−)).\nBecauseθrcommutes with h, there exists α(z)/\\e}atio\\slash= 0 such that D(τm,z)˜θr(˜E−) =\nα(z)˜E−. We thus obtain:\nD(τm,z)f(˜E−) =α(z)D˜θr(τm,z))f(˜E−).\nFor the same reasons, there exists β(z)/\\e}atio\\slash= 0 such that\nD(τm,z)f(˜E+) =β(z)D˜θr(τm,z))f(˜E+).28 CHARLES FRANCES\nGoing back to z=z0,θr(z) =z1, we see that ( D(τm,z1)f)−1◦D(τm,z0)fis a linear\nisometry, preserving the orientation, and sending the direct fram e (˜E−,˜E+,∂\n∂t) at\n(τm,z0) to the frame ( α(z0)˜E−,β(z0)˜E+,−∂\n∂t) at (τm,z1). The isometric condition\nyieldsα(z0)β(z0) = 1 and the orientation-preserving condition yields −α(z0)β(z0).\nThis provides the desired contradiction.\nOnce we know that fis one-to-one in restriction to ˜Στm, we get that f(˜Στm) is\na Lorentz surface of M, to which we can again apply the normal flow. This results\ninto an extension of fto a smooth immersion defined on a domain (0 ,τm+ǫ)×Σ0.\nIff(˜Στm) does not meet Σ 0, it is easily checked that for ǫ >0 small enough, fis\none-to-one on (0 ,τm+ǫ)×Σ0, contradicting the definition of τm.\nWe infer that there exist z1andz2in Σ0such thatf(τm,z1) =f(0,z2). Observe\nthatD(τm,z1)f(T˜Στm) andD(0,z2)f(T˜Σ0) can not intersect transversely, otherwise f\nwould not be one-to-one on (0 ,τm)×Σ0. We can recast this property saying that\nγz1(τm) =z2, andγ′\nz1(τm) is orthogonal to Tz2Σ0. In other words, z1belongs to the\nfirst return set Ω r, and the return time is τr=τm. It means in particular that for\nallz∈Ωr,f(τm,z)∈Σ0. By density of Ω rin Σ0, we finally get that fmaps˜Στm\nisometrically and diffeomorphically onto Σ 0.\n♦\n5.6. End of proof of Theorem 5.2. — Let us just recollect what we did so far.\nFirst, showing that the normal flow φtis defined on ( −ǫ,τm+ǫ) withφτm(Σ0) = Σ0\nimmediately implies that φtis defined for every t∈R. Equivalently, the map fis\ndefined on all of N=R×Σ0.\nNext, Proposition 5.5 implies that ( φτm)∗g0=a(τm)g0, witha(τm)>0. Because\nthe global Lorentz volume of Σ 0must be preserved, we get a(τm) = 1. The trans-\nformationφτmis a Lorentz isometry of (Σ 0,g0) commuting with h: It must be either\n±Idor a linear hyperbolic transformation. The possibility φτm=−Idis ruled out\nby the assumption that ( M,g) is time-orientable.\nIn the following, we denote by Athe transformation φτm. We just showed that\nt/ma√sto→a(t) isτm-periodic, and thanks to Propositions 5.5 and 5.8, we get that f:\n(N,˜g)→(M,g) is an isometric immersion. Let us call ϕ:N→Nthe transformation\nϕ(t,x) = (t+1,A−1x). Thenϕacts isometrically for ˜ g, andf◦ϕ=f. Calling Γ the\ncyclic group generated by ϕ, we finally see that finduces an isometry between Γ \\N\n(endowed with the metric induced by ˜ g) and (M,g). This shows the topological part\nof Theorem 5.2.\nSince Σ 0is a flat torus, the universal cover ( ˜M,˜g) is isometric to R3endowed\nwith the metric dt2+2a(t)dudv. Affine transformations preserving the planes t=t0\nand acting by Lorentz isometries on the Minkowski ( u,v)-plane, provide an isometric\naction of SOL on ( ˜M,˜g). This shows points 2) and 3) of Theorem 5.2.\n6. The local geometry of manifolds with no hyperbolic compon ent\nWe keep going in our study of closed 3-dimensional Lorentz manifolds (M,g), such\nthat Iso(M,g) is not compact. Thanks to sections 4 and 5, we can prove Theorem ALORENTZ DYNAMICS ON CLOSED 3-MANIFOLDS 29\nwhen all the components of the integrability locus Mintare locally homogeneous, or\nwhen there exists at least one hyperbolic component. Looking at th e posibilities for\nthe different components listed in Section 3.5, it only remains to invest igate the case\nwhere all the components are either parabolic or of constant curv ature, and there is\nat least one non locally homogeneous component. This section is devo ted to a careful\ngeometric study of such manifolds, and our aim is to prove the\nTheorem 6.1 . —Let(M,g)be a3-dimensional Lorentz manifold. If all the com-\nponents of Mintare of constant curvature or parabolic, and if (M,g)is not locally\nhomogeneous, then (M,g)is conformally flat.\nRecall that ( M,g) is said to be conformally flat if each sufficiently small open\nneighborhood of Mis conformally diffeomorphic to an open subset of Minkowski\nspace. Hence, Theorem 6.1 tells us that at the conformal level , our structure ( M,g)\nis locally homogeneous. This local information will be decisive to recove r the global\nproperties of ( M,g), both topologically and geometrically, a task that will be carried\nover in Section 8.\n6.1. More on the geometry of parabolic components. — Parabolic compo-\nnents split into two categories, the locally homogeneous ones for wic h the Killing\nalgebra is 4-dimensional, and the others for which it is 3-dimensional.\n6.1.1. Locally homogeneous parabolic components . — The study of locally homoge-\nneous parabolic components was made in [ F2], and can be summarized as follows:\nProposition 6.2 . — [F2, Proposition4.3] LetMbe a component of the integrability\nlocusMintof a3-dimensional Lorentz manifold (M,g), which is locally homogeneous\nand parabolic. Then:\n1. If the scalar curvature of Mis0, the Lie algebra killloc(M)is isomorphic to a\nsemi-direct product R⋉heis,whereheisstands for the 3-dimensional Heisenberg\nLie algebra.\n2. If the scalar curvature is nonzero on M, thenkillloc(M)is isomorphic to\nsl(2,R)⊕R.\nActually, thestatementof[ F2]isslightlymoreprecisesinceitdescribeswhichsemi-\ndirect products R ⋉heiscan occur. However, we won’t need this extra information\nhere. For the sequel, it will be important to notice that in the first ca se of Proposition\n6.2, the Lie subalgebra heiscontains the isotropy algebra at each points, hence acts\nwith 2-dimensional pseudo-orbits (this follows from the computatio ns done in [ F2,\nSection 4.3.3]).\n6.1.2. Parabolic components which are not locally homogene ous. — We investigate\nnow the geometry of parabolic components which are not locally homo geneous.\nProposition 6.3 . —LetM ⊂Mintbe a parabolic component which is not locally\nhomogeneous.\n1. The Lie algebra killloc(M)is isomorphic to the 3-dimensional Heisenberg algebra\nheis.30 CHARLES FRANCES\n2. Thekillloc-orbits on Mare totally geodesic, lightlike surfaces.\n3. The scalar curvature σvanishes on M.\nThe proof of Proposition 6.3 involves quite a bit of computations, tha t we defer to\nAnnex A, at the end of the text. Let us mention an important corolla ry which will\nbe crucial later on.\nCorollary 6.4 . —LetMbe a parabolic component which is not locally homoge-\nneous. Letx∈ M. Then the Isloc-orbit ofxis a submanifold of Mintwhich is closed\ninMint.\nProof: We already know (Theorem 3.1) that the Isloc-orbit ofxis a submanifold Σ\nofMint. WehavetoshowthatΣisclosedin Mint. Wethusconsiderasequence( xk)of\nΣconvergingtoapoint x∞∈Mint. Let ˆxbealift ofxinˆM. Werecallthegeneralized\ncurvature map Dκ:ˆM→ W(see Section 3.1). Let us call w=Dκ(ˆx), andO.wthe\norbit ofwunder the action of O(1 ,2) onW. SinceMis a parabolic component,\nO.wis 2-dimensional and the isotropy at wis a 1-parameter unipotent subgroup of\nO(1,2). Let (ˆxk) be a sequence of ˆMlifting (xk), such that ˆ xk→ˆx∞. Sincexk∈Σ\nfor allk, we have Dκ(ˆxk)∈ O.wfor allk, and in particular Dκ(ˆx∞) belongs to the\nclosureO.w. The action of O(1 ,2) onWis algebraic, hence orbits of O.w\\ O.w\nhave dimension <2. Since there are no 1-dimensional orbits in finite dimensional\nrepresentations of O(1 ,2), we conclude that if Dκ(ˆx∞) belongs to O.w\\ O.w, then\nthe stabilizer of Dκ(ˆx∞) is 3-dimensional. Since ˆ x∞∈ˆMint, this means that the\nisotropy algebra Is(x∞) is 3-dimensional, isomorphic to o(1,2) (Fact 3.2). Let M′\nthe component containing x∞. The points xkbelong to M′forklarge enough, what\nshowskillloc(M′)≃killloc(M)≃heis(3). This contradicts Is(x∞)≃o(1,2).\nWe infer that Dκ(ˆx∞)∈ O.w. Hence, replacing the sequence ˆ xkby ˆxk.pkfor a\nbounded sequence ( pk) ofP, we may assume that Dκ(ˆxk) =wfor allk. By the\ndiscussion following Theorem 3.1, Dκ−1(w)∩ˆMintis a submanifold, the connected\ncomponent of which are killloc-orbits. We conclude that for klarge enough, ˆ x∞and\nˆxkarein the same killloc-orbit. The same is thus true for x∞andxk, and the corollary\nis proved.\n♦\n6.2. Conformal flatness. — Under the standing assumptions stated at the begin-\ning of Section 6, the only non locally homogeneous components in Mare parabolic.\nIt follows from Proposition 6.3 that the scalar curvature of gis constant on each com-\nponent, and equal to zero on the non locally homogeneous ones. As a consequence,\nthe scalar curvature vanishes identically on M, which implies that components of\nconstant sectional curvature are actually flat, hence conforma lly flat. It thus remains\nto show that all parabolic components (locally homogeneous or not) are conformally\nflat. Observe that conformal flatness is given by a tensorial cond ition, namely the\nvanishing of the Cotton-York tensor in dimension 3, so that ( M,g) will be confor-\nmally flat as soon as a dense open subset of Mis. Observe also that the vanishing of\nthe scalar curvature says that locally homogeneous parabolic comp onents are exactlyLORENTZ DYNAMICS ON CLOSED 3-MANIFOLDS 31\nthose described by the first point of Proposition 6.2. This fact, tog ether with Propo-\nsition 6.3 and the remark after Proposition 6.2 reduces the proof of Theorem 6.1 to\nthe following general observation:\nProposition 6.5 . —Let(N,h)be a3-dimensional Lorentz manifold. Assume that\nthere exists on Na Lie algebra nof Killing fields which is isomorphic to heis(3), and\nwhose pseudo-orbits have dimension ≤2. Then all pseudo-orbits are 2-dimensional\nand lightlike, and (N,h)is conformally flat.\nProof: Our hypothesis that all pseudo-orbits have dimension ≤2 implies that the\nisotropy algebra at each point x∈Nis a nontrivial subalgebra of n. This isotropy\nalgebraisthusisomorphicto R,R2orheis(3). Becausethereisnosubalgebraof o(1,2)\nisomorphic to R2orheis(3), the isotropy algebra of nis 1-dimensional at each point,\nand all pseudo-orbits of nhave dimension 2. Let us consider X,Y,Zthree Killing\nfields generating n, and satisfying the relations [ X,Y] =−Zand [X,Z] = [Y,Z] = 0.\nWe are going to look at the subalgebra aspanned by YandZ. Because no subalgebra\nofo(1,2) is isomorphic to R2, pseudo-orbits of ahave dimension 1 or 2. We claim\nthat the open subset Ω where the pseudo-orbits of aare 2-dimensional is dense in\nN. To see this, let us consider ∆ a 1-dimensional pseudo-orbit of a, and letx∈∆.\nThe isotropy, in a, of the point xis spanned by an element U=aY+bZ. There is\nanother vector vector field V=cY+dZsuch thatv:=V(x)/\\e}atio\\slash= 0. Since UandV\ncommute,Uactually vanishes at each point of ∆. Let t/ma√sto→γ(t) be a geodesic for the\nmetrich, satisfying γ(0) =x,h(γ′(0),v)/\\e}atio\\slash= 0, andγ′(0)/\\e}atio\\slash∈R.v. Clairault’s equation\nensures that for t >0 small enough, h(γ′(t),U(γ(t))) = 0 and h(γ′(t),V(γ(t)))/\\e}atio\\slash= 0.\nObserve that U(γ(t))/\\e}atio\\slash= 0, because locally, the zero set of a nontrivial Killing field\non a 3-dimensional manifold is a submanifold of dimension ≤1. We thus get that\nγ(t)∈Ω fort>0 small, ensuring the density of Ω.\nThis density property shows that we will be done if we show that Ω is co nformally\nflat. To this aim, we consider a point x0∈Ω. Since, [Y,Z] = 0 andY,Zspan a 2-\ndimensional space at each point of Ω, there exist local coordinates (x1,x2,x3) around\n(0,0,0) such that Z=∂\n∂x1andY=∂\n∂x2. Because the orbits of nare 2-dimensional,\nXis of the form λ∂\n∂x1+µ∂\n∂x2for some functions λandµ. The bracket relations\n[X,Z] = 0 and [ X,Y] =−Zlead to 0 =∂λ\n∂x1=∂µ\n∂x1=∂µ\n∂x2and∂λ\n∂x2= 1. Hence we\ncan write\nX= (x2+a(x3))∂\n∂x1+b(x3)∂\n∂x2.\nObserve that replacing XbyX−a(0)Z−b(0)Ywon’t affect the bracket relations\nbetweenX,YandZ, so that we will assume in the following that a(0) =b(0) = 0.\nLet us consider a point p= (p1,p2,p3). The vector field U=X−(p2+a(p3))Z−\nb(p3)Yis nonzero and vanishes at p. We compute that at p:\n[U,∂\n∂x1] = 0,[U,∂\n∂x2] =−∂\n∂x1,[U,∂\n∂x3] =−a′(p3)∂\n∂x1−b′(p3)∂\n∂x2.32 CHARLES FRANCES\nSinceUbelongs to n, hence is Killing for the metric h, we infer that the matrix\nA=\n0−1a′(p3)\n0 0b′(p3)\n0 0 0\nmust be antisymmetric for the Lorentz scalar product hp.\nIt is readily checked that a rank 1 nilpotent matrix never has this pro perty (basically\nbecause exp( tA) would be a nontrivial 1-parameter group in (a conjugate of) O(1 ,2)\nfixing pointwise a 2-plane, which is impossible). We thus infer that the d erivativeb′\nis nowhere 0. Because b(0) = 0, this implies thatb(x3)\nx3(orb′(0) ifx3= 0) is nowhere\n0. The transformation\nϕ: (x1,x2,x3)/ma√sto→(b(x3)\nx3x1,b(x3)\nx3x2−a(x3),x3).\nthus yields a local diffeomorphism fixing the origin. Applying ϕ∗toX,Y,Z, we get\nX=x2∂\n∂x1+x3∂\n∂x2, Y=x3\nb(x3)∂\n∂x2,andZ=x3\nb(x3)∂\n∂x1.\nLet againp= (p1,p2,p3) be a point in our coordinate chart. The vector field\nU=X−p2∂\n∂x1−p3∂\n∂x2vanishes at p, and is a Killing field for hbecauseU=\nX−p2b(p3)\np3Z−b(p3)Y. A straigthforward computation yields\n[U,∂\n∂x1] = 0,[U,∂\n∂x2] =−∂\n∂x1,and [U,∂\n∂x3] =−∂\n∂x2.\nIt follows that the matrix\n0 1 0\n0 0 1\n0 0 0\nmust be antisymmetric with respect to hp.\nThis allows to see that the matrix of hpin the frame (∂\n∂x1,∂\n∂x2,∂\n∂x3) is of the form\n0 0 −β(p)\n0β(p) 0\n−β(p) 0γ(p)\n, withβ(p)>0. Now,ZandYbeing Killing fields for h,\nwe see that βandγonly depend on the variable x3, and we conclude that the metric\nhwrites as :\n−2β(x3)dx1dx3+β(x3)dx2\n2+γ(x3)dx2\n3.\nNow, ifx3/ma√sto→ζ(x3) is a primitive of−γ(x3)\n2β(x3), a change of coordinates\n(x1,x2,x3)/ma√sto→(x1+ζ(x3),x2,x3)\nshows that his locally isomorphic to −2β(x3)dx1dx3+β(x3)dx2\n2, hence is conformally\nflat.\n♦\n7. Geometry on Einstein’s universe\nLorentz conformally flat structures in dimension n= 3 are examples of ( G,X)-\nstructures in the sense of Thurston. In particular, there is a univ ersal space among\nthose structures, called Einstein’s universe Ein3, such that if ( M,g) is Lorentz andLORENTZ DYNAMICS ON CLOSED 3-MANIFOLDS 33\nconformally flat, there exists a conformal immersion δ:˜M→Ein3, which is equiv-\nariant under a representation of π1(M) into Conf( Ein3) (see Section 7.1.3 below).\nThe proof of Theorem A for manifolds ( M,g) satisfying hypotheses of Theorem 6.1\nand with a noncompact isometry group, will rely in a crucial way on the study of\nthis developing map δ. This study will be carried over in the next section 8, and it\nwill require a deeper knowledge of the geometry of Ein3. That’s why we dedicate\nthe present section to studying Ein3in more details. The reader eager to learn more\nabout the geometry of Ein3is refered to [ F1] or [BCDGM ].\n7.1. Basics on Einstein’s universe. — Einstein’s universe is the Lorentz ana-\nlogue of the Riemannian conformal sphere. We recall its construct ion, sticking to\ndimension 3, which is the relevant one for our purpose.\nLetR2,3be the space R5endowed with the quadratic form\nQ2,3(x0,...,x 4) = 2x0x4+2x1x3+x2\n2\nWe consider the null cone\nN2,3={x∈R2,3|Q2,3(x) = 0}\nand denote by /hatwideN2,3the cone N2,3with the origin removed. The projectivization\nP(/hatwideN2,3) is a smooth submanifold of RP4, and inherits from the pseudo-Riemannian\nstructure of R2,3a Lorentz conformal class (more details can be found in [ F1],\n[BCDGM ]). We call the 3-dimensional Einstein universe , denoted Ein3this com-\npact manifold P(/hatwideN2,3) with this conformal structure. One can check that a 2-fold\ncover ofEin3is conformally diffeomorphic to the product ( S1×S2,−gS1⊕gS2).\nThe orthogonal group of Q2,3, isomorphic to O(2 ,3), acts naturally on the 4-\ndimensional projective space, preserving Ein3and its conformal structure. It turns\nout (see Theorem 7.1 below) that PO(2 ,3) is the full conformal group of Ein3. Ob-\nserve that Ein3is homogeneous under the action of PO(2 ,3).\n7.1.1. Photons and lightcones . — It is a remarkable fact of Lorentz geometry that all\nthe metrics of a given conformal class have the same lightlike geodes ics (as sets but\nnot as parametrized curves). In the case of Einstein’s universe, t he lightlike geodesics\nare the projections on Ein3of totally isotropic 2-planes P⊂R2,3(namely planes\nPon whichQ2,3vanishes identically). We will rather use the term photonfor the\nlightlike geodesics of Einstein’s universe. Observe that all photons o fEin3are simple\nclosed curves, and are endowed with a natural class of projective parametrizations.\nGiven a point pinEin3, thelightcone with vertex p, denoted by C(p), is the union\nof all photons containing p. Ifp∈Ein3is the projection of u∈ N2,3, the lightcone\nC(p) is just P(u⊥∩ N2,3). The lightcone C(p) is singular (from the differentiable\nviewpoint) at its vertex p, andC(p)\\{p}is topologically a cylinder. The entire cone\nC(p) has the topology of a 2-torus pinched at p.\n7.1.2. Stereographic projection . — There is for Ein3a generalized notion of stere-\nographic projection, which shows that Ein3is a conformal compactification of the\nMinkowski space.34 CHARLES FRANCES\nLetus call R1,2the space R3endowedwith the quadraticform Q1,2(x,x) = 2x1x3+\nx2\n2. Consider ϕ:R1,2→Ein3given in projective coordinates of P(R2,3) by\n(5) ϕ:x= (x1,x2,x3)/ma√sto→[−1\n2Q1,2(x,x) :x1:x2:x3: 1]\nThenϕis a conformal embedding of R1,2intoEin3, called the inverse stereographic\nprojection with respect to p0:= [e0]. The image ϕ(R1,2) is a dense open set of Ein3\nwith boundary the lightcone C(p0). Observe that this proves the fact (rather hard to\nvisualize): The complement of a lightcone C(p) inEin3is connected.\n7.1.3. Developing conformally flat structures into Einstei n’s universe . — It is a stan-\ndard fact that Einstein’s universe satisfies an analogue of the class ical Liouville’s\ntheorem on the sphere. Namely:\nTheorem 7.1 (Liouville’s theorem for Ein 3). —LetU⊂Ein3be a connected\nnonempty open set. Let f:U→Ein3be a conformal immersion. Then fis the\nrestriction to Uof an unique element of PO (2,3).\nThe existence of the stereographic projection (5), and the tran sitivity of the action\nof PO(2,3) onEin3shows that Ein3is conformally flat. Liouville’s theorem 7.1\nshows that any 3-dimensional, conformally flat Lorentz structure (M,g) is actually a\n(PO(2,3),Ein3)-structure, in the sense of Thurston.\nAs a consequence, for every conformally flat Lorentz structure (M,g), there exists\na conformal immersion\nδ: (˜M,˜g)→Ein3\ncalledthe developing map of the structure. Here, ˜Mis the universal cover of the\nmanifoldM, and ˜gis the lifted metric. This developing map comes with a holonomy\nmorphismρ: Conf(˜M,˜g)→PO(2,3) satisfying the equivariance relation:\n(6) δ◦h=ρ(h)◦δ\navailable for every h∈Conf(˜M,˜g).\n7.2. More geometry on Ein 3. —\n7.2.1. The foliation F∆. — We refer here to the notations introduced in Section 7.\nLetPbe the plane in R2,3spanned by the vectors e0ande1. The form Q2,3vanishes\nidentically on P, hence the projection of PonEin3defines a photon that we will\ndenote by ∆. The open subset obtained by removing ∆ to Ein3will be called Ω ∆.\nGiven a point p∈∆, we consider the lightcone C(p) with vertex p. Since ∆\nis a photon, we have ∆ ⊂C(p). Now, the intersection of C(p) with Ω ∆, namely\nC(p)\\∆ is a lightlike hypersurface of Ω ∆, diffeomorphic to a plane. We call it F∆(p).\nWe now make the observation that in Ein3, there is no nontrivial lightlike triangle,\nnamely if two photons ∆ 1and ∆ 2intersect ∆ transversely at two distinct points,\nthen ∆ 1∩∆2=∅. This is the geometric counterpart of the following algebraic fact:\nInR2,3, there are no 3-dimensional spaces on which Q2,3vanish identically. It follows\nthat ifp/\\e}atio\\slash=p′are points of ∆, C(p)∩C(p′) = ∆, or in other words F∆(p)∩F∆(p′) =∅.\nThis shows that {F∆(p)}p∈∆are the leaves of a codimension 1 lightlike foliation ofLORENTZ DYNAMICS ON CLOSED 3-MANIFOLDS 35\nΩ∆, that we will call F∆. The space of leaves of F∆is naturally identified with ∆.\nForx∈Ω∆, we will adopt the notation F∆(x) for the leaf of F∆containingx.\n7.2.2. Symetries of the foliation F∆. — Letuscall G∆thestabilizerof∆inPO(2 ,3).\nObviously, G∆preserves Ω ∆and the foliation F∆.\nIt is readily checked that this group is a semi-direct product\nG∆≃PGL(2,R)⋉N,\nwhere the group Nis isomorphic to the 3-dimensional Heisenberg group Heis(3), and\ngiven in PO(2 ,3) by the matrices:\n(7) N(x,y,z) :=\n1 0−x−(z+xy)−x2\n2\n0 1−y−y2\n2z\n0 0 1 y x\n0 0 0 1 0\n0 0 0 0 1\nx,y,z∈R.\nThe factor PGL(2 ,R) is the subgroup of PO(2 ,3) corresponding to matrices:\n(8) RA:=\nA0 0\n0 1 0\n0 0A\ndet(A)\nA∈PGL(2,R).\nObserve that the action of G∆on the space of leaves of F∆corresponds to the\nprojective action of the factor PGL(2 ,R) on ∆. The subgroup S∆⊂G∆which\npreserves individually all the leaves of F∆is a semi-direct product\nS∆≃R∗\n+⋉N,\nwhere the factor R∗\n+corresponds to matrices:\n(9) Rλ:=\nλ0\n0λ\n1\n1\nλ0\n01\nλ\n, λ∈R∗\n+.\nLet us end this algebraic parenthesis by giving more details about the action of\nthe groupN. Obviously, Nfixes the point p0= [e0]∈Ein3, hence if we perform a\nstereographic projection given by formula (5), the group Nbecomes a subgroup of\nconformal transformations of R1,2. These transformations are affine, given by\n(10) N(x,y,z) =\n1−y−y2\n2\n0 1y\n0 0 1\n+\nz\nx\n0\n36 CHARLES FRANCES\nInside the group N, there is a 2-dimensional subgroup of translations, denoted T,\ncomprising all transformations of the form\nT(x,z) :=Id+\nz\nx\n0\n, x,z∈R.\nIn PO(2,3), such transformations take the matricial form:\nT(x,z) =\n1 0−x−z−x2\n2\n0 1 0 0 z\n0 0 1 0 x\n0 0 0 1 0\n0 0 0 0 1\n.\nFrom this matricial representation, it is straigthforward to check the following\nFact 7.2 . —1. The set of fixed points for the action of the group N(resp.T) on\nEin3is exactly ∆.\n2. For every x∈Ω∆, theN-orbit ofxis the leafF∆(x)\n3. The action of Tis free on Ω∆\\F∆(p0), and orbits of Ton this open set coincide\nwith leaves of F∆.\n4. OnF∆(p0), orbits of Tare1-dimensional and coincide with the photons of\nC(p0), withp0removed.\nIn the rest of the paper, we will adopt the notations g∆,s∆,n,tfor the Lie subal-\ngebras of o(2,3) corresponding to the groups G∆,S∆,N,T.\n7.3. Standard Heisenberg algebras in o(2,3). —The Lie group Nadmits a Lie\nalgebran⊂o(2,3) that will be called the standard Heisenberg algebra ofo(2,3).\nIt is not true that all subalgebras of o(2,3) which are isomorphic to heis(3) are\nconjugated to the standard algebra n. There is however the following useful charac-\nterization:\nLemma 7.3 . —Leth⊂o(2,3)be a Lie subalgebra isomorphic to heis(3), andH⊂\nPO(2,3)the corresponding connected Lie subgroup. Assume there exi sts a nonempty\nopen set of Ein3where the orbits of Hare2-dimensional and lightlike. Then his\nconjugated in PO(2,3)to the standard Heisenberg algebra n.\nProof: As any solvable Lie subalgebra of o(2,3),hmust leave invariant a line R.v\nor a 2-plane PinR2,3. Such a vector vcan not be timelike or spacelike, otherwise\nthe decomposition R2,3=R.v⊕v⊥would lead to an embedding of hin one of the Lie\nalgebras R⊕o(1,3) orR⊕o(2,2)≃R⊕sl(2,R)⊕sl(2,R). But none of those algebras\ncontains a subalgebra isomorphic to heis(3). Similarly, Pcan not be of signature\n(+,+), (+,−) or (−,−), otherwise the decomposition R2,3=P⊕P⊥would lead to\nan embedding of hintoo(2)⊕o(2,1)≃R⊕o(1,2),o(1,1)⊕o(1,2)≃R⊕o(1,2)\noro(2)⊕o(3)≃R⊕o(3). One checks as above that this is not possible. The only\npossibilities are then:LORENTZ DYNAMICS ON CLOSED 3-MANIFOLDS 37\na) The vector vis lightlike or Phas signature (0 ,+) (resp. (0 ,−)). This means that\nHhas a global fixed point in Ein3, that we can assume to be p0after conjugating\ninto PO(2,3).\nb) The form Q2,3vanishes identically on P, in which case Hhas an invariant photon\nthat we can assume to be ∆.\nWe firstdeal with case a). After consideringa stereographicprojectionofpole p0,h\nbecomes a subalgebra of Conf(R1,2)≃(R⊕o(1,2))⋉R3. Here the normal subalgebra\nR3integrates into the subgroup of translations. Let us consider the projectionπ:\n(R⊕o(1,2))⋉R3→o(1,2). Since o(1,2) does not have any subalgebra isomorphic to\nheis(3) orR2, the rank of π|his 0 or 1. Because R⋉R3(withRacting by homothetic\ntransformations on R3) does not contain a copy of heis(3), this rank is actually 1,\nhence the kernel of π|h, denoted a, has dimension 2 in h, hence is abelian. The only\nsubalgebras isomorphic to R2inR⋉R3are actually contained in R3.\nOur hypothesis on the orbits of the group Himplies that the translation vectors in\naspan a lightlike hyperplane, hence after conjugating into Conf( R1,2), we can assume\na=t, wheretwas introduced at the end of Section 7.2.2.\nThe first point of Fact 7.2 implies that since Hcentralizes t,H⊂G∆. The\nhypothesis on the orbits of Hsays that on some open set, H-orbits and T-orbits\ncoincide. Points 3 and 4 of Fact 7.2 imply that the action of Hon ∆ is trivial on\nsome nonempty open set, hence trivial. This yields H⊂S∆. Because the normalizer\noftinS∆isN, we finally get H=N, and the proof is completed in this case.\nConsider now case b). Because Hleaves ∆ invariant, His a subgroup of G∆. As\nabove, we can look at the morphism\nπ:g∆≃(R⊕sl(2,R))⋉n→sl(2,R).\nThe same arguments as above show that the kernel of π|his a 2-dimensional abelian\nLie subalgebra a⊂h. Observe that a⊂s∆, and the only 2-dimensional abelian\nsubalgebras of s∆are included in n. After conjugating into G∆, we can ensure a=t.\nWe then finish the proof as in the first case.\n♦\n8. The global geometry of manifolds without hyperbolic comp onents\nThis section is devoted to establishing Theorem A in the only remaining c ase to be\nstudied, namely that of closed 3-dimensional Lorentz manifolds ( M,g) which are not\nlocally homogeneous, such that Mintdoes not admit any hyperbolic component, and\nwith a noncompact isometry group Iso( M,g). By Theorem 6.1, those manifolds are\nconformally flat, and we consider δ:˜M→Ein3the corresponding developing map.\nWe also recall the holonomy morphism ρ: Conf(˜M,˜g)→PO(2,3).\nWhat we will really show in this section is:\nTheorem 8.1 . —Let(M,g)be a closed, orientable and time-orientable, 3-\ndimensional Lorentz manifold, such that Iso(M,g)is noncompact. We assume\nthat(M,g)is not locally homogeneous, and that Mintdoes not admit any hyperbolic\ncomponent. Then:38 CHARLES FRANCES\n1. The manifold Mis homeomorphic to a 3-torus, or a parabolic torus bundle T3\nA.\n2. There exists a metric g′=e2σgin the conformal class of gwhich is flat, and\nwhich is preserved by Iso(M,g).\n3. There exists a smooth, positive, periodic function a:R→(0,∞)such that the\nuniversal cover (˜M,˜g)is isometric to R3endowed with the metric\n˜g=a(v)(dt2+2dudv).\n4. There is an isometric action of Heison(˜M,˜g).\nThis result clearly implies Theorem A in the case under study. Its proo f will be the\naim of Sections 8.1 to 8.5 below. In all those sections, ( M,g) satisfies the asumptions\nof Theorem 8.1.\n8.1. Approximately stable foliation on M. —So far, we saw that ( M,g) is an\nagregate of (possibly infinitely many) components, the local geome try of which we\nunderstand fairly well. But we need a global object which allows to und erstand how\nthose component fit together. This global object turns out to be a foliation provided\nby the noncompactness of Iso( M,g) as follows.\nConsider a sequence ( fn) in Iso(M,g) which tends to infinity, and call AS(fn) the\nsubset ofTMcomprising all vectors v∈TMfor which there exists a sequence ( vn) in\nTMconverging to v, such that |Dfn(vn)|is bounded (where |.|is the norm associated\nto an auxiliary Riemannian metric on M). In [Z3], A. Zeghib proved the following\nresult :\nTheorem 8.2 . — [Z3, Theorem 1.2] Let(M,g)be a closed Lorentz manifold, and\n(fn)a sequence of Iso(M,g)tending to infinity. Replacing if necessary (fn)by a\nsubsequence, the set AS(fn)is a codimension 1, lightlike, Lipschitz distribution in\nTM, which integrates into a codimension 1, totally geodesic, lightlike foliation.\nThe foliation given by Theorem 8.2 is called the approximately stable foliation of\n(fn).\nIn the particular case of a 3-dimensional manifold, codimension 1, to tally geodesic,\nlightlike foliations have very nice properties that were studied by A. Z eghib in [ Z5].\nHe proved in particular:\nTheorem 8.3 . — [Z5, Theorem 11] Let(M,g)be a3-dimensional closed Lorentz\nmanifold. Let Fbe aC0, codimension 1, totally geodesic, lightlike foliation of M.\nThen:\n1. A leaf of Fis homeomorphic to a plane, a cylinder or a torus.\n2. The foliation Fhas no vanishing cycles.\nWe now choose a sequence ( fn) tending to infinity in Iso( M,g), and after consid-\nering a suitable subsequence, we denote the approximatively stable foliation of ( fn)\nbyF. By Theorem 8.3, the leaves of Fare planes, cylinders or tori. Our main aim,\nand a decisive step to prove Theorem 8.1 will be to show that all leaves of Fare\ntori, yielding the torus bundle structure of M. It will be convenient in the sequel to\nconsider the lift of Fto the universal cover ˜M. We will call ˜Fthis lifted foliation.LORENTZ DYNAMICS ON CLOSED 3-MANIFOLDS 39\n8.2. Leaves of Fcoincide with killloc-orbits on non locally homogeneous\ncomponents. — The aim of this section is to show:\nProposition 8.4 . —LetM ⊂Mintbe a component which is not locally homoge-\nneous. Then killloc-orbits in Mcoincide with leaves of F. In particular, any such\ncomponent is saturated by leaves of F.\n8.2.1. The pullback foliation ˜F∆and its geometric properties . — We consider the\ndeveloping map δ:˜M→Ein3, and take the pullback by δof the foliation F∆\ndefined in Section 7.2.1. We get in this way a (singular) foliation ˜F∆on˜M. Actually,\n˜F∆is a genuine foliation by lightlike hypersurfaces on the open set ˜Ω∆=δ−1(Ω∆).\nSingularities occur on the complement of ˜Ω∆in˜M, namely ˜∆ :=δ−1(∆). This\nsingular set is either empty (in which case ˜F∆is a regular foliation on ˜M), or a\n1-dimensional lightlike manifold.\nLet us emphasize the fact that a priori, we don’t have any invariance property\nfor˜F∆under the action of the fundamental group π1(M). In particular, there is no\nreason for ˜F∆to define any foliation on M.\nIn the following, we will identify o(2,3) with the Lie algebra of conformal vector\nfields ofEin3(see Theorem 7.1). We can pull back the vector fields of the Lie algeb ra\nnby the developing map δ:˜M→Ein3, getting a Lie algebra ˜nof conformal vector\nfields on ( ˜M,˜g). By Fact 7.2, the pseudo-orbits of ˜ncoincide with the leaves of ˜F∆.\n8.2.2. Foliation ˜F∆andkillloc-orbits. — A first important feature of the foliation ˜F∆\nis its relation to the killloc-orbits in ˜M. To see this, let us fix a parabolic component\nM ⊂Mintas in Proposition 8.4. We lift this component to the universal cover ˜Mof\nM, and call ˜Ma connected component of this lift.\nLemma 8.5 . —Replacing if necessary the developing map δbyg◦δ, for some\ng∈PO(2,3), the restriction to ˜Mof any vector field of ˜nis a Killing field for ˜g.\nConversely, any local Killing field defined on some open set U⊂˜Mis the restriction\nof a vector field in ˜n.\nProof: We pickx∈˜MandUa 1-connected neighborhood of xon which the\ndevelopingmap δisinjective. If Uischosensmallenough, theLiealgebra kUofKilling\nfields onUcoincides with killloc(x). Einstein’s universe Ein3satisfies a generalization\nof Liouville’s theorem: Any conformal Killing field defined on some conne cted open\nset ofEin3is the restriction of a global one. Thus the algebra δ∗(kU) is a subalgebra\nofo(2,3) isomorphic to the 3-dimensional Heisenberg algebra. The pseudo -orbits of\nδ∗(kU) onδ(U) are 2-dimensional and lightlike by the second point of Proposition 6.3 .\nLemma 7.3 applies and says that post-composing δby an element of PO(2 ,3), we\nmay assume δ∗(kU) =n. We thus get that any Killing field on Uis the restriction of\na vector field of ˜n. Since˜nandkUhave same dimension, the restriction to Uofany\nvector field X∈˜nmust be Killing. Let us call Y=X|U. Let us pick an arbitrary\ny∈˜M, and draw a simple curve γjoiningytoxinside˜M. Let us consider V\na 1-connected open neighborhood of γcontained in ˜Mand containing U. Because\nthe dimension of killloc(z) is constant on ˜M, the vector field Ycan be extended by40 CHARLES FRANCES\nanalytic continuation to a Killing field (still denoted Y) defined on V. But now, Y\nandX|Vare two conformal Killing fields on V, which coincide on U. They must\nthen coincide on V, showing that Xis Killing in a neighborhood of y. The same\ndimentional argument as above shows that conversely, a Killing field d efined in a\nsufficiently small neighborhood of yis the restriction of a field in ˜n.\n♦\nCorollary 8.6 . —The component ˜Mis included in ˜Ω∆.\nProof: Points of ˜∆ are singularities for the vector fields of ˜n. Hence if a point x∈˜∆\nbelongs to ˜M, Lemma 8.5 will provide a Lie subalgebra of Killing fields vanishing at\nxand isomorphic to heis(3). The isotropy representation then yields an embedding\nof Lie algebras heis(3)→o(1,2). This is impossible. ♦\nWe conclude this paragraph with the following important lemma.\nLemma 8.7 . —Letxbe a point of ˜M, and˜F∆(x)the leaf of ˜F∆throughx. Then\n˜F∆(x)is included in ˜M, and coincides with the killloc-orbit ofx.\nProof: Let us consider a leaf ˜F∆having a nonempty intersection with ˜M. Assume\nfor a contradiction that V=˜F∆∩˜Mis not all of ˜F∆. It means that Vis an open\nsubset of ˜F∆having a nontrivial boundary ∂Vinside˜F∆. Of course, ∂V⊂∂˜M(this\nlast boundary is taken in ˜M). Since ˜F∆is a pseudo orbit of ˜n, it is easy to show that\nthere exists y∈∂V, a vector field X∈˜nand a point x∈Vsuch that the local orbit\nt/ma√sto→φt\nX.xis defined on [0 ,1],φt\nX.xbelongs toVfort∈[0,1/2) butφ1/2\nX.x∈∂V. We\ndenote by ˆRthe bundle of frames on ˜M, and exceptionnaly in this proof, we adopt\nthe notation ˆMfor the bundle of orthonormal frames of ˜M(and not of M). The\nlocal action of φt\nXlifts naturally to ˆR. We pick ˆx∈ˆMin the fiber of x, and look\nat the orbit t/ma√sto→φt\nX.ˆxinˆR. BecauseXis Killing on ˜M(Lemma 8.5), this orbit is\ncontained in ˆMfort∈[0,1/2), and the same is true for t∈[0,1/2] because ˆMis\nclosed in ˆR. We now look at the generalized curvature map Dκ:ˆM→ W, and its\nderivative that we see as a map DDκ:ˆM→Hom(g,W). The map t/ma√sto→DDκ(φt\nX.ˆx)\nmakes sense for t∈[0,1/2], and is constant on this interval because Xis Killing on\n˜M. In particular, the kernel of DDκ(φt\nX.ˆx) is the same for all t∈[0,1/2], hence the\nrank ofDκis the same at ˆ xand atφ1/2\nX.ˆx. We get that the rank of Dκatφ1/2\nX.xis\n3, but we already observed in the proof of Lemma 3.6, that all points whereDκhas\nrank 3 are contained in ˜Mint. We inferφ1/2\nX.x∈˜Mint, contradicting φ1/2\nX.x∈∂M.\nThe last part of the lemma follows easily. Lemma 8.5, together with Cor ollary 8.6\nensures that for every x∈˜M, thekillloc-orbit ofxcoincides with ˜F∆(x)∩˜M. But\n˜F∆(x)∩˜M=˜F∆(x) by the first part of the proof. ♦\n8.2.3. Proof of Proposition 8.4 . — We keep the notations of the previous paragraph.\nWe also lift the foliation Fto a foliation ˜Fon the universal cover ˜M. For eachx∈˜M,\nwe denote by ˜F(x) the leaf of ˜Fcontainingx.\nThanks to Lemma 8.7, Proposition 8.4 will be a simple consequence of:\nLemma 8.8 . —For everyx∈˜M, one has ˜F∆(x) =˜F(x).LORENTZ DYNAMICS ON CLOSED 3-MANIFOLDS 41\nProof: We workon ˜M, and weconsiderthe two1-dimensionallightlikedistributions\n˜D∆=T˜F∆⊥and˜D=T˜F⊥. Our aim is to show that those distributions coincide on\n˜M. For every x∈˜M, let us introduce the set C(x), comprising all lightlike directions\nu∈P(Tx˜M) such that there exists a lightlike totally geodesic hypersurface Σ t hrough\nx, withTxΣ⊥=u. Let us recall a key observation made in [ Z4]:\nLemma 8.9 . — [Z4, Proposition 2.4] If the set C(x)spansTx˜M, then the sectional\ncurvature at xis constant.\nIf at some point xof˜M, the directions ˜D∆and˜Ddo not coincide, then Lemma\n8.9 implies that they must both be fixed by the local flow generated by the isotropy\nalgebraIs(x). But a nontrivial parabolic 1-parameter flow in O(1 ,2) has only one\ninvaraint direction: Contradiction.\nWe are thus led to ˜F∆(x)⊂˜F(x) for every x∈˜M. This inclusion can not be\nproper, otherwise ˜F∆could be extended in a smooth way to points of ˜∆.♦\nRemark 8.10 . —The previous proof shows actually that on a non locally homog e-\nneous parabolic component ˜M, any lightlike, totally geodesic, codimension 1foliation\nhas to coincide with ˜F.\n8.3. Existence of toral leaves for F. —We keep going in the geometric study\nof the foliation F(and simultaneously in the understanding of the developing map\nδ), by proving the existence of one toral leaf for F.\n8.3.1. Injectivity properties of δ. — We keep the notationsofSection 8.2.1, and recall\nthe open set ˜Ω∆where the foliation ˜F∆is defined.\nLemma 8.11 . —Letx∈˜Ω∆, and assume that the leaves ˜F(x)and˜F∆(x)coincide.\nThenδis injective in restriction to ˜F(x).\nProof: Considering if necessary a finite cover of M(what won’t change ˜M), there\nexistsWa vector field on M, tangent to Fand satisfying g(W,W) = 1. We lift W\nto a vector field ˜Won˜M, which is tangent to ˜F. Notice that ˜Wis complete. By\nassumption, ˜Wis tangent to ˜F(x). The proof follows now closely the arguments of\n[Z2, Proposition 6.5]. Let us ˜Dthe 1-dimensional foliation integrating ˜F⊥. A funda-\nmental remark made in [ Z5, Proposition 2] is that because ˜F(x) is totally geodesic,\nany vector field Utangent to ˜Dacts as a Killing field on the degenerate surface\n(˜F(x),˜g). It follows easily that if γandηare two curves on ˜F(x) parametrized by\n[0,1], such that γ(0) andη(0) belong to the same leaf of ˜D, and ifγandηhave the\nsame length with respect to ˜ g, thenγ(1) andη(1) also belong to a same leaf of ˜D.\nApplying this remark to the integral curves of the flow {ψt}generated by ˜W, we\nobtain that ψtmaps leaves of ˜Dto leaves of ˜D. Given ˜D0a leaf of ˜Din˜F(x), the\nunionU(˜D0) =/uniontext\nt∈Rψt(˜D0) is open in ˜F(x), and two such open sets either coincide,\nor are disjoint, so that ˜F(x) =U(˜D0). By hypothesis, the leaves ˜F∆(x) and˜F(x)\ncoincide, so that the developing map δsends˜F(x) toF∆=F∆(δ(x))⊂Ein3. Let\nγ:I→˜F(x) be an injective parametrization of the leaf of ˜Dthroughx. We observe42 CHARLES FRANCES\nthat for every s∈I,δis injective on the curve t/ma√sto→ψt(γ(s)), because in F∆, there is\nno closed curve transverse to photons of F∆. Also, for every t∈R,δis injective in\nrestriction to s/ma√sto→ψt(γ(s)), because no photon in F∆is closed. The injectivity of δ\non˜F(x) follows. ♦\n8.3.2. The group Iso(M,g)is not a torsion group . — Ournoncompactnesshypothesis\non the groupIso( M,g) does not prevent a prioriIso(M,g) from being a torsion group.\nIn particular, we still don’t know if there exists a single element h∈Iso(M,g) such\nthat{hk}is infinite discrete. The aim of this paragraph is to show it is indeed the\ncase, and to prove the stronger statement:\nProposition 8.12 . —LetM ⊂Mintbe a component which is not locally homoge-\nneous. Let F⊂ Mbe a leaf of Fcontaining at least one recurrent point. Let SFbe\nthe stabilizer of FinIso(M,g). There exists h∈SFsuch that the group {hk}is not\nrelatively compact in Iso(M,g).\nRecall (Proposition 8.4) that non locally homogeneous components o fMintare\nsaturated by the leaves of F.\nThe proof of Proposition 8.12 will require the intermediate lemmas 8.13 and 8.14\nbelow. We lift Ftoaleaf ˜F⊂˜MandcallS˜Fthe stabilizerof ˜Fin Iso(˜M,˜g). Observe\nthatS˜Fprojects surjectively on SFunder the epimorphism Iso( ˜M,˜g)→Iso(M,g).\nLemma 8.13 . —For every leaf F⊂ Mcontaining recurrent points, the groups SF\nandS˜Fare closed, noncompact subgroups of Iso(M,g)andIso(˜M,˜g)respectively.\nProof: We first prove that SFis closed in Iso( M,g). If (fk) is a sequence of\nSFwhich converges to f∞∈Iso(M,g), then for kvery large, f−1\nkf∞belongs to the\nidentity component Isoo(M,g). BecauseFcoincides with a killloc-orbit of M(Lemma\n8.8), we thus have f−1\nkf∞(F) =F, which in turns implies f∞(F) =fk(f−1\nkf∞(F)) =\nfk(F) =F.\nLet us now check that SFis noncompact. By assumption on F, there is a recur-\nrent pointxinF. It means that there exists a sequence ( fk) tending to infinity in\nIso(M,g) such that fk(x)→x. Because the Isloc-orbit ofxis a 2-dimensional sub-\nmanifold, the connected components of which are killloc-orbits (see Theorem 3.1), we\nget thatfk(x)∈Fforklarge enough. In particular, SFis a noncompact subgroup\nof Iso(M,g).\nThe corresponding assertions on S˜Fare then straigthforward. ♦\nLemma 8.14 . —LetF⊂ Mbe a leaf of F, and˜Fa lift ofFto˜M.\n1. The holonomy morphism ρmaps the group S˜Finto the group S∆. In particular,\nany element of S˜Fleaves invariant the leaves of ˜Fwhich are sufficiently close\nto˜F.\n2. The morphism ρ:S˜F→S∆is injective and proper.\nProof: We heavily use the notations introduced in Section 7.2. We choose a\ntransversal I⊂˜Mto the foliation ˜F, that cuts ˜Fatx. We assume that Iis smallLORENTZ DYNAMICS ON CLOSED 3-MANIFOLDS 43\nenough, sothat δsendsIinjectivelyonatransversal JofF∆. Shrinking Iifnecessary,\nJmeets each leaf of F∆at most once, so that by Lemma 8.11, Imeets each leaf of ˜F\nat most once. We call Vthe open subset obtained by saturating Iby leaves of ˜F. By\nwhat we just said, Iis the space of leaves of V, and the map ϕ:=π∆◦δ:I→∆ gives\nan identification of IwithJ′=π∆(J), the space of leaves of the foliation induced by\nF∆onδ(V). Under this identification, the point xis sent to a point p∈J′.\nLet˜hbe an element of S˜F. Its action on the space of leaves of ˜Fyields a germ h\nof diffeomorphism of Ifixingx. The equivariance relation δ◦˜h=ρ(˜h)◦δshows that\nρ(˜h) permutes leaves of F∆nearδ(˜F). In particular, ρ(˜h) mapsJ′to an interval of\n∆ containing p. We infer that ρ(˜h) preserve ∆, what yields ρ(˜h)∈G∆. Moreover,\ndenotingl:G∆≃PGL(2,R)⋉N→PGL(2,R), we get the equivariance relation\nϕ◦h=l(ρ(˜h))◦ϕ. Now,l(ρ(˜h)) acts as an element of PGL(2 ,R) on ∆, admitting\npas fixed point. We know the local dynamics of a M¨ obius transformat ion around\none of its fixed points: If l(ρ(˜h)) is nontrivial, we can choose q∈J′,q/\\e}atio\\slash=p, such\nthatl(ρ(˜hk))(q) belongs to J′for allk≥0, and lim k→∞l(ρ(˜hk))(q) =p. This means\nthat if˜F′is a leaf corresponding to ϕ−1(q), the iterates ˜hk(˜F′) will accumulate on ˜F.\nBut˜F′is akillloc-orbit by Proposition 8.4, and closeness of the Isloc-orbit of ˜F′in\n˜Mint(Corollary 6.4) says that ˜Fand all the hk(˜F′),k∈N, belong to the same Isloc-\norbit. This accumulation phenomenon then contradicts the fact th at Isloc-orbits are\nsubmanifolds in ˜Mint(see Theorem 3.1). We conclude that l(ρ(˜h)) is trivial, which\nimplies that ρ(˜h)∈S∆. Moreover, his trivial, which means that all leaves of ˜Fclose\nto˜Fare left invariant by ˜h.\nWe now prove the second point of the Lemma. Let ˜h∈S˜Fsuch thatρ(h) =id.\nEquivariance relation δ◦˜h=ρ(˜h)◦δ, together with Lemma 8.11, shows that the\naction of ˜hon˜F. The following fact then implies ˜h=id.\nFact 8.15 . —LetNbe a Lorentz manifold and Σ⊂Na lightlike hypersurface.\ndenote bySΣthe stabilizer of ΣinIso(N). Then the restriction map r:SΣ→\nHomeo(Σ) is injective and proper.\nProof: The proof relies on the fact that the map, which to an isometry asso ciates\nits 1-jet at a given point, is injective and proper, and that restrict ing elements of\nO(1,n−1) to a lightlike hyperplane is also injective and proper. Details can be f ound\nin [Z2, Prop 3.6], for instance. ♦\nProperness of the map ρ:S˜F→S∆follows the same lines. If ( ˜hk) is a sequence of\nS˜Fsuch thatρ(˜hk) is relatively compact in S∆. Thenρ(˜hk)|δ(˜F)is relativelycompact,\nhence the retriction of ˜hkto˜Fis relatively compact by Lemma 8.11. Fact 8.15 yields\nthat (˜hk) is relatively compact in S˜F.\n♦\nWe can now proceed to the proof of Proposition 8.12. We know from T heorem 8.3,\nthat the leaves of Fare discs, cylinders or tori, and there are no vanishing circles. It\nmeans that the leaf ˜Fis a disc, and the stabilizer Γ Fof˜Finπ1(M) is either trivial,\nor a discrete subgroup isomorphic to ZorZ2. On the other hand, we also know that\n˜SF/ΓFis noncompact, because of Lemma 8.13.44 CHARLES FRANCES\na)Case where Fis a disk. We choose a nontrivial ˜h∈S˜F. It restricts to a nontrivial\ntransformation of ˜F(Fact 8.15), hence ρ(˜h) is a nontrivial element of S∆, by\nLemma 8.14. Every nontrivial element of S∆generates an infinite discrete group.\nIn particular, {ρ(˜h)k}k∈Zis not relatively compact in S∆. Second point of Lemma\n8.14 says that {hk}is not relatively compact in Iso( ˜M,˜g). Fact 8.15 thus implies\nthat{˜hk\n|˜F}is not relatively compact in Homeo( ˜F), hence the same is true for\n{hk\n|F}, because the projection π:˜F→Fis a diffeomorphism in the case we are\nconsidering. Finally, {hk}is not relatively compact in Iso( M,g).\nb)Case where Fis a cylinder. Because Fdoes not have vanishing cycles, Γ F:=\nS˜F∩Γ is nontrivial, generated by a single element γ. After considering an index\n2 subgroup of S˜F, we may assume that γis centralized by all elements of S˜F. We\nobserve that ρ(γ) is nontrivial by Lemma 8.14, and consider its centralizer in S∆.\nTwo cases can then occur:\n- The group ρ(S˜F) is contained in a 1-paramater subgroup of S∆. In this case,\nρ(S˜F)/<ρ(γ)>is relatively compact in S1. This implies that ( SF)|Fis relatively\ncompact, hence SFis relatively compact in Iso( M,g) (again Fact 8.15). This is\nruled out by Lemma 8.13.\n- If we are not in the previous case, we can find ˜h∈S˜Fsuch that the group\ngeneratedby ρ(˜h) andρ(γ) isdiscreteisomorphicto Z2. Asabove, applyingsecond\npoint of Lemma 8.14 and Fact 8.15, one gets that ˜hprojects to h∈SF, such that\n{hk}is infinite discrete in Iso( M,g).\nc)Case where Fis a torus. This time, Γ Fis isomorphic to Z2and generated by γ1\nandγ2. Lemma 8.14 ensures that τ1:=ρ(γ1) andτ2:=ρ(γ2) generate a discrete\nsubgroup of S∆isomorphic to Z2. Such a subgroup must be included in N, and\nafter conjugating ρintoG∆(what amounts to post-compose δby some element\nofG∆), we have that < τ1,τ2>⊂T. We must have ρ(S˜F)⊂NbecauseS˜F\nnormalizes Γ F, and because SFis noncompact, ρ(S˜F)/\\e}atio\\slash⊂T. Picking ˜h∈S˜Fsuch\nthatρ(˜h)/\\e}atio\\slash∈T, we get an element h∈SFwhich, by similar arguments as above,\ngenerates an infinite discrete group {hk} ⊂Iso(M,g).\n8.3.3. Existence of a toral leaf . — We now consider an element h∈Iso(M,g) given\nby Proposition 8.12, namely {hk}is not relatively compact in Iso( M,g). Theorem 8.2\nprovides an approximately stable foliation Fhassociated to a subsequence of {hk},\nand since all what we did before did not assume anything special on F, we can decide\nthat now F=Fh.\nProposition 8.16 . —Every leafFofFwhich is included in Mis a torus.\nLetFbe a leaf of Fincluded in M, such that almost every point of Fis recurrent\nfor{hk}. We liftFto˜F⊂˜M, and we will also assume that δ(˜F) is not included in\nthe leafF∆(p0) (see Fact 7.2). Observe that since Mis closed, Poincar´ e recurrence\nensures that almost every point of ( M,g) is recurrent for {hk}. It follows that for\nalmost every leaf of F, almost every point is recurrent (leaves of Mcoincide with\nkillloc-orbits hence are locally closed, so there is nothing tricky in desinteg rating the\nvolume form in Malong those leaves). Hence almost every leaf of FinMis anFLORENTZ DYNAMICS ON CLOSED 3-MANIFOLDS 45\nwith the properties above. We claim that Fis a torus. To see this, we lift hto an\nelement˜h∈S˜F, and our assumption is that the set of points in ˜F, which arerecurrent\nunder the group <˜h,ΓF>have full measure in ˜F. By Lemma 8.11, almost all points\nofδ(˜F) must be recurrentfor the group ρ(<˜h,ΓF>). We now goback to the analysis\nmade at the end of Section 8.3.2. If Fis not a torus, we are in cases a) orb) of this\ndiscussion. In case a), ΓFis trivial. The action of ρ(˜h) onδ(˜F) is conjugated to\nthat of an affine transformation on (an open subset of) the plane ( see Section 7.2 and\nformula (10)). The set of recurrent points of ρ(˜h) has thus zero measure on δ(˜F). A\ncontradiction.\nIn caseb), we saw that the group ρ(<˜h,ΓF>) is conjugated to a lattice in the\nclosed subgroup T. Since by assumption, δ(˜F) is not included in the leaf F∆(p0),\npoint 3) of Fact 7.2 shows that Thas no recurrent point on δ(˜F). We reach a new\ncontradiction.\nThe arguments above show that almost all leaves F⊂ Mare tori. Now, for a\ncodimension 1 foliation on a closed manifold, the union of all compact lea ves is itself\ncompact (see [ God, Chap II, Corollary 3.10]). Proposition 8.16 follows.\n8.4. All leaves of Fare tori. — We keep the notations of the last section. We\nstill consider h∈Iso(M,g) such that {hk}is not relatively compact. We consider the\napproximately stable foliation Fassociated to some sequence ( hnk), withnk→ ∞.\nProposition 8.16 and its proof provide us with a leaf F0which is an h-invariant torus,\nand is included in a non locally homogeneous component M. We liftF0to˜F0⊂˜M,\nandhto˜h∈Iso(˜M,˜g) preserving ˜F0. The group Γ 0=S˜F0∩π1(M) is discrete and\nisomorphic to Z2, generated by two elements γ1andγ2. Lemma 8.14 ensures that\nτ1:=ρ(γ1) andτ2:=ρ(γ2) generate a discrete subgroup of S∆isomorphic to Z2.\nAfter conjugating into G∆, we may assume that τ1andτ2are elements of T.\nWe call in the sequel ˜Hthe subgroup generated by ˜h,γ1andγ2. Lemma 8.14\nensures that ρ(˜h)∈S∆=R∗\n+⋉N(see Section 7.2.2). Elements of S∆which are not\ninNact onNwith nontrivial dilation. They can not preserve any lattice in T. This\nsays that because ˜hnormalizes Γ 0, we must have ρ(˜H)⊂N. There are thus three\nelementsX,Y,Zin the Lie algebra nsuch thatτ1=eX,τ2=eZandρ(˜h) =eY. The\ncenter of nis included in Span( X,Z), and we pick Z0/\\e}atio\\slash= 0 in this center.\nWe now pull back the four vector fields X,Y,Z,Z 0onEin3by the developing map\nδ:˜M→Ein3. This way, we get vector fields ˜X,˜Y,˜Z,˜Z0onEin3. Observe that\nτ∗\n1X=X,τ∗\n2Z=Z,ρ(˜h)∗Y=Y, andρ(˜h)∗Z0=τ∗\n1Z0=τ∗\n2Z0=Z0, τ∗\n1Z=τ∗\n2Z=\nZimply the relations\n(11)\nγ∗\n1˜X=˜X, γ∗\n2˜Z=˜Z,˜h∗˜Y=˜Y,and˜h∗˜Z0=γ∗\n1˜Z0=γ∗\n2˜Z0=˜Z0, γ∗\n1˜Z=γ∗\n2˜Z=˜Z\non˜M. After introducing those notations, we can state what will be the la st technical\nstep of our study:\nProposition 8.17 . —For everyx∈˜M, we have:\ni) The point xbelongs to ˜Ω∆and˜F(x) =˜F∆(x).46 CHARLES FRANCES\nii) The leaf ˜F(x) =˜F∆(x)is˜H-invariante.\niii) The restriction of the 3vector fields ˜X,˜Y,˜Zare complete on ˜F∆(x), and the\nequalitiesφ1\n˜X=γ1, φ1\n˜Z=γ2, φ1\n˜Y=˜hhold on ˜F∆(x).\nThe proposition will show that ˜Fcoincide ˜F∆on˜M, and Γ 0-invariance of the\nleaves of ˜Fwill easily imply that leaves of Fare all tori. In the next section 8.5 we\nwill derive more consequences from this equality E=˜M, and prove Theorem 8.1.\nWe are going to consider the set E ⊂˜M, comprising all points x∈˜Msatisfying the\nthree conditions of Proposition 8.17, and show that Eis nonempty, open and closed\nin˜M, yielding E=˜M.\n8.4.1. The set Eis nonempty . — We check here that every point x∈˜F0belongs to\nE. Recallρ: Conf(˜M)→PO(2,3) the holonomy morphism.\nLemma 8.18 . —Letx∈˜Ω∆such that ˜F(x) =˜F∆(x). Assume moreover that\n˜F∆(x)is invariant by a subgroup Λ⊂π1(M), isomorphic to Z2and such that\nρ(Λ)⊂T. Then the map δis a diffeomorphism from ˜F∆(x)toF∆(δ(x)). More-\noverF∆(δ(x))/\\e}atio\\slash=F∆(p0).\nProof: Lemma8.11ensuresthat δisadiffeomorphismfrom ˜F∆(x) toanopensubset\nU⊂F∆(δ(x)). The group Λ is isomorphic to Z2and acts properly discontinuously\non the disk ˜F(y) =˜F∆(y). By a cohomological dimension argument, the action\nmust be cocompact. The group ρ(Λ) is thus a lattice in T, and must act properly\nand cocompactly on U. Last point of Fact 7.2 says that the action of ρ(Λ) can not\nbe proper on any open subset of F∆(p0). We thus infer that F∆(δ(x))/\\e}atio\\slash=F∆(p0). In\nparticular, againby Fact 7.2, the actionof ρ(Λ) is properand cocompact on F∆(δ(x)).\nWe then must have U=F∆(δ(x)).♦\nThe completeness of ˜X,˜Yand˜Zon˜F0follows from Lemma 8.18, applied for\nΛ = Γ 0, becauseX,Y,Zare complete on leaves of F∆. The relations τ1=eX,\nτ2=eZandρ(˜h) =eYimply that the relations φ1\n˜X=γ1, φ1\n˜Z=γ2, φ1\n˜Y=˜hhold on\n˜F0. We infer that ˜F0⊂ E.\n8.4.2. The set Eis open. — We begin by stating a lemma that we will use repeatedly\nin the sequel.\nLemma 8.19 . —LetU⊂˜Ω∆be a connected open set. Let f,g:U→˜Mtwo\nconformal immersions. Assume that for some x∈˜Ω∆,˜F∆(x)∩U/\\e}atio\\slash=∅, and thatf\nandgcoincide on ˜F∆(x)∩U, thenfandgcoincide on U.\nProof: ShrinkingUif necessary and looking at δ(U)⊂Ein3, we are reduced to\nthe situation of two transformations g1andg2of PO(2,3) which coincide on some\nopen subset of a lightcone in Ein3. At level of linear algebra, it means that those\ntwo transformations of PO(2 ,3) must coincide on a lightlike hyperplane of R2,3. This\neasily implies g1=g2.♦\nLet us start with x∈ E. Vector fields ˜X,˜Yand˜Zare complete in restriction to\n˜F∆(x) =˜F(x), so given ǫ >0, we can choose U⊂˜Ω∆a small neighborhood of xLORENTZ DYNAMICS ON CLOSED 3-MANIFOLDS 47\nsuch thatφt\n˜X.yis defined on [ −ǫ,1+ǫ] for everyy∈U, and for every t∈[−ǫ,1+ǫ]\nand everyy∈U,φs\n˜Z.φt\n˜X.yis defined for every s∈[−ǫ,1+ǫ]. Lemma 8.19 says that\nidentityφt\n˜X.y=γ1.yholds onU, because it holds on U∩˜F∆(x). It follows easily\nfrom the property γ∗\n1˜X=˜Xthat for every y∈U,φt\n˜X.yis defined for t∈R. Relation\nγ∗\n1˜Z=˜Znow implies that φs\n˜Z.φt\n˜X.ymakes sense for every t∈R,s∈[−ǫ,1+ǫ], and\ny∈U. Let us call U={φt\n˜X.y|t∈R,y∈U}. This is an open set on which φ1\n˜Zis\ndefined. Relation φ1\n˜Z=γ2holds on U ∩˜F∆(x), hence on Uby Lemma 8.19. Together\nwith the property γ∗\n2˜Z=˜Z, this implies that φs\n˜Z.φt\n˜X.ymakes sense for every y∈U,\nands,t∈R.\nNow, Lemma 8.18 says that F∆(δ(x))/\\e}atio\\slash=F∆(p0). IfUwas chosen small enough,\nδ(y)/\\e}atio\\slash∈F∆(p0) for everyy∈U. It follows that ( t,s)/ma√sto→esZ.etX.δ(y) is a diffeomorphic\nparametrization of F∆(δ(y)). In other words, for every y∈U,{φs\n˜Z.φt\n˜X.y|(s,t)∈R2}\ncoincides with the leaf ˜F∆(y), and the developing map δ:˜F∆(y)→F∆(δ(y)) is\na diffeomorphism. Completeness of vector fields ˜X,˜Y,˜Zon˜F∆(y) follows, because\nvector fields X,Y,Zare complete on leaves of F∆.\nMoreover, Lemma 8.19 says that relations φ1\n˜X=γ1, φ1\n˜Z=γ2andφ1\n˜Y=˜hhold on\nUbecause they hold on U ∩˜F∆(x). In particular, for y∈U, the leaf ˜F∆(y) is stable\nbyγ1, γ2and˜h, hence by ˜H.\nTo conclude that U⊂ E, it remains to check that ˜F∆(y) coincides with ˜F(y)\nfor everyy∈U. A first observation is that ˜F∆(y) is diffeomorphic to F∆(δ(y)),\nhence is a disk. It follows from a cohomological dimension argument th at because\nΓ0≃Z2, the quotient Σ = Γ 0\\˜F∆(y) is a torus in M. Recall from (11) the relation\n˜h∗˜Z0=γ∗\n1˜Z0=γ∗\n2˜Z0=˜Z0. Remember also that ˜Z0is a linear combination of\n˜Xand˜Z, hence tangent to ˜F∆(y). On the torus Σ, ˜Z0thus induces a vector field\nZ0which ish-invariant. Notice that Z0is lightlike because Z0is lightlike on Ein3,\nandZ0is nonsingular because the singularities of Z0are exactly the points of ∆,\nand˜F∆(y)⊂˜Ω∆. Hence, for every z∈Σ,Z0(z)⊥=TzΣ. On the other hand,\nequalityDzhnk(Z0(z)) =Z0(hnk(z)) shows that Z0(z) belongs to the aproximately\nstable distribution of hnk(see the definition of this distribution in Section 8.1). The\naproximatelystabledistribution hascodimension1andislightlike, soth at itcoincides\nwithZ0(z)⊥=TzΣ for allz∈Σ. We conclude that Σ is a leaf of F, what proves\n˜F(y) =˜F∆(y).\n8.4.3. The set Eis closed. — We consider a sequence ( xk) ofEconverging to x∞∈\n˜M. The leaf ˜F(x∞) is accumulated by the sequence of leaves ˜F(xk) =˜F∆(xk). In\nparticular, the vector fields ˜X,˜Y,˜Zbeing tangent to ˜F∆(xk) for allk, they are also\ntangent to ˜F(x∞). Point 2) of Fact 7.2 then says that ˜F(x∞)\\˜∆ is a union of leaves\nof˜F∆. If the set ˜F(x∞)∩˜∆ is not empty, those leaves of ˜F∆might be prolongated\nsmoothly accros the singular set ˜∆, a contradiction. We infer that ˜F(x∞)⊂˜Ω∆, and\n˜F(x∞) =˜F∆(x∞).\nThe union of the compact leaves of Fis a compact subset of M(see [God, Chap\nII, Corollary 3.10]). Since Fhas no vanishing cycles, ˜F(x∞) is left invariant by a48 CHARLES FRANCES\ndiscrete subgroup Λ 1⊂π1(M) which is isomorphic to Z2. We choose I⊂M, a small\ntransversal to the foliation Fcontaining the point π(x∞). Following the loops of\nF(π(x∞)) defining Λ 1in the neighboring leaves, we get a corresponding holonomy\nmorphism:\nhol : Λ 1→Diff(I).\nIfγ∈Λ1, and if for z∈Iwe have hol( γ2).z/\\e}atio\\slash=z, then the pseudo-orbit hol( γn).z\nis infinite and the leaf F(z) cannot be closed. Because F(π(xk)) is a torus for each\nk∈N, it follows that replacing Λ 1by some index 2 subgroup, we may assume that\nΛ1leaves invariant ˜F(xk) forklarge enough.\nThis property also shows that ρ(Λ1), which is a priorinot a subgroup of G∆, leaves\ninvariant infinitely many leaves of F∆, hence infinitely many points on ∆. It follows\nthatρ(Λ1) leaves∆ invariant: ρ(Λ1)⊂G∆. A M¨ obius transformation fixing infinitely\nmany points on the circle must be trivial, hence ρ(Λ1)⊂S∆.\nThe group generated by Λ 1and Γ0is included in π1(M), hence must act properly\ndiscontinuously on each ˜F(xk). It follows that Λ = <Λ1,Γ0>is a discrete group\nisomorphic to Z2. In particular Λ 1commutes with Γ 0, andρ(Λ1)⊂T. We can then\napplyLemma 8.18to Λ, and weget that δisa diffeomorphismfrom ˜F(x∞) =˜F∆(x∞)\ntoF∆(x∞), andF∆(δ(x∞))/\\e}atio\\slash=F∆(p0). It follows that ˜X,˜Yand˜Zare complete in\nrestriction to ˜F∆(x∞).\nLety∞∈˜F(x∞), andletU⊂˜Ω∆be asmallneighborhoodof y∞. Bycompleteness\nof˜X,˜Yand˜Zin restriction to ˜F∆(x∞), and shrinking Uif necessary, the local\ndiffeomorphisms φ1\n˜X,φ1\n˜Yandφ1\n˜Zare defined on U. Forklarge,U∩˜F∆(xk)/\\e}atio\\slash=∅, and\nidentitiesφ1\n˜X=γ1,φ1\n˜Y=˜handφ1\n˜Z=γ2hold onU∩˜F∆(xk). Lemma 8.19 says\nthat those identities hold on U. Finallyy∞was arbitrary in ˜F∆(x∞) so that these\nidentities hold on ˜F∆(x∞). This proves x∞∈ E.\n8.5. Proof of Theorem 8.1. — Let us draw further conclusions from Proposition\n8.17. The coincidence of the foliations ˜Fand˜F∆implies that ˜F∆isπ1(M)-invariant.\nMoreover, the Γ 0-invariance of each leaf ˜F∆, together with Lemma 8.18 implies that\nδis injective on each leaf ˜F∆, and thatδ(˜M)⊂Ω∆\\F∆(p0).\nAlso, it follows from Proposition 8.17 that Γ 0is exactly the subgroup of π1(M)\nleaving each leaf of ˜Finvariant. It follows that Γ 0is normal in π1(M). We claim\nthat Γ 0is also normalized by Nπ1, the normalizer of π1(M) in Iso( ˜M,˜g). Indeed, if\nf∈Nπ1, thenfΓ0f−1leaves each leaf of f(˜F) invariant. Now f(˜F) is a lightlike,\ntotally geodesic, codimension 1 foliation. Remark8.10ensures that o n anynon locally\nhomogeneous component ˜M,f(˜F) coincides with ˜F. In particular, fΓ0f−1coincide\nwith Γ 0on˜M, hencefΓ0f−1= Γ0.\nThe groupρ(Nπ1) normalizes ρ(Γ0), henceTsinceρ(Γ0) is Zariski-dense in T. By\nfact 7.2, the lightcone C(p0) can be characterized as the set of points where the orbits\nofρ(Γ0) are contained in a photon of Ein3. It follows that C(p0) is left invariant by\nNor(T), the normalizer of Tin PO(2,3). Applying the stereographic projection ϕof\npolep0(see Section 7.1.2) we can see ρ(Nπ1) as a subgroup of Conf( R1,2). We then\nshow:LORENTZ DYNAMICS ON CLOSED 3-MANIFOLDS 49\nLemma 8.20 . —Seen in Conf(R1,2), the elements of ρ(Nπ1)are included in the\ngroup\nG:=\n\n\nǫ1−ǫ1y−ǫ1\n2y2\n0ǫ2ǫ2y\n0 0 ǫ1\n+\nz\nx\nt\n, x,z,y,t ∈R, ǫi=±1\n\n.\nIn particular, we have the inclusion ρ(Nπ1)⊂Iso(R1,2).\nProof: After performing the stereographic projection ϕ, the foliation F∆restricted\ntoEin3\\C(p0) becomes a foliation of R1,2. Formula (5) for ϕreadily shows that this\nis the foliation by affine planes of direction Span( e1,e2). Recall (see Section 7.2.2)\nthat the group Tcorresponds to the group of translations of vectors v∈Span(e1,e2).\nSince Nor(T), henceρ(Nπ1), must preserve this foliation (this is a consequence of\nFact 7.2), we see that elements of ρ(Nπ1) belong to the subgroup G′⊂Conf(R1,2)\ncomprising all elements of the form:\n(12)\nλµ−λµy−λµ\n2y2\n0µµ\nµy\n0 0µ\nλ\n+\nz\nx\nt\n, x,y,z,t ∈Rλ,µ∈R∗.\nIf a matrix\nλµ−λµy−λµ\n2y2\n0µµ\nµy\n0 0µ\nλ\npreserves a lattice in Span( e1,e2), then the\ndeterminant of its restriction to Span( e1,e2) is±1. It follows that µ=±1√\n|λ|.\nWe saw that ρ(˜h) belongs to the group N, hence has the form:\nρ(˜h) =\n1−y−y2\n2\n0 1y\n0 0 1\n+\nz\nx\n0\n, y/\\e}atio\\slash= 0.\nIn particular, because ˜hnormalizes Γ 0, ifτ∈ρ(Γ0), and if we see τas a translation\nof vectorv∈Span(e1,e2), thenρ(Γ0) will also contain v′=v−A.v, whereA=/parenleftbigg\n1y\n0 1/parenrightbigg\n. In other words ρ(Γ0) contains a translation of vector αe1,α/\\e}atio\\slash= 0. The\nfact thatρ(Nπ1) normalizes the discrete group ρ(Γ0), leads to the relation λµ=±1\nin (12). Together with the relation µ=±1√\n|λ|, this leads to |µ|=|λ|= 1, and the\nLemma follows.\n♦\nLemma 8.20 says that our ( Ein3,PO(2,3))-structure is actually a ( R1,2,Iso(R1,2))-\nstructure. We conclude that there exists g′in the conformal class of gwhich is flat,\nand which is preserved by Iso( M,g). We can thus apply the results of Section 4.2.\nTheorem 4.5 and Proposition 4.6 say that ( M,g′) is the quotient of R3, Heis or SOL\nbyalattice. But ρ(π1(M))⊂GbyLemma8.20, and Gdoesnotcontainanysubgroup\nisomorphic to SOL. We thus get that Mis homeomorphic to T3or to a torus bundle\nT3\nAwithA⊂SL(2,Z) parabolic. This proves points 1) and 2) of Theorem 8.1.50 CHARLES FRANCES\nFinally, Carri` ere’s completeness result [ Ca] says that δ:˜M→R1,2is a conformal\ndiffeomorphism. It follows that if the coordinates associated to ( e1,e2,e3) inR1,2are\n(u,t,v), the metric ˜ gis of the form\na(u,t,v)(dt2+2dudv).\nIt remains to check that the function adepends only on v. First, the foliation by\nplanes with direction Span( e1,e2) is totally geodesic. If ˜∇denotes the Levi-Civita\nconnection of ˜ g, we thus have:\n0 = ˜g(˜∇∂t∂t,∂u) =−1\n2∂u.˜g(∂t,∂t) =−1\n2∂a\n∂u.\nIdentifying ˜handρ(˜h), we saw that\n˜h=\n1−y−y2\n2\n0 1y\n0 0 1\n+\nµ0\nν0\n0\n\nwherey/\\e}atio\\slash= 0.\nIt follows that ρ(˜h) acts on each hyperplane v=v0by the affine transformation:\n/parenleftbigg\nu\nt/parenrightbigg\n/ma√sto→/parenleftbigg\nu−yt+µ(v0)\nt+ν(v0)/parenrightbigg\n,\nwhereµ(v0) =µ0−y2\n2v0andν(v0) =ν0+yv0.\nThe group Γ 0is generated by two translations τ1,τ2of (linearly independant)\nvectors/parenleftbigg\na\nb/parenrightbigg\nand/parenleftbigg\nc\nd/parenrightbigg\nrespectively. The w-coordinate of ˜hk◦τm\n1◦τn\n2/parenleftbigg\nu\nt/parenrightbigg\nis\nt+kν(v0) +mb+nd. Because ˜hk◦τm\n1◦τn\n2acts isometrically for ˜ gthis leads to\na(t,v) =a(t+kν(v) +mb+nd,v) for every ( k,m,n)∈Z3. Sincebanddcan not\nbe both zero (let say b/\\e}atio\\slash= 0), and because ν(v) andbare rationally independant for\nalmost every value of w(becausey/\\e}atio\\slash= 0), we get that for almost every w,t/ma√sto→a(t,v)\nis constant. As a consequence, a=a(v), and the fact that it is a periodic function\nfollows easily from the compactness of M.\nFinally the group Nwhich comprises transformations of the form\n\n1−y−y2\n2\n0 1y\n0 0 1\n+\nz\nx\n0\n, x,y,z ∈R\nis isomorphic to Heis and acts isometrically on ( ˜M,˜g). This concludes the proof of\nTheorem 8.1.\n9. Conclusions\nThe study made in Section 4, as well as Theorems 5.2 and 8.1 provide all possible\ntopologiesforaclosed3-dimensional,orientableandtime-orientable ,Lorentzmanifold\nwith a noncompact isometry group. Those are the 3-dimensional to rus, hyperbolicLORENTZ DYNAMICS ON CLOSED 3-MANIFOLDS 51\nor parabolic torus bundles, and compact quotients Γ \\/tildewidestPSL(2,R). Together with the\nexamples provided in Section 2, this yields Theorem A.\nLet us now look at the geometries which can occur on those manifolds , and prove\nTheoremC. The manifoldsΓ \\/tildewidestPSL(2,R) occuronly in Proposition4.7. Hence the only\nmetrics on such manifolds which admit a noncompact isometry group a re covered\nby/tildewidestPSL(2,R), endowed with a Lorentzian, non-Riemannian, left-invariant metr ic.\nIn particular those manifolds ( M,g) are locally homogeneous and ( ˜M,˜g) admits an\nisometric action of /tildewidestPSL(2,R).\nParabolic torus bundles appear in Proposition 4.6 and Theorem 8.1. We saw there\nthat the universal cover is isometric to R3endowed with a metric\na(v)(dt2+2dudv),\nwithasmooth and periodic. This universal cover admits an isometric action of Heis.\nIf the manifold ( M,g) is locally homogeneous, Proposition 4.6 ensures that gis flat\nor locally isometric to the Lorentz-Heisenberg metric.\nHyperbolic torus bundles appear only in Proposition 4.6 Theorem 5.2. W e saw\nthat the universal cover is isometric with R3endowed with a metric dt2+2a(t)dudv,\nwithasmooth and periodic. There is an isometric action of SOL on this univer sal\ncover. The manifold ( M,g) is locally homogeneous if and only if it is flat.\nFinally, 3-toriappearin Proposition4.6Theorem 5.2and Theorem 8.1. The metric\non the universal cover ˜Mis provided by those two last theorems, and there is always\nan isometric action of Heis or SOL on ( ˜M,˜g). Finally, ( M,g) is locally homogeneous\nif and only if it is flat.\nThose results alltogether prove Theorem C and Corollary D.\n10. Annex A: Some computations\nWe present here the necessary computations leading to Propositio n 6.3.\n10.0.1. The curvature module . — We consider on R3the Lorentzian form, with ma-\ntrix in a basis e,h,fgiven byJ=\n0 0 1\n0 1 0\n1 0 0\n.\nWe call O(1 ,2) the subgroup of GL(3 ,R) preserving the bilinear form determined\nbyJ. Its Lie algebra is denoted by o(1,2), and admits the following basis :\nE=\n0 1 0\n0 0−1\n0 0 0\n,H=\n1 0 0\n0 0 0\n0 0−1\n,F=\n0 0 0\n1 0 0\n0−1 0\n.\nWe thus have the commutation relations [ H,E] =E,[H,F] =−Fand [E,F] =H.\nLet (M,g) be 3-dimensional Lorentz manifold, and denote by ˆMits bundle\nof orthonormal frames. At each ˆ x∈ˆM, the curvature κ(ˆx) is an element of\nHom(∧2(R3),o(1,2)). Because of Bianchi’s identities, the curvature module is ac-\ntually a 6-dimensional submodule of Hom( ∧2(R3),o(1,2)). Choosing e∧h,e∧f,\nh∧fas a basis for ∧2(R3), andE,H,F as a basis for o(1,2), an element of52 CHARLES FRANCES\nHom(∧2(R3),o(1,2)) is merely given by a 3 ×3 matrix. The action of O(1 ,2) on\nHom(∧2(R3),o(1,2)) corresponds to the conjugation on matrices.\nScalar matrices are O(1 ,2)-invariant, and form a 1-dimensional irreducible sub-\nmodule (corresponding to constant sectional curvature).\nTheotherirreduciblesubmoduleofthecurvaturemoduleis5-dimens ional,spanned\nby the matrices:\n\n0 0 1\n0 0 0\n0 0 0\n,\n0 1 0\n0 0 1\n0 0 0\n,\n1 0 0\n0−2 0\n0 0 1\n,\n0 0 0\n1 0 0\n0 1 0\n,\n0 0 0\n0 0 0\n1 0 0\n.\nWe callκ0the element of Hom( ∧2(R3),o(1,2)) corresponding to the identity ma-\ntrix, namely κ0mapse∧htoE,e∧ftoHandh∧ftoF. We also call κ1the\nelement of Hom( ∧2(R3),o(1,2)) corresponding to the matrix\n0 0 1\n0 0 0\n0 0 0\n.\nThe two dimensional vector space spanned by κ0andκ1is the set of fixed points\nof the action of {etE}t∈Ron the curvature module.\n10.0.2. Identification of the killloc-algebra. — We consider a parabolic component\nMwhich is not locally homogeneous. In such a component, the points ar e either\nparabolic, or points where the isotropy algebra is 3-dimensional and the sectional\ncurvature is constant. The set of parabolic points is thus a dense o pen set Ω ⊂ M.\nObservethat at a parabolicpoint x∈Ω, ifXa localKilling field around x, generating\nthe isotropy Is(x), the 1-parameter group Dxϕt\nXis unipotent in O( TxM). In a\nsuitable basis ( u1,u2,u3) ofTxMsatisfyingg(u1,u3) = 1 =g(u2,u2) and all the\nother products are 0, the matrix of Dxϕt\nXreads\n\n1t−t2/2\n0 1−t\n0 0 1\n\nWe quickly check that the only 2-plane stable by Dxϕt\nXis spanned by u1andu2,\nso that on Ω, the killloc-orbits must be lightlike surfaces.\nLet us now fix a point x∈Ω. We work in the fiber bundle ˆM(and lift all local\nKilling fields there). After multiplying Xby a suitable constant, we can find ˆ x∈ˆM\nin the fiber of xsuch thatω(X(ˆx)) =E. We now choose ZandYtwo local Killing\nfields around xsuch thatZ(x) =u1andY(x) =u2. After adding to ZandYa\nsuitable multiple of X, we can write, at ˆ x:\nω(Z) =e+βH+γFandω(Y) =h+αH+νF.\nThe curvature κ(ˆx) is Ad(etE)-invariant, hence is of the form κ=σκ0+bκ1. In\nparticular, the following identities hold at ˆ x:\n(13) κ(e∧h) =σE, κ(e∧f) =σH, κ(h∧f) =bE+σF.\nNotice that σ,b,α,β,γ,νdepend on xand ˆx, but since those points are fixed, there\nwill be considered as constant in the sequel.LORENTZ DYNAMICS ON CLOSED 3-MANIFOLDS 53\nCartan’s formula LXω=ιXdω+d(ιXω) shows that whenever U,Vare two Killing\nfields on ˆM, the following relation holds:\n(14) ω([U,V]) =K(U,V)−[ω(U),ω(V)]\nIn the sequel, we will call Hthe span of ω(Z),ω(X),ω(Y) at ˆx, and we are going\nto write Equation (14) at ˆ x, using identities (13), when UandVrange over Z,X,\nY. For instance, the first equation is:\nω([Z,X]) =−[ω(Z),ω(X)] =−[e+βH+γF,E]\n=−βE+γH=ω(−βX)+γH.\nThe fact that killloc(x) is a Lie algebra, together with the property H/\\e}atio\\slash∈ Hforces\nγto vanish. Next, two Killing fields which coincide at ˆ xmust be equal (by freeness\nof the action of isometries on the orthonormal frames), which implie s [Z,X] =−βX.\nTo summarize:\n(15) γ= 0 and [Z,X] =−βX.\nWe proceed exactly in the same way for the two other equations:\nω([X,Y]) =−[ω(X),ω(Y)] =−[E,h+αH+νF]\n=−e+αE−νH=ω(−Z+αX)+(β−ν)H.\nleads to:\n(16) β=νand [X,Y] =−Z+αX.\nFinally\nω([Z,Y]) =κ(e,h)−[e+νH,h+αH+νF] =σE+αe+νh+ν2F\n=ω(αZ+σX+νY)−2ανH.\nimplies:\n(17) αν= 0 and [Z,Y] =αZ+σX+νY.\nNotice that establishing (15) and (16), we have actually shown that ad(X) is a\nnilpotent endomorphism of killloc(x). This property did not use anything special on\nx, so that we actually have:\nFact 10.1 . —At eachz∈Ω, ifUis a local Killing field around zgenerating the\nisotropy at z, thenad(U)is a nilpotent endomorphism of killloc(z).\nAtx,Z(x) is lightlike and nonzero and Y(x) is spacelike, orthogonal to Z(x).\nThe orthogonal to Y(x) atxis a Lorentzian plane spanned by Z(x) and another\nvectorw∈TxM. Let us call t/ma√sto→γ(t) the geodesic through xsatisfying ˙γ(0) =w.\nClairault’s equation ensures that the quantities g(˙γ(t),Z(γ(t))),g(˙γ(t),X(γ(t))) and\ng(˙γ(t),Y(γ(t))) do not depend on t. In particular, for t >0, bothY(γ(t)) and\nX(γ(t)) are orthogonal to ˙ γ(t) whileZ(γ(t)) is not. For t>0 small enough, Y(γ(t))\nis still spacelike, hence nonzero, and γ(t) belongs to Ω. In particular, the killloc-orbit\natγ(t) is 2-dimensional, so that YandXmust be colinear at γ(t). One then has54 CHARLES FRANCES\nX(γ(t)) =λtY(γ(t)), for some real λt. Observe finally that wis not fixed by Dxφt\nX,\nhence is transverse to the set where Xvanishes. In particular, for t≥0 small,\nX(γ(t)) = 0 only for t= 0, and thus λt/\\e}atio\\slash= 0 ift/\\e}atio\\slash= 0.\nWe claim that those considerations lead necessarily to α= 0. Indeed, using the\nbracket relations (16),(17) and (15), we compute\nTrace(ad(λtY−X)) =−2λtα.\nFort≥0 small,X,Y, Zgenerate killloc(γ(t)), hence −2λtα= 0 because of Fact 10.1.\nSinceλt/\\e}atio\\slash= 0 ift/\\e}atio\\slash= 0, we get α= 0. Injecting this data in equation (17) and (15), we\nfind that the matrix of ad( λtY−X) in the basis Y,Z,Xis:\n\n0−λtν0\n1 0 λt\n0−λtσ−ν0\n.\nThe characteristic polynomial of ad( λtY−X) is\nQ(x) =−x3−λtx(λtσ+2ν).\nHence, the nilpotency of ad( λtY−X) (Fact 10.1) implies\n(18) λtσ+2ν= 0,\nIfσ/\\e}atio\\slash= 0, we get that t/ma√sto→λtis constant, which is not the case since we observed that\nλ0= 0 butλt/\\e}atio\\slash= 0 fort >0 small. We end up with the equality σ=ν= 0. The\nvector fields Z,X,Ythen satisfy the bracket relations:\n[Z,X] = 0 = [Z,Y],and [X,Y] =−Z,\nshowing that Lie algebra killloc(x) is isomorphic to heis. We also proved that σ, the\nscalar curvature at x, vanishes, but since xwas arbitrary in the open set Ω, we finally\nget the vanishing of the scalar curvature on Ω, and then on Mby density.\n10.0.3. Description of the killloc-orbits. — The fact that the local Killing algebra is\nisomorphic to heisshows that no point in Mhas a 3-dimensional isotropy algebra.\nIndeed, the isotropy representation at those points would yield an embedding heis→\no(1,2), what is impossible. We thus get Ω = M, and all the killloc-orbits on Mare\n2-dimenional and lightlike.\nOn the other hand, since the isotropy algebra Is(x) generates a parabolic 1-\nparameter subgroup of O(1 ,2) at eachx, there is a totally geodesic lightlike hyper-\nsurfaceF(x), whose tangent space is left invariant by the isotropy (see [ DZ, Lemma\n3.5] and its proof). We already observed that at x∈ M, the local isotropy preserves\nonly one 2-plane of TxM. This implies that the killloc-orbits are everywhere tangent\nto a leaf of a totally geodesic foliation of M, hence the killloc-orbits are themselves\ntotally geodesic. This concludes the proof of Proposition 6.5.\nReferences\n[AS] S. Adams, G. Stuck, The isometry group of a compact Loren tz manifold. I, II. Invent.\nMath.129(1997), no. 2, 239–261, 263–287.LORENTZ DYNAMICS ON CLOSED 3-MANIFOLDS 55\n[BCDGM] T. Barbot, V. Charette, T. Drumm, W. Goldman, K. Meln ick, A primer on the\n(2+1) Einstein universe. Recent developments in pseudo-Riemannian geometry , 179–229,\nESI Lect. Math. Phys., Eur. Math. Soc., Zrich, 2008.\n[Ca] Y. Carri` ere, Autour de la conjecture de L. Markus sur le s vari´ et´ es affines. Invent. Math. ,\n95, (1989), 615–628.\n[DA] G. D’Ambra, Isometry groups of Lorentz manifolds. Invent. Math. 92(1988), no. 3,\n555–565.\n[DM] S. Dumitrescu, K. Melnick, Quasihomogeneous three-di mensional real-analytic\nLorentz metrics do not exist. Geom. Dedicata 179(2015), 229–253.\n[DZ] S. Dumitrescu, A. Zeghib, G´ eom´ etries Lorentziennes de dimension 3: classification et\ncompl´ etude. Geom. Dedicata 149(2010), 243–273.\n[F1] C. Frances, G´ eom´ etrie et dynamique lorentziennes co nformes. Th` ese, ENS Lyon (2002).\navailable at http://irma.math.unistra.fr/ ∼frances/.\n[F2] C. Frances, Variations on Gromov’s open-dense orbit th eorem. arXiv:1605.05755. To\nappear in Bulletin de la SMF .\n[FG] D. Fried, W. Goldman, Three-dimensional affine crystall ographic groups. Adv. Math. ,\n47(1), (1983), 1–49.\n[FZ] D. Fisher, R.J Zimmer, Geometric lattice actions, entr opy and fundamental groups.\nComment. Math. Helv. 77(2002), no. 2, 326–338.\n[God] C. Godbillon, Feuilletages. (French) tudes gomtriqu es. Progress in Mathematics, 98.\nBirkh¨ auser Verlag, Basel, 1991.\n[GK] W. Goldman, Y. Kamishima, The fundamental group of a com pact flat Lorentz space\nform is virtually polycyclic. J. Differential Geom. ,19(1), (1984), 233–240.\n[GuK] F. Gueritaud, F. Kassel, Maximally stretched laminat ions on geometrically finite\nhyperbolic manifolds. Geom. Topol. 21(2017), no. 2, 693–840.\n[Gr] M. Gromov, Rigid transformation groups, G´ eom´ etrie D iff´ erentielle, (D. Bernard et\nChoquet-Bruhat Ed.), Travaux en cours , Hermann, Paris, 33, (1988), 65–141.\n[Kl] B. Klingler, Compl´ etude des vari´ et´ es lorentzienne s courbure constante. Math. Ann.\n306(1996), no. 2, 353–370.\n[KN] S. Kobayashi, K. Nomizu, Foundations of differential ge ometry I (New York : Inter-\nscience Publishers, 1963).\n[KR] R. Kulkarni, F. Raymond, 3-dimensional Lorentz space- forms and Seifert fiber spaces.\nJ. Differential Geometry. 21no. 2, 1985, 231–268.\n[M] K. Melnick, A Frobenius theorem for Cartan geometries, w ith applications.\nL’Enseignement Math´ ematique (S´ erie II) 57(2011), no. 1-2, 57–89.\n[MS] S. B Myers, N. E Steenrod, The group of isometries of a Rie mannian manifold. Ann.\nof Math. (2)40(1939), no. 2, 400–416.\n[No] K. Nomizu, On local and global existence of Killing field s.Ann. of Math. 72(1960),\nno. 2, 105–112.\n[Or] P. Orlik, Seifert Manifolds, Lecture Notes in Math., 291. Springer-Verlag, 1972.\n[P] V. P´ ecastaing, On two theorems about local automorphis ms of geometric structures.\nAnn. Inst. Fourier 66(2016), no. 1, 175–208.\n[Sa1] F. Salein, Vari´ et´ es anti-de Sitter de dimension 3 po ss´ edant un champ de Killing non\ntrivial.C. R. Acad. Sci. Paris Sr. I Math. 324(1997), no. 5, 525–530.\n[Sa2] F. Salein, Vari´ et´ es anti-de Sitter de dimension 3. T h` ese de doctorat, ENS Lyon, 1999.\nhttp://www.umpa.ens-lyon.fr/ ∼zeghib/these.salein.pdf.56 CHARLES FRANCES\n[Sh] R.W. Sharpe, Differential Geometry: Cartan’s generalization of Klein’s E rlangen Pro-\ngram. New York, Springer, 1997.\n[Sco] P. Scott, The geometries of 3-manifolds. Bull. London Math. Soc. 15(1983), no. 5,\n401–487.\n[Si] I. Singer, Infinitesimally homogeneous spaces. Comm. Pure Appl. Math. 13(1960), 685–\n697.\n[Tho] N. Tholozan, Uniformisation des vari´ et´ es pseudo-r iemanniennes locale ment ho-\nmog` enes . PhD thesis, Universit´ e de Nice Sophia-Antipoli s, 2014.\n[Z1] A. Zeghib, Sur les espaces-temps homog` enes. (French) [On homogeneous space-times]\nThe Epstein birthday schrift, 551–576, Geom. Topol. Monogr., 1, Geom. Topol. Publ.,\nCoventry, 1998.\n[Z2] A. Zeghib, Killing fields in compact Lorentz 3-manifold s,J. Differential Geom. ,43,\n(1996), 859–894.\n[Z3] A. Zeghib, Isometry groups and geodesic foliations of L orentz manifolds. I. Foundations\nof Lorentz dynamics. Geom. Funct. Anal ,9(1999), no. 4, 775–822.\n[Z4] A. Zeghib, Isometry groups and geodesic foliations of L orentz manifolds. II. Geometry\nof analytic Lorentz manifolds with large isometry group Geom. Funct. Anal ,9(1999), no.\n4, 823–854.\n[Z5] A. Zeghib, Geodesic foliations inLorentz3-manifolds .Comment. Math. Helv. 74(1999),\nno. 1, 1–21.\n[Z] R.J Zimmer, On the automorphism group of a compact Lorent z manifold and other\ngeometric manifolds. Invent. Math. 83(1986), no. 3, 411–424.\nCharles Frances , IRMA, 7 rue Ren´ e Descartes, 67000 Strasbourg.\nE-mail : cfrances@math.unistra.fr"    },    {        "title": "0908.3821v1.Influence_of_an_external_magnetic_field_on_forced_turbulence_in_a_swirling_flow_of_liquid_metal.pdf",        "content": "arXiv:0908.3821v1  [physics.flu-dyn]  26 Aug 2009Influence of an external magnetic field on forced turbulence i n a\nswirling flow of liquid metal\nBasile Gallet, Michael Berhanu, and Nicolas Mordant\nLaboratoire de Physique Statistique,\nEcole Normale Sup´ erieure & CNRS,\n24 Rue Lhomond, 75231 PARIS Cedex 05, France\n(Dated: October 29, 2018)\nAbstract\nWe report an experimental investigation on the influence of a n external magnetic field on forced\n3D turbulence of liquid gallium in a closed vessel. We observ e an exponential damping of the\nturbulent velocity fluctuations as a function of the interac tion parameter N(ratio of Lorentz force\nover inertial terms of the Navier-Stokes equation). The flow structures develop some anisotropy\nbut do not become bidimensional. From a dynamical viewpoint , the damping first occurs homo-\ngeneously over the whole spectrum of frequencies. For large r values of N, a very strong additional\ndamping occurs at the highest frequencies. However, the inj ected mechanical power remains inde-\npendent of the applied magnetic field. The simultaneous meas urement of induced magnetic field\nand electrical potential differences shows a very weak correl ation between magnetic field and ve-\nlocity fluctuations. The observed reduction of the fluctuati ons is in agreement with a previously\nproposed mechanism for the saturation of turbulent dynamos and with the order of magnitude of\nthe Von K´ arm´ an Sodium dynamo magnetic field.\n1Situations were a magnetic field interacts with a turbulent flow are fo und in various\ndomains of physics, including molten metals processing, laboratory fl ows, and astrophysics.\nThe motion of electrically conducting fluid in a magnetic field induces elec trical currents,\nwhich in turn react on the flow through the Lorentz force. The pow er injected in the flow\nis thus shared between two dissipative mechanisms: viscous friction and ohmic dissipation\nof the induced currents. On the one hand, the situation where the flow is laminar is very\nwell understood, and the geometry of the velocity field and of the in duced currents can\nbe computed analytically. On the other hand, several questions re main open in the fully\nturbulent situation: when a statistically steady state is reached, is the mean injected power\nhigher or lower than in the nonmagnetic case? What controls the rat io of ohmic to viscous\ndissipation? How is the turbulent cascade affected by the magnetic fi eld? To adress some\nof these questions, we have designed an experimental device which allows to apply a strong\nmagnetic field on a fully turbulent flow.\nWhen an electrically conducting fluid is set into motion, a magnetic Reyn olds number\nRmcan be defined as the ratio of the ohmic diffusive time to the eddy-tur nover time. This\nnumber reaches huge values in galactic flows, but can hardly exceed one in a laboratory\nexperiment. The common liquids of high electrical conductivity (gallium , mercury, sodium)\nhave a very low kinematic viscosity: their magnetic Prandtl number Pm(ratio of the kine-\nmatic viscosity over the magnetic diffusivity) is less than 10−5. A flow with Rmof order\none is turbulent and thus requires a high power input to be driven. Fo r this reason most\nexperimental studies have been restricted to low Rm. They were conducted mostly in chan-\nnel flows and grid generated turbulence [1, 2]. The most general ob servation is that the\napplication of a strong magnetic field leads to a steeper decay of the power spectra of the\nturbulent velocity fluctuations at large wavenumbers: the decay g oes from a classical k−5/3\nscaling without magnetic field to k−3ork−4for the highest applied fields. In the meantime,\nsome anisotropy is developed leading to larger characteristic scales along the applied mag-\nnetic field. The phenomenology of these transformations is quite we ll understood in terms\nof the anisotropy of the ohmic dissipation [3]. One difficulty arises from the fact that the\nboundary conditions can have a strong effect on the turbulence lev el. In channel flows the\nchoice between conducting and insulating walls strongly impacts the fl ow. A strong external\nmagnetic field perpendicular to the boundaries leads to an increase o f the turbulence level\n2due to modifications of the boundary layers [1]. On the contrary, Ale many et al. observed in\ngrid generated decaying turbulence an enhanced decay of the tur bulent fluctuations. As far\nas forced turbulence is concerned, Sisan et al. studied the influenc e of a magnetic field on a\nflow of liquid sodium inside a sphere [4]. However, the flow again goes thr ough a variety of\ninstabilities which prevents a study solely focussed on the impact of t he magnetic field on\nthe turbulent fluctuations: the geometry of the mean flow and of t he induced magnetic field\nkeeps changing as the magnetic field is increased.\nSeveral numerical simulations of this issue have been conducted [5 , 6, 7, 8]. Despite\nthe rather low spatial resolution available for this problem, these wo rks show the same phe-\nnomenology: development of anisotropy, steepening ofthe spect ra andtrend to bidimension-\nalization of the flow as the magnetic field increases. The direct numer ical simulations (DNS)\nalso allow to compute the angular flux of energy from the energy-co ntaining Fourier modes\n(more or less orthogonal to the applied field) to the modes that are preferentially damped\nby ohmic dissipation. Once again, the magnetic field can affect the larg e scale structure of\nthe velocity field: the interaction between forcing and magnetic field in the DNS of Zikanov\n& Thess [5, 6] leads to an intermittent behavior between phases of r oughly isotropic flow\nand phases of bidimensional columnar vortices that eventually get u nstable. Vorobev et\nal.[7] and Burratini et al.[8] performed DNS in a forced regime which are closely related to\nour experiment. However, the maximum kinetic Reynolds number rea ched in their studies\nremains orders of magnitude below what canbe achieved inthe labora tory. Anexperimental\ninvestigation at very high kinetic Reynolds number thus remains nece ssary to characterize\nthe effect of a strong magnetic field on a fully developped turbulent c ascade.\nThe flow under study in our experiment resembles the Von K´ arm´ an geometry: counter-\nrotating propellers force a flow inside a cylindrical tank. The large sc ale flow is known to\nhave a strong shear layer in the equatorial plane where one observ es the maximum level of\nturbulence [9]. We do not observe any bifurcation of the flow as the m agnetic field increases\nnor any drastic change of the large scale recirculation imposed by th e constant forcing.\nThese ingredients strongly differ from the previously reported exp eriments and allow to\nstudy the influence of an applied magnetic field on the turbulent casc ade in a given flow\ngeometry. This issue arises in the framework of turbulent laborato ry dynamo studies such\nas the Von K´ arm´ an Sodium (VKS) experiment [10] which flow’s geome try is similar to ours.\nIn the turbulent dynamo problem, one issue is to understand the pr ecise mechanism for the\n3FIG. 1: Sketch of the experimental setup. The curved arrows r epresent the average large-scale\nmotion of the fluid.\nsaturation of the magnetic field. Once the dynamo magnetic field get s strong enough, it\nreacts on the flow through the Lorentz force. This usually reduce s the ability of the flow to\nsustain dynamo action. An equilibrium can be reached, so that the ma gnetic field saturates.\nThis backreaction changes the properties of the bulk turbulence. In a situation where an α\neffect takes part in the generation of the dynamo, i.e. if the turbule nt fluctuations have a\nmean field effect, then the changes in the statistics of turbulence s hould be involved in the\nsaturation mechanism of the dynamo.\nI. DESCRIPTION OF THE EXPERIMENT\nA. The flow\nOur turbulent flow resembles the Von K´ arm´ an geometry, which is w idely used in exper-\niments on turbulence and magnetohydrodynamics ([9, 10] for exam ple). 8 liters of liquid\ngallium are contained in a closed vertical cylinder of diameter 20 cm and height 24 cm. The\nthickness of the cylindrical wall is 7.5 mm, and that of the top and bot tom walls is 12 mm\n(Fig. 1). This tank is made of stainless steel. Gallium melts at about 30◦C. Its density is\nρ= 6090 kg/m3and its kinematic viscosity is ν= 3.1110−7m2/s. Its electrical conductivity\nat our operating temperature (45◦C) isσ= 3.9106Ω−1m−1. The main difference with the\nusual Von K´ arm´ an setup is the use of two propellers to drive the fl ow, rather than impellers\nor disks. The two propellers are coaxial with the cylinder. They are m ade of 4 blades\n4inclined 45 degrees to the axis. The two propellers are counter-rot ating and the blades are\nsuch that both propellers are pumping the fluid from the center of t he tank towards its end\nfaces. The large scale flow is similar to the traditional counter-rota ting geometry of the Von\nK´ arm´ an setup. The propellers are 7 cm in diameter and their rotat ion ratefrotranges from\n3 Hz to 30 Hz. They are entrained at a constant frequency by DC mo tors, which drives a\nlarge scale flow consisting in two parts : first the fluid is pumped from t he center of the\nvessel on the axis of the cylinder by the propellers. It loops back on the periphery of the\nvessel and comes back to the center in the vicinity of the equatoria l plane. In addition to\nthis poloidal recirculation, the fluid is entrained in rotation by the pro pellers. Differential\nrotation of the fluid is generated by the counter rotation of the pr opellers, which induces a\nstrong shear layer in the equatorial plane where the variousprobe s arepositioned. The shear\nlayer generates a very high level of turbulence [9]: the velocity fluc tuations are typically of\nthe same order of magnitude as the large scale circulation.\nA stationary magnetic field B0is imposed by a solenoid. The solenoid is coaxial with\nthe propellers and the cylinder. B0is mainly along the axis of the cylinder with small\nperpendicular components due to the finite size of the solenoid (its le ngth is 32 cm and its\ninner diameter is 27 cm). The maximum magnetic field imposed at the cen ter of the tank\nis 1600 G.\nB. Governing equations and dimensionless parameters\nThe fluid is incompressible and its dynamics is governed by the Navier-S tokes equations:\nρ/parenleftbigg∂v\n∂t+(v·∇)v/parenrightbigg\n=−∇p+ρν∆v+j×B (1)\n∇·v= 0 (2)\nwherevis the flow velocity, pis the pressure, Bis the magnetic field and jis the electrical\ncurrent density. The last term in the r.h.s. of (1) is the Lorentz for ce.\nA velocity scale can be defined using the velocity of the tip of the blade s. The radius of\nthe propeller being R= 3.5 cm, the velocity scale is 2 πRfrot. A typical length scale is the\nradius of the cylinder L= 10 cm. The kinetic Reynolds number is then\nRe=2πRLfrot\nν. (3)\n5At the maximum speed reported here it reaches 2106. The flow is then highly turbulent and\nremains so even for a rotation rate ten times smaller.\nIn the approximations of magnetohydrodynamics, the temporal e volution of the magnetic\nfield follows the induction equation:\n∂B\n∂t=∇×(v×B)+η∆B. (4)\nHereη= (µ0σ)−1is the magnetic diffusivity, µ0being the magnetic permeability of vacuum\n(η= 0.20 m2s−1for gallium so that Pm=ν/η= 1.610−6) [11]. The first term in the\nr.h.s describes the advection and the induction processes. The sec ond term is diffusive and\naccounts for ohmic dissipation. A magnetic Reynolds number can be d efined as the ratio of\nthe former over the latter:\nRm=µ0σ2πRLfrot, (5)\nIt is a measure of the strength of the induction processes compar ed to the ohmic dissipation\n[11]. For Rm→0 the magnetic field obeys a diffusion equation. For Rm≫1, it is\ntransported and stretched by the flow. This can lead to dynamo ac tion, i.e. spontaneous\ngeneration of a magnetic field sustained by a transfer of kinetic ene rgy from the flow to\nmagnetic energy. In our study, Rmis about 3 at the maximum speed reported here ( frot=\n30 Hz): Induction processes are present but do not dominate ove r diffusion. As frotgoes\nfrom 3 to 30 Hz, the magnetic Reynolds number is of order 1 in all case s.\nIn our range of magnetic Reynolds number, the induced magnetic fie ldb=B−B0is\nweak relative to B0(of the order of 1%). The Lorentz force is j×B0wherejis the current\ninduced by the motion of the liquid metal (no exterior current is applie d). From Ohm’s\nlaw, this induced current is j∼σv×B0and is thus of order 2 πσRfrotB0. One can define\nan interaction parameter Nthat estimates the strength of the Lorentz force relative to the\nadvection term in the Navier-Stokes equation:\nN=σLB2\n0\n2πρRfrot, (6)\nBecause of the 1 /frotfactor, this quantity could reach very high values for low speeds,\nbut the flow would not be turbulent anymore. In order to remain in a t urbulent regime,\nthe smallest rotation rate reported here is 3 Hz. The maximum intera ction parameter\n(built with the smallest rotation frequency and the strongest applie d magnetic field) is then\napproximately 2 .5. Many studies on magnetohydrodynamics use the Hartmann numb er to\n6quantify the amplitude and influence of the magnetic field. This numbe r measures the ratio\nof the Lorentz force over the viscous one. It is linked to the intera ction parameter by the\nrelationHa=√\nNRewhich gives the typical value Ha≃700 for the present experiment.\nThe influence of the electrical boundary conditions at the end face s is determined by this\nnumber and the conductivity ratio K=σwlw\nσL, whereσwis the electrical conductivity of\nthe walls and lwtheir thickness. A very detailed numerical study on this issue has be en\nperformed in the laminar situation on a flow which geometry is very simila r to the present\none in [12]. As far as turbulent flows are concerned, Eckert et al. st ress the importance of\nthe product KHa, which measures the fraction of the electrical current that leaks from the\nHartmann layers into the electrically conducting walls. With 12 mm thick end faces Kis\nabout 0.05 so that KHa≃35: the walls have to be considered as electrically conducting.\nThese boundary effects are of crucial importance in experiments w here the boundary layers\ncontrol the rate of turbulence of the flow. However, in the prese nt experimental setup the\nfluid is forced inertially and we have checked that the magnetic field ha s only little influence\non the mean flow (see section IIB below). Furthermore, the high va lue of the Hartmann\nnumber implies that the dominant balance in the bulk of the flow is not be tween the Lorentz\nand viscous forces but between the Lorentz force and the inertia l term of the Navier-stokes\nequation: in the following, it is the interaction parameter and not the Hartmann number\nthat leads to a good collapse of the data onto a single curve. For the se reasons the present\nexperimental device allows to study the influence of a strong magne tic field on the forced\nturbulence generated in the central shear layer, avoiding any bou ndary effect.\nC. Probes and measurements\nThe propellers are driven at constant frequency. The current pr ovided to the DC motors\nis directly proportional to the torques they are applying. It is reco rded to access the\nmechanical power injected in the fluid.\nAs liquid metals are opaque, the usual velocimetry techniques such a s Laser Doppler\nVelocimetry or Particle Image velocimetry cannot be used. Hot wire a nemometry is difficult\nto implement in liquid metals even if it has been used in the past [2]. Other v elocimetry\ntechniques have been developed specifically for liquid metals. Among t hose are the potential\n7FIG. 2: Schematics of the potential probes. These probes are vertical and the electrodes are\npositioned in the mid-plane of the tank, where the shear laye r induces strong turbulence.\nprobes. They relyonthemeasurement ofelectric potentialdiffere nces induced bythemotion\nof the conducting fluid in a magnetic field [1, 13, 14]. The latter can be a pplied locally with\na small magnet or at larger scale as in our case. We built such probes w ith 4 electrodes\n(Fig. 2). The electrodes are made of copper wire, 1 mm in diameter, a nd insulated by a\nvarnish layer except at their very tip. The electrodes are distant o fl∼3 mm. The signal\nis first amplified by a factor 1000 with a transformer model 1900 fro m Princeton Applied\nResearch. It is further amplified by a Stanford Research low noise p reamplifier model SR560\nand then recorded by a National Instrument DAQ. Note that beca use of the transformer the\naverage potential cannot be accessed, so that our study focus es on the turbulent fluctuations\nof the velocity field.\nFor a steady flow, and assuming j=0, Ohm’s law gives ∇φ=v×Bso that the electric\npotential difference between the electrodes is directly related to t he local velocity of the\nfluid. One gets δφ=/integraltext\nv×B·dlintegrating between the two electrodes. If Bis uniform\nthenδφ=v⊥Blwherev⊥is the component of the velocity orthogonal to both the magnetic\nfield and the electrodes separation. In the general time-depende nt case, the link is not so\ndirect. Using Coulomb’s gauge and taking the divergence of Ohm’s law, one gets :\n∆φ=ω·B0 (7)\nwhereω=∇×vis the local vorticity of the flow [14]. The measured voltage depends o n\n8(a)00.5 11.5 22.5−30−20−10010203040\ntime [s]δφ [µV]\n(b)10010110210−810−610−410−2\nf [Hz]PSD\nFIG. 3: (a) Time series from a potential probe for frot= 20 Hz and an applied magnetic field\nof 178 G. (b) Corresponding power spectrum density. The dash ed line is a f−5/3scaling and the\nmixed line a f−11/3scaling. These straight lines are drawn as eye guides.\nthe vorticity component parallel to the applied magnetic field, i.e. to g radients of the two\ncomponents ofvelocity perpendicular to B0. The relationbetween the flowand thepotential\nis not straightforward but the potential difference can be seen as a linearly filtered measure-\nment of the velocity fluctuations. For length scales larger than the electrode separation, the\npotential difference can be approximated by the potential gradien t, which has the dimension\nofvB0. The spectral scaling of ∇φ/B0is expected to be the same as that of the velocity,\ni.e. thek−5/3Kolmogorov scaling. For smaller scales, some filtering results from th e finite\nsize of the probe. For scales smaller than the separation l, the values of the potential on the\ntwo electrodes are likely to be uncorrelated: if these scales are also in the inertial range, one\nexpects the spectrum of the potential difference to scale as the p otential itself, i.e. k−11/3\ndue to the extra spatial derivative. Assuming sweeping of the turb ulent fluctuations by the\n9average flow or the energy containing eddies [15] and a k−5/3Kolmogorov scaling for the\nvelocity spectrum, we expect the temporal spectrum of the meas ured potential difference to\ndecay as f−5/3for intermediate frequencies and as f−11/3for high frequencies in the inertial\nrange.\nAn example of a measured time series of the potential difference is sh own in figure 3\ntogether with its power spectrum density. For this dataset, one e xpects a signal of the order\nof 2πRfrotB0l∼20µV which is the right order of magnitude. Because of the quite large\nseparation of the electrodes, the cutoff frequency between the f−5/3and thef−11/3behaviors\nis low so that no real scaling is observed but rather trends. At the h ighest frequencies, the\ndecay is faster than f−11/3.\nBolonov et al. [16] had a rather empirical approach to take into acco unt the filtering\nfrom the probe. Assuming that the velocity spectrum should decay asf−5/3they observed a\nspectral response of the potential probe which displays an expon ential decay exp( −lf/0.6u).\nThe factor 0 .6 is most likely dependent on the geometry. We reproduced their ana lysis in\nfigure 4. In the inset is displayed the spectrum of figure 3 multiplied by the expected\nf−5/3scaling. The decay is seen to be exponential from about 25 Hz to 300 Hz (at higher\nfrequencies the signal does not overcome the noise). The charac teristic frequency of the\ndecay can be extracted and plotted as a function of the rotation f requency of the propeller\n(the local velocity is expected to scale as Rfrot). A clear linear dependence is observed, in\nagreement with the results of Bolonov et al.. The cutoff frequency is about twice frot. It\nis not very high due to the rather large size of the probe. Neverthe less the most energetic\nlength scales are resolved in our measurement. In the following we sh ow only direct spectra,\nand no correction of the filtering is attempted.\nWe can conclude that although some filtering is involved, the measure ment of potential\ndifferences gives an image of the spectral properties of the velocit y fluctuations. Any change\nin the spectral properties of the flow in the vicinity of the probe will t hus be visible on the\nspectrum of the potential difference.\nThe potential probes are quite large in order to fit a gaussmeter Ha ll probe in the vicinity\nof the electrodes (see Fig. 2). These probes are connected to an F.W.Bell Gaussmeter model\n7030 that allows to measure the induced magnetic field down to a few t enths of Gauss. The\nproximity between thevelocity andmagnetic fieldmeasurements allow s tostudy thepossible\ncorrelations between these two fields.\n10(a)10010110210−810−610−410−2\nf [Hz]PSDf−5/3\n010020030040050010−410−310−210−1PSD × f5/3\n(b)051015202530010203040506070\nfrot [Hz]cut−off frequency [Hz]\nFIG. 4: (a) Filtering effect due to the probe geometry. – Inset: semilog plot of the power spectrum\ndensity (PSD) of the potential difference compensated by f5/3. The dashed line is an exponential\nfit. – Main figure: lower curve, PSD; upper solid line: PSD corr ected from the exponential decay\nfitted in the inset. Dashed line: f−5/3decay. Same data set as that of the previous figure. (b)\nCutoff frequency of theproberesponsefor the probeused in (a ) and for various rotation frequencies\nof the propeller. The dashed line is a linear fit. The applied m agnetic field is 178 G.\nII. EFFECT OF THE APPLIED MAGNETIC FIELD ON THE TURBULENCE\nLEVEL\nA. Velocity field\nWe first focus on the fluctuation level of the velocity field accessed through measurements\nofpotentialdifferences. Theevolutionofthe rmsvalueofthepotentialdifference isdisplayed\nin figure 5 asa function of theapplied magnetic field and of the rotatio nrate. Fromequation\n110 500 1000 150002468x 10−5\nB0 [G]δφrms [V]\nFIG. 5: Evolution of the rms value of the potential difference a s a function of the applied vertical\nmagnetic field, for different values of the rotation frequency .•:frot= 5 Hz,△: 10 Hz, /square: 15 Hz,\n⋄: 20 Hz and ⋆: 30 Hz. The solid lines correspond to the azimuthal potentia l difference and the\ndashed line to the radial potential difference for the same fou r electrode potential probe.\n(2), the potential difference should behave as δφrms∝B0v/l. For low values of the applied\nmagnetic field, the interaction parameter Nis low: the magnetic field has almost no effect\non the flow, and the velocity scales as Rfrot. Thus, for high frotand lowB0(i.e. low N),\nδφrmssould be linear in both frotandB0. The upper curve corresponds to the highest\nvelocityfrot= 30 Hz. For low B0there is a linear increase of δφrms. Then it seems to\nsaturate. For smaller rotation rates, the linear part gets smaller a nd the saturation region\ngets wider. Eventually, for frot= 5 Hz, the potential decays for the highest values of the\napplied magnetic field. This demonstrates that there is a strong inte raction between the\nmagnetic field and the flow.\nTo investigate in more details the scaling properties of the potential difference, δφrms/frot\nis displayed versus B0in figure 6. One can clearly see that as frotdecreases, the potential\ndeviates from the linear trend for lower and lower values of the magn etic field.\nFigure 7 shows δφrms/B0as a function of frot. Here a linear trend is observed for large\nfrot. As the external magnetic field is increased, the fluctuations of th e potential are damped\nand the linear trend is recovered for increasingly high values of the r otation rate.\nAll that information can be synthesized by plotting the dimensionless potential\nσ⋆\nv=δφrms\nB0lRfrot. (8)\n120 500 1000 150000.511.522.53x 10−6\nB0 [G]δφrms / frot\nFIG. 6: Evolution of the rms value of the azimuthal potential difference normalized by the rotation\nfrequency, as a function of the applied vertical magnetic fie ld.•:frot= 5 Hz,△: 10 Hz, /square: 15 Hz,\n⋄: 20 Hz and ⋆: 30 Hz. The dashed line is a linear trend fitted on the first four points of the 30\nHz data.\n05101520253000.20.40.60.81x 10−3\nfrot [Hz]δφrms / B0\nFIG. 7: Evolution of the rms value of the azimuthal potential difference normalized by the imposed\nvertical magnetic field, as a function of the rotation freque ncy of the propeller. •:B0= 356 G,\n△: 712 G, /square: 1070 G, ⋄: 1420 G. The parallel dashed lines are used as eye guides. The upper one\ncorresponds to a linear fit of the data at B0= 356 G.\nThis quantity can be understood as the velocity fluctuations of the flow normalized by the\nforcing velocity of the propeller. This quantity is displayed as a funct ion of the interaction\nparameter N=σLB2\n0\n2πρRfrotin figure 8. In this representation, the data collapses fairly well on\na single master curve for high rotation rates. The damping of the tu rbulent fluctuations\n13(a)0 0.5 1 1.510−210−1\nNσv*\n  \n5 Hz\n10 Hz\n15 Hz\n20 Hz\n30 Hz\n(b)0 0.5 1 1.510−210−1\nNσv*\n  \n3 Hz\n5Hz\n7.5 Hz\n10 Hz\nFIG. 8: Evolution of the dimensionless potential σ⋆\nv(see text) as a function of the interaction\nparameter N. (a) and (b) correspond to the azimuthal potential difference taken from two different\ndatasets. The potential probes are similar but not perfectl y identical. The rotation frequencies\nreach the highest values in (a) and the lowest in (b). The repr esentation is semilogarithmic.\ncan reach one order of magnitude for Nclose to 1. In fig. 8(b) - that corresponds to lower\nrotation rates - there is a slight and systematic drift of the curves with the rotation rate.\nThis indicates a slight dependence in Reynolds number that comes likely from the fact\nthat the flow is not fully similar with frotfor low values of this rotation rate. The main\ndependence is clearly in the interaction parameter. The velocity fluc tuations are seen to\ndecay exponentially with Nfor values of Nup to order 1. For higher Nthe data is affected\nby the noise. The velocity fluctuations are so strongly damped that the signal to noise ratio\ndecreases significantly, ascanbeseen onthespectra inthefollowin gsections. Thedecayrate\nis about 2 .5 in fig. 8(a) and 3 .5 in (b). The difference may come from geometrical factors of\n14the probes which are not exactly similar in both datasets, or froma s light mismatch between\nthe positions of the probes in the two datasets. We observed a simila r collapse of potential\ndata with Nin a different experiment [17]. This older experiment was smaller in size, with\nonly one propeller and smaller magnetic field. Only the beginning of the e xponential decay\ncould be observed in that case.\nB. Induced magnetic field\n(a)0 500 1000 1500051015\nB0 [G]b       [G]r rms\n(b)00.20.40.60.8 110−210−1\nNbr rms / ( B0 Rm )\n  \n5 Hz\n10 Hz\n15 Hz\n20 Hz\n30 Hz\nFIG. 9: (a) Evolution of the rms value of the induced radial ma gnetic field as a function of the\napplied vertical magnetic field, for different values of the ro tation frequency. •:frot= 5 Hz, △:\n10 Hz,/square: 15 Hz, ⋄: 20 Hz and ⋆: 30 Hz. (b) Evolution of brrms/B0Rmas a function of the\ninteraction parameter N. The representation is semilogarithmic.\nThe induced magnetic field is of the order of one percent of the applie d field. This low\nvalue is due to the low magnetic Reynolds number which is at best of ord er 1. For low Rm\n15the induction equation reduces to\nB0·∇v+η∆b= 0 (9)\n(assumingthat B0isuniform). Theinducedmagneticfieldthusreflectsthevelocitygra dients\nin the direction of the applied magnetic field. We use it as a second tool to investigate the\nstatistical properties of the flow.\nThe fluctuation level of the radial magnetic field brrmsis displayed in figure 9(a). Its\nevolution with B0andfrotis strongly similar to that of the potential differences, as expected\nfrom the previous arguments. The azimuthal component displays t he same behavior (not\nshown).\nFor low values of the applied magnetic field, the flow is not affected ver y much by the\nLorentz force and from equation (9) one expects the induced mag netic field amplitude to\nscale as: b∝B0Rm(see [18] for example). From the previous section, the amplitude of\nthe velocity field decays exponentially with N. We thus expect brms/B0Rmto have the\nsame qualitative behavior. This is what is observed in figure 9(b). The various datasets\nare seen to collapse on a single master curve for Nup to 0.5. For higher N, the sensitivity\nof our gaussmeter is not high enough for the signal to overcome th e electronic noise. As\nfor the velocity, the curve for the lowest frotis below all the others, which confirms the\nslight dependence with Rmobserved on the potential measurements. At a given value of\nN, the collapse of the measurements shows that the fluctuations of the induced magnetic\nfield are indeed linear both in applied magnetic field and in magnetic Reyno lds number:\nbrms≃0.1B0Rm.\nWe have also measured the average value of the induced magnetic fie ld. From equation\n(9) it is linked to the vertical gradients of the time-averaged velocit y field. ∝angb∇acketleftbr∝angb∇acket∇ight/B0Rm\nand∝angb∇acketleftbθ∝angb∇acket∇ight/B0Rmare shown in figure 10 as a function of N. For both components, this\nrepresentation leads to a good collapse of the datasets. At a given value ofN, the collapse\nshows again that the average induced magnetic field is linear in RmandB0with∝angb∇acketleftbi∝angb∇acket∇ight ≃\n0.01B0Rmfor the data shown here. In an ideal Von K´ arm´ an experiment, wh en the two\npropellers counter-rotate at the same speed, the time averaged flow is invariant to a rotation\nof angleπaround a radial unit vector taken in the equatorial plane (denoted as/vector eron figure\n1). If the applied field were perfectly symmetric and the probe posit ioned exactly in the\nequatorial plane, this symmetry should lead to ∝angb∇acketleftbr∝angb∇acket∇ight= 0. However, the introduction of the\n16(a)00.20.40.60.81−0.01−0.00500.0050.010.015\nN<br> / ( B0 Rm )\n  \n5 Hz\n10 Hz\n15 Hz\n20 Hz\n30 Hz\n(b)00.20.40.60.8 100.0020.0040.0060.0080.010.012\nN<bθ> / ( B0 Rm )\n  \n5 Hz\n10 Hz\n15 Hz\n20 Hz\n30 Hz\nFIG. 10: (a) Evolution of∝angb∇acketleftbr∝angb∇acket∇ight\nB0Rmas a function of the interaction parameter N. (b) same for\n∝angb∇acketleftbθ∝angb∇acket∇ight\nB0Rm.\nprobe breaks the symmetry and neither the mecanical device nor t he applied magnetic field\nare perfectly symmetric. It has been observed in our setup and in t he VKS experiment\nthat the measurement of ∝angb∇acketleftbr∝angb∇acket∇ightis extremely sensitive to the position of the probe (private\ncommunicationfromtheVKScollaboration). Forexample, aslightmism atchofthepropeller\nfrequencies or of the relative positions of the probe and mid-plane s hear layer can lead to\nstrong changes in the mean value of the magnetic field. This may be th e reason why ∝angb∇acketleftbr∝angb∇acket∇ight\nis not zero here. Nevertheless there is a systematic change of the average with Nwhich is\nconsistent across the different values of the velocity and of the ap plied magnetic field. It\nindicates a change of the time-averaged flow in the vicinity of the pro be. To the shear layer\nlying in the equatorial plane corresponds strong radial vorticity, o rthogonal to the applied\nmagnetic field. The applied strong magnetic field will impact the shear la yer, as it tends to\n17elongate the flow structures along its axis. Even a small change in th e shear layer geometry\naffects strongly the measured ∝angb∇acketleftbr∝angb∇acket∇ight. Here we see that its sign is changed at high N.\nThe time-averaged induced azimuthal magnetic field ∝angb∇acketleftbθ∝angb∇acket∇ightis due to ω-effect from the\ndifferential rotation of the propellers [11, 18]. It is seen to decay slig htly (about 35%) with\nN. This effect may be due to some magnetic braking that leads to an elon gation of the shear\nlayer and thus to weaker differential rotation in the vicinity of the mid -plane. Nevertheless\nthis effect is limited and we expect the average large scale structure of the flow to remain\nalmost unchanged. The small change of the large scale flow is not the reason for the strong\ndamping of the turbulent fluctuations by an order of magnitude.\nC. Injected mechanical power\n0 5 10 15 20050100150200250300\nfrot [Hz]ε [W]\n  \n101100102\nfrot [Hz]ε [W]\n  \nFIG. 11: Injected mechanical power as a function of the rotat ion frequency of\nthe propellers. The circles correspond to measurements wit hout magnetic field, and\nthe triangles correspond to measurements at constant rotat ion frequency and B0=\n90,180,530,705,880,1000,1230,1240,1320,1410, and 1500 Gauss. The solid line has equation\ny= 0.032f3\nrotwhich comes from the turbulent scaling law. The inset is a log -log representation.\nAn interesting issue in MHD turbulence is to understand how the injec ted mechanical\npower is shared between ohmic and viscous dissipations. The torque sT1andT2provided\nby the motors can be accessed through measurements of the cur rent delivered to them. The\ninjected mechanical power is then ǫ= (T1+T2)2πfrot. This quantity has been measured as a\n18function of the rotation frequency with and without applied magnet ic field. The results are\ndrawn on figure 11: without magnetic field, the injected power follow s the turbulent scaling\nlawǫ∼f3\nrot. Moresurprisingly, wenoticethatatagivenrotationfrequencyth eexperimental\npointscorresponding todifferent amplitudes ofthemagnetic fieldar eindistinguishable. This\nobservation confirms that no change of the global structure of t he flow is induced by the\nmagnetic field. The fact that the injected power is independent of t he applied magnetic\nfield seems to be in contradiction with results from numerical simulatio ns where an ohmic\ndissipation of the same order of magnitude as the viscous one is obse rved when a strong\nmagnetic field is applied (the ohmic dissipation is around three times the viscous dissipation\nforN= 1inBurattinietal.[8]). Onecouldarguethattheremaybealargeohm icdissipation\ncompensated by a drop in the viscous dissipation: the energy flux in t he turbulent cascade\nwould then be dissipated mainly through ohmic effect without changing the overall injected\npower. However, a rough estimate of the ohmic dissipation Djgives values which are rather\nlow: with the maximum value of the induced magnetic field b≃0.1B0Rm, and assuming\nthat the magnetic field is dissipated mostly at large scale, one gets:\nDj∼j2\nσL3∼1\nσ(0.1RmB0)2\nL2µ2\n0L3≃4W (10)\nwhere we used the values Rm= 1, and B0= 1500G. This estimate is much lower than\nthe injected power and we may expect ohmic dissipation to remain neg ligible compared\nto viscous dissipation, even at order one interaction parameter. H owever, one also needs\nto know the current that leaks through the boundaries to evaluat e the additional ohmic\ndissipation that takes place inside the walls. As these effects are diffic ult to quantify, we are\nnot able to determine the ratio of ohmic to viscous dissipation. Never theless it is interesting\nto notice that the injected mechanical power remains the same whe n a strong magnetic field\nis applied, although velocity fluctuations are decreased by a factor 10 in the mid-plane of\nthe tank.\nD. Development of anisotropy\nThe application of a uniform magnetic field on a turbulent flow is known t o elongate\nthe flow structures in the direction of the applied field [3]. In decaying turbulent flows,\nthis effect eventually leads to the bidimensionalization of the flow [19]. As far as forced\n1900.20.40.60.8 10123456\nNa\n  \n5 Hz\n10 Hz\n15 Hz\n20 Hz\n30 Hz\nFIG. 12: Evolution of the anisotropy parameter a=ηbrrms\nδφrmsas a function of the interaction\nparameter.\nturbulence is concerned, only numerical simulations have demonstr ated this effect [7]. The\nmeasurement of both the induced magnetic field and the electric pot ential allows to quantify\nthe elongation of the turbulent structures in the zdirection: on the one hand, equation (7)\nlinks the electric potential to the vertical component of vorticity, i.e. to horizontal gradients\nof velocity. On the other hand the induced magnetic field is related to vertical gradients of\nvelocity through equation (9). We expect then that\n∆b\n∆φ∼1\nη∂||v\n∂⊥v(11)\nwhere∂||and∂⊥denote derivatives in directions parallel and perpendicular to the ap plied\nmagneticfield. Aswecannotaccessexperimentally theorderofmag nitudeoftheLaplacians,\nwe define the quantity a=ηbrrms\nδφrmswhich we expect to give a crude estimate of the ratio of\nthe vertical to the horizontal gradients of velocity. It is somewha t related to the parameter\nG1defined in Vorobev et al. [7]. This anisotropy parameter ais represented as a function\nof the interaction parameter Nin fig. 12 for different values of the rotation frequency: it\ndecreases from about 4 until it saturates around 1.5. This decrea se of the parameter aby a\nfactor about 3 when a strong magnetic field is applied provides eviden ce for the elongation\nof the flow structures in the zdirection: the derivatives of the velocity field in the direction\nofB0are much smaller than its derivatives in directions perpendicular to B0. Although the\nturbulence becomes moreanisotropic, itremains threedimensional even forthehighest value\nof the interaction parameter reached in this experiment. This is due to the fact that the\n20forcing imposed by the propellers is three dimensional and prevents the flow from becoming\npurely two dimensional.\nIII. TEMPORAL DYNAMICS\nWe observe that the turbulent fluctuations are being damped by ma gnetic braking when\na strong magnetic field is applied to homogeneous and nearly isotropic turbulence. An\ninteresting question is to know how this damping is shared among scale s. In this section we\nstudy the evolution of the power spectrum densities of the potent ial difference and induced\nmagnetic field.\nA. Potential\nWe show in figure 13 the power spectrum densities (PSD) of the dimen sionless potential\nv⋆=φ/B0lRfrot. For a rotation frequency of 20 Hz, the spectra are seen to deca y as the\nmagnetic field is increased, the shape of the different spectra rema ining the same (fig. 13(c)).\nThe decay of the PSD is at most of a factor 10. This dataset reache s a maximum value of the\ninteraction parameter about 0.4. For the lowest displayed value of t he rotation frequency\nfrot= 5 Hz (fig. 13(a)), Nreaches 1.5. As the interaction parameter increases, two distinct\nregimes are observed: first an overall decay of the PSD and secon d a change in the shape\nof the PSD. The highest frequencies are overdamped compared to the lowest ones: for the\nhighest value of N, the PSD decays by about 6 orders of magnitude at twice the rotat ion\nfrequency, whereas it decays by only two orders of magnitude at lo w frequency. We already\nobserved the first regime in a previous experiment performed on a d ifferent flow [17]. The\ninteraction parameter was not high enough in this experiment to obs erve the second regime.\nTo quantify more precisely the relative decay, we plot in figure 14 the ratio of the PSD\nofv⋆over the PSD at B0= 178 G (and at the same rotation rate of the propeller). At\nthe smallest values of N, the ratio weakly changes across the frequencies but it decays wit h\nN. WhenNis increased over approximately 0 .1 an overdamping is observed at the highest\nfrequencies. This extra damping can be qualitatively characterized by a cutoff frequency,\nwhich decreases extremely rapidly and seems to reach the rotation frequency for N≃0.3.\nAbove the cuto��� frequency, the decay rate seems to behave as a power law of the frequency.\n21(a)10−110010110−2100102104\nf / frotPSD(v*) frot\n(b)10−110010110−2100102104\nf / frotPSD(v*) frot\n(c)10−110010110−2100102104\nf / frotPSD(v*) frot\nFIG. 13: Evolution of the power spectrum density of the dimen sionless potential v⋆=δφ\nB0lRfrot\nfor the radial potential difference. The subfigures correspon d to different rotation rates of the\npropellers: (a) frot= 5 Hz, (b) 10 Hz, (c) 15 Hz. In each subfigure, the different curve s correspond\nto the different values of the applied magnetic field B0= 178, 356, 534, 712, 890, 1070, 1250, 1420\nand 1600 G. They are naturally ordered from top to bottom as th e magnetic field is increased. The\nnoise part of the spectra has been removed to improve the clar ity of the figures.\nThe exponent of this power law gets more and more negative as Nincreases and seems to\nreach−5 at the highest value of Ndisplayed here ( frot= 5 Hz,B0= 1600 G, N= 1.5).\nIn figure 15, we gathered the dimensionless spectra at various rot ation rates that corre-\nspond to the same interval of N. The spectra are collapsing fairly well onto each other. A\nlittle bit of scatter is observed, most likely due to the slight dependen ce onRmdescribed\npreviously. The shape of the spectra is essentially a function of the interaction parameter.\nIn experiments on the influence of a magnetic field on decaying turbu lence, the velocity\nspectrum goes from an f−5/3to anf−3behavior as the interaction parameter is increased.\nThis−3 exponent is attributed either to two-dimensional turbulence or t o a quasi-steady\n22(a)10−110010110−610−410−2100\nf / frotratio of PSDs\n(b)10−110010110−410−310−210−1100\nf / frotratio of PSDs\n(c)10−110010110−310−210−1100\nf / frotratio of PSDs\nFIG. 14: Evolution of the shape of the dimensionless potenti al’s power spectrum. The spectra of\nfigure 13 are divided by the spectrum obtained at the smallest value of the applied magnetic field\nB0= 178 G and at the same frot. The subfigures correspond to different rotation rates of the\npropellers: (a) frot= 5 Hz, (b) 10 Hz, (c) 15 Hz.\nequilibrium between velocity transfer and ohmic dissipation. As far as forced turbulence\nis concerned, we observe a strong steepening of the velocity spec trum, with slopes already\nmuch steeper than f−3forN= 1. We do not observe any signature of this quasi-steady\nequilibrium or of 2D turbulence. Once again, this comes from the thre e-dimensional forcing\nof the propellers which prevents the flow from becoming purely 2D.\nUsing the full ensemble of datasets at the various rotation rates o ne can interpolate the\nevolution of the shape of the dimensionless spectra as a function of bothf/frotandN. The\nresult is shown in figure 16. This representation summarizes all prev ious observations. As\nNincreases, first there is a self similar decay of the spectra. When Nreaches approximately\n0.1, an additional specific damping of the high frequencies is observe d. The high frequency\nspectrum gets extremely steep. This extremely steep regime cove rs the full inertial range\n2310−110010110−21001021041060 < N < 0.13PSD( v* )   frot\n10−110010110−21001021041060.13 < N < 0.26\n10−110010110−21001021041060.26 < N < 0.38PSD( v* )   frot\n10−110010110−21001021041060.38 < N < 0.51\n10−110010110−21001021041060.51 < N < 0.64\nf / frotPSD( v* )   frot\n10−110010110−21001021041060.64 < N < 0.77\nf / frot\nFIG. 15: Evolution of the shape of the normalized potential’ s power spectrum as a function of the\ninteraction parameter. Colors corresponds to the different r otation speeds: black 3 Hz, blue 5 Hz,\ngreen 7.5 Hz and red 10 Hz. Each subfigure corresponds to data r estricted to the specified interval\nofN. The dashed line corresponds to the spectrum at the smallest non zero value of N.\n10−1\n100\n10110−2\n10−1\n100−1012345\nf / frot NPSD(v*)   frot\nFIG. 16: Evolution of the shape of the power spectrum of the no rmalized potential as a function\nof the interaction parameter. Data for 5, 10, 15 and 20 Hz have been used for this representation.\n24forNclose to 1.\nB. Induced magnetic field\n(a)10−110010−2100102104\nf / frotPSD(br)   frot / ( B0 Rm )\n(b)10−110010−2100102104\nf / frotPSD(br)   frot / ( B0 Rm )\n(c)10−110010−2100102104\nf / frotPSD(br)   frot / ( B0 Rm )\nFIG. 17: Evolution of the power spectrum of the dimensionles s induced magnetic fieldbr\nB0Rm.\nThe subfigures correspond to different rotation rates of the pr opellers: (a) 10 Hz, (b) 15 Hz and (c)\n20 Hz. In each subfigure, the various curves correspond to diffe rent applied magnetic fields. The\ncurves are naturally ordered from top to bottom as the magnet ic field is increased. The results\nfor the following values of B0are displayed: B0= 178, 356, 534, 712, 890, 1070, 1250, 1420 and\n1600 G. The noise part of the spectra has been removed for the c larity of the figures.\nThe same analysis can be performed on the induced magnetic field. Th e dimensionless\nspectra of brare shown in figure 17. One concern is that the induced magnetic field is low, so\nthat the signal to noise ratio of the gaussmeter is not as goodas th at of the potential probes.\nThe magnetic field spectrum reaches the noise level at a frequency which is approximately\n2frot. This is also related to the fact that the scaling law expected from a K olmogorov-like\nanalysis for the magnetic field is much steeper than the one of the ve locity (f−11/3in the\n25dissipative range of the magnetic field, at frequencies correspond ing to the inertial range of\na velocity scaling as f−5/3). Nevertheless, the same two regimes are observed: at low N\nthe spectra are damped in a self similar way. At large Nthe high frequencies seem to be\noverdamped. However, the picture is not so clear because of the lo w signal to noise ratio.\n(a)10−110010−310−210−1100\nf / frotratio of PSDs\n(b)10−110010−210−1100\nf / frotratio of PSDs\n(c)10−110010−1100\nf / frotratio of PSDs\nFIG. 18: Evolution of the shape of the power spectrum of the in duced magnetic field br. The\nspectra of figure 17 are divided by the spectrum obtained at th e smallest value of the applied\nmagnetic field B0= 178 G and at the same frot. The subfigures correspond to different rotation\nrates of the propellers: (a) 10 Hz, (b) 15 Hz and (c) 20 Hz.\nThis last point is better seen on the ratio of the spectra in figure 18. The ratio corre-\nsponding to the highest applied magnetic field is decaying at large freq uencies.\nC. Coherence between velocity and magnetic field\nThe experiment has been designed to record the potential differen ce and the magnetic\nfield in the vicinity of the same point. Figure 19 displays the spectral c oherence between\nthese two quantities. In this figure we have identified the azimuthal potential difference with\n26(a)10010110200.050.10.150.20.25\nf [Hz]coherence of vr, vθ with br\nvθvr 101102PSDvθ\nbr\n(b)10010110200.050.10.150.2\nf [Hz]coherence of vr, vθ with bθ\nvθ\nvr101102PSD\nbθvr\nFIG. 19: Spectral coherence between the velocity and the ind uced magnetic field. (a) coherence\nbetween brandvrorvθ. The inset recalls the spectra of brandvθ. (b) coherence between bθand\nvrorvθ. The inset recalls the spectra of bθandvr. The data correspond to frot= 15 Hz and\nB0= 356 G.\nthe radial component of velocity vrand the radial potential difference with the azimuthal\ncomponent of velocity vθ, even though this identification may be somewhat abusive. We\nrecall that the coherence is one when both signals are fully correlat ed at a given frequency\nand zero if they are uncorrelated at this frequency. We have plott ed in the insets the\npower spectrum of each signal. The magnetic field spectrum falls belo w the noise level at\nabout 2frot, which explains why all coherence curves go to zero above 2 frot(there is no\nmagnetic signal at these frequencies). For lower frequencies, a s mall but non-zero value of\nthe coherence is observed for the following pairs: ( br,vr) with a coherence level above 0.1,\n(br,vθ) with a coherence close to 0.05 and ( bθ,vθ) with a coherence level barely reaching 0.1.\nThe coherence value changes a little for other values of frot, but the global picture remains\nthe same, with ( br,vr) being the most coherent.\nThe time correlations are shown in figure 20. No clear correlation is ob served except for\nthe pair ( br,vr) which displays a small peak reaching 0.1. The weak coherence obser ved in\nthe previous figure is barely seen here, probably because of an insu fficient convergence of\nthe correlation function.\nAt the low values of magnetic Reynolds number attained in our experim ent, the magnetic\nfield is diffused through Joule effect, so that its structure is mainly lar gescale. The Reynolds\nnumber being large, the velocity fluctuations develop down to much s maller scales. Because\nof the strong ohmic diffusion, the magnetic field is expected to be sen sitive mostly to large-\n27−5 0 5−0.100.1\ntime (s)Cb\nrv\nθ , Cb\nrv\nr vrvθ\n−5 0 5−0.100.1\ntime (s)Cb\nθv\nθ , Cb\nθv\nr\nvθvr\nFIG. 20: Correlation coefficients between the velocity and th e induced magnetic field. (a) Correla-\ntion between brandvr(upper curve) or vθ(lower curve). (b) Correlation between bθandvr(lower\ncurve) or vr(upper curve). The data correspond to frot= 15 Hz and B0= 356 G.\nscale and low frequency fluctuations of the velocity field. For examp le one could expect\nto observe a correlation or coherence between vθandbθ: fluctuations of the differential\nrotation can induce fluctuations in the conversion of the axial B0into azimuthal magnetic\nfield through ωeffect. As far as bris concerned, the poloidal recirculation bends the vertical\nfield lines towards the exterior of the tank in the vicinity of the mid-pla ne, a process which\ninduces radial magnetic field from the vertical applied field B0. Fluctuations of this poloidal\nrecirculation thus directly impacts the radial induced magnetic field, hence the correlation\nbetween vrandbr. Although such low frequency coherence is indeed observed in figur e 19\nits amplitude remains very low, so that there is almost no correlation b etween the velocity\nfield and the magnetic field measured in the vicinity of the same point.\nIV. DISCUSSION AND CONCLUSION\nThe effect of a strong magnetic field on forced turbulence is studied experimentally with\npotential probes and induced magnetic field mesurements. The velo city fluctuations are\nstrongly damped as the applied magnetic field increases: for N≃0.6, the turbulence inten-\nsity in the mid-plane of the tank is decreased by an order of magnitud e. As a consequence,\nthe standard deviation of the induced magnetic field - normalized by B0Rm- also diminishes\nby a factor ten. The spectrum of the non-dimensional potential v∗is affected in two differ-\n28ent ways by the magnetic field: for low values of the interaction para meter the spectrum\ndecays uniformly at all frequencies, its shape remaining the same. F or higher values of N,\nwe observe an overdamping of the high frequencies. The same effec t is seen on the induced\nmagnetic field spectra.\nWe identify several features which highlight the very different beha viors of forced and\ndecaying turbulence when they are subject to a strong magnetic fi eld: decaying turbulence\nis thought to evolve towards a bidimensional structure. Its velocit y spectrum displays a −3\nexponent which can be attributed either to this bidimensionalization o r to a quasi-steady\nequilibrium between velocity transfer and ohmic dissipation. In the pr esent experiment,\nno such−3 exponent is observed, and the spectra are much steeper (expo nent−5 to−6)\nfor values of Nhigher than 0 .5. Moreover, we have introduced a parameter a=ηbrms\nδφrmsto\nquantify the anisotropy of the turbulence in the mid-plane of the ta nk. When Nincreases,\nthe decrease of this parameter by a factor 3 is the signature of th e elongation of the flow\nstructures along the applied magnetic field. However, the flow alway s remains 3D since ais\nnon-zero even for the highest value of the interaction parameter reached in this experiment.\nThese differences between forced anddecaying turbulence come f romthe 3Dforcing imposed\nby the propellers, which rules out the possibility of a 2D statistically st eady state of the flow.\nIt would be interesting to perform the same kind of experimental st udy of the anisotropy\nwith other forcing mechanisms, such as current-driven MHD flows o r turbulent thermal\nconvection in a liquid metal.\nWe have studied the evolution of the injected mechanical power as t he applied magnetic\nfield increases and found almost no influence of the latter: the injec ted mechanical power\nremains the same although velocity fluctuations are decreased by a n order of magnitude in\nthe central shear layer.\nFinally, we stress the poor level of correlation between the velocity and induced magnetic\nfields measured in the vicinity of the same point, and attribute it to th e scale separation\nbetween the two fields.\nThe strong damping of turbulent fluctuations by the magnetic field c an be invoked as\na saturation mechanism for turbulent dynamos: in a dynamo experim ent, one observes\nspontaneous generation of magnetic field when the magnetic Reyno lds number is above a\ncritical value Rmc. If the turbulent fluctuations are involved in the generating proce ss of the\nmagnetic field (through αωorα2mechanisms for instance), there is a critical level of rms\n29turbulent fluctuations σvcabovewhich magneticfield isgenerated( σv=σvcforRm=Rmc).\nForRm > Rm c, the initial level of turbulent fluctuations is above σvc, andthe magnetic field\ngrows exponentially from a small perturbation: the interaction par ameter increases, and the\nturbulent fluctuations are damped according to figure 8. An equilibr ium is reached when the\ndamping is such that the rms turbulent fluctuations are reduced to σvc. For small values of\nNweobserved that σv(N) =σv(N= 0)e−γN≃σv(N= 0)(1−γN), with2.5< γ <3.5. The\nsaturated value NsatofNfollows from the equality σv(Nsat) =σvc=σv(N= 0)(1−γNsat).\nAsσvis proportional to Rm, we getNsat=Rm−Rmc\nγRmc, hence the following scaling law for the\nmagnetic field:\nB2\nsat=2πρRfrot\nσLγRm−Rmc\nRmc(12)\nThis is the turbulent scaling law for the saturation of a dynamo, which was originally\ndescribed by P´ etr´ elis et al. (see [20] for instance). Although the exact geometry of the\nexperimental setup and large scale magnetic fields are different fro m that of the present\nexperimental study, results from the VKS dynamo can be used to t est this relationship:\nusingR= 15.5 cm,L= 21 cm, frot= 16 Hz, ρ= 930 kgm−3,σ= 9.5106Ω−1m−1and\nγ= 3, the computed magnetic field amplitude is Bsat≃290 G forRm−Rmc\nRmc=1\n3. This is the\nright order of magnitude: the amplitude of the VKS dynamo field meas ured in the vicinity\nof the axis of the cylinder is approximately 150 Gfor this value of Rm([21]: figure 3(b)).\nTheauthorswouldlike tothankF.P´ etr´ elisforhiscomments andfo rhishelpinthedesign\nof the experimental setup, and S Fauve for insightful discussions . This work is supported\nby ANR BLAN08-2-337433.\n[1] S. Eckert, G. Gerbeth, W. Witke and H. Langenbrunner, “MH D turbulence measurements in\na sodium channel flow exposed to a transverse magnetic field,” Int. J. Heat Fluid Flow 22,\np. 358-364, (2001).\n[2] A. Alemany, R. Moreau, P.L. Sulem and U. Frisch, “Influenc e of an external magnetic field\non homogeneous MHD turbulence,” J. M´ ecanique 18, 2, (1979).\n[3] B. Knaepen, R. Moreau, Magnetohydrodynamic turbulence at low magnetic Reynolds n umber,\nAnn. Rev. Fluid Mech. 40, p. 25-45 (2008).\n30[4] D. R. Sisan, W. L. Shew and D. P. Lathrop, “Lorentz force effe cts in magneto-turbulence,”\nPhys. Earth Planet. Int. 135, p. 137-159, (2003).\n[5] O. Zikanov and A. Thess, “Direct numerical simulation of forced MHD turbulence at low\nmagnetic Reynolds number,” J. Fluid Mech. 358, p. 299-333, (1998).\n[6] T. Boeck, D. Krasnov, A. Thess and O. Zikanov, “Large-Sca le Intermittency of Liquid-Metal\nChannel Flow in a Magnetic Field,” Phys. Rev. Lett. 101, 244501, (2008).\n[7] A. Vorobev, O. Zikanov, P. A. Davidson and B. Knaepen, “An isotropy of magnetohydrody-\nnamic turbulence at low magnetic Reynolds number,” Phys. Fl uids17, 125105, (2005).\n[8] P. Burattini, M. Kinet, D. Carati and B. Knaepen, “Anisot ropy of velocity spectra in qua-\nsistatic magnetohydrodynamic turbulence,” Phys. Fluids 20, 065110, (2008).\n[9] L. Mari´ e, F. Daviaud, “Experimental measurement of the scale-by-scale momentum transport\nbudget in a turbulent shear flow,” Phys. Fluids 16, p. 457-461, (2004).\n[10] R. Monchaux, M. Berhanu, M. Bourgoin, M. Moulin,P. Odie r, J.-F. Pinton, R. Volk, S. Fauve,\nN. Mordant, F. P´ etr´ elis, A. Chiffaudel, F. Daviaud, B. Dubru lle, C. Gasquet, L. Mari´ e and F.\nRavelet, “Generation of a Magnetic Field by Dynamo Action in a Turbulent Flow of Liquid\nSodium,” Phys. Rev. Lett. 98, 044502, (2007).\n[11] H. K. Moffatt, Magnetic Field Generation in Electrically conducting Flui ds, (Cambridge\nUniversity Press, 1978).\n[12] A. Kharicha, A. Alemany, D. Bornas, “Influence of the mag netic field and the conductance\nratio on the mass transfer rotating lid driven flow,” Int. J. H eat Mass Transfer 47, p. 1997-\n2014, (2004).\n[13] R. Ricou and C. Vives, “Local velocity and mass transfer measurements in molten metals\nusing an incorporated magnet probe,” Int. J. Heat Mass Trans fer25, p. 1579-1588, (1982).\n[14] A. Tsinober, E. Kit, M. Teitel, “On the relevance of the p otential-difference method for\nturbulence measurements,” J. Fluid Mech. 175, p. 447-461, (1987).\n[15] H. Tennekes and J.L. Lumley, A First Course in Turbulence , (MIT Press, Cambridge, 1972).\n[16] N. I. Bolonov, A. M. Kharenko, A. E. ´Eidel’man, “Correction of spectrum of turbulence in the\nmeasurement by a conduction anemometer,” Inzhernerno-Fiz icheskii Zhurnal 31, 243, (1976).\n[17] M. Berhanu, B. Gallet, N. Mordant and S. Fauve, “Reducti on of velocity fluctuations in a\nturbulent flow of liquid gallium by an external magnetic field ,” Phys. Rev. E 78, 015302,\n(2008).\n31[18] P. Odier, J.-F. Pinton and S. Fauve, “Advection of a magn etic field by a turbulent swirling\nflow,” Phys. Rev. E 58, p. 7397-7401, (1998).\n[19] J. Sommeria and R. Moreau, “Why, how, and when, MHD turbu lence becomes two-\ndimensional,” J. Fluid Mech. 118, p. 507-518, (1982).\n[20] F. P´ etr´ elis, N. Mordant, S. Fauve, “On the magnetic fie lds generated by experimental dy-\nnamos,” Geophysical and Astrophysical Fluid Dynamics 101, p. 289-323, (2007).\n[21] R. Monchaux, M. Berhanu, S. Aumaˆ ıtre, A. Chiffaudel, F. D aviaud, B. Dubrulle, F. Ravelet,\nS. Fauve, N. Mordant, F. P´ etr´ elis, M. Bourgoin, P. Odier, J .-F. Pinton, N. Plihon and R.\nVolk, “The Von K´ arm´ an Sodium experiment: turbulent dynam ical dynamos,” Phys. Fluids,\n21, 035108 , (2009).\n32"    },    {        "title": "2309.15885v1.Dispersion_and_damping_of_ion_acoustic_waves_in_the_plasma_with_a_regularized_kappa_distribution.pdf",        "content": " \nDispersion and damping of ion-acoustic wa ves in the plasma with a regularized \nkappa-distribution \n \nHuo Rui, Du Jiulin \nDepartment of Physics, School of Science, Tianjin University, Tianjin 300350, China\n \nAbstract The dispersion and damping of ion-acoustic waves in the plasma with a regularized \nkappa-distribution are studied. The generalized dispersion relation and damping rate are derived, \nwhich both depend significantly on the parameters α and κ. The numerical analyses show that the \nwave frequency ( ωr /ωpi) and the  damping rate ( -γ /ωr) of ion-acoustic waves in the plasma with \nthe regularized kappa-distribution  are both generally less than those in the plasma with the kappa- \ndistribution , and if κ < 3/2 the ion-acoustic waves and their damping rate exist in the plasma with \nthe regularized kappa-distribution . \nKeywords : kappa-distribution, ion-acoustic waves, complex plasma, nonextensive statistics  \n1. Introduction \nIon-acoustic waves are basic low frequency oscillation waves, which can be observed in \nspace and laboratory plasmas. The kinetic model is  a basic theoretical study of ion-acoustic wave \nin plasmas, it is a statistical model based on the kinetic equations and the distribution functions of \nplasma components. Usually, one assumes the plasma components to be a Maxwellian distribution \nand studies the dispersion and damping rate of ion-acoustic waves from the kinetic model [1]. However, for the systems with long-range interact ion, long memory or multifractal structure, the \nvelocity distribution of particles is not a Maxwellian one, but often has a long power-law tail, a non-Maxwell distribution. Non-Maxwellian distributions are very common in nonequilibrium astrophysical and space plasmas, such as the q-distribution in nonextensive statistics [2], the \nkappa-distribution [3], the non-thermal α-distribution [4], the Vasyliunas–Cairns   distribution [5], \nand so on. The plasma dispersion function was modified when the plasma is a kappa-distribution [6]. The  α-distribution was used to simulate the non-thermal electrons observed by Freja and \nViking satellites in the space plasma [7]. The ion-acoustic waves of the plasmas with the \npower-law q-distribution were studied in the framework of nonextensive statistics [8, 9]. T he \nLandau damping of electrostatic modes with a α-distribution was presented [10]. The ion-acoustic \nwaves in the plasma with a non-thermal Vasyliu nas–Cairns distribution was also studied [11]. \nThe kappa-distribution and the related property have been widely applied to describe the \ncollective behaviors of the plasmas such as the solar wind [12], the flare [13], the Earth \nmagnetopause and magnetosheath [14], the interstellar medium and some planetary magneto- \nspheres [15], etc.  But there might be a mathematical issue in the kappa-distribution function due to  \n1 \n  \nκ > 3/2, i.e., existence of the velocity moments  is restricted. Recently, a regularized kappa- \ndistribution function was defined [16], with which the exponential cutoff of high energy tail made \nall velocity moments valid for all κ > 0. If one used the regularized kappa-distribution to replace \nthe standard kappa-distribution, the properties of plasmas might not be the same. For example, it \nwas employed to study moments in magneto-hydrodynamics [17], solitary ion-acoustic waves [18], \nand Langmuir waves [19]. In this paper, we study the dispersion and damping rate of ion-acoustic \nwaves in the plasma with the regularized kappa-distribution.  \nThe paper is organized as follows. In section 2, the dispersion relation and the damping rate \nof ion-acoustic waves are derived from the kinetic theory. In section 3, numerical analyses are made. And in section 4, the conclusion is given. \n2. The dispersion and damping of ion-acoustic waves \nWe now introduce the theory of ion-acoustic waves for the unmagnetized and collisionless \nplasma. The kinetic model is the Vlasov equation [20],  \n()ss s s\nsff q f\ntm∂∂ ∂+⋅ + +× ⋅ =∂∂ ∂vE v Brv0,                         ( 1 )  \nwhere fs is the distribution function of the plasma components ( s = i for ions and s=e for electrons); \nms and qs are respectively mass and charge of a partic le. If there is a weak disturbance around a \nstate, the linear perturbation in field E, B and distribution function was considered as \n01 (, ,) (, ,) (, ,)ss sf tf t f t=+ rv rv rv , \n01 (,) (,) (,)tt=+ Er E r E r t\nt, \n            .                            ( 2 )  01 (,) (,) (,)tt=+ Br B r B r\nBut, in the case of electrostatic oscillation, it is  unnecessary to consider the magnetic field, and the \ndisturbing electric field is only electrostatic, i.e.,  \n    E 0 = B B0 = 0      1 ϕ =−∇ E .                          ( 3 )  \nWe assume that fs0 and E0 are known and satisfy the Vlasov equation and Poisson equation: \n 00 0\n0 0ss s s\nsff q f\ntm∂∂ ∂+⋅+ ⋅=∂∂ ∂vErv,                          ( 4 )  \n0\n01\nss\nsqf dε∇⋅ = ∑ 0∫E v.                             ( 5 )  \nSubstituting Eqs. (2)-(5) into Eq .(1), we obtain  the equation, \n11 00ss s s\nsff q f\ntmϕ ∂∂ ∇ ∂+⋅− =∂∂ ∂vrv,                        (6) \nand \n                         22\n1\n01\nss\nssqf d ϕϕε∇= ∇ = −∑∑ s∫v.                        ( 7 )  \nIf frequency and wave vector is ω and k, and, without lose of generality, the x-axis is along \nthe direction of wave vector, then the solution of fs1 and φ has the following form: \n1( , , ) ( )exp( )exp( )ss k f tf i i t ω =⋅ − xv v k x ( , ) exp( )exp( )k tii t , ϕω=⋅ −xk x .           ( 8 )  ϕ\n2 \n  \nFrom Eq. (6) and Eq.(7) we can get that \n0 (/ ) ( )ss s\nskqm ffkϕω⋅∇=−−⋅vk\nkv,                          ( 9 )  \nand \n2\n2 00\n2\n0ˆ/ 1(1 ) 0/ss s x\nk\ns sxqn f vkmk vkϕεω∞\n−∞∂∂+−∑ ∫ xdv=.                     ( 1 0 )  \nTherefore, the dispersion function of ion-acous tic waves is obtained from the equation [20], \n                      2\n0\n2ˆ/(, ) 1 0/ps sx\nx\ns xfvDk d vkv kωωω∞\n−∞∂∂=+−∑∫=,                     ( 1 1 )  \nwhere 0ˆ\nsfis the normalized distribution function, 00ˆ /0 s ss f fn= , and\n 2\n2 0\n0ss\nps\nsqn\nmωε= is the natural \noscillation plasma frequency. \nFrom Eq.(11) we see that the dispersion relation of ion-acoustic waves is determined by the \nvelocity distribution function of particles in the plasma. Thus if one used the regularized kappa \n-distribution to replace the standard kappa-distribution, the property of the ion-acoustic waves \nwould be not the same. The regularized kappa-distribution  was defined by introducing a parameter \nα > 0 [16] by  \n                      122\n2\n00 2() 1 e x pss\nTs Tsvvfv n Nvκ\nακ ακ−−⎛⎞ ⎛⎞=+ − ⎜⎟ ⎜\n⎝⎠ ⎝⎠2v⎟,               ( 1 2 )  \nwhere ( )3/221 2( 3/2 ,3/2 ; )Ts Nv Uακ πκ−−= κακ − is the normalized constant [19], vTs=s s Bm T k/ is the \nthermal speed, κ>0 and α ≥0 are two non-thermal parameters. U(a, b; z) in Nακ is a Tricomi \nfunction (or a Kummer U function), defined by \n                 1\n01( , ; ) exp( ) (1 )()ab aUa bz z tt t d ta∞−− −=− +Γ∫1,                     ( 1 3 )  \nwhere Г (a) is a Gamma function. When we take α=0, (12) becomes the kappa-distribution, and \nwhen we take α=0 and κ→∞, (12) becomes to a Maxwellian distribution. \nBy integrating the function (12) over vy and vz , one can obtain the 1-dimensional expression \nof the regularized kappa-distribution [18,19],  \n22\n22 2\n00 22() 1 , 1 ; ( 1 ) 1xx\nsx s T s\nTs Ts Tsvvf v n N v U expvvκ\nακπκ κ α κ ακκ−⎛ ⎞ ⎛⎞ ⎛=− + + ⎜ ⎟ ⎜⎟ ⎜\n⎝ ⎠ ⎝⎠ ⎝⎠2\n2xv\nv⎞− ⎟.           ( 1 4 )  \nWhen we take α=0, the regularized kappa-distribution (14) recovers to the 1-dimensional  kappa- \ndistribution, \n()\n()2\n0\n22 1\n2() 1s x\nx\ntstsn vfvv vκ\nκκ\nκ πκ κ−Γ ⎛⎞= ⎜\nΓ− ⎝⎠+ ⎟,                               ( 1 5 )  \nwhich is equivalent to the q-distribution in nonextensive statistics when it is applied to study \nastrophysical and space plasmas [21,22]. \nNow using the new distribution function and substituting (14) into Eq. (11), one gets that  \n3 \n  \n1 2 22 2 2\n, 22\n11 2 22\n22 2\n22 2exp( / )(,) 12 1/\nexp( / )          1 2 1 exp( / ) 1/ps xx x T s\nx\ns x Ts\nps xx\nxx Ts x\ns x Ts Tsvv v vDk N d vkv kv\nvvNd v v v d vkk kv vκ\nακ α κ\nκκ\nακω αωπω κ\nω α ωπαω κκ−−\n∞\n−∞\n−− −−\n∞∞\n−∞ −∞⎛⎞ −−=+ +⎜⎟− ⎝⎠\n⎡ ⎛⎞ ⎛⎞ −=+ + − − +⎜⎟ ⎜⎟− ⎝⎠ ⎝⎠ ⎣∑ ∫\n∑ ∫∫22 2\nx Ts vv\nv\n1 2 2 22 2 11\n22\n22 2 2 2 33 11\n22 222( , ; ) exp( / ) /         1 1 ( ) 1/ (, ; ) (, ; )ps xx x T s\ns Ts x Ts TsU vv v v kdvk kv U v Uκω κακ α ω\nω κκ α κ κ πκ κ α κ−−\n∞\n−∞⎤\n⎢⎥\n⎢⎥⎦\n⎡⎤ ⎛⎞ − −⎢⎥ =+ − +⎜⎟− − − ⎢⎥ ⎝⎠ ⎣⎦∑ ∫2\nv\n    ()2 2 11\n22\n, 22 2 33\n22(, ; ) 21(, ; )ps\nss\ns TsUZkv Uακω κακ1ξ ξκκ α κ−⎡ =+ −⎣−∑ ⎤⎦,                                  ( 1 6 )  \nwhere ξs = ω/(kvTs) is the ratio of the phase velocity ω/k to the thermal speed vTs , and Z(ξs) is the \ngeneralized dispersion function,  \n         ()( )1 222\n, 2 11\n22exp 11\n(, ; )s\nsx xZd xx Uκ\nακα\nξκξ πκ κ α κ−−\n∞\n−∞− ⎛⎞=+ ⎜⎟− − ⎝⎠∫  .       ( 1 7 )  \nObviously, there exists a singularity in (17) at x = ξs ; According to Plemelj formula [20], if the \nfrequency is written as the complex form, ω = ωr + iγ , then Eq. (17) can be expressed as \n( )1 222\n,2 11\n22exp 1() P r 1\n(, ; )s\nsx xZ dxx Uκ\nακα\nξκξ πκ κ α κ−−\n∞\n−∞− ⎛⎞=+ ⎜⎟− − ⎝⎠∫\n  \n(12\n22\n2 11\n2211e x p(, ; )s\ns iUκξ π). αξκκ κακ−−⎛⎞−+ ⎜⎟− ⎝⎠−              ( 1 8 )  \nThen Eq. (16) can be written as \n( )1 22 2 2 2 11\n22\n, 22 2 2 33 11\n22 22exp (, ; ) 2(,) 1 1 P r 1(, ; ) (, ; )ps s\ns s Tsx U xDk d xx kv U Uκ\nακα ω κακ ξωκξ κκ α κ πκ κ α κ−−\n∞\n−∞− ⎡ −⎛=+ − + ⎢ ⎜⎟− − − ⎢ ⎝⎠ ⎣∑ ∫⎞\n \n(12\n22\n2 11\n22/1e x p(, ; ).s\ns iUκξ πκξκ κακ−−⎤ ⎛⎞)sαξ⎥ −+ ⎜⎟−−\n⎥ ⎝⎠ ⎦                         ( 1 9 )  \nFor the ion-acoustic waves, the relationship between the phase velocity and the thermal \nvelocities is usually that vTi < ω / k < v Te, so the modified dispersion function can be expanded in \nsuch a way with a small variable expansion ( ξe <<1) for electrons but with a large variable \nexpansion (ξ i >>1) for ions. In this way, making series expansion for the integrand in Eq. (19) we \nhave that (see Appendix), \n22 2 2 2 22 55 11\n22 22\n, 24 2 2 2 33 33\n22 2232 (, ; ) (, ; )(,) 12 ( ,; ) ( ,;pi pi Ti pe\nk\nTekv UUDkUv kUαωκ ω ω\n2)κακ κακωω ωκ α κ κ κ−−=− − +−− α κ \n1 2 22\n32 3 3 2 2 33\n2221e xp(, ; )ps\ns Ts Ts TsiUk v k vκω πω ω α\nκκ α κ κ−−⎛⎞ ⎛ −++ ⎜⎟ ⎜− ⎝⎠ ⎝∑2\n22kvω⎞\n⎟\n⎠.          ( 2 0 )  \n4 \n  \nInserting ω = ωr + iγ in Eq. (20) and, according the disp ersion equation (11), making the real \npart of (20) zero, we obtain that \n     22 2 2 2 22 55 11\n22 22\n24 22 2 2 33 33\n22 2232 (, ; ) (, ; )102 ( ,; ) ( ,; )pi pi Ti pe\nrr Tekv UU\nUv kUωω ω κκ α κ κ α κ\nωω κ α κ κ κ α κ−−−− + =−−.        ( 2 1 )  \nFurther, we use 22/22\nDsT s pvs λ ω = , and write Eq.(21) as \n22 2 2 22 55\n2 22\n22 2 33 11\n22 22 22\n2 33\n223( , ; )1,(, ; ) 2 (, ; )\n(, ; )De pi Ti\nr\nr\nDek U kv\nUUkUλω κκ α κωκακ ω κακλκκ α κ⎛⎞ −=+ ⎜⎟− ⎝⎠+−−   (22) \n        Because /Ti rkv ωis very small and the second term in the above bracket is also very small, we \ncan approximately apply this expression, \n          22 2\n2\n2 11\n22 22\n2 33\n22(, ; )\n(, ; )De pi\nr\nDek\nUkUλωωκακλκ κακ≈−+−,                          (23) \nto the right-hand side of Eq.( 22) [18], and then we derive the generalized dispersion relation, \n        2 2 55\n22 2 22\n22 33 11\n22 22 22\n2 33\n223( , ; /\n(, ; ) 2 (, ; )\n(, ; )Be i\nrT i\nDeU kkT mkvUUkU) κ κακωκακ κακλκκ α κ−=+−+−−,            ( 2 4 )  \nWhen we take α = 0, this expression (24) can correctly recover to the dispersion relation of the \nplasma with the q-distribution (or the κ-distribution) in nonextensive statistics [8]. \nFor a weak Landau damping, the damping rate can be obtained [18] by the equation, \n            ,\n,Im ( , ),Re ( , ) /r\nrrDk\nDkακ\nακωγω ω=−∂∂                             ( 2 5 )  \nwhere from Eq.(20) we have that \n    12 22\n, 32 3 3 2 2 33\n222Im ( , ) 1 exp ,(, ; )ps rr\nr\ns Ts Ts TsDkUk v k vκ\nακω ωω πωκκ α κ κ−−⎛⎞ ⎛=+ ⎜⎟ ⎜− ⎝⎠ ⎝∑2\n22r\nkvαω⎞− ⎟\n⎠  \n22 2 2 2 22 55 11\n22 22\n, 24 2 2 2 2 33 33\n22 2232 (, ; ) (, ; )Re ( , ) 1 .2 ( ,; ) ( ,;pi Ti pi pe\nr\nrr Tekv UUDkUk vUακωω ω\n)κ κακ κακωω ω κακ κ κακ−−=− − +−− \nAnd then we derive the damping rate of the ion-acoustic waves, \n142 2 3\n23 3 2 2 2 2 33\n22/1e xp(, ; )rr\nTi Ti Ti Uk v k vκ2\nr\nkvω ωα πκγκακ κ−−⎡⎛⎞ ⎛⎢ =− + − ⎜⎟ ⎜− ⎢⎝⎠ ⎝⎣ω⎞\n⎟\n⎠\n 12 3 22\n2 3 22 221e xppe Ti r\npi Te Te Tev\nvk v k vκω ωα\nωκ−−⎤ ⎛⎞ ⎛2\nrω⎞⎥ ++ − ⎜⎟ ⎜ ⎟⎥ ⎝⎠ ⎝ ⎦⎠.                               ( 2 6 )  \n5 \n  \n   In order to do numerical analyses, we need to express Eq.(26) as a function of temperatures \nand masses of the plasma components. By using Eq.(15), we have that \n2 2 55\n22\n2 22 2 11 33\n22 22 22\n2 33\n223( , ; 1\n(, ; ) 22 ( ,;\n(, ; )e r\nTi i\nDeU T\nU kv T UkU)\n)κ κακ ω\nκακ κακλκκ α κ−=+− −+−,                ( 2 7 )  \n    2 22 55\n22\n2 22 22 2 11 33\n22 22 22\n2 33\n223( , ; ) 1\n(, ; ) 2(\n(, ; )ei e i rr\nTe Ti i e i e\nDeU mT m T\nU kv kv m T m T UkU, ; )κ κακ ωω\nκακ κακλκκ α κ⎛⎞\n⎜⎟−⎜⎟ == +⎜⎟ − −+⎜⎟− ⎝⎠.  (28) \nSubstituting Eq.(27), Eq.(28) and ()2 3\n3/2\n32/pe Ti\nie e i\nTe pivTT m mvω\nω= / into Eq.(26), it becomes \n() (3/2 1133 2 2\n22 22\n2 33\n22/1 exp 1 exp(, ; )ie\nr eiTm\nUT mκκγπ κ χχ ζαχ αζωκ α κ κ κ−− −−⎡⎤ ⎛⎞ ⎛⎞ ⎛⎞⎢⎥ =− + − + + − ⎜⎟ ⎜⎟ ⎜⎟− ⎢⎥⎝⎠ ⎝⎠ ⎝⎠ ⎣⎦),   (29) \nwhere we have denoted that \n1/2\n2 55\n22\n2 2 11 33\n22 22 22\n2 33\n223( , ; ) 1\n(, ; ) 22 ( ,; )\n(, ; )e r\nTi i\nDeU T\nU kv T UkUκκ α κ ωχκακ κακλκκ α κ⎛⎞\n⎜⎟−⎜⎟ ≡= +⎜⎟ − −+⎜⎟− ⎝⎠,             ( 3 0 )  \n1/2\n2 55\n22\n2 2 11 33\n22 22 22\n2 33\n223( , ; ) 1\n(, ; ) 2( ,\n(, ; )e i r\nTe i e\nDeU mT\nU kv m U TkUκκ α κ ωζκακ κακλκκ α κ⎛⎞\n⎜⎟−⎜⎟ ≡= +⎜⎟ − −+⎜⎟− ⎝⎠; ).        ( 3 1 )  \nWhen we take α = 0, this expressions (16) and (29) can correctly recover to those of the plasma \nwith the q-distribution (or the κ-distribution) in nonextensive statistics [8]. \n3. Numerical analyses \nIn this section, we make numerical analyses of the dispersion relation in Eq. (24)  and the \ndamping rate in Eq. (29)  to show the effect of the parameter α on the ion acoustic waves. To do \nthe numerical analyses convenie ntly, we may write Eq.(24) as \n2 2 55\n22 22\n2 2 2 11 33\n22 22 22\n2 33\n22(, ; ) 13(, ; ) (, ; )\n(, ; )i r\nDe\npi e\nDeUTkU TUkUκακ ωλκκακ ω κακλκκ α κ⎛⎞\n⎜⎟−⎜⎟ =+⎜⎟ − −+⎜⎟− ⎝⎠,            ( 3 2 )  \nwhere we have used 22/Dep i B e i kT m λω = . \nBased on Eq.(32) and Eq.(29), the numerical analyses of the dispersion relation and damping \nrate are made respectively as a function of the wave number kλDe for several different α-parameter, \n6 \n  \nwhere the basic  plasma physical quan tities are chosen as and . / 1837iemm = /0 . 0ieTT = 1\nIn Fig.1, based on the dispersion relation (32), the wave frequency ( ωr / ω pi) of ion-acoustic \nwaves is showed as a function of the wave number kλ De and for different values of the \nα-parameter, where Fig.1(a) is for the parameter κ = 2, and the line with α=0 is corresponding to \nthe case of the plasma with the kappa-distribution for κ = 2; Fig.1(b) is for the parameter κ = 1( as \nan example of κ < 3/2) and α ≠ 0.  \nFig.1 shows that with increase of the wave number kλDe , the wave frequency ( ωr / ω pi) will \nincrease monotonically and will decrease as the α-parameter increases. Therefore, the wave \nfrequency ( ωr /ωpi) of ion-acoustic waves in the plasma with the regularized kappa-distribution (12)  \nis generally less than that with the kappa-distribution (15) . And as expected, Fig.1(b) also shows \nthat the ion-acoustic waves exist in the plasma with the regularized kappa-distribution if κ < 3/2.  \n \n \n \n(a) For  κ=2 \n \n \n(b) For κ =1 and α ≠ 0. \nFigure 1.  The dispersion relation  for different values of the α-parameter . \n \n \n7 \n  \n \n(a) For κ = 4 \n \n \n(b) For κ  = 1 and α ≠ 0 \nFigure 2. The damping rate  for different values of the α-parameter. \n \nIn Fig.2, based on Eq. (29), the  damping rate ( -γ / ωr) of ion-acoustic waves is also showed as \na function of the wave number kλDe and for four different values of the α-parameter, where Fig.2(a) \nis for the parameter κ =4, and the line with α=0 is corresponding to the case of the plasma with the \nkappa-distribution for κ =4. Fig.2 (b) is for the parameter κ = 1( as an example of κ < 3/2) and α  ≠ \n0.  \nFig.2 shows that with increase of the wave number kλDe , the  damping rate ( -γ /ωr) will \nincrease monotonically and will also decrease as the α-parameter increases. Therefore, the  \ndamping rate ( -γ /ωr) of ion-acoustic waves in the plasma with the regularized kappa-distribution \n(12) is also generally less than that with the kappa-distribution (15) . And as expected, Fig.2 (b) \nalso shows that the  damping rate of ion-acoustic waves  exist in the plasma with the regularized \nkappa-distribution if κ < 3/2.  \n4. Conclusion \nIn conclusion, we have studied the dispersion relation and the damping rate of the ion- \nacoustic waves in the plasma with the so-called regularized kappa-distribution given by Eq.(12), \n8 \n  \nan optimized kappa-distribution which is not only suitable for the parameter κ > 3/2, but also \nsuitable  for 0< κ < 3/2 . We have derived the dispersion relation and the damping rate, which are \ngiven by Eq.(15) and Eq.(17), respectively. We find that the dispersion relation Eq.(15) and the \ndamping rate Eq.(17) both depend significantly on the parameters α and κ. \n   The numerical analyses showed that the wave frequency (ω r /ωpi) and the  damping rate ( -γ /ωr) \nof ion-acoustic waves in the plasma with the regularized kappa-distribution  are both generally less \nthan those in the plasma with the kappa-distribution , and as expected, the ion-acoustic waves and \ntheir damping rate exist in the plasma with the regularized kappa-distribution if κ < 3/2.  \n \nAppendix \nFor ions ( ξi >>1), one has that \n()()11 2222\n22\n234exp 11e xp 1 (\niix xxdx x dxxκκα\nακξ κ ξ ξ ξ ξ−− −−\n∞∞\n−∞ −∞− ⎛⎞ ⎛⎞+= − + + + ⎜⎟ ⎜⎟− ⎝⎠ ⎝⎠∫∫2 3\n),\ni i ixxx+        ( A . 1 )  \nwhere  \n()12\n22 2 11\n22111e xp ( , ;\niixdx x Uκ\n), α πκ κ α κκξ ξ−−\n∞\n−∞⎛⎞+− = −⎜⎟\n⎝⎠∫   \n()122\n22 3 / 2 2 33\n22 3311e xp ( , ;2iixxdx x Uκ\n), α πκ κ α κκξ ξ−−\n∞\n−∞⎛⎞+− = −⎜⎟\n⎝⎠∫   \n()124\n22 5 / 2 2 55\n22 5531e xp ( , ;4iixxdx x Uκ\n) α πκ κ α κκ ξξ−−\n∞\n−∞⎛⎞+− = −⎜⎟\n⎝⎠∫ . \nFor electrons ( ξe <<1), one lets exξη −= and \n()()\n()1 1 222 2\n22\n122\n22 2exp ()11 ex\n21e xp ( 2 )e\ne\ne\nee\neex xddxx\ndκ κ\nκα ξη ηαξ ηκξ ηκ\nξξ η η ηαξ ξ ηηηκ−− −−\n∞∞\n−∞ −∞\n−−\n∞\n−∞− ⎛⎞ ⎛⎞ ++= − + −⎜⎟ ⎜⎟− ⎝⎠ ⎝⎠\n⎛⎞ ++=− + − + + ⎜⎟\n⎝⎠∫∫\n∫p ( )+\n  \n()12\n221e x p 0dκηηαηηκ−−\n∞\n−∞⎛⎞≈− + − = ⎜⎟\n⎝⎠∫.                                         ( A . 2 )  \nEq.(19) becomes \n( )1 22 2 2 2 11\n22\n, 22 2 2 33 11\n22 22\n22 2 55\n22\n22 2 2 2 33\n22exp 2( , ; ) / 1(,) 1 1 P r 1(, ; ) (, ; )\n2 (, ; ) 13                =1 12( ,; )ps s\ns s Ts\npi pe\nTi i ix U xDk d xx Uk v U\nU\nkv Uκ\nακα ω κακ ξπ κωκξ κκ α κ κ α κ\nωω κακ κ\nξξ κ α κ−−\n∞\n−∞⎡ ⎤ − −⎛ ⎞⎢ ⎥ =+ − +⎜⎟− −− ⎢ ⎥ ⎝⎠⎣ ⎦\n⎛⎞ −−+ + ⎜⎟− ⎝⎠∑ ∫\n2 11\n22\n22 2 33\n22(, ; )\n(, ; )TeU\nkv Uκακ\nκκ ακ−−  \n1 2 22\n32 3 3 2 2 33\n2221e xp(, ; )ps\ns Ts Ts TsiUk v k vκω πω ω α\nκκ α κ κ−−⎛⎞ ⎛ −++ ⎜⎟ ⎜− ⎝⎠ ⎝∑2\n22kvω⎞\n⎟\n⎠.                ( A . 3 )  \n9 \n  \nSubstituting i kvTi ξω= into Eq.(A.1), one obtains Eq.(20). \nAcknowledgements \nThis work is supported by the National Natural scien ce foundation of China under Grant No.  \n11775156. \n \nReferences \n[1] B.D. Fried, Roy W. Gould, Phys. Fluids 4 (1961) 139. \n[2] J. Du, Phys. Lett. A 329 (2004) 262. \n[3] V. Vasyliunas, J. Geophys. Res. 73 (1968) 2839. \n[4] R.A. Cairns, R. Bingham, et al., J. Phys. 5 (1995) C6-43. \n[5] A. A. Abid, S. Ali, J. Du, an d A. A. Mamun, Phys. Plasmas 22 (2015) 084507 . \n[6] D. Summers, R.M. Thorne, Phys. Fluids B 3 (1991) 1835. \n[7] P.O. Dovner, A.I. Eriksson, et al., Geophys. Res. Lett. 21 (1994) 1827; R. Bostrom, IEEE \nTrans. Plasma Sci. 20 (1992) 756. \n[8] L. Liu, J. Du, Physica A 387 (2008) 4821. \n[9] Z. Liu, L. Liu, J. Du, Phys. Plasmas 16 (2009) 07211. \n[10] F. Hadi, Ata-ur-Rahman,  Anisa Qamar. Phys. Plasmas 24 (2017) 104503. \n[11] M. A. Shahzad, Aman-ur-Rehman, S. Mahmood, M. Bilal, M. Sarfraz , Phys. J. Plus 137 \n(2022) 236. \n[12] G. Nicolaou, et al., Astrophys. J. 864 (2018) 3. \n[13] S. R. Cranmer, Astrophys. J. Lett. 791 (2014) L31. \n[14] F. Xiao, et al., J. Geophys. Res. 113 (2008) A05203; K. Ogasawara,  et al., J. Geophys. Res. \n118 (2013) 3126; I. P. Kirpichev, E. E. Antonova, Astrophys. J. 891 (2020) 35. \n[15] M. R. Collier, D. C. Hamilton, Geophys. Res. Lett. 22 (1995) 303; P. Schippers, et al., J. \nGeophys. Res. 113 (2008) A07208. \n[16] K. Scherer, H. Fichtner, M. Lazar EPL 120 (2017) 50002. \n[17] K. Scherer, et al., Astrophys. J. 880 (2019) 118. \n[18] Y. Liu , AIP Advances 10 (2020) 085022. \n[19] Y. Liu, X. Chen, Plasma Sci. Technol. 24 (2022) 015301. \n[20] F. F. Chen, Introduction to Plasma Physics and Controlled Fusio n, 3rd edn. Springer, \nSwitzerland (2016); Xu Jialuan, Jin Shangxian, Plasma Physics , Nuclear Energy Press, \nBeijing (1981). \n[21] Y. Wang, J. Du, Physica A 566 (2021) 125623. \n[22] H. Yu, J. Du, EPL 116 (2016) 60005. \n \n10 \n "    },    {        "title": "0804.0269v1.Lorentz_Violation__Electrodynamics__and_the_Cosmic_Microwave_Background.pdf",        "content": "arXiv:0804.0269v1  [hep-ph]  1 Apr 2008Lorentz Violation, Electrodynamics, and the Cosmic Microw ave\nBackground∗\nMatthew Mewes\nPhysics Department, Marquette University, Milwaukee, WI 5 3201\nVacuumbirefringenceisasignatureofLorentz-symmetryvi olation. Herewereport\non a recent search for birefringence in the cosmic microwave background. Polariza-\ntion data is used to place constraints on certain forms of Lor entz violation.\nI. INTRODUCTION\nThe properties of light have proved to be a valuable testing ground f or special relativity\nfor more than a century. Contemporary experiments are motivat ed in part by a possible\nbreakdown of special relativity with origins in Planck-scale physics [1, 2, 3]. These exper-\niments include modern versions of the classic Michelson-Morley and Ke nnedy-Thorndike\nexperiments that use highly stable resonant cavities to search for violations of rotation and\nboost symmetries [4]. However, the highest sensitivities to relativity violations in electro-\ndynamics are found in searches for vacuum birefringence in light fro m very distant sources\n[5, 6, 7, 8]. Birefringence studies take advantage of the extremely long baselines that allow\nthe miniscule effects of a Lorentz violation to accumulate to (potent ially) detectable levels\nover the billions of years it takes for the light to reach Earth. The co smic microwave back-\nground (CMB) is the oldest light available to observation and therefo re provides an excellent\nsource for birefringence searches. Here we summarize a recent s earch for signals of Lorentz\nviolation using CMB polarimetry [8].\nGeneral Lorentz violation is described by a framework known as the Standard Model\nExtension (SME) [3]. The SME provides the theoretical backbone fo r studies in a number\nof areas [2], including photons [4, 6, 7, 8, 9]. Most tests of Lorentz v iolation focus on the\nminimal SME, which assumes usual gauge invariance and energy-mom entum conservation\nandrestricts attentionto superficially renormalizable operators. Operatorsofdimension d≤\n4 are of renormalizable dimension and are included the minimal SME. Two types of Lorentz-\n∗Presented at the Fourth Meeting on CPT and Lorentz Symmetry, B loomington, Indiana, August, 2007.2\nviolating operatorsappear in theminimal SME, CPT-oddoperatorswithcoefficients ( kAF)κ\nandCPT-even operators with coefficients ( kF)κλµν.\nIn this work, we also consider non-minimal higher-dimensional opera tors in the photon\nsector with d >4. In general there are an infinite number of possible operators th at emerge\nwhen we relax the renormalizable condition. These operators are ph enomenologically and\ntheoretically relevant in that they help provide a connection to the u nderlying Planck-scale\nphysics. They also add a number of new and interesting signals for Lo rentz violation that\nmay be tested experimentally.\nII. THEORY\nGeneral Lorentz-violating electrodynamics is given by a lagrangian t hat takes the same\nbasic form as the minimal-SME photon sector [8]:\nL=−1\n4FµνFµν+1\n2ǫκλµνAλ(ˆkAF)κFµν−1\n4Fκλ(ˆkF)κλµνFµν. (1)\nWe assume a linear theory and impose the usual U(1) gauge invarianc e. The key differ-\nence between this theory and the minimal-SME photon sector is that here the ˆkAFandˆkF\ncoefficients are differential operators. The effects of these oper ators mimic the effects of\na permeable medium whose activity depends on the photon energy an d momentum. This\nintroduces a plethora of new effects not found in either the conven tional Lorentz-conserving\ncase or the minimal SME. These include drastically different frequenc y dependences and\ndirection-dependent propagation of light.\nExpandingthe ˆkAFandˆkFoperatorsinthe4-momentum pµ=i∂µleadstotheexpressions\n(ˆkAF)κ=/summationdisplay\n(k(d)\nAF)κα1...α(d−3)∂α1...∂α(d−3), (2)\n(ˆkF)κλµν=/summationdisplay\n(k(d)\nF)κλµνα 1...α(d−4)∂α1...∂α(d−4). (3)\nThecoefficientsforLorentzviolationassociatedwiththedimension doperatorsarenowgiven\nby (k(d)\nAF)κα1...α(d−3)and (k(d)\nF)κλµνα 1...α(d−4). TheˆkAFexpression contains all CPT-breaking\neffects, and the sum is restricted to odd-dimensional operators, d= odd. The ˆkFcoefficients\ncontrol all CPT-even violations and have d= even. Imposing gauge invariance places\nvarious constraints on these coefficients. For k(d)\nAFcoefficients, any trace of the that involves\nthe first index vanishes identically. For k(d)\nF, the antisymmetrization on any three indices3\nvanishes. Standard group theory techniques allow a counting of th e independent coefficients\nforLorentzviolations[10]. Foragivendimension d, wefind1\n2(d+1)(d−1)(d−2) independent\nk(d)\nAFcoefficients in the CPT-odd case and ( d+ 1)d(d−3) independent k(d)\nFcoefficients in\ntheCPT-even case.\nForstudiesofLorentz-violationinducedbirefringence, certainline arcombinationsofthese\ngeneralcoefficientsareimportant. Theyresultfromaspherical-h armonicexpansionofplane-\nwaves propagating in the vacuum. This plane-wave expansion is best characterized using the\nlanguage of Stokes parameters. We begin by defining a Stokes vect ors= (s1,s2,s3)T. The\ndirection in which this vector points in the abstract 3-dimensional St okes space uniquely\ncharacterizes the polarization of the radiation. Stokes vectors ly ing in the s1-s2plane cor-\nrespond to all possible linear polarizations, while Stokes vectors par allel and antiparallel to\nthes3axis give the two circular polarizations. General right-handed elliptic al polarizations\npoint in the upper-half Stokes space, s3>0, while left-handed are given by the lower half,\ns3<0.\nThis formalism provides an intuitive picture of birefringence. It can b e shown that bire-\nfringence causes a rotation of the Stokes vector sabout some axis ς. This occurs whenever\nthe usual degeneracy between the various polarizations in broken . Formally, we solve the\nmodified equations of motion. We find that some types of violations lea d to two propagating\neigenmodes that have slightly different velocities. They also differ in po larization, and light\nof an arbitrary polarization is a superposition of the two eigenmodes . This superposition is\naltered as the eigenmodes propagate at different velocities, causin g an oscillatory effect that\nreveals itself as a rotation of the Stokes vector. The rotation tak es the form\nds/dt= 2ως×s, (4)\nwhereωis the wave frequency, and the rotation axis ςcorresponds to the Stokes vector of\nthe faster eigenmode. In general, ςmay depend on both the direction of propagation and\nthe frequency.\nThe basic idea of a birefringence test is to examine light from a distant polarized source\nfor the above rotation. To do this we need to express the rotation axisςin terms of the\ncoefficients for Lorentz violation. The general result is rather com plicated, but can be\nwritten in a relatively simple form in terms of a set of “vacuum” coefficie nts, which are\nlinear combinations of the general coefficients. The calculation involv es decomposing ςinto4\nspin-weighted spherical harmonics. The result takes the form\nς1∓iς2=/summationdisplay\ndlmωd−4(k(d)\n(E)lm±ik(d)\n(B)lm)±2Ylm(ˆn), (5)\nς3=/summationdisplay\ndlmωd−4k(d)\n(V)lm0Ylm(ˆn), (6)\nwhere sYlmis a spin-weighted spherical harmonic with spin-weight s, andˆnis the radial unit\nvector pointing toward the source on the sky. The vacuum coefficie ntsk(d)\n(V)lm,k(d)\n(E)lm, and\nk(d)\n(B)lmrepresent the minimal combinations of coefficients for Lorentz viola tion that cause\nbirefringence and affect polarization. The designations EandBrefer to the parity of the\ncoefficient and is borrowed from radiation theory. In the next sect ions, we describe a search\nfor these effects in existing CMB polarization data.\nIII. CMB\nThe CMB is conventionally parameterized by a spin-weighted spherica l-harmonic expan-\nsion similar to the expansion of ςgiven above [11, 12]. The complete characterization of\nradiation from a given point on the sky includes the temperature T, the linear polarization,\ngiven by Stokes parameters s1ands2, and the circular polarization, given by s3. The global\ndescription is given by the expansion\nT=/summationdisplay\na(T)lm0Ylm(ˆn), s3=/summationdisplay\na(V)lm0Ylm(ˆn),\ns1∓is2=/summationdisplay\n(a(E)lm±ia(B)lm)±2Ylm(ˆn). (7)\nOne then constructs various power spectra,\nCX1X2\nl=1\n2l+1/summationdisplay\nm/angbracketlefta∗\n(X1)lma(X2)lm/angbracketright, (8)\nwhereX1,X2=T,E,B,V . These spectra quantify the angular size variations in each mode\nand any correlation between different modes. Smaller lcorrelates to larger angular size on\nthe sky.\nWithin conventional physics, we expect a nearly isotropic ( l= 0) temperature distribu-\ntion. However, tiny fluctuations in temperature during recombinat ion not only introduce\nhigher-order multipole moments ( l >0) but also provide the necessary anisotropies to pro-\nduce a net polarization. Only linear polarizations are expected since n o circular polarization5\nis produced in Thomson scattering. Furthermore, E-type polarization is expected to domi-\nnate and be correlated with the temperature. No correlation is exp ected between the much\nsmallerBpolarization and temperature. This general picture agrees with ob servation to\nthe extent to which CMB radiation has been measured [13, 14].\nAbreakdown of Lorentzsymmetry may alter these basic features . Someof thenew effects\ncan be readily understood as consequences of the Stokes rotatio ns. For example, the CPT-\nodd coefficients k(d)\n(V)lmlead to a Stokes rotation axis that points along the s3direction. The\nresulting local rotations in polarization leave the circularly polarized c omponent unchanged.\nHowever, it does lead to a rotation in the linear components, causing a simple change in\nthe polarization angle at each point on the sky. Globally this causes a m ixing between the\nEandBpolarization. This could introduce an unusually large Bcomponent, which gives\na potential signal of CPTand Lorentz violation. Similar mixing can arise from the k(d)\n(E)lm\nandk(d)\n(B)lmcoefficients. However, since these give a rotation axis that lies in the s1-s2plane,\nthe rotations in this case also introduce circular polarization. So a lar ge circularly polarized\ncomponent in the CMB might indicate a CPT-even violation of Lorentz invariance.\nAll except the d= 3 coefficients result in frequency-dependent rotations. Also, on ly\nthel= 0 coefficients cause isotropic rotations that are uniform across t he sky. As a re-\nsult, the coefficient k(3)\n(V)00provides a simple isotropic frequency-independent special case.\nA calculation shows that this case causes a straightforward rotat ion between CEE\nl,CBB\nl,\nandCEB\nl, as well as between CTE\nlandCTB\nl[15]. In contrast, more general anisotropic and\nfrequency-dependent cases cause very complicated mixing betwe en the various CX1X2\nland\nrequire numerical integration of the rotation (4) over the sky and frequency range.\nIV. RESULTS\nTo illustrate the kinds of sensitivities that are possible in CMB searche s for birefringence,\nwe next examine the results of the BOOMERANG experiment [14]. This balloon-based ex-\nperiment made polarization measurements in a narrow band of frequ encies at approximately\n145 GHz. This relatively high frequency implies that BOOMERANG is well s uited to bire-\nfringence tests since for all violations, except those with d= 3, higher photon energy implies\na larger rotation in polarization. The small frequency range is also he lpful since we can\napproximate all frequencies as ∼145 GHz.6\n-30 -20 -10 0 10 20 3001\n-30 -20 -10 0 10 20 3001\n-60 -40 -20 0 20 40 6001\n-60 -40 -20 0 20 40 6001\n-40 -20 0 20 4001\n-40 -20 0 20 4001\nk(3)\n(V)00(10−43GeV) k(3)\n(V)10(10−43GeV) k(3)\n(V)11(10−43GeV)\n-40 -20 0 20 4001\n-40 -20 0 20 4001\n-40 -20 0 20 4001\n-40 -20 0 20 4001\n-10 -5 0 5 1001\n-10 -5 0 5 1001\nk(4)\n(E)20(10−31) k(4)\n(B)20(10−31) k(5)\n(V)00(10−20GeV−1)\n-20 -15 -10 -5 0 5 10 15 2001\n-20 -15 -10 -5 0 5 10 15 2001\n-20 -15 -10 -5 0 5 10 15 2001\n-20 -15 -10 -5 0 5 10 15 2001\n-20 -15 -10 -5 0 5 10 15 2001\n-20 -15 -10 -5 0 5 10 15 2001\nk(5)\n(V)10(10−20GeV−1) k(5)\n(V)20(10−20GeV−1) k(5)\n(V)30(10−20GeV−1)\n-30 -20 -10 0 10 20 3001\n-30 -20 -10 0 10 20 3001\n-30 -20 -10 0 10 20 3001\n-30 -20 -10 0 10 20 3001\n-30 -20 -10 0 10 20 3001\n-30 -20 -10 0 10 20 3001\nk(6)\n(E)20(10−10GeV−2) k(6)\n(E)30(10−10GeV−2) k(6)\n(E)40(10−10GeV−2)\nFIG. 1: Relative likelihood versus coefficients for Lorentz v iolation. Boxes indicate numerically\ncalculated values, and the curve is the smooth extrapolatio n of these points. The dark-gray shaded\nregion indicates the 68% confidence level, and the light-gra y shows the 95% level.\nInour calculation, we assume conventional polarizationisproduced during recombination\nand numerically determine the rotated polarization for points acros s the sky. The resulting\nCX1X2\nlfor various values of coefficients for Lorentz violation are determin ed and compared\nto published BOOMERANG results. Figure 1 shows the calculated relat ive likelihood for\na sample of 12 coefficients for Lorentz violation. In each case, we va ry the value of one\ncoefficient, setting all other coefficients to zero. The 1 σand 2σregions are shown.\nSome generic features are seen in our survey. In each case, the c oefficient is nonzero\nat the 1σlevel, hinting at possible Lorentz violation. However, since this occur s in every\ncase, it is likely that this indicates some systemic feature of the BOOM ERANG data or our\nanalysis. We also see that each case is consistent with no Lorentz vio lation at the 2 σlevel,\ngiving conservative upper bounds on the 12 coefficients in Figure 1.\nThese results demonstrate the potential of the CMB for testing L orentz invariance. Due\nto the long propagation times, the sensitivities to d= 3 coefficients afforded by the CMB\nare near the limit of what can be expected in birefringence tests. Ho wever, for d≥4, bet-\nter sensitivities might be obtained using high-frequency sources like gamma-ray bursts [7].\nRegardless, because of its all-sky nature, the CMB provides a usef ul probe that can simul-\ntaneously probe large portions of coefficient space, which is difficult in searches involving a7\nhandful of point sources that access a limited number of propagat ion directions.\nAcknowledgments\nThis work was supported in part by the Wisconsin Space Grant Conso rtium.\n[1] V.A. Kosteleck´ y and S. Samuel, Phys. Rev. D 39, 683 (1989); V.A. Kosteleck´ y and R. Potting,\nNucl. Phys. B 359, 545 (1991).\n[2] For a recent review of various experimental and theoreti cal approaches to Lorentz violation,\nsee, for example, V.A. Kosteleck´ y, ed., CPT and Lorentz Symmetry III, World Scientific,\nSingapore, 2005.\n[3] D. Colladay and V.A. Kosteleck´ y, Phys. Rev. D 55, 6760 (1997); 58, 116002 (1998); V.A.\nKosteleck´ y, Phys. Rev. D 69, 105009 (2004).\n[4] J. Lipa et al., Phys. Rev. Lett. 90, 060403 (2003); H. M¨ uller et al., Phys. Rev. Lett. 91,\n020401 (2003); P. Antonini et al., Phys. Rev. A 71, 050101 (2005); S. Herrmann et al., Phys.\nRev. Lett. 95, 150401 (2005); P.L. Stanwix et al., Phys. Rev. D 74, 081101 (2006).\n[5] S.M. Carroll, G.B. Field, and R. Jackiw, Phys. Rev. D 41, 1231 (1990).\n[6] V.A. Kosteleck´ y and M. Mewes, Phys. Rev. Lett. 87, 251304 (2001); Phys. Rev. D 66, 056005\n(2002).\n[7] V.A. Kosteleck´ y and M. Mewes, Phys. Rev. Lett. 97, 140401 (2006).\n[8] V.A. Kosteleck´ y and M. Mewes, Phys. Rev. Lett. 99, 011601 (2007).\n[9] H. M¨ uller et al., Phys. Rev. D 67, 056006 (2003); R. Lehnert and R. Potting, Phys. Rev. Lett.\n93, 110402 (2004); Q.G. Bailey and V.A. Kosteleck´ y, Phys. Rev . D70, 076006 (2004); M.\nMewes and A. Petroff, Phys. Rev. D 75, 056002 (2007); A.J. Hariton and R. Lehnert, Phys.\nLett. A367, 11 (2007); B. Altschul, Phys. Rev. D 75, 105003 (2007); A. Kobakhidze and\nB.H.J. McKellar, Phys. Rev. D 76, 093004 (2007).\n[10] V.A. Kosteleck´ y and M. Mewes, in preparation.\n[11] For review of current CMB theory and experiment, see, fo r example, Particle Data Group\n(http://pdg.lbl.gov), S. Eidelman et al., Phys. Lett. B 592 , 1 (2004).8\n[12] For a review of CMB polarimetry, see, for example, W. Hu a nd M. White, New Astron. 2,\n323 (1997).\n[13] G.F. Smoot et al., Ap. J. 396, L1 (1992); J. Kovac et al., Nature 420772 (2002); A.C.S.\nReadnead et al., Science 306, 836 (2004); D. Barkats et al., Ap. J. 619, L127 (2005); E.M.\nLeitchet al., Ap. J.624, 10 (2005); W.C. Jones et al., Ap. J.647, 823 (2006); G. Hinshaw et\nal., Ap. J. Supp. 170, 288 (2007); L. Page et al., Ap. J. Suppl. 170, 335 (2007).\n[14] T.E. Montroy et al., Ap. J.647, 813 (2006); F. Piacentini et al., Ap. J.647, 833 (2006).\n[15] B. Feng et al., Phys. Rev. Lett. 96, 221302 (2006); P. Cabella et al., Phys. Rev. D 76, 123014\n(2007)."    },    {        "title": "0801.0549v1.Spin_orbit_precession_damping_in_transition_metal_ferromagnets.pdf",        "content": "arXiv:0801.0549v1  [cond-mat.mtrl-sci]  3 Jan 2008Spin-orbit precession damping in transition metal ferroma gnets\nK. Gilmore1,2, Y.U. Idzerda2, and M.D. Stiles1\n1National Institute of Standards and Technology, Gaithersb urg, MD 20899-6202\n2Physics Department, Montana State University, Bozeman, MT 59717\n(Dated: November 1, 2018, Journal of Applied Physics )\nWe provide a simple explanation, based on an effective field, f or the precession damping rate due\nto the spin-orbit interaction. Previous effective field trea tments of spin-orbit damping include only\nvariations of the state energies with respect to the magneti zation direction, an effect referred to\nas the breathing Fermi surface. Treating the interaction of the rotating spins with the orbits as\na perturbation, we include also changes in the state populat ions in the effective field. In order to\ninvestigate the quantitative differences between the dampi ng rates of iron, cobalt, and nickel, we\ncompute the dependence of the damping rate on the density of s tates and the spin-orbit parameter.\nThere is a strong correlation between the density of states a nd the damping rate. The intraband\nterms of the damping rate depend on the spin-orbit parameter cubed while the interband terms are\nproportional to the spin-orbit parameter squared. However , the spectrum of band gaps is also an\nimportant quantity and does not appear to depend in a simple w ay on material parameters.\nI. INTRODUCTION\nMagnetic memory devices are useful if they can be re-\nliably switched between two stable states. The fidelity of\nthis switching process depends sensitively on the damp-\ning rate of the system. Despite decades of research and\nthe relentless industrial push toward smaller and faster\ndevices, many questions about the damping process re-\nmain unanswered, particularly for metallic ferromagnets.\nRecent experimental efforts have investigated the extent\nto which the damping rate of NiFe alloys can be tuned\nthrough doping, particularly with the addition of rare\nearth [1] and transition metal elements [2]. While these\ninvestigations found a general trend suggesting damp-\ning increases with increasing spin-orbit coupling of the\ndopant, the details behind this effect remained elusive.\nTo aid this effort, this article provides a simple descrip-\ntion of the damping process and investigates how some\nmaterial properties affect the damping rate.\nPrecession damping in metallic ferromagnets results\npredominantly from a combined effort of spin-orbit cou-\npling and electron-lattice scattering [3, 4]. The role of\nlattice scattering was studied in early experimental work\nthrough the temperature dependence of damping rates\n[5, 6]. Measurement of damping rates versus tempera-\nture revealed two primary contributions to damping, an\nexpected part that increased with temperature, and an\nunexpected part that decreased with temperature. In\ncobalt these two opposing contributions combine to pro-\nduce a minimum damping rate near 100 K, for nickel the\nincreasing term is weaker leading to a temperature inde-\npendent damping rate above 300 K, while for iron the\ndamping rate becomes independent of temperature be-\nlow room temperature. Heinrich et al. later noted that\nthe temperature dependence of the increasing and de-\ncreasingcontributionsmatchedthat ofthe resistivityand\nconductivity, respectively [7, 8], and so dubbed the two\ncontributions conductivity-like for the decreasing piece\nand resistivity-like for the increasing part.\nAmong the many theories on intrinsic precessiondamping[3, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], Kam-\nbersk´ y’storque-correlationmodel [3] is unique in qualita-\ntivelymatchingtheobservednon-monotonictemperature\ndependence that we just described. We recently evalu-\nated this model for iron, cobalt, and nickel, and showed\nthat it accurately predicts the precession damping rates\nofthesesystems[4]. Whilethis modelsucceedsincaptur-\ning the important physical effects involved in precession\ndamping, it does not easily identify the important physi-\ncal processes or give insight into how one might alter the\ndampingratethroughsamplemanipulation. In sectionII\nwe briefly describe the torque-correlation model. In or-\nder to provide a more tangible explanation of precession\ndamping we rederive the damping rate from an effective\nfield approach in section III. This discussion is followed\nin section IV by a quantitative analysis of the effect on\nthe damping rate of tuning the density of states and the\nspin-orbit parameter.\nII. TORQUE-CORRELATION MODEL\nKambersk´ y’s theory describes damping in terms of the\nspin-orbit torque correlation function, finding a damping\nrate of\nλ=π/planckover2pi1γ2\nµ0/summationdisplay\nnm/integraldisplayd3k\n(2π)3|Γ−\nmn(k)|2\n×/integraldisplay\ndǫ1Ank(ǫ1)Amk(ǫ1)η(ǫ1). (1)\nThe gyromagneticratio is γ=gµ0µB//planckover2pi1,gis the Land´ e g\nfactor,µ0is the permeability of space, nandmare band\nindices, and kis the electronwavevector. The matrix ele-\nments|Γ−\nmn(k)|2describe a torque between the spin and\norbital moments that arises as the spins precess. η(ǫ)\nis the derivative of the Fermi function −df/dǫ, which\nis a positive distribution peaked about the Fermi level\nthat restricts scattering events to the neighborhood of\ntheFermi surface. The electronspectralfunctions Ank(ǫ)2\nareLorentziansin energyspace centeredat band energies\nwithwidthsdeterminedbythescatteringrate. Theyphe-\nnomenologically account for electron-lattice scattering.\nEquation (1) includes two processes: the decay of\nmagnons into electron-hole pairs and the scattering of\nthe electrons and holes with the lattice. This expres-\nsion is similar in structure to sp-dmodels that have\nproven successful in describing dissipation in semicon-\nductors [20]. However, the physics of the magnon decay\nprocess is very different. In the present case, there is no\ndistinction between spanddelectrons. The spin-orbit\ntorque annihilates a uniform mode magnon and gener-\nates an electron-hole pair. The electron-hole pair is then\ncollapsed through lattice scattering. The electron and\nhole are dressed through lattice interactions and are best\nthought of as a single quasiparticle with indeterminant\nenergy and a lifetime given by the electron-lattice scat-\ntering time. The dressed electron and hole can occupy\nthe same band ( m=n), which we call an intraband\ntransition, or two different bands ( m/negationslash=n), aninter-\nbandtransition. For intraband transitions, the integra-\ntion over the spectral functions is proportional to the\nscattering time, just like the conductivity. For interband\ntransitions, the intregration over the spectral functions\nis roughly inversely proportional to the scattering time,\nas is the resistivity. Therefore, the intraband terms in\nEq. (1) give the conductivity-like contributions to damp-\ning that decrease with temperature while the interband\nterms yield the resistivity-likecontributionsthat increase\nwith temperature.\nIII. EFFECTIVE FIELD DERIVATION\nAn effective field for the magnetization dynamics is de-\nfinedasthevariationoftheelectronicenergywithrespect\nto the magnetization direction µ0Heff=−∂E/∂M. The\nmagnitude of the magnetization Mis considered con-\nstant within the Landau-Lifshitz formulation, only the\ndirection ˆMof the magnetization changes. The total\nelectronic energy of the system can be approximated by\nE=/summationtext\nnkρnkǫnk, which is a summation over the single\nelectron energies ǫnkweighted by the state occupancies\nρnk. If the state occupancies are held at their equilbrium\nvalues, the resulting effective field is equivalent to that of\nthe magnetocrystalline anisotropy [21], which describes\nreversible processes. If however, the state occupancies\nare allowed to deviate from the equilibrium populations\ninresponsetotheoscillatingperturbation, anirreversible\ncontribution also arrises, which we show produces the\ndamping in Eq. (1).\nAs the magnetization precesses the energies of the\nstates change through variations in the spin-orbit con-\ntribution and transitions between states occur. These\ntwo effects, the changing energies of the states and the\ntransitions between states, produce a contribution to theeffective field\nHeff=−1\nµ0M/summationdisplay\nnk/bracketleftbigg\nρnk∂ǫnk\n∂ˆM+∂ρnk\n∂ˆMǫnk/bracketrightbigg\n.(2)\nThe first term in the brackets describes the variation in\nthe spin-orbit energies of the states as the magnetization\ndirection changes. This effect, which has been discussed\nand evaluated before [10, 13, 22], is generally referred\nto as the breathing Fermi surface model. The spin-orbit\ntorque does not cause transitions between states in this\npicture, but does cause the Fermi surface to swell and\ncontract as the magnetization precesses. We will show\nthat this portion of the effective field gives the intraband\ntermsofEq.(1). Thesecondterminthebracketshaspre-\nviously been neglected in effective field treatments, but\naccounts for changes in the system energy due to tran-\nsitions between states. This term does not change the\nenergies of the states, but does create electron-hole pairs\nby exciting electrons from lower bands to higher bands.\nThis process can be pictured as a bubbling of individ-\nual electrons on the Fermi surface. We will demonstrate\nthat this portion of the effective field gives the interband\nterms of Eq. (1).\nA. Intraband terms\nThe first term in the effective field Eq. (2) accounts for\nthe effects of the breathing Fermi surface (bfs) model.\nSince this model has previously been discussed in detail\n[10, 13, 22] we will give only a very brief review of it here,\nfocusing instead on connecting it to the intraband terms\nof Eq. (1).\nAs the magnetization precesses the spin-orbit energy\nof each state changes. Some occupied states originally\njust below the Fermi level get pushed above the Fermi\nlevel and simultaneously some unoccupied state origi-\nnally above the Fermi level may be pushed below it.\nThis process takes the system, which was originally in\nthe ground state, and drives it out of equilibrium into an\nexcited state creating electron-hole pairs in the absence\nof any scattering events. Scattering, which occurs with a\nrate given by the inverse of the relaxation time τ, brings\nthe system to a new equilibrium. The relaxation time\napproximation determines how far from equilibrium the\nsystem can get.\nρnk=fnk−τdfnk\ndt. (3)\nThe occupancy ρnkof each state ψnkdeviates from its\nequilibrium value fnkby an amount proportional to the\nscattering time. How quickly the system damps depends\non the magnitude of this deviation.\nThe rate of change of the equilibrium distribution\ndfnk/dtdepends on how much the distribution changes\nas the energy of the state changes dfnk/dǫnk, how much\nthe state energy changes as the precession angle changes3\nFIG. 1: Schematic description of precession geometry. With in\nthebreathingFermi surface model (a) thedampingrate is cal -\nculatedas themagnetization passesthroughaspecific point in\na given direction. The torque correlation model (b) gives th e\ndamping rate for precessing about a given direction. Dashed\ncurves indicate the precession trajectory.\ndǫnk/dˆM, and how quickly the spin direction is precess-\ningdˆM/dt. These can be combined with a chain rule\ndfnk\ndt=dfnk\ndǫnkdǫnk\ndˆMdˆM\ndt. (4)\nCombining this result with the relaxation time approxi-\nmation Eq. (3) and substituting these state occupancies\ninto the first term of the effective field in Eq. (2) gives\nHeff\nbfs=Hani\nbfs+Hdamp\nbfs, (5)\nHani\nbfs=−1\nµ0M/summationdisplay\nnkfnk∂ǫnk\n∂ˆM, (6)\nHdamp\nbfs=−1\nµ0M/summationdisplay\nnkτ/parenleftbigg\n−dfnk\ndǫnk/parenrightbigg/parenleftbiggdǫnk\ndˆM/parenrightbigg2dˆM\ndt.(7)\nHani\nbfsis a contribution to the magnetocrystalline\nanisotropy field and Hdamp\nbfsthe damping field from the\nbreathing Fermi surface model. When we compare this\ndamping field to the damping field postulated by the\nLandau-Lifshitz-Gilbert equation\nHdamp\nLLG=−λ\nγ2MdˆM\ndt(8)\nwe find that the damping rate is\nλbfs=τγ2\nµ0/summationdisplay\nnkη(ǫnk)/parenleftbigg∂ǫnk\n∂ˆM/parenrightbigg2\n. (9)\nAs in Eq. (1), η(ǫ) is the negative derivative of the Fermi\nfunction and is a positive distribution peaked about the\nFermi energy.\nAs described in Fig. (1a), the result of the breathing\nFermi surface model Eq. (9) describes the damping rate\nof a material as the magnetization rotates through a par-\nticular point ˆ zabout a given axis ˆϑ. When ˆMis instan-\ntaneously aligned with ˆ zthe direction of the change in\nthe magnetization dˆMwill be perpendicular to ˆ z, in the\nˆx-ˆyplane. On the other hand, the torque correlationmodel Eq. (1) gives the damping rate when the magne-\ntization is undergoing small angle precession about the\nˆzdirection (see Fig.(1b)). When ˆ zis a high symmetry\ndirection the change in the magnetization will stay in the\nˆx-ˆyplane. In each scenario – rotating ˆMthroughˆzin the\nbreathing Fermi surface model and rotating ˆMaboutˆz\nin the torque correlation model – dˆMis confined to the\nˆx-ˆyplane. Therefore, rotating through ˆ zand rotating\nabout ˆzare equivalent in the small angle limit when ˆ z\nis a high symmetry direction. With this observation we\nnow show that the intraband contributions of the torque\ncorrelation model are equivalent to the breathing Fermi\nsurface result under these conditions.\nThe only energy that changes as the magnetization\nrotates is the spin-orbit energy Hso. As the spin of the\nstate|nk/angbracketrightrotates about the ˆϑdirection by angle ϑits\nspin-orbit energy is given by\nǫ(ϑ) =/angbracketleftnk|eiσ·/vectorϑHsoe−iσ·/vectorϑ|nk/angbracketright (10)\nwhere/vectorϑ=ϑˆϑ. Taking the derivative of this energy with\nrespect toϑin the limit that ϑgoes to zero shows that\nthe energy derivatives are\n∂ǫ\n∂ϑ=i/angbracketleftnk|[σ·ˆϑ,Hso]|nk/angbracketright. (11)\nFigure (1) shows that the derivative ∂ǫ/∂ϑis identical to\n∂ǫ/∂ˆMandthatwhen ˆM= ˆztherotationdirection ˆϑlies\nin thex−yplane. The two components of the transverse\ntorque operatorΓxand Γycan be obtained (up to factors\nofi) by setting ˆϑequal to ˆxor ˆy, respectively. From this\nobservation we find\n|/angbracketleftnk|Γ−|nk/angbracketright|2=/parenleftbigg∂ǫ\n∂x/parenrightbigg2\n+/parenleftbigg∂ǫ\n∂y/parenrightbigg2\n.(12)\nWhen the magnetization direction ˆ zis pointed along a\nhigh symmetry direction the transverse directions ˆ xand\nˆyare equivalent and |Γ−|2= 2(∂ǫ/∂ˆM)2.\nSubstituting the torque matrix elements for the energy\nderivatives in Eq.(9) gives a damping rate of\nλbfs=τγ2\n2µ0/summationdisplay\nnk/vextendsingle/vextendsingleΓ−\nn(k)/vextendsingle/vextendsingle2η(ǫnk). (13)\nFor the intraband terms in Eq. (1) the integration over\nthe spectralfunctions reduces to τη(ǫnk)/2π/planckover2pi1so we find\nλbfs=π/planckover2pi1γ2\nµ0/summationdisplay\nn/integraldisplayd3k\n(2π)3/vextendsingle/vextendsingleΓ−\nn(k)/vextendsingle/vextendsingle2\n×/integraldisplay\ndǫ1Ank(ǫ1)Ank(ǫ1)η(ǫ1),(14)\nwhich matches the intraband terms of Eq. (1).4\nB. Interband terms\nAs the magnetization precesses, the spins rotate and\nthe spin-orbit energy changes. This variation acts as a\ntime dependent perturbation\nV(t) =eiσ·ϕ(t)Hsoe−iσ·ϕ(t)−Hso(0)≈i[σ·ϕ(t),Hso].\n(15)\nThis approximation results from linearizing the expo-\nnents, which is appropriate in the small angle limit.\nThe time dependence of the spin direction is ˆ ϕ(t) =\ncosωtˆx+sinωtˆy, up to a phase factor. This perturba-\ntion causes band transitions between the states ψnkand\nψmk. The initial and final states have the same wavevec-\ntor because these transitions are caused by the uniform\nprecession, which has a wavevector of zero. The transi-\ntion rate between states due to this perturbation is\nWmn(k) =2π\n/planckover2pi1/vextendsingle/vextendsingleΓ−\nmn(k)/vextendsingle/vextendsingle2δ(ǫmk−ǫnk−/planckover2pi1ω).(16)\nThe variations of the occupancies of the states with\nrespect to the magnetization direction are given by the\nmaster equation\n∂ρnk\n∂t=/summationdisplay\nm/negationslash=nWmn(k)[ρmk−ρnk].(17)\nThe second term in the effective field Eq. (2) contains\nthe factor∂ρnk/∂ˆMwhich is (∂ρnk/∂t)/(∂ϕ/∂t) where\n∂ϕ/∂t=ω. Inserting these expressions into the second\nterminthe effectivefield andrearrangingthe sumsgives\nHeff=−1\n2µ0M/summationdisplay\nnk/summationdisplay\nm/negationslash=nWmn(k)\nω2[ρnk−ρmk][ǫmk−ǫnk]dˆM\ndt.\n(18)\nComparing this result to the effective field predicted by\nthe Landau-Lifshitz-Gilbertequation(8) wefind adamp-\ning rate of\nλ=γ2\n2µ0/summationdisplay\nnk/summationdisplay\nm/negationslash=nWmn(k)[ρnk−ρmk]\nω[ǫmk−ǫnk]\nω.(19)\nThe finite lifetime of the states is introduced with the\nspectral functions\nλ=/planckover2pi12γ2\n2µ0/summationdisplay\nnk/summationdisplay\nm/negationslash=n/integraldisplay\ndǫ1Ank(ǫ1)/integraldisplay\ndǫ2Amk(ǫ2)\n×Wmn(k)[f(ǫ1)−f(ǫ2)]\n/planckover2pi1ω[ǫ2−ǫ1]\n/planckover2pi1ω. (20)\nInserting the transition rate Eq. (16), integrating over ǫ2,\nand taking the limit that ωgoes to zero leaves\nλ=π/planckover2pi1γ2\nµ0/summationdisplay\nn/summationdisplay\nm/negationslash=n/integraldisplayd3k\n(2π)3/vextendsingle/vextendsingleΓ−\nmn(k)/vextendsingle/vextendsingle2\n×/integraldisplay\ndǫ1Ank(ǫ1)Amk(ǫ1)η(ǫ1), (21)which are the interband terms of Eq. (1).\nIn this derivation of the bubbling Fermi surface contri-\nbution to the damping we have ignored an additional, re-\nversible term that contributes to the magnetocrystalline\nanisotropy. This contribution arises from changes in the\nequilibrium state occupancies as the magnetization di-\nrection changes. This contribution to the magnetocrys-\ntalline anisotropy is localized to the Fermi surface while\nthe contribution dicussed in IIIA is spread over all of the\noccupied levels.\nIV. TUNING THE DAMPING RATE\nWe have previously demonstrated that the mechansim\nof thetorque correlation model Eq. (1) accounts for the\nmajority of the precession damping rates of the transi-\ntion metals iron, cobalt, and nickel [4]. In the present\nwork we have so far shown that this expression for the\ndamping rate can be described simply within an effective\nfield picture. We now investigate the degree to which\nthe damping rate may be modified by adjusting certain\nmaterial parameters. Inspection of Eq. (1) reveals that\nthe damping rate depends on the convolution of two fac-\ntors: thetorquematrixelementsandtheintegraloverthe\nspectral functions. We separate the quantitative analy-\nsis of the damping rates into their dependencies on these\ntwo factors, beginning with the spectral weight.\nThe calculations for the damping rate of Eq. (1) dis-\ncussed below are performed using the linear augmented\nplanewave method in the local spin density approxima-\ntion. The details of the computational technique may be\nfound in [4], [21], and the included references.\nA. Spectral overlap\nFor the intraband terms, the integral over the spec-\ntral functions is essentially proportional to the density of\nstates at the Fermi level. Therefore, it appears reason-\nable to suspect that the intraband contribution to the\ndamping rate of a given material should be roughly pro-\nportional to the density of states of that material at the\nFermi level. To test this claim numerically, we artificially\nvaried the Fermi level of the metals within the d-bands\nand calculated the intraband damping rate as a function\nof the Fermi level. The results of these calculations are\nsuperimposed on the calculated densities of states of the\nmaterials in Fig. 2. The correlation between the damp-\ning rates and the densities of states, while not exact, is\ncertainly strong, indicating that increasing the density\nof states of a system at the Fermi level will generally\nincrease the intraband contribution to damping.\nThe dependence of the interband terms on the spectral\noverlap is more complicated than that of the intraband\nterms. The spectral overlap depends on the energy dif-\nferencesǫm−ǫn, which can vary significantly between\nbands and over k-points. When the scattering rate /planckover2pi1/τ5\nFIG. 2: Intraband damping rate versus Fermi level superim-\nposed upon density of states. A strong correlation between\nthe intraband damping rate versus Fermi level ( •) and the\ndensity of states (solid curves) is observed. Vertical blac k\nlines indicates true Fermi energy calculated by density fun c-\ntional theory.\nis much less than these energy gaps the interband terms\nare proportional to the scattering rate. However, this\nproportionality only holds at low scattering rates when\nthe interband contribution is much less than the intra-\nband contribution. The proportionality breaks down at\nhigher scattering rates when /planckover2pi1/τbecomes comparable to\nthe band gaps. After this point the damping rate gradu-\nally plateaus with respect to the scattering rate. Unfor-\ntunately, this complicated functional dependence of the\nspectral overlap on the scattering rate makes it difficult\nto obtain a simple description of the effect of the spec-\ntral overlap on the interband damping rate in terms of\nmaterial parameters.B. Torque matrix elements\nThe damping rate also depends on the square of the\ntorquematrixelements. Agoalofdopingisoftentomod-\nify the effective spin-orbit coupling of a sample. While\ndoping does more than this, such as intoducing strong lo-\ncal scattering centers, it is nevertheless useful to estimate\nthe dependence of the matrix elements on the spin-orbit\nparameter ξ. We begin with pure spin states ψ0\nnand\ntreat the spin-orbit interaction V=ξV′as a perturba-\ntion. The states can be expanded in powers of ξas\nψn=ψ0\nn+ξψ1\nn+ξ2ψ2\nn+... . (22)\nThe superscripts refer to the unperturbed wavefunction\n(0) and the additions ( i) due to the perturbation to the\nith order while the subscript nis the band index, which\nincludes the spin direction, up or down. Since the torque\noperator also contains a factor of the spin-orbit parame-\nter the matrix elements haveterms in everyorderof ξbe-\nginning with the first order. Therefore, the squared ma-\ntrix elements have contributions of order ξ2and higher.\nTo determine the importance of these terms we arti-\nficially tune the spin-orbit interaction from zero to full\nstrength, calculating the damping rate over this range.\nWe then fit the intraband and interband damping rates\nseparately to polynomials. In each material, this fitting\nshowed that for the intraband terms the ξdependence of\nthe damping rate was primarily third order, with smaller\ncontributions from the second and fourth order terms.\nRestricting the fit to only the third order term produced\na very reasonable result, shown in Fig. (3). For the in-\nterband terms, polynomial fitting was dominated by the\nsecond order term, with all other powers contributing\nonly negligibly. The second order fit is shown in Fig. (3).\nTo understand the difference in the ξdependence of\nthe intraband and interband contributions it is useful to\ndefine the torque operator\nΓ−=ξ(ℓ−σz−ℓzσ−). (23)\nThe torque operator lowers the angular momentum of\nthe state it acts on. This can be accomplished either by\nlowering the spin momentum ℓzσ−, a spin flip, or low-\nering the orbital momentum ℓ−σz, an orbital excitation.\nTherefore, both the intraband and interband contribu-\ntionseachhavetwosub-mechanism: spinflipsandorbital\nexcitations.\nThe second order terms for the intraband case are\nξ2|/angbracketleftψ0\nn|(ℓzσ−−ℓ−σz)|ψ0\nn/angbracketright|2. Sincetheunperturbedstates\nψ0\nnare pure spin states the spin flip part ℓzσ−of the\ntorque returns zero. Therefore, only the orbital excita-\ntions exist to lowest order in ξ, reducing the strength\nof the second order term in the intraband case. How-\never, the interband terms contain matrix elements be-\ntween several states, some with the same spin direction,\nbut others with opposite spin direction. Therefore, both\nspin flips and orbital excitations contribute in second or-\nder to the interband contribution.6\nFIG. 3: ξdependence of intraband and interband damping\nrates. Damping rates were calculated for a range of spin-orb it\ninteraction strengths between off ( ξ= 0) and full strength\n(ξ= 1).ξ2fits were made to the interband damping rates\n(left axes and ◭symbols) and ξ3fits to the intraband rates\n(right axes and ◮symbols).\nV. CONCLUSIONS\nThebreathingFermisurfacemodelhasprovidedasim-\nple and understandable effective field explanation of pre-\ncession damping in metallic ferromagnets. However, it is\nonlyapplicabletoverypuresystemsatlowtemperatures.\nOn the other hand, the torque correlation model accu-\nrately predicts damping rates of systems with imperfec-\ntions from low temperatures to above room temperature.\nThe shortcoming of the torque correlation model is that\nit does not illuminate the phyiscal mechanisms responsi-\nble for damping. We have pointed out that the breath-\ning Fermi surface model accounts for only one of the two\nterms in the effective field. By constructing an effective\nfield with the previously studied breathing Fermi surface\ncontribution and also the new bubbling effect we haveshown that this simpler picture may be mapped onto the\ntorque correlation model such that the breathing terms\nmatchtheintrabandcontributionandthebubblingterms\nmatch the interband contribution.\nSince there is considerable interest in understanding\nhow to manipulate the damping rates of materials we\ninvestigated the dependence of the intraband and inter-\nband damping rates on both the spectral overlap inte-\ngral and the torque matrix elements. For the intraband\nterms, the spectral overlap is proportional to the density\nof states and we found a strong correlation between the\nintraband damping rate and the density of states of the\nmaterial. The interband case is significantly complicated\nby the range of band gaps present in materials. No sim-\nple relation was found between the strength or scattering\nratedependence ofthe interband terms and common ma-\nterial parameters. The importance of the torque matrix\nelements to the damping rates was characterizedthrough\ntheir dependence on the spin-orbitparameter. The intra-\nband damping rates were found to vary as the spin-orbit\nparametercubed while the interbanddamping rateswent\nas the spin-orbit parameter squared. This difference was\nexplained by noting that the torque operatorchanges the\nangular momentum of states either through spin flips, or\nby changing their orbital angular momentum. Spin-flip\nexcitations do not occur to second order in ξfor the in-\ntraband terms, but do contribute at second order for the\ninterband terms.\nIt is desirable to understand the relative differences in\ndamping rates amoung various materials, such as why\nthe damping rate for nickel is higher than that for cobalt\nand iron. We have shown that the relative damping rates\nof these materials depend in part on the differences of\ntheirdensitiesofstatesandspin-orbitcouplingstrengths.\nHowever, they also depend in an intricate way on the en-\nergy gap spectra of each metal. For the interband terms\nthe dependence on the gap spectrum enters through the\nspectral overlap integral. For the intraband terms the\nenergy gaps appear in the denominators of the matrix\nelements. Therefore, states with very small splittings\ncan dominate the k-space convolution. The abundance\nofsuchstates in nickel appearsto contribute to the larger\ndamping rate in this material [23].\nDoping is a common technique for modifying damp-\ning rates. Doping has a number of consequences on a\nsample and these effects vary with the method of dop-\ning. Dopants can increase the electron-lattice scattering\nrate, introduce magnetic inhomogeneities that act as lo-\ncal scattering centers, alter the density of states, and\nchange the effective spin-orbit parameter. We have in-\nvestigated the consequences of modifying the densities of\nstatesandspin-orbitparameteronthe dampingrate, and\npreviously demonstrated the scattering rate dependence\nof the damping rate; however, it is not clear what new\ndamping mechanisms arise when rare-earth elements are\nadded to a transition metal host.\nThis work was supported in part by the Office of Naval\nResearch through grant N00014-03-1-0692 and through7\ngrant N00014-06-1-1016.\n[1] W. Bailey, P. Kabos, F. Mancoff, and S. Russek, IEEE\nTrans. Mag. 37, 1749 (2001).\n[2] J. Rantschler, R. McMichael, A. Castiello, A. Shapiro,\nW.F. Egelhoff, Jr., B. Maranville, D. Pulugurtha,\nA. Chen, and L. Conners, J. Appl. Phys. 101, 033911\n(2007).\n[3] V. Kambersk´ y, Czech. J. Phys. B 26, 1366 (1976).\n[4] K. Gilmore, Y. Idzerda, and M. Stiles, Phys. Rev. Lett.\n99, 027204 (2007).\n[5] S. Bhagat and P. Lubitz, Phys. Rev. B 10, 179 (1974).\n[6] B. Heinrich and Z. Frait, Phys. Stat. Sol. 16, K11 (1966).\n[7] B. Heinrich, D. Meredith, and J. Cochran, J. Appl. Phys.\n50, 7726 (1979).\n[8] J.F.Cochran and B. Heinrich, IEEE Trans. Magn. 16,\n660 (1980).\n[9] B. Heinrich, D. Fraitov´ a, and V. Kambersk´ y,\nPhys. Stat. Sol. 23, 501 (1967).\n[10] V. Kambersk´ y, Can. J. Phys. 48, 2906 (1970).\n[11] V. Korenman and R. Prange, Phys. Rev. B 6, 2769\n(1972).\n[12] V. Kambersk´ y and C. Patton, Phys. Rev. B 11, 2668\n(1975).\n[13] J. Kuneˇ s and V. Kambersk´ y, Phys. Rev. B 65, 212411(2002).\n[14] J. Ho, F. Khanna, and B. Choi, Phys. Rev. Lett. 92,\n097601 (2004).\n[15] Y. Tserkovnyak, G. Fiete, and B. Halperin,\nAppl. Phys. Lett. 84, 5234 (2004).\n[16] E. Rossi, O. G. Heinonen, and A. H. MacDonald,\nPhys. Rev. B 72, 174412 (2005).\n[17] B. Heinrich, Ultrathin magnetic structures III (Springer,\nBerlin, 2005).\n[18] M. F¨ ahnle and D. Steiauf, Phys. Rev. B 73, 184427\n(2006).\n[19] H.J. Skadsem, Y. Tserkovnyak, A. Brataas, and\nG.E.W. Bauer, Phys. Rev. B 75, 094416 (2007).\n[20] J. Sinova, T. Jungwirth, X. Liu, Y. Sasaki, J. Furdyna,\nW. Atkinson, and A. MacDonald, Phys. Rev. B 69,\n085209 (2004).\n[21] M. Stiles, S. Halilov, R. Hyman, and A. Zangwill,\nPhys. Rev. B 64, 104430 (2001).\n[22] D. Steiauf and M. F¨ ahnle, Phys. Rev. B 72, 064450\n(2005).\n[23] V. Kambersk´ y, Phys. Rev. B 76, 134416 (2007)."    },    {        "title": "1606.00756v2.Local_Lorentz_transformations_and_Thomas_effect_in_general_relativity.pdf",        "content": "arXiv:1606.00756v2  [gr-qc]  14 Jun 2016Local Lorentz transformations and Thomas effect in general\nrelativity\nAlexander J. Silenko∗\nResearch Institute for Nuclear Problems,\nBelarusian State University, Minsk 220030, Belarus\nand Bogoliubov Laboratory of Theoretical Physics,\nJoint Institute for Nuclear Research, Dubna 141980, Russia\nAbstract\nThe tetrad method is used for an introduction of local Lorent z frames and a detailed analysis\nof local Lorentz transformations. A formulation of equatio ns of motion in local Lorentz frames is\nbased on the Pomeransky-Khriplovich gravitoelectromagne tic fields. These fields are calculated in\nthe most important special cases and their local Lorentz tra nsformations are determined. Thelocal\nLorentz transformations and the Pomeransky-Khriplovich g ravitoelectromagnetic fields are applied\nfor a rigorous derivation of a general equation for the Thoma s effect in Riemannian spacetimes and\nfor a consideration of Einstein’s equivalence principle an d the Mathisson force.\nPACS numbers: 04.20.Cv, 04.25.-g\n∗Email: alsilenko@mail.ru\n1I. INTRODUCTION\nMethods of description of gravitational phenomena based on an int roduction of tetrads\nare often used in contemporary gravity. Any tetrad can charact erize a local Lorentz frame\n(LLF) attributed to some observer. The LLFs applied in the classica l monograph [1] for\ndiscussing Einstein’s equivalence principle are also very convenient fo r a description of spin\neffects including quantum mechanical analysis with the covariant Dira c equation (see, e.g.,\nthe reviews [2, 3]). Since the metric of LLFs is locally Minkowskian, a tra nsition from one\nLLF (also called a coframe or a tetrad frame) to another one is defin ed by an appropriate\nLorentz transformation. Such Lorentz transformations have b een considered in Refs. [4, 5].\nBasic tetrads satisfy the Schwinger gauge [6, 7] (see also [5]) while ot her tetrads can also be\nused. Tetrads which do not satisfy this gauge are carried by obser vers moving in a described\nspacetime. For example, a tetrad with e/hatwide0\ni/ne}ationslash= 0, e0\n/hatwidei/ne}ationslash= 0 in a Schwarzschild field is attributed\nto an observer moving relative to the source. All tetrads are equiv alent and the use of any\ntetrad is possible. Nevertheless, Hamiltonians and equations of mot ion of a test particle do\nnot coincide even for different tetrads belonging to the Schwinger g auge [8, 9] (see Subsec.\nIVB).\nGreat success achieved in description of electromagnetic phenome na stimulated a search\nfor direct analogies between the electrodynamics and gravity. Ind eed, the Newton and\nCoulomb laws as well as the Coriolis and Lorentz forces seems to be sim ilar. This is the rea-\nsonwhy oneoftenappliesaconceptionofgravitoelectromagnetism basedontheintroduction\nof scalar and vector potentials of a gravitoelectromagnetic field. T hen one introduces grav-\nitoelectic and gravitomagnetic fields connected each with others by Maxwell-like equations\n(see Refs. [10–12] and references therein). Since mathematical tools of the electrodynamics\nand gravity significantly differ, this approach ensures one only an ap proximate description\nof gravitational phenomena.\nA new big step in formulation of more exact equations of gravitoelect romagnetic fields\nhas been made by Pomeransky and Khriplovich [13]. They have starte d from standard\nequations stating the zero values of covariant derivatives of the f our-spin and four-velocity.\nA following transition to a LLF has allowed them to derive general equa tions for the tetrad\ncomponents of the four-spin and four-velocity. The obtained equ ations are pretty similar to\nthe corresponding equations in electrodynamics, namely, to the Th omas-Bargmann-Michel-\n2Telegdi equation for a Dirac particle and to the equation of motion of a charged particle.\nThis similarity has madeit possible toderive general formulas fortheg ravitoelectromagnetic\nfields [13] defined in an anholonomic tetrad frame and describing a relativistic particle in an\narbitrarily strong gravitational field or in a noninertial frame. It is important to mentio n\nthat this approach is based on the equivalence principle extended on the spin by Kobzarev\nand Okun [14] (see also Ref. [15]).\nThe most general description of motion of a spinning particle in gener al relativity (GR)\nis provided by the Mathisson-Papapetrou (MP) equations [16, 17]. T hese equations predict\nthe violation of the weak equivalence principle for pointlike spinning par ticles (see Refs.\n[18–20] and references therein). Nevertheless, the MP and Pome ransky-Khriplovich (PK)\nequations agree when one can neglect the mutual influence of part icle and spin motion\nleading to the aforementioned violation [4]. This circumstance substa ntiates the results\nobtainedbyPomeransky andKhriplovich andbringsapossibility ofawid eapplicationofthe\napproach based on the PK gravitoelectromagnetic fields. However , the original method [13]\nused the symmetric gauge which is inconvenient and may stimulate a wr ong interpretation\nof results obtained (see Refs. [4, 5]). In particular, the formula fo r the angular velocity\nof spin rotation in a rotating frame derived by this method differs fro m the well-known\nGorbatsevich-Mashhoon formula [21]. Therefore, we use the Schw inger gauge.\nA set of previously obtained results [4, 5, 22, 23] has shown the app licability of the con-\nception of gravitoelectromagnetism based on the PK gravitoelectr omagnetic fields and on\nthe Schwinger gauge. While the PK equations define the general for m of the gravitoelec-\ntromagnetic fields, the weak-field approximation happens to be rat her convenient to obtain\nsimple expressions for the gravitoelectromagnetic fields clarifying t heir physical meaning.\nFor stationary spacetimes, such expressions have been deduced in Ref. [4]. In the present\nwork, we derive general formulas for the gravitoelectromagnetic fields in this approximation.\nThe formulas obtained are applicable for a time-dependent metric. W e exactly calculate the\ngravitoelectromagnetic fields in several important special cases. We determine the local\nLorentz transformations of these fields. We also use the gravitoe lectromagnetic fields for an\nanalysisoffundamentalproblemsoftheMathissonforceandEinst ein’s equivalenceprinciple.\nThe very important problem of spin physics is the Thomas precession . The general for-\nmula for the Thomas precession in electrodynamics is based on specia l relativity [24] and\nis perfectly substantiated [25–29]. However, manifestations of th e Thomas effect in GR\n3are much less clear. In the present work, we fulfill the general des cription of the Thomas\nprecession in gravity with the use of the local Lorentz transforma tions and the gravitoelec-\ntromagnetic fields.\nThe paper is organized as follows. The detailed analysis of Lorentz tr ansformations in\ncoframes is carried out in Sec II. In Sec. III, we expound the conc eption based on the\nPK gravitoelectromagnetic fields and use it for a derivation of the eq uations of motion in\ncoframes. We calculate the gravitoelectromagnetic fields in the mos t important special cases\nin Sec. IV. The local Lorentz transformations of these fields are d etermined in Sec. V. The\nvalidity of these transformations is shown in the case of the uniform ly accelerated frame.\nIn Sec. VI, we apply the gravitoelectromagnetic fields for a demons tration of a difference\nbetween a behavior of spinning particles in the uniformly accelerated frame and in the\nSchwarzschild spacetime. The Mathisson force is considered in Sec. VII. We rigorously\nderive the general formula for the Thomas precession in arbitrary Riemannian spacetimes\nin Sec. VIII. The results obtained are summarized in Sec. IX.\nWe denote world and spatial indices by Greek and Latin letters α,µ,ν,... =\n0,1,2,3, i,j,k,... = 1,2,3, respectively. Tetrad indices are denoted by Latin letters from\nthe beginning of the alphabet, a,b,c,... = 0,1,2,3. Temporal and spatial tetrad indices are\ndistinguished by hats. The signature is (+ −−−). Commas and semicolons before indices\ndenote partial and covariant derivatives, respectively.\nII. LORENTZ TRANSFORMATIONS IN COFRAMES\nLet us consider Lorentz transformations in coframes. Our explan ation partially follows\nRefs. [4, 5].\nOne of the most powerful methods in GR is an introduction of tetrad s. They define the\nLLF characterized by the Minkowski metric ds2=ηabdxadxb, ηab= diag(1 ,−1,−1,−1).\nThe metric tensor of a given spacetime can be split into tetrads ea\nµsatisfying the relations\nea\nµeaν=gµν, eaµeµ\nb=ηab, ea\nµeµ\nb=δa\nb, ea\nµeν\na=δν\nµ. (1)\nAs usual, the world and tetrad indices (which all run from 0 to 3) are r aised and lowered\nwith the metric and Minkowski tensors, gµνandηab, respectively.\nAny tetrad can be attributed to an observer. Observers carryin g different tetrads may\n4moverelative toeachother. AlocalityofaLorentzframedefinedby sometetradiscausedby\nnonzero derivatives of the metric tensor. First derivatives define forces like the Newton force\nwhile second derivatives define the spacetime curvature and tidal f orces. These forces are\nfelt by the observer and can be detected in the observer’s lab. In p articular, the velocity of\nlight is equal to cnear the observer but the light can be accelerated (due to the New ton-like\nforce) and can undergo a deflection.\nIn the present work, we apply the approach based on the LLFs for a derivation of the\nequations of motion of a spinning particle. We use the approximation d isregarding effects\nconditioned by second derivatives of the metric tensor. In particu lar, we do not consider\nthe spin-curvature coupling. To study such effects, some other a pproaches seem to be\nmore convenient. For example, one can describe the spin-curvatu re coupling with the MP\nequations (see Ref. [23]).\nLet us consider two observers and two LLFs in the same area, i.e., in t he vicinity of\nsome point ( x0\n(0),x1\n(0),x2\n(0),x3\n(0)). Sincedxa=ea\nµdxµ, dx′a=e′a\nµdxµ, the connection between\ncoordinates in the two frames is given by [4]\ndxa=Ta\nbdx′b, (2)\nwhere\nTa\nb=ea\nµe′µ\nb. (3)\nWe mention that the both tetrads are bound with the same metric te nsorgµν=eaµea\nν=\ne′\naµe′a\nν.\nThe connection between the four-velocities in the two frames has t he form\nua≡dxa\ndτ=Ta\nbu′b, (4)\nwhereτ=s/cis the proper time.\nWe should underline that these relations are not valid beyond the loca l area.\nCertainly, the connection between two LLFs is realized by local Lore ntz transformations.\nIf the axes in the two frames are parallel and the direction of the ve locityVof the primed\nframe in the unprimed one is arbitrary, this connection is given by\ndx′/hatwide0=γ(dx/hatwide0−β·d/hatwider), d/hatwider′=d/hatwider+γ2\nγ+1β(β·dr)−βγdx/hatwide0,\ndx/hatwide0=γ(dx′/hatwide0+β·d/hatwider′), d/hatwider=d/hatwider′+γ2\nγ+1β(β·d/hatwider′)+βγdx′/hatwide0,β=V\nc,(5)\n5whereγ= (1−β2)−1/2is the Lorentz factor. We need to specify that the axes of the two\nframes remain to be parallel. Unlike the usual Lorentz transformat ions, the relative motion\nof the LLFs may be accelerated ( ˙V/ne}ationslash= 0).\nThe present analysis demonstrates that the quantities dxaanduaare four-vectors relative\nto the local Lorentz transformations. The four-momentum pa=−∂S/(∂xa) (Sis an action)\nand other four-vectors with tetrad components possess the sa me property. The tetrad eµ\na\nis a world four-vector when ais fixed and is a four-vector relative to the local Lorentz\ntransformations when µis fixed.\nHowever, we should note that the quantity eµ\naAamay not be a covariant four-vector even\nifAais a four-vector relative to the local Lorentz transformations. E xamples of such a\nsituation are given in Sec. IVD of Ref. [30].\nIt is important to consider the case when the unprimed frame is at re st relative the world\none. In this case, the unprimed tetrad satisfies the Schwinger gau ge [4–7] ( e/hatwide0\ni= 0, e0\n/hatwidei= 0).\nWe can mention that the Schwinger gauge defines an infinite set of te trads carrying by ob-\nservers immobileintheworldframe. These tetradsareconnectedb yspatialtransformations.\nDifferent tetrads (satisfying the Schwinger gauge) lead to differen t equations of motion [8].\nThe corresponding Hamiltonians also differ [8, 9]. However, appropria te coordinate trans-\nformations establish connections between them.\nEquation (3) reduces in the weak-field approximation when gravitat ional and noninertial\nfields are weak [ |gµν−ηµν| ≪1 (µ,ν= 0,1,2,3)]. Since the quantities e0\n/hatwide0andei\n/hatwideiare close to\n1, the equation ea\nµeν\na=δν\nµ= 0 (µ/ne}ationslash=ν) results in e/hatwideν\nµ=−eν\n/hatwideµ. Here hats point out the tetrad\nindices. The following relations are valid:\ng0i=e/hatwide0\ni−e/hatwidei\n0, g0i=e′/hatwide0\ni−e′/hatwidei\n0=e′i\n/hatwide0−e′0\n/hatwidei, T/hatwide0\n/hatwidei=e/hatwide0\ni+e′0\n/hatwidei, T/hatwidei\n/hatwide0=e/hatwidei\n0+e′i\n/hatwide0.(6)\nEvidently, T/hatwide0\n/hatwidei−T/hatwidei\n/hatwide0= 0.\nEquation (5) can be presented in the form\ndx′a=La\nbdxb, (7)\nwhereLa\nbis the Lorentz tensor. Therefore,\nTa\nb=La\nb, (8)\nwhere\nL/hatwide0\n/hatwide0=γ, L/hatwide0\n/hatwidei=L/hatwidei\n/hatwide0=−βiγ, L/hatwidej\n/hatwidei=δj\ni+γ2\nγ+1βiβj. (9)\n6Equations (3), (5), (8), and (9) define the dependence of the re lative motion of observers\non the tetrads carrying by them.\nThe connection between the two tetrads can also be obtained in an e xplicit form. Equa-\ntion (7) and the definition of tetrads result in\ne′a\nµ=La\nbeb\nµ. (10)\nThe tetrads can be used even for two observers moving in the Minko wski spacetime.\nIf we suppose the first observer to be at rest, the tetrad carrie d by him has only trivial\ncomponents. In this case, dx/hatwideµ=dxµand Eq. (10) takes the form\ne′a\nµ=La\nµ, e′/hatwide0\n0=γ, e′/hatwide0\ni=e′/hatwidei\n0=−βiγ, e′/hatwidei\nj=δi\nj+γ2\nγ+1βiβj. (11)\nThe tetrad (11) satisfies the requirements (1).\nLet us also consider an accelerated frame. The metric tensor is give n by\ngµν=/parenleftbigg/bracketleftBig\n1+a·r\nc2/bracketrightBig2\n,−1,−1,−1/parenrightbigg\n,\nwhereais an acceleration. For the observer at rest, the only nontrivial te trad component is\ne/hatwide0\n0= 1+a·r/c2. To simplify the analysis, we can suppose that the second observer moves\nin the LLF of the first observer along the x1axis with the velocity Vand the first observer\nis at rest in the world frame. In this case, the nontrivial tetrad com ponents are\ne′/hatwide0\n0=γe/hatwide0\n0, e′/hatwide1\n0=−βγe/hatwide0\n0, e′/hatwide0\n1=−βγ, e′/hatwide1\n1=γ. (12)\nIII. EQUATIONS OF MOTION IN COFRAMES\nIn the present work, we explain and develop the conception of grav itoelectromagnetism\nfirst proposed by Pomeransky and Khriplovich [13]. This conception h as been advanced in\nseveral works [4, 5, 22, 23, 30–32].\nCertainly, there are other conceptions of gravitoelectromagnet ism. The conventional\nconception of gravitoelectromagnetism (see Refs. [10–12]) is bas ed on a four-potential of\ngravitoelectromagnetic field expressed in terms of components of the metric tensor. This\nconception can be applied for a nonrelativistic particle in the weak-fie ld approximation and\ndoes not work in the relativistic case. Of course, this is nothing but t he simplest way to\nintroduce the gravitoelectromagnetic fields. Some other approac hes provide a description of\n7dynamics of spinning particles beyond the nonrelativistic approximat ion and the weak-field\none. We can mention the known papers by Bailey and Israel [33] and b y Yee and Ban-\nder [34]. Important results have been obtained [35] with the canonic al formulation of GR\nbased on the Arnowitt-Deser-Misner parametrization. In this con nection, Ref. [36] can also\nbe noticed. The results obtained allow one to draw a parallel between electromagnetism\nand gravity. In particular, some approaches connecting this para llel have been proposed\nin Refs. [37–39]. Established relations between electromagnetism an d gravity are of in-\nterest. Nevertheless, only the gravitoelectromagnetic fields intr oduced by Pomeransky and\nKhriplovich allow one to reveal a deep analogybetween spinning partic les in electromagnetic\nand gravitational/inertial fields.\nWe can note that a canonical method has also been applied in Refs. [5, 22, 23, 30, 40, 41]\nfor a derivation of quantum mechanical equations of motion. In Ref . [23], this method has\nalso been used to obtain the corresponding classical equations. In the present work, we\nbasically follow the original method by Pomeransky and Khriplovich [13 ]. The distinctive\nfeature of the PK fields is their introduction in the coframes.\nThe equation of motion of a pointlike spinless particle in GR is given by\nDuµ= 0. (13)\nThis equation defines the particle motion on a geodesic line which is pert urbed by the\nMathisson force for spinning particles and by tidal forces for exte nded ones. As a rule, the\nMathisson and tidal forces are relatively small and can be neglected in the present study.\nThe motion of spinning particles with allowance for a particle deflection from the geodesic\nline is defined by the Mathisson-Papapetrou equations [16, 17].\nThe orthogonality condition interconnects the spin four-vector aµwith either the four-\nvelocity or the four-momentum. The three-component spin ζis defined in the particle rest\nframe, when aµ= (0,ζ). The Mathisson-Pirani [16, 42] condition connects the spin with\nthe four-velocity,\nuµaµ= 0, (14)\nwhile the Tulczyjev condition [43] joins the spin with the four-moment um,\npµaµ= 0. (15)\nA choice of specific condition does not influence the next derivations . However, we can\n8note that there exist situations when the results following from the MP equations with the\nsupplementary condition (15) are not satisfactory from the phys ical point of view [44].\nWhen Eq. (13) is satisfied,\nDua=D(uµea\nµ) =uµea\nµ;νdxν. (16)\nTherefore,\nDua\ndτ=ea\nµ;νuµuν=eµ\nbea\nµ;νeν\ncubuc= Γa\nbcubuc. (17)\nHere Γ abc=−Γbac=eµ\nbeν\nceaµ;νare the Lorentz connection coefficients (Ricci rotation coeffi-\ncients). They can also be presented in the form\nΓabc=1\n2(λabc+λbca−λcab), λabc=−λacb=eµ\nbeν\nc(eaµ,ν−eaν,µ). (18)\nSinceua=uµea\nµis a world scalar, Dua=duaand Eq. (17) takes the form [13]\ndua\ndτ= Γa\nbcubuc. (19)\nSinceD(uµaµ) = 0, Eqs. (13) and (14) result in\nDaµ= 0. (20)\nEquations (13) and (20) can be considered as a mathematical form ulation of the equiva-\nlence principle for the particle and spin.\nThe same derivation as above leads to the equation of spin motion in LL Fs. Since\nDaa\ndτ=ea\nµ;νuµuν=eµ\nbea\nµ;νeν\ncubuc= Γa\nbcabuc, (21)\nwe finally obtain\ndaa\ndτ= Γa\nbcabuc. (22)\nEquations (19) and (22) have been first obtained in Ref. [13]. We can note that the use of\nLLFs for a description of spin dynamics is quite natural. The three-c omponent spin which\nevolution is commonly described is defined in the particle rest frame. T his frame can be\nattributed to some observer and the local Lorentz transformat ion of the spin (pseudo)vector\ncoincides with its transformation in special relativity (see, e.g., Ref. [45]):\naa= (a/hatwide0,/hatwidea),/hatwidea=ζ+γ2β(β·ζ)\nγ+1, a/hatwide0=β·/hatwidea=γβ·ζ. (23)\n9In this case,\nua= (u/hatwide0,/hatwideu) = (γ,γβ). (24)\nLet us remember that V=βcis the velocity of relative motion of the two LLFs. Therefore,\nthe quantity ζdefines here the three-component spin in the instantaneously acc ompanying\nframe. The spin motion in the nonrotating instantaneously accompa nying frame and in the\nparticle rest frame differs due to the Thomas effect [24]. This differen ce is defined by [24, 25]\n/parenleftbigg∂ζ\n∂t/parenrightbigg\nnonrot=/parenleftbigg∂ζ\n∂t/parenrightbigg\nrestframe+ωT×ζ, (25)\nwhereωTis the angular velocity of the Thomas precession. In electrodynamic s and special\nrelativity,\nωT=−γ\nγ+1/parenleftbigg\nβ×dβ\ndτ/parenrightbigg\n. (26)\nEquations (19) and (22) are similar to the corresponding equations of motion in electro-\ndynamics:\nduµ\ndτ=e\nmFµνuν, (27)\ndsµ\ndτ=e\nmFµνsν. (28)\nEquation (28) describes the spin motion of a particle with the Dirac magnetic moment\n(g= 2).\nThe electromagnetic field tensor can be expressed in terms of the e lectric and magnetic\nfields,Fµν= (E,B). A similarity between ( e/m)Fµνand Γ abcuchas allowed Pomeransky\nandKhriplovichtointroducethegravitoelectricandgravitomagnet icfields,cΓabcuc= (E,B).\nExplicitly [13]\nE/hatwidei=cΓ0icuc,B/hatwidei=−c\n2eiklΓklcuc. (29)\nFor the gravitoelectromagnetic fields EandB, we do not make a difference between upper\nand lower indices.\nAs a result of comparison of the foregoing equations of motion in gra vitational/inertial\nand electromagnetic fields, Pomeransky and Khriplovich [13] have o btained the following\nequation of motion for the three-component spin:\ndζ\ndτ=Ω′×ζ,Ω′=−B+/hatwideu×E\nu/hatwide0+1. (30)\nThe time dilation is given by\ndt=u0dτ, d/hatwidet=u/hatwide0dτ. (31)\n10As a result,\ndζ\ndt=Ω×ζ,Ω=1\nu0/parenleftbigg\n−B+/hatwideu×E\nu/hatwide0+1/parenrightbigg\n. (32)\nThe validity of Eqs. (30) and (32) has been substantiated in Refs. [4 , 5, 32]. It has been\ndemonstrated [4, 5] that the LLF which is at rest relative to the wor ld frame satisfies the\nSchwinger gauge. Other gauges define the equations of motion in fr ames moving relative to\nthe world frame. This important property was not taken into accou nt in Refs. [13, 31, 32].\nThe equations of particle motion can be presented as follow [13]:\nd/hatwideu\ndτ=u/hatwide0E+/hatwideu×B,du/hatwide0\ndτ=E·/hatwideu. (33)\nWe can underline that the equation of motion (19) is essentially nonline ar and Eq. (22)\ncontains the four-velocity. These properties differ the equations of motion in GR from the\ncorresponding equations in electrodynamics [Eqs. (27) and (28)]. T herefore, the gravitoelec-\ntromagnetic fields depend on the four-velocity and they are effect ive fields. In particular,\none cannot use these fields to construct a free gravitoelectroma gnetic field.\nSince the particle velocity in the coframe is equal to /hatwidev≡d/hatwider/d/hatwidet=c/hatwideu/slashBig/radicalbig\n1+/hatwideu2and\nd/d/hatwidet= (1/u/hatwide0)(d/dτ), the corresponding acceleration is given by\n/hatwidew≡d/hatwidev\nd/hatwidet=c\nu/hatwide0/bracketleftBigg\nd/hatwideu\nd/hatwidet−/hatwideu\n/parenleftbig\nu/hatwide0/parenrightbig2/parenleftbigg\n/hatwideu·d/hatwideu\nd/hatwidet/parenrightbigg/bracketrightBigg\n=c\nu/hatwide0/bracketleftBig\nE+/hatwideβ×B−/hatwideβ/parenleftBig\nE·/hatwideβ/parenrightBig/bracketrightBig\n,(34)\nwhere/hatwideβ=/hatwidev/c.\nWe should take into account that Eqs. (22), (32), and (33), (34) are not equally useful.\nEquations(22)and(32)areperfectforadescriptionofthespinm otion. Itisoftenadmissible\nto disregard an inhomogeneity of the gravitoelectromagnetic fields and their dependence on\nthe four-velocity. One can also take into account the aforementio ned inhomogeneity and the\nevolution of the four-velocity in the LLF. As a contrary, Eqs. (33) and (34) do not define\nmeasurable dynamics of the four-velocity. First of all, one cannot ig nore the nonlinearity\ncaused by the dependence of the gravitoelectromagnetic fields on the four-velocity. Second,\nan observer needs a description of dynamics of the four-velocity in the world frame. In\nparticular, the conventional force which governs the particle mot ion is defined in this frame\nand has the form fi=mc(dui/dt). An example of the description of the particle motion in\nthe world frame will be presented in Sec. VI.\nThus, the introduction of the gravitoelectromagnetic fields defined in LLFs simplifies the\ndescription of dynamics of spinning particles in GR. Equation (29) sho ws that the important\n11specific feature of the gravitoelectromagnetic fields is their depen dence on the particle four-\nvelocity. However, the results presented do not define some othe r fundamental properties\nof gravitoelectromagnetic fields. It is very important to find a tran sformation law of these\nfields. This problem will be solved in Sec. V.\nIV. GRAVITOELECTROMAGNETIC FIELDS IN SOME IMPORTANT SPECIAL\nCASES\nIt is instructive to calculate the gravitoelectromagnetic fields in the most important spe-\ncial cases. Certainly, we will determine these fields in the LLFs which a re at rest relative\nto the corresponding world frames. We will consider, among others , several examples of a\nnonstationary (time-dependent) metric, the importance of which for analyzing fundamental\nproblems of GR has been recently affirmed in Ref. [8].\nThe general form of the line element of an arbitrary gravitational fi eld can be given by\n[22, 46]\nds2=V2c2dt2−δ/hatwidei/hatwidejW/hatwidei\nkW/hatwidej\nl(dxk−Kkcdt)(dxl−Klcdt). (35)\nAn analysis of the most important metrics can be simplified with the use of the isotropic\n(more exactly, Cartesian-like isotropic) coordinates. In this case , the line element takes the\nform (see Ref. [5])\nds2=V2c2dt2−W2δij(dxi−Kicdt)(dxj−Kjcdt). (36)\nA. General noninertial frame\nThe accelerated and rotating noninertial frame presents the gen eral case of inertial fields\nin a flat spacetime. The acceleration aand the angular velocity of rotation ωof an observer\nare independent of the spatial coordinates but may depend arbitr arily on time. The exact\nmetric of the general noninertial frame has the form (36), where\nV= 1+a(t)·r\nc2, W= 1,K=−1\nc(ω(t)×r). (37)\nExplicitly, the metric is given by [47]\nds2=/bracketleftBigg/parenleftbigg\n1+a(t)·r\nc2/parenrightbigg2\n−[ω(t)×r]2\nc2/bracketrightBigg\nc2dt2−2[ω(t)×r]·drdt−δijdxidxj.(38)\n12The nonzero Lorentz connection coefficients have the same forms for time-dependent and\ntime-independent inertial fields,\nΓ/hatwidei/hatwidej/hatwide0=−ceijkωk(t)\nc2+a(t)·r,Γ/hatwide0/hatwidei/hatwide0=−Γ/hatwidei/hatwide0/hatwide0=−ai(t)\nc2+a(t)·r. (39)\nThe gravitoelectromagnetic fields are given by\nE=−ca(t)\nc2+a(t)·ru/hatwide0,B=c2ω(t)\nc2+a(t)·ru/hatwide0. (40)\nThere is only the gravitoelectric field in the uniformly accelerated fra me,\nE=−ca(t)\nc2+a(t)·ru/hatwide0,B= 0. (41)\nIn the rotating frame, there is only the gravitomagnetic field:\nE= 0,B=ω(t)u/hatwide0. (42)\nIt is important that the gravitoelectric field does not depend on /hatwideu. All equations presented\nin this subsection are exact.\nThe use of Eqs. (32) and (33) allows one to reproduce known formu las for the particle\nmotion and the spin rotation in the general noninertial frame.\nB. Cylindrical coordinate system\nWhen the cylindrical coordinate system is used, the spacetime is flat . However, the\nmetric tensor is nontrivial and has the form gµν= diag(1,−1,−ρ2,−1). The simplest tetrad\nsatisfying the Schwinger gauge has the only nontrivial component e/hatwideφ\nφ≡e/hatwide2\n2=ρ. As a result,\nthe nonzero Lorentz connection coefficients are\nΓ/hatwide2/hatwide1/hatwide2=−Γ/hatwide1/hatwide2/hatwide2=1\nρ. (43)\nThe gravitoelectromagnetic fields are given by [48]\nE= 0,B/hatwideρ≡ B/hatwide1= 0,B/hatwideφ≡ B/hatwide2= 0,B/hatwidez≡ B/hatwide3=u/hatwide2\nρ=uφ≡u2. (44)\nEquations (33) and (44) show that the force determined by the gr avitomagnetic field\nactinginthecylindricalcoordinatesystemisananalogueoftheLore ntzforce. Itsappearance\nis a consequence of the fact that, if the azimuthal angle of the par ticle changes by d/hatwideφ, the\n13horizontal axes of the cylindrical and Cartesian systems of coord inates rotate by the same\nangle with respect to each other. Thus, the cylindrical coordinate system rotates with an\ninstantaneous angular velocity −d/hatwideφ/dt=−v/hatwideφ/ρwith respect to the Cartesian one [48].\nThe nonzero gravitomagnetic field leads to forces acting on particle s and torques rotating\nspins. However, theseforcesandtorquesarefictitious. Theirap pearanceiscausedbythefact\nthat particle trajectories have different shapes in the Cartesian a nd cylindrical coordinate\nsystems. The aforementioned forces and torques are not felt by an observer. In contrast, the\nobserver feels the acceleration force and the centrifugal one ac ting in the general noninertial\nframe.\nThe result can differ for another tetrad even if it also satisfies the S chwinger gauge.\nLet us consider an observer using the Cartesian coordinates ( x/hatwide0=x0, x/hatwide1=ρcosφ, x/hatwide2=\nρsinφ, x/hatwide3=z). In this case, the nontrivial tetrad components are given by\ne/hatwide1\n1= cosφ, e/hatwide1\n2=−ρsinφ, e/hatwide2\n1= sinφ, e/hatwide2\n2=ρcosφ.\nIt is easy to obtain that the gravitoelectromagnetic fields are equa l to zero ( E= 0,B= 0).\nThis result is natural for the Cartesian coordinate system in the Min kowski spacetime.\nThis comparison of the two tetrads satisfying the Schwinger gauge elucidates the state-\nment made in Ref. [8]. A tetrad field in the Schwinger gauge is not unique and different\ntetrads may lead to different equations of motion. The correspond ing Hamiltonians do not\ncoincide either [9, 48]. However, the forces and torques caused by a difference of the tetrads\nare fictitious and are not felt by an observer.\nC. Gravitoelectromagnetic fields in the weak-field approximation\nThe weak-field approximation can often be used. To find gravitoelec tromagnetic fields in\nthis approximation, we use the Schwinger gauge and suppose that t he tetrad components\ne/hatwideµ\nµ(µ=/hatwideµ, µ= 0,1,2,3) are close to unit. Under these conditions, some important gener al\nrelations can be obtained,\neν\n/hatwideµ+e/hatwideν\nµ=δν\nµ, gµν=e/hatwideµν+e/hatwideνµ,\nΓabc=1\n2/parenleftBig\ne/hatwideab,c−e/hatwideba,c+gbc,a−gac,b/parenrightBig\n.(45)\n14As a result, nonzero Lorentz connection coefficients are given by\nΓ/hatwide0/hatwidei/hatwide0=−1\n2g00,i,Γ/hatwide0/hatwidei/hatwidej=−1\n2(g0i,j+g0j,i−gij,0),Γ/hatwidei/hatwidej/hatwide0=1\n2(g0j,i−g0i,j),\nΓ/hatwidei/hatwidej/hatwidek=1\n2(gjk,i−gik,j).(46)\nThe gravitoelectromagnetic fields are equal to\nE/hatwidei=−c\n2/bracketleftBig\ng00,iu/hatwide0+(g0i,j+g0j,i−gij,0)u/hatwidej/bracketrightBig\n,\nB/hatwidei=c\n4eijk/bracketleftBig\n(g0j,k−g0k,j)u/hatwide0+(gjl,k−gkl,j)u/hatwidel/bracketrightBig\n.(47)\nIn Ref. [4], Eqs. (46) and (47) have been obtained for a stationary metric. Only the\ngravitoelectric field explicitly depends on the time derivative.\nWe can conclude that the equations of motion in coframes (32) and ( 33) become very\nsimple in the weak-field approximation. In this case, they contain only first derivatives of\nthe metric tensor and can be easily derived. The equations obtained are relativistic.\nD. Lense-Thirring metric\nLense and Thirring have discovered in 1918 that rotating bodies “dr ag” the spacetime\naround themselves (frame dragging [49]). In other words, they h ave demonstrated the sim-\nilarity between rotating frames and spacetimes created by rotatin g bodies.\nThe Lense-Thirring (LT) metric [49] defines a gravitational field of a rotating source in\nthe weak-field approximation. It can be obtained from the Kerr met ric when the distance\nfrom the source is much large than the gravitational radius. The st atic part of the LT metric\ncharacterizes the Schwarzschild field of a distant source. It is con venient to transform the\nLT metric to the isotropic coordinates [5],\nV= 1−GM\nc2r, W= 1+GM\nc2r,K=ω×r\nc,\nω=2G\nc2r3J=/parenleftbigg\n0,0,2GMa\ncr3/parenrightbigg\n.(48)\nHereJ=Mcaezis the total angular momentum of the source and Mis its mass.\nFor this metric, the gravitoelectromagnetic fields read\nE=−Gm\ncr3ru/hatwide0+3G\nc2r5[r(J·(r×/hatwideu))−(r×J)(/hatwideu·r)],\nB=−Gm\ncr3r×/hatwideu−G\nc2r3/bracketleftbigg3(r·J)r\nr2−J/bracketrightbigg\nu/hatwide0.(49)\n15Because (see Ref. [5])\n(r×/hatwideu)(J·(r×/hatwideu))+(/hatwideu×(r×J))(/hatwideu·r) = 2(r×/hatwideu)(J·(r×/hatwideu))\n+(/hatwideu×(r×/hatwideu))(r·J)+r2(/hatwideu×(/hatwideu×J)),\nequations (32) and (49) lead to the following equivalent equations of spin motion [5]:\nΩ=1\nu0/braceleftBigg\nGM\ncr3·2u/hatwide0+1\nu/hatwide0+1r×/hatwideu+G\nc2r5/bracketleftbig\n3r(r·J)−r2J/bracketrightbig\nu/hatwide0\n−3G\nc2r5·1\nu/hatwide0+1[(r×/hatwideu)(J·(r×/hatwideu))+(r·/hatwideu)(/hatwideu×(r×J))]/bracerightBigg\n,(50)\nΩ=1\nu0/braceleftBigg\nGM\ncr3·2u/hatwide0+1\nu/hatwide0+1r×/hatwideu+G\nc2r5/bracketleftbig\n3r(r·J)−r2J/bracketrightbig\nu/hatwide0\n−3G\nc2r5·1\nu/hatwide0+1/bracketleftbig\n2(r×/hatwideu)(J·(r×/hatwideu))+(/hatwideu×(r×/hatwideu))(r·J)+r2(/hatwideu×(/hatwideu×J))/bracketrightbig/bracerightBigg\n.(51)\nThese equations describe the geodetic precession and the LT one f or therelativistic particle.\nEquations (50) and (51) agree with the approximate formula by How ever, one of impor-\ntant preferences of the relativistic approach is the discovery of a dditional dependence of E\nandBon the nondiagonal and diagonal components of the metric tensor , respectively. The\ncorresponding contributions to the angular velocity of the spin pre cession do not follow from\nthe nonrelativistic approximation.\nE. Static gravitational fields in isotropic coordinates\nThe static metric in the isotropic coordinates is defined by Eq. (36) a t condition that\nK= 0 and has the form\nds2=V2c2dt2−W2(dr·dr). (52)\nThe respective gravitoelectromagnetic fields are given by\nE=−cu/hatwide0∇V,B=c∇W×/hatwideu. (53)\nAn appearance of nonzero Bdepending on Wis a new property as compared with the\nnonrelativistic approximation.\nThe most important examples of static fields are the Schwarzschild, de Sitter, and anti-de\nSitter spacetimes. We use the weak-field approximation.\n16For the Schwarzschild metric in the isotropic coordinates, VandWare given by Eq.\n(48). In the weak-field approximation, the gravitoelectromagnet ic fields take the form\nE=−GMr\ncr3u/hatwide0=gu/hatwide0\nc,B=−GM\ncr3r×/hatwideu=g×/hatwideu\nc, (54)\nwheregis the Newtonian acceleration. We can state the significant differenc e between the\ngravitoelectromagnetic fields in the uniformly accelerated frame an d in the Schwarzschild\nspacetime. The gravitomagnetic field in the Schwarzschild spacetime , contrary to the uni-\nformly accelerated frame, is nonzero.\nThe four-dimensional de Sitter metric can be presented in the form\nds2=/parenleftbigg\n1−r2\nα2/parenrightbigg\nc2dt2−/parenleftbigg\n1−r2\nα2/parenrightbigg−1\ndr2−r2(dθ2+sin2θdφ2). (55)\nThere is a cosmological horizon at r=α.\nThe de Sitter spacetime is an Einstein manifold since the Ricci tensor is proportional to\nthe metric:\nRµν=4\nα2gµν.\nThis means that the de Sitter spacetime is a vacuum solution of the Ein stein equation with\nthe cosmological constant Λ = 3 /α2and the scalar curvature R= 4Λ = 12 /α2.\nThe anti-de Sitter metric can be obtained with the substitution α→ik.\nThewell-knowncoordinatetransformationmayreducedeSitteran danti-deSittermetrics\nto isotropic forms. For the de Sitter metric, this transformation is given by\nr=ρ\n1+ρ2\n4α2. (56)\nThe metric takes the form\nds2=k2\n−k−2\n+c2dt2−k−2\n+/bracketleftbig\ndρ2+ρ2(dθ2+sin2θdφ2)/bracketrightbig\n, k±= 1±ρ2\n4α2. (57)\nAs a result, the isotropic Cartesian coordinates can be used.\nThe gravitoelectromagnetic fields read\nE=c\nα2ρu/hatwide0,B=−c\nα2ρ×/hatwideu. (58)\nFor the anti-de Sitter metric, the gravitoelectromagnetic fields ca n be obtained with the\nsubstitution α→ikand are equal to\nE=−c\nk2ρu/hatwide0,B=c\nk2ρ×/hatwideu. (59)\nAn existence of the gravitomagnetic field in static spacetimes is very important for an\nevolution of momentum and spin. This field is not weak, at least for rela tivistic particles.\n17V. LOCAL LORENTZ TRANSFORMATIONS OF GRAVITOELECTROMAG-\nNETIC FIELDS\nThe properties of the local Lorentz transformations allow us to de termine the general\ndependence of the gravitoelectromagnetic fields from the choice o f a tetrad. The results\nobtained in Sec. II and the explicit expression of Γ abcucin terms of tetrads (18) lead to the\nconclusion that this quantity is an antisymmetric tensor relative to t he local Lorentz trans-\nformations. Therefore, the gravitoelectric and gravitomagnetic fields,EandB, transform\nlike the electric and magnetic ones,\nE′=γ/bracketleftbigg\nE−γ\nγ+1β(β·E)+β×B/bracketrightbigg\n,\nB′=γ/bracketleftbigg\nB−γ\nγ+1β(β·B)−β×E/bracketrightbigg\n.(60)\nThis property also establishes a great similarity between electromag netism and gravity.\nWhen the local Lorentz transformations (60) are used, one need s to take into account the\ndependence of the gravitoelectromagnetic fields on the four-velo city. To express E′andB′\nin terms of the four-velocity in the primed frame, one needs to tran sform the components\nofucentering into Eq. (29) as follows:\nu/hatwide0=γ(u′/hatwide0+β·/hatwideu′),/hatwideu=/hatwideu′+γ2\nγ+1β(β·/hatwideu′)+βγu′/hatwide0. (61)\nLet us consider, as an example, the local Lorentz transformation s of the gravitoelectro-\nmagnetic fields in the uniformly accelerated frame. Let us suppose t hat the acceleration is\nconstant and has nonzero projections onto the axes e1ande2(a=a(1)e1+a(2)e2). Let\nus suppose that the second observer moves in the LLF of the first observer along the x1\naxis with the velocity Vand the first observer is at rest in the world frame. In this case,\nthe nontrivial tetrad components are given by Eq. (12). The nont rivial components of the\ninverse tetrad are equal to\ne′0\n/hatwide0=γc2\nc2+a·r, e′0\n/hatwide1=βγc2\nc2+a·r, e′1\n/hatwide0=βγ, e′1\n/hatwide1=γ (62)\nand the nonzero Lorentz connection coefficients (Γ abc=−Γbac) read\nΓ/hatwide1/hatwide0/hatwide0=γa(1)\nc2+a·r,Γ/hatwide2/hatwide0/hatwide0=γ2a(2)\nc2+a·r,Γ/hatwide1/hatwide0/hatwide1=βγa(1)\nc2+a·r,\nΓ/hatwide0/hatwide2/hatwide1=−βγ2a(2)\nc2+a·r,Γ/hatwide2/hatwide1/hatwide0=βγ2a(2)\nc2+a·r,Γ/hatwide1/hatwide2/hatwide1=−β2γ2a(2)\nc2+a·r.(63)\n18The use of Eqs. (29) and (61) results in\nE′\n1=−a(1)c\nc2+a·ru/hatwide0,E′\n2=−γa(2)c\nc2+a·ru/hatwide0,B′\n3=βγa(2)c\nc2+a·ru/hatwide0. (64)\nSincethesameresult canbeeasilyobtainedwithEqs. (41)and(60), thisderivationconfirms\nthe validity of the general equation (60).\nThe results presented in this section explicitly show the possibility of t he local Lorentz\ntransformations of the gravitoelectromagnetic fields and the equ ivalence of all tetrads. How-\never, the tetrads belonging the Schwinger gauge are much more co nvenient.\nVI. CONNECTION BETWEEN EQUATIONS OF MOTION IN THE UNI-\nFORMLY ACCELERATED FRAME AND IN THE SCHWARZSCHILD FIELD\nAND ITS RELATION TO EINSTEIN’S EQUIVALENCE PRINCIPLE\nIn this section, we use the results presented for a comparison of m otion of a spinning\nparticle in the uniformly accelerated frame and in the Schwarzschild fi eld. This problem\nis directly related to Einstein’s equivalence principle which is a cornerst one of GR. In this\nsection, we do not make a difference between xwith upper and lower indices.\nA. Previously obtained results\nEquation (41) and (54) shows the difference between the gravitoe lectromagnetic fields in\nthe uniformly accelerated frame andinthe Schwarzschild metric inth e isotropic coordinates.\nThe corresponding angular velocities of the spin precession in the no nrelativistic limit ( v≪\nc) are equal to [50, 51]\nΩ(a)=−a×/hatwidev\n2c2,Ω(i)=3g×/hatwidev\n2c2. (65)\nOwing to the difference between the two angular velocities on the con dition that a=−g,\nit has been claimed in Refs. [50, 52] that Einstein’s equivalence princip le is violated. This\nclaim has been based on a noncoincidence of the quantum mechanical Hamiltonians for a\nnonrelativisticDiracparticleintheuniformlyacceleratedframe[47]a ndintheSchwarzschild\nmetric in the isotropic coordinates. The latter Hamiltonian has been fi rst derived in Ref.\n[53]. In Ref. [52], the nonrelativistic quantum mechanical Hamiltonians has been obtained\n19for the Schwarzschild metric in the Cartesian coordinates but its derivation contains an\nerror. This error has been corrected in Ref. [51].\nThe Schwarzschild metric in the Cartesian coordinates has the form (35) where\nV= 1−Φ, W/hatwidei\nk=δi\nk+Φxixk\nr2,K= 0,Φ =GM\nc2r. (66)\nThe appropriate Schwinger tetrad is given by [23]\ne/hatwide0\nµ=Vδ0\nµ, e/hatwidei\nµ=W/hatwidei\nkδk\nµ. (67)\nThe corrected Hamiltonian for the Schwarzschild metric in the Carte sian coordinates\n[51] leads to the angular velocity of the spin precession coinciding with that defined by the\nDonoghue-Holstein Hamiltonian [53],\nΩ(C)=3g×/hatwidev\n2c2. (68)\nIn Ref. [51], the relativistic expression for the angular velocity of th e spin precession has\nalso been obtained.\nTherefore, the angular velocity of the spin precession of the nonr elativistic Dirac particle\nis three times bigger in the Schwarzschild field than in the uniformly acc elerated frame. It\nhas been shown in Ref. [51] that the part of the effect in the Schwa rzschild spacetime caused\nby the temporal component of the metric is equal to the total effe ct in the accelerated frame\nand the additional effect in the Schwarzschild spacetime is due to the spatial components of\nthe metric. The classical equations for the spin precession in the Sc hwarzschild spacetime\n[51, 54] and in the uniformly accelerated frame [51, 55] fully agree wit h the corresponding\nquantum mechanical ones. In the general case, the perfect agr eement between classical and\nquantum mechanical equations of motion has been proven in Refs. [2 3, 30].\nIn Ref. [40], the known results has been generalized to a relativistic D irac particle. The\nrelativistic Foldy-Wouthuysen transformation has been performe d and quantum mechanical\nand semiclassical equations of motion for the momentum and spin hav e been derived. It\nhas been shown that all equations of motion are different for partic les in the Schwarzschild\nspacetime and in the uniformly accelerated frame. The semiclassical equations of motion in\nthe Schwarzschild field are given by [40]\ndp\ndt=2γ2−1\nγmg,dζ\ndt=−2γ+1\nc2(γ+1)(g×v)×ζ, (69)\n20wherep=−{pi}is the generalized momentum. The corresponding equations of motio n for\nthe accelerated frame read [40]\ndp\ndt=−γma,dζ\ndt=γ\nc2(γ+1)(a×v)×ζ. (70)\nThe weak-field approximation and the isotropic coordinates are use d in Eqs. (69) and (70).\nWhena=−g, these equations significantly differ.\nEquation (69) and (70) demonstrate a difference between the par ticle motion in the\nSchwarzschild spacetime and in the uniformly accelerated frame. Fo r example, the light\ndeflection in the Schwarzschild field in the isotropic coordinates ( g=−a) defined by Eq.\n(69) seems to be twice as much as in the uniformly accelerated frame . This problem will be\nconsidered below in details. An origin of the aforementioned effects is the nonzero spatial\npart of the Schwarzschild metric. Equations (41) and (54) show th at the gravitoelectric field\nin the uniformly accelerated frame is the same as in the Schwarzschild spacetime. However,\nthe spatial part of the Schwarzschild metric generates the gravit omagnetic field B=g×/hatwideu/c\nwhich is absent in the uniformly accelerated frame.\nNevertheless, we do not share the statement about the violation o f Einstein’s equiva-\nlence principle presented in Refs. [50, 52]. It is incorrect to suppose that the equivalence\nprinciple as formulated by Einstein and successors states the comp lete equivalence of static\ngravitational fields and uniformly accelerated frames. In Einstein’s papers [56], the equiv-\nalence principle has been formulated only relative to constant uniform gravitational fields.\nThe Schwarzschild field (as well as other real gravitational fields) is nonuniform . Since the\nequivalence principle is one of fundamental principles of GR, we will con sider the problem\nof the importance of a field inhomogeneity in detail.\nWeshouldmentionthatthepresenceoftidalandMathissonforces alwaysdiffersthestatic\ngravitational field from the uniformly accelerated frame. The tidal forces are proportional\nto derivatives of the Newtonian acceleration, i.e. to secondderivatives of the metric tensor.\nThe Mathisson force, which defines the spin-curvature coupling an d is also proportional to\nsecond derivatives of the metric tensor, will be considered in the ne xt section.\n21B. Comparison of equations of motion in the Schwarzschild field in the Cartesian\nand isotropic coordinates\nTo demonstrate an influence of a spatial inhomogeneity of the Schw arzschild field on\nthe particle motion, we can compare the equations of motion in the Ca rtesian and isotropic\ncoordinates. To establish a difference in these equations of motion, it is sufficient to consider\na field of a distant source and to use the weak-field approximation. I n this approximation,\nthe metric tensors of the Schwarzschild field can be given by [57]\ng(C)\n00= 1−rg\nr, g(C)\n0i= 0, g(C)\nij=−/parenleftBig\nδij−rgxixj\nr3/parenrightBig\n(71)\nand\ng(i)\n00= 1−rg\nr, g(i)\n0i= 0, g(i)\nij=−/parenleftBig\n1−rg\nr/parenrightBig\nδij (72)\nin the Cartesian and isotropic coordinates, respectively. Here rg= 2GM/c2is the gravita-\ntional radius.\nTheuseofthegeneralequation(47)forthemetric(71)resultsin thefollowingexpressions\nfor the gravitoelectromagnetic fields:\nE(C)=−crgr\n2r3u/hatwide0=gu/hatwide0\nc,B(C)=−crg\n2r3r×/hatwideu=g×/hatwideu\nc. (73)\nA comparison of Eqs. (54) and (73) shows that the gravitoelectro magnetic fields in the\nCartesian coordinates are the same as in the isotropic ones. There fore, the equations of the\nspin motion in the Schwarzschild field have the same form in the Cartes ian and isotropic\ncoordinates. However, the opposite situation takes place for the particle motion. While the\nmetrics (71) and (72) characterize the gravitational field of the s ame source, they belong\nto different kinds of the spatial inhomogeneity. As a result, the equ ations of the particle\nmotion in the Cartesian and isotropic coordinates significantly differ.\nLet us use the conventional equations of the particle motion\nduµ\nds−1\n2gνλ,µuνuλ= 0 (74)\nand\nduµ\nds+{µ\nνλ}uνuλ= 0, (75)\nwhere\n{µ\nνλ}=1\n2gµρ(gρν,λ+gρλ,ν−gνλ,ρ) (76)\n22are the Christoffel symbols. The equations of the particle motion in t he Cartesian coordi-\nnates take the form\ndui\nds=(u0)2rg\n2r3/braceleftbigg\nxi/bracketleftbigg\n1+3(β·r)2\nr2/bracketrightbigg\n−2βi(β·r)/bracerightbigg\n=−(u0)2\nc2/braceleftbigg\ngi/bracketleftbigg\n1+3(β·r)2\nr2/bracketrightbigg\n−2βi(β·g)/bracerightbigg\n,du0\nds= 0,(77)\ndui\nds=−(u0)2rg\n2r3xi/bracketleftbigg\n1+2β2−3(β·r)2\nr2/bracketrightbigg\n=(u0)2\nc2gi/bracketleftbigg\n1+2β2−3(β·r)2\nr2/bracketrightbigg\n,\ndu0\nds=−(u0)2rg(β·r)\nr3= 2(u0)2\nc2(β·g).(78)\nWe do not make a difference between the upper and lower indices for t he Newtonian accel-\nerationg.\nThe corresponding equations in the isotropic coordinates read\ndui\nds=(u0)2rg\n2r3xi/parenleftbig\n1+β2/parenrightbig\n=−(u0)2\nc2gi/parenleftbig\n1+β2/parenrightbig\n,du0\nds= 0, (79)\ndui\nds=−(u0)2rg\n2r3/bracketleftbig\nxi(1+β2)−2βi(β·r)/bracketrightbig\n=(u0)2\nc2/bracketleftbig\ngi(1+β2)−2βi(β·g)/bracketrightbig\n,\ndu0\nds=−(u0)2rg(β·r)\nr3= 2(u0)2\nc2(β·g).(80)\nThese equationscanbecomparedwiththerelatedequationsforth euniformlyaccelerated\nframe,\ndui\nds=(u0)2ai\nc2,du0\nds= 0, (81)\ndui\nds=−(u0)2ai\nc2,du0\nds=−2(u0)2\nc2(β·a). (82)\nThe comparison of Eqs. (77) – (82) shows that the terms different for the Schwarzschild\nfield and the uniformly accelerated frame ( g=−a) are of the same order of magnitude as\nthe corresponding terms different for the Schwarzschild field in the Cartesian and isotropic\ncoordinates. We can conclude that the spatial inhomogeneity signifi cantly influences the\nform of the equations of motion. Therefore, the results present ed do not give a reason for\nthe assertion about a violation of Einstein’s equivalence principle.\nWe should mention that the forces conditioned by the mismatched te rms in Eqs. (77) –\n(82) are proportional to firstderivatives of the spatial components of the metric tensor.\n23VII. MATHISSON FORCE\nOneof Mathissons great achievements was thediscovery of anadd itional forceacting ona\nspinning particle in a curved spacetime. The Mathisson force is similar t o the Stern-Gerlach\none in electrodynamics. The gravitoelectromagnetism allows us to ex plain the main differ-\nence between thetwo forcesasa result ofa specific dependence o f thegravitoelectromagnetic\nfields on the particle four-velocity. An analysis of the Mathisson for ce can be fulfilled in the\ngeneral case (see, e.g., Refs. [18–20, 23, 44]). Nevertheless, a d erivation of a simple ex-\npression in the weak-field approximation seems to be rather importa nt. In particular, this\nexpression shows that a similarity between the gravity and electrom agnetism exists even for\neffects depending on curvature.\nTo take into account the influence of the spin on the particle motion, we may use the\nHamiltonian method and may add the Hamiltonian of a spinless particle in a gravitational\nfield [58] by the term ζ·Ω,\nH=H0+ζ·Ω. (83)\nThe possibility of this addition has been mentioned in Ref. [13]. In Refs. [22, 23], this\naddition has been rigorously substantiated not only in framework of classical gravity but\nalso for Dirac particles.\nIt follows from Eqs. (32) and (83) that the additional force acting on a spinning particle\nis given by\nfM=−∇(H−H 0) =∇/parenleftbigg1\nu0ζ·/bracketleftbigg\nB−/hatwideu×E\nu/hatwide0+1/bracketrightbigg/parenrightbigg\n. (84)\nIn electrodynamics, the Stern-Gerlach force acting on a Dirac particle (g= 2) has a similar\nform,\nfSG=e\nmc∇/parenleftbigg1\nu0ζ·/bracketleftbigg\nB−u×E\nu0+1/bracketrightbigg/parenrightbigg\n. (85)\nEquation (85) is obtained in the classical limit. The most important diffe rence between\nEqs. (84) and (85) consists of the dependence of the gravitoelec tromagnetic fields on the\nfour-velocity.\nSince the Stern-Gerlach and Mathisson forces are proportional t o gradients of scalars,\nthey define the additions only to the electric and gravitoelectric for ces, respectively. This\nproperty has been proved in Ref. [23] for arbitrarily strong gravit ational fields. As a result,\n∇×fSG= 0,∇×fM= 0.\n24The Mathisson force violates the weak equivalence principle [18–20] b ecause particles\nwithdifferent spindirectionsmoveondifferenttrajectories. Sincet hegravitoelectromagnetic\nfields are proportional to first derivatives of the metric, the Math isson force is proportional\nto second derivatives of the metric, i.e., to the curvature (see Ref . [23]).\nThe resulting force acting on a spinning particle in a LLF is given by\nF=mcu/hatwide0\nu0/parenleftbigg\nE+/hatwideu×B\nu/hatwide0/parenrightbigg\n+fM. (86)\nIn Ref. [23], the Mathisson force has been obtained in an explicit form . Evidently, Eq.\n(86) presents the next-order approximation as compared with th e PK equations.\nVIII. THOMAS PRECESSION IN GENERAL RELATIVITY\nAmazingly, the use of the local Lorentz transformations and the P K fields allow us to\nderive the formula for the Thomas precession in inertial and gravita tional fields. For this\npurpose, we apply the method developed in electrodynamics and pre sented in Refs. [25, 29].\nEquation (25) is in fact the definition of the Thomas effect. We calcula te spin dynamics and\nseparate contributions from the local Lorentz transformations and from the Thomas effect\nto the angular velocity of the spin precession.\nTodeterminethecontributionfromthelocalLorentztransforma tions, weneedtocompare\nthe spin motion in the two LLFs connected with the chosen tetrad an d with the observer\ninstantaneously accompanying the test particle. At a given moment of time, the velocity\nof the test particle in zero. In the general case, the test particle can be accelerated in this\nframe. It is more convenient to present Eq. (30) in the form\ndζ\nd/hatwidet=/tildewideΩ×ζ,/tildewideΩ=1\nu/hatwide0/parenleftbigg\n−B+/hatwideu×E\nu/hatwide0+1/parenrightbigg\n. (87)\nIn the instantaneously accompanying frame, the angular velocity o f the spin motion is\ndefined only by the gravitomagnetic field,\n/tildewideΩ(0)=−B(0). (88)\nTheconnectionbetween theangularvelocities ofthespinmotioninth eLLFwhichrelates\nto the chosen tetrad and in the instantaneously accompanying fra me is defined by the time\n25dilation. As follows from Eqs. (24), (31), (60), and (88), this conn ection is given by\n/tildewideΩL=/tildewideΩ(0)\nu/hatwide0=−B(0)\nu/hatwide0=−B+1\nu/hatwide0(u/hatwide0+1)/hatwideu(/hatwideu·B)+/hatwideu×E\nu/hatwide0,\nΩL=u/hatwide0\nu0/bracketleftbigg\n−B+1\nu/hatwide0(u/hatwide0+1)/hatwideu(/hatwideu·B)+/hatwideu×E\nu/hatwide0/bracketrightbigg\n.(89)\nEquation (89) would be sufficient for a description of the spin preces sion if the three-\ncomponent spin were defined in the instantaneously accompanying f rame. However, it is\ndefined in the particle rest frame. As a result, we need to take into a ccount the Thomas\nprecession. For this purpose, we follow Refs. [25, 29]. It is convenie nt to denote\nFab=cΓabcuc= (E,B).\nWith the use of Eq. (19), Eq. (22) can be presented in the form\ndaa\ndτ=Fabab−uaFbcubac−uadub\ndτab. (90)\nThe next derivations can be made similarly to Refs. [25, 29]. We use th e denotation\nΦa=Fabab−uaFbcubac. (91)\nEvidently, Φa= (Φ/hatwide0,/hatwideΦ) is a four-vector relative to the local Lorentz transformations.\nSinceuaΦa=u/hatwide0Φ/hatwide0−/hatwideu·/hatwideΦ= 0, it satisfies the relation Φ/hatwide0= (/hatwideu·/hatwideΦ)/u/hatwide0=β·/hatwideΦ, whereβ\nis defined by Eq. (24). The similar relation for the four-spin is given by Eq. (23). We can\nperform the following transformation:\nabdub\ndτ=a/hatwide0du/hatwide0\ndτ−/hatwidea·βdu/hatwide0\ndτ−u/hatwide0/hatwidea·dβ\ndτ=−u/hatwide0/hatwidea·dβ\ndτ, uadub\ndτab=−uau/hatwide0/hatwidea·dβ\ndτ.(92)\nThus, Eq. (90) leads to\nda/hatwide0\ndτ= Φ/hatwide0+/parenleftBig\nu/hatwide0/parenrightBig2\n/hatwidea·dβ\ndτ,d/hatwidea\ndτ=/hatwideΦ+/parenleftBig\nu/hatwide0/parenrightBig2\nβ/parenleftbigg\n/hatwidea·dβ\ndτ/parenrightbigg\n. (93)\nNow we can calculate the equation of motion for the rest frame spin ζwith the use of\nthe relations\nζ=/hatwidea−u/hatwide0\nu/hatwide0+1β(β·/hatwidea),d\ndτ/parenleftBigg\nu/hatwide0\nu/hatwide0+1β/parenrightBigg\n=u/hatwide0\nu/hatwide0+1dβ\ndτ+/parenleftBig\nu/hatwide0/parenrightBig3\n(u/hatwide0+1)2β/parenleftbigg\nβ·dβ\ndτ/parenrightbigg\n.\n26The needed equation has the form (cf. Refs. [25, 29])\ndζ\ndτ=/hatwideΦ−u/hatwide0β\nu/hatwide0+1Φ/hatwide0+/parenleftBig\nu/hatwide0/parenrightBig2\nu/hatwide0+1ζ×/parenleftbigg\nβ×dβ\ndτ/parenrightbigg\n. (94)\nThe transformation of the given four-vector Φato the instantaneously accompanying\nframe results in/parenleftbig\nΦ(0)/parenrightbiga=/parenleftbig\n0,/hatwideΦ(0)/parenrightbig\n, where\n/hatwideΦ(0)=/hatwideΦ−u/hatwide0\nu/hatwide0+1β(β·/hatwideΦ) =/hatwideΦ−u/hatwide0β\nu/hatwide0+1/hatwideΦ0.\nAs follows from Eq. (31), the derivation of /hatwideΦ(0)from Eq. (91) brings the equation of the\nspin motion to the form\ndζ\nd/hatwidet=−B(0)\nu/hatwide0×ζ+u/hatwide0\nu/hatwide0+1ζ×/parenleftbigg\nβ×dβ\ndτ/parenrightbigg\n. (95)\nThe angular velocity of the spin precession is given by\n/tildewideΩ=−B(0)\nu/hatwide0−u/hatwide0\nu/hatwide0+1/parenleftbigg\nβ×dβ\ndτ/parenrightbigg\n,Ω=u/hatwide0\nu0/tildewideΩ. (96)\nSince/tildewideΩ=/tildewideΩL+/tildewideΩT, the angular velocity of the Thomas precession is equal to\n/tildewideΩT=−u/hatwide0\nu/hatwide0+1/parenleftbigg\nβ×dβ\ndτ/parenrightbigg\n=−1\nu/hatwide0(u/hatwide0+1)/parenleftbigg\n/hatwideu×d/hatwideu\ndτ/parenrightbigg\n. (97)\nExplicitly,\n/tildewideΩT=−/hatwideu×E(0)\nu/hatwide0(u/hatwide0+1)=−1\nu/hatwide0+1/bracketleftbigg\n/hatwideu×E+/hatwideu×(/hatwideu×B)\nu/hatwide0/bracketrightbigg\n=u/hatwide0−1\nu/hatwide0B−1\nu/hatwide0(u/hatwide0+1)/hatwideu(/hatwideu·B)−/hatwideu×E\nu/hatwide0+1.(98)\nEquation (87) can be presented in terms of the rest frame fields:\n/tildewideΩ=−B(0)\nu/hatwide0−/hatwideu×E(0)\nu/hatwide0(u/hatwide0+1). (99)\nEquations (87), (89), (98), and (99) are consistent.\nThat, the use of the PK fields has allowed us to derive Eq. (97) for th e Thomas effect in\nRiemannian spacetimes. We have rigorously proven that this equatio n has practically the\nsameformasthecorresponding equation(26)defining theThomas effect inelectrodynamics.\nHowever, the equation for the Thomas effect in the world frame (wit h a substitution of the\nvelocity and the acceleration in the world frame for the correspond ing LLF quantities) may\nbe different.\n27Let us calculate the contributions from the local Lorentz transfo rmations and from the\nThomas effect to the angular velocity of the spin precession for the uniformly accelerated\nframe and for the Schwarzschild field in the isotropic coordinates. W e can use the weak-field\napproximation. For the uniformly accelerated frame, these contr ibutions are given by\n/tildewideΩ(a)\nL=−c/hatwideu×a(t)\nc2+a(t)·r,/tildewideΩ(a)\nT=u/hatwide0\nu/hatwide0+1·c/hatwideu×a(t)\nc2+a(t)·r. (100)\nIn the weak-field approximation, this equation takes the form\n/tildewideΩ(a)\nL=−/hatwideu×a(t)\nc,/tildewideΩ(a)\nT=u/hatwide0\nc(u/hatwide0+1)/hatwideu×a(t). (101)\nThe corresponding relations for the Schwarzschild field in the isotro pic coordinates are\ngiven by\n/tildewideΩ(i)\nL=2/hatwideu×g\nc,/tildewideΩ(i)\nT=−2/parenleftBig\nu/hatwide0/parenrightBig2\n−1\ncu/hatwide0(u/hatwide0+1)/hatwideu×g.(102)\nWhena=const=−g, Eqs. (101) and (102) significantly differ. We have discussed the\norigin of this difference in Sec. VI. We should underline that neither th e contributions\nfrom the local Lorentz transformations nor those from the Thom as effect vanish in the\nnonrelativistic limit. It has been claimed in Ref. [50] that the spin rotat ion in the uniformly\naccelerated frame is caused only by the Thomas effect. The fallacy o f this claim has been\nshown in Ref. [51].\nWe can also specify the two contributions to the angular velocity of t he spin precession\nin the rotating frame. In this case, the both contributions are also nonzero and are given by\n/tildewideΩL=−ωu/hatwide0+/hatwideu(/hatwideu·ω)\nu/hatwide0+1,/tildewideΩT=−/hatwideu×(/hatwideu×ω)\nu/hatwide0+1=ω(u/hatwide0−1)−/hatwideu(/hatwideu·ω)\nu/hatwide0+1.(103)\nThe quantity\nO=−1\nu0(u0+1)/parenleftbigg\nu×du\ndτ/parenrightbigg\n(104)\nfrequently used for a specification of the Thomas precession in cur ved spacetimes can be\nobtained with Eq. (75). In the weak-field approximation ( /hatwideu≈u, u/hatwide0=u0), the quantity O\nis twice as much as /tildewideΩT,\n/tildewideΩT=−u×(u×ω)\nu0+1, (105)\nO=−2u×(u×ω)\nu0+1. (106)\n28This example unambiguously shows that the use of the quantity (104 ) for the determi-\nnation of the Thomas precession in curved spacetimes is incorrect. We should nevertheless\nmention that Eqs. (97), (98), (103), and (105) describe the pre cession of the spin pseudovec-\ntor defined in the LLF. These equations cannot be directly applied to the Thomas precession\nof a segment of a rapidly rotating disk which has been considered in Re f. [59].\nIX. DISCUSSION AND SUMMARY\nThe introduction of the gravitoelectromagnetic fields being effectiv e fields in an anholo-\nnomic tetrad frame (coframe) significantly simplifies a description of motion of spinning\nparticles in GR. When one neglects the spin-curvature coupling and t he mutual influence of\nparticle and spin motion, dynamics of the four-velocity and spin is defi ned by Eqs. (32) and\n(33) similar to corresponding equations in electrodynamics. Howeve r, the equations of mo-\ntion for the four-velocity and spin are not equally useful. The conve ntional three-component\nspin is defined in the particle rest frame which is one of LLFs. In contr ast, the four-velocity\nis the world vector and its evolution should be defined in the world fram e. Certainly, the\ntransition to the world coordinates is not difficult because\nuµ=eµ\naua,duµ\ndτ=eµ\nadua\ndτ+uadeµ\na\ndτ.\nMoreover, theinvestigationoftheparticlemotionissimplified whenth etrajectoryisinfinite.\nIn this case, one can apply the fact that the quantities uµanduacoincide at the initial and\nfinal parts of the particle trajectory because of the very large d istance to the field source\n[22]. However, the use of Eq. (75) seems to be more straightforwa rd.\nFor a derivation of the equations of motion, canonical methods bas ed on the use of\nHamiltonians and Lagrangians can also be successfully applied.\nBasic tetrads satisfy the Schwinger gauge while other tetrads are also applicable. Tetrads\nwhich do not satisfy this gauge arecarried by observers moving ina d escribed spacetime. All\ntetrads are equivalent and the use of any tetrad is possible. Differe nt tetrads are connected\nby local Lorentz transformations. This connection is determined in Sec. II. In the general\ncase, the gravitoelectromagnetic fields differ in different coframes . In accordance with Refs.\n[8, 9], these fields do not coincide even for different tetrads belongin g to the Schwinger gauge\n(see the example given in Sec. IVB).\n29In the present work, we explain and develop the conception of grav itoelectromagnetism\nfirst proposed by Pomeransky and Khriplovich [13]. Wecalculate the g ravitoelectromagnetic\nfields in the most important special cases (Sec. IV) and determine t heir local Lorentz\ntransformations (Sec. V). The validity of these transformations is demonstrated for the case\nof the uniformly accelerated frame.\nWe apply the gravitoelectromagnetic fields for a comparison of inert ia and gravity and of\nthe Mathisson and Stern-Gerlach forces. In agreement with Refs . [40, 50–52], the uniformly\naccelerated frame cannot completely imitate the gravitational field of the Schwarzschild\nsource. The forces conditioned by the mismatched terms in the equ ations of motion for the\nSchwarzschild field in the Cartesian and isotropic coordinates and fo r the uniformly accel-\nerated frame are proportional to firstderivatives of the spatial components of the metric\ntensor while the tidal and Mathisson forces are defined by secondderivatives of the metric\ntensor. However, we cannot support the claim made in Refs. [50, 52 ] that these properties\nviolate Einstein’s equivalence principle. This principle has been formulat ed only relative to\nconstant uniform gravitational fields. The Schwarzschild field (as well as other real g ravi-\ntational fields) is nonuniform . We have shown that the spatial inhomogeneity significantly\ninfluences the form of the equations of motion. The terms different for the Schwarzschild\nfield and the uniformly accelerated frame ( g=−a) are of the same order of magnitude as\nthe corresponding terms different for the Schwarzschild field in the Cartesian and isotropic\ncoordinates. Therefore, the difference between the equations o f motion in the Schwarzschild\nfield and the uniformly accelerated frame does not violate Einstein’s e quivalence principle.\nAn expression of the Mathisson force in terms of the gravitoelectr omagnetic fields allows\nus to state that the deep similarity between the gravity and electro magnetism exists even\nfor effects depending on curvature. It is well known that the Math isson force violates the\nweak equivalence principle.\nProbably the most exciting result of the use of the local Lorentz tr ansformations and\nthe gravitoelectromagnetic fields is the general description of the Thomas precession in GR\ncarried out in Sec. VIII. Amazingly, Eq. (97) defining the angular ve locity of the Thomas\nprecession in LLFs is analogous to the corresponding formula [24–29 ] of special relativity.\nEquations (98) and (99) show the convenience of the gravitoelect romagnetic fields for a\ndescription of spin effects in GR and detach the Thomas effect.\nWe underline a great importance of the Thomas effect for a better u nderstanding of spin\n30dynamics in inertial and gravitational fields. Experimental investiga tions of this dynamics\nin turn arevery important to determine fundamental properties o f gravity. In particular, the\nGravity Probe B experiment [60] has confirmed the theoretical pre diction [54] (see also Sec.\nIVD) for the spin precession due to the geodetic and LT effects. Th e LT effect for orbiting\nbodies (frame dragging) has been certified in experiments with the L AGEOS satellites [61].\nExperiments with atomic and nuclear spins in Earth’s rotating frame [6 2, 63] have verified\nthe behavioral equivalence of quantum mechanical spins and classic al gyroscopes [30, 41].\nExperimental constraints for equivalence principles and new intera ctions have been analyzed\nin Ref. [64].\nWe can conclude that the conception of gravitoelectromagnetism u sed here perfectly\ndescribes the evolution of the spin of a relativistic particle in general noninertial frames and\narbitrarily strong gravitational fields. The main distinctive feature s of this conception are\ncomparatively simple equations of motion and the clear analogy betwe en electromagnetism\nand gravity. 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We f ound that the both methods are\ngood for the Beliaev damping at zero temperature and Landau d amping at very low temperature,\nhowever, at high temperature, the hydrodynamical approach overestimates the Landau damping\nand the HFB gives a better approximation. This result shows t hat the comparison of the theoretical\ncalculation using the hydrodynamical approach and the expe rimental data for high temperature\ndone by Vincent Liu (PRL 214056 (1997)) is not proper. For two-dimensional systems, we show\nthat the Beliaev damping rate is proportional to k3and the Landau damping rate is proportional\ntoT2for low temperature and to Tfor high temperature. We also show that in two dimensions\nthe hydrodynamical approach gives the same result for zero t emperature and for low temperature\nas HFB, but overestimates the Landau damping for high temper ature.\nI. INTRODUCTION\nThe experimental realization of Bose-Einstein conden-\nsation(BEC)inmagneticallytrappedakaliatoms[1, 2, 3]\nprovides a good tool to study the properties of 3D dilute\nBose gases. Furthermore, with the anisotropic of new\ntraps [4, 5], one can further confine condensate atoms\nin a quasi-two-dimensional regime [5, 6]. Most theoreti-\ncal work (for 3D and 2D systems, see Review [7, 8], re-\nspectively) has focused on the dynamics of condensates\nand the zero-temperature behavior, which can be ob-\ntained by solving the non-linear Gross-Pitaevskii (GP)\nequation. However, the finite-temperature behavior has\nstill remained difficult to study, where experiments have\nshown damping of the condensates modes in the presence\nof a significant noncondensate component [9, 10]. Damp-\ning mechanism associated with collective excitations of\nBose condensed atoms interacting with a non-condensed,\nthermal component is not well understood and still rep-\nresents a challenging problem in theoretical physics. The\ndamping of collective modes can have various origins.\nThere are two distinct contributions to the total decay\nrateγ=γB+γL: One arises at T= 0 from the pro-\ncess of decay of a quantum of excitation into two or more\nexcitations with lower energy. This mechanism was ��rst\nstudied by Baliaev [11]in 3D uniform Bose superfluids\nand is known as Balieav damping γB. At finite tempera-\nture, a different mechanism of damping (known as Lan-\ndaudamping, γL)comesfromtheprocessofonequantum\nof excitation decays due to coupling with transitions as-\nsociated with other elementary excitations and occurring\natthe samefrequency. Landaudampingisnot associated\nwith thermalization process and can be well described in\nthe framework of mean field theory [12, 13, 14]. The\nsubject of Landau damping in dilute BECs has been ex-\nplored by several authors. Landau damping in an uni-\nform Bose gas at low temteratures was first investigated\nby Popov [15] Hohenberg and Martin [16], while at high\ntemperatures was first investigated by Sz‘´ epfalusy and\nKondor [17]. The relevance of Landau damping to ex-plain experimental data of trapped Bose gas was pro-\nposed by Liu and Schieve [18] and developed by Liu [19]\nusing the Popov’shydrodynamicalapproach[15]. On the\nothehand, Pitaevskiiand Stringari[12] investigatedLan-\ndau damping in a weakly interacting uniform as well as\nnon-uniform Bose gas by means of semi-classical theory.\nThey showedthat forthe uniform Bosegas, it reproduces\nknown results for both the low temperature asymptotic\nbehaviour of the phonon coupling. However, for the high\ntemperature, Liu showed higher Landau damping rate\nthan those obtained by Sz´ epfalusy and Kondor, while\nthe hign-temperature behavior could be reproduced by\nPitaevskii and Stringari.\nHowever all investigations of the damping rate has\nbeen done for a 3D Bose gas. 2D Bose gases are in-\nteresting as their low temperature physics is governed\nby strong long-range fluctuations. These fluctuations in-\nhibit the formation of true long-range order, which is a\nkey concept of phase transition theory in 3D. Thus a\n2D uniform interacting Bose gas does not undergo Bose-\nEinstein condensation at finite temperatures. However,\nthis system turns superfluid below the BKT (Berezinski,\nKosterlitz and Thouless) temperature TKT[20, 21]. The\nexperiment indication of the BKT transition in weakly\ninteracting Bose system has even been shown in Ref. [6].\nDamping in a 2D Bose gas is an open question which was\nrecently addressed by the authors for a uniform Bose gas\n[23] using the hydrodynamical theory of Popov [15]. In\nthis work, we show that the hydrodynamical approach\nactually overestimates the damping rate at high tem-\nperatures, both for 3D and 2D systems and we calcu-\nlate the Balieav and Landau damping rates for a 2D\nuniform Bose gas using the semi-classical Hartree-Fock-\nBogoliubov (HFB) approach. In the limit of low temper-\natures, the results of this approach is in good agreement\nwith that found from hydrodynamical approach. Con-\ntrary to earlier work [19], we show that for the 3D case\nin the high temperature limit, the hydrodynamical ap-\nproach cannot be used to explain the experimental data.\nThis paper is organized as follows. In Sec. II we dis-2\ncuss the relation between the atom-atom interaction and\nthe scattering length for 2D and 3D dilute gases. In\nSec. III we first introduce the hydrodynamical approach\ndeveloped by Popov[15], and then calculate the Beliaev\ndamping and Landau damping for 3D and 2D gases. The\nmistake using this approach for high temperature is also\ndiscussed. In Sec. IV the HFB approximation is devel-\noped to calculate 3D and 2D Beliaev and Landau damp-\ning.\nII. ATOM-ATOM INTERACTION AND\nSCATTERING LENGTH\nThe standard Hamiltonian of an interacting Bose gas\nis\nH=/integraldisplay\nddr1\n2∇ψ†(r)∇ψ†(r)+Vext(r)ψ†(r)ψ(r)\n+1\n2/integraldisplay\nddrddr′ψ†(r)ψ†(r′)U(r−r′)ψ(r)ψ(r′),(1)\nwhereUis the atom-atom interaction and Vextis the\nexternal potential. For uniform Bose gas, Vext= 0. The\ntrueinteractionbetweenatomsisverycomplicatedwhere\none has to consider the fine structure of atoms. However,\nthe scattering process can offer a effective potential to\nsimplify the ineraction. In order to do that, one has to\nintroduce Green functions for bosonic systems with con-\ndensate. The difficulty of doing so arises from the fact\nthat the terms containing the odd number of annihila-\ntion operators do not vanish for a Bose gas after averag-\ning the ground state due to the existence of condensate,\nwhich unfortunately destroys the hope to apply the nor-\nmal technique of Feynman diagrams to the system. This\ndifficulty was successfully resolved by Beliaev [11, 24].\nHe separated the operators with zero momentum, which\nsemi-classically can be regarded as a c-number, and the\nother operators with nonzero momenta. In this way, the\nFeynman diagrams can be used for the Bose gas.\nBeliaev considered a three-dimensional system with\nshort-range, central interaction potential with radius\n1/a3andthen calculatedthe renormalizedatom-atomin-\nteraction in the presence of the condensate between two\nparticles with non-zero momenta, which one should sum\nover all ladder diagrams. In this way, one can obtain\nthe renormalized interaction in terms of the s-wave scat-\ntering amplitude according to the elementary scattering\ntheory [7, 25, 26]. Therefore the effective potential can\nbe written as\nU(r) =4πa3\nmδ(r), (2)\nwith the atom mass m, and the momentum dependence\nof the scattering amplitude can be ignored in the low\ntemperature limit. In the rest of the paper we define the\natom-atom interaction strength g3as\ng3=4πa3\nm. (3)For two dimensions, Schick followed the methods de-\nveloped by Beliaev and examined a two-dimensional sys-\ntems of hard-core bosons with a diameter a2at low den-\nsity and zero temperature. Unlike the three-dimensional\nsystems, where ladder diagrams are independent of the\ndimensionless parameter na3\n3, hence, it is natural to take\nit as the small perturbation terms to expand the quan-\ntities. For two-dimensional systems, contributions from\nthe ladder diagrams depends logarithmically on na2\n2, the\ndimensionless parameter for 2D systems, but not directly\nonna2\n2itself. In particular, the renormalized interaction\nis proportional to 1 /ln1/na2\n2:\ng2=4π\nmln(1/na2\n2). (4)\nSchick concluded that the 1 /ln1/na2\n2plays a role of a\nsmall parameter in the two-dimensional dilute systems,\nand other quantities, like damping rate in this paper, can\nbe expanded in terms of it.\nIII. HYDRODYNAMICAL APPROACH\nIn the low temperature and low energy limit, Popov\n[15] developed a hydrodynamical approach to find an\neffective Hamiltonian for a nonideal Bose gas. In or-\nder to do that, one has to separate the order parameter\noverrapidlyandslowlyoscillatingfield, andthehydrody-\nnamical Hamiltonian can be obtained by integrating the\nfunctional over rapidly oscillating field. The theory de-\nscribes then the hydrodynamicalHamiltonian in terms of\ntwo slowly varying fields : phase φ(x) and density fluc-\ntuationπ(x) =n(x)−n0withn0: the density of the\nground state. Here the four-dimension Euclidean space\nx= (x,τ) is used with the imaginary time τ. The hydro-\ndynamical action for a d−dimensional nonideal Bose gas,\naccording to Popov, can be written in the form (notice\nthat/planckover2pi1= 1 throughout the paper)\nS[φ,π] =/integraldisplayβ\n0dτ/integraldisplay\nddx/braceleftbigg\ni∂2p\n∂µ∂nπ∂τφ−1\n2m∂p\n∂µ(∇φ)2\n−1\n2∂2p\n∂µ2(∂τφ)2+1\n2∂2p\n∂n2π2−(∇π)2\n8mn0−π(∇φ)2\n2m/bracerightbigg\n(5)\nwith the atom mass m, the pressure of a homogeneous\nsystemp, the chemical potential µand the atom density\nn. The fields φandπare periodic in imaginary time\nτwith the period β= 1/(kBT). For very low tempera-\nture, as long as the non-condensate part can be neglected\ncompared to the condensate, the pressure p(µ,n) at zero\ntemperature can be a good approximation. Therefore we\ncan use the expression of a weakly interacting dilute gas\nasp=µn−gd\n2n2\n0,wheregdis the atom-atom interaction\nrelated to the scattering length. It follows that\n∂2p\n∂µ∂n= 1;∂p\n∂µ=n≃n0;∂2p\n∂µ2= 0;∂2p\n∂n2=−gd,\n(6)3\nFIG. 1: Green’s functions and vertex\nand the action (5) takes the form\n/integraldisplay\ndτddx/parenleftbigg\niπ∂τφ−(n0+π)\n2m(∇φ)2−gd\n2π2−(∇π)2\n8mn0/parenrightbigg\n.\n(7)\nThe action contains all quadratic functions except the\nterm 1/2mπ(∇φ)2, considered as an interacting poten-\ntial. The Hamiltonian can be derived from the effective\naction (7) as\n/integraldisplay\nddx/parenleftbiggm\n2nv2+gd\n2(n−n0)2+(∇n)2\n8mg0/parenrightbigg\n,(8)\nwhere the field of velocities is defined as v= 1/m∇φ.\nThis Hamiltonian is consistent with one particular real-\nization of the Landau hydrodynamical Hamiltonian.\nFourier transforming the fields φandπ, the effective\naction (8) can be written in the form\n−1\n2/summationdisplay\nν/integraldisplay\nddkn0\nmk2φ(k)φ(−k)+2ωνφ(k)π(−k)\n+(gd+k2\n4mn0)π(k)π(−k)\n−1√βV/summationdisplay\nk1+k2+k3=0k1·k2\n2mφ(k1)φ(k2)π(k3),(9)\nwherekisthe vector( k,iων)and theMatsubarafrequen-\nciesων= 2πν/βwith integers ν. From the action of the\nFourier transformation(9) one can extract the important\ninformation needed for the perturbation calculations us-\ning diagrammatictechnique. First of all, the free Green’s\nfunctions is defined as follows\nG0(k) =/parenleftbigg\n/angb∇acketle{tφ(k)φ(−k)/angb∇acket∇ight0/angb∇acketle{tφ(k)π(−k)/angb∇acket∇ight0\n/angb∇acketle{tπ(k)φ(−k)/angb∇acket∇ight0/angb∇acketle{tπ(k)π(−k)/angb∇acket∇ight0,/parenrightbigg\n(10)\nwhere/angb∇acketle{t···/angb∇acket∇ight0denotes the expectation value of fields cal-\nculated only with the quadratic action. From the action\n(9), the inverse of the free Green’s function can be found\nas\nG−1\n0(k) =/parenleftbiggn0\nmk2ων\n−ωνgd+k2\n4mn0/parenrightbigg\n.(11)\nTherefore,\nG0(k) =/parenleftBigggd+k2/(4mn0)\nω2ν+ǫ2(k)−ων\nω2ν+ǫ2(k)\nων\nω2ν+ǫ2(k)(n0/m)k2\nω2ν+ǫ2(k)/parenrightBigg\n,(12)\nFIG. 2: One-loop diagrams for the self energy.\nwhere\nǫ(k) =/radicalbigg\n(k2\n2m)2+c2k2 (13)\nwithc≡/radicalbig\ngdn0/m. We represent the relation be-\ntween the free Green’s functions and the Feyman di-\nagrams in Fig.1. The cubic term of the action\n(9) is known as phonon-phonon interaction in the\nlow temperature region, giving rise to a vertex of\nδd(k1+k2+k3)δν1+ν2+ν3,0(k1·k2)\nmrepresented by the\nlast diagram of Fig. 1.\nThe exact Green’s function has to involve the phonon-\nphonon interaction, given by the Dyson equation G(k) =\nG0(k)+G0(k)Σ(k)G(k), where Σ( k) represents the self-\nenergy matrix. The low-frequency spectrum of collective\nmodes can be obtained by the poles of the exact Green’s\nfunction as\ndetG−1(k) = det[G−1\n0(k))−Σ(k)] = 0,(14)\nthrough the analytical continuation iων=ω+iη(η=\n0+) after the Matsubara frequency sum. The complex\nfrequencyω=E−iγ(k) represents the energy spectrum\nEand the damping rate γ. Neglecting the matrix Σ,\nthe zero order approximation for the Eqs. (14) gives the\nsquare of the Bogoliubov energy spectrum:\nE2= (k2\n2m)2+c2k2=ǫ(k)2(15)\nWhen the phonon-phonon interaction is considered, the\nimaginary part appears in the spectrum. Fig. 2 shows\nthe one-loop diagrams for the self energy Σ. The contri-\nbution to the imaginary part of the spectrum is given by\nthe last five Figs. 2(b) - 2(f). The damping rate can be\nobtained from Eqs. (11) and (14) [19] as\nγ(k) =1\n2E/bracketleftbigg\n(gd+k2\n4mn0)ImΣφφ(k,ω+iη)\n+n0k2\nmImΣππ(k,ω+iη)/bracketrightbigg\n−ReΣφπ(k,ω+iη).(16)4\nReplacing (15) into Eqs. (16) and calculating the dia-\ngrams, we obtain γ(k) =γB(k)+γL(k),whereγBis the\nBeliaev damping as\nγB(k) =1\n2d+2πd−1/integraldisplay\nddk′δ(ǫ(k)−ǫ(k′)−ǫ(k−k′))\n[f0(ǫ(k′))−f0(−ǫ(k−k′))]/braceleftbigg(k−k′)2(k·k′)2ǫ(k′)ǫ(k)\n2mn0k2k′2ǫ(k−k′)\n+(k·k′)(k·(k−k′))ǫ(k)\n2mn0k2\n+k2(k′·(k−k′))2ǫ(k′)ǫ(k−k′)\n4mn0k′2(k−k′)2ǫ(k)\n+(k′·(k−k′))(k·k′)ǫ(k′)\nmn0k′2/bracerightbigg\n(17)\nandγLis known as Landau damping as\nγL(k) =1\n2d+2πd−1/integraldisplay\nddk′δ(ǫ(k)+ǫ(k′)−ǫ(k+k′))\n[f0(ǫ(k′))−f0(ǫ(k+k′))]/braceleftBigg\nǫ(k)\n2mn0/bracketleftBigg\nk′2(k·(k+k′))2ǫ(k+k′)\nk2(k+k′)2ǫ(k′)\n+(k+k′)2(k·k′)2ǫ(k′)\nk2k′2ǫ(k+k′)/bracketrightbigg\n+(k·k′)(k·(k+k′))ǫ(k)\nmn0k2\n+k2(k′·(k+k′))2ǫ(k′)ǫ(k+k′)\n2mn0k′2(k+k′)2ǫ(k)+(k′·(k+k′))\nmn0\n×/bracketleftbigg(k·k′)ǫ(k′)\nk′2+(k·(k+k′))ǫ(k+k′)\n(k+k′)2/bracketrightbigg/bracerightbigg\n,\n(18)\nwhere the bosonic distribution function f0(ǫ) =\n1/[exp(βǫ)−1].\nA. Quantum regime ck≫kBT\nAtT= 0, the Landau damping disappears and the\nBeliaev damping contributes to the damping rate. In the\nBeliaev damping mechanism the momenta of the three\nexcitations are comparable, |k| ≃ |k′| ≃ |k−k′|. Then\nthe Eqs. (17) yields\nγB(k) =9c\n2d+4πd−1mn0/integraldisplay\nddk′|k||k′||k−k′|\nδ(ǫ(k)−ǫ(k′)−ǫ(k−k′)).(19)\nIn three dimensions the damping rate for small k(k≪\nmc) is\nγd=3\nB(k) =3k5\n640πmn0, (20)\nknown as Beliaev’s result[11].For two-dimensional systems, the Eqs. (19) can be\nwritten as\nγd=2\nB(k) = 2/bracketleftbigg9\n64πmn0/integraldisplay\ndk′|k′|(|k−k′|)2\nsinθ/bracketrightbigg\n,(21)\nwhereθis the angle between kandk′. The factor two\nin front of the bracket comes from the fact that there are\ntwo angles corresponding to the energy conservation for\nthe Beliaev damping ( ǫ(k)−ǫ(k′) =ǫ(k−k′)): sinθ≃\n±√\n3|k−k′|\n2mc[22], and the Beliaev damping rate for a 2-D\nBose gas has the form\nγd=2\nB=√\n3c\n32πn0k3. (22)\nThis result corrects the wrong result previously given by\nChung and Bhattacherjee [23] and the factor two will ap-\npear naturally in the two-dimensional Landau damping.\nB. Thermal Regime ck≫kBT\nFor finite temperature and small momenta such that\ncq≪kBTandcq≪n0gd, the Beliaev damping is much\nsmaller than the Landau damping. In three dimensions,\nthe damping rateto the lowestorderin kcanbe obtained\nfrom Eq. (18) as\nγd=3\nL(k)\nǫ(k)=k5\n0\n16πmn0kBTI3(τ), (23)\nwhere\nI3(τ) =1\n4/integraldisplay∞\n0dzz2sech2(z\n2τ)\n[1\n2+3\n2(z2+1)+2z2\n(z2+1)2−2\n(z2+1)2],(24)\nwithk0=√mn0g3andτ≡kBT/n0g3. This result was\nfirst obtained by V. Liu [19]. For kBT≪mc2, Eq.24 is\nreduced to the Hohenberg and Martin’s result [16]\nγd=3\nL(k) =3π3k(kBT)4\n40mn0c4. (25)\nThis low temperature limit gives the same result as that\nusing the HFB approach, which will be introduced in\nthe next section. For high temperature kBT≫mc2,\nI(τ)∼38.735τ, and the damping rate is approximated\nby\nγd=3\nL(k)\nǫ(k)≃9.648kBTa3\nc(26)\nwith the three-dimensional scattering length a3=\nmg3/4π. Unfortunately this result is different than that\ninvestigated by Sz´ epfalusy and Kondor [17], which reads\nγd=3\nL(k)\nǫ(k)≃3π\n8kBTa3\nc. (27)5\nTherefore the hydrodynamical approach is no longer cor-\nrect for the high temperature. The reason is that in the\nhydrodynamical Hamiltonian only the slow oscillating\nfields are considered by integrating out the fast oscillat-\ning fields. For high temperature the fast oscillating fields\nshould also be considered to reduce the damping rate.\nWe can conclude that the hydrodynamical approach is\nvery good for low temperature, however, for high tem-\nperature, other method should be introduced. We will\ndiscuss that in the next section.\nIn Fig. 3 the three-dimensional Landau damping per\nunit energy using the hydrodynamical approach (dashed\nline) and HFB (solid line) is plotted as a function of τ.\nAlso shown are the asymptotic behavior at high tem-\nperature (the dashed-dot line) and the low temperature\nlimit (the dashed-dot-dot line). We can see that the hy-\ndrodynamical approach gives very good agreement with\nthe low temperature limit for τ≤0.5, however, it goes\ntoo large at high temperature and it does not approach\nthe asymptotic value given by Szepfalusy and Kondor.\nTherefore the conclusion made by Liu in Ref. [19] that\nthe results of the hydrodynamical approach can fit the\nexperimental data is not proper.\nIn two dimensions, the damping rate reads\nγd=2\nL(k)\nǫ(k)=√\n2k4\n0\n16mn0kBTI2(τ), (28)\nwhere\nI2(τ) =1\n4/integraldisplay∞\n0dzz2sech2(z\n2τ)/radicalBig√\nz2+1−1−z2\n2(z2−1)\n[1\n2+3\n2(z2+1)+2z2\n(z2+1)2−2\n(z2+1)2].(29)\nIn the low temperature limit: kBT≪mc2,I2(τ)→√\n6π2τ2, therefore the damping coefficient is given by\nγd=2\nL\nǫ(k)=√\n3π\n8(kBT)2\nn0c2. (30)\nIn this low temperature regime, the damping rate is pro-\nportional to T2. As far as we know, this quadratic de-\npendence of the temperature for the damping rate in the\nlow temperature is found for the first time in this paper.\nFor the high temperature, as the three-dimensional\ncase, the hydrodynamic Hamiltonian overestimates the\ndamping. In the next section, we will use the\nHartree-Fock-Bogoliubov approximation to obtain the\ntwo-dimensional damping at high temperature.\nFig. (4) shows the two-dimensional Landau damp-\ning rate per unit energy using both hydrodynamical\napproach (dashed line) and HFB method (solid line).\nIn this figure the low temperature limit (dashed-dot-\ndot line) and the asymptotic value at high temperature\n(dashed-dot line) are also shown. The hydrodynamical\nresult is in agreement with the low temperature limit\nforτ <0.2, however, it will not approach the asymp-\ntotic value for high temperature similar to the three-\ndimensional case.0 0.5 1 1.5 2 2.5 3\nτ024681012\nγL/ε(k)\nFIG. 3: Landau damping rate per unit energy versus τin\nthree dimensions. The unit of γL/ǫ(k) isp\na3\n3no. Solid\nblack line represents the result obtained by HFB method,\ndashed blue line by the hydrodynamical approach. The high-\ntemperature asymptotic behavior and low-temperature limi t\nare also shown by dashed-dot green line and dashed-dot-dot\nred line, respectively.\nIV. HARTREE-FOCK-BOGOLIUBOV\nAPPROACH\nIn this section we represent a semi-classical method:\nHartree-Folk-Bogolubov (HFB) . We will see that in the\nlow-temperature regime this approach is in a good agree-\nment with the hydrodynamical approach, while for the\nhigh temperature, on the contrary to the hydrodynamic\napproach,HFBgivesabetterapproximationtothedecay\nrate.\nWe start with the method by Giorgini[13]. The grand-\ncanonical Hamiltonian of a system with a nonuniform\nexternal field Vext(r) reads\nK=H−µN=/integraldisplay\nddrψ†(r,t)H0ψ(r,t)\n+gd\n2/integraldisplay\ndrψ†(r,t)ψ†(r,t)ψ(r,t)ψ(r,t),(31)\nwhere\nH0=−∇2\n2m+Vext(r)−µ (32)\nandψ†(r,t) andψ(r,t) are the creation and annihilation\nfield operators. Since the system is in the regime where\nthe condensate exists, we define a time-dependent con-\ndensate wave function Φ( r,t) [16]\nΦ(r,t) =/angb∇acketle{tψ(r,t)/angb∇acket∇ight (33)\nwith the average /angb∇acketle{t···/angb∇acket∇ightusing the grand-canonical Hamil-\ntonian (31). We have to notice that the Eq. (33) can\nalways be used if the condensate exists, however, for a\nhomogeneous two-dimensional system, i.e. Vext(r) = 0,\nthe condensate does not exist at finite temperature. In\nthis case the long-range order disappears, it remains the6\nquasi-long-range order for a two-dimensional homoge-\nneous Bose gas. That means, though a macroscopic oc-\ncupation number of a single state does not exist, there\nexists a small value of kcin the momentum space where\na macroscopic occupation number of the states k < k c\nstill forms a quasi-condensate. Therefore the bracket\nin Eq. (33) should count all the states in the quasi-\ncondensate. We can see that Φ( r,t) allows us to describe\nthe oscillating condensate awayfrom the equilibrium. To\navoid the confusion for the notation, we define here the\nstationary value of the condensate in its equilibrium as\nΦ0(r)\nΦ0(r) =/angb∇acketle{tψ(r)/angb∇acket∇ight0, (34)\nwhere/angb∇acketle{t···/angb∇acket∇ight0denotes the time-independent averageofthe\ncondensate in its equilibrium. The particle field can be\ndecomposedinto acondensateandanoncondensatecom-\nponent\nψ(r,t) = Φ(r,t)+˜ψ(r,t). (35)\nBy the definition of the condensate (33), the nonconden-\nsate component has to satisfied the condition:\n/angb∇acketle{t˜ψ(r,t)/angb∇acket∇ight= 0. (36)\nBy applying the decomposition (34) to the grand-\ncanonicalHamiltonian, itcanbeseparatedtoaquadratic\nand a quartic term: K=K2+K4, where\nK2=/integraldisplay\nddr/parenleftBig\n˜ψ†(r,t)H0˜ψ(r,t)\n+2gd|Φ(r,t)|2˜ψ†(r,t)˜ψ(r,t)\n+gd\n2Φ2˜ψ†(r,t)˜ψ†(r,t)\n+gd\n2Φ⋆2˜ψ(r,t)˜ψ(r,t)/parenrightBig\n,(37)\nand\nK4=gd\n2/integraldisplay\nddr˜ψ†(r,t)˜ψ†(r,t)˜ψ(r,t)˜ψ(r,t).(38)\nWe are interested in the regime where the condensate\nslightly differs from the equilibrium state, that is,\nΦ(r,t) = Φ0(r)+δΦ(r,t) (39)\nwith a small fluctuation δΦ(r,t). Expanding K2up to\nthe linear term in δΦ(r,t), we can rewrite it as K2=\nK(0)\n2+K(1)\n2, whereK(0)\n2is the zero order term of the\ncondensate\nK(0)\n2=/integraldisplay\nddr˜ψ†(r,t)(H0+2gdn0(r))˜ψ(r,t)\n+gd\n2n0(r)/parenleftBig\n˜ψ†(r,t)˜ψ†(r,t)+˜ψ(r,t)˜ψ(r,t)/parenrightBig\n(40)with the condensate density n0(r) =|Φ0(r)|2, whileK(1)\n2\nis the linear term in the fluctuation:\nK(1)\n2=/integraldisplay\nddr2gdΦ0(r)[δΦ0(r,t)+δΦ⋆\n0(r,t)]\n˜ψ†(r,t)˜ψ(r,t)+gdΦ0(r)/bracketleftBig\nδΦ0(r,t)˜ψ†(r,t)˜ψ†(r,t)\n+δΦ⋆\n0(r,t)˜ψ(r,t)˜ψ(r,t)/bracketrightBig\n.\n(41)\nAs for the quartic term K4, the mean-field decompo-\nsition is first used\n˜ψ†˜ψ†˜ψ˜ψ= 4˜n˜ψ†˜ψ+ ˜m˜ψ†˜ψ†+ ˜m⋆˜ψ˜ψ,(42)\nwhere\n˜n(r,t) =/angb∇acketle{t˜ψ†(r,t)˜ψ(r,t)/angb∇acket∇ight\n˜m(r,t) =/angb∇acketle{t˜ψ(r,t)˜ψ(r,t)/angb∇acket∇ight. (43)\nasthenormalandabnormaltime-dependentdensity. Un-\nder the linearization in the fluctuation (39), the normal\nandabnormaldensityarealsodisplacedwithasmallfluc-\ntuation as\n˜n(r,t) = ˜n0(r)+δ˜n(r,t),\n˜m(r,t) = ˜m0(r)+δ˜m(r,t), (44)\nexpanding around their stationary values ˜ n0(r) =\n/angb∇acketle{t˜ψ†(r,t)˜ψ(r,t)/angb∇acket∇ight0and ˜m0(r) =/angb∇acketle{t˜ψ(r,t)˜ψ(r,t)/angb∇acket∇ight0. In the\nliterature, often referred to so-called Popov approxima-\ntion\n˜m0(r) = 0 (45)\nhas been used in the mean-field treatment. Atually, this\napproximation was never suggested by Popov, as indi-\ncated by Yukalov [27], but was first proposed by Shohno\n[28] and we will refer it to Shohno approximation or\nShohno Ansatz in the remaining paper. The Shohno ap-\nproximation is necessary in the treatment because with-\nout that the elementary excitation would have a gap,\nwhich disobeys the gapless spectrum of the Goldstone\nmodes caused by the continuous gauge symmetry break-\ning in the ground state. However, the using of Shohno\napproximation is still under debate. Several attempts\nhave been done to go beyond the Shohno approximation\n(for example, see Ref. [27, 29, 30]). The Popov approx-\nimation is needed in the mean-field factorization (42).\nBy avoiding factorization, for example, using the per-\nturbation or random-phase approximation to calculate\nthe quartic terms, the gapless dispersion can be obtained\neven without Popov approximation.\nInserting Eqs.(39), (42), (44) and the Shohno Ansatz\n(45) into (38), the quartic term K4can be expanded up\nto the first order terms K4=K(0)\n4+K(1)\n4in fluctuations\nδ˜nandδ˜mas\nK(0)\n4= 2gd/integraldisplay\nddr˜n0(r)˜ψ†(r,t)˜ψ(r,t) (46)7\nis the zero order term, which represents the coupling to\nthe condensate from the quartic term, while the first or-\nder term reads\nK(1)\n4=gd\n2/integraldisplay\nddr/parenleftBig\n4δ˜n(r,t)˜ψ†(r,t)˜ψ(r,t)\n+δ˜m(r,t)˜ψ†(r,t)˜ψ†(r,t)+δ˜m⋆(r,t)˜ψ(r,t)˜ψ(r,t)/parenrightBig\n.\n(47)\nUnlikeK(1)\n2represents the coupling between the fluctu-\nations of the condensate and the noncondensate parti-\ncles,K(1)\n4is related to the coupling of the noncondensate\nparticles to the normal and abnormal densities. If the\ndensity of the noncondensate particles is much smaller\nthan the density of the condensate, K(1)\n2is more im-\nportant than K(1)\n4, therefore K(1)\n4can be neglected and\nK=K(0)\n2+K(0)\n4+K(1)\n2.\nIn the case of large occupation number of particles in\nthecondensate, K(1)\n2ismuchsmallerthan K(0)\n2andK(0)\n4,\nwe can useK(0)\n2+K(0)\n4as basis to develop a perturbation\nexpansion in terms of K(1)\n2. To diagonalize K(0)\n2+K(0)\n4,\none can apply a Bogoliubov transformation\n˜ψ(r,t) =/summationdisplay\njuj(r)αj(t)+v⋆\nj(r)α†\nj(t)\n˜ψ†(r,t) =/summationdisplay\nju⋆\nj(r)α†\nj(t)+vj(r)αj(t),(48)\nwith the quasi-particle creation and annihilation oper-\natorsα†\nj,αjobeying the Bose commutation relations\n[αi,α†\nj] =δij, which gives the normalization condition\nfor the functions uj(r,t),vj(r,t) as\n/integraldisplay\nddr[u⋆\ni(r)uj(r)−v⋆\ni(r)vj(r)] =δij.(49)\nTherefore the operator K(0)\n2+K(0)\n4can be diagonalized\nif the Bogoliubov-de Genes equations are satisfied:\nLuj(r)+gn0(r)vj(r) =ǫjuj(r),\nLvj(r)+gn0(r)uj(r) =−ǫjvj(r),(50)\nwhere a Hermitian operator is introduced as\nL=H0+2gdn(r) (51)\nwith the total density n(r) defined as the sum of the con-\ndensate density and normal density in the equilibrium:\nn(r) =n0(r) +n0(r). As a result, the grand-canonical\nHamiltonian (31) becomes\nK=K2+K4=/summationdisplay\njǫjα†\njα+K(1)\n2 (52)\nwith the eigenvalues ǫjobtained from the Bogoliubov-de\nGennes equations (50).In order to obtain the decay rate, we have to find\nthe time evolution of the fluctuation of the condensate:\nδΦ(r,t). The equation of motion:\ni∂\n∂tψ(r,t) = [ψ(r,t),K] (53)\nleads to the result\ni∂\n∂tψ(r,t) =H0ψ(r,t)+gdψ†(r,t)ψ(r,t)ψ(r,t).(54)\nInsertingthe decomposition(35) into the equationofmo-\ntion (54), it reads\ni∂\n∂tΦ(r,t) =/parenleftbig\nH0+gd|Φ(r,t)|2/parenrightbig\nΦ(r,t)\n+2gdΦ(r,t)˜n(r,t)+gdΦ⋆(r,t)˜m(r,t)(55)\nHere we assume that the cubic product of the noncon-\ndensate contributes very little to the dynamics of the\ncondensate, therefore the average value is set equal to\nzero:/angb∇acketle{t˜ψ†˜ψ˜ψ/angb∇acket∇ight= 0. The wavefunction in the equilibrium\ncan be obtained by setting ∂Φ0/∂t= 0, which leads to\nthe stationary equation\n/parenleftbig\nH0+gd/bracketleftbig\nn0(r)+2˜n0(r)/bracketrightbig/parenrightbig\nΦ0(r) = 0.(56)\nBy inserting Eq.(39) and the stationary equation (56)\ninto the equation of motion for the condensate (55), the\nequation of motion for the small amplitude δΦ reads\ni∂\n∂tδΦ(r,t) =(H0+2gdn(r))δΦ(r,t)+gdn0(r)δΦ⋆(r,t)\n+2gdΦ0δ˜n(r,t)+gdΦ0(r)δ˜m(r,t).\n(57)\nApplying the Bogoliubov transformation (48) to Eq.\n(57), the Eq. (57) gives the final form:\ni∂\n∂tδΦ(r,t) =(H0+2gdn(r))δΦ(r,t)+gdn0(r)δΦ⋆(r,t)\n+gdΦ0(r)/summationdisplay\nij{2[u⋆\niuj+v⋆\nivj+v⋆\niuj]fij(t)\n+[2viuj+uiuj]gij(t)+[2v⋆\niu⋆\nj+u⋆\niu⋆\nj]g⋆\nij(t)/bracerightbig\n,\n(58)\nwherefij(t)≡ /angb∇acketle{tα†\ni(t)αj(t)/angb∇acket∇ightandgij≡ /angb∇acketle{tαi(t)αj(t)/angb∇acket∇ightare\nnormal and anomalous quasiparticle distribution func-\ntions.\nTo calculate the normal and anomalous quasiparticle\ndistribution functions using the perturbation Hamilto-\nnian (52), we have to use the equation of motion :\ni∂\n∂tfij(t) =/angb∇acketle{t[α†\ni(t)αj(t),K]/angb∇acket∇ight\ni∂\n∂tgij(t) =/angb∇acketle{t[αi(t)αj(t),K]/angb∇acket∇ight. (59)\n(60)8\nTo the first order, the Fourier transform of fijandgijat\nthe frequency ωis given by\nfij(ω) =2gdf0\ni−f0\nj\nω+(ǫi−ǫj)+i0+/integraldisplay\nddr\n×Φ0/bracketleftbig\nδΦ1(r,ω)(uiu⋆\nj+viv⋆\nj+viu⋆\nj)\n+δΦ2(r,ω)(uiu⋆\nj+viv⋆\nj+uiv⋆\nj)/bracketrightbig\n;(61)\ngij(ω) =2gd1+f0\ni+f0\nj\nω−(ǫi+ǫj)+i0+/integraldisplay\nddr\n×Φ0/bracketleftbig\nδΦ1(r,ω)(u⋆\niv⋆\nj+v⋆\niu⋆\nj+u⋆\niu⋆\nj)\n+δΦ2(r,ω)(u⋆\niv⋆\nj+v⋆\niu⋆\nj+v⋆\niv⋆\nj)/bracketrightbig\n,(62)\nwhereδΦ1(r,ω) andδΦ2(r,ω)are the Fourier transform\nofδΦ(r,t) andδΦ⋆(r,t):\nδΦ1(r,ω) =/integraldisplay\ndte−iωtδΦ(r,t)\nδΦ2(r,ω) =/integraldisplay\ndte−iωtδΦ⋆(r,t), (63)\n(64)\nandf0\njis the bosonic distribution function\nf0\nj=1\n[eβǫj−1](65)\nwithβ= 1/kBT. Fourier transforming the equation of\nmotion (58) and replacing Eqs. (61) and (62) into it, we\nobtain the perturbed eigenfrequency:\nω=ω0+4g2\nd/summationdisplay\nij(f0\ni−f0\nj)|Aij|2\nω0+(ǫi−ǫj)+i0+\n+2g2\nd/summationdisplay\nij/parenleftbigg\n1+f0\ni+f0\nj)|Bij|2\nω0−(ǫi+ǫj)+i0+\n−|˜Bij|2\nω0+(ǫi+ǫj)+i0+/parenrightBigg\n,(66)\nwheretheunpeturbedeigenfrequency ω0isobtainedfrom\nthe unperturbed RPA equation [31]\n/parenleftbigg\nLgdn0\n−gdn0−L/parenrightbigg/parenleftbigg\nδΦ0\n1\nδΦ0\n2/parenrightbigg\n=ω0/parenleftbigg\nδΦ0\n1\nδΦ0\n2/parenrightbigg\n(67)\nwith the normalization condition\n/integraldisplay\nddr(|δΦ0\n1|2−|δΦ0\n2|2) = 1, (68)andAij,Bijand˜Bijare defined as\nAij=/integraldisplay\nddrΦ0/bracketleftbig\nδΦ0\n1(uiu⋆\nj+viv⋆\nj+viu⋆\nj)\n+δΦ0\n2(uiu⋆\nj+viv⋆\nj+uiv⋆\nj)/bracketrightbig\n,\nBij=/integraldisplay\nddrΦ0/bracketleftbig\nδΦ0\n1(u⋆\niv⋆\nj+v⋆\niu⋆\nj+u⋆\niu⋆\nj)\n+δΦ0\n2(u⋆\niv⋆\nj+v⋆\niu⋆\nj+v⋆\niv⋆\nj)/bracketrightbig\n˜Bij=/integraldisplay\nddrΦ0/bracketleftbig\nδΦ0\n1(uivj+viuj+uiuj)\n+δΦ0\n2(uivj+viuj+vivj)/bracketrightbig\n.(69)\nThe real part of the right-hand side (Eq. (66)) gives the\neigenenergy of the system and the imaginary part tells\nus the damping coefficient γ. Using the relation\n1\nx+i0+=P1\nx−iπδ(x), (70)\nwe can divide the damping rate into two different types:\none comes from the process that one phonon with the\nfrequencyω0is absorbed by a thermal excitation ǫi\njumping to another thermal excitation with the energy\nǫj=ǫi+ω0. This mechanism is so-called Landau damp-\ning given by the second term on the right-hand side of\nEq. (66),\nγL= 4πg2\nd/summationdisplay\nij|Aij|2(f0\ni−f0\nj)δ(ω0+ǫi−ǫj).(71)\nThis process happens mostly at finite temperature, it is\ntherefore a thermal process. Another kind of decay arises\nfrom the process of a long wave-length phonon decaying\ninto two phonons, as indicated by Beliaev, and it can\nbe obtained by the imaginary part of the first term in\nbrackets on the right-hand side of Eq. (66):\nγB= 2πg2\nd/summationdisplay\nij|Bij|2(1+f0\ni+f0\nj)δ(ω0−ǫi−ǫj).(72)\nThis process occurs mostly at zero temperature, which\nis a pure quantum effect. The total damping rate is the\nsum of the two damping coefficients: γ=γB+γL.\nIn this paper we are interested in homogeneous Bose\ngases, i.e. Vext(r) = 0. For homogeneous systems\nthe condensate density remains the same throughout the\nspace: Φ 0=√n0, while the excitations and the fluctua-\ntions satisfying Eq. (50) and Eq. (67) can be described\nby the plane waves\n/parenleftbigg\nδΦ1(r)\nδΦ2(r)/parenrightbigg\n=1√\nV/integraldisplay\nddkeik·r/parenleftbigg\nuk\nvk/parenrightbigg\n,(73)\n/parenleftbigg\nuk′(r)\nvk′(r)/parenrightbigg\n=1√\nV/integraldisplay\nddk′eik′·r/parenleftbigg\nuk′\nvk′/parenrightbigg\n,(74)\nwhereukandvksatisfy the Bogoliubov relations:\nu2\nk= 1+v2\nk=(ǫ2(k)+g2\ndn2\n0)1/2+ǫ(k)\n2ǫ(k)\nukvk=−gdn0\n2ǫ(k)(75)9\n0 0.2 0.4 0.6 0.8 1τ00.10.2\nγL/ε(k)\nFIG. 4: Landau damping rate per unit energy in two dimen-\nsions intheunitof8 π/|lnna2\n2|. Theblacksolid linerepresents\nthe result from HFB and the blue dashed line from the hy-\ndrodynamical approach. The high-temperature asymptotic\nbehavior and the low-temperature limit are also reported as\nthe green dashed-dot line and the red dashed-dot-dot line,\nrespectively.\nandǫ(k) is the Bogoliubov energy (13).\nA. Quantum regime\nAs mentioned in the last section, the dacay rate is\nmostly contributed by Beliaev damping, which can be\nobtained by setting f0\ni=f0\nj= 0 in Eq. (72). The matrix\nelementBk′−k,k′reads\nBk′−k,k′=/radicalbiggn0\nV[(uk(uk′vk′−k+vk′uk′−k+uk′uk′−k)\n+vk(uk′vk′−k+vk′uk′−k+vk′vk′−k)],\n(76)\nand other elements are zero. At zero temperature, only\nthe momenta with long wavelength are involved in the\nBeliaev damping process, i.e. k′∼k∼ |k′−k| ≪mc.\nTherefore the long-wavelength approximation for the en-\nergyǫ(k) and the wave functions ukandvkcan be used:\nǫ(k)≃ck+k3\n8m2c, (77)\nuk≃/parenleftBigmc\n2k/parenrightBig1/2\n+1\n2/parenleftbiggk\n2mc/parenrightbigg1/2\n+1\n8/parenleftbiggk\n2mc/parenrightbigg3/2\n−1\n8/parenleftbiggk\n2mc/parenrightbigg5/2\nvk≃ −/parenleftBigmc\n2k/parenrightBig1/2\n+1\n2/parenleftbiggk\n2mc/parenrightbigg1/2\n−1\n8/parenleftbiggk\n2mc/parenrightbigg3/2\n−1\n8/parenleftbiggk\n2mc/parenrightbigg5/2\n. (78)Substituting (77) and (78) in Eq. (76), we obain the\nresult\nBk′−k,k′=/radicalbiggn0\nV3\n4√\n2|k||k′||k′−k|\n(mc)3/2.(79)\nInserting Eq. (79) to Eq. (72) and summerizing all the\nmomenta kandk′, one obtain the same result (19) as\nthat using hydrodynamical approach. Therefore we re-\nproduces the results for 3-D and 2-D decay as Eq. (20)\nand Eq. (22).\nB. Thermal Regime\nIn the theraml regime where ω0≃kBT, a long-\nwavelength Goldstone mode with the eigenfrequency\n(ω0≃ck) describes the behavior of the condensate in\nthe thermal clouds. In this limit, the uandvfunctions\ncan be expanded as\nuk≃/parenleftbiggmc2\n2ǫ(k)/parenrightbigg1/2\n+1\n2/parenleftbiggǫ(k)\n2mc2/parenrightbigg1/2\n,\nvk≃ −/parenleftbiggmc2\n2ǫ(k)/parenrightbigg1/2\n+1\n2/parenleftbiggǫ(k)\n2mc2/parenrightbigg1/2\n.(80)\nUsing this expansion,the long-wavelength behavior for\nthe nonzero elements of the matrix Acan be expressed\nas\nAk′,k′+k=√n0√\nV/parenleftbiggǫ(k)\n2mc2/parenrightbigg1/2/parenleftbig\nu2\nk′+v2\nk′+uk′vk′\n−vg\nccosθ2u2\nk′v2\nk′\nu2\nk′+v2\nk′/parenrightbigg\n,(81)\nwith the angle θbetween the vectors k′andk, and the\ngroup velocity of the excitation defined as vg=∂ǫk/∂k.\nIn three dimensions, the damping rate can be obtained\nby inserting the nonzero coefficients (81) into (71) and\nintegrating out the angle θas follows:\nγd=3\nǫ(k)≃γd=3\nL\nǫ(k)= (a3\n3n0)1/2F(τ) (82)\nwhereτ=kBT/mc2as dimensionless temperature, a3\nis the three-dimensional scattering length, and F(τ) is\ndefined in the following:\nF(τ) =√π\nτ/integraldisplay\ndzsech2(z\n2τ)(1−1\n2u−1\n2u2)2(83)\nwith the definition u=√\n1+z2. This expression was\nfirst found by Pitaevskii and Stringari [12].\nFor low twmperature kBT≪mc2, i.e.τ≪1, the\nfunctionFtakes its limit F≈3π9/2τ4/5 and one finds\nthe Hohenbergand Martin’sresult (25). As mentioned in\nthe last section, the hydrodynamical approach and HFB\ngivethe same limit at low temperature. However, at high10\ntemperature, the hydrodynamic approach fails. For tem-\nperatureτ≫1., i.e.kBT≫mc2, the function Ftakes\nthe asymptotic limit F→3π3/2τ/4, and the damping\nrate approaches the Sz´ epfalusy and Kondor result (27).\nThereforetheHFB givescorrectasymptoticvalueathigh\ntemperature.\nIn Fig. 3 the famous result for the three-dimensional\ndamping rate using HFB method obtained by Stringari\nand Pitaevskii [12], and then recovered by Giorgini [13]\nis shown as the solid line. We can see that the three-\ndimensional damping rate leaves the low-temperature\nlimitverysoon, whileitapproachesthehigh-temperature\nlinear law very slowly.\nIn two dimensions, HFB offers a good approximation\nfor all regime of temperature. After inserting the matrix\nelements (81) into 71 and then integrating out the angle\nθ, one obtains the damping rate:\nγd=2\nǫ(k)≃γd=2\nL\nǫ(k)= 2mg2G(τ), (84)\nwhere\nG(τ) =√\n2\nπτ/integraldisplay∞\n0dzsech2(z\n2τ)/radicalBig√\nz2+1−1−z2\n2(z2−1)\n/parenleftbigg\n1−1\n2u−1\n2u2/parenrightbigg2\n.(85)\nIn the low temperature limit τ≪1, the function Gtakes\nthe limit: G(τ)→√\n3πτ2/16, the damping rate goes\nto the result (30). As in three dimensions HFB gives\nthe same result as the hydrodynamical approach at low\ntemperature in two dimensions.\nFor high temperature kBT≫mc2, the function G\ntakes its asymptotic value: G(τ)→0.013τ, and the\ndamping rate is given by\nγd=2\nǫ(k)≃0.026mkBT\nn0. (86)\nTherefore the damping rate itself reads\nγd=2≃0.0138π\n|ln(1/na2\n2)|kkBT\nmc. (87)\nIn Fig. 4, the two-dimensional damping rate per unit en-\nergy using HFB method is also shown as a function of τ\n(solid line). We can see that the two-dimensional damp-\ningrateapproachesthehightemperaturelinearlawmuch\nsooner than the three-dimensional case. That means,\nthe two dimensional systems go to the high-temperature\nasymptoticvalueatlowervalueoftemperaturecompared\nto that in three dimensions. This behavior has been\nfoundbyGuilleumasandPitaevskiistudyingaquasitwo-\ndimensional system [32] . In this figure one can also see\nthat the hydrodynamical approach can only give good\nresults in the regime where τ <0.5.V. CONCLUSION\nIn this work, we have compared the hydrodynamical\napproach and the HFB approach to calculate the damp-\ning rate in 2D and 3D bose gas. The hydrodynamical\napproach is a powerful tool due to the fact that one can\nuse the Green’s-function technique based on the effective\naction(5). This worksverywellat zeroand lowtempera-\ntures. However, this method truncates the rapid oscillat-\ning fields, which is not the case for the high temperature\nregime,therefore it overestimates the Landau damping.\nOn the other hand, the HFB approximation can explain\neither low temperature or high temperature damping. It\nseems that the mean-field approach (HFB) is a better\nmethod for Bose gases. The HFB approach based on the\nmeanfieldmethod factorizesthe quartictermsandthere-\nfore Shohno Ansatz has to be used to avoid anomalous\nbehavior. In the absence of the Shohno Ansatz, there\nwould exist a gap in the low excitation spectrum. There-\nfore the mean field approach cannot guarantee a zero\nenergy gap. From a physical point of view, the existence\nof a gapless excitation is a general rule for Bose systems.\nIn order to avoid errors in the higher order calculations,\nthe mean field approach should be very carefully used.\nTherefore we can see the benefit of the hydrodynamical\napproach for the low temperature regime. The low en-\nergy excitation is always gapless using hydrodynamical\napproach. Another benefit of using the hydrodynami-\ncal approach, as indicated by Popov [15], is that it can\navoid the strange divergence at high and low momenta,\nso-called ultraviolet and infrared catastrophe, which can\nbe causedby the perturbation theory based on the mean-\nfield approach.\nWehavefoundforthefirsttimethattheBeliaevdamp-\ningrate isproportionalto k3at zerotemperatureand the\nLandau damping rate for the 2D bose gas is proportional\ntoT2for lowtemperature and to Tfor high temperature.\nThe behavior of the 2D damping is also totally different\nfrom the 3D damping. While the 3D Landau damping\napproaches the linear regime very slowly with increasing\ntemperature, the 2D damping become linear very fast.\nThe linear regime symbolizes the classical high tempera-\nture behavior, therefore the two dimensional systems go\nto the high-temperature asymptotic value at lower value\nof temperature compared to that in three dimensional\nsystem. This behavior was also found in Ref. [32] with\nnumerical calculation for a quasi-2D system.\nAcknowledgments\nWe especially thank A. G. Semenov for the correction\nof the two-dimensional damping rate, and V. I. Yokalov\nfor the kindly reminding of the unjust using of the con-\ncept “Popov approximation”. Ming-Chiang Chung is\nsupported by the National Science Coucil of Taiwan,\nR.O.C. under grant No:NSC95-2112-M001-054-MY3.11\n[1] M. H. Anderson, J. R. Ensher, M. R. Mattews, C. E.\nWieman, and E. A. Cornell, Science 269, 198 (1995).\n[2] K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van\nDruten, D. S. Durfee, D. M. 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Beliaev, Soviet Phys. JETP 34, 299 (1958).\n[12] L. P. Pitaevskii and S. Stringari, Phys. Lett. A 235, 398\n(1997).\n[13] S. Giorgini, Phys. Rev. A 37, 2949 (1998).\n[14] M. Guilleumas and L. P. Pitaevskii, Phys. Rev. A 61,\n013602 (1999).\n[15] V. N. Popov, Functional Integrals and Collective excit a-\ntions (Cambridge University Press, Cambridge, 1987)\n[16] P. C. Hohenberg and P. C. Martin, Ann. Phys. (N.Y.)34, 291 (1965).\n[17] P. Sz´ epfalusy and I. Kondor, Ann. Phys. 82, 1 (1974).\n[18] W. V. Liu and W. C. Schieve, arXiv: cond-mat/9702122.\n[19] W. Vicent Liu. Phys. Rev. Lett. 79, 4056 (1997).\n[20] V. L. Berezinskii, Sov. Phys. JETP, 34, 610 (1971).\n[21] J. M. Kosterlitz and D. J. Thouless, J. Phys. C, 6, 1181\n(1973).\n[22] In this place we would like to thank Professor Semenov\nfor his kindly reminding of the factor two in two di-\nmensions. He compared his unpublished two-dimensional\ndamping results for zero and low temperature with ours\nand found the error.\n[23] M.-C. Chung and A. B. Bhattacherjee, Phys. Rev.Lett.\n101, 070402 (2008).\n[24] S. T. Beliaev, Soviet Phys. JETP 34, 289 (1958).\n[25] A. L. Fetter and J. D. Walecka Quantum Theory of\nMany-Particle Systems (McGraw-Hill 2003)\n[26] A. J. Leggett, Rev. Mod. Phys. 73, 307 (2001).\n[27] V. I. Yukalov, Phys. Lett. A 359, 712 (2006); Ann. Phys.\n323, 461 (2008).\n[28] A. Shohno, Prog. Theor. Phys. 31, 513 (1964).\n[29] S. A. Morgan, J. Phys. B: At. Mol. Opt. Phys. 333, 3847\n(2000).\n[30] J. Wachter, R. Walser, J. Cooper, and M. Holland, Phys.\nRev. A 64, 053612 (2001); cond-mat/0212432.\n[31] D. A. W. Hutchinson et. al. Phys. Rev. Lett. 78, 1842\n(1997).\n[32] M. Guilleumas and L. P. Pitaevskii, Phys. Rev. A 67,\n053607 (2003)"    },    {        "title": "0704.2568v2.Probing_non_standard_decoherence_effects_with_solar_and_KamLAND_neutrinos.pdf",        "content": "arXiv:0704.2568v2  [hep-ph]  18 Jul 2007Probing non-standard decoherence effects\nwith solar and KamLAND neutrinos\nG.L. Fogli1, E. Lisi1, A. Marrone1, D. Montanino2, and A. Palazzo3,1\n1Dipartimento di Fisica and Sezione INFN di Bari,\nVia Amendola 173, 70126, Bari, Italy\n2Dipartimento di Fisica and Sezione INFN di Lecce\nVia Arnesano, 73100 Lecce, Italy\n3Astrophysics, Denys Wilkinson Building,\nKeble Road, OX1 3RH,\nOxford, United Kingdom\nAbstract\nIt has been speculated that quantum gravity might induce a “f oamy” space-time structure at\nsmallscales, randomlyperturbingthepropagation phaseso ffree-streamingparticles (suchaskaons,\nneutrons, or neutrinos). Particle interferometry might th en reveal non-standard decoherence ef-\nfects, in addition to standard ones (due to, e.g., finite sour ce size and detector resolution.) In this\nwork we discuss the phenomenology of such non-standard effect s in the propagation of electron\nneutrinos in the Sun and in the long-baseline reactor experi ment KamLAND, which jointly provide\nus with the best available probes of decoherence at neutrino energies E∼few MeV. In the solar\nneutrino case, by means of a perturbative approach, decoher ence is shown to modify the standard\n(adiabatic) propagation in matter through a calculable dam ping factor. By assuming a power-law\ndependence of decoherence effects in the energy domain ( Enwithn= 0,±1,±2), theoretical\npredictions for two-family neutrino mixing are compared wi th the data and discussed. We find\nthat neither solar nor KamLAND data show evidence in favor of non-standard decoherence effects,\nwhose characteristic parameter γ0can thus be significantly constrained. In the “Lorentz-inva riant”\ncasen=−1, we obtain the upper limit γ0<0.78×10−26GeV at 95% C.L. In the specific case\nn=−2, the constraints can also be interpreted as bounds on possi ble matter density fluctuations\nin the Sun, which we improve by a factor of ∼2 with respect to previous analyses.\nPACS numbers: 14.60.Pq, 26.65.+t, 03.65.Yz, 04.60.-m\n1I. INTRODUCTION\nAlthough a satisfactory theory of quantum gravity is still elusive, it has been speculated\nthat it should eventually entail violations of basic quantum mechanics , including the sponta-\nneousevolutionofpurestatesintomixed(decoherent)states[1] throughunavoidableinterac-\ntions with a pervasive and “foamy” space-time fabric at the Planck s cale [2]. The pioneering\npaper [3] showed that such hypothetical source of decoherence might become manifest in os-\ncillating systems which propagateover macroscopical distances, t hroughadditional smearing\neffects in the observable interferometric pattern (besides the us ual smearing effects due, e.g.,\nto the finite source size and the detector resolution). However, la cking an “ab initio” theory\nof quantum gravity decoherence, its effects can only be paramete rized in a model-dependent\n(and somewhat arbitrary) way. Searches with neutral kaon oscilla tions [3, 4, 5, 6], neutron\ninterferometry [3, 7] and, more recently, neutrino oscillations [8, 9, 10, 11, 12, 13, 14], have\nfound no evidence for such effects so far, and have placed bounds on model parameters.\nQuantumgravityeffectsinneutrino systems havebeeninvestigate d withincreasing atten-\ntion in the last decade, as a result of the evidence for neutrino flavo r oscillations. Early at-\ntemptstriedtointerpretthesolarneutrinopuzzle[8, 9, 10]orth eatmospheric νanomaly[10]\nin terms of decoherence. After the first convincing evidence for a tmospheric neutrino oscil-\nlations [15], a quantitative analysis was performed in [11], considering possible decoherence\neffects in the νµ→νµchannel (see also [12]). The phenomenology of terrestrial neutrin o\nexperiments has also been investigated in [13, 14]. More recently, pr ospective studies have\nfocused on decoherence effects in high energy neutrinos [16, 17, 1 8, 19, 20], observable in\nnext generation neutrino telescopes. Furthermore, more forma l aspects of quantum gravity\ndecoherence in neutrino systems have been developed [21, 22, 23 , 24, 25, 26, 27]. This is\nonly a fraction of the related literature, which testifies the wide and increasing interest in\nthe subject.\nDespite this interest, to our knowledge such decoherence effects have not been systemat-\nically investigated in the light of the solar neutrino experiments perfo rmed in the last few\nyears. Solar neutrino oscillations [28] dominated by matter effects [2 9, 30] are currently well\nestablished by solar neutrino experiments [31, 32, 33, 34, 35, 36, 3 7, 38, 39, 40, 41] and have\nbeen independently confirmed by the long-baseline reactor experim ent KamLAND [42, 43].\nThe striking agreement between solar and KamLAND results determ ines a unique solution\nin the mass-mixing parameter space [the so-called Large Mixing Angle ( LMA) solution, see\ne.g. [44, 45]], provides indirect evidence for matter effects with stan dard amplitude [46], and\ngenerally (although not always [47]) implies that additional, non-stand ard physics effects\nmay play only a subleading role, if any. In particular, the KamLAND colla boration has\nexploited the observation of half oscillation cycle in the energy spect rum [43] to exclude\ndecoherence as a dominant explanation of their data.\nThe main purpose of this paper is then to study decoherence as a subdominant effect\nin solar and KamLAND neutrino oscillations. Modifications of the stand ard oscillation\nformulae in the presence of decoherence, and qualitative bounds o n decoherence parameters,\narediscussed inSec. IIandIIIforKamLANDandsolarneutrinos, r espectively. Quantitative\nbounds on subdominant decoherence effects from a joint analysis o f solar and KamLAND\ndata are studied in Sec. IV. Implications for decoherence induced b y matter fluctuations in\nthe Sun are discussed in Sec. V. The main results are finally summarize d in Sec. VI. The\nsolar neutrino flavor evolution in the presence of standard matter effects plus non-standard\ndecoherence is discussed in a technical Appendix.\n2II. OSCILLATIONS WITH(OUT) DECOHERENCE IN KAMLAND\nHere and in the following, we assume the standard notation [48] for n eutrino mixing, and\nset the small mixing angle θ13to zero for the sake of simplicity. For θ13= 0, oscillations in\ntheνe→νechannel probed by long-baseline reactor (KamLAND) and by solar n eutrinos\nare driven by only two parameters: the mixing angle θ12and the neutrino squared mass\ndifference δm2=m2\n2−m2\n1. In particular, the standard νesurvival probability over a baseline\nLin KamLAND reads:\nPee= 1−1\n2sin22θ12/parenleftBigg\n1−cos/parenleftBiggδm2L\n2E/parenrightBigg/parenrightBigg\n. (1)\nIn the presence of additional decoherence effects, the oscillating factor is exponentially\nsuppressed, as shown in [11] for the atmospheric νµ→νµchannel. By changing the ap-\npropriate parameters for the KamLAND νe→νechannel, the results of [11] lead to the\nfollowing modification of the previous equation,\nPee= 1−1\n2sin22θ12/parenleftBigg\n1−e−γLcos/parenleftBiggδm2L\n2E/parenrightBigg/parenrightBigg\n, (2)\nwhere the dimensional parameter γrepresents the inverse of the decoherence length after\nwhich the neutrino system gets mixed.1Equation (2) includes the limiting cases of pure\noscillations ( γ= 0 and δm2∝negationslash= 0) and of pure decoherence ( γ∝negationslash= 0 and δm2= 0).\nUnfortunately, lacking a fundamental theory for quantum gravit y, the dependence of γ\non the underlying dynamical and kinematical parameters (most not ably the neutrino energy\nE) is unknown. Following common practice, such ignorance is paramete rized in a power-law\nform\nγ=γ0/parenleftbiggE\nE0/parenrightbiggn\n, (3)\nwhereE0is anarbitrary pivot energy scale, which we set as E0= 1 GeV in order to facilitate\nthe comparison with limits on γ-parameters investigated in other contexts (as reviewed, e.g.,\nin [49]). We shall consider only five possible integer exponents,\nn= 0,±1,±2, (4)\nwhich include the following cases of interest: The “energy independe nt” case ( n= 0); the\n“Lorentz invariant” case [11] ( n=−1); the case n= +2 that can arise in some D-brane\nor quantum-gravity models, in which γ0∼O(E2\n0/MPlanck)∼10−19GeV is expected (see,\ne.g. [50]); andthe case where decoherence might be induced by “mat ter density fluctuations”\nrather than by quantum gravity ( n=−2, see Sec. V).\nAs previously remarked, the KamLAND collaboration [43] (see also [5 1]) has ruled out\npure decoherence in the Lorentz invariant case ( n=−1). In our statistical χ2analysis, we\nalso find that this case is rejected at 3 .6σ(i.e., ∆χ2= 13 with respect to pure oscillations).\nIn addition, we find that the other exponents in Eq. (4) arealso rej ected at>3σfor the pure\ndecoherence case. Therefore, decoherence effects can only be subdominant in KamLAND,\nnamely\nγL≪1. (5)\n1Units: [γ] = 1/length = energy. Conversion factor: (1 km)−1= 1.97×10−19GeV.\n3FortypicalKamLANDneutrinoenergies( E∼few MeV)andbaselines( L∼2×102km), the\nabove inequality implies upper bounds on γ0, which range from γ0≪10−26GeV (n=−2)\ntoγ0≪10−16GeV (n= +2). We do not refine the analysis of such bounds (placed by\nKamLAND alone), since they are superseded by solar data constra ints, as shown in the next\nSection.\nIII. OSCILLATIONS WITH(OUT) DECOHERENCE EFFECTS IN SOLAR NEU-\nTRINOS\nThe survival probability describing standard adiabatic νetransitions in the solar matter\nis given by the simple formula (up to small Earth matter effects)\nP⊙\nee=1\n2/parenleftBig\n1+cos2˜θ12(r0)cos2θ12/parenrightBig\n, (6)\nwhere˜θ12(r0) is the energy-dependent effective mixing angle in matter at the pro duction\nradiusr0(see, e.g., [52] and references therein).\nIn the presence of non-standard decoherence effects, we find t hat the energy dependent\nterm is modulated by an exponential factor,\nP⊙\nee=1\n2/parenleftBig\n1+e−γ⊙R⊙cos2˜θ12(r0)cos2θ12/parenrightBig\n, (7)\nwhereR⊙= 6.96×105km is the Sun radius, while γ⊙is defined as\nγ⊙=γ0gn(E), (8)\nwhere the dimensionless function gn(E) embeds, besides the power-law dependence En, also\nthe information about the solar density profile [which is instead absen t in Eq. (6)]. The\nreader is referred to the Appendix for a derivation of Eq. (7) and f or details about the\nfunction gn(E).\nEquation (7) includes the subcase of pure oscillations ( γ0= 0 andδm2∝negationslash= 0), but not the\nsubcase of pure decoherence ( γ0∝negationslash= 0 and δm2= 0), since the limit δm2→0 would entail\nstrongly nonadiabatic transitions, and thus a breakdown of the ad iabatic approximation\nassumed above. However, as noted in the previous section, the Ka mLAND data exclude the\nlimitδm2→0; moreover, they require δm2values which are high enough to guarantee the\nvalidity of the adiabatic approximations, with or without subleading de coherence effects (as\nwe have numerically verified). Therefore, for solar neutrinos, the only phenomenologically\nrelevant cases are those including oscillations plus decoherence.2\nWe have analyzed all the available solar neutrino data with ( δm2,θ12,γ0) taken as free\nparameters. It turns out that, despite the allowance for an extr a degree of freedom ( γ0), the\ndata always prefer the pure oscillations ( γ0= 0) as best fit, independently of the power-law\nindex in Eq. (4). Since there are no indications in favor of decoheren ce effects, the exponent\nin Eq. (7) is expected to be small,\nγ⊙R⊙≪1. (9)\n2For the sake of curiosity, we have anyway calculated P⊙\neefor the pure decoherence case, by numerically\nsolving the neutrino evolution equations (discussed in the Appendix) forδm2= 0 and γ∝negationslash= 0. We always\nfindP⊙\nee>1/2, which is forbidden by8B solar neutrino data [46].\n4TABLE I: Upper limits on the decoherence parameter γ0obtained for different values of nfrom a\nglobal fit to solar and KamLAND data, after marginalization o f the mass-mixing parameters. The\nlimits refer to 95% C.L. (i.e., 2 σ, or ∆χ2= 4).\nn γ0(GeV)\n−2 <0.81×10−28\n−1 <0.78×10−26\n0 <0.67×10−24\n+1 <0.58×10−22\n+2 <0.47×10−20\nFor typical solar neutrino energies E∼10 MeV, it turns out that gn(E)∼0.2×10−2n(see\nthe Appendix), and the above inequality can be translated into uppe r bounds on γ0, which\nrange from γ0≪10−28GeV (for n=−2) toγ0≪10−20GeV (for n= +2). Such bounds\nare two to four orders of magnitude stronger than those placed b y KamLAND alone (see\nthe end of the previous Section). A useful complementarity then e merges between solar and\nKamLAND data in joint fits: The former dominate the constraints on decoherence effects,\nwhile the latter fix the mass-mixing parameters independently of (ne gligible) decoherence\neffects.\nIV. COMBINATION OF SOLAR AND KAMLAND DATA: RESULTS AND DIS-\nCUSSION\nWehaveperformedajointanalysisofsolarandKamLANDdata3inthe(δm2,sin2θ12, γ0)\nparameter space for the five power-law exponents n= 0,±1,±2. The main results are: (i)\nγ0= 0isalwayspreferredatbestfit, i.e., thereisnoindicationinfavorof decoherence effects;\n(ii) the best fit values and the marginalized bounds for ( δm2,sin2θ12) do not appreciably\nchange from those obtained in the pure oscillation case, namely, δm2= (7.92±0.71)×10−5\neV2and sin2θ12= (0.314+0.057\n−0.047) at±2σ[44]; (iii) significant upper bounds can be set on\nthe decoherence parameter γ0. Our limits on γ0are given numerically in Table I (at the 2 σ\nlevel, ∆χ2= 4) and graphically in Fig. 1 (at 2 σand 3σlevel). Such limits are consistent\nwith those discussed qualitatively after Eq. (9) in the previous Sect ion.\nFigure 1 clearly shows that the bounds on γ0scale with nalmost exactly as a power law,\nchanging by about two decades for |∆n|= 1. The reason is that the bounds are dominated\nby solar neutrino data, and in particular by data probing the8B neutrinos in a relatively\nnarrow energy range around E∼10 MeV; the power-law dependence assumed in Eq. (3)\nand embedded in the function gn(E) then implies that the parameter γ0scales roughly as\n(E/E0)−n∼102n.\nAlthough the case of no decoherence ( γ0= 0) is preferred, it makes sense to ask what\none should observe for decoherence effects as large as currently allowed by the data at, say,\nthe 2σlevel. Figure 2 compares the P⊙\neeenergy profile for the cases of pure oscillations (left\n3The details of the data set, of the solar model used [53] and of the statistical χ2analysis have been\nreported in [44] and are not repeated here.\n5panel) andofoscillationsplus decoherence (right panel), where γ0istakenequal totheupper\nbound at 2 σ, as taken from Table I.4It can be seen that decoherence effects, to some extent,\nmimic the effects of a larger mixing. For instance, the curve with n= 0 in the right panel is\nnot much different from the curve at sin2θ12= 0.371 (upper 2 σvalue) in the left panel. As\na consequence, one expects some degeneracy between the para metersγ0and sin2θ12when\nfitting the data. The degeneracy is only partial however, because decoherence effects can\nsignificantly change both the shape and the slope of the energy pro file within the current\n2σbounds, as evident in the right panel. Therefore, future measure ments of the (currently\nnot well constrained) solar neutrino energy spectrum will provide f urther important probes\nof decoherence effects.\nThe variation of P⊙\needue to subdominant decoherence effects [see Eqs. (6) and (7)] is\ngiven, in first approximation, by\n∆P⊙\nee≃ −1\n2γ⊙R⊙cos2˜θ12cos2θ12, (10)\nand changes sign with cos2 ˜θ12. As the energy increases, the value of cos2 ˜θ12changes from\ncosθ12>0 (low-energy, vacuum-dominated regime) to −1 (high-energy, matter-dominated\nregime), the transition being located around 2 MeV for the8B neutrino curves shown in\nFig. 2. This fact explains the general increase of P⊙\neeforE>∼2 MeV in the right panel of\nFig. 2. More detailed features depend instead on the energy behav ior of the function gn(E),\nwhich modulates decoherence effects (see the Appendix). In gene ral,gn(E) grows rapidly\nwith increasing energy for n >0 (which explains the high-energy upturn of the curves with\nn= +1 and n= +2), while it vanishes withdecreasing energy forall n∝negationslash=−2 (which explains\nthe low-energy equality of all curves but for n=−2).5The “bunching” of the curves around\nE∼10 MeV in the right panel of Fig. 2 is in part a data selection effect, sinc e this energy\nregion is strongly constrained by precise8B neutrino data. Further spectral8B data will\nbe very useful to constrain the slope of the energy spectrum and thus also the sign of the\npower-law index n.\nFigure 3 illustrates the partial degeneracy between decoherence effects and mixing angle,\nas a shift in the allowed regions for fixed γ0∝negationslash= 0 in three representative cases (from left\nto right, n=−2,0,+2). In each panel, the thin dotted curves enclose the mass-mixing\nparameter regions allowed at 2 σby the standard oscillation fit of solar data (larger region)\nand by solar plus KamLAND data (smaller region). The thick solid curve s refer to the\nsame data, but fixing a priori the decoherence parameter γ0at the 2σupper limit value\nin Table I. In all cases in Fig. 3, the curves with γ0∝negationslash= 0 are shifted to lower values of the\nmixing angle, as compared to pure oscillations; this means that decoh erence effects can be\npartlytraded fora smaller value ofthe mixing angleinsolar neutrino os cillations. Therefore,\nshould future solar neutrino data prefer smaller (larger) values of sin2θ12with respect to\nKamLAND data, there would be more (less) room for possible subdom inant decoherence\neffects. As already remarked, the degeneracy between γ0and sin2θ12is only partial, and\nfuture neutrino spectroscopy will provide a further handle to bre ak it, should decoherence\neffects (if any) be found.\n4For definiteness, Fig. 2 shows the daytime probability of8B neutrinos, averaged over their production\nregion in the Sun.\n5Constraints on the specific case n=−2 might thus benefit of sub-MeV solar neutrino observations in\nBorexino [54].\n6We conclude this section by confronting the bounds in Table I with tho se derivable from\nthe analysis of atmospheric neutrino data (which, by themselves, e xclude pure decoherence,\nat least in the n=−1 case [55]). In principle, a direct comparison is not possible, since the\nγ0parameter introduced here for the solar νe→νechannel does not need to be the same as\nfortheatmospheric νµ→νµchannel. However, ifthe γ0’sforthese two channels are assumed\nto be roughly equal in size, then it is easy to realize that solar+KamLA ND neutrinos set\nstronger (weaker) bounds than atmospheric neutrinos for n <0 (n >0), as a consequence\nof the different neutrino energy range probed. In fact, due to th e assumed power-law energy\ndependence of decoherence effects, negative (positive) values o fnare best probed by low-\nenergy solar (high-energy atmospheric) neutrino experiments. F or the intermediate case\n(n= 0), it turns out that matter effects render solar neutrinos more sensitive to γ0than\natmospheric neutrinos. Just to make specific numerical examples: forn= (−1,0,+2) one\nroughly gets the bounds γ0<∼(0.7×10−21,0.4×10−22,0.9×10−27) GeV from atmospheric\n[11] (plus accelerator [12]) neutrino data, to be compared with the c orresponding limits from\nsolar+KamLAND data from Table I, γ0<(0.78×10−26,0.67×10−24,0.47×10−20) GeV.\nSolar neutrinos clearly win over atmospheric neutrinos for n≤0. This comparison must be\ntaken with a grain of salt, since it can radically change by assuming eith er independent γ0’s\nin different oscillation channels, or functional forms of γ(E) different from power laws.\nFinally we observe that in the n= +2 case, motivated by some “quantum-gravity” or\n“string-inspired” models [50], the solar+KamLAND limit on γ0is one order of magnitude\nlower than the theoretical expectation [ γ0∼O(E2\n0/MPlanck)∼10−19GeV]. In the case of\natmospheric neutrinos, this bound is even stronger ( γ0<0.9×10−27GeV). Consequently,\nthese models appear strongly disfavored, at least in the neutrino s ector.\nV. RECOVERING THE CASE OF DENSITY FLUCTUATIONS IN THE SUN\nDecoherence effects in solar neutrino oscillations can be induced not only by quantum\ngravity, but also by more “prosaic” sources, such as matter dens ity fluctuations—possibly\ninduced by turbulence in the innermost regions of the Sun. This topic has been widely\ninvestigated in the literature [56, 57, 58, 59, 60, 61, 62, 63, 64, 6 5], and quantitative upper\nlimits have already been set [61, 62, 63, 64] by combining solar data an d first KamLAND\nresults.\nIt turns out that stochastic density fluctuations lead (with appro priate redefinition of\nparameters) to effects which have the same functional form as th ose induced by quantum\ngravity in the n=−2 case. More precisely, let us consider fluctuations of the solar elec tron\ndensityNearound the average value ∝angbracketleftNe∝angbracketrightpredicted by the standard solar model,\nNe(r) = (1+ βF(r))∝angbracketleftNe(r)∝angbracketright, (11)\nwhereF(r) is a random variable describing fluctuations at a given radius r, andβrepresents\ntheir fractional amplitude around the average. It is customary to assume a delta-correlated\n(white) noise,\n∝angbracketleftF(r1)F(r2)∝angbracketright= 2τδ(r1−r2), (12)\nwhereτis the correlation length of fluctuations along the ( ∼radial) neutrino direction.\nAs shown in [59] through a perturbative method (which inspired our a pproach to deco-\nherence in the Appendix), the effect of delta-correlated noise on a diabatic neutrino flavor\n7transitions can beembedded throughan exponential damping fact or asinEq. (7). The func-\ntional form turns out to be the same as for the n=−2 case, provided that one makes—in\nour notation—the replacement\nγ0→β2τ/parenleftBiggδm2\n2E0/parenrightBigg2\n. (13)\nBy using the bound γ0<0.81×10−28GeV (Table I, case n=−2) and the best-fit value\nδm2= 7.92×10−5eV2, one gets the following upper limit\nβ2τ <1.02×10−2km (2σ), (14)\non the parameter combination β2τwhich is relevant [59, 63] for density fluctuation effects\non neutrino propagation.\nAs stressed in [63], care must be taken in extracting an upper limit on t he fractional am-\nplitudeβfor fixed correlation length τ. Indeed, the delta-correlated noise is an acceptable\napproximation only if the correlation length τis much smaller than the oscillation wave-\nlength in matter—a condition that becomes critical for low neutrino e nergies. In particular,\nassuming a reference value τ= 10 km as in [63], such condition is violated for energies\nE<∼1 MeV. Following [63], we have thus excluded low-energy, radiochemica l solar neutrino\ndata form the solar+KamLAND data fit, and obtained a slightly weake r (but more reliable)\nupper bound from the analysis of8B neutrino data only,\nβ2τ <1.16×10−2km (2σ), (15)\nwhich, for τ= 10 km, translates into an upper limit on the fractional fluctuation a mplitude,\nβ <3.4% (2σ). (16)\nThis limit improves the previous one derived in [63] ( β <6.3% at 2σ) by a factor of ∼2,\nessentially as a result of the inclusion of the most recent solar and Ka mLAND neutrino data\nappeared after [63].6\nVI. SUMMARY AND CONCLUSIONS\nIn this paper, we have investigated hypothetical decoherence eff ects (e.g., induced by\nquantum gravity) in the νe→νeoscillation channel explored by the solar and KamLAND\nexperiments. In both kinds of of experiments, decoherence effec ts can be embedded through\nexponential damping factors, proportional to a common paramet erγ0, which modulate the\nenergy-dependent part of the νesurvival probability. By assuming that the (unknown)\nfunctional form of decoherence effects is a power-law in energy ( En), we have studied the\nphenomenological constraints on the main decoherence paramete r (γ0) forn= 0,±1,±2.\nIt turns out that both solar and KamLAND data do not provide indica tions in favor of\ndecoherence effects and prefer the standard oscillation case ( γ0= 0) for any index n. By\ncombining the two data sets, n-dependent upper bounds (dominated by solar neutrino data)\n6We have verified that, by adopting the same (older) data set and st andard solar model as used in [63], we\nrecover the same 2 σupper limit, β<∼6%.\n8have been derived on γ0, as reported in Table I and shown in Fig. 1. In the “Lorentz-\ninvariant” case n=−1, we obtain the upper limit γ0<0.78×10−26GeV at 95% C.L. For\nn=−2, the results can also be interpreted as limits on the amplitude of pos sible (delta-\ncorrelated) density fluctuations in the Sun, which we improve by a fa ctor of two [Eqs. (15)\nand (16)] with respect to previous bounds.\nFurther progress might come from a better determination of the e nergy profile of solar\nneutrino flavor transitions as well as from more precise measureme nts of sin2θ12(which\nis partly degenerate with γ0), attainable with KamLAND and future long-baseline reactor\nneutrino experiments [66].\nAPPENDIX: DECOHERENCE AND MATTER EFFECTS IN SOLAR NEUTRI-\nNOS\nInthissection we discuss aperturbative calculation ofdecoherenc e effects forsolar neutri-\nnos, where matter effects are known to be relevant. The approac h, inspired by the work [59],\ndraws on the formalism and the notation introduced in [11] for the ca se of decoherence in\ntheνµ→νµchannel, here adapted to the νe→νechannel. In the following, the the notation\nis made more compact by setting θ=θ12,c2θ= cos2θ12,s2θ= sin2θ12,Pee=P⊙\neeetc.\nDecoherence effects in the flavor evolution of the ( νe,νa) system (where a=µ,τ) along\nthe space coordinate r(≃t)7can be described in terms of the neutrino density matrix,\nobeying a modified master equation of the form [67]\ndρ\ndr=−i[Hv+Hm(r),ρ]−γ[D,[D,ρ]], (17)\nwhereHv(Hm) is the “vacuum” (matter) Hamiltonian, and the operator Dembeds de-\ncoherence effects with amplitude γ, parameterized as in Eq. (3): γ=γ0(E/E0)nwith\nE0= 1 GeV. While unitarity is preserved (i.e., Tr ρ(r) = 1), coherence is lost in the propa-\ngation (d\ndrTrρ2≤0). Equation (17) satisfies the conditions of complete positivity [68 ] and\nnon-decreasing entropy in the νsystem evolution [69].\nIn the flavor basis the standard oscillation terms read\nHv=−k\n2Uθσ3U†\nθ=k\n2/bracketleftBigg\n−c2θs2θ\ns2θc2θ/bracketrightBigg\n, (18)\nHm=−V(r)\n2σ3, (19)\nwhereσ3is the third Pauli matrix, k=δm2/2Eis the vacuum wavenumber, and V(r) =√\n2GFNe(r) is the interaction potential in matter.\nAs in [11], we assume energy conservation for evolution in vacuum, i.e., Tr[Hvρ(r)] = con-\nstant. This condition is satisfied if [ Hv,D] = 0 [4, 70], namely, in a two-dimensional system,\nifD∝Hv. We can thus take\nDθ=1\n2/bracketleftBigg\n−c2θs2θ\ns2θc2θ/bracketrightBigg\n, (20)\n7Note that rdoes not necessarily coincide with the radial coordinate, due to the extended neutrino pro-\nduction region (which is taken into account in our analysis.)\n9without loss of generality, since any overall factor can be absorbe d inγ0. We also make the\nplausible assumption that Eq. (20) is not altered for evolution in matt er, since decoherence\ninduced by quantum gravity is unrelated to electroweak matter effe cts.\nAs usual, Eq. (17) can be written in terms of a “polarization” vector Pwith components\nPi=1\n2Tr[ρσi] (Bloch equation):\ndP\ndr= [kn+V(r)e3]×P−γP⊥\n=H(r)P−γDθP (21)\nwheren= [s2θ,0,−c2θ]T,P⊥=P−(P·n)n, and:\nH(r) =\n0−V(r)+kc2θ0\nV(r)−kc2θ0−ks2θ\n0 ks2θ 0\n, (22)\nDθ=\nc2\n2θ0c2θs2θ\n0 1 0\nc2θs2θ0s2\n2θ\n. (23)\nNote that for V= 0 (vacuum propagation), the solution of the above Bloch equation leads\nto the survival probability in Eq. (2), (see [11] for details.)\nThe matrix Hhas eigenvalues λ0= 0 and λ±=±˜k, corresponding to the eigenvectors:\nu0=\ns2˜θ\n0\n−c2˜θ\n,u±=1√\n2\nc2˜θ\n±i\ns2˜θ\n. (24)\nIn the above equations, a “tilde” marks effective parameters in mat ter:˜kis the oscillation\nwavenumber in matter, defined through ˜k/k= [1−2Vc2θ/k+(V/k)2]1/2, while˜θis the\nmixing angle in matter, defined through s2˜θ=ks2θ/˜kandc2˜θ= (kc2θ−V)/˜k. The matrix H\nis diagonalized through the matrix R(˜θ) = [u0,u+,u−]:R†(˜θ)·H·R(˜θ) = diag[0 ,+i˜k,−i˜k].\nIn the absence of decoherence effects, the adiabatic solution of E q. (21) appropriate for\ncurrent solar neutrino phenomenology is\nP(R⊙) =R(θ)·diag/bracketleftBigg\n1, e+i/integraltextR⊙\nr0dr˜k(r), e−i/integraltextR⊙\nr0dr˜k(r)/bracketrightBigg\n·R†(˜θ0)P(r0) (25)\nwhereP(r0) =T[0,0,1] for an initial νestate,˜θ0is the mixing angle in matter at the\nproduction point r0, and˜θ=θis taken at r=R⊙. After averaging on the fast oscillating\nterms (a “standard” decoherence effect), one recovers the us ual adiabatic formula [Eq. (6)]\nfor the survival probability Pee\nPee= Tr[ρ|νe∝angbracketright∝angbracketleftνe|] =1+P3(R⊙)\n2=1+c2θc2˜θ0\n2. (26)\nLet us now treat the term −γDθPin Eq. (21) as a perturbation [59]. The corrections to\nthe eigenvectors lead to variations of PeeofO(γ/k)<∼10−3(for the range of γ/kallowed a\n10posteriori by the fit to solar neutrino data) and can be neglected. The corrections to the\ntwo eigenvalues λ±can also be neglected, since they would only lead to a further damping\nof the fast oscillating terms, which are already averaged out.8\nThe first-order correction to the eigenvalue λ0(whose unperturbed value is zero) is the\nonly relevant one,\nδλ0=−γu†\n0Dθu0=−γsin22(˜θ−θ). (27)\nand leads to the following correction to Eq. (26):\nPee=1+e−Γc2θc2˜θ0\n2, (28)\nwhere\nΓ =γ/integraldisplayR⊙\nr0/bracketleftBiggV(r)s2˜θ(r)\nk/bracketrightBigg2\ndr . (29)\nEquation (7) is then recovered by setting γ⊙=γ0gn(E) and by defining the dimensionless\nfunction gn(E) as\ngn(E) =/parenleftbiggE\nE0/parenrightbiggn/integraldisplayR⊙\nr0/bracketleftBiggV(r)s2˜θ(r)\nk/bracketrightBigg2dr\nR⊙. (30)\nThe function gn(E) depends mostly on the neutrino energy Eand, to some extent, on\nthe parameters r0,δm2, and sin2θ12. Figure 4 shows this function as calculated for r0= 0,\nδm2= 7.92×10−5eV2, and sin2θ12= 0.314. For E→0 the function gn(E) (and the\nassociated decoherence effect) vanishes, except for the case n=−2, where the factors E−2\nandk−2cancel out and provide a finite limit g−2(0)∝negationslash= 0.\nWe have tested the analytical Eq. (28) against the results of a num erical integration of\nthe Bloch equation, for many representative points in the paramet er space relevant for solar\nνphenomenology, and we find very good agreement ( δPee<10−4) for all values of n∝negationslash=−2.\nOnly in the case n=−2, the comparison of analytical and numerical results is slightly\nworse (but still very good, δPee<10−3) at the lowest detectable energies ( ∼0.1 MeV),\ndue to the breakdown of perturbation theory for E→0. For practical purposes, however,\nthe modified adiabatic Eq. 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B 314, 52 (1993).\n14FIG. 1: Upper bounds on the decoherence parameter γ0as a function of the power-law index n,\nas obtained from a combined analysis of solar and KamLAND dat a. The solid and dotted curves\nrefer to 2 σand 3σconfidence level, respectively.\n15FIG. 2: Energy profile of the (daytime) survival probability of8B neutrinos, averaged over their\nproduction region in the sun: Comparison of the effects produc ed by variations of sin2θ12for\nγ0= 0 (left panel) and by γ0∝negationslash= 0 at fixed sin2θ12(right panel). In the left panel, sin2θ12is varied\nwithin its ±2σlimits. In the right panel, for each index n= [−2,−1,0,+1,+2] the value of γ0is\ntaken equal to the corresponding 2 σupper limit (reported in Table I), which in units of 10−24GeV\ncorresponds respectively to: 0 .81×10−4(n=−2), 0.78×10−2(n=−1), 0.67 (n= 0), 0.58×102\n(n= +1), 0 .47×104(n= +2). The curve corresponding to standard oscillations ( γ0= 0) is also\nshown in the right panel as a guide to the eye. In all cases, δm2is fixed at its best-fit value.\n16FIG. 3: Constraints on the mass-mixing parameters for γ0= 0 (thin dotted curves) and for γ0\nfixed at its 2 σupper limit in Table I (thick solid curves). The three panels refer, from left to\nright, to the three cases n=−2, 0, and +2. The smaller (larger) allowed regions refer to th e\nsolar+KamLAND (solar only) data analysis.\n17FIG. 4: Energy profile of the auxiliary function gn(E), which modulates the exponent of the\ndamping factor induced by decoherence in solar neutrino osc illations. The function is multiplied\nby 103nfor a better graphical view. The shown function refers to a ne utrino produced at the Sun\ncenter and to best-fit oscillation parameters; altering thi s choice would induce minor variations.\nSee the text for details.\n18"    },    {        "title": "2307.12182v1.Damping_of_strong_GHz_waves_near_magnetars_and_the_origin_of_fast_radio_bursts.pdf",        "content": "Draft version July 25, 2023\nTypeset using L ATEXtwocolumn style in AASTeX63\nDamping of strong GHz waves near magnetars and the origin of fast radio bursts\nAndrei M. Beloborodov1, 2\n1Physics Department and Columbia Astrophysics Laboratory, Columbia University, 538 West 120th Street New York, NY 10027,USA\n2Max Planck Institute for Astrophysics, Karl-Schwarzschild-Str. 1, D-85741, Garching, Germany\nABSTRACT\nWe investigate how a GHz radio burst emitted near a magnetar propagates through its magnetosphere\nat radii r= 107-109cm. Bursts propagating near the magnetic equator behave as magnetohydrody-\nnamic (MHD) waves if they have luminosity L≫1040erg/s. The waves develop plasma shocks in each\noscillation and dissipate at r∼3×108L−1/4\n42cm. GHz waves with lower Lor propagation directions\ncloser to the magnetic axis do not obey MHD. Instead, they interact with individual particles, which\nrequires a kinetic description. The kinetic interaction quickly accelerates particles to Lorentz factors\n104-105at the expense of the wave energy, which again results in strong damping of the wave. In\neither regime of wave propagation, MHD or kinetic, the magnetosphere acts as a pillow absorbing the\nGHz burst and re-radiating the absorbed energy in X-rays. We conclude that a GHz source confined\nin the inner magnetosphere would be blocked by the outer magnetosphere at practically all relevant\nluminosities and viewing angles. This result constrains the origin of observed fast radio bursts (FRBs).\nWe argue that observed FRBs come from magnetospheric explosions ejecting powerful outflows.\nKeywords: X-ray transient sources (1852); Neutron stars (1108); Magnetars (992); Radiative processes\n(2055); Radio bursts (1339); Plasma astrophysics (1261)\n1.INTRODUCTION\nFast radio bursts (FRBs) are among the most mys-\nterious astrophysical phenomena. They are detected at\nGHz frequencies from large cosmological distances. The\nbursts have huge luminosities up to ∼1043erg/s and\nmillisecond durations (Petroff et al. 2019).\nThe short durations suggest that FRBs are generated\nby compact objects. In particular, magnetars are natu-\nral candidates, as they are well known as prolific X-ray\nbursters (Kaspi & Beloborodov 2017). Evidence for the\nmagnetar-FRB association came from the recent detec-\ntion of millisecond GHz bursts from SGR 1935+2154, a\nknown magnetar in our Galaxy (Bochenek et al. 2020;\nThe CHIME/FRB Collaboration et al. 2020), although\nthe bursts were weaker than the cosmological FRBs.\nThe radio bursting mechanism is not established (see\nLyubarsky (2021); Zhang (2022) for a review).\nUseful constraints on the FRB origin can be found\nby examining propagation of radio waves through the\nplasma magnetosphere surrounding magnetars. In par-\nticular, if the GHz source sits in the ultrastrong inner\nmagnetosphere, the observed emission must be able to\nescape through the surrounding outer magnetosphere.\nCan the radio wave actually escape?\namb@phys.columbia.eduA dangerous region for the wave is where the back-\nground dipole magnetic field Bbgdecreases to ∼E0(E0\nis the wave amplitude). In particular, calculation of the\nplasma response to a sine radio wave with E0> B bg\nshows strong damping (Beloborodov 2021, 2022a). The\nwave quickly accelerates plasma particles up to the ra-\ndiation reaction limit, and the particles radiate the re-\nceived energy in the gamma-ray band. Effectively, the\nplasma scatters the radio wave to gamma rays, and then\nits energy converts to an avalanche of e±pairs. This\ncalculation did not address how the oscillating wave\nreached the outer magnetosphere where Bbg< E 0, but\ndemonstrated that if it did, it would not survive.\nThe present paper investigates the full evolution of\nGHz waves emitted at small radii (where Bbg≫E0)\nand propagating to the outer region where Bbg< E 0.\nAt small radii the wave has no problem with propaga-\ntion — it is well described as a vacuum electromagnetic\nwave superimposed on the dipole background. This de-\nscription fails where Bbg/E0decreases to ∼1. Here, the\nelectromagnetic invariant B2−E2approaches zero, and\na dramatic transition occurs in the wave evolution.\nKinetic plasma simulations of this transition show\nthat the wave launches shocks in the background plasma\n(Chen et al. 2022). One can demonstrate the forma-\ntion of shocks and track their evolution using the MHD\nframework (Beloborodov 2022b, hereafter Paper I). The\nMHD description holds for waves of sufficiently low fre-arXiv:2307.12182v1  [astro-ph.HE]  22 Jul 20232\nquencies, and then the radio wave behaves as a compres-\nsive MHD mode, called “fast magnetosonic.” Paper I\nfocused on kHz magnetosonic waves and showed that\nthey evolve into monster radiative shocks, with Lorentz\nfactors exceeding 105.\nRemarkably, MHD description also holds for GHz\nwaves of sufficiently high power L > L MHD. As shown\nin the present paper, LMHD depends on the propagation\nangle θrelative to the magnetic dipole axis and falls\nin the range relevant for FRBs. In particular, near the\nmagnetic equator, the condition L > L MHD is satisfied\nby typical FRBs. We use this fact to solve for the wave\npropagation in the equatorial region. Then, we examine\nwaves around the magnetic axis, where LMHD > Land\na kinetic description is required. We find that in both\nregimes the GHz waves are nearly completely damped.\nOur method for tracking wave evolution in the MHD\nregime is described in Section 2. Section 3 presents nu-\nmerical results for GHz waves in the equatorial plane\nθ=π/2, and explains analytically the observed damp-\ning by shocks and radiative losses. Section 4 describes\nshock formation at polar angles θ̸=π/2. In Section 5,\nwe find the region on the L-θplane where waves do not\nobey MHD. In this region, wave damping switches to the\nkinetic regime and occurs through particle acceleration,\nas described in Section 6. The results are discussed in\nSection 7.\n2.RADIO WAVES IN MHD REGIME\n2.1. Formulation of the problem\nSuppose a GHz wave packet is emitted near the mag-\nnetar and then expands to larger radii rwhere the mag-\nnetosphere is initially unperturbed. We are interested in\nthe evolution of the packet at r∼108−109cm where E2\napproaches B2and the linear vacuum-like propagation\nends. These radii are still well inside the light cylinder\nRLC=c/Ω∼1010cm, so rotation of the magnetosphere\nis slow, Ω r≪c, and may be neglected. The magneto-\nsphere here may well be described as a dipole Bbg.\nThe unperturbed outer magnetosphere in front of the\nwave is populated with mildly relativistic electrons and\npositrons (their speeds are reduced by drag exerted by\nthe magnetar radiation, see Beloborodov (2013)). Thus,\nthe energy density of the background plasma is compa-\nrable to its rest mass-density ρbgc2. This density enters\nthe definition of the magnetization parameter,\nσbg≡B2\nbg\n4πρbgc2≈D\nr3, D =µ2\n4πNmc2,(1)\nwhere mis the electron mass, and µis the magnetic\ndipole moment of the magnetar. The dimensionless pa-\nrameter N ≡ r3ρbg/mis approximately constant with\nradius r; its typical expected value is N ∼ 1037(Be-\nloborodov 2020).\nWe will consider a wave packet far from its source. It\noccupies a thin shell δr/r≪1 and has a nearly radialwavevector k, so the packet behaves locally as part of\nan axisymmetric wave ( ∂ϕ≈0). Magnetosonic waves\nhave a toroidal electric field E∥k×Bbg, and we define\nE≡ −Eϕ, (2)\nusing the normalized basis er,eθ,eϕof the spherical\ncoordinate system r, θ, ϕ with the polar axis along the\nmagnetospheric dipole moment µ. Our calculation will\ntrack the propagation of the spherically expanding wave\npacket. As a concrete example, we will consider a radio\nwave launched with an initial sine profile\nE(ξ) =E0sin(ωξ), 0< ξ≡t−r\nc< τ. (3)\nThe packet has a short duration τ<∼1 ms, and we are\ninterested in its propagation at radii r≫cτ.\nAs long as the magnetospheric particles exposed to the\nwave remain magnetized, i.e. their Larmor frequency\nfar exceeds the wave frequency ω, the radio wave obeys\nMHD and can be thought of as a fast magnetosonic wave\n(the validity of MHD description is discussed in detail\nin Section 5). Particle motion in the MHD wave can\nbe thought of as the drift of the Larmor orbit. The e±\ndrift velocity β±has a charge-symmetric (MHD) com-\nponent βand a small antisymmetric component ±βp\n(polarization drift), which sustains the electric current\nj=enβp. There is no need for explicit calculations of\nthese drifts in response to the electromagnetic wave. In-\nstead, MHD describes the evolution of fields EandB\nby treating the plasma as a perfectly conducting fluid,\nwhich satisfies E+v×B/c= 0. The fluid is described\nby its velocity v=βcand mass density ρ=mn. The\nunperturbed static background corresponds to E= 0,\nB=Bbg,v= 0, and ρ=ρbg.\nAt small radii, where E0/Bbg≪1, the wave propa-\ngates without deformation. It has the speed vwave/c=\n1−σ−1\nbg≈1, and the MHD wave is equivalent to a vac-\nuum electromagnetic wave superimposed on Bbg. The\nlinear propagation ends where the linear superposition\nhits the condition E2=B2, which corresponds to v→c\n(Paper I). In the equatorial plane, this occurs at radius\nR×where E0=Bbg/2,\nR×=\u0012cµ2\n8L\u00131/4\n≈2.47×108µ1/2\n33\nL1/4\n42cm. (4)\nHere, L=cr2E2\n0/2 is the wave power.\nWe wish to find the nonlinear evolution of the electro-\nmagnetic wave as it crosses radius R×. Note that the\nevolution occurs at a very high magnetization parameter\nσbg. In particular, at R×one finds\nσ×≡σbg(R×)≈6.4×108µ1/2\n33L3/4\n42\nN37. (5)\nNumerical examples shown below will assume a mag-\nnetar with a typical magnetic dipole moment µ=\n1033G cm3and plasma density parameter N= 1037.3\n2.2. Nonlinear wave equation\nBefore describing the full problem of GHz waves in a\nhot plasma (heated by shocks), we start with waves in\na cold plasma. This gives a quick introduction to the\ncalculation method using characteristics.\nThe nonlinear evolution equation for magnetosonic\nwaves with a spherical wave front (far from the source)\nis derived in Paper I. At all polar angles θ, the wave\nexcites a pure toroidal current jwhile sustaining charge\ndensity ρe= 0. Plasma motion in the wave obeys the\nmomentum and energy equations,\nρc2du\ndt=j×B, ρc2dγ\ndt=E·j, (6)\nwhere u=γβ,γ= (1−β2)−1/2, and the time derivative\nis taken along the fluid streamline: d/dt =∂t+v· ∇.\nConservation of particle number (neglecting e±creation\nand annihilation) is stated by the continuity equation,\n∂αFα=∂tn+∇ ·(nv) = 0 , (7)\nwhere uα= (γ,u) is the fluid four-velocity, Fα= ˜nuα\nis the four-flux of particle number, and ˜ n=n/γis the\nproper density.\nWe are interested here in wave packets with many os-\ncillations and a short length cτ≪r. It is convenient to\nuse coordinates ( t, ξ, θ, ϕ ), so that the fast oscillation is\nisolated in the single coordinate ξ=t−r/c(and varia-\ntions with tandrat fixed ξare slow). The continuity\nequation in the short-wave limit gives\nFξ= (c−vr)n=const =cnbg, (8)\nand energy conservation can be cast into the following\nform (Paper I):\n∂t(r2E2) =−4πr2ρbgc2\u0014\n∂ξγ+r∂tγ+βθ∂θγ\nr(1−βr)\u0015\n.(9)\nHere, the derivative ∂tis taken at fixed ξ, θ(i.e. along\nthe radial ray r=ct+const ), and the derivative ∂ξ\nis taken at fixed t, θ. The second term in the square\nbrackets is small compared to ∂ξγunless γapproaches\nσ1/3\nbg(Paper I). This does not occur in GHz waves (which\ndevelop less extreme γcompared to kHz waves), as will\nbe shown below. Therefore, the energy equation for GHz\nwaves simplifies to\n∂t(r2E2) =−4πr2ρbgc2∂ξγ. (10)\nIt describes the coupled evolution of E(t, ξ, θ ) and\nγ(t, ξ, θ ) for the waves propagating in the MHD regime.\nWe now focus on waves in the equatorial plane ( θ=\nπ/2), assuming equatorial symmetry. Waves at different\npolar angles will be investigated in Section 4.2.3. Equatorial waves\nIn the equatorial wave, the plasma oscillates with a\nradial drift speed β=E×B/B2=βrer, since vθ= 0 by\nsymmetry. Besides E≡ −Eϕwe will use the following\nnotation:\nB≡Bθ, β ≡vr\nc=E\nB. (11)\nEquation (8) gives the plasma compression factor in\nshort waves, n= (1−β)nbg. The magnetic field is\nfrozen in the fluid and compressed by the same factor,\nn\nnbg=ρ\nρbg=B\nBbg= (1−β)−1. (12)\nAll MHD quantities in the equatorial wave can now be\nexpressed in terms of β, including the electric field,\nE=βB=βBbg\n1−β. (13)\nSubstituting Equation (13) into Equation (10) and using\ndγ=γ3βdβ, one obtains\n2σbg∂tγ\nγ3(1−β)3+∂ξγ=4c σbgβ2\nr(1−β)2. (14)\nA convenient MHD variable is the compression of\nproper density ˜ ρ=ρ/γrelative to its background value\nρbg,\nκ≡˜ρ\nρbg=˜B\nBbg=σ\nσbg=s\n1 +β\n1−β, (15)\nwhere ˜ ρand˜Bare measured in the fluid rest frame, and\nσ≡˜B2/4π˜ρc2. Equation (14) rewritten in terms of κ\nbecomes\n2σbgκ3∂tκ+∂ξκ=2c\nrσbgκ2(κ2−1). (16)\nUsing the method of characteristics, we express this\nequation as\ndκ\ndt\f\f\f\f\nC+=c\nr\u0000\nκ−κ−1\u0001\n, (17)\nwhere the derivative is taken along curves C+(charac-\nteristics) determined by the ratio of the coefficients of\n∂tξand∂tκin Equation (16),\ndξ+\ndt=1\n2σbgκ3. (18)\nThe characteristics ξ+(t) can also be described by their\nradial speed β+=c−1dr+/dt= 1−dξ+/dt.\nRecall that Equations (17) and (18) are obtained as-\nsuming cold plasma. Appendix A gives a more formal\nderivation using the stress-energy tensor of electromag-\nnetic field + plasma, and shows that Equation (17) also\nholds when the cold approximation is relaxed, i.e. the\nplasma is allowed to be relativistically hot. The shape of4\nC+characteristics in this more general case is described\nby\ndξ+\ndt=1\n2γ2sκ2, (19)\nwhere γsis the magnetosonic Lorentz factor defined in\nAppendix A. Its value in a hot plasma is given by (see\nAppendix A.5)\n1\nγ2s=1\nk2σ\u0014\n(k2−1)ε+1\nε3\u0015\n. (20)\nHere, ε= (˜ρc2+Up)/˜ρc2is the dimensionless spe-\ncific plasma energy (including rest-mass and thermal en-\nergy); k= 3 if the thermal motions of plasma particles\nare isotropic in the fluid frame, and k= 2 if the ther-\nmal velocities are confined to the plane perpendicular\ntoB. As shown below, the plasma is heated in shocks,\nwhich are mediated by Larmor rotation, and so heating\noccurs in the plane perpendicular to B. It is uncertain\nwhether the plasma becomes isotropic far downstream\nof the shock; therefore, we allow both possibilities k= 2\nandk= 3.\nFor waves in a cold plasma, ε= 1 and γ2\ns=σ=κσbg.\nIn this case, Equation (19) is reduced to Equation (18).\n2.4. Bending of characteristics\nShocks form because the C+characteristics in space-\ntime are bent from straight lines, leading to collisions\nbetween them. This bending is described by dξ+/dt̸= 0,\nand one can see from Equation (19) that it is strongest\nwhen κis small. Note that κ=γ(1+β) = [γ(1−β)]−1is\nsmallest where βapproaches −1 (i.e. the plasma drifts\nwith a maximum Lorentz factor γmaxtoward the star),\nwhich occurs where E2approaches B2.\nAs explained in Paper I (and in Section 3.3 below),\nγmaxandκmin≈(2γmax)−1are set by the ratio of the\nelectromagnetic energy in one wave oscillation, L/ν, to\nthe plasma rest mass in the magnetosphere, ∼4πNmc2.\nThis ratio scales with the wave frequency as ν−1, and\nhere one can see the first big difference between GHz and\nkHz waves: γmaxis much lower in GHz waves. In partic-\nular, γstays far below γs, and γsκ≫1 holds across the\nwave. This implies that the C+characteristics prop-\nagate with dξ+/dt≪1 (see Equation 19), i.e. their\nspeeds dr+/dtstay close to c. Thus, the bending of\ncharacteristics is a small parameter.\nThis feature of GHz waves allows one to easily find the\nevolution of κalong C+. Using dt= (1−dξ+/dt)−1dr/c\nanddξ+/dt≪1, we find from Equation (17),\ndκ\ndlnr=\u0000\nκ−κ−1\u0001\u0014\n1 +O\u0012dξ+\ndt\u0013\u0015\n. (21)\nThis equation implies that the radial dependence of κ\nalong each C+has the functional shape,\nκ=p\n1 + 2 Kr2, (22)where K=const . The constant Kis different on dif-\nferent C+and set by the initial profile of the wave. Us-\ningE=βB=βBbg/(1−β) and substituting β=\n(κ2−1)/(κ2+1) we obtain the solution for Ealong C+,\nE=µK\nr. (23)\nIt is the same as in a vacuum wave, E∝r−1. We\nconclude that the presence of plasma influences the GHz\nwave propagation by slightly changing the shape of C+\ncharacteristics while the evolution of E(r) along each\nC+remains unchanged from the vacuum solution.\nNote that the small bending of characteristics,\ndξ+/dt≪1, can strongly deform the oscillations with\nwavelength λ≪r. This occurs when the small deviation\nofC+from straight lines, δr+∼c(dξ+/dt)t≪r, reaches\na fraction of λ. Then, characteristics collide, forming a\ndiscontinuity of the MHD quantities – a shock.\n2.5. Coupling of wave evolution to thermal balance\nNext, we note another essential difference between\nkHz and of GHz waves. In kHz waves, the monster\nshocks have ultra-fast radiative losses. As a result, it\nturns out sufficient to use the cold approximation ε≈1,\nwhich gives γ2\ns≈σ=κσbg. By contrast, for GHz waves,\nthe plasma cooling time exceeds the wave oscillation pe-\nriod. This leads to accumulation of a large εalong the\nwave train. It affects γs, and the wave evolution becomes\ncoupled to the plasma thermal balance. Thus, the wave\nproblem requires a self-consistent solution for κ(t, ξ) and\nε(t, ξ). The evolution of ε(t, ξ) is governed by heating in\nshocks and synchrotron cooling, as described below.\n2.6. Shock heating\nThe plasma speed βis discontinuous at the shock, as\nthe upstream and downstream characteristics bring to\nthe shock different values of β:βu̸=βd(hereafter sub-\nscripts “u” and “d” refer to the immediate upstream\nand immediate downstream of the shock). The Lorentz\nfactor of the upstream plasma relative to the down-\nstream plasma is related to the shock compression factor\nq= ˜ρd/˜ρu=κd/κu(Appendix B),\nΓrel=γuγd(1−βuβd) =1\n2\u0000\nq+q−1\u0001\n. (24)\nIn Appendix B we describe the shock jump conditions\nand derive the plasma energy per unit mass immediately\ndownstream of the shock:\nεd=b+p\nb2−(3k−1)q4+ 2(k−1)q2+k+ 1\n(3k−1)q2+k+ 1,(25)\nb≡q\n2\u0002\n(k+ 1)q2+ 3k−1\u0003\nεu−q(q2−1)\n2εu,\nwhere k= 3 for isotropic plasma and k= 2 when par-\nticles are heated only in the plane perpendicular to B.5\nNote that radiative losses do not affect the shock jump\nconditions, as the plasma cools on a timescale much\nlonger than the Larmor time that sets the shock width\n(the opposite regime occurs in kHz waves, see Paper I).\nThe jump conditions also determine the shock speed,\n1−βsh=(k+ 1)εu−ε−1\nu−[(k+ 1)εd−ε−1\nd]/q\nk σbgκ3u(q2−1).(26)\nFor relativistic shocks with q≫1, this simplifies to\n1−βsh≈2(k−1)εd\nk σbgκ3uq3, ε d≈q[(k+ 1)εu−ε−1\nu]\n3k−1.(27)\n2.7. Radiative losses\nThe shock-heated plasma gradually loses energy to\nsynchrotron emission. Thermal evolution of the plasma\nbehind a shock obeys the first law of thermodynamics\nalong the fluid streamline:\ndε=−Ppd1\n˜ρc2+dεs=p dlnκ−dεs, (28)\nwhere Ppis the plasma pressure,\np≡Pp\n˜ρc2=1\nk\u0000\nε−ε−1\u0001\n, (29)\nmc2dεsis the energy loss due to synchrotron emission,\ndεs=σT\nmc˜B2\n2πk(ε2−1)d˜t, (30)\nand ˜dt=dt/γ is the proper time of the fluid element.\nWe here approximated the particle distribution function\nin the fluid frame as mono-energetic (each particle has\nthe Lorentz factor ˜ γe=ε). Then, using the relations\n˜B=κBbgandd˜t=κ dξalong the fluid streamline, we\nobtain the equation for ε(ξ),\ndε=ε2−1\nkεdlnκ−σTB2\nbgκ3(ε2−1)\n2πk mcdξ, (31)\nwhich can be integrated numerically along a given wave\nprofile κ(ξ).\nEnergy emitted in the lab frame is dEs=γ dεsmc2\nper particle. The number of particles passing through\nthe wave per unit time is 4 πr2(ρbg/m)c(we have multi-\nplied by 4 πr2to define the isotropic equivalent). Energy\nEsradiated per particle is distributed over ξin the wave\nasdEs/dξ=γ mc2dεs/dξ. This gives the following dis-\ntribution of the synchrotron power over ξ,\ndLs\ndξ=σTµ4γκ3(ε2−1)\n2πk mD r7. (32)\nSolutions for MHD quantities along characteristics de-\nscribed by Equations (17) and (19), with local γscalcu-\nlated under the adiabatic assumption (Equation 20), be-\ncome inaccurate if the plasma radiates a significant frac-\ntion of its energy εduring one wave oscillation. We willmonitor for this condition, which limits the applicabil-\nity of our simulation method. Note that Equations (17)\nmay hold even when radiative losses have a strong net\neffect on the wave. For instance, the GHz wave packet\nwith 30,000 oscillations (simulated below) eventually\nloses most of its energy to synchrotron emission, however\nit is approximately adiabatic in each oscillation. Radia-\ntive losses impact εthat enters Equation (19) through\nγs, however the local fast-magnetosonic speed remains\napproximately adiabatic and determined by the local ε\naccording to Equation (20).\n2.8. Numerical implementation\nOne advantage of using characteristics, compared with\ngrid-based MHD solvers, is the ability to track waves in\na plasma with any large magnetization parameter σ. In\naddition, low computational costs of tracking charac-\nteristics allows one to follow radio bursts with a large\nnumber of oscillations N.\nThe calculation starts at small radii where Bbgfar\nexceeds the wave electric field E, and the plasma os-\ncillates with small |β| ≪1, which implies a negligible\nmodulation of plasma density, |κ−1| ≪1. In this in-\nner zone, the C+characteristics propagate with speed\nβ+= 1− O(σ−1\nbg), and so each characteristic keeps a\nconstant coordinate ξ=t−r/c=ξi. The initial wave\nprofile E(ξi) is well defined in this inner zone of nearly\nvacuum propagation with E∝r−1. It is conveniently\ndescribed by the function,\nK(ξi)≡rE\nµ=K0sin(ωξi), K 0≡rE0\nµ. (33)\nWhen launching the wave, we set up an initially uniform\ngrid in ξiof size N+, and then use the N+characteristics\nto track the wave evolution. Typically, 500 characteris-\ntics per wave oscillation are sufficient (convergence has\nbeen verified by varying N+).\nAt each timestep dt, the displacement dξ+of each\ncharacteristic ξiis determined by dξ+/dt(Equation 19),\nwhich is controlled by the evolving values of κ(t, ξi) and\nε(t, ξi) onC+. The compression κ(t, ξi) evolves accord-\ning to Equation (17), and the plasma specific energy\nε(t, ξi) is found by integrating the ordinary differential\nequation (31) in ξwhen scanning though the array of\nN+characteristics.1The downstream energy εdof each\nshock is found from the jump conditions (Section 2.6).\nThe propagation speed of each shock is determined by\nEquation (26).\nAfter each timestep, the code examines the updated\npositions or the characteristics and any existing shocks,\n1Recall that we consider short wave packets, so that the plasma\ncrosses the wave faster than the wave evolves. Then, the profile\nofε(ξ) can be calculated at fixed t=const . This approximation\nloses accuracy in low-power waves, which have extremely fast\nevolution at R×(the model with L= 1040erg/s shown below),\nhowever this weakly affects the final result.6\nand determines which characteristics terminate at the\nshocks. The code also constantly watches for any new\ncrossings of characteristics to detect formation of new\nshocks. We use an adaptive timestep to resolve any fast\nevolution in MHD quantities near R×. We have also im-\nplemented sub-timesteps in the leading oscillation of the\nwave, which is coldest and reaches the lowest κ, leading\nto more demanding timestep requirements. Note also\nthat the density of characteristics dN+/dξdrops ahead\nof shocks, where κis lowest and the high dξ+/dtresults\nin stretching the array of C+inξ. To maintain suffi-\ncient spatial resolution everywhere in the wave, we use\nadaptive mesh refinement in ξiwithout changing the to-\ntal number of 500 active (not terminated) characteristics\nin each oscillation. These technical tricks allow one to\nsignificantly speed up the simulation and trace the evo-\nlution of long wave trains. Sample wave trains presented\nbelow have N= 3×104oscillations, traced on a grid\nwith N+= 500 N= 1.5×107characteristics.\nThe wave evolution should conserve the total energy\n(electromagnetic + plasma + synchrotron losses), which\nprovides a simple test. The simulations passed this test.\n3.SIMULATION RESULTS\nWe have calculated the propagation of waves with fre-\nquency ν=ω/2π= 0.3 GHz and duration τ= 0.1 ms.\nTwo models are presented below: waves with initial\npower L= 1042erg/s and L= 1040erg/s. Figure 1\nshows the evolution of the wave power with radius found\nin the two simulations.\n3.1. Model I: L= 1042erg/s\nAs one can see in Figure 1, the wave experiences strong\ndamping near radius R×≈2.5×108cm. The devel-\nopment of shocks in each oscillation results in plasma\nheating and synchrotron losses, reducing the wave en-\nergy by a factor of ∼10 between R×and 2 R×. Then,\nthe electromagnetic packet evolves into a smooth, al-\nmost uniform, Poynting flux with unchanged duration\nτ= 0.1 ms and strongly damped oscillations. The wave\npower Lνthat is carried by the alternating component\nof the electromagnetic field with frequency νis reduced\nbelow 10−3of its original value.\nThe evolution of the wave profile is shown in Figure 2.\nShowing the entire profile with 3 ×104oscillations would\nbe impractical, so we limited the figure to the first 20\noscillations in the packet; this is sufficient to see the evo-\nlution. It has three phases:\n(1) Cold oscillations at r < R ×.As the plasma flows\nthrough the wave, it performs N=τνsmall-amplitude,\nharmonic oscillations with frequency ν, and exits behind\nthe packet. This simple behavior ends when the packet\nreaches rc≈0.998R×. Then, caustics form, launch-\ning shocks in each oscillation. The shocks appear at\nthe oscillation phases where Eis close to its minimum\n(E≈ −E0) and κ≪1.\n(2) Main dissipation phase at r>∼R×.The shocks\nFigure 1. Evolution of the wave power Lwith radius r, for\ntwo wave packets (with initial L= 1042erg/s and 1040erg/s).\nBoth waves have frequency ν= 0.3 GHz. Total Poynting flux\nL(isotropic equivalent) is shown by black curves, and its os-\ncillating component Lνis shown by blue curves. The calcula-\ntions have been performed assuming k= 3 (isotropic plasma,\ndotted curves) and k= 2 (thermal motions perpendicular to\nB; solid curves). Shocks form at radius rcindicated by the\nred dot; it is slightly smaller than R×(see text).\nreach a maximum strength at r≈1.1R×. The shock\nin the first oscillation of the wave train is strongest (be-\ncause it has a cold upstream), reaching the compression\nfactor q=κd/κu∼102(Figure 3). Subsequent shocks\ndown the wave train occur in the plasma already heated\nin the leading shocks; therefore, they have smaller q.\nFigure 5 shows the evolution of a typical shock located\n1000 oscillations away from the leading edge of the wave.\nThe oscillation of compression factor κ(Figure 3)\nmodulates the plasma temperature by adiabatic heat-\ning/cooling and, in addition, there is dissipative heating\nat each shock. In general, dissipation breaks periodic-\nity: as the plasma moves through the wave train, it can\naccumulate heat εdissgained in each of the Nshocks.\nHowever, synchrotron losses offset the gradual growth\nofεand make the wave train approximately periodic,\nwith εoscillating about a flat ¯ ε(Figure 4). The value of\n¯ε≈102is determined by the heating = cooling balance,\nas described in Section 3.3.2 below.\n(3) Evolution toward a uniform Poynting flux at\nr≫R×.By the time the packet reaches r= 2R×the\nshocks have erased the low- κregions, and the plasma\noscillations now have γ∼1 throughout the wave. The\nshock strength becomes sub-relativistic, Γ rel∼1. The7\nFigure 2. Evolution of the wave profile E(ξ) in Model I ( L=\n1042erg/s, ν= 0.3 GHz, isotropic plasma). The snapshots\nwere taken when the packet reached r/R×= 0.9, 1.5, 2.6,\nand 29. For clarity only the leading 20 oscillations are shown.\nElectric field Eis normalized to the amplitude Evac\n0that the\nwave would have if it propagated in vacuum. Evac\n0→E0?\nFigure 3. Evolution of the plasma compression profile κ(ξ)\nin Model I (same model and snapshot times as in Figure 2).\nThe vertical jump observed in each oscillation is a shock.\nelectric field Eoscillates with a decreasing amplitude\nFigure 4. Evolution of the plasma internal energy ε(ξ) in\nModel I (same snapshot times as in Figure 3). The larger\nnumber (200) of oscillations are shown to demonstrate the\nheating by the shock train and the saturation of εwhen\nsynchrotron cooling offsets shock heating in each oscillation.\nabout a positive average value ¯E∼0.3E0. Atr>∼30R×,\nE(ξ) becomes nearly uniform across the entire wave.\nWe conclude that the oscillating GHz wave is absorbed\nin the magnetosphere. Part of its energy and momentum\n(∼10%) is used to eject the outer magnetospheric lay-\ners, forming the Poynting flux that continues to expand\nfreely. Most of the absorbed wave energy ( ∼90%) con-\nverts to synchrotron emission from the heated plasma.\nThe simulation also demonstrates the gradual steepen-\ning of the wave profile at the leading edge of the packet.\nThis leads to the formation of a strong forward shock\nwhen the packet reaches RF≈7×108cm. It is consis-\ntent with the analytical expectation (Paper I):\nRF=\u00128cσ×\nωR×\u00131/6\nR×≈7×108cm. (34)\nThe forward shock is the leading edge of the Poynt-\ning flux ejected from the magnetosphere, i.e. the wave\npacket effectively has become an ultrarelativistic blast\nwave that continues to expand into the external medium.\nThe blast has thickness cτand carries ∼10% of the orig-\ninal wave energy.\n3.2. Model II: L= 1040erg/s\nThe main difference of the low-power wave is seen from\nFigure 1: it is damped in a narrow range of radii δr\nwhen the wave packet approaches R×≈7.8×108cm.\nShocks and damping develop somewhat before R×, at8\nFigure 5. Evolution of a typical shock (measured 1000 os-\ncillations from the leading edge of the wave train) in Model I.\nAll curves begin at the shock formation radius rc≈0.998R×.\nThe shown parameters are the shock compression factor q,\ndownstream specific energy εd, and specific dissipated energy\nεdiss. In addition, the figure shows Larmor frequency of the\nplasma particles ωLnormalized to the wave frequency ω;ωL\noscillates in the wave and the figure shows the evolution of\nits minimum value.\nrc≈0.943R×. The final outcome is that about 95% of\nthe wave energy is radiated away in synchrotron X-rays,\nand the remaining 5% forms a smooth Poynting flux of\nduration τ, with no GHz oscillations.\nEvolution of the wave profile is qualitatively similar\nto Model I. There are two main quantitative differences:\n(1) The shocks are weaker (Figure 6), which implies a\nlower dissipated fraction εdiss/εdin each shock. (2) The\nplasma passing through the wave develops a much higher\ntemperature ( εreaches ∼2×103, see Figure 7). This oc-\ncurs because dissipation takes place at a larger R×where\nthe magnetic field is weaker and synchrotron cooling is\nmuch slower. This leads to εdiss∼200 during the main\ndissipation phase, much higher than in Model I (com-\npare Figures 8 and 5). Therefore, the wave damping\noccurs much faster.\nOur numerical method loses accuracy when radiative\nlosses during one oscillation become comparable to the\nplasma energy (then the speed of characteristics βssig-\nnificantly deviates from its adiabatic value). This does\nnot occur in Model II. Model I is less accurate, in par-\nticular near the dissipation onset, when the plasma ra-\ndiates in one oscillation ∼30% of its thermal energy, in\nbalance with heating εdiss/εd∼0.3 (Figure 5).\nFigure 6. Oscillations of the plasma compression κ(ξ) in\nModel II ( L= 1040erg/s, ν= 0.3 GHz, isotropic plasma).\nThree snapshots are shown, when the wave packet reached\nr/R×= 0.946R×, 1, 2.6. The figure shows the leading 100\noscillations (the simulated packet has 3 ×104oscillations).\nFigure 7. Evolution of the plasma internal energy ε(ξ) in\nModel II (same snapshot times as in Figure 6). The larger\nnumber (600) of oscillations are shown to demonstrate the\nheating by the shock train and the saturation of εwhen\nsynchrotron cooling offsets shock heating in each oscillation.9\nFigure 8. Evolution of a typical shock (measured 1000 oscil-\nlations from the leading edge of the wave train) in Model II.\nThe shocks appear at rc≈0.943R×, and the figure focuses\non the narrow range δrwhere the wave energy is dissipated.\nThe shown parameters are the shock compression factor q,\ndownstream specific energy εd, and specific dissipated energy\nεdiss. In addition, the figure shows Larmor frequency of the\nplasma particles ωLnormalized to the wave frequency ω;ωL\noscillates in the wave and the figure shows the evolution of\nits minimum value.\n3.3. Analytical description\nThe wave behavior shown by the numerical models\ncan be understood analytically and described by approx-\nimate formulae. An important dimensionless parameter\nof the problem is\nζ≡ωR×\nc σ×=π2mc2Nν\nL≈8×10−2N37L−1\n42ν9.(35)\nAs shown in Paper I, the plasma exposed to waves with\nζ≪1 develops Lorentz factor γ∼ζ−1atr∼R×. The\nparameter ζis tiny in powerful kHz waves investigated\nin Paper I, and GHz waves have less extreme ζ. Our\nfirst sample model ( ν= 0.3 GHz and L= 1042erg/s)\nhasζ≈2.4×10−2. The second model ( ν= 0.3 GHz and\nL= 1040erg/s) has ζ >1, which creates only mildly rel-\nativistic plasma motions in the wave. Note also that the\nplasma speed is related to compression κ(Equation 15),\nwhich implies γ= (κ2+ 1)/2κ.\n3.3.1. Shock formation\nThe caustic forms quickest on characteristics with\nK < 0, which develop the smallest κ. The first shockforms in the first (leading) oscillation in the initially\ncold plasma. It occurs at coordinate ξcwith plasma\ncompression factor κcat time tc, all of which can be\nderived analytically. For waves with ζ≪1 the result is\n(Paper I)\nκc≈√ζ\n241/4, ωξ c≈3π\n2+16√ζ\n243/4+c\nR×ω,(36)\nctc\nR×−1≈ −ζ\n3√\n24+8√ζ\n243/4c\nR×ω+c2\n4R2\n×ω2.(37)\nNote that ζ≫c/ωR ×in the GHz waves. In particu-\nlar, for the 0.3-GHz wave with L= 1042erg/s, we find\nctc/R×−1≈ −ζ/3√\n24≈ −1.6×10−3, in agreement\nwith the numerical simulation. Note also that Model II\nis in the opposite regime, ζ >1. In this case, the caustic\nforms without a strong drop in κ.\nSimilar shocks develop in each oscillation of the wave\ntrain and eventually dissipate the wave energy into heat,\nmost of which is radiated away.\n3.3.2. Thermal balance in the shock-heated wave train\nEach shock in the wave train heats the plasma pass-\ning through the wave. Without synchrotron cooling,\nthe plasma specific entropy would monotonically grow\nwith ξfrom the leading edge of the shock train toward\nits end. This growth occurs over many oscillations and\ngradually pushes the specific energy εto so high values\nthat synchrotron cooling Q∝ε2becomes important.\nIndeed, one can see in Figures 4 and 7 that at large\nξthe growth of ε(ξ) stops. Thus, the plasma passing\nthrough the wave enters a thermal balance: shock heat-\ning in each oscillation is offset by synchrotron cooling.\nThe thermal balance may be stated by equating the\nspecific energy dissipated at the shock εdissto the syn-\nchrotron losses (Equation 31) integrated over one oscil-\nlation,I\n2π/ωσTB2\nbgκ3(ε2−1)\n2πk mcdξ=εdiss. (38)\nThe shock dissipates energy εdiss=εd−εad, where εdis\ngiven by Equation (25) and εad=qα−1εuaccounts for\nadiabatic heating by shock compression qwith adiabatic\nindex α. The plasma flowing through the train of many\nshocks sustains ε≫1 and α= 1 + k−1. Equation (25)\nforεdthen simplifies, and we find its dissipation part:\ng≡εdiss\nεd= 1−[(3k−1)q2+k+ 1]q1/k\n(k+ 1)q3+ (3k−1)q(εu≫1).\n(39)\nAt large q≫10,gapproaches unity. In the opposite\nweak-shock limit, expansion in q−1 gives\ng≈(3k−1)(k2−1)\n12k3(q−1)3(q−1≪1).(40)10\nSynchrotron losses peak downstream of each shock,\nand we estimate the thermal balance (38) as\nσTB2\nbgκ3\ndε2\nd\n2πk mc ω∼εdiss. (41)\nThis gives\nεd∼g(q)2πkmc ω\nσTB2\nbgκ3\nd. (42)\nThis balance determines εthat shocks with compression\nfactors qcan sustain against radiative losses. Note that\nεdissis a small fraction of εdforq <5, and that εos-\ncillates in the wave within a modest range εu<∼ε<∼εd\n(Figures 4 and 7).\nIn Equation (42), one can use B2\nbg=B2\n×/x6where\nx=r/R×,B2\n×=µ2/R6\n×=µ−1(8L/c)3/2, and Lis the\ninitial power of the wave (before the dissipation). Then,\nwe obtain\nεd∼1.7kgx6\nκ3\ndmc5/2νµ\nσTL3/2. (43)\nMain dissipation occurs near R×, atx≈0.94-0.95 for\nL= 1040erg/s and x≈1.3-1.5 for L= 1042erg/s. The\ntwo waves also have different shock strengths, q≈2 and\n5, which give kg≈0.2 and 1.2, respectively. The value\nofκd∼1.5-1.8 is close to the peak κin the oscillation.\nUsing these values, one can see that the estimate (43)\nwell explains εdobserved in the simulation: we find εd∼\n200 for L= 1042erg/s (Figure 5), and εd∼3×103for\nL= 1040erg/s (Figure 8).\n3.3.3. Duration of the main dissipation phase\nAs one can see in Figure 1, the main dissipation phase\nof the powerful wave, L= 1042erg/s, extends over a sig-\nnificant range of radii δr∼R×. By contrast, the weaker\nwave with L= 1040erg/s dissipates quite suddenly near\nR×, in a narrow range δr≪R×.\nThe length δris related to the number of particles\npassing through the wave, δN ∼ 4πN(δr/r). This num-\nber (and hence δr) can be estimated using energy con-\nservation. The wave energy contained in one oscillation\nisL/ν, and each particle receives energy εdissmc2from\nthe shock. Hence, the number of particles it takes to\ndamp the wave is\nδNdamp∼L\nεdissmc2ν=L\ngεdmc2ν, (44)\nand the corresponding damping length is\nδrdamp\nr∼κ3\ndσTL5/2\n20kg2x6m2c9/2µNν2, (45)\nwhere we substituted εdfrom Equation (42). In particu-\nlar, for the wave with L= 1040erg/s this estimate gives\nδrdamp/r∼2×10−3, consistent with the simulation re-\nsults (Figure 8). Equation (45) holds if δrdamp/R×≪1.\nNote that δrdamp/R×grows with Land saturates at ∼1\nfor waves with large L(as in Model I) or low ν.3.3.4. Validity of MHD description\nThe MHD description of waves fails when the particles\nbecome unmagnetized, i.e. their Larmor timescale be-\ncomes comparable to the fluid dynamical timescale mea-\nsured in the fluid rest frame. The demagnetization can\nhappen in kHz waves when they accelerate the plasma to\nhuge Lorentz factors (Paper I). In GHz waves, the fluid\nLorentz factors are modest, and demagnetization may\noccur for a different reason: the ultra-relativistic tem-\nperature of the plasma increases its Larmor timescale.\nIn both presented simulations, the MHD requirement\nωL≈ωB/ε≫ωholds throughout the evolution of the\nwave. The ratio ωL/ωis shown in Figures 5 and 8. For\nwaves with lower power L <1040erg/s and the same ν=\n0.3 GHz the condition ωL≫ωwould become violated.\nOne can estimate ωL≈eBbg/εmc during the main\ndissipation phase using ε∼εdgiven by Equation (43):\nωL\nω∼83/4κ3\nd\n6πgx9eσTL9/4\nm2c17/4µ3/2ν2, (46)\nwhere we used Bbg=µ(xR×)−3=x−3µ−1/2(8L/c)3/4.\nIn particular, for the wave with ν= 0.3 GHz and L=\n1040erg/s (Model II) with isotropic plasma ( k= 3),\none can substitute the numerical factors g∼0.1 and\nx9∼0.6 evaluated above. The result is approximately\nconsistent with the minimum ωL/ω∼7 observed in the\nsimulation (Figure 8).\n4.SHOCK FORMATION OUTSIDE THE\nEQUATORIAL PLANE\nConsider a spherical wave front expanding through\nthe dipole magnetosphere. It has a radial wavevector k.\nNote that Bbg⊥konly in the equatorial plane θ=π/2.\nThe angle αbetween Bbgandkis given by\ntanα=Bθ\nbg\nBr\nbg=1\n2tanθ. (47)\nShock formation is expected at radius r×(θ), which can\nbe found from the condition E2=B2with EandB\nevaluated for the wave propagation in vacuum (Paper I):\nr×(θ) =R×\u00124−3 sin2θ\nsinθ\u00131/2\n=R×sin1/2θ\nsinα.(48)\nThe density parameter of the magnetosphere at θ̸=\nπ/2 is defined similarly to that in the equatorial plane,\nNθ≡r3nbg(r, θ). (49)\nWe have added subscript “ θ” to highlight a possible vari-\nation of density with θ. It may be estimated assuming\napproximately uniform e±loading in the inner magneto-\nsphere, where Bbg∼1013G (Beloborodov 2013, 2020).\nThe created pairs outflow along Bbgwith mildly rela-\ntivistic speeds and annihilate when they approach the\nequatorial plane. This picture implies\nnbg∝Bbg,Nθ\nN=sinθ\nsinα. (50)11\n4.1. Plasma velocity profile in the wave\nThe simplicity of equatorial waves described in the\nprevious sections is due to the simple relation between\nthe wave electric field Eand the plasma speed β, which\nallows one to formulate the energy equation (10) for a\nsingle unknown function. In the equatorial wave, the\nplasma executes the radial E×Bdrift with β=E/B =\nE/(Bbg+E). By contrast, outside the equatorial plane\nthe oblique Bbgimplies that the wave will move the\nplasma in both randθ. Furthermore, in addition to the\nE×Bdrift, the plasma can slide along the oblique B.\nDespite this complication one can still find analytically\nthe relation between βandEacross the wave profile.\nThe MHD condition E2< B2implies existence of a\n“drift frame” ˜K(moving with velocity βD=E×B/B2)\nin which ˜E= 0. In this frame, the plasma has a pure\nsliding motion along ˜Bwith some speed ˜βand Lorentz\nfactor ˜ γ= (1−˜β2)−1/2. Transformation of the plasma\nfour-velocity uα= (γ,u) from frame ˜Kto the lab frame\ngives\nγ= ˜γγD,u=˜u+ ˜γuD, (51)\nwhere\nγD=B√\nB2−E2,uD=E×B\nB√\nB2−E2, (52)\nandB=p\nB2r+B2\nθ. Equation (51) shows the decompo-\nsition of uinto components parallel and perpendicular\ntoB:u∥=˜uandu⊥= ˜γuD.\nThe relation between the plasma Lorentz factor γ=\nγD(1+ ˜u2)1/2and the wave electric field Ewill be found\nif we solve for ˜ u(E). This can be done using\nd\ndt(˜uB) =d\ndt(u·B) =u·dB\ndt≈uθdBθ\ndt, (53)\nwhere the derivative d/dtis taken along the fluid stream-\nline, and we used B·du/dt= 0 (implied by Equation 7).\nThe last (approximate) equality in Equation (53) makes\nuse of dBr/dt≪dBθ/dt, which holds for short waves.\nIndeed, the derivative dB/dt≈dBw/dtis dominated\nby the fast oscillation of the wave field Bw≡B−Bbg\non top of the slowly varying Bbg, and Bwin a short\nwave satisfies2\nBr\nw≪Bθ\nw≈E. (54)\nRelations BdB/dt ≈BθdBθ/dtand ˜uθ= ˜uBθ/Bgive\n˜udB\ndt≈˜uθdBθ\ndt, (55)\n2Vector potential Awfor a short axisymmetric wave satisfies\n∂ξAw≫∂tAw\f\f\nξwhich implies Br\nw≪Bθ\nwandBθ\nw≈E(Paper I).\nThe small difference Bw−Eis not negligible only in terms with\nthe large derivative ∂ξ; in particular, ∂ξ(E−Bw) =∂tBw|ξ+E/r,\nas follows from induction equation ∂tE=−c∇ ×Bw.and Equation (53) becomes\nBd˜u\ndt= ˜γuθ\nDdBθ\ndt=−˜γ EBr\nbg\nB√\nB2−E2dE\ndt. (56)\nThis gives a differential equation for ˜ u(E):\nd˜γ\n˜u=d˜u\n˜γ=−EBr\nbgdE\n(B2\nbg+ 2Bθ\nbgE+E2)q\nB2\nbg+ 2Bθ\nbgE.\n(57)\nIt can be integrated with Bbg≈const across the short\nwave profile,\nln (˜u+ ˜γ) =I≡ −ZE/B bg\n0s(α, z)dz, (58)\nwhere z=E/B bg, tan α=Bθ\nbg/Br\nbg, and\ns≡EBr\nbgBbg\nB2√\nB2−E2=zcosα\n(1 + 2 zsinα+z2)√1 + 2 zsinα.\n(59)\nNote that z→ − (2 sin α)−1corresponds to E2→B2.\nIn this limit, we find\n˜γ=1\nsinαwhen E2→B2. (60)\nThe obtained ˜ u(E) determines γ(E) andu(E) accord-\ning to Equation (51). This solution describes the plasma\nvelocity profile in a short wave.\n4.2. Shock formation\nAs long as the plasma oscillating in the wave stays\ncold (i.e. until shock formation), the wave evolution\nobeys Equation (10), which states ( ∂tE)ξin terms of\n(∂ξγ)t. One can use the relation between Eandγfound\nin Section 4.1 to express ∂tEin terms of ∂tγ, and then\nEquation (10) becomes a differential equation for γ(t, ξ),\n∂tγ\nf+ 2πρbgc2∂ξγ=2cE2\nr, (61)\nwhere function f(E,Bbg) is given by (see Appendix C),\nf≡(B2\nbg+Bθ\nbgE)γ3−˜uBr\nbgBγ2\nB4˜γ2. (62)\nEquation (61) is the generalization of Equation (14) to\npolar angles θ̸=π/2. It contains θas a parameter and\nwill show how the wave develops caustics at each θ. Its\ncharacteristics ξ+(t) are determined by the coefficients\nof∂tγand∂ξγ,\ndξ+\ndt= 2πρbgc2f, (63)\nand the evolution of γalong characteristics is given by\ndγ\ndt\f\f\f\f\nC+=2cE2\nrf. (64)12\nIn the equatorial plane, these equations reproduce the\nresults of the previous sections. Then, f=Bbgγ3/B3=\n(κ3B2\nbg)−1anddξ+/dtis reduced to Equation (18) while\n(dγ/dt )C+becomes equivalent to Equation (17) (taking\ninto account the relation γ= (κ2+ 1)/2κ).\nThe evolution equation for rEalong C+has the form\n(see Appendix C),\n1\nEd(rE)\ndt\f\f\f\f\nC+=O\u0012γ3\nσbg\u0013\n. (65)\nThus, waves with γ3≪σbgsatisfy rE≈const along the\ncharacteristics. This feature was demonstrated in the\nprevious sections at θ=π/2, and it also holds outside\nthe equatorial plane.\nShocks appear at caustics in the flow of characteristics.\nThe caustic location rc(θ) and the plasma Lorentz factor\nat the caustic γc(θ) are calculated in Appendix C. The\ncalculation can be completed analytically for waves with\nζ≪1 and ζ≫1. In these two limits, we find\nrc\nr×≈\n\n1 ζ≪1\u00128\nζsin3α\nsinθ\u00131/6\nζ≫1(66)\nγc≈\n\n p\n3/2\nζsinθsinα!1/2\nζ≪1\n1 ζ≫1(67)\nwhere α(θ) is given by Equation (47), and the parameter\nζis defined in the equatorial plane (Equation 35).\n5.TRANSITION TO THE KINETIC REGIME\n5.1. Shock heating and de-magnetization of particles\nShock heating of the plasma to ε≫1 reduces the Lar-\nmor frequency ωL=ωB/ε=eBbg/mcε . IfωLbecomes\ncomparable to ω, the MHD description of the wave fails.\nThen, a kinetic description will be required, where parti-\ncles individually interact with the wave. In Section 3.3.4\nwe estimated ωL/ωin the equatorial waves. Below we\nevaluate this ratio in waves outside the equatorial plane,\nand find the boundary between the two damping regimes\n(kinetic vs. MHD shocks) on the L-θplane.\nThe plasma internal energy εis controlled by the bal-\nance between synchrotron cooling (averaged over one\noscillation) and shock heating. This thermal balance is\ndescribed in Section 3.3.2 for waves in the equatorial\nplane, and a similar balance can be stated at θ̸=π/2.\nIn particular, we can use Equation (42) to evaluate ε.\nThen, we find\nωL\nω∼eσTB3\nbg\n6πmc2ω2g, (68)\nwhere the numerical factor g=εdiss/εddepends on\nthe shock compression factor q. For the perpendicularshocks at θ=π/2,g(q) is stated in Equation (39). The\nchoice of k= 2 vs. k= 3 weakly affects the numerical\ncoefficient in Equation (68); for definiteness, we will as-\nsume isotropic plasma ( k= 3). Note also that g(θ)≈1\nholds for waves with ζ≪1, which develop relativistic\nshocks during the main damping phase.\nThe plasma response to the wave may be described\nas MHD drift if the full Larmor period 2 π/ω Lis shorter\nthan the timescale for a large change of the field, which\nwe take as 1/4 of the wave period. Thus, we roughly es-\ntimate that the transition between the MHD and kinetic\nregimes occurs at ωL/ω∼4. This corresponds to\nBbg∼BMHD≈4\u0012\ngm2c2ω2\neσT\u00131/3\n≈2×107g1/3ν2/3\n9G.\n(69)\nThe same condition may be stated as\nωB∼2\u0012\ngc\nreω2\u00131/3\n, (70)\nwhere re=e2/mc2. Damping of the wave through MHD\nshocks begins when caustics form (at radius rc(θ) given\nby Equation (66)), and most of the damping occurs at\nr∼rc. The approximate condition for the wave to be\ndamped in the MHD regime at a given polar angle θis\nBbg(rc)> B MHD (shock damping) . (71)\nIn particular, waves with ζ <1 have rc≈r×(Equa-\ntion 66) and g∼1, and then we find\nBbg(rc)\nBMHD∼\u0012L\nc\u00133/4sin2α√µsinθ\u0010eσT\nm2c2ω2\u00111/3\n,(72)\nwhere we used Equation (48) for r×. One can see that\nthe MHD regime Bbg(rc)> B MHD holds at a given angle\nθif the wave power Lexceeds a critical value LMHD,\nLMHD(θ)∼cµ2/3\u0012m2c2ω2\neσT\u00134/9sin2/3θ\nsin8/3α(ζ<∼1)\n∼2×1040µ2/3\n33ν8/9\n9sin2/3θ\nsin8/3αerg\ns. (73)\nAt small polar angles θ≪1, one can use α≈θ/2 and\nsee that LMHD∝θ−2. The MHD damping condition\nL > L MHD(θ) may also be written as a condition on θ\nfor a given L:θ > θ MHD(L). For example, waves with\nfrequency ν= 1 GHz and power L= 1042erg/s will\nexperiences shock damping in the MHD regime at polar\nangles θ > θ MHD∼0.3 (Figure 9).\n5.2. Charge starvation?\nAnother condition for wave propagation in the MHD\nregime is a sufficiently high plasma density, n > j/ec ,\ncapable of sustaining electric current jdemanded by13\nFigure 9. Two regions in the L-sinθparameter space: wave\ndamping in the MHD regime (through shocks) and in the ki-\nnetic regime (through stochastic particle acceleration). Their\napproximate boundary LMHD(θ) [orθMHD(L)] is plotted here\nfor waves with frequency ν= 1 GHz propagating in the mag-\nnetosphere with dipole moment µ= 1033G cm3. The scaling\nofLMHD(θ) with νandµis given in Equation (73).\nMHD. The electric current in an axisymmetric wave is\ntoroidal, j= (0,0, jϕ), and satisfies the relation,\n−Ejϕ=ρc2dγ\ndt=ρc2(1−βr)dγ\ndξ. (74)\nFor waves with γ3≪σbgone can use dγ/dξ ≈∂ξγ, so\njϕ\nenc≈ −mc(1−βr)\neE∂ξγ. (75)\nIn particular, consider waves in the equatorial plane\nwith ζ≪1. Large ∂ξγdevelops at caustic formation\nwith β≈ −1 and γgrowing to γc≈241/4ζ−1/2in the\nnarrow interval δξ∼ζ1/2/ω, which gives\n|∂ξγ| ∼241/4ω\nζ. (76)\nA similar gradient of γis sustained later ahead of the\ndeveloped shock, when the upstream γapproaches its\nmaximum γ∼ζ−1while the width of the pre-shock\nacceleration region grows to δξ∼ω−1(see Paper I).\nUsing Equation (76), we obtain a rough estimate for the\ncurrent density during the MHD evolution of the wave:\nj\nenc∼10\nωBωζ, (77)where we used E≈Bbg/2 near R×andωB≡eBbg/mc.\nWe conclude that charge starvation does not prevent the\nMHD evolution if\nc σ×\nωBR×<∼0.1. (78)\nUsing σ×=ω2\nB/ω2\np, one can also rewrite this condition\nascωB/ω2\npR×>∼0.1. The same condition for shock for-\nmation was found in PIC simulations of magnetosonic\nwaves (Chen et al. 2022).\nIf the wave had only one oscillation, with a single\nshock, it would heat the plasma up to ε∼ζ−1at the\npeak of the shock strength. Then, the MHD condition\nωL>∼4ωwould be similar to j/enc < 1 derived above.\nIn a wave train with many shocks, the plasma is heated\ntoε≫ζ−1. Therefore, the condition ωL>∼4ωexamined\nin Section 5.1 becomes more demanding than j/enc < 1.\n6.DAMPING IN THE KINETIC REGIME\nRadio waves propagating at a polar angle θwith\npower isotropic equivalent L(θ)< L MHD(θ) cannot be\ndamped by shocks, because the magnetospheric par-\nticles become unmagnetized in the wave, i.e. reach\nωL∼ω, before the wave experiences significant losses.\nThe unmagnetization transition occurs close to the ra-\ndiusr×with the background gyro-frequency ωB(r×) =\n(eµ/mcr3\n×) sinθ/sinα. The transition happens when\nshock heating increases the plasma internal energy ε\n(thermal Lorentz factor) to\nεtr∼ωB(r×)\nω=ωB(R×)\nωsin2α\nsin1/2θ. (79)\nThe breaking of MHD description (i.e. breaking of\nthe drift description of plasma response to the oscil-\nlating electromagnetic field) implies that the condition\nE2< B2is no longer enforced. The part of the wave\ndeveloping E2> B2is no longer “shaved off” by the\nshock as described in Paper I, because the shocks dis-\nsolve when ωL∼ω.\nSubsequent evolution will further energize plasma par-\nticles as described below, leading to ωL< ω(note that\ndamping of a small fraction of the wave energy is suf-\nficient for plasma heating to ωL< ω). In a wave with\nL≪LMHD, radiative losses are negligible at the transi-\ntionωL∼ω(εtris below the radiative ceiling), however\nlosses will become important with increasing particle en-\nergies, leading to efficient radiative damping of the wave.\nThe main damping will develop in the regime of ωL< ω,\naway from the cyclotron resonance ωL=ω.\nWhen the wave exits the MHD regime at r≈r×and\ncontinues its propagation to r > r ×with the intact sine\nprofile (which includes parts with E2> B2), it interacts\nwith the plasma very differently from the MHD regime.\nAs shown in Beloborodov (2022a) (hereafter B22), mag-\nnetospheric particles exposed to the train of wave os-\ncillations experience quick stochastic acceleration.This14\nprocess is convenient to view in frame K′boosted along\nBbgso that the wavevector k′becomes perpendicular to\nB′\nbg=Bbg.3This frame moves along Bbgwith speed\nβF= cos αand Lorentz factor\nγF=1\nsinα. (80)\nHereafter all quantities with primes are measured in\nframe K′. Note that\nω′=ωsinα, a′\n0=a0,B′\nbg=Bbg. (81)\nThe transition radius r×corresponds to 2 E′\n0=Bbgor\na0=ωB/2ω′.\nThe particle interaction with the radio wave affects\nthe four-velocity component u′\n⊥perpendicular to Bbg.\nIndeed, in frame K′particles experience no force com-\nponent along Bbg, since E′⊥BbgandB′∥Bbg. As\nstochastic acceleration pumps u′\n⊥, it tends to increase\nthe particle pitch angle relative to Bbg. At the same\ntime, radiative losses reduce all components of u′, as\nthe relativistic particle radiates along u′. Thus, the\ncontinued pumping of u′\n⊥combined with persistent ra-\ndiative losses will drive the particle pitch angle toward\nπ/2 in frame K′. Note also that γ′≈u′\n⊥for the ultra-\nrelativistic particle.\nThe description of particle acceleration by the wave\nin B22 applies in frame K′. Stochastic acceleration may\nbe viewed as diffusion in γ′with a diffusion coefficient\nD∼ω′\nL\u0012γ′a3\n0ω′\nωB\u00132/3\n, (82)\nwhere ω′\nL=ωB/γ′, and\nωB=eBbg\nmc≈1.8×1013µ33\nr3\n9sinθ\nsinαrad\ns. (83)\nMean expectation for the energy gain rate is given by\n(omitting a numerical factor ∼1),\n⟨˙γ′⟩ ∼D\nγ′∼a2\n0ω1/3\nBω′2/3\nγ′4/3. (84)\nIn a quasi-steady state, stochastic acceleration be-\ncomes offset by radiative losses (which are Lorentz-\ninvariant),\n⟨˙γem⟩=2re\n3ca2\n0ω′2⟨γ′2⟩. (85)\n3In this frame, plasma has no bulk motion perpendicular to Bbg,\nafter averaging over Larmor rotation. Note that in the kinetic\nregime, ωL< ω, averaging over Larmor rotation also removes the\nwave oscillation, as the wave period is smaller than the Larmor\nperiod. (By contrast, in the MHD regime, ωL≫ω, the fluid\nvelocity is defined on scales smaller than ω−1and oscillates with\nthe wave period.)Then, the mean expectation for the particle Lorentz fac-\ntor⟨γ′⟩may be estimated by balancing ⟨˙γ′⟩with⟨˙γem⟩,\n⟨γ′⟩ ∼\u0012c\nreω′\u00133/10\u0010ωB\nω′\u00111/10\n. (86)\nIts value for GHz waves is a few times 104. One can also\ncheck the condition ⟨γ′⟩> γ⋆≡(a3\n0ω′/ωB)1/2that is\nexpected in stochastic acceleration (see B22). Note that\n⟨γ′⟩ ∝r−3/10, and γ⋆=a0(r×)/√\n2 is constant with\nradius. After some algebra, we find (at θ < θ MHD <1):\n⟨γ′⟩\nγ⋆∼\u0012θMHD\n0.06\u00133/5\u0012θMHD\nθ\u00133/4\u0010r×\nr\u00113/10\n.(87)\nDirect calculation of unmagnetized particle motions in\nthe wave demonstrates that they quickly develop chaos,\nforming a quasi-steady distribution around ⟨γ′⟩. B22\nshowed that the distribution extends from ∼γ⋆up to\nthe radiation-reaction limit,\nγ′\nRRL≈\u0012c\nreω′a0\u00133/8\u0010ωB\nω′\u00111/4\n≈\u0012c\nreω′\u00133/8\u00144\na0(r×)\u00151/8\u0010r×\nr\u00113/8\n.(88)\nFor typical FRB parameters, γ′\nRRL/⟨γ′⟩ ∼a few.\nThe characteristic timescale for particle acceleration\nto⟨γ′⟩is\nt′\nacc∼⟨γ′⟩\n⟨˙γ′⟩∼⟨γ′⟩7/3\na2\n0ω1/3\nBω′2/3, (89)\nwhich gives\nω′t′\nacc∼\u0012c\nreω′\u00137/10\u0012ω′\nωB\u00131/10\na−2\n0. (90)\nThe Lorentz-invariant quantity Nacc=ω′t′\nacc/2πis the\nnumber of wave oscillations that the plasma should cross\nto approach the quasi-steady ⟨γ′⟩(see Figure 4 in B22).\nThe value of a0≡eE0/mcω may be expressed as\na0=ωB(r×)\n2ω′r×\nr=25/4e\nmcµ1/2ω\u0012L\nc\u00133/4sinα\nsin1/2θ\n≈9×104L3/4\n42\nµ1/2\n33ν9sinα\nsin1/2θr×\nr. (91)\nThen, we find\nω′t′\nacc∼0.1µ21/20\n33ν7/5\n9\nL63/40\n42sin21/20θ\nsin14/5α\u0012r\nr×\u001317/10\n.(92)\nThe wave train in a typical FRB with duration τ∼0.1-\n1 ms has N=ωτ/2π∼105-106≫Nacc. So, after the\ntransition to the kinetic regime (which occurs near r×),15\nparticles exposed to the wave almost immediately de-\nvelop a quasi-steady distribution with the mean expec-\ntation ⟨γ′⟩(Equation 86), and continue to move through\nthe wave train with ⟨γ′⟩.\nThe quasi-steady particle distribution established in\nthe wave reflects the balance between stochastic accel-\neration and radiative losses. The radiative losses are\nirreversible and occur at the expense of the electromag-\nnetic wave energy. The rate of energy loss per particle\n(Equation 85) in the quasi-steady distribution is\n⟨˙γem⟩ ∼2\n3a2\n0\u0010re\nc\u00112/5\nω6/5ω1/5\nBsin6/5α. (93)\nAs the plasma crosses the wave train, it radiates the fol-\nlowing energy per particle (measured in the lab frame):\n∆Ee\nmc2∼ ⟨˙γem⟩tcross, (94)\nwhere tcrossis the time it takes the plasma to cross the\nwave train of duration τ. The crossing time is deter-\nmined by the radial component of the fluid velocity,\nβr\nF=βFcosα= cos2α. The fluid speed relative to\nthe wave front, is 1 −βr\nF= sin2α, and so\ntcross∼τ\nsin2α. (95)\nAs the spherical wave propagates a radial distance δr,\nit interacts with δN ≈ 4πNθδr/r of magnetospheric\nparticles. The number of particles sufficient to damp\nthe wave is δNdamp =Lτ/∆Ee, and we find\nδNdamp\nNθ≈Lτ\nNθ∆Ee∼2πr2sin4/5α\nσTNθ\u0012r3\neω4\nc3ωB\u00131/5\n∼3×10−3r13/5\n9ν4/5\n9\nµ1/5\n33N37sin2α\nsin6/5θ, (96)\nwhere we used L/a2\n0=mc ω2r2/2reand substituted\nNθ=Nsinθ/sinα(Equation 50). One can see that\nδNdamp <Nθfor the relevant radii r>∼r×, which im-\nplies quick damping of the wave near r×.\nOne caveat in the above calculation is that at θ≪1\nthe plasma may fail to cross the entire wave train\non the wave expansion timescale r/c. This happens\nwhere tcross> r/c , which corresponds to α < α cross∼\n(cτ/r)1/2. For the typical parameters, αcross∼0.1.\nThus, near the magnetic axis, the plasma becomes\n“stuck” in the wave train and surfs its leading part in-\nstead of crossing it. The damping process cannot be\ncompleted without plasma filling the entire wave train,\nand one might conclude that the wave will escape in the\ncone of θ≈2α <2αcross.\nHowever, there is an additional process that efficiently\nfills the wave train with plasma. The accelerated parti-\ncles surfing the leading part of the wave emit gamma-\nrays, which freely propagate across the wave train andload it with e±pairs via photon-photon collisions. This\nprocess triggers an e±avalanche (Beloborodov 2021).\nIt is easy to verify that the spectrum of curvature\nphotons emitted by the accelerated particles extends to\nthe gamma-ray band. The characteristic frequency of\ncurvature photons in frame K′is given in B22,\nω′\nc≈2a0γ′2ω′. (97)\nParticles with the average ⟨γ′⟩(Equation 86) radiate\nphotons of characteristic frequency\n⟨ω′\nc⟩ ∼2a0\u0012c\nre\u00133/5\n(ω′ωB)1/5. (98)\nAtr≈r×, one can substitute a0≈ωB/2ω′(Equa-\ntion 91) and find the characteristic photon energy,\nϵ′\nc(r×)≡ ⟨ℏω′\nc\nmc2⟩ ∼103L9/10\n42\nµ3/5\n33ν4/5\n9sin8/5α\nsin3/5θ. (99)\nThis gives ϵ′\nc(r×)∼30−100 at polar angles where wave\ndamping occurs in the kinetic regime. It is not far from\nthe maximum ϵ′\ncestimated in B22 for the most energetic\nparticles in the distribution, near the radiation-reaction\nlimit γ′\nRRL,\nℏω′\nc\nmc2\f\f\f\f\nRRL≈1\nαf\u0012reω2\nBa0\ncω′\u00131/4\n, (100)\nwhere αf=e2/ℏc≈1/137. In particular, at r≈r×,\none can use a0(r×)≈ωB/2ω′to get\nℏω′\nc\nmc2\f\f\f\f\nRRL≈a3/4\n0\nαf\u00124reω′\nc\u00131/4\n≈500L9/16\n42\nµ3/8\n33ν1/2\n9sinα\nsin3/8θ.\n(101)\nThe curvature radiation spectrum emitted by each\nparticle extends to lower energies with the power-law in-\ndex 1 /3. Therefore, a significant fraction of the radiated\npower is in the MeV range, where photons collide and\nconvert to e±with a large cross section ∼0.1σT. Using\nestimates similar to Beloborodov (2021), one can verify\nthat this process converts a significant fraction of the\nwave energy into secondary e±pairs, loading the wave\nwith a large number of particles, far exceeding the initial\nparticle number in the background magnetosphere. The\ne±loading of the wave implies its inevitable damping.\nOur choice of frame K′neglected the fact that\nthe wave exerts pressure on the background magneto-\nsphere, driving its bulk acceleration and compression\n(see Beloborodov (2021)). This additional effect slightly\nchanges the fluid rest frame used to calculate stochas-\ntic particle acceleration (in this frame E′\nbg= 0 and the\nplasma motion vanishes after averaging over Larmor ro-\ntation). This effect is moderate at radii of interest,\nr∼r×(θ), where the wave becomes damped and de-\nposits radial momentum into the magnetosphere. Near16\nradius R×=r×(π/2), deposition of wave momentum\nE/cinto the magnetosphere results in its bulk accel-\neration to speed βp≈ E/R3\n×B2\nbg≈τc/8R×∼10−2\nfor typical parameters. At small θ, bulk acceleration is\nstronger. However, at all polar angles, the damping ra-\ndiusr≈r×(θ) corresponds to E′\n0≈Bbg/2, i.e. damping\noccurs where the wave energy density in frame K′is com-\nparable to B2\nbg/8π. Therefore, at r∼r×the wave is at\nbest capable of mildly relativistic bulk acceleration and\nmoderate compression of the background magnetic field.\nThe corresponding change of frame K′will not change\nour conclusion that GHz waves are efficiently damped\nby stochastically accelerating particles.\nQu et al. (2022) argued that near the magnetic\ndipole axis the wave damping should be considered on\nopen magnetic field lines with the background magneto-\nspheric plasma flowing with a Lorentz factor γbg>∼103.\nThey suggested that the high γbgwould render the\ndamping mechanism inefficient and that radio bursts\ncould escape in a broad solid angle around the magnetic\naxis. This possibility is, however, problematic:\n(1) Solid angle δΩ where damping develops on open\nfield lines is not broad. Damping of waves near the mag-\nnetic axis begins at r×(θ)≈2R×/θ1/2. The open field\nlines occupy θ < θ open(r)≈(r/R LC)1/2. One can see\nthat damping develops on the open field lines for waves\npropagating at angles θ <(2R×/RLC)4/5. The corre-\nsponding solid angle is\nδΩ≈1\n2\u00122R×\nRLC\u00138/5\n∼10−2P−8/5µ4/5\n33L−2/5\n42,(102)\nwhere Pis the magnetar rotation period in seconds. All\nknown local magnetars have periods P > 2 s, as ex-\npected from their fast spindown due to the strong fields.\nYoung hyper-active magnetars proposed as sources of\nrepeating FRBs have P∼1 s (Beloborodov 2017).\n(2) Assuming a large γbg∼102−103would be reason-\nable for open field lines in ordinary (rotation-powered)\npulsars, but not in magnetars. The e±plasma in the\nouter magnetosphere, on both open and closed field\nlines, experiences drag exerted by resonant scattering of\ndense radiation flowing from the magnetar (Beloborodov\n2013). Therefore, the e±flow in the outer magneto-\nsphere is expected to be mildly relativistic. The speed\nand density of e±on the open field lines of magnetars is\nestimated in Beloborodov (2020).\n(3) Particles tend to forget their pre-wave motion in\nthe magnetosphere once they become exposed to the\nwave. As shown above, particles interacting with the\nwave in the kinetic regime ( ωL< ω) quickly establish\na quasi-steady momentum distribution, with stochastic\nmotions becoming perpendicular to Bbgin frame K′de-\nfined by the condition k′⊥B′\nbg=Bbg.\nQu et al. (2022) also argued that the wave pressure\non the magnetosphere could stretch out the magneto-\nspheric field lines, making them more radial, so that theangle between Bbgand the wavevector kis reduced, po-\ntentially helping the wave to escape. In fact, the wave\npressure cannot significantly change the direction of Bbg\nin the wave. Note that near the damping surface r×(θ)\nthe wave pressure perpendicular to Bbgis comparable to\nB2\nbg/8π, so the background field resists strong changes.\nFurthermore, even a much stronger wave pressure would\nbe unable to stretch radially the magnetic field lines in-\nside the spherical wave packet, because its thickness is\nfar smaller than radius, cτ≪r.\n7.DISCUSSION\n7.1. Summary of main results\nOur main conclusion is that FRBs are unable to es-\ncape through the static dipole magnetosphere surround-\ning the magnetar at radii 108cm< r < R LC∼1010cm.\nGHz waves with power Lbecome damped near the sur-\nface defined by\nr×(θ)≈2.5×108µ1/2\n33L−1/4\n42s\n4−3 sin2θ\nsinθcm,(103)\nwhere µis the magnetic dipole moment of the magne-\ntar. Damping develops because the wave approaches the\ncondition E2≈B2which leads to particle energization.\nWe have investigated this process in detail and found\nthat it develops in two different regimes near the mag-\nnetic axis and near the equator. Our estimate for the\nboundary θMHD(L) between the two regimes is shown in\nFigure 9.\n(1) In the equatorial region sin θ >sinθMHD the en-\ntire wave evolution is well described by MHD. The MHD\nsolution demonstrates that at r≈r×the wave train de-\nvelops shocks in each oscillation. The resulting shock\ntrain heats the plasma to an ultra-relativistic tempera-\nture (specific internal energy ε∼102−103) at which\nheating becomes offset by synchrotron cooling. We have\nfollowed the wave evolution with a detailed simulation\nin the equatorial plane ( θ=π/2) and also described it\nanalytically. The results show nearly complete damping\nof the GHz oscillations (Figure 1). The alternating com-\nponent of the electromagnetic field gets suppressed by\na factor of ∼10−3, and the wave train becomes trans-\nformed into a smooth and week electromagnetic pulse\nof the same duration and the wiped-out oscillating com-\nponent. We have also examined the MHD evolution at\nθ̸=π/2 and verified that it leads to similar shocks.\nOur method for solving this MHD problem employed\ncharacteristics C±. It allows one to find the solution\nwith realistic parameters of the magnetosphere (where\nmagnetization σbgcan exceed 108, see Equation (1)).\nWe also exploited the fact that the wavelength λ=c/ν\nis far shorter than radius r(the variation scale of Bbg),\nand the wave duration τ<∼1 ms satisfies τ��r/cat\nradii of interest. This feature facilitates the solution, as\nit gives a simple integral along C−across the wave.17\n(2) In the polar regions sin θ < sinθMHD, the wave\ndamping also begins with MHD shocks developing at r×.\nHowever, here the shock heating quickly ends, because\nthe heated particles become unmagnetized in the wave,\ni.e. their Larmor frequency ωLdrops below the wave\nfrequency ω. This transition happens before damping,\nwith the practically intact wave profile E(t−r/c). As\nthe wave continues to propagate to r > r ×, it devel-\nops regions of E2> B2in each oscillation, triggering\nstochastic particle acceleration described in B22: mag-\nnetospheric particles exposed to the GHz wave train de-\nvelop a quick random walk in energy. The particles are\nthen forced into a quasi-steady energy distribution (with\nLorentz factors 104-105), in which stochastic accelera-\ntion is balanced by radiative losses, quickly draining the\nwave energy. This damping effect may be formulated\nas a large particle cross section for scattering the GHz\nwave to the gamma-ray band, σsc>∼108σT(B22). The\ngamma-rays emitted by the accelerated particles pro-\nduce copious secondary e±pairs, which fill the entire\nradio wave and assist its damping.\nIn both kinetic and MHD regimes, the magnetosphere\natr>∼R×effectively acts as a pillow absorbing the\nwave, with most of the wave energy converted to hard\nradiation and a residual fraction feeding a low-energy\nmagnetic explosion, ejecting the outer layers of the mag-\nnetosphere.\n7.2. Comparison with kHz waves\nThe strong GHz waves in the magnetosphere evolve\ndifferently from kHz waves studied in Paper I.\n(a) In the MHD propagation regime, radio waves ac-\ncelerate the plasma to a bulk Lorentz factor γ∝ν−1,\nwhich differs by ∼106between kHz and GHz waves.\nThe moderate γin the GHz waves leads to moderately\nstrong relativistic shocks, different from the monster\nshocks described in Paper I.\n(b) The number of oscillations in the GHz wave train\nisN=τ/ν= 106(τ/1 ms) ν−1\n9. Shocks develop in each\noscillation and their large number produces a huge cu-\nmulative damping effect on the wave, nearly completely\nerasing the GHz oscillations. By contrast, in kHz waves\nthe monster shocks erase half of each oscillation.\n(c) Radiative (synchrotron) cooling of the shock-\nheated plasma in GHz waves occurs on timescales longer\nthan the wave period, but much faster than the wave du-\nration, so a thermal balance is established in the wave\npacket. In kHz waves, the monster shock radiates the\ndissipated energy almost instantaneously, in the shock\nitself.\n(d) The critical wave power LMHD(θ) for the transi-\ntion to the kinetic regime scales as ν8/9(Equation 73).\nThe transition is relevant for GHz waves and irrelevant\nfor kHz waves. MHD fails in powerful kHz waves differ-\nently: when the plasma is accelerated to extremely high\nLorentz factors its motion transitions to the two-fluid\nregime as explained in Paper I.7.3. Mechanism of observed FRBs\nAn observed FRB power Lrequires a source with en-\nergy density U∼L/4πr2cηwhere η≪1 is the effi-\nciency of GHz emission. The energy density around a\nmagnetar is U(r)∼µ2/8πr6, and then the condition\nU≫L/4πr2crequires a source of size r≪(cµ2/2L)1/4.\nIt is tempting to picture a compact GHz source confined\ninside the ultra-strong magnetosphere (e.g. Lu et al.\n2020), However, our results imply trouble for this sce-\nnario: the condition r≪(cµ2/2L)1/4is nearly the same\nasr≪R×, and we find that the emitted waves ex-\nperience strong damping at r>∼R×. Damping occurs\nin both propagation regimes (MHD and kinetic) and so\nholds practically for the entire range of relevant FRB\nluminosities Land propagation angles θ.\nTherefore, emission of observed FRBs must involve\nviolent events that relocate energy from radii r≪R×\nto outside the magnetosphere, where GHz waves can be\nreleased. This is accomplished by magnetospheric ex-\nplosions, which produce ultra-relativistic ejecta. The\nexplosion transports a large magnetic energy Efar out-\nside the magnetosphere, e.g. E ∼ 1044erg is expected\nin repeating FRBs from hyper-active, flaring magnetars\n(Beloborodov 2017). It has been shown that the blast\nwave from the explosion can emit a GHz burst with en-\nergyEFRB∼(10−4−10−5)Eand sub-millisecond dura-\ntion as radii r∼1 AU (Beloborodov 2017, 2020). The\nemission is generated by the well studied mechanism of\n“shock maser precursor” (e.g. Sironi et al. 2021). A\nvariation of the blast wave model involving a slow ion\nwind ahead of the explosion is discussed in Metzger et al.\n(2019) and Beloborodov (2020). In addition, Thompson\n(2023) recently proposed that the blast wave may emit\nradio waves via another mechanism if it expands into a\nturbulent medium (a pre-explosion magnetar wind car-\nrying a spectrum of perturbations).\nEjecta from powerful magnetospheric explosions may\nthemselves carry magnetosonic fluctuations with radio\nfrequencies.4At large radii, the fluctuations may decou-\nple and leave the ejecta as free waves, forming a GHz\nburst. A model of this type was proposed by Lyubarsky\n(2020) and further investigated by Mahlmann et al.\n(2022). These works invoked the explosion interaction\nwith the current sheet near the light cylinder as a source\nof ejecta fluctuations.\nAnother possibility for FRB production is the precur-\nsor emission from the magnetospheric monster shocks\ndescribed in Paper I. The precursor will ride on top of\nthe parent kHz wave that forms the monster shock, not\nin a static dipole magnetosphere, and therefore it could\n4The ejecta serve as a new background Bbg∝r−1for small oscil-\nlations with amplitude E0≪Bbg. Small oscillations can remain\nfrozen in the ejecta for a long time, as they expand from the\nmagnetosphere with E0∝Bbg∝r−1, keeping E0/Bbg≪1.18\nescape from small radii. This possibility is further dis-\ncussed elsewhere.\nObservational diagnostics for FRB models include the\nburst spectra, temporal structure, and polarization (see\nfor example a recent discussion of polarization in Qu &\nZhang (2023)). The observed properties can be changed\nby the burst propagation through the magnetar wind(Sobacchi et al. 2022) and the surrounding nebula (Mar-\ngalit & Metzger 2018; Vedantham & Ravi 2019; Gruzi-\nnov & Levin 2019). The propagation effects will need\nto be disentangled from the intrinsic emission properties\nbefore conclusions can be made.\nThis work is supported by NSF AST 2009453, NASA\n21-ATP21-0056 and Simons Foundation #446228.\nAPPENDIX\nA.CHARACTERISTICS IN RELATIVISTIC MHD\nA.1. MHD stress-energy tensor\nMHD fluid is described by the plasma mass density ρ, velocity v=cβ, magnetic field B, and electric field E.\nThroughout this Appendix we will use the units of c= 1. The stress-energy tensor Tµνof the MHD fluid includes\ncontributions from the electromagnetic field ( Tµν\nf) and plasma ( Tµν\np). Explicit expressions for Tµν\nfin terms of E,B\nand for Tµν\npin terms of ρ,vare given below. Energy and momentum conservation in MHD is expressed by\nTµν\n;ν=1√−g∂ν\u0000√−gTν\nµ\u0001\n−1\n2Tαβ∂µgαβ=−Qµ, Tµν=Tµν\nf+Tµν\np, (A1)\nwhere semicolon denotes covariant derivative, gαβis the spacetime metric, and g≡detgαβ.Qµrepresents radiative\nlosses of the plasma. The losses typically have the form Qµ=−Quµ, where uµ= (γ, γβ) is the plasma four-velocity.\nThe electromagnetic stress-energy tensor is Tαβ\nf=FαµFβ\nµ/4π−gαβFµνFµν/16π(e.g. Landau & Lifshitz 1975),\nFµν=∂µAν−∂νAµ,FµνFµν= 2(B2−E2), and Aµis the four-potential of the electromagnetic field. We express all\nfield components in the normalized basis ( er,eθ,eϕ) in Minkowski space with coordinates xµ= (t, r, θ, ϕ ). This gives\nTµν\nf=\nE2+B2\n8π−EϕBθ\n4πEϕBr\n4πr0\n−EϕBθ\n4πE2−B2\nr+B2\nθ\n8π−BθBr\n4πr0\nEϕBr\n4πr−BrBθ\n4πrE2+B2\nr−B2\nθ\n8πr2 0\n0 0 0B2−E2\n8πr2sin2θ\n(A2)\nThe stress-energy tensor of the plasma treated as an ideal (isotropic) fluid has the form,\nTµν\np=Hpuµuν+gµνPp, (A3)\nwhere Ppis the plasma pressure, and Hpis its relativistic enthalpy density (including the fluid rest mass density ˜ ρ).\nHeating by Larmor-mediated shocks can result in a two-dimensional (2D) plasma, with e±thermal speeds βe⊥˜B.\nThen, one can calculate Tµν\npas follows. First, find the (diagonal) stress-energy tensor in the fluid rest frame ˜Kby\nviewing the plasma as a collection of cold e±streams with different u˜µ\neand proper densities d˜ρ/γe,\nT˜µ˜ν\np= ˜ρ⟨u˜µ\neu˜ν\ne\nγe⟩, (A4)\nwhere ⟨...⟩means averaging over the distribution of u˜µ\ne. The stress-energy tensor in the lab frame is Tµν\np= Λµ\n˜µΛν\n˜νT˜µ˜ν\np\nwhere Λµ\n˜µis the Lorentz matrix for the boost from the fluid frame to the lab frame. This gives the general Tµν\npfor\nplasmas with any anisotropy; in the isotropic case it is reduced to Equation (A3).\nTo avoid unnecessary distraction, our derivation of MHD characteristics will assume isotropic plasma. However,\nlooking at the derivation, one will see that only the t, rcomponents of the plasma stress-energy tensor Tµν\npaffect\nthe final result, so only radial pressure Pp=T˜r˜r\npenters the wave propagation problem. The calculation of Tµν\npfor\nanisotropic plasma in the equatorial plane gives the t, rcomponents of Tµν\npof the same form as in Equation (A3), with\nPp=T˜r˜r\npinstead of isotropic pressure. Therefore, the final equations for characteristics hold for anisotropic plasma.\nThe only important effect of anisotropy is that it changes the plasma equation of state — the relation between energy\ndensity and radial pressure. This relation enters through γs, which is given in Section A.5.19\nA.2. Equatorial waves\nWe now focus on the wave dynamics in the equatorial plane θ=π/2. By symmetry, Br= 0 and vθ= 0 at θ=π/2.\nWe will use the following notation:\nE≡ −Eϕ, B ≡Bθ. (A5)\nThese definitions imply E2=E2andB2=B2atθ=π/2;EandBmay be positive or negative. The plasma\nfour-velocity has the form,\nuα= (γ, u,0,0), u =γβ, β ≡βr=E\nB, γ2=1\n1−β2=B2\nB2−E2\u0010\nθ=π\n2\u0011\n. (A6)\nIn the equatorial plane, there are two relevant components of the dynamical equation Tµν\n;ν=−Quµwith µ=t, r.\nInstead of these two components we will use two projections:\nuµTµν\n;ν=Q, (utuµ+gtµ)Tµν\n;ν= 0. (A7)\nThe vanishing of Brandvθin the equatorial plane implies Tθ\nt=Tθ\nr= 0. However, their θ-derivatives are not zero and\nwill enter the conservation laws. For instance, divergence of the Poynting flux ∇ ·(E×B)/4πincludes a term with\n∂θBr̸= 0, as Brchanges sign across the equatorial plane.\nFor the plasma stress-energy tensor (Equation A3), one finds\nuµTµν\np ;ν=−Hpuν\n;ν−uµ∂µ(Hp−Pp), (utuµ+gtµ)Tµν\np ;ν=−Hpuν∂νγ−γuν∂νPp+∂tPp. (A8)\nThe term uµ∂µ(Hp−Pp) may be written as the sum of adiabatic part Hpuµ∂µln ˜ρand radiative part −Q. Note that\nuµTµν\np ;ν=Qregardless of the presence of Tµν\nfin the system. This condition states the first law of thermodynamics\n(and for a cold flow it is reduced to conservation of the plasma rest mass). The divergence of four-velocity is\nuν\n;ν=1√−g∂ν\u0000√−guν\u0001\n=∂tγ+1\nr2∂r(r2u) +1\nr∂θuθ\u0010\nθ=π\n2\u0011\n, (A9)\nwhere the component uθis taken in the normalized basis ( er,eθ,eϕ).\nNext, consider the electromagnetic stress-energy tensor Tµν\nf(Equation A2). Direct calculation yields at θ=π/2:\nuµTµν\nf ;ν=−√\nB2−E2\n4π\u0012\n∂tB+∂rE+E\nr\u0013\n, (utuµ+gtµ)Tµν\nf ;ν=−E\n4π\u0012\n∂tE+∂rB+B\nr+2A\nr2\u0013\n. (A10)\nThe identity ∂t∇×A=∇×∂tAimplies ∂tB+∂rE+E/r= 0 and uµTµν\nf ;ν= 0. For our purposes it will be convenient\nto rewrite Tµν\nfusing the effective pressure Pfand the effective inertial mass density Hfdefined by\nPf=B2−E2\n8π=˜B2\n8π, H f=˜B2\n4π= 2Pf. (A11)\nThis allows one to cast the t,rcomponents of Tµν\nfin the form similar to ideal fluid:\nTtt\nf=γ2Hf−Pf, Ttr\nf=γuH f, Trr\nf=u2Hf+Pf. (A12)\nOther relevant components of Tµν\nfare\nTθt\nf=−EBr\n4πr, Tθr\nf=−BBr\n4πr, Tϕ\nfϕ=Hf\n2=−Tθ\nfθ. (A13)\nThen, we find in the equatorial plane\nuµTµν\nf ;ν=−Hf(∂tγ+∂ru)−uµ∂µ(Hf−Pf)−Hfu\nr, (A14)\n(utuµ+gtµ)Tµν\nf ;ν=−Hfuµ∂µγ−γuν∂νPf+∂tPf−Hfuγ\nr+E ∂θBr\n4πr. (A15)\nNote that Hfuν\n;ν+uµ∂µ(Hf−Pf) =√Hf(√Hfuν);ν= (˜B/4π)(˜Buν);ν. For short waves, the flow oscillation is nearly\nplane parallel, and the equations may be simplified: magnetic flux freezing gives ˜B∝˜ρand the continuity equation\nimplies ( ˜Buν);ν= 0; in the same approximation one can use uµ∂µ(Hf−Pf) =Hfuµ∂µln ˜ρ.20\nSubstitution of Equations (A8), (A14), and (A15) into Equations (A7) gives\n−H(∂tγ+∂ru)−uµ∂µ(H−P)−Hfu\nr−Hp\nr(2u+∂θuθ) =Q, (A16)\n−Huµ∂µγ−γuν∂νP+∂tP−Hfuγ\nr+E∂θBr\n4πr= 0, (A17)\nwhere\nP=Pf+Pp, H =Hf+Hp. (A18)\nWe will use dγ=βduto express all derivatives of uνin terms of derivatives of u=γβ. Equations (A16) and (A17)\nalso contain derivatives of PandH−P. One can retain only derivatives of Pby defining\nβ2\ns≡uµ∂µP\nuµ∂µ(H−P)=dP\nd(H−P), (A19)\nwhere differential dis taken along the worldline of a fluid element. The quantity βs(and the characteristics C±below)\nwill be defined in the adiabatic approximation, Q≈0. Then, Equations (A16) and (A17) become\nH(β∂tu+∂ru) +γ\nβ2s(∂tP+β∂rP) =−Hfu\nr−Hp\nr(2u+∂θuθ), (A20)\nH(∂tu+β∂ru) +γ(β∂tP+∂rP) =−Hfγ\nr+B ∂θBr\n4πrγ. (A21)\nWe multiply Equation (A20) by βsand add/subtract it from Equation(A21). This yields\n(1±βsβ) \n∂±lns\n1 +β\n1−β±∂±P\nβsH!\n=Hf\nrH\u0012∂θBr\nB−1∓ββs\u0013\n∓βsHp\nγrH(2u+∂θuθ), (A22)\nwhere we used the identity\ndu\nγ=dlns\n1 +β\n1−β, (A23)\nand defined\n∂±≡∂t+β±∂r, β ±≡β±βs\n1±ββs. (A24)\nThe radial speed β±in the lab frame corresponds to propagation with speed ±βsrelative to the fluid. The derivatives\n∂±are taken along the characteristics C±. The characteristics are defined as the curves r±(t) that satisfy dr±/dt=β±.\nEquation (A22) is the MHD generalization of equations given by Johnson & McKee (1971) and McKee & Colgate\n(1973), which were derived for one-dimensional relativistic hydrodynamics. It is easy to verify that their hydrody-\nnamical equations are recovered in the limit of a weak electromagnetic field E, B→0. In this limit, Hf/Hp= 0,\nHp/H= 1, and ∂θuθ= 0 if the flow is spherically symmetric. Then, Equation (A22) becomes equation II.b.20 in\nMcKee & Colgate (1973). We are interested in the opposite, field-dominated, regime H≈Hf≫Hp.\nA.3. Magnetically dominated limit ( Hf≫Hp)\nIn the magnetically dominated regime, one can simplify the MHD equations. Equation (A22) becomes\nr(1±ββs) \n∂±lns\n1 +β\n1−β±∂±P\nβsH!\n=−1∓β−2Bbg\nB+O\u0012Hp\nHf\u0013\n, (A25)\nwhere on the r.h.s. we used 1 −βs=O(Hp/Hf). We also find\n∂±P\nβsH=∂±Pf\nHf\u0014\n1 +O\u0012Hp\nHf\u0013\u0015\n=∂±ln˜B\u0014\n1 +O\u0012Hp\nHf\u0013\u0015\n, (A26)\nwhere we used Hf= 2Pf=˜B2/4π. Thus, Equation (A22) simplifies to\n∂±J±=−1∓β+∂θBr/B+O(Hp/Hf)\nr(1±ββs), J ±= lns\n1 +β\n1−β±ln˜B\u0014\n1 +O\u0012Hp\nHf\u0013\u0015\n. (A27)\nIn the denominator we did not use the expansion 1 ±ββs= 1±β+O(Hp/Hf), because 1+ βcan approach zero during\nthe wave evolution. The numerator is never close to zero, since ∂θBrhas a finite negative value.21\nA.4. Short waves\nWe are interested in short wave packets with wavelength λmany orders or magnitudes shorter than r. The wave\nelectromagnetic potential Aw=A−Abgis related to the wave magnetic field Bw=B−BbgbyrBθ\nw=−∂r(rAw)\nandrBr\nw=∂θAw. In short waves, ∂rAw≫r−1∂θAw, and so Br\nw≪Bθ\nw. This implies Br≈Br\nbgand\n∂θBr≈ −2Bbg\u0010\nθ=π\n2\u0011\n. (A28)\nThen, only derivatives ∂±are left in Equation (A27), i.e. the problem is reduced to ordinary differential equations.\nThis enables simple integration for J±along C±.\nThe C−characteristics propagate radially inward, and cross the short wave packet on a timescale tcross\n−≪r.\nTherefore, the change of J−across the wave is small, |∆J−/J−| ≪1, i.e. J−is approximately uniform across the wave\nand weakly changed from its value in the unperturbed background just ahead of the wave, Jbg\n−≈ −lnBbg. This gives\nthe following relations:\nJ−=−lnBbg ⇒ J+= lnκ2+ lnBbg, κ ≡˜B\nBbg=s\n1 +β\n1−β, B =Bbg\n1−β. (A29)\nThe relation B=Bbg/(1−β) states the compression of the magnetic field in the lab frame by the factor (1 −β)−1.\nPlasma density is compressed by the same factor (Equation 8), consistent with magnetic flux freezing: B/ρ=Bbg/ρbg.\nMagnetic flux freezing also implies σ/σ bg=κ, where σ≡Hf/˜ρ.\nThe equation for J+evolution along C+(Equation A27) and the definition of C+(∂+r=β+) give two coupled\nequations for β(t) and r(t) along C+:\n∂+lns\n1 +β\n1−β=2β+O(Hp/Hf)\nr(1 +ββs), ∂ +r=β+= 1−(1−β)\n1 +ββs(1−βs), (A30)\nwhere we used ∂+lnBbg=−3β+/r. These equations still contain the fast magnetosonic speed βs, which is close to\nunity. Setting βs= 1 would correspond to FFE. It is the small term 1 −βs≈(2γ2\ns)−1=O(Hp/Hf) that controls\nthe MHD correction to FFE, and it is retained in the leading order in Equations (A30). In particular, it controls the\ndeviation of β+from unity, bending the C+characteristics from straight lines in spacetime. This is the main effect\nresponsible for the deformation of the wave profile. When 1 −β+≪1 (satisfied in GHz waves), one can simplify\n1 +ββs≈1 +βin the denominators in Equations (A30). Retaining βsin the denominators is required in kHz waves\n(see Paper I) because in that case β+significantly decreases below unity and even changes sign.\nSubstituting β= (κ2−1)/(κ2+ 1), one can state Equations (A30) in terms of κ. Using γ2\ns−1≈γ2\ns, we obtain\n∂+lnκ=1−κ−2\nr[1 + (2 γsκ)−2], ∂ +r= 1−2\n(2γsκ)2+ 1. (A31)\nFor a cold plasma γ2\ns=σ=κσbg(see below), and then Equations (A31) reproduce Equations (43), (45) in Paper I.\nA.5. Fast magnetosonic speed\nFrom the definition of βs(Equation A19), one finds\n1\nγ2s= 1−β2\ns=d(H−2P)\nd(H−P)=d(Hp−2Pp)\nd(Pf+Hp−Pp), (A32)\nwhere we used Hf= 2Pf. It is convenient to express γsin terms of σ=Hf/˜ρ, where ˜ ρ=ρ/γis the proper rest-mass\ndensity of the plasma. For a cold plasma Hp= ˜ρandPp= 0, and Equation (A32) gives\nγs=√\n1 +σ, β s=rσ\n1 +σ. (A33)\nFor a hot plasma,\nHp= ˜ρ+Up+Pp, (A34)\nwhere Upis the thermal energy density (measured in the fluid frame). Note that Pphere is the plasma pressure in the\nradial direction; this fact becomes important if the plasma is anisotropic.22\nA useful analytical expression for γscan be derived in the limit of Hf/Hp≫1, which corresponds to γs≫1. Then,\nEquation (A32) simplifies to\n1\nγ2s=d(Hp−2Pp)\ndPf\u0014\n1 +O\u0012Hp\nH\u0013\u0015\n. (A35)\nIt can be rewritten using the magnetic flux freezing condition for short waves, ˜B/˜ρ=Bbg/ρbg, which implies\ndlnPf\ndln ˜ρ= 2dln˜B\ndln ˜ρ= 2 ⇒1\nγ2s=˜ρ\nHfd\nd˜ρ(˜ρ+Up−Pp) ( γs≫1). (A36)\nIt can be further simplified using the equation of state that relates Upand the radial pressure Pp. For a hot plasma\nwith mono-energetic particles this relation is\nPp=Up\nk\u0012\n1 +1\nε\u0013\n, ε ≡˜ρ+Up\n˜ρ, (A37)\nwhere k= 2 if the thermal velocity distribution is two-dimensional (confined to the plane perpendicular to B), and\nk= 3 if the plasma is isotropic. For a Maxwellian plasma, the same expression holds at all εwith better than 5%\naccuracy. Equation (A36) now becomes\n1\nγ2s=˜ρ\nkHf\u0014\n(k−1)ε+1\nε+\u0012\nk−1−1\nε2\u0013dε\ndln ˜ρ\u0015\n. (A38)\nWhen radiative losses are small during each wave oscillation, one can use the adiabatic law d(Up/˜ρ) =−Ppd(1/˜ρ)\nalong the plasma streamline. Plasma compression in a short wave is one-dimensional (in the radial direction), so only\nthe radial pressure Ppenters the adiabatic law. It gives\ndε\ndln ˜ρ=Pp=1\nk\u0012\nε−1\nε\u0013\n⇒1\nγ2s=˜ρ\nk2Hf\u0014\n(k2−1)ε+1\nε3\u0015\n(γs≫1). (A39)\nB.JUMP CONDITIONS FOR PERPENDICULAR SHOCKS\nJump conditions are formulated in the shock rest frame. In this section, it will be denoted by K′, and quantities\nmeasured in frame K′are denoted with a prime. Indices “u” and “d” will refer to the plasma in the immediate\nupstream and immediate downstream of the shock. The jump conditions state the continuity of particle flux F′= ˜nu′,\nenergy flux Tt′r′, and momentum flux Tr′r′. Magnetic flux freezing ˜B∝˜nimplies that proper density ˜ ρ=m˜nand\nmagnetization parameter σ≡˜B2/4π˜ρ∝˜ρjump at the shock by the same factor (denoted as q). The continuity of F′\nyields\nmF′=u′\nd˜ρd=u′\nu˜ρu,σd\nσu=q≡˜ρd\n˜ρu. (B40)\nThe ultra-relativistic motion of the shock relative to the plasma implies |u′\nu|>|u′\nd| ≫1. Since we know the evolution\nof ˜ρ=κρbgalong each C+in the simulation, the shock compression factor q(t) will be known if we keep track which\ncharacteristics C+\nuandC+\ndterminate at the shock at time t. Note also that q=κd/κu. The colliding characteristics\nC+\nuandC+\ndalso define the Lorentz factor of the upstream relative to the downstream, Γ rel=γuγd(1−βuβd). It can\nbe expressed in the shock frame as Γ rel= (u′\nu/u′\nd+u′\nd/u′\nu)/2 (using |u′\nu|,|u′\nd| ≫1), which gives Γ rel= (q+q−1)/2.\nThet′r′andr′r′components of the total stress-energy tensor (plasma + electromagnetic field) have the ideal-fluid\nform, same as in the lab frame (Equations (A3) and (A12)) but with the fluid four-velocity measured in frame K′,\nuα′= (γ′, u′,0,0). The continuity of F′,Tt′r′, and Tr′r′gives\nTt′r′\nmF′=γ′\ndhd=γ′\nuhu,Tr′r′\nmF′=u′\ndhd+pd\nu′\nd+σd\n2u′\nd=u′\nuhu+pu\nu′u+σu\n2u′u, (B41)\nwhere p≡Pp/˜ρandh≡ε+p+σ. We wish to find the downstream specific energy εdin terms of the upstream param-\neters and the shock compression factor q. We use the continuity of Tt′r′to express hd=huγ′\nu/γ′\nd, and Equation (B40)\nto exclude σd=qσuandu′\nd=u′\nu/q. Then, the continuity of Tr′r′gives\npd=\u00121\nq−γ′\nu\nq2γ′\nd\u0013\nu′\nu2hu−\u0012\nq−1\nq\u0013σu\n2+pu\nq. (B42)23\nRatio γ′\nu/γ′\ndcan be expressed in terms of q=u′\nu/u′\ndandu′\nu. The calculation simplifies for magnetically dominated\nshocks, since we can use |u′\nu|,|u′\nd| ≫1 and expand γ′\nu/γ′\nd=q(1 + u′\nu−2)1/2(1 + u′\nd−2)−1/2in the small parameter\nu′−2≪1. We expand up to the second-order terms ∼u′−4; this is needed because larger terms get canceled with the\nterm ( q−q−1)σu/2 in Equation (B42). As a result, we find\n2qpd= (q2−1)εu+ (q2+ 1)pu−(3q2+ 1)ψ, ψ ≡hu(q2−1)\n4u′u2. (B43)\nIn the relation hd=huγ′\nu/γ′\ndit is sufficient to expand γ′\nu/γ′\ndup to the liner order in u′−2. This gives, after cancelation\nof two large terms proportional to σu,\nεd+pd=q(εu+pu−2ψ). (B44)\nNote that pandεare not independent — they are related by the equation of state p= (ε−ε−1)/k(Equation A37).\nTherefore, Equations (B43) and (B44) form a closed set for two unknowns εdandψ. We use Equation (B44) to express\nψin terms of εd, substitute it into Equation (B43), and obtain a quadratic equation for εd:\n\u0002\n(3k−1)q2+k+ 1\u0003\nε2\nd−\u001a\u0002\n(k+ 1)q2+ 3k−1\u0003\nεu+1−q2\nεu\u001b\nq εd+q2−1 = 0 . (B45)\nOne should choose the larger root of the quadratic equation, as this branch satisfies εd=εuatq= 1.\nNext, we find the shock speed in the lab frame βshusing the relation βsh= (βu−β′\nu)/(1−βuβ′\nu). It gives\n1−βsh=(1−βu)(1 + β′\nu)\n1−βuβ′u≈1−βu\n1 +βu(1 +β′\nu)≈1\n2u′u2κ2u, (B46)\nwhere we used 1 + β′\nu≈1/2u′\nu2≪1 +βu. Using the definition of ψ(Equation B43) and Equation (B44), we find\n1−βsh≈2ψ\nκ2uhu(q2−1)=1\nκ2uhu(q2−1)\u0012\nεu+pu−εd+pd\nq\u0013\n. (B47)\nHere, one can substitute hu≈σu=κuσbg≫εu+puandp= (ε−ε−1)/kto obtain the final expression (Equation 26)\nfor 1−βshin terms of κu,εu,q, and the found εd.\nNote also that the shock four-velocity relative to the upstream is equal to −u′\nuand related to βshby Equation (B46).\nThe shock four-velocity relative to the downstream equals −u′\nd=−u′\nu/q. In the case of a cold upstream εu= 1 and\nq≫1, the above relations give u′\nd2=σu(3k−1)/[4(k−1)].\nC.WAVE PROPAGATION AND SHOCK FORMATION OUTSIDE THE EQUATORIAL PLANE\nC.1. Equations for γ(t, ξ)andE(t, ξ)\nAt a given polar angle θ, the coupled oscillation of γ(t, ξ) and E(t, ξ) in a cold GHz wave (before shock heating)\nis described by Equation (10). One can use it to obtain a wave equation containing only derivatives of γand no\nderivatives of E. This can be accomplished using the relation between Eandγfound in Section 4.1. As a first step,\nexpress ∂tEin terms of ∂tγDusing\nγ2\nD−1 =E2\nB2−E2≈E2\nB2\nbg+ 2Bθ\nbgE. (C48)\nWe take ∂t|ξof both sides, use ∂tBbg=−3cBbg/r, and find (in this section, we do not use the units of c= 1 and so\nretain cin all equations)\nE ∂tE=B4\nB2\nbg+Bθ\nbgE∂tγD\nγ3\nD−3cE2\nr. (C49)\nHere, we substitute\n∂tγD=∂tγ−γD∂t˜γ\n˜γ, (C50)\nwhich follows from γ= ˜γγD. It remains to evaluate ∂t˜γ. Note that ˜ γis a function of angle αandE/B bg(Equation 58),\nandα(θ) is constant in ∂t˜γ, so we find\n∂t˜γ\n˜u=∂t˜u\n˜γ=−s ∂t\u0012E\nBbg\u0013\n⇒ γD∂t˜γ=−γ2\nD˜uEBr\nbg\nB3\u0012\n∂tE+3cE\nr\u0013\n. (C51)24\nSubstitution of Equations (C50) and (C51) into Equation (C49) gives\nE ∂tE=∂tγ\nf−3cE2\nr,where f≡(B2\nbg+Bθ\nbgE)γ3−˜uBr\nbgBγ2\nB4˜γ2. (C52)\nSubstituting this result into Equation (10), we obtain the equation for γ(t, ξ) stated in the main text (Equation 61).\nDerivation of the equation for w=rEinvolves rewriting dγ/dξ on the r.h.s. of Equation (10) in terms of ∂twand\n∂ξw. We here outline the steps of the derivation, omitting the algebra details, and give the final result. For waves with\nγ3≪σbg, it is reduced to a simple statement: w≈const along C+. The formal derivation can start from γ= ˜γγD, use\nthe expressions for dγD/dξandd˜γ/dξ in terms of dE/dξ anddB/dξ , and substitute d/dξ =∂ξ+(1−βr)−1[∂t+(vθ/r)∂θ]\nfor the derivative along the fluid streamline. B∂ξB=Bθ∂ξBθ+Br∂ξBrcan be expressed in terms of the derivatives\nofE(orw) using the induction equation ∂tB|r=−c∇ ×E|trewritten in coordinates ( t, ξ). The final result is\n1\nEd(rE)\ndt\f\f\f\f\nC+=B2\nbg\n2σbgE2\u0000\ncE2f−rγ3\nDW\u0001\n=O\u0012γ3\nσbg\u0013\n, (C53)\nwhere\nW=˜γE2\nB3\u0014cEB θ\nrB−∂tB+cBr∂θ(Esinθ)\nrBsinθ\u0015\n+E\nB2(1−βr)\u0014\u0012\n˜γ−˜uBr\nbg\nγDB\u0013\u0010\n∂tE+vθ\nr∂θE\u0011\n−˜γE\nB\u0010\n∂tB+vθ\nr∂θB\u0011\u0015\n.\nC.2. Shock formation\nThe ratio of Equations (63) and (64) that govern the C+flow gives\ndξ+\ndγ=πmc n bgr\nE2=πmcNθ\nµ2K2⇒ ξ+=ξi+πmcNθ\nµ2K2(γ−1), (C54)\nwhere we substituted E≈µK/r along C+(which holds for waves with γ3≪σbg) and used the initial condition γ= 1\natξ+=ξi. The stated relation between ξ+andγholds along each C+. Note that the plasma Lorentz factor γ= ˜γγD\nis a known function of EandBbg, and E=µK(ξi)/r. Thus, one can express ξ+in terms of ξi,r, and θ.\nVacuum wave propagation would correspond to ξ+=ξiandr=c(t−ξi) along C+. The MHD correction ξ+−ξi\nmay be evaluated using iteration, by substituting the vacuum solution for r,\nr=rvac=c(t−ξi). (C55)\nThen, the r.h.s. of Equation (C54) becomes a known function of ξiandt(and θ, which is constant along C+).\nDeformation of the C+flow with time, which eventually leads to shocks, is described by ( ∂ξ+/∂ξi)t. Viewing γas\na composite function γ[E(t, ξi),Bbg(t−ξi)], we can write\n∂γ\n∂ξi\f\f\f\f\nt=∂γ\n∂E\f\f\f\f\nBbg∂E\n∂ξi\f\f\f\f\nt+∂γ\n∂Bbg\f\f\f\f\nE·∂Bbg\n∂ξi\f\f\f\f\nt=∂γ\n∂E\f\f\f\f\nBbg \nµ˙K\nr+µK\nr2!\n+∂γ\n∂Bbg\f\f\f\f\nE·3cBbg\nr. (C56)\nHere ˙K≡dK/dξ i, and ( ∂γ/∂E )Bbgcan be found from Equations (57) and (C48),\n∂γ\n∂E\f\f\f\f\nBbg=Ef, (C57)\nwhere fis defined in Equation (C52). For short waves, the term containing ˙Kis dominant in Equation (C56), and\nthe other terms are negligible. Thus, we find\n∂ξ+\n∂ξi\f\f\f\f\nt= 1 +πmcNθ˙K\nµ2K3\u0002\nE2f−2(γ−1)\u0003\n. (C58)\nA caustic appears on C+that first reaches the condition ( ∂ξ+/∂ξi)t= 0. It can be found by calculating time tv(ξi)\nat which ( ∂ξ+/∂ξi)tvanishes, and then identifying the characteristic ξc\niwith the minimum tv. The result will also\ndetermine the caustic time tc=tv(ξc\ni) and the plasma Lorentz factor at the caustic γc. The calculation can be done\nnumerically. Below we derive the result analytically in two limits, γc≫1 and γc−1≪1.25\nC.2.1. Caustics with γc≫1\nThe limit of γ≫1 corresponds to γD≫1 and E2→B2. Note that the ratio γ/γD= ˜γremains finite: ˜ γ−1≈sinα.\nEquation (54) implies\nB2−E2≈B2\nbg+ 2Bθ\nbgE. (C59)\nHence, E2→B2corresponds to B≈ −E≈B2\nbg/2Bθ\nbg. In this limit, the function fgiven in Equation (C52) simplifies:\nE2f≈(B2\nbg+Bθ\nbgE)γ3\nE2˜γ2≈2γ3sin2α\n˜γ2≈2γ3sin4α(γ≫1), (C60)\nand Equations (63) simplify to\ndγ\ndt\f\f\f\f\nC+≈4c γ3sin4α\nr,dξ+\ndt≈4γ3sin6α\nσbg. (C61)\nWaves with γ3≪σbghave dξ+/dt≪1, which implies dt≈dr/c, so Equation (C61) can be integrated for γ(r),\n1\nγ2≈8 sin4αlnr⋆\nr≈8xsin4α, x ≡r⋆−r\nr⋆, (C62)\nwhere we used ln[(1 + x)−1] =x+O(x2) for x≪1, which corresponds to γ≫1. Substitution of the obtained γ(r)\nandr=c(t−ξ+) into Equation (C54) gives a cubic equation for r(t, ξi). Its solution verifies that rvac−r≪r⋆−r\nwhen γ3≪σbg, and so one can use r=rvac(Equation C55) in x, i.e.\nx=r⋆−c(t−ξi)\nr⋆. (C63)\nThe integration constant r⋆in Equation (C62) defines the radius where γwould diverge, however the characteristic\nwill become terminated at the shock before reaching r=r⋆. The radius r⋆(ξi) can be found for each C+with K < 0\nfrom B2−E2≈B2\nbg+ 2Bθ\nbgE= 0 using rE=µK:\nrB2\nbg+ 2Bθ\nbgµK= 0 ⇒ r2\n⋆=−sinθ\n2Ksin2α. (C64)\nSubstituting Equation (C60) into Equation (C58) and noting that E2f≫2(γ−1) when γ≫1, we find\n∂ξ+\n∂ξi\f\f\f\f\nt= 1 +2πmcNθ˙Kγ3sin4α\nµ2K3. (C65)\nTheC+characteristic reaches ( ∂ξ+/∂ξi)t= 0 when\nγ3=−µ2K3\n2πmcNθ˙Ksin4α. (C66)\nUsing the obtained solution for γ(x) (Equation C62) we find that ( ∂ξ+/∂ξi)t= 0 is reached at time\ntv(ξi) =ξi+r⋆\nc−r⋆\n27/3cK2 \nπmcNθ˙K\nµ2sin2α!2/3\n. (C67)\nThe minimum of tvcan be found from dtv/dξi= 0. Using 2 dr⋆/r⋆=−dK/K (Equation C64), we obtain\ndtv\ndξi= 1−(t−ξi)˙K\n2K−r⋆\n24/3c\u0012πmcNθ\nµ2sin2α\u00132/3K¨K−3˙K2\n3K3˙K1/3. (C68)\nHere, ( t−ξi)˙K/2K≈r⋆˙K/2cK≫1, and the condition dtv/dξi= 0 becomes\n21/3˙K4/3=\u0012πmcNθ\nµ2sin2α\u00132/33˙K2−K¨K\n3K2. (C69)26\nThis equation determines the Lagrangian coordinate ξc\niof the caustic in the C+flow with a given K(ξi). In particular,\nfor a wave with an initial sine profile, K=K0sin(ωξi), it gives\ncos4/3(ωξc\ni) =\u0012πmcNθω√\n2µ2K2\n0sin2α\u00132/3\u00141\n3+ cos2(ωξc\ni)\u0015\n≈1\n3\u0012πmcNθω√\n2µ2K2\n0sin2α\u00132/3\n. (C70)\nThe last (approximate) equality took into account the condition γc≫1 (E≈ −B), which implies that ξc\niis close to\nthe minimum of the sine profile of K(ξi), and so cos( ωξc\ni)≪1. The obtained ξc\nidetermines ˙K(ξc\ni) =ωK0cos(ωξc\ni),\nand then from Equation (C66) we find the plasma Lorentz factor at the caustic,\nγc≈\u00123\n2\u00131/4\u0012µ2K2\n0\nπmcNθωsin2α\u00131/2\n≈(3/2)1/4\n√ζsinθsinα. (C71)\nIn the last equality we subsituted Nθgiven by Equation (50) and used the parameter ζ=µ2K2\n0/πmcNdefined in\nthe equatorial plane (Equation 35). At θ=π/2, Equation (C71) reproduces γc= (2κc)−1given in Equation (36) and\nderived in Paper I. The obtained extension to θ̸=π/2 shows that γcincreases outside the equatorial plane. Recall that\nthis result was derived assuming γc≫1. This regime holds for waves with ζ≪1, as one can see from Equation (C71).\nC.2.2. Caustics with γc−1≪1\nTheC+flow with γ−1≪1 can be described using expansion in variable z≡E/B bg,|z| ≪1. From Equations (58)\nand (C59) we find\n˜u=O(z2), ˜γ= 1 + O(z4), B =Bbg\u0002\n1 +zsinα+O(z2)\u0003\n. (C72)\nThis determines βD=E/B,γD, and\nγ= ˜γγD= 1 +z2\n2(1−2zsinα) +O(z4). (C73)\nThen, Equation (62) gives\nE2f=z2(1−3zsinα) +O(z4). (C74)\nSubstituting these expansions into Equation (C58), we obtain\n∂ξ+\n∂ξi\f\f\f\f\nt= 1−πmcNθ˙K\nµ2K3z3sinα= 1−πmcNθ˙K\nµ2sin4α\nsin3θr6. (C75)\nHere, we substitute z≡E/B bg=µK/rB bg=r2Ksinα/sinθand find that ∂ξ+/∂ξivanishes when the characteristic\nreaches the radius\nrv(ξi) =\u0012µ2sin3θ\nπmcNθ˙Ksin4α\u00131/6\n. (C76)\nThe caustic appears where rvis minimum, i.e. where ˙K(ξi) reaches its maximum ˙K=ωK0(which occurs at ξi= 0).\nThus, we find that the caustic appears when the wave reaches the radius\nrc=\u0012µ2sin3θ\nπmcNθωK0sin4α\u00131/6\n. (C77)\nUsing the definition of ζ(Equation 35), r×(Equation 48), Nθ(Equation 50), and K−1\n0= 2R2\n×, one can rewrite rcas\nstated in Equation (66).27\nREFERENCES\nBeloborodov, A. M. 2013, ApJ, 777, 114,\ndoi: 10.1088/0004-637X/777/2/114\n—. 2017, ApJL, 843, L26, doi: 10.3847/2041-8213/aa78f3\n—. 2020, ApJ, 896, 142, doi: 10.3847/1538-4357/ab83eb\n—. 2021, ApJL, 922, L7, doi: 10.3847/2041-8213/ac2fa0\n—. 2022a, PhRvL, 128, 255003,\ndoi: 10.1103/PhysRevLett.128.255003\n—. 2022b, arXiv e-prints, arXiv:2210.13509,\ndoi: 10.48550/arXiv.2210.13509\nBochenek, C. D., Ravi, V., Belov, K. 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K., & Ravi, V. 2019, MNRAS, 485, L78,\ndoi: 10.1093/mnrasl/slz038\nZhang, B. 2022, arXiv e-prints, arXiv:2212.03972,\ndoi: 10.48550/arXiv.2212.03972"    },    {        "title": "0905.3242v1.Eigenvalue_asymptotics__inverse_problems_and_a_trace_formula_for_the_linear_damped_wave_equation.pdf",        "content": "arXiv:0905.3242v1  [math.SP]  20 May 2009EIGENVALUE ASYMPTOTICS, INVERSE PROBLEMS\nAND A TRACE FORMULA FOR THE LINEAR\nDAMPED WAVE EQUATION\nDENIS BORISOV AND PEDRO FREITAS\nAbstract. We determine the general form of the asymptotics for\nDirichlet eigenvalues of the one–dimensional linear damped wave\noperator. Asaconsequence,weobtainthatgivenaspectrumcor re-\nsponding to a constant damping term this determines the damping\nterm in a unique fashion. We also derive a trace formula for this\nproblem.\n1.Introduction\nConsider the one–dimensional linear damped wave equation on the\ninterval (0 ,1), that is,\n(1.1)\n\nwtt+2a(x)wt=wxx+b(x)w, x ∈(0,1), t >0\nw(0,t) =w(1,t) = 0, t > 0\nw(x,0) =w0(x), wt(x,0) =w1(x), x∈(0,1)\nThe eigenvalue problem associated with (1.1) is given by\nuxx−(λ2+2λa−b)u= 0, x∈(0,1), (1.2)\nu(0) =u(1) = 0, (1.3)\nand has received quite a lot of attention in the literature since the pa -\npers of Chen et al. [CFNS] and Cox and Zuazua [CZ]. In the first of\nthese papers the authors derived formally an expression for the a symp-\ntotic behaviour of the eigenvalues of (1.2), (1.3) in the case of a zer o\npotential b, whichwaslaterprovedrigorouslyinthesecondoftheabove\nDate: November 17, 2018.\n2000Mathematics Subject Classification. Primary 35P15; Secondary 35J05.\nD.B. was partially supported by RFBR (07-01-00037) and gratefully acknowl-\nedges the support from Deligne 2004 Balzan prize in mathematics. D.B . is\nalso supported by the grant of the President of Russia for young s cientist and\ntheir supervisors (MK-964.2008.1) and by the grant of the Preside nt of Russia\nfor leading scientific schools (NSh-2215.2008.1) P.F. was partially sup ported by\nFCT/POCTI/FEDER. .\n12 DENIS BORISOV AND PEDRO FREITAS\npapers. Following this, there were several papers on the subject which,\namong other things, extended the results to non–vanishing b[BR], and\nshowed that it is possible to design damping terms which make the\nspectral abscissa as large as desired [CC]. In [F2] the second autho r of\nthe present paper addressed the inverse problem in arbitrary dime n-\nsion giving necessary conditions for a sequence to be the spectrum of\nan operator of this type in the weakly damped case. As far as we\nare aware, these are the only results for the inverse problem asso ciated\nwith (1.2), (1.3). Other results for the n−dimensional problem include,\nfor instance, the fact that in that case the decay rate is no longer de-\ntermined solely by the spectrum [L], a study of some particular case s\nwhere the role of geometric optics is considered [AL], the asymptotic\nbehaviour of the spectrum [S] and the study of sign–changing damp ing\nterms [F1].\nThe purpose of the present paper is twofold. On the one hand,\nwe show that problem (1.2), (1.3) may be addressed in the same way\nas the classical Sturm–Liouville problem in the sense that, although\nthis is not a self–adjoinf problem, the methods used for the former\nproblemmaybeappliedherewithsimilarresults. Thisideawasalready\npresent in both [CFNS] and [CZ]. Here we take further advantage of\nthis fact to obtain the full asymptotic expansion for the eigenvalue s\nof (1.2), (1.3) (Theorem 1). Based on these similarities, we were also\nled to a (regularized) trace formula in the spirit of that for the Stur m–\nLiouville problem (Theorem 4).\nOn the other hand, the idea behind obtaining further terms in the\nasyptotics was to use this information to address the associated in verse\nspectral problem of finding all damping terms that give a certain spe c-\ntrum. Our main result along these lines is to show that in the case of\nconstant damping there is no other smooth damping term yielding the\nsame spectrum (Corollary 2). Namely, we obtain the criterion for th e\ndamping term to be constant. Note that this is in contrast with the\ninverse (Dirichlet) Sturm–Liouville problem, where for each admissible\nspectrum there will exist a continuum of potentials giving the same\nspectrum [PT]. In particular this result shows that we should expect\nthe inverse problem to be much more rigid in the case of the wave\nequation than it is for the Sturm–Liouville problem. This should be\nunderstood in the sense that, at least in the case of constant dam ping,\nit will not be possible to perturb the damping term without disturbing\nthe spectrum, as is the case for the potential in the Sturm–Liouville\nproblem.\nThe plan of the paper is as follows. In the next section we set the\nnotation and state the main results of the paper. The proof of theEIGENVALUE ASYMPTOTICS FOR THE DAMPED WAVE EQUATION 3\nasymptotics of the eigenvalues is done in Sections 3 and 4, where in\nthe first of these we derive the form of the fundamental solutions of\nequation (1.2), while in the second we apply a shooting method to\nthese solutions to obtain the formula for the eigenvalues as zeros o f an\nentire function – the idea is the same as that used in [CZ]. Finally, in\nSection 5 we prove the trace formula.\n2.Notation and results\nIt is easy to check that if λis an eigenvalue of the problem (1.2),\n(1.3), then λis also an eigenvalue of the same problem. In view of this\nproperty, we denote the eigenvalues of this problem by λn,n/ne}ationslash= 0, and\norder them as follows\n.../lessorequalslantImλ−2/lessorequalslantImλ−1/lessorequalslantImλ1/lessorequalslantImλ2/lessorequalslant...\nwhile assuming that λ−n=λn. We also suppose that possible zero\neigenvalues are λ±1=λ±2=...=λ±p= 0. Ifp= 0, the problem\n(1.2), (1.3) has no zero eigenvalues. For any function f=f(x) we\ndenote/an}bracketle{tf/an}bracketri}ht:=/integraltext1\n0f(x)dx.\nTheorem 1. Suppose a∈Cm+1[0,1],b∈Cm[0,1],m/greaterorequalslant1. The\neigenvalues of (1.2), (1.3) have the following asymptotic b ehaviour as\nn→ ±∞:\nλn=πni+m−1/summationdisplay\nj=0cjn−j+O(n−m), (2.1)\nwere the cj’s are numbers which can be determined explicitly. In par-\nticular,\nc0=−/an}bracketle{ta/an}bracketri}ht, c1=/an}bracketle{ta2+b/an}bracketri}ht\n2πi, (2.2)\nc2=1\n2π2/bracketleftbigg\n/an}bracketle{ta(a2+b)/an}bracketri}ht−/an}bracketle{ta/an}bracketri}ht/an}bracketle{ta2+b/an}bracketri}ht+a′(1)−a′(0)\n2/bracketrightbigg\n. (2.3)\nA straightforward consequence of the fact that the spectrum d eter-\nmines the average as well as the L2norm of the damping term (as-\nsumingbfixed) is that the spectrum corresponding to the constant\ndamping determines this damping uniquely.\nCorollary 2. Assume that a∈C3[0,1],λnare the eigenvalues of the\nproblem (1.2), (1.3), the function b∈C2[0,1]is fixed, and the formula\n(2.1) gives the asymptotics for these eigenvalues. Then the function\na(x)is constant, if and only if\nc2\n0= 2πic1−/an}bracketle{tb/an}bracketri}ht,4 DENIS BORISOV AND PEDRO FREITAS\nin which case a(x)≡ −c0.\nIn the same way, the asymptotic expansion allows us to derive other\nspectral invariants in terms of the damping term a. However, these\ndo not have such a simple interpretation as in the case of the above\nconstant damping result.\nCorollary 3. Suppose b≡0,ai(x) =a0(x) +/tildewideai(x),i= 1,2, where\na0(1−x) =a0(x),/tildewideai(1−x) =−/tildewideai(x),/tildewideai,a0∈C4[0,1], and for a=ai\nthe problems (1.2), (1.3) have the same spectra. Then\n/an}bracketle{t/tildewidea2\n1/an}bracketri}ht=/an}bracketle{t/tildewidea2\n2/an}bracketri}ht,/an}bracketle{ta0/tildewidea2\n1/an}bracketri}ht=/an}bracketle{ta0/tildewidea2\n2/an}bracketri}ht\nis valid.\nFrom Theorem 1 we have that the quantity Re( λn−c0) behaves as\nO(n−2) asn→ ∞. This means that the series\n∞/summationdisplay\nn=−∞\nn/negationslash=0(λn−c0) = 2∞/summationdisplay\nn=1Re(λn−c0)\nconverges. In the following theorem we express the sum of this ser ies in\nterms of the function a. This is in fact the formula for the regularized\ntrace.\nTheorem 4. Leta∈C3[0,1],b∈C2[0,1]. Then the identity\n∞/summationdisplay\nn=−∞\nn/negationslash=0(λn−c0) =a(0)+a(1)\n2−/an}bracketle{ta/an}bracketri}ht\nholds.\n3.Asymptotics for the fundamental system\nInthissectionweobtaintheasymptoticexpansionforthefundame n-\ntal system of the solutions of the equation (1.2) as λ→ ∞,λ∈C. This\nis done by means of the standard technique described in, for instan ce,\n[E, Ch. IV, Sec. 4.2, 4.3], [Fe, Ch. II, Sec. 3].\nWe begin with the formal construction assuming the asymptotics to\nbe of the form\n(3.1) u±(x,λ) = e±λx±xR\n0φ±(t,λ)dt\n,\nwhere\n(3.2) φ±(x,λ) =m/summationdisplay\ni=0φ(±)\ni(x)λ−i+O(λ−m−1), m/greaterorequalslant1.EIGENVALUE ASYMPTOTICS FOR THE DAMPED WAVE EQUATION 5\nIn what follows we assume that a∈Cm+1[0,1],b∈Cm[0,1].\nWe substitute the series (3.1), (3.2) into (1.2) and equate the coef -\nficients of the same powers of λ. It leads us to a recurrent system of\nequations determining φ(±)\niwhich read as follows:\nφ(±)\n0=a, (3.3)\nφ(±)\n1=−1\n2(±a′+a2+b), (3.4)\nφ(±)\ni=−1\n2/parenleftBigg\n±φ(±)\ni−1′+i−1/summationdisplay\nj=0φ(±)\njφ(±)\ni−j−1/parenrightBigg\n, i/greaterorequalslant2. (3.5)\nThe main aim of this section is to prove that there exist solutions to\n(1.2) having the asymptotics (3.1), (3.2). In other words, we are g oing\nto justify these asymptotics rigorously. We will do this for u+, the case\nofu−following along similar lines.\nLet us write\nUm(x,λ) = eλx+mP\ni=0λ−ixR\n0φ(+)\ni(t)dt\n.\nIn view of the assumed smoothness for aandbwe conclude that Um∈\nC2[0,1]. It is also easy to check that\n(3.6)U′′\nm−λ2Um−2λaUm+bUm=λ−meλxfm(x,λ), x∈[0,1],\nUm(0) = 1, U′\nm(0) =λ+m/summationdisplay\ni=0φ(+)\ni(0)λ−i,\nwhere the function fmsatisfies the estimate\n|fm(x,λ)|/lessorequalslantCm\nuniformly for large λandx∈[0,1]\nWe consider first the case Re λ/greaterorequalslant0. Differentiating the function u+\nformally we see that\nu′\n+(0,λ) =λ+φ+(0,λ) =λ+m/summationdisplay\ni=0φ(+)\ni(0)λ−i+O(λ−m−1).\nLet\nA0(λ) =λ+m/summationdisplay\ni=0φ(+)\ni(0)λ−i,\nandu+(x,λ) be the solution to the Cauchy problem for the equation /diamondsolid\n(1.2) subject to the initial conditions\nu+(0,λ) = 1, u′\n+(0,λ) =A0(λ).6 DENIS BORISOV AND PEDRO FREITAS\nWe introduce one more function wm(x,λ) =u+(x,λ)/Um(x,λ). This\nfunction solves the Cauchy problem\n(U2\nmw′\nm)′+λ−mUmeλxfmwm= 0, x∈[0,1],\nwm(0,λ) = 1, w′\nm(0,λ) = 0.\nThe last problem is equivalent to the integral equation\nwm(x,λ)+λ−m(Km(λ)wm)(x,λ) = 1,\n(Km(λ)wm)(x,λ) :=x/integraldisplay\n0U−2\nm(t1)t1/integraldisplay\n0Um(t2)eλt2fm(t2,λ)wm(t2,λ)dt2dt1.\nSince Re λ/greaterorequalslant0 for 0/greaterorequalslantt2/greaterorequalslantt1/greaterorequalslant1, the estimate\n|U−2\nm(t1,λ)Um(t2,λ)eλt1|/lessorequalslantCm\nholds true, where the constant Cmis independent of λ,t1,t2. Hence,\nthe integral operator Km:C[0,1]→C[0,1] is bounded uniformly in λ\nlarge enough, Re λ/greaterorequalslant0. Employing this fact, we conclude that\nwm(x) = 1+O(λ−m), λ→ ∞,Reλ/greaterorequalslant0,\nin theC2[0,1]-norm. Hence, the formula (3.1), where\n(3.7) φ+(x,λ) =m−1/summationdisplay\ni=0φ(+)\ni(x)λ−i+O(λ−m),\ngives the asymptotic expansion for the solution of the Cauchy prob lem\n(1.2), (3.6) as λ→ ∞, Reλ/greaterorequalslant0.\nSuppose now that Re λ/lessorequalslant0. LetA1(λ),A2(λ) be functions having\nthe asymptotic expansions\nA1(λ) =λ+m/summationdisplay\ni=0λ−i1/integraldisplay\n0φ(+)\ni(x)dx, A 2(λ) =λ+m/summationdisplay\ni=0φ(+)\ni(1)λ−i.\nWe define the function /tildewideu+(x,λ) as the solution to the Cauchy problem\nfor equation (1.2) subject to the initial conditions\n/tildewideu+(1,λ) = eA1(λ),/tildewideu′\n+(1,λ) =A2(λ)eA1(λ).\nIn a way analogous to the arguments given above, it is possible to\ncheck that the function /tildewideu+has the asymptotic expansion (3.1) in the\nC2[0,1]-norm as λ→+∞, Reλ/lessorequalslant0. Hence,/tildewideu+(0,λ) = 1 +O(λ−m)\nfor eachm/greaterorequalslant1. In view of this identity we conclude that the function\nu+(x,λ) :=/tildewideu+(x,λ)//tildewideu+(0,λ) is a solution to (1.2), satisfies the condi-\ntionu+(0,λ) = 1, and has the asymptotic expansion (3.1), where the\nasymptotics for φ+is given in (3.7).EIGENVALUE ASYMPTOTICS FOR THE DAMPED WAVE EQUATION 7\nFor convenience we summarize the obtained results in\nLemma 3.1. Leta∈Cm+1[0,1],b∈Cm[0,1]. There exist two linear\nindependent solutions to the equation (1.2) satisfying the initial con-\nditionu±(0,λ) = 1and having the asymptotic expansions (3.1) in the\nC2[0,1]-norm as λ→ ∞,λ∈C, where\nφ±(x,λ) =m−1/summationdisplay\ni=0φ(±)\ni(x)λ−i+O(λ−m).\n4.Asymptotics of the eigenvalues\nThis section is devoted to the proof of Theorem 1 and Corollaries 2\nand 3. We assume that a∈Cm+1[0,1],b∈Cm[0,1],m/greaterorequalslant1.\nLetu=u(x,λ) be the solution to (1.2) subject to the initial condi-\ntionsu(0,λ) = 0,u′(0,λ) = 1. Denote γ0(λ) :=u(1,λ). The function\nγ0is entire, and its zeros coincide with the eigenvalues of the problem\n(1.2), (1.3). It follows from Lemma 3.1 that, for λlarge enough the\nfunction u(x,λ) can be expressed in terms of u±by\nu(x,λ) =u+(x,λ)−u−(x,λ)\nu′\n+(0,λ)−u′\n−(0,λ).\nThe denominator is non-zero, since due to (3.1)\nu′\n+(0,λ)−u′\n−(0,λ) = 2λ+2/an}bracketle{ta/an}bracketri}ht+O(λ−1), λ→ ∞.\nThus, for λlarge enough\n(4.1) γ0(λ) =u+(1,λ)−u−(1,λ)\nu′\n+(0,λ)−u′\n−(0,λ).\nLemma 4.1. Fornlarge enough, the set\nQ:={λ:|Reλ|< πn+π/2,|Imλ|< πn+π/2}\ncontains exactly 2neigenvalues of the problem (3.1), (3.3).\nProof.Letγ1(λ) :=γ0(λ)eλ+/angbracketlefta/angbracketright. The zeros of γ1are those of γ0(λ). For\nλlarge enough we represent the function γ1(λ) as /diamondsolid\nγ1(λ) =γ2(λ)+γ3(λ), γ2:=e2(λ+a(0))−1\n2(λ+a(0)),\nγ3(λ) =−γ2(λ)/tildewideφ+(0,λ)+/tildewideφ−(0,λ)+2(1+ λ−1a(0))(1−eλ−1/angbracketlefteφ+(·,λ)/angbracketright)\n2λ(λ+a(0))+/tildewideφ+(0,λ)+/tildewideφ−(0,λ)\n+eλ−1/angbracketlefteφ+(·,λ)/angbracketright−e−λ−1/angbracketlefteφ−(·,λ)/angbracketright\n2(λ+a(0))+λ−1(/tildewideφ+(0,λ)+/tildewideφ−(0,λ)),8 DENIS BORISOV AND PEDRO FREITAS\n/tildewideφ±(x,λ) :=λ−1(φ±(x,λ)−a(x)).\nItisclear thatfor λlargeenoughthefunction γ3(λ) satisfies anuniform\ninλestimate\n|γ3(λ)|/lessorequalslantC|λ|−2/parenleftbig\n|γ2(λ)|+1/parenrightbig\n.\nOne can also check easily that /diamondsolid\n|γ2(λ)|/greaterorequalslantC|λ|, λ∈∂K,\nifnis large enough. These two last estimates imply that |γ3(λ)|/lessorequalslant\n|γ2(λ)|asλ∈∂K, ifnislargeenough. ByRouch´ etheoremweconclude\nthat for such nthe function γ1has the same amount of zeros inside\nQas the function γ2does. Since the zeros of the latter are given by\nπni−/an}bracketle{ta/an}bracketri}ht,n/ne}ationslash= 0, this completes the proof. /square\nProof of Theorem 1. Assume first that a∈C2[0,1],b∈C1[0,1]. As\nwas mentioned above, the eigenvalues of problem (1.2), (1.3) are th e\nzeros of the function γ0(λ) = 0. It follows from Lemma 4.1 that these\neigenvalues tend to infinity as n→ ∞. By Lemma 3.1, for λlarge\nenough the equation γ0(λ) = 0 becomes\ne2λ+/angbracketleftφ+(·,λ)+φ−(·,λ)/angbracketright= 0\nwhich may be rewritten as\n(4.2) 2 λ+/an}bracketle{tφ+(·,λ)+φ−(·,λ)/an}bracketri}ht= 2πni, n∈Z.\nIf we now replace φ±by the leading terms of their asymptotic expan-\nsions we obtain\n2λ+2/an}bracketle{ta/an}bracketri}ht+O(λ−1) = 2πni, (4.3)\nλ=πni−/an}bracketle{ta/an}bracketri}ht+o(1), n→ ∞.\nHence, the eigenvalues behave as λ∼πni−/an}bracketle{ta/an}bracketri}htfor large n. Moreover,\nit follows from Lemma 4.1 that it is exactly the eigenvalue λnwhich\nbehaves as\nλn=πni−/an}bracketle{ta/an}bracketri}ht+o(1), n→ ∞.\nIt follows from this identity and (4.3) that\nλn=πni−/an}bracketle{ta/an}bracketri}ht+O(n−1), n→ ∞,\nand we complete the proof in the case m= 1. Ifm= 2, we substitute\nthe above identity and (3.1) into (4.2) and get\nλn+/an}bracketle{ta/an}bracketri}ht+1\nλn/an}bracketle{tφ(+)\n1+φ(−)\n1/an}bracketri}ht+O(λ−2\nn) =πni,\nλn=πni−/an}bracketle{ta/an}bracketri}ht−/an}bracketle{tφ(+)\n1+φ(−)\n1/an}bracketri}ht\nπni+O(n−2).EIGENVALUE ASYMPTOTICS FOR THE DAMPED WAVE EQUATION 9\nThe last formula and the identities (3.4) yield formulas (2.2) for c0and\nc1. Repeating the described procedure one can easily check that the\nasymptotics (2.1), (2.2) hold true. /square\nProof of Corollary 2. The coefficients c0,c1in the asymptotics (2.1)\nare determined by the formulas (2.2) and, by the Cauchy-Schwarz in-\nequality, we thus obtain\nc2\n0=/an}bracketle{ta/an}bracketri}ht2/lessorequalslant/an}bracketle{ta2/an}bracketri}ht= 2πic1−/an}bracketle{tb/an}bracketri}ht,\nwith equality if and only if a(x) is a constant function. This fact\ncompletes the proof. /square\nProof of Corollary 3. Itfollowsfrom(2.2),(2.3)that /an}bracketle{ta2\n1/an}bracketri}ht=/an}bracketle{ta2\n2/an}bracketri}ht,/an}bracketle{ta3\n1/an}bracketri}ht=\n/an}bracketle{ta3\n2/an}bracketri}ht. Now we check that\n/an}bracketle{ta2\ni/an}bracketri}ht=/an}bracketle{ta2\n0/an}bracketri}ht+/an}bracketle{t/tildewidea2\ni/an}bracketri}ht,/an}bracketle{ta3\ni/an}bracketri}ht=/an}bracketle{ta3\n0/an}bracketri}ht+3/an}bracketle{ta0/tildewidea2\ni/an}bracketri}ht, i= 1,2,\nand arrive at the statement of the theorem. /square\n5.Regularized trace formulas\nIn this section we prove Theorem 4. We follow the idea employed in\nthe proof of the similar trace formula for the Sturm-Liouville operat ors\nin [LS, Ch. I, Sec. 14].\nWe begin by defining the function\nΦ(λ) :=λ2p∞/productdisplay\nn=p+1/parenleftbigg\n1−λ\nλn/parenrightbigg/parenleftbigg\n1−λ\nλn/parenrightbigg\n.\nThe above product converges, since\n/parenleftbigg\n1−λ\nλn/parenrightbigg/parenleftbigg\n1−λ\nλn/parenrightbigg\n= 1+λ2−2λReλn\n|λn|2,\nand by Theorem 1 we have\n(5.1)|λn|2=π2n2−2πic1+c2\n0+O(n−2),\nReλn=c0+O(n−2)\nasn→+∞. Proceeding in the same way as in the formulas (14.8),\n(14.9) in [LS, Ch. I, Sec. 14], we obtain\nΦ(λ) =C0Ψ(λ)sinhλ\nλ,\nΨ(λ) :=∞/productdisplay\nn=1/parenleftbigg\n1−π2n2−|λn|2+2λReλn\nπ2n2+λ2/parenrightbigg\n,10 DENIS BORISOV AND PEDRO FREITAS\nC0:= (πn)2p∞/productdisplay\nn=p+1π2n2\n|λn|2.\nIn what follows we assume that λis real, positive and large. In the\nsame way as in [LS, Ch. I, Sec. 14] it is possible to derive the formula\n(5.2) lnΨ( λ) =−∞/summationdisplay\nk=11\nk∞/summationdisplay\nn=1/parenleftbiggπ2n2−|λn|2+2λReλn\nπ2n2+λ2/parenrightbiggk\n.\nOur aim is to study the asymptotic behaviour of lnΨ( λ) asλ→+∞.\nEmploying the same arguments as in the proof of Lemma 14.1 and in\nthe equation (14.11) in [LS, Ch. I, Sec. 14], we arrive at the estimate\n∞/summationdisplay\nn=1/parenleftbiggπ2n2−|λn|2+2λReλn\nπ2n2+λ2/parenrightbiggk\n/lessorequalslantckλk∞/summationdisplay\nn=11\n(π2n2+λ2)k\n/lessorequalslantckλk+∞/integraldisplay\n0dt\n(π2t2+λ2)k=ck\nλk+∞/integraldisplay\n0dz\n(π2z2+1)k/lessorequalslantck+1\nλk,\n∞/summationdisplay\nk=31\nk∞/summationdisplay\nn=1/parenleftbiggπ2n2−|λn|2+2λReλn\nπ2n2+λ2/parenrightbiggk\n=O(λ−3), λ→+∞, (5.3)\nwherecis a constant independent of kandn. Let us analyze the\nasymptotic behaviour of the first two terms in the series (5.2). As\nk= 1, we have\n(5.4)∞/summationdisplay\nn=1π2n2−|λn|2+2λReλn\nπ2n2+λ2=∞/summationdisplay\nn=1π2n2−|λn|2−2πic1+c2\n0\nπ2n2+λ2\n+/parenleftbig\n2πic1−c2\n0+2λc0/parenrightbig∞/summationdisplay\nn=11\nπ2n2+λ2\n+2λ−1S−2λ−1∞/summationdisplay\nn=1π2n2(Reλn−c0)\nπ2n2+λ2,\nwhereS:=∞/summationdisplay\nn=1(Reλn−c0).\nTaking into account (5.1), we have\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay\nn=1π2n2−|λn|2−2πic1+c2\n0\nπ2n2+λ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslantC∞/summationdisplay\nn=11\nn2(π2n2+λ2)\n=π2\n63+λ2−3cothλ\nλ4/lessorequalslantCλ−2,EIGENVALUE ASYMPTOTICS FOR THE DAMPED WAVE EQUATION 11\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay\nn=1(Reλn−c0)π2n2\nπ2n2+λ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslantC∞/summationdisplay\nn=11\nπ2n2+λ2/lessorequalslantCλ−1,\nwhere the constant Cis independent of λ. Here we have also used the\nformula\n(5.5)∞/summationdisplay\nn=11\nπ2n2+λ2=λcothλ−1\n2λ2=λ−1−λ−2\n2+O(λ−1e−2λ)\nasλ→+∞. We employ this formula to calculate the remaining terms\nin (5.4) and arrive at the identity\n(5.6)\n∞/summationdisplay\nn=1π2n2−|λn|2+2λReλn\nπ2n2+λ2=c0+/parenleftbigg\n2S−c0−c2\n0\n2+iπc1/parenrightbigg\nλ−1+O(λ−2),\nasλ→+∞. Fork= 2 we proceed in the similar way,\n∞/summationdisplay\nn=1/parenleftbiggπ2n2−|λn|2+2λReλn\nπ2n2+λ2/parenrightbigg2\n=∞/summationdisplay\nn=1(π2n2−|λn|2)2\n(π2n2+λ2)2\n−2λ∞/summationdisplay\nn=1(π2n2−|λn|2)Reλn\n(π2n2+λ2)2+4λ2c2\n0∞/summationdisplay\nn=11\n(π2n2+λ2)2\n+4λ2∞/summationdisplay\nn=1(Reλn)2−c2\n0\n(π2n2+λ2)2.\nBy differentiating (5.5) we obtain\n∞/summationdisplay\nn=11\n(π2n2+λ2)2=λcothλ−2−λ2(1−coth2λ)\n4λ4.\nThis identity and (5.1) yield that as λ→+∞\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay\nn=1(π2n2−|λn|2)2\n(π2n2+λ2)2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslantC∞/summationdisplay\nn=11\n(π2n2+λ2)2/lessorequalslantCλ−3,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay\nn=1(π2n2−|λn|2)Reλn\n(π2n2+λ2)2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslantC∞/summationdisplay\nn=11\n(π2n2+λ2)2/lessorequalslantCλ−3,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay\nn=1(Reλn)2−c2\n0\n(π2n2+λ2)2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslantC∞/summationdisplay\nn=11\n(π2n2+λ2)2/lessorequalslantCλ−3.\nHence,\n(5.7)∞/summationdisplay\nn=1/parenleftbiggπ2n2−|λn|2+2λReλn\nπ2n2+λ2/parenrightbigg2\n=c2\n0λ−1+O(λ−2)12 DENIS BORISOV AND PEDRO FREITAS\nasλ→+∞. It follows from (5.2), (5.3), (5.6), (5.7) that\nlnΨ(λ) =−c0−(2S−c0+iπc1)λ−1+O(λ−2),\nΦ(λ) =C0e−c0sinhλ\nλ/bracketleftbig\n1−(2S−c0+iπc1)λ−1+O(λ−2)/bracketrightbig\n(5.8)\nasλ→+∞. It follows from (4.1) and Lemma 3.1 that for λlarge\nenough the estimate /diamondsolid\n|γ0(λ)|/lessorequalslantC|λ|−1e|λ|\nholds true. Hence, the order of the entire function γ0(λ) is one. In view\nof Theorem 1 we also conclude that the series∞/summationtext\nn=p+1|λn|−2converges\nand therefore the genus of the canonical product associated wit hγ0is\none. We apply Hadamard’s theorem (see, for instance, [Le, Ch. I, Sec.\n10, Th. 13]) and obtain that\nγ0(λ) = eP(λ)Φ(λ), P(λ) =α1λ+α0+2∞/summationdisplay\nn=p+1|λn|−2Reλn,\nwhereα1,α0are some numbers. Hence, due to (5.8), it follows that γ0\nbehaves as\nγ0(λ) =C0eP(λ)sinhλ\nλ/bracketleftbig\n1−(2S−c0+iπc1)λ−1+O(λ−2)/bracketrightbig\n,\nasλ→+∞. On the other hand, Lemma 3.1 and (4.1) imply that\nγ0(λ) =eλ+/angbracketlefta/angbracketright\n2λ/bracketleftbig\n1+(/an}bracketle{tφ(+)\n1/an}bracketri}ht−a(0))λ−1+O(λ−2)/bracketrightbig\n+O(λ−1e−λ),\nasλ→+∞. Comparing the last two identities yields α1= 0,\nC0eα0−c0+2∞P\nn=1|λn|−2Reλn= e/angbracketlefta/angbracketright\nand\n−(2S−c0+iπc1) =/an}bracketle{tφ(+)\n1/an}bracketri}ht−a(0).\nIt now follows from (2.2), (3.4) that\n∞/summationdisplay\nn=−∞\nn/negationslash=0(λn−c0) = 2S=c0+a(0)−/an}bracketle{tφ(+)\n1/an}bracketri}ht−iπc1=a(0)+a(1)\n2−/an}bracketle{ta/an}bracketri}ht,\ncompleting the proof of Theorem 4.\nAcknowledgments\nThis work was done during the visit of D.B. to the Universidade de\nLisboa; he is grateful for the hospitality extended to him. P.F. would\nlike to thank A. Laptev for several conversations of this topic.EIGENVALUE ASYMPTOTICS FOR THE DAMPED WAVE EQUATION 13\nReferences\n[AL] M. Asch and G. Lebeau, The spectrum of the damped wave oper ator for\na bounded domain in R2,Exp. Math. 12(2003), 227-241.\n[BR] A. Benaddi and B. Rao, Energy decay rate of damped wave equ ations\nwith indefinite damping, J. Differential Eq. 161(2000), 337–357.\n[CC] C. Castro and S. Cox, Achieving arbitrarily large decay in the dam ped\nwave equation, SIAM J. Control Optim. 39(2001), 1748–1755.\n[CFNS] G. Chen, S.A. Fulling, F.J. Narcowich and S. Sun. Exponential d ecay of\nenergy of evolution equations with locally distributed damping. SIAM J.\nAppl. Math. 51(1991), 266–301.\n[CZ] S. Cox and E. Zuazua, The rate at which energy decays in a damp ed\nstring.Comm. Part. Diff. Eq. ,19(1994), 213–243.\n[E] A. Erd´ elyi. Asymptotic expansions. Dover Publications Inc., N.Y. 1956.\n[Fe] M.V. Fedoryuk, Asymptotic analysis: linear ordinary differential equa-\ntions. Berlin: Springer-Verlag. 1993.\n[F1] P. Freitas, On some eigenvalueproblems relatedto the waveequ ation with\nindefinite damping, J. Differential Equations ,127(1996), 320–335.\n[F2] P.Freitas,Spectralsequencesforquadraticpencilsandthe inversespectral\nproblem for the damped wave equation, J. Math. Pures Appl. 78(1999),\n965–980.\n[L] G. Lebeau, ´Equations des ondes amorties, S´ eminaire sur les ´Equations\naux D´ eriv´ ees Partielles, 1993–1994,Exp. No. XV, 16 pp., ´Ecole Polytech.,\nPalaiseau, 1994.\n[Le] B.Ya. Levin. Distribution of zeros of entire functions. Providen ce, R.I.:\nAmerican Mathematical Society. 1964.\n[LS] B.M. Levitan, I.S. Sargsjan. Introduction to spectral theor y: Selfadjoint\nordinarydifferentialoperators.TranslationsofMathematicalMo nographs.\nVol. 39. Providence, R.I.: American Mathematical Society. 1975.\n[PT] J. P¨ oschel and E. Trubowitz, Inverse spectran theory, Pu re and Applied\nMathematics, Vol. 130, Academic Press, London, 1987.\n[S] J.Sj¨ ostrand, Asymptoticdistributionofeigenfrequenciesfo rdampedwave\nequations, Publ. Res. Inst. Math. Sci. 36(2000), 573–611.\nDepartment of Physics and Mathematics, Bashkir State Pedag ogi-\ncal University, October rev. st., 3a, 450000, Ufa, Russia\nE-mail address :borisovdi@yandex.ru\nDepartment ofMathematics, Faculdade de Motricidade Human a (TU\nLisbon) andGroup of Mathematical Physics of the University of Lis-\nbon, Complexo Interdisciplinar, Av. Prof. Gama Pinto 2, P-1 649-003\nLisboa, Portugal\nE-mail address :freitas@cii.fc.ul.pt"    },    {        "title": "1605.05358v2.Are_Maxwell_s_equations_Lorentz_covariant_.pdf",        "content": "arXiv:1605.05358v2  [physics.class-ph]  6 Apr 2023Are Maxwell’s equations Lorentz-covariant?\nD V Redˇ zi´ c\nFaculty of Physics, University of Belgrade, PO Box 44, 11000 Beogr ad, Serbia\nE-mail:redzic@ff.bg.ac.rs\nAbstract. The statement that Maxwell’s electrodynamics in vacuum is already\ncovariant under Lorentz transformations is commonplace in the lite rature. We analyse\nthe actual meaning of that statement and demonstrate that Max well’s equations are\nperfectly fit to be Lorentz-covariant; they become Lorentz-co variant if we construct\nto be so, by postulating certain transformation properties of field functions. In\nAristotelian terms, the covariance is a plain potentiality, but not nec essarily entelechy.\n1. Introduction\nLorentz-covariance of Maxwell’s equations is certainly the key link be tween classical\nelectrodynamics and special relativity. While there is a clear consens us in the literature\nthat ‘the electrodynamic foundation of Maxwell–Lorentz’s theory is in agreement with\nthe principle of relativity,’ and thus that Maxwell’s equations are Lore ntz-covariant,\nthe true meaning of that statement appears to be somewhat elusiv e. Generally, it is\ndemonstrated that Maxwell’s equations are Lorentz-covariant if a nd only if the electric\nand magnetic fields and charge and current densities appearing in th em transform\naccording to some specific transformation laws. As is well known, th is can be done\nbasically in two ways: either transforming directly Maxwell’s equations (‘steep and\ndifficult mountaineer’s path’) as Einstein originally did [1, 2, 3, 4], or emplo ying the\npowerful and elegant, almost dazzling, tensorial approach in Minko wski space-time.\nNeither way is very transparent to the student.\nOn the other hand, the student of relativity encounters frequen tly some potentially\nconfusing locutions onLorentz-covariance of Maxwell’s equations w hich, in thelong run,\nmight lead the student to think that ‘requirement of form–invarianc e is automatically\nfulfilledforMaxwell’s fundamentalequationsofelectrodynamics in vacuo.’ Forexample,\nin his classic book, Møller [5] states: ‘we saw that it is necessary to cha nge the\nfundamental equations of mechanics in order to bring them into acc ordance with the\nprinciple of relativity. This is not so with the equations of electrodyna mics in vacuum,\nthe Maxwell equations, which, as we shall see, are already covarian t under Lorentz\ntransformations [...].’ In the same vein, Rindler [6] writes: ‘Having examin ed and\nrelativistically modified Newtonian particle mechanics, it would be natur al to look\nnext with the same intentions at Maxwell’s electrodynamics, at first in vacuum. ButAre Maxwell’s equations Lorentz-covariant? 2\nthat theory turns out to be already “special-relativistic”. In othe r words, its basic\nlaws, as summarized by the four Maxwell equations plus Lorentz’s fo rce law, are form-\ninvariant under Lorentz transformations, i. e. under transform ations from one inertial\nframe to another.’ Similarly, Mario Bunge [7] asserts that relativist ic electrodynamics\n‘is not a new theory but a reformulation of CEM [classical electromagn etism], which\nwas relativistic without knowing it.’ Also, in his fine book [8], Ugarov affir ms: ‘It is\nremarkable that the system of Maxwell’s equations formulated fifty years prior to the\nadvent of the special theory of relativity proved to be covariant w ith respect to the\nLorentz transformation, i.e. it retains its appearance, with the ac curacy of variables’\ndesignations, under the Lorentz transformation. This signifies th at the system of\nMaxwell’s equations retains its appearance in any inertial frame of re ference, and the\nprinciple of relativity holds automatically.’ As the last characteristic e xample, I quote\nfrom a recent book by Christodoulides [9]: ‘It is obvious that electro magnetic theory,\nas expressed by Maxwell’s equations, is a relativistic theory, whose e quations needed no\nmodification in order to become compatible with the Theory of Relativit y, at least as\nthese apply to the vacuum.’\nRecently, I pointed out that the above statements should be take ncum grano salis :\nLorentz-covariance of Maxwell’s equations is notfulfilled automatically [10]. I noted\nthat, for example, the so-called source-free Maxwell’s equations, curlE=−∂B/∂tand\ndivB= 0, are Lorentz-covariant if one definesE′andB′viaEandBas given by\nthe well-known transformation rules. A complete but succinct discu ssion of the issue\nis given in [11]. However, taking into account a possible relevance of th e issue for the\nstudent of relativity, it is perhaps worthwhile to discuss in some deta il what the above-\nmentioned authors actually meant by ‘Maxwell’s equations are already covariant under\nLorentz transformations.’\n2. Lorentz-covariance of Maxwell’s equations\n2.1. Mathematical prelude\nThe problem of Lorentz-covariance of Maxwell’s equations is basically a mathematical\nquestion. In this Subsection, for the convenience of the reader, we recall some familiar\nresults in the simple way, from vectorial perspective.\nWe begin by writing a set of coupled partial differential equations\n∂Ez\n∂y−∂Ey\n∂z=−∂Bx\n∂t, (1)\n∂Ex\n∂z−∂Ez\n∂x=−∂By\n∂t, (2)\n∂Ey\n∂x−∂Ex\n∂y=−∂Bz\n∂t, (3)\n∂Bx\n∂x+∂By\n∂y+∂Bz\n∂z= 0, (4)Are Maxwell’s equations Lorentz-covariant? 3\nwhereEi=Ei(x,y,z,t) andBi=Bi(x,y,z,t),istands for subscripts x,y,z, are\nfunctions of the mutually independent variables x,y,zandt. Introduce another set\nof the mutually independent variables x′,y′,z′andt′, and let them be the following\nfunctions of x,y,zandt\nx′=γ(x−vt),y′=y,z′=z,t′=γ(t−vx/c2), (5)\nwhereγ≡(1−v2/c2)−1/2,c≡/radicalbig\n1/ǫ0µ0,ǫ0andµ0are positive constants, and vis a\nnonnegative constant satisfying 0 ≤v < c.\nAs is well known, expressing unprimed by primed variables in equations (1)-(4),\nemploying the standard procedure which involves the chain rule for d ifferentiation, after\nsome manipulations one obtains that the following primed equations ap ply (a detailed\nderivation is found, e.g., in [2], Section 8.2):\n∂E′\nz\n∂y′−∂E′\ny\n∂z′=−∂B′\nx\n∂t′, (6)\n∂E′\nx\n∂z′−∂E′\nz\n∂x′=−∂B′\ny\n∂t′, (7)\n∂E′\ny\n∂x′−∂E′\nx\n∂y′=−∂B′\nz\n∂t′, (8)\n∂B′\nx\n∂x′+∂B′\ny\n∂y′+∂B′\nz\n∂z′= 0, (9)\nwhere\nE′\nx≡Ex B′\nx≡Bx\nE′\ny≡γ(Ey−vBz) B′\ny≡γ(By+v\nc2Ez)\nE′\nz≡γ(Ez+vBy) B′\nz≡γ(Bz−v\nc2Ey)\n\n(10)\nIn equations (10) E′\ni=E′\ni(x′,y′,z′,t′) andEi=Ei[γ(x′+vt′),y′,z′,γ(t′+vx′/c2)] and\nanalogously for B′\niandBi. Obviously, equations (6)-(9) have the same form as equations\n(1)-(4). Thus, transforming equations (1)-(4) by transforma tion of variables (5), one\nreveals that those equations imply that, in the primed variables, equ ations (6)-(9) of\nthe same form apply under the proviso that E′\niandB′\nitherein be given by identities\n(10). Consequently, if EiandBisatisfy unprimed equations (1)-(4), one knows that E′\ni\nandB′\nidetermined by identities (10) satisfy primed equations (6)-(9).\nNote that, fromequations (10) and (5), mutatis mutandis , one obtains the following\ninverse identities:\nEx≡ E′\nx Bx≡ B′\nx\nEy≡γ(E′\ny+vB′\nz) By≡γ(B′\ny−v\nc2E′\nz)\nEz≡γ(E′\nz−vB′\ny) Bz≡γ(B′\nz+v\nc2E′\ny)\n\n(11)\nwhich of course are obtained quickly by interchanging primed and unp rimed quantities\nand replacing vby−vin (10).Are Maxwell’s equations Lorentz-covariant? 4\nAssume now that functions EiandBi, in addition to equations (1)-(4), must also\nsatisfy another set of equations:\n∂Bz\n∂y−∂By\n∂z=µ0ρux+1\nc2∂Ex\n∂t, (12)\n∂Bx\n∂z−∂Bz\n∂x=µ0ρuy+1\nc2∂Ey\n∂t, (13)\n∂By\n∂x−∂Bx\n∂y=µ0ρuz+1\nc2∂Ez\n∂t, (14)\nρ=ǫ0/parenleftbigg∂Ex\n∂x+∂Ey\n∂y+∂Ez\n∂z/parenrightbigg\n≡̺E, (15)\nwhereρ(x,y,z,t) is the ‘charge density’ (without attaching any physical meaning to\nit), andui=ui(x,y,z,t) are Cartesian components of the velocity field of the ‘charge.’\nThe only constraint imposed by equations (12)-(15) on ρanduiis the ‘equation of\ncontinuity,’\n∂(ρux)\n∂x+∂(ρuy)\n∂y+∂(ρuz)\n∂z+∂ρ\n∂t= 0, (16)\nwhich is a necessary condition for the validity of equations (12)-(15 ). Thus, the equation\nof continuity may apply even if equations (12)-(15) do not apply [12].\nIntroduce symbol\n̺′\nE≡ǫ0/parenleftbigg∂E′\nx\n∂x′+∂E′\ny\n∂y′+∂E′\nz\n∂z′/parenrightbigg\n, (17)\nwhereE′\niare given by identities (10). Employing the standard procedure, on e finds that\n̺′\nEtransforms according to equation\n̺′\nE=γ/bracketleftbigg\n̺E−ǫ0v/parenleftbigg∂Bz\n∂y−∂By\n∂z−1\nc2∂Ex\n∂t/parenrightbigg/bracketrightbigg\n, (18)\nwherefrom using equations (12) and (15) one obtains\n̺′\nE=γ/parenleftBig\nρ−v\nc2ρux/parenrightBig\n, (19)\nMaking use of the familiar transformations for velocity field compone nts,\nu′\nx=ux−v\n1−uxv/c2, u′\ny=uy\nγ(1−uxv/c2), u′\nz=uz\nγ(1−uxv/c2),(20)\nand their inverse, from eq. (19) one gets\nρ=γ/parenleftBig\n̺′\nE+v\nc2̺′\nEu′\nx/parenrightBig\n. (21)\nTransforming eq. (12) through eqs. (11) yields directly\n∂B′\nz\n∂y′−∂B′\ny\n∂z′=µ0/parenleftbiggρux\nγ−̺′\nEv/parenrightbigg\n+1\nc2∂E′\nx\n∂t′, (22)Are Maxwell’s equations Lorentz-covariant? 5\nwhich using eq. (19) and the first formula (20) gives\n∂B′\nz\n∂y′−∂B′\ny\n∂z′=µ0/bracketleftBig\nγ/parenleftBig\nρ−v\nc2ρux/parenrightBig/bracketrightBig\nu′\nx+1\nc2∂E′\nx\n∂t′, (23)\nTransforming in the same way eqs. (13)-(15), and using formulas ( 20) and eq. (23),\none gets\n∂B′\nx\n∂z′−∂B′\nz\n∂x′=µ0/bracketleftBig\nγ/parenleftBig\nρ−v\nc2ρux/parenrightBig/bracketrightBig\nu′\ny+1\nc2∂E′\ny\n∂t′, (24)\n∂B′\ny\n∂x′−∂B′\nx\n∂y′=µ0/bracketleftBig\nγ/parenleftBig\nρ−v\nc2ρux/parenrightBig/bracketrightBig\nu′\nz+1\nc2∂E′\nz\n∂t′, (25)\nγ/parenleftBig\nρ−v\nc2ρux/parenrightBig\n=̺′\nE, (26)\nrespectively. Eq. (26) is identical with eq. (19), as it should be.\nEquations (1)-(4) and (12)-(15) can obviously be recast into the compact form\n∇×E=−∂B\n∂t,∇·B= 0,∇×B=µ0ρu+ǫ0µ0∂E\n∂t,∇·E=ρ\nǫ0, (27)\nand the transformed equations (6)-(9) and (23)-(26) can be re cast into\n∇′×EEE′=−∂BBB′\n∂t′,∇′·BBB′= 0,∇′×BBB′=µ0̺′u′+ǫ0µ0∂EEE′\n∂t′,∇′·EEE′=̺′\nǫ0, (28)\nwhere̺′≡γ/parenleftbig\nρ−v\nc2ρux/parenrightbig\n.\nThus, transforming equations (27) by transformation of variable s (5), one obtains\nthat,inprimedvariables, equations(28)ofthesameformapplyund ertheprovisothat EEE′\nandBBB′therein be defined by identities (10). Consequently, if EandBsatisfy unprimed\nequations (27), one knows that EEE′andBBB′defined by identities (10) satisfy primed\nequations (28). This is all one can extract from the unprimed Maxwe ll’s equations\n(27), transforming them by the Lorentz transformation (5).\n2.2. Are Maxwell’s equations Lorentz-covariant?\nComparing equations (27) and (28), the following conclusion is readily reached: in\norder that Maxwell’s equations which apply in unprimed variables, hold a lso in primed\nvariables, (that is, in standard parlance, in order that Maxwell’s equ ations be Lorentz-\ncovariant), it suffices to defineE′andB′(by which we mean counterparts ofEandB\nin primed variables) by equations\nE′d=EEE′,B′d=BBB′, (29)\nandthe ‘charge density’ in primed variables, ρ′, by equation\nρ′d=γ/parenleftBig\nρ−v\nc2ρux/parenrightBig\n. (30)Are Maxwell’s equations Lorentz-covariant? 6\nSince transformation properties of E,Bandρare not known beforehand, E′,B′and\nρ′have to be definedby equations (29) and (30), i.e. through transformation laws. †\nIntroducing those definitions ‘is an inconspicuous but indispensable s tep, aconditio\nsine qua non for Lorentz-covariance of Maxwell’s equations in the strict sense o f the\nword’ [11].\nNow we are armed with all the facts necessary to answer our query , are Maxwell’s\nequations Lorentz-covariant. That is, do Maxwell’s equations reta in their form under\ntransformation of variables (5)? The correct answer appears to be: the Maxwell\nequations are ready-made to be Lorentz-covariant, but they ar e actually Lorentz-\ncovariant only if we construct to be so (cf, e.g., [13, 14]). As was demonstrated\nabove, what exactly is sufficient to be postulated for the covarianc e emanates from\nthe equations themselves. In this sense, and in this sense only, one can speak about ‘a\nmiracle[that]Maxwell, fullyunawareofrelativity, hadnevertheless w rittenhisequations\nin a relativistically covariant form straight away’ [15]. However, the covariance is not\nfulfilledautomatically; thereisnocovariancewithoutpostulatingspe cifictransformation\nproperties of the quantities appearing in the equations (with the ex ception of course of\npurely geometric quantities, such as velocity and acceleration, whic h are already defined\nin both unprimed and primed coordinates, and whose transformatio n properties follow\nfromthedefinitions). InAristotelianterms, Lorentz-covariance iscontainedinMaxwell’s\nequations as a plain potentiality, but not as entelechy. One should ke ep this in mind. ‡\nEinstein’s original demonstration that ‘the electrodynamic foundat ion of Lorentz’s\ntheory of the electrodynamics of moving bodies agrees with the prin ciple of relativity,’\n[1] is basically mathematics disguised as physics. Einstein postulates t hat Maxwell’s\nequations conform to the principle of relativity and thus that both e qs. (27) and\nequations\n∇′×E′=−∂B′\n∂t′,∇′·B′= 0,∇′×B′=µ0ρ′u′+ǫ0µ0∂E′\n∂t′,∇′·E′=ρ′\nǫ0,(31)\nhold; basically, he thus postulates that Maxwell’s equations are Lore ntz–covariant.\nComparing (mutually equivalent) equations (28) and (31), he deduc es the necessary\nconditions for the covariance (eqs. (29) and (30) regarded as tr ansformation laws).\n3. Concluding comments\nTake now that the Lorentz transformation (5) has its received ph ysical meaning, i.e.,\nassume that it relates space and time coordinates of an event in a giv en inertial frame S\nwith the space and time coordinates of the same event in an inertial f rameS′which is in\na standard configuration with S. As is well known, assuming the validity of Maxwell’s\n†While it appears that a multiplicative factor Ψ( v) could be included in eqs. (29) and (30), a simple\nanalysis demonstrates that Ψ( v) must equal one [1].\n‡Incidentally, recall that the equation of continuity (16) itselfis ready-made to be not only Lorentz-\ncovariant but also Galilei-covariant (cf, e.g., [16, 17]). Whichever co variance is preferred on physical\ngrounds, the remaining one then becomes a purely mathematical pr operty.Are Maxwell’s equations Lorentz-covariant? 7\nequations in the given frame S, and also taking that E′,B′andρ′aredefinedby\nequations (29) and (30) (achieving thus Lorentz-covariance of M axwell’s equations),\nwould ensure the validity of Maxwell’s equations in any reference fram eS′in uniform\ntranslation with respect to S, if the special theory of relativity is valid. In this context,\ndefinitions (29) and (30) express the electric and magnetic fields an d charge density\ninS′, and thus, basically, represent a fundamental physical assumpt ion. However, as\nBartocci and Mamone Capria pointed out, the plain possibility of achieving Lorentz-\ncovariance of Maxwell’s equations can be regarded as nothing more t han an interesting\nmathematical property devoid of any physical contents [18]. It is p erhaps instructive to\nrecognize that the formalcovariance can be employed as a handy tool, quite outside the\nrelativistic framework [10].\nTo summarize, in the context of physics, if Maxwell’s equations descr ibe physical\nfields in an inertial frame S, and Lorentz transformations relate space and time\ncoordinates of the same event as observed in two inertial frames SandS′in relative\nmotion, Lorentz-covariance of Maxwell’s equations expresses a fu ndamental physical\nassumption that the same (primed!) Maxwell’s equations describe the physical fields\nalso in the S′frame. On the other hand, from the mathematical side, what is late nt\nin Maxwell’s equations is, first, that they are ready-made to be Lore ntz-covariant, and,\nsecond, the precise ‘recipe’ how to achieve that they actually be Lo rentz-covariant.\nShortly, Maxwell’s equations are Lorentz-covariant if we construc t to be so, but they\nneed not be. /bardblHowever, it was indeed a miracle that Maxwell had written his equation s\nin a form perfectly fit to be Lorentz-covariant. From this perspec tive, Heinrich Hertz’s\nfeelingthat Maxwell’s equations ‘give back to us more than was originally put int o\nthem,’ proved prophetic.\nFinally, note that, as is well known, analysis of Maxwell’s equations can be often\nmade much easier in terms of potentials. For the sake of completene ss, a brief discussion\nofLorentz-covarianceofMaxwell’s equationsfromtheperspectiv e ofpotentials, skipping\nthe familiar details, is given in Appendix.\nAcknowledgments\nI thank Vladimir Onoochin for stimulating correspondence. My work is supported by\nthe Ministry of Science and Education of the Republic of Serbia, proj ect No. 451-03-\n47/2023-01/200162.\nAppendix\nFirst of all, recall that Maxwell’s equations imply the equation of contin uity (16):\n∂ρ\n∂t+∂(ρux)\n∂x+∂(ρuy)\n∂y+∂(ρuz)\n∂z= 0, (A.1)\n/bardblThus,Rindler’sformulationthatMaxwell’sequations‘fit perfectly intotheschemeofspecialrelativity’\n[19], should perhaps be amended as ‘ canfitperfectly into the scheme of special relativity.’Are Maxwell’s equations Lorentz-covariant? 8\nwhich is a necessary condition for the validity of Maxwell’s equations an d thus it may\nbe valid even if Maxwell’s equations do not apply. Recall also that Maxwe ll’s equations\npossess, inter alia , a nice property that they allow themselves to be considerably\nsimplified mathematically by expressing EiandBiin terms of potentials Φ and Ai\nintroduced by\nEx=−∂Φ\n∂x−∂Ax\n∂tBx=∂Az\n∂y−∂Ay\n∂z\nEy=−∂Φ\n∂y−∂Ay\n∂tBy=∂Ax\n∂z−∂Az\n∂x\nEz=−∂Φ\n∂z−∂Az\n∂tBz=∂Ay\n∂x−∂Ax\n∂y\n\n(A.2)\nAssuming that the potentials satisfy the Lorenz gauge condition,\n1\nc2∂Φ\n∂t+∂Ax\n∂x+∂Ay\n∂y+∂Az\n∂z= 0, (A.3)\nit follows that the potentials then should satisfy the inhomogeneous d’Alembert type\nequations:\n/squareΦ =−ρ\nǫ0, (A.4)\n/squareAx=−µ0ρux,/squareAy=−µ0ρuy,/squareAz=−µ0ρuz, (A.5)\nwhere\n/square≡∂2\n∂x2+∂2\n∂y2+∂2\n∂z2−1\nc2∂2\n∂t2, (A.6)\nNow transform the continuity equation (A.1) replacing unprimed by p rimed variables\naccording to equations (5), employing formulae for changing partia l differential\ncoefficients\n∂\n∂t=γ/parenleftbigg∂\n∂t′−v∂\n∂x′/parenrightbigg∂\n∂x=γ/parenleftbigg∂\n∂x′−v\nc2∂\n∂t′/parenrightbigg\n,∂\n∂y=∂\n∂y′,∂\n∂z=∂\n∂z′.(A.7)\nOne obtains\n∂\n∂t′γ(ρ−v\nc2ρux)+∂\n∂x′γ(ρux−ρv)+∂\n∂y′(ρuy)+∂\n∂z′(ρuz) = 0. (A.8)\nUsing equations (20), one has\n∂\n∂t′γ(ρ−v\nc2ρux)+∂\n∂x′γρu′\nx(1−uxv\nc2)+∂\n∂y′γρu′\ny(1−uxv\nc2)+∂\n∂z′γρu′\nz(1−uxv\nc2) = 0.(A.9)\nInspecting the last equation, the following conclusion is readily reach ed: in order that\nequation of continuity (A.1) implies equation of the same form and con tent in primed\nvariables, it suffices to definethe charge density in primed coordinates, ρ′, by\nρ′=γ/parenleftBig\nρ−v\nc2ρux/parenrightBig\n. (A.10)\nWith that definition, equation (A.9) obviously reduces to\n∂ρ′\n∂t′+∂(ρ′u′\nx)\n∂x′+∂(ρ′u′\ny)\n∂y′+∂(ρ′u′\nz)\n∂z′= 0, (A.11)Are Maxwell’s equations Lorentz-covariant? 9\nwhich is identical with equation (A.1), except for primes. Functions ρ′u′\niare Cartesian\ncomponents of the convection current density in the S′frame, as ρuiare in the Sframe.\nClearly,ρc,ρux,ρuyandρuztransform according to the rules\nρ′c=γ(ρc−v\ncρux), ρ′u′\nx=γ(ρux−v\ncρc), ρ′u′\ny=ρuy, ρ′u′\nz=ρuz,(A.12)\nunder the Lorentz transformation (5).\nNow transform in the same way another simple equation, the Lorenz gauge\ncondition (A.3). One obtains automatically\n1\nc2∂\n∂t′γ(Φ−vAx)+∂\n∂x′γ(Ax−v\nc2Φ)+∂\n∂y′Ay+∂\n∂z′Az= 0. (A.13)\nObviously, in order that equation (A.3) be Lorentz-covariant, it su ffices to define\nfunctions of primed variables Φ′,A′\nx,A′\nyandA′\nzby equations\nΦ′=γ(Φ−vAx), A′\nx=γ(Ax−v\nc2Φ), A′\ny=Ay, A′\nz=Az, (A.14)\nthe result that Poincar´ e reached a long time ago by a different path , postulating charge\ninvariance and invariance of the inhomogeneous d’Alembert type equ ations for the\npotentials [20]. Since the primed functions are definedby equations (A.14), it follows\nthat Φ,Ax,AyandAza fortiori transform according to equations (A.14) under the\nLorentz transformation (5). For convenience, recast the tran sformation rules into\nΦ′\nc=γ/parenleftbiggΦ\nc−v\ncAx/parenrightbigg\n, A′\nx=γ/parenleftbigg\nAx−v\ncΦ\nc/parenrightbigg\n, A′\ny=Ay, A′\nz=Az. (A.15)\nA glance at equations (A.12) reveals that, with definition (A.10) of ρ′density,ρc,\nρux,ρuyandρuzbecome contravariant components of a 4-vector of Minkowski sp ace-\ntime; equation (A.15) shows that the analogous conclusion applies to Φ/c,Ax,Ayand\nAz. Thus, 4-current density Jµand 4-potential Aµare constructed.\nNow we arrived at the familiar, wide and well trodden path, and no nee d to go\nfurther. Namely, as is well known, in the tensorial notation, with 4- vectorsJµand\nAµ, Lorentz-covariance of Maxwell’s equations is an obvious fact, offe red as on a plate.\nWhat is perhaps less obvious is that, instead of simply asserting that Φ andAtogether\nconstitute a 4-vector, it would be more correct to specify that no w Φ and Atogether\nconstitute a 4-vector per definitionem , namely, we constructed to be so. ¶The same\nremark applies to ρandρu. Of course, in the latter case, the transformation rules\n(A.12) can be obtained without construction, as a consequence of the principle of charge\ninvariance.\nThus, in the language of potentials and 4-tensors, our main conclus ion is reached in\na simpler andmoretransparent way. Maxwell’s equations areperfec tly fit tobe Lorentz-\ncovariant; they become Lorentz-covariant only if we define the pr imed potentials and\n¶As Rindler ([6], p 155) notes, ‘[...] we can construct a tensor by specify ing its components arbitrarily\ninonecoordinate system, say {xi}, and then using the transformation law [expressing the familiar\ninformal definition of tensors] to define its components in all other systems, or, in the case of a qualified\ntensor, in all those systems which are mutually connected by trans formations belonging to the chosen\nsubgroup.’Are Maxwell’s equations Lorentz-covariant? 10\ncharge density so that (Φ /c,A) and (ρc,ρu) be 4-vectors of Minkowski space-time.\nDeciding that (Φ /c,A) and (ρc,ρu) be contravariant components of 4-vectors ensures\nLorentz-covariance of Maxwell’s equations, enabling us to recast t hose equations in an\nexplicitly Lorentz-covariant form.\nThus, we can agree with Sommerfeld’s [21] simile that ‘the true mathem atical\nstructure of these entities [Φ and A] will appear only now [in the language of 4-\ntensors], as in a mountain landscape when the fog lifts,’ only in the fra mework of the\ninterpretation given above. Sommerfeld’s claim that‘by reducing the Maxwell equations\nto the four-vector [ Aµ,Jµand the operators ∂µ,/square=−∂µ∂µ] we have demonstrated at\nthe same time their general validity , independent of the coordinate system’ is, strictly\nspeaking, incorrect. Basically, we have assumed a four-vector character of (Φ /c,A) and\n(ρc,ρu) and thus we have constructed that ‘the Maxwell equations satisf y the relativity\npostulate from the very beginning.’ This constructional aspect of Lorentz-covariance of\nMaxwell’s equations, clearly enunciated by Einstein [22] and by Bergma nn [23], seems\nto be understated in the literature.\nReferences\n[1] Einstein A 1905 Zur Elektrodynamik bewegter K¨ orper Ann. Phys., Lpz. 17891–921\n[2] Rosser W G V 1964 An Introduction to the Theory of Relativity (London: Butterworths)\n[3] Schwartz H M 1977 Einstein’s comprehensive 1907 essay on relativ ity, part II Am. J. Phys. 45\n811–7\n[4] Resnick R 1968 Introduction to Special Relativity (New York: Wiley)\n[5] Møller C 1972 The Theory of Relativity 2nd edn (Oxford: Clarendon)\n[6] Rindler W 1991 Introduction to Special Relativity 2nd edn (Oxford: Clarendon)\n[7] Bunge M 1967 Foundations of Physics (Berlin: Springer)\n[8] Ugarov V A 1979 Special Theory of Relativity (Moscow: Mir)\n[9] Christodoulides C 2016 The Special Theory of Relativity: Foundations, Theory, Ver ification,\nApplications (Cham: Springer)\n[10] Redˇ zi´ c D V 2014 Force exerted by a moving electric current o n a stationary or co-moving charge:\nMaxwell’s theory versusrelativistic electrodynamics Eur. J. Phys. 35045011\n[11] Redˇ zi´ c D V 2017 Are Maxwell’s equations Lorentz–covariant? Eur. J. Phys. 38015602\n[12] Heras J A 2009 How to obtain the covariant form of Maxwell’s equa tions from the continuity\nequation Eur. J. Phys. 30845–54\n[13] Strauss M 1969 Corrections to Bunge’s Foundations of Physics (1967)Synthese 19433–42\n[14] Freudenthal H 1971 More about Foundations of Physics Found. Phys. 1315–23\n[15] Feinberg E L 1997 Special theory of relativity: how good-faith d elusions come about Physics-\nUspekhi40433–5\n[16] Redˇ zi´ c D V 1993 Comment on “Some remarks on classical elect romagnetism and the principle\nof relativity,” by Umberto Bartocci and Marco Mamone Capria [Am. J. Phys. 59, 1030-1032\n(1991)]Am. J. Phys. 611149\n[17] Colussi V and Wickramasekara S 2008 Galilean and U(1)-gauge sy mmetry of the Schr¨ odinger field\nAnn. Phys. 3233020–36\n[18] Bartocci U and Capria M M 1991 Symmetries and asymmetries in cla ssical and relativistic\nelectrodynamics Found. Phys. 21787–801\n[19] Rindler W 2006 Relativity: Special, General and Cosmological 2nd edn (Oxford: Oxford University\nPress)\n[20] Poincar´ e H 1906 Sur la dynamique de l’´ electron Rend. Circ. Mat. Palermo 21129–75Are Maxwell’s equations Lorentz-covariant? 11\n[21] Sommerfeld A 1952 Electrodynamics (translated by E G Ramberg) (New York: Academic) pp\n212–4\n[22] Einstein A 1922 The Meaning of Relativity (Princeton: Princeton University Press) p 43\n[23] Bergmann P G 1976 Introduction to the Theory of Relativity (New York: Dover) pp 111-3"    },    {        "title": "2211.13486v1.Influence_of_non_local_damping_on_magnon_properties_of_ferromagnets.pdf",        "content": "In\ruence of non-local damping on magnon properties of ferromagnets\nZhiwei Lu,1,\u0003I. P. Miranda,2,\u0003Simon Streib,2Manuel Pereiro,2Erik Sj oqvist,2\nOlle Eriksson,2, 3Anders Bergman,2Danny Thonig,3, 2and Anna Delin1, 4\n1Department of Applied Physics, School of Engineering Sciences, KTH Royal\nInstitute of Technology, AlbaNova University Center, SE-10691 Stockholm, Swedeny\n2Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden\n3School of Science and Technology, Orebro University, SE-701 82, Orebro, Sweden\n4SeRC (Swedish e-Science Research Center), KTH Royal Institute of Technology, SE-10044 Stockholm, Sweden\n(Dated: November 28, 2022)\nWe study the in\ruence of non-local damping on magnon properties of Fe, Co, Ni and Fe 1\u0000xCox\n(x= 30%;50%) alloys. The Gilbert damping parameter is typically considered as a local scalar\nboth in experiment and in theoretical modelling. However, recent works have revealed that Gilbert\ndamping is a non-local quantity that allows for energy dissipation between atomic sites. With\nthe Gilbert damping parameters calculated from a state-of-the-art real-space electronic structure\nmethod, magnon lifetimes are evaluated from spin dynamics and linear response, where a good\nagreement is found between these two methods. It is found that non-local damping a\u000bects the\nmagnon lifetimes in di\u000berent ways depending on the system. Speci\fcally, we \fnd that in Fe, Co,\nand Ni the non-local damping decreases the magnon lifetimes, while in Fe 70Co30and Fe 50Co50an\nopposite, non-local damping e\u000bect is observed, and our data show that it is much stronger in the\nformer.\nINTRODUCTION\nIn recent years, there has been a growing interest in\nmagnonics, which uses quasi-particle excitations in mag-\nnetically ordered materials to perform information trans-\nport and processing on the nanoscale. Comparing to the\nconventional information device, the magnonics device\nexhibits lower energy consumption, easier integrability\nwith complementary metal-oxide semiconductor (CMOS)\nstructure, anisotropic properties, and e\u000ecient tunability\nby various external stimuli to name a few [1{10]. Yttrium\niron garnet (YIG) [11] as well as other iron garnets with\nrare-earth elements (Tm, Tb, Dy, Ho, Er) [12] are very\npromising candidates for magnonics device applications\ndue to their low energy dissipation properties and, thus,\nlong spin wave propagation distances up to tens of \u0016m.\nContrary, the damping of other materials for magnonics,\nlike CoFeB, is typically two orders of magnitude higher\ncompared to YIG [12], leading to much shorter spin wave\npropagation distances. A clear distinction can be made\nbetween materials with an ultra-low damping parame-\nter, like in YIG, and those with a sign\fciantly larger,\nbut still small, damping parameter. Materials like YIG\nare insulating, which hinders many of the microscopic\nmechanisms for damping, resulting in the low observed\ndamping parameter. In contrast, materials like CoFeB\nare metallic. In research projects that utilize low damp-\ning materials, YIG and similar non-metallic low damping\nsystems are typically favored. However, metallic systems\nhave an advantage, since magnetic textures can easily by\nin\ruenced by electrical currents. Hence, there is good\n\u0003These two authors contributed equally\nyCorresponding author: zhiweil@kth.sereason to consider metallic systems for low damping ap-\nplications, even though their damping typically is larger\nthan in YIG. One can conclude that Gilbert damping\nis one of the major bottlenecks for the choice of mate-\nrial in magnonics applications and a detailed experimen-\ntal as well as theoretical characterisation is fundamen-\ntal for this \feld of research, especially for metallic sys-\ntems. Thus, a more advanced and detailed understand-\ning of Gilbert damping is called for, in order to overcome\nthis obstacle for further development of magnonics-based\ntechnology.\nWhereas most studies consider chemical modi\fcations\nof the materials in order to tune damping [13, 14], only a\nfew focus on the fundamental physical properties as well\nas dependencies of the Gilbert damping. Often Gilbert\ndamping is considered as a phenomenological scalar pa-\nrameter in the equation of motion of localized atom-\nistic magnetic moments, i.e. the Landau-Lifshitz-Gilbert\n(LLG) equation [15]. However, from using the general\nRayleigh dissipation function in the derivation proposed\nby Gilbert [16], it was theoretically found that the Gilbert\ndamping should be anisotropic, a tensor, and non-local.\nFurthermore, it depends on the temperature and, thus,\non underlying magnon as well as phonon con\fgurations\n[17{20]. This is naturally built into the multiple theoret-\nical methods developed to predict the damping parame-\nter, including breathing Fermi surface model [21], torque\ncorrelation model [22], and linear response formulation\n[23]. For instance, the general Gilbert damping tensor\nas a function of the non-collinear spin con\fguration has\nbeen proposed in Ref. 24.\nNonetheless, an experimental veri\fcation is still miss-\ning due to lacking insights into the impact of the gen-\neralised damping on experimental observables. In a re-\ncent experiment, however, the anisotropic behavior of the\ndamping has been con\frmed for Co 50Fe50thin \flms andarXiv:2211.13486v1  [cond-mat.mtrl-sci]  24 Nov 20222\nwas measured to be of the order of 400% [25], with respect\nto changing the magnetization direction. Changes of\nGilbert damping in a magnetic domain wall and, thus, its\ndependency on the magnetic con\fguration was measured\nin Ref. [26] and \ftted to the Landau-Lifshitz-Baryakhtar\n(LLBar) equation, which includes non-locality of the\ndamping by an additional dissipation term proportional\nto the gradient of the magnetisation [27{29]. However,\nthe pair-wise non-local damping \u000bijhas not yet been\nmeasured.\nThe most common experimental techniques of evaluat-\ning damping are ferromagnetic resonance (FMR) [30] and\ntime-resolved magneto-optical Kerr e\u000bect (TR-MOKE)\n[31]. In these experiments, Gilbert damping is related\nto the relaxation rate when (i)slightly perturbing the\ncoherent magnetic moment out of equilibrium by an ex-\nternal magnetic \feld [32] or (ii)when disordered mag-\nnetic moments remagnetise after pumping by an ultrafast\nlaser pulse [33]. Normally, in case (i)the non-locality is\nsuppressed due to the coherent precession of the atomic\nmagnetic moments. However, this coherence can be per-\nturbed by temperature, making non-locality in principle\nmeasurable. One possible other path to link non-local\ndamping with experiment is magnon lifetimes. Theoret-\nically, the magnon properties as well as the impact of\ndamping on these properties can be assessed from the\ndynamical structure factor, and atomistic spin-dynamics\nsimulations have been demonstrated to yield magnon dis-\npersion relations that are in good agreement with exper-\niment [34]. In experiment, neutron scattering [35] and\nelectron scattering [36] are the most common methods for\nprobing magnon excitations, where the linewidth broad-\nening of magnon excitations is related to damping and\nprovides a way to evaluate the magnon lifetimes [37]. It is\nfound in ferromagnets that the magnon lifetimes is wave\nvector (magnon energy) dependent [38{40]. It has been\nreported that the magnon energy in Co \flms is nearly\ntwice as large as in Fe \flms, but they have similar magnon\nlifetimes, which is related to the intrinsic damping mech-\nanism of materials [41]. However, this collective e\u000bect of\ndamping and magnon energy on magnon lifetimes is still\nan open question. The study of this collective e\u000bect is of\ngreat interest for both theory and device applications.\nHere, we report an implementation for solving the\nstochastic Landau-Lifshitz-Gilbert (SLLG) equation in-\ncorporating the non-local damping. With the dynamical\nstructure factor extracted from the spin dynamics sim-\nulations, we investigate the collective e\u000bect of non-local\ndamping and magnon energy on the magnon lifetimes.\nWe propose an e\u000ecient method to evaluate magnon life-\ntimes from linear response theory and verify its validity.\nThe paper is organized as follows. In Sec. I, we give\nthe simulation details of the spin dynamics, the adiabatic\nmagnon spectra and dynamical structure factor, and the\nmethodology of DFT calculations and linear response.\nSec. II presents the non-local damping in real-space, non-\nlocal damping e\u000bects on the spin dynamics and magnon\nproperties including magnon lifetimes of pure ferromag-nets (Fe, Co, Ni), and Fe 1\u0000xCox(x= 30%;50%) alloys.\nIn Sec. III, we give a summary and an outlook.\nI. THEORY\nA. Non-local damping in atomistic spin dynamics\nThe dynamical properties of magnetic materials at \f-\nnite temperature have been so far simulated from atom-\nistic spin dynamics by means of the stochastic Landau-\nLifshitz-Gilbert equation with scalar local energy dissipa-\ntion. Here, the time evolution of the magnetic moments\nmi=mieiat atom site iis well described by:\n@mi\n@t=mi\u0002\u0012\n\u0000\r[Bi+bi(t)] +\u000b\nmi@mi\n@t\u0013\n;(1)\nwhere\ris the gyromagnetic ratio. The e\u000bective \feld Bi\nacting on each magnetic moment is obtained from:\nBi=\u0000@H\n@mi: (2)\nThe here considered spin-Hamiltonian Hconsists of a\nHeisenberg spin-spin exchange:\nH=\u0000X\ni6=jJijei\u0001ej: (3)\nHere,Jij{ the Heisenberg exchange parameter { cou-\nples the spin at site iwith the spin at site jand is cal-\nculated from \frst principles (see Section I C). Further-\nmore,\u000bis the scalar phenomenological Gilbert damp-\ning parameter. Finite temperature Tis included in\nEq. (1) via the \ructuating \feld bi(t), which is modeled\nby uncorrelated Gaussian white noise: hbi(t)i= 0 and\nb\u0016\ni(t)b\u0017\nj(t0)\u000b\n= 2D\u000eij\u000e\u0016\u0017\u000e(t\u0000t0), where\u000eis the Kro-\nnecker delta, i;jare site and \u0016;\u0017=fx;y;zgCartesian\nindices. Furthermore, the \ructuation-dissipation theo-\nrem givesD=\u000bkBT\n\rmi[42], with the Boltzman constant\nkB.\nA more generalized form of the SLLG equation that\nincludes non-local tensorial damping has been reported\nin previous studies [20, 43, 44] and is:\n@mi\n@t=mi\u00020\n@\u0000\r[Bi+bi(t)] +X\nj\u000bij\nmj@mj\n@t1\nA;(4)\nwhich can be derived from Rayleigh dissipation func-\ntional in the Lagrange formalism used by Gilbert [16].\nIn the presence of non-local damping, the Gaussian \ruc-\ntuating \feld ful\flls [43, 45, 46]\n\nb\u0016\ni(t)b\u0017\nj(t0)\u000b\n= 2D\u0016\u0017\nij\u000e(t\u0000t0); (5)\nwithD\u0016\u0017\nij=\u000b\u0016\u0017\nijkBT\n\rmi. The damping tensor \u000b\u0016\u0017\nijmust be\npositive de\fnite in order to be physically-de\fned. Along3\nwith spatial non-locality, the damping can also be non-\nlocal in time, as discussed in Ref. [47]. To prove the\n\ructuation-dissipation theorem in Eq. (5), the Fokker-\nPlanck equation has to be analysed in the presence of\nnon-local damping, similar to Ref. [15]. This is, however,\nnot the purpose of this paper. Instead, we will use the\napproximation \u000b\u0016\u0017\nij=1\n3Trf\u000biig\u000eij\u000e\u0016\u0017within the di\u000busion\nconstantD. Such an approximation is strictly valid only\nin the low temperature limit.\nTo solve this SLLG equation incorporating the non-\nlocal damping, we have implemented an implicit mid-\npoint solver in the UppASD code [48]. This iterative\n\fx-point scheme converges within an error of 10\u000010\u0016B,\nwhich is typically equivalent to 6 iteration steps. More\ndetails of this solver are provided in Appendix A. The\ninitial spin con\fguration in the typical N= 20\u000220\u000220\nsupercell with periodic boundary conditions starts from\ntotally random state. The spin-spin exchange interac-\ntions and non-local damping parameters are included up\nto at least 30 shells of neighbors, in order to guarantee\nthe convergence with respect to the spatial expansion of\nthese parameters (a discussion about the convergence is\ngiven in Section II A). Observables from our simulations\nare typically the average magnetisation M=1\nNPN\nimi\nas well as the magnon dispersion.\nB. Magnon dispersion\nTwo methods to simulate the magnon spectrum are\napplied in this paper: i)the dynamical structure factor\nandii)frozen magnon approach.\nFor the dynamical structure factor S(q;!) at \fnite\ntemperature and damping [34, 49], the spatial and time\ncorrelation function between two magnetic moments iat\npositionrandjat positionr0as well as di\u000berent time 0\nandtis expressed as:\nC\u0016(r\u0000r0;t) =hm\u0016\nr(t)m\u0016\nr0(0)i\u0000hm\u0016\nr(t)ihm\u0016\nr0(0)i:(6)\nHereh\u0001idenotes the ensemble average and \u0016are Carte-\nsian components. The dynamical structure factor can be\nobtained from the time and space Fourier transform of\nthe correlation function, namely:\nS\u0016(q;!) =1p\n2\u0019NX\nr;r0eiq\u0001(r\u0000r0)Z1\n\u00001ei!tC\u0016(r\u0000r0;t)dt:\n(7)\nThe magnon dispersion is obtained from the peak\npositions of S(q;!) along di\u000berent magnon wave vectors\nqin the Brillouin zone and magnon energies !. It\nshould be noted that S(q;!) is related to the scattering\nintensity in inelastic neutron scattering experiments [50].\nThe broadening of the magnon spectrum correlates to\nthe lifetime of spin waves mediated by Gilbert damping\nas well as intrinsic magnon-magnon scattering processes.\nGood agreement between S(q;!) and experiment hasbeen found previously [34].\nThe second method { the frozen magnon approach\n{ determines the magnon spectrum directly from the\nFourier transform of the spin-spin exchange parameters\nJij[51, 52] and non-local damping \u000bij. At zero tempera-\nture, a time-dependent external magnetic \feld is consid-\nered,\nB\u0006\ni(t) =1\nNX\nqB\u0006\nqeiq\u0001Ri\u0000i!t; (8)\nwhereNis the total number of lattice sites and B\u0006\nq=\nBx\nq\u0006iBy\nq. The linear response to this \feld is then given\nby\nM\u0006\nq=\u001f\u0006(q;!)B\u0006\nq: (9)\nWe obtain for the transverse dynamic magnetic suscep-\ntibility [53, 54]\n\u001f\u0006(q;!) =\u0006\rMs\n!\u0006!q\u0007i!\u000bq; (10)\nwith saturation magnetization Ms, spin-wave frequency\n!q=E(q)=~and damping\n\u000bq=X\nj\u000b0je\u0000iq\u0001(R0\u0000Rj): (11)\nWe can extract the spin-wave spectrum from the imagi-\nnary part of the susceptibility,\nIm\u001f\u0006(q;!) =\rMs\u000bq!\n[!\u0006!q]2+\u000b2q!2; (12)\nwhich is equivalent to the correlation function S\u0006(q;!)\ndue to the \ructuation-dissipation theorem [55]. We\n\fnd that the spin-wave lifetime \u001cqis determined by the\nFourier transform of the non-local damping (for \u000bq\u001c1),\n\u001cq=\u0019\n\u000bq!q: (13)\nThe requirement of positive de\fniteness of the damping\nmatrix\u000bijdirectly implies \u000bq>0, since\u000bijis diago-\nnalized by Fourier transformation due to translational\ninvariance. Hence, \u000bq>0 is a criterion to evaluate\nwhether the damping quantity in real-space is physically\nconsistent and whether \frst-principles calculations are\nwell converged. If \u000bq<0 for some wave vector q, energy\nis pumped into the spin system through the correspon-\ndent magnon mode, preventing the system to fully reach\nthe saturation magnetization at su\u000eciently low temper-\natures.\nThe e\u000bective damping \u000b0of the FMR mode at q=\n0 is determined by the sum over all components of the\ndamping matrix, following Eqn.11,\n\u000btot\u0011\u000b0=X\nj\u000b0j: (14)\nTherefore, an e\u000bective local damping should be based\non\u000btotif the full non-local damping is not taken into\naccount.4\nC. Details of the DFT calculations\nThe electronic structure calculations, in the framework\nof density functional theory (DFT), were performed us-\ning the fully self-consistent real-space linear mu\u000en-tin\norbital in the atomic sphere approximation (RS-LMTO-\nASA) [56, 57]. The RS-LMTO-ASA uses the Haydock\nrecursion method [58] to solve the eigenvalue problem\nbased on a Green's functions methodology directly in\nreal-space. In the recursion method, the continued frac-\ntions have been truncated using the Beer-Pettifor termi-\nnator [59], after a number LLof recursion levels. The\nLMTO-ASA [60] is a linear method which gives precise\nresults around an energy E\u0017, usually taken as the center\nof thes,panddbands. Therefore, as we calculate \fne\nquantities as the non-local damping parameters, we here\nconsider an expression accurate to ( E\u0000E\u0017)2starting\nfrom the orthogonal representation of the LMTO-ASA\nformalism [61].\nFor bcc FeCo alloys and bcc Fe we considered LL= 31,\nwhile for fcc Co and fcc Ni much higher LLvalues (51 and\n47, respectively), needed to better describe the density of\nstates and Green's functions at the Fermi level.\nThe spin-orbit coupling (SOC) is included as a l\u0001s\n[60] term computed in each variational step [62]. All\ncalculations were performed within the local spin den-\nsity approximation (LSDA) exchange-functional (XC) by\nvon Barth and Hedin [63], as it gives general magnetic\ninformation with equal or better quality as, e.g., the\ngeneralized gradient approximation (GGA). Indeed, the\nchoice of XC between LSDA and GGA [64] have a mi-\nnor impact on the onsite damping and the shape of the\n\u000bqcurves, when considering the same lattice parame-\nters (data not shown). No orbital polarization [65] was\nconsidered here. Each bulk system was modelled by a\nbig cluster containing \u001855000 (bcc) and\u0018696000 (fcc)\natoms located in the perfect crystal positions with the re-\nspective lattice parameters of a= 2:87\u0017A (bcc Fe and bcc\nFe1\u0000xCox, su\u000eciently close to experimental observations\n[66]),a= 3:54\u0017A (fcc Co [20, 67]), and a= 3:52\u0017A (fcc\nNi [68]). To account for the chemical disorder in the\nFe70Co30and Fe 50Co50bulks, the electronic structure\ncalculated within the simple virtual crystal approxima-\ntion (VCA), which has shown to work well for the fer-\nromagnetic transition metals alloys (particularly for el-\nements next to each other in the Periodic Table, such\nas FeCo and CoNi) [69{76], and also describe in a good\nagreement the damping trends in both FeCo and CoNi\n(see Appendix C).\nAs reported in Ref. [77], the total damping of site\ni, in\ruenced by the interaction with neighbors j, can\nbe decomposed in two main contributions: the onsite\n(fori=j), and the non-local (for i6=j). Both can be\ncalculated, in the collinear framework, by the followingexpression,\n\u000b\u0016\u0017\nij=\u000bCZ1\n\u00001\u0011(\u000f)Tr\u0010\n^T\u0016\ni^Aij(^T\u0017\nj)y^Aji\u0011\nd\u000fT!0K\u0000\u0000\u0000\u0000!\n\u000bCTr\u0010\n^T\u0016\ni^Aij(\u000fF+i\u000e)(^T\u0017\nj)y^Aji(\u000fF+i\u000e)\u0011\n;\n(15)\nwhere we de\fne ^Aij(\u000f+i\u000e) =1\n2i(^Gij(\u000f+i\u000e)\u0000^Gy\nji(\u000f+i\u000e))\nthe anti-Hermitian part of the retarded physical Green's\nfunctions in the LMTO formalism, and \u000bC=g\nmti\u0019a\npre-factor related to the i-th site magnetization. The\nimaginary part, \u000e, is obtained from the terminated con-\ntinued fractions. Also in Eq. 15, ^T\u0016\ni= [\u001b\u0016\ni;Hso] is the\nso-called torque operator [20] evaluated in each Cartesian\ndirection\u0016;\u0017=fx;y;zgand at site i,\u0011(\u000f) =\u0000@f(\u000f)\n@\u000fis\nthe derivative of the Fermi-Dirac distribution f(\u000f) with\nrespect to the energy \u000f,g= 2\u0010\n1 +morb\nmspin\u0011\ntheg-factor\n(not considering here the spin-mixing parameter [78]),\n\u001b\u0016are the Pauli matrices, and mtiis the total magnetic\nmoment of site i(mti=morbi+mspini). This results\nin a 3\u00023 tensor with terms \u000b\u0016\u0017\nij. In the real-space bulk\ncalculations performed in the present work, the \u000bij(with\ni6=j) matrices contain o\u000b-diagonal terms which are can-\ncelled by the summation of the contributions of all neigh-\nbors within a given shell, resulting in a purely diagonal\ndamping tensor, as expected for symmetry reasons [15].\nTherefore, as in the DFT calculations the spin quanti-\nzation axis is considered to be in the z([001]) direction\n(collinear model), we can ascribe a scalar damping value\n\u000bijas the average \u000bij=1\n2(\u000bxx\nij+\u000byy\nij) =\u000bxx\nijfor the\nsystems investigated here. This scalar \u000bijis, then, used\nin the SLLG equation (Eq. 1).\nThe exchange parameters Jijin the Heisenberg\nmodel were calculated by the Liechtenstein-Katsnelson-\nAntropov-Gubanov (LKAG) formalism [79], according to\nthe implementation in the RS-LMTO-ASA method [61].\nHence all parameters needed for the atomistic LLG equa-\ntion have been evaluated from ab-initio electronic struc-\nture theory.\nII. RESULTS\nA. Onsite and non-local dampings\nTable I shows the relevant ab-initio magnetic prop-\nerties of each material; the TCvalues refer to the Curie\ntemperature calculated within the random-phase approx-\nimation (RPA) [80], based on the computed Jijset. De-\nspite the systematic \u000btotvalues found in the lower limit\nof available experimental results (in similar case with,\ne.g., Ref. [81]), in part explained by the fact that we\nanalyze only the intrinsic damping, a good agreement\nbetween theory and experiment can be seen. When the\nwhole VCA Fe 1\u0000xCoxseries is considered (from x= 0%\ntox= 60%), the expected Slater-Pauling behavior of5\nthe total magnetic moment [73, 82] is obtained (data not\nshown).\nFor all systems studied here, the dissipation is domi-\nnated by the onsite ( \u000bii) term, while the non-local pa-\nrameters (\u000bij,i6=j) exhibit values at least one order of\nmagnitude lower; however, as it will be demonstrated in\nthe next sections, these smaller terms still cause a non-\nnegligible impact on the relaxation of the average magne-\ntization as well as magnon lifetimes. Figure 1 shows the\nnon-local damping parameters for the investigated ferro-\nmagnets as a function of the ( i;j) pairwise distance rij=a,\ntogether with the correspondent Fourier transforms \u000bq\nover the \frst Brillouin Zone (BZ). The \frst point to no-\ntice is the overall strong dependence of \u000bon the wave\nvectorq. The second point is the fact that, as also re-\nported in Ref. [20], \u000bijcan be an anisotropic quantity\nwith respect to the same shell of neighbors, due to the\nbroken symmetry imposed by a preferred spin quantiza-\ntion axis. This means that, in the collinear model and for\na given neighboring shell, \u000bijis isotropic only for equiva-\nlent sites around the magnetization as a symmetry axis.\nAnother important feature that can be seen in Fig. 1\nis the presence of negative \u000bijvalues. Real-space neg-\native non-local damping parameters have been reported\npreviously [20, 77, 97]. They are related to the decrease\nof damping at the \u0000-point, but may also increase \u000bqfrom\nthe onsite value in speci\fc qpoints inside the BZ; there-\nfore, they cannot be seen as ad hoc anti-dissipative con-\ntributions. In the ground-state, these negative non-local\ndampings originate from the overlap between the anti-\nHermitian parts of the two Green's functions at the Fermi\nlevel, each associated with a spin-dependent phase factor\n\b\u001b(\u001b=\";#) [20, 80].\nFinally, as shown in the insets of Fig. 1, a long-range\nconvergence can be seen for all cases investigated. An\nillustrative example is the bcc Fe 50Co50bulk, for which\nthe e\u000bective damping can be \u001860% higher than the con-\nverged\u000btotif only the \frst 7 shells of neighbors are con-\nsidered in Eq. 14. The non-local damping of each neigh-\nboring shell is found to follow a1\nr2\nijtrend, as previously\nargued by Thonig et al. [20] and Umetsu et al. [97].\nExplicitly,\n\u000bij/sin(k\"\u0001rij+ \b\") sin(k#\u0001rij+ \b#)\njrijj2; (16)\nwhich also qualitatively justi\fes the existence of negative\n\u000bij's. Thus, the convergence in real-space is typically\nslower than other magnetic quantities, such as exchange\ninteractions ( Jij/1\njrijj3) [80], and also depends on the\nimaginary part \u000e(see Eq. 15) [20]. The di\u000berence in the\nasymptotic behaviour of the damping and the Heisenberg\nexchange is distinctive; the \frst scales with the inverse of\nthe square of the distance while the latter as the inverse\nof the cube of the distance. Although this asymptotic\nbehaviour can be derived from similar arguments, both\nusing the Greens function of the free electron gas, the\nresults are di\u000berent. The reason for this di\u000berence issimply that the damping parameter is governed by states\nclose to the Fermi surface, while the exchange parameter\ninvolves an integral over all occupied states [20, 79].\nFrom bcc Fe to bcc Fe 50Co50(Fig. 1(a-f)), with in-\ncreasing Co content, the average \frst neighbors \u000bijde-\ncreases to a negative value, while the next-nearest neigh-\nbors contributions reach a minimum, and then increase\nagain. Similar oscillations can be found in further shells.\nAmong the interesting features in the Fe 1\u0000xCoxsystems\n(x= 0%;30%;50%), we highlight the low \u000bqaround the\nhigh-symmetry point H, along the H\u0000PandH\u0000N\ndirections, consistently lower than the FMR damping.\nBoth\u000bvalues are strongly in\ruenced by non-local con-\ntributions &5 NN. Also consistent is the high \u000bqob-\ntained forq=H. For long wavelengths in bcc Fe, some\n\u000bqanisotropy is observed around \u0000, which resembles the\nsame trait obtained for the corresponding magnon dis-\npersion curves [80]. This anisotropy changes to a more\nisotropic behavior by FeCo alloying.\nFar from the more noticeable high-symmetry points,\n\u000bqpresents an oscillatory behavior along BZ, around the\nonsite value. It is noteworthy, however, that these oscil-\nlatory\u000bqparameters exhibit variations up to \u00182 times\n\u000bii, thus showing a pronounced non-local in\ruence in\nspeci\fcqpoints.\nIn turn, for fcc Co (Fig. 1(g,h)) the \frst values are\ncharacterized by an oscillatory behavior around zero,\nwhich also re\rects on the damping of the FMR mode,\n\u000bq=0. In full agreement with Ref. [20], we compute a\npeak of\u000bijcontribution at rij\u00183:46a, which shows\nthe long-range character that non-local damping can ex-\nhibit for speci\fc materials. Despite the relatively small\nmagnitude of \u000bij, the multiplicity of the nearest neigh-\nbors shells drives a converged \u000bqdispersion with non-\nnegligible variations from the onsite value along the BZ,\nspecially driven by the negative third neighbors. The\nmaximum damping is found to be in the region around\nthe high-symmetry point X, where thus the lifetime of\nmagnon excitations are expected to be reduced. Simi-\nlar situation is found for fcc Ni (Fig. 1(i,j)), where the\n\frst neighbors \u000bijare found to be highly negative, con-\nsequently resulting in a spectrum in which \u000bq> \u000bq=0\nfor everyq6= 0. In contrast with fcc Co, however, no\nnotable peak contributions are found.\nB. Remagnetization\nGilbert damping in magnetic materials determines the\nrate of energy that dissipates from the magnetic to other\nreservoirs, like phonons or electron correlations. To ex-\nplore what impact non-local damping has on the energy\ndissipation process, we performed atomistic spin dynam-\nics (ASD) simulations for the aforementioned ferromag-\nnets: bcc Fe 1\u0000xCox(x= 0%;30%;50%), fcc Co, and\nfcc Ni, for the (i)fully non-local \u000bijand (ii)e\u000bective\n\u000btot(de\fned in 14) dissipative case. We note that, al-\nthough widely considered in ASD calculations, the adop-6\nTABLE I. Spin ( mspin) and orbital ( morb) magnetic moments, onsite ( \u000bii) damping, total ( \u000btot) damping, and Curie temper-\nature (TC) of the investigated systems. The theoretical TCvalue is calculated within the RPA. In turn, mtdenotes the total\nmoments for experimental results of Ref. [82].\nmspin(\u0016B)morb(\u0016B)\u000bii(\u000210\u00003) \u000btot(\u000210\u00003) TC(K)\nbcc Fe (theory) 2.23 0.05 2.4 2.1 919\nbcc Fe (expt.) 2.13 [68] 0 :08 [68] \u0000 1:9\u00007:2 [33, 83{89] 1044\nbcc Fe 70Co30(theory) 2.33 0.07 0.5 0.9 1667\nbcc Fe 70Co30(expt.) mt= 2:457 [82] \u0000 0:5\u00001:7a[33, 83, 90] 1258 [92]\nbcc Fe 50Co50(theory) 2.23 0.08 1.5 1.6 1782\nbcc Fe 50Co50(expt.) mt= 2:355 [82] \u0000 2:0\u00003:2b[25, 33, 83] 1242 [93]\nfcc Co (theory) 1.62 0 :08 7.4 1.4 1273\nfcc Co (expt.) 1 :68(6) [94] \u0000 \u0000 2:8(5) [33, 89] 1392\nfcc Ni (theory) 0 :61 0 :05 160.1 21.6 368\nfcc Ni (expt.) 0 :57 [68] 0 :05 [68] \u0000 23:6\u000064 [22, 83, 87{89, 95, 96] 631\naThe lower limit refers to polycrystalline Fe 75Co2510 nm-thick \flms from Ref. [33]. Lee et al. [90] also found a low Gilbert damping in\nan analogous system, where \u000btot<1:4\u000210\u00003. For the exact 30% of Co concentration, however, previous results [33, 84, 91] indicate\nthat we should expect a slightly higher damping than in Fe 75Co25.\nbThe upper limit refers to the approximate minimum intrinsic value for a 10 nm-thick \flm of Fe 50Co50jPt (easy magnetization axis).\ntion of a constant \u000btotvalue (case (ii)) is only a good ap-\nproximation for long wavelength magnons close to q= 0.\nFirst, we are interested on the role of non-local damp-\ning in the remagnetization processes as it was already\ndiscussed by Thonig et al. [20] and as it is important\nfor,e.g., ultrafast pump-probe experiments as well as all-\noptical switching. In the simulations presented here, the\nrelaxation starts from a totally random magnetic con-\n\fguration. The results of re-magnetization simulations\nare shown in Figure 2. The fully non-local damping (i)\nin the equation of motion enhances the energy dissipa-\ntion process compared to the case when only the e\u000bective\ndamping (ii)is used. This e\u000bect is found to be more pro-\nnounced in fcc Co and fcc Ni compared to bcc Fe and bcc\nFe50Co50. Thus, the remagnetization time to 90% of the\nsaturation magnetisation becomes \u00185\u00008 times faster\nfor case (i)compared to the case (ii). This is due to\nthe increase of \u000bqaway from the \u0000 point in the whole\nspectrum for Co and Ni (see Fig. 1), where in Fe and\nFe50Co50it typically oscillates around \u000btot.\nFor bcc Fe 70Co30, the e\u000bect of non-local damping on\nthe dynamics is opposite to the data in Fig. 2; the re-\nlaxation process is decelerated. In this case, almost the\nentire\u000bqspectrum is below \u000bq=0, which is an interest-\ning result given the fact that FMR measurements of the\ndamping parameter in this system is already considered\nan ultra-low value, when compared to other metallic fer-\nromagnets [33]. Thus, in the remagnetization process of\nFe70Co30, the majority of magnon modes lifetimes is un-\nderestimated when a constant \u000btotis considered in the\nspin dynamics simulations, which leads to a faster overall\nrelaxation rate.\nAlthough bcc Fe presents the highest Gilbert damp-ing obtained in the series of the Fe-Co alloys (see Table\nI) the remagnetization rate is found to be faster in bcc\nFe50Co50. This can be explained by the fact that the ex-\nchange interactions for this particular alloy are stronger\n(\u001880% higher for nearest-neighbors) than in pure bcc\nFe, leading to an enhanced Curie temperature (see Table\nI). In view of Eq. 13 and Fig. 1, the di\u000berence in the\nremagnetization time between bcc Fe 50Co50and elemen-\ntal bcc Fe arises from \u000bqvalues that are rather close,\nbut where the magnon spectrum of Fe 50Co50has much\nhigher frequencies, with corresponding faster dynamics\nand hence shorter remagnetization times.\nFrom our calculations we \fnd that the sum of non-local\ndamping\u0010P\ni6=j\u000bij\u0011\ncontributes with \u000013%,\u000081%,\n\u000087%, +80%, and +7% to the local damping in bcc Fe,\nfcc Co, fcc Ni, bcc Fe 70Co30, and bcc Fe 50Co50, respec-\ntively. The high positive ratio found in Fe 70Co30indi-\ncates that, in contrast to the other systems analyzed, the\nnon-local contributions act like an anti-damping torque,\ndiminishing the local damping torque. A similar anti-\ndamping e\u000bect in antiferromagnetic (AFM) materials\nhave been reported in theoretical and experimental in-\nvestigations ( e.g., [98, 99]), induced by electrical current.\nHere we \fnd that an anti-damping torque e\u000bect can have\nan intrinsic origin.\nTo provide a deeper understanding of the anti-damping\ne\u000bect caused by a positive non-local contribution, we an-\nalytically solved the equation of motion for a two spin\nmodel system, e.g. a dimer. In the particular case when\nthe onsite damping \u000b11is equal to the non-local con-\ntribution\u000b12, we observed that the system becomes un-\ndamped (see Appendix B). As demonstrated in Appendix\nB, ASD simulations of such a dimer corroborate the re-7\nFIG. 1. Non-local damping ( \u000bij) as a function of the nor-\nmalized real-space pairwise ( i;j) distance computed for each\nneighboring shell, and corresponding Fourier transform \u000bq\n(see Eq. 11) from the onsite value ( \u000bii) up to 136 shells of\nneighbors (136 NN) for: (a,b) bcc Fe; (c,d) bcc Fe 70Co30;\n(e,f) bcc Fe 50Co50in the virtual-crystal approximation; and\nup to 30 shells of neighbors (30 NN) for: (g,h) fcc Co; (i,j) fcc\nNi. The insets in sub\fgures (a,c,e,g,i) show the convergence\nof\u000btotin real-space. The obtained onside damping values are\nshown in Table I. In the insets of the left panel, green full\nlines are guides for the eyes.\nsult of undamped dynamics. It should be further noticed\nthat this proposed model system was used to analyse\nthe stability of the ASD solver, verifying whether it can\npreserve both the spin length and total energy. Full de-\ntail of the analytical solution and ASD simulation of a\nspin-dimer and the anti-damping e\u000bect are provided inAppendix B.\nFIG. 2. Remagnetization process simulated with ASD, con-\nsidering fully non-local Gilbert damping ( \u000bij, blue sold lines),\nand the e\u000bective damping ( \u000btot, red dashed lines), for: (a) fcc\nNi; (b) fcc Co; and (c) bcc Fe 1\u0000xCox(x= 0%;30%;50%).\nThe dashed gray lines indicate the stage of 90% of the satu-\nration magnetization.\nC. Magnon spectra\nIn order to demonstrate the in\ruence of damping on\nmagnon properties at \fnite temperatures, we have per-\nformed ASD simulations to obtain the excitation spectra\nfrom the dynamical structure factor introduced in Sec-\ntion I. Here, we consider 16 NN shells for S(q;!) calcula-\ntions both from simulations that include non-local damp-\ning as well as the e\u000bective total damping (see Appendix\nD for a focused discussion). In Fig. 3, the simulated\nmagnon spectra of the here investigated ferromagnets are\nshown. We note that a general good agreement can be\nobserved between our computed magnon spectra (both\nfrom the the frozen magnon approach as well as from the\ndynamical structure factor) and previous theoretical as\nwell as experimental results [34, 52, 80, 100{103], where\ndeviations from experiments is largest for fcc Ni. This\nexception, however, is well known and has already been\ndiscussed elsewhere [104].\nThe main feature that the non-local damping causes to\nthe magnon spectra in all systems investigated here, is in\nchanges of the full width at half maximum (FWHM) 4q\nofS(q;!). Usually,4qis determined from the super-\nposition of thermal \ructuations and damping processes.\nMore speci\fcally, the non-local damping broadens the\nFWHM compared to simulations based solely on an e\u000bec-\ntive damping, for most of the high-symmetry paths in all\nof the here analyzed ferromagnets, with the exception of\nFe70Co30. The most extreme case is for fcc Ni, as \u000bqex-\nceeds the 0:25 threshold for q=X, which is comparable\nto the damping of ultrathin magnetic \flms on high-SOC\nmetallic hosts [105]. As a comparison, the largest di\u000ber-\nence of FWHM between the non-local damping process\nand e\u000bective damping process in bcc Fe is \u00182 meV, while\nin fcc Ni the largest di\u000berence can reach \u0018258 meV. In\ncontrast, the di\u000berence is \u0018\u00001 meV in Fe 70Co30and the8\nlargest non-local damping e\u000bect occurs around q=N\nand in the H\u0000Pdirection, corroborating with the dis-\ncussion in Section II A. At the \u0000 point, which corresponds\nto the mode measured in FMR experiments, all spins in\nthe system have a coherent precession. This implies that\n@mj\n@tin Eq. 4 is the same for all moments and, thus, both\ndamping scenarios discussed here (e\u000becive local and the\none that also takes into account non-local contributions)\nmake no di\u000berence to the spin dynamics. As a conse-\nquence, only a tiny (negligible) di\u000berence of the FWHM\nis found between e\u000bective and non-local damping for the\nFMR mode at low temperatures.\nThe broadening of the FWHM on the magnon spec-\ntrum is temperature dependent. Thus, the e\u000bect of non-\nlocal damping to the width near \u0000 can be of great in-\nterest for experiments. More speci\fcally, taking bcc Fe\nas an example, the di\u000berence between width in e\u000bective\ndamping and non-local damping process increases with\ntemperature, where the di\u000berence can be enhanced up to\none order of magnitude from T= 0:1 K toT= 25 K.\nNote that this enhancement might be misleading due to\nthe limits of \fnite temperature assumption made here.\nThis temperature dependent damping e\u000bect on FWHM\nsuggests a path for the measurement of non-local damp-\ning in FMR experiments.\nWe have also compared the di\u000berence in the imaginary\npart of the transverse dynamical magnetic susceptibility\ncomputed from non-local and e\u000bective damping. De\fned\nby Eq. 12, the imaginary part of susceptibility is re-\nlated to the FWHM [15]. Similar to the magnon spectra\nshown in Fig. 3, the susceptibility di\u000berence is signi\f-\ncant at the BZ boundaries. Taking the example of fcc\nCo, Im\u001f\u0006(q;!) for e\u000bective damping processes can be\n11:8 times larger than in simulations that include non-\nlocal damping processes, which is consistent to the life-\ntime peak that occurs at high the symmetry point, X,\ndepicted in Fig. 4. In the Fe 1\u0000xCoxalloy, and Fe 70Co30,\nthe largest ratio is 1 :7 and 2:7 respectively. The intensity\nat \u0000 point is zero since \u000bqis independent on the coupling\nvector and equivalent in both damping modes. The ef-\nfect of non-local damping on susceptibility coincides well\nwith the magnon spectra from spin dynamics. Thus, this\nmethod allows us to evaluate the magnon properties in a\nmore e\u000ecient way.\nD. Magnon lifetimes\nBy \ftting the S(q;!) curve at each wave vector with\na Lorentzian curve, the FWHF and hence the magnon\nlifetimes,\u001cq, can be obtained from the simple relation\n[15]\n\u001cq=2\u0019\n4q: (17)\nFigure 4 shows the lifetimes computed in the high-\nsymmetry lines in the BZ for all ferromagnets here in-vestigated. As expected, \u001cqis much lower at the qvec-\ntors far away from the zone center, being of the order\nof 1 ps for the Fe 1\u0000xCoxalloys (x= 0%;30%;50%),\nand from\u00180:01\u00001 ps in fcc Co and Ni. In view of\nEq. 13, the magnon lifetime is inversely proportional to\nboth damping and magnon frequency. In the e\u000bective\ndamping process, \u000bqis a constant and independent of\nq; thus, the lifetime in the entire BZ is dictated only by\n!q. The situation becomes more complex in the non-\nlocal damping process, where the \u001cqis in\ruenced by the\ncombined e\u000bect of changing damping and magnon fre-\nquency. Taking Fe 70Co30as an example, even though\nthe\u000bqis higher around the \u0000, the low magnon frequency\ncompensates the damping e\u000bect, leading to an asymp-\ntotically divergent magnon lifetime as !q!0. However,\nthis divergence becomes \fnite when including e.g. mag-\nnetocrystalline anisotropy or an external magnetic \feld\nto the spin-Hamiltonian. In the H\u0000Npath, the magnon\nenergy of Fe 70Co30is large, but \u000bqreaches\u00184\u000210\u00004\natq=\u00001\n4;1\n4;1\n2\u0001\n, resulting in a magnon lifetime peak of\n\u001810 ps. This value is not found for the e\u000bective damping\nmodel.\nIn the elemental ferromagnets, as well as for Fe 50Co50,\nit is found that non-local damping decreases the magnon\nlifetimes. This non-local damping e\u000bect is signi\fcant in\nboth Co and Ni, where the magnon lifetimes from the \u000bij\nmodel di\u000ber by an order of magnitude from the e\u000bective\nmodel (see Fig. 4). In fact, considering \u001cqobtained from\nEq. 13, the e\u000bective model predicts a lifetime already\nhigher by more than 50% when the magnon frequencies\nare\u001833 meV and\u001814 meV in the K\u0000\u0000 path ( i.e.,\nnear \u0000) of Ni and Co, respectively. This di\u000berence mainly\narises, in real-space, from the strong negative contriu-\ntions of\u000bijin the close neighborhood around the refer-\nence site, namely the NN in Ni and third neighbors in Co.\nIn contrast, due to the \u000bqspectrum composed of almost\nall dampings lower than \u000btot, already discussed in Section\nII A, the opposite trend on \u001cqis observed for Fe 70Co30:\nthe positive overall non-local contribution guide an anti-\ndamping e\u000bect, and the lifetimes are enhanced in the\nnon-local model.\nAnother way to evaluate the magnon lifetimes is from\nthe linear response theory. As introduced in Section I B,\nwe have access to magnon lifetimes at low temperatures\nfrom the imaginary part of the susceptibility. The \u001cq\ncalculated from Eq. 13 is also displayed in Fig. 4. Here\nthe spin-wave frequency !qis from the frozen magnon\nmethod. The magnon lifetimes from linear response have\na very good agreement with the results from the dynam-\nical structure factor, showing the equivalence between\nboth methods. Part of the small discrepancies are re-\nlated to magnon-magnon scattering induced by the tem-\nperature e\u000bect in the dynamical structure factor method.\nWe also \fnd a good agreement on the magnon lifetimes\nof e\u000bective damping in pure Fe with previous studies\n[106]. They are in the similar order and decrease with\nthe increasing magnon energy. However, their results\nare more di\u000bused since the simulations are performed at9\nFIG. 3. Magnon spectra calculated with non-local Gilbert damping and e\u000bective Gilbert damping in: (a) bcc Fe; (b) bcc\nFe70Co30; (c) bcc Fe 50Co50; (d) fcc Co; and (e) fcc Ni. The black lines denote the adiabatic magnon spectra calculated from\nEq. 7. Full red and open blue points denote the peak positions of S(q;!) at each qvector for\u000btotand\u000bijcalculations,\nrespectively, at T= 0:1 K. The width of transparent red and blue areas corresponds to the full width half maximum (FWHM)\non the energy axis \ftted from a Lorentzian curve, following the same color scheme. To highlight the di\u000berence of FWHM\nbetween the two damping modes, the FWHMs shown in the magnon spectrum of Fe 1\u0000xCox, Co, and Ni are multiplied by 20,\n5, and1\n2times, in this order. The triangles represent experimental results: in (a), Fe at 10 K [102] (yellow up) and Fe with\n12% Si at room-temperature [101] (green down); in (d), Co(9 ML)/Cu(100) at room-temperature [103] (green down); in (e) Ni\nat room-temperature (green down) [100]. The standard deviation of the peaks are represented as error bars.\nroom-temperature.\nIII. CONCLUSION\nWe have presented the in\ruence of non-local damping\non spin dynamics and magnon properties of elemental fer-\nromagnets (bcc Fe, fcc Co, fcc Ni) and the bcc Fe 70Co30\nand bcc Fe 50Co50alloys in the virtual-crystal approxima-\ntion. It is found that the non-local damping has impor-\ntant e\u000bects on relaxation processes and magnon prop-\nerties. Regarding the relaxation process, the non-local\ndamping in Fe, Co, and Ni has a negative contribution\nto the local (onsite) part, which accelerates the remagne-\ntization. Contrarily, in\ruenced by the positive contribu-\ntion of\u000bij(i6=j), the magnon lifetimes of Fe 70Co30and\nFe50Co50are increased in the non-local model, typically\nat the boundaries of the BZ, decelerating the remagneti-\nzation.\nConcerning the magnon properties, the non-local\ndamping has a signi\fcant e\u000bect in Co and Ni. More\nspeci\fcally, the magnon lifetimes can be overestimated\nby an order of magnitude in the e\u000bective model for these\ntwo materials. In real-space, this di\u000berence arises as a\nresult of strong negative non-local contributions in theclose neighborhood around the reference atom, namely\nthe NN in Ni and the third neighbors in Co.\nAlthough the e\u000bect of non-local damping to the\nstochastic thermal \feld in spin dynamics is not included\nin this work, we still obtain coherent magnon lifetimes\ncomparing to the analytical solution from linear response\ntheory. Notably, it is predicted that the magnon lifetimes\nat certain wave vectors are higher for the non-local damp-\ning model in some materials. An example is Fe 70Co30, in\nwhich the lifetime can be \u00183 times higher in the H\u0000N\npath for the non-local model. On the other hand, we\nhave proposed a fast method based on linear response\nto evaluate these lifetimes, which can be used to high-\nthroughput computations of magnonic materials.\nFinally, our study provides a link on how non-local\ndamping can be measured in FMR and neutron scat-\ntering experiments. Even further, it gives insight into\noptimising excitation of magnon modes with possible\nlong lifetimes. This optimisation is important for any\nspintronics applications. As a natural consequence of\nany real-space ab-initio formalism, our methodology and\n\fndings also open routes for the investigation of other\nmaterials with preferably longer lifetimes caused by non-\nlocal energy dissipation at low excitation modes. Such\nmaterials research could also include tuning the local10\nFIG. 4. Magnon lifetimes \u001cqof: (a) bcc Fe; (b) bcc Fe 70Co30; (c) bcc Fe 50Co50; (d) fcc Co; and (e) fcc Ni as function of q,\nshown in logarithmic scale. The color scheme is the same of Fig. 3, where blue and red represents \u001cqcomputed in the e\u000bective\nand non-local damping models. The transparent lines and opaque points depict the lifetimes calculated with Eq. 13 and by\nthe FWHM of S(q;!) atT= 0:1 K (see Eq. 17). The lifetime asymptotically diverges around the \u0000-point due to the absence\nof anisotropy e\u000bects or external magnetic \feld in the spin-Hamiltonian.\nchemical environments by doping or defects.\nIV. ACKNOWLEDGMENTS\nFinancial support from Vetenskapsr\u0017 adet (grant num-\nbers VR 2016-05980 and VR 2019-05304), and the\nKnut and Alice Wallenberg foundation (grant number\n2018.0060) is acknowledged. Support from the Swedish\nResearch Council (VR), the Foundation for Strategic Re-search (SSF), the Swedish Energy Agency (Energimyn-\ndigheten), the European Research Council (854843-\nFASTCORR), eSSENCE and STandUP is acknowledged\nby O.E. . Support from the Swedish Research Coun-\ncil (VR) is acknowledged by D.T. and A.D. . The\nChina Scholarship Council (CSC) is acknowledged by\nZ.L.. The computations/data handling were enabled by\nresources provided by the Swedish National Infrastruc-\nture for Computing (SNIC) at the National Supercom-\nputing Centre (NSC, Tetralith cluster), partially funded\nby the Swedish Research Council through grant agree-\nment No. 2016-07213.\n[1] A. Barman, G. Gubbiotti, S. Ladak, A. O. Adeyeye,\nM. Krawczyk, J. Gr afe, C. Adelmann, S. Cotofana,\nA. Naeemi, V. I. Vasyuchka, et al. , J. Phys. Condens.\nMatter 33, 413001 (2021).\n[2] P. Pirro, V. I. Vasyuchka, A. A. Serga, and B. Hille-\nbrands, Nat. Rev. Mater 6, 1114 (2021).\n[3] B. Rana and Y. 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Sakuma, J. Appl. Phys. 117, 013912 (2015).13\nAppendix A: Numerical solver\nIn this Appendix, the numerical method to solve Eq.\n1 is described. In previous studies, several numerical\napproaches have been proposed to solve the local LLG\nequations, including HeunP method, implicit midpoint\nmethod, Depondt-Merten's method [107], semi-implicit\nA (SIA) and semi-implicit B (SIB) methods [42]. To solve\nthis non-local LLG equation, we use the \fxed-point iter-\nation midpoint method. We have done convergence tests\non this method and \fnd that it preserve the energy and\nspin length of the system, which is demonstrated in Fig.\n5 for the case of a dimer. With stable outputs, the solver\nallows for a relatively large time step size, typically of\nthe order of \u0001 t\u00180:1\u00001 fs.\nFollowing the philosophy of an implicit midpoint\nmethod, the implemented algorithm can be described as\nfollows. Let mt\nibe the magnetic moment of site iat a\ngiven time step t. Then we can de\fne the quantity mmidand the time derivative of mi, respectively, as\nmmid=mt+1\ni+mt\ni\n2;\n@mi\n@t=mt+1\ni\u0000mt\ni\n\u0001t:(A1)\nUsing this de\fnition in Eq. 4, the equation of motion\nof thei-th spin becomes:\n@mi\n@t=mmid\u00020\n@\u0000\r[Bi(mmid) +bi(t)] +X\nj\u000bij\nmj@mj\n@t1\nA:\n(A2)\nThus, with a \fxed-point scheme, we can do the follow-\ning iteration\nmt+1(k+1)\ni =mt\ni+ \u0001t0\n@ \nmt+1(k)\ni +mt\ni\n2!\n\u00020\n@\u0000\r\"\nBi \nmt+1(k)\ni +mt\ni\n2!\n+bi(t)#\n+X\nj\u000bij\nmjmt+1(k)\nj\u0000mt\nj\n\u0001t1\nA1\nA:\n(A3)\nIfmt+1(k+1)\ni\u0019mt+1(k)\ni , the self-consistency con-\nverges. Typically, about 6 iteration steps are needed.\nThis solver was implemented in the software package Up-\npASD [48] for this work.\nAppendix B: Analytical model of anti-damping in\ndimers\nIn the dimer model, there are two spins on site 1 and\nsite 2 denoted by m1andm2, which are here supposed\nto be related to the same element { so that, naturally,\n\u000b11=\u000b22>0. Also, let's consider a su\u000eciently low\ntemperature so that bi(t)!0, which is a reasonable\nassumption, given that damping has an intrinsic origin\n[108]. This simple system allows us to provide explicit\nexpressions for the Hamiltonian, the e\u000bective magnetic\n\felds and the damping term. From the analytical solu-\ntion, it is found that the dimer spin system becomes an\nundamped system when local damping is equal to non-\nlocal damping, i.e.the e\u000bective damping of the system\nis zero.\nFollowing the de\fnition given by Eq. 4 in the main\ntext, the equation of motion for spin 1 reads:\n@m1\n@t=m1\u0002\u0012\n\u0000\rB1+\u000b11\nm1@m1\n@t+\u000b12\nm2@m2\n@t\u0013\n;(B1)\nand an analogous expression can be written for spin 2.\nFor sake of simplicity, the Zeeman term is zero and thee\u000bective \feld only includes the contribution from Heisen-\nberg exchange interactions. Thus, we have B1= 2J12m2\nandB2= 2J21m1. Withj\u000bijj\u001c 1, we can take the\nLL form@mi\n@t=\u0000\rmi\u0002Bito approximate the time-\nderivative on the right-hand side of the LLG equation.\nLetm1=m2and\u000b12=\u0015\u000b11. SinceJ12=J21and\nm1\u0002m2=\u0000m2\u0002m1, then we have\n@m1\n@t=\u00002\rJ12m1\u0002\u0014\nm2+ (1\u0000\u0015)\u000b11\nm1(m1\u0002m2)\u0015\n:\n(B2)\nTherefore, when \u000b12=\u000b21=\u000b11(i.e.,\u0015= 1), Eq. B1\nis reduced to:\n@m1\n@t=\u00002\rJ12m1\u0002m2; (B3)\nand the system becomes undamped. It is however\nstraightforward that, for the opposite case of a strong\nnegative non-local damping ( \u0015=\u00001), Eq. B2 describes\na common damped dynamics. A side (and related) con-\nsequence of Eq. B2, but important for the discussion in\nSection II B, is the fact that the e\u000bective onsite damp-\ning term\u000b\u0003\n11= (1\u0000\u0015)\u000b11becomes less relevant to the\ndynamics as the positive non-local damping increases\n(\u0015!1), or, in other words, as \u000btot= (\u000b11+\u000b12) strictly\nincreases due to the non-local contribution. Exactly the\nsame reasoning can be made for a trimer, for instance,\ncomposed by atoms with equal moments and exchange\ninteractions ( m1=m2=m3,J12=J13=J23), and\nsame non-local dampings ( \u000b13=\u000b12=\u0015\u000b11).14\nThe undamped behavior can be directly observed from\nASD simulations of a dimer with \u000b12=\u000b11, as shown in\nFig. 5. Here the magnetic moment and the exchange are\ntaken the same of an Fe dimer, m1= 2:23\u0016BandJ12=\n1:34 mRy. Nevertheless, obviously the overall behavior\ndepicted in Fig. 5 is not dependent on the choice of\nm1andJ12. Thezcomponent is constant, while the x\nandycomponents of m1oscillate in time, indicating a\nprecessing movement.\nIn a broader picture, this simple dimer case exempli\fes\nthe connection between the eigenvalues of the damping\nmatrix\u000b= (\u000bij) and the damping behavior. The occur-\nrence of such undamped dynamics has been recently dis-\ncussed in Ref. [109], where it is shown that a dissipation-\nfree mode can occur in a system composed of two sub-\nsystems coupled to the same bath.\n0.00 0.02 0.04 0.06 0.08 0.10\nt(ps)0.2\n0.00.20.40.60.81.0Magnetization\n3.0\n2.5\n2.0\n1.5\n1.0\n0.5\n0.0\nEnergy(mRy)mxmymzmEnergy\nFIG. 5. Spin dynamics at T= 0 K of an undamped dimer\nin which\u000b12=\u000b21=\u000b11(see text). The vector m1is\nnormalized and its Cartesian components are labeled in the\n\fgure asmx,myandmz. The black and grey lines indicate\nthe length of spin and energy (in mRy), respectively.\nAppendix C: E\u000bective and onsite damping in the\nFeCo and CoNi alloys\nAs mentioned in Section I, the simple VCA model al-\nlows us to account for the disorder in 3 d-transition-metal\nalloys in a crude but e\u000ecient way which avoids the use\nof large supercells with random chemical distributions.\nWith exactly the same purpose, the coherent potential\napproximation (CPA) [110] has also been employed to\nanalyze damping in alloys ( e.g., in Refs. [84, 111, 112]),\nshowing a very good output with respect to trends, when\ncompared to experiments [33, 81]. In Fig. 6 we show\nthe normalized calculated local (onsite, \u000bii) and e\u000bec-\ntive damping ( \u000btot) parameters for the zero-temperature\nVCA Fe 1\u0000xCoxalloy in the bcc structure, consistent with\na concentration up to x\u001960% of Co [33]. The computed\nvalues in this work (blue, representing \u000bii, and red points,\nrepresenting \u000btot) are compared to previous theoretical\nCPA results and room-temperature experimental data.\nThe trends with VCA are reproduced in a good agree-ment with respect to experiments and CPA calculations,\nshowing a minimal \u000btotwhen the Co concentration is\nx\u001930%. This behavior is well correlated with the local\ndensity of states (LDOS) at the Fermi level, as expected\nby the simpli\fed Kambersk\u0013 y equation [113], and the on-\nsite contribution. Despite the good agreement found, the\nvalues we have determined are subjected to a known error\nof the VCA with respect to the experimental results.\nThis discrepancy can be partially explained by three\nreasons: ( i) the signi\fcant in\ruence of local environ-\nments (local disorder and/or short-range order) to \u000btot\n[25, 77]; ( ii) the fact that the actual electronic lifetime\n(i.e., the mean time between two consecutive scattering\nevents) is subestimated by the VCA average for random-\nness in the FeCo alloy, which can have a non-negligible\nimpact in the damping parameter [22, 114]; and ( iii) the\nin\ruence on damping of noncollinear spin con\fgurations\nin \fnite temperature measurements [54, 115]. On top of\nthat, it is also notorious that damping is dependent on\nthe imaginary part of the energy (broadening) [22, 114],\n\u000e, which can be seen as an empirical quantity, and ac-\ncounts for part of the di\u000berences between theory and ex-\nperiments.\n 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035\n 0  10  20  30  40  50  60 0 5 10 15 20 25Damping value\nDOS at EF (states/Ry−atom)\nCo concentration (%)onsite  (αii)\ntotal  (αtot)\nTurek et al.  (αtot)\nMankovsky  et al.  [2013]  (αtot)\nMankovsky  et al.  [2018]  (αtot)\nSchoen et al.\nn(EF)\nFIG. 6. (Color online) Left scale : Computed Gilbert e\u000bec-\ntive (\u000btot, red circles) and onsite ( \u000bii, blue squares) damping\nparameters as a function of Co the concentration ( x) for bcc\nFe1\u0000xCoxbinary alloy in the virtual-crystal approximation.\nThe values are compared with previous theoretical results us-\ning CPA, from Ref. [84] (gray full triangles), Ref. [54] (black\nopen rhombus), Ref. [112] (yellow open triangles), and room-\ntemperature experimental data [33]. Right scale : The calcu-\nlated density of states (DOS) at the Fermi level as a function\nofx, represented by the black dashed line.\nIn the spirit of demonstrating the e\u000bectiveness of the\nsimple VCA to qualitatively (and also, to some extent,\nquantitatively) describe the properties of Gilbert damp-\ning in suitable magnetic alloys, we also show in Fig. 7 the\nresults obtained for Co xNi1\u0000xsystems. The CoNi alloys15\nare known to form in the fcc structure for a Ni concen-\ntration range of 10% \u0000100%. Therefore, here we mod-\neled CoxNi1\u0000xby a big fcc cluster containing \u0018530000\natoms in real-space with the equilibrium lattice parame-\nter ofa= 3:46\u0017A. The number of recursion levels consid-\nered isLL= 41. A good agreement with experimental\nresults and previous theoretical calculations can be no-\nticed. In particular, the qualitative comparison with the-\nory from Refs. [81, 84] indicates the equivalence between\nthe torque correlation and the spin correlation models\nfor calculating the damping parameter, which was also\ninvestigated by Sakuma [116]. The onsite contribution\nfor each Co concentration, \u000bii, is omitted from Fig. 7\ndue to an absolute value 2 \u00004 times higher than \u000btot,\nbut follows the same decreasing trend. Again, the over-\nall e\u000bective damping values are well correlated with the\nLDOS, and re\rect the variation of the quantity1\nmtwith\nCo concentration (see Eq. 15).\n 0 0.005 0.01 0.015 0.02 0.025\n 0  10  20  30  40  50  60  70 10 15 20 25 30Damping value\nDOS at EF (states/Ry−atom)\nCo concentration (%)total  (αtot)\nMankovsky  et al.  [2013]  (αtot)\nStarikov et al.  (αtot)\nSchoen [2017] et al.\nn(EF)\nFIG. 7. (Color online) Left scale : Computed Gilbert e\u000bective\n(\u000btot, red circles) damping parameters as a function of the Co\nconcentration ( x) for fcc Co xNi1\u0000xbinary alloy in the virtual-\ncrystal approximation. The values are compared with previ-\nous theoretical results using CPA, from Ref. [84] (gray full\ntriangles), Ref. [81] (gold full circles), and room-temperature\nexperimental data [89]. Right scale : The calculated density of\nstates (DOS) at the Fermi level as a function of x, represented\nby the black dashed line.\nAppendix D: E\u000bect of further neighbors in the\nmagnon lifetimes\nWhen larger cuto\u000b radii ( Rcut) of\u000bijparameters are\nincluded in ASD, Eq. A3 takes longer times to achieve a\nself-consistent convergence. In practical terms, to reach a\nsizeable computational time for the calculation of a given\nsystem,Rcutneeds to be chosen in order to preserve the\nmain features of the magnon properties as if Rcut!1 .\nA good quantity to rely on is the magnon lifetime \u001cq,as it consists of both magnon frequency and q-resolved\ndamping (Eq. 13). In Section II C, we have shown the\nequivalence between Eq. 13 and the inverse of FWHM\non the energy axis of S(q;!) for the ferromagnets inves-\ntigated here. Thus, the comparison of two \u001cqspectra for\ndi\u000berentRcutcan be done directly and in an easier way\nusing Eq. 13.\nFIG. 8. (Color online) Magnon lifetimes calculated using Eq.\n13 for: (a) bcc Fe; and (b) bcc Fe 50Co50, using a reduced set\nof 16 NN shells (opaque lines), and the full set of 136 NN\nshells (transparent lines).\nAn example is shown in Figure 8 for bcc Fe and bcc\nFe50Co50. Here we choose the \frst 16 NN ( Rcut\u00183:32a)\nand compare the results with the full calculated set of\n136 NN (Rcut= 10a). It is noticeable that the reduced\nset of neighbors can capture most of the features of the\n\u001cqspectrum for a full NN set. However, long-range in-\n\ruences of small magnitudes, such as extra oscillations\naround the point q=Hin Fe, can occur. In particu-\nlar, these extra oscillations arise mainly due to the pres-\nence of Kohn anomalies in the magnon spectrum of Fe,\nalready reported in previous works [52, 80]. In turn, for\nthe case of Fe 50Co50, the long-range \u000bijreduces\u000btot, and\ncauses the remagnetization times for non-local and e\u000bec-16\ntive dampings to be very similar (see Fig. 2). For the\nother ferromagnets considered in the present research,comparisons of the reduced Rcutwith analogous quality\nwere reached."    },    {        "title": "1210.1552v1.QED_with_chiral_nonminimal_coupling__aspects_of_the_Lorentz_violating_quantum_corrections.pdf",        "content": "arXiv:1210.1552v1  [hep-th]  4 Oct 2012QED with chiral nonminimal coupling: aspects of the Lorentz -violating\nquantum corrections\nA. P. Baˆ eta Scarpelli(a)∗\n(a) Setor T´ ecnico-Cient´ ıfico - Departamento de Pol´ ıcia F ederal\nRua Hugo D’Antola, 95 - Lapa - S˜ ao Paulo - Brazil\n(Dated: November 13, 2018)\nAn effective model for QED with the addition of a nonminimal co upling with a chiral character is\ninvestigated. This term, which is proportional to a fixed 4-v ectorbµ, violates Lorentz symmetry and\nmay originate a CPT-even Lorentz breaking term in the photon sector. It is shown that this Lorentz\nbreaking CPT-even term is generated and that,in addition, t he chiral nonminimal coupling requires\nthis term is present from the beginning. The nonrenormaliza bility of the model is invoked in the\ndiscussion of this fact andthe result is confronted with the one from amodel with aLorentz-violating\nnonminimal coupling without chirality.\nPACS numbers: 11.30.Cp, 11.30.Er, 11.30.Qc, 12.60.-i\nI. INTRODUCTION\nLorentz violating models are being object of intensive investigation s ince the beginning of the nineties\n[1]-[50]. The Standard Model Extension (SME) [1]-[4] proposes a wide r ange of possibilities in this\ncontext, concerning both the gauge and fermion sectors. There are two particularly interesting terms\nin the photon sector of the SME: the CPT-odd term of Carroll-Field- Jackiw [17] and a CPT-even term\nwhich is quadratic in the field strength [2]. In the context of the SME, there have been great interest\nin the quantum induction of the Carroll-Field-Jackiw term [22]-[36]. On t he other hand, the radiative\ngeneration of the CPT-even one (an aether-like term [5]) has been in vestigated in the context of effective\nmodels which include a Lorentz violating nonminimal (magnetic) coupling [6], [7], [8]. The perturbative\ngeneration of higher derivative Lorentz-breaking terms from the nonminimal coupling has also been\ninvestigated [9].\nThis kind ofmodel hasbeen consideredbefore in papers[10]-[13] in t he contextofRelativistic Quantum\nMechanics. It was used in the calculation of corrections to the Hydr ogen spectrum, from which very\nstringent bounds have been set up in the magnitude of the Lorentz violating parameter [12]. It was also\nused to study the magnetic moment generation from the nonminimal coupling, since a tiny magnetic\ndipole moment of elementary neutral particles might signal Lorentz symmetry violation [13].\nIn [6] and [7] the magnetic coupling has been considered in the place of the minimal coupling in a\ngauge violating model. The coefficient of the induced aether-like term was shown to be regularization\ndependent. The inclusion of the minimal coupling together with the ma gnetic one was considered in\npaper [8]. In this case, if a gauge invariant regularization prescriptio n is used, there is no quantum\ninduction of Lorentz-violating terms in the gauge sector (CPT-eve n or -odd) at one-loop order. However,\na gauge violating regularization technique followed by a symmetry res toring counterterm in the Lorentz\nsymmetric sector opens the possibility for such inductions.\nIn this paper, another kind of nonminimal coupling is considered, in wh ich the background vector bµ\nappears coupled to the gauge field by means of a term of chiral char acter. It is shown that, in this\ncase, there is no induction of CPT-odd terms. On the other hand, t he one-loop photon self-energy will\ninclude an aether-like CPT-even part, with a divergent coefficient. T his shows that this aether-like part\nmust be included from the beginning in the gauge sector. This fact ca n be understood in terms of the\nnonrenormalizability of the model. If higher order contributions are considered, this term is just the first\n∗scarpelli.apbs@dpf.gov.br2\none of an infinite list that should be included from the beginning. We disc uss this fact is the context of\nan effective model, in which the determination of a physical cutoff is an essential step.\nThispaperisorganizedasfollows: thesecondsectionisdedicatedto ageneraldiscussiononnonminimal\ncouplingmodels; thethirdsectionpresentstheone-loopcalculation ofthephotonself-energyforthemodel\nwith a chiral nonminimal coupling; the conclusions are drawn in section four.\nII. DISCUSSION ON QED WITH NONMINIMAL COUPLINGS\nThe action of a QED model with a nonminimal coupling is written as\nΣ =/integraldisplay\nd4x/braceleftbigg\n−1\n4FµνFµν+¯ψ/parenleftbig\ni∂ /−m−eA /−gεµναβΓβbµFνα/parenrightbig\nψ/bracerightbigg\n, (1)\nwhere Γβ=γβfor the simple magnetic coupling and Γ β=γ5γβfor the chiral nonminimal coupling.\nConventional QED is recovered in the limit g→0. The action of equation (1) describes a gauge invariant\nmodel with some interesting classical features as investigated in [10 ]. Indeed, it is shown for the case with\nΓβ=γβ, that the 3-vector /vectorbplays the role of a kind of magnetic dipole moment ( /vector µ=g/vectorb). Besides, in this\ncase there is an induction of a Aharonov-Casher (A-C) effect. On t he other hand, the chiral nonminimal\ncoupling contributes to the interaction energy without inducting a A -C phase.\nWe present now a general discussion on the one-loop correction to the photon self-energy for a QED\nwith nonminimal coupling. First, a great simplification in the calculations is obtained if we write Bβ=\nεµναβbµFνα. In the caseofthe simple nonminimal coupling, if the purpose is onlyth e one-loopcalculation\nof the vacuum polarization tensor, it is yet possible to define the field ˜Aµ=eAµ+gBµ, so that the new\nlagrangian density for the fermion sector can be written as\nLψ=¯ψ/parenleftBig\ni∂ /−m−˜A //parenrightBig\nψ. (2)\nIn terms of ˜Aµ, the one-loop correction to the photon two-point function is ident ical to the one for the\nconventional QED. In momentum space, we have\nTµν(p) = tr/integraldisplay\nkγνs(p+k)γµs(k), (3)\nin whichs(k) is the fermion propagator and/integraltext\nkstands for/integraltext\nd4k/(2π)4. The corrections to the photon\nsector in the lagrangian density will be given by\n−1\n2˜AµTµν(x)˜Aν\n=−1\n2Aµe2Tµν(x)Aν−1\n2Aµ2egTµν(x)Bν−1\n2Bµg2Tµν(x)Bν. (4)\nIn this case in which chirality is absent, treated in [8], the discussion of the induced Lorentz-violating\nterms can be performed in general grounds. A general expressio n forTµνobtained by means of some\nregularization technique, not necessarily gauge invariant, compat ible with its Lorentz structure is\nTµν=/parenleftbig\npµpν−p2gµν/parenrightbig\nΠ(p2)+αm2gµν+βpµpν, (5)\nin whichαandβare dimensionless constants. In eq. (4), the first term is the trad itional QED one. The\nsecond is a Lorentz-violating CPT-odd Chern-Simons-like term, whic h in the on-shell limit is given by\nLCS=−αegm2εµναβbµAνFαβ. (6)\nThe third term in eq. (4) is a CPT-even term, which in the on-shell limit is written as\nLeven=−αm2g2b2FµνFµν+2αm2g2(bµFµν)2, (7)3\nwhere\nBµBµ= 2b2FµνFµν−4(bµFµν)2(8)\nhas been obtained with the help of some properties of the L´ evi-Civit ` a tensor. We recognize in the second\nterm ofLeventhe Lorentz-violating aether term.\nA comment is in order. The value of the constant αis determined by the regularization procedure used\nin the calculation. If a regularization technique which preserves gau ge invariance of the original QED is\nused,αwill be null. However, it is always possible to choose a gauge breaking pr ocedure and then restore\nthe symmetry by means of a non-symmetric counterterm. In such case, since the Lorentz violating and\nLorentz preserving parts are independent on each other, the Lo rentz breaking terms would survive. Such\nprocedure is equivalent to use different regularization techniques in different sectors. Nevertheless, the\nnatural framework is using an unique regularization in the calculation of integrals which contribute to\nthe same amplitude. In this case, the calculation with a gauge preser ving method would not induce, at\none-loop order, these two Lorentz-violating terms.\nThe discussion above will not apply to the case of chiral nonminimal int eraction, as it will be presented\nin the next section.\nIII. QUANTUM CORRECTIONS TO THE QED WITH CHIRAL NONMINIMAL\nCOUPLING\nWe now turn our attention to the the one-loop quantum correction s to the photon sector in a model\nwith a nonminimal interaction with a chiral character. We now have th e following lagrangian density for\nthe fermion sector:\nLψ=¯ψ(i∂ /−m−eA /−gγ5B /)ψ. (9)\nThis fermion lagrangian in terms of the vector and axial-vector fields AµandBµ, in the case in which\nthe fields do not depend on each other, has been vastly investigate d (see, for example, [52] and [53]). The\none-loop corrections to the photon sector will be given by\n−1\n2Aµe2TVV\nµν(x)Aν−1\n2Aµ2egTAV\nµν(x)Bν−1\n2Bµg2TAA\nµν(x)Bν, (10)\nwhere the upper indices AandVrefer to the axial and vectorial vertices, so that TVV\nµν=Tµν. In\nmomentum space, we have\nTAV\nµν(p) =/integraldisplay\nktr{γνs(p+k)γ5γµs(k)} (11)\nand\nTAA\nµν(p) =/integraldisplay\nktr{γ5γνs(p+k)γ5γµs(k)}. (12)\nThe second term in (10), which would be CPT-odd, is actually identically null, since in eq. (11), we have\nin the numerator\ntr{γ5γρ(p /+k /+m)γµ(k /+m)}= tr{γ5γρ(p /+k /)γµk /}+\n+m2tr{γ5γργµ}+mtr{γ5γρ(p /+k /)γµ}+mtr{γ5γργµk /}\n= 4iερκµλ(p+k)κkλ= 4iερκµλpκkλ. (13)\nUnder integration, this will vanish due to the antisymmetry of the L´ evi-Civit` a tensor.\nWe are left with the CPT-even term of (10). First, we can write\ntr/braceleftbig\nγ5γδ(p /+k /+m)γ5γρ(k /+m)/bracerightbig\n= tr/braceleftbig\nγδ(p /+k /+m)γρ(k /+m)/bracerightbig\n−8m2gρδ, (14)4\nso that\nTAA\nµν=Tµν−8m2gµνI (15)\nwhere\nI=/integraldisplayΛd4k\n(2π)41\n(k2−m2)[(p−k)2−m2](16)\nis a divergent integral and Λ is to indicate that some regularization pr escription is applied. We have to\nnote that the contribution of the first term in (15) is identical to th e one calculated with the nonminimal\ncoupling without chirality. The divergent integral Ican be evaluated by any regularization method.\nIn order to make the regularization independence manifest, we may write equation (16) in a way that\ndivergences are expressed in terms of the loop momentum only, as in Implicit Regularization [54]:\nI=Ilog/parenleftbig\nλ2/parenrightbig\n−bZ0(p2,m2,λ2), (17)\nwhere\nIlog/parenleftbig\nλ2/parenrightbig\n=/integraldisplayΛd4k\n(2π)41\n(k2−λ2)2, (18)\nZ0(p2,m2,λ2) =/integraldisplay1\n0dxln/parenleftbiggp2x(1−x)−m2\n(−λ2)/parenrightbigg\n, (19)\nb=i/(4π)2andλ2is an arbitrary ultraviolet mass scale. Since we are interested in the o n-shell limit, we\nwill be left with\nTAA\nµν=m2gµν/bracketleftbigg\n−8Ilog/parenleftbig\nλ2/parenrightbig\n+α+8bln/parenleftbiggm2\nλ2/parenrightbigg/bracketrightbigg\n≡F/parenleftbig\nm2,λ2/parenrightbig\ngµν, (20)\nwhereαis defined in equation (5). So, the CPT-even term will be given by\nLeven=−1\n2F/parenleftbig\nm2,λ2/parenrightbig\nBµBµ\n=−F/parenleftbig\nm2,λ2/parenrightbig/bracketleftBig\nb2FµνFµν−2(bµFµν)2/bracketrightBig\n. (21)\nThe Lorentz-violating second term above can be mapped in the CPT- even term proposed in [2],\nLeven=−1\n4κµναβFµνFαβ, (22)\nas long as we establish the relation\nκµναβ=−2F/parenleftbig\nm2,λ2/parenrightbig\n(gµαbνbβ−gναbµbβ+gνβbµbα−gµβbνbα). (23)\nWe are now in position to discuss the result of equation (21). First, a s discussed in the last section, the\nconstantαvanishes if a gauge invariant regularization technique is used, althou gh the gauge symmetry\nof the model could be preserved if the Lorentz invariant and Loren tz-violating sectors are treated inde-\npendently. However, the value of αis irrelevant here, since it can be absorbed in the other finite term,\nwhich depend on an arbitrary mass scale parameter.\nSecond, the presence of a divergent term in the CPT-even coefficie nt is an important point to be\nanalyzed. This indicates that the original classical action must cont ain such a term. In other words, the\ninclusion of a chiral nonminimal coupling in a modified QED requires the pr esence of the aether term\nfrom the beginning. This divergent factor multiplies also a Maxwell ter m. This means that the Lorentz\npreserving sector is also affected by the presence of the nonminima l interaction of chiral character. This5\nis in contrast with the case of standard nonminimal coupling treated in papers [6], [7] and [8]. In that\ncase, the correction to the Maxwell term (and also the induced aet her term) is finite and arbitrary, with\nthe possibility of being set to zero.\nLast but not least, we must take into account that our model is non renormalizable. We have carried\nout an one-loopcalculation and, at this order in the perturbative ex pansion, it has been shown that a new\ntermwhichviolatesLorentzsymmetryandisCPT-evenshouldbe inclu dedintheclassicalaction. Ifwego\nbeyondtheone-looporder,certainlynewothertermswillhavetob econsidered. Thenonrenormalizability\nof the model tells us that there is no a finite number of counterterm s that will be sufficient to renormalize\nthe theory. So, if we would like to deal with this effective model, we will h ave to stop at one-loop\norder. For this, it is necessary to find a cutoff energy Λ. This can be done like in the case of the simple\nnonminimal coupling, discussed bellow.\nWe have to note that higher order terms in the coupling constant will allow for higher power contribu-\ntions in the Lorentz violating parameter, with an increasing of the de gree of divergence of the integrals.\nHowever, as demonstrated in [12] for the vectorial nonminimal co upling, the magnitude of the back-\nground vector is extremely small. We can impose the reasonable cond ition for the recovering of QED,\n|b2|Λ2<<1. Since the effect of the divergences can be seen in a simplified form b y substituting m2by\nΛ2in the coefficients, higher order calculations will furnish us higher pow ers of|b2|Λ2. So, although we\ncan not prevent the proliferation of new terms beyond one-loop or der, the predictability of such effective\nmodel is assured by the cutoff inequality above. This happens becau se at the same loop order each\nnonminimal vertex contributes with one factor of bµ, whereas higher loop orders with a fixed number of\nnonminimal vertices are controlled by the smallness of the coupling co nstant.\nIn an effective model, the cutoff energy is an important parameter w hich should be established on\nphysical grounds. Important features of a Quantum Field Theory , like causality and stability, can be\nlost at high energies [15], [16]. The condition imposed by the inequality |b2|Λ2<<1 is so that higher\npower terms in bµbecome less significant. In [12], it has been established a bound such t hatg· |bµ|<\n10−32(eV)−1for the simple magnetic coupling. This bound and the inequality we prop ose above assure\nthat such effective model is not considered at energies beyond the Planck scale.\nSo, it is desirable that a calculation such that of [12] would be perform ed in order to establish a bound\nfor the Lorentz violating parameter in the case of a chiral nonminima l coupling. In this case, we could\nconsider the one-loop calculation meaningful.\nIV. CONCLUDING COMMENTS\nAn effective model for QED with the addition of a chiral nonminimal cou pling has been investigated.\nThis term, which is proportional to a fixed 4-vector bµ, violates Lorentz symmetry and originates a CPT-\neven Lorentz breaking term in the photon sector. Besides, since t he coefficient of this quantum correction\nis divergent, such a model requires the presence of this aether te rm from the beginning in the classical\naction. This divergent factor multiplies also a Maxwell term. This mean s that the Lorentz preserving\nsectoris alsoaffected by the presence ofthe nonminimal interactio nof chiralcharacter. This is in contrast\nwith the case of standard nonminimal coupling treated in the papers [6], [7] and [8]. In that case, the\ncorrection to the Maxwell term is finite and arbitrary, with the poss ibility of being set to zero.\nThefactthattheaethertermshouldbepresentintheclassicalac tionfromthebeginningisproblematic.\nIf we go beyond one-loop order, the nonrenormalizability of the mod el tells us that there is no a finite\nnumber of counterterms that will be sufficient to renormalize the th eory. So, if we would like to deal with\nthis effective model, we will have to stop at one-loop order. For this, it is necessary to find a cutoff energy\nΛ. This can be done like in the case of the simple nonminimal coupling. In a n effective model, the cutoff\nenergy is an important parameter which should be established on phy sical grounds.Important features of\na Quantum Field Theory, like causality and stability, can be lost at high e nergies [15], [16]. The condition\nimposed by the inequality |b2|Λ2<<1 is so that higher power terms in bµbecome less significant. In\n[12], it has been established a bound such that g·|bµ|<10−32(eV)−1for the simple magnetic coupling.\nThis bound and the inequality we propose above assure that such eff ective model is not considered at\nenergies beyond the Planck scale.\nSo, it is desirable that a calculation such that of [12] would be perform ed in order to establish a bound6\nfor the Lorentz violating parameter in the case of a chiral nonminima l coupling. In this case, we could\nconsider the one-loop calculation meaningful.\nAcknowledgements\nThis work was partially supported by CNPq. The author wish to thank Marcos Sampaio and Prof. J.\nA. 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Battistel, PhD thesis , Federal\nUniversity of Minas Gerais (2000)."    },    {        "title": "1109.6043v2.Current_induced_forces_in_mesoscopic_systems__a_scattering_matrix_approach.pdf",        "content": "Current-induced forces in mesoscopic systems: a scattering matrix approach\nNiels Bode,1Silvia Viola Kusminskiy,1Reinhold Egger,2and Felix von Oppen1\n1Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universit at Berlin, 14195 Berlin, Germany\n2Institut f ur Theoretische Physik, Heinrich-Heine-Universit at, D-40225 D usseldorf, Germany\n(Dated: August 7, 2021)\nNanoelectromechanical systems are characterized by an intimate connection between electronic\nand mechanical degrees of freedom. Due to the nanoscopic scale, current \rowing through the\nsystem noticeably impacts the vibrational dynamics of the device, complementing the e\u000bect of\nthe vibrational modes on the electronic dynamics. We employ the scattering matrix approach to\nquantum transport to develop a uni\fed theory of nanoelectromechanical systems out of equilibrium.\nFor a slow mechanical mode, the current can be obtained from the Landauer-B uttiker formula in\nthe strictly adiabatic limit. The leading correction to the adiabatic limit reduces to Brouwer's\nformula for the current of a quantum pump in the absence of the bias voltage. The principal result\nof the present paper are scattering matrix expressions for the current-induced forces acting on the\nmechanical degrees of freedom. These forces control the Langevin dynamics of the mechanical\nmodes. Speci\fcally, we derive expressions for the (typically nonconservative) mean force, for the\n(possibly negative) damping force, an e\u000bective \"Lorentz\" force which exists even for time reversal\ninvariant systems, and the \ructuating Langevin force originating from Nyquist and shot noise of the\ncurrent \row. We apply our general formalism to several simple models which illustrate the peculiar\nnature of the current-induced forces. Speci\fcally, we \fnd that in out of equilibrium situations the\ncurrent induced forces can destabilize the mechanical vibrations and cause limit-cycle dynamics.\nI. INTRODUCTION\nScattering theory has proved a highly successful method for treating coherent transport in mesoscopic systems [1].\nPart of its appeal is rooted in its conceptual simplicity: transport through a mesoscopic object can be described\nin terms of transmission and re\rection of electronic waves which are scattered by a potential. This approach was\nintroduced by Landauer [2,3] and generalized by B uttiker et al. [4] and leads to their well-known formula for the\nconductance of multi-terminal mesoscopic conductors. For time-dependent phenomena, scattering matrix expressions\nhave been obtained for quantum pumping [5,6], a process by which a direct current is generated through temporal\nvariations of relevant parameters of the system, such as a gate voltage or a magnetic \feld. The case of pumping in\nan out-of-equilibrium, biased system has remained largely unexplored so far [7,8].\nThe purpose of the present paper is to further develop the scattering matrix approach into a simple, unifying\nformalism to treat nanoelectromechanical systems (NEMS). The coupling between mechanical and electronic degrees\nof freedom is the de\fning characteristic of NEMS [9,10], such as suspended quantum dots [11], carbon nanotubes or\ngraphene sheets [12,13], one-dimensional wires [14], and molecular junctions [15,16]. For these systems, a transport\ncurrent can excite mechanical modes, and vice versa, the mechanical motion a\u000bects the transport current.The reduced\nsize and high sensitivity of the resulting devices make them attractive for applications such as sensors of mass or\ncharge, nanoscale motors, or switches [17]. On a more fundamental level, the capability of cooling the system via\nback-action allows one to study quantum phenomena at the mesoscopic level, eventually reaching the quantum limit\nof measurement [18,19].\nAll of these applications require an understanding of the mechanical forces that act on the nanoelectromechanical\nsystem in the presence of a transport current. These are referred to as current-induced forces , and have been observed\nin seminal experiments [20,21]. Recently we have shown that it is possible to fully express the current-induced forces\nin terms of a scattering matrix formalism, for arbitrary (albeit adiabatic) out of equilibrium situations [22], thus\nproviding the tools for a systematic approach to study the interplay between electronic and mechanical degrees of\nfreedom in NEMS.\nIn the context of NEMS, two well de\fned limits can be identi\fed for which electronic and mechanical time scales\ndecouple, and which give rise to di\u000berent experimental phenomena. On one side, when the electronic time scales are\nslow compared with the mechanical vibrations, drastic consequences can be observed for the electronic transport, such\nas side bands due to phonon assisted tunneling [23,24] or the Frank-Condon blockade e\u000bect, a phononic analog of\nthe Coulomb blockade in quantum dots [25{27]. In the opposite regime, electrons tunnel through the nanostructure\nrapidly, observing a quasistatic con\fguration of the vibrational modes, but a\u000becting their dynamics profoundly at the\nsame time [18{21]. It is on this regime that our present work focuses. We treat the vibrational degrees of freedom as\nclassical entities embedded in an electronic environment: pictorially, many electrons pass through the nanostructure\nduring one vibrational period, impinging randomly on the modes. In this limit, it is natural to assume that the\ndynamics of the vibrational modes, represented by collective coordinates X\u0017, will be governed by a set of coupledarXiv:1109.6043v2  [cond-mat.mes-hall]  10 Apr 20122\nLangevin equations\nM\u0017X\u0017+@U\n@X\u0017=F\u0017\u0000X\n\u00170\r\u0017\u00170_X\u00170+\u0018\u0017: (1)\nHere we have grouped the purely elastic contribution on the left hand side (LHS) of Eq. (1), M\u0017being the e\u000bective\nmass of mode \u0017andU(X) an elastic potential. On the right hand side (RHS) we collected the current-induced forces:\nthe mean force F\u0017, a term proportional to the velocity of the modes \u0000P\n\u00170\r\u0017\u00170_X\u00170, and the Langevin \ructuating\nforces\u0018\u0017. The main result of our work are expressions for the current-induced forces in terms of the scattering matrix\nand its parametric derivatives. These are given by Eq. (39) for the mean force F\u0017(X), Eq. (42) for the correlator\nD\u0017\u00170(X) of the stochastic force \u0018\u0017, and Eqs. (47) and (50) for the two kinds of forces (dissipative friction force and\ne\u000bective \\Lorentz\" force, as we discuss below) encoded by the matrix \r\u0017\u00170(X).\nTheoretically, these forces have been studied previously within di\u000berent formalisms. The case of one electronic level\ncoupled to one vibrational mode has been studied with a Green's function approach in Refs. [28,29], where the authors\nshowed that the current-induced forces can lead to a bistable e\u000bective potential and consequently to switching. In\nRef. [30], the authors studied the case of multiple vibrational modes within a linear approximation, \fnding a Lorentz-\nlike current-induced force arising from the electronic Berry phase [31]. In simple situations, the current-induced\nforces have been also studied within a scattering matrix approach in the context of quantum measurement backaction\n[32] (see also [33]), momentum transfer statistics [34], and of magnetic systems to describe Gilbert damping [35].\nCurrent induced forces have been shown to be of relevance near mechanical instabilities [36{38] and to drive NEMS\ninto instabilities and strong non-linear behavior [39{41]. Our formalism allows us to retain the nonlinearities of the\nproblem, which is essential for even a qualitative description of the dynamics, while turning the problem of calculating\nthe current-induced forces into a scattering problem for which standard techniques can be applied.\nIn what follows we develop these ideas in detail, giving a thorough derivation of the expressions in terms of the\nscattering matrix for the current-induced forces found in Ref. [22], and include several applications to speci\fc systems.\nMoreover, we extend the theoretical results of Ref. [22] in two ways. We treat a general coupling between the collective\nmodesX\u0017and the electrons, generalizing the linear coupling expressions obtained previously. We also allow for an\narbitrary energy dependence in the hybridization between the leads and the quantum dot, allowing more \rexibility\nfor modeling real systems. In Section II we introduce the theoretical model, and derive the equations of motion of the\nmechanical degrees of freedom starting from a microscopic Hamiltonian. We show how the Langevin equation, Eq.\n(1), emerges naturally from a microscopic model when employing the non-equilibrium Born Oppenheimer (NEBO)\napproximation, appropriate for the limit of slow vibrational dynamics, and derive the current induced forces in terms\nof the microscopic parameters. In Section III we show that the current induced forces can be written in terms of\nparametric derivatives of the scattering matrix (S-matrix) of the system, and state general properties that can be\nderived from S-matrix symmetry considerations. In Section IV we complete the discussion of nanoelectromechanical\nsystems in terms of scattering matrices by providing a corresponding expression for the charge current. In Section V\nwe apply our formalism to simple models of increasing complexity, namely a single resonant level, a two-level model,\nand a two-level/two-mode model. We conclude in Section VI. For better readability, we have relegated part of some\nlengthy calculations to the Supplementary Material, together with a list of useful relations that are used throughout\nthe main text.\nII. MICROSCOPIC DERIVATION OF THE LANGEVIN EQUATION\nA. Model\nWe model the system as a mesoscopic quantum dot connected to multiple leads and coupled to vibrational degrees\nof freedom. Throughout this work we consider non-interacting electrons and we set ~= 1. The Hamiltonian for the\nfull system reads\nH=HD+HX+HL+HT; (2)\nwhere the di\u000berent terms are introduced in the following.\nWe describe the quantum dot by Melectronic levels coupled to Nslow collective degrees of freedom ^X=\n(^X1;:::; ^XN). This is contained in the dot's Hamiltonian\nHD=X\nmm0dy\nmh\nh0(^X)i\nmm0dm0 (3)3\nwhich describes the electronic levels of the dot and their dependence on the collective modes' coordinates ^X\u0017(\u0017=\n1;:::;N ) by the hermitian M\u0002Mmatrixh0(^X). The operator dy(d) creates (annihilates) an electron in the dot and\nthe indices m,m0(= 1;:::;M ) label the electronic levels. Note that here we generalize our previous results obtained\nfor a linear coupling in ^X[22], and allow h0to be a general function of ^X. Our analysis is valid for any coupling\nstrength. The free evolution of the `mechanical' degrees of freedom of the dot is described by the Hamiltonian\nHX=X\n\u0017\"^P2\n\u0017\n2M\u0017+U(^X)#\n: (4)\nThe leads act as electronic reservoirs kept at \fxed chemical potentials \u0016\u000band are described by\nHL=X\n\u0011(\u000f\u0011\u0000\u0016\u000b)cy\n\u0011c\u0011; (5)\nwhere we represent the electrons in the leads by the creation (annihilation) operators cy(c). The leads' electrons\nobey the Fermi-Dirac distribution f\u000b(\u000f) =\u0002\n1 +e(\u000f\u0000\u0016\u000b)=kT\u0003\u00001. The leads are labeled by \u000b= 1;:::;L , each containing\nchannelsn= 1;:::;N\u000b. We combine \u0011= (\u000b;n) into a general `lead' index, \u0011= 1;:::;N 0withN0=P\n\u000bN\u000b.\nFinally, the Hamiltonian HTrepresents the tunneling between the leads and the levels in the dot,\nHT=X\n\u0011;m(cy\n\u0011W\u0011mdm+ h:c:): (6)\nB. Non-equilibrium Born-Oppenheimer approximation\nWe use as a starting point the Heisenberg equations of motion for the mechanical modes which can be cast as\nM\u0017^X\u0017+@U\n@^X\u0017=\u0000X\nn;n0dy\nnh\n\u0003\u0017(^X)i\nnn0dn0; (7)\nwhere we have introduced the ^X-dependent matrices\n\u0003\u0017\u0010\n^X\u0011\n=@h0\n@^X\u0017: (8)\nThe RHS of (7) contains the current-induced forces, expressed through the electronic operators dof the quantum\ndot. We now proceed to calculate these forces within a non-equilibrium Born-Oppenheimer (NEBO) approximation,\nin which the dynamics of the collective modes is assumed slow. In this limit, we can treat the mechanical degrees of\nfreedom as classical, acting as a slow classical \feld on the fast electronic dynamics.\nThe NEBO approximation consists of averaging the RHS of Eq. (7) over times long compared to the electronic\ntime scale, but short in terms of the oscillator dynamics. In this approximation, the force operator is represented by\nits (average) expectation value hdy\u0003diX(t), evaluated for a given trajectory X(t) of the mechanical degrees of freedom,\nplus \ructuations containing both Johnson-Nyquist and shot noise. These \ructuations give rise to a Langevin force\n\u0018\u0017. Hence Eq. (7) becomes\nM\u0017X\u0017+@U\n@X\u0017= tr[i\u0003\u0017G<(t;t)] +\u0018\u0017; (9)\nwhere the trace \\tr\" is taken over the dot levels, and we have introduced the lesser Green's function\nG<\nnn0(t;t0) =ihdy\nn0(t0)dn(t)iX(t): (10)\nThe variance of the stochastic force \u0018\u0017is governed by the symmetrized \ructuations of the operator dy\u0003d. Given that\nthe electronic \ructuations happen on short time scales, \u0018\u0017is locally correlated in time,\nh\u0018\u0017(t)\u0018\u00170(t0)i=D\u0017\u00170(X)\u000e(t\u0000t0): (11)\n(An alternative but equivalent derivation, is based on a saddle point approximation for the Keldysh action, see e.g.\nRef. [42]). Since we are dealing with non-interacting electrons, D(X) can be expressed in terms of single particle\nGreen's functions using Wick's theorem. This readily yields\nh\u0018\u0017(t)\u0018\u00170(t0)i= trf\u0003\u0017G>(t;t0)\u0003\u00170G<(t0;t)gs; (12)4\nwhere\nG>\nmm0(t;t0) =\u0000ihdm(t)dy\nm0(t0)iX(t) (13)\nis the greater Green's function. These expressions for the current-induced forces show that we need to evaluate the\nelectronic Green's function for a given classical trajectory X(t). In doing so, we can exploit that the mechanical\ndegrees of freedom are assumed to be slow compared to the electrons. Thus, we can approximate the Green's function\nby its solution to \frst order in the velocities _X(t). We now proceed with this derivation, starting with the Dyson\nequation for the retarded Green's function\nGR\nmm0(t;t0) =\u0000i\u0012(t\u0000t0)hfdm(t);dy\nm0(t0)giX(t): (14)\nHeref:;:gindicates the anti-commutator. We note that since we consider non-interacting electrons, we can restore the\nlesser and greater Green's functions (or the advanced Green's function GA) at the end of the calculation by standard\nmanipulations.\nThe hybridization with the leads is taken into account through the self-energy [43]\n\u0006R(\u000f) =\u0000iX\n\u000b\u0000\u000b(\u000f); (15)\nwhich is given in terms of the width functions\n\u0000\u000b(\u000f) =\u0019Wy(\u000f)\u0005\u000bW(\u000f): (16)\nHere we have de\fned \u0005 \u000bas a projection operator onto lead \u000band absorbed square root factors of the density of\nstates in the leads into the coupling matrix Wfor notational simplicity. Note that we allow Wto depend on energy.\n(Compare with the wide-band limit discussed in Ref. [22], which employs an energy-independent hybridization \u0000.)\nDyson's equation for the retarded Green's function can then be written, in matrix form, as\n\u0000i@t0GR(t;t0) =\u000e(t\u0000t0) +Z\ndt1GR(t;t1)\u0006R(t1;t0) +GR(t;t0)h0(X): (17)\nTo perform the adiabatic expansion, it is convenient to work in the Wigner representation, in which fast and slow\ntime scales are easily identi\fable. The Wigner transform of a function A(t1;t2) depending on two time arguments is\ngiven by\n~A(t;\u000f) =Z\nd\u001cei\u000f\u001cA(t+\u001c=2;t\u0000\u001c=2): (18)\nUsing this prescription for the Green's function GR, the slow mechanical motion implies that GRvaries slowly with\nthe central time t=t1+t2\n2and oscillates fast with the relative time \u001c=t1\u0000t2. The Wigner transform of a convolution\nC(t1;t2) =R\ndt3A(t1;t3)B(t3;t2) is given by\n~C= exp\u0014i\n2\u0010\n@~A\n\u000f@~B\nt\u0000@~A\nt@~B\n\u000f\u0011\u0015\n~A~B\n'~A~B+i\n2@\u000f~A@t~B\u0000i\n2@t~A@\u000f~B; (19)\nwhere we have dropped higher order derivatives in the last line, exploiting the slow variation with t. Therefore, using\nEq. (19) we can rewrite the Dyson equation Eq. (17) as\n1\u0019GR\u0000\n\u000f\u0000\u0006R\u0000h0\u0001\n\u0000i\n2@\u000fGR@th0\u0000i\n2@tGR\u0000\n1\u0000@\u000f\u0006R\u0001\n; (20)\nwhere the Green's functions are now in the Wigner representation. Unless otherwise denoted by explicitly stating the\nvariables, here and in the following all functions are in the Wigner representation. Finally, with the help of Eqs. (A5)\n-(A6) from Supp. Mat. A, we obtain\nGR'GR+i\n2X\n\u0017_X\u0017\u0000\n@\u000fGR\u0003\u0017GR\u0000GR\u0003\u0017@\u000fGR\u0001\n; (21)5\nin terms of the strictly adiabatic Green's function\nGR(\u000f;X) =\u0002\n\u000f\u0000h0(X)\u0000\u0006R(\u000f)\u0003\u00001: (22)\nOur notation is such that Gdenotes fullGreen's functions, while Gdenotes the strictly adiabatic (or frozen ) Green's\nfunctions that are evaluated for a \fxed value of X(so that all derivatives with respect to central time in Eq. (20)\ncan be dropped). From now on, G(R;A;<;> )denote the Green functions in the Wigner representation, with arguments\n(\u000f;t), andGA= (GR)y.\nUsing Langreth's rule (see e.g.Ref. [43])\nG<(t;t0) =Z\ndt1Z\ndt2GR(t;t1)\u0006<(t1;t2)GA(t2;t0); (23)\nwe can relateG<withGR. In Eq. (23) we have introduced the lesser self energy \u0006<, which in the Wigner representation\ntakes the form\n\u0006<(\u000f) = 2iX\n\u000bf\u000b(\u000f)\u0000\u000b(\u000f): (24)\nNote that \u0006<depends only on \u000fand is independent of the central time. Expanding Eq. (23) up to the leading\nadiabatic correction according to Eq. (19), we obtain G<to \frst order in _X,\nG<=G<+i\n2X\n\u0017_X\u0017\u0002\n(@\u000fG<)\u0003\u0017GA\u0000GR\u0003\u0017@\u000fG<+ (@\u000fGR)\u0003\u0017G<\u0000G<\u0003\u0017@\u000fGA\u0003\n; (25)\nwithG<=GR\u0006<GA.\nC. Current-induced forces in terms of Green's functions\nWe can now collect the results from the previous section and identify the current-induced forces appearing in the\nLangevin equation (1). Except for the stochastic noise force, the current induced forces are encoded in tr( G<\u0003\u0017). In\nthe strictly adiabatic limit, i.e., retaining only the \frst term on the RHS of Eq. (25), G<'G<, we obtain the mean\nforce\nF\u0017(X) =\u0000Zd\u000f\n2\u0019itr\u0002\n\u0003\u0017G<\u0003\n: (26)\nThe leading order correction in Eq. (25) gives a velocity-dependent contribution to the current induced forces, which\ndetermines the tensor \r\u0017\u00170. After integration by parts, we \fnd\n\r\u0017\u00170=Zd\u000f\n2\u0019tr\u0000\nG<\u0003\u0017@\u000fGR\u0003\u00170\u0000G<\u0003\u00170@\u000fGA\u0003\u0017\u0001\n:\nThis tensor can be split into symmetric and anti-symmetric contributions, \r=\rs+\ra, which de\fne a dissipative\nterm\rsand an orbital, e\u000bective magnetic \feld \rain the space of the collective modes. The latter interpretation is\nbased on the fact that the corresponding force takes a Lorentz-like form. Using Eq. (A1) in the Supp. Mat. A and\nnoting that 2R\nd\u000fG<@\u000fG<=R\nd\u000f@\u000f(G<)2= 0, we obtain the explicit expressions\n\rs\n\u0017\u00170(X) =Zd\u000f\n2\u0019tr\b\n\u0003\u0017G<\u0003\u00170@\u000fG>\t\ns; (27)\n\ra\n\u0017\u00170(X) =\u0000Zd\u000f\n2\u0019tr\b\n\u0003\u0017G<\u0003\u00170@\u000f\u0000\nGA+GR\u0001\t\na: (28)\nHere we have introduced the notation\nfA\u0017\u00170gs;a=1\n2(A\u0017\u00170\u0006A\u00170\u0017)\nfor symmetric and anti-symmetric parts of an arbitrary matrix A.6\nAt last, the stochastic force \u0018\u0017is given by the thermal and non-equilibrium \ructuations of the force operator\n\u0000dy\u0003\u0017din Eq. (7). As indicated by the \ructuation-dissipation theorem, the \ructuating force is of the same order\nin the adiabatic expansion as the velocity dependent force. Thus, we can evaluate the expression for the correlator\nD\u0017\u00170(X) of the \ructuating force given in Eq. (12) to lowest order in the adiabatic expansion, so that\nD\u0017\u00170(X) =Zd\u000f\n2\u0019tr\b\n\u0003\u0017G<\u0003\u00170G>\t\ns: (29)\nThis formalism gives the tools needed to describe the dynamics of the vibrational modes in the presence of a bias for\nan arbitrary number of modes and dot levels. When expressions (26) - (28) are inserted back in Eq. (1), they de\fne\na non-linear Langevin equation due to their non-trivial dependences on X(t) [28,29].\nIII. S-MATRIX THEORY OF CURRENT-INDUCED FORCES\nA. Adiabatic expansion of the S-matrix\nScattering matrix approaches to mesoscopic transport generally involve expressions in terms of the elastic S-matrix.\nFor our problem, the S-matrix is elastic only in the strictly adiabatic limit, in which it is evaluated for a \fxed value\nofX,\nS(\u000f;X) = 1\u00002\u0019iW(\u000f)GR(\u000f;X)Wy(\u000f): (30)\nAs pointed out by Moskalets and B uttiker [8,44], this is not su\u000ecient for general out of equilibrium situations, even\nwhen X(t) varies in time adiabatically. In their work, they calculated, within a Floquet formalism, the leading\ncorrection to the strictly adiabatic S-matrix. We follow here the same approach, rephrased in terms of the Wigner\nrepresentation. The full S-matrix can be written as [45] (note that, in line with the notation established before for\nthe Green's functions, the strictly adiabatic S-matrix is denoted by Swhile the full S-matrix is denoted by S)\nS(\u000f;t) = 1\u00002\u0019i\u0002\nWGRWy\u0003\n(\u000f;t): (31)\nTo go beyond the frozen approximation, we expand Sto leading order in _X,\nS(\u000f;t)'S(\u000f;X(t)) +X\n\u0017_X\u0017(t)A\u0017(\u000f;X(t)): (32)\nThus, the leading correction de\fnes the matrix A, which, similar to S, has de\fnite symmetry properties. In particular,\nif the system is time-reversal invariant, the adiabatic S-matrix is even under time reversal while Ais odd. For a given\nproblem, the A-matrix has to be obtained along with S.\nWe can now derive a Green's function expression for the matrix A[46,47]. Comparing Eq. (32) with the expansion\nto the same order of Sin terms of adiabatic Green's functions (obtained straightforwardly by performing explicitly\nthe convolution in Eq. (31) and keeping terms up to _X) we obtain\nA\u0017(\u000f;X) =\u0019@\u000f\u0002\nW(\u000f)GR(\u000f;X)\u0003\n\u0003\u0017(X)GR(\u000f;X)Wy(\u000f)\n\u0000\u0019W(\u000f)GR(\u000f;X)\u0003\u0017(X)@\u000f\u0002\nGR(\u000f;X)Wy(\u000f)\u0003\n:(33)\nCurrent conservation constrains both the frozen and full scattering matrices to be unitary. From the unitarity of the\nfrozen S-matrix, SyS=1, we obtain the useful relation\n@Sy\n@X\u0017S+Sy@S\n@X\u0017= 0: (34)\nWe will make use of Eq. (34) repeatedly in the following sections. On the other hand, unitarity of the full S-matrix,\nSyS=1, imposes a relation between the A-matrix and the frozen S-matrix. To \frst order in the velocity _Xwe have\n1=SSy+SAy+ASy+i\n2\u0012@S\n@\u000f@Sy\n@t\u0000@S\n@t@Sy\n@\u000f\u0013\n(35)\nwhereA(\u000f;X) =P\n\u0017A\u0017(\u000f;X)_X\u0017. Therefore, SandAare related through\nA\u0017Sy+SAy\n\u0017=i\n2\u0012@S\n@X\u0017@Sy\n@\u000f\u0000@S\n@\u000f@Sy\n@X\u0017\u0013\n: (36)\nIn the next section we will see that the A-matrix is essential to express the current-induced dissipation and \\Lorentz\"\nforces, Eqs. (27) and (28).7\nB. Current-induced forces\n1. Mean Force\nThe mean force exerted by the electrons on the oscillator is given by Eq. (26). Writing Eq. (26) explicitly and\nusing Eq. (A2) in Supp. Mat. A, we can express G<in terms of GRandGAand obtain\nF\u0017(X) =\u0000Z\nd\u000fX\n\u000bf\u000btr\u0000\n\u0003\u0017GRWy\u0005\u000bWGA\u0001\n=\u0000Z\nd\u000fX\n\u000bf\u000btr\u0000\n\u0005\u000bWGA\u0003\u0017GRWy\u0001\n;(37)\nwhere the second equality exploits the cyclic invariance of the trace. Noting that, by Eq. (A7) in Supp. Mat. A,\nWGA\u0003\u0017GRWy=\u00001\n2\u0019iSy@S\n@X\u0017; (38)\nEq. (37) can be expressed directly in terms of scattering matrices S(\u000f;X) as\nF\u0017(X) =X\n\u000bZd\u000f\n2\u0019if\u000bTr\u0012\n\u0005\u000bSy@S\n@X\u0017\u0013\n: (39)\nNote that now the trace (denoted by \\Tr\") is over lead-space.\nAn important issue is whether this force is conservative ,i.e., derivable from a potential. A necessary condition for\nthis is a vanishing \\curl\" of the force,\n\n\u0017\u00170\u0011@F\u00170\n@X\u0017\u0000@F\u0017\n@X\u00170=X\n\u000bZd\u000f\n\u0019if\u000bTr\u0012\n\u0005\u000b@Sy\n@X\u0017@S\n@X\u00170\u0013\na: (40)\nFrom Eq. (40) it is seen that the mean force is conservative in thermal equilibrium, where Eq. (40) can be turned\ninto a trace over a commutator of \fnite-dimensional matrices: Indeed, in equilibrium the sum over the lead indices\ncan be directly performed since f\u000b=ffor all\u000b, andP\n\u000b\u0005\u000b= 1. Using the unitarity of the S-matrix and the cyclic\nproperty of the trace, we obtain:\n\n\u0017\u00170=Zd\u000f\n2\u0019ifTr\u0012@Sy\n@X\u0017@S\n@X\u00170\u0000@Sy\n@X\u00170SSy@S\n@X\u0017\u0013\n=Zd\u000f\n2\u0019ifTr\u0012@Sy\n@X\u0017@S\n@X\u00170\u0000@S\n@X\u00170@Sy\n@X\u0017\u0013\n= 0;(41)\nwhere in the last line we have used Eq. (34). In general, however, the mean force will be non-conservative in out-of-\nequilibrium situations, providing a way to exert work on the mechanical degrees of freedom by controlling the external\nbias potential [30,48,49].\n2. Stochastic Force\nNext, we discuss the \ructuating force \u0018\u0017with variance D\u0017\u00170given by Eq. (29). Following a similar path as described\nin the previous subsection for the mean force F\u0017, we can also express the variance Eq. (29) of the \ructuating force\nin terms of the adiabatic S-matrix,\nD\u0017\u00170(X) =X\n\u000b\u000b0Zd\u000f\n2\u0019F\u000b\u000b0Tr(\n\u0005\u000b\u0014\nSy@S\n@X\u0017\u0015y\n\u0005\u000b0Sy@S\n@X\u00170)\ns; (42)\nwhere we have introduced the function F\u000b\u000b0(\u000f) =f\u000b(\u000f) [1\u0000f\u000b0(\u000f)]. From Eq. (42) it is straightforward to show that\nD\u0017\u00170is positive de\fnite. By performing a unitary transformation to a basis in which D\u0017\u00170is diagonal, using \u0005 \u000b= \u00052\n\u000b\nand the cyclic invariance of the trace, we obtain the expression\nD\u0017\u0017(X) =X\n\u000b\u000b0Zd\u000f\n2\u0019F\u000b\u000b0Tr(\u0012\n\u0005\u000b0Sy@S\n@X\u0017\u0005\u000b\u0013y\n\u0005\u000b0Sy@S\n@X\u0017\u0005\u000b)\ns: (43)\nwhich is evidently positive.8\n3. Damping Matrix\nSo far, we were able to express quantities in terms of the frozen S-matrix only. This is no longer the case for the\n\frst correction to the strictly adiabatic approximation, given by Eqs. (27) and (28). We start here with the \frst of\nthese terms, the symmetric matrix \rs, which is responsible for dissipation of the mechanical system into the electronic\nbath.\nThe manipulations to write the dissipation term as a function of S-matrix quantities are lengthy and the details are\ngiven in the Supp. Mat. B. The damping matrix can be split into an \\equilibrium\" contribution, \rs;eq, and a purely\nnon-equilibrium contribution \rs;ne, as\rs=\rs;eq+\rs;ne. We \frst treat \rs;eq. By the calculations given in Supp. Mat.\nB, we obtain\n\rs;eq\n\u0017\u00170=1\n4X\n\u000b\u000b0Zd\u000f\n2\u0019@\u000f(f\u000b+f\u000b0)Tr\u001a\n\u0005\u000bSy@S\n@X\u0017\u0005\u000b0Sy@S\n@X\u00170\u001b\ns\n=1\n2X\n\u000bZd\u000f\n2\u0019(\u0000@\u000ff\u000b)Tr\u0012\n\u0005\u000b@Sy\n@X\u0017@S\n@X\u00170\u0013\n;(44)\nwhere we have used thatP\n\u000b0\u0005\u000b0= 1 ,SyS= 1, and Eq. (34) in the last line. Note that in general, \rs;eqalso\ncontains non-equilibrium contributions, but gives the only contribution to the damping matrix when in equilibrium.\nEq. (44) is analogous to the S-matrix expression obtained for dissipation in ferromagnets in thermal equilibrium,\ndubbed Gilbert damping [35].\nTo express \rs;nein terms of S-matrix quantities, we have to make use of the A-matrix de\fned in Eq. (33). Again\nthe details are given in the Supp. Mat. B, where we \fnd after lengthy manipulations that\n\rs;ne\n\u0017\u00170=Zd\u000f\n2\u0019iX\n\u000bf\u000bTr\u001a\n\u0005\u000b\u0012@Sy\n@X\u0017A\u00170\u0000Ay\n\u00170@S\n@X\u0017\u0013\u001b\ns: (45)\nThis quantity vanishes in equilibrium, as can be shown using the properties of the SandAmatrices. Since the sum\nover leads can be directly performed in equilibrium, expression (45) involves\nTr\u001a@Sy\n@X\u0017A\u00170\u0000Ay\n\u00170@S\n@X\u0017\u001b\ns=\u0000Tr\u001a@S\n@X\u0017Sy\u0010\nA\u00170Sy+SAy\n\u00170\u0011\u001b\ns\n=\u0000i\n2Tr\u001a@S\n@X\u0017Sy\u0012@S\n@X\u00170@Sy\n@\u000f\u0000S@Sy\n@\u000f@S\n@X\u00170Sy\u0013\u001b\ns= 0 (46)\nwhere we have used the unitarity of Sand the cyclic invariance of the trace multiple times. In the \frst equality, we\ninsertedSyS= 1 and used Eq. (34), the second equality follows by inserting the identity (36) and using again (34).\nFinally, combining all terms we obtain an S-matrix expression for the full damping matrix \rs,\n\rs\n\u0017\u00170(X) =\u0000X\n\u000bZd\u000f\n4\u0019@\u000ff\u000bTr\u001a\n\u0005\u000b@Sy\n@X\u0017@S\n@X\u00170\u001b\ns\n+X\n\u000bZd\u000f\n2\u0019if\u000bTr\u001a\n\u0005\u000b\u0012@Sy\n@X\u0017A\u00170\u0000Ay\n\u00170@S\n@X\u0017\u0013\u001b\ns:(47)\nNote that in equilibrium, by the relation \u0000@\u000ff=f(1\u0000f)=Tand using Eq. (34), the \ructuating force Dand\ndamping\rsare related via\nD\u0017\u00170= 2T\rs;eq\n\u0017\u00170= 2T\rs\n\u0017\u00170 (48)\nas required by the \ructuation-dissipation theorem.\nFollowing a similar set of steps as shown above for the variance D\u0017\u00170in Eq. (43), \rs;eq\n\u0017\u00170has positive eigenvalues. On\nthe other hand, the sign of \rs;ne\n\u0017\u00170is not \fxed, allowing the possibility of negative eigenvalues of \rs. The possibility\nof negative damping is, therefore, a pure non-equilibrium e\u000bect. Several recent papers found negative damping in\nspeci\fc out of equilibrium models [22,40,50,51].9\n4. Lorentz force\nWe turn now to the remaining term, the antisymmetric contribution \ragiven in Eq. (28), which acts as an e\u000bective\nmagnetic \feld. Using Eq. (A2) in Supp. Mat. A, it can be written as\n\ra\n\u0017\u00170=iZ\nd\u000fX\n\u000bf\u000bTr\b\n\u0005\u000bWGA\u0003\u0017\u0000\n@\u000fGR+@\u000fGA\u0001\n\u0003\u00170GRWy\t\na: (49)\nIn order to relate this to the scattering matrix, we use the Supp. Mat. A Eq. (A10), which allows us to write \rain\nterms of the S-matrix as\n\ra\n\u0017\u00170(X) =X\n\u000bZd\u000f\n2\u0019if\u000bTr\u001a\n\u0005\u000b\u0012\nSy@A\u0017\n@X\u00170\u0000@Ay\n\u0017\n@X\u00170S\u0013\u001b\na: (50)\nIf the system is time-reversal invariant, \ravanishes in thermal equilibrium. The latter impliesP\n\u000b\u0005\u000bf\u000b=f, so\nthat Eq. (50) involves only\nTr\u001a\nSy@A\u0017\n@X\u00170\u0000@Ay\n\u0017\n@X\u00170S\u001b\n= Tr\u001a@AT\n\u0017\n@X\u00170S\u0003\u0000ST@A\u0003\n\u0017\n@X\u00170\u001b\n= Tr\u001a\n\u0000@A\u0017\n@X\u00170Sy+S@Ay\n\u0017\n@X\u00170\u001b\n;\nyielding\ra= 0 due to the cyclic invariance of the trace. In the last equality, we have used S=STandA=\u0000ATas\nimplied by time-reversal invariance.\nOut of equilibrium, \ragenerally does not vanish even for time reversal symmetric conductors, since the current\ne\u000bectively breaks time reversal symmetry.\nIV. CURRENT\nSo far we have focused on the e\u000bect of the electrons on the mechanical degrees of freedom. For a complete picture, we\nalso need to consider the reverse e\u000bect of the mechanical vibrations on the electronic current. In the strictly adiabatic\nlimit, this obviously has to reduce to the Landauer-B uttiker formula for the transport current. Considering the leading\nadiabatic correction to the current in equilibrium is closely related to the phenomenon of quantum pumping, and we\nwill see that our results in this limit essentially reduce to Brouwer's S-matrix formula for the pumping current [5].\nOur full result is, however, more general since it gives the leading adiabatic correction to the current in arbitrary\nnon-equilibrium situations [8].\nThe current through lead \u000bis given by [43]:\nI\u000b=\u0000eh_N\u000bi=ieX\nn;\u00112\u000bW\u0011nhcy\n\u0011(t)dn(t)i+ h:c: (51)\nwithN\u000b=P\n\u00112\u000bcy\n\u0011c\u0011. Using the expressions for the self-energies this can be expressed in terms of the dot's Green's\nfunctions and self-energies,\nI\u000b(t) =eZ\ndt0tr\b\nGR(t;t0)\u0006<\n\u000b(t0;t) +G<(t;t0)\u0006A\n\u000b(t0;t)\t\n+ h:c:: (52)\nAgain we use the separation of time scales and go to the Wigner representation, yielding\nI\u000b=eZd\u000f\n2\u0019tr\b\nGR\u0006<\n\u000b+G<\u0006A\n\u000b\u0000i\n2\u0000\n@tGR@\u000f\u0006<\n\u000b+@tG<@\u000f\u0006A\n\u000b\u0001\t\n+ h:c:: (53)\nWe split the current into an adiabatic contribution I0\n\u000band a term proportional to the velocity _X\u0016:\nI\u000b=I0\n\u000b+I1\n\u000b: (54)\nWe will express these quantities in terms of the scattering matrix.10\nA. Landauer-B uttiker current\nThe strictly adiabatic contribution to the current is given by\nI0\n\u000b(X) =eZd\u000f\n2\u0019tr\b\u0000\nGR\u0000GA\u0001\n\u0006<\n\u000b+G<\u0000\n\u0006A\n\u000b\u0000\u0006R\n\u000b\u0001\t\n; (55)\nwhere we have collected the purely adiabatic terms from Eqs. (21) and (25). Inserting the expressions for the\nself-energies Eqs. (15) and (24), we can express this as\nI0\n\u000b(X) =eZd\u000f\n2\u0019X\n\ff\f2\u0019iTr\b\nW\u0002\n\u000e\u000b\f(GR\u0000GA) + 2\u0019iGRWy\u0005\fWGA\u0003\nWy\u0005\u000b\t\n; (56)\nwhere we used Supp. Mat. A Eq. (A2). Inserting the adiabatic S-matrix, Eq. (30) yields\nI0\n\u000b(X) =eZd\u000f\n2\u0019X\n\ff\fTr\b\u0002\n\u000e\u000b\f\u0000S\u0005\fSy\u0003\n\u0005\u000b\t\n(57)\n=eZd\u000f\n2\u0019X\n\f(f\u000b\u0000f\f) Tr\b\nS\u0005\fSy\u0005\u000b\t\n; (58)\nwhere we usedP\n\fS\u0005\fSy= 1 in the last line. We hence recover the usual expression for the Landauer-B uttiker\ncurrent [4]. Note that the total adiabatic current depends implicitly on time through X(t), and is conserved at every\ninstant of time,P\n\u000bI0\n\u000b(X) = 0. To obtain the dccurrent, we need to average this expression over the Langevin\ndynamics of the mechanical degrees of freedom. Alternatively, we can average the current expression with the prob-\nability distribution of X, which can be obtained from the corresponding Fokker-Planck equation. Similar remarks\nwould apply to calculations of the current noise.\nB. First order correction\nNow we turn to the \frst order correction to the adiabatic approximation [8], restricting our considerations to the\nwide-band limit. The contribution to the current (53) which is linear in the velocity reads\nI1\n\u000b(X) =eZd\u000f\n2\u0019iX\n\u0016_X\u0016tr\b\n(@\u000fGR)\u0003\u0016GR\u0006<\n\u000b+\u0002\u0000\n@\u000fG<\u0001\n\u0003\u0016GA\u0000GR\u0003\u0016(@\u000fG<)\u0003\n\u0006A\n\u000b\t\n+ h:c:; (59)\nafter integration by parts. Again, we insert Eq. (A2) from Supp. Mat. A for the lesser Green's function, and expres-\nsions (15) and (24) for the self-energies. In the wide band limit, the identity ( i=2)@\u000f@X\u0017S+A\u0017=W(@\u000fGR)\u0003\u0017GRWy\nholds, so that we can write\nI1\n\u000b(X) =\u0000eZd\u000f\n2\u0019_X\u0001X\n\ff\fTr\u0014\u0012i\n2@2S\n@\u000f@X+A\u0013\n\u0005\fSy\u0005\u000b\u0015\n+ h:c: (60)\nafter straightforward calculation. After integration by parts, we can split this expression as\nI1\n\u000b(X) =\u0000e\n2\u0019Z\nd\u000f_X\u0001X\n\f@\u000ff\fImTr\u001a\n\u0005\u000b@S\n@X\u0005\fSy\u001b\n+e\n2\u0019Z\nd\u000f_X\u0001X\n\ff\fReTr\u001a\ni\u0005\u000b@S\n@X\u0005\f@Sy\n@\u000f\u00002\u0005\u000bA\u0005\fSy\u001b\n:(61)\nIn equilibrium, the second term vanishes due to the identity Eq. (36) and the \frst term agrees with Brouwer's formula\nfor the pumping current [5]. As for the strictly adiabatic contribution, the dccurrent is obtained by averaging over\nthe probability distribution of X.11\nV. APPLICATIONS\nA. Resonant Level\nTo connect with the existing literature, as a \frst example we treat the simplest case within our formalism: a resonant\nlevel coupled to a single vibrational mode and attached to two leads on the left ( L) and right ( R). This model has\nbeen discussed in detail for zero temperature in references [28,29], and it provides a simple description on how current-\ninduced forces can be used to manipulate a molecular switch. Here we derive \fnite-temperature expressions for the\ncurrent-induced forces for a generic coupling between electronic and mechanical degrees of freedom, starting from\nthe scattering matrix of the system, and show how they reduce to the known results for zero temperature and linear\ncoupling.\nWe consider N=M= 1, denoting the mode coordinate by X, the energy of the dot level by ~ \u000f(X), and the number\nof channels in the left and right leads by NLandNR, respectively. The Hamiltonian of the dot can then be written\nas\nHD= ~\u000f(X)dyd (62)\nand the hybridization matrix as Wy=\u0000\nwL;wR\u0001y, with w\u000b= (w\u000b\n1;:::w\u000b\nN\u000b) and\u000b=L; R. Hence the frozen S-matrix,\nEq. (30), is given by\nS=1\u00002\u0019i\nL \nwL\u0000\nwL\u0001ywL\u0000\nwR\u0001y\nwR\u0000\nwL\u0001ywR\u0000\nwR\u0001y!\n; (63)\nwhereL(\u000f;X) =\u000f\u0000~\u000f(X)+i\u0000, \u0000 = \u0000L+\u0000R, and \u0000\u000b=\u0019(w\u000b)y\u0001w\u000b. Rotating to an eigenbasis of the lead channels, this\nS-matrix does not mix channels within the same lead, and hence we can project the S-matrix into a single non-trivial\nchannel in each lead, to obtain\nS=1\u00002i\nL\u0012\n\u0000Lp\u0000L\u0000Rp\u0000L\u0000R \u0000R\u0013\n: (64)\nTo calculate the mean force from Eq. (39), we need an explicit expression for Eq. (A7) in Supp. Mat. A. This can\nbe easily calculated to be\nSy@S\n@X=\u0000@~\u000f\n@X2i\njLj2\u0012\n\u0000Lp\u0000L\u0000Rp\u0000L\u0000R \u0000R\u0013\n(65)\nand hence\nF(X) =\u0000Zd\u000f\n\u0019\"\nfL\u0000L+fR\u0000R\njLj2#\n@~\u000f\n@X: (66)\nAnalogously, the variance of the stochastic force, Eq. (42), becomes\nD(X) = 2Zd\u000f\n\u0019X\n\u000b\u000b0\u0000\u000b\u0000\u000b0F\u000b\u000b0\njLj4\u0014@~\u000f\n@X\u00152\n: (67)\nIt only remains to calculate the dissipation coe\u000ecient \r. Since there is only one collective mode, \u0017= 1,\ris a scalar\nand hence\ra= 0. Moreover, for energy-independent hybridization we have that @\u000fGR=\u0000G2\nR, and the A-matrix\n(33) can be written as [22]\nA\u0017=\u0000\u0019WGR[GR;\u0003\u0017]GRWy: (68)\nBeing the commutator of scalars, in this case A1= 0 and from Eq. (47), \rsmust be positive and is given by Eq.\n(44). (For an alternative derivation of the positiveness of the friction coe\u000ecient in a resonant-level system, see Ref.\n[52]). After some algebra, we obtain\n\u0012@S\n@X\u0013y@S\n@X= 4\u0014@~\u000f\n@X\u00152\u0000\njLj2\u0012\n\u0000Lp\u0000L\u0000Rp\u0000L\u0000R \u0000R\u0013\n: (69)12\nand hence the damping coe\u000ecient becomes\n\r(X) =\u0000Zd\u000f\n\u0019\u0000\u0000L@\u000ffL+ \u0000R@\u000ffR\njLj4\u0014@~\u000f\n@X\u00152\n: (70)\nWe can evaluate the remaining integrals analytically in the zero-temperature limit [28,29]. In the following we\nassume\u0016L\u0015\u0016R. The average force is given by\nF(X) =\u00001\n\u0019@~\u000f\n@XX\n\u000b\u0000\u000b\n\u0000\u0014\narctan\u0012\u0016\u000b\u0000~\u000f\n\u0000\u0013\n+\u0019\n2\u0015\n: (71)\nSimilarly we obtain the dissipation coe\u000ecient\n\rs(X) =\u0000\n\u0019\u0014@~\u000f\n@X\u00152X\n\u000b\u0000\u000bh\n(\u0016\u000b\u0000~\u000f)2+ \u00002i2; (72)\ntogether with the \ructuation kernel\nD(X) =\u0000L\u0000R\n\u0019\u00003\u0014@~\u000f\n@X\u00152\"\narctan\u0012\u0016\u0000~\u000f\n\u0000\u0013\n+\u0000(\u0016\u0000~\u000f)\n(\u0016\u0000~\u000f)2+ \u00002#\f\f\f\f\f\u0016=\u0016L\n\u0016=\u0016R(73)\nThe position of the dot electronic level can be adjusted by an external gate voltage\neVgate=\u0016L+\u0016R\n2\u0000\u000f0; (74)\nwhere the factor ( \u0016L+\u0016R)=2 is included for convenience, to measure energies from the center of the conduction\nwindow. The di\u000berence in chemical potential between the leads is adjusted viaa bias voltage\neVbias=\u0016L\u0000\u0016R: (75)\nFor a single vibrational mode, the average current-induced force is necessarily conservative and we can de\fne a\ncorresponding potential. Restricting now our results to linear coupling, we write the local level as ~ \u000f(X) =\u000f0+\u0015X.\nIn Fig. 1, we show the e\u000bective potential ~U(X) =M\n2!2\n0X2\u0000R\ndXF (X) which describes both the elastic and the\ncurrent-induced forces at zero temperature and various bias voltages. Already this simple example shows that the\ncurrent-induced forces can a\u000bect the mechanical motion qualitatively [29]. Indeed, the e\u000bective potential ~U(X) can\nbecome multistable even for a purely harmonic elastic force and depends sensitively on the applied bias voltage.\nAlternative expressions of the current-induced forces for the resonant level model, in terms of phase shifts and\ntransmission coe\u000ecients, are given in the Supplementary Material C.\nB. Two-level model\nFor the resonant level model discussed so far, the A-matrix vanishes and the damping is necessarily positive. We\nnow consider a model which allows for negative damping [53]. Our toy model could be inspired by a double dot on\na suspended carbon nanotube, or an H 2molecule in a break junction. The model is depicted schematically in Fig.\n2. The bare dot Hamiltonian corresponds to degenerate electronic states \u000f0, localized on the left and right atoms or\nquantum dots, with tunnel coupling tin between,\nH0=\u0012\n\u000f0t\nt \u000f0\u0013\n: (76)\nWe consider a single oscillator mode with coordinate Xthat couples linearly to the di\u000berence in the occupation of the\nlevels. In our previous notation, this means \u0003 1=\u00151\u001b3, where we denote by \u001b\u0016, with\u0016= 0;:::; 3, the Pauli matrices\nacting in the two-site basis. The shift of the electronic levels is given by ~ \u000f\u0006(X) =\u000f0\u0006\u00151X. The hybridization\nmatrices are given by \u0000\u000b=1\n2\u0000\u000b(\u001b0\u0006\u001b3), where the +(\u0000) refers to\u000b=L(R). We can deduce the tunneling matrix\nWin terms of the hybridization matrices,\nW= 1=2p\n\u0000L=\u0019(\u001b0+\u001b3) + 1=2p\n\u0000R=\u0019(\u001b0\u0000\u001b3): (77)13\n−0.3−0.2−0.10˜U(x)\n−1 −0.5 0 0.5\nxeVbias=0\neVbias=0.4\neVbias=0.8\nFIG. 1: Resonant level. The shape of the e\u000bective potential ~U(X) can be tuned by the bias voltage. We consider the parameters\neVgate= 0, ~!0= 0:01 and \u0000 = 0 :1. The dimensionless coordinate is x= (M!2\n0=\u0015)Xand energies are measured in units of\n\u00152=(M!2\n0).\nFIG. 2: Sketch of the two-level model. Electrons tunnel through two degenerate energy levels between left and right leads. The\nsystem is modulated by the coupling to the vibrational modes.\nIn the wide-band limit, we approximate Wand \u0000\u000bto be independent of energy. The retarded adiabatic GF takes the\nform\nGR(\u000f;X) =1\n\u0001\u0012\n\u000f\u0000~\u000f++i\u0000Rt\nt \u000f\u0000~\u000f\u0000+i\u0000L\u0013\n; (78)\nwith \u0001(X) = (\u000f\u0000~\u000f\u0000+i\u0000L)(\u000f\u0000~\u000f++i\u0000R)\u0000t2.\nFor simplicity, we restrict our attention to symmetric couplings to the leads, \u0000 L= \u0000R= \u0000=2. Hence the frozen\nS-matrixS(\u000f;X) becomes\nS(\u000f;X) = 1\u0000i\u0000\n\u0001\u0012\n\u000f\u0000~\u000f++i\u0000=2t\u0000\nt\u0000\u000f\u0000~\u000f\u0000+i\u0000=2\u0013\n; (79)\nwhile the A-matrix takes the form\nA(\u000f;X) =i\u00151\u0000t(\u000f\u0000\u000f0+i\u0000=2)2+i\u0002\n(\u00151X)2\u0000t2\u0003\n\u00013\u001b2: (80)\nWe can now give explicit expressions for the current-induced forces. The explicit expressions are lengthy and are\ngiven in Supp. Mat. D, Eqs. (D1) and (D2) for the mean force and damping matrix, respectively. The variance of\nthe \ructuating force can be calculated accordingly.\nThe average force given in Eq. (D1) of Supp. Mat. D combines with the elastic force to give rise to the e\u000bective\npotential ~U(X) depicted, for zero temperature, in Fig. 3. As in the case studied in the previous section, the system\ncan exhibit various levels of multistability when changing the bias.\nThe results for the friction coe\u000ecient, given in Supp. Mat. D Eq. (D2), are shown in Fig. 4 as a function of\nthe dimensionless oscillator coordinate x, for zero temperature. The contribution \rs;eqto the friction coe\u000ecient is14\n(a)\n−0.100.10.2˜U(x)\n−1−0.5 0 0.5 1\nxeVbias=0.2\neVbias=0.4\neVbias=0.8\n(b)\n00.050.10.15˜U(x)\n−1−0.5 0 0.5 1\nxeVbias=0.2\neVbias=0.4\neVbias=0.8\n(c)\n00.050.10.15˜U(x)\n−1−0.5 0 0.5 1\nxeVbias=0.2\neVbias=0.4\neVbias=0.8\nFIG. 3: E\u000bective potential for the mechanical motion in the two-level model. The shape of the potential can be tuned by\nchanging the bias and gate voltages: (a) eVgate= 0, (b)eVgate= 0:2 and (c)eVgate= 0:4. We consider the parameters\n~!0= 0:01,t= 0:1 and \u0000 = 0 :1. The dimensionless coordinate is x= (M!2\n0=\u00151)Xand energies are measured in units of\n\u00152\n1=(M!2\n0).\npeaked ateVgate\u0006eVbias=2 =\u0006p\n(\u00151X)2+t2, as depicted in Figs. 4 (a) and (c). Neglecting the coupling to the\nleads, our toy model can be considered as a two-level system with level-spacing 2p\n(\u00151X)2+t2. Thus, the peaks\noccur when one of the dot's electronic levels enters the conduction window. When this happens, small changes in\nthe oscillator coordinate Xcan have a large impact on the occupation of the levels. This e\u000bect is more pronounced\nwhen the dots' levels pass the Fermi levels that they are directly attached to [corresponding to X > 0 for current\n\rowing from left to right, see Fig. 4 (a) and Fig. 5 (a), (b)]. The broadening of the peaks is due to the hybridization\nwith the leads, \u0000 =2. WheneVgate= 0, two peaks are expected symmetrically about X= 0, as shown in Fig. 4\n(a) [see also Figs. 5 (a) and (b)]. The e\u000bect of a \fnite gate voltage eVgateis two-fold: it shifts the non-interacting\nelectronic levels of the dot away from the middle of the conduction window, and hence the shifted levels ~ \u000f\u0006pass the\nFermi levels of right and left leads at di\u000berent values of X, Figs. 5 (c) and (d). Therefore in this case four peaks are\nexpected, with two larger peaks located at X > 0, and two smaller peaks located at X < 0. This is shown in Fig. 4\n(c). The height of the peaks in this case is reduced with respect to the case eVgate= 0, since for a given peak, only\none of the dot's levels is in resonance with one of the leads. Note that four real values of Xcan be obtained only if\n(eVgate\u0006eVbias=2)2>t2. A situation with ( eVgate\u0000eVbias=2)2<t2while (eVgate+eVbias=2)2>t2is shown in 4 (c)\n(red-dotted line), where a big peak is observed for X= 1=\u00151q\n(eVgate+eVbias=2)2\u0000t2, a corresponding small peak\nforX=\u00001=\u00151q\n(eVgate+eVbias=2)2\u0000t2[not displayed in Fig. 4 (c)], plus a peak at X= 0.\nFor this model, the A-matrix is generally non-vanishing, which can result in negative damping for out-of-equilibrium\nsituations. This is due to a negative contribution of \rs;neto the total damping. This is visualized in Figs. 4 (b)\nand (d). Negative damping is possible when both dot levels are inside the conduction window, restricting the region\ninXover which negative damping can occur. Indeed, when only one level is within the conduction window, the\nsystem e\u000bectively reduces to the resonant level model for which, as we showed in the previous subsection, the friction\ncoe\u000ecient\rsis always positive. When current \rows from left to right, negative damping occurs only for positive\nvalues of the oscillator coordinate X, as shown in Figs. 4 (b) and (d). This is consistent with a level-inversion picture,\nas discussed recently in Ref. [51]. Pictorially, the electron-vibron coupling causes a splitting in energy of the left\nand right levels. When X > 0, electrons can go \\down the ladder\" formed by the energy levels by passing energy\nto the oscillator and hence amplifying the vibrations. For X < 0, electrons can pass between the two dots only by\nabsorbing energy from the vibrations, causing additional non-equilibrium damping. For small broadening of the dot15\n00.511.522.5γs,eq(x)/(Mω0)\n−0.4−0.2 0 0.2 0.4\nxeVbias=0\neVbias=0.4\neVbias=0.8(a)\n012γs(x)/(Mω0)\n−0.4−0.2 0 0.2 0.4\nxeVbias=0\neVbias=0.4\neVbias=0.8(b)\n00.511.522.5γs,eq(x)/(Mω0)\n−0.5 0 0.5\nxeVgate=0\neVgate=0.2\neVgate=0.4(c)\n012γs(x)/(Mω0)\n−0.5 0 0.5\nxeVgate=0\neVgate=0.2\neVgate=0.4(d)\nFIG. 4: Damping vs.mechanical displacement in the two-level model. (a) Contribution \rs;eqto the friction coe\u000ecient for\nvarious bias voltages at \fxed gate voltage eVgate= 0. (b) At the same gate voltage, the total damping exhibits a region of\nnegative damping due to the contribution of \rs;ne. (c)\rs;eqfor various gate voltages with the bias voltage eVbias= 0:8. Note\nthat for both eVgate= 0:2 andeVgate= 0:4, one small peak for negative xfalls outside of the shown range of x. (d) Again,\nthe full damping \rsexhibits regions of negative damping. We choose ~!0= 0:01, \u0000 = 0:1 andt= 0:1. The dimensionless\ncoordinate is x= (M!2\n0=\u00151)Xand energies are measured in units of \u00152\n1=(M!2\n0).\nlevels due to the coupling to the leads, this e\u000bect is expected to be strongest when the vibration-induced splitting\n\u00151Xbecomes of the same order as the strength of the hopping t. WhenXgrows further, the increasing detuning of\nthe dot levels reduces the current and hence the non-equilibrium damping [see Figs. 4 (b) and (d) and Figs. 6 (a),\n(b)]. The coexistence of a multistable potential together with regions of negative damping can lead to interesting\nnonlinear behavior for the dynamics of the oscillator. In particular, and as we show in the next example, limit-cycle\nsolutions are possible, in the spirit of a Van der Pol oscillator [54].\nWe can also calculate the current. The pumping contribution is proportional to the velocity _Xand thus small.\nTherefore we show here results only for the dominant adiabatic part of the current. This is given by\nI0=e\nhZ\nd\u000f2t2\u00002(fL\u0000fR)\nj\u0001j2: (81)\nFor zero temperature, the behavior of the current is shown in Fig. 6 as a function of various parameters. Figs. 6 (a)\nand (b) show the current as a function of the (dimensionless) oscillator coordinate xfor two di\u000berent values of gate\npotential for which the system exhibits multistability by developing several metastable equilibrium positions. For\nVgate= 0 and independently of bias, the current shows a maximum at the local minimum of the e\u000bective potential\nx= 0, whileI0\u00190 for another possible local minimum, x\u00190:5 (compare with Fig. 3 (a)). The true equilibrium value\nofxcan be tuned viathe bias potential, showing the possibility of perfect switching. For \fnite gate potential however,\nthe current is depleted from x= 0 with diminishing bias. Figs. 6 (c) to (d) show the current as a function of gate or\nbias voltage for \fxed representative values of the oscillator coordinate x. The current changes stepwise as the number\nof levels inside the conduction window changes, coinciding with the peaks in the friction coe\u000ecient illustrated in Fig.\n4. In an experimental setting, the measured dccurrent would involve an average over the probability distribution of\nthe coordinate x, given by the solution of the Fokker-Planck equation associated to the Langevin equation (1).16\nFIG. 5: Cartoon of the positions of the electronic levels in the dot with respect to the Fermi levels of the leads, depending on\nthe sign of xand the existence of a gate voltage. The levels are broadened due to the hybridization with the leads \u0000. When\nx>0, \\left\" and \\right\" levels approach the Fermi levels of left and right leads respectively, (a) for eVgate= 0 the levels align\nsimultaneously for left and right, (c) a \fnite eVgateproduces an assymmetry between left and right. For x<0 the alignment\nof the levels is inverted, (b) eVgate= 0, (d) \fnite eVgate.\nC. Two vibrational modes\nAs a \fnal example, we present a simple model which allows for both a non-conservative force and an e\u000bective\n\\Lorentz\" force, in addition to negative damping. For this it is necessary to couple the two electronic orbitals of the\nprevious example, see Eq. (76), to at least two oscillatory modes which we assume to be degenerate. The relevant\nvibrations in this case can be thought of as a center-of-mass vibration X1between the leads, and a stretching mode\nX2. (It should be noted that this is for visualization purposes only. In reality, for an H 2molecule, the stretching\nmode is a high energy mode when compared to a transverse and a rotational mode, see Ref. [55]. Nevertheless, the\nH2molecule does indeed have two near-degenerate low energy vibrational modes, corresponding to rigid vibrations\nbetween the leads and a rigid rotation relative to the axis de\fned by the two leads.) The stretch mode modulates the\nhopping parameter,\nt!~t(X2) =t+\u00152X2; (82)\nwhile the center of mass mode X1is modeled as coupling linearly to the density,\n\u000f0!~\u000f(X1) =\u000f0+\u00151X1; (83)\nhence \u0003 1=\u00151\u001b0and \u0003 2=\u00152\u001b1. We work in the wide-band limit, but allow for asymmetric coupling to the leads.\nThe retarded Green's function becomes\nGR(\u000f;X 1;X2) =1\n\u0001\u0012\n\u000f\u0000~\u000f+i\u0000R~t\n~t \u000f\u0000~\u000f+i\u0000L\u0013\n; (84)\nwhere now \u0001( X1;X2) = (\u000f\u0000~\u000f+i\u0000L)(\u000f\u0000~\u000f+i\u0000R)\u0000~t2. The frozen S-matrix can be easily calculated to be\nS(\u000f;X 1;X2) = 1\u00002i\n\u0001\u0012\n(\u000f\u0000~\u000f+i\u0000R) \u0000L~tp\u0000L\u0000R\n~tp\u0000L\u0000R (\u000f\u0000~\u000f+i\u0000L) \u0000R\u0013\n: (85)\nThe A-matrices also take a simple form for this model. Since \u0003 1is proportional to the identity operator,\nA1(\u000f;X 1;X2) =\u0000\u0019\u00151WGR[GR;\u001b0]GRWy= 0: (86)\nOn the other hand, the A-matrix associated with X2is non-zero and given by\nA2(\u000f;X 1;X2) =\u0000i\u00152p\u00001\u00002\n\u00012\u001b2: (87)17\n(a)\n0510152025I0/(eω0)\n−0.5−0.25 0 0.25 0.5\nxeVbias=0.2\neVbias=0.4\neVbias=0.8\n(b)\n02.557.51012.5I0/(eω0)\n−0.5−0.25 0 0.25 0.5\nxeVbias=0.2\neVbias=0.4\neVbias=0.8\n(c)\n0510152025I0/(eω0)\n0 0.25 0.50.75 1 1.25\neVbiasx=0x=0.2x=0.5\n(d)\n0510152025I0/(eω0)\n0 0.25 0.50.75 1 1.25\neVbiasx=0x=0.2x=0.5\n(e)\n0510152025I0/(eω0)\n0 0.5 1 1.5 2\neVbiaseVgate=0\neVgate=0.2\neVgate=0.4\n(f)\n00.250.50.7511.25I0/(eω0)\n0 0.5 1 1.5 2 2.5\neVbiaseVgate=0\neVgate=0.2\neVgate=0.4\nFIG. 6: Dependence of the current in the two-level model on various parameters. Current as function of mechanical displacement\nfor (a)Vgate= 0 and (b) Vgate= 0:4; as function of bias for (c) Vgate= 0, (d)Vgate= 0:4, (e)x= 0 and (f) x= 0:5. We\nchoose ~!0= 0:01, \u0000 = 0:1 andt= 0:1. The dimensionless coordinate is x= (M!2\n0=\u00151)Xand energies are measured in units\nof\u00152\n1=(M!2\n0).\nFrom this we can compute the average force, damping, pseudo-Lorentz force, and noise terms. These are listed in\nSupp. Mat. E. At zero temperature, it is possible to obtain analytical expressions for these current-induced forces.\nStudying the dynamics of the modes X1;2(t) implies solving the two coupled Langevin equations given by Eq. (1),\nafter inserting the expressions for the forces given in Supp. Mat. E. Within our formalism we are able to study\nthe full non-linear dynamics of the problem, which brings out a plethora of new qualitative behavior. In particular,\nanalyses which linearize the current-induced force about a static equilibrium point would predict run-away modes due\nto negative damping and non-conservative forces [30]. Taking into account nonlinearities allows one to \fnd the new\nstable attractor of the motion. Indeed, we \fnd that these linear instabilities typically result in dynamical equilibrium,\nnamely limit-cycle dynamics [22]. We note in passing that limit cycle dynamics in a nanoelectromechanical system\nwas also discussed recently in Ref. [53].\nWe have studied the zero-temperature dynamics of our two-level, two-mode system for di\u000berent ranges of parameters.\nIn Fig. (7) we map out the values of the curl of the mean force, ( r\u0002F)?, indicating that the force is non-conservative\nthroughout parameter space. We also plot one of the two eigenvalues of the dissipation matrix \rs, showing that it\ncan take negative values in some regions of the parameter space. We \fnd that it is possible to drive the system into\na limit cycle by varying the bias potential. The existence of this limit cycle is shown in Fig. 8 (a), where we have\nplotted various Poincar\u0013 e sections of the non-linear system without \ructuations. The \fgure shows the trajectory in\nphase space of the (dimensionless) oscillator coordinate x1after the dynamical equilibrium is reached, for several cuts18\n(a)\n−4 −2 0x1−0.500.5\nx20 0 .16 0 .32 0 .48 0 .64 0 .8(∇ ×F)⊥/(Mω2\n0)\n(b)\n−4 −3 −2x1−0.6−0.4−0.200.20.40.6\nx2−0.003 −0.001 0.001γ/(Mω 0)\nFIG. 7: Curl of the average force and damping coe\u000ecient for the model with two vibrational modes: (a) The curl of the current-\ninduced mean force Fis, in a non-equilibrium situation, generally non-zero, indicating that the force is non-conservative. (b)\nOne of the two eigenvalues of \rs. Remarkably, it undergoes sign changes. A dissipation matrix \rswhich is non-positive de\fnite\nimplies destabilization of the static equilibrium solution found at lower bias potentials, in this case driving the system into a\nlimit cycle, see main text and Fig. 8. The parameters used are such that \u00151=\u00152= 3=2. The elastic modes are degenerate\nwith ~!0= 0:014, \u0000 L;R=1\u00060:8\n2(\u001b0\u0006\u001bz), and the hopping between the orbitals is t= 0:9. The dimensionless coordinates are\nxi= (M!2\n0=\u0015)Xiand energies are in units of \u00152=(M!2\n0), where\u0015= (\u00151+\u00152)=2.\n(a)\neVbias=10 −101˙x1\n−3 −2 −1 0\nx1x2=0.35\nx2=0.3\nx2=0.2\nx2=0.1\nx2=0.\nx2=−0.1\nx2=−0.2\nx2=−0.3\nx2=−0.35\n(b) eVbias=10\n−0.5−0.2500.250.5x2\n−4 −3 −2 −1 0\nx1\nFIG. 8: Limit-cycle dynamics for the model with two vibrational modes. (a) At large bias voltages, Poincar\u0013 e sections of the\nfour dimensional phase space show the presence of a limit cycle in the Langevin dynamics without \ructuating force. (b) Several\nperiods of typical trajectories (for di\u000berent initial conditions after a transient) in the presence of the \ructuating forces \u0018are\nshown. The same general parameters as in Fig. 7 are used here.\nof the (dimensionless) coordinate x2. Each cut shows two points in x1phase space, indicating the entry and exit of\nthe trajectory. Each point in the plot actually consists of several points that fall on top of each other, corresponding\nto every time the coordinate x2has the value indicated in the legend of Fig. 8 (a). This shows the periodicity of the\nsolution of the non-linear equations of motion for x1; x2for the particular bias chosen. Surveying over the various\nvalues ofx2reveals a closed trajectory in the parametric coordinate space x1; x2.\nRemarkably, signatures of the limit cycle survive the inclusion of the Langevin force. Fig. 8 (b) depicts typical\ntrajectories in the oscillator's coordinate space x1; x2in the presence of the stochastic force, showing \ructuating\ntrajectories around the stable limit cycle.\nExperimentally, the signature of the limit cycle would be most directly re\rected in the current-current correlation\nfunction, as depicted in Fig. 9. We \fnd that in the absence of a limit cycle the system is dominated by two\ncharacteristic frequencies, shown by the peaks in Fig. 9. These frequencies correspond to the shift in energy of the\ntwo degenerate vibrational modes due to the average current-induced forces F1andF2. When the bias voltage is\nsuch that the system enters a limit cycle, the current-current correlation shows instead only one peak as a function\nof frequency. This result, as shown in Fig. 9, is fairly robust to noise, making the onset of limit-cycle dynamics\nobservable in experiment.19\n0.8 0.9 1ω/ω000.40.8<|I(ω)/e|2>1/2Vbias=10Vbias=5Vbias=2.5*3\nFIG. 9: Current-current correlation function in the presence of noise for the system with two vibrational modes. The limit cycle\nis signaled by a single peak ( Vbias= 10, see Fig. 8), as opposed to two peaks in the absence of a limit cycle ( Vbias= 2:5;5).\nIncreasing the bias potential increases the noise levels but the peaks are still easily recognizable. The results are obtained by\naveraging over times long enough compared with the characteristic oscillation times. The same general parameters as in Fig.\n7 are used here.\nVI. CONCLUSIONS\nWithin a non-equilibrium Born-Oppenheimer approximation, the dynamics of a nanoelectromechanical system can\nbe described in terms of a Langevin equation, in which the mechanical modes of the mesoscopic device are subject to\ncurrent-induced forces. These forces include a mean force, which is independent of velocity and due to the average\nnet force the electrons exert on the oscillator, a stochastic Langevin force which takes into account the thermal and\nnon-equilibrium \ructuations with respect to the mean force value, and a force linear in the velocity of the modes.\nThis last, velocity dependent force, consists of a dissipative term plus a term that can be interpreted as an e\u000bective\n\\Lorentz\" force, due to an e\u000bective magnetic \feld acting in the parameter space of the modes.\nIn this work we have expressed these current-induced forces through the scattering matrix of the coherent mesoscopic\nconductor and its parametric derivatives, extending the results found previously in Ref. [22]. Our results are now valid\nfor a generic coupling between the electrons and the vibrational degrees of freedom, given by a matrix h0(X), and\nfor energy-dependent hybridization with the leads, given by the matrix W(\u000f). We have shown that expressing allthe\ncurrent-induced forces in terms of the S-matrix is only possible by going beyond the strictly adiabatic approximation,\nand it is necessary to include the \frst order correction in the adiabatic expansion. This introduces a new fundamental\nquantity into the problem, the A-matrix, which needs to be calculated together with the frozen S-matrix for a given\nsystem.\nThere are several circumstances in which the \frst non-adiabatic correction, encapsulated in the A-matrix, is nec-\nessary. While the average as well as the \ructuating force can be expressed solely in terms of the adiabatic S-matrix,\nthe A-matrix enters both the frictional and the Lorentz-like force. In equilibrium, the frictional force reduces to an\nexpression in terms of the adiabatic S-matrix. Out of equilibrium, however, an important new contribution involving\nthe A-matrix appears. In contrast, the A-matrix is always required to express the Lorentz-like force, even when the\nsystem is in thermal equilibrium.\nThe expressions for the current-induced forces in terms of the scattering matrix allow us to extract important prop-\nerties from general symmetry arguments. Driving the nanoelectromechanical system out of equilibrium by imposing a\nbias results in qualitatively new features for the forces. We have shown that the mean force is non-conservative in this\ncase, and that the dissipation coe\u000ecient acquires a non-equilibrium contribution that can be negative. We have also\nshown that when considering more than one mechanical degree of freedom, a pseudo Lorentz force is present even for\na time-reversal invariant system, unless one also imposes thermal equilibrium on top of the time-reversal condition.\nOur model allows one to study, within a controlled approximation, the non-linear dynamics generated by the\ninterplay between current and vibrational degrees of freedom, opening up the path for a systematic study of these\ndevices. By means of simple model examples, we have shown that it is possible to drive a nanoelectromechanical\nsystem into interesting dynamically stable regimes such as a limit cycle, by varying the applied bias potential. In\na limit cycle, the vibrational modes vary periodically in time, which can be the operating principle for a molecular20\nmotor. On the other hand, the possibility of non-conservative forces could also allow one to extract energy from the\nsystem, providing a controllable tool for cooling. The study of these kinds of phenomena for realistic systems is an\ninteresting application of the formalism presented in this paper.\nVII. ACKNOWLEDGMENTS\nWe acknowledge discussions with P. W. Brouwer, G. Zarand, and L. Arrachea as well as support by the DFG\nthrough SPP 1459, SFB TR/12, and SFB 658.21\nAppendix A: Useful relations\nHere we list a set of useful relations for the derivations in the main text.\n1. Green's functions relations\nThe Green's functions are related via\nGR\u0000GA=G>\u0000G<: (A1)\nThe lesser and larger Green's functions are given by\nG<=GR\u0006<GA= 2iX\n\u000bf\u000bGR\u0000\u000bGA= 2\u0019iX\n\u000bf\u000bGRWy\u0005\u000bWGA(A2)\nG>=G<+GR\u0000GA=\u00002\u0019iX\n\u000b(1\u0000f\u000b)GRWy\u0005\u000bWGA: (A3)\nFrom (22) it is easy to see that\nWyW=1\n2\u0019i[(GR)\u00001\u0000(GA)\u00001]; (A4)\n@X\u0017GR=GR\u0003\u0017GR(A5)\nand\n@\u000fGR=\u0000GR(1\u0000@\u000f\u0006R)GR: (A6)\n2. Green's functions and S-matrix relations\nNoting that (for given t)@X\u0017GR=GR\u0003\u0017GR, we \fnd using Eq.(A4):\nSy@S\n@X\u0017=\u00002\u0019i(1 + 2\u0019iWGAWy)WGR\u0003\u0017GRWy=\u00002\u0019iWGA\u0003\u0017GRWy: (A7)\nThis holds for arbitrary magnitude of X\u0017.\nIn the main text we use\n1\n\u0019@Sy\n@X\u0017A\u00170= 2\u0019iWGA\u0003\u0017GAWy@\u000f(WGR)\u0003\u00170GRWy\u0000WGA\u0003\u0017(GA\u0000GR)\u0003\u00170@\u000f(GRWy); (A8)\n\u0019 \u0014\nSy@S\n@X\u0017;WGA\u0003\u00170@(GRWy)\n@\u000f\u0000@(WGA)\n@\u000f\u0003\u00170GRWy\u0015\n\u0000!\ns=\u0012@Sy\n@X\u0017A\u00170\u0000Ay\n\u00170@S\n@X\u0017\u0013\ns(A9)\nand\n\u0014\nSy@A\u0017\n@X\u00170\u0015\na=\u00002\u0019\u0002\nWGA\u0003\u0017(@\u000fGR)\u0003\u00170GRWy\u0003\na: (A10)\nFor energy-independent \u0000\u000b, we can use (A6) so that also\nSy@S\n@\u000f= 2\u0019iWGAGRWy; (A11)\n@\u000f\u0012\nSy@S\n@X\u0017\u0013\n= 2\u0019iWGA\u0000\nGA\u0003\u0017+ \u0003\u0017GR\u0001\nGRWy(A12)\nand (A8) simpli\fes to\n@Sy\n@X\u0017A\u00170=\u0019WGA\u0003\u0017\u0000\nGA\u0000GR\u0001\u0000\n\u0003\u00170GR\u0000GR\u0003\u00170\u0001\nGRWy: (A13)22\nAppendix B: S-matrix derivation of the damping matrix\nThe expression for \rsgiven in Eq. (27) can be written explicitly in terms of retarded and advanced Green's functions\nas\n\rs\n\u0017\u00170= 2\u0019X\n\u000b\u000b0Z\nd\u000ff\u000btr\b\n\u0003\u0017GRWy\u0005\u000bWGA\u0003\u00170@\u000f\u0002\n(1\u0000f\u000b0)GRWy\u0005\u000b0WGA\u0003\t\ns: (B1)\nWe split Eq. (B1) into two terms, the \frst due to the derivative acting on the Fermi function, the second from the\nrest,\rs=\rs(I)+\rs(II). The \frst term is given by\n\rs(I)\n\u0017\u00170= 2\u0019X\n\u000b\u000b0Z\nd\u000ff\u000b(\u0000@\u000ff\u000b0)tr\b\n\u0005\u000b0WGA\u0003\u0017GRWy\u0005\u000bWGA\u0003\u00170GRWy\t\ns(B2)\nwhere we have used the cyclic invariance of the trace. Similar to the derivation for the mean force, by means of\nexpression (A7) in Supp. Mat. A, Eq. (B2) can be expressed in terms of the frozen S-matrix as\n\rs(I)\n\u0017\u00170=\u0000X\n\u000b\u000b0Zd\u000f\n2\u0019f\u000b(\u0000@\u000ff\u000b0)Tr\u001a\n\u0005\u000bSy@S\n@X\u0017\u0005\u000b0Sy@S\n@X\u00170\u001b\ns: (B3)\nThe second contribution, in terms of GRandGA, reads\n\rs(II)\n\u0017\u00170= (2\u0019)2X\n\u000b\u000b0Zd\u000f\n2\u0019F\u000b\u000b0tr\b\n\u0003\u0017GRWy\u0005\u000bWGA\u0003\u00170@\u000f\u0000\nGRWy\u0005\u000b0WGA\u0001\t\ns: (B4)\nIt is instructive to split the factor F\u000b\u000b0into a symmetric and an antisymmetric part under exchange of the lead\nindices,F\u000b\u000b0=Fs\n\u000b\u000b0+Fa\n\u000b\u000b0, with\nFs\n\u000b\u000b0\u00111\n2(f\u000b+f\u000b0\u00002f\u000bf\u000b0)\nFa\n\u000b\u000b0\u00111\n2(f\u000b\u0000f\u000b0):(B5)\nCorrespondingly, we split \rs(II)into symmetric\u0002\n\rs(IIs)\u0003\nand antisymmetric\u0002\n\rs(IIa)\u0003\nparts in the lead indices: \rs(II)=\n\rs(IIs)+\rs(IIa). Due to its symmetries, \rs(IIs)can be easily expressed in terms of the S-matrix,\n\rs(IIs)\n\u0017\u00170=\u0019X\n\u000b\u000b0Z\nd\u000fFs\n\u000b\u000b0@\u000ftr\b\n\u0003\u0017GRWy\u0005\u000bWGA\u0003\u00170GRWy\u0005\u000b0WGA\t\ns\n=\u0000\u0019X\n\u000b\u000b0Z\nd\u000f(@\u000fFs\n\u000b\u000b0) tr\b\n\u0003\u0017GRWy\u0005\u000bWGA\u0003\u00170GRWy\u0005\u000b0WGA\t\ns\n=1\n4\u0019X\n\u000b\u000b0Z\nd\u000f(@\u000fFs\n\u000b\u000b0) tr\u001a\n\u0005\u000bSy@S\n@X\u0017\u0005\u000b0Sy@S\n@X\u00170\u001b\ns(B6)\nwhere in the second line we have integrated by parts since Fsvanishes for \u000f!\u00061 , and in the last line we have used\nEq. (A7) from App. A once again.\n1. \\Equilibrium\" dissipative term \rs;eq\nSince in equilibrium Fa\n\u000b\u000b0=Fa\n\u000b\u000b= 0,\rs(IIa)\f\f\neq= 0 and we can now regroup terms into an \\equilibrium\" contri-\nbution,\rs;eq=\rs(I)+\rs(IIs), and a purely non-equilibrium contribution \rs;ne\u0011\rs(IIa):\n\rs=\rs;eq+\rs;ne: (B7)\nBy adding up expressions (B3) and (B6), it is straightforward to obtain Eq. (44) for \rs;eqgiven in the main text.23\n2. Non-equilibrium dissipative term \rs;ne\nTo obtain\rs;nein terms of S-matrix quantities we start from the expression\n\rs;ne\n\u0017\u00170= 2\u0019X\n\u000b\u000b0Z\nd\u000fFa\n\u000b\u000b0tr\b\n\u0003\u0017GRWy\u0005\u000bWGA\u0003\u00170@\u000f\u0000\nGRWy\u0005\u000b0WGA\u0001\t\ns; (B8)\nand exploitingP\n\u000b\u0005\u000b= 1 and the identity (A7) in Supp. Mat. A, we note that Eq. (B8) can be written as\n\rs;ne\n\u0017\u00170=\u0000i\n2Z\nd\u000fX\n\u000bf\u000btr\u001a\n\u0005\u000b\u0014\nSy@S\n@X\u0017;WGA\u0003\u00170@(GRWy)\n@\u000f\u0000@(WGA)\n@\u000f\u0003\u00170GRWy\u0015\u001b\ns; (B9)\nwhere [:; :] indicates the commutator. Calculating each term in the commutator separately we obtain\nSy@S\n@X\u0017\u0014\nWGA\u0003\u00170@(GRWy)\n@\u000f\u0000@(WGA)\n@\u000f\u0003\u00170GRWy\u0015\n=\u0000WGA\u0003\u0017(GA\u0000GR)\u0003\u00170@(GRWy)\n@\u000f\n+ 2\u0019iWGA\u0003\u0017GRWy@(WGA)\n@\u000f\u0003\u00170GRWy\n\u0014\nWGA\u0003\u00170@(GRWy)\n@\u000f\u0000@(WGA)\n@\u000f\u0003\u00170GRWy\u0015\nSy@S\n@X\u0017=\u0000@(WGA)\n@\u000f\u0003\u00170(GA\u0000GR)\u0003\u0017GRWy\n\u00002\u0019iWGA\u0003\u00170@(GRWy)\n@\u000fWGA\u0003\u0017GRWy;(B10)\nwhere we have used Eq. (A4) from Supp. Mat. A. Finally, with help of the identity (A8) in Supp. Mat. A, the\nnon-equilibrium term can be expressed as Eq. (45) in the main text.\nAppendix C: Resonant level forces: alternative expressions\nTo calculate the current-induced forces for the resonant level model presented in Sec. V, we can alternatively start\nwith the popular S-matrix parametrization [1,32]\nS=\u0012p\n1\u0000Tei\u0012p\nTei\u0011p\nTei\u0011\u0000p\n1\u0000Tei(2\u0011\u0000\u0012)\u0013\n; (C1)\nwhere the transmission coe\u000ecient Tand the phases \u0011;\u0012depend onX. We present here the results for linear coupling,\n~\u000f(X) =\u000f0+\u0015X. We can then identify the transmission probability\nT(\u000f;X) =4\u0000L\u0000R\n(\u000f\u0000\u000f0\u0000\u0015X)2+ \u00002(C2)\nand the phases\n\u0011(\u000f;X) =\u0000\u0019\n2\u0000arctan\u0012\u0000\n\u000f\u0000\u000f0\u0000\u0015X\u0013\n\u0012(\u000f;X) =\u0019\n2+\u0011+ arctan\u0012\u0000R\u0000\u0000L\n\u000f\u0000\u000f0\u0000\u0015X\u0013\n:\nWe can now relate the current-induced forces to this S-matrix parametrization. The result for the average force can\nbe split into a non-equilibrium force Fneand an equilibrium force Feq,i.e.,F=Fne+Feqwith\nFne(X) =Zd\u000f\n2\u0019(fL\u0000fR)(1\u0000T)@(\u0012\u0000\u0011)\n@X(C3)\nFeq(X) =Zd\u000f\n2\u0019(fL+fR)@\u0011\n@X:24\nThe amplitude of the \ructuating force can be obtained from Eq. (42) and is given by\nD(X) =Zd\u000f\n2\u0019X\n\u000b\u000b0Fs\n\u000b\u000b0Y\u000b\u000b0; (C4)\nwhere we have de\fned\nYLL=\u0014\n(1\u0000T)@(\u0011\u0000\u0012)\n@X\u0000@\u0011\n@X\u00152\nYRR=\u0014\n(1\u0000T)@(\u0011\u0000\u0012)\n@X+@\u0011\n@X\u00152\nYLR=YRL=1\n4T(1\u0000T)\u0012@T\n@X\u00132\n+T(1\u0000T)\u0012@(\u0011\u0000\u0012)\n@X\u00132\n:\nAfter some algebra, we also obtain\n\rs(X) =1\n2T\u0014\nD(X)\u0000Zd\u000f\n2\u0019(fL\u0000fR)2YLR\u0015\n: (C5)\nThis last expression corresponds to \rs;eqgiven in Eq. (44). (As we pointed out previously, \rs;nevanishes in this\ncase). Here we have isolated a term that vanishes in equilibrium, showing explicitly that there is a non-equilibrium\ncontribution in (44).\nAppendix D: Current-induced forces for the two-level model\nThe mean force is given by\nF(X) =\u0000\u00151\u0000Zd\u000f\n2\u0019\"\n(fL+fR)2\u00151X(\u000f\u0000\u000f0)\nj\u0001j2+ (fL\u0000fR)(\u000f\u0000\u000f0)2+ (\u00151X)2\u0000t2+ (\u0000=2)2\nj\u0001j2#\n: (D1)\nThe friction coe\u000ecient \rs=\rs;eq+\rs;nereads\n\rs;eq=\u00152\n1\u00002\n4\u0019Z\nd\u000f(\n\u0000@\u000ffL+@\u000ffR\nj\u0001j4h\u0000\n(\u000f\u0000\u000f0)2+ (\u0000=2)2+ (\u00151X)2+t2\u00012+ (2(\u000f\u0000\u000f0)\u00151X)2\n\u0000(2(\u000f\u0000\u000f0)t)2i\n+@\u000ffR\u0000@\u000ffL\nj\u0001j4\u0002\n4(\u000f\u0000\u000f0)\u00151X\u0000\n(\u000f\u0000\u000f0)2+ (\u0000=2)2+ (\u00151X)2\u0000t2\u0001\u0003)\n;\n\rs;ne=2\u00152\n1\u00002t2\u00151X\n\u0019Z\nd\u000ffR\u0000fL\nj\u0001j6h\u0000\n(\u000f\u0000\u000f0)2\u0000(\u00151X)2\u0000t2\u00012\n+2(\u0000=2)2\u0000\n(\u000f\u0000\u000f0)2+ (\u00151X)2+t2\u0001\n+ (\u0000=2)4i\n: (D2)\nAppendix E: Current-induced forces for the two vibrational modes model\nHere we list the current-induced forces quantities, calculated from Eqs. (39), (42), (47) and (50) for the two-modes\nexample discussed in the main text. For convenience, we de\fne the following quantities:\ng\u000b0(\u000f) =(\u000f\u0000~\u000f)2+~t2+ \u00002\n1\u0000\u000b\nj\u0001j2(E1)\ng\u000b1(\u000f) =2~t(\u000f\u0000~\u000f)\nj\u0001j2(E2)\ng\u000b2(\u000f) =\u0006\u00002~t\u00001\u0000\u000b\nj\u0001j2(E3)\ng\u000b3(\u000f) =\u0006(\u000f\u0000~\u000f)2+ \u00002\n1\u0000\u000b\u0000~t2\nj\u0001j2(E4)25\nwhere the +(\u0000) refers to\u000b=L(R) and with 1\u0000\u000b=R(L) for\u000b=L(R), and \u0001(X1;X2) = (\u000f\u0000~\u000f+i\u0000L)(\u000f\u0000~\u000f+i\u0000R)\u0000~t2.\n1. Mean force\nF1=\u00002Zd\u000f\n2\u0019\u00151X\n\u000bf\u000b(\u000f)\u0000\u000b\u0000\n(\u000f\u0000~\u000f)2+~t2+ \u00002\n1\u0000\u000b\u0001\n\u0002\n(\u000f\u0000~\u000f)2\u0000~t2\u0000\u0000L\u0000R\u00032+ [(\u0000L+ \u0000R)(\u000f\u0000~\u000f)]2(E5)\nF2=\u00004Zd\u000f\n2\u0019\u00152~t(\u000f\u0000~\u000f) (fL(\u000f)\u0000L+fR(\u000f)\u0000R)\n\u0002\n(\u000f\u0000~\u000f)2\u0000~t2\u0000\u0000L\u0000R\u00032+ [(\u0000L+ \u0000R)(\u000f\u0000~\u000f)]2(E6)\n2. Fluctuating force\nD11= 2 (\u00151)2Zd\u000f\n2\u0019X\n\u000b\ff\u000b(\u000f)\u0000\u000b(1\u0000f\f(\u000f)) \u0000\fX\n\u0016g\u000b\u0016g\f\u0016 (E7)\nD12= 2\u00151\u00152Zd\u000f\n2\u0019X\n\u000b\ff\u000b(\u000f)\u0000\u000b(1\u0000f\f(\u000f)) \u0000\f(g\u000b0g\f1+g\u000b1g\f0) (E8)\nD22= 2 (\u00152)2Zd\u000f\n2\u0019X\n\u000b\ff\u000b(\u000f)\u0000\u000b(1\u0000f\f(\u000f)) \u0000\f(g\u000b0g\f0+g\u000b1g\f1\u0000g\u000b2g\f2\u0000g\u000b3g\f3) (E9)\n3. 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B 71, 161402\n(2005)."    },    {        "title": "0803.1246v1.The_Impact_of_Stochastic_Primordial_Magnetic_Fields_on_the_Scalar_Contribution_to_Cosmic_Microwave_Background_Anisotropies.pdf",        "content": "arXiv:0803.1246v1  [astro-ph]  8 Mar 2008The Impact of Stochastic Primordial Magnetic Fields on the S calar Contribution to\nCosmic Microwave Background Anisotropies\nFabio Finelli∗\nINAF/IASF-BO, Istituto di Astrofisica Spaziale e Fisica Cos mica di Bologna\nvia Gobetti 101, I-40129 Bologna - Italy\nINAF/OAB, Osservatorio Astronomico di Bologna, via Ranzan i 1, I-40127 Bologna - Italy and\nINFN, Sezione di Bologna, Via Irnerio 46, I-40126 Bologna, I taly\nFrancesco Paci†\nDipartimento di Astronomia,\nUniversit` a degli Studi di Bologna,\nvia Ranzani, 1 – I-40127 Bologna – Italy\nINAF/IASF-BO, Istituto di Astrofisica Spaziale e Fisica Cos mica di Bologna\nvia Gobetti 101, I-40129 Bologna - Italy and\nINFN, Sezione di Bologna, Via Irnerio 46, I-40126 Bologna, I taly\nDaniela Paoletti‡\nDipartimento di Fisica, Universit` a degli Studi di Ferrara ,\nvia Saragat, 1 – I-44100 Ferrara – Italy\nINAF/IASF-BO, Istituto di Astrofisica Spaziale e Fisica Cos mica di Bologna\nvia Gobetti 101, I-40129 Bologna - Italy and\nINFN, Sezione di Bologna, Via Irnerio 46, I-40126 Bologna, I taly\nWe study the impact of a stochastic background of primordial magnetic fields on the scalar contri-\nbutionofCMB anisotropies andonthematter powerspectrum. Wegive thecorrect initial conditions\nfor cosmological perturbations and the exact expressions f or the energy density and Lorentz force\nassociated to the stochastic background of primordial magn etic fields, given a power-law for their\nspectra cut at a damping scale. The dependence of the CMB temp erature and polarization spectra\non the relevant parameters of the primordial magnetic fields is illustrated.\nPACS numbers: 98.80.Cq\nI. INTRODUCTION\nLarge scale magnetic fields are almost everywhere in\nthe universe, from galaxies up to those present in galaxy\nclusters and in the intercluster medium [1]. The origin of\nthese magnetic fields depends on the size of the objects\nand may become mysterious for the largest ones. The\ndynamo mechanism provides a mechanism to explain the\nobserved magnetic field associated to galaxies, whereas\nthose associated to clusters may be generated by gravi-\ntational compression starting from an initial seed.\nThe requirement of an initial seed for magnetic fields\nobserved in galaxies and galaxy clusters leads directly to\nquestiontheexistenceofprimordialmagneticfieldsinthe\nearly universe. Cosmology described by an homogeneous\nand isotropic expanding metric neither supports a uni-\nform magnetic field nor a gravitational amplification of\ngauge fields because of conformal invariance; the genera-\ntion of large scale magnetic fields has therefore generated\nalotofinterest. Astochasticbackground(SB) ofprimor-\n∗Electronic address: finelli@iasfbo.inaf.it\n†Electronic address: paci@iasfbo.inaf.it\n‡Electronic address: paoletti@iasfbo.inaf.itdial magnetic fields (PMF) can provide the initial seeds\nfor the large-scale magnetic fields observed and can leave\nimprints on different observables, as the CMB pattern of\ntemperature and polarization anisotropies [2, 3] and the\nmatter power spectrum.\nA SB of PMF carries zero energy and pressure at ho-\nmogeneous level in a Robertson-Walkermetric. It carries\nhowever perturbations, of any kind, i.e. scalar, vector\nand tensor, and it is usually studied in a quasi-linear ap-\nproximation, i.e. its EMT - quadratic in the magnetic\nfield amplitude - is considered at the same footing as\nfirst order terms in a perturbative series expansion. Vec-\ntor[4, 5] and tensor[4, 6, 7] metric perturbationssourced\nby a PMF SB have been object of several investigations;\nbeyond the technical simplicity of vector and tensor over\nscalar, a perfect fluid cannot support vector and tensor\nperturbations at linear order and therefore represent a\nkey prediction of a PMF SB. We know howeverthat tem-\nperature and polarization anisotropies sourced by scalar\nfluctuations with adiabatic initial conditions are a good\nfit to the whole set of observations; it is therefore crucial\nto investigate how a PMF SB can modify these scalar\nfluctuations. Analytic [8] and numerical [9, 10, 11] works\nin this direction have already been made. However a\ndetailed analysis which takes into account the Lorentz\nforce on baryons, a careful treatment of initial conditions2\nand an accurate treatment of the Fourier spectra of PMF\nenergy-momentum tensor is still lacking. As is clear in\nthe following, our work address carefully both these is-\nsues.\nThe goal of this paper is to investigate the impact\nof a stochastic background (SB) of primordial magnetic\nfields (PMF) on scalar cosmological perturbations and in\nparticular on CMB temperature anisotropies and matter\npower spectrum. Our paper is organized as follows. In\nSection II we reviewhow to add a fully inhomogenousSB\nof PMF treated in the one-fluid plasma description [2] to\nthe Einstein-Boltzmann system of equations. In Sections\nIII and IV we review the baryons evolution and we give\nthe initial conditions for cosmological perturbations in\na form suitable to be plugged in most of the Einstein-\nBotzmann codes. In Section V we give the PMF energy\ndensity andLorentzforcepowerspectraand compareour\nresults with the ones given in the literature. In Sections\nVI-VIII we showthe resultsobtained by our modification\noftheEinstein-BoltzmanncodeCAMB[18]forcosmolog-\nical scalar perturbations, CMB spectrum of temperature\nand polarization, matter power spectrum, respectively.\nIn the Appendix we show the detailed calculations for\nthe convolution integrals leading to the energy density\nand Lorentz force, starting from a power-law spectrum\nsharply cut at a given scale for the PMF.\nII. STOCHASTIC MAGNETIC FIELDS AND\nCOSMOLOGICAL SCALAR PERTURBATIONS\nWe model a SB of PMFs as a fully inhomogenous com-\nponent, considering B2at the same level of metric and\ndensity fluctuations in a perturbative expansion∗. Al-\nthough a SB of PMFs carries no energy at the homoge-\nneouslevel, it affectsscalarcosmologicalperturbationsin\nthree different ways. First, inhomogeneous PMFs carry\nenergy density and pressure and therefore gravitate at\nthe levelofperturbations. Second, inhomogeneousPMFs\nhave anisotropic stress - differently from perfect fluids -\nwhich adds to the photon and neutrino ones, with the\ncaveat that the photon anisotropic stress is negligible be-\nfore the decoupling epoch. Last, but not least, the in-\nduced Lorentz force acting on baryons, affects also pho-\ntons during the tight coupling regime.\nSince the EMT of PMF at homogeneous level is zero,\nat linear order PMFs evolve like a stiff source and there-\nfore it is possible to discard all the back reactions of the\nfluid or gravity onto the SB of PMF. Before the decou-\npling epoch the electric conductivity of the primordial\n∗Note that in such a way we do not take into account the modifica-\ntion of the sound speed of baryons induced by PMFs, pionereed\nin [12], since it would be technically of second order in the e qua-\ntions of motion. However, since the baryons speed of sound go es\nrapidly to zero in the matter dominated era, this effect, lead ing\nto a shift in the Doppler peaks, may be anyway important.plasma is very large, therefore it is possible at the first\norder to consider the infinite conductivity limit. In this\nlimit the induced electric field is zero. Within the infinite\nconductivity limit the SB of PMF time evolution simply\nreduces to : B(x,τ) =B(x)/a(τ)2.†\nThe evolution of the metric perturbations in the pres-\nence of PMF is governed by the Einstein equations:\nGµν= 8π(Tµν+τPMF\nµν), (1)\nInthe approximationin whichtheinduced electricfieldis\nvanishing (i.e. the infinite conductivity limit) the energy\nmomentum tensor of the electromagnetic field becomes:\nτ0PMF\n0=−ρB=−|B(x)|2\n8πa4, (2)\nτ0PMF\ni= 0, (3)\nτiPMF\nj=1\n4πa4/parenleftbigg|B(x)|2\n2δi\nj−Bj(x)Bi(x)/parenrightbigg\n.(4)\nIn the Fourier space‡the Einstein equations with the\ncontribution of PMF in the synchronous gauge are:\nk2η−1\n2H˙h= 4πGa2(Σnρnδn+ρB),\nk2˙η= 4πGa2Σn(ρn+Pn)θn,\n¨h+2H˙h−2k2η=−8πGa2(Σnc2\nsnρnδn\n+ρB\n3),\n¨h+6¨η+2H(˙h+6˙η)−2k2η=−24πGa2×\n[Σn(ρn+Pn)σn+σB],(6)\nwhere by nwe mean the number of components, i.e.\nbaryons, cold dark matter (CDM), photons and neutri-\nnos. The conservation of the PMF EMT - ∇µτµPMF\nν= 0\n- simply reduces to :\nσB=ρB\n3+L, (7)\nwhereσBrepresents the PMFs anisotropic stress and L\nthe Lorentz force. The energy density of PMF evolves\nlike radiation: ρB(x,τ) =ρB(x,τ0)/a(τ)4.\nIII. BARYONS EVOLUTION\nThe presence of PMFs in a plasma which contains\ncharged particles induces a Lorentz force on these par-\nticles, that, in the primordial plasma, are baryons. The\n†We choose the standard convention in which at present time t0,\na(t0) = 1.\n‡As Fourier transform and its inverse, we use - in agreement wi th\n[13] -:\nY(/vectork,τ) =Zd3x\n(2π)3e−i/vectork·/vector xY(/vector x,τ), Y(/vector x,τ) =Z\nd3kei/vectork·/vector xY(/vectork,τ).\n(5)\nwhereYis a generic function.3\ngeneral expression for the Lorentz force is [8]:\nLi(x,τ0) =1\n4π/bracketleftbigg\nBj(x)∇jBi(x)−1\n2∇iB2(x)/bracketrightbigg\n,(8)\nwhereL(x,τ) =L(x,τ0)\na4.\nWeareinterestedonlyinthescalarperturbationsandthe\nscalarpartofthe Lorentzforcedefined as ∇2L(S)≡ ∇iLi\nis therefore:\n∇2L(S)=1\n4π/bracketleftBig\n(∇iBj(x))∇jBi(x)−1\n2∇2B2(x)/bracketrightBig\n.(9)\nIn the presence of an electromagnetic source the con-\nservation equations of the baryon component of the pri-\nmordial fluid becomes:\n∇µδTµνbaryons∝FµνJµ (10)\nwhereJµis the quadrivector of the density current and\nFµνis the Maxwell tensor. The primordial plasma can\nbe considered globally neutral, this leads to J0= 0\nand therefore to the fact that the energy conservation\nof baryons is not modified by the presence of the Lorentz\nterm. The Euler equation for baryons in instead affected\nby the Lorentz force and the scalar part is therefore [2]:\n˙θb=−Hθb+k2c2\nsbδb−k2L\nρb. (11)\nNow we study how the tight-coupling regime is modified\nby the presence of a SB of PMF [14]. The Euler equation\nfor photons during the tight-coupling regime is:\n˙θγ=k2/parenleftBigδγ\n4−σγ/parenrightBig\n+aneσT(θb−θγ) (12)\nCombining the photons and baryons equations gives:\n˙θb=−Hθb+c2\nsk2δb+k2R/parenleftBig\nδγ\n4−σγ/parenrightBig\n+R(˙θγ−˙θb)−k2L\nρb\n(1+R),\nwith:\n˙θb−˙θγ=2R\n(1+R)H(θb−θγ)+τ\n(1+R)/parenleftbigg\n−¨a\naθb+\n−Hk2\n2δγ+k2/parenleftBig\nc2\ns˙δb−˙δγ\n4/parenrightBigg\n+Hk2L\nρb/parenrightBig\n,\nThe photon Euler equation in tight coupling regime\ninstead is:\n˙θγ=−R−1/parenleftBig\n˙θb+Hθb−c2\nsk2δb+k2L\nρb/parenrightBig\n+k2/parenleftBigδγ\n4−σγ/parenrightBig\n(13)\nWe note that there is a term depending on the Lorentz\nforce which disappears when the tight coupling ends,\nleaving the normal Euler equation for the photon veloc-\nity.IV. INITIAL CONDITIONS\nIn order to study the effect of a PMF SB on scalar\ncosmological perturbations, the initial conditions for the\nlatterdeepintheradiationeraarerequired(see[2]forthe\nresults in the longitudinal gauge). The magnetized adi-\nabatic mode initial conditions in the synchronous gauge\nare given by [15]:\nh=C1(kτ)2\nη= 2C1−5+4Rν\n6(15+4Rν)C1(kτ)2+\n−/bracketleftbiggΩB(1−Rν)\n6(15+4Rν)+LB\n2(15+4Rν)/bracketrightbigg\n(kτ)2\nδγ=−ΩB−2\n3C1(kτ)2+/bracketleftbiggΩB\n6+LB\n2(1−Rν)/bracketrightbigg\n(kτ)2\nδν=−ΩB−2\n3C1(kτ)2−/bracketleftbiggΩB(1−Rν)\n6RνLB\n2Rν/bracketrightbigg\n(kτ)2\nδb=−3\n4ΩB−C1\n2(kτ)2+/bracketleftbiggΩB\n8+3LB\n8(1−Rν)/bracketrightbigg\n(kτ)2\nδc=−C1\n2(kτ)2\nθγ=−C1\n18k4τ3+/bracketleftbigg\n−ΩB\n4−3\n4LB\n(1−Rν)/bracketrightbigg\nk2τ\n+k/bracketleftbiggΩB\n72+LB\n24(1−Rν)/bracketrightbigg\n(kτ)3\nθb=θγ\nθc= 0\nθν=−(23+4Rν)\n18(15+4 Rν)C1k4τ3+/bracketleftbiggΩB(1−Rν)\n4Rν+3\n4LB\nRν/bracketrightbigg\nk2τ\n−/bracketleftbigg(1−Rν)(27+4Rν)ΩB\n72Rν(15+4Rν)+(27+4Rν)LB\n24Rν(15+4Rν)/bracketrightbigg\nk4τ3\nσν=4C1\n3(15+4Rν)(kτ)2−ΩB\n4Rν−3\n4LB\nRν\n+/bracketleftbigg(1−Rν)\nRν(15+4Rν)ΩB\n2+3\n2LB\nRν(15+4Rν)/bracketrightbigg\n(kτ)2,(14)\nwhereRν=ρν/(ρν+ργ) andC1is the constant which\ncharacterize the regular growing adiabatic mode as given\nin [13]. We have checked that the result reported in [11]\nand ours [15] agree.\nNote how the presence of a SB of PMFs induces a new\nindependent mode in matter and metric perturbations,\ni.e. the fully magnetic mode. This new independent\nmode is the particular solution of the inhomogeneous\nsystem of the Einstein-Botzmann differential equations:\nthe SB of PMF treated as a stiff source acts indeed as\na force term in the system of linear differential equa-\ntions. Whereas the sum of the fully magnetic mode with\nthe curvature one can be with any correlation as for an\nisocurvaturemode, thenatureofthefully magneticmode\n- and therefore its effect - is different: the isocurvature\nmodesaresolutionsofthehomogeneoussystem(inwhich\nall the species have both backgroundand perturbations),4\nwhereas the fully magnetic one is the solution of the in-\nhomogeneous system sourced by a fully inhomogeneous\ncomponent.\nIt is interesting to note the magnetic contribution\ndrops from the metric perturbation at leading order, al-\nthough is actually larger than the adiabatic solution for\nphotons, neutrinos and baryons (the latter being tightly\ncoupled to photons deep in the radiation era). This is\ndue to a compensation which nullifies the sum of the\nleading contributions (in the long-wavelength expansion)\nin the single species energy densities and therefore in the\nmetric perturbations. A similar compensation exists for\na network of topological defects, which does not carry\na background energy-momentum tensor as the PMF SB\nstudied here§.\nV. MAGNETIC FIELD POWER SPECTRA\nPower spectra for the amplitude and the EMT of\nSB of PMF have been subject of several investigation\n[4, 6, 8, 9]. We shall work in the Fourier space accord-\ning to Eq. (5). We shall consider PMFs with a power\nlaw power spectrum, which therefore are characterize by\ntwo parameters: an amplitude Aand a spectral index\nnB. PMFs are suppressed by radiation viscosity on small\nscales: we approximate this damping by introducing an\nultraviolet cut-off in the power spectrum at the (damp-\ning) scale kD.\nThe two-point correlation function for a statistically\nhomogeneous and isotropic field is\n∝angb∇acketleft/vectorB∗\ni(/vectork)/vectorBj(/vectork′)∝angb∇acket∇ight=δ3(/vectork−/vectork′)/bracketleftbigg\n(δij−ˆkiˆkj)PB(k)\n2+\nǫijlkl\nkPH(k)/bracketrightbigg\n,(15)whereǫijlis the totally antisymmetric tensor, PBand\nPHare the non-helical and helical part of the spectrum\nfor theBamplitude, respectively. Scalar cosmological\nperturbations only couple to the non-helical part of the\nspectrum and we shall therefore consider only PBin the\nfollowing.\nA. Magnetic Energy Density\nAs is clear from Eqs. (1-3), the EMT for PMF is\nquadratic in the field amplitude. The PMF energy den-\nsity spectrum is [9]:\n|ρB(k)|2=1\n128π2a8/integraldisplay\nd3pPB(/vector p)PB(|/vectork−/vector p|)(1+µ2),\n(16)\nwhereµ=/vector p(/vectork−/vector p)\np|/vectork−/vector p|=kcosθ−p√\n(k2+p2−2kpcosθ). As for the two-\npoint function in the coincidence limit, for several physi-\ncal spectral indexes and PMF configurations such convo-\nlution is not finite. There are in general problems both\non large and short scales. Since the spectrum of the com-\nponents of PMF EMT are relevant for the final impact\non cosmological perturbations and CMB anisotropies, it\nis better to address this point in much more detail with\nrespect to what is present in literature.\n§Note however that a network of topological defects does not s cale\nwith radiation and interacts only gravitationally with the rest ofmatter, i.e. a Lorentz term is absent.\nThe usual choice in the literature is to modify the scalar part two-po int function of Eq. (15) for zero helicity as [8]:\n∝angb∇acketleft/vectorB∗\ni(/vectork)/vectorBj(/vectork′)∝angb∇acket∇ight=/braceleftbigg\nδ3(/vectork−/vectork′)(δij−ˆkiˆkj)PB(k)\n2fork < kD\n0 for k > kD,\nwith\nPB(k) =A/parenleftbiggk\nk∗/parenrightbiggnB\n, (17)\nwherek∗is a reference scale. With such choice the two-\npoint function in the coincident limit (the mean square\nof the magnetic field) is:\n∝angb∇acketleftB2(x)∝angb∇acket∇ight=/integraldisplay\nk<kDd3kPB(k)\n=4πA\nnB+3knB+3\nD\nknB∗. (18)It is also usual in the literature to give the amplitude of\nBat a given smearing scale kSby imposing a Gaussian\nfilter :\n∝angb∇acketleftB2(x)∝angb∇acket∇ightkS=/integraldisplay\nd3kPB(k)e−k2/k2\nS\n= 2πAknB+3\nS\nknB∗Γ/parenleftbiggnB+3\n2/parenrightbigg\n.(19)5\n/Minus2/Minus1 0 1 2 3 40.00.51.01.52.0\n/Minus2/Minus1 0 1 2 3 40.00.51.01.52.0\nFIG. 1: Plots of the different ways of computing the magnetic\npower spectra(inunitsof4 πAknB+3\nD/knB∗)versusnBforkS=\nkD/2 (left) and kS=kD(right)./angbracketleftB2/angbracketright,/angbracketleftB2/angbracketrightkS,/angbracketleftB2/angbracketrightcut,kSare\nrepresented by solid, dotted and dashed lines, respectivel y.\nBy smearing the magnetic power spectrum and integrat-\ning fork < kD, one gets:\n∝angb∇acketleftB2(x)∝angb∇acket∇ightcut\nkS=/integraldisplay\nk<kDd3kPB(k)e−k2/k2\nS\n= 2πAknB+3\nD\nknB∗/bracketleftbigg\nΓ/parenleftbiggnB+3\n2/parenrightbigg\n−Γ/parenleftbiggnB+3\n2,k2\nD\nk2\nS/parenrightbigg/bracketrightbigg\n,(20)\nwhere the incomplete Gamma function Γ( ...,...) [17] has\nbeen introduced. Note how nB>−3 in order to prevent\ninfrared divergencies either in the mean square field or\nthe amplitude of the field smeared at a given scale. In\nthe following by ∝angb∇acketleftB2∝angb∇acket∇ightwe mean the value given by Eq.\n(18). Fig. (1) shows how ∝angb∇acketleftB2(x)∝angb∇acket∇ightkSmay be much larger\nthan∝angb∇acketleftB2∝angb∇acket∇ightfornB>0.\nThe exact result for the Fourier convolution leading to\nthe magnetic energy density Fourier square amplitude is\none of the new main results of this paper. The convolu-\ntion involves a double integral, one in the angle between\nkandpand one in the modulus of p. The integral in\nthe angle, often omitted in the literature, is the reason\nfor having the result for |ρB(k)|2non vanishing only for\nk <2kD. Thedetailedcalculationsfortheenergydensity\nconvolutionsaregiveninAppendixAforseveralvaluesof\nnB. The generic behaviour for k << k DandnB>−3/2\nis white noise with amplitude\n|ρB(k)|2≃A2k2n+3\nD\n16πk2n∗(3+2nB)(21)\nand then goes to zero for k= 2kD, which is a result\nobtained by performing correctly the integral. The pole\nfornB=−3/2 in Eq. (21) is replaced by a logarithmicdiveregence in kin the exact result; for nB<−3/2 the\nspectrum is no more white noise for k << k D. Fig. (2)\nshows the dependence of k3|ρB(k)|2onnBat fixed∝angb∇acketleftB2∝angb∇acket∇ight.\nOur result are different from the one reported in the\nliterature [8], which is\n|ρB(k)|2\nKR=3A2k2nB+3\nD\n64πk2nB∗(3+2nB)/bracketleftBigg\n1+nB\nnB+3/parenleftbiggk\nkD/parenrightbigg2nB+3/bracketrightBigg\n,\n(22)\n0.0 0.5 1.0 1.5 2.0012345\nFIG. 2: Plot of magnetic energy density power spectrum\nk3|ρB(k)|2in units of /angbracketleftB2/angbracketright2/(1024π3) versus k/kDfor dif-\nferentnBfor fixed /angbracketleftB2/angbracketright. The different lines are for nB=\n−3/2,−1,0,1,2,3,4 ranging from the solid to the longest\ndashed.\n0.0 0.5 1.0 1.5 2.005101520\nFIG. 3: Comparison of magnetic energy density convolution\nk3\nD|ρB(k)|2obtained in this paper (dotted, solid) in units\nof/angbracketleftB2/angbracketright2/(1024π3) and the one in Eq. (22) (dashed, long-\ndashed) versus k/kDfornB= 2,−3/2 with fixed /angbracketleftB2/angbracketright.\nand is not limited in k. In Fig. (3) we showthe difference\nbetween the literature result [8] and our result for nB=\n2,−3/2.\nB. Lorentz Force\nAs is clear from previous sections, we also need the Lorentz force\n|L(k)|2=1\n128π2a8/integraldisplay\nd3p PB(p)PB(|k−p|)[1+µ2+4γβ(γβ−µ)], (23)6\nand the magnetic anisotropic stress\n|σB(k)|2=1\n288π2a8/integraldisplay\nd3pPB(p)PB(|k−p|)[9(1−γ2)(1−β2)−6(1+γµβ−γ2−β2)(1+µ2)],(24)\nwhereγ=ˆk·ˆp,β=/vectork·(/vectork−/vector p)/(k|/vectork−/vector p|) andµ=/vector p·(/vectork−/vector p)/(p|/vectork−/vector p|).\n0.0 0.5 1.0 1.5 2.001234\nFIG. 4: Plot of the Lorentz force power spectrum k3|L(k)|2\nin units of /angbracketleftB2/angbracketright2/(1024��3) versus k/kDfor different nBfor\nfixed/angbracketleftB2/angbracketright. The different lines are for nB=−3/2,−1,0,1,2,3\nranging from the solid to the longest dashed.\nWe decide to compute the spectrum of the Lorentz\nforce and obtain the anisotropic stress by Eq. (7). The\nexact computation for the Lorentz force power spectrum\nis given in Appendix B for several values of nB. A\nterm−ρBcan be easily identified in Eq. (9); since we\nknow from the exact computation that the integral of\nPB(p)PB(|k−p|)(1 +µ2) is larger than the remaining\npiece in Eq. (23) we chose the the signs for ρB(k) and\nL(k) as opposite.\nFig. (4) shows the dependence of k3|L(k)|2onnBat\nfixed∝angb∇acketleftB2∝angb∇acket∇ight. Fig. (5) compares the approximation L(k)≃\n−ρB(k) suggested in Ref. [8] with the exact calculation.\nAs can be checked in Appendix B, our exact calculations\nfor the values of nBstudied here show that\n|L(k)|2≃11\n15|ρB(k)|2fork << k D.(25)\nVI. RESULTS FOR COSMOLOGICAL\nPERTURBATIONS\nIn order to study the effects of a SB of PMFs on CMB\nanisotropies and matter power spectrum we modified the\nCAMBEinstein-Boltzmanncode[18](June2006version)\nby introducing the PMF contribution in the Einstein\nequations, in the evolution equation for baryons and ini-\ntial conditions, along Eqs. (6,26,14).\nWe note that implementing the baryons evolution as\nfromEq. (26), theMHDapproximationinagloballyneu-\ntralplasmais used up to the presenttime: this makesthe\nLorentz term non-vanishing up to the present time. Al-\nthoughthe term L/ρbin Eq. (26) decreaseswith time, its0.0 0.5 1.0 1.5 2.00123\nFIG. 5: Comparison of the magnetic energy density and\nLorentz force power spectra versus k/kDfor fixed /angbracketleftB2/angbracketright. The\nsolid (medium dashed) and long-dashed (short-dashed) line s\nare respectively for k3|ρB(k)|2andk3|L(k)|2fornB= 2\n(nB=−3/2).\neffect on the baryon velocity is crucial. Much later than\nthe larger between the decoupling time and the time at\nwhich sound speed of baryons is effectively zero, baryons\nvelocity can be approximated as:\nθlate\nb≃ −k2/parenleftbiggLa\nρb/parenrightbiggτ\na(26)\nduringthematterdominatedera. OurmodifiedEinstein-\nBoltzmann code reproduces correctly this asymptotic\nregime for different wavelengths, as can be seen by Fig.\n(6). The corresponding effects on the density contrasts\nfor the same wavelengths are shown in Fig. (7). In Fig.\n(8) the effects due to the pure magnetic mode and due\nto the correlation with the adiabatic mode are shown.\nFig. (9) displays the importance of the Lorentz term\ncompared to the purely gravitational effect.\nVII. RESULTS FOR CMB TEMPERATURE\nAND POLARIZATION POWER SPECTRA\nIn this section we show the results on the CMB tem-\nperature and polarization pattern obtained by our mod-\nifications of the CAMB code. Fig. (10) shows the var-\nious contributions to the total CMB temperature and\npolarization angular power spectra from the pure mag-\nnetic mode and its correlation with the adiabatic mode.\nFig. (11) shows the dependence of the total temperature\npower spectrum on the spectral index nB.\nAs is clear from the previous section, the Lorentz force\nof a fully correlated magnetic contribution decreases the7\nFIG.6: Evolution ofbaryonsvelocity for4differentwavenum -\nbers with (dashed) and without (solid) PMF. k2L/ρb(dot-\ndashed line) and the solution θlate\nb(dotted line) are also plot-\nted: note how the numerics agree with θlate\nbat late times.\nThe cosmological parameters of the flat Λ CDMmodel are\nΩbh2= 0.022, Ω ch2= 0.123,τ= 0.04,ns= 1,H0=\n72kms−1Mpc−1.\nFIG. 7: Evolution of baryons (dotted), CDM (solid) and pho-\ntons (dashed) density contrast for 4 different wavenumbers\nwith fully correlated (blue) and without (black) PMF. The\ncosmological parameters are the same of Fig. (6).\ndensity contrasts and therefore the CMB APS in an in-\ntermediate range of multipoles before high ℓincrease.\nVIII. RESULTS FOR THE MATTER POWER\nSPECTRA\nIn Fig. (13) we present the results for the linear CDM\npower spectrum evaluated at present time in presence\nof SB of PMF. By analyzing Fourier spectra we have\nchecked that the adiabatic results are recovered for k >FIG. 8: Time evolution of baryons (left) and CDM (right)\ndensity contrasts with vanishing PMF (solid), fully correl ated\n(dashed), fully anti-correlated (dotted) and purely magne tic\ninitial conditions (dot-dashed). The other cosmological p a-\nrameters are the same as Fig. (6).\nFIG. 9: Time evolution of baryons (dashed) and CDM (solid)\ndensity contrasts for purely adiabatic with vanishing PMF\n(black), fully correlated (left panel) and anti-correlate d (right\npanel) PMF with vanishing (blue) and non vanishing (red)\nLorentz force for k= 1Mpc−1. These figures show clearly\nthat the Lorentz force and the gravitational contribution a re\nof opposite sign, and the Lorentz term is more important.\nThe cosmological parameters are the same of Fig. (6).\n2kD. We compare the results obtained by neglecting or\nby taking into account the Lorentz term. By considering\nthe equations evolved and the previous figures, it is clear\nhow the Lorentz term treated as in Eq. (26) is a leading\ncontribution for baryons which gives rise to a long-time\neffectasshowinFig. 6. ThroughgravityCDMisaffected\nas shown in Figs. (7-9) and therefore a large feature is\npresent in the linear CDM matter PS.\nIX. CONCLUSIONS\nWe have investigated the impact of a SB of PMF on\nscalarcosmologicalperturbationsanditsimpactonCMB\nanisotropies and matter power spectrum. The effects\non the CMB angular power spectrum is one of the dis-\ntinctive features of stochastic PMF together with non-\ngaussianities and Faraday rotation [19]: future missions\nasPlanck [20] will greatly improve the present con-\nstraints [10, 16].\nWehaveanalyzedtheSBofPMFintheone-fluidMHD\napproximation [2] as a source for cosmological pertur-\nbations and we have inserted such modifications in the\nCAMB code [18]. Our numerical code improves previ-\nous studies [10, 11] for the treatment of initial conditions8\nFIG. 10: CMB temperature angular power spectra obtained\nwithp\n/angbracketleftB2/angbracketright= 3×10−7Gauss,nB=−1,kD=πin compar-\nison with the adiabatic spectrum with vanishing PMF (solid\nline): TT, EE, TE are displayed in the top, middle, bottom\npanel, respectively. The purely magnetic, correlation, fu lly\ncorrelated, fully anti-correlated and uncorrelated spect ra are\nrepresented as triple dotted - dashed, dashed, dotted, dot -\ndashed and long dashed lines, respectively. The other cosmo -\nlogical parameters are the same as Fig. (6).FIG. 11: In the top panel variation of the CMB temper-\nature angular power spectrum with nBin comparison to\nthe case with vanishing PMF (solid line). In the top figurep\n/angbracketleftB2/angbracketright= 8×10−7Gauss,kD= 2πand fully correlated ini-\ntial conditions are considered. The spectral indexes plott ed\narenB=−3/2,−1,1,2 (dotted, dot-dashed, dashed, long-\ndashed lines, respectively). In the bottom panel, variatio n of\nthe CMB angular power spectrum with kDin comparison to\nthe case with vanishing PMF (solid line). In the bottom fig-\nurep\n/angbracketleftB2/angbracketright= 3×10−7Gauss,nB=−1 andkD= 2π,π,π/2\n(dotted, dot-dashed, dashed, respectively). In both panel s\nthe initial conditions are fully correlated and the other co s-\nmological parameters are the same as Fig. (6).\nand exact convolutions for the PMF energy-momentum\ntensor. Note that the present constraints [10, 16] used\nneither the correct initial conditions nor the correct con-\nvolutions for the PMF energy density and Lorentz force\npower spectra. Ref. [11] uses the correct initial condi-\ntions, but a power spectrum for the PMF energy -density\nwith a spectral index which is twice the one for the power\nspectrum of the magnetic field. We have shown exten-\nsively in Sect. VI and Appendix A,B that this is not the\ncase.\nWe have also shown how the Lorentz term for baryons9\nFIG. 12: In the left panel CMB temperature power spectrum\nobtained with fully correlated PMF with (dashed line) and\nwithout (dotted line) Lorentz term in comparison with the\nvanishing PMF (solid line). As is clear from the previous\nsection, the Lorentz force of a fully correlated magnetic co n-\ntribution decreases the density contrasts and therefore th ere\nis range in which the CMB TT APS is decreased respect to\nthe adiabatic case. In the right panel the same figure with\nuncorrelated spectra. In the figuresp\n/angbracketleftB2/angbracketright= 3×10−7Gauss,\nkD= 2πandnB= 2 are considered. The other cosmological\nparameters are the same as Fig. (6).\nFIG. 13: Linear cold dark matter power spectrum obtained\nwith fully correlated PMF with (dashed line) and without\n(dotted line) Lorentz term, with uncorrelated PMF and the\nLorentzforce (dot-dashedline)incomparison with thevani sh-\ning PMF (solid line). In the figurep\n/angbracketleftB2/angbracketright= 3×10−8Gauss,\nkD= 2πandnB= 2 are considered. The other cosmological\nparameters are the same as Fig. (6).\nin the one-fluidplasmadescription [2] may leadto along-\ntime effect which we have described analytically in Eq.\n(26). This last point deserves further investigation.Acknowledgments\nWe are grateful to Chiara Caprini, Ruth Durrer and\nJose Alberto Rubino-Martin for conversations and dis-\ncussions on magnetic fields. This work has been done in\nthe framework of the Planck LFI activities and is par-\ntially supported by ASI contract Planck LFI Activity of\nPhase E2. We thank INFN IS PD51 for partial support.\nF. F. is partially supported by INFN IS BO 11.\nX. NOTE ADDED\nWhile this paper was about to be completed, an arti-\ncle [21] which computes numerically the convolution in-\ntegralsby taking into account the angularpart appeared.\nRef. [21] does not display the dependence on kof these\nconvolution integrals and therefore we cannot compare\nour analytical expressions with their results.\nAPPENDIX A: ENERGY DENSITY\nThe convolution which gives the magnetic energy den-\nsity spectrum is given in Eq. (16) with the parametriza-\ntion for the PS for the magnetic field given in Eq. (17).\nLet us compute the energy density of magnetic field ap-\nplying the sharp cut-off to both spectra in the convolu-\ntion according to Eq. (VA). In this case we have two\nconditions of existence to take into account:\np < kD, |/vectork−/vector p|< kD(A1)\nThis second condition poses a k-dependence on the an-\ngular integration domain and, together with the first\none, allows the energy power spectrum to be defined for\n0< k <2kD. For simplicity of notation we normalize\nthe Fourier wavenumber to kDand perform the integra-\ntion with this choice. The double integral (over γand\noverp) we have to compute must therefore be splitted in\nthree parts depending on the γandplower and upper\nk-depending bounds. This splitting is well displayed in\n(k,p) plane as showed in Fig.(14): in region athe angu-\nlar integration has to be done between −1 and 1, while\nwithinbandcregions between ( k2+p2−1)/2kpand 1.10\nA sketch of the integration is thus the following:\n1) 0< k <1\n/integraldisplay1−k\n0dp/integraldisplay1\n−1dγ···+/integraldisplay1\n1−kdp/integraldisplay1\nk2+p2−1\n2kpdγ··· ≡/integraldisplay1−k\n0dpIa(p,k)+/integraldisplay1\n1−kdpIb(p,k)\n2) 1< k <2\n/integraldisplay1\nk−1dp/integraldisplay1\nk2+p2−1\n2kpdγ··· ≡/integraldisplay1\nk−1dpIc(p,k) (A2)\nThe angular integrals can be performed as:\nIa=/integraldisplay1\n−1pn+2/bracketleftBigg\n2−k2(1−γ2)\nk2+p2−2kpγ/bracketrightBigg\n(k2+p2−2kpγ)n/2dγ\n=2pn−1\nkn(2+n)(4+n)/bracketleftBig\n(k+p)n+2/parenleftBig\nk2−k(2+n)p+(1+4n+n2)p2/parenrightBig\n−|k−p|n+2/parenleftBig\nk2+k(2+n)p+(1+4n+n2)p2/parenrightBig/bracketrightBig\n, (A3)\nIb=Ic=/integraldisplay1\nk2+p2−1\n2kppn+2/bracketleftBigg\n2−k2(1−γ2)\nk2+p2−2kpγ/bracketrightBigg\n(k2+p2−2kpγ)n/2dγ\n=pn−1\n4kn(2+n)(4+n)/bracketleftBig\n8k4+2n−8k2n+6k4n+n2−2k2n2+k4n2\n−16k2p2+24np2−12k2np2+6n2p2−2k2n2p2+8p4+6np4+n2p4\n−8|k−p|n+2/parenleftbig\nk2+k(2+n)p+(1+4n+n2)p2/parenrightBig/bracketrightBig\n(A4)\n0.20.40.60.8 11.21.4p\n-0.50.511.522.5k\nabc\nFIG. 14: Integration domains in ( k,p) plane\nNote that the divergent terms at the denominator n\nandn+2 simply means that the above formulae are not\napplicable for n= 0 and n=−2 (logarithmic terms\nappear in these cases).\nA special care must be taken in the radial integral.In\nparticular the presence of the term |k−p|n+2in both\nintegrands, when considering odd spectral indexes,\nmakes necessary to divide the two cases: p < k and\np > k.This leads to a further division of the integration\ndomain.The scheme of the radial integration is then:\n/integraldisplay(1−k)\n0dp→\n\nk <1/2/braceleftBigg/integraltextk\n0dp... withp < k/integraltext(1−k)\nkdp...withp > k\nk >1/2/integraltext(1−k)\n0dp...withp < k/integraldisplay1\n(1−k)dp→\n\nk <1/2/integraltext1\n(1−k)dp... withp > k\nk >1/2/braceleftBigg/integraltextk\n(1−k)dp...withp < k/integraltext1\nkdp... withp > k\n/integraldisplay1\n(k−1)dp→/braceleftBig\n1< k <2/integraltext1\n(k−1)dp...withp < k\nIt is important to study some relevant behaviour of the\nintegrands in p. Forp∼0:\nIa∼8\n3knpn+2, (A5)11\nForp∼kthe above integrands behave as\nIa∼2kn−2\nn(n+2)(n+4)/bracketleftbigg\n2n+2kn+4n(n+3)−/parenleftbig\n(k−p)2/parenrightbign+2\n2(n+1)(n+4)k2/bracketrightbigg\n, (A6)\nIb∼kn−2\n4n(n+2)(n+4)/bracketleftbigg\nn/parenleftbig\n4(n+4)k2+n+2/parenrightbig\n−8/parenleftbig\n(k−p)2/parenrightbign+2\n2(n+1)(n+4)k2/bracketrightbigg\n. (A7)\nForIc,p∼kcannot be obtained since 1 < k <2. It is important to stress that for n >−3 the divergences in p∼0\nandp∼kare integrable. The coefficients of both leading terms are proportio nal tokn.\nFollowing the scheme (A2) we can perform the integration over p. Our exact results are given for particular values\nofnB.\n1.nB= 4\n|ρB(k)|2\nnB=4=A2k11\nD\n64πk8∗/bracketleftBigg\n4\n11−˜k+4\n3˜k2−˜k3+8\n21˜k4−˜k5\n24−˜k7\n192+˜k11\n9856/bracketrightBigg\n(A8)\n2.nB= 3\n|ρB(k)|2\nnB=3=A2k9\nD\n64πk6∗/braceleftBigg\n4\n9−˜k+20\n21˜k2−5\n12˜k3+4\n75˜k4+4\n315˜k6−˜k9\n525for 0≤˜k≤1\n(2−˜k)2264−436˜k+863˜k2−528˜k3+48˜k5+48˜k6+16˜k7+4˜k8\n6300kfor 1≤˜k≤2\n3.nB= 2\n|ρB(k)|2\nnB=2=A2k7\nD\n64πk4∗/bracketleftBigg\n4\n7−˜k+8\n15˜k2−˜k5\n24+11\n2240˜k7/bracketrightBigg\n(A9)\n4.nB= 1\n|ρB(k)|2\nnB=1=A2k5\nD\n64πk2∗/braceleftBigg\n4\n5−˜k+1\n4˜k3+4\n15˜k4−1\n5˜k5for 0≤˜k≤1\n(2−˜k)28−4˜k−˜k2+4˜k4\n60kfor 1≤˜k≤2\n5.nB= 0\n|ρB(k)|2\nnB=0=A2k3\nD\n64π\n\n1\n96˜k/bracketleftBig\n˜k(116−102˜k−84˜k2+˜k3(53+4π2))+\n12log(1−˜k)(−1+4˜k2−3˜k4+4˜k4log˜k)−\n48˜k4PolyLog[2 ,−1+˜k\n˜k]/bracketrightBig\nfor 0≤˜k≤1\n1\n96˜k/bracketleftBig\n116˜k−102˜k2−84˜k3+53˜k4+\nlog[−1+˜k](−12+48˜k2−36˜k4+24˜k4log˜k)+\n24˜k4PolyLog[2 ,1\n˜k]−24˜k4PolyLog[2 ,−1+˜k\n˜k]/bracketrightBig\nfor 1≤˜k≤212\n6.nB=−1\n|ρB(k)|2\nnB=−1=A2kDk2\n∗\n64π/braceleftBigg\n4−5˜k+4˜k2\n3+˜k3\n4for 0≤˜k≤1\n((−2+˜k)2(8−4˜k+3˜k2))\n12˜kfor 1≤˜k≤2\n7.nB=−3/2\n|ρB(k)|2\nn=−3/2=A2k3\n∗\n64π\n\n1\n45/bracketleftbigg\n8(−33+29˜k−4˜k2+8˜k3)√\n1−˜k˜k+264\n˜k+60˜k+5˜k3\n−90π+360log[1+√\n1−k]−180log˜k/bracketrightBig\nfor 0≤˜k≤1\n1\n45/bracketleftbigg\n−(8(−33+29˜k−4˜k2+8˜k3)√\n−1+˜k˜k+264\n˜k+60k+5k3\n−180arctan[1√\n−1+˜k]+180arctan[/radicalbig\n−1+˜k]/bracketrightbigg\nfor 1≤˜k≤2\nAPPENDIX B: LORENTZ FORCE\nIn order to obtain the complete estimate of the contri-\nbution of PMFs to the perturbation evolution is neces-\nsary to solve the convolution for the Lorentz Force powerspectrum. Theanisotropicstresscanbeobtaineddirectly\nfrom its relation with the Lorentz force and the magnetic\nenergy density. The convolution which gives the Lorentz\nforce is given in Eq. (23) with the parametrization for\nthe PS for the magnetic field given in Eq. (17).\n1.nB= 4\n|L(k)|2\nnB=4=A2k11\nD\n64πk8∗/braceleftBigg\n4\n15−16˜k\n15+317˜k2\n135−10˜k3\n3+45˜k4\n14−1481˜k5\n720+4˜k6\n5−977˜k7\n6720+1357˜k11\n2661120for 0≤˜k≤1\n248\n1155−2˜k\n3+137˜k2\n135−5˜k3\n6+5˜k4\n14−41˜k5\n720−17˜k7\n6720+41˜k11\n532224for 1≤˜k≤2\n2.nB= 3\n|L(k)|2\nnB=3=A2k9\nD\n64πk6∗\n\n44\n135−7˜k\n6+547˜k2\n245−8\n3˜k3+3293˜k4\n1575−˜k5+2363˜k6\n10395−131˜k9\n33075for 0≤˜k≤1\n1\n727650/parenleftBig\n−344960+1920\n˜k5−12320\n˜k3+133056\n˜k+582120 ˜k−585090˜k2+323400 ˜k3+\n−66066˜k4−3710˜k6+319˜k9/parenrightBig\nfor 1≤˜k≤2\n3.nB= 2\n|L(k)|2\nnB=2=A2k7\nD\n64πk4∗/braceleftBigg\n44\n105−4˜k\n3+11˜k2\n5−13˜k3\n6+4˜k4\n3−33˜k5\n80+7˜k7\n320for 0≤˜k≤1\n32\n105−2˜k\n3+3˜k2\n5−˜k3\n6−˜k5\n80+19˜k7\n6720for 1≤˜k≤213\n4.nB= 1\n|L(k)|2\nn=1=A2k5\nD\n64πk2∗/braceleftBigg\n44\n75−5˜k\n3+761˜k2\n315−2˜k3+659˜k4\n630−41˜k5\n150for 0≤˜k≤1\n128−480k2+2240k4−4368k5+4200k6−1310k7−145k9+77k10\n3150k5 for 1≤˜k≤2\n5.nB= 0\n|L(k)|2\nnB=0=A2k3\nD\n64π\n\n1\n1152˜k5/bracketleftBig\n12(−1+˜k2)3(3+˜k2)log[1−˜k]+\n12(−3+8˜k2−30˜k4+64˜k5−48˜k6+9˜k8)log[1−˜k]+\n˜k(−72−36˜k+168˜k2+78˜k3+744˜k4−2484˜k5+4728˜k6−2869˜k7+1536˜k7log˜k)/bracketrightBig\nfor 0≤˜k≤1\n1\n1152k5[(2−k)k(−36+k(−36+k(66+k(72+k(152+k(−14+53k))))))+\n24(−3+k2(8+k2(−18+k(32−24k+5k3))))log(−1+˜k)/bracketrightBig\nfor 1≤˜k≤2\n6.nB=−1\n|L(k)|2\nnB=−1=A2kDk2\n∗\n64π/braceleftBigg\n44\n15−83˜k2\n35+4˜klog˜k for 0≤˜k≤1\n−64+˜k2(112+˜k3(112+˜k(−140+39 ˜k)))\n105˜k5 ; for 1 ≤˜k≤2\n7.nB=−3/2\n|L(k)|2\nnB=−3/2=A2k3\n∗\n64π\n\n8\n9−2048\n2925˜k5+128\n135˜k3+8\n5˜k−4˜k\n3−2π\n15+88\n15log[1+/radicalbig\n1−˜k]\n−44log˜k\n15−8(−768+384 ˜k+1136˜k2−472˜k3+1655˜k4+5455˜k5−10160˜k6+2770˜k7)\n8775√\n1−˜k˜k5for 0≤˜k≤1\n4\n8775√\n−1+˜k˜k5/bracketleftBig\n−1536−1536/radicalbig\n−1+˜k+768˜k+2272˜k2+\n2080/radicalbig\n−1+˜k˜k2−944˜k3+3310˜k4+3510/radicalbig\n−1+˜k˜k4−4690˜k5+\n1950/radicalbig\n−1+˜k˜k5−820˜k6−2925/radicalbig\n−1+˜k˜k6+1640˜k7\n−585/radicalbig\n−1+˜k˜k5arctan[1√\n−1+˜k]+585/radicalbig\n−1+˜k˜k5arctan[/radicalbig\n−1+˜k]/bracketrightbigg\nfor 1≤˜k≤2\n[1] D. 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D 77(2008) 043005"    },    {        "title": "1309.0629v1.Hidden_Lorentz_symmetry_of_the_Horava___Lifshitz_gravity.pdf",        "content": "arXiv:1309.0629v1  [gr-qc]  3 Sep 2013Hidden Lorentz symmetry of the Hoˇ rava - Lifshitz gravity\nJ. Rembieli´ nski∗\nDepartment of Theoretical Physics, University of Lodz, Pom orska 149/153, 90-236 /suppress L´ od´ z, Poland\n(Dated: October 9, 2018)\nIn this letter it is shown that the Hoˇ rava-Lifshitz gravity theory admits Lorentz symmetry pre-\nserving preferred global time foliation of the spacetime.\nPACS numbers: 04.60.Bc, 04.50.Kd\nThe possibility that gravity may exhibit a preferred\nfoliation at its most fundamental level has attracted a lot\nof attention recently, mainly due to the Hoˇ rava’s papers\n[1–3] devoted to gravity models characterized by certain\nspecific anisotropic scaling between space and time. The\nleading idea of the Hoˇ rava approach to the quantization\nof gravity is to achieve power-counting renormalizability\nby modifying the graviton propagator. This is obtained\nby adding to the action terms containing higher order\nspatial derivatives of the metric which, in turn, naturally\nleads to the preferred co-dimension one foliation Fof\nspace-time manifold Mtopologically equivalent [1–3] to\nR1×Σ. The resulting theory, known as the Hoˇ rava-\nLifshitz (HL) gravity, is then invariant under a group of\ndiffeomorphisms Diff(F,M) preserving this foliation\n˜t=˜t(t),˜xi= ˜xi(t,xi) (1)\nwherei= 1,2,...,D . The above mentioned anisotropic\nscaling characterizing HL gravity is of the form\nt−→bzt,x−→bx. (2)\nThus the (momentum) dimension [ t] =−z, [xi] =−1, so\nthe light velocity chas the dimension [ c] =z−1. When z\nequals the number of spatial dimensions Dthe theory be-\ncomes power-counting renormalizable provided all terms\nallowed are compatible with the gauge symmetries in the\naction.\nThe HL theory is naturally described by the ADM\ndecomposition [4] of the relativistic metric, namely by\nthe lapse function N([N] = 0), the shift vector Ni\n([Ni] = [Ni] =z−1) and the metrics γij([γij] = 0) on the\nspacial slices Σ. In the HL gravity the lapse N=N(t)\nis only a function of time twhich is constant along Σ\nwhereas the shift vector Nidepends on the spacetime\npoint (t,x). In terms of the ADM variables the metrics\ncan be written as\nds2=gµνdxµdxν\n=−c2N2dt2+γij( dxi+Nidt)( dxj+Njdt) (3)\nThe HL action, respecting the symmetries Diff(F,M)\nis [1–3]\nS=2\nκ2/integraldisplay\ndtdDx√γN/bracketleftbig/parenleftbig\nKijKij−λK2/parenrightbig\n−V/bracketrightbig\n,(4)whereK=Ki\ni,λis a dimensionless coupling constant\nand\nKij=1\n2N(∂tγij−∇iNj−∇jNi) (5)\nis the extrinsic curvature of the leaves hypersurface Σ. A\nscalar potential function Vis built out of the spatial met-\nrics, the spatial Riemann tensor and its covariant spatial\nderivatives but is independent of the time derivatives of\nfields. For a review and extensions of the Hoˇ rava’s ap-\nproach see [5–8]. In the following we restrict ourselves to\nthe physically important z=D= 3 case.\nOne of the problems of the Hoˇ rava-Lifshitz gravity\nis that this theory does not exhibit Lorentz symmetry.\nA proposed way out of this situation is an appropriate\npreparation of the potential to restore dynamically local\nLorentz invariance in the low-energy limit [1–3]. How-\never, for each finite energy scale the Lorentz symmetry\nis in fact broken. In this letter we suggest a way to over-\ncome the difficulty with the Lorentz symmetry in the\nHoˇ rava-Lifshitz gravity in a physically acceptable way.\nTo do this let us consider a coordinate independent so-\nlution to the model defined by the action (4) where the\npotential Vis chosen as in Ref. [1] with the cosmological\nconstant equal to zero. Namely, let us choose the shift\nvectorNas\nN=−cǫ\n1−ǫ2(6)\nwith 0≤ǫ2<1, while the lapse Nis given by\nN=1√\n1−ǫ2. (7)\nFurthermore, the space metrics is chosen as\nγ= (I−ǫ⊗ǫT), (8)\nwhere T denotes transposition of the coordinate indepen-\ndent dimensionless column vector ǫ= (ǫa),a= 1,2,3.\nWith help of the classical equations of motion [9] it can\nbe verified that the equations (6-8) define the flat solu-\ntion to the HL theory determined by (4). The spacetime\nmetrics (3) takes the form\nds2=ζαβdxαdxβ\n=−c2dt2−2cǫ·dxdt+ dxT(I−ǫ⊗ǫT) dx(9)2\nwith the metric tensor\nζαβ=/parenleftbigg−1−ǫT\n−ǫI−ǫ⊗ǫT/parenrightbigg\n. (10)\nHereα,β = 0,1,2,3.It is easy to see that the metrics\nform (9) is related to the Minkowski spacetime as well\nas the space geometry is Euclidean. Now, let us consider\nthe standard rotations\nt′=t,x′=Rx,ǫ′=Rǫ, (11)\nwhereRbelongs to the group of orthogonal matrices, and\nthe transfromations defined by\nt′=t\na+a·ǫ, (12a)\nx′=/parenleftbigg\nI+a⊗ǫT+a⊗aT\n1 +a/parenrightbigg\nx+act, (12b)\nǫ′=1\na+a·ǫ/bracketleftbigg\nǫ+a/parenleftbigg\n1 +a·ǫ\n1 +a/parenrightbigg/bracketrightbigg\n, (12c)\nwhereaparametrizes the standard Lorentz boost L(a)\nL(a) =/parenleftBigg\naaT\naI+a⊗aT\n1+a/parenrightBigg\n, (13)\nwitha=√\n1 +a2. It can be shown that the transfor-\nmations (11-12c) taken together form the realization of\nthe Lorentz group and it is obvious that they do not\ndestroy the foliation F. Consequently, they do not af-\nfect absolute simultaneity and causality characteristic to\nthe Hoˇ rava-Lifshitz model. Moreover, the metrics (9) is\nform invariant under the transformations (11-12c). We\npoint out that in view of (11-12c) the above transforma-\ntions form a nonlinear realization of the Lorentz group\n[10, 11]. Nonlinearity affects the coordinate independent\nvectorǫonly, whereas xandttransform linearly. The\nnonlinear realization (11-12c) was firstly introduced in a\ndifferent context and form in [12] and was applied to the\nlocalization problem in the relativistic quantum mechan-\nics [13, 14] as well as to a Lorentz-covariant formulation of\nthe statistical physics [15]. There is a simple relationship\nbetween the standard Lorentz transformations and those\ngiven by (11-12c). Indeed, introducing the new time co-\nordinate by the affine transformation (not belonging to\ntheDiff(F,M))\ntE=t+ǫ·x\nc(14)\nwe arrive at the standard Minkowski form of the met-\nrics (9). Moreover, we can easily recover for xandtE\nthe standard Lorentz transformations in the standard\npseudoorthogonal frame. Thus the time redefinition (14)\nshould be interpreted as the change of distant clock syn-\nchronization [16–20]. Consequently, the vector ǫplaysthe role of the Reichenbach synchronization coefficient\n[16, 21].\nNow, it is not difficult to apply the above Lorentz co-\nvariant flat solution as the local reference frame in a gen-\neral case. This can be done by introducing the tetrad\nfieldsωα=λα\nµdxµsatisfying\nζαβωαωβ=gµνdxµdxν, (15)\nwithζαβandgµνgiven by (9) and (3) respectively. The\nsolution has the form\nω0= (cN−ǫaea\niNi) dt−ǫaea\nidxi, (16a)\nωa=ea\ni( dxi+Nidt), (16b)\nwhere the triads ea\nidetermine the space metrics ea\niea\nj=\nγij. The tetrads ωαtransform with respect to the in-\ndexαaccording to the law (11-12c) treated as the frame\ntransformations. Notice, that in general the synchro-\nnization vector ǫis frame dependent because it trans-\nforms from frame to frame according to the formula (12c).\nIn particular, we can specify the boost parameter ato\nobtain the synchronization vector ǫequal to zero in a\ndistinguished frame. In this peculiar frame related to\nthe preferred foliation Fthe Einstein synchronization\nconvention applies. It can be shown [13, 14] that the\nsynchronization vector ǫcan be related to the velocity\nof the preferred frame. Finally, let us stress that the\nsynchronization change (14) does not affect the physi-\ncal content of theory on the classical level because of the\nconventionality of the synchronization procedure [16–21].\nHowever, it breaks the quantization procedure essential\nto the Hoˇ rava approach. This can indicate that result\nof quantization depends on the adapted synchronization\nscheme. Concluding, the Hoˇ rava-Lifshitz gravity admits\nLorentz symmetry preserving preferred global time foli-\nation of the spacetime. This symmetry can be related to\nthe standard Lorentz transformations by the frame de-\npendent change of synchronization (14) to the Einstein\none. However, (14) breaks the preferred foliation of the\nHL gravity. Thus the HL theory forces Lorentz symme-\ntry realized in the synchronization scheme related to the\ntransformation laws (11-12c). Our observation can be\nalso applied to the causal dynamical triangulation the-\nory [22], where the global time foliation is assumed too\n(however see [23]).\nThe author is grateful to Bogus/suppress law Broda and\nKrzysztof Kowalski for discussion and to Jerzy Ju-\nrkiewicz for helpful remarks concerning the causal dy-\nnamical triangulation theory.\n∗jaremb@uni.lodz.pl\n[1] P. Hoˇ rava, Phys. Rev. D 79, 084008 (2009).\n[2] P. Hoˇ rava, Phys. Rev. Lett. 102, 161301 (2009).3\n[3] P. Hoˇ rava and C. M. Melby-Thompson, Phys. Rev. D 82,\n064027 (2010).\n[4] R. Arnowitt, S. Deser, and C. W. Misner, in Gravitation:\nAn Introduction to Current Research , editedbyL. Witten\n(Wiley, 1962).\n[5] D. Blas, O. Pujolas, and S. Sibiryakov, JHEP 0910, 029\n(2009).\n[6] D. Blas, O. Pujolas, and S. Sibiryakov, Phys. Rev. Lett.\n104, 181302 (2010).\n[7] D. Blas, O. Pujolas, and S. Sibiryakov, JHEP 04, 018\n(2011).\n[8] T. P. Sotiriou, M.Visser, and S.Weinfurtner, JHEP\n0910, 033 (2009).\n[9] E.KiritsisandG.Kofinas,Nucl.Phys.B 821,467(2009).\n[10] S. Coleman, J. Wess, and B. Zumino, Phys. Rev. 177,\n2239 (1969).\n[11] A. Salam and J. Strathdee, Phys. Rev. 184, 1750 (1969).\n[12] J. Rembieli´ nski, Phys. Lett. 78A, 33 (1980).[13] P. Caban and J. Rembieli´ nski, Phys. Rev. A. 59, 4187\n(1999).\n[14] J. Rembieli´ nski and K. A. Smoli´ nski, Phys. Rev. A. 66,\n052114 (2002).\n[15] K.Kowalski, J. Rembieli´ nski, andK.A.Smoli´ nski, Ph ys.\nRev. D. 76, 045018 (2007).\n[16] H. Reichenbach, Axiomatisation of the Theory of Rela-\ntivity(University of California Press, Berkeley, 1969).\n[17] P. Havas, Gen. Rel. Grav. 19, 435 (1987).\n[18] Y. Z. Zhang, Gen. Rel. Grav. 27, 475 (1994).\n[19] R. Anderson, I. Vetharaniam, and G. E. Stedman, Phys.\nRep.295, 93 (1998).\n[20] C. Lammerzahl, Ann. Phys. 14, 71 (2005).\n[21] M. Jammer, Concepts of Simultaneity: From Antiquity\nto Einstein and beyond (Johns Hopkins University Press,\nBerkeley, 2006).\n[22] J. Ambjorn, A. Gorlich, S. Jordan, J. Jurkiewicz, and\nR. Loll, Phys. Rev. B 690, 413 (2010).\n[23] S. Jordan and R. Loll, arXiv:1307.5469 (2013)."    },    {        "title": "1602.07325v1.Experimental_Investigation_of_Temperature_Dependent_Gilbert_Damping_in_Permalloy_Thin_Films.pdf",        "content": "1    Experimental Investigation of Temperature-Dependent Gilbert \nDamping in Permalloy Thin Films  \nYuelei Zhao1,2†, Qi Song1,2†, See-Hun Yang3, Tang Su1,2, Wei Yuan1,2, Stuart S. P. Parkin3,4, Jing \nShi5*, and Wei Han1,2*   \n1International Center for Quantum Materials,  Peking University, Beijing, 100871, P. R. China \n2Collaborative Innovation Center of Quantum Matter, Beijing 100871, P. R. China \n3IBM Almaden Research Center, San Jose, California 95120, USA \n4Max Planck Institute for Microstructu re Physics, 06120 Halle (Saale), Germany \n5Department of Physics and Astronomy, Univers ity of California, Riverside, California 92521, \nUSA \n†These authors contributed equally to the work \n*Correspondence to be addressed to: jing.shi @ucr.edu (J.S.) and weihan@pku.edu.cn (W.H.) \n \n \nAbstract \nThe Gilbert damping of ferromagnetic materials is arguably the most important but least \nunderstood phenomenological parameter that dictates real-time magnetization dynamics. \nUnderstanding the physical origin of  the Gilbert damping is highly relevant to developing future \nfast switching spintronics devices such as magnetic sensors and magnetic random access memory. Here, we report an experimental stud y of temperature-dependent Gilbert damping in \npermalloy (Py) thin films of varying thicknesses  by ferromagnetic resonance. From the thickness \ndependence, two independent cont ributions to the Gilbert damping are identified, namely bulk \ndamping and surface damping. Of particular inte rest, bulk damping decreases monotonically as \nthe temperature decreases, while surface da mping shows an enhancement peak at the 2 temperature of ~50 K. These results provide an important insight to the physical origin of the \nGilbert damping in ultr athin magnetic films.  \n \nIntroduction \nIt is well known that the magnetization dynamics  is described by the Landau-Lifshitz-Gilbert \nequation with a phenomenological parameter called the Gilbert damping ( α),1,2:   \n eff\nSdM dMMH Mdt M dtαγ=− × + × \n   (1) \nwhere M\nis the magnetization vector,  γis the gyromagne tic ratio, and SM M=\n is the saturation \nmagnetization. Despite intense theore tical and experimental efforts3-15, the microscopic origin of \nthe damping in ferromagnetic (FM) metallic ma terials is still not well understood. Using FM \nmetals as an example, vanadium doping decreases the Gilbert damping of Fe3 while many other \nrare-earth metals doping increase s the damping of permalloy (Py)4-6,16. Theoretically, several \nmodels have been developed to explain some  key characteristics. For example, spin-orbit \ncoupling is proposed to be the intrinsic or igin for homogenous time-varying magnetization9. The \ns-d exchange scattering model assumes that damp ing results from scattering of the conducting \nspin polarized electrons with the magnetization10. Besides, there is the Fermi surface breathing \nmodel taking account of the spin scattering with  the lattice defects ba sed on the Fermi golden \nrule11,12. Furthermore, other damping mechanisms in clude electron-electron scattering, electron-\nimpurity scattering13 and spin pumping into the adjacent nonmagnetic layers14, as well as the two \nmagnon scattering model, which refers to that pa irs of magnon are scatte red by defects, and the \nferromagnetic resonance (FMR) mode  moves into short wavelength spin waves, leading to a 3 dephasing contribution to the linewidth15. In magnetic nanostructu res, the magnetization \ndynamics is dictated by the Gilbert damping of the FM materials which can be simulated by \nmicromagnetics given the boundaries and dimens ions of the nanostructures. Therefore, \nunderstanding the Gilbert damping in FM materials is particularly important for characterizing \nand controlling ultrafast responses in magnetic  nanostructures that ar e highly relevant to \nspintronic applications such as  magne tic sensors and magnetic random access memory17.  \nIn this letter, we report an expe rimental investigation of the G ilbert damping in Py thin films \nvia variable temperature FMR in a modified  multi-functional insert  of physical property \nmeasurement system with a coplanar waveguide (see methods for details). We choose Py thin \nfilms since it is an interesting FM metallic material for spintronics due to its high permeability, nearly zero magnetostriction, low coercivity, a nd very large anisotropi c magnetoresistance. In \nour study, Py thin films are gr own on top of ~25 nm SiO\n2/Si substrates with a thickness ( d) range \nof 3-50 nm by magnetron sputtering (see methods  for details). A capping layer of TaN or Al 2O3 \nis used to prevent oxidation of the Py during m easurement. Interestingly, we observe that the \nGilbert damping of the thin Py films ( d <= 10 nm) shows an enhanced peak at ~ 50 K, while \nthicker films ( d >= 20 nm) decreases monotonically as the temperature decreases. The distinct \nlow-temperature behavior in the Gilbert dampi ng in different thickness regimes indicates a \npronounced surface contribution in the thin limit. In  fact, from the linear  relationship of the \nGilbert damping as a function of the 1/ d, we identify two contribu tions, namely bulk damping \nand surface damping. Interestingl y, these two contributions show  very different temperature \ndependent behaviors, in whic h the bulk damping decreases m onotonically as the temperature \ndecreases, while the surface damping indicates an  enhancement peak at ~ 50 K. We also notice \nthat the effective magnetization sh ows an increase at the same temperature of ~50 K for 3 and 5 4 nm Py films.  These observations could be all related to the magnetization reorientation on the \nPy surface at a certain temperatur e. Our results are important for theoretical investigation of the \nphysical origins of Gilbert damping and also us eful for the purpose of designing fast switching \nspintronics devices.  \nResults and Discussion \nFigure 1a shows five representative curves  of the forward amplitude of the complex \ntransmission coefficients (S 21) vs. in plane magnetic field meas ured on the 30 nm Py film with \nTaN capping at the frequencies of 4, 6, 8, 10 an d 12 GHz and at 300 K after renormalization by \nsubtracting a constant background. These experiment al results could be fitted using the Lorentz \nequation18: \n 2\n21 0 22()\n() ( )resHSSHH HΔ∝Δ+ −  (2) \nwhere S0 is the constant describing the coefficient for the transmitted microwave power, H is the \nexternal magnetic field, Hres is the magnetic field under the resonance condition, and ΔH is the \nhalf linewidth. The extracted ΔH vs. the excitation frequency ( f) is summarized in Figures 1b and \n1c for the temperature of 300 K and 5 K respect ively. The Gilbert damp ing could be obtained \nfrom the linearly fitted curves (red lin es), based on the following equation:  \n 02()H fHπαγΔ= + Δ   (3) \nin which γ is the geomagnetic ratio and ΔH0 is related to the inhom ogeneous properties of the \nPy films. The Gilbert damping at 300 K and 5 K is calculated to be 0.0064 ± 0.0001 and 0.0055 \n± 0.0001 respectively.  5 The temperature dependence of the Gilbert damp ing for 3-50 nm Py films with TaN capping \nlayer is summarized in Figure 2a. As d decreases, the Gilbert damping increases, indicative of \nthe increasing importance of the film surfaces. Interestingly, fo r thicker Py films (e.g. 30 nm), \nthe damping decreases monotonically as the temper ature decreases, which is expected for bulk \nmaterials due to suppressed sca ttering at low temperature. As d decreases down to 10 nm, an \nenhanced peak of the damping is obser ved at the temperature of ~ 50 K. As d decreases further, \nthe peak of the damping becomes more pronounce d. For the 3 nm Py film, the damping shows a \nslight decrease first from 0.0126 ± 0.0001 at 3 00 K to 0.0121 ± 0.0001 at 175 K, and a giant \nenhancement up to 0.0142 ± 0.0001 at 50 K, and then a sharp decrease back down to 0.0114 ± \n0.0003 at 5 K.  \nThe Gilbert damping as a function of the Py th icknesses at each temperature is also studied. \nFigure 2b shows the thickness dependence of the Py damping at 300 K. As d increases, the \nGilbert damping decreases, which indicates a surface/interface enhanced damping for thin Py \nfilms19. To separate the damping due to the bul k and the surface/interface contribution, the \ndamping is plotted as a function of 1/ d, as shown in Figure 2c, and it follows this equation as \nsuggested by theories19-21. \n 1()BSdαα α=+   (4) \nin which the Bα and Sα represent the bulk and surface da mping, respectively. From these \nlinearly fitted curves, we are able to separate  the bulk damping term and the surface damping \nterm out. In Figure 2b, the best fitted parameters for Bα and Sα are 0.0055 ± 0.0003 and 0.020 ± \n0.002 nm. To be noted, there are two insulating mate rials adjacent to the Py films in our studies. 6 This is very different from previous studies on Py/Pt bilayer systems, where the spin pumping \ninto Pt leads to an enhanced magnetic dampi ng in Py. Hence, the enhanced damping in our \nstudies is very unlikely resulti ng from spin pumping into SiO 2 or TaN. To our knowledge, this \nsurface damping could be related to  interfacial spin f lip scattering at the interface between Py \nand the insulating layers, which ha s been included in a generalized  spin-pumping theory reported \nrecently21. \nThe temperature dependence of the bulk damp ing and the surface damping are summarized \nin Figures 3a and 3b. The bulk damping of Py is  ~0.0055 at 300 K. As the temperature decreases, \nit shows a monotonic decrea se and is down to ~0.0049 at 5 K. Th ese values are consistent with \ntheoretical first principle calculations21-23 and the experimental valu es (0.004-0.008) reported for \nPy films with d ≥ 30 nm24-27. The temperature dependence of the bulk damping could be \nattributed to the magnetization rela xation due to the spin-lattice scattering in the Py films, which \ndecreases as the temperature decreases. \nOf particular interest, the surface damping sh ows a completely different characteristic, \nindicating a totally different mechanism from th e bulk damping. A strong enhancement peak is \nobserved at ~ 50 K for the surface damping. Could this enhancement of this surface/interface \ndamping be due to the strong spin-orbit coupli ng in atomic Ta of Ta N capping layer? To \ninvestigate this, we measure the damping of the 5 nm and 30 nm Py films with Al 2O3 capping \nlayer, which is expected to exhibit much lo wer spin-orbit coupling compared to TaN. The \ntemperature dependence of the Py damping is su mmarized in Figures 4a and 4b. Interestingly, \nthe similar enhancement of the damping at ~ 50 K is observed for 5 nm Py film with either Al 2O3 \ncapping layer or TaN layer, whic h excludes that the origin of the feature of the enhanced 7 damping at ~50 K results from th e strong spin-orbit coupling in TaN layer. These results also \nindicate that the mechanism of this feature is most likely related to the common properties of Py \nwith TaN and Al 2O3 capping layers, such as the crysta lline grain boundary and roughness of the \nPy films, etc. \nOne possible mechanism for the observed peak of  the damping at ~50 K could be related to a \nthermally induced spin reorientation transition  on the Py surface at that temperature. For \nexample, it has been show n that the spin reorientation of Py  in magnetic tunnel junction structure \nhappens due to the competition of different magne tic anisotropies, which c ould give rise to the \npeak of the FMR linewidth around the temperature of ~60 K28. Furthermore, we measure the \neffective magnetization ( Meff) as a function of temperature. Meff is obtained from the resonance \nfrequencies ( fres) vs. the external magnetic field via the Kittel formula29:  \n 12() [ ( 4 ) ]2res res res efffH H Mγππ=+   (4) \nin which Hres is the magnetic field at the resonance condition, and Meff is the effective \nmagnetization which contains the saturation ma gnetization and other anisotropy contributions. \nAs shown in Figures 5a and 5b, the 4π*M eff for 30 nm Py films w ith TaN capping layer are \nobtained to be ~10.4 and ~10.9 kG at 300 K and 5 K respectively. The temperature dependences \nof the 4π*M eff for 3nm, 5 nm, and 30 nm Py films are s hown in Figures 6a-6c. Around ~50 K, an \nanomaly in the effective magnetization for thin Py films (3 and 5 nm) is observed. Since we do \nnot expect any steep change in Py’s saturation magnetization at this temperature, the anomaly in \n4π*M eff should be caused by an anisot ropy change which coul d be related to a sp in reorientation. \nHowever, to fully understand the underlying mechan isms of the peak of the surface damping at ~ \n50 K, further theoretical and e xperimental studies are needed. 8 Conclusion  \nIn summary, the thickness and temperature dependences of the Gilbert damping in Py thin \nfilms are investigated, from which the contributio n due to the bulk damping and surface damping \nare clearly identified. Of particular interest, the bulk damping decreases monotonically as the \ntemperature decreases, while the surface damping develops an enhancement peak at ~ 50 K, \nwhich could be related to a thermally induced spin reorientation for the surface magnetization of \nthe Py thin films. This model is also consistent  with the observation of an enhancement of the \neffective magnetization below ~50 K. Our expe rimental results will contribute to the \nunderstanding of the intrinsic and ex trinsic mechanisms of the Gilber t damping in FM thin films.  \n \nMethods   \nMaterials growth. The Py thin films are deposited on ~25 nm SiO 2/Si substrates at room \ntemperature in 3×10- 3 Torr argon in a magnetron sputtering sy stem with a base pressure of ~ \n1×10-8 Torr. The growth rate of the Py is ~ 1  Å/s. To prevent ex situ  oxidation of the Py film \nduring the measurement,  a ~ 20 Å TaN or Al 2O3 capping layer is grown in situ environment. The \nTaN layer is grown by reactive sputtering of a Ta target in an argon-nitrogen gas mixture (ratio: \n90/10). For Al 2O3 capping layer, a thin Al (3 Å) layer is deposited first, and the Al 2O3 is \ndeposited by reactive spu ttering of an Al target in an ar gon-oxygen gas mixture (ratio: 93/7).  \nFMR measurement.  The FMR is measured using the vector network analyzer (VNA, Agilent \nE5071C) connected with a coplanar wave guide30 in the variable temperature insert of a \nQuantum Design Physical Properties Measuremen t System (PPMS) in the temperature range \nfrom 300 to 2 K. The Py sample is cut to be 1 × 0.4 cm and attached to the coplanar wave guide 9 with insulating silicon paste. For each temper ature from 300 K to 2 K, the forward complex \ntransmission coefficients (S 21) for the frequencies between 1 - 15 GHz are recorded as a function \nof the magnetic field sweeping from ~2500 Oe to 0 Oe.  \n \nContributions  \nJ.S. and W.H. proposed and supervised the studies. Y.Z. and Q.S. performed the FMR \nmeasurement and analyzed the data. T.S. and W.Y.  helped the measurement. S.H.Y. and S.S.P.P. \ngrew the films. Y.Z., J.S. and W.H. wrote the manuscript. All authors commented on the \nmanuscript and contributed to its final version. \n \nAcknowledgements   \nWe acknowledge the fruitful discussions with  Ryuichi Shindou, Ke Xia, Ziqiang Qiu, Qian \nNiu, Xincheng Xie and Ji Feng and the support of National Basic Research Programs of China \n(973 Grants 2013CB921903, 2014CB920902 and 2015 CB921104). Wei Han also acknowledges \nthe support by the 1000 Talents Program for Young Scientists of China. \n \nCompeting financial interests \nThe authors declare no compe ting financial interests. \n \n \nReferences: \n \n1 Landau, L. & Lifshitz, E. On the theory of the dispersion of magnetic permeability in \nferromagnetic bodies. Phys. Z. Sowjetunion  8, 153 (1935). \n2 Gilbert, T. L. A phenomenological theory  of damping in ferromagnetic materials. \nMagnetics, IEEE Transactions on  40, 3443-3449, doi:10.1109/TMAG.2004.836740 \n(2004). 10 3 Scheck, C., Cheng, L., Barsukov, I., Frait, Z.  & Bailey, W. E. 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E ., García-Hernández, M., Mompean, F., Rozada, \nR., Chubykalo-Fesenko, O., Snoeck, E., Miao, G. X., Moodera, J. S. & Aliev, F. G. \nInterface and Temperature Dependent Magnetic  Properties in Permalloy Thin Films and \nTunnel Junction Structures. Journal of Nanoscience and Nanotechnology  11, 7653-7664 \n(2011). \n29 Kittel, C. On the Theory of Ferromagnetic Resonance Absorption. Phys. Rev.  73, 155 \n(1948). \n30 Kalarickal, S. S., Krivosik, P., Wu, M., Patt on, C. E., Schneider, M. L., Kabos, P., Silva, \nT. J. & Nibarger, J. P. Ferromagnetic reso nance linewidth in metallic thin films: \nComparison of measurement methods. J. Appl. Phys.  99, 093909 (2006). \n 12  \nFigure Captions  \n \nFigure 1.  Measurement of Gilbert damping in Py thin films via ferromagnetic resonance \n(Py thickness = 30 nm).   a, Ferromagnetic resonance spectra of the absorption for 30 nm Py thin \nfilms with TaN capping layer at gigahertz frequencies of 4, 6, 8, 10 and 12 GHz at 300 K after \nnormalization by background subtraction. b, c, The half linewidths as a function of the resonance \nfrequencies at 300 K and 5 K respectively. The red solid lines indicate the fitted lines based on \nequation (3), where the Gilbert damp ing constants could be obtained.  \n \nFigure 2. Temperature dependence of the Gilber t damping of Py thin films with TaN \ncapping.  a, The temperature dependence of the Gilbert damping fo r 3, 5, 10, 15, 20, 30, and 50 \nnm Py films. b, The Gilbert damping as a function of the Py thickness, d, measured at 300 K. c, \nThe Gilbert damping as a function of 1/ d measured at 300 K. The linear fitting corresponds to \nequation (4), in which the slope and the intercep t are related to the surf ace contribution and bulk \ncontribution to the total Gilber t damping. Error bars correspond to one standard deviation. \n Figure 3. Bulk and surface damping of Py  thin films with TaN capping layer.   a, b,  The \ntemperature dependence of the bulk damping an d surface damping, respectively. The inset table \nsummarizes the experimental values reported in early studies. Error bars correspond to one \nstandard deviation.  \nFigure 4. Comparison of the Gilbert damping of Py films with different capping layers.  a, \nb, Temperature dependence of the Gilbert dampi ng of Py thin films with TaN capping layer 13 (blue) and Al 2O3 capping layer (green) for 5 nm Py a nd 30 nm Py, respectively. Error bars \ncorrespond to one standard deviation.  \nFigure 5. Measurement of effective magnetizat ion in Py thin films via ferromagnetic \nresonance (Py thickness = 30 nm).   a, b, The resonance frequencies vs. the resonance magnetic \nfield at 300 K and 5 K, respectively. The fitted li nes (red curves) are obtained using the Kittel \nformula.   \nFigure 6. Effective magnetization of Py fi lms as a function of the temperature.  a, b, c, \nTemperature dependence of the effective magnetizati on of Py thin films of a thickness of 3 nm,  \n5 nm and 30 nm Py respectively. In b, c, the blue/green symbols correspond to the Py with \nTaN/Al\n2O3 capping layer.  \n \n 0\n500\n1000\n1500\n2000\n-0.3\n-0.2\n-0.1\n0.0\n0.1\n   4 \n   6 \n   8\n 10 \n 12 \n \nS\n21\n (dB) \n \nH (Oe)\nT=300 K\nf\n (GHz)\n0\n4\n8\n12\n16\n0\n10\n20\n30\n \n\nH (Oe)\n \nf (GHz)\nT=300 K\n0\n4\n8\n12\n16\n0\n10\n20\n30\n \n\nH (Oe)\n \nf (GHz)\nT=5 K\nb\nc\na\nFigure 10\n50\n100\n150\n200\n250\n300\n0.006\n0.008\n0.010\n0.012\n0.014\nd\n (nm)\n 3      \n 15        \n 5      \n 20\n 10    \n 30    \n               \n 50 \n \na\n \nTemperature (K)\n0.0\n0.1\n0.2\n0.3\n0.004\n0.006\n0.008\n0.010\n0.012\n0.014\n \na\n \n \n1/\nd\n (nm\n-1\n)\n0\n10\n20\n30\n0.006\n0.008\n0.010\n0.012\n0.014\n \n \nd\n (nm)\n \na\na\nb\nc\nFigure \n20\n50\n100\n150\n200\n250\n300\n0.0040\n0.0045\n0.0050\n0.0055\n0.0060\nTheory\n Ref. 21, 22\n Ref. 23\n Temperature (K)\n \na\nB\n \na\na\nExp.\n0.006\nRef. 24\n0.004\n-\n0.008\nRef.\n25\n0.007\nRef.\n26\n0.0067\nRef. 27\n0\n50\n100\n150\n200\n250\n300\n0.016\n0.018\n0.020\n0.022\n0.024\n0.026\n0.028\n0.030\n Temperature (K)\na\nS\n (nm)\n \nb\nFigure \n30\n50\n100\n150\n200\n250\n300\n0.004\n0.006\n0.008\n0.010\n 5 nm Py/TaN\n 5 nm Py/Al\n2\nO\n3\n Temperature (K)\na\n \n \n0\n50\n100\n150\n200\n250\n300\n0.004\n0.005\n0.006\n0.007\n 30 nm Py/TaN\n 30 nm Py/Al\n2\nO\n3\n Temperature (K)\na\n \n \na\nb\nFigure \n4a\nb\n0\n500\n1000\n1500\n2000\n0\n4\n8\n12\n16\n \nf\n (GHz)\n \nH\n (Oe)\nT=300 K\n0\n500\n1000\n1500\n2000\n0\n4\n8\n12\n16\n \nf\n (GHz)\n \nH (Oe)\nT=5 K\nFigure \n58.6\n8.8\n9.0\n9.2\n9.4\n9.6\n4\n\nM\neff\n (kG) \n 5 nm Py/TaN\n 5 nm Py/Al\n2\nO\n3\n \n6.2\n6.3\n6.4\n6.5\n6.6\n6.7\n6.8\n6.9\n4\n\nM\neff\n (kG) \n 3 nm Py/TaN\n \n0\n50\n100\n150\n10.6\n10.7\n10.8\n10.9\n11.0\n 30 nm Py/TaN\n 30 nm Py/Al\n2\nO\n3\n4\n\nM\neff\n (kG) \n \na\nb\nc\nTemperature (K) \nFigure \n6"    },    {        "title": "0903.2047v2.Generalized_Lorentz_Dirac_Equation_for_a_Strongly_Coupled_Gauge_Theory.pdf",        "content": "arXiv:0903.2047v2  [hep-th]  13 Mar 2009Generalized Lorentz-Dirac Equation for a Strongly-Couple d Gauge Theory\nMariano Chernicoff, J. Antonio Garc´ ıa, and Alberto G¨ uijosa\nDepartamento de F´ ısica de Altas Energ´ ıas, Instituto de Ci encias Nucleares,\nUniversidad Nacional Aut´ onoma de M´ exico, Apdo. Postal 70 -543, M´ exico D.F. 04510\nWe derive a semiclassical equation of motion for a ‘composit e’ quark in strongly-coupled large- Nc\nN= 4 super-Yang-Mills, making use of the AdS/CFT corresponde nce. The resulting non-linear\nequation incorporates radiation damping, and reduces to th e standard Lorentz-Dirac equation for\nexternal forces that are small on the scale of the quark Compt on wavelength, but has no self-\naccelerating or pre-accelerating solutions. From this equ ation one can read off a non-standard\ndispersion relation for the quark, as well as a Lorentz covar iant formula for its radiation rate.\nPACS numbers: 11.25.Tq,11.15.-q,41.60.-m,12.38.Lg\nEnergy conservation implies that a radiating charge\nmust experience a damping force, originating from its\nself-field. In the context of classicalelectrodynamics, this\neffect is incorporated in the (Abraham-)Lorentz-Dirac\nequation [1],\nm/parenleftbiggd2xµ\ndτ2−re/bracketleftbiggd3xµ\ndτ3−d2xν\ndτ2d2xν\ndτ2dxµ\ndτ/bracketrightbigg/parenrightbigg\n=Fµ,(1)\nwithτthe proper time and Fµ≡γ(/vectorF·/vector v,/vectorF) the four-\nforce. The electron here is modeled as a vanishingly\nsmall spherically symmetric charge distribution. The\ncharacteristic time/size associated with radiation damp-\ning,re≡2e2/3m, is set by the classical electron ra-\ndius. The second term within the square brackets is the\nnegative of the rate at which four-momentum is carried\naway from the charge by radiation, so it is only this term\nthat can properly be called radiation reaction. The first\ntermwithin thesquarebrackets,usuallycalledtheSchott\nterm, is known to arise from the effect of the charge’s\n‘bound’ (as opposed to radiation) field [2, 3].\nThe appearance of a third-order term in (1) leads to\nunphysical behavior, including pre-accelerating and self-\naccelerating (or ‘runaway’) solutions. These deficiencies\nare known to originate from the assumption that the\ncharge is pointlike. For a charge distribution of small\nbut finite size l, the above equation is corrected by an\ninfinite number of higher-derivative terms ( ld/dt)n, and\nis physically sound as long as l > re[3, 4]. Upon shifting\nattention to the quantum case, one intuitively expects\nthe pointlike ‘bare’ electron to acquire an effective size\nof order the Compton wavelength λC≡1/m, due to its\nsurrounding cloud of virtual particles. Indeed, in [5] it\nwas shown non-relativistic QED leads to a generalization\nof (the non-relativistic version of) (1) where the charge\ndevelops a characteristic size l=λC.\nGoing further to the quantum non-Abelian case is a\nserious challenge. Nevertheless, it is the purpose of this\nletter to show that the AdS/CFT correspondence [6, 7]\nallows us to address this question rather easily in cer-\ntain strongly-coupled non-Abelian gauge theories. In the\ncontext of this duality, the quark corresponds to the end-\npoint of a string, whose body codifies the profile of thenon-Abelian (bound and radiation) fields sourced by the\nquark, including, as we will see, the effect of radiation\ndamping. We expect this basic story to apply gener-\nallytoallinstancesofthe gauge/stringduality(including\ncaseswith finite temperature orchemicalpotentials), but\nfor simplicity we will concentrate on the best understood\nexample: quark motion in the vacuum of N= 4 super-\nYang-Mills (SYM). Besides the gauge field, this maxi-\nmally supersymmetric and conformally invariant theory\ncontains 6 real scalar fields and 4 Weyl fermions, all in\ntheadjointrepresentation of the gauge group.\nIt is by now well-known that N= 4SU(Nc) SYM\nwith coupling gYMis, despite appearances, completely\nequivalent [6] to Type IIB string theory on a background\nthat asymptotically approaches the five-dimensional [8]\nanti-de Sitter (AdS) geometry\nds2=GMNdxMdxN=R2\nz2/parenleftbig\n−dt2+d/vector x2+dz2/parenrightbig\n,(2)\nwhereR4/l4\ns=g2\nYMNc≡λdenotes the ’t Hooft cou-\npling, and lsis the string length. The radialdirection zis\nmappedholographicallyintoavariablelengthscaleinthe\ngauge theory, in such a way that z→0 andz→ ∞are\nrespectively the ultraviolet and infrared limits [9]. The\ndirections xµ≡(t,/vector x) are parallel to the AdS boundary\nz= 0andaredirectlyidentifiedwiththe gaugetheorydi-\nrections. The state of IIB string theory described by the\nunperturbed metric (2) correspondsto the vacuum of the\nN= 4 SYM theory, and the closed string sector describ-\ning(small orlarge)fluctuationson topofit fully captures\nthe gluonic (+ adjoint scalarand fermionic) physics. The\nstring theory description is under calculational control\nonly for small string coupling and low curvatures, which\ntranslates into Nc≫1,λ≫1.\nIt is also known that one can add to SYM Nfflavorsof\nmatter in the fundamental representation of the SU(Nc)\ngauge group by introducing an open string sector asso-\nciated with a stack of NfD7-branes [10]. We will refer\nto these degrees of freedom as ‘quarks,’ even though, be-\ningN= 2 supersymmetric, they include both spin 1 /2\nand spin 0 fields. For Nf≪Nc, the backreaction of the2\nD7-branes on the geometry can be neglected; in the field\ntheory this corresponds to a ‘quenched’ approximation.\nAn isolated quark of mass mis dual to an open string\nthat extends radially from the location\nzm=√\nλ\n2πm(3)\non the D7-branes to the AdS horizon at z→ ∞. The\nstring dynamics is governed by the Nambu-Goto action\nSNG=−1\n2πl2s/integraldisplay\nd2σ/radicalbig\n−detgab≡/integraldisplay\nd2σLNG,(4)\nwheregab≡∂aXM∂bXNGMN(X) (a,b= 0,1) denotes\nthe induced metric on the worldsheet. We can exert an\nexternal force /vectorFon the string endpoint by turning on an\nelectric field F0i=Fion the D7-branes. This amounts to\nadding to (4) the usual minimal coupling, which in terms\noftheendpoint/quarkworldline xµ(τ)≡Xµ(τ,zm)reads\nSF=/integraldisplay\ndτ Aµ(x(τ))dxµ(τ)\ndτ. (5)\nNotice that the string is being described (as is cus-\ntomary) in first-quantized language, and, as long as it is\nsufficiently heavy, we are allowed to treat it semiclassi-\ncally. In gauge theory language, then, we are coupling\na first-quantized quark to the gluonic (+ other SYM)\nfield(s), and then carrying out the full path integral over\nthe strongly-coupled field(s) (the result of which is cod-\nified by the AdS spacetime), but treating the path in-\ntegral over the quark trajectory xµ(τ) in a saddle-point\napproximation.\nVariation of SNG+SFimplies the standard Nambu-\nGoto equation of motion for all interior points of the\nstring, plus the boundary condition\nΠz\nµ(τ)|z=zm=Fµ(τ)∀τ , (6)\nwhere Πz\nµ≡∂LNG/∂(∂zXµ) is the worldsheet (Noether)\ncurrent associated with spacetime momentum, and Fµ=\n−Fνµ∂τxν= (−γ/vectorF·/vector v,γ/vectorF) the Lorentz four-force.\nFor the interpretation of our results it will be crucial\nto keep in mind that the quark described by this string\nis not ‘bare’ but ‘composite’ or ‘dressed’. This can be\nseen most clearly by working out the expectation value\nof the gluonic field surrounding a static quark located at\nthe origin. For m→ ∞(zm→0), the result is just the\nCoulombic field expected (by conformal invariance) for\na pointlike charge [11]. For finite mthe profile is still\nCoulombic far away from the origin but in fact becomes\nnon-singular at the location of the quark [12]. The char-\nacteristicthicknessoftheimpliednon-Abelianchargedis-\ntribution is precisely the length scale zmdefined in (3).\nThis is then the size of the ‘gluonic cloud’ which sur-\nrounds the quark, or in other words, the analog of the\nCompton wavelength for our non-Abelian source.We will take as our starting point the results obtained\nin a remarkable paper by Mikhailov [13], which we now\nvery briefly review (more details can be found in [14]).\nThis author considered an infinitely massive quark, and\nwas able to find a solution to the equation of motion\nfor the dual string on AdS 5, for an arbitrary timelike\ntrajectory xµ(τ) of the string endpoint:\nXµ(τ,z) =zdxµ(τ)\ndτ+xµ(τ). (7)\nThis solution is ‘retarded’, in the sense that the behav-\nior at time t=X0(τ,z) of the string segment located\nat radial position zis completely determined by the be-\nhavior of the string endpoint at an earliertimetret(t,z)\nobtained by projecting back toward the boundary along\nthe null line at fixed τ.\nUsing (7), Mikhailov was able to rewrite the total\nstring energy in the form\nE(t) =√\nλ\n2��/integraldisplayt\n−∞dtret/vector a2−[/vector v×/vector a]2\n(1−/vector v2)3+Eq(/vector v(t)),(8)\nwith/vector v≡d/vector x/dx0and/vector a≡d/vector v/dx0the velocity and ac-\nceleration of the endpoint/quark. The first term codifies\nthe accumulated energy lostby the quark over all times\nprior to t, and, surprisingly, has the same form as the\nstandardLienardformulafromclassicalelectrodynamics.\nThe second term in (8) arises from a total derivative on\nthe string worldsheet, and gives the expected expression\nfor the energy intrinsic to the quark [14],\nEq(/vector v) =√\nλ\n2π/parenleftbigg1√\n1−/vector v21\nz/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsinglezm=0\n∞=γm . (9)\nThe total spatial momentum of the string /vectorPcan be sim-\nilarly split [13, 14]. We see then that, in spite of the\nnon-linear nature of the system, Mikhailov’s procedure\nleads to a clean separation between the quark (including\nits bound field) and its gluonic radiation field.\nWe will now exploit this to study in more detail the\ndynamics of the quark with finite mass, zm>0, where\nour non-Abelian source is no longerpointlike but has size\nzm. As in [14], the embeddings ofinterest tous canbe re-\ngarded as the z≥zmportions of the Mikhailov solutions\n(7). These are parametrized by (now merely auxiliary)\ndata at the AdS boundary z= 0, which we will hence-\nforth denote with tildes to distinguish them from the ac-\ntual physical data associated with the endpoint/quark at\nz=zm. In this notation, (7) reads\nXµ(˜τ,z) =zd˜xµ(˜τ)\nd˜τ+ ˜xµ(˜τ). (10)\nRepeated differentiation with respect to ˜ τand evaluation\natz=zm(where we can read off the quark trajectory\nxµ(˜τ)≡Xµ(˜τ,zm)) leads to the recursive relations\ndnxµ\nd˜τn=zmdn+1˜xµ\nd˜τn+1+dn˜xµ\nd˜τn∀n≥1.(11)3\nAdding the n≥2 equations respectively multiplied by\n(−zm)n−2, we can deduce that\nd2˜xµ\nd˜τ2=d2xµ\nd˜τ2−zmd3xµ\nd˜τ3+z2\nmd4xµ\nd˜τ4−... . (12)\nNext we wish to rewrite d˜τandd2˜xµ/d˜τ2in terms of\nquantities at the actual string boundary z=zm. The\nfirst task is easy: from (10) it follows that\ndXµ=dzd˜xµ\nd˜τ+d˜τ/parenleftbigg\nzd2˜xµ\nd˜τ2+d˜xµ\nd˜τ/parenrightbigg\n,\nwhich when evaluated at fixed z=zmimplies\ndτ2≡ −dxµdxµ=d˜τ2/bracketleftBigg\n1−z2\nm/parenleftbiggd2˜x\nd˜τ2/parenrightbigg2/bracketrightBigg\n.(13)\nTo arrive at this last equation, we have made use of\nthe fact that ˜ τis by definition the proper time for the\nauxiliary worldline at z= 0, so ( d˜x/d˜τ)2=−1 and\n(d˜x/d˜τ)·(d2˜x/d˜τ2) = 0.\nFor the remaining task, we note first that, upon sub-\nstituting the solution (10), the worldsheet momentum\ncurrent evaluated at z=zmsimplifies to\n2π√\nλ˜Πz\nµ=1\nzmd2˜xµ\nd˜τ2+/parenleftbiggd2˜x\nd˜τ2/parenrightbigg2d˜xµ\nd˜τ.(14)\nThe tilde in the left-hand side does not indicate eval-\nuation at z= 0 (as all other tildes do), but the fact\nthat this current is defined as charge (momentum) flow\nper unit ˜ τ. The corresponding flow per unit τis just\nΠz\nµ= (∂˜τ/∂τ)˜Πz\nµ, and it is this object which according\nto (6) must equal the external force Fµ. Using this, (13)\nand (11) in (14), one can deduce that\nd2˜xµ\nd˜τ2=1/radicalbig\n1−z4m/F2/parenleftbigg\nzm/Fµ−z3\nm/F2dxµ\ndτ/parenrightbigg\n,(15)\nwhere we have used the abbreviation /Fµ≡(2π/√\nλ)Fµ.\nSinceF·(dx/dτ) = 0 (no work is done on the quark in its\ninstantaneousrestframe), this implies that( d2˜x/d˜τ2)2=\nz2\nm/F2, which allows (13) to be simplified into\nd˜τ=dτ/radicalbig\n1−z4m/F2. (16)\nUsing (15) and (16), we can rewrite (12) purely in\nterms of quark data. The result is an equation with an\ninfinite number of higher derivatives, which we omit here\nfor brevity. A more manageable form of the equation of\nmotion can be obtained by goingback to (12) and adding\nto it its ˜τ-derivative multiplied by zm, yielding\nd2xµ\nd˜τ2=d2˜xµ\nd˜τ2+zmd3˜xµ\nd˜τ3. (17)Through (15), (16) and (3), this can be reexpressed as\nd\ndτ\nmdxµ\ndτ−√\nλ\n2πmFµ\n/radicalBig\n1−λ\n4π2m4F2\n=Fµ−√\nλ\n2πm2F2dxµ\ndτ\n1−λ\n4π2m4F2,(18)\nwhich is our main result.\nThis equation correctly reduces to md2xµ/dτ2=Fµ\nin the pointlike limit m→ ∞. Specializing to the case of\nmotion along one dimension, it is also possible to show\nthat (18) correctly reproduces the energy (and momen-\ntum) split between quark and radiation field deduced at\nfinitemin [14], and in fact encodes the Lorentz-covariant\ngeneralization of that split. To make this explicit, we\nrewrite (18) in the form\ndPµ\ndτ≡dpµ\nq\ndτ+dPµ\nrad\ndτ=Fµ, (19)\nrecognizing Pµas the total string (= quark + radiation)\nfour-momentum,\npµ\nq=mdxµ\ndτ−√\nλ\n2πmFµ\n/radicalBig\n1−λ\n4π2m4F2(20)\nas the intrisic momentum of the quark, and\ndPµ\nrad\ndτ=√\nλF2\n2πm2/parenleftBigg\ndxµ\ndτ−√\nλ\n2πm2Fµ\n1−λ\n4π2m4F2/parenrightBigg\n(21)\nas the rate at which momentum is carried away from\nthequarkbychromo-electromagneticradiation. (Insome\ncases, this vacuum radiation rate can also be relevant for\nquark motion within a thermal plasma [14, 15].)\nWe can immediately deduce from (20) the mass-shell\ncondition p2\nq=−m2, which shows in particular that the\nsplitPµ=pµ\nq+Pµ\nraddefined in (19)-(21) is correctly\nLorentz covariant. As we indicated above, Pµ\nradrepre-\nsents the portion of the total four-momentum stored at\nany given time in the purely radiative part of the gluonic\nfield set up by the quark. The remainder, pµ\nq, includes\nthe contribution of the boundfield sourced by our par-\nticle, or in quantum mechanical language, of the gluonic\ncloud surrounding the quark, which gives rise to the de-\nformed dispersion relation seen in (20). In other words,\npµ\nqis the four-momentum of the composite quark. Sur-\nprisingly, all of this is closely analogous to the classical\nelectromagnetic case, and in particular, to the covariant\nsplitting of the energy-momentum tensor achieved in [2].\nAs noticed already in [14] for the case of linear motion,\na prominent feature of the equation of motion (18), as\nwell as the dispersion relation (20) and radiation rate\n(21), is the presence of a divergence when F2=F2\ncrit,\nwhereF2\ncrit= 4π2m4/λis the critical value at which the\nforcebecomes strongenough to nucleatequark-antiquark\npairs (or, in dual language, to create open strings).4\nLet us now examine the behavior of a quark that is\nsufficiently heavy, or is forced sufficiently softly, that the\ncondition/radicalbig\nλ|F2|/2πm2≪1 (i.e.,|F2| ≪ |F2\ncrit|) holds.\nIt is then natural to expand the equation of motion in a\npower series in this small parameter. To zeroth order in\nthisexpansion,wehavethepointlikeresult m∂2\nτxµ=Fµ,\nas we had already mentioned above. If we instead keep\nterms up to first order, we find\nmd\ndτ/parenleftBigg\ndxµ\ndτ−√\nλ\n2πm2Fµ/parenrightBigg\n=Fµ−√\nλ\n2πm2F2dxµ\ndτ.\nIn theO(√\nλ) terms it is consistent, to this order, to\nreplaceFµwith its zeroth order value, thereby obtaining\nm/parenleftBigg\nd2xµ\ndτ2−√\nλ\n2πmd3xµ\ndτ3/parenrightBigg\n=Fµ−√\nλ\n2πd2xν\ndτ2d2xν\ndτ2dxµ\ndτ.\n(22)\nInterestingly, this coincides exactlywith the Lorentz-\nDirac equation (1). As expected from the preceding dis-\ncussion, on the left-hand side we find the Schott term\n(associated with the bound field of the quark) arising\nfrom the modified dispersion relation (20). On the right-\nhand side we see the radiation reaction force given by\nthe covariant Lienard formula, as expected from the re-\nsult (8) [13], which is the pointlike limit of the radiation\nrate (21). Moreover, by comparing (1) and (22) we learn\nthat it is the Compton wavelength (3) that plays the role\nof characteristic size refor the composite quark. This is\nindeed the natural quantum scale of the problem.\nWe can continue this expansion procedure to arbitrar-\nily high order in/radicalbig\nλ|F2|/2πm2. At order nin this pa-\nrameter,wewouldobtainanequationwith derivativesup\nto order n+2. Our full equation (18) is thus recognized\nas an extension of the Lorentz-Dirac equation that au-\ntomatically incorporates the size zmof our non-classical,\nnon-pointlike and non-Abelian source.\nIt is curious to note that (18), which incorporates the\neffect of radiation damping on the quark, has been ob-\ntained from (7), which does notinclude such damping\nfor the string itself. The supergravity fields set up by\nthe string are of order 1 /N2\nc, and therefore subleading at\nlargeNc. Even more curious [16] is the fact that it is\nprecisely these suppressed fields that encode the gluonic\nfield profile generated by the quark, as has been explored\ningreatdetailin recentyears[17]. It wouldbeinteresting\nto explore how the split into bound and radiation fields\nis achieved from this perspective.\nThe passage from (22) to (18), which can be viewed\nintuitively as the addition of an infinite number of higher\nderivative terms ( zmd/dτ)n, has a profound impact on\nthe space of solutions. Here we will limit ourselves to\ntwo general observations, leaving the search for specific\nexamples of solutions to a more extensive report [18].\nThe first is to notice that, unlike its classical electrody-\nnamic counterpart (1), our composite quark equation ofmotion has no pre-accelerating or self-accelerating solu-\ntions. In the (continuous) absence of an external force,\n(18) uniquely predicts that the four-acceleration of the\nquark must vanish. Our second observation, however, is\nthat the converse to this last statement is not true: con-\nstant four-velocity does not uniquely imply a vanishing\nforce. This is again a consequence of the extended, and\nhence deformable, nature of the quark.\nAll in all, then, we have in (18) a physically sensible\nand interesting description of the dynamics of a com-\nposite quark in SYM. It is truly remarkable that the\nAdS/CFT correspondence grants us such direct access\nto this piece of strongly-coupled non-Abelian physics.\nAcknowledgments. We thank David Mateos and\nMat´ ıasMorenoforusefuldiscussions. Thisworkwassup-\nportedbyMexico’sNationalCouncilofScienceandTech-\nnology (CONACyT) grant 50-155I and DGAPA-UNAM\ngrant IN116408.\n[1] P. A. M. Dirac, Proc. Roy. Soc. Lond. A 167(1938) 148.\n[2] C. Teitelboim, Phys.Rev.D 1(1970)1572 [Erratum-ibid.\nD2(1970) 1763].\n[3] F. Rohrlich, Classical Charged Particles , 2nd. ed. (Ad-\ndison Wesley, Redwood City, California, 1990); Am. J.\nPhys.65(1997) 1051.\n[4] J. D.Jackson, Classical Electrodynamics , 2nd.ed.(Wiley,\nNew York, 1975).\n[5] E. J. Moniz and D. H. Sharp, Phys. Rev. D 15(1977)\n2850.\n[6] J. M. Maldacena, Adv. Theor. Math. Phys. 2,\n231 (1998) [Int. J. Theor. Phys. 38, 1113 (1999)]\n[arXiv:hep-th/9711200].\n[7] O. Aharony, S. S. Gubser, J. M. Maldacena,\nH. Ooguri and Y. Oz, Phys. Rept. 323, 183 (2000)\n[arXiv:hep-th/9905111].\n[8] The relevant background is actually AdS 5×S5, with a\nconstant dilaton and Ncunits of Ramond-Ramond five-\nform flux through the five-sphere, but these additional\nfeatures will play no direct role in our analysis.\n[9] L. Susskind and E. Witten, arXiv:hep-th/9805114.\n[10] A. Karch and E. Katz, JHEP 0206(2002) 043\n[arXiv:hep-th/0205236].\n[11] U. H. Danielsson, E. Keski-Vakkuri and M. Kruczenski,\nJHEP9901, 002 (1999) [arXiv:hep-th/9812007].\n[12] J. L. Hovdebo, M. Kruczenski, D. Mateos, R. C. Myers\nand D. J. Winters, Int. J. Mod. Phys. A 20(2005) 3428.\n[13] A. Mikhailov, arXiv:hep-th/0305196.\n[14] M. Chernicoff and A. G¨ uijosa, JHEP 0806, 005 (2008)\n[arXiv:0803.3070 [hep-th]]; arXiv:0903.0306.\n[15] Y. Hatta, E. Iancu and A. H. Mueller, arXiv:0803.2481\n[hep-th]; K. B. Fadafan et al., arXiv:0809.2869 [hep-ph].\n[16] C. G. Callan and A. G¨ uijosa, Nucl. Phys. B 565, 157\n(2000) [arXiv:hep-th/9906153].\n[17] S. S. Gubser et al., arXiv:0902.4041.\n[18] M. Chernicoff, J. A. Garc´ ıa and A. G¨ uijosa, to appear."    },    {        "title": "1011.5175v2.Magnetohydrodynamic_kink_waves_in_two_dimensional_non_uniform_prominence_threads.pdf",        "content": "arXiv:1011.5175v2  [astro-ph.SR]  27 Jul 2011Astronomy& Astrophysics manuscriptno.ms c/ci∇cleco√y∇tESO 2018\nOctober12,2018\nMagnetohydrodynamic kink wavesintwo-dimensional non-un iform\nprominence threads\nI. Arregui1,R.Soler2,J. L.Ballester1, and A.N.Wright3\n1Departament de F´ ısica,Universitat de les IllesBalears,E -07122, Palma de Mallorca, Spain\ne-mail:[inigo.arregui,joseluis.ballester]@uib.es\n2CentreforPlasmaAstrophysics,DepartmentofMathematics ,KatholiekeUniversiteitLeuven,Celestijnenlaan200B,3 001Leuven,\nBelgium\ne-mail:roberto.soler@wis.kuleuven.be\n3School of Mathematics and Statistics,Universityof St.And rews, St.Andrews, KY169SS,UK\ne-mail:andy@mcs.st-and.ac.uk\nReceived ,;accepted\nABSTRACT\nAims.Weanalysetheoscillatorypropertiesofresonantlydamped transversekinkoscillationsintwo-dimensionalprominen cethreads.\nMethods. The fine structures are modelled as cylindrically symmetric magnetic flux tubes witha dense central part withprominence\nplasmapropertiesandanevacuatedpart,bothsurroundedby coronalplasma.Theequilibriumdensityisallowedtovaryn on-uniformly\ninboththetransverseandthelongitudinaldirections.Wee xaminetheinfluenceoflongitudinaldensitystructuringon periods,damping\ntimes, and damping rates for transverse kink modes computed by numerically solving the linear resistive magnetohydrod ynamic\n(MHD) equations.\nResults.The relevant parameters are the length of the thread and the d ensity in the evacuated part of the tube, two quantities that\nare difficult to directly estimate from observations. We find that bot h of them strongly influence the oscillatory periods and damp ing\ntimes,andtoalesserextent thedampingratios.Theanalysi softhespatialdistributionofperturbations andoftheene rgyfluxintothe\nresonances allows us toexplain the obtained damping times.\nConclusions. Implications for prominence seismology, the physics of res onantly damped kink modes in two-dimensional magnetic\nfluxtubes, and the heatingof prominence plasmas are discuss ed.\nKey words. Magnetohydrodynamics (MHD) –Waves–Sun: filaments,promin ences\n1. Introduction\nQuiescent filaments /prominences are cool and dense magnetic\nand plasma structures suspended against gravity by forces\nthoughttobe ofmagneticorigin.Inspite oftheirphysicalp rop-\nerties, with temperatures and densities that are akin to tho se\nin the chromosphere, some as yet not well determined mecha-\nnismsprovidethe requiredthermalisolation fromthe surro und-\ning coronal plasma and mechanical support during typical li fe-\ntimesfromfew daysto weeks. The magneticfield that pervades\nthese structures is believed to play a key role in the nature a nd\nthe thermodynamic and mechanical stability of prominences .\nEarly observations carried out with good seeing conditions\npointed out that prominences consist of fine threads (deJage r,\n1959; Kuperus&Tandberg-Hanssen, 1967). More recent high-\nresolution Hαobservations obtained with the Swedish Solar\nTelescope (SST) in La Palma (Linet al., 2005) and the Dutch\nOpen Telescope (DOT) in Tenerife (Heinzel&Anzer, 2006)\nhaveallowedtofirmlyestablishthefilamentsub-structurin gand\nthe basic geometrical and physical properties of threads (s ee\nalsoEngvold,1998;Linet al.,2005,2008; Lin,2010).Thesu b-\nstructureofquiescentprominencesisoftencomposedbya my r-\niad of horizontal,dark and fine threads, made of cool absorbi ng\nmaterial,believedtooutlinemagneticfluxtubes(Engvold, 1998,\n2008;Lin,2004;Linetal.,2005,2008;Martinet al.,2008). The\ntubes are only partially filled with cool and dense plasma and\ntheir total length is probably much larger ( ∼105km) than thethreads themselves. The measured average width of resolved\nthreads is about 0.3 arcsec ( ∼210 km) while their length is be-\ntween5and40arcsec( ∼3500-28000km).Theabsorbingcool\nmaterialisusuallyvisibleforupto20minutes(Linet al.,2 005).\nThe measured widths are close to the current resolution limi t,\n∼0.16 arcsec at the SST, hence thinner structures are likely t o\nexist.\nSmall amplitude oscillations in prominence threads are fre -\nquently observed (see reviews by Oliver& Ballester, 2002;\nEngvold, 2004; Wiehr, 2004; Ballester, 2006; Banerjeeet al .,\n2007; Engvold, 2008;Oliver,2009;Ballester, 2010).Early two-\ndimensional observations of filaments (Yi &Engvold, 1991;\nYi et al., 1991) revealed that individual threads or groups o f\nthem oscillate with periods that range between 3 and 20 min-\nutes. Recent relevant examples are traveling waves propaga t-\ning along a number of threads with average phase speed of 12\nkm s−1, wavelength of 4 arcsec, and oscillatory periods that\nvary from 3 to 9 minutes (Linet al., 2007), the both propa-\ngating and standing oscillations detected over large areas of\nprominences by Terradaset al. (2002) and Lin (2004), as well\nas observations from instruments onboard space-crafts, su ch as\nSoHO (Blancoetal., 1999; R´ egnieretal., 2001; Pougetet al .,\n2006)andHinode(Okamotoetal.,2007;Terradaset al.,2008 b;\nNingetal., 2009). The transverse oscillation nature of som e of\nthese events has been clearly established by Linetal. (2009 )\nby combining H αfiltergrams in the plane of the sky with H α\nDopplergrams which allow to detect oscillations in the line -2 I.Arregui et al.:Magnetohydrodynamic kink waves intwo-d imensional non-uniform prominence threads\nof-sight direction. A recurrently observed property of pro mi-\nnence oscillations is their rapid temporal damping, with pe r-\nturbations decaying in time-scales of only a few oscilla-\ntory periods (Landmanet al., 1977; Tsubaki& Takeuchi, 1986 ;\nTsubaki, 1988; Wiehret al., 1989; Molowny-Horasetal., 199 9;\nTerradaset al.,2002; Lin, 2004; Ningetal.,2009).\nTransverse thread oscillations are commonly interpreted i n\ntermsofstandingorpropagatingmagnetohydrodynamic(MHD )\nkink waves. The measured periods are of the order of a few\nminutes and the wavelengths are in between 3000 - 20 000\nkm, although Okamotoet al. (2007) report larger wavelength s\nmoreconsistentwiththestandingwaveinterpretation.The mea-\nsured wave quantities allow us to derive phase speeds that\nare consistent with the kink speed in magnetic and plasma\nconfigurations with typical properties of prominence plasm as.\nThe MHD wave interpretation of thread oscillations has al-\nlowed the development of theoretical models (see Ballester ,\n2005,2006,forrecentreviews).Joarderet al.(1997);D´ ıa z et al.\n(2001, 2003) considered the MHD eigenmodes supported by a\nfilament thread modelled in Cartesian geometry. More realis -\ntic studies using cylindrical configurations have extended the\ninitial investigations (D´ ıazet al., 2002; Dymova& Ruderm an,\n2005; D´ ıaz &Roberts, 2006). These studies have determined\nthe frequencies and confinement properties of the perturba-\ntions as functions of the length and the width of the threads.\nTheoreticaldampingmechanismshavealsobeendeveloped(s ee\nBallester, 2010; Arregui&Ballester, 2010, for recent revi ews).\nA systematic comparative study of di fferent mechanisms has\nbeen presented by Soler (2010), who assesses the ability of\neach mechanism to reproduce the observed attenuation time-\nscales. The considered mechanisms include non-ideal e ffects,\nsuch as radiation and thermal conduction, partial ionisati on\nthrough ion-neutral collisions, ion-electron collisions , and res-\nonant absorption due to coupling to Alfv´ en and slow waves.\nNon-idealeffectsdonotseemtoprovidetherequiredattenuation\ntime-scales for kink oscillations (Ballai, 2003; Carbonel let al.,\n2004;Terradaset al.,2001,2005;Soleret al.,2007,2008). Soler\n(2010) finds that resonant absorption in the Alfv´ en contin-\nuum is the only mechanism able to produce the observed at-\ntenuation time-scales (see Soleret al., 2008, 2009a,b,c, f or de-\ntails). Resonant damping was first considered in this contex t by\nArreguietal.(2008),andhasbeensubsequentlystudiedinc om-\nbination with damping in the slow continuum and partial ioni -\nsationbySoleret al.(2009a,b,c).Thesestudieshaveconsi dered\none-dimensional models with a density variation in the tran s-\nverse direction only. On the other hand, theoretical studie s that\ntake into account the longitudinal density structuring, he nce in-\ncorporating the fact that magnetic tubes supporting thread s are\nonly partially filled with cool plasma, consider piece-wise ho-\nmogeneous models in the transverse direction, and hence rul e\noutresonantdamping.\nArecentinvestigationbySoleret al.(2010)istheexceptio n.\nThese authors have obtained analytical and semi-analytica l ap-\nproximationsforperiods,dampingtimes,anddampingratio sof\nstanding kink modes in a two-dimensional prominence thread\nmodel. However, the analysis by Soleret al. (2010) is restri cted\nto the thin tube and thin boundary approximations,radial in ho-\nmogeneityis constrainedtothe densecoolpartofthetubeon ly,\nandthedensityintheevacuatedpartofthetubeistakenequa lto\nthe coronaldensity,furtherlimitingthe prominencethrea dsthat\ncan be theoretically modelled. These limitations are remov ed\nin the present investigation. We adopt a fully inhomogeneou s\ntwo-dimensional prominence thread model. The density dist ri-\nbution is allowed to vary non-uniformly in both the transver seandlongitudinaldirections.Wealsoaddanotherrelevantp aram-\neter with seismological implications, namely, the density in the\nevacuatedpartofthetube.Bycombiningthetwoparameterst hat\ncharacterise the longitudinal structuring in prominence t hreads,\nthe length of the thread and the density in the evacuated part of\nthetube,awiderangeofprominencethreadswithverydi fferent\nphysical conditions can be modelled and their oscillatory p rop-\nerties characterised. In addition, a general parametric st udy is\nnumericallyperformed,thatallowsustogobeyondthethint ube\nandthinboundaryapproximations.\nBesides obtaining the influence of longitudinal structur-\ning on periods and damping times for kink modes in two-\ndimensional threads, and extracting conclusionsabout the ir im-\nplicationsforprominenceseismology,ourstudyaimsatexp lain-\ning the obtainedresults. For this reason, we have performed the\nanalysis of the spatial distribution of perturbations and t he en-\nergy flux into the resonances. These two analyses, which are\nnovelin the contextof prominenceoscillations, provideus with\na comprehensiveexplanationfortheobtainedparametricre sults\nandaddfurtherinsightstothephysicsofresonantlydamped kink\nmodesintwo-dimensionalequilibriumstates.\nThe paper is organised as follows. Section 2 describes\nthe thread model, the relevant parameters introduced by its\ntwo-dimensional character, the linear MHD wave equations t o\nbe solved, and the numerical method used for that purpose.\nSection 3 presents our analysis and results. We first show com -\nputations of periods, damping times, and damping ratios as a\nfunctionofthelengthofthethreadandthedensityintheeva cu-\natedpartofthetube.Implicationsforthedeterminationof phys-\nical parameters in prominences are discussed. Next, a quali ta-\ntive explanation of the obtained results is given by analysi ng\nthespatialdistributionofperturbations.Finally,wedes cribeour\nenergy analysis, that in combination with the spatial struc ture\nof eigenfunctionsfully explains the obtained dampingtime s. In\nSection4, ourconclusionsarepresented.\n2. Threadmodel,linearMHD waveequations,and\nnumericalmethod\nWe consider an individual and isolated prominence thread in\na gravity-free static equilibrium in the zero plasma- βapproxi-\nmation. The fine structure is modelled by means of a cylindri-\ncally symmetric flux tube of radius aand length L. In a sys-\ntemofcylindricalcoordinates( r,ϕ,z)withthe z-axiscoinciding\nwith the axis of the tube, the magnetic field is pointing in the\nz-direction and has a uniform field strength. Because of the as -\nsumed zero-βapproximation the density profile can be chosen\narbitrarily. The non-uniform thread is then modelled as a de n-\nsity enhancement with a two-dimensional distribution of de n-\nsity,ρ(r,z) (see Fig. 1). The density distribution has two non-\nuniform layers, with length landlzin ther- andz- directions,\nrespectively. The first is introduced so as to study the reson ant\ndamping of oscillations. The second produces irrelevant ph ys-\nical results, but enables us to avoid contact discontinuiti es and\nprovidesuswithacontinuousbackgrounddensity.Surfacep lots\nofthe densitydistributionfordi fferentthreadmodelsare shown\nin Figure 2, where we have made use of the symmetry of the\nsystem in the r- andz- directions and only plot their positive\nvalues. The dense part of the tube with prominence condition s,\ni.e., the thread, has a density ρfand occupies only part of the\nlarger magnetic flux tube. It extends over a length Lthreadin the\nz- direction.The rest of theinternalpart ofthe tube,with le ngth\nL−Lthreadinthelongitudinaldirection,isfilledwithplasmawith\na densityρev, with the subscript “ev” indicating the evacuatedI.Arregui et al.:Magnetohydrodynamic kink waves intwo-di mensional non-uniform prominence threads 3\nFig.1.Schematic representation of the model configuration adopte d in this work. The two-dimensional cylindrically symmetri c\nline-tied structure consist of a dense part, the thread, wit h lengthLthreadand densityρf, and two evacuated parts, with density ρev.\nBothregionsareseparatedbyanon-uniformlayerofwidth lzinthelongitudinaldirection.Transversenon-uniformity isconsidered\nonalayerofwidth l. Thestructureissurroundedbyplasmawithcoronalpropert iesanddensityρc.\nFig.2.Density distribution in the ( r,z)-plane in the domain r∈[0,rmax],z∈[0,L/2] for prominence thread models with non-\nuniform radial and longitudinal structuring. The threads a re defined by a cool and dense part with density, ρfand length Lthread\nand an evacuated part of the tube with density ρevand length L−Lthread. These two regions connect non-uniformlyto the coronal\nsurrounding medium with density ρc. Depending on the value of ρev, three different situations are possible: (a) ρev=ρc, (b)\nρc≤ρev≤ρf, and(c)ρev≤ρc.\npart of the tube. All lengths are normalised by taking a=1. In\ncontrast to Soleret al. (2010), the value of ρevcan be different\nfromthecoronaldensity, ρc, andmayhavevalueslowerthan ρc\nup to a valueequal to the filament density, ρf, case in which the\nfulltubeis occupiedwith densecoolplasmaandwe recoverth e\none-dimensional case. Both regions along the axis of the tub e\nare connected by means of a non-uniform transitional layer o f\nlengthlzto produce a smooth longitudinal profile. The density\nvariationat thislayer,with Lthread−lz/2≤z≤Lthread+lz/2,can\nbeexpressedas\nρz(z)=ρf\n2/bracketleftBigg/parenleftBigg\n1+ρev\nρf/parenrightBigg\n−/parenleftBigg\n1−ρev\nρf/parenrightBigg\nsinπ(z−Lthread)\nlz/bracketrightBigg\n, (1)\nforr≤a−l/2. As for the radial direction, the internal filament\nplasma, with density ρf, is connected to the external coronal\nplasma,with density ρc, bymeansofa non-uniformtransitional\nlayerofthickness l,definedintheinterval[ a−l/2,a+l/2],that\ncan vary in between l/a=0 (homogeneous tube) and l/a=2\n(fully non-uniform tube). In contrast to the one-dimension al\nmodel used by Arreguiet al. (2008) the dense plasma of the\nthread does not occupy the full length of the tube, hence theradial variation of the plasma density for z≤Lthread−lz/2 and\na−l/2≤r≤a+l/2isgivenby\nρr(r)=ρf\n2/bracketleftBigg/parenleftBigg\n1+ρc\nρf/parenrightBigg\n−/parenleftBigg\n1−ρc\nρf/parenrightBigg\nsinπ(r−a)\nl/bracketrightBigg\n. (2)\nRadial non-uniformity is not restricted to the dense part, a s in\nSoleret al. (2010), but can be present also in the evacuatedp art\nofthetube.Asaconsequence,thereisanoverlapregionofra dial\nand longitudinal non-uniform layers, for a−l/2≤r≤a+l/2\nandLthread−lz/2≤z≤Lthread+lz/2,wherethe densityis given\nby\nρrz(r,z)=ρz(z)\n2/bracketleftBigg/parenleftBigg\n1+ρc\nρz(z)/parenrightBigg\n−/parenleftBigg\n1−ρc\nρz(z)/parenrightBigg\nsinπ(r−a)\nl/bracketrightBigg\n.(3)\nIntheevacuatedpartofthetube,for z≥Lthread+lz/2anda−l/2≤\nr≤a+l/2,theradialdensityprofileisgivenby\nρr(r)=ρev\n2/bracketleftBigg/parenleftBigg\n1+ρc\nρev/parenrightBigg\n−/parenleftBigg\n1−ρc\nρev/parenrightBigg\nsinπ(r−a)\nl/bracketrightBigg\n. (4)\nNote that since the case ρev≤ρcis not excluded, the slope of\ntheradialdensityprofileintheevacuatedpartofthetubeca nbe4 I.Arregui et al.:Magnetohydrodynamic kink waves intwo-d imensional non-uniform prominence threads\npositive (ifρev<ρev), negative (ifρev>ρev) or zero (ifρev=\nρev). Finally, for r≥a+l/2 we reach the coronal medium and\nρ(r,z)=ρc.Themodeladoptedinthispaperremovesunrealistic\ndiscontinuitiesinthedensitydistribution,consideredi nprevious\nworks,andallowsthetheoreticalmodellingofalargenumbe rof\nthreadsbysimplyconsideringdi fferentgeometricalandphysical\nparameter values for, e.g., the three di fferent situations outlined\ninFigure2.\nThis paper is concernedwith standing kink waves in promi-\nnence threads. To study small amplitude thread oscillation s, we\nconsider the linear resistive MHD wave equations for pertur ba-\ntionsoftheform f(r,z)∼exp(i(ωt+mϕ))withconstantresistiv-\nity,η. Heremis the azimuthal wave-numberand ω=ωR+iωI,\nthe complexoscillatory frequency.For resonantlydampeds olu-\ntions, the real part of the frequency gives the period of the o s-\ncillation, P=2π/ωR, while the imaginary part is related to the\ndamping time,τd=1/ωI. Magnetic diffusion is only included\nhere to avoid the singularity of the MHD equations at the reso -\nnantposition,butdi ffusionhasnoeffectontheresonantdamping\ntime-scales. Oscillations are then governedby the followi ng set\nof partial differential equations for the two components of the\nvelocityperturbation, vrandvϕ,andthethreecomponentsofthe\nperturbedmagneticfield, br,bϕ,andbz,\niωvr=B\nµρ/parenleftBigg∂br\n∂z−∂bz\n∂r/parenrightBigg\n, (5)\niωvϕ=−B\nµρ/parenleftBiggim\nrbz−∂bϕ\n∂z/parenrightBigg\n, (6)\niωbr=B∂vr\n∂z+η/bracketleftBigg∂2br\n∂r2−m2\nr2br+∂2br\n∂z2+1\nr∂br\n∂r\n−2im\nr2bϕ−br\nr2/bracketrightBigg\n, (7)\niωbϕ=B∂v��\n∂z+η/bracketleftBigg∂2bϕ\n∂r2−m2\nr2bϕ+∂2bϕ\n∂z2+1\nr∂bϕ\n∂r\n+2im\nr2br−bϕ\nr2/bracketrightBigg\n, (8)\niωbz=−B/parenleftBigg∂vr\n∂r+vr\nr+im\nrvϕ/parenrightBigg\n+η/bracketleftBigg∂2bz\n∂r2−m2\nr2bz\n+∂2bz\n∂z2+1\nr∂bz\n∂r/bracketrightBigg\n. (9)\nEquations (5)–(9), together with the appropriate boundary con-\nditions, define an eigenvalue problem for resonantly damped\nmodes.Astheplasma- β=0,theslowmodeisabsentandthereare\nnomotionsparalleltotheequilibriummagneticfield, vz=0.We\nfurther concentrate on perturbations with m=1, which repre-\nsent kink waves that producethe transverse displacementof the\naxis of the tube. The MHD kink wave represents a wave mode\nwith mixed fast and Alfv´ en character, its Alfv´ enic nature being\ndominant in and around the resonant position (Goossenset al .,\n2009). The problem can be further simplified by making use of\nthedivergence-freeconditionfortheperturbedmagneticfi eld\n1\nr∂(rbr)\n∂r+1\nr∂bϕ\n∂ϕ+∂bz\n∂z=0, (10)\nwhichreducesthesystemofequationstobesolvedtofour,up on\nexpressing bϕintermsof brandbz.Solutionstotheseequations\nare obtained by performing a normal mode analysis. Becauseof the complexityof the problem when a two-dimensional den-\nsityρ(r,z) is considered, numerical solutions to the frequency\nand spatial structure of eigenfunctions in the ( r,z)- plane are\ncomputed using PDE2D (Sewell, 2005), a general-purposepar -\ntial differential equation solver. The code uses finite elements\nand allows the use of non-uniformly distributed grids, whic h\nare needed to properly resolve the large gradients that aris e in\nthevicinityofresonantpositions.Di fferentgridresolutionshave\nbeen tested so as to assure the proper computation of the res-\nonant eigenfunctions. Magnetic dissipation has no influenc e on\nthe dampingof kinkmodesdue toresonantabsorption,a condi -\ntion that has to be checked in all numerical computations, bu t\nallows us to properly compute the spatial distribution of pe r-\nturbations in the resonance. We have made use of the symme-\ntry of the system and solutions are computed in the domain\nr∈[0,rmax],z∈[0,L/2].Non-uniformgridsareusedinbothdi-\nrections,toproperlyresolvetheregionswith r∈[a−l/2,a+l/2]\nandz∈[Lthread/2−lz/2,Lthread/2+lz/2]. The first is located so\nas to include the non-uniformtransitional layer in the radi al di-\nrection,while thesecondembracesthenon-uniformtransit ional\nlayer along the tube. As for the boundaryconditions, we equa te\nthem to the spatial distribution of perturbations for the fu nda-\nmental kink mode. The two perturbed velocity components, vr\nandvϕand the compressive component of the perturbed mag-\nnetic field, bzhave vanishing longitudinal derivatives at z=0,\nwhich correspondsto the apex of the flux tube, while they van-\nish atz=L/2, because of the line-tying boundary condition at\nthe photosphere. In the radial direction, they also have van ish-\ning radial derivative at the axis of the tube, r=0, while we\nimpose the vanishing of the perturbed velocity components f ar\nawayfromthe tubein theradialdirection,hence( vr,vϕ)→0as\nr→∞, a condition that is accomplished by setting the pertur-\nbations equal to zero at r=rmax, wherermax, the upper limit of\nthe domain in the radial direction, has to be chosen to be su ffi-\ncientlyfartoproperlycomputethedrop-o ffrateofperturbations\nin the radial direction. In all our computations,we have con sid-\neredrmax=20a.\n3. Analysisandresults\nArreguietal. (2008), in their analysis of the damping of kin k\noscillations in one-dimensional thread models, showed tha t the\nparameters that determine the temporal attenuation of osci lla-\ntions are the density contrast, ρf/ρcand the width of the non-\nuniform transitional layer, l/a. The damping ratio is rather de-\npendent on the first parameter for low values of the density\ncontrast, but stops being dependent in the high contrast rat io\nregime, typical of prominenceplasmas. The strongest influe nce\ncomes from the width of the transitional layer, with the damp -\ning time rapidly decreasing for increasing values of l/a. Before\nwe deal with the additional parameters introduced by the lon -\ngitudinal density structuring in two-dimensionalthread m odels,\nsolutions to Equations (5)–(9) have been first obtained by us -\ningatwo-dimensional(2D)densitydistributionwith Lthread=L.\nThe purpose of these numerical experiments has been to check\nthe correctbehaviourof the code by reproducingthe results ob-\ntained by Arreguiet al. (2008). In addition, we have also con -\nsidered the magnetic Reynolds number, Rm=vAfa/η, which\nshould not affect the computed damping times, in the limit of\nlarge Reynoldsnumbers. Figure 3 displays the obtained resu lts.\nThedampingtimeofresonantlydampedkinkwavesisindepen-\ndentofthemagneticReynoldsnumber,aslongasthisquantit yis\nlargeenoughforresonanceabsorptiontobetheoperatingda mp-\ningmechanism.Thisregime(seetheplateauregions)isobta inedI.Arregui et al.:Magnetohydrodynamic kink waves intwo-di mensional non-uniform prominence threads 5\nFig.3.Dampingtime,inunitsoftheinternalfilamentAlfv´ encross ingtime,τAf=a/vAf,asfunctionofthethreerelevantparameters\nfor kink oscillations in one-dimensionalthread models: (a ) the magnetic Reynolds number, Rm=vAfa/η, (b) the density contrast,\nρf/ρc, and (c) the transverse inhomogeneity length-scale, l/a. Solid lines indicate the 1D solution, while symbols repres ent the\nnumerical2D solutionsobtainedwith Lthread=L. In (a) and (c) the density contrast is ρf/ρc=200.In (b), l/a=0.2.In (b) and(c)\nRm=106.Allcomputationshavebeenperformedinatwo-dimensional gridwith Nr=401andNz=51points,with250grid-points\nintheradialtransitionallayer.Lengthsarenormalisedto a=1 andL=50a.\nFig.4.(a)-(c): period, damping time, and damping ratio as a functi on of the length of the thread for thread models with l/a=0.2\nandfortwovaluesofthedensitycontrast.(d)-(f):thesame quantitiesforthreadmodelswith ρf/ρc=200andforthreevaluesofthe\ntransverseinhomogeneitylength-scale.Inall figures a=1,Rm=106,ρev=ρc, andlz=a.Timesareshowninunitsofthe internal\nfilament Alfv´ encrossingtime, τAf=a/vAf. Symbolscorrespondto fullynumerical2D computationswhi le thedifferentline styles\nrepresenttheapproximate2DsolutionsobtainedbySoleret al.(2010).Allcomputationshavebeenperformedinatwo-di mensional\ngridwith Nr=401andNz=51points,with250grid-pointsin theresonantlayer.Lengt hsarenormalisedto a=1andL=50a.\nfor different values of Rmwhen different transitional layers are\nconsidered. Figure 3a shows a perfect agreement between the\n1D resultsandthecurrentcomputationsusingthe 2D code.Th e\nperfect correspondence between 1D and 2D computations for\nthe dampingtime as a functionof the density contrast in the r a-\ndial directionis shownin Figure3b.Finally,the most impor tant\nparameter that determines the damping of transverse thread os-\ncillationsisthewidthofthenon-uniformtransitionallay erinthe\nradialdirection.Thedampingtimestronglydecreaseswhen this\nparameterisincreased,ascanbeseenin Figure3c,whichaga in\nshows a very good agreement between the values computed in\n2D and the previous 1D computations. These results were inagreementwithpreviousworksinthecontextofthedampingo f\ncoronal loop transverse oscillations (e.g. Goossenset al. , 1992;\nRuderman&Roberts, 2002; Goossensetal., 2002)\nOnce we are confident about the goodness of the code, we\nconsiderthenewingredientsintroducedbythetwo-dimensi onal\nnatureoftheprominencethreadmodelsconsideredinthiswo rk.\nThese new ingredients are the length of the thread, Lthread, i.e.,\nthe lengthof the part of the magneticflux tube filled with dens e\nabsorbing plasma, and the density in the evacuated part of th e\ntube,ρev. Thenon-uniformtransitionallayerin densitybetween\nbothregionsalongthetube, lz,hasanirrelevante ffectontheos-\ncillatory period for low harmonicsin the longitudinaldire ction,6 I.Arregui et al.:Magnetohydrodynamic kink waves intwo-d imensional non-uniform prominence threads\nas shown by D´ ıazet al. (2008). Our computations (not shown\nhere) confirmthe findingby D´ ıaz etal. (2008) on the negligib le\nimportanceofthisparameterfortheoscillatoryperiodand show\na similar irrelevance concerning the damping time by resona nt\nabsorption. For this reason, we have concentrated our analy sis\nontheremainingtwo parameters, Lthreadandρev.\n3.1. Periodsanddampingtimes\nWe first consider the influence of the length of the thread on\nthe periodand dampingof resonantlydampedtransverse thre ad\noscillations. An initial analysisonthis subject was prese ntedby\nSoleret al.(2010),whoconsideredthethintubeandthinbou nd-\nary approximations(TTTB) in a two-dimensional thread mode l\nwithtransverseinhomogeneityonlyinthedensepartofthet ube.\nOur analysis goes beyondthe TTTB approximationsby consid-\nering a fully inhomogeneoustwo-dimensional density distr ibu-\ntion and combining its influence with that of the density in th e\nevacuatedpartofthetube, ρev.\nWehavefirstanalysedthevariationofperiod,dampingtime,\nand damping ratio, τd/P, by settingρev=ρc, so that we mimic\nthecasestudiedbySoleret al.(2010)analytically.Westar twith\nafullyfilledtubeandgraduallydecreasethelengthoftheth read.\nThe obtained results, for two values of the density contrast be-\ntweenthefilamentandcoronalplasma,areshowninFigures4a -\nc.Theperiodisstronglydependentonthelengthofthethrea d.It\ndecreasesbyalmosta factoroftwowhengoingfrom Lthread=L\ntoLthread=0.1L. Figure 4a also shows that the oscillatory pe-\nriod is almost independentof the density contrast, once thi s pa-\nrameter is large enough. As for standing kink waves in one-\ndimensional thread models (Arreguiet al., 2008), the kink f re-\nquency is a weighted mean of the internal and external Alfv´ e n\nfrequencies.Regardless of the density contrast, the perio d is al-\nlowed to vary in a a narrow range determined by a factor that\ngoes from√\n2 to 1, when going from ρf/ρc=1 toρf/ρc→∞.\nFor typical density contrasts in prominence plasmas, the pe -\nriod can be considered independentof the density contrast. The\ndampingtimeproducedbyresonantabsorption(Fig.4b)also de-\ncreases remarkably when the length of the cool and dense part\nof the tube is decreased. The decrease is also around a factor\nof two in the considered range of values for Lthread. Soleret al.\n(2010) find that in the TTTBlimit the dependenceof the period\nandthedampingtimewiththelengthofthethreadisexactlyt he\nsame, hence any influence on the damping ratio, τd/P, is can-\ncelled out. Outside the TTTB approximations, we find that thi s\nis not the case (see Fig. 4c), although the damping ratio is al -\nmostindependentofthelengthofthreadandonlyforverysho rt\nthreadsaslight increaseinthedampingratioisfoundwhenf ur-\nther decreasing this parameter. In Figure 4 we overplot resu lts\nobtained by Soleretal. (2010) by solving their dispersion r ela-\ntion. We see that there is a very good agreement and the di ffer-\nences,that are duetothe simplifyingassumptionsofthe ana lyt-\nical treatment, are rather small. One can observe an anomalo us\nbehaviouronthedampingtimecomputedbySoleret al.(2010) ,\nforsmallvaluesof Lthread.Thisisduetothesimplifyingassump-\ntionsconsideredto obtain the semi-analyticsolution,tha t might\nnot be entirely valid outside the long-wavelengthlimit. Ov erall,\nourresults confirmthe validity of the analytical approxima tions\nobtained by Soler etal. (2010) concerning the influence of th e\nlength of the thread on periods and damping times. In physi-\ncal terms, the shortening of the length of the thread produce s\nshorter period oscillations, since the physical system is e quiv-\nalent to a fully filled tube, with the wavelength of oscillati onsreplaced by a shorter e ffective wavelength. A physical explana-\ntionofthedampingtimedependenceonthelengthofthethrea d\nisprovided,byusingenergyarguments,inSect. 3.3.\nSimilar conclusions can be extracted from the computa-\ntions we have performed by fixing the density contrast and for\nthree different values of the width of the inhomogeneouslayer.\nFigures 4d-f show the obtained results. They clearly show ho w\nstronglythe dampingtime and the dampingratio are influence d\nby the width of the transitional layer, also in 2D models, whi le\nthe period of the oscillations is almost una ffected by the value\nofl/a.In view ofthe results displayedin Figure 4, we conclude\nthatthelengthofthethreadisaveryimportantparameter.W hen\nallowing to vary from the limit of fully filled tube to 10% fille d\ntube, periodsand dampingtimes are decreasedas much as 50%\npercent. This is very relevant in connectionto prominences eis-\nmology.We mustnotethat,in principle,thelengthofthethr ead\ncanbeestimateddirectlyfromobservations.However,thel ength\nofthesupportingmagneticfluxtubeismuchmoredi fficulttoes-\ntimate,sinceits endpointsareusuallyunobservable.\nOur generaldensity modelenable us to analyse the e ffect of\nthe density in the evacuated part of the tube on the oscillato ry\nproperties, as well. The study of the influence of this parame ter\nwas not undertaken by Soleret al. (2010), and requires a full y\nnumerical approach. We allow for ρevto be different from the\ncoronaldensity,ρc. Whenchangingthe densityinthe evacuated\npart of the tube, in addition to the radial non-uniformlayer that\nconnectstheprominencematerialtothecoronaweareintrod uc-\ning an additional radial non-uniformlayer in between the ev ac-\nuated part of the tube and the corona.The densityprofile in th is\nlayer and its slope depend on the relative values of ρevandρc,\nas givenin Equation(4) andshown in Figs. 2b and c. Insteadof\nanalysingtheeffectofρevontheoscillatorypropertiesinasepa-\nratemanner,wehaveselectedthreerepresentativevaluesf orthe\nlengthofthethreadandhavecomputedperiods,dampingtime s,\nand dampingratios as a functionof the density in the evacuat ed\npart of the tube, measured in units of the coronal density. We\nhavesplitouranalysisintotwoparts.\nFirst, we consider that ρc≤ρev≤ρf, hence the parameter\nis allowed to vary in between the coronal and prominence den-\nsities. Fig.5displaystheobtainedresults.For ρev=200ρc=ρf,\nthe tube is fully filled with cool and dense plasma. Then, as we\ndecreaseρev, periods and damping times have a marked linear\ndecrease.When 40% of the tube is filled with cool plasma, they\ndecrease by about a 15%. When a 10% filled tube is consid-\nered a decrease of up to a 50% is obtained. In physical terms,\nthe period decrease when the density in the evacuated part of\nthe tube is gradually decreased can be understood if we think\nabout the different inertia of the system, with or without dense\nplasma at those locationsalongthe tube. For explainingthe dif-\nferent damping time-scales, energy argumentsthat combine the\nenergyof the mode and the energyflux into the resonance have\ntobeconsidered,seeSect.3.3.Thedecreaseinperiodandda mp-\ning time is very similar, but not exactly the same. For instan ce,\nFig.5cshowsthatthedampingratioisslightlydependenton the\ndensity in the evacuated part of the tube, for all the conside red\nvaluesintherange ρc≤ρev≤ρf.\nNext, we have considered the possibility of the density in\nthe evacuated part of the tube being lower than the coronal\ndensity. For instance, D´ ıazet al. (2002) considered a valu e of\nρev=0.6ρc. The computations shown in Fig. 5 are extended to\nlowervaluesofρev.Theobtainedresultsaredisplayedinthein-\nsetplotsofFig.5andshowthatthedensityintheevacuatedp art\nof the tube is irrelevant in relation to the period of the osci lla-I.Arregui et al.:Magnetohydrodynamic kink waves intwo-di mensional non-uniform prominence threads 7\nFig.5.Period,dampingtime,anddampingratioasafunctionofthed ensityintheevacuatedpartofthetubeforprominencethrea ds\nwithρf/ρc=200,a=1,Rm=106,lz=a, andl/a=0.2 and three values of the length of the thread. The main plots a re for\nρc≤ρev≤ρf, while the inset plots correspond to the range 0 .1ρc≤ρev≤ρc. Times are shown in units of the internal filament\nAlfv´ en crossing time, τAf=a/vAf. All computationshave been performed in a two-dimensional grid with Nr=401 andNz=51\npoints,with 250grid-pointsin theresonantlayer.Lengths arenormalisedto a=1andL=50a.\nFig.6.Longitudinalandradialdependenceoftheeigenfunctionst ransversevelocitycomponent, vr,azimuthalvelocitycomponent,\nvϕ, and compressive magnetic field component, bz, in prominence threads with ρf/ρc=200,l/a=0.2,a=1,ρev=ρc, and\nRm=106, for different values of the length of the thread. All computations ha ve been performed in a two-dimensional grid with\nNr=401andNz=51points,with250grid-pointsintheresonantlayer.\ntions and the dampingby resonant absorption in the consider ed\nrangeofvalueswith0 .1ρc≤ρev≤ρc.\nThese results show that the density in the evacuated part of\nthetubeis alsoa relevantparameterforprominencethreads eis-\nmology,because of its e ffect on periodsand dampingtimes and\nthe difficulty in being measured by direct means. When consid-\nered in combinationwith e ffects due to the length of the thread,\nout computations enable us to perform a more accurate promi-\nnenceseismology,applicabletoa largenumberofthreads.\n3.2. Spatialdistributionof eigenfunctions\nChanges in the longitudinal density structuring have a dire ct\nimpact on the oscillatory period, which is easy to understan d,\nbut also on the damping time by resonant absorption. Besidesobtaining the parametric behaviour of kink mode periods and\ndampingtimesasafunctionofthelongitudinaldensitystru ctur-\ning, we aim to explain the obtained results. As a first step, we\nhave analysed the spatial structure of eigenfunctions. The rel-\nevant perturbed quantities are the radial and azimuthal vel oc-\nity components, vrandvϕ, and the compressive component of\nthe perturbedmagneticfield, bz, directlyrelatedto the magnetic\npressureperturbation, PT=Bbz/µ.\nWe first considertheinfluenceof thelengthof the threadon\nthe profiles of the eigenfunctions in the radial and longitud inal\ndirections.Resultsaregivenintermsofthemodulusofthec om-\nplex eigenfunctions. Fig. 6 shows one-dimensional cuts alo ng\nthe longitudinal and radial directions of the eigenfunctio ns for\ndifferentvaluesofthelengthofthethread.Thelongitudinalpr o-\nfiles are shown at the axis ( r=0) forvr, and at the mean radius8 I.Arregui et al.:Magnetohydrodynamic kink waves intwo-d imensional non-uniform prominence threads\nofthetube( r=a)forvϕandbz. Inthelongitudinaldirectionall\nthreeeigenfunctionsdisplaya trigonometricdependencew ithz,\nwhenLthread=L, i.e., the case that mimics the one-dimensional\nthread.When the lengthof the thread is decreasedseveralin ter-\nesting effects occur. First, both perturbed velocity components\ndisplay a slightly improved confinement. The maximum values\nof the velocity perturbations still occur in the dense part o f the\ntube, but the drop-o ffrate changes, becomingalmost linear out-\nside the threadso that theysatisfy the boundaryconditiona t the\nfoot-point of the tube. When eigenfunctions are normalised to\nthevalueof|vr|attheapexofthetube,itisseenthatthedecrease\nof the lengthof the thread produceslargeramplitudesof the az-\nimuthalvelocitycomponent(seeFig.6b),relatedtotheAlf v´ enic\ncharacter of the mode. This amplitude almost doubles its val ue\nwhengoingfrom Lthread=LtoLthread=0.04L.Thez-component\nofthemagneticfieldperturbationgivesanindicationofthe com-\npressibility of the normal mode. Our results indicate that w hen\nthelengthofthethreadisdecreased,thelongitudinalprofi leofbz\n(Fig.6c)becomesstrictly confinedto thedensepart ofthe tu be,\nwhere it reachesits maximumvalue. As the thread gets shorte r,\nthe maximum value of bzincreases, hence the compressibility\nbecomes larger at the apex of the tube for shorter threads osc il-\nlations, though the kink mode remains being an almost incom-\npressible wave mode. Kink modes in fully filled magnetic flux\ntubes are almost incompressible. Longitudinal density str uctur-\ningbytheinclusionofadensecentralpartincreases(decre ases)\nkink mode compressibility in the dense (evacuated) parts of\nthe tube, in comparison to the one-dimensional flux tube kink\nmodes.\nA close look at the spatial structure of eigenfunctionsin th e\nradialdirection(Figs.6d-f)providesuswithaqualitativ eexpla-\nnation on why the change of the length of the thread a ffects the\ndamping times computed in the previous subsection (and also\nthosepresentedbySoleret al.2010).First,thechangeinth epe-\nriod produced by the change in the length of the thread a ffects\ntheresonantpositionintheradialdirectionandhencethed amp-\ning time. However, the slope of the density profile in the tran -\nsitional layer is very large, because of the high density con trast\nratiotypicaloffilamentthreadsandthise ffectisnotveryimpor-\ntant.Second,theeigenfunctionsintheresonantlayerarea ffected\nby the value of the length of the thread. Both e ffects are clearly\nseen in Figs. 6d-f, where the radial dependenceof vr,vϕ, andbz\nis plotted for different thread lengths. We first note that the per-\nturbed velocity components at the resonance have a larger am -\nplitude the shorter the length of the thread. We interpret th is as\nanindicationofanimprovede fficiencyoftheresonantdamping\nmechanism.Thisisparticularlyclearifwelookattheazimu thal\nvelocity profile in the resonant layer (Fig. 6e). Now, not onl y\nthe increase of the amplitude for shorter lengths is evident , but\nalso the transverselengthscale is seen to decreasewhen sho rter\nthreads are considered, hence thinner resonance widths are ob-\ntained. This detail of the resonance also shows a slight shif t to-\nwards the left hand side which is due to the change of the reso-\nnantpositionfordi fferentlengthsofthethreads(hencedi fferent\noscillatoryperiods).Althoughtheresonancesarenotexac tlylo-\ncated atrA=a, our numericalresultsindicate that this approxi-\nmation,usedbySoleret al.(2010),isfullyjustified.Final ly,the\nradialprofileofthecompressivecomponentofthemagneticfi eld\nperturbation (Fig. 6f) shows the increase in its amplitude a t the\nresonant position mentioned above, when considering short er\nthreads. This is anotherindication of the improvede fficiencyof\nresonant damping. Notice that although our study is limited to\nlinear MHD waves, in a realistic situation perturbations in side\ntheresonantlayerbecomenonlinear,asshowninstudiesbye .g.,Terradaset al. (2008a); Clack et al. (2009); Ballai & Ruderm an\n(2011).\nWe have next examined the spatial distribution of the rel-\nevant perturbed quantities as a function of the density in th e\nevacuated part of the tube. Fig. 7 shows the obtained profiles\nin the longitudinal and radial directions for a fixed value fo r\nthe length of the thread. As before, the longitudinal profile s are\nshown at the axis ( r=0) forvr, and at the mean radius of the\ntube (r=a) forvϕandbz. Our results indicate that the den-\nsity in the evacuated part of the tube has also a direct impact\nontheradialandlongitudinalprofilesofeigenfunctions.F orthe\nlongitudinal profiles, as we decrease the value of ρevfromρf,\nwe have a slightly improvedconfinementof the velocitypertu r-\nbations and a strict confinement of the longitudinal compone nt\noftheperturbedmagneticfieldto thedensepartofthe tube(s ee\nFigs.7a-c).Theamplitudeof bzattheapexofthetubeincreases,\nwhile in the evacuated part of the tube, and for a fixed length\nof the thread, compressibility decreases as ρevis decreased (see\nFig. 7c). Taking a look at the eigenfunctions in the radial di -\nrection (Figs. 7d-f), the amplitude of both perturbed veloc ity\ncomponentsincreases when the density in the evacuatedpart of\nthe tube is decreased. The increase in the resonance peak and\nthe shorteningof the transverse spatial scale are mainly ev ident\nfortheazimuthalvelocitycomponent,whichgivesitsreson antly\ndampedandAlfv´ eniccharactertothemode.Wefindanincreas e\nin the compressibility of the normalmode (Fig. 7f) in the den se\nprominenceplasmaregionasthedensityintheevacuatedpar tof\nthetubeisdecreased.\nOverall, the decrease of the density in the evacuated part\nof the tube, starting from a fully filled tube, produces simil ar\nqualitative effects on the radial and longitudinal profiles for the\neigenfunctions as the ones produced by the shortening of the\nlengthof thethread.In the parameterrangestudiedin this w ork\nthoseeffectsarequantitativelymoreimportantinthecaseofthe\nchanges of the length of the thread. The properties of the spa -\ntial structure of eigenfunctionsgive us a qualitative expl anation\nonwhychangesinthelongitudinaldensitystructuringinpr omi-\nnence threads have a significant e ffect on the damping time of\nkinkoscillationsin two-dimensionalthreadmodels.\n3.3. Energyanalysis\nOur analysis of the spatial structure of eigenfunctionsind icates\nthat the decrease of both the length of the thread and the den-\nsity in the evacuatedpart of the tube producea strengthenin gof\ntheresonanceabsorptionprocessthatbecomesapparentthr ough\nthe appearance of more pronounced resonant profiles and thin -\nner resonance widths in the perturbed velocity components a t\nthe resonance. This would explain the marked decrease of the\ndampingtimesfoundin Section3.1whenvaryingthese twopa-\nrameters.Thisresult showsthat resonantdampingmaystron gly\ndepend on the details of the longitudinal density structuri ng in\nrathergeneraltwo-dimensionaldensitymodels,suchasthe ones\nusedheretomodelprominencethreads.\nIn order to obtain a quantitativeexplanationan energyanal -\nysis is carried out. Our analysis involves the energy of the k ink\nmode and the energyflux into the resonance. The energyof the\nmode can directly be computed from the numerical eigenfunc-\ntionsas\nE=1\n2/parenleftBig\nρv2+b2/parenrightBig\n. (11)\nThis expression has to be integrated over the entire ( r,z)-\nplane except for the resonance layer, easily recognisable f romI.Arregui et al.:Magnetohydrodynamic kink waves intwo-di mensional non-uniform prominence threads 9\nFig.7.Longitudinalandradialdependenceoftheeigenfunctionst ransversevelocitycomponent, vr,azimuthalvelocitycomponent,\nvϕ, and compressive magnetic field component, bzin prominence threads with ρf/ρc=200,l/a=0.2,a=1,Lthread=0.2L,\nandRm=106for different values of the density in the evacuated part of the tube. All computations have been performed in a\ntwo-dimensionalgridwith Nr=401andNz=51points,with250grid-pointsin theresonantlayer.\nthe eigenfunctions displayed in Figs. 6 and 7, as this region\nwould include a contribution from the Alfv´ en waves. In one-\ndimensional equilibrium models, the energy flux into the res -\nonance is proportional to the magnetic pressure perturbati on\nsquared(Andrieset al., 2000; Andries&Goossens, 2001), a r e-\nsult that was used by Arreguiet al. (2007b) to analyse the infl u-\nenceof the internalstructuringof coronalloopson the damp ing\nby comparing the e fficiency of the process at internal and ex-\nternal layers. In two-dimensional equilibrium states, the energy\nfluxabsorbedataparticularfieldlineisproportionaltothe over-\nlap integral between PT(rA,z), the profile of PTalong the tube\nat the resonantposition,andthe resonantAlfv´ eneigenfun ctions\n(Thompson&Wright, 1993; Tirry&Goossens, 1995). This is\nwhythelongitudinalprofilesof bzandvϕ,andtheirmodification\nduetochangesinthelongitudinaldensitydistributionare sorel-\nevantindeterminingkinkmodedampingtimes.Ageneralmath -\nematicalexpression,validinthethree-dimensionalcase, isgiven\nby Wright& Thompson (1994). When adapted to our cylindri-\ncal equilibrium,this expression gives the energy flux at the res-\nonanceperunitϕintheform\nF=rA/integraldisplay/bracketleftbigSr(r−\nA)−Sr(r+\nA)/bracketrightbigdz (12)\n=πm2\n4µ2\n0×B2\nrA(/integraltext\nφbzdz)2\n/integraltext\nφρφdz×/parenleftBiggdωA\ndr/parenrightBigg−1\nrA.\nIn this expression, Sr(r−\nA) andSr(r+\nA) represent the values of the\nradial component of the time-averaged Poynting vector to th e\nleft and to the right of the resonance position, that we appro x-\nimate by rA=a,φis the Alfv´ en eigenfunction along the tube\nat the resonant position, and ωA(r) the Alfv´ en continuum fre-\nquency.In order to evaluate the integrals as well as the slop e oftheAlfv´ encontinuumattheresonantpositioninexpressio n(12),\nwe have first solved the following equation for the Alfv´ en co n-\ntinuummodes\nd2φ(rA,z)\ndz2+ω2\nA(rA)\nv2\nA(rA,z)φ(rA,z)=0, (13)\nwithboundaryconditions\ndφ\ndz(z=0)=0,φ(z=L/2)=0.\nFor each value of rA, this equation is solved for di fferent val-\nuesfortherelevantparameters Lthreadandρev,thatinturndefine\ndifferent profiles for v2\nA(rA,z). The Alfv´ en continua arise from\nrepeating the procedure for di fferent values of rAin the range\n[a−l/2,a+l/2].Oncethisisdone,theslopeoftheAlfv´ encon-\ntinuaattheresonantpositionandtheAlfv´ eneigenfunctio nsthat\ncorrespondto eachthreadmodel can be evaluated.By using th e\nnumerically computedprofiles for bz(rA,z), in order to evaluate\ntheoverlapintegralinEquation(12),andmultiplyingby2 π,the\nrequired total energy flux is obtained. The ratio of the energ y\nof the kink mode to the time-averaged energy flux into the res-\nonance gives the required damping time, for every considere d\nvalueofLthreadandρev.\nFigure 8 shows the obtained results. They have been nor-\nmalised to the value of the damping time for the fully filled\ntube.Weseeanexcellentagreementbetweenthedampingtime s\ncomputedthroughthe energyanalysisexplainedin detail ab ove\nand the numerically computed ones for both cases in which we\nchange the length of the thread and the density in the evacu-\nated part of the tube. The results obtained in Section 3.1 can\ntherefore be explained in terms of the energetics of the mode s\nand the resonant energy transfer. These results also show th e10 I.Arregui et al.:Magnetohydrodynamic kink waves intwo- dimensional non-uniform prominence threads\nFig.8.(a) Damping time, normalised to the full tube damping\ntime, as a function of the length of the thread, for ρev=ρc.\n(b) Dampingtime, normalised to the full tube dampingtime, a s\na function of the density in the evacuated part of the tube, fo r\nLthread=0.2L. In both figures lines correspond to the numeri-\ncally computed damping times and the symbols are the values\nobtainedthroughtheenergyanalysisdescribedin Section3 .3.\naccuracy and utility of the analytical expression derived b y\nWright&Thompson (1994) for the time-averaged energy flux\nin terms of eigenfunctions. Furthermore, we have just demon -\nstrated that damping time estimates can be obtained, using e n-\nergy arguments, without the need to solve the full non-unifo rm\nproblem, but instead by solving the much more simpler piece-\nwise uniformproblemin orderto obtain the real part of the fr e-\nquency and the longitudinal profiles for the eigenfunctions for\nkink modes in combinationwith the non-uniformcomputation s\nfortheAlfv´ encontinuummodes,usingthisinformationinc om-\nbination with Equations (11) and (12). This is possible beca use\ntherealpartofthefrequencyandthelongitudinalprofileso fthe\neigenfunctions are only slightly a ffected by resonant coupling,\nandbecausetheenergyinEquation(11)hastobeintegratede x-\ncludingtheresonantlayer.Ofcourse,thefullnumericalso lution\nprovides us with more accurate results, but as Fig. 8 illustr ates\ntheagreementbetweenbothmethodsisexcellent.\nAconvenientwaytofurtherunderstandtheenergyfluxatthe\nresonanceandthedynamicsofresonantlydampedkinkmodesi s\ntocomputethespatialdistributionofthetime-averagedPo ynting\nflux in our two-dimensional domain by making use of the gen-\neralexpression\n<S>=1\n2Re(E×b∗), (14)Fig.9.(a) Spatial distribution in the ( r,z)-plane of the time-\naveraged Poynting flux given by Equations (15) and (16). (b)\nOhmicheatingdistribution,in arbitraryunits. Theseresu ltscor-\nrespondtothetransverseoscillationofaprominencethrea dwith\nρf/ρc=200,ρev=ρc,l/a=0.2,a=1, andLthread=0.2L. The\nmagneticReynoldsnumberis Rm=106.\nwithE=−(v×B)+ηjtheperturbedelectricfield, b∗=(b∗\nr,b∗\nϕ,b∗\nz)\nthe complex conjugate of the perturbed magnetic field, and j=\n(∇×b)/µthe current. In contrast to the energy analysis above,\nwe now make use of the full numerical computations. In terms\nof the wave fields analysed in Section 3.2, the two components\nof the time-averaged Poynting vector in the ( r,z)- plane can be\ncast as\n<Sr>=1\n2B\nµRe(vrb∗\nz)+η\n2µ2Re/bracketleftBigg\nb∗\nz/parenleftBigg∂br\n∂z−∂bz\n∂r/parenrightBigg\n−b∗\nϕ/parenleftBigg∂bϕ\n∂r+bϕ\nr−im\nrbr/parenrightBigg/bracketrightBigg\n, (15)\n<Sz>=−1\n2B\nµRe(vrb∗\nr+vϕb∗\nϕ)+η\n2µ2Re/bracketleftBigg\nb∗\nϕ/parenleftBiggim\nrbz−∂bϕ\n∂z/parenrightBigg\n−b∗\nr/parenleftBigg∂br\n∂z−∂bz\n∂r/parenrightBigg/bracketrightBigg\n. (16)\nBy making use of the divergence-free condition given by\nEquation (10) to compute bϕ, we have produced an example of\nthe two-dimensional distribution of the time-averagedPoy nting\nfluxaroundtheresonantlayer,fora partiallyfilled thread.\nThe arrow plot in Fig. 9a shows that energy is fed into the\nresonant layer by concentrating it over the dense thread sec -\ntion. The amount of energy flux in the radial direction is de-\ntermined by the jump in <Sr>(see Stenuitet al., 1999, for aI.Arregui et al.:Magnetohydrodynamic kink waves intwo-di mensional non-uniform prominence threads 11\none-dimensional example in ideal MHD). The resistive layer is\non a scale where both the ideal and resistive terms of the per-\nturbed electric field can be of similar magnitude, but each of\nthem could be dominant over di fferent regions of the domain.\nFor instance, the jump in <Sr>is determined by the ideal\nterm in Equation (15) which is dominant in the thread region\nand zero in the evacuatedpart of the tube. Hence the energyin -\nflow towards the resonance in the radial direction comes from\nthe interior of the tube and is determined by the spatial profi les\nof the transverse velocitycomponentand the compressivema g-\nneticfieldperturbation.Notethattheamplitudeofthekink mode\nis smaller outside the tube compared to inside and also that t he\nevacuatedpartofthetubeismuchlessdensethanthe thread.\nOnce in the layer,the energyflow is divertedalong the field\nlinesinamannerdeterminedby <Sz>,withasmallradialcon-\ntributionthatisduetotheresistivetermsinEquation(15) .Such\nas correspondsto the Alfv´ enic character of the mode inside the\nresonant layer, the dominant term in Equation (16) involves the\nazimuthal velocity and magnetic field perturbations, vϕb∗\nϕ. This\nquantity decreases linearly along the field lines, producin g the\nlessening in the parallel Poynting flux towards the foot-poi nt.\nIn our example, the radial non-uniformlayer is restricted t o the\ndensepartofthefluxtube,since ρev=ρc.However,energycon-\ncentratedin the resonantlayer ofthe threadbecauseof reso nant\nwavedampingcanflowalongthefieldlinesand,eventually,su p-\nplyheatingintheevacuatedregion,wherefieldalignedcurr ents\nare dominant. Although the energy inflow into the resonance,\ngiven by<Sr>, and its subsequent divergence along the field\nlines, given by<Sz>, are mostly determined by ideal terms\nin Equations(15) and (16), this does not mean that resistivi ty is\nnot important. For instance, the amount of heating, in the fo rm\nof Ohmic dissipation, will be determined by those currents, re-\nsistivity, and their spatial distribution. For this partic ular case,\nheatingis distributedin a constantmannerin the evacuated part\nof the tube, even if there is no resonant layer in that region ( see\nFig.9b).\n4. Summaryand Conclusions\nQuiescent filament fine structures are only partially filled w ith\ncold and dense absorbing material. The length of the threads\ncan in principle be measured in events showing transverse os -\ncillations, provided the lifetime of threads is su fficiently large\ncomparedtotheoscillatoryperiod.Thelengthofthesuppor ting\nmagneticfluxtubesare howevermuchlarger,andcannotbe ob-\nserved.Densitymeasurements,bothinthethreadasintheev ac-\nuated part of the supportingmagnetictube, are also challen ging\nfrom the observational point of view. It is therefore import ant\nto quantify the variations on wave properties due to changes in\nthese equilibrium parameters if we aim to perform an accurat e\nprominence seismology. It is essential to have computation s of\nperiods and damping times for a wide range of thread models\nto include regimes in which the applicability of simple anal yti-\ncal models could be of limited extent. For this reason we have\ncomputedtheoscillatorypropertiesofresonantlydampedt rans-\nverse kink oscillations in rather general two-dimensional fully\nnon-uniformprominencethreadmodels.Thisallowsforabro ad\nrange of prominence threads with very di fferent physical con-\nditions to be modelled and their oscillatory properties cha rac-\nterised.\nThelengthofthethreadandthedensityintheevacuatedpart\nof the tube definetheir longitudinaldensity structuring.W e find\nthat the length of the thread strongly influences the period a nd\ndampingtimeoftransversekinkoscillations,whilethedam pingratioisratherinsensitivetothisparameter.Theseresult sconfirm\nthevalidityoftheanalyticalapproximationsmadebySoler et al.\n(2010). In addition, our modelling has allowed us to identif y\na new physical parameter with seismological implications, the\ndensityintheevacuatedpartofthethread.Thisquantityal soin-\nfluencesperiodsanddampingtimes,andtoalesserextentdam p-\ning ratios, and must be taken into account in the inversion of\nphysicalparametersinthe contextofprominenceseismolog y.\nCurrently available inversion schemes for one-dimensiona l\ncoronal loops and prominence threads (Arreguiet al., 2007a ;\nGoossenset al., 2008; Arregui& Ballester, 2010; Soleret al .,\n2010) make use of observed periods and damping ratios. The\nfirst, influence the inferred values for the Alfv´ en speed, wh ile\nthe second determine the transverse density structuring. B ased\non our results, we can conclude that ignorance on the length o f\nthe thread, the length of the supporting magnetic flux tube, a nd\nthe density in the evacuated part of the tube will have a signi f-\nicant impact on the inferred values for the Alfv´ en speed (he nce\nmagneticfield strength)in thethread,dependingonwhether we\nusethoseone-dimensionalinversionschemesortheresults from\ntwo-dimensionalmodelshereobtained.Onthecontrary,bec ause\nof the smaller sensitivity of the damping ratio to changes in the\nlongitudinal density structuring, seismological estimat es of the\ntransverse density structuring will be less a ffected by our igno-\nrance about the longitudinal density structuring of promin ence\nthreads.\nOur study provides additional insight to the physics of res-\nonantly damped kink modes in two-dimensional equilibrium\nstates, by extending previous applications (e.g, Andriese t al.,\n2005; Arreguiet al., 2005) to morecomplexnon-separablede n-\nsity distributions. It also provides an example of the metho ds\nandusesofcombiningtheinformationfromthespatialdistr ibu-\ntionofeigenfunctionswiththatobtainedfromenergyargum ents.\nIn particular, our energy analysis has allowed us to explain the\ndecrease in damping times for shorter thread lengths found b y\nSoleret al. (2010). The length of the thread influences the en -\nergyof the kink mode,andhence its oscillatory period,but a lso\naffects the damping by resonant absorption, through the energy\nflux into the resonance. In an analogous way, the value of the\ndensityintheevacuatedpartofthetubealsodeterminesper iods\nand dampingtimes, since both the energyof the kink mode and\ntheenergyfluxintotheresonancevary.Thismeansthatchang es\nintheequilibriumconfigurationinanon-resonantdirectio npro-\nducevariationsinthedampingpropertiesofkinkmodes,are sult\nthatwasqualitativelyexplainedbyadetailedexamination ofthe\nradial and longitudinal profiles of the eigenfunctions. Bot h the\nshortening of the length of the thread and the decrease of the\ndensity in the evacuated part of the tube produce more marked\nresonances, with the amplitude of the velocity perturbatio ns at\nthe resonance and the compressibility of the mode in the thre ad\nbeing larger. Inside the resonant layers shorter transvers e spa-\ntial scales for the Alfv´ enic velocity component are obtain ed. In\ncombination with the analysis of the energy of the kink modes\nand the energy flux into the resonances a quantitative explan a-\ntion was obtained for both the damping properties obtained i n\nourstudyandthoseinSoleret al.(2010).\nThe damping of kink oscillations in two-dimensional fully\nnon-uniformequilibriumconfigurationscanbecomputedbyu s-\ningenergyargumentstogetherwiththesolutionofsimplerp rob-\nlems for kink mode and Alfv´ en continuum modes. This aspect\nis worth to be considered in future studies of resonant absor p-\ntion in 2D/3D models of solar atmospheric magnetic structures\ninvolvingchangesofequilibriumparametersthata ffecttheden-\nsity structuring in a non-resonant direction. The use of ene rgy12 I.Arregui et al.:Magnetohydrodynamic kink waves intwo- dimensional non-uniform prominence threads\nargumentswould allow to have a first indication about how im-\nportantagivenparameterthatmodifiestheequilibriuminan on-\nresonant direction is, while avoiding to solve the full prob lem\nuntilwe areinterestedin thedetails.\nThe two-dimensional distribution of the time-averaged\nPoynting flux shows how energy is fed into the resonance and\nsubsequently flows along the field lines by properties of wave\nfieldsassociatedtoidealprocesses.Thisisthe reasonwhyr eso-\nnantdampingisamechanismforwaveenergytransferinwhich\ntime-scales are independent of resistivity, in the limit of high\nmagnetic Reynolds numbers. Magnetic di ffusion plays its role\nonce energy is concentrated at small spatial scales, by prov id-\ning heating at locations that, as in our example for a partial ly\nfilled thread, are distributed in regions where no resonant l ayer\nis present. This result o ffers additional insights to the dynamics\nofresonantlydampedkinkmodesandtheheatingofprominenc e\nplasmas by wave transformation processes. It must be consid -\nered in detail and extended to density models relevant to oth er\nsolaratmosphericstructures.\nAcknowledgements. IA, RS, and JLB acknowledge the funding provided under\ntheprojectAYA2006-07637bySpanishMICINNandFEDERFunds andthedis-\ncussionswithintheISSITeamonSolarProminenceFormation andEquilibrium:\nNew Data, New Models. RS acknowledges support from a Marie Cu rie Intra-\nEuropeanFellowship withintheEuropeanCommission7thFra mework Program\n(PIEF-GA-2010-274716). 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Universidad\nde Valencia, Burjassot–46100, Valencia, Spain\n2Departamento de F´ ısica, Universidade Federal da Para´ ıba,\nCaixa Postal 5008, 58051-970, Jo˜ ao Pessoa, Para´ ıba, Brazil.\n3Department of Physics, University of Hradec Kr ´alov´e,\nRokitansk ´eho 62, 500 03 Hradec Kr ´alov´e, Czechia.\n4Physics Department, Shahrood University of Technology, Shahrood, Iran\nAbstract\nThis work explores various manifestations of bumblebee gravity within the metric–affine for-\nmalism. We investigate the impact of the Lorentz violation parameter, denoted as X, on the\nmodification of the Hawking temperature. Our calculations reveal that as Xincreases, the values\nof the Hawking temperature attenuate. To examine the behavior of massless scalar perturbations,\nspecifically the quasinormal modes, we employ the WKB method. The transmission and reflec-\ntion coefficients are determined through our calculations. The outcomes indicate that a stronger\nLorentz–violating parameter results in slower damping oscillations of gravitational waves. To com-\nprehend the influence of the quasinormal spectrum on time–dependent scattering phenomena, we\npresent a detailed analysis of scalar perturbations in the time–domain solution. Additionally, we\nconduct an investigation on shadows, revealing that larger values of Xcorrespond to larger shadow\nradii. Furthermore, we constrain the magnitude of the shadow radii using the EHT horizon–scale\nimage of SgrA∗. Finally, we calculate both the time delay and the deflection angle.\n∗Electronic address: dilto@fisica.ufc.br\n†Electronic address: hha1349@gmail.com\n‡Electronic address: heidari.n@gmail.com\n§Electronic address: jan.kriz@uhk.cz\n¶Electronic address: soroushzrg@gmail.com\n1arXiv:2305.18871v2  [gr-qc]  26 Dec 2023I. INTRODUCTION\nThe problem of consistently implementing Lorentz symmetry breaking (LSB) within\nthe gravitational framework is fundamentally different from constructing Lorentz–breaking\nextensions for non–gravitational field theories. While flat spacetimes can accommodate\nLorentz–breaking additive terms like Carroll–Field–Jackiw [1], aether time [2], and others\n(see, for example, [3]), these constructions cannot be readily applied to curved spacetimes.\nIn flat spacetime, constant tensors can be well–defined, allowing for simple conditions such\nas∂µkν= 0. However, in curved spacetimes, these conditions cannot be introduced in an\nanalogous manner. The non–covariant condition ∂µkν= 0 is not suitable, and its covariant\nextension ∇µkν= 0 imposes severe restrictions on spacetime geometries (known as the no–\ngo constraints [4]). Therefore, the most natural way to incorporate local Lorentz violation\ninto gravitational theories is through the mechanism of spontaneous symmetry breaking. In\nthis case, Lorentz/CPT violating coefficients (operators) arise as vacuum expectation values\n(VEV) of dynamical tensor fields, which are driven by nontrivial potentials.\nThe generic effective field framework that describes all possible coefficients for\nLorentz/CPT violation is the well–known Standard Model Extension (SME) [5]. The grav-\nitational sector of SME is defined in a Riemann–Cartan manifold, where the torsion is\ntreated as a dynamic geometric quantity alongside the metric. However, despite the non–\nRiemannian background, most studies in bumblebee gravity, for example, have been con-\nducted within the metric approach of gravity, where the metric is the only dynamical geo-\nmetric field. These investigations focus on obtaining exact solutions for different models that\nincorporate LSB in curved spacetimes, including bumblebee gravity [6–13], parity–violating\nmodels [14–18], and Chern–Simons modified gravity [19–21].\nWhile the majority of works in the literature on modified theories of gravity employ the\nmetric approach, it is interesting to explore more general geometric frameworks. Motiva-\ntions for considering theories of gravity in Riemann–Cartan background include, allowing\nfor the possibility of having gravitational topological terms [22]. Another intriguing non–\nRiemannian geometry that has garnered attention is Finsler geometry [23], which has been\nlinked to LSB in recent years through various studies [24–28].\nHowever, the most compelling generalization of the metric approach is the metric–affine\nformalism, where the metric and connection are treated as independent dynamic geometric\n2quantities. Despite this formalism significance, LSB remains relatively unexplored in the\nliterature. Recent works involving bumblebee gravity have started to address this gap within\nthe metric–affine approach [29–33].\nUnderstanding gravitational waves and their characteristics is essential for studying vari-\nous physical processes, including cosmological events in the early universe and astrophysical\nphenomena like the evolution of stellar oscillations [34–36] and binary systems [37–40]. Grav-\nitational waves exhibit a range of intensities and characteristic modes, and their spectral\nproperties depend on the phenomena that generate them [41]. Of particular interest is the\nemission of gravitational waves from black holes. When a black hole forms through the\ngravitational collapse of matter, it enters a perturbed state and emits radiation that in-\ncludes a bundle of characteristic frequencies unrelated to the collapse process [42]. These\nperturbations with distinct frequencies are referred to as quasinormal modes [43, 44].\nThe investigation of quasinormal modes of black holes has extensively utilized the weak\nfield approximation in the literature, both in general relativity (GR) [42, 44–57] and in\nalternative gravity theories such as Ricci–based theories [58–60], Lorentz violation [61, 62],\nand related fields [63–67].\nSignificant progress has been made in the development of gravitational wave detectors, en-\nabling the detection of gravitational waves emitted from various physical phenomena [68–71].\nGround–based interferometers like VIRGO, LIGO, TAMA–300, and EO–600 have played\ncrucial roles in these detections [72–75]. Over the years, these detectors have improved their\naccuracy, approaching the desired sensitivity range [76]. These detections have provided\nvaluable insights into the composition of astrophysical objects, including boson stars and\nneutron stars.\nThe detection of gravitational waves has a significant connection to black hole physics.\nThe emitted gravitational radiation from a perturbed black hole carries a unique signature\nthat allows for direct observations of their existence [77]. Early studies on black hole pertur-\nbations were conducted by Regge and Wheeler, who examined the stability of Schwarzschild\nblack holes [78], followed by Zerilli’s work on perturbations [79, 80].\nThe study of gravitational solutions involving scalar fields has received significant at-\ntention in recent years due to their notable and peculiar characteristics. These intriguing\naspects have led to the emergence of various astrophysical applications. Notably, black\nholes (BHs) with nontrivial scalar fields challenge the well–known “no–hair theorem” [81].\n3Additionally, the existence of long–lived scalar field patterns [82], the formation of boson\nstars [83–85], and the exploration of exotic astrophysical scenarios in the form of gravastars\n[86–88] have all stemmed from this line of research. Moreover, by considering Klein–Gordon\nscalar fields on curved backgrounds, a wide range of phenomena can be derived, including\nblack hole bombs [89–91] and superradiance [92].\nIn this study, we thoroughly investigate bumblebee gravity within the metric–affine for-\nmalism, examining its diverse manifestations and shedding light on its intriguing properties.\nOur analysis encompasses the modification of the Hawking temperature, the behavior of\nquasinormal modes, the influence on time–dependent scattering phenomena, the analysis\nof shadows, the exploration of time delay. We reveal the correlation between bumblebee\ngravity and thermodynamic properties, with the Hawking temperature attenuating as the\nLorentz violation parameter increases. Furthermore, our investigation highlights the dis-\ntinct influence of bumblebee field on gravitational wave damping, with slower oscillations\nobserved under stronger Lorentz violation. The analysis of shadows provides valuable in-\nsights, demonstrating larger shadow radii associated with higher values of the parameter.\nAdditionally, we provide the analysis of the time delay in this scenario.\nII. THE GENERAL SETUP AND HAWKING TEMPERATURE\nIn this section, we begin by introducing a metric–affine generalization of the gravitational\nsector found in the SME [5]. Like the metric formulation, the action of this sector can be\nexpressed in the following manner\nS=1\n2κ2ˆ\nd4x√−g\b\n(1−u)R(Γ) + sµνRµν(Γ) + tµναβRµναβ(Γ)\t\n+Smat(gµν, ψ) +\n+Scoe(gµν, u, sµν, tµναβ). (1)\nHere, we have gµνas the covariant metric tensor, gµνas the contravariant metric tensor,\nand the determinant of the metric is given by g= det( gµν). In this formulation, we express\nthe action as follows, incorporating various geometric quantities. The Ricci scalar R(Γ)≡\ngµνRµν(Γ), Ricci tensor Rµν(Γ), and Riemann tensor Rµ\nναβ(Γ) play key roles. Additionally,\nSmatrepresents the action that accounts for the contributions of matter sources, which are\n4assumed to be coupled solely to the metric1. As previously mentioned, we adopt the metric–\naffine formalism, wherein the metric and connection are regarded as independent dynamical\nentities a priori .\nMoreover, the coefficients (fields) u=u(x),sµν=sµν(x), and tµναβ=tµναβ(x) play a\ncrucial role in explicitly breaking local Lorentz symmetry, as extensively discussed in Ref.\n[5]. It is important to note that the background field sµνpossesses the same symmetries as\nthe Ricci tensor. However, for the purposes of this study, we assume it to be a symmetric\nsecond–rank tensor, denoted as sµν=s(µν). Therefore, it couples exclusively to the sym-\nmetric component of the Ricci tensor. On the other hand, tµναβshares the symmetries of\nthe Riemann tensor. Lastly, the final term in Eq. (1), denoted as Scoe, encompasses the\ndynamical contributions stemming from the Lorentz–violating coefficients.\nIn this study, our primary focus lies on investigating the nontrivial consequences of\nLorentz symmetry breaking, which involve both the Ricci tensor and scalar. Consequently,\nwe impose constraints such that the coefficients sµνandutake nonzero values, while setting\ntµναβ= 0. This choice is motivated by the fact that the connection equation cannot be\nstraightforwardly solved through a simple metric redefinition when a nontrivial parameter\ntµναβis present. This issue is commonly referred to as the “t–puzzle” [93, 94]. Thus, the\naction of interest can be expressed as follows:\nS=1\n2κ2ˆ\nd4x√−g{(1−u)R(Γ) + sµνRµν(Γ)}+Smat+Scoe. (2)\nIt is important to emphasize that the aforementioned action exhibits invariance under pro-\njective transformations of the connection\nΓµ\nνα−→Γµ\nνα+δµ\nαAν. (3)\nUnder the this transformation described by Eq. (3), where Aαrepresents an arbitrary vector,\nit can be readily verified that the Riemann tensor undergoes the following change:\nRµ\nναβ−→Rµ\nναβ−2δµ\nν∂[αAβ]. (4)\nConsequently, it follows that the symmetric part of the Ricci tensor remains invariant under\nthe transformation described by Eq. (3), along with the entire action (2). The model\n1It is worth noting that fermions naturally couple to the connection. However, for convenience, we omit\nconsideration of spinors in this discussion and focus solely on bosonic matter sources that are minimally\ncoupled to the metric.\n5presented by the action (2) belongs to a broader class of gravitational theories known as\nRicci–based theories [95, 96]. It has been demonstrated that within this class of models, the\nproperty of projective invariance prevents the appearance of ghost–like propagating degrees\nof freedom in gravity.\nThe static and spherically symmetric solution in metric–affine traceless bumblebee model\nis given by [33]\nds2\n(g)=−\u0000\n1−2M\nr\u0001\ndt2\nq\u0000\n1 +3X\n4\u0001\u0000\n1−X\n4\u0001+dr2\n\u0000\n1−2M\nr\u0001vuut\u0000\n1 +3X\n4\u0001\n\u0000\n1−X\n4\u00013+r2\u0000\ndθ2+ sin2θdϕ2\u0001\n,(5)\nwhere we have used the shorthand notation: X=˜ξb2,bµis the vacuum expectation value\nof the bumblebee field Bµ, i.e., < B µ>=bµ, and ˜ξis a dimensionless parameter. Note that\nthe line element in Eq. (5) describes a LSB modified Schwarzschild metric.\nThe historical foundations of the thermodynamic interpretation of gravity trace back to\nthe pioneering research of the mid–1970s, notably by Bekenstein [97–99] and Hawking [100]\non the thermodynamics of black holes. In 1995, Jacobson made a significant advancement\nin our understanding by demonstrating that the Einstein field equations, the fundamental\nmathematical framework characterizing relativistic gravitation, can be derived from the\namalgamation of overarching thermodynamic principles and the foundational concept of the\nequivalence principle [101]. This groundbreaking revelation marked a pivotal milestone in the\nsynthesis of gravitational theory and thermodynamics. Subsequent to this critical research,\nPadmanabhan further explored the link between gravity and the concept of entropy [102].\nIn the context of the thermodynamic properties, the unique geometric quantity that will\nbe affected is the Hawking temperature\nT=1\n4π√g00g11dg00\ndr\f\f\f\f\nr=rs≈1\n8πM−X\n16(πM), (6)\nwhere rsis the Schwarzschild radius. This thermal quantity, as shown in Fig. 1, illustrates\nthe relationship between Lorentz–violating parameters and the respective temperatures. It\nis evident that as Xincreases, the corresponding Hawking temperatures become increasingly\nattenuated. It is important to mention that the thermodynamic aspects were calculated for\na variety contexts, including cosmological scenarios [103–114] and more [115–118].\n60.2 0.4 0.6 0.8 1.00.00.10.20.30.40.5Figure 1: The modifications of Hawking temperature caused by different values of X.\nIII. THE QUASINORMAL MODES\nDuring the ringdown phase, an intriguing phenomenon known as quasinormal modes be-\ncomes evident. These modes exhibit characteristic oscillation patterns that are independent\nof the initial perturbations and instead reflect the intrinsic properties of the system. This\nremarkable behavior arises from the fact that the quasinormal modes correspond to the free\noscillations of spacetime itself, unaffected by the specific initial conditions.\nIn contrast to the normal modes, which are associated with closed systems, the quasi-\nnormal modes are linked to open systems. Consequently, these modes gradually lose energy\nby radiating gravitational waves. Mathematically, the quasinormal modes can be described\nas the poles of the complex Green function.\nTo determine the frequencies of the quasinormal modes, we need to find solutions to\nthe wave equation for a system described by a background metric gµν. However, analytical\nsolutions for such modes are generally unattainable.\nNumerous techniques have been proposed in scholarly works to derive solutions for the\nquasinormal modes. Among these methods, the WKB (Wentzel–Kramers–Brillouin) ap-\nproach stands out as one of the most renowned. Its inception can be traced back to Will\nand Iyer [119, 120], and subsequent advancements up to the sixth order were made by Kono-\nplya [121]. For our calculations, we specifically focus on investigating perturbations utilizing\nthe scalar field. Consequently, we formulate the Klein–Gordon equation in the context of a\ncurved spacetime\n1√−g∂µ(gµν√−g∂νΦ) = 0 . (7)\n7While the exploration of backreaction effects is intriguing in this particular scenario,\nthis manuscript focuses on other aspects and does not delve into this feature. Instead, our\nattention is directed towards studying the scalar field as a minor perturbation. Furthermore,\ngiven the presence of spherical symmetry, we can exploit the opportunity to decompose the\nscalar field in a specific manner, as outlined below:\nΦ(t, r, θ, φ ) =∞X\nl=0lX\nm=−lr−1Ψlm(t, r)Ylm(θ, φ), (8)\nwhere the spherical harmonics are represented by Ylm(θ, φ). By substituting the scalar field\ndecomposition, as given in Eq. (8), into Eq. (7), we arrive at a Schr¨ odinger–like equation.\nThis transformed equation exhibits wave–like characteristics, making it suitable for our\nanalytical investigation\n−∂2Ψ\n∂t2+∂2Ψ\n∂r∗2+Veff(r∗)Ψ = 0 . (9)\nThe potential Veffis commonly referred to as the Regge–Wheeler potential, also known as\nthe effective potential. It contains valuable information regarding the geometry of the black\nhole. Additionally, we introduce the tortoise coordinate r∗(running r∗→ ±∞ all over the\nspacetime) as d r∗=p\n−g11/g00dr, which reads\nr∗=s\u0000\n1 +3X\n4\u0001\n\u0000\n1−X\n4\u0001(r+ 2Mln|r−2M|). (10)\nAfter performing several algebraic manipulations, we can express the effective potential\nexplicitly as follows:\nVeff(r) =\u0012\n1−2M\nr\u0013\n\u0000\n1−X\n4\u0001\n\u0000\n1 +3X\n4\u00012M\nr3+l(l+ 1)\nr2q\u0000\n1 +3X\n4\u0001\u0000\n1−X\n4\u0001\n. (11)\nIn the left panel of Fig. 2, we depict the behavior of the effective potential Veffas a\nfunction of r∗for different values of lwhile maintaining a fixed value of X(X= 0.5). On\nthe right panel of Fig. 2, we present the behavior for a constant value of l(l= 1) while\nvarying the parameter Xacross different values.\nThe figure also showcases the modifications induced by parameter X, which triggers the\nLorentz violation. It is worth noting that the tortoise coordinate r∗is a transcendental\nfunction of r, and making a “parametric plot” is the only way to visualize Veffagainst r∗.\nAs it is expected, if we consider the limit where X→0, we recover the simplest spherically\nsymmetric black hole, i.e., the Schwarzschild case.\n8-20 -10 0 10 20 300.000.050.100.150.200.25\n-20 -10 0 10 20 30 400.000.020.040.060.080.10Figure 2: The modifications of Veffcaused by different values of l(left) and X(right).\nA. The WKB method\nIn this section, our objective is to derive stationary solutions, characterized by the as-\nsumption that Ψ( t, r) can be written as Ψ( t, r) =e−iωtψ(r), where ωrepresents the fre-\nquency. By adopting this assumption, we can effectively separate the time–independent\ncomponent of Eq. (9) in the following manner:\n∂2ψ\n∂r∗2−\u0002\nω2−Veff(r∗)\u0003\nψ= 0. (12)\nIn order to solve Eq. (12), it is crucial to take into account the appropriate boundary\nconditions. In our particular scenario, the solutions that satisfy the desired conditions are\nthose which exhibit pure ingoing behavior near the horizon. This concept is illustrated\nbelow:\nψin(r∗)∼\n\nCl(ω)e−iωr∗(r∗→ −∞ )\nA(−)\nl(ω)e−iωr∗+A(+)\nl(ω)e+iωr∗(r∗→+∞),(13)\nThe complex constants Cl(ω),A(−)\nl(ω), and A(+)\nl(ω) play a significant role in the following\ndiscussion. In general, the quasinormal modes of a black hole can be defined as the fre-\nquencies ωnlfor which A(−)\nl(ωnl) = 0. This condition ensures that the modes correspond to\na purely outgoing wave at spatial infinity and a purely ingoing wave at the event horizon.\nHere, the integers nandlrepresent the overtone and multipole numbers, respectively. The\nspectrum of quasinormal modes is determined by the eigenvalues of Eq. (12). To analyze\nthese modes, we employ the WKB method, i.e., a semi–analytical approach that capitalizes\non the analogy with quantum mechanics.\nThe application of the WKB approximation to calculate the quasinormal modes in the\ncontext of particle scattering around black holes was initially introduced by Schutz and\n9Will [122]. Subsequently, Konoplya made further advancements in this technique [121, 123].\nHowever, it is important to note in order to make this method valid, the potential must\nexhibit a barrier–like shape, approaching constant values as r∗→ ±∞ . By fitting the power\nseries of the solution near the turning points of the maximum potential, the quasinormal\nmodes can be obtained [46]. The Konoplya formula for calculating them is expressed as\nfollows:\ni(ω2\nn−V0)p\n−2V′′\n0−6X\nj=2Λj=n+1\n2. (14)\nIn the aforementioned expression, Konoplya’s formula for the quasinormal modes incorpo-\nrates several key elements. The term V′′\n0represents the second derivative of the potential\nevaluated at its maximum point r0, while Λ jconstants that depend on the effective potential\nand its derivatives at the maximum. Notably, it is important to mention that a recent devel-\nopment in the field introduced a 13th–order WKB approximation, as proposed by Matyjasek\nand Opala in the literature [124].\nTables I, II, and III present the quasinormal frequencies calculated using the sixth–order\nWKB method for different values of the parameter X. The tables are specifically divided\nbased on the multipole number l, with Table I corresponding to l= 0, Table II to l= 1,\nand Table III to l= 2.\nImportantly, it is noteworthy that the modes associated with the scalar field exhibit\nnegative values in their imaginary part. This indicates that these modes decay exponentially\nover time, representing the dissipation of energy through scalar waves. These findings align\nwith previous studies investigating scalar, electromagnetic, and gravitational perturbations\nin spherically symmetric geometries [42, 43, 48, 125].\nIn our investigation, we have observed a decrease in the absolute values of imaginary\nparts of the quasinormal modes as Lorentz–violating parameter increases in general. The\nparameter Xplays a critical role in this scenario as it governs the damping of the scalar\nwaves. Depending on the its value, the damping can occur at a faster or slower rate.\nIn Fig. 3, we present a plot showcasing the real and imaginary parts of the quasinormal\nfrequencies as a function of the WKB order. Throughout the plot, we maintain the multipole\nand overtone numbers fixed at l= 0 and n= 0, respectively. It is evident that in all cases,\nthe real and imaginary parts of ω0exhibit convergence as the WKB order increases. This\nconvergence serves as an indication of the effectiveness of such an approach in providing\n10accurate approximations for the quasinormal modes.\nNevertheless, it is crucial to acknowledge that increasing the number of WKB orders does\nnot guarantee convergence in general, as the WKB series converges asymptotically. Thus,\nwhile the method proves reliable in this particular scenario, caution should be exercised\nwhen applying it to other cases.\nX ω 0 ω1 ω2\n0.1 0.1991 - 0.1963 i0.1554 - 0.6856 i0.4191 - 0.8840 i\n0.2 0.1782 - 0.1900 i0.1339 - 0.6811 i0.4743 - 0.8021 i\n0.3 0.1581 - 0.1826 i0.1134 - 0.6757 i0.5555 - 0.7121 i\n0.4 0.1381 - 0.1737 i0.0935 - 0.6695 i0.6678 - 0.6229 i\n0.5 0.1180 - 0.1628 i0.0740 - 0.6614 i0.8115 - 0.5445 i\n0.6 0.9754 - 0.1494 i0.0547 - 0.6519 i0.9754 - 0.4856 i\n0.7 0.0738 - 0.1326 i0.0353 - 0.6411 i1.1491 - 0.4453 i\n0.8 0.0457 - 0.1108 i0.0154 - 0.6282 i1.3302 - 0.4185 i\n0.9 0.0037 - 0.0835 i0.0050 - 0.6145 i1.5189 - 0.4010 i\nTable I: The quasinormal frequencies by using sixth–order WKB approximation for\ndifferent values of Lorentz–violating parameter X. In this case, the multipole number is\nl= 0.\nAn additional crucial aspect to consider is the utilization of the WKB method for studying\nscattering, which necessitates appropriate boundary conditions. In this context, our objec-\ntive is to determine the reflection and transmission coefficients, which bear resemblance to\nthose encountered in quantum mechanics for tunneling phenomena. To compute these coef-\nficients, we take advantage of the fact that ( ω2\nn−V0) is purely real and derive the following\nexpression:\nΥ =i(˜ω2−V0)p\n−2V′′\n0−6X\nj=2Λj(Υ). (15)\nTo determine the reflection and transmission coefficients, we analyze the scattering process\nusing the semi–classical WKB approach, which has been the subject of recent research\nin the field [126–128]. These coefficients are associated with the effective potential and\n11X ω 0 ω1 ω2\n0.1 0.5719 - 0.1947 i0.5138 - 0.6118 i0.4464 - 1.0841 i\n0.2 0.5596 - 0.1941 i0.5004 - 0.6106 i0.4326 - 1.0839 i\n0.3 0.5487 - 0.1934 i0.4884 - 0.6095 i0.4203 - 1.0836 i\n0.4 0.5389 - 0.1928 i0.4778 - 0.6085 i0.4093 - 1.0834 i\n0.5 0.5303 - 0.1923 i0.4683 - 0.6075 i0.3995 - 1.0831 i\n0.6 0.5226 - 0.1918 i0.4598 - 0.6066 i0.3907 - 1.0828 i\n0.7 0.5157 - 0.1913 i0.4522 - 0.6057 i0.3828 - 1.0825 i\n0.8 0.5096 - 0.1908 i0.4455 - 0.6048 i0.3757 - 1.0821 i\n0.9 0.5043 - 0.1904 i0.4395 - 0.6040 i0.3694 - 1.0817 i\nTable II: The quasinormal frequencies by using sixth–order WKB approximation for\ndifferent values of Lorentz–violating parameter X. In this case, the multipole number is\nl= 1.\nX ω 0 ω1 ω2\n0.1 0.9517 - 0.1932 i0.9115 - 0.5906 i0.8438 - 1.0172 i\n0.2 0.9380 - 0.1929 i0.8973 - 0.5901 i0.8289 - 1.0171 i\n0.3 0.9260 - 0.1927 i0.8848 - 0.5896 i0.8158 - 1.0169 i\n0.4 0.9155 - 0.1925 i0.8738 - 0.5892 i0.8043 - 1.0168 i\n0.5 0.9062 - 0.1922 i0.8641 - 0.5887 i0.7941 - 1.0166 i\n0.6 0.8981 - 0.1921 i0.8556 - 0.5883 i0.7852 - 1.0164 i\n0.7 0.8910 - 0.1919 i0.8483 - 0.5879 i0.7775 - 1.0162 i\n0.8 0.8849 - 0.1917 i0.8419 - 0.5876 i0.7708 - 1.0160 i\n0.9 0.8797 - 0.1916 i0.8365 - 0.5872 i0.7650 - 1.0157 i\nTable III: The quasinormal frequencies by using sixth–order WKB approximation for\ndifferent values of Lorentz–violating parameter X. In this case, the multipole number is\nl= 2.\n12Figure 3: Real (top lines) and imaginary (bottom lines) parts of the quasinormal modes to\ndifferent WKB orders, considering different values of X(for fixed l= 0).\nare represented by complex functions denoted as Λ j(Υ), where Υ is a purely imaginary\nquantity, and ˜ ωcorresponds to the real frequency associated with the quasinormal modes.\nThe expressions for the reflection and transmission coefficients are given as follows:\n|R|2=|A(+)\nl|2\n|A(−)\nl|2=1\n1 +e−2iπΥ, (16)\n|T|2=|Cl|2\n|A(−)\nl|2=1\n1 +e+2iπΥ. (17)\nwhere, A(+)\nl,A(−)\nlandClare the complex numbers which can be found according to the\nboundary condition in Eq. (13). In Figs. 4 and 5, the transmission and reflection co-\nefficients are displayed, respectively, for various values of the Lorentz–violating parameter\nand multipole numbers. Notably, it is observed that for a fixed value of X, increasing the\nmultipole number lresults in a rightward shift of the transmission coefficients. Conversely,\nwhen lis kept constant and Xis increased, the reflection coefficients shift in the opposite\ndirection.\nHowever, it is important to highlight that the 6th–order WKB approximation does not\n130.2 0.4 0.6 0.8 1.0 1.20.00.20.40.60.81.0\n0.2 0.4 0.6 0.8 1.0 1.20.00.20.40.60.81.0Figure 4: Transmission coefficients for distinct values of landX.\n0.2 0.4 0.6 0.8 1.0 1.20.00.20.40.60.81.0\n0.2 0.4 0.6 0.8 1.0 1.20.00.20.40.60.81.0\nFigure 5: Reflection coefficients for distinct values of landX.\nyield accurate results for l= 0 in the context of transmission and reflection coefficients.\nThis observation aligns with findings from previous studies [43, 127, 128].\nIV. TIME–DOMAIN SOLUTION\nTo thoroughly investigate the impact of the quasinormal spectrum on time–dependent\nscattering phenomena, a detailed analysis of scalar perturbations in the time domain is\nnecessary. However, given the intricacy of our effective potential, a more precise approach\nis required to gain deeper insights. In this regard, we employ the characteristic integration\nmethod developed by Gundlach and collaborators [129] as an effective tool to study the\nproblem effectively. By utilizing this approach, we can acquire valuable insights into the\nrole of quasinormal modes in time–dependent scattering scenarios, which have significant\nimplications for the investigation of black holes and related phenomena.\nThe methodology presented in Ref. [129] revolves around the utilization of light–cone\n14coordinates, represented by u=t−xandv=t+x. These coordinates enable the re-\nformulation of the wave equation in a more suitable manner, allowing for a comprehensive\nexamination of the system\n\u0012\n4∂2\n∂u∂v+V(u.v)\u0013\nΨ(u, v) = 0 . (18)\nTo achieve efficient integration of the aforementioned expression, a discretization scheme can\nbe employed, making use of a simple finite–difference method and numerical techniques. This\napproach allows for the effective numerical integration of the equation, providing accurate\nand efficient results\nΨ(N) =−Ψ(S) + Ψ( W) + Ψ( E)−h2\n8V(S)[Ψ(W) + Ψ( E)] +O(h4), (19)\nwhere S= (u, v),W= (u+h, v),E= (u, v+h), and N= (u+h, v+h), where hrepresents the\noverall grid scale factor. The null surfaces u=u0andv=v0are of particular significance,\nas they serve as locations where the initial data are specified. In our investigation, we have\nchosen to employ a Gaussian profile centered at v=vcwith a width of σ, which is selected\non the null surface u=u0\nΨ(u=u0, v) =Ae−(v−v∗)2/2σ2,Ψ(u, v0) = Ψ 0. (20)\nAtv=v0, a constant initial condition Ψ( u, v0) = Ψ 0was imposed, and without loss of\ngenerality, we assume Ψ 0= 0. The integration process then proceeds along the u=const.\nlines in the direction of increasing v, once the null data are specified.\nFurthermore, the results of our investigation into the scalar test field are presented in\nthis study. For the sake of convenience, we set m= 1, and the null data were defined by\na Gaussian profile centered at u= 10 on the u= 0 surface, with a width of σ= 3, and\nΨ0= 0. The grid was established to cover the ranges u∈[0,200] and v∈[0,200], with grid\npoints sampled to yield an overall grid factor of h= 0.1.\nTo validate our findings, we present Fig. 6, which showcases the typical evolution profiles\nfor different combinations of Xandl, allowing for a visual comparison and analysis of the\nresults.\n150 50 100 150 200-15-10-50\n0 50 100 150 200-15-10-505\n0 50 100 150 200-15-10-505Figure 6: Time domain profiles of scalar perturbations at r∗= 10Mare presented for\nvarious values of the Lorentz–violating parameter Xandl.\nV. SHADOWS\nTo investigate the effects of the Lorentz violation of Eq. (5) in the shadows, we consider\n∂S\n∂τ=−1\n2gµν∂S\n∂τµ∂S\n∂τν(21)\nwhere Sis the Jacobi action and τis the arbitrary affine parameter. Also, above expression\ncan be decomposed as\nS=1\n2m2τ−Et+Lϕ+Sr(r) +Sθ(θ). (22)\nWe can mathematically represent Sr(r) and Sθ(θ) as functions that depend on the variables\nrandθ, respectively. Since we are interested in the photon behavior, as a consequence, the\nenergy Eand angular momentum Lbecome constants of motion. By utilizing Eqs. (21) and\n(22), we can derive the equations governing the path of the photon, known as null geodesic\nequations\ndt\ndτ=s\u00123X\n4+ 1\u0013\u0012\n1−X\n4\u0013E\nf(r), (23)\n16dr\ndτ=p\nR(r)\nr2, (24)\ndθ\ndτ=±p\nQ(r)\nr2, (25)\ndφ\ndτ=Lcsc2θ\nr2, (26)\nwhere R(r) andQ(θ) defined as\nR(r) =\u0012\n1−X\n4\u00132\nE2r4−(K+L2)r2f(r)vuut\u0000\n1−X\n4\u00013\n3X\n4+ 1(27)\nQ(θ) =K −L2cot2θ. (28)\nThe symbol Krepresents the Carter constant [130]. In Eq. (25), the plus and minus signs\ncorrespond to the photon’s motion in the outgoing and ingoing radial directions, respectively.\nFor simplicity and without loss of generality, we fix the angle θtoπ/2 and restrict our analysis\nto the equatorial plane. With this simplification, our attention turns to the radial equation,\nwhere we introduce the concept of an effective potential Veff(r)\n\u0012dr\ndτ\u00132\n+Veff(r) = 0 . (29)\nHere, it is defined\nVeff(r) = (L2+K)vuut\u0000\n1−X\n4\u00013\n3X\n4+ 1f(r)\nr2−\u0012\n1−X\n4\u00132\nE2, (30)\nand we introduce two parameters in order to better accomplish our analysis\nξ=L\nEandη=K\nE2. (31)\nThe so–called critical radius, rc(photon sphere), can be found by considering\nVeff|r=rc=dVeff\ndr\f\f\f\f\nr=rc= 0. (32)\nIn addition, by using Eqs. (30) and (31) and considering Veff|r=rc= 0, we have\nξ2+η=s\u0012\n1 +3X\n4\u0013\u0012\n1−X\n4\u0013r2\nc\nf(rc). (33)\n17To determine the shadow radius, we will utilize the celestial coordinates αandβ[131],\nwhich are connected to the constants of motion as follows: α=−ξandβ=±√η. With\nthese coordinates, the shadow radius can be expressed as:\nRShadow =p\nξ2+η=p\nα2+β2=rcp\nf(rc)\u0012\u0012\n1 +3X\n4\u0013\u0012\n1−X\n4\u0013\u00131/4\n. (34)\nIn this manner, by consideringdVeff\ndr\f\f\f\nr=rc= 0, and the using the fact that f(rc) = 1−2M\nrc\none can obtain rc= 3M. Therefore, the equation for the shadow radius reads explicitly\nRShadow = 3√\n3M4s\u0012\n1 +3X\n4\u0013\u0012\n1−X\n4\u0013\n. (35)\nIt is obvious that the Eq. (35) recovers the shadow radius for ordinary Schwarzschild black\nhole by considering X= 0. Based on the EHT horizonscale image of SgrA∗, two constraints\nhave been proposed for shadow radius [132], as 4 .55< R sh<5.22 and 4 .21< R sh<5.56\nconsidered as 1 σand 2 σ, respectively.\nTherefore, we have examined these conditions to explore the limits on the Xvalue. For\nbetter visualization the shadow radius versus Xis plotted for M= 1 and the experimental\nconstraints are shown by two pairs of horizontal lines in Fig. 7.\n0.0 0.2 0.4 0.6 0.8 1.03.54.04.55.05.56.06.5\nXRShadow /M\nFigure 7: Shadow radius versus XforM= 1, the dashed lines represente the experimental\nconstraints of 4 .21, 5.56 and 4 .55, 5.22\nAs depicted in Fig. 7, the constraint derived for the parameter Xfalls within the range of\n0< X < 0.037 at the 1 σconfidence level and extends to 0 < X < 0.86 at the 2 σconfidence\nlevel.\n18-10 -5 0 5 10-10-50510\n-15 -10 -5 0 5 10 15-15-10-5051015Figure 8: (left) Shadows with mass M= 1 for diverse values of parameter\nX= 0.1,0.2,0.5,0.8,0.9. On the other hand, (right) shadows with X= 0.5, considering a\nvariety of mass: M= 0.75,1.00,1.25,1.50.\nThen, in Fig. 8, we display an analysis of the shadows of our black hole for a range\nofXvalues on the left side. Remarkably, the shadow radius demonstrates a noticeable\naugmentation as the parameter Xincreases. On the right–hand side, we investigate the\ninfluence of mass on the same Lorentz–violating parameter, namely, X= 0.5. As illustrated\nin the figure, the shadow radius experiences a notable enlargement when the initial mass M\ntransitions from 0 .75 to 1 .50. In addition, since we are dealing with a real positive defined\nvalues of radius, RShadow turns out to be bounded from above, i.e., having its maximum\nvalue for X= 0.86.\nVI. TIME DELAY AND DEFLECTION ANGLE\nThe investigation of time delay in Lorentz–violating scenarios holds a profound signif-\nicance within the realm of theoretical physics. By delving into this exploration, we gain\nprofound insights into the fundamental aspects of spacetime and Lorentz symmetry break-\ning. In this regard, the determination of time delay can be accomplished by rewriting the\nLagrangian for null geodesics using Eq. (29) and Eq. (30). This mathematical approach\nallows us to accurately quantify the temporal delay experienced by particles, shedding light\n19on the intricate dynamics of spacetime as follows\n\u0012dr\ndt\u00132\n=f2(r)1−X\n4\n1 +3X\n4\u0012\n1−r2\nminf(r)\nr2f(rmin)\u0013\n(36)\nMoreover, we know that d r/dt= 0 at r=rmin, resulting in\ndt\ndr=r\n1+3X\n4\n1−X\n4\nf(r)q\n1−(rmin\nr)2f(r)\nf(rmin). (37)\nIn this way, the time delay is given by\nt(r, rmin) =ˆ∞\nrminr\n1+3X\n4\n1−X\n4\nf(r)q\n1−(rmin\nr)2f(r)\nf(rmin)dr. (38)\nMoreover, the deflection of light as it passes through curved spacetime serves as a tool\nto analyze the physics of a source of gravitation [133–136]. The calculation of the deflection\nangle can be carried out by applying the following equation [137]\nˆα= 2ˆ∞\nrmin\nvuut\u0000\n1−X\n4\u00013\n3X\n4+ 1\u0012r2\nr2\nminf(rmin)−f(r)\u0013\n−1\n2\ndr\nr−π. (39)\nWithin the scope of our analysis, we employ the symbol ˆ αto represent the deflection\nangle. Table. IV provides a systematic evaluation of the time delay and deflection angle for\na range of Xvalues. Our method involves initially determining the minimum radius along\nthe trajectory for each specific X.\nTo quantify the time delay, we establish a reference point using t0as the baseline for the\ntime delay, which corresponds to the original Schwarzschild black hole scenario with X= 0.\nWe then calculate the time delay ratio for each parameter within the bumblebee theory.\nThe deflection angles are similarly derived through the application of Eq. (39).\nThe results in Table. IV reveal that while the minimum radius diminishes with increas-\ningX, it is important to note that the time delay ratio, the actual time delay, and the\ndeflection angle demonstrate distinct behaviors. Specifically, an increase in the value of Xis\nassociated with longer time delays and larger deflection angles, underscoring the heightened\ngravitational effect corresponding to higher values of X.\n20Table IV: Variation of rmin, time delay ratio and deflection angle for various X,M= 1 and\nimpact parameter ( L/E) = 10\nX 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0\nrmin8.7889 8.6684 8.5632 8.4713 8.3914 8.3222 8.2627 8.2123 8.1702 8.1359 8.1092\nt\nt01.0000 1.0500 1.1002 1.1508 1.2019 1.2536 1.3061 1.3596 1.4142 1.4701 1.5275\nˆα1.8660 1.9421 2.0192 2.0977 2.1778 2.2599 2.3442 2.4311 2.5209 2.6139 2.7105\nVII. CONCLUSION\nThis study delved into the intriguing realm of bumblebee gravity, employing the metric–\naffine formalism to explore its various signatures. By investigating the effects of the Lorentz\nviolation parameter, X, we observed significant modifications in the renowned Hawking tem-\nperature. Interestingly, as Xincreased, the values of the Hawking temperature experienced\na notable attenuation.\nOur analysis extended to the examination of quasinormal modes through the utilization\nof the WKB method. Notably, we found that a stronger Lorentz–violating parameter led to\na deceleration in the damping oscillations of gravitational waves, showcasing the intricate\ninterplay between bumblebee gravity and Lorentz violation. In addition, we calculated the\ntransmission and reflection coefficients in this context.\nTo better comprehend the implications of the quasinormal spectrum on time–dependent\nscattering phenomena, we conducted a meticulous exploration of scalar perturbations in the\ntime–domain solution. This analysis provided a comprehensive understanding of the impact\nand behavior of scalar perturbations under varying conditions.\nFurthermore, our investigation extended to the concept of shadows, where we discovered\na compelling relationship between the Lorentz violation parameter Xand shadow radii.\nSpecifically, larger values of Xwere found to correspond to increased shadow radii. More\nso, they were bounded from above, i.e., X= 0.86. Lastly, our research delved into the realm\nof time delay and deflection angle within this scenario, shedding light on additional aspects\nof the intricate interplay between bumblebee gravity and its implications.\n21Acknowledgements\nThe authors express their gratitude to the diligent referees who meticulously reviewed\nthe manuscript and provided invaluable feedback to aid in its improvement. Most of the\ncalculations were performed by using the Mathematica software. A. A. Ara´ ujo Filho is\nsupported by Conselho Nacional de Desenvolvimento Cient´ ıfico e Tecnol´ ogico (CNPq) and\nFunda¸ c˜ ao de Apoio ` a Pesquisa do Estado da Para´ ıba (FAPESQ) – [200486/2022-5] and\n[150891/2023-7]. The authors would like to thank R. Oliveira, R. Konoplya for providing\ntheMathematica notebook to perform our numerical calculations, and P. J. Porf´ ırio for the\nfruitiful discussions.\nVIII. DATA AVAILABILITY STATEMENT\nData Availability Statement: No Data associated in the manuscript\n[1] M. Chaichian, A. Tureanu, and G. Zet, “Corrections to schwarzschild solution in noncom-\nmutative gauge theory of gravity,” Physics Letters B , vol. 660, no. 5, pp. 573–578, 2008.\n[2] G. Zet, V. Manta, and S. Babeti, “Desitter gauge theory of gravitation,” International Jour-\nnal of Modern Physics C , vol. 14, no. 01, pp. 41–48, 2003.\n[3] N. Seiberg and E. 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Pontremoli\", University of Milan, via Celoria 16, 20133 Milan, Italy.\nDepartment of Chemical Engineering and Biotechnology,\nUniversity of Cambridge, Philippa Fawcett Drive, CB30AS Cambridge, U.K.\nCavendish Laboratory, University of Cambridge, JJ Thomson Avenue, CB30HE Cambridge, U.K.\nA theory of superconductivity is presented where the e\u000bect of anharmonicity, as entailed in the\nacoustic, or optical, phonon damping, is explicitly considered in the pairing mechanism. The gap\nequation is solved including di\u000busive Akhiezer damping for longitudinal acoustic phonons or Klemens\ndamping for optical phonons, with a damping coe\u000ecient which, in either case, can be directly related\nto the Gr uneisen parameter and hence to the anharmonic coe\u000ecients in the interatomic potential.\nThe results show that the increase of anharmonicity has a strikingly non-monotonic e\u000bect on the\ncritical temperature Tc. The optimal damping coe\u000ecient yielding maximum Tcis set by the velocity\nof the bosonic mediator. This theory may open up unprecedented opportunities for material design\nwhereTcmay be tuned via the anharmonicity of the interatomic potential, and presents implications\nfor the superconductivity in the recently discovered hydrides, where anharmonicity is very strong\nand for which the anharmonic damping is especially relevant.\nI. INTRODUCTION\nAtomic vibrations in solids are inevitably a\u000bected by\nthe shape of the interatomic potential. For all real ma-\nterials, the shape of the interatomic potential is far from\nbeing quadratic, i.e. harmonic. The intrinsic anhar-\nmonicity of solids has many well known consequences\nsuch as thermal expansion, soft modes and instabili-\nties, sound absorption, identi\fcation of stable crystalline\nphases etc. [1] A well established approach to anhar-\nmonicity is the self-consistent method introduced by\nBorn and Hooton [2], leading to the concept of renormal-\nization of phonon frequencies in the quasiharmonic or\nself-consistent phonon approximation, where the renor-\nmalized phonon frequencies arise from an e\u000bective vibra-\ntional dynamics within a region about equilibrium, which\ntakes anharmonic terms of the potential into account via\nadjustable parameters obtained from a self-consistent so-\nlution to the many-body problem [3].\nHowever, the e\u000bect of anharmonicity extends to far\ngreater areas, including electron-phonon coupling, where\ntraditionally the e\u000bect of anharmonic damping has al-\nways been neglected, and where instead recent \frst-\nprinciple calculations demonstrate an important e\u000bect of\nanharmonicity on band-structure [4, 5].\n\u0003settychandan@gmail.com\nymatteo.baggioli@uam.es\nzalessio.zaccone@unimi.itIn the context of high Tcsuperconductors, the e\u000bect\nof anharmonic enhancement on Tchas been studied in\nthe early days following the discovery of high- Tcsuper-\nconductivity in the cuprates. In particular, several works\nby Plakida and others have studied the e\u000bect of anhar-\nmonicity on Tcfor the case of structurally unstable lat-\ntices or deformed lattice potentials [6{8]. Even more re-\ncent works on the high- Tchydrides [9{19] only take into\nconsideration phonon energy renormalizations due to an-\nharmonicity but neglect anharmonic damping.\nHowever, a fundamental understanding of the e\u000bect\nof anharmonic damping on phonon-mediated supercon-\nductivity and e.g. on Tcis absent due to the lack of\nanalytical approaches to this problem. Yet, this is a fun-\ndamental issue in the context of high-T superconductors\nwhere anharmonicity becomes important due to the sig-\nni\fcant temperature values, since in general anharmonic-\nity in solids grows roughly linear in T[1]. Even more\nurgent is the problem of the e\u000bect of anharmonicity in\nhydrogen-based materials, which have recorded the high-\nestTcvalues so far: in these systems the presence of\na light element such as hydrogen induces a huge anhar-\nmonicity due to the large oscillation amplitudes of the\nhydrogen atoms [13, 18, 20{23].\nNumerical studies and \frst-principle calculations can\nassess the e\u000bect of anharmonicity in an empirical way for\na speci\fc material by benchmarking against harmonic\ncalculations, but a systematic fundamental understand-\ning of the role of anharmonic damping on conventional\nsuperconductivity is missing. This would be highly bene-arXiv:2003.06220v2  [cond-mat.supr-con]  20 Nov 20202\n\fcial to obtain system-independent guidelines to not only\nestimate the e\u000bect of anharmonic damping in general\ncases, but also to develop generic guidelines for material\ndesign. For example, by relating anharmonic damping\nto the interatomic potential it could become possible to\ndesign materials with ad-hoc or tunable electron-phonon\ncoupling and superconducting properties.\nHere we take a \frst step in this direction by studying\nthe e\u000bect of phonon Akhiezer and Klemens damp-\ning on superconductivity beyond the quasi-harmonic\napproximation. We do this by explicitly taking into\naccount the phonon damping due to anharmonicity in\nthe mediator for the electron pairing. The theory shows\nthat, unexpectedly, the e\u000bect of the anharmonicity\n(as represented by the damping coe\u000ecient) on Tcis\nnon-monotonic, i.e. Tc\frst increases then goes through\na maximum and then decreases upon increasing the\nanharmonic damping. This occurs because electron-\nphonon scattering processes involving energy-loss and\nenergy-gain (Stokes and anti-Stokes) act constructively\nto increase the e\u000bective attraction driving the formation\nof Cooper pairs. The enhancement is most e\u000ecient for\na window of critical damping parameter ( Dmax) set by\nthe bosonic velocity and correlated with the Io\u000be-Regel\nscale. Outside this window, the strength of pairing\ndeteriorates leading a reduction in Tc. These results are\nvalid for both cases of acoustic and optical phonons, as\nshown in in the Appendix A.3 below.\nII. THE THEORETICAL FRAMEWORK\nThe displacement \feld of an anharmonic solid obeys\nthe following dynamical equation [24]:\n\u001a@2ui\n@t2=CT\nijkl@2uk\n@xj@xl\u0000CT\nijkl\u000bkl@\u0001T\n@xj+\u0017ijkl@2_uk\n@xj@xl\n(1)\nwhich is coupled to Fourier's law for heat transfer and\nto the energy balance equation for the thermal gradient\n\u0001T. In Eq. (1) , uidenotes the i-th Cartesian component\nof the atomic displacement \feld, CT\nijklis the isothermal\nelastic constant tensor, \u000bklis the thermal expansion ten-\nsor, and\u0017ijklis the viscosity tensor. The dot indicates\nderivative with respect to time of the elastic \feld ukin\nthe last dissipative term.\nFor solids, where acoustic excitations can be split into\nlongitudinal (LA) and transverse (TA), Eq. (1) can be\nsplit into two decoupled equations for LA and TA dis-\nplacements, leading to the following Green's function in\nFourier space [25]:\nG\u0015(!;q) =1\n!2\u0000\n2\n\u0015(q) +i!\u0000\u0015(q)(2)\nwhere\u0015=TA;LA is the branch label, and \u0000 \u0015(q) =Dq2\nrepresents the Akhiezer damping, which coincides with\nthe acoustic absorption coe\u000ecient [24], while \n \u0015(q) =v\u0015qis the acoustic eigenfrequency, already renormalized\nto account for the shift induced by anharmonicity [26],\nwithv\u0015the speed of sound for branch \u0015.\nThe quadratic dependence \u0000 \u0015(q) =Dq2of the damp-\ning stems directly from the viscous term in Eq. (1) and\nis typical of Akhiezer damping [24, 27]. In particular, it\nhas been shown [27] that \u0000 takes the following general\nform for longitudinal excitations (see also [28]):\n\u0000L=q2\n2\u001a\"\u00124\n3\u0011+\u0010\u0013\n+\u0014T\u000b2\u001a2v2\nL\nC2p\u0012\n1\u00004v2\nT\n3v2\nL\u00132#\n:(3)\nwhere\u0011\u0011\u0017xyxy is the shear viscosity, \u0010is the bulk vis-\ncosity,\u001ais the solid density, \u0014is the thermal conductivity,\n\u000bis the longitudinal thermal expansion coe\u000ecient, and\nCpis the speci\fc heat at constant pressure. The second\nterm in Eq. (3), \u0018\u000b2, represents the phonon damping\ndue to heat exchange between the compressed and the\nrare\fed regions of the longitudinal wave. This second\ncontribution, in practice, represents only a few percent\nof the \frst viscous contribution in Eq. (3) and is there-\nfore negligible.\nThe above derivation follows a hydrodynamic approach\n[29]; by comparing with the result of a microscopic ap-\nproach based on the Boltzmann transport equation for\nphonons, it has been shown that [24]\nDL=CvT\u001c\n2\u001a\u00124\n3h\r2\nxyi\u0000h\rxyi2\u0013\n\u0019CvT\u001c\n2\u001ah\r2\nxyi(4)\nwhere we neglected the contribution from bulk viscosity\n\u0010, since normally \u0011\u001d\u0010. Furthermore,h:::iindicates\naveraging with respect to the Bose-Einstein distribution\nas a weight, while \rxyis thexycomponent of the tensor\nof Gr uneisen constants. Also, Cvis the speci\fc heat at\nconstant volume, while \u001cis the phonon life-time. Since\n\u001c\u0018T\u00001(which is an experimental observation for most\nsolids [24, 30]), the di\u000busion constant DLis independent\nof temperature, i.e. a well-known experimental fact [30].\nA substantially equivalent expression for the damping\nof longitudinal phonons, in terms of an average Gr uneisen\nconstant of the material \rav, was derived by Boemmel\nand Dransfeld [30]\nDL\u0019CvT\u001c\n2\u001a\r2\nav (5)\nand provides a good description of the Akhiezer damping\nmeasured experimentally in quartz at T >60K[30].\nIn turn, the Gr uneisen constant \r, or at least the lead-\ning term [31] of \ravor\rxyabove, can be directly related\nto the anharmonicity of the interatomic potential. For\nperfect crystals with pairwise nearest-neighbour interac-\ntion, the following relation holds [31]\n\r=\u00001\n6V000(a)a2+ 2[V00(a)a\u0000V0(a)]\nV00(a)a+ 2V0(a)(6)\nwhereais the equilibrium lattice spacing between\nnearest-neighbours, and V000(a) denotes the third deriva-\ntive of the interatomic potential V(r) evaluated in r=a.3\nHence, the phonon damping coe\u000ecient DLcan be\ndirectly related to the anharmonicity of the interatomic\npotential via the Gr uneisen coe\u000ecient and Eq. (6).\nIII. RESULTS\nBecause in crystals momentum is always conserved\nduring electron-phonon scattering events, only longitu-\ndinal phonons contribute to pairing [32, 33], therefore we\nwill focus on the LA phonon, \u0015=LA, and we will drop\nthe\u0015index in the following. According to Eq. (2) we\nthus choose a phonon propagator written in Matsubara\nfrequency of the form\n\u0005(i\nn;q) =1\nv2q2+ \n2n+ \u0000(q) \nn; (7)\nwith \u0000( q) =Dq2being the Akhiezer damping discussed\nabove, and vis the phonon velocity. We de\fne the\nBosonic Matsubara frequency \n n= 2n\u0019T wherenis\nan integer number and Tthe temperature. The super-\nconducting gap equation for a generic gap at momentum\nkand Fermionic Matsubara frequency !n= (2n+ 1)\u0019T\ntakes the form (see Ref. [34] or [35])\n\u0001(i!n;k) =g2\n\fVX\nq;!m\u0001(i!m;k+q)\u0005(q;i!n\u0000i!m)\n!2m+\u00182\nk+q+ \u0001(i!m;k+q)2;\n(8)\nfor a constant attractive interaction gand volume V.\nHere\u0018kis the free electron dispersion which we choose\nto be quadratic with a chemical potential \u0016. The inverse\ntemperature is denoted by \fand we work in simpli\fed\nunits where twice the electron mass is set to unity. For\nanalytical tractability, we also choose an isotropic gap\nfunction independent of frequency, i.e., \u0001( i!m;k+q)\u0011\n\u0001. Converting the momentum summation into energy\nintegral with variable \u0018and assuming a constant density\nof states, the gap equation reduces to\n1 =X\n!mZ1\n\u0000\u0016\u0015Td\u0018\n[(v2\u0000D!m)(\u0018+\u0016) +!2m] [!2m+\u00182+ \u00012]\n(9)\nwhere\u0015=N(0)g2andN(0) is the density of states at\nthe Fermi level. To begin the discussion, we con\fne our-\nselves to small Dso that we can ignore D\u0016\u001cT\u0018Tc\neven though the chemical potential is allowed to be large\ncompared to Tc. This implies that the linear term in !m\ncan be neglected. The remaining constant \u0016v2acts like\na mass term and reduces Tcfor allD[36]. As this e\u000bect\nis only quantitative, this term can also be ignored, as a\n\frst approximation, without a\u000becting the central claims\nof the paper. The full e\u000bect of the chemical potential\nterm will be included in the upcoming paragraphs. With\nthese assumptions and using the energy integral identityR1\n\u00001d\u0018\n(z\u0018+s)(\u00182+r2)=\u0019s\nr(s2+z2r2), we obtain\n1 =X\n!m\u0015\u0019T!2\nmp\n!2m+ \u00012\u0010\n!4m+ (!2m+ \u00012)(v2\u0000D!m)2\u0011:\n(10)\nTo determine the condition for Tc, we set the supercon-\n0.5 1.4 v\n0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5D 0.10.20.30.40.5Tc\n0.9 1.0 1.1 1.2 1.3 1.4v0.51.01.5Dmax\nFigure 1. Top: The dimensionless critical temperature \u0016Tc\nas a function of the damping constant Dat di\u000berent dimen-\nsionless speeds \u0016 v2[0:5;1:4].Bottom: The position of\nthe maximum temperature as a function of the dimensionless\nlongitudinal sound speed. In both plots we \fxed \u0016 \u0016= 0:1.\nducting gap \u0001 = 0. We can then perform the in\fnite\nsum over Matsubara frequencies (see the Appendix A.1\nfor more details) to obtain the simpli\fed gap equation\n1 =\u00001\n\u0016v4\"\n \u00121\n2\u0013\n+i(i+D)\n4 \u00121\n2\u0000\u0016v2\n2\u0019\u0016Tc(i+D)\u0013\n+i(i+D)\n4 \u00121\n2+\u0016v2\n2\u0019\u0016Tc(i+D)\u0013\n+c:c#\n; (11)\nwhere, henceforth, the barred quantities are normalized\nbyp\n\u0015, i.e., \u0016v=v=p\n\u0015and (x) is the digamma function.\nA solution for \u0016Tccan be obtained from Eq. (11) and is\nplotted in Fig. 1 (Top) as a function of the anharmonic\ndamping parameter D. The plot shows that \u0016Tcis\nenhanced quadratically for small D, reaches a maximum\nat an optimal anharmonicity parameter Dmax (set by\nthe dimensionless phonon velocity \u0016 v), and falls o\u000b as a\npower law for larger D. The optimal parameter Dmax\nincreases with \u0016 vas shown in Fig. 1 (Bottom). In the4\n0.01 1.5μ0.0 0.5 1.0 1.5 2.0 2.5 3.0D 0.100.150.200.250.300.350.40Tc\n0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4μ1.301.351.401.451.50Dmax\nFigure 2. Top: The dimensionless critical temperature \u0016Tcin\nfunction of the di\u000busion constant Dat di\u000berent dimensionless\nchemical potentials \u0016 \u00162[0:01;1:5].Bottom: The position\nof the maximum temperature in function of the dimensionless\nchemical potential. In both plots we \fxed \u0016 v= 1:2.\nAppendix A.2 (see also Refs. [37{42] quoted therein),\nwe discuss the behavior of Dmax for larger values of\n\u0016vwhere it saturates to a value Dmax\u0018v2=Tc(not\nshown in Fig. 1). This condition for resonance can be\nobtained from the denominator in Eq. 10. Note that\nthe enhancement of the transition temperature occurs\nonly above a critical value of the phonon velocity that\nis set by the interaction parameterp\n\u0015. The reason forthe non-monotonic behavior of \u0016Tccan be understood\nfrom Eq.(10) and the anti-symmetry in !of the phonon\ndamping term. Because of this property, Stokes and\nanti-Stokes processes ( !m<and>0, respectively) add\nup constructively to increase the e\u000bective attraction\ndriving the formation of Cooper pairs. This constructive\ninterference grows with Dwhich gets to the numerator\nupon adding the two processes. Eventually, however, for\nsu\u000eciently large anharmonic damping D\u001dv2=!m, the\nquadratic term\u0018D2!2\nmin the denominator of Eq.(10)\nbecomes the dominant contribution, the Stokes and\nanti-Stokes processes now add up in a destructive way\nand superconductivity gets suppressed. In the regime\nwherevis very small, the last term in the denominator of\nEq.(10) can be approximated as ( v2\u0000D!m)2\u0018D2!2\nm\nand the non-monotonicity is absent even at small D\nvalues (see dark lines in Fig.1).\nIV. CHEMICAL POTENTIAL EFFECTS\nIn the following paragraphs, we relax the assumptions\nmade previously on the chemical potential. We restrict\nourselves to the BCS/quasi-BCS regime where the chem-\nical potential is positive and not below the band bottom.\nThis assumption ignores e\u000bects where the pairing scale\nbecomes comparable to the band-width and hence keep-\ning the BCS-BEC cross-over regime inaccessible. Follow-\ning the same steps of the previous section, we obtain the\nsimpli\fed formula\n1 =X\n!m\u0015\u0019Tc\u0000\n!2\nmc+ (v2\u0000D!mc)\u0016\u0001\nj!mcj\u00001\n(!2mc+ (v2\u0000D!mc)\u0016)2+!2mc(v2\u0000D!mc)2\n(12)\nwhere!mcis the Fermionic Matsubara frequency at T=\nTc. After algebraic manipulations of the Matsubara sum,\nas shown in the Appendix A.1, the \fnal equation for \u0016Tc\nwith a \fnite chemical potential reduces to\n1 =1\n2\u0016v2\u0016\u0016\"\n\u0000 \u00121\n2\u0013\n+(\nb+\u0000a\u0003\n2(b+\u0000b\u0000) \u00121\n2\u0000b+\u0013\n+a\u0003\u0000b\u0000\n2(b+\u0000b\u0000) \u00121\n2\u0000b\u0000\u0013\n+c:c)#\n+ [D$\u0000D]; (13)\nwhere we have the de\fnitions a\u0011z\nD+i,\nb\u0006\u0011z\u0003\u0006r\nz2+4\u0016v2\u0016\u0016\n(2\u0019\u0016Tc)2\n2(D\u0000i)andz\u0011\u0016v2\n2\u0019\u0016Tc+iD\u0016\u0016\n2\u0019\u0016Tc. A\nplot of the numerical solution for \u0016TcversusDis shown\nin Fig. 2 (Top). Many of the features appearing in\nFig. 1 (Top) are reproduced when the chemical po-\ntential is introduced { a non-monotonic dependence\non the anharmonicity parameter, a quadratic rise and\npower-law fall o\u000b for small and large Drespectively.\nThis rea\u000erms the assumptions made on the chemicalpotential in deriving Eq. (11). However, the chemical\npotential has an additional non-trivial e\u000bect of reducing\n\u0016Tcat small and large D, but enhances its peak value at\noptimalD. Furthermore, the \u0016Tcpeak position ( Dmax)\nchanges substantially for small \u0016 \u0016and remains virtually\nunchanged for larger \u0016 \u0016. A plot of Dmaxas a function of\n\u0016\u0016is shown in Fig. 2 (Bottom).5\nV. DISCUSSION\nMuch attention has been devoted to the role of dis-\norder induced damping on superconducting Tc(see [43]\nand references therein); however, only a few theoretical\nworks have examined directly the e\u000bects of damping on\nthe superconducting properties, mostly in terms of glassi-\nness [36, 44{46]. Ref. [45] \fnds an enhancement of super-\nconducting transition driven by a spin-glass phase formed\nfrom paramagnetic spins interacting through Ruderman-\nKittel-Kasuya-Yosida exchange couplings. On the other\nhand, Ref. [44] \fnds that a glassy phase leads to mono-\ntonically decreasing Tcbut does not take into account\nthe role of anharmonic phonon damping explicitly. The\ndissipative aspect of the glass phase was considered at\na phenomenological level in Ref. [36] in the context of\nthe under-doped high- Tccuprates. While a similar non-\nmonotonic behavior in Tcis found, its mechanism does\nnot arise from the time-reversal symmetry breaking in\nthe dissipation term. This is re\rected in the linear rise\nof\u0016Tcfor small damping as opposed to the quadratic rise\nas found in this work. Furthermore, as alluded to ear-\nlier, the parameter Dis a characteristic of anharmonic\ndamping and originates from the viscous damping term\nin Eq. (1) describing anharmonic phonons. It can be di-\nrectly related to the Gr uneisen constant, which, in turn,\ncan be determined via \frst-principle calculations of the\ninter-atomic potential through Eq. (6); therefore, this re-\nlation provides a microscopic handle for tuning Dgiving\none signi\fcant control in designing real materials.\nVI. CONCLUSION\nTo conclude, we have developed superconducting gap\nequations which account for the e\u000bect of anharmonic\ndamping of phonons. The phonon viscosity parameter\nDcan be related directly to the Gr uneisen coe\u000ecient\nand to the shape of the interatomic potential. Upon\nsolving the gap equation, it is found that the Tcdepends\nnon-monotonically upon the anharmonic damping\nparameterDand features a maximum as a function of\nD. The value of the critical damping parameter ( Dmax)\naround which Cooper pairing is the strongest is set by\nthe velocity vof the phonon. Within this optimal range\nof damping, Stokes and anti-Stokes electron-phonon\nscattering processes act constructively to increase the\ne\u000bective coupling constant. Outside this window, the\nstrength of pairing deteriorates leading to a reduction in\nTc. The prominence of the peak is enhanced when the\nFermi energy is large compared to the electron-phonon\ncoupling. Since the phonon damping corresponds to\nthe phonon linewidth, these predictions may be further\ntested and investigated experimentally. The same results\n(anharmonic enhancement of Tcand non-monotonicity\nwith damping) and the same resonance mechanism (this\ntime due to Klemens damping [47]) apply in the case of\npairing mediated by optical phonons, as shown in theAppendix A.3 below. Hence, the presented framework\nmay lead to new guidelines for material design to\noptimizeTcin conventional superconductors, including\nhigh-T hydrides.\nAcknowledgements { Useful discussions with Boris\nShapiro are gratefully acknowledged. M.B. acknowledges\nthe support of the Spanish MINECO's \\Centro de Exce-\nlencia Severo Ochoa\" Programme under grant SEV-2012-\n0249. CS is supported by the U.S. DOE grant number\nDE-FG02-05ER46236.\nAppendix A: Details of the derivations\n1. Theoretical framework\nTo obtain Eq. 9 from the gap equation (Eq. 8; see\nFig. 3 for the associated self-energy diagram) we make\nthe assumption of an isotropic gap function independent\nof frequency, i.e., \u0001( i!m;k+q)\u0011\u0001. This allows us\nto cancel the order parameter in the numerator on both\nsides of Eq. 8 and eliminate the !ndependence to yield\n1 =g2\n\fVX\nq;!m1\n((v2\u0000D!m)q2+!2m) (!2m+\u00182q+ \u00012):\n(A1)\nWe can now convert the qmomentum sum into an in-\ntegral by replacing1\nVP\nq!1\n(2\u0019)dR\nddq!R\nN(\u0018)d\u0018,\nwhereN(\u0018) is the density of states at energy \u0018. For\nquadratic bands with chemical potential \u0016, we have\n\u0018q=q2\u0000\u0016written in units stated in the main text.\nWe now further assume a featureless density of states\nand approximate N(\u0018)'N(0) as in a BCS supercon-\nductor. This is exact in two dimensions and works well\nwhen the chemical potential is far away from the band\nbottom in three dimensions. De\fning \u0015=g2N(0), we\n\fnally obtain Eq. 9.\nTo obtain Eq. 11 from Eq. 10, we can simplify the\nMatsubara sum by summing over only positive frequen-\ncies and writing the equation for Tcas\n1 =\u0015\n2(2\u0019Tc)21X\nm=0\"\n1\nx(x2+ (v20+Dx)2)\n+1\nx(x2+ (v20\u0000Dx)2)#\n(A2)\nwherex\u0011m+1\n2and the primed quantities are di-\nmensionless variables normalized by 2 \u0019Tc(i.e,v20=\nv2=2\u0019Tc). One can then use partial fractions to sim-\nplify the denominators and use the identity  (z) =\nlimk!1n\n\u0000Pk\u00001\nn=01\nn+z+ lnko\n. The logarithmic terms\ncancel to yield Eq. 11. Similarly, one can obtain Eq. 13\nfrom Eq. 12 by shifting the summation over positive fre-\nquencies and writing the equation for Tcas6\n1 =\u0015\u0019Tc\n(2\u0019Tc)31X\nm=0\"\u0010\nx2+ (v20\u0000Dx)\u00160\u0011\nx\u00001\nh\n(x2+ (v20\u0000Dx)\u00160)2+x2(v20\u0000Dx)2i+\u0010\nx2+ (v20+Dx)\u00160\u0011\nx\u00001\nh\n(x2+ (v20+Dx)\u00160)2+x2(v20+Dx)2i#\n:(A3)\n\u0005(q;i!n\u0000i!m)\ng(k+q;i!m)\nFigure 3. Feynman diagram for the anomalous self-energy.\nIn the weak coupling BCS limit, the anomalous self-energy\nreduces to the gap function. The solid (zig-zag) line is the\nelectron (boson) Green function in the superconducting state.\nWe again expand the summand above in partial frac-\ntions by factoring the denominators. Performing the\nremaining integer summations using the identity for\n (x) de\fned above, we obtain Eq. 13.\n2. The resonance condition\nIn this paragraph, we provide more details about the\nresonance condition discussed in the main text. The idea\nis that at a speci\fc frequency, sometimes referred to as\ntheIo\u000be-Regel frequency [37], the boson mediator for the\nphonons undergoes a crossover from a ballistic propaga-\ntion to a di\u000busive incoherent motion. More precisely, this\nhappens at:\n!IR\u0018v2\n\u0019D(A4)\nThis value is of fundamental importance in the realm of\namorphous systems, because of its correlation with the\nboson peak frequency, where the vibrational density of\nstates (VDOS), normalized by the Debye law \u0018!2, dis-\nplays a maximum value [38{40]. The same boson peak\nphenomenology, however, is also at play in strongly an-\nharmonic crystals [41, 42].\nPhysically, this means that the density of the boson\nmediators is maximal around the boson peak frequency.\nAs a consequence, one would expect the e\u000bects of the\nmediators to be enhanced at such energy scale. By esti-\nmating that:\n!IR\u0018Tc (A5)\nwe arrive at the following phenomenological resonance\n1.0 1.2 1.4 1.6v0.20.40.60.81.0Dmax\nv2πTcFigure 4. A validation of the resonance condition (A6) using\nthe data of \fg.1.\ncondition:\nTc\u0018v2\n\u0019Dmax(A6)\nwhich is quoted in the main text. Here Dmaxis the value\nof the phonon viscosity at which Tcis maximized.\nIn order to validate this expression, we plot the ratio\n\u0019DTc=v2in \fgure 4 for the same curves shown in the\nmain text in \fg.1. We observe, that, especially for large\nvalues of the sound speed (compared to the phonon vis-\ncosityD), the resonance condition (A6) holds to good\naccuracy. This observation provides a useful correlation\nbetween the energy scale of the boson peak (induced by\nanharmonicity) and the maximum critical temperature\nthat can be reached.\n3. Pairing mediated by anharmonic optical\nphonons\nIn the main text we focused our attention on the case\nof pairing mediated by acoustic phonons, where the an-\nharmonic damping is di\u000busive, \u0000 \u0018q2, according to the\nAkhiezer mechanism. In this section, we consider the\ncase of pairing mediated by optical phonons. In the case\nof optical phonons, the anharmonic damping is mainly\nrelated to the decay process of the optical phonon into\ntwo acoustic phonons. The damping coe\u000ecient \u0000 is in-\ndependent of q, in this case, and was famously calculated\nby Klemens using perturbation theory [47]. As shown by\nKlemens, the damping parameter \u0000 for optical phonons is\nproportional to the square of the Gr uneisen constant \rof\nthe material. Hence, also in this case the Tc-enhancement\ncould be tuned via the interatomic potential of the pa-\nrameter through \r, in a material-by-design perspective.\nHence, we take a typical dispersion relation for optical7\n1 2 3 4Γ0.81.01.21.41.6T˜\nc\n1 2 3 4 5Γ0.10.20.30.40.50.60.70.8T˜\nc\nFigure 5. The dimensionless critical temperature ~Tc\u0011\n2\u0019Tc=p\n\u0015in function of the constant damping \u0000. Top: In-\ncreasing the mass gap of the optical mode !2\n0from orange\nto purple. Bottom: Increasing the curvature of the optical\ndispersion relation \u000bfrom yellow to black.\nphonons,\n\nopt(q) =!0+\u000bq2(A7)with Klemens damping given a constant \u0000. We imple-\nment this model of optical phonons into the Green's func-\ntion Eq. (2) of the main article, this time with damping\n\u0000 =const independent of q[47], leading to the following\nform of the Bosonic propagator:\n\u0005(i\nn;q) =1\n[!2\n0+ 2!0\u000bq2+O(q4) ] + \n2n\u0000\u0000 \nn:\n(A8)\nUpon implementing this propagator in the theoretical\nframework above, we obtain the theoretical predictions\nforTcas a function of anharmonic damping constant\n\u0000 for pairing mediated by optical phonons, reported in\nFig.5 above.\nThese predictions align well with the e\u000bect of Tc-\nenhancement due to anharmonic damping at low damp-\ning, followed by a peak and subsequent decrease of Tc,\nthat was shown in the main article for acoustic phonons.\nAlso, in this case, clearly, the anharmonic damping can\nlead to a substantial increase of Tc, by at least a factor\nthree. Furthermore, theory predicts that the damping-\ninduced enhancement, and the peak, become larger upon\nincreasing the optical phonon energy gap !0, as shown in\nthe top panel of Fig.5. 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The transformations of t he decay laws at rest and\nof the intermediate times at rest, which are induced by the ch ange of reference frame,\nare obtained by decomposing the modulus of the survival ampl itude at rest into purely\nexponential and exponentially damped oscillating modes. T he mass distribution density\nis considered to be approximately symmetric with respect to the mass of resonance. Under\ndetermined conditions, the modal decay widths at rest, Γj, and the modal frequencies\nof oscillations at rest, Ωj, reduce regularly, Γj/γandΩj/γ, in the laboratory reference\nframe. Consequently, the survival probability at rest, the intermediate times at rest and,\nif the oscillations are periodic, the period of the oscillat ions at rest transform regularly\nin the laboratory reference frame according to the same time scaling, over a determined\ntime window. The time scaling reproduces the relativistic d ilation of times if the mass of\nresonance is considered to be the effective mass at rest of the moving unstable quantum\nsystem with relativistic Lorentz factor γ.\n1 Introduction\nIn recent years the experimental work which has been devoted to analyze the decays of unstable\nsystems has shown various oscillating behaviors of the deca y laws [1, 2, 3]. Oscillations of\nthe decay probability density, or, equivalently, the decay rate, have been detected over short\ntimes in the electron capture decays of Hydrogen-like ions [ 1, 2]. These short-time oscillations\nare superimposed on the canonical exponential decay law and are known as the GSI anomaly.\nRefer to [1, 2] for details. Over much longer time scale, smal l periodical deviations are observed\nin the exponential decay of the unstable nuclei32Si[3]. The period of the superimposed\noscillations is one year, while the half-life of the unstabl e state is approximately 170years.\nRefer to [3] for details. Oscillating decay laws of unstable quantum systems are obtained\nin Refs. [4, 5] by introducing deviations from the Breit-Wign er mass (energy) distribution\ndensity. These deviations are proposed as an explanation of the GSI anomaly. See Refs. [4, 5]\nfor details.\nDecay processes are often detected in laboratory reference frames where the unstable\nsystems are moving at relativistic or ultrarelativistic ve locities. For example, the relativistic\nLorentz factor of the Hydrogen-like ions which are produced in the GSI experiment is equal to\n1.43. See Table 2 of Ref. [1] for details. Due to the change of refer ence frame, the decay laws\nat rest are detected as transformed in the laboratory refere nce frame. This transformation\nis described via quantum theory and special relativity in ca se the unstable system moves\nwith constant linear momentum in the laboratory reference f rame [6, 7, 8, 9, 10, 11, 12, 13,\n114, 15, 16, 17]. The survival amplitude at rest transforms in the laboratory reference frame\naccording to an integral form which involves the MDD and depe nds on the linear momentum\nof the moving unstable quantum system.\nThe appearance of the relativistic dilation of times in the d ecay laws of moving unstable\nparticles remains a matter of central interest. See Refs. [9 , 10, 18, 11, 19, 20, 13, 14, 15, 16, 17],\nto name but a few. Recent analysis has shown that the long-tim e inverse-power-law decays\nat rest of moving unstable quantum systems transform in the l aboratory reference frame,\napproximately, according to a scaling relation [21]. The sc aling factor is determined by the\n(non-vanishing) lower bound of the mass spectrum and by the l inear momentum, and consists\nin the ratio of the asymptotic value of the instantaneous mas s and of the instantaneous mass\nat rest of the moving unstable quantum system [13, 16, 22, 23, 24, 25]. The scaling relation\nreproduces, approximately, the relativistic dilation of t imes if the (non-vanishing) lower bound\nof the mass spectrum is considered to be the effective mass at r est of the moving unstable\nquantum system. See Ref. [21] for details. Oscillating beha viors of the survival probability\nappear if the unstable quantum system is prepared in the supe rposition of two eigenstates\nof the Hamiltonian with different eigenvalues [11, 26]. In th e laboratory reference frame the\ntransformed survival probability shows a dilation of times which, in general, is not Einsteinian.\nSee Refs. [11, 26] for details. The relativistic dilatation of times is found, approximately, if the\ninitial superposition consists in two, approximately orth ogonal, unstable quantum states with\nBreit-Wigner forms of MDDs and the two masses of resonance are approximately equal [10].\nIf the two MDDs are bounded from below and exhibit thresholds , the survival probability is\ndescribed in the laboratory reference frame by damped oscil lations, in case the lower bounds\nof the two mass spectra differ [27]. The frequency of the dampe d oscillations diminishes if\nit is compared to the frequency of the oscillations at rest. T he transformed frequency tends\nto vanish in the ultrarelativistic limit. The lower bounds o f the mass spectra and the linear\nmomentum of the moving unstable quantum state determine the ratio of the two frequencies.\nRecently, the decay laws of moving unstable quantum systems have been studied over\nintermediate times by decomposing the modulus of the surviv al amplitude at rest into su-\nperpositions of exponential modes via the Prony analysis [2 8]. The transformation of the\nintermediate times is described by expressing the survival probability Pp(t), which is detected\nin the laboratory reference frame Sp, where the unstable system moves with constant linear\nmomentum p, as the transformed form P0(ϕp(t))of the survival probability at rest P0(t),\nwhich is detected in the rest reference frame S0. Over a determined time window, this func-\ntion grows linearly. The corresponding scaling law reprodu ces the relativistic dilation of times,\napproximately, over the time window if the mass of resonance of the MDD is interpreted as\nthe effective mass at rest of the moving unstable quantum syst em. Refer to [28] for details.\nAs a continuation of the above-described scenario, here, we consider moving unstable\nquantum systems which exhibit in the rest reference frame S0oscillating decay rates. We\nintend to evaluate the transformed decay laws and the transf ormation of times in case the\ndecay laws at rest are decomposed into superpositions of pur ely exponential modes and expo-\nnentially damped oscillating modes. We aim to find the condit ions under which the relativistic\ndilation of times appears in the transformation of times.\nThe paper is organized as follows. Section 2 is devoted to the decay laws of moving\nunstable quantum systems and to the general transformation which is due to the change of\nreference frame. In Section 3, the transformation of the dec ay laws with oscillating decay\nrates is determined via the transformed exponential and exp onentially damped oscillating\nmodes. Section 4 is devoted to the appearance of the relativi stic time dilation in the genera\ntransformation of times. Summary and conclusions are repor ted in Section 5. Demonstrations\nof the results are provided in appendix.\n22 Moving unstable quantum systems and oscillating decay rat e\nFor the sake of clarity, we report below some details about th e general transformation of the\ndecay laws at rest which provides the decay laws in the labora tory reference frame Sp. The\ndescription is performed by following Ref. [16]. Let the sta te kets|m,p/an}bracketri}htbelong to the Hilbert\nspaceHof the quantum states of the unstable system and be the common eigenstates of the lin-\near momentum Pand of the Hamiltonian Hself-adjoint operator. The corresponding eigenval-\nues are respectively p, i.e.,P|m,p/an}bracketri}ht=p|m,p/an}bracketri}ht, andE(m,p), i.e.,H|m,p/an}bracketri}ht=E(m,p)|m,p/an}bracketri}ht, for\nevery value mwhich belongs to the continuous spectrum of the Hamiltonian . Let|φ/an}bracketri}htbe the ini-\ntial state of the unstable quantum system. This state ket bel ongs to the Hilbert space Hand is\nrepresented in terms of the eigenstates |m,0/an}bracketri}htof the Hamiltonian, |φ/an}bracketri}ht=´∞\nµ0/an}bracketle{t0,m||φ/an}bracketri}ht|m,0/an}bracketri}htdm,\nwhere/an}bracketle{t0,m|is the bra of the state ket |m,0/an}bracketri}ht.\nIn the rest reference frame S0of the moving unstable quantum system, the survival\namplitude at rest A0(t)is given by the following form, A0(t) =/an}bracketle{tφ|e−ıHt|φ/an}bracketri}ht, whereıis the\nimaginary unit. The completeness of the eigenstates of the H amiltonian leads to the following\nintegral expression of the survival amplitude at rest [16, 1 3, 22, 23, 29],\nA0(t) =ˆ\nspec{H}ω(m)exp(−ımt)dm. (1)\nThe domain of integrations, spec{H}, is the spectrum of the Hamiltonian H, which is assumed\nto be continuous. The function ω(m)represents the MDD of the unstable quantum system,\nω(m) =|/an}bracketle{t0,m||φ/an}bracketri}ht|2. The initial state and the Hamiltonian of the system determi nes the\nMDD via the eigenstates |m,0/an}bracketri}ht. In the rest reference frame S0of the moving unstable\nquantum system, the non-decay or survival probability is gi ven by the square modulus of\nthe survival amplitude at rest, P0(t) =|A0(t)|2, and is referred to as survival probability\nat rest. Expression (1) of the survival amplitude provides o scillations of the decay rate,\nwhich are superimposed to the canonical exponential decay l aw, in case the Breit-Wigner\nform of the MDD is cut on the left and right side of the peak [4, 5 ]. The high-mass cutoff\ndetermines the oscillatory behavior of the decay rate and do es not alter significantly the\nexponential decay, while the low-mass cutoff determines the transition from the exponential\nto the inverse-power-law regime of the decay. The MDD is symm etric with respect to the\nmass of resonance, in general, in case the mass-dependent cu toff of the Breit-Wigner form\nexhibits the same symmetry. In particular, the Breit-Wigner MDD is symmetric if it is cut\nby the smooth Fermi-like cutoff function. See Refs. [4, 5] for details.\nIn the laboratory reference frame Spthe unstable system is described by the state ket |φp/an}bracketri}ht\nwhich is an eigenstate of the linear momentum Pwith eigenvalue p. The transformed survival\namplitude Ap(t)is given by the following expression, /an}bracketle{tφp|e−ıHt|φp/an}bracketri}ht, and is represented by the\nintegral form below,\nAp(t) =ˆ\nspec{H}ω(m)exp/parenleftBig\n−ı/radicalbig\np2+m2t/parenrightBig\ndm. (2)\nThe survival probability, Pp(t), is provided by the square modulus of the above expression,\nPp(t) =|Ap(t)|2. The integral expression (2) of the survival probability ha s been obtained in\nvarious ways by adopting quantum theory and special relativ ity. See Refs. [9, 10, 11, 14, 15,\n16, 17] for details.\nAn analytical form of the transformed survival amplitude Ap(t)is obtained in Ref. [10]\nfor the truncated form of the Breit-Wigner MDD which vanishes for negative values of the\nmass spectrum. Over sufficiently long times the survival ampl itude results in a superposition\nof a dominant exponential decay and a dominant inverse power law. If the decay width is\nvery small compared to the mass of resonance the exponential term of the survival amplitude\ndominates over the inverse power law for various lifetimes [ 10, 29] and the survival probability\n3exhibits an approximate exponential decay over these times . Under determined conditions,\nthe relativistic dilation of times is approximately recove red. See Ref. [10] for details.\nIn Ref. [28] the transformed decay laws are described analyt ically over intermediate times\nby approximating the modulus of the survival amplitude at re st with superpositions of expo-\nnential modes via the Prony analysis. This description reli es upon the following assumptions.\nOver intermediate times, the decay laws are mainly determin ed by the behavior of the MDD\naround the mass of resonance Mand the contribution of the unphysical negative values of th e\nmass spectrum to the decay laws is negligible. The MDD is cons idered to be approximately\nsymmetric with respect to the mass of resonance,\nω/parenleftbig\nM+m′/parenrightbig\n=ω/parenleftbig\nM−m′/parenrightbig\n. (3)\nThese assumptions are founded on the following observation s [30, 31, 32, 4, 14, 15]. Purely\nexponential decays of the survival probability are obtaine d from Lorentzian MDDs through the\ncomplex-valued and simple pole of the Lorentzian function. Lorentzian MDDs are symmetric\nwith respect to the mass of resonance. The support of these MD Ds is the whole real line\nand the contribution of the unphysical negative values of th e mass spectrum to the survival\namplitude is negligible. This condition is satisfied in case the decay widths of the exponential\nmodes are small compared to the mass of resonance. This appro ach provides an integral form\nof the MDD in term of the modulus of the survival amplitude at r est,\nω/parenleftbig\nM+m′/parenrightbig\n=1\nπ/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ∞\n0/radicalbig\nP0(t′)cos/parenleftbig\nm′t′/parenrightbig\ndt′/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (4)\nIn this way, the transformed survival probability is relate d to the modulus of the survival\namplitude at rest by the integral form below,\nPp(t) =/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nπˆ∞\n0exp/parenleftBig\n−ı/radicalbig\np2+m2t/parenrightBig\ndmˆ∞\n0/radicalbig\nP0(t′)cos/parenleftbig\nMt′/parenrightbig\ncos/parenleftbig\nmt′/parenrightbig\ndt′/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n. (5)\nIn the rest reference frame S0, the exponential-like regime represents the condition ove r which\nthe decay is purely exponential, or slower, but has not yet be come the inverse power law.\nRelying on the above-reported assumptions, the modulus of t he survival amplitude at rest is\napproximated by a superposition of a finite number of exponen tial modes over the intermediate\ntimes. The above approximation is provided by the Prony anal ysis of the modulus of the\nsurvival amplitude at rest. The transformed expression of t he survival probability is evaluated\nvia Eq. (5) and from the analysis which is performed in Ref. [1 0]. In the laboratory reference\nframeSp, the survival probability Pp(t)is described, under determined conditions, by a\nsuperposition of exponential modes over an estimated time w indow. The survival probability\nPp(t)is approximately related to the survival probability at res tP0(t)by the scaling relation\nwhich reproduces, approximately, the relativistic dilati on of times, if the mass of resonance\nMof the MDD is interpreted as the effective mass at rest of the un stable quantum systems\nwhich moves with constant linear momentum p. See Ref. [28] for details.\nIn the following, we intend to use Eq. (5) in order to evaluate the survival probability\nPp(t), which is detected in the laboratory reference frame Spover intermediate times, by\ndecomposing the modulus of the survival amplitude at rest/radicalbig\nP0(t)into purely exponential\nand exponentially damped oscillating modes. It is assumed t hat the MDD is approximately\nsymmetric with respect to the mass of resonance, Eq. (3). Thi s assumption relies on the\nobservations which are reported at the end of the second para graph and at the beginning of\nthe fifth paragraph of the present Section.\n3 Transformation of oscillating decay laws\nAt this stage, we start our analysis by considering decay law s at rest with oscillating decay\nrates. For the sake of shortness, here, these decay laws are r eferred to as oscillating. We\n4intend to study how this kind of decays transforms by changin g the reference frame. We\nstart from the simplest form of oscillating decay law at rest which is obtained by adding an\nexponentially damped oscillating term to the exponentiall y decaying modulus of the survival\namplitude,\n/radicalbig\nP0(t) = exp/parenleftbigg\n−Γ\n2t/parenrightbigg\n(1−a+acos(Ωt)). (6)\nThis form is generalized in the next Section by approximatin g the modulus of the survival\namplitude at rest via the superposition of an arbitrary finit e number of purely exponential\nmodes and exponentially damped oscillating modes. The rest rictions1/2> a >0,Ω< M,\nΓ/(M−Ω)≪1andΓ>2aΩ/√1−2aare required for the parameter a, for the frequency\nΩof the oscillations and for the decay width Γ. In this way, the survival probability at rest\nP0(t)fulfills the canonical proprieties. The function P0(t)is positive, decreases monotonically,\n˙P0(t)<0for every t >0, and takes the value P0(0) = 1 , in addition to P0(∞) = 0. The\nsurvival probability at rest P0(t)is the sum of the purely exponential term P(exp)\n0(t)and of\nthe exponentially damped oscillating term P(osc)\n0(t),\nP0(t) =P(exp)\n0(t)+P(osc)\n0(t), (7)\nwhere\nP(exp)\n0(t) =/parenleftbigg\n(1−a)2+a2\n2/parenrightbigg\nexp(−Γt), (8)\nP(osc)\n0(t) = exp( −Γt)/parenleftbigg\n2a(1−a)cos(Ωt)+a2\n2cos(2Ωt)/parenrightbigg\n. (9)\nThe decay probability density at rest, or the decay rate at re st, is defined as the opposite of\nthe derivative of the survival probability at rest and is pro vided by the oscillating form below,\n˙P0(t) =−exp(−Γt)(1−a+acos(Ωt))(λ1+λ2cos(Ωt−β)). (10)\nThe coefficients λ1andλ2, and the angle βread\nλ1= Γ(1−a), λ2=a/radicalbig\nΓ2+4Ω2, β= arccosΓ√\nΓ2+4Ω2.\nAgain, we stress that Eq. (6) can not approximate the surviva l probability over very short and\nvery long times. In fact, Eq. (10) provides a negative value o f the decay rate, ˙P0(0) =−Γ,\ninstead of vanishing, for t= 0.\nIn the laboratory reference frame Spthe survival probability Pp(t)is approximated by\nthe expression below,\nPp(t)≃/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleK(M,Γ,p,Ω,a,t)+ıpΓ\nπM2Φ(M,p,Ω,a,t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n, (11)\nover times tsuch that either t >1/(10Γ) or(M−Ω)t≫1. The function K(M,Γ,p,Ω,a,t)\nandΦ(M,p,Ω,a,t)are defined as below,\nK(M,Γ,p,Ω,a,t) = (1−a)exp/parenleftbigg\n−1\n2Υ(M,Γ,p)t/parenrightbigg\n+a\n2/parenleftBigg\nexp/parenleftbigg\n−1\n2Υ(M−Ω,Γ,p)t/parenrightbigg\n+exp/parenleftbigg\n−1\n2Υ(M+Ω,Γ,p)t/parenrightbigg/parenrightBigg\n, (12)\nΦ(M,p,Ω,a,t) = (1−a)Ξ(M,p,t)+a\n2/parenleftBigg\nΞ(M−Ω,p,t)\n(1−Ω/M)2+Ξ(M+Ω,p,t)\n(1+Ω/M)2/parenrightBigg\n. (13)\n5The function Υ(M,Γ,p)is defined for every value of the mass of resonance M, of the decay\nwidthΓand of the linear momentum p, by the following expression [28],\nΥ(M,Γ,p) = Λ−(M,Γ,p)+ıΛ+(M,Γ,p), (14)\nwhere\nΛ∓(M,Γ,p) =/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt2\n/radicalBigg/parenleftbigg\nM2−Γ2\n4+p2/parenrightbigg2\n+M2Γ2∓/parenleftbigg\nM2−Γ2\n4+p2/parenrightbigg\n. (15)\nThe function Ξ(M,p,t)is defined, for every value of the mass of resonance M, of the decay\nwidthΓand of the linear momentum p, via the Bessel Function J1(pt), the modified Bessel\nfunction Y1(pt)and the Struve function H1(pt)as below [33, 34, 35],\nΞ(M,p,t) =π\n2(H1(pt)−ıJ1(pt))−1+1−p2/M2\n(1+p2/M2)2\n×/parenleftBig\n1+π\n2(Y1(pt)−H1(pt))/parenrightBig\n. (16)\nDue to the oscillations, additional terms appear in the expr ession of the transformed survival\nprobability. These terms are related to the shifted masses o f resonance M−andM+which\nare determined by the frequency Ωof the oscillations,\nM∓=M∓Ω.\nWe remind that the condition Ω< Mis required.\n3.1 Time window for exponentially damped oscillating decay s in the lab-\noratory reference frame\nAt this stage, we search for times over which the oscillating decay laws are exponentially\ndamped in the laboratory reference frame Sp. Here, we refer to these times, if they exist, as\nthe exponential times of the transformed oscillating decay s. For the sake of convenience, we\nintroduce the shifted relativistic Lorentz factors γ+andγ−and the shifted decay widths Γ+\nandΓ−,\nγ∓=/radicalBigg\n1+p2\n(M∓)2,Γ∓=γ\nγ∓Γ.\nThe function Π(M,Γ,p,Ω,a,t), which appears in Eq. (11), provides purely exponential dec ays\nwith decay widths Γ−/γ,Γ/γ,Γ+/γ, and exponentially damped oscillations with frequencies\n|M−γ−−Mγ|,|M+γ+−Mγ|,|M−γ−−M+γ+|, due to the condition Γ/M−≪1.\nIn case the frequency of the oscillations at rest is small com pared to the mass of resonance,\nΩ≪M, and if the parameter ais not too close to the value 1/2, the modulus of the\ntransformed survival amplitude is approximated via the sim ple form below,\nPp(t)≃exp/parenleftbigg\n−Γ\nγt/parenrightbigg/parenleftbigg\n1−a+acos/parenleftbiggΩ\nγt/parenrightbigg/parenrightbigg2\n, (17)\nover the time window\n2ζmax\nΓγ/greaterorsimilart/greaterorsimilar2ζmin\nΓγ, (18)\n6on condition that the following constraints are fulfilled, M−t≫1ort >1/(10Γ), andpt≫1.\nThe allowed values of the parameter aare estimated by studying the order of magnitude of\nthe parameter ξ′which is defined as below,\nξ′=ΓW(M,Ω,a)\n2M(1−2a)/radicalBigg\nΓ\nπM/radicalbigg\n1−1\nγ2. (19)\nThe function W(M,Ω,a)is defined in appendix. If ξ′≪10−2andΩ≪M, the transformed\nsurvival probability is given by Eq. (17) over the times tof the exponential time window (18)\nsuch that either t >1/(10Γ) orM−t≫1, andpt≫1. This technique is adopted in Ref.\n[28] in order to estimate the set of times over which the decay laws consist, approximately, in\nsuperpositions of exponential modes in the laboratory refe rence frame Sp. For example, let\nthe order of magnitude of the ratio Γ/Mbe equal to or less than (−3)and let the parameter\nafulfill the constraint 0< a </parenleftbig\n1−10−1/parenrightbig\n/2. The condition t >1/(10Γ1)is fulfilled over the\ntime window (18) if 20ζminγ >1. If this condition is not realized, the constraint M−t≫1\nis fulfilled over the time window if 2ζminγ≫Γ/M−. The condition pt≫1holds over the\ntime window if 2ζminγ/radicalbig\nγ2−1≫Γ/M. The nonrelativistic limit, γ→1+, is excluded.\nUsually, the above constraint hold except for strong decays and nonrelativistic regime, γ≃1\nor, equivalently, p≪M. If these constraints do not hold, a different order of magnit ude of\nthe parameter ξ′is chosen and the values of the parameters ζminandζmaxmust be changed\naccordingly. Refer to [28] for details.\nIn caseΩ≪Mand if the parameter ais not too close to the value 1/2in such a way\nthatξ′≪10−2, the analysis which is performed in the previous paragraph s uggests that the\nsurvival probability at rest transforms according to the fo llowing scaling relation,\nPp(t)≃ P0/parenleftbiggt\nγ/parenrightbigg\n, (20)\nover the times tof the window (18) such that either M−t≫1ort >1/(10Γ), andpt≫1.\nOver the selected times both the exponential-like term P(exp)\np(t)and the damped oscillating\ntermP(osc)\np(t)transform, independently and regularly, according to the s ame scaling relation,\nPp(t)≃ P(exp)\np(t)+P(osc)\np(t), (21)\nwhere\nP(exp)\np(t)≃ P(exp)\n0/parenleftbiggt\nγ/parenrightbigg\n, (22)\nP(osc)\np(t)≃ P(osc)\n0/parenleftbiggt\nγ/parenrightbigg\n. (23)\nThe decay width Γand the frequency Ωof the oscillating decay law at rest (6) transform reg-\nularly, by changing reference frame, in the values Γ/γandΩ/γ, respectively. The transformed\ndecay width, Γ/γ, and the transformed frequency of the oscillation, Ω/γ, tend to vanish in the\nultrarelativistic limit, p≫M. Approximately, the transformed period Tpof the oscillations\ndepends regularly on the period of oscillation T0at rest,\nTp≃γT0, (24)\nwhereT0= 2π/Ω. Obviously, the transformed period diverges in the ultrare lativistic limit,\np≫M.\nNumerical analysis of the survival probability Pp(t)is performed in Figure 1 in case the\nsurvival probability at rest P0(t)is obtained from Eq. (6). The damped oscillating behavior\nwhich is displayed in Figure 2 is in accordance with the asymp totic analysis which is described\nby Eqs. (17)-(24). The transformed periods of the oscillati ons agree with Eq. (24).\n7Figure 1: (Color online) Transformed survival probability Pp(t)versusΓt, for2≤Γt≤11,\nand different values of the ratios p/ΓandM/Γand of the corresponding Lorentz factor γ.\nThe survival probability at rest P0(t)is described via Eq. (6) with different values of the\nfrequency Ωand of the amplitude a. Curve (a)corresponds to p/Γ = 150 ,M/Γ = 30 ,\nγ≃5.0990,Ω/Γ = 5 anda= 0.09. Curve (b)corresponds to p/Γ = 200 ,M/Γ = 80 ,\nγ≃2.6926,Ω/Γ = 10 anda= 0.04. Curve (c)corresponds to p/Γ = 210 ,M/Γ = 100 ,\nγ≃2.3259,Ω/Γ = 10 anda= 0.04. Curve (d)corresponds to p/Γ = 200 ,M/Γ = 150 ,\nγ≃1.6667,Ω/Γ = 40 anda= 0.01. Curve(e)corresponds to p/Γ = 100 ,M/Γ = 100 ,γ0√\n2,\nΩ/Γ = 10 anda= 0.04.\nFigure 2: (Color online) Quantity exp(Γt/γ−)Pp(t)versusΓt, for2≤Γt≤15, and different\nvalues of the ratios p/ΓandM/Γand of the corresponding Lorentz factor γ. The survival\nprobability at rest P0(t)is described via Eq. (6) for different values of the frequency Ωand of\nthe amplitude a. Curve(a)corresponds to p/Γ = 200 ,M/Γ = 80 ,γ≃2.6926,Ω/Γ = 10 and\na= 0.04. Curve(b)corresponds to p/Γ = 100 ,M/Γ = 100 ,γ=√\n2,Ω/Γ = 10 anda= 0.04.\nCurve(c)corresponds to p/Γ = 210 ,M/Γ = 100 ,γ≃2.3259,Ω/Γ = 10 anda= 0.04. Curve\n(d)corresponds to p/Γ = 150 ,M/Γ = 30 ,γ≃5.0990,Ω/Γ = 5 anda= 0.09.\n83.2 Transformation of times and relativistic time dilation in the oscillating\ndecay laws\nAt this stage, we study the transformation of times in case th e oscillating decay law at rest\nis obtained from Eq. (6). This transformation is described b y the function ϕp(t), which is\nintroduced in Ref. [28] via the following relation,\nPp(t) =P0(ϕp(t)). (25)\nThe survival probability at rest P0(t)is an invertible function of time since it is canonically\ndecreasing. Therefore, the function ϕp(t)is properly defined by the following expression,\nϕp(t) =P−1\n0(Pp(t)), (26)\nfor every t≥0. In general, for the oscillating decay law (6) the inverse fu nctionP−1\n0(r)is\ncomputed numerically. Once the function P−1\n0(r)is obtained, the function ϕp(t)is evaluated\nunder the conditions Ω< M,Γ/M−≪1, and either M−t≫1ort >1/(10Γ), via Eqs.\n(11)-(13),\nϕp(t)≃ P−1\n0/parenleftBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleK(M,Γ,p,Ω,a,t)+ıpΓ\nπM2Φ(M,p,Ω,a,t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/parenrightBigg\n. (27)\nIn case the frequency Ωof the oscillations at rest is small compared to the mass of re so-\nnance,Ω≪M, and ifξ′≪10−2, the survival probability transforms according to the scal ing\nrelation (20) over the times twhich belong to the time window (18) and such that either\nM−t≫1ort >1/(10Γ), andpt≫1. The nonrelativistic regime, p≪M, is excluded.\nAccording to the scaling relation (20), the function ϕp(t)is approximately linear over the\nexponential times,\nϕp(t)≃t\nγ. (28)\nIf the mass of resonance Mis considered to be the effective mass at rest of the unstable\nquantum systems which moves with constant linear momentum p, the scaling relation (20)\nreproduces, approximately, the relativistic dilation of t imes. Over these times, the period of\nthe oscillations is the relativistic dilation of the period T0of the oscillations at rest.\nIn summary, we have considered decay laws at rest which repro duce an oscillating decay\nrate, Eq. (6). We have described the oscillating decay laws i n the laboratory frame, via Eqs.\n(11)-(13), the exponential-like regime of the oscillating decay laws, via Eq. (17) and Eqs.\n(21)-(23), the time window over which the oscillating decay laws are exponentially damped,\nvia Eq. (18), and the transformed modal decay width and frequ ency. Under special conditions\nthe survival probability and the period of the damped oscill ations transforms according to\nthe relativistic dilation of times, Eq. (20) and Eq. (24). Th ese descriptions and properties\nconstitute the first central result of the paper.\nNumerical analysis of the function ϕp(t)is displayed in Figure 3 in case the survival\nprobability is described by the oscillating decay law (6). T he linear growth of the function\nϕp(t), which is predicted theoretically over the time window (18) by the scaling laws (21)-(23),\nis confirmed numerically by the approximate linear behavior s which are observed in Figure 3.\n4 Transformation of decay laws with more exponentially damp ed\noscillating modes\nThe analysis which is performed in the previous Section is ge neralized in the present Section\nto decay laws which are composed by more oscillating modes. I n fact, the modulus of the\n9Figure 3: (Color online) Function ϕp(t)versusΓt, for2≤Γt≤120, and different values of the\nratiosp/ΓandM/Γand of the corresponding Lorentz factor γ. The survival probability at\nrestP0(t)is described via Eq. (6) with different values of the frequenc yΩand of the amplitude\na. Curve(a)corresponds to p/Γ = 100 ,M/Γ = 100 ,γ=√\n2,Ω/Γ = 10 anda= 0.04. Curve\n(b)corresponds to p/Γ = 200 ,M/Γ = 150 ,γ≃1.6667,Ω/Γ = 40 anda= 0.01. Curve\n(c)corresponds to p/Γ = 210 ,M/Γ = 100 ,γ≃2.3259,Ω/Γ = 10 anda= 0.04. Curve\n(d)corresponds to p/Γ = 200 ,M/Γ = 80 ,γ≃2.6926,Ω/Γ = 10 anda= 0.04. Curve (e)\ncorresponds to p/Γ = 150 ,M/Γ = 30 ,γ≃2.6926,Ω/Γ = 5 anda= 0.09.\nsurvival amplitude at rest is approximated by the superposi tion of an arbitrary finite number\nof purely exponential modes and exponentially damped oscil lating modes with decay widths\nΓ1,...,ΓN, and frequencies of oscillations Ω1,...,ΩN,\n/radicalbig\nP0(t) =N/summationdisplay\nj=1wjexp/parenleftbigg\n−Γj\n2t/parenrightbigg\n(1−aj+ajcos(Ωjt)), (29)\nwhere0<Γ1< ... <ΓN. The constraints wj>0,0< aj<1/2,Ωj< M,Γj/(M−Ωj)≪\n1,2aΩj//radicalbig1−2aj<Γjare required to hold for every j= 1,...,N . The weights are normal-\nized to unity,/summationtextN\nj=1wj= 1. In this way, the survival probability is canonically decre asing,\n˙P0(t)<0for every t >0, and the canonical value P0(0) = 1 is recovered, in addition to the\nasymptotic limit P0(∞) = 0. The survival probability at rest P0(t), which is obtained from\nEq. (29), is the sum of the exponential-like term P(exp)\n0(t)and of the damped oscillating term\nP(osc)\n0(t),\nP0(t) =P(exp)\n0(t)+P(osc)\n0(t), (30)\nwhere\nP(exp)\n0(t) =N/summationdisplay\nj=1N/summationdisplay\nl=1wjwl/parenleftBigg\n(1−aj)(1−al)+υ′\nj,l\n2ajal/parenrightBigg\nexp/parenleftbigg\n−Γj+Γl\n2t/parenrightbigg\n, (31)\nP(osc)\n0(t) =N/summationdisplay\nj=1N/summationdisplay\nl=1wjwlajexp/parenleftbigg\n−Γj+Γl\n2t/parenrightbigg/parenleftBigg\n2(1−al)cos(Ω jt)\n+al\n2/parenleftBigg\ncos((Ω j+Ωl)t)+υ′′\nj,lcos((Ω j−Ωl)t)/parenrightBigg/parenrightBigg\n. (32)\nThe coefficient υ′\nj,lis unity for j=l, and for j/ne}ationslash=lifΩj= Ωl, and vanishes otherwise. The\ncoefficient υ′′\nj,lvanishes for j=l, and for j/ne}ationslash=lifΩj= Ωl, and is unity otherwise. The decay\n10rate at rest results to be the following product,\n˙P0(t) =−/radicalbig\nP0(t)N/summationdisplay\nj=1wjexp/parenleftbigg\n−Γj\n2t/parenrightbigg\n(λ1,j+λ2,jcos(Ωjt−βj)), (33)\nwhich consists in a superposition of purely exponential mod es and exponentially damped\noscillating modes with more frequencies of oscillations. T he involved coefficients are defined\nby the expressions below,\nλ1,j= Γj(1−aj), λ2,j=aj/radicalBig\nΓ2\nj+4Ω2\nj, βj= arccosΓj/radicalBig\nΓ2\nj+4Ω2\nj,\nfor every j= 1,...,N . Again, we stress that the survival probability at rest can n ot be\napproximated by Eq. (29) over very short and very long times. In fact, according to Eq. (33),\nthe decay rate is negative, ˙P0(0) =−/summationtextN\nj=1wjΓj, instead of vanishing, for t= 0.\nIf the modulus of the survival amplitude at rest is given by Eq . (29) with the required\nconditions, the survival probability Pp(t)is approximated in the laboratory reference frame\nSpby the expression below,\nPp(t)≃/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleN/summationdisplay\nj=1wj/parenleftbigg\nK(M,Γj,p,Ωj,aj,t)+ıpΓj\nπM2Φ(M,p,Ωj,aj,t)/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n, (34)\nover times tsuch that either t >1/(10Γj)or(M−Ωj)t≫1, for every j= 1,...,N . For the\nsake of simplicity, expression (34) certainly approximate s the transformed survival probability\nPp(t)over times tsuch that either t >1/(10Γ1)or(M−Ωmax)t≫1. The frequency Ωmax\nis the maximum among the frequencies Ω1,...,ΩN.\n4.1 Time window for decay laws with more exponentially dampe d oscil-\nlating modes in the laboratory reference frame\nAt this stage, we estimate the exponential times in the labor atory reference frame Sp. We\nrefer to these times, if they exist, as the times over which th e modulus of the transformed\nsurvival amplitude results in superpositions of purely exp onential modes and exponentially\ndamped oscillating modes. For the sake of convenience, we in troduce the shifted masses\nM+\n1,...,M+\nNandM−\n1,...,M−\nN, the shifted decay widths Γ+\n1,...,Γ+\nNandΓ−\n1,...,Γ−\nN, and the\nshifted relativistic Lorentz factors γ+\n1,...,γ+\nNandγ−\n1,...,γ−\nN, via the following expressions,\nM∓\nj=M∓Ωj,Γ∓\nj=γ\nγ∓\njΓj, γ∓\nj=/radicaltp/radicalvertex/radicalvertex/radicalbt1+p2\n/parenleftBig\nM∓\nj/parenrightBig2,\nfor every j= 1,...,N . Exponentially damped oscillations are provided to the exp ression (34)\nof the survival probability by the functions K(M,Γ1,p,Ω1,a1,t),...,K(M,ΓN,p,ΩN,aN,t).\nSee appendix for the description of the exponentially dampe d oscillations of the transformed\nsurvival probability Pp(t)and for details.\nThe transformation of the oscillating decay laws is simplifi ed in case the modal frequencies\nof oscillations are small compared to the mass of resonance, Ωmax≪M. In the laboratory\nreference frame Spthe exponential times are estimated by studying the order of magnitude\nof the parameter ξ′\njwhich is defined as below,\nξ′\nj=/radicalBigg\nΓj\nπM/radicalbigg\n1−1\nγ2N/summationdisplay\nl=1wlΓlW(M,Ωl,al)\n2Mwj(1−2aj), (35)\n11for every j= 1,...,N . The indexes j1,...,jn1are chosen among the indexes 1,...,N , in such\na way that ξ′\njl≪10−2for every l= 1,...,n 1. Notice that this constraint is not fulfilled in the\nlimiting case ajl→1/2−. The chosen indexes are sorted in increasing order, j1< ... < j n1,\nwith1≤n1≤N. The set I′\npof the exponential times is given by the following union,\nI′\np=n1/uniondisplay\nl=1Ip,l, (36)\nwhere\nIp,l=/bracketleftbigg2ζminγ\nΓjl,2ζmaxγ\nΓjl/bracketrightbigg\n, (37)\nfor every l= 1,...,n 1. The set I′\npis represented by one single closed interval if n1= 1and\nin other cases. If n1>1, let the condition\nΓjl\nΓjl+1>ζmin\nζmax≃1.83×10−5, (38)\nhold for every l= 1,...,n 1−1. Then, in the laboratory reference frame Sp, the setI′\npcoincides\nwith the closed interval/bracketleftbig\n2ζminγ/Γjn1,2ζmaxγ/Γj1/bracketrightbig\n. Consequently, the following time window,\n2ζmax\nΓj1γ/greaterorsimilart/greaterorsimilar2ζmin\nΓjn1γ, (39)\nestimates the exponential times in the laboratory referenc e frameSp, in case Ωmax≪M.\nThe present approach requires that the functions Φ(M,p,Ω1,a1,t),...,Φ(M,p,ΩN,aN,t)\nprovide the inverse-power-law behavior 1/√ptto the expression (34) of the transformed sur-\nvival probability over the exponential times. Consequentl y, over the set I′\npof the exponential\ntimes the following constraints must hold, either t >1/(10Γjl)orM−\njlt≫1, for every\nl= 1,...,n 1, andpt≫1. These constraints are certainly fulfilled if either t >1/(10Γ1)or\n(M−Ωmax)t≫1, andpt≫1. The constraint t >1/(10Γ1)is fulfilled over the set I′\npif\n20ζminγ >Γjn1/Γ1. If this constraint is not realized, the condition (M−Ωmax)t≫1holds\nover the set I′\npif2ζminγ≫Γjn1/(M−Ωmax). The condition pt≫1holds over the set I′\np\nif2ζminγ/radicalbig\nγ2−1≫Γjn0/M. The nonrelativistic limit, γ→1+, is excluded. Generally, the\nabove constraints are fulfilled except for strong decays and in the nonrelativistic regime. If the\nabove constraints do not hold, the method which is reported i n the previous paragraph can\nbe adjusted by choosing a different order of magnitude for the parameter ξjand by changing\nthe values of the parameters ζminandζmaxaccordingly. See Ref. [28] for details.\nThe transformed survival probability Pp(t)is approximated by the form\nPp(t)≃/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nl′wjlexp/parenleftbigg\n−Γjl\n2γt/parenrightbigg/parenleftbigg\n1−ajl+ajlcos/parenleftbiggΩjl\nγt/parenrightbigg/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n, (40)\nover the set I′\npof the exponential times, or over the time window (39) if n1= 1or ifn1>1\nand condition (38) is fulfilled for every l= 1,...,n 1−1. The sum/summationtext\nl′depends on time as\nfollows. For every instant twhich belongs to the set I′\npof the exponential times, the sum is\nperformed over every index lwhich is selected among the indexes 1,...,n 1, in such a way\nthatt∈Ip,l. Expression (40) of the survival probability Pp(t)results to be the sum, Eq. (21),\nof the exponential-like term P(exp)\np(t)and of the damped oscillating term P(osc)\np(t), which are\n12given by the following expressions,\nP(exp)\np(t)≃1\n2/summationdisplay\nl′/summationdisplay\nl′′υ′\njl,jl′wjlwjl′ajlajl′exp/parenleftBigg\n−Γjl+Γjl′\n2γt/parenrightBigg\n+/parenleftBigg/summationdisplay\nl′wjl(1−ajl)exp/parenleftbigg\n−Γjl\n2γt/parenrightbigg/parenrightBigg2\n, (41)\nP(osc)\np(t)≃2/summationdisplay\nl′/summationdisplay\nl′′wjlwjl′ajl′(1−ajl)exp/parenleftBigg\n−Γjl+Γjl′\n2γt/parenrightBigg\ncos/parenleftBigg\nΩjl′\nγt/parenrightBigg\n+1\n2/summationdisplay\nl′/summationdisplay\nl′′wjlwjl′ajlajl′exp/parenleftBigg\n−Γjl+Γjl′\n2γt/parenrightBigg\n×/parenleftBigg\ncos/parenleftBigg\nΩjl+Ωjl′\nγt/parenrightBigg\n+υ′′\njl,jl′cos/parenleftBigg\nΩjl−Ωjl′\nγt/parenrightBigg/parenrightBigg\n. (42)\nIn the rest reference frame S0consider the set of times I′\n0which is defined by the following\nunion,\nI′\n0=n1/uniondisplay\nl=1I0,l, (43)\nwhere\nI0,l=/bracketleftbigg2ζmin\nΓjl,2ζmax\nΓjl/bracketrightbigg\n, (44)\nfor every l= 1,...,n 1. By substituting the ratio t/γwithtin Eq. (40), the technique which\nis described in the second paragraph of the present subsecti on selects the dominant modes\nof the modulus of the survival amplitude at rest over the set I′\n0. Refer to [28] for details.\nConsequently, Eq. (21) and the scaling laws (20), (22), (23) hold over the set I′\npof the\nexponential times, in the laboratory reference frame Sp. In the rest reference frame S0the\nscaling laws holds over the set I′\n0of times. If n1= 1, or ifn1>1and condition (38) is fulfilled\nfor every l= 1,...,n 1−1, the setI′\n0coincides with the closed interval/bracketleftbig\n2ζmin/Γjn1,2ζmax/Γj1/bracketrightbig\nand represents the time windows,\n2ζmax\nΓj1/greaterorsimilart/greaterorsimilar2ζmin\nΓjn1, (45)\nin the rest reference frame S0. Consider the set I′\npof the exponential times in the laboratory\nreference frame Sp. Over these times the transformed decay laws are composed by transformed\npurely exponential modes and transformed exponentially da mped oscillating modes. The\nmodal decay width at rest Γjltransforms in the laboratory reference frame in the reduced\ndecay width Γjl/γ, and the modal frequency of oscillation at rest Ωjltransforms in the reduced\nfrequency Ωjl/γ, for every l= 1,...,n 1.\nLet the interval [tp,0,tp,1]exist in the set I′\npof the exponential times such that the modes\nwhich compose the sum/summationtext\nl′do not change for every time t∈[tp,0,tp,1]. Let the maximum\nmodal frequency involved be multiple of the remaining modal frequencies of oscillations. This\nmeans that relation Ωjl= Ω′\nmax/kl, whereklis a nonvanshing natural number, holds for every\nindexlover which the sum/summationtext\nl′is performed. The frequency Ω′\nmaxis the maximum among\nthe frequencies Ωjlof the exponentially damped oscillating modes which are sel ected for the\nevaluation of the sum/summationtext\nl′over the interval [tp,0,tp,1]. Under the above-mentioned conditions,\n13expression (40) of the survival probability is approximate ly periodic over the interval [tp,0,tp,1]\nin the laboratory reference frame Sp. The period T′\npof these oscillations is\nT′\np≃γ2π\nΩ′\nmax, (46)\nand diverges in the ultrarelativistic limit, p≫M. Similarly, the damped oscillations of the\nsurvival probability at rest P0(t)are approximately periodic over the interval [t0,0,t0,1], where\nt0,0=tp,0/γandt0,1=tp,1/γ. The period T′\n0of these oscillations reads\nT′\n0≃2π\nΩ′\nmax. (47)\nRelations (46) are (46) are examined below by considering th e relativistic dilation of times.\n4.2 Transformation of times and relativistic time dilation in decay laws\nwith more exponentially damped oscillating modes\nAt this stage, we describe the transformation of times by eva luating the function ϕp(t)which\nis defined in Section 3.2 via Eqs. (25) and (26). The survival p robability at rest P0(t)is\ngiven by Eq. (29) and is a canonically decreasing function of time for the selected values of\nthe involved parameters. In general, the function P−1\n0(r)is evaluated numerically for the\noscillating decay laws under study. Once the function P−1\n0(r)is obtained, the function ϕp(t)\nis estimated by the following expression,\nϕp(t)≃ P−1\n0/parenleftBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleN/summationdisplay\nj=1wj/parenleftbigg\nK(M,Γj,p,Ωj,aj,t)+ıpΓj\nπM2Φ(M,p,Ωj,aj,t)/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/parenrightBigg\n, (48)\nover times tsuch that either M−\njt≫1ort >1/(10Γj), for every j= 1,...,N .\nIfΩmax≪Mthe survival probability transforms, approximately, acco rding the scaling law\n(20) over the set I′\npof the exponential times in the laboratory reference frame Sp. This scaling\nlaw reproduces the relativistic dilation of times in case th e mass of resonance M is considered to\nbe the rest mass of the moving unstable system. The function ϕp(t)exhibits the approximately\nlinear growth (28) over the set I′\npof the exponential times, or over the time window (39) if n1=\n1or ifn1>1and the constraint (38) is fulfilled for every l= 1,...,n 1−1. The constraints\nwhich are described in the second paragraph of Section 4.1 ar e required to hold over the\nexponential times. Under the above-reported conditions, t he transformed survival probability\nis approximately periodic in the interval [tp,0,tp,1]. Approximately periodic oscillations of the\nsurvival probability at rest appear in the rest reference fr ame over the interval [t0,0,t0,1]. The\nperiod of the transformed oscillations T′\np, given by Eq. (46), is related to the period of the\napproximately periodic oscillations at rest T′\n0, given by Eq. (47), by the following relation,\nT′\np≃γT′\n0. (49)\nThis relation represents the relativistic time dilation of the period T′\n0of the damped oscilla-\ntions at rest if the mass of resonance Mis considered to be the mass at rest of the unstable\nquantum system which moves with constant linear momentum p. Notice that the duration of\nthe periodic oscillations, (tp,1−tp,0), in the laboratory reference frame Spis the relativistic\ntime dilation of the periodic oscillations at rest, (t0,1−t0,0), in the rest reference frame S0.\nIn summary, we have considered oscillating decay laws at res t which consist in superpo-\nsitions of an arbitrary finite number of purely exponential m odes and exponentially damped\noscillating modes with different frequencies of oscillatio ns, Eq. (29). We have described the\noscillating decay laws in the laboratory reference frame, v ia Eq. (34), the exponential-like\n14regime of the oscillating decay laws, via Eqs. (40)-(42), th e exponential times, via Eqs. (35)-\n(39), and the decay widths and the frequencies of oscillatio ns of the transformed modes. Under\nspecial conditions, the purely exponential term, Eq. (31), and the exponentially damped os-\ncillating term, Eq. (32), of the survival probability trans form according to the same scaling\nlaw, Eqs. (22) and (23) and Eqs. (41) and (42), respectively. Under determined conditions\nthe transformed survival probability exhibits damped peri odic oscillations. If the mass of\nresonance is interpreted as the mass at rest of the moving uns table quantum system, the\nsurvival probability and the period of the oscillations, tr ansform, approximately, according to\nthe relativistic dilation of times, Eq. (20) and Eq. (49), re spectively. These descriptions and\nproperties constitute the the last of the main results of the paper.\n5 Summary and conclusions\nThe appearance of oscillations in the decay rate of unstable systems is a peculiar phenomenon\nwhich has attracted a great deal of attention. This interest follows, especially, the detection\nin the GSI experiment of oscillations which are superimpose d on the canonical exponential\ndecay laws [1, 2]. This kind of oscillations are obtained, th eoretically, in the decay laws of\nunstable quantum system by introducing deviations in the Bre it-Wigner form of the MDD\n[4, 5]. Usually, unstable systems move in the laboratory ref erence frame where the decay\nlaws are detected. The general transformation of the decay l aws at rest, which is induced\nby the change of reference frame, is described via quantum th eory and special relativity\n[9, 10, 11, 14, 15, 16, 17]. The transformed survival amplitu de consists in an integral form\nwhich involves the model-independent MDD and the linear mom entum of the moving unstable\nquantum system. This integral form was adopted to study the t ransformation of oscillating\ndecay laws over short and long times in case the decaying syst em is initially prepared in\nsuperpositions of two, approximately orthogonal, unstabl e quantum states [10, 27]. The\ncorresponding MDDs are bounded from below and are represent ed by truncated Breit-Wigner\nforms [10] or exhibit thresholds in the (different) non-vani shing lower bounds of the mass\nspectra [27]. The transformation of the period of the oscill ations is determined by the features\nof the MDDs and by the linear momentum of the moving unstable q uantum system.\nAs a continuation of the above-mentioned scenario, in the pr esent research work we have\nconsidered general forms of the modulus of the survival ampl itude at rest which consist in su-\nperpositions of purely exponential and exponentially damp ed oscillating modes. These forms\nprovide general expressions of the survival probability at rest which decay monotonically\nand exhibit oscillating decay rates. These expressions app roximate the oscillating decay laws\nat rest of unstable quantum systems over intermediate times . We have studied the trans-\nformation of the oscillating decay laws at rest which is indu ced by the change of reference\nframe. The transformed decay laws and the transformed times have been determined in the\nlaboratory reference frame where the unstable quantum syst em moves with constant linear\nmomentum, by assuming that the MDD is approximately symmetr ic with respect to the mass\nof resonance. By considering the modal frequencies of oscill ations to be small compared to\nthe mass of resonance, time intervals are determined over wh ich the transformed decay laws\nconsist in the superposition of transformed purely exponen tial modes and transformed expo-\nnentially damped oscillating modes. The modal decay widths at rest,Γj, transform regularly\nin reduced decay widths, Γj/γ, and the modal frequencies at rest, Ωj, transform regularly in\nreduced frequencies, Ωj/γ, in the laboratory reference frame. Equivalently, over the selected\ntimes, both the purely exponential modes and the exponentia lly damped oscillating modes\nat rest transform, independently and regularly, according to the same time scaling. The time\nintervals constitute one single time window if the modal dec ay widths fulfill determined con-\nditions. If the oscillations of the decay laws at rest are app roximately periodic over the time\nwindow, the transformed decay laws are approximately perio dic in the laboratory reference\n15frame over the transformed time window and the period of the o scillations transforms, reg-\nularly, according to the time scaling. The relativistic dil ation of times is reproduced by the\ntime scaling, over the time window, if the mass of resonance o f the MDD is considered to be\nthe mass at rest of the moving unstable quantum system with re lativistic Lorentz factor γ. By\nadopting this interpretation, the survival probability at rest, the duration of the time window\nin the rest reference frame, the period of the damped oscilla tions at rest, if the oscillations are\nperiodic, and the modal frequencies of the oscillations tra nsform according to the relativistic\ndilation of times, by changing reference frame.\nIn conclusion, the decay laws of moving unstable systems wit h oscillating decay rates\nexhibit, over determined intermediate times, some regular ities in the transformation which is\ninduced by the change of reference frame. Interpreting the e xperimental works on oscillating\ndecay rates of moving unstable systems is beyond the purpose s of the present paper. However,\nwe believe that further insight about the description of the oscillating decay laws via quantum\ntheory and special relativity may be provided by decomposin g the detected oscillating decay\nlaws into the above-mentioned purely exponential and expon entially damped oscillating modes\nand searching for the above-mentioned regularities.\nA Details\nThe transformation of the oscillating decay law at rest whic h is described by Eq. (6) is\nobtained from Eq. (5). The involved integrals are studied in Ref [10]. In this way, Eqs.\n(11)-(16) are determined. The function K(M,Γ,p,Ω,a,t), which appears in the form (11)\nof the transformed survival probability, is properly appro ximated by the following sum of\nexponential terms,\nK(M,Γ,p,Ω,a,t)≃a\n2exp/parenleftbigg\n−t\n2/parenleftbiggΓ−\nγ+2ıM−γ−/parenrightbigg/parenrightbigg\n+(1−a)\n×exp/parenleftbigg\n−t\n2/parenleftbiggΓ\nγ+2ıMγ/parenrightbigg/parenrightbigg\n+a\n2exp/parenleftbigg\n−t\n2/parenleftbiggΓ+\nγ+2ıM+γ+/parenrightbigg/parenrightbigg\n,(50)\nas the condition Γ/M−≪1holds. Instead, the function Φ(M,Γ,p,Ω,a,t)contributes to the\nform (11) with an inverse power law over times tsuch that pt≫1,\nıpΓ\nπM2Φ(M,p,Ω,a,t)≃exp(ı(pt−3π/4))√2πptpΓ\nM2W(M,Ω,a). (51)\nAccording to Eq. (50), the function K(M,Γ,p,Ω,a,t)provides to the expression (11) purely\nexponential decays with transformed decay widths Γ−/γ,Γ/γ,Γ+/γ, and exponentially\ndamped oscillations with frequencies |M−γ−−Mγ|,|M+γ+−Mγ|,|M−γ−−M+γ+|.\nThe relations which are reported below help to estimate the e xponential times. The\nfollowing inequalities,\n1\n2</radicalBigg\n1+p2/M2\n1+4p2/(M2)<γ\nγ−<1<γ\nγ+</radicalBigg\n1+p2/M2\n1+4p2/(9M2)<3\n2, (52)\nhold forΩ< M/2,0< a <1/2and for every value of the linear momentum p. The following\napproximations of the transformed decay widths,\nΓ∓\nγ=Γ\nγ∓≃Γ\nγ/parenleftbigg\n1∓p2Ω\nγ2M3/parenrightbigg\n, (53)\nand of the transformed frequencies of the oscillations,\nM∓γ∓−Mγ≃ ∓Ω\nγ/parenleftbigg\n1∓p2Ω\n2γ2M3/parenrightbigg\n, M+γ+−M−γ−≃Ω\nγ/parenleftbigg\n2−p2Ω2\nM4γ4/parenrightbigg\n, (54)\n16hold forΩ≪M. The function W(M,Ω,a), which appears in Eq. (19), is defined as below,\nW(M,Ω,a) = 1+aΩ2\nM23−Ω2/M2\n(1−Ω2/M2)2, (55)\nfor every value of the mass of resonance Mand for every allowed value of the frequency Ω\nand of the parameter a. The inequality\n1< W(M,Ω,a)<29\n18, (56)\nholds for Ω< M/2,0< a <1/2and for every value of the mass of resonance M.\nIn caseΩ≪M, the function K(M,Γ,p,Ω,a,t)is approximated by the form below,\nK(M,Γ,p,Ω,a,t)≃exp/parenleftbigg\n−t\n2/parenleftbiggΓ\nγ+2ıMγ/parenrightbigg/parenrightbigg/parenleftbigg\n1−a+acos/parenleftbiggΩ\nγt/parenrightbigg/parenrightbigg\n. (57)\nThis expression suggests that Eq. (17) properly approximat es the transformed decay law over\nthe time window (18) in the laboratory reference frame Sp. These times are obtained by\nrequiring the following condition,\n(1−2a)exp/parenleftbigg\n−Γt\n2γ/parenrightbigg\n≫pΓ\nπM2|Φ(M,p,Ω,a,t)|, (58)\nto hold for pt≫1. The above relation is studied by analyzing the order of magn itude of the\nparameter ξ′which is given by Eq. (19). See Ref. [28] for details. The comp arison between\nEq. (6) and Eq. (17) leads to Eqs. (21)-(24).\nThe transformation of the oscillating decay law at rest whic h is described by Eq. (29) is\nobtained from Eq. (5) and results in Eq. (34). The transforme d purely exponential modes\nand the transformed exponentially damped oscillating mode s are provided by the functions\nK(M,Γ1,p,Ω1,a1,t),...,K(M,ΓN,p,ΩN,aN,t), as the condition Γj/M−\nj≪1holds for\neveryj= 1,...,N ,\nN/summationdisplay\nj=1wjK(M,Γj,p,Ωj,aj,t)≃N/summationdisplay\nj=1wj/parenleftBigg\naj\n2exp/parenleftBigg\n−t\n2/parenleftBigg\nΓ−\nj\nγ+2ıM−\njγ−\nj/parenrightBigg/parenrightBigg\n+(1−aj)exp/parenleftBigg\n−t\n2/parenleftBigg\nΓj\nγ+2ıMγ/parenrightBigg/parenrightBigg\n+aj\n2exp/parenleftBigg\n−t\n2/parenleftBigg\nΓ+\nj\nγ+2ıM+\njγ+\nj/parenrightBigg/parenrightBigg/parenrightBigg\n. (59)\nInstead, the functions Φ(M,Γ1,p,Ω1,a1,t),...,Φ(M,ΓN,p,ΩN,aN,t)contribute to the ex-\npression (34) of the survival probability with the inverse p ower law 1/√ptforpt≫1,\nıN/summationdisplay\nj=1wjpΓj\nπM2Φ(M,p,Ωj,aj,t)≃exp(ı(pt−3π/4))√2πptN/summationdisplay\nj=1wjpΓj\nM2W(M,Ωj,aj). (60)\nDifferently form the decay widths Γ1,...,ΓN, the frequencies Ω1,...,ΩNare not sorted in\nincreasing or decreasing order and, in general, can even coi ncide. Therefore, some simplifica-\ntions are performed by introducing the parameters w′\n1,...,w′\nN���,Γ′\n1,...,Γ′\nN′andM′\n1,...,M′\nN′.\nThese parameters and the natural number N′are defined via the following equality,\nN/summationdisplay\nj=1wj/parenleftBigg\naj\n2exp/parenleftBigg\n−t\n2/parenleftBigg\nΓ−\nj\nγ+2ıM−\njγ−\nj/parenrightBigg/parenrightBigg\n+(1−aj)exp/parenleftBigg\n−t\n2/parenleftBigg\nΓj\nγ+2ıMγ/parenrightBigg/parenrightBigg\n+aj\n2exp/parenleftBigg\n−t\n2/parenleftBigg\nΓ+\nj\nγ+2ıM+\njγ+\nj/parenrightBigg/parenrightBigg/parenrightBigg\n=N′/summationdisplay\ni=1w′\njexp/parenleftBigg\n−t\n2/parenleftBigg\nΓ′\nj\nγ+2ıM′\njγ/parenrightBigg/parenrightBigg\n, (61)\n17by sorting the decay widths in increasing order, Γ′\n1< ... <Γ′\nN′. Therefore, the transformed\ndecay widths are determined by simplifying and ordering the decay widths Γ−\n1,Γ1,Γ+\n1,...,\nΓ−\nN,ΓN,Γ+\nN. The right side of Eq. (61) contributes to the expression (34 ) of the survival\nprobability with purely exponential modes and exponential ly damped oscillations modes. The\ndominant purely exponential mode exp(−Γ′\n1t/(2γ))and the fastest purely exponential mode\nexp/parenleftbig\n−Γ′\nN′t/(2γ)/parenrightbig\nare determined by the following decay widths,\nΓ′\n1= min/braceleftBig\nΓ−\nj,∀j= 1,...,N/bracerightBig\n,Γ′\nN′= max/braceleftBig\nΓ+\nj,∀j= 1,...,N/bracerightBig\n.\nThe weights w′\n1,...,w′\nN′are normalized to unity,/summationtextN′\ni=1w′\nj= 1. The selected frequencies of\nthe oscillations are chosen among the non-vanishing values of the form γ/vextendsingle/vextendsingleM′\nl−M′\nl′/vextendsingle/vextendsingle, which\nappears in the right side of Eq. (61).\nThe exponential times are obtained by requiring that the con dition\nwjl(1−2ajl)exp/parenleftbigg−Γjlt\n2γ/parenrightbigg\n≫1√2πptN/summationdisplay\ni=1wipΓi\nM2W(M,Ωi,ai), (62)\nholds over times tsuch that either t >1/(10Γ1)or(M−Ωmax)≫1, andpt≫1. In this\nway, the jlth term of the sum/summationtext\nl′, which appear in Eq. (40), dominates the inverse power\nlaw (60) over the set I′\np,lof the exponential times. The parameter ξ′\njis defined via Eq. (35)\nby following Ref. [28]. Relation (62) is equivalent to the in equality ξ′\njl≪10−2and holds\nover the time interval Ip,lwhich is given by Eq. (37). The set I′\npis defined via Eq. (36)\nand the exponential time window is approximated via Eq. (39) in casen1= 1, orn1>1\nand condition (38) is hold for every l= 1,...,n 1−1. The comparison between Eq. (40) and\nEq. (29) suggests the scaling properties of the survival pro bability, of the purely exponential\ntermP(exp)\np(t)and of the oscillating term P(osc)\np(t). The form (48) of the function ϕp(t)is\nobtained from expression (34) of the survival probability Pp(t). If the survival probability is\napproximated by Eq. (40), the scaling law (20) holds and the l inear growth (28) appears over\nthe setI′\npof the exponential times. This concludes the demonstration of the present results.\nReferences\n[1] Yu.A. Litvinov et al. 2008 Phys. Lett. B 664162\n[2] P. Kienle et al. 2013 Phys. Lett. B 726638\n[3] D.E. Alburger, G. Harbottle and E.F. Northon 1986 Earth a nd Planetary Science Letters\n78168\n[4] F. Giacosa and G. Pagliara 2013 Quantum Matter 2013 254\n[5] F. Giacosa and G. Pagliara 2012 PoS BORMIO 2012 028\n[6] L.A. Khalfin 1958 Sov. Phys. JETP 61053\n[7] B. Bakamjian 1961 Phys. Rev. 1211849\n[8] P. Exner 1983 Phys. Rev. D 28, 2621\n[9] E.V. Stefanovich 1996 Int. J. Theor. Phys. 352539\n[10] M.I. Shirokov 2004 Int. J. Theor. Phys. 431541\n[11] M.I. Shirokov 2006 Concepts Phys. 3193\n18[12] E.V. Stefanovich arXiv: 0603043\n[13] K. Urbanowski 2014 Phys. Lett. B 737346\n[14] F. Giacosa 2016 Acta Phys. Pol. B 472135\n[15] F. Giacosa 2017 Acta Phys. Pol. B 481831\n[16] K. Urbanowski 2017 Acta Phys. Pol. B 481411\n[17] F. Giacosa 2018 Adv. High En. Phys. ID 4672051\n[18] J. Bailey et al. 1977 Nature 268301\n[19] M.I. Shirokov 2009 Phys. Part. Nuclei Lett. 614\n[20] G.N. Fleming arXiv: 1104.1815\n[21] F. Giraldi 2018 Adv. High En. Phys. ID 7308935\n[22] K. Urbanowski 2009 Eur. Phys. J. D 5425\n[23] K. Urbanowski 2009 Centr. Eur. J. Phys. 7696\n[24] K. Urbanowski 1994 Phys. Rev. A 502847\n[25] F. Giraldi, Eur. Phys. J D 2015 695; Eur. Phys. J D 2016 70229\n[26] M.I. Shirokov and V.A. Naumov 2007 Concepts Phys. 4127\n[27] F. Giraldi 2018 J. Phys. A 51435303\n[28] F. Giraldi arXiv: 1902.05210\n[29] L. Fonda, G.C. Ghirardi and A. Rimini 1978 Rep. Prog. Phy s.41587\n[30] K.M. Sluis and E.A. Gislason 1991 Phys. Rev. A 434581\n[31] D.S. Onley and A. Kumar 1992 Am. J. Phys. 60432\n[32] H. Jakobovits, Y. Rothschild and J. Levitan 1995 Am. J. P hys.63439\n[33] F.W.J. Olver, D.W. Loizer, R.F. Boisvert, C. W. Clark 201 0The NIST Handbook of\nMathematical Functions (New York Cambridge University Press)\n[34] I.S. Gradshteyn and I.M. Ryzhik 2007 Table of Integral, Series and Products seventh ed.\n(Academic Press Orlando Florida)\n[35] M. Abramowitz and I. Stegun 1964 Handbook of Mathematical Functions with Formulas,\nGraphs and Mathematical Tables (Dover, New York)\n19"    },    {        "title": "0907.0167v1.Modal_approximations_to_damped_linear_systems.pdf",        "content": "arXiv:0907.0167v1  [math-ph]  1 Jul 2009Modal approximations to damped linear\nsystems\nKreˇ simir Veseli´ c∗\nAbstract\nWe consider a finite dimensional damped second order system a nd\nobtain spectral inclusion theorems for the related quadrat ic eigenvalue\nproblem. The inclusion sets are the ’quasi Cassini ovals’ wh ich may\ngreatly outperform standard Gershgorin circles. As the unp erturbed\nsystem we take a modally damped part of the system; this inclu des\nthe known proportionally dampedmodels, but may give much sh arper\nestimates. These inclusions are then applied to derive some easily cal-\nculablesufficient conditionsfortheoverdampednessofagiv en damped\nsystem.\n1 Introduction and preliminaries\nA damped linear system without gyroscopic forces is governed by th e differ-\nential equation\nM¨x+C˙x+Kx=f(t). (1)\nHerex=x(t) is anRn-valued function of time t∈R;M,C,K are real\nsymmetric matrices of order n. Typically M,Kare positive definite whereas\nCis positive semidefinite. The physical meaning of these objects is\nx(t) position or displacement\nM mass\nC damping\nK stiffness\nf(t) external force\n∗Fakult¨ at f¨ ur Mathematik und Informatik, Fernuniversit¨ at Hag en, Postf 940, 58084\nHagen, Germany\n1If in the homogeneous equation above we insert x(t) =eλtx,xconstant, we\nobtain\n(λ2M+λC+K)x= 0 (2)\nwhich is called the quadratic eigenvalue problem, attached to (1), λis an\neigenvalue and xa corresponding eigenvector.\nThe quadratic eigenvalue problem may have poor spectral theory in spite\nof the hermiticity and positive (semi)definiteness of M.C,K. There always\nexists a non-singular matrix Φ such that\nΦTMΦ =I,ΦTKΦ = Ω = diag( ω2\n1,...,ω2\nn). (3)\nIf the matrix Φ can be chosen such that also\nD= ΦTCΦ (4)\nis diagonal then the system is called modally damped .\nWhile (3) is the standard spectral decomposition of a symmetric pos itive\ndefinite matrix pair. a simultaneous achieving of (4) is rather an exce ption\nbeing equivalent to the generalised commutativity property\nCK−1M=MK−1C. (5)\nHowever, as an approximation, modal damping is attractive since it is han-\ndled by the standard theory and numerics of Hermitian matrices. Th e aim\nof this paper is to assess modal approximations of general damped systems.\nMore precisely, we will derive spectral inclusion theorems for eigenv alues\nwhere the unperturbed system is modally damped. There is some hier archy\namong various modal approximations of a given damped system and w e will\ninvestigate this issue as well. Our inclusion sets will not be circles, we will\ncall them quasi Cassini ovals. We will show that our ovals outdo class ical\nGershgorin circles. A special case are overdamped systems the eig envalues of\nwhich are particularly well behaved, there ovals reduce to intervals and inclu-\nsions of Wielandt-Hoffman type will be derived. Finally, we will derive new\ncalculable sufficient conditions for the overdampedness of a given sy stem.\n2 Modal approximation\nSome, rather rough, facts on the positioning of the eigenvalues ar e given\nin [4]. Further, more detailed, information is obtained by the perturb ation\n2theory. A simplest thoroughly known system is the undamped one. N ext to\nthis lie the modally damped systems.\nA simplest eigenvalue inclusion for a general matrix Aclose to a matrix\nA0is\nσ(A)⊆G1={λ:/bardbl(A−A0)(A0−λI)−1/bardbl<1} (6)\nObviously G1⊆G2with\nG2={λ:/bardbl(A0−λI)−1/bardbl−1≤ /bardbl(A−A0)/bardbl}. (7)\nThis is valid for any matrices A,A0. Using Φ, Ω from (3) we set\ny1= ΩΦ−Tx, y2=λΦ−Tx,\nso the quadratic eigenvalue equation (2) is equivalent to\nAy=λy. (8)\nHere we have set\nA=/bracketleftbigg0 Ω\n−Ω−D/bracketrightbigg\n, A0=/bracketleftbigg0 Ω\n−Ω 0/bracketrightbigg\n. (9)\nHence\nA−A0=/bracketleftbigg0 0\n0−D/bracketrightbigg\n.\nThematrix A0isskew-symmetric andthereforenormal, so /bardbl(A0−λI)−1/bardbl−1=\ndist(λ,σ(A0)) hence\nG2={λ: dist(λ,σ(A0)≤ /bardbl(A−A0)/bardbl} (10)\nwhere\n/bardbl(A−A0)/bardbl=/bardblD/bardbl=/bardblL−1\n2CL−T\n2/bardbl= maxxTCx\nxTMx(11)\nis the largest eigenvalue of the matrix pair C,M. We may say that here ’the\nsize of the damping is measured relative to the mass’.\nThus, the perturbed eigenvalues are contained in the union of the d isks\nof radius /bardblD/bardblaroundσ(A0).\n3Remark 2.1. In fact,σ(A) is also contained in the union of the disks\n{λ:|λ∓iωj| ≤Rj} (12)\nwith\nRj=n/summationdisplay\nk=1|dkj|. (13)\n(Replace the spectral norm in (6) by the norm /bardbl·/bardbl1).\nThe bounds obtained above are, in fact, too crude, since we have n ot\ntaken into account the structure of the perturbation A−A0which has a\nremarkable zero pattern.\nInstead of working with the matrix Awe may turn back to the original\nquadratic eigenvalue problem in the representation in the form (see (3) and\n(9))\ndet(λ2I+λD+Ω2) = 0.\nThe inverse\n(λ2I+λD+Ω2)−1=\n(λ2I+Ω2)−1(I+λD(λ2I+Ω2)−1)−1\nexists, if\n/bardblD(λ2I+Ω2)−1/bardbl|λ|<1 (14)\nwhich is implied by\n/bardbl(λ2I+Ω2)−1/bardbl/bardblD/bardbl|λ|=/bardblD/bardbl|λ|\nminj(|λ−iωj||λ+iωj|)<1 (15)\nThus,\nσ(A)⊆ ∪jC(iωj,−iωj,/bardblD/bardbl), (16)\nwhere the set\nC(λ+,λ−,r) ={λ:|λ−λ+||λ−λ−| ≤ |λ|r} (17)\nwill be called quasi Cassini ovals with foci λ±and extension r. This is in\nanalogy with the standard Cassini ovals where on the right hand side instead\nof|λ|rone has just r2. (The latter also appear in eigenvalue bounds in\nsomewhat different context.) We note the obvious relation\nC(λ+,λ−,r)⊂C(λ+,λ−,r′),whenever r < r′. (18)\n4The quasi Cassini ovals are qualitatively similar to the standard ones ; they\ncan consist of one or two components; the latter case occurs whe nris suf-\nficiently small with respect to |λ+−λ−|. In this case the ovals in (16) are\napproximated by the disks\n|λ±iωj| ≤/bardblD/bardbl\n2(19)\nand this is one half of the bound in (10), (11).\nRemark 2.2. σ(A) is also contained in the union of the ovals\nC(iωj,−iωj,/bardblΩ−1DΩ−1/bardblω2\nj). (20)\nIndeed, instead of inverting λ2I+λD+Ω2invertλ2Ω−2+λΩ−1DΩ−1+I.\nRemark 2.3. σ(A) is also contained in the union of the ovals\nC(iωj,−iωj,Rj) (21)\nand also\nC(iωj,−iωj,ρjω2\nj) (22)\nwith\nρj=n/summationdisplay\nk=1\nk/negationslash=j|dkj|\nωkωj. (23)\nThe just considered undamped approximation was just a prelude to the\nmain topic of this section, namely the modal approximation. The moda lly\ndamped systems are so much simpler than the general ones that pr actitioners\noften substitute the true damping matrix by some kind of ’modal app roxi-\nmation’. Most typical such approximations in use are of the form\nCprop=αM+βK (24)\nwhereα,βare chosen in such a way that Cpropbe in some sense as close as\npossible to C, for instance,\nTr[(C−αM−βK)W(C−αM−βK)] = min, (25)\nwhereWis some convenient positive definite weight matrix. This is a propor-\ntional approximation . In general such approximations may go quite astray\n5and yield thoroughly false predictions. We will now assess them in a mor e\nsystematic way.\nAmodal approximation to the system (1) isobtained by first repres enting\nit in modal coordinates by the matrices D, Ω and then by replacing Dby its\ndiagonal part\nD0= diag(d11,...,d nn). (26)\nThe off-diagonal part D′=D−D0is considered a perturbation. Again\nwe can work in the phase space or with the original quadratic eigenva lue\nformulation. In the first case we can make perfect shuffling to obta in\nA= (Ai,j), Aii=/bracketleftbigg0ωi\n−ωidii/bracketrightbigg\n, Aij=/bracketleftbigg0 0\n0dij/bracketrightbigg\n(27)\nA0= diag(A11,...,A nn). (28)\nSo, forn= 3\nA=\n0ω10 0 0 0\n−ω1−d110−d120−d13\n0 0 0ω20 0\n0−d12−ω2−d220−d23\n0 0 0 0 0ω3\n0−d130−d23−ω3−d33\n.\nThen\n/bardbl(A0−λI)−1/bardbl−1= max\nj/bardbl(Ajj−λI)−1/bardbl−1.\nEven for 2 ×2-blocks any common norm of ( Ajj−λI)−1seems complicated\nto express in terms of disks or other simple regions, unless we diagon alise\neachAjjas\nS−1\njAjjSj=/bracketleftbiggλj\n+0\n0λj\n−/bracketrightbigg\n, λj\n±=−djj±/radicalBig\nd2\njj−4ω2\nj\n2.(29)\nAs is directly verified,\nκ(Sj) =/radicalBigg\n1+θ2\nj\n|1−θ2\nj|, θj=djj\n2ωj.\n6with\nθj=djj\n2ωj.\nSetS= diag(S11,...,S nn) and\nA′=S−1AS=A′\n0+A′′\nthen\nA′\n0= diag(λ1\n±,...,λn\n±),\nA′′\njk=S−1\njA′\njkSk, A′′=S−1A′S\nNow the general perturbation bound (10), applied to A′\n0,A′′, gives\nσ(A)⊆ ∪j,±{λ:|λ−λj\n±| ≤κ(S)/bardblD′/bardbl}. (30)\nThere is a related ’Gershgorin-type bound’\nσ(A)⊆ ∪j,±{λ:|λ−λj\n±| ≤κ(Sj)rj} (31)\nwith\nrj=n/summationdisplay\nk=1\nj/negationslash=i/bardbldjk/bardbl. (32)\nTo show this we replace the spectral norm /bardbl · /bardblin (6) by the norm /bardbl| · /bardbl|1,\ndefined as\n/bardbl|A/bardbl|1:= max\nj/summationdisplay\nk/bardblAkj/bardbl\nwhere the norms on the right hand side are spectral. Thus, (6) will h old, if\nmax\nj/summationdisplay\nk/bardbl(A−A0)kj/bardbl/bardbl(Ajj−λI)−1/bardbl<1\nTaking into account the equality\n/bardbl(A−A0)kj/bardbl=/braceleftbigg|dkj|, k/ne}ationslash=j\n0k=j\nλ∈σ(A) implies\nrj≥ /bardbl(Ajj−λI)−1/bardbl ≥min{|λ−λj\n+|,|λ−λj\n−|}\nκ(Sj)\n7and this is (31).\nNote that the bounds (30) and (31) are poor whenever the modal approx-\nimation is close to a critically damped eigenvalue.\nBetter bounds are expected, if we work directly with the quadratic eigen-\nvalue equation. The inverse\n(λ2I+λD+Ω2)−1=\n(λ2I+λD0+Ω2)−1(I+λD′(λ2I+λD0+Ω2)−1)−1\nexists, if\n/bardblD′(λ2I+λD0+Ω2)−1/bardbl|λ|<1 (33)\nwhich is insured, if\n/bardbl(λ2I+λD0+Ω2)−1/bardbl/bardblD′/bardbl|λ|=/bardblD′/bardbl|λ|\nminj(|λ−λj\n+||λ−λj\n−|)<1 (34)\nThus,\nσ(A)⊆ ∪jC(λj\n+,λj\n−,/bardblD′/bardbl). (35)\nThese ovals will always have both foci either real or complex conjug ate. If\nr=/bardblD′/bardblissmallwithrespectto |λj\n+−λj\n−|=/radicalBig\n|d2\njj−4ω2\nj|theneither |λ−λj\n+|\nor|λ−λj\n−|is small. In the first case the inequality |λ−λj\n+||λ−λj\n−| ≤ |λ|r\nis approximated by\n|λ−λj\n+| ≤|λj\n+|r\n|λj\n+−λj\n−|=r\n\nωj√\nd2\njj−4ω2\njdjj<2ωj\ndjj−√\nd2\njj−4ω2\nj√\nd2\njj−4ω2\njdjj>2ωj(36)\nand in the second\n|λ−λj\n−| ≤|λj\n−|r\n|λj\n+−λj\n−|=r\n\nωj√\nd2\njj−4ω2\njdjj<2ωj\ndjj+√\nd2\njj−4ω2\nj√\nd2\njj−4ω2\njdjj>2ωj.(37)\nThis is again a union of disks. If djj≈0 then their radius is ≈r/2. If\ndjj≈2ωji.e.λ−=λ+≈ −djj/2 the ovals look like a single circular disk.\n8For large djjthe oval around the absolutely larger eigenvalue is ≈r(the\nsamebehaviouraswith(31))whereasthesmallereigenvaluehasthe diameter\n≈2rω2\nj/d2\njjwhich is drastically better than (31).\nIn the same way as before the Gershgorin type estimate is obtained\nσ(A)⊆ ∪jC(λj\n+,λj\n−,rj). (38)\nWe have called D′amodal approximation to Dbecause the matrix D\nis not uniquely determined by the input matrices M,C,K. Different choices\nof the transformation matrix Φ give rise to different modal approxim ations\nD′but the differences between them are mostly non-essential. To be m ore\nprecise, let Φ and ˜Φ both satisfy (3). Then\nM= Φ−TΦ−1=˜Φ−T˜Φ−1,\nK= Φ−TΩ2Φ−1=˜Φ−TΩ2˜Φ−1\nimplies that U= Φ−1˜Φ is an orthogonal matrix which commutes with\nΩ = diag( ω1In1,...,ω sIns), ω1<···< ωs. (39)\nHence\nU0= diag(U11,...,U ss),\nwhere each Ujjis an orthogonal matrix of order njfrom (26). Now,\n˜D=˜ΦTC˜Φ =UTΦTCΦU=UTDU, (40)\n˜Dij=UT\niDijUj (41)\nand hence\n˜D′=UTD′U. (42)\nNow, if the undamped frequencies are all simple, then Uis diagonal and the\nestimates (11) or (16)–(17) remain unaffected by this change of c oordinates.\nOtherwise we replace diag( d11,...,d nn) by\nD0= diag(D11,...,D ss) (43)\nwhereD0commutes with Ω. In fact, a general definition of a modal approx-\nimation is that it\n1. is block-diagonal and\n92. commutes with Ω.\nThe modal approximation with the coarsest possible partition — this is the\none whose block dimensions equal the multiplicities in Ω — is called a max-\nimal modal approximation . Accordingly, we say that C0= Φ−1D0Φ−Tis a\nmodal approximation to C(and also M,C0,KtoM,C,K).\nProposition 2.4. Each modal approximation to Cis of the form\nC0=s/summationdisplay\nk=1P∗\nkCPk (44)\nwhereP1,....P sis anM-orthogonal decomposition of the identity (that is\nP∗\nk=MPkM−1) andPkcommute with the matrix\n√\nM−1K=M−1/2√\nM−1/2KM−1/2M1/2\nProof.Use the formula\nD0=s/summationdisplay\nk=1P0\nkDP0\nk\nwith\nP0\nk= diag(0 ...,Ink,...,0, D= ΦTCΦ, D0= ΦTC0Φ\nand setPk= ΦP0\nkΦ−1. Q.E.D.\nIt is obvious that the maximal approximation is the best among all mod al\napproximations in the sense that\n/bardblD−D0/bardblE≤ /bardblD−ˆD0/bardblE, (45)\nwhere\nˆD0= diag(ˆD11,...,ˆDzz) (46)\nandD= (ˆDij) is any block partition of Dwhich is finer than that in (43).\nWe will now prove that the inequality (45) is valid for the spectral nor m also.\nWe shall need the following\nProposition 2.5. LetH= (Hij)be any partitioned Hermitian matrix such\nthat the diagonal blocks Hiiare square. Set\nH0= diag(H11,...,H ss), H′=H−H0.\n10Then\nλk(H)−λn(H)≤λk(H′)≤λk(H)−λ1(H) (47)\nwhereλk(·)denotes the non-decreasing sequence of the eigenvalues of a ny\nHermitian matrix.\nProof.By the monotonicity property (Wielandt’s theorem) we have\nλk(H)−max\njmaxσ(Hjj)≤λk(H′)≤λk(H)−min\njminσ(Hjj).\nBy the interlacing property,\nλ1(H)≤σ(Hjj)≤λn(H).\nTogether we obtain (47). Q.E.D.\nFrom (47) some simpler estimates immediately follow:\n/bardblH′/bardbl ≤λn(H)−λ1(H) =: spread( H) (48)\nand, ifHis positive (or negative) semidefinite\n/bardblH′/bardbl ≤ /bardblH/bardbl. (49)\nNow (45) for the spectral norm immediately follows from (49). So, a best\nboundin(35) isobtained, if D0=D−D′isa maximal modal approximation.\nProposition 2.6. Any modal approximation is better than any proportional\none.\nProof.With\nDprop=αI+βΩ\nwe have\n|(D−Dprop)ij| ≥ |(D−D0)ij|=|D0\nij|\nwhich implies\n/bardblD−Dprop/bardbl ≥ /bardblD′/bardbl.\nQ.E.D.\nIfD0is block diagonal and the corresponding D′=D−D0is inserted in\n(35) then the values djjfrom (29) should be replaced by the corresponding\n11eigenvalues of the diagonal blocks Djj. But in this case we can further\ntransform Ω and Dby a unitary similarity\nU= diag(U1,...,U s)\nsuch that each of the blocks Djjbecomes diagonal (Ω stays unchanged).\nWith this stipulation we may retain the formula (35) unaltered. This sh ows\nthat taking just the diagonal part D0ofDcovers, in fact, all possible modal\napproximations , when Φ varies over all matrices performing (3).\nSimilar extension can be made with the bound (38) but then no improve -\nments in general can be guaranteed although they are more likely th an not.\nBy the usual continuity argument it is seen that the number of the e igen-\nvalues in each component of ∪iC(λi\n+,λi\n−,ri) is twice the number of involved\ndiagonals. In particular, if we have the maximal number of 2 ncomponents,\nthen each of them contains exactly one eigenvalue.\nA strengthening in the sense of Brauer is possible as well. We will show\nthat the spectrum is contained in the union of double ovals , defined as\nD(λp\n+,λp\n−,λq\n+,λq\n−,rprq) =\n{λ:|λ−λp\n+||λ−λp\n−||λ−λq\n+||λ−λq\n−| ≤rprq|λ|2}, (50)\nwhere the union is taken over all pairs p/ne}ationslash=qandλp\n±are the solutions of\nλ2+dppλ+ω2\np= 0 and similarly for λq\n±. The proof just mimics the standard\nBrauer’s one. The quadratic eigenvalue problem is written as\n(λ2+λdpp+ω2\ni)xi=−λn/summationdisplay\nj=1\nJ/negationslash=idijxj, (51)\n(λ2+λdii+ω2\ni)xi=−λn/summationdisplay\nj=1\nJ/negationslash=idijxj, (52)\nwhere|xp| ≥ |xq|are the two absolutely largest components of x. Ifxq= 0\nthenxj= 0 for all j/ne}ationslash=pand trivially λ∈D(λp\n+,λp\n−,λq\n+,λq\n−,rprq). Ifxq/ne}ationslash= 0\nthen multiplying the equalities (51) and (52) yields\n|λ−λp\n+||λ−λp\n−||λ−λq\n+||λ−λq\n−||xp||xq| ≤\n12|λ|2n/summationdisplay\nj=1\nj/negationslash=pn/summationdisplay\nk=1\nk/negationslash=q|dpj||dqk||xj||xk|.\nBecause in the double sum above there is no term with j=k=pwe always\nhave|xj||xk| ≤ |xp||xq|, hence the said sum is bounded by\n|λ|2|xp||xq|n/summationdisplay\nj=1\nj/negationslash=p|dpj|n/summationdisplay\nk=1\nk/negationslash=q|dqk|.\nThus, our inclusion is proved. As it is immediately seen, the union of all\ndouble ovals is contained in the union of all quasi Cassini ov als.\nThe simplicity of the modal approximation suggests to try to extend it\nto as many systems as possible. A close candidate for such extensio n is any\nsystem with tightly clustered undamped frequencies, that is, Ω is clo se to an\nΩ0from (39). Starting again with\n(λ2I+λD+Ω2)−1=\n(λ2I+λD0+(Ω0)2)−1(I+(λD′+Z)+(λ2I+λD0+(Ω0)2)−1)−1\nwithZ= Ω2−(Ω0)2we immediately obtain\nσ(A)⊆ ∪jˆC(λj\n+,λj\n−,/bardblD′/bardbl,/bardblZ/bardbl). (53)\nwhere the set\nˆC(λ+,λ−,r,q) ={λ:|λ−λ+||λ−λ−| ≤ |λ|r+q} (54)\nwill be called modified Cassini ovals with foci λ±and extensions r,q.\nRemark 2.7. The basis of any modal approximation is the diagonalisation\nof the matrix pair M,K. An analogous procedure with similar results can\nbe performed by diagonalising the pair M,KorC,K.\n3 Modal approximation and overdampedness\nIf the systems in the previous section are all overdamped then est imates are\ngreatly simplified as ovals become just intervals. But before going int o this a\n13−0.22−0.2−0.18−0.16−0.14−0.12−0.1−0.08−0.06−0.04−0.02−1.5−1−0.500.511.5\n−1−0.9−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.10−1.5−1−0.500.511.5\nFigure 1: Ovals for ω= 1;d= 0.1,1;r= 0.3\n14−1.6−1.4−1.2−1−0.8−0.6−0.4−0.20−1.5−1−0.500.511.5\n−2 −1.5 −1 −0.5 0−1.5−1−0.500.511.5\n−2 −1.5 −1 −0.5 0−1.5−1−0.500.511.5\nFigure 2: Ovals for ω= 1;d= 1.7,2.3,2.2;r= 0.3,0.3,0.1\n15more elementary — and more important — question arises: Can the mo dal\napproximation help to decide the overdampedness of a given system ?\nWe begin with some obvious facts the proofs of which are left to the\nreader.\nProposition 3.1. If the system M,C,K is overdamped, then the same is\ntrue of the projected system\nM′=X∗MX, C′=X∗CX, K′=X∗KX (55)\nwhereXis any injective matrix. Moreover, the definiteness interva l of the\nformer is contained in the one of the latter.\nProposition 3.2. Let\nM= diag(M11,...,M ss)\nC= diag(C11,...,C ss)\nK= diag(K11,...,K ss).\nThen the system M,C,K is overdamped, if and only if each of Mjj,Cjj,Kjj\nis overdamped and their definiteness intervals have a non tri vial intersection\n(which is then the definiteness interval of M,C,K)\nCorollary 3.3. If the system M,C,K is overdamped, then the same is true\nof any of its modal approximations.\nObviously, if a maximal modal approximation is overdamped, then so a re\nall others.\nIn the following we shall need some well known sufficient conditions for\nnegative definiteness of a general Hermitian matrix A= (aij); these are:\najj<0 (56)\nfor alljand either\n/bardblA−diag(a11,...,a nn)/bardbl<−max\njajj (57)\n(norm-diagonal dominance) or\nn/summationdisplay\nk=1\nk/negationslash=j|akj|<−ajjfor allj (58)\n(Gershgorin-diagonal dominance).\n16Theorem 3.4. LetΩ,D,rjbe from (3), (4), (32), respectively and\nD0= diag(d11,...,d nn), D′=D−D0.\nLet either\n∆j= (djj−/bardblD′/bardbl)2−4ω2\nj>0for allj (59)\nand\np−:= max\nj−djj+/bardblD′/bardbl−/radicalbig\n∆j\n2<min\nj−djj+/bardblD′/bardbl+/radicalbig\n∆j\n2=:p+(60)\nor\nˆ∆j= (djj−rj)2−4ω2\nj>0for allj (61)\nand\nˆp−:= max\nj−djj+rj−/radicalBig\nˆ∆j\n2<min\nj−djj+rj+/radicalBig\nˆ∆j\n2=: ˆp+.(62)\nThen the system M,C,K is overdamped. Moreover, the interval (p−,p+),\n(ˆp−,ˆp+), respectively, is contained in the definiteness interval of M,C,K.\nProof.Letp−< µ < p +. The negative definiteness of\nµ2I+µD+Ω2=µ2I+µD0+Ω2+µD′\nwill be insured by norm-diagonal dominance, if\n−µ/bardblD′/bardbl<−µ2−µdjj−ω2\njfor allj,\nthat is, if µlies between the roots of the quadratic equation\nµ2+µ(djj−/bardblD′/bardbl)+ω2\nj= 0 for all j\nand this is insured by (59) and (60). The conditions (61) and(62) ar e treated\nanalogously. Q.E.D.\nWe are now prepared to adapt the spectral inclusion bounds from t he\nprevious section to overdamped systems. Recall that in this case t he defi-\nniteness interval divides the 2 neigenvalues into two groups: J-negative and\nJ-positive.\n17Theorem 3.5. If (59) and (60) hold then the J-negative/ J-positive eigen-\nvalues are contained in\n∪j(µj\n−−,µj\n−+),∪j(µj\n+−,µj\n++), (63)\nrespectively, with\nµj\n++\n−−=−djj−/bardblD′/bardbl±/radicalBig\n(djj+/bardblD′/bardbl)2−4ω2\nj\n2(64)\nµj\n+−\n−+=−djj+/bardblD′/bardbl±/radicalBig\n(djj−/bardblD′/bardbl)2−4ω2\nj\n2. (65)\nAn analogous statement holds, if (61) and (62) hold and µj\n++\n−−,µj\n+−\n−+is replaced\nbyˆµj\n++\n−−,ˆµj\n+−\n−+where in (64,65) /bardblD′/bardblis replaced by rj.\nProof.Allspectraarereal,sowehavetofindtheintersectionof C(λj\n+,λj\n−,r)\nwith the real line the foci λj\n+,λj\n−from (29) being also real. This intersection\nwill be a union of two intervals. For λ < λj\n−and also for λ > λj\n+thej-th\novals are given by\n(λj\n−−λ)(λj\n+−λ)≤ −λr\ni.e.\nλ2−(λj\n++λj\n−−r)λ+λj\n+λj\n−≤0\nwhereλj\n++λj\n−=−djjandλj\n+λj\n−=ω2\nj. Thus, theleft andthe right boundary\npoint of the real ovals are µj\n++\n−−.\nForλj\n−< λ < λj\n+the ovals will not contain λ, if\n(λ−λj\n−)(λj\n+−λ)≤ −λr\ni.e.\nλ2+(djj−r)λ+ω2\nj<0\nwith the solution\nµj\n−+< λ < µj\n+−.\nNow take r=/bardblD′/bardbl. The same argument goes with r=rj. Q.E.D.\n18Note the inequality\n(µj\n−−,µj\n−+)<(µk\n+−,µk\n++) (66)\nfor allj,k.\nMonotonicity-based bounds. As it is known for symmetric matrices\nmonotonicity-based bounds for the eigenvalues (Wielandt-Hoffman n bounds\nforasinglematrix)haveanimportantadvantageoverGershgorin- typebounds:\nWhile the latter are merely inclusions, that is, the eigenvalue is contain ed\nin a union of intervals the former tell more: there each interval con tains ’its\nown eigenvalue’. even if it intersects other intervals.\nIn this section we will derive bounds of this kind for overdamped syst ems.\nA basic fact is the following theorem\nTheorem 3.6. With overdamped systems the eigenvalues go asunder under\ngrowing viscosity. More precisely, Let\nλ−\nn−m≤ ··· ≤λ−\n1< λ+\n1≤ ··· ≤λ+\nm<0\nbe the eigenvalues of an overdamped system M,C,K. IfˆM,ˆC,ˆKis more\nviscous that is, ˆM≤M,ˆC≥C,ˆK≤Kin the sens of forms then its\ncorresponding eigenvalues ˆλ±\nksatisfy\nˆλ−\nk≤λ−\nk, λ+\nk≤ˆλ+\nk (67)\nA possible way to prove this theorem is to use the Duffin’s minimax\nprinciple [1], moreover, the following formulae hold\nλ+\nk= min\nSkmax\nx∈Skp+(x), λ−\nk= max\nSkmin\nx∈Skp−(x). (68)\nwhereSkis anyk-dimensional subspace. Now the proof of Theorem 3.6 is\nimmediate, if we observe that\nˆp±(x)>\n<p±(x) (69)\nfor anyx.\nAs a natural relative bound for the system matrices we assume\n|xTδMx| ≤ǫxTMx,|xTδCx| ≤ǫxTCx,|xTδKx| ≤ǫxTKx,(70)\n19with\nδM=ˆM−M, δC =ˆC−C, δH =ˆK−K, ǫ < 1.(71)\nWe suppose that the system M,C,K is overdamped and modally damped.\nOne readily sees that the overdampedness of the perturbed syst emˆM,ˆC,ˆK\nis insured, if\nǫ <d−1\nd+1, d= min\nxxTCx\n2√\nxTMxxTKx. (72)\nSo, the following three overdamped systems\n(1+ǫ)M,(1−ǫ)C,(1+ǫ)K;ˆM,ˆC,ˆK; (1−ǫ)M,(1+ǫ)C,(1−ǫ)K\nare ordered in growing viscosity. The first and the last system are o ver-\ndamped and also modally damped and their eigenvalues areknown andg iven\nby\nλ±\nk/parenleftbigg1−ǫ\n1+ǫ/parenrightbigg\n, λ±\nk/parenleftbigg1+ǫ\n1−ǫ/parenrightbigg\n,\nrespectively, where\nλ±\nk(η) =−djjη±/radicalBig\nd2\njjη2−4ω2\nj\n2,\nare the eigenvalues of the system M,ηC,K . We suppose that the unper-\nturbed eigenvalues λ±\nk=λ±\nk(1) are ordered as\nλ−\nn≤ ··· ≤λ−\n1< λ+\n1≤ ··· ≤λ+\nn.\nBy the monotonicity property the corresponding eigenvalues are b ounded as\n˜λ+\nk/parenleftbigg1−ǫ\n1+ǫ/parenrightbigg\n≤ˆλ+\nk≤˜λ+\nk/parenleftbigg1+ǫ\n1−ǫ/parenrightbigg\n, (73)\nwhere˜λ+\nk(η) are obtained by permuting λ+\nk(η) such that\n˜λ+\n1(η)≤ ··· ≤˜λ+\nn(η)\nfor allη >0. It is clear that each ˜λ+\nk(η) is still non-decreasing in η. An\nanalogous bound holds for ˆλ−\nkas well.\n20References\n[1] Duffin, R. J., A minimax theory for overdamped networks, J.\nRational Mech. Anal. 4(1955), 221–233.\n[2] Gohberg, I., Lancaster, P., Rodman, L., Matrices and indefinite\nscalar products, Birkh¨ auser, Basel 1983.\n[3] Gohberg, I., Lancaster, P., Rodman, L., Matrix polynomials,\nAcademic Press, New York, 1982.\n[4] Lancaster, P., Lambda-matrices and vibrating systems, Perga -\nmon Press Oxford 1966.\n[5] Lancaster, P., Quadratic eigenvalue problems, LAA 150(1991)\n499–506.\n21"    },    {        "title": "2301.13180v1.Massive_Gravity_and_Lorentz_Symmetry.pdf",        "content": "arXiv:2301.13180v1  [gr-qc]  30 Jan 2023Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022\n1\nMassive Gravity and Lorentz Symmetry\nR. Potting1,2\n1Departamento de F´ ısica, Faculdade de Ciˆ encias e Tecnolog ia,\nUniversidade do Algarve, 8005-139 Faro, Portugal\n2Centro de Astrof´ ısica e Gravita¸ c˜ ao, Instituto Superior T´ ecnico,\nUniversidade de Lisboa, Avenida Rovisco Pais, 1049-001 Lis bon, Portugal\nWe consider Lorentz-symmetry properties ofthe ghost-free massive gravity the-\nory proposed by de Rham, Gabadadze, and Tolley. In particula r, we present\npotentially observable effects in gravitational-wave prop agation and in New-\nton’s law, including Lorentz-violating signals.\n1. The de Rham–Tolley–Gabadadze action\nOne of the motivations for considering massive gravity is the possibilit y\nthat modifications of General Relativity over large distances may yie ld a\nsolution to the cosmological constant problem. A ghost-free theo ry of non-\ninteractinggravitonswasconstructedin1939byFierzandPauli.1However,\nattempts to generalize it to the nonlinear level failed during decades , with\nthe work of Boulware and Deser showing that generically such theor ies will\nsuffer from ghost instabilities.2Recently, however, de Rham, Gabadadze,\nandTolley(dRGT)showedthatthereexistsanon-linearextensiono fFierz–\nPauli massive gravity that does not suffer from ghosts.3It can be shown\nthat their model can be maximally extended to the action:4\nS=1\n2κ/integraldisplay\nd4x√−g/parenleftBig\nR−2m24/summationdisplay\nn=0βnen(X)/parenrightBig\n, (1)\nwhereRis the Ricci scalar, βiare free parameters and κ= 8πG. The\n4×4 matrix Xµνequals (/radicalbig\ng−1f)µν, where, in addition to the usual phys-\nical metric gµν, one assumes a given, nondynamical “fiducial” background\nmetricfµν. The invariant polynomials en(X) are defined through the re-\nlation det( /BD+λX) =/summationtext4\nn=0λnen(X), yielding e0(X) = 1,e1(X) = [X],\ne2(X) =1\n2([X]2−[X2]), ....Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022\n2\nBy expressing the metric in ADM form,5using as the dynamical vari-\nables the spatial metric together with lapse and shift variables it can be\nshown that the equations of motion arising from the action (1) yield fi ve\nlocal degrees of freedom, corresponding to the helicity states of a massive\nspin-2 particle. Crucial in the counting is the presence of the so-ca lled\nHamiltonian constraint.\n2. Spacetime symmetries\nIn this talk I will report on recent work with Alan Kosteleck´ y6in which we\nstudiedtheroleofLorentzsymmetryindRGTmassivegravity. Defin ingthe\nvierbein ea\nµas usual through gµν=ηabea\nµeb\nν, one can identify local Lorentz\ntransformations as well as diffeomorphisms. As was explained in the t alk\nby Alan Kosteleck´ y at this conference, suitable local Lorentz tra nsforma-\ntions and diffeomorphisms can be combined to yield the so-called manifo ld\nLorentz transformations, defined by\nxµ→(Λ−1)µνxν, g µν→(Λ−1)ρµ(Λ−1)σνgρσ,\neµa→(Λ−1)ρµΛabeρb, f µν→fµν, (2)\nfor spacetime-independent Lorentz transformations Λ. These a re the ana-\nlogues in approximately Minkowski spacetime of global Lorentz tran sfor-\nmations in Minkowski spacetime. The dRGT action is invariant under\nmanifold Lorentz transformations if fµν∝ηµν, but is otherwise Lorentz\nviolating7due to the presence of the background fµν.\nIn Ref. [6] a careful study was done of the static solutions of the d RGT\npotential, for flat fiducial metrics. It was shown that the four-pa rameter\npotential has a highly nontrivial structure of extrema and saddle p oints,\ndepending on the values of the parameters. Stability of these solut ions\nwas investigated by using the technique of bordered Hessians. The surface\ngenerated by the Hamiltonian constraint and the positions of the so lutions\non its connected sheets were used to establish global and absolute stability\nproperties. We concluded that extrema of the potential are invar iant un-\nder manifold Lorentz transformations, while the saddle point solutio ns are\nLorentz violating, with maximally four broken Lorentz generators.\n3. Linearized massive gravity\nThe action (1) yields the equations of motion\nGµν+m2\n23/summationdisplay\nn=0(−1)nβn/parenleftBig\ngµαYα\n(n)ν+gναYα\n(n)µ/parenrightBig\n=κTµν,(3)Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022\n3\nwhereY(n)(X) =/summationtextn\nk=0(−1)kXn−kek(X). Writing gµν=ηµν+hµνand\nfµν=ηµν+δfµνit follows that, to first order in hµνandδfµν,\nX≈ /BD+1\n2η−1δf−1\n2η−1h+1\n8η−1δf η−1h−3\n8η−1hη−1δf.(4)\nThe equations of motion (3) become\nGL\nµν+m2\n23/summationdisplay\nn=0βn/braceleftbigg\n2/parenleftbigg3\nn/parenrightbigg/parenleftbig\nηµν+hµν/parenrightbig\n+/parenleftbigg2\nn−1/parenrightbigg/parenleftbig\nhµν−δfµν−ηµν/bracketleftbig\nh−δf/bracketrightbig/parenrightbig\n+/parenleftbigg\n1\n2/parenleftbigg1\nn−2/parenrightbigg\n+/parenleftbigg2\nn−1/parenrightbigg/parenrightbigg/bracketleftbig\nδf/bracketrightbig\nhµν+1\n2/parenleftbigg1\nn−2/parenrightbigg/bracketleftbig\nh/bracketrightbig\nδfµν\n−1\n2/parenleftbigg1\nn−1/parenrightbigg/bracketleftbig\nδf η−1h/bracketrightbig\nηµν−1\n2/parenleftbigg1\nn−2/parenrightbigg/bracketleftbig\nδf/bracketrightbig/bracketleftbig\nh/bracketrightbig\nηµν\n−/parenleftbigg\n3\n4/parenleftbigg2\nn−1/parenrightbigg\n−1\n2/parenleftbigg1\nn−1/parenrightbigg/parenrightbigg/parenleftbig\nhη−1δf+δf η−1h/parenrightbig\nµν/bracerightbigg\n=κTµν,(5)\nwhereGL\nµνis the linearized Einstein tensor and [ X] =ηµνXµν. It is usual\nto require that hµν=δfµν=Tµν= 0 satifies the equations of motion (5),\nyielding the constraint β0+ 3β1+ 3β2+β3= 0. Moreover, we normalize\nthe mass msuch that β1+2β2+β3= 1.\nConsider now a nontrivial fiducial background metric δfµν∝negationslash= 0, it fol-\nlows that spacetime now has nonzero curvature in the absence of m atter!\nIn order to simplify the further analysis we will assume a special cons tant\nbackground energy–momentum tensor\nκTµν=−m2\n2(δfµν−ηµν[δf]). (6)\nFor this special choice, hµν= 0 solves the equations of motion.\n4. Gravitational waves\nIn order to investigate the propagation of gravitational waves, w e define\nthe Fourier transform of hµνthrough hµν(x) = (2π)−4/integraltext\nd4pe−ip·x˜hµν(p).\nForδfµν= 0,˜hµνsatisfies the conditions ˜hµαpα= 0 and ˜hµµ= 0, thus\nyielding 10 −5 = 5 propagating modes. These can be identified with the\nhelicity eigenstates of a massive spin-two field, ˜h(n)\nµν, withn= 0,±1,±2.\nAll modes satisfy the massive dispersion relation ( p2+m2)˜h(n)\nµν= 0.\nFor nonzero δfµνthe situation becomes more complicated. The equa-\ntions of motion (5) can be cast in the form\n(p2+m2)˜hµν=c2m2\n2Sµναβ˜hαβ (7)Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022\n4\nwherec2=/summationtext\nnβn/parenleftBig/parenleftbig1\nn−1/parenrightbig\n−3\n2/parenleftbig2\nn−1/parenrightbig/parenrightBig\n. The quantities Sµναβare tensor\ncoefficients depending on the momentum pµand onδfµν. It is convenient\nto expand them in a momentum-dependent orthonormal basis span ned by\npµand three other, spacelike vectors. It is then straightforwardt o workout\nthe expressions Sµναβ˜h(n)\nαβfor any helicity mode n, which are well-defined\nlinear combinations of the five helicity modes.\nEquation(7)canbesolvedbyconstructingtheeigenstatesof Sµναβ. For\ngeneralδf, we find a “pentarefringence” effect: each of these eigenstates\nsolves Eq. (7) with a (slightly) different dispersion relation. These pe ntare-\nfringence effects are momentum (and direction) dependent. The m odes can\nbe sub- or superluminal, a result typical of Lorentz-violating theor ies. For\ndetails, see Ref. [6].\n5. Corrections to Newton’s law\nNext westudy theeffects ofthe extraterms inthe equationofmot ion(5) on\nNewton’s law. Writing the momentum-space linearized modified Einstein\nequation as ˜Oµναβ˜hαβ= 0, the corresponding propagator is defined to\nsatisfy˜Dµνστ˜Oσταβ=δα\n(µδβ\nν). At first order in δfit has the form\n˜Dµναβ=1\np2+m2/bracketleftbigg\nδα\n(µδβ\nν)−1\n3ηµνηαβ+2\nm2p(µp(αδβ)\nν)\n−1\n3m2/parenleftbigg\npµpνηαβ+ηµνpαpβ−2\nm2pµpνpαpβ/parenrightbigg/bracketrightbigg\n−m2\n(p2+m2)2/bracketleftBig\nρ1δfα\n(µδβ\nν)+ρ2δfµνηαβ+ρ4pµpνδfαβ+.../bracketrightBig\n,\n(8)\nwhereρi=ρi(p2) are momentum-dependent scalars (parentheses in the\nindices indicate symmetrization). For given energy–momentum tens or the\nlinearized solution for the metric can then be expressed as\nhµν(x) = 2κ/integraldisplayd4p\n(2π)4e−ip·xDµναβ˜Tαβ(p), (9)\nwhereTµν(x) = (2π)−4/integraltext\nd4pe−ip·x˜Tµν(p).\nConsider now the gravitationalpotential energy between two sta tionary\npoint masses with energy–momentum tensors\nTµν\n1(x) =M1δµ\n0δν\n0δ3(/vector x), Tµν\n2(x) =M2δµ\n0δν\n0δ3(/vector x−/vector r).(10)\nAt linear order in hµνthe matter Lagrangian corresponding to a given\nenergy–momentum tensor is Lm≈ −1\n2hµνTµν. From this it follows thatProceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022\n5\nthe potential energy corresponding to the energy–momentum te nsors (10)\nis given by U(/vector r) =/integraltext\nd3xhµν\n1(/vector x)T2,µν(/vector x) =M2h1,00(/vector r).Using Eqs. (8)\nand (9) we obtain\nU(/vector r) =−GM1M2e−mr\n9r/bracketleftbigg\n24−δf00/parenleftbig\n(4c2+9)mr+8c2+8/parenrightbig\n−δfii/parenleftbigg\n(4c2−9)mr+2c2+4+2\nmr+4\nm2r2/parenrightbigg\n−δfijxixj\nr2/parenleftbigg\n2c2mr+2c2−4−6\nmr−12\nm2r2/parenrightbigg/bracketrightbigg\n+2κM1M2\n9m2δ3(/vector r)/bracketleftbig\nδf00−2\n3δfii/bracketrightbig\n. (11)\nThe exponential suppression factor e−mris as expected due to the massive\ngraviton. The term independent of δfµνis scaled by 4/3 relative to the\ngravitational potential in General Relativity, in concordance with t he van\nDam–Veltman–Zakharov discontinuity. Moreover, note that U(/vector r) acquires\nterms that generally violate rotational invariance.\nAcknowledgments\nThis work was supported in part by the Portuguese Funda¸ c˜ ao pa ra\na Ciˆ encia e a Tecnologia under grants SFRH/BSAB/150324/2019 a nd\nUID/FIS/00099/2019, and by the Indiana University Center for S pacetime\nSymmetries.\nReferences\n1. M. Fierz, Helv. Phys. Acta 12, 3 (1939); M. Fierz and W. Pauli, Proc. Roy.\nSoc. (London) A 173, 211 (1939).\n2. D.G. Boulware and S. Deser, Phys. Rev. D 6, 3368 (1972).\n3. C. de Rham, G. Gabadadze, and A.J. Tolley, Phys. Rev. Lett. 106, 231101\n(2011); Phys. Lett. B 711, 190 (2012).\n4. S.F. Hassan and R.A. Rosen, JHEP 04, 123 (2012); Phys. Rev. Lett. 108,\n041101 (2012); S.F. Hassan, R.A. Rosen and A. Schmidt-May, J HEP02, 026\n(2012).\n5. R. Arnowitt, S. Deser, and C. Misner, Phys. Rev. 116, 1322 (1959).\n6. V.A. Kosteleck´ y and R. Potting, Phys. Rev. D 104, 104046 (2021).\n7. V.A. Kosteleck´ y, Phys. Rev. D 69, 105009 (2004).\n8. H. van Dam and M. Veltman, Nucl. Phys. 22, 397 (1970).\n9. V.I. Zakharov, JETP Lett. (Sov. Phys.), 12, 312 (1970)."    },    {        "title": "1707.06747v2.Quasinormal_ringing_of_black_holes_in_Einstein_aether_theory.pdf",        "content": "arXiv:1707.06747v2  [gr-qc]  24 Oct 2017Quasinormal ringing of black holes in Einstein aether theor y\nChikun Ding∗\nDepartment of Physics, Hunan University of Humanities,\nScience and Technology, Loudi, Hunan 417000, P. R. China\nKey Laboratory of Low Dimensional Quantum Structures and Qu antum Control of Ministry of Education,\nand Synergetic Innovation Center for Quantum Effects and Appl ications,\nHunan Normal University, Changsha, Hunan 410081, P. R. Chin a\nAbstract\nThe gravitational consequence of local Lorentz violation ( LV) should show itself in derivation of\nthe characteristic quasinormal ringing of black hole merge rs from their general relativity case. In\nthis paper, we study quasinormal modes (QNMs) of the scalar a nd electromagnetic field perturba-\ntions to Einstein aether black holes. We find that quasinorma l ringing of the first kind aether black\nhole is similar to that of another Lorentz violation model—t he QED-extension limit of standard\nmodel extension. These similarities between completely di fferent backgrounds may imply that LV\nin gravity sector and LV in matter sector have some connectio ns between themself: damping quasi-\nnormal ringing of black holes more rapidly and prolonging it s oscillation period. By compared to\nSchwarzschild black hole, both the first and the second kind a ether black holes have larger damping\nrate and smaller real oscillation frequency of QNMs. And the differences are from 0.7 percent to 35\npercents, those could be detected by new generation of gravi tational antennas.\nPACS numbers: 04.50.Kd, 04.70.Dy, 04.30.-w\nI. INTRODUCTION\nAfter the first discovery of gravitational wave (GW) on Septembe r 14, 2015 (GW150914) [1], Laser Inter-\nferometer Gravitational wave Observatory (LIGO) has detecte d GW for the third time, on January 4, 2017\n(GW170104) [2]. It provides a direct confirmation for the existence of a black hole and, confirms that black\nhole mergers are common in the universe, and will be observed in large numbers in the near future. The\ndetections of GW give us opportunity and the ideal tool to stress t est general relativity (GR) [3]. Some of\nthem are used to test alternative theories of gravity where Loren tz invariance (LI) is broken which affects the\ndispersion relation for GW [2]. For the first time, they used GW170104 to put upper limits on the magnitude\nof Lorentz violation tolerated by their data and found that the bou nds are important.\nWhy consider Lorentz violation (LV)? Because that Lorentz invaria nce may not be an exact symmetry at all\nenergies [4]. Condensed matter physics, which has an analog of LI, s uggests some scenarios of LV: a) LI is an\napproximate symmetry emerging at low energies and violated at ultra high energies [5]; b) LI is fundamental\nbut broken spontaneously [6]. Any effective description must break down at a certain cutoff scale, which\nsigns the emergence of new physical degrees of freedom beyond t hat scale. Examples of this include the\nhydrodynamics, Fermi’s theory of beta decay [7] and quantization o f GR [8] at energies beyond the Planck\nenergy. Lorentz invariance also leads to divergences in quantum fie ld theory which can be cured with a short\ndistance of cutoff that breaks it [9].\n∗Email: dingchikun@163.com; Chikun Ding@huhst.edu.cn2\nThus, the study of LV is a valuable tool to probe the foundations of GR without preconceived notions of the\nnumerical sensitivity [10]. These studies include LV in the neutrino sec tor [11], the standard-model extension\n[12], LV in the non-gravity sector [13], and LV effect on the formatio n of atmospheric showers [14]. A more\nrecent area for searching for LV is in the pure gravity sector, suc h as gravitational Cerenkov radiation [15]\nand gravitational wave dispersion [16]. Einstein-aether theory can be considered as an effective description of\nLorentz symmetry breaking in the gravity sector and has been ext ensively used in order to obtain quantitative\nconstraints on Lorentz-violating gravity[17]. On another side, viola tions of Lorentz symmetry have been used\nto construct modified-gravity theories that account for dark-m atter phenomenology without any actual dark\nmater [18].\nEinstein-aether theory [17] is originated from the scalar-tensor t heory [19]. In Einstein-aether theory, the\nbackground tensor fields break the Lorentz symmetry only down t o a rotation subgroup by the existence of\na preferred time direction at every point of spacetime, i.e., existing a preferred frame of reference established\nby aether vector ua. The introduction of the aether vector allows for some novel effec ts, e.g., matter fields\ncan travel faster than the speed of light [20], dubbed superlumina l particle. It is the universal horizons that\ncan trap excitations traveling at arbitrarily high velocities. In 2012, two exact black hole solutions and some\nmechanics of universal horizons in Einstein aether theory were fou nd by Berglund et al[21]. In 2015, two\nexact charged black hole solutions and their Smarr formula on both u niversal and Killing horizons were found\nby Ding et al[22]. In 2016, two exact black hole solutions and their Smarr formula o n universal horizons in\n3-dimensional spacetime were found by Ding et al[23]. Other studies on universal horizons can be found in\n[24, 25].\nIn Ref. [25], Ding et alstudied Hawking radiation from the charged Einstein aether black ho le and found\nthat i) the universal horizon seems to be no role on the process of r adiating luminal or subluminal particles;\nwhile ii) the Killing horizon seems to be no role on superluminal particle rad iation. Since up to date, the\nparticles with speed higher than vacuum light speed aren’t yet found , we here consider only subluminal or\nluminal particles perturbation to these LV black holes. In 2007, Kon oplyaet al[26] studied the perturbations\nof the non-reduced Einstein aether black holes and found that bot h the real part and the absolute imaginary\npart of QNMs increase with the aether coefficient c1.\nOur goal here is to study on a perturbed black hole in Einstein aether theory. Perturbations of black\nholes in GR or alternative theories of gravity carry signatures of th e effective potential around them and\none could look for them. Once a black hole is perturbed, it responds t o perturbations by emitting GWs [27]\nwhich are dominated by quasinormal ringing. The GW signal can in gene ral be divided into three stages:\n(i) a prompt response at early times, which depends strongly on the initial conditions; (ii) an exponentially\ndecaying “ringdown” phase at intermediate times, where quasinorm al modes (QNMs) dominate the signal,\nwhich depends entirely on the final black hole’s parameters and (iii) a la te-time tail [28]. QNMs can be used\nin the analysis of a gravitational wave signal to provide a wealth of inf ormation: the masses and radii of\nthe perturbed objects [29]. Recent QNMs study show that at the current precision of GW detections, there\nremains some possibility for alternative theories of gravity [30].\nSo by studying these LV black holes’ QNMs, we can obtain some signal of LV from future GW events. By\nusing QED-extension limit of standard model extension ( SME, see Ap pendix for more detail), Chen et al[31]\nhas studied the influence of LV on Dirac field perturbation to Schwar zschild black hole, and one will find that\nits properties have some similarities to our result. The plan of rest of our paper is organized as follows. In\nSec. II we review briefly the Einstein aether black holes and the third order WKB method (A recent study\non semianalytic technique appears in [32]). In Sec. III we adopt to t he third order WKB method and obtain\nthe perturbation frequencies of the first kind Einstein aether blac k holes. In Sec. IV, we discuss the QNMs\nfor the second kind Einstein aether black hole. In Sec. V we present a summary. Appendix is for introducing\nSME and the accuracy of WKB method.3\nII. EINSTEIN AETHER BLACK HOLES AND WKB METHOD\nThe general action for the Einstein-aether theory can be constr ucted by assuming that: (1) it is general\ncovariant; and (2) it is a functional of only the spacetime metric gaband a unit timelike vector ua, and\ninvolvesno morethan twoderivativesofthem, sothat the resulting field equationsaresecond-orderdifferential\nequations of gabandua. Then, the Einstein aether theory to be studied in this paper is desc ribed by the\naction,\nS=/integraldisplay\nd4x√−g/bracketleftBig1\n16πGæ(R+Læ)/bracketrightBig\n, (2.1)\nwhereGæis the aether gravitational constant, Læis the aether Lagrangian\n−Læ=Zab\ncd(∇auc)(∇bud)−λ(u2+1) (2.2)\nwith\nZab\ncd=c1gabgcd+c2δa\ncδb\nd+c3δa\ndδb\nc−c4uaubgcd, (2.3)\nwhereci(i= 1,2,3,4) are coupling constants of the theory. The aether Lagrangian is therefore the sum\nof all possible terms for the aether field uaup to mass dimension two, and the constraint term λ(u2+ 1)\nwith the Lagrange multiplier λimplementing the normalization condition u2=−1. There are a number of\ntheoretical and observational bounds on the coupling constants ci[17, 33, 34]. Here, we impose the following\nconstraints[47],\n0≤c14<2,2+c13+3c2>0,0≤c13<1, (2.4)\nwherec14≡c1+c4, and so on.\nThe static, spherically symmetric metric for Einstein aether black ho le spacetime can be written in the form\nds2=−f(r)dt2+dr2\nf(r)+r2(dθ2+sin2θdφ2). (2.5)\nThere are two kinds of exact solutions [21, 22]. In the first case c14= 0, c123/negationslash= 0 (termed the first kind aether\nblack hole), the metric function is\nf(r) = 1−2M\nr−I/parenleftBig2M\nr/parenrightBig4\n, I=27c13\n256(1−c13). (2.6)\nIf the coefficient c13= 0, then it reduces to Schwarzschild black hole. The quantity Mis the mass of the black\nhole spacetime[48]. Its location of the Killing horizon is the largest root off(r) = 0, which is given by [22]\nrKH=M/parenleftBigg\n1\n2+L+/radicalbigg\nN−P+1\n4L/parenrightBigg\n, L=/radicalbigg\n1\n4+P,\nP=21/3·4I\nH+H\n3·21/3, H=/parenleftBig\n27I+3√\n3I√\n27−256I/parenrightBig1/3\n. (2.7)\nIn the second case c14/negationslash= 0, c123= 0 (termed the second kind aether black hole), the metric function is\nf(r) = 1−2M\nr−J/parenleftBigM\nr/parenrightBig2\n, J=c13−c14/2\n1−c13. (2.8)\n[47] Note the slight difference between the constraints impo sed here and the ones imposed in [21], as in this paper we also r equire\nthat vacuum Cerenkov radiation of gravitons is forbidden [3 5].\n[48] The total mass of the given spacetime is MGæ= (1−c14/2)r0/2. And the constant Gæis related to Newton’s gravitational\nconstant GNbyGæ= (1−c14/2)GN, which can be obtained by using the weak field/slow-motion li mit of the Einstein-aether\ntheory [22, 36, 37]. Therefore we can always set r0= 2MGNregardless of the coefficient c14.4\nIts Killing horizon locates at\nrKH=M/parenleftBig\n1+√\nJ+1/parenrightBig\n. (2.9)\nIf the coefficient c13=c14/2, it also reduces to Schwarzschild black hole.\nThere is an universal horizon in these black hole spacetimes behind th eir Killing horizons even for aether\ncoefficientc13= 0 orc13=c14/2 (see [21, 22] for more detail). The universal horizon can trap par ticles\nwith arbitrary high velocity, i.e., super-luminal particles. The killing hor izons are invisible to these super-\nluminal particles. In another side, the universal horizon seems has no role on radiating luminal or sub-luminal\nparticles during Hawking radiation [25]. For this reason, to luminal or sub-luminal particles perturbation, we\nhere wouldn’t consider the role of the universal horizon at present .\nTo scalar and electromagnetic fields perturbation, we shall neglect interaction of these fields with aether\nfor simplicity and use general covariant wave equations. Then, the wave equations for test scalar Φ and\nelectromagnetic Aµfields are\n1√−g∂µ(√−ggµν∂νΦ) = 0,\n1√−g∂µ(√−gFµν) = 0, (2.10)\nwithFµν=∂µAν−∂νAµ. They can be reduced to Schrodinger like equations:\nd2Ψi\ndr2∗+[ω2−Vi(r)]Ψi= 0, dr∗=f(r)dr, (2.11)\nfor scalar field Ψ sand electromagnetic one Ψ e. The effective potentials take the form as:\nVi=f(r)/bracketleftbiggl(l+1)\nr2+β\nrdf(r)\ndr/bracketrightbigg\n, (2.12)\nwhereβ= 1 for the scalar field potential Vs,β= 0 for the electromagnetic one Ve, respectively. The\neffective potentials Videpend on the value r, angular quantum number (multipole momentum) land the\naether coefficient c13.\nFrom the potential formula (2.12), the effective potential for the first kind aether black hole is\nVi=/parenleftbig\n1−2M\nr/parenrightbig/bracketleftbiggl(l+1)\nr2+2Mβ\nr3/bracketrightbigg\n+16M4I\nr6/bracketleftbigg\n2β/parenleftbig\n2−5M\nr−32IM4\nr4/parenrightbig\n−l(l+1)/bracketrightbigg\n, (2.13)\nwhere the the first two terms are Schwarzschild potential, the res ts are the aether modified terms, shown\nin Fig. 1. In FIG. 1, it is the effective potential of scalar and electrom agnetic field perturbations near the\n0 2 4 6 8 10r/Slash1M0.0050.0100.0150.0200.0250.030Vs/LParen1r/RParen1\nc13/Equal0.8c13/Equal0.6c13/Equal0\n/GothicL/Equal0\n0 2 4 6 8 10r/Slash1M0.020.040.060.080.100.12Vs/LParen1r/RParen1\nc13/Equal0.8c13/Equal0.6c13/Equal0\n/GothicL/Equal1\n0 2 4 6 8 10r/Slash1M0.020.040.060.080.100.12Ve/LParen1r/RParen1\nc13/Equal0.8c13/Equal0.6c13/Equal0\n/GothicL/Equal1\nFIG. 1: The left both figures are the effective potential of sca lar field perturbations Vsnear the first kind aether black\nhole (M= 1) with different coefficients c13. The third figure is for the electromagnetic field perturbati onsVe.\nfirst kind aether black hole. Obviously, if c13= 0, the effective potentials Vican be reduced to those of\nthe Schwarzschild black hole. For l= 0, the peak value of the scalar potential barrier gets higher with c135\nincreasing. On the contrary, for l >0 the peak value gets lower with c13. This contrariness is similar to the\ncase of the deformed Hoˇ rava-Lifshitz black hole [38] where the pe ak gets lower for l= 0 and higher for l>0\nwith the parameter αincrease. In Einstein-Maxwell theory, i.e., Reissner-Norstr¨ om bla ck hole, the electric\nchargeQincreases the peak for all l. In the Einstein-Born-Infeld theory, the Born-Infeld scale para meterb\ndecreases the peak for all l. These properties of the potential will imply that the quasinormal m odes posses\nsome different behavior from those black holes.\nFrom the potential formula (2.12), the effective potential for the second kind aether black hole is\nVi=/parenleftbig\n1−2M\nr/parenrightbig/bracketleftbiggl(l+1)\nr2+2Mβ\nr3/bracketrightbigg\n+M2J\nr4/bracketleftbigg\n2β/parenleftbig\n1−3M\nr−JM2\nr2/parenrightbig\n−l(l+1)/bracketrightbigg\n, (2.14)\nwhere the the first two terms are Schwarzschild potential, the res ts are the aether modified terms, shown in\nFig. 2. In FIG. 2, it is the effective potential of the scalar and electr omagnetic field perturbation near the\n0 2 4 6 8 10r/Slash1M0.0050.0100.0150.0200.0250.030Ve/LParen1r/RParen1\nc13/Equal0.8c13/Equal0.6c13/Equal0.1\n/GothicL/Equal0\n0 2 4 6 8 10r/Slash1M0.020.040.060.080.100.12Ve/LParen1r/RParen1\nc13/Equal0.8c13/Equal0.6c13/Equal0.1\n/GothicL/Equal1\n0 2 4 6 8 10r/Slash1M0.020.040.060.080.100.12Ve/LParen1r/RParen1\nc13/Equal0.8c13/Equal0.6c13/Equal0.1\n/GothicL/Equal1\nFIG. 2: The left both figures are the effective potential of sca lar field perturbations Vsnear the second kind aether black\nhole (M= 1) with different coefficients c13and fixed coefficient c14= 0.2. The third figure is for the electromagnetic\nfield perturbations Ve.\nsecond kind aether black hole. It is easy to see that for all l, the peak value of the potential barrier gets lower\nwithc13increasing just like the Born-Infeld scale parameter bin the Einstein-Born-Infeld theory.\nThe Schr¨ odinger-like wave equation (2.11) with the effective poten tial (2.12) containing the lapse function\nf(r) related to the Einstein aether black holes is not solvable analytically. Since then we now use the third-\norder WKB approximation method to evaluate the quasinormal mode s of massless scalar and electromagnetic\nfield perturbation to the first and second kind aether black holes. T his semianalytic method has been proved\nto be accurate up to around one percent for the real and the imag inary parts of the quasinormal frequencies\nfor low-lying modes with n < l. Due to its considerable accuracy for lower lying modes, this method has\nbeen used extensively in evaluating quasinormal frequencies of var ious black holes. In this approximation, the\nformula for the complex quasinormal frequencies in this approximat ion is given by [40, 41, 45]\nω2=/bracketleftBig\nV0+/radicalbig\n−2V′′\n0Λ/bracketrightBig\n−i/parenleftBig\nn+1\n2/parenrightBig/radicalbig\n−2V′′\n0(1+Ω), (2.15)\nwhere\nΛ =1/radicalbig\n−2V′′\n0/bracketleftBigg\n1\n8/parenleftBigV(4)\n0\nV′′\n0/parenrightBig/parenleftBig1\n4+α2/parenrightBig\n−1\n288/parenleftBigV(3)\n0\nV′′\n0/parenrightBig2\n(7+60α2)/bracketrightBigg\n,\nΩ =1\n−2V′′\n0/bracketleftBig5\n6912/parenleftBigV(3)\n0\nV′′\n0/parenrightBig4/parenleftBig\n77+188α2/parenrightBig\n−1\n384/parenleftBigV′′′2\n0V(4)\n0\nV′′3\n0/parenrightBig\n(51+100α2)\n+1\n2304/parenleftBigV(4)\n0\nV′′\n0/parenrightBig2\n(67+68α2)1\n288/parenleftBigV′′′\n0V(5)\n0\nV′′2\n0/parenrightBig\n(19+28α2)\n−1\n288/parenleftBigV(6)\n0\nV′′\n0/parenrightBig\n(5+4α2)/bracketrightBig\n, (2.16)\nand\nα=n+1\n2, V(m)\n0=dmVi\ndrm∗/vextendsingle/vextendsingle/vextendsingle\nr∗(rp), (2.17)6\nTABLE I: The lowest overtone ( n= 0) quasinormal frequencies of the massless scalar field in t he first kind aether\nblack hole spacetime.\nc13 ω(l= 0) ω(l= 1) ω(l= 2) ω(l= 3) ω(l= 4)\n0.00 0.104647-0.115197 i0.291114-0.098001 i0.483211-0.096805 i0.675206-0.096512 i0.867340-0.096396 i\n0.15 0.103976-0.117446 i0.289524-0.099256 i0.480578-0.097877 i0.671547-0.097541 i0.862658-0.097409 i\n0.30 0.101739-0.120032 i0.287271-0.100767 i0.477014-0.099179 i0.666633-0.098790 i0.856385-0.098640 i\n0.45 0.096768-0.123153 i0.283882-0.102601 i0.471917-0.100790 i0.659659-0.100340 i0.847507-0.100168 i\n0.60 0.087386-0.127661 i0.278310-0.104812 i0.463995-0.102821 i0.648904-0.102303 i0.833850-0.102107 i\n0.75 0.072016-0.136350 i0.267685-0.107300 i0.449780-0.105389 i0.629746-0.104808 i0.809568-0.104588 i\n0.90 0.051721-0.155269 i0.239468-0.108510 i0.414053-0.107887 i0.581804-0.107315 i0.748823-0.107086 i\nTABLE II: The lowest overtone ( n= 0) quasinormal frequencies of the electromagnetic field in the first kind aether\nblack hole spacetime.\nc13 ω(l= 1) ω(l= 2) ω(l= 3) ω(l= 4) ω(l= 5)\n0.00 0.245870-0.093106 i0.457131-0.095065 i0.656733-0.095631 i0.853018-0.095865 i1.047870-0.095984 i\n0.15 0.243928-0.094312 i0.454325-0.096122 i0.652964-0.096654 i0.848255-0.096875 i1.042100-0.096987 i\n0.30 0.241266-0.095728 i0.450551-0.097388 i0.647915-0.097889 i0.841883-0.098098 i1.034390-0.098205 i\n0.45 0.237420-0.097401 i0.445196-0.098932 i0.640772-0.099409 i0.832880-0.099610 i1.023490-0.099712 i\n0.60 0.231411-0.099372 i0.436956-0.100837 i0.629806-0.101314 i0.819066-0.101516 i1.006780-0.101620 i\n0.75 0.220681-0.101581 i0.422371-0.103156 i0.610386-0.103695 i0.794587-0.103926 i0.977154-0.104044 i\n0.90 0.194630-0.102849 i0.386513-0.105074 i0.562260-0.105885 i0.733694-0.106235 i0.903309-0.106415 i\nnis overtone number and rpis the turning point value of polar coordinate rat which the effective potential\nreaches its maximum (2.12). Substituting the effective potential Vi(2.12) into the formula above, we can\nobtain the quasinormal frequencies for the scalar and electromag netic field perturbations to Einstein aether\nblack holes. In the next sections, we obtain the quasinormal modes for the both kinds of Einstein aether black\nholes and analyze their properties.\nIII. QUASINORMAL MODES FOR THE FIRST KIND AETHER BLACK HOLE\nIn this section, we study the scalar and electromagnetic field pertu rbations to the first kind Einstein aether\nblack hole. The scalar field perturbations are shown in Tab. I and Fig. 3 to 5. The electromagnetic field\nperturbations are shown in Tab. II and Fig. 6.\nIn Tab. I, we list the lowest overtone quasinormal modes of massles s scalar field for some lwith different\naether coefficient c13. Tab. I shows that, for fixed c13, the real part of frequencies increase and, the absolute\nimaginarypartofthemdecreasewiththeangularquantumnumber l. Forlargel, theimaginarypartsapproach\na fixed value. These properties are similar to the usual black holes an d, are also shown in Fig. 4.\nTab I also shows the derivations from Schwarzschild black hole. For l= 1, the decrease in Re ωis about\nfrom 0.7 percent to 17 percents, while the increase in −Imωis about from 1 percent to 11 percents, and could\nbe detected by new generation of gravitational antennas. It will h elp us to seek LV information in nature in\nlow energy scale.\nIn another side, for the fixed angular number l, or overtone number nwith different c13(smallc13), Tab.\nI, Fig. 3 and Fig. 5 show that the real part of frequencies decreas e, and the absolute imaginary ones increase\nwithc13firstly toc13= 0.87 and then decrease on the contrary, which is different from that of the non-reduced\naether black hole [26], where both all increase with c1.\nFor different overtone numbers n, Fig. 3 and Fig. 5 show that the real parts decrease and the absolu te\nimaginary ones increase with n, which is the same as that of Schwarzschild black hole.7\nΒ/Equal1,/GothicL/Equal1, n/Equal0,c13is from0.96 to 0c13/Equal0.960.930.900.87\n0.22 0.24 0.26 0.280.0980.1000.1020.1040.1060.108\nReΩ/MinusImΩ\nn/Equal0n/Equal1n/Equal2n/Equal3n/Equal4\nΒ/Equal1,/GothicL/Equal5,c13is from0.96 to 00.96\n0.96\n0.96\n0.96\n0.96\n0.6 0.7 0.8 0.9 1.00.00.20.40.60.81.0\nReΩ/MinusImΩ\nFIG. 3: The relationship between the real and imaginary part s of quasinormal frequencies of the scalar field in the\nbackground of the first kind aether black hole with the decrea sing ofc13.\nΒ/Equal1, n/Equal0\n/GothicL/Equal0/GothicL/Equal1/GothicL/Equal2\n0.0 0.2 0.4 0.6 0.80.00.10.20.30.40.5\nc13ReΩΒ/Equal1, n/Equal0\n/GothicL/Equal0\n/GothicL/Equal1\n/GothicL/Equal2\n0.0 0.2 0.4 0.6 0.80.100.110.120.130.140.150.16\nc13/MinusImΩ\nFIG. 4: The real (left) and imaginary (right) parts of quasin ormal frequencies of the scalar field in the background of\nthe first kind aether black hole with different c13.\nFor angular number l= 0 andl>0, Fig. 4 shows an unusual behavior. When l>0, the absolute imaginary\npart of frequencies increases for small c13, and then decrease in the region of large c13. However when l= 0,\nit increases for all c13. This behavior is similar to the case of deformed Hoˇ rava-Lifshitz bla ck hole where the\nreal part one increases for l >0 and only decreases for l= 0 with the coefficient α[38]. It is related to its\nunusual potential behavior Fig. 1.\nBy compared to Reissner-Norstr¨ om black hole, Fig. 3 and 4 show us a similar behavior that the absolute\nimaginary part of frequencies increases for small parameter c13orQ, and then decrease in the region of large\nparameter [26]. The only difference is that the real part decreases here for all c13and increases there for all\nQ.\nFor fixedl, Tab. II and Fig. 6 shows us that the behavior of electromagnetic p erturbation frequencies is\nsimilar to that of the scalar case. The real part of frequencies dec reases for all c13and, the absolute imaginary\none increases for small c13, and then decrease in the region of large c13. For fixed c13, Tab. II shows that\nboth the real and the absolute imaginary parts increase with l. And the real and absolute imaginary parts of\nelectromagnetic field are smaller than those corresponding value of scalar field.\nBy compared to another LV model — the QED-extension limit of stand ard model extension (SME, see\nAppendix for more detail), the above scalar and electromagnetic fie ld QNMs properties with c13are similar\nto Dirac field QNMs with LV coefficient b[31], i.e., the real part decreases while the absolute imaginary part\nincreases with the given LV coefficient. In the theory of QED-exten sion limit of SME, local LV coefficient bµis\nintroduced in a matter sector, while in Einstein aether theory, local LV is in a gravity sector. For the former,\nLV matter perturbs to LI black hole — Schwarzschild black hole and pr oduces QNMs. For the latter, LI\nmatter perturbs to LV black hole — Einstein aether black hole and the n produces QNMs. These similarities8\nn/Equal0\nn/Equal1\nn/Equal2\nn/Equal3\nn/Equal4\nΒ/Equal1,/GothicL/Equal5\n0.0 0.2 0.4 0.6 0.80.60.70.80.91.0\nc13ReΩ\nn/Equal0n/Equal1n/Equal2n/Equal3n/Equal4\nΒ/Equal1,/GothicL/Equal5\n0.0 0.2 0.4 0.6 0.80.00.20.40.60.81.0\nc13/MinusImΩ\nFIG. 5: The real (left) and imaginary (right) parts of quasin ormal frequencies of the scalar field in the background of\nthe first kind aether black hole with different c13.\nbetween different backgrounds may imply some common property of LV coefficient on QNMs, i.e., in presence\nof LV, the perturbation field oscillation damps more rapidly, and its pe riod becomes longer.\nΒ/Equal0,/GothicL/Equal1, n/Equal0,c13is from0.96 to 0c13/Equal0.960.930.900.87\n0.18 0.20 0.22 0.240.0940.0960.0980.1000.102\nReΩ/MinusImΩ\nn/Equal0n/Equal1n/Equal2n/Equal3n/Equal4\nΒ/Equal0,/GothicL/Equal5,c13is from0.96 to 00.96\n0.96\n0.96\n0.96\n0.96\n0.6 0.7 0.8 0.9 1.00.00.20.40.60.81.0\nReΩ/MinusImΩ\nFIG. 6: The relationship between the real and imaginary part s of quasinormal frequencies of the electromagnetic field\nin the background of the first kind aether black hole with the d ecreasing of c13.\nIV. QUASINORMAL MODES FOR THE SECOND KIND AETHER BLACK HOLE\nIn this section, we study the scalar and electromagnetic field pertu rbations to the second kind of Einstein\naether black hole with fixed c14= 0.2. The scalar field perturbations are shown in Tab. III, Fig. 7 and Fig .\n8. The electromagnetic field perturbations are shown in IV and Fig. 9 .\nIn Tab. III, we list the lowest overtone quasinormal modes of mass less scalar field for some lwith different\naether coefficient c13. Tab. III shows that, for fixed c13, the real part of frequencies increase and, the absolute\nimaginarypart ofthem decreasewith the angular quantum number l. For largel, the imaginarypart approach\na fixed value which is similar to the usual black holes and, also shown in Fig . 8.\nTab III also shows the derivations from Schwarzschild black hole. Fo rl= 1, the decrease in Re ωis about\nfrom 3 percents to 35 percents, while the decrease in −Imωis about from 1 percent to 21 percents, both\nare bigger than the first kind aether black hole, and could be detect ed by new generation of gravitational\nantennas.\nIn another side, for the fixed angular number l, or overtone number n, Tab. III, Fig. 7 and Fig. 9 show\nthat both the real and the absolute imaginary parts of frequencie s all decrease with c13increasing, which is\ncompletely different from that of the non-reduced aether black ho le [26], where both all increase with c1. This\nproperty of both decrease is similar to that of Einstein-Born-Infe ld black hole [38, 42].9\nTABLE III: The lowest overtone ( n= 0) quasinormal frequencies of the massless scalar field in t he second kind aether\nblack hole spacetime with fixed c14= 0.2.\nc13 ω(l= 0) ω(l= 1) ω(l= 2) ω(l= 3) ω(l= 4)\n0.10 0.104647-0.115197 i0.291114-0.098001 i0.483211-0.096805 i0.675206-0.096512 i0.867340-0.096396 i\n0.25 0.100755-0.114893 i0.281760-0.096962 i0.468061-0.095718 i0.654122-0.095413 i0.840293-0.095293 i\n0.40 0.095828-0.114071 i0.269748-0.095309 i0.448616-0.094016 i0.627061-0.093699 i0.805577-0.093575 i\n0.55 0.089374-0.112234 i0.253524-0.092567 i0.422340-0.091228 i0.590489-0.090899 i0.758658-0.090770 i\n0.70 0.080354-0.108123 i0.229737-0.087618 i0.383738-0.086246 i0.536739-0.085908 i0.689690-0.085775 i\n0.85 0.065688-0.097327 i0.188631-0.076846 i0.316653-0.075488 i0.443244-0.075155 i0.569686-0.075023 i\nTABLE IV: The lowest overtone ( n= 0) quasinormal frequencies of the electromagnetic field in the second kind aether\nblack hole spacetime with fixed c14= 0.2.\nc13 ω(l= 1) ω(l= 2) ω(l= 3) ω(l= 4) ω(l= 5)\n0.10 0.245870-0.093106 i0.457131-0.095065 i0.656733-0.095631 i0.853018-0.095865 i1.047870-0.095984 i\n0.25 0.236985-0.091929 i0.442267-0.093924 i0.635855-0.094505 i0.826131-0.094746 i1.014980-0.094868 i\n0.40 0.225711-0.090122 i0.423262-0.092160 i0.609107-0.092759 i0.791659-0.093008 i0.972796-0.093135 i\n0.55 0.210705-0.087219 i0.397700-0.089304 i0.573042-0.089924 i0.745134-0.090183 i0.915833-0.090314 i\n0.70 0.189117-0.082138 i0.360372-0.084257 i0.520194-0.084899 i0.676866-0.085167 i0.832193-0.085304 i\n0.85 0.152828-0.071437 i0.296066-0.073499 i0.428661-0.074143 i0.558382-0.074414 i0.686885-0.074552 i\nFor different overtone numbers n, Fig. 7 and Fig. 9 show that the real parts decrease and the absolu te\nimaginary ones increase with n, which is the same as that of Schwarzschild black hole.\nΒ/Equal1,/GothicL/Equal1, n/Equal0,c13is from0.97 to 0.1\n0.970.940.910.880.85\n0.15 0.20 0.250.050.060.070.080.09\nReΩ/MinusImΩ\nn/Equal0n/Equal1n/Equal2n/Equal3n/Equal4Β/Equal1,/GothicL/Equal5,c13is from0.97 to 0.1\n0.970.970.970.970.97\n0.4 0.5 0.6 0.7 0.8 0.9 1.00.00.20.40.60.8\nReΩ/MinusImΩ\nFIG. 7: The relationship between the real and imaginary part s of quasinormal frequencies of the scalar field in the\nbackground of the second kind aether black hole with the decr easing of c13.\nFor fixedl, Tab. IV and Fig. 9 shows us that the behavior of electromagnetic p erturbation frequencies\nis similar to that of the scalar case, i.e., both the real part and the ab solute imaginary one of frequencies\ndecreases for all c13. The only difference is that the real and absolute imaginary parts of electromagnetic field\nare smaller than those corresponding value of scalar field. For fixed c13, both the real and the imaginary parts\nincrease for all l.\nV. SUMMARY\nThe gravitational consequence of local Lorentz violation should sh ow itself in radiative processes around\nblack holes. The significant difference between Einstein and Einstein a ether theories can show itself in deriva-\ntion of the characteristic QNMs of black hole mergers from their Sch warzschild case.10\nΒ/Equal1, n/Equal0/GothicL/Equal1/GothicL/Equal2/GothicL/Equal3/GothicL/Equal4\n0.0 0.2 0.4 0.6 0.80.00.20.40.60.8\nc13ReΩ\nΒ/Equal1, n/Equal0/GothicL/Equal1\n/GothicL/Equal2/GothicL/Equal3\n0.0 0.2 0.4 0.6 0.80.050.060.070.080.09\nc13/MinusImΩ\nFIG. 8: The real (left) and imaginary (right) parts of quasin ormal frequencies of the scalar field in the background of\nthe second kind aether black hole with different c13.\nΒ/Equal0,/GothicL/Equal1, n/Equal0,c13is from0.97 to 0.1\n0.970.940.910.880.85\n0.10 0.15 0.200.050.060.070.080.09\nReΩ/MinusImΩ\nn/Equal0n/Equal1n/Equal2n/Equal3n/Equal4Β/Equal0,/GothicL/Equal5,c13is from0.97 to 0.1\n0.970.970.970.970.97\n0.4 0.5 0.6 0.7 0.8 0.9 1.00.00.20.40.60.8\nReΩ/MinusImΩ\nFIG. 9: The relationship between the real and imaginary part s of quasinormal frequencies of the electromagnetic field\nin the background of the second kind aether black hole with th e decreasing of c13\nIn this paper, we study on QNMs of the scalar and electromagnetic fi eld perturbations to Einstein aether\nblack holes. There exist a series of single parameter c13black holes solutions: the first and the second kind\naether black hole, instead of four coefficients there in Einstein aeth er theory.\nFor the effective potential, when c13increases, its turning point always becomes larger, and the value of its\npeak becomes lower except a special case that for the first aethe r black hole with angular quantum number\nl= 0, where its peak becomes higher.\nFor the three kinds, the first, the second and the non-reduced a ether black holes, their QNMs are different\nfrom each other that show their complexity. For the non-reduced aether hole [26], both real part and the\nabsolute value of imaginary part of QNMs (both scalar and electroma gnetic fields) increase with c1. On the\ncontrary for the second kind aether black hole, both decrease wit hc13, that are similar to the scalar field\nQNMs of Einstein-Born-Infeld black holes with the Born-Infeld para meterb[38].\nFor the first kind aether black hole, the real part of scalar and elec tromagnetic QNMs becomes smaller\nwith allc13increase. The absolute value of imaginary part of QNMs becomes bigg er with small c13increase,\nand then decreases with big c13. These properties with c13are similar to the behaviors of Dirac QNMs with\nLV coefficient b[31]. These similarities between different backgrounds may imply some connections between\nEinstein aether theory and the QED-extension limit of SME, i.e., LV in gr avity sector and LV in matter sector\nwill make quasinormal ringing of black holes damping more rapidly and its period becoming longer.\nCompared to Schwarzschild black hole, both the first and the secon d kind aether black holes have larger\ndamping rate and smaller real oscillation frequency of QNMs. And the differences are from 0.7 percent to\n35 percents, those could be detected by new generation of gravit ational antennas. If the breaking of Lorentz\nsymmetry is not very small, the derivation of QNMs from Schwarzsch ild values might be observed in the near11\nfuture gravitational wave events.\nAcknowledgments\nThis work was supported by the National Natural Science Foundat ion of China (grant No. 11247013),\nHunan Provincial Natural Science Foundation of China (grant No. 2 015JJ2085), and the fund under grant\nNo. QSQC1203.\nAppendix A: Standard Model Extension\nInvariance under Lorentz transformation is one of the pillars of bo th Einstein’s General Relativity (GR)\nand the Standard Model (SM) of particle physics. GR describes gra vitation at the classical level, while SM\nencompasses all other phenomena involving the basic particles and f orces down to the quantum level [12].\nThey are expected to merge at the Plank scale into a single unified and quantum-consistent description of\nnature. Theunderlyingunified quantumgravitytheoryindicatesth eLorentzsymmetryisonlyanapproximate\nsymmetry emerging at low energies and will be violated at ultrahigh ene rgies. Any observable signals of LV\ncan be described using effectively field theory. An effective field theo ryconstructionknown asthe gravitational\nstandard model extension (SME) providessuch a comprehensivef rameworkthat contains known physics along\nwith all possible LV effects. So that SME isn’t a specific model, but a con struction ideally suited for a broad\nsearch and having the power to predict the outcome of relevant ex periments [43].\nThe background of SME is Riemann-Cartan geometry which allows for nonzero vacuum quantities that\nviolate local Lorentz invariance but preserve general coordinate invariance. The usual Riemann spacetime of\nGR can be recovered in the zero torsion limit. SME can be constructe d from the action of GR and SM by\nadding all local LV and coordinated-independent terms, perhaps t ogether with suppressed higher-order terms,\nand they incorporategeneralCPT (Charge, Parityand Time symme try) violation[44]. EachSME term comes\nwith a coefficient for LV governing the size of the associated experim ental signals. The action SSMEfor the\nfull SME can be expressed as a sum of partial actions. These action s can be split into a matter sector and a\ngravity sector,\nSSME=Smatter+Sgravity+···. (A1)\nIn curved spacetime, the matter sector naturally couples to grav itation. The matter sector is given by\nSmatter=SSM+Sψ+SA. (A2)\nThe termSSMis the SM action, modified by the addition ofgravitationalcouplingsan d containingall Lorentz-\nand CPT-violating terms that involve SM fields and dominate at low ener gies. SM fields include the three\ncharged leptons, the three neutrinos, the six quark flavors, Higg s, gauge and Yukawa couplings. The terms Sψ\nandSAare the QED-extension (Quantum Electrodynamics) limit of SME, Dira c fermionψand the photon\nAµ, respectively. The term Sgravityrepresents the pure-gravity sector incorporating possible Lore ntz and CPT\nviolation. It is easy to see that Einstein aether theory is a specifical model ofSgravity. The ellipsis represents\ncontributions to SSMEthat are of higher order at low energies.\nThe fermion sector Sψin (A2) can be explicitly expressed as [12, 31]\nSψ=/integraldisplay\nd4x√−g(1\n2ieµ\naψΓa← →Dµψ−ψM∗ψ), (A3)\nwhereeµ\nais the inverse of the vierbein ea\nµ. The symbols ΓaandM∗are\nΓa≡γa−cµνeνaeµ\nbγb−dµνeνaeµ\nbγ5γb−eµeµa−ifµeµaγ5−1\n2gλµνeνaeλ\nbeµ\ncσbc, (A4)12\nand\nM∗≡m+im5γ5+aµeµ\naγa+bµeµ\naγ5γa+1\n2Hµνeµ\naeν\nbσab. (A5)\nThe first terms of Eqs.(A4) and (A5) lead to the usual Lorentz inva riant kinetic term and mass for the Dirac\nfield. The parameters aµ,bµ,cµν,dµν,eµ,fµ,gλµν,Hµνare Lorentz violating coefficients which arise from\nnonzerovacuum expectationvalues oftensorquantities andcomp rehensivedescribeeffects ofLorentzviolation\non the behavior of particles coupling to these tensor fields.\nFrom the action (A3), we can obtain that due to presence of Loren tz violating coefficients, Dirac equation\nmust be modified. According to the variation of ψin the action (A3), we find that the massless Dirac equation\nonly containing the CPT and Lorentz covariance breaking kinetic ter m associated with an axial-vector bµfield\nin the curve spacetime can be expressed as\n[iγaeµ\na(∂µ+Γµ)−bµeµ\naγ5γa]Ψ = 0, (A6)\nwhere\nγ0=/parenleftbigg\nI0\n0−I/parenrightbigg\n, γi=/parenleftbigg\n0σi\n−σi0/parenrightbigg\n, γ5=iγ0γ1γ2γ3=/parenleftbigg\n0I\nI0/parenrightbigg\n. (A7)\nIt is reasonable for us to assume that the axial-vector bµfield does not change the background metric since the\nLorentz violation is very small. For convenience, we take bµas a non-zero timelike vector (b\nr2,0,0,0), where\nbis a constant. The vierbein of Schwarzschild spacetime can be define d as\nea\nµ= (/radicalbig\n1−2M/r,1/radicalbig\n1−2M/r, r, rsinθ). (A8)\nAnd then byusingWKB method, Chen et al[31]obtained the influence ofLVonDiracQNMs inSchwarzschild\nspacetime. They found that at fundamental overtone, the real part decreases linearly as the parameter b\nincreases, while for the larger multiple moment k, the absolute imaginary part increases with b, which means\nthat presence of Lorentz violation makes Dirac field damps more rap idly. These behaviors with LV coefficient\nare similar to those of the first kind Einstein aether black hole.\nAppendix B: Accuracy of Wkb method\nThe Schr¨ odinger-like wave equation (2.11) with the effective poten tial (2.12) is not solvable analytically. A\nnumerical integration of it requires selecting a value for the complex frequency, then integrating and checking\nwhether the boundary conditions for QNMs are satisfied. Since tho se conditions are not met in general, the\ncomplex frequency plane must be surveyed for discrete values. So this technique is time consuming and costly.\nHowever for a semianalytic method, there is an accuracy problem wh ich will be discussed in this section.\nAs is well known, the accuracy of the WKB approximation should incre ase with multipole momentum l,\nthen only the low lying modes of the lowest overtone are considered h ere. Firstly the lowest three modes\nof the scalar and electromagnetic field perturbations of Schwarzs child black hole are shown in Tab. V via\nthree methods: the Schutz-Will approximation (the first order WK B) [45], the third order WKB [40, 41]\nand the numerical technique [46]. The formula for the complex quasin ormal frequencies in the Schutz-Will\napproximation is given by\nω2=V0−i/parenleftBig\nn+1\n2/parenrightBig/radicalbig\n−2V′′\n0, (B1)\nwith\nV(m)\n0=dmVi\ndrm∗/vextendsingle/vextendsingle/vextendsingle\nr∗(rp), (B2)13\nTABLE V: The three lowest modes ( n= 0) quasinormal frequencies of the massless scalar (above) and electromagnetic\n(lower) field in the Schwarzschild black hole spacetime. The numerical results are from [46]\nl1st WKB 3rd WKB numerical\n0 0.189785-0.098240 i0.104647-0.115197 i0.1105-0.1049 i\n1 0.329434-0.096256 i0.291114-0.098001 i0.2929-0.0977 i\n2 0.506317-0.096123 i0.483211-0.096805 i0.4836-0.0968 i\n1 0.287050-0.091235 i0.245870-0.093106 i0.2483-0.0925 i\n2 0.480754-0.094354 i0.457131-0.095065 i0.4576-0.0950 i\n3 0.673438-0.095258 i0.656733-0.095631 i0.6569-0.0956 i\nTABLE VI: The three lowest modes ( n= 0) quasinormal frequencies of the massless scalar (above) and electromagnetic\n(lower) field in the first kind aether black hole spacetime via the 1st WKB method.\nl ω(c13= 0) ω(c13= 0.15)ω(c13= 0.3)ω(c13= 0.45)ω(c13= 0.6)ω(c13= 0.75)ω(c13= 0.9)\n0 0.1898-0.0982 i0.1910-0.0995 i0.1925-0.1012 i0.1945-0.1036 i0.1971-0.1070 i0.2007-0.1119 i0.2035-0.1188 i\n1 0.3294-0.0963 i0.3286-0.0972 i0.3274-0.0983 i0.3257-0.0998 i0.3229-0.1019 i0.3176-0.1047 i0.3025-0.1085 i\n2 0.5063-0.0961 i0.5042-0.0971 i0.5013-0.0982 i0.4971-0.0997 i0.4907-0.1015 i0.4789-0.1040 i0.4485-0.1067 i\n1 0.2871-0.0912 i0.2859-0.0920 i0.2844-0.0930 i0.2822-0.0942 i0.2787-0.0956 i0.2723-0.0974 i0.2554-0.0985 i\n2 0.4808-0.0944 i0.4785-0.0953 i0.4754-0.0964 i0.4710-0.0977 i0.4642-0.0995 i0.4521-0.1016 i0.4211-0.1035 i\n3 0.6734-0.0953 i0.6700-0.0962 i0.6655-0.0974 i0.6591-0.0988 i0.6491-0.1006 i0.6314-0.1029 i0.5867-0.1051 i\nwherenis overtone number and rpis the turning point value of polar coordinate rat which the effective\npotential reaches its maximum (2.12).\nFrom Tab. V, one can see that the results via the 3rd WKB method ha ve higher accuracy than the 1st one.\nThen to the correctness of the present data in the maintext, the simplest way is to compare them to those\nobtained by the 1st WKB method. So the three lowest modes for bot h types of the aether black hole and for\nthe scalar and electromagnetic perturbations via this method are lis ted in Tab. VI and VII.\nTo the first kind aether black hole, by comparing Tab. VI to Tabs. I a nd II, one can see that the behavior\nof these data is the same as those, i.e., for fixed c13, the real part of frequencies increase and, the absolute\nimaginary parts decrease for the scalar field, increase for the elec tromagnetic one with the angular quantum\nnumberl; for the fixed angular number l(exceptl= 0 case), the real part of frequencies decrease, and the\nabsolute imaginary ones increase with c13. To the case of fixed l= 0, the behavior of the real part of scalar\nperturbation modes is different from Tab. I, which may be due to its u nusual potential.\nTo the second kind aether black hole, by comparing Tab. VII to Tabs . III and IV, one can see that the\nbehavior of these data is the same as those, i.e., for fixed c13, the real part of frequencies increase and, the\nabsolute imaginary part of them decrease for the scalar field, incre ase for the electromagnetic field with l; for\nthe fixed angular number l, both the real part and the absolute imaginary of frequencies dec rease withc13.\nTherefore, both methods give approximately the same modes ω.\n[1] B. P. Abbott et al, Phys. Rev. Lett. 116, 061102 (2016).\n[2] J. Driggers and S. Vitale, Newsletter, arXiv: 1706.0618 3.\n[3] B. P. Abbott et al, Phys. Rev. Lett. 116, 221101 (2016).\n[4] D. Mattingly, Living Rev. Rel. 8, 5 (2005).\n[5] S. Chadha and H.B. Nielsen, Nucl. Phys. B 217, 125 (1983).14\nTABLE VII:The threelowest modes ( n= 0) quasinormal frequenciesofthemassless scalar (above) andelectromagnetic\n(lower) field in the second kind aether black hole spacetime v ia the 1st WKB method.\nl ω(c13= 0.1)ω(c13= 0.25)ω(c13= 0.4)ω(c13= 0.55)ω(c13= 0.7)ω(c13= 0.85)\n0 0.1898-0.0982 i0.1869-0.0975 i0.1827-0.0963 i0.1764-0.0940 i0.1657-0.0895 i0.1438-0.0792 i\n1 0.3294-0.0963 i0.3206-0.0952 i0.3091-0.0936 i0.2932-0.0909 i0.2692-0.0861 i0.2260-0.0756 i\n2 0.5063-0.0961 i0.4914-0.0950 i0.4722-0.0933 i0.4460-0.0905 i0.4072-0.0856 i0.3386-0.0749 i\n1 0.2871-0.0912 i0.2786-0.0900 i0.2678-0.0882 i0.2529-0.0853 i0.2309-0.0802 i0.1920-0.0697 i\n2 0.4808-0.0944 i0.4661-0.0932 i0.4473-0.0914 i0.4218-0.0886 i0.3842-0.0835 i0.3183-0.0728 i\n3 0.6734-0.0953 i0.6527-0.0941 i0.6261-0.0924 i0.5901-0.0895 i0.5370-0.0845 i0.4443-0.0738 i\n[6] F. R. Klinkhamer and G. E. Volovik, JETP Lett. 94, 673 (2011); A. Jenkins, Phys. Rev. D 69, 105007 (2004).\n[7] J. 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A 402, 285 (1985)."    },    {        "title": "2104.04490v1.Kinetic_instability_in_inductively_oscillatory_plasma_equilibrium.pdf",        "content": "Kinetic instability in inductively oscillatory plasma equilibrium\nF. Cruz,\u0003T. Grismayer, and L. O. Silva\nGoLP/Instituto de Plasmas e Fus~ ao Nuclear, Instituto Superior T\u0013 ecnico,\nUniversidade de Lisboa, 1049-001 Lisboa, Portugal\n(Dated: April 12, 2021)\nA uniform in space, oscillatory in time plasma equilibrium sustained by a time-dependent current\ndensity is analytically and numerically studied resorting to particle-in-cell simulations. The disper-\nsion relation is derived from the Vlasov equation for oscillating equilibrium distribution functions,\nand used to demonstrate that the plasma has an in\fnite number of unstable kinetic modes. This\ninstability represents a new kinetic mechanism for the decay of the initial mode of in\fnite wave-\nlength (or equivalently null wavenumber), for which no classical wave breaking or Landau damping\nexists. The relativistic generalization of the instability is discussed. In this regime, the growth rate\nof the fastest growing unstable modes scales with \r\u00001=2\nT, where\rTis the largest Lorentz factor of\nthe plasma distribution. This result hints that this instability is not as severely suppressed for large\nLorentz factor \rows as purely streaming instabilities. The relevance of this instability in inductive\nelectric \feld oscillations driven in pulsar magnetospheres is discussed.\nPlasma equilibria are usually found as solutions to a\ncombination of kinetic or \ruid equations and Maxwell's\nequations in the stationary limit. Remarkably, time-\ndependent equilibrium conditions can also be determined\nfor certain systems consisting of a plasma and one or\nmore waves. Such systems are often unstable to para-\nmetric instabilities [1], determined by the properties of\nthe waves supporting the equilibrium. Parametric mode\nexcitation can be generally understood as the excitation\nof two or more plasma waves from a pump wave of \fnite\namplitudeE0, frequency !0and wavenumber k0. The\npump can be purely electromagnetic, e.g. a laser [2{9],\nof Alfv\u0012 enic nature [10{13], or even electrostatic [14, 15].\nThe scope of application of parametric decay is vast, as\nunstable parametric modes have been explored in the\ncontext of laser-plasma interactions [7, 8], inertial [2, 3]\nand magnetic [9] con\fnement fusion, laser beam ampli\f-\ncation schemes [16] as well as of space and astrophysical\nplasmas [10, 17{19].\nIn this work, we address the stability of a uniform in\nspace, oscillatory in time plasma equilibrium. In this\nequilibrium, ( !0;k0= 0) electric \feld oscillations, are in-\nductively supported by repeated reversals of the plasma\ncurrent. This con\fguration is similar to the oscillat-\ning two-stream instability [20], that has been historically\naddressed as a parametric instability, but with a time-\ndependent relative drift velocity [21]. This is a regime\nof interest in pulsar magnetospheres, where inductive os-\ncillations are excited [22, 23] following electron-positron\npair cascades in strong \felds [24, 25]. In this regime,\na fundamental analytical description of the instability is\ndi\u000ecult to obtain because di\u000berent !0harmonics are cou-\npled. Here, we present a theoretical analysis of this insta-\nbility, and show that it acts as a fundamental plasma pro-\ncess for the transfer of energy from the inductive pump\nwave to smaller and smaller plasma kinetic scales.\nThis Letter is organized as follows: \frst, we de-\nscribe the plasma equilibrium supporting these oscilla-tions. We then present the dispersion relation of elec-\ntrostatic plasma waves developed in this equilibrium for\nan initial waterbag distribution function, and analyse it\nboth theoretically and numerically resorting to particle-\nin-cell (PIC) simulations. Finally, the conclusions of this\nwork are outlined, and their relevance in pulsar magne-\ntospheres is discussed.\nWe consider a uniform unmagnetized pair plasma [26]\nin the presence of a uniform electric \feld E=E^ x=E0^ x\nwhich we assume, without loss of generality, to be posi-\ntive. In the presence of this \feld, electrons and positrons\nare accelerated in opposite directions, driving a current\nj=j^ xthat inductively reduces E. The plasma current\nis maximum when Evanishes, thus reversing the elec-\ntric \feld. Electrons and positrons are decelerated and\njdecreases in magnitude until it is reversed. The in-\nverse process occurs and the system re-establishes the\ninitial conditions. Since all dynamics is one-dimensional\nin space, we hereafter restrict our analysis to the ^ xcom-\nponents of \felds, currents and particle trajectories. To\ndetermine the evolution of the E, we \frst take the time\nderivative of Amp\u0012 ere's law,\n@2E\n@t2=\u00004\u0019e@j\n@t=\u00008\u0019eZ\ndp v@f+\n0\n@t; (1)\nwhere we have assumed that the plasma current is driven\nby counter-propagating positrons and electrons with uni-\nform density n0and average velocity \u0006hv+irespec-\ntively, i.e.j= 2en0hv+i(eis the elementary charge).\nWe have also used the de\fnition of average velocity\nhv+i=R\ndp vf+\n0=R\ndp f+\n0, wheref+\n0=f+\n0(p;t) is\nthe positron momentum distribution function, normal-\nized asR\ndpf+\n0(p;t) =n0(the same applies for the elec-\ntron distribution function f\u0000\n0). From the Vlasov equa-\ntion describing this equilibrium, we can write @f+\n0=@t=\n\u0000eE@f+\n0=@p, and perform the integral in Eq. (1) by partsarXiv:2104.04490v1  [physics.plasm-ph]  9 Apr 20212\nto obtain\n@2E\n@t2=\u00008\u0019e2\nmeEZ\ndpf+\n0(t;p)\n\r3\u0011\u0000\u001c8\u0019e2n0\nme\r3\u001d\nE ; (2)\nwhere we have used the relationship between momen-\ntum and velocity, p=\rmev, wheremeis the elec-\ntron mass and \r= 1=p\n1\u0000v2=c2. The natural os-\ncillation frequency of the electric \feld is then !0=p\nh8\u0019e2n0=me\r3i. In the non relativistic limit, !0is a\nconstant in time and Eis purely harmonic. If the electric\n\feld amplitude is large enough to accelerate particles to\nrelativistic velocities, the oscillation may be, in general,\nmore complex, with the \feld changing purely linearly in\ntime between crests and troughs ( i.e.with a triangular\nshape) [22]. For simplicity, we present here an analytical\ndescription of these waves in non relativistic regime, and\nthen discuss the generalization to the relativistic regime.\nIn the non relativistic limit, the momentum conservation\nequation of positrons/electrons in the equilibrium de\fned\nin Eq. (2) can be integrated, and their unperturbed orbits\nare\nv\u0006(t) =v0\u0006\u000evcos(!0t+\u0012)\u0011v0\u0006\u000evcos\u001e ; (3)\nwhere\u000ev=eE0=me!0and\u0012is a phase factor such that\nv\u0006=v0at a reference time t=t0. We may look for\nunstable plasma modes k=kex, withk6= 0, that can\ndevelop and grow exponentially, by integrating the lin-\nearized Vlasov equation along the unperturbed orbits in\nEq. (3). This is an approach similar to the derivation of\nBernstein waves [27], extensively documented in the liter-\nature [28, 29]. In this Letter, we discuss the \fnal disper-\nsion relation, and present all the details of the derivation\nin the Supplemental Material.\nWe consider a plasma where the equilibrium distribu-\ntion function of both electrons and positrons is a wa-\nterbag,f\u0006\n0(v;t) =n0=\u0001v(H(v+vT\u0007\u000evcos\u001e))\u0000H(v\u0000\nvT\u0007\u000evcos\u001e)), where \u0001 v= 2vTandH(x) is the Heavi-\nside function. The dispersion relation is\n1\u0000+1X\nn=\u00001J2\nn\u0012k\u000ev\n!0\u0013!2\n0\n(!\u0000n!0)2\u0000k2v2\nT= 0;(4)\nwhereJn(x) are Bessel functions of the \frst kind and\nordern. The dispersion relation in Eq. (4) readily in-\ndicates that an in\fnite number of branches !(k) exists.\nThese branches correspond to regions in the ( !;k) space\nwhere each term (in n) on the right-hand side of Eq. (4)\ndominates the series.\nGiven the complex form of Eq. (4), general analyti-\ncal calculations of the unstable modes and their growth\nrates are di\u000ecult to obtain. Solving Eq. (4) with each se-\nries term individually yields purely real branches, !n\u0006=\nn!0\u0006p\n!2\n0J2n+k2v2\nT, whereJn\u0011Jn(k\u000ev=! 0). The\nbranches withjnj \u0014 2, relevant for kvT=!0.1 at\nfrequencies near small multiples of !0, are plotted inFig. 1(a) as a function of k. An in\fnite number of cross-\nings between branches exists, with those between consec-\nutive branches ( n;n+1) occurring at lower wavenumbers.\nSymmetric branches ( n;\u0000n) cross roughly at k'n!0=vT\nand!r\u0011<(!) = 0 (as expected from symmetry). Writ-\ning!=!r+i\u0000 and assuming that \u0000 \u001c!0, we can\nshow that the terms of the series in Eq. (4) decrease\nwithn, and thus we can keep only the small nterms\nof the series to solve the dispersion relation. Here, we\npresent two analytical solutions of Eq. (4) yielding un-\nstable modes, corresponding to the interaction between\nbranches i) n= 0;\u00061 at small k, and ii)n=\u00061 at\nkvT=!0'1. The latter mode only exists for \fnite vT,\nwhereas the former exists even when vT= 0. For this\nreason, we hereafter refer to these modes as thermal and\n\ruid, respectively.\nTo determine the properties of the thermal mode, we\nlook for solutions to Eq. (4) with !r= 0 with only terms\n\u0006n, and then take the particular case n= 1. Tak-\ning advantage of the Bessel functions' symmetry J\u0000n=\n(\u00001)nJn, we obtain\n!2\nn=n2!2\n0+F(k)\u0006q\n!4\n0J4n+ 4n2!2\n0F(k);(5)\nwhereF(k) =!2\n0J2\nn+k2v2\nT. The solutions in Eq. (5)\nare unstable ( !2\nn<0) if 2!2\n0J2\nn> n2!2\n0\u0000k2v2\nT, which\nis satis\fed for wavenumbers kvT=!02(p\nn2\u0000J2n;n).\nForn= 1 in particular, the symmetric branches cross\natkvT=!0= 1, and \u0000 1;max\u0011max(=(!1))'!0J2\n1=2.\nIfk\u000ev=! 0>1, the large argument asymptotic expan-\nsion of Bessel functions applies, and we \fnd \u0000 1;max=!0'\nme!0vT=\u0019eE 0'vT=\u0019\u000ev . In Fig. 1(b), we plot the imag-\ninary component of !nas a function of kforn= 1;2,\nshowing that unstable modes indeed occur close to cross-\nings between the corresponding branches, and that their\ngrowth rate decays with n.\nThe \ruid mode couples branches n= 0;\u00061 at small\nk. ForvT= 0 andk\u000ev=! 0\u001c1, the dispersion relation\nreduces to\n(\n\u00001)3\n3\n\u00001=K ; (6)\nwhere \n = ( !=! 0)2andK= (1=2)(k\u000ev=! 0)2. Eq. (6) has\nunstable solutions with !r=!0'\u00061\u0007(1=4)(k\u000ev=! 0)2=3\nand growth rate \u0000 =!0'(p\n3=4)(k\u000ev=! 0)2=3.\nTo con\frm our theoretical \fndings, we have performed\na set of 1D PIC simulations with OSIRIS [30, 31] con-\nsidering a uniform pair plasma of density n0and a wa-\nterbag distribution function in momentum with \u0001 v=c=\n0:05\u00000:3. The plasma is subject to an initial electric \feld\nE0=(mec!p=e) = 0:2, where!2\np= 4\u0019e2n0=me=!2\n0=2.\nThe simulation domain has a length L=(c=!0)'70,\nand is discretized in N= 5000 cells, with 500 parti-\ncles/cell/species. The simulation time step is \u0001 t!p=\n0:005. When the simulations start, the plasma undergoes\nthe oscillations described by Eq. (2). Unstable modes3\nFIG. 1. Theoretical prediction of real and imaginary com-\nponents of wave frequencies excited with \u000ev=c = 0:14 and\n\u0001v=c= 0:1: (a) shows the purely real solutions obtained from\nindividual branches of the full dispersion relation in Eq. (4).\nLines labeled with n\u0006correspond to solutions where only term\nnwas kept in the series and where @!r=@k?0 fork\u001d1,\nrespectively; (b) illustrates the growth rate of unstable modes\nobtained by combining the symmetric branches of Eq. (4).\ngrow on top of the k= 0 oscillations. These oscillations\nremain initially stable, but are then damped as energy\nis transferred to unstable modes and into particle kinetic\nenergy.\nFig. 2 shows the electric \feld pro\fle and the electron\nphase space at (a) the beginning of the simulation, (b)\nduring the linear stage of the instability and (c) at a\ntime when the instability has saturated for a simula-\ntion with \u0001 v=c= 0:1. In Fig. 2(b), we observe that\nsmall perturbations start growing on top of the oscillat-\ning electric \feld, modifying the initial waterbag velocity\ndistribution. The initial electromagnetic energy density\nE02=8\u0019is converted into unstable modes until they satu-\nrate. The initial particle distribution is then strongly dis-\ntorted, extending well beyond the initial thermal spread\nvT=c= 0:05 for long times, see Fig. 2(c).\nThe time evolution of the electric \feld Fourier spec-\ntrum is presented in Fig. 3(a), showing that well-de\fned\nunstable modes grow exponentially during the linear\nstage of the instability. We observe multiple unstable\nthermal modes: the mode with lowest kis the \frst togrow (region R2), followed by higher kmodes (regions\nR3-4). We conjecture that these unstable modes are cou-\npled, as discussed in the Supplemental Material. The\noverall growth rate and total energy stored in unsta-\nble modes is dominated by the lowest kthermal mode,\nas shown in Fig. 3(b), illustrating the electric \feld en-\nergy stored in regions R1-4 of the kspace identi\fed in\nFig. 3(a). It is also possible to observe in Fig. 3(b) that\nthe energy in R1, corresponding to k\u00180, decreases with\ntime at the expense of the growth of all the other modes.\nBoth the bandwidth and maximum growth rate of the\nmost unstable modes are in good agreement with the\nsolutions of Eq. (5). We observe \ruid modes grow at\nk\u000ev=! 0\u001c1 but only weakly at early times, saturating\nat levels that do not play any dynamic role on the evo-\nlution of the system. When the unstable thermal modes\nreach a \fnite amplitude, particle acceleration occurs and\ncauses a saturation of the instability. This is followed by\nelectrostatic turbulence, which causes a strong distortion\nof the distribution (see Fig. 2(c)). The numerical disper-\nsion relation of the plasma in this simulation is presented\nin Fig. 3(c), which was obtained by Fourier transforming\nthe data in Fig. 3(a) in time during the linear stage of the\ninstability. The numerical dispersion relation shows that\nthe inductive mode is present at !=!0for lowkand\nthat, in general, unstable modes have a real frequency\ncomponent multiple of !0.\nFor all simulations performed with \u000ev=vT&1, the\ngrowth rate of the fastest growing modes is \u0000 max=!0\u0018\n0:1 and decreases with increasing \u000ev=vT. For\u000ev=vT.1,\nwe have veri\fed that the growth rate decreases with\nincreasing vT. This is also veri\fed for pair plasmas\nwith Maxwellian distributions with thermal velocities\nvth=c= 0:05\u00000:2. A detailed discussion of the scal-\ning of the growth rate with \u000ev=vTis presented in the\nSupplemental Material. We have found that the fastest\ngrowing modes in all simulations with \u000ev=vT&1 have\nkvT=!0'0:5, slightly lower than that predicted from\nlinear theory, k=(!0=vT)'0:6\u00001. We attribute this\ndi\u000berence to i) an average performed in the derivation\nto obtain a dispersion relation independent of time (see\nSupplemental Material) and to ii) the weakly nonlinear\nregime in which the thermal modes develop. Regarding\nii), we note that \ruid modes develop from early times,\nwith which the distribution may interact and broaden\nslightly, modifying the properties of the thermal mode.\nFor\u000ev=vT<1, simulations show that the fastest growing\nmodes have kvT=!0\u0018\u000ev=vT, a result that the dispersion\nrelation presented in this work fails to explain. Surpris-\ningly, however, the growth rate of these modes is well\ndescribed by linear theory.\nWe now discuss the generalization to relativistic con-\nditions, i.e.E0=(mec!0=e)\u001d1. In general, under these\nconditions, the unperturbed orbits in velocity space be-\ncome nearly perfect square waves, oscillating between \u0006c,\ni.e.the trajectories x(t) are triangular waves. It is well4\n0 20 40 60\nx[c/ω 0]−0.6−0.4−0.20.00.20.40.6E[mecω0e−1],v[c](a)Time = 0 [1 /ω0]\n0 100 200# electrons [arb. u.]E\n0 20 40 60\nx[c/ω 0](b)Time = 85 [1 /ω0]\n0 20 40 60\nx[c/ω 0](c)Time = 170 [1 /ω0]\nFIG. 2. Temporal evolution of electron phase space (in color) and electric \feld (in black lines) for a simulation with\nE0=(mec!0=e) =\u000ev=c = 0:14 and \u0001v=c= 0:1: (a) initial plasma con\fguration, (b) linear stage of the instability, showing\nelectric \feld perturbations grown on top of its uniform oscillatory component, (c) \fnal plasma state, after the instability has\nsaturated.\nknown that these can be described as a series of sinu-\nsoidal functions that is well approximated by the lowest\norder term. The same approximation can be applied to\nthe electric \feld pro\fle in regimes where \u000ep=pT\u001d1,\nwhere\u000ep=eE0=!0andpT=\rTmevTis the momentum\nthermal spread. In this regime, we \fnd that h1=\r3iis\nnot constant, and the electric \feld oscillation is instead\nwell described by a triangular shape [22]. We also per-\nformed simulations in the relativistic regime (considering\nE0=(mec!0=e)'14 and \u0001p=mec= 2\u0000800), where all\nconclusions outlined in this Letter for the non relativistic\nregime hold qualitatively and quantitatively.\nIn the relativistic regime, we \fnd also that \u0000 max=!0\u0018\n0:1, where!0/ h1=\r3i1=2, according to Eq. (2).\nFor waterbag and exponential distributions of the type\nexp(\u0000\r=\rT), we can show that h1=\r3i=\r\u00001\nTand\n(2\rT)\u00001, respectively, and thus !0/\r\u00001=2\nT. Hence, the\nunstable modes studied in this work are not as severely\nsuppressed for large \rTas streaming instabilities, for\nwhich \u0000 max/\r\u00003=2\nT, and may be more easily excited in\nextreme astrophysical settings where the Lorentz factor\nof the plasma \rows is very large. In particular, we \fnd\nfor typical pulsar parameters (surface \feld B'1012G,\nperiod 0:1 s) that!0\u00181\u000010 GHz [23, 32], such\nthat the instability develops on a typical time 1 =\u0000\u0018\n10 (pT=\u000ep)2=!0\u001810 (pT=\u000ep)2ns for the pT=\u000ep < 1\nregime expected in these scenarios (see Supplemental Ma-\nterial). This should be compared to the lifetime of the\ninductive plasma waves, which is well approximated by\nthe time between pair production bursts, Tb\u00181\u0016s [33].\nWe \fnd that 1 =\u0000< T bforpT=\u000ep < 10, which may\nbe achieved in the plasma trail behind pair production\nfronts, where the electric \feld oscillates inductively, but\nat an amplitude that is not enough to trigger a consider-\nable number of pair production events [32]. This insta-bility may also perturb particle trajectories in inductive\nwaves, previously identi\fed as a source of linear acceler-\nation emission [34, 35]. We note that the strong longitu-\ndinal magnetic \feld typical of these environments is not\nexpected to play a signi\fcant role in the development of\nthis instability (contrary to e.g. \flamentation modes), as\nparticle trajectories are purely one-dimensional.\nIn conclusion, we have studied the plasma waves para-\nmetrically excited in an inductively oscillatory plasma\nequilibrium. Our results show that the energy in the in-\nductive pump wave is transferred to other plasma modes\nvia an oscillating two-stream instability. Since the pump\nwave has in\fnite wavelength, other fundamental wave\ndepletion mechanisms such as wave breaking or Landau\ndamping do not operate, regardless of the value of its\namplitudeE0. We have presented the dispersion relation\nof these waves for a waterbag equilibrium distribution\nfunction, which captures thermal e\u000bects and is analyti-\ncally tractable. In general, in\fnite branches !(k) exist,\neach branch being a purely real mode when far from other\nbranches in the ( !;k) space. However, coupling between\ndi\u000berent branches yields unstable modes, with the max-\nimum growth rate \u0000 max=!0\u00180:1. All analytical results\nhave been con\frmed using PIC simulations. We have\nalso investigated the relevance of this instability in pulsar\nmagnetospheres, and determined that it may be excited\nover short distances following pair production bursts in\nstrong \felds. Furthermore, we speculate that it may be\nrelevant in other astrophysical scenarios, e.g. black hole\nmagnetospheres [36{38], where strong rotation-powered\nelectric \felds are also self-consistently screened by plasma\ncurrents.\nF. Cruz and T. Grismayer contributed equally to this\nwork. This work was supported by the European Re-\nsearch Council (ERC-2015-AdG Grant 695088) and FCT\n(Portugal) (grant PD/BD/114307/2016) in the frame-5\n−4−2024k[ω0/vT]\nR1R2R3R4(a)\n10−2101104|Ek|[E0]\n0 25 50 75 100 125 150\nTime [1 /ω0]10−710−510−310−1101/summationtextE2\nk[E2\n0](b)R1\nR2\nR3\nR4\n0 1 2 3 4 5 6\nk[ω0/vT]−10−50510ω[ω0](c)\n10−1102105|Eω,k|[E0]\n−4−2024k[ω0/vT]\nR1R2R3R4(a)\n10−2101104|Ek|[E0]\n0 25 50 75 100 125 150\nTime [1 /ω0]10−710−510−310−1101/summationtextE2\nk[E2\n0](b)R1\nR2\nR3\nR4\n0 1 2 3 4 5 6\nk[ω0/vT]−10−50510ω[ω0](c)\n10−1102105|Eω,k|[E0]\n−4−2024k[ω0/vT]\nR1R2R3R4(a)\n10−2101104|Ek|[E0]\n0 25 50 75 100 125 150\nTime [1 /ω0]10−710−510−310−1101/summationtextE2\nk[E2\n0](b)R1\nR2\nR3\nR4\n0 1 2 3 4 5 6\nk[ω0/vT]−10−50510ω[ω0](c)\n10−1102105|Eω,k|[E0]\nFIG. 3. Fourier analysis of unstable modes in the simulation\nillustrated in Fig. 2: (a) and (b) show respectively the time\nevolution of the electric \feld Fourier transform and of the\nenergy inkbands corresponding to the main oscillatory mode\n(R1) and to the three unstable modes with lowest k(R2-4); (c)\nillustrates the numerical plasma dispersion relation, obtained\nby taking the Fourier transform of (a) in time during the\nlinear stage of the instability.\nwork of the Advanced Program in Plasma Science and\nEngineering (APPLAuSE, FCT grant PD/00505/2012).\nWe acknowledge PRACE for granting access to MareNos-\ntrum, Barcelona Supercomputing Center (Spain), where\nthe simulations presented in this work were performed.6\nSupplemental Material\nDERIVATION OF THE DISPERSION RELATION\nWe start by linearizing the Vlasov equation for positrons and electrons (superscripts \u0006, respectively),\n@f\u0006\n1\n@t+v\u0006@f\u0006\n1\n@x\u0006eE0\nme@f\u0006\n1\n@v=\u0007eE1\nme@f\u0006\n0\n@v; (S1)\nwhere subscripts 0 (1) correspond to zeroth (\frst) order, time-varying quantities. The left hand side of Eq. (S1) can\nbe interpreted as a total time derivative of f\u0006\n1in phase space. Thus, f\u0006\n1can be obtained by integrating the right\nhand side of Eq. (S1),\nf\u0006\n1(x;v;t ) =\u0007e\nmeZt\n\u00001dt0E1(x0;t0)@f\u0006\n0(x0;v0;t0)\n@v0; (S2)\nwhere all primed quantities are taken along the unperturbed trajectories. The solution in Eq. (S2) can then be used\nin Poisson's equation to \fnd the dispersion relation. Using an anzats of the type A1\u0018\u0016A1exp(i(kx\u0000!t)) for \frst\norder quantities, we can write\nik\u0016E1=\u0000X\ns=\u00064\u0019e2\nmeZ\ndvZt\n\u00001dt0\u0016E1@fs\n0\n@v0exp [ik(x0\ns\u0000xs)\u0000i!(t0\u0000t)]]: (S3)\nWe now attempt to calculate the integrals in Eq. (S3) along the unperturbed trajectories. These can be written, in\nthe non relativistic regime, as\nv\u0006(t) =v0\u0006\u000evcos(!0t+\u0012)\u0011v0\u0006\u000evcos\u001e ; (S4)\nwhere the phase \u0012is such that v=v0at some reference time t0, and is the same for all particles in the distribution.\nEq. (S4) allows us to write the velocity at any time t0as\nv0\n\u0006\u0011v\u0006(t0) =v0\u0006\u000evcos(!0(t0\u0000t) +\u001e) =v\u0006\u0006\u000ev[cos(!0(t0\u0000t) +\u001e)\u0000cos\u001e]; (S5)\nand the di\u000berence in positions between times t0andtas\nx0\n\u0006\u0000x\u0006=Zt0\ntd\u001cv(\u001c) =v0(t0\u0000t)\u0006\u000ev\n!0[sin(!0(t0\u0000t) +\u001e)\u0000sin\u001e]: (S6)\nUsing the di\u000berence in Eq. (S6), we can write the exponential term in Eq. (S3) for positrons as\nexp [\u0000i(!\u0000kv0)(t0\u0000t)] exp [ik\u000ev=! 0(sin(!0(t0\u0000t) +\u001e)\u0000sin\u001e)]: (S7)\nFor the electron species, this exponential term has the same shape, with \u000ev!\u0000\u000ev. This di\u000berence does not introduce\nany change in the \fnal result, so we proceed with the derivation focusing on the positron species terms. Let us focus\non the last exponential in Eq. (S7). Using the Bessel identity\neiasinx=+1X\nn=\u00001Jn(a)einx;\nwhereJnis the Bessel function of the \frst kind and order n, we can write the last exponential in Eq. (S7) as\n+1X\nn=\u00001+1X\nm=\u00001Jn\u0012k\u000ev\n!0\u0013\nJm\u0012k\u000ev\n!0\u0013\nexp[in!0(t0\u0000t)] exp[i(n\u0000m)\u001e]: (S8)7\nThis can be plugged back in Eq. (S3) to obtain\nik=\u0000X\ns4\u0019e2\nmeZ\ndvZt\n\u00001dt0@fs\n0\n@v0X\nn;mJnJmexp [\u0000i(!\u0000n!0\u0000kv0)(t0\u0000t)]An;m(\u001e); (S9)\nwhere we have written An;m(\u001e) = exp [i(n\u0000m)\u001e] and where we have omitted the limits of the sums in nandm\nand the arguments of the Bessel functions JnandJm. We now focus on evaluating the integrals in Eq. (S9), by\nconsidering, for simplicity, the zeroth order distribution functions as waterbags,\nf\u0006\n0\u0011f\u0006\n0(v;t) =n0\n\u0001v[H(v+vT\u0007\u000evcos\u001e)\u0000H(v\u0000vT\u0007\u000evcos\u001e)]: (S10)\nHere, \u0001v= 2vTis the thermal spread of the distribution and H(x) is the Heaviside function. The derivative with\nrespect tovof the distribution function reads @f\u0006\n0=@v=n0=\u0001v(\u000e(v+vT\u0007\u000evcos\u001e)\u0000\u000e(v\u0000vT\u0007\u000evcos\u001e)), where\n\u000e(x) = dH(x)=dx.\nTo evaluate the integral in vin Eq. (S9), we note from Eq. (S4) and (S10), that the Dirac deltas in @f0=@v0are of\nthe type\n\u000e(v0+vT\u0007\u000evcos(!0(t0\u0000t) +\u001e)) =\u000e(v+vT\u0007\u000evcos\u001e); (S11)\nand thus@f0=@v0=@f0=@v. We can \fnally replace v0=v\u0007\u000evcos\u001efor positrons and electrons, respectively, and the\nintegration in vcan be easily performed. The dispersion relation becomes, at this point,\nik=\u00008\u0019e2n0\nme\u0001vZt\n\u00001dt0X\nn;mJnJmAn;m(\u001e)\u0002\n\b\nexp [\u0000i(!\u0000n!0+kvT)(t0\u0000t)]\u0000exp [\u0000i(!\u0000n!0\u0000kvT)(t0\u0000t)]\t\n; (S12)\nwhere we have performed the sum over species. We now change the time integration variable to \u001c=t\u0000t0. In this\ncase, we have\nik=8\u0019e2n0\nme\u0001vZ+1\n0d\u001cX\nn;mJnJmAn;m(\u001e)\u0002\n\b\nexp [i(!\u0000n!0+kvT)\u001c]\u0000exp [i(!\u0000n!0\u0000kvT)\u001c]\t\n: (S13)\nThe integrals in Eq. (S13) can be readily evaluated, and we \fnd\n1\u0000X\nnJn!2\n0\n(!\u0000n!0)2\u0000k2v2\nTX\nmJmAn;m(\u001e) = 0: (S14)\nTo obtain a dispersion relation that does not depend on \u001e, we take an average over one period of the driving oscillation\nofAn;m,\nhAn;m(\u001e)i=1\n2\u0019Z2\u0019\n0d\u001eAn;m(\u001e) =1\n2\u0019Z2\u0019\n0d\u001eexp[i(n\u0000m)\u001e] =\u000en;m; (S15)\nwhere we have used the de\fnition of the Kronecker delta \u000en;m. Plugging this in Eq. (S14), the dispersion relation\nreduces to\n1\u0000+1X\nn=\u00001J2\nn\u0012k\u000ev\n!0\u0013!2\n0\n(!\u0000n!0)2\u0000k2v2\nT= 0: (S16)\nMODE COUPLING EQUATIONS\nThe dispersion relation obtained analytically and numerically for the instability described in this manuscript hints\nthat unstable modes are coupled. Here we derive a closed set of equations in the \ruid limit that puts in evidence\nthis coupling, and use it to interpret why some modes may dominate in the simulations presented in this work. For8\nsimplicity, we present this derivation in the non relativistic regime. We attribute the subscripts 0, 1 to zeroth order\n(equilibrium) and \frst order quantities and the superscripts \u0006to quantities associated with positrons and electrons,\nrespectively. We start by linearizing the continuity equations,\n@n\u0006\n1\n@t\u0006v0(t)@n\u0006\n1\n@x+n0@v\u0006\n1\n@x= 0: (S17)\nHere,v0(t) is the time dependant zeroth order \ruid velocity developed by both species in the oscillating zeroth order\nelectric \feld E0(t). Taking the partial time derivative of Eq. (S17), we get\n@2n\u0006\n1\n@t2\u0006@v0\n@t@n\u0006\n1\n@x\u0006v0@2n\u0006\n1\n@t@x+n0@2v\u0006\n1\n@t@x= 0: (S18)\nChanging the order of the derivatives in the last term, and using the momentum equation, we have\n@2n\u0006\n1\n@t2\u0006@v0\n@t@n\u0006\n1\n@x\u0006v0@2n\u0006\n1\n@t@x+n0@\n@x\u0012\n\u00001\nn0me@p\u0006\n@x\u0006e\nmeE1\u0007v0@v\u0006\n1\n@x\u0013\n= 0; (S19)\nwherep\u0006is the positron/electron \ruid pressure. The last two terms of Eq. (S19) can be expressed as a function of\nn\u0006\n1by using Gauss's law and the continuity equations, respectively,\n@2n\u0006\n1\n@t2\u0006@v0\n@t@n\u0006\n1\n@x\u0006v0@2n\u0006\n1\n@t@x\u0006!2\np(n+\n1\u0000n\u0000\n1)\u0000\r\u0006T\u0006\nme@2n\u0006\n1\n@x2\u0007v0@\n@x\u0012\n\u0000@n\u0006\n1\n@t\u0007v0@n\u0006\n1\n@x\u0013\n= 0; (S20)\nwhere we have written !2\np= 4\u0019e2n0=meandp\u0006=\r\u0006n\u0006T\u0006, with\r\u0006andT\u0006being the adiabatic index and\ntemperature of each \ruid, respectively. Eq. (S20) can be simpli\fed to\n@2n\u0006\n1\n@t2+\u0010\nv2\n0\u0000\r\u0006v\u0006\nth2\u0011@2n\u0006\n1\n@x2\u0006@v0\n@t@n\u0006\n1\n@x\u00062v0@2n\u0006\n1\n@t@x\u0006!2\np(n+\n1\u0000n\u0000\n1) = 0; (S21)\nwithv\u0006\nth2=T\u0006=me. Eq. (S21) describes two non-trivially forced and coupled oscillators, n\u0006\n1. Assuming now that\nv0(t) =\u000evcos(!0t) =\u000ev=2(exp(i!0t) + exp(\u0000i!0t)), we can look for wave solutions to Eq. (S21), which is a general-\nization of Mathieu's equation. For a thorough review of the stability properties of Mathieu's equation, see ref. [39].\nA particularly insightful equation can be obtained by taking the Fourier transform in space and time of Eq. (S21).\nAfter some algebra, we obtain\nA\u0006\n0n\u0006\n1(!)\u0006A1+n\u0006\n1(!+!0)\u0006A1\u0000n\u0006\n1(!\u0000!0)\u0000A2\u0002\nn\u0006\n1(!+ 2!0) +n\u0006\n1(!\u00002!0)\u0003\n=!2\npn\u0007\n1(!); (S22)\nwheren\u0006\n1(!) =n\u0006\n1(!;k) is the Fourier mode with frequency !and wave vector k, and where A\u0006\n0=!2\np+\r\u0006k2v\u0006\nth2\u0000\nk2\u000ev2=2\u0000!2,A1\u0006= (k\u000ev=2)(2!\u0006!0) andA2= (k\u000ev=2)2. Eq. (S22) shows that each mode !is coupled to its\nneighbours !\u0006!0and!\u00062!0. This result is similar to that obtained in other works describing multiple light/plasma\nwave interactions originally inspired by ref. [1].\nA possible approach to obtain a dispersion relation from Eq. (S22) is to solve the system of coupled equations for\nneighbour modes, e.g. !,!\u0006!0and!\u00062!0. This yields a system of linear equations that can be truncated to any\ndesired neighbour mode order. This truncation corresponds to an ordering condition in k\u000ev=! 0, sinceA1\u0006/k\u000ev=! 0\nandA1/(k\u000ev=! 0)2. We have veri\fed numerically that keeping only modes !and!\u00061 yields a system of linear\nequations whose solution recovers the dispersion relation in Eq. (6), and in particular the scaling \u0000 =!0/(k\u000ev=! 0)2=3.\nSCALING OF INSTABILITY GROWTH RATE WITH TEMPERATURE\nWe discuss here the scaling with temperature of the growth rate of the instability presented in this manuscript. For\nsimplicity, we adopt the normalization \u000ev!\u000ev=c,vT!vT=c,k!kvT=!0and!!!=! 0. The dispersion relation\nin Eq. (4) reads\n1\u0000X\nnJ2\nn\u0012\nk\u000ev\nvT\u00131\n(!\u0000n)2\u0000k2= 0: (S23)9\n0.50.60.70.80.91.0k[ω0/vT](a)\n0.00 0.05 0.10 0.15 0.20Γ [ω0]\n0 5 10 15 20\nδv/vT0.000.050.100.150.20Γmax [ω0]\n(b)\nFIG. S1. Instability growth rate. (a) shows the wavenumber and \u000ev=v Tratio dependence of the growth rate, and (b) illustrates\nthe dependence of the maximum growth rate for \fxed \u000ev=v Tas a function of the wavenumber. Dashed-dotted and dashed\nblack lines in (b) represent theoretical asymptotic limits of the maximum growth rate, whereas black dots represent simulation\nresults.\nAs mentioned in the manuscript, keeping only the terms n=\u00061 in this series is a reasonable approximation to\ndetermine the growth rate of unstable thermal modes. The solution to Eq. (S23) is, in this approximation, given by\n!2= 1 +F(k)\u0006q\nJ4\n1(k\u000ev=vT) + 4F(k); (S24)\nwithF(k) =J2\nn(k\u000ev=vT) +k2. The imaginary component of this solution, =(!)\u0011\u0000, is plotted in Figure S1(a) as a\nfunction of kand\u000ev=vT, and its maximum value for a given \u000ev=vTratio, \u0000 max, is plotted in Figure S1(b) as a function\nofk. In Figure S1(b) we show also in dashed-dotted and dashed black lines the analytical estimate in the asymptotic\nlimits of small and large \u000ev=vT. These lines are obtained by taking asymptotic limits of the Bessel function J1, with\nwhich we can estimate \u0000 max'!0J2\n1=2. For\u000ev=vT\u001d1, we have \u0000 max=!0'vT=\u0019\u000ev , whereas for \u000ev=vT\u001c1 we \fnd\n\u0000max=!0'(\u000ev=vT)2=8. Figure S1(b) also shows the growth rate of the fastest growing modes in several simulations\nwith di\u000berent \u000ev=vTratios in black dots, showing a good agreement with theoretical predictions. All simulations were\nperformed with \fxed \u000ev=c'0:14 and varying vT. 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[1], but in which the pump and all other coupled modes have a similar frequency. This should not be\nconfused with other misleading de\fnitions of the oscillating two-stream instability, discussed in Ref. [15].\n[21] H. Qin and R. C. Davidson, Phys. Plasmas 21, 064505 (2014).\n[22] A. Levinson, D. Melrose, A. Judge, and Q. Luo, Astrophys. J. 631, 456 (2005).\n[23] A. Philippov, A. Timokhin, and A. Spitkovsky, Phys. Rev. Lett. 124, 245101 (2020).\n[24] P. A. Sturrock, Astrophys. J. 164, 529 (1971).\n[25] M. A. Ruderman and P. G. Sutherland, Astrophys. J. 196, 51 (1975).\n[26] The inductive waves and the instability presented in this work also develop for a single oscillating electron population in an\nion background. We choose to present the derivation for pair plasmas due to their applicability to pulsar magnetospheres.\n[27] I. B. Bernstein, Phys. Rev. 109, 10 (1958).\n[28] T. H. Stix, Waves in Plasmas (AIP Press, New York, 1992).\n[29] D. A. Keston, E. W. Laing, and D. A. Diver, Phys. Rev. E 67, 036403 (2003).\n[30] R. A. Fonseca et al. , OSIRIS: A three-dimensional, fully relativistic particle in cell code for modeling plasma based\naccelerators, in Computational Science | ICCS 2002 , edited by P. M. A. Sloot, A. G. Hoekstra, C. J. K. Tan, and J. J.\nDongarra (Springer Berlin Heidelberg, Berlin, Heidelberg, 2002) pp. 342{351.\n[31] R. A. Fonseca, S. F. Martins, L. O. Silva, J. W. Tonge, F. S. Tsung, and W. B. Mori, Plasma Phys. Control. Fusion 50,\n124034 (2008).\n[32] F. Cruz, T. Grismayer, and L. O. Silva, Astrophys. J. 908, 149 (2021).\n[33] A. N. Timokhin, Mon. Not. R. Astron. Soc. 408, 2092 (2010).\n[34] D. B. Melrose and Q. Luo, Astrophys. J. 698, 124 (2009).\n[35] B. Reville and J. G. Kirk, Astrophys. J. 715, 186 (2010).\n[36] A. Levinson and B. Cerutti, Astron. Astrophys. 616, A184 (2018).\n[37] A. Y. Chen and Y. Yuan, Astrophys. J. 895, 121 (2020).\n[38] S. Kisaka, A. Levinson, and K. Toma, Astrophys. J. 902, 80 (2020).\n[39] I. Kovacic, R. Rand, and S. Mohamed Sah, Appl. Mech. Rev. 70, 10.1115/1.4039144 (2018)."    },    {        "title": "1310.1981v1.Fidelity_of_Mobius_matrices_related_with_Lorentz_boosts.pdf",        "content": "arXiv:1310.1981v1  [math-ph]  8 Oct 2013FIDELITY OF M ¨OBIUS MATRICES RELATED WITH\nLORENTZ BOOSTS\nSEJONG KIM\nAbstract. In this article we consider the extended version of a real cou nterpart of qubit\ndensity matrices, called the M¨ obius matrix, and we see that it is a normalized Lorentz\nboost. Using the isomorphic gyrogroup structures between t he setPof all Lorentz boosts\nand the Einstein gyrogroup on the open unit ball BofRnwe give an explicit formula of\nthe fidelity for M¨ obius matrices in terms of Lorentz gamma fa ctors.\nPACS(2010): 03.30.+p, 02.20.-a, 06.20.F-\nKeywords : M¨ obius matrix, Lorentz boost, gyrogroup, Lorentz gamma f actor, fidelity\n1.Introduction\nA qubit density matrix is a 2 ×2 positive semidefinite Hermitian matrix with trace 1. It\ncan be described by a Bloch vector\nv=\nv1\nv2\nv3\n,/bardblv/bardbl ≤1,\nwhere/bardbl·/bardblis the Euclidean norm. In details,\nρv=1\n2/parenleftigg\n1+v3v1−iv2\nv1+iv21−v3/parenrightigg\n=1\n2(v1σx+v2σy+v3σz),\nwhere\nσx=/parenleftigg\n0 1\n1 0/parenrightigg\n,σy=/parenleftigg\n0−i\ni0/parenrightigg\n,σz=/parenleftigg\n1 0\n0−1/parenrightigg\nare Pauli matrices. It is known that all qubit pure states are parameterized by the unit\nsphere, while all qubit mixed states (or invertible density matrices) are parameterized by\nthe open unit ball in R3. In the following we denote the open unit ball in RnasBnand\nconsider column vectors vinRn.\nIn general, it is difficult to extend the qubit mixed state ρvto a density matrix that is\nparametrized by an n-dimensional Bloch vector v= (v1,v2,...,vn)∈Bnforn >3. On the\n12 SEJONG KIM\nother hand, A. A. Ungar has suggested in [7, Section 9.5] the r eal counterpart of ρvthat\nshares similar properties with ρvand its extended version such as\nµn,v=2γ2\nv\n(n−3)+4γ2v/parenleftigg\n1−1\n2γ2vvT\nv1\n2γ2vIn+vvT/parenrightigg\n=2γ2\nv\n(n−3)+4γ2v\n1−1\n2γ2vv1 v2 v2···vn\nv11\n2γ2v+v2\n1v1v2v1v3···v1vn\nv2v1v21\n2γ2v+v2\n2v2v3···v2vn\nv3v1v3v2v31\n2γ2v+v2\n3···v3vn\n..................\nvnv1vnv2vnv3vn···1\n2γ2v+v2\nn\n,\nwhereInis then×nidentity matrix and γv=1/radicalbig\n1−/bardblv/bardbl2is known as the Lorentz gamma\nfactor. Thisiscalleda M¨ obius matrix parameterizedbythevector v= (v1,v2,...,vn)∈Bn.\nAlthough it is not a natural extension of the qubit density ma trix, it is meaningful that we\nexploreµn,vas a density matrix in the study of higher-level quantum stat es. The aim of\nthis paper is to see the fidelity of M¨ obius matrices, as one of the known measurements.\nLorentz was seeking the transformation under which Maxwell ’s equations were invariant\nwhen transformed from the ether to a moving frame. In 1905 Hen ri Poincar´ e recognized\nthat the transformation has the properties of a mathematica l group and named it after\nLorentz. Later in the same year Albert Einstein derived the L orentz transformation under\nthe assumption of the principle of relativity and the consta ncy of the speed of light in any\ninertial reference frame. Lorentz transformation of the re lativistically admissible vector is\ncurrently an important tool in special relativity, since it enables us to study relativistic\nmechanics in hyperbolic geometry. It also may include a rota tion of space, and especially\na rotation-free Lorentz transformation is called a Lorentz boost . The Lorentz boost is a\npositive definite member of the Lorentz group O(1,n), the group (under composition) of all\nlinear transformations preserving the Lorentz form Ldefined by\nL/an}bracketle{t(s,x1,...,x n),(t,y1,...,yn)/an}bracketri}ht=−st+n/summationdisplay\ni=1xiyi.\nIndeed, the Lorentz boost is a member of the restricted Loren tz group SO+(1,n), the\nidentity component of the Lorentz group consisting of all pr oper orthochronous maps.\nIn this paper, we can see the interesting result that the M¨ ob ius matrix is a normalized\nLorentz boost. So the study of M¨ obius matrices will be assoc iated with the algebraic\nstructure of Lorentz boosts. In Section 2 we review a non-ass ociative algebra structureFIDELITY OF M ¨OBIUS MATRICES RELATED WITH LORENTZ BOOSTS 3\n(called a gyrogroup ) on the set of Lorentz boosts and provide an isomorphism with the\nEinstein gyrogroup on the open unit ball Bn. In Section 3 we show that the M¨ obius matrix\nisa normalized Lorentz boostviaa diagonalization, andin S ection 4we calculate thefidelity\nof M¨ obius matrices and give an explicit formula in terms of L orentz gamma factors.\n2.Gyrogroup for Lorentz boosts\nWe review first the Einstein’s relativistic sum of admissibl e velocities of which magnitude\nis less than the speed of light c/equaldotleftright3×105km/sec. In our purpose of this article, we assume\nthe speed of light is normalized by the value 1, so that the adm issible vectors are in the\nopen unit ball\nBn:={v∈Rn:/bardblv/bardbl<1}.\nThen the relativistic sum of two admissible vectors uandvinBnis given by\nu⊕v=1\n1+uTv/braceleftbigg\nu+1\nγuv+γu\n1+γu(uTv)u/bracerightbigg\n, (2.1)\nwhereγuis the well-known Lorentz factor\nγu=1/radicalbig\n1−/bardblu/bardbl2. (2.2)\nNote that uTvis just the Euclidean inner product of uandvwritten in matrix form.\nDefinition 2.1. The formula (2.1) defines a binary operation, called the Einstein velocity\naddition, on the open unit ball BnofRn.\nRemark 2.2. The Einstein addition u⊕vof two admissible vectors uandvinBnmay\nbe alternatively obtained by applying the Lorentz boost\nB(u) =/parenleftigg\nγuγuuT\nγuuI+γ2\nu\n1+γuuuT/parenrightigg\n(2.3)\nto/parenleftigg\nγv\nγvv/parenrightigg\nand obtaining\nB(u)/parenleftigg\nγv\nγvv/parenrightigg\n=/parenleftigg\nγu⊕v\nγu⊕v(u⊕v)/parenrightigg\n,\nwhere we use the gamma identity γu⊕v=γuγv(1+uTv).\nTo abstractly analyze Einstein velocity addition in the the ory of special relativity, A. A.\nUngar has introduced and studied in several papers and books structures that he has called\ngyrogroups; see [7] and its bibliography. His algebraic axi oms are reminiscent of those for\na group, but a gyrogroup operation is neither associative no r commutative in general.4 SEJONG KIM\nDefinition 2.3. A triple ( G,⊕,0) is agyrogroup if the following axioms are satisfied for all\na,b,c∈G.\n(G1) 0⊕a=a⊕0 =a(existence of identity);\n(G2)a⊕(−a) = (−a)⊕a= 0 (existence of inverses);\n(G3) There is an automorphism gyr[ a,b] :G→Gfor each a,b∈Gsuch that\na⊕(b⊕c) = (a⊕b)⊕gyr[a,b]c(gyroassociativity);\n(G4) gyr[0 ,a] = idG;\n(G5) gyr[ a⊕b,b] = gyr[a,b] (loop property).\nA gyrogroup ( G,⊕) isgyrocommutative if it satisfies\na⊕b= gyr[a,b](b⊕a) (gyrocommutativity).\nA gyrogroup is uniquely 2-divisible if for every b∈G, there exists a unique a∈Gsuch that\na⊕a=b.\nThe map gyr[ a,b] is called the gyroautomorphism orThomas gyration generated by aand\nb, which is analogous to the precession map in a loop theory. It has been shown in [6] that\ngyrocommutative gyrogroups are equivalent to Bruck loops w ith respect to the same op-\neration. It follows that uniquely 2-divisible gyrocommuta tive gyrogroups are equivalent to\nB-loops, uniquely 2-divisible Bruck loops. J. Lawson and Y. L im have recently introduced\ndyadic symmetric sets in [3] and showed the equivalence with uniquely 2-divisible gyrocom-\nmutative gyrogroups. In our purpose of this article we follo w the notion of gyrogroups.\nA. A. Ungar has shown in [7, Chapter 3] by computer algebra tha t Einstein addition\non the open unit ball Bnis a gyrocommutative gyrogroup operation, and the gyroauto -\nmorphisms are orthogonal transformations preserving the E uclidean inner product and the\ninherited norm. We call ( Bn,⊕) theEinstein (gyrocommutative) gyrogroup , where ⊕is\ndefined by the equation (2.1).\nRemark 2.4. We note that the Einstein gyrogroup ( Bn,⊕) is uniquely 2-divisible; for any\nv∈Bnthere exists a unique\nw=γv\n1+γvv∈B\nsuchthat w⊕w=v(seetheequation (6.297) of[7]). We denoteit simplyby w:= (1/2)⊗v,\norv= 2⊗w.\nWe now see the gyrogroup structure on the set Pof all Lorentz boosts given in the\nequation (2.3). From the polar decomposition of B(u)B(v) foru,v∈Bnwe have the\nrelation\nB(u⊕v) =/parenleftbig\nB(u)B(v)2B(u)/parenrightbig1/2, (2.4)FIDELITY OF M ¨OBIUS MATRICES RELATED WITH LORENTZ BOOSTS 5\nsee [2] for more details. Hence, we obtain\nTheorem 2.5. The Lorentz boost map Bis an isomorphism from (Bn,⊕,0)to(P,⋆,I),\nwhere\nB(u)⋆B(v) =/parenleftbig\nB(u)B(v)2B(u)/parenrightbig1/2.\nFurthermore, the powers and roots in (P,∗)agree with those of matrix multiplication.\nOn the cone Ω of positive definite Hermitian matrices, the squ aring map D: Ω→Ω,\nD(A) =A2gives us a different algebraic structure on the set P. We note that the squaring\nmapDis a bijection since any positive definite Hermitian matrix h as a unique square root\nin ��.\nTheorem 2.6. The composition D◦B: (Bn,⊕,0)→(P,∗,I)is also an isomorphism,\nwhere\nB(u)∗B(v) =B(u)1/2B(v)B(u)1/2.\nRemark 2.7. From Theorem 2.5 and Theorem 2.6 we see that both ( P,⋆,I) and (P,∗,I)\nare uniquely 2-divisible gyrocommutative gyrogroups. Mor eover, we have\nB(2⊗v) =B(v)2, B((1/2)⊗v) =B(v)1/2\nfor anyv∈Bn.\n3.M¨obius matrices and Lorentz boosts\nFirst of all, we see the M¨ obius matrix parameterized by the v ectorv∈Bn,n≥3, as an\nextended version of the real counterpart of qubit density ma trices:\nµn,v=2γ2\nv\n(n−3)+4γ2v/parenleftigg\n1−1\n2γ2vvT\nv1\n2γ2vIn+vvT/parenrightigg\n. (3.5)\nThis is an ( n+1)×(n+1) symmetric matrix, and we verify a diagonalization of M¨ o bius\nmatrix.\nTheorem 3.1. For each v∈Bnthere exist an orthogonal matrix Ovand a diagonal matrix\nDv\nOv=/parenleftigg1√\n2−1√\n20···0\n1√\n2/bardblv/bardblv1√\n2/bardblv/bardblv u1···un−1/parenrightigg\n,\nDv=1\n(n−3)+4γ2v\nλ20\n01\nλ20\n0In−1\n6 SEJONG KIM\nsuch that µn,v=OT\nvDvOv, where {uj:vTuj= 0for allj= 1,2,...,n−1}is an\northonormal set obtained by the Gram-Schmidt process, and\nλ=/radicaligg\n1+/bardblv/bardbl\n1−/bardblv/bardbl>1.\nProof.Let\nA=(n−3)+4γ2\nv\n2γ2vµn,v=/parenleftigg\n1−1\n2γ2vvT\nv1\n2γ2vIn+vvT/parenrightigg\nIt is enough to show that\nA=OT\nv·1\n2γ2v\nλ20\n01\nλ20\n0In−1\n·Ov.\nIndeed,\nA/parenleftigg1√\n2\n1√\n2/bardblv/bardblv/parenrightigg\n=\n1√\n2/parenleftig\n1−1\n2γ2v/parenrightig\n+1√\n2/bardblv/bardblvTv\n1√\n2v+/parenleftig\n1\n2γ2vIn+vvT/parenrightig\n1√\n2/bardblv/bardblv\n\n=/parenleftigg1\n2√\n2(1+/bardblv/bardbl)2\n1\n2√\n2/bardblv/bardbl(1+/bardblv/bardbl)2v/parenrightigg\n=(1+/bardblv/bardbl)2\n2/parenleftigg1√\n2\n1√\n2/bardblv/bardblv/parenrightigg\n.\nHere,(1+/bardblv/bardbl)2\n2=1\n2γ2v·1+/bardblv/bardbl\n1−/bardblv/bardbl=λ2\n2γ2v. Similarly,\nA/parenleftigg\n−1√\n2\n1√\n2/bardblv/bardblv/parenrightigg\n=(1−/bardblv/bardbl)2\n2/parenleftigg\n−1√\n2\n1√\n2/bardblv/bardblv/parenrightigg\n=1\n2γ2vλ2/parenleftigg\n−1√\n2\n1√\n2/bardblv/bardblv/parenrightigg\n.\nFinally for each j= 1,2,...,n−1\nA/parenleftigg\n0\nuj/parenrightigg\n=/parenleftigg\nvTuj/parenleftig\n1\n2γ2vIn+vvT/parenrightig\nuj/parenrightigg\n=1\n2γ2v/parenleftigg\n0\nuj/parenrightigg\nsincevTuj= 0. /square\nRemark 3.2. By Theorem 3.1 we have that the matrix µn,vis positive definite,\ntrµn,v=1\n(n−3)+4γ2v/parenleftbigg\nλ2+1\nλ2+n−1/parenrightbigg\n= 1,FIDELITY OF M ¨OBIUS MATRICES RELATED WITH LORENTZ BOOSTS 7\nand\ndetµn,v=/parenleftbigg1\n(n−3)+4γ2v/parenrightbiggn+1\n=/parenleftbigg1−/bardblv/bardbl2\n(n+1)−(n−3)/bardblv/bardbl2/parenrightbiggn+1\n>0.\nWe proved the equation (9.85) in [7] and that µn,vis an (n+1)×(n+1) real mixed state.\nIn [2, Theorem 5.6] it has been shown that\nB(v) =OT\nv\nλ0\n01\nλ0\n0In−1\nOv, (3.6)\nwhereOvis the same orthogonal matrix in Theorem 3.1. So we obtain the interesting result\nthat M¨ obius matrix is the normalized Lorentz boost generat ed by the vector 2 ⊗v, since\ntrB(v)2=λ2+1\nλ2+n−1 = 2γ2\nv(1+/bardblv/bardbl2)+n−1 = (n−3)+4γ2\nv.\nProposition 3.3. For each v∈Bn,\nµn,v=1\ntrB(v)2B(v)2=1\ntrB(2⊗v)B(2⊗v).\n4.Fidelity\nIt has been issued how to measure the distance of quantum stat es represented by density\nmatrices, i.e., positive semidefinite Hermitian matrices w ith trace 1. The fidelityis one of\ncrucial measurements although it is actually not a metric fo r quantum states. On the other\nhand, it is a measure of the closedness of two quantum states, that is, the fidelity is 1 if and\nonly if two quantum states are identical. Moreover, it does g ive rise to a useful metric and\nis able to apply for a variety of research areas in quantum inf ormation and computation\ntheory; see [5] and [4, Section 9.2.2].\nThe fidelity for density matrices ρandσis defined by\nF(ρ,σ) := tr/radicalig\nρ1/2σρ1/2. (4.7)\nWe review some basic properties of the fidelity.\nLemma 4.1. The following are satisfied for any density matrices ρandσ.\n(i) 0≤F(ρ,σ)≤1.\n(ii)F(ρ,σ) = 1if and only if ρ=σ.\n(iii)F(ρ,σ) =F(σ,ρ).\n(iv)F(UρU∗,UσU∗) =F(ρ,σ)for any unitary U.\nThe property (iv) of Lemma 4.1 is called the invariance under unitary congruence trans-\nformation, so that the fidelity is basis-independent.8 SEJONG KIM\nRemark 4.2. The fidelity Fcan be quite difficult to calculate, but it takes a simple form\nfor the 2-by-2 density matrices ρandσ: see the equation (8.52) in [1],\nF(ρ,σ)2= tr(ρσ)+2/radicalbig\ndet(ρ)det(σ). (4.8)\nFrom the equations (9.64) and (9.68) in [7] we have alternati ve expression of the fidelity for\nthe 2-by-2 density matrices ρuandρv, whereu,v∈B3:\nF(ρu,ρv)2=1+γu⊕v\n2γuγv=1\n2/braceleftig\n1+uTv+/radicalbig\n1−/bardblu/bardbl2/radicalbig\n1−/bardblv/bardbl2/bracerightig\n. (4.9)\nOne can verify that two equations (4.8) and (4.9) are the same .\nWe introduce a normalized Lorentz boost to be a density matri x, especially the M¨ obius\nmatrix. Let us denote\nˆB(v) :=1\ntrB(v)B(v)\nfor anyv∈Bn. Indeed, ˆB(v) must be a mixed state, a positive definite density matrix,\nand by Proposition 3.3 we have µn,v=ˆB(2⊗v).\nWe see useful properties to show the main result.\nLemma 4.3. For any positive semidefinite matrices A,Bandα,β >0,\nF(αA,βB) =/radicalbig\nαβF(A,B).\nLemma 4.4. For any u,v∈(Bn,⊕)\ntr[B(u)B(v)2B(u)]1/2= 2γu⊕v+n−1.\nProof.From the equation (3.6) we have\ntrB(v) =λ+1\nλ+n−1 = 2γv+n−1.\nSo it is proved by the equation (2.4). /square\nWe now see an explicit formula of the fidelity for normalized L orentz boosts in terms of\nLorentz factors.\nTheorem 4.5. For any u,v∈(Bn,⊕)\nF(ˆB(u),ˆB(v)) =2γw+n−1/radicalbig\n(2γu+n−1)(2γv+n−1),\nwhere\nw=1\n2⊗u⊕1\n2⊗v.FIDELITY OF M ¨OBIUS MATRICES RELATED WITH LORENTZ BOOSTS 9\nProof.Letu′:= (1/2)⊗uandv′:= (1/2)⊗v. Then\nF(ˆB(u),ˆB(v)) =F(B(u),B(v))/radicalbig\ntrB(u)trB(v)=tr[B(u′)B(v′)2B(u′)]1/2\n/radicalbig\ntrB(u)trB(v)\n=2γu′⊕v′+n−1/radicalbig\n(2γu+n−1)(2γv+n−1).\nThe first equality follows from Lemma 4.3, the second follows from Remark 2.7, and the\nlast follows from Lemma 4.4. /square\nFor anyu,v∈Bn, in general,\n(1/2)⊗u⊕(1/2)⊗v/ne}ationslash= (1/2)⊗(u⊕v),\nsee [7, Chapter 6] for more details. On the other hand, we give an formula of Lorentz\nfactor for w= (1/2)⊗u⊕(1/2)⊗v, so that the fidelity for M¨ obius matrices can be simply\ncalculated.\nLemma 4.6. For any u,v∈(Bn,⊕),\nγw=1√1+2γu1√1+2γv((1+γu)(1+γv)+γuγvuTv),\nwherew= (1/2)⊗u⊕(1/2)⊗v.\nProof.Letu′:= (1/2)⊗uandv′:= (1/2)⊗v. By Remark 2.4\nγv′=1/radicalbig\n1−/bardblv′/bardbl2=1/radicalbig\n1−γ2v/(1+γv)2=1+γv√1+2γv.\nApplying the gamma identity γu⊕v=γuγv(1+uTv) tou′andv′, it is proved. /square\nRemark 4.7. Directly from the Einstein velocity addition we have\n2⊗v=v⊕v=2γ2\nv\n2γ2v−1v,\nso thatγ2⊗v= 2γ2\nv−1. Hence, the result of Theorem 4.5 reduces to\nF(µn,u,µn,v) =F(ˆB(2⊗u),ˆB(2⊗v)) =γu⊕v+n−1/radicalbig\n(4γ2u+n−3)(4γ2v+n−3).\nEspecially, if n= 3,\nF(µ3,u,µ3,v) =1+γu⊕v\n2γuγv=F(ρu,ρv)2.10 SEJONG KIM\nReferences\n[1] Stephen M. Barnett, Quantum Information, Oxford Univer sity Press, 2009.\n[2] S. Kim and J. Lawson, Unit balls, Lorentz boosts, and hype rbolic geometry, Results Math. 63(2013),\n1225-1242.\n[3] J. Lawson and Y. Lim, Symmetric sets with midpoints and al gebraically equivalent theories, Results\nMath.46(1-2), 37-56 (2004).\n[4] Z. Ma, F.-L. Zhang, and J.-L. Chen, Fidelity induced dist ance measures for quantum states, Phys. Lett.\nA373(2009), 3407-3409.\n[5] M. Nielsen and I. Chuang, Quantum Computation and Quantu m Information, Cambridge, 2010.\n[6] L. V. Sabinin, L. L. Sabinina, and L. V. Sbitneva, On the no tion of a gyrogroup, Aeq. Math. 56(1998),\n11-17.\n[7] A. A. Ungar, Analytic hyperbolic geometry and Albert Ein stein’s special theory of relativity, World\nScientific Press, 2008.\nSejong Kim, Department of Mathematics, Chungbuk National U niversity, Cheongju 361-763,\nKorea\nE-mail address :skim@chungbuk.ac.kr"    },    {        "title": "1608.04071v1.Mechanical_energy_and_mean_equivalent_viscous_damping_for_SDOF_fractional_oscillators.pdf",        "content": "arXiv:1608.04071v1  [physics.class-ph]  14 Aug 2016Mechanical energy and mean equivalent viscous\ndamping for SDOF fractional oscillators\nJian Yuan1,∗, Bao Shi, Mingjiu Gai, Shujie Yang\nInstitute of System Science and Mathematics, Naval Aeronau tical and Astronautical\nUniversity, Yantai 264001, P.R.China\nAbstract\nThispaperaddressesthetotalmechanical energyofasingledegr eeoffreedom\nfractional oscillator. Based on the energy storage and dissipation properties\nof the Caputo fractional derivatives, the expression for total m echanical en-\nergy in the single degree of freedom fractional oscillator is firstly pr esented.\nThe energy regeneration due to the external exciting force and t he energy\nloss due to the fractional damping force during the vibratory motio n are an-\nalyzed. Furthermore, based on the mean energy dissipation of the fractional\ndamping element in steady-state vibration, a new concept of mean e quiva-\nlent viscous damping is suggested and the value of the damping coeffic ient is\nevaluated.\nKeywords: Fractional oscillators, linear viscoelasticity, fractional\nconstitutive relations, mechanical energy, mean equivalent viscou s damping\n1. Introduction\nViscoelasticmaterialsanddampingtreatmenttechniqueshavebeen widely\napplied in structural vibration control engineering, such as aeros pace indus-\ntry, military industry, mechanical engineering, civil and architectu ral engi-\nneering[1]. Describingtheconstitutive relationsforviscoelastic mat erialsisa\ntop priority to seek for the dynamics of the viscoelastically damped s tructure\nand to design vibration control systems.\n∗Corresponding author. Tel.: +8613589862375.\nEmail address: yuanjianscar@gmail.com (Jian Yuan)\nPreprint submitted to Elsevier September 19, 2018Recently, theconstitutiverelationsemploying fractionalderivativ es which\nrelatestressandstraininmaterials, alsotermedasfractionalvisc oelasticcon-\nstitutive relations, have witnessed rapid development. They may be viewed\nasanaturalgeneralizationoftheconventional constitutiverelat ionsinvolving\ninteger order derivatives or integrals, and have been proven to be a power-\nful tool of describing the mechanical properties of the materials. Over the\nconventional integer order constitutive models, the fractional o nes have vast\nsuperiority. The first attractive feature is that they are capable of fitting ex-\nperimental results perfectly and describing mechanical propertie s accurately\nin both the frequency and time domain with only three to five empirical pa-\nrameters[2]. Thesecondisthattheyarenotonlyconsistent withth ephysical\nprinciples involved [3] and the molecule theory [4], but also represent t he fad-\ning memory effect [2] and high energy dissipation capacity [5]. Finally, fr om\nmathematical perspectives the fractional constitutive equation s and the re-\nsulting fractional differential equations of vibratory motion are co mpact and\nanalytic [6].\nNowadays many types of fractional order constitutive relations h ave been\nestablished via a large number of experiments. The most frequently used\nmodels include the fractional Kelvin-Voigt model with three paramet ers [2]:\nσ(t) =b0ε(t)+b1Dαε(t), the fractional Zener model with four parameters\n[3]:σ(t)+aDασ(t) =b0ε(t)+b1Dαε(t), and the fractional Pritz model with\nfive parameters [7]: σ(t)+aDασ(t) =b0ε+b1Dα1ε(t)+b2Dα2ε(t).\nFractionaloscillators, orfractionallydampedstructures, aresy stemswhere\nthe viscoelastic damping forces in governing equations of motion are de-\nscribed by constitutive relations involving fractional order derivat ives [8].\nThe differential equations of motion for the fractional oscillators a re frac-\ntional differential equations. Researches on fractional oscillator s are mainly\nconcentratedontheoreticalandnumericalanalysisofthevibrat ionresponses.\nInvestigations on dynamical responses of SDOF linear and nonlinear frac-\ntional oscillators, MDOF fractional oscillators and infinite-DOF frac tional\noscillators have been reviewed in [8]. Asymptotically steady state beh avior\nof fractional oscillators have been studied in [9, 10]. Based on the fu nc-\ntional analytic approach, the criteria for the existence and the be havior of\nsolutions have been obtained in [11-13], and particularly in which the im-\npulsive response function for the linear SDOF fractional oscillator is derived.\nThe asymptotically steady state response of fractional oscillator s with more\nthan one fractional derivatives have been analyzed in [14]. Consider ing the\nmemory effect and prehistory of fractional oscillators, the histor y effect or\n2initialization problems for fractionally damped vibration equations has been\nproposed by Fukunaga, M. [15-17] and Hartley, T.T., and Lorenzo, C.F. [18,\n19].\nStability synthesis for nonlinear fractional differential equations h ave re-\nceived extensive attention in the last five years. Mittag-Leffler sta bility theo-\nrems [20, 21]andtheindirect Lyapunov approach[22] based onthe frequency\ndistributed model are two main techniques to analyze the stability of non-\nlinear systems, though there is controversy between the above t wo theories\ndue to state space description and initial conditions for fractional systems\n[23]. In spite of the increasing interest in stability of fractional differ ential\nequations, there’s little results on the stability of fractionally dampe d sys-\ntems. For the reasons that Lyapunov functions are required to c orrespond to\nphysical energy and that there exist fractional derivatives in the differential\nequations of motion for fractionally damped systems, it is a primary t ask to\ndefine the energies stored in fractional operators.\nFractional energy storage and dissipation properties of Riemann- Liouville\nfractional integrals is defined [24, 25] utilizing the infinite state appr oach.\nBased on the fractional energies, Lyapunov functions are propo sed and sta-\nbility conditions of fractional systems involving implicit fractional der iva-\ntives are derived respectively by the dissipation function [24, 25] an d the\nenergy balance approach [26, 27]. The energy storage properties of frac-\ntional integrator and differentiator in fractional circuit systems h ave been\ninvestigated in [28-30]. Particularly in [29], the fractional energy for mula-\ntion by the infinite-state approach has been validated and the conv entional\npseudo-energy formulations based on pseudo state variables has been inval-\nidated. Moreover, energy aspects of fractional damping forces described by\nthe fractional derivative of displacement in mechanical elements ha ve been\nconsidered in [31, 32], in which the effect on the energy input and ener gy\nreturn, as well as the history or initialization effect on energy respo nse has\nbeen presented.\nOn the basis of the recently established fractional energy definitio ns for\nfractional operators, our main objective in this paper is to deal wit h the\ntotal mechanical energy of a single degree of freedom fractional oscillator.\nTo this end, we firstly present the mechanical model and the differe ntial\nequation of motion for the fractional oscillator. Then based on the energy\nstorage and dissipation in fractional operators, we provide the ex pression of\ntotal mechanical energy in the single degree of freedom fractiona l oscillator.\nFurthermore, we analyze the energy regeneration due to the ext ernal exciting\n3force and the energy loss due to the fractional damping force in th e vibration\nprocesses. Finally, based on the mean energy dissipation of the fra ctional\ndampingelementinsteady-statevibration, weproposeanewconce ptofmean\nequivalent viscous damping and determine the expression of the dam ping\ncoefficient.\nThe rest of the paper is organized as follows: Section 2 retrospect some\nbasic definitions and lemmas about fractional calculus. Section 3 intr oduces\nthe mechanical model and establishes the differential equation of m otion for\nthe single degree of freedom fractional oscillator. Section 4 provid es the\nexpression of total mechanical energy for the SDOF fractional o scillator and\nanalyzes the energy regeneration and dissipation in the vibration pr ocesses.\nSection 5 suggests a new concept of mean equivalent viscous dampin g and\nevaluatesthevalueofthedamping coefficient. Finally, thepaperisco ncluded\nin section 6 with perspectives.\n2. Preliminaries\nDefinition 1. The Riemann-Liouville fractional integral for the function\nf(t) is defined as\naIα\ntf(t) =1\nΓ(α)/integraldisplayt\na(t−τ)α−1f(τ)dτ, (1)\nwhereα∈R+is an non-integer order of the factional integral, the subscripts\naandtare lower and upper terminals respectively.\nDefinition 2. The Caputo definition of fractional derivatives is\naDα\ntf(t) =1\nΓ(n−α)/integraldisplayt\naf(n)(τ)dτ\n(t−τ)α−n+1,n−1< α < n. (2)\nLemma 1. The frequency distributed model for the fractional integra tor [33-\n35]Theinput ofthe Riemann-Liouvilleintegralis denotedb yv(t)andoutput x(t),\nthenaIα\ntv(t)is equivalent to\n/braceleftBigg\n∂z(ω,t)\n∂t=−ωz(ω,t)+v(t),\nx(t) =aIα\ntv(t) =/integraltext+∞\n0µα(ω)z(ω,t)dω,(3)\nwithµα(ω) =sin(απ)\nπω−α.\n4System (3) is the frequency distributed model for fractiona l integrator,\nwhich is also named as the diffusive representation.\nLemma 2. The following relation holds[26]\n/integraldisplay∞\n0ωµα(ω)\nω2+Ω2dω=sinαπ\n2Ωαsinαπ\n2. (4)\n3. Differential equation of motion for the fractional oscill ator\nThis section will establish the differential equation of motion for a sin-\ngle degree of freedom fractional oscillator, which consists of a mas s and a\nspring with one end fixed and the other side attached to the mass, d epicted\nin Fig.1. The spring is a solid rod made of some viscoelastic material with\nthe cross-sectional area Aand length L, and provides stiffness and damping\nfor the oscillator.\nm\nFigure 1: Mechanical model for the SDOF fractional oscillator.\nIn accordance with Newton’s second law, the dynamical equation fo r the\nSDOF fractional oscillator is\nm¨x(t)+fd(t) =f(t),fd(t) =Aσ(t), (5)\nwherefd(t) is the forceprovided by the viscoelastic rodandcan be separated\ninto two parts: the resilience and the damping force. f(t) is the vibration\nexciting force acted on the mass.\nThe kinematic relation is\nε(t) =x(t)\nL. (6)\n5As for the constitutive equation of viscoelastic material, the followin g frac-\ntional Kelvin-Voigt model (7) with three parameters will be adopted\nσ(t) =b0ε(t)+b1Dα\ntε(t), (7)\nwhereα∈(0,1) is the order of fractional derivative, b0andb1are positive\nconstant coefficients.\nThe above three relations (5) (6) and (7) form the following differen tial equa-\ntion of motion for the single degree of freedom fractional oscillator\nm¨x(t)+cDαx(t)+kx(t) =f(t), (8)\nwherec=Ab1\nL,k=Ab0\nL.\nFor the reason that the Caputo derivative is fully compatible with the clas-\nsical theory of viscoelasticity on the basis of integral and different ial consti-\ntutive equations [36], the adoption of the Caputo derivative appear s to be\nthe most suitable choice in the fractional oscillators. For the simplific ation\nof the notation, the Caputo fractional-order derivativeC\n0Dα\ntis denoted as Dα\nin this paper.\nComparing the forms of differential equations for the fractional o scillator (8)\nwith the following classical ones\nm¨x(t)+c˙x(t)+kx(t) =f(t), (9)\nonecansee thatthefractionalone(8)isthegeneralizationofthe classical one\n(9)byreplacing thefirst orderderivative ˙ xwiththefractionalorderderivative\nDαx. However, the generalization induces the following essential differe nces\nbetween them.\n•In view of the formalization of the mechanical model, the classical os -\ncillator is composed of a mass, a spring and a dashpot, where kis the\nstiffness coefficient of the spring offering restoring force kxandcis\nthe damping coefficient of the dashpot offering the damping force c˙x.\nThe fractional oscillator is formed by a mass and an viscoelastic rod.\nThe rod offers not only resilience but also damping force. In fraction al\ndifferential equation(8), the coefficient candkare determined by both\nthe constitute equation (7) for the viscoelastic material and the g eo-\nmetrical parameters for the rod, which can be interpreted respe ctively\nas the fractional damping coefficient and the stiffness coefficient. A s a\n6result, the physical meaning of candkin the fractional oscillator (8)\nandtheclassical one(9) aredifferent. Thefractional damping for cecan\nbe viewed as a parallel of a spring component kx(t) and a springpot\ncomponent cDαx(t) which is termed in [37] and illustrated in Fig.2.\nThe hysteresis loop of the fractional damping force is dipicted in Fig.3 .\nmkx\ncD xD\nFigure 2: Abstract mechanical model for the SDOF fractional osc illator.\n−2−1.5 −1−0.5 00.5 11.5−4−3−2−101234\nx(t)Dalpha x(t)\nFigure 3: Hysteresis loop of the fractional damping force.\n•Fractional operators are characterized by non-locality and memo ry\nproperties, so fractional oscillators (8) also exhibit memory effect and\nthe vibration response is influenced by prehistory. While the classica l\none (9) has no memory effect and the vibration response is irrelevan t\nwith prehistory.\n•In the aspect of mechanical energy, the fractional term Dαxin (8) not\nonly stores potential energy but also consumes energy due to the fact\n7that fractional operators exhibit energy storage and dissipation simul-\ntaneously [24]. As a result, the total mechanical energy in fraction al\noscillator consists of three parts: the kinetic energy1\n2m˙x2stored in the\nmass, the potential energy corresponding to the spring element1\n2kx2,\nand the potential energy e(t) stored in the fractional derivative. How-\never, in [38] the potential energy e(t) stored in the fractional term\nDαxhas been neglected and the expression1\n2m˙x2+1\n2kx2for the total\nmechanical energy is incomplete.\n4. The total mechanical energy\nGiven the above considerations, we present the total mechanical energy\nof the SDOF fractional oscillator (8)in this section. The fractional system is\nassumed to be at rest before exposed to the external excitation . We firstly\nanalyze the energy stored in the Caputo derivative, based on which the ex-\npression for total mechanical energy is derived. Then we obtain th e energy\nregeneration due to external excitation and the energy dissipatio n due to the\nfractional viscoelastic damping.\nBydefinitions(1)and(2),theCaputoderivativeiscomposedofone Riemann-\nLiouville fractional order integral and one integer order derivative ,\nDαx(t) =I1−α˙x(t).\nIn view of Lemma 1, the frequency distributed model for the Caput o deriva-\ntive is /braceleftBigg\n∂z(ω,t)\n∂t=−ωz(ω,t)+ ˙x,\nDαx(t) =/integraltext∞\n0µ1−α(ω)z(ω,t)dω.(10)\nIn terms of the fractional potential energy expression for the f ractional inte-\ngral operator in [24], the stored energy in the Caputo derivative is\ne(t) =1\n2/integraldisplay∞\n0µ1−α(ω)z2(ω,t)dω. (11)\nThe total mechanical energy of the SDOF fractional oscillator is th e sum of\nthe kinetic energy of the mass1\n2m˙x2, the potential energy corresponding to\nthe spring element1\n2kx2, and the potential energy stored in the fractional\nderivative ce(t)\nE(t) =1\n2m˙x2+1\n2kx2+c\n2/integraldisplay∞\n0µ1−α(ω)z2(ω,t)dω. (12)\n8To analyze the energy consumption in the fractional viscoelastic os cillator,\ntaking the first order time derivative of E(t),one derives\ndE(t)\ndt=m˙x¨x+kx˙x+c/integraldisplay∞\n0µ1−α(ω)z(ω,t)∂z(ω,t)\n∂tdω.(13)\nSubstituting the first equation in the frequency distributed model (10) into\nthe third term of the above equation (13), one derives\ndE(t)\ndt=m˙x¨x+kx˙x+c/integraldisplay∞\n0µ1−α(ω)z(ω,t)[−ωz(ω,t)+ ˙x]dω\n=m˙x¨x+kx˙x+c˙x/integraldisplay∞\n0µ1−α(ω)z(ω,t)dω\n−c/integraldisplay∞\n0ωµ1−α(ω)z2(ω,t)dω. (14)\nSubstituting the second equation in the frequency distributed mod el (10)\ninto the second term of the above equation (14), one derives\ndE(t)\ndt=m˙x¨x+kx˙x+c˙xDαx−c/integraldisplay∞\n0ωµ1−α(ω)z2(ω,t)dω\n= ˙x[m¨x+cDαx+kx]−c/integraldisplay∞\n0ωµ1−α(ω)z2(ω,t)dω. (15)\nSubstituting the differential equation of motion (8) for the fractio nal oscilla-\ntor into the first term of the above equation (15), one derives\ndE(t)\ndt=f(t) ˙x(t)−c/integraldisplay∞\n0ωµ1−α(ω)z2(ω,t)dω. (16)\nFrom Eq. (16) it is clear that the energy regeneration in the fractio nal\noscillator due to the work done by the external excitation in unit time is\nP(t) =f(t) ˙x(t). (17)\nOn the other hand, the energy consumption or the Joule losses due to the\nfractional viscoelastic damping is\nJ(t) =c/integraldisplay∞\n0ωµ1−α(ω)z2(ω,t)dω. (18)\n9The mechanical energy changes in the vibration process can be obs erved\nthrough the following numerical simulations. Parameters in the frac tional\noscillator (8) are taken respectively as m= 1,c= 0.4,k= 2,α= 0.56, the\nexternal force are assumed to be f(t) = 30cos6 t. Fig.4 shows the fractional\npotential energy ce(t); Fig.5 shows comparison between the fractional energy\nce(t)and the total mechanical energy E(t); Fig.6 illustrates the mechanical\nenergy consumption J(t).\n0 5 10 15 20 25 3000.050.10.150.20.250.30.350.4\nTime(sec)Fractional Energy Ec(t)\nFigure 4: Fractional energy of the SDOF fractional oscillator.\n0 5 10 15 20 25 30051015202530354045\nTime(sec)Energy\nFigure 5: Comparison between the fractional energy and the tota l mechanical energy.\n100 5 10 15 20 25 3000.511.522.533.54\nTime(sec)Energy Consumption J (t)\nFigure 6: Mechanical energy consumption in the SDOF fractional os cillator.\nRemark 1. IfthefollowingmodifiedfractionalKelvin-Voigtconstituteequa-\ntion (19)which is proposed in [39] is taken to describe the viscoelastic stress-\nstrain relation\nσ(t) =b0ε(t)+b1Dα1ε(t)+b2Dα2ε(t), (19)\nwithα1,α2∈(0,1), the differential equation of motion for the SDOF frac-\ntional oscillator is\nm¨x(t)+c1Dα1x(t)+c2Dα2x(t)+kx(t) =f(t), (20)\nwherec1=Ab1\nL,c2=Ab2\nL,k=Ab0\nL.\nIn view of the following equivalences (21) and (22) between the Capu to\nderivatives and the frequency distributed models\nDα1x(t) =I1−α1˙x(t)⇔/braceleftBigg\n∂z1(ω,t)\n∂t=−ωz1(ω,t)+ ˙x(t)\nDα1x(t) =/integraltext∞\n0µ1−α1(ω)z1(ω,t)dω(21)\nand\nDα2x(t) =I1−α2˙x(t)⇔/braceleftBigg\n∂z2(ω,t)\n∂t=−ωz21(ω,t)+ ˙x(t)\nDα2x(t) =/integraltext∞\n0µ1−α2(ω)z2(ω,t)dω(22)\n11the total mechanical energy of the fractional oscillator (20) is ex pressed as\nE(t) =1\n2m˙x2+1\n2kx2+c1\n2/integraldisplay∞\n0µ1−α1(ω)z2\n1(ω,t)dω\n+c2\n2/integraldisplay∞\n0µ1−α2(ω)z2\n2(ω,t)dω. (23)\nIn the above expression (23) for the total mechanical energy,\nP1(t) =c1\n2/integraldisplay∞\n0µ1−α1(ω)z2\n1(ω,t)dω\nrepresents the potential energy stored in Dα1x(t), whereas\nP2(t) =c2\n2/integraldisplay∞\n0µ1−α2(ω)z2\n2(ω,t)dω\nrepresents thepotential energystored in Dα2x(t). Taking the first order time\nderivative of E(t) in Eq.(23), one derives\n˙E(t) =f(t) ˙x(t)−c1/integraldisplay∞\n0ωµ1−α1(ω)z2\n1(ω,t)dω\n−c2/integraldisplay∞\n0ωµ1−α2(ω)z2\n2(ω,t)dω.\nIt is clear that the energy dissipation due to the fractional viscoela stic damp-\ningc1Dα1xis\nJα1(t) =c1/integraldisplay∞\n0ωµ1−α1(ω)z2\n1(ω,t)dω, (24)\nand the energy dissipation due to the fractional viscoelastic dampin gc2Dα2x\nis\nJα2(t) =c2/integraldisplay∞\n0ωµ1−α2(ω)z2\n2(ω,t)dω. (25)\n5. The mean equivalent viscous damping\nThe resulting differential equations of motion for structures incor porating\nfractional viscoelastic constitutive relations to dampen vibratory motion are\nfractional differential equations, which are strange and intricate ly to tackled\nwithfor engineers. In engineering, complex descriptions fordampin g areusu-\nallyapproximately represented by equivalent viscous damping to simp lify the\n12theoretical analysis. Inspired by this idea, we suggest a new conce pt of mean\nequivalent viscous damping based on the expression of fractional e nergy (18).\nUsing thismethod, fractionaldifferential equations aretransfor med into clas-\nsical ordinary differential equations by replacing the fractional da mping with\nthe mean equivalent viscous damping. The principle for the equivalenc y is\nthat the mean energy dissipation due to the desired equivalent damp ing and\nthe fractional viscoelastic damping are identical.\nTo begin with, some comparisons of the energy dissipation between t he frac-\ntional oscillator (8) and the classical one (9) are made in the following .In\nview of the concept of work and energy in classical physics, the wor k done\nby any type of damping force is expressed as\nW(t) =/integraldisplayt\n0fc(τ)dx(τ), (26)\nwherefc(t) is some type of damping force, x(t) is the displacement of the\nmass.\nIn the classical oscillators, the viscous damping force is\nfc1(t) =c˙x(t).\nThe work done by the viscous damping force is\nW1(t) =/integraldisplayt\n0c˙x(τ)dx(τ) =/integraldisplayt\n0c˙x2(τ)dτ. (27)\nIt is well known that the energy consumption in unit time is\nJ1(t) =c˙x2(t), (28)\nwhich is equal to the rate of the work done by the viscous damping fo rce\nJ1(t) =dW1(t)\ndt.\nObviously, the entire work done by the viscous damping force is conv erted\nto heat energy.\nHowever, the case in the fractional oscillators is different. As a mat ter fact,\nthe fractional damping force is\nfc2(t) =cDαx(t).\n13The work done by the fractional damping force is\nW2(t) =/integraldisplayt\n0cDαx(τ)dx(τ).\nDue to the property of energy storage and dissipation in fractiona l deriva-\ntives, the entire work done by the fractional damping force W2is converted\nto two types of energy: one of which is the heat energy\nJ(t) =c/integraldisplay∞\n0ωµ1−α(ω)z2(ω,t)dω,\nand the other is the fractional potential energy\nP(t) =c\n2/integraldisplay∞\n0µ1−α(ω)z2(ω,t)dω.\nHowever, in [40] the equivalent viscous damping coefficient was obtain ed by\nthe equivalency\n/contintegraldisplay\ncDαx(τ)dx(τ) =/contintegraldisplay\nceq˙x(τ)dx(τ)\nBythisequivalencythepropertiesoffractionalderivativehavebe enneglected\nand the work done by the fractional damping force is considered to be con-\nvertedintotheheatentirely. Asaresult, theaboveequivalencyisp roblematic\nand the value of the derived equivalent viscous damping coefficient is la rger\nthan the actual value.\nIn terms of the energy consumption (18), (24) and (25) due to th e fractional\ndamping force, we suggest a new the concept of mean equivalent vis cous\ndamping and evaluate the expression of the damping coefficient.\nAssuming the steady-state response of the fractional oscillator (8) is\nx(t) =XejΩt,\nwhereXis the amplitude and Ω is the vibration frequency.\nStep1. We firstly need to calculate the mean energy consumption due to\nthe fractional viscoelastic damping element, i.e.\nJα(t) =c/integraldisplay∞\n0ωµ1−α(ω)z(ω,t)2dω. (29)\n14To this end, we evaluate the mean square of z(ω,t), i.e.z(ω,t)2.\nIn terms of the first equation in the diffusive representation of Cap uto deriva-\ntive (10)\n˙z(ω,t) =−ωz(ω,t)+ ˙x(t),\nwe get\nz(ω,t) =˙x(t)\nω+jΩ=jΩxejΩt\n√\nω2+Ω2ejθ,\nwhereθ= arctanΩ\nω.\nFurthermore we get\nz(ω,t)2=1\n2z(ω,t)z(ω,t)∗=1\n2Ω2x2\nω2+Ω2, (30)\nwherez(ω,t)∗is the complex conjugate of z(ω,t).\nSubstituting Eq. (30) into Eq.(29), one derives\nJα(t) =c/integraldisplay∞\n0ωµ1−α(ω)z(ω,t)2dω\n=c\n2Ω2X2/integraldisplay∞\n0ωµ1−α(ω)\nω2+Ω2dω. (31)\nApplying the relation (4) in Lemma 2 ,one derives\n/integraldisplay∞\n0ωµ1−α(ω)\nω2+Ω2dω=sin(1−α)π\n2Ωαsin/parenleftbig1−α\n2/parenrightbig\nπ. (32)\nSubstituting Eq.(32) into Eq. (31) one derives\nJα(t) =c\n4Ω1+αX2sin(1−α)π\nsin/parenleftbig1−α\n2/parenrightbig\nπ. (33)\nStep2. Now we calculate the mean energy loss due to the viscous damping\nforce in the classical oscillator. From the relation(28), we have\nJ(t) =cmeq˙x2(t),\nwherecmeqis denoted as the mean equivalent viscous damping coefficient for\nthe fractional viscoelastic damping.\n15Then the mean of the energy loss is derived as\nJ(t) =cmeq˙x2=1\n2cmeq˙x˙x∗=1\n2cmeqΩ2X2. (34)\nStep3. Letting Jα(t) =J(t) and from the relations (33) and (34) one\nderives\nc\n4Ω1+αX21\n2sin(1−α)π\nsin/parenleftbig1−α\n2/parenrightbig\nπ=1\n2cmeqΩ2X2.\nConsequently, we obtain the mean equivalent viscous damping coeffic ient for\nthe fractional viscoelastic damping\ncmeq=c\n2Ωα−1sin(1−α)π\nsin/parenleftbig1−α\n2/parenrightbig\nπ. (35)\nIt is clear from (35) that the mean equivalent viscous damping coeffic ient for\nthe fractional viscoelastic damping is a function of the vibration fre quency\nΩ and the order αof the fractional derivative. To this point, the fractional\ndifferential equations for the SDOF fractional oscillator (8) is appr oximately\nsimplified to the following classical ordinary differential equation\nm¨x(t)+cmeq˙x(t)+kx(t) =f(t). (36)\nWith the aid of numerical simulations, we compare the vibration respo nses\nof the approximate integer-order oscillator (36) with the fraction al one (8).\nThe coefficients are respectively taken as m= 1,c= 0.4,k= 2,α= 0.56,\nthe external force is taken asthe form f=FcosΩt, whereF= 30, Ω = 6. In\nterms of Eq.(35), we derive the mean equivalent viscous damping coe fficient\ncmeq= 0.14.\n160510152025303540−2−1.5−1−0.500.511.5\nTime(sec)x(t)\nFigure 7: The mean equilvalent damping coefficient for the SDOF fract ional oscillator of\nKelvin-Voigt type.\nRemark 2. By the above procedure, we can furthermore evaluate the mean\nequivalent viscous damping coefficient for the SDOF fractional oscilla tor (20)\ncontaining two fractional viscoelastic damping elements. Letting Jα1(t) +\nJα2(t) =J(t) and from the relations (24) (25) and (34), we get\nc1\n4Ωα1+1X2sin(1−α1)π\nsin/parenleftbig1−α1\n2/parenrightbig\nπ+c2\n4Ωα2+1X2sin(1−α2)π\nsin/parenleftbig1−α2\n2/parenrightbig\nπ\n=1\n2c(α1,α2,Ω)Ω2X2.(37)\nFrom (37) we obtain the mean equivalent viscous damping coefficient\ncmeq=c1\n2Ωα1−1sin(1−α1)π\nsin/parenleftbig1−α1\n2/parenrightbig\nπ+c2\n2Ωα2−1sin(1−α2)π\nsin/parenleftbig1−α2\n2/parenrightbig\nπ.(38)\nWith the aid of numerical simulations, we compare the vibration respo nses\nof the approximate integer-order oscillator (36) with the fraction al one(20).\nThe coefficients are respectively taken as m= 1,c1= 0.4,c2= 0.2,k= 2,\n17α1= 0.56,α2= 0.2, the external force is taken as the form f=FcosΩt,\nwhereF= 30, Ω = 6. In terms of Eq.(38), we derive the mean equivalent\nviscous damping coefficient cmeq= 0.56.\n0510152025303540−2−1.5−1−0.500.511.5\nTime(sec)x(t)\nFigure 8: The mean equilvalent damping coefficient for the SDOF fract ional oscillator of\nmodified Kelvin-Voigt type.\n6. Discussion\nThe total mechanical energy in single degree of freedom fractiona l oscil-\nlators has been dealt with in this paper. Based on the energy storag e and\ndissipation properties of the Caputo fractional derivative, the to tal mechani-\ncal energy is expressed asthe sum ofthe kinetic energy of themas s1\n2m˙x2, the\npotential energy corresponding to the spring element1\n2kx2, and the potential\nenergy stored in the fractional derivative e(t) =1\n2/integraltext∞\n0µ1−α(ω)z2(ω,t)dω.\nThe energy regeneration and loss in vibratory motion have been ana lyzed by\nmeans of the total mechanical energy. Furthermore, based on t he mean en-\nergy dissipation of the fractional damping element in steady-state vibration,\na new concept of mean equivalent viscous damping has been suggest ed and\nthe expression of the damping coefficient has been evaluated.\n18By virtue of the total mechanical energy in SDOF fractional oscillat ors,\nit becomes possible to formulate Lyapunov functions for stability an alysis\nand control design for fractionally damped systems as well as othe r types\nof fractional dynamic systems. As for the future perspectives, our research\nefforts will be focused on fractional control design for fractiona lly damped\noscillators and structures.\nAcknowledgements\nThe author Yuan Jian expresses his thanks to Prof. Dong Kehai fr om\nNaval Aeronautical and Astronautical University, and Prof. Jian g Jianping\nfrom national University of Defense technology. All the authors a cknowledge\nthe valuable suggestions from the peer reviewers. 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Aircraft Engineering and Aeros pace\nTechnology, 2012, 84(2): 103-108\n23"    },    {        "title": "2201.05168v1.Damping_of_Alfvén_waves_in_MHD_turbulence_and_implications_for_cosmic_ray_streaming_instability_and_galactic_winds.pdf",        "content": "DRAFT VERSION JANUARY 17, 2022\nTypeset using L ATEX default style in AASTeX63\nDamping of Alfv ´en waves in MHD turbulence and implications for cosmic ray streaming instability and galactic winds\nALEX LAZARIAN1, 2AND SIYAO XU3\n1Department of Astronomy, University of Wisconsin, 475 North Charter Street, Madison, WI 53706, USA; lazarian@astro.wisc.edu\n2Centro de Investigaci ´on en Astronom ´ıa, Universidad Bernardo O’Higgins, Santiago, General Gana 1760, 8370993,Chile\n3Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA; sxu@ias.edua\nABSTRACT\nAlfv ´enic component of MHD turbulence damps Alfv ´enic waves. The consequences of this effect are impor-\ntant for many processes, from cosmic ray (CR) propagation to launching outflows and winds in galaxies and\nother magnetized systems. We discuss the differences in the damping of the streaming instability by turbulence\nand the damping of a plane parallel wave. The former takes place in the system of reference aligned with the\nlocal direction of magnetic field along which CRs stream. The latter is in the reference frame of the mean mag-\nnetic field and traditionally considered in plasma studies. We also compare the turbulent damping of streaming\ninstability with ion-neutral collisional damping, which becomes the dominant damping effect at a sufficiently\nlow ionization fraction. Numerical testing and astrophysical implications are also discussed.\n1.PROPAGATION OF ALFV ´EN WA VES IN MHD TURBULENCE\nAstrophysical media are turbulent and magnetized (see a collection of relevant reviews in Lazarian et al. 2015a). The propa-\ngation of Alfv ´en waves in turbulent magnetized media is an important astrophysical problem that influences fundamental astro-\nphysical processes (see e.g., Uhlig et al. 2012, Wiener, Oh & Guo 2013, van der Holst et al. 2014, Lynch et al. 2014). This\nreview focuses on the damping of Alfv ´en waves in MHD turbulence. The Alfv ´en waves can arise from instabilities induced by\ncosmic rays (CRs), e.g. from the streaming of CRs (Lerche 1967, Kulsrud & Pearce 1969, Wentzel 1969, Skilling 1971), and the\ngyroresonance instability related to the compression of magnetic field and CRs (see Lazarian & Beresnyak 2006). They can also\nbe generated by large scale perturbations of magnetic field (see Konigl 2009 and ref. therein, Suzuki 2013).\nTurbulent damping of Alfv ´en waves causes heating of, e.g. coronal gas in Solar atmosphere (e.g. Arber, Brady & Shelyag\n2016, Reep & Russell 2016). In the case of the streaming instability, turbulent damping suppresses its growth and affects the\nstreaming speed of CRs. As a result, turbulent damping of streaming instability is important for studies on the diffusion and\nacceleration of CRs in shocks, galaxies, and galaxy clusters (Bell 1978, Kulsrud 2005, Ensslin et al. 2011, Blasi et al. 2012,\nWiener et al. 2013, Badruddin, & Kumar, A. 2016, Xu & Lazarian 2022), stellar wind launching (e.g. Suzuki & Inutsuka 2005,\nvan Ballegooijen, & Asgari-Targhi 2016), and galaxy evolution (e.g., Hopkins et al. 2021).\nIt should be noted that the well-known study of Alfv ´en wave damping by turbulent plasmas performed by Silimon & Sudan\n(1989) employed an unrealistic model of isotropic MHD turbulence. Later, turbulent damping of Alfv ´en waves was mentioned\nas a process for suppressing CR streaming instability in Yan & Lazarian (2002, henceforth YL02). This process was quantified\nby Farmer & Goldreich (2004, henceforth FG04), where the Goldreich & Sridhar (1995; henceforth, GS95) model of Alfv ´enic\nturbulence with scale-dependent anisotropy was adopted. The limitation of the aforementioned study was that for the calculations\nit was assumed that turbulence is injected isotropically with the turbulent velocity uLexactly equal to the Alfv ´en velocityVA,\ni.e. Alfv ´en Mach number MAequal to unity. In addition, only turbulent damping of streaming instability was considered.\nFollowing the study in Lazarian (2016), we will seperately discuss the turbulent damping of Alfv ´en waves that are generated\nby streaming instability and by large-scale magnetic perturbations. We will demonstrate the strong dependence of turbulent\ndamping on MAin various turbulence regimes and astrophysical media with different levels of medium magnetization. In §2\nwe provide the derivation of the Alfv ´enic turbulent scaling. In §3 we describe the turbulent damping of Alfv ´en waves generated\nby streaming instability in the reference system aligned with the local direction of turbulent magnetic field. In §4 we discuss the\nturbuelnt damping of Alfv ´en waves induced by large-scale magnetic perturbations in a global system of reference. We compare\nthe turbulent damping with ion-neutral collisional damping of streaming instability in a partially ionized medium in §5. The\nnumerical testing of the theoretical predictions is provided in §6. The discussion of the astrophysical implications on propagation\nof CRs in galaxies and launching of winds follows in §7. The summary is given in §8.\naHubble FellowarXiv:2201.05168v1  [astro-ph.GA]  13 Jan 20222\n2.DERIV ATION OF ALFV ´ENIC TURBULENT SCALING\nIn Alfv ´enic turbulence the relative perturbations of velocities and magnetic fields are related as follows:\n\u000eBl\nB=\u000eBl\nBLBL\nB=ul\nuLMA=ul\nVA; (1)\nwhereBlis the fluctuation of the magnetic field Bat scalel,BLis the fluctuation of the magnetic field at the driving scale\nLof turbulence. Correspondingly, ulis the turbulent velocity fluctuation at the scale landuLis the turbulent velocity at L.\nMA=uL=VAis the Alfv ´en Mach number.\nOne way to understand the non-linear interactions of Alfv ´en waves within the MHD turbulent cascade is to consider colliding\nAlfv ´en wave packets with parallel scales lkand perpendicular scales l?. The collision of a wave packet induces an energy change\n\u0001E\u0018(du2\nl=dt)\u0001t; (2)\nwhere the term in brackets manifests the change of the energy of a wave packet induced by its interaction with the oppositely\nmoving Alfv ´en wave packet. The time of this interaction is equal to the time of the passage of these wave packets through each\nother. As the size of the packet is lk, the interaction time is simply \u0001t\u0018lk=VA.\nThe rate of turbulent energy cascade is related to the rate of structure change of the oppositely moving wave packet. The latter\nisul=l?. As a result, Eq. (2) provides\n\u0001E\u0018ul\u0001_ul\u0001t\u0018(u3\nl=l?)(lk=VA); (3)\nThe fractional change of packet energy taking place per collision is \u0001E=E . This characterises the strength of the nonlinear\nturbulent interaction:\nf\u0011\u0001E\nu2\nl\u0018ullk\nVAl?: (4)\nIn Eq. (4),fis the ratio of the shearing rate of the wave packet, i.e. ul=l?, to its propagation rate, i.e. VA=lk.\nOne can identify two distinct cases. If f\u001c1, the shearing rate is significantly smaller than the propagation rate, and the\ncascade presents a random walk process. Therefore\n@=f\u00002(5)\nsteps are required for the energy cascade, and therefore the cascading time is\ntcas\u0018@\u0001t: (6)\n@>1corresponds to the weak turbulent cascade. Naturally, @cannot become less than unity. Therefore, the limiting case is\n@\u00191. This is the case of strong MHD turbulence.\nTraditionally, the wavevectors are defined in the system of reference related to the mean field. However, the system of reference\nrelated to a wave packet with given parallel and perpendicular dimensions is more relevant when dealing with strong MHD\nturbulence. We take this into account by considering Alfv ´en wave packets having the dispersion relation !=VAjkkj, where we\nusekk\u0018l\u00001\nkas the component of wavevector parallel to the local background magnetic field. As the result of interaction the\nincrease ofk?\u0018l\u00001\n?occurs. In the rest of the discussion we use lkandl?that are defined in the local frame of wave packets.\nIn weak turbulence, the decrease of l?whilelkdoes not change signifies the increase of the energy change per collision. This\nforces@to be of the order of unity. In this case one gets\null\u00001\n?\u0019VAl\u00001\nk(7)\nin strong turbulence, which signifies the cascading time being equal to the wave period \u0018\u0001t. Any further decrease of l?\ninevitably results in the corresponding decrease of lkand Eq. (7) is still satisfied. The change of lkentails the increase of the\nfrequencies of interacting waves. This is compatible with the conservation of energy condition above, as the cascade introduces\nthe uncertainty in wave frequency !of the order of 1=tcas.\nThe cascade of turbulent energy satisfies the relation (Batchelor 1953):\n\u000f\u0019u2\nl=tcas=const; (8)\nwhich for the hydrodynamic cascade provides\n\u000fhydro\u0019u3\nl=l\u0019u3\nL=L=const; (9)3\nwhere the relation for the cascading time tcas\u0019l=ulis employed.\nFor the weak turbulent cascade with @\u001d 1, we have (LV99)\n\u000fw\u0019u4\nl\nV2\nA\u0001t(l?=lk)2\u0019u4\nL\nVAL; (10)\nwhere Eqs. (8) and (6) are used. The isotropic turbulence injection at scale Lresults in the second relation in Eq. (10). Taking\ninto account that for the weak turbulence lkis constant, it is easy to see that Eq. (10) provides\nul\u0018uL(l?=L)1=2; (11)\nwhich is different from the hydrodynamic \u0018l1=3scaling.1\nIt was shown in LV99 that for turbulence with isotropic injection at scale LwithVL<VAthe transition to the strong regime\ncorresponding to@\u00191happens at the scale\nltrans\u0018L(uL=VA)2\u0011LM2\nA: (12)\nAs a result, the inertial range of weak turbulence is limited, i.e. [L;LM2\nA], and atltrans the turbulence transits into the regime of\nstrong MHD turbulence. At the transition, the velocity is\nutrans\u0019VAltrans\nL\u0019VAM2\nA; (13)\nwhich follows from @\u00191condition given by Eqs. (4) and (5).\nThe scaling relations for the strong turbulence with VL< VAcan be easily obtained. The turbulence is strong and cascades\nover one wave period, which according to Eq. (7) is equal to l?=ul. Substituting the latter in Eq. (8) one gets\n\u000fs\u0019u3\ntrans\nltrans\u0019u3\nl\nl=const: (14)\nThe latter energy cascading rate is analogous to that in an ordinary hydrodynamic Kolmogorov cascade. However, this cascading\ntakes place in the direction perpendicular to the local direction of the magnetic field.2\nThis strong MHD turbulence cascade starts at ltrans and its injection velocity is given by Eq. (13). This provides another way\nto obtain the Alfv ´enic turbulent scaling in strong turbulence regime (LV99)\nul\u0019VA\u0012l?\nL\u00131=3\nM4=3\nA; (15)\nwhich can be rewritten in terms of the injection velocity uL(see Eq. (15) )\n\u000eul\u0019uL\u0012l?\nL\u00131=3\nM1=3\nA: (16)\nSubstituting this in Eq. (7) we get the relation between the parallel and perpendicular scales of the eddies (LV99):\nlk\u0019L\u0012l?\nL\u00132=3\nM\u00004=3\nA: (17)\nThe relations Eq. (17) and (15) reduce to the GS95 scaling for transAlfv ´enic turbulence if MA\u00111.\nIn the opposite case we deal with superAlfv ´enic turbulence, i.e. with uL>VA. As a result, at scales close to the injection scale\nthe turbulence is essentially hydrodynamic as the influence of magnetic forces is marginal. Therefore, the velocity is Kolmogorov\nul=uL(l=L)1=3: (18)\n1Using the relation kE(k)\u0018u2\nkit is easy to show that the energy spectrum of weak turbulence is Ek;weak \u0018k\u00002\n?(LV99, Galtier et al. 2000).\n2There is an intuitive way of presenting the Alfv ´enic cascade in terms of eddies mixing the magnetic field in the direction perpendicular to the magnetic field\nsurrounding the eddies. The existence of such magnetic eddies is possible due to the fact that, as shown in LV99, the turbulent magnetic reconnection happens\nwithin one eddy turnover. As a result, the existence of magnetic field does not constrain magnetic eddies, if they are aligned with the magnetic field in their\nvicinity, i.e. with the local magnetic field. This eddy representation of MHD turbulence vividly demonstrates the importance of the local system of reference,\nwherel?andlkare defined.4\nThe magnetic field becomes more important at smaller scales and the cascade changes its nature at the scale\nlA=LM\u00003\nA; MA>1; (19)\nat which the turbulent velocity becomes equal to the Alfv ´en velocity (Lazarian 2006). The rate of cascade for l<lAis:\n\u000fsuperA\u0019u3\nl=l\u0019M3\nAV3\nA=L=const: (20)\nUnlike the case of subAlfv ´enic turbulence, the case of superAlfv ´enic turbulence can be reduced to the case of transAlfv ´enic\nturbulence, but with lAacting as the injection scale. At scales l<lA\nlk\u0019L\u0012l?\nL\u00132=3\nM\u00004=3\nA; (21)\nul\u0019uL\u0012l?\nL\u00131=3\nM1=3\nA: (22)\nThe relations for subAlfv ´enic and superAlfv ´enic tubulence that we obtain above coincide with the expressions first obtained in\nLazarian & Vishniac (1999) using a different approach. These expressions will be used below in our discussion on turbulent\ndamping of Alfv ´en waves.\n3.TURBULENT DAMPING OF STREAMING INSTABILITY\nLinear Alfv ´en waves undergo non-linear cascading when they propagate through Alfv ´enic turbulence. This process is of MHD\nnature and the non-linear damping of Alfv ´en waves does not depend on plasma microphysics. The interaction between CR-driven\nAflven waves and turbulence is similar to that of oppositely moving wave packets of turbulent cascade.\nThe Alfv ´en waves emitted parallel to the local magnetic field experiences the least distortions from the oppositely moving\neddies. Thus the least distorted Alfv ´en waves are those with the largest value of l?. Indeed, the larger l?, the longer time it\ntakes for the evolution of the oppositely moving wave packets. For instance, for strong GS95 turbulence the time corresponds to\nl?=vl\u0018l2=3\n?.\nThe case of Alfv ´en waves parallel to the local direction of magnetic field corresponds to streaming and gyroresonance insta-\nbilities. In what follows, we will focus on the streaming instability. The dispersion of magnetic field directions with respect\nto the mean magnetic field determines the corresponding l?. Naturally, the turbulent damping of Alfv ´en waves is different for\nweak turbulence and strong turbulence. Thus we will separately discuss turbulent damping of streaming instability in different\nturbulence regimes.\n3.1. Streaming instability and local system of reference\nThe streaming instability of CRs happens as CR particles moving in one direction scatter back from a magnetic field perturba-\ntion and thus increase the amplitude of the perturbation. The induced perturbations are Alfv ´en waves. If the Alfv ´en waves are\nseverely damped, the CR particles can stream freely along the magnetic field.\nPhysically, the generation of Alfv ´en waves takes place as CRs stream along the local magnetic field. During the process the\nsampling scale for the magnetic field is the CR Larmor radius rL. In this setting one should consider the process in the system of\nreference related to the local direction3of the wondering magnetic field (LV99, Cho & Vishniac 2000, Maron & Goldreich 2001,\nCho, Lazarian & Vishniac 2002).\nIn the direction parallel to the local magnetic field, the growth rate of the streaming instability is (see Kulsrud & Pearce 1969):\n\u0000cr\u0019\nBncr(>\r)\nni\u0012vstream\nVA\u00001\u0013\n; (23)\nwhere \nB=eB=mc is the nonrelativistic gyrofrequency, ncris the number density of CRs with gyroradius rL>\u0015=\rmc2=eB,\nand\ris the Lorentz factor. If the growth rate given by Eq. (23) is less than the rate of turbulent damping, the streaming instability\nis suppressed.\n3The fact that MHD turbulence is formulated in terms of the local quantities is required for describing the interaction of MHD turbulence with CRs. Indeed,\nperturbations in the local system of reference are exactly what CRs interact with.5\n3.2. Damping by SubAlfv ´enic strong turbulence\nOur first approach is based on calculating the distortion of Alfv ´en waves by MHD turbulence as the waves propagate along\nmagnetic field. The cause of the wave distortion is the field line wandering over angle \u0012x. This angle is determined by the\namplitude of magnetic field fluctuations \u000eBxthat are induced by turbulent eddies with perpendicular scale x. One can see that\nthe distortion induced during the time tis\n\u000ex\u0019VAtsin2\u0012x\u0019VAt\u0012\u000eBx\nB\u00132\nt; (24)\nwhere the fluctuation induced by turbulence evolves as\n\u0012\u000eBx\nB\u0013\nt\u0019\u0012ux\nVA\u0013 \u0012t\nx=ux\u0013\n: (25)\nIn the above expression uxdenotes the velocity corresponding to the magnetic field fluctuation \u000eBx. The timetin Eq. (25) is\nchosen to be less than the eddy turnover time x=ux. As a result, the ratio reflects the partial sampling of the magnetic perturbation\nby the wave. By using the velocity scaling of strong subAlfv ´enic turbulence for uxin Eq. (25), it is easy to rewrite Eq. (24) as\n\u000ex\u0019V3\nAM16=3\nAt3\nx2=3L4=3: (26)\nThe wave damping corresponds to the “resonance condition” \u000ex=\u0015, where\u0015is the wavelength. Inserting this in Eq. (26) we\nobtain the perpendicular scale of the “resonance” magnetic fluctuations that distort the Alfv ´en waves:\nx\u0019V9=2\nAt9=2M8\nA\n\u00153=2L2: (27)\nThe time required to damp the Alfv ´en waves is equal to the turnover time of the “resonant” eddy:\nt\u0019x\nul\u0019V2\nAt3M4\nA\n\u0015L: (28)\nThis provides the rate of non-linear damping of the Alfv ´en waves,\n\u0000subA;s\u0019t\u00001; (29)\nor\n\u0000subA;s\u0019VAM2\nA\n\u00151=2L1=2; (30)\nwhere the subsscript “s” denotes “strong turbulence”. For transAlfv ´enic turbulence, i.e. MA= 1, this result was obtained in\nFG04. The square of the Alfv ´en Mach number dependence presented in Eq. (30) means a significant change of the damping rate\ncompared to the transAlfv ´enic case.4\nIf the injection of turbulence is isotropic, the maximal perpendicular scale of strong subAlfv ´enic motions is xmax=LM2\nA.\nSubstituting this in Eq. (27) and using Eq. (29) and Eq. (30) to express t, we get\n\u0015max;s\u0019LM4\nA: (31)\nThe streaming CRs generate Alfv ´en waves at a scale comparable to the gyroradius rL. Thus it requires that\nrL<LM4\nA; (32)\nwhich is a notable limitation on CR energy if MAis small. The CRs with larger energies interact with weak turbulence as we\nwill discuss in Section 3.3.\nDue to the importance of turbulent damping of streaming instability, it is advantageous to provide another derivation of Eq. (30).\nThis alternative derivation is based on the picture of propagating wave packets that we used while obtaining Eq. (3). Consider\n4We note that in FG04 the injection scale for turbulence was defined not as the actual injection scale, but the scale at which the turbulent velocity becomes equal\nto the Alfv ´en one. Such scale does not exist for subAlfv ´enic turbulence.6\ntwo oppositely moving Alfv ´en wave packets with the perpendicular scale x0\u0018k0\u00001\n?. As we discussed earlier, each wave packet\ninduces the distortion \u00120\nxof the oppositely moving waves. Consider an Alfv ´en wave with wavenumber k\u00001\nk\u0018\u0015moving parallel\nto the local direction of magnetic field. Such a wave is mostly distorted when interacting with turbulent perturbations with the\nperpendicular wavenumber k?\u0018kksin\u00120\nx. The interactions are most efficient if they are “resonant”, i.e. a wave with k?interacts\nwith the oppositely moving packets and k0\n?=k?.5Thus the perpendicular scale of the “resonant” wave packet is determined by\nthe relationkksin\u0012x=k?, which results in\n\u0015\u0019xsin\u0012x\u0019x\u000eBx\nB: (33)\nUsing the scaling in Eqs. (15) and (25), we derive the “resonant” perpendicular scale x:\nx=L1=4\u00153=4M\u00001\nA: (34)\nThis can be used to determine the rate of damping defined as \u0000subA;s\u0019ux=x. This coincides with the earlier result given by Eq.\n(30). Then the maximal wavelength of the non-linearly damped Alfv ´en waves can be obtained from Eq. (33) if the scale ltrans is\nused instead of x, i.e.\n\u0015max;s\u0019\u0012utrans\nVA\u0013\nltrans\u0019LM4\nA: (35)\nNaturally, the latter coincides with the result given by Eq. (31). The minimal scale of non-linearly damped waves depends on the\nperpendicular scale of the smallest Alfv ´enic eddieslmin. The full range of rLfor which turbulent damping is essential can be\nobtained by using Eq. (33) and the scaling of strong turbulence given by Eq. (15):\nl4=3\nmin\nL1=3M4=3\nA<rL<LM4\nA: (36)\nThe value of lmin depends on the particular damping process of MHD turbulent cascade, which can be relatively large in a\nweakly ionized gas (see Xu & Lazarian 2017). Due to the differences of rLfor protons and electrons, Eq. (36) presents a\npossible situation when the streaming instability of CR electrons is not damped by turbulence, while it is damped for CR protons.\nWe note that the turbulent damping of streaming instability for rL<l4=3\nmin\nL1=3M4=3\nAis still present, although it is reduced. We can\nget an estimate of it by considering the distortion \u000ex\u001c\u0015given by Eq. (26) for the time period of the wave \u0015=VA. This time\nis significantly less than the period of the eddy at the scale lmin,teddy\u0019l2=3\nminL1=3=(VAM4=3\nA). The distortions act in a random\nwalk fashion with the time step given by teddy. The damping requires \u0015=\u000exsteps, which induces the damping rate\n\u0000sub;s;r L\u001clmin\u0019M12\nAVAr4\nL\nl2\nminL3: (37)\nThe latter clearly illustrates the inefficiency of damping when turbulence has the perpendicular scale larger than the “resonant”\nscale.\n3.3. Damping by subAlfv ´enic weak turbulence\nIn many instances the weak turbulence is not important. It has a limited inertial range and transfers to strong turbulence at\nsmaller scales. However, as we show below, this may not be true for wave damping by turbulence. For wavelengths longer than\n\u0015max;s the wave is non-linearly damped through interactions with the wave packets of the weak turbulence, having perpendicular\nscales given by Eq. (33). Naturally, the scaling of weak turbulence given by Eq. (11) should be used. This provides the relation\nbetween the Alfv ´en wave wavelength and the perpendicular scale of the “resonant” weak turbulence perturbation\n\u0015=l?\u0012l?\nL\u00131=2\nMA: (38)\nThis delivers the perpendicular scale\nl?\u0019\u00152=3L1=3M\u00002=3\nA: (39)\n5Simple estimates demonstrate that the interactions with smaller and larger turbulent scales are subdominant compared with the interaction with the “resonant”\nscale.7\nAccording to Eqs (5) and (6), the weak turbulence packets cascade @times slower compared to the case of strong turbulence.\nTaking into account that the parallel scale of weak turbulence wave packets is equal to the injection scale L, we have\n@\u0019\u0012VAl?\nulL\u00132\n: (40)\nThe rate of turbulent damping of the wave is therefore\n\u0000subA;w\u0019(@\u0001t)\u00001=@\u00001VA\nL; (41)\nwhich gives\n\u0000subA;w\u0019VAM8=3\nA\n\u00152=3L1=3; (42)\nwhere the subsscript “w” denotes “weak turbulence”. Note that compared to the case of damping given by Eq. (30) we now have\na stronger dependence on MA, as well as a different scaling with the wavelength \u0015.\nThe maximal wavelength of the Alfv ´en waves that is cascaded by the weak cascade we derive by substituting l?=L, i.e.\nusing the energy injection scale in Eq. (38). This gives:\n\u0015max;w\u0019LMA: (43)\nThus for CRs that generate Alfv ´en waves, their rLshould satisfy\nLM4\nA<rL<LMA (44)\nto interact with weak Alfv ´enic turbulence. The underlying assumption here is that LM4\nAis larger than the turbulent damping\nscalelmin. Otherwise the lower boundary in Eq. (44) is determined by lmin.\nWaves with\u0015>\u0015max;w can interact with the turbulent motions at the injection scale L. The cascade of such waves is induced\nby the largest wave packets at a rate @\u00001VA\nL, i.e.\n\u0000outer\u0019@\u00001VA\nL\u0019M2\nAVA\nL; (45)\nwhich does not depend on wavelength. Physically, this means that all waves in the range LMA<\u0015<L decay at the same rate\nthat is determined by the restructuring of the magnetic field at the injection scale.\nThe above expression is valid for \u0015<L . In the case of \u0015\u001dLthe rate is reduced due to the random walk, which results in a\nfactor (L=\u0015)2, i.e.\n\u0000outer;extreme\u0019@\u00001VA\nLL2\n\u00152\u0019M2\nAVA\nLL2\n\u00152: (46)\nThe latter result is relevant for the damping induced by turbulence injected at scales smaller than the wavelength.\nIn terms of the dependence of damping rate on \u0015for subAlfv ´enic turbulence, we observe that the dependence becomes stronger\nwith the increase of \u0015up to\u0015=LMA. For\u0015less thanLM4\nA, the waves interact with strong Alfv ´enic turbulence and the damping\nrate\u0000is proportional to \u0015\u00001=2. The scaling changes for waves longer than LM4\nAbut shorter than LMA. The scaling of \u0000gets\nto\u0015\u00002=3as Alfv ´en waves interact with weak turbulence. For smaller MAthe range for which weak Alfv ´enic turbulence damps\nAlfv ´en waves increases. A further increase of the wavelength, i.e. for \u0015fromLMAtoL, introduces a flat regime of damping,\ni.e., no dependence on \u0015. The damping at this regime is determined by the evolution of turbulence at the injection scale. In its\nturn, this regime proceeds untill \u0015gets of the order of L. Finally, if\u0015is much larger than L, the damping modifies further that\nit transfers to \u0015\u00002. In that regime the Alfv ´en waves have so large \u0015that they only feel the distortions that are introduced by\nturbulence at the outer scale. In comparison, the FG04 study considered only transAlfv ´enic turbulence and provided only \u0015\u00001=2\nscaling for all scales.\nIn terms of the dependence of turbulent damping rate on MA, it changes from M2\nAfor strong turbulence to M8=3\nAfor weak\nturbulence. The case of no damping naturally follows as MA!0. As for FG04 study, it only provided the result for MA= 1.8\n3.4. Damping by SuperAlfv ´enic turbulence\nAs we discussed earlier, if turbulence is superAlfv ´enic, at large scales the effects of magnetic field are marginal and turbulence\nis hydrodynamic-like. However, the turbulent velocity decreases with the decreasing scale and at a scale lAbecomes equal to\nthe Alfv ´en velocity. This scale can be considered as the injection scale of transAlfv ´enic turbulence. Therefore, the case of\nAlfv ´en wave damping by superAlfv ´enic turbulence at scales less than lAcan be related to the case of damping by transAlfv ´enic\nturbulence considered in FG04. Indeed, a simple substitution of LbylAprovides the required rate of magnetic structure evolution\non scales less than lA. This gives:\n\u0000super\u0019VA\nl1=2\nA\u00151=2=VAM3=2\nA\nL1=2\u00151=2: (47)\nTreatinglAas the effective injection scale and using Eq. (31), it is easy to obtain the maximal wavelength up to which our\ntreatment of the non-linear damping is applicable:\n\u0015max;super\u0019lA=LM\u00003\nA: (48)\nAssociating \u0015withrL, we define the corresponding range of rL\nl4=3\nmin\nL1=3MA<rL<LM\u00003\nA; (49)\nassuming that the minimal/damping scale of turbulent motions lminis less thanLM\u00003\nA.\nFor Alfv ´en waves with \u0015larger than that given by Eq. (48) and therefore for rL> LM\u00003\nA, the damping is induced by\nKolmogorov-type isotropic hydrodynamic turbulence. The characteristic damping rate in this case is expected to coincide with\nthe eddy turnover rate, i.e.\n\u0000hydro\u0019u\u0015\n\u0015\u0019VA\nl1=3\nA\u00152=3\u0019VAMA\nL1=3\u00152=3; (50)\nwhere we use Eq. (18).\nSimilar to the case of sub-Alfv ´enic turbulence, in superAlfv ´enic case, we observe the change of the rate of Alfv ´en wave\ndamping changing from \u0015\u00001=2for short wavelengths to \u0015\u00002=3for\u0015longer thanLM\u00003\nA. The turbulent damping rate of Alfv ´en\nwaves increases with MA.\n3.5. Other forms of presenting our results\nThe scaling of weak turbulence is different from that of strong turbulence that starts at the transition scale ltrans =LM2\nAof\nsubAlfv ´enic turbulence. However, what is the same in the two regimes of turbulence is the cascading rate. Indeed, the energy\ncascades at the same rate without accumulating at any scale and dissipates only at the small dissipation scale. Therefore, by\nexpressing the dissipation rate of Alfv ´en waves through the cascading rate of turbulence, we will demonstrate a higher degree of\nuniversality of the obtained expressions.\nThe cascading rate of the weak turbulence is given by Eq. (10) and we can write it as\n\u000fw\u0019V3\nAM4\nA\nL: (51)\nThis reflects the decrease of energy dissipation by M4\nAcompared to the case of transAlfv ´enic turbulence in FG04. If rL<LM4\nA,\nthe rate of Alven wave damping can be obtained by combining Eq. (51) and Eq. (30):\n\u0000subA;s\u0019\u000f1=2\nw\nV1=2\nAr1=2\nL: (52)\nThe peculiar feature of Eq. (52) is that if one formally substitutes instead of \u000fwthe cascading rate of strong turbulence, one will\nget the expression in FG04. This is exactly the universality of expressions that we sought. Nevertheless, this analogy is only\nformal as the cascading rate for weak turbulence is M4\nAtimes lower compared to the transAlfv ´enic case. Therefore, the obtained\ndamping rate for subAlfv ´enic turbulence is M2\nAtimes less than the case of trans-Alfv ´enic turbulence (see also Eq. (30)).\nFor wavelengths in the range LM4\nA< \u0015 < LM Athe weak turbulence is responsible for the Alfv ´en wave damping. Thus,\nexpressingMAfrom Eq. (51) and substituting it in Eq. (42) we can get\n\u0000subA;w\u0019\u000f1=3\nwL1=3\nVAr2=3\nL\u0019\u000f1=3\nwM4=3\nA\nr2=3\nL: (53)9\nThe expression given by Eq. (53) demonstrates a slower damping rate in comparison to Eq. (52). The decrease of damping rate\nby the factor M8=3\nA(see Eq. (42)) is significant. It is important to note that for MA\u001c1it provides the smooth transition to the\nregime of marginal Alfv ´en wave damping as the magnetic field perturbations get smaller and smaller. Naturally, this expression\nis very different from that in FG04, as the latter study did not consider the damping induced by weak Alfv ´enic turbulence.\nFor damping of Alfv ´en waves generated by CRs with larger rL, i.e.LMA<rL<L (see Eq. (38)) we obtain:\n\u0000outer\u0019\u000f1=2\nw\nL1=2V1=2\nA: (54)\nFor superAlfv ´enic turbulence at scales less than lA, by expressing MAfrom Eq. (20) and substituting it in Eq. (47), we obtain\n\u0000super\u0019\u000f1=2\nsuper\nV1=2\nAr1=2\nL: (55)\nFormally, the above expression coincides with the expression for the damping by subAlfv ´enic strong turbulence given by Eq.\n(52). Nevertheless, the subAlfv ´enic turbulence demonstrates the significant reduction of the cascading rate compared to the\ntransAlfv ´enic turbulence. On the contrary, the superAlfv ´enic strong MHD turbulence corresponds to a significant increase of\ndissipation rate in comparison with the transAlfv ´enic case. Thus, for the same injection scale Land the same injection velocity\nVL, the damping of Alfv ´en waves depends on the magnetization of media. At a lower magnetization, e.g., for superAlfv ´enic\nturbulence, the damping of Alfv ´en waves is more efficient than that at a higher medium magnetization, i.e. for the subAlfv ´enic\ncase.\nAs we discussed earlier, in superAlfv ´enic turbulence, the long Alfv ´en waves with \u0015larger thanlA=LM\u00003\nAinteract with\nhydrodynamic turbulence and the corresponding damping rate is\n\u0000hydro\u0019\u000f1=3\nhydro\nr2=3\nL; (56)\nwhere the hydrodynamic dissipation rate is \u000fhydro\u0019V3\nL=L.\nBelow we present a few more forms of presenting the damping rates that we obtained above. For instance, it could be sometimes\nuseful to rewrite the expressions given by Eq. (30) and (42) in terms of \u0015max;s given by Eq. (35). We remind the reader that the\nphysical meaning of \u0015max;s is the longest wavelength that still interacts with strong turbulent cascade. Then,\n\u0000subA;s\u0019VA\nL\u0012\u0015max;s\nrL\u00131=2\n; rL<\u0015max;s; (57)\nand\n\u0000subA;w\u0019VA\nL\u0012\u0015max;s\nrL\u00132=3\n; rL>\u0015max;s: (58)\nIt is easy to see that Eq. (57) demonstrates that the damping by strong MHD turbulence \u0000subA;s happens faster than the Alfv ´en\ncrossing rate of the injection scale eddies. In the case of weak turbulence, Eq. (58) demonstrates that \u0000subA;w is slower than the\nabove rate.\n4.TURBULENT DAMPING OF ALFV ´EN WA VES GENERATED IN THE GLOBAL SYSTEM OF REFERENCE\nThe turbulent damping of Alfv ´en waves generated by streaming CRs is an important special case of turbulent damping as the\nstreaming instability induces Alfv ´en waves that are aligned with the local direction of magnetic field. Another case arises if we\nconsider the damping of a flux of Alfv ´en waves generated by an extended source. The difference between the two cases is that\nin the latter setting the waves are generated irrespectively to the local direction of magnetic field. Therefore, such Alfv ´en waves\nshould be viewed in the global system of reference related to the mean magnetic field. As a result, our earlier treatment of the\nAlfv ´en wave damping by MHD turbulence should be modified.\n4.1. Case of Strong SubAlfv ´enic turbulence10\nConsider an Alfv ´en wave generated at an angle \u0012\u001d\u000eB=B with respect to the global mean magnetic field. In this situation it\nis natural to disregard the dispersion of angles that arises from magnetic wandering induced by turbulence.6To distinguish these\ntwo cases we use sin\u0012instead of sin\u0012xin Eq. (33). In this case the perpendicular scale of eddies that the waves interact with is\ngiven by:\nx\u0019\u0015\nsin\u0012: (59)\nFor strong turbulence the rate of the wave damping is equal to the turnover rate of subAlfv ´enic eddies. Therefore using Eq. (59),\nwe find\n\u0000subA;s;\u0012\u0019VAM4=3\nAsin2=3\u0012\n\u00152=3L1=3: (60)\nThis provides the non-linear damping rate of an Alfv ´en wave moving at the angle \u0012with respect to the mean field.\nUsing the expression of weak turbulent cascading rate \u000fw(see Eq. (10)), one can write:\n\u0000subA;s;\u0012\u0019\u000f1=3\nwsin2=3\u0012\n\u00152=3: (61)\nThe turbulent damping given by Eq.(61) is applicable to\nlminsin\u0012<\u0015<LM2\nAsin\u0012; (62)\nwherelminis the perpendicular damping scale, and LM2\nA=ltrans is the transition scale from strong to weak MHD turbulence.\nNaturally, the adopted approximation \u0012\u001d\u000eB=B fails if the wave is launched parallel to the mean magnetic field. The\ndirections of the local magnetic field deviates from the mean field and this makes the actual \u00120different from zero. In the global\nsystem of reference the dispersion is dominated by the magnetic field variations presented at the injection scale (see Cho et al.\n2002). Therefore\n\u00120\u0019BL\nB\u0019MA: (63)\nSubstituting this into Eq. (60) we get\n\u0000subA;s; 0\u0019\u000f1=3\nwM2=3\nA\n\u00152=3: (64)\nThe above expression is different from Eqs. (30) and (52). The difference stems from the different properties of Alfv ´en waves\ngenerated in the local system versus the global system of reference. The damping rate in Eq. (64) is applicable to the range of\nwavelength\nlminMA<\u0015<LM3\nA; (65)\nthe latter result trivially follows from Eqs.(62) and (63).\n4.2. The case of Weak SubAlfv ´enic turbulence\nFor weak subAlfv ´enic turbulence, in the case \u0012\u001d\u000eB=B , one should use Eq. (59) to relate \u0015to the scale of perpendicular\nmotions that the wave strongly non-linearly interacts with. To obtain the damping rate, Eq. (41) should be used:\n\u0000weak;global;\u0012\u0019VAsin\u0012M2\nA\n\u0015\u0019\u000f1=2L1=2sin\u0012\nV3=2\nA\u0015; (66)\nwhere we use the weak cascading rate \u000fw. The range of wavelength for this type of damping is\nLM2\nAsin\u0012<\u0015<LM Asin\u0012: (67)\nThe last inequality is obtained by substituting the maximal perpendicular eddy scale LMAforxin Eq.(59).\nIn the case of Alfv ´en wave propagation along the mean magnetic field, one should use Eq. (63) to get\n\u0000weak;global; 0\u0019VAM3\nA\n\u0015\u0019VA\u000f3=4L3=4\n\u0015V5=4\nA: (68)\nUsing Eqs.(63) and (67), we find the range of wavelength that is subject to the turbulent damping:\nLM3\nA<\u0015<LM2\nA: (69)\n6In the case\u0012\u0018\u000eB=B , one should average the final expressions over the \u0012dispersion that arises from magnetic field wandering.11\n4.3. Other cases\nAfter illustrating the difference of non-linear damping for waves generated in the local reference system of magnetic field and\nin the global reference system of the mean field, we can provide results for other cases. More detailed discussion was presented\nin Lazarian (2016). For instance, for superAlfv ´enic turbulence, there is\n\u0000super;global;\u0012\u0019VAMAsin2=3\u0012\n\u00152=3L1=3; (70)\nwhere in superAlfv ´enic turbulence angle \u0012varies from one turbulent eddy of size lAto another. As a result, the correspond-\ning averaging over such changing directions should be performed. For the random distribution of the relevant directions, the\ncorresponding geometric factor is hsin2=3\u0012i= 3=5.\nOn scales larger than lA, MHD turbulence is marginally affected by magnetic fields. As a result, no difference between local\nand global frames is present in terms of Alfv ´en wave damping. This difference also disappears for the damping by turbulent\nfluctuations at the injection scale.\n4.4. Summary of main results in Sections 3 and 4 on turbulent damping\nSome of our results for non-linear turbulent damping of Alfv ´en waves in different turbulence regimes are summarized in Table\n1. We show results relevant both to the damping of waves in the local system of reference, e.g. corresponding to the waves\ngenerated by streaming instability (fifth column in Table 1 with the name “Instability damping rate”), and the damping of waves\ngenerated by external sources parallel to the mean magnetic field (sixth column in Table 1 with the name “Wave damping rate).\nThe table illustrates that the rate of damping and the ranges of wavelengths for which damping is applicable are very different\nfor the two situations. At the first glance, this seems strange. However, the difference stems from the fact that in the case of\nstreaming instability the waves are aligned with the local magnetic field, while the waves generated by an extended source are\nsent parallel to the mean magnetic field.\nWe did not cover in Table 1 the general case of Alfv ´en waves generated at an arbitrary angle relative to the mean magnetic\nfield, as well as damping of Alfv ´en waves by outer-scale turbulence. It is also necessary to stress again the important role of\nweak turbulence for the suppression of streaming instability at low MA. While the weak turbulence is frequently disregarded due\nto its limited inertial range [LM2\nA;L], it can affect CR streaming for rLin the range [LM4\nA;LMA], which can be extensive for\nsufficiently small MA.\n5.ION-NEUTRAL COLLISIONAL DAMPING OF STREAMING INSTABILITY\nIn the presence of partial ionization, an additional effect of damping by ion-neutral collisions becomes important. This effect\nwas discussed originally by Kulsrud & Pearce (1965) for Alfv ´en waves. The damping of turbulent motions in partially ionized\ngas was recently summarized in Xu & Lazarian (2017).\nIn the presence of neutrals, a slippage between them and ions induces the dissipation. In a mostly neutral medium, at wave\nfrequencies !=VAkkless than the neutral-ion collisional frequency \u0017ni, both species move together and the dissipation is\nminimal. As the wave frequency increases, not all neutrals get the chance to collide with ions and the relative motions of ions\nand neutrals induce significant dissipation. For strongly coupled ions and neutrals, the ion-neutral collisional (IN) damping rate\nis (Kulsrud & Pearce 1969)\n\u0000IN=\u0018nV2\nAk2\nk\n2\u0017ni; (71)12\nwhere\u0018n=\u001an=\u001a, and\u001anand\u001aare the neutral and total mass densities. For weakly coupled ions and neutrals with !=VAikk>\n\u0017in, whereVAiis the Alfv ´en speed in ions and \u0017inis the ion-neutral collisional frequency, there is\n\u0000IN=\u0017in\n2: (72)\nWe note that both turbulent and wave motions are subject to the IN damping. Strong Alfv ´enic turbulence injected in the strong\ncoupling regime cannot cascade into the weak coupling regime due to the severe damping effect (Xu et al. 2015, 2016).\nIN damping is sensitive to the ionization fraction and becomes weak at a high ionization fraction. For strongly coupled ions\nand neutrals with VAkk<\u0017in,\u0000INis still given by Eq. (71). For decoupled ions with VAikk>\u0017in, there is (Xu et al. 2016)\n\u0000IN=\u0017ni\u001fV2\nAik2\nk\n2\u0002\n(1 +\u001f)2\u00172\nni+V2\nAik2\nk\u0003; (73)\nwhere\u001f=\u001an=\u001aiand\u001aiis the ion mass density. Furthermore, when neutrals are also decoupled from ions with VAikk> \u0017ni,\nthe above expression is reduced to Eq. (72). Because of the weak damping effect, Alfv ´enic cascade in a highly ionized medium\nis not dissipated by IN damping (Xu & Lazarian 2022).\nNaturally, to understand whether turbulent damping or IN damping is more important for damping the streaming instability,\n\u0000INshould be compared with the turbulent damping rate \u0000that we provided earlier. This comparison has been recently carried\nout in detail by Xu & Lazarian (2022). Here we selectively review some of their results.\nIn a weakly ionized interstellar medium, e.g., molecular clouds, CR-driven Alfv ´en waves are likely in the weak coupling regime\nwith\nVAi\nrL\u0017in\u00192\u0002103\u0010B0\n10\u0016G\u00112\u0010nH\n100cm\u00003\u0011\u00003\n2\u0010ne=nH\n10\u00004\u0011\u00001\n2\u0010ECR\n1GeV\u0011\u00001\n\u001d1; (74)\nwhereB0is the mean magnetic field strength, neandnHare number densities of electrons and atomic hydrogen, and ECRis\nthe CR energy. As already mentioned above, strong Alfv ´enic turbulence injected at a large scale in the strong coupling regime is\nseverely damped and its cascade cannot persist in the weak coupling regime. Therefore, there is\n\u0000<\u0000IN=\u0017in\n2: (75)\nSo the damping of streaming instability in a weakly ionized medium is dominated by IN damping.\nIn a highly ionized interstellar medium, e.g., the warm ionized medium, CR-generated Alfv ´en waves are still in the weak\ncoupling regime and have\nVAi\nrL\u0017ni= 7:6\u0002103\u0010B0\n1\u0016G\u00112\u0010ni\n0:1cm\u00003\u0011\u00003\n2\u0010ECR\n1GeV\u0011\u00001\n\u001d1: (76)\nTo have the turbulent damping dominate over IN damping, there should be\n\u0000\n\u0000IN=\u0000\n\u0017in\n2>1; (77)\nwhich can be rewritten as\nMA>\u0010\u0017in\n2V\u00001\nAiL1\n2r1\n2\nL\u00112\n3\n= 0:2\u0010B0\n1\u0016G\u0011\u00001\u0010ni\n0:1cm\u00003\u00111\n3\u0010nn\n0:01cm\u00003\u00112\n3\u0010L\n100pc\u00111\n3\u0010ECR\n1GeV\u00111\n3(78)\nfor superAlfv ´enic turbulence, where niandnnare the number densities of ions and neutrals, and\nMA>\u0010\u0017in\n2V\u00001\nAiL1\n2r1\n2\nL\u00111\n2\n= 0:3\u0010B0\n1\u0016G\u0011\u00003\n4\u0010ni\n0:1cm\u00003\u00111\n4\u0010nn\n0:01cm\u00003\u00111\n2\u0010L\n100pc\u00111\n4\u0010ECR\n1GeV\u00111\n4(79)\nfor subAlfv ´enic turbulence. We see that the condition in Eq. (78) is naturally satisfied for superAlfv ´enic turbulence. In a highly\nionized medium, as the IN damping is weak, streaming instability is predominantly damped by the turbulent damping.13\nFigure 1. The damping time-scale \u0000\u00001of Alfv ´en waves that are injected at kk= 10 in 3D MHD turbulence, where the parallel direction is\nchosen with respect to the mean magnetic field. In one approach the Alfv ´en wave energy Ewdecays in the turbulent medium over the time scale\n\u001c1= ln(E(t1)=E(t2))=(t2\u0000t1). The values of \u001c1are given by triangular symbols. In the other approach the wave energy is continuously\ninjected atkk= 10 until it reaches a saturation level Ew. The corresponding damping time scale is given by \u001c2=Ew=\u000fdriving , where\u000fdriving\nis the wave energy injection rate. \u001c2is denoted by diamond symbols. The two measurements are both consistent with k\u00002=3scaling. From Cho\n& Lazarian 2022.14\n6.NUMERICAL TESTING OF TURBULENT DAMPING OF ALFV ´EN WA VES\nNumerical testing of Lazarian (2016) is essential in a variety of regimes. By using 3D MHD turbulence simulations (Cho et\nal. 2002), the results of numerical testing on turbulent damping of externally driven Alfv ´en waves are presented in Figure 1. The\nobserved scaling is consistent with Lazarian (2016) predictions, but inconsistent with FG04 prediction.\nThe reason for this difference arises from the global reference frame adopted in the numerical experiment. Launching of Alfv ´en\nwaves with respect to the local direction of magnetic field is complicated in turbulent fluid. Therefore, the testing presented in\nFigure 1 was carried out with Alfv ´en waves launched with respect to the mean magnetic field. This is the setting corresponding\nto turbulent damping of Alfv ´en waves generated in the global system of reference that we considered in §4. As a result, the\nnumerical simulations confirmed the scaling of inverse of damping rate \u0000\u00001, i.e., damping time scale, which is measured at\ndifferent\u0015as\u00152=3\u0018k\u00002=3\nk. This result is different from the prediction of \u0000\u00001\u0018k\u00001=2\nkof streaming instability in FG04 for\ntransAlfv ´enic turbulence and in Lazarian (2016) for the strong Alfv ´enic turbulence part of the cascade for a wide range of MA.\nNumerical testing on turbulent damping of streaming instability in the local reference frame requires a more complicated setup\nand has not been performed so far.\n7.ASTROPHYSICAL IMPLICATIONS\n7.1. Propagation of CRs\nFor decades the study on CR propagation was performed within a simple model, the so-called “leaky box model” (see Longair\n2011). In this model Galactic CRs propagate freely within the partially ionized disk of the Galaxy. The Alfv ´en waves experience\ndamping in the partially ionized gas (Kulsrud & Pearce 1969, Lithwick & Goldreich 2001, Xu et al. 2016, 2017) and thus the\nstreaming instability is suppressed. On the contrary, in fully ionized plasmas of the Galactic halo, the damping of Alfv ´en waves is\nsignificantly reduced and the streaming instability is present. Therefore, in this classical simplistic picture that ignores turbulence,\nGalactic CRs stream freely through the Galactic disk and are scattered backwards in the Galactic halo.\nThis classical “leaky box model” is problematic, as it is well known now that the Galactic disk is not fully filled with partially\nionized gas. In fact, a significant fraction of the Galactic disk material is warm ionized gas (McKee & Ostriker 1977, Draine\n2011). Therefore, CRs cannot zoom through the Galactic disk due to the streaming instability.\nFG04 quantified the idea of turbulent damping of streaming instability mentioned in Yan & Lazarian (2002) and came to a para-\ndoxical conclusion by applying their theory to the propagation of CRs in the Galaxy. By assuming homogeneous transAlfv ´enic\nturbulence in the Galaxy, they found significant turbulent damping of streaming instability and thus poor confinement of CRs.\nThis would entail problems with explaining e.g., the observed isotropy of CRs and their residence time in the Galaxy.\nIn Lazarian (2016) the gist of the “leaky box model” was preserved, but instead of damping by ion-neutral collisional friction,\nthe study appealed to the turbulent damping of streaming instability in the Galactic disk and proposed a “turbulent leaky box\nmodel”. Different from FG04, by considering inhomogeneous turbulence properties in the Galaxy and the strong MAdependence\nof turbulent damping, they found that the damping by weak subAlfv ´enic turbulence is marginal in the Galactic halo and thus CRs,\neven at high energies, can still be confined by streaming instability.\nIn a recent study by Xu & Lazarian (2022), they identified the important role of turbulent damping of streaming instability in\nthe warm ionized medium (WIM). Fig. 2 shows the diffusion coefficient Dof streaming CRs. The MAdependence comes from\nboth turbulent damping of streaming instability and wandering of turbulent magnetic field lines. In particular, the smaller Din\nsuperAlfv ´enic turbulence is caused by the tangling of turbulent magnetic fields, which results in an effective mean free path lA\nof the CRs streaming along turbulent magnetic fields (Brunetti & Lazarian 2007).\nTheMA-dependent diffusion of CRs is important for a realistic modeling of inhomogeneous CR diffusion in the Galaxy (Xu\n2021). The actual values of MAin the Galaxy can be measured from observations using a newly developed gradient technique\n(Lazarian et al. 2018, see also Xu & Hu 2021) or with more traditional magnetic field and turbulent velocity measurements.\n7.2. Launching of winds and heating\nWhile the damping of Alfv ´en waves by turbulence is an accepted process in the field of CR research, we would like to point\nout that the turbulent damping of Alfv ´en waves can be responsible for many fundamental astrophysical processes. For instance,\ndifferent processes of damping were discussed for heating of stellar corona by Alfv ´en waves, as well as for launching of stellar\nwinds (see Suzuki & Inutsuka 2005, Verdini et al. 2005, Evans et al. 2009, Vidotto & Jatenco-Pereira 2010, Verdini et al. 2010,\nSuzuki 2015). It is clear that the turbulent damping of Alfv ´en waves can be very important in these settings. More recently,\nlaunching galactic winds by turbulent damping of the Alfv ´en waves generated by galactic activity was considered in Suzuki &\nLazarian (2017). Accounting for the dependence of turbulent damping on MAis important for the quantitative modeling of the\nprocess. A similar process is important for launching winds from other types of active disk systems, e.g. circumstellar disks.15\n100101102ECR [GeV]1026102710281029D [cm2 s-1]MA = 0.7MA = 1.4 ECR1.1\nFigure 2. Diffusion coefficient vs. ECRof streaming CRs in super and subAlfv ´enic turbulence in the WIM. From Xu & Lazarian (2022).\nApart from launching galactic winds by turbulent damping of Alfv ´en waves generated by the galaxy, the turbulent damping\nof streaming instability also plays a very important role in coupling CRs and magnetized galactic matter. The pressure of CRs\nin galactic settings is significant and it can modify interstellar dynamics. Galactic winds driven by CRs present an important\nexample of this modification.\nIn general, the importance of galactic winds is easy to understand. For galaxies of the Milky Way luminosity, about 20 percent\nof baryons are accounted for when matching the observed luminosity to the halo mass function. Observing absorption lines in\nspectra of background quasars testifies for the efficient expulsion of galactic baryons from the galaxies. In fact there is evidence\nthat galaxies with significant star formation can drive mass outflows up to 10 times the rate of star formation (Brand-Hawthorn\net al. 2007).\nNumerical simulations have demonstrated that CRs indeed influence the generation of global outflows and the local structure of\nthe interstellar medium (ISM) (see Ruszkowski et al. 2017). The exact properties of the simulated outflows depend sensitively on\nhow CR transport is modeled. Recent simulations by Holguin et al. (2019) employed Lazarian (2016) model of turbulent damping\nand obtained the results that differ significantly from the earlier modeling in e.g., Ruszkowski et al. (2017). The difference\nstemmed from the fact that the earlier calculations employed the model by FG04, which is only applicable to transAlfv ´enic\nturbulence, i.e. MA= 1. However, the actual MAof gas can vary significantly in simulations.\nThe results of the numerical simulations in Holguin et al. (2019) are presented in Figure 3. Some of the implications include,\nfist of all, when turbulent damping of CR streaming instability is included, there is an increase of star formation rate, and the\nincrease is more significant at a higher level of turbulence. The reason is that the turbulent damping increases the average CR\nstreaming speed. This allows CRs to leave the dense mid-plane, reducing the pressure support from CRs to the gas. As a result,\nthe gas in the disk collapses and stars form more efficiently. Furthermore, the higher efficiency of star formation results in more\nCRs produced in the mid-plane. The increased streaming speed of CRs leads to a more extended CR distribution away from the\nmid-plane. It is also important that the escape of CRs from the dense regions allows them to interact with lower-density gas. This\nwidens the gas distribution in height and accelerates the gas to form CR-driven galactic winds.\nIn addition, the theory of Alfv ´en wave damping by turbulence suggests that Alfv ´en waves can propagate across longer distances\nin highly magnetized regions of solar atmosphere (small MA) compared to the regions with higher MA. This prediction can be\nobservationally tested. This effect should be accounted in both modelling of solar wind launching and modelling of plasma\nheating. For instance, it is likely that the turbulent damping can be important in order to explain the observed “unexpected”\ndamping of Alfv ´en waves in the regions above the Sun’s polar coronal holes (Hahn et al. 2012).\n8.SUMMARY\nAlfv ´en waves are damped in turbulent media and the damping depends on the Alfv ´en Mach number MAof the turbulence. At\nthe same wavelength, the wave damping depends on whether the waves are generated in the local reference system of magnetic\neddies by the CR streaming or they are injected at an angle relative to the large-scale mean magnetic field from an extended\nastrophysical source. The latter is, e.g., the case of the Alfv ´en waves arising from magnetic reconnection, or oscillations in16\nni[ cm-3]\nncr[ cm-3]5\n-505\n0\n-5-1-111-11-11\nFigure 3. Simulations of the galactic ISM evolution in the presence of star formation and CR driven outflows. The figure shows the gas ( ni)\nand CR (ncr) density slices \u00065kpc alongzdirection perpendicular to the midplane obtained in two simulations over time 200 Myr. The\nCR streaming is affected by turbulent damping of streaming instability with the turbulent velocity \u001b= 10 km/s. The results obtained in the\nabsence of turbulent damping on the left side of each pair of plots are clearly different from those with turbulent damping on the right side. The\ndistribution of both gas and CRs is more extended in the presence of turbulent damping. From Hoguin et al. (2019).\naccretion disks and stellar atmospheres. The difference in their damping rates arises from the difference between the local and\nglobal systems of reference where the Alfv ´en waves are generated.\nThe dependence of damping rate on the wavelength \u0015of the Alfv ´en waves in the local system of reference is \u0015\u00001=2, as opposed\nto a stronger dependence \u0015\u00002=3for the waves in the global reference system.\nThe turbulent damping also depends on whether Alfv ´en waves interact with weak or strong Alfv ´enic turbulence. For MA<1,\nthe turbulence from the injection scale Lto the scaleLM2\nAis weak and is strong at smaller scales. Weak turbulence can play an\nimportant role in turbulent damping of streaming instability driven by high-energy CRs at a small MA.\nIn a partially ionized gas, the turbulent damping still dominates the damping of streaming instability when the ionization\nfraction is sufficiently high, e.g., in the warm ionized medium (Xu & Lazarian 2022). In star burst galaxies, the ionization\nfraction is low and the ion-neutral collisional damping can be more important (e.g., Krumholz et al. 2020).\nThe turbulent damping of streaming instability has important implications on propagation of CRs in the Galaxy, star formation,\ncoupling between CRs and magnetized gas and thus driving galactic winds. In addition, the turbulent damping of Alfv ´en waves\nresults in heating of the medium and transfer of the momentum from Alfv ´enic flux to the medium. The latter is also important\nfor launching winds.17\nACKNOWLEDGMENTS\nThe research is supported by NASA TCAN 144AAG1967. 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By changing the relative phase between th e two frequency components of the\nbichromatic laser field, an ultrarelativistic electron bun ch colliding head-on with the laser pulse can\nbe deflected in a controlled way, with the deflection angle bei ng independent of the initial electron\nenergy. The effect is predicted to be observable with laser po wers and intensities close to those of\ncurrent state-of-the-art petawatt laser systems.\nPACS numbers: 41.20.-q, 41.60.-m, 41.75.Ht, 41.75.Jv\nI. INTRODUCTION\nThe rapid progress of high-power laser systems has\npavedthe wayforthe investigationofunexploredregimes\nof laser-matter interaction with a number of appli-\ncations, e.g., in extreme field physics [ 1,2], nuclear\nphysics [ 3], hadron-therapy [ 4,5] and relativistic labo-\nratory astrophysics [ 6]. Next-generation 10-PW optical\nlaser systems are expected to achieve intensities beyond\n1023W/cm2[2,7], and laser pulses with power beyond\n100 PW and intensity up to 1025W/cm2are envisaged\nat the Extreme Light Infrastructure (ELI) [ 8] and at the\neXawatt Center for Extreme Light Studies (XCELS) [ 9].\nAt such ultrahigh intensities, an electron becomes rela-\ntivistic in a fraction of the laser period and its dynamics\nis dominated by radiation reaction (RR) effects, i.e., by\nthe back reaction on the electron’s motion of the radia-\ntion emitted by the electron itself while being accelerated\nby the laser pulse [ 10]. Hence, a deep understanding of\nRR effects is crucial for the design and the interpretation\nof future laser-matter experiments in the ultrarelativis-\ntic regime. Indeed, RR effects have several important\nimplications ranging from the generation of high-energy\nphoton [11–13], electron [ 14–16] and ion [ 17–21] beams,\nto the determination of bounds on particle acceleration\nin relativistic astrophysics [ 22,23].\nAt available and upcoming laser intensities, RR effects\nbecome large for ultrarelativistic electrons, where the\nRR force basically amounts to a strongly nonlinear and\nanisotropic friction-like force [ 19]. This explains why all\nthe proposalsto experimentallytest the underlyingequa-\ntion of motion [the so called Landau-Lifshitz (LL) [ 10]\nequation] rely on the RR-driven damping of the elec-\ntron motion when an ultrarelativistic electron beam col-\nlides head-on with an intense laser pulse [ 11,24–27].\nHowever, the research to date has focused on revealing\nRR effects and understanding their fundamental features\n∗matteo.tamburini@mpi-hd.mpg.derather than exploiting them in a possibly beneficial and\ncontrolled way.\nIn this paper, we show that RR effects can provide a\nroute to the control ofthe electron dynamics via the non-\nlinear interplay between the Lorentz and the RR force.\nThis is achievedin a setup where an ultrarelativistic elec-\ntron is exposed to a strongeither few-cycle [ 28] or bichro-\nmatic [29] laser pulse. Our exact analytical calculations\nfor a plane-wave pulse and our more realistic numerical\nsimulationsforafocusedlaserpulseshowthat, alreadyat\nthe intensities achievable with state-of-the-art laser sys-\ntems, an ultrarelativistic electron colliding head-on with\na bichromatic laser pulse can be deflected in an ultrafast\nand controlled way within a cone of about 8◦aperture\nindependently of the initial electron energy as long as\nquantum effects remain small. At still higher intensities,\nthe interplay between the RR and the Lorentz force can\neven overcome the radiation losses themselves, resulting\ninaRRassistedelectronaccelerationinsteadofdamping.\nII. ELECTRON DYNAMICS IN AN\nARBITRARY PLANE-WAVE FIELD\nThe LL equation of an electron (mass mand charge e)\nin the presence of an external electromagnetic field Fµν,\nis [10]:\nduµ\ndτ=−Fµνuν+rR/bracketleftbig\nFµνFναuα−(FβνuβFναuα)uµ/bracketrightbig\n,\n(1)\nwhereτis the proper time, uµ≡dxµ/dτand where\nrR= 4πe2/3mc2λ≈1.18×10−8/λµm, withλbeing\na typical length scale, conveniently chosen as the wave-\nlength of a Ti:sapphire laser, i.e., λ= 0.8µm. In Eq. ( 1)\ndimensionless units have been employed, such that time\nis in units of ω−1≡λ/2πc, length is in units of ω−1c,\nand fields are in units of E∗≡mωc/|e|. Note that the\nterm of the RR force containing the derivatives of the\nfield tensor Fµν[10] has been neglected in Eq. ( 1) since\nits contribution is smaller than quantum effects [ 19] and2\nit does not appreciably influence the electron dynamics\nin the regime of interest here.\nModeling the laser pulse as a plane wave propagating\nalong the direction /vector n, the LL equation can be solved ex-\nactly for any plane-wave electromagnetic field which is\nanarbitrary function of the phase of the wave ϕ=nµxµ\nonly, where nµ≡(1,/vector n)andnµnµ= 0[30]. Hereafter, the\nsubscripts0and frefertotheinitialandfinalvalueofthe\ncorresponding quantity, respectively. In order to analyze\nthe origin of each term in the solution, we first omit the\nlasttermontheright-handsideofEq.( 1)(Larmorterm).\nIn this case dτ/dϕ= 1/ρ0, whereρ0≡nµuµ\n0is the initial\nDoppler factor and uµ\n0is the initial four-velocity. In-\nclusion of the Larmor term renders the relation between\nthe proper time and the phase nonlinear [ 30]:dτ/dϕ=\nh(ϕ)/ρ0whereh(ϕ) = 1+rRρ0/integraltextϕ\nϕ0[/vectorE(φ)×/vectorB(φ)]·/vector ndφ,\nwith/vectorE(φ) and/vectorB(φ) being the plane wave electric and\nmagnetic field, respectively. For an arbitrary plane wave,\nEq. (1) written as a function of the phase ϕbecomes:\nd˜uµ\ndϕ=−h\nρ0Fµν˜uν+h\nρ2\n0dh\ndϕnµ, (2)\nwhere ˜uµ≡dxµ/dϕ. InEq.( 2), theonlyeffectoftheLar-\nmor term is to multiply the terms in the right-hand side\nbyh(ϕ). SincenµFµν= 0, the solution of Eq. ( 2) is the\nsum of the two solutions obtained considering each term\non the right-hand side of Eq. ( 2) separately. The second\nterm in Eq. ( 2) results in a contribution proportional to\n(h2−1)nµ. This term accounts for the effect of the radi-\nation pressure [ 10] and leads to a small energy gain when\nan electron at rest is swept by a laser pulse [ 31,32]. Fi-\nnally, the first term on the right-hand side of Eq. ( 2) is\na strongly nonlinear effective Lorentz force. This term\ncan be integrated analytically, and the exact solution of\nEq. (2) for the dimensionless four-momentum pµ= (ε,/vector p)\nas a function of ϕis [30]:\nε=ε0\nh+2/vectorI ·/vector p0+(h2−1)+/vectorI2\n2ρ0h, (3)\n/vector p=/vector p0+/vectorI\nh+2/vectorI ·/vector p0+(h2−1)+/vectorI2\n2ρ0h/vector n,(4)\nwhere/vectorI(ϕ) =−/integraltextϕ\nϕ0h(φ)/vectorE(φ)dφ. Since /vectorE·/vector n= 0, in\nEq. (4) the vectors directed along /vector nand/vectorIcorrespond to\nthe longitudinal and transverse momentum gain, respec-\ntively.\nLet us consider a bichromatic plane-wave pulse propa-\ngatingalongthepositive z-axisandpolarizedalongthe x-\naxis with Ex(ϕ) =g(ϕ)[ξ1sin(ϕ+θ1)+ξ2sin(2ϕ+θ2)],\nwhereg(ϕ) is a smooth temporal envelope identically\nvanishing for ϕoutside the interval ( ϕ0,ϕf),ξ1,ξ2are\nthe field amplitudes of each frequency component, and\nθ1,θ2are two constant initial phases. After the electron\npasses through the laser beam, the relevant functions\nin the electron four-momentum are: hf= 1 +rRρ0Ψ,\nIy,f= 0 and Ix,f=−rRρ0∆, where Ψ ≡/integraltextϕf\nϕ0dφE2\nx(φ),∆≡/integraltextϕf\nϕ0dφEx(φ)/integraltextφ\nϕ0dϑE2\nx(ϑ). For simplicity, in the\nfollowing we assume a pulse envelope g(ϕ) = sin2(ϕ/2N)\nin the interval (0 ,ϕf) (i.e.ϕ0= 0), where N=ϕf/2πis\nthe total integer number of cycles of the pulse.\nFor the sake of comparison, we first consider a quasi-\nmonochromatic plane wave ( ξ2= 0 and N≫1). In\nthis case, two frequencies are basically present in E2\nx(ϕ),\nwhich arise from sin2(ϕ+θ1). After integrating E2\nx(ϕ),\nonly the zero-frequency component provides a net con-\ntribution to Ψ. Analogously, the integrand of ∆ only\ncontains frequencies which are odd multiples of the cen-\ntral frequency ω, and ∆ averages out to zero for a quasi-\nmonochromatic plane wave. In fact, in our case\n∆ =3πξ3\n1Ncos(θ1)\n16(N2−1)(N≥4), (5)\nwhich tends to zero for N→ ∞. The situation is essen-\ntially different for the bichromatic plane wave considered\nabove. Here, a zero-frequency term arises in the inte-\ngrand of ∆, such that ∆ diverges in the limit N≫1:\n∆≈15π\n64ξ2\n1ξ2Ncos(θ2−2θ1) (N≫1).(6)\nRecalling that Ix,f=−rRρ0∆, Eqs. ( 4) and6) al-\nready show in general that the electron dynamics can be\ncontrolled either by changing the constant initial phase\n(θ2−2θ1) or the field amplitudes ξ1, ξ2, and the effect\ndramatically increases for increasing ξ1, ξ2, N. Indeed,\nforN≫1 a different pulse envelope g(ϕ) only alters the\nnumerical factor on the right side of Eq. ( 6). Finally, we\nmention that Ix,fcan become large also for ultraintense\nnearly one-cycle laser pulses [ 28]. However, in this case\nIx,fis sensitive both to the carrierenvelope phase θ1and\nto the precise shape of the pulse g(ϕ).\nPhysically, without RR the electron transverse mo-\nmentum /vector p⊥(ϕ) =/vector p(ϕ)−[/vector n·/vector p(ϕ)]/vector noscillates with the\nsame frequencies as the plane-wave field [see Eq. ( 4) with\nh(ϕ) = 1]. Hence, the cumulative effect of the force\neventually averages out to zero. However, the energy\nloss associated with the RR force modulates the posi-\ntion of the electron within the plane-wave field. For a\nquasi-monochromatic plane wave, there is no control on\nthis modulation and thus no net transverse momentum\ngain [24,27], as the modulation is intrinsically related\nto the frequency of the driving field. On the contrary,\nif a higher-frequency field is also included, its frequency\nand absolute phase can be chosen in such a way that\na Fourier component is nonlinearly generated in the re-\nsulting modulation, which resonantly oscillates with the\nlower-frequency field. In turn, this resonance can result\nin a net transverse momentum gain δpx=Ix,f/hf, and\nthe interplay of the two components of the bichromatic\nfield is indeed reflected in Eq. ( 6).3\nIII. ELECTRON DYNAMICS CONTROL\nLet us consider the effects arising from the interac-\ntion of an ultrarelativisticelectron collidinghead-on with\na second-harmonic enriched laser pulse. Hereafter, the\nterm (h2−1) in the numerator of Eqs. ( 3) and (4) is ne-\nglected in the analytical results, since it does not appre-\nciably affect our conclusions. Eq. ( 4) indicates that the\ninitially counterpropagating electron is deflected in the\nxz-plane asymmetrically. Since ρ0≈2|/vector p0|, the deflection\nangle with respect to the initial propagation direction is\nζ≈ −arctan/parenleftbigg2rR∆\n1−r2\nR∆2/parenrightbigg\n(7)\nifrR|∆|<1,ζ+πifrR∆<−1 andζ−πifrR∆>1\nindependently of the initial electron energy. Again in\nthe ultrarelativistic regime, for rR|∆|>1 the electron is\nback reflected by the plane-wave pulse. We stress that\nthisconditionisindependentoftheinitialelectronenergy\nbecause higher initial energies imply higher RR effects,\nthe functions hfand/vectorIfbeing proportional to the initial\nDopplerfactor ρ0. In otherwords,for rR|∆|>1the laser\npulsebehaveslikeaperfectlyreflectingelectron“mirror”,\ni.e., it reflects back all the electrons with arbitrarily high\ninitialenergy,aslongastheonsetofquantumeffectsdoes\nnot severely alter the predictions of classical electrody-\nnamics (see below). In addition, from Eq. ( 3) it follows\nthat if the initial electron energy ε0is less than rR∆2/2Ψ\nthen a surprising circumstance occurs: the final electron\nenergy is largerthan its initial energy. In fact, although\nthe direct effect of the RR force is to reduce the elec-\ntron energy, it also alters the temporal electron evolu-\ntion, such that the electron’s world line with RR effects\ndiffers from the electron’s world line without them. As\na result, while without RR effects the Lorentz force can-\nnot perform a net work on the electron [see Eq. ( 3) with\nh(ϕ) = 1], with RR effects the Lorentz force can per-\nform a positive work along the RR-altered electron world\nline. Hence, the dissipative RR force indirectly allows\nthe Lorentz force to accelerate the electron, and when\nε0< rR∆2/2Ψ the indirect energy gain is larger than\nthe direct energy loss. In order to observe this effect, an\nintensity beyond 1023W/cm2and a waist radius of the\norder of some tens of micrometers are required, resulting\nin a power of the order of a few exawatts. Although such\npowers are well beyond those currently available, they\nmay be achieved employing coherent beam superposition\ntechniques [ 9,33,34].\nIV. NUMERICAL RESULTS FOR A FOCUSED\nLASER PULSE\nThe above analytical predictions are exact if the laser\nfield is modeled as a plane wave. In order to test them in\na more realistic set-up, we solve Eq. ( 1) numerically for\na focused laser pulse interacting with an electron bunch.\n−20−1001020px/mc\n03ne(a)\n−10010px/mc\n05ne(b)\n−190−170−150\npz/mc−20−1001020px/mc\n03ne(c)\n−110−90−70\npz/mc−10010px/mc\n05ne(d)\nFIG.1. (Color online)Electron densitydistribution ne(pz,px)\nas a function of the longitudinal pzand transverse pxmomen-\ntum after the interaction of 400 electrons with a bichromati c\nlaser pulse. Panel (a): cos( θ2) = 0 without RR. Panel (b):\ncos(θ2) = 0 with RR. Panel (c): cos( θ2) = 1 without RR.\nPanel (d): cos( θ2) = 1 with RR. See the text for further nu-\nmerical details.\nOur simulations show that the plane-wave and the fo-\ncused pulse results are in good agreement already with\na 5µm waist radius (see below). Following Refs. [ 35,36],\na hyperbolic secant temporal envelope and a Gaussian\ntransverse profile with terms up to the fifth order in the\ndiffraction angle are employed to accurately describe the\nlaser pulse, which reaches its maximal focusing at the\norigin with waist radius wO. According to the notation\nemployed so far, the laser beam stems from two pulses\nwith wavelengths 0 .8µm and 0.4µm, respectively, and\nwith peak field amplitudes ξ1andξ2, respectively. Here-\nafter, for simplicity we set the constant phase θ1= 0.\nThe electrons are initially distributed according to a six-\ndimensional Gaussian probability distribution\nf(/vector x,/vector p) =Nee−/bracketleftbig\nx2+y2\n2σ2\nT+(z−z0)2\n2σ2\nL/bracketrightbig\n−/bracketleftbigp2\nx+p2\ny\n2σ2pT+(pz−pz,0)2\n2σ2pL/bracketrightbig\n(2π)3σ2\nTσ2pTσLσpL,(8)\nwithNebeing the total number of electrons and σTand\nσL(σpTandσpL) being the transverse and the longitu-\ndinal position (momentum) widths, respectively.\nA. Simulation setup\nIn our simulation, the laser pulse is 70 fs long between\nits first and last half maximal intensity with ξ1= 40\n(3.4×1021W/cm2),ξ2= 28 (1.7×1021W/cm2) and\nthe waist radius is wO= 5µm. Hence, the total inten-\nsity and power are 5 .1×1021W/cm2and 2 PW, respec-\ntively. Initially, the electron bunch has mean momentum\npz,0=−165mcwith standard deviations σT= 0.2µm,\nσL= 0.5µm,σpT= 1mcandσpL= 12mc. The\nelectron average density is 3 ×1015cm−3so that the elec-\ntron bunch contains about 400 electrons. The above-\nmentioned laser parameters are similar to those of avail-\nable petawatt laser systems [ 2,7]. Much larger effects4\ncan be achieved at higher intensities, since the transverse\nmomentum gain increases rapidly with rising laser field\namplitudes ξ1, ξ2[see Eq. ( 6)]. In addition, the electron\ndeflection can be controlled by changing either the phase\nθ2or the amplitudes ξ1, ξ2. The latter approach can\nbe exploited tuning the ratio between ξ1andξ2by con-\ntrolling the second-harmonic conversion efficiency, e.g.,\nby changing the tilt angle in a tilted-crystal configura-\ntion [37]. To date, frequency-doubling efficiencies up to\n73% at 2 TW/cm2intensity have been demonstrated ex-\nperimentally for femtosecond pulses [ 29]. Also, phase-\ncontrol of bichromatic laser pulses has been employed\nat intensities of the order of 1014W/cm2to steer the\nelectron dynamics in nonrelativistic atomic physics [ 38].\nSimilar techniques might be extended to higher inten-\nsities via coherent beam superposition of multiple laser\nbeams [9,33,34], since a relatively compact optics can\nbe employedforeachamplificationchannel. Finally, elec-\ntron bunches with the same parameters as in our simula-\ntion have been generated experimentally employing stan-\ndard multiterawatt optical lasers [ 39]. Such relatively\nlow-power pulses can also be generated by extracting a\nfraction of energy from the initial strong pulse before the\nfrequency-doubling.\nB. Results and discussion\nFigure1reports the electron density distribution\nne(pz,px) as a function of the longitudinal pzand trans-\nversepxmomentum for the interaction of 400 electrons\nwith the focused laser pulse both for cos( θ2) = 0 and\ncos(θ2) = 1, with and without RR. No appreciable dif-\nference between cos( θ2) = 0 and cos( θ2) = 1 is found\nif only the Lorentz force is taken into account. Fur-\nthermore, if the RR force is neglected, the mean of the\nmomentum distribution remains unaltered after the elec-\ntron bunch has passed through the laser pulse ¯ px≈0\nand ¯pz≈ −165mc[see Figs. 1(a),1(c)]. However, if\nthe RR force is taken into account, for cos( θ2) = 0 the\nelectrons still move along their initial propagation direc-\ntion and are distributed symmetrically in the transverse\nmomentum space with ¯ px≈0 and ¯pz≈ −82mc[see\nFig.1(b)] in good agreement with the plane wave predic-\ntionpx,f≈0 andpz,f≈ −79mc. On the other hand, for\ncos(θ2) = 1 all the electrons are deflected in the trans-\nverse direction independently of their initial energy, the\nmean of the momentum distributions being ¯ px≈ −7mc\nand ¯pz≈ −82mc[see Fig. 1(d)]. For the correspond-\ning plane-wave pulse, we obtain px,f≈ −5.8mcand\npz,f≈ −79mc, in good agreement with the above men-\ntioned focused pulse results.\nThe effect of QED corrections to the classical predic-\ntion has been estimated by introducing a quantum cor-\nrected RR force, which accounts for the reduction of the\nemitted powerin the quantum case comparedto the clas-\nsicalone[ 40]. Thepresentapproachisvalidaslongasthe\nquantum parameter χ=|e|/planckover2pi1/radicalbig\n|[Fµνpν]2|/m3c4(Gaus-\n0.00.40.8x[µm](a) cos(θ2) =−1/√\n2\n−0.040.000.04x[µm](b) cos(θ2) = 0\n−0.8−0.40.0x[µm](c) cos(θ2) = 1/√\n2\n−15 −10 −5 0 5\nz[µm]−0.8−0.40.0x[µm](d) cos(θ2) = 1\nFIG. 2. (Color online) Trajectory of an electron colliding\nhead-on with a bichromatic laser pulse without (blue dashed\nline) and with (red solid line) RR force included. The cor-\nresponding plane wave result with RR (green dotted line) is\nalso reported for comparison. In all cases θ1= 0. Panel (a):\ncos(θ2) =−1/√\n2. Panel (b): cos( θ2) = 0. Panel (c):\ncos(θ2) = 1/√\n2. Panel (d): cos( θ2) = 1. See the text for\nfurther numerical details.\nsian units) remains much smaller than unity [ 2,40]. In-\ndeed, in our simulations we found χ/lessorsimilar0.04. Moreover,\ndue to RR effects, χremains significantly smaller com-\npared to the case without RR, especially at higher laser\npulse intensities. In our simulation, quantum corrections\ndonot qualitativelyaffect the resultsbut induce acorrec-\ntion to the final mean momenta of the electron distribu-\ntion, with ¯ px≈0 and ¯pz≈ −87mcfor cos(θ2) = 0, and\n¯px≈ −6mcand ¯pz≈ −88mcfor cos(θ2) = 1. Finally,\nstochasticity effects in quantum RR may broaden the fi-\nnal electron distribution but do not significantly alter its\nmean value [ 41].\nFigure2displays the trajectory of an electron injected\ninto the focus of the bichromatic laser pulse with initial\nmomentum /vector p0= (0,0,−165mc) without (blue dashed\nline) and with (red solid line) RR effects included [the\ncorresponding plane wave result with RR (green dotted\nline) is also shown for comparison]. In all cases, the\nelectron passes through the laser pulse without chang-\ning its initial propagation direction when RR effects are\nneglected. When RR effects are included, for cos( θ2) = 0\nthe electron goes through the laser pulse without sig-\nnificantly deviating from its initial propagation direction\n[seeFig.2(b)], whereasit isquicklydeflectedinthetrans-\nverse direction for cos( θ2)/negationslash= 0 [see Figs. 2(a),2(c) and\n2(d)]. From Eq. ( 7) with cos( θ2) = 1 [cos( θ2) =∓1/√\n2],\nthepredicteddeflectionangleforthe plane-wavebecomes5\nζ≈ −4.2◦(ζ≈ ±3◦) in fair agreement with the focused\npulse result ζ≈ −5.4◦(ζ≈ ±3.8◦). Quantum effects\nlead to relatively small corrections, the deflection an-\ngle being ζ≈ −3.6◦(ζ≈ ±2.5◦) for the plane wave\nwith cos( θ2) = 1 [cos( θ2) =∓1/√\n2] andζ≈ −4.5◦\n(ζ≈ ±3.2◦) for the focused pulse.ACKNOWLEDGMENTS\nWe acknowledge useful discussions with N. Neitz,\nG. Sarri, and A. M. Sergeev.\n[1] F. Ehlotzky, K. Krajewska, and J. Z. Kami´ nski, Rep.\nProg. Phys. 72, 046401 (2009) .\n[2] A. Di Piazza, C. M¨ uller, K. Z. Hatsagortsyan, and C. H.\nKeitel,Rev. Mod. Phys. 84, 1177 (2012) .\n[3] K. W. D. Ledingham, P. McKenna, and R. P. Singhal,\nScience300, 1107 (2003) .\n[4] S. D. Kraft, C. Richter, K. Zeil, M. Baumann,\nE. Beyreuther, S. Bock, M. Bussmann, T. E. Cowan,\nY. Dammene, W. Enghardt, U. Helbig, L. Karsch,\nT. Kluge, L. Laschinsky, E. Lessmann, J. Metzkes,\nD. Naumburger, R. 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Crocker1,y\n1Department of Chemical and Biomolecular Engineering,\nUniversity of Pennsylvania, Philadelphia, Pennsylvania\nSeveral seemingly di\u000berent soft materials, including foams, cells, and many complex \ruids, exhibit\nremarkably similar rheological properties and microscopic dynamics, termed soft glassy mechanics.\nHere, we show that such behavior emerges from a simple model of a damped ripening foam, for\nsu\u000eciently weak damping. In particular, we observe intermittent avalanchey dynamics, bubble\nsuper-di\u000busion, and power-law rheology that vary as the damping factor is changed. In the limit\nof weak damping, the dynamics are determined by the tortuous low-lying portions of the energy\nlandscape, as described in a recent study. For strong damping the viscous stresses cause the system\ncon\fguration to evolve along higher energy paths, washing out small-scale tortuosity and produc-\ning motion with an increasingly ballistic character. Using a microrheological approach, the linear\nviscoelastic response of the model can be e\u000eciently calculated. This resembles the power-law rhe-\nology expected for soft glassy mechanics, but unexpectedly, is only weakly sensitive to the damping\nparameter. Lastly, we study the reported memory e\u000bect in foams after large perturbations and \fnd\nthat the timescale of the memory goes to zero as the damping parameter vanishes, suggesting that\nthe e\u000bect is due to viscous stress relaxation rather than slow structural changes stabilized by the\nenergy landscape.\nI. INTRODUCTION\nSoft glassy materials (SGMs) [1{3] such as foams and\nemulsions exhibit complex physical and rheological prop-\nerties that continue to defy explanation. Moreover, the\nsimilarity of soft glassy mechanics to that of living cells\n[4{6] and glassy materials [7] has long been noted. Pre-\nvious experimental and theoretical models have captured\ndi\u000berent aspects of such systems while falling short of a\ncomplete physical picture. For foams, rheological exper-\niments have shown con\ricting results | showing weak\n[8, 9] or no [10] power-law frequency dependence of the\ndynamic shear modulus. Modeling e\u000borts have largely\nfocused on the now canonical `bubble model' [11, 12],\nbut the dynamic shear modulus of this model has not\nbeen reported. While a more recent study did report\npower-law rheology [13] it used a simpli\fed system with-\nout damping. Further, experiments have shown memory\ne\u000bects [14, 15] in which a deformed foam shows perturbed\nmechanics which relaxes back to the unperturbed trend\nafter a long time. The physical origin of this memory\ne\u000bect remains poorly understood.\nHere, we study the soft glassy mechanics and rheology\nof foams, as well as their recovery from mechanical per-\nturbation using a 3-D bubble model [11, 12, 16] with a\nsimple damping law [7, 12], driven by simulated Ostwald\nripening [17]. Previous stress-strain simulations [16, 18]\nof a 2-D bubble model without ripening have indicated\na transition to avalanchey dynamics with reduced ap-\nplied strain rate. We look for a similar e\u000bect in our\nripening foam model by changing the damping param-\neter\u0018, e\u000bectively changing the relative rates of ripening\n\u0003rrig@seas.upenn.edu\nyjcrocker@seas.upenn.eduand viscous relaxation. This however requires the com-\nputationally expensive integration of the bubble model's\nequation of motion at low \u0018. We \fnd that for su\u000eciently\nlow damping (or equivalently slow ripening), the system\ndynamics are determined by the tortuous character of the\nenergy landscape, as observed in a damping-free model\n[13], leading to avalanches in energy, super-di\u000busive bub-\nble motion, and fractal con\fguration-space paths. For\nstronger damping, this behavior disappears, being re-\nplaced by a more continuous motion having a ballistic\ncharacter. We use a microrheological approach to deter-\nmine the dynamic shear modulus of our model from its in-\ntrinsic, non-thermal \ructuations [19, 20], and \fnd that it\ngenerically has power-law rheology resembling recent ex-\nperimental measurements [8, 9]. The rheology exponent\nis, unexpectedly, only a weak function of damping, pro-\nviding new insights into the origin of power-law rheology\nin SGMs. Lastly, we study foam's recovery from mechan-\nical perturbation by randomly scrambling the locations\nof bubbles in our model, \fnding that scrambling leads to\nperturbed mechanics that slowly return to the (average)\nunperturbed baseline, resembling experimental reports of\nmechanical memory in foams [14, 15]. The foam recov-\ners to the baseline more quickly as the damping factor\nis reduced, and does so immediately when damping is\nremoved, indicating that the memory e\u000bect is controlled\nby viscous stress relaxation, and not due to activation\nbetween energy minima.\nII. DAMPED SGM MODEL\nA. Coarsening bubble dynamics\nWe model a coarsening foam using the bubble model\n[11, 12] with a simpli\fed damping rule and simulated\nOstwald ripening. While the bubble model has been tra-arXiv:2301.13400v1  [cond-mat.soft]  31 Jan 20232\nditionally used to simulate foams [11, 18], it also serves as\nan e\u000bective model for many other SGMs [7, 13, 21]. The\nconstituent bubbles of foam in this model are treated as\nsoft-sphere particles that can overlap and interact via a\npairwise repulsive potential when overlapping:\nV(rij) =8\n<\n:\u000f\n2\u0010\n1\u0000krijk\nai+aj\u00112\n;ifkrijk<ai+aj\n0; otherwise:(1)\nwithrijbeing the distance between two bubbles of radii\naiandaj, andV(rij) being the corresponding potential.\nThe positions of the bubbles interacting via the pair-\nwise potential in Eq. 1 are evolved using an (over-\ndamped) equation of motion [11]. Notably, we consider\na simpli\fed version of the viscous force, Fi=\u0018vi[7, 12],\non each bubble to reduce computational overload while\npreserving relevant model physics.\n\u0018dri\ndt=\u0000nnX\nj@V(rij)\n@ri(2)\nwith the right-hand side representing a summation over\nneighboring bubbles that contribute to the force on bub-\nblei. Meanwhile, the left side is the damping force with\n\u0018being the e\u000bective viscous damping factor.\nTo simulate the mass exchange between bubbles due\nto Ostwald ripening [17], the bubble radii are allowed to\nevolve while keeping total bubble volume constant (pre-\nserving notional mass). Ripening causes larger bubbles\nto grow and smaller ones to shrink over time via a pair-\nwise mass \rux. We model this process with a \row rate\nthat depends on the degree of overlap between neighbor-\ning bubbles (over the overlap cross-section), along with\na mean-\feld \rux that \rows through the connected phase\nmedium.\nQi=\u0000\u000b1\u001annX\nj\u00121\nai\u00001\naj\u0013\nAoverlap\n|{z }\nneighbor-neighbor\n\u0000\u000b2\u001a\u00121\nai\u00001\nhai\u0013\nai\n|{z}\nmean-\feld(3)\nAs indicated, the \frst term represents the pair-wise mass-\n\rux between neighboring bubbles over the cross-section\nof overlap and the second represents the mean \feld con-\ntribution. The values for \u000b1(= 0:05) and\u000b2(= 0:002)\nwere chose akin to our previous study [13]. When a bub-\nble's volume turns negative over the course of the simula-\ntion, we remove it from the simulation box, while ensur-\ning that the mass of the deleted bubble and its neighbors\nis conserved.To realize the overall evolution of the system, we inde-\npendently evolve the system using the equation of motion\n(Eq. 2) and ripening rules (Eq. 3) for all bubbles at every\ntime step. We use a simple explicit Euler scheme with\nsmalldtvalues, to numerically integrate the equations of\nmotion (more details in Appendix A 2. The use of other\nintegrators like a second-order Runge-Kutta numerical\ndiscretization led to similar results. While we note that\nthe equations of motion can be physically unstable at\nvery high energies (when there is a signi\fcant overlap\nbetween bubbles), we verify that such large overlaps are\nnot present at the energy levels presented here. Further,\nwhile using small step-sizes within the range of numerical\nstability (more details in Appendix A 2), we ensure the\nsimulation has converged by cross-validating with smaller\nstep-sizes (dt). It may be noted here that smaller step-\nsizes (dt) are required for lower values of \u0018, and thus are\ncomputationally more expensive to simulate.\nBubble model simulations [13] when initialized ran-\ndomly, eventually reach a dynamic steady state with a\ncharacteristic bubble size distribution and various sys-\ntematic trends in properties such as total energy or\nmean bubbles size, as with experiments [22]. We ini-\ntialize a system of N\u00181000 bubbles at a volume frac-\ntion of\u001e= 0:75 (just above its jamming volume frac-\ntion [13, 23]) with a Weibull bubble radius distribution,\nP(a)\u0018(k=\u0015)(x=\u0015)k\u00001, wherek= 1:75;\u0015= 0:73, and\nlet the system evolve as a function of time. This distri-\nbution is representative of the steady state bubble size\ndistribution that a Gaussian initialized system reaches\nwhen evolved in the quasi-static limit ( \u0018!0) [13]. Us-\ning this as a starting point for all our simulations, we\nmodel over a range of damping factors ( \u0018) and calculate\nvarious physical quantities of interest. It must be noted\nthat we end our simulations when bubbles grow consid-\nerably large leading to multiple ( >1) overlaps between 2\nbubbles. This happens much earlier in larger \u0018systems\nthus producing shorter simulation trajectories overall.\nFluctuations in the system's total potential energy\nchange signi\fcantly for di\u000berent simulated viscosities.\nTwo distinct limits are observed, as shown in Fig. 1.\nLow viscosity simulations ( \u0018\u00140:001), produce large \ruc-\ntuations in \u0001 U(\u0001t= 1)=U(t) (see Fig. 1a, indicative\nof avalanchey, intermittent dynamics. These are sug-\ngestive of the system following the `bumpy' lower lev-\nels of the energy landscape. Conversely, with higher \u0018\nvalues, the system no longer moves from minimum to\nminimum of the underlying energy landscape but evolves\nin a dynamic force balance between the larger interac-\ntion forces and viscous stresses. This allows the system\nto \ry over the barriers and rugged features of the energy\nlandscape, with a higher time average potential energy\nfor the system (see Fig. 6a). This change in the \ructua-\ntions is shown more clearly in the distribution of energy\ndrops, Fig. 1b, which becomes more heavy-tailed at lower\n\u0018. Further, a similar trend can also be seen in Fig. 1c,\nwhere the average coordination number (over a system\ncon\fguration) is higher at higher viscosities, indicating3\n(a)\n(b) (c)\nFIG. 1. (a) Traces of relative energy di\u000berences \u0001 U=U(t) (for\n\u0001t= 1 or simulation points spaced by 1 time unit) are sensi-\ntive to intermittent dynamics. For lower damping, \u0018.0:01,\nthe relative change in energy shows abrupt peaks character-\nistic of intermittent motion. (b) Lower \u0018simulations show\na heavy-tailed probability distribution of energy \ructuations,\ntypical of an avalanchey system. As the system becomes more\nviscous, the energy \ructuations become more Gaussian. (c)\nThe average system coordination number hziremains low for\nlower\u0018simulations, characteristic of lower energy con\fgura-\ntions close to potential energy minima on the landscape. zC\nis the critical coordination for jamming, with zC= 6 in 3\u0000D\n[21].\nthat viscous stresses are shifting the foam structure away\nfrom the minimum energy states, and farther from jam-\nming, de\fned as coordination with hzi'zc[21]. Thus\nfoam con\fgurations formed at low damping explore lower\nand more tortuous portions of the energy landscape and\nones with higher damping cruise through higher and ap-\nparently smoother portions of the energy landscape.\nTo characterize the system's high dimensional motion\nover the energy landscape, we look at the path traversed\nby the system through con\fguration space for the range\nof viscosities studied. The di\u000berent time points on a\nsimulation trajectory in con\fguration space are analyzed\nfor end-to-end distances (\u0001 R2) and path contour lengths\n(\u0001s). This serves as a measure of the tortuosity of the 3 N\ndimensional con\fgurational trajectory taken by the sys-\ntem over time. As expected from our conclusions above,\nwe observe that lower viscosities yield fractal, self-similar\n(a)\n(b)\nFIG. 2. Analysis of bubble motion in 3- Ndimensional space\nand real space shows a mixture of fractal and ballistic motion.\n(a) The di\u000berent simulation points in high dimensional (3 N)\nspace are analyzed for end-to-end distances \u0001 R2and contour\nlengths \u0001sto study fractal scaling over di\u000berent length scales.\nSimulations with larger \u0018values give almost ballistic scaling in\nhyperspace. However, lower \u0018lead to a more tortuous trajec-\ntory characteristic of a fractal path, leading to super-di\u000busive\nscaling. The grey dashed line is a reference with ballistic\nscaling \u0001R2\u0018\u0001sthroughout. All data points above rep-\nresent values pooled over 4 simulations and log-bin-averaged\nover contour distances. (b) Time and ensemble (4) averaged\nmean-squared displacement for an ensemble of bubbles that\nremain \fnite sized throughout our simulations, plotted for\nthe di\u000berent \u0018values shows ballistic motion rolling over to a\nsuper-di\u000busive form for lower \u0018simulations. In comparison,\nmore viscous simulations show more ballistic behavior over a\nlarger range of \u001c.\nscaling at large lengthscales with a fractal dimension\nofDf\u00182=1:38 (slope at large distances) '1:45 (see\nFig. 2a) { capturing the intrinsic fractal physics of the\nlandscape [13]. Simulations with higher damping show\nalmost no large lengthscale fractal character, indicative\nof their ability to avoid lower energy portions of the en-\nergy landscape. Alternatively, the slight bends on Fig. 2\nmay be interpreted as a shift in the lengthscale (as a func-\ntion of\u0018) over which a fractal slope would be observed.\nThis, however, is further evidence for the self-similar frac-\ntal nature of the landscape and indicates that one would\nhave to examine considerable lag times (or con\fguration4\ndistances traveled by the system) to observe soft-glassy\nmechanics for systems with larger damping. Here, it\nmust be noted that when particles shrink to zero size\nas a result of ripening, we \fx their positions in space,\nthus conserving the number of dimensions (3 N) used to\ncalculate \u0001 R2. In Fig. 2b, we compute the ensemble and\ntime-averaged mean-squared displacement as a function\nof lag time ( \u001c). These curves show a functional form that\nis similar to that of the \u0001 R2above because the mean-\nsquared displacement is a projection of those curves to\n3-D space; the slight di\u000berence in the exponent is due\nto the calculation being done on a slightly di\u000berent en-\nsemble of bubbles that remain \fnite sized throughout the\nsimulation.\nHere, one may also identify a dimensionless group of\ninterest called the Deborah number De, which can be\nexpressed as the ratio of time scales associated with re-\nlaxation and the mode of driving|the two relevant dy-\nnamic processes for this system. Here, that would be\nthe damped relaxation time from Eq. 2 ( \u001cR=\u0018hai2=\u000f)\nand timescale associated with changing bubble radii ( a)\nimparted by the ripening process, Eq. 3 ( \u001cC=hai2=\u000b1\nwhen\u000b1> \u000b 2). This gives us a ripening Deborah num-\nber, (De\u000b=\u0018\u000b1=\u000f) which is a ratio of the relaxation\n(\u001cR) and coarsening ( \u001cC) times (typically ranging be-\ntween 10\u00003\u000010\u00006for our simulations). This dimension-\nless group presumably depends on the system's volume\nfraction\u001eand its proximity to the jamming volume frac-\ntion\u001eJ[11, 18].\nThis dimensionless group formalism can be a useful\nway to explain many previous experimental and simula-\ntion results [12, 16, 18]. We begin by noting that the\navalanchey dynamics and intermittent rearrangements\nobserved in our simulations resemble previous studies\nof similar systems [11] driven by shear strain instead\nof coarsening. Various comprehensive studies [16, 18]\nusing 2Dshear strain point out a similar transition to\navalanchey rearrangement events below a certain shear\nstrain rate. Thus, our results can be interpreted as a\ntransition in landscape physics as a function of De\u000bwhile\nshear simulation results [16] can be explained using a cor-\nresponding shear Deborah number ,De\r.\nFor a foam experiment, we note that the energy scale\nand damping factor vary as the system evolves: \u000f'\u001bhai2\n[11] and\u0018/hai, while\u000b1is e\u000bectively independent of\nhai. Thus experimentally, De\u000b/hai3changes for dy-\nnamically aging foam where haiincreases as a function\nof time [13, 22](see Fig. 5). This keeps pushing the aging\nsystem away from the landscape-dominated regime, po-\ntentially explaining the issue associated with the shifting\ncut-o\u000b [1], and tending to produce behavior akin to high\n\u0018simulations.\nB. Rheology of SGMs\nThe rheology of soft-glassy systems is typically found\nto be weakly frequency dependent (solid-like), often witha power-law form, while di\u000berent experiments on foams\n[8, 10] yield apparently con\ricting results. Computation-\nally, capturing low-frequency responses to applied strains\ncan be very expensive, making the determination of rhe-\nology di\u000ecult [13]. Here, we provide a numerical proce-\ndure that derives its essentials from a microrheological\napproach [19, 24] that computes the power spectra of the\nactive, \ructuating shear strain and stress from the par-\nticle motions, and computes the dynamic shear modulus\nfrom their ratio.\nWe begin by noting that one can relate the stress ( \u001b(t))\nand strain ( \r(t)) to the creep compliance ( J(t)) using the\ntheory of linear response [25, 26] and the Boltzmann su-\nperposition principle, relating them through a convolu-\ntion:\nJ(t)~_\u001b(t) =\r(t)\nZt\n\u00001J(t\u0000t0) _\u001b(t0)dt0=\r(t)(4)\nWhile this basic constitutive equation represents the\nrelation between the macroscopic stress and strain for a\nlinear material, we extend this formalism to its micro-\nrheological version wherein each bubble/ particle can be\ntreated as a tracer moving in a homogeneous viscoelas-\ntic continuum (formed by all the other bubbles) driven\nby active \ructuating stresses. Thus, typically one can\nuse the bubbles' positional vectors describing their mo-\ntion in the e\u000bective medium to describe the local, time-\ndependent strain in the e\u000bective medium [24]. Similarly,\nthe local \ructuating active stress acting on each bubble\nin the system can be computed as follows [27, 28]:\n\u001b(ri) =\u00000\n@nnX\njrij\nFij1\nA\u000e(r\u0000ri) (5)\nwhererijandFijrepresent the inter-particle displace-\nments and forces between particles iandj.\nApplying the above equations directly to the data\nwould be impractical because the \u001b(t) and\r(t) signals\nfor each bubble are random functions of time. Instead,\nwe transform the equation described in Appendix B 1 to\na relation between the ensemble-averaged mean squared\ndi\u000berences (MSD), the stress, and strain. The stress\nMSD is calculated by considering the squared di\u000berence\nbetween the three o\u000b-diagonal elements of the bubble-\nwise symmetric tensor (see Eq. 5). Further, we consider\nthe ensemble average over all bubbles in our system and\nover similar lag times to get a statistically consistent\nMSD. Meanwhile, the strain MSD can be estimated using\nthe positional MSD or mean-squared displacement intro-\nduced earlier (see Fig. 2a). These quantities can further\nbe related using the modi\fed Fourier transformed (FT)\nversion of the above equation [13, 19, 20]:5\njG\u0003(!)j2'g\u0001\u001b2(!)\n3\u0019haig\u0001r2(!)(6)\nTo avoid assumptions and approximations related to\ncomputing Fourier transforms of these MSDs over a \fnite\nrange of lag times, [13, 20], we consider the exact convolu-\ntional relation described in Appendix B 1. This equation\ncan be further modi\fed using the Wiener-Khinchin the-\norem and the relationship between autocorrelation and\nMSD for the stress and strain, giving us the following\nequation:\n2J2(0)h\u001b2i+Z\u001ci\n0f(\u001ci\u0000t0)(h\u001b2i\u0000h\u0001\u001b2i(t0)=2)dt0\n= (h\r2i\u0000h\u0001\r2i(\u001ci)=2)\n'3\u0019hai(hr2i\u0000h\u0001r2i(\u001ci)=2)=hai3\nwheref(\u001ci) is de\fned as follows,\nf(\u001ci) =\u0012Z\u001ci\n0_J(\u001ci\u0000t\")_J(t\")dt\" + 2J(0)_J(\u001ci)\u0013(7)\nwhere \u0001\u001b2(\u001c) and \u0001\r2(\u001c) represent the time-averaged,\nmean-squared di\u000berence of the bubbles' stress and\nstrains, in our analyses. We approximate the strain using\nthe position vector, r[19, 24] as discussed above. Further,\nwe ensemble average our MSDs over 4 simulation runs.\nFinally, to represent the creep compliance, we use a mod-\ni\fed version of the model suggested by Lavergne and co-\nauthors in Ref. [8]: J(t) = 1=G1+kD=G1[(1+t=\u001c0)\f\u00001]\n(more details in Appendix B 2). Using this as a model\nfor the viscoelastic rheology for the foam, we undertake a\nsimultaneous \ftting operation for the parameters of the\nmodel, i.e., G1;kDand\f, at various lag times or \u001ciin\nthe convolutional integral equation shown above (Eq. 7).\nUsing the optimal parameters from the \ft gives us the\ncreep compliance and, subsequently, the complex modu-\nlusG\u0003(!) using the relation, G\u0003(!)J(!) = 1=i!. Further\ndetails of the derivation and mathematics of the numer-\nical procedure are provided in Appendix B 1. It may be\nfurther noted that attempts to model the rheology using\na Maxwell model produced inferior solutions to Eq. 7,\nwith the power-law model cited above providing signi\f-\ncantly better \fts.\nThe results from the computed creep compliance and\ndynamic shear moduli are summarized in Fig. 3. G\u0003(!)\nexhibits a power-law regime over the !range of inter-\nest and is characteristic of behavior predicted in theory\n[1], simulations [13] and observed in experiments [8{10].\nRecent experiments and our simulation results here (see\nFig. 3), evaluated with a robust numerical approach pro-\nvide clear evidence in support of the existence of power-\nlaw rheology in SGMs. Fig. 3a, shows the \fts for the J(t)\nmodel described above, with a family of curves with simi-\nlar power-low exponents. Considering the semi-analytical\nFT to obtain G\u0003(!) gives us the viscoelastic moduli with\n(a)\n(b)\nFIG. 3. We compute the viscoelastic moduli for the dynamic\nviscous simulations considered from the \ructuating stresses\nand displacements of bubbles in the simulation, as described\nin the text. \u0018values with data over a signi\fcant \u001crange were\nconsidered for the calculation. The dotted grey lines indicate\nthe\u001crange of the MSD data used for the above calculation.\n(a) Fitting the model explained in the text to simulation data\ngives us suitable \fts with a family of curves with power-law\nbehavior. The creep compliance scales as J(\u001c)\u0018t\fin the\nlag time range shown above. (b) G\u0003(!) obtained from J(\u001c),\ngives us power-law rheology in !i.e.,G\u0003(!)\u0018!\f. This\nbehavior is observed at all \u0018values calculated above. (inset)\nThe predicted \fvalues, indicative of the log-slope for the\ncurves in (b), hover consistently in the range \u00180:15\u00000:2,\nsimilar to previously observed values in simulation [13], and\nexperiments [8, 9].\na power regime de\fned by G\u0003(!)\u0018!\f, with weak depen-\ndence of the exponent \fon damping, showing that this\nis a universal feature for foams, regardless of damping.\nC. Memory and recovery in perturbed SGMs\nThe SGM system shows a signi\fcant downhill descent\nin energy as the largest bubbles coarsen and grow. As this\ndownward trend continues, the system reaches a dynam-\nical scaling steady state [13]. While it is unclear whether\ncon\fgurations in this regime form an ergodic ensemble\nover some characteristic time, the bubbles show stable\ntrends in various structural quantities like average co-6\n(a)\n(b)\nFIG. 4. Scrambling a quasi-static system shows an imme-\ndiate return to trend. (a) We scramble the con\fgurational\npositions of a system in steady state (at t= 400) in a quasi-\nstatic (\u0018= 0) simulation. Surprisingly, the system always\n\fnds a 'new' steady state right away, as indicated by the co-\nordination number ( z\u0000zC) measured here, and continues to\nevolve with similar dynamic properties. The dark and light\nsymbols represent the scrambled and unscrambled simulation,\nrespectively. (b) Running multiple ( \u0018100) such scrambles at\nt= 400, gives us a Gaussian distribution of hzias shown\nabove. This overlaps well with hzivalues obtained at t= 400\nfor 10 di\u000berent realizations of the same simulation as indicated\nby the mean\u0006standard deviations. This tells us that the\nscrambled simulation returns to the newly found steady state\ninstantaneously. (inset) Moreover, the temporal autocorre-\nlations for these zensembles - scrambled and unscrambled -\nprovide similar decorrelation times. These \fndings indicate\nsimilar dynamic properties for the scrambled and unscram-\nbled simulation.\nordination number, mean bubble radius, normalized ra-\ndial distribution, etc. Bubbles initially move around to\nreach the steady state, de\fned by the dynamical scaling\n`attractor' on the energy landscape, and then continue\nto evolve in this steady state ensemble. Any perturba-\ntion away from the attractor would thus lead the system\nback to a 'new' steady state as de\fned by the structural\nand dynamical properties of the attractor and the sys-\ntem landscape. Experiments have observed [14, 15] that\na strain-perturbed foam relaxed back to its unperturbed\nsteady state after an unexpectedly long waiting time, and\nFIG. 5. Scrambling a quasi-static system shows no change to\nripening evolution. Here, we look at the structure through\nthe radial distribution formed at steady state for a scrambled\nand unscrambled system. As previously in Fig. 4, the system\ninstantaneously continues in steady state. As can be noticed,\nthe slope changes for the scrambled simulation at t= 400 (in-\ndicated by arrows), indicative of a new foam initiation time\n[13]; however, the trend remains linear, consistent with dy-\nnamic scaling state behavior hai2\u0018tage.\nhave described this as a memory phenomenon or measure\nof history dependence. The consensus [14, 15] on the ori-\ngin of this memory is that coarsening mediated excita-\ntions are needed to enable the system to overcome local\nminima that the perturbed system relaxes into. Thus,\nthe long waiting time has been considered a result of\nslow coarsening.\nTo study this phenomenon's structural and dynamical\nsigni\fcance computationally, we run a set of simulations\nusing our modi\fed damped model over various \u0018values.\nWe consider the theoretical extreme of a perturbation\nby introducing positional scrambles in our system. To\ndo so, we begin with a typical steady-state system and\nrandomly scramble the various 3 Npositions of the bub-\nbles. This scramble randomly assigns a point in hyper-\nspace for the system of soft spheres, providing a ran-\ndom structural perturbation. We then continue with the\nrelaxation-coarsening procedure described previously in\nSection II A. It must be noted here that for the quasi-\nstatic case when \u0018= 0, we relax the system to its \frst en-\nergy minimum (i.e. mechanical equilibrium) using FIRE\n[29] instead of using Eq. 2.\nFor the quasi-static case, we see that the system, upon\none (or even multiple) scrambles, returns to the earlier\ndynamical scaling steady trend (see Fig. 4a) immediately.\nIndicators likehziandhai2show no signi\fcant change\nfrom steady-state behavior, as can be seen in Fig. 4 which\nplots the scrambled (at t= 400) and unscrambled aver-\nage coordination number as a function of time. Here the\nscrambled system experiences no barriers to reaching this\n'new' steady state with FIRE traversing the large con\fg-\nurational distance on a relatively smooth portion of the\nenergy landscape (at higher energies) to \fnd the nearest\n(primary) minima. It may be noted that the scramble7\nmoves the system to a random 3 N-dimensional con\fgu-\nration on the energy landscape. The system thus evolves\nin the particular meta-basin corresponding to the scram-\nbled positions going forward. So while a single simula-\ntion might not erogodically explore all portions in con-\n\fguration space, these di\u000berent hyperspaces on the en-\nergy landscape have similar structural properties. Small\nchanges in moving ensemble averages indicate the slight\nvariations in di\u000berent regions or metabasins of the energy\nlandscape. E\u000bectively, all these primary minima that\nthe minimizer \fnds belong to the 'steady-state ensem-\nble' of the particular foam radii distribution realization\natt= 400.\nTo test whether the scrambled simulation actually re-\nturns to the 'steady-state ensemble' for a similar foam\nat the same age, we compare the average coordination\nnumbers at t= 400 for 10 di\u000berent quasi-static simu-\nlations (di\u000berent positional initializations at t= 0) in\nFig. 4b; with a pool of coordination numbers obtained\nfrom scrambling the same test simulation (from Fig. 4a)\ncon\fguration at t= 400, a 100 di\u000berent ways and quench-\ning them using FIRE. As seen in Fig. 4b, the distribution\nof average coordination numbers for a quenched minima\nfrom the test simulation lies in the range of expected\nsteady-statehzivalues for a similar foam simulation at\nthe same age. Lastly, looking at the inherent tempo-\nral correlations in the coordination number further tells\nus that similar correlations get rebuilt after the system\nevolves on the randomly chosen 'metabasin' on the en-\nergy landscape Fig. 4a(inset). Thus we conclude that\nwhile the system doesn't return to the exact con\fgu-\nrational hyperspace on the energy landscape, the foam\nexhibits a 'memory' e\u000bect in various physical and dy-\nnamical properties.\nInterestingly, this also doesn't a\u000bect the coarsening\nmediated bubble size distribution reached at dynamic\nscaling as seen in Fig. 5. This can be seen in the av-\nerage system radii measured over time in Fig. 5. While\nthere is a noticeable change in the rate of radial change or\nthe slope, the trajectory remains in steady state as indi-\ncated by the linear gradient. Additionally, the scrambled\nsimulation continues to evolve with similar moments of\nthe radii distribution (see Fig. 4a(inset)). Overall, this\nshows that while any steady state structure built in by\nthe dynamics and coarsening before the scramble gets ru-\nined by the perturbation, it gets restored immediately by\nquenching to the nearest minima (using FIRE).\nRepeating the same computational experiment at \f-\nnite\u0018provides insight into the mechanism of the mem-\nory phenomenon. In agreement with previous experi-\nments [10, 15], we see that the system requires a sur-\nprisingly long time to recover to its former steady state\ntrend (see Fig. 6). However, unlike previous suggestions\nof this time-scale being coarsening-mediated, we observe\na\u0018dependent phenomenon. This viscous time scale dic-\ntates the time the system takes to relax any energetic\nstress built in by the overlaps caused by the positional\nscramble. Larger \u0018leads to a longer time for the sys-\n(a)\n(b)\nFIG. 6. Scrambled foams with damping show very slow relax-\nation towards their prior trends in energy and coordination.\nFinite\u0018simulations are scrambled at t= 400 and evolved\nusing the dynamical equation Eq. 2. (a) The potential en-\nergies post scrambling show a progressive trend in reaching\nthe steady state. Fitting the energies (b) The mean coordi-\nnation numberhziof the system shows similar \u0018dependent\ntrends. Interestingly, the initial dynamics directs the system\nto lowerhzicon\fgurations before relaxing to the appropriate\nsteady-state value.\ntem to relax these unstable overlaps. Fig. 6 shows the\nprogression of energy and average coordination number\ntowards equilibrium after a positional scramble. Inter-\nestingly, the zvalues shoot below the steady state line\npost scramble before trending back to steady state, much\nlike previous experiments measuring rearrangement rate\n[14]. Finally, one may note here that while the waiting\ntimes seem to be damping dependent, they are, however,\nmuch larger than \u001cR=\u0018hai=\u000f\u0018O(\u0018). This discrepancy\nmight be due to the extreme nature of positional pertur-\nbation introduced in our simulation { which introduces\nmany large and small perturbations for all Nbubbles\naway from the nearest steady-state ensemble con\fgura-\ntion. Thus, the waiting time is a compounded sum of all\nthese di\u000berent distances that the bubbles must traverse\nto reach the 'new' steady state.8\nIII. CONCLUSIONS\nWe have shown that the `bubble model' with sim-\nple damping and simulated ripening recreates many of\nthe exciting phenomena reported for soft-glassy materi-\nals. Speci\fcally, this model exhibits avalanchey, inter-\nmittent dynamics at low viscosities with non-Brownian\nsuper-di\u000busive motion. Considered in high-dimensional\ncon\fguration space, such motion occurs along a fractal\ncon\fgurational path that is constrained to the lowest en-\nergy portions of the potential energy landscape. We \fnd\nthat the energy minima of the bubble model are clus-\ntered together in con\fguration space, scattered along\nthe con\fguration path followed by the model (at low\nviscosity). In the practical absence of viscous stresses,\nthe system hops from each (ripening destabilized) en-\nergy minimum to a nearby, adjacent energy minimum.\nThis landscape-dominated motion subsequently produces\nthe observed super-di\u000busive motion and stress and strain\n\ructuations corresponding to power-law rheology. As the\nsimulated viscosity is increased, the system shows pro-\ngressively smoother dynamics and motion with a more\nballistic character. In this case, viscous stresses cause the\ncon\fguration never to explore the true potential energy\nminima but instead evolve along a path that is adjacent\nto the cluster of energy minima. The system e\u000bectively\nstays at higher potential energy, and the \fner details of\nthe fractal con\fguration path seen at lower viscosity are\nwashed out, leading to a straighter path and more ballis-\ntic motion. Between these two limits, one can \fnd a gra-\ndation of properties, where the system displays increasing\ncharacteristic length and time scales above which the low\nviscosity behavior may still be observed.\nThis model also successfully generates power-law rheol-\nogy, which previously has not been reported for a damped\nbubble model. However, unlike other properties, power-\nlaw rheology seems to be a consistent feature in SGMs\nover a wide range of viscosity values. This suggests an\nextended fractal nature for the energy landscape, con-\nsistently producing power-law rheology even when the\ncon\fgurations are at energy somewhat above the energy\nminima. Further, our microrheology-based approach pro-\nvides a robust and reliable way to compute viscoelas-\ntic moduli from force and strain \ructuations measure-\nments of constituent particles and is free of any system-\natic truncation errors associated with earlier microrheo-\nlogical methods.\nLastly, we investigate the `memory' of the ripening\nbubble model for mechanical perturbations by randomly\nscrambling the bubble positions. We \fnd that scram-\nbled con\fgurations (e\u000bectively a random point in con\fg-\nuration space) must relax a long con\fguration distance\nbefore reaching their \frst potential energy minimum.\nMoreover, those \frst energy minima are indistinguish-\nable (statistically) from the ensemble of con\fgurations\nexplored by other ripening simulations of the same age.\nThis is most clearly shown by the quasi-static simulation\nthat immediately recovers its earlier (ensemble averaged)baseline properties when it reaches its \frst energy min-\nimum. For \fnite viscosity, the system can take a long\ntime to traverse the required con\fguration distance to\nreturn to the vicinity of the energy minima cluster and\nrecover its earlier mechanical properties. In viscous sys-\ntems, this recovery time (and e\u000bective `memory' time) is\nproportional to the viscous relaxation time of the model.\nThis viscosity-mediated recovery process is contrary to\nprevious experimental inferences and a consequence of\nthe barrier-free potential energy landscape at higher en-\nergies that the perturbed system must traverse.\nFuture work would include developing a model that\ncaptures other long-time characteristics of these SGMs.\nWe hope that such a description will provide a more com-\nplete and practical model for SGMs. Modeling the phys-\nical properties of the many materials categorized under\nSGMs could be of potential use in \felds ranging from\nmaterial science (foams and complex \ruids) to biology\n(living cells [4, 5]).\nAUTHOR CONTRIBUTIONS\nA.T., R.A.R., and J.C.C. designed research; A.T. per-\nformed research and analyzed data; A.T., R.A.R., and\nJ.C.C. wrote the paper. R.A.R. and J.C.C. contributed\nequally to this work.\nACKNOWLEDGMENTS\nWe are grateful for valuable conversations with Dou-\nglas Durian, Fran\u0018 cois Lavergne, Andrea Liu, Talid Sinno,\nand V\u0013 eronique Trappe. This work was supported by\nNSF-DMR 1609525 and 1720530 and computational re-\nsources provided by XSEDE through TG-DMR150034.\nAppendix A: Damped SGM Model\n1. Underdamped limit of an overdamped equation\nThe equation used for the simulations, as described in\nthe main text, is:\n\u0018dri\ndt=Fi\n=\u0000nnX\nj@V(rij)\n@ri(A1)\nIt can be seen that the equation is similar to an over-\ndamped equation of motion. However, it must be noted\nthat though the equation resembles and has the charac-\nteristics of an overdamped equation of motion, the same\nis not due to a large viscosity but rather the non-inertial9\nnature of the constituent particles considered in the sys-\ntem. One may recall that dynamics for a mass attached\nto a damped spring are mediated by the damping factor\n\u0010=b=(2p\nkm). That can be evaluated for our system\nof interest as follows \u0010'\u0018=p\n\u000f\u001ahai. Since overdamped\ndynamics is achieved when \u0010\u00151, we see that the non-\ninertial particles ( \u001a!0) in our case give rise to the\nso-called overdamped equation of motion. Meanwhile,\nwe continue to operate with a \fnite value of \u0018.\n2. Integration and stability\nThe simulation can be summarized as a numerical in-\ntegration of the two equations { Eq. 2 and eq. 3 using a\nnumerical integration technique. Due to the sti\u000b nature\nof Eq. 2 (especially at small \u0018values, one needs to choose\nappropriate dtvalues to ensure any error perturbations\ndon't diverge as the simulation proceeds and that the\nsolution is a converged one. We use a simple Explicit\nEuler scheme to perform our integration here. We note\nthat other methods, like implicit Euler and second-order\nRunge Kutta scheme, provide more extensive stability\nregimes for dtand are more accurate but can have more\nsigni\fcant computational overload associated with the\nintegration scheme. Below, we perform a simple numeri-\ncal stability test.\nWe start by considering Eq. 2 for all Nparticles or\n3Ndegrees of freedom, i.e., i2f1;2;:::3Ng, which can\nbe expressed in terms of the Hessian for using a Taylor\nexpansion as follows: and\n\u0018dr\ndt=F\n=F0\u0000Hr(A2)\nwhere r,Fare 3Ndimensional vectors and His a\n3N\u00023Nmatrix or the Hessian of the potential \feld.\nWe may note here that for most \u0018simulations, the sys-\ntem con\fgurations are close to mechanical equilibrium,\nso for our stability analysis, we may approximate this us-\ningF0'0. Further one may note that the any error \u000fi\nwould propagate via an equation similar to Eq. A2:\n\u0018d\u000f\ndt=\u0000H\u000f (A3)\nNow, using the Explicit Euler formalism, for time steps\nn+ 1 andn, we get:\n\u0018\u000fn+1\u0000\u000fn\ndt=\u0000H\u000fn\n=\u0000\u0015\u000fn\njj\u000fn+1jj\njj\u000fnjj=jjI\u0000\u0015jjdt=\u0018(A4)where\u0015is a matrix containing all eigenvalues of H.\nEnforcing the criteria of stability on the equation above\nwe have,\njj\u000fn+1jj\njj\u000fnjj\u00141\njjI\u0000\u0015jjdt=\u0018\u00141\n0\u0014jj\u0015maxjjdt=\u0018\u00142(A5)\nwhere\u0015maxis the largest eigenvalue for H. Since all\neigenvalues would be real for this physical system, we\nnow have,\n0\u0014jj\u0015maxjjdt=\u0018\u00142\ndt\u00142\u0018=\u0015max(A6)\nFor most con\fgurations, explored in our system simu-\nlation the\u0015maxvaries around\u00181\u000010. This gives us that\ndt6\u0018=5 is the condition for stability. Here, we choose\ndt=\u0018=10, as the step size for all simulations reported\nin this study. Since we have Eq. 3, which also controls\noverall dynamics, this choice of time-step was validated\nfor convergence. We have veri\fed that our explicit Euler\nscheme was converged by checking other smaller values\nofdt. Other schemes like the RK-2 also produced similar\nresults. Further, for \u0018 >0:01, we stuck with the use of\ndt= 0:001, as the system moves further away from me-\nchanically stable states and the above approximation in\nEq. A3 fails to strictly hold.\n3. Dimensionless group analysis: Deborah number\nApart from evaluating the Deborah number Deas\nthe ratio of the damped relaxation time from Eq. 2\n(\u001cR=\u0018hai2=\u000f) and probing time associated with chang-\ning bubble radii ( a) imparted by the coarsening process,\nEq. 3 (\u001cC=hai2=\u000b1, we can do a simple Buckingham Pi\nanalysis to determine the relevant \u0005 group. Below, we\npresent the analysis to derive the Deborah number as a\n\u0005 group.\nOne can re-model the system through an experimental\nlens and pose the problem statement as measuring the\naverage radiihaias a function of time. Intuitively, this\nmight be in\ruenced by system properties like \u000f,\u001a,\u000b1,\u0018.\nThese 4 quantities along with hai, are comprised of the\ndimensions M,LandT. Thus 2 \u0005 groups can be made\nusing these variables for every combination of 3 repeating\nvariables being chosen. Here, we choose \u001a,\u000b, and\u0018as\nare repeating variables.\n\u00051=f(\u000f;\u001a;\u000b 1;\u0018)\n=\u000f\u001ax\u000by\n1\u0018z (A7)\nSolving for x,y, andzso that \u0005 1is dimensionless, we\nget \u0005 1=\u000f=(\u000b1\u0018) orDe=\u0018\u000b1=\u000f.10\nAppendix B: Rheology\n1. Analytical Derivation\nHere, we provide a derivation for the integral equation\nEq. 7, which we used to compute the viscoelastic moduli\nfor our simulation. We start by noting that the theory of\nviscoelasticity for linear materials [25, 26] shows that the\ncreep compliance J, can be related to the stress \u001band\nstrain\ras follows:\nZt\n\u00001J(t\u0000t0) _\u001b(t0)dt0=\r(t) (B1)\nHere we may note that J,\u001band\rare = 08t2(\u00001;0)\nand\u001508t2[0;1). Thus, we can extend the integral\nlimits by doing the following:\nZ1\n\u00001J(t\u0000t0) _\u001b(t\u0000t0)dt0=\r(t)\nZ1\n\u00001_J(t\u0000t0)\u001b(t\u0000t0)dt0=\r(t)\nusing the product rule\nZt\n0_J(t\u0000t0)\u001b(t\u0000t0)dt0+J(0)\u001b(t) =\r(t)(B2)\nTaking the Fourier Transform of the non-decomposed\nequation above and applying the convolution theorem\ngives us,\ne_Je\u001b=e\r (B3)\nWhile one could potentially work with Eq. B2 or\nEq. B3, the numerical inaccuracies associated with an\nFT [13, 20] and the statistical noise in a trajectory func-\ntion like\u001b(t) or\r(t), would make the procedure more\ndi\u000ecult. Thus, we use the Wiener{Khinchin theorem\nand further transform the auto-correlation into its mean\nsquared version as follows:\ne_Je_J=jje\rjj2\njje\u001bjj2\n=gR\r\r\ngR\u001b\u001b\n=^h\r2i\u0000h\u0001\r2i=2\n^h\u001b2i\u0000h\u0001\u001b2i=2(B4)\nReshu\u000fing this equation and taking the inverse FT\nyields an integral equation. We decompose the limits to\nstay between 0 and t, which adds a few boundary terms\nfor the step function jump in Jand\u001batt= 0. Further,we change our notation for tto\u001c, to be consistent with\nthe MSDs which are calculated as averages over lag times.\n2J2(0)h\u001b2i+Z\u001c\n0f(\u001c\u0000t0)(h\u001b2i\u0000h\u0001\u001b2i(t0)=2)dt0\n= (h\r2i\u0000h\u0001\r2i(\u001c)=2)\nwheref(\u001c) is de\fned as follows,\nf(\u001c) =\u0012Z\u001c\n0_J(\u001c\u0000t\")_J(t\")dt\" + 2J(0)_J(\u001c)\u0013(B5)\nThis equation can be approximated, using similar\nmathematical approximations as used earlier in Ref. [20].\nZ\u001c\n0g(\u001c\u0000t0)h\u0001\u001b2i(t0)dt0=h\u0001\r2i(\u001c)\nwhereg(\u001c) is de\fned as follows,\ng(\u001c) =\u0012Z\u001c\n0_J(\u001c\u0000t\")_J(t\")dt\"\u0013(B6)\nWe approximate the right-hand side of this equa-\ntion using the bubble portions r[19, 24], giving: '\n3\u0019hai(hr2i\u0000h\u0001r2i(\u001ci)=2)=hai3. However, it may be noted\nthat this equation is not well de\fned at \u001c= 0. Thus we\nevaluate this only for lag times greater than zero. To get\nan accurate solution, we consider the above equation at\nvarious \fnite lag time values or \u001ciand solve a set of simul-\ntaneous equations to \fnd the appropriate creep compli-\nance,J(t). Speci\fcally, we choose \u001ci2f\u001c1;\u001c2;\u001c3:::\u001cmaxg.\nHere,\u001c1can be as small as dt. We report here results\nfor\u001c1= 1. This choice, however, brings in some nu-\nmerical error due to the integrals going from 0 !\u001c. It\nmay be noted that this equation is mathematically exact\nfor8\u001c > 0 and that the upper limit of our observation\n{\u001cmax, does not a\u000bect the numerical procedure, e\u000bec-\ntively avoiding a source of truncation error present in\nmany earlier approaches.\n2. Choice of Fitting Model\nOne may notice that solving Eq. B5 or Eq. B6 requires\na model for J(t). Here we choose a modi\fed version of\nthe model suggested in Ref. [8]. The original model put\nforth in the above study has a terminal mode of relax-\nation at long times, given by t=\u0011R, and has been observed\npreviously in experiments [10]. In our simulations, we,\nhowever, do not observe any terminal relaxation and thus\nignore the additional term mentioned above. We consid-\nered a modi\fed version of the model given as follows.\nJ(t) = 1=G1+kD=G1[(1 +t=\u001c0)\f\u00001] (B7)11\n[1] P. Sollich, F. Lequeux, P. H\u0013 ebraud, and M. E. 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Tyler Mall, Tempe, AZ 85287-1504 USA\nbDeutsches Elektronen-Synchrotron (DESY), Platanenallee 6,\nD-15738 Zeuthen, Germany\nMay 18, 2017\nAbstract\nWe study the production of high energy neutrinos in jets from the tidal disruption of stars\nby supermassive black holes. The di\u000buse neutrino \rux expected from these tidal disruption\nevents (TDEs) is calculated both analytically and numerically, taking account the depen-\ndence of the rate of TDEs on the redshift and black hole mass. We \fnd that \u001810% of\nthe observed di\u000buse \rux at IceCube at an energy of about 1 PeV can come from TDEs\nif the characteristics of known jetted tidal disruption events are assumed to apply to the\nwhole population of these sources. If, however, plausible scalings of the jet Lorentz factor\nor variability timescale with the black hole mass are taken into account, the contribution\nof the lowest mass black holes to the neutrino \rux is enhanced. In this case, TDEs can\naccount for most of the neutrino \rux detected at IceCube, describing both the neutrino \rux\nnormalization and spectral shape with moderate baryonic loadings. While the uncertainties\non our assumptions are large, a possible signature of TDEs as the origin of the IceCube\nsignal is the transition of the \rux \ravor composition from a pion beam to a muon damped\nsource at the highest energies, which will also result in a suppression of Glashow resonance\nevents.\naEmail: cecilia.lunardini@asu.edu\nbEmail: walter.winter@desy.dearXiv:1612.03160v2  [astro-ph.HE]  17 May 20171 Introduction\nIt is an established fact that supermassive black holes (SMBH) inhabit the center of most\nor all galaxies. The physics of these objects is still mysterious in many ways, and can be\nstudied by observing the e\u000bects that the enormous gravitational \feld of a SMBH produces\non the surrounding gas and stellar matter.\nA particularly dramatic e\u000bect is a Tidal Disruption Event (TDE), the phenomenon in which\na star passing within a critical distance from the SMBH is torn apart by its extremely\nstrong tidal force. The accretion of the disrupted stellar matter on the SMBH can generate\nobservable \rares of radiation in the thermal, UV and X-rays that might last for days,\nmonths, or even years [1{4]. These \rares have the potential to reveal important information\non the innermost stellar population of a galaxy, and on the physics of SMBH. TDEs are\nespecially valuable as probes of SMBH that are normally quiet { as opposed to the Active\nGalactic Nuclei { and therefore more di\u000ecult to study. In a recent catalogue [5] (see\nalso Ref. [6]), 66 TDE candidates have been identi\fed with various degrees of con\fdence.\nRoughly, observations con\frm the general theory of tidal disruption, whereas they leave\nmany open questions on the energetics and dynamics of these phenomena { and possible\nselection biases in their detection, see e.g.Ref. [7].\nInterestingly, a subset of all the observed TDEs show evidence for a relativistic jet, and\nexhibit a signi\fcantly higher luminosity in X-rays. The best observed jetted TDE is Swift\nJ1644+57 [8]; others are Swift J2058.4+0516 [9] and Swift J1112.2-8238 [10] (the latter\nbeing somewhat atypical, see [5]). Other transient events have been proposed to be jetted\nTDEs [11, 12], although their interpretation is less robust. It has also been suggested that\njetted TDEs might have have been observed in the past in gamma rays, as a new class of\nUltra-Long Gamma Ray Bursts (ULGRBs) [13].\nIf they indeed generate jets, TDEs are candidate sources of cosmic rays. This was \frst\ninvestigated by Farrar and Gruzinov [14], who showed that TDEs naturally meet all the\nnecessary criteria to accelerate protons to energy E\u00181020eV, and might be su\u000eciently\nabundant to account for the observed ultra-high energy cosmic ray \rux. Following works\n[15] discussed this result for TDEs with parameters compatible with the Swift J1644+57\nevent, suggesting that they could explain the recently observed cosmic ray hotspot; see also\nRef. [16].\nUnder the hadronic hypothesis, jetted TDEs are also sources of neutrinos, via proton-photon\ninteractions. A prediction of the neutrino \rux from a TDE was \frst published by Wang et\nal. [17] for parameters motivated by Swift J1644+57. The corresponding number of events\nat the IceCube detector was estimated. A follow up study [18] (see also [19]) shows that\nTDEs could be hidden neutrino sources, lacking a photon counterpart due to the jet choking\ninside an envelope made of the debris of the disrupted star1.\nThe topic of TDEs as neutrino sources is especially timely. Indeed, IceCube has discovered a\ndi\u000buse \rux of high-energy astrophysical neutrinos of dominantly extragalactic origin [20]. So\nfar, no class of objects which could power most of this \rux has been identi\fed. Speci\fcally,\nthe contributions from Active Galactic Nuclei (AGN) blazars [21] and Gamma-Ray Bursts\n1We do not consider this possibility in our work.\n1(GRBs) [22,23] have been strongly constrained by stacking the information from many dif-\nferent gamma-ray sources. Furthermore, optically thin sources with neutrino production via\nproton-proton interactions are constrained by the related gamma-ray production from \u00190\ndecay and its contribution to the di\u000buse extragalactic gamma-ray backgrounds. Compar-\nisons with observations have shown that starburst galaxies cannot be the dominant source\npowering the di\u000buse neutrino \rux [24,25]. This favors a photohadronic origin for the neutri-\nnos, because mechanisms with proton-photon interactions can reproduce the spectral shape\nand \ravor composition of the observed neutrinos [26]. It is plausible that photohadronic\nsources might be hidden in GeV-TeV gamma-rays, because the parameters that cause a\nhigh neutrino production e\u000eciency at the same time produce high opacity to gamma-rays\nat the highest energies [27].\nTo summarize, if one single class of sources dominates the observed high-energy neutrino\n\rux, it likely obeys the following criteria: (i) the neutrino production occurs by photo-\nhadronic interactions, (ii) the photon counterpart, if any, is more likely to be found in the\nKeV-MeV energy bands than in the GeV-TeV bands, and (iii) the sources should be abun-\ndant enough in the universe, so that each of them individually is su\u000eciently weak to evade\nconstraints from neutrino multiplets searches [28{30].\nIn this work, we study the di\u000buse \rux of neutrinos from TDEs, which may address these\nthree criteria. Speci\fcally, we compute the neutrino production from a single TDE, and use\nthe existing information on the TDE demographics to compute the di\u000buse \rux of neutrinos\nfrom TDEs. Parameters motivated by Swift J1644+57 observations will be used, with\nemphasis on the physical scenarios that could reproduce the observed neutrino signal at\nIceCube. Signatures of a TDE neutrino \rux that could be relevant for future observations\nwill be discussed as well.\nThe paper is structured as follows. In Sec. 2 the physics of tidal disruption, and the cosmo-\nlogical rate of TDEs are discussed. Sec. 3 presents the details of the numerical calculation\nof the neutrino \rux from a single TDE, with results for several illustrative combinations of\nparameters. In Sec. 4 results are given for the di\u000buse \rux expected at Earth from a cosmo-\nlogical population of TDEs. A discussion on the compatibility with the IceCube data, and\nfuture prospects, is given in Sec. 5.\n2 Physics of TDEs\n2.1 Tidal disruption and jet formation\nThe basic physics of tidal disruption of star by a SMBH was \frst discussed in the 1970s and\n1980s, in a number of seminal papers [1{4]; see Refs. [31,32] for more recent examples. Here\nwe summarize the main aspects for a star of solar mass and radius, m=M\f'1:99\u00021033g\nandR=R\f'6:96\u00021010cm. LetMbe the mass of the SMBH.\nAs it moves closer to the SMBH, the star can be deformed, and ultimately destroyed by\ntidal forces. This happens when the star reaches a distance close enough to the SMBH\nso that the force on a mass element (inside the star) due to the self-gravity of the star is\n2comparable to the force produced on the same element by the SMBH. This distance is the\ntidal radius\nrt=\u00122M\nm\u00131=3\nR'8:8\u00021012cm\u0012M\n106M\f\u00131=3R\nR\f\u0012m\nM\f\u0013\u00001=3\n; (1)\nand the orbital period at such radius is\n\u001ct= 2\u0019\u0012r3\nt\n2MG\u00131=2\n'104s\u0012R\nR\f\u00133=2\u0012m\nM\f\u0013\u00001=2\n: (2)\nIt is useful to compare these quantities with the SMBH Schwarzschild radius\nRs=2MG\nc2'3\u00021011cm\u0012M\n106M\f\u0013\n; (3)\nand the corresponding time scale\n\u001cs\u00182\u0019Rs=c'63 s\u0012M\n106M\f\u0013\n: (4)\nHere\u001csis a good approximation of the orbital period at the innermost stable circular orbit,\nfor a Schwarzschild black hole (in the observer's frame, see e.g.Ref. [33]).\nComparing rtwithRsshows that the star will be swallowed whole, with no prior disruption,\nifM>\u0018Mmax'108M\f. Here a more conservative value, Mmax'107:2M\fwill be used,\nmotivated by Ref. [7]. As will be clear from Sec. 4.1, our results for the di\u000buse neutrino \rux\ndepend weakly on Mmax.\nIn the case where disruption occurs, the main phenomenology can be described analytically\nin terms of basic physics arguments [2]. About \u00181=2 of the mass of the disrupted star\nbecomes bound to the SMBH and is ultimately accreted on it. Therefore, an upper limit to\nthe energy emitted in this event is\nEmax\u0018M\fc2=2'9\u00021053erg; (5)\nassuming that the change in the SMBH's own internal energy is negligible. After a dark\ninterval ofO(10) days { the time-scale of infall of the tightest bound debris { rapid accretion\nof matter on the SMBH begins. In circumstances where the mass infall rate is su\u000eciently\nhigh { depending on the detailed dynamics of the stellar debris (see e.g. Refs. [34,35]) { a\n\rare is generated, with super-Eddington luminosity that declines with time as \u0018t\u00005=3. The\n\rare vanishes rapidly after a time \u0001 T\u0018O(0:1\u00001) yr , when the infall rate drops below\nthe Eddington rate.\nExtreme, highly super-Eddington \rares are expected if a relativistic jet is launched. The\nbest known jetted TDE is Swift J1644+57. Its X-ray \rare had an isotropic equivalent\nluminosityLX'1047:5erg s\u00001over a time interval \u0001 T'106s, for a total energy in X-rays\nEX=LX\u0001T'3\u00021053erg. Note that the applicable luminosity is arguable, as the average\n(versus peak) luminosity depends on the time interval considered because the luminosity\n3drops with time, see e.g.Ref. [5]. Here we choose the time window and luminosity which\nwe \fnd most appropriate for neutrino production.\nA minimum variability time tv\u0018102s was observed in the X-ray luminosity, and a Lorenz\nfactor \u0000\u001810 for the jet was inferred from the data [8]. The energy EXis therefore\nwell below the (beaming factor-corrected) maximal isotropic equivalent energy \u00182\u00002Emax.\nParameters motivated by Swift J1644+57 are considered typical, as they are overall similar\nto those of the other well established jetted TDE, Swift J2058.4+0516 [9].2They were used\nin [17] for neutrino \rux estimation, and will be used here as well as benchmark (see Sec. 3.2).\n2.2 Rate of TDE\nThe cosmological rate of TDEs is given by the product of the the rate of TDEs per black\nhole _NTD, the SMBH mass function \u001e(z;M), de\fned as the number of black holes per\ncomoving volume and per unit mass at redshift z, and the occupation fraction, focc(M),\nwhich represents the probability that a SMBH is located at the center of a host galaxy:\n_\u001a(z;M) = _NTD(M)focc(M)\u001e(z;M): (6)\nWe describe these quantities following mainly Shankar et al. [36], Stone and Metzger [37],\nand Kochanek [7]. In [36] the black hole mass function is calculated for M\u0015105M\f, using\ninformation from quasar luminosity functions, and estimates of merger rates to model the\ngrowth of black holes. Constraints from local estimates of the black hole mass function are\ntaken into account as well. It is found that \u001e(z;M) declines with z{ roughly as (1 + z)\u00003\n{ and scales approximately like M\u00003=2for allzand for 105M\f<\u0018M<\u0018107:5.\nThe occupation fraction focccan be modeled in \frst approximation as a step function, with\nfocc'1 (focc'0) above (below) a cuto\u000b mass Mmin. Below this mass a number of e\u000bects\nsuppress the probability that low mass SMBH are found in the center of galaxies. For\nexample, a low mass SMBH is more likely to be ejected from the host galaxy, see e.g., [38].\nIn [37] several possibilities are discussed for the cuto\u000b, with Mmin\u0018(2\u0002105\u00007\u0002106)M\f.\nInstead, in [7] the entire mass range 105M\f<\u0018M<\u0018107:5M\fis used for the calculation\nof TDE rates. Even lower mass values, down to M'104:5M\fare considered in [38],\nmotivated by the values of Mreported in recent observation of dwarf galaxies [39,40]. We\nconsider the Shankar et al. mass function, extrapolated at M < 105M\f, and use, for\nillustration, several values in the interval Mmin= (104:5\u0000106:5)M\ffor the cuto\u000b mass.\nThe rate of tidal disruptions (jetted and non-jetted) per SMBH decreases weakly with\nincreasingM; here we use _NTD'10\u00003:7(M=106M\f)\u00000:1yr\u00001[7], which is close to the upper\nlimit obtained from the ASAS-SN data [41]. We consider only the total rate of disruptions\nper SMBH, neglecting their distribution in the mass of the disrupted star m. As shown\nin [7], this distribution ranges in the interval m\u0018(0:1\u00002)M\f, withm'0:3M\fa typical\nvalue. Variations of min this interval would only produce weak e\u000bects (less than a factor\nof\u00182) in our calculations (see Eqs. (1) and (2)). These e\u000bects are subdominant compared\nto those ofMvarying over two orders of magnitude, and therefore they are neglected here.3\n2Note however that for Swift J2058.4+0516, noise limited the sensitivity to time variability to scales\n410510610710-10\n10-11\n10-12\n10-13\n10-14\n10-15\nM/M⊙ρ(Mpc-3yr-1M⊙-1)\nz=0.1z=1z=2\n0 1 2 3 4 5 610-910-810-710-610-5\nzR(z) (Mpc-3yr-1)104.5M⊙105M⊙105.5M⊙106M⊙106.5M⊙Figure 1: Left panel: The di\u000berential rate of TDEs (jetted and not jetted) _ \u001a(z;M) as a\nfunction of Mfor selected values of z(labels on curves). The mass interval [ Mmin;Mmax] =\n[104:5M\f;107:2M\f] is used here. Right panel: The total volumetric rate of TDEs, R(z),\nforMmax= 107:2M\fand di\u000berent values of Mmin(labels on curves).\nFig. 1 shows the di\u000berential TDE rate, _ \u001a(z;M)/M\u00001:6(left), and the total rate R(z) =RMmax\nMmin_\u001a(z;M)dM, as a function of z(right). As expected, R(z) is dominated by the lowest\nmass SMBH, decreasing by a factor \u0018100:6'4 whenMminis increased by an order of\nmagnitude.\nConsider the e\u000bective rate of observable jetted TDEs ~R, which can be estimated as ~R=\nR\u0011=(2\u00002) with the beaming factor 1 =(2\u00002) and the fraction \u0011of all TDEs producing a jet.\nUsing \u0000'10 [8] and\u0011\u00180:1 [8], the suppression factor between observable jetted and all\nTDEs can be estimated to be \u00185\u000110\u00004. Consequently, the local rate of observable jetted\nTDEs is expected to be ~R(0)'0:35\u000010 Gpc\u00003yr\u00001, depending on Mmin. Note that this\nrate is still subject to possible selection biases if one compares it to data.\nIt is interesting to compare the expected jetted TDE rate ~R(0) to constraints from current\nIceCube data [28{30]. For example, Ref. [29] discusses the case of transient sources, under\nthe assumption that they contribute to most of the astrophysical neutrino \rux observed at\nIceCube. The main result is that rare but powerful transients, with a local rate ~R(0)<\n10 Gpc\u00003yr\u00001, can be excluded within \fve years of operation (corresponding to present\ndata) from the non-observation of multiplets. These bounds apply to short transients (like\nGamma Ray Bursts); they relax somewhat for longer lived sources like TDEs.4However,\nit is already evident from that estimate that a di\u000buse neutrino \rux from TDEs describing\nlarger than\u0018103s. The smallest time scale of variability observed was at the level of \u0018104s [9].\n3Note also that the mass of the disrupted star, m, does not directly enter our calculation, where we use\nobserved values of the X-ray luminosity and total energy, LXandEX, as inputs; see Sec. 3.1.\n4The other references come to similar conclusions, with some dependence on source evolution history,\nspectral shape etc.\n5Table 1: Parameters used in this work, unless noted explicitly otherwise. These parameters\napply to the SMBH frame.\nSymbol De\fnition Standard value\ntv Variability timescale 102s\n\u0000 Lorentz factor 10\n\u0018p Baryonic loading (energy in protons versus X-rays) 10\n\u0018B Magnetic loading (energy in magnetic \feld versus X-rays) 1\nkp Proton spectral index 2\nEX Isotropic equivalent energy in X-rays 3 \u00011053erg\n\u0001T Duration of X-ray \rare 106s\n\"X;br Observed X-ray break energy 1 keV\n\u000b Lower X-ray spectral index \"<\"X;br 2=3\n\f Higher X-ray spectral index \">\"X;br 2\n\u0011 Fraction of TDEs with jet formation (used for di\u000buse \rux) 0.1\nIceCube data must be dominated by the low mass part of the SMBH mass function in order\nto avoid the tension with these constraints. A next generation instrument [42] will be more\nsensitive, being able to identify sources that are more frequent but less bright. A bound\nmight be as strong as ~R(0)<103Gpc\u00003yr\u00001, which can clearly test the TDE hypothesis.\nWe will discuss another way to test of the TDE hypothesis, using the \rux \ravor composition,\nin Sec. 4.2.\n3 Neutrino production in a TDE-generated jet\n3.1 Photohadronic processes and neutrino emission\nFor the computation of the neutrino \rux from a single TDE, we follow the relativistic wind\ndescription in Ref. [17]. We apply however methods as they have been used in state-of-\nthe-art calculations for relativistic winds in Gamma-Ray Bursts before [43{45] using the\nNeuCosmA software. A comparison between the numerical computation used in this study\nand the analytical estimate can be found in App. A. Our standard parameter values are\nsummarized in Table 1. We note that the approach in this section could be also easily\napplied to TDE stacking analyses, for which the required input is listed in Table 1 (except\nfrom\u0011, but including z); in fact, a very similar method has been used for Gamma-Ray\nBurst stacking in Ref. [46].\nThe photon spectrum is assumed to \ft the observed spectral energy distributions of TDEs\ndescribed as a broken power law with a spectral break, parameterized in the shock rest\n6frame (SRF) by (we use primed quantities for the SRF)\nN0\n\r(\"0) =C0\n\r\u00018\n>><\n>>:\u0010\n\"0\n\"0\nX;br\u0011\u0000\u000b\n\"0\nX;min\u0014\"0<\"0\nX;br\u0010\n\"0\n\"0\nX;br\u0011\u0000\f\n\"0\nX;br\u0014\"0<\"0\nX;max\n0 else; (7)\nwhereC0\n\ris a normalization factor. Typical values can be found in Table 1, where \"0\nX;br=\n\"X;br(1 +z)=\u0000, and\"0\nX;minand\"0\nX;maxcan be translated from the observed energy band cor-\nrespondingly. We use the Swift energy band with \"X;min'0:4 keV and\"X;max'13:5 keV [8]\nto de\fne the target photon spectrum, unless noted otherwise. Note that one may de\fne a\nbolometric correction to that (such as one may extend the target photon spectrum beyond\nthat range), but the increase of the neutrino \rux would be small as long as the break energy\nwas su\u000eciently well covered.\nThe proton spectrum is assumed to be a cut-o\u000b power law with a spectral index kp'2\nexpected from Fermi shock acceleration\nN0\np(E0\np) =C0\np\u0001(\u0010\nE0\np\nGeV\u0011\u0000kp\n\u0001exp\u0010\n\u0000E02\np\nE02p;max\u0011\nE0\np\u0015E0\np;min\n0 else: (8)\nThe maximal proton energy E0\np;maxis determined automatically by balancing the accelera-\ntion rate with synchrotron loss and adiabatic5cooling rates and comes from the cuto\u000b from\nacceleration, and we choose E0\np;min'1 GeV. However, for the neutrino production in TDEs\nthe maximal proton energy is not so important as long as E0\np;max&108GeV, because the\nmagnetic \feld e\u000bects on the pions and muons will dominate the maximal neutrino energies\nand the energy budget only logarithmically depends on E0\np;minandE0\np;maxforkp= 2. Con-\nsequently, the chosen shape of the (super-exponential) cuto\u000b, which may be relevant for the\ndescription of ultra-high energy cosmic rays, does not have any impact on the neutrino \rux\ncomputation.\nThe isotropic equivalent energy EX(in erg) is given in the SMBH frame6as\nEX=4\u0019d2\nL\n(1 + z)SX: (9)\nin terms of the X-ray \ruence SX(in units of erg cm\u00002). Note again that this \ruence is\nassumed to be measured in the energy band from 0.4 to 13.5 keV . The isotropic energy EX\ncan be obtained from the X-ray luminosity by EX=LX\u0001Tobs=(1 +z). Here we see already\none known subtlety: redshift enters here because the observed duration \u0001 Tobsis de\fned\nin the observer's frame and EXandLXin the SMBH frame. For the computation of the\ndi\u000buse \rux, it will be most convenient to de\fne all quantities in the SMBH frame, including\n5The adiabatic cooling timescale is chosen to be similar to the dynamical timescale, which means that\nit is implied that the dynamical timescale can limit the maximal energy.\n6A clari\fcation is due on the de\fnition of frames of reference used here. For brevity, the wording \\SMBH\nframe\" will be used to indicate a frame of reference of an observer at rest with respect to the SMBH and\nlocated at a distance Lfrom it such that Rs\u001cL\u001cc=H 0. Instead, \\observer's frame\" indicates the frame\nof reference of Earth. Energies in the two frames di\u000ber by redshift e\u000bects.\n7\u0001T,tv,\"X;br,\"X;min, and\"X;max, which means that all TDEs with the same parameters will\nbe alike in the SMBH frame. Practically, we implement that by computing the neutrino\n\ruxes for a TDE that takes place at a very small z\u001c1 (where the oberver's frame is\nbasically identical with the SMBH frame). This computation gives the neutrino \ruence,\ni.e., the number of neutrinos of a given \ravor that reach Earth per unit energy per unit\narea,F\u000b(E) (\u000b=e;\u0016;\u001c ). For future use, it is convenient to also consider the number of\nproduced neutrinos of a given \ravor (after oscillations) per unit energy Q\u000b(E), which is\nrelated to the \ruence by F\u000b(E) =Q\u000b(1 +z)3=(4\u0019d2\nL) (which is F\u000b(E) =Q\u000b=(4\u0019L2) for\nsmallzwith the lookback distance L'dL). FromF\u000bandQ\u000b, the \ruence of neutrinos\nfor a generic TDE at any redshift zis obtained by the appropriate re-scaling. We checked\nthat the di\u000berence between the two methods (all parameters alike in SMBH frame versus\noberver's frame) is small.\nThe isotropic energy can be easily boosted into the SRF by E0\nX=EX=\u0000. Assuming that the\nemitted photons are coming from synchrotron emission of electrons (or mainly interact with\nelectrons), the amount of energy in electrons and photons should be roughly equivalent. In\na baryonically dominated relativistic wind, we have\nE0\np'E0\nX\u0018p (10)\nwhere\u0018pis the ratio between proton and X-ray energy { referred to as \\baryonic loading\".\nWe compute the photon and proton densities in the SRF de\fning an \\isotropic volume\"\nV0\niso, which is the volume of the interaction region in the source frame assuming isotropic\nemission of the engine. Thus, the assumption of isotropic emission will cancel in the density.\nSimilarly,V0\nisois an equivalent volume in the SRF where only the radial direction is boosted,\nwhich is given by\nV0\niso= 4\u0019R2\nC\u0001\u0001d0(11)\nwith shell width \u0001 d0'\u0000ctv=(1+z) obtained from the variability timescale, and the collision\nradiusRC'2 \u00002ctv=(1+ z). From Eq. (11) we can then estimate the size of the interaction\nregion asV0\niso/\u00005t3\nv, which means that it strongly depends on the \u0000 factor.\nBecause of the intermittent nature of TDEs, the total \ruence is assumed to be coming\nfromN'\u0001T=tvsuch interaction regions. Now one can determine the normalization of the\nphoton spectrum in Eq. (7) from\nZ\n\"0N0\n\r(\"0)d\"0=E0\nX\nNV0\niso: (12)\nif one assumes that the target photons can escape from the source. Note that EX=N'LXtv,\nwhich means that one can use LXequivalently to de\fne the target photon density or pion\nproduction e\u000eciency { as we do in the analytical approach in App. A.\nSimilarly, one can compute the normalization of the proton spectrum in Eq. (8) by\nZ\nE0\npN0\np(E0\np) dE0\np=\u0018p\u0001E0\nX\nNV0\niso: (13)\n8Given that the ratio between magnetic \feld and X-ray energies is \u0018B, one has in addition7\nU0\nB=\u0018B\u0001E0\nX\nNV0\nisoorB0=s\n8\u0019\u0001\u0018B\u0001E0\nX\nNV0\niso: (14)\nOnce the proton and photon densities and the magnetic \feld are determined, the rest of the\ncomputation is straightforward. We solve the time-dependent di\u000berential equation system\nfor the pion and consequent muon densities, including photo-meson production based on\nSOPHIA [47] (with an updated method similar to Ref. [48] and \frst used in Ref. [49]).\nWe also include the helicity-dependent muon decays [50] and the leading kaon production\nmode. The radiation processes of the secondary pions, muons and kaons include synchrotron\nlosses, adiabatic losses, and escape through decay, which lead to characteristic cooling breaks\ndi\u000berent for pions, muons, and kaons, and a transition in the \ravor composition [51]; see\nApp. A for an analytical discussion.\n3.2 Neutrino \ruence from a tidal disruption event\n3.2.1 Modeling the jet: inputs and assumptions\nConsidering that the masses of the black holes responsible for TDEs may vary over more\nthan two orders of magnitude, it is natural to expect a certain degree of diversity in the\njetted TDEs. Here we estimate how certain parameters of the jet may depend on the\nSMBH mass. For the sake of generality, we choose parameter scalings that either have an\nobservational basis, or a direct connection to fundamental physics.\nLet us \frst discuss three parameters that most in\ruence the neutrino \rux, the minimum\nvariability time tv, the Lorenz factor \u0000, and the luminosity LX. Observations of Active\nGalactic Nuclei indicate a mild dependence of \u0000 on Mwhich is best \ft by [52]\n\u0000 =\u0012M\n10M\f\u00130:2\n: (15)\nThis corresponds to \u0000 \u00186;10;and 16 forM= 105;106;and 107M\f, respectively, which are\ncompatible with observational estimates for Swift J1644+57. The relationship in Eq. (15) is\nconsistent [52] with the magnetically arrested accretion \row model [53, 54], which predicts\n\u0000 to depend on the square of the SMBH spin, which in turn increases with M. Still, it is\nnot known if Eq. (15) applies to the broader set of galaxies (most of them not hosting an\nactive nucleus) of interest here. Therefore, it should be considered as a mere possibility,\nalthough theoretically substantiated.\nFor the minimum variability time, tv, it is reasonable to make the hypothesis that it be\nrelated to the smallest possible time scale available in a black hole, the Schwarzschild \\time\"\n\u001cs, Eq. (4). Typical values of \u001csare consistent with the variability seen in Swift J1644+57,\n7With this de\fnition of \u0018B, the magnetic loading is slightly di\u000berent from Ref. [17], who de\fne the\nmagnetic energy with respect to the wind luminosity (which is a factor of three higher than the radiated\nenergy). Their magnetic loading is therefore e\u000bectively a factor of three higher than ours.\n9and indeed the hypothesis of a connection to the Schwarzschild time is used in interpretations\nof Swift J1644+57 data to infer the mass of the parent SMBH [8].\nLastly, we can expect some dependence of LXonM. Combined data on jetted and non-\njetted TDEs are well \ft by a luminosity function that scales like the inverse square of\nLX[55]:8\n_\u001a(M)\n\u00002(M)/L\u00002\nX; (16)\nwhich implicitly gives a scaling LX=LX(M)/\u0000 _\u001a\u00001=2. Eq. (16) is just a a possibility.\nIndeed, it is also possible that X-ray \rares from jetted TDEs do not follow the same trend as\nthe ones from non-jetted TDEs (see Fig. 11 of Ref. [55]). There are also di\u000berent scalings,\nresulting from theoretical relationships between X-ray luminosity and SMBH mass ( e.g.\nRef. [31,32]) such as coming from a possible connections between the peak X-ray luminosity\nand the rate of accretion.\nFor the purpose of illustrating possible di\u000berent degrees of dependence of the neutrino \rux\non the SMBH mass, M, we present results for four scenarios (see Table 2):\n\u000fBase case . Here no dependence on Mis considered at all, and a single set of jet\nparameters is assumed to describe all jetted TDE. The parameters are the same as\nin [17], see Table 1.\n\u000fWeak scaling case. Here the weak dependence of \u0000 on M, motivated by AGN obser-\nvations, is included, Eq. (15). All other parameters are as in the base case, except\nthe variability time, which is taken to be tv= 103s (which is more conservative for\nneutrino production). This value is meant to illustrate a di\u000berent possibility, relative\nto the base value in Table 1, and is motivated by the median (rather than minimum)\nscale of time variability observed in Swift J1644+57.\n\u000fStrong scaling case. Here both \u0000 and tv\u0018\u001csscale withMas given in Eqs. (4) and\n(15). This means that, in addition to \u0000 scaling in the Weak case, it is assumed that\nthe time variability of the jet is correlated with the period of the lowest stable orbit\nof the star disrupted by the SMBH.\n\u000fLumi scaling case. Here the same scalings as the Strong case are used, and additionally\nthe scaling of LXis included, as in Eq. (16). Explicitly, considering that _ \u001a(M)/M\u00001:6\nand \u00002/M0:4(cf., Eq. (15)), Eq. (16) implies that LX/M. We therefore take\nLX= 3\u00011047M=(106M\f) erg s\u00001, such that the luminosity is the same as in the\nStrong case for the benchmark SMBH mass M= 106M\f.\nThe general e\u000bect of the scalings proposed here can be understood by embedding Eqs. (4),\n(15), and (16) in the analytical formalism in App. A. Considering that \u0000 and/or tvincrease\nwithM, we expect: (i) a decrease of the pion production e\u000eciency fp\r, and therefore of the\nneutrino production, with the increase of M. Indeed,fp\r/\u0000\u00004t\u00001\nv(Eq. (20)), which implies\n8One can substitute the integral over Mby one inLXin the di\u000buse \rux Eq. (17) for LX/M\u000b, in\nwhich case one observes that ( _ \u001a(M)=\u00002(M) corresponds to the luminosity distribution function.\n10Table 2: Our standard scaling scenarios.\nCase tv[s] \u0000 LX[erg/s] Reference(s)\nBase case 10210 3\u00011047Table 1, Ref. [17]\nWeak scaling case 103\u0000 =\u0010\nM\n10M\f\u00110:2\n3\u00011047Eq. (15), Ref. [52]\nStrong scaling case tv'63M\n106M\f\u0000 =\u0010\nM\n10M\f\u00110:2\n3\u00011047Eqs. (4), (15), Ref. [52]\nLumi scaling case tv'63M\n106M\f\u0000 =\u0010\nM\n10M\f\u00110:2\n3\u00011047M\n106M\fRef. [55] (see text)\nfp\r/M\u00000:8(fp\r/M\u00001:8) in the Weak (Strong) case; similar results can be found from\nEq. (11). This means that smaller SMBH masses imply higher pion production e\u000eciencies.\n(ii) An increase of the proton, pion and muon break energies with M, resulting in a hardening\nof the neutrino spectrum. This is because these energies scale as, respectively, Ep;br/\u00002,\nE\u0019;br/\u00004tv,E\u0016;br/\u00004tv(Eqs. (21), (22) and (23)). This means that Ep;br/M0:4in both\nscalings, and E\u0019;br;E\u0016;br;/M0:8(E\u0019;br;E\u0016;br;/M1:8) in the Weak (Strong) case.\nWhen the dependence of LXonMis included (Lumi case), the neutrino \rux roughly scales\nas\u001e/LXfp\r/M0:2, thus increasing slightly with M, contrary to the other scenarios\nconsidered here.\nAll our neutrino \rux calculations include neutrino oscillations in vacuum, with the exception\nof Fig. 5 in App. A. We have checked that a possible envelope ahead of the jet, caused\nby debris of the disrupted star, is su\u000eciently thin that matter-driven \ravor conversion is\nnegligible. Due to the extremely long propagation distance, only the e\u000bect of averaged\noscillations is observable in a detector, and the corresponding \ravor conversion probabilities\ndepend only on the mass-\ravor mixing matrix, U:P(\u0017\u000b!\u0017\f) =P3\ni=1jU\u000bij2jU\fij2, where\n\u000b;\f=e;\u0016;\u001c andi= 1;2;3 runs over the neutrino mass eigenstates. We use the standard\nparameterization of the mixing matrix, with the following values of the mixing angles:\nsin2\u001212= 0:308, sin2\u001223= 0:437, sin2\u001213= 0:0234 (see e.g.Ref. [56]).\n3.2.2 Neutrino \ruence at Earth\nFig. 2 (left column) shows the \ruence, E2F\u0016, of\u0017\u0016+ \u0016\u0017\u0016(after oscillations) for a single\nTDE, for the Base case and for the scaling scenarios outlined above and for di\u000berent values\nofM. The Base case result does not depend on Mbecause the neutrino \rux is computed\nusingLXand \fxed values of \u0000 and t\u0017.\nThe Weak case with M= 106M\fdi\u000bers from the Base one only for the value of tv,\nwhich is one order of magnitude larger. The neutrino \ruence is suppressed by roughly the\nsame factor, as expected from the suppression of the pion production e\u000eciency (Sec. 3.2.1,\nApp. A). From the scaling of \u0000 (Eqs. (15) and (20)) a suppression by \u0018100:8\u00187 is expected\nfor every decade of increase of M; this matches the behavior in Fig. 2 well at low energy.\nThe \fgure also shows the expected hardening of the neutrino spectrum with increasing M,\n1110410510610710810910-510-40.0010.010.11\nEGeVE2FmHELHGeV cm-2L\n1041051061071081090.460.480.500.520.54\nEGeVFmHFe+FtL\nPion beamMuon damped\n10410510610710810910-510-40.0010.010.11\nEGeVE2FmHELHGeV cm-2L\n105M\n106M\n107M\n1041051061071081090.460.480.500.520.54\nEGeVFmHFe+FtL\nPion beamMuon damped\n10410510610710810910-510-40.0010.010.11\nEGeVE2FmHELHGeV cm-2L105M\n106M\n107M\n1041051061071081090.460.480.500.520.54\nEGeVFmHFe+FtL\nPion beamMuon damped\n10410510610710810910-510-40.0010.0100.1001\nE/GeVE2F��(E) (GeV cm-2)\n105M⊙106M⊙107M⊙\n1041051061071081090.460.480.500.520.54\nE/GeVFμ/(Fe+Fτ)\nPion beamMuon dampedFigure 2: The \ruence of \u0017\u0016+ \u0016\u0017\u0016(left panels) and the \ravor ratio (right panels), including\n\ravor mixing, as a function of the neutrino energy, for a single TDE at z= 0:35. Shown are\nresults for the Base, Weak, Strong, and Lumi cases (from top to bottom panes). In each\n\fgure, the curves correspond to di\u000berent SMBH mass, M= 105M\f(solid),M= 106M\f\n(dashed),M= 107M\f(dotted). For the \ravor ratio, the horizontal lines show the values\nexpected for the standard pre-oscillation compositions ( F0\ne:F0\n\u0016:F0\n\u001c) = (1;2;0) (pion beam)\nand (F0\ne:F0\n\u0016:F0\n\u001c) = (0;1;0) (muon damped source).\n12due to the increase of the break energies (Sec. 3.2.1).\nFor the Strong case, the result for M= 106M\fis very similar to the Base case, due to the\nnearly identical parameters. As discussed in Sec. 3.2.1, here the degree of enhancement with\nthe decrease of Mis stronger; this means that low mass black holes will strongly dominate\nthe neutrino \rux because the time variability of the jet is assumed to be correlated with\nthe period of the innermost stable orbit { which is shorter for smaller SMBH masses. We\nalso observe the expected stronger broadening of the spectrum towards higher energies due\nto the stronger scaling of the pion and muon break energies. In the Lumi scaling scale, the\nenhancement from small Mis compensated by the scaling LX/M, which means that the\n\rux actually increases with M{ as discussed above.\nIn summary, for a single TDE, in most cases, the quantity E2F\u0016(withF\u0016the \ruence of\nmuon neutrinos) peaks at E\u0018106\u0000107GeV, with a maximum value E2F\u0016\u001810\u00005\u0000\nfew\u000210\u00003GeVcm\u00002. In the scenarios with parameter scaling, the neutrino \rux increases\nwith decreasing M. In the case of Strong scaling, the \ruence can be as high as E2F\u0016\u0018\n10\u00001GeVcm\u00002atE\u0018few\u0002105GeV.\nFig. 2 (right column) also illustrates the \ravor ratio f\u0016=F\u0016=(Fe+F\u001c) of neutrino \rux\nafter \ravor mixing in the three models considered here, and selected values of M. This\n\ravor ratio roughly corresponds to the observable ratio between muon tracks and cascades\nin IceCube, and has been widely used in the literature to study the impact of a change of\nthe \ravor composition. For all models there is an energy window where f\u0016'0:48; this is\nthe region where \u0016decay proceeds unimpeded, E\u0016<\u0018E\u0016;br. In this regime, for each pion in\nthe jet, two muon neutrinos and one electron neutrinos are expected, so that the original\n(before oscillations) \ravor composition of the neutrino \rux is ( F0\ne:F0\n\u0016:F0\n\u001c) = (1 : 2 : 0).\nAt higher energy, E\u0016>\u0018E\u0016;br, the ratiof\u0016increases to f\u0016'0:536, re\recting the transition\nto the regime where \u0016absorption dominates over decay, so that the original \ravor content\nis (F0\ne:F0\n\u0016:F0\n\u001c)'(0;1;0) WhileE\u0016;bris nearly the same for the Base and Weak cases,\nit is much lower for the Strong and Lumi scenarios at lower M, so that for M= 105M\f,\nalready at energies of a few PeV the neutrino \rux enters the muon damped regime.\n4 Di\u000buse neutrino \rux from TDE\n4.1 Di\u000buse \rux prediction: Spectrum and \ravor composition\nThe di\u000buse \rux of neutrinos of a given \ravor \u000bfrom TDEs { di\u000berential in energy, time,\narea and solid angle { is obtained by convolving the neutrino emission of a single TDE with\nthe cosmological rate of TDEs (see, e.g. [57] for the formalism):\n\b\u000b(E) =c\n4\u0019H0ZMmax\nMmindM\u0011\n2\u00002(M)Zzmax\n0dz_\u001a(z;M)Q\u000b(E(1 +z);M)p\n\nM(1 +z)3+ \n \u0003; (17)\nwhereQ\u000bis the number of neutrinos emitted per unit energy in the SMBH frame, and\nE0=E(1 +z) is the neutrino energy in the same frame; Eis the energy observed at Earth.\n13Here\u0011is the fraction of TDEs that generate relativistic jets, which is assumed to be a\nconstant,\u0011'0:1. This value has been suggested as plausible on the basis of a possible\nsimilarity with AGN [8]. The beaming factor 1 =(2\u00002(M)) accounts for the fraction of jets\nalong our line of sight. Its use in Eq. (17) is consistent with the fact that the same equation\ncontains the physical comoving rate of TDEs (not corrected for beaming or observational\nbiases). Due to the decline of _ \u001a(z;M) withz(Fig. 1), the \rux \b \u000bdepends only weakly\nonzmax; here we take zmax= 6, which is the maximum value considered in SMBH mass\nfunction calculation of Shankar et al. [36].\nFig. 3 shows the di\u000buse muon neutrino \rux, E2\b\u0016(E), for the four scaling scenarios of\ninterest, and Mmin= 105;106M\f(solid curves). The shaded area models the uncertainty\nonMmin, which is varied in the interval Mmin= [104:5;106:5]M\f. In all cases, the spectrum\nresembles the spectrum of a single TDE with M\u0018Mminandz\u001c1 (Fig. 2), as expected\nsince the TDE rate is a decreasing function of zandM(Fig. 1). For the Base, Weak\nand Lumi cases (with Mmin= 105M\f), the di\u000buse \rux has a maximum of E2\b\u0016(E)\u0018\n10\u00009GeVcm\u00002s\u00001sr\u00001between 1 PeV and 10 PeV. For the Weak case, we note the stronger\ncontribution of the lowest mass SMBH, M= 105\u0000106M\f, re\recting the more powerful\nneutrino emission as Mdecreases (Sec. 3.2). The same features are observed for the Strong\ncase, with an even more enhanced contribution of the lowest mass SMBH, which causes the\n\rux to peak at lower energy, E\u00180:3 PeV. The dependence on Mminis, on the other hand,\nrelatively mild for the Base and Lumi cases.\nThe post-oscillation \ravor ratio for the di\u000buse \rux is shown in Fig. 3 (right column). Like\nthe \ruence, it mainly follows the corresponding quantity for a single TDE with lowest M\nand lowest z(Fig. 2). In the Lumi case, the contributions from lower and higher Mare\ncomparable at about 10 PeV. The observational implications of the energy dependence of\nthe \ravor composition will be discussed in the next section.\nBefore closing this section, let us brie\ry comment on constraints on TDEs from X-ray\nsurveys. As a consistency check, in App. B we present the di\u000buse X-ray \rux corresponding\nto the four scaling scenarios in Table 2. This \rux is found to be consistent with observations\n(see Appendix).\n4.2 Comparison to IceCube data\nLet us now discuss the impact of current and future IceCube data on the search for neutrinos\nfrom TDEs. After about 6 years of data taking, IceCube has established that the Earth\nreceives a \rux of astrophysical neutrinos which is di\u000buse in nature, in \frst approximation,\nand at the level of E2\b\u0018few\u000210\u00008GeVcm\u00002s\u00001sr\u00001, at observed energies between\n\u001830 TeV and\u0018PeV. Even accounting for large and poorly known uncertainties { which\ndepend in part on the model of the candidate sources { this measurement appears to be in\ntension with the most extreme \rux predictions in Fig. 3. In particular, for the parameters\nof reference used in this work, the Strong scaling scenario with Mmin\u0018104:5\u0000105M\f\nshould be already strongly disfavored by the IceCube data, whereas all the other cases are\ncompatible with IceCube observations.\nOne should consider, however, that the jet parameters have a wide range of plausible values,\n1410410510610710810910-1210-1110-1010-910-810-710-6\nEGeVE2FmHELHGeV cm-2s-1sr-1L\n105M\n106M\n1041051061071081090.460.480.500.520.54\nEGeVFmHFe+FtL\nPion beamMuon damped\nGlashow res.\n10410510610710810910-1210-1110-1010-910-810-710-6\nE/GeVE2Φμ(E) (GeV cm-2s-1sr-1)\n105M⊙106M⊙\n1041051061071081090.460.480.500.520.54\nEGeVFmHFe+FtL\nPion beamMuon damped\nGlashow res.\n10410510610710810910-1210-1110-1010-910-810-710-6\nE/GeVE2Φμ(E) (GeV cm-2s-1sr-1)\n105M⊙106M⊙\n1041051061071081090.460.480.500.520.54\nEGeVFmHFe+FtL\nPion beamMuon damped\nGlashow res.\n10410510610710810910-1210-1110-1010-910-810-710-6\nE/GeVE2Φμ(E) (GeV cm-2s-1sr-1)\n105M⊙106M⊙\n1041051061071081090.460.480.500.520.54\nE/GeVΦμ/(Φe+Φτ)\nPion beamMuon damped\nGlashow res.Figure 3: The di\u000buse \rux of \u0017\u0016+ \u0016\u0017\u0016(left panes) and the corresponding \ravor ratio (right\npanes) at Earth, including \ravor mixing, as a function of the neutrino energy, for the Base,\nWeak, Strong, and Lumi scaling cases (top to bottom), for Mmin= 105M\f(solid), and\nMmin= 106M\f(dashed). In the \rux plots, the shaded regions show the variation corre-\nsponding to varying Mminin the interval Mmin= [104:5;106:5]M\f. In the \ravor ratio \fgures,\nthe horizontal lines show the values expected for the standard pre-oscillation compositions\n(\u001e0\ne:\u001e0\n\u0016:\u001e0\n\u001c) = (1;2;0) (pion beam) and ( \u001e0\ne:\u001e0\n\u0016:\u001e0\n\u001c) = (0;1;0) (muon damped source).\nThe energy of the Glashow resonance in the \ravor composition panels is marked by a dotted\nline.\n15\n\n\n⊥⊥\n⊥⊥⊥ ⊥⊤⊤\n⊤⊤⊤\n⊤⊤\n⊤\n⊤\n3 4 5 6 7 8 9-9.0-8.5-8.0-7.5-7.0-6.5-6.0\nLog[E/GeV]Log[E2Φ(E)/(GeVs-1cm-2sr-1)]WeakscalingStrongscalingLumiscalingnoscalingFigure 4: The spectra for the di\u000buse all-\ravor \rux, for the Strong, Weak, Lumi and Base\n(i.e., no scaling) models (labels on curves), for Mmin= 105M\f. The overall constant\nG=\u0018p\u0002\u0011has been adjusted to saturate the measured IceCube \rux at E'PeV (shown,\ndata points [58,59]), and takes the values G= 0:2;10:9;4:3;8:1 for the Strong, Weak, Lumi\nand Base cases, respectively.\nwhich leads to a a more quantitative question: can TDEs account for most of the IceCube\n\rux, and for what values of the parameters? From Eqs. (13) and (17) we see that \b \u0016scales\ndirectly with\nG\u0011\u0018p\u0002\u0011'10\u00020:1'1; (18)\nevaluated for our standard assumptions. In principle, the neutrino \rux also scales with the\nX-ray luminosity (as both initial proton and the pion production e\u000eciency are proportional\ntoLX), the beaming factor, the minimal and maximal proton energies (as the proton total\nenergy is distributed over that energy range), etc.. However, these scaling factors are less\ntrivial to treat, as, for instance, a higher luminosity will not only increase the pion pro-\nduction, but also the magnetic \feld and therefore the secondary cooling { which partially\ncompensates for that. We therefore do not include them directly in G.\nFig. 4 shows the all-\ravor \ruence for the di\u000buse \rux, for Mmin= 105M\f, with the factor\nGadjusted (values in the \fgure caption) to saturate the IceCube data (shown as well) at\nE'1 PeV. Note that we do not perform a statistical analysis of the data, but rather the\nnormalization of the predicted \rux is chosen so that the data point at 1 PeV is exactly\nreproduced. We see that the Base and Weak cases can describe the data in the 0.1-1 PeV\nenergy range, although at the price of invoking parameter values G\u00188\u000011. Such an\nincrease of the neutrino \rux could come from a factor of ten higher baryonic loading than\nanticipated in Table 1, i.e.,\u0018p\u0018100, or, equivalently, from a higher value of \u0011.\nSuch a large baryonic loading may on the one hand not be unreasonable, as similar values\nare found for gamma-ray bursts \ftting the UHECR data [60]. In our notation, one can easily\ncompare the energy in baryons with the constraint on the energy Eq. (5): For the chosen EX\nand the conservative estimate \u0000 &6 (forM > 105M\f), one \fnds \u0018p.2\u00002Emax=EX\u0018200\nin order to not to violate the constraint on the maximal emitted energy. This constraint\n16is satis\fed here, but the jets will have to be dominated by baryons for low mass black\nholes. However, increasing \u0018pversus\u0011has the problem that it increases the tension with\nthe multiplet constraints in IceCube, whereas increasing \u0011versus\u0018pchanges the fraction of\njetted versus non-jetted TDEs. In addition to requiring a somewhat extreme value of \u0018p,\nthe Base and Weak cases are in overall tension with the data due to their relatively hard\nspectrum, which overestimates the \rux above PeV while underestimating it at lower energy.\nOur nominal assumptions may therefore be more plausible. In di\u000berent words, for the Base\ncase, we \fnd that 1 =8'12% of the observed \rux in IceCube can be described by TDEs,\nwhereas we \fnd 1 =11'9% for the Weak case at an energy of about 1 PeV.\nInstead, the Strong case describes the data best: it reproduces the observed energy spectrum\nwell, with only some tension with the data in the second lowest energy bin, and only slightly\noverestimating the \rux at the highest energy; see Fig. 4. The normalization leads to G'0:2,\ni.e., parameters even more conservative than the reference reference values used in Eq. (18).\nFor example, one may choose \u0018p= 2 and\u0011= 0:1, or\u0018p= 10 and\u0011= 0:02. Regarding\nthe spectral properties, one should keep in mind that in this scenario the neutrino \rux is\nvastly dominated by the lowest mass black holes (see Fig. 3, lower left panel), with a strong\ndependence on Mmin. For smaller Mmin,Mmin\u0018104:5M\f, the description of the data\nbecomes better due to the softening of the neutrino spectrum, and the value of Grequired\nto saturate the measured \rux decreases further. The opposite e\u000bect (worse description of\ndata) is expected for larger Mmin.\nThe Lumi scaling case is found to be in between these two scenarios. At the nominal\nprediction, 23% of the observed IceCube \rux can be described by TDEs. The \rux can,\nhowever, be saturated if the baryonic loading or \u0011are slightly adjusted, such as \u0018p'40 and\n\u0011= 0:1. The spectrum describes the IceCube data with slightly larger cuto\u000b energy.\nIt is interesting to compare our prediction to current IceCube data \fts. For a global analysis\nof data [59], relatively soft spectral indices of the neutrinos are found: \u000b'2:5 for a power\nlaw \ft. A recent through-going muon analysis, however, indicates a spectral index \u000b'\n2 [61]. These \fndings, together with information on the spatial distribution of the events,\nsuggest the possibility that at low energies a softer, possibly Galactic contribution dominates\n(cf., low energy datapoints in Fig. 4, which cannot be reproduced), whereas at high energy,\nan extragalactic component dominates and the spectrum becomes harder [62, 63]. The\ndi\u000buse \rux from TDEs is an example for such an extragalactic hard component.\nIt is especially noteworthy that the Strong and Lumi scaling cases have a unique signature,\napart from the good description of the spectral shape: in this case the \ravor composition\nchanges from a pion beam to a muon damped source at E\u0018PeV (see lower right panels\nof Fig. 3). This indicates the transition to a regime, as the energy increases, where muons\ncool faster by synchrotron losses than they can decay, see Eq. (23). While this e\u000bect can\nprobably not be seen in the current IceCube experiment, it might be visible at the planned\nvolume upgrade IceCube-Gen2 [42]; see Ref. [64] (Figs. 3 and 9 there).\nPerhaps even easier to test is the fact that the di\u000buse \rux becomes muon damped at the\nGlashow resonance, see vertical lines in right panels of Fig. 3. This issue is discussed in\nRef. [65]: if the spectrum is hard enough, Glashow events must be seen in the current\nIceCube experiment after about 10 years of operation even in the p\rcase under realistic\n17assumptions for the photohadronic interactions. The most plausible scenario which can\nevade this constrained is a muon-damped source at the Glashow resonance at 6.3 PeV, for\nwhich the \u0016\u0017eat Earth can only come from oscillated \u0016 \u0017\u0016from the\u0019\u0000contamination at the\nsource. A non-observation of Glashow event rates in IceCube may therefore be a smok-\ning gun signature for a muon damped source at the Glashow energy, and therefore TDEs\nas dominant source class - or alternatives, such as low-luminosity gamma-ray bursts [66],\nmicroquasars [67], or AGN nuclei [26]. IceCube-Gen2 can then be used for more detailed\nsource diagnostics.\n5 Summary and conclusions\nWe have studied the production of high energy neutrinos in baryonic jets generated in\nthe tidal disruption events (TDEs) of stars by supermassive black holes (SMBH). Using the\nNeuCosmA numerical package, detailed results have been obtained for the \ruence and \ravor\ncomposition of a neutrino burst from an individual TDE, and for the di\u000buse \rux of neutrinos\nof each \ravor from all cosmological TDEs. Jet parameters motivated by observations have\nbeen used, and variations of these parameters over the diverse population of parent SMBH {\nin the form of scalings with the SMBH mass M{ have been studied. Four scaling scenarios\nhave been considered, ranging from no scaling at all (all TDE being identical in the SMBH\nframe) to a strong scaling, where the bulk Lorentz factor \u0000 has been varied in a way\nmotivated by AGN observations, and the variability time scale tvhas been assumed to be\ncorrelated with the innermost stable orbital period of the SMBH. We have also considered\na possible luminosity distribution function, related to the SMBH mass distribution. The\ndependence on the occupation fraction of SMBH { in the form of the minimum mass Mmin\nof SMBH that can be found in the core of galaxies { has been studied as well.\nIn summary, we \fnd that:\n\u000fThe largest contribution to the di\u000buse neutrino \rux is expected from the SMBH with\nlower mass located at low redshift z<\u00181. This is because the rate of TDEs decreases\nwithM, and withzas well. The dominance of low MTDEs is stronger in the scenarios\nwith parameter scaling, as discussed above, and weakened if the luminosity scales with\nSMBH mass. In all cases, the spectral features and \ravor composition of the di\u000buse\n\rux generally re\rect the quantities of the lowest mass SMBH, and therefore are very\nsensitive to the cuto\u000b of the SMBH occupation fraction, Mmin.\n\u000fFor the jet parameters of reference (Table 1), and in cases with weak or no scaling,\nTDEs can be responsible for \u001810% of the observed neutrino \rux at IceCube at an\nenergy of about 1 PeV. Instead, for the same parameters, strong scaling and Mmin<\u0018\n105M\f, the nominal neutrino \rux would exceed the IceCube measurement, which\nmeans that this extreme situation is already disfavored by current data, and IceCube\nconstrainsMmin, the baryonic loading \u0018p, and the fraction \u0011of TDE producing jets.\n\u000fAs a consequence, more moderate parameters can be chosen for the strong scaling case\n{ which can describe both normalization and spectral shape of the observed di\u000buse\n18\rux at the highest energies. Examples are \u0011= 0:1 and\u0018p= 2, or\u0011= 0:02 and\n\u0018p= 10. Note that more frequent TDEs with lower baryonic loadings can release\na possible tension with constraints from the non-observation of neutrino multiplets.\nWe also \fnd that a second, possibly softer contribution to the \rux of di\u000berent origin\n(possibly Galactic) is needed to account for the lowest energy neutrino events.\n\u000fFor the strong scaling case, which describes the spectral shape best, the \ravor compo-\nsition changes with increasing energy, and approaches a muon damped source at E>\u0018\nPeV. This signature may be detectable in the next generation upgrade IceCube-Gen2.\nIn addition, recall that so far IceCube has not observed any events at the Glashow\nresonance. If this became a statistically signi\fcant suppression in the future, it could\nbe a smoking gun signature for TDEs as dominant source, because it is expected for\na muon damped source.\n\u000fIf the luminosity of jetted TDEs scales with the SMBH mass { in addition to the\nscalings of the Lorentz factor and of the variability time scale { an intermediate case\nis found which describes the spectrum very well, which exhibits a \ravor composition\nchange at the Glashow resonance, and which can describe about one fourth of the ob-\nserved di\u000buse IceCube \rux at its nominal prediction { or saturate the di\u000buse \rux with\na slight increase of \u0018por\u0011. This case corresponds to a X-ray luminosity distribution\nfunction/L\u00002\nX(Sec. 3.2.1).\nOverall, we \fnd TDEs to be an attractive possibility to explain, at least in part, the still\nelusive origin of the observed neutrino \rux. Indeed, they naturally \ft a hypothesis that has\nrecently emerged from data analyses: that the IceCube signal might be due to relatively\nfrequent, transient photohadronic sources with photon counterparts at sub-MeV energies\nonly. Upcoming, higher statistics data at IceCube and its future evolutions (such as IceCube-\nGen2) could substantiate the TDE hypothesis in a number of ways. One could be dedicated\nsearches for time- and space- correlations of neutrino events with known TDEs, possibly to\nbe done in collaboration with astronomical surveys. Another way is more detailed studies\nof the di\u000buse \rux, that could show transitions, as the energy increases, in the spectral index\nand \ravor composition of the \rux, thus indicating the presence of a distinct component at\n\u0018PeV, of di\u000berent origin than the lower energy events.\nOf course, one should consider the large uncertainties that a\u000bect the prediction of the\nneutrino \rux from TDEs. We illustrated some of them, especially those due to the uncertain\nlow mass cuto\u000b (i.e., the lower mass end of the SMBH occupation fraction), and those\nassociated to the scaling of time variability or of the Lorentz factor of the jet with the\nSMBH mass, the baryonic loading of the jet, the fraction of TDEs producing jets, and the\nuncertain luminosity distribution. These quantities are likely to become better known as\nmore astronomical data are gathered on TDEs.\nIt is also fascinating that neutrino detectors themselves might contribute to our learning of\nthe physics of tidal disruption. Indeed, \rux constraints from neutrino data could establish\nimportant upper limits on the energetics, baryon content and and frequency of TDEs;\nthese limits would be complementary to astronomical observations, which are more strongly\na\u000bected by absorption and limited sky coverage. Alternatively, a discovery of TDEs as\n19neutrino sources would most likely give upper limits on the low black hole mass cuto\u000b,\nand would distinguish among di\u000berent scaling models for the jet parameters with the black\nhole mass. By probing tidal disruption events, neutrino data would therefore contribute\nto answering unresolved questions on the fundamental physics of black holes, on the birth\nand evolution of supermassive black holes, and on the dynamics of galactic cores that are\nusually quiet and are only \\illuminated\" occasionally by tidal disruption.\nNote added. During completion of this study, Refs. [68,69] have appeared. Their conclu-\nsions (about 5\u000010% of the di\u000buse \rux observed at IceCube could be consistent with the\nTDE hypothesis) is roughly consistent with our result for the Base case (1 =8:12'12%).\nRef. [68] compute event rates from individual TDEs and derive a constraint on the di\u000buse\nIceCube \rux. Ref. [69] also considers neutrinos from chocked jets, and conclude that the\ncontribution must be sub-dominant. Note that their ~\u0018crcorresponds to our \u0018p. Compared to\nRefs. [68,69], our work includes a fully numerical computation of the di\u000buse \rux including\n\ravor e\u000bects, and the scaling assumptions with the SMBH mass function are unique to our\nwork.\nAcknowledgments\nWe thank M. Ahlers, L. Dai, A. Franckowiak, M. Kowalski, K. Murase, A. Stasik and\nN. L. Strotjohann for useful discussions. CL is grateful to the DESY Zeuthen laboratory\nfor hospitality when this work was initiated. She acknowledges funding from Deutscher\nAkademischer Austausch Dienst (German Academic Exchange Service), the National Sci-\nence Foundation grant number PHY-1205745, and the Department of Energy award DE-\nSC0015406. 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[17].\nWe assume the same parameters as in Table 1 and the main text, unless explicitly stated.\nThe following relationships among the observables are assumed to hold at z= 0, where the\noberver's frame corresponds to the SMBH frame.\nIt is reasonable to approximate the pion, muon and and neutrino energies as \fxed fractions\nof the parent proton energy, E\u0019\u00180:2Ep,E\u0016\u00180:15EpandE\u00180:05Ep. The neutrino\n\ravor \ruences (without \ravor mixings) can be modeled analytically as [17]\nE2F0\n\u0016(E) =1\n32\u0019d2\nLEX\u0018p\nln (Ep;max=Ep;min)fp\r\u0010\u0019(1 +\u0010\u0016);\nE2F0\ne(E) =1\n32\u0019d2\nLEX\u0018p\nln (Ep;max=Ep;min)fp\r\u0010\u0019\u0010\u0016: (19)\nThe pion production e\u000eciency fp\ris the average fraction of energy deposited into pion\nproduction. It is, similar to gamma-ray bursts [70,71], given by\nfp\r'0:35\u0012LX\n1047:5erg s\u00001\u0013\u0012\u0000\n10\u0013\u00004\u0012tv\n102s\u0013\u00001\u0010\u000fb\nKeV\u0011\u00001\n\u0002(\n(Ep=Epb)\f\u00001forEp<Ep;br\n(Ep=Epb)\u000b\u00001forEp\u0015Ep;br:\n(20)\nHereLX=EX=\u0001Tis the average X-ray luminosity, and Ep;bris the proton energy leading\nto photo-pion production at the \u0001-resonance corresponding to the X-ray break energy, i.e.,\nEp;br= 1:5 107GeV\u0012\u0000\n10\u00132\u00121 KeV\n\u000fX;br\u0013\n: (21)\nIn Eq. (19), the factors \u0010\u0019and\u0010\u0016are suppression factors that account for pion and muon\npropagation (energy losses and decay). The quantity \u0010\u0019can be expressed in terms of a\nspectral break energy E\u0019;br, beyond which the timescale of synchrotron losses becomes\nsmaller than the pion lifetime:\n\u001a\u0010\u0019= 1 for E\u0019<\u0018E\u0019;br\n\u0010\u0019/E\u00002\n\u0019forE\u0019>\u0018E\u0019;br\nwithE\u0019;br'5:8\u0002108GeV\u0012LX\n1047:5erg s\u00001\u0013\u00001\n2\u0012\u0018B\n1\u0013\u00001\n2\u0012\u0000\n10\u00134\u0012tv\n102s\u0013\n:(22)\nA similar expression holds for \u0010\u0016, for with\nE\u0016;br'3:1\u0002107GeV\u0012LX\n1047:5erg s\u00001\u0013\u00001\n2\u0012\u0018B\n1\u0013\u00001\n2\u0012\u0000\n10\u00134\u0012tv\n102s\u0013\n: (23)\nA comparison between the analytical technique and the numerical techniques is given in\nFig. 5. First of all, it is noteworthy that our numerical and analytical techniques match\nrelatively well in terms of both shape and normalization.\n2610410510610710810910-510-40.0010.010.11\nEGeVE2F0HGeV cm-2LFigure 5: The \ruence E2F0\n\u000bfor\u0017\u0016+ \u0016\u0017\u0016(\u000b=\u0016, thick) and \u0017e+ \u0016\u0017e(\u000b=e, thin), for a TDE\natz= 0:35. Dashed: analytical approximation; solid: NeuCosmA numerical result. Flavor\nmixing is not included here, hence the factor of \u00182 di\u000berence in the \ruence compared to\nFig. 2.\nThe slightly di\u000berent shape comes mostly from high-energy photohadronic processes, see\nRef. [43], from kaon production (at the highest energies), and from the di\u000berent treatment of\nthe photo-production threshold (at the breaks) [45]. At low neutrino energies, the analytical\ncurves are simply extrapolated with Eq. (20) using the high-energy photon spectral index\n\f, whereas the numerical computation cuts o\u000b at a maximal photon energy given by the\nobserved energy window (relevant for the minimal neutrino energy).\nThe somewhat lower numerical versus analytical normalization is rather a coincidence of two\ncompeting processes: additional (to the \u0001-resonance) high-energy photomeson production\nmodes enhance the pion production compared to the analytical estimate and make the\nspectral peaks more pronounced [43], whereas several reduction factors have been identi\fed\nin Refs. [45, 72]. One example for the \rux reduction is the over-estimation of the pion\nproduction e\u000eciency using the break energy in Eq. (20) instead of integrating over the\nwhole spectrum. The relative normalization between analytical and numerical computation\ndepends on the type of assumptions made for the analytical computation and the parameters\n(such as photon break energy and spectral indices).\nWe observe that the analytical method matches the numerical computation relatively well\n(within about a factor of two), whereas for GRBs, the analytical computation typically\noverestimates the neutrino \rux much more signi\fcantly.\n27B Predicted X-ray \rux\nAs a consistency check of our results, we calculated the X-ray \rux expected for the di\u000berent\nscaling scenarios (Table 2), following the same formalism as in Eq. (17). The results are\nshown in Fig. 6. As expected from the scalings of \u0000 and LX, the contribution of the lower\nmass SMBH is largest in the Weak and Strong cases, and suppressed in the Lumi case. Note\nthat the Weak and Strong cases di\u000ber only by tv, which a\u000bects the neutrino production but\nnot the X-ray \rux. In all scenarios, the \rux is consistent with observations, being at\nleast one order of magnitude below the di\u000buse extragalactic soft X-ray \rux as measured at\nE'0:25 KeV by ROSAT: E2\bX'5\u00009 KeV cm\u00002s\u00001sr\u00001[73]. AtE>\u0018KeV, the observed\ndi\u000buse \rux is even larger (see e.g., [74] and references therein), thus strongly outshining the\npredicted TDE \rux.\n280.20.51.02.05.010.020.00.0010.0050.0100.0500.1000.500\nEKeVE2FXHELHKeV cm-2s-1sr-1L\n105M\n106M\n0.20.51.02.05.010.020.00.0010.0050.0100.0500.1000.500\nEKeVE2FXHELHKeV cm-2s-1sr-1L\n105M\n106M\n0.20.51.02.05.010.020.00.0010.0050.0100.0500.1000.500\nEKeVE2FXHELHKeV cm-2s-1sr-1L\n105M\n106MFigure 6: The di\u000buse X-ray \rux predicted in the Base case (upper pane), Weak and Strong\ncase (middle pane) and Lumi case (lower pane), for Mmin= 105M\f(solid), and Mmin=\n106M\f(dashed). The shaded regions show the variation corresponding to varying Mminin\nthe interval Mmin= [104:5;106:5]M\f.\n29"    },    {        "title": "1409.7452v2.An_ultimate_storage_ring_lattice_with_vertical_emittance_generated_by_damping_wigglers.pdf",        "content": "arXiv:1409.7452v2  [physics.acc-ph]  7 Oct 2014An ultimate storage ring lattice with vertical emittance\ngenerated by damping wigglers\nXiaobiao Huang\nSLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025\nAbstract\nWediscuss theapproachofgeneratingroundbeamsforultimatest oragerings\nusing vertical damping wigglers (with horizontal magnetic field). The verti-\ncal damping wigglers provide damping and excite vertical emittance. This\neliminates the need to generate large linear coupling that is impractica l with\ntraditional off-axis injection. We use a PEP-X compatible lattice to de mon-\nstrate the approach. This lattice uses separate quadrupole and s extupole\nmagnets with realistic gradient strengths. Intrabeam scattering effects are\ncalculated. The horizontal and vertical emittances are 22.3 pm and 10.3 pm,\nrespectively, for a 200 mA, 4.5 GeV beam, with a vertical damping wigg ler\nof a total length of 90 meters, peak field of 1.5 T and wiggler period of 100\nmm.\nKeywords: ultimate storage ring, vertical emittance, damping wiggler\n1. Introduction\nInpresent daythirdgenerationlightsources, theverticalemitta nceisusu-\nally small compared to the horizontal emittance. It is typically a few p ercent\nof the latter or below without coupling correction and can reach pico -meter\nlevel with coupling correction. For ultimate storage rings (USR), it is not\nadvisable to maintain the same level vertical-to-horizontal emittan ce ratio.\nThis is because the horizontal emittance will already be diffraction-lim ited\nand hence there is no need to make the vertical emittance any smalle r. In\naddition, a smaller vertical emittance will cause significant emittance growth\ndue to intrabeam scattering (IBS) and also severe Touschek beam loss. Most\nEmail address: xiahuang@slac.stanford.edu (Xiaobiao Huang)\nPreprint submitted to Elsevier June 2, 2021USR designs to-date (such as PEP-X [3]) assume the vertical emitta nce to\nbe equal to the horizontal emittance, resulting in a round beam.\nVertical emittance in a storage ring can be generated with linear cou pling\nor vertical dispersion. A round beam can be achieved with 100% linear cou-\npling, in which case the horizontal and vertical emittances are 50% o f the\nnatural emittance. The reduction of horizontal emittance by a fa ctor of 2\nis a significant benefit of this approach. However, large coupling bet ween\nthe two transverse directions will cause injection difficulties for off- axis injec-\ntion. The injected beam, initially at a large horizontal offset, will take large\nvertical oscillation and likely get lost to small vertical apertures suc h as the\nsmall-gap insertion devices. Effectively, large coupling with small vert ical\napertures causes the dynamic aperture to decrease. This is expe rimentally\ndemonstrated on the SPEAR3 storage ring as is shown in Figure 1, wh ich\nshowsthattheinjectionefficiency dropstozeroatorbeforethec ouplingratio\nis increased to 26%. Large linear coupling may also reduce Touschek lif etime\nsince the horizontal oscillation of the Touschek particles will be coup led to\nthe vertical plane which usually has smaller apertures.\n00.050.10.150.20.250.30.3500.20.40.60.81\ncoupling ratioinjection efficiency\nFigure 1: Injection efficiency vs. coupling ratio at SPEAR3.\nThe second approach to generate vertical emittance is to create vertical\ndispersion inside dipole magnets. This will not cause severe injection a nd\nlifetime difficulties, but will lose the benefit of horizontal emittance re duc-\ntion. And since strong skew quadrupoles are needed to create larg e vertical\ndispersion, it may be inevitable to introduce large linear coupling.\nWe have studied a third approach which can mitigate the negative effe cts\nof both of the above approaches. In this approach we use vertica l damp-\ning wigglers (with horizontal magnetic field) to achieve both the redu ction\n2of horizontal emittance and the generation of vertical emittance . Damping\nwigglers are usually required for USRs because in USRs the dipole bend ing\nradius is large and hence the radiation energy loss from dipole magnet s is\ntoo small for sufficient damping, which is required for controlling collec tive\neffects such as intrabeam scattering and beam instabilities. Usually d amp-\ning wigglers have vertical magnet field that causes wiggling beam motio n on\nthe horizontal plane. The horizontal dispersion generated by the damping\nwiggler itself contributes to an increase of the horizontal emittanc e. The\nrelative emittance increase can be significant when the natural emit tance is\nsmall. Choosing to use small period damping wigglers alleviates the emit-\ntancegrowthproblem to someextent. But it putsa challenge to the damping\nwiggler design and increases the cost. A vertical damping wiggler doe s not\nincrease the horizontal emittance and in the same time generates t he desir-\nable vertical emittance. Therefore it is reasonable to use vertical damping\nwigglers for USRs. The idea of using vertical damping wigglers to gene rate\nvertical emittance has been independently proposed in Refs. [1, 2 ].\nIn this study we demonstrate this approach with a lattice that is com -\npatible with the PEP tunnel at SLAC National Accelerator Laborato ry. In\nsection 2 we use a simple model to calculate and compare the emittanc es\nwiththehorizontalandvertical dampingwiggler approaches. Inse ction3the\nPEP compatible lattice with vertical damping wigglers is presented. Em it-\ntance parameters with intrabeam scattering effects are given in se ction 4.\nThe conclusions are given in section 5.\n2. Theoretic calculation\nThe effects of vertical damping wigglers can be analytically estimated .\nSuppose the wiggler peak field is Bw, its length is Lw, and the horizontal\nfield is given by\nBx=Bwcoshkxcosks, B y= 0, (1)\nwherek= 2π/λwandλwis the wiggler period, then the vertical closed orbit\ninside the damping wiggler (DW) is [4]\nyco=1\nρwk2(1−cosks), y′\nco=1\nρwksinks, (2)\n3whereρw=Bρ/B wis the minimum bending radius. The vertical dispersion\ngenerated by the DW itself is\nDy=−1\nρwk2(1−cosks), D′\ny=−1\nρwksinks, (3)\nConsequently the radiation integral contributions are\nI2w=Lw\n2ρ2w, I 3w=4Lw\n3πρ3w,\nI4wy=3Lw\n8πρ4wk2, I5wy=4< βy> Lw\n15πρ5wk2, (4)\nwhere< βy>is the average vertical beta function across the DW. The\nemittances and momentum spread are given by\nσ2\nδ=γ2CqI3+I3w\nI2+I2w1\n2+Dx+Dy, (5)\nǫx=γ2CqI5\nI2+I2w1\n1−Dx, (6)\nǫy=γ2CqI5wy\nI2+I2w1\n1−Dy, (7)\nwhereI2−5are radiation integrals for the bare lattice and\nDx=I4\nI2+I2w,Dy=I4wy\nI2+I2w. (8)\nWe now consider a PEP-X compatible lattice at 4.5 GeV (see section 3).\nThe relevant radiation integrals without DWs are\nI2= 0.1026m−1, I3= 1.674×10−3m−2,\nI4=−0.1215m−1, I5= 3.092×10−7m−1. (9)\nAssuming the average beta function over the DW is 10 m, the emittan ces\nas a function of wiggler length is calculated and compared to the case with\na regular horizontal damping wiggler for various sets of peak magne tic field\nand wiggler period values. The results are shown in Figure 2. Clearly th e\nvertical DW provides damping of the horizontal emittance and in the mean-\ntime generates vertical emittance. The total emittance is only sligh tly larger\nthan the case with a regular horizontal DW. The difference is smaller f or\nsmaller wiggler periods.\n40 20 40 60 80 100010203040\nDW length (m)emittances (pm)\n  \nB0 = 1.2 T, λ = 200 mmεx VDW\nεy VDW\nεx,y HDW\n0 20 40 60 80 100010203040\nDW length (m)total emittance (pm)\n  \nB0 = 1.2 T, λ = 200 mmHDW\nVDW\n0 20 40 60 80 100010203040\nDW length (m)emittances (pm)\n  \nB0 = 1.2 T, λ = 100 mmεx VDW\nεy VDW\nεx,y HDW\n0 20 40 60 80 100010203040\nDW length (m)total emittance (pm)\n  \nB0 = 1.2 T, λ = 100 mmHDW\nVDW\n0 20 40 60 80 100010203040\nDW length (m)emittances (pm)\n  \nB0 = 1.5 T, λ = 100 mmεx VDW\nεy VDW\nεx,y HDW\n0 20 40 60 80 100010203040\nDW length (m)total emittance (pm)\n  \nB0 = 1.5 T, λ = 100 mmHDW\nVDW\nFigure 2: Comparison of the emittances of a PEP-X ring with vertical or horizontal\ndamping wigglers. Top row with Bw= 1.2 T and λw= 200 mm; middle row with\nBw= 1.2 T and λw= 100 mm; bottom tow with Bw= 1.5 T and λw= 100 mm. A 100%\ncoupling is assumed for the regular horizontal DW case.\n53. Application to a PEP-X compatible lattice\nWe have implemented the vertical DW approach for a PEP-X compatib le\nlattice with the design beam energy at 4.5 GeV. This lattice is similar to th e\nPEP-X USR design as it adopts the same MBA and fourth order achro mat\napproach [3]. The 2.2-km long PEP tunnel has a hexagonal geometry . There\nare six 120-m long straight sections which can be used to host long da mping\nwigglers. The lattice has 6 arcs, each consists of 8 MBA (with M= 7) cells.\nAn MBA cell is composed of 5 identical TME cells in the middle and two\nmatching cells at the ends. The MBA cell and the TME cell are shown in\nFigure 3. The TME and 7BA cell lengths are 3.12 m and 30.4 m, respec-\ntively. The TME dipole magnet is 1.12 m in length and its bending angle\nis 1.0475◦. This dipole is a combined-function magnet with a defocusing\nquadrupole component and the normalized gradient is −0.7989 m−2. The\nfocusing quadrupole (QF) is split into two halves to put the SF sextup ole in\nbetween. The length of each half is 0.18 m. The length of SF is 0.30 m. On e\nSD sextupole magnet is put at each end of the dipole. Its length is 0.21 m.\nThe matching dipole has no quadrupole gradient. Its length is 8% longe r\nthan the TME dipole. At each end of the MBA cell, outside of the match -\ning dipole, there is a quadrupole triplet. Three harmonic sextupoles a re put\nbetween these magnets. The minimum edge-to-edge distance for m agnets\nis 8 cm to accommodate coils and BPMs [5]. The quadrupole strength is\nbelow 51 T/m and the sextupole strength is below 7500 T/m2. With a bore\nradius of 12.5 mm, the pole tip magnetic field would be below 0.64 T for\nquadrupoles and below 0.59 T for sextupoles.\nThe insertion device straight sections between the MBA cells are 5 me ter\nlong. The horizontal and vertical beta functions at the centers o f these\nstraight sections are 0.8 m and 2.0 m, respectively. The horizontal b eta\nfunction is made very small to provide better matching of the electr on and\nphoton optics. But we keep the vertical beta at a level close to half of the\nID straight length to allow small gap insertion devices [6].\nThe 120-m long straight sections are filled with FODO cells. One of the\nlong straight sections houses the damping wigglers. The wiggler sect ions are\n4.06 meter long and are put between the quadrupoles of the FODO ce lls.\nThe optics functions are shown in Figure 4 for a FODO cell for the cas e with\nwiggler period at 200 mm and peak field at 1.2 T. Optics function for one\nhalf of the long straight section is as shown in Figure 5.\nThe ring lattice parameters for three wiggler settings are compare d in\n60.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0\ns (m)\nδE/ p0c = 0.\nTable name = TWISSTME\nWin32 version 8.51/15 08/03/13  13.40.34\n0.01.2.3.4.5.6.7.8.9.10.β(m)\n0.00.010.020.030.040.050.060.070.08\nDx(m) βxβyDx\n0.0 5. 10. 15. 20. 25. 30. 35. 40.\ns (m)\nδE/ p0c = 0.\nTable name = TWISSStandard cell.\nWin32 version 8.51/15 08/03/13  13.40.34\n0.02.55.07.510.012.515.017.520.022.525.0β(m)\n-0.010.00.010.020.030.040.050.060.070.08\nDx(m) βxβyDx\nFigure 3: The TME cell (left) and 7BA cell (right) for the PEP-X comp atible lattice.\n0.0 2. 4. 6. 8. 10.\ns (m)\nδE/ p0c = 0.\nTable name = TWISSFODO w/ wiggler\nWin32 version 8.51/15 08/03/13  13.40.34\n0.02.55.07.510.012.515.017.520.022.525.0β(m)\n-1.00-0.75-0.50-0.250.00.250.500.751.00\nDy(m) [*10**( - 3)]βxβyDy\nFigure 4: One FODO lattice period with damping wigglers.\n70.0 10. 20. 30. 40. 50. 60. 70. 80.\ns (m)\nδE/ p0c = 0.\nTable name = TWISSTME2FODOW\nWin32 version 8.51/15 08/03/13  13.40.34\n0.05.10.15.20.25.30.β(m)\n-0.010.00.010.020.030.040.050.060.070.08\nDx(m) βxβyDx\nFigure 5: Half of the DW straight section.\nTable 1. The parameters were calculated with MAD8 [7]. The results ag ree\nwith the prediction given in Figure 2. For the vertical DW sets of (1.2 T ,\n200 mm) and (1.5 T, 100 mm), the horizontal and vertical emittance s are\nnearly equal, with values down to 17/17 pm and 13/10 pm, respective ly.\nTable 1: Ring parameters with or without vertical damping wigglers\nparameters no DW VDW-1 VDW-2 VDW-3\nEnergy (GeV) 4.5\nCircumference (m) 2199.3\nνx,y 130.15/73.30\nαc 0.38×10−4\nVDW length (m) 0 90.8 89.4 89.4\nVDWBw(T) 1.2 1.2 1.5\nVDWλw(m) 0.2 0.1 0.1\nU0(MeV) 0.59 2.30 2.26 3.19\nǫx(pm) 36.5 16.5 16.7 12.7\nǫy(pm) 0 17.1 4.3 10.2\nσδ(×0.001) 0.77 1.05 1.05 1.10\ndamping time τx(ms) 47 21 22 17\ndamping time τy(ms) 112 29 29 21\ndamping time τs(ms) 176 17 18 12\n84. Intrabeam scattering calculation\nIntrabeam scattering (IBS) can significantly increase the emittan ce and\nenergy spread for very low emittance beams at high current. To ex amine\nhow the IBS effects may differ for the two approaches of generatin g vertical\nemittance, i.e., with full coupling or with vertical damping wiggler, we did\nIBS calculation for both cases for the PEP-X compatible lattice with t he\nhigh energy approximation model [3, 8]. Similar to PEP-X, we assume th e\nCoulomb log function is (l og) = 11. For the full coupling case, a horizontal\ndamping wiggler is put into the model to reduce emittance. The wiggler\nparameters are the same as the vertical damping wiggler. For the c ase cor-\nresponding to VDW-3 in Table 1 (i.e., with peak field 1.5 T, wiggler period\n100 mm and wiggler length 89.4 m), the emittances are 10.5 pm for both\nplanes. For the vertical wiggler case, a small linear coupling ratio of 0 .001\nis assumed. The vertical emittance is almost entirely generated with the\nvertical damping wiggler.\nThe emittances vs. beam current for the two cases with IBS effect s are\nshown in Figure 6. The bunch length is assumed to be σz= 2.7 mm, corre-\nsponding to an RF gap voltage of 6 MV with the 476.0 MHz rf system. Th e\ntotalnumber ofbunches isassumed tobe3300. Forthevertical D Wcase, the\nhorizontal emittance has a significant increase. But the vertical e mittance\ngrowth is very small. This is because the vertical dispersion is confine d to in-\nside the damping wiggler, which constitutes only a small fraction of th e ring,\nwhile the horizontal dispersion is present at all arc areas. In additio n, the\nvertical dispersion is much smaller than the horizontal dispersion wh ile the\nhorizontal and vertical emittances at zero current are nearly th e same. This\nis because the average bending field in the damping wiggler is much stro nger\nthan in the bending magnets. Overall the vertical IBS growth rate is much\nsmaller thanthehorizontal planebecause theIBS growthrateispr oportional\nto the dispersion invariant averaged over the ring circumference.\nThe distribution of the IBS growth rate for the vertical DW case is s hown\nin Figure 7 for the case with a 200 mA total current. For this case, t he\naverage IBS growth rates for x,y,pdirections are 26.1 s−1, 0.20 s−1and\n8.4 s−1, respectively. The corresponding emittances are ǫx= 22.3 pm and\nǫy= 10.3 pm and the momentum spread is σδ= 1.16×10−3. If harmonic\ncavities are used to lengthen the bunch to σz= 5 mm, the x,y,pIBS\ngrowth rates become 18.8 s−1, 0.13 s−1and 5.5 s−1, respectively, and the\nemittances and the momentum spread become ǫx= 18.4 pm,ǫy= 10.25 pm\n90 50 100 150 200 250051015202530\nI0 (mA)emittance (pm)\n  \nεx,y HDW\nεx VDW\nεy VDW\nFigure 6: Emittance growth vs total beam current, assuming a unif orm current distribu-\ntion in 3300 bunches and a bunch length of 2.7 mm.\nandσδ= 1.13×10−3.\n0 500 1000 1500 2000 2500−10010203040506070\ns (m)δ (1/T), (s−1)\n  \nδ (1/Tp)\nδ (1/Tx)\nδ (1/Ty)\nFigure 7: The distribution of local IBS growth rate for the three dim ensions for a beam\ncurrent of 200 mA in 3300 bunches.\n5. Conclusion\nFor ultimate storage rings that require off-axis injection, we propo se the\nuse of vertical damping wiggler (with horizontal magnetic field) to ge nerate\n10vertical emittance in order to obtain round beams. This approach is better\nthan generating round beams with 100% coupling because it does not couple\nthe large amplitude horizontal oscillation of the injected beam to the vertical\nplane and therefore the small vertical apertures in the ring does n ot pose\nsevere limitation to the dynamic aperture. It is shown that for damp ing\nwigglers with reasonably small wiggler period (e.g., λw= 100 mm), the total\nemittances of the two approaches are nearly equal.\nA PEP-X compatible lattice is designed to demonstrate this approach . It\nconsists of 6 arcs, each made up of 8 MBA cells. The beamenergy is as sumed\nto be 4.5 GeV. The quadrupoles and sextupoles are separate funct ion mag-\nnets, with strengths below 51 T/m and 7500 T/m2, respectively. The bare\nlattice horizontal emittance is 36.5 pm. The beta functions at the mid dle of\nstraights are0.8m horizontal and2.0 mvertical, which allows goodmat ching\nto the photon optics and supports the use of small gap insertion de vices.\nWhen a 90-m long vertical damping wiggler with peak field at 1.5 T and\nwiggler period at 100 mm is put into one of the long straight sections, t he\nhorizontal and vertical emittances are 13 pm and 10 pm, respectiv ely. The\nrms momentum spread is 1.1 ×10−3. Intrabeam scattering is calculated for\nthis case. For 200 mA beam current in 3300 bunches and a bunch leng th\nofσz= 2.7 mm, the horizontal and vertical emittance become 22.3 pm and\n10.3 pm, respectively. The rms momentum spread is 1 .16×10−3.\nAcknowledgment\nThe study is supported by DOE Contract No. DE-AC02-76SF00515 .\nReferences\n[1] X. Huang, “A PEP-X lattice with vertical emittance by damping wig-\nglers”, SSRL-AP-note 51, July 2013.\n[2] A. Bogomyagkov, et al, presentation at the 4th Low Emittance R ings\nWorkshop, September 2014.\n[3] Y. Cai, et al, Phys. Rev. ST Accel. Beams, 054002 (2012).\n[4] S. Y. Lee, Accelerator Physics , World Scientific (1999).\n11[5] A 7.5 cm minimum magnet separation is reserved in MAX-IV\ndesign. See Detailed Design Report on the MAX IV Facility,\nhttps://www.maxlab.lu.se/node/1136, August 2010, chapter 2.\n[6] T. Rabedeau, private communications (2013).\n[7] H. Grote, F.C. Iselin, The MAD Program User’s Manual , CERN/SL/90-\n13 (1990).\n[8] K. Bane, EPAC 2002, Paris, France (2002)\n12"    },    {        "title": "0811.4431v1.The_Lorentz_Condition_is_Equivalent_to_Maxwell_Equations.pdf",        "content": "                       The Lorentz  Condition is Equivalent to Maxw ell Equations \n \nEdm und A. Di Marzio \nBio-Poly-Ph ase, 142 05 Parkvale Ro ad, Rockv ille, MD 20853 \nReceived 26 Novem ber 2008 \n \nIt is sh own that the Lorentz con dition which is a conservation law on  the electro magnetic fou r-vector-\ndensity , plus the Lorentz transformation, taken together, are equivalent to the microsco pic Maxwell’s \nequat ions. µA\n \n   Jackson in his book “Classical E lectrodynam ics” derives Eqs 1 and 2 from the \nmicroscopic Maxwell’s equations1. They are, in Gaussian units, \n \n0=\n∂∂\nµµ\nxA                                                                                                             (1) \n \nµ µπjcA4=                                                                                                      (2) \n \nAlso, it is straightforward to derive the m icroscopic Maxwell’s equations from  Eqs. 1 and \n2.  This m eans that the above equations ar e equivalent to the microscopic Maxwell’s \nequations.  Eq. 1 is the Lorentz condition (LC) and  is (m inus) the inv ariant \nd’Alem bertian. \n \n= )\n)( )( )( )((232\n222\n212\n202\nx x x x ∂∂−\n∂∂−\n∂∂−\n∂∂                                                            (3) \n \nwhere, . The electrom agnetic four-vector-density is ,  ),,,(),,,(3 2 1 0zyxct xxxx =µA\n),,,(),,,(3 2 1 3 2 1 0AAA AAAA φ= . The charge-current-density is , \n, where ρ is charge density. One s hould also note that the  µj\n),,,(),,,(3 2 1 3 2 1 0jjjc jjjj ρ=\nLC is a conservation law  on the four-vector density . Eq. 1 is also invariant to the \nLorentz gauge. µA\n    A m ore precise s tatement of the claim  of this paper is that the LC plus the Lorentz \ntransform ation (LT), taken together, have th e same formal structure (syn tax) as th e \nmicroscopic Maxwell’s equations. \n   Our derivation uses the f act that the  d’Alem bertian, is an inv ariant oper ator so tha t \nwhen it operates on a any vector th e result is alw ays a vector. \n   We now a rgue that (the “one”) Eq. 1 im plies (the four) Eqs. 2. Since  is a function \nof the tim e and space variables we can operate  on it by th e inv ariant d’Alembertian to \nobtain  µA\n \nµ µπkcA4=                                                                                                       (4)  \nwhere  is so me four-vector-den sity.                                                             µk\n                                                                   1   We make the following 3 points. First,  and  have the sam e dimensions. Second, µkµj\nthe divergen ce µx∂∂ comm utes with  so we have from Eqs 2 and 4 \nµ µµπ\nx xj\nc∂∂=\n∂∂4=µA 0=\n∂∂\nµµ\nxA                                                                          (5)  \n \nµ µµπ\nx xk\nc∂∂=\n∂∂4=µA 0=\n∂∂\nµµ\nxA                                                                          (6) \n \nwhich are conservation laws on the four-vector-densities  and  . Third, we also have  µjµk\n \n''')'(0 0dzdydxj dxdydzj=                                                                                        (7) \n \n'                                                                                      (8) '')'(0 0dzdydxk dxdydzk=\n \nwhere the prim ed coordinates are connected to the unprim ed coordinates by a LT.  For \ncharge this is the statem ent that the charge is an invar iant so that, no m atter with what \nspeed we approach a charge distribution, the amount of charge does not change. It is a \nmatter of experien ce that charg e is both a conserved quantity and an invariant quantity. \n   Since, as we see, ther e is nothing that distinguishes  from  they are identical.                                  µkµj\n   At this point a discussion of the differe nce between syntax and sem antics is needed.  \nConsider the heat diffusion and the particle di ffusion equations. The syntax is the same;  \nthat is to say the equations have the sam e form. But the sem antics, or meaning is m uch  \ndifferent. What we have done a bove is show that we can deve lop from  the LC alone a set \nof equations that are identical in form to th e microscopic Maxwell’s equations. That is to \nsay, we have obtained equations with the same syntax as Maxwell’s equations. Our \nderivation also shows that any 4-vector wh atsoev er for which  the 4-d ivergence is zero \nhas the sam e syntax as Maxwell’s equations.  \n   Note that we have not derived Maxwell’s equations from  first prin ciples. A com plete \nderivation of Maxwell’s equations from  first principles requires th at prior m eaning be \nassigned to . Then Maxwell’s equations would be derivative both by syntax and by \nmeaning. But how can we cloth them with m eaning?  The experim ents of Faraday and  µA\nAmpere and the genius of Maxwell have l ong ago clothed the Maxw ell equations with \nmeaning. At  this point in tim e we can say that we have derived the Maxw ell equations \nfrom  the LC only by appropriating this m eaning.  \n \n     W e make the following observations. \n  \n [1] Various rela tivists h ave observed that the constraint  of cov ariance is a severe \nconstraint on the possible for ms of t he laws  of electricity and m agnetism . We give 3 \nexam ples. \n   Misner, T horne and Wheeler2 show that the hom ogeneous equation div B=0 plus \nrelativity im plies the ho mogeneous equation curl E +∂B/c∂t =0.  \n                                                                   2    Using relativity Kobe3 is able to derive Maxwell’ s equations using only Coulom b’s \nLaw and conservation of electric charge. \n   Anderson4 observes th at the assumption th at the electrom agnetic field be an asymm etric \ntensor and also that the equations b e linear , implies that in free space th e equations Fµν ,µ \n= 0,  F [µν,ρ]  =0 (his notation) are the only possible ones.\n   Perhaps then our derivation is not so surprising. \n \n[2] We note the Aharonov-Bohm  papers5 which show a deep connection of  to \nquantum  mechanics. The quantum  mechan ical w ave function is dependent on  even \nwhen  is a constant so that E and B are equal to zero; B = curl A, E = -gradφ-∂A/c∂t. µA\nµA\nµA\nAt a bare m inimum this m eans that  has a physical reality  above and beyond the \nformulas relating it to E and B since it can exis t even when E and B do not. This requires \nus to seek the m eaning of the electrom agnetic 4-vector not from  E and B but from  \nsomething more prim al. Since both the LT and the LC were each requ ired in our \nderivation of Maxwell’s equations we need to understand each of them  at a deeper level. µA\n \n[3] Some tim e ago an attem pt was made to  derive the LT and the LC from m ore \nfunda mental considerations and thereby obtai n Maxwell’s equations in both their syntax \nand their sem antics.  \n   First, the LT was derived in a m anner that  took no note of Maxwell’s equations, nor of \nthe fact th at the speed of  light is a co nstant, no r even that ligh t existed6. The arbitrary \nconstant c with the dim ensions of velocity that ap pears in the de rivation is given the \nnumerical v alue of the s peed of light becau se, as is eas ily sho wn, the LT implies that \nsomething tr aveling with  the speed c in one c oordinate system  trave ls with  the speed c in \nevery coordinate system . Light has this exact property. This means that although \nMaxwell’s equations are in no way implied by the LT alone, they are sub ject to it, and  \ntheir f unctio nal form is severe ly lim ited by the constraint of covariance. At a bare \nminimum it is im plied th at the n atural stru cture or textu re of the atom s comprising th e \nmeter sticks and clocks involves light in a most fundam ental way since they were the \nonly things assum ed in our derivation. As an aside we rem ark that relatively few in the \nphysics com munity are aware that the LT can be derived without assum ing the existence \nof light7.   \n    Second, the fact that the LC has the for m of  a conservation law fo rces us to search for \na set of basic equations with a symm etry w hose associated conservation  law is th e LC.  \nIt is an easy m atter to show that assum ing two unique speed s is not a cova riant concep t \nsince on perfor ming a boost only th e  speed c w ill rem ain c, the other speed will ch ange; \nsimilarly for n distin ct unique speed s. Occam ’s razor forces us to the bo ld, but sim ple \nview that th ere ex ists on ly one speed  in thes e basic equations, that of light in a vacuum. It \nwas hypothesized8 that there exists a funda mental field ρµ(xν, Ω) from which all the \nvarious laws of physics are derivable.  At each space-tim e point this field w hich is a null \n4-vecto r-den sity travels in each of th e direction s Ω, always with the speed of light . The \nconservation law on ρµ(xν, Ω)  arises from  the fact that the total am ount of stuff in the \nUniverse is a constan t. This m eans that Qµ = ∫ ρµ(xν, Ω) dΩ obeys a conservation law  and \nby identif ying it with th e electrom agnetic 4-vector Aµ, Eq. 1 obtains8. \n   Even before obtaining the basic equations governing the behavior of ρµ(xν, Ω) we see 3 \nconcordances with reality. First, the con cept is manif estly co varian t; null vectors \n                                                                   3 transform  to null vectors. Second, the concept of  mass arises from  the simple fact that the \nsum of null vectors is no t a null vector. Third the ρµ(xν, Ω) field serv es to transf er len gth \nfrom  point to point so that W eyl geom etry reduces to Riem annian geometry. \n   In order for the ρµ(xν, Ω) field (pa rticles, e lectric f ield, gravity) to persist at a given \nplace s tuff must scatter stuff, otherwise it would explosiv ely leave the pla ce with th e \nspeed of lig ht. Because action at a d istance is meaningless in relativ ity, the princip le of \ncontiguous action requires that ρµ(xν, Ωi) scatters ρµ(xν, Ωj) at the sam e space-tim e point. \nWe argued for the un iqueness of a simple dot pro duct. Equations for the s pace-tim e \nvariation of ρµ(xν, Ω) were obtained8. This procedure will be meaningful only if various \nmeasures of ρµ(xν, Ω) can be identified with the cons erved quantities of physics such as \nmass, energy, and spin. This has been done, for a toy world of (1+1) dim ensions6,9,10 but \nnot yet for our real world of (3+1) dim ensions. \n  \n[4] The ques tion can be asked whether Eqs.1 plus 2 are an overdeterm ined system  since \nwe have m ore equations than variables. But because Eq. 2 has been derived from  Eq 1 it \nis obvious that we have  a consistent set. \n  How can one equation (the Lorentz condi tion) lead to m any equations (Maxwells  \nEquations)?  We stress here that the L orentz c ondition is in fact m any equations. This is  \nbecause it embodies both the Loren tz transf ormation and the condition form ulated by \nEinstein that, in special relativity, the la ws of physics have the sam e form in all \ncoordinate system s moving with constant rela tive velocity . Th e transform ation from  the \nunprim ed  and ( ct, x. y, z ) to the p rimed  and ( ct, x.  y, z) involves six continuous \nparam eters, the three components of the velocity and the thre e angles of rotation of the \ncoordinate system . So there is a sense for which the Lorentz conditi on really corresponds \nto an infinite num ber of equations. µAµA\n  \n[5] Another question has to do with causalit y. Eq. 2 can be solved by Green function  \ntechniques and the result is that there are bot h advanced and retarded solutions. I t is the  \nretarded solution that is usua lly thought to correspond to re ality since the response to a \nchange in the source m ust occur later than the change  in the source. If we know we can \nuse Green’s functions (plus bounda ry conditions) to determ ine ; but if we know  \nwe can use Eq.2 to determ ine . Each im plies the other.  Also, there are m eaningful \nsolution s to Maxwell’ s equations wh ich requ ire both the retarded and advanced solutions.  \nWe see no reason to give  priority over , or  priority over . µj\nµAµA\nµj\nµjµAµAµj\n \n[6] In lim iting ourselve s to the m icroscopic Ma xwell equa tions (tha t is to say we assu me \nE = D and B = H ) we ig nore the p roperties of Materials. Because of caus ality D can lag \nE ; alternatively the d ielectric cons tant can  be frequency dependent and the Kram ers-\nKronig relations obta in. W e will not try to extend  our resu lts to the f ull Maxwell \nequations here, but we expect the full Maxw ell’s equation s to be valid in a ll the \ncoordinate system s of special relativity. \n \n[7] The Lore ntz condition (Lorentz gauge) is a covariant concept. The Coulom b gauge, \nthough useful is not a covari ant concept and is not rele vant to our discussion.  \n \n                                                                   4  [8] Can our equations which are covariant to special relativity be m ade generally \ncovariant? If we interp ret  as a vector-density rath er than a vector then Eq. 1 is \ngenerally covariantµA\n11. Equation 2 is not generally cova riant. T he generally covariant \nMaxwell equations would require a gene rally co varian t d’Al embertian. Several \ncandidates exist12 .  \n   \nIn conclusion, the wave equation, Eq. 2 has been  derived from Eq. 1; it did not need to  \nbe assum ed. Therefore the syntax of Maxwe ll’s equations is derived from the Lorentz \ncondition alone (but und er the aegis of speci al relativity ). Th at the Lo rentz Condition \nalone im plies the m icroscopic Maxw ell’s equations  is a rem arkable result.  \n    \n  \n References : \n \n(1) Jackson, J. D., Classical  Electrodynam ics, second edition, John W ily and Sons, New \nYork, NY (1975); Pages 549 and 220. \n \n (2) Misner, C.W ., Thorne, K.S., W heeler, J.A.: G ravitation, W. H. Free man and Co., \nNew York, (1973); Chapter 3. \n \n{3] Kobe, H. D.: Generalization of Coulom b’s Law to Maxwell’s equations using Special \nRelativity, Am . J. Phys. 54 631-636 (1986). \n \n (4) Anderson, J.L.: Principl es of Relativity Physics, Academ ic Press, New York, NY \n(1967); Page 76. \n \n(5) Y. Aharonov, Y., Bohm , D.: Significance of the Electrom agnetic Potentials in the \nQuantum  Theory, Phys. Rev. 115: 485-491 ( 1959): Further Cons iderations of  \nElectrom agnetic Po tentia ls in th e Qua ntum Theory,  Phys. Rev 123 : 1511-1524 (1961). \n \n(6)  Di Marzio, E.A.: A Unified Theory of Matter. I. The Fundam ental Idea, Foundations \nof Physics, 7, 511-528 (1977). \n \n(7) Mitchell Fiegenbaum, The Theory  of Relativity-Galileo ’s Child, \narXiv.org/abs/0806.1234v1, (2008) gives a com plete proof and lists ten other relevant \nreferences. Wolfgang Pauli, Theory of Re lativity, Pergam an Press, NY  (1958), p11, \ngives th e original referen ces from  the year 1911 and outlin es a derivation.  \n \n(8)  Di Marzio, E.A.: A Unified Theory of  Matter. II. Derivation of the Fundam ental \nPhysical Law, Foundations of Physics, 7, 885-905 (1977) \n \n(9)  Di Marzio, E.A.: Field Theory, C urdling, L imit Cycles and Cellular Autom ata, J. \nStat. Phys., 36, 897-907 (1984). \n \n(10) E. A. Di Marzio, A New Paradigm  for Ob taining Physical Law, Preprint available. \n \n                                                                   5 (11) J. L. Synge. and A. Schild, Tensor Calcul us, University of Toronto Press, Toronto,  \n(1956); Page 252.  \n \n(12) Reference 2, Chapter 22. \n \n \n \n \n \n                                                                   6 "    },    {        "title": "1710.07690v2.Tidal_dissipation_in_rotating_fluid_bodies__the_presence_of_a_magnetic_field.pdf",        "content": "MNRAS 000, 1{14 (2017) Preprint 23 November 2021 Compiled using MNRAS L ATEX style \fle v3.0\nTidal dissipation in rotating \ruid bodies: the presence of a\nmagnetic \feld\nYufeng Lin?and Gordon I. Ogilvie\nDepartment of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences,\nWilberforce Road, Cambridge CB3 0WA, UK\nAccepted XXX. Received YYY; in original form ZZZ\nABSTRACT\nWe investigate e\u000bects of the presence of a magnetic \feld on tidal dissipation in rotating\n\ruid bodies. We consider a simpli\fed model consisting of a rigid core and a \ruid en-\nvelope, permeated by a background magnetic \feld (either a dipolar \feld or a uniform\naxial \feld). The wavelike tidal responses in the \ruid layer are in the form of magnetic-\nCoriolis waves, which are restored by both the Coriolis force and the Lorentz force. En-\nergy dissipation occurs through viscous damping and Ohmic damping of these waves.\nOur numerical results show that the tidal dissipation can be dominated by Ohmic\ndamping even with a weak magnetic \feld. The presence of a magnetic \feld smooths\nout the complicated frequency-dependence of the dissipation rate, and broadens the\nfrequency spectrum of the dissipation rate, depending on the strength of the back-\nground magnetic \feld. However, the frequency-averaged dissipation is independent\nof the strength and structure of the magnetic \feld, and of the dissipative parame-\nters, in the approximation that the wave-like response is driven only by the Coriolis\nforce acting on the non-wavelike tidal \row. Indeed, the frequency-averaged dissipation\nquantity is in good agreement with previous analytical results in the absence of mag-\nnetic \felds. Our results suggest that the frequency-averaged tidal dissipation of the\nwavelike perturbations is insensitive to detailed damping mechanisms and dissipative\nproperties.\nKey words:\n1 INTRODUCTION\nTidal interactions may have played an important role in the\nevolution of short-period exoplanetary systems, binary stars\nand planet-satellite systems. The e\u000eciency of tidal dissipa-\ntion, which is usually parameterized as the tidal quality fac-\ntorQ(Goldreich & Soter 1966), will determine the fate of\nthese systems. With the accumulation of observations over\nseveral decades, the observational constraint of the tidal\nquality factor is now possible, especially in hot Jupiter sys-\ntems (e.g. Wilkins et al. 2017; Patra et al. 2017). However,\nit is still very challenging to theoretically predict the tidal\nquality factor owing to intrinsic di\u000eculties and uncertainties\nof the problem.\nTidal responses can be generally separated into two\nparts: the equilibrium tide and the dynamic tide (e.g. Ogilvie\n2014). The equilibrium tide is a quasi-hydrostatic deforma-\ntion to the gravitational pulling of the orbital companion.\nThe dynamic tide is usually associated with di\u000berent kinds\nof internal waves such as internal gravity waves (Zahn 1975;\n?E-mail: yl552@cam.ac.ukSavonije & Papaloizou 1983; Goldreich & Nicholson 1989;\nGoodman & Dickson 1998; Barker & Ogilvie 2010; Essick\n& Weinberg 2016) and inertial waves (Ogilvie & Lin 2004,\n2007; Wu 2005; Ogilvie 2009, 2013; Goodman & Lackner\n2009; Rieutord & Valdettaro 2010; Papaloizou & Ivanov\n2010). The tidal dissipation associated with these hydro-\ndynamic waves exhibits very complicated dependences on\nthe tidal frequency (Ogilvie 2014). The non-linear e\u000bect and\nthe interactions with convection and magnetic \felds may\nwash out some of the frequency-dependence (Ogilvie 2013),\nyet these e\u000bects on tides remain to be elucidated. Magnetic\n\felds are ubiquitous in astrophysical bodies, where they can\ninteract with tidal \rows in electrically conducting \ruid lay-\ners and lead to additional Ohmic dissipation. The magnetic\n\feld will modify the propagation and dissipation of hydrody-\nnamic waves through the coupling with Alfv\u0013 en waves. The\npresent study aims to investigate the magnetic e\u000bects on\ntidally forced inertial waves. In the presence of a magnetic\n\feld and rotation, wavelike motions are hybrid inertial waves\nand Alfv\u0013 en waves, which we collectively refer to as magnetic-\nCoriolis waves (Finlay 2008).\nMagnetic-Coriolis (MC) waves were \frst investigated\nc\r2017 The AuthorsarXiv:1710.07690v2  [astro-ph.EP]  21 Nov 20172 Y. Lin and G. I. Ogilvie\nby Lehnert (1954). He showed the rotation can split the\nAlfv\u0013 en waves into two groups: fast waves and slow waves, al-\nthough this separation is vague in some parameter regimes.\nMC waves have been studied in di\u000berent geophysical and\nastrophysical contexts since the pioneering study of Lehnert\n(1954). In geophysics, slow MC waves (also called magne-\ntostrophic waves) are of particular interest, as these waves\nare thought to be important in the generation and secular\nvariations of the Earth's magnetic \feld through the dynamo\nprocess in the liquid outer core (Hide 1966; Malkus 1967;\nJault 2008; Finlay 2008; Bardsley & Davidson 2017). The\nslow MC waves usually have frequencies much lower than\nthe rotation frequency and are quasi-geostrophic, i.e. nearly\ninvariant along the rotation axis. In astrophysics, studies of\nstellar oscillations have shown that MC waves can be excited\nin rotating magnetized stars (Lander et al. 2010; Abbassi\net al. 2012). MC waves have also been observed recently\nin liquid metal experiments (Nornberg et al. 2010; Schmitt\n2010; Schmitt et al. 2013). However, tidally forced MC waves\nand the energy dissipation associated with MC waves have\nnot been well studied, except for a recent study using a pe-\nriodic box (Wei 2016). Bu\u000bett (2010) calculated the Ohmic\ndissipation of tidally driven (free inner-core nutation) iner-\ntial waves in the Earth's outer core, but the Lorentz force is\ndropped in his calculations.\nThe present paper studies the propagation and the\nenergy dissipation of tidally forced MC waves in spher-\nical shells. We use a simpli\fed model consisting of a\nrigid core and a homogeneous \ruid envelope, of which the\nhydrodynamic responses have previously been considered\n(Ogilvie 2009, 2013; Rieutord & Valdettaro 2010). Tidal\nresponses are decomposed into non-wavelike and wavelike\nparts (Ogilvie 2013), and we focus on the latter in this study.\nThe wavelike perturbations are MC waves in the presence\nof a magnetic \feld, where both the Lorentz force and the\nCoriolis force act as the restoring force. The linearized equa-\ntions describing the wavelike perturbations are numerically\nsolved using a pseudo-spectral method. The total energy dis-\nsipation can be contributed to by both viscous damping and\nOhmic damping, but is dominated by the latter in most\ncases. We investigate the dependence of the total dissipa-\ntion rate on the magnetic \feld strength, the \feld structure,\nthe dissipative parameters and the tidal frequency. We also\nexamine the frequency-averaged dissipation, which has been\nused to study tidal dissipation and evolution of stars re-\ncently (Guenel et al. 2014; Mathis 2015; Bolmont & Mathis\n2016; Gallet et al. 2017; Bolmont et al. 2017). Remark-\nably, the frequency-averaged dissipation quantity is in good\nagreement with previous analytical results in the absence of\nmagnetic \felds (Ogilvie 2013). Our results suggest that the\nfrequency-averaged tidal dissipation is insensitive to the de-\ntailed damping mechanisms, at least for the wavelike tides.\nThis may have important implications for studying the long-\nterm tidal evolution, as the detailed damping mechanism\nand dissipative properties are not well constrained in stars\nand planets.\nThe paper is organized as follows. Section 2 introduces\nthe simpli\fed model, the basic equations and the numeri-\ncal method. Section 3 presents numerical results. Section 4\nsummarizes the key \fndings of this study.\nΩ\nB0\nRigid  insulator\nConducting fluid\nInsulatingFigure 1. Illustration of the model. The dashed lines are a\nschematic representation of the background magnetic \feld B0.\n2 THE SIMPLIFIED MODEL\nOur model essentially builds upon a hydrodynamic model\nconsidered by Ogilvie (2009, 2013). We consider a uniformly\nrotating spherical body consisting of a rigid inner core and\na homogeneous incompressible \ruid envelope. In order to\ntake into account magnetic e\u000bects, we assume that the whole\nbody is permeated by a steady axisymmetric magnetic \feld\nand the \ruid is electrically conducting (see Fig. 1). We fo-\ncus on the wavelike tidal perturbations in the \ruid envelope,\nwhich may be regarded as an idealized model of the convec-\ntive zones of stars and giant planets as it lacks stable strat-\ni\fcation. This section introduces basic equations describing\nthe wavelike perturbations and the numerical method we\nused to solve these equations.\n2.1 Basic equations\nA perfectly rigid inner core of radius r=\u000bRis enclosed by\na homogeneous, incompressible and electrically conducting\n\ruid shell with a free surface at r=R. We assume that\nthe rigid core and the \ruid envelope have the same density\n\u001a. The whole body uniformly rotates at \n= \n^zand is\npermeated by a steady magnetic \feld B0. For simplicity, we\nassume that the background magnetic \feld B0is either a\ndipolar \feld\nB0=B0\"\n^r\u0012R\nr\u00133\ncos\u0012+^\u0012\u0012R\nr\u00133sin\u0012\n2#\n; (1)\nor a uniform axial \feld\nB0=B0^z=B0(^rcos\u0012\u0000^\u0012sin\u0012); (2)\nwhere we have used spherical coordinates ( r;\u0012;\u001e ). Note that\nfor both cases, B0is a potential \feld, i.e. r\u0002B0= 0.\nWe consider the linear responses of the \ruid enve-\nlope to a tidal potential \t = A(r=R)lYm\nl(\u0012;\u001e)e\u0000i!t, where\nMNRAS 000, 1{14 (2017)Tidal dissipation: the presence of a magnetic \feld 3\nYm\nl(\u0012;\u001e) is a spherical harmonic and !is the tidal frequency\nin the rotating frame. The tidal responses can be decom-\nposed into non-wavelike and wavelike parts as introduced\nby Ogilvie (2013). In the absence of a magnetic \feld, the\nnon-wavelike part represents the instantaneous response to\nthe tidal potential, while the wavelike part is in the form of\ninertial waves excited by the e\u000bective force (Ogilvie 2013)\nf= 2\n\u0002r[Xl(r)Ym\nl(\u0012;\u001e)]e\u0000i!t; (3)\nwhereXlis associated with the non-wavelike motions. This\ne\u000bective force is not generally irrotational and results from\nthe failure of the non-wavelike tide to satisfy the equation\nof motion when the Coriolis force is included. For a homo-\ngeneous incompressible \ruid of the same density as that of\nthe rigid core, Xl(r) is given as (Ogilvie 2013)\nXl(r) =Cl\"\u0010r\nR\u0011l\n+\u000b2l+1l\nl+ 1\u0012R\nr\u0013l+1#\n: (4)\nThe constant Clis given by\nCl=i!(2l+ 1)R3A\n2l(l\u00001)(1\u0000\u000b2l+1)GM; (5)\nwhereGis the gravitational constant and Mis the total\nmass of the body. In this paper, we consider only the domi-\nnant tidal component of l= 2 andm= 2, unless otherwise\nspeci\fed.\nIn the presence of a magnetic \feld, we assume that the\nlarge-scale non-wave-like part is unchanged, and that the\nwave-like perturbations are still driven by the e\u000bective force\ngiven in equation (3), but we allow the magnetic \feld to\na\u000bect the small-scale wave-like motions through the induc-\ntion equation and the Lorentz force. The linearized equations\ngoverning the wavelike velocity perturbation uand magnetic\n\feld perturbation bin the rotating frame can be written as\n\u0000i!u+2\n\u0002u=\u00001\n\u001arp+1\n\u001a\u00160(r\u0002b)\u0002B0+\u0017r2u+f;(6)\n\u0000i!b=r\u0002(u\u0002B0) +\u0011r2b; (7)\nr\u0001u= 0; (8)\nr\u0001b= 0; (9)\nwhere\u0017is the \ruid viscosity, \u00160is the magnetic permeability\nand\u0011is the magnetic di\u000busivity. Equation (6) is the Navier-\nStokes equation including the Coriolis force and the Lorentz\nforce. Equation (7) is the magnetic induction equation.\nUsingR, \n\u00001,B0as units of length, time and magnetic\n\feld strength, we get the non-dimensional equations:\n\u0000i!u+2^z\u0002u=\u0000rp+Le2(r\u0002b)\u0002B0+Ekr2u+f;(10)\n\u0000i!b=r\u0002(u\u0002B0) +Emr2b; (11)\nwhere (and whereafter) !,u,p,b,B0andfdenote the\ncorresponding non-dimensional quantities. The dimension-\nless parameters in equations (10-11) are the Lehnert num-\nberLe, the Ekman number Ekand the magnetic Ekman\nnumberEm:\nLe=B0p\u001a\u00160\nR; Ek=\u0017\n\nR2; Em=\u0011\n\nR2: (12)The Lehnert number Lemeasures the strength of the back-\nground magnetic \feld with respect to rotation, i.e. the ratio\nbetween the Alf\u0013 ven velocity B0=p\u001a\u00160and the rotation ve-\nlocity \nRat the equator. The Ekman number Ekis the\nratio between the rotation time scale \n\u00001and the viscous\ntime scaleR2=\u0017, while the magnetic Ekman number Emis\nthe ratio between the rotation time scale \n\u00001and the mag-\nnetic di\u000busion time scale R2=\u0011. The last two parameters are\nrelated by the magnetic Prandtl number\nPm=Ek\nEm=\u0017\n\u0011: (13)\nWe set the dimensionless forcing term as\nf=^z\u0002r[(r2+\u000b5=r3)Y2\n2(\u0012;\u001e)](1\u0000\u000b5)\u00001e\u0000i!t; (14)\nforl= 2 andm= 2.\nIn order to minimize the viscous and electromagnetic\ncouplings between the \ruid layer and the rigid core, we use\nthe stress-free boundary condition for the velocity uand\nthe insulating boundary condition for the magnetic \feld b\nat both inner and outer boundaries. The stress-free bound-\nary condition implies that the tangential component of the\nviscous stress should vanish:\nur=@\n@r\u0010u\u0012\nr\u0011\n=@\n@r\u0010u\u001e\nr\u0011\n= 0 (15)\nThe insulating boundary condition indicates that no electric\ncurrent can go through the boundaries:\n(r\u0002b)r= 0; (16)\nand the magnetic \feld in the conducting \ruid should also\nmatch a potential \feld Be=\u0000rPin the exterior insulating\nregions, where Pis a scalar potential. This condition can\nbe readily expressed in terms of spherical harmonics (see\nAppendix A).\nThe viscous dissipation rate in dimensionless form is\nDvis=1\n2EkZ\nVRe[(r2u)\u0001u\u0003]dV; (17)\nand the dimensionless Ohmic dissipation rate is\nDohm=1\n2Le2EmZ\nVjr\u0002bj2dV; (18)\nwhere integrals are evaluated over the \ruid domain. It can\nbe shown from the integrated energy equation that the total\nenergy dissipation rate equals the power input by the tidal\nforcingf\nDvis+Dohm\u00111\n2Z\nVRe[f\u0001u\u0003]dV; (19)\nwhereu\u0003is the complex conjugate of u.\n2.2 Numerical method\nEquations (10-11) are solved using a pseudo-spectral\nmethod. We use a spheroidal-toroidal decomposition and\nthen project the equations on to spherical harmonics in a\nsimilar way as in Rincon & Rieutord (2003). The velocity\nand magnetic \feld perturbations are expanded as:\nu=X\num\nl(r)Rm\nl+X\nvm\nl(r)Sm\nl+X\nwm\nl(r)Tm\nl; (20)\nb=X\nam\nl(r)Rm\nl+X\nbm\nl(r)Sm\nl+X\ncm\nl(r)Tm\nl; (21)\nMNRAS 000, 1{14 (2017)4 Y. Lin and G. I. Ogilvie\nwith the summation is carried out over integers l\u0015m\u00150.\nHereRm\nl,Sm\nl,Tm\nlare vector spherical harmonics:\nRm\nl=Ym\nl(\u0012;\u001e)^r;Sm\nl=rrYm\nl(\u0012;\u001e);Tm\nl=rr\u0002Rm\nl:\n(22)\nFor the divergence-free \felds, the \frst two terms in equations\n(20-21) are also referred to as the poloidal part and the last\nterm is the toroidal part. The divergence-free conditions of\nuandb(equations 8-9) are satis\fed by\nvm\nl=1\nl(l+ 1)rd(r2um\nl)\ndr; (23)\nbm\nl=1\nl(l+ 1)rd(r2am\nl)\ndr: (24)\nThe equations projected on to spherical harmonics are given\nin Appendix A. We can see that equations (A1-A4) are de-\ncoupled for each mowing to the axisymmetric rotation and\nmagnetic \feld B0. The Coriolis force and the magnetic \feld\nonly couple the neighbouring spherical harmonics l\u00001 and\nl+ 1. Numerically, the system is truncated at a spherical\nharmonical degree L. In radial direction, we use Chebyshev\ncollocation on N+ 1 Gauss-Lobatto nodes. We use a typ-\nical truncation of L=N= 400 in most of calculations,\nbut higher resolutions up to L=N= 600 are also used\nfor a few more demanding calculations (when Em\u001410\u00005\nandLe\u001410\u00004). The numerical discretization leads to linear\nequations involving a large block-tridiagonal matrix, which\nis solved using the standard direct method based on LU fac-\ntorization.\nThe stress-free boundary condition at the inner and\nouter boundaries becomes\num\nl=d\ndr\u0012vm\nl\nr\u0013\n=d\ndr\u0012wm\nl\nr\u0013\n= 0: (25)\nThe insulating boundary condition requires vanishing\ntoroidal \feld, i.e. cm\nl= 0, at the inner and outer bound-\naries. The poloidal \feld needs to match a potential \feld,\nleading to (see Appendix A)\ndam\nl\ndr\u0000l\u00001\nram\nl= 0; (26)\nat the inner boundary and\ndam\nl\ndr+l+ 2\nram\nl= 0; (27)\nat the outer boundary.\nThe dissipation rates in equations (17-18) can be re-\nduced to integrals in radius only using the orthogonality of\nspherical harmonics:\nDvis=1\n2EkZ1\n\u000bLX\nl=ml(l+ 1)\f\f\f\fum\nl(r) +r2d[vm\nl(r)=r]\ndr\f\f\f\f2\n+l(l+ 1)\f\f\f\fr2d[wm\nl(r)=r]\ndr\f\f\f\f2\n+ 3\f\f\f\frdum\nl(r)\ndr\f\f\f\f2\n+ (l\u00001)l(l+ 1)(l+ 2)\u0000\njvm\nl(r)j2+jwm\nl(r)j2\u0001\ndr;(28)Dohm=1\n2Le2EmZ1\n\u000bLX\nl=ml2(l+ 1)2jcm\nl(r)j2\n+l(l+ 1)\f\f\f\fd[rcm\nl(r)]\ndr\f\f\f\f2\n+l(l+ 1)\f\f\f\fam\nl(r)\u0000d[rbm\nl(r)]\ndr\f\f\f\f2\ndr:\n(29)\nThe integrals are evaluated by a Chebyshev quadrature\nformula which uses the function values at the collocation\npoints. We also calculate the total dissipation rate using\nequation (19), which involves only spectral coe\u000ecients of\nl= 1 andl= 3. This is because the e\u000bective forcing term\nprojected onto spherical harmonics has only l= 1 andl= 3\ncomponents for the l= 2 tidal forcing (see equations (A33-\nA35) in Appendix A), and because of the orthogonality\nof spherical harmonics. The identity (19) can be used to\ncheck the numerical accuracy and convergence. Our numer-\nical code is also validated by comparing with some of the\nresults in Rincon & Rieutord (2003) and Ogilvie (2009).\n2.3 Dispersion relation of magnetic-Coriolis waves\nBefore presenting our numerical results, let us brie\ry recall\nthe dispersion relation of magnetic-Coriolis waves, which is\nuseful for the discussion of some results. Substituting the\nplane wave ansatz u;b/ei(k\u0001r\u0000!t)into equations (6-7),\nand neglecting the di\u000busive terms and the forcing term, we\ncan obtain the dispersion relation of magnetic-Coriolis waves\nin a uniform \feld B0(e.g. Finlay 2008):\n!=\u0006\n\u0001k\njkj\u0006\u0012(\n\u0001k)2\njkj2+(B0\u0001k)2\n\u001a\u00160\u00131=2\n: (30)\nIn the absence of a magnetic \feld, i.e. B0= 0, equation (30)\nrecovers the dispersion relation of inertial waves\n!=\u00062\n\u0001k\njkj; (31)\nwhich exist only when j!j<2\n. The group velocity of iner-\ntial waves is\nVg=\u00062k\u0002(\n\u0002k)\njkj3: (32)\nIn the absence of rotation, i.e. \n= 0, the dispersion relation\nof Alfv\u0013 en waves is obtained:\n!=Va\u0001k; (33)\nwhereVa=B0=p\u001a\u00160is the group velocity.\nThe propagation of MC waves is more complicated, de-\npending on the Lehnert number which measures the impor-\ntance of the magnetic \feld with respect to the rotation. The\ndispersion relation (30) in dimensionless form can be written\nas\n!=\u0006^z\u0001k\njkj\u0006\u0012(^z\u0001k)2\njkj2+Le2k2\nB\u00131=2\n; (34)\nwherekBis the wavenumber along the magnetic \feld B0.\n3 RESULTS\n3.1 Overview\nIn this section, we show a general overview of the spatial\nstructure and the dissipation rate of tidally forced MC waves\nMNRAS 000, 1{14 (2017)Tidal dissipation: the presence of a magnetic \feld 5\n(a)\n(b)\n(c)\n(d)\n(e)\n(f)\nFigure 2. Structure of the velocity perturbation jujand the mag-\nnetic \feld perturbation jbjin the meridional plane with an axial\n\feldB0at di\u000berent Le.Ek= 1:0\u000210\u00009,Em= 1:0\u000210\u00005,\n!= 1:1,\u000b= 0:5. Note that the colour scales may be di\u000berent for\ndi\u000berentLe.by varying the Lehnert number Le. We consider a case of\nthe radius ratio \u000b= 0:5 and the tidal frequency != 1:1,\nof which the hydrodynamic response has been studied in\ndetail (Ogilvie 2009). In the absence of a magnetic \feld, in-\nertial waves propagate along the characteristics (at a \fxed\nangle with respect to the rotation axis) and form two sim-\nple wave attractors after multiple re\rections (see Fig. 9 in\nOgilvie (2009)). The dissipation rate associated with iner-\ntial wave attractors is independent of the viscosity (the Ek-\nman number) provided that the Ekman number is asymp-\ntotically small (Ogilvie 2005, 2009). We use this case as a\nreference, mainly because of its relatively simple hydrody-\nnamic responses, to examine the e\u000bects of magnetic \felds.\nFig. 2 shows the velocity perturbation jujand the mag-\nnetic \feld perturbation jbjin the meridional plane in the\npresence of an axial magnetic \feld with various values of the\nLehnert number Le. The dissipative parameters are \fxed at\nEk= 10\u00009andEm= 10\u00005, meaning that the magnetic\nPrandtl number Pm= 10\u00004. When the Lehnert number is\nsu\u000eciently small ( Le\u0014O(E2=3\nm) as we shall show later),\nMC waves retain the rays of inertial waves, leading to the\nwave attractors as for purely inertial waves (Fig 2 a). Weak\nmagnetic \feld perturbations are induced along the attrac-\ntors by the velocity perturbations, but the Lorentz force\nhas negligible in\ruence on the propagation of waves. As the\nLehnert number is gradually increased, the perturbations do\nnot concentrate on the wave attractors any more, because\nthe Lorentz force starts to play a part. In Fig 2 (b), however,\nwe can still see the predominant e\u000bect of the rotation as the\nperturbations are mainly organized along the characteristics\nof inertial waves, but slightly modi\fed.\nAs we increase the Lehnert number further, the e\u000bect of\nrotation becomes less visible and the perturbations spread\nout to the whole \ruid domain (Fig 2 c). At certain values of\nthe Lehnert number, e.g. Le= 0:1 for this case, we observe\nsome large-scale structures in the polar region in Fig 2 (d).\nThese structures may be associated with eigen-modes of the\nsystem. We shall show more examples of such structures at\ndi\u000berent frequencies in section 3.2.\nAt relatively large values of Le, i.e.Le> 0:1, the mag-\nnetic e\u000bect become predominant as we can see from Fig. 2(e-\nf) that the perturbations concentrate along certain magnetic\n\feld lines. In this regime, the perturbations are essentially\nin the form of Alfv\u0013 en waves, where the magnetic tension acts\nas the restoring force. Each magnetic \feld line can be anal-\nogous to a string, which has a natural frequency depending\non the length and strength of the \feld line. Perturbations\nmainly concentrate along certain \feld lines, where a reso-\nnance may occur if the tidal frequency matches the natural\nfrequency of the \feld line.\nFig. 3 shows the structure of the perturbations as in\nFig. 2 but for a dipolar \feld B0. The spatial structures vary\nin a similar way as in the case of an axial \feld, as we grad-\nually increase Le. However, there are some local di\u000berences\nbecause the dipolar \feld B0is spatially non-uniform, being\nstronger at the polar regions than at the equator and de-\ncaying as a function of the radius. For instance, in Fig. 3(c),\nthe perturbations are less in\ruenced by the magnetic \feld\nnear the equator compared to other regions. Nevertheless,\nthe general picture is qualitatively similar to Fig. 2, from\ninertial wave attractors at small Leto nearly Alfv\u0013 en wave\nperturbations at relatively large Le.\nMNRAS 000, 1{14 (2017)6 Y. Lin and G. I. Ogilvie\n(a)\n(b)\n(c)\n(d)\n(e)\n(f)\nFigure 3. Same as Fig. 2 but for a dipolar \feld B0. Dashed lines\nshow some \feld lines of B0.\n10-510-410-310-210-1100\nL e10-710-610-510-410-310-210-1Dissipation rate\nViscous\nOhmic\nTotal(a)\n10-510-410-310-210-1100\nL e10-810-710-610-510-410-310-210-1100Dissipation rate\nViscous\nOhmic\nTotal\n(b)\nFigure 4. Dissipation rate versus the Lehnert number Lefor (a)\nan axial \feld, (b) a dipolar \feld. Vertical dash lines represent\nLe=E2=3\nm.Ek= 1:0\u000210\u00009,Em= 1:0\u000210\u00005,!= 1:1,\u000b= 0:5.\nBlack triangles correspond to cases shown in Figs. 2-3.\nWe have mentioned that the perturbations retain the\nray dynamics of inertial waves when Le\u0014O(E2=3\nm). We can\nderive this scaling by simply comparing several typical time\nscales in the system when Le\u001c1,Ek\u001c1,Em\u001c1 and\nPm\u001c1. The inertial wave propagation time in the \ruid\ndomain is\n\u001ci=L\njVgj\u0018\u0012l\nL\u0013\u00001\n\n\u00001: (35)\nThe time scale for Alfv\u0013 en waves to transversely cross the\ninertial wave beams is\n\u001ca=l\njVaj\u0018l\nLLe\u00001\n\u00001: (36)\nThe magnetic di\u000busion time across the beams is\n\u001c\u0011=l2\n\u0011\u0018\u0012l\nL\u00132\nE\u00001\nm\n\u00001; (37)\nand the viscous di\u000busion time is\n\u001c\u0017=l2\n\u0017\u0018\u0012l\nL\u00132\nE\u00001\nk\n\u00001: (38)\nHerelis the typical width of the wave beams, whereas L\nMNRAS 000, 1{14 (2017)Tidal dissipation: the presence of a magnetic \feld 7\nis the domain size, i.e. L=R. The viscous time scale is\nirrelevant here because of the assumption Pm\u001c1. In order\nto keep the perturbations within the inertial wave beams,\nthe crossing time of Alfv\u0013 en waves \u001cashould be longer than\nthe inertial wave propagation time \u001ci:\n\u001ca\u0015\u001ci: (39)\nMeanwhile, the width of the wave beams is set by the di\u000bu-\nsion time across the beams:\n\u001ci=\u001c\u0011: (40)\nCombining equations (39-40) and using equations (35-37),\nwe obtain\nLe\u0014O(E2=3\nm): (41)\nNote that this scaling analysis is merely heuristic. The\nprefactor of the scaling (41) varies depending on the fre-\nquency and the structure of background \felds. Neverthe-\nless, the above scaling provides an approximate threshold,\nabove which the presence of a magnetic \feld would modify\nthe propagation of inertial waves. This scaling is also clearly\nevidenced from the dissipation rate in Fig. 4.\nFig. 4 shows the viscous dissipation rate, the Ohmic\ndissipation rate and the total dissipation rate as a function\nofLefor an axial \feld (a) and a dipolar \feld (b). When\nLe\u0014O(E2=3\nm), the dissipation rate due to the Ohmic damp-\ning grows and then saturates as Leincreases, while the vis-\ncous dissipation rate drops and eventually becomes negligi-\nble. However, the total dissipation rate remains unchanged\nin the range of Le\u0014O(E2=3\nm). This observation is remi-\nniscent of the analytical result by Ogilvie (2005), who has\nshown that the total rate of energy dissipation of a wave\nattractor is independent of the dissipative properties and\nthe detailed damping mechanisms. The theoretical analysis\nhas been con\frmed by hydrodynamic calculations in spher-\nical shells (Ogilvie 2009; Rieutord & Valdettaro 2010). Here\nwe show that the theory is still valid in the presence of a\nmagnetic \feld as long as the wave attractors are retained.\nWhenLe > O (E2=3\nm), the total dissipation is almost\ntotally contributed by the Ohmic damping, whereas the vis-\ncous dissipation is negligible. The dissipation rate \ructuates\nas a function of Le, exhibiting several peaks and troughs. In\nthis range of parameters, the magnetic \feld modi\fes the\npropagation of waves depending on the Lehnert number.\nResonance may occur at certain values of Lefor a given fre-\nquency, leading to enhanced dissipation. Indeed, these peaks\nin the dissipation rate usually correspond to either large-\nscale perturbations, e.g. Fig 2(d) and Fig 3 (d), or pertur-\nbations concentrating on certain \feld lines, e.g. Fig 2 (f) and\nFig 3 (e).\nNote that we have restricted our investigations in the\nrange ofLe\u00141. We found that strong magnetic bound-\nary layers arise when Le\u001d1, and thus the energy dissi-\npation is mainly contributed by the boundary layers, which\nmay be not realistic because of our idealized boundary con-\nditions. For instance, a rigid core of \fnite electric conduc-\ntivity (rather than insulating) would relax the accumula-\ntion of the electrical current near the inner boundary. Any-\nway, the Lehnert number should be smaller than unity in\nmost stars and planets, although the speci\fc value is di\u000e-\ncult to estimate owing to the uncertainties of the magnetic\n\feld strength and \ruid properties. The Lehnert number forthe Sun is estimated to be around 10\u00005, if we assume the\ntypical magnetic \feld strength is a few 10\u00003T (Charbon-\nneau 2014), and use the mean density of the Sun. Mean-\nwhile, the magnetic Ekman number is also very small for\nthe Sun, i.e. Em<10\u000010, as the magnetic di\u000busion time is\naround 1010year (Charbonneau 2014). The magnetic \feld\nis strong enough to modify the propagation of inertial waves\nasLe > O (E2=3\nm) for the Sun, despite the small Lehnert\nnumber.\n3.2 Frequency-dependence\nWe now investigate the frequency-dependence of the dissi-\npation rate. Fig 5 shows a frequency scan of the total dissi-\npation rate in a frequency range of \u00003\u0014!\u00143 at various\nvalues ofLewith an axial \feld, and \u000b= 0:5,Ek= 10\u00008\nandEm= 10\u00004. For comparison, we show also the dissipa-\ntion rate in the absence of a magnetic \feld, i.e. Le= 0, in\nthe frequency range of inertial waves. In the presence of the\nmagnetic \feld, we show only the cases when Le>O (E2=3\nm),\nbecause the total dissipation rate shows similar behaviour\nto that of inertial waves when Le\u0014O(E2=3\nm).\nFig 5 (a) shows the dissipation rate at Le= 3:2\u000210\u00003\n(the blue curve) and Le= 2:4\u000210\u00002( the red curve). We\ncan see that these curves are very bumpy, likewise the curve\nin the absence of a magnetic \feld (the black dashed line).\nThe peaks and troughs are closely related to those the black\ndashed line. These cases can be regarded as weakly modi\fed\ninertial waves. In particular, we can see from the red curve\nthat the peaks shift to higher frequencies, which is in line\nwith the dispersion relation of magnetic-Coriolis waves.\nFig 5 (b) shows the dissipation rate at Le= 0:1 (the\nblue curve) and Le= 0:42 ( the red curve). These two curves\nare signi\fcantly smoothed out by the presence of a mag-\nnetic \feld compared to the black dashed line. In addition,\nthe spectra of the dissipation rate are broadened beyond\nthe frequency range of inertial waves, as expected from the\ndispersion relation (34).\nAlthough the dissipation rate curves become smooth at\nrelatively large Le, they still exhibit a few peaks in the fre-\nquency range we have shown. Fig. 6 shows structures of the\nperturbations at some peak frequencies of the blue curve in\nFig. 5 (b). We can see that all these cases feature smooth\nlarge-scale structures, in particular in the polar region. We\nalso note that the number of nodes in the vertical direc-\ntion increases as the frequency increases, which is reminis-\ncent of the dispersion relation of magnetic-Coriolis waves.\nWe have mentioned that such smooth structures may be\neigen-modes of the system, which are resonantly excited at\nthe eigen-frequencies, leading to the enhanced energy dissi-\npation. However, the theoretical analysis of the eigen value\nproblem is beyond the scope of this paper.\nFor the case of a dipolar \feld, the frequency-dependence\nof the total dissipation rate is qualitatively similar to that\nof an axial \feld, although details are di\u000berent, such as the\npeak frequencies.\n3.3 Frequency-averaged dissipation rate\nThe tidal dissipation leads to a long-term evolution of the\nspin and orbital parameters through the exchange of angu-\nlar momentum. As the system evolves, the tidal frequency\nMNRAS 000, 1{14 (2017)8 Y. Lin and G. I. Ogilvie\n-3 -2 -1 0 1 2 3\nFrequency10-610-510-410-310-210-1100D i s s i p a t i o n r a t e\n(a)\n-5-4-3-2-1012345\nFrequency10-610-510-410-310-210-1100D i s s i p a t i o n r a t e\n(b)\nFigure 5. Total dissipation rate versus the tidal forcing frequency. The background magnetic \feld is set to be an axial \feld. \u000b= 0:5,\nEk= 10\u00008andEm= 10\u00004. (a)Le= 3:2\u000210\u00003(blue) and Le= 2:4\u000210\u00002(red); (b)Le= 0:1 (blue) and Le= 0:42 (red). Black\ndashed lines represent the dissipation rate in the absence of a magnetic \feld. Blue triangles correspond to cases shown in Fig 6.\nvaries over time. It is very di\u000ecult to estimate the instanta-\nneous tidal dissipation owing to the complicated frequency-\ndependence. However, the frequency-averaged dissipation\nrate can be useful to study the long-term evolution of the\nsystem (Mathis 2015; Bolmont & Mathis 2016; Gallet et al.\n2017; Bolmont et al. 2017). Ogilvie (2013) has shown that\nthe frequency-averaged dissipation rate of inertial waves is\nindependent of the dissipative properties, but strongly de-\npends on the size of the rigid core. Here we examine the\nfrequency-averaged dissipation in the presence of a magnetic\n\feld.\nThe frequency-averaged dissipation can be measured by\nthe dimensionless quantity (Ogilvie 2013)\n\u0003 =Z1\n\u00001Im[Km\nl(!)]d!\n!; (42)\nwhere Im[Km\nl] is the imaginary part of the potential Lovenumber, and is related to the dissipation rate by\n^D=(2l+ 1)R\n8\u0019GA2\n!Im[Km\nl(!)]: (43)\nwhereAis the tidal amplitude with the unit of gravitational\npotential and ^Dis the dimensional dissipation rate with the\nunit of power. With our normalization of the forcing in equa-\ntion (14) for l= 2 andm= 2, the frequency-averaged quan-\ntity becomes\n\u0003 =Z1\n\u00001Im[K2\n2(!)]d!\n!=15\n2\u000f2Z1\n\u00001D(!)d!; (44)\nwhere\u000f2= \n2R3=GM andD(!) =Dvis+Dohmis the di-\nmensionless total dissipation rate as shown in Fig. 5. Note\nthat\u000fis a small parameter for astrophysical bodies, but we\nsimply set \u000f= 1 in our linear calculations of the wavelike\nperturbations. The above integral can be carried out only in\na \fnite frequency range numerically. For the hydrodynamic\ncase (Le= 0), the integral is evaluated over the frequency\nMNRAS 000, 1{14 (2017)Tidal dissipation: the presence of a magnetic \feld 9\n(a)\n(b)\n(c)\n(d)\n(e)\n(f)\nFigure 6. Structure of the velocity perturbation jujand the mag-\nnetic \feld perturbation jbjin the meridional plane with an axial\n\feldB0at di\u000berent tidal frequencies. Le= 0:1,Ek= 10\u00008,\nEm= 10\u00004,\u000b= 0:5.\n10-310-210-1100\nL e10-310-210-1100$ H y d r o\nU n i f o r m B 0 , E k = 1 0! 8, E m = 1 0! 4\nU n i f o r m B 0 , E k = 1 0! 1 0, E m = 1 0! 5\nD i p o l a r B 0 , E k = 1 0! 8, E m = 1 0! 4Figure 7. Frequency-averaged quantity \u0003 versus the Lehnert\nnumberLefor various di\u000berent parameters but for \fxed inner\ncore radius \u000b= 0:5.\nrange of inertial waves, i.e. \u00002<!< 2, as in Ogilvie (2013).\nThe spectrum of MC waves is unlimited according to the dis-\npersion relation (34). However, the major dissipation still oc-\ncurs in a \fnite frequency range in our calculations of Le< 1.\nThe integral is typically evaluated over the frequency range\nof\u00003< ! < 3 whenLe\u00140:1, and\u00005< ! < 5 when\nLe> 0:1. We also doubled the frequency range at large Le\nand found that the results of the integral are converged. Fig.\n7 shows the frequency-averaged quantity \u0003 at several di\u000ber-\nent parameters but with the \fxed inner core size \u000b= 0:5.\nWe can see that the frequency-averaged dissipation rate is\nindependent of the strength and the structure of the mag-\nnetic \feld, and the dissipative parameters EkandEm. Sim-\nilar results have been observed in a previous study using a\nperiodic box (Wei 2016). Also, the frequency-averaged dissi-\npation rate in the presence of a magnetic is nearly the same\nas that in the absence of a magnetic \feld. The small discrep-\nancies are likely due to the errors of numerical integrals.\nHowever, the frequency-averaged dissipation rate\nstrongly depends on the size of the inner core. Ogilvie\n(2013) derived an analytical expression by considering low-\nfrequency hydrodynamic responses to an impulsive forcing,\nwhich is given as\n\u0003 =Z1\n\u00001Im[K2\n2(!)]d!\n!=100\u0019\n63\u000f2\u0012\u000b5\n1\u0000\u000b5\u0013\n; (45)\nfor a homogeneous \ruid of the same density as that of the\nrigid core, and for the tidal component of l=m= 2. Equa-\ntion (45) has been veri\fed numerically (Ogilvie 2013), and\nresembles the scaling of \u000b5for the small inner core size found\nby previous hydrodynamic studies (Goodman & Lackner\n2009; Ogilvie 2009; Rieutord & Valdettaro 2010).\nFig. 8 show the frequency-averaged quantity \u0003 as a\nfunction of the radius ratio \u000bat di\u000berent values of Le. Re-\nmarkably, this frequency-averaged quantity in the presence\nof a magnetic \feld is still in very good agreement with equa-\ntion (45), which is derived in the absence of magnetic \felds.\nIn the presence of a magnetic \feld, the energy is mainly\ndissipated through the Ohmic damping of magnetic-Coriolis\nMNRAS 000, 1{14 (2017)10 Y. Lin and G. I. Ogilvie\n00.2 0.4 0.6 0.8 1\n,10-510-410-310-210-1100101102$\nL e = 0 ( a n a l y t i c a l )\nL e = 0 : 0 1\nL e = 0 : 1\nFigure 8. Frequency-averaged quantity \u0003 versus the radius ratio\n\u000bat di\u000berent values of Lewith an axial \feld and Ek= 10\u00008,\nEm= 10\u00004. The solid line represents the analytical expression\nfrom Ogilvie (2013).\nwaves, which occur over a wider frequency range, but the\nfrequency-averaged dissipation rate is the same as that of\nviscous dissipation of inertial waves. This suggests that the\nfrequency-averaged dissipation rate is independent of the de-\ntailed damping mechanisms. Indeed, we show in Appendix\nB that the analytical results on the frequency-averaged dis-\nsipation in Ogilvie (2013) are not altered by the presence\nof a magnetic \feld, in the approximation that the wave-like\nresponse is driven only by the Coriolis force acting on the\nnon-wavelike tidal \row, if we assume that the frequencies\nof MC waves are small compared to those of acoustic and\nsurface gravity waves. This is the case in our numerical cal-\nculations of the Lehnert number Le\u00141 (see Fig. 5). In real\nastrophysical \ruid bodies, this assumption may be justi\fed\nby the fact the tidal frequency is usually small compared to\nthe frequencies of acoustic and surface gravity waves.\n3.4 Obliquity tide\nSo far, we considered only the tidal component of l= 2\nandm= 2, which mainly determines the orbital evolution\nand synchronization. In this section, we brie\ry consider an-\nother important tidal component of l= 2 andm= 1, the\nso-called obliquity tide, which exists only in spin-orbit mis-\naligned systems and mainly determines the evolution of the\nspin-orbit angle. The obliquity tide is peculiar because the\ntidal frequency in the rotating frame is always equal to \u0000\nregardless of the orbital frequency, i.e. !=\u00001 in dimension-\nless form. In addition, the obliquity tide is responsible for\nthe precessional motion of the spin axis and the orbital nor-\nmal around the total angular momentum vector. We have\nshown that dissipative inertial waves can be excited by the\nobliquity tide on top of precession in a hydrodynamic study\n(Lin & Ogilvie 2017). To examine the magnetic e\u000bect on the\nwavelike responses of the obliquity tide, we need to replace\nthe forcing in equation (3) by\nf= 2\n\u0002r[Xl(r)Y1\n2(\u0012;\u001e)]ei\nt+ (\n\u0002\np)\u0002r; (46)\n0.2 0.4 0.6 0.8\n,10-610-410-2100102Dissipation rateL e = 0\nL e = 0 : 0 1\nL e = 0 : 1Figure 9. Total dissipation rate of the obliquity tide as a function\nof radius ratio \u000bat di\u000berent values of Lewith an axial \feld.\nEk= 10\u00007,Em= 10\u00004.\nwhere the last term arises from the precessional motion\naround the total angular momentum vector. The precession\nfrequency is determined by the tidal amplitude and given as\n(Lin & Ogilvie 2017)\n\np=\u000015\n8r\n5\n6\u0019R3\nA\n(1\u0000\u000b5)GMsini; (47)\nwhereiis the angle between the spin angular momentum\nand the total angular momentum.\nFig. 9 shows the total dissipation as a function of the\nradius ratio \u000bat di\u000berent value of Lewith an axial magnetic\n\feld. For comparison, we show also the dissipation rate in\nthe absence of a magnetic \feld, which exhibits complicated\ndependence of the core size owing to varied ray dynamics of\ninertial waves, especially when \u000b>0:5 (Lin & Ogilvie 2017).\nThe major e\u000bect of a magnetic \feld is, again, to smooth out\nthe dissipation rate curves, but the overall level is similar to\nthat of the hydrodynamic case. Note that the frequency of\nthe obliquity tide is always !=\u00001 in the rotating frame,\nso we did not explore the frequency-dependence and the\nfrequency-averaged quantity for the obliquity tide.\n4 CONCLUSIONS\nWe have investigated the magnetic e\u000bects on the tidal dis-\nsipation in rotating \ruid bodies using a simpli\fed model.\nThe tidal responses are decomposed into the non-wavelike\nand wavelike parts, but we have focused on the latter in\nthis study, which is in the form of magnetic-Coriolis waves.\nThe linearized wave equations are numerically solved using\na pseudo-spectral method. The major e\u000bects of the presence\nof a magnetic \feld can be summarized as follows.\n(i) When the magnetic \feld is very weak, namely\nLe\u0014O(E2=3\nm), the wavelike perturbations retain the ray\ndynamics of inertial waves, while the energy can be dissi-\npated through the viscous damping and the Ohmic damping.\nMNRAS 000, 1{14 (2017)Tidal dissipation: the presence of a magnetic \feld 11\nWhenLe>O (E2=3\nm), the magnetic \feld start to modify the\npropagation of waves, and the dissipation of energy occurs\nmainly through the Ohmic damping.\n(ii) The magnetic \feld smooths out the complicated de-\npendence of the total dissipation rate on the tidal frequency,\nand broadens the frequency spectrum of the dissipation rate,\ndepending on the Lehnert number Le.\n(iii) However, the frequency-averaged dissipation quan-\ntity is independent of the magnetic \feld strength, the \feld\nstructure and the dissipative parameters, but increases as\nthe relative size of the rigid core in our simpli\fed model. In\nmore realistic models, this frequency-averaged quantity may\ndepend on other properties of the internal structure such as\nthe density pro\fle. Indeed, the frequency-averaged quantity\nis in very good agreement with previous analytical results in\nthe absence of magnetic \felds (Ogilvie 2013). In Appendix\nB, we show that the magnetic \feld has no e\u000bect on the\nfrequency-averaged dissipation, in the approximation that\nthe wave-like response is driven only by the Coriolis force\nacting on the non-wavelike tidal \row, if we assume that the\nfrequencies of magnetic-Coriolis waves are small compared\nto those of acoustic and surface gravity waves.\nOwing to the complicated frequency-dependence of the\ndissipation rate of inertial waves, it is very di\u000ecult to di-\nrectly apply the instantaneous tidal dissipation to astro-\nphysical bodies. Alternatively, the frequency-averaged dis-\nsipation rate has been used to study tidal dissipation and\nevolution of stars (Mathis 2015; Bolmont & Mathis 2016;\nGallet et al. 2017; Bolmont et al. 2017). It has been con-\njectured that other e\u000bects such as non-linear interactions,\nconvection, di\u000berential rotations and magnetic \felds may\nwash out some of the complicated frequency-dependence of\npurely inertial waves (Ogilvie 2013). Indeed, our numeri-\ncal results show a smoothing e\u000bect of the magnetic \feld\non the frequency-dependence. More complicated processes\ncould lead to smoother dissipation curves. Therefore, the\nfrequency-averaged dissipation quantity is probably a useful\nindicator of the e\u000eciency of tidal dissipation. In addition,\nour results suggest that the frequency-averaged quantity is\ninsensitive to the detailed damping mechanisms and dissi-\npative properties. If this is still the case in more realistic\nmodels, it would be very useful in applications as these de-\ntails are not well understood in stars and planets. Note that\nthe frequency-averaged dissipation does depend on the in-\nternal structure of the bodies, which is merely determined\nby the size of the rigid core in our simpli\fed model.\nIt is worthwhile to mention that we considered only the\nmagnetic e\u000bects on the wave-like tidal perturbations due\nto the Coriolis force in this study. The magnetic \feld can\nalso interact with large-scale non-wave-like motions to pro-\nduce a further wave-like response, which remains to be stud-\nied. Our tentative investigations based on a radially forced\nmodel (Ogilvie 2009) suggest that the interactions between\na magnetic \feld and the non-wavelike motions may be not\nnegligible when the Lehnert number Le > 0:1. However,\nthe boundary conditions probably need to be treated more\ncarefully with a magnetic \feld, as the non-wavelike part is\nassociated with the instantaneous tidal deformation.ACKNOWLEDGEMENTS\nWe would like to thank the anonymous referee for a set of de-\ntailed and constructive comments, which helped improve the\npaper. YL acknowledges the support of the Swiss National\nScience Foundation through an advanced Postdoc.Mobility\nfellowship.\nREFERENCES\nAbbassi S., Rieutord M., Rezania V., 2012, MNRAS, 419, 2893\nBardsley O., Davidson P., 2017, Geophysical Journal Interna-\ntional, 210, 18\nBarker A. J., Ogilvie G. I., 2010, MNRAS, 404, 1849\nBolmont E., Mathis S., 2016, Celestial Mechanics and Dynamical\nAstronomy, 126, 275\nBolmont E., Gallet F., Mathis S., Charbonnel C., Amard L., Al-\nibert Y., 2017, A&A, 604, A113\nBu\u000bett B. A., 2010, Nature, 468, 952\nCharbonneau P., 2014, ARA&A, 52, 251\nEssick R., Weinberg N. N., 2016, ApJ, 816, 18\nFinlay C. 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J., Deming D., Hamilton D.,\nGillon M., Jehin E., 2017, ApJ, 836, L24\nWu Y., 2005, ApJ, 635, 674\nZahn J.-P., 1975, A&A, 41, 329\nMNRAS 000, 1{14 (2017)12 Y. Lin and G. I. Ogilvie\nAPPENDIX A: EQUATIONS PROJECTED ON\nTO SPHERICAL HARMONICS\nTaking the curl and the curl of the curl of equation (10),\ntaking the curl of equation (11) and using the orthogonality\nof the vector spherical harmonics (Rieutord 1987), we obtain\nthe following projected equations. Similar equations were\ngiven in the appendix of Rincon & Rieutord (2003), but\nwithout the Coriolis term. The projection of the Coriolis\nforce can be found in, e.g. Rieutord (1987). The projected\nequations are given as\n\u0000i!lDl(um\nl) =\u00002\fm\nlDw\nl\u00001(wm\nl\u00001)\u00002\fm\nl+1Dw\nl+1(wm\nl+1)\n+Le2[imLc\nl(cm\nl) +\u000bm\nlLa\nl\u00001(am\nl\u00001) +\u000bm\nl+1La\nl+1(am\nl+1)]\n+EDu\nl(um\nl) +fl;(A1)\n\u0000i!lwm\nl=\u00002\u000bm\nlDu\nl\u00001(um\nl\u00001)\u00002\u000bm\nl+1Du\nl+1(um\nl+1)\n+Le2[imLa\nl(am\nl) +\u000bm\nlLc\nl\u00001(cm\nl\u00001) +\u000bm\nl+1Lc\nl+1(cm\nl+1)]\n+EDw\nl(wm\nl) +fl\u00001+fl+1;(A2)\n\u0000i!am\nl= imLw\nl(wm\nl) +\u000bm\nlLu\nl\u00001(um\nl\u00001) +\u000bm\nl+1Lu\nl+1(um\nl+1)\n+EmDa\nl(am\nl);(A3)\n\u0000i!cm\nl= imLu\nl(um\nl) +\u000bm\nlLw\nl\u00001(wm\nl\u00001) +\u000bm\nl+1Lw\nl+1(wm\nl+1)\n+EmDc\nl(cm\nl);(A4)\nwhereDandLdenote linear di\u000berential operators:\nDl=rd2\ndr2+ 4d\ndr\u0000(l2+l\u00002)1\nr; (A5)\nDw\nl\u00001=d\ndr\u0000(l\u00001)1\nr; (A6)\nDw\nl+1=d\ndr+ (l+ 2)1\nr(A7)\nDu\nl=rd4\ndr4+ 8d3\ndr3\u00002(lp\u00006)1\nrd2\ndr2\n\u00004lp1\nr2d\ndr+lp(lp\u00002)1\nr3;(A8)\nLc\nl=Brd2\ndr2+\u00122Br\nr+dBr\ndr\u0013d\ndr\n+dBr\ndr\u0000lpd\ndr\u0012B\u0012\nr\u0013\n;(A9)\nLa\nl\u00001=lp\u0014\nrBrd3\ndr3+\u0012\nlB\u0012+rdBr\ndr+ 6Br\u0013d2\ndr2\u0015\n\u0000lp\u0014\n(l+ 2)(l\u00003)Br\nr\u00004dBr\ndr\u00004lB\u0012\nr\u0015d\ndr\n\u0000lp(l+ 1)(l\u00002)\u0012\nlB\u0012\nr2+1\nrdBr\ndr\u0013\n;(A10)\nLa\nl+1=lp\u0014\nrBrd3\ndr3\u0000\u0012\n(l+ 1)B\u0012\u0000rdBr\ndr\u00006Br\u0013d2\ndr2\u0015\n\u0000lp\u0014\n(l+ 4)(l\u00001)Br\nr\u00004dBr\ndr\u00004(l+ 1)B\u0012\nr\u0015d\ndr\n+lpl(l+ 3)\u0012\n(l+ 1)B\u0012\nr2\u00001\nrdBr\ndr\u0013\n;(A11)Du\nl\u00001=rd\ndr\u0000(l\u00002); (A12)\nDu\nl+1=rd\ndr+ (l+ 3); (A13)\nDw\nl=d2\ndr2+2\nrd\ndr\u0000lp1\nr2; (A14)\nLa\nl=\u0000Br\nl2p\u0012\nrd2\ndr2+ 4d\ndr\u0000(lp\u00002)1\nr\u0013\n; (A15)\nLc\nl\u00001=l(l\u00001)\u0012\nBrd\ndr+Br\nr+lB\u0012\nr\u0013\n; (A16)\nLc\nl+1= (l+ 1)(l+ 2)\u0012\nBrd\ndr+Br\nr\u0000(l+ 1)B\u0012\nr\u0013\n;(A17)\nDa\nl=d2\ndr2+4\nrd\ndr+ (2\u0000lp)1\nr2; (A18)\nLw\nl=Br\nr; (A19)\nLu\nl\u00001=lp\u0012\nBrd\ndr+ 2Br\nr+lB\u0012\nr\u0013\n; (A20)\nLu\nl+1=lp\u0012\nBrd\ndr+ 2Br\nr\u0000(l+ 1)B\u0012\nr\u0013\n; (A21)\nlp=l(l+ 1) (A22)\nDc\nl=d2\ndr2+2\nrd\ndr+lp1\nr2; (A23)\nLu\nl=\u00001\nl2p\u0014\nrBrd2\ndr2+\u0012\n4Br+rdBr\ndr\u0013d\ndr\u0015\n1\nl2p\u0012\nlpdB\u0012\ndr\u0000lpB\u0012\nr\u0000dBr\ndr\u0000Br\nr\u0013\n;(A24)\nLw\nl\u00001=l(l\u00001)\u0012\nBrd\ndr+Br\nr+dBr\ndr+lB\u0012\nr\u0013\n; (A25)\nLw\nl+1= (l+ 1)(l+ 2)\u0012\nBrd\ndr+Br\nr+dBr\ndr\u0000(l+ 1)B\u0012\nr\u0013\n:\n(A26)\nIn above operators, we have used the following notations:\n!l=!+2m\nl(l+ 1); (A27)\nlp=l(l+ 1); (A28)\nMNRAS 000, 1{14 (2017)Tidal dissipation: the presence of a magnetic \feld 13\nqm\nl=\u0012l2\u0000m2\n4l2\u00001\u00131=2\n; (A29)\n\u000bm\nl=1\nl2qm\nl; \fm\nl= (l2\u00001)qm\nl; (A30)\nandBrandB\u0012represent the radial dependence of the back-\nground magnetic \feld B0. For the dipolar \feld, we have\nBr=1\nr3; B\u0012=1\n2r3; (A31)\nwhile for the uniform vertical \feld\nBr= 1; B\u0012=\u00001: (A32)\nThe vortical tidal forcing fcan be also project on to spher-\nical harmonics\nfl=\u00002im\nl(l+ 1)\u0012\nl(l+ 1)Xl(r)\nr\u0000rd2Xl(r)\ndr2\u00002dXl(r)\ndr\u0013\n;\n(A33)\nfl\u00001=\u00002qm\nl\nl\u0012dXl(r)\ndr+ (l+ 1)Xl(r)\nr\u0013\n; (A34)\nfl+1= 2qm\nl+1\nl+ 1\u0012dXl(r)\ndr\u0000lXl(r)\nr\u0013\n: (A35)\nFor a homogeneous \ruid, fl\u00110 because Xl(r) satis\fes\n(equation (86) in Ogilvie 2013)\n1\nr2d\nr\u0012\nr2dXl(r)\ndr\u0013\n\u0000l(l+ 1)\nr2Xl(r) = 0: (A36)\nThe boundary conditions can be also projected onto\nspherical harmonics. The stress-free boundary condition (15)\nleads to\num\nl=d\ndr\u0012vm\nl\nr\u0013\n=d\ndr\u0012wm\nl\nr\u0013\n= 0: (A37)\nThe insulating boundary condition ( r\u0002b)rleads tocm\nl=\n0. The poloidal magnetic \feld continuously extends to the\ninsulating regions as a potential \feld Be=\u0000rP, where the\nscalar potential Psatis\fes Laplace's equation\nr2P= 0: (A38)\nThe solution of the above equation is\nP=XX\ngm\nlrlYm\nl(\u0012;\u001e); (A39)\nin the rigid inner core and\nP=XX\nhm\nlr\u0000(l+1)Ym\nl(\u0012;\u001e); (A40)\noutside the body. Substituting those solutions into Be=\n\u0000rPand comparing with the spherical harmonics expan-\nsion ofbin the \ruid region, we obtain the boundary condi-\ntions\nam\nl(r)\u0000lbm\nl(r) = 0; (A41)\nat the inner boundary and\nam\nl(r) + (l+ 1)bm\nl(r) = 0; (A42)\nat the outer boundary. Using the divergence-free conditionofbin equation (24), we can write the boundary condi-\ntions (A41-A42) as\ndam\nl\ndr\u0000l\u00001\nram\nl= 0; (A43)\nat the inner boundary and\ndam\nl\ndr+l+ 2\nram\nl= 0; (A44)\nat the outer boundary.\nAPPENDIX B: FREQUENCY-AVERAGED\nDISSIPATION IN THE PRESENCE OF A\nMAGNETIC FIELD\nOgilvie (2013) found a way of calculating a certain\nfrequency-average of the tidal response of a slowly and uni-\nformly rotating barotropic \ruid body to harmonic forcing.\nIn this Appendix we consider how the argument and re-\nsults presented in Section 4 of that paper are a\u000bected by the\npresence of a magnetic \feld. All the section numbers and\nequation numbers used below refer to Ogilvie (2013).\nThe equilibrium condition (21) is modi\fed to\n0 =\u0000r(h+ \b g+ \b c) +1\n\u00160\u001a(r\u0002B)\u0002B (B1)\nand the linearized equations are modi\fed to\n\u0018+ 2\n\u0002_\u0018=\u0000rW+F\u0018; (B2)\nW=h0+ \b0+ \t; (B3)\n\u001a0=\u0000r\u0001(\u001a\u0018); (B4)\nr2\b0= 4\u0019G\u001a0; (B5)\nwhereFis the linearized Lorentz force operator, which is\nself-adjoint with respect to a mass-weighted inner product\nand is given by\nF\u0018=1\n\u00160\u001a\u0002\n(r\u0002B0)\u0002B+ (r\u0002B)\u0002B0\u0003\n; (B6)\nwhere\nB0=r\u0002(\u0018\u0002B): (B7)\nIn making the low-frequency asymptotic analysis in Sec-\ntion 4.4, we wish to assume that the frequencies of Alfv\u0013 en\nand slow magnetoacoustic waves are, like those of the in-\nertial waves, small compared to those of acoustic (or fast\nmagnetoacoustic) and surface gravity waves. Unlike the in-\nertial waves, however, the Alfv\u0013 en and slow magnetoacoustic\nwaves are not bounded in frequency if we allow ourselves to\nconsider arbitrarily short wavelengths. We therefore need to\napply a high-wavenumber cuto\u000b to the response in order to\ncontain the spectrum of low-frequency oscillations. This may\nbe justi\fed by assuming that the tidal response is smooth or\nby appealing to resistivity to eliminate disturbances of small\nscale.\nThe relevant scaling assumptions are then that the\nMNRAS 000, 1{14 (2017)14 Y. Lin and G. I. Ogilvie\nLehnert number is O(1) (or smaller) and that the pertur-\nbations are of large scale. Formally this can be achieved by\nsaying that both \n and BareO(\u000f). With the arbitrary nor-\nmalization \t = O(1), we then have (as before) \u0018=O(1),\nW=O(\u000f2),h0=O(1), \b0=O(1) and nowB0=O(\u000f). Our\nreduced system of linearized equations at leading order is\nthen\n\u0018+ 2\n\u0002_\u0018=\u0000rW+F\u0018; (B8)\nh0+ \b0+ \t = 0; (B9)\n\u001a0=\u0000r\u0001(\u001a\u0018); (B10)\nr2\b0= 4\u0019G\u001a0; (B11)\nand is to be solved on a spherically symmetric, hydrostatic\nbasic state una\u000bected by rotation or magnetic \felds.\nWe again decompose the perturbations into non-\nwavelike and wavelike parts, satisfying respectively\n\u0018nw=\u0000rWnw; (B12)\nh0\nnw+ \b0\nnw+ \t = 0; (B13)\n\u001a0\nnw=\u0000r\u0001(\u001a\u0018nw); (B14)\nr2\b0\nnw= 4\u0019G\u001a0\nnw; (B15)\nand\n\u0018w+ 2\n\u0002_\u0018w=\u0000rWw+F\u0018w+f; (B16)\nr\u0001(\u001a\u0018w) = 0; (B17)\nwhere\u001a0\nw=h0\nw= \b0\nw= 0 and\nf=\u00002\n\u0002_\u0018nw+F\u0018nw (B18)\nis the e\u000bective force per unit mass driving the wavelike part\nof the solution. As before, the non-wavelike tide may be\nassumed to be instantaneously related to the tidal potential\nthrough\n\u0018nw=\u0000rX; (B19)\nwhereXis the solution of the elliptic equation (61). The\nenergy equation for the wavelike part is\nd\ndt\u00141\n2Z\n\u001a\u0000\njuwj2\u0000\u0018w\u0001F\u0018w\u0001\ndV\u0015\n=Z\n\u001auw\u0001fdV; (B20)\nwhereuw=_\u0018wis the wavelike velocity and the second term\nin the integral on the left-hand side is the magnetic energy\nassociated with the wavelike displacement.\nTurning now to the impulsive forcing analysed in Sec-\ntion 4.6, we again consider a tidal potential of the form\n\t = ^\t(r)H(t); (B21)whereH(t) is the Heaviside step function. This implies that\n\u0018nw=^\u0018nw(r)H(t); (B22)\nleading to an e\u000bective force\nf=^f(r)\u000e(t) +~f(r)H(t); (B23)\nwhere\u000e(t) is the Dirac delta function,\n^f=\u00002\n\u0002^\u0018nw (B24)\nderives solely from the Coriolis force and\n~f=F^\u0018nw (B25)\nderives solely from the Lorentz force. Therefore the impul-\nsive contibution to the e\u000bective force comes only from the\nCoriolis force and not from the Lorentz force. The solution\nof equations (B16) and (B17) in this case involves a wave-\nlike displacement \u0018wthat is continuous in tbut has a dis-\ncontinuous \frst derivative at t= 0. The wavelike velocity\nimmediately after the impulse is again\n^uw=^f\u0000r^Ww; (B26)\nwhere ^Wwis chosen to satisfy the anelastic constraint\nr\u0001(\u001a^uw) = 0 and the boundary conditions ^ uw;r= 0. The\nenergy transferred in the impulse is equal to the kinetic en-\nergy immediately after the event,\n^E=1\n2Z\n\u001aj^uwj2dV: (B27)\nThere is no change in the wavelike magnetic energy at t= 0\nbecause\u0018wis continuous there. Since ^uwderives solely from\nthe Coriolis force, we conclude that the magnetic \feld has\nno e\u000bect either on the impulsive energy transfer associated\nwith this term, or on the frequency-averaged dissipation re-\nlated to it through equation 100. We note, however, that\nthe Lorentz part of the e\u000bective force in equation (B23),\nwhich is neglected in the main part of this paper, could al-\nter the energy of the wave-like disturbance after the impulse\natt= 0 and therefore make an additional contribution to\nthe frequency-averaged dissipation, which requires further\ninvestigation.\nThis paper has been typeset from a T EX/LATEX \fle prepared by\nthe author.\nMNRAS 000, 1{14 (2017)"    },    {        "title": "2112.06835v2.Rotons_and_their_damping_in_elongated_dipolar_Bose_Einstein_condensates.pdf",        "content": "Rotons and their damping in elongated dipolar Bose-Einstein condensates.\nS. I. Matveenko,1, 2M. S. Bahovadinov,2, 3M. A. Baranov,4, 5and G. V. Shlyapnikov2, 6, 7, 8\n1L. D. Landau Institute for Theoretical Physics, Chernogolovka, Moscow region 142432, Russia\n2Russian Quantum Center, Skolkovo, Moscow 143025, Russia\n3Physics Department, National Research University Higher School of Economics, Moscow, 101000, Russia\n4Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria\n5Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria\n6Moscow Institute of Physics and Technology, Inst. Lane 9, Dolgoprudny, Moscow Region 141701, Russia\n7Universit\u0013 e Paris-Saclay, CNRS, LPTMS, 91405 Orsay, France\n8Van der Waals-Zeeman Institute, Institute of Physics, University of Amsterdam,\nScience Park 904, 1098 XH Amsterdam, The Netherlands\n(Dated: March 17, 2022)\nWe discuss \fnite temperature damping of rotons in elongated Bose-condensed dipolar gases, which\nare in the Thomas-Fermi regime in the tightly con\fned directions. The presence of many branches\nof excitations which can participate in the damping process, is crucial for the Landau damping\nand results in signi\fcant increase of the damping rate. It is found, however, that even rotons with\nenergies close to the roton gap may remain fairly stable in systems with the roton gap as small as\n1nK.\nI. INTRODUCTION\nThe spectrum of elementary excitations is strongly\nin\ruenced by the character of interparticle interactions\nand is a key concept for understanding the behavior of\nquantum many-body systems. For Bose-condensed sys-\ntems with a short-range interparticle interaction the low-\nenergy part of the spectrum represents phonons with a\nlinear energy-momentum dependence. In some cases, the\nexcitation spectrum has an energy minimum at rather\nlarge momenta with roton excitations around it, which\nis separated by a maximum (maxon excitations) from\nthe low-energy phonon part. The roton-maxon excita-\ntion was \frst observed in liquid4He, and intensive dis-\ncussions during decades arrived at the conclusion that\nthe presence of the roton is related to the tendency to\nform a crystalline order [1, 2]. The presence of rotons in\nthe excitation spectrum of dipolar Bose-Einstein conden-\nsates was \frst predicted in Refs. [3, 4] and is considered\nas a precursor of the formation of a supersolid phase (for\na review on the supersolid phase and its experimental\nmanifestations see, for example, Ref. [5]).\nDuring last several years, supersolid phases were ob-\nserved experimentally in systems of ultracold trapped\nbosonic magnetic atoms (Dy, Er) [6{8], as well as the\npresence of the roton excitations and their role in the for-\nmation of the supersolid state [9{12]. In systems of mag-\nnetic atoms, the formation of the supersolid state and\nthe appearance of the roton excitations are attributed\nto the magnetic dipole-dipole interaction. The present\ntheoretical description of the excitations is based on nu-\nmerical solutions of the three-dimensional Bogoliubov -\nde Gennes equations in a trapped geometry at zero tem-\nperature [11{13], and is focused on the real part of their\ndispersion, without addressing the question of the exci-\ntation damping. The damping, however, strongly a\u000bects\nthe system response to external perturbations which are\nused to probe the system properties (see, for example,[14]). Therefore, studies of the excitation damping and\nits temperature dependence have not only theoretical in-\nterest, but also direct experimental relevance. These\nstudies should also indicate how stable are rotonic excita-\ntions, in particular for building up roton-induced density\ncorrelations in non-equilibrium systems. [15]. Another\nissue is the spatial roton con\fnement in trapped Bose-\nEinstein condensates [16].\nDamping of rotons in quasi-1D dipolar Bose-condensed\ngases has been discussed in Refs. [17{19]. In this pa-\nper we investigate the damping of rotons in an elongated\nBose-condensed polarized dipolar gas, which is in the\nThomas-Fermi (TF) regime in the tightly con\fned di-\nrections [20, 21]. In this case, there is a large number of\nbranches in the excitation spectrum, and many of them\ncan contribute to the Landau damping, which is the lead-\ning damping mechanism at \fnite temperatures [22]. This\nmay signi\fcantly increase the damping rate and make ro-\ntons unstable.\nThe paper is organized as follows. In Section II we\npresent general relations for calculating the condensate\nwave function, excitation spectrum, and damping rates\nfor rotons. Sections III and IV are dedicated to the ap-\nproximation of cylindrically isotropic consdensate and\ncontains analytical expressions for excitation energies\nand the resulting damping rates. In Section V we present\nthe results of direct numerical calculations of these quan-\ntities and it is con\frmed that the approximation of iso-\ntopic condensate gives a qualitatively correct picture.\nOur concluding remarks are given in Section VI.\nII. GENERAL RELATIONS\nWe consider an elongated Bose-Einstein condensate\nof polarized dipolar particles (magnetic atoms or polar\nmolecules). The motion in the zdirection is free, and in\nthex;ydirections it is harmonically con\fned with fre-arXiv:2112.06835v2  [cond-mat.quant-gas]  16 Mar 20222\nquency!. We consider the case where the dipoles are\npolarized perpendicularly to the zaxis (let say, are alongthexdirection). The ground state condensate wave func-\ntion \t 0(r) obeys the Gross-Pitaevskii (GP) equation:\n\u0014\n\u0000~2\n2mr2+m!2\u001a2\n2+Z\nd3r0V(r\u0000r0)j\t0(r0)j2\u0015\n\t0(r) =\u0016\t0(r); (1)\nwhere\u001a2=x2+y2,\u0016is the chemical potential of the\nsystem, and\nV(r) =g\u000e(r) +Vd(r); (2)\nwithgbeing the coupling constant of the short-range\n(contact) interaction, and Vd(r) the potential of the\ndipole-dipole interaction between two atoms. For the\ndipoles doriented in one and the same direction we haveVd(r) =d2r2\u00003(dr)2\nr5: (3)\nRepresenting the \feld operator of the non-condensed\npart of the system as \t0(r;t) =P\n\u0017[u\u0017(r)^b\u0017\u0000v\u0003\n\u0017(r)by\n\u0017],\nwhere the index \u0017labels eigenstates of the excitations and\nb\u0017,by\n\u0017are their creation and annihilation operators, we\nhave the Bogoliubov-de Gennes equations for the func-\ntionsuandv:\n\u0000~2\n2mr2u(r) +m!2\u001a2\n2u(r)\u0000\u0016u(r) + [^V\t2\n0(\u001a)]u(r) + [^V\t0(\u001a)u(r)] \t0(\u001a)\u0000[^V\t0(\u001a)v(r)] \t0(\u001a) =Eu(r);(4)\n\u0000~2\n2mr2v(r) +m!2\u001a2\n2v(r)\u0000\u0016v(r) + [^V\t2\n0(\u001a)]v(r) + [^V\t0(\u001a)v(r)] \t0(\u001a)\u0000[^V\t0(\u001a)u(r)] \t0(\u001a) =\u0000Ev(r);(5)\nwhere\n[^Vf(r)]\u0011Z\nV(r\u0000r0)f(r0)d3r0; (6)\nfor any function f, andEis the excitation energy.\nAt \fnite temperatures, the leading damping mecha-\nnism for the roton excitation is the Landau damping.\nIn particular, a roton with energy E0(q) and momen-\ntumqinteracts with a thermal low-momentum ( jpj\u001cq)sound type excitation with energy Ej(p). Both get anni-\nhilated, and an excitation with a higher energy El(q+p)\nis created, where j;lare excitation branch numbers j;l=\n0;1;2:::. We calculate the damping rate for the lowest ro-\ntonic excitation which has momentum k. The damping\nrate is given by the Fermi golden rule\n1\n\u001c=X\nj;l1\n\u001cjl; (7)\n1\n\u001cjl=2\u0019\n~Z1\n\u00001dp\n2\u0019\u0002\njhq+p;ljHintjq;fp;jgij2\u0000jhq;fp;jgjHintjq+p;lij2\u0003\n\u000e(Eq+p;l\u0000Eq;0\u0000Ep;j): (8)\nThe Hamiltonian Hintresponsible for the damping represents the interaction between excitations and is given by (see,\ne. g. [22])\nHint=Z\nd3rd3r0\u0002\n\t0(r)\t0y(r0)V(r\u0000r0)\t0(r0)\t1(r) + \t0y(r)\t0y(r0)V(r\u0000r0)\t0(r0)\t0(r)\u0003\n: (9)\nWe use the representation of functions u;vin the non-\ncondensed operator \t0in the form u\u0017(r) =uk;j(~ \u001a)eikz\nand similarly for v\u0017(r), wherekis the wave vector, and jis the number of the excitation branch. Then the Landau3\ndamping of rotons acquires the form\n1\n\u001c=2\u0019\nL~X\nk;n1;n2jAq\nk;k+qj2(Nk\u0000Nk+q)\u000e(Ek;n1+Eq;0\u0000Ek+q;n2);\n(10)\nwhereNk= 1=(eEk=T\u00001) are excitation occupation num-\nbers, and the matrix element Aq\nk;k+qis the sum of the\nintegrals:\nAq\nk;k+q=Z\nd\u001ad\u001a0h\n\t\u0003\n0(\u001a)u\u0003\nk+q;n1(\u001a0)~V(\u001a\u0000\u001a0;k)uk;n2(\u001a)uq;0(\u001a0) + \t\u0003\n0(\u001a)u\u0003\nk+q;n1(\u001a0)~V(\u001a\u0000\u001a0;q)uq;0(\u001a)uk;n2(\u001a0)\n+\t\u0003\n0(\u001a)vk;n1(\u001a0)~V(\u001a\u0000\u001a0;q)uq;0(\u001a)v\u0003\nk+q;n2(\u001a0) + \t\u0003\n0(\u001a)vq;0(\u001a0)~V(\u001a\u0000\u001a0;k)uk;n1(\u001a)v\u0003\nk+q;n2(\u001a0)\n+\t\u0003\n0(\u001a)vk;n1(\u001a0)~V(\u001a\u0000\u001a0;k+q)v\u0003\nk+q;n2(\u001a)uq;0(\u001a0) + \t\u0003\n0(\u001a)vq;0(\u001a0)~V(\u001a\u0000\u001a0;k+q)v\u0003\nk+q;n2(\u001a)uk;n1(\u001a0)\n\u0000\t0(\u001a0)u\u0003\nk+q;n1(\u001a)vk;n2(\u001a0)~V(\u001a\u0000\u001a0;k)uq;0(\u001a)\u0000\t0(\u001a0)u\u0003\nk+q;n1(\u001a)vq;0(\u001a0)~V(\u001a\u0000\u001a0;q)uk;n2(\u001a)\n\u0000\t0(\u001a0)vk;n1(\u001a)vq;0(\u001a0)~V(\u001a\u0000\u001a0;q)v\u0003\nk+q;n2(\u001a)\u0000\t0(\u001a0)vq;0(\u001a)vk;n1(\u001a0)~V(\u001a\u0000\u001a0;q)v\u0003\nk+q;n2(\u001a)\n\u0000\t0(\u001a0)vk;n1(\u001a)u\u0003\nk+q;n2(\u001a0)~V(\u001a\u0000\u001a0;k+q)uq;0(\u001a)\u0000\t0(\u001a0)vq;0(\u001a)u\u0003\nk+q;n2(\u001a0)~V(\u001a\u0000\u001a0;k+q)uk;n1(\u001a)i\n:(11)\nThe Fourier transform of the interaction potential is\nequal to\n~V(\u001a;k) =Z\ndzV(r)eikz=\u00003g\u0011\n2\u0019@2\n@2xK0(k\u001a)+g(1\u0000\u0011)\u000e(~ \u001a);\n(12)\nwithK0being the modi\fed Bessel function of the second\nkind, and\n\u0011=4\u0019d2\n3g=gd\ng: (13)\nGenerally speaking, the condensate wave function is\nanistropic in the x;yplane. However, in order to gain\ninsight into the physical picture it is \frst reasonable to\nassume that \t2\n0is isotropic. This allows us to get analyt-\nical expressions for excitations energies and use them for\n\fnding the damping rates of rotons. Direct numerical\ncalculations presented in Section V show that the ap-\nproximation of isotropic condensate gives a qualitatively\ncorrect physical picture and reasonable results.\nIII. APPROXIMATION OF ISOTROPIC\nCONDENSATE. EXCITATION SPECTRUM.\nThe condensate wave function \t 0isz-independent and\nin this section we assume that it depeneds only on \u001a,\ni.e it is symmetric in the x;yplane. In the Thomas-\nFermi regime the kinetic energy of the condensate is omit-\nted, and one expects that \t2\n0has the shape of inverted\nparabola. Then we obtain\nZ\nV(r\u0000r0)\t2\n0(r0)d3r0\u0019g(1 +\u0011=2)\t2\n0(\u001a): (14)Equation (1) then takes the form\nm!2\u001a2\n2 0(\u001a) +g(1 +\u0011=2)\t3\n0(\u001a) =\u0016\t0(\u001a); (15)\nand, hence, the condensate wavefunction is given by\n\t2\n0(\u001a) =n0\u0012\n1\u0000\u001a2\nR2\u0013\n\u0002(R\u0000\u001a); (16)\nwhere \u0002(x) is the Heaviside step function, and\nn0=2\n\u00191\nR2n1D; (17)\nwithn1Dbeing the one-dimensional density (the number\nof particles per unit length in the zdirection). The radius\nof the condensate in the x;yplane is given by\nR2=2\u0016\nm!2=2\u0016\n~!l2\nH; (18)\nwherelH=q\n~\nm!is the harmonic oscillator length, and\nthe relation for \u0016=~!is given below in Eq. (40). The\nchemical potential and density are related to each other\nas\n\u0016= (g+gd=2)2\n\u0019R2n1D=n0g(1 +\u0011=2); (19)\nand the validity of the TF regime requires the chemi-\ncal potential (interaction between particles) to be much\nlarger than the level spacing between the trap levels,\n\u0016=~!\u001d1. Turning to the functions f\u0006=u\u0006vand\nrepresenting\nf\u0006(r) =f\u0006(\u001a)eikz; (20)4\nwherekis the momentum of the motion along the zaxis.\nUsing the GP equation (1) we transform Eqs. (4) and (5)\nto\n~2\n2m \n\u0000r2\n\u001a+k2+r2\n\u001a\t0\n\t0!\nf+=Ef\u0000; (21)\n~2\n2m \n\u0000r2\n\u001a+k2+r2\n\u001a\t0\n\t0!\nf\u0000+2[^V\t0f\u0000(r)]\t0=Ef+:\n(22)When acting with operator (12) on the function that\ndepends only on \u001ait is useful to explicitly di\u000berentiate in\nEq.(12) and make an average over the azimuthal angle,\nat least for small and large k. This gives\nZ\nV(r)eikzdz\u0019g(1 +\u0011\n2)\u000e(~ \u001a)\u00003g\u0011\n2\u0019k2\n2K0(k\u001a)\u0011A(\u001a):\n(23)\nIn the TF regime we omit the \frst and third terms\nin the round brackets in the l.h.s of Eq. (22). We then\nexpressf\u0000throughf+from Eq. (21) and substitute it\ninto Eq. (22). This yields\n~4k2\n4m2 \nk2\u0000r2\n\u001a+r2\n\u001a\t0\n\t0!\nf++ 2~2\n2mZ\nd2r0A(j~ \u001a\u0000~ \u001a0j)\t0(\u001a0) \nk2\u0000r2\n\u001a0+r2\n\u001a0\t0\n\t0!\nf+(~ \u001a0)\t0(\u001a) =E2f+(~ \u001a):(24)\nIn the low momentum limit, kR\u001c1, we omit the \frst\nterm of Eq. (24) and angular momentum dependent\nterms in the expression (23) for A(\u001a\u0000\u001a0). Representing\nf+=W(\u001a)p\n1\u0000\u001a2=R2, for excitations with zero orbital\nmomentum (of the motion around the zaxis) we \fnd\n(1\u0000~\u001a2)(~k2\u0000r2\n~\u001a)W(~\u001a) + 2~\u001adW(~\u001a)\nd~\u001a~k= 2\u000f2W(~\u001a);(25)\nwhere we turned to dimensionless momenta, energy, and\ncoordinates: ~k=kR,\u000f=E=~!, and ~\u001a=\u001a=R. In terms\nof the variable s= ~\u001a2equation (25) becomes\ns(1\u0000s)d2W\nds2+ (1\u00002s)dW\nds+\"\n\u000f2\n2\u0000~k2\n4+s~k2\n4#\nW= 0:\n(26)\nOmitting the term ~k2s=4, Eq. (25) is nothing else than\nthe hypergeometric equation. This term will be taken\ninto account later in a perturbative approach. The so-\nlution which is regular at the origin and \fnite at s!1\n(\u001a!R) reads\nWj=CF(\u0000j;j+ 1;1;s); j = 0;1;2;:::; (27)\nwherejis a non-negative integer, and Cis the normal-\nization constant. The related energy spectrum is given\nby\u000f2=h\n~k2=2 + 2j(j+ 1)i\n. From Eq. (22) we obtain\n2\u0016\nE(1\u0000~\u001a2)f\u0000\u0019f+, and, hence, the normalization con-ditionR\nd3rf+\u0003f\u0000= 1 gives\nf+=r\n2\u0016\n~!\u000fp\n1\u0000~\u001a2WqR\nd2~\u001aW2eikz; (28)\nf\u0000=s\n~!\u000f\n2\u0016W\np\n1\u0000~\u001a2qR\nd2~\u001aW2eikz: (29)\nWe now take into account the omitted term ~k2s=4 pertur-\nbatively. The \frst order correction to \u000f2is\u000e\u000f2=\u0000~k2=4,\nfor anyj. Higher order corrections are proportional to\nhigher powers of ~kand can be omitted. Thus, we have\nfor the spectrum in the original units\nEj=~!p\n(kR)2=4 + 2j(j+ 1): (30)\nFor the lowest branch of the spectrum (j=0) the excita-\ntion energy has a linear dependance on k:\nE0=~!R\n2k: (31)\nIn the opposite limit, kR\u001d1, we keep the term\n(~2k2\nz=2m)2in Eq. (24). In this limiting case the main\ncontribution to the integral over d2\u001a0in equation (24)\ncomes from distances ~\u001a0very close to ~ \u001a, and this equa-\ntion takes the form (for zero orbital momentum):\n\u0012~2k2\n2m\u00132\nf+(\u001a) + 2g(1\u0000\u0011)~2\n2m\t0 \nk2\u0000r2\n\u001a+r2\n\u001a\t0(\u001a)\n\t0(\u001a)!\nf+(\u001a)\t0(\u001a)\u00006g\u0011~2\n2m\t0(\u001a)d2f+(\u001a)\t0\nd\u001a2=E2f+(\u001a):(32)\nRepresenting f+(\u001a) = (1\u0000\u001a2=R2)W, in terms of dimensionless variables s,\u000fequation (32) reads\ns(1\u0000s)W00+ (1\u00003s)W0+\"\n\u000f2\u0000\u000f2\n\u0003(~k)\n3\u0011(1 +\u0011=2)\u0000(\u0011\u00001)~k2\n6\u0011s#\nW= 0; (33)5\n1 2 3 4ϵ\n5 6 7 8k R1234567\nFIG. 1. Excitation spectrum \u000f(~k) as a function of ~k=kRfor\n\u0016=~!= 5:9 (\f= 50),\u0011= 1:8 (R=lH= 3:45;\u0001 = 1:5nK;! =\n280Hz;\u0016 = 68nK). Two vertical dashed lines de\fne a region\nwhere analytical form of the excitation spectrum does not\nexist.\nwhere\n\u000f2\n\u0003(~k) =\u0012~!\n4\u0016\u00132\n~k4\u00001\n2\u0011\u00001\n1 +\u0011=2~k2+3\u0011\n1 +\u0011=2:(34)\nOmitting the term \u0000(\u0011\u00001)~k2s\n6\u0011W(which will be taken\ninto account perturbatively later), equation (33) becomes\na hypergeometric equation. The solution regular at the\norigin and \fnite for s!1 is\nWj=~CF(\u0000j;j+ 2;1;s); (35)\nwherejis a non-negative integer, and ~Cis the normal-\nization constant. The related energy spectrum is given\nby\u000f2\nj=\u000f2\n\u0003(q) +3\u0011\n1+\u0011=2j(j+ 2). From equations (21) and\n(22) in the limit of kR\u001d1 we have\nf+\u00194\u0016\u000f\n~!~k2f\u0000; (36)\nand, hence,\nf+=r\n4\u0016\u000f\n~!~k2p\n1\u0000~\u001a2WqR\nd2~\u001a(1\u0000~\u001a2)W2eikz; (37)\nf\u0000=s\n~!~k2\n4\u0016\u000fp\n1\u0000~\u001a2WqR\nd2~\u001a(1\u0000~\u001a2)W2eikz: (38)\nAdding the \frst order correction to \u000f2\njfrom the omitted\nterm\u0000(\u0011\u00001)~k2s\n6\u0011W, we obtain\n\u000f2\nj=\u000f2\n\u0003(~k) +1\n2\u0011\u00001\n1 +\u0011=2~k2xj+3\u0011\n1 +\u0011=2j(j+ 2);(39)\nwherex0= 1=3,x1= 7=15,x2= 17=35,x3= 31=63,...\nNote that in the considered approximation the roton\nspectrum exists only at \u0011 > 1. The typical spectrum\nis shown in Fig.1 for \u0011= 1:8. The lowest branch (in\nblue) has a roton.The character of the spectrum is determined by two pa-\nrameters:\u0011and\u0016=~!. Having in mind direct numerical\ncalculations given below, instead of \u0016=~!we will use the\nparameter\f= 2nr\u0003, wherer\u0003=md2=~2is the so called\ndipolar length. For isotropic condensate wave function\nwe have\n\u0016=~!=p\n(2 +\u0011)\f=3\u0011: (40)\nIV. APPROXIMATON OF ISOTROPIC\nCONDENSATE. DAMPING OF ROTONS.\nSubstituting solutions (28), (29), (37), (38) into\nEq. (10). and integrating over the momentum pand\ncoordinates, we obtain for the damping rate of a roton\nwith momentum kthe expression\n1\n\u001cjl=n0g2\n4~R2Np;j\u0000Nq+p;l\njE0\nq+p;l\u0000E0\np;jjZjl; (41)\nZjl=h\nG1(q)\u0016f\u0000\nq;0f+\np;j\u0016f+\nq+p;l+G2(q)\u0016f\u0000\nq;0\u0016f\u0000\np;j\u0016f\u0000\nq+p;l\n+G3(p)(\u0016f\u0000\np;j\u0016f+\nq;0\u0016f+\nq+p;l+\u0016f\u0000\np;j\u0016f\u0000\nq;0\u0016f\u0000\nq+p;l)\n\u0000G1(q+p)\u0016f\u0000\nq+p;l\u0016f+\nq;0\u0016f+\np;j+G2(q+p)\u0016f\u0000\nq+p;l\u0016f\u0000\np;j\u0016f\u0000\nq;0i2\n;\nwhereE0\nk;l=dEk;l=dk, and the momentum pis found\nfrom the energy conservation law:\nEq+p;l=Eq;0+Ep;j: (42)\nThe functions \u0016f\u0006are coe\u000ecients in Eqs. (28), (29), (37),\nand (38):\n\u0016f\u0006\np;j=\u00142\u0016\n~!\u000fp;j\u0015\u00061\n2\n;\u0016f\u0006\nq;l=\u00144\u0016\u000fq;l\n~!~q2\u0015\u00061\n2\n:\nFor the functions Giwe obtain by the use of Eqs. (23) -\n(24) the following expressions:\nG1(k) =(\u000f2\nk;l\u0000\u0010\n~!\n4\u0016\u00112~k4)(2 +\u0011)\n~k2\n\u0002R1\n0(1\u0000s)F1F2dsqR\nF2\n1dsqR\n(1\u0000s)F2\n2ds;(43)\nG2(k) =(\u000f2\nk;l\u0000\u0010\n~!\n4\u0016\u00112~k4)(2 +\u0011)\n~k2\n\u0002R1\n0F1F2dsqR\n(1\u0000s)F2\n1dsqR\n(1\u0000s)F2\n2ds;(44)6\nG3(p) = (1 +\u0011=2)R1\n0(1\u0000s)F1F2dsqR\nF2\n1dsqR\n(1\u0000s)F2\n2ds;(45)\nwhereF1;2are the hypergeometric functions: F1\u0011\nF(\u0000j;j+ 1;1;s),F2\u0011F(\u0000l;l+ 2;1;s), and we assume\nin Eqs. (43) - (45) that pR\u001c1,kR\u001d1. Hence, the\nquantityZjlis a function of the parameters \u0011and\u0016=~!.\nAt temperatures greatly exceeding ~!the occupation\nnumbers are Nk\u0019T=Ek, and, consequently, the damp-\ning rates are linear in T:\n1\n\u001cjl=kBT\n~mg\n~2R\u000bjlZjl; (46)\nwhere\n\u000bjl=\u00141\n\u000fp;j\u00001\n\u000fq+p;l\u00151\nj\u000f0\nq+p;l\u0000\u000f0\np;jj(47)\nand this quantity is responsible for the increase of the\ndamping rate in the limit of vanishingly small roton gap\n\u0001\u0011\u000f0(q). For example, \u000b00/1=\u0001 at \u0001!0. For\na very small gap the roton excitation is unstable due to\nstrong decay processes. Its energy becomes of the order\nof~=\u001cor even smaller.\nWe estimate now the damping rate for some values of\ndimensionless parameters \fand\u0011for dysprosium atoms\n(r\u0003= 200 \u0017A).\nFor\u0016=~!= 7:2 (\f= 70),\u0011= 1:6,!= 340Hzwe\nhave \u0001 = 0 :9nK;\u0016 = 101nK;R = 1:6\u000110\u00004cm. The\nmain decay channel in this case is the one with j= 0,\nl= 0. The damping rate reaches the value1\n\u001c\u001810s\u00001at\nT\u0018100nK.\nA decrease of \u0016=~!may increase the number of decay\nchannels. In particular, for \u0016=~!= 5:9 (\f= 50),\u0011= 1:8,\n!= 280Hz, we have \u0001 = 1 :5nK;\u0016 = 68nK;R = 1:6\u0001\n10\u00004cmand there are two channels: j= 0,l= 0; and\nj= 3,l= 2. The total damping rate at T\u0018100nK\nbecomes of the order of1\n\u001c\u0018102s\u00001.\nFor all data the damping rate is much smaller than the\nroton energy: ~=(\u0001\u001c)\u001c1. Therefore, the roton is a well\nde\fned excitation.\nV. DIRECT NUMERICAL CALCULATIONS\nA. Condensate wave function and excitation\nspectrum\nIn this section we present our numerical results for the\nground state wave function and excitation spectrum of\nthe condensate. We numerically solved the GPE equation\n(1) for \t 0(x;y) using the imaginary-time evolution algo-\nrithm in 2D Cartesian grid. The Bogoliubov-de Gennes\nequations (4) - (5) for the excitation spectrum are solved\nusing the large-scale Krylov-Schur eigensolver.\nIn Figs. 2 and 3 we present the condensate density dis-\ntribution and the excitation spectrum in the transverse\nFIG. 2. The density of the condensate plotted in the x;y\nplane for\f= 50 and\u0011= 1:231. The dashed red line is the\ncountour plot at half-width of the condesate density. The\ncondensate form is anistropic with elongation in the direction\nof dipoles.\n0 0.5 1 1.5 2 2.5 300.511.522.533.54\nϵ  (  )n\nFIG. 3. Low-lying excitation branches \u000fn(k) as a function of\nklHfor\f= 50 and\u0011= 1:231. The branches are labelled with\nan indexnbased on their value at k= 0 starting from n= 0.\nOnly the lowest branch \u000f0(k) has a roton-type excitation with\nthe rotonic gap \u0001 \u00190:13~!. The largest contribution to the\nroton damping comes from the intraband transitions in the\nthe excitation branches \u000f2(k) and\u000f6(k) (red solid curves).\ndirection for the dimensionless parameters \f= 50 and\n\u0011= 1:231. In Fig. 2 the length scale is in units of har-\nmonic oscillator length lH. Our numerical results for the\ncondensate wavefunction show its anisotropic form with\nelongation in the direction of dipoles, as shown in Fig. 2.\nThe red dashed curve in the \fgure marks the countour\nplot at half-width of the condesate density.\nThe excitation spectrum Ekconsists of an in\fnite num-7\nber of branches. We considered only eight low-lying\nbranches, which is enough for our purposes. As clear\nfrom Fig. 3, only the lowest band contains a roton type\nexcitation with the rotonic gap \u0001 \u00190:13~!atklH\u00191:33\nfor the dimensionless parameters given above. At a \fxed\nparameter\f= 50, increasing/decreasing \u0011modi\fes the\nexcitation spectrum and decreases/increases the rotonic\ngap \u0001. For \u0011.1:2 the rotonic minimum disappears, and\nthere are no rotonic excitations. On the other hand, for\n\u0011c\u00191:2325 the rotonic gap goes to zero, and for larger\n\u0011the condensate uniform in the zdirection is unstable.\nFor comparison, for a larger value \f= 104 the rotonic ex-\ncitation appears at \u0011\u00191:12 and the rotonic gap vanishes\nat\u0011c\u00191:147.\nB. Damping rate for rotons\nUsing the solutions of the Bogoliubov-de Gennes equa-\ntions we calculate the damping rate for the roton excita-\ntions with the help of expression (10). Due to the broken\nrotational symmetry, the projection of angular momen-\ntum on the z-axis is not a conserved quantity and does\nnot serve as a good quantum number. Thus, both intra-\nband (n1=n2in Eq. (10)) and interband ( n16=n2in\nEq. (10)) transitions can contribute to the damping rate\nof the rotonic excitation.\nWe \frst present the rates for \f= 50 and\u0011= 1:231 and\n\fx the trap frequency != 280Hz. Our numerical results\nshow that the main contribution to the damping rate is\ngiven by the intraband transition within the third band,\ni.e the transition with n1=n2= 2 with 1=\u001c2;2= 75s\u00001\natT= 100nK. To compare, the damping rate due to\nthe transition in the rotonic branch is 1 =\u001c0;0= 23:9s\u00001.\nSuch large contribution of 1 =\u001c2;2results from relatively\nlarge matrix elements Aq\nk;k+qand large density of states\n(vanishingly small velocity di\u000berence in Eq. (41)). The\ncontributions of intraband transitions in higher bands are\nsuppressed due to small matrix elements. As an example,\nthe next largest contribution to the damping rate from\nthe intraband transition is 1 =\u001c6;6= 4:8s\u00001\u001c1=\u001c2;2. The\ntotal damping rate from all other intraband transitions\nis smaller than 1 =\u001c6;6.\nSimilarly, our result for the total damping rate due\nto all possible interband transitions is smaller than\n5s\u00001. Although such transitions are not prohibited,\npractically typical matrix elements are smaller by or-\nders of magnitude. We also considered the damping\nrates in three other regimes, achieved by varying \u0011at\n\f= 50. We estimated the damping rates at \u0011=\nf1:225;1:231;1:232;1:2323gand presented our \fnal re-\nsults in Table I. The total damping rate at T= 100nK\nincreases with decreasing the rotonic gap from 1 =\u001c=\n20:4s\u00001at \u0001 = 0:28~!up to 1=\u001c= 214:5s\u00001at \u0001 =\n0:06~!. Finally, we plot the temperature dependance of\nthe damping rates in Fig. 4.TABLE I. Damping rates \u001c\u00001\nn;nin units of s\u00001given atT=\n100nKat di\u000berent system parameters \u0011with the rotonic gaps\n\u0001. The values \f= 50 and!= 280Hzare kept \fxed.\n\u0011 \u0001=~!\u001c\u00001\n0;0\u001c\u00001\n2;2\u001c\u00001\n6;6\u001c\u00001\n1:225 0:28 4:9 15:2 0:3 20:4\n1:231 0:13 23:9 75 2:4 101:3\n1:232 0:09 29:4 129:6 4:8 163:8\n1:2323 0:06 30:6 178:2 5:7 214:5\n0 20 40 60 80 100050100150200250\n1=0.13\n=0.09\n=0.06\nFIG. 4. The total damping rates 1 =\u001cin units ofs\u00001versus\ntemperature Tgiven innKfor!= 280Hzand\f= 50 in\nthree di\u000berent regimes with [ \u0011= 1:231, \u0001 = 0:13~!(red) ], [\n\u0011= 1:232, \u0001 = 0:09~!(blue) ] and [ \u0011= 1:2323, \u0001 = 0 :06~!\n(black) ].\nVI. CONCLUSIONS\nIn this paper we have calculated the \fnite temperature\ndamping rate for rotons in an elongated Bose-condensed\ngas of polarized dipolar particles, which is in the Thomas-\nFermi regime in the tightly con\fned directions. The im-\nportant feature of this case is the presence of a large\nnumber of excitation branches which can contribute to\nthe damping process. We found out that this leads to\na signi\fcant increase of the damping rate. Nevertheless,\nour calculations show that, even in this regime, rotons in\nsystems with the roton energy gap of the order of 1 nK\nare su\u000eciently long-living and can be observed as well-\nde\fned peaks in the excitation spectrum and contribu-8\ntions in response functions.\nACKNOWLEDGMENTS\nWe thank Francesca Ferlaino for fruitful discus-\nsions. This research was supported by the Russian Sci-ence Foundation Grant No. 20-42-05002 and by the\njoint-project grant from the FWF (Grant No. I4426\nRSF/Russia 2019). We also acknowledge support of this\nwork by Rosatom.\n[1] E. P. Gross, Classical theory of boson wave \felds, Ann.\nPhys. (N.Y.) 4, 57 (1958).\n[2] P. Nozi`eres, J. Low Temp. Phys. 137, 45 (2004).\n[3] D. H. J. O'Dell, S. Giovanazzi, and G. Kurizki, Phys.\nRev. Lett. 90, 110402 (2003).\n[4] L. Santos, G. V. Shlyapnikov, and M. Lewenstein, Phys.\nRev. Lett. 90, 250403 (2003).\n[5] M. 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A\n97, 063610 (2018).\n[20] D.H.J. O'Dell, S. Giovanazzi, and C. Eberlein, Phys. Rev.\nLett. 92, 250401 (2004).\n[21] C. Eberlein, S. Giovanazzi, and D.H.J. O'Dell, Phys. Rev.\nA71, 033618 (2005).\n[22] L. Pitaevskii and S. Stringari, Bose-Einstein Condensa-\ntion and Super\ruidity (Oxford University Press, Oxford,\n2016), Vol. 164."    },    {        "title": "1310.2481v1.Dissipation_in_Relativistic_Outflows__A_Multisource_Overview.pdf",        "content": "arXiv:1310.2481v1  [astro-ph.HE]  9 Oct 2013Dissipation in Relativistic Outflows:\nA Multisource Overview\nChristopher Thompson\nPhysics and Astronomy, University of North Carolina,\nChapel Hill, NC 27599\nAbstract: Relativistically expanding sources of X-rays and γ-rays cover an enormous\nrange of (central) compactness and Lorentz factor. The underlying phys ics is discussed,\nwith an emphasis on how the dominant dissipative mode and the emergent spectrum\ndepend on these parameters. Photons advected outward from high optical de pth are a\npotentially important source of Compton seeds. Their characteristic e nergy is bounded\nbelow by ∼1 MeV in pair-loaded outflows of relatively low compactness, and remains\nnear∼1 MeV at very high compactness and low matter loading. This is compared\nwith the characteristic energy of O(1) MeV observed in the rest frame spectra of many\nsources, including γ-ray bursts, OSSE jet sources, MeV Blazars, and the intense initi al\n0.1 s pulse of the March 5 event. Additional topics discussed includ e the feedback of pair\ncreation on electron heating and the formation of non-thermal spectra, the ir effective-\nness at shielding the dissipative zone from ambient photons, direct Compton damping of\nirregularities in the outflow, the relative importance of various soft p hoton sources, and\nthe softening of the emergent spectrum that results from heavy matt er loading. The im-\nplications of this work for X-ray and optical afterglow from GRB’s are brie fly considered.\nDirect synchrotron emission behind the forward shock is inhibite d by the extremely low\nenergy density of the ambient magnetic field. Mildly relativistic e jecta off axis from the\nmainγ-ray emitting cone become optically thin to scattering on a timescal e of∼1 day\n(E/1052erg)1/2, and can be a direct source of afterglow radiation.\n1 Introduction: Variety of Sources and Spectral Behavior\nX-ray and γ-ray emission from relativisticoutflows is powered by the co nversion of\nbulk kinetic energy and Poynting luminosity, by a variety of possible mechanisms.\nHowever, the assumed values of key parameters such as the Lor entz factor γof\nthe outflow, the compactness of the central source, as well as the optical depth\nand size of the dissipative zone, vary dramatically between the different classes of\nsources. For example, ℓc∼1015andγ∼102−103are inferred for cosmological\nγ-ray burst (GRB) sources vs. ℓc∼10−102andγ∼3−30 for Blazars (Fig. 1).\nThis motivates a more global analysis of how dissipation of k inetic and magnetic2 Christopher Thompson\nenergy is achieved, which provides some interesting new per spectives on particular\nsources.\nγHARD PULSEMARCH 5GRBs\nBLAZARS\n3 210 10 105101510\n10\n10cl\nFig.1.Blazars, γ-ray burst sources and the initial 0.1 sec hard pulse of the March\n5, 1979 burst occupy distinct regions of the plane defined by source compac tnessℓcand\nasymptotic Lorentz factor γ∞. GRB outflows may contain lower- γ∞ejecta that manifests\nitself as softer tails, precursors and sub-pulses (T94, T96). The Loren tz factors of Blazar\nsources are only indirectlyconstrained by superluminal motionoutsi de theγ-rayemitting\nregion (cf. Wagner, these proceedings).\nAnother key point is that sources (or types of sources) exhib it different spectral\nstates. GRBs are predominantly non-thermal, but some contain subl uminous pre-\ncursors, tails, and sub-pulses with distinctly thermal hig h energy cutoffs (Yoshida\net al. 1989; Pendleton et al. 1996). Blazars are occasionall y observed with emission\npeaked strongly at ∼1 MeV, in distinction to the more usual extended power-law\nbehavior (Bloemen et al. 1995; Blom et al. 1995). And the rema rkable 5 March\n1979 burst was initiated by an extremely intense ∼0.1 sec flare whose luminosity\nexceeded that of the remainder of the burst by a factor ∼300 and showed much\nmore pronounced spectral softening (Fenimore et al. 1996).\nSpectral states with sharp high energy cutoffs have a simple i nterpretation\nas the residue of an optically thick outflow. Indeed, at very high ℓc, the outflow\nis self-shielding from the central radiation source, as wel l as from external (e.g.\nside-scattered) radiation. Radiation advected outward fr om large scattering depth\nthen becomes an important source of seeds for extended non-t hermal spectra – in\naddition to the more familiar optically thin synchrotron-s elf-Compton mechanism.\nThis means that the inner boundary conditions on the flow are m uch more impor-\ntant than is usually supposed. And this raises an interestin g question: are sources\nof relatively low compactness (e.g. Blazars) ever self-shi elding in this manner?Dissipation in Relativistic Outflows: A Multisource Overview 3\nIt is intriguing to note, in this regard, that the characteri stic energy ∼1 MeV\nappears in a number of sources (of widely varying parameters ): the spectral breaks\nobserved in OSSE jet sources ( ∼1−3 MeV after compensating for cosmological\nredshift; e.g. McNaron-Brown et al. 1995) and in classical G RBs (∼100 keV −1\nMeV at peak luminosity, without compensating for redshift; Mallozzi et al. 1995);\nin MeV Blazars; as well as the initial hard spike of the March 5 burst (∼300\nkeV). Of course, the possibility that selection effects narr ow the observed spectral\nbreak distribution should be considered carefully. This ha ppens in the case of the\nGRB sources (Piran & Narayan 1996) only if the total burst ene rgy is constrained\nto yield an inverse correlation between break energy and flux – in distinction to\nthe strong positive correction observed within individual bursts. Most plausible\ncosmological GRB sources release as much as 1053−1054erg, which allows for a\nfraction of very energetic and hard bursts.\nAfter careful consideration of various photon sources, it t urns out that an ad-\nvected Wien peak maintains a characteristic energy of ∼1 MeV over a wide range\nof central compactness, if the flow is sufficiently relativist ic (Sects. 1.2; 3.1). Pair\ncreation by photon collisions, γ+γ→e++e−, which has traditionally been\nviewed as inimical to high energy gamma-ray production, can in fact play an es-\nsential role by i) increasing the efficiency of leptonic dissi pative modes; ii) reducing\nthe lower cut-off energy of the non-thermal pair distributio n toγmin∼1, which\nyields a break in the spectral distribution of Compton-upsc attered Wien photons\nnear the position of the original Wien peak; and iii) selecti ng non-thermal over\nthermal spectra at Comptonizing hotspots a high- γoutflow (Sect. 3.3). Since the\nphoton collision cross section is comparable to Thomson, fe edback from pair cre-\nation works most effectively near τT∼1. This contrasts with the inhomogeneous\nexternal Comptonization model (Blandford and Levinson 199 5, hereafter BL95),\nwhere the non-thermal high energy continuum emerges well ou tside the scattering\nphotosphere. Indeed, the position of the scattering photos phere in a pair-loaded\noutflow is sensitive to the amount of continuous heating, and pair creation can\nsignificantly broaden the transition zone between opticall y thick and thin flows\n(Sects. 1.2, 2.2).\nThus, by focussing on those aspects of the physics that are sp ecial to the\nlarge-ℓcGRB regime, and then considering how these vary with compact ness,\ninteresting new insights can be obtained on both the GRB and B lazar problems.\nWhen constructing GRB models, it is sobering to realize that Blazars are still far\nfrom being understood, even withthemuch broader spectral i nformationavailable.\nIn these notes, I will explore the following additional poin ts:\n•Thedependence ofthedominantdissipativemodeonopticald epth.At τT>1\ndirectComptonizationbybulkfluidmotionsismosteffective ;whereasnon-thermal\n(e.g. Fermi) particle acceleration is a crucial ingredient of any radiative model at\nlowτT. Strong-wave acceleration can be excluded if enough scatte ring charges are\npresent to generate an observable flux of Comptonized high en ergy photons.\n•What is the relative importance of double Compton emission a nd cyclo-\nsynchrotron emission as seeds for Comptonization (at large ℓc)?\n•How does the influence of geometrical effects (e.g. beaming) v ary with γ∞?4 Christopher Thompson\n•High energy cutoffs to extended power-law spectra are extrem ely diagnostic.\nIn GRB sources these are very poorly constrained. Measureme nts by EGRET in\nthe 30 MeV - 10 GeV range indicate a significant decorrelation with the 1 MeV\nflux, with the high energy emission often being significantly delayed(Hurley et al.\n1994).\n1.2 The ℓc-γ∞Plane\nThevarietyofpossibledissipativemodesisneatlysummari zedinatwo-dimensional\nplane labeled by (Fig. 2)\nℓc=LrelσT\n4πmec3R0;γ∞=Lrel\n˙Mc2. (1)\nThe radius R0of the central engine is identified with the Alfv´ en radius or light-\ncylinder radius as appropriate. At risk of oversimplificati on I will usually assume\nthatγhas attained the limiting value γ∞=Lrel/˙Mc2due to matter loading ˙M\n(when discussing delayed dissipation at large distances fr om the central engine).\nThe radius at which dissipation takes place is limited signi ficantly by causality\nat largeγ, as long as γ(r) grows faster with radius than r1/2near the base of the\noutflow. Thus, it is convenient to define the compactness ℓ∆t=ℓc×(R0/c∆t) asso-\nciatedwithvariabilityona timescale ∆t. A plausiblevalue ofthe dissipativeradius\nis∼2γ2\n∞c∆tfor a variety of dissipative modes, including magnetic reco nnection\nand MHD turbulence (Romanova & Lovelace 1992, hereafter RL9 2; Thompson\n1994, 1996, hereafter T94, T96; Levinson, these proceeding s) and shocks powered\nby variations in ˙M(Pacyz´ nski & Xu 1994, hereafter PX94; Rees and M´ esz´ aros\n1994, hereafter RM94). The radius of the scattering photosp here of the wind is\nrelated to Rdissby a very strong function of γ∞,\nRe−p\nτ=1\n2γ2∞c∆t∼me\nmpℓ∆t\nγ5∞(np≫ne+), (2)\nwhen the scattering depth is dominated by the advected elect ron-ion contaminant.\nA non-thermal photon tail extending above energy mec2in the rest frame will\ngreatly increase the density of scatterers, with the result\nRe±\nτ=1\n2γ2∞c∆t∼ℓ∆t\nγ5∞(ne+≫np) (3)\nforaphotonindex β∼ −2characteristicofGRBs.Theeffectsofpairsarediscussed\nfurther in Sect. 3.\nOutflows with Lorentz factor γ∞< γe−p= (me/mp)1/5ℓ1/5\n∆tdissipate well in-\nsidetheelectron-ionphotosphere. Theyoccupy theupper le ftportionofFig.1,and\nhave a possible association with the soft X-ray precursors, tails and quasi-thermal\nsub-pulses of GRBs (T96). Outflows with γe−p< γ∞< γe±=ℓ1/5\n∆tdissipate inside\nthe the pair photosphere, if the high energy continuum exten ds up to ∼mec2inDissipation in Relativistic Outflows: A Multisource Overview 5\nLTE (R=10 km)\n-(soft cutoff)--p photosphereedissipation inside\nHEATINGCONTINUOUSHEATINGDELAYEDamplifier\nthinoptically\nMeV peakdirect\nloaded- +\n+\n3 210 10 10 1e\ne\nγcl16\n14\n12\n10\n8\n6\n4\n2\n110\n10\n10\n10\n10\n10\n1010\nFig.2.The variety of dissipative regimes in relativistic outflows as summar ized in the\nℓc-γ∞plane. Flows with heavy matter loading and asymptotic Lorentz factor γ∞< γe−p\noccupy the top left region. They are a plausible source of soft subcompon ents of GRBs.\nFlows with γe−p< γ∞< γe±can become pair loaded, and occupy the adjoining stripe.\nFlows with low matter loading, γ∞> γe±, dissipate at low optical depth. Above the\nupper horizontal dashed line, local thermodynamic equilibrium is achieved at the base\nof the flow; whereas below this line the photon distribution is Wien . All of the flows in\nthis region are assumed to undergo delayeddissipation after thermal pairs freeze out, at\na radius where inhomogeneities (such as reconnecting magnetic fields and shocks) regain\ncausal contact. By contrast, flows in the lower portion of the diagram are assu med to\ndissipate continuously and remain pair loaded out to the scattering photosphere. This\nis expected in collimated flows with lower asymptotic Lorentz factor s (such as Blazars).\nLarge scattering depths cannot be maintained below the lower horizont al dashed line.\nthe rest frame. And, outflows with γ∞> γe±dissipate at low scattering depth, in-\ndependent oftheefficiencyofpaircreation.Thespectralcon sequences ofvariations\nin the matter loading are discussed further in Sect. 3.\nFlows in which the irregularities maintain causal contact w ill undergo con-\ntinuous heating while optically thick. An interesting exam ple of such a flow is\nthe low-γsheath of a relativistic jet, in which the luminosity of entr ained pho-\ntons increases with radius as the kinetic energy of the highe r-γcore of the jet\nis dissipated. Although the relation between scattering de pth and internal tem-\nperature of the flow is sensitive to the high energy distribut ions of the pairs\nand photons, the mean energy of the emergent photon spectrum is regulated to\nnear∼γmec2. In the case where the photon distribution is Wien, the stron g\nT-dependence ne±/nγ= (π/2)1/2(T/mec2)−3/2exp(−mec2/T) of the equilibrium\npair density guarantees thatassoonas Tdrops much below mec2intherest frame,\nthe flow becomes optically thin. This yields a simple relatio n between the observed6 Christopher Thompson\n(Lorentz-boosted) temperature and the temperature T0at the base of the flow,\nTobs=T0(Lγ/Lγ0), (4)\nin terms of the growth of the photon luminosity from Lγ0toLγ.\nThe pair density resulting from direct heating of the photon s by MHD tur-\nbulence can be less sensitive to T. I assume that the bulk of the photon energy\nlies in a Wien bump, but that wave energy is excited in transie nt surges (e.g. by\nreconnection) with an equivalent temperature Twsomewhat in excess of mec2.\nThen a significant fraction of the wave energy is converted to photons above the\npair production threshold. Balancing this source against p air annihilation, the\nscattering depth in the direction perpendicular to a jet (of opening angle θ) is\nτT⊥=1\n2neσRθ∼(γℓγ)1/2, whereℓγis the compactness (1) in the advected pho-\ntons at radius R. AtτT>1 (ℓγ>1), freshly created pairs Compton cool before\nannihilating (the cooling timescale being shorter by a fact or∼τ−1\nT/bardbl) and carry a\nfraction ∼τ−1\nT/bardblof the total energy of the flow. This regulates the Compton pa-\nrameter induced by mildly relativistic pairs to y≃τ−1\nT/bardbl·τT/bardbl∼1. The annihilation\nphotons Compton downscatter off the cooled pairs to an energy ∼mec2/τT/bardblin\nthe rest frame. The photon spectrum emerging at the pair phot osphere then peaks\nat an energy ∼γmec2. This implies only a modest increase in the mean energy\nper photon along the jet, by a factor ∼γmec2/3T0, which is easily supplied from\nthe high- γcore to the low- γsheath.\nContinuous heating by relativistic electrons at low optica l depth has been con-\nsidered by Sikora et al. (1997) as a model for the MeV Blazars. They relate the\npeak energy to the observed variability timescales, but sin ce this energy is not\ndirectly tied to a microphysical scale, it could be expected to lie well below ∼1\nMeV in some sources. By contrast, direct Compton damping of m ildly relativistic\nturbulence in an optically thick jet yields a Wien peak energ y that is bounded\nbelow by ∼1 MeV (although still dependent on bulk γand the energy transferred\nfrom the bulk motion to the photons).\n2 Dissipative Mechanisms\n2.1 Sources of Free Energy\nThe internal sources of free energy in a relativistic flow can be broadly divided\ninto two categories: those associated with radial and angul ar inhomogeneities.\nBoth appear to be important in jet sources, and while both hav e also been con-\nsidered in GRB sources, radial inhomogeneities are probabl y more important in\nthe large- γcontext. Reconnection surfaces and variations in the ratio of particle\npressure to magnetic pressure will lead to internal heating of strongly-magnetized\noutflows (RL92; T94), as will kinetic energy fluctuations in p article-dominated\noutflows (PX94; RM94). However, interactions with an extern al medium (Rees\nand M´ esz´ aros 1992) can become significantly non-spherica l as the outflow in a\nγ-ray burst source decelerates (Sect. 4).Dissipation in Relativistic Outflows: A Multisource Overview 7\nWhich of these energy sources dominates depends on the stren gth of the mag-\nnetic field in the outflow and the radius of the dissipative zon e. It appears that\nγ∞∼100−300 can be achieved in an outflow of luminosity ∼1051erg s−1only\nif it is Poynting-flux dominated at the source. Indeed, anytriggering scenario for\na GRB that produces an object with the density of nuclear matt er and a rotation\nperiodof ∼10−3splausibly involvesmagneticfields as strong as ∼1015Gthrough\ndynamo amplification and thus leads to a rotationally-drive n MHD outflow of lu-\nminosity LP∼1050−1051erg s−1(T94; see also Duncan & Thompson 1992; Usov\n1992, 1994; Vietri 1996; M´ esz´ aros and Rees 1997 for partic ular models). The alter-\nnative mechanism of ν−¯νannihilation into e±pairs has an efficiency of ∼10−3\n(Jaroszynski 1993), and so the neutrino luminosity require d to power a γ-ray flux\nof∼1051erg s−1drives a mass flux (by absorption on nucleons) that exceeds th e\ntolerable value by ∼106(cf. Duncan, Shapiro and Wasserman 1986).\nThis advected magnetic field almost certainly has an importa nt effect on the\ndissipativemechanism.Quasi-perpendicularshocksaresi gnificantlyweakenedeven\nwhen the magnetic field carries a few percent of the energy flux ; this in turn\nsteepends non-thermal particle spectra arising from first- order Fermi acceleration.\nAlthough particle acceleration at oblique relativistic sh ocks can be quite efficient\n(as discussed by Kirk, these proceedings), one expects that internalshocks in GRB\noutflows with γ∼100−300 are approximately radial. Thus, GRB models based on\ninternal shocks may unfortunately require complicated dep artures from spherical\nsymmetry.\n2.2 Geometrical Effects: Pair Cocoons\nThe huge compactness of a GRB source ( ∼1015) causes it to be self-shielding . Not\nonly is the central engine hidden from the dissipative zone, but the optical depth\nτ⊥perpendicular to the axis of the flow exceeds the parallel dep th by the very\nlarge factor,τ⊥\nτ/bardbl∼γ2θ∼104−5θ, (5)\nfor reasonable values of the opening angle θ.\nThis raises the question: as ℓcdecreases toward the values typical of AGN, at\nwhat point does the central engine become visible? One intri guing possibility is\nthat the engine remains shielded in some Blazars, with the hi gh-γcores of the jet\nbeing surrounded by a pair cocoon (Fig 3). Because Lorentz dilation causes τ/bardblto\ndecrease in proportion to γ−2(Ljet/˙Mjetc2)−1∝γ−3, the high- γcore of a jet can\nbecome optically thin along its axis well inside the photosp here of the γ∼1−2\nsheath. This lies at a radius\nRe±\nτ=1∼2×1018/parenleftbiggLjet\n1047erg s−1/parenrightbigg /parenleftbiggθ\n0.1/parenrightbigg−1\ncm, (6)\nassuming that the sheath acquires a non-negligible fractio n of the kinetic energy\nat this radius. All that is required (Sect. 1) is that the shea th be i) pair-loaded and\noptically thick at its base, and ii) continuously heated (e.g. by Kelvin-Helmholtz8 Christopher Thompson\ninstabilities with higher- γmaterial) at a sufficient rate to overcome the effects of\nadiabatic cooling. Condition ii) is plausibly satisfied in t he cores of superluminal\nsources, and i) is satisfied if the outflow is strongly turbule nt at its base (Sect.\n3). Below this threshold heating rate, the position of the pa ir photosphere is very\nsensitive to the amount of heating.\n- - + +e e\n//τ   = 1Γ=5−30\nΓ=1−2 Γ=1−2\nFig.3.A sheath of low- γmaterial surrounding the high- γcore of a jet can remain pair\nloaded out to a large distance (6) from the central engine. The scattering photosphere\nof thispair cocoon will in general lie far outside the radius at which the core becomes\ntransparent to high energy γ-rays.\n2.3 Diverse Photon Sources\nWhen the flow is self-shielding in this manner, advected quasi-thermal radiation\nbecomes an important source of Compton seeds (T94, T96). Eve n though external\nradiation exerts a stronger drag force than internally gene rated radiation by a\nfactorγ2(Sikora, Begelman, & Rees 1994; BL95), it cannot penetrate t he high-\nγcomponent of the flow. This also disfavors models involving a cceleration and\nheating of the jet by central continuum radiation propagati ng along the jet axis\n(Dermer and Schlieckeiser 1993; Marcowith et al. 1995). The main competing\nsource of seed photons is then synchrotron radiation (e.g. M araschi et al. 1992;\nM´ esz´ aros, Rees and Papathanassiou 1994, hereafter MRP94 ).\nIn quantifying the relative importance of these two radiati on sources, one must\nconsider separately thermal and non-thermal scatterers. T he advected radiation\ncharacteristically has a much higher frequency than the cyc lotron frequency, and if\nits energy density Ua\nγis comparable to B2/8π, then it will dominate the magnetic\nfield as a coolant of thermal electrons, due to self-absorpti on near the cyclotron\nresonance. Parametrizing the frequency at which the cyclo- synchrotron radiationDissipation in Relativistic Outflows: A Multisource Overview 9\nbecomes self-absorbed in terms of a cyclotron harmonic Nsa, one deduces from\nKirchoff’s law that C-S cooling rate is smaller than the Compt on cooling rate off\nthe advected radiation by the factor\nνjν\n4σTnec(T/mec2)Uγ=2αem\nπN3\nsa\nτT/parenleftbiggB\nBQED/parenrightbiggB2/8π\nUγ,/parenleftbigg\nν=NsaeB\n2πmec/parenrightbigg\n(7)\nwhereαem= 1/137 and BQED= 4.4×1013G. For example, one expects\nB/BQED<10−8near the scattering photospheres of GRB outflows, so that C-S\ncooling can be neglected even if Nsa∼300.\nAnother effect of the advected radiation is to limit the Compt on parameter\ndy/dlnR≃4(T/mec2)(1 + 4T/mec2)dτT/dlnRin optically thick regions of the\nflow. The magnitude of this effect depends on the mean energy pe r photon ∝angb∇acketlefthν∝angb∇acket∇ight,\nand the bulk Lorentz factor γ. If the flow is photon rich, with ∝angb∇acketlefthν∝angb∇acket∇ight/γ≪mec2in\nits rest frame, then the advected photons limit yto a value ∼ln(∝angb∇acketleftν∝angb∇acket∇ight/ν0), where ν0\nis peak frequency before the advected photons undergo (dela yed) reheating. This\nprevents very low energy C-S photons from being upscattered much in frequency.\nA stronger limit y∼1 applies in outflows that are continuously Compton heated\nby MHD waves and shocks (Sect. 1.2). Nonetheless, a much larg ery-parameter\nwill be maintained at the base of the outflow, where the photon flux has a net\ndivergence (Sect. 3.1).\nSynchrotron radiation from Blazar sources is usually ascri bed to the same high\nenergy electrons/positrons that are responsible for the X- ray/γ-ray emission. The\nhigh energy cutoff of the synchrotron peak has been ascribed t o a suppression of\nthe non-thermal particle density inside the scattering pho tosphere (e.g. Levinson\n1996), but mildly relativistic pairs are only suppressed by a factor ∼τ−1\nT. Thus,\nthe particle spectrum must itself steepen considerably at τT>1. Rather large\nminimum non-thermal Lorentz factors γminare sometimes conjectured in GRB\nsources, sufficient to place the synchrotron peak energy dire ctly in the MeV range\n(e.g. MRP94). However, synchrotron absorption becomes muc h more important in\nthat context, if the outflow becomes sufficiently pair loaded t hatγmin∼1 (Sect.\n3.3).\nAdvected radiation can also be the dominant coolant at external relativistic\nshocks. Consider a shell of relativistic matter of initial w idthc∆t. After the shell\nbecomes optically thin to scattering, the radiation moves a head of the shell, but\ncontinues to overlap inside a radius R >2γ2c∆t. The point is that when the exter-\nnal medium is cold (with a sound or Alfv´ en speed much less tha nc), the external\nmagnetic field Bexthat is swept up and compressed in between the forward shock\nand the contact discontinuity typically has a muchsmaller energy density than\nthe radiation, by a factor\nB2/8π\nUγ∼2×10−10/parenleftbiggBex\n3×10−6G/parenrightbigg2/parenleftbiggLγ\n1050erg s−1/parenrightbigg−1/parenleftBigγ\n102/parenrightBig8/parenleftbigg∆t\n10 s/parenrightbigg2\n(8)\natR= 2γ2c∆t(assuming a compression factor of 7). This strongly suggests that\ndirect synchrotron radiation from the forward shock is notthe mechanism primarily10 Christopher Thompson\nresponsible for delayed optical and X-ray emission from GRB s.Direct radiation\nfrom a low- γcomponent of the ejecta is considered in Sect. 4.\n2.4 Dissipation at τT>1\nMHD waves, turbulent motions and shocks can in principle tap a significant frac-\ntion of the energy of the outflow. A periodic excitation of fre quencyωkinvolving a\ndisplacement ξkofthefluid depositsitsenergy directlyinthephotonsviaCompton\ndrag when the scattering depth across ξkis less than ∼c/ωkξ(T94). For example,\nrelativistic Alfv´ en waves have a damping time\ntdragωk∼(δB)2\nk/8π\nUγ, (9)\nwhich isshorter than thewaveperiodas longasthe photongas has ahigher energy\ndensity than waves in the relevant range of wavenumbers (Thompson and Blaes\n1997, hereafter TB97). This mechanism is particularly effec tive near the scattering\nphotosphere, and is also effective even at large τT, as long as higher wavenumber\nturbulence is generated via a turbulent cascade. The wave am plitude generally\ndecreases withwavenumber insuch a cascade, with the result that (δB)2\nk/8π≪Uγ\nat dissipative scales even if ( δB)2\nk/8π∼Uγat the outer scale. Shocks also transfer\nenergy tothephotonfluidviacompression anddirect first-or der Fermiacceleration\n(Blandford and Payne 1981), although energy transfer is slo wed significantly when\nthe photon pressure becomes comparable to the material ram p ressure ahead of\nthe shock.\n2.5 Dissipation at τT<1\nThe high energy emission in Blazars covers a wide range of ene rgies, up to 10 GeV\n(or higher in the TeV sources) and hence must be powered by non -thermal particle\ndistributions. The most familiar possibilities are first-o rder Fermi acceleration at\nshocks (Blandford and Eichler 1987) and electrostatic acce leration (e.g. RL92).\nPhoto-pion production on protons can more easily accomodat e the TeV sources\n(Mannheim 1993), but requires a supplementary ∼MeV emission mechanism in\nsources with soft high energy spectral states, because of th e much greater cross-\nsection for γ+γ→e±.\nThe relevant physics is treated in sufficient depth elsewhere that I will focus\non two questions here.\n1.Does reconnection deposit energy primarily in thermal or no n-thermal parti-\ncles, and in electrons or ions? This problem isfar from being understood from first\nprinciples, but Solar flares do provide clear evidence that t he efficiency of electron\nacceleration can (at least in a non-relativistic plasma) be quite high. However,\nType III radio bursts (which are powered by flare particles th at escape the Sun\nalong open magnetic field lines) also provide direct evidenc e that most of the flare\nenergy is dissipated in the form of bulk heating of electrons to energies of 10-100Dissipation in Relativistic Outflows: A Multisource Overview 11\nkeV, with only a small fraction being deposited in a relativi stic, non-thermal tail\n(Lin 1990).\n2.What is the effectiveness of electrostatic and strong-wave a cceleration in a\nrelativistic, Comptonizing medium, as compared to shock ac celeration? This ques-\ntion highlights a crucial difference between the large scale relativistic outflows\nassociated with Blazars and GRBs, and a laboratory system su ch as a Tokomak\n(or even smaller scale astrophysical systems such as corona l loops and arcades).\nA magnetic field Bwith gradient scale ℓB=B/|∇B|requires a minimal charge\ndensitync,min=B/4πeℓBto support the associated current; otherwise the dis-\nplacement current cannot be neglected and charges are accel erated to relativistic\nenergies. The ratio of the actual electron density to nc,mincan be expressed in\nterms of the scattering depth across ℓB,\nne\nnc,min∼τT\nαem/parenleftbiggB\nBQED/parenrightbigg−1\n. (10)\nThis works out to ne/nc,min∼1013for parameters appropriate to inhomogeneous\nexternalComptonBlazarmodels(BL95).Ithasbeenhypothes izedthatthebound-\nary layers of jets may be partially evacuated and sites for st rong-wave acceleration\n(Bisnovaty-Kogan & Lovelace 1997), but in fact the degree of evacuation must be\nextraordinary for such a mechanism to be important.\nAnother approach to this problem is to re-express Bin terms of the plasma\nβe= 8πneT/B2,\nne\nnc,min∼βe/parenleftbiggB\nBQED/parenrightbigg /parenleftbiggT\nmec2/parenrightbigg−1ℓB\nℓme, (11)\nwhereℓme= 4×10−11cm is the Compton wavelength of the electron. This is\nne/nc,min∼104βe, 108βe, and 1021βefor parameters appropriate to Tokomaks,\nSolar flares, and Blazar jets. This suggests that much higher wavenumber distor-\ntions of the magnetic field are required to provide efficient el ectron acceleration\nthrough reconnection in relativistic outflows. By contrast , direct Comptonization\nof a background photon fluid is moreeffective in jets and GRBs due to the higher\nscattering depth (T94).\n3 Spectral Consequences\nMost modelling of high energy emission from relativisticou tflows ignores the inner\nboundary condition on the outflow. The flow is hypothesized to dissipate outside\nthe pair annihilation radius, and flow conditions interior t o that radius are as-\nsumed not to influence the emergent high energy spectrum. See d radiation for\nComptonization is assumed to originate in a central accreti on diskexteriorto the\nvolume of the outflow.\nWe have already seen, however, that the emergent spectrum ca n be dominated\nby advected radiation if the outflow is optically thick at its base (T94, T96).12 Christopher Thompson\nMoreover, the inner boundary conditions are very well define d in flows of large\ncentral compactness, such asGRBs( ℓc∼1015).Themeanphotonenergy emerging\nfrom the flow is ∝angb∇acketlefthν∝angb∇acket∇ight ∼LP/˙Nγ, when the asymptotic Lorentz factor lies near the\ncritical value γe±(orγe−pif pairs are absent). Thermalization is rapid near the\nbase of the flow, the photon gas is very close to black body, and ∝angb∇acketlefthν∝angb∇acket∇ightis directly\nrelated to the effective temperature at the light cylinder,\n∝angb∇acketlefthν∝angb∇acket∇ight ∼Teff= 0.8/parenleftbiggLγ\n1050/parenrightbigg1/4/parenleftbiggP\n10−3s/parenrightbigg−1/2\nMeV. (12)\nThis is remarkably close to the observed range of spectral br eak energies, after\nallowing for cosmological redshift.\nIn this regime, the ratio of photon luminosity Lγto (ordered) Poynting lumi-\nnosityLPat the base of the wind is a key parameter. It should be emphasi zed that\nthe baryon loading is tolerably small only if Lγ<10−2LPat the neutrinosphere.\nIn other words, a key requirement of this model is that the win d be reheated from\nLγ≪LPtoLγ∼LPwell outside the neutrinosphere. This is plausibly accom-\nplished by a MHD cascade to high wavenumber, even though phot ons and pairs\nare tightly coupled on macroscopic scales (Sect. 2.4; TB97) .\nExpression (12)alsoleads toaninteresting question: how d oesthe meanenergy\nper photon change as one decreases the central compactness? As we now show,\nthe trend of decreasing mean photon energy with decreasing c ompactness in fact\ncan be reversed, with ∝angb∇acketlefthν∝angb∇acket∇ightapproaching ∼mec2atℓc∼102.\n3.1 Direct MeV Wien Peak\nA magnetized outflow that is strongly turbulent will trigger a cascade to high\nwavenumber that mustdissipate inside the Alfv´ en radius. We look for an opticall y\nthick equilibrium state in which the dissipative zone is shi elded from external\nphotons and soft photons are generated internally. In appli cations to AGN the ad-\nvectedmatter contributes negligibleopticaldepth, andso theoutflow isnecessarily\nhot and pair loaded. Damping via resonant couplings between high wavenumber\nturbulence and cosmic ray particles has been considered by D ermer, Miller, & Li\n(1996). However, direct Compton drag of bulk turbulent moti ons is an effective\ndamping mechanism at high compactness, typically at much lo wer wavenumbers\n(TB97). I now estimate the temperature T0of the flow at its base, focussing on\ntwo photon sources.\n1. Double Compton Emission. This dominates bremsstrahlung emission when\nT0∼mec2and the outflow is photon rich, nγ≫ne=ne++ne−. In a Wien photon\ngas,\n˙ndC=16Λ\nπαemnenγσTc/parenleftbiggT0\nmec2/parenrightbigg2\n, (13)\nwhereΛ≃ln(T/hνmin) and the photon gas approaches a Planckian distribution\nat frequency νmin.\nDouble Compton dominates cyclotron emission when the magne tic field in the\ncentral engine exceeds BQED= 4.4×1013G, so that thermal pairs do not populateDissipation in Relativistic Outflows: A Multisource Overview 13\nexcited Landau levels. Such strong fields have indeed been as sociated with SGR\n0526-66, which emitted the March 5, 1979 superburst. The ini tial 0.1 s pulse of\nthat burst appears to have approached a luminosity of ∼107times the Eddington\nluminosity (Fenimore et al. 1996), and has the appearance of an expanding pair\nfireball (Thompson&Duncan 1995;Fatuzzo& Melia1996).Ifth eoutflow isdriven\nby a magnetic field that is also strongly turbulent, then Comp ton drag can raise\nthe photon energy density 3 Tnγclose toB2/8πnear the light cylinder (TB97).\nEquating ˙ ndCwith the photon loss rate, one deduces an equilibrium scatte ring\ndepth\nτT=π\n16Λαem/parenleftbiggT\nmec2/parenrightbigg−2\n. (14)\nRe-expressing this in terms of the equilibrium pair density yields the relation\nbetween T0and compactness ℓcshown in Fig. 4.\nFig.4.Mean photon energy /angbracketlefthν/angbracketrightversus central compactness ℓcfor flows in which double\nCompton emission (solid line) and cyclo-synchrotron emission (long-das hed line) is the\ndominant soft photon source. The minimum temperature for an optical thic k flow (as-\nsuming a Wien photon distribution) is labelled by the short-dashed line. Note the breaks\nin the curves at the transition from Wien to black-body photon distri butions.\n2. Cyclo-Synchrotron Emission in a Non-thermal Pair Plasma . Such a very\nhigh optical depth and compactness cannot be maintained in w eaker magnetic\nfields. Cyclo-synchrotron photons are created rapidly, and their energy rapidly ex-\nponentiates. (For related calculations with non-thermal p article distributions, see\nGhisellini, Guilbert, & Svensson 1988.) To show this, I para metrize by NceB/m ec\nthe critical frequency at which Compton scattering increas es the frequency of a\nC-S photon at the same rate as it is absorbed, dy/dt=cαν. From Kirchoff’s law,14 Christopher Thompson\n˙nC−S\n˙ndC≃2N2\nc\nΛ/parenleftbiggB2/8π\nnγmec2/parenrightbigg\n. (15)\nSinceNc≫1for3T∼mec2(Mahadevan,Narayan,&Yi1996)thisyields ˙ nC−S≫\n˙ndC.\nIs the energy released by a turbulent cascade at the base of th e outflow dissi-\npated at large or small optical depth? If even a tiny fraction of the bulk kinetic\nenergy is carried by particles, then the wave energy must cas cade to very high\nwavenumber before charges are accelerated either electros tatically – as the turbu-\nlence becomes charge starved – or by resonant interactions. The relative impor-\ntance of these two effects is determined by the ratio\nkz\neB/m ec2∼/parenleftbiggℓ\nℓme/parenrightbigg−1τT\nαem/parenleftbiggB\nBQED/parenrightbigg−2\nin the case of sheared Alfv´ en waves of wavenumber kz(TB97). The cascade can,\nhowever, be cut off by Compton drag much closer to the outer sca le. The MHD\nwavemotionsareonlymildlyrelativistic,andsoalargesca tteringdepthisrequired\nto provide the Compton parameter y∼10 needed to upscatter the bulk of the C-S\nphotons to high energy. The mean energy per photon is given by\n3T0\nmec2≃B2/8π\nmec2˙nC−S(R/c)∼π\n32N2cαemτ−1\nT/parenleftbiggT0\nmec2/parenrightbigg−2\n, (16)\nwhere I approximate the bulk of the photon distribution as Wi en. Assuming that\nthermal and turbulent velocities are comparable, one has T0/mec2≃y/8τTand\nτT∼13/parenleftBigy\n10/parenrightBig3/2/parenleftbiggNc\n20/parenrightbigg\n. (17)\nThis gives T0/mec2= 0.095(y/10)−1/2(Nc/20)−1, as shown in Fig. 4.\nThis scattering depth lies above the thermal value, and so mu st be maintained\nby an extended non-thermal tail to the pair distribution fun ction. This is naturally\nprovided by MHD wave heating, as discussed in Sect. 1.2. The m inimum central\ncompactness needed to support this optically thick state is , on energetic grounds,\nℓc∼10 (Fig 2).\n3.2 Dissipation at Large Compactness and Low γ∞\nA much wider range of peak energies is possible if enough matt er is advected by\nthe outflow to make it optically thick. This is possible only f or a rather higher\ncompactness, ℓc> mp/me∼2000, than is expected in Blazar sources, but well\nwithin the range of cosmological GRB sources (cf. Paczy´ nsk i 1990). If the matter\nloadingisheavy enoughthat γ∞< γe−p= 0.2ℓ1/5\n∆t, thendissipationonanobserved\ntimescale ∆toccurs inside the e−pscattering photosphere. Thisprovides a natural\nexplanation for the soft tails and precursors to GRBs seen by Ginga, and the soft\nsub-pulses seen by BATSE (T94, T96).Dissipation in Relativistic Outflows: A Multisource Overview 15\nUnlike pair-dominated outflows with negligible matter (Sec t. 3.1) the emer-\ngent temperature is very sensitive to the amount of continuo us heating. (In the\ncase of spherical flows with radial inhomogeneities, this is equivalent to a broad\npower spectrum of inhomogeneities.) I further assume that t he flow expands rel-\nativistically at its base, with γ(r)∝rαwithα >1\n2. Then inhomogeneities in\nthe flow fall out of causal contact,1and dissipation on timescale ∆tisdelayed\nto radius Rdis≃2γ2c∆t. The corresponding luminosity L(Rdis) is then depleted\nby adiabatic expansion out to the scattering photosphere. A ssuming that photon-\nnumber changing processes freeze out before dissipation ta kes place, the emergent\ntemperature is a very strong function of matter loading,\nTobs\nT0= 0.7L(Rdis)\nLγ(R0)/parenleftbiggγ\nγe−p/parenrightbigg10/3\n∝(∆t)2/3. (18)\nThe correction for adiabatic losses (the last factor in this expression) has a very\nstrong dependence on γ, but a much weaker dependence on the (more directly\nobservable) variability timescale ∆t. This means that the power spectrum of in-\nhomogeneities must cover a very wide range of frequencies fo r continuous heating\nto overcome the effects of adiabatic cooling in a radialflow. The situation is quite\ndifferent in collimated flows, where energy can be continuous ly extracted from\nangular (e.g. Kelvin-Helmholtz) instabilities (Sect. 2.2 ).\nThe number of advected photons must also be compared with the number of\nfresh cyclo-synchrotron photons generated in the flow outsi de the central engine.\nFollowing Sect 3.1, this is\n˙nC−St\nnγ∼8αemN2\nsa\nπγdy\ndlnt/parenleftbiggT\nmec2/parenrightbigg\n. (19)\nHere all quantities refer to the rest frame of the outflow. The advected photons\nsuppress ˙ nC−Sat large τTby soaking up energy from the electrons and holding\ndowny. We conclude that ˙ nC−S/nγistypicallyless thanunity for high- γoutflows.\n3.3e+−e−Amplifier in Relativistic Outflows\nThe efficiency with which the available energy (in shocks, MHD waves, and re-\nconnecting magnetic fields) is deposited in high energy phot ons depends directly\non the fraction εeof the energy deposited in electrons and pairs. Pair creatio n\n(via photon collisions γ+γ→e++e−) has traditionally been used to constrain\nthe emission region in high energy sources (e.g. Cavallo & Re es 1978; Baring &\nHarding 1997), but one would like to emphasize an opposing po int of view here:\nthat the high energy photon flux from a relativistic outflow ca n be significantly\n1In contrast to the model of MeV Blazars discussed in Sects. 2.2 and 3.1, i n which the\ncollimated, lower- γflow is assumed to be heated continuously by non-radial Kelvin-\nHelmholtz instabilities.16 Christopher Thompson\nraisedby pair creation, due to an increase in εe. Energization of positrons by high-\nharmonic proton synchrotron maser radiation behind a shock (Hoshino & Arons\n1991) provides an example of such a leptonic acceleration me chanism.\nPair creationhas, ofcourse, been included for sometimeinm odelsofBHaccre-\ntion disk coronae (e.g. Stern et al. 1996 and references ther ein), but in that context\nthe formation of a power law high energy continuum does not de pend essentially\non the presence of pairs. For example, direct Comptonizatio n by MHD motions in\nthe corona will effectively heat the photons, independently of the relative amounts\nof rest energy in baryons and leptons (T94). The situation is quite different in\na relativistic outflow that expands sufficiently rapidly at it s base that inhomo-\ngeneities fall out of causal contact. If these inhomogeneit ies cover a wide range of\nspatial frequencies kmin< k < k max(as is needed in GRB models to accomodate\nthe broad power spectra of the bursts) then pair creation at w ave number kmax\n(radius∼4πγ2k−1\nmax) will increase the radius of the scattering photosphere by a\nfactor\nRe±\nτ=1\nRe−p\nτ=1∼/parenleftbiggmp\nme/parenrightbigg /parenleftbiggEbr\nγmec2/parenrightbigg2−β\n, (20)\ngiven a high energy photon index β. Thispair amplifier allows a much wider range\nof wavenumbers to be dissipated directly by Compton drag, an d hence causes re-\ngions of the wind with non-thermal high energy spectra to be s ignificantly brighter\nthan regions with thermal spectra (T96).\nAt this point I should distinguish between pair amplificatio n that is linear(the\nprobability of energization Peof a newly created pair is the same as that of a seed\nelectron) and non-linear (the pairs feed back on Pe). For example, the amplifier\noperating at a shock is non-linear if fresh pairs all act as su prathermal seeds for\nfirst-order Fermi acceleration; whereas it is linear if the p airs cool down to the\ntemperature of the background thermal plasma before intera cting resonantly with\nplasma waves.2The corresponding leptonic efficiencies are\nεe=Pe+2ne+/np\nPp+Pe+2ne+/np(non−linear);\n=Pe(1+2ne+/np)\nPp+Pe(1+ne+/np)(linear),(21)\nwherePpis the probability of energization of a proton and εe=Pe/(Pe+Pp) in\nthe absence of pairs.\nThe pair amplifier operating at Comptonizing hotspots (Sect . 3.4) is non-linear\nindifferentsense: paircreationregulatesthehighenergyi ndexβtotheappropriate\nvalue to yield a Thomson depth τT∼1\n4(T/mec2)−1within individual hotspots.\nThis second-order Fermi acceleration mechanism therefore yields a power-law high\nenergy spectrum over a wider range of matter loadings ( γe−p< γ∞< γe±=\n4.5γe−p) than does synchrotron cooling of shock-accelerated pairs .\n2The gyroperiod of a relativistic electron is orders of magnitude shorte r than its cooling\ntime if the magnetic field contributes an appreciable fraction of the pr essure of the\noutflow.Dissipation in Relativistic Outflows: A Multisource Overview 17\nPair amplification via photon collisions is also non-local : photons upscattered\nabove energy mec2in one hotspot will raise the pair density in another portion of\nthe flow. A nice example is provided by an expanding shell of ma tter and photons\nof radial width c∆t. The bulk Lorentz factor of the photon gas is not limited by\nthe inertia of the matter when γ > γe±. Outside a radius ∼2γ2c∆tphotons with\nenergies greater than mec2in the wind frame stream ahead of the forward shock,\nand sidescatter against seed electrons. The pair density gr ows exponentially (at\nfirst) inside a radius ∼ℓcR0, untilne+/npreaches unity and the material ahead\nof the shock is accelerated to a limiting Lorentz factor ∼ℓ1/2\nγ(for a hard incident\nphoton spectrum with β=−2). The feedback of pair creation on the structure of\na relativisticshock is an interesting problem that has not b een properly addressed.\nOne immediate spectral consequence of the pair amplifier is a suppression of\nthe minimum leptonic Lorentz factor γmin, and hence a suppression of the mini-\nmum synchrotron frequency3Esync(min)∼γ2\nmineB/m ec. In fact, γmin→1 as the\ninertia of the pairs becomes comparable to that of the proton s. To give an exam-\nple of the potential importance of this effect, consider a Poy nting-flux dominated\noutflow that approaches its limiting Lorentz factor γ∞. The high energy photon\nindex is taken to be β=−2 out to a rest frame energy mec2. Near the scattering\nphotosphere of the wind, the cyclotron energy is\n¯heB\nmec=8πec2\nσT(2LPc)1/2γ3\n∞= 0.04/parenleftBigγ∞\n300/parenrightBig3/parenleftbiggLP\n1051erg s−1/parenrightbigg−1/2\neV.(22)\nThe efficiency of electron acceleration can be increased by pa ir creation, but at the\ncost of suppressing Esync(min) far below the observed range of break energies in\nGRB spectra. As a result, the primary emission process must be inverse Com pton.\n3.4 Delayed Inhomogeneous Comptonization:\nBroken Power-law Spectra with a Thermal Photon Source\nLet us now consider the photon spectrum that results from del ayed reheating of a\nrelativistic outflow at large ℓc, outside the electron-ion photosphere ( γ∞∼γe−p).\nAt large ℓc, the photons are adiabatically cooled in between the centra l engine\nand the causal contact radius, where they are reheated to a lu minosity Lγ∼\n(δB/B)2LP(when the outflow is Poynting-flux dominated). The mean photo n\nenergy is restored to a value\n∝angb∇acketlefthν∝angb∇acket∇ight ∼0.7Lγ/Lγ0, (23)\ngiven that photon number is conserved at this radius (Sect. 3 .2).\nAs before, we consider a broad power spectrum of inhomogenei ties,kmin< k <\nkmax; the corresponding (radial) size of a hot spot is ∆∼π/k. Let us suppose that\n3The feedback of pair creation on the formation MeV breaks in high energy syn chro-\nCompton cascades above accretion disks has been considered by Done, Ghi sellini and\nFabian (1990).18 Christopher Thompson\nwavenumbers k⋆< k < k maxdissipate inside the electron-ion photosphere, and\nkmin< k < k ⋆outside. Individual hotspots are assumed to release their e nergy\nwhen the causal propagation distance R/2γ2\n∞begins to exceed ǫ−1·∆.\nHotspots with k > k⋆dissipate when the scattering depth of the flow is τT=\nk/k⋆(duetoseedelectrons). Thescatteringdepth acrossanindi vidualspot τspot\nTis\nsmallerby ε. Theresultant spectrum isWienwhen τspot\nT≫1.When τspot\nT<1(but\nthe flow itselfis stillopticallythick) cold seed photons es cape the spot before being\nupscattered, and one may use the standard loss-probability formalism (Shapiro,\nLightman, & Eardley 1976). Since the seed photons have adiab atically cooled\nby a factor ∼(2γ∞)−2/3(kR0/2π)2/3, the accumulated y-parameter required to\nupscatter them is large, and the resulting photon index is\nα=1\n2−/radicalbig\n(9/4)+(4/y)≃ −1. (24)\nThis power law distribution [extending up to a mean energy (2 3)], with a superim-\nposed Wien peak at energy (23), is the net result of this first s tage of Comptoniza-\ntion. It compares favorably with the low energy spectra of GR Bs (e.g. Cohen et\nal. 1996).\nAs dissipation continues at wavenumber k∼k⋆, Compton drag regulates the\ny-parameter to a value near unity. Hotspots with temperature Tw∼mec2will\nupscatter photons above energy ∝angb∇acketlefthν∝angb∇acket∇ightin a non-thermal tail that extends to the\npair creation threshold in the wind rest frame. This in turn g reatly amplifies the\nnumber of scattering charges, since photons greatly outnum ber electrons in the\noutflow (by a factor ∼γ∞(mp/me)(mec2/∝angb∇acketlefthν∝angb∇acket∇ight).\nThe keypoint here isthat theresulting expansionofthe scat tering photosphere\nfeeds back directly on the shape of the high energy continuum (T96). If the photon\ncompactness ℓγ=LγσT/4πγ3mec3R≫1atthisradius,thenahighenergyphoton\nindex as hard as β=−2 generates τT≫1 within individual hotspots, which in\nturn prevents theformation ofan extended high energy conti nuum. For example, if\nthe heating is triggered by reconnection (T94) then this req uires only that VA∼c\nin the wind rest frame, so that individual reconnection even ts induce bulk mass\nmotions at velocities close to the speed of light. As the comp actness drops, β\nrises to maintain τT∼1\n4y(Tw/mec2) within individual hotspots.4In other words,\nthe feedback works primarily through the scattering depth, rather than through\na balance between the time-averaged heating and cooling rat es (y= 1) as in\naccretion disk corona models (Shapiro et al. 1976; Haardt & M araschi 1993).\nThe net result isthat hotspots inthe wind with the right prop erties to generate\npairs are observed to be much brighter in X-rays and γ-rays than are regions of\nthe wind with thermal spectra, because a much wider range of w avenumbers is\ndissipated by Compton drag (Sect. 3.3).\nThis mechanism does not require fine-tuning of the Lorentz fa ctor if the range\nof wavenumbers is broad, kmax≫kmin. Nonetheless, one expects that γ∞is a\nstrong function of time in any GRB source involving an optica lly thick neutron\n4As long as ǫ≪1 photons are able to diffuse freely between spots, but not escape the\nwind entirely.Dissipation in Relativistic Outflows: A Multisource Overview 19\ntorusorneutronstarthatemitsneutrinos(T94).Theneutri noluminosityplausibly\npasses through the critical value at which the neutrino driv en mass-loss rate is\n˙M=LP/c2γe−p; indeed the total Poynting luminosity LPfrom a centrifugally\nsupported torus is limited to LP∼1051erg s−1in this manner.\nA related model uses strong shocks to directly accelerate th e photons via the\nfirst-order Fermi process (Blandford and Payne 1981). If the photons pass succes-\nsively through several strong shocks separated by adiabati c cooling, then it can be\nshown that the number index converges to a value −1 (Melrose & Pope 1993) up\nto an energy ∼LP/˙Nγ.\n3.5 A Hybrid Model:\nComptonization of an MeV Bump by Non-thermal Pairs\nAn advected Wien photon gas with temperature T0∼mec2can seed Comptoniza-\ntion by non-thermal pairs below the scattering photosphere . If the distribution\nof relativistic pairs has a lower cutoff γmin=O(1), then the resultant spectrum\nbreaks in the MeV range . Such a low cutoff results from a high energy pair cascade\nin a compact photon source (BL95), and results even for steep er pair spectra if\npair creation feeds back on the leptonic acceleration efficie ncy to load the outflow\nheavily with pairs, nemec2∼(δB)2/8π(Sect. 3.3).\nA further benefit of heavy pair loading is that the outflow beco mes photon-\nstarved when 3 Tapproaches mec2in the rest frame, so that the advected bump is\nstrongly depleted by Compton upscattering above ∼1 MeV. The resultant break\nenergy is then (for νFν∼const above 1 MeV)\nhνbr∼Lγ\n˙Nγ/bracketleftbigg\nln/parenleftbigghνmax\nmec2/parenrightbigg/bracketrightbigg−1\n∼3T0/parenleftbiggLγ\nLγ0/parenrightbigg /bracketleftbigg\nln/parenleftbigghνmax\nmec2/parenrightbigg/bracketrightbigg−1\n.(25)\nIndeed, a narrow bump near 1 MeV appears to be the exception ra ther than\nthe rule in Blazar spectra, although PKS 0208-512 does provi de a spectacular\nexception (von Montigny et al., these proceedings).\nAs a model for Blazar spectra, this has a number of advantages over models\ninvolving i) direct Comptonization of photons from the cent ral source (Dermer\nand Schliekieser 1993); and ii) Comptonization of side-sca ttered photons (Sikora,\nBegelman, & Rees 1994; BL95). First, a Comptonized UV bump (S ikora et al.\n1997) is avoided because the flow is self-shielding (Sect. 2. 2); second, the advected\nWien peak has a high enough temperature that the photon sourc e is depleted\nduring creation of the power-law γ-ray spectrum; and, third, observations of both\nMeV power-law breaks and isolated MeV bumps in Blazar spectr a are directly\ntied to the electron rest energy. The duality between these two spectral states is\nascribed to the presence or absence of a strong non-thermal e±component.\nComptonization of an advected MeV bump can be powered either by thermal\nor non-thermal particles. Is it reasonable to expect that qu asi-thermal motions\nshould be the dominant Compton heat source in GRB sources, bu t non-thermal\nparticles in Blazars? The key difference between these sourc es, aside from the\ncentral compactness, appears to be the degree of relativist ic expansion. Shocks20 Christopher Thompson\nin a GRB outflow should (locally) more closely approximate sp herical surfaces,\nwith the result that first-order Fermi acceleration is stron gly suppressed in the\nrelativistic limit (cf. Kirk, these proceedings).\n4 Conclusions: Optically Thick vs. Thin Sources\nWe have studied dissipation in relativistic outflows that ar e sufficiently compact\n(ℓc>10) to be optically thick at the center. Advected MeV radiati on provides\nan interesting new source of Compton seeds in this regime. In teracting with bulk\nturbulent motions and non-thermal pairs near the scattering photosphere , it can\nmanifest itself either as an MeV Blazar, or as an extended pow er law state when\nmost of the available energy is converted to non-thermal pai rs. The absence of a\nprominent MeV bump in most Blazar sources can be explained si nce the outflow is\nphoton starved . The spectral signature is expected to be different when ℓc<1−10,\nor when the advected radiative flux is low. The high energy spe ctral break energy\ncan cover a wider range of frequencies when synchrotron phot ons are the dominant\nseeds (MRP94; Ghisellini, these proceedings; Takahara, th ese proceedings).\nElectron-positron pairs play a crucial role here by i) maint aining a large scat-\ntering depth near the base of the outflow and shielding the hig h-γcore of a jet\nfrom ambient radiation; ii) maintaining the mean energy of t he advected radiation\nabove∼1 MeV; iii) enhancing the efficiency of leptonic dissipative m odes; and\niv) reducing the minimum energy of the non-thermal pair popu lation to γmin∼1,\nwhich keeps the minimum energy of the Comptonized MeV photon s in the MeV\nrange. In a large- γ(GRB) outflow, pairs also feed back on the emergent spectrum\nby expanding the scattering photosphere, and thus greatly i ncreasing the range of\nwavenumbers that are damped by Compton drag off advected radi ation.\nHeavy Matter Loading and GRB Afterglow. One should also consider the ef-\nfects of matter opacity in GRB outflows with extremely high ce ntral compactness.\nAlthough a core of the outflow (e.g. near the rotation axis of t he central engine)\nmust attain very high γ∞∼100−300, material off axis may not.5Material ex-\npanding with γ∞∼1−2 becomes optically thin to scattering on a timescale ∼1\nday (E/1052erg)1/2. This is comparable to the timescale on which the optical\nafterglow detected from GRB970508 reached a maximum (Bond 1 997). This is\ntelling, since direct synchrotron emission between the con tact discontinuity and\nthe forward shock probably is strongly suppressed due to the relative weakness of\nthe ambient magnetic field (Sect. 2.2). A further motivation forsimultaneous opti-\ncal observations of GRBs comes from the observation that the minimum frequency\nNceB/m ec∼(102−103)eB/m ecfor optically thin cyclo-synchrotron emission lies\nnear∼1 eV (Sect. 3.3) near the e±scattering photosphere and at a bulk Lorentz\nfactor of ∼102.\n5For example, if the central engine is a rapidly-rotating neutron star or neutron torus,\nthen neutrino emission can easily power mass loss rates as high as (1051erg s−1/c2∼\n10−3g s−1.Dissipation in Relativistic Outflows: A Multisource Overview 21\nGRB Time Profiles: FRED vs. Chaotic. The light-curves of GRBs show a be-\nwildering variety of shapes (Meegan et al. 1996), but at leas t two main classes can\nbe identified: bursts with smooth, asymmetric pulses (‘Fast Rise and Exponential\nDecay’ or FRED); and more ‘chaotic’ bursts in which narrow an d wide pulses are\noften superimposed and in which the asymmetry of individual pulses is usually\nless clearly defined.\nChaotic bursts are most easily explained if the dissipation in a burst is driven\nby local physics within the outflow (PX94; T94; RM94; Sari & Pi ran 1997), rather\nthan by interaction with an external medium. This leads to a s imple discriminant\nbetween the two classes: FRED bursts arise from shells of eje cta that come into\ncausal contact before the γ-raysescape, and vice versa for chaotic bursts. However,\nif the FRED bursts are also powered by local physics on a scale smaller than the\nwidth of the shell of ejecta, then the smoothness of the light curves indicates that\ntheγ-ray emitting zone lies at scattering depths τT>1. Given the lack of a clear\nspectral distinction between the two classes, one reaches t he same conclusion for\nchaotic bursts. Indeed, the transition zone between large a nd small τTcan be\nconsiderably broadened by pair creation, and pairs are most effective at enhancing\nleptonic dissipative modes near the scattering photospher e (Sect 3.3). In sum,\nthis leads to the following simple model: a burst is smooth (F RED) or chaotic\ndepending on whether the scattering photosphere lies outside or inside the causal\ncontact radius 2 γ2\n∞c∆t.\nRange of Temporal Frequencies and GRB Soft Tails. A flow will, in general, be\nvariable on a range of timescales ∆tand so dissipation can occur over a range of\noptical depths. The emergent spectrum varies considerably depending on whether\nmost of the available energy resides at long timescales or sh ort.\nIn this regard, it is interesting to note that the extended so ft bump follow-\ning GRB 870303 (a chaotic burst) detected by Ginga had a quasi -thermal cutoff\n(Yoshida et al. 1989),whereas the extended soft emission in GRB 960720 (a FRED\nburst) detected by BeppoSAX is closer to an extended powerla w (with the pos-\nsibility of a cutoff at ∼30 keV; Piro et al. 1997). This spectral difference could\nbe explained if the ejecta that produced the soft tail of GRB 8 70303 dissipated\natτe−p\nT≫1 while they were causally disconnected from the primary pul se of\nejecta. High energy cut-offs to GRB afterglow are in general v ery diagnostic: the\nhigh energy spectral index is attracted to β=−2 below an energy ∼(γ/τT)mec2\n(T94), and so the presence of a high energy spectral break yie lds information\nabout a combination of bulk Lorentz factor and scattering de pth (see also Baring\n& Harding 1997).22 Christopher Thompson\nReferences\nBaring, M.G. & Harding, A.K. 1997, Ap. J. 481, L85\nBisnovaty-Kogan, G.S. & Lovelace, R.V.E. 1997, preprint\nBlandford, R.D. & Payne, D.G. 1981, M.N.R.A.S., 194, 1041\nBlandford, R.D. & Eichler, D. 1987, Phys. Rep., 154 1\nBlandford, R.D. & Levinson, A. 1995, Ap. J. 441, 79 (BL95)\nBloemen, H., et al. 1995, A. A., 293, L1\nBlom, J.J., et al. 1995, A. A., 298, L33\nBond, H.E. 1997, IAU circular 6654; see also circulars 6655-6658\nCavallo, G. & Rees, M.J. 1978, M.N.R.A.S., 183, 359\nCohen, E., Katz, J.I., Piran, T., Sari, R., Preece, R.D., & Band, D. L. 1996, preprint.\nDermer, C.D. & and Schlickeiser, R. 1993, Ap. J., 416, 458\nDermer, C.D., Miller, J.A., & Li, H. 1996, Ap. J. 456, 106\nDone, C., Ghisellini, G., & Fabian, A.C. 1990, M.N.R.A.S., 245, 1\nDuncan, R.C., Shapiro, S.L., & Wasserman, I. 1986, Ap. J. 309, 141\nDuncan, R.C. & Thompson, C. 1992, Ap. J. 392, L9\nFatuzzo, M. & Melia, F. 1996, Ap. J. 464, 316\nFenimore, E.E., Klebesadel, R.W., & Laros, J.G. 1996, Ap. J., 460, 964\nGhisellini, G., Guilbert, P.W., & Svensson, R. 1988, Ap. J., 334, L5\nHaardt, F. & Maraschi, L. 1993, Ap. J. 413, 680\nHoshino, M. & Arons, J. 1991, Phys. Fluids B, 3, 818\nHurley, K., et al. 1994, Nat., 372, 652\nJaroszynski, M. 1993, A. A., 43, 183\nLevinson A. 1996, Ap. J., 459, 520\nLin, R.P. 1990, in Basic Plasma Processes on the the Sun , ed. E.R. Priest & V. Krishan,\np. 467\nMahadevan, R., Narayan, R. & Yi, I. 1996, Ap. J. 465, 327\nMallozzi, et al. 1995, Ap. J. 454, 597\nMannheim, K. 1993, A. A. 269, 67\nMaraschi, Ghisellini, & Celotti, 1992, Ap. J. 397, L5\nMarcowith, A., Henri, G., & Pelletier, G. 1995, M.N.R.A.S., 277, 681\nMcNaron-Brown, et al. 1995, Ap. J. 451, 575\nMeegan, C.A., et al. 1996, Ap. J. 106, 65\nMelrose, D.B. & Pope, M.H. 1993, Proc. Ast. Soc. Aust., 10, 222\nM´ esz´ aros, P., Rees, M.H. & Papathanassiou, H. 1994, Ap. J. 432, 181 (MRP94)\nM´ esz´ aros, P. & Rees, M.J. 1997, Ap. J. 482, 29\nPaczy´ nski, B. 1990, Ap. J., 363, 218\nPaczy´ nski, B. & Xu, G. 1994, Ap. J. 427, 708 (PX94)\nPapathanassiou, H. & M´ eszar´ os, P. 1996, Ap. J. 471, L91\nPendleton, G.N., et al., 1996, in proceedings of the Third Huntsville Symposium on\nGamma-Ray Bursts, ed. C. Kouveliotou, M.S. Briggs & G.J. Fishman, p. 228\nPiran, T. & Shemi, A. 1993, Ap. J. 403, L67\nPiran, T. & Narayan, R. 1996, in proceedings of the Third Huntsville Sym posium on\nGamma-Ray Bursts, ed. C. Kouveliotou, M.S. Briggs & G.J. Fishman, p. 233\nPiro, L., et al. 1997, preprint\nRees, M.J. & M´ esz´ aros, P. 1992, M.N.R.A.S., 258, 41\nRees, M.J. & M´ esz´ aros, P. 1994, Ap. J. 430, 93 (RM94)\nRomanova, M.M. & Lovelace, R.V.E. 1992, A. A., 262, 26Dissipation in Relativistic Outflows: A Multisource Overview 23\nSari, R. & Piran, T. 1997, Ap. J., 485, 270\nShapiro, S.L., Lightman, A.P., & Eardley, D.M. 1976, Ap. J., 204, 187\nSikora, M., Begelman, M.C., & Rees, M.J. 1994, Ap. J. 421, 153\nSikora, M., Madejski, G., Moderski, R., & Poutanen, J. 1997, Ap. J. 484, 108\nStern, B., Poutanen, J., Svensson, R., Sikora, M., & Begelman, M.C. 1995, 449, L13\nThompson, C. 1994, M.N.R.A.S., 270, 480 (T94)\nThompson, C. & Duncan R.C. 1995, M.N.R.A.S., 275, 255\nThompson, C. 1996, in proceedings of the Third Huntsville Symposium on G amma-Ray\nBursts, ed. C. Kouveliotou, M.S. Briggs & G.J. Fishman, p. 802 (T96)\nThompson, C. & Blaes, O. 1997, submitted to Phys. Rev. D (TB97)\nUsov, V.V. 1992, Nat., 357, 472\nUsov, V.V. 1994, M.N.R.A.S., 267, 1035\nVietri, M. 1996, Ap. J. 471, L95\nYoshida, A., et al. 1989, P.A.S.P., 41, 509\nThis book was processed by the author using the T EX macro package from Springer-\nVerlag."    },    {        "title": "2109.14018v3.A_robust_and_efficient_line_search_for_self_consistent_field_iterations.pdf",        "content": "A robust and e\u000ecient line search for self-consistent \feld iterations\nMichael F. Herbst\u0003\nApplied and Computational Mathematics, RWTH Aachen University, Schinkelstr. 2, 52062 Aachen, Germany.\nAntoine Levitty\nInria Paris and CERMICS, \u0013Ecole des Ponts, 6 & 8 avenue Blaise Pascal, 77455 Marne-la-Vall\u0013 ee, France.\nWe propose a novel adaptive damping algorithm for the self-consistent \feld (SCF) iterations\nof Kohn-Sham density-functional theory, using a backtracking line search to automatically adjust\nthe damping in each SCF step. This line search is based on a theoretically sound, accurate and\ninexpensive model for the energy as a function of the damping parameter. In contrast to usual\nSCF schemes, the resulting algorithm is fully automatic and does not require the user to select\na damping. We successfully apply it to a wide range of challenging systems, including elongated\nsupercells, surfaces and transition-metal alloys.\nI. INTRODUCTION\nAb initio simulation methods are standard practice\nfor predicting the chemical and physical properties of\nmolecules and materials. At the level of simulating elec-\ntronic structure the majority of approaches are either\ndirectly based upon Kohn-Sham density-functional the-\nory (DFT) or Hartree-Fock (HF) or use these techniques\nas starting points for more accurate post-DFT or post-\nHF developments. Both HF and DFT ground states are\ncommonly found by solving the self-consistent \feld (SCF)\nequations, which for both type of methods are very sim-\nilar in structure. Being thus fundamental to electronic-\nstructure simulations substantial e\u000bort has been devoted\nin the past to develop e\u000ecient and widely applicable SCF\nalgorithms. We refer to Woods et al.1and Lehtola et al.2\nfor recent reviews on this subject.\nHowever, the advent of both cheap computational\npower as well as the introduction of data-driven ap-\nproaches to materials modelling has caused simulation\npractice to change noticeably. In particular in domains\nsuch as catalysis or battery research where experiments\nare expensive or time-consuming, it is now standard prac-\ntice to perform systematic computations on thousands to\nmillions of compounds. The aim of such high-throughput\ncalculations is to either (i) generate data for training\nsophisticated surrogate models or to (ii) directly screen\ncomplete design spaces for relevant compounds. The de-\nvelopment of such data-driven strategies has already ac-\ncelerated research in these \felds and enabled the discov-\nery of novel semiconductors, electrocatalysts, materials\nfor hydrogen storage or for Li-ion batteries3{5.\nCompared to the early years where the aim was to per-\nform a small number of computations on hand-picked sys-\ntems, high-throughput screening approaches have much\nstronger requirements. In particular the key bottleneck\nis the required human time to set up and supervise com-\nputations. To minimize manual e\u000bort state-of-the-art\nhigh-throughput frameworks6{8provide a set of heuris-\ntics which automatically select computational parame-\nters based on prior experience. In case of a failing calcu-\nlation such heuristics may also be employed for param-eter adjustment and automatic rescheduling. While this\nempirical approach manages to take care of the major-\nity of failures automatically, it is far from perfect. First,\nstate-of-the-art heuristic approaches cannot capture all\ncases, and keeping in mind the large absolute number\nof calculations already a 1% fraction of cases that re-\nquire human attention easily equals hundreds to thou-\nsands of calculations. This causes idle time and severely\nlimits the overall throughput of a study. Second, any fail-\ning calculation, whether automatically caught by a high-\nthroughput framework or not, needs to be redone, im-\nplying wasted computational resources that contributes\nto the already noteworthy environmental footprint of su-\npercomputing9,10. The objectives for improving the algo-\nrithms employed in high-throughput work\rows is there-\nfore to increase the inherent reliability as well as re-\nduce the number of parameters, which need to be cho-\nsen. Ideally each building block of a simulation work\row\nwould be entirely black-box and automatically self-adapt\nto each simulated system. To some extent this amounts\nto taking the existing empirical wisdom already imple-\nmented in existing high-throughput frameworks and con-\nverting it into simulation algorithms with convergence\nguarantees using a mixture of both mathematical and\nphysical arguments.\nWith this objective in mind, this work will focus on\nimproving the robustness of self-consistent \feld (SCF)\nalgorithms, as mentioned above one of the most funda-\nmental components of electronic-structure simulations.\nOur main motivation and application are DFT simula-\ntions discretized in plane wave or \\large\" basis sets, for\nwhich it is only feasible to store and compute with or-\nbitals, densities and potentials, and not the full density\nmatrix or Fock matrix. In this setting, the standard SCF\napproach are damped, preconditioned self-consistent it-\nerations. Using an approach based on potential-mixing\nthe next SCF iterate is found as\nVnext=Vin+\u000bP\u00001(Vout\u0000Vin); (1)\nwhereVinandVoutare the input and output potentials to\na simple SCF step, \u000bis a \fxed damping parameter and\nPis a preconditioner. It is well-known that simple SCFarXiv:2109.14018v3  [cond-mat.mtrl-sci]  1 Mar 20222\niterations (where Pis the identity) can converge poorly\nfor many systems due to a number of instabilities11. Ex-\namples are the large-wavelength divergence due to the\nCoulomb operator leading to the \\charge-sloshing\" be-\nhavior in metals or the e\u000bect of strongly localized states\nnear the Fermi level, e.g. due to surface states or d- or\nf-orbitals. To accelerate the convergence of the SCF iter-\nation despite these instabilities, one typically aims to em-\nploy a preconditioner Pmatching the underlying system.\nDespite some recent progress towards cheap self-adapting\npreconditioning strategies11for the charge-sloshing-type\ninstabilities, choosing a matching preconditioner is still\nnot a straightforward task for other types of instabilities.\nFor example currently no cheap preconditioner is avail-\nable to treat the instabilities due to strongly localized\nstates near the Fermi level, such that in such systems\nusing a suboptimal preconditioning strategy is unavoid-\nable. While convergence acceleration techniques are usu-\nally crucial in such cases, these also complicate the choice\nof an appropriate damping parameter \u000bto achieve the\nfastest and most reliable convergence. As we will detail in\na series of example calculations on some transition metal\nsystems the interplay of mismatching preconditioner and\nconvergence acceleration can lead to a very unsystematic\npattern between the chosen damping parameter \u000band\nobtaining a successful or failing calculation. Especially\nfor such cases \fnding a good combination of precondi-\ntioning strategy and damping parameter can require sub-\nstantial trial and error.\nAs an alternative approach to a \fxed damping selected\nby a user a priori Canc\u0012 es and Le Bris suggested the op-\ntimal damping algorithm (ODA)12,13. In this algorithm\nthe damping parameter is obtained automatically by per-\nforming a line search along the update suggested by a\nsimple SCF step. Following this strategy, the ODA en-\nsures a monotonic decrease of the energy, which leads\nto strong convergence guarantees. This can be improved\nusing the history to improve convergence, such as in the\nEDIIS method14, or trust-region strategies15,16. These\napproaches are successfully employed for SCF calcula-\ntions on atom-centered basis sets, where an explicit rep-\nresentation of the density matrix is possible. However,\ntheir use with plane-wave DFT methods, where only or-\nbitals, densities and potentials are ever stored, does not\nappear to be straightforward, in particular in conjunction\nwith accelerated methods.\nAnother development towards \fnding an DFT ground\nstate in a mathematically guaranteed fashion are ap-\nproaches based on a direct minimization of the DFT en-\nergy as a function of the orbitals and occupations, not\nusing the self-consistency principle (see Reference 17 for\na mathematical comparison). Although direct minimiza-\ntion methods are often quite e\u000ecient for gapped systems,\ntheir use for metals requires a minimization over occupa-\ntion numbers18,19, which is potentially costly and unsta-\nble. For this reason such approaches seem to be less used\nthan the SCF schemes in solid-state physics.\nIn the realm of self-consistent iterations, variable-stepmethods have been successfully used20,21to increase ro-\nbustness. These methods are based on a minimization\nof the residual. Although this often proves e\u000ecient in\npractice, this has a number of disadvantages. First, the\nresidual might go up then down on the way to a solution\nmaking it rather hard to design a linesearch algorithm.\nSecond, this forces an algorithm to select an appropriate\nnotion of a residual norm, with results potentially sen-\nsitive to this choice. Third, there is the possibility of\ngetting stuck in local minima of the residual, or a sad-\ndle point of the energy. By contrast, we aim to \fnd a\nscheme ensuring energy decrease as an important ingredi-\nent to ensure robustness. Indeed, under mild conditions,\na scheme that decreases the energy monotonically is guar-\nanteed to converge to a solution of the Kohn-Sham equa-\ntions (see Theorems 1 and 2 below). This is in contrast to\nresidual-based schemes, which a\u000bord no such guarantee.\nThe very good practical performance of these schemes,\ndespite the lack of global theoretical guarantees, is an\ninteresting direction for future research.\nOur goal in this work is to design a mixing scheme that\n(a) is applicable to plane-wave DFT, and involves only\nquantities such as densities and potentials; (b) is based\non an energy minimization, to ensure robustness; (c) is\nbased on the self-consistent iterations; (d) is compati-\nble with acceleration and preconditioning. Our scheme\nis based on a minimal modi\fcation of the damped pre-\nconditioned iterations (1). Similar to the ODA approach\nwe employ a line search procedure to choose the damp-\ning parameter automatically. Our algorithm builds upon\nideas of the potential-based algorithm of Gonze22to con-\nstruct an e\u000ecient SCF algorithm. In combination with\nAnderson acceleration on challenging systems we show\nour adaptive damping scheme to be less sensitive than\nthe approach based on a \fxed damping parameter. In\ncontrast to the \fxed damping approach the scheme does\nnot require a manual damping selection from the user.\nThe outline of the paper is as follows. Section II\npresents the mathematical analysis of the self-consistent\n\feld iterations justifying our algorithmic developments.\nIn particular it presents a justi\fcation for global conver-\ngence of the SCF iterations. The proofs for the results\npresented in this section are given in the appendix. Sec-\ntion III discusses the adaptive damping algorithm itself\nfollowed by numerical tests (Section IV) to illustrate and\ncontrast cost and performance compared to the standard\n\fxed-damping approach. Concluding remarks and some\noutlook to future work is given in Section V.\nII. ANALYSIS\nA. Preliminaries\nWe use similar notation to those in Canc\u0012 es et al.17, ex-\ntend the analysis in that paper to the \fnite-temperature\ncase23, and introduce the potential mixing algorithm. We\nwork in the grand-canonical ensemble : we \fx a chemical3\npotential (or Fermi level) \u0016and an inverse temperature\n\f. In particular, the number of electrons is not \fxed.\nThis is for mathematical convenience: \fxing the number\nof electrons Ninstead of\u0016does not change our results.\nWe assume that space has been discretized in a \fnite-\ndimensional orthogonal basis (typically, plane-waves) of\nsizeNb, and will not treat either spin or Brillouin zone\nsampling explicitly for notational simplicity, although of\ncourse the formalism can be extended easily. In this sec-\ntion we will work with the formalism of density matrices,\nself-adjoint operators Psatisfying 0\u0014P\u00141. Such op-\nerators can be diagonalized as\nP=NbX\ni=1fij\u001eiih\u001eij: (2)\nThe numbers 0\u0014fi\u00141 are the occupation numbers, and\n\u001eiare the orbitals. Either density matrices or the set of\noccupation numbers and orbitals can be taken as the pri-\nmary unknowns in the self-consistency problem. Density\nmatrices are impractical numerically in plane-wave ba-\nsis sets, since they are Nb\u0002Nb; however, they are very\nconvenient to formulate and analyze algorithms. Accord-\ningly, we will use them in this theoretical section, but\nimplement the resulting algorithms using orbitals only.\nWe work on the sets\nH=fH2RNb\u0002Nb;HT=Hg (3)\nP=fP2H;0<P < 1g (4)\nof Hamiltonians and density matrices, equipped with\nthe standard Frobenius metric. Here and in the fol-\nlowing, inequalities between matrices are understood\nin the sense of symmetric matrices. The closure\nP=fP2H;0\u0014P\u00141gis compact. LetE0be a twice\ncontinuously di\u000berentiable function on P: we aim to solve\nthe problem\nmin\nP2PE0(P): (5)\nLet\nHKS(P) =rE0(P) (6)\nbe its gradient, and\nK(P) =d2E0(P) =drE0(P) (7)\nbe its Hessian. We will denote in bold \\super-operators\"\nor \\four-point operators\", operators from HtoH. Lets\nbe the fermionic entropy\ns(p) =\u0000(plogp+ (1\u0000p) log(1\u0000p)); (8)\nwith derivatives\ns0(p) = log\u00121\u0000p\np\u0013\n; s00(p) =\u00001\np(1\u0000p):(9)Let\nE(P) =E0(P)\u00001\n\fTr(s(P))\u0000\u0016TrP: (10)\nbe the free energy of a density matrix, where here and\nin the following we use functional calculus implicitly to\nde\fnes(P)2 H .Ediverges on the boundary of P,\nwhose closure is compact, and therefore Ehas at least\none minimizer in P. The \frst-order optimality condition\nrE(P) = 0 gives\nHKS(P)\u0000\u0016\u00001\n\fs0(P) = 0; (11)\nand therefore\nP=fFD(HKS(P)); (12)\nwhere we de\fne the Fermi-Dirac map fFDby\nfFD(H) =1\n1 +e\f(H\u0000\u0016): (13)\nHere we have used the equation s0(fFD(\")) =\f(\"\u0000\u0016),\nwhich will also be useful in the following. Although we\nuse the Fermi-Dirac smearing function for concreteness,\nour results apply just as well to Gaussian smearing for\ninstance; however, they don't apply to schemes with non-\nmonotonous occupations such as the Methfessel-Paxton\nscheme24.\nB. The dual energy\nReformulating the ideas in Gonze22, we now de\fne a\n\\dual\" energy\nI(H) =E(fFD(H)): (14)\nSince the map fFDis a bijection from HtoP, we have\nmin\nH2HI(H) = min\nP2PE(P): (15)\nThis is analogous to convex duality since the unknown in\nthis formulation is now H=rE(P).\nWe can compute the derivative \u001f0=dfFDoffFD(see\nLemma 1 in the Appendix for details):\n\u001f0 NbX\ni=1\"ij\u001eiih\u001eij!\n\u0001\u000eH\n=NbX\ni=1NbX\nj=1fFD(\"i)\u0000fFD(\"j)\n\"i\u0000\"jh\u001ei;\u000eH\u001ejij\u001eiih\u001ejj\n(16)\nThe linear map \u001f0is a \\four-point\" generalization of\nthe independent-particle polarizability. It describes the\nchange to the density matrix of a system of independent\nelectrons to a change in Fock matrix.4\nWe then have\nrI(H) =\u001f0(H)rE(fFD(H))\n=\u001f0(H)\u0010\nHKS(fFD(H))\u0000\u0016\u00001\n\fs0(fFD(H))\u0011\n=\u001f0(H)(HKS(fFD(H))\u0000H)\n(17)\nwhere again we used s0(fFD(\")) =\f(\"\u0000\u0016).\nThe Hessian of Iis a complicated object due to\nthe derivative of \u001f0(H). However, at a solution of\nHKS(fFD(H\u0003)) =H\u0003, this term vanishes, and we have\nthe simple result\nd2I(H\u0003) =\u0000\u001f0(H\u0003)(1\u0000K(H\u0003)\u001f0(H\u0003)) (18)\nTo better understand this object, we compute the Hes-\nsian ofE. FromrE(P) =HKS(P)\u00001\n\fs0(P)\u0000\u0016we get\nd2E(P)\u0001\u000eP=K(HKS(P))\u0001\u000eP\n\u00001\n\fNbX\ni=1NbX\nj=1s0(pi)\u0000s0(pj)\npi\u0000pjh\u001ei;\u000eP\u001ejij\u001eiih\u001ejj:\n(19)\nDe\fning\n\n NbX\ni=1\"ij\u001eiih\u001eij!\n\u0001\u000eP\n=\u0000NbX\ni=1NbX\nj=1\"i\u0000\"j\nfFD(\"i)\u0000fFD(\"j)h\u001ei;\u000eP\u001ejij\u001eiih\u001ejj\n(20)\nwe get\nd2E(P) =K(HKS(P)) +\n(f\u00001\nFD(P)): (21)\nThe point of this formula is to recognize now that\n\n(H) =\u0000\u001f0(H)\u00001. This links the Hessians of Eand\nI: at a \fxed point H\u0003=HKS(fFD(H\u0003)),\nd2E(fFD(H\u0003)) =\n(H\u0003)d2I(H\u0003)\n(H\u0003): (22)\nSince \nis self-adjoint and positive de\fnite, both Hes-\nsians have the same inertia (number of negative eigen-\nvalues).\nC. Hamiltonian mixing\nThe very simplest Hamiltonian mixing algorithm is\nHn+1=HKS(fFD(Hn)): (23)\nAs already recognized in Reference 22, (17) makes it pos-\nsible to reinterpret this simple algorithm in a new light:\nit is a gradient descent algorithm on Iwith step 1, pre-\nconditioned by \u001f0(Hn)\u00001. It is natural to use a smaller\nstepsize to try to ensure convergence, and indeed this is\nguaranteed to work:Theorem 1. LetH02H. There is \u000b0>0such that,\nfor all 0<\u000b<\u000b 0, the algorithm\nHn+1=Hn+\u000b(HKS(fFD(Hn))\u0000Hn) (24)\nsatis\fesHKS(fFD(Hn))\u0000Hn!0. If furthermore E\nis analytic, Hnconverges to a solution of the equation\nHKS(fFD(H)) =H.\nAdaptive-step schemes can also ensure guaranteed con-\nvergence:\nTheorem 2. FixH02H, and constants 0<\u000b max<1,\n0<c< 1;0<\u001c < 1. Consider the algorithm\nHn+1=Hn+\u000bn(HKS(fFD(Hn))\u0000Hn) (25)\nwhere\u000bnis chosen in the following way: starting from\n\u000bmax, decrease\u000bmaxby a factor \u001cwhile the Armijo line\nsearch condition\nI(Hn+\u000bn(HKS(fFD(Hn))\u0000Hn))\n\u0014I(Hn)\u0000\u000bnch\n(fFD(Hn))rI(Hn);rI(Hn)i(26)\nis not veri\fed. Then this algorithm satis\fes\nHKS(fFD(Hn))\u0000Hn!0. If furthermore Eis an-\nalytic,Hnconverges to a solution of the equation\nHKS(fFD(H)) =H.\nThe proofs of both these statements are found in the\nAppendix.\nThe adaptive-step scheme above however su\u000bers from\ntwo important drawbacks. First, it is costly (requiring\nseveral SCF steps per iteration). Second, it is imcompat-\nible with preconditioned or accelerated schemes because,\nin contrast to the SCF direction, there is no guarantee in\nthese cases that the chosen direction is a descent direc-\ntion to the energy. This would make a straightforward\nimplementation of the above algorithm uncompetitive for\n\\easy\" systems, and therefore motivates the search for a\ncompromise algorithm that tries to recover some robust-\nness properties while not sacri\fcing performance.\nD. Potential mixing\nWe now specialize the above discussion to our\ncase of interest of semi-local density-functional the-\nory (DFT) models. We introduce the operators\ndiag :RNb\u0002Nb!RNband diagm : RNb!RNb\u0002Nb. The\ndiag operator takes the diagonal (in real space) of a den-\nsity matrix, yielding a density. The diagm operator con-\nstructs a Fock matrix contribution with a given local po-\ntential. Both operators are adjoint of each other. With\nthese notations, the energy function takes the form\nE0(P) = Tr(H0P) +g(diag(P)); (27)\nwhereH0is a given operator (the core Hamiltonian)\nandgis a nonlinear function (the Hartree-exchange-\ncorrelation energy). For these models the gradient of5\nE0(P) (the Fock matrix) depends on diag( P) (the den-\nsity) only:\nHKS(P) =H0+ diagm(V(diag(P))); (28)\nwith the potential\nV(\u001a) =rg(\u001a)2RNb: (29)\nBased on (13) and the de\fnition of the density we de\fne\nthe potential-to-density mapping\n\u001a(V) = diag(fFD(H0+ diagm(V))); (30)\nwhich allows to solve the self-consistency problem (12)\nby iteration in the potential Vonly:\nVn+1=Vn+\u000b\u000eVn; (31)\nwhere we de\fned the search direction\n\u000eVn=V(\u001a(Vn))\u0000Vn: (32)\nThe corresponding energy functional minimized by this\n\fxed-point problem is\nI(V) =E(fFD(H0+ diagm(V))): (33)\nCompared to an algorithm based on Kohn-Sham Hamil-\ntonians as suggested in Section II C this formulation has\nthe advantage that only vector-sized potentials Vnin-\nstead of matrix-sized quantities need to be handled.\nThe analysis of the previous sections carries forward\nstraightforwardly to the potential mixing setting. In par-\nticular one identi\fes as the analogue of Kthe Hessian of\ng, i.e. the (two-point) Hartree-exchange-correlation ker-\nnelK, and as the analogue of \u001f0the derivative of V(\u001a),\nwhich is the independent-particle susceptibility \u001f0. The\nlatter becomes apparent by comparing (16) to the Adler-\nWiser formula for \u001f025,26\n\u001f0(V) =NbX\ni=1NbX\nj=1fFD(\"i)\u0000fFD(\"j)\n\"i\u0000\"jj\u001e\u0003\ni\u001ejih\u001e\u0003\ni\u001ejj(34)\nin which (\"i;\u001ei) denotes the eigenpairs of H0+diagm(V).\nBothKand\u001f0arise naturally when considering the Ja-\ncobian matrix\nJ\u000b= 1\u0000\u000b\u0000\n1\u0000K(V\u0003)\u001f0(V\u0003)\u0001\n(35)\nof the potential-mixing SCF iteration (31) near a \fxed\npointV\u0003. If the eigenvalues of J\u000bare between\u00001 and 1\nthe potential-mixing SCF iterations converge. By anal-\nogy with Hamiltonian mixing, Theorem 1 guarantees that\nglobal convergence can always be ensured by selecting\n\u000bsmall enough. In this respect our results from Sec-\ntion II C strengthen a number of previous results17,22,27,\nwhich established local convergence for su\u000eciently small\n\u000b.E. Improving the search direction \u000eVn:\nPreconditioning and acceleration\nThe Jacobian matrix (35) involves the dielectric ma-\ntrix\u000f(V) = 1\u0000K(V)\u001f0(V), which can become badly\nconditioned for many systems. In such cases, a very\nsmall step must be employed to ensure stability (small-\nest eigenvalue of J\u000blarger than\u00001), which slows down\nconvergence (largest eigenvalue of J\u000bvery close to 1) to\na level too slow to be practical. A solution is to improve\nthe search direction \u000eVnto ensure faster convergence1.\nThis is usually achieved by a combination of techniques\njointly referred to as \\mixing\", which amend \u000eVnusing\nboth preconditioning as well as convergence acceleration.\nEmploying a preconditioned search direction\n\u000eVn=P\u00001[V(\u001a(Vn))\u0000Vn] (36)\nin a damped SCF iteration, the corresponding Jacobian\nbecomes\nJ\u000b= 1\u0000\u000bP\u00001\u000f(V): (37)\nProvided that the inverse P\u00001approximates the inverse\ndielectric matrix \u000f\u00001su\u000eciently well, the spectrum of\nP\u00001\u000fis close to 1, so that a larger damping \u000band and\nfaster iteration is possible. While suitable cheap precon-\nditionersPare not yet known for all sources of bad condi-\ntioning in SCF iterations, a number of successful strate-\ngies have been suggested. Examples include Kerker mix-\ning28to improve SCF convergence in metals or LDOS-\nbased mixing11to tackle heterogeneous metal-vacuum or\nmetal-insulator systems. For a more detailed discussion\non this matter we refer the reader to Reference 11.\nAn additional possibility to speed up convergence\nis to use black-box convergence accelerators. These\ntechniques build up a history of the previous iterates\nV1;:::;Vnas well as the previous preconditioned residu-\nalsP\u00001R1;:::;P\u00001Rn(withRn=V(\u001a(Vn))\u0000Vn) and\nuse this information to obtain the next search direction\n\u000eVn. The most frequently used acceleration technique in\nthis context is variously known as Pulay/DIIS/Anderson\nmixing/acceleration, which we will refer to as Anderson\nacceleration. This method obtains the search direction\nas a linear combination\n\u000eVn=P\u00001Rn\n+1\n\u000bn\u00001X\ni=1\fi\u0000\nVi+\u000bP\u00001Ri\u0000Vn\u0000\u000bP\u00001Rn\u0001(38)\nwhere the expansion coe\u000ecients \fiare found by mini-\nmizing\n\r\r\r\r\rP\u00001Rn+n\u00001X\ni=1\fi\u0000\nP\u00001Ri\u0000P\u00001Rn\u0001\r\r\r\r\r: (39)\nIn practice, it is impractical to keep a potentially large\nnumber of past iterates, and only the last 10 iterates are6\ntaken into account. Furthermore, the associated linear\nleast squares problem can become ill-conditioned29. We\nuse the simple strategy of discarding past iterates to en-\nsure a maximal conditioning of 106.\nThis method is known to be equivalent to a multise-\ncant Broyden method. In the linear regime and with in-\n\fnite history Anderson acceleration is further equivalent\nto the well-known GMRES method to solve linear equa-\ntions. For details see Reference30and References therein.\nProvided that nonlinear e\u000bects are negligible, Anderson\nacceleration typically inherits the favorable convergence\nproperties of Krylov methods31, explaining their frequent\nuse in the DFT context. However, especially at the be-\nginning of the SCF iterations or when treating systems\nthat feature many close SCF minima, nonlinear e\u000bects\ncan become important. In such cases the behavior of\nAnderson is more complex and mathematically not yet\nfully understood. In particular the dependence of the\nconvergence behavior on numerical parameters such as\nthe chosen damping can become less regular and harder\nto interpret, as we will see in our numerical examples in\nSections IV B and IV C.\nIII. ADAPTIVE DAMPING ALGORITHM\nUp to now we have assumed that the step size \u000bis\nconstant, re\recting common practice in plane-wave DFT\ncomputations. We now describe the main contribution\nof this paper, an algorithm to adapt this step size to in-\ncrease robustness and minimize user intervention into the\nconvergence process. At step nof the algorithm, given a\ntrial potential Vn, we compute the search direction \u000eVn\nthrough (38), and look for a step \u000bnto take as\nVn+1=Vn+\u000bn\u000eVn (40)\nNote that the de\fnition of \u000eVnitself in (38) depends on\na stepsize\u000b; since our scheme will adapt \u000bnto\u000eVn, we\ncannot just take \u000b=\u000bnin (38), and so we use for \u000ba\ntrial damping e\u000b(to be discussed in Section III C).\nTo select\u000bn, we could try to minimize I(Vn+1), or\nemploy an Armijo line-search strategy. However, each\nevaluation ofIis very costly, and it is therefore desir-\nable to obtain e\u000ecient approximate schemes. The energy\nI(V+\u000b\u000eVn), can be expanded as\nI(Vn+\u000b\u000eVn) =I(Vn) +\u000bh\u001f0(Vn)Rn;\u000eVni\n+1\n2\u000b2h\u000eVn;d2I(Vn)\u0001\u000eVni+O(\u000b3k\u000eVnk3);\n(41)\nwhere we have used rI(Vn) =\u001f0(Vn)Rn. This approx-\nimation is good for small dampings \u000band/or close to\nthe solution, when \u000eVnis small. The object d2Iis com-\nplicated and expensive to compute in general. However,\nclose to a \fxed point, we can use the expression (18) to\nwrited2I(Vn)\u0019\u001f0(Vn)(1\u0000K(Vn))\u001f0(Vn). We can thenapproximate the terms in (41), leading to the model\n'n(\u000b) =I(Vn) +\u000bhRn;\u001f0(Vn)\u000eVni\n\u00001\n2\u000b2\n\u001f0(Vn)\u000eVn;\u0002\n1\u0000K(Vn)\u001f0(Vn)\u0003\n\u000eVn\u000b(42)\nfor the energy, where we have used the self-adjointness of\n\u001f0(Vn) to make it act only on \u000eVn. To compute the coe\u000e-\ncients in this model, we still need to compute \u001f0(Vn)\u000eVn,\na costly operation. However, for all \u000bwe have to \frst\norder\n\u000b\u001f0(Vn)\u000eVn=\u001a(Vn+\u000b\u000eVn)\u0000\u001a(Vn) +O(\u000b2k\u000eVnk2):\n(43)\nNote that if we set Vn+1=Vn+\u000bn\u000eVnand then proceed\nalong the iterative algorithm, we will have to compute\n\u001a(Vn+1) in any case. An approximation to the coe\u000ecients\nof the model 'ncan therefore be constructed without any\nextra diagonalization.\nThis is the basis of the adaptive damping scheme de-\nscribed in Algorithm 1. Since \u001a(Vn) is already known\n(it is needed to construct \u000eVn), the only expensive step\nin this algorithm is the computation of \u001a(Vn+1), which\noccurs only once per loop iteration. In particular, when\nset to always accept Vn+1, this algorithm reduces to the\nstandard damped SCF algorithm. Notice that the algo-\nrithm only allows \u000bnto shrink between iterations. As a\nresult (i) the model 'nprovides better and better damp-\ning predictions and (ii) keeping in mind our analysis of\nSection II C the proposed tentative steps Vn+1become\nmore likely to be accepted.\nAlgorithm 1 Adaptive damping algorithm\nInput: Current iterate Vn, search direction \u000eVn, trial damp-\ninge\u000b\nOutput: Damping\u000bn, next iterate Vn+1\n1:\u000bn e\u000b\n2:loop\n3: Make tentative step Vn+1=Vn+\u000bn\u000eVn\n4: Compute\u001a(Vn+1);I(Vn+1) (the expensive step)\n5: ifacceptVn+1(see Section III A) then\n6: break\n7: else\n8: Build the coe\u000ecients of the model 'n\n9: ifmodel'nis good (see Section III B) then\n10: \u000bn argmin\u000b'n(\u000b)\n11: Scale\u000bnto ensurej\u000bnjis strictly decreasing\n12: else\n13: \u000bn \u000bn\n2\n14: end if\n15: end if\n16:end loop\nWe complete the description of the algorithm by spec-\nifying some practical points: when to accept a step, how\nto determine whether a model is good, how to select the\ninitial trial step e\u000band how to integrate adaptive damping\nwith Anderson acceleration.7\nA. Step acceptance\nWe accept the step as soon as the proposed next iterate\nVn+1=Vn+\u000bn\u000eVnsatis\fes\nI(Vn+1)<I(Vn) or\r\rP\u00001Rn+1\r\r<\r\rP\u00001Rn\r\r;\n(44)\ni.e. if either the energy or the preconditioned residual de-\ncreases. Although accepting steps higher in energy may\ndecrease the robustness of the algorithm, we found in\npractice that accepting steps that decrease the residual\nhelps keeping the method e\u000bective in the later stages of\nconvergence, when the Anderson acceleration is able to\ntake e\u000ecient steps that may slightly increase the energy\nbut are not worth reverting.\nB. Quality of the model 'n\nOur model 'nmakes various assumptions that might\nnot hold in practice, especially far from convergence.\nHowever, by comparing the actual energy I(Vn+\u000bn\u000eVn)\nto the prediction 'n(\u000bn), we can inexpensively check the\nquality of the model. We do this by computing the ratio\nrn=jI(Vn+\u000bn\u000eVn)\u0000'n(\u000bn)j\njI(Vn+\u000bn\u000eVn)\u0000I(Vn)j; (45)\nwhich should be small if the model is accurate. We deem\nthe model good enough if\nrn<0:1 and 'nhas a minimum : (46)\nNotice that the minimizer of 'nmay not necessarily be\npositive. For particularly accurate models ( rn<0:01)\nwe additionally allow backward steps (i.e. \u000bn<0), which\nturned out to overall improve convergence in our tests.\nC. Choice of the trial step e\u000b\nTo ensure that as many as possible SCF steps only re-\nquire a single line search step, we dynamically adjust e\u000b\nbetween two subsequent SCF steps. A natural approach\nis to reuse the adaptively determined damping \u000bnas the\ne\u000bin the next line search, which e\u000bectively shrinks e\u000bbe-\ntween SCF steps. However, the algorithm may need small\nvalues ofe\u000bin the initial stages of convergence, and keep-\ning these small values for too long limits the eventual\nconvergence rate. To counteract the decreasing trend,\nwe allowe\u000bto increase if a line search was immediately\nsuccessful (i.e. \u000bn=e\u000b). In this case we again use the\nmodel'n. If it is su\u000eciently good (as described in Sec-\ntion III B), we set\ne\u000b max\u0010\ne\u000b;1:1\u0001argmin\n\u000b'n(\u000b)\u0011\n: (47)\nOtherwisee\u000bis left unchanged.As an additional measure, to prevent the SCF from\nstagnating we enforce e\u000bto not undershoot a minimal\ntrial damping e\u000bmin. We used mostly e\u000bmin= 0:2 as a\nbaseline, and report varying this parameter in the nu-\nmerical experiments.\nWith this dynamic adjustment of e\u000b, we checked that its\ninitial valuee\u000b0, i.e. the value used in the \frst SCF step,\nhas little in\ruence on the overall convergence behavior.\nHowever, in well-behaved cases, too small values for this\nparameter lead to an unnecessary slowdown of the \frst\nfew SCF steps. We therefore settled on e\u000b0= 0:8 similar\nto standard recommendations for the default damping32.\nIV. NUMERICAL TESTS\nThe adaptive damping algorithm described in Section\nIII was compared against a conventional preconditioned\ndamped potential-mixing SCF scheme featuring only a\n\fxed damping. For this we employed three kinds of\ntest problems. The \frst are calculations on aluminium\nsystems of various size including cases with an unsuit-\nable computational setup, i.e. where charge sloshing is\nnot prevented by employing the Kerker preconditioner.\nThese are discussed in more detail in Section IV A. The\nsecond, discussed in Section IV B, is a gallium arsenide\nsystem which we previously found to feature strongly\nnonlinear behavior in the initial SCF steps11. Lastly in\nsection Section IV C we will consider Heusler systems and\nother transition-metal compounds, which are generally\nfound to be di\u000ecult to converge.\nFor our tests we used the implementation of the\nadaptive damping algorithm available in the density-\nfunctional toolkit (DFTK)33,34, a \rexible open-source\nJulia package for plane-wave density-functional theory\nsimulations. For all calculations we used Perdew-Burke-\nErnzerhof (PBE) exchange-correlation functional35as\nimplemented in the libxc36library, and Goedecker-Teter-\nHutter pseudopotentials37. Depending on the system, a\nkinetic energy cuto\u000b between 20 and 45 Hartree as well\nas an unshifted Monkhorst-Pack with a maximal k-point\nspacing of at most 0 :14 inverse Bohrs was used. For the\nHeusler systems this was reduced to at most 0 :08 inverse\nBohrs. With the exception of the gallium arsenide system\na Gaussian smearing scheme with width of 0.001 Hartree\nwas employed. For the systems containing transition-\nmetal elements collinear spin polarization was allowed\nand the initial guess was constructed assuming ferromag-\nnetic spin ordering except when otherwise noted. Notice,\nthat this initial guess is generally not close to the \fnal\nspin ordering, see Section IV C for discussion regarding\nthis choice. The full computational details for each sys-\ntem (including the employed structures) as well as in-\nstructions how to reproduce all results of this paper can\nbe found in our repository of supporting information38.\nTable I summarizes the required number of Hamilto-\nnian diagonalizations to converge the SCF energy to an\nerror of 10\u000010Hartree for various \fxed dampings \u000bas8\nSystem Precond. \fxed damping \u000b adaptive\n0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 damping\nAl8supercell Kerkera\u000258 37 27 21 16 13 1112 18 17\nAl8supercell Nonea\u000252\u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0002 24\nAl40supercell Kerker 19 15 14 12 1112 12 12 12 12 12\nAl40supercell None 3840 40 39 44 50 49 \u000276\u0002 44\nAl40surface Kerker \u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0002\nAl40surface None 4648 50 49 51 60 61 66 89 \u0002 49\nGa20As20supercell None 2633 40 42 45 44 70 70 65 76 26\nCoFeMnGa Kerker \u0002 \u0002 \u0002 \u0002 282124 28 22 22 30\nFe2CrGa Kerker \u0002 \u0002 \u0002 27\u0002 \u0002 1925\u000222 39\nFe2MnAl Kerker \u000248\u0002 \u0002 \u0002 20 21 17 16 15 34\nFeNiF 6 Kerker\u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0002 23 22 21 24\nMn2RuGa Kerker \u0002 \u0002 \u0002 \u0002 37 24 23 2223 23 36\nMn3Si Kerker \u0002 \u0002 \u0002 \u0002 26 30 22 20\u0002 \u0002 \u0002\nMn3Si (AFM)bKerker\u0002 \u0002 58 29 31 30 2022 26 28 35\nCr19defect Kerker \u0002 \u0002 \u0002 74 46 48 46 4147 53 48\nFe28W8multilayer Kerker 3234 37 34 38 43 41 48 \u0002 \u0002 37\nawithout Anderson acceleration\nbinitial guess with antiferromagnetic spin ordering\nTABLE I. Number of Hamiltonian diagonalizations required to obtain convergence in the energy to 10\u000010with a cross (\u0002)\ndenoting a failure to converge within 100 diagonalizations. Except where otherwise noted Anderson acceleration has been\nemployed and for the transition-metal systems (third/fourth group of compounds) a ferromagnetic initial guess has been\nused. On supercells atomic positions were slightly randomised. Computational details are given in the text. Notice that the\ntransition-metal systems may not converge to the same SCF solution for each calculation.\nwell as the adaptive damping algorithm. We carefully\nveri\fed the obtained solutions to be stationary points by\nmonitoring the SCF residual Rn. Note that for the adap-\ntive damping procedure the number of SCF steps is not\nidentical to the number of Hamiltonian diagonalizations,\nsince multiple tentative steps might be required until a\nstep is accepted. Since iterative diagonalization overall\ndominates the cost of the SCF procedure, the number\nof diagonalizations provides a better metric to compare\nbetween the cost of both damping strategies.\nA. Inadequate preconditioning: Aluminum\nTo investigate the in\ruence of the choice of a subopti-\nmal preconditioner on the convergence for both the \fxed\ndamping and adaptive damping strategies we considered\nthree aluminium test systems: two elongated bulk su-\npercells with 8 or 40 atoms as well as a surface with 40\natoms and a portion of vacuum of identical size. For the\nelongated supercells both the initial guess as well as the\natomic positions were slightly rattled.\nThe results are summarized in the \frst segment of Ta-\nble I. For the small Al8system, where Anderson accel-\neration was not used, representative convergence curves\nare shown in Figure 1. Due to the well-known charge-\nsloshing behavior, SCF iterations on such metallic sys-\ntems are ill-conditioned. Without preconditioning small\fxed damping values \u000bare thus required to obtain con-\nvergence, with only a small window of damping values be-\ning able to achieve a convergence within 100 Hamiltonian\napplications. On the other hand in combination with the\nmatching Kerker preconditioning strategy28large \fxed\ndamping values generally converge more quickly.\nIn contrast the adaptive damping strategy is much\nless sensitive to the choice of the minimal trial damping\ne\u000bmin. Moreover, it leads to a much improved convergence\nfor the case without suitable preconditioning while still\nmaintaining similar costs if Kerker mixing is employed.\nThese observations carry over to cases including An-\nderson acceleration and larger aluminium systems, see\nFigure 2 for a representative computation on an alu-\nminium surface. Notice that Kerker mixing is extremely\nbadly suited for the large aluminium surface, such that\nconvergence is not obtained in 100 Hamiltonian for any\nof the damping strategies, see Reference 11 for a better\npreconditioner in such inhomogeneous systems. Overall\nemploying adaptive damping therefore makes the reliabil-\nity and e\u000eciency of the SCF less dependent on the choice\nof the preconditioning strategy, while not requiring the\nuser to manually select a damping parameter.9\nFIG. 1. SCF convergence for a randomized aluminium su-\npercell (8 atoms) without Anderson acceleration and using\nsimple mixing (top) as well as Kerker mixing(bottom). The\nadaptive scheme converges robustly even in the unprecondi-\ntioned case, without requiring the manual selection of a step.\nFIG. 2. SCF convergence without preconditioning for an\nelongated supercell of an aluminium surface with 40 alu-\nminium atom and a vacuum portion of equal size. A \fxed\nstep of\u000b= 0:1 was optimal here, but the adaptive scheme\ngets very close performance without manual stepsize selection.\nFIG. 3. Elongated gallium arsenide supercell (20 gallium and\n20 arsenide atoms) with slightly randomized atomic positions.\nFor all cases, simple mixing and Anderson acceleration are\nemployed.\nB. Strong nonlinear e\u000bects: Gallium arsenide\nIn previous work we identi\fed elongated supercells of\ngallium arsenide with slightly perturbed atomic positions\nto be a simple system that still exhibits strong nonlinear\ne\u000bects when the SCF is far from convergence11. In the\nconvergence pro\fles of these systems this manifests by\nthe error shooting up abruptly with Anderson failing to\nquickly recover. In Figure 3, for example, the error in-\ncreases steeply between Hamiltonian diagonalizations 3\nand 6 for the \fxed-damping approaches with stepsizes\nbeyond 0:1. It should be noted that this behavior is an\nartefact of the interplay of Anderson acceleration and\ndamped SCF iterations on these systems, which is not\nobserved in case Anderson acceleration is not employed.\nFor more details see the discussion of the gallium arsenide\ncase in Ref. 11.\nFor the calculations employing a \fxed damping strat-\negy only small damping values of \u000b= 0:1 are able to\nprevent this behavior. Already slightly larger damping\nvalues noticably increase the number of Hamiltonian di-\nagonalizations required to reach convergence (compare\nTable I), and thus a careful selection of the damping\nvalue is in order for such systems. In contrast the pro-\nposed adaptive damping strategy with our baseline min-\nimal trial damping of e\u000bmin= 0:2 automatically detects\nthe unsuitable Anderson steps and downscales them. As\na result an optimal or near-optimal cost is obtained with-\nout any manual parameter tuning. For comparison, we\nalso display in Figure 3 the results with a large value of\ne\u000bmin= 0:5, which prevents the damping algorithm to\navoid the nonlinear e\u000bects.10\nFIG. 4. Convergence of the Fe 2CrGa Heusler alloy with\nKerker mixing without Anderson acceleration (top) and with\nAnderson acceleration (bottom). Notice that the SCF cal-\nculations do not necessarily converge to the same local SCF\nminimum.\nC. Challenging transition-metal compounds\nIn this section we discuss two types of transition-metal\nsystems. First, we consider a selection of smaller prim-\nitive unit cells, including the mixed iron-nickel \ruoride\nFeNiF6as well a number of Heusler-type alloy struc-\ntures, see the third group of Table I. These structures\nwere found found in the course of high-throughput com-\nputations to be di\u000ecult to converge39. Moving to larger\nsystems we considered an elongated chromium supercell\nwith a single vacancy defect as well as a layered iron-\ntungsten system, see the fourth group of Table I. Both\ntest cases were taken from previous studies20,40on SCF\nalgorithms.\nIn particular the Heusler compounds are known to ex-\nhibit rich and unusual magnetic and electronic proper-\nties. From our test set, for example, Fe2MnAl shows\nhalfmetallic behaviour, i.e. a vanishing density of states\nat the Fermi level in only the minority spin channel41.\nOther compounds, such as Mn2RuGa or CoFeMnGa\nshow an involved pattern of ferromagnetic and antifer-\nromagnetic coupling of the neighboring transition-metalsites42,43. Such e\u000bects are closely linked to the d-orbitals\nforming localized states near the Fermi level44,45and im-\nply that there are a multiple accessible spin con\fgura-\ntions, which are close in energy. Unfortunately these\ntwo properties also make Heusler compounds di\u000ecult to\nconverge using standard methods. First, localized states\nnear the Fermi level are a source of ill-conditioning for\nthe SCF \fxed-point problem11, with no cheap and widely\napplicable preconditioning strategy being available. Sec-\nond, the abundance of multiple spin con\fgurations im-\nplies a more involved SCF energy landscape with multi-\nple SCF minima. On such a landscape convergence may\neasily \\hesitate\" between di\u000berent local minima or sta-\ntionary points. Furthermore the setup of an appropriate\ninitial guess, which ideally guides the SCF towards the\n\fnal spin ordering requires human expertise and is hard\nto automatise in the high-throughput setting. Albeit not\nfully appropriate for the systems we consider, we followed\nthe guess setup, which has also been used in the afore-\nmentioned high-throughput procedure39, namely to start\nthe calculations with an initial guess based on ferromag-\nnetic (FM) spin ordering.\nAs a result in our tests, calculations on Heusler sys-\ntems without Anderson acceleration require very small\n\fxed damping values below 0 :1 even if the Kerker precon-\nditioner is used, see Figure 4 (a). The adaptive damping\nstrategy improves the convergence behavior and in agree-\nment with our previous results partially corrects for the\nmismatch in preconditioner and initial guess. Still, con-\nvergence is extremely slow.\nAn acceptable convergence is only accessible in com-\nbination with Anderson acceleration. However, the\nAnderson-accelerated \fxed-damping SCF is very suscep-\ntible to the chosen damping \u000b, see Figure 4 (b). In par-\nticular the lowest-energy SCF minimum is only found\nwithin 100 Hamiltonian diagonalizations for \u000b= 0:4,\n\u000b= 0:7,\u000b= 0:8 and\u000b= 1:0. Other \fxed damping\nvalues initially converge, but then convergence stagnates\nand the error only reduces very slowly beyond around 30\ndiagonalizations.\nWe investigated the source of this pathological behav-\nior by restarting the iterations after stagnation. This did\nnot noticeably alter the behavior, eliminating the possi-\nbility that the history of the iterates within the Anderson\nacceleration scheme somehow \\jam\" the SCF into stag-\nnation in the strongly nonlinear regime | as can happen\nfor instance in nonlinear conjugate gradient methods46.\nAnother possibility is that the iterations somehow got\ninto a particularly rough region of SCF energy landscape\nbetween multiple stationary points, which is simply hard\nto escape. However, this is not the case either. For in-\nstance on the Fe2CrGa system with a \fxed damping of\n0:3, the restarted iterations did converge quickly using an\nAnderson scheme with a small maximal conditioning of\n102for the linear least squares problem. It would there-\nfore appear that this phenomenon is due to inadequate\nregularization of the least squares problem. We expect\nmore sophisticated techniques for controlling the Ander-11\nFIG. 5. Convergence of the Mn 3Si Heusler compound with\nKerker mixing and Anderson acceleration and starting from\na ferromagnetically ordered initial guess. Small dampings are\nsusceptible to stagnation induced by Anderson instabilities.\nson history30to be worth investigating for such systems\nin the future.\nBecause of this stagnation issue we found Anderson-\naccelerated SCF iterations to become unreliable for our\ntransition-metal test systems: for \fxed damping values\nbelow\u000b= 0:5, hardly any calculation converges. No-\ntably, due to the non-trivial interplay with the Anderson\nscheme, this result is the exact opposite to our theoret-\nical developments on damped SCF iterations in Section\nII C, which suggested to reduce the damping to achieve\nreliable convergence.\nOverall the transition-metal cases emphasize the di\u000e-\nculty in manually choosing an appropriate \fxed damping.\nFor a number of cases the window of converging damping\nvalues is rather narrow, e.g. consider Fe2CrGa, FeNiF6\nor Mn3Si (with the FM guess) and in our tests only a\nsingle damping value of \u000b= 0:8 fortitiously manages to\nconverge all systems.\nIn contrast the proposed adaptive damping strategy\nwith our baseline value of e\u000bmin= 0:2 is less suscepti-\nble to the stagnation issue. Across the unit cells and\nextended transition-metal systems we considered we ob-\nserved only a convergence failure in one test case, namely\nthe Mn3Si Heusler alloy with the FM guess, see Figure 5.\nFor some test cases, adaptive damping did cause a notice-\nable computational overhead: in extreme cases (such as\nFe2CrGa or Fe2MnAl) the number of required Hamilto-\nnian diagonalizations almost doubles. However, it should\nbe emphasized that no user adjustments were needed\nto obtain these results, even though adaptive damping\nhas been constructed on the here invalid principle that\nsmaller damping increases reliability.\nYet even on cases where Anderson instabilities cause\nnon-convergence of the adaptive scheme, \fne-tuning is\npossible. Increasing the minimal trial damping from\ne\u000bmin= 0:2 toe\u000bmin= 0:5, for example, increases the\nminimal step size and thus lowers the risk of Ander-son stagnation. For all transition-metal cases we con-\nsiderede\u000bmin= 0:5 strictly reduces the number of diag-\nonalizations required to reach convergence compared to\ne\u000bmin= 0:2, see for example Figure 4 (b). Moreover this\nparameter adjustment even resolves the convergence is-\nsues of Mn3Si, see Figure 5. If manual intervention is\npossible another option is to incorporating prior knowl-\nedge of the \fnal ground state electronic structure into\nthe initial guess. For the Mn3Si case, for example, an im-\nproved initial guess based on an antiferromagnetic spin\nordering (AFM) between adjacent manganese layers sim-\npli\fes the SCF problem, such that both a larger range\nof \fxed damping values as well as the adaptive damping\nstrategy give rise to converging calculations.\nV. CONCLUSION\nWe proposed a new linesearch strategy for SCF com-\nputations, based on an e\u000ecient approximate model for\nthe energy as a function of the damping. Our algorithm\nfollows four general principles: (a) the algorithm should\nneed no manual intervention from the user; (b) it should\nbe combinable with known e\u000bective mixing techniques\nsuch as preconditioning and Anderson acceleration; (c)\nin \\easy\" cases where convergence with a \fxed damping\nis satisfactory it should not slow down too much; and\n(d) it should be possible to relate it to schemes with\nproved convergence guarantees. We demonstrated that\nour proposed scheme ful\flls all these objectives. With\nour default parameter choice of e\u000bmin= 0:2 the resulting\nadaptively damped SCF algorithm is able to converge all\nof the \\easy\" cases faster or almost as fast as the \fxed-\ndamping method with the best damping. Simultaneously\nit is more robust than the \fxed-damping method on the\n\\hard\" cases we considered, such as elongated bulk met-\nals and metal surfaces without proper preconditioning or\nHeusler-type transition-metal alloys. In particular the\nlatter kind of systems feature a very irregular conver-\ngence behavior with respect to the damping parameter,\nmaking a robust manual damping selection very challeng-\ning. In practice the classi\fcation between \\easy\" and\n\\hard\" cases may well depend on the considered system\nand the details of the computational setup, e.g. the em-\nployed mixing and acceleration techniques. However, our\nscheme makes no assumptions about the details how a\nproposed SCF step has been obtained. We therefore be-\nlieve adaptive damping to be a black-box stabilisation\ntechnique for SCF iterations, which applies beyond the\nAnderson-accelerated setting we have considered here.\nStill, our results on these \\hard\" cases also highlight\npoorly-understood limitations of the commonly used An-\nderson acceleration process. For example, despite fol-\nlowing standard recommendations to increase Anderson\nrobustness, we frequently observe SCF iterations to stag-\nnate. A more thorough understanding of this e\u000bect would\nbe an interesting direction for future research.\nOur scheme was applied to semilocal density function-12\nals in a plane-wave basis set. It is not speci\fc to plane-\nwave basis sets, and we expect it to be similarly e\u000ecient\nin other \\large\" basis sets frequently used in condensed-\nmatter physics. For atom-centered basis sets, like those\ncommon in quantum chemistry, direct mixing of the den-\nsity matrix is feasible, and likely more e\u000ecient. Our\nscheme does not apply directly to hybrid functionals,\nwhere orbitals or Fock matrices have to be mixed also; an\nextension to this case would be an interesting direction\nfor future research.\nACKNOWLEDGEMENTS\nThis project has received funding from the European\nResearch Council (ERC) under the European Union's\nHorizon 2020 research and innovation program (grant\nagreement No 810367). We are grateful to Marnik Bercx\nand Nicola Marzari for pointing us to the challenging\ntransition-metal structures that stimulated most of the\npresented developments. Fruitful discussions with Eric\nCanc\u0012 es, Xavier Gonze and Benjamin Stamm and pro-\nvided computational time at Ecole des Points and RWTH\nAachen University are gratefully acknowledged.\nAPPENDIX: MATHEMATICAL PROOFS\nLemma 1. Let\nH\u0003=NbX\ni=1\"ij\u001eiih\u001eij (48)\nwith orthonormal \u001eiand non-decreasing \"i. Letf:\nR!Rbe a real analytic function in a neighborhood\nof[\"1;\"Nb]. Then the map H7!f(H)is analytic in\na neighborhood of H\u0003, and\ndf(H\u0003)\u0001\u000eH=NbX\ni=1NbX\nj=1f(\"i)\u0000f(\"j)\n\"i\u0000\"jh\u001ei;\u000eH\u001ejij\u001eiih\u001ejj\n(49)\nwith the convention thatf(\"i)\u0000f(\"i)\n\"i\u0000\"i=f0(\"i).\nProof of Lemma 1. This is a classical result, known\nas the Daleckii-Krein theorem in linear algebra; see for\ninstance Higham47. To keep this paper self-contained,\nwe follow here the proof in Levitt48in the analytic case.\nSincefis analytic on [ \"1;\"Nb], it is analytic in a complex\nneighborhood. Let Cbe a positively oriented contour\nenclosing [\"1;\"Nb]. Then, for Hclose enough to H\u0003, we\nhave\nf(H) =1\n2\u0019iI\nCf(z)1\nz\u0000Hdz (50)and analyticity of ffollows. For \u000eHsmall enough,\nf(H\u0003+\u000eH)\n=1\n2\u0019iI\nCf(z)1\nz\u0000H\u0003\u0000\u000eHdz\n\u0019f(H\u0003) +1\n2\u0019iI\nCf(z)1\nz\u0000H\u0003\u000eH1\nz\u0000H\u0003dz\n=f(H\u0003) +1\n2\u0019iI\nCNbX\ni=1NbX\nj=1f(z)h\u001ei;\u000eH\u001eji\n(z\u0000\"i)(z\u0000\"j)j\u001eiih\u001ejjdz\n=f(H\u0003) +NbX\ni=1NbX\nj=1f(\"i)\u0000f(\"j)\n\"i\u0000\"jh\u001ei;\u000eH\u001ejij\u001eiih\u001ejj\n(51)\nwhere\u0019means up to terms of order O\u0000\nk\u000eHk2\u0001\n.\nProof of Theorem 1. If\u000b0\u00141,Hnbelongs to the\nconvex hull spanned by H0andfHKS(fFD(H));H2Hg .\nOn this compact set X,fFD,Iand their derivatives are\nbounded. We have for all H2X\nI(H+\u000b(HKS\u0000H))\n=I(H)\u0000\u000bh\n\u00001(HKS\u0000H);(HKS\u0000H)i+O(\u000b2)\n=I(H)\u0000\u000bh\nrI(H);rI(H)i+O(\u000b2)\n(52)\nwhere in this expression the functions \nandHKSare\nevaluated at fFD(H), and the constant in the O(\u000b2) term\nis uniform in n. It follows that for \u000b0small enough, there\nisc>0 such that\nI(Hn+1)\u0014I(Hn)\u0000\u000bckrI(Hn)k2; (53)\nand therefore rI(Hn)!0, so that\nHKS(fFD(Hn))\u0000Hn!0.\nWe now proceed as in Levitt49. LetI\u0003=\nlimn!1I(Hn). The set \u0000 = fH2X;\nI(H) = limn!1I(Hn)gis non-empty and compact.\nFurthermore, d(Hn;\u0000)!0; if this was not the case,\nwe could extract by compactness of Xa subsequence\nat \fnite distance of \u0000 converging to a H\u00032Xsatis-\nfyingI(H\u0003) = limn!1I(Hn), which would imply that\nH\u00032\u0000, a contradiction.\nAt every point Hof \u0000, by analyticity there is a neigh-\nborhood of HinHsuch that the  Lojasiewicz inequality\njI(H0)\u0000I\u0003j1\u0000\u0012H\u0014\u0014HkrI(H0)k (54)\nholds for some constants \u0012H2(0;1=2],\u0014H>049,50. By\ncompactness, we can extract a \fnite covering of these\nneighborhoods, and obtain a  Lojasiewicz inequality with\nuniversal constants \u00122(0;1=2];\u0014> 0 in a neighborhood\nof \u0000. Therefore, for nlarge enough, using the concavity\ninequalityx\u0012\u0014y\u0012+\u0012y\u0012\u00001(x\u0000y) withx=I(Hn+1)\u0000I\u0003,13\ny=I(Hn)\u0000I\u0003, we get\nkrI(Hn)k2\u00141\n\u000bcI(Hn)\u0000I(Hn+1)\n\u00141\n\u0012\u000bc(I(Hn)\u0000I\u0003)1\u0000\u0012\n\u0001h\n(I(Hn)\u0000I\u0003)\u0012\u0000(I(Hn+1)\u0000I\u0003)\u0012i\n\u0014\u0014\n\u0012\u000bckrI(Hn)k\n\u0001h\n(I(Hn)\u0000I\u0003)\u0012\u0000(I(Hn+1)\u0000I\u0003)\u0012i\nkrI(Hn)k\u0014\u0014\n\u0012\u000bch\n(I(Hn)\u0000I\u0003)\u0012\u0000(I(Hn+1)\u0000I\u0003)\u0012i\n(55)\nIt follows thatkrI(Hn)kis summable, and therefore\nthatkHn+1\u0000Hnkis; this implies convergence of Hnto someH\u0003. When\u0012= 1=2 (or, in light of (22), when\nd2E(fFD(H\u0003)) is positive de\fnite), we can get exponen-\ntial convergence49.\nNote that the bounds used in the proof of the above\nstatement (for instance, on \u000b0) are extremely pessimistic,\nsince they rely on the fact that the set of possible Pis\nbounded, and therefore all density matrices of the form\nfFD(HKS(P)) have occupations bounded away from 0\nand 1, which results in bounded derivatives for Tr( s(P)).\nA more careful analysis is needed to obtain better bounds\n(for instance, bounds that are better behaved in the zero\ntemperature limit).\nProof of Theorem 2. From (52) it is easily seen that\nthe linesearch process stops in a \fnite number of iter-\nations, independent on n. This ensures that there is\n\u000bmin>0 such that \u000bmin\u0014\u000bn\u0014\u000bmax. 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Lojasiewicz, Lectures Notes IHES (Bures-sur-Yvette)\n(1965)."    },    {        "title": "2402.08240v2.Forecasts_for_Constraining_Lorentz_violating_Damping_of_Gravitational_Waves_from_Compact_Binary_Inspirals.pdf",        "content": "Forecasts for Constraining Lorentz-violating Damping of Gravitational Waves from\nCompact Binary Inspirals\nBo-Yang Zhanga,b,c,∗Tao Zhub,c,†Jing-Fei Zhanga,‡and Xin Zhanga,d,e§\naKey Laboratory of Cosmology and Astrophysics (Liaoning) & College of Sciences,\nNortheastern University, Shenyang 110819, China\nbInstitute for Theoretical Physics and Cosmology,\nZhejiang University of Technology, Hangzhou, 310032, China\ncUnited Center for Gravitational Wave Physics (UCGWP),\nZhejiang University of Technology, Hangzhou, 310032, China\ndKey Laboratory of Data Analytics and Optimization for Smart Industry (Ministry of Education),\nNortheastern University, Shenyang 110819, China\neNational Frontiers Science Center for Industrial Intelligence and Systems Optimization,\nNortheastern University, Shenyang 110819, China\nViolation of Lorentz symmetry can result in two distinct effects in the propagation of the gravi-\ntational waves (GWs). One is a modified dispersion relation and another is a frequency-dependent\ndamping of GWs. While the former has been extensively studied in the literature, in this pa-\nper we concentrate on the frequency-dependent damping effect that arises from several specific\nLorentz-violating theories, such as spatial covariant gravities, Hoˇ rava-Lifshitz gravities, etc. This\nLorentz-violating damping effect changes the damping rate of GWs at different frequencies and\nleads to an amplitude correction to the GW waveform of compact binary inspiral systems. With\nthis modified waveform, we then use the Fisher information matrix to investigate the prospects of\nconstraining the Lorentz-violating damping effect with GW observations. We consider both ground-\nbased and space-based GW detectors, including the advanced LIGO, Einstein Telescope, Cosmic\nExplorer (CE), Taiji, TianQin, and LISA. Our results indicate that the ground-based detectors in\ngeneral give tighter constraints than those from the space-based detectors. Among the considered\nthree ground-based detectors, CE can give the tightest constraints on the Lorentz-violating damping\neffect, which improves the current constraint from LIGO-Virgo-KAGRA events by about 8 times.\nI. INTRODUCTION\nSince the landmark discovery of the first gravitational\nwave (GW) event, GW150914, resulting from the coa-\nlescence of two massive black holes by the LIGO-Virgo\ncollaboration in 2015 [1], the field of GW astronomy\nhas rapidly evolved. To date, approximately 90 events\nhave been meticulously identified by the LIGO-Virgo-\nKAGRA (LVK) scientific collaborations [2–5]. On May\n24, 2023, the advanced LIGO (aLIGO) initiated the\nObserving Run 4 (O4) project. Following LVK, the\nforthcoming third-generation ground-based GW detec-\ntors, such as Einstein Telescope (ET) [6] and Cosmic\nExplorer (CE) [7], are presently in the design phase,\nwith a specific emphasis on detecting high-redshift GW\nevents ( z >10). Concurrently, a new cohort of space-\nbased detectors (Taiji [8–10], TianQin [11–15], and LISA\n[16, 17]) is designed to explore the low-frequency GW\nsignals ( f∼10−4Hz). We anticipate that these detec-\ntors will play a crucial role in the era of GW astronomy\n[18–22].\nGeneral relativity (GR) remains the preeminent the-\nory for explaining gravitational phenomena. Yet, it faces\n∗zhangby@stumail.neu.edu.cn\n†Corresponding author: zhut05@zjut.edu.cn\n‡Corresponding author: jfzhang@mail.neu.edu.cn\n§zhangxin@mail.neu.edu.cnchallenges in accounting for enigmatic concepts such as\ndark matter and dark energy, and reconciling them with\nquantum mechanics, particularly in the contexts of sin-\ngularities and the quantization of gravity. To address\nthese issues, a plethora of experiments have been devised\nto rigorously test GR’s predictions. Regrettably, the ma-\njority of these experiments have been limited to investi-\ngating the weak-field regime [23–25]. GWs, one of the\nfundamental predictions of GR, are produced in the tu-\nmultuous environments of strong gravitational fields and\ninteract only weakly with matter, making them pristine\nmessengers of the dynamics of space-time. The detec-\ntion of GWs, especially those originating from the co-\nalescences of compact binary systems [26–28], has thus\nheralded a new era for testing the robustness of GR under\nextreme conditions. These observations offer a powerful\ntool for probing the strong-field regime of gravity, poten-\ntially unlocking answers to the persistent questions that\nchallenge the current understanding within the frame-\nwork of GR.\nIn the theoretical realm, various modified theories of\ngravity have been proposed to address challenges within\nGR, see refs. [29–39] and references therein. A subset\nof these theories has garnered significant attention for\ndeviating from a fundamental principle of GR — the\nLorentz invariance. At high-energy levels, it is widely\nbelieved that this invariance will be broken when grav-\nity is quantized. Various modified gravity theories have\nbeen proposed to explore the nature of Lorentz violationarXiv:2402.08240v2  [gr-qc]  10 Apr 20242\nin gravity, including the Einstein-Æther theory [40–47],\nHoˇ rava-Lifshitz theories of quantum gravity [48–51], and\nspatial covariant gravities [52–56]. A phenomenological\nframework, the standard model extension, has also been\nextensively studied in the literature for exploring the pos-\nsible properties of Lorentz violations in the gravitational\nsector [57–63].\nThe violation of Lorentz symmetry in gravity can in-\ntroduce deviations from GR in the propagation of GWs.\nThese deviations manifest in two distinct ways, influenc-\ning the propagation behavior of GWs in the cosmological\nbackground. First, with Lorentz violation, the conven-\ntional linear dispersion relation of GWs can be modified\ninto a nonlinear one, which in turn changes the phase\nvelocities of GWs at different frequencies. This effect\ncan arise from a large number of Lorentz-violating theo-\nries. Second, Lorentz violation can introduce frequency-\ndependent friction into the propagation equation of GWs,\nresulting in varying damping rates for GWs of different\nfrequencies during their propagation. This effect nor-\nmally arises from those theories with mixed temporal\nand spatial derivatives of the spacetime metric in the\nmodified theories of gravity with spatial covariance, for\nexample, the Hoˇ rava-Lifshitz gravity [64], the spatial co-\nvariant gravities [56, 65], etc. Here we would like to note\nthat the possible Lorentz violations could also lead to\nsource-dependence on the speed of GWs [66].\nTesting Lorentz symmetry of gravitational interac-\ntion by using the observational data from GW events\nin LIGO-Virgo-KAGRA catalogs and future GW detec-\ntors has been carried out in a lot of works, see Refs.\n[21, 22, 65, 67–70] and references therein. In most of\nthese works, the effects due to the nonlinear dispersion\nrelation have been extensively considered. Recently, the\nconstraint on the Lorentz-violating damping effects from\nGW events in LIGO-Virgo-KAGRA catalogs was first ob-\ntained [71].\nIn this paper, we detail the Lorentz-violating damp-\ning effects in the propagation of GW in a cosmological\nbackground. Decomposing the GWs into left-hand and\nright-hand circular polarization modes, we observe that\nthe Lorentz-violating damping effects manifest through\nexplicit modifications in the GW amplitude. We derive\ncorrections to the waveform of the compact binary inspi-\nral system accordingly. With this modified waveform, we\nuse the Fisher information matrix (FIM), which is widely\nused in cosmology and astrophysics [72–89] to investigate\nthe prospects of constraining the Lorentz-violating damp-\ning effect with GW observations of compact binary sys-\ntems. We consider both ground-based and space-based\nGW detectors, including aLIGO, CE, ET, Taiji, Tian-\nQin, and LISA. Our results indicate that the ground-\nbased detectors in general give tighter constraints than\nthose from the space-based detectors. Among the con-\nsidered three ground-based detectors, CE can give the\ntightest constraints on the Lorentz-violating damping ef-\nfect, which improves the current constraint from LIGO-\nVirgo-KAGRA events by about 8 times [71].Our paper is organized as follows. In Sec. II, We\npresent a very brief introduction to the Lorentz-violating\ndamping effect and calculate the modified waveform of\nGWs of compact binary inspiral systems with the ef-\nfect. Sec. III summarizes the application of the FIM for\nconstraining the modified waveform parameters of GWs.\nThe main results of our analysis are discussed in Sec. IV.\nFinally, Sec. V provides a summary and further discus-\nsion of our work in this paper.\nII. MODIFIED WAVEFORM OF GWS WITH\nLORENTZ-VIOLATING DAMPING EFFECT\nIn this section, we present a brief introduction to\nthe modified waveform of GWs with Lorentz-violating\ndamping effect. As we mentioned, such Lorentz-violating\ndamping effect can modify the amplitude damping rates\nof the two tensorial modes of GWs, which arise from\nseveral specific Lorentz-violating theories of gravity, for\ninstance, the spatial covariant gravities [56, 65] and\nHoˇ rava-Lifshitz gravity [64].\nA. Propagating equation of GWs with\nLorentz-violating damping effects\nLet us investigate the propagation of GWs with\nLorentz-violating damping effect on a flat Friedmann-\nRobertson-Walker spacetime. Treating this spacetime as\na background, GWs can be described by the tensor per-\nturbations of the metric, where the metric is expressed\nin the form of\nds2=a2(τ)h\n−dτ2+ (δij+hij)dxidxji\n, (2.1)\nwhere a(τ) is the scale factor of the expanding Universe\nandτrepresents the conformal time. One can transform\nthe conformal time τto the cosmic time tbydt=a(τ)dτ.\nThroughout this paper, we set the present expansion fac-\ntora0= 1. hijdenote GWs, which are transverse and\ntraceless, i.e.,\n∂ihij= 0 = hi\ni. (2.2)\nFor later convenience, let us expand hijover spatial\nFourier harmonics,\nhij(τ, xi) =X\nA=R,LZd3k\n(2π)3hA(τ, ki)eikixieA\nij(ki),(2.3)\nwhere eA\nijis the circular polarization tensor and obeys\nthe following rules\nϵijknieA\nkl=iρAejA\nl(2.4)\nwith ρR= 1 and ρL=−1.3\nFIG. 1. The noise spectral density of the six detectors considered in this paper. Both ground-based and space-based detectors\nare included in the picture.\nTo study the Lorentz-violating damping effect on the\npropagation of GWs, let us first write the modified prop-\nagation equation of motions of the two GW modes in the\nfollowing parametrized form [71, 90],\nh′′\nA+ (2 + ¯ ν+νA)Hh′\nA+ (1 + ¯ µ+µA)k2hA= 0,\n(2.5)\nwhere a prime denotes the derivative concerning the con-\nformal time τandH=a′/a. The four parameters, ¯ ν,\nνA, ¯µ, and µAlabel the new effects on the propagation of\nGWs arising from theories beyond GR. As mentioned in\nRef. [71], such parametrization provides a general frame-\nwork for exploring possible modified GW propagations\narising from a large number of modified theories of grav-\nity. Different parameters correspond to different effects\non the propagation of GWs. The parameters νAandµA\nrepresent the effects of parity violations, while the pa-\nrameters ¯ νand ¯µ, if frequency-dependent, can originate\nfrom other potential modifications involving Lorentz vi-\nolations. For ¯ νand ¯µ, the former provides an amplitude\nmodulation of the GW waveform, while the latter one ¯ µ\ndetermines the phase velocities of the GWs.\nIn this paper, we will only concentrate on the case of\nthe Lorentz-violating damping effect, and for this case\none has\n¯ν̸= 0, νA= 0,¯µ= 0 = µA. (2.6)\nIn general, due to whether the parameter ¯ νis frequency-\nindependent or not, the effects of ¯ νhave two possibilities.\nWhen ¯ νis frequency-independent and time-dependent, it\ncan be related to a time-dependent Planck mass M∗(t)by writing [91]\nH¯ν=HdlnM2\n∗\nlna. (2.7)\nSee ref. [65] as well for a specific example with an explicit\naction for nonzero ¯ νand its relation to the running of the\nPlanck mass M∗(t). Another possibility corresponds to\na frequency-dependent ¯ ν, which represents the Lorentz-\nviolating damping effect we studied in this paper. For\nthis case, following Ref. [71], one can further parametrize\n¯νin the form of\nH¯ν=\"\nα¯ν(τ)\u0012k\naMLV\u0013β¯ν#′\n, (2.8)\nwhere β¯νis an arbitrary number, α¯νis an arbitrary func-\ntion of time, and MLVdenotes the energy scale of the\nLorentz violation1. The parameters α¯νandβ¯νdepend\non the specific modified theories of gravity. This case\ncan arise from the mixed temporal and spatial deriva-\ntives of the spacetime metric in the modified theories of\ngravity with spatial covariance [56, 64, 65]. In the next\nsubsection, we present a specific example that induces\nthe Lorentz-violating damping effect in the propagation\nof GWs.\n1In ref. [90], a different symbol MPVis used to represent the\nLorentz-violating energy scale. Note that M−2\nLVin the above\nparametrization is also directly related to the coefficient G2/G0\nused for parametrizing the modified GW propagarions in ref. [56].4\nB. A specific example with Lorentz-violating\ndamping effect\nTo illustrate the Lorentz-violating damping effects\nclearly, let us consider a specific example, with the mixed\nterm∇kKij∇kKij, which can appear in both the spa-\ntial covariant gravity [56] and Hoˇ rava-Lifshitz gravity\n[64], where ∇kdenotes the covariant derivative associ-\nated with the spatial metric gijandKijis the extrin-\nsic curvature tensor. It is also shown that by includ-\ning mixed derivative terms, the non-protectable Hoˇ rava-\nLifshitz gravity could be power-counting renormalizable\nand free of ghosts [64]. With this mixed term, one can\nwrite down the general action of the gravitational part\nwith spatial covariance in the form of [65]\nS=M2\nPl\n2Z\ndtd3x√gN(KijKij+R−K2)\n+M2\nPl\n2Z\ndtd3x√gNc 1(∇kKij∇kKij−RijRij),\n(2.9)\nwhere the first term represents the Einstein-Hilbert ac-\ntion of GR in the 3 + 1 form, the second term signifies\none of the modifications to GR, c1is the coupling coef-\nficient which is a function of the lapse function Nand\ntime t, and MPlis the reduced Planck mass. Here we\nwould like to mention that in the second term of the ac-\ntion, we also include RijRijto eliminate the effect of the\nmixed derivative term ∇kKij∇kKijin the disperation\nof GWs, such that the GWs propagate at the speed of\nlight. Note that in writting the above action, we adopt\nthe Arnowitt-Deser-Misner (ADM) form [92]. In eq.(2.9),\nNis the lapse function, ∇kdenotes the covariant deriva-\ntive associated with the spatial metric gijandKijis the\nextrinsic curvature tensor in the ADM form. Then, the\naction of GWs with the c1term up to the quadratic order\ncan be written in the form [56],\nS(2)=M2\nPl\n8Z\ndtd3xa3\u0012\n˙hij˙hij+hij△\na2hij\n−c1˙hij△\na2˙hij+c1hij△2\na2hij\u0013\n.\n(2.10)\nHere△ ≡ δij∂i∂jwith δijbeing the Kronecker delta and\na dot denotes the derivative respect to the cosmic time t.\nVariation of the quadratic action (2.10) with respect\ntohij, one obtains the equation of motion for hijas\n\u0012\n1−c1∂2\na2\u0013\nh′′\nij+\u0014\n2H −c′\n1∂2\na2\u0015\nh′\nij\n−\u0012\n1−c1∂2\na2\u0013\n∂2hij= 0. (2.11)\nThen using the Fourier harmonics (2.3), the above equa-\ntion can be cast into the form of eq. (2.5) as\nh′′\nA+ (2 + ¯ ν)Hh′\nA+k2hA= 0, (2.12)where\nH¯ν=\u0014\nln\u0012\n1 +c1k2\na2\u0013\u0015′\n. (2.13)\nConsidering that the effect from c1term is small, one\napproximately has\nH¯ν≃\u0012\nc1k2\na2\u0013′\n. (2.14)\nThen, one can connect the coupling coefficient c1in the\naction (2.9) to the parameters α¯νandMLVvia\nc1(τ) =α¯ν(τ)\nM2\nLV, (2.15)\nwith β¯ν= 2.\nIn this paper, we consider the case with β¯ν= 2 and\nderive the corresponding modified waveform of GWs. We\nthen explore the potential constraints on this modified\nwaveform using proposed GW detectors such as aLIGO,\nCE, ET, Taiji, TianQin, and LISA. This case is induced\nby∇kKij∇kKijwhich contains two time derivatives and\ntwo spatial derivatives. The cases with β¯ν>2 are also\npossible if one added terms with two time derivatives\nand more than two spatial derivatives in the gravitational\naction. However, these higher spatial derivative terms\nare expected to be suppressed, comparing to the leading-\norder case with β¯ν= 2. Therefore, in this paper, we only\nfocus on the leading order one with β¯ν= 2.\nC. Amplitude modulation of GWs with\nLorentz-violating damping effect\nThe nonzero parameter νprovides a frequency-\ndependent damping of GW amplitudes during propaga-\ntion. This means that GWs with different frequencies will\nexperience different damping rates. This damping rate\neffect induces an amplitude modification in the GWs.\nTo study the modified waveform of GWs with this\nfrequency-dependent damping of GW amplitudes, follow-\ning the derivations in Refs. [90, 93], let us decompose hA\nin Eq. (2.12) as\nhA=hGR\nAe−iθ(τ), hGR\nA=AGR\nAe−iΦGR(τ).(2.16)\nHere hGR\nAis the solution of Eq. (2.12) when ¯ ν= 0.AGR\nA\nand ΦGR(τ) are the amplitude and phase of hGR\nArespec-\ntively. With this decomposition, θ(τ) encodes the cor-\nrection arising from ¯ νwhich characterizes the Lorentz-\nviolating damping effect. We would like to mention that,\nto obtain a waveform model with the propagation effects\ndue to both the Lorentz-violating damping effect, we as-\nsume that the waveform extracted in the binary’s local\nwave zone is well-described by a waveform in GR. The\nsame assumption has also been used in the analysis for\ntesting the propagation effects in [21, 22]. In this way, one5\ncan calculate both the amplitude and phase corrections\ndue to the propagation effects to the GR-based wave-\nform by using the stationary phase approximation (SPA)\nduring the inspiral phase of the binary system [90].\nPlugging the second equation of decomposition Eq.\n(2.16) into Eq. (2.12) with ¯ ν= 0, one finds\niΦ′′+ Φ′2+ 2iHΦ′−k2= 0. (2.17)\nSimilarly, plugging the first equation of the decomposi-\ntion Eq. (2.16) into Eq. (2.12), one gets\ni(θ′′+ Φ′′) + (Φ′+θ′)2+i(2 + ¯ν)H(θ′+ Φ′)−k2= 0.\n(2.18)\nIn GR, the time derivative of the phase Φ′∼k. Here the\nwavenumber kis connected to the frequency of GWs by\nk= 2πf/a 0. Since the amplitude correction θis induced\nby the expansion of the universe, one has θ′∼ H and\nθ′′∼ H2. Considering k≫ H andθ′′≪Φ′θ′∼kθ′, Eq.\n(2.18) can be simplified into\n2θ′+iH¯ν≃0. (2.19)\nSolving this equation gives\nθ=−i\n2Zτ0\nτeH¯νdτ. (2.20)\nThe Lorentz-violating damping effect in the phase θis\npurely imaginary, indicating that it modifies the ampli-\ntude of the GWs during their propagation. Considering ¯ ν\nis also frequency-dependent, such amplitude modulation\ndepends on the frequency of GWs as well.\nSpecifically, with the above solution, one can write the\nwaveform of GWs with Lorentz-violating effect as\nhA=hGR\nAeδh2, (2.21)\nwhere\nδh2=−1\n2Zτ0\nτeH¯νdτ\n=−1\n2\"\nα¯ν\u0012k\naMLV\u0013β¯ν#\f\f\f\f\fa0\nae. (2.22)It can be further rewritten in the form\nδh2=−1\n2\u00122u\nMLVM\u0013β¯νh\nα¯ν(τ0)−α¯ν(τe)(1 + z)β¯νi\n.\n(2.23)\nHere we define u=πMf, where M = (1 +\nz)Mcrepresents the measured chirp mass, and Mc≡\n(m1m2)3/5/(m1+m2)1/5denotes the chirp mass of the\nbinary system with component masses m1andm2.\nD. Amplitude modification to the waveform of\nGWs\nTo derive the modified waveform of GWs, we consider\nthe GWs produced during the inspiral stage of the com-\npact binaries. To directly contact with the observations,\nit is convenient to analyze the GWs in the Fourier do-\nmain. In this approach, under the stationary phase ap-\nproximation, the responses of the detectors to the GW\nsignal ˜h(f) can be written in the form of\n˜h(f) =h\nF+h+(f) +F×h×(f)i\ne−2πif∆t,(2.24)\nwhere F+andF×denote the beam pattern functions of\nGW detectors, which depend on the GW source’s loca-\ntion and polarization angle [94, 95]. The two polariza-\ntions of GWs, h+(f) and h×(f), are related to the left-\nand right-handed polarization modes, hL(f) and hR(f),\nvia\nh+=hL+hR√\n2, h×=hL−hR√\n2i. (2.25)\nThen using Eq. (2.21), after tedious calculations, one\nobtains the following restricted form for the waveform\nof GWs in the Fourier domain as a function of the GW\nfrequency f, i.e.,\n˜h(f) =AGRf−7/6eiΨGR(f)eδh2, (2.26)\nwhere AGRand Ψ GRrepresent the amplitude and phase\nof GWs of a compact binary inspiral signal in GR, and\neδh2with δh2being given by Eq. (2.23) is the amplitude\ncorrection to the waveform of GWs in GR. In the post-\nNewtonian approximation, the amplitude and the phase\nof GWs in GR can be expressed as [96]\nAGR=2\n5×r\n5\n24π−2/3M5/6\nDL, (2.27)\nand6\nΨGR(f) = 2 πftc−Φc−π\n4+3\n128ηu−5/3\u001a\n1 +\u00123715\n756+55\n9η\u0013\nu2/3−16πu\n+\u001215293365\n508032+27145\n504η+3085\n72η2\u0013\nu4/3+π\u001238645\n756−65\n9η\u0013\n(1 + ln u)u5/3\n+\u001411583231236531\n4694215680−640\n3π2−6848\n21γE−6848\n63ln(64 u) +\u0012\n−15737765635\n3048192+2255\n12π2\u0013\nη\n+76055\n1728η2−127825\n1296η3\u0015\nu2+π\u001277096675\n254016+378515\n1512η−74045\n756η2\u0013\nu7/3\u001b\n, (2.28)\nTABLE I. Characteristics of six GW detectors.\nDetector Configuration flower [Hz]fupper [Hz] Reference\naLIGO Rightangle 10 5000 Refs. [96, 97]\nET Rightangle 1 10000 Ref. [98]\nCE Rightangle 5 4000 Ref. [99]\nTaiji Triangle 0.0001 0.1 Ref. [100]\nTianQin Triangle 0.0001 1 Ref. [101]\nLISA Triangle 0.0001 0.1 Refs. [100, 102]\nwhere the luminosity distance DLis expressed as\nDL(z) =1 +z\nH0Zz\n0dz′\np\nΩM(1 +z′)3+ Ω Λ.(2.29)\nHere we adopt the ΛCDM model with Hubble constant\nH0≈67.4 km s−1Mpc−1, matter density fractoin Ω m≈\n0.315, and vacuum energy density fraction Ω Λ≈0.685\n[103, 104]. tcand Φ care time and phase at coalescence,\nγEis the Euler constant, and η=m1m2/(m1+m2)2is\nthe symmetric mass ratio.\nIII. ANALYSIS FRAMEWORK WITH FISHER\nINFORMATION MATRIX\nA. General considerations\nIn this section, we provide a brief overview of the\nmatched-filter analysis with the FIM approach, which\nfollows the method outlined for compact binary inspiral\nin Refs. [105–107]. We calculate the noise-weighted in-\nner product between the partial derivatives of each GW\nwaveform parameter and the one-sided power spectral\ndensity (PSD) of the detector noises. This calculation\nyields the FIM. Inverting the FIM provides the variance-\ncovariance matrix, where the diagonal elements represent\nthe square root of the mean squared error for the esti-\nmated parameters of the signal. Previous studies haveshowcased the precision and utility of the FIM approach,\nparticularly in situations with high signal-to-noise ratios\n(SNRs).\nTo be specific, we first give the noise-weighted inner\nproduct of two signals h1andh2\n(h1|h2) = 2Zfmax\nfmin˜h∗\n1(f)˜h2(f) +˜h∗\n2(f)˜h1(f)\nSn(f)d f, (3.1)\nwhere ˜h1(f) and ˜h2(f) are the Fourier transformation of\nGW signal h(t),Sn(f) is the PSD of the detector’s noise,\nand the star superscript stands for complex conjugation.\nIn the above expression, fmax(fmin) represents the in-\nstrumental maximum (minimum) threshold frequency.\nFor a given signal h, the SNR is defined as\nρ= (h|h)1/2. (3.2)\nThe modified waveform of GW from binary inspirals, in-\nfluenced by the Lorentz-violating damping effect as de-\nscribed in Eq. (2.26), are generally characterized by a\nset of parameters θi. In this context, one can define the\nFIM as\nFij= \n∂˜h\n∂θi\f\f\f\f\f∂˜h\n∂θj!\n. (3.3)\nHere θiand θjrepresent the elements in the\nset of modified waveform parameters of GWs\n{lnA,lnM,lnη, ϕc, tc, Cν}, where Cν=M−2\nLVchar-\nacterizes the Lorentz-violating damping effect in the\nwaveform. Then one can calculate each element of the\nFIM, which are given respectively in Appendix A.\nIn the large SNR approximation, if the noise is station-\nary and Gaussian, the probability that the GW signal\nh(t) can be characterized by a given set of values of the\nparameters θi,\np(θi|h) =p(0)(θi) exp\u0014\n−1\n2Fij∆θi∆θj\u0015\n, (3.4)\nwhere p(0)(θi) represents the distribution of prior infor-\nmation. Then, the standard deviations ∆ θiin measuring\nthe parameter θi, which mean 1 σbounds on parameters,\ncan be calculated in the large SNR approximation. This\ncan be obtained by taking the square root of the corre-\nsponding diagonal elements in the inverse of FIM,\n∆θi=p\n(F−1)ii, (3.5)7\nwhere F−1is the inverse of FIM.\nOur main purpose in employing FIM analysis here is\nto gauge the potential of future GW detectors in con-\nstraining the energy scale MLVassociated with Lorentz\nviolation, which induces the Lorentz-violating damping\neffect in the propagation of GWs. We consider two types\nof GW detectors, ground-based GW detectors including\naLIGO, CE, and ET, and space-based GW detectors, in-\ncluding Taiji, TianQin, and LISA. We summarize the\ninformation of all the six GW detectors in Table I.\nTo investigate the variation tendency of MLVin GW\nevents, we choose the reasonable astrophysical horizon\nof each GW detector as the sources’ property and con-\nstrain MLVby FIM. To maintain the effectiveness of FIM,\nthe ranges of redshift zand total mass Mare suitable\nenough to satisfy the SNR threshold ( ρ >8 for ground-\nbased detectors and ρ >15 for space-based detectors).\nFor ground-based GW detectors, their detectable fre-\nquency bands are several Hz to thousand Hz. As the\nthird generation GW detector, the frequency band of CE\ncan reach [5 −4000] Hz. For space-based detectors, with\nhuge arm-lengths so that they can detect the GWs from\ncompact binary systems in a very low-frequency band\n[10−4−10−1] Hz. The compact binary systems in this\nlow-frequency band usually consist of supermassive black\nholes and their total mass range is [104−106] M⊙.\nWhen calculating the integrals of the inner product, it\nis necessary to use the appropriate limits of integration.\nThe minimum frequency fminin Eq. (3.1) is taken as the\ninstrumental minimum threshold frequency of the GW\ndetector as shown in Table I. The upper cut-off frequency\nfmaxis chosen from min {fupper, fISCO}, where fupper is\nthe upper frequency of detectors and fISCO is usually\nestimated by the innermost stable circular orbit (ISCO)\n[68]\nfISCO = 6−3/2π−1η3/5M−1. (3.6)\nWhen calculating the FIM, we set Cν= 0 as the fiducial\nvalue. Additionally, the values of tcandϕcdo not impact\nthe constraints on other parameters; hence, we set tc= 0\nandϕc= 0.\nB. Noise power spectral density of detector\nIn Fig. 1, we illustrate the noise PSD of both ground-\nbased and space-based detectors. The three ground-\nbased detectors’ PSD are from the official data files\nquoted in Table I. The other three PSD data are from\nthe theoretical formula introduced below.\nThe sensitivity curve of Taiji can be obtained by the\nformula [100]\nSn(f) =10\n3L2\u0012\nPdp+ 2(1 + cos2(f/f∗))Pacc\n(2πf)4\u0013\n×\u0012\n1 + 0 .6(f\nf∗)2\u0013\n, (3.7)where\nPdp=\u0000\n8×10−12m\u00012 \n1 +\u00122 mHz\nf\u00134!\nHz−1,(3.8)\nPacc=\u0000\n3×10−15m s−2\u00012 \n1 +\u00120.4 mHz\nf\u00132!\n× \n1 +\u0012f\n8 mHz\u00134!\nHz−1. (3.9)\nHere Pdpis the PSD of the displacement noise and Paccis\nthe PSD of the acceleration noise. f∗= 1/(2πL) and Lis\nthe arm-length of the detector. For Taiji L= 3×109m.\nFor LISA, L= 2.5×109m and the displacement noise\ncan be written as [100, 108]\nPdpL=\u0000\n15×10−12m\u00012 \n1 +\u00122 mHz\nf\u00134!\nHz−1.\n(3.10)\nThe sensitivity curve for TianQin can be modeled by\nthe following equation [101]\nSn(f) =10\n3L2\u0014\nSx+4Sa\n(2πf)4\u0012\n1 +10−4Hz\nf\u0013\u0015\n×\"\n1 + 0 .6\u0012f\nf∗\u00132#\n, (3.11)\nwhere displacement measurement noise S1/2\nxand residual\nacceleration noise S1/2\naare defined as\nS1/2\nx= 1×10−12m/Hz1/2, (3.12)\nand\nS1/2\na= 1×10−15m s−2/Hz1/2. (3.13)\nf∗=c/(2πL) is the transfer frequency and L=√\n3×\n108m.\nIV. RESULTS AND DISCUSSION\nIn this section, we present the results of the poten-\ntial constraints on the parameters of the modified wave-\nform of GWs with the Lorentz-violating damping ef-\nfect. Among these parameters, we focus on the energy\nscale MLVof Lorentz violation which characterizes the\nfrequency-dependent damping of GWs during their prop-\nagations. To conduct a comprehensive and reliable anal-\nysis of GW parameter estimation, we consider the fre-\nquency bands of both ground-based and space-based de-\ntectors. We employ simulated GW data to constrain the\nLorentz-violating damping effect through the FIM ap-\nproach. For the simulated GW data, we refer to the\nastrophysical horizons of the detectors from Ref. [7] (for8\nTABLE II. The best constraints of the simulated single GW events and the combination of joint events from each GW detector.\nThe results from the single event are the best constraint which are from Fig. 3. The redshift ranges and the total mass ranges\nare only for the joint events. The number of joint events contains 100 (for both ground-based and space-based detectors)\nsimulated GW events. We choose ρ >8 and ρ >15 as the threshold for ground-based detectors and space-based detectors\nrespectively.\nDetector Single event [Gev] Joint redshift range Joint total mass range [ M⊙] Joint number Joint event [Gev]\naLIGO 8 .54×10−22[0.01-0.5] [3-10] 100 2 .39×10−21\nCE 3 .02×10−21[0.01-0.5] [3-10] 100 8 .84×10−21\nET 2 .39×10−21[0.01-0.5] [3-10] 100 7 .18×10−21\nTaiji 2 .10×10−24[5-10] [4 .2−9.2]×103100 5 .86×10−24\nTianQin 1 .89×10−24[0.01-5] [1 .5−2]×104100 4 .56×10−24\nLISA 1 .52×10−24[5-10] [8 .3−13.3]×103100 4 .37×10−24\nFIG. 2. The results of MLVand SNR of GW150914-like and GW170817-like events in three ground-based detectors. GW170817-\nlike event is the compact binary system of two neutron stars and GW150914-like event is a system containing two black holes.\naLIGO, ET), Refs. [7, 109] (for CE), Ref. [100] (for\nTaiji), Ref. [101] (for TianQin), and Ref. [102] (for\nLISA). We set ρ > 8 and ρ > 15 as the thresholds\nfor ground-based detectors and space-based detectors, re-\nspectively. To fully exploit the high event rates of future\nGW detectors, we also decide to conduct a multi-event\njoint constraint analysis. The results of constraining both\nindividual GW events and their combinations are de-\ntailed in Table II. The constraint results and SNR from\nGW150914-like and GW170817-like events are shown in\nFig. 2. We depict the dependence of the lower bound of\nMLVin Fig. 3.\nA. MLVfrom ground-based detectors\nFirst, let us analyze the constraints on MLVfrom the\nthree ground-based detectors. In Fig. 2, we illustrate the\nconstraints on MLV(right panel) and SNRs (left panel)\nof two examples of GW events, the GW150914-like andGW170817-like events with three different ground-based\ndetectors. As observed in the left panel of Fig. 2, the\nlower bounds on MLVfrom both CE and ET extend to\n≳10−21GeV. CE also gives the best constraints on MLV\nfor both GW150914-like and GW170817-like events. No-\ntably, under similar redshift conditions, the GW170817-\nlike event, characterized by a smaller total mass, attains\na higher lower bound of MLV. Moving to the right panel\nof Fig. 2, CE exhibits a distinct advantage in SNR, sug-\ngesting its effectiveness in ensuring the detection of such\nGW events. There appears to be an indicative trend im-\nplying that GW events with a greater total mass could\npotentially yield higher SNRs.\nIn Table II, it is evident that ground-based detectors,\ncapable of observing high-frequency GWs, exhibit supe-\nrior performance compared to their space-based coun-\nterparts. Among ground-based detectors, the strongest\nconstraint for MLVis achieved by the third-generation\ndetector CE at 3 .02×10−21GeV. The result from the\nET is marginally smaller than that of CE. For aLIGO,9\nFIG. 3. Dependence of the lower bound of MLVon the total mass and redshift of the binary inspiral systems for different\ndetectors, aLIGO (top left), CE (top middle), ET (top and right), Taiji (bottom left), TianQin (bottom middle), and LISA\n(bottom right).\nthe constraint can reach 8 .54×10−22Gev.\nConsidering the prospect of observing a large number\nof GW events in the future, we also do research on the\njoint analysis for individual detectors. As anticipated,\nthe outcomes from each detector exhibit a marked en-\nhancement. The most notable improvement comes from\nCE, yielding the optimal result with MLV>8.84×10−21\nGeV. We combine 100 simulated GW events within the\nredshift range [0.01-0.5] Hz and total mass range [3-\n10]M⊙of sources. aLIGO attains a result of MLV>\n2.39×10−21GeV. This value is approximately twice as\nhigh as the result reported in Ref. [71], which means that\naLIGO continues to be a powerful tool in testing the\nLorentz-violating damping effect. It is crucial to high-\nlight that the arm-length of ET is 10 km. Meanwhile,\nCE is designed with arm-lengths of 20 km and 40 km.\nDue to their long arm-lengths and enhanced sensitivity,\nCE and ET can provide values of MLVthat are roughly\nmore than double the value obtained by aLIGO. The ex-\npected detection rate, as reported in Refs. [110, 111], is\nconservatively estimated at 100.\nB. MLVfrom space-based detectors\nFor the space-based detectors, including Taiji, Tian-\nQin, and LISA, the constraints on MLVare about three\norders of magnitude weaker than those from the ground-\nbased detectors. As shown in Table II, the constraints\nonMLVis roughly at ≳2×10−24GeV for single GW\nevent. When considering a joint analysis of 100 simulated\nGW events, the constraints from all three detectors areroughly at ≳5×10−24GeV. Among the three detectors,\nTaiji achieves the best result, with MLV>5.86×10−24\nGeV. The reason why the constraints from the space-\nbased detectors are weaker than those from the ground-\nbased detectors is easy to understand. As one can see\nfrom Eq. (2.23), the amplitude correction to the wave-\nform due to the Lorentz-violating damping effect is pro-\nportional to the square of the GW frequencies, which\nimplies this effect is more sensitive to the higher GW fre-\nquencies, and thus the ground-based detectors can give\nstronger constraints than space-based detectors. We note\nthat to estimate the number of events within our joint\nredshift and total mass range, we employ the data and\nmethods from Refs. [112, 113].\nC. Trends of MLV\nIn Fig. 3, we illustrate how the lower bound of the\nLorentz-violating parameter, MLV, varies with the to-\ntal mass and redshift of binary inspiral systems across\na selection of detectors. Specifically, for aLIGO, CE,\nand ET, we focus on a redshift range of [0 .01,0.5] Hz\nand a total mass range of [3 ,50]M⊙. For Taiji, Tian-\nQin, and LISA, we extend the redshift range to [0 .01,10]\nHz. The rationale behind selecting different total mass\nranges for these latter three detectors is linked to ensur-\ning that the SNR, ρ, exceeds 15 when the redshift value\nis maximized. This approach is designed to sharpen the\nvisibility of the MLVtrend, highlighting the impact of\nhigh redshift values. As illustrated in Fig. 3, the high-\nest values of the Lorentz violation energy scale, MLV,10\nare achieved for sources that are both nearest and of the\nlowest mass, as observed in the cases of aLIGO, CE, ET,\nand TianQin. Conversely, for Taiji and LISA, the trend\ndeviates. Here, MLVdoes not attain its maximum in\nregions characterized by lower total masses and smaller\nredshifts. This discrepancy arises because the innermost\nstable circular orbit frequency, fISCO, in this domain,\nsurpasses the upper-frequency limit of these detectors.\nThis is indicated by Eq. (3.6) and illustrated in Fig. 4.\nConsequently, this suggests that Taiji and LISA might be\nless efficient in detecting the final inspiral phase in com-\npact binary systems that have lower redshifts and smaller\ntotal masses.\nThe choice to initiate the redshift ( z) analysis from 0.01\nstems from the observation that the lowest redshift value\nrecorded in the LVK event catalog is 0.01. Considering\nthat the mass of a neutron star is typically around 1.5\nM⊙, we adopt this figure as the minimum mass thresh-\nold for our total mass range. We cap the redshift at\nz= 0.5 and set the maximum total mass at 50 M⊙,\nwhich corresponds to the median GW source mass re-\nported in the LVK events. To streamline our analysis,\nwe concentrate on binary systems comprising either two\nblack holes (BBH) or two neutron stars (BNS), assuming\nequal mass for both components.\nV. CONCLUSION\nWith the advent of future detectors, GWs are poised\nto play a pivotal role in testing gravity in the strong field\nregime. Both ground-based and space-based detectors\nare designed to capture GWs across different frequency\nbands, spanning from 10−4to 104Hz. In this study,\nwe delve into the investigation of the Lorentz-violating\ndamping effect, which influences the propagation of GWs.\nWe aim to evaluate the capability of both ground-based\n(aLIGO, CE, and ET) and space-based (Taiji, TianQin,\nand LISA) detectors in constraining this effect. We be-\ngin by formulating the modified equations of motion for\nthe two polarizations of GWs. We then proceed to de-\nrive the altered GW waveform in the Fourier domain,\nincorporating the Lorentz-violating damping effect. Uti-\nlizing the FIM, we set out to quantify the constraints on\nthe energy scale MLV, showcasing its projected sensitiv-\nity for each detector. For the FIM analysis, we estab-\nlish detection thresholds for ground-based detectors and\nspace-based detectors at ρ >8 and ρ >15, respectively.\nAdditionally, we conduct a joint analysis of MLVusing\nsimulated GW events to further our understanding of the\nconstraints achievable with future GW observations.\nFor ground-based detectors, the tightest constraint\nfrom a single event is set by CE, with MLV>3.02×10−21\nGeV. When conducting a joint analysis of 100 GW\nevents, this constraint improves to MLV>8.84×10−21\nGeV. Regarding space-based detectors, the results are\nin line with our expectations. For a single event, the\nconstraints on MLVfrom these detectors are approxi-\nFIG. 4. The distribution of fISCO for Taiji and LISA is de-\npicted within the same astrophysical horizon, as illustrated in\nFig. 3.\nmately ≳2×10−24GeV, which is roughly three or-\nders of magnitude less stringent than those obtained from\nground-based detectors. Upon performing a joint anal-\nysis of 100 simulated GW events, the constraints from\nthe space-based detectors converge to approximately ≳\n5×10−24GeV.\nIn our conservative estimation of event numbers, we\nnote an improvement of more than twofold over the\nconstraints obtained from individual events. We posit\nthat, with the ongoing accumulation of observational\ndata, even more stringent constraints on this effect will\nbe achievable. Our analysis indicates that targeting\nthe high-frequency band offers a more efficacious ap-\nproach for constraining the Lorentz-violating damping\neffect. This suggests that ground-based detectors are\nmore adept at imposing rigorous constraints on the ef-\nfect in comparison to space-based detectors. Within this\ngroup of detectors, CE distinguishes itself by offering the\nmost stringent lower bound on MLV.\nOur analysis of the data distribution reveals that an ef-11\nfective strategy for aLIGO, CE, ET, and TianQin to con-\nstrain the Lorentz-violating damping effect involves con-\ncentrating on compact binary systems characterized by\nboth a small total mass and low redshift. Conversely, for\nTaiji and LISA, targeting GW sources that have a small\ntotal mass but are situated at higher redshifts proves to\nbe more appropriate. This strategic focus is informed by\nthe differential sensitivity of these detectors to the fre-\nquency and amplitude of GW signals, which in turn af-\nfects their capability to place constraints on the Lorentz-\nviolating effect.\nACKNOWLEDGMENTS\nWe thank Hong-Chao Zhang, Chao Zhang, and\nPeng-Ju Wu for useful discussions. Tao Zhu and\nBo-Yang Zhang are supported by the National Key\nResearch and Development Program of China underGrant No.2020YFC2201503, the National Natural Sci-\nence Foundation of China under Grants No.12275238 and\nNo. 11675143, the Zhejiang Provincial Natural Science\nFoundation of China under Grants No.LR21A050001\nand No. LY20A050002, and the Fundamental Research\nFunds for the Provincial Universities of Zhejiang in China\nunder Grant No. RF-A2019015. Jing-Fei Zhang and Xin\nZhang are supported by the National Natural Science\nFoundation of China (Grants Nos. 11975072, 11875102,\nand 11835009), the National SKA Program of China\n(Grants Nos. 2022SKA0110200 and 2022SKA0110203),\nand the National 111 Project (Grant No. B16009).\nAPPENDIX A: PARTIAL DERIVATIVES OF THE\nWAVEFORM OF BINARY INSPIRAL\nIn this appendix, we present the partial\nderivatives of the GW waveform parameters\n{lnA,lnM,lnη, ϕc, tc, Cν}as follows,\n∂˜h(f)\n∂lnA=˜h(f), (A.1)\n∂˜h(f)\n∂lnM=5\n6˜h(f), (A.2)\n∂˜h(f)\n∂lnη=1\n2+iη(\n−3\n128η−2u−5/3CΨ+3\n128η−1u−5/3\"\n55\n9u2/3+\u001227145\n504+3085\n36η\u0013\nu4/3−65π\n9(1 + ln u) u5/3\n+\u0012\n−15737765635\n3048192+2255\n12π2+76055\n864η−127825\n432η2\u0013\nu2+π\u0012378515\n1512−74045\n378η\u0013\nu7/3#)\n, (A.3)\n∂˜h(f)\n∂ϕc=−i˜h(f), (A.4)\n∂˜h(f)\n∂tc= (2 πif)˜h(f), (A.5)\n∂˜h(f)\n∂Cν=h1\n2(2πf)2((1 + z)2−1)i\n˜h(f), (A.6)\nwhere CΨin Eq. 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D 93,\n024003 (2016) [arXiv:1511.05581 [gr-qc]]."    },    {        "title": "1211.7129v1.Tests_of_Lorentz_and_CPT_violation_with_neutrinos.pdf",        "content": "arXiv:1211.7129v1  [hep-ph]  30 Nov 2012Testsof LorentzandCPTviolationwithneutrinos\nTeppeiKatori∗\nLaboratoryforNuclearScience,\nMassachusettsInstituteofTechnology,\nCambridge,USA\nE-mail:katori@mit.edu\nLorentz violation is a predicted phenomenonfrom the Planck scale physics. Although the three\nactive massive neutrino framework with the Standard Model ( SM), so-called the neutrino Stan-\ndard Model ( nSM), is successful, series of signals not understood within thenSM suggest neu-\ntrino physicsmay be the first place to see the physicsbeyondt he SM, such as Lorentz violation.\nEspecially,neutrinooscillationsarethenaturalinterfe rometerandtheyaresensitivetotheLorentz\nviolationwithcomparablesensitivity withpreciseoptica lexperiments.\nThe LSND oscillation signal was analyzed under the Standard Model Extension (SME) frame-\nwork,anditwasfoundthattheoscillationdatawasconsiste ntwithnoLorentzviolation,butdata\ncannot reject Lorentz violation hypothesis with order ∼10−17. By assuming LSND signal was\ndue to the Lorentz violation, a global phenomenologicalmod el was made to describe all known\noscillation data includingthe LSND signal. The model also p redictedthe signal for MiniBooNE\nat thelowenergyregion.\nLater,MiniBooNEannouncedaneventexcessat thelow energy region. However,the oscillation\ncandidatesignalsfromMiniBooNEwereconsistentwithnoLo rentzviolation. Thelimitobtained\nbyMiniBooNEandMINOSonthe e−msectorrejectthesimplescenariotoexplainLSNDsignal\nwithLorentzviolation.\nMeantime, MINOS and IceCube set tight limits on the m−tsector Lorentz violation. The last\nuntestedchannel,the e−tLorentzviolatingmixing,wastestedusingreactordisappe arancedata\nfrom Double Chooz. However, Double Chooz data was consisten t with flat, and sidereal time\ndependent Lorentz violation hypothesis is rejected. Combi nations of all oscillation data from\nLSND, MiniBooNE, MINOS, IceCube, and Double Chooz provide v ery tight constraint for a\npossibleLorentzviolationintheneutrinosectorinterres triallevel.\n36thInternationalConferenceonHighEnergyPhysics,\nJuly 4-11,2012\nMelbourne,Australia\n∗Onbehalf of LSND,MiniBooNE, andDouble Chooz collaboratio ns\nc/circlecopyrtCopyrightownedbytheauthor(s)underthetermsoftheCreat ive CommonsAttribution-NonCommercial-ShareAlikeLicen ce. http://pos.sissa.it/Tests of LorentzandCPT violationwith neutrinos\nFigure1: Anillustrationofspontaneoussymmetrybreaking(SSB).Fi gureistakenfrom [1].\n1. Introduction\n1.1 SpontaneousLorentz Symmetry Breaking(SLSB)\nLorentz violation is a predicted phenomenon from the Planck scale physics. Especially if it\nwere made by the spontaneous process, quantum field theory an d general relativity would require\nno modifications. Figure 1 shows a cartoon of this situation. When the universe is hot, the scalar\nfield preserves the perfect symmetry (Fig. 1a). But once it ge ts cold, there is a chance that the\npotential of the scalar field shifts to the \"Mexican hat\" pote ntial (Fig. 1b), and nonzero field value\nis more stable, i.e.the vacuum acquires the vacuum expectation value of this sca lar field. If the\nscalar fieldhasanyquantum numbers, saySU(2)charge, suchq uantum numbers arenot preserved\nin the vacuum, namely chirality is not conserved for massive particles in the SM vacuum. This is\nthe spontaneous symmetry breaking (SSB)intheSM,andthis y ear isthe great year for this mech-\nanism since the strong candidate of Higgs particle isdiscov ered, and presented at this ICHEP2012\nconference!\nThis process can be extended to any field with Lorentz indices beyond the scalar field. For\nsimplicity I discuss only the vector field. When the universe is hot, vector field keeps the perfect\nsymmetry (Fig. 1c). But again, when universe gets cold, the v ector field could generate nonzero\nvacuum expectation values (Fig 1d). This is the situation of the spontaneous Lorentz symmetry\nviolation (SLSB)[2], and Lorentz symmetry is spontaneousl y broken. In this case, the universe is\nfilled with the background vector fields represented by the ar rows. If the SM particles couple with\nthose fields, their physics depends onthe orientation of the arrows or direction.\nSince such background fieldsof theuniverse are fixedin thesp ace, presence of such coupling\nimplies the direction dependent physics. In particular, ro tation of the Earth (period 86164 .1 sec)\ncauses sidereal time dependent physics for terrestrial mea surements. Therefore, sidereal time de-\n2Tests of LorentzandCPT violationwith neutrinos\nFigure 2: An illustration of the Particle Lorentz transformation and the Observer Lorentz transformation.\nFigureistakenfrom[1].\npendence of physics observables isthe smoking gun of the Lor entz violation.\n1.2 Particle andObserver Lorentz transformation\nLorentz violation is more precisely the violation of the Par ticle Lorentz transformation. This\nsituation is described in Figure 2. In Fig. 2a, aSM particle i s moving upward in two-dimensional\nspace wherethehypothetical background vector fieldsatura tes thespace (depicted byarrows), and\nEinstein represents a local observer.\nThechangeinmotionofaSMparticlewithinafixedcoordinate systemisdescribedbyParticle\nLorentz transformation (Fig. 2b). Since the background fiel d is unchanged, as a consequence, a\ncoupling between the SM particle and vector field is not prese rved. In general, Lorentz violation\nmeans Particle Lorentz violation.\nOn the other hand, local observer’s inverse coordinate chan ge can also generate change in\nthe motion of a SM particle (Fig. 2c). In the theory without Lo rentz violation, this Observer\nLorentz transformation coincides with the Particle Lorent z transformation. However, as you see,\nthis corresponds to mere coordinate transformation and the coupling of the SM particle and the\nbackground field is preserved. Therefore, Lorentz-violati ng effect is conserved by the coordinate\ntransformation, and it can be studied in anyframe or coordin ate.\n1.3 Test of Lorentz violation withneutrinooscillations\nLorentzviolation isrealizedasacoupling ofSMparticles a ndthebackground fieldoftheuni-\nverse. AlthoughtheLorentz-violating phenomenoniscoord inateindependent, weneedtochoosea\ncoordinatesystemsothatwecanreportmeasurementsofthec oefficientsofsuchfields[3]. Figure3\nshowsourscheme. First,themotionoftheEarthisdescribed intheSun-centeredcoordinatesystem\n(Fig. 3a). This coordinate system provides the bases for the Lorentz violating fields to specify the\n3Tests of LorentzandCPT violationwith neutrinos\nFigure3: Thecoordinatesystemusedbythisanalysis,theSun-center edcoordinates(a),theEarth-centered\ncoordinates (b), and the local polar coordinate system (c). The time zero is defined when the experiment\nsite isatmidnightneartheautumnalequinox,inotherwords ,whenthelarge“Y”andsmall“y”axesalmost\nalign(d). Figureistakenfrom[5].\ncoefficients. Thenthelocationoftheexperimental siteiss pecifiedintheEarth-centered coordinate\nsystem (Fig. 3b). Finally, the direction of SM particles ( i.e., direction of the beam) is described in\nthe local polar coordinate system (Fig. 3c).\nTheStandardModelExtension(SME)[4]isconstructedasage neralframeworktoanalyzethe\ndata to search possible Lorentz violation. In the SME,Loren tz violating interactions are described\nby theperturbative termsintheLagrangian, ontop of theSMt erms. Weareparticularly interested\nin the test of Lorentz violation using neutrino oscillation s. Since neutrino oscillation is a natural\ninterferometer, small couplings of neutrinos with Lorentz violating fields could cause phase shifts\nandcouldresultinneutrinooscillations. Thesensitivity ofneutrinooscillationstoLorentzviolation\nis comparable with precise optical experiments. To analyze the neutrino data we use the neutrino\nsector SME[6]1,\nL=1\n2i¯yAGm\nAB↔\nDmyB−¯yAMAByB+h.c., (1.1)\nGn\nAB≡gndAB+cmn\nABgm+dmn\nABg5gm+en\nAB+ifn\nABg5+1\n2glmn\nABslm, (1.2)\n1Inthis article,we limitoutselves withinthe renormalizab le SME only. However, the results inthis articlecan also\nset the limitson the nonrenormalizable SME coefficients [7] .\n4Tests of LorentzandCPT violationwith neutrinos\nMAB≡mAB+im5ABg5+am\nABgm+bm\nAB+1\n2Hmn\nABsmn. (1.3)\nHere, the ABsubscripts represent flavor space. The first term of Eq. 1.2 an d the first and the\nsecond terms of Eq. 1.3 are the only nonzero terms in the SM. Th e rest of the terms are from the\nSME. These SME coefficients can be classified into two groups: em\nAB,fm\nAB,gmnl\nAB,am\nAB, andbm\nABare\nCPT-oddSMEcoefficients, and cmn\nAB,dmn\nAB,andHmn\nABareCPT-evenSMEcoefficients.\nInthisway, physical observables can bewrittendowninclud ing Lorentzviolation. Byassum-\ningthebaselineisshortenoughfortheoscillationlengtht heneutrinooscillationprobability a→b\ncan be written as follows [8]2,\nPa→b≃L2\n(¯hc)2|(C)ab+(As)absinw⊕T⊕+(Ac)abcosw⊕T⊕\n+(Bs)absin2w⊕T⊕+(Bc)abcos2w⊕T⊕|2. (1.4)\nHere,w⊕isthe sidereal time angular frequency ( w⊕=2p\n86164.1rad/s). Theneutrino oscillation\nprobability is described by the function of the sidereal tim eT⊕with five amplitudes. (C)abis\nthe sidereal time independent amplitude, and (As)ab,(Ac)ab,(Bs)ab, and(Bc)abare the side-\nreal time dependent amplitudes. Therefore, an analysis of L orentz and CPT violation in neutrino\noscillation data involves fitting the data with Eq. 1.4 to find nonzero sidereal time dependent am-\nplitudes. These amplitudes are written by a combination of t he SME coefficients, and the explicit\nexpressions are given at elsewhere [8].\n2. Lorentz violationanalysisonLSND experiment\nLSND is an appearance neutrino oscillation experiment at Lo s Alamos. A low energy ¯nm\nbeam (∼40 MeV) was made by pion decay-at-rest. The detector was loca ted∼30 m away from\nthetarget. LSNDobserved theexcessof ¯necandidate eventsfromthe ¯nmbeam[10],andthisresult\nis not understood within the nSM. Thus, LSND signal may be the signal of new physics, such as\nsterile neutrino oscillations. However, it may be the first s ignal of the Lorentz violation. If this is\nthe case, interference pattern, i.e., the number of the oscillation candidates, would depend on t he\nsidereal time.\nWeanalyzed the LSNDoscillation candidate data with afunct ion of the sidereal time [5]. We\nfitEq.1.4tofindthebest parameter set byanunbinned likelih ood method. Fig.4showstheresult.\nThedataisconsistentwithaflathypothesis(=nosiderealti medependence), howeversmallLorentz\nviolation (=sidereal time dependent solution) is not rejec ted. If that is the case, LSND oscillation\ncandidate is explained by order 10−17CPT-odd Lorentz violation and/or order 10−19GeV CPT-\neven Lorentz violation [11].\n3. Globalneutrino oscillationmodel withLorentz violatio n\nAlthough LSND signal could be described by the Lorentz viola tion, naively such new pa-\nrameters would be forbidden by other oscillation experimen t data. We examined the possible phe-\n2Ifthisisnotthecase,Lorentzviolationcanbestudiedaspe rturbationsofstandardoscillations[9]. Undermentioned\nMINOS far detector and IceCube analyses are based onthis sch eme.\n5Tests of LorentzandCPT violationwith neutrinos\nsidereal time (secs)beam-on events\n0510\n0 20000 40000 60000 80000\nFigure 4: Sidereal time distribution of the LSND oscillation candida te events (marker). Data is fit with\noscillation models including sidereal time dependence (do tted and dot-dashed curves) and flat hypothesis\n(solidline). Thebackgroundisassumedto beflat (dashedlin e). Figureistakenfrom[5].\nnomenological modelstodescribetheworldoscillationdat aincludingLSND.The\"tandem\"model\nwas made in such concept [12] (as an extension of the \"bicycle \" model [13]). The tandem model\nsatisfies all requirements as an alternative oscillation mo del. One of the very attractive features\nof this model is that it only uses three parameters to describ e four oscillation signals; solar, at-\nmospheric, KamLAND, and LSND signals. In 2006, the nSM had four parameters (two mass\ndifferences and two mixing angles), so the tandem model was a more economical phenomeno-\nlogical model than the nSM. Fig. 5 shows our prediction on the short baseline experim ents. It\nreproduces a ∼0.1% level oscillation signal at LSND. On the other hand, the signal at KARMEN\nis smaller to be consistent with the observation. It also pre dicted an oscillation signal at the low\nenergy region of MiniBooNE, both neutrinos and antineutrin os.\nLater more advanced global oscillation models had been prop osed [14], but all of them failed\nto reproduce recent reactor neutrino results [15], which is another great discovery of this year, and\nit isalso being presented at this ICHEP2012conference!\n4. Lorentz violationanalysisonMiniBooNE experiment\n4.1 MiniBooNEexperiment\nMiniBooNE is the neutrino and antineutrino appearance expe riment designed to confirm or\nreject LSNDsignals under thetwomassiveneutrino oscillat ion hypothesis. Theneutrino (antineu-\ntrino) beamsarecreatedbytheBoosterNeutrinoBeamline(B NB)[16]. The8GeVprotonbeamis\nextracted from the Fermilab Booster, and it is sent to the tar get where the collision with the target\nmakesshowerofmesons. Themagneticfocusinghornsurround ing thetargetselectseitherpositive\nmesons or negative mesons and their decay-in-flight make ∼800 MeV nmor∼600MeV ¯nmbeam.\n6Tests of LorentzandCPT violationwith neutrinos\n10-410-310-210-1\n10-410-310-210-1\n110102103110102103P(nm → ne)\nP(nm → ne)\nP(n–\nm → n–\ne)\nE (MeV)P(nm → ne)\nE (MeV)P(nm → ne)\nE (MeV)P(nm → ne)\nE (MeV)KARMEN LSND\nOscSNS MiniBooNE\nFigure5: Oscillationprobabilitiesasafunctionofenergyforneutr ino(solidline)andantineutrino(dashed\nlines). Figureistakenfrom[12].\nThe MiniBooNE detector is located 541 m away from the target [ 17]. It is a 12.2 m diameter\nspherical Cherenkov detector, filled withthe mineral oil, a nd lined with 1,280 8-inch PMTson the\nwall toobserve the Cherenkov radiation from the charged par ticles.\nThetimeandchargeinformation oftheCherenkovringsfromt hechargedparticlesareusedto\nreconstruct charged particle momentum and particle type [1 8]. Byassuming interaction ischarged\ncurrent quasielastic (CCQE) and the target nucleon is at res t, the neutrino energy is reconstructed\n(QE assumption) [19]. It is vital to be able to reconstruct th e neutrino energy for the neutrino\noscillation physics.\n4.2 MiniBooNEoscillation analysis results\nThesignatureofthe nm→ne(¯nm→¯ne)oscillationisthesingle,isolatedelectron-likeCheren kov\nring produced bythe CCQEinteraction.\nnmoscillation−→ne+n→e−+p,\n¯nmoscillation−→¯ne+p→e++n.\nThe cuts are designed to select such events. Both neutrino an d anti-neutrino mode observed\nthe event excesses. For the neutrino mode, MiniBooNE observ ed the event excess only in the low\n7Tests of LorentzandCPT violationwith neutrinos\nsidereal time (sec)0 10000 20000 30000 40000 50000 60000 70000 80000-osc candidate eventsn1020304050607080 data\nflat solution\nbackground\n3 parameter fit\n5 parameter fit\nFigure 6: The fit results for the MiniBooNE neutrinomode low energyreg ion. The plot shows the curves\ncorresponding to the flat solution (dotted line), sidereal t ime dependent fits (solid and dash-dotted curves),\ntogether with binned data (solid marker). Here the fitted bac kground is shown as a dashed line. Figure is\ntakenfrom [23].\nenergyregion[20]. Theobservedexcessescannotbedescrib ed bythe nSM,soitmaybethesignal\nof new physics, such as Lorentz violation. Onthe other hand, for the antineutrino mode, the event\nexcess is seen in the entire energy region [21]. Again, the ob served excesses cannot be described\nby thenSM3.Since CPTviolation naturally arises within Lorentz viola tion, thisdifferent patterns\nof excesses between neutrino mode and antineutrino mode is n atural if it were caused by Lorentz\nviolation. Therefore it is very interesting to take a look at their sidereal time distributions to find a\npossible Lorentz violation.\n4.3 MiniBooNELorentz violation analysis results\nWefit Eq. 1.4 to MiniBooNE neutrino mode low energy necandidate excess and antineutrino\nmodel excess events. Figure 6 shows the neutrino mode low ene rgy region fit result. As you see,\ndata is quite consistent with a flat hypothesis. We construct ed a fake data set without signal (=flat\nhypothesis) to evaluate the compatibility with a flat soluti on over the fit result by the Dc2method.\nIt turns out data iscompatible withaflat solution over a26 .9%, and it concludes necandidate data\nare consistent withno sidereal timedependence.\nFigure 7shows the antineutrino mode fitresult. Thefit result ismore interesting here because\nthe fit favors a sidereal time dependent solution. We again co nstructed a fake data set to find the\nsignificance of this solution, and it turns out that the compa tibility with a flat solution is now only\n3.0%. Although this is interesting, the significance isnot hig h enough toclaim the discovery.\nSince we didn’t find the Lorentz violation, we set limits to fit parameters (=sidereal time\ndependent and independent amplitudes) [23]. From these lim its one can extrapolate the limits to\n3This analysis was done when only the half of the all antineutr ino data set was available. Recently published full\nantineutrino mode data shows a somewhat different shape of t he excess events [22].\n8Tests of LorentzandCPT violationwith neutrinos\nsidereal time (sec)0 10000 20000 30000 40000 50000 60000 70000 80000-osc candidate eventsn510152025303540data\nflat solution\nbackground\n3 parameter fit\n5 parameter fit\nFigure 7: The fit results for the MiniBooNE antineutrino mode data. Not ations are the same as previous\nfigure. Figureistakenfrom [23].\nthe SME coefficients [1]. It turns out these limits indeed exc lude the SME coefficients needed to\nexplain the LSND signal. Therefore, there is no simple scena rio to explain the LSND signal by\nLorentz violation.\n5. Lorentz violationanalysisonDouble Choozexperiment\nFrom the MiniBooNE data analysis, we set limits on the e−moscillation channel SME co-\nefficients. MINOS near detector analysis [24] also sets seve re limits to some of these coefficients.\nMeantime, MINOSfar detector data [25] and IceCube data [26] set tight limits on the m−toscil-\nlationchannelSMEcoefficients. Thelastuntestedchannel i sthee−tsector, andthiscanbetested\nby the reactor ¯nedisappearance data, because nonzero Lorentz violating neu trino oscillation in the\ne−tchannel would contribute to the reactor neutrino disappear ance. We analyzed data from the\nDouble Chooz reactor experiment, where ∼4 MeV reactor ¯neare detected by the detector located\nat∼1050 maway.\nFigure 8 shows the result. Since the reactor power varies wit h a day-night cycle, the neutrino\nfluxisafunction ofthesolartime(period86400 .0 sec) anditcanmimicthesidereal timevariation\neffect (period 86164 .1 sec), unless data taking is continuous in all one year. This is not the case\nfor Double Chooz. However, the reactor cycle effect is simul ated and taken into account in our\nanalysis. We found that the data over simulation is flat and th e data is consistent with no sidereal\ntime dependence. Therefore weset limits on e−tsector SMEcoefficients.\nWith the addition of this work, most of SME coefficients of all neutrino oscillation channels\nareconstraint. Sinceneutrino oscillation isaninterfere nce experiment, asopposed totimeofflight\n(TOF) which is a kinematic measurement, neutrino oscillati on experiment is far more sensitive\nto small phenomena such as Lorentz violation. Therefore it i s difficult to explain superluminal\nneutrinos observed bytheOPERAexperiment [28]whilekeepi ng allnullLorentzviolation signals\n9Tests of LorentzandCPT violationwith neutrinos\nFigure 8: The fit results forthe reactorantineutrinodata at DoubleCh ooz far detector. The raio of data to\nsimulationisoverlaidwith thebestfit curvesofmodelswith Lorentzviolation. Figureistakenfrom [27].\nin neutrino oscillation experiments. Therefore, it will be challenging to detect Lorentz violation in\nthe neutrino sector in any terrestrial experiments. In the f uture, astrophysical neutrinos [29] may\nimprove sensitivity toLorentz violation bymany orders of m agnitude comapared to these limits.\n6. Conclusions\nLorentz and CPTviolation has been shown tooccur in Planck-s cale physics. There is aworld\nwideefforttotestLorentzviolationwithvariousstate-of -art technologies, includingneutrinooscil-\nlations. LSNDandMiniBooNEdatasuggest Lorentzviolation isaninterestingsolutiontoneutrino\noscillations. MiniBooNE neutrino data prefer a sidereal ti me independent solution, and Mini-\nBooNEantineutrino datapreferasiderealtimedependent so lution, althoughstatistical significance\nis not high. Limits from MiniBooNE exclude simple Lorentz vi olation motivated scenario for\nLSND. Finally, MiniBooNE, LSND, MINOS, IceCube, and Double Chooz set sringent limits on\nLorentz violation inneutrino sector interrestrial level.\nAcknowledgements\nI thank Jennifer Dickson for a careful reading of this manusc ript. I also thank Jorge Díaz for\nvaluable comments. Finally I thank to the ICHEP 2012 organiz ers and IUPAP C11 committee for\nthe invitation tothe ICHEP2012conference.\nReferences\n[1] TeppeiKatori,Mod.Phys.Lett.A 27,1230024(2012).\n[2] Forexample,see, V. A.KosteleckýandS. Samuel,Phys.Re v.D39,683(1989).\n[3] V. A.KosteleckýandN. Russell, Rev.Mod.Phys. 83,11(2011).\n10Tests of LorentzandCPT violationwith neutrinos\n[4] D.ColladayandV.A. Kostelecký,Phys.Rev.D 55,6760(1997); 58,116002(1998);\nV. A.Kostelecký,Phys.Rev.D 69,105009(2004).\n[5] L.B. Auerbach etal.,Phys.Rev.D 72,076004(2005).\n[6] V. A.KosteleckýandM. Mewes,Phys.Rev.D 69,016005(2004).\n[7] V. A.KosteleckýandM. Mewes,Phys.Rev.D 85,096005(2012).\n[8] V. A.KosteleckýandM. Mewes,Phys.Rev.D 70,076002(2004).\n[9] J. S.Díaz, V.A. Kostelecký,andM.Mewes,Phys.Rev.D 80,076007(2009).\n[10] A.A. Aguilar et al.,Phys.Rev.D 64,112007(2001).\n[11] TeppeiKatori,arXiv:1008.0906[hep-ex].\n[12] T.Katori,V.A. Kostelecký,andR. Tayloe,Phys.Rev.D 74,105009(2006).\n[13] V. A.KosteleckýandM. Mewes,Phys.Rev.D 70,031902(R) (2004).\n[14] J. S.Díaz andV.A. Kostelecký,Phys.Lett. B 700,70025(2011);Phys.Rev.D 85,016013(2012).\n[15] Y. Abe et al.,Phys.Rev.Lett. 108,131801(2012);F. P.An etal.,Phys.Rev.Lett. 108,171803\n(2012);J. K.Ahn et al.,Phys.Rev.Lett. 108,191802(2012).\n[16] A.A. Aguilar-Arevalo etal.,Phys.Rev.D 79,072002(2009).\n[17] A.A. Aguilar-Arevalo etal.,Nucl.Instrum.Meth.A 599,28(2009).\n[18] R. B. Patterson et al.,Nucl.Instrum.Meth.A 608,206(2009).\n[19] A.A. Aguilar-Arevalo et al.,Phys.Rev.Lett. 100,032301(2008).\n[20] A.A. Aguilar-Arevalo etal.,Phys.Rev.Lett. 102,101802(2009).\n[21] A.A. Aguilar-Arevalo etal.,Phys.Rev.Lett. 105,181801(2010).\n[22] A.A. Aguilar-Arevalo etal.,arXiv:1207.4809[hep-ex].\n[23] A.A. Aguilar-Arevalo etal.,arXiv:1109.3480[hep-ex].\n[24] P.Adamson et al.,Phys.Rev.Lett. 101,151601(2008);Phys.Rev.D 85,031101(2012).\n[25] P.Adamson et al.,Phys.Rev.Lett. 105,151601(2010).\n[26] R. Abbasi et al.,Phys.Rev.D 82,112003(2010).\n[27] Y. Abe et al.,arXiv:1209.5810[hep-ex].\n[28] T.Adam etal.,JHEP1210,093(2012).\n[29] R. Abbasi et al.,Nature,484,351(2012).\n11"    },    {        "title": "2210.00234v1.The_Lorentz_Process_with_a_Nearly_Periodic_Distribution_of_Scatterers.pdf",        "content": "THE LORENTZ GAS WITH A NEARLY PERIODIC DISTRIBUTION OF\nSCATTERERS\nBERNT WENNBERG(1)\n(1) Department of Mathematical Sciences,\nChalmers University of Technology and University of Gothenburg\nSE 41296 Göteborg, Sweden\nemail: wennberg@chalmers.se\nABSTRACT . We consider the Lorentz gas in a distribution of scatterers which microscopi-\ncally converges to a periodic distribution, and prove that the Lorentz gas in the low density\nlimit satisfies a linear Boltzmann equation. This is in contrast with the periodic Lorentz\ngas, which does not satisfy the Boltzmann equation in the limit.\n1. I NTRODUCTION\nThe Lorentz gas was introduced in [9] to give a new understanding of phenomena such\nas electric resistivity and the Hall effect. Lorentz introduced many simplifications to admit\n“rigorously exact solutions” to some questions, which has made the model very attractive\nin the mathematical community.\nThe Lorentz gas can be described as follows: Let Xbe a point set in R2(actually most\nof what is said in this paper could equally well have been set in Rnwithn\u00152). The point\nset can be expressed as a locally finite counting measure,\n(1) X(A) =#fXi2Ag;\nthe number of points in the set A, or the empirical measure\n(2) X=X\n\u000eXj;\nExamples of interest are the periodic set X=Z2, or random point processes such as a\nPoisson distribution. The Lorentz process is the motion of a point particle with constant\nspeed in the plane with an obstacle of a fixed radius rat each point X2X . Given an\ninitial position and velocity of the point particle, (x0;v0)2R2\u0002S1, its position at time t\nis given by\n(x(t);v(t)) =Tt\nX;r((x0;v0)) =\u0010\nx0+MX\nj=1(tj\u0000tj\u00001)vj\u00001+ (t\u0000tM)vM;vM\u0011\n; (3)\nwheret0= 0, andft1;:::;tMgis the set of times where the trajectory of the point particle\nhits an obstacle, and where vjis the new velocity that results from a specular reflection on\nthe obstacle. We also set xj=x(tj). The notation is clarified in Fig. 1. For a fixed point\nsetXthis is a deterministic motion, but when Xis random, so is the motion of the point\nparticle.\nFor a given point set \u001fwe also consider the rescaled set X\u000f=p\u000fX, so that for any set\nA\u001aR2\nX\u000f(A) =X(Ap\u000f): (4)\n1arXiv:2210.00234v1  [math-ph]  1 Oct 20222 BERNT WENNBERG(1)\nFIGURE 1. A trajectory of the Lorentz process\nWe think of \u001fas describing the domain of the Lorentz gas at a microscopic scale, and \u001f\u000f\nas the macroscopic scale. Expressed in the macroscopic scale, we will assume that for any\nopen setA\u001aR2\n(5) lim\n\u000f!0\u000f#fx2\u001f\u000f\\Ag=cm(A);\nwherem(A)is the Lebesgue measure of the set Aandcis a positive constant. Some\nadditional condition is needed to ensure that the motion is (almost certainly) well defined,\nand that the number of encounters with obstacles, M, is finite. This is true, for example, if\nthe point set is a Delone set, and hence satisfies bounds on the minimal distance between\nthe points as well as on the density of points.\nFor the rest of the paper the obstacle radius is fixed to be equal to \u000fin the macroscopic\nscaling, and thereforep\u000fin the microscopic scaling.\nIn the macroscopic scale, the time t1of the first encounter with an obstacle for a typical\ntrajectoryTt\nX\u000f;\u000f(x0;v0)satisfiest1=O(1=c),i.e.themean free path-length of a typical\ntrajectory is of the order 1=c. This is known as the low density limit, or the Boltzmann-\nGrad limit.\nConsider next an initial density of point particles f02L1\n+(R2\u0002S1), and its evolu-\ntion under the Lorentz process. The density at a later time tis thenf\u000f;t=f\u000f(x;v;t ) =\nE[f0(T\u0000t\nX\u000f;\u000f(x;v))], where the expectation is taken with respect to the probability distribu-\ntion of the point set X\u000fin case it is random.\nEquivalently (because for a fixed realization of X\u000f, the orbitTt\nX\u000f;\u000fis reversible), for\neachg2C0(R2\u0002S1)\n(6)Z\nR2\u0002S1f\u000f(x;v;t )g(x;v)dxdv =Z\nR2\u0002S1f0(x;v)E[g(Tt\nX\u000f;\u000f(x;v))]dxdv:\nGallavotti [5, 6] proved that when Xis a Poisson process with unit intensity, and \u000f\nconverges to zero, then f\u000f;tconverges to a density ftwhich satisfies the linear Boltzmann\nequation:\n(7)@tf(x;v;t ) +v\u0001rxf(x;v;t ) =\u00002f(x;v;t ) +Z\nS1\n\u0000f(x;v0;t)jv\u0001!jd!:\nHereS1\n\u0000=S1\n\u0000(v) =f!2S1jv\u0001! < 0gandv0=v\u00002(!;v)!. Spohn has proven a\nrelated, and more general, result in [21].\nOn the other hand, it is also known that when X=Z2(or for that matter many other\nregular point sets, such as quasi crystals), then the Lorentz process is not Markovian inTHE LORENTZ GAS WITH A NEARLY PERIODIC DISTRIBUTION OF SCATTERERS 3\nthe limit and therefore (6) does not hold, see [1, 8]. For a periodic distribution of scatter-\ners, Caglioti and Golse [3] proved that there is a limiting kinetic equation in an enlarged\nphase space, and more general results of the same kind were obtained by Marklof and\nStrömbergsson [11, 12].\nMarklof and Strömbergsson has studied this problem in several papers [14, 13, 15, 16],\nand present in [17] a very general theorem concering the Boltzmann-Grad limit of the\nLorentz process, where they give a concise set of conditions for a point set X, such that\nthe Lorentz process Tt\nX\u000f;\u000f(x;v)converges to a free flight process. The point set Xhere\nis a fixed point set, that could be for example a periodic set, or one given realization of a\nrandom point process, and hence, for a fixed \u000fthe particle flow is deterministic and not the\nexpectation over the point process as in Eq. (6).\nThe problem studied in this paper is the following: Let Xbe a given point process, and\nletY\u000fbe a family of point processes converging to Xin the sense that the Lévy-Prohorov\ndistance\u0019(X;Y\u000f)between the measures Y\u000fandXconverges to zero ( a.s. ) when \u000f!0.\nRecall here the definition of the Lévy-Prohorov distance:\n(8)\u0019(\u0017;\u0016) = inf\u001a\n\">0\f\f\f\f\u0016(A)\u0014\u0017(A\") +\"and\u0017(A)\u0014\u0016(A\") +\"for allA2B\u001b\n:\nHereBis the Borel sigma algebra (of R2in our case), and\n(9) A\"=[\nx2AB\"(x)\nwhereB\"(x)is the ball of radius \"and centre at x2A. This convergence is thus assumed\nto take place at the microscopic level. To study this in the Boltzmann-Grad limit, we set\n(10) Y\u000f;\u000f=p\u000fY\u000f andX\u000f=p\u000fX:\nWe are then interested in comparing the limits of the corresponding Lorentz processes,\nTt\nY\u000f;\u000fandTt\nX\u000f, assuming that the obstacle radius is \u000f. The main result of the paper is the\nconstruction of a family of point sets Y\u000fthat converges to the periodic distribution, and yet\nthe Lorentz process Tt\nY\u000f;\u000f(x;v)converges to the free flight process generated by the linear\nBoltzmann equation (6), contrary to the limit of Tt\nX\u000f(x;v). It is in a sense a non-stability\nresult for the periodic Lorentz gas, and while not proven in this paper it seems very likely\nthat ifXis a Poisson process with constant intensity, (almost) any approximation Y\u000fwould\nresult in convergence of the two Lorentz processes to the same limit. The proof follows\nquite closely the construction in [2] and [20], and consists in constructing a third process,\neTt\nY\u000f;\u000fwhich which can be proven to be path-wise close the free flight process, and also\nto the Lorentz process. The full statement of the result, together with the main steps of\nthe proof are given in Section 2. Section 3 gives the somewhat technical proof that eTt\nY\u000f;\u000f\nconverges to the Boltzmann process, and Section 4 contains a proof that the probability\nthat an orbit of Tt\nY\u000f;\u000fcrosses itself near an obstacle is negligible in the limit as \u000f!0,\nwhich then used to prove that eTt\nY\u000f;\u000fandTt\nY\u000f;\u000fwith large probability are path-wise close.\nThe scaling studied here for the Lorentz process is not the only one studied in literature.\nA more challenging problem is the long time limit, where the process is studied over a\ntime interval of [0;t\u000f[, wheret\u000f!1 when\u000f!0. Recent results of this kind have been\nobtained in [10]. Other related results can be found in [19] and [18].\n2. T HE MAIN RESULT AND THE PRINCIPAL STEPS OF ITS PROOF\nLetX=Z2and defineY\u000fas a perturbation of Xin the following way: Let \u001ebe a\nrotationally symmetric probability density supported in jxj<1, fix\u00172]1=2;1[, set\n(11)Y\u000f=\b\n(j;k) +\u000f1\u0000\u0017\u0018j;kj(j;k)2Z2;\u0018j;ki.i.d. with density \u001e\t\n:4 BERNT WENNBERG(1)\nand let\n(12) Y\u000f;\u000f=p\u000fY\u000f:\nThus each obstacle has its center in a disk of radius \u000f1\u0000\u0017centered at an integer coordinate,\n(j;k). This disk will be called the obstacle patch below. We have almost trivially that the\nLévy-Prohorov distance between XandY\u000fis smaller than \u000f1\u0000\u0017for all realizations of the\nrandom setY\u000f. The obstacle itself reaches at most a distance \u000f1\u0000\u0017+\u000f1=2from the same\ninteger coordinate; this larger disk will be called an obstacle range below. The ratio of the\nobstacle range and the support of the center distribution is thus 1+\u000f\u0017\u00001=2, which converges\nto1when\u000f!0, and to simplify some notation the radius of the obstacle patch will be\nused instead of the radius of the obstacle range, and the difference will be accounted for\nwith a constant in the estimates.\nClearlyY\u000fconverges in law to the periodic distribution in R2. Nevertheless we have the\nfollowing theorem:\nTHEOREM 2.1. LetTt\nY\u000f;\u000f(x;v)be the Lorentz process obtained by placing a circular\nobstacle of radius \u000fat each point ofY\u000f;\u000f. Let\u0016t>0, and letf0(x;v)be a probability density\ninR2\u0002S1. Definef\u000f(x;v;t )as the function such that for all t2[0;\u0016t]g2C(R2\u0002S1),\n(13)Z\nR2\u0002S1f\u000f(x;v;t )g(x;v)dxdv =Z\nR2\u0002S1f0(x;v)E[g(Tt\nY\u000f;\u000f(x;v))]dxdv:\nThen there is a density f(x;v;t )2C([0;\u0016t];L1(R2\u0002S1)such that for all t\u0014\u0016t,\nf\u000f(x;v;t )!f(x;v;t )inL1(R2\u0002S1)when\u000f!0, and such that f(x;v;t )satisfies the\nlinear Boltzmann equation\n(14)@tf(x;v;t ) +v\u0001rxf(x;v;t ) =\u0014 Z\nS1\n\u0000f(x;v0;t)jv\u0001!jd!\u00002f(x;v;t )!\n:\nThe constant \u0014depends on\u001eandS1\n\u0000andv0are defined as in Eq. (7).\nRemark 2.2. To simplify notation, point particles are allowed to start inside obstacles, and\nto cross the obstacle boundary from the inside without any change of velocity.\nBoth the statement in this theorem and the proof are very similar to the main results\nof [2] and [20], but the implication is anyway quite different. In those papers the point\nprocessesY\u000fare constructed as the thinning of a periodic point set:\n(15)Y\u000f=\b\n\u000f\u0017(j;k)j\u0018j;k= 1; \u0018j;ki.i.d. Bernoulli with P[\u0018j;k= 1] =\u000f2\u0017\t\n:\nHere it is clear that this Y\u000fconverges in law to the Poisson process with intensity one, and\nin hindsight it is perhaps not surprising that the limit of the Lorentz process is the same as\nfor the Lorentz process generated by a Poisson distribution of the obstacles. Theorem 2.1\nin the present paper states that the limit of the Lorentz process is the free flight process of\nthe Boltzmann equation in some cases also if the limiting obstacle density is periodic, and\nraises the question as to which point processes Xare stable to perturbation when it comes\nto the low density limit of the corresponding Lorentz processes.\nIn the proof we consider three processes: The Lorentz process Tt\nY\u000f;\u000f(x;v), the free flight\nprocessTt\nB(x;v)generated by the Boltzmann equation (13), and an auxiliary Markovian\nLorentz process, eTt\nY\u000f;\u000f(x;v).\nThe Boltzmann process is the random flight process (x(t);v(t)) =Tt\nB(x;v)generated\nby Eq. (1). Let 0 =t0<t1<;:::;tn<tbe a sequence of times generated by independent,\nexponentially distributed increments tj\u0000tj\u00001with intensity 2, letv0=vdefinevj=\nvj\u00001\u00002(!j\u0001vj\u00001)!j, where the !j2S1are independent and uniformly distributed.THE LORENTZ GAS WITH A NEARLY PERIODIC DISTRIBUTION OF SCATTERERS 5\nFinally set\nx(t) =xn(t) =x+t1v+ (t2\u0000t1)v1+\u0001\u0001\u0001+ (tn\u0000tn\u00001)vn\u00001+ (t\u0000tn)vn\nv(t) =vn: (16)\nOf course the number nin the sum is then Poisson distributed. The solution of Eq. (1) may\nbe defined weakly by\n(17)Z\nR2\u0002S1f0(x;v)g(x;v;t )dxdv =Z\nR2\u0002S1f(x;v;t )g(x;v)dxdv:\nThe function g(x;v;t )can then be expanded as a sum of terms, each representing the paths\nwith a fixed number of jumps:\n(18) g(x;v;t ) =Vtg(x;v) =1X\nn=0gn(x;v;t );\nwith\ngn(x;v;t ) = (Vtg)n(x;v) =e\u00002tZt\n0Zt\nt1\u0001\u0001\u0001Zt\ntn\u00001dt1\u0001\u0001\u0001dtn\u0002\nZ\n(S1\n\u0000)nd!1\u0001\u0001\u0001d!nnY\nk=1j!k\u0001vk\u00001jg(xn(t);vn): (19)\nThe Markovian Lorentz process here is similar to the corresponding process in [2, 20],\nin the way that re-collisions, i.e.the event that a particle trajectory meets the same obstacle\na second time, are eliminated. The construction is explained in Fig. 2. The red thin circles\nmark the support of the distribution of obstacle center in each cell, and the blue circle\nindicates the reach of an obstacle, the obstacle range. In each cell there is an obstacle,\nfixed from the start in the Lorentz model, but changing in the Markovian Lorentz model.\nNote that the size of both the obstacle radius and the obstacle support are very small,\nand decreasing to zero with \u000f, but drawn large here, for clarity. The path meets the same\nobstacle patch twice in the boxed cell. In the Lorentz case, the orbit simply traverses the\ncell the second time, because it misses the obstacle. In the Markovian case the obstacle is\nat a new, random position, drawn in blue color, the second time the orbit enters the support,\nand there is a positive probability that the path collides with the obstacle, as shown with\nthe blue trajectory. In both cases the trajectory is surely well defined, but the probability of\nrealizing an orbit is different.\nThe process can be defined as in Eq. (17), with the function g(x;v;t )replaced by\n~g\u000f(x;v;t ) =Eh\ng(eTt\nY\u000f;\u000f(x;v))i\n=1X\nn=0~g\u000f;n(x;v;t ); (20)\n\u0010\neVt\n\u000f;n\u0011\ng(x;v) = ~g\u000f;n(x;v;t ) (21)\nwhere ~g\u000f;n(x;v;t )is the contribution of trajectories having exactly nvelocity jumps in the\ninterval [0;t]. All these terms can be computed rather explicitly by counting the number of\ntimes a trajectory crosses the obstacle range of one cell.\nTo simplify notation these three processes are hereafter denoted z\u000f(t),z(t)and~z\u000f(t).\nAs described in [2, 20] all these processes belong with probability one to the Skorokhod\nspaceD[0;\u0016t](R2\u0002S1)ofcàdlàg functions on R2\u0002S1equiped with the distance\n(22)dS(x;y) = inf\n\u00152\u0003(\nsup\nt2[0;\u0016t]kx(t)\u0000y(\u0015(t))kR2\u0002S1+ sup\nt2[0;\u0016t]jt\u0000\u0015(t)j)\n;\nwhere\n(23) \u0003 =f\u00152C([0;\u0016t]) :t>s =)\u0015(t)>\u0015(s);\u0015(0) = 0;\u0015(\u0016t) =\u0016tg:6 BERNT WENNBERG(1)\nFIGURE 2. Trajectories for the Lorentz process and the Markovian\nLorentz process. The obstacle size and the support for the probability\ndistributions are exaggerated.\nAnyz2C(D[0;\u0016t](R2\u0002S1))induces a measure \u0016onD[0;\u0016t](R2\u0002S1)which is first\ndefined on cylindrical continuous functions F,i.e.functionsF2C(D[0;\u0016t](R2\u0002S1))of\nthe formF(z) =Fn(z(t1);z(t2);:::;z (tn))whereFn2C((R2\u0002S1)n)and0\u0014t1<\nt2<\u0001\u0001\u0001<tn\u0014\u0016t. For such functions \u0016is defined by\n(24)Z\nF(z)\u0016(dz) =Z\nf0(z)Ptn;:::;t 1;0(z1;z2;\u0001\u0001\u0001;znjz0)dz0dz1\u0001\u0001\u0001dzn;\nwherePtn;:::;t 1;0(z1;z2;\u0001\u0001\u0001;znjz0)is the joint probability density of z(t1);z(t2);:::;z (tn)\ngiven the starting point z0. For a Markov process, such as the Boltzmann process, this is\n(25)Z\nF(z)\u0016(dz) =Z\nf0(z)Pt1;0(z1jz0)Pt2;t1(z2jz1)\u0001\u0001\u0001Ptn;tn\u00001(znjzn\u00001)dz0dz1\u0001\u0001\u0001dzn;\nwherePtj;tj\u00001(zjjzj\u00001)is the transition probability of going from state zj\u00001to statezjin\nthe time interval from tj\u00001totj. Then by a density argument the measure \u0016is defined for\nallF2C(D[0;\u0016t](R2\u0002S1)).\nProof: (of Theorem 2.1) Let \u0016,\u0016\u000fand~\u0016\u000fbe the measures induced by z(t),z\u000f(t)and~z\u000f(t).\nJust like in in [20], one may prove that for every F\n(26) lim\n\u000f!0Z\nF(z)~\u0016\u000f(dz)!Z\nF(z)\u0016(dz):\nThe argument uses a theorem from Gikhman and Shorokhod(1974) [7], and relies on an\nequicontinuity condition and on the convergence of the marginal distributions of ~z\u000f(t).\nBoth the equicontinuity condition and the convergence of the marginal distributions are\nconsequences of Proposition 3.5 which shows that the number of jumps of z\u000fin small time\nintervals is not too large, and Theorem 3.1, which states that the one-dimensional marginals\nof~z\u000f(t)converge to the marginals of z(t).\nThe next step is to prove that (\u0016\"\u0000~\u0016\u000f)*0when\u000f!0,i.e.\n(27)\f\f\f\fZ\nF(z)\u0016\u000f(dz)\u0000Z\nF(z)~\u0016\u000f(dz)\f\f\f\f!0THE LORENTZ GAS WITH A NEARLY PERIODIC DISTRIBUTION OF SCATTERERS 7\nwhen\u000f!0, and this is done by a coupling argument. Given that z\u000f(0) = ~z\u000f(0), the\ntwo processes have the same marginal distributions up to the first time t\u0003wherez\u000f(t) =\n~z\u000f(t)returns to an obstacle range it already visited. For the process z\u000f(t), the position of\nthe obstacle within its support is then fixed, whereas for the process ~z\u000f(t)a new random\nposition of the obstacle is chosen when the trajectory arrives. Hence\f\f\f\fZ\nF(z)\u0016\u000f(dz)\u0000Z\nF(z)~\u0016\u000f(dz)\f\f\f\f=\f\f\f\fZ\nK\u000fF(z)\u0016\u000f(dz)\u0000Z\nK\u000fF(z)~\u0016\u000f(dz)\f\f\f\f\n\u0014supjFj(\u0016\u000f(K\u000f) + ~\u0016\u000f(K\u000f)); (28)\nwhereK\u000f\u001aD[0;\u0016t](R2\u0002S1)is the set of trajectories that contain at least one such re-\nencounter for t\u0014\u0016t. Proposition 4.1 states that the righthand side of (28) converges to zero\nwhen\u000f!0. This together with (26) implies that\n(29) \u0016\u000f!\u0016\nweakly when \u000f!0, and this concludes the proof of Theorem 2.1. \u0003\nWe end the section with a comment on the propagation of chaos for these procesess.\nFor the Boltzmann process z(t)and for the process ~z(t)it is clear that if a pair of initial\nconditions (z1(0);z2(0)) are chosen with joint density f1\n0(x;v)f2\n0(x;v), then joint densi-\nties of (z1(t);z2(t))and(~z1\n\u000f(t);~z2\n\u000f(t))also factorize: a chaotic initial state is propagated\nby the flow, because the two paths are independent, they never interact. The same is not\ntrue for (z1\n\u000f(t);z2\n\u000f(t)), because there is a positive probability that the two paths will meet\nin the obstacle density support of one obstacle, which creates correlations. However, the\nsame kind of estimates as the ones used to prove that the probability of re-encounters for\none trajectory becomes negligible when \u000f!0can be used to prove that also the proba-\nbility that two different trajectories meet inside an obstacle range becomes small, and such\nestimates could be performed for any finite number of trajectories. The calculations are\ncarried out in some detail in Section 4, and formlated as Theorem 4.2.\n3. T HEMARKOVIAN LORENTZ PROCESS\nThe Markovian process may be described using the underlying periodic structure. Over\na time interval [0;t], the particle traverses O\u0000\n\u000f\u00001=2\u0001\nlattice cells, and when the path is\nsufficiently close to the cell center to cross the obstacle range, there is a positive probability\nthat it is reflected by an obstacle. Because the obstacle position is chosen independently\neach time the trajectory enters a cell, this may all be computed rather explicitly.\nWe write\n(30) ~g\u000f(x;v;t ) =Eh\ng(eTt\nY\u000f;\u000f(x;v))i\n=eVt\n\u000fg(x;v) =1X\nn=0(eV\u000f)t\nng(x;v);\nThis is the test function evaluated along the path of a point particle. It defines a semi-\ngroupeVt\n\u000facting on the test function, and the terms in the sum express the contribution to\nthis semigroup from paths with exactly nvelocity jumps. Obviously these terms are not\nsemigroups in their own right.\nTHEOREM 3.1. LeteTt\nY\u000f;\u000f(x;v)be the Markovian Lorentz process as defined above. Fix\n\u0016t>0. For any density f0(x;v)inR2\u0002S1, let~f\u000f(x;v;t )be the unique function such that\n(31)Z\nR2\u0002S1~f\u000f(x;v;t )g(x;v)dxdv =Z\nR2\u0002S1f0(x;v)g(eTt\nY\u000f;\u000f(x;v))dxdv;\nfor allt2[0;\u0016t], and anyg(x;v)2C(R2\u0002S1). Then there is a function f(x;v;t )such\nthat for allt\n(32) ~f\u000f(x;v;t )!f(x;v;t );\nandf(x;v;t )satisfies the linear Boltzmann equation , Eq. (7).8 BERNT WENNBERG(1)\nProof: Letg2C0(R2\u0002S1), andt<\u0016t. We must prove that\n(33)Z\nR2\u0002S1\u0010\n~f\u000f(x;v;t )\u0000f(x;v;t )\u0011\ng(x;v)dxdv!0when\u000f!0;\nor, equivalently,\n(34)Z\nR2\u0002S1f0(x;v)\u0010\nVtg(x;v)\u0000~Vt\n\u000fg(x;v)\u0011\ndxdv!0:\nThe expression in (34) is bounded by\n(35)kgkL1Z\nR2\u0002S1f0\u0000\n1f0>\u0015+ 1jxj\u0015M\u0001\ndxdv +\u0015k(Vtg\u0000~Vt\n\u000fg) 1jxj<MkL1;\nand choosing \u0015andMlarge enough, depending on f0, the first term can be made arbitrarily\nsmall, smaller than \"0=2, say, where \"0is taken arbitrarily small. It the remains to show\nthat the second term can be made smaller than \"0=2by choosing \u000fsmall enough. Both\nVtandeVt\n\u000fare bounded semigroups, and therefore, after dividing the interval [0;t]intoN\nequal sub-intervals, [jt=N; (j+ 1)t=N]one gets\n(36)Vtg(x;v)\u0000eVt\n\u000fg(x;v) =N\u00001X\nj=0eVjt=N\n\u000f\u0010\nVt=N\u0000eVt=N\n\u000f\u0011\nV(N\u00001\u0000j)t=Ng(x;v):\nThe two semigroups are contractions in L1\\L1, and therefore it is enough to prove that\nforN=N\u000fappropriately chosen\n(37) N\u000f\r\r\rVt=N\u000fg(x;v)\u0000eVt=N\u000f\n\u000fg\r\r\r\nL1!0\nwhen\u000f!0. SettingN\u000f=t=\u001c\u000ffor a suitable \u001c\u000f, we find\n\f\f\fV\u001c\u000fg(x;v)\u0000eV\u001c\u000f\n\u000fg(x;v)\f\f\f=\f\f\f\f\f1X\nk=0\u0010\n(V\u001c\u000f)kg(x;v)\u0000(eV\u001c\u000f\n\u000f)kg(x;v)\u0011\f\f\f\f\f\n\u0014\f\f\f(V\u001c\u000f)0g(x;v)\u0000(eV\u001c\u000f\n\u000f)0g(x;v)\f\f\f+\f\f\f(V\u001c\u000f)1g(x;v)\u0000(eV\u001c\u000f\n\u000f)1g(x;v)\f\f\f+\n+\f\f\f\f\f1X\nk=2(V\u001c\u000f)kg(x;v)\f\f\f\f\f+\f\f\f\f\f1X\nk=2(eV\u001c\u000f\n\u000f)kg(x;v)\f\f\f\f\f:(38)\nDenote the four terms in the right-hand side by RI;RII;RIII, andRIV, and setr\u000f=\n\u000f(2\u0017\u00001)=2(1 + log(t=p\u000f)). From Proposition 3.4 it follows that the first term, accounting\nfor paths with no jump in the given interval, satisfies\n(39) N\u000fZ\nR2\u0002S1RI(x;v)dxdv\u0014t\n\u001c\u000fCM2r1=2\n\u000f;\nwhich converges to zero with \u000fifr1=2\n\u000f=\u001c\u000fdoes. The second term accounts for paths with\nexacly one jump, and according to Proposition 3.6,\n(40)N\u000fZ\nR2\u0002S1RII(x;v)dxdv\u0014CM2(!(\u001c\u000f;g) +r\u000f+kgk1pr\u000f+kgk1r\u000f=\u001c\u000f):\nFor this term it is enough that r\u000f=\u001c\u000fconverges to zero, but the rate of convergence may\ndepend on the modulus of continuity of the test function g.\nWe also have that if \u001c\u000f>r1=2\n\u000f\n(41) N\u000fZ\nR2\u0002S1(RIII(x;v) +RIV(x;v))dxdv\u0014CM2kgk1t\u001c\u000f:\nThis follows because for the Boltzmann process, the probability of having more than two\njumps in an interval of length \u001c\u000fis of the order \u001c2\n\u000f, and Proposition 3.5 states that the same\nis true for the Markovian Lorentz process. Therefore, in conclusion, the convergence statedTHE LORENTZ GAS WITH A NEARLY PERIODIC DISTRIBUTION OF SCATTERERS 9\nin eq. (33) holds with a rate depending on f0but which can be controlled by entropy and\nmoments, and on the modulus of continuity of the test function g.\n\u0003\nThe expression (36) implies that it is enough to study the processes in very short in-\ntervals, and Proposition 3.5 below shows that it is then enough to consider two cases: a\nparticle path starting at (x;v)2R2\u0002S1,i.e.with position x2R2in the direction v,\nmoves without changing velocity during the whole interval, or, hits an obstacle at some\npointx0and continues from there in the new direction v0, but suffers no more collisions.\nThe three propositions used in the proof of Theorem 3.1 all depend on Lemma 3.2 be-\nlow, were the underlaying periodic structure is used to analyze the particle paths in de-\ntail up to and just after the first collision with an obstacle. To set the notation for the\nfollowing results, we consider a path starting in the direction vfrom a point xin the di-\nrection ofv. It is sometimes convenient to denote the velocity by v2S1by an angle\n\f2[0;2\u0019[, and both notations are used below without further comment. When a collision\ntakes place, the the new velocity v0depends onvand on the impact parameter ras shown\nin Fig. 4, where r2[\u00001;1]when scaled with respect to the obstacle radius. In this scaling,\ndr= cos(\f0=2)d\f0=2, when\f0is given as the change of direction.\nWithout loss of generality we may assume that the path is in the upward direction with\nan angle\f2[0;\u0019=4]to the vertical axis as in Fig. 3, because all other cases can be treated\nin the same way just by finite number of rotations and reflections of the physical domain.\nLety1be the first time the path enters the lower boundary of the lattice cell, and then let\nyj=y1+ (j\u00001) tan(\f) mod 1 be the consecutive points of entry to the lattice cell.\nSettingy0=\u0000tan(\f)=2, the signed distance between the particle path through the cell\nand the cell center is \u001aj= (yj\u0000y0) cos\f, and the probability that the path is reflected at\nthej-th passage and given the sequence \u001aj, the events of scattering in cell passage nr jare\nindependent.\nDefine\np0(x;v;t ) =Ph\nno collision in the interval [0;t[\nfor a path starting at (x;v)2R2\u0002S1i\n: (42)\nThe probability that a scattering event takes place in the j-passage of a lattice cell is a\nfunction of the distance from the path to the centre of the cell, \u001aj, and is zero outside\nj\u001ajj< \u000f1\u0000\u0017+\u000f1=2. This probability will be denoted pj=pj(x;v), and depends only\non the initial position xand the direction v. Given that a scattering event takes place,\nthe outcome of this event depends on the scattering parameter, i.e.the distance between\nthe path and the (random) center of the obstacle. Let Ax;v;t be any event that depends\non a position xand direction vof a path segment of length t. Then the probability that\nA=Ax0;v0;t0is realized after the first collision can be computed as\nP[A] =nX\nj=1p0(x;v;tj\u0000)pj(x;v)P[Ax0;v0;t0jj]: (43)\nHereP[Ax0;v0;t0jj]denotes the conditional probability of the event Ax0;v0;t0given that\nthe collision takes place when the particle crosses cell number jalong the path, and this\nalso depends on the xandv. The time when the path enters cell number jis denotedtj\u0000,\nand the terms in the sum contains a factor p0(x;v;tj\u0000)for the probability that no collision\nhas taken place earlier. Similarly tj+denotes the time when the trajectory leaves the cell;\nthat is a random number, but we always have 0<tj+\u0000tj\u00001<2p\u000f.\nIn (43), (x0;v0;t0)are random, so P[A]involves also an integral over these variables,\nand the notation P[Ax0;v0;t0jj]is intended to include this integration. The expectation\nE[ ]of some function  is computed in the same way with P[Ax0;v0;t0jj]replaced by\nE[Ax0;v0;t0jj].10 BERNT WENNBERG(1)\nGiven the density \u001efor the position of the obstacle define\n(44) '0(x1) =Z\nR\u001e0(x1;x2)dx2;\nwhere (x1;x2)is an arbitrary coordinate system in R2. Because\u001eis assumed to be ro-\ntationally symmetric, the resulting function '0is independent of the orientation of this\ncoordinate system. It is a smooth probability density with support in [\u00001;1], and then\n'\u000f(r)\u0011\u000f\u0017\u00001'0(\u000f\u0017\u00001r)is also a smooth probability density with support in jrj\u0014\u000f1\u0000\u0017.\nFor eachkthe obstacle center is chosen randomly inside a circle of radius \u000f1\u0000\u0017around\nthe cell center, and one can then compute the probability that the path is reflected at step k.\nLEMMA 3.2. Consider a lattice cell in the microscopic scale, so that the cell side has\nlength one, and assume that there is an obstacle of radius \u000f1=2with center distributed with\na density of the form \u001e(x) =\u000f2\u0017\u00002\u001e0(\u000f\u0017\u00001x), where\u001e02C1(R2)has support in the\nunit ball of R2. Let'\u000fbe defined as in Eq. (44) and rescaled with \u000f. Let 02L1(R)with\n 0(x) = 0 forjxj>1, and set \u000f(x) = 0(x=p\u000f). Consider a particle path traversing\nthe cellntimes with an angle \f2[0;\u0019=4]to the vertical cell sides, and set y1;:::;ynto\nbe the consecutive points at the lower edge of the cell. We have y12[\u00001=2;1=2[and\nyk=\u00001=2 + (1=2 + (k\u00001) tan(\f) mod 1) . Then\n(45) p[ \u000f](y;\f) =Z\nR \u000f(x1)\u001e\u000f(ycos(\f) +x1)dx1\nsatisfies\n(46)nX\nk=1p[ \u000f](yk;\f) =p\u000fn\ncos(\f)Z1\n1 0(x)dx+Ra(y1;\f):\nThe remainder term Raalso depends on nand\u000f. It is bounded by a function \u0016RA(\f), that\ndepends on\u001e,nand\u000fbut not on , and that satisfies\njRa(n;\u000f;y 1;\f)j\u0014k 0kL1\u0016RA(\f) where (47)\nZ\n\u0016RA(\f)d\f\u0014C\u000f\u0017\u00001=2(1 + logn): (48)\nProof: The result is a small extension of the corresponding proposition in [20], and also\nthe proof follows closely that paper. For any fixed \f, the expression p[ \u000f](y;\f)has support\nin an interval of length 2(\u000f1\u0000\u0017+\u000f1=2)=cos(\f). Extend this function to be a one-periodic\nfunction ofy. For simplicity we assume that nis odd, and set n= 2m+ 1. Withnlarge,\nthis assumption would only make a very small contribution from adding one extra term to\nthe sum in case nwere even. Then\nnX\nj=1p[ \u000f](yj;\f) =mX\nj=\u0000mp[ \u000f](y1+ (j+m\u00001) tan(\f);\f)\n=1X\nk=\u00001^pkmX\nj=\u0000me2\u0019ik(y1+(j+m\u00001) tan(\f))\n=1X\nk=\u00001^pke2\u0019ik(y1+(m\u00001) tan(\f))mX\nj=\u0000me2\u0019ikj tan(\f): (49)\nIn this expression ^pkis thek-th Fourier coefficient of the periodic function p[ \u000f](\u0001;\f), and\nthe sum to the right can be evaluated as the Dirichlet kernel of order mwith argument\nktan(\f):\n(50) Dm(x) =sin((2m+ 1)\u0019x)\nsin(\u0019x):THE LORENTZ GAS WITH A NEARLY PERIODIC DISTRIBUTION OF SCATTERERS 11\nTherefore\n(51)nX\nj=1p[ \u000f](yj;\f) = (2m+ 1)^p0+Ra\nwhere\n(52) Ra=X\nk6=0^pke2\u0019ik(y1+(m\u00001) tan(\f))Dm(ktan\f):\nThe Fourier coefficients ^pkcan be computed as the Fourier transform of p[ \u000f](\u0001;\f)evalu-\nated at the integer points k,\n(53) ^pk=Z1\n\u00001e\u00002\u0019ikyp[ \u000f](y;\f)dy=p\u000f\ncos(\f)b 0(\u0000p\u000fk)b'0(\u000f1\u0000\u0017k):\nHereb'0andb 0are the Fourier transforms of '0and 0. Note the factorp\u000fwhich is due\nto the definition of  \u000f, which is not rescaled to preserve the L1-norm. We have\n(54) ^p0=p\u000f\ncos(\f)Z\n 0(x)dx;\nand because  02L1and'0is smooth, the coefficients ^pkdecay rapidly: For any a>0\nthere is a constant cadepending on '0such that\n(55) j^pkj\u0014p\u000fk 0kL1ca\n1 +j\u000f1\u0000\u0017kja:\nBecause\f2[0;\u0019], the dependence on \fcan be absorbed into the constant ca. Let\n(56) \u0016RA(\f) =X\nk6=0cap\u000f\n1 +j\u000f1\u0000\u0017kj1jDm(ktan\f)j\nThe remainder term Ra=Ra(n;\u000f;y 1;\f)is therefore bounded by\n(57) jRaj\u0014k 0kL1\u0016RA(\f):\nUsing a standard estimate of the L1-norm of the Dirichlet kernel,\nZ\u0019=4\n0jDm(ktan\f)jd\f=Z1\n0jDm(k\u0018)j\n1 +\u00182d\u0018\n\u00141\nkZk\n0jDm(\u0018)jd\u0018\u0014C(1 + logm): (58)\nTherefore these integrals are independent of k, and so\nZ\nS1\f\f\u0016RA(\f)\f\fd\f\u0014Cp\u000f(1 + log(m))X\nk6=0cak 0kL1\n1 +j\u000f1\u0000\u0017kja\n\u0014Ck 0kL1\u000f\u0017\u00001=2(1 + log(n)); (59)\nbecause the sum is bounded by CR1\n0(1+(\u000f1\u0000\u0017x)a)\u00001dx, and replacing log(m)bylog(n)\nonly modifies the constant. This concludes the proof.\n\u0003.\nRemark 3.3. The assumption that  02L1is actually much stronger than needed. The\nproof works equally well as long as the Fourier transform of  0is not increasing too fast,\nand a Dirac mass with support in ]\u00001;1[, for example, would give the same kind of error\nestimate.12 BERNT WENNBERG(1)\nPROPOSITION 3.4. The terms (Vt)0g(x;v)and(eVt\n\u000f)0g(x;v)are given by\n(Vt)0g(x;v) =e\u00002tg(x+vt;v) and (60)\n(eVt\n\u000f)0g(x;v) =p0(x;v;t )g(x+vt;v): (61)\nwherep0(x;v;t )is the probability that a trajectory starting at x2R2in directionvdoes\nnot hit an obstacle in the interval [0;t]; this can be computed explicitly. The function\np0(x;v;t )satisfiesp0(x;v;t )\u0000e\u00002t=Rb(x;v;t )with\njRb(x;v;t )j\u0014\u0016RB(\f) (62)\nfor a function \u0016RB(\f)that satisfies\nZ\nS1\u0016RB(\f)dv\u0014C\u000f2\u0017\u00001)=4q\n1 + log(t=p\u000f); (63)\nand consequently, for any g2C\u0000\nR2\u0002S1\u0001\nwith support injxj\u0014M,\n(64)\r\r\r(Vt)0g\u0000(eVt\n\u000f)0g\r\r\r\nL1(R2\u0002S1)\u0014CM2kgkL1\u000f(2\u0017\u00001)=4q\n1 + log(t=p\u000f):\nFIGURE 3. A particle path traversing the lattice cells. The cell size,\ndiameter of the distribution of the obstacle center and obstacle size are\ngiven in the macroscopic scale to the left and microscopic scale to the\nright\nFIGURE 4. The figure illustrates a collision with an obstacle and the\nnotation used to describe this.THE LORENTZ GAS WITH A NEARLY PERIODIC DISTRIBUTION OF SCATTERERS 13\nProof: Consider a path starting at x2R2in the direction of v. There is no restriction in\nassuming that the direction vis clockwise rotated with an angle \fas illustrated in Fig. 3,\nwhich shows one lattice cell, with an obstacle patch indicated with a red circle, and a\nblue circle indicating the maximal range for an obstacle, and the red solid disk shows one\npossible position of the obstacle. In the macroscopic scale, the lattice size is \u000f1=2, the\nobstacle has radius \u000f, and the obstacle patch has radius \u000f3=2\u0000\u0017. These values are indicated\nwithin parenthesis, and the microsopic scale, in which the lattice size is 1is indicated to\nthe left. When \u000f!0the obstacle patch in microscopic scale shrinks to a point, but is is\ndrawn large in the image for clarity. The path under consideration enters a new lattice cell\nfor the first time at y1, and enters the second lattice cell at y2, drawn in the same image. A\npath of length tin macroscopic scale will traverse a number nof lattice cells in the vertical\ndirection. We set m=btcos\f\n2\u000f1=2c. Then\n(65) n= 2m+ 1 =tcos\f\n\u000f1=2+\u0010;\nwhere\u00102]\u00001;1], and depending on whether where the path starts and ends, the path may\ntouch one additional cell. The error coming from not knowing the start and end points can\nbe taken into account by allowing \u00102[\u00002;2]. Then\np0(x;v;t ) =P[no collision in the interval [0;t[ ]\n=nY\nj=1(1\u0000p(yj;\f)) = exp\u0012nX\nj=1log(1\u0000p(yj;\f))\u0013\n(66)\nwherep(y;\f)is the probability that a trajectory entering a lattice cell at ywith angle\f\nalong the lower cell boundary meets an obstacle before leaving the cell on the top. Because\np(yj;\f)<c\u000f\u0017\u00001=2!0when\u000f!0we may assume that p(yj;\f)<1=2, and therefore\n(67)\u0000(1 +c\u000f\u0017\u00001=2)p(yj;\f)<log(1\u0000p(yj;\f))<\u0000p(yj;\f);\nwhich provides an asymptotic expression for p0(x;v;t )once the sumPn\nj=1p(yj;\f)has\nbeen evaluated. That (Vt)0g(x;v)has the form (60) is evident, and hence it remains to\nprove the estimate (64). From (67) we get\n\f\fe\u00002t\u0000p0(x;v;t )\f\f\u0014\f\f\fe\u00002t\u0000ePn\nj=1p(y1\f)\f\f\f+\f\f\fe\u0000(1+c\u000f\u0017\u00001=2)Pn\nj=1p(y1;\f)\u0000e\u0000Pn\nj=1p(yj;\f)\f\f\f\n\u0014\f\f\fe\u00002t\u0000e\u0000Pn\nj=1p(y1;\f)\f\f\f+c\u000f\u0017\u00001=2: (68)\nReferring to the notation in Lemma 3.2 the probability p(yj;\f)can be computed as\n(69) p(yj;\f) =p[ 1\u000f](y;\f);\nthat is, 0is set to one in the interval [\u00001;1]and zero outside this interval. The same\nlemma then gives\n(70)nX\nj=1p(y1;\f) =2p\u000fn\ncos(\f)+Ra(n;\u000f;y 1;\f) = 2t+^Ra;\nwhereRasatisfies the estimate (48), and jbRa\u0000Raj\u00142p\u000f. Hencej^Ra(n;\u000f;y 1;\f)j\u0014\nC\u0016RA(\f)for some constant C, and Markov’s inequality says that for any \u0015 > 0, the\ninequalitym(fv2S1j\u0016RA>\u0015g)\u0014C\n\u0015\u000f\u0017\u00001=2(1 + log(t=p\u000f)holds. Therefore\nZ\nS1\r\re\u00002t\u0000p0(\u0001;v;t)\r\r\nL1dv\u0014Z\nS1\r\r\re\u00002t\u0000e\u00002t+^R3\r\r\r\nL1dv+ 2\u0019c\u000f\u0017\u00001=2\n\u0014C\u0010\nm(fv2S1j\u0016RA>\u0015g) +\u0015+\u000f\u0017\u00001=2\u0011\n\u0014\n\u0014C\u000f\u0017\u00001=2(1 + log(t=p\u000f))\n\u0015+C\u0015+C\u000f\u0017\u00001=2\u0014C\u000f(2\u0017\u00001)=4q\n1 + log(t=p\u000f): (71)14 BERNT WENNBERG(1)\nwhere the norms inside the integrals are taken with respect to the variable x. Then (64)\nfollows after integrating the expression over R2because\n(72)\r\r\r(Vt)0g\u0000(eVt\n\u000f)0g\r\r\r\nL1(R2\u0002S1)\u0014kgkL1(R2\u0002S1)\u0019M2Z\nS1\r\rp0(\u0001;v;t)\u0000e\u00002t\r\r\nL1dv;\nwheng= 0forjxj>M .\n\u0003\nPROPOSITION 3.5. Lett>\u000fawitha<(2\u0017\u00001)=4. Then ifg= 0forjxj>M ,\n(73)\r\r\r\r\r1X\nk=2(eVt\n\u000f)kg\r\r\r\r\r\nL1(R2\u0002S1)\u0014CkgkL1M2t2:\nProof: LetBMbe the ball of radius MinR2, and consider a path starting at (x;v)2\nBM\u0002S1, and setJk=Jk(x;v;t )be the event that this path has exactly kvelocity jumps\nin the time interval [0;t]. Similarly, let Jk+denote the same for the case of at least kjumps.\nThen\n(74)\r\r\r\r\r1X\nk=2(eVt\n\u000f)kg\r\r\r\r\r\nL1(R2\u0002S1)\u0014kgkL1(R2\u0002S1)Z\nBM\u0002S1P[J2+j(x;v)]dxdv;\nthat is, the conditional probability that a path has at least two velocity jumps given the\ninitial position and velocity (x;v)is integrated over xandv. Considering J2+(x;v;t )for\none octant of S1at a time, we may represent vby an angle \f2[0;\u0019=4]as in the proof\nof Lemma 3.2. A path in the direction of \fstarting atxwill traverse (almost exactly)\nn=tcos(\f)=p\u000fif it is not reflected on an obstacle along the way, and so\nP[J2+j(x;v)] =nX\nj=1p0(c;v;tj\u0000)pjP[J1+(x0;v0;t\u0000tj+)j(j;x0;v0)] (75)\nthat is, as a sum of terms conditioned on the event that the first jump takes place at the j-th\npassage of a lattice cell. Clearly\nP[J2+j(x;v)]\u0014nX\nj=1pjP[J1+(x0;v0;t)j(j;x0;v0)]; (76)\nand the terms pjP[J1+(x0;v0;t)j(j;x0;v0)]can be expressed as p[ \u000f](y;\f)with\n(77)  \u000f=\u0010\n1\u0000p0(x0;v0(r=\u000f))\u0011\n1jrj\u0014\u000f\nin Lemma 3.2; p0(x0;v0(r=\u000f);t)is the probability that there is no collision in a path of\nlengthtstarting atx0in the direction of v0, and this direction is given as the outcome of a\ncollision with an obstacle with impact parameter r=\u000f. Using Proposition 3.4 we find that\f\fp0(x0;v0(r=\u000f);t)\u0000e\u00002t\f\f\u0014\u0016RB(\f0(r=\u000f)), where\f0is the angular direction correspond-\ning tov0. Rescaling \u000fgives\n\f\f\f\fZ1\n\u00001 0dr\u00002\u0000\n1\u0000e\u00002t\u0001\f\f\f\f\u0014Z1\n\u00001\u0016RB(v0(r);t)dr\n=Z\nS1\u0016RB(\u000f;v0)cos(\f0=2)\n2dv0; (78)\nwhere\f0is the scattering angle of v0with respect to the velocity before scattering, v. It\nfollows that\n(79)k 0kL1\u00142(1\u0000e\u00002t) +C\u000f(2\u0017\u00001)=4q\n1 + log(t=p\u000f):THE LORENTZ GAS WITH A NEARLY PERIODIC DISTRIBUTION OF SCATTERERS 15\nTherefore, using Lemma 3.2, the righthand side of Eq. 75 is bounded by\n(80)p\u000fn\ncos(\f)k 0kL1+Ck 0kL1\u0016RA(\f);\nand integrating with respect to voverS1and thenxoverBM, gives\nkgkL1M2C\u0010\n1\u0000e\u00002t+r1=2\n\u000f\u0011\u0012p\u000fn\ncos(\f)+r\u000f\u0013\n\u0014kgkL1M2C\u0010\nt2+t(r1=2\n\u000f+r(t;\u000f)) +r3=2\n\u000f\u0011\n(81)\nas a bound for the right hand side of Eq. (74). Here r\u000f=\u000f(2\u0017\u00001)=2(1 + log(t=p\u000f), andt\nis assumed to be small. The proof is concluded by comparing the terms when t>\u000fawith\na<(2\u0017\u00001)=4.\n\u0003\nThe following proposition concerns the terms (Vt)1g(x;v)and\u0010\neVt\n\u000f\u0011\n1g(x;v)defined\nby\n(82)\u0000\nVt\u0001\n1g(x;v) =e\u00002tZt\n0Z\nS1\n\u0000g(x+\u001cv+ (t\u0000\u001c)v0;v0)jv\u0001!jd!d\u001c\nand\n(83)\u0010\neVt\n\u000f\u0011\n1g(x;v) =E\u0002\ng(x+ ~\u001cv+ (t\u0000~\u001c)v0;v0) 1J1(x;v;t)\u0003\n;\nwhere as before, J1(x;v;t ))is the event that there is exactly one jump on the trajectory\nstarting atxin the direction of v. The ~\u001c2]0;t[andv02S1are the random jump time and\nvelocity of the particle after the jump.\nPROPOSITION 3.6. Let!(\u000e;g)be the modulus of continuity for g,i.e.a function such\nthatjg(x;v)\u0000g(x1;v1)j\u0014!(jx\u0000x1j+jv\u0000v1j;g). Then\n\r\r\r\u0000\nVt\u0001\n1g\u0000\u0010\neVt\n\u000f\u0011\n1g\r\r\r\nL1(R2\u0002S1)\n\u0014CM2\u0010\nt!(t;g) +r\u000f!(t;g) +kgkL1r\u000f+kgkL1tr1=2\n\u000f\u0011\n(84)\nwhere the support of gis contained inf(x;v)jjxj<Mgandr\u000f=\u000f\u0017\u00001=2(1 + logn).\nProof: First, because gis assumed to have compact support, the modulus of continuity\nexists, and!(\u000e;g)!0when\u000e!0. Definegx(v) =g(x;v),i.e.a function depending\nonly onvbut withxas a parameter. Because jvj=jv0j= 1, we then havejgx(v0)\u0000g(x+\n\u001cv+ (t\u0000\u001c)v0;v0)j\u0014!(t;g), and therefore\n(85)\f\f\u0000\nVt\u0001\n1gx\u0000\u0000\nVt\u0001\n1g\f\f\u0014t!(t;g);\nwhere the factor tmultiplying !(t;g)comes from the integral in the definition of Vt\n1.\nSimilarly\n(86)\f\f\f\u0010\neVt\n\u000f\u0011\n1gx\u0000\u0010\neVt\n\u000f\u0011\n1g\f\f\f\u0014(1\u0000p0(x;v;t ))!(t;g)\u0014\u0000\n1\u0000e\u00002t+\u0016RA(\f)\u0001\n!(t;g);\nwherep0and\u0016RAare defined as above. It remains to compare\n(87)\u0000\nVt\u0001\n1gx(v) =te\u00002tZ\nS1\n\u0000gx(v0)jv\u0001!jd!\nand\n(88)\u0010\neVt\n\u000f\u0011\n1gx(v) =E\u0002\ngx(v0) 1J1(]x;v;t)\u0003\n:16 BERNT WENNBERG(1)\nThe operators (Vt)1and\u0010\neVt\n\u000f\u0011\n1act only on the variable vofgx,xbeing considered as a\nparameter, and it is possible to use Lemma 3.2. By conditioning on the event that a velocity\njump takes place in the j-the passage of a cell,\nE\u0002\ngx(v0) 1J1(x;v;t)\u0003\n=nX\nj=1pjE\u0002\ng(v0)p0(x;v;tj\u0000)p0(xj;v0;t\u0000tj+)\f\fj\u0003\n(89)\nwheretj\u0000andtj+denote the time points when the trajectory enters and leaves cell number\njalong the path, xjis the (random) point of reflection of the trajectory inside cell j,v0the\nrandom new velocity, and finally, E[fjj]is shorthand for the expectation of fsubject to\nthe event that the jump takes place in cell number jalong the path. Then\nE\u0002\ngx(v0) 1J1(x;v;t)\u0003\n=e\u00002tnX\nj=1pjE\u0002\ng(v0)\f\fj\u0003\n+nX\nj=1pjEh\ng(v0)p0(x;v;tj\u00001)\u0010\np0(xj;v0;t\u0000tj\u0000)\u0000e\u00002(t\u0000tj+)\u0011\f\fji\n+nX\nj=1pjEh\ng(v0)\u0000\np0(x;v;tj\u00001)\u0000e\u00002tj\u0000\u0001\ne\u00002(t\u0000tj+)\f\fji\n+e\u00002tnX\nj=1pjEh\ng(v0)\u0010\ne2(tj+\u0000tj\u0000)\u00001)\u0011\f\fji\n(90)\nUsing Lemma 3.2 with  0(r) =gx(v0(r)) 1jrj\u00141one finds that the first term is\n(91) tZ1\n\u00001gx(v0(r))dr+Ra(n;\u000f;x;\f );\nwithRabounded bykgkL1\u0016RA(\f)andR\nS1\u0016RA(\f)dv0\u0014C\u000f\u0017\u00001=2(1 + log(t=p\u000f)). The\nsecond term is bounded by\n(92)kgkL1nX\nj=1pjE\u0002\u0016RB(\f0(r=\u000f))\f\fj\u0003\n=kgkL1tk\u0016RB(\f(\u0001)kL1+kgkL1Rb(n;\u000f;x;v );\nwhere, in the same way as before, jRbj\u0014k \u0016RB(\f(\u0001)kL1\u0016RA(\f). Similarly the third term\nis bounded by\nkgkL1nX\nj=1\f\fp0(x;v;tj\u00001)\u0000e\u00002tj\u0000\f\fpj\u0014CkgkL1\u0016RB(\f)\u0000\nt+C\u0016RA(\f)\u0001\n;\n\u0014CkgkL1t\u0016RB(\f) +CkgkL1\u0016RA(\f); (93)\nand the last term by\n(94) Cp\u000f\u0000\nt+\u0016RA(\f)\u0001\n:\nAdding the error estimates from (85), (86), (91), (92), (93),and (94) and integrating over \f\ngives\nZ\nS1\f\f\f\u0000\nVt\u0001\n1g(x;v)\u0000\u0010\neVt\n\u000f\u0011\n1g(x;v)\f\f\fdv\n\u0014C\u0010\nt!(t;g) +r\u000f!(t;g) +kgkL1r\u000f+kgkL1tr1=2\n\u000f\u0011\n; (95)\nagain writing r\u000ffor the expression \u000f\u0017\u00001=2(1 + logn). This concludes the proof, because\nthe integration over the support of ginxis trivial.\n\u0003THE LORENTZ GAS WITH A NEARLY PERIODIC DISTRIBUTION OF SCATTERERS 17\n4. E QUIVALENCE OF PROCESSES AND PROPAGATION OF CHAOS\nThe Lorentz process and the Markovian Lorentz process are equivalent on the set of\ntrajectories that don’t return to the same obstacle patch, and the purpose of this section\nis to prove that the probability that a path in the Markovian process returns to the same\npatch vanishes in the limit as \u000f!0. This is a geometrical construction which consists\nin estimating the measure of the set B\u000f\u001aR2\u0002S1of initial values that result in a path\nreturning to the same obstacle patch. This subset depends on the realization of the obstacle\npositions, but the estimates of the measure m(B\u000f)of this set do not, and m(B\u000f)!0\nwhen\u000f!0. A very similar calculation is then made to prove that the probability that\na pair of trajectories i the Markovian process meet in an obstacle range also converges to\nzero when\u000fdoes. The same calculation could be repeated for any number of simultaneous\ntrajectories, and from that one can conclude that propagation of chaos holds for this system.\nConsider then a path passing through an obstacle range, and then returning to the same\npatch. All such paths can be enumerated by giving the relative (integer) coordinates of the\nlattice cells where the path changes direction, that is \u0018j2Z2denotes the difference of the\ninteger coordinates of the end point and starting point of a straight line segment of the path.\nThis is illustrated in Fig. 5, where the starting point pof the loop is indicated with a black\ndot in cell nr. 0, which in this case is a point where the path is reflected, but in most cases\nit would not be. The point is not the starting point of the particle path, which may have a\nlong history before entering this particular loop. For a loop with ncollisions, we therefore\nhave\n(96) \u00181+\u00182+:::+\u0018n+1= 02Z2:\nFor a given time interval t2[0;\u0016t[, we must have\n(97) j\u0018ij+j\u00182j+\u0001\u0001\u0001+j\u0018n+1j\u0014\u0016t\u000f\u00001=2;\nbecause the size of a lattice cell isp\u000fin the scaling we are considering here. The notation\nis indicated in Fig. 5, where the first few and the last segment of a loop are indicated with\na solid line, the segments joining the lattice centers with dashed lines, and angles \fjof the\npath segments, as well as the integer coordinates.\nFIGURE 5. A path making a loop by returning to the same obstacle\nrange in the lattice cell that here is indexed by 02Z218 BERNT WENNBERG(1)\nWhen\u000fis small, the path segments are almost parallel to the corresponding segment\njoining the lattice cell centers, and the same holds for the segment lengths. In the worst\ncase, with a segment joining two neighboring cells, the relative error at most of order\nO(\u000f1\u0000\u0017). In the following computation, all lengths are expressed in terms of the inte-\nger coordinates, and the error coming from this approximation is taken into account by a\nconstant denoted c.\nA loop returning to the same obstacle range is not uniquely determined by the integer\nsequence\u00181;:::;\u0018n+1, because there is some freedom in setting the initial position and\nangle of the path. The last angle \fn+1must belong to an interval of width \u0001\fn+1that\nsatisfies\n(98) \u0001\fn+1\u0014c\u000f3=2\u0000\u0017\np\u000fj\u0018n+1j\nwhere the enumerator is the diameter of the obstacle range, andp\u000fj\u0018n+1jis the length of\nthe last segment. And it is an easy argument to see that then the angle of the n-th segment\nmust belong to an interval of width smaller than\n(99) \u0001\fn\u0014c\u000fp\u000fj\u0018nj\u0001\fn+1:\nFollowing the path backwards gives\n(100) \u0001\f1\u0014cn\u000fn=2\nj\u00181jj\u00182j\u0001\u0001\u0001\u0001\u0001j\u0018nj\u0001\fn+1\u0014cn+1\u000fn=2\nj\u00181jj\u00182j\u0001\u0001\u0001\u0001\u0001j\u0018nj\u000f1\u0000\u0017\nj\u0018n+1j:\nThe history of the path leading up to the point pis at most \u0016t, and hence the phase space vol-\nume spanned by the possible histories leading to a loop indexed by a sequence (\u00181;:::;\u0018n)\nis bounded by \u0016tc\u000f3=2\u0000\u0017\u0001\f1, the length of the path times the diameter of of the obstacle\nrange times the angular interval. The constant cis there to account for the difference be-\ntween the diameter of the obstacle range and obstacle patch, and can be taken as close to\n1 as one wishes. Of course the history is not likely to be one straight line segment, but\ndifferent histories leading a particular loop may be very different, with none or many re-\nflections. However, it is well-known that the so called billiard map is measure preserving.\nLet@\n =S\nz2p\u000fZ2fx2R2jx\u0000z=\u000f3=2\u0000\u0017g,i.e.the union of all obstacle boundaries. If\n(x;v)2@\n\u0002S1\n+, a point on the boundary of an obstacle, with velocity pointing out from\nthe obstacle, then the billiard map is the map (x;v)7!(x1;v1)where (x1;v1)are the po-\nsition and outgoing velocity after the trajectory hits the next obstacle. This map preserves\nthe measurejn\u0001vjdxkdv, wherenis the normal point out from the obstacle at xanddxkis\nthe length mesure of the obstacle boundary, see e.g. [4]. This means exactly that even if the\nset of histories leading leading to the loop indexed by (\u00181;:::;\u0018n)splits into a complicated\nform, the phase space volume in the full space still satisfies the same bound. Because the\nloop is indexed with coordinates relative to the starting point, all periodic translates of a\npoint of the history maps into a loop of with the same index sequence, the fraction of points\n(x;v)2B\u0002S1, whereBis any one lattice cell, that leads to a loop with the given index\nis bounded by\n\u0016tc\u000f3=2\u0000\u0017\u0001\f1m(B)\u00001\u0014\u0016tc\u000f3=2\u0000\u0017cn+1\u000fn=2\nj\u00181jj\u00182j\u0001\u0001\u0001\u0001\u0001j\u0018nj\u000f1\u0000\u0017\nj\u0018n+1j\u000f\u00001\n=\u0016tccn+1\u000f2(1\u0000\u0017)+(n\u00001)=2 1\nj\u00181jj\u00181j\u0001\u0001\u0001j\u0018n+1j: (101)\nTo compute an estimate of the fraction of (x;v)2B\u0002S1that leads to any loop with n+1\nsegments it is enough to sum this expression over the set of (\u00181;:::;\u0018n;\u0018n+1)satisfying\n\u00181+\u00182+:::+\u0018n+1= 0;; \u0018j6= 0;\nj\u00181j+j\u00182j+:::+j\u0018n+1j\u0014\u0016t=p\u000f:THE LORENTZ GAS WITH A NEARLY PERIODIC DISTRIBUTION OF SCATTERERS 19\nUsingj\u00181j\u00001j\u0018n+1j\u00001\u00141\n2\u0000\nj\u00181j\u00002+j\u0018n+1j\u00002\u0001\n, we find that the sum is bounded by\nX 1\nj\u00181j2j\u00182j\u0001\u0001\u0001j\u0018nj\n\u0014CZ\njx1j\u0014\u0016t=p\u000f1\njx1j2dx1Z\njx2j+:::+jxnj\u0014\u0016t=p\u000f\u0000jx1j1\njx2j\u0001\u0001\u0001jxn+1jdx2\u0001\u0001\u0001dxn\n\u0014C(2\u0019)nZ\u0016t=p\u000f\n11\nsds(\u0016t=p\u000f)n\u00001\n(n\u00001)!=C(2\u0019)nlog(\u0016t=p\u000f)\u0016tn\u00001\n(n\u00001)!\u000f(n\u00001)=2:(102)\nThe last line has been obtained by changing to polar coordinates in R2, and taking the\nsimplex in the inner integral to span over the full length T=p\u000f. Therefore, summing the\nestimate in (101) first over the loops of length n+1and then over n= 1\u0001\u0001\u00011 the following\nestimate for the fraction of initial points in phase space that result in a loop along the path:\n(103) \u0016tCec\u0016tlog(\u0016t=p\u000f)\u000f2(\u0017\u00001):\nThis calculation may be summarized as a proposition:\nPROPOSITION 4.1. LetTt\nY\u000f;\u000fbe the Lorentz process as in Theorem 2.1. Denote the\nevent that there is a loop along the path of length \u0016tstarting at (x;v)byL(x;v). Then\n(104)Z\nR2\u0002S1f0(x;v)Eh\ng(Tt\nY\u000f;\u000f(x;v)) 1L(x;v)i\ndxdv!0\nwhen\u000f!0\nProof: It is enough to see that the integral is bounded by\n(105) kgk1Z\nBM\u0002S1f0(x;v) 1L(x;v)dxdv\nwhereBMis the ball of radius MinR2, and assumed to contain the support of g. This is\nbounded by\nkgk1Z\nBM\u0002S1f0(x;v) 1ff0>\u0015gdxdv +kgk1\u0015Z\nBM\u0002S11L(x;v)[ff<\u0015gdxdv\n\u0014kgk1Z\nBM\u0002S1f0(x;v) 1ff0>\u0015gdxdv +kgk1\u0015\u0016tCec\u0016tlog(\u0016t=p\u000f)\u000f2(\u0017\u00001): (106)\nThis expression can be made arbitrarily small by first choosing \u0015large enough to make the\nfirst term as one wishes, and then the second term can be made equally small by choosing\n\u000fsufficiently small. All this is uniform in the random positions of the obstacles.\n\u0003\nThe proof of propagation of chaos is similar in many ways. Consider two paths of the\nMarkovian Lorentz model with independent initial conditions (x;v)and(x0;v0). Because\nthere is no interaction between the two particles, the particles remain independent until\nthey meet the same obstacle range, if ever. As in the previous calculation, take a fixed\nrealization of the random configuration of obstacles in each obstacle range. Let A(x;z)\nbe the set of angles v2S1such that there is possible path from the point x2R2to the\nobstacle range with center at z2p\u000fZ2. The set of angles such that the path reaches the\npatch before hitting an obstacle is bounded by \u000f3=2\u0000\u0017=jx\u0000zj, and the measure of angles v\nsuch that the path meets the obstacle patch after at least one collision with an obstacle can\nbe computed as above. The result is that\n(107) m(A(x;z))\u0014C\u000f3=2\u0000\u0017\u00121\njx\u0000zj+eC\u0016tlog(\u0016t=p\u000f)\u0013\n;20 BERNT WENNBERG(1)\nand the set of angles vandv0such that both the path Tt(x;v)andTt(x0;v0)meet the same\nobstacle range is bounded by\nm\u0000\b\n(v;v0)2(S1)2jv2A(x;z);v02A(x0;z)\t\u0001\n\u0014C\u000f3\u00002\u0017\u00121\njx\u0000zj+eC\u0016tlog(\u0016t=p\u000f)\u0013\u00121\njx0\u0000zj+eC\u0016tlog(\u0016t=p\u000f)\u0013\n: (108)\nThis expression should now be summed over all possible z2p\u000fZ2, and because the speed\nof a particle is equal to one, these zbelong to a ball of diameter \u0016t. As above we find\nX\nz2Z2;jzj<\u0016t=p\u000f1\njx\u0000zj1\njx\u0000zj\u00141\n2X\nz2Z2;jzj<\u0016t=p\u000f\u00121\njx\u0000zj2+1\njx0\u0000zj2\u0013\n\u0014Clog(\u0016t=p\u000f); (109)\nand\nX\nz2Z2;jzj<\u0016t=p\u000f\u00121\njx\u0000zj+1\njx\u0000zj\u0013\n\u0014C\u0016t=p\u000f: (110)\nThe dominating term when summing the expression in (108) comes from the part that does\nnot depend on z. We get\n(111)X\nz2Z2;jzj<\u0016t=p\u000fm\u0000\b\n(v;v0)2(S1)2jv2A(x;z);v02A(x0;z)\t\u0001\n\u0014C(1+\u0016t2)eC\u0016tlog(\u0016t=p\u000f)\u000f2(1\u0000\u0017);\nwhich again decreases to zero when \u000fdoes. The maps Tt\nY\u000fandeTt\nY\u000fextend to pairs of\nparticles in a natural way, so that\n(x(t);v(t);x0(t);v0(t)) =Tt\nY\u000f(x0;v0;x0\n0;v0\n0)and\n(~x(t);~v(t);~x0(t);~v0(t)) =eTt\nY\u000f(x0;v0;x0\n0;v0\n0) (112)\ndenote the position in phase space of a pair of particles evolving with the Lorentz process\nand the Markovian Lorentz process respectively. In the latter case the obstacle position\ninside an obstacle patch is determined independently for the two particles, and every time\na path meets an obstacle range, and therefore\n(113) eTt\nY\u000f(x0;v0;x0\n0;v0\n0) = (eTt\nY\u000f(x0;v0);eTt\nY\u000f(x0\n0;v0\n0)):\nIn the Lorenz evolution this is breaks down as soon as there is a loop for one of the particle\npaths, or when the two particle paths meet in one and the same obstacle range. However,\nthe computation above leads to the following theorem:\nTHEOREM 4.2. LetTt\nY\u000f;\u000fbe the Lorentz process as in Theorem 2.1. Then for g1;g22\nC(BM\u0002S1)\nlim\n\u000f!0Z\n(R2\u0002S1)2f0(x;v)f0(x0;v0)\u0010\nEh\ng1\u000eg2(Tt\nY\u000f;\u000f(x;v;x0;v0))i\n\u0000\nEh\ng1(Tt\nY\u000f;\u000f(x;v))i\nEh\ng2(Tt\nY\u000f;\u000f(x0;v0))i\u0011\ndxdvdx0dv0= 0: (114)\nThis says that the evolution of two particles in the Lorentz gas become independent in the\nlimit as\u000f!0.\nProof: LetL2(x;v;x0;v0)denote the event that the two particle paths starting at (x;v)and\n(x0;v0)and evolving in the same random obstacle configuration meet in an obstacle range.\nThen\nEh\ng1\u000eg2(Tt\nY\u000f;\u000f(x;v;x0;v0))(1\u0000 1L2(x;v;x0;v0))i\n=Eh\ng1(Tt\nY\u000f;\u000f(x;v))g2(Tt\nY\u000f;\u000f(x0;v0))(1\u0000 1L2(x;v;x0;v0))i\n(115)THE LORENTZ GAS WITH A NEARLY PERIODIC DISTRIBUTION OF SCATTERERS 21\nTherefore the integral in (114) is bounded by\n(116)kg1k1kg2k1E\"Z\n(R2\u0002S1)2f(x;v)f(x0;v0) 1B(x;v;x0;v0);dxdvdx0dv0#\n:\nFix\">0arbitrary, and take M > 0and\u0015>0so large that\n(117)Z\nR2\u0002S1f(x;v)( 1jxj>M+ 1f>\u0015)dxdv<\":\nWe find that (114) is bounded by\n(118) kg1k1kg2k12\"+C(\u0016t)\u00152R2log(\u0016t=p\u000f)\u000f2(1\u0000\u0017)\nwhich can be made smaller then 2\"by choosing \u000fsufficiently small. Here C(\u0016t)is a\u0016t\ndepending constant coming from the expression (111). This concludes the proof because \"\nwas arbitrarily small.\n\u0003\nTheorem 4.2 could have been proven for tensor products of any order with exactly the\nsame kind of computations, and therefore this really is a proof that propagation of chaos\nholds. The rate of convergence in the theorem depends on the density f0, and also on the\ntime interval. The dependence of fcould have been replaced by bounds of the moments\nand entropy. The dependence of the time interval is much more difficult to get passed, but\ncould maybe be addressed as in [10] which deals with the Lorentz gas in a Poisson setting.\nREFERENCES\n[1] Jean Bourgain, François Golse, and Bernt Wennberg. On the distribution of free path lengths for the periodic\nLorentz gas. Comm. Math. Phys. , 190(3):491–508, 1998.\n[2] E. Caglioti, M. Pulvirenti, and V . Ricci. Derivation of a linear Boltzmann equation for a lattice gas. Markov\nProcess. Related Fields , 6(3):265–285, 2000.\n[3] Emanuele Caglioti and François Golse. On the Boltzmann-Grad limit for the two dimensional periodic\nLorentz gas. J. Stat. Phys. , 141(2):264–317, 2010.\n[4] Nikolai Chernov and Roberto Markarian. Chaotic billiards , volume 127 of Mathematical Surveys and\nMonographs . American Mathematical Society, Providence, RI, 2006.\n[5] Giovanni Gallavotti. Rigorous theory of the Boltzmann equation in the Lorentz gas. Nota Interna, Istituto\ndi Fisica, Università di Roma , (358), 1972.\n[6] Giovanni Gallavotti. Statistical mechanics . Texts and Monographs in Physics. Springer-Verlag, Berlin, 1999.\nA short treatise.\n[7] Iosif I. Gikhman and Anatoli V . Skorokhod. The theory of stochastic processes. I . Classics in Mathematics.\nSpringer-Verlag, Berlin, 2004. Translated from the Russian by S. Kotz, Reprint of the 1974 edition.\n[8] François Golse. On the periodic Lorentz gas and the Lorentz kinetic equation. Ann. Fac. Sci. Toulouse Math.\n(6), 17(4):735–749, 2008.\n[9] Hendrik Lorentz. Le mouvement des électrons dans les méteaux. Arch. Néerl. , 10:336–371, 1905.\n[10] Christopher Lutsko and Bálint Tóth. Invariance principle for the random Lorentz gas—beyond the\nBoltzmann-Grad limit. Comm. Math. Phys. , 379(2):589–632, 2020.\n[11] Jens Marklof and Andreas Strömbergsson. Kinetic transport in the two-dimensional periodic Lorentz gas.\nNonlinearity , 21(7):1413–1422, 2008.\n[12] Jens Marklof and Andreas Strömbergsson. The distribution of free path lengths in the periodic Lorentz gas\nand related lattice point problems. Ann. of Math. (2) , 172(3):1949–2033, 2010.\n[13] Jens Marklof and Andreas Strömbergsson. The Boltzmann-Grad limit of the periodic Lorentz gas. Ann. of\nMath. (2) , 174(1):225–298, 2011.\n[14] Jens Marklof and Andreas Strömbergsson. The periodic Lorentz gas in the Boltzmann-Grad limit: asymp-\ntotic estimates. Geom. Funct. Anal. , 21(3):560–647, 2011.\n[15] Jens Marklof and Andreas Strömbergsson. Free path lengths in quasicrystals. Comm. Math. Phys. ,\n330(2):723–755, 2014.\n[16] Jens Marklof and Andreas Strömbergsson. Generalized linear Boltzmann equations for particle transport in\npolycrystals. Appl. Math. Res. Express. AMRX , (2):274–295, 2015.\n[17] Jens Marklof and Andreas Strömbergsson. Kinetic theory for the low-density Lorentz gas.\narXiv:1910.04982, to appear in Memoirs of the AMS , 2019.\n[18] Ian Melbourne, Françoise Pène, and Dalia Terhesiu. Local large deviations for periodic infinite horizon\nLorentz gases, 2021.22 BERNT WENNBERG(1)\n[19] Françoise Pène and Dalia Terhesiu. Sharp error term in local limit theorems and mixing for Lorentz gases\nwith infinite horizon. Communications in Mathematical Physics , 382(3):1625–1689, feb 2021.\n[20] Valeria Ricci and Bernt Wennberg. On the derivation of a linear Boltzmann equation from a periodic lattice\ngas.Stochastic Process. Appl. , 111(2):281–315, 2004.\n[21] Herbert Spohn. The Lorentz process converges to a random flight process. Comm. Math. Phys. , 60(3):277–\n290, 1978."    },    {        "title": "2402.16133v1.Variable_martingale_Hardy_Lorentz_Karamata_spaces_and_their_applications_in_Fourier_Analysis.pdf",        "content": "VARIABLE MARTINGALE HARDY-LORENTZ-KARAMATA SPACES\nAND THEIR APPLICATIONS IN FOURIER ANALYSIS\nZHIWEI HAO\nSchool of Mathematics and Computing Science, Hunan University of Science and Technology,\nXiangtan 411201, People’s Republic of China\ne-mail: haozhiwei@hnust.edu.cn\nXINRU DING\nSchool of Mathematics and Computing Science, Hunan University of Science and Technology,\nXiangtan 411201, People’s Republic of China\ne-mail: dingxinru0724@163.com\nLIBO LI\nSchool of Mathematics and Computing Science, Hunan University of Science and Technology,\nXiangtan 411201, People’s Republic of China\ne-mail: lilibo@hnust.edu.cn\nFERENC WEISZ∗\nDepartment of Numerical Analysis, E¨ otv¨ os L. University,\nH-1117 Budapest, P´ azm´ any P. s´ et´ any 1/C, Hungary\ne-mail: weisz@inf.elte.hu\nAbstract. In this paper, we introduce a new class of function spaces, which unify and\ngeneralize Lorentz-Karamata spaces, variable Lorentz spaces and other several classical\nfunction spaces. Based on the new spaces, we develop the theory of variable martingale\nHardy-Lorentz-Karamata spaces and apply it to Fourier Analysis. To be precise, we\ndiscuss the basic properties of Lorentz-Karamata spaces with variable exponents. We in-\ntroduce five variable martingale Hardy-Lorentz-Karamata spaces and characterize them\nvia simple atoms as well as via atoms. As applications of the atomic decompositions, dual\ntheorems and the generalized John-Nirenberg theorem for the new framework are pre-\nsented. Moreover, we obtain the boundedness of σ-sublinear operator defined on variable\nmartingale Hardy-Lorentz-Karamata spaces, which leads to martingale inequalities and\nthe relation of the five variable martingale Hardy-Lorentz-Karamata spaces. Also, we in-\nvestigate the boundedness of fractional integral operators in this new framework. Finally,\nwe deal with the applications of variable martingale Hardy-Lorentz-Karamata spaces in\nFourier analysis by using the previous results. More precisely, we show that the partial\nsums of the Walsh-Fourier series converge to the function in norm if f∈Lp(·),q,bwith\n1< p−≤p+<∞. The Fej´ er summability method is also studied and it is proved that the\nmaximal Fej´ er operator is bounded from variable martingale Hardy-Lorentz-Karamata\nspaces to variable Lorentz-Karamata spaces. As a consequence, we get conclusions about\nalmost everywhere and norm convergence of Fej´ er means. The results obtained in this\npaper generalize the results for martingale Hardy-Lorentz-Karamata spaces and variable\nmartingale Hardy-Lorentz spaces. Especially, we remove the condition that bis nonde-\ncreasing in previous literature.\n2020 Mathematics Subject Classification. Primary: 60G42; Secondary: 42C10, 60G46, 46E30.\nKey words and phrases. variable Lorentz-Karamata space, atomic decomposition, martingale inequal-\nity, Walsh-Fourier series, Fej´ er means, maximal Fej´ er operator.\n∗Corresponding author, email: weisz@inf.elte.hu.\n1arXiv:2402.16133v1  [math.FA]  25 Feb 20242\n1.Introduction\nLet 0 < p < ∞, 0< q≤ ∞ andbbe a slowly varying function. The Lorentz-\nKaramata space Lp,q,b(Ω,F,P), or briefly Lp,q,b, is defined to be the set of all measurable\nfunctions fon a probability space (Ω ,F,P) such that ∥f∥p,q,b<∞, where\n∥f∥p,q,b:=\n\n\u0014Z∞\n0\u0000\nP(|f|> t)1/pγb(P(|f|> t))t\u0001qdt\nt\u00151\nq\n,if 0< q < ∞,\nsup\nt>0tP(|f|> t)1/pγb(P(|f|> t)), ifq=∞,\nwhere γb(t) =b(t−1), 0< t < 1. This class of function spaces was introduced in 2000 by\nEdmunds et al. in [19] and it was used to investigate the important problem of the optimal\nSobolev embeddings on regular domains in the Euclidean space. By taking different p,\nqandb, these spaces generalize the classical Lebesgue spaces, Lorentz spaces, Zygmund\nspaces, Lorentz-Zygmund spaces and the generalized Lorentz-Zygmund spaces. The theory\nof Lorentz-Karamata spaces is not only a pursuit of mere generality, but it shows the\nessential points of the issues while we are less likely to be overwhelmed by the technical\ndetails. Some significant results of the Lorentz-Karamata spaces are studied in [18, 38,\n44, 52, 58, 69, 85] and the references therein.\nAnother interesting topic we shall touch in our work is a kind of function spaces\nwith variable exponents. The variable Lebesgue space Lp(·)is an important class of non-\nrearrangement invariant spaces and it is defined by the (quasi)-norm\n∥f∥p(·):= inf(\nλ >0 :Z\nRn\u0012|f(x)|\nλ\u0013p(x)\ndx≤1)\n.\nRecently, this topic has a great development and appears in the research on endpoint\nanalysis and in different applications, such as elasticity, fluid dynamics, variational calculus\nand partial differential equations (see [2, 4, 6, 7, 17, 34, 61, 62, 73] and so on). Such\nspaces were firstly introduced by Orlicz [59] in 1931. Kov` aˇ cik and R´ akosn´ ık [49], Fan and\nZhao [20] investigated various properties of variable Lebesgue spaces and variable Sobolev\nspaces. Since some well known properties do not hold, such as the boundedness of the\ntranslation operator and the maximal inequality in a modular form, the theory of variable\nfunction spaces is difficult. A fundamental breakthrough in this topic is due to Diening,\nwho introduced the so-called log-H¨ older continuity condition as follows:\n|p(x)−p(y)|≲1\nlog(1 /|x−y|)\nand\n|p(x)−p∞|≲1\nlog(e+|x|)\nfor all x, y∈Rn, see [14, 15]. Heavily relying on this condition, the theory of variable\nHardy spaces has got a rapid development. One of the most important results states that\nthe classical Hardy-Littlewood maximal operator is bounded on variable Lebesgue spaces\n(see Cruz-Uribe et al. [10, 12], Nekvinda [57], Cruz-Uribe and Fiorenza [11] and Diening et\nal. [16]). Nakai and Sawano [56] firstly introduced the Hardy space Hp(·)(Rn) and discussed\nthe atomic decompositions. They also studied the duality and the boundedness of singular\nintegral operators as applications of atomic decompositions. Under some weaker conditionsVariable martingale Hardy-Lorentz-Karamata spaces 3\nonp(·), Cruz-Uribe and Wang [13] and Sawano [65] also characterized the variable Hardy\nspaces Hp(·)(Rn) and extended the conclusions in [56]. Later, Yan et al. [89] considered\nthe variable weak Hardy spaces Hp(·),∞(Rn) and characterized these spaces by the radial\nmaximal functions, atoms and Littlewood-Paley functions. Meanwhile, Jiao et al. [47]\ninvestigated the variable Hardy-Lorentz spaces Hp(·),q(Rn). The relevant conclusions on\nthe anisotropic Hardy spaces can be found in Liu et al. [50, 51] and the references therein.\nVery recently, Weisz [83] introduced a new fractional maximal operator and showed that\nit is bounded from variable Hardy spaces to variable Lebesgue spaces. He also proved\nthat a new type of maximal operators is bounded from variable Hardy-Lorentz spaces to\nvariable Lorentz spaces in [82].\nMotivated by the study of Lorentz-Karamata spaces and variable Lorentz spaces, we\nintroduce a new type of function spaces, the so-called variable Lorentz-Karamata space\nLp(·),q,bby the quasi-norm\n∥f∥p(·),q,b=\n\n\u0014Z∞\n0\u0000\nt∥χ{|f|>t}∥p(·)γb(∥χ{|f|>t}∥p(·))\u0001qdt\nt\u00151\nq\n,if 0< q < ∞,\nsup\nt>0t∥χ{|f|>t}∥p(·)γb(∥χ{|f|>t}∥p(·)), ifq=∞.\nThe variable Lorentz-Karamata spaces go back to the Lorentz-Karamata spaces and vari-\nable Lorentz spaces when p(·)≡pis a constant and b≡1, respectively. Moreover, we\nprove that the variable Lorentz-Karamata space is a quasi-Banach space. We also give\nsome fundamental geometric properties for these new spaces.\nInspired by the progress of real theory of variable Hardy spaces, the martingale theory\nof variable Hardy spaces has gained widespread attentions. The major difficulty in mar-\ntingale theory is that the probability space has no natural metric structure compared with\nEuclidean space Rn, which means that the above-mentioned log-H¨ older continuity condi-\ntion can no longer hold. In the last decade, Hao and Jiao [31] in 2015 (see also [42, 46])\novercame the last obstacle, and found a suitable replacement of the above log-H¨ older con-\ntinuity condition. That is, suppose that there exists a constant Kp(·)≥1 depending only\nonp(·) such that\nP(A)p−(A)−p+(A)≤Kp(·),∀A∈[\nn≥0A(Fn), (1.1)\nwhere Fnis generated by countable many atoms and A(Fn) denotes the family of all atoms\ninFn. Under this condition, Jiao et al. [42, 46] proved the Doob maximal inequality and\ndeveloped the martingale theory in the framework of variable exponent setting. Since\nthen, this branch of martingale theory has become a very active area of research, see\n[33, 39, 40, 42, 45, 80, 81, 83, 84]. Martingale Musielak-Orlicz Hardy spaces were considered\nin Xie et al. [35, 36, 41, 86, 87, 88].\nWe continue this line of investigation. We introduce a new type of Hardy spaces, the\nso called variable martingale Hardy-Lorentz-Karamata spaces, which are much more wider\nthan the martingale Hardy-Lorentz-Karamata spaces and variable martingale Hardy-\nLorentz spaces. The purpose of this paper is to make a systematic study of these new\nspaces. Under the condition (1.1), we discuss atomic decompositions, dual theorems, mar-\ntingale inequalities and the relation of the different martingale Hardy spaces. Compared\nwith the previous literature, we obtain some stronger conclusions even for martingale\nHardy-Lorentz-Karamata spaces and variable martingale Hardy-Lorentz spaces. In detail,4 Z. Hao, X. Ding, L. Li and F. Weisz\nwe remove the restriction that bis a nondecreasing slowly varying function in [44, 52, 85]\nand extend the range of qin [38, 40, 44, 48, 52]. Moreover, we give some applications\nof variable martingale Hardy-Lorentz-Karamata spaces in Fourier analysis. For example,\nunder some conditions, we show the convergence of the partial sums of the Walsh-Fourier\nseries and the boundedness of the maximal Fej´ er operator. As a consequence, we get\nconclusions about almost everywhere and norm convergence of Fej´ er means.\nThis paper is organized as follows. In Section 2, we give the definition of variable\nLorentz-Karamata spaces and some properties which will be used in the subsequent sec-\ntions. We also introduce five types of variable martingale Hardy-Lorentz-Karamata spaces,\nwhich generalize the variable martingale Hardy-Lorentz spaces and martingale Hardy-\nLorentz-Karamata spaces.\nThe target of Section 3 is to establish the atomic decompositions of variable mar-\ntingale Hardy-Lorentz-Karamata spaces. We generalize several known inequalities from\nmartingale Hardy-Lorentz-Karamata spaces and variable martingale Hardy-Lorentz spaces\nto variable martingale Hardy-Lorentz-Karamata spaces. More exactly, we remove the re-\nstriction of 0 < p≤1 and 0 < q≤pin [38, Theorem 5.3]; the condition that bis\nnondecreasing in [44, Theorem 3.1]. Moreover, it is worth noting that in [38, 44], they\nonly considered the atomic decomposition theorems with “ ∞-atoms”. We discuss the\natomic decompositions via “simple r-atoms” and “ ∞-atoms”, respectively. We extend\nand complete the relevant conclusions in [38, 40, 42, 44].\nThe objective of Section 4 is to show the martingale inequalities between different vari-\nable martingale Hardy-Lorentz-Karamata spaces. If {Fn}n≥0is regular, the equivalence of\ndifferent variable martingale Hardy-Lorentz-Karamata spaces is proved. The way we used\nis to find a sufficient condition for a σ-sublinear operator to be bounded from variable\nmartingale Hardy-Lorentz-Karamata spaces to variable Lorentz-Karamata spaces. The\nproof depends on the atomic decompositions of the variable martingale Hardy-Lorentz-\nKaramata spaces. We point out that, the martingale inequalities extend Theorem 4 .11 in\n[42] and Theorems 1 .2, 1.3 in [85].\nIn Section 5, we present the dual theorems of Hp(·),q,bas an application of the atomic\ndecomposition theorems. We give the definitions of the generalized martingale spaces\nBMO r,b(α(·)) and BMO r,q,b(α(·)), the special case of which are the BMO spaces in [40,\n44, 75]. We firstly get that the dual space of Hs\np(·),q,bisBMO 2,b(α(·)) if 0 < p−≤p+<1\nand 0 < q≤1. The result extends the dual theorems in [32, 38, 40, 44, 75]. Next we\nconsider the case 0 < p−≤p+<2, 0< q < ∞and prove that the dual of Hs\np(·),q,bis\nBMO 2,q,b(α(·)). For q=∞, the essential difficulty is that Lpis not dense in Lp,∞for\n0< p < ∞. To overcome this difficulty, we introduce a closed subspace of Hs\np(·),q,b, namely\nHs\np(·),∞,band verify that L2is dense in Hs\np(·),∞,b. We show that the dual space of Hs\np(·),∞,b\nisBMO 2,∞,b(α(·)). Furthermore, we note that in the dual theorems, bis not necessarily\nnondecreasing. Hence, we further extend the relevant conclusions in [38, 40, 44, 52].\nSection 6 is devoted to the John-Nirenberg theorems on variable Lorentz-Karamata\nspaces when the stochastic basis {Fn}n≥0is regular. For a constant exponent, the well\nknown classical John-Nirenberg theorem can be found in [23]. Jiao et al. [44] showed\nthe John-Nirenberg theorem for BMO r,q,b(α). Moreover, Jiao et al. [43] discussed the\nJohn-Nirenberg theorem for BMO r,q(α). For the generalized BMO martingale spaces\nassociated with variable exponents, the John-Nirenberg theorems were proved in [40, 90].\nIn this section, we generalize these theorems to variable Lorentz-Karamata spaces.Variable martingale Hardy-Lorentz-Karamata spaces 5\nIn Section 7, we deal with the boundedness of fractional integrals in HM\np(·),q,b. In\nmartingale theory, Chao and Ombe [9] firstly introduced the fractional integrals for dyadic\nmartingales. The boundedness of fractional integrals on martingale Hardy spaces for\n0< p≤1 was proved by Sadasue [63]. In [5, 43, 52, 55], the notion of fractional integrals\nwas extended to more general martingales. For fractional integrals associated with variable\nexponents, Hao and Jiao [31] considered the boundedness on variable martingale Hardy\nspaces. The boundedness of fractional integrals studied in Section 7 goes back to Liu and\nZhou [52, Theorem 4.4] if p(·)≡pis a constant.\nIn the last section, we investigate some applications of variable martingale Hardy-\nLorentz-Karamata spaces in Fourier analysis. Martingale theory has extensive applica-\ntions in dyadic harmonic analysis and in summability of Walsh-Fourier series, for constant\nexponents, see [24, 25, 26, 27, 28, 29, 54, 60, 66, 67, 68, 70, 71, 72, 78, 79] and the refer-\nences therein. Using dyadic martingale theory, Weisz [78] verified that the maximal Fej´ er\noperator σ∗is bounded from Hp,qtoLp,qunder the condition of p >1\n2. Motivated by\nthis result, Jiao et al. [42] discussed the boundedness of σ∗from Hp(·)toLp(·)and from\nHp(·),qtoLp(·),q. In Section 8, we show that the partial sums of the Walsh-Fourier series\nconverges to the function in norm if f∈Lp(·),q,bwith 1 < p−≤p+<∞. The boundedness\nof the maximal Fej´ er operator from variable martingale Hardy-Lorentz-Karamata spaces\nto variable Lorentz-Karamata spaces is also proved. As a consequence, we get conclusions\nabout almost everywhere and norm convergences of Fej´ er means. Furthermore, the con-\nclusions we get here extend the results in [42, 78] when b≡1. Note that these results are\nnew even if p(·) is a constant and b̸≡1.\nFinally, we end this section by making some conventions on notation. Throughout\nthis paper, we denote by ZandNthe set of integers and the set of nonnegative integers,\nrespectively. We denote by ca positive constant, which can vary from line to line. The\nsymbol A≲Bstands for the inequality A≤cB. If we write A≈B, then it means\nA≲B≲A. We use χIto denote the indicator function of a measurable set I.\n2.Preliminaries\nIn this section, we introduce a new class of function spaces, namely the variable\nLorentz-Karamata spaces. Meanwhile, some fundamental geometric properties of these\nspaces are given, including completeness and dominated convergence theorems. Based on\nthe new spaces, five types of variable martingale Hardy-Lorentz-Karamata spaces are given\nin the last subsection. Except where otherwise stated, we always assume that (Ω ,F,P) is\na complete probability space.\n2.1.Variable Lebesgue spaces. A measurable function p(·) : Ω→(0,∞) is called a\nvariable exponent. For a measurable subset A⊂Ω, we write\np+(A) := ess sup\nω∈Ap(ω) and p−(A) := ess inf\nω∈Ap(ω).\nFor brevity, we use the abbreviations\np+:=p+(Ω) and p−:=p−(Ω).\nDenote by P(Ω) the collection of all variable exponents p(·) such that 0 < p−≤p+<∞.\nThroughout the paper, given a variable exponent p(·), the conjugate variable exponent6 Z. Hao, X. Ding, L. Li and F. Weisz\np′(·) ofp(·) is defined pointwise by\n1\np(ω)+1\np′(ω)= 1, ω ∈Ω.\nThe variable Lebesgue space Lp(·):=Lp(·)(Ω) is the set of all measurable functions f\ndefined on (Ω ,F,P) such that for some λ >0,\nρ(f/λ) =Z\nΩ\u0012|f(·)|\nλ\u0013p(·)\ndP<∞.\nThe variable Lebesgue space equipped with the (quasi)-norm\n∥f∥p(·):= inf\b\nλ >0 :ρ(f/λ)≤1\t\nbecomes a (quasi)-Banach function space. We present some basic properties of functional\n∥ · ∥ p(·), which can be found in [11, 16, 56].\nRemark 2.1. For any f, g∈Lp(·), the following properties hold:\n1.∥f∥p(·)≥0,∥f∥p(·)= 0if and only if f= 0.\n2.∥cf∥p(·)=|c|∥f∥p(·)for any c∈C.\n3. for 0< l≤p:= min {p−,1},\n∥f+g∥l\np(·)≤ ∥f∥l\np(·)+∥g∥l\np(·) and ∥f+g∥p(·)≤21/l−1\u0000\n∥f∥p(·)+∥g∥p(·)\u0001\n.\nMoreover, we have the following lemmas for variable Lebesgue spaces, which will be\nuseful in the whole paper.\nLemma 2.2 ([40]).Letp(·)∈ P(Ω)with p+≤1. For any positive functions f, g∈Lp(·),\nwe have\n∥f∥p(·)+∥g∥p(·)≤ ∥f+g∥p(·).\nLemma 2.3 ([13]).Letp(·)∈ P(Ω)ands >0. Then for all f∈Lsp(·), there is\n∥|f|s∥p(·)=∥f∥s\nsp(·).\nLemma 2.4 ([11] or [20]) .Letp(·)∈ P(Ω). For any f∈Lp(·), we have\n(1)∥f∥p(·)<1 (resp. = 1, >1)if and only if ρ(f)<1 (resp. = 1, >1);\n(2)if∥f∥p(·)>1, then\nρ(f)1/p+≤ ∥f∥p(·)≤ρ(f)1/p−;\n(3)if0<∥f∥p(·)≤1, then\nρ(f)1/p−≤ ∥f∥p(·)≤ρ(f)1/p+.\nLemma 2.5 ([11]).Letp(·), q(·)∈ P(Ω). Ifp(·)≤q(·), then\n∥f∥p(·)≤2∥f∥q(·)for every f∈Lq(·).Variable martingale Hardy-Lorentz-Karamata spaces 7\n2.2.Slowly varying functions. For a function f: [1,∞)→(0,∞), we say that fis\nequivalent to a nondecreasing (resp. nonincreasing) function giff≈g. In order to define\nthe Lorentz-Karamata spaces with variable exponents, we recall the definition of slowly\nvarying functions.\nDefinition 2.6 ([18]).A Lebesgue measurable function b: [1,∞)→(0,∞)is said to be\na slowly varying function if for any given ε > 0, the function tεb(t)is equivalent to a\nnondecreasing function and the function t−εb(t)is equivalent to a nonincreasing function\non[1,∞).\nExample 2.7. Clearly, η(t)≡1and1 + log tare slowly varying functions, respectively\n(see[58]). Let 0< p < ∞,m∈Nandα= (α1, α2, . . . , α m)∈Rm. Define the family of\npositive functions {ℓk}m\nk=0on(0,∞)by\nℓ0(t) = 1 /tand ℓk(t) = 1 + log\u0000\nℓk−1(t)\u0001\n,0< t≤1,1≤k≤m.\nMoreover, define\nΘm\nα(t) =mY\nk=1ℓαk\nk(t).\nIt is easy to see that Θm\nαis a slowly varying function (see[18]). Moreover, it follows from\n[19]that(e+ log t)α(log(e+ log t))β(α, β∈R)andexp(√logt)are also slowly varying\nfunctions.\nLetbbe a slowly varying function. We define γbon (0 ,∞) by\nγb(t) =b\u0000\nmax{t, t−1}\u0001\n, t > 0.\nThis definition is from [18]. The following proposition shows some properties of the slowly\nvarying functions. We refer to [18, 58, 69] for more information of slowly varying functions.\nProposition 2.8. Letbbe a slowly varying function on [1,∞).\n(1)Ifbis a nondecreasing function, then γbis nonincreasing on (0,1].\n(2)For any given ε >0, the function tεγb(t)is equivalent to a nondecreasing function\nand the function t−εγb(t)is equivalent to a nonincreasing function on (0,∞).\n(3)Ifεandrare positive numbers, then there exists positive constants cεandCεsuch\nthat\ncεmin{rε, r−ε}b(t)≤b(rt)≤Cεmax{rε, r−ε}b(t), t > 0.\n(4)For any a >0, denote b1(t) =b(ta)on[1,∞). Then b1also is a slowly varying\nfunction.\n(5)For any given r∈R, the function bris a slowly varying function and γbr=γr\nb.\n(6)Let0< p≤ ∞ . For any positive constants αandβ, we have\n(α+β)pγb(α+β)≲αpγb(α) +βpγb(β).\nWe refer to [44, 69] for the proof of (1); (2) and (3) come from [58]; (4) and (5) were\nshowed in [18]; (6) can be found in [38].\nFurthermore, we extend Proposition 2.8 (6) to more general situations.\nLemma 2.9. Letbbe a slowly varying function and αi(i∈N)be positive constants. If\n1< p < ∞, thenX\ni∈Nαp\niγb(αi)≲\u0010X\ni∈Nαi\u0011p\nγb\u0010X\ni∈Nαi\u0011\n.8 Z. Hao, X. Ding, L. Li and F. Weisz\nIf0< p < 1, then\u0010X\ni∈Nαi\u0011p\nγb\u0010X\ni∈Nαi\u0011\n≲X\ni∈Nαp\niγb(αi).\nProof. We first consider the case of 1 < p < ∞. Let 0 < ε < p −1. Then p−ε >1 and\nwe know that\nX\ni∈Nαp−ε\ni≤\u0010X\ni∈Nαi\u0011p−ε\n. (2.1)\nSince tεγb(t) is equivalent to a nondecreasing function, there is\nαp−ε\nk\u0010X\ni∈Nαi\u0011ε\nγb\u0010X\ni∈Nαi\u0011\n≳αp\nkγb(αk),∀k∈N.\nCombining (2.1) with the above inequality, we have\nX\nk∈Nαp\nkγb(αk)≲X\nk∈Nαp−ε\nk\u0010X\ni∈Nαi\u0011ε\nγb\u0010X\ni∈Nαi\u0011\n≤\u0010X\ni∈Nαi\u0011p−ε\u0010X\ni∈Nαi\u0011ε\nγb\u0010X\ni∈Nαi\u0011\n.\nIf 0< p < 1, we set 0 < θ < 1−p. Then we obtain that θ+p <1 and\n\u0010X\ni∈Nαi\u0011θ+p\n≤X\ni∈Nαθ+p\ni.\nSince t−θγb(t) is equivalent to a nonincreasing function, there is\nαθ+p\nk\u0010X\ni∈Nαi\u0011−θ\nγb\u0010X\ni∈Nαi\u0011\n≲αp\nkγb(αk),∀k∈N.\nThis yields that\n\u0010X\nk∈Nαθ+p\nk\u0011\u0010X\ni∈Nαi\u0011−θ\nγb\u0010X\ni∈Nαi\u0011\n≲X\nk∈Nαp\nkγb(αk).\nThen we conclude that\n\u0010X\ni∈Nαi\u0011p\nγb\u0010X\ni∈Nαi\u0011\n=\u0010X\nk∈Nαk\u0011θ+p\u0010X\ni∈Nαi\u0011−θ\nγb\u0010X\ni∈Nαi\u0011\n≤\u0010X\nk∈Nαθ+p\nk\u0011\u0010X\ni∈Nαi\u0011−θ\nγb\u0010X\ni∈Nαi\u0011\n≲X\nk∈Nαp\nkγb(αk).\n□\nLemma 2.10. Letp(·)∈ P(Ω)andbbe a slowly varying function. If 0< θ < p , we have\n\r\r\r\rX\ni∈NχAi\r\r\r\rθ\np(·)γθ\nb\u0012\r\r\r\rX\ni∈NχAi\r\r\r\r\np(·)\u0013\n≲X\ni∈N∥χAi∥θ\np(·)γθ\nb\u0000\n∥χAi∥p(·)\u0001\n,Variable martingale Hardy-Lorentz-Karamata spaces 9\nwhere Ai∈ F,i∈Nare arbitrary sets. If p+< θ < ∞and(Ai)i∈Nare disjoint, then we\nhave\nX\ni∈N∥χAi∥θ\np(·)γθ\nb\u0000\n∥χAi∥p(·)\u0001\n≲\r\r\r\rX\ni∈NχAi\r\r\r\rθ\np(·)γθ\nb\u0012\r\r\r\rX\ni∈NχAi\r\r\r\r\np(·)\u0013\n. (2.2)\nMoreover, if bis nonincreasing and p+≤θ <∞, then (2.2)also holds.\nProof. For the case of 0 < θ < p , setθ\np≤α <1 and b1(t) =b(tα\nθ) for t∈[1,∞). From\nProposition 2.8 (4), (5) and Lemma 2.3, we know that bθ\n1is a slowly varying function and\nγθ\nb\u0000\n∥χAi∥p(·)\u0001\n=γbθ\u0000\n∥χAi∥p(·)\u0001\n=γbθ\u0010\n∥χAi∥α\nθ\np(·)α\nθ\u0011\n=γbθ\n1\u0000\n∥χAi∥p(·)α\nθ\u0001\n.\nThen according to Lemma 2.9, we get that\nX\ni∈N∥χAi∥θ\np(·)γθ\nb\u0000\n∥χAi∥p(·)\u0001\n=X\ni∈N∥χAi∥α\np(·)α\nθγbθ\n1\u0000\n∥χAi∥p(·)α\nθ\u0001\n≳\u0012X\ni∈N∥χAi∥p(·)α\nθ\u0013α\nγbθ\n1\u0012X\ni∈N∥χAi∥p(·)α\nθ\u0013\n=\u0012X\ni∈N∥χAi∥θ\nα\np(·)\u0013α\nγbθ\n1\u0012X\ni∈N∥χAi∥θ\nα\np(·)\u0013\n.\nSince 0 <θ\nα≤p, we see that ∥ · ∥ p(·)is aθ\nα-norm and\n\r\r\r\rX\ni∈NχAi\r\r\r\rθ\nα\np(·)≤X\ni∈N∥χAi∥θ\nα\np(·).\nThen\nX\ni∈N∥χAi∥θ\np(·)γθ\nb\u0000\n∥χAi∥p(·)\u0001\n≳\r\r\r\rX\ni∈NχAi\r\r\r\rθ\np(·)γbθ\n1\u0012\r\r\r\rX\ni∈NχAi\r\r\r\rθ\nα\np(·)\u0013\n=\r\r\r\rX\ni∈NχAi\r\r\r\rθ\np(·)γθ\nb\u0012\r\r\r\rX\ni∈NχAi\r\r\r\r\np(·)\u0013\n.\nNext we discuss the situation of p+< θ < ∞. Let 1 < β≤θ\np+andb2(t) =b(tβ\nθ) for\nt∈[1,∞). Obviously, bθ\n2is a slowly varying function and γθ\nb\u0000\n∥χAi∥p(·)\u0001\n=γbθ\n2\u0000\n∥χAi∥p(·)β\nθ\u0001\n.\nThen it follows from Lemma 2.9 thatX\ni∈N∥χAi∥θ\np(·)γθ\nb\u0000\n∥χAi∥p(·)\u0001\n=X\ni∈N∥χAi∥β\np(·)β\nθγbθ\n2\u0000\n∥χAi∥p(·)β\nθ\u0001\n≲\u0012X\ni∈N∥χAi∥p(·)β\nθ\u0013β\nγbθ\n2\u0012X\ni∈N∥χAi∥p(·)β\nθ\u0013\n.\nSince\u0000p(·)β\nθ\u0001\n+≤1, then we have the following estimation by Lemma 2.2:\nX\ni∈N∥χAi∥p(·)β\nθ≤\r\r\r\rX\ni∈NχAi\r\r\r\rp(·)β\nθ.10 Z. Hao, X. Ding, L. Li and F. Weisz\nHence, we get by the disjointness of ( Ai)i∈Nthat\nX\ni∈N∥χAi∥θ\np(·)γθ\nb\u0000\n∥χAi∥p(·)\u0001\n≲\r\r\r\rX\ni∈NχAi\r\r\r\rβ\np(·)β\nθγbθ\n2\u0012\r\r\r\rX\ni∈NχAi\r\r\r\rp(·)β\nθ\u0013\n=\r\r\r\rX\ni∈NχAi\r\r\r\rθ\np(·)γθ\nb\u0012\r\r\r\rX\ni∈NχAi\r\r\r\r\np(·)\u0013\n.\nIfbis nonincreasing and p+≤θ <∞, it follows from the definition of γband Lemma\n2.2 that\nX\ni∈N∥χAi∥θ\np(·)γθ\nb\u0000\n∥χAi∥p(·)\u0001\n≲X\ni∈N∥χAi∥θ\np(·)γθ\nb\u0012\r\r\r\rX\ni∈NχAi\r\r\r\r\np(·)\u0013\n=X\ni∈N∥χAi∥p(·)\nθγθ\nb\u0012\r\r\r\rX\ni∈NχAi\r\r\r\r\np(·)\u0013\n≤\r\r\r\rX\ni∈NχAi\r\r\r\rp(·)\nθγθ\nb\u0012\r\r\r\rX\ni∈NχAi\r\r\r\r\np(·)\u0013\n=\r\r\r\rX\ni∈NχAi\r\r\r\rθ\np(·)γθ\nb\u0012\r\r\r\rX\ni∈NχAi\r\r\r\r\np(·)\u0013\n.\nHence, this completes the proof. □\nMotivated by Lemma 2.10, we have the following lemma as another deduction of\nProposition 2.8 (6).\nLemma 2.11. Letp(·)∈ P(Ω)andbbe a slowly varying function. For any sets A, B∈ F,\nthere is\n∥χA∪B∥p(·)γb\u0000\n∥χA∪B∥p(·)\u0001\n≲∥χA∥p(·)γb\u0000\n∥χA∥p(·)\u0001\n+∥χB∥p(·)γb\u0000\n∥χB∥p(·)\u0001\n.\nProof. According to Remark 2.1, we have\n∥χA∪B∥p(·)≤ ∥χA+χB∥p(·)≲∥χA∥p(·)+∥χB∥p(·).\nSince tγb(t) is equivalent to a nondecreasing function, it follows from Proposition 2.8 (6)\nthat\n∥χA∪B∥p(·)γb\u0000\n∥χA∪B∥p(·)\u0001\n≲\u0000\n∥χA∥p(·)+∥χB∥p(·)\u0001\n·γb\u0000\n∥χA∥p(·)+∥χB∥p(·)\u0001\n≲∥χA∥p(·)γb\u0000\n∥χA∥p(·)\u0001\n+∥χB∥p(·)γb\u0000\n∥χB∥p(·)\u0001\n.\nTherefore, the proof is finished. □\n2.3.Variable Lorentz-Karamata spaces. In this subsection, we introduce a new class\nof function spaces as a generalization of Lorentz-Karamata spaces and variable Lorentz\nspaces.\nDefinition 2.12. Letp(·)∈ P(Ω),0< q≤ ∞ andbbe a slowly varying function. The\nvariable Lorentz-Karamata space Lp(·),q,b:=Lp(·),q,b(Ω)consists of all measurable functionsVariable martingale Hardy-Lorentz-Karamata spaces 11\nfwith a finite functional ∥f∥p(·),q,bgiven by\n∥f∥p(·),q,b=\n\n\u0014Z∞\n0\u0000\nt∥χ{|f|>t}∥p(·)γb(∥χ{|f|>t}∥p(·))\u0001qdt\nt\u00151\nq\n,if 0< q < ∞,\nsup\nt>0t∥χ{|f|>t}∥p(·)γb(∥χ{|f|>t}∥p(·)), ifq=∞.\nRemark 2.13. These spaces coincide with the Lorentz-Karamata spaces Lp,q,b 1when\np(·)≡pis a positive constant and b1(t) =b(t1/p)fort∈[1,∞). In fact, b1is a slowly\nvarying function by Proposition 2.8 (4) . Moreover, when p(·)≡p, for any f∈Lp(·),q,b, we\nhaveZ∞\n0\u0000\nt∥χ{|f|>t}∥pγb(∥χ{|f|>t}∥p)\u0001qdt\nt=Z∞\n0\u0002\ntP(|f|> t)1/pγb\u0000\nP(|f|> t)1/p\u0001\u0003qdt\nt\n=Z∞\n0\u0002\nt∥χ{|f|>t}∥pγb1\u0000\nP(|f|> t)\u0001\u0003qdt\nt,\nwhich means that ∥f∥p(·),q,b=∥f∥p,q,b 1for0< q < ∞. Similarly, it is obvious that\n∥f∥p(·),∞,b=∥f∥p,∞,b1. For more details of Lorentz-Karamata spaces, we refer the readers\nto[18, 37, 38, 58] and the references therein. Moreover, variable Lorentz-Karamata spaces\nbecome variable Lorentz spaces when b≡1. For an introduction to variable Lorentz spaces,\nwe refer to [11, 16, 42, 48] .\nIn order to discuss whether variable Lorentz-Karamata space is a (quasi)-normed\nspace, we shall show the embedding relationship on variable Lorentz-Karamata spaces.\nThis result extends the embedding relationship on Lorentz-Karamata spaces (see [18,\nTheorem 3 .4.45]).\nLemma 2.14. Suppose that p(·)∈ P(Ω),0< q 1≤q2≤ ∞ and the slowly varying\nfunctions b1, b2satisfy sup\n1≤t<∞b2(t)\nb1(t)<∞. Then Lp(·),q1,b1⊂Lp(·),q2,b2.\nProof. Since sup\n1≤t<∞b2(t)\nb1(t)<∞, there exists a constant C > 0 such that b2(t)≤Cb1(t) for\n1≤t <∞. Hence, we have\nγb2(t) =b2(1/t)≲b1(1/t) =γb1(t) (2.3)\nfor 0 < t≤1. When q1=∞, it yields q2=∞. For α >0, it is easy to see that\nα∥χ{|f|>α}∥p(·)γb2\u0000\n∥χ{|f|>α}∥p(·)\u0001\n≲α∥χ{|f|>α}∥p(·)γb1\u0000\n∥χ{|f|>α}∥p(·)\u0001\n.\nTaking the supremum for all α >0, we get Lp(·),∞,b1⊂Lp(·),∞,b2.\nNow we consider the case of 0 < q1< q2=∞. Suppose that f∈Lp(·),q1,b1andα >0.\nWe have\nh\nα∥χ{|f|>α}∥p(·)γb2\u0000\n∥χ{|f|>α}∥p(·)\u0001iq1=q1Zα\n0sq1−1ds·h\n∥χ{|f|>α}∥p(·)γb2\u0000\n∥χ{|f|>α}∥p(·)\u0001iq1\n≲Zα\n0h\ns∥χ{|f|>s}∥p(·)γb2\u0000\n∥χ{|f|>s}∥p(·)\u0001iq1ds\ns\n≲Zα\n0h\ns∥χ{|f|>s}∥p(·)γb1\u0000\n∥χ{|f|>s}∥p(·)\u0001iq1ds\ns.12 Z. Hao, X. Ding, L. Li and F. Weisz\nThis means\n∥f∥p(·),∞,b2= sup\nα>0α∥χ{|f|>α}∥p(·)γb2\u0000\n∥χ{|f|>α}∥p(·)\u0001\n≲\u0012Z∞\n0h\ns∥χ{|f|>s}∥p(·)γb1\u0000\n∥χ{|f|>s}∥p(·)\u0001iq1ds\ns\u00131/q1\n=∥f∥p(·),q1,b1.\nHence, we get Lp(·),q1,b1⊂Lp(·),∞,b2.\nFinally, we suppose that 0 < q1≤q2<∞. It follows from (2.3) that\n∥f∥q2\np(·),q2,b2=Z∞\n0tq2∥χ{|f|>t}∥q2\np(·)γq2\nb2\u0000\n∥χ{|f|>t}∥p(·)\u0001dt\nt\n=Z∞\n0tq2−q1∥χ{|f|>t}∥q2−q1\np(·)γq2−q1\nb2\u0000\n∥χ{|f|>t}∥p(·)\u0001\n×tq1∥χ{|f|>t}∥q1\np(·)γq1\nb2\u0000\n∥χ{|f|>t}∥p(·)\u0001dt\nt\n≲Z∞\n0tq2−q1∥χ{|f|>t}∥q2−q1\np(·)γq2−q1\nb2\u0000\n∥χ{|f|>t}∥p(·)\u0001\n×tq1∥χ{|f|>t}∥q1\np(·)γq1\nb1\u0000\n∥χ{|f|>t}∥p(·)\u0001dt\nt\n≤ ∥f∥q2−q1\np(·),∞,b2· ∥f∥q1\np(·),q1,b1≲∥f∥q2\np(·),q1,b1.\nTherefore, Lp(·),q1,b1⊂Lp(·),q2,b2holds again. □\nWe also recall the next embedding relationship for Lorentz-Karamata spaces.\nLemma 2.15 ([18]).Letp1, p2∈(0,∞), q1, q2∈(0,∞]with p2< p1andb1, b2be slowly\nvarying functions. Then Lp1,q1,b1⊂Lp2,q2,b2.\nBased on the above embedding relationship of variable Lorentz-Karamata spaces, we\ncollect the following properties of the functional ∥ · ∥ p(·),q,b.\nLemma 2.16. Letp(·)∈ P(Ω),0< q≤ ∞ andbbe a slowly varying function. The\nfollowing properties hold.\n(1)∥f∥p(·),q,b≥0,∥f∥p(·),q,b= 0if and only if f≡0.\n(2)∥cf∥p(·),q,b=|c|∥f∥p(·),q,bfor any c∈C.\n(3)For every fixed λ >0, the functional ∥ · ∥ p(·),q,bsatisfies the equivalence\n∥f∥p(·),q,b≈\n\n\u0012X\nk∈Zh\n2k∥χ{|f|>λ2k}∥p(·)γb\u0000\n∥χ{|f|>λ2k}∥p(·)\u0001iq\u00131\nq\n,if 0< q < ∞,\nsup\nk∈Z2k∥χ{|f|>λ2k}∥p(·)γb\u0000\n∥χ{|f|>λ2k}∥p(·)\u0001\n, ifq=∞.\n(4)For any set A∈ F withP(A)>0, we have\n∥χA∥p(·),q,b=\n\n\u00101\nq\u00111/q\n∥χA∥p(·)γb(∥χA∥p(·)),if 0< q < ∞,\n∥χA∥p(·)γb(∥χA∥p(·)), ifq=∞.Variable martingale Hardy-Lorentz-Karamata spaces 13\n(5)Lets >0andb1(t) =b(ts)fort∈[1,∞). For any f∈Lsp(·),sq,b1/s\n1, we have\n∥|f|s∥p(·),q,b=\n\ns1\nq∥f∥s\nsp(·),sq,b1/s\n1,if 0< q < ∞,\n∥f∥s\nsp(·),∞,b1/s\n1, ifq=∞.\nProof. (1) This fact can be checked by direct calculation.\n(2) Fix c̸= 0. For any f∈Lp(·),q,bwith 0 < q < ∞, we have\n∥cf∥q\np(·),q,b=Z∞\n0tq−1∥χ{|cf|>t}∥q\np(·)γq\nb(∥χ{|cf|>t}∥p(·))dt\n=Z∞\n0tq−1∥χ{|f|>t/|c|}∥q\np(·)γq\nb(∥χ{|f|>t/|c|}∥p(·))dt\n=|c|qZ∞\n0\u0010t\n|c|\u0011q−1\n∥χ{|f|>t/|c|}∥q\np(·)γq\nb(∥χ{|f|>t/|c|}∥p(·))d\u0010t\n|c|\u0011\n=|c|q∥f∥q\np(·),q,b.\nForf∈Lp(·),∞,b, it is obvious that\nt∥χ{|cf|>t}∥p(·)γb(∥χ{|cf|>t}∥p(·)) =t∥χ{|f|>t/|c|}∥p(·)γb(∥χ{|f|>t/|c|}∥p(·))\n=|c|t\n|c|∥χ{|f|>t/|c|}∥p(·)γb(∥χ{|f|>t/|c|}∥p(·)).\nTaking the supremum for all t >0, we have\n∥cf∥p(·),∞,b=|c| · ∥f∥p(·),∞,b.\n(3) When 0 < q < ∞, according to Proposition 2 .8 (2), we have\n∥f∥q\np(·),q,b=Z∞\n0\u0000\nt∥χ{|f|>t}∥p(·)γb(∥χ{|f|>t}∥p(·))\u0001qdt\nt\n=λqZ∞\n0\u0000\ns∥χ{|f|>λs}∥p(·)γb(∥χ{|f|>λs}∥p(·))\u0001qds\ns\n=λqX\nk∈ZZ2k+1\n2k\u0000\ns∥χ{|f|>λs}∥p(·)γb(∥χ{|f|>λs}∥p(·))\u0001qds\ns(2.4)\n≲X\nk∈ZZ2k+1\n2k\u0000\ns∥χ{|f|>λ2k}∥p(·)γb(∥χ{|f|>λ2k}∥p(·))\u0001qds\ns\n=X\nk∈Z∥χ{|f|>λ2k}∥q\np(·)γq\nb(∥χ{|f|>λ2k}∥p(·))Z2k+1\n2ksq−1ds\n=2q−1\nqX\nk∈Z2kq∥χ{|f|>λ2k}∥q\np(·)γq\nb(∥χ{|f|>λ2k}∥p(·)).\nOn the other hand, by (2.4)\n∥f∥q\np(·),q,b=λqX\nk∈ZZ2k\n2k−1\u0000\ns∥χ{|f|>λs}∥p(·)γb(∥χ{|f|>λs}∥p(·))\u0001qds\ns14 Z. Hao, X. Ding, L. Li and F. Weisz\n≳X\nk∈ZZ2k\n2k−1\u0000\ns∥χ{|f|>λ2k}∥p(·)γb(∥χ{|f|>λ2k}∥p(·))\u0001qds\ns\n=X\nk∈Z∥χ{|f|>λ2k}∥q\np(·)γq\nb(∥χ{|f|>λ2k}∥p(·))Z2k\n2k−1sq−1ds\n=1\nq\u0010\n1−1\n2q\u0011X\nk∈Z2kq∥χ{|f|>λ2k}∥q\np(·)γq\nb(∥χ{|f|>λ2k}∥p(·)).\nHence,\n∥f∥p(·),q,b≈\u0012X\nk∈Zh\n2k∥χ{|f|>λ2k}∥p(·)γb\u0000\n∥χ{|f|>λ2k}∥p(·)\u0001iq\u00131\nq\n,0< q < ∞.\nNow, we consider the case of q=∞. For any t >0 and λ >0, there exist k∈Zsuch\nthatλ2k< t≤λ2k+1. Let t=λs. It follows from Proposition 2 .8 (2) that\nt∥χ{|f|>t}∥p(·)γb\u0000\n∥χ{|f|>t}∥p(·)\u0001\n=λs∥χ{|f|>λs}∥p(·)γb\u0000\n∥χ{|f|>λs}∥p(·)\u0001\n≲s∥χ{|f|>λ2k}∥p(·)γb\u0000\n∥χ{|f|>λ2k}∥p(·)\u0001\n≤2k+1∥χ{|f|>λ2k}∥p(·)γb\u0000\n∥χ{|f|>λ2k}∥p(·)\u0001\n.\nOn the other hand, for any t >0 and λ >0, there exist k∈Zsuch that λ2k−1≤t < λ 2k.\nWe have\nt∥χ{|f|>t}∥p(·)γb\u0000\n∥χ{|f|>t}∥p(·)\u0001\n=λs∥χ{|f|>λs}∥p(·)γb\u0000\n∥χ{|f|>λs}∥p(·)\u0001\n≳s∥χ{|f|>λ2k}∥p(·)γb\u0000\n∥χ{|f|>λ2k}∥p(·)\u0001\n≥2k−1∥χ{|f|>λ2k}∥p(·)γb\u0000\n∥χ{|f|>λ2k}∥p(·)\u0001\n.\nHence,\n∥f∥p(·),∞,b≈sup\nk∈Z2k∥χ{|f|>λ2k}∥p(·)γb\u0000\n∥χ{|f|>λ2k}∥p(·)\u0001\n.\n(4) It is obvious that χ{χA>t}isχAif 0< t < 1 and it is 0 if t≥1. When 0 < q < ∞,\nwe have\n∥χA∥q\np(·),q,b=Z∞\n0tq−1∥χ{χA>t}∥q\np(·)γq\nb\u0000\n∥χ{χA>t}∥p(·)\u0001\ndt\n=Z1\n0tq−1∥χ{χA>t}∥q\np(·)γq\nb\u0000\n∥χ{χA>t}∥p(·)\u0001\ndt\n=Z1\n0tq−1∥χA∥q\np(·)γq\nb\u0000\n∥χA∥p(·)\u0001\ndt\n=1\nq∥χA∥q\np(·)γq\nb\u0000\n∥χA∥p(·)\u0001\n.\nThe statement can be proved similarly for q=∞.\n(5) For any s >0 and 0 < q < ∞, it follows from Lemma 2 .3 that\n∥|f|s∥q\np(·),q,b=Z∞\n0\u0000\nt∥χ{|f|s>t}∥p(·)γb(∥χ{|f|s>t}∥p(·))\u0001qdt\nt\n=Z∞\n0tq∥χ{|f|>t1/s}∥q\np(·)γq\nb(∥χ{|f|>t1/s}∥p(·))dt\ntVariable martingale Hardy-Lorentz-Karamata spaces 15\n=Z∞\n0tq∥χ{|f|>t1/s}∥sq\nsp(·)γq\nb(∥χ{|f|>t1/s}∥s\nsp(·))dt\nt\n=Z∞\n0tq∥χ{|f|>t1/s}∥sq\nsp(·)γq\nb1(∥χ{|f|>t1/s}∥sp(·))dt\nt,\nwhere b1(t) =b(ts) fort∈[1,∞). Notice that b1is a slowly varying function by Proposition\n2.8 (4). Set m=t1/s, by Proposition 2 .8 (5), there is\n∥|f|s∥q\np(·),q,b=Z∞\n0msq∥χ{|f|>m}∥sq\nsp(·)γq\nb1(∥χ{|f|>m}∥sp(·))dms\nms\n=sZ∞\n0msq∥χ{|f|>m}∥sq\nsp(·)γsq\nb1/s\n1(∥χ{|f|>m}∥sp(·))dm\nm\n=s∥f∥sq\nsp(·),sq,b1/s\n1.\nSimilarly, we obtain the result for q=∞:\n∥|f|s∥p(·),∞,b= sup\nt>0t∥χ{|f|s>t}∥p(·)γb(∥χ{|f|s>t}∥p(·))\n= sup\nt>0t∥χ{|f|>t1/s}∥p(·)γb(∥χ{|f|>t1/s}∥p(·))\n= sup\nm>0ms∥χ{|f|>m}∥p(·)γb(∥χ{|f|>m}∥p(·))\n= sup\nm>0ms∥χ{|f|>m}∥s\nsp(·)γs\nb1/s(∥χ{|f|>m}∥s\nsp(·))\n= sup\nm>0ms∥χ{|f|>m}∥s\nsp(·)γs\nb1/s\n1(∥χ{|f|>m}∥sp(·))\n=∥f∥s\nsp(·),∞,b1/s\n1.\nThe proof is complete now. □\nBy virtue of Lemma 2.11, we state the quasi-triangle inequality for the Lorentz-\nKaramata spaces with variable exponents.\nLemma 2.17. Letp(·)∈ P(Ω),0< q≤ ∞ andbbe a slowly varying function. For any\nf, g∈Lp(·),q,b, there is\n∥f+g∥p(·),q,b≲∥f∥p(·),q,b+∥g∥p(·),q,b.\nProof. Suppose that f, g∈Lp(·),q,b. It is easy to see that\n{|f+g|> t} ⊂ {| f|> t/2} ∪ {| g|> t/2},∀t >0.\nWhen 0 < q < ∞, it follows from Proposition 2.8 (2) and Lemma 2 .11 that\n∥f+g∥q\np(·),q,b=Z∞\n0tq−1∥χ{|f+g|>t}∥q\np(·)γq\nb(∥χ{|f+g|>t}∥p(·))dt\n≲Z∞\n0tq−1∥χ{|f|>t/2}∪{|g|>t/2}∥q\np(·)γq\nb(∥χ{|f|>t/2}∪{|g|>t/2}∥p(·))dt\n≲Z∞\n0tq−1\u0010\n∥χ{|f|>t/2}∥p(·)γb(∥χ{|f|>t/2}∥p(·))\n+∥χ{|g|>t/2}∥p(·)γb(∥χ{|g|>t/2}∥p(·))\u0011q\ndt16 Z. Hao, X. Ding, L. Li and F. Weisz\n≲Z∞\n0tq−1\u0010\n∥χ{|f|>t/2}∥q\np(·)γq\nb(∥χ{|f|>t/2}∥p(·))\n+∥χ{|g|>t/2}∥q\np(·)γq\nb(∥χ{|g|>t/2}∥p(·))\u0011\ndt\n= 2q\u0010\n∥f∥q\np(·),q,b+∥g∥q\np(·),q,b\u0011\n.\nWhen q=∞, we obtain that for t >0,\nt∥χ{|f+g|>t}∥p(·)γb(∥χ{|f+g|>t}∥p(·))\n≲t∥χ{|f|>t/2}∪{|g|>t/2}∥p(·)γb(∥χ{|f|>t/2}∪{|g|>t/2}∥p(·))\n≲t\u0010\n∥χ{|f|>t/2}∥p(·)γb(∥χ{|f|>t/2}∥p(·)) +∥χ{|g|>t/2}∥p(·)γb(∥χ{|g|>t/2}∥p(·))\u0011\n.\nBy taking the supremum for all t >0, we have\n∥f+g∥p(·),∞,b≲∥f∥p(·),∞,b+∥f∥p(·),∞,b.\nThe proof is finished. □\nBy Lemma 2 .16 (1), (2) and Lemma 2 .17, we obtain that the variable Lorentz-\nKaramata spaces are quasi-normed spaces. Next, we will show that these spaces are also\nquasi-Banach spaces. Before that, we state the Aoki-Rolewicz theorem, which is proved\nin [3].\nLemma 2.18 ([3]).Let(X,∥ · ∥ X)be a quasi-normed space, that is, for any x, y∈X,\n∥x+y∥ ≤K(∥x∥+∥y∥).\nThen for α∈(0,1]defined by (2K)α= 2, we have\n∥x1+···+xn∥α\nX≤4(∥x1∥α\nX+···+∥xn∥α\nX)\nfor all x1,···, xn(n≥1)inX.\nRemark 2.19. It follows from Lemmas 2.17and2.18that there exists α∈(0,1]such that\n∥f1+···+fn∥α\np(·),q,b≤4(∥f1∥α\np(·),q,b+···+∥fn∥α\np(·),q,b)\nfor all f1,···, fn(n≥1)inLp(·),q,b. We denote by N ∈ (0,1]the supremum of the α’s\nfor which the above inequality holds.\nLemma 2.20. Letp(·)∈ P(Ω),0< q≤ ∞ andbbe a slowly varying function. Then\nLp(·),q,bis a quasi-Banach space.\nProof. Let ( hn)n≥1⊂Lp(·),q,bbe a Cauchy sequence with respect to ∥ · ∥ p(·),q,b. We choose\na subsequence ( hnk)k≥1such that\n∥hnk+1−hnk∥p(·),q,b≤1\n22k, k≥1\nandhn0:= 0. Define the function\ng(ω) :=∞X\nk=0|hnk+1(ω)−hnk(ω)|.\nIt is obvious that\nχ{g>λ}≤∞X\nk=0χ{|hnk+1−hnk|>λ\n2k+1}.Variable martingale Hardy-Lorentz-Karamata spaces 17\nLet 0 < β < p −and 0 < ε≤min{p−−β,1}. Then\n∥χ{g>λ}∥ε\np−−β≤\r\r\r\r∞X\nk=0χ{|hnk+1−hnk|>λ\n2k+1}\r\r\r\rε\np−−β\n≤∞X\nk=0\r\rχ{|hnk+1−hnk|>λ\n2k+1}\r\rε\np−−β\n≤∞X\nk=02(k+1)ε\nλε∥hnk+1−hnk∥ε\np−−β.\nIt follows from Lemma 2.15 that\n∥f∥p−−β=∥f∥p−−β,p−−β,1≲∥f∥p−,q,b\nfor any measurable function f, which implies\n∥χ{g>λ}∥ε\np−−β≲∞X\nk=02(k+1)ε\nλε∥hnk+1−hnk∥ε\np−,q,b\n≲∞X\nk=02(k+1)ε\nλε∥hnk+1−hnk∥ε\np(·),q,b≤∞X\nk=02(k+1)ε\nλε22k.\nHence, ∥χ{g>λ}∥p−−β→0 asλ→ ∞ andgis finite almost everywhere. Set\nh(ω) :=∞X\nk=0\u0000\nhnk+1(ω)−hnk(ω)\u0001\nandeh(ω) :=h(ω)−hn1(ω), ω∈Ω.\nThen handehconverge almost everywhere.\nNext we shall verify that h∈Lp(·),q,band ( hn)n≥1converges to hinLp(·),q,b. LetNbe\nthe constant in Remark 2.19. We have\n\r\r\reh\r\r\rN\np(·),q,b=\r\r\r\r∞X\nk=1hnk+1−hnk\r\r\r\rN\np(·),q,b≤4∞X\nk=1∥hnk+1−hnk∥N\np(·),q,b≤4∞X\nk=11\n22kN<∞,\nwhich means that eh∈Lp(·),q,b. Combining this with Lemma 2.17, we have h=eh+hn1∈\nLp(·),q,b. Since ( hn)n≥1⊂Lp(·),q,bis a Cauchy sequence, for any ε >0, there exists Nsuch\nthat for any n, m > N ,\n∥hn−hm∥p(·),q,b< ε.\nClearly, there exists k0≥1 such that nk0> N and 2−2k0<(1−2−2N)1/N·ε. It follows\nfrom Lemma 2.17 and Remark 2 .19 that, for any n > N ,\n∥hn−h∥p(·),q,b≲\r\r\r\rhn−k0−1X\nk=0(hnk+1−hnk)\r\r\r\r\np(·),q,b+\r\r\r\r∞X\nk=k0(hnk+1−hnk)\r\r\r\r\np(·),q,b\n=∥hn−hnk0∥p(·),q,b+\r\r\r\r∞X\nk=k0(hnk+1−hnk)\r\r\r\rN1\nN\np(·),q,b\n≲ε+\u0012∞X\nk=k0∥(hnk+1−hnk)∥N\np(·),q,b\u00131\nN18 Z. Hao, X. Ding, L. Li and F. Weisz\n≤ε+\u0012∞X\nk=k02−2kN\u00131\nN\n=ε+\u00121\n1−2−2N\u00131\nN\n2−2k0<2ε.\nTherefore, hn→hasn→ ∞ with respect to ∥ · ∥ p(·),q,b. Hence, Lp(·),q,bis complete. □\nAt the end of this subsection, we establish dominated convergence theorems in variable\nLorentz-Karamata spaces. We begin with the definition of absolutely continuous quasi-\nnorm. Let ( X,∥ · ∥ X) be a (quasi)-normed space. A function f∈Xis said to have\nabsolutely continuous (quasi)-norm in X, if\nlim\nn→∞∥fχAn∥X= 0\nfor every sequence ( An)n≥0satisfying lim\nn→∞P(An) = 0.\nLemma 2.21. Letp(·)∈ P(Ω),0< q < ∞andbbe a slowly varying function. Then\nevery f∈Lp(·),q,bhas absolutely continuous quasi-norm.\nProof. Since f∈Lp(·),q,b, for any ε >0, there exists N1∈Nsuch that\n\u0012∞X\nk=N12kq∥χ{|f|>2k}∥q\np(·)γq\nb\u0000\n∥χ{|f|>2k}∥p(·)\u0001\u00131\nq\n< ε.\nLet the sequence ( An)n≥0satisfy lim\nn→∞P(An) = 0. There exists N2∈Nsuch that for\nn≥N2,P(An)<\u0000ε\n2N1γb(1)\u00012p+. Now let n≥N2. By Proposition 2.8 (2) and Lemma 2.5,\nwe have\n∥fχAn∥q\np(·),q,b=∞X\nk=N12kq∥χ{|fχAn|>2k}∥q\np(·)γq\nb\u0000\n∥χ{|fχAn|>2k}∥p(·)\u0001\n+N1−1X\nk=−∞2kq∥χ{|fχAn|>2k}∥q\np(·)γq\nb\u0000\n∥χ{|fχAn|>2k}∥p(·)\u0001\n≲∞X\nk=N12kq∥χ{|f|>2k}∥q\np(·)γq\nb\u0000\n∥χ{|f|>2k}∥p(·)\u0001\n+N1−1X\nk=−∞2kq∥χAn∥q\np(·)γq\nb\u0000\n∥χAn∥p(·)\u0001\n< εq+N1−1X\nk=−∞2kq∥χAn∥q/2\np(·)∥χAn∥q/2\np(·)γq\nb\u0000\n∥χAn∥p(·)\u0001\n≲εq+N1−1X\nk=−∞2kq∥χAn∥q/2\np(·)γq\nb(1)\n≤εq+N1−1X\nk=−∞2kq(2∥χAn∥p+)q/2γq\nb(1)\n< εq+ 2q/2\u0010ε\n2N1γb(1)\u0011q\nγq\nb(1)N1−1X\nk=−∞2kq=\u0010\n1 +2q/2\n2q−1\u0011\nεq,\nwhich yields that ∥fχAn∥p(·),q,b≲ε. Hence, fhas absolutely continuous quasi-norm. □Variable martingale Hardy-Lorentz-Karamata spaces 19\nNext, we give the dominated convergence theorem for two cases: 0 < q < ∞and\nq=∞, which extend the results in [42, 52].\nLemma 2.22 (Dominated convergence theorem) .Letp(·)∈ P(Ω),0< q < ∞andbbe\na slowly varying function. Suppose that hn, f, g∈Lp(·),q,b. Ifhn→fasn→ ∞ a.e. and\n|hn| ≤ga.e. for every n≥1, then\nlim\nn→∞∥hn−f∥p(·),q,b= 0.\nProof. Since g∈Lp(·),q,bhas absolutely continuous quasi-norm, then for any ε >0, there\nexists N1such that ∥gχ{g>N 1}∥p(·),q,b< ε. On{g≤N1}, it is easy to see that |hn−f| ≤2N1.\nBy Lemma 2.16 (4), there is\n∥(hn−f)χ{g≤N1}∥p(·),q,b=∥(hn−f)χ{g≤N1}χ{hn̸=f}∥p(·),q,b\n≤2N1∥χ{hn̸=f}∥p(·),q,b= 2N1\u00101\nq\u00111/q\n∥χ{hn̸=f}∥p(·)γb\u0000\n∥χ{hn̸=f}∥p(·)\u0001\n.\nSince hn→fasn→ ∞ a.e., then χ{hn̸=f}→0 asn→ ∞ . By dominated convergence\ntheorem in Lp(·)(see [11]), it follows from χ{hn̸=f}≤1 that ∥χ{hn̸=f}∥p(·)→0. Hence there\nexists N2such that for n≥N2,\n∥(hn−f)χ{g≤N1}∥p(·),q,b< ε.\nTherefore, for n≥N2, we have\n∥hn−f∥p(·),q,b≲∥(hn−f)χ{g≤N1}∥p(·),q,b+∥(hn−f)χ{g>N 1}∥p(·),q,b\n≲ε+ 2∥gχ{g>N 1}∥p(·),q,b<3ε,\nwhich completes the proof. □\nFor the case q=∞, we shall introduce a subspace of Lp(·),∞,bas follows.\nDefinition 2.23. Letp(·)∈ P(Ω)andbbe a slowly varying function. Denote by Lp(·),∞,b\nthe set of all f∈Lp(·),∞,bhaving absolutely continuous quasi-norm, that is\nLp(·),∞,b={f∈Lp(·),∞,b: lim\nn→∞∥fχAn∥p(·),q,b= 0},\nwhere the sequence (An)n≥0satisfies lim\nn→∞P(An) = 0 .\nLemma 2.24. Letp(·)∈ P(Ω)andbbe a slowly varying function. Then Lp(·),∞,bis a\nclosed subspace of Lp(·),∞,b.\nProof. Let (gn)n≥1⊂ L p(·),∞,bbe a Cauchy sequence with respect to quasi-norm ∥·∥p(·),∞,b.\nThen there exists f∈Lp(·),∞,bsuch that\nlim\nn→∞∥gn−f∥p(·),∞,b= 0.\nFor any given sequence ( An)n≥0satisfying lim\nn→∞P(An) = 0, it follows from Lemma 2.17\nthat\n∥fχAn∥p(·),∞,b≲∥gnχAn∥p(·),∞,b+∥(gn−f)χAn∥p(·),∞,b\n≲∥gnχAn∥p(·),∞,b+∥gn−f∥p(·),∞,b.\nSince gn∈ L p(·),∞,b, we get\n∥fχAn∥p(·),∞,b→0,asn→ ∞ .\nHence, it implies that f∈ L p(·),∞,b. □20 Z. Hao, X. Ding, L. Li and F. Weisz\nSimilarly to the proof of Lemma 2.22, we get the dominated convergence theorem for\nq=∞, the proof is left to the readers.\nLemma 2.25. Letp(·)∈ P(Ω)andbbe a slowly varying function. Suppose that hn, f∈\nLp(·),∞,bandg∈ L p(·),∞,b. Ifhn→fasn→ ∞ a.e and |hn| ≤g a.e. for every n≥1,\nthen\nlim\nn→∞∥hn−f∥p(·),∞,b= 0.\n2.4.Martingale and variable martingale Hardy-Lorentz-Karamata spaces. Let\n{Fn}n≥0be a nondecreasing sequence of sub- σ-algebras of Fsuch that F=σ\u0010S\nn≥0Fn\u0011\n.\nThe expectation operator and the conditional expectation operator relative to Fnare\ndenoted by EandEn, respectively. A sequence of measurable functions f= (fn)n≥0⊂L1\nis called a martingale with respect to {Fn}n≥0if\nEn(fn+1) =fn for every n≥0.\nDenote by Mthe set of all martingales f= (fn)n≥0relative to {Fn}n≥0such that f0= 0.\nForf∈ M , we define the martingale differences by\nd0f:= 0, d nf:=fn−fn−1 (n >0).\nLetTbe the set of all stopping times relative to {Fn}n≥0. For f∈ M andτ∈ T, the\nstopped martingale fτ= (fτ\nn)n≥0is defined by\nfτ\nn:=nX\nm=0χ{τ≥m}dmf.\nDefine the maximal function, the square function and the conditional square function\nof a martingale f, respectively, as follows:\nMm(f) = sup\nn≤m|fn|, M (f) = sup\nn≥0|fn|;\nSm(f) =\u0012mX\nn=0|dnf|2\u00131\n2\n, S(f) =\u0012∞X\nn=0|dnf|2\u00131\n2\n;\nsm(f) =\u0012mX\nn=0En−1|dnf|2\u00131\n2\n, s(f) =\u0012∞X\nn=0En−1|dnf|2\u00131\n2\n.\nLetp(·)∈ P(Ω), 0 < q≤ ∞ andbbe a slowly varying function. Denote by Λ p(·),q,b\nthe collection of all sequences λ= (λn)n≥0of nondecreasing, nonnegative and adapted\nfunctions with λ∞:= lim\nn→∞λn∈Lp(·),q,b.\nDefinition 2.26. Letp(·)∈ P(Ω),0< q≤ ∞ andbbe a slowly varying function. The\nvariable martingale Hardy spaces associated with variable Lorentz-Karamata spaces are\ndefined by\nHM\np(·),q,b=\b\nf∈ M :∥f∥HM\np(·),q,b=∥M(f)∥p(·),q,b<∞\t\n;\nHs\np(·),q,b=\b\nf∈ M :∥f∥Hs\np(·),q,b=∥s(f)∥p(·),q,b<∞\t\n;\nHS\np(·),q,b=\b\nf∈ M :∥f∥HS\np(·),q,b=∥S(f)∥p(·),q,b<∞\t\n;\nQp(·),q,b=\b\nf∈ M :∃λ= (λn)n≥0∈Λp(·),q,bs.t.Sn(f)≤λn−1\tVariable martingale Hardy-Lorentz-Karamata spaces 21\nwith∥f∥Qp(·),q,b= inf\nλ∈Λp(·),q,b∥λ∞∥p(·),q,b;\nPp(·),q,b=\b\nf∈ M :∃λ= (λn)n≥0∈Λp(·),q,bs.t.|fn| ≤λn−1\t\nwith∥f∥Pp(·),q,b= inf\nλ∈Λp(·),q,b∥λ∞∥p(·),q,b.\nWe define HM\np(·),∞,bas the space of all martingales such that M(f)∈ L p(·),∞,b. Replacing\nM(f)bys(f)orS(f), we can define Hs\np(·),∞,bandHS\np(·),∞,b, respectively.\nRemark 2.27. (1)It is easy to check that HM\np(·),∞,b,Hs\np(·),∞,bandHS\np(·),∞,bare closed\nsubspaces of HM\np(·),∞,b,Hs\np(·),∞,bandHS\np(·),∞,b, respectively.\n(2)If we take p(·)≡pin Definition 2.26, we get back the martingale Hardy-Lorentz-\nKaramata spaces introduced in [38, 44] . Moreover, the variable martingale Hardy-Lorentz\nspace defined in [40, 42] is also a special case of the variable martingale Hardy-Lorentz-\nKaramata space when b≡1.\nWe recall the definition of regularity. The stochastic basis {Fn}n≥0is said to be\nregular, if for n≥0 and A∈ F n, there exists B∈ F n−1such that A⊂BandP(B)≤\nRP(A), where Ris a positive constant independent of n. Equivalently, there exists a\nconstant R>0 such that\nfn≤ Rfn−1 (2.5)\nfor all nonnegative martingales ( fn)n≥0adapted to the stochastic basis {Fn}n≥0. We refer\nto [53, Chapter 7] for more details about regular basis.\nRecall that B∈ F nis an atom if A⊂Bwith A∈ F nsatisfies P(A)<P(B), then\nP(A) = 0. Denote by A(Fn) the set of all atoms in Fn. In the sequel of this paper, we\nalways assume that every σ-algebra Fnis generated by countable many atoms. There are\nmany examples for σ-algebras generated by countable atoms, see [42, 75].\nWe need the following Doob maximal inequality on variable Lebesgue spaces, which\nwas proved by Jiao et al. [42].\nLemma 2.28. Letp(·)∈ P(Ω)satisfy (1.1)with 1< p−≤p+<∞. Then for any\nf∈Lp(·), we have\n∥M(f)∥p(·)≲∥f∥p(·).\nMoreover, by applying the inequality above and interpolation results for variable\nLorentz-Karamata spaces, we obtain the next generalization of Doob’s maximal inequality.\nLemma 2.29 ([30]).Letp(·)∈ P(Ω)satisfy (1.1)with 1< p−≤p+<∞,0< q≤ ∞\nand let bbe a slowly varying function. Then\n∥M(f)∥p(·),q,b≲∥f∥p(·),q,b for any f∈Lp(·),q,b.\n3.Atomic Decompositions\nThe atomic decomposition is a powerful tool for dealing with duality theorems and\nsome fundamental inequalities both in martingale theory and harmonic analysis. In this\nsection, we characterize the five variable martingale Hardy-Lorentz-Karamata spaces by\natomic decompositions. Note that there are two notions of atoms in this paper, one is a\nmeasurable set described in Section 2, the other is a measurable function defined in this\nsection (see Definitions 3.1 and 3.17).22 Z. Hao, X. Ding, L. Li and F. Weisz\n3.1.Atomic Decompositions via Simple Atoms. In this subsection, we use simple\natoms.\nDefinition 3.1 ([40]).Letp(·)∈ P(Ω)and1< r≤ ∞ . A function ais called a simple\n(p(·), r)s-atom (resp. simple (p(·), r)S-atom or simple (p(·), r)M-atom )if there exists I∈\nA(Fi)such that\n(1)supp(a)⊂I;\n(2)∥s(a)∥r(resp.∥S(a)∥ror∥M(a)∥r)≤P(I)1/r\n∥χI∥p(·);\n(3)Ei(a) = 0 .\nRemark 3.2. (i)If we take p(·)≡p, then the simple (p(·), r)∗-atom (∗=s, S, M )is the\nsame as the classical definition in Weisz [75].\n(ii)Letp(·)∈ P(Ω)and1< r≤ ∞ . Ifais a simple (p(·), r)∗-atom (∗=s, S, M )\nassociated with I∈A(Fi)for some i∈N, then\ns(a)χI=s(a), S (a)χI=S(a)and M(a)χI=M(a).\nWe refer to [40]for the proof.\nDefinition 3.3. Letp(·)∈ P(Ω),0< q, r ≤ ∞ and let bbe a slowly varying function.\nThe atomic martingale Lorentz-Karamata space with variable exponent Hs−at,r,1\np(·),q,b\u0000\nresp.\nHs−at,r,2\np(·),q,borHs−at,r,3\np(·),q,b\u0001\nis defined to be the set of all f= (fn)n≥0∈ M such that for each\nn≥0,\nfn=X\nk∈Zn−1X\ni=0X\njµk,i,jEn(ak,i,j), (3.1)\nwhere (ak,i,j)k∈Z,i∈N,jis a sequence of simple (p(·), r)s-atoms (resp. simple (p(·), r)S-atoms\nor simple (p(·), r)M-atoms )associated with (Ik,i,j)k∈Z,i∈N,j, the sets Ik,i,jare disjoint for\nany fixed k,µk,i,j= 3·2k∥χIk,i,j∥p(·)and\n\r\r\r\r\r(\r\r\r\r∞X\ni=0X\njµk,i,j∥χIk,i,j∥−1\np(·)χIk,i,j\r\r\r\r\np(·)γb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013)\nk∈Z\r\r\r\r\r\nlq<∞.\nEndow Hs−at,r,1\np(·),q,b\u0000\nresp. Hs−at,r,2\np(·),q,borHs−at,r,3\np(·),q,b\u0001\nwith the (quasi )-norm\n∥f∥Hs−at,r, 1\np(·),q,b\u0000\nresp.∥f∥Hs−at,r, 2\np(·),q,bor∥f∥Hs−at,r, 3\np(·),q,b\u0001\n= inf\r\r\r\r\r(\r\r\r\r∞X\ni=0X\njµk,i,j∥χIk,i,j∥−1\np(·)χIk,i,j\r\r\r\r\np(·)γb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013)\nk∈Z\r\r\r\r\r\nlq,\nwhere the infimum is taken over all decompositions of fas above.\nIn order to present the atomic decomposition theorems for variable Lorentz-Karamata\nspaces, we need the following lemmas. First of all, we shall show a useful tool to verify\nwhether a function belongs to variable Lorentz-Karamata spaces. Its original idea comes\nfrom Abu-Shammala and Torchinsky [1].\nLemma 3.4. Letp(·)∈ P(Ω),0< q≤ ∞ andbbe a slowly varying function. Assume\nthat the nonnegative sequence {2kµk}k∈Z∈lq. Suppose that the nonnegative function φ\nverifies the following property: there exist 0< m < ∞and0< ε < 1such that, given anVariable martingale Hardy-Lorentz-Karamata spaces 23\narbitrary integer k0, we have φ≤ψk0+ηk0, where ψk0is essentially bounded and satisfies\n∥ψk0∥∞≲2k0andηk0satisfies\n2k0mε∥χ{ηk0>2k0}∥m\np(·)γm\nb\u0000\n∥χ{ηk0>2k0}∥p(·)\u0001\n≲∞X\nk=k0(2kεµk)m. (3.2)\nThen φ∈Lp(·),q,band\n∥φ∥p(·),q,b≲∥{2kµk}k∈Z∥lq.\nProof. For given k0∈Z, letψk0andηk0be as above. Since ∥ψk0∥∞≲2k0, there exists a\npositive constant Csuch that P(ψk0> C2k0) = 0. Then we have\n{φ > λ 2k0} ⊂ { ψk0> C2k0} ∪ {ηk0>2k0}={ηk0>2k0},\nwhere λ=C+ 1. Thus, it clearly suffices to verify that\n\r\r\b\n2k0∥χ{ηk0>2k0}∥p(·)γb\u0000\n∥χ{ηk0>2k0}∥p(·)\u0001\t\nk0∈Z\r\r\nlq≲∥{2kµk}k∈Z∥lq.\nWhen q=∞, the hypothesis (3.2) gives that\n2k0∥χ{ηk0>2k0}∥p(·)γb\u0000\n∥χ{ηk0>2k0}∥p(·)\u0001\n≲2k0\u0012\n2−k0mε∞X\nk=k0\u0000\n2kεµk\u0001m\u00131\nm\n= 2k0(1−ε)\u0012∞X\nk=k02km(ε−1)(2kµk)m\u00131\nm\n≲2k0(1−ε)2k0(ε−1)sup\nk≥k02kµk= sup\nk≥k02kµk.\nIt follows that\n\r\r\b\n2k0∥χ{ηk0>2k0}∥p(·)γb\u0000\n∥χ{ηk0>2k0}∥p(·)\u0001\t\nk0∈Z\r\r\nl∞≲∥{2kµk}k∈Z∥l∞.\nWhen 0 < q < ∞, let 0 < m < ∞, 0< ε < 1 and θ=1−ε\n2. Then\n∞X\nk=k0(2kεµk)m=∞X\nk=k02−kθm\u0000\n2k(1−θ)µk\u0001m. (3.3)\nSetting r=q\nm, we consider the right side of (3.3) for two cases: 0 < r < 1 and 1 ≤r <∞.\nIf 0< m≤q, by using H¨ older’s inequality with1\nr+1\nr′= 1, we have\n∞X\nk=k02−kθm\u0000\n2k(1−θ)µk\u0001m≤\u0012∞X\nk=k02−kθmr′\u00131\nr′\u0012∞X\nk=k0\u0000\n2k(1−θ)µk\u0001q\u00131\nr\n=\u00121\n1−2θmr′\u00131/r′\n2−k0θm\u0012∞X\nk=k0\u0000\n2k(1−θ)µk\u0001q\u0013m\nq\n.\nWhen 0 < q≤m, we get a similar estimation by simply observing that\n∞X\nk=k02−kθm\u0000\n2k(1−θ)µk\u0001m≤2−k0θm\u0012∞X\nk=k0\u0000\n2k(1−θ)µk\u0001m\u0013r·1\nr\n≤2−k0θm\u0012∞X\nk=k0\u0000\n2k(1−θ)µk\u0001q\u0013m\nq\n.24 Z. Hao, X. Ding, L. Li and F. Weisz\nCombining with (3.2) and (3.3), we deduce that\n2k0mε∥χ{ηk0>2k0}∥m\np(·)γm\nb\u0000\n∥χ{ηk0>2k0}∥p(·)\u0001\n≲2−k0θm\u0012∞X\nk=k0\u0000\n2k(1−θ)µk\u0001q\u0013m\nq\n.\nIt yields that\n2k0q∥χ{ηk0>2k0}∥q\np(·)γq\nb\u0000\n∥χ{ηk0>2k0}∥p(·)\u0001\n≲2k0(1−ε−θ)q∞X\nk=k0\u0000\n2k(1−θ)µk\u0001q= 2k0θq∞X\nk=k0\u0000\n2k(1−θ)µk\u0001q.\nMoreover, according to Abel’s transformation, there is\nX\nk0∈Z2k0q∥χ{ηk0>2k0}∥q\np(·)γq\nb\u0000\n∥χ{ηk0>2k0}∥p(·)\u0001\n≲X\nk0∈Z2k0θq∞X\nk=k0\u0000\n2k(1−θ)µk\u0001q=X\nk∈Z\u0000\n2k(1−θ)µk\u0001qkX\nk0=−∞2k0θq\n=1\n1−2−ε(1−ε−θ)X\nk∈Z\u0000\n2k(1−θ)µk\u0001q2kθq=X\nk∈Z2kqµq\nk.\nHence, we have\n∥φ∥p(·),q,b≈\r\r\b\n2k0∥χ{φk0>λ2k0}∥p(·)γb\u0000\n∥χ{φk0>λ2k0}∥p(·)\u0001\t\nk0∈Z\r\r\nlq\n≲\r\r\b\n2k0∥χ{ηk0>2k0}∥p(·)γb\u0000\n∥χ{ηk0>2k0}∥p(·)\u0001\t\nk0∈Z\r\r\nlq\n≲∥{2kµk}k∈Z∥lq,\nwhich completes the proof. □\nAlso, we need the next lemma, its proof can be found in [40].\nLemma 3.5. Letp(·)∈ P(Ω)satisfy (1.1)andmax{p+,1}< r < ∞. Take 0< ε < p\nand1< L < min{r\np+,1\nε}. If for a sublinear operator Tand all simple (p(·), r)∗-atoms ak,i,j\n(∗=s, S, M ), there is\n∥T(ak,i,j)∥r≲∥χIk,i,j∥r\n∥χIk,i,j∥p(·),\nthen\r\r\r\r∞X\ni=0X\njh\n∥χIk,i,j∥p(·)T(ak,i,j)χIk,i,jiLε\r\r\r\rp(·)\nε≲\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rp(·)\nε.\nTheorem 3.6. Letp(·)∈ P(Ω)satisfy (1.1),0< q≤ ∞ ,max{p+,1}< r≤ ∞ and let b\nbe a slowly varying function. Then\nHs\np(·),q,b=Hs−at,r,1\np(·),q,b\nwith equivalent (quasi )-norms.\nProof. Letf∈Hs\np(·),q,b. For k∈Zandn∈N, define\nτk:= inf{n∈N:sn+1(f)>2k}.Variable martingale Hardy-Lorentz-Karamata spaces 25\nIt is easy to check that {τk}k∈Zis a nondecreasing sequence of stopping times. For each\nn∈N, there is\nfn=X\nk∈Z(fτk+1\nn−fτk\nn)a.e.\nIt is obvious that for fixed k, i, there exist disjoint atoms ( Ik,i,j)j⊂A(Fi) such that\n[\njIk,i,j={τk=i} ∈ F i.\nThus the sets Ik,i,jare disjoint for each fixed k. Then for each n∈N, we have\nfn=X\nk∈Z(fτk+1\nn−fτk\nn)χ{τk<n}\n=X\nk∈Zn−1X\ni=0(fτk+1\nn−fτk\nn)χ{τk=i}\n=X\nk∈Zn−1X\ni=0X\nj(fτk+1\nn−fτk\nn)χIk,i,j\nand∞S\ni=0S\njIk,i,j={τk<∞}.Set\nµk,i,j:= 3·2k∥χIk,i,j∥p(·)and ak,i,j\nn:=fτk+1n−fτkn\nµk,i,jχIk,i,j\n(ifµk,i,j= 0, then let ak,i,j\nn= 0). Since fτk+1n=nP\nm=0χ{τk+1≥m}dmfandIk,i,j⊂ {τk=i}, we\nhave\n(fτk+1\nn−fτk\nn)χIk,i,j=χIk,i,jnX\nm=0χ{τk+1≥m>τ k}dmf (3.4)\n=χIk,i,jnX\nm=i+1χ{τk+1≥m>τ k}dmf.\nHence\nEi(ak,i,j\nn) = 0 ,Z\nIk.i.jak,i,j\nn= 0,\nand for fixed k, i, j , (ak,i,j\nn)n≥0is a martingale. By the definition of τk, we have\ns\u0000\n(ak,i,j\nn)n\u0001\n≤s(fτk+1n) +s(fτkn)\nµk,i,j≤2k+1+ 2k\nµk,i,j=1\n∥χIk,i,j∥p(·).\nThus ( ak,i,j\nn)n≥0is an L2-bounded martingale. Moreover, there exists ak,i,j∈L2such that\nEn(ak,i,j) =ak,i,j\nnand\n∥s(ak,i,j)∥r≤P(Ik,i,j)1/r 1\n∥χIk,i,j∥p(·).\nTherefore, ak,i,jis a simple ( p(·), r)s-atom and (3.1) holds.26 Z. Hao, X. Ding, L. Li and F. Weisz\nFor the case of 0 < q < ∞, it follows from∞S\ni=0S\njIk,i,j={τk<∞}={s(f)>2k}and\nProposition 2.8 (2) that\nX\nk∈Z\r\r\r\r∞X\ni=0X\njµk,i,j∥χIk,i,j∥−1\np(·)χIk,i,j\r\r\r\rq\np(·)γq\nb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013\n=X\nk∈Z\r\r\r\r∞X\ni=0X\nj3·2kχIk,i,j\r\r\r\rq\np(·)γq\nb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013\n= 3qX\nk∈Z2kq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rq\np(·)γq\nb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013\n= 3qX\nk∈Z2kq∥χ{s(f)>2k}∥q\np(·)γq\nb\u0000\n∥χ{s(f)>2k}∥p(·)\u0001\n≲X\nk∈ZZ2k\n2k−1∥χ{s(f)>2k}∥q\np(·)γq\nb\u0000\n∥χ{s(f)>2k}∥p(·)\u0001\ntq−1dt\n≲X\nk∈ZZ2k\n2k−1∥χ{s(f)>t}∥q\np(·)γq\nb\u0000\n∥χ{s(f)>t}∥p(·)\u0001\ntq−1dt\n=Z∞\n0∥χ{s(f)>t}∥q\np(·)γq\nb\u0000\n∥χ{s(f)>t}∥p(·)\u0001\ntq−1dt\n=∥f∥q\nHs\np(·),q,b.\nSimilarly, when q=∞, we have\nsup\nk∈Z\r\r\r\r∞X\ni=0X\njµk,i,j∥χIk,i,j∥−1\np(·)χIk,i,j\r\r\r\r\np(·)γb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013\n= 3 sup\nk∈Z2k\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)γb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013\n= 3 sup\nk∈Z2k∥χ{s(f)>2k}∥p(·)γb\u0000\n∥χ{s(f)>2k}∥p(·)\u0001\n= 3 sup\nk∈Zsup\nt>0χ{2k−1<t≤2k}t∥χ{s(f)>2k}∥p(·)γb\u0000\n∥χ{s(f)>2k}∥p(·)\u0001\n≲sup\nk∈Zsup\nt>0χ{2k−1<t≤2k}t∥χ{s(f)>t}∥p(·)γb\u0000\n∥χ{s(f)>t}∥p(·)\u0001\n= sup\nt>0t∥χ{s(f)>t}∥p(·)γb\u0000\n∥χ{s(f)>t}∥p(·)\u0001\n=∥s(f)∥p(·),∞,b.\nHence, we obtain that\n∥f∥Hs−at,r, 1\np(·),q,b≲∥f∥Hs\np(·),q,b.Variable martingale Hardy-Lorentz-Karamata spaces 27\nFor the converse part, assume that the martingale fhas a decomposition (3.1). For\nan arbitrary k0, let\nT1=k0−1X\nk=−∞∞X\ni=0X\njµk,i,js(ak,i,j) and T2=∞X\nk=k0∞X\ni=0X\njµk,i,js(ak,i,j).\nIt follows from the sublinearity of the conditional square operator sthat\ns(f)≤X\nk∈Z∞X\ni=0X\njµk,i,js(ak,i,j) =T1+T2.\nLet 0 < ε < min{p, q}, 1< L < min{r\np+,1\nε}and 0 < σ < 1−1\nL. According to H¨ older’s\ninequality, we have\nT1=k0−1X\nk=−∞∞X\ni=0X\njµk,i,js(ak,i,j)\n=k0−1X\nk=−∞2kσ2−kσ∞X\ni=0X\njµk,i,js(ak,i,j)\n≤\u0012k0−1X\nk=−∞2kσL′\u00131\nL′\u0014k0−1X\nk=−∞2−kσL\u0012∞X\ni=0X\njµk,i,js(ak,i,j)\u0013L\u00151\nL\n≈2k0σ\u0014k0−1X\nk=−∞2−kσL\u0012∞X\ni=0X\njµk,i,js(ak,i,j)\u0013L\u00151\nL\n,\nwhere L′is the conjugate of L. According to Lemma 2.3 and Remark 2.1 (3), we have\n∥χ{T1>2k0}∥p(·)≤2−k0L∥TL\n1∥p(·)\n≲2−k0L\r\r\r\r\r2k0σLk0−1X\nk=−∞2−kσL\u0012∞X\ni=0X\njµk,i,js(ak,i,j)\u0013L\r\r\r\r\r\np(·)\n≈2k0L(σ−1)\r\r\r\r\rk0−1X\nk=−∞2kL(1−σ)\u0012∞X\ni=0X\nj∥χIk,i,j∥p(·)s(ak,i,j)\u0013L\r\r\r\r\r\np(·)\n= 2k0L(σ−1)\r\r\r\r\r\u0014k0−1X\nk=−∞2kL(1−σ)\u0012∞X\ni=0X\nj∥χIk,i,j∥p(·)s(ak,i,j)\u0013L\u0015ε\r\r\r\r\r1\nε\np(·)\nε\n≤2k0L(σ−1)\r\r\r\r\rk0−1X\nk=−∞2kLε(1−σ)\u0012∞X\ni=0X\nj∥χIk,i,j∥p(·)s(ak,i,j)\u0013Lε\r\r\r\r\r1\nε\np(·)\nε\n≤2k0L(σ−1)\r\r\r\r\rk0−1X\nk=−∞2kLε(1−σ)∞X\ni=0X\nj\u0010\n∥χIk,i,j∥p(·)s(ak,i,j)\u0011Lε\r\r\r\r\r1\nε\np(·)\nε\n≲2k0L(σ−1) k0−1X\nk=−∞2kLε(1−σ)\r\r\r\r∞X\ni=0X\nj\u0010\n∥χIk,i,j∥p(·)s(ak,i,j)\u0011Lε\r\r\r\rp(·)\nε!1\nε\n.28 Z. Hao, X. Ding, L. Li and F. Weisz\nSince ( ak,i,j)k∈Z,i∈N,jare simple ( p(·), r)s-atoms, it follows from Lemma 3.5 that\n\r\r\r\r∞X\ni=0X\nj\u0010\n∥χIk,i,j∥p(·)s(ak,i,j)\u0011Lε\r\r\r\rp(·)\nε≲\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rp(·)\nε.\nThen we have\n∥χ{T1>2k0}∥p(·)≲2k0L(σ−1) k0−1X\nk=−∞2kLε(1−σ)\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rp(·)\nε!1\nε\n(3.5)\n= 2k0L(σ−1) k0−1X\nk=−∞2kε(L(1−σ)−δ)2kεδ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rε\np(·)!1\nε\n,\nwhere δ >0 satisfying 1 < δ < L (1−σ).\nFirstly, we consider the case of 0 < q < ∞. By H¨ older’s inequality withε\nq+q−ε\nq= 1,\nwe obtain\n∥χ{T1>2k0}∥p(·)≲2k0L(σ−1) k0−1X\nk=−∞2kε(L(1−σ)−δ)q\nq−ε!q−ε\nqε k0−1X\nk=−∞2kqδ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rq\np(·)!1\nq\n≲2−k0δ k0−1X\nk=−∞2kδq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rq\np(·)!1\nq\n= k0−1X\nk=−∞2(k−k0)δq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rq\np(·)!1\nq\n.\nThen this yields that\nX\nk0∈Z2k0q∥χ{T1>2k0}∥q\np(·)γq\nb\u0000\n∥χ{T1>2k0}∥p(·)\u0001\n≲X\nk0∈Z2k0qk0−1X\nk=−∞2(k−k0)δq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rq\np(·)γq\nb\" k0−1X\nk=−∞2(k−k0)δq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rq\np(·)!1\nq#\n.\nDefine b1(t) =b\u0000\nt1\nq\u0001\nfort∈[1,∞). Set 0 < θ < 1, it follows from Lemma 2.9 that\nX\nk0∈Z2k0q∥χ{T1>2k0}∥q\np(·)γq\nb\u0000\n∥χ{T1>2k0}∥p(·)\u0001\n≲X\nk0∈Z2k0qk0−1X\nk=−∞2(k−k0)δq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rq\np(·)γbq\n1\u0012k0−1X\nk=−∞2(k−k0)δq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rq\np(·)\u0013\n=X\nk0∈Z2k0q\"\u0012k0−1X\nk=−∞2(k−k0)δq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rq\np(·)\u0013θ\n×γbθq\n1\u0012k0−1X\nk=−∞2(k−k0)δq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rq\np(·)\u0013#1\nθVariable martingale Hardy-Lorentz-Karamata spaces 29\n≲X\nk0∈Z2k0q\"k0−1X\nk=−∞2(k−k0)δθq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rθq\np(·)γbθq\n1\u0012\n2(k−k0)δq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rq\np(·)\u0013#1\nθ\n=X\nk0∈Z2k0q\"k0−1X\nk=−∞2(k−k0)δθq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rθq\np(·)γθq\nb\u0012\n2(k−k0)δ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013#1\nθ\n.\nLet 0 < β <δ−1\nδ. By applying H¨ older’s inequality with 1 −θ+θ= 1, we have\nX\nk0∈Z2k0q∥χ{T1>2k0}∥q\np(·)γq\nb\u0000\n∥χ{T1>2k0}∥p(·)\u0001\n≲X\nk0∈Z2k0q\"k0−1X\nk=−∞2(k−k0)δθqβ2(k−k0)(1−β)δθq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rθq\np(·)\n×γθq\nb\u0012\n2(k−k0)δ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013#1\nθ\n≤X\nk0∈Z2k0q\u0012k0−1X\nk=−∞2(k−k0)δθqβ/ (1−θ)\u00131−θ\nθk0−1X\nk=−∞2(k−k0)(1−β)δq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rq\np(·)\n×γq\nb\u0012\n2(k−k0)δ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013\n≲X\nk0∈Z2k0qk0−1X\nk=−∞2(k−k0)(1−β)δq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rq\np(·)γq\nb\u0012\n2(k−k0)δ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013\n.\nSetting 0 < z <δ−δβ−1\nδ, we obtain that for k≤k0−1,\nγb\u0012\n2(k−k0)δ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013\n(3.6)\n=\u0012\n2(k−k0)δ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013−z\n×\u0012\n2(k−k0)δ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013z\nγb\u0012\n2(k−k0)δ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013\n≲\u0012\n2(k−k0)δ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013−z\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rz\np(·)γb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013\n= 2−(k−k0)δzγb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013\n.\nHence it follows from Abel’s transformation that\nX\nk0∈Z2k0q∥χ{T1>2k0}∥q\np(·)γq\nb\u0000\n∥χ{T1>2k0}∥p(·)\u000130 Z. Hao, X. Ding, L. Li and F. Weisz\n≲X\nk0∈Z2k0qk0−1X\nk=−∞2(k−k0)(1−β)δq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rq\np(·)2−(k−k0)δzqγq\nb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013\n=X\nk∈Z2k(1−β−z)δq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rq\np(·)γq\nb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013∞X\nk0=k+12k0q[1+δ(β−1)+δz]\n≲X\nk∈Z2kq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rq\np(·)γq\nb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013\n.\nNext we discuss the case of q=∞. According to (3.5) and H¨ older’s inequality with\nε+ 1−ε= 1, we obtain that\n∥χ{T1>2k0}∥p(·)\n≲2k0L(σ−1) k0−1X\nk=−∞2kε(L(1−σ)−δ)/(1−ε)!(1−ε)/εk0−1X\nk=−∞2kδ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\n≲2−k0δk0−1X\nk=−∞2kδ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)=k0−1X\nk=−∞2(k−k0)δ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·).\nThen it follows from Lemma 2.9 and H¨ older’s inequality with 1 −θ+θ= 1 that\nsup\nk0∈Z2k0∥χ{T1>2k0}∥p(·)γb(∥χ{T1>2k0}∥p(·))\n≲sup\nk0∈Z2k0k0−1X\nk=−∞2(k−k0)δ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)γb k0−1X\nk=−∞2(k−k0)δ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)!\n≲sup\nk0∈Z2k0\"k0−1X\nk=−∞2(k−k0)δθ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rθ\np(·)γbθ\u0012\n2(k−k0)δ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013#1\nθ\n= sup\nk0∈Z2k0\"k0−1X\nk=−∞2βδθ(k−k0)2(k−k0)(1−β)δθ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rθ\np(·)\n×γθ\nb\u0012\n2(k−k0)δ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013#1\nθ\n≤sup\nk0∈Z2k0 k0−1X\nk=−∞2βδθ(k−k0)/(1−θ)!1−θ\nθk0−1X\nk=−∞2(k−k0)(1−β)δ\n×\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)γb\u0012\n2(k−k0)δ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013\n≲sup\nk0∈Z2k0k0−1X\nk=−∞2(k−k0)(1−β)δ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)γb\u0012\n2(k−k0)δ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013\n.Variable martingale Hardy-Lorentz-Karamata spaces 31\nFurthermore, by (3.6),\nsup\nk0∈Z2k0∥χ{T1>2k0}∥p(·)γb(∥χ{T1>2k0}∥p(·))\n≲sup\nk0∈Z2k0k0−1X\nk=−∞2(k−k0)(1−β)δ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)2−(k−k0)δzγb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013\n= sup\nk∈Z2kδ(1−β−z)\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)γb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013∞X\nk0=k+12k0(1−δ+βδ+δz)\n≲sup\nk∈Z2k\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)γb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013\n.\nTo sum up, for 0 < q≤ ∞,\n∥T1∥p(·),q,b≲\r\r\r\r\r(\n2k\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)γb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013)\nk∈Z\r\r\r\r\r\nlq.\nNow, we estimate T2. Since s(ak,i,j) = 0 on {τk=∞}, we have\n{T2>2k0} ⊂ { T2>0} ⊂∞[\nk=k0{s(ak,i,j)>0} ⊂∞[\nk=k0∞[\ni=0[\njIk,i,j.\nLet 0 < m < p −and 0 < ε < min{p, q}. It follows from Lemma 2.10 that\n2k0mε∥χ{T2>2k0}∥m\np(·)γm\nb\u0000\n∥χ{T2>2k0}∥p(·)\u0001\n≲2k0mε\r\r\r\r∞X\nk=k0∞X\ni=0X\njχIk,i,j\r\r\r\rm\np(·)γm\nb\u0012\r\r\r\r∞X\nk=k0∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013\n≲2k0mε∞X\nk=k0\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rm\np(·)γm\nb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013\n≲∞X\nk=k02kmε\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rm\np(·)γm\nb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013\n.\nObviously, according to Lemma 3.4, we obtain\n∥T2∥p(·),q,b≲\r\r\r\r\r(\n2k\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)γb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013)\nk∈Z\r\r\r\r\r\nlq.\nHence, it follows from Lemma 2.17 that\n∥f∥Hs\np(·),q,b=∥s(f)∥p(·),q,b≤ ∥T1+T2∥p(·),q,b\n≲∥T1∥p(·),q,b+∥T2∥p(·),q,b\n≲\r\r\r\r\r(\n2k\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)γb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013)\nk∈Z\r\r\r\r\r\nlq32 Z. Hao, X. Ding, L. Li and F. Weisz\n≈\r\r\r\r\r(\r\r\r\r∞X\ni=0X\njµk,i,j∥χIk,i,j∥−1\np(·)χIk,i,j\r\r\r\r\np(·)γb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013)\nk∈Z\r\r\r\r\r\nlq\n≈ ∥f∥Hs−at,r, 1\np(·),q,b.\nThe proof is complete. □\nRemark 3.7. The sumlP\nk=m∞P\ni=0P\njµk,i,jak,i,jconverges to finHs\np(·),q,basm→ −∞ ,l→ ∞ ,\nwhen p(·)∈ P(·),0< q < ∞andbis a slowly varying function. Indeed,\nlX\nk=m∞X\ni=0X\njµk,i,jak,i,j=lX\nk=m(fτk+1−fτk)∞X\ni=0X\njχIk,i,j\n=lX\nk=m(fτk+1−fτk)χ{τk<∞}=fτl+1−fτm.\nBy the sublinearity of the conditional square operator sand Lemma 2.17, we have\n\r\r\r\rf−lX\nk=m∞X\ni=0X\njµk,i,jak,i,j\r\r\r\r\nHs\np(·),q,b=\r\rs(f−fτl+1+fτm)\r\r\np(·),q,b\n≤\r\rs(f−fτl+1) +s(fτm)\r\r\np(·),q,b≲\r\rs(f−fτl+1)\r\r\np(·),q,b+\r\rs(fτm)\r\r\np(·),q,b.\nSince s(f−fτl+1)2=s(f)2−s(fτl+1)2, then s(f−fτl+1)≤s(f),s(fτm)≤s(f)and\ns(f−fτl+1)→0, s(fτm)→0a.e. as m→ −∞ , l→ ∞ . Hence, by dominated\nconvergence theorem (Lemma 2.22), we have\n\r\rs(f−fτl+1)\r\r\np(·),q,b→0,\r\rs(fτm)\r\r\np(·),q,b→0,asm→ −∞ , l→ ∞ ,\nwhich means that\n\r\r\r\rf−lX\nk=m∞X\ni=0X\njµk,i,jak,i,j\r\r\r\r\nHs\np(·),q,b→0,asm→ −∞ , l→ ∞ .\nFurthermore, for each k∈Z,ak,i,j= (ak,i,j\nn)n≥0isL2-bounded. If p+<2, we have Hs\n2⊂\nHs\np(·),q,b. Thus Hs\n2=L2is dense in Hs\np(·),q,bwhen p+<2. Similarly,lP\nk=m∞P\ni=0P\njµk,i,jak,i,j\nconverges to finHs\np(·),∞,basm→ −∞ ,l→ ∞ andHs\n2=L2is dense in Hs\np(·),∞,bwhen\np+<2.\nNext, we discuss the atomic decomposition theorems in HM\np(·),q,b, HS\np(·),q,b,Pp(·),q,band\nQp(·),q,b, respectively.\nTheorem 3.8. Letp(·)∈ P(Ω)satisfy (1.1),0< q≤ ∞ ,max{p+,1}< r≤ ∞ and let b\nbe a slowly varying function. If {Fn}n≥0is regular, then\nHS\np(·),q,b=Hs−at,r,2\np(·),q,band HM\np(·),q,b=Hs−at,r,3\np(·),q,b\nwith equivalent (quasi )-norms.Variable martingale Hardy-Lorentz-Karamata spaces 33\nProof. We only give the proof of HM\np(·),q,b, since it can be proved similarly for HS\np(·),q,b. Take\nf∈HM\np(·),q,b. Define the stopping times with respect to {Fn}n≥0by\nvk:= inf{n∈N:|fn|>2k}, k∈Z.\nDefine\nFk\nn:=n\nEn−1\u0000\nχ{vk=n}\u0001\n≥1\nRo\n∈ F n−1, n∈N.\nSince{Fn}n≥0is regular, we obtain that\n{vk=n} ⊂Fk\nnandP(Fk\nn)≤ RP(vk=n).\nDefine another set of stopping times by\nτk(ω) := inf {n∈N:ω∈Fk\nn+1}.\nIt is obvious that {τk}k∈Zis a nondecreasing sequence of stopping times and τk(ω)≤n−1\nwhen vk(ω) =n. In other words, τk< vkon the set {vk̸=∞}. Moreover, by Lemma 3.9,\n∥χ{τk<∞}∥p(·)≲∥χ{vk<∞}∥p(·)=∥χ{M(f)>2k}∥p(·)≤2−k∥M(f)∥p(·). (3.7)\nThen∥χ{τk<∞}∥p(·)→0 ask→ ∞ , which implies that\nlim\nk→∞P(τk=∞) = 1 .\nHence lim\nk→∞τk=∞a.e.and lim\nk→∞fτkn=fna.e.forn∈N. Define µk,i,jandak,i,j\nnin the\nsame as in the proof of Theorem 3.6. By the definition of τk, (3.4) holds and\nM\u0000\n(ak,i,j\nn)n≥0\u0001\n≤M(fτk+1n) +M(fτkn)\nµk,i,j≤2k+1+ 2k\nµk,i,j=1\n∥χIk,i,j∥p(·).\nThus ( ak,i,j\nn)n≥0is an L2-bounded martingale. Moreover, there exists ak,i,j∈L2such that\nEn(ak,i,j) =ak,i,j\nnand\n∥M(ak,i,j)∥r≤P(Ik,i,j)1/r 1\n∥χIk,i,j∥p(·).\nTherefore, ak,i,jis a simple ( p(·), r)M-atom and (3.1) holds. It is easy to get from (3.7)\nthat\n\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)≲\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r1/ε\np(·)/ε=\r\rχ{M(f)>2k}\r\r1/ε\np(·)/ε=\r\rχ{M(f)>2k}\r\r\np(·).\nSince tγb(t) is equivalent to a nondecreasing function, for 0 < q < ∞, we obtain\nX\nk∈Z\r\r\r\r∞X\ni=0X\njµk,i,j∥χIk,i,j∥−1\np(·)χIk,i,j\r\r\r\rq\np(·)γq\nb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013\n= 3qX\nk∈Z2kq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rq\np(·)γq\nb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013\n≲X\nk∈Z2kq∥χ{M(f)>2k}∥q\np(·)γq\nb\u0000\n∥χ{M(f)>2k}∥p(·)\u0001\n≈ ∥M(f)∥q\np(·),q,b=∥f∥q\nHM\np(·),q,b.34 Z. Hao, X. Ding, L. Li and F. Weisz\nSimilarly, for the case of q=∞,\n\r\r\r\r∞X\ni=0X\njµk,i,j∥χIk,i,j∥−1\np(·)χIk,i,j\r\r\r\r\np(·)γb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013\n= 3·2k\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)γb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013\n≲2k∥χ{M(f)>2k}∥p(·)γb\u0000\n∥χ{M(f)>2k}∥p(·)\u0001\n≲∥M(f)∥p(·),∞,b=∥f∥HM\np(·),∞,b.\nHence, we have ∥f∥Hs−at,r, 3\np(·),q,b≲∥f∥HM\np(·).q.b.\nWe omit the proof of the converse part, since it is similar to the proof of Theorem\n3.6. Thus, the proof is complete. □\nThe next lemma is used in the proof of the previous theorem and we refer the readers\nto Lemma 3.6 in [42] for its proof.\nLemma 3.9. Letp(·)∈ P(Ω)satisfy (1.1)and{Fn}n≥0be regular. Taking the same\nstopping times τkandvkas in the proof of Theorem 3.8, we have\n∥χ{τk<∞}∥p(·)≲∥χ{vk<∞}∥p(·).\nRemark 3.10. Letp(·)∈ P(·)andbbe a slowly varying function. Similarly to Remark\n3.7, we know that the sumlP\nk=m∞P\ni=0P\njµk,i,jak,i,jconverges to finHM\np(·),q,b(resp. HS\np(·),q,b)if\n0< q < ∞and in HM\np(·),∞,b(resp.HS\np(·),∞,b), asm→ −∞ ,l→ ∞ .\nNext we will prove that the conclusion in Theorem 3.6 is also valid for Qp(·),q,band\nPp(·),q,bwith simple ( p(·),∞)S-atoms and simple ( p(·),∞)M-atoms, respectively.\nTheorem 3.11. Letp(·)∈ P(Ω)satisfy (1.1),0< q≤ ∞ andbbe a slowly varying\nfunction. Then\nQp(·),q,b=Hs−at,∞,2\np(·),q,bandPp(·),q,b=Hs−at,∞,3\np(·),q,b\nwith equivalent (quasi )-norms.\nProof. The proof is similar to that of Theorem 3.6, so we only sketch it. Let f= (fn)n≥0∈\nQp(·),q,b(resp. Pp(·),q,b). For k∈Z, define the stopping times\nτk:= inf{n∈N:λn>2k},\nwhere ( λn)n≥0is the sequence in the definition of Qp(·),q,b(resp. Pp(·),q,b). Let the definitions\nofµk,i,jandak,i,j\nnbe the same as in the proof of Theorem 3.6. Then we get\nfn=X\nk∈Zn−1X\ni=0X\njµk,i,jEn(ak,i,j),\nwhere ( ak,i,j)k∈Z,i∈N,jis a sequence of simple ( p(·),∞)S-atoms (resp. simple ( p(·),∞)M-\natoms). Moreover, Hs−at,∞,2\np(·),q,b⊃ Q p(·),q,b(resp. Hs−at,∞,3\np(·),q,b⊃ P p(·),q,b).Variable martingale Hardy-Lorentz-Karamata spaces 35\nFor the converse part, let\nλn=X\nk∈Zn−1X\ni=0X\njµk,i,jχIk,i,j∥S(ak,i,j)∥∞\n\u0012\nresp. λn=X\nk∈Zn−1X\ni=0X\njµk,i,jχIk,i,j∥M(ak,i,j)∥∞\u0013\n,\nwhere ( ak,i,j)k∈Z,i∈N,jis a sequence of simple ( p(·),∞)S-atoms (resp. simple ( p(·),∞)M-\natoms) associated with ( Ik,i,j)k∈Z,i∈N,jandµk,i,j= 3·2∥χIk,i,j∥p(·). Then ( λn)n≥0∈Λp(·),q,b\nwith Sn+1(f)≤λn(resp. Mn+1(f)≤λn) for any n≥0. For any given integer k0, let\nλ∞=λ(1)\n∞+λ(2)\n∞,\nwhere\nλ(1)\n∞=k0−1X\nk=−∞∞X\ni=0X\njµk,i,jχIk,i,j∥S(ak,i,j)∥∞,\nλ(2)\n∞=∞X\nk=k0∞X\ni=0X\njµk,i,jχIk,i,j∥S(ak,i,j)∥∞\n\u0012\nresp. λ(1)\n∞=k0−1X\nk=−∞∞X\ni=0X\njµk,i,jχIk,i,j∥M(ak,i,j)∥∞,\nλ(2)\n∞=∞X\nk=k0∞X\ni=0X\njµk,i,jχIk,i,j∥M(ak,i,j)∥∞\u0013\n.\nBy replacing T1andT2in the proof of Theorem 3.6 with λ(1)\n∞andλ(2)\n∞, respectively, we\nobtain f∈ Q p(·),q,b(resp.Pp(·),q,b) andQp(·),q,b⊃Hs−at,∞,2\np(·),q,b(resp. Pp(·),q,b⊃Hs−at,∞,3\np(·),q,b).□\nRemark 3.12. Letp(·)∈ P(·),0< q < ∞andbbe a slowly varying function. The sum\nlP\nk=m∞P\ni=0P\njµk,i,jak,i,jconverges to finQp(·),q,b(resp.Pp(·),q,b)asm→ −∞ ,l→ ∞ . The\nproof is similar to Remark 3.7and left to the reader.\nEspecially, if we set p(·)≡pin Theorems 3.6, 3.8 and 3.11, then the following result\nholds.\nCorollary 3.13. Let0< p < ∞,0< q≤ ∞ andbbe a slowing varying function. We\nobtain the atomic decomposition theorems of martingale Hardy-Lorentz-Karamata spaces\nHs\np,q,b,Qp,q,bandPp,q,b, respectively. Moreover, if {Fn}n≥0is regular, we have the atomic\ndecompositions of HM\np,q,bandHS\np,q,b, respectively.\nRemark 3.14. The atomic decomposition theorems of Hs\np,q,b,Qp,q,bandPp,q,bcan be found\nin[44]under the condition that 0< p < ∞,0< q≤ ∞ andbis a nondecreasing slowly\nvarying function. Notice that the slowly varying function bis not necessarily nondecreasing\nin Corollary 3.13. Hence our results extend the atomic decomposition theorems of Hs\np,q,b,\nQp,q,bandPp,q,bin[44]and also give the atomic decomposition theorems of HM\np,q,band\nHS\np,q,b.\nParticularly, we have the following conclusion if b≡1 in Theorems 3.6, 3.8 and 3.11.36 Z. Hao, X. Ding, L. Li and F. Weisz\nCorollary 3.15. Letp(·)∈ P(Ω)satisfy (1.1)and0< q≤ ∞ . We get the atomic\ndecomposition theorems of the variable martingale Hardy-Lorentz spaces Hs\np(·),q,Pp(·),qand\nQp(·),q, respectively. Furthermore, if {Fn}n≥0is regular, we have the atomic decomposition\ntheorems of HM\np(·),qandHS\np(·),q.\nRemark 3.16. Corollary 3.15was proved in [40, 42] .\n3.2.Atomic Decompositions via Atoms. In this subsection, we give a slightly differ-\nent atomic decomposition for the five variable martingale Hardy-Lorentz-Karamata spaces.\nDefinition 3.17. Letp(·)∈ P(Ω). A measurable function ais called a (p(·),∞)s-atom\n(resp. (p(·),∞)S-atom or (p(·),∞)M-atom )if there exists a stopping time τsuch that\n(1)En(a) = 0 ,∀n≤τ;\n(2)∥s(a)∥∞(resp.∥S(a)∥∞or∥M(a)∥∞)≤\r\rχ{τ<∞}\r\r−1\np(·).\nDefinition 3.18. Letp(·)∈ P(Ω),0< q≤ ∞ andbbe a slowly varying function. The\natomic martingale Lorentz-Karamata space with variable exponent Hat,∞,1\np(·),q,b\u0000\nresp. Hat,∞,2\np(·),q,b\norHat,∞,3\np(·),q,b\u0001\nis defined as the set of all f= (fn)n≥0∈ M such that for each n≥0,\nfn=X\nk∈ZµkEn(ak),\nwhere (ak)k∈Zis a sequence of (p(·),∞)s-atoms\u0000\nresp. (p(·),∞)S-atoms or (p(·),∞)M-atoms\u0001\nassociated with stopping times (τk)k∈Zand\n\r\r\r\b\nµkγb\u0000\n∥χ{τk<∞}∥p(·)\u0001\t\nk∈Z\r\r\r\nlq<∞\nwith µk= 3·2k∥χ{τk<∞}∥p(·)for each k∈Z. Endow Hat,∞,1\np(·),q,b\u0000\nresp. Hat,∞,2\np(·),q,borHat,∞,3\np(·),q,b\u0001\nwith the (quasi )-norm\n∥f∥Hat,∞,1\np(·),q,b\u0000\nresp.∥f∥Hat,∞,2\np(·),q,bor∥f∥Hat,∞,3\np(·),q,b\u0001\n= inf\r\r\r\b\nµkγb\u0000\n∥χ{τk<∞}∥p(·)\u0001\t\nk∈Z\r\r\r\nlq,\nwhere the infimum is taken over all decompositions of fas above.\nWith the help of the atom adefined in Definition 3.17, we obtain another kind of\natomic decompositions for Hs\np(·),q,b,Qp(·),q,b,Pp(·),q,b,HM\np(·),q,bandHS\np(·),q,b. The proofs are\nsimilar to those of Theorems 3.6, 3.8 and 3.11, so we leave them to the reader.\nTheorem 3.19. Letp(·)∈ P(Ω)satisfy (1.1),0< q≤ ∞ and let bbe a slowly varying\nfunction. Then\nHs\np(·),q,b=Hat,∞,1\np(·),q,b,Qp(·),q,b=Hat,∞,2\np(·),q,bandPp(·),q,b=Hat,∞,3\np(·),q,b\nwith equivalent (quasi )-norms.\nTheorem 3.20. Letp(·)∈ P(Ω)satisfy (1.1),0< q≤ ∞ and let bbe a slowly varying\nfunction. If {Fn}n≥0is regular, then\nHS\np(·),q,b=Hat,∞,2\np(·),q,band HM\np(·),q,b=Hat,∞,3\np(·),q,b\nwith equivalent (quasi )-norms.Variable martingale Hardy-Lorentz-Karamata spaces 37\n4.Martingale Inequalities\nIn this section, we show some martingale inequalities between different variable mar-\ntingale Hardy-Lorentz-Karamata spaces. The method we use here is to establish a suf-\nficient condition for a σ-sublinear operator to be bounded from the variable martingale\nHardy-Lorentz-Karamata spaces to the variable Lorentz-Karamata spaces.\nLet us recall that an operator T:X→Yis said to be σ-sublinear, if for any constant\nc,\f\f\f\fT\u0012∞X\nk=1fk\u0013\f\f\f\f≤∞X\nk=1|T(fk)|and|T(cf)|=|c||T(f)|,\nwhere Xis a martingale space and Yis a measurable function space.\nLemma 4.1. Letp(·)∈ P(Ω)satisfy (1.1),0< q≤ ∞ and let bbe a slowly varying\nfunction. Suppose that max{p+,1}< r < ∞. IfT:Hs\nr→Lris a bounded σ-sublinear\noperator and\n{|T(a)|>0} ⊂I\nfor every simple (p(·), r)s-atom aassociated with I, then for f∈Hs\np(·),q,b,\n∥T(f)∥p(·),q,b≲∥f∥Hs\np(·),q,b.\nProof. Letf∈Hs\np(·),q,b. According to Theorem 3.6, there exist a sequence of simple\n(p(·), r)s-atoms ( ak,i,j)k∈Z,i∈N,jandµk,i,j= 3·2k∥χIk,i,j∥p(·)such that\nf=X\nk∈Z∞X\ni=0X\njµk,i,jak,i,j\nand\r\r\r\r\r(\r\r\r\r∞X\ni=0X\njµk,i,j∥χIk,i,j∥−1\np(·)χIk,i,j\r\r\r\r\np(·)γb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013)\nk∈Z\r\r\r\r\r\nlq��∥f∥Hs\np(·),q,b.\nFor an arbitrary integer k0, set\nD1:=k0−1X\nk=−∞∞X\ni=0X\njµk,i,j|T(ak,i,j)|and D2:=∞X\nk=k0∞X\ni=0X\njµk,i,j|T(ak,i,j)|.\nThen by the σ-sublinearity of T, there is\n|T(f)| ≤X\nk∈Z∞X\ni=0X\njµk,i,j|T(ak,i,j)|=D1+D2.\nLet 0 < ε < min{p, q}, 1< L < min{r\np+,1\nε}and 0 < σ < 1−1\nL. It follows from H¨ older’s\ninequality that\nD1=k0−1X\nk=−∞∞X\ni=0X\njµk,i,j|T(ak,i,j)|\n≤\u0012k0−1X\nk=−∞2kσL′\u00131\nL′\u0014k0−1X\nk=−∞2−kσL\u0012∞X\ni=0X\njµk,i,j|T(ak,i,j)|\u0013L\u00151\nL38 Z. Hao, X. Ding, L. Li and F. Weisz\n≈2k0σ\u0014k0−1X\nk=−∞2−kσL\u0012∞X\ni=0X\njµk,i,j|T(ak,i,j)|\u0013L\u00151\nL\n,\nwhere L′is the conjugate of L. According to Lemma 2.3 and Remark 2.1 (3), we have\n∥χ{D1>2k0}∥p(·)≤2−k0L∥DL\n1∥p(·)\n≲2−k0L\r\r\r\r\r2k0σLk0−1X\nk=−∞2−kσL\u0012∞X\ni=0X\njµk,i,j|T(ak,i,j)|\u0013L\r\r\r\r\r\np(·)\n≈2k0L(σ−1)\r\r\r\r\rk0−1X\nk=−∞2kL(1−σ)\u0012∞X\ni=0X\nj∥χIk,i,j∥p(·)|T(ak,i,j)|\u0013L\r\r\r\r\r\np(·)\n= 2k0L(σ−1)\r\r\r\r\r\u0014k0−1X\nk=−∞2kL(1−σ)\u0012∞X\ni=0X\nj∥χIk,i,j∥p(·)|T(ak,i,j)|\u0013L\u0015ε\r\r\r\r\r1\nε\np(·)\nε\n≤2k0L(σ−1)\r\r\r\r\rk0−1X\nk=−∞2kLε(1−σ)\u0012∞X\ni=0X\nj∥χIk,i,j∥p(·)|T(ak,i,j)|\u0013Lε\r\r\r\r\r1\nε\np(·)\nε\n≤2k0L(σ−1)\r\r\r\r\rk0−1X\nk=−∞2kLε(1−σ)∞X\ni=0X\nj\u0010\n∥χIk,i,j∥p(·)|T(ak,i,j)|\u0011Lε\r\r\r\r\r1\nε\np(·)\nε\n≲2k0L(σ−1) k0−1X\nk=−∞2kLε(1−σ)\r\r\r\r∞X\ni=0X\nj\u0010\n∥χIk,i,j∥p(·)|T(ak,i,j)|\u0011Lε\r\r\r\rp(·)\nε!1\nε\n.\nSince T:Hs\nr→Lris bounded, we have that for a simple ( p(·), r)s-atom ak,i,j,\n∥T(ak,i,j)∥r≲∥s(ak,i,j)∥r≤∥χIk,i,j∥r\n∥χIk,i,j∥p(·).\nThen using Lemma 3.5, we obtain\n\r\r\r\r∞X\ni=0X\nj\u0010\n∥χIk,i,j∥p(·)|T(ak,i,j)|\u0011Lε\r\r\r\rp(·)\nε≲\r\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\rp(·)\nε.\nHence we get that\n∥χ{D1>2k0}∥p(·)≲2k0L(σ−1) k0−1X\nk=−∞2kLε(1−σ)\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rp(·)\nε!1\nε\n(4.1)\n= 2k0L(σ−1) k0−1X\nk=−∞2kLε(1−σ)\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rε\np(·)!1\nε\n= 2k0L(σ−1) k0−1X\nk=−∞2kε(L(1−σ)−δ)2kεδ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rε\np(·)!1\nε\n,\nwhere δ >0 satisfies 1 < δ < L (1−σ).Variable martingale Hardy-Lorentz-Karamata spaces 39\nFirstly, we consider the case of 0 < q < ∞. By H¨ older’s inequality withε\nq+q−ε\nq= 1,\nwe obtain\n∥χ{D1>2k0}∥p(·)≲2k0L(σ−1) k0−1X\nk=−∞2kε(L(1−σ)−δ)q\nq−ε!q−ε\nqε k0−1X\nk=−∞2kqδ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rq\np(·)!1\nq\n≲2−k0δ k0−1X\nk=−∞2kδq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rq\np(·)!1\nq\n= k0−1X\nk=−∞2(k−k0)δq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rq\np(·)!1\nq\n.\nThis yields that\nX\nk0∈Z2k0q∥χ{D1>2k0}∥q\np(·)γq\nb\u0000\n∥χ{D1>2k0}∥p(·)\u0001\n≲X\nk0∈Z2k0qk0−1X\nk=−∞2(k−k0)δq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rq\np(·)γq\nb\" k0−1X\nk=−∞2(k−k0)δq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rq\np(·)!1\nq#\n.\nDefine b1(t) =b\u0000\nt1\nq\u0001\nfort∈[1,∞) and let 0 < θ < 1. It follows from Lemma 2.9 that\nX\nk0∈Z2k0q∥χ{D1>2k0}∥q\np(·)γq\nb\u0000\n∥χ{D1>2k0}∥p(·)\u0001\n≲X\nk0∈Z2k0qk0−1X\nk=−∞2(k−k0)δq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rq\np(·)γbq\n1\u0012k0−1X\nk=−∞2(k−k0)δq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rq\np(·)\u0013\n=X\nk0∈Z2k0q\"\u0012k0−1X\nk=−∞2(k−k0)δq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rq\np(·)\u0013θ\n×γbθq\n1\u0012k0−1X\nk=−∞2(k−k0)δq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rq\np(·)\u0013#1\nθ\n≲X\nk0∈Z2k0q\"k0−1X\nk=−∞2(k−k0)δθq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rθq\np(·)γbθq\n1\u0012\n2(k−k0)δq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rq\np(·)\u0013#1\nθ\n=X\nk0∈Z2k0q\"k0−1X\nk=−∞2(k−k0)δθq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rθq\np(·)γθq\nb\u0012\n2(k−k0)δ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013#1\nθ\n.\nLet 0 < β <δ−1\nδ. By applying H¨ older’s inequality with 1 −θ+θ= 1, we have\nX\nk0∈Z2k0q∥χ{D1>2k0}∥q\np(·)γq\nb\u0000\n∥χ{D1>2k0}∥p(·)\u0001\n≲X\nk0∈Z2k0q\"k0−1X\nk=−∞2(k−k0)δθqβ2(k−k0)(1−β)δθq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rθq\np(·)40 Z. Hao, X. Ding, L. Li and F. Weisz\n×γθq\nb\u0012\n2(k−k0)δ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013#1\nθ\n≤X\nk0∈Z2k0q\u0012k0−1X\nk=−∞2(k−k0)δθqβ/ (1−θ)\u00131−θ\nθk0−1X\nk=−∞2(k−k0)(1−β)δq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rq\np(·)\n×γq\nb\u0012\n2(k−k0)δ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013\n≲X\nk0∈Z2k0qk0−1X\nk=−∞2(k−k0)(1−β)δq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rq\np(·)γq\nb\u0012\n2(k−k0)δ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013\n.\nSet 0 < z <δ−δβ−1\nδ, it follows from (3.6) that\nγb\u0012\n2(k−k0)δ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013\n≲2−(k−k0)δzγb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013\n.\nHence we get the following by Abel’s transformation:\nX\nk0∈Z2k0q∥χ{D1>2k0}∥q\np(·)γq\nb\u0000\n∥χ{D1>2k0}∥p(·)\u0001\n≲X\nk0∈Z2k0qk0−1X\nk=−∞2(k−k0)(1−β)δq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rq\np(·)2−(k−k0)δzqγq\nb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013\n=X\nk∈Z2k(1−β−z)δq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rq\np(·)γq\nb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013∞X\nk0=k+12k0q[1+δ(β−1)+δz]\n≲X\nk∈Z2kq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rq\np(·)γq\nb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013\n.\nNext we discuss the case of q=∞. According to (4.1) and H¨ older’s inequality with\nε+ 1−ε= 1, we obtain that\n∥χ{D1>2k0}∥p(·)≲2k0L(σ−1) k0−1X\nk=−∞2kε(L(1−σ)−δ)/(1−ε)!(1−ε)/εk0−1X\nk=−∞2kδ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\n≲2−k0δk0−1X\nk=−∞2kδ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)=k0−1X\nk=−∞2(k−k0)δ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·).\nThen it follows from Lemma 2.9 and H¨ older’s inequality with 1 −θ+θ= 1 that\nsup\nk0∈Z2k0∥χ{D1>2k0}∥p(·)γb(∥χ{D1>2k0}∥p(·))\n≲sup\nk0∈Z2k0k0−1X\nk=−∞2(k−k0)δ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)γb k0−1X\nk=−∞2(k−k0)δ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)!Variable martingale Hardy-Lorentz-Karamata spaces 41\n= sup\nk0∈Z2k0\" k0−1X\nk=−∞2(k−k0)δ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)!θ\n×γbθ k0−1X\nk=−∞2(k−k0)δ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)!# 1\nθ\n≲sup\nk0∈Z2k0\"k0−1X\nk=−∞2(k−k0)δθ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rθ\np(·)γbθ\u0012\n2(k−k0)δ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013#1\nθ\n= sup\nk0∈Z2k0\"k0−1X\nk=−∞2βδθ(k−k0)2(k−k0)(1−β)δθ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rθ\np(·)\n×γθ\nb\u0012\n2(k−k0)δ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013#1\nθ\n≤sup\nk0∈Z2k0 k0−1X\nk=−∞2βδθ(k−k0)/(1−θ)!1−θ\nθk0−1X\nk=−∞2(k−k0)(1−β)δ\n×\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)γb\u0012\n2(k−k0)δ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013\n≲sup\nk0∈Z2k0k0−1X\nk=−∞2(k−k0)(1−β)δ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)γb\u0012\n2(k−k0)δ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013\n.\nInequality (3.6) implies\nsup\nk0∈Z2k0∥χ{D1>2k0}∥p(·)γb(∥χ{D1>2k0}∥p(·))\n≲sup\nk0∈Z2k0k0−1X\nk=−∞2(k−k0)(1−β)δ\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)2−(k−k0)δzγb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013\n= sup\nk∈Z2kδ(1−β−z)\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)γb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013∞X\nk0=k+12k0(1−δ+βδ+δz)\n≲sup\nk∈Z2k\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)γb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013\n.\nTherefore, we conclude that\n∥D1∥p(·),q,b≲\r\r\r\r\r(\n2k\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)γb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013)\nk∈Z\r\r\r\r\r\nlq.42 Z. Hao, X. Ding, L. Li and F. Weisz\nNow, we estimate D2. Since s(ak,i,j) = 0 on the set∞S\ni=0S\njIk,i,j, we have\n{D2>2k0} ⊂ { D2>0} ⊂∞[\nk=k0{s(ak,i,j)>0} ⊂∞[\nk=k0∞[\ni=0[\njIk,i,j.\nLet 0 < m < p andεbe the same as before. It follows from Lemma 2.10 that\n2k0mε∥χ{D2>2k0}∥m\np(·)γm\nb\u0000\n∥χ{D2>2k0}∥p(·)\u0001\n≲2k0mε\r\r\r\r∞X\nk=k0∞X\ni=0X\njχIk,i,j\r\r\r\rm\np(·)γm\nb\u0012\r\r\r\r∞X\nk=k0∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013\n≲2k0mε∞X\nk=k0\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rm\np(·)γm\nb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013\n≲∞X\nk=k02kmε\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rm\np(·)γm\nb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013\n.\nObviously, according to Lemma 3.4, we obtain\n∥D2∥p(·),q,b≲\r\r\r\r\r(\n2k\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)γb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013)\nk∈Z\r\r\r\r\r\nlq.\nHence, by Lemma 2.17, there is\n∥T(f)∥p(·),q,b≲∥D1∥p(·),q,b+∥D2∥p(·),q,b\n≲\r\r\r\r\r(\n2k+1\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)γb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013)\nk∈Z\r\r\r\r\r\nlq\n≲∥f∥Hs\np(·),q,b.\nThe proof is complete now. □\nSimilarly, with Theorems 3.8 and 3.11, we get the following results.\nLemma 4.2. Letp(·)∈ P(Ω)satisfy (1.1),0< q≤ ∞ andbbe a slowly varying function.\nSuppose that {Fn}n≥0is regular and max{p+,1}< r < ∞. IfT:HS\nr→Lr(resp. T:\nHM\nr→Lr)is a bounded σ-sublinear operator and\n{|T(a)|>0} ⊂I\nfor every simple (p(·), r)S-atom (resp. simple (p(·), r)M-atom )aassociated with I, then\nforf∈HS\np(·),q,b(resp. f∈HM\np(·),q,b),\n∥T(f)∥p(·),q,b≲∥f∥HS\np(·),q,b\u0000\nresp.∥T(f)∥p(·),q,b≲∥f∥HM\np(·),q,b\u0001\n.\nLemma 4.3. Letp(·)∈ P(Ω)satisfy (1.1),0< q≤ ∞ and let bbe a slowly varying\nfunction. Suppose that max{p+,1}< r < ∞. IfT:HS\nr→Lr(resp. T:HM\nr→Lr)is a\nbounded σ-sublinear operator and\n{|T(a)|>0} ⊂IVariable martingale Hardy-Lorentz-Karamata spaces 43\nfor every simple (p(·), r)S-atom (resp. simple (p(·), r)M-atom )aassociated with I, then\nforf∈ Q p(·),q,b(resp. f∈ P p(·),q,b),\n∥T(f)∥p(·),q,b≲∥f∥Qp(·),q,b\u0000\nresp.∥T(f)∥p(·),q,b≲∥f∥Pp(·),q,b\u0001\n.\nTo get the martingale inequalities for variable martingale Hardy-Lorentz-Karamata\nspaces, we also require the following lemma.\nLemma 4.4 ([23, 75]) .Letfbe a martingale. Then\n∥M(f)∥2≤2∥S(f)∥2= 2∥s(f)∥2≤2∥M(f)∥2;\n∥s(f)∥r≤rr\n2∥M(f)∥r, r≥2;\n∥s(f)∥r≤rr\n2∥S(f)∥r, r≥2.\nMoreover, if the stochastic basis {Fn}n≥0is regular, then\n∥M(f)∥r≈ ∥S(f)∥r≈ ∥s(f)∥r,0< r < ∞.\nNow, we show the main conclusion of this section.\nTheorem 4.5. Letp(·)∈ P(Ω)satisfy (1.1),0< q≤ ∞ and let bbe a slowly varying\nfunction. Then\n∥f∥HM\np(·),q,b≲∥f∥Hs\np(·),q,b,∥f∥HS\np(·),q,b≲∥f∥Hs\np(·),q,b,0< p−≤p+<2; (4.2)\n∥f∥HM\np(·),q,b≲∥f∥Pp(·),q,b,∥f∥HS\np(·),q,b≲∥f∥Qp(·),q,b; (4.3)\n∥f∥HS\np(·),q,b≲∥f∥Pp(·),q,b,∥f∥HM\np(·),q,b≲∥f∥Qp(·),q,b; (4.4)\n∥f∥Hs\np(·),q,b≲∥f∥Pp(·),q,b,∥f∥Hs\np(·),q,b≲∥f∥Qp(·),q,b; (4.5)\n∥f∥Pp(·),q,b≲∥f∥Qp(·),q,b≲∥f∥Pp(·),q,b. (4.6)\nIf{Fn}n≥0is regular, then\nHS\np(·),q,b=Qp(·),q,b=Pp(·),q,b=HM\np(·),q,b=Hs\np(·),q,b.\nProof. It is obvious that the operators s, M, S areσ-sublinear and\n{s(a1)>0} ⊂I1,{M(a2)>0} ⊂I2and{S(a3)>0} ⊂I3,\nwhere ai(i= 1,2,3) is ( p(·), r)∗-atom ( ∗=s, M, S ) associated with Ii(i= 1,2,3),\nrespectively. Lemma 4.4 gives that the operators M:Hs\n2→L2andS:Hs\n2→L2\nare bounded. Hence, using Lemma 4.1, we know that the inequalities in (4.2) hold if\n0< p−≤p+<2.\nThe inequalities in (4.3) follow directly from the definitions of Pp(·),q,bandQp(·),q,b.\nAccording to Burkholder-Davis-Gundy’s inequality (see [75, Theorem 2.12]), namely,\ncr∥f∥HSr≤ ∥f∥HMr≤Cr∥f∥HSr,1≤r <∞\nand Doob’s maximal inequality, we obtain that the operators M:HS\nr→LrandS:\nHM\nr→Lrare bounded. Hence, using Lemma 4.3, the inequalities in (4.4) hold.\nLet max {p+,2}< r < ∞. It follows from Lemma 4.4 that s:HM\nr→Lrand\ns:HS\nr→Lrare bounded. Then by Lemma 4.3, we obtain the inequalities in (4.5).44 Z. Hao, X. Ding, L. Li and F. Weisz\nIn order to prove (4.6), if we take f= (fn)n≥0∈ Q p(·),q,b, then there exists ( λ(1)\nn)n≥0∈\nΛp(·),q,bsuch that Sn(f)≤λ(1)\nn−1with λ(1)\n∞∈Lp(·),q,b. Since\n|fn| ≤ |fn−fn−1|+|fn−1| ≤Sn(f) +Mn−1(f)≤λ(1)\nn−1+Mn−1(f)\nand ( λ(1)\nn+Mn(f))n≥0∈Λp(·),q,b, by the second inequality of (4.4), we get\n∥f∥Pp(·),q,b≤ ∥λ(1)\n∞+M(f)∥p(·),q,b≲∥M(f)∥Hp(·),q,b+∥λ(1)\n∞∥p(·),q,b≲∥f∥Qp(·),q,b.\nIff= (fn)n≥0∈ P p(·),q,b, then there exists ( λ(2)\nn)n≥0∈Λp(·),q,bsuch that |fn| ≤λ(2)\nn−1with\nλ(2)\n∞∈Lp(·),q,b. Since\nSn(f)≤Sn−1(f) +|fn−fn−1| ≤Sn−1(f) + 2Mn(f)≤Sn−1(f) + 2λ(2)\nn\nand ( λ(2)\nn+Sn(f))n≥0∈Λp(·),q,b, by the first inequality of (4.4), we have\n∥f∥Qp(·),q,b≤ ∥λ(2)\n∞+S(f)∥p(·),q,b≲∥S(f)∥Hp(·),q,b+∥λ(2)\n∞∥p(·),q,b≲∥f∥Pp(·),q,b.\nHence, the inequality (4.6) holds and this yields that Pp(·),q,b=Qp(·),q,b.\nIf{Fn}n≥0is regular, combining Theorem 3.8 with Theorem 3.11, we conclude that\nHM\np(·),q,b=Pp(·),q,band HS\np(·),q,b=Qp(·),q,b.\nAccording to [75, p.33], there is\nSn(f)≤ R1\n2sn(f).\nThen by the definition of Qp(·),q,b, we obtain\n∥f∥Qp(·),q,b≲∥s(f)∥p(·),q,b=∥f∥Hs\np(·),q,b,\nsince sn(f)∈ F n−1. Hence, it follows from (4.5) and (4.6) that\nHs\np(·),q,b=Qp(·),q,b=Pp(·),q,b.\nTherefore,\nHS\np(·),q,b=Qp(·),q,b=Hs\np(·),q,b=Pp(·),q,b=HM\np(·),q,b\nand the proof of this theorem is complete. □\nRemark 4.6. When p(·)≡pis a constant, Theorem 4.5removes the condition that bis\nnondecreasing in [85, Theorems 1 .2 and 1 .3]. When b≡1, Theorem 4.5reduces to [42,\nTheorem 4 .11]. Moreover, if p(·)≡pis a constant and b≡1, we obtain the relation of\nfive martingale Hardy-Lorentz spaces, see [75].\nMoreover, combining Theorem 4.5 with Theorem 3.19, we have the following result.\nCorollary 4.7. Letp(·)∈ P(Ω)satisfy (1.1),0< q≤ ∞ and let bbe a slowly varying\nfunction. If {Fn}n≥0is regular, then\nHS\np(·),q,b=Qp(·),q,b=Pp(·),q,b=HM\np(·),q,b=Hs\np(·),q,b=Hat,∞,i\np(·),q,b,\nwhere i= 1,2,3.Variable martingale Hardy-Lorentz-Karamata spaces 45\nAs another application of Theorem 4.5, we can obtain the boundedness of the mar-\ntingale transform for variable Lorentz-Karamata spaces. The martingale transform was\nintroduced by Burkholder in [8] and he obtained the boundedness of the martingale trans-\nform generated by uniformly bounded sequence on the Lebesgue spaces. Jiao et al. [42]\nextended this result to the variable exponents setting, and studied the boundedness of\nmartingale transform on variable Lebesgue spaces and variable Lorentz spaces. Firstly, we\nrecall the definition of martingale transform.\nDefinition 4.8. For any martingale f∈ M , the martingale transform of fis defined by\n(Tvf)n=nX\nk=0vk−1dkf, n ∈N,\nwhere for each k,vkisFk-measurable and ∥vk∥∞< L for some L >0.\nTheorem 4.9. Letp(·)∈ P(Ω)satisfy (1.1)with 1< p−≤p+<∞,0< q≤ ∞ and let\nbbe a slowly varying function. Then we have\n∥Tvf∥p(·),q,b≲∥f∥p(·),q,b for any f∈Lp(·),q,b.\nProof. Since∥vk∥∞< Lfor each kand some L >0, we have\nS(Tvf) =\u0012∞X\nn=0\f\fdn(Tvf)\f\f2\u00131\n2\n=\u0012∞X\nn=0|vn−1dnf|2\u00131\n2\n< L\u0012∞X\nn=0|dnf|2\u00131\n2\n=LS(f).\nHence, by using Theorem 4.5 and Lemma 2.29, we obtain that\n∥Tvf∥p(·),q,b≲∥M(Tvf)∥p(·),q,b≲∥S(Tvf)∥p(·),q,b\n≲∥S(f)∥p(·),q,b≲∥M(f)∥p(·),q,b≲∥f∥p(·),q,b.\nThe proof is complete. □\n5.Dual Results\nIn this section, our main objective is to characterize the dual spaces of variable martin-\ngale Hardy-Lorentz-Karamata spaces. To this end, we introduce the following generalized\nBMO martingale spaces.\nDefinition 5.1. Letα(·) + 1∈ P(Ω),1≤r <∞and let bbe a slowly varying function.\nThe space BMO r,b(α(·))consists of those functions f∈Lrfor which\n∥f∥BMO r,b(α(·))= sup\nn≥0sup\nI∈A(Fn)∥χI∥−1\n1\nα(·)+1P(I)1−1\nrγ−1\nb\u0000\n∥χI∥ 1\nα(·)+1\u0001\n∥(f−En(f))χI∥r\nis finite.\nRemark 5.2. Ifb≡1, then the above generalized martingale space is the same as\nBMO r(α(·))in[40].BMO r,b(α)given by Jiao et al. [44]is a special case of Defini-\ntion 5.1when α(·)≡α > 0is a constant. Moreover, if b≡1andα(·)≡0, then the\ngeneralized martingale space goes back to the classical martingale BMO rspace in [75].\nNow we introduce another generalization of BMO spaces.46 Z. Hao, X. Ding, L. Li and F. Weisz\nDefinition 5.3. Letα(·)+1∈ P(Ω),1≤r <∞,0< q≤ ∞ and let bbe a slowly varying\nfunction. The generalized martingale space BMO r,q,b(α(·))is defined by\nBMO r,q,b(α(·)) =\b\nf∈Lr:∥f∥BMO r,q,b(α(·))<∞\t\n,\nwhere\n∥f∥BMO r,q,b(α(·))= supP\nk∈Z∞P\ni=0P\nj2kP(Ik,i,j)1−1\nr∥(f−Ei(f))χIk,i,j∥r\n\r\r\r\r\u001a\n2k\r\r\r∞P\ni=0P\njχIk,i,j\r\r\r1\nα(·)+1γb\u0010\r\r\r∞P\ni=0P\njχIk,i,j\r\r\r1\nα(·)+1\u0011\u001b\nk∈Z\r\r\r\r\nlq\nand the supremum is taken over all sequences of atoms (Ik,i,j)k∈Z,i∈N,jsuch that Ik,i,jare\ndisjoint if kis fixed, Ik,i,jbelong to Fiand\n(\n2k\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r1\nα(·)+1γb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r1\nα(·)+1\u0013)\nk∈Z∈lq.\nRemark 5.4. Ifb≡1,BMO r,q,b(α(·))reduces to BMO r,q(α(·))in[40].BMO r,q,b(α)\ndefined by Jiao et al. [44]is a special case of Definition 5.3when α(·)≡α > 0is a\nconstant. Moreover, if b≡1andα(·)≡0, then the generalized martingale space goes back\nto classical martingale BMO r,qspace in [75].\nThe following result gives the connection of BMO r,b(α(·)) and BMO r,q,b(α(·)).\nProposition 5.5. Letα(·) + 1∈ P(Ω),1≤r <∞,0< q < ∞and let bbe a slowly\nvarying function. Then\n∥f∥BMO r,b(α(·))≤ ∥f∥BMO r,q,b(α(·)).\nIf we suppose that α−>0and0< q≤1, then\n∥f∥BMO r,b(α(·))≈ ∥f∥BMO r,q,b(α(·)). (5.1)\nIn addition, if α−≥0andbis nonincreasing, we also have (5.1).\nProof. If we take the supremum in the definition of BMO r,q,b(α(·)) only for one atom, then\nwe get back to the definition of BMO r,b(α(·)). Hence the first inequality is obvious. For\nthe second inequality, it clearly suffices to verify that ∥f∥BMO r,q,b(α(·))≲∥f∥BMO r,b(α(·)).\nFor any f∈BMO r,b(α(·)), by the definition of BMO r,b(α(·)), there is\nP(I)1−1\nr∥(f−Ei(f))χI∥r≤ ∥χI∥ 1\nα(·)+1γb\u0000\n∥χI∥ 1\nα(·)+1\u0001\n∥f∥BMO r,b(α(·))\nfor any I∈A(Fi).Since α−>0 and 0 < q≤1, then it follows from Lemma 2.10 that\nP\nk∈Z∞P\ni=0P\nj2kP(Ik,i,j)1−1\nr∥(f−Ei(f))χIk,i,j∥r\n\u0012P\nk∈Z2kq\r\r\r∞P\ni=0P\njχIk,i,j\r\r\rq\n1\nα(·)+1γq\nb\u0010\r\r\r∞P\ni=0P\njχIk,i,j\r\r\r1\nα(·)+1\u0011\u00131\nq\n≤P\nk∈Z∞P\ni=0P\nj2k∥χIk,i,j∥ 1\nα(·)+1γb\u0000\n∥χIk,i,j∥ 1\nα(·)+1\u0001\n∥f∥BMO r,b(α(·))\n\u0012P\nk∈Z2kq\r\r\r∞P\ni=0P\njχIk,i,j\r\r\rq\n1\nα(·)+1γq\nb\u0010\r\r\r∞P\ni=0P\njχIk,i,j\r\r\r1\nα(·)+1\u0011\u00131\nqVariable martingale Hardy-Lorentz-Karamata spaces 47\n≤ ∥f∥BMO r,b(α(·))P\nk∈Z∞P\ni=0P\nj2k∥χIk,i,j∥ 1\nα(·)+1γb\u0000\n∥χIk,i,j∥ 1\nα(·)+1\u0001\nP\nk∈Z2k\r\r\r∞P\ni=0P\njχIk,i,j\r\r\r1\nα(·)+1γb\u0010\r\r\r∞P\ni=0P\njχIk,i,j\r\r\r1\nα(·)+1\u0011\n≲∥f∥BMO r,b(α(·))P\nk∈Z2k\r\r\r∞P\ni=0P\njχIk,i,j\r\r\r1\nα(·)+1γb\u0010\r\r\r∞P\ni=0P\njχIk,i,j\r\r\r1\nα(·)+1\u0011\nP\nk∈Z2k\r\r\r∞P\ni=0P\njχIk,i,j\r\r\r1\nα(·)+1γb\u0010\r\r\r∞P\ni=0P\njχIk,i,j\r\r\r1\nα(·)+1\u0011\n=∥f∥BMO r,b(α(·)).\nHence, ∥f∥BMO r,q,b(α(·))≲∥f∥BMO r,b(α(·))and this completes the proof of (5.1). If α−≥0\nandbis nonincreasing, the proof is similar. □\nWe give the main result of this section.\nTheorem 5.6. Letp(·)∈ P(Ω)satisfy (1.1)with 0< p−≤p+<2,0< q < ∞and let b\nbe a slowly varying function. Then\n\u0000\nHs\np(·),q,b\u0001∗=BMO 2,q,b(α(·)), α(·) =1\np(·)−1.\nProof. Letg∈BMO 2,q,b(α(·))⊂L2. Define the functional ℓgas\nℓg(f) :=E(fg),∀f∈L2.\nWe shall prove that ℓgis a bounded linear functional on Hs\np(·),q,b. By Remark 3.7, we see\nthatL2⊂Hs\np(·),q,b. Moreover,\nf=X\nk∈Z∞X\ni=0X\njµk,i,jak,i,j\nholds also in L2, where µk,i,j= 3·2k∥χIk,i,j∥p(·)and ( ak,i,j)k∈Z,i∈N,jis a sequence of simple\n(p(·),2)s-atoms. In addition,\n X\nk∈Z\r\r\r\r∞X\ni=0X\njµk,i,j∥χIk,i,j∥−1\np(·)χIk,i,j\r\r\r\rq\np(·)γq\nb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013!1\nq\n≲∥f∥Hs\np(·),q,b.\nHence,\nℓg(f) =E(fg) =X\nk∈Z∞X\ni=0X\njµk,i,jE(ak,i,jg).\nBy the definition of ak,i,j, there is\nE(ak,i,jg) =E\u0000\n(ak,i,j−Ei(ak,i,j))g\u0001\n=E\u0000\nak,i,j(g−Ei(g))\u0001\n.\nAccording to H¨ older’s inequality, we obtain that\n|ℓg(f)|=\f\f\f\fX\nk∈Z∞X\ni=0X\njµk,i,jE(ak,i,j(g−Ei(g)))\f\f\f\f\n≤X\nk∈Z∞X\ni=0X\njµk,i,j∥ak,i,j∥2∥(g−Ei(g))χIk,i,j∥248 Z. Hao, X. Ding, L. Li and F. Weisz\n≲X\nk∈Z∞X\ni=0X\njµk,i,jP(Ik,i,j)1/2\n∥χIk,i,j∥p(·)∥(g−Ei(g))χIk,i,j∥2\n= 3X\nk∈Z∞X\ni=0X\nj2kP(Ik,i,j)1/2∥(g−Ei(g))χIk,i,j∥2\n≲ X\nk∈Z2kq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\rq\np(·)γq\nb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013!1\nq\n· ∥g∥BMO 2,q,b(α(·))\n≲∥f∥Hs\np(·),q,b· ∥g∥BMO 2,q,b(α(·)).\nSince L2is dense in Hs\np(·),q,b, the linear functional ℓgcan be uniquely extended to a bounded\nlinear functional on Hs\np(·),q,b.\nConversely, let ℓ∈\u0000\nHs\np(·),q,b\u0001∗. We shall show that there exists g∈BMO 2,q,b(α(·))\nsuch that ℓ=ℓgand\n∥g∥BMO 2,q,b(α(·))≲∥ℓ∥.\nSince L2can be embedded continuously to Hs\np(·),q,b, then there exists g∈L2such that\nℓ(f) =E(fg),∀f∈L2.\nLet{Ik,i,j}k∈Z,i∈N,jbe an arbitrary sequence of atoms such that Ik,i,jare disjoint if kis\nfixed, Ik,i,jbelong to Fiand\n(\n2k\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r1\nα(·)+1γb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r1\nα(·)+1\u0013)\nk∈Z∈lq.\nSet\nhk,i,j=(g−Ei(g))χIk,i,j∥χIk,i,j∥2\n∥(g−Ei(g))χIk,i,j∥2· ∥χIk,i,j∥ 1\nα(·)+1.\nThen hk,i,jis a simple ( p(·),2)s-atom and\nE\u0000\nhk,i,j(g−Ei(g))\u0001\n=∥χIk,i,j∥2· ∥(g−Ei(g))χIk,i,j∥2\n∥χIk,i,j∥p(·).\nBy Theorem 3.6, we find that\nf=X\nk∈Z∞X\ni=0X\nj3·2k∥χIk,i,j∥p(·)hk,i,j∈Hs\np(·),q,b\nand\n∥f∥Hs\np(·),q,b≲ X\nk∈Z2kq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rq\np(·)γq\nb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013!1/q\n. (5.2)\nHence\nℓ(f) =E(fg) = 3X\nk∈Z∞X\ni=0X\nj2k∥χIk,i,j∥p(·)E(hk,i,jg).Variable martingale Hardy-Lorentz-Karamata spaces 49\nFurthermore, we have\nX\nk∈Z∞X\ni=0X\nj2kP(Ik,i,j)1\n2∥(g−Ei(g))χIk,i,j∥2\n=X\nk∈Z∞X\ni=0X\nj2k∥χIk,i,j∥p(·)E\u0000\nhk,i,j(g−Ei(g))\u0001\n=X\nk∈Z∞X\ni=0X\nj2k∥χIk,i,j∥p(·)E(hk,i,jg)\n=1\n3ℓ(f)≤1\n3∥f∥Hs\np(·),q,b· ∥ℓ∥.\nThus, applying (5.2) and the definition of ∥ · ∥ BMO 2,q,b(α(·)), we obtain\n∥g∥BMO 2,q,b(α(·))≲∥ℓ∥.\n□\nRemark 5.7. For the dual theory of martingale Hardy spaces, the researchers always split\nthe range of qto0< q≤1and1< q < ∞and discuss these two cases separately, see\n[32, 38, 40, 44, 75] . In this paper, we get the duality theorem when 0< q < ∞, which\nunifies the two cases.\nAs a deduction of Proposition 5.5 and Theorem 5.6, we obtain the dual theorem\nassociated with BMO 2,b(α(·)).\nCorollary 5.8. Letp(·)∈ P(Ω)satisfy (1.1)with 0< p−≤p+<1,0< q≤1and let b\nbe a slowly varying function. Then\n\u0000\nHs\np(·),q,b\u0001∗=BMO 2,b(α(·)), α(·) =1\np(·)−1. (5.3)\nMoreover, if 0< p−≤p+≤1andbis nonincreasing, (5.3)also holds.\nParticularly, Theorem 5.6 reduces to the following result when b≡1.\nCorollary 5.9. Letp(·)∈ P(Ω)satisfy (1.1)with 0< p−≤p+<2and0< q < ∞.\nThen\n\u0000\nHs\np(·),q\u0001∗=BMO 2,q(α(·)), α(·) =1\np(·)−1.\nRemark 5.10. Jiao et al. [40]got the dual theorem of Hs\np(·),qin the situation that p(·)∈\nP(Ω)satisfies (1.1)with 0< p−≤p+<2and1< q < ∞. Notice that, the condition of q\nis0< q < ∞in Corollary 5.9, therefore, our result extends the dual theorem in [40].\nSpecially, if α(·)≡αis a constant, then we have the next conclusion.\nCorollary 5.11. Let0< p < 2,0< q < ∞andbbe a slowly varying function. Then\n\u0000\nHs\np,q,b\u0001∗=BMO 2,q,b(α), α =1\np−1.50 Z. Hao, X. Ding, L. Li and F. Weisz\nRemark 5.12. We refer to Jiao et al. [44]for the duality of Hs\np,q,bif0< p≤1,1< q < ∞\nandbis a nondecreasing slowly varying function. We extend the range of pandqto\n0< p < 2and0< q < ∞, respectively. Moreover, the slowly varying function bis\nnot necessarily nondecreasing in Corollary 5.11. Hence, Corollary 5.11improves the dual\ntheorem in [44].\nNext we consider q=∞. This case is different to 0 < q < ∞, since Lpis not dense\ninLp,∞for 0 < p < ∞. We refer to [77] for this fact. From Remark 2.27 and Theorem\n3.6, we know that L2is dense in Hs\np(·),∞,b. Hence, similarly to the proof of Theorem 5.6,\nwe get the dual space when q=∞.\nTheorem 5.13. Letp(·)∈ P(Ω)satisfy (1.1)with 0< p−≤p+<2andbbe a slowly\nvarying function. Then\n\u0000\nHs\np(·),∞,b\u0001∗=BMO 2,∞,b(α(·)), α(·) =1\np(·)−1.\nWhen b≡1, we obtain the dual spaces of Hs\np(·),∞as follows.\nCorollary 5.14. Letp(·)∈ P(Ω)satisfy (1.1)with 0< p−≤p+<2. Then\n\u0000\nHs\np(·),∞\u0001∗=BMO 2,∞(α(·)), α(·) =1\np(·)−1.\nRemark 5.15. Obviously, Corollary 5.14can reduce to the result in [40].\nIfα(·)≡αis a constant, we know that the next result holds.\nCorollary 5.16. Let0< p < 2andbbe a slowly varying function. Then\n\u0000\nHs\np,∞,b\u0001∗=BMO 2,∞,b(α), α =1\np−1.\nRemark 5.17. The duality of Hs\np,∞,bwas proved by Liu and Zhou in [52]if0< p≤1\nandbis a nondecreasing slowly varying function. Notice that Corollary 5.16improves the\nrange of pto0< p < 2and shows that the nondecreasing condition of bis unnecessary.\nHence, Corollary 5.16extends the dual theorem in [52].\n6.John-Nirenberg Theorem\nIn this section, we prove the John-Nirenberg theorem for variable Lorentz-Karamata\nspaces when the stochastic basis {Fn}n≥0is regular.\nTheorem 6.1. Letp(·)∈ P(Ω)satisfy (1.1)with 0< p−≤p+≤1,0< q < ∞,\n1< r < ∞and let bbe a slowly varying function. If the stochastic basis {Fn}n≥0is\nregular, then\n\u0000\nHs\np(·),q,b\u0001∗=BMO r,q,b(α(·)), α(·) =1\np(·)−1.\nProof. The proof is very similar to that of Theorem 5.6, so we only sketch it. Let r′be\nthe conjugate number of r. Firstly we claim that Lr′⊂Hs\np(·),q,b. Indeed, for any f∈Lr′,\nit follows from Lemma 2.15 and Theorem 4.5 that\n∥f∥Hs\np(·),q,b=∥s(f)∥p(·),q,b≲∥s(f)∥1,q,b≲∥s(f)∥r′,r′,1=∥s(f)∥r′≈ ∥f∥r′.Variable martingale Hardy-Lorentz-Karamata spaces 51\nHence, for any f∈Lr′, on the basis of Theorem 3.6, there exist a sequence of simple\n(p(·), r)s-atoms ( ak,i,j)k∈Z,i∈N,jandµk,i,j= 3·2k∥χIk,i,j∥p(·)such that\nf=X\nk∈Z∞X\ni=0X\njµk,i,jak,i,j\ninLr′and\n X\nk∈Z\r\r\r\r∞X\ni=0X\njµk,i,j∥χIk,i,j∥−1\np(·)χIk,i,j\r\r\r\rq\np(·)γq\nb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013!1\nq\n≲∥f∥Hs\np(·),q,b.\nFor any given g∈BMO r,q,b⊂Lr, define\nℓg(f) :=E(fg),∀f∈Lr′.\nIn the same way as in the proof of Theorem 5.6, we get that ℓgcan be uniquely extended\nto a bounded linear functional on Hs\np(·),q,b.\nTo prove the converse, let ℓ∈\u0000\nHs\np(·),q,b\u0001∗. Since\nLr′=Hs\nr′⊂Hs\np(·),q,b=Hs−at,∞,1\np(·),q,b⊂Hs−at,r′,1\np(·),q,b,\nwe have\u0000\nHs−at,r′,1\np(·),q,b\u0001∗⊂\u0000\nHs\np(·),q,b\u0001∗⊂Lr.\nHence there exists g∈Lrsuch that\nℓ(f) =ℓg(f) =E(fg),∀f∈Lr′.\nLet{Ik,i,j}k∈Z,i∈N,jbe an arbitrary sequence of atoms such that Ik,i,jare disjoint if kis\nfixed, Ik,i,jbelong to Fiand\n(\n2k\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r1\nα(·)+1γb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r1\nα(·)+1\u0013)\nk∈Z∈lq.\nSet\nhk,i,j=|g−Ei(g)|r−1sgn(g−Ei(g))χIk,i,j∥χIk,i,j∥r′\n∥(g−Ei(g))χIk,i,j∥r−1\nr· ∥χIk,i,j∥p(·).\nThen hk,i,jis a simple ( p(·), r)s-atom and\nE\u0000\nhk,i,j(g−Ei(g))\u0001\n=∥χIk,i,j∥r′· ∥(g−Ei(g))χIk,i,j∥r\n∥χIk,i,j∥p(·).\nBy replacing hk,i,jin the proof of Theorem 5.6 by this new definition, we can obtain the\nconclusion and the proof is complete. □\nFor the case of r= 1, we need some new insight. Let the dual space of Pp(·),q,bbe\nP∗\np(·),q,b. Denote by\u0000\nP∗\np(·),q,b\u0001\n1those elements ℓ∈ P∗\np(·),q,bfor which there exists g∈L1such\nthatℓ(f) =E(fg),f∈L∞. That is,\n\u0000\nP∗\np(·),q,b\u0001\n1:=\b\nℓ∈ P∗\np(·),q,b:∃g∈L1s.t.ℓ(f) =E(fg),∀f∈L∞\t\n.52 Z. Hao, X. Ding, L. Li and F. Weisz\nTheorem 6.2. Letp(·)∈ P(Ω)satisfy (1.1)with 0< p−≤p+≤1,0< q < ∞and let b\nbe a slowly varying function. Then\n\u0000\nP∗\np(·),q,b\u0001\n1=BMO 1,q,b(α(·)), α(·) =1\np(·)−1.\nProof. Letg∈BMO 1,q,b(α(·))⊂L1. Define the functional ℓgas\nℓg(f) :=E(fg),∀f∈L∞.\nBy Theorem 3.8, there exist a sequence of simple ( p(·),∞)M-atoms ( ak,i,j)k∈Z,i∈N,jand\nµk,i,j= 3·2k∥χIk,i,j∥p(·)such that\nf=X\nk∈Z∞X\ni=0X\njµk,i,jak,i,j\nand\n X\nk∈Z\r\r\r\r∞X\ni=0X\njµk,i,j∥χIk,i,j∥−1\np(·)χIk,i,j\r\r\r\rq\np(·)γq\nb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013!1\nq\n≲∥f∥Pp(·),q,b.\nHence, it follows from dominated convergence theorem and H¨ older’s inequality that\n|ℓg(f)| ≤X\nk∈Z∞X\ni=0X\njµk,i,jE(ak,i,jg)\n≤X\nk∈Z∞X\ni=0X\njµk,i,j∥ak,i,j∥∞∥(g−Ei(g))χIk,i,j∥1\n= 3X\nk∈Z∞X\ni=0X\nj2k∥(g−Ei(g))χIk,i,j∥1\n≲ X\nk∈Z2kq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rq\np(·)γq\nb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013!1\nq\n∥g∥BMO 1,q,b(α(·))\n≲∥f∥Pp(·),q,b∥g∥BMO 1,q,b(α(·)).\nThen we get that ℓgcan be extended to a bounded linear functional on Pp(·),q,bandℓg∈\u0000\nP∗\np(·),q,b\u0001\n1.\nConversely, let ℓ∈\u0000\nP∗\np(·),q,b\u0001\n1. Then there exists g∈L1such that\nℓ(f) =E(fg),∀f∈L∞.\nLet ( Ik,i,j)k∈Z,i∈N,jbe an arbitrary sequence of atoms such that Ik,i,jare disjoint if kis\nfixed, Ik,i,jbelong to Fiand\n(\n2k\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)γb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013)\nk∈Z∈lq.\nLet\nφk,i,j= sign( g−Ei(g))χIk,i,jVariable martingale Hardy-Lorentz-Karamata spaces 53\nand\nhk,i,j=φk,i,j−Ei(φk,i,j)\n2∥χIk,i,j∥p(·).\nThen hk,i,jis a simple ( p(·),∞)M-atom and\nE\u0000\nhk,i,j(g−Ei(g))\u0001\n=∥(g−Ei(g))χIk,i,j∥1\n2∥χIk,i,j∥p(·).\nSetting µk,i,j= 3·2k∥χIk,i,j∥p(·), it follows from Theorem 3.11 that\nf=X\nk∈Z∞X\ni=0X\njµk,i,jhk,i,j∈ P p(·),q,b\nand\n∥f∥Pp(·),q,b≲ X\nk∈Z2kq\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rq\np(·)γq\nb\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013!1\nq\n.\nFor an arbitrary positive integer N, set\n(f)N=NX\nk=−N∞X\ni=0X\njµk,i,jhk,i,j.\nThen\nNX\nk=−N∞X\ni=0X\nj2k∥(g−Ei(g))χIk,i,j∥1\n= 2NX\nk=−N∞X\ni=0X\nj2k∥χIk,i,j∥p(·)E\u0000\nhk,i,j(g−Ei(g))\u0001\n= 2NX\nk=−N∞X\ni=0X\nj2k∥χIk,i,j∥p(·)E(hk,i,jg)\n=2\n3E((f)Ng) =2\n3ℓ((f)N)≤2\n3∥ℓ∥∥(f)N∥Pp(·),q,b.\nBy the definition of BMO 1,q,b(α(·)) and by taking the supremum in N, we get that\n∥g∥BMO 1,q,b(α(·))≲∥ℓ∥.\nThe proof is finished now. □\nTheorem 6.3. Letp(·)∈ P(Ω)satisfy (1.1) with 0< p−≤p+≤1,0< q < ∞and let b\nbe a slowly varying function. If the stochastic basic {Fn}n≥0is regular, then\n\u0000\nP∗\np(·),q,b\u0001\n1=P∗\np(·),q,b.\nProof. Since{Fn}n≥0is regular, we have Pp(·),q,b≈Hs\np(·),q,b. Thus L2can also be embedded\ncontinuously in Pp(·),q,b. Then P∗\np(·),q,b⊂L∗\n2=L2. For ℓ∈ P∗\np(·),q,b, there exists g∈L2⊂L1\nsuch that ℓ=ℓg. It is obvious that ℓ∈\u0000\nP∗\np(·),q,b\u0001\n1, which means P∗\np(·),q,b⊂\u0000\nP∗\np(·),q,b\u0001\n1. By\nthe definition of\u0000\nP∗\np(·),q,b\u0001\n1, we have\u0000\nP∗\np(·),q,b\u0001\n1⊂ P∗\np(·),q,b. Hence\u0000\nP∗\np(·),q,b\u0001\n1=P∗\np(·),q,b.□\nCombining Theorem 6 .2 with Theorem 6 .3, we have the dual theorem of Pp(·),q,b.54 Z. Hao, X. Ding, L. Li and F. Weisz\nCorollary 6.4. Letp(·)∈ P(Ω)satisfy (1.1)with 0< p−≤p+≤1,0< q < ∞and let b\nbe a slowly varying function. If the stochastic basic {Fn}n≥0is regular, then\nP∗\np(·),q,b=BMO 1,q,b(α(·)), α(·) =1\np(·)−1.\nNow, we state the John-Nirenberg theorem for variable Lorentz-Karamata spaces.\nTheorem 6.5. Letα(·) + 1∈ P(Ω)satisfy (1.1)with α−≥0,0< q < ∞and let bbe a\nslowly varying function. If the stochastic basis {Fn}n≥0is regular, then\nBMO r,q,b(α(·)) =BMO 2,q,b(α(·))\nwith equivalent norms for all 1≤r <∞.\nProof. When 1 < r < ∞, it follows from Theorems 5.6 and 6.1 that\nBMO r,q,b(α(·)) =BMO 2,q,b(α(·)).\nNext we consider the case of r= 1. Since the stochastic basis {Fn}n≥0is regular, we have\nPp(·),q,b≈Hs\np(·),q,b, where p(·) =1\nα(·)+1. Hence, according to Theorem 5.6 and Corollary\n6.4, we obtain that\nBMO 1,q,b(α(·)) =BMO 2,q,b(α(·)).\n□\nWhen α(·)≡αis a constant, we have the following result.\nCorollary 6.6. Letα≥0,0< q < ∞and let bbe a slowly varying function. If the\nstochastic basis {Fn}n≥0is regular, then\nBMO r,q,b(α) =BMO 2,q,b(α)\nwith equivalent norms for all 1≤r <∞.\nRemark 6.7. In2015, Jiao et al. [44]have showed the John-Nirenberg theorem on\nLorentz-Karamata spaces. However, they need the condition that the slowly varying func-\ntionbis nondecreasing. Hence, Corollary 6.6extends [44, Theorem 1.6] .\nWhen b≡1, we obtain the John-Nirenberg theorem associated with BMO r,q(α(·)).\nCorollary 6.8. Letα(·) + 1∈ P(Ω)satisfy (1.1)with α−≥0and0< q < ∞. If the\nstochastic basis {Fn}n≥0is regular, then\nBMO r,q(α(·)) =BMO 2,q(α(·))\nwith equivalent norms for all 1≤r <∞.\n7.Boundedness of fractional integrals\nIn this section, without loss of generality, we always suppose that the constant in\n(2.5) satisfies R ≥ 2. Firstly, we introduce the definition of the fractional integral and\ngive some necessary results.\nDefinition 7.1 ([55, 63]) .Forf= (fn)n≥0∈ M andα > 0, the fractional integral\nIαf=\u0000\n(Iαf)n\u0001\nn≥0offis defined as\n(Iαf)n=nX\nk=1bα\nk−1dkf,Variable martingale Hardy-Lorentz-Karamata spaces 55\nwhere bkis anFk-measurable function such that for all B∈A(Fk)andω∈B,bk(ω) =\nP(B).\nRemark 7.2. We point out that Iαfis a martingale and (Iαf)nis a martingale transform\nintroduced by Burkholder. Especially, if Ω = [0 ,1]and the σ-algebra Fnis generated by the\ndyadic intervals of [0,1], then Iαfis closely related to a class of multiplier transformations\nof Walsh-Fourier series, see [74]. We refer to [64]for the classical fractional integrals.\nLemma 7.3 ([31, 46]) .Letp(·), q(·)∈ P(Ω)satisfy (1.1). For any set A∈S\nn≥0A(Fn), we\nhave\n∥χA∥r(·)≈ ∥χA∥p(·)∥χA∥q(·),\nwhere1\nr(ω)=1\np(ω)+1\nq(ω),∀ω∈Ω.\nLemma 7.4 ([63]).Let{Fn}n≥0be regular, f∈ M ,α >0and let Rbe the constant in\n(2.5). If there exists B∈ F such that M(f)≤χB, then there exists a positive constant\nCα= 2 +R+1\n1−(1+1\nR)α−1independent of fandBsuch that\nM(Iαf)≤CαP(B)αχB.\nLemma 7.5. Letp1(·), p2(·)∈ P(Ω)satisfy (1.1),α > 0,{Fn}n≥0be regular, bbe a\nnondecreasing slowly varying function and let Rbe the constant in (2.5). If0< p 1(·)<\np2(·)<∞,α≥sup\nω∈Ω\u00001\np1(ω)−1\np2(ω)\u0001\n,0< q≤ ∞ andais a simple (p1(·),∞)M-atom, then\n∥Iαa∥HM\np2(·),q,b≲Cαγb(∥χI∥p1(·)),\nwhere Cαis the same as in Lemma 7.4andI∈A(Fn)is associated with a.\nProof. LetI∈A(Fn) be associated with the simple ( p1(·),∞)M-atom a. We have M(a)≤\n∥χI∥−1\np1(·)χIand\nM\u0000\n∥χI∥p1(·)a\u0001\n=∥χI∥p1(·)M(a)≤χI.\nIt follows from Lemma 7.4 that\nM\u0000\nIα(∥χI∥p1(·)a)\u0001\n≤CαP(I)αχI.\nForp1(·), p2(·)∈ P(Ω) with p1(·)< p 2(·), we can find a variable exponent r(·)∈ P(Ω)\nsuch that1\nr(ω)=1\np1(ω)−1\np2(ω),∀ω∈Ω.\nThen sup\nω∈Ω1\nr(ω)≤α, which means r−(Ω)≥1\nα. By applying Lemma 7.3, we have\n∥χI∥p1(·)≈ ∥χI∥p2(·)∥χI∥r(·)≥ ∥χI∥p2(·)∥χI∥1\nα=∥χI∥p2(·)P(I)α.\nHence there is\nM(Iαa)≤CαP(I)α∥χI∥−1\np1(·)χI≲Cα∥χI∥−1\np2(·)χI.\nSince bis nondecreasing, we know that γbis nonincreasing on (0 ,1]. Then it follows from\nLemma 2.5 that, for p1(·)< p2(·),\nγb\u0000\n∥χI∥p2(·)\u0001\n≲γb\u0000\n∥χI∥p1(·)\u0001\n.56 Z. Hao, X. Ding, L. Li and F. Weisz\nIt yields that\n∥Iαa∥HM\np2(·),q,b=∥M(Iαa)∥p2(·),q,b\n≲Cα∥χI∥−1\np2(·)∥χI∥p2(·),q,b\n≈Cα∥χI∥−1\np2(·)∥χI∥p2(·)γb\u0000\n∥χI∥p2(·)\u0001\n≲Cαγb\u0000\n∥χI∥p1(·)\u0001\n.\nTherefore, the proof is complete. □\nNow, we show the main conclusion of this section.\nTheorem 7.6. Let{Fn}n≥0be regular, p(·), q(·)∈ P(Ω)satisfy (1.1),0< s≤ ∞ ,α >0\nandb1, b2be slowly varying functions. Assume that Nis the constant in Remark 2.19. If\n0< p(·)< q(·)<∞,α≥sup\nω∈Ω\u00001\np(ω)−1\nq(ω)\u0001\n,b1is nondecreasing and sup\n1≤t<∞b2(t)\nb1(t)<∞, then\nfor any constant rwithp+< r≤min{s,N}, there is\n∥Iαf∥HM\nq(·),s,b2≲∥f∥HM\np(·),r,b1\nfor all f∈HM\np(·),r,b1.\nProof. Letf∈HM\np(·),r,b1. Since {Fn}n≥0is regular, there exist a sequence of simple\n(p(·),∞)M-atoms ( ak,i,j)k∈Z,i∈N,jandµk,i,j= 3·2k∥χIk,i,j∥p(·)such that for all n∈N,\nf=X\nk∈Z∞X\ni=0X\njµk,i,jak,i,ja.e.\nand\r\r\r\r\r(\r\r\r\r∞X\ni=0X\njµk,i,j∥χIk,i,j∥−1\np(·)χIk,i,j\r\r\r\r\np(·)γb1\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013)\nk∈Z\r\r\r\r\r\nlr≲∥f∥HM\np(·),r,b1.\nAccording to the sublinearity of M, we have\n∥Iαf∥HM\nq(·),s,b2=∥M(Iαf)∥q(·),s,b2=\r\r\r\r\rM \nIα\u0012X\nk∈Z∞X\ni=0X\njµk,i,jak,i,j\u0013!\r\r\r\r\r\nq(·),s,b2\n≤\r\r\r\rX\nk∈Z∞X\ni=0X\nj|µk,i,j|M\u0000\nIα(ak,i,j)\u0001\r\r\r\r\nq(·),s,b2.\nIt follows from Remark 2.19 that for any constant rwith p+< r≤min{s,N},\n\r\r\r\rX\nk∈Z∞X\ni=0X\nj|µk,i,j|M\u0000\nIα(ak,i,j)\u0001\r\r\r\rr\nq(·),s,b2≤4r\nNX\nk∈Z∞X\ni=0X\nj\r\r|µk,i,j|M\u0000\nIα(ak,i,j)\u0001\r\rr\nq(·),s,b2.\nHence, according to Lemmas 2.14, 7.5 and 2.10, we have\n∥Iαf∥r\nHM\nq(·),s,b2≤4X\nk∈Z∞X\ni=0X\nj|µk,i,j|r\r\rM\u0000\nIα(ak,i,j)\u0001\r\rr\nq(·),s,b2\n≲4X\nk∈Z∞X\ni=0X\nj|µk,i,j|r\r\rM\u0000\nIα(ak,i,j)\u0001\r\rr\nq(·),l,b1Variable martingale Hardy-Lorentz-Karamata spaces 57\n≲Cr\nαX\nk∈Z∞X\ni=0X\nj|µk,i,j|rγr\nb1(∥χIk,i,j∥p(·))\n=Cr\nαX\nk∈Z3r·2kr∞X\ni=0X\nj∥χIk,i,j∥r\np(·)γr\nb1(∥χIk,i,j∥p(·))\n≲X\nk∈Z3r·2kr\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\rr\np(·)γr\nb1\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013\n=\r\r\r\r\r(\r\r\r\r∞X\ni=0X\njµk,i,j∥χIk,i,j∥−1\np(·)χIk,i,j\r\r\r\r\np(·)γb1\u0012\r\r\r\r∞X\ni=0X\njχIk,i,j\r\r\r\r\np(·)\u0013)\nk∈Z\r\r\r\r\rr\nlr\n≲∥f∥r\nHM\np(·),r,b1.\nThe proof of this theorem is complete now. □\nWhen p(·)≡pandq(·)≡q, Theorem 7.6 reduces to [52, Theorem 4.4]. When\nb1≡b2≡1, we obtain the boundedness of fractional integral on variable Hardy-Lorentz\nspaces.\nCorollary 7.7. Let{Fn}n≥0be regular, p1(·), p2(·)∈ P(Ω)satisfy (1.1),0< q 2≤ ∞\nandα > 0. Suppose that Lp2(·),q2is anN-normed space. If 0< p 1(·)< p 2(·)<∞and\nα≥sup\nω∈Ω\u00001\np1(ω)−1\np2(ω)\u0001\n, then for any constant q1with (p1)+< q1≤min{q2,N}, there is\n∥Iαf∥HM\np2(·),q2≲∥f∥HM\np1(·),q1\nfor all f∈HM\np1(·),q1.\n8.Applications in Fourier Analysis\nAs an application of the previous results in Fourier analysis, we will investigate the\nboundedness of the maximal Fej´ er operator on variable Hardy-Lorentz-Karamata spaces.\nWe consider the probability space ([0 ,1),F, dx), where Fdenotes the Lebesgue measurable\nsets. A dyadic interval means an interval of the form [ k2−n,(k+ 1)2−n) for some k, n∈N,\n0≤k <2n. Given any n∈Nandx∈[0,1), denote In(x) the dyadic interval of length 2−n\nwhich contains x. The σ-algebra Fngenerated by the dyadic intervals {In(x) :x∈[0,1)}\nis called the nth dyadic σ-algebra. In this section, we always assume that {Fn}n≥0is\na sequence of dyadic σ-algebras. Obviously, {Fn}n≥0is regular. According to Theorem\n4.5, the five variable martingale Hardy-Lorentz-Karamata spaces are equivalent. Hence,\ndenote by Hp(·),q,bone of them.\n8.1.Walsh system and partial sums. For any n∈Nandx∈[0,1), let\nrn(x) := sgn sin(2nπx).\nThe set of {rn(x)}n∈Nis called to be the system of Rademacher functions. The product\nsystem generated by the Rademacher functions is the Walsh system:\nwn:=∞Y\nk=0rnk\nk, n∈N,58 Z. Hao, X. Ding, L. Li and F. Weisz\nwhere\nn=∞X\nk=0nk2k, n k= 0,1. (8.1)\nIff∈L1, thenbf(n) :=E(fwn) (n∈N) is said to be the n-th Walsh-Fourier coefficient\noff. We remember that lim\nk→∞Ek(f) =fin the L1-norm. Hence,\nbf(n) = lim\nk→∞E\u0000\n(Ekf)wn\u0001\n, n∈N.\nIff= (fk)k≥0is a martingale, then the Walsh-Fourier coefficients of fare defined by\nbf(n) := lim\nk→∞E(fkwn), n∈N.\nSince wnisFk-measurable for n <2k, it is easy to see that this limit does exist. Moreover,\nthe Walsh-Fourier coefficients of f∈L1are the same as those of the martingale\u0000\nEk(f)\u0001\nk≥0\nobtained from f.\nFrom [21], we recall that the Walsh-Dirichlet kernels\nDn:=n−1X\nk=0wk, n∈N\nsatisfy\nD2n(x) =(\n2n,ifx∈[0,2−n),\n0,ifx∈[2−n,1),n∈N.\nDenote by snfthen-th partial sum of the Walsh-Fourier series of a martingale f,\nthat is\nsnf:=n−1X\nk=0bf(k)wk.\nIff∈L1, then\nsnf=Z1\n0f(t)Dn(x˙+t)dt, n ∈N,\nwhere ˙+ denotes the dyadic addition (see [68] or [78]). We get immediately that\ns2nf=fn, n∈N.\nBy martingale results, when f∈Lpwith 1 ≤p <∞, then\nlim\nn→∞s2nf=fin the Lp-norm .\nThis result was extended by Schipp et al. [68] to the partial sums snf: when f∈Lpwith\n1< p < ∞, then\nlim\nn→∞snf=fin the Lp-norm .\nRecently, Jiao et al. [42] generalized this result to variable Lebesgue spaces and vari-\nable Lorentz spaces. In this subsection, we extend these conclusions to variable Lorentz-\nKaramata spaces.Variable martingale Hardy-Lorentz-Karamata spaces 59\nTheorem 8.1. Letp(·)∈ P(Ω)satisfy (1.1)with 1< p−≤p+<∞,0< q≤ ∞ and let\nbbe a slowly varying function. If f∈Lp(·),q,b, then\nsup\nn∈N∥snf∥p(·),q,b≲∥f∥p(·),q,b.\nProof. Set\nT0f:=∞X\nk=1nk−1dkf,\nwhere the binary coefficients nkare defined in (8.1). It follows from [68] or [75] that\nsnf=wnT0(fwn).\nObviously, T0is a martingale transform and S(T0f)≤S(f). Since {Fn}n≥0is regular, it\nfollows from Theorem 4.9 that\n∥snf∥p(·),q,b=∥T0f∥p(·),q,b≲∥f∥p(·),q,b.\n□\nCorollary 8.2. Letp(·)∈ P(Ω)satisfy (1.1)with 1< p−≤p+<∞,0< q < ∞and let\nbbe a slowly varying function. If f∈Lp(·),q,b, then\nlim\nn→∞snf=f in the L p(·),q,b-norm.\nProof. Since the Walsh polynomials are dense in Lp(·),q,bby Lemma 2.22, the proof follows\nfrom the usual density argument. □\nSimilarly, we have the following result.\nCorollary 8.3. Letp(·)∈ P(Ω)satisfy (1.1)with 1< p−≤p+<∞and let bbe a slowly\nvarying function. If f∈ L p(·),∞,b, then\nlim\nn→∞snf=f in the Lp(·),∞,b-norm .\n8.2.The maximal Fej´ er operator σ∗.Notice that for p−= 1, the results in Subsection\n8.1 are not true anymore. To be able to extend these theorems, we introduce the Fej´ er\nsummability method. The Fej´ er mean of order n∈Nof the Walsh-Fourier series of a\nmartingale fis given by\nσnf:=1\nnnX\nk=1skf.\nCompared to skf,σnfhas better convergence properties. The maximal operator σ∗is\ndefined by\nσ∗f:= sup\nn∈N|σnf|.\nIn this part, we mainly discuss the boundedness of σ∗from variable Hardy-Lorentz-\nKaramata spaces to variable Lorentz-Karamata spaces. For classical Hardy spaces, Fujii\n[22] found that σ∗is bounded from H1toL1, see also [67]. In 1996, Weisz [76] generalized\nthe above result and proved that σ∗is bounded from HptoLpwith1\n2< p < ∞and\nfrom Hp,qtoLp,qwith1\n2< p < ∞, 0< q≤ ∞ . Recently, Jiao et al. [42] considered the\nmaximal Fej´ er operator on Hp(·)andHp(·),q. We refer to [22, 25, 67, 70, 72, 76] for more\ndetails.60 Z. Hao, X. Ding, L. Li and F. Weisz\nTheorem 8.4. Letp(·)∈ P(Ω)satisfy (1.1),0< q≤ ∞ ,bbe a slowly varying function\nand let max{p+,1}< r≤ ∞ . Suppose that T:Lr→Lris a bounded σ-sublinear operator\nand for some 0< β < 1and every (p(·),∞)M-atom aassociated with stopping time τ,\nthere is\r\r|T(a)|βχ{τ=∞}\r\r\np(·)≲∥χ{τ<∞}∥1−β\np(·). (8.2)\nThen for f∈Hp(·),q,b,\n∥T(f)∥p(·),q,b≲∥f∥Hp(·),q,b.\nProof. Letf∈Hp(·),q,b. According to Theorem 3.19 and Corollary 4.7, there exist a\nsequence ( ak)k∈Zof (p(·),∞)M-atoms associated with the stopping times ( τk)k∈Zand a\nsequence ( µk)k∈Z=\u0000\n3·2k∥χ{τk<∞}∥p(·)\u0001\nk∈Zof positive numbers such that\nf=X\nk∈Zµkak.\nFor an arbitrary integer k0, set\nD1:=k0−1X\nk=−∞µk|T(ak)|and D2:=∞X\nk=k0µk|T(ak)|.\nThen by the σ-sublinearity of T, there is\n|T(f)| ≤X\nk∈Zµk|T(ak)|=D1+D2.\nFirstly, we consider the case of r=∞. Since T:L∞→L∞is bounded, we have\n∥D1∥∞≤k0−1X\nk=−∞µk∥T(ak)∥∞≤k0−1X\nk=−∞µk∥M(ak)∥∞\n≤k0−1X\nk=−∞µk∥χ{τk<∞}∥−1\np(·)≤3·2k0.\nWe decompose D2intoX1+X2, where\nX1:=∞X\nk=k0µk|T(ak)|χ{τk<∞}and X2:=∞X\nk=k0µk|T(ak)|χ{τk=∞}.\nIt is obvious that\n{|X1|>2k0} ⊂ {| X1|>0} ⊂∞[\nk=k0{τk<∞}.\nHence we have\n∥χ{|X1|>2k0}∥p(·)≤\r\r\r\r∞X\nk=k0χ{τk<∞}\r\r\r\r\np(·).\nLet 0 < m < p and 0 < ε < 1. It follows from Lemma 2.10 that\n2k0mε∥χ{|X1|>2k0}∥m\np(·)γm\nb\u0000\n∥χ{|X1|>2k0}∥p(·)\u0001\n≲2k0mε\r\r\r\r∞X\nk=k0χ{τk<∞}\r\r\r\rm\np(·)γm\nb\u0012\r\r\r\r∞X\nk=k0χ{τk<∞}\r\r\r\r\np(·)\u0013Variable martingale Hardy-Lorentz-Karamata spaces 61\n≲2k0mε∞X\nk=k0∥χ{τk<∞}∥m\np(·)γm\nb\u0000\n∥χ{τk<∞}∥p(·)\u0001\n≲∞X\nk=k02kmε∥χ{τk<∞}∥m\np(·)γm\nb\u0000\n∥χ{τk<∞}∥p(·)\u0001\n.\nObviously, according to Lemma 3.4 and Corollary 4.7, we obtain\n∥D1+X1∥p(·),q,b≲\r\r\r\b\n2k∥χ{τk<∞}∥p(·)γb\u0000\n∥χ{τk<∞}∥p(·)\u0001\t\nk∈Z\r\r\r\nlq≲∥f∥Hp(·),q,b.\nNext, we investigate X2. Let 0 < ξ < min{p, q}. It follows from (8.2) that for any\n0< β < 1,\n∥χ{|X2|>2k0}∥p(·)≤1\n2βk0\r\r|X2|β\r\r\np(·)(8.3)\n≲2−k0β\r\r\r\r∞X\nk=k0µβ\nk|T(ak)|βχ{τk=∞}\r\r\r\rξ1\nξ\np(·)\n≲2−k0β\u0012∞X\nk=k0µβξ\nk\r\r|T(ak)|βχ{τk=∞}\r\rξ\np(·)\u00131\nξ\n≲2−k0β\u0012∞X\nk=k0µβξ\nk∥χ{τk<∞}∥ξ−βξ\np(·)\u00131\nξ\n≈2−k0β\u0012∞X\nk=k02kβξ∥χ{τk<∞}∥ξ\np(·)\u00131\nξ\n.\nFirstly, we consider the case of 0 < q < ∞. Let β < α < 1. According to H¨ older’s\ninequality, we obtain\n∥χ{|X2|>2k0}∥p(·)≲2−k0β\u0012∞X\nk=k02k(β−α)ξq\nq−ξ\u0013q−ξ\nqξ\u0012∞X\nk=k02kqα∥χ{τk<∞}∥q\np(·)\u00131\nq\n≲2−k0α\u0012∞X\nk=k02kqα∥χ{τk<∞}∥q\np(·)\u00131\nq\n.\nThen we get\n∥X2∥q\np(·),q,b≈X\nk0∈Z2k0q∥χ{|X2|>2k0}∥q\np(·)γq\nb\u0000\n∥χ{|X2|>2k0}∥p(·)\u0001\n≲X\nk0∈Z2k0q∞X\nk=k02(k−k0)qα∥χ{τk<∞}∥q\np(·)γq\nb\u0014\u0012 ∞X\nk=k02(k−k0)qα∥χ{τk<∞}∥q\np(·)\u00131\nq\u0015\n.\nDefine b1(t) =b\u0000\nt1\nq\u0001\nfort∈[1,∞). Set 0 < θ < 1, it follows from Lemma 2.9 that\n∥X2∥q\np(·),q,b\n≲X\nk0∈Z2k0q∞X\nk=k02(k−k0)αq∥χ{τk<∞}∥q\np(·)γbq\n1\u0012∞X\nk=k02(k−k0)αq∥χ{τk<∞}∥q\np(·)\u001362 Z. Hao, X. Ding, L. Li and F. Weisz\n=X\nk0∈Z2k0q\"\u0012∞X\nk=k02(k−k0)αq∥χ{τk<∞}∥q\np(·)\u0013θ\nγbθq\n1\u0012∞X\nk=k02(k−k0)αq∥χ{τk<∞}∥q\np(·)\u0013#1\nθ\n≲X\nk0∈Z2k0q ∞X\nk=k02(k−k0)αθq∥χ{τk<∞}∥θq\np(·)γbθq\n1\u0000\n2(k−k0)αq∥χ{τk<∞}∥q\np(·)\u0001!1\nθ\n=X\nk0∈Z2k0q ∞X\nk=k02(k−k0)αθq∥χ{τk<∞}∥θq\np(·)γθq\nb\u0000\n2(k−k0)α∥χ{τk<∞}∥p(·)\u0001!1\nθ\n.\nLet 0 < z <1−α\nα. By the same method as inequality (3.6), we obtain that for k≥k0,\nγb\u0000\n2(k−k0)α∥χ{τk<∞}∥p(·)\u0001\n≲2(k−k0)αzγb\u0000\n∥χ{τk<∞}∥p(·)\u0001\n.\nHence, there is\n∥X2∥q\np(·),q,b≲X\nk0∈Z2k0q\"∞X\nk=k02(k−k0)αθq(1+z)∥χ{τk<∞}∥θq\np(·)γθq\nb\u0000\n∥χ{τk<∞}∥p(·)\u0001#1\nθ\n.\nSet 0 < ζ <1−α−αz\nα. By applying H¨ older’s inequality with 1 −θ+θ= 1 again, we have\n∥X2∥q\np(·),q,b\n≲X\nk0∈Z2k0q\"∞X\nk=k02−(k−k0)αθqζ2(k−k0)αθq(1+z+ζ)∥χ{τk<∞}∥θq\np(·)γθq\nb\u0000\n∥χ{τk<∞}∥p(·)\u0001#1\nθ\n≤X\nk0∈Z2k0q\u0012∞X\nk=k02−(k−k0)αθqζ/ (1−θ)\u00131−θ\nθ∞X\nk=k02(k−k0)αq(1+z+ζ)∥χ{τk<∞}∥q\np(·)γq\nb\u0000\n∥χ{τk<∞}∥p(·)\u0001\n≲X\nk0∈Z2k0q∞X\nk=k02(k−k0)αq(1+z+ζ)∥χ{τk<∞}∥q\np(·)γq\nb\u0000\n∥χ{τk<∞}∥p(·)\u0001\n.\nHence, it follows from Abel’s transformation and Theorem 3.19 that\n∥X2∥q\np(·),q,b≈X\nk0∈Z2k0q∥χ{X2>2k0}∥q\np(·)γq\nb\u0000\n∥χ{X2>2k0}∥p(·)\u0001\n=X\nk∈Z2kαq(1+z+ζ)∥χ{τk<∞}∥q\np(·)γq\nb\u0000\n∥χ{τk<∞}∥p(·)\u0001kX\nk0=−∞2k0q[1−α(1+z+ζ)]\n≲X\nk∈Z2kq∥χ{τk<∞}∥q\np(·)γq\nb\u0000\n∥χ{τk<∞}∥p(·)\u0001\n≲∥f∥q\nHp(·),q,b.\nNext, we discuss the case of q=∞. According to (8.3) and H¨ older’s inequality, we\nhave\n∥χ{|X2|>2k0}∥p(·)≲2−k0β\u0012∞X\nk=k02k(β−α)ε2kεα∥χ{τk<∞}∥ε\np(·)\u00131/εVariable martingale Hardy-Lorentz-Karamata spaces 63\n≤2−k0β\u0012∞X\nk=k02k(β��α)ε\u00131/ε\nsup\nk≥k02kα∥χ{τk<∞}∥p(·)\n= sup\nk≥k02(k−k0)α∥χ{τk<∞}∥p(·),\nwhere 0 < β < α < 1. Since t−νγb(t) is equivalent to a nonincreasing function for ν >0\nandt∈(0,∞), we have that for any lsatisfying l≥k0,\n∥χ{|X2|>2k0}∥p(·)γb\u0000\n∥χ{|X2|>2k0}∥p(·)\u0001\n≲sup\nk≥k02(k−k0)α∥χ{τk<∞}∥p(·)γb\u0010\nsup\nk≥k02(k−k0)α∥χ{τk<∞}∥p(·)\u0011\n=\u0010\nsup\nl≥k02(l−k0)α∥χ{τl<∞}∥p(·)\u00111+ν\n×\u0010\nsup\nk≥k02(k−k0)α∥χ{τk<∞}∥p(·)\u0011−ν\nγb\u0010\nsup\nk≥k02(k−k0)α∥χ{τk<∞}∥p(·)\u0011\n≲sup\nl≥k02(1+ν)(l−k0)α∥χ{τl<∞}∥1+ν\np(·)·2−(l−k0)αν∥χ{τl<∞}∥−ν\np(·)γb\u0000\n2(l−k0)α∥χ{τl<∞}∥p(·)\u0001\n= sup\nl≥k02(l−k0)α∥χ{τl<∞}∥p(·)γb\u0000\n2(l−k0)α∥χ{τl<∞}∥p(·)\u0001\n.\nWith the help of Corollary 4.7, we obtain that\n∥X2∥p(·),∞,b≈sup\nk0∈Z2k0∥χ{|X2|>2k0}∥p(·)γb\u0000\n∥χ{|X2|>2k0}∥p(·)\u0001\n≲sup\nk0∈Z2k0·sup\nl≥k02(l−k0)α∥χ{τl<∞}∥p(·)γb\u0000\n2(l−k0)α∥χ{τl<∞}∥p(·)\u0001\n≲sup\nk0∈Z2k0·sup\nl≥k02(l−k0)α(1+z)∥χ{τl<∞}∥p(·)γb\u0000\n∥χ{τl<∞}∥p(·)\u0001\n= sup\nl∈Z2lα(1+z)∥χ{τl<∞}∥p(·)γb\u0000\n∥χ{τl<∞}∥p(·)\u0001\nsup\nk0≤l2k0(1−α(1+z))\n≲sup\nl∈Z2l∥χ{τl<∞}∥p(·)γb\u0000\n∥χ{τl<∞}∥p(·)\u0001\n≲∥f∥Hp(·),∞,b.\nCombining the above inequalities, we conclude that\n∥T(f)∥p(·),q,b≲∥D1+X1∥p(·),q,b+∥X2∥p(·),q,b≲∥f∥Hp(·),q,b.\nNow consider max {p+,1}< r < ∞. The estimation of D2is the same as above. For\nD1, similarly to the proof of atomic decomposition theorem, we obtain\n∥D1∥p(·),q,b≲∥f∥Hp(·),q,b.\nHence, the proof of this theorem is complete. □\nTheorem 5.39 in [42] shows that the operator σ∗satisfies the condition of the above\ntheorem. More exactly,\nLemma 8.5. Letp(·)∈ P(Ω)satisfy (1.1). If1\n2< p−≤p+<∞and\n1\np−−1\np+<1. (8.4)64 Z. Hao, X. Ding, L. Li and F. Weisz\nFor some 0< β < 1and every (p(·),∞)M-atom aassociated with stopping time τ, there\nis\r\r|σ∗a|βχ{τ=∞}\r\r\np(·)≲∥χ{τ<∞}∥1−β\np(·).\nNow, the boundedness of σ∗from variable Hardy-Lorentz-Karamata spaces to variable\nLorentz-Karamata spaces follows from Theorem 8.4.\nCorollary 8.6. Letp(·)∈ P(Ω)satisfy (1.1)and(8.4),0< q≤ ∞ and let bbe a slowly\nvarying function. If1\n2< p−≤p+<∞, then\n∥σ∗f∥p(·),q,b≲∥f∥Hp(·),q,b.\nRemark 8.7. (1)Ifb≡1, then (8.4)is also a sufficient condition (see[42]).\n(2)When b≡1, for p(·)≡pwith p≤1\n2or with p=q≤1\n2, Theorem 8.6does not\nhold any more, see [25, 70, 72] .\nBy virtue of Theorem 8.6, we consider the convergence of σnf. The proofs are omitted,\nsince they are similar to [42].\nCorollary 8.8. Letp(·)∈ P(Ω)satisfy (1.1)and(8.4),0< q≤ ∞ and let bbe a slowly\nvarying function. If1\n2< p−≤p+<∞andf∈Hp(·),q,b, then σnfconverges almost\neverywhere on [0.1)and in the Lp(·),q,b-norm.\nLetIbe an atom of Fk. Define the restriction of a martingale fto the atom Iby\nfχI:= (En(f)χI, n≥k).\nCorollary 8.9. Letp(·)∈ P(Ω)satisfy (1.1)and(8.4),0< q≤ ∞ and let bbe a slowly\nvarying function. Suppose that1\n2< p−≤p+<∞andf∈Hp(·),q,b. If there exists a dyadic\ninterval Isuch that fχI∈L1(I), then σnfconverges to falmost everywhere on Iand in\ntheLp(·),q,b(I)-norm.\nMoreover, for the case of 1 ≤p−<∞, we have the next result.\nCorollary 8.10. Letp(·)∈ P(Ω)satisfy (1.1)and(8.4),0< q≤ ∞ and let bbe a slowly\nvarying function. If 1≤p−≤p+<∞andf∈Hp(·),q,b, then σnfconverges almost\neverywhere on [0,1)and in the Lp(·),q,b-norm.\nFor the operator σ2n, we do not need the condition of1\n2< p−≤p+<∞.\nLemma 8.11 ([42]).Letp(·)∈ P(Ω)satisfy (1.1)and(8.4). Then for some 0< β < 1\nand every (p(·),∞)M-atom aassociated with stopping time τ, there is\n\r\rsup\nn∈N|σ2na|βχ{τ=∞}\r\r\np(·)≲∥χ{τ<∞}∥1−β\np(·).\nSimilarly, we deduce the boundedness of σ2nand some properties of convergence can\nbe showed as follows.\nCorollary 8.12. Letp(·)∈ P(Ω)satisfy (1.1)and(8.4),0< q≤ ∞ and let bbe a slowly\nvarying function. Then for f∈Hp(·),q,b,\n∥sup\nn∈N|σ2nf|∥p(·),q,b≲∥f∥Hp(·),q,b.\nCorollary 8.13. Letp(·)∈ P(Ω)satisfy (1.1)and(8.4),0< q≤ ∞ and let bbe a slowly\nvarying function. If f∈Hp(·),q,b, then σ2nfconverges almost everywhere on [0,1)and in\ntheLp(·),q,b-norm.Variable martingale Hardy-Lorentz-Karamata spaces 65\nCorollary 8.14. Letp(·)∈ P(Ω)satisfy (1.1)and(8.4),0< q≤ ∞ and let bbe a slowly\nvarying function. If f∈Hp(·),q,band there exists a dyadic interval Isuch that fχI∈L1(I),\nthen σ2nfconverges to falmost everywhere on Iand in the Lp(·),q,b(I)-norm.\nRemark 8.15. Note that the convergence results in Corollaries 8.8,8.9,8.10,8.13and\n8.14hold also for Hp,∞,band to Lp,∞,b.\nRemark 8.16. The corresponding conclusions of the above theorems and lemmas for vari-\nable Hardy spaces and variable Hardy-Lorentz spaces can be found in Jiao et al. [42], while\nthey are new for Hardy-Lorentz-Karamata spaces.\nDeclarations\nEthical approval Not applicable.\nCompeting interests The authors declare that there is no competing interests.\nAuthor Contributions All authors wrote the main manuscript text and also reviewed\nthe manuscript.\nFunding Zhiwei Hao is supported by the NSFC (No. 11801001) and Hunan Provincial\nNatural Science Foundation (No. 2022JJ40145), Libo Li is supported by the NSFC (No.\n12101223) and Hunan Provincial Natural Science Foundation (No. 2022JJ40146).\nAvailability of data and materials Data sharing not applicable to this article as no\ndatasets were generated or analysed during the current study.\nReferences\n[1] W. Abu-Shammala, A. Torchinsky, The Hardy-Lorentz Spaces Hp,q(Rn), Studia Math. 2007, 182:\n283-294.\n[2] E. Acerbi, G. Mingione, Regularity results for a class of functionals with non-standard growth, Arch.\nRation. Mech. Anal. 2001, 156: 121-140.\n[3] T. 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Lett. 2014,\n86: 68-73."    },    {        "title": "2207.00624v1.Particle_acceleration_and_radiation_reaction_in_a_strongly_magnetized_rotating_dipole.pdf",        "content": "Astronomy &Astrophysics manuscript no. pousseur ©ESO 2022\nJuly 5, 2022\nParticle acceleration and radiation reaction in a strongly\nmagnetized rotating dipole\nJ. Pétri1\nUniversité de Strasbourg, CNRS, Observatoire astronomique de Strasbourg, UMR 7550, F-67000 Strasbourg, France.\ne-mail: jerome.petri@astro.unistra.fr\nReceived ; accepted\nABSTRACT\nContext. Neutron stars are surrounded by ultra-relativistic particles e \u000eciently accelerated by ultra strong electromagnetic fields.\nThese particles copiously emit high energy photons through curvature, synchrotron and inverse Compton radiation. However so far,\nno numerical simulations were able to handle such extreme regimes of very high Lorentz factors and magnetic field strengths close or\neven above the quantum critical limit of 4,4 · 109T.\nAims. It is the purpose of this paper to study particle acceleration and radiation reaction damping in a rotating magnetic dipole with\nrealistic field strengths of 105T to 1010T typical of millisecond and young pulsars as well as of magnetars.\nMethods. To this end, we implemented an exact analytical particle pusher including radiation reaction in the reduced Landau-Lifshitz\napproximation where the electromagnetic field is assumed constant in time and uniform in space during one time step integration. The\nposition update is performed using a velocity Verlet method. We extensively tested our algorithm against time independent background\nelectromagnetic fields like the electric drift in cross electric and magnetic fields and the magnetic drift and mirror motion in a dipole.\nEventually, we apply it to realistic neutron star environments.\nResults. We investigated particle acceleration and the impact of radiation reaction for electrons, protons and iron nuclei plunged\naround millisecond pulsars, young pulsars and magnetars, comparing it to situations without radiation reaction. We found that the\nmaximum Lorentz factor depends on the particle species but only weakly on the neutron star type. Electrons reach energies up to\n\re\u0019108\u0000109whereas protons energies up to \rp\u0019105\u0000106and iron up to \r\u0019104\u0000105. While protons and irons are not a \u000bected\nby radiation reaction, electrons are drastically decelerated, reducing their maximum Lorentz factor by 2 orders of magnitude. We also\nfound that the radiation reaction limit trajectories fairly agree with the reduced Landau-Lifshitz approximation in almost all cases.\nKey words. magnetic fields – methods: analytical – stars: neutron – stars: rotation – pulsars: general\n1. Introduction\nNeutron stars are known to harbour ultra-strong magnetic fields\nclose to or even above the quantum critical limit of Bc\u0019\n4;4 · 109T. The subclass of magnetars usually sustains field\nstrengths well above this value of Bc. These stars are therefore\nable to accelerate leptons and hadrons to extremely relativistic\nregimes of very high Lorentz factors \r\u0019109. In such an ex-\ntreme environment, radiation reaction is expected to drastically\nperturb their trajectory compared to the pure Lorentz force mo-\ntion. High energy and very high energy photons are produced\nand sometimes detected on Earth by Cerenkov telescopes.\nNevertheless, so far a quantitatively accurate study of this\nacceleration and radiation reaction mechanisms has failed due to\nthe incapability of current numerical algorithms to handle such\nstrong fields. The problem is circumvent by artificially decreas-\ning the magnetic field strength and other relevant physical pa-\nrameters like the Lorentz factor and meanwhile increasing the\nassociated Larmor radius. Unfortunately, the highly non linearity\nof the problem renders any extrapolation to realistic fields risky.\nThe only satisfactory results must come from faithfull simula-\ntions employing appropriate length and time scales met around\nneutron stars.\nThe combination of strong fields and large Lorentz fac-\ntors leads naturally to strong radiation reaction damping of the\ncharged particle motion. Those trajectories have been computedin the past for test particles like for instance by Finkbeiner\net al. (1989) in the pulsar vacuum field. Finkbeiner et al.\n(1990) discussed the validity of the Lorentz-Dirac equation and\nthe Landau-Lifshitz approximation used in such computations.\nHerold et al. (1985) integrated the equation of motion with ra-\ndiation reaction in the ultra-relativistic regime and showed the\ndi\u000berence between radiative damping and no damping for an\naligned rotator. They also gave an estimate of the maximum\nLorentz factor.\nExact analytical solutions of the Landau-Lifshitz equations\nhave been found for monochromatic plane wave as reported by\nPiazza (2008) and Hadad et al. (2010). For constant and uniform\nelectromagnetic fields, solutions are known since the work of\nHeintzmann & Schrüfer (1973). The latter are special solutions\nfound by removing the temporal and spatial derivatives from the\nLandau-Lifshitz approximation. This simplified version is some-\ntimes called the reduced Landau-Lifshitz equation (LLR). We\nwill use this approximation to advance in time the position and\nvelocity of charged particles.\nPusher based on exact analytical solutions have been im-\nplemented by several authors. For instance Laue & Thiel-\nheim (1986) evolved particles in an orthogonal magnetic dipole\nwhereas Ferrari & Trussoni (1974) investigated particle motion\nin a dipole field, neglecting the displacement current. Recently\nPétri (2020) developed an algorithm to evolve particles in a\nstrong electromagnetic field. Tomczak & Pétri (2020) applied it\nArticle number, page 1 of 18arXiv:2207.00624v1  [astro-ph.HE]  1 Jul 2022A&A proofs: manuscript no. pousseur\nto a magnetic dipole associated to strongly magnetized rotating\nneutron stars. Gordon et al. (2017b) and Gordon et al. (2017a)\nshowed how to implement a fully covariant particle pusher and\ngave some hints to include radiation reaction. Later Gordon &\nHafizi (2021) developed a special unitary pusher for extreme\nfields achieving computation costs comparable to the Boris al-\ngorithm (Boris 1970).\nIn the ultra-relativistic regime, radiation reaction almost ex-\nactly balance the electric field acceleration leading to a particle\nvelocity only depending on the local electromagnetic field con-\nfiguration. As shown by Mestel et al. (1985), the Lorentz fac-\ntor can then be deduced from the trajectory curvature. Kelner\net al. (2015) carefully studied the synchro-curvature radiation of\nultra-relativistic particles evolving in a strongly curved electro-\nmagnetic field. The pitch angle plays a central role in controlling\nthe synchrotron versus curvature regime.\nSeveral di \u000berent but not equivalent approaches have been\ndesigned to include radiation reaction in a particle pusher for\nultra strong electromagnetic fields. Vranic et al. (2016) o \u000bers a\ncomprehensive study of the most widely used techniques to im-\nplement the radiation reaction force in standard Lorentz force\npushers. However, numerical algorithms solving explicitly the\nLandau-Lifshitz equation face some issues to satisfy conserva-\ntion laws for long time runs. Nevertheless time-symmetric im-\nplicit methods seem to give better results (Elkina et al. 2014).\nInterestingly, exact analytical solutions of the reduced Landau-\nLifshitz equation have been found several decades ago by\nHeintzmann & Schrüfer (1973) for a constant electromagnetic\nfield. These expressions are used by Li et al. (2021) for imple-\nmentation in a PIC code following a projection onto an electric\nand a magnetic sub-space (Boghosian 1987). Pétri (2021) also\napplied this exact solution to the acceleration of particles in a\nlow frequency strong amplitude electromagnetic plane wave as\nthat launched by a strongly magnetized rotating neutron star.\nIn this paper we study particle acceleration in a realistic neu-\ntron star environment, using the exact scaling between the neu-\ntron star spin and the cyclotron frequency. In section 2 we re-\ncall the equation of motion as derived by Landau-Lifshitz and its\nexact analytical solution, the appropriate normalization and the\nalgorithm. Section 3 presents extensive tests of our algorithm\nin static fields showing its second order in time convergence.\nSection 4 describes an astrophysical application to neutron star\nelectrodynamics and the upper limit of particle acceleration e \u000e-\nciency. Section 5 compares the radiation reaction limit regime to\nthe exact motion. Eventually conclusions are drawn in section 6.\n2. Equation of motion\nThe self-force produced by an accelerated charge is usually de-\nscribed by the Lorentz-Abraham-Dirac equation (LAD) (Abra-\nham 1902, 1904; Lorentz 1916; Dirac 1938). Unfortunately this\nself-force leads to runaway solutions because the associated\nequation of motion is of third order in time. Several remedies\nhave been found to remove this unacceptable solutions. See for\ninstance Rohrlich (2007) for some discussions. One approach\noften quoted in the literature is the Landau-Lifshitz formulation,\na perturbative expansion of the LAD equation (Landau & Lif-\nchitz 1989). In the remainder of this paper, we adopt this point\nof view.\n2.1. Landau-Lifshitz approximation\nIn order to get rid of the LAD flaw, Landau & Lifchitz (1989)\nderived an approximation valid in most configurations met inastrophysical applications. This new equation of motion is free\nof runaway instabilities and is largely employed in the plasma\ncommunity. Their formulation leads to the following equation of\nmotion\ndui\nd\u001c=q\nmFikuk+q\u001cm\nmgi(1a)\ngi=@`Fikuku`+q\nm \nFikFk`u`+(F`mum) (F`kuk)ui\nc2!\n(1b)\nwhere qandmare the particle charge and rest mass, uiits 4-\nvelocity,\u001cits proper time, Fikthe electromagnetic or Faraday\ntensor, cthe speed of light and \u001cmthe light crossing time across\nthe particle classical radius rm(within a factor unity)\n\u001cm=q2\n6\u0019\"0m c3: (2)\nIt is advantageous to express it in term of the electron classical\nradius recrossing time amounting to\n\u001ce=2\n3re\nc=6;26 · 10\u000024s: (3)\nThe typical time scale for the radiation reaction is therefore\n\u001cm=2\n3rm\nc= q2=e2\nm=me!\n\u001ce: (4)\nFor instance for protons, this time is three orders of magnitude\nless than for leptons\n\u001cp=me\nmp\u001ce=3;41 · 10\u000027s: (5)\nInterestingly, exact analytical solutions have been computed\nfor eq.(1) in some special configurations of electromagnetic\nfields, time dependent or time independent. We succinctly recall\nthe useful results required for the present work.\n2.2. Exact analytical solutions\nAn exact solution for LLR is based on the eigensystem expan-\nsion of the electromagnetic tensor Fik. Earlier results were given\nby Heintzmann & Schrüfer (1973). Here we follow the notation\nof Li et al. (2021). Starting from the Lorentz force written as\ndu\nd\u001c=G u (6)\nwhere the electromagnetic tensor Fhas been replaced by G=\nq F=mto absorb the charge over mass ratio, we decompose the 4-\nvelocity uin a magnetic and an electric part denoted respectively\nbyuBanduEsuch that u=uE+uB. The real eigenvalues of Gik\nare\u0006\u0015Ewhereas the imaginary eigenvalues are \u0006i\u0015B,\u0015Eand\n\u0015Bbeing real and positive numbers, with dimensions similar to\npulsation thus in 1 =s. Then, each vector uEanduBremains in a\neigen-subspace satisfying\nG u E=\u0006\u0015EuE (7a)\nG u B=\u0006i\u0015BuB: (7b)\nThe vector components uEanduBare obtained by defining the\nprojection operators onto the sub-spaces EandBby (Boghosian\n1987)\nP=\u00152\nBI+G2\n\u00152\nE+\u00152\nB(8a)\nQ=\u00152\nEI\u0000G2\n\u00152\nE+\u00152\nB(8b)\nArticle number, page 2 of 18J. Pétri : Particle acceleration and radiation reaction in a strongly magnetized rotating dipole\nwhere Iis the identity matrix. These operators are well defined\nonly if\u00152\nE+\u00152\nB,0. If both electromagnetic invariants vanish,\nwe retrieve a null-like field which requires a di \u000berent treatment\nas given for instance by Pétri (2021). In the non null-like field\nwe get\nuE=P u (9a)\nuB=Q u: (9b)\nThe equation of motion decouples into two parts given by\nd2uE\nd\u001c2= +\u00152\nEuE (10a)\nd2uB\nd\u001c2=\u0000\u00152\nBuB: (10b)\nThe exact analytical solutions with initial conditions u0\nE=P u0\nandu0\nB=Q u0are\nuE(\u001c)=u0\nEcosh(\u0015E\u001c)+G u0\nEsinh(\u0015E\u001c)\n\u0015E(11a)\nuB(\u001c)=u0\nBcos(\u0015B\u001c)+G u0\nBsin(\u0015B\u001c)\n\u0015B: (11b)\nAdding the radiation reaction in the LLR limit leads to the exact\nexpression\nuE(\u001c)\nc=u0\nEcosh(\u0015E\u001c)+G u0\nEsinh(\u0015E\u001c)=\u0015Eq\nju0\nEj2+ju0\nBj2e\u00002\u000b\u001c(12a)\nuB(\u001c)\nc=u0\nBcos(\u0015B\u001c)+G u0\nBsin(\u0015B\u001c)=\u0015Bq\nju0\nBj2+ju0\nEj2e2\u000b\u001c(12b)\nwith\u000b=\u001cm(\u00152\nE+\u00152\nB). These expressions are similar to the orig-\ninal formulas found by Heintzmann & Schrüfer (1973). The ra-\ndiation reaction e \u000bect becomes perceptible after a time \u001c\u00191=\u000b.\nThe component uEis associated to the accelerating motion in-\nduced by the electric field whereas the uBcomponent is related to\nthe gyro-motion in the magnetic field. When \u000bvanishes, the radi-\nation reaction e \u000bect disappears. The denominators in uEanduB\nreduce to unity and the solution to the Lorentz force 4-velocity\ncomponents are recovered.\n2.3. Normalisation\nThe relevant physical parameters determining the particle trajec-\ntory is decided through some normalisation procedure incrimi-\nnating the following useful quantities in order to write the equa-\ntion of motion without dimensions. These primary fundamental\nvariables are\n–the speed of light c.\n–a typical frequency !involved in the problem.\n–the particle electric charge q.\n–the particle rest mass m.\nFrom these quantities we derive a typical time and length scale\nas well as electromagnetic field strengths such that\n–the length scale L0=c=!.\n–the time scale T0=1=!.\n–the magnetic field strength B0=m!=q.\n–the electric field strength E0=c B0.Normalized quantities will be overlaid with a tilde symbol.\nThe two important parameters defining the family of solu-\ntions are the field strength parameters aBandaEand the radiation\nreaction e \u000eciency!\u001c maccording to the following definitions\naB=B\nB0=!B\n!(13a)\naE=E\nE0=!E\n!(13b)\nb=!\u001c m: (13c)\nIntroducing the weighted and normalized electromagnetic field\ntensor by ˜Fik=q Fik=m!and a normalized time ˜ \u001c=!\u001c, the\nLandau-Lifshitz equation (1) is rewritten without dimensions as\nd˜ui\nd˜\u001c=˜Fik˜uk+b˜gi(14a)\n˜gi=˜@`˜Fik˜uk˜u`+\u0010˜Fik˜Fk`˜u`+(˜F`m˜um) (˜F`k˜uk) ˜ui\u0011\n: (14b)\nThe normalised and reduced Landau-Lifshits equation reads\nd˜ui\nd˜\u001c=˜Fik˜uk+b\u0010˜Fik˜Fk`˜u`+(˜F`m˜um) (˜F`k˜uk) ˜ui\u0011\n: (15)\nThe particle 4-velocity depends only on the strength parame-\ntersaBandaEand on the radiation reaction strength parameter b.\nTherefore it is unnecessary to compute trajectories for di \u000berent\nparticles possessing the same numbers aB;aE;b. The only di \u000ber-\nences reflect in the physical time and space scales involved.\nAs a rule of thumb, we admit that radiation reaction is neg-\nligible whenever the time scale of damping, given by 1 =\u000bbe-\ncomes larger than the characteristic time scale of our system,\nthat is 1=!. Expressed in quantities without dimension, we get\n\u001cm!(a2\nE+a2\nB)=b(a2\nE+a2\nB)\u001c1. Therefore the relevant param-\neter to quantify radiation reaction is not bbut the combination of\nband the strength parameters aBandaE. Specific examples will\nbe given in the test section 3.\n2.4. Algorithm\nFor the remainder of this paper, we use a Cartesian coordinate\nsystem labelled by ( x;y;z) and the corresponding Cartesian or-\nthonormal basis ( ex;ey;ez).\nThe velocity vector is integrated analytically following the\nprevious discussion. Unfortunately, for the position vector, there\nexists no simple analytical expression, although some formulas\ncan be found involving hypergeometric 2F1functions with com-\nplex arguments, see section 3 for an example in a constant mag-\nnetic field. The update in particle position is therefore performed\nby the velocity-Verlet algorithm namely\nun+1=2=L(\u0001\u001c=2;un;E(xn);B(xn)) (16a)\nxn+1=xn+un+1=2\u0001\u001c (16b)\nun+1=L(\u0001\u001c=2;un+1=2;E(xn+1);B(xn+1)): (16c)\nThe subscript nrefers to the proper time \u001cn=n\u0001\u001cand the same\nfor half integer subscript \u001cn+1=2=(n+1=2)\u0001\u001c. We found this\nmethod more robust than the full analytical update in velocity\nand position. Indeed for particles trapped in a dipole magnetic\nfield, undergoing bouncing motion with banana orbits typical of\nmagnetic confinement devices for thermonuclear fusion reactors\nor in Earth magnetosphere known as Van Allen belt, the stability\nand convergence properties of the velocity-Verlet algorithm is\nsuperior.\nArticle number, page 3 of 18A&A proofs: manuscript no. pousseur\nBefore using our code to compute particle acceleration and\nradiation in the ultra-strong electromagnetic field of a dipole ro-\ntating in vacuum, we test it against exact analytical solutions in\nsimple geometric configurations but with very high Lorentz fac-\ntors and /or very high fields. Results will also be compared to the\nradiation reaction limit regime which is much less time consum-\ning from a computational point of view but also less accurate in\nsome configurations, section 5.\n2.5. Radiation reaction limit\nIn ultra-strong electromagnetic fields as such present around\nneutron stars, radiation reaction plays an important role. In the\nasymptotic limit of ultra-relativistic motions, assuming that the\nradiation damping exactly balances the electric field accelera-\ntion, there exists a simple analytical expression for the particle\nvelocity depending only on the local values of the fields (Mestel\net al. 1985). This velocity is decomposed into an electric drift\nmotion, interpreted as the velocity required to switch to a frame\nwhere the electric and magnetic field are aligned, and a motion\nalong this common direction in this new frame. Denoting the\nvelocity vector for positive charges as v+and that for negative\ncharges as v\u0000, we find\nv\u0006=E^B\u0006(E0E=c+c B0B)\nE2\n0=c2+B2: (17)\nIt corresponds to particles moving exactly at the speed of light.\nE0andB0are the strength of the electric and magnetic field in\nthe frame where they are aligned. They are obtained from the\nelectromagnetic invariants I1=E2\u0000c2B2=E2\n0\u0000c2B2\n0and\nI2=cE·B=c E0B0. Imposing E0\u00150 we find\nE2\n0=1\n2(I1+q\nI2\n1+4I2\n2) (18a)\nc B0=sign(I2)q\nE2\n0\u0000I 1: (18b)\nWe will compared the simulation results obtained from this sim-\nple prescription with the exact integration of the equation of mo-\ntion according to LLR.\nApplying this radiation reaction limit to neutron star mag-\nnetospheres, the velocity in eq. (17) can be slightly simplified\nbecause of the presence of a plasma, the parallel electric field\ncomponent (with respect to the magnetic field direction) being\ne\u000eciently screened. In such a configuration, jI2j \u001c jI 1jand\nI1<0. The velocity then reduces to\nv\u0006\u0019E^B\nB2\u0006sign(E·B)p\nc2B2\u0000E2\nB2B: (19)\nThe first term corresponds to the electric drift speed whereas\nthe second term is associated to the motion along the magnetic\nfield lines, the particle gyro-motion being absent in this picture.\nWe note that the velocity component along the magnetic field\nreverses sign when crossing a point where E·Bchanges sign.\nThese regions are able to trapped particles depending on their\ncharge and on the ( E·B) configuration in the neighbourhood of\nthis surface (Finkbeiner et al. 1989).\nSeveral limiting cases are also useful to discuss. First, if the\nelectric field vanishes, E0=0, the radiated power vanishes too\nand the particle moves along the field lines with v\u0006=\u0006cB=B.\nSecond if the electric field is orthogonal to the magnetic field,E·B=0 and E<c B, the particle motion is decomposed into\nan electric drift and a motion along Bsuch that\nv\u0006=E^B\nB2\u0006p\nc2B2\u0000E2\nB2B: (20)\nThis expression holds well within the light-cylinder of a force-\nfree magnetosphere.\n3. Tests\nWe checked our algorithm against simple electromagnetic field\nconfigurations containing either only an electric field or a mag-\nnetic field or a cross electromagnetic field. Although the ex-\nact solutions are simple expressions, from a numerical point of\nview it is of paramount importance to ensure the code to be\nable to handle very high strength parameters and Lorentz fac-\ntors as those met in neutron star magnetospheres, that is about\naB\u00191020and\r\u00191010. Our main purpose in this section is to\ncheck that the results are not a \u000bected by round o \u000berrors.\n3.1. Constant electric field\nIn a constant electric field, a charged particle is permanently ac-\ncelerated in the direction of the electric field while it loses en-\nergy. Specializing the general solution (12) to a pure electric field\naligned with the zaxis such that E=Eezwe get\nut\nc=\r(\u001c)=\r0ccosh(!E\u001c)+u0\nzsinh(!E\u001c)\nq\n\r2\n0c2\u0000u2\nk\u0000u2\n?e\u00002\u000b\u001c(21a)\nux\nc=u0\nxq\n(\r2\n0c2\u0000u2\nk)e2\u000b\u001c\u0000u2\n?(21b)\nuy\nc=u0\nyq\n(\r2\n0c2\u0000u2\nk)e2\u000b\u001c\u0000u2\n?(21c)\nuz\nc=u0\nzcosh(!E\u001c)+\r0csinh(!E\u001c)\nq\n\r2\n0c2\u0000u2\nk\u0000u2\n?e\u00002\u000b\u001c(21d)\nwith uk=u0\nzthe initial 4-velocity component along E,u?the\ninitial 4-velocity component perpendicular to Eand\u000b=\u001cm!2\nE.\nFor a particle starting at rest, uk=u?=0 and\r0=1, the\n4-velocity simplifies drastically into\nui=c(cosh(!E\u001c);0;0;sinh(!E\u001c)): (22)\nThis 4-velocity does not depend on the radiation reaction inten-\nsity. It accelerates as if it only experiences the Lorentz force.\nThis peculiar situation is well known and discussed in length by\nFulton & Rohrlich (1960) for a charge and its related classical\nradiation in an uniformly accelerating field.\nThe 4-position is given by introducing the two complex\nfunctions with help on the hypergeometric functions 2F1(Olver\n2010) such that\nJ1(\u001c)=e(\u000b+!E)\u001c\n\u000b+!E2F1 1\n2;1+1\n2b;3\n2+1\n2b;\r2e2\u000b\u001c\n\r2\u00001!\n(23a)\nJ2(\u001c)=e(\u000b\u0000!E)\u001c\n\u000b\u0000!E2F1 1\n2;1\u00001\n2b;3\n2\u00001\n2b;\r2e2\u000b\u001c\n\r2\u00001!\n: (23b)\nArticle number, page 4 of 18J. Pétri : Particle acceleration and radiation reaction in a strongly magnetized rotating dipole\nThen the time and position are given by\nt=\u0000\rp\n\r2\u0000e2\u000b\u001c\u00001\u0001+1\n2\u0000\r2\u00001\u0001 (J2(\u001c)+J1(\u001c))+C0 (24a)\nx=c=0 (24b)\ny=c=1\n\u000barctan0BBBBBBB@s\n\r2\u0000e2\u000b\u001c\u00001\u0001+1\n\r2\u000011CCCCCCCA+C2 (24c)\nz=c=\rp\n\r2\u0000e2\u000b\u001c\u00001\u0001+1\n2\u0000\r2\u00001\u0001 (J2(\u001c)\u0000J1(\u001c))+C3 (24d)\nwith C0;C2;C3complex constants of integration to satisfy the\ninitial conditions.\nReturning to eq. (21), the typical electric acceleration time\nscales is\u001cacc\u00181=!E. On the other side, the radiation damping\ntime scale is \u001crad\u00181=\u000b. The ratio between both time scales is\ntherefore\u001cacc=\u001crad\u0018\u001cm!E=b. As expected, for small damping\nparameters b\u001c1, the acceleration time is much shorter than the\nradiative damping and the particle accelerated as if it would not\nradiate, until the time \u001crad\u0018\u001cacc=b\u001d\u001cacc. Note that this rough\nestimate needs to be corrected by taking into account the initial\nLorentz factor as discussed below.\nFor performing the simulations, we use the characteristic fre-\nquency!Eas normalisation leading to a normalized proper time\n˜\u001c=!E\u001c. Therefore the only relevant parameter apart from the\ninitial conditions is b=\u001cm!Eand\u000b\u001c=b˜\u001c. Particles starting\nat rest or possessing an initial velocity directed along the electric\nfield do not su \u000ber from the radiative force. Consequently, as a\ntypical example, particles are starting with an initial velocity per-\npendicular to the electric field, meaning uk=0 and u?,0. We\nchose di \u000berent initial Lorentz factors in the set log \r0=f0;4;8g.\nThe damping factor is given by log b=f0;\u00005;\u000010;\u000015g. As\noutput of the simulations, we plot the Lorentz factor increasing\naccording to eq. (21a) and shown in Fig. 1. For a particle start-\ning at rest, whatever the damping parameter b, the solution is al-\nways equal to (22) with an acceleration arising around the time\n!E\u001c\u00181. This configuration is very particular and is not im-\npacted by radiation reaction. More interesting are the particles\nstarting with a substantial kick velocity and high Lorentz fac-\ntors\r0\u001d1. The time derivative of the Lorentz factor is always\nnegative for \r0>1 because\nd\r(\u001c)\nd\u001c=\u0000\u000b\r0(\r2\n0\u00001)<0 (25)\nmeaning that the particle first decelerates due to the radiative\nfriction. At large times, when \u000b\u001c\u001d1 and!E\u001c\u001d1, the Lorentz\nfactor behaves as \r(\u001c)\u0019cosh(!E\u001c), loosing its information\nabout the initial state. It resembles to the motion of particle start-\ning at rest, independently of \r0. This is because the perpendicular\nmotion is strongly damped, lim\n\u001c!+1u?!0 and only the parallel\nvelocity uksurvives at large times with lim\n\u001c!+1uk!csinh(!E\u001c).\nIn between, the normalized time remains small and the Lorentz\nfactor can be approximated by\n\r(\u001c)\u0019\r0cosh(!E\u001c)q\n1+2\r2\n0\u000b\u001c: (26)\nTherefore, before the acceleration phase starts, there is a decel-\neration step arising at time !E\u001c\u00181=2\r2\n0b. These values agree\nwith the curves in figure 1. If \r2\n0b.1, the radiation reaction\nforce has no time to set in and the motion tends to a purely ac-\ncelerated regime given by eq.(22). This is for instance the case\nFig. 1. Increase of the Lorentz factor due to radiation reaction for dif-\nferent initial Lorentz factor log \r0=f0;4;8gand di \u000berent damping fac-\ntor log b=f0;\u00005;\u000010;\u000015g. The vertical lines show the time when\nthe damping sets in before the electric acceleration phase starts. Dot-\nted colour points show the simulation results and the black solid lines\ncorrespond to the exact analytical solutions.\nwith\r0=104andb=10\u000010, orange dots, or \r0=108and\nb=10\u000015which is just on the edge of this condition, showing a\nweak deceleration right before the electric boost, blue dots. The\nperpendicular momentum decrease is not necessarily significant\nbefore the acceleration, it is controlled by band\r0because at\ntime!E\u001c\u00191 it braked to a momentum\nu?(\u001c)\u0019u0\n?q\n1+2\r2\n0b: (27)\nThus again, radiation reaction impacts the motion if \r2\n0b&1.\nConsequently, it is the combination \r2\n0bthat controls the damp-\ning e \u000eciency, not balone found from the simple arguments\nabove.\nBecause the 4-position of the particle is computed numeri-\ncally and not analytically according to eq. (24), it is important to\nestimate the convergence rate of our scheme. To this end Fig. 2\nshows the error in the yandzposition and time twith decreasing\nproper time step \u0001\u001cfor log\r0=4 and log b=\u00005 in blue, or-\nange, green and red respectively. The second order expectations\nare depicted by the green line. We conclude that the decrease in\nthe relative error follows a second order in time scheme as ex-\npected from the velocity-Verlet algorithm exposed in section 2.\n3.2. Constant magnetic field\nA charged particle orbiting in a constant magnetic field loses\nenergy and decays until it rests. The rate of decay is controlled\nby the magnetic field strength only. The exact solution for the\n4-velocity in a magnetic field directed along the zaxis with B=\nArticle number, page 5 of 18A&A proofs: manuscript no. pousseur\nFig. 2. Relative error of the position y;zand time tas shown in the\nlegend. The error decreases with second order in \u0001\u001cas given by the\ngreen line \u0001\u001c2for log\r0=4 and log b=\u00005.tandzerrors overlap and\nare undistinguishable.\nBezis given by\nut\nc=\r(\u001c)=\r0cq\n\r2\n0c2\u0000u2\nk\u0000u2\n?e\u00002\u000b\u001c(28a)\nux\nc=u0\nxcos(!B\u001c)+u0\nysin(!B\u001c)\nq\n(\r2\n0c2\u0000u2\nk)e2\u000b\u001c\u0000u2\n?(28b)\nuy\nc=u0\nycos(!B\u001c)\u0000u0\nxsin(!B\u001c)\nq\n(\r2\n0c2\u0000u2\nk)e2\u000b\u001c\u0000u2\n?(28c)\nuz\nc=u0\nzq\n\r2\n0c2\u0000u2\nk\u0000u2\n?e\u00002\u000b\u001c(28d)\nwith uk=u0\nzthe initial 4-velocity component along B,u?the\ninitial 4-velocity component perpendicular to Band\u000b=\u001cm!2\nB.\nApart from the change in the gyro-frequency, the magnetic field\nstrength impacts only the time scale for the decay via the expo-\nnential terms of arguments 2 \u000b\u001c.\nFor performing simulations, we use the characteristic fre-\nquency!Bas normalisation with ˜ \u001c=!B\u001c. Therefore the only\nrelevant parameter apart from the initial conditions is b=\u001cm!B\nand\u000b\u001c=b˜\u001c. The length scale is therefore given in units of the\nnon-relativistic Larmor radius\nrB=c\n!B: (29)\nIntegrating the 4-velocity vector, an exact analytical expres-\nsion for the 4-position is computed with help on the hypergeo-\nmetric functions 2F1. Introducing the complex functions\nH1(\u001c)=e+i!B\u001c\n2F1 1\n2;+i\n2b; 1+i\n2b;\r2e\u00002\u000b\u001c\n\r2\u00001!\n(30a)\nH2(\u001c)=e\u0000i!B\u001c\n2F1 1\n2;\u0000i\n2b; 1\u0000i\n2b;\r2e\u00002\u000b\u001c\n\r2\u00001!\n: (30b)\nFig. 3. Particle orbit in an uniform and constant magnetic field and sub-\nject to radiation reaction. The initial Lorentz factor is log \r0=4. The\ninset shows the strong damped motion in green and even stronger damp-\ning in red where the spiralling is not seen.\nthe solution reads\nt=1\n\u000btanh\u000010BBBBB@\re\u000b\u001c\np\n\r2\u0000e2\u000b\u001c\u00001\u0001+11CCCCCA+C0 (31a)\nx=rB=H1(\u001c)+H2(\u001c)\n2i+C1 (31b)\ny=rB=H1(\u001c)\u0000H2(\u001c)\n2+C2 (31c)\nz=rB=0 (31d)\nwhere the Ciwith i2[0::2] are complex constants of integration\nto satisfy the initial conditions.\nThe particle trajectory follow a spiral as shown in Fig. 3. The\nparticle comes to rest after a typical time !B\u001c1\u001d1=b. The cor-\nresponding Lorentz factor decreases according to eq. (28a) and\nis shown in Fig. 4. The time when damping sets in is given ap-\nproximately by 2 \u000b\r2\n0\u001c\u00191. These times are shown as coloured\nvertical lines in the Fig. 4. If the particle moves along the field\nline, it experiences no damping and keeps a uniform motion.\nA comparison between the analytical trajectory in red solid\nline and the numerical integration in blue dots is shown in Fig. 5.\nA more quantitative agreement is proven in Fig. 6 where the rela-\ntive error decreases with respect to the proper time step \u0001\u001c. Here\nalso the method is second order in time as expected.\n3.3. Cross electric and magnetic field\nThe cross electric and magnetic field configuration is a stringent\ntest for an ultra relativistic particle pusher. If the electric field\nstrength is less than the magnetic field strength E<c B, then\nan appropriate Lorentz transform brings the problem to a frame\nwhere the electric field vanishes. We therefore return to the sit-\nuation of the last section with a constant and uniform magnetic\nfield. For su \u000ecient long time, the only remaining motion is the\nelectric drift at speed vE=E^B=B2. Therefore the velocity is\nArticle number, page 6 of 18J. Pétri : Particle acceleration and radiation reaction in a strongly magnetized rotating dipole\nFig. 4. Decrease of the Lorentz factor due to radiation reaction in an\nuniform and constant magnetic field associated to the orbits shown in\nfig. 3. Dotted colour points show the simulation results and the black\nsolid lines correspond to the exact analytical solutions.\nFig. 5. Comparison between the analytical solution, eq. (31), in red solid\nline and the numerical simulation in blue dots for log \r0=4 and log b=\n\u00005.\n\fE=vE=c=E=cBand the corresponding final Lorentz factor\n\r1=(1\u0000\f2\nE)\u00001=2.\nWe performed simulations with E=cB=0:999 and initial\nLorentz factors log \r0=f0;4;8g. The final Lorentz factor is\n\r1\u001922:3. Fig. 7 shows the Lorentz factor with colour dots\ncompared to the analytical expression shown in black solid lines.\nThe agreement is excellent and demonstrates the high e \u000eciency\nof our algorithm to capture ultra-relativistic motion with high\nprecision.\nThe quantitative agreement is checked by transforming the\ntrajectory to the electric drift frame denoted by the coordi-\nnates ( x0;y0). In this frame the orbital radius is decreasing as\nshown in Fig. 8 for log \r0=4 and log b=\u00005, using di \u000ber-\nent proper time steps such as log( !B\u0001\u001c)=f\u00001;\u00002grespectively\nin orange and blue dots. The analytical solution found from ap-\nFig. 6. Relative error of the position xandyas shown in the legend. The\nerror decreases with second order in \u0001\u001cas given by the violet line \u0001\u001c2\nfor log\r0=4 and log b=f\u00005;\u000010g.\nFig. 7. Decrease of the Lorentz factor due to radiation reaction in a\ncross electric and magnetic field. Dotted colour points show the simula-\ntion results and the black solid lines correspond to the exact analytical\nsolutions.\npropriate parameters in eq. (31) is overlapped in red solid lines.\nFig. 9 shows the relative error in the x0andy0position depend-\ning on the proper time step \u0001\u001c. The scheme converges to second\norder in proper time step.\n3.4. Parallel electric and magnetic field\nAnother interesting configuration not reducible to any of the pre-\nvious one is a parallel electric and magnetic field. In this case the\nsecond electromagnetic invariant does not vanish I2,0. Con-\nsequently, there exist no frame where either the electric or mag-\nnetic field vanishes. The electric and magnetic velocity compo-\nnents uEanduBdecouple into an acceleration along the common\ndirection and a gyration around the same direction. Assuming\nArticle number, page 7 of 18A&A proofs: manuscript no. pousseur\nFig. 8. Orbit in the electric drift frame ( x0;y0) for log\r0=4 and log b=\n\u00005 for di \u000berent proper time steps log( !B\u0001\u001c)=f\u00001;\u00002g.\nFig. 9. Relative error of the particle position associated to the electric\ndrift motion shown in Fig. 8.\nthis direction to be ez, the 4-velocity reads\nut\nc=\r(\u001c)=\r0ccosh(!E\u001c)+u0\nzsinh(!E\u001c)\nq\n\r2\n0c2\u0000u2\nk\u0000u2\n?e\u00002\u000b\u001c(32a)\nux\nc=u0\nxcos(!B\u001c)+u0\nysin(!B\u001c)\nq\n(\r2\n0c2\u0000u2\nk)e2\u000b\u001c\u0000u2\n?(32b)\nuy\nc=u0\nycos(!B\u001c)\u0000u0\nxsin(!B\u001c)\nq\n(\r2\n0c2\u0000u2\nk)e2\u000b\u001c\u0000u2\n?(32c)\nuz\nc=u0\nzcosh(!E\u001c)+\r0csinh(!E\u001c)\nq\n\r2\n0c2\u0000u2\nk\u0000u2\n?e\u00002\u000b\u001c: (32d)\nWe recognize the special cases of a pure electric field for ut;uz\nand a pure magnetic field for ux;uz, the only di \u000berence being the\nvalue of\u000b=\u001cm(\u00152\nE+\u00152\nB), including a non vanishing electric and\nmagnetic contribution.\nFig. 10. Analytic evolution of the Lorentz factor with initial condi-\ntion log\r0=8, di\u000berent damping factor log b=f0;\u00005;\u000010;\u000015gand\ndi\u000berent electric field strength E0relative to B0such that log( E0=B0)=\nf\u00002;0;2gin solid lines, dashed lines and dotted lines respectively.\nAfter a transition time, the gyro-motion has been signifi-\ncantly damped and the electric acceleration has directed the ve-\nlocity along its direction. The initial conditions are washed out\nand the particle moves like in a constant electric field with an al-\nmost constant acceleration leading to a hyperbolic motion, well\nknow in special relativity kinematics. Let us estimate the dura-\ntion of this transient stage. Either the particle is drastically ac-\ncelerated before the gyration is damped or vice versa the orbit\nshrinks significantly before the electric field accelerated sensibly\nthe particle. The situation depends on ordering of the eigenvalues\n\u0015Eand\u0015B.\nFig. 10 shows the analytic evolution of the Lorentz for a\nparticle starting with only a perpendicular velocity component\nsuch that log \r0=8. The damping factor is set to log b=\nf0;\u00005;\u000010;\u000015gand the electric field strength E0is varied rel-\native to B0such that log( E0=B0)=f\u00002;0;2gand depicted in\nsolid lines, dashed lines and dotted lines respectively. For a weak\nelectric field E0\u001cB0the particle trajectory follows a spiral\nmotion similar to the previous case of a pure magnetic field\nuntil it almost rests. At later times the electric field starts to\naccelerate it quickly to ultra-relativistic speeds on a timescale\n1=!E\u001d1=!B. The particle performs many orbits before be-\ning deflected along the parallel direction ( ez). Increasing E0will\ndecrease this time scale and the particle follow the common E\nandBdirection before performing many gyrations. In the oppo-\nsite limit of a strong electric field E0\u001dB0electric acceleration\nquickly sets in. Fig. 11 shows some results of numerical simula-\ntions with a weak electric field log( E0=B0)=\u00002, pertinent for al-\nmost force-free neutron star magnetospheres, and initial Lorentz\nfactors log\r0=f0;4;8gand log b=f0;\u00005;\u000010;\u000015g.\nWhenever there exist a magnetic field aligned electric field\ncomponent, at late times particles are always accelerated along\nthe common direction. The timescale required is estimated by\nreckoning the proper time at which the parallel 4-velocity com-\nponent becomes comparable to the perpendicular 4-velocity\ncomponent for an initial velocity perpendicular to the magnetic\nfield line. This represents the worst case, useful to be compared\nto the radiation reaction limit regime.\nArticle number, page 8 of 18J. Pétri : Particle acceleration and radiation reaction in a strongly magnetized rotating dipole\nFig. 11. Evolution of the Lorentz factor for a parallel electromagnetic\nfield configuration with initial Lorentz factors log \r0=f0;4;8g, an elec-\ntric field strength log( E0=B0)=\u00002 and log b=f0;\u00005;\u000010;\u000015g. The\nblack solid lines correspond to the exact analytical solutions.\n3.5. Almost cross field\nIn a plasma filled magnetosphere, the electric field is e \u000eciently\nscreened meaning that the parallel component of the electric\nfield is negligible with respect to its perpendicular component,\nEk\u001cE?. As an application towards this configuration, we com-\nputed the motion of a particle in an almost cross electric and\nmagnetic field with log( Ek\u001cE?)=f\u00001;\u00002;\u00003;\u00004gand dif-\nferent ratio E?=c B=f0:1;0:999g. The particle starts with an\ninitial velocity in a plane perpendicular to Band Lorentz fac-\ntor log\r0=4 in a field with log b=\u00005. Fig. 12 shows the\nevolution to alignment of the velocity vector with the radiation\nreaction limit direction for a weak parallel electric field compo-\nnent as reported in the legend. Note that the angle \u0012should not\nbe interpreted as the angle between the velocity vector and the\nmagnetic field direction because the velocity in eq.(17) is not\nalong B. We observe that the time required for alignment is in-\nsensitive to the ratio E?=c Bbut depends strongly on the ratio\nEk=E?. As expected a weak parallel component tends to align\nslower the trajectory compared to a strong parallel component.\nIf this alignment occurs on a length scale smaller than the mag-\nnetic field curvature radius, the radiation reaction limit could be\nused without significant loss of accuracy.\n3.6. Dipole magnetic field\nAs a step towards realistic configurations, we also investigate\nparticle motion in a static magnetic dipole. Unfortunately, no\nsimple exact analytical expressions are available for checking the\nalgorithm therefore no quantitative accurate converge test can be\nperformed. First we study the magnetic drift in the equatorial\nplane of a dipole field. Next we look at trapped particles due to\nthe mirror e \u000bect.\n3.6.1. Magnetic drift\nThe magnetic drift motion in the equatorial plane of a dipole field\nis an interesting example to test our algorithm. The characteristic\nfrequency is again !Band the particle initial Lorentz factor is\n\r0=104. The damping parameter is log b=f0;\u00005;\u000010;\u000015g.\nFig. 13 shows the time evolution of the Lorentz factor that\ntends asymptotically to unity meaning the particle will rest. The\nFig. 12. Alignment of the velocity vector with the radiation reaction\nlimit direction given by \u0012=0\u000efor di \u000berent strengths of the parallel\nelectric field component Ekcompared to the perpendicular component\nE?forE?=cB=0:1 in dashed lines and E?=cB=0:999 in solid lines.\nThe damping parameter is log b=\u00005 and the initial Lorentz factor\nlog\r0=4.\nFig. 13. Decrease of the Lorentz factor due to radiation reaction when\ndrifting in a dipole magnetic field with initial Lorentz factor log \r0=4\nand di \u000berent damping constants.\nparticle returns to rest after a typical time controlled by band\nshown as coloured vertical lines. Fig. 14 highlights the cor-\nresponding particle trajectory in the equatorial plane. For the\nweakest damping, the motion remains circular for the guiding\ncentre. For the strongest damping, in green and red, the parti-\ncle su \u000bers from drastic radiative friction and tends to rest on a\nvery short time scale compared to the drifting motion and orbital\nmotion.\n3.6.2. Magnetic mirror\nDue to the mirror e \u000bect, particles remain trapped in the dipole\nmagnetic field of a star like around Earth in the van Allen belt.\nHowever when some dissipation occurs as for instance through\nradiation reaction, for high damping parameters particles quickly\ncrash onto the surface of the magnetic object.\nFig. 15 shows the evolution of the Lorentz factor for par-\nticles moving in the magnetic dipole field. For weak damping\nparameters log( \u001cm!B).\u000010, radiation reaction remains negli-\ngible and the particle motion is almost adiabatic with the three\ncharacteristic periodic motions: gyration around the magnetic\nfield, bouncing between north and south magnetic pole and pre-\nArticle number, page 9 of 18A&A proofs: manuscript no. pousseur\nFig. 14. Particle trajectory in the equatorial plane of a dipole magnetic\nfield and associated to Fig. 13. The inset shows the strong damped mo-\ntion in green and even stronger damping in red where the spiralling is\nnot seen.\nFig. 15. Decrease of the Lorentz factor of particles subject to the mirror\ne\u000bect.\ncession in the azimuthal direction, see blue and orange lines in\nFig. 16. For log( \u001cm!B)=\u00005, the cyclotron motion is rapidly\ndamped and the particle falls onto the star, green trajectory. For\nlog(\u001cm!B)=0, the damping is even faster and the particle\ncrashes onto the stellar surface, following a trajectory similar to\nthe previous case, see red solid line in Fig. 16.\n4. Application to neutron stars\nAfter checking and testing our new algorithm, we are ready to\napply it to realistic extreme cases of rotating magnetized neutron\nstars. The neutron star radius is fixed to R\u0003=12 km. The accu-\nrate configuration of the electromagnetic field is taken from a ro-\ntating magnetic dipole in vacuum and given by Deutsch (1955).\nFig. 16. Particle trajectories in the dipole magnetic field and associated\nto Fig. 15. For small damping parameter, orange and blue lines, the par-\nticle is trapped for a long time in the dipole, whereas for larger damping\nit quickly crashes onto the stellar surface.\n4.1. Relevant parameters without dimension\nAs a typical frequency we choose the stellar angular fre-\nquency!= \n\u0003and consider three populations of neutron stars:\nyoung pulsars with period P\u0003=1 s and surface magnetic field\nstrengths B\u0003=108T, millisecond pulsars with period P\u0003=5 ms\nandB\u0003=105T and magnetars with period P\u0003=10 s and\nB\u0003=1010T. These quantities and their associated normalised\nstrength and damping parameters aB,aEandbfor electrons, pro-\ntons and irons are summarized in table 1. The normalization fre-\nquency is arbitrary but from a microscopic point of view, the\nmost relevant frequencies are related to the electromagnetic ten-\nsor eigenfrequencies. Therefore the low value of bshould not be\nmisinterpreted as a weak feedback of radiation reaction. It is an\nartefact of the chosen typical frequency associated to the stellar\nrotation and which is many orders of magnitude smaller than the\ncyclotron frequency.\nWe distinguish three kind of particles: a first group crashing\nonto the stellar surface, a second group of trapped particles and a\nthird group of escaping particles, all accelerated to high energies.\nParticles are considered trapped when they still have not crashed\nonto the surface or not yet escaped the light cylinder. Particles\nare placed regularly within the light-cylinder, starting at rest or\nwith an initial velocity vector oriented along the magnetic field\nline, directed toward the star or towards infinity, with a Lorentz\nfactor equal to \r0=103or starting at rest. The neutron star obliq-\nuity is set to \u001f=f0\u000e;30\u000e;60\u000e;90\u000e;120\u000e;150\u000e;180\u000eg. We found\nthat the final results are not very sensitive to the initial Lorentz\nfactor because charges are immediately accelerated in the direc-\ntion of the electric field and therefore loose memory about their\ninitial state. Our simulation results are thus summarized for par-\nticles starting at rest only. We simulated a total number of 48\nparticles for each neutron star type and each obliquity, spread\naround three radii r0, right at the surface R\u0003, approximately half-\nway between the surface and the light-cylinder (a geometric av-\nArticle number, page 10 of 18J. Pétri : Particle acceleration and radiation reaction in a strongly magnetized rotating dipole\nNeutron star P\u0003(s) log B\u0003(T) log aB logaE\u0000logb\nmillisecond 0.005 5 13.1 /9.9/9.5 11.8 /8.6/8.2 20.1 /23.4/22.5\nyoung 1 8 18.4 /15.2/14.8 14.8 /11.6/11.2 22.4 /25.7/24.8\nmagnetar 10 10 21.4 /18.2/17.8 16.8 /13.6/13.2 23.4 /26.7/25.8\nTable 1. Typical period and surface magnetic field strength of millisecond pulsars, young pulsars and magnetars. The relevant parameters without\ndimension are given by the strength parameters for the magnetic field aBand for the electric field aEand the damping parameter bfor electrons /\nprotons /iron nuclei.\nNeutron star log \rfull\nmax=log\rpc\nmax\nelectron proton iron\nmillisecond 10.5 /9.3 7.3 /6.0 7.0 /5.7\nyoung 11.2 /7.7 8.0 /4.4 7.7 /4.1\nmagnetar 12.2 /7.7 9.0 /4.4 8.7 /4.1\nTable 2. Maximum Lorentz factor orders of magnitude from conserva-\ntive arguments about neutron star magnetospheres. Values for full po-\ntential drops are given on the left of the \" /\" symbol and for polar cap\npotential drops on the right in logarithmic scale.\nerage) and at the light cylinder, thus r0=fR\u0003;pR\u0003rL;rLg. For\ncomparison we performed simulations with and without radia-\ntion reaction.\n4.2. Orders of magnitude\nBefore presenting the accurate numerical simulations of particle\ntrajectories and their radiation reaction in the vicinity of neu-\ntron stars, we remind the orders of magnitude of the maximum\nLorentz factors expected when charges are accelerated in the\nelectric potential produced by a rotating magnetized perfectly\nconducting star. The most optimistic view adopts the full poten-\ntial drop between the pole and the equator as an estimate of the\naccelerating field thus\n\rfull\nmax\u0019q\n\u0003B\u0003R2\n\u0003\nm c2=R\u0003\nrLR\u0003\nrB(33)\nwhere rB=c=!Bis the non relativistic Larmor radius. If the ac-\ncelerating potential is only available across the polar caps as ex-\npected from nearly force-free magnetosphere models, the maxi-\nmum energy corresponds to\n\rpc\nmax\u0019q\n2\n\u0003B\u0003R3\n\u0003\nm c3= R\u0003\nrL!2R\u0003\nrB\u0019R\u0003\nrL\rfull\nmax (34)\nwhich is a factor R\u0003=rLsmaller than for the former case. Table 2\nsummarizes the maximum Lorentz factors for electrons, protons\nand irons around millisecond pulsars, young pulsars and magne-\ntars. The values reported in this table for \rfull\nmaxare at best upper\nlimits for the vacuum case. Only an accurate numerical integra-\ntion of the equation of motion gives robust results as we now\nshow.\n4.3. Escaping particles\nParticles reaching distances larger than 10 rLare reputed to be\nleaving the neutron star magnetosphere. The run halts when the\nparticle reaches larger distances. Fig. 17 shows the histogram\nof Lorentz factor for electrons in green, protons in red and iron\nnuclei in blue, irrespective of the magnetic field inclination an-\ngle\u001f. The left column corresponds to a motion with radiation\nreaction (RR) whereas the right column to motion without radi-\nation reaction. First, electrons are the most e \u000bectively acceler-\nated particles reaching final Lorentz factors up to \rf\u0018109in theLLR approximation for millisecond pulsars. This is however two\norders of magnitude less than without radiation reaction where\n\rf\u00181011. Second, as expected protons and iron nuclei acquire\nmuch less energy, only about \rf\u0018106for millisecond pulsars,\nwherever LLR is used or not. For young pulsars, electrons also\nreach\rf\u0018109in the LLR regime instead of \rf\u00181011for the\npure Lorentz force. Protons and iron nuclei are much less sub-\nject to radiation reaction, showing no impact on the maximum\nLorentz factor remaining at \rf\u0018104\u0000104:5. For magnetars, ra-\ndiation reaction remains negligible irrespective of the nature of\neach species. Electrons reach energies up to \rf\u0018107:5whereas\nprotons and iron nuclei \rf\u0018103\u0000103:5. Therefore, radiation re-\naction does not significantly perturb the trajectories of particles\nwith lower charge over mass ratio q=m. Contrary to electrons,\nprotons and irons do not su \u000ber from radiation friction.\n4.4. Crashed particles\nCloser to the star, most species quickly crash onto the surface in\na time much shorter than the neutron star spin period. Particles\ncrashing onto the neutron star surface are easily recognized by\nthe fact that their final position lies inside the star. Compared to\nescaping particles, the situation is now reversed, magnetars of-\nfering the highest energetic particles heating the surface and mil-\nlisecond pulsars the lowest energetic particles, see left column of\nFig. 18. This is accounted for by the lower surface magnetic field\nof millisecond pulsars, being three to five orders of magnitude\nlower than young pulsars or magnetars respectively. Neglecting\nradiation reaction , electrons are able to reach Lorentz factors up\nto\rf\u00181011for magnetars but only \rf\u0018108:5for millisecond\npulsars. The radiation reaction impact is strongest for magnetars.\nHowever, protons and irons are not perturbed by radiation re-\naction except sensibly for magnetars. Nevertheless, we observe\nthat with radiation reaction protons remain the most energetic\nparticles with final Lorentz factors about \rf\u0018107:5\u0000108:5irre-\nspective of the neutron star nature, millisecond, young or mag-\nnetar. For electrons the situation is drastically di \u000berent. They\nradiate copiously, decreasing they Lorentz factor by three or-\nders of magnitude comparing to the no radiation reaction case\nin the magnetar environment. The decrease is less pronounced\nfor young or millisecond pulsars but still perceptible.\n4.5. Trapped particles\nBy default we assume that trapped particles are those not crash-\ning onto the neutron star and not escaping to large distances\noutside the light cylinder within the simulation time span corre-\nsponding to several neutron star periods. Fig. 19 summarizes the\ndistribution of Lorentz factors for electrons, protons and irons\nin the LLR approximation and without radiation reaction. Pro-\ntons and irons are still insensitive to radiation reaction except for\nmagnetars. Electrons are much more sensitive to radiation reac-\ntion, decreasing their Lorentz factor by four orders of magnitude\nfor millisecond pulsars, young pulsars and magnetars. Millisec-\nArticle number, page 11 of 18A&A proofs: manuscript no. pousseur\nFig. 17. Histogram of escaped particles for millisecond pulsars on the top row, for young pulsar on the middle row and for magnetars on the bottom\nrow. Electron Lorentz factors are shown in green, proton in red and iron nuclei in blue. The left column includes radiation reaction (RR) whereas\nthe right panel does not.\nond pulsars produces trapped protons and irons with energies\nabout\rf\u0018107whereas young pulsars and magnetars one decade\nmore up to\rf\u0018108, no matter if radiation reaction is included or\nnot. Electrons are trapped with similar Lorentz factor although\nslightly less for millisecond pulsars.4.6. Maximum Lorentz factor\nFor escaping particles, in the wave zone, the gain in energy\nis limited by the spherical nature of the electromagnetic field,\nmeaning decreasing in strength with distance like 1 =r. For a null\nlike electromagnetic field Pétri (2021) showed that this severely\nArticle number, page 12 of 18J. Pétri : Particle acceleration and radiation reaction in a strongly magnetized rotating dipole\nFig. 18. Same as Fig. 17 but for crashed particles.\nlimits the maximum Lorentz factor to values of\n\rmax\u00192 (aBL=\u0019)2=3(35)\naBLbeing the strength parameter as measured at the light cylin-\nder. Radiation reaction remains also negligible in this wave zone.\nTable 3 summarizes the relevant parameters at the light cylinder\nfor the three kinds of neutrons stars. As a rule of thumb, wefound no particle with Lorentz factor exceeding \rf\u0019109:1in\nthe LLR regime. Because the vacuum electromagnetic used in\nour simulations corresponds to the one producing the strongest\nparallel electric field (with respect to the magnetic field), no par-\nticle should be created and moving with Lorentz factor higher\nthan 109:1within the magnetosphere. Eq. (35) is satisfied for non\nnull like electromagnetic waves as those launched by a rotating\nArticle number, page 13 of 18A&A proofs: manuscript no. pousseur\nFig. 19. Same as Fig. 17 but for trapped particles.\nNeutron star log aBL log\rmax\nCrashed Trapped Escaped\nmillisecond 9.3 /6.0/5.7 7.1 /7.1/6.8 6.5 /7.1/6.8 9.1 /6.0/5.6\nyoung 7.6 /4.4/4.1 6.0 /7.8/7.5 7.8 /7.8/7.5 8.0 /4.1/3.7\nmagnetar 7.6 /4.4/4.1 6.4 /8.1/7.7 8.2 /8.1/7.7 7.1 /3.2/2.9\nTable 3. Maximum Lorentz factors \rmaxfor the three kind of particles: electrons /protons /iron nuclei. The value of the strength parameter at the\nlight cylinder is also given.\nArticle number, page 14 of 18J. Pétri : Particle acceleration and radiation reaction in a strongly magnetized rotating dipole\nmagnetic dipole. Instead we found a simple linear relation re-\nlating the strength parameter aBLto the Lorentz factor such that\n\rmax\u0019aBL: (36)\nThis increase in the acceleration e \u000eciency is imputed to the pres-\nence of a still strong radial component of the electric field which\nwas absent in the study of Pétri (2021). The simulations per-\nformed in this section only followed a small number of parti-\ncles due to the stringent computation time required to accurately\nevolve the particle velocity and position. Describing the plasma\nfeedback onto the electromagnetic field would require a much\nlarger number of particles coupled to the evolution of the elec-\ntromagnetic field via Maxwell equations, leading to a particle-\nin-cell code. So far, although PIC codes exist and have been\nadapted to simulate neutron star magnetospheres, none was yet\nable to handle the parameter space explored in the present work.\nTherefore let us contrast our results in light of existing kinetic\ndescriptions of the magnetosphere.\n4.7. Comparison to previous works\nSeveral investigations of particle acceleration and radiation reac-\ntion around neutron stars have been attempted in the literature.\nVery rare are however studies employing realistic field strengths\nfor the neutron star due to severe numerical limitations. Let us\nmention however the pioneer work of Finkbeiner et al. (1989)\nand Finkbeiner et al. (1990) who employed a single test particle\napproach with radiation reaction and found acceleration around\nthe neutron star up to Lorentz factors of about \rf.109for the\nCrab parameters. In a similar manner, at very large distances,\nin the wind zone, Michel & Li (1999) studied particle acceler-\nation without radiation reaction and found asymptotic values of\n\rf.109. The flaw of these studies is that particles evolve in\na prescribed external field without possible feedback due to the\nplasma around the neutron star. A fully kinetic description of the\nplasma and field started only recently using PIC schemes, ear-\nlier simulations having not taken into account radiation reaction.\nUnfortunately, the flaw of this approach is the use of unrealisti-\ncally low field strength. Let us however mention some of these\nworks.\nCerutti et al. (2015) studied acceleration for an axisymmetric\nmagnetosphere without radiation reaction. They got a maximum\nenergy for leptons \rf.103related linearly to the magnetisa-\ntion parameter in the plasma. Thereafter Cerutti et al. (2016)\nincluded the radiation reaction force and got maximum ener-\ngies one order of magnitude less with \rf.102. Dissipation in\nthe striped wind due to magnetic reconnection led Cerutti et al.\n(2020) to the same conclusion. Other PIC simulations performed\nby Kalapotharakos et al. (2018) using similar algorithms with\nradiation reaction found similar results with \rf.103for pairs.\nNevertheless, these authors extrapolated to realistic energies by\nusing rescaling techniques for field strengths, time and space\nscales. How e \u000bective and consistent this rescaling operates is not\nclear as the problem is highly non-linear in a significant radiation\nreaction regime. General relativity does not significantly change\nthese conclusions as shown by Philippov & Spitkovsky (2018)\nwho including frame-dragging e \u000bects and found \rf.500 by ex-\ntending their special relativistic results in Philippov et al. (2015).\nWhen focusing on the near field of a dipole Ferrari &\nTrussoni (1974) found an asymptotic Lorentz factor for electron\nabout 108and slight larger for proton almost 109but for faster\nrotation in a field of an oblique rotating dipole with strength5 · 106T. When radiation reaction remains irrelevant, their re-\nsults agree with those of Kulsrud (1972), demonstrating a lin-\near growth with the field strength parameter. Laue & Thielheim\n(1986) investigates the special case of an orthogonal rotator with\nradiation reaction and for typical neutron star parameters, they\nfound maximum energy for electrons about 109and for protons\nabout 106.\nHadron acceleration has been much less discuss in this con-\ntext but is equally important to understand the origin of ultra-\nhigh energy cosmic rays. To this end Guépin et al. (2020) inves-\ntigated proton and pair acceleration in an aligned neutron star\nmagnetosphere with radiation reaction. They drastically reduced\nthe neutron star radius and the proton over electron mass ratio\nfor computational purposes, therefore they found highest ener-\ngies for pairs only about \rf.700 and for protons about \rf.40.\nThese state of the art results emphasize the di \u000eculty to tend to-\nwards a realistic and self-consistent description of neutron star\nmagnetospheres. The central bottleneck is the particle pusher, re-\nquiring to resolve temporally the gyromotion. This drawback is\ncircumvent by employing an approximation called the radiation\nreaction limit, summarized in section 2. It is therefore important\nto assess quantitatively the accuracy and e \u000eciency of this alter-\nnative approach as done in the next section.\n5. Comparison with the radiation reaction limit\nThe results obtained in this section rely on the numerical inte-\ngration of the LLR equation accounting for realistic parameters\nintroducing a huge gap between the gyro-frequency and the neu-\ntron star rotation period. The question arises then on how to im-\nprove our algorithm or to speed up the computation by several\ndecades. To this end, in a last section we compare the LLR re-\nsults to the radiation reaction limit regime to assert the usefulness\nof the latter.\nIntegrating the exact LLR equations requires to resolve the\ngyro-frequency which is very stringent and impossible to use for\na large sample of particles as required to perform kinetic simu-\nlations such as those done in PIC or Vlasov codes. We therefore\nchecked the accuracy of the much faster radiation reaction limit\napproximation where the particle velocity is expressed in terms\nof the local electromagnetic field, eq. (17). To this aim, we com-\nputed trajectories for electrons and protons in the field of a mil-\nlisecond pulsar for di \u000berent magnetic moment inclination angles\nand di \u000berent initial particle positions. Because by construction\nthe speed in eq. (17) is equal to the speed of light v\u0006=c, in\nthe LLR approach particles are kicked with high initial Lorentz\nfactors\r0=103and a velocity parallel to v\u0006in order to have\ncomparable initial conditions for both sets of runs.\nFig. 20 shows a sample of electron trajectories and demon-\nstrates the reasonable results obtained by this asymptotic regime\nfor a millisecond pulsar. However the precision depends on the\nparticle initial position. For motions starting at the surface, up-\nper row in Fig. 20, some trajectories, in blue, orange and yellow\nare well reproduced by the radiation reaction limit regime. The\naccuracy is less good for the brown and green paths although\nthe general trend is conserved. When starting at larger distances\nfrom the surface, atpR\u0003rLlike in the middle row of Fig. 20, we\nobserve better agreement between both regimes. The best results\nare obtained for particles well away from the surface, starting at\nr=rL, lower row in Fig. 20. All trajectories computes in the ra-\ndiation reaction limit regime overlap with the LLR integration.\nComparison of trajectories for protons are shown in fig. 21.\nHere the agreement is satisfactory within the light cylinder, close\nArticle number, page 15 of 18A&A proofs: manuscript no. pousseur\nFig. 20. A sample of trajectories for electrons obtained with LLR in the\nfield of a millisecond pulsar, in solid lines, and in the radiation reaction\nlimit, marked with dotted symbols. Trajectories are projected along the\nxyplane on the left column and on the xzplane on the right column.\nParticles are launched from the stellar surface in the upper row, at a\ndistancepR\u0003rLin the middle row and at a distance r=rLon the bottom\nrow.\nto the surface, upper row, and at intermediate distances, middle\nrow. For protons starting at the light cylinder radius r=rLthe\nresults are more contrasted, some trajectories being well repro-\nduced, in blue, yellow and orange colours and some being false\nlike the brown and green colour motions, expected to crash on\nthe surface but escaping in the radiation reaction limit regime.\nIrons show trajectories very similar to protons because of the\nnearly identical mass over charge ratio q=m, figures are there-\nfore not shown in this almost identical case.\nThe radiation reaction limit regime is less accurate than the\nexact LLR integration scheme but this is partially compensated\nby the drastic decrease in computational time, lowered by sev-\neral orders of magnitude in this approximation. Expression (17)\ncould certainly be improved by carefully investigating theses\nproblematic cases but we do not pursue this aim in this work.\nWe demonstrated however that the velocity in eq.(17) o \u000bers a\nvaluable compromise between a time consuming full integration\nof the equation of motion in LLR and a artificial and unrealistic\nFig. 21. Same as fig. 20 but for protons.\ndown scaling of the major physical parameters making a neutron\nstar a neutron star.\nA convergence analysis of the radiation reaction limit inte-\ngration scheme is shown in Fig. 22 for the relative error. We\nsimulated a sample of twelve particles starting at di \u000berent lo-\ncations within the magnetosphere and compared their last posi-\ntion to a reference solution. As no exact analytical solutions are\nknown, we use as a reference numerical solution the one with\nthe smallest time step. The integration scheme oscillates between\nfirst and second order in time depending on the initial position\nof the particle as shown by the number r0=rLin the legend. For\nreference the \u0001t2behaviour is shown in blue filled circles. In the\nprevious simulations, we fixed the time step to \u0001t\u001910\u00004thus\nexpecting a precision better that 3 digits in all cases. The discrep-\nancy between Landau-Lifshitz and radiation reaction regime can\ntherefore not be explained by some discretization e \u000bect. We also\nchecked that the initial condition on the velocity does not im-\npact the trajectory in Landau-Lifshitz . The explanation must be\nsearch in the deficiency of the radiation reaction regime to satis-\nfactorily account for all possible trajectories. Indeed, this regime\nassumes a radiative friction force opposite to the 3-velocity vec-\ntor. However, the 3D version of the LLR equation also contains\ncomponents along E^Band along EandBwhen the linear\nterm in velocity is retained. The main discrepancy arises from\nArticle number, page 16 of 18J. Pétri : Particle acceleration and radiation reaction in a strongly magnetized rotating dipole\nFig. 22. Convergence of the radiation reaction limit regime showing\nthe method of integration to be between first and second order in time\ndepending on the initial position of the particle given by the number\nr0=rLin the legend. The \u0001t2decrease in shown in blue filled circles.\nthe neglect of this linear term. In order to proof this argument,\nwe designed a simplified version of the Landau-Lifshitz equation\nby only retaining in the new analytical solution the part of the ra-\ndiation reaction force directed along the velocity (which is valid\nfor ultra-relativistic speeds). The expressions for the 4-velocity\nthen become\nuE=\u0015Bu0\nEcosh(\u0015E\u001c)+˜F u0\nEsinh(\u0015E\u001c)=\u0015Eq\n(\u00152\nE+\u00152\nB)ju0\nEj2+(\u00152\nB\u0000(\u00152\nE+\u00152\nB)ju0\nEj2)e\u00002\u00152\nB\u001c0\u001c\n(37a)\nuB=\u0015Eu0\nBcos(\u0015B\u001c)+˜F u0\nBsin(\u0015B\u001c)=\u0015Bq\n(\u00152\nE+\u00152\nB)ju0\nBj2+(\u00152\nE\u0000(\u00152\nE+\u00152\nB)ju0\nBj2)e2\u00152\nE\u001c0\u001c\n(37b)\nreplacing equation (12). The results are shown in Fig. 23 for the\nexact Landau-Lifshitz equation in solid lines, the approximated\nLandau-Lifshitz equation in dashed lines and the radiation re-\naction regime in dots. We observe some significant di \u000berences\nnotably, in the middle right panel. We stress however that radia-\ntion reaction regime gives accurate results at low computational\ntime expense for the majority of cases.\n6. Conclusions\nStrongly magnetized rotating neutron stars are powerful and e \u000e-\ncient particle accelerators able to accelerate leptons and hadrons\nto Lorentz factors as high as 109for the former and slightly less\nfor the latter. This upper limit remains largely independent on\nthe nature of neutron star: millisecond pulsar, young pulsar or\nmagnetar. We achieved these results by implementing realistic\nparameters in our particle pusher based on the exact solution of\nthe LLR approximation of the equation of motion. Through ex-\ntensive numerical tests, we show that our scheme is second order\nin proper time.\nThe simulation results are accurate and robust but at the ex-\npense of high computational cost because of the need to resolve\nthe gyro-motion which is many decades smaller than the neu-\ntron star spin period. The radiation reaction limit regime o \u000bers\na good compromise between accuracy and computational cost\nbut the simplistic expression used is unable to reproduce all tra-\njectories satisfactorily. Nevertheless, it could be conceivable to\nFig. 23. Comparison between the radiation reaction limit, in dot mark-\ners, the LLR, in thick solid lines and the approximated LLR motion, in\nthin dashed lines.\nimprove this expression by taking into account a finite Lorentz\nfactor and special electromagnetic field configuration when the\nradiation reaction is negligible due to a weak accelerating elec-\ntric field. Nevertheless this extension is left for future work.\nA straightforward implementation of the above pusher into\na PIC code or codes is prevented by the fact that LLR uses the\nproper time as integration parameter. Its conversion into an iner-\ntial observer time is however feasible as shown by Pétri (2020).\nThe next logical step would then be to shift from the test particle\nmotion to a fully kinetic plasma simulation where the particle\ncharge and current densities retroact to the electromagnetic field\nvia Maxwell equations.\nAcknowledgements. I am grateful to the referee for helpful comments and sug-\ngestions that helped to improved this work. This work has been supported by the\nCEFIPRA grant IFC /F5904-B /2018 and ANR-20-CE31-0010.\nReferences\nAbraham, M. 1902, Annalen der Physik, 315, 105\nAbraham, M. 1904, Annalen der Physik, 319, 236\nBoghosian, B. 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Plasma Phys., 86, 825860402, publisher: Cambridge University\nPress\nPétri, J. 2021, Monthly Notices of the Royal Astronomical Society, 503, 2123\nRohrlich, F. 2007, Classical Charged Particles, 3rd edn. (Singapore ; Hacken-\nsack, NJ: World Scientific Pub Co Inc)\nTomczak, I. & Pétri, J. 2020, J. Plasma Phys., 86, 825860401, publisher: Cam-\nbridge University Press\nVranic, M., Martins, J. L., Fonseca, R. A., & Silva, L. O. 2016, Computer Physics\nCommunications, 204, 141\nArticle number, page 18 of 18"    },    {        "title": "1003.5389v1.Undamped_energy_transport_by_collective_surface_plasmon_oscillations_along_metallic_nanosphere_chain.pdf",        "content": "arXiv:1003.5389v1  [cond-mat.mes-hall]  28 Mar 2010PACS No: 73.21.-b, 36.40.Gk, 73.20.Mf, 78.67.Bf\nUndamped energy transport by collective surface plasmon os cillations along metallic\nnanosphere chain\nW. Jacak1, J. Krasnyj1,2, J. Jacak1, A. Chepok2, L. Jacak1, W. Donderowicz1, D. Z. Hu3, and D. M. Schaadt3\n1Institute of Physics, Wroc/suppress law University of Technology,\nWyb. Wyspia´ nskiego 27, 50-370, Wroc/suppress law, Poland;\n2Theor. Phys. Group., International University, Fontanska ya Doroga 33, Odessa, Ukraine;\n3Institute of Applied Physics/DFG-Center for Functional Na nostructures, University of Karlsruhe, Karlsruhe, German y\nThe random-phase-approximation semiclassical scheme for description of plasmon excitations in\nlarge metallic nanospheres (with radius 10–100 nm) is devel oped for a case of presence of dynamical\nelectric field. The spectrum of plasmons in metallic nanosph ere is determined including both surface\nand volume type excitations and their mutual connections. I t is demonstrated that only surface\nplasmons of dipole type can be excited by a homogeneous dynam ical electric field. The Lorentz\nfriction due to irradiation of e-m energy by plasmon oscilla tions is analysed with respect to the\nsphere dimension. The resulting shift of resonance frequen cy due to plasmon damping is compared\nwith experimental data for various sphere radii. Collectiv e of wave-type oscillations of surface\nplasmons in long chains of metallic spheres are described. T he undamped region of propagation of\nplasmon waves along the chain is found in agreement with some previous numerical simulations.\nI. INTRODUCTION\nExperimental and theoretical invesitigations of plasmon excitation s in metallic nanocrystals rapidly grew mainly\ndue to perspectives of possible applications in photovoltaics and micr oelectronics. A significant enhancement of\nabsorption of the incident light in photodiode-systems with active su rface covered with nano-dimension metallic\nparticles (of Au, Ag or Cu) with planar density ∼108/cm2was observed1–7. This is due to a mediating role in light\nenergy transport played by surface plasmon oscillations in metallic na no-compounds on semiconductor surface. These\nfindings are of practical importance towardsenhancement of sola rcell efficiency especially for thin film cell technology.\nHybridized states of the surface plasmons and photons result in pla smon-polaritons8which are of high importance\nfor applications in photonics and microelectronics9,10, in particular for transportation of energy in metallic modified\nstructures in nano-scale11,12.\nSurface plasmons in nanoparticles were widely investigated since the ir classical description by Mie13. Many partic-\nular studies, including numerical modelling of multi-electron clusters, have been carried out14–18. They were develop-\nments of Kohn-Sham attitude in form of LDA (Local Density Approa ch) or TDLDA (Time Dependent LDA)14,15,17 ?\naddressed, however, to small metallic clusters, up to ca. 200 elect rons (as limetted by severe numerical constraints).\nThe random phase approximation (RPA) was formulated19for description of volume plasmons in bulk metals and\nutilised also for confined geometry mainly in numerical or semi-numeric al manner16. Usually, in these analyses the\njelliummodel was assumed for description of positive ion background in met al and dynamics was addressed to elec-\ntron system only16–18, and such an attitude is preferable for clusters of simple metals, inc luding noble metals (also\ntransition and alkali ones).\nIn the present paper we generalise the bulk RPA description19, using semiclassical approach, for a large metallic\nnanosphere (with radius of several tens nm, and with 105− −107electrons) in an all analytical calculus version20.\nThe plasmon oscillations of compresional and traslational type, res ulting in excitations inside the sphere and on its\nsurface, respectively, are analysed and referred to volume and s urface plasmons. Damping effects of plasmons via\nelectron scattering processes and radiation losses are included, t he latter ones, via Lorentz friction force. The shift\nof the resonance frequency of dipole-type surface plasmons (on ly plasmons induced by homogeneous time-dependent\nelectric field), due to damping phenomena, is compared with the expe rimental data for various nanosphere radii.\nCollective surface dipole-type plasmon oscillations in the linear chain of metallic nanospheres are analysed and wave-\ntype plasmon modes are described. A coupling in near field regime betw een oscillating dipoles of surface plasmons\ntogether with retardation effects of energy irradiation lead to a po ssibility of undamped propagation of plasmon waves\nalong the chain in the experimentally realistic region of parameters (o f separation of spheres in the chain and their\nradii). These effect would be of particular significance for by plasmon arranged transport of energy along metallic\nchains for application in nanoelectronics.\nThe paper is organised as follows. In the next section the standard RPA theory in quasiclassicallimit, is generalised\nfor the confined system of spherical shape. The resulting equatio ns for volume and surface plasmons are solved in2\nthe following section (with particularities of calculus shifted to the Ap pendix). The next section contains description\nof the Lorentz friction for surface plasmons oscillations of the dipo le-type. The analysis of the collective wave-type\nsurfaceplasmonoscillationsinthe chainofmetallicnanospheresis pre sentedin the lastsection. Besidesthe theoretical\nmodel the comparison of the characteristic nano-scale plasmon be haviour with available experimental data, including\nown measurements, is presented.\nII. RPA APPROACH TO ELECTRON EXCITATIONS IN METALLIC NANOSP HERE\nA. Derivation of RPA equation for local electron density in a confined spherical geometry\nLet us consider a metallic sphere with a radius alocated in the vacuum, ε= 1, µ= 1 and in the presence of\ndynamical electric field (magnetic field is assumed to be zero). We will c onsider collective electrons in the metallic\nmaterial. The model jellium16–18is assumed in order to account for screening background of positiv e ions in the form\nof static uniformly distributed over the sphere positive charge:\nne(r) =neΘ(a−r), (1)\nwherene=Ne/Vandne|e|is the averagedpositive chargedensity, Nethe number of collectiveelectronsin the sphere,\nV=4πa3\n3the sphere volume, and Θ is the Heaviside step-function. Neglecting the ion dynamics within jelliummodel,\nwhich is adopted in particular for description of simple metals, e.g. nob le, transition and alkali metals, we deal with\nthe Hamiltonian for collective electrons,\nˆHe=Ne/summationdisplay\nj=1/bracketleftBigg\n−¯h2∇2\nj\n2m−e2/integraldisplayne(r0)d3r0\n|rj−r0|+eϕ(R+rj,t)/bracketrightBigg\n+1\n2/summationdisplay\nj/negationslash=j′e2\n|rj−rj′|+∆E, (2)\nwhererjandmare the position (with respect to the dot center) and the mass of t hejthelectron, Ris the position of\nthe metallic sphere center, ∆ Erepresents electrostatic energy contribution from the ion ’jellium’, ϕ(r,t) is the scalar\npotential of the external electric field. The corresponding electr ic fieldE(R+rj,t) =−gradjϕ(R+rj,t). Assuming\nthat space-dependent variation of Eis weak on the scale of the sphere radius athenE(R+rj,t)≃E(R,t), i.e the\nelectric field is homogeneous over the sphere (it holds for |R| ≫a). Thenϕ(R+rj,t)≃ −E(R,t)·R+ϕ1(R+rj,t),\nwhereϕ1(R+rj,t) =−rj·E(R,t). Hence, one can rewrite the Hamiltonian (2) in the form:\nˆHe=ˆH′\ne−eNE(R,t)·R, (3)\nwhere\nˆH′\ne=Ne/summationdisplay\nj=1/bracketleftBigg\n−¯h2∇2\nj\n2m−e2/integraldisplayne(r)d3r0\n|rj−r0|+eϕ(rj,t)/bracketrightBigg\n+1\n2/summationdisplay\nj/negationslash=j′e2\n|rj−rj′|+∆E, (4)\nthe corresponding wave function can be represented as,\nΨ(re,t) = Ψ′(re,t)eieN\n¯h/integraltext\nE(R,t)·Rdt(5)\nwithi¯h∂Ψ′\n∂t=ˆH′\neΨ′,re= (r1,r2,...,rN).\nA local electron density can be written as follows19:\nρ(r,t) =<Ψ(re,t)|/summationdisplay\njδ(r−rj)|Ψ(re,t)>=<Ψ′(re,t)|/summationdisplay\njδ(r−rj)|Ψ′(re,t)>, (6)\nwith the Fourier picture:\n˜ρ(k,t) =/integraldisplay\nρ(r,t)e−ik·rd3r=<Ψ′(re,t)|ˆρ(k)|Ψ′(re,t)>, (7)\nwhere the ’operator’ ˆ ρ(k) =/summationtext\nje−ik·rj.3\nUsing the above notation one can rewrite ˆH′\nein the following form, in analogy to the bulk case21:\nˆH′\ne=Ne/summationtext\nj=1/bracketleftBig\n−¯h2∇2\nj\n2m/bracketrightBig\n−e2\n4π2/integraltext\nd3k˜ne(k)1\nk2/parenleftBig\nˆρ+(k)+ ˆρ(k)/parenrightBig\n+e2\n16π3/integraltext\nd3k˜ϕ1(k,t)/parenleftBig\nˆρ+(k)+ ˆρ(k)/parenrightBig\n+e2\n4π2/integraltext\nd3k1\nk2/bracketleftBig\nˆρ+(k)ˆρ(k)−Ne/bracketrightBig\n+∆E,(8)\nwhere: ˜ne(k) =/integraltext\nd3rne(r)e−ik·r,4π\nk2=/integraltext\nd3r1\nre−ik·r, ˜ϕ1(k) =/integraltext\nd3rϕ1(r,t)e−ik·r,t.\nUtilizing this form of the electron Hamiltonian one can write the secod t ime-derivative of ˆ ρ(k):\nd2ˆρ(k,t)\ndt2=1\n(i¯h)2/bracketleftBig/bracketleftBig\nˆρ(k),ˆH′\ne/bracketrightBig\n,ˆH′\ne/bracketrightBig\n, (9)\nwhich resolves itself into the equation:\nd2δˆρ(k,t)\ndt2=−/summationtext\nje−ik·rj/braceleftBig\n−¯h2\nm2(k·∇j)2+¯h2k2\nm2ik·∇j+¯h2k4\n4m2/bracerightBig\n−e2\nm2π2/integraltextd3q˜ne(k−q)k·q\nq2δˆρ(q)−e\nm8π3/integraltextd3q˜ne(k−q)(k·q)˜ϕ1(q,t)\n−e\nm8π3/integraltext\nd3qδˆρ(k−q)(k·q)˜ϕ1(q,t)−e2\nm2π2/integraltext\nd3qδˆρ(k−q)k·q\nq2δˆρ(q),(10)\nwhereδˆρ(k) = ˆρ(k)−˜ne(k) is the ’operator’ of local electron density fluctuations beyond th e uniform distribution.\nTaking into account that: δ˜ρ(k,t) =<Ψ′(t)|δˆρ(k)|Ψ′(t)>= ˜ρ(k,t)−˜ne(k) we find:\n∂2δ˜ρ(k,t)\n∂t2=<Ψ′|−/summationtext\nje−ik·rj/braceleftBig\n−¯h2\nm2(k·∇j)2+¯h2k2\nm2ik·∇j+¯h2k4\n4m2/bracerightBig\n|Ψ′>\n−e2\nm2π2/integraltext\nd3q˜ne(k−q)k·q\nq2δ˜ρ(q,t)−e\nm8π3/integraltext\nd3q˜ne(k−q)(k·q)˜ϕ1(q,t)\n−e\nm8π3/integraltextd3qδ˜ρ(k−q,t)(k·q)˜ϕ1(q,t)−e2\nm2π2/integraltextd3qk·q\nq2<Ψ′|δˆρ(k−q)δˆρ(q)|Ψ′>,(11)\nOne can simplify the above equation upon the assumption that δρ(r,t) =1\n8π3/integraltext\neik·rδ˜ρ(k,t)d3konly weakly varies on\nthe interatomic scale, and hence three components of the first te rm in right-hand-side of Eq. (11) can be estimated as:\nk2v2\nFδ˜ρ(k),k3vF/kTδ˜ρ(k) andk4v2\nF/k2\nTδ˜ρ(k), respectively, with 1 /kTthe Thomas-Fermi radius19,kT=/radicalBig\n6πnee2\nǫF,\nǫFthe Fermi energy, and vFthe Fermi velocity. Thus the contribution of the second and the th ird components\nof the first term can be neglected in comparison to the first compon ent. Small and thus negligible is also the last\nterm in right-hand-side of Eq. (11), as it involves a product of two δ˜ρ(which we assumed small δ˜ρ/ne<<1).\nThis approach corresponds to random-phase-approximation (RP A) attitude formulated for bulk metal19,21(note that\nδˆρ(0) = 0 and the coherent RPA contribution of interaction is comprise d by the second term in Eq. (11)). The last\nbut one term in Eq. (11) can also be omitted if one confines it to linear t erms with respect to δ˜ρand ˜ϕ1. Next, due\nto spherical symmetry, <Ψ′|/summationtext\nje−ik·rj¯h2\nm2(k·∇j)2|Ψ′>≃2k2\n3m<Ψ′|/summationtext\nje−ik·rj¯h2∇2\nj\n2m|Ψ′>. Performing the inverse\nFourier transform, Eq. (11) attains finally the form:\n∂2δρ(r,t)\n∂,t2=−2\n3m∇2<Ψ′|/summationtext\njδ(r−rj)¯h2∇2\nj\n2m|Ψ′>\n+ω2\np\n4π∇/braceleftBig\nΘ(a−r)∇/integraltext\nd3r11\n|r−r1|δρ(r1,t)/bracerightBig\n+ene\nm∇{Θ(a−r)∇ϕ1(r1,t)}.(12)\nAccording to Thomas-Fermi approximation19the RPA averaged kinetic energy can be represented as follows:\n<Ψ′|−/summationdisplay\njδ(r−rj)¯h2∇2\nj\n2m|Ψ′>≃3\n5(3π2)2/3¯h2\n2mρ5/3(r,t) =3\n5(3π2)2/3¯h2\n2mn5/3\neΘ(a−r)/bracketleftbigg\n1+5\n3δρ(r,t)\nne+.../bracketrightbigg\n.(13)\nTaking then into account the above approximation and that ∇Θ(a−r) =−r\nrδ(a−r) =−r\nrlimǫ→0δ(a+ǫ−r) as\nwell as that ϕ1(R,r,t) =−r·E(R,t), one can rewrite Eq. (12) in the following manner:\n∂2δρ(r)\n∂t2=/bracketleftbig2\n3ǫF\nm∇2δρ(r,t)−ω2\npδρ(r,t)/bracketrightbig\nΘ(a−r)\n−2\n3m∇/braceleftbig/bracketleftbig3\n5ǫFne+ǫFδρ(r,t)/bracketrightbigr\nrδ(a+ǫ−r)/bracerightbig\n−/bracketleftBig\n2\n3ǫF\nmr\nr∇δρ(r,t)+ω2\np\n4πr\nr∇/integraltext\nd3r11\n|r−r1|δρ(r1,t)+ene\nmr\nr·E(R,t)/bracketrightBig\nδ(a+ǫ−r).\n(14)4\nIn the above formula ωpis the bulk plasmon frequency, ω2\np=4πnee2\nm, andδ(a+ǫ−r) = lim ǫ→0δ(a+ǫ−r). The\nsolution of Eq. (14) can be decomposed into two parts related to th e domain:\nδρ(r,t) =/braceleftbigg\nδρ1(r,t), forr<a,\nδρ2(r,t), forr≥a,(r→a+),(15)\ncorrespondingtothevolumeandsurfaceexcitations,respective ly. Thesetwopartsoflocalelectrondensityfluctuations\nsatisfy the equations:\n∂2δρ1(r,t)\n∂t2=2\n3ǫF\nm∇2δρ1(r,t)−ω2\npδρ1(r,t), (16)\nand\n∂2δρ2(r,t)\n∂t2=−2\n3m∇/braceleftbig/bracketleftbig3\n5ǫFne+ǫFδρ2(r,t)/bracketrightbigr\nrδ(a+ǫ−r)/bracerightbig\n−/bracketleftBig\n2\n3ǫF\nmr\nr∇δρ2(r,t)+ω2\np\n4πr\nr∇/integraltext\nd3r11\n|r−r1|(δρ1(r1,t)Θ(a−r1)+δρ2(r1,t)Θ(r1−a))+ene\nmr\nr·E(R,t)/bracketrightBig\nδ(a+ǫ−r).\n(17)\nIt is clear from Eq. (16) that the volume plasmons are independent o f surface plasmons. However, surface plasmons\ncan be excited by volume plasmons due to the last term in Eq. (17), wh ich expresses a coupling between surface and\nvolume plasmons in the metallic nanosphere within RPA semiclassical pict ure. It is in fact a surface tail of volume\ncompressional-type excitations, while surface traslational-type e xctations have no a volume tail.\nIn a dielectric medium in which the metallic sphere can be embedded, the electrons on the surface interact with\nforcesε(dielectric susceptibility constant) times weaker in comparison to ele ctrons inside the sphere. To account for\nit, one substitutes Eqs (16) and (17) with the following ones:\n∂2δρ1(r,t)\n∂t2=2\n3ǫF\nm∇2δρ1(r,t)−ω2\npδρ1(r,t), (18)\nand\n∂2δρ2(r,t)\n∂t2=−2\n3m∇/braceleftbig/bracketleftbig3\n5ǫFne+ǫFδρ2(r,t)/bracketrightbigr\nrδ(a+ǫ−r)/bracerightbig\n−/bracketleftBig\n2\n3ǫF\nmr\nr∇δρ2(r,t)+ω2\np\n4πr\nr∇/integraltextd3r11\n|r−r1|/parenleftbig\nδρ1(r1,t)Θ(a−r1)+1\nεδρ2(r1,t)Θ(r1−a)/parenrightbig\n+ene\nmr\nr·E(R,t)/bracketrightBig\nδ(a+ǫ−r).\n(19)\nLet us also assume that both volume and surface plasmon oscillations are damped with the time ratio τ0which can\nbe phenomenologically accounted for via the additional term, −2\nτ0∂δρ(r,t)\n∂t, to the right-hand-side of above equations.\nThey attain the form:\n∂2δρ1(r,t)\n∂t2+2\nτ0∂δρ(r,t)\n∂t=2\n3ǫF\nm∇2δ˜ρ1(r,t)−ω2\npδρ1(r,t), (20)\nand\n∂2δρ2(r,t)\n∂t2+2\nτ0∂δρ(r,t)\n∂t=−2\n3m∇/braceleftbig/bracketleftbig3\n5ǫFne+ǫFδρ2(r,t)/bracketrightbigr\nrδ(a+ǫ−r)/bracerightbig\n−/bracketleftBig\n2\n3ǫF\nmr\nr∇δ˜ρ2(r,t)+ω2\np\n4πr\nr∇/integraltext\nd3r11\n|r−r1|/parenleftbig\nδρ1(r1,t)Θ(a−r1)+1\nεδρ2(r1,t)Θ(r1−a)/parenrightbig\n+ene\nmr\nr·E(R,t)/bracketrightBig\nδ(a+ǫ−r).\n(21)\nFrom Eqs (20) and (21) it is noticeable that the homogeneous electr ic field does not excite the volume-type plasmon\noscillations but only contributes to surface plasmons.\nB. Solution of RPA equations: volume and surface plasmons fr equencies\nEqs (20) and (21) can be solved upon imposing the boundary and sym metry conditions—cf. Appendix A. Let us\nwrite the both parts of the electron fluctuation in the following mann er:\nδρ1(r,t) =ne[f1(r)+F(r,t)], forr<a,\nδρ2(r,t) =nef2(r)+σ(Ω,t)δ(r+ǫ−a), forr≥a,(r→a+),(22)\nand let us choose the convenient initial conditions F(r,t)|t=0= 0, σ(Ω,t)|t=0= 0, (Ω = ( θ,ψ)—the spherical angles),\nmoreover (1+ f1(r))|r=a=f2(r)|r=a(continuity condition), F(r,t)|r=a= 0,/integraltext\nρ(r,t)d3r=Ne(neutrality condition).5\nWe thus arrive at the explicit form of the solutions of Eqs (20) and (2 1) (as it is described in the Appendix A):\nf1(r) =−kTa+1\n2e−kT(a−r)1−e−2kTr\nkTr, for r<a,\nf2(r) =/bracketleftbig\nkTa−kTa+1\n2/parenleftbig\n1−e−2kTa/parenrightbig/bracketrightbige−kT(r−a)\nkTr, for r≥a,(23)\nwherekT=/radicalBig\n6πnee2\nǫF=/radicalbigg\n3ω2p\nv2\nF. For time-dependent parts of electron fluctuations we find:\nF(r,t) =∞/summationdisplay\nl=1l/summationdisplay\nm=−l∞/summationdisplay\nn=1Almnjl(knlr)Ylm(Ω)sin(ω′\nnlt)e−t/τ0, (24)\nand\nσ(Ω,t) =∞/summationtext\nl=1l/summationtext\nm=−lYlm(Ω)/bracketleftbigBlm\na2sin(ω′\n0lt)e−t/τ0(1−δ1l)+Q1m(t)δ1l/bracketrightbig\n+∞/summationtext\nl=1l/summationtext\nm=−l∞/summationtext\nn=1Almn(l+1)ω2\np\nlω2p−(2l+1)ω2\nnlYlm(Ω)nea/integraltext\n0dr1rl+2\n1\nal+2jl(knlr1)sin(ω′\nnlt)e−t/τ0,(25)\nwherejl(ξ) =/radicalBig\nπ\n2ξIl+1/2(ξ) is the spherical Bessel function, Ylm(Ω) is the spherical function, ωnl=ωp/radicalbigg\n1+x2\nnl\nk2\nTa2\nare the frequencies of electron volume free self-oscillations (volum e plasmon frequencies), xnlare nodes of the Bessel\nfunctionjl(ξ),ω0l=ωp/radicalBig\nl\nε(2l+1)are the frequencies of electron surface free self-oscillations (su rface plasmon frequen-\ncies), andknl=xnl/a;ω′=/radicalBig\nω2−1\nτ2\n0are the shifted frequencies for all modes due to damping. The coeffi cientsBlm\nandAlmnare determined by the initial conditions. As we have assumed that δρ(r,t= 0) = 0, we get Blm= 0 and\nAlmn= 0, except for l= 1 in the former case (of Blm) which corresponds to homogeneous electric field excitation.\nThis is described by the function Q1m(t) in the general solution (25). The function Q1m(t) satisfies the equation:\n∂2Q1m(t)\n∂t2+2\nτ0∂Q1m(t)\n∂t+ω2\n1Q1m(t)\n=/radicalBig\n4π\n3ene\nm/bracketleftbig\nEz(R,t)δm0+√\n2(Ex(R,t)δm1+Ey(R,t)δm−1)/bracketrightbig\n,(26)\nwhereω1=ω01=ωp√\n3ε(it is a dipole-type surface plasmon Mie frequency13). Only this function contributes the\ndynamical response to the homogeneous electric field (for the ass umed initial conditions). From the above it follows\nthus that local electron density (within RPA attitude) has the form :\nρ(r,t) =ρ0(r)+ρ1(r,t), (27)\nwhere the RPA equilibrium electron distribution (correcting the unifo rm distribution ne):\nρ0(r) =/braceleftbigg\nne[1+f1(r)], for r<a,\nnef2(r), for r≥a, r→a+(28)\nand the nonequilibrium, of surface plasmon oscillation type for the ho mogeneous forcing field:\nρ1(r,t) =\n\n0, for r<a,\n1/summationtext\nm=−1Q1m(t)Y1m(Ω)for r≥a, r→a+.(29)\nIn general, F(r,t) (volume plasmons) and σ(Ω,t) (surface plasmons) contribute to plasmon oscillations. However, in\nthe case homogeneous perturbation, only the surface l= 1 mode is excited.\nFor plasmon oscillations given by Eq. (29) one can calculate the corre sponding dipole,\nD(R,t) =e/integraldisplay\nd3rrρ(r,t) =4π\n3eq(R,t)a3, (30)\nwhereQ11(R,t) =/radicalBig\n8π\n3qx(R,t),Q1−1(R,t) =/radicalBig\n8π\n3qy(R,t),Q10(R,t) =/radicalBig\n4π\n3qx(R,t) andq(R,t) satisfies the\nequation (cf. Eq. (26)),\n/bracketleftbigg∂2\n∂t2+2\nτ0∂\n∂t+ω2\n1/bracketrightbigg\nq(R,t) =ene\nmE(R,t). (31)6\nIII. LORENTZ FRICTION FOR NANOSPHERE PLASMONS\nConsidering the nanosphere plasmons induced by the homogeneous electric field, as described in the above para-\ngraph, one can note that these plasmons are themselves a source of the e-m radiation. This radiation takes away\nthe energy of plasmons resulting in their damping, which can be descr ibed as the Lorentz friction22. This e-m wave\nemission causes electron friction which can be described as the addit ional electric ���eld22,\nEL=2\n3εv2∂3D(t)\n∂t3, (32)\nwherev=c√εis the light velocity in the dielectric medium, and D(t) the dipole of the nanosphere. According to Eq.\n(30) we arrive at the following:\nEL=2e\n3εv24π\n3a3∂3q(t)\n∂t3. (33)\nSubstituting it into Eq. (31) we get\n/bracketleftbigg∂2\n∂t2+2\nτ0∂\n∂t+ω2\n1/bracketrightbigg\nq(R,t) =ene\nmE(R,t)+2\n3ω1/parenleftBigω1a\nv/parenrightBig3∂3q(t)\n∂t3. (34)\nIf rewrite the above equation (for E=0) in the form\n/bracketleftbigg∂2\n∂t2+ω2\n1/bracketrightbigg\nq(R,t) =∂\n∂t/bracketleftbigg\n−2\nτ0+2\n3ω1/parenleftBigω1a\nv/parenrightBig3∂2q(t)\n∂t2/bracketrightbigg\n, (35)\nthus the zeroth order approximation (neglecting attenuation) co rresponds to the equation;\n/bracketleftbigg∂2\n∂t2+ω2\n1/bracketrightbigg\nq(R,t) = 0. (36)\nIn order to solve Eq. (35) in the next step of perturbation, in the r ight-hand-side of this equation one can subsitute\n∂2q(t)\n∂t2by−ω2\n1q(t) (acc. to Eq. (36)).\nTherefore, if one assumes the above estimation,∂3q(t)\n∂t3≃ −ω2\n1∂q(t)\n∂t, then one can include the Lorentz friction in a\nrenormalised damping term:\n/bracketleftbigg∂2\n∂t2+2\nτ∂\n∂t+ω2\n1/bracketrightbigg\nq(R,t) =ene\nmE(R,t), (37)\nwhere\n1\nτ=1\nτ0+ω1\n3/parenleftBigω1a\nv/parenrightBig3\n≃vF\n2λB+CvF\n2a+ω1\n3/parenleftBigω1a\nv/parenrightBig3\n, (38)\nwhere we used for1\nτ0≃vF\n2λB+CvF\n2a(λBis the free path in bulk, vFthe Fermi velocity, and C≃1 a constant)23,24\nwhich corresponds to inclusion of plasmon damping due to electron sc attering on other electrons and on nanoparticle\nboundary. The renormalised damping causes the change in the shift of self-frequencies of free surface plasmons,\nω′\n1=/radicalBig\nω2\n1−1\nτ2.\nUsing Eq. (38) one can determine the radius a0corresponding to minimal damping,\na0=√\n3\nωp/parenleftbig\nvFc3√ε/2/parenrightbig1/4. (39)\nFor nanoparticles of gold, silver and copper in air, in water and in a collo idal solution, one can find a0≤10nm\n(cf. Tab. 1), which corresponds to the experimental data26–28. Fora > a 0damping increases due to Lorentz\nfriction (proportional to a3) but fora < a0damping due to electron scattering dominates and causes also damp ing\nenhancement (with lowering a, as∼1\na, cf. Fig. 1),\nwhich agrees with experimental observations23,26.7\nFIG. 1: Effective damping ratio for surface plasmon oscillat ions, Eqs (37), (38), the upper (blue) curve is the sum of both\nterms,∼1\na(red) and ∼a3(green); the minimum corresponds to minimal damping for rad iusa0, Eq. (39), left—for Ag in the\nair, right—for Au in colloidal water solution\nTab. 1.a0—nanosphere radius corresponding to minimal damping\nrefraction rate of the surrounding medium, n0Au,a0[nm] Ag,a0[nm] Cu,a0[nm]\n(air) 1 8.8 8.44 8.46\n(water) 1.4 9.14 9.18 9.20\n(colloidal solution) 2 9.99 10.04 10.04\nSurface plasmon oscillations cause attenuation of the incident e-m r adiation where the maximum of attenuation is\nat the resonant frequency20ω1=/radicalBig\nω2\n1−1\nτ2. This frequency diminishes with rise of a, fora > a0according to Eq.\n(38), which agrees with experimental observations for Au and Ag p resented in Fig. 2, and Tab. 2 (Au) and Tab 3\n(Ag).\nTab. 2. Resonant frequency for e-m wave attenuation in Au nanos pheres\nradius of nanosheres [nm] 10 15 20 25 30 40 50\n¯hω′\n1(experiment) [eV] 2.371 2.362 2.357 2.340 2.316 2.248 2.172\n¯hω′\n1(theory) [eV], n0= 1.43.721 3.720 3.716 2.702 3.666 3.415 2.374\n¯hω′\n1(theory) [eV], n0= 22.604 2.603 2.600 2.590 2.565 2.388 1.656\nTab. 3. Resonant frequency for e-m wave attenuation in Ag nanos pheres\nradius of nanosheres [nm] 10 20 30 40\n¯hω′\n1(experiment) [eV] 3.024 2.911 2.633 2.385\n¯hω′\n1(theory) [eV], n0= 1.43.707 3.702 3.654 3.410\n¯hω′\n1(theory) [eV], n0= 22.595 2.591 2.557 2.384\nIV. PLASMON-MEDIATED ENERGY TRANSFER THROUGH A CHAIN OF MET ALLIC\nNANOSPHERES\nLet us consider a linear chain of metallic nanospheres with radii ain a dielectric medium with dielectric constant ε.\nWe assume that spheres are located along z-axis direction equidistantly with separation of sphere centers d>2a23,25.\nAt timet= 0 we assume the excitation of plasmon oscillation via a Dirac delta ∼δ(t) shape signal of electric field.\nTaking into account the mutual interaction of induced surface plas mons on the spheres via the radiation of dipole\noscillations, we aim to determine the stationary state of the whole infi nite chain. For separation dmuch shorter than\nthe wavelength λof the e-m wave corresponding to surface plasmon self-frequenc y, the dipole type plasmon radiation\ncan be treated within near-field regime, at least for nearest neighb ouring spheres. In the near-field region a<R 0<λ,\nthe radiation of the dipole D(t) is not a planar wave (as for far-field region, R0≫λ) but of only electric field type8\nFIG. 2: Extinction spectra for nanospheres of Au (a) and Ag (b ) in colloidal water solution for various sphere radii\nin retarded form (without magnetic field)22:\nE(R,R0,t) =1\nεR3\n0/bracketleftbigg\n3n/parenleftbigg\nn·D/parenleftbigg\nR,t−R0\nv/parenrightbigg/parenrightbigg\n−D/parenleftbigg\nR,t−R0\nv/parenrightbigg/bracketrightbigg\n, (40)\nR—position of the sphere (center) irradiating e-m energy due to its d ipole surface plasmon oscillations, R0position\nof another sphere (center), with respect to the center of the f ormer one, where the field E(R,R0,t) is given by the\nabove formula, R0<λ,n=R0/R0,v=c/√ε=c/n0.\nWhen both vectors RandR0are along the z-axis (the linear chain) the above equation can be resolved as:\nEα(R,R0,t) =σα\nεR3\n0Dα/parenleftbigg\nR,t−R0\nv/parenrightbigg\n, (41)\nwhereα= (x,y,z),σx=σy=−1 andσz= 2. Assuming that the z-axis origin coincides with the center of one\nsphere in the chain, for the lthsphere located in the point Rl= (0,0,ld), an electric field caused by neighbouring\nspheres, E(Rm,Rml,t), and the Lorentz friction force caused by self-radiation, EL(Rl,t), has to be considered. By\nvirtue of Eq. (31) the equation for surface plasmon oscillation of th elthsphere is\n/bracketleftbigg∂2\n∂t2+2\nτ0∂\n∂t+ω2\n1/bracketrightbigg\nq(Rl,t) =ene\nmm=∞/summationdisplay\nm=−∞, m/negationslash=l, Rml<λE(Rm,Rml,t)+ene\nmEL(Rl,t), (42)9\nprovided that the dipole field of the mthsphere can be treated as homogeneous over the lthsphere and the sum over\nmis confined by the distance of mthsphere from lthsphere not exceeding the near-field range ( ∼λ). In the case of\nthe equidistant chain, Rl=ldandRml=|l−m|d, and using Eqs (41), (30) and 33), one can rewrite Eq. (42) in the\nform:\n/bracketleftbigg∂2\n∂t2+2\nτ0∂\n∂t−2\n3ω1/parenleftBigω1a\nv/parenrightBig3∂3\n∂t3+ω2\n1/bracketrightbigg\nqα(ld,t) =σαa3\nd3m=∞/summationdisplay\nm=−∞, m/negationslash=l,|l−m|d<λqα/parenleftbig\nmd,t−d\nv|l−m|/parenrightbig\n|l−m|3,(43)\nhereα=x,y, which describe the transversal plasmon modes and α=z, which describes the longitudinal one. The\naboveequation coincides with the appropriateone from Refs [23,25 ], if one assumes that4π\n3a3ne=N= 1 and neglects\nthe retardation of the field.\nTaking into account the periodicity of the infinite chain, one can cons ider the solution of the above equation in the\nform\nq(ld,t) = ˜q(k,t)e−ikld. (44)\nThe right-hand-side term in Eq. (43) attains the form\nm=∞/summationtext\nm=−∞, m/negationslash=lqα(md,t−d\nv|l−m|)\n|l−m|3=l−1/summationtext\nm=−∞qα(md,t−d\nv|l−m|)\n|l−m|3+m=∞/summationtext\nm=l+1qα(md,t−d\nv|l−m|)\n|l−m|3\n= 2e−ikld∞/summationtext\nm=1cos(mkd)\nm3˜q(k,t−md/v).\nThus the Eq. (43) can be written as follows:\n/bracketleftbigg∂2\n∂t2+2\nτ0∂\n∂t−2\n3ω1/parenleftBigω1a\nv/parenrightBig3∂3\n∂t3+ω2\n1/bracketrightbigg\n˜qα(k,t) =σαω1a3\nd3∞/summationdisplay\nm=1, md<λcos(mkd)\nm3˜q(k,t−md/v) (45)\nThis equation is linear and therefore we look for the solutions of the s hape: ˜qα(k,t) =˜Qα(k)eiωαt, and\n−ω2\nα+2iωα\nτα(ωα)+ ˜ω2\nα(ωα) = 0, (46)\nwhere\n˜ω2\nα(ωα) =ω1\n1−2σαa3\nd3∞/summationdisplay\nm=1, md<λcos(mkd)\nm3cos/parenleftbiggωαmd\nv/parenrightbigg\n (47)\nand\n1\nτα(ωα)=1\nτ0+ω2\n1a\n3v/parenleftBigωαa\nv/parenrightBig2\n+σαω2\n1a3\nd3∞/summationdisplay\nm=1, md<λcos(mkd)\nm3sin/parenleftbigωαmd\nv/parenrightbig\nωα. (48)\nIf we confine the sum in the Eq. (47) to m= 1 (the nearest neighbour approximation) we get\n˜ω2\nα(ωα)≃ω1/bracketleftbigg\n1−2σαa3\nd3cos(kd)cos/parenleftbiggωαd\nv/parenrightbigg/bracketrightbigg\n(49)\nand from Eq. (48),\n1\nτα(ωα)=1\nτ0+ω2\n1a3\n4vd2/bracketleftBigg/parenleftbiggωαd\nv/parenrightbigg2\n−(kd−π)2+π2\n3/bracketrightBigg\n, forα =x,y (50)\nand\n1\nτz(ωz)=1\nτ0+ω2\n1a3\n2vd2/bracketleftBigg/parenleftbiggωzd\nv/parenrightbigg2\n+(kd−π)2−π2\n3/bracketrightBigg\n, forα =z. (51)10\nIn the derivation of two above formulae the following summation was p erformed29:\n1\nωα∞/summationtext\nm=1cos(mkd)\nm3sin/parenleftbigωαmd\nv/parenrightbig\n=1\n2ωα∞/summationtext\nm=11\nm3[sin(kmd+ωαmd/v)−sin(kmd−ωαmd/v)]\n=d\nv/bracketleftBig\nπ2\n6−π\n2kd+k2d2\n4+ω2\nαd2\n2v2/bracketrightBig\n,\nas the terms in the sum drop quickly to zero then the above formula w ell approximates the sum with limitation\nmd<λ.\nAssuming now ωα=ω′\nα+iω′′\nαthe Eq. (46) gives the dependence of ω′\nαandω′′\nαonk. The general solution of Eq.\n(43) attains the form,\nqα(ld,t) =Ns/summationdisplay\nn=1˜Qα(kn)ei(ω′\nα(kn)t−knld)−ω′′\nα(kn)t, (52)\nwherekn=2πn\nNsd,L=Nsdis assumed length of the chain with Nsspheres, and periodic (of Born-Karman type)\nboundary condition imposed. The components of Eq. (52) describe monochromatic waves with wavelength λn=\n2π\nkn=L\nn, which are analogous to planar waves in crystals, when damping is not big, i.e., when ω′′\nα≪ω′\nα. Provided\nthis inequality one can approximate:\nfor transversal modes ( α=x,y)\n(ω′\nα)2= (˜ωα)2=ω2\n1/bracketleftbigg\n1+2a3\nd3cos(kd)cos(ω′\nαd/v)/bracketrightbigg\n, (53)\nω′′\nα=1\nτα=1\nτ0+ω2\n1a3\n4vd2\n/parenleftBigg\nω′\nαd\nv/parenrightBigg2\n−(kd−π)2+π2\n3\n, (54)\nand for longitudinal mode ( α=z)\n(ω′\nz)2= (˜ωz)2=ω2\n1/bracketleftbigg\n1−4a3\nd3cos(kd)cos(ω′\nzd/v)/bracketrightbigg\n, (55)\nω′′\nz=1\nτz=1\nτ0+ω2\n1a3\n2vd2\n/parenleftBigg\nω′\nαd\nv/parenrightBigg2\n+(kd−π)2−π2\n3\n. (56)\nFrom the Eqs (54) and (56) it follows that ω′′\nαcan change its sign. In the case of ω′′\nα<0 the oscillations are\ndestabilized, which could be avoided by inclusions of some nonlinear ter ms neglected in the expression for the Lorentz\nfriction, whichinmoreaccurateform22includesalsoasmallnonlineartermwithrespectto D, asidefromthetermwith\n∂3D\n∂t3. Including of it will result in damping of too highly rising oscillations leading to stable amplitude of oscillations.\nDue to this stabilisation caused by nonlinear effects, undamped wave modes of dipole oscillations will propagate in\nthe chain in the region of parameters where ω′′\nα≤0 (and with fixed amplitude accommodated by nonlinear term).\nThe condition ω′′\nα=1\nτα= 0, for critical parameters, resolves into:\n/parenleftbiggωαd\nv/parenrightbigg2\n= (kd−π)2−π2\n3−4vd2\nτ0ω2\n1a3, (57)\nforα= (x,y) and forα=z,\n/parenleftbiggωzd\nv/parenrightbigg2\n=−(kd−π)2+π2\n3−2vd2\nτ0ω2\n1a3. (58)\nObtained from the above equations/parenleftbigωαd\nv/parenrightbig\nleads to determination of the dependence of wave vector kwith respect to\nparameters danda, via Eqs (53)-(56). Solution for this equations, found numerically f or the chain of Ag nanospheres,\nis depicted in the Fig. 3.\nThe undamped plasmon waves in the chain appear if d < d maxand havek=π/d,dmax= 98.5(68.8) nm for\ntransversal(longitudinal) modes. For example, for Ag spheres with radiusa= 20nm and separation d= 60 nm, the\nundamped transversal modes appear for 0 ≤kd≤π/4 or 3π/4≤kd≤2πand longitudinal for 3 π/4≤kd≤5π/4.\nLet us underline that the determined undamped plasmon oscillation wa ve modes explain the numerically observed\nsimilar behaviour30.11\nFIG. 3: Wave vector kdversus sphere separation d, at constant d/a= 3 (a—sphere radius) (upper), for a= 20 nm (lower) for\nAg sphere chain, transversal modes—red, longitudinal mode —blue\nV. CONCLUSIONS\nIn the present paper we analysed plasmons in large metallic nanosphe res induced by homogeneous time-dependent\nelectric field. Within all-analytical RPA quasiclassical approach the vo lume and surface plasmons are described and a\nproof that only dipole-type of surface plasmons can be induced by a homogeneous field (while none of volume modes)\nis given. An irradiation of energy by plasmon oscillations is described wit hin the Lorentz friction effect. Its scaling\nwith the nanosphere dimension leads to sphere radius dependent sh ift of resonant frequency, similarly as observed in\nexperiments. The description of surface dipole-type plasmon oscilla tions in single nanospheres is applied to analysis of\ncollective oscillation in linear chain of metallic nanospheres. The wave-t ype collective plasmon oscillations in the chain\nare also considered. The undamped region of wave energy transpo rt through the chain is found for a certain sphere\nseparation in the chain with corresponding appropriate wavelength of plasmon waves. This phenomenon confirms a\nsimilar behaviour observed by numerical simulations30.\nAcknowledgments\nSupported by the Polish KBN Project No: N N202 260734 and the FNP Fellowship Start (W. J.), as well as DFG\ngrant SCHA 1576/1-1 (D. S. and D. Z. Hu)\nAppendix A: Analytical solution of plasmon equations for th e nanosphere\nLet us solve first the Eq. (20), assuming a solution in the form:\nδρ1(r,t) =ne[f1(r)+F(r,t)], forr<a, (A1)12\nEq. (20) resolves thus into:\n∇2f1(r)−k2\nTf1(r) = 0,\n∂2F(r,t)\n∂t2+2\nτ0∂F(r,t)\n∂t=v2\nF\n3∇2F(r,t)−ω2\npF(r,t),(A2)\nThe solution for function f1(r) (nonsingular at r= 0) has thus the form:\nf1(r) =αe−kTa\nkTr/parenleftbig\ne−kTr−ekTr/parenrightbig\n, (A3)\nwhereα—const.,kT=/radicalBig\n6πnee2\nǫF=/radicalbigg\n3ω2p\nv2\nF(kT—inverse Thomas-Fermi radius), ωp=/radicalBig\n4πnee2\nm(bulk plasmon\nfrequency).\nSince we assume F(r,0) = 0, then for function F(r,t) the solution can be taken as,\nF(r,t) =Fω(r)sin(ω′t)e−τ0t(A4)\nwhereω′=/radicalbig\nω2+1/τ2\n0.Fω(r) satisfies the equation (Helmholtz equation):\n∇2Fω(r)+k2Fω(r) = 0, (A5)\nwithk2=ω2−ω2\np\nv2\nF/3. A solution of the above equation, nonsingular at r= 0, is as follows:\nFω(r) =Ajl(kr)Ylm(Ω), (A6)\nwhereA—constant, jl(ξ) =/radicalbig\nπ/(2ξ)Il+1/2(ξ)—the spherical Bessel function [ In(ξ)—the Bessel function of the first\norder],Ylm(Ω)—the spherical function (Ω—the spherical angle). Owing to the boundary condition, F(r,t)|r=a= 0,\none has to demand jl(ka) = 0, which leads to the discrete values of k=knl=xnl/a, (wherexnl, n= 1,2,3..., are\nnodes ofjl), and next to the discretisation of self-frequencies ω:\nω2\nnl=ω2\np/parenleftbigg\n1+x2\nnl\nk2\nTa2/parenrightbigg\n. (A7)\nThe general solution for F(r,t) has thus the form\nF(r,t) =∞/summationdisplay\nl=0l/summationdisplay\nm=−l∞/summationdisplay\nn=1Almnjl(knlr)Ylm(Ω)sin(ω′\nnlt)e−τ0t. (A8)\nA solution of Eq. (21) we represent as:\nδρ2(r,t) =nef2(r)+σ(Ω,t)δ(r+ǫ−a), forr≥a,(r→a+,i.e.ǫ→0). (A9)\nThe neutrality condition,/integraltext\nρ(r,t)d3r=Ne, withδρ2(r,t) =σ(ω,t)δ(a+ǫ−r)+nef2(r),(ǫ→0), can be rewritten\nas follows: −a/integraltext\n0drr2f1(r) =∞/integraltext\nadrr2f2(r),a/integraltext\n0d3rF(r,t) = 0,/integraltext\ndΩσ(Ω,t) = 0. Taking into account also the continuity\ncondition on the spherical particle surface, 1 + f1(a) =f2(a), one can obtain: f2(r) =βe−kT(r−a)/(kTr) and it is\npossible to fit α(cf. Eq. (A3)) and βconstants:α=kTa+1\n2,β=kTa−kTa+1\n2/parenleftbig\n1−e−2kTa/parenrightbig\n—which gives Eqs (23).\nFrom the conditiona/integraltext\n0d3rF(r,t) = 0 and from Eq. (A8) it follows that A00n= 0, (because of/integraltext\ndΩYlm(ω) =\n4πδl0δm0).\nTo remove the Dirac delta functions we integrate both sides of the E q. (21) with respect to the radius (∞/integraltext\n0r2dr...)\nand then we take the limit to the sphere surface, ǫ→0. It results in the following equation for surface plasmons:\n∂2σ(Ω,t)\n∂t2+2\nτ0∂σ(Ω,t)\n∂t=−∞/summationtext\nl=0l/summationtext\nm=−lω2\n0lYlm(Ω)/integraltext\ndΩ1σ(Ω1,t)Y∗\nlm(Ω1)\n+ω2\npne∞/summationtext\nl=0l/summationtext\nm=−l∞/summationtext\nn=1Almnl+1\n2l+1Ylm(Ω)a/integraltext\n0dr1rl+2\n1\nal+2jl(knlr1)sin(ωnlt),\n+ene\nm/radicalbig\n4π/3/bracketleftbig\nEz(R,t)Y10(Ω)+√\n2Ex(R,t)Y11(Ω)+√\n2Ey(R,t)Y1−1(Ω)/bracketrightbig\n,(A10)13\nwhereω2\n0l=ω2\npl\n2l+1. In derivation of the above equation the following formulae were exp loited, (for a<r1):\n∂\n∂a1/radicalbig\na2+r2\n1−2ar1cosγ=∂\n∂a∞/summationdisplay\nl=0al\nrl+1\n1Pl(cosγ) =∞/summationdisplay\nl=0lal−1\nrl+1\n1Pl(cosγ), (A11)\nwherePl(cosγ) is the Legendre polynomial [ Pl(cosγ) =4π\n2l+1l/summationtext\nm=−lYlm(Ω)Y∗\nlm(Ω1)],γis an angle between vectors\na=aˆrandr1, and (fora>r1):\n∂\n∂a1/radicalbig\na2+r2\n1−2ar1cosγ=∂\n∂a∞/summationdisplay\nl=0rl\n1\nal+1Pl(cosγ) =−∞/summationdisplay\nl=0l/summationdisplay\nm=−l4πl+1\n2l+1rl\n1\nal+2Ylm(Ω)Y∗\nlm(Ω1).(A12)\nTaking into account the spherical symmetry, one can assume the s olution of the Eq. (A10) in the form:\nσ(Ω,t) =∞/summationdisplay\nl=0l/summationdisplay\nm=−lqlm(t)Ylm(Ω). (A13)\nFrom the condition/integraltext\nσ(ωt)dΩ = 0 it follows that q00= 0. Taking into account the initial condition σ(ω,0) = 0 we\nget (forl≥1),\nqlm(t) =Blm\na2sin(ω′\n0lt)e−t/τ0(1−δl1)+Q1m(t)δl1\n+∞/summationtext\nn=1Almn(l+1)ω2\np\nlω2p−(2l+1)ω2\nnlnea/integraltext\n0dr1rl+2\n1\nal+2jl(knlr1)sin(ω′\nnlt)e−t/τ0,(A14)\nwhereω′\n0l=/radicalbig\nω2\n0l−1/τ2\n0andQ1m(t) satisfies the equation:\n∂2Q1m(t\n∂t2+2\nτ0∂Q1m(t)\n∂t+ω2\n01Q1m(t) =ene\nm/radicalbig\n4π/3/bracketleftBig\nEz(R,t)δm0+√\n2Ex(R,t)δm1+√\n2Ey(R,t)δm−1/bracketrightBig\n.(A15)\nThusσ(ω,t) attains the form:\nσ(Ω,t) =∞/summationtext\nl=2l/summationtext\nm=−lYlm(Ω)Blm\na2sin(ω′\n0lt)e−t/τ0+1/summationtext\nm=−1Q1m(t)Y1m(Ω)\n+∞/summationtext\nl=1l/summationtext\nm=−l∞/summationtext\nn=1Anlm(l+1)ω2\np\nlω2p−(2l+1)ω2\nnlYlm(Ω)nea/integraltext\n0dr1rl+2\n1\nal+2jl(knlr1)sin(ω′\nnlt)e−t/τ0.(A16)\n1S. Pillai, K. B. Catchpole, T. Trupke, G. Zhang, J, Zhao, and M . A. Green, Appl. Phys. Let., 88, 161102 (2006)\n2M. Westphalen, U. Kreibig, J. Rostalski, H. L¨ uth, and D. Mei ssner, Sol. Energy Mater. Sol. 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B: Lasers Opt. 64, 269 (1997)"    },    {        "title": "1402.2610v2.Radiation_reaction_at_the_level_of_the_action.pdf",        "content": "Prepared for submission to JHEP\nRadiation reaction at the level of the action\nOfek Birnholtz, Shahar Hadar and Barak Kol\nRacah Institute of Physics, Hebrew University, Jerusalem 91904, Israel\nofek.birnholtz@mail.huji.ac.il ,shaharhadar@phys.huji.ac.il ,\nbarak.kol@mail.huji.ac.il\nAbstract: The aim of this paper is to highlight a recently proposed method for the\ntreatment of classical radiative e\u000bects, in particular radiation reaction, via e\u000bective \feld\ntheory methods. We emphasize important features of the method, and in particular the\ndoubling of \felds. We apply the method to two simple systems: a mass{rope system\nand an electromagnetic charge{\feld system. For the mass{rope system in 1+1 dimensions\nwe derive a double-\feld e\u000bective action for the mass which describes a damped harmonic\noscillator. For the EM charge{\feld system, i.e. the system of an accelerating electric charge\nin 3+1 d, we show a reduction to a 1+1 d radial system of an electric dipole source coupled\nto an electric dipole \feld (analogous to the mass coupled to the rope). For this system we\nderive a double-\feld e\u000bective action, and reproduce in an analogous way the leading part\nof the Abraham{Lorentz{Dirac force.arXiv:1402.2610v2  [hep-th]  12 Mar 2014Contents\n1 Introduction 1\n2 Mass{rope system 4\n3 Electromagnetic charge{\feld system 6\nA Gauge-invariant spherical waves for electromagnetism 10\nB Bessel functions 12\n1 Introduction\nAn action formulation for radiation reaction (RR) was presented in [1]. While [1] focused\non the post-Newtonian approximation to the two body problem in Einstein's gravity, it\nstressed the method's generality, and presented detailed calculations also for radiation\nreaction of scalar and electromagnetic (EM) \felds. Following demand, this paper highlights\nthe method by focusing on the well-known electromagnetic Abraham{Lorentz{Dirac (ALD)\nforce [2{7].\nThe method o\u000bers certain advantages over the more standard approach. First, the ac-\ntion formulation allows the application of e\u000ecient tools including the elimination of \felds\nthrough Feynman diagrams, and allows ready formulation and systematic computation of\nanalogous quantities in non-linear theories such as gravity. Secondly, a single e\u000bective ac-\ntion would be seen to encode several ALD-like forces and its form would stress a connection\nbetween the source and target not seen in the usual force expression.\nWe start in this section by surveying the background and reviewing the general method\nand its key ingredients before proceeding to concrete demonstrations in the following sec-\ntions. In section 2 the method is illustrated in a simple set-up { a mass attached to\nan in\fnite rope. Section 3 describes the application of the method to the leading non-\nrelativistic ALD force (also known as the Abraham-Lorentz force), i.e. the force acting\non an accelerating electric charge due to its own EM \feld. The derivation has a strong\nanalogy with the mass{rope system, especially after a reduction to a radial system through\nthe use of spherical waves; many technicalities of the reduction are deferred to Appendix\nA, and may be skipped in a \frst reading.\nBackground . Feynman diagrams and e\u000bective \feld theories were \frst introduced in the\ncontext of quantum \feld theories, but they turn out to be applicable already in the context\nof classical (non-quantum) \feld theories. One such theory is the E\u000bective Field Theory\n(EFT) approach to general relativity (GR) introduced in [8], see also contributions in\n[9, 10].\n{ 1 {Similarly, the development of an action method for dissipative systems was \frst devel-\noped in the early 60's in the context of quantum \feld theory, and is known as the Closed\nTime Path formalism [11]. The essential idea is to formally introduce a doubling of the\n\felds which can account for dissipation. Here too relevant non-quantum problems exist,\nsuch as radiation reaction, and hence it was natural to seek a theory for the non-quantum\nlimit. This was discussed in the context of the EFT approach to GR in [12] and was pro-\nmoted to general classical non-conservative systems in [13]. We also recommend comparing\nthe mass{rope system presented here with the case of coupling a mass to in\fnitely many\noscillators, presented in [13], and with the action formulations of a damped harmonic os-\ncillator in [14]. Essential elements of the theory were reformulated in [1]. This formulation\nwas extended from 4d to general dimensions in [15].\nBrief review of method\nHere we present the tools generally, before demonstrating their use in the following sections.\nField doubling . The theory in [1] was formulated for classical (non-quantum) \feld\ntheories with dissipative e\u000bects, such as radiation reaction. The source for the departure\nfrom the standard theory was identi\fed to be a non-symmetric (or directed) propagator ,\nsuch as the retarded propagator. For such theories the \felds are doubled together with the\nsources, so that the propagator always connects a \feld and its double and thereby assigns\na direction to it. Here the term \feld should be understood to refer also to dynamical\nvariables in mechanics (viewed as a \feld theory in 0+1 d).\nA generic theory may be described by a \feld \u001e=\u001e(x) together with its source \u001a=\u001a(x)\nand the equation of motion for \u001e, 0 =EOM\u001e(x), where the subscript notation readily\ngeneralizes to a multi-\feld set-up. The double \feld action is given by [1]\n^Sh\n\u001e;^\u001e;\u001a;^\u001ai\n:=Z\nddx\u0014\nEOM\u001e(x)^\u001e(x) +Z\nddy\u000eEOM\u001e(y)\n\u000e\u001a(x)^\u001a(x)\u001e(y)\u0015\n; (1.1)\nwhere ^\u001eis a doubled auxiliary \feld, ^ \u001ais the corresponding source, and ddx\u0011dd\u00001xdtis a\nspace-time volume element. The action is best described in terms of functional variations,\nbut in our examples it will amount to simple concrete expressions shown in the following\nsections. The expression is constructed such that \u000e^S=\u000e^\u001e=EOM\u001e, namely that the equa-\ntion of motion with respect to ^\u001ereproduces the original EOM\u001e. The \frst part introduces\nterms of the form \u001e(x)^\u001e(x),\u001a(x)^\u001e(x), while the second part introduces terms of the form\n^\u001a(x)\u001e(x).\nThis formulation is applicable to arbitrary equations of motion, not necessarily associ-\nated with an action. When they canbe derived from an action S=S[\u001e;\u001a], such as in the\ncase of radiation reaction, namely if EOM\u001e=\u000eS\n\u000e\u001efor someS, the double \feld action is\n^Sh\n\u001e;^\u001e;\u001a;^\u001ai\n:=Z\nddx\u0014\u000eS[\u001e;\u001a]\n\u000e\u001e(x)^\u001e(x) +\u000eS[\u001e;\u001a]\n\u000e\u001a(x)^\u001a(x)\u0015\n: (1.2)\nHere ^\u001e;^\u001acan be assigned the following meaning: ^\u001eis the linearized \feld perturbation,\nsourced by reverse (advanced) propagation from a generic source ^ \u001a. They correspond to\n{ 2 {the Keldysh basis of the Closed Time Path formulation and ^Sis a linearization of the CTP\naction with respect to ^\u001e. We remark that this procedure determines the action explicitly,\nand in fact gives a recipe for the function Kmentioned in [13]. It also generalizes the\nspeci\fc \feld doubling method (using complex conjugation) of [14].\nZone separation . The EFT approach is characterized by a hierarchy of scales leading\nto multiple zones (and a matched asymptotic expansion). In particular when the velocities\nof the sources are small with respect to the velocity of outgoing waves, such as in the\nPost-Newtonian approximation to GR, one de\fnes a system zone and a radiation zone.\nElimination . Given ^S, one may proceed to eliminate \felds through Feynman dia-\ngrams. Radiation sources Q[x] are de\fned diagrammatically by\n\u0000Q[x] :=\n (1.3)\nwhere the double heavy line denotes the whole system zone. Their de\fnition involves the\nelimination of the system zone \feld and matching, see [1] eq. (2.49), but this shall mostly\nnot be required in the present paper. The doubled radiation sources are given by\n^Q[x;^x] =Z\ndt\u000eQ[x]\n\u000exi(t)^xi(t): (1.4)\nwhich is the linearized perturbation of the source and as such does not require the compu-\ntation of new diagrams beyond (1.3). Here too this general formula will be seen to reduce\nto simple expressions in the examples.\nNext the radiation reaction e\u000bective action ^SRRmay be computed through elimination\nof the radiation zone, see [1] eq. (2.53). This action describes the e\u000bects of dissipation and\nradiation reaction.\nGauge invariant spherical waves in the radiation zone . When the spatial\ndimensionD> 1 (as in the ALD problem) each zone respects an enhanced symmetry and\ncorresponding \feld variables and gauge should be chosen:\n\u000fIn the system zone time-independence (stationarity) is an approximate symmetry, due\nto the slow velocity assumption, and hence non-Relativistic \felds (and a compatible\ngauge) should be used [10].\n\u000fIn the radiation zone the system appears point-like and hence it was recognized in\n[1] that this zone is spherically symmetric and accordingly gauge invariant spherical\nwaves (see [16]) should be chosen as the \feld variables.\nMatching within action . Subsection 2.3 of [1] demonstrated how the matching\nequations which are a well-known, yet sometime sticky element of the e\u000bective \feld theory\napproach, can be promoted to the level of the action. This is achieved by introducing an\naction coupling between the two zones, whereby the matching equations become ordinary\nequations of motion with respect to novel \feld variables. These are termed two-way mul-\ntipoles, and reside in the overlap region. This is an important ingredient of our method,\nbut is not used directly in this paper.\n{ 3 {k\nxy\nT, λmFigure 1 . The mass{rope system.\n2 Mass{rope system\nAs a \frst example we consider a system shown in \fg. 1 composed of a rope stretched along\nthexaxis with linear mass density \u0015and tension T(hence the velocity of waves along it\nisc:=p\nT=\u0015). Its displacement along the yaxis is given by \u001e(x). The rope is attached\natx= 0 to a point mass mwhich is free to move along the yaxis and is connected to the\norigin through a spring with spring constant k. We assume there are no incoming waves\nfrom in\fnity. Our goal is to \fnd an e\u000bective action describing the evolution of the point\nmass in time without direct reference to the \feld { that is, after the \feld's elimination. The\neliminated \feld carries waves, and therefore energy, away from the mass; we show directly\nhow this elimination gives a generalized (non-conservative) e\u000bective action describing the\ndynamics of the mass, which is that of a damped harmonic oscillator.\nThe action describing the system (mass+rope) is given by\nS=1\n2Z\ndt\u0002\nm_y2\u0000ky2\u0003\n+T\n2Z\ndtdx (@\u001e)2\u0000Z\ndtQ[\u001e(0)\u0000y]; (2.1)\nwhere (@\u001e)2=1\nc2_\u001e2\u0000(@x\u001e)2and _y:=@ty. The action is composed of the mass's action, a\ndynamical rope term, and a coupling term. This coupling enforces the boundary condition\ny\u0011\u001e(x= 0) at the level of the action and introduces the Lagrange multiplier Q(t), where\nthe choice of notation will be explained later.\nWe wish to obtain a long-distance e\u000bective action for the rope, and for that purpose\nwe analyze the equations of motion. Varying the action with respect to \u001eandQwe \fnd\n\u0003\u001e=\u0000Q\nT\u000e(x); (2.2)\ny=\u001e(0); (2.3)\nwhere \u0003:=1\nc2@2\nt\u0000@2\nx. Analysis of (2.2) near x= 0 gives [@x\u001e] =Q=T where [@x\u001e] :=\n(@x\u001e)j0+\u0000(@x\u001e)j0\u0000denotes the jump at the origin. On the other hand the equations of\nmotion and the outgoing wave condition imply\n\u001e(x;t) =(\ny(t\u0000x=c)x>0\ny(t+x=c)x<0(2.4)\n{ 4 {and hence [ @x\u001e] =\u00002 _y=c. Altogether we obtain\nQ\nT= [@x\u001e] =\u00002\nc_y : (2.5)\nAt this point we can substitute (2.5) in the eq. of motion for yand obtain a damped\nharmonic oscillator. Yet, here we wish to demonstrate how such a dissipative system may\nbe described by a (double \feld) action. For that purpose we proceed and \fnd that the\nfollowing action implies the same equations of motion1including (2.5)\nS=1\n2Z\ndt\u0002\nm_y2\u0000ky2\u0003\n+T\n2Z\ndtdx (@\u001e)2+ 2ZZ\ndt_y\u001e(0); (2.6)\nwhereZ:=p\nT\u0015is the rope's impedance. We interpret Sas a long-distance e\u000bective action\nfor\u001e(as it incorporates the solution (2.4) including the asymptotic boundary conditions).\nComparing (2.6) with a standard origin source term Sint=\u0000R\n\u001e(0;t)Q(t)dtjusti\fes the\nnotationQ. In fact,Qis the force exerted on the mass by the rope.\nWe double the \feld \u001eas well as its source yin (2.6) as described in (1.2) and \fnd the\ndouble \feld action\n^S=TZ\ndtdx@\u001e@ ^\u001e+ 2ZZ\ndth\n_y^\u001e(0) + _^y\u001e(0)i\n; (2.7)\nTransforming to frequency domain with the convention \u001e(t) =Rd!\n2\u0019\u001e(!)e\u0000i!t, complex\nconjugate \felds appear such as \u001e\u0003(!) and we obtain the Feynman rules. The directed\npropagator from ^\u001e\u0003to\u001eis\nLx\nx′\n=G!(x0;x) =\u0000cei!\ncjx\u0000x0j\ni!Z; (2.8)\nand the sources for the \felds ^\u001e\u0003; \u001eare (compare (1.3),(1.4))\n\u0000Q!\u0011\n =\u0000i!Zy!;\u0000^Q\u0003\n!\u0011\n =i!Z ^y\u0003\n!: (2.9)\nSimilar Feynman rules hold for \u001e\u0003;^\u001e.\nNow that we have the Feynman rules we can proceed to compute the outgoing radiation\nand the radiation reaction e\u000bective action. Radiation away from the source, for x>0, is\ngiven by\n\u001e!(x) =\nx =cy!ei!x=c; (2.10)\n1In fact there is a slight di\u000berence: the e\u000bective action (2.6) implies _\u001e(0) = _ywhich is not exactly the\nsame as (2.3).\n{ 5 {which in the time domain becomes\n\u001e(t;x) =Zd!\n2\u0019y!e\u0000i!(t\u0000x\nc)=y(t\u0000x\nc); (2.11)\nreproducing (2.4).\nThe radiation reaction e\u000bective action is found by eliminating the \feld \u001e\n^SRR=\n +c:c:=Zd!\n2\u0019^Q\u0003\n!G!(0;0)Q!+c:c:=\n=Zd!\n2\u0019i!Z ^y\u0003\n!y!+c:c:=\u00002ZZ\n^y_ydt : (2.12)\nThus the full generalized e\u000bective action for the mass becomes\n^Stot= ^y\u000e\n\u000ey\u001a1\n2Z\ndt\u0002\nm_y2\u0000ky2\u0003\u001b\n\u00002ZZ\n^y@tydt (2.13)\nand by taking the Euler{Lagrange equation with respect to ^ ywe obtain\nmy=\u0000ky\u00002Z_y ; (2.14)\nwhich is the equation of a damped harmonic oscillator, as expected.\nWe remark that the 1 + 1 mass{rope system can be regarded as a speci\fc case of a\nscalar point charge coupled to a scalar \feld in d=D+ 1 dimensions, and thus falls under\nthe general considerations of [15], for d= 2. Excluding relativistic, retardation and higher-\nmultipole e\u000bects (irrelevant in our 1 spatial dimension), we recover the form given there\nfor the radiation-reaction force (eq. 2.50), with Gq2=Z, and a factor of 2 because the\nrope has two sides.\n3 Electromagnetic charge{\feld system\nThe physical problem of the self-force (radiation-reaction force) on an accelerating electric\ncharge has been treated for over 100 years [2{7] in di\u000berent methods. We wish to show\nhow in its simplest form and to leading order, the Abraham{Lorentz force can be easily\nfound in essentially the same method as in the mass{rope system. We start likewise with\nthe standard Maxwell electromagnetic action2\nSfull=\u00001\n16\u0019Z\nd4xF\u0016\u0017F\u0016\u0017\u0000Z\nd4xA\u0016J\u0016; (3.1)\nwhere the \feld of the rope \u001ehas been replaced by the EM \feld A\u0016(withF\u0016\u0017=@\u0017A\u0016\u0000@\u0016A\u0017)3\nand the mass on a spring is replaced by a point-charge qwith trajectory xp(t) and current\ndensityJ\u0016=qdx\u0016\np\nd\u001c\u000e(x\u0000xp).\n2We use Gaussian units, the speed of light c= 1, and the metric signature is (+ ;\u0000;\u0000;\u0000).\n3As usualF\u0016\u0017encodes the electric and magnetic \felds through Ei=F0i,Bi=\u00001\n2\u000fijkFjk.\n{ 6 {As a 3+1 d problem, this appears more complicated than the mass-rope system. How-\never, in the radiation zone the system appears point-like and the problem becomes spher-\nically symmetric. As dissipation and reaction are related to the waves propagating to\nspatial in\fnity, we can reduce the problem using the spherical symmetry to an e\u000bective\n1+1 dimensional ( r;t) system, which can be treated analogously to the mass{rope system.\nPhysically, the reduction amounts to working with spherical wave variables which are de-\nscribed in appendix A. For the purpose of the leading non-relativistic ALD force it su\u000eces\nto consider the electric dipole ( `= 1) sector. We denote the corresponding \feld variable\nbyA=A(r;t) and the source by \u001a=\u001a(r;t), both de\fned in Appendix A. In this sector\nthe action, reduced to 1+1 dimensions, is given by (A.13). In the time domain it becomes\nS=Z\ndtZ\ndr\u0014\n\u0000r4\n12A\u0001\u0003A\u0000A\u0001\u001a\u0015\n; (3.2)\nwhere now \u0003:=@2\nt\u0000@2\nr\u00004\nr@r. This \feld content is very similar to that of the rope: the\ncoordinate rreplacesx, the \feld A(r) replaces the \feld \u001e(x). As in (2.1) the action has\nboth a kinetic \feld term and a source-coupling term, though their form here di\u000bers. The\ndouble-\feld action is found using (1.2) as in the mass{rope system, and is given by\n^S=Z\ndtZ\ndr\u0014\n\u0000r4\n6^A\u0001\u0003A\u0000\u0010\n^A\u0001\u001a+A\u0001^\u001a\u0011\u0015\n: (3.3)\nThe method thus proceeds similarly to \fnd the Feynman propagator for the \feld and\nthe expressions for the source vertices - non-hatted and hatted. To obtain the propagator\nforAwe transform to the !frequency domain and consider the homogenous part of the\n\feld equation (A.11), which in dimensionless variables x:=!r, becomes\n[@2\nx+4\nx@x+ 1]~b3=2= 0: (3.4)\nIts solutions are the origin-normalized Bessel functions\n~b3=2:= \u0000(5\n2) 23=2B3=2\u000b(x)\nx3=2; (3.5)\nwhereB\u0011fJ;Y;H\u0006gincludes the Bessel functions J;Y and the Hankel functions, H\u0006=\nJ\u0006iY. The origin-normalized Bessel ~j3=2= 1 +O\u0000\nx2\u0001\nis smooth at the origin and\n~h+\n3=2=\u00003\nx2eix\u0000\n1 +O\u00001\nx\u0001\u0001\nis an outgoing wave. More details on these Bessel functions\nare given in Appendix B.\nThus the propagator for spherical waves is\nLr\nr′\n=G(r0;r) =\u00002i!3\n3~j3=2(!r1)~h+\n3=2(!r2) ; (3.6)\nr1:= minfr0;rg; r2:= maxfr0;rg:\n{ 7 {In this sector, the source in the radiation zone is nothing but the electric dipole Q\u0011D.\nIts form is identi\fed through matching (1.3) the full theory with the radiation zone\nAfull(r)=Z\ndr0\u001a(r0)\u0012\u00002i!3\n3~j3=2(!r0)~h+\n3=2(!r)\u0013\n=Q\u00002i!3\n3~h+\n3=2(!r)=Arad(r): (3.7)\nUsing (1.3,1.4,3.4, A.3, A.12) and integration by parts, we read from (3.7) the electric\ndipole source vertex Qat leading non-relativistic order\n=Q=Z\ndr0~j3=2(!r0)\u001a(r0)=\u0000q!\n2Z\nd3x0~j3=2(!r0)n0\u0002\nr02\u000e(x0\u0000xp)\u00030=qxp+\u0001\u0001\u0001\u0011 D\n=^Q=\u000eQ\n\u000exi^xi=q^xp+\u0001\u0001\u0001\u0011 ^D (3.8)\nWe remark that this is merely the static electric dipole moment Dof the source, where the\nellipsis denote relativistic corrections.\nWith the Feynman rules at hand we can proceed to determine the outgoing radia-\ntion and the radiation reaction e\u000bective action. Radiation away from the source is given\ndiagrammatically by (compare 2.10)\nA(r) =\nr =\u0000QG(0;r) =\u00002i!Qei!r\nr2; (3.9)\nwhere we have used (3.6,3.8,B.4). In the time domain, we \fnd\nA(x;t) =2\nr@tQ(t\u0000r): (3.10)\nThe EM radiation reaction e\u000bective action is\n^SEM=\n +c:c: =1\n2Zd!\n2\u0019^Q\u0003G(0;0)Q+c:c:\n=2\n3Z\ndt^Q\u0001@3\ntQ (3.11)\nwhere the propagator was evaluated at r=r0= 0, as in (2.12), and we regulated ~h+(0)!\n~j(0) = 1. In the process of computing ^SEMthe \felds were eliminated and only the particle's\ndipole remains. Note that the third time derivative originates from the !3term in (3.6),\nwhich in turn originates from the behavior of the Bessel function near its origin.\nFor a single charge we substitute (3.8) in (3.11) to obtain\n^SEM=2\n3q2Z\ndt^x\u0001@3\ntx: (3.12)\n{ 8 {This can now be used to \fnd the radiation-reaction (self) force, similarly to (2.13,2.14),\nthrough the Euler-Lagrange equation with respect to ^x\nFRR=2\n3q2...x: (3.13)\nThis of course matches the Abraham{Lorentz result [2{4], and is the leading order term in\nthe fully relativistic result of Dirac [5] (given in similar form in [1], eq. (3.67)).\nIn addition our approach o\u000bers some bene\fts. Rewriting (3.11) with the notation\nD\u0011Qwe have\n^SEM=2\n3Z\ndt^D\u0001@3\ntD (3.14)\nThis expression applies not only to single charge, but also to a system of charges if only\nwe set\nD[xa] :=X\naqaxa (3.15)\nwhere the sum is over all the particles in the system. To get the radiation reaction force\non any speci\fc charge in the system we need only vary the single object ^S\nFi\nRR;a =\u000e^S\n\u000exia(3.16)\nMoreover, the form (3.14) and the Feynman diagram (3.11) reveal that the dipole\nappears in the self-force twice: once in an obvious way as D, the source of the radiation\nand reaction \felds, and a second, less obvious, time as ^Dwhich is the \\target\" coupling\nthrough which the reaction \feld acts back on the charges. In this sense the source and the\ntarget are seen to be connected.\nGoing beyond the non-relativistic limit, an expression for all the relativistic corrections\nwas given in eq. (3.65) of [1]. When expanded, the \frst relativistic correction includes the\nelectric quadrupole term, the magnetic dipole term, and relativistic corrections to the\nelectric dipole. These were shown to con\frm the expansion of Dirac's formula to next-to-\nleading order (eq. (3.68) there).\nConclusions\nInspired by the (quantum) Closed Time Path formalism, we have shown how dissipative\nsystems can be treated with an action principle in classical contexts. The explicit algorithm\nto \fnd the generalized action incorporates \feld doubling, zone separation and spherical\nwaves. They were demonstrated by deriving the generalized (dissipative) action for a\nclassical oscillator attached to a rope (2.12) and for the Abraham{Lorentz{Dirac EM self-\nforce (3.12). While these two problems may seem remote from each other, and involve\ndi\u000berent dimensionality, \felds, and sources, the treatment of their radiation follows a very\nsimilar path.\n{ 9 {Acknowledgments\nWe thank B. Kosyakov for encouragement and many helpful comments and B. Remez for\ncommenting on a draft.\nThis research was supported by the Israel Science Foundation grant no. 812/11 and it\nis part of the Einstein Research Project \"Gravitation and High Energy Physics\", which is\nfunded by the Einstein Foundation Berlin. OB was partly supported by an ERC Advanced\nGrant to T. Piran.\nA Gauge-invariant spherical waves for electromagnetism\nIn this appendix we provide details on spherical electromagnetic waves and the de\fnition\nofA;\u001awhich are used in the main text and represent the \feld and source, respectively, of\nthe electric dipole sector. While it is rather technical in nature, its main purpose is simple:\nreducing the 3+1 d EM action (3.1) - the action of a 4-vector \feld in 4d - to the action\nof a single \feld in 1+1 d, in complete analogy with the mass{rope system. The single\n\feld will be labeled A(r;t), and it will be coupled to the source \u001a(r;t). The reduction\nis performed by \frst singling out the radial coordinate rfrom the spherical coordinates\nand then using the tool of gauge invariant spherical \felds, mentioned in Sec. 1. The\ntechnicalities themselves are best returned to after a full reading.\nTo utilize spherical symmetry, we de\fne the basis of symmetric trace free (STF) mul-\ntipolesXL`=x<k1xk1\u0001\u0001\u0001xk`>=\u0000\nxk1xk1\u0001\u0001\u0001xk`\u0001STF 4, which satisfy\n\u0001\nxL=\u0000`(`+ 1)\nr2xL;Z\nxL`xL0\n`0d\n =4\u0019r2`\n(2`+ 1)!!\u000e``0\u000eL0\n`0\nL`: (A.1)\nBy also Fourier-transforming over time, we express the \felds and sources as\nAt=r=Zd!\n2\u0019X\nLAL!\nt=rxLe\u0000i!t; A \n=Zd!\n2\u0019X\nL\u0000\nAL!\nS@\nxL+AL!\nVxL\n\n\u0001\ne\u0000i!t;\nJt=r=Zd!\n2\u0019X\nLJt=r\nL!xLe\u0000i!t; J\n=Zd!\n2\u0019X\nL\u0000\nJS\nL!@\nxL+JV\nL!x\nL\u0001\ne\u0000i!t;(A.2)\nwhere the four general functions comprising the \feld A\u0016(t;x) have been replaced by\nAt\nL!(r);Ar\nL!(r);AS\nL!(r);AV\nL!(r), which are each 1-dimensional. The leading order expres-\nsion inv=c\u001c1 (3.13), i.e. the Abraham{Lorentz force, requires only the dipole ( `= 1)\nelectric components of the \feld and source5, i.e. only the 1-dimensional vector \felds6\nAt(r);Ar(r);AS(r) and source vectors Jt(r);Jr(r);JS(r), where\nJt(r) =3!!\n4\u0019r2Z\n\u001a!(x)xd\n =3\n4\u0019r2ZZ\ndtei!t\u001a(x;t)xd\n;\nJr(r) =3\n4\u0019r2Z\u0010\n~Jw(~ r)\u0001^r\u0011\nxd\n =3\n4\u0019r2ZZ\ndtei!t\u0010\n~J(~ r;t)\u0001^r\u0011\nxd\n; (A.3)\n4For completeness we mention that there are also divergence-less vector multipoles xL\n\n= (~ r\u0002~rxL)\n,\nbut they do not contribute at leading order at v=c; see [1].\n5Higher multipole orders (of both the scalar (electric) and vector (magnetic) \felds) correspond to\nrelativistic corrections of higher order in v=c, and must be included to reproduce the full ALD force [1].\n6We henceforth drop the index !, it will be implied. Note A\u0000!=A\u0003\n!;J\u0000!=J\u0003\n!sinceA\u0016;J\u0016are real.\n{ 10 {andJS(r) is replaced using the equation of current conservation\n0 =D\u0016J\u0016=i!Jt+ (@r+3\nr)Jr\u00002JS: (A.4)\nSubstituting (A.2) into Maxwell's action (3.1) and using (A.1) we obtain (at leading order)\nS=1\n6Zd!\n2\u0019Z\ndrr4(\f\f\f\fi!Ar\u00001\nr(rAt)0\f\f\f\f2\n+2\nr2ji!AS\u0000Atj2\n\u00002\nr2\f\f\f\f1\nr(AS)0\u0000Ar\f\f\f\f2\n\u00004\u0019\u0002\nAr\u0001Jr\u0003+At\u0001Jt\u0003+ 2AS\u0001JS\u0003+c:c:\u0003)\n;(A.5)\nwhere0:=d\ndr. We notice that Aris an auxiliary \feld, which means its derivative A0\nrdoes\nnot appear in (A.5). Hence, its EOM is algebraic and is readily solved,\nAr=\u00001\n!2\u00002\nr2\u0014i!\nr(rAt)0+2\nr3(rAS)0\u00004\u0019Jr\u0015\n: (A.6)\nSubstituting this algebraic solution for the \feld back into the action and using (A.4) for the\nsources, we see that the 1-dimensional action depends in fact on a single gauge invariant\n1-dimensional \feld ~Awith a single source ~\u001a:\nS=1\n6Zd!\n2\u0019Z\ndrr4\"\n1\n1\u0000\u0003\f\f\f\f1\nr(r~A)0\f\f\f\f2\n+cs\nr2\f\f\f~A\f\f\f2\n+ 4\u0019(~A\u0001~\u001a\u0003+c:c:)#\n; (A.7)\nwhere\n~A:=At\u0000i!AS;~\u001a:=\u0000Jt+i\n!r3\u0012\nr3\u0003\n\u0003\u00001Jr\u00130\n; (A.8)\nand \u0003 :=!2r2\n2. For the single charge qwith trajectory xp(t), the second summand on the\nright hand side of ~\u001a(A.8) is sub-leading\u0000\n\u0018(!r)2\u0001\nand can be dropped, leaving\n~\u001a=\u0000Jt=\u00003q\n4\u0019r2Z\nd\nx\u000e(x\u0000xp): (A.9)\nIn order to bring the action (A.7) closer to canonical form we will de\fne a new \feld A,\nwhich is essentially the momentum conjugate to ( r~A), via the transformation\n\u0012r2A\n3\u0013\n:=@L\n@(r~A\u0003)0=r2(r~A)0\n3(1\u0000\u0003);\u0012r2A\n3\u00130\n:=@L\n@(~A\u0003)=2\n3(r~A) +4\u0019r3~\u001a\n3;(A.10)\nand by di\u000berentiating once more w.r.t rand combining these two equations, we \fnd\n0 =r4\n6\u0012\n!2+@2\nr+4\nr@r\u0013\nA\u0000\u001a; (A.11)\nwhere\n\u001a=2\u0019r2(r3~\u001a)0\n3=\u0000qr\n2Z\nd\nx\u0002\nr2\u000e(x\u0000xp)\u00030; (A.12)\nas the new \feld's equation and source term.\n{ 11 {Altogether in this sector the 1-dimensional action is given by\nS=Zd!\n2\u0019Z\ndr\u0014r4\n6A\u0003\u0001\u0012\n!2+@2\nr+4\nr@r\u0013\nA\u0000(A\u0003\u0001\u001a+A\u0001\u001a\u0003)\u0015\n: (A.13)\nThis 1+1 d action is (to leading order) the EM analog of the mass{rope action, and serves\nas the starting point for \feld doubling and solution ((3.2) and on).\nB Bessel functions\nEquation (3.4) is a special case of\n\u0014\n@2\nx+2\u000b+ 1\nx@x+ 1\u0015\n~b\u000b(x) = 0; (B.1)\nfor order\u000b= 3=2. As described in [15] (see eq. B.6), it matches the case of dipole ( `= 1)\nwaves ind= 4 spacetime, as \u000b=`+d\u00003\n2. The solutions are given by the origin-normalized\nBessel functions\n~b\u000b:= \u0000(\u000b+ 1) 2\u000bB\u000b(x)\nx\u000b; (B.2)\nwhereB\u0011fJ;Y;H\u0006gincludes Bessel's functions of the \frst and second kind J;Y and\nHankel's functions, H\u0006=J\u0006iY. With this de\fnition, ~j\u000bbehaves smoothly in the\nvicinity of the origin x= 0,\n~j\u000b(x) =1X\np=0(\u0000)p(2\u000b)!!\n(2p)!!(2p+ 2\u000b)!!x2p= 1 +O\u0000\nx2\u0001\n; (B.3)\nand the Hankel functions' asymptotic form for x!1 is given by\n~h\u0006\n\u000b(x)\u0018(\u0007i)\u000b+1=22\u000b+1=2\u0000(\u000b+ 1)p\u0019e\u0006ix\nx\u000b+1=2: (B.4)\nFor further details see Appendix A.2 of [1] or Appendix B.2 of [15]. Note the conventions\nfor the subscripts of ~jare slightly di\u000berent; we adopt those of [15].\nReferences\n[1] O. Birnholtz, S. Hadar and B. 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Tiglio, \\Radiation reaction and gravitational waves in\nthe e\u000bective \feld theory approach,\" Phys. Rev. D 79, 124027 (2009) [arXiv:0903.1122\n[gr-qc]]. C. R. Galley and A. K. Leibovich, \\Radiation reaction at 3.5 post-Newtonian order\nin e\u000bective \feld theory,\" Phys. Rev. D 86, 044029 (2012) [arXiv:1205.3842 [gr-qc]].\n[13] C. R. Galley, \\The classical mechanics of non-conservative systems,\" Phys. Rev. Lett. 110,\n174301 (2013) [arXiv:1210.2745 [gr-qc]].\n[14] B. P. Kosyakov, \\Introduction to the classical theory of particles and \felds,\" pp 88-90,\nBerlin, Germany: Springer (2007).\n[15] O. Birnholtz and S. Hadar, \\An action for reaction in general dimension,\" Phys. Rev. D 89,\n045003 (2014) [arXiv:1311.3196 [hep-th]].\n[16] V. Asnin and B. Kol, \\Dynamical versus auxiliary \felds in gravitational waves around a\nblack hole,\" Class. Quant. Grav. 24, 4915 (2007) [hep-th/0703283].\n{ 13 {"    },    {        "title": "0908.0675v3.Transplanckian_bremsstrahlung_and_black_hole_production.pdf",        "content": "arXiv:0908.0675v3  [hep-ph]  16 Dec 2010LAPTH-1344/09\nCCTP-2010-18\nTransplanckian bremsstrahlung and black hole production\nDmitry V. Gal’tsov,1,2,∗Georgios Kofinas,3,†Pavel Spirin,1,4,‡and Theodore N. Tomaras3,§\n1Department of Theoretical Physics, Moscow State Universit y, 119899, Moscow, Russia\n2Laboratoire de Physique Th´ eorique LAPTH (CNRS, Universit ´ e de Savoie),\nB.P.110, F-74941 Annecy-le-Vieux cedex, France\n3Department of Physics and Institute of Theoretical and Comp utational Physics, University of Crete, 71003 Heraklion, G reece\n4Bogolubov Laboratory of Theoretical Physics, JINR, Joliot -Curie 6, Dubna, Russia\nClassical gravitational bremsstrahlung in particle colli sions at transplanckian energies is studied in\nM4×Td. The radiation efficiency ǫ≡Erad/Einitialis computed in terms of the Schwarzschild radius\nrS(√s), the impact parameter band the Lorentz factor γcmand found to be ǫ=Cd(rS/b)3d+3γ2d+1\ncm,\nlarger than previous estimates by many powers of γcm≫1. This means that in the ultrarelativistic\ncase radiation loss becomes significant for b≫rS, so radiation damping must be taken into account\nin estimates of black hole production at transplanckian ene rgies. The result is reliable for impact\nparameters in the overlap of γνrS< b < b c, ν= 1/2(d+ 1),andb > λ C, withbcmarking (for\nd/negationslash= 0) the loss of the notion of classical trajectories and λC≡/planckover2pi1/mcthe Compton length of the\nscattered particles.\nBlack hole (BH) production in LHC, predicted [1] in models with TeV-sc ale gravity and large extra dimensions\n[2–4] about ten years ago, has been the subject of intense theor etical study and numerical simulations (for a review\nsee [5]). The prediction is based on the assumption that for impact pa rameters of the order of the horizon radius\ncorresponding to the CM collision energy 2 E=√s\nrS=1√π/bracketleftBigg\n8Γ/parenleftbigd+3\n2/parenrightbig\nd+2/bracketrightBigg1\nd+1/parenleftbiggGD√s\nc4/parenrightbigg1\nd+1\n(1)\nan event horizon should form due to the non-linear nature of gravit y. TheD= 4 +ddimensional gravitational\nconstant is GD=/planckover2pi1d+1/(Md+2\n∗cd−1), withM∗theD-dimensional Planck mass. It is related to the four-dimensional\nPlanck mass MPl≡M4viaM2\nPl=Md+2\n∗V, V= (2πR)d,whereRis the large compactification radius.\nThis classical, essentially, picture of BH formation is justified for tra nsplanckian energies s≫G−2/(d+2)\nD =M2\n∗.\nIndeed [6], in this case the D-dimensional Planck length l∗=/parenleftbig\n/planckover2pi1GD/c3/parenrightbig1/(d+2)=/planckover2pi1/M∗cand the de Broglie length of\nthe collision λB=/planckover2pi1c/√ssatisfy the classicality conditionλB≪l∗≪rS.Furthermore, gravity is believed to be the\ndominant force in the transplanckian region. Thus, for BH masses la rge compared to M∗, the use of classical Einstein\ntheory is well justified. Moreover, it seems that formation of BHs in four dimensions is predicted by string theory\n[7]. Thus, in spite of the fact that there are issues which require fur ther study [8], a consensus has been reached that\nthe prediction of BHs in ultra-high energy collisions is robust and is sum marized in the widely accepted four-stage\nprocess of formation and evaporation of BHs in colliders [1, 9], namely (i) formation of a closed trapped surface (CTS)\nin the collision of shock waves modeling the head-on particle collision, (ii) the balding phase, during which the BH\nemits gravitational waves and relaxes to the Myers-Perry BH, (iii) H awking evaporation and superradiance phase in\n∗Electronic address: galtsov@physics.msu.ru\n†Electronic address: gkofin@phys.uoa.gr\n‡Electronic address: salotop@list.ru\n§Electronic address: tomaras@physics.uoc.gr2\nwhich the experimental signatures are supposed to be produced, and (iv) the quantum gravity stage, where more\nfundamental theory like superstrings is important. This scenario w as implemented in computer codes [10] to simulate\nthe BH events in LHC, where they are expected to be produced at a rate of several per second, and in ultra high\nenergy cosmic rays.\nHere we focus on stage (i). Replacing the field of an ultrarelativistic p article by that of a black hole in the infinite\nmomentum frame strictly speaking is only valid in the linearized level, while the associated non-linear phenomena for\nparticles may be quite different from those in colliding waves. A typical non-linear effect for particles is gravitational\nbremsstrahlung, which is an important inelastic process in transplan ckian collisions. Apart from [11], where gravita-\ntionalbremsstrahlungofsoft photonswasstudied inthe context ofstringtheory, the existingestimatesofgravitational\nradiation either refer to phase (ii), or are based on the assumption of an already existing BH (e.g. radiation from par-\nticles falling into the BH [12]), on results of linearized theory relevant only to the case of non-gravitational scattering\n[8], on weakly relativistic numerical simulations [13] or again on collisions o f waves in 4D [14]. For a related discussion\nsee also [15]. However, a detailed study of gravitational bremsstra hlung in the transplanckian regime in the ADD\nscenario was missing. The purpose of this note is to present the res ults of such a study (a more detailed account will\nfollow [16]). The emitted energy is found to be larger than earlier estim ates (see e.g. [6]) by a dimension dependent\npower of the Lorentz factor of the collision. According to this resu lt, multidimensional gravitational radiation loss in\ntransplanckian collisions becomes significant already for impact para meters much larger than the gravitational radius\nof the black hole. This means that radiation reaction [17] becomes e ssential and the picture of colliding waves, which\nmodels particle motion with constant speed, may not be adequate.\nThe standard ADD model assumes that empty space-time is the pro duct of the four-dimensional Minkowski space\nM4(the brane) and a d-dimensional torus Tdand treats gravity in the linear approximation gMN=ηMN+κDhMN\n(ηMNhas mostly minuses). To calculate gravitational bremsstrahlung cla ssically one has to extend the ADD setup\nbeyond the linearized level. For this, one expands the metric furthe r ashMN→hMN+δhMNand adds the cubic\ninteraction terms to the Fierz-Pauli lagrangian. Equivalently, as in t he standard theory of gravitational radiation in\nfour dimensions, one may expand the D-dimensional Einstein tensor up to quadratic terms:\nGMN=−κD\n2✷(ψMN+δψMN)−1\n2κ2\nDSMN, (2)\nwhere the last term plays the role of a gravitational stress-tenso r, whose form is dimension independent. Here\nh≡ηMNhMN,ψMN≡hMN−ηMNh/2 is the trace-reversed metric perturbation, ✷≡ηMN∂M∂Nis theD-\ndimensional d’Alembertian and the coupling constant κDis defined by κ2\nD≡16πGD. Symmetrization and alternation\nover indices is understood with 1 /2, while raising/lowering of indices is meant with the flat metric.\nSinceD-dimensional Einstein equations imply the D-dimensional Bianchi identities, which in turn imply the D-\ndimensional geodesic equations for the particles, going beyond the linearized theory might contradict the assumption\nof matter confinement on the brane. However, it turns out that it is enough to assume that to zeroth order in κD\nparticles move on the brane. Then, the corresponding zeroth ord er stress-tensor also lies on the brane\nTMN(xP) =ηµ\nMην\nNTµν(x)δd(y). (3)\nIts first orderperturbation δTMN, due to the linearizedgravitationalinteraction, is alsoconfined on t he brane. Indeed,\nto first order the wave equation in the flat harmonic gauge ∂MψMN= 0 is\n✷ψMN=−κDTMN. (4)\nThe source term is constructed neglecting gravity, hence it is flat- space divergenceless too. Since particles move freely\nin this order, this equation describes non-radiative Lorentz-cont racted gravitational potentials. Radiation appears in\nsecond order in κDand is described by the field δψMNsatisfying (again in the flat harmonic gauge) the equation\n✷δψMN=−κDτMN, τMN=δTMN+SMN, (5)3\nwhereSMNis quadratic in the first order gravitational potentials. The right ha nd side of (5) is divergenceless by\nvirtue of the expanded Einstein equations. Therefore, the effect ive source of radiation is the sum of the perturbation\nδTMNof the matter tensor, caused by the first order gravitational int eraction, and the gravitational stress tensor\nSMN, constructed from the first order metric perturbations. This te nsor is not confined on the brane, but extends\nto the bulk. Note that δTMNandSMNare not separately gauge invariant. Thus, beyond the linearized lev el the\neffective source of radiation is not any more confined on the brane.\nDenote the D-dimensional coordinates as xM= (xµ, yk), wherexµ, µ= 0,1,2,3,lie on the brane and yk, k=\n1,...,dlabel the points of Td. Imposing periodicity conditions hMN(x,yk+2πR) =hMN(x,yk) we obtain an infinite\nnumber of four-dimensional massive modes\nhMN(xP) =1√\nV/summationdisplay\nn∈Zdhn\nMN(x)einkyk/R(6)\nwith masses m2=q2\nk,whereqk=nk/Ris the quantized momentum on the torus, and which can be regroupe d into\nspin 2,1,0massive fields [4, 18]. But, being interested in the total radiationin all modes, it is more convenientto think\nabout the metric perturbation as a D-dimensional massless field with discrete momenta in the extra dimens ions. The\nfour-dimensional fields hn\nMN(xµ) are further expanded into Fourier integrals defined by Ψ( x)=/integraltext\nΨ(q)e−iq·xd4q/(2π)4.\nConsider the small-angle collision of two point masses m, m′. Assuming that to zeroth order in κDtheir world-lines\nlie on the brane xM=zM(τ), x′M=z′M(τ′), one verifies that, as expected, they remain on the brane even a fter the\ngravitational interaction is switched on. Therefore, zM=zµδM\nµ,\nzµ(τ) =bµ+pµ\nmτ+δzµ, z′µ(τ) =p′µ\nm′τ+δz′µ, (7)\nwherepµandp′µare momentum parameters. Although the true initial momenta Pµ=mlim\nτ→−∞˙zµ(τ) andP′µ=\nm′lim\nτ→−∞˙z′µ(τ) differ from pµ,p′µ, they still satisfy s= (P+P′)2= (p+p′)2[19]. It is convenient to work in the\nrest frame of one of the particles, m′, choosing the coordinate axes on the brane so that p′µ=m′(1,0,0,0), pµ=\nmγ(1,0,0,v), γ= 1/√\n1−v2. Also, with no loss of generality one may set bµ= (0,b,0,0) andb′µ= 0, so that b\nis the impact parameter. Finally, one may think of brane localized vect ors asD-dimensional vectors with zero bulk\ncomponents, e.g. pM= (pµ,0,...,0).\nIn the linearized theory the superposition principle implies that both t he zeroth and the first order energy-\nmomentum tensor is the sum of the contributions of the two masses , i.e.TMN=m\nTMN+m′\nTMNandδTMN=m\nδTMN\n+m′\nδTMN. The stress term SMN, on the other hand, represents a collective contribution, which ca nnot be attributed\nto any one of the particles. Solving the wave equation (5), one finds that the radiation amplitude consists of three\nterms, which correspondto radiationfrom massesand from their g ravitationalstresses. This is similar to the structure\nof the Born amplitude in quantum theory [20], which in the case without extra dimensions and in the low frequency\nlimit is known to coincide with the classical result. However, like in elastic scattering [19] in ADD ( d/ne}ationslash= 0), the classical\ntreatment is essentially non-perturbative in the quantum sense.\nThe spectral-angular distribution of the emitted energy in gravitat ional bremsstrahlung during the collision is given\nby\ndErad\ndωdΩ=GDω2\n2π2V/summationdisplay\nn∈Zd/summationdisplay\npol|τMN(k)εMN\npol|2, (8)\nwhere\nτMN(k) =m\nδTMN(k)+m′\nδTMN(k)+SMN(k) (9)\nis the Fourier transform of the right hand side of (5). The first two terms have only the brane components M,N=\n0,1,2,3, while the third is truly multidimensional. Introduce next the massive four-dimensional wave-vector kµ≡4\n(√\nω2+M2, ω˜n),with M2=κ2, κi=ni/R. The three-dimensional unit vector ˜nwill be parameterized by the\nspherical angles θ, ϕ, in the usual way. Alternatively, one can think of the radiation wave vector as the D-dimensional\nnullvectorkM= (kµ, κi). Thepolarizationtensorsofthe emittedradiationarechosenacc ordingtothe D-dimensional\npicture. They are D(D��3)/2 symmetric transverse traceless tensors orthogonal to kMand to each other. To\nconstruct them start with D−4 unit space-like mutually orthogonal vectors eM\na,a= 3,4,...,D−2, orthogonal also\ntopM, p′M, kMandbM. Their contractions withm\nδTMN(k),m′\nδTMN(k) andSMN(k) will vanish. Take in addition\nthe two vectors\neM\n1=˜N−1/bracketleftbigg\n(k·p)p′M−(k·p′)pM+/parenleftBig\np·p′−m′2k·p\nk·p′/parenrightBig\nkM/bracketrightbigg\neM\n2=˜N−1ǫMM1M2M...M D−1pM1p′\nM2kM3e3M4...eD−2MD−1, (10)\n(with˜N2=−[p(k·p′)−p′(k·p)]2) satisfying e2·p=e2·p′=e1·p′= 0.Using those, one builds two chiral graviton\npolarizations, which have direct four-dimensional analogs:\nεMN\n±=eM\n±eN\n±, eM\n±= (eM\n1±ieM\n2)/√\n2, (11)\nand a third one which also give non-zero contribution in our case,\nεMN\n3=N/parenleftBig/summationdisplay\naeM\naeN\na−(D−4)e(M\n+eN)\n−/parenrightBig\n, (12)\nwhereNis a normalization factor. The remaining polarization tensors contain at least one vector from the set {eM\na}\nand they give zero being contracted with τMN.\nThe computation ofthe bremsstrahlungradiationproceedsas follo ws. One finds the retardedfields generatedby the\nunperturbed particle trajectories and substitutes to the partic le geodesic equations to get the first order corrections\nto the particles’ motion. These are used to build the perturbations of the particle energy-momentum tensors, whose\nFourier transforms are the first two terms in the radiation source (9). The result is expressed in terms of Macdonald\n(modified Bessel) functions with argument wn= [w2\n0+m2b2]1/2, w0=k·pb/(mγv) and reads\nm\nδTµν(k) =−mm′κ2\nD\n2πγv3Veik·b/summationdisplay\nn∈Zd/bracketleftBigg\niˆK1(wn)\n(w0)2σµν+2p(µp′\nν)\nmm′K0(wn)γ+pµpν\nm2K0(wn)/parenleftbigg\nγw′\n0\nw0−1/parenrightbigg/bracketrightBigg\n,(13)\nwherew′\n0≡k·p′b/(m′γv),σµν= [pµpνk·b−2k·pp(µbν)]/m2andˆKλ(w)≡wλKλ(w). The second termm′\nδTµν(k)\nis obtained from (13) by the substitution ( m,m′,pµ,p′\nµ)→(m′,m,p′\nµ,pµ) and eik·b→1.\nNext, one projects over polarizations. Only the first three contr ibute in the chosen gauge, and all of them have zero\ncontractions with the m′term in (9). So, the total amplitude receives two contributions: (a ) from the moving mass\n(13), and (b) from the stress term in (9), whose significance will be discussed shortly.\nThe next step is to sum over the interaction modes. Assuming that a large number of modes is excited, one can\nreplace summation over niby integration:\n/summationdisplay\nnfn≈VdΩd−1\n(2π)d/integraldisplay\nf(q)qd−1dq, (14)\nwhereqi=ni/R, where we have taken into account that the summand depends only onn2\ni. Performing integration\nwe get [16]:\n/summationdisplay\nn∈ZdˆKλ(wn)≈/parenleftbig\n2πR2/b2/parenrightbigd/2ˆKλ+d/2(w0). (15)\nSummation over modes in the amplitude is the classical counterpart o f integration over transverse momenta of virtual\ngravitons, which leads to tree-level divergences in ADD [4]. However , classically the result is finite like in elastic5\nscattering [19]. The right hand side of 15 is precisely the same as we co uld obtain considering bremsstrahlung in\nan uncompactified D-dimensional Minkowski space. Namely, the par t of the radiation amplitude corresponding to\nthe fast particle is expressed in terms of the Macdonald functions o f the argument w0. Properties of the Macdonald\nfunctions imply that the corresponding radiation amplitude is concen trated in a narrow cone θ <1/γwith a high\nfrequency classical cutoff at ωc∼2γ2/b. This happens also in electromagnetic bremsstrahlung in flat space.\nHowever, in the case of gravitational interaction at hand the spec tral-angular distribution is very different, exactly\nbecause of the special role of SMN. Already in 4d gravity it was shown that the contributions of (13) an dSµνcancel\nin the above spectral-angular region [20]. Physically, this is due to th e fact that the radiation, emitted by particles\nfollowing ultrarelativistic time-like geodesics, follows null geodesics, w hich are close to the former and give an effective\nformation length of radiation in a given direction γtimes larger than in flat space [21]. It is not surprising that\nthe situation is similar in M4×Tdin the kinematical regime under discussion. The contribution from th e stresses\nSMN, which is the classical counterpart of the amplitude involving the thr ee-graviton vertex, is rather complicated.\nNevertheless, it can be shown [16] that like for d= 0 the leading term in the SMN-amplitude for γ≫1 exactly\ncancels (13). The next to leading term, integrated over the trans verse virtual momenta, is also expressed in terms of\nMacdonald functions, with argument w′\n0=ωb/γ, which does not depend on angles. Therefore, the radiation does n ot\nexhibit sharp anisotropy and the frequency cutoff (from the cond itionw′\n0∼1) isω/lessorequalslantωcr=γ/b.\nMore thorough analysis shows that only light emission modes give the le ading contribution of the total emitted\nenergy in the ultrarelativistic collision. For them the projection of th e total amplitude over polarizations ε±leads to\n(the third polarization ε3gives a similar contribution):\nτ±=κ2\nDmm′(γξ)−1\n4(2π)d/2+1bd/bracketleftbigg\n(cos2ϕ∓2i)cosθ′ˆKd/2(w′\n0)+sinθ′\nw′\n0/parenleftBig\n(icos3ϕ∓sin2ϕ)ˆKd/2+1(w′\n0)−sin2ϕ(isinϕ±1)ˆKd/2(w′\n0)/parenrightBig/bracketrightbigg\n,\nwhereξ= 1−vcosθandθ′is the radiation polar angle in the rest frame of m(sinθ′= sinθ/γξ).\nFinally, upon integration of (8) over Ω and ωone obtains the radiated energy\nErad=˜CDm2m′2κ6\nD\nb3d+3γd+3(16)\nwith a known dimension-dependent coefficient. Qualitatively the depe ndence on bandγcan be understood as\nfollows. The averaged over emission angles amplitude-squared is /an}bracketle{t|τ±|2/an}bracketri}ht ∼b−2d, andErad∼ /an}bracketle{t|τ±|2/an}bracketri}htω3\ncrNeff, where\nNeff∼(Rγ/b)dis the effective number of emitted light modes [16].\nThe expression (16) was obtained in the rest frame of the particle m′. To pass to the CM frame, we calculate the\nrelative energy loss (radiation efficiency) ǫ≡Erad/E, and expresses the result in terms of the Lorentz factor in the\nCM frame via (for m=m′)γ2\ncm= (1+γ)/2:\nǫ=Cd/parenleftBigrS\nb/parenrightBig3(d+1)\nγ2d+1\ncm. (17)\nThe two new features of (17) are: (a) the large factor γ2d+1\ncmdue to the large number of light KK\nmodes involved both in the gravitational force and in the radiation, a nd (b) the growing with dcoefficient:\nd123 4 5 6\nCd7.5711016802.6×1044.1×1056×106\nrS3.451.881.46 1.29 1.21 1.17\nbc1967.903.15 2.11 1.72 1.53\nThe Tablehas rSandbcin TeV−1evaluatedfor M∗≃1TeVand√s≃14TeV. The classicaldescriptionofsmallangle\nultrarelativistic scattering is strictly speaking valid for impact param eter in the region γνrS≪b≪bc, ν= 1/2(d+1),\nwherebc≡π−1/2/bracketleftbig\nΓ(d/2)GDs//planckover2pi1c5/bracketrightbig1/d∼rS(rS/λB)1/disthescalebeyondwhich(for d/ne}ationslash= 0)theclassicalnotionofpar-\nticle trajectory is lost [6]. Another restriction comes from the quan tum bound on the radiation frequency /planckover2pi1ωcr<mγ,\nwhich is equivalent to b>λC≡/planckover2pi1/(mc). Ford/ne}ationslash= 0 the two conditions overlap provided λC<bc. A rough estimate6\nofǫcan be obtained in the case γνrS≪λC≪bcby settingb=λCin (17), which becomes ǫ=Bd(sm/M3\n∗)d+2\n(Bd= 7.4,0.6,0.36,0.45,0.81,2.1,7.0, ford= 0,1,...,6, respectively). Thus, a simple condition for strong radiation\ndamping is\nsm/greaterorsimilarM3\n∗, (18)\nwhich may well hold for heavy point particles with LHC energies. For ex ample, in collisions of particles with m∼\nO(100GeV) and energy√s∼ O(10TeV) all conditions for extreme gravitational bremsstrahlung are sat isfied for\nd= 2.\nTo conclude, a classical computation was presented of the gravita tional bremsstrahlung radiation in massive point-\nparticle collisions with transplanckian energies in 4+d dimensions. The p resence of the powers of γfactors in our\nformula (17), which, incidentally, agrees with the one obtained for d= 0 in [22], implies enhanced and even extreme\ngravitational bremsstrahlung and strong radiation damping in tran splanckian collisions. However, even though it is\ntempting and physically reasonable to do so, one cannot strictly spe aking apply (17) to the case of proton-proton\ncollisions in LHC. Protons are not point-particles, while their constitu ents are too light and lie outside the region of\nvalidity of our approximation. For the same reason formula (17) can not be applied to massless particle collisions,\nwhich seem to require a different treatment [11]. Nevertheless, (a) one may have to include the reaction force [17] in\nthe study of processes such as BH production, which might even ex clude the formation of a CTS. Note that there\nare indications that gravitational collapse of an oscillating macrosco pic string does not take place, once gravitational\nradiation is taken into account [23]. Also, (b) bremsstrahlung, is a s trong process leading to missing energy signatures\nin transplanckian collisions, which may further constrain the ADD par ameters.\nWork supported in part by the EU grants INTERREG IIIA (Greece- Cyprus) and FP7-REGPOT-2008-1-\nCreteHEPCosmo-228644. DG and PS are grateful to the ITCP of t he U. of Crete for its hospitality in the early\nstages of this work. Their work was supported by the RFBR projec t 08-02-01398-a. 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Iengo, hep-th/0702087."    },    {        "title": "2304.09633v1.Hamiltonian_dynamics_on_the_symplectic_extended_phase_space_for_autonomous_and_non_autonomous_systems.pdf",        "content": "arXiv:2304.09633v1  [math-ph]  19 Apr 2023Hamiltonian dynamics on the symplectic extended phase\nspace for autonomous and non-autonomous systems\nJ¨ urgen Struckmeier\nGesellschaft f¨ ur Schwerionenforschung (GSI), Planckstr asse 1, 64291 Darmstadt, Germany\nE-mail:j.struckmeier@gsi.de\nAbstract. We will present a consistent description of Hamiltonian dyn amics on the\n“symplectic extended phase space” that is analogous to that of a time-in dependent Hamiltonian\nsystem on the conventional symplectic phase space. The exte nded Hamiltonian H1and the\npertaining extended symplectic structure that establish t he proper canonical extension of a\nconventional Hamiltonian Hwill be derived from a generalized formulation of Hamilton’ s\nvariational principle. The extended canonical transforma tion theory then naturally permits\ntransformations that also map the time scales of original an d destination system, while\npreserving the extended Hamiltonian H1, and hence the form of the canonical equations\nderived from H1. The Lorentz transformation, as well as time scaling transf ormations in\ncelestial mechanics, will be shown to represent particular canonical transformations in the\nsymplectic extended phase space. Furthermore, the general ized canonical transformation\napproach allows to directly map explicitly time-dependent Hamiltonians into time-independent\nones. An “extended” generating function that defines transf ormations of this kind will be\npresented for the time-dependent damped harmonic oscillat or and for a general class of\nexplicitly time-dependent potentials. In the appendix, we will reestablish the proper form\nof the extended Hamiltonian H1by means of a Legendre transformation of the extended\nLagrangian L1.\nPACS numbers: 45.20.-d, 05.45.-a, 45.50.Jf\nPublished in: J. Phys. A: Math. Gen. 38(2005) 1257–1278\n1. Introduction\nThe modern description of time-in dependent Hamiltonian systems on symplectic manifolds\nis well established (e.g. Abraham and Marsden 1978, Arnold 1 989, Jos´ e and Saletan 1998,\nMarsden and Ratiu 1999, Frankel 2001). If the Hamiltonian His explicitly time dependent,\nthe carrier manifold of the Hamiltonian is of odd dimension, and the symplectic description is\nno longer appropriate. However, many features of the symple ctic description can be extended\nto the “presymplectic description” on an odd-dimensional c ontact manifold (Abraham and\nMarsden 1978, Marsden and Ratiu 1999, Frankel 2001). For ins tance, the theory of canonical\ntransformations, hence mappings within a symplectic manif old that preserve its symplectic\nstructure, can indeed be generalized on a presymplectic geo metry. Nevertheless, the canonical\ntransformation theory within the presymplectic context su ffers from the restriction that the\ntransformation must preserve time (Abraham and Marsden 197 8, p 384). This means that\nboth the original and the destination system are always corr elated at the same instant of\ntheir respective time scales. Mappings, such as the Lorentz transformation, that necessitateHamiltonian dynamics on T∗Q1 2\na time shift between original and destination systems thus e scape a description in terms of\na canonical transformation within the presymplectic forma lism. Furthermore, regularization\ntransformations in celestial mechanics dating back to L. Eu ler (Siegel and Moser 1971), as\nwell as transformations of non-linear, explicitly time-de pendent Hamiltonian systems into\ntime-independent systems (Struckmeier and Riedel 2001) ar e well-known to require non-\ntrivial mappings of the time scales.\nVarious approaches were made to describe these transformat ions within the context\nof a generalized canonical transformation theory (Lanczos 1949, Synge 1960, Szebehely\n1967, Kuwabara 1984, Asorey et al 1983, Cari˜ nena et al 1987, Cari˜ nena et al 1988). The\nunderlying idea is to develop a generalized Hamiltonian for malism on a “symplectic extended\nphase space” in analogy to the symplectic description on the conventional phase space of an\nautonomous Hamiltonian system ( ∂H/∂t= 0). Specifically, in the extended formalism, the\ntimetis treated as an ordinary canonical function t(s)≡qn+1(s)of a new superordinated\nsystem evolution parameter, s. Its canonically conjugate function pn+1(s)will constitute\nthe additional coordinate that renders the carrier manifol d even-dimensional — and hence\neligible for a symplectic description. The dynamics of the g iven system is then determined\nby an extended Hamiltonian H1with∂H1/∂s= 0. As a consequence, all properties of\nHamiltonian systems on symplectic manifolds can then be sim ilarly reformulated within the\nextended Hamiltonian formalism. For example, time-depend ent symmetries of explicitly\ntime-dependent Hamiltonian systems can be treated like usu al symmetries of autonomous\nHamiltonian systems. Moreover, it is possible to define cano nical transformations within\nthe symplectic extended phase space that are more general th an those within the lower-\ndimensional presymplectic description. This formalism wi ll then naturally permit generating\nfunctions of extended canonical transformations that also define a non-trivial mapping of\nthe time-scales of the original and the destination systems . Of course, an analogy with the\nconventional canonical transformation theory will requir e the extended Hamiltonian H1to be\npreserved under extended canonical transformations, and h ence the form of the transformed\ncanonical equations derived from H1.\nFollowing the pioneering works of C. Lanczos (Lanczos 1949, p 189) and J. L. Synge\n(Synge 1960, p 143), the extended Hamiltonian HLSis commonly defined as the energy\nsurfaceHLS=H−e= 0, withedenoting the value of the conventional Hamiltonian H(e.g.\nStiefel and Scheifele 1971, Thirring 1977, Asorey et al 1983, Kuwabara 1984, Lichtenberg\nand Lieberman 1992, Stump 1998, Tsiganov 2000, Struckmeier and Riedel 2002a). Yet, the\nso-defined extended Hamiltonian HLSfails to meet the requirement of being preserved under\nextended canonical transformations that define a non-trivi al time mapping t(s)/mapsto→t′(s). As\na consequence, the Hamiltonian HLSdoes not preserve the form of the canonical equations\nunder non-trivial time transformations, but satisfies only the weaker condition of preserving\nthe canonical form of the Hamilton-Jacobi equations (Synge 1960, Tsiganov 2000). With\na missing canonical transformation rule for HLS, the extended canonical transformation\nformalism based on HLSisincomplete .\nIn order to consistently construct an extended Hamiltonian on the symplectic extended\nphase space, other approaches (e.g. Gotay 1982, Cani˜ nena et al 1987, Cani˜ nena et al 1988)\npursued the idea of a “coisotropic embedding” of the presymp lectic geometry of a time-\ndependentHinto the geometry of the symplectic extended phase space as t he carrier manifold\nof the extended Hamiltonian. This geometric reasoning led t o a rather general form of an\nextended Hamiltonian HCthat is not necessarily physical. Specifically, the propose d extended\nHamiltonian HC=f(H−e), withfan arbitrary function of the canonical variables and time\n(Cani˜ nena et al 1987, Cani˜ nena et al 1988), is not compatible with Hamilton’s variational\nprinciple and does not yield an analogous description of the system’s dynamics.Hamiltonian dynamics on T∗Q1 3\nSo, despite the fact that the idea of a generalized formulati on of Hamiltonian dynamics\non an extended symplectic manifold is long-established, we can state that a consistent\nformulation that is analogous to the conventional symplect ic description has not yet been\nworked out. With this article, we aim to provide a consistent symplectic theory of the\nextended phase space that is based on a canonically invarian t extended Hamiltonian function\nH1. The derivation of the extended Hamiltonian H1from a generalized formulation of\nHamilton’s principle will, therefore, be the starting poin t of our analysis in Section 2.1.\nIt will turn out that our extended Hamiltonian H1=k(H−e), withk= dt/dscoincides\nwith the Hamiltonian of the well-known Poincar´ e time trans formation (Siegel and Moser\n1971, p 35), also referred to as the symplectic time rescalin g method in the realm of\nmolecular dynamics. In conjunction with the extended sympl ectic2-form, we will show\nin Section 2.2 that it is exactly this extended Hamiltonian H1that permits a description of\nHamiltonian dynamics on the symplectic extended phase spac e that is completely analogous\nto the conventional symplectic description of time-indepe ndent Hamiltonian systems on the\nnon-extended, conventional phase space.\nIn Section 2.3, we will derive a consistent formalism of exte nded canonical transfor-\nmations within the symplectic extended phase space that pre serve the extended Hamiltonian\nH1and, therefore, the form of the canonical equations. In this description, the evolution\nparametersserves as the independent variable that is common to both the original and the\ndestination system. In this respect, the new evolution para metersplays exactly the role of\nthe timetin the conventional canonical transformation theory. The g eneralized formulation\nof Hamilton’s variational principle thus establishes the b asis for the definition of extended\ngenerating functions forfinite canonical transformations that make it possible to relate b oth\nsystems at different instants of their respective time scales, t(s)andt′(s). We will furthermore\nformulate the extended version of Liouville’s theorem that applies to the symplectic extended\nphase space. In addition, the restrictions will be worked ou t that are to be imposed on the\nfunctional structure of extended generating functions in o rder for the transformed time t′(s)\nto sustain the meaning of t(s)as a common parameter for all canonical variables p′\niandq′i.\nAs a first example of an extended canonical transformation, w e will show in Section 3.1\nthat the Lorentz transformation can be formulated as a parti cular canonical transformation in\nthe extended phase space. Since its generating function doe s not explicitly depend on s, we\nwill encounter the interesting result that the extended Ham iltonianH1is Lorentz-invariant.\nIn celestial mechanics, regularization transformations o f many-body systems are known\nto require a replacement of the physical time tby a “fictitious” time t′. We will show in\nSection 3.2 that these transformations can be formulated as finite canonical transformations in\nthe symplectic extended phase space that preserve the exten ded Hamiltonian H1. We thereby\nintegrate the useful and well-established regularization techniques of celestial mechanics into\nthe framework of a now consistent generalized formulation o f the canonical transformation\ntheory.\nIn Section 3.3, an extended generating function will be pres ented that defines a canonical\nmapping of a general class of explicitly time-dependent Ham iltonian systems into time-\nindependent ones. Demanding the transformed system to be auto nomous then determines\nthe time correlation of both systems. For the simple but impo rtant case of a time-dependent\ndamped harmonic oscillator, the extended canonical transf ormation yields the well known\ninvariant given by Leach (Leach 1978). For the general class of non-linear and explicitly\ntime-dependent Hamiltonian systems treated in Section 3.4 , we will show that the canonical\ntransformation establishes a linear mapping of the system’s global quantities energy e(t)and\nsecond moments q(t)p(t)andq2(t)into their respective initial values e0,q0p0andq2\n0.\nIn Appendix A, the concept of a parameterization of time will be reviewed for Lagrange’sHamiltonian dynamics on T∗Q1 4\nformulation of dynamics. By means of a Legendre transformat ion of the extended Lagrangian\nL1, we will reestablish our extended Hamiltonian H1of Section 2.1. We thereby confirm that\nit is precisely this extended Hamiltonian that is compatibl e with the extended formulation\nof Hamilton’s variational principle, and hence with a consi stent formulation of Hamiltonian\ndynamics on the symplectic extended phase space.\n2. Hamiltonian formalism in the symplectic extended phase s pace\n2.1. Hamilton’s variational principle, extended Hamilton ian\nWe consider an explicitly time-dependent Hamiltonian Hthat is defined on a finite-\ndimensional contact manifold T∗Q×Rwith its closed, generally degenerate contact 2-form\nωH=ω−dH∧dt. Herein,ωstands for a symplectic, i.e. closed, non-degenerate and\nantisymmetric 2-form that renders the manifold T∗Qsymplectic. On an exact symplectic\nmanifold, there exists a 1-formλwith exterior derivative dλ=ω. Hamilton’s variational\nprinciple states that the actual system trajectory C0⊂T∗Q×Ris the critical point of a path\nmapS:C→RwithdS(C0) = 0 . The map Sis defined as the integral along a path\nC⊂T∗Q×Rover the contact 1-formλ−Hdt, hence\nS(C) =/integraldisplay\nC(λ−Hdt),dS(C0) = 0. (1)\nIn canonical coordinates, we have λ=pdq, with(q,p)∈T∗Qthe pair ofn-component\nvectors and covectors. The actual variation of Cis performed on the presymplectic manifold\nT∗Q×R. As the basis for a symplectic description, we reformulate E quation (1) by\ntreating the time t=t(s)≡qn+1(s)as an ordinary canonical variable that now depends,\nlike all other canonical variables, on a newly introduced su perordinated system evolution\nparameters. To this end, we first introduce formally the extended configu ration manifold as\nthe product manifold Q1:=Q×R, whose elements, in coordinate representation, comprise\nthe vectors q1≡(q,t)∈Q1. The extended Hamiltonian H1is then to be defined as a\ndifferentiable function on the cotangent bundle T∗Q1as the carrier manifold. Following\nthe usual nomenclature of T∗Qas the “phase space”, we refer to the symplectic manifold\nT∗Q1as the “symplectic extended phase space”. With respect to a c anonical basis of a chart\nU1⊂T∗Q1andp1≡(p,pn+1)∈T∗\nq1Q1, the extended Hamiltonian H1(q1,p1)thus maps\nall pairs of (n+1) -component vectors and covectors (q1,p1)∈U1intoR. Of course, T∗Q1\nembodies a cotangent bundle, hence a symplectic manifold, i f and only if the symplectic\nstructureωonT∗Qcan be “extended” to a symplectic, i.e. non-degenerate stru ctureΩon\nT∗Q1. This is achieved by a proper choice of pn+1.\nWithS1:˜C→Rdenoting a mapping of paths ˜C⊂T∗Q1intoR, Hamilton’s\nvariational principle of Equation (1) can be written equiva lently in terms of the extended\n1-formΛ =λ+pn+1dtas the integral\nS1(˜C) =/integraldisplay\n˜C(Λ−H1ds),dS1(˜C0) = 0, (2)\nwith\nH1ds= (H+pn+1)dt. (3)\nIn canonical coordinates, the extended 1-form is given by Λ =p1dq1. The variation of the\naction integral (2) is now to be performed by varying ˜C⊂T∗Q1. We will see later that the\ncritical path ˜C0is compatible with the path C0following from (1).Hamiltonian dynamics on T∗Q1 5\nIn the case of an autonomous system, the Hamiltonian H= const defines a (2n−1)-\ndimensional hypersurface in T∗Q. The corresponding requirement for H1— considering\nthat the Hamiltonian is only determined up to an arbitrary ad ditive constant — then suggests\nto defineH1as an implicit function H1= 0, which then defines a (2n+ 1) -dimensional\nhypersurface in T∗Q1. With regard to Equation (3), this, in turn, implies to define −pn+1as\nthevalueeof the system’s Hamiltonian H\n−pn+1(s)≡e(s) =H/parenleftbig\nq(s),p(s),t(s)/parenrightbig\n. (4)\nThe notation eis used to distinguish the Hamiltonian functionH, defined on T∗Q×R, from\nitsvaluee(s)∈Ras a news-dependent canonical variable. Provided that the Hamilton ian\nrepresents the sum of the system’s kinetic and potential ene rgies,e(s)quantifies the system’s\ninstantaneous energy content. The negative system energy c ontent−ethus embodies the\ncanonical variable that is conjugate to the canonical varia ble timet. As will be shown\nin Appendix A.2, the result pn+1(s) =−e(s)follows also from the derivative of the extended\nLagrangianL1with respect to the fibre dt/ds.\nIn canonical coordinates, the transition from the presympl ectic carrier manifold of Hto\nthe symplectic extended phase space is thus given by the map ψ:T∗Q×R→T∗Q1\n(q,p,t)ψ/mapsto→(q,p,t,e), e=H(q,p,t).\nThe inverse mapping ψ−1:T∗Q1→T∗Q×Rthus replaces the e-terms by the Hamiltonian\nfunctionH. The general, coordinate-free equation that uniquely rela tes a given Hamiltonian\nH:T∗Q×R→R, to its extension H1:T∗Q1→Ris thus obtained from (3) as the equation\nH1ds= (H−e)dt. (5)\nWe emphasize that H1isuniquely determined by the setting pn+1=−eand by the\nrequirement that the conventional form of Hamilton’s varia tional principle from Equation (1)\nbe equivalent to its extended formulation in Equation (2).\nWith the extended covector p1≡(p,−e), the canonical coordinate representation of the\nextended Hamiltonian H1is given by\nH1(q1,p1) =k/bracketleftbig\nH(q,p,t)−e/bracketrightbig\n, k=dt\nds. (6)\nThe canonical representation of the extended symplectic st ructureΩ = dΛ onT∗Q1is then\nwith the additional pair (qn+1,pn+1)≡(t,−e)of canonical coordinates\nΩ =n+1/summationdisplay\ni=1dpi∧dqi=n/summationdisplay\ni=1dpi∧dqi−de∧dt. (7)\nRemarkably, the extended Hamiltonian H1in the form of Equation (6) was first introduced by\nPoincar´ e (Siegel and Moser 1971, p 35). Comparing this form of the extended Hamiltonian\nH1with those frequently found in literature (Lanczos 1949, p 1 89, Synge 1960, p 143,\nSzebehely 1967, p 329, Stiefel and Scheifele 1971, Thirring 1977, Asorey et al 1983,\nKuwabara 1984, Lichtenberg and Lieberman 1992, Wodnar 1995 , Stump 1998, Tsiganov\n2000, Struckmeier and Riedel 2002a), we notice the addition al scaling factor k= dt/ds. As\nwill become clear in the context of extended canonical trans formations, this factor is crucial\nto ensure the form-invariance of H1under non-trivial canonical time transformations. On\nthe other hand, we observe that the scaling factor kmust not be defined as an arbitrary\ndifferentiable function on T∗Q1in order to be compatible with Hamilton’s variational\nprinciple. We will discuss this issue and its implications f or the canonical transformation\nformalism in Section 3.2.Hamiltonian dynamics on T∗Q1 6\nAsestands for the value of H, the extended Hamiltonian H1of Equation (6) occurs as\ntheimplicit function H1(q1,p1) = 0 . In view of the assertion that the extended Hamiltonian\nH1vanishes identically (Lanczos 1949, p 186), we observe that this is only true for th e\nrepresentation of H1on the(2n+ 1) -dimensional presymplectic submanifold T∗Q×R,\nwhich is obtained by replacing ein (6) with the Hamiltonian function H. However, on the\n(2n+2) -dimensional symplectic extended phase space T∗Q1, the extended Hamiltonian H1\ndoes notvanish identically but constitutes the implicit constrain t functionH1(p1,q1) = 0 ,\nwhich defines a (2n+1)-dimensional hypersurface in T∗Q1on which the system’s evolution\ntakes place — analogously to the (2n−1)-dimensional hypersurface in T∗Qthat defines a\nregular energy surface through the holonomic constraint fu nctionH(q,p)−e0= 0 in the\ncase of an autonomous Hamiltonian system.\nWe note furthermore that in the Lagrangian description on th e extended tangent bundle\nTQ1, reviewed in Appendix A, the factor dt/dsis the(n+ 1) -th element of the extended\ntangent vector dq1/ds∈Tq1Q1, and hence appears as an independent variable in the\nargument list of the extended Lagrangian L1, defined onTQ1. In the Hamiltonian description\non the extended cotangent bundle T∗Q1, the scaling factor k= dt/dsis no longer an\nindependent function of the system evolution parameter, bu t takes on the role of a parameter\nfunction.\n2.2. Canonical equations, Poisson brackets\nWith the canonical coordinate representation Λ =p1dq1of the extended 1-formΛ, we obtain\nthes-parameterization of the variational integral (2) as\nδ/integraldisplays2\ns1/bracketleftiggn+1/summationdisplay\ni=1pi(s)dqi(s)\nds−H1/parenleftbig\nq1(s),p1(s)/parenrightbig/bracketrightigg\nds= 0. (8)\nThis representation of Hamilton’s variational principle f orH1(q1,p1)formally agrees with\nthe conventional description for a time-independent Hamil tonianH(q,p). Therefore, the\ncritical path within the extended phase-space (q1(s),p1(s))⊂T∗Q1is constituted by the\nsolution of the extended set of canonical equations\ndqi\nds=∂H1\n∂pi,dpi\nds=−∂H1\n∂qi, i= 1,...,n+1. (9)\nIn contrast to the total time derivative of the original Hami ltonianH, the totalsderivative\nofH1obviously vanishes identically by virtue of Equations (9): dH1/ds≡0. Thus,\nthe extended Hamiltonian H1(q1,p1)from Equation (6) formally converts any given non-\nautonomous system H(q,p,t)into an autonomous system in T∗Q1. InsertingH1from (6),\nwe may express the extended set of canonical equations (9) in terms of the conventional\nHamiltonian H(q,p,t)as\ndqi\nds=k∂H\n∂pi,dpi\nds=−k∂H\n∂qi,dt\nds=k,de\nds=k∂H\n∂t. (10)\nThe leftmost two equations are simply the conventional cano nical equations with sthe\nindependent variable instead of t. This shows that the critical paths C0and˜C0from\nEquations (1) and (2) are equivalent, as required. The right most equation from (10) states that\nthe partial time derivative of Hnow constitutes a regular canonical equation — the equation of\nmotion fore(s). Yet the conjugate equation of motion for t(s)merely constitutes an identity .\nThis reflects the fact that the variational principle of Equa tion (2) does not provide additional\ninformation, compared to its formulation in Equation (1). T he parameterization of the timeHamiltonian dynamics on T∗Q1 7\nt=t(s)thus remains undetermined. As a consequence, the critical p ath˜C0that is given\nas the solution of the extended set of canonical equations sa tisfies Hamilton’s variational\nprinciple (2) for alldifferentiable parameterizations of time t=t(s). According to (10),\ninstants ofswithk= 0simply mean that the system “freezes” at those points. A nega tivek\ndescribes a backward time flow as sincreases. This is also a valid parameterization t=t(s)\nas Hamilton’s variational principle is invariant with resp ect to the time reversal transformation\nt/mapsto→ −t.\nThe extended set of canonical equations formally (9) agrees with those derived by\nLanczos (Lanczos 1949, p 189) and Synge (Synge 1960, p 144). Y et, inserting the ad hoc\napproach of an extended Hamiltonian HLS=H−eof Lanczos and Synge into (9) yields the\ncanonical equation dt/ds=−∂HLS/∂e≡1in place of the identity dt/ds≡kin (10). The\nextended Hamiltonian HLSthus restricts t=t(s)to a fixed function of the system evolution\nparameter,s, and hence abolishes the idea of t(s)≡qn+1(s)as an ordinary canonical\nvariable. It is now evident why the factor k= dt/dsin the extended Hamiltonian H1from\nEquation (6) is important. As we will see in the context of ext ended canonical transformations\n— which are associated with non-trivial time mappings t(s)/mapsto→t′(s)— the a priori fixation\noft(s)is inadequate as dt/ds≡1anddt′/ds≡1cannot simultaneously hold true in the\noriginal and the transformed system. Therefore, an extende d Hamiltonian HLS=H−edoes\nnot meet the requirement of conserving the form of the canoni cal equations under extended\ncanonical transformations.\nObviously, the vector analysis operations on T∗Q1with its symplectic 2-formΩfrom\nEquation (7) are analogous to the corresponding operations on(T∗Q,ω). LetFdenote a\ndifferentiable function on F:T∗Q1→R. We can associate to Fa unique vector field XF\nonT∗Q1by means of the interior product\nXF/rightangleseΩ =−dF.\nIn canonical coordinate description, this means that we hav e along the integral curves of XF\ndqi\nds=∂F\n∂pi,dpi\nds=−∂F\n∂qi, i= 1,...,n+1.\nIdentifyingFwith the extended Hamiltonian H1, the related vector field that generates the\nsystem’s dynamical evolution is then the extended Hamilton ian vector field XH1\nXH1=k/bracketleftiggn/summationdisplay\ni=1/parenleftbigg∂H\n∂pi∂\n∂qi−∂H\n∂qi∂\n∂pi/parenrightbigg\n+∂H\n∂t∂\n∂e+∂\n∂t/bracketrightigg\n, (11)\nsatisfying\nXH1/rightangleseΩ =−k(dH−de) =−dH1.\nObviously, the (2n+ 1) -dimensional submanifold of T∗Q1, defined by H1= 0 of\nEquation (6), is invariant under the flow of XH1. The restriction of XH1to the presymplectic\nmanifoldT∗Q×Rthen yields the scaled representation k˜XHof the vector field ˜XH(Abraham\nand Marsden 1976, p 376) that describes the dynamics of a time -dependent Hamiltonian\nsystem(T∗Q×R,ωH,H).\nLetFandGnow denote two differentiable functions on T∗Q1, withXFandXGthe\nrelated dynamical vector fields. The extended symplectic 2-form induces a corresponding\nextended Poisson bracket {.,.}evia\n−Ω(XF,XG)≡ {F,G}e=n/summationdisplay\ni=1/parenleftbigg∂F\n∂qi∂G\n∂pi−∂F\n∂pi∂G\n∂qi/parenrightbigg\n−∂F\n∂t∂G\n∂e+∂F\n∂e∂G\n∂t. (12)Hamiltonian dynamics on T∗Q1 8\nAs is easily verified, the fundamental Poisson brackets are\n{qi,qj}e= 0{pi,pj}e= 0{qi,pj}e=δi\nji,j= 1,...n+1.\nThe extended set of canonical equations (9) yields for diffe rentiable functions on T∗Q1, hence\nfunctionsFthat do not explicitly depend on s\n{F,H1}e=dF\nds.\nThe Poisson bracket representation of the canonical equati ons are thus obtained as\n{pi,H1}e=dpi\nds,{qi,H1}e=dqi\nds, i= 1,...n+1.\nThis shows again that the canonical equations emerging from the Hamiltonian H1are just the\nconventional canonical equations, expressed in terms of th e system evolution parameter s.\nFrom Equation (12), we easily confirm that the extended 2-formΩis non-degenerate.\nGiven two differentiable functions GonT∗Q1, then{F,G}e= 0for allGobviously implies\nF= 0. Consequently, the extended Hamiltonian vector field XH1from Equation (11) is\nuniquely determined by the extended Hamiltonian H1. This establishes the complete analogy\nof our extended description of explicitly time-dependent H amiltonian systems (T∗Q1,Ω,H1)\nwith the conventional description of time-in dependent Hamiltonian systems (T∗Q,ω,H).\n2.3. Canonical transformations, Liouville’s theorem\nBy definition, the subset of diffeomorphisms φ:T∗Q1→T∗Q1that preserve the symplectic\nstructureΩare referred to as symplectic, or, synonymously, as canonic al. This means that the\ninduced map (pull-back) φ∗that acts on Ωmust satisfy\nφ∗Ω = Ω. (13)\nAs the closed canonical 2-formΩis locally exact ( Ω = dΛ ), and the pull-back commutes\nwith the exterior derivative, it can be concluded that φ∗Λmay differ from Λat most by an\nexact1-form. With a differentiable “generating function” F1, we thus obtain\nφ∗Λ−Λ+dF1= 0.\nThe condition that the integrand of the extended variationa l principle (2) must remain form-\ninvariant is then expressed in canonical coordinates for a f unctionF1:Q1×Q1×R→R\nas\nn/summationdisplay\ni=1pidqi−edt−H1ds=n/summationdisplay\ni=1p′\nidq′i−e′dt′−H′\n1ds+dF1(q,q′,t,t′,s). (14)\nThe requirement (14) thus automatically ensures the form-i nvariance of the canonical\nequations switching from the unprimed to the primed variabl es. Comparing the coordinates\nof the1-formdF1\ndF1=n/summationdisplay\ni=1/parenleftbigg∂F1\n∂qidqi+∂F1\n∂q′idq′i/parenrightbigg\n+∂F1\n∂tdt+∂F1\n∂t′dt′+∂F1\n∂sds\nwith (14), we obtain the transformation rules ( i= 1,...,n )\npi=∂F1\n∂qi, p′\ni=−∂F1\n∂q′i, e=−∂F1\n∂t, e′=∂F1\n∂t′, H′\n1=H1+∂F1\n∂s.\nWe immediately conclude that the extended Hamiltonian H1is preserved if and only if the\ngenerating function F1does not explicitly depend on s\nH′\n1(q′,p′,t′,e′) =H1(q,p,t,e)⇐⇒∂F1/∂s= 0. (15)Hamiltonian dynamics on T∗Q1 9\nIn order for the description in the new set of coordinates to b e equivalent to the original set of\nunprimed coordinates, the transformation must be invertib le. This is assured if and only if the\nHessian condition\ndet/parenleftbigg∂F1\n∂qi∂q′j/parenrightbigg\n/ne}ationslash= 0 (16)\nof the(n+1)×(n+1) matrix of second partial derivatives of F1is satisfied along s.\nWith the help of the Legendre transformation\nF2(q,p′,t,e′,s) =F1(q,q′,t,t′,s)+n/summationdisplay\ni=1q′ip′\ni−t′e′, (17)\nan equivalent generating function F2can be defined that depends on the original configuration\nspace and the new momentum coordinates. If we compare the coe fficients pertaining to the\nrespective 1-formsdqi,dp′\ni,dt,de′, andds, we find the following coordinate transformation\nrules to apply for each index i= 1,...,n :\npi=∂F2\n∂qi, q′i=∂F2\n∂p′\ni, e=−∂F2\n∂t, t′=−∂F2\n∂e′, H′\n1=H1+∂F2\n∂s. (18)\nEquivalent transformation rules are induced by generating functions\nF3(q′,p,t′,e,s) =F1(q,q′,t,t′,s)−n/summationdisplay\ni=1qipi+te\np′\ni=−∂F3\n∂q′i, qi=−∂F3\n∂pi, e′=∂F3\n∂t′, t=∂F3\n∂e, H′\n1=H1+∂F3\n∂s, (19)\nand\nF4(p,p′,e,e′,s) =F3(q′,p,t′,e,s)+n/summationdisplay\ni=1q′ip′\ni−t′e′\nqi=−∂F4\n∂pi, q′i=∂F4\n∂p′\ni, t=∂F4\n∂e, t′=−∂F4\n∂e′, H′\n1=H1+∂F4\n∂s.\nThe extended Hamiltonian H1is again preserved if the F2,3,4do not explicitly depend on s.\nOf course, the Hessian condition (16) must apply similarly f orF2,3,4in order to assure the\ninvertibility of the generated symplectic map.\nThe transformation rule for the conventional (non-extende d) Hamiltonians H(q,p,t)\nandH′(q′,p′,t′)follows finally from the common transformation rule for the e xtended\nHamiltonians H1(q1,p1)andH′\n1(q′\n1,p′\n1),\nH′\n1=H1+∂F\n∂s,\nwithFstanding for one of the particular generating functions F1,2,3,4. We rewrite this\nrule by expressing the extended Hamiltonians H1,H′\n1according to Eq. (6) in terms of the\nconventional ones,\n/bracketleftig\nH′(q′,p′,t′)−e′/bracketrightigdt′\nds=/bracketleftig\nH(q,p,t)−e/bracketrightigdt\nds+∂F\n∂s. (20)\nIf the generating function Fis not explicitly s-dependent, the evolution parameter scan be\neliminated from Eq. (20) to yield the more specific transform ation rule for the Hamiltonians\nH,H′under generalized canonical transformations\n/bracketleftig\nH′(q′,p′,t′)−e′/bracketrightig∂t′\n∂t=H(q,p,t)−e. (21)Hamiltonian dynamics on T∗Q1 10\nAs we will see in the following, the rules of Eqs. (20) and (21) generalize the transformation\nrule for the Hamiltonians of the conventional canonical tra nsformation theory.\nWith the set of extended canonical transformations providi ng a superset of the conven-\ntional ones, it is not astonishing that a conventional gener ating function f2(q,p′,t)can always\nbe reformulated as a particular extended generating functi onF2(q,p′,t,e′)by means of\nF2(q,p′,t,e′) =f2(q,p′,t)−te′. (22)\nThe related transformation rules follow from Equations (18 ) as\npi=∂f2\n∂qi, q′i=∂f2\n∂p′\ni, e=e′−∂f2\n∂t, t′=t, H′\n1=H1.\nAs the generating function from Equation (22) does not expli citly depend on s, the non-\nextended Hamiltonians transform according to Eq. (21), whi ch yields with the above\ntransformation rules for eande′\nH′=H+∂f2\n∂t.\nThe conventional transformations — generated by f2— thus coincide with the particular\nsubset of general transformations generated by F2from Equation (22). In other words, the\nconventional canonical transformations distinguish them selves by the fact that the system\nevolution parameter scan be replaced by the time tas the common independent variable of\nboth the original and the destination systems. In this respe ct, the transformations generated by\nf2are the time-dependent canonical transformations on the pr esymplectic contact manifold\nT∗Q×Rof definition 5.2.6 of Abraham and Marsden 1978.\nAccording to the fourth rule of Equations (18), the “extende d” generating functions F2in\ngeneral define non-trivial time transformations t′/ne}ationslash=t. As will become clear in the following\nexample section, it is this freedom to relate a given system to a destination system at different\ninstants of their respective time scales that enables us to f ormulate the Lorentz transformation\nas a particular canonical transformation in the extended ph ase space. Furthermore, we\nwill show that only the generalized canonical transformati on approach allows us to directly\ntransform an explicitly time-dependent Hamiltonian syste m into a time-in dependent one.\nThe extended 2-formΩhas the highest non-vanishing power\nΩn+1= (n+1)!·(−1)n(n+1)/2·dp1∧···∧dpn+1∧dq1∧···∧dqn+1.\nThe invariance of Ωwith respect to canonical transformations thus induces the invariance of\nthe extended volume form V1\nV1= dp1∧···∧dpn+1∧dq1∧···∧dqn+1= dp′\n1∧···∧dp′\nn+1∧dq′1∧···∧dq′n+1.\nThis is, in canonical coordinates, the extended phase-spac e formulation of Liouville’s\ntheorem.\nIt is obvious that an extended canonical transformation can only sustain the character of\ntimetas a common parameter for all particles if the transformed ti met′does not depend on\nthe coordinates of different particles, i.e. if\n∂t′\n∂qi=∂t′\n∂pi= 0. (23)\nThus, the conditions from Equation (23) impose restriction s on the functional dependence\nof extended generating functions in the case of multi-parti cle systems. We will encounter\nthis restriction in the context of the Lorentz transformati on, to be discussed in the following\nsection.Hamiltonian dynamics on T∗Q1 11\nMoreover, we easily convince ourselves that a “space-time d ecomposition” T∗Q1=\nT∗Q×T∗Rof the symplectic extended phase space is preserved if in add ition to (23)\n∂t′\n∂e=∂q′i\n∂e=∂p′\ni\n∂e= 0. (24)\nWith the conditions (23) and (24) fulfilled, the extended can onical transformation can be\nfactorized as a conventional canonical transformation tim es a pure time scaling transformation\nof Section 3.2. In contrast to an assertion of Asorey et al 1983, this is not necessarily the case:\nthe Lorentz transformation, regarded as a particular canon ical transformation in the extended\nphase space, does not satisfy all conditions, and hence cann ot be factorized.\nFurthermore, Liouville’s theorem applies simultaneously in the subspace T∗Q, hence in\nthe conventional phase space if\n∂t′\n∂t∂e′\n∂e= 1 (25)\nholds in conjunction with Equations (23) and (24).\n3. Examples of canonical transformations in the extended ph ase space\n3.1. Lorentz transformation\nHere we consider two Cartesian frames of reference (x,y,z)and(x′,y′,z′)that move\nwith respect to each other at a constant velocity v. For simplicity, we first set up the\ncoordinate system to be aligned so that the relative motion o ccurs along the x-axis. Under\nthese circumstances, the y- andz-coordinates are not affected by the Lorentz transformatio n\n(y′=y,z′=z). As this transformation necessarily involves a non-trivi al mapping of the\nrespective time scales t/mapsto→t′, it cannot be described in terms of a canonical transformati on\nin the conventional phase space. Nevertheless, in the exten ded phase space, a generating\nfunctionF2exists that exactly yields the Lorentz transformation rule s,\nF2(x,p′\nx,t,e′) =γ/bracketleftbigg\np′\nx(x−βct)−e′\nc(ct−βx)/bracketrightbigg\n. (26)\nWithcdenoting the speed of light, the common notation is used to ex press the scaled relative\nvelocity in the abbreviated form β=v/c. As usual,γstands for the relativistic factor, defined\nbyγ−2= 1−β2.\nFor the particular generating function (26), the general ru les for a canonical transfor-\nmation in the extended phase space from Equations (18) speci alize to\npx=∂F2\n∂x=γ/parenleftbigg\np′\nx+βe′\nc/parenrightbigg\n, e=−∂F2\n∂t=γ(e′+βcp′\nx),\nx′=∂F2\n∂p′x=γ(x−βct), t′=−∂F2\n∂e′=γ/parenleftbigg\nt−β\ncx/parenrightbigg\n,\nH′\n1(x′,p′\nx,t′,e′) =H1(x,px,t,e), H=γH′+βγcp′\nx.\nAs required, the transformation rule for the Hamiltonians HandH′, given by Eq. (21), agrees\nwith the corresponding rule for their respective values, eande′. In complex notation, the\ncoordinate transformation rules take on the familiar form o f an orthogonal linear mapping\n\nx′\nict′\np′\nx\nie′/c\n=\nγ iβγ 0 0\n−iβγ γ 0 0\n0 0 γ iβγ\n0 0 −iβγ γ\n\nx\nict\npx\nie/c\n. (27)Hamiltonian dynamics on T∗Q1 12\nWe observe that the generating function (26) provides both t he transformation rules for the\n(x,ct)coordinates — which commonly refer to the Lorentz transform ation — and the related\nrules for the conjugate coordinates momentum and energy (px,e/c). This is not astonishing,\nas a canonical transformation always maintains the symplec tic structure of the Hamiltonian in\nquestion — which requires the transformation rules for all c anonical variables to be uniquely\ndefined.\nFor the general case that both frames of reference are not ali gned, their relative scaled\nvelocity is expressed by the 3-component vector β= (βi). Withq= (x,y,z) = (qi)and\np′= (p′\nx,p′\ny,p′\nz) = (p′\ni), the general form of the generating function F2for the Lorentz\ntransformation is then given by\nF2(q,p′,t,e′) =γe′\nc/bracketleftigg3/summationdisplay\ni=1βiqi−ct/bracketrightigg\n+3/summationdisplay\ni=1p′\ni/bracketleftigg3/summationdisplay\nk=1/parenleftigg\nδik+(γ−1)βiβk\n|β|2/parenrightigg\nqk−γctβi/bracketrightigg\n(28)\nwhich simplifies to the generating function (26) for the alig ned caseβ1=βx=β,\nβ2=βy= 0, andβ3=βz= 0.\nThe generating function (28) of the Lorentz transformation has the particular property\n−∂2F2\n∂qi∂e′=∂t′\n∂qi=−γ\ncβi, (29)\nwhich does not vanish for βi/ne}ationslash= 0. This is the reason why it is impossible to maintain the\nmeaning of the transformed time t′as a global parameter for the transformed coordinates p′\ni\nandq′iin multi-particle systems. Because of (29), the transforme d timet′depends on qi,\nwhich means that individual particles in the transformed sy stem no longer carry the same\ntimet′. Furthermore, a factorization of the symplectic extended p hase spaceT∗Q1is not\npreserved as the conditions (23) and (24) are not satisfied. H ence, the Liouville volume form\nV1is preserved on T∗Q1, but not the volume forms of any of its subspaces.\nIn order to relativistically describe multi-particle syst ems, Sorge et al 1989 introduced\nan8N-dimensional phase space, in which the positions and moment a ofNparticles are\ndescribed as two 4-vectors that depend on a common evolution parameter. A more general,\nfield-theoretical approach has been developed by Gotay (Got ayet al 1997) on a “multiphase\nspace” that is endowed with a “multisymplectic” (n+2)-form as the covariant generalization\nof the symplectic 2-form of Hamiltonian mechanics.\nWe furthermore observe that the generating function (28) do es not explicitly depend\nons. According to (15), this means that extended Hamiltonians H1from (5) are always\nLorentz-invariant — in contrast to non-extended Hamiltoni ansH. Of particular interest are,\ntherefore, those non-extended Hamiltonians Hthat are form-invariant under the canonical\ntransformation generated by F2from Equation (28). As an example of how to convert a given\nnon-Lorentz-invariant Hamiltonian HNLinto a Lorentz-invariant form, HL, we consider the\nHamiltonian of a particle with mass mand chargeζwithin an electromagnetic field, defined\nby the potentials (A,φ)\nHNL(q,P,t) =[P−ζA(q,t)]2\n2m+ζφ(q,t). (30)\nAs only expressions of the form q2−c2t2and(P−ζA)2−(e−ζφ)2/c2are invariant under the\northogonal transformation (27), the Hamiltonian (30) is ob viously not Lorentz-invariant. InHamiltonian dynamics on T∗Q1 13\nthe extended phase space, however, a Lorentz-invariant for m of (30) can easily be constructed\nby adding the required e-term\nHL(q,P,t,e) =1\n2m/bracketleftigg\n(P−ζA)2−(e−ζφ−mc2)2\nc2/bracketrightigg\n+ζφ+mc2. (31)\nWe thus have chosen the usual normalization to define HL=e=mc2for the particular\ncaseP= 0 and zero field. The addition of the mc2terms merely describes a Lorentz\ninvariant shift of the origin. According to (4), the Hamilto nian (31) on T∗Q1is mapped\ninto a Hamiltonian on the usual carrier manifold T∗Q×Rby replacing the Hamiltonian’s\nvalueewith the Hamiltonian HLitself,\nHL(q,P,t) =1\n2m/bracketleftigg\n(P−ζA)2−(HL−ζφ−mc2)2\nc2/bracketrightigg\n+ζφ+mc2. (32)\nSolving (32) for HL, we obtain the well known result\nHL(q,P,t) =/radicalig\nc2[P−ζA(q,t)]2+m2c4+ζφ(q,t),\ni.e. the Lorentz-invariant form of the Hamiltonian Hfor a particle within an electromagnetic\nfield.\n3.2. Euler’s time scaling transformation\nIn the context of the regularizing transformation of the thr ee-body problem (Siegel and\nMoser 1971), we encounter from heuristic reasoning a replac ement of the time tby a new\nindependent variable, t′, defined by\nt′(t) =/integraldisplayt\nt0dτ\nξ(τ).\nWe shall see in the following that this particular transform ation constitutes a simple canonical\ntransformation in the extended phase space. Namely, a canon ical transformation that defines\nan identical mapping in qandpbut merely “scales” the extended variables t(s)ande(s)is\ninduced by the extended generating function\nF2/parenleftbig\nq,p′,t,e′/parenrightbig\n=qp′−e′/integraldisplayt\nt0dτ\nξ(τ), (33)\nwithξ(t)denoting an arbitrary function of time only. The particular transformation rules\nemerge from the general rules (18) as\nq′i=qi, p′\ni=pi, t′=/integraldisplayt\nt0dτ\nξ(τ), e′=ξ(t)e, H′\n1=H1.(34)\nFromH′\n1(q′\n1,p′\n1) =H1(q1,p1), hence from H−e= (H′−e′)∂t′/∂t, we directly conclude\nH′/parenleftbig\nq(t′),p(t′),t′/parenrightbig\n=ξ(t)H/parenleftbig\nq(t),p(t),t/parenrightbig\n.\nNot surprisingly, the Hamiltonians HandH′follow again the same transformation rule as\ntheir respective values, eande′. Because of\n∂2F2\n∂qi��e′=∂2F2\n∂p′\ni∂e′= 0,Hamiltonian dynamics on T∗Q1 14\nthe transformed time t′retains the character of a global parameter that is common to all\ncoordinatesq′iandp′\niin the transformed system. A factorization of the extended p hase space\nT∗Q1=T∗Q×T∗Ris preserved, since additionally\n∂t′\n∂e=∂q′i\n∂e=∂p′\ni\n∂e= 0.\nBecause furthermore\n∂t′\n∂t∂e′\n∂e= 1,\nthe volume forms on the subspaces T∗QandT∗Rare separately conserved by means of the\ncanonical transformation generated by (33), hence\nV= dp1∧···∧dpn∧dq1∧···∧dqn=V′,dt∧de= dt′∧de′.\nFollowing from the fact that ξ(t)is an arbitrary function of time only, it can be freely\nidentified with any combination of the canonical coordinate sq(t)andp(t), regarded as the\ntime functions that are obtained from integrating the canonical equations . Of course, this does\nnot mean that ξ(t)acquires an explicit dependence on the coordinates qandp; hence an\nexpression of ξ(t)in terms of q(t)andp(t)must not be inserted back into the Hamiltonian!\nTherefore, the identification of ξ(t)with functions of the canonical coordinates q(t)andp(t)\nis admissible only after all differentiations with respect to the canonical coordin ates have been\naccomplished.\nA simple example of how to formulate a time-scaling transfor mation as an extended\ncanonical transformation will be presented in the followin g for Euler’s regularization of the\nKepler equation of motion in one dimension. The Hamiltonian of this problem is given in\nnormalized form by (Stiefel and Scheifele 1971).\nH(x,p) =1\n2p2−K2\nx, K2= Γ·(M+m),\nwithΓthe gravitational constant, M,mthe masses, and xthe distance of the collision\npartners. As Hdoes not explicitly depend on time, we have de/dt=∂H/∂t= 0, hence\ne=1\n2p2−K2\nx= const. (35)\nThe resulting equation of motion for xis obviously singular at the point of collision at x= 0,\nd2x\ndt2+K2\nx2= 0.\nIt was L. Euler who first worked out a transformation of the tim e scales that regularizes this\nequation of motion. The canonical transformation in the sym plectic extended phase that\ndefines this regularization transformation is generated by the function F2of Equation (33)\nand the subsequent coordinate transformation rules (34). T he transformed Hamiltonian H′is\nthen\nH′(x,p,t′) =ξ(t′)/parenleftbigg\n1\n2p2−K2\nx/parenrightbigg\n, (36)\nwith the coordinates x′=x,p′=pandξnow understood as functions of t′. As a result of\nthe fact that the transformation is canonical, the form of th e canonical equations is preserved\nin the transformed system,\ndx\ndt′=∂H′\n∂p=ξ(t′)p,dp\ndt′=−∂H′\n∂x=−ξ(t′)K2\nx2.Hamiltonian dynamics on T∗Q1 15\nThe equation of motion in terms of t′and the energy conservation relation from Equation (35)\nfollow as\nd2x\ndt′2−1\nξdξ\ndt′dx\ndt′+K2ξ2\nx2= 0,/parenleftbiggdx\ndt′/parenrightbigg2\n= 2ξ2/parenleftbigg\ne+K2\nx/parenrightbigg\n. (37)\nHaving worked out the transformed equations of motion, we ar e now free to identify the as\nyet undetermined function ξ(t′)with an arbitrary function of the canonical variables x(t′)and\np(t′), regarded as functions of time t′, respectively. In doing so, we fix the correlation of the\n“fictitious” time t′with the physical time t. In the present case we define\nξ(t′)≡x(t′) =⇒t(t′) =/integraldisplayt′\n0x(τ)dτ.\nThe equation of motion and the energy conservation relation from Equation (37) now reduce\nto\nd2x\ndt′2−1\nx/parenleftbiggdx\ndt′/parenrightbigg2\n+K2= 0,/parenleftbiggdx\ndt′/parenrightbigg2\n= 2ex2+2K2x.\nBy finally inserting the energy conservation relation into t he equation of motion, we obtain\nEuler’s regularized equation of motion\nd2x\ndt′2−2ex=K2.\nSummarizing, we can state that the time scaling transformat ion from Equation (34) can\nbe considered as a finite canonical transformation in the symplectic extended phase space.\nHowever, in order to maintain the consistency of the canonic al transformation approach, the\nidentification of the arbitrary time function ξ(t′)with a suitable combination of the time\nfunctions q(t′)andp(t′)can only be performed after having worked out the transformed\ncanonical equations. In other words, the identification ξ(t′)≡x(t′)must not be inserted back\ninto the Hamiltonian H′of Equation (36) since ξ(t′)continues to be a function of time only\nand does not “acquire” an explicit dependence on the canonic al variables. This contrasts with\nprocedures sometimes found in literature (Cari˜ nena et al 1987, Cari˜ nena et al 1988, Tsiganov\n2000).\n3.3. Time-dependent damped harmonic oscillator\nThe time-dependent harmonic oscillator model is frequentl y used as the first-order approxi-\nmation for non-linear, explicitly time-dependent Hamilto nian systems. We shall demonstrate\nin the following that a system of nparticles that is confined within a time-dependent harmonic\noscillator potential and that is subject to linear time-dep endent damping forces can be mapped\ninto a conventional undamped time-independent harmonic os cillator system by means of a\nsingle canonical transformation in the extended phase spac e. The Hamiltonian of the original\nsystem is given by\nH(q,p,t) =1\n2e−F(t)p2+1\n2eF(t)ω2(t)q2. (38)\nwhich yields the equations of motion\n¨qi+f(t) ˙qi+ω2(t)qi= 0, i= 1,...,n, f (t) =˙F(t). (39)\nThe destination Hamiltonian H′— witht′its independent variable — shall be the autonomous\nsystem\nH′(q′,p′) =1\n2p′2+1\n2ω2\n0q′2. (40)Hamiltonian dynamics on T∗Q1 16\nA one-parameter family of functions F2(q,p′,t,e′)that generates the mapping of the\nHamiltonian (38) into the Hamiltonian (40) has been found to be\nF2/parenleftbig\nq,p′,t,e′/parenrightbig\n=/radicaligg\neF(t)\nξ(t)qp′+1\n4eF(t)/bracketleftigg˙ξ(t)\nξ(t)−f(t)/bracketrightigg\nq2−e′/integraldisplayt\n0dτ\nξ(τ),(41)\nwith the parameter ξ(t), for the moment, an undetermined differentiable function o f time.\nBecause of the quadratic dependence on the canonical coordi nates, the transformation\nrules (18) yield the linear mapping\n/parenleftiggq′i\np′\ni/parenrightigg\n=/parenleftigg/radicalbig\neF(t)/ξ(t) 0\n−1\n2(˙ξ−ξf)/radicalbig\neF(t)/ξ(t)/radicalbig\nξ(t)/eF(t)/parenrightigg/parenleftiggqi\npi/parenrightigg\n.(42)\nThe transformations of time t, energye, and extended Hamiltonian H1between both systems\nemerge from the generating function (41) as\nt′=/integraldisplayt\n0dτ\nξ(τ),\ne′=ξe−1\n2/parenleftig\n˙ξ−ξf/parenrightig\nqp+1\n4eF(t)/parenleftig\n¨ξ−˙ξf−ξ˙f/parenrightig\nq2, (43)\nH′\n1=H1.\nWe observe that the time-shift transformation between both systems is determined by the as\nyet unknown function ξ(t). In any case, the transformed time t′does not depend on individual\nparticles coordinates, hence retains the character of tas a global parameter for all particles.\nThe correlation of the Hamiltonians HandH′follows from Eq. (21). With ∂t′/∂t=ξ−1(t)\nand inserting the Hamiltonian (38), we obtain H′as\nH′(q′,p′,t′) =1\n2p′2+1\n2q′2/bracketleftig\n1\n2ξ¨ξ−1\n4˙ξ2+ξ2/parenleftig\nω2−1\n2˙f−1\n4f2/parenrightig/bracketrightig\n,\nhaving replaced all unprimed variables according to the rul es (42) and (43). Hence, the\ndestination Hamiltonian (40) indeed emerges if we identify\n1\n2ξ¨ξ−1\n4˙ξ2+ξ2/parenleftig\nω2(t)−1\n2˙f−1\n4f2/parenrightig\n=ω2\n0= const. (44)\nThe primed system’s potential is not time-dependent if and o nly ifdω2\n0/dt′= 0. This\ncondition yields the linear third-order equation\n...\nξ(t)+˙ξ/parenleftig\n4ω2(t)−2˙f(t)−f2(t)/parenrightig\n+ξ/parenleftig\n4ω˙ω−¨f−f˙f/parenrightig\n= 0. (45)\nAs a result of this requirement, the function ξ(t)is now determined — and hence the\ntime correlation t′(t)of both systems. It is precisely the extended canonical-tra nsformation\napproach that enables us to properly adjust this time correl ation. With ξ(t)a solution\nof (45), the Hamiltonian H′does not depend on time explicitly. Expressed in the unprime d\ncoordinates, the value e′ofH′then yields an invariant of the original system (38)\ne′=1\n2e−Fξp2−1\n2/parenleftig\n˙ξ−ξf/parenrightig\nqp+1\n4eF(t)/parenleftig\n¨ξ−˙ξf−ξ˙f+2ξω2(t)/parenrightig\nq2= const. (46)\nIn terms ofρ(t) =/radicalbig\nξ(t), the invariant (46) agrees with the invariant found by Leach (Leach\n1978) for the case n= 1. Moreover, the invariant e′can be rendered a function of qand\nponly for this linear dynamical system. We easily verify that a particular solution ξ(t)of\nEquation (45) is given by\nξ(t) =eF(t)q2(t), (47)Hamiltonian dynamics on T∗Q1 17\nof course provided that q(t)is a solution vector of the equations of motion (39). Inserti ng (47)\nand its first and second time derivatives into (46), the invar iante′takes on the simple form\ne′=q2p2−(qp)2=1\n2n/summationdisplay\ni,j=1/parenleftbig\npiqj−pjqi/parenrightbig2.\nWe immediately conclude that the individual sum terms\nIj\ni=piqj−pjqi, i,j= 1,...,n\nare also invariant. These quantities correspond to the cons erved angular momenta in central\nforce fields (Leach 1977). Consequently, the antisymmetric tensor(Ij\ni)is a non-trivial\ninvariant of the n-particle time-dependent harmonic oscillator with time-d ependent linear\ndamping force (39).\nThe canonical equivalence of (38) and (40) is physical as long as the transformation\nrules (42) describe the correlation of realparticles coordinates, hence as long as ξ(t)>0. In\norder to show that ξ(t)remains positive in the course of its time evolution if ξ(0)>0, we\nmake use of Equation (44) to eliminate ¨ξ(t)in (46), which yields\n2e′e−F(t)ξ(t) =ω2\n0q2+/bracketleftig\nξe−Fp−1\n2(˙ξ−ξf)q/bracketrightig2\n.\nSince the right hand side is obviously always positive, we im mediately find ξ(t)>0for allt.\nThus, for a time-dependent damped harmonic oscillator syst em (38) there always exists an\nequivalent genuine physical system of a time-independent u ndamped harmonic oscillator (40).\n3.4. General time-dependent potential\nAs generalization of the previous example, we will now trans form ann-degree-of-free-\ndom Hamiltonian system with a general non-linear time-depe ndent potential into a time-\nindependent one. Let the Hamiltonian of the original system be given by\nH(q,p,t) =1\n2p2+V(q,t). (48)\nAgain, we require a destination system H′of the same form , but with a potential V′that does\nnotexplicitly depend on the system’s independent variable, t′,\nH′(q′,p′) =1\n2p′2+V′(q′). (49)\nThe most general “extended” generating function F2that retains both the quadratic momen-\ntum dependence of H′and a momentum-independent potential V′turns out to be precisely\nthe generating function (41) from the previous example (Str uckmeier and Riedel 2002a\n2002b), setting F(t)≡0, and hence f(t)≡0, as the actual system (48) does not include\ndamping forces. The transformed Hamiltonian H′is then again obtained from the particular\ntransformation rule from Eq. (21) as H′=ξ(H−e)+e′. Inserting the Hamiltonian Hof (48)\nand the transformation rule for e′of (43), and replacing the unprimed coordinates, we find\nH′(q′,p′,t′) =1\n2p′2+1\n4q′2/parenleftbig\nξ¨ξ−1\n2˙ξ2/parenrightbig\n+ξV/parenleftbig/radicalbig\nξq′,t/parenrightbig\n.\nThus, a Hamiltonian H′of the form of (49) turns out if the transformed potential V′is\nidentfied with\nV′/parenleftbig\nq′,t′/parenrightbig\n=1\n4q′2/parenleftig\nξ¨ξ−1\n2˙ξ2/parenrightig\n+ξV/parenleftbig/radicalbig\nξq′,t/parenrightbig\n. (50)\nWe can now make use of the freedom to appropriately adjust the time correlation t′(t)between\nthe original system (48) and the destination system (49) by r equiring the new potential V′to\nbe independent of its time t′explicitly\n∂V′\n∂t′!= 0. (51)Hamiltonian dynamics on T∗Q1 18\nBy this requirement, we now determine ξ(t)— which was initially defined in the generating\nfunction (41) as an arbitrary differentiable function of ti me. For the potential (50), the\ncondition (51) evaluates to\n...\nξq2+4˙ξ/bracketleftbigg\nV(q,t)+1\n2q∂V\n∂q/bracketrightbigg\n+4ξ∂V\n∂t= 0. (52)\nThe linear and homogeneous third-order differential equat ion (52) is equivalent to the linear\nsystem\nd\ndt\nξ\n˙ξ\n¨ξ\n=A(t)\nξ\n˙ξ\n¨ξ\n, A(t) =\n0 1 0\n0 0 1\n−g1(t)−g2(t) 0\n (53)\nwith the coefficients g1andg2defined by\ng1(t) =4\nq2∂V\n∂t, g 2(t) =4\nq2/bracketleftbigg\nV(q,t)+1\n2q∂V\n∂q/bracketrightbigg\n.\nAs by definition ξ=ξ(t)embodies a function of tonly, the coefficients g1andg2must\nalso be functions of time only if the system (53) is to be solva ble. This means that all\nspatial (q-)dependencies in g1andg2must be conceived of as implicit time-dependencies via\nq=q(t). In other words, the trajectory q=q(t)as the solution of the equations of motion\nmust be known in advance. Equation (53) should, therefore, b e regarded as an extension of\nthe system of canonical equations. In conjunction with the f ull set of canonical equations, the\nsystem (53) is closed and its functional dependence is uniqu ely determined.\nRegarding the system matrix A(t), we observe that its trace is always zero. Hence, the\nWronski determinant of any 3×3solution matrix Ξ(t)of (53) is always constant, regardless of\nthe particular form of the system’s potential V(q,t). With the 3×3unit matrix as a particular\ninitial condition ( Ξ(0) = 1), we thus obtain\nΞ(t) =\nξ1ξ2ξ3\n˙ξ1˙ξ2˙ξ3\n¨ξ1¨ξ2¨ξ3\n,Ξ(0) = 1,detΞ(t)≡1. (54)\nThe transformation rule (43) now provides an integral of mot ionIfor the original system (48)\nif and only if ξ(t)and its time derivatives represent a linear combination of t he three linearly\nindependent vectors of the solution matrix Ξ(t),\ne′=I=ξ(t)e−1\n2˙ξ(t)qp+1\n4¨ξ(t)q2= const. (55)\nWith the normalization Ξ(0) = 1, the three invariants, i.e. the three integration constant s of\nthe third order system (53), can be written in matrix form in t erms of the transpose solution\nmatrixΞT(t),\n\ne0\n−1\n2q0p0\n1\n4q2\n0\n=\nξ1˙ξ1¨ξ1\nξ2˙ξ2¨ξ2\nξ3˙ξ3¨ξ3\n\ne\n−1\n2qp\n1\n4q2\n\nt. (56)\nThe particular normalization Ξ(0) = 1thus induces the invariants to represent the initial\nvalues of the Hamiltonian eand of the scalar products qpandq2. One might have expected\nthis result as the generic Hamiltonian system of Equation (4 8) cannot have invariants other\nthan its initial conditions and, trivially, combinations t hereof. Nevertheless, what is actually\nsurprising with Equation (56) is the fact that the particula r vector/parenleftbig\ne,−1\n2qp,1\n4q2/parenrightbig\nalways\ndepends linearly on its initial state, and that this mapping is associated wit h aunit determinant .Hamiltonian dynamics on T∗Q1 19\nIf the given system (48) is autonomous ( ∂V/∂t≡0), then the linear equation (52)\nobviously has the particular solution ξ1(t)≡1. With regard to (56), this solution simply\nexpresses the fact that the value of the Hamiltonian is a constant of motion ( e(t) =e0) if\nHdoes not depend on time explicitly. This well known feature o f autonomous Hamiltonian\nsystems thus appears in our analysis in a more global context . Particularly, we observe that\ntwo other invariants always exist for autonomous systems th at are associated with the non-\nconstant solutions ξ2(t)andξ3(t).\nThe physical meaning of Equation (56) is expressed by the time evolution of the elements\nof the “transfer matrix” ΞT(t). As was shown by Struckmeier 2006, the properties of this map\nyield information with regard to the regularity of the system’s time evolution.\n3.5. Kustaanheimo-Stiefel (KS) transformation\nA transformation to “Kustaanheimo-Stiefel” variables con stitutes a generalization of a\ntransformation to Levi-Civita variables (Stiefel and Sche ifele 1971). It has the properties (i)\nto ensure the regularization of the equations of motion, (ii ) to permit a uniform treatment all\nthree types of Keplerian motion, and (iii) to transform the e quations of the two-body problem\ninto a linear form. It is also associated with a mapping of the physical time tinto a fictitious\ntimet′. The KS-transformation constitutes a canonical point tran sformation in the extended\nphase space, generated by an extended function of type F3,\nF3/parenleftbig\nq′,p,t′,e/parenrightbig\n=/parenleftbig\n−q′2\n1+q′2\n2+q′2\n3−q′2\n4/parenrightbig\np1−2(q′\n1q′\n2−q′\n3q′\n4)p2\n−2(q′\n1q′\n3+q′\n2q′\n4)p3+e/integraldisplayt′\n0ξ(τ)dτ. (57)\nAccording to (19), the subsequent mapping q′/mapsto→qof the vector of the new “spatial”\ncoordinates q′= (q′\n1,q′\n2,q′\n3,q′\n4)into the old physical spatial coordinates q= (q1,q2,q3,0)is\ngiven by\nq1=q′2\n1−q′2\n2−q′2\n3+q′2\n4\nq2= 2q′\n1q′\n2−2q′\n3q′\n4\nq3= 2q′\n1q′\n3+2q′\n2q′\n4\nq4= 0.\nThe generating function (57) yields the following associat ed transformation rules into the\nvector of transformed momentum p′,\np′\n1= 2q′\n1p1+2q′\n2p2+2q′\n3p3\np′\n2=−2q′\n2p1+2q′\n1p2+2q′\n4p3\np′\n3=−2q′\n3p1−2q′\n4p2+2q′\n1p3\np′\n4= 2q′\n4p1−2q′\n3p2+2q′\n2p3.\nThe transformations of energy e, timet, and extended Hamiltonian H1follow as\ne′=eξ(t), t=/integraldisplayt′\n0ξ(τ)dτ, H′\n1=H1.\nAs in the previous example, the arbitrary time function ξ(t)can be identified with any\nfunction of the canonical variables. Yet, this identificati on ofξ(t)with time evolution of some\nfunction the canonical variables does notmean thatξ(t)acquires an explicit dependence on\nthe canonical variables.\nFromH′\n1=H1, we have\nH(q′(t′),p′(t′),t′) =ξ(t)H(q(t),p(t),t).Hamiltonian dynamics on T∗Q1 20\n4. Conclusions\nWith the present paper, we have provided a consistent reform ulation of the classical\nHamiltonian theory on the symplectic extended phase space. The extended description is\nbased on a generalized understanding of Hamilton’s variati onal principle by conceiving the\ntimet(s) =qn+1(s)and the negative value −e(s) =pn+1(s)of the Hamiltonian Has\nan additional pair of canonically conjugate variables that depends, like all other pairs of\ncanonically conjugate variables, on a superordinated syst em evolution parameter s. With\nω=/summationtext\nidpi∧dqithe canonical coordinate representation of the symplectic 2-form onT∗Q,\nthe corresponding extended symplectic 2-formΩonT∗Q1is then given by Ω =ω−de∧dt.\nThe extended 2-formΩwas shown to be non-degenerate. From Hamilton’s variationa l\nprinciple, the general form of the extended Hamiltonian H1was derived, and its uniquely\ndetermined relation H1ds= (H−e)dtto the conventional Hamiltonian Hwas established.\nThe result can now be summarized as follows:\nThe symplectic Hamiltonian system (T∗Q1,Ω,H1), withQ1=Q×R,Ω =\nω−de∧dt,H1= (H−e)dt/ds, andHpossibly time-dependent is the proper\ncanonical extension of the symplectic Hamiltonian system (T∗Q,ω,H)with time-\nindependent Hamiltonian H.\nNeither the frequently cited extended Hamiltonian HLS=H−eof Lanczos and Synge nor\nCari˜ nena’s extended Hamiltonian HC=f(H−e), withf∈C∞(T∗Q1), yield a formulation\nof dynamics on ( T∗Q1,Ω)that is analogous to that of (T∗Q,ω,H). In the first case, the\nsubsequent canonical equation dt/ds≡1implies that HLSis not preserved under non-trivial\ntime transformations t(s)/mapsto→t′(s). In the second case, the obtained extended set of canonical\nequations cannot be derived from Hamilton’s variational pr inciple.\nThe canonically invariant form of the extended Hamiltonian H1that is consistent with\nHamilton’s variational principle turned out to coincide wi th the Hamiltonian of Poincar´ e’s\ntransformation of time. The well-known feature of Poincar´ e’s approach to preserve the\ndescription of the system’s dynamics was reflected by the fac t that the extended set of\ncanonical equations is equivalent to the conventional set o f canonical equations.\nIn contrast, the formulation of extended canonical transfo rmations on T∗Q1was shown\nto generalize the conventional presymplectic canonical tr ansformation theory. Specifically,\nconventional canonical transformations were shown to cons titute the particular subset of\nextended ones for which the system evolution parameter scan be replaced by the time t\nas a common independent variable of both the original and the destination system. With\nFdenoting an extended generating function for an extended ca nonical transformation, we\nshowed that the extended Hamiltonian H1is preserved if Fdoes not explicitly depend\nons. The extended Hamiltonian H1now meets the requirement to preserve the form of\nthe canonical equations under extended canonical transfor mations generated by F.\nWe have furthermore worked out the restrictions that are to b e imposed on extended\ngenerating functions in order for the transformed time t′to retain the meaning of tas a\ncommon parameter for all coordinates p′\niandq′i. In a similar way, the conditions were\nobtained for Liouville’s volume form to be separately conse rved in the subspace T∗Q, i.e. in\nthe conventional phase space.\nIn the first example, we demonstrated that the Lorentz transf ormation represents a\nparticular canonical transformation in the symplectic ext ended phase space, which preserves\nH1, for its generating function does not explicitly depend on s. The Lorentz transformation\nwas shown to represent a particular extended canonical tran sformation that cannot be\ndecomposed into a conventional canonical transformation t imes a canonical time scalingHamiltonian dynamics on T∗Q1 21\ntransformation. This canonical mapping clearly reveals th e conditions for the non-extended\nHamiltonian Hto be also Lorentz-invariant. In demonstrating this in the c ase of a particle\nwithin an electromagnetic field, we obtain a guideline for co nverting non-Lorentz-invariant\nHamiltonians Hinto Lorentz-invariant ones.\nIn the realm of celestial mechanics, the transformation of t he physical time tto a\n“fictitious” time t′is a long-established technique for regularizing singular equations of\nmotion. With the theory of extended canonical transformati ons, we can now conceive\nregularization transformations of celestial mechanics as finite canonical transformations in\nthe symplectic extended phase space that preserve H1. This was demonstrated explicitly for\nEuler’s regularization transformation of the one-dimensi onal Kepler motion.\nMoreover, the generalized concept of canonical transforma tions permits a direct mapping\nof Hamiltonian systems with explicitly time-dependent pot entials into time-independent\nHamiltonian systems. An “extended” generating function of typeF2that defines a canonical\nmapping such as this was presented for both the time-depende nt harmonic oscillator\nwith time-dependent damping and for a general time-depende nt potential. Similar to the\nregularization transformations, this generating functio n was defined to depend on an arbitrary\ntime function ξ(t). The freedom to finally commit oneself to a particular ξ(t)was then utilized\nto render the destination system autonomous. The fundament al solution of the subsequent\nlinear third-order differential equation for ξ(t)was shown to provide information on the\nirregularity of the system’s time evolution (Struckmeier 2 006).\nTo conclude, the symplectic description of possibly time-d ependent Hamiltonians H\non the symplectic extended phase space (T∗Q1,Ω)establishes a generalization of the usual\npresymplectic description on (T∗Q×R,ωH). With the extended symplectic 2-formΩ,\nthe induced extended Poisson bracket should then provide th e means for a generalized Lie-\nalgebraic description of dynamical systems with explicitl y time-dependent Hamiltonians H\non the symplectic extended phase space.\nAppendix A. Extended Lagrangian description\nAppendix A.1. Extended Euler-Lagrange equations\nA time-independent Lagrangian Lis defined as the mapping of the tangent bundle TQintoR.\nIf the Lagrangian Lis explicitly time-dependent, then its domain is TQ×R, with Rdenoting\nthe time axis. In local coordinates (qi,˙qi), the actual system path (qi(t),˙qi(t))⊂TQis given\nas the solution of the variational problem\nδ/integraldisplayt2\nt1L/parenleftbig\nq1,...,qn,˙q1,...,˙qn,t/parenrightbig\ndt!= 0.\nThe variational integral can be expressed equivalently in p arametric form if one replaces the\ntimetas the independent variable with a new system evolution para meter,s. With\nqn+1=t,˙qi=dqi/ds\ndqn+1/ds,\nwe obtain (Lanczos 1949, Arnold 1989)\nδ/integraldisplays2\ns1L/parenleftbigg\nq1,...,qn+1,dq1/ds\ndqn+1/ds,...,dqn/ds\ndqn+1/ds/parenrightbiggdqn+1\ndsds!= 0. (A.1)\nThe integrand of (A.1) thus defines the extended Lagrangian L1:TQ1→R,\nL1/parenleftbigg\nq1,dq1\nds/parenrightbigg\n=L/parenleftbigg\nq1,...,qn+1,dq1/ds\ndqn+1/ds,...,dqn/ds\ndqn+1/ds/parenrightbiggdqn+1\nds, (A.2)Hamiltonian dynamics on T∗Q1 22\nwithq1= (q,t)∈Q×R=Q1the extended configuration space vector. The local coordina te\nrepresentation of the actual system path (qi(s),˙qi(s))⊂TQ1is now given as the solution of\nthe variational problem\nδ/integraldisplays1\ns0L1/parenleftbigg\nq1(s),dq1(s)\nds/parenrightbigg\nds!= 0. (A.3)\nAs in the case of the conventional variational problem with a LagrangianL(q,˙q), we find that\n(A.3) is globally fulfilled if the extended set of Euler-Lagr ange equations is satisfied,\n∂L1\n∂q1−d\nds/parenleftbigg∂L1\n∂(dq1/ds)/parenrightbigg\n= 0. (A.4)\nThe following identities are readily derived from (A.2)\n∂L1\n∂q=dt\nds∂L\n∂q,∂L1\n∂(dq/ds)=∂L\n∂˙q, (A.5)\n∂L1\n∂t=dt\nds∂L\n∂t,∂L1\n∂(dt/ds)=L−˙q∂L\n∂˙q, (A.6)\nwhich make it possible to rewrite (A.4) in terms of the conven tional Lagrangian L\ndt(s)\nds/bracketleftbigg∂L\n∂q−d\ndt/parenleftbigg∂L\n∂˙q/parenrightbigg/bracketrightbigg\n= 0,dq(s)\nds/bracketleftbigg∂L\n∂q−d\ndt/parenleftbigg∂L\n∂˙q/parenrightbigg/bracketrightbigg\n= 0. (A.7)\nWe observe that both equations (A.7) are fulfilled if and only if the equations in brackets\n— the conventional Euler-Lagrange equations — are satisfied . Thus, the extended set of\nEuler-Lagrange equations (A.4) is equivalent to the conven tional set and does notprovide\nan additional equation of motion for t=t(s). This result corresponds to the observation\nfrom (10) that the extended set of canonical equations does n ot furnish a substantial canonical\nequation for dt/ds, thus leaving the parameterization of time undetermined. N evertheless,\nwe may take advantage of having introduced the extended Lagr angianL1: it is now possible\nto mapL1by means of a Legendre transformation into an extended Hamil tonianH1whose\ndomain is the symplectic manifold T∗Q1.\nAppendix A.2. Extended Hamiltonian H1as the Legendre transform of the extended\nLagrangian L1\nReplacing all derivatives dqi/dswithcdqi/ds,c∈R, we realize that L1from Equation (A.2)\nis a homogeneous form of first order in the n+ 1variables dq1/ds,...,dqn+1/ds. Hence,\nEuler’s theorem on homogeneous functions yields the identi ty (Lanczos 1949)\nL1≡n+1/summationdisplay\ni=1∂L1\n∂(dqi/ds)dqi\nds. (A.8)\nFor the indices i= 1,...,n , the partial derivatives of L1along the fibres dqi/dsdefine the\ngeneralized canonical momenta pi,\n∂L1\n∂(dqi/ds)≡∂L\n∂˙qi≡pi, i= 1,...,n. (A.9)\nThe partial derivative of L1with respect to dqn+1/dsfollows from its definition in (A.2) as\n∂L1\n∂(dqn+1/ds)≡L−n/summationdisplay\ni=1pi˙qi≡ −H(q,p,t). (A.10)Hamiltonian dynamics on T∗Q1 23\nInserting (A.9) and (A.10) into the identity (A.8), the exte nded Lagrangian L1takes on the\nform\nL1≡n/summationdisplay\ni=1pidqi\nds−H(q,p,t)dqn+1\nds. (A.11)\nWith the extended Hamiltonian H1as the Legendre transform of L1\nH1≡n+1/summationdisplay\ni=1pidqi\nds−L1,\nwe find, inserting the identity for L1from (A.11), that the index n+ 1 furnishes the only\nremaining term\nH1≡/bracketleftig\nH(q,p,t)+pn+1/bracketrightigdqn+1\nds. (A.12)\nFrom Equation (A.10), we conclude that in the description of the extended phase space T∗Q1\nthe canonical variable pn+1(s)∈R, i.e. the derivative of L1along the fibre dt/ds, isuniquely\ndetermined by the negative value −e(s)of the Hamiltonian H:T∗Q×R→R\npn+1(s)≡ −e(s)/negationslash≡=−H(q(s),p(s),t(s)). (A.13)\nWithpn+1≡ −eandqn+1≡t, the extended Hamiltonian H1from (A.12) is finally obtained\nas the implicit function\nH1(q,p,t,e)≡/bracketleftig\nH(q(s),p(s),t(s))−e(s)/bracketrightigdt(s)\nds/negationslash≡= 0. (A.14)\nThe extended Hamiltonian (A.14) coincides with the Hamilto nianH1previously obtained\nin (6). As the extended Hamiltonian H1does not vanish identically inT∗Q1, the partial\nderivatives of (A.14) are non-zero in general. Therefore, i n contrast to the assertion of\nLanczos (Lanczos 1949, p 187), the extended Hamiltonian H1must not be eliminated from\nthe integrand of the generalized variational problem (8).\nAcknowledgment\nThe author is indebted to C. Riedel (GSI) for his essential co ntributions to this work.\nReferences\nAbraham R and Marsden J E 1978 Foundations of Mechanics 2nd edn (Boulder, CO: Westview Press)\nArnold V I 1989 Mathematical Methods of Classical Mechanics 2nd edn (New York: Springer)\nAsorey M, Cari˜ nena J F and Ibort L A 1983 J. Math. Phys. 242745\nCari˜ nena J F and Ibort L A 1987 Nuovo Cimento 98172\nCari˜ nena J F, Ibort L A and Lacomba E A 1988 Celes. Mech. 42201–13\nFrankel Th 2001 The Geometry of Physics (Cambridge: Cambridge University Press)\nGotay M J 1982 Proc. Am. Math. Soc. 84111\nGotay M J, Isenberg J, Marsden J E, and Montgomery R 1997 Momentum Maps and Classical Relativistic Fields:\nPart I. Covariant Field Theory online at http://www.math.hawaii.edu/˜gotay/GIMMsyI.p df\nJos´ e J V and Saletan E J 1998 Classical Dynamics (Cambridge: Cambridge University Press)\nKuwabara R 1984 Rep. Math. Phys. 1927–38\nLanczos C 1949 The Variational Principles of Mechanics (Toronto, Ontario: University of Toronto Press) Reprint\n1986 4th edn (New York: Dover Publications)\nLeach P G L 1977 J. Math. Phys. 181608\nLeach P G L 1978 SIAM J. Appl. Math. 34496\nLichtenberg A J and Lieberman M A 1992 Regular and Chaotic Motion (New York: Springer)\nMarsden J E and Ratiu T S 1999 Introduction to Mechanics and Symmetry (New York: Springer)Hamiltonian dynamics on T∗Q1 24\nSiegel C L and Moser J K 1971 Lectures on Celestial Mechanics (Berlin: Springer)\nSorge H, St¨ ocker H and Greiner W 1989 Nucl. Phys. A 498567c\nStiefel E L and Scheifele G 1971 Linear and Regular Celestial Mechanics (Berlin: Springer)\nStruckmeier J and Riedel C 2001 Phys. Rev. E 64026503\nStruckmeier J and Riedel C 2002a Ann. Phys. Lpz. 1115–38\nStruckmeier J and Riedel C 2002b Phys. Rev. E 66066605\nStruckmeier J 2006 Phys. Rev. E 74026209\nStump D R 1998 J. Math. Phys. 393661\nSynge J L 1960 Encyclopedia of Physics V ol 3/1 ed S Fl¨ ugge (Berlin: Springer)\nSzebehely V 1967 Theory of Orbits (New York: Academic Press)\nThirring W 1977 Lehrbuch der Mathematischen Physik (Wien: Springer)\nTsiganov A V 2000 J. Phys. A: Math. Gen. 334169–82\nWodnar K 1995 Symplectic mappings and Hamiltonian systems Perturbation theory and chaos in nonlinear dynamics\nwith emphasis to celestial mechanics ed J Hagel, M Cunha and R Dvorak R (Portugal: Universidade da M adeira,\nFunchal)"    },    {        "title": "2201.08229v2.Derivation_of_the_linear_Boltzmann_equation_from_the_damped_quantum_Lorentz_gas_with_a_general_scatterer_configuration.pdf",        "content": "DERIVATION OF THE LINEAR BOLTZMANN EQUATION FROM THE DAMPED\nQUANTUM LORENTZ GAS WITH A GENERAL SCATTERER CONFIGURATION\nJORY GRIFFIN\nAbstract . It is a fundamental problem in mathematical physics to derive macroscopic transport equa-\ntions from microscopic models. In this paper we derive the linear Boltzmann equation in the low-density\nlimit of a damped quantum Lorentz gas for a large class of deterministic and random scatterer config-\nurations. Previously this result was known only for the single-scatterer problem on the flat torus, and\nfor uniformly random scatterer configurations where no damping is required. The damping is critical\nin establishing convergence – in the absence of damping the limiting behaviour depends on the exact\nconfiguration under consideration, and indeed, the linear Boltzmann equation is not expected to appear\nfor periodic and other highly ordered configurations.\n1. Introduction\nThe quantum Lorentz gas is a model of conductivity in which a single quantum particle (electron)\nevolves in the presence of a potential given by an infinite collection of compactly supported profiles\nplaced on a discrete point set P\u001aRd. These profiles, called scatterers from here on, represent the\nrelatively heavy molecules of the background material. The point set one should choose, and the\nlimiting behaviour one should expect, is thus dependent on the microscopic structure of the material\nin question. A fundamental question is whether one can, for a given P, derive a macroscopic\ntransport equation, e.g. the linear Boltzmann equation, from this microscopic model.\nSome reasonable choices for Pare (i) a realisation of a (Poisson) point process to model disordered\nmaterials or an environment with random impurities, (ii) a lattice, union of lattices, or other periodic\nset to model metals and heavily ordered materials, (iii) aperiodic point sets to model quasicrystals.\nIn the classical (non-quantum) setting, the pioneering papers [8, 20, 1] established convergence of\nthe Liouville equation to the linear Boltzmann equation in the low-density (Boltzmann-Grad) limit,\nprovided the scatterer configuration Pis random, e.g. given by a homogeneous Poisson point\nprocess. More recent work has shown that in the case of crystals [2, 15] or other point sets with\nlong-range correlations (e.g. quasicrystals) [16], different transport equations will emerge in the\nBoltzmann-Grad limit due to correlations that arise between consecutive collisions. These findings\nare somewhat mirrored in the quantum setting: On one hand, Eng and Erdös [5] proved convergence\nto the linear Boltzmann equation for random potentials in the low-density limit, following analogous\nresults in the weak-coupling limit by Spohn [19] and Erdös and Yau [6]; on the other hand, recent\nevidence suggests that a different transport law emerges in the same scaling limit when the potential\nis periodic [9, 10].\nThe motivation for the work of the present paper is Castella’s striking observation [3, 4] that the\nspace-homogeneous linear Boltzmann equation can be obtained as the limit of the von Neumann\nequation on the flat torus with a small scatterer if some damping is introduced. In particular, the\nevolution for ‘diagonal’ terms is undamped (where incoming and outgoing momenta are equal), and\nthe evolution for ‘nondiagonal’ terms is exponentially damped in time (where incoming and outgo-\ning momenta differ). This exponential damping of nondiagonal terms models phenomenologically\nthe interaction of the system with, for example, a bath of photons or phonons, see [3] and references\ntherein, in particular [21, Chapter 7-3]. (Also [14, 11]). In a rough sense, interactions with a ‘noisy’\nexternal environment can lead to ‘random’ perturbations of the momenta. When the incoming and\noutgoing momenta are equal, these random perturbations tend to cancel one another out, but when\nthe incoming and outgoing momenta are distinct, these random perturbations persist and lead to\nexponential decay. Here we will show, using such a damping mechanism, that the full(position\nResearch supported by EPSRC grant EP/S024948/1.\n1arXiv:2201.08229v2  [math-ph]  8 Aug 20222 JORY GRIFFIN\ndependent) linear Boltzmann equation can be obtained as a limit of the quantum Lorentz gas in Rd\nfor a general class of scatterer configurations which includes both periodic and disordered examples.\nThe proof differs from that of the main Theorem in [3, 4] in a number of ways. If the problem is\nrestricted to the torus one has discrete momenta, and this allows Castella to (i) introduce a damping\nwhich is constant on all nondiagonal terms, but zero for diagonal terms, and then (ii) derive a\ntransport equation for the diagonal part of the density matrix before taking any scaling limit to\neliminate the nondiagonal terms - the convergence is then established on the level of this transport\nequation. If one instead considers the problem in Rdthe momenta are continuous and this approach\nno longer works. Instead, we (i) introduce a smooth damping function which is zero for diagonal\nterms and approaches some constant value smoothly as one moves away from the diagonal, and (ii)\ncompute the limit of the full Duhamel expansion, separating damped and undamped regions using\na combinatorial argument, and then show that the resulting expression satisfies the linear Boltzmann\nequation. The damping function in particular must be carefully chosen to scale in the correct way in\nthe small scatterer limit in order to obtain this limiting behaviour, and one must be careful in dealing\nwith the intermediate regime between the undamped and fully damped terms.\nWe assume in the following that d\u00153. The time evolution of the quantum Lorentz gas is described\nby the Schrödinger equation\n(1.1)ih\n2p¶ty(t,x) =Hh,ly(t,x),\nwhere\n(1.2) Hh,l=\u0000h2\n8p2D+å\nq2Pl(rd\u00001q)W(r\u00001(x\u0000q)).\nThe single-site potential Wis assumed to be in the Schwartz class S(Rd),r>0 is the effective radius\nof each scatterer, and the lis a cut-off function which we assume to be smooth with compact support\ncontained within the unit ball. The classical mean free path length is O(r1\u0000d), solhas the effect of\ntruncating the potential on the macroscopic scale. The assumption that lis compactly supported is\na technical one to avoid infinite summation and it’s possible that it can be weakened siginificantly.\n(For example, one may ideally wish to take l(q)constant.)\nWe assume thatP \u001aRdis a uniformly discrete point set with asymptotic density one. This\ntechnical requirement is introduced so that Pprovides a suitable set over which a d-dimensional\nRiemann sum can be computed, and that this Riemann sum converges with an explicit error term.\nIn particular, we require that there exists bP,cP>0 such thatkq\u0000q0k>bPfor all q,q02P with\nq6=q0and for every g2C¥\nc(Rd), 0<e<1 we have\n(1.3) edå\nq2Pg(eq)=Z\nRdg(x)dx+O(ecPkrgk).\nDeterministic examples of Pthat satisfy these assumptions are lattices (e.g. P=Zd) and large\nclasses of quasicrystals (e.g. the vertices of a Penrose tiling). For random examples one can take the\nso-called Matérn processes [17] in which a realisation of a homogeneous Poisson point process is then\nthinned to remove clusters, or a random displacement model, in which each point in a deterministic\nset (e.g. a lattice) is randomly perturbed by a small amount. (As long as the random perturbation\nis small enough the resulting point set will be uniformly discrete provided the initial point set is\nuniformly discrete). The restriction to uniform discreteness likely can be weakened. For example,\none may wish to take bPto depend onkqkandkq0k, or insist that that kq\u0000q0k>bPholds only for\nalmost all pairs of points in P. In both cases we expect the same results to hold.\nTo study the quantum transport and the Boltzmann-Grad limit, it is convenient to move to the\nequivalent Heisenberg picture and study the quantum Liouville equation (or von Neumann equa-\ntion/ backward Heisenberg equation)\n(1.4) ¶trt=\u00002pi\nh[Hh,l,rt]DERIVATION OF THE LINEAR BOLTZMANN EQUATION 3\nfor a density operator rt. We introduce damping to the system by considering the a-damped von\nNeumann equation (in momentum representation):\n(1.5) ¶tbrt(y,y0) =\u00002pi\nh[bHh,l,brt](y,y0)\u0000ad\nh\u0000\n1\u0000G(ah1\u0000d(y\u0000y0))\u0001brt(y,y0),\nwhere a\u00150 is the strength of the damping and G2C¥\nc(Rd)with values in [0, 1]so that G(y) = 1\nin some neighbourhood of the origin and G(y) = 0 forkyk>1. Eq. (1.5) describes the averaged\nquantum dynamics of a particle subject to white noise in momentum where G(y)is the covariance\nfunction of the corresponding Gaussian random field. We refer the reader to [7] for detailed rigorous\ntreatment of white noise perturbations in phase space, and to [12, 13, 18] for the more standard\nsetting in position space.\nIn order to establish the convergence of the damped von Neumann equation (1.5) to the linear\nBoltzmann equation, we need to carefully prepare the initial condition of rtrelative to a classical\nphase space density a. Following the approach in [9], we achieve this by the rescaled Weyl quantisa-\ntion Opr,h(a)of a classical phase-space symbol a:\n(1.6) Opr,h(a)f(x) =rd(d\u00001)/2hd/2Z\nR2da(1\n2rd\u00001(x+x0),hy)e((x\u0000x0)\u0001y)f(x0)dx0dy,\nwith the shorthand e (x):=e2pix. This means we measure momenta on the semi-classical scale,\nand position on the scale of the classical mean free path. Although other scalings are possible, we\nwill here focus on the case when r=h. This will ensure that scattering remains truly quantum in\nthe limit r!0, and that we see the full quantum T-operator in the limit. For the single scatterer\nHamiltonian\n(1.7) Hm=\u00001\n8p2D+mW(x),\nwe define the T-operator at energy Eto be the operator satisfying\n(1.8) Tm(E) =mOp1,1(W)\u0012\n1+1\nE\u0000H0+i0+Tm(E)\u0013\nand write Tm(y,y0)for its integral kernel in momentum representation at energy E=1\n2kyk2. We\nhave the explicit expansion (understood in terms of distributions)\nTm(y,y0) =mbW(y\u0000y0)\n+¥\nå\n`=1(\u00002pi)`m`+1Z\nRd`bW(y\u0000y1)\u0001\u0001\u0001bW(y`\u0000y0)\n\u0002[`\nÕ\ni=1Z¥\n0e(1\n2(kyk2\u0000kyik2)u)du]dy1\u0001\u0001\u0001dy`(1.9)\nwhere\n(1.10) bW(y):=Z\nRdW(x)e(\u0000x\u0001y)dx.\nTheorem 1. Let a ,b be in the Schwartz class S(Rd\u0002Rd). Ifrtis a solution of the a-damped von Neumann\nequation (1.5) subject to the initial condition r0=Opr,h(a), then for t >0\n(1.11) lim\na!0lim\nr=h!0Tr(rr1\u0000dtOpr,h(b)) =Z\nR2df(t,x,y)b(x,y)dxdy\nwhere f (t,x,y)solves the linear Boltzmann equation\n(1.12)8\n<\n:\u0000\n¶t+y\u0001rx\u0001\nf(t,x,y) =Z\nRd\u0002\nSl(x)(y,y0)f(t,x,y0)\u0000Sl(x)(y0,y)f(t,x,y)\u0003\ndy0\nf(0,x,y) =a(x,y)\nwith the collision kernel\n(1.13) Sm(y,y0) =8p2jTm(y,y0)j2d(kyk2\u0000ky0k2).4 JORY GRIFFIN\nNote that the limits a!0 and r!0 do not commute. Indeed if one first takes the limit a!0\nfollowed by r!0 one is back in the situation of [5, 9, 10] where the limit depends on the precise\nnature ofP. The striking feature of Theorem 1 is that the limit is the same for all admissible scatterer\nconfigurationsP, from periodic to highly disordered.\nIn Section 2 we perform the Duhamel expansion of the solution to the damped Heisenberg equa-\ntion, this allows us to obtain an explicit formal expansion for the solution as a power series in l(x).\nIn Section 3 we perform a carefully chosen partition of unity which allows us to isolate the damped\nand undamped regions. In Section 4 we perform the low-density followed by the zero damping\nlimit on this reorganised series. This Section constitutes the bulk of the paper: we first show that the\nsum of all nondiagonal terms converges, and then vanishes in the limit; then we show that the sum\nof all diagonal terms converges, and hence that the entire series converges to some f(t,x,y)given\nexplicitly as an expansion in l. In Section 5 we prove that our limiting expression coincides with a\nsolution of the linear Boltzmann equation using [4].\n2. Deriving a Formal Expansion\nIn the momentum representation, the kernel of the Hamiltonian (1.7) reads\n(2.1) bHh,l(y,y0) =h2\n2kyk2d(y\u0000y0) +cOp(V)(y,y0)\nwhere\n(2.2) cOp(V)(y,y0) =rdå\nq2Pl(rd\u00001q)e(q\u0001(y0\u0000y))bW(r(y\u0000y0)).\nInserting these into (1.5) yields, after a suitable variable substitution,\n¶tbrt(y,y0) =\u0000 \npih(kyk2\u0000ky0k2) +ad\nh(1\u0000G(ah1\u0000d(y\u0000y0)))!\nbrt(y,y0)\n\u00002pi\nhrdå\nq2Pl(rd\u00001q)Z\nRddze(\u0000q\u0001z)bW(rz) [brt(y\u0000z,y0)\u0000brt(y,y0+z)].(2.3)\nFollowing Castella [3], it will be convenient to write\n(2.4) brt(y\u0000z,y0)\u0000brt(y,y0+z) =\u0000å\ng2f0,1g(\u00001)gbrt(y\u0000gz,y0+¯gz)\nwith ¯g:=1\u0000g. The Duhamel principle for (2.3) yields\nbrt(y,y0) =e(\u0000h\n2(kyk2\u0000ky0k2)t)e\u0000ad\nh(1\u0000G(ah1\u0000d(y0\u0000y)))tbr0(y,y0)\n+2pi\nhrdå\nq2Pl(rd\u00001q)Z\nRddze(\u0000q\u0001z)bW(rz)å\ng2f0,1g(\u00001)g\n\u0002Zt\n0e(\u0000h\n2(kyk2\u0000ky0k2) (t\u0000s))e\u0000ad\nh(1\u0000G(ah1\u0000d(y0\u0000y))) ( t\u0000s)brs(y\u0000gz,y0+¯gz)ds.(2.5)DERIVATION OF THE LINEAR BOLTZMANN EQUATION 5\nIterating this expression and making the substitutions u0=t\u0000s1and uj=sj\u0000sj+1forj\u00151 we\nobtain the formal expansion\nbrt(y,y0) =e(\u0000h\n2(kyk2\u0000ky0k2)t)e\u0000ad\nh(1\u0000G(ah1\u0000d(y0\u0000y)))tbr0(y,y0)\n+¥\nå\nm=1(2pih\u00001rd)må\nq1,\u0001\u0001\u0001,qm2Pl(rd\u00001q1)\u0001\u0001\u0001l(rd\u00001qm)\n\u0002Z\nRmddz1\u0001\u0001\u0001dzme(\u0000q1\u0001z1\u0000\u0001\u0001\u0001\u0000 qm\u0001zm)bW(rz1)\u0001\u0001\u0001bW(rzm)\n\u0002å\ng1,\u0001\u0001\u0001,gm2f0,1g(\u00001)g1+\u0001\u0001\u0001+gmZ\n4m(t)du0\u0001\u0001\u0001dum\n\u0002\"\nm\nÕ\nj=0e(\u0000h\n2(ky\u0000j\nå\ni=1gizik2\u0000ky0+j\nå\ni=1¯gizik2)uj)#\n\u0002\"\nm\nÕ\nj=0e\u0000ad\nh(1\u0000G(ah1\u0000d(y0\u0000y+åj\ni=1zi)))uj#\nbr0(y\u0000m\nå\ni=1gizi,y0+m\nå\ni=1¯gizi)(2.6)\nwhere4m(t)\u001aRm+1is the set\n4m(t) =f(u0, . . . , um)2Rm+1\n+ju0+\u0001\u0001\u0001+um=tg.\nWe now wish to compute Tr (rr1\u0000dtOpr,r(b)) = Tr(brr1\u0000dtcOpr,r(b)), where rtsolves the damped von\nNeumann equation (1.5) with initial condition r0=Opr,r(a). The kernel of Opr,h(a)as defined in\n(1.6) reads in momentum representation\n(2.7) cOpr,h(a)(y,y0) =r\u0000d(d\u00001)/2hd/2˜a(r1\u0000d(y\u0000y0),h\n2(y+y0))\nwhere ˜a(x,y) =R\nRda(x,y)e(\u0000x\u0001x)dx. Inserting these in (2.6) yields the expansion\n(2.8) Tr (rr1\u0000dtOpr,r(b)) =¥\nå\nm=0(2pi)mAa,r\nm(t)\nwhere\n(2.9)Aa,r\n0(t) =r\u0000d(d\u00002)Z\nR2ddydhe(\u00001\n2r2\u0000d(kyk2\u0000khk2)t)e\u0000adr\u0000d(1\u0000G(ar1\u0000d(h\u0000y)))t\n\u0002˜a(r1\u0000d(y\u0000h),r\n2(y+h))˜b(r1\u0000d(h\u0000y),r\n2(h+y)),\nand for m\u00151\nAa,r\nm(t) =r(m\u0000d)(d\u00001)+då\nq1,\u0001\u0001\u0001,qm2Pl(rd\u00001q1)\u0001\u0001\u0001l(rd\u00001qm)\n\u0002å\ng1,\u0001\u0001\u0001,gm2f0,1g(\u00001)g1+\u0001\u0001\u0001+gmZ\nR2ddydhZ\nRmddz1\u0001\u0001\u0001dzm\n\u0002\"\nm\nÕ\ni=1e(\u0000qi\u0001zi)bW(rzi)#Z\n4m(r1\u0000dt)du0\u0001\u0001\u0001dum\"\nm\nÕ\nj=0e\u0000ad\nr(1\u0000G(ar1\u0000d(h\u0000y+åj\ni=1zi)))uj#\n\u0002\"\nm\nÕ\nj=0e(r\n2(kh+j\nå\ni=1¯gizik2\u0000ky\u0000j\nå\ni=1gizik2)uj)#\n\u0002˜a(r1\u0000d(y\u0000h\u0000m\nå\ni=1zi),r\n2(y+h\u0000m\nå\ni=1(gi\u0000¯gi)zi))˜b(r1\u0000d(h\u0000y),r\n2(h+y)).(2.10)6 JORY GRIFFIN\nWe first make the substitution h!y+rd\u00001h. Then, make the substitution y!r\u00001yand for all j,\nmake the substitutions uj!ruj,zj!r\u00001zj. This yields the expression\nAa,r\nm(t) =å\nq1,\u0001\u0001\u0001,qm2Pl(rd\u00001q1)\u0001\u0001\u0001l(rd\u00001qm)å\ng1,\u0001\u0001\u0001,gm2f0,1g(\u00001)g1+\u0001\u0001\u0001+gm\n\u0002Z\nR2ddydhZ\nRmddz1\u0001\u0001\u0001dzm\"\nm\nÕ\ni=1e(\u0000r\u00001qi\u0001zi)bW(zi)#\n\u0002Z\n4m(r\u0000dt)du0\u0001\u0001\u0001dum\"\nm\nÕ\nj=0e(xjuj)e\u0000ad(1\u0000G(a(h+r\u0000dåj\ni=1zi)))uj#\n\u0002˜a(\u0000h\u0000r\u0000dm\nå\ni=1zi,y\u0000m\nå\ni=1gizi+1\n2rdh+1\n2m\nå\ni=1zi))˜b(h,y+1\n2rdh)(2.11)\nwhere xjis given by\nxj=1\n2(ky+j\nå\ni=1¯gizi+rdhk2)\u0000ky\u0000j\nå\ni=1gizik2)\n= (y\u0000j\nå\ni=1gizi)\u0001(j\nå\ni=1zi+rdh) +1\n2kj\nå\ni=1zi+rdhk2.(2.12)\nThe limit of the first term can be computed immediately.\nProposition 1.\n(2.13) lim\na!0lim\nr!0Aa,r\n0(t) =Z\nR2ddxdya(x\u0000ty,y)b(x,y).\nProof. We have that\n(2.14)Aa,r\n0(t) =Z\nR2ddydhe(y\u0001ht+rdkhk2t)e\u0000r\u0000dad(1\u0000G(ah))t˜a(\u0000h,y+1\n2rdh)˜b(h,y+1\n2rdh).\nThe functions ˜aand ˜bare rapidly decaying so this is uniformly bounded as r!0. By dominated\nconvergence we thus obtain\n(2.15) lim\nr!0Aa,r\n0(t) =Z\nR2ddydhe(y\u0001ht)˜a(\u0000h,y)˜b(h,y)1[G(ah) =1].\nAgain, by the rapid decay of ˜aand ˜bthis converges in the limit a!0 and we obtain\n(2.16) lim\na!0lim\nr!0Aa,r\n0(t) =Z\nR2ddydhe(y\u0001ht)˜a(\u0000h,y)˜b(h,y) =Z\nR2ddxdya(x\u0000ty,y)b(x,y).\n\u0003\n3. M anipulating the Expansion\nFor the higher order terms we perform a partitioning of the ziintegration region. To see why,\nnote that (2.11) has a product of factors of the form\n(3.1) e\u0000ad(1\u0000G(a(h+r\u0000dåj\ni=1zi)))uj.\nIf the argument a(h+r\u0000dåj\ni=1zi)is large, then this entire factor becomes e\u0000aduj, and hence the uj\nintegral is exponentially damped. Our partition will be precisely into these damped and undamped\nregions. LetS=fs1,\u0001\u0001\u0001,spg\u001af 0,\u0001\u0001\u0001,mgwith s1=0 and sp=mand write Pmfor the set of all\nsuchS. Define cS:Rd(m\u00001)!Rby\n(3.2) cS(z1, . . . , zm\u00001) =\"p\u00001\nÕ\nj=2c(zsj)#\"\nÕ\nj/2S(I\u0000c)(zj)#\nwhere c2C¥\nc(Rd!R)is decreasing inkzksuch that c(z) =1 for allkzk<1 and c(z) =0 for all\nkzk>2. This implies the bound\n(3.3) kckL1\u0014vol(B2)DERIVATION OF THE LINEAR BOLTZMANN EQUATION 7\nwhereBris the d-ball of radius r. Note that cSforms a partition of unity: åS2PmcS=1; and also\nthat by assumption on the support of G\n(3.4) (1\u0000c(az))e\u0000ad(1\u0000G(az))u= (1\u0000c(az))e\u0000adu.\nWe put g= (g1, . . . , gm)and rewrite (2.11) as\n(3.5) Aa,r\nm(t) =å\ng2f0,1gm(\u00001)g1+\u0001\u0001\u0001+gmå\nS2PmAa,r\ng,S(t)\nwhere\nAa,r\ng,S(t) =å\nq1,\u0001\u0001\u0001,qm2Pl(rd\u00001q1)\u0001\u0001\u0001l(rd\u00001qm)Z\nR(m+2)ddydhdz1\u0001\u0001\u0001dzm\n\u0002bW(z1)\u0001\u0001\u0001bW(zm)e(\u0000r\u00001q1\u0001z1\u0000\u0001\u0001\u0001\u0000 r\u00001qm\u0001zm)\n\u0002Z\n4m(r\u0000dt)du0\u0001\u0001\u0001dum\"\nm\nÕ\nj=0e(xjuj)e\u0000ad(1\u0000G(a(h+r\u0000dåj\ni=1zi)))uj#\n\u0002cS(a(h+r\u0000dz1), . . . , a(h+r\u0000d(z1+\u0001\u0001\u0001+zm\u00001)))\n\u0002˜a(\u0000h\u0000r\u0000dm\nå\ni=1zi,y\u0000m\nå\ni=1gizi+1\n2(rdh+m\nå\ni=1zi))˜b(h,y+1\n2rdh).(3.6)\nNote that all elements in the complement of Soccur injSj\u0000 1=p\u00001 contiguous blocks (possibly\nof size zero). Write ki=si+1\u0000si\u00001\u00150 for the number of elements in the ithblock. To simplify\nnotation we will use double subscripts to refer to the jth element of the ith block, e.g. zij:=zsi+j\nwhere 0\u0014j\u0014ki. When j=0 we will write zsiorzi0interchangeably. We then put h=h1and for\ni=2,\u0001\u0001\u0001,pwe make the change of coordinates for z(i+1)0by\nhi+1=r\u0000d(z(i+1)0+ki\nå\nj=1zij).\nThis gives a factor of rd2(p\u00001). In these new coordinates we have that\n(3.7) q1\u0001z1+\u0001\u0001\u0001+qm\u0001zm=p\u00001\nå\ni=1(rdq(i+1)0\u0001hi+1+ki\nå\nj=1(qij\u0000q(i+1)0)\u0001zij).\nThe product of potentials can be written\n(3.8) bW(z1)\u0001\u0001\u0001bW(zm) =p\u00001\nÕ\ni=1bW(rdhi+1\u0000ki\nå\nj=1zij)ki\nÕ\nj=1bW(zij).\nBy convention let us assume that gs1=g0=0. For i=1,\u0001\u0001\u0001,pwe have that xsi=rdziwhere\n(3.9) zi= (y\u0000i\u00001\nå\nk=1kk\nå\n`=1(gk`\u0000g(k+1)0)zk`)\u0001(i\nå\nk=1hk) +1\n2rd(ki\nå\nk=1¯gskhkk2\u0000ki\nå\nk=1gskhkk2).\nFori=1, . . . , p\u00001 and j=1, . . . , kiwe have that\n(3.10) xij=1\n2(ky+i\u00001\nå\nk=1kk\nå\n`=1(¯gk`\u0000¯g(k+1)0)zk`+j\nå\n`=1¯gi`zi`+rdi\nå\nk=1¯gk0hkk2\n\u0000ky\u0000i\u00001\nå\nk=1kk\nå\n`=1(gk`\u0000g(k+1)0)zk`\u0000j\nå\n`=1gi`zi`\u0000rdi\nå\nk=1gk0hkk2).\nThe functions ˜aandcSbecome\n(3.11) ˜a(\u0000h\u0000r\u0000dm\nå\ni=1zi,y\u0000m\nå\ni=1gizi+1\n2(rdh+m\nå\ni=1zi))\n=˜a(\u0000p\nå\ni=1hi,y\u0000p\u00001\nå\ni=1ki\nå\nj=1(gij\u0000g(i+1)0)zij\u00001\n2rdp\nå\ni=1(gi0\u0000¯gi0)hi),8 JORY GRIFFIN\nand\n(3.12) cS(a(h+r\u0000dz1), . . . , a(h+r\u0000d(z1+\u0001\u0001\u0001+zm\u00001)))\n= p\u00001\nÕ\ni=2c(ai\nå\nk=1hk)! p\u00001\nÕ\ni=1ki\nÕ\nj=1(I\u0000c)(a(i\nå\nk=1hk+r\u0000dj\nå\n`=1zi`))!\n.\nWe write H= (h1,\u0001\u0001\u0001,hp)and\nZS= (z11, . . . , z1k1, . . . , z(p\u00001)1, . . . , z(p\u00001)kp\u00001)\nfor the collection of remaining zivariables. Make the substitution usi=r\u0000dni, then equation (3.6)\ncan now be written\nAa,r\ng,S(t) =rd(d\u00001)(p\u00001)å\nq1,\u0001\u0001\u0001,qm2Pl(rd\u00001q1)\u0001\u0001\u0001l(rd\u00001qm)Z\nR(m+2)ddydHdZS\n\u0002Fr\ng,S(ZS,H,y)\"p\u00001\nÕ\ni=1ki\nÕ\nj=1(I\u0000c)(a(i\nå\nk=1hk+r\u0000dj\nå\n`=1zi`))#\n\u0002\"p\u00001\nÕ\ni=1e \n\u0000rd\u00001q(i+1)0\u0001hi+1\u0000r\u00001ki\nå\nj=1(qij\u0000q(i+1)0)\u0001zij!#\n\u0002\"p\u00001\nÕ\ni=2c(ai\nå\nk=1hk)#Z\nRm+1\n+d(n1+\u0001\u0001\u0001+np+rdå\ni/2Sui\u0000t)\n\u0002\"p\nÕ\ni=1e(zini)e\u0000ad(1\u0000G(aåi\nk=1hk))r\u0000dnidni#\"\nÕ\ni/2Se(xiui)e\u0000aduidui#(3.13)\nwhere Fr\ng,Sis defined by\nFr\ng,S(ZS,H,y) =\"p\u00001\nÕ\ni=1\u0012\nbW(rdhi+1\u0000ki\nå\nj=1zij)ki\nÕ\nj=1bW(zij)\u0013#\n˜b(h1,y+1\n2rdh1)\n\u0002˜a(\u0000p\nå\ni=1hi,y\u0000p\u00001\nå\ni=1ki\nå\nj=1(gij\u0000g(i+1)0)zij\u00001\n2rdp\nå\ni=1(gi0\u0000¯gi0)hi).(3.14)\n4. Computing the Limit r!0\nWe first separate diagonal and nondiagonal terms by writing\n(4.1) Aa,r\ng,S(t) =Aa,r\nd,g,S(t) +Aa,r\nnd,g,S(t)\nwhereAr,a\nd,g,S(t)is defined by restricting (3.13) to the diagonal qij=q(i+1)0for all i=1, . . . , p\u00001\nand j=1, . . . , ki. The nondiagonal term contains the remainder of the summation.\n4.1.Nondiagonal Terms.\nProposition 2 (Upper bound on nondiagonal terms) .Fora,t>0, there exists a constant C >0\ndepending on a,t,W,a and b such that\n\f\f\fAa,r\nnd,g,S(t)\f\f\f\u0014Cmrlog2(1+r1\u0000db\u00001\nP)klkm\n1,¥. (4.2)\nwhereklk1,¥=maxfklkL1,klkL¥g.\nThe idea of the proof is simple: we note that (3.13) has the form of a Fourier transform in the\nzijvariables; if we can show that this function, as well as the partial derivative Õp\u00001\ni=1Õki\nj=1Õd\nk=1¶zijk\nof this function, is in L1(Rd(m+1\u0000p)), then the Fourier transform is bounded and decays at least\nlinearly in each coordinate direction. This will allow us to sum over the nondiagonal terms andDERIVATION OF THE LINEAR BOLTZMANN EQUATION 9\nobtain the logarithmic bound needed. The only issue is in taking this partial derivative. Note that\n(3.13) contains factors of the form\n(4.3)ki\nÕ\nj=1(I\u0000c) \na(i\nå\nk=1hk+r\u0000dj\nå\n`=1zi`)!\n.\nTaking the partial derivative Õki\nj=1Õd\nk=1¶zijkof this factor alone yields (ki!)dterms by the product\nrule. Recall that kimay be as large as m\u00001, so this would preclude us from obtaining an upper\nbound of the form Cmas is needed. The solution to this is to first perform a carefully chosen variable\nsubstitution. Write Bi:=f1, . . . , kigand define\ntij=j\nå\nk=1gik,\nmij=ki+1\u0000j\nå\nk=1¯gik.(4.4)\nWe also write ti=tikiandmi=miki– observe that mi=ti+1.\nLemma 1. Let M :Rdm!R(m+1\u0000p)be defined component-wise for i =1, . . . , p\u00001and j =1, . . . , kiby\n(4.5) M(q1, . . . , qm)ij=(\nqi(j+1)\u0000qij+q(i+1)1\u0000q(i+1)(ki+1+1)j=si,i=1, . . . , p\u00002\nqi(j+1)\u0000qijotherwise.\nThen, we have that\nAa,r\nnd,g,S(t) =rd(d\u00001)(p\u00001)å\nQ2Pm\nM(Q)6=0l(rd\u00001q1)\u0001\u0001\u0001l(rd\u00001qm)bJr\nq1m1\u0001\u0001\u0001q(p\u00001)mp\u00001(M(r\u00001q1, . . . , r\u00001qm))\n(4.6)\nwhere the hat denotes the usual Fourier transform and\nJr\nq1m1\u0001\u0001\u0001q(p\u00001)mp\u00001(YS) =Z\nR(p+1)ddydH˜a(\u0000p\nå\ni=1hi,y(p\u00001)sp\u00001\u00001\n2rdp\nå\ni=1(gi0\u0000¯gi0)hi)˜b(h1,y+1\n2rdh1)\n\u0002\"p\u00001\nÕ\ni=1\u0012\n[ti\nÕ\nj=1bW(yi(j\u00001)\u0000yij)]bW(yiti\u0000yimi+rdhi+1)[ki\nÕ\nj=mibW(yij\u0000yi(j+1))]\u0013#\n\u0002\"p\u00001\nÕ\ni=1ki\nÕ\nj=1(I\u0000c)(a(i\nå\nk=1hk+r\u0000d(yimij\u0000yitij)))#\"p\u00001\nÕ\ni=2c(ai\nå\nk=1hk)#\n\u0002\"p\u00001\nÕ\ni=1e\u0010\n\u0000rd\u00001qimi\u0001hi+1\u0011#Z\nRm+1\n+d(n1+\u0001\u0001\u0001+np+rdå\ni/2Sui\u0000t)\n\u0002\"p\nÕ\ni=1e(z0\nini)e\u0000ad(1\u0000G(aåi\nk=1hk))r\u0000dnidni#\"\nÕ\ni/2Se(x0\niui)e\u0000aduidui#\n.(4.7)\nProof. Permute the indices in each block, so that all those indices si+jwith gij=1 come first, in\ntheir original order, and all those indices with gij=0 come last, in reverse order. The equation (3.13)10 JORY GRIFFIN\ncan be written\nAa,r\ng,S(t) =rd(d\u00001)(p\u00001)å\nq1,\u0001\u0001\u0001,qm2Pl(rd\u00001q1)\u0001\u0001\u0001l(rd\u00001qm)Z\nR(m+2)ddydHdZS\n\u0002Fr\ng,S(ZS,H,y)2\n4p\u00001\nÕ\ni=1ki\nÕ\nj=1(I\u0000c)(a(i\nå\nk=1hk+r\u0000dtij\nå\n`=1zi`+r\u0000dki\nå\n`=mijzi`))3\n5\n\u0002\"p\u00001\nÕ\ni=1e \n\u0000rd\u00001q(i+1)0\u0001hi+1\u0000r\u00001ki\nå\nj=1(qij\u0000q(i+1)0)\u0001zij!#\n\u0002\"p\u00001\nÕ\ni=2c(ai\nå\nk=1hk)#Z\nRm+1\n+d(n1+\u0001\u0001\u0001+np+rdå\ni/2Sui\u0000t)\n\u0002\"p\nÕ\ni=1e(zini)e\u0000ad(1\u0000G(aåi\nk=1hk))r\u0000dnidni#\"\nÕ\ni/2Se(x\u0003\niui)e\u0000aduidui#(4.8)\nwhere ziand Fr\ng,Sare defined as before, and\n(4.9) x\u0003\nij=1\n2(ky+i\u00001\nå\nk=1kk\nå\n`=1(¯gk`\u0000¯g(k+1)0)zk`+ki\nå\n`=mijzi`+rdi\nå\nk=1¯gk0hkk2\n\u0000ky\u0000i\u00001\nå\nk=1kk\nå\n`=1(gk`\u0000g(k+1)0)zk`\u0000tij\nå\n`=1zi`\u0000rdi\nå\nk=1gk0hkk2).\nWe now perform the substitutions\n(4.10) zij=(\nyi(j\u00001)\u0000yijj\u0014ti\nyij\u0000yi(j+1)j\u0015mi\nwith the convention yi0=yi(ki+1)=y(i\u00001)si\u00001andsi=ti+g(i+1)0. Note that\n(4.11)ki\nå\nj=1(g(i+1)0\u0000gij)zij=yisi\u0000y(i\u00001)si\u00001.\nWe thus have\nAa,r\ng,S(t) =rd(d\u00001)(p\u00001)å\nq1,\u0001\u0001\u0001,qm2Pl(rd\u00001q1)\u0001\u0001\u0001l(rd\u00001qm)Z\nR(m+2)ddydHdYS\n\u0002Gr\ng,S(YS,H,y)\"p\u00001\nÕ\ni=1ki\nÕ\nj=1(I\u0000c)(a(i\nå\nk=1hk+r\u0000d(yimij\u0000yitij)))#\n\u0002\"p\u00001\nÕ\ni=1e\u0010\n\u0000rd\u00001q(i+1)0\u0001hi+1+r\u00001q(i+1)0\u0001(yimi\u0000yiti)\u0011#\n\u0002\"p\u00001\nÕ\ni=1e\u0010\n\u0000r\u00001(qi1\u0001yi0+ (qi2\u0000qi1)\u0001yi1+\u0001\u0001\u0001+ (qiti\u0000qi(ti\u00001))\u0001yi(ti\u00001)\u0000qiti\u0001yiti)\u0011#\n\u0002\"p\u00001\nÕ\ni=1e\u0010\n\u0000r\u00001(qimi\u0001yimi+ (qi(mi+1)\u0000qimi)\u0001yi(mi+1)+\u0001\u0001\u0001+ (qiki\u0000qi(ki\u00001))\u0001yiki\u0000qiki\u0001yi(ki+1))\u0011#\n\u0002\"p\u00001\nÕ\ni=2c(ai\nå\nk=1hk)#Z\nRm+1\n+d(n1+\u0001\u0001\u0001+np+rdå\ni/2Sui\u0000t)\n\u0002\"p\nÕ\ni=1e(z0\nini)e\u0000ad(1\u0000G(aåi\nk=1hk))r\u0000dnidni#\"\nÕ\ni/2Se(x0\niui)e\u0000aduidui#(4.12)DERIVATION OF THE LINEAR BOLTZMANN EQUATION 11\nwhere\nz0\ni=y(i\u00001)si\u00001\u0001(i\nå\nk=1hk) +1\n2rd(ki\nå\nk=1¯gskhkk2\u0000ki\nå\nk=1gskhkk2),\nx0\nij=1\n2(kyimij+rdi\nå\nk=1¯gk0hkk2\u0000kyitij\u0000rdi\nå\nk=1gk0hkk2),(4.13)\nand\nGr\ng,S(YS,H,y) =\"p\u00001\nÕ\ni=1\u0012\n[ti\nÕ\nj=1bW(yi(j\u00001)\u0000yij)]bW(yiti\u0000yimi+rdhi+1)[ki\nÕ\nj=mibW(yij\u0000yi(j+1))]\u0013#\n\u0002˜a(\u0000p\nå\ni=1hi,y(p\u00001)sp\u00001\u00001\n2rdp\nå\ni=1(gi0\u0000¯gi0)hi)˜b(h1,y+1\n2rdh1).(4.14)\nFinally, we relabel the qiindices according to the map\n(4.15) si+j7!8\n><\n>:si+j 1\u0014j\u0014ti\nsi+j+1mi\u0014j\u0014ki\nsi+mi j=ki+1.\nWe thus obtain\nAa,r\ng,S(t) =rd(d\u00001)(p\u00001)å\nq1,\u0001\u0001\u0001,qm2Pl(rd\u00001q1)\u0001\u0001\u0001l(rd\u00001qm)Z\nR(m+2)ddydHdYS\n\u0002Gr\ng,S(YS,H,y)\"p\u00001\nÕ\ni=1ki\nÕ\nj=1(I\u0000c)(a(i\nå\nk=1hk+r\u0000d(yimij\u0000yitij)))#\n\u0002\"p\u00001\nÕ\ni=1e \n\u0000rd\u00001qimi\u0001hi+1\u0000r\u00001(qi1\u0000qiki)\u0001y(i\u00001)si\u00001\u0000r\u00001ki\nå\nj=1(qi(j+1)\u0000qij)\u0001yij!#\n\u0002\"p\u00001\nÕ\ni=2c(ai\nå\nk=1hk)#Z\nRm+1\n+d(n1+\u0001\u0001\u0001+np+rdå\ni/2Sui\u0000t)\n\u0002\"p\nÕ\ni=1e(z0\nini)e\u0000ad(1\u0000G(aåi\nk=1hk))r\u0000dnidni#\"\nÕ\ni/2Se(x0\niui)e\u0000aduidui#\n.(4.16)\nThe result then follows. \u0003\nLemma 2. There exists a constant C J>0such that\n(4.17)\f\f\f\fbJr\nq1m1\u0001\u0001\u0001q(p\u00001)mp\u00001(x1, . . . ,xm+1\u0000p)\f\f\f\f\u0014Cm\nJhti(d+1)p\u00001\n(p\u00001)!1\nad(m\u00001+p)\nkWkm\n2dkckp\u00002\nL1kak\u0003\ndkbkL1m+1\u0000p\nÕ\ni=1d\nÕ\nj=1minf1,x\u00001\nijg\nwhere\nkWkN=sup\np\u00151sup\nkb1k,kb2k\u0014Nkzb1¶b2zWkLp,\nkak\u0003\nN= sup\nkb1k,kb2k\u0014NZ\nRd\u0012Z\nRdjyb1¶b2y˜a(h,y)j2dy\u00131/2\ndh,(4.18)\nandb1,b2are multi-indices.12 JORY GRIFFIN\nProof. We first prove that Jis in L1(Rd(m+1\u0000p))and hence that the Fourier transform is well defined.\nTaking absolute values inside the integral yields\nkJr\nq1m1\u0001\u0001\u0001q(p\u00001)mp\u00001kL1\u0014Z\nR(m+2)ddydHdYS\f\f\f\f\f˜a(\u0000p\nå\ni=1hi,y(p\u00001)sp\u00001\u00001\n2rdp\nå\ni=1(gi0\u0000¯gi0)hi)˜b(h1,y+1\n2rdh1)\f\f\f\f\f\n\u0002\f\f\f\f\fp\u00001\nÕ\ni=1\u0012\n[ti\nÕ\nj=1bW(yj\u00001\u0000yj)]bW(yiti\u0000yimi+rdhi+1)[ki\nÕ\nj=mibW(yij\u0000yi(j+1))]\u0013\f\f\f\f\f\n\u0002\"p\u00001\nÕ\ni=2c(ai\nå\nk=1hk)#Z\nRm+1\n+d(n1+\u0001\u0001\u0001+np\u0000t)dn1\u0001\u0001\u0001dnp\"\nÕ\ni/2Se\u0000aduidui#\n.(4.19)\nIntegrating over nanduyields\nkJr\nq1m1\u0001\u0001\u0001q(p\u00001)mp\u00001kL1\u0014tp\u00001\n(p\u00001)!1\nad(m+1\u0000p)Z\nR(m+2)ddydHdYS\f\f\f˜b(h1,y+1\n2rdh1)\f\f\f\n\u0002\f\f\f\f\f˜a(\u0000p\nå\ni=1hi,y(p\u00001)sp\u00001\u00001\n2rdp\nå\ni=1(gi0\u0000¯gi0)hi)\f\f\f\f\f\"p\u00001\nÕ\ni=2c(ai\nå\nk=1hk)#\n\u0002\f\f\f\f\fp\u00001\nÕ\ni=1\u0012\n[ti\nÕ\nj=1bW(yj\u00001\u0000yj)]bW(yiti\u0000yimi+rdhi+1)[ki\nÕ\nj=mibW(yij\u0000yi(j+1))]\u0013\f\f\f\f\f.(4.20)\nThe ithblock ofbWfactors has the form\n(4.21)\nbW(y(i\u00001)si\u00001\u0000yi1)bW(yi1\u0000yi2)\u0001\u0001\u0001bW(yiti\u0000yimi+rdhi+1)\u0001\u0001\u0001bW(yi(ki\u00001\u00001)\u0000yiki)bW(yiki\u0000y(i\u00001)si\u00001).\nBy a series of substitutions this can be written\n(4.22) bW(yi1)\u0001\u0001\u0001bW(yiki)bW(rdhi+1\u0000yi1\u0000\u0001\u0001\u0001\u0000 yiki).\nHence, after applying Cauchy-Schwarz to the y(p\u00001)sp\u00001andh1integrals we obtain\nkJr\nq1m1\u0001\u0001\u0001q(p\u00001)mp\u00001kL1\u0014tp\u00001\n(p\u00001)!1\nad(m\u00001)kbWkp\u00001\nL¥kbWkL2kbWkm\u0000p\nL1kckp\u00002\nL1kak\u0003kbkL1 (4.23)\nwhere\n(4.24) kak\u0003=Z\nRd\u0012Z\nRdj˜a(h,y)j2dy\u00131/2\ndh.\nNext we prove that differentiating once with respect to each component of each yijvariable yields\na function which is also in L1, and hence we can conclude that not only does the Fourier transform\nexist, it decays at least linearly in each coordinate direction.\nThe first step is to bound the number of terms we obtain when applying this partial derivative.\nThe function ˜adepends only on y(p\u00001)sp\u00001which appears once. The product of bWdepends on all\nyijvariables, with each one appearing either twice, if j6=si, or four times if j=si. The number\nof terms this generates is thus bounded above by 4(m+1\u0000p)d. The product of (I\u0000c)factors is more\nsubtle. Each factor has the form\n(4.25) (I\u0000c) \na(i\nå\nk=1+r\u0000d(yimij\u0000yitij))!\n,\ni.e. it is a function of two yijvariables. In passing from one factor to the next, when gij=1\nwe increase the index of the second variable by one, and when gij=0 we decrease the index\nof the first variable by one. If the block consists of alternating sequences of ones and zeroes of\nlengths `1, . . . ,`nwith`1+\u0001\u0001\u0001+`n=kiand n\u0014kithen we have n\u00001 variables which appear\n`2+1, . . . ,`n+1 times respectively, and the remaining variables appear only once. For n\u00152 this\nyields ((`2+1)\u0001\u0001\u0001(`n+1))dterms which is bounded above by (1+ki\nn)nd. This is increasing, and\nhence the maximum number of terms from each block is bounded above by 2kid, and from the entire\nproduct is 2(m+1\u0000p)d. The product of e (x0\nijuij)is similar. Finally, each zidepends only on y(i\u00001)si\u00001.DERIVATION OF THE LINEAR BOLTZMANN EQUATION 13\nIn total then, there exists a constant C1such that the number of terms is bounded above by Cm\n1. Each\ntime a derivative is applied to the factor e (z0\nini)we obtain a multiplying factor of ni(åi\nk=1hk). By the\ncompact support of c(and the rapid decay of ˜a,˜b) this is essentially bounded above by ta\u00001. Each\ntime a derivative is applied to the factor e (x0\nijuij)we obtain a multiplying factor of \u0006uij. There are\nat most 2 dderivatives which act on each of these factors so these factors can be uniformly bounded\nabove by e.g. Õi/2Shuii2d. Proceeding as before, there thus exists a uniform constant C2>1 such\nthat\nk[p\u00001\nÕ\ni=1ki\nÕ\nj=1d\nÕ\nk=1¶\n¶yijk]Jr\nq1m1\u0001\u0001\u0001q(p\u00001)mp\u00001kL1\u0014Cm\n2hti(d+1)p\u00001\n(p\u00001)!1\nad(m\u00001+p)kWkm\n2dkckp\u00002\nL1kak\u0003\ndkbkL1. (4.26)\nThe result then follows. \u0003\nWe can now prove Proposition 2.\nProof of Proposition 2. By Lemmas 1 and 2 we have that\n\f\f\fAa,r\ng,S(t)\f\f\f\u0014Cm\n3hti(d+1)p\u00001\n(p\u00001)!1\nad(m\u00001+p)kWkm\n2dkckp\u00002\nL1kak\u0003\ndkbkL1\n\u0002rd(d\u00001)(p\u00001)å\nQ2Pml(rd\u00001q1)\u0001\u0001\u0001l(rd\u00001qm)p\u00001\nÕ\ni=1ki\nÕ\nj=1d\nÕ\nk=1minf1,M(r\u00001Q)\u00001\nijkg.(4.27)\nWe are summing over the nondiagonal terms, so there exists an iand jsuch that qij6=qimi. In\nparticular this implies that at least one of the M(r\u00001Q)ijkis nonzero. By the compact support of l,\n(4.28) å\nq2P\nq6=q0l(rd\u00001q)d\nÕ\nj=1minf1,r(qj\u0000q0\nj)\u00001g\n\u0014klkL¥ å\ni2Zd\n\u00150nf0g\nkik1<log2(1+r1\u0000db\u00001\nP)å\nq2Pd\nÕ\nj=11\"\n2ij\u00001<jqj\u0000q0\njj\nbP<2ij+1\u00001#\nminf1,r(qj\u0000q0\nj)\u00001g.\nThe number of points in a region of volume Vis bounded above by Vb\u0000d\nPso we conclude\n(4.29) å\nq2P\nq6=q0l(rd\u00001q)d\nÕ\nj=1minf1,r(qj\u0000q0\nj)\u00001g\u0014 2dklkL¥ å\ni2Zd\n\u00150nf0g\nkik1<log2(1+r1\u0000db\u00001\nP)d\nÕ\nj=12ijminf1,r\n2ij\u00001g.\nIn fact this can be written more simply: for r<2,\n(4.30) 2iminf1,r\n2i\u00001g=(\n1 i=0\nr2i\n2i\u00001jij>0\nWe partition the sum into 2dregions according to whether ijis zero or nonzero. The region which\ngives the largest contribution to the sum as r!0 is the one where all but one ijare zero. Using this\nupper bound we obtain\n(4.31) å\nq2P\nq6=q0l(rd\u00001q)d\nÕ\nj=1minf1,r(qj\u0000q0\nj)\u00001g\u0014 22d+1klkL¥rlog2(1+r1\u0000db\u00001\nP).\nHence, we may write\n\f\f\fAa,r\nnd,g,S(t)\f\f\f\u00142rlog2(1+r1\u0000db\u00001\nP)hti(d+1)p\u00001\n(p\u00001)!Cm\nJ\nad(m\u00001+p)kWkm\n2dkckp\u00002\nL1kak\u0003\ndkbkL1\n\u0002(4dklkL¥)m+1\u0000prd(d\u00001)(p\u00001)å\nq1m1,...,q(p\u00001)mp\u000012Pl(rd\u00001q1m1)\u0001\u0001\u0001l(rd\u00001q(p\u00001)mp\u00001)(4.32)14 JORY GRIFFIN\nand the result follows from our assumption (1.3). \u0003\nTheorem 2 (Sum of nondiagonal terms vanishes) .There exists a constant l0>0depending on a,t,W,a\nand b such that for all lwithklk1,¥<l0\n(4.33)¥\nå\nm=1(2pi)må\ng2f0,1gm(\u00001)g1+\u0001\u0001\u0001+gmå\nS2PmAa,r\nnd,g,S(t) =O\u0010\nrlog2(1+r1\u0000db\u00001\nP)\u0011\n.\nProof. Begin from the result of Proposition 2. Using the fact that jPmj=2m\u00001we see that the left\nhand side of (4.33) is bounded above by\n(4.34) rlog2(1+r1\u0000db\u00001\nP)¥\nå\nm=0(8pCklk1,¥)m\nwhich converges for klk1,¥<(8pC)\u00001.\n\u0003\n4.2.Diagonal Terms.\nProposition 3 (Convergence of diagonal terms) .\nlim\nr!0Aa,r\nd,g,S(t) =Z\nR(m+2)ddydHdZSblk1(h2)\u0001\u0001\u0001blkp\u00001(hp)Fg,S(ZS,H)\n\u0002\"\nÕ\ni/2SZ\nR+e(x0\niu)e\u0000adudu#Z\nRp\n+dnd(n1+\u0001\u0001\u0001+np\u0000t)\n\u0002\"p\nÕ\ni=1e((y\u0000i\u00001\nå\nk=1kk\nå\n`=1(gk`\u0000g(k+1)0)zk`)\u0001(i\nå\nk=1hk)ni)1[G(ai\nå\nk=1hk) =1]#\n.(4.35)\nwhereblk(h) =R\nRd[l(x)]ke(\u0000x\u0001h)dx,\nFg,S(ZS,H) =\"p\u00001\nÕ\ni=1bW(\u0000ki\nå\nj=1zij)ki\nÕ\nj=1bW(zij)#\n\u0002˜a(\u0000p\nå\ni=1hi,y\u0000p\u00001\nå\ni=1ki\nå\nj=1(gij\u0000g(i+1)0)zij)˜b(h1,y),(4.36)\nand for i =1, . . . , p\u00001and j =1, . . . , ki\n(4.37) x0\nij=1\n2(ky+i\u00001\nå\nk=1ki\nå\n`=1(¯gk`\u0000¯g(k+1)0)zk`+j\nå\n`=1¯gi`zi`k2\n\u0000ky\u0000i\u00001\nå\nk=1ki\nå\n`=1(gk`\u0000g(k+1)0)zk`\u0000j\nå\n`=1gi`zi`k2).\nProof. From (3.13) and the definition of the diagonal terms we have\nAa,r\nd,g,S(t) =rd(d\u00001)(p\u00001)å\nqs2,\u0001\u0001\u0001,qsp2Pl(rd\u00001qs2)k1\u0001\u0001\u0001l(rd\u00001qsp)kp\u00001Z\nR(m+2)ddydHdZS\n\u0002Fr\ng,S(ZS,H,y)\"p\u00001\nÕ\ni=1ki\nÕ\nj=1(I\u0000c)(a(i\nå\nk=1hk+r\u0000dj\nå\n`=1zi`))#\n\u0002\"p\u00001\nÕ\ni=1e\u0010\n\u0000rd\u00001q(i+1)0\u0001hi+1\u0011#\n\u0002\"p\u00001\nÕ\ni=2c(ai\nå\nk=1hk)#Z\nRm+1\n+d(n1+\u0001\u0001\u0001+np+rdå\ni/2Sui\u0000t)\n\u0002\"p\nÕ\ni=1e(zinsi)e\u0000ad(1\u0000G(aåi\nk=1hk))r\u0000dnidni#\"\nÕ\ni/2Se(xiui)e\u0000aduidui#(4.38)DERIVATION OF THE LINEAR BOLTZMANN EQUATION 15\nBy assumption (1.3), we have that\n(4.39) rd(d\u00001)å\nq2P[l(rd\u00001q)]ke\u0010\n\u0000rd\u00001q\u0001h\u0011\n=blk(h) +O(r(d\u00001)cPkhk).\nWe thus obtain the upper bound\n\f\f\fAa,r\nd,g,S(t)\f\f\f\u0014tp\u00001\n(p\u00001)!1\nad(m+1\u0000p)Z\nR(m+2)ddydHdZS\f\f\fFr\ng,S(ZS,H,y)\f\f\f\n\u0002\"p\nÕ\ni=2\u0010\nblki\u00001(hi) +O(r(d\u00001)cPkhik)\u0011\nc(ai\nå\nk=1hk)#\n.(4.40)\nSince cis compactly supported, this integral converges and we can apply dominated convergence.\nThe result then follows by taking the pointwise limit r!0, using the fact that for all c>0\n(4.41) lim\nr!0e\u0000c(1\u0000G(z))r\u0000d=1[G(z) =1]\nand that for all z6=0 we have\n(4.42) lim\nr!0c(a(i\nå\nk=1hk+r\u0000dz)) = 0.\n\u0003\nTheorem 3 (Sum of diagonal terms converges) .Fora,t>0fixed, there exists a constant l0>0such\nthat for all lwithklk1,¥<l0, the series\n(4.43)¥\nå\nm=1(2pi)må\ng2f0,1gm(\u00001)g1+\u0001\u0001\u0001+gmå\nS2PmAa,r\nd,g,S(t)\nis absolutely convergent, uniformly as r !0.\nProof. Proceeding as in the Proof of Proposition 3, we may write\n\f\f\fAa,r\nd,g,S(t)\f\f\f\u0014tp\u00001\n(p\u00001)!1\nad(m+1\u0000p)Z\nR(m+2)ddydHdZS\f\f\fFr\ng,S(ZS,H,y)\f\f\f\n\u0002\"p\nÕ\ni=2\u0010\nblki\u00001(hi) +O(r(d\u00001)cPkhik)\u0011\nc(ai\nå\nk=1hk)#\n.(4.44)\nSince cis compactly supported and ˜aand ˜bare rapidly decaying, the integral over Hconverges. By\nthe definition of Fr\ng,Sthere exists a constant C>0 such that\f\f\fAa,r\nd,g,S(t)\f\f\f<Cm+1. Equation (4.43) can\nthus be bounded above by\n(4.45)¥\nå\nm=0(8pklk1,¥C)m\nwhich converges for klk1,¥<(8pC)\u00001.\n\u0003\n4.3.The zero-damping limit.\nProposition 4 (Convergence of diagonal terms) .\n(4.46) lim\na!0lim\nr!0Aa,r\nd,g,S(t) =Z\nR2ddxdyfg,S(t,x,y)b(x,y)16 JORY GRIFFIN\nwhere\nfg,S(t,x,y) =Z\nR(m+1\u0000p)ddZSZ\nRp\n+dnd(n1+\u0001\u0001\u0001+np\u0000t)\n\u00022\n4p\nÕ\ni=2\"\nl \nx\u0000i\u00001\nå\nj=1\u0012\ny\u0000j\u00001\nå\nk=1kk\nå\n`=1(gk`\u0000g(k+1)0)zk`\u0013\nnj!#ki\u000013\n5\n\u0002\"p\u00001\nÕ\ni=1bW(\u0000ki\nå\nj=1zij)ki\nÕ\nj=1bW(zij)#\"p\u00001\nÕ\ni=1ki\nÕ\nj=1Z\nR+e(x0\nijuij)duij#\n\u0002a(x\u0000ty+p\nå\ni=1i\u00001\nå\nk=1kk\nå\n`=1(gk`\u0000g(k+1)0)zk`ni,y\u0000p\u00001\nå\ni=1ki\nå\nj=1(gij\u0000g(i+1)0)zij).\nProof. We begin from the statement of Proposition 3, and claim that the uintegral converges uni-\nformly for a\u00150. Using the same substitutions as in the proof of Proposition 2 we can write\nlim\nr!0Aa,r\nd,g,S(t) =Z\nR(m+2)ddydHdZSblk1(h2)\u0001\u0001\u0001blkp\u00001(hp)Gg,S(YS,H)\n\u0002\"p\u00001\nÕ\ni=1ki\nÕ\nj=1Z\nR+e(1\n2(kyimijk2\u0000kyitijk2)u)e\u0000adudu#Z\nRp\n+dnd(n1+\u0001\u0001\u0001+np\u0000t)\n\u0002\"p\nÕ\ni=1e(y(i\u00001)si\u00001\u0001(i\nå\nk=1hk)ni)1[G(ai\nå\nk=1hk) =1]#(4.47)\nwhere\nGg,S(YS,H,y) =\"p\u00001\nÕ\ni=1ki+1\nÕ\nj=1bW(yi(j\u00001)\u0000yij)#\n˜a(\u0000p\nå\ni=1hi,y(p\u00001)sp\u00001)˜b(h1,y) (4.48)\nand we have the convention yi0=yi(ki+1)=y(i\u00001)si\u00001. Considering only the yijintegration this has\nthe form\n(4.49)Z\nR(m+1\u0000p)dg(YS)\"p\u00001\nÕ\ni=1ki\nÕ\nj=1Z\nR+e(1\n2(kyimijk2\u0000kyitijk2)u)e\u0000adudu#\ndYS\nwhere gis Schwartz class uniformly in a. Consider just the first block fg1, . . . , gk1g, and suppose\nthat it consists of sub-blocks of `1ones, followed by `2zeroes, followed by `3ones, and so on. If\nthere are 2 kof these sub-blocks in total then\n(4.50)k1\nÕ\nj=1Z\nR+e(1\n2(kyimijk2\u0000kyitijk2)u)e\u0000adudu\n=Z\nRk1+du1\u0001\u0001\u0001duk1 `1\nÕ\ni=1e(1\n2(kyk2\u0000kyik2)ui)e\u0000adui! k1\nÕ\ni=k1+1\u0000`2e(1\n2(kyik2\u0000ky`1k2)ui)e\u0000adui!\n\u0002\u0001\u0001\u0001\u00020\n@`1+\u0001\u0001\u0001+`2k\u00001\nÕ\ni=`1+\u0001\u0001\u0001+`2k\u00003+1e(1\n2(kyk1+1\u0000`2\u0000\u0001\u0001\u0001\u0000`2k\u00002k2\u0000kyik2)ui)e\u0000adui1\nA\n\u0002 k1\u0000`2\u0000\u0001\u0001\u0001\u0000`2k\u00002\nÕ\ni=k1+1\u0000`2\u0000\u0001\u0001\u0001\u0000`2ke(1\n2(kyik2\u0000ky`1+\u0001\u0001\u0001+`2k\u00001k2)ui)e\u0000adui!\n.\nLetg02S(Rd)be Schwartz class, then by stationary phase one obtains\n(4.51)Z\nRdg0(y)e(1\n2kyk2s)dy\u001chsi\u0000d/2\nwhere A\u001cBmeans there exists a constant csuch that A<cB. Most of the yiappear in only\none factor, as e (\u00061\n2kyik2ui), which after integrating over yiagainst the Schwartz function ggives aDERIVATION OF THE LINEAR BOLTZMANN EQUATION 17\nfactorhuii\u0000d/2. Ifi=`1+\u0001\u0001\u0001+`2j\u00001ork1\u0000`2\u0000\u0001\u0001\u0001\u0000 `2jfor some jthen it appears with a more\ncomplicated coefficient. For example, y`1appears as the exponential factor\n(4.52) e (\u00001\n2ky`1k2(u`1+uk1+1\u0000`2+\u0001\u0001\u0001+uki)).\nAfter integrating over y`1this yields a factor of hu`1+uk1+1\u0000`2+\u0001\u0001\u0001+ukii\u0000d/2, but since all the ui\nare nonnegative, this can be bounded above by hu`1i\u0000d/2. In other words, we have that\n(4.53)Z\nR(m+1\u0000p)dg(YS)\"p\u00001\nÕ\ni=1ki\nÕ\nj=1Z\nR+e(1\n2(kyimijk2\u0000kyitijk2)u)e\u0000adudu#\ndYS\n\u001cZ\nRm+1\u0000p\n+p\u00001\nÕ\ni=1ki\nÕ\nj=1huiji\u0000d/2duij\nuniformly for all a\u00150. These integrals converge for all d\u00153, and the result then follows by\nintegrating over Hand setting a=0 in (4.35). \u0003\nTheorem 4 (Convergence of the full series) .There exists a constant l0>0depending on a,t,W,a and b\nsuch that for all lwithklk1,¥<l0\n(4.54) lim\na!0lim\nr!0Tr(rr1\u0000dtOpr,h(b)) =Z\nR2ddxdyf(t,x,y)b(x,y)\nwhere\n(4.55) f(t,x,y) =a(x\u0000ty,y) +¥\nå\nm=1(2pi)må\ng2f0,1gm(\u00001)g1+\u0001\u0001\u0001+gmå\nS2Pmfg,S(t,x,y).\nProof. We begin from the definition\n(4.56) Tr (rr1\u0000dtOpr,h(b)) =¥\nå\nm=0(2pi)mAa,r\nm(t).\nThe m=0 term converges by Proposition 1. Separating the remaining terms into diagonal and\nnondiagonal parts gives\n(4.57)¥\nå\nm=1(2pi)mAa,r\nm(t) =¥\nå\nm=1(2pi)må\ng2f0,1gm(\u00001)g1+\u0001\u0001\u0001+gmå\nS2PmAa,r\nd,g,S(t)\n+¥\nå\nm=1(2pi)må\ng2f0,1gm(\u00001)g1+\u0001\u0001\u0001+gmå\nS2PmAa,r\nnd,g,S(t)\nApplying Theorems 2 and 3 tells us that for lsmall enough, the first term on the right hand side\nconverges, and that the second vanishes in the limit r!0. Following the proof of Proposition 4\nthere exists a constant C>0 such that\n(4.58)\f\f\f\flim\nr!0Aa,r\nd,g,S(t)\f\f\f\f<Cm\nuniformly for all a\u00150. The series thus converges uniformly for klk1,¥<(8pC)\u00001and the result\nfollows from Proposition 4. \u0003\n5. Extracting the Linear Boltzmann Equation\nWe are now ready to prove Theorem 1, namely that the weak limit, f(t,x,y), in Theorem 4 coin-\ncides with a solution of the linear Boltzmann equation. We first show that it satisfies an auxiliary\ntransport equation.18 JORY GRIFFIN\nProposition 5. The expression f (t,x,y)in(4.55) satisfies\n(¶t+y\u0001rx)f(t,x,y)\n=2 Re\u001a¥\nå\nn=2(\u00002pil(x))nå\ng2f0,1gn\u00001(\u00001)g1+\u0001\u0001\u0001+gn\u00001\n\u0002Z\nRd(n\u00001)dz1\u0001\u0001\u0001dzn\u00001bW(\u0000z1)\u0001\u0001\u0001bW(\u0000zn\u00001)bW(z1+\u0001\u0001\u0001+zn\u00001)\n\u0002\u0014n\u00001\nÕ\ni=1Z\nR+e(1\n2(ky\u0000i\nå\nj=1gjzjk2\u0000ky+i\nå\nj=1¯gjzjk2)u)du\u0015\nf(t,x,y\u0000n\u00001\nå\ni=1gizi)\u001b(5.1)\nProof. Note that everyS2Pmcan be ‘decomposed’ into two pieces: if S=f0,n, . . . , mgwe decom-\npose it into the pieces f0,ngandfn, . . . , mg. Through this decomposition the function f(t,x,y)can\nbe written recursively as\nf(t,x,y) =a(x\u0000ty,y)\n+¥\nå\nn=1(2pi)nå\ng2f0,1gn(\u00001)g1+\u0001\u0001\u0001+gnZt\n0dn\u0014\nl\u0012\nx\u0000(t\u0000n)y\u0013\u0015n\n\u0002Z\nR(n\u00001)ddz1\u0001\u0001\u0001dzn\u00001bW(z1)\u0001\u0001\u0001bW(zn\u00001)bW(\u0000z1\u0000\u0001\u0001\u0001\u0000 zn\u00001)\n\u0002\u0014n\u00001\nÕ\ni=1Z\nR+e(1\n2(ky+i\nå\nj=1¯gjzjk2\u0000ky\u0000i\nå\nj=1gjzjk2)u)du\u0015\n\u0002f(n,x\u0000(t\u0000n)y,y\u0000(g1\u0000gn)z1\u0000\u0001\u0001\u0001\u0000 (gn\u00001\u0000gn)zn\u00001).(5.2)\nThe n=1 term vanishes – g1=1 and g1=0 yield the same expression with opposite signs.\nApplying the operator (¶t+y\u0001rx)to both sides yields\n(¶t+y\u0001rx)f(t,x,y)\n=¥\nå\nn=2(2pil(x))nå\ng2f0,1gn(\u00001)g1+\u0001\u0001\u0001+gn\n\u0002Z\nR(n\u00001)ddz1\u0001\u0001\u0001dzn\u00001bW(z1)\u0001\u0001\u0001bW(zn\u00001)bW(\u0000z1\u0000\u0001\u0001\u0001\u0000 zn\u00001)\n\u0002\u0014n\u00001\nÕ\ni=1Z\nR+e(1\n2(ky+i\nå\nj=1¯gjzjk2\u0000ky\u0000i\nå\nj=1gjzjk2)u)du\u0015\n\u0002f(t,x,y\u0000(g1\u0000gn)z1\u0000\u0001\u0001\u0001\u0000 (gn\u00001\u0000gn)zn\u00001)(5.3)\nBy summing over gnwe obtain\n(¶t+y\u0001rx)f(t,x,y)\n=¥\nå\nn=2(2pil(x))nå\ng2f0,1gn\u00001(\u00001)g1+\u0001\u0001\u0001+gn\u00001\n\u0002Z\nR(n\u00001)ddz1\u0001\u0001\u0001dzn\u00001bW(z1)\u0001\u0001\u0001bW(zn\u00001)bW(\u0000z1\u0000\u0001\u0001\u0001\u0000 zn\u00001)\n\u0002\u0014n\u00001\nÕ\ni=1Z\nR+e(1\n2(ky+i\nå\nj=1¯gjzjk2\u0000ky\u0000i\nå\nj=1gjzjk2)u)du\u0015\n\u0002 \nf(t,x,y\u0000n\u00001\nå\ni=1gizi)\u0000f(t,x,y+n\u00001\nå\ni=1¯gizi)!\n.(5.4)\nFor the second term, we replace giby¯giand make the variable substitutions zi!\u0000 zi. This allows\nus to combine the two terms and the result follows.\n\u0003DERIVATION OF THE LINEAR BOLTZMANN EQUATION 19\nProof of Theorem 1. Define the distribution\n(5.5) D(n,p):=Z¥\n0expfi(knk2\u0000kpk2)sgds,\nand putbV(y) =\u00002bW(\u0000y). Then, (5.1) can be written\n(¶t+y\u0001rx)f(t,x,y)\n=2pRe\u001a¥\nå\nn=2l(x)nå\ng2f0,1gn\u00001(\u00001)g1+\u0001\u0001\u0001+gn\u00001\n\u0002Z\nRd(n\u00001)dz1\u0001\u0001\u0001dzn\u00001[ibV(z1)]\u0001\u0001\u0001[ibV(zn\u00001)][ibV(\u0000z1\u0000\u0001\u0001\u0001\u0000 zn\u00001)]\n\u0002\u0014n\u00001\nÕ\ni=1D(y\u0000i\nå\nj=1gjzj,y+i\nå\nj=1¯gjzj)\u0015\nf(t,x,y\u0000n\u00001\nå\ni=1gizi)\u001b\n.(5.6)\ni.e.\n(¶t+y\u0001rx)f(t,x,y) =p¥\nå\n`=1l(x)`+1Q`(gx)(t,y)\nwithQ`as in [4, Eq (2.7)] and gx:(t,y)7!f(t,x,y). In view of [4, Lemma 3 & Theorem 2] (Recall\nthat Wand the initial data aare both Schwartz, so certainly satisfy the weaker regularity assumptions\nmade in [4]) we obtain\n(¶t+y\u0001rx)f(t,x,y) =pZ\nRd[Sld(y,y0)f(t,x,y0)\u0000Sld(y0,y)f(t,x,y)]dy0(5.7)\nwhere\n(5.8) Sld(y,y0) =2pd(kyk2\u0000ky0k2)jT(y0,y)j2\nand (see [4, (2.5)])\nT(y0,y) =l(x)bV(y0\u0000y)\n\u0000i¥\nå\n`=1(il(x))`+1Z\nR`dbV(y0\u0000k1)\u0001\u0001\u0001bV(k`\u0000y)D(y,k1)\u0001\u0001\u0001D(y,k`)dk1\u0001\u0001\u0001dk`\n=\u00002l(x)bW(y\u0000y0)\n\u00002¥\nå\n`=1l(x)`+1(\u00002pi)`Z\nR`dbW(y\u0000k1)\u0001\u0001\u0001bW(k`\u0000y0)\n\u0002\"Z¥\n0`\nÕ\ni=1e(1\n2(kyk2\u0000kkik2)u)du#\ndk1\u0001\u0001\u0001dk`.(5.9)\nHence,T(y0,y) =\u00002T(y,y0)and\n(5.10) pSld(y,y0) =8p2d(kyk2\u0000ky0k2)jTl(x)(y,y0)j2\nwith Tmas in (1.9).\n\u0003\nRemark: The relation T(y0,y) =\u00002T(y,y0)is due to a number of minor differences between the\npresent set-up and Castella’s work [3, 4]: (i) the Fourier transforms are normalised differently, (ii)\nthe Schrödinger operator is normalised differently, (iii) the initial von Neumann equation (1.4) has y\nandy0interchanged.\nReferences\n[1] C. Boldrighini, L.A. Bunimovich and Y.G. Sinai, On the Boltzmann equation for the Lorentz gas. J. Stat. Phys. 32 (1983),\n477–501.\n[2] E. Caglioti and F. Golse, On the Boltzmann-Grad limit for the two dimensional periodic Lorentz gas. J. Stat. Phys. 141\n(2010), 264–317.\n[3] F. Castella. From the von Neumann equation to the quantum Boltzmann equation in a deterministic framework. J. Stat.\nPhys., 104:387–447, 2001.20 JORY GRIFFIN\n[4] F. Castella. From the von Neumann equation to the quantum Boltzmann equation. II. Identifying the Born series. J. Stat.\nPhys., 106:1197–1220, 2002.\n[5] D. Eng and L. Erdös. The linear Boltzmann equation as the low density limit of a random Schrödinger equation. Reviews\nin Mathematical Physics, 17(06):669–743, 2005.\n[6] L. Erdös and H.-T. Yau, Linear Boltzmann equation as the weak coupling limit of the random Schrödinger equation,\nComm. Pure Appl. Math. LIII (2000) 667–735.\n[7] W. Fischer, H. Leschke, and P . Müller. \"On the averaged quantum dynamics by white-noise hamiltonians with and\nwithout dissipation.\" Annalen der Physik 510.2 (1998): 59-100.\n[8] G. Gallavotti, Divergences and approach to equilibrium in the Lorentz and the Wind-tree-models, Physical Review 185\n(1969), 308–322.\n[9] J. Griffin and J. Marklof. Quantum transport in a low-density periodic potential: homogenisation via homogeneous flows,\nPure and Applied Analysis. 1 (2019), no. 4, 571–614.\n[10] J. Griffin and J. Marklof. \"Quantum Transport in a Crystal with Short-Range Interactions: The Boltzmann–Grad Limit.\"\nJournal of Statistical Physics 184.2 (2021): 1-46.\n[11] M. Hensel and H. J. Korsch. \"Dissipative quantum dynamics: solution of the generalized von Neumann equation for the\ndamped harmonic oscillator.\" Journal of Physics A: Mathematical and General 25.7 (1992): 2043.\n[12] P . D. Hislop, K. Kirkpatrick, S. Olla and J. Schenker. \"Transport of a quantum particle in a time-dependent white-noise\npotential.\" Journal of Mathematical Physics 60.8 (2019): 083303.\n[13] A. M. Jayannavar and N. Kumar. \"Nondiffusive quantum transport in a dynamically disordered medium.\" Physical\nReview Letters 48.8 (1982): 553.\n[14] H. J. Korsch and H. Steffen. \"Dissipative quantum dynamics, entropy production and irreversible evolution towards\nequilibrium.\" Journal of Physics A: Mathematical and General 20.12 (1987): 3787.\n[15] J. Marklof and A. Strömbergsson, The Boltzmann-Grad limit of the periodic Lorentz gas, Annals of Math. 174 (2011)\n225–298.\n[16] J. Marklof and A. Strömbergsson, Kinetic theory for the low-density Lorentz gas, arXiv 1910.04982 (106pp)\n[17] B. Matern, Spatial Variation, Springer Science & Business Media 36 (2013)\n[18] A. Madhukar and W. Post. \"Exact solution for the diffusion of a particle in a medium with site diagonal and off-diagonal\ndynamic disorder.\" Physical Review Letters 39.22 (1977): 1424.\n[19] H. Spohn, Derivation of the transport equation for electrons moving through random impurities, J. Stat. Phys. 17 (1977)\n385–412.\n[20] H. Spohn, The Lorentz process converges to a random flight process, Comm. Math. Phys. 60 (1978), 277–290.\n[21] M. Sargent III, M. O. Scully and W. E. Lamb Jr. \"Laser Physics\", Addison-Wesley (1974).\nJory Griffin , School of Mathematics , University of Bristol , Bristol BS8 1TW, UK\nEmail address :j.griffin@bristol.ac.uk"    },    {        "title": "0812.1570v1.Landau_Damping_and_Alfven_Eigenmodes_of_Neutron_Star_Torsion_Oscillations.pdf",        "content": "arXiv:0812.1570v1  [astro-ph]  8 Dec 2008Landau Damping and Alfven Eigenmodes of Neutron Star Torsio n Oscillations\nAndrei Gruzinov\nCCPP, Physics Department, New York University, 4 Washington P lace, New York, NY 10003\nABSTRACT\nTorsion oscillations of the neutron star crust are Landau da mped by the Alfven\ncontinuum in the bulk. For strong magnetic fields (in magneta rs), undamped Alfven\neigenmodes appear.\n1. Introduction and Conclusion\nIt is thought that Israel et al (2005) have detected torsion o scillations of the neutron star crust.\nHere we show that crustal modes are strongly affected by the Alf ven continuum in the bulk. For\nweek magnetic fields, there is Landau damping of crustal eige nmodes. At stronger magnetar-like\nmagnetic fields undamped Alfven eigenmodes appear. In this r egime, the Alfven waves in the bulk\ncontrol the torsion oscillations of the crust.\nIt might be that Israel et al (2005) have actually measured th e Alfven speed in the bulk\ncA, rather than the torsion speed in the crust ct. If so, the measured ν≈20Hz frequency gives\ncA≈2νR≈4×107cm/s (assuming that νis the fundamental Alfven eigenmode).\nWe calculate the Landau damping of the crustal torsion modes and Alfven eigenmode (un-\ndamped) frequencies. The thin-crust uniform-field constan t-density model is used, and we only\nconsider the antisymmetric axisymmetric modes. This is obv iously not good enough for serious\nneutron star seismology. One must include: (i) the non-axis ymmetric modes, (ii) the toroidal mag-\nnetic field, (iii) the realistic density profile. But the qual itative results – Landau damping and\nemergence of undamped Alfven eigenmodes – should be robust.\nThis paper is a formal mathematical development of the ideas of Levin (2007).\n2. Results\nWe use the following units and notations:\n•R= 1 is the stellar radius\n•cA= 1 is the Alfven speed in the bulk– 2 –\n•3Mis the crust to bulk mass ratio.\n•ctis the torsion sound speed in the crust (in units of cA).\nThe calculated frequencies ω(in units of cA/R) of the antisymmetric axisymmetric torsion\nmodes (l= 2,4,6,...,m= 0) are given in the table for various mass ratios Mand speed ratios ct.\nThe eigenmodes (and cuts, see §3 ) are also shown in the figure for M= 0.1,ct= 2.\n3. The eigenmode equation and the dispersion relation\nToroidal displacements of the crust launch Alfven waves int o the bulk. For an axisymmetric\nantisymmetric torsion mode, the toroidal displacement in t he bulk is ∝e−iωtsin(ωz), wherezis\nthe Cartesian coordinate along the magnetic field, with the e quatorial plane z= 0. Including the\ncorresponding force into the thin-crust eigenmode equatio n, we get\nc2\nt/parenleftbig\n(1−x2)f′′−4xf′/parenrightbig\n+ω2f=ωxcot(ωx)\nMf. (1)\nHerex≡cosθ,θis the spherical angle with θ= 0 alongz,fsinθis the toroidal displacement of\nthe crust, prime is the x-differentiation. The detailed deriv ation is given in the Appendix.\nFor the antisymmetric mode, the boundary condition at x= 0 isf= 0. The boundary\ncondition at x= 1 is given by the equation itself: −4c2\ntf′+ω2f=ωcot(ω)\nMf.\nTo obtain the dispersion relation in the form F(ω) = 0, choose some f(1) andf′(1) satisfying\nthex= 1 boundary condition. Then integrate (1) form x= 1 tox= 0 and put F(ω) =f(0).\nAccording to the Landau rule, one calculates F(ω) in the upper half-plane of complex ω. The\neigenmodes are given by the zeros of F(ω) on the real axis (undamped modes) and in the lower\nhalf plane (Landau damped modes). The value of Fin the lower half-plane should be obtained by\nthe analytic continuation from the upper half-plane.\nTable 1: Eigenfrequencies in units of cA/R.\nM= 0.1,ct= 1 2.10, 2.74, 3.03, 3.12, 5.05-0.001i, 5.75-0.002i, ...\nM= 0.1,ct= 3 3.06, 6.18-0.11i, 9.35-0.04i, ...\nM= 0.1,ct= 10 21.7-4.01i, 46.3-6.19i, ...\nM= 0.1,ct= 100 200-3.14i, 424-3.19i, ...\nM= 0.01,ct= 10 3.11, 6.26-0.005i, 9.41-0.006i, ...\nM= 0.01,ct= 30 103-76i, 198-144i, ...\nM= 0.01,ct= 100 219-55i, 490-102i, ...– 3 –\nThe analytic continuation of Fto the lower half-plane can be accomplished by complexifyin g\nxand integrating (1) from x= 1 tox= 0 along an arbitrary contour in the upper half-plane of x.\nThe shape of the x-contour determines the shape of the cuts in the lower half-plane of ω. We chose\nthe semicircle |x−0.5|= 0.5. This gives the cuts in the lower half-plane of ωrunning straight down\nfromω=πk.\nThe dispersion function Fwas calculated numerically. The eigenmodes, both the Landa u-\ndamped crustal modes and the undamped Alfven eigenmodes are given in the table. The real\nentries of the table (what we call Alfven eigenmodes), were c onfirmed by a straightforward real-\nnumbers integration of (1).\nDue to the cuts, the late-time asymptotic of the crustal moti on will also have algebraically\ndamped modes with frequencies ω=πk. The late time asymptotic is determined solely by the\nlocation of the tips of the cuts and is not affected by the choice of the x-contour. The zeros of F\nare also independent of the x-contour.\nFor generic parameters Mandct, numerical integration seems to be the only way. But there\nare limiting cases which can be treated analytically. These may serve to confirm that equation (1)\nactually makes sensible predictions and also to check the nu merical results:\n•ct≪1,M≫1 gives undampedcrustal modes ω2= (l2+l−2)c2\nt+1\nM,l= 2,4,6,.... This case\nisobvious–magneticfieldandthefluidbulkjustfollow thesl owandheavy crust, providingan\nadditional elasticity and therefore increasing the freque ncy. This case is unphysical, because\nthe crust is actually lighter than the bulk.\n•ct≫1,Mct≫1 gives Landau damped crustal modes ω=√\nl2+l−2ct−ial\nM,l= 2,4,6,...,\nalare calculable dimensionless numbers (of order unity for lo wl). This case is physical, it\noccurs for small enough magnetic fields.\nThe calculation is as follows. By the Euler’s formula,\ncot(ωx) =/summationdisplay\nk1\nωx−πk→/integraldisplay\ndk1\nωx−πk=−i. (2)\nNow (1) becomes\nc2\nt/parenleftbig\n(1−x2)f′′−4xf′/parenrightbig\n+ω2f=−iωx\nMf. (3)\nWe solved it using the first order perturbation theory in 1 /(Mct). Forl= 2, one calculates\na2= 5/16 = 0.313 – in agreement with the M= 0.1,ct= 100 entry of the table.\nI thank Yuri Levin for showing me the problem and for useful di scussions.\nThis work was supported by the David and Lucile Packard found ation.– 4 –\nA. Torsion oscillations of the elastic crust with the magnetized bulk\nConsider a neutron star with a thin crust. Choose the zaxis along the uniform magnetic field\nB. The toroidal displacement of the fluid bulk ξis described by the Alfven wave equation\n∂2\ntξ=c2\nA∂2\nzξ, (A1)\nwherec2\nA=B2\n4πρis the Alfven speed in the bulk of density ρ.\nThe toroidal displacement of the crust ψis described by the torsion wave equation with the\nmagnetic force from the bulk\n∂2\ntψ=c2\nt\nR2/parenleftbigg\n∂2\nθψ+cosθ\nsinθ∂θψ−1\nsin2θψ+2ψ/parenrightbigg\n+cosθTφz\nσ. (A2)\nHerectis the torsion sound speed in the crust, Ris the radius of the star, θis the polar angle with\nrespect toz,Tφzis the toroidal-vertical component of the Maxwell stress te nsor,σis the surface\ndensity of the crust.\nThe Maxwell stress is given by Tφz=−BBφ\n4π,Bφ=B∂zξ, which should be calculated at the\nboundary of the bulk, where ξ=ψ. We get:\n∂2\ntψ=c2\nt\nR2/parenleftbigg\n∂2\nθψ+cosθ\nsinθ∂θψ−1\nsin2θψ+2ψ/parenrightbigg\n−ρc2\nA\nσcosθ∂zξ|z=cosθ. (A3)\nAs a check, one confirms that this system conserves energy and angular momentum:\nE=1\n2/integraldisplay\nρdV/parenleftbig\n(∂tξ)2+c2\nA(∂zξ)2/parenrightbig\n+1\n2/integraldisplay\nσdA/parenleftbigg\n(∂tψ)2+c2\nt\nR2(sinθ∂θ(ψ\nsinθ))2/parenrightbigg\n(A4)\nL=/integraldisplay\nρdV rsinθ ∂tξ+/integraldisplay\nσdA Rsinθ ∂tψ, (A5)\nwhere/integraltext\ndVand/integraltext\ndAare volume and surface integrals and ris the radius.\nREFERENCES\nIsrael, G. L. et al, ApJL, 628, 53, 2005\nLevin, Y., Mon.Not.Roy.Astron.Soc, 377, 159, 2007\nThis preprint was prepared with the AAS L ATEX macros v5.2.– 5 –\n0 2 4 6 8 10-0.2-0.100.1\nFig. 1.—M= 0.1,ct= 2. Zeros and cuts of the dispersion function F(ω) in the complex ωplane."    },    {        "title": "1604.06738v1.Damping_Evaluation_for_Free_Vibration_of_Spherical_Structures_in_Elastodynamic_Acoustic_Interaction.pdf",        "content": "DAMPING EVALUATION FOR FREE VIBRATION OF\nSPHERICAL STRUCTURES IN ELASTODYNAMIC-ACOUSTIC\nINTERACTION\nHADY K. JOUMAA\nAbstract. This paper discusses the free vibration of elastic spherical struc-\ntures in the presence of an externally unbounded acoustic medium. In this\nvibration, damping associated with the radiation of energy from the con\fned\nsolid medium to the surrounding acoustic medium is observed. Evaluating\nthe coupled system response (solid displacement and acoustic pressure) and\ncharacterizing the acoustic radiation damping in conjunction with the media\nproperties are the main objectives of this research. In this work, acoustic\ndamping is demonstrated for two problems: the thin spherical shell and the\nsolid sphere. The mathematical approach followed in solving these coupled\nproblems is based on the Laplace transform method. The linear under-damped\nharmonic oscillator is the reference model for damping estimation. The damp-\ning evaluation is performed in frequency as well as in time domains; both\ninvestigations lead to identical damping factor expressions.\n1.Introduction\nCoupled \ruid-solid interaction (FSI) problems are in general complex. They\nrequire advanced computational methods to handle them. Various studies have\npreviously investigated the vibration of structures while interacting with a sur-\nrounding acoustic medium [1, 2, 3, 4, 5, 6, 7]. Experimental investigations were also\nconducted on lightweight aerospace structures to estimate their acoustic radiation\ndamping under a wide band of frequency excitations [8]. During this interaction,\nthe structure's energy is continuously radiated into the surrounding \ruid medium,\nresulting in a damped structure's response. In [9], a general expression for the to-\ntal loss factor exhibited in the vibration of a solid body when immersed in a \ruid\nmedium is given as\n(1) \u0011t=\u0011struc+\u0011aero+\u0011rad\nThe total loss factor \u0011t, consists of three components: the structural loss factor\n\u0011struc, which represents damping associated with the intrinsic material properties\nof the structure (e.g., viscoelasticity), the aerodynamic loss factor \u0011aero, which is\ndue to the presence of a non-zero mean \row over the structure, and \fnally, the\nradiation loss factor \u0011rad, the focus of our work, which results from the radiation of\nsound as a consequence of the structure's vibration. Suppressing the structural loss\nby assuming a non-dissipative material model and eliminating the aerodynamic loss\nby considering a perturbing \row in a quiescent \ruid medium, the acoustic radiation\nDate : Feb 3rd, 2016.\n1991 Mathematics Subject Classi\fcation. Primary 74J05, 74H45; Secondary 35Q74.\nKey words and phrases. Acoustic damping, Elastodynamics, Acoustic radiation, Spherical\nstructures.\n1arXiv:1604.06738v1  [physics.class-ph]  4 Feb 20162 HADY K. JOUMAA\nbecomes the sole source of damping pertaining to the problem. Obviously, when the\nstructure is vibrating in vacuum, no acoustic radiation loss is encountered and thus,\n\u0011rad= 0. In many situations, acoustic radiation e\u000bects can be reasonably neglected;\nhowever, in some cases, particularly for thin lightweight structures, acoustic damp-\ning can be an order of magnitude higher than its structural counterpart [8]. The\nmajor objective of this research is to formulate and validate a closed-form expres-\nsion for the acoustic radiation damping factor ( \u0011rad) revealing its dependence on\nthe physical parameters of the coupled problem. The development of this mathe-\nmatical expression constitutes the real novelty of our work since no such evaluation\nhas been conducted in previous acoustic radiation damping investigations.\nThe generalized acoustic-structure interaction problem, whose schematic layout\nis shown in Figure 1, is comprised of a solid medium \n sthat is wholly immersed\nin an acoustic medium \n a. For our work, we adopt the simplest possible models\nfor both \n sand \n ain hope of making analytical solutions feasible. Therefore,\nthe small-strain Hookean model whose elastodynamics is governed by the Navier\nequation [10], and the inviscid compressible \ruid in which the acoustic wave prop-\nagation is described by the well-known wave equation [11], are applied in \n sand\n\narespectively. The two media have a common interface \u0000 inon which essential\nboundary conditions (BC) are maintained to complete the problem's description.\nAs a physical constraint, no interpenetration is allowed through \u0000 in, thus the nor-\nmal displacement of the structural medium must equate its \ruid counterpart; this\nensures a continuous displacement \feld throughout. In addition, the equilibrium\non the common surface requires balancing the solid normal traction to the acoustic\npressure. Finally, at the in\fnite boundary \u0000 1, the non-re\rection (wave absorp-\ntion) condition is applied to emulate the absence of any re\recting surface within\nthe acoustic medium at far \feld. In summary, the governing equations that consti-\ntute the mathematical core of the general coupled problem are presented as follows\n(2a)1\n1\u00002\u0017uj;ji+ui;jj=2(1 +\u0017)\u001as\nEuiin \n s\n(2b) p;kk=1\n#2apin \n a\n(2c)  uknk=\u00001\n\u001aap;jnjon \u0000 in\n(2d) \u001bijnj=\u0000pnion \u0000 in\n(2e) lim\njrj!1jrj\u0012@p\n@r+1\n#a@p\n@t\u0013\n= 0 on \u00001\nwhereE,\u0017, and\u001asare respectively the Young's modulus, Poisson's ratio, and\nmass density of the solid medium, while #aand\u001aaare respectively the celerity\nof sound and mass density corresponding to the acoustic medium. The above\nmodel was applied in [12] where an FSI problem is solved computationally. The\nunknowns of Eq. (2) are, to a great importance, the solid displacement u, and\nless importantly, the acoustic pressure p. In our work, we analytically solve the\nelastodynamic-acoustic interaction problem described by Eq. (2) for two structures:\nthe thin spherical shell and the solid sphere. In both problems, the exact solutionsACOUSTIC DAMPING IN SPHERICAL STRUCTURES 3\nR\nbfractal \nmedium, D\nthin elastic \nshell/uni221E\nin \ns\nan\nFigure 1. Schematic layout for the general acoustic-structure\nproblem with shown outward normal vector on the common in-\nterface\nfor both displacement and pressure \felds are presented. Consequently, acoustic\ndamping is demonstrated and the damping factor expression is derived from the\nfrequency domain solution. As for validation, the e\u000bects of the media properties,\nproblem geometry, and modal wavenumbers are highlighted through the damping\nextraction algorithms applied on transient solutions.\nIn previous works, the acoustic-structure interaction problem was considered in\nthe framework where the dynamics is triggered by an external source of excitation\n(e.g., applied step load on the structure as in [3] or pressure pulse in the acoustic\nmedium as in [4]). In addition, the work presented in [7] predicts the acoustic radi-\nation damping of \rexible panels where the discrepancy in results corresponding to\nthe uniform/nonuniform pressure theories is revealed. More interestingly, the work\nof [6] treats a spherical structure problem; nevertheless, the approach in solving\nthe problem relies on considering a potential \row in the acoustic part, leading to\na rather challenging di\u000berential functional equation that was never solved. As a\nresult, no rigorous mathematical expression for the acoustic damping could be pro-\nduced. The distinction of our work lies in considering the situation of self-excitation,\nwhere the initially deformed structure is the sole source of energy pertaining to the\ncoupled system. In such a case, the structure's response has no choice but to decay\ncontinuously with time due to the lack of any external energy feed that could com-\npensate for the radiated acoustic energy. In consequence, under self-excitation, the\ncoupled system reveals its intrinsic damping characteristics allowing for a direct\ncorrelation between the natural response and the acoustic damping behaviour to\nbe established.\nThe paper \frst brie\ry overviews the applied damping extraction methods asso-\nciated with the transient analysis of the linear under-damped harmonic oscillator.\nNext, we analyse the thin shell problem, which is characterized by its mathematical\nsimplicity with regard to solution formulation and damping estimation. Then, we4 HADY K. JOUMAA\ndiscuss the more challenging solid sphere problem in which we present the radial\nmode of oscillation and its corresponding energy calculations, to later solve the\ncoupled problem and verify the damping expression. Finally, we conduct a quali-\ntative comparison between the radiation damping resistances of our problems and\nthose of some previously analysed problems, to draw meaningful conclusions about\ngeneral acoustic damping estimation.\n2.Damping extraction methods\nThe coupled elastodynamic-acoustic problem is linear as depicted in Eq. (2),\ntherefore, the damping behaviour of the coupled system is expected to mimic that\nof the simplest linear dynamic model: the ideal under-damped harmonic oscillator.\nThis model is widely explored in [13] and in other references. Its natural vibration\nis governed by\n(3)  u+ 2\u0010!n_u+!2\nnu= 0\nwhich admits a normalized displacement solution u(t) given by\n(4) u(t) = exp(\u0000\u0010!nt)\"\ncos(!dt) +\u0010p\n1\u0000\u00102sin(!dt)#\nThe total energy stored in the system, e(t)>0, is the sum of kinetic and potential\nenergies. It is expressed in normalized form as\ne(t) =u2+_u2\n!2n\n= exp(\u00002\u0010!nt)\"\n1\u0000\u00102cos(2!dt)\n1\u0000\u00102+\u0010sin(2!dt)p\n1\u0000\u00102# (5)\nwhere the the damped frequency !dis de\fned as\n(6) !d=!np\n1\u0000\u00102\nIn account of the above analysis, mathematical procedures for evaluating the\ndamping property of a dynamic system are established. The log decrement method\nexplained in [14], is a statistical algorithm that extracts a characteristic damping\nratio from the transient response, portraying the intrinsic dissipation of a damped\noscillator. Mathematically, for a typical under-damped oscillatory displacement\nresponse, consistent with Eq. (4) and shown in Fig. 2(a), the algorithm predicts an\naverage displacement-based damping ratio, \u0016\u0010dis, computed as follows\n(7) \u0016\u0010dis=1\nNN\u00001X\nn=0log\u0012jun+1j\njunj\u0013\ns\nlog2\u0012jun+1j\njunj\u0013\n+\u00192\nwhereunis thenth extremum value of u(t), andNis the total number of all suc-\ncessive extrema used in the evaluation. For the algorithm to generate a meaningful\ndamping ratio, the displacement peak values must steadily decay with time, i.e.\njun+1j<junjfor alln. In some cases, this condition fails (as will be shown in\nthe solid sphere problem), rendering the displacement-based log decrement method\nerroneous. By necessity to overcome this failure, we extend the applicability of the\nlog decrement method into the energy response, and a corresponding energy-basedACOUSTIC DAMPING IN SPHERICAL STRUCTURES 5\nd0d\nktu\nt=\n()k k u u t =\nk\ndktπω=\n2\n2d d 0d d \nk kt te e \nt t = = ()k k e e t =k\ndktπω=t1t\n2tkt3t\n1u3u2u\nku0u\ne1e\n2e0eu\n(a)\n2\n2d d 0d d \nk kt te e \nt t = = ()k k e e t =k\ndktπω=1u3uku\nte\n1t2t3tkt1e\n2e\n3e\nke0e (b)\nFigure 2. Transient response for the under-damped harmonic os-\ncillator; labelled are (a): the extrema points of the displacement\nresponse and (b): the in\rection points of the energy response\nalgorithm is developed. Considering the energy response expressed in Eq. (5) and\nplotted in \fgure 2(b), we observe a non-increasing or \\staircase\" behaviour that is\ntypical for this type of dynamic systems. The interesting points to consider for this\nalgorithm are those where the response \rats out, i.e. in\rection points on which the\n\frst and second time derivatives vanish. The average energy-based damping ratio,\n\u0016\u0010en, can be evaluated using the following formula\n(8) \u0016\u0010en=1\nNN\u00001X\nn=0log\u0012en+1\nen\u0013\ns\nlog2\u0012en+1\nen\u0013\n+ 4\u00192\nThis algorithm requires the identi\fcation of Nconsecutive in\rection points en, the\n\frst one occurs at initial time. The obvious condition for the algorithm's success\nis the steady decay of energy, i.e. en+1< enfor alln, and this is always satis\fed\nfor a damped system subject to natural excitations. Hence, the energy-based log\ndecrement method supersedes its displacement-based counterpart by possessing an\nunrestricted applicability.\nIn addition to damping extraction procedures based on transient analysis, meth-\nods based on the frequency (Laplace) response are applied in this work too. Since\nour analysis is primarily conducted in the frequency domain, the frequency-based\ndamping extraction method generates a meaningful estimate for the damping ra-\ntio bypassing all errors involved in the Laplace transform inversion. In addition,\nthis method expresses the damping ratio in closed form by explicitly revealing its\nphysical dependence on the problem's governing parameters. In comparison, the\nlog decrement method simply provides a \\numerical\" value for the damping ratio\nwithout unveiling its mathematical structure. In our work, we achieve the damping6 HADY K. JOUMAA\nω( )ˆU iω\npωpˆU\nFigure 3. Frequency response plot for the natural response of the\nunder-damped harmonic oscillator.\ncharacterization by \frst applying the frequency-based method to obtain the closed-\nform expression of the damping ratio, and then verifying it through log-decrement\nmethods. The mathematical procedure of the frequency method is presented as\nfollows. The Laplace form of u(t) is expressed as\n(9) ^U(s) =s+ 2\u0010!n\ns2+ 2\u0010!ns+!2n\nThe Bode plot of this expression is shown in Fig. 3. The response peaks near the\nnatural frequency where\n(10a) !p\u0018=!n\u0000\n1\u00004\u00104\u0001\n(10b)\f\f\f^U\f\f\f\np\u0018=1 + 2\u00102\n2\u0010!n\nInverting Eq. (10b) to solve for \u0010, we obtain\n(11) \u0010f\u00191\n2r\f\f\f^U\f\f\f2\np!2n\u00001\nThe above equations are simpli\fed from a more complex form where an approxima-\ntion for small \u0010is appreciated. Acoustic radiation problems are in general lightly\ndamped; thus the approximation is valid and the analysis is accurate.\n3.Pulsating Thin Spherical Shell\nIn this section, we present the \frst and simplest coupled problem: radial pul-\nsation of a thin spherical shell. Consider the elastic thin spherical shell of mean\nradiusRand thickness bwhereb\u001cR. The shell's inner core is void and its center\nresides on the origin of the spherical coordinate system. The spherical symmetry\nis achieved by considering a uniform and wholly radial displacement \feld through-\nout the shell surface, thus u\u0011u(t). This symmetry extends to the surrounding\nacoustic part, where pspatially depends on the radial distance only, i.e. p\u0011p(r;t).\nInitially, the acoustic medium is quiescent and the shell is deformed by u0. TheACOUSTIC DAMPING IN SPHERICAL STRUCTURES 7\ngoverning simpli\fed elastodynamic equation describing the radial oscillation of the\nshell under this prescription is given as (also applied in [3])\n(12)d2u\ndt2+\u0012#s\nR\u00132\nu=\u0000pout\nb\u001as\nwherepoutis the uniform acoustic pressure applied on the shell's outer surface, i.e.\np(R;t). The structural wave speed is de\fned as\n(13) #s=s\n2E\n\u001as(1\u0000\u0017)\nOn the acoustic side, the acoustic wave equation (D'Alembertian form) expressed\nin spherical coordinates reduces to\n(14) r2@2p\n@r2+ 2r@p\n@r\u0000r2\n#2a@2p\n@t2= 0r\u0015R\nOn the shell surface, the no-interpenetration condition for small shell radial dis-\nplacement is expressed as\n(15)d2u\ndt2=\u00001\n\u001aa@p\n@rjr=R\nAnd \fnally, at far \feld, the Sommerfeld radiation condition becomes\n(16) lim\nr!1r\u0012@p\n@r+1\n#a@p\n@t\u0013\n= 0\nMathematically, equations (12), (14), (15), and (16) describe the coupled problem.\nThe inversion into the Laplace domain is applied to solve for the shell's response.\nWhen the wave equation is transformed into the Laplace domain, we obtain the\n0th order spherical Bessel equation in radmitting the spherical Hankel functions\nof \frst and second kind as fundamental solutions. Thus\n(17) ^P(r;s) =C(s)h(1)\n0\u0012\n\u0000isr\n#a\u0013\n+D(s)h(2)\n0\u0012\n\u0000isr\n#a\u0013\nr\u0015R\nwhereC(s) andD(s) are determined by applying the necessary BC and i=p\u00001.\nFor the Sommerfeld condition in (16) to be satis\fed, the \frst kind Hankel function\nh(1)\n0must not appear in the pressure general solution, this is achieved by setting\nC(s)\u00110. The reader is advised to review appendix A.4 of [11] or the appendix of\n[15] to better understand the asymptotic behaviour of Hankel functions and their\nderivatives. Substituting for ^Pand its derivative in the Laplace versions of Eq. (12)\nand Eq. (15) and then taking the ratio of these two equations to eliminate D(s),\nwe obtain an expression for ^Ugiven by\n(18)^U(s)\nu0=s2+s(#a\nR+#a\nb\u001aa\n\u001as)\ns3+s2(#a\nR+#a\nb\u001aa\n\u001as) +s#2s\nR2+#2s#a\nR3\nIf we normalize the displacement with u0, time withR\n#sand consequently the fre-\nquency (and Laplace variable, s) with#s\nR, the above expression in normalized form\nbecomes\n(19) ^U(s) =s2+s(#a\n#s+ 2\u00100)\ns3+s2(#a\n#s+ 2\u00100) +s+#a\n#s8 HADY K. JOUMAA\nc0u\nu\n-1.0 -0.6 -0.2 0.2 0.6 1.0 \n0 10 20 30 40 = 0.025 \n= 0.050 \n= 0.100 \nRts\nFigure 4. Transient solution for the shell displacement for di\u000ber-\nent\u00100\nwhere\u00100is de\fned as\n(20) \u00100=1\n2\u001aa\n\u001as#a\n#sR\nb\nApplying the frequency method to extract the damping expression \u0010f, we obtain\n(21) \u0010f=\u00100q\n1 + 4\u00100#a\n#s\nIn real applications, both#a\n#sand\u00100\u001c1, thus\u00100constitutes a \frst order approx-\nimation for \u0010f. Note that \u00100is inversely proportional to \u001asandb, con\frming that\nacoustic damping intensi\fes for thin lightweight structures.\nSubstituting the expression of ^Uinto the Laplace form of Eq. (15) to solve for\nD(s), the acoustic pressure (which can be normalized by the characteristic pressure\n\u001aa#a#su0\nR) is then obtained as follows\n(22)^P(r;s)\n\u001aa#a#su0\nR=\u0000R\nrsexp[\u0000#s\n#a(r\nR\u00001)s]\ns3+s2(#a\n#s+ 2\u00100) +s+#a\n#s\nNote that both pressure and displacement solutions possess the same characteristic\nequation (denominator of their Laplace form), thus they exhibit the same dynamic\nbehaviour, in particular, damping behaviour. The Laplace inversion of the displace-\nment and pressure expressions into the time domain is performed via the method\nof partial fractions (consult chapter 1 of [16] and appendix B.2 of [10] for under-\nstanding the inversion technique). Plots of the displacement and pressure solutions\nare shown in Fig. 4 and Fig. 5 respectively.\nConcerning the pressure response, we mark two observations. First, a period of\nsilence i.e. ( p= 0) occurs everywhere in the acoustic domain; its duration increasesACOUSTIC DAMPING IN SPHERICAL STRUCTURES 9\n-1.0 -0.6 -0.2 0.2 0.6 1.0 \n0 10 20 30 40 50 r\nR= 1.0r\nR= 3.0r\nR= 1.5\nRts\na\ns\nau0pR\nFigure 5. Transient solution for the acoustic pressure at di\u000berent\nradial locations for \u00100= 0:025 and#a\n#s= 0.1\nTable 1. Results of the damping factors (in %) for all three cases.\nNote the strong similarity at low damping.\ncase 1 case 2 case 3\n\u00100% 2.500 5.000 10.000\n\u0010f% 2.487 4.903 9.285\n\u0016\u0010dis% 2.466 4.842 8.560\nRel. err. % 0.851 1.260 8.470\nlinearly with the distance from the shell's surface according to\n(23) Tsilence =r\u0000R\n#a\nSecond, the maximum amplitude of the acoustic pressure, being inversely propor-\ntional to the radial distance, decays as we move away from the shell's surface to\neventually become zero at in\fnity. The displacement simulations are conducted\nat \fxed density ratio\u001aa\n\u001as= 0:05 and \fxed geometryR\nb= 10, but variable wave\nspeed ratio#a\n#s= 0.1, 0.2, and 0.4 resulting in three di\u000berent values for \u00100and\nconsequently, three cases 1, 2, and 3 respectively. It is shown that as the ratio\n#a\n#sdecreases, the relative error between the transient damping value (obtained by\nlog decrement method) and the closed form value (obtained by frequency method)\ndecreases, and both values approach \u00100corroborating the validity of the asymptotic\nexpression of Eq. (21) at low ratio of wave speeds. The results of the three cases\nare presented in Table 1.10 HADY K. JOUMAA\n4.Vibrating Solid Sphere\nIn this section, we demonstrate acoustic damping in the vibrating solid sphere\nproblem which, in concept, coincides with that of the thin spherical shell, but\npresents some mathematical challenges. The complexity arises because of the pres-\nence of modes of vibrations for the solid sphere whose elastodynamics is described\nby a partial di\u000berential equation instead of an ordinary one as is the case for the\nthin shell. Our \frst insight about this problem is that the acoustic damping will be\nmode dependent. Indeed, the closed form expression of the damping factor, along\nwith the results of the transient simulations, corroborate this conjecture. Spher-\nical symmetry will again be enforced to simplify this three-dimensional problem\nwhereby the displacement and the pressure will be solely dependent on the radial\ndistance. Before discussing the coupled problem, we will brie\ry present the solid\nsphere radial vibration in vacuum in which we overview the modal analysis and\ndetermine the mode shapes and their corresponding natural frequencies. We also\nintroduce the energy calculation which will be applied in the damping evaluation\nprocedures.\n4.1.Elastodynamic overview. Consider the elastic solid sphere of radius R, cen-\ntered at the origin of the spherical coordinate system. We consider radial modes of\nvibration, thus the transcendental and the azimuthal displacements are suppressed\nto zero (u\u0012\u00110,u\u001e\u00110, andur\u0011u(r;t)). Equations listed in appendix A.9.2 of\n[10] help understanding the following analysis. The spherical version of the Navier\nequation relevant to our problem is\n(24) r2@2u\n@r2+ 2r@u\n@r\u00002u=r2u\n#2s\nin which the wave speed #sis de\fned as\n(25) #s=s\nE(1\u0000\u0017)\n\u001as(1 +\u0017)(1\u00002\u0017)\nTo \fnd the mode shapes um, we setu(r;t) =um(r)ei!mt. Substituting this form\ninto Eq. (24), the following eigenvalue problem is obtained\n(26) r2d2um\ndr2+ 2rdum\ndr+um\"\u0012r!m\n#s\u00132\n\u00002#\n= 0\nThis is a \frst order spherical Bessel equation admitting the spherical Bessel func-\ntions of \frst and second kind as general solutions thus\n(27) um(r) =C1j1\u0012r!m\n#s\u0013\n+C2y1\u0012r!m\n#s\u0013\nThe BC relevant to this problem are: \fnite displacement at the origin ( r= 0) and\ntraction-free outer surface, i.e. \u001brrjR= 0. The \frst BC forces C2to vanish because\ny1is singular at zero. For simplicity, we set C1= 1 and we de\fne the normalized\nmodal wave number \u000emas\n(28) \u000em=!mR\n#sACOUSTIC DAMPING IN SPHERICAL STRUCTURES 11\n-0.2 0.0 0.2 0.4 0.6 \n0 0.2 0.4 0.6 0.8 1 r\nRmu\n4th mode, \n= 12.486 2nd mode, \n= 6.117 1stmode, 1= 2.744 3rd mode, = 9.317 \nFigure 6. Modal shapes for the solid sphere radial vibration for\n\f=\u00001 with modal wave numbers indicated\nHence, the mode shape becomes\n(29) um(r) =j1\u0010\n\u000emr\nR\u0011\nDe\fne a modi\fed Poisson's ratio \fas follows\n(30) \f=4\u0017\u00002\n1\u0000\u0017\u00002<\f < 0\nThe application of the second BC results in the following transcendental equation\nfor\u000em\n(31) tan \u000em=\f\u000em\n\u000e2m+\f\nThis nonlinear equation is solved numerically via a root \fnding algorithm (e.g.,\nbisection) to determine the eigenvalues of \u000em. A meaningful asymptotic approxi-\nmation for \u000emat large integer mis given by \u000em\u0019m\u0019+\f\nm\u0019. The plot in Fig. 6\nshows the \frst four modal functions corresponding to \f=\u00001 or\u0017=1\n3.\n4.2.Energy calculations. We now introduce the energy calculations correspond-\ning to the radial deformation of this spherical body. We encourage the reader to\nrefer to appendix A.8 and A.9.2 of [10] to better understand the mathematical anal-\nysis. For the general case, the stored strain energy per unit volume of an elastic\nbody is de\fned as\n(32) ep=Z\n\u001bijd\u000fij\nIn spherical problems, and for the case of radial displacement \feld depending solely\nonr, the expression for epreduces to\n(33) ep(r;t) =\u001as#2\ns\n2\u0014\u0012\n1 +\f\n2\u0013\n\u000fii\u000fjj\u0000\f\n2\u000fij\u000fij\u001512 HADY K. JOUMAA\nClearly, this energy form is space dependent. In order to observe global e\u000bects\nwithin the entire sphere (example: energy decay), we introduce the volume average\nstrain energy de\fned as follows\n(34) \u0016 ep(t) =1\nVZ\nVep(r;t)dV=3\nR3RZ\n0ep(r;t)r2dr\nIn a similar approach, the kinetic energy per unit volume is de\fned as\n(35) ek(r;t) =\u001as\n2\u0012@u\n@t\u00132\nand for the same reason (spatial dependency), we introduce the volume average\nkinetic energy de\fned as\n(36) \u0016 ek(t) =1\nVZ\nVek(r;t)dV=3\nR3RZ\n0ek(r;t)r2dr\nFinally, we de\fne the normalized volume average total energy stored in the solid\nsphere by summing the strain and kinetic energy and normalizing the resulting sum\nwith its initial value. As a result we have\n(37) \u0016 et=\u0016ep+ \u0016ek\n\u0016epjt=0+ \u0016ekjt=0\nFor free vibrations in vacuum, the energy form of (37) remains constant (identically\none) perpetually. In such a case, an exchange between kinetic and strain energy\noccurs without any external dissipation or addition of energy. However, in the\ncoupled situation where an interaction between the sphere and its surrounding\n\ruid medium takes place, the total energy decays due to acoustic radiation. The\nunderstanding of this decay helps characterizing the acoustic damping, the topic of\nour investigation.\n4.3.Coupled problem formulation. In this section, we consider the coupled\nsolid sphere problem, where an acoustic medium \flls the entire space surrounding\nthe solid sphere. The goal is to evaluate the acoustic damping corresponding to\neach mode of vibration. Thus, a single mode is excited in each case study, and\nthe resulting damping factor is recorded. In brief, the trend to obtain a veri\fed\nexpression for the acoustic damping ratio is explained in these steps:\n(1) solve the IBVP (single-mode excitation) and obtain the Laplace form of the\ndisplacement and acoustic pressure\n(2) apply the frequency method to extract the closed form expression of the\ndamping ratio\n(3) invert the Laplace form of the displacement into time domain\n(4) evaluate the velocity and strain \felds to solve for the total energy\n(5) apply the log decrement methods and verify the results with the expression\nobtained in step (2)\nFor the structural domain, the IBVP is formulated by considering Eq. (24) with\nmodi\fed BC on the sphere's outer surface. Initially, the sphere is deformed in\naccordance with the mode of interest; thus the modal function um(r) appears inACOUSTIC DAMPING IN SPHERICAL STRUCTURES 13\nthe Laplace form of Eq. (24), which is an inhomogeneous spherical Bessel equation\ninrexpressed as\n(38) r2@2^U\n@r2+ 2r@^U\n@r+^U\"\u0012isr\n#s\u00132\n\u00002#\n=\u0000sr2\n#2sum(r)\nThe full solution of ^Uconsists of the homogeneous part (spherical Bessel function)\nin addition to the particular one imposed by the right-hand side term. The full\nsolution, satisfying the \fnite displacement BC at the sphere's centre, admits the\nform\n(39) ^U(r;s) =C(s)j1\u0012isr\n#s\u0013\n+um(r)s\ns2+!2m\nConcerning the acoustic medium, the general solution for the pressure is identical to\nthat of the pulsating thin shell problem. This is intuitively explained by observing\nthat the acoustic wave propagation is indi\u000berent to the inner core of the spherical\ndomain whether it is void or \flled. As a matter of fact, Eq. (14) and Eq. (16) are\nindependent from any solid in\ruence. Therefore, the acoustic pressure's general\nsolution is given by\n(40) ^P(r;s) =D(s)h(2)\n0\u0012\u0000isr\n#a\u0013\nEquations (39) and (40) contain two unknown constants C(s) andD(s); they are\ndetermined by applying the coupled BC on the common interface, the sphere's\nouter surface. The no-interpenetration BC is no di\u000berent from that of the thin\nshell problem expressed in Eq. (15). Rewriting it in Laplace domain, we obtain\n(41) \u001aah\ns2^U(R;s)\u0000sum(R)i\n=\u0000@^P\n@rjR\nThe traction BC balances the radial stress with the acoustic pressure. This BC is\nexpressed in (a) time and (b) Laplace domain as follows\n(42a) \u001brrjR=\u0000pjR\n(42b) \u001as#2\ns\"\n@^U\n@rjR+2\u0017\n(1\u0000\u0017)R^U(R;s)#\n=\u0000^P(R;s)\nIn account of Eq. (41) and Eq. (42b), we can solve for C(s) andD(s) after an\nextensive algebraic work. Hence the Laplace form of the sphere's displacement and\nthe acoustic pressure are obtained.\nWe introduce the dimensionless parameter q\u0011\u001aa\n\u001as#a\n#s, and we note that q\u001c1 for\nmost real applications. This parameter appears in the expression of the damping\ncoe\u000ecient as will be shown. We simplify the expressions of the coupled solution by\n\frst normalizing the Laplace variable swith!mand the displacement with um(R),\nand second, by substituting the spherical Bessel functions of complex argument with\ntheir equivalent hyperbolic functions to obtain the solution of the displacement \feld\nas follows\n(43a) ^U(r;s) =s\ns2+ 1\"^U1,num (s)\n^U1,den(s)\u0001^U2(r;s) +um(r)\num(R)#\n(43b) ^U1,num (s) =q\u000e2\nm[\u000ems\u0000tanh (\u000ems)]14 HADY K. JOUMAA\n10.0 12.0 q%% ζfreq. ζLD ,ener.  \n1 0.173 0.171 \n5 0.864 0.860 \n10 1.727 1.757 \n q%% ζfreq. ζLD ,ener.  \n1 0.496 0.495 \n5 2.482 2.477 \n10 4.963 4.864 \n  ζfreq. ζLD ,disp.  \n1 0.496 0.491 \n5 2.482 2.457 \n10 4.963 4.816 \n q%%0612 18 24 30 \n0.0 0.4 0.8 1.2 1.6 2.0 ()ˆU iω\nmωω1stmode 2nd mode \nFigure 7. Bode plot of the sphere's outer surface displacement\ncorresponding to the \frst two modal excitations\n^U1,den(s) =q(\u000ems)3+\f(\u000ems)2+\f#a\n#s\u000ems\n+ tanh (\u000ems)\u0014\n(\u000ems)3+\u0012#a\n#s\u0000q\u0013\n(\u000ems)2\u0000\f\u000ems\u0000\f#a\n#s\u0015 (43c)\n(43d) ^U2(r;s) =R\nr\u000emscosh\u0000r\nR\u000ems\u0001\n\u0000R\nrsinh\u0000r\nR\u000ems\u0001\n\u000emscosh (\u000ems)\u0000sinh (\u000ems)\nand that of the acoustic pressure \feld as\n(44a)^P(R;s)\n\u001aa#a#sum(R)\nR=\u000e2\nms\n#a\n#s+\u000ems\u0014s\ns2+ 1\u0010\n^U1(s) + 1\u0011\n\u00001\u0015\n(44b) ^P(r;s) =R\nr^P(R;s) exp\u0014\n\u0000\u000em#s\n#a\u0010r\nR\u00001\u0011\ns\u0015\nClearly, these Laplace expressions are not at all familiar; their inversion technique\nwill be discussed in the upcoming subsection.\n4.4.Damping evaluation. The frequency method is applied on the Laplace form\nof the displacement presented in Eq. (43). On the outer surface ( r=R),^U2\u00111\nand the expression of ^Ureduces to\n(45) ^U(R;s) =s\ns2+ 1h\n^U1(s) + 1i\nThe amplitude of ^U(R;s) is plotted in Fig. 7. Note that this frequency response\nis behaviourally identical to that of the harmonic oscillator shown in Fig. 3. The\nmajor peak of the response occurs at frequency slightly below the natural frequency\n(!d\u0019!m). The value of the peak increases as we go higher in modes, hinting to\ndamping attenuation for higher modes. In addition, second and higher modes\nexhibit secondary peaks occurring far from the frequency of excitation; these peaksACOUSTIC DAMPING IN SPHERICAL STRUCTURES 15\nare irrelevant to damping evaluation but a\u000bect the transient response as will be\nrealized. Solving analytically for the peak frequency and the corresponding response\namplitude is not feasible due to the mathematical complexity involved in the ^U(R;s)\nexpression. We will approximate the peak of the amplitude response by the value\noccurring at the natural frequency, in other words\f\f\f^Up\f\f\f\u0018=\f\f\flim\ns!i^U(R;s)\f\f\f\n=\f\f\f\f\u000e2\nm+\f2+ 3\f\n2q\u000em\u0000i\u0012\n1 +#a\n#s\u000e2\nm+\f2+ 3\f\n2q\u000e2m\u0013\f\f\f\f(46)\nThe damping value can be then obtained using Eq. (11), thus\n(47) \u0010m,f\u0018=q\u000em\n\u000e2m+\f2+ 3\f1r\n1 +\u0010\n#a\n\u000em#s\u00112\n+ 4q\n\u000e2m+\f2+3\f#a\n#s\nNote that the square root term in Eq. (47) can be safely disregarded for small#a\n#s\nand high modes (large \u000em). In spite of the approximation applied to obtain the\n\fnal form of \u0010m,f, the latter still meaningfully evaluates the acoustic damping as is\ncon\frmed by the transient analysis results shown in Table 2.\nThe displacement u(r;t) is obtained by inverting the Laplace expression of Eq.\n(43). As such, we have\n(48) u(r;t) = cos(t)\u0003u1(t)\u0003u2(r;t) +um(r)\num(R)cos(t)\nwhere\u0003refers to the convolution operator. On the outer surface, the displacement\nsimpli\fes to\n(49) u(R;t) = cos(t)\u0003u1(t) + cos(t)\nThe Laplace inversion is performed using the Bromwich method explained in ap-\npendix B.2 of [10] and chapter 2 of [16]. In the general case, we have\n(50)u(t) =1\n2\u0019i\r+i1Z\n\r\u0000i1exp (st)^U(s) ds=NpX\nk=1Res\u0010\n^U(s);sk\u0011\nexp (skt)\nwhereskis thekth pole of ^U(s) andNpis the total number of poles. In this\nproblem, ^U1(s) has in\fnitely many poles (one is real and the remaining are com-\nplex conjugates), thus if the sum in Eq. (50) were to converge, it can be truncated.\nAsymptotic analysis shows that the complex part of the poles increases inde\fnitely\nrendering their residues insigni\fcant. We can therefore safely disregard the contri-\nbution of these large poles and consider the \frst pole pairs with the smallest complex\npart. We have indeed considered the \frst \ffty conjugate pairs of poles. The same\nprocedure applies to ^U2(r;s). The poles were obtained numerically via the two-\ndimensional bisection method (a root \fnding algorithm of assured convergence).\nThe convolution integral is performed numerically using the cubic splines method.\nThe displacement response at the outer surface for the \frst three modes at three\ndi\u000berent values of qis shown in Fig. 8. For real applications, the solid is primarily\nmetallic and the \ruid is ambient air; thus a qvalue of 10\u00004\u001810\u00003is reasonable.\nFor undersea applications, 10\u00001would be a realistic value for q. But in this case,\nthe viscous dissipation in water, especially at low speed (low Reynolds number),16 HADY K. JOUMAA\nTable 2. Summarized damping evaluation results. Clear match-\ning is noted at low q\nq \u0010 f% \u0016\u0010dis%\u0016\u0010en%\n\frst mode 10\u000030.049 0.049 0.049\n10\u000020.496 0.491 0.494\n10\u000014.942 4.816 4.849\nsecond mode 10\u000030.017 NA 0.017\n10\u000020.173 NA 0.171\n10\u000011.726 NA 1.730\nthird mode 10\u000030.010 NA 0.011\n10\u000020.110 NA 0.108\n10\u000011.098 NA 1.191\nwould intensify, spoiling the major assumption of its negligence when evaluating\nacoustic damping. In any case, the system is simulated at q2f10\u00003;10\u00002;10\u00001g\nin order to test the validity of the \u0010m,fexpression over a wide range of q. In all\ncases, we \fxed#a\n#s= 0:1. The transient solution of the acoustic pressure at three\nspots in the \ruid domain is shown in Fig. 9.\nThe transient response due to the \frst modal excitation reveals damping char-\nacteristics similar to those of the harmonic oscillator where the magnitude of the\nextrema points decay steadily with time. However, for the second and higher modes,\nthis steady decay property is lost. This observation is justi\fed by the presence of\nthe secondary (minor) peaks in the frequency response shown in Fig. 7. The pres-\nence of these minor peaks points to a slight contribution into the transient response\nfrom an additional frequency, the issue that leads to an intermittent perturbation\nof the steady decay of the values of the extrema points. Being the case, the dis-\nplacement version of the log decrement method fails to predict the damping ratio\nfor other than the \frst mode; the energy version comes into e\u000bect to ful\fl this\nprediction. The strain \feld, along with the velocity \feld are obtained from the\ndisplacement solution via direct numerical di\u000berentiation with respect to space and\ntime. The strain and the kinetic energy are consequently evaluated in conjunc-\ntion with Eq. (34) and Eq. (36). The total energy is then obtained for the \frst\nthree modes at the three chosen values of qas shown in Fig. 10. The energy-based\nlog decrement method is \fnally applied and the extracted damping coe\u000ecients are\nlisted in Table 2. Strong agreement between the various coe\u000ecients is observed\nparticularly at small qand low modes.\n5.Radiation Resistance Analysis\nIn this section, we conduct a qualitative comparison with regard to acoustic\ndamping characteristics between the coupled systems discussed in this paper and\nsome other previously analysed ones. Indeed, a signi\fcant resemblance is noticed\nbetween the \fndings, validating the analysis performed in this work and corrobo-\nrating the dependence of acoustic damping on the physical properties of the coupled\nsystem. The key parameter on which the following analysis is centred is the acous-\ntic radiation resistance Rrad, which is nothing but the equivalent of the damper in\nthe under-damped harmonic oscillator.ACOUSTIC DAMPING IN SPHERICAL STRUCTURES 17\n-1.0 -0.5 0.0 0.5 1.0 \n0 5 10 15 20 u\n1t\nωq= 10 -3\nq= 10 -2\nq= 10 -1\n(a) \frst mode\n2t\nω-1.0 \n0 5 10 15 20 \n-1.0 -0.5 0.0 0.5 1.0 \n0 5 10 15 20 \n2t\nωu\nq= 10 -3\nq= 10 -2\nq= 10 -1\n(b) second mode\n-1.0 -0.5 0.0 0.5 1.0 \n0 5 10 15 20 \n3t\nωu\nq= 10 -3\nq= 10 -2\nq= 10 -1\n(c) third mode\nFigure 8. Transient solutions of the sphere's outer surface nor-\nmalized displacement for the \frst three modal excitations.18 HADY K. JOUMAA\n1t\nω-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 \n0 5 10 15 20 25 30 35 r\nR= 1.00 r\nR= 1.50 r\nR= 1.25 asaum(R)pR\nFigure 9. Transient solution of the normalized acoustic pressure\nat three di\u000berent radial locations for the \frst modal excitation,\nq= 10\u00002,#a\n#s= 0:1\nIn [17], an expression for the acoustic radiation resistance of a one-sided \rat\nplate subject to excitation at frequency fhigher than a certain critical frequency\nfc, is given by\n(51) Rrad,plate =\u001aa#aAq\n1\u0000fc\nfforf >f c\nFor our problems, the radiation resistance is obtained through the following evalu-\nation\n(52) Rrad= 2m\u0010!\nApplying Eq. (52) on the spherical systems, we obtain after simplifying the expres-\nsions of\u0010by suppressing higher order terms\n(53a) Rrad,shell =\u001aa#aA\n(53b) Rrad,sphere =2\n3\u001aa#aA\nwhereAis the surface area in contact with the acoustic medium. The direct\ndependence of the resistance value on the \u001aa#aAfactor is noticed in the spherical\nproblems as is the case with the \rat plate one. As such, we conjecture that acoustic\ndamping in both natural and forced excitations for general structures is dominated\nby this factor. For the case of forced excitations, frequency e\u000bects arise but can be\nlumped in a secondary proportionality factor as shown in Eq. (51).\n6.Conclusion\nAcoustic damping is demonstrated in ideal spherical structural-acoustic models\nand an analytical expression for this damping is formulated and veri\fed. The under-\ndamped linear harmonic oscillator accurately depicts the energy dissipation in the\ncoupled system. The matching of damping coe\u000ecients obtained by various dampingACOUSTIC DAMPING IN SPHERICAL STRUCTURES 19\n0.95 1.00 \ne3rd mode 0.980 0.985 0.990 0.995 1.000 \n0.0 5.0 10.0 15.0 20.0 te\nmt\nω1stmode 2nd mode 3rd mode \n(a)q= 10\u00003\n0.980 \n0.0 5.0 10.0 15.0 20.0 \nmt\nω\n0.80 0.85 0.90 0.95 1.00 \n0.0 5.0 10.0 15.0 20.0 \nmt\nωte\n1stmode 2nd mode 3rd mode \n(b) q= 10\u00002\n0.0 0.2 0.4 0.6 0.8 1.0 \n0.0 5.0 10.0 15.0 20.0 \nmt\nωte\n1stmode 2nd mode 3rd mode \n(c)q= 10\u00001\nFigure 10. Transient solution of the sphere's volume averaged\nnormalized total energy20 HADY K. JOUMAA\nextraction methods corroborates the accuracy of the analysis. The availability\nof analytical solutions in both solid and acoustic domains permits the e\u000ecient\nvalidation of expensive FSI simulations while considering a single problem. Finally,\nthe future trend of this work is portrayed in two courses; \frst is the exploration of\nthe modal superposition e\u000bects on acoustic damping, and second is the prediction\nof the coupled system forced response when subject to both acoustic and structural\nloading. These crucial studies, if integrated to the work achieved in this paper,\nprovide a considerable advancement in understanding real multi-physics problems\nwith ultimate aim to simplify the design procedures of FSI applications.\nAcknowledgments. This research was made possible by the support from Sandia\nNational Laboratories { DTRA (grant HDTRA1-08-10-BRCWMD) and the NSF\n(grant CMMI-1030940). The author thanks Dr. C.A. Morales for his support in\npublishing this manuscript.\nReferences\n1. R. W. Lewis, P. Bettess, and E. Hinton, Numerical Methods in Coupled Systems . John Wiley\n& Sons, New York 1984.\n2. M. C. Junger and D. Feit, Sound, Structures, and Their Interaction . The Acoustical Society\nof America, 1993.\n3. T. A. Du\u000by, and J. N. Johnson, \\Transient response of a pulsed spherical shell surrounded by an\nin\fnite elastic medium\". International Journal of Mechanical Sciences ,23(10), pp. 589{593,\n1981.\n4. B. S. Berger, \\Vibration of the hollow sphere in an acoustic medium\". Journal of Applied\nMechanics ,36(2), pp. 330{333, 1969.\n5. M. J. Forrestal and M.J. Sagartz, \\Radiated pressure in an acoustic medium produced by\npulsed cylindrical and spherical shells\". Journal of Applied Mechanics ,38(4), pp. 1057{1060,\n1971.\n6. H. P. W. Gottlieb, \\Acoustical radiation damping of vibrating solids\". Journal of Sound and\nVibration ,40(4), pp. 521{533, 1975.\n7. R. A. Mangiarotty, \\Acoustic radiation damping of vibrating structures\". The Journal of the\nAcoustical Society of America ,35(3), pp. 369{377, 1963.\n8. B. L.Clarkson and K. T. Brown, \\Acoustic radiation damping\". Journal of Vibration, Acous-\ntics, Stress, and Reliability in Design ,107(4), pp. 357{360,1985.\n9. J. F. Wilby, 6thedition, Vibration of Structures Induced by Sound . Chap. 32, in A. G. Piersol\nand T. L. Paez Harris Shock and Vibration Handbook , McGraw Hill, New York, 2010.\n10. K. F. Gra\u000b, Wave Motion in Elastic Solids , Dover Publications, New York, 1991.\n11. L. E. Kinsler and A. R. Frey, Fundamentals of Acoustics , John Wiley & Sons, New York,\n2000.\n12. L. F. Kallivokas and J. Bielak, \\An element for the analysis of transient exterior \ruid-structure\ninteraction problems using the FEM\". Finite Element in Analysis and Design ,15, pp. 69{81,\n1993.\n13. J. P. Hartog, Mechanical Vibrations . McGraw-Hill Book Company, New York, 1956.\n14. E. E. Ungar, Vibration Isolation and Damping . Chap. 71 in M.J. Crocker Encyclopedia of\nAcoustics , John Wiley & Sons, New York, 1997.\n15. C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers .\nSpringer, 1999.\n16. A. Cohen, Numerical Methods for Laplace Transform Inversion . Springer, New York, 2007.\n17. R. H. Lyon, Statistical Energy Analysis on Dynamical Systems: Theory and Applications .\nThe MIT Press, Cambridge, Massachusetts, 1975.\nE-mail address :hkj@alum.mit.edu, hady.joumaa@gmail.com"    },    {        "title": "1612.00590v2.Two_fluid_model_of_the_pulsar_magnetosphere_represented_as_an_axisymmetric_force_free_dipole.pdf",        "content": "arXiv:1612.00590v2  [astro-ph.HE]  14 May 2017Prepared for submission to JCAP\nTwo-fluid model of the pulsar\nmagnetosphere represented as an\naxisymmetric force-free dipole\nS. A. Petrova\nInstitute of Radio Astronomy of the NAS of Ukraine,\nMystetstv Str., 4, Kharkiv 61002, Ukraine\nE-mail:petrova@rian.kharkov.ua\nAbstract. Based on the exact dipolar solution of the pulsar equation th e self-consistent two-\nfluid model of the pulsar magnetosphere is developed. We conc entrate on the low-mass limit\nof the model, taking into account the radiation damping. As a result, we obtain the particle\ndistributions sustaining the dipolar force-free configura tion of the pulsar magnetosphere in\ncase of a slight velocity shear of the electron and positron c omponents. Over most part of\nthe force-free region, the particles follow the poloidal ma gnetic field lines, with the azimuthal\nvelocities being small. Close to the Y-point, however, the p article motion is chiefly azimuthal\nandtheLorentz-factor grows unrestrictedly. Thismay resu ltinthevery-high-energy emission\nfromthevicinity of theY-point andmay alsoimplythemagnet ocentrifugal formationof ajet.\nAs for the first-order quantities, the longitudinal acceler ating electric field is found to change\nthe sign, hinting at coexistence of the polar and outer gaps. Besides that, the components of\nthe plasma conductivity tensor are derived and the low-mass analogue of the pulsar equation\nis formulated as well.\nKeywords: pulsar magnetosphere, magnetohydrodynamics, radio pulsa rs, neutron starsContents\n1 Introduction 1\n2 General picture of the dipolar force-free magnetosphere 3\n3 Two-fluid model in the force-free approximation 4\n4 Low-mass limit of the two-fluid model 5\n4.1 Force balance along the magnetic field 7\n4.2 Particle trajectories 8\n4.3 Accelerating electric field 9\n4.4 Force balance across the force-free magnetic field 9\n4.5 Plasma conductivity 10\n5 Discussion 11\n6 Conclusions 12\n1 Introduction\nPulsars are known to be rotating magnetized neutron stars, w ith the stellar magnetic field\nbeing roughly dipolar (for a more advanced conception of the magnetic field structure of the\nneutron star see, e.g., [ 1,2]). The pulsar magnetosphere, however, cannot be adequatel y\nrepresented by the vacuum rotating dipole due to the presenc e of a dense electron-positron\nplasma which should affect the electromagnetic fields. The sel f-consistent treatment of the\ncurrents and fields in the pulsar magnetosphere proceeds fro m the model of an ideal axisym-\nmetric force-free rotator containing enough plasma to scre en the accelerating longitudinal\nelectric field and to provide the electromagnetic force bala nce. In this case, the pulsar mag-\nnetosphere is described by the well-known pulsar equation [ 3,4]. In its most general form\n[5], the pulsar equation relates the magnetic flux, the poloida l current, and the transverse\npotential drop causing differential rotation of the magnetos phere.\nThe first exact solution of the pulsar equation was guessed at once [3], and for long\nyears it was thought to be indirectly relevant to the pulsar m agnetosphere. The solution\ncorresponds to the monopolar magnetic field rigidly corotat ing with the neutron star. Al-\nthough the monopolar solution allows straightforward gene ralization to the arbitrary profile\nof the differential rotation velocity (e.g., [ 6]), this was used in the literature only occasionally\n(e.g., [7]), mainly because of the lack of an idea as to the realistic fo rm of such a profile. Of\ncourse, direct application of the monopolar solution to the neutron star would be somewhat\nunnatural, but its extensions to the cases of a split [ 8] and offset [ 6] monopole are supposed\nto mimic the magnetospheric structure far from the neutron s tar. It is the split-monopole\npicture that prompted the set of boundary conditions used tr aditionally in the numerical\nsimulations of the pulsar magnetosphere [ 9]: the magnetic field is assumed to be dipolar at\nthe neutron star, monopolar-like at infinity and continuous across the light cylinder. The\nsubsequent numerical studies proved the uniqueness of such a solution and made a number\nof advances (see [ 10] for a recent review).\n– 1 –Being merely hinted by the exact split-monopole solution an d based on the general con-\nsiderations as to smoothness of the plasma flow, the customar y set of boundary conditions\n[9] involved in the numerical studies of the pulsar magnetosph ere for long time escaped phys-\nical corroboration. As was shown in [ 11], the quasi-monopolar structure at infinity, smooth\npassage through the light cylinder and zero poloidal curren t at the magnetic axis exclude\nthe purely dipolar condition at the stellar surface. Recent ly, the ab initio simulations of the\npair cascade in the pulsar force-free magnetosphere have de monstrated that the customary\nset of boundary conditions does not provide intense enough p lasma production to supply the\nforce-free regime over most part of the magnetosphere [ 12,13]. This questioned the existing\nmagnetospheric picture as a whole. Scarce alternative magn etospheric pictures suggesting\nmodifications in the equatorial [ 6,14] and axial [ 15–17] regions are conjectural as well.\nObviously, the pulsar equation should have other exact solu tions which may also give\nvaluable insights as to the boundary conditions for the puls ar magnetosphere. Recently we\nhave serendipitously found the exact dipolar solution of th e pulsar equation and constructed\nthe magnetospheric model on its basis [ 18]. It appears to differ substantially from what is\ncommonly thought of the pulsar force-free magnetosphere (s ee also Sect. 2 below for details)\nand, surprisingly, seems to have some observational suppor t. In the present paper, we elab-\norate our model of an axisymmetric force-free dipole by findi ng the particle distributions\nwhich may sustain this magnetospheric configuration.\nThe force-free treatment and the resultant pulsar equation involve only the plasma\nchargeandcurrentdensities, leavingasidetheparticledi stributionfunctions. Fortheclassical\nmonopolar case, however, the degenerate picture is well kno wn [3]: the particles stream\nalong the poloidal magnetic field lines at a speed of light. La ter on this was generalized\nfor arbitrary force-free electromagnetic fields [ 19], in which case the particle speed-of-light\nmotion is directed so as to preclude the radiation damping.\nOf course, for the sake of studying the physical processes an d the resultant emission in\nthepulsarmagnetosphere, therealisticnon-degeneratepa rticledistributionsareofmuchmore\ninterest. They can be obtained within the framework of the se lf-consistent two-fluid model.\nAlthough such a model is generally too complicated and allow s only simplistic numerical\nillustrations [ 20], the low-mass limit of the model (which is indeed suitable f or pulsars)\nmay well be addressed analytically. For the monopolar case, this was done in [ 21,22].\nIn the present paper, we generalize our previous formalism [ 22] for an arbitrary force-free\nconfiguration and apply it to the axisymmetric force-free di pole in order to find the particle\ndistribution functions and trajectories as well as the comp onents of the plasma conductivity\ntensor in the realistic pulsar magnetosphere.\nNote that the classical quasi-monopolar numerically const ructed models proceed from\nthe minimal set of boundary conditions providing smoothnes s of the solution over the whole\nspace. Within the framework of such a problem, the global for ce-free solution obtained is\nknown to exclude any physical processes in the magnetospher e. In our case, it is the purely\ndipolar analytic solution that dictates the boundary condi tions, the boundaries themselves\nand, ultimately, the corresponding magnetospheric physic s. As is shown in [ 18] (see also\nSect. 2 and Fig. 1below), in the purely dipolar case, the force-free regime ho lds only in\na certain region inside the magnetosphere rather than over t he whole space. But it is the\nforce-free region that is believed to be responsible for the pulsar radio emission and most\nprobably for the non-thermal high-energy emission as well f or geometrical and also for phys-\nical reasons, since the energetics of these emissions is muc h lower than that of the particle\nflow. Furthermore, specification of the physical conditions at the boundary of the force-free\n– 2 –zone should provide a proper basis for studying the physics o f the neighbouring regions of the\nmagnetosphere and the processes at the boundaries themselv es. Thus, our present research\nhas strong enough motivations.\nThe plan of the paper is as follows. In Sect. 2, we sketch a gene ral scheme of the pulsar\nmagnetosphere in the form of an axisymmetric force-free dip ole. Sect. 3 is devoted to the\ntwo-fluid model with the particle inertia being ignored. The low-mass limit of the two-fluid\nmodel is addressed in Sect. 4. Our results are discussed in Se ct. 5 and briefly summarized in\nSect. 6.\n2 General picture of the dipolar force-free magnetosphere\nLet us consider the pulsar magnetosphere based on the model o f an axisymmetric force-free\ndipole. As is found in [ 18], the magnetic flux function fin the form of a pure dipole,\nf= sin2θ/r, (2.1)\n(wherer,θare the coordinates in the spherical system with the axis alo ng the pulsar axis) is\nthe exact solution of the pulsar equation, with the correspo nding poloidal current function\ng(f) and angular rotation velocity µ(f) being\ng=/radicalBigg\n4C2\n1\nf2+C2, µ=C1\nf2, (2.2)\nwhereC1andC2are arbitrary constants. Note that throughout the paper we t akec=e=\nΩ∗=M= 1, where cis the speed of light, ethe electron charge, Ω ∗andMthe rotational\nvelocity and magnetic moment of the neutron star. Of course, in the closest vicinity of the\nneutron star surface one expects µ≡1, but the force-free region is separated from the stellar\nsurface by the sheet of current closure and pair formation. A s is shown in [ 18], the step in µ\nacross the sheet is indeed consistent with the moment of the c losing current dictated by the\nmagnetosphere.\nAsµchanges with f, the characteristic surface, where the linear velocity of r otation\nequals the speed of light, µrsinθ= 1, is no longer the light cylinder. With µ(f) given by\nEq. (2.2), the light surface is described by the equation r=C−1/3\n1sinθand represents the\ntorus whose small and large radii are equal (see Fig. 1and [18] for details).\nEquilibrium condition for the separatrix [ 5,23,24] and its stability [ 7] dictate the choice\nof constants C1= 1 and C2=−4 [18]. Note that in the closed field line region, where g≡0\nandµ≡1, the force-free flux function should differ from the purely di polar one. In the open\nfield line region, for the solution ( 2.1)-(2.2) we have B2−E2>0 everywhere, except for the\nY-point ( r= 1,θ=π/2), where it turns into zero. Therefore we assume terminatio n of the\nforce-free regime and current closure at the light torus, in which case in the axial region,\nwhere both gandµgiven by Eq. ( 2.2) show discontinuity, the force-free regime is excluded\nnaturally.\nWith such a general magnetospheric picture in hand, in the pr esent paper we consider\nthe segment of the force-free region lying between the light torus surface and the closed field\nline zone (see Fig. 1). Most probably, it is the place of generation of the pulsar r adio and\nnon-thermal high-energy emissions. Let us also note in pass ing that the light torus surface\nitself is suggestive as a very-high-energy emission region . On the whole, all the boundaries of\nthe region considered (i.e. the pair formation front close t o the neutron star, the separatrix\n– 3 –bounding the closed field line zone and the light torus surfac e with the closing current sheet)\nare rich in physics. Thus, our present study is believed to pr ovide a basis for diversiform\nadvances in pulsar research.\n3 Two-fluid model in the force-free approximation\nFirst let us address the self-consistent two-fluid model of t he pulsar magnetosphere in the\nforce-free approximation. Inapplication to themonopolar configuration of themagnetic field,\nthis was already donein [ 22]. Herewe generalize that treatment to thecase of arbitrary force-\nfree fields in order to derive some general features and make o ur formalism applicable to the\npurely dipolar case.\nIn the spherical coordinate system ( r,θ,φ) with the axis along the pulsar axis, the\naxisymmetric force-free magnetic and electric field streng ths,BandE, can be presented as\nB=∇f×eφ\nrsinθ+g\nrsinθeφ,E=−µ∇f . (3.1)\nWith the electromagnetic fields in the form ( 3.1), the Maxwell’s equations ∇ ·B= 0 and\n∇×E= 0 as well as the ideality condition E·B= 0 are fulfilled automatically. The two-fluid\ndescription also involves the other two Maxwell’s equation s,\nn+v+−n−v−=∇×B, (3.2)\nn+−n−=∇·E, (3.3)\n(wheren±are the electron and positron number densities and v±the corresponding veloci-\nties), the continuity equations for the two particle specie s,\n∇·(n±v±) = 0, (3.4)\nand the force-free equation of motion,\nE+v±×B= 0. (3.5)\nIn the latter equation, the effects of particle inertia, press ure and gravitation are neglected.\nFurthermore, although the radiation damping is generally s trong, it does not contribute to\nthe force-free approximation (see below) and, correspondi ngly, the radiation reaction force is\nalso ignored.\nBecause of the axisymmetry of the problem, we have ( ∇f,eφ) = 0, and the scalar\nproduct of the equation of motion ( 3.5) byeφyields\n(v±,∇f) = 0. (3.6)\nThus, the poloidal component of the particle velocity, vp±, is directed along the magnetic\nfield lines, precluding radiation damping. The explicit exp ression for the poloidal velocity\ncan be obtained by taking the scalar product of the equation o f motion ( 3.5) by∇fand\nkeeping in mind Eq. ( 3.6),\nvp±=∇f×eφvφ±−µrsinθ\ng, (3.7)\nwherevφ±are the azimuthal components of the particle velocities.\n– 4 –With the magnetic field in the form ( 3.1), the poloidal current density reads\n(∇×B)p=∇g×eφ\nrsinθ. (3.8)\nSubstituting Eqs. ( 3.7) and (3.8) into the poloidal component of Eq. ( 3.2) and taking into\naccount that for the multipolar magnetic fields ( ∇×B)φ≡0 yields\nn+−n−=−1\nµr2sin2θgdg\ndf. (3.9)\nOn the other hand, from Eq. ( 3.3) we have\nn+−n−=−µ∆f−dµ\ndf(∇f)2. (3.10)\nCombining Eqs. ( 3.9) and (3.10), we arrive at the pulsar equation for the case of multipolar\nmagnetic fields.\nFrom the azimuthal component of Eq. ( 3.2) and Eq. ( 3.3) one can obtain the particle\nnumber densities,\nn±=vφ∓∇·E\nvφ−−vφ+. (3.11)\nAnd finally, making use of Eqs. ( 3.3), (3.7), (3.9) and (3.11) in the continuity equation ( 3.4),\none can find thatw+\nw−=η(f), (3.12)\nwherew±≡(vφ±−µrsinθ)/vφ±andηis an arbitrary function of f.\nNote that if one of the functions vφ±were known, the other one would be given by\nEq. (3.12), in which case the poloidal velocities ( 3.7) and number densities ( 3.11) would be\ndetermined as well. As the force-free approximation actual ly leaves an uncertainty in the\nparticle characteristics, it is the low-mass limit of the tw o-fluid model which is believed to\nfurther constrain the force-free quantities.\n4 Low-mass limit of the two-fluid model\nTaking into account the particle inertia, the equation of mo tion is written as\nξ(v±·∇)γ±v±=±(E+v±×B), (4.1)\nwhereγ±≡(1−v2\n±)−1/2are the Lorentz-factors of the two particle species and ξ∝mis the\nnumerical factor,\nξ≡2Ω∗\nωGL, (4.2)\nwithωGL≡eBL/(mc) being the particle gyrofrequency and BLthe magnetic field strength\nat a distance c/Ω∗from the neutron star (for details see [ 22]).\nThe numerical estimate of ξreads\nξ= 10−7P2/parenleftbiggB∗\n1012G/parenrightbigg−1/parenleftbiggR∗\n106cm/parenrightbigg−3\n, (4.3)\n– 5 –wherePis the pulsar period in seconds, B∗the magnetic field strength at the neutron star\nsurface and R∗the stellar radius. One can see that the left-hand side of the equation of\nmotion ( 4.1) is small on condition\nξγc≪1, (4.4)\nwhereγcis the characteristic Lorentz-factor of the particles. Alt hough the current models of\npulsar gamma-ray emission incorporate high enough Lorentz -factors, γc∼105−106[25–27],\nthe inequality ( 4.4) is satisfied for typical pulsar parameters. Hence, the quan tities entering\nthe equation of motion ( 4.1) can be presented as\nv±=v0±+ξv1±+... ,\nE±=E0±+ξE1±+... ,\nB±=B0±+ξB1±+... , (4.5)\nwherev0±,E0±andB0±are the force-free quantities involved into the formalism o f the\npreceding section; hereafter the subscripts ’0’ are omitte d. Below we consider the first ap-\nproximation in ξin order to further restrict the zeroth-order force-free qu antities from the\nsolvability condition for the first-order ones.\nStrictly speaking, the force balance in the equation of moti on should be supplemented\nwith the radiation reaction force [ 28]\nFrad=ξαγ[(v·∇)E+v×(v·∇)B]\n+α[E×B+B×(B×v)+E(v·E)]\n−αγ2v/bracketleftBig\n(E+v×B)2−(v·E)2/bracketrightBig\n, (4.6)\nwhere the relevant normalization of the quantities is alrea dy introduced, α≡reωGL/c,re≡\ne2/(mc2) is the classical electron radius and, correspondingly, α≪1. One can see that for\nthe force-free quantities obeying Eq. ( 3.5) all the three terms in Eq. ( 4.6) are identically zero.\nThis signifies that the equation Frad= 0 has a more general solution than that generally\nknown, where the particles merely slide along the straight fi eld lines at a speed of light.\nThus, the force-free approximation generally takes into ac countFradself-consistently.\nTo the first order in ξ, the first and third terms of Fradare absent as well. Hence,\nhereafter we consider only the second term of the radiation r eaction force. At the conditions\nrelevant to the pulsar magnetosphere, the radiation dampin g can be strong, especially close\nto the neutron star surface (e.g., [ 19,29]). In our case, it can be quantified as\nα∗\nξγc= 103P−2γ−1\nc/parenleftbiggB∗\n1012G/parenrightbigg2\n, (4.7)\nwhereα∗corresponds to the stellar surface. As is shown in Sect. 4.4 b elow, it is impossible to\nconstructthe self-consistent two-fluid model of a force-fr ee dipolewithouttaking into account\nthe radiation damping.\n– 6 –4.1 Force balance along the magnetic field\nLet us turn to the equation of motion linearized in ξand consider its projection onto the\nforce-free magnetic field. Keeping in mind the above conside rations and making use of the\nforce-free equation of motion ( 3.5), the linearized radiation reaction force can be written as\nFrad1=α(E1+v×B1+v1×B)×B\n+αE(v1·E+v·E1). (4.8)\nThen, taking into account the force-free ideality conditio nE·B≡0, one can see that Frad1·\nB≡0. The scalar product of the linearized Lorentz force by BreadsFL1=E1·B+E·B1,\nwheretheright-handsidecanberecognizedasthefirst-orde rlongitudinalaccelerating electric\nfield. The inertial term of the equation of motion can also be s implified proceeding from the\nequality\n(v·∇)γv=−v×(∇×γv)+∇γ\nand applying the vector identity\n∇·(x×y) =y×(∇×x)−x×(∇×y),\nwherexandyare arbitrary vectors. After some manipulation, the longit udinal projection\nof the linearized equation of motion finally takes the form\nBp·∇/bracketleftbig\nγ±/parenleftbig\n1−µrsinθvφ±/parenrightbig/bracketrightbig\n=±(E·B1+B·E1). (4.9)\nIf the first-order longitudinal electric field were zero, the expression in the square brack-\nets in Eq. ( 4.9) would be a function of fand could be written as\nw±+a/radicalBig\n(1−b)w2\n±+2w±+a=C±(f), (4.10)\nwherea≡1−µ2r2sin2θ,b≡µ2r2sin2θ(∇f)2/g2andC±(f) signify theinitial Lorentz-factor\ndistributions at r→0 for the two particle species. The solution of Eq. ( 4.10) with respect\ntow±reads\nw±≡w0=C2\n±−a±C±/radicalBig/parenleftbig\nC2\n±−a/parenrightbig\n(1−a+ab)\n1+C2\n±(b−1), (4.11)\nand in case of a force-free dipole (see Eqs. ( 2.1)-(2.2))\na= 1−r6\nsin6θ, b=4−3sin2θ\n4(1−sin4θ/r2). (4.12)\nAsw±are positively defined quantities, one should choose the sig n ’+’ before the square root\nin Eq. (4.11).\nAs can be seen from Eq. ( 3.3) and the azimuthal component of Eq. ( 3.2), the azimuthal\nvelocity of the two particle species should be different. At th e same time, the velocities\nobtained from Eq. ( 4.11) atC+∝negationslash=C−do not obey the continuity condition ( 3.4). Hence, the\nself-consistent two-fluid model should necessarily involv e the non-zero first-order accelerating\nelectric field.\n– 7 –In general, noticing that the sum of the right-hand sides of E q. (4.9) for the two particle\nspecies is zero, we have\nw++a/radicalBig\n(1−b)w2\n++2w++a+w−+a/radicalBig\n(1−b)w2\n−+2w−+a= 2C(f). (4.13)\nThe solution of such an equation, however, would be too compl icated. Below we dwell on\nthe physically meaningful case of a slight distinction of th e electron and positron distribution\nfunctions. With the choice\nw±=w0[1+ε(f)], (4.14)\n(whereε≪1 over the whole range of fconsidered) both the continuity condition ( 3.4) and\nEq. (4.13) are satisfied, since in the latter the terms ∼εare cancelled, whereas w0is the\nsolution of Eq. (24). The particle number densities then rea d\nn±=∇·E\n2/parenleftbigg1+w0\n2εw0±1/parenrightbigg\n, (4.15)\nand the quantity ε(f) can be interpreted as the inverse multiplicity of the plasm a. Thus,\nfor given initial Lorentz-factor γi≡C(f) and multiplicity ε−1(f) of the plasma particles the\nforce-free velocity and number density distributions of th e two particle species are completely\ndetermined. As in our consideration ξ≪ε≪1 andξis typically negligible (see Eq. ( 4.3)),\nthe force-free distributions are believed to be a proper rep resentation of the plasma flow\ncharacteristics in the region considered.\nFigure2,a shows the isolines of the quantity vφ≡µrsinθ/(1 +w0) in the force-free\nregion of the dipolar magnetosphere of a pulsar. The corresp onding distribution of the\ninverse square of the Lorentz-factor, 1 −v2\nφ−v2\np, is plotted in Fig. 2,b. The quantities vφ,vp\nandγi/γat the light torus surface ( r= sinθ) are presented in Fig. 2,c as functions of the\npolar angle θ. Note an increase of vφandγwith distance from the neutron star and from\nthe magnetic axis. Over most part of the region considered, vφremains small and γ/γi≤10,\nwhereas in the vicinity of the Y-point both quantities incre ase drastically (cf. Fig. 2,c) with\nvφ→1 andγ→ ∞.\nThe isolines of the logarithm of the particle number density are shown in Fig. 3,a and\nits distribution at the light torus in Fig. 3,b. One can see that the number density is large\nnot only close to the neutron star but also in the vicinity of t he Y-point. Thus, the latter\nregion is suggestive as a site of the pulsar very-high-energ y emission. Note that for display\npurposes, in Figs. 2,3we have presented the ’effective’ quantities based on w0, keeping in\nmind that the actual distributions of the two particle speci es differ slightly.\n4.2 Particle trajectories\nWith the force-free velocities in hand, one can find the parti cle trajectories from the standard\nset of equations\ndr±\ndt=vr±,\nr±dθ±\ndt=vθ±,\nr±sinθ±dφ±\ndt=vφ±, (4.16)\n– 8 –wheretcan be regarded as a parameter along the trajectory. Excludi ngtfrom the first and\nsecond equations of the set ( 4.16) and making use of Eq. ( 3.7) yield\ndr±\ndθ±=−∂f/∂θ\n∂f/∂r. (4.17)\nThus, in the poloidal plane the electron and positron trajec tories exactly follow the magnetic\nfield lines. In the dipolar case, Eq. ( 4.17) results in sin2θ±=fr±.\nFrom the first and the third equations of the set ( 4.16) one finds\ndφ±=dr±\n2f/radicalbig\n1−fr±g(f)\nw±(f,r), (4.18)\nwherew±is given by Eqs. ( 4.11), (4.14) and is expressed in terms of f,r. The results of\nnumerical integration of Eq. ( 4.16) withw0instead of w±for different fin the dipolar case\nare presented in Fig. 4, where the azimuthal coordinate is shown as a function of s≡r/sinθ.\nThe quantity sis the parameter along a magnetic field line and is defined as s≡S/SL, where\nS≡√\ncosθ/rsatisfies the condition ∇S· ∇f≡0 and takes the value SL=√\ncosθ/sinθ\nat the light torus surface r= sinθ. As can be seen form Fig. 4, stronger variation of φ\ncorresponds to larger f. However, for any fthe azimuthal twist of the particle trajectory\nis much weaker than that of the force-free magnetic line. Thi s is analogous to the case of\na force-free monopole (cf. figure 5 in [ 22]). Recall that according to the classical result of\n[3] the ’massless’ particles strictly follow the poloidal mag netic lines without any azimuthal\nmotion at all.\n4.3 Accelerating electric field\nSubstituting Eq. ( 4.14) into Eq. ( 4.9) and retaining the terms ∼εyield the longitudinal\nelectric field in the form\nE1·B+E·B1=εγ3\ni(f)Bp·∇/bracketleftbiggw0(ab−a+1)\n(a+w0)3/bracketrightbigg\n. (4.19)\nFurther manipulation using Eq. ( 4.11) leads to\nE1·B+E·B1=εγ3\ni(f)Bp·∇\n1−a/γ2\ni\na\n1−/radicalBig\n1−a/γ2\ni√\nab−a+1\n\n.(4.20)\nFor the dipolar case, the isolines of the logarithm of the abs olute value of the longitudinal\nelectric field are shown in Fig. 5,a. Note that at f≈0.9 the quantity E1·B+E·B1changes\nthe sign. The line of zero longitudinal electric field is show n in Fig. 5,a in light gray and also\nin Fig.5,b in the coordinates ( f,s). As can be seen from the latter figure, for the field lines\nin the range f= 0.88−0.92 the sign of the longitudinal electric field changes twice.\n4.4 Force balance across the force-free magnetic field\nTo supplement the picture, it is necessary to examine the for ce balance across B. Given\nthat the radiation reaction force is neglected, the transve rse projections of the resultant\nequation of motion immediately give the first-order velocit y components perpendicular to\nthe force-free magnetic field, v1·Eandv1·(E×B). Using them in the linearized analogue\n– 9 –of Eq. (3.2) together with the force-free quantities n±andv±found in Sect. 4.1 then leads\ntoB1,E1∼1/ε→ ∞. Thus, the self-consistent two-fluid model of a force-free d ipole cannot\nbe constructed without taking into account some other const ituents. The radiation damping\nseems a proper effect to be included into the model.\nIn case of efficient radiation damping, the transverse compon ents of the linearized equa-\ntion of motion take the form\nFL1·E= 0,FL1·(E×B) = 0, (4.21)\n(whereFL1is the linearized Lorentz force) and after some manipulatio n are reduced to\nv1·E+v·E1=v·B(E·B1+E1·B)\nB2−E2, (4.22)\nFL1·E= 0. (4.23)\nCombiningEq.( 3.2)withitslinearizedcounterpartandincorporatingEqs.( 4.22), (4.23)\nyield, respectively,\n(E+ξE1)·∇×((B+ξB1)) =ξ(E1·B+E·B1)B·∇×B\nB2−E2, (4.24)\n(E1·B+E·B1)·∇×B−E2∇·E1= (B×E)·∇×B1. (4.25)\nIn terms of the total fields ˆB≡B+ξB1andˆE≡E+ξE1, Eq. (4.25) is compactly rewritten\nas\nE·/braceleftBig\nˆE∇·ˆE+(∇׈B׈B)/bracerightBig\n= 0. (4.26)\nThe latter equation, being direct consequence of Eq. ( 4.23) and implying force balance along\nE, can be recognized as the analogue of the pulsar equation for the total fields ˆBandˆE.\n4.5 Plasma conductivity\nNow wefinallyaddresstheplasmaconductivity inthelow-mas s two-fluid modelofaforce-free\ndipole. The relation between the electric field and current i s generally presented as\nj=σ0E·b+σPE⊥−σHE×b, (4.27)\nwherebis the unit vector along the magnetic field, E⊥the electric field component perpen-\ndicularto themagnetic field, σ0,σPandσHaretheparallel, PedersenandHall conductivities,\nrespectively.\nIn the case considered, the parallel conductivity,\nσ0≈j·B\nˆE·ˆB,\ntakes the form\nσ0=(∇f)2dg/df\nr2sin2θˆE·ˆB. (4.28)\nOne can see that σ0∼(εξ)−1→ ∞.\nWith Eq. ( 4.24), the Pedersen conductivity,\nσP≈j·E\nE2,\n– 10 –is reduced to\nσP=B·(∇×B)ˆE·ˆB\nE2(B2−E2)(4.29)\nand appears as small as ( εξ).\nThe Hall conductivity,\nσH=Bj·(B×E)\n(B×E)2,\nreads\nσH=gdg/df\nµrsinθ/radicalbig\n(∇f)2+g2. (4.30)\nIt is of order unity and is related to the particle drift in the crossed electric and magnetic\nfields.\nThe conductivities ( 4.28)-(4.30) in case of a force-free dipole are shown in Fig. 6. Note\nthe complicated character of the conductivity distributio ns in the region considered.\n5 Discussion\nWe have considered the two-fluid model of a force-free dipole with an eye to describing\nthe physical picture in the segment of the pulsar magnetosph ere enclosed between the light\ntorus surface and the closed field line region and believed to contain copious electron-positron\nplasma. Starting from the force-free formalism for ’massle ss’ particles, we have developed the\nfirst-order approximation in particle mass, ξ≪1, taking into account the radiation damping.\nThe solvability condition for the first-order quantities al lowed to further constrain the force-\nfree distributions of the two particle species as well as to d erive the particle trajectories and\nthe components of the plasma conductivity tensor. Besides t hat, the first-order longitudinal\nelectric field is found and the inertial analogue of the force -free pulsar equation is obtained.\nAs could be expected in advance, our present results have muc h in common with those\nfound for the case of a force-free monopole (see [ 22]). In particular, the difference in the\nelectron and positron velocities, v+−v−∝negationslash= 0 and the presence of the first-order longitudinal\nelectric field, ˆE·ˆB∝negationslash= 0, appear inherent features of the self-consistent two-flu id model. We\nconcentrate on the physically meaningful assumption of a sm all velocity shear, |v+−v−| ∼ε\n(withξ≪ε≪1), in which case ˆE·ˆB∼ξεandε=ε(f) characterizes the inverse multiplicity\nof the pulsar plasma. Then for given multiplicity and Lorent z-factor distributions at the pair\nformation front, ε−1(f) andγi(f), our description yields unambiguously the velocities and\nnumber densities of the two particle species at each point of the region considered.\nAs was already pointed out for the monopolar case, the poloid al and azimuthal velocity\nshear of the electrons and positrons is suggestive of the two -stream and diocotron instabilities\nin the pulsar plasma, which may underlie the radio emission m echanism and the subpulse\ndrift phenomenon, respectively. Furthermore, the relatio n between the velocity shear and the\nlongitudinal electric field may manifest itself as a correla tion of the radio and high-energy\nemissions of a pulsar.\nSimilarly to the monopolar case, in the dipolar force-free m agnetosphere the particle\ntrajectories follow exactly the poloidal magnetic field lin es, while their azimuthal twist is too\nsmall as compared to that of the field lines. At the same time, t he characteristic features\nof the particle motion in the monopolar and dipolar cases are distinct. In the monopolar\ncase with its cylindrical geometry, the particle motion dep ends only on the distance to the\n– 11 –magnetic axis and acceleration becomes significant far from the light cylinder, at distances\napproximately γitimes larger. In the dipolar case, however, the particle mot ion is a function\nofbothcoordinatesinthepoloidalplane. Overmostpartoft heregionconsidered, theparticle\nacceleration is moderate, with the Lorentz-factor ≤10γi. An exception is the vicinity of the\nY-point ( r= 1,θ=π/2), where γgrows drastically, tending to infinity, and the azimuthal\nvelocity component becomes the dominant one. Thus, the vici nity of the Y-point seems\npromising as a site of the pulsar very-high-energy emission observed over the range up to 1.5\nTeV [30]. Our result is in a general agreement with the current studi es of acceleration in the\npulsar magnetosphere (see, e.g., [ 31–34]). Together with the X-type field line structure in\nthe equatorial region beyond the light torus (for details se e [18]), our picture of the particle\nmotion close to the Y-point is suggestive of the magnetocent rifugal formation of a jet (see\n[35] for general theory and [ 32] for pulsar applications).\nThe longitudinal electric field structure also exhibits a su bstantial distinction from the\nmonopolar case. In the dipolar magnetosphere, ˆE·ˆBappears to change sign, roughly at\nf= 0.9, making the region 0 .9≤f≤1 suspicious of being controlled by the outer gap. A\nsimilar behaviour of the accelerating electric field was als o found in [ 36] by means of particle\nsimulation of the magnetospheric structure. Note, however , that in contrast to the previous\nattempts at including the outer gap into the force-free magn etosphere [ 6,36], in our case the\nfield lines at 0 .9≤f≤1 carry direct rather than return poloidal electric current . (The same\npicture is also preserved for other values of C1/C2in the current function ( 2.2)). Our result\nmay well be understood if one keep in mind that the particle mo tion, and, in particular,\nacceleration, is chiefly determined by the force-free elect romagnetic fields rather than by\nthe first-order longitudinal electric field. It is the force- free fields that dictate the necessary\nvelocity at each spatial point. Of course, our present consi deration does not include the\nmagnetospheric gaps, but it does yield the boundary conditi ons for such a problem and,\nhopefully, will facilitate its treatment in the future.\nThe first-oder longitudinal electric field, being proportio nal to the inverse multiplicity,\nformally determines the components of the plasma conductiv ity tensor. In contrast to the\ndissipative models assuming ad hoc zeroth-order longitudi nal electric field [ 37–39], in our\nconsiderationtheconductivitytensorisdeterminedunamb iguously. Theparallelconductivity\nappears ∼(ˆE·ˆB)−1, the Pedersen conductivity ∼ˆE·ˆBand the Hall conductivity ∼(ˆE·\nˆB)0. The former two quantities change sign together with ˆE·ˆB, demonstrating that the\nelectric current is actually determined by the force-free c ondition rather than by the first-\norder longitudinal electric field.\n6 Conclusions\nThe low-mass limit of the self-consistent two-fluid model of a force-free dipole is considered.\nWe have found the particle distributions sustaining the dip olar force-free configuration of the\npulsar magnetosphere in case of high plasma multiplicity, o r, equivalently, small distinction\nof the electron and positron distributions. The only free pa rameters are the initial Lorentz-\nfactor,γi(f), and multiplicity, ε−1(f), which are believed to be determined by the physics of\nthe pair production cascade at the boundary of the force-fre e zone.\nIn the poloidal plane, the particles of both species move str ictly along the magnetic\nfield lines and experience moderate acceleration, γ/γi≤10. The azimuthal velocities in\nthe plasma flow generally appear too low for the particles to f ollow the field line rotation.\nHowever, close to the Y-point the particle motion becomes pr edominantly azimuthal, with\n– 12 –the Lorentz-factor growing unrestrictedly. This may under lie the pulsar very-high-energy\nemission and the magnetocentrifugal acceleration of a jet.\nThe first-order longitudinal electric field proved to be an in herent constituent of the\nself-consistent two-fluid model. In contrast to the monopol ar case, in the region considered\nthis quantity changes the sign. This happens roughly at f≈0.9, but the line ˆE·ˆB= 0\ndoes not coincide with a field line and the lines with f= 0.88−0.92 intersect it twice. The\nreverse field beyond the line ˆE·ˆB= 0 can be speculated to be a residual field of the outer\ngap. Note, however, that the poloidal electric current has t he same direction on both sides of\nthe lineˆE·ˆB= 0. This can be understood if one take into account that the pa rticle motions\nare chiefly determined by the force-free fields rather than by ˆE·ˆB.\nWehavealsofoundthecomponentsoftheformalconductivity tensorrelatingtheelectric\nfield and current in the model considered. The parallel condu ctivity is ∼(ˆE·ˆB)−1, the\nPedersen conductivity is ∼ˆE·ˆBand the Hall conductivity is of order unity. The low-mass\nanalogue of the pulsar equation is derived as well.\nOn the whole, our present results are believed to provide a ba sis for extensive studies\nof pulsar physics both inside and outside of the region consi dered.\nReferences\n[1] Pons, J.A., Miralles, J.A., Geppert, U., Magneto-thermal evolution of neutron stars ,Astron.\nAstrophys. 496(2009) 207.\n[2] Ibanez-Mejia, J.C., Braithwaite, J., Stability of toroidal magnetic fields in stellar interiors ,\nAstron. Astrophys. 578(2015) A5.\n[3] Michel, F. C., Rotating magnetospheres: an exact 3-D solution ,Astrophys. 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Soc. 404(2010) 767.\n[26] Gruzinov, A., Pulsar Emission Spectrum , arXiv1309.6974.\n[27] Harding, A.K., Kalapotharakos, C., Synchrotron Self-Compton Emission from the Crab and\nOther Pulsars ,Astrophys. J. 811(2015) 63.\n[28] Landau, L.D., Lifshitz, E.M., Theory of Field , Oxford, Pergamon Press, (1971).\n[29] Finkbeiner, B., Herold, H., Ertl, T., Ruder, H., Effects of radiation damping on particle motion\nin pulsar vacuum fields ,Astron. Astrophys. 225(1989) 479.\n[30] Ansoldi, S. et al., Teraelectronvolt pulsed emission from the Crab Pulsar dete cted by MAGIC ,\nAstron. Astrophys. 585(2016) A133.\n[31] Bednarek W., On the origin of sub-TeV gamma-ray pulsed emission from rota ting neutron\nstars,Mon. Notic. R. Astron. Soc. 424(2012) 2079.\n[32] Bogovalov, S.V., Magnetocentrifugal acceleration of bulk motion of plasma i n pulsar\nmagnetosphere ,Mon. Notic. R. Astron. Soc. 443(2014) 2197.\n[33] Hirotani, K., Does a strong particle accelerator arise very close to the li ght cylinder in a pulsar\nmagnetosphere? ,Mon. Notic. R. Astron. Soc. 442(2014) L43.\n[34] Hirotani, K., Three-dimensional Non-vacuum Pulsar Outer-gap Model: Loc alized Acceleration\nElectric Field in the Higher Altitudes ,Astrophys. J. 798(2015) L40.\n[35] Blandford, R.D., Payne, D.G., Hydromagnetic flows from accretion discs and the production of\nradio jets ,Mon. Not. R. Astron. Soc. 199(1982) 883.\n[36] Yuki, S., Shibata, S., A Particle Simulation for the Pulsar Magnetosphere: Relati onship of\nPolar Cap, Slot Gap, and Outer Gap ,PASJ64(2012) 43.\n[37] Gruzinov, A., Dissipative pulsar magnetospheres ,J. Cosmol. Astropart. Phys. 11(2008) 2.\n[38] Kalapotharakos, C., Kazanas, D., Harding, A., Contopoulos, I., Toward a Realistic Pulsar\nMagnetosphere ,Astrophys. J. 749(2012) id. 2.\n– 14 –[39] Li, J., Spitkovsky, A., Tchekhovskoy, A., Resistive Solutions for Pulsar Magnetospheres ,\nAstrophys. J. 746(2012) id. 60.\n– 15 –ρz\n  \n0.10.20.30.40.50.60.70.80.910.050.10.150.20.250.30.350.40.450.5\n0.10.20.30.40.50.60.70.8\nlight torus\nclosed field line region\nFigure 1 . Geometry of the force-free region considered in the paper (see text for details). Bold lines\nshow the light torus surface and the boundary of the open field line r egion. Also plotted are the\nmagnetic field lines corresponding to the magnetic flux function levels from 0.1 to 0.9 with the step\n0.1.\n– 16 –ρz\n  \n0.10.20.30.40.50.60.70.80.90.050.10.150.20.250.30.350.40.45\n0.10.20.30.40.50.60.70.80.9\nρz\n  \n0.10.20.30.40.50.60.70.80.90.050.10.150.20.250.30.350.40.45\n12345678910x 10−5\n010203040506070809000.10.20.30.40.50.60.70.80.91\nθ, deg  \nvφ\nvp\nγi/γ\nFigure 2 . Velocity characteristics of the particles sustaining dipolar force- free configuration of the\npulsar magnetosphere, γi= 100; a) distribution of the azimuthal velocity vφ; b) distribution of the\ninverse square of the Lorentz-factor, 1 −v2\np−v2\nφ; c) poloidal and azimuthal velocity components\ntogether with the Lorentz-factor at the light torus as functions of the polar angle.– 17 –ρz\n  \n0.10.20.30.40.50.60.70.80.90.050.10.150.20.250.30.350.40.45\n11.522.533.544.55\n0102030405060708090024681012\nθ, deglog(n ε)\nFigure 3 . Number density of the particles sustaining dipolar force-free con figuration of the pulsar\nmagnetosphere, γi= 100; a) distribution of log( nε); b) log( nε) at the light torus surface as a function\nof the polar angle.\n– 18 –0.10.20.30.40.50.60.70.80.9100.10.20.30.40.50.60.7\nsφ\nFigure 4 . Particle azimuthal coordinate as a function of the parameter sfor different magnetic field\nlines,γi= 100 (see text for details); the curves (from bottom to top) cor respond to the flux function\nvalues from 0.1 to 0.9 with the step 0.1.\n– 19 –ρz\n  \n0.10.20.30.40.50.60.70.80.90.050.10.150.20.250.30.350.40.45\n−2−1.5−1−0.500.511.522.53\nfs\n0.88 0.9 0.92 0.94 0.96 0.98 10.10.20.30.40.50.60.70.80.91\nFigure 5 . Structure of the first-order longitudinal electric field in units of εγ3\ni; a) distribution of\nlog|ˆE·ˆB|; the line ˆE·ˆB= 0 is shown in light gray; b) the line ˆE·ˆB= 0 in the coordinates ( f,s).\n– 20 –ρz\n  \n0.10.20.30.40.50.60.70.80.90.050.10.150.20.250.30.350.40.45\n−5−4−3−2−1012345\nρz\n  \n0.10.20.30.40.50.60.70.80.90.050.10.150.20.250.30.350.40.45\n12345678\nρz\n  \n0.10.20.30.40.50.60.70.80.90.050.10.150.20.250.30.350.40.45\n0.40.50.60.70.80.9\nFigure 6 . Distributions of the plasma conductivity components, γi= 100,ε= 0.1; a) log σ0; b)\nlogσP; c) logσH.\n– 21 –"    },    {        "title": "1605.05063v1.Simultaneous_Identification_of_Damping_Coefficient_and_Initial_Value_in_PDEs_from_boundary_measurement.pdf",        "content": "arXiv:1605.05063v1  [math.AP]  17 May 2016Simultaneous Identification of Damping Coefficient and\nInitial Value for PDEs from Boundary Measurement\nZhi-Xue Zhaoa,bM.K. Bandab, and Bao-Zhu Guoc,d∗\naSchool of Mathematical Sciences,\nTianjin Normal University, Tianjin 300387, China\nbDepartment of Mathematics and Applied Mathematics,\nUniversity of Pretoria, Pretoria 0002, South Africa\ncAcademy of Mathematics and Systems Science,\nAcademia Sinica, Beijing 100190, China,\ndSchool of Computer Science and Applied Mathematics,\nUniversity of the Witwatersrand, Johannesburg, South Afri ca\nAbstract\nIn this paper, the simultaneous identification of damping or anti-dam ping coefficient and\ninitial value for some PDEs is considered. An identification algorithm is p roposed based on the\nfact that the output of system happens to be decomposed into a p roduct of an exponential func-\ntion and a periodic function. The former contains information of the damping coefficient, while\nthe latter does not. The convergence and error analysis are also d eveloped. Three examples,\nnamely an anti-stable wave equation with boundary anti-damping, th e Schr¨ odinger equation\nwith internal anti-damping, and two connected strings with middle jo int anti-damping, are in-\nvestigated and demonstrated by numerical simulations to show the effectiveness of the proposed\nalgorithm.\nKeywords: Identification; damping coefficient; anti-stable PDEs; anti-damping coefficient.\nAMS subject classifications: 35K05, 35R30, 65M32, 65N21, 15A22.\n1 Introduction\nLetHbe a Hilbert space with the inner product /an}bracketle{t·,·/an}bracketri}htand inner product induced norm /bardbl·/bardbl, and\nletY=R(orC). Consider the dynamic system in H:\n/braceleftBigg\n˙x(t) =A(q)x(t), x(0) =x0,\ny(t) =Cx(t)+d(t),(1.1)\n∗Corresponding author. Email: bzguo@iss.ac.cn\n1whereA(q) :D(A(q))⊂H→His the system operator depending on the coefficient q, which is\nassumed to be a generator of C0-semigroup Tq= (Tq(t))t∈R+onH,C:H→Yis the admissible\nobservation operator for Tq([20]),x0∈His the initial value, and d(t) is the external disturbance.\nVarious PDE control systems with damping mechanism can be fo rmulated into system (1.1),\nwhereqis the damping coefficient. For a physical system, if the dampi ng is produced by material\nitself that dissipates the energy stored in system, then the system keeps stable. The identification\nof damping coefficient has been well considered for distribut ed parameter systems like Kelvin-Voigt\nviscoelastic damping coefficient in Euler-Bernoulli beam in vestigated in [4], and a more general\ntheoretical framework for various classes of parameter est imation problems presented in [5]. In\nthese works, the inverse problems are formulated as least sq uare problems and are solved by finite\ndimensionalization. For more revelent works, we can refer t o the monograph [6]. Sometimes,\nhowever, the source of instability may arise from the negati ve damping. One example is the\nthermoacoustic instability in duct combustion dynamics an d the other is the stick-slip instability\nphenomenon in deep oil drilling, see, for instance, [7] and t he references therein. In such cases,\nthe negative damping will result in all the eigenvalues loca ted in the right-half complex plane, and\nthe open-loop plant is hence “anti-stable” (exponentially stable in negative time) and the qin such\nkind of system is said to be the anti-damping coefficient.\nAwidely investigated probleminrecent years isstabilizat ion foranti-stable systemsbyimposing\nfeedback controls. A breakthrough on stabilization for an a nti-stable wave equation was first\nreached in [19] where a backstepping transformation is prop osed to design the boundary state\nfeedback control. By the backstepping method, [11] general izes [19] to two connected anti-stable\nstrings with joint anti-damping. Very recently, [12, 13] in vestigate stabilization for anti-stable\nwave equation subject to external disturbance coming throu gh the boundary input, where the\nsliding mode control and active disturbance rejection cont rol technology are employed. It is worth\npointing out that in all aforementioned works, the anti-dam ping coefficients are always supposed\nto be known.\nOn the other hand, a few stabilization results for anti-stab le systems with unknown anti-\ndamping coefficients are also available. In [16], a full state feedback adaptive control is designed for\nan anti-stable wave equation. By converting thewave equati on into acascade of two delay elements,\nan adaptive output feedback control and parameter estimato r are designed in [7]. Unfortunately,\nno convergence of the parameter update law is provided in the se works.\nIt can be seen in [7, 16] that it is the uncertainty of the anti- damping coefficient that leads to\ncomplicated design for adaptive control and parameter upda te law. This comes naturally with the\nidentification of unknown anti-damping coefficient. To the be st of our knowledge, there are few\nstudies on this regard. Our focus in the present paper is on si multaneous identification for both\nanti-damping (or damping) coefficient and initial value for s ystem (1.1), where the coefficient qis\nassumed to be in a prior parameter set Q= [q,q] (qorqmay be infinity) and the initial value is\nsupposed to be nonzero.\nWe proceed as follows. In Section 2, we propose an algorithm t o identify simultaneously the\n2coefficient and initial value through the measured observati on. The system may not suffer from\ndisturbance or it may suffer from general bounded disturbance . In Section 3, a wave equation\nwith anti-damping term in the boundary is discussed. A Schr¨ odinger equation with internal anti-\ndamping term is investigated in Section 4. Section 5 is devot ed to coupled strings with middle joint\nanti-damping. In all these sections, numerical simulation s are presented to verify the performance\nof the proposed algorithms. Some concluding remarks are pre sented in Section 6.\n2 Identification algorithm\nBefore giving the main results, we introduce the following w ell known Ingham’s theorem [14, 15, 23]\nas Lemma 2.1.\nLemma 2.1. Assume that the strictly increasing sequence {ωk}k∈Zof real numbers satisfies the\ngap condtion\nωk+1−ωk≥γfor allk∈Z, (2.1)\nfor someγ >0. Then, for all T >2π/γ, there exist two positive constants C1andC2, depending\nonly onγandT, such that\nC1/summationdisplay\nk∈Z|ak|2≤/integraldisplayT\n0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nk∈Zakeiωkt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndt≤C2/summationdisplay\nk∈Z|ak|2, (2.2)\nfor every complex sequence (ak)k∈Z∈ℓ2, where\nC1=2T\nπ/parenleftbigg\n1−4π2\nT2γ2/parenrightbigg\n, C2=8T\nπ/parenleftbigg\n1+4π2\nT2γ2/parenrightbigg\n. (2.3)\nTo begin with, we suppose that there is no external disturban ce in system (1.1), that is,\n/braceleftBigg\n˙x(t) =A(q)x(t), x(0) =x0,\ny(t) =Cx(t).(2.4)\nThe succeeding Theorem 2.1 indicates that identification of the coefficient qand initial value x0\ncan be achieved exactly simultaneously without error for A(q) with some structure.\nTheorem 2.1. LetA(q)in system (2.4) generate a C0-semigroup Tq= (Tq(t))t∈R+and suppose\nthatA(q)andCsatisfy the following conditions:\n(i).A(q)has a compact resolvent and all its eigenvalues {λn}n∈N(or{λn}n∈Z) admit the\nfollowing expansion:\nλn=f(q)+iµn,···<µn<µn+1<···, (2.5)\nwheref:Q→Ris invertible, µnis independent of q, and there exists an L>0such that\nµnL\n2π∈Zfor alln∈N. (2.6)\n(ii). The corresponding eigenvectors {φn}n∈Nform a Riesz basis for H.\n3(iii). There exist two positive numbers κandKsuch thatκ≤ |κn| ≤Kfor alln∈N, where\nκn:=Cφn, n∈N. (2.7)\nThen both coefficient qand initial value x0can be uniquely determined by the output y(t),t∈[0,T],\nwhereT >2L. Precisely,\nq=f−1/parenleftBigg\n1\nLln/bardbly/bardblL2(T1,T2)\n/bardbly/bardblL2(T1−L,T2−L)/parenrightBigg\n, (2.8)\nfor anyL<T1<T2−L, and\nx0=1\nL/summationdisplay\nn∈N1\nκn/parenleftbigg/integraldisplayL\n0y(t)e−λntdt/parenrightbigg\nφn. (2.9)\nProof.Since{φn}n∈Nforms a Riesz basis for H, there exists a sequence {ψn}n∈Nof eigenvectors\nofA(q)∗, which is biorthogonal to {φn}n∈N, that is, /an}bracketle{tφn,ψm/an}bracketri}ht=δnm. In this way, we can express\nthe initial value x0∈Hasx0=/summationtext\nn∈N/an}bracketle{tx0,ψn/an}bracketri}htφn,and the solution of system (2.4) as\nx(t) =Tq(t)x0=/summationdisplay\nn∈Neλnt/an}bracketle{tx0,ψn/an}bracketri}htφn. (2.10)\nBy (2.6), there exists an increasing sequence {Kn} ⊂Zsuch that\nµn=2πKn\nL, n∈N, (2.11)\nwhich implies that {µn}satisfies the following gap condition\nµn+1−µn=2π(Kn+1−Kn)\nL≥2π\nL/definesγ. (2.12)\nIn addition, by the assumption that {φn}forms a Riesz basis for Hand|Cφn|is uniformly bounded\nwith respect to n, it follows from Proposition 2 of [8] or Theorem 2 of [9] that Cis admissible for\nTq. So generally, we have\ny(t) =Cx(t) =/summationdisplay\nn∈Neλnt/an}bracketle{tx0,ψn/an}bracketri}htCφn=ef(q)t/summationdisplay\nn∈Neiµnt/an}bracketle{tx0,ψn/an}bracketri}htCφn/definesef(q)tPL(t),(2.13)\nwherePL(t) =/summationtext\nn∈Neiµnt/an}bracketle{tx0,ψn/an}bracketri}htCφnis well-defined and it follows from (2.11) that PL(t) is a\nY-valued function of period L. For anyT2−L>T1>L,\n/integraldisplayT2\nT1|y(t)|2dt=e2f(q)L/integraldisplayT2−L\nT1−L|y(t)|2dt, (2.14)\nthat is,\n/bardbly/bardblL2(T1,T2)=ef(q)L/bardbly/bardblL2(T1−L,T2−L). (2.15)\nTo obtain (2.8), we need to show that /bardbly/bardblL2(T1,T2)/ne}ationslash= 0 forT2−T1> L. Actually, it follows from\n(2.13) that\n/bardbly/bardbl2\nL2(T1,T2)=/integraldisplayT2\nT1/vextendsingle/vextendsingle/vextendsingleef(q)tPL(t)/vextendsingle/vextendsingle/vextendsingle2\ndt≥C3/integraldisplayT2\nT1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nn∈Neiµnt/an}bracketle{tx0,ψn/an}bracketri}htCφn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndt, (2.16)\n4whereC3= min/braceleftbig\ne2T1f(q),e2T2f(q)/bracerightbig\n>0. By Lemma 2.1 and the gap condition (2.12), it follows\nthat forT2−T1>2π\nγ=L,\n/integraldisplayT2\nT1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nn∈Neiµnt/an}bracketle{tx0,ψn/an}bracketri}htCφn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndt≥C1κ2/summationdisplay\nn∈N|/an}bracketle{tx0,ψn/an}bracketri}ht|2, (2.17)\nwhere\nC1=2(T2−T1)\nπ/parenleftbigg\n1−L2\n(T2−T1)2/parenrightbigg\n>0 forT2−T1>L.\nThe inequality (2.16) together with (2.17) gives\n/bardbly/bardbl2\nL2(T1,T2)≥C1C3κ2/summationdisplay\nn∈N|/an}bracketle{tx0,ψn/an}bracketri}ht|2. (2.18)\nNotice that {φn}n∈Nforms a Riesz basis for Hand so does {ψn}n∈NforH, there are two positive\nnumbersM1andM2such that\nM1/summationdisplay\nn∈N|/an}bracketle{tx0,ψn/an}bracketri}ht|2≤ /bardblx0/bardbl2≤M2/summationdisplay\nn∈N|/an}bracketle{tx0,ψn/an}bracketri}ht|2. (2.19)\nCombining (2.18) with (2.19) yields\n/bardbly/bardblL2(T1,T2)≥C/bardblx0/bardbl>0, (2.20)\nwhereC=κ/radicalBig\nC1C3\nM2>0. The identity (2.8) then follows from (2.15).\nThe inequality (2.20) means that system (2.4) is exactly obs ervable for T2−T1> L. So the\ninitial value x0can be uniquely determined by the output y(t),t∈[T1,T2]. We show next how to\nreconstruct the initial value from the output.\nActually, it follows from (2.11) that\n1\nL/integraldisplayL\n0ei(µm−µn)tdt=δnm, (2.21)\nHence,/integraldisplayL\n0y(t)e−λntdt=/integraldisplayL\n0/parenleftBigg/summationdisplay\nm∈Nei(µm−µn)t/an}bracketle{tx0,ψm/an}bracketri}htCφm/parenrightBigg\ndt=κnL·/an}bracketle{tx0,ψn/an}bracketri}ht,(2.22)\nTherefore the initial value x0can be reconstructed by\nx0=/summationdisplay\nn∈N/an}bracketle{tx0,ψn/an}bracketri}htφn=1\nL/summationdisplay\nn∈N1\nκn/parenleftbigg/integraldisplayL\n0y(t)e−λntdt/parenrightbigg\nφn. (2.23)\nThis completes the proof of the theorem.\nRemark 2.1. Clearly, (2.8) and (2.9) provide an algorithm to reconstruc tqandx0from the\noutput. It seems that the condition (2.6) is restrictive but it is satisfied by some physical systems\ndiscussed in Sections 3-5. Condition (2.6) is only for ident ification of q. For identification of initial\nvalue only, this condition can be removed. From numerical st andpoint, the function PL(t) in (2.13)\n5can be approximated by the finite series in (2.13) with the firs tNterms for sufficiently large N.\nHence condition (2.6) can be relaxed in numerical algorithm to be\nC1.There exists an Lsuch that: everyµnL\n2πis equal to (or close to) some integer for n∈\n{1,2,···,N},for some sufficiently large N.\nObviously, the relaxed condition C1 can still ensure that PL(t) is close to a function of period\nL. In this case, some points µnmay be very close to each other and the corresponding Riesz ba sis\nproperty of the family of divided differences of exponentials eiµntdeveloped in [1, Section II.4] and\n[2, 3] can be used. For the third condition, |Cφn| ≤Kimplies that Cis admissible for Tqwhich\nensures that the output belongs to L2\nloc(0,∞;Y), and|Cφn| ≥κimplies that system (2.4) is exactly\nobservable which ensures the unique determination of the in itial value. It is easily seen from (2.15)\nthat the coefficient qcan always be identified as long as /bardbly/bardblL2(T1,T2)/ne}ationslash= 0 for some time interval\n[T1,T2], which shows that the identifiability of coefficient qdoes not rely on the exact observability\nyet approximate observability.\nRemark 2.2. The condition T2−T1>Lin Theorem 2.1 is only used in application of Ingham’s\ninequalityin(2.17)toensurethat /bardbly/bardblL2(T1,T2)/ne}ationslash= 0. Inpracticalapplications, however, thiscondition\nis not always necessary. Actually, any L<T1<T2is applicable in (2.8) as long as /bardbly/bardblL2(T1,T2)/ne}ationslash= 0.\nSimilar remark also applies for Theorem 2.2 below.\nRemark 2.3. It should be noted that for identification of damping coefficie nt in [4, 5, 6], the\ndistributed observations are always required. In Theorem 2 .1, however, we use only boundary\nmeasurement and our identification algorithm utilizes phys ics of the system that the anti-damping\ncoefficient can make the measurement have an exponential term in (2.13). We should also point\nout that identification of the damping or anti-damping coeffic ientqin Theorem 2.1 does not rely on\nthe knowledge of the initial value. Actually, after qbeing estimated, there are various methods for\ninitial value reconstruction, see, e.g. [18, 21] and the ref erences therein. The idea of the algorithm\nfor reconstruction of the initial value here is borrowed fro m the Riesz basis approach proposed in\n[21].\nNow we come to the system with external disturbance which is i nevitable in many situations.\nSuppose that system (1.1) is corrupted by an unknown general bounded disturbance d(t) in obser-\nvation. It should be noted that system (1.1) is supposed to be anti-stable in Theorem 2.2 below\nwhereas in Theorem 2.1, there is no constraint on the stabili ty of system.\nTheorem 2.2. Suppose that system (1.1) is anti-stable and all the conditi ons in Theorem 2.1 are\nsatisfied. If the inverse of f(q)is continuous and the disturbance d(t)is bounded, i.e. |d(t)| ≤M\nfor someM >0and allt≥0, then for any T2−L>T1>L,\nlim\nT1→+∞qT1=q,lim\nT1→+∞/bardblˆx0T1−x0/bardbl= 0, (2.24)\nwhere\nqT1=f−1/parenleftBigg\n1\nLln/bardbly/bardblL2(T1,T2)\n/bardbly/bardblL2(T1−L,T2−L)/parenrightBigg\n, (2.25)\n6and\nˆx0T1=1\nL/summationdisplay\nn∈N1\nκn/parenleftbigg/integraldisplayT1+L\nT1y(t)e−λntdt/parenrightbigg\nφn, T1≥0. (2.26)\nMoreover, for sufficiently large T1, the errors |f(qT1)−f(q)|and/bardblˆx0T1−x0/bardblsatisfy\n|f(qT1)−f(q)|<4\nLM√T2−T1\n/bardbly/bardblL2(T1−L,T2−L)−M√T2−T1, (2.27)\nand\n/bardblˆx0T1−x0/bardbl ≤CM\nκ√\nLe−f(q)T1for someC >0. (2.28)\nProof.Introduce\nye(t) =CTq(t)x0=y(t)−d(t) =ef(q)tPL(t), (2.29)\nwherePL(t) is defined in (2.13). We first show that\nlim\nT1→+∞/bardblye/bardblL2(T1,T2)= +∞. (2.30)\nSince system (1.1) is anti-stable, the real part of the eigen valuesf(q)>0. It then follows from\n(2.29) that\n/bardblye/bardbl2\nL2(T1,T2)=/integraldisplayT2\nT1/vextendsingle/vextendsingle/vextendsingleef(q)tPL(t)/vextendsingle/vextendsingle/vextendsingle2\ndt≥e2f(q)T1/integraldisplayT2\nT1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nn∈Neiµnt/an}bracketle{tx0,ψn/an}bracketri}htCφn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndt. (2.31)\nUsing the same arguments as (2.16)-(2.20) in the proof of The orem 2.1, we have\n/bardblye/bardblL2(T1,T2)≥Cef(q)T1/bardblx0/bardbl, (2.32)\nwhereC=κ/radicalBig\nC1\nM2>0. Sincef(q)>0, x0/ne}ationslash= 0,(2.30) holds. Therefore for sufficiently large T1,\n/bardbly/bardblL2(T1,T2)\n/bardbly/bardblL2(T1−L,T2−L)=/bardblye+d/bardblL2(T1,T2)\n/bardblye+d/bardblL2(T1−L,T2−L)≤/bardblye/bardblL2(T1,T2)+/bardbld/bardblL2(T1,T2)\n/bardblye/bardblL2(T1−L,T2−L)−/bardbld/bardblL2(T1−L,T2−L).(2.33)\nSince|d(t)| ≤M, for any finite time interval I,\n/bardbld/bardblL2(I)=/parenleftbigg/integraldisplay\nI|d(t)|2dt/parenrightbigg1\n2\n≤M/radicalbig\n|I|, (2.34)\nwhere|I|represents the length of the time interval I. Hence\n/bardbly/bardblL2(T1,T2)\n/bardbly/bardblL2(T1−L,T2−L)≤eLf(q)+ε(T1,T2)\n1−ε(T1,T2), (2.35)\nwhere\nε(T1,T2) =M√T2−T1\n/bardblye/bardblL2(T1−L,T2−L). (2.36)\nSimilarly,\n/bardbly/bardblL2(T1,T2)\n/bardbly/bardblL2(T1−L,T2−L)≥eLf(q)−ε(T1,T2)\n1+ε(T1,T2). (2.37)\n7It is clear from (2.30) and (2.36) that lim T1→+∞ε(T1,T2) = 0. This together with (2.35) and (2.37)\ngives\nlim\nT1→+∞/bardbly/bardblL2(T1,T2)\n/bardbly/bardblL2(T1−L,T2−L)=eLf(q). (2.38)\nSincef−1(q) is continuous,\nlim\nT1→+∞qT1=f−1/parenleftBigg\n1\nLln lim\nT1→+∞/bardbly/bardblL2(T1,T2)\n/bardbly/bardblL2(T1−L,T2−L)/parenrightBigg\n=q.\nWe next show convergence of the initial value. Similarly wit h the arguments (2.21)-(2.23) in\nthe proof of Theorem 2.1, we have\nx0=1\nL/summationdisplay\nn∈N1\nκn/parenleftbigg/integraldisplayT1+L\nT1ye(t)e−λntdt/parenrightbigg\nφn,∀T1≥0.\nIt then follows from (2.26) that for arbitrary T1≥0,\nˆx0T1−x0=1\nL/summationdisplay\nn∈N1\nκn/parenleftbigg/integraldisplayT1+L\nT1d(t)e−λntdt/parenrightbigg\nφn. (2.39)\nIn view of the Riesz basis property of {φn}, it follows that\n/bardblˆx0T1−x0/bardbl2=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble1\nL/summationdisplay\nn∈N1\nκn/parenleftbigg/integraldisplayT1+L\nT1d(t)e−λntdt/parenrightbigg\nφn/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n≤M2\nL2κ2e−2f(q)T1/summationdisplay\nn∈N/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayL\n0/parenleftBig\nd(t+T1)e−f(q)t/parenrightBig\ne−iµntdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n,(2.40)\nwhereM2>0 is introduced in (2.19). To estimate the last series in (2.4 0), we need the Riesz\nbasis (sequence) property of the exponential system Λ :=/braceleftbig\nfn=eiµnt/bracerightbig\nn∈N. There are two cases\naccording to the relation between the sets {Kn}n∈Nintroduced in (2.11) and integers Z:\nCase 1: {Kn}n∈N=Z, that is, Λ =/braceleftBig\nei2nπ\nLt/bracerightBig\nn∈Z. In this case, since/braceleftbig\neint/bracerightbig\nn∈Zforms a Riesz\nbasis forL2[−π,π], Λ forms a Riesz basis for L2[−L\n2,L\n2].\nCase 2: {Kn}n∈N/subsetnoteql Z. In this case, it is noted that the exponential system/braceleftbig\neiµnt/bracerightbig\nn∈Nforms\na Riesz sequence in L2[−L\n2,L\n2].\nIn each case above, by properties of Riesz basis and Riesz seq uence (see, e.g., [23, p. 32-35,\np.154]), there exists a positive constant C4>0 such that\n/summationdisplay\nn∈N|(g,fn)|2≤C4/bardblg/bardbl2\nL2[−L\n2,L\n2], (2.41)\nfor allg∈L2[−L\n2,L\n2], where ( ·,·) denotes the inner product in L2[−L\n2,L\n2].\nWe return to the estimation of /bardblˆx0T1−x0/bardbl. By variable substitution of t=L\n2−sin (2.40),\ntogether with (2.41), we have\n/bardblˆx0T1−x0/bardbl2≤M2\nL2κ2e−2f(q)T1/summationdisplay\nn∈N/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayL\n2\n−L\n2/bracketleftbigg\nd/parenleftbigg\nT1+L\n2−s/parenrightbigg\nef(q)(s−L\n2)e−iµnL\n2/bracketrightbigg\neiµnsds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n≤M2C4\nL2κ2e−2f(q)T1/integraldisplayL\n2\n−L\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingled/parenleftbigg\nT1+L\n2−s/parenrightbigg\nef(q)(s−L\n2)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nds\n≤M2M2C4\nLκ2e−2f(q)T1.\n8Therefore,\n/bardblˆx0T1−x0/bardbl ≤/radicalbigg\nM2C4\nLM\nκe−f(q)T1, (2.42)\nwhich implies that /bardblˆx0T1−x0/bardblwill tend to zero as T1→+∞forf(q)>0. The inequality (2.28)\nwith the positive number C=√M2C4is also concluded.\nFinally, we estimate |f(qT1)−f(q)|. SettingT1large enough so that ε(T1,T2)<1, it follows\nfrom (2.25) and (2.35) that\nLf(qT1)≤lneLf(q)+ε(T1,T2)\n1−ε(T1,T2)<Lf(q)+2ε(T1,T2)\n1−ε(T1,T2). (2.43)\nSimilarly, for T1large enough so that ε(T1,T2)≤1\n4, it follows from (2.25) and (2.37) that\nLf(qT1)>Lf(q)−4ε(T1,T2)\n1+ε(T1,T2). (2.44)\nCombining (2.43) and (2.44), and setting T1large enough so that ε(T1,T2)≤1\n4, we have\n|f(qT1)−f(q)|<4ε(T1,T2)\nL.\nThe error estimation (2.27) comes from the fact\nε(T1,T2)≤M√T2−T1\n/bardbly/bardblL2(T1−L,T2−L)−M√T2−T1. (2.45)\nWe thus complete the proof of the theorem.\nRemark 2.4. Theorem 2.2 shows that when system (1.1) is anti-stable, the nqT1defined in (2.25)\ncan be regarded as an approximation of the coefficient qwhenT1is sufficiently large. Roughly\nspeaking, the ε(T1,T2) defined in (2.36) reflects the ratio of the energy, in L2norm, of the distur-\nbanced(t) which is an unwanted signal, with the energy of the real outp ut signalye(t). We may\nregard 1/ε(T1,T2) as signal-to-noise ratio (SNR) which is well known in signa l analysis. Theorem\n2.2 indicates that qT1defined in (2.25) is an approximation of the coefficient qwhen SNR is large\nenough. However, if system (1.1) is stable, i.e.f(q)<0, similar analysis shows that the output\nwill be exponentially decaying oscillation, which implies that the unknown disturbance will account\nfor a large proportion in observation and the SNR can not be to o large. In this case, it is difficult\nto extract enough useful information from the corrupted obs ervation as that with large SNR.\nRemark 2.5. Theanti-stability assumptionin Theorem 2.2 is almost nece ssary since otherwise, we\nmay have the case of y(t) =Cx(t)+d(t)≡0 for which we cannot obtain anything for identification.\nRemark 2.6. It is well known that the inverse problems are usually ill-po sed in the sense of\nHadamard, that is, arbitrarily small error in the measureme nt data may lead to large error in\nsolution. Theorem 2.2 shows that if system (1.1) is anti-sta ble, our algorithm is robust against\nbounded unknown disturbance in measurement data. Actually , similar to the analysis in Theorem\n2.2, it can be shown that when system (1.1) is not anti-stable , the algorithm in Theorem 2.1 is also\nnumerically stable in the presence of small perturbations i n the measurement data, as long as the\nperturbation is relatively small in comparison to the outpu t. Some numerical simulations validate\nthis also in Example 3.1 in Section 3.\n93 Application to wave equation\nIn this section, we apply the algorithm proposed in previous section to identification of the anti-\ndamping coefficient and initial values for a one-dimensional vibrating string equation described by\n([7, 16])\n\nutt=uxx, 0<x<1, t>0,\nu(0,t) = 0, ux(1,t) =qut(1,t), t≥0,\ny(t) =ux(0,t)+d(t), t ≥0,\nu(x,0) =u0(x), ut(x,0) =u1(x),0≤x≤1,(3.1)\nwherexdenotes theposition, tthetime, 0 <q/ne}ationslash= 1theunknownanti-dampingcoefficient, u0(x) and\nu1(x) the unknown initial displacement and initial velocity, re spectively, and y(t) is the boundary\nmeasured output corrupted by the disturbance d(t).\nLetH=H1\nE(0,1)×L2(0,1), whereH1\nE(0,1) ={f∈H1(0,1)|f(0) = 0}, equipped with the\ninner product /an}bracketle{t·,·/an}bracketri}htand the inner product induced norm\n/bardbl(f,g)/bardbl2=/integraldisplay1\n0/bracketleftbig\n|f′(x)|2+|g(x)|2/bracketrightbig\ndx.\nDefine the system operator A:D(A)(⊂ H)→ Has\n/braceleftBigg\nA(f,g) = (g,f′′),\nD(A) =/braceleftbig\n(f,g)∈H2(0,1)×H1\nE(0,1)|f(0) = 0,f′(1) =qg(1)/bracerightbig\n,(3.2)\nand the observation operator CfromHtoCas\nC(f,g) =f′(0),(f,g)∈D(A). (3.3)\nIt is indicated in [21] that the operator Agenerates a C0-group on H.\nLemma 3.1. [21] LetAbe defined by (3.2) and let q/ne}ationslash= 1. Then the spectrum of Aconsists of all\nisolated eigenvalues given by\nλn=1\n2ln1+q\nq−1+inπ, n∈Z, if q> 1, (3.4)\nor\nλn=1\n2ln1+q\n1−q+i2n+1\n2π, n∈Z, if0<q<1, (3.5)\nand the corresponding eigenfunctions Φn(x)are given by\nΦn(x) =/parenleftbiggsinhλnx\nλn,sinhλnx/parenrightbigg\n,∀n∈Z. (3.6)\nMoreover, {Φn(x)}n∈Zforms a Riesz basis for H.\nLemma 3.2. [21] Let Abe defined by (3.2) and let q/ne}ationslash= 1. Then the adjoint operator A∗ofAis\ngiven by\n/braceleftBigg\nA∗(v,h) =−(h,v′′),\nD(A∗) =/braceleftbig\n(v,h)∈H2(0,1)×H1(0,1)|v(0) = 0,v′(1)+qh(1) = 0/bracerightbig\n,(3.7)\n10andσ(A∗) =σ(A). The eigenvector Ψn(x)ofA∗corresponding to λnis given by\nΨn(x) =/parenleftbiggsinhλnx\nλn,−sinhλnx/parenrightbigg\n,∀n∈Z. (3.8)\nIt is easy to verify that for any n,m∈Z,/an}bracketle{tΦn,Ψm/an}bracketri}ht=δnm.System (3.1) can be written as the\nfollowing evolutionary equation in H:\ndX(t)\ndt=AX(t), t>0, X(0) = (u0,u1), (3.9)\nwhereX(t) = (u(·,t),ut(·,t)), and the solution of (3.9) is given by\nX(t) =/summationdisplay\nn∈Zeλnt/an}bracketle{tX(0),Ψn/an}bracketri}htΦn. (3.10)\nThus\ny(t) =/summationdisplay\nn∈Zeλnt/an}bracketle{tX(0),Ψn/an}bracketri}ht+d(t). (3.11)\nIt can be seen from Lemma 3.1 that when q= 1, the real part of the eigenvalues is + ∞, while\nfor 0<q/ne}ationslash= 1, the real part is finite positive. Hence, we suppose 1 /∈Qas usual (see, e.g., [7, 16]),\nwhereQ= [q,q] is the prior parameter set.\nWe takeq∈Q= (1,+∞) as an example to illustrate how to apply the algorithms prop osed in\nprevious section to simultaneous identification for the ant i-damping coefficient qand initial values.\nThe following Corollaries 3.1-3.2 are the direct consequen ces of Theorem 2.1 and Theorem 2.2,\nrespectively, by noticing that for system (3.1), the releva nt function and parameters now are\nf(q) =1\n2lnq+1\nq−1, µn=nπ, L= 2, κn= 1. (3.12)\nCorollary 3.1. Suppose that d(t) = 0in system (3.1). Then both the coefficient qand initial values\nu0(x)andu1(x)can be uniquely determined by the output y(t),t∈[0,T], whereT >4. Specifically,\nqcan be recovered exactly from\nq=/bardbly/bardblL2(T1,T2)+/bardbly/bardblL2(T1−2,T2−2)\n/bardbly/bardblL2(T1,T2)−/bardbly/bardblL2(T1−2,T2−2),2≤T1<T2−2, (3.13)\nand the initial values u0(x)andu1(x)can be reconstructed from\nu0(x) =1\n2/summationdisplay\nn∈Z/parenleftbigg/integraldisplay2\n0y(t)e−λntdt/parenrightbiggsinhλnx\nλn, u1(x) =1\n2/summationdisplay\nn∈Z/parenleftbigg/integraldisplay2\n0y(t)e−λntdt/parenrightbigg\nsinhλnx.(3.14)\nNotice that in (3.14), the observation interval [0 ,2] is the minimal time interval for observation\nto identify the initial values for any identification algori thm.\nCorollary 3.2. Suppose that q∈Q= (1,+∞)in system (3.1) and the disturbance is bounded, i.e.\n|d(t)| ≤Mfor someM >0and allt≥0. Then for any T2−2>T1≥2,\nlim\nT1→+∞qT1=q,lim\nT1→+∞/bardbl(ˆu0T1,ˆu1T1)−(u0,u1)/bardbl= 0, (3.15)\n11where\nqT1=/bardbly/bardblL2(T1,T2)+/bardbly/bardblL2(T1−2,T2−2)\n/bardbly/bardblL2(T1,T2)−/bardbly/bardblL2(T1−2,T2−2), (3.16)\nand\nˆu0T1(x) =1\n2/summationdisplay\nn∈Z/parenleftbigg/integraldisplayT1+2\nT1y(t)e−λntdt/parenrightbiggsinhλnx\nλn,\nˆu1T1(x) =1\n2/summationdisplay\nn∈Z/parenleftbigg/integraldisplayT1+2\nT1y(t)e−λntdt/parenrightbigg\nsinhλnx.(3.17)\nTo end this section, we present some numerical simulations f or system (3.1) to illustrate the\nperformance of the algorithm.\nExample 3.1. The observation with random noises when system (1.1) is stabl e.\nA simple spectral analysis together with Theorem 2.1 shows t hat Corollary 3.1 is also valid for\nq∈Q= (−∞,−1). In this example, the damping coefficient qand initial values u0(x),u1(x) are\nchosen as\nq=−3, u0(x) =−3sinπx, u 1(x) =πcosπx. (3.18)\nIn this case, the output can be obtained from (3.11) (with d(t) = 0), where the infinite series is\napproximated by a finite one, that is, {n∈Z}is replaced by {n∈Z|−5000≤n≤5000}. Some\nrandom noises are added to the measurement data and we use the se data to test the algorithm\nproposed in Corollary 3.1.\nLetT1= 2,T2= 2.5. Then the damping coefficient qcan be recovered from (3.13), and the\ninitial values u0(x) andu1(x) can be reconstructed from (3.14). Table 1 lists the numeric al results\nfor the damping coefficients (the second column in Table 1) and Figure 1(a)-1(c) for the initial\nvalues in various cases of noise levels. In Table 1, the absol ute errors of the real damping coefficient\nand the recovered ones, and the L2-norm of the differences between the exact initial values and t he\nreconstructed ones are also shown.\nIt is worth pointing out that in reconstruction of the initia l values from (3.14), the infinite series\nis approximated by a finite one once again, that is, {n∈Z}is replaced by {n∈Z| |n| ≤1000},\nwhich accounts for the zero value of the reconstructed initi al velocity at the left end. This is also\nthe reason that the errors of the initial velocity (the last c olumn in Table 1) are relatively large\neven if there is no random noise in the measured data.\nTable 1: Absolute errors with different noise levels\nNoise Level Recovered qErrors forqErrors foru0(x) Errors for u1(x)\n0 -3.0000 9.3259E-15 1.1744E-08 2.2215E-01\n1% -2.9994 6.2498E-04 1.1618E-03 2.2748E-01\n3% -2.9979 2.0904E-03 3.5604E-03 2.6662E-01\n120 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1í3í2.5í2í1.5í1í0.50\nxinitial displacementreal u0(x)\nrecovered\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1í4í3í2í101234\nxinitial velocityreal u1(x)\nrecovered\n(a) without random noise0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1í3.5í3í2.5í2í1.5í1í0.50\nxinitial displacementreal u0(x)\nrecovered\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1í4í3í2í101234\nxinitial velocityreal u1(x)\nrecovered\n(b) with 1% random error\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1í3.5í3í2.5í2í1.5í1í0.50\nxinitial displacementreal u0(x)\nrecovered\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1í4í3í2í101234\nxinitial velocityreal u1(x)\nrecovered\n(c) with 3% random error\nFigure 1: The initial values: initial displacement (upper) and initial velocity (lower)\nExample 3.2. The observation with general bounded disturbance when syste m (1.1) is anti-stable.\nThe anti-damping coefficient and initial values are chosen as\nq= 3, u0(x) = 3sinπx, u 1(x) =πcosπx. (3.19)\nand the observation is corrupted by the bounded disturbance :\nd(t) = 2sin1\n1+t+3cos10t. (3.20)\nThe relevant parameters in Corollary 3.2 are chosen to be T2=T1+3, and let T1be different values\nincreasing from 2 to 10. The corresponding anti-damping coe fficientsqT1recovered from (3.16) are\ndepicted in Figure 2. It is seen that qT1converges to the real value q= 3 asT1increases. Setting\nT1= 0,3,7 in (3.17) and reconstructing the initial values produce re sults in Figure 2 from which\nwe can see that the reconstructed initial values become clos er to the real ones as T1increases.\n132 3 4 5 6 7 8 9 102.9533.053.13.153.2\nT1Real and recovered antidamping coefficient\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 101234\nxInitial dispacement u0(x)\nreal u0(x)\nT1=0\nT1=3\nT1=7\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1í505\nxInitial velocity u1(x)\nreal u1(x)\nT1=0\nT1=3\nT1=7\nFigure 2: anti-damping coefficient qand initial values u0, u1\n4 Application to Schr¨ odinger equation\nIn this section, we consider a quantum system described by th e following Schr¨ odinger equation:\n\n\nut=−iuxx+qu, 0<x<1, t>0,\nux(0,t) = 0, u(1,t) = 0, t≥0,\ny(t) =u(0,t)+d(t), t ≥0,\nu(x,0) =u0(x), 0≤x≤1.(4.1)\nwhereu(x,t) is the complex-valued state, iis the imaginary unit, and the potential q>0 andu0(x)\nare the unknown anti-damping coefficient and initial value, r espectively.\nLetH=L2(0,1) be equipped with the usual inner product /an}bracketle{t·,·/an}bracketri}htand the inner product induced\nnorm/bardbl·/bardbl.Introduce the operator Adefined by\n/braceleftBigg\nAφ=−iφ′′+qφ,\nD(A) =/braceleftbig\nφ∈H2(0,1)|φ′(0) =φ(1) = 0/bracerightbig\n.(4.2)\nA straightforward verification shows that such defined Agenerates a C0-semigroup on H.\nLemma 4.1. [17] Let Abe defined by (4.2). Then the spectrum of Aconsists of all isolated\neigenvalues given by\nλn=q+i/parenleftbigg\nn−1\n2/parenrightbigg2\nπ2, n∈N, (4.3)\n14and the corresponding eigenfunctions φn(x)are given by\nφn(x) =√\n2cos/parenleftbigg\nn−1\n2/parenrightbigg\nπx, n∈N. (4.4)\nIn addition, {φn(x)}n∈Nforms an orthonormal basis for H.\nSystem (4.1) can be rewritten as the following evolutionary equation in H:\ndX(t)\ndt=AX(t), t>0, X(0) =u0, (4.5)\nand the solution of (4.5) is given by\nX(t) =/summationdisplay\nn∈Neλnt/an}bracketle{tX(0),φn/an}bracketri}htφn. (4.6)\nThus\ny(t) =√\n2/summationdisplay\nn∈Neλnt/an}bracketle{tu0,φn/an}bracketri}ht+d(t). (4.7)\nThe relevant function and parameters in Theorems 2.1-2.2 fo r system (4.1) are\nf(q) =q, µn=/parenleftbigg\nn−1\n2/parenrightbigg2\nπ2, L=8\nπ, κn=√\n2.\nParallel toSection 3, wehavetwocorollaries correspondin gtotheexact observation andobservation\nwith general bounded disturbance, respectively, for syste m (4.1). Here we only list the latter one\nand the former is omitted.\nCorollary 4.1. Suppose that q∈Q= (0,+∞)in system (4.1) and the disturbance is bounded, i.e.\n|d(t)| ≤Mfor someM >0and allt≥0. Then for any T2−8\nπ>T1>8\nπ,\nlim\nT1→+∞qT1=q,lim\nT1→+∞/bardblˆu0T1−u0/bardbl= 0, (4.8)\nwhere\nqT1=π\n8ln/bardbly/bardblL2(T1,T2)\n/bardbly/bardblL2(T1−8\nπ,T2−8\nπ),8\nπ<T1<T2−8\nπ, (4.9)\nand\nˆu0T1(x) =π\n8/summationdisplay\nn∈Z/parenleftBigg/integraldisplayT1+8\nπ\nT1y(t)e−λntdt/parenrightBigg\ncos/parenleftbigg\nn−1\n2/parenrightbigg\nπx. (4.10)\nWe also give a numerical simulation to test the algorithm pro posed in Corollary 4.1 for system\n(4.1), where the anti-damping coefficient qand initial value u0(x) are chosen as\nq= 0.7, u0(x) = sinπx+icosπx, (4.11)\nand the observation is corrupted by the disturbance\nd(t) = 2sint\n10+t+3icos20t. (4.12)\n15The observation can be obtained from (4.7) by a finite series a pproximation, that is, {n∈N}\nis replaced by {n∈N|n≤5000}. The relevant parameters in Corollary 4.1 are chosen to be\nT2=T1+1, andT1increasing from 2 .55 to 10. The corresponding anti-damping coefficients qT1\nrecovered from (4.9) are shown in Figure 3. It is obvious that qT1is convergent to the real value\nq= 0.7 asT1increases. Setting T1= 0,3,7, respectively, in (4.10), the reconstructed initial valu es\nare shown in Figure 3 from which it is seen that the errors betw een the reconstructed initial values\nand the real ones become smaller as T1increases.\n3 4 5 6 7 8 9 100.60.650.7\nT1Real and recovered antidamping coefficient\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1í0.500.511.5\nxReal part of initial value\nRe(u0)\nT1=0\nT1=3\nT1=7\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1í1.5í1í0.500.511.5\nxImaginary part of initial value\nIm(u0)\nT1=0\nT1=3\nT1=7\nFigure 3: coefficient (upper), real part (middle) and imagina ry part (lower) of initial value u0(x)\n5 Application to coupled strings equation\nIn this section, we consider the following two connected ant i-stable strings with joint anti-damping\ndescribed by \n\nutt(x,t) =uxx(x,t),x∈(0,1\n2)∪(1\n2,1), t>0,\nu/parenleftBig\n1\n2−,t/parenrightBig\n=u/parenleftBig\n1\n2+,t/parenrightBig\n, t≥0,\nux/parenleftBig\n1\n2−,t/parenrightBig\n−ux/parenleftBig\n1\n2+,t/parenrightBig\n=qut(1,t), t≥0,\nu(0,t) =ux(1,t) = 0, t≥0,\nu(x,0) =u0(x),ut(x,0) =u1(x),0≤x≤1,\ny(t) =ux(0,t)+d(t), t≥0,(5.1)\nwhereq >0,q/ne}ationslash= 2 is the unknown anti-damping constant. System (5.1) model s two connected\nstrings with joint vertical force anti-damping, see [10, 11 , 22] for more details.\n16LetH=H1\nE(0,1)×L2(0,1) be equipped with the inner product /an}bracketle{t·,·/an}bracketri}htand its induced norm\n/bardbl(u,v)/bardbl2=/integraldisplay1\n0/bracketleftbig\n|u′(x)|2+|v(x)|2/bracketrightbig\ndx,\nwhereH1\nE(0,1) =/braceleftbig\nu|u∈H1(0,1),u(0) = 0/bracerightbig\n. Then system (5.1) can be rewritten as an evolution-\nary equation in Has follows:\nd\ndtX(t) =AX(t), (5.2)\nwhereX(t) = (u(·,t),ut(·,t))∈ HandAis defined by\nA(u,v) = (v(x),u′′(x)), (5.3)\nwith the domain\nD(A) =\n\n(u,v)∈H1(0,1)×H1\nE(0,1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleu(0) =u′(1) = 0, u|[0,1\n2]∈H2(0,1\n2),\nu|[1\n2,1]∈H2(1\n2,1), u′(1\n2−)−u′(1\n2+) =qv(1\n2),\n\n,(5.4)\nwhereu|[a,b]denotes the function u(x) confined to [ a,b].\nWe assume without loss of generality that the prior paramete r set forqisQ= (2,+∞) since\nthe case for Q= (0,2) is very similar.\nLemma 5.1. [22] LetAbe defined by (5.3)-(5.4) and q∈Q= (2,+∞). ThenA−1is compact on\nHand the eigenvalues of Aare algebraically simple and separated, given by\nλn=1\n2lnq+2\nq−2+inπ, n∈Z. (5.5)\nThe corresponding eigenfunctions Φn(x)are given by\nΦn(x) = (φn(x),λnφn(x)),∀n∈Z, (5.6)\nwhere\nφn(x) =\n\n√\n2\nλncoshλn\n2sinhλnx, 0<x<1\n2,\n√\n2\nλnsinhλn\n2coshλn(1−x),1\n2<x<1.\nand{Φn(x)}n∈Zforms a Riesz basis for H. In addition, Agenerates a C0-semigroup on H.\nLemma 5.2. [22] LetAbe defined by (5.3)-(5.4) and q∈Q= (2,+∞). Then the adjoint operator\nA∗ofAis given by\nA∗(u,v) =−(v,u′′), (5.7)\nwith the domain\nD(A∗) =\n\n(u,v)∈H1(0,1)×H1\nE(0,1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleu(0) =u′(1) = 0, u|[0,1\n2]∈H2(0,1\n2),\nu|[1\n2,1]∈H2(1\n2,1), u′(1\n2−)−u′(1\n2+) =−qv(1\n2)\n\n,\nandσ(A∗) =σ(A). The eigenfunctions Ψn(x)ofA∗corresponding to λnare given by\nΨn(x) =/parenleftbig\nφn(x),−λnφn(x)/parenrightbig\n,∀n∈Z. (5.8)\n17A direct calculation shows that {Ψn(x)}is biorthogonal to {Φn(x)}. Hence, the solution of\n(5.2) can be expressed as\nX(t) =/summationdisplay\nn∈Zeλnt/an}bracketle{tX(0),Ψn/an}bracketri}htΦn. (5.9)\nDefine the observation operator CfromHtoCto be\nC(f,g) =f′(0),(f,g)∈D(A). (5.10)\nThen\ny(t) =/summationdisplay\nn∈Zeλnt/an}bracketle{tX(0),Ψn/an}bracketri}htCΦn+d(t). (5.11)\nThe succeeding Corollary 5.1 is a direct consequence of Theo rem 2.2 by noticing that for system\n(5.1), the relevant function and parameters now are\nf(q) =1\n2lnq+2\nq−2, µn=nπ, L= 2, κn=√\n2coshλn\n2,\nand a simple calculation shows that\n√\n2\n2/bracketleftBigg/parenleftbiggq+2\nq−2/parenrightbigg1\n4\n−/parenleftbiggq−2\nq+2/parenrightbigg1\n4/bracketrightBigg\n≤ |κn| ≤√\n2\n2/bracketleftBigg/parenleftbiggq+2\nq−2/parenrightbigg1\n4\n+/parenleftbiggq−2\nq+2/parenrightbigg1\n4/bracketrightBigg\n,∀n∈Z.\nThe corollaries corresponding to Theorem 2.1 are omitted he re.\nCorollary 5.1. Suppose that q∈Q= (2,+∞)in system (5.1) and the disturbance is bounded, i.e.\n|d(t)| ≤Mfor someM >0and allt≥0. Then for any T2−2>T1≥2,\nlim\nT1→+∞qT1=q,lim\nT1→+∞/bardbl(ˆu0T1,ˆu1T1)−(u0,u1)/bardbl= 0,\nwhere\nqT1=2/parenleftbig\n/bardbly/bardblL2(T1,T2)+/bardbly/bardblL2(T1−2,T2−2)/parenrightbig\n/bardbly/bardblL2(T1,T2)−/bardbly/bardblL2(T1−2,T2−2), (5.12)\nand\nˆu0T1(x) =\n\n1\n2/summationdisplay\nn∈Z/parenleftbigg/integraldisplayT1+2\nT1y(t)e−λntdt/parenrightbiggsinhλnx\nλn, 0<x<1\n2,\n1\n2/summationdisplay\nn∈Z/parenleftbigg/integraldisplayT1+2\nT1y(t)e−λntdt/parenrightbigg\ntanhλn\n2coshλn(1−x)\nλn,1\n2<x<1.\nˆu1T1(x) =\n\n1\n2/summationdisplay\nn∈Z/parenleftbigg/integraldisplayT1+2\nT1y(t)e−λntdt/parenrightbigg\nsinhλnx, 0<x<1\n2,\n1\n2/summationdisplay\nn∈Z/parenleftbigg/integraldisplayT1+2\nT1y(t)e−λntdt/parenrightbigg\ntanhλn\n2coshλn(1−x),1\n2<x<1.(5.13)\nAs before, we present some numerical simulations for system (5.1) to showcase the effectiveness\nof the algorithm proposed in Corollary 5.1. The anti-dampin g coefficient qand initial values u0(x)\nandu1(x) are chosen to be\nq= 3, u0(x) = sinx, u1(x) = cosx, (5.14)\n18and the observation is corrupted by the disturbance\nd(t) = sint2\n10+t+cos10t. (5.15)\nTheobservation is obtained from(5.11) by afiniteseries app roximation, that is, {n∈N}is replaced\nby{n∈N| |n| ≤5000}. The relevant parameters in Corollary 5.1 are chosen to be T2=T1+1, and\nT1increases from 2 to 8. The corresponding anti-damping coeffic ientsqT1recovered from (5.12)\nare plotted in Figure 4. It can be seen that qT1converges to the real value q= 3 asT1increases.\nLetT1= 0,3,7 in (5.13), and the reconstructed initial values are shown i n Figure 4, from which\nit is seen that the reconstructed initial values become clos er to the real ones as T1increases.\n2 3 4 5 6 7 82.9533.053.13.15\nT1Real and recovered antidamping coefficient\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.20.40.60.81\nxInitial dispacement u0(x)\nreal u0(x)\nT1=0\nT1=3\nT1=7\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1í10123\nxInitial velocity u1(x)\nreal u1(x)\nT1=0\nT1=3\nT1=7\nFigure 4: anti-damping coefficient q(upper), initial values u0(middle) and u1(lower)\n6 Concluding remarks\nIn this paper, we propose an algorithm to reconstruct simult aneously an anti-damping coefficient\nand an initial value for some anti-stable PDEs. When the meas ured output is exact, the recovered\nvalues are exact whereas if the measured output suffers from bo unded unknown disturbance, the\napproximated values of the anti-damping coefficient and init ial value can also be obtained. Some\nnumericalexamplesarecarriedouttovalidatetheeffectiven ess ofthealgorithm. Itisverypromising\nto apply the algorithm presented here to stabilization of an ti-stable systems with an unknown anti-\ndamping coefficient.\n19Acknowledgements\nThis work was supported by the National Natural Science Foun dation of China and the National\nResearch Foundation of South Africa.\nReferences\n[1] S.A. Avdonin and S.A. Ivanov, Families of Exponentials: The Method of Moments in Control-\nlability Problems for Distributed Parameter Systems , Cambridge University Press, 1995.\n[2] S.A. Avdonin and W. Moran, Ingham-type inequalities and Riesz bases of divided differences,\nInt. J. Appl. Math. Comput. Sci. , 11(2001), 803–820.\n[3] S.A. Avdonin and S.A. Ivanov, Riesz bases of exponential s and divided differences, St. Peters-\nburg Math. J. , 13(2002), 339–351.\n[4] H.T. Banks and I.G. Rosen, Computational methods for the identification of spatially varying\nstiffness and damping in beams, Control Theory Adv. Tech. , 3(1987), 1–32.\n[5] H.T. Banks and K. Ito, A unified framework for approximati on in inverse problems for dis-\ntributed parameter systems, Control Theory Adv. Tech. , 4(1988), 73–90.\n[6] H.T. Banks and K. Kunisch, Estimation Techniques for Distributed Parameter System ,\nBirkh¨ auser, Boston, 1989.\n[7] D. Bresch-Pietri and M. Krstic, Output-feedback adapti ve control of a wave PDE with bound-\nary anti-damping, Automatica , 50(2014), 1407–1415.\n[8] B.Z. Guo and Y.H. Luo, Controllability and stability of a second-order hyperbolic system with\ncollocated sensor/actuator, Systems Control Lett. , 46(2002), 45–65.\n[9] B.Z. Guo and Y.H. Luo, Riesz basis property of a second-or der hyperbolic system with collo-\ncated scalar input-output, IEEE Trans. Autom. Control , 47(2002), 693–698.\n[10] B.Z. Guo and W.D. Zhu, On the energy decay of two coupled s trings through a joint damper,\nJ. Sound Vib. , 203(1997), 447–455.\n[11] B.Z. Guo and F.F. Jin, Arbitrary decay rate for two conne cted strings with joint anti-damping\nby boundary output feedback, Automatica , 46(2010), 1203–1209.\n[12] B.Z. Guo and F.F. Jin, Sliding mode and active disturban ce rejection control to stabilization\nof one-dimensional anti-stable wave equations subject to d isturbance in boundary input, IEEE\nTrans. Autom. Control , 58(2013), 1269–1274.\n[13] B.Z. Guo and F.F. Jin, Output feedback stabilization fo r one-dimensional wave equation sub-\nject to boundary disturbance, IEEE Trans. Autom. Control , 60(2015), 824–830.\n20[14] A.E. Ingham, Sometrigonometrical inequalities with a pplications in thetheory of series, Math.\nZ., 41(1936), 367–379.\n[15] V. Komornik and P. Loreti, Fourier Series in Control Theory , Springer-Verlag, New York,\n2005.\n[16] M. Krstic, Adaptive control of an anti-stable wave PDE, Dyn. Contin. Discrete Impuls. Syst.\nSer. A. Math. Anal. , 17(2010), 853–882.\n[17] M. Krstic, B.Z. Guo, and A. Smyshlyaev, Boundary contro llers and observers for the linearized\nSchr¨ odinger equation, SIAM J. Control Optim. , 49(2011), 1479–1497.\n[18] K. Ramdani, M. Tucsnak, and G. Weiss, Recovering the ini tial state of an infinite-dimensional\nsystem using observers, Automatica, 46(2010), 1616–1625.\n[19] A. Smyshlyaev and M. Krstic, Boundary control of an anti -stable wave equation with anti-\ndamping on the uncontrolled boundary, Systems Control Lett. , 58(2009), 617–623.\n[20] G. Weiss, Admissible observation operators for linear semigroups, Israel J. Math. , 65(1989),\n17-43.\n[21] G.Q. Xu, State reconstruction of a distributed paramet er system with exact observability, J.\nMath. Anal. Appl. , 409(2014), 168–179.\n[22] G.Q. Xu and B.Z. Guo, Riesz basis property of evolution e quations in Hilbert spaces and\napplication to a coupled string equation, SIAM J. Control Optim. , 42(2003), 966–984.\n[23] R. M. Young, An Introduction to Nonharmonic Fourier Series , Academic Press, New York,\n1980.\n21"    },    {        "title": "1503.04126v1.A_one_step_optimal_energy_decay_formula_for_indirectly_nonlinearly_damped_hyperbolic_systems_coupled_by_velocities.pdf",        "content": "arXiv:1503.04126v1  [math.OC]  13 Mar 2015A one-step optimal energy decay formula for indirectly\nnonlinearly damped hyperbolic systems coupled by velociti es\nFatiha Alabau-Boussouira∗, Zhiqiang Wang†and Lixin Yu‡\nJuly 7, 2021\nAbstract\nIn this paper, we consider the energy decay of a damped hyperbolic system of wave-\nwave type which is coupled through the velocities. We are interested in the asymptotic\nproperties of the solutions of this system in the case of indirect non linear damping, i.e.\nwhen only one equation is directly damped by a nonlinear damping. We pr ove that the\ntotal energy of the whole system decays as fast as the damped sin gle equation. Moreover,\nwe give a one-step general explicit decay formula for arbitrary non linearity. Our results\nshows that the damping properties are fully transferred from the damped equation to\nthe undamped one by the coupling in velocities, different from the cas e of couplings\nthrough displacements as shown in [1, 2, 3, 13] for the linear dampin g case, and in [7] for\nthe nonlinear damping case. The proofs of our results are based on multiplier techniques,\nweightednonlinearintegralinequalitiesandthe optimal-weightconve xitymethod of[5, 8].\nKeywords Energy decay, nonlineardamping, hyperbolicsystems, wave equation, plate equa-\ntion, weighted nonlinear integral inequality, optimal-we ight convexity method.\n1 Introduction\nLet Ω be a bounded subset of Rnwith a smooth boundary denoted by Γ. We consider\nthe following wave system\n\n\nu′′−∆u+α(x)v′+ρ(x,u′) = 0 in Ω ×(0,+∞),\nv′′−∆v−α(x)u′= 0 in Ω ×(0,+∞),\nu=v= 0 on Γ ×(0,+∞),\n(u,u′)(0) = (u0,u1),(v,v′)(0) = (v0,v1) in Ω.(1.1)\n∗IECL, Universit´ e de Lorraine and CRNS (UMR 7502), 57045 Met z, France. E-mail:\nfatiha.alabau@univ-lorraine.fr .\n†School of Mathematical Sciences and Shanghai Key Laborator y for Contemporary Applied Mathematics,\nFudan University, Shanghai 200433, China. E-mail: wzq@fudan.edu.cn .\n‡School of Mathematics and Information Sciencs, Yantai Univ ersity, Yantai 264005, China. E-mail:\nfdylx01@sina.com.cn .\n1To a strong solution of this system, we associate the energy d efined by\nE(t) =1\n2/integraldisplay\nΩ(|u′|2+|∇u|2+|v′|2+|∇v|2)dx. (1.2)\nOne can show that the energy of the strong solutions of this sy stem satisfies\nE′(t) =−/integraldisplay\nΩu′ρ(x,u′)dx, (1.3)\nIn the sequel, the assumption on ρwill ensure that ρ(·,s)s/greaterorequalslant0 a.e. in Ω and for all s∈R,\nso that the energies of the strong solutions satisfy E′(t)≤0. Hence the nonlinear term in\nour coupled system is indeed a damping term, so that one expec ts the energies to decay to 0\nat infinity.\nLet us recall that for the scalar damped wave equation, that i s for\nu′′−∆u+ρ(x,u′) = 0 in Ω ×(0,+∞),u= 0 on Γ ×(0,+∞).\nIt is well known that when the damping term is linear, i.e. whe nρ(.,u′) =a(·)u′, where\na/greaterorequalslant0 a.e. on Ω, the energy of the solution decays exponentially u nder some some geometric\nconditions on the support of a(see [15, 24]). When the damping term ρ(·,·) is nonlinear\nwith respect to the second variable, [5] and [8], [11] give, u nder some suitable geometric\nconditions, a one-step explicit energy decay formula in ter ms of the behavior of the nonlinear\nfeedback close to the origin. These results rely on a general weighted nonlinear integral\ninequality together withanoptimal-weight convexity meth oddevelopedin[5]. Ifnogeometric\nassumptions on the damping region are made, the decay is know n to be of logarithmic type\nfor a linear damping (see e.g. [16, 18]).\nLet us go back to the above wave-wave coupled system (1.1). At this stage, four main\nfeatures characterize this system:\n•only the first equation is damped\n•the damping ρmay have an arbitrary growth around 0 with respect to the seco nd\nvariable\n•the coupling coefficient αmay vanish in some parts of Ω\n•the coupling is acting through the velocities.\nLet us now comment on these features.\n•The fact that only one equation of the coupled system is dampe d refers to the so-\ncalled class of ”indirect” stabilization problems initiat ed and studied in [1, 2, 3] and further\nstudied in [6, 12, 13]. Indeed, when dealing with coupled sys tems, it may be impossible\nor too expensive to damp each equation. Such an example is pro vided for instance by the\nTimoshenko system [7, 20, 30]. More generally, coupled syst em involving some undamped\nequations, are said to be indirectly damped. From the point o f view of applications in control\n2theory, a challenging question is to determine whether the s ingle feedback is sufficient to\nguarantee that the energy of the full system decays to 0 at infi nity and to determine at which\nrate. In this latter case, the lack of feedback on the second e quation is compensated by the\ncoupling effects.\n•The case of general damping feedbacks, that is with arbitrar y growth close to 0, has\nreceived a lot of attention since more than a decade. The first result in this direction has\nbeen derived in [22], however no general simple explicit for mula was provided except for\nlinearly or polynomially growing dampings close to 0. Such fi rst examples of explicit general\nformula are given in [27] (see also [26]), but this formula do es not allow to recover in a single\nstep the expected quasi-optimal energy decay rate in the pol ynomial case (or for polynomial-\nlogarithmicgrowth). Asfarasweknow, thefirstresultgivin gageneralone-stepquasi-optimal\nsemi-explicit formula is given in [5]. A further analysis ba sed on a suitable and original\nclassification of the feedback growth has been introduced in [8]. This classification gives\na very simple one-step explicit energy decay formula for gen eral feedbacks growth, provided\nthat this growth is not close to a linear behavior. A more comp lex semi-explicit formula holds\nin the general case including feedbacks with a growth close t o a linear one around 0. By a\none-step formula, we mean here, a formula which gives a decay rate depe nding explicitly on\nthefeedback inasimpleexplicit way. Inparticular, thisfo rmuladoesnot requirefurthersteps\nas in most of the existing literature to lead to explicit expr essions. Moreover the optimality of\nthe formula is proved for the corresponding finite dimension al systems or for semi-discretized\nscalar wave or plate equations, whereas optimality results are proved for some examples in\nthe infinite dimensional case in [5], using results of [31, 32 ]. For previous results and also\nresults for semilinear damped equations, one can see [17, 21 , 29, 33, 34].\n•When the coupling coefficient is bounded below by a strictly po sitive constant, the\ncoupling is active in the whole domain Ω, so that the equation s are coupled in the whole\ndomain. If αis nonnegative on Ω but is allowed to vanish on a subset of Ω, th e equations\nare ”uncoupled” in Ω \\supp{α}, so that in this region the second equation is decoupled from\nthe first equation and is undamped. Such cases are harder to ha ndle. A first study in this\nframework, but for linearly damped wave-type equations cou pled indisplacements is given\nin [13].\n•When indirect damping occurs through displacements, that i s for systems coupled in\ndisplacements, as for instance\n\n\nu′′−∆u+α(x)v+a(x)u′= 0 in (0 ,∞)×Ω,\nv′′−∆v+α(x)u= 0 in (0 ,∞)×Ω,\nu=v= 0 on (0 ,∞)×Γ,\n(u(0),u′(0)) = (u0,u1),(v(0),v′(0)) = (v0,v1),in Ω,\nit has been shown in [2] that even for constant coefficients α, the energy of this linearly\ndamped system never decays exponentially, but decays only p olynomially with a decay rate\n3depending on the smoothness of the initial data and a general lemma announced in [1] (see\n[2, 3] for a proof). These results are based on the method of hi gher order energies initiated\nin [1] and developped in [2] and [3] for the indirect boundary damping cases in an abstract\nsetting and applied to various examples. The result of [2] ha s been generalized to coefficients\nαthat vanishes on a subset of Ω in [13], under certain assumpti ons on the supports of α\nanda(roughly speaking they are both supposed to satisfy the Geom etric Control Condition\n(GCC) of [15]). Further results on coupled models with distr ibuted dampings but satisfying\nhybrid boundary conditions have been obtained in [12]. Shar per results have been obtained\nthrough interpolation techniques extending the first resul ts of [1]. Let us further mention\nthat in [19], the author shows that the energy decays logarit hmically under the assumption\nsupp{α} ∩supp{a} /\\e}atio\\slash=∅. Hence when the coupling acts through displacements, indir ect\nstabilization occurs but in a weaker form than the one of the c orresponding scalar case, since\nexponential stabilization does not hold even for a linearly damped case.\nThe goal of this paper is to generalize the quasi-optimal ene rgy decay formula given for\nscalarwave-type systems in [8] (see also [5]), to the case of coupled systems in velocities,\nunder the above four features. More precisely, we prove, und er some geometric conditions\non the localized damping domain and the localized coupling d omain, that the energy of\nthese kinds of system decays as fast as that of the correspond ing scalar nonlinearly damped\nequation. Hence, the coupling through velocities allows a f ull transmission of the damping\neffects, quite different from the coupling through the displace ments.\nThe optimality of the above estimates has been proved for fini te dimensional equations,\nincluding the semi-discretized wave equations in [8]. In th e infinite dimensional setting, lower\nenergy estimates or optimality are open questions. Optimal ity has been only proved in the\nparticular case of one-dimensional wave equation with boun dary damping (see [5, 31, 32]).\nLower energy estimates have been established in [8, 9, 10] fo r scalar one-dimensional wave\nequations, scalar Petrowsky equations in two-dimensions a nd Timoshenko systems. We use\nthe comparison method developed in [10] to extend these resu lts to one-dimensional wave\nsystems coupled by velocities.\nRemark 1.1. The method presented here is general and can easily be adapte d to handle\ncorresponding coupled systems of plate equations, elastic ity models, and more complex ex-\namples in the spirit of the general approach given in [5]. Our aim through this paper is to\ngive a general methodology on a concrete PDE example to show t hat if the damping effects\nare suitably transferred through the coupling operators, t hen indirect stabilization can pro-\nduce damping mechanisms of the quality of a direct damping fo r the corresponding scalar\nequation.\nNote that one can also present, with no additional mathemati cal originality and no gain\nwith respect to applications, all our results by means of a mo re ”Lyapunov” presentation. In\nthis case, it is sufficient not to integrate over the time inter val and to handle terms of the\n4formd\ndt/bracketleftbig/integraltext\nΩu′2dx/bracketrightbig\nfor instance, and multiply afterwards by a weight function, which can be\na weaker (and less good weight function) than in the original method introduced for the first\ntime in [5] and announced in [4].\nThe rest of this paper is organized as follows: In Section 2, w e give some basic preliminar-\nies, assumptions and notations. The main results, includin g Theorems 3.1 on well-posedness\nof (1.1) , Theorem 3.2 on energy decay for polynomially growi ng damping case, Theorem\n3.3 on energy decay for general nonlinear damping case, Theo rem 3.4 on lower energy es-\ntimates for one dimensional system (3.7), are presented in S ection 3. Explicit decay rates\ncorresponding to some typical dampings are also provided in Subsection 3.4. As the main\ntool of deriving the quasi-optimal one-step explicit energ y decay formula, the optimal-weight\nconvexity method together with general weighted nonlinear integral inequalities are intro-\nduced in Section 4. The proof of the main results, as well as th e decay rates of Example\n3.1-3.4 are given in Section 5. Finally, Section 6 is used to p rove Lemma 5.1 on weighted\nenergy estimates for a single non homogeneous wave equation by the multiplier method.\n2 Preliminaries, assumptions and notations\nNotations\nFor the simplicity of statement, we denote in the whole paper L2(Ω) byL2,H1\n0(Ω)\nbyH1\n0,H2(Ω) byH2. Moreover, we say that the initial data are in the energy spac e\nwhenever (u0,u1)∈H1\n0×L2and (v0,v1)∈H1\n0×L2and the initial data are smooth if\n(u0,u1)∈(H2∩H1\n0)×H1\n0and (v0,v1)∈(H2∩H1\n0)×H1\n0.\nGeometric conditions\nAs already mentioned in the introduction, stabilization re sults for wave-like systems\nrequire geometric conditions on the region where the feedba ck is active. In the sequel,\nwe shall consider the so-called Piecewise Multipliers Geom etric Condition (denoted by\nPMGC, in short):\nDefinition 2.1 (PMGC). We say that a subset ω⊂Ω satisfies the PMGC, if there\nexist subsets Ω j⊂Ω having Lipschitz boundaries and points xj∈RN, j= 1,···,J,\nsuch that Ω i∩Ωj=∅fori/\\e}atio\\slash=jandωcontains a neighborhood in Ω of the set\n∪J\nj=1γj(xj)∪(Ω\\∪J\nj=1Ωj), whereγj(xj) ={x∈∂Ωj: (x−xj)·νj(x)/greaterorequalslant0}andνjis\nthe outward unit normal to ∂Ωj.\nAssumptions on the coupling coefficient\n5We assume that the coupling function α∈C(¯Ω) satisfies\n\n\n∃α+>0,α−>0 such that\nα+/greaterorequalslantα(x)/greaterorequalslant0,∀x∈Ω,\nα(x)/greaterorequalslantα−>0,∀x∈ωc⊂Ω,(HC)\nwhere andωcis an open subset of Ω with positive measure.\nAssumptions on the feedback\nWeconsiderfeedbacks ρwithanarbitrarygrowthcloseto0. However, togivetheread er\na better insight of the scope and challenge of one-step expli cit general quasi-optimal\nenergy decay formulas, we first provide the result and proof f or polynomially growing\nfeedbacks, for which the proofs are easier and then the gener al result for arbitrary\ngrowing feedbacks. Hence, we detail below the two sets of ass umptions: the one for the\npolynomial case, then those for the general case.\nThe assumptions in the case of polynomially growing feedbacks is as follows\n\n\nρ∈C(Ω×R),ρ(x,0) = 0∀x∈Ω,\ns/ma√sto→ρ(x,s) is nondecreasing ∀x∈Ω,\n∃c>0 andp/greaterorequalslant1,∃a∈C(Ω) such that\na(x)|s|/lessorequalslant|ρ(x,s)|/lessorequalslantca(x)|s|,∀x∈Ω,|s|/greaterorequalslant1,\na(x)|s|p/lessorequalslant|ρ(x,s)|/lessorequalslantca(x)|s|1\np,∀x∈Ω,|s|/lessorequalslant1,where\na/greaterorequalslant0 on Ω,∃a−>0 such that a(x)/greaterorequalslanta−,∀x∈ωd⊂Ω,(HFp)\nwhereωdis an open subset of Ω with positive measure.\nThe assumptions in the case of arbitrary growing feedbacks is as follows\n\n\nρ∈C(Ω×R),ρ(x,0) = 0∀x∈Ω,\ns/ma√sto→ρ(x,s) is nondecreasing ∀x∈Ω,\n∃c>0,∃a∈C(Ω) and∃g∈ C1(R) such that\na(x)|s|/lessorequalslant|ρ(x,s)|/lessorequalslantca(x)|s|,∀x∈Ω,|s|/greaterorequalslant1,\na(x)g(|s|)/lessorequalslant|ρ(x,s)|/lessorequalslantca(x)g−1(|s|),∀x∈Ω,|s|/lessorequalslant1,where\na/greaterorequalslant0 on Ω,∃a−>0 such that a(x)/greaterorequalslanta−,∀x∈ωd⊂Ω,\ngis a strictly increasing and odd function .(HFg)\nRemark 2.1. Thanks to the hypotheses (HF p) or (HF g), we have\nρ(x,s)s/greaterorequalslant0,∀x∈Ω,∀s∈R, (2.1)\nwhichensuresthattheenergyofthesolutionsoftheabovewa vesystemisnonincreasing.\nRemark 2.2. Note that we can infer from ( HFg) that for very ε∈(0,1), there exists\nconstantsc1>0,c2>0 such that\n/braceleftBigg\nc1a(x)|s|/lessorequalslant|ρ(x,s)|/lessorequalslantc2a(x)|s|,∀x∈Ω,|s|/greaterorequalslantε,\nc1a(x)g(|s|)/lessorequalslant|ρ(x,s)|/lessorequalslantc2a(x)g−1(|s|),∀x∈Ω,|s|/lessorequalslantε.(2.2)\n6Convexity assumptions on the feedback and some definitions\nAssumethat (HF g) holds. Then, following [5, 8], we assumethat thefunction Hdefined\nby\nH(x) =√xg(√x), (2.3)\nis strictly convex in a right neighborhood of 0, i.e. on [0 ,r2\n0] for some sufficiently small\nr0∈(0,1]. We define the function /hatwideHonRby/hatwideH(x) =H(x) for every x∈[0,r2\n0] and\nby/hatwideH(x) = +∞otherwise, and we define the function Lon [0,+∞) by\nL(y) =\n\n/hatwideH⋆(y)\nyify>0,\n0 ify= 0,(2.4)\nwhere/hatwideH⋆is the convex conjugate function of /hatwideH, defined by\n/hatwideH⋆(y) = sup\nx∈R{xy−/hatwideH(x)}.\nBy construction, the function L: [0,+∞)→[0,r2\n0) is continuous, one-to-one, onto and\nincreasing, moreover it is easy to check that\n0<L(H′(r2\n0))<r2\n0 (2.5)\nholds (see [5, 8] for a complete proof). We also define the func tion ΛHon (0,r2\n0] by\nΛH(x) =H(x)\nxH′(x). (2.6)\nRemark 2.3. ThefunctionΛ Hhasbeenintroducedforthefirsttimebythefirstauthor\nin [8]. It is an essential tool to classify the feedback growt hs around 0 and to simplify\nthe decay estimate formula given in [5] –without loosing opt imality properties– for the\nfeedbacks having a growth around 0 which is not close to a line ar one (as explained\nbelow).\nRemark 2.4. Note that due to our convexity assumptions, we have\nΛH([0,1])⊂[0,1].\nWheng(x) =x(linear feedback case), Λ H(.)≡1. If we now set for instance, g(x) =\nx(ln(1/x))−qin (0,ε] withq >0 andε >0, then limsup\nx→0+ΛH(x) = 1. Many other\nexamples of feedbacks such that limsup\nx→0+ΛH(x) = 1 can be given, they characterize a\ngrowth which is close to a linear one. This leads to the follow ing definition:\nDefinition 2.2. We say that a feedback ρsatisfying (HF g) has a growth close to a\nlinear one in a neighborhood of 0, if it is such that the functi onHdefined by (2.3)\nsatisfies limsup\nx→0+ΛH(x) = 1. Otherwise, one says that the feedback ρis away from a\nlinear growth.\n7On the opposite side, for functions gwhich converge very fast to 0 as xgoes to 0,\nsuch as for instance g(x) =e−1/xforx∈(0,ε] (and many other examples), one has\nlimsup\nx→0+ΛH(x) = 0.\nFor polynomially growing feedbacks, e.g. when g(x) =xpwithp >1, we have\nΛH(.)≡2\np+1. For feedbacks such as g(x) =xp(ln1/x)qwithp >1,q >0, we still\nhave limsup\nx→0+ΛH(x) =2\np+1.\nWe will seelater on, in Theorem 3.3, that thecase of feedback s close toa linear behavior\nasxgoes to 0 has to be distinguished from the other cases.\nFinally, we define, for x≥1/H′(r2\n0),\nψ0(x) =1\nH′(r2\n0)+/integraldisplayH′(r2\n0)\n1/x1\nθ2(1−ΛH((H′)−1(θ)))dθ. (2.7)\nRemark 2.5. Note that when g′(0)/\\e}atio\\slash= 0,ghas a linear growth close to 0. Therefore,\nthis case is similar to the linear case which is already well- known. We thus focus in the\nsequel on the cases where g′(0) = 0.\n3 Main results\n3.1 Well-posedness\nWe setH= (H1\n0×L2)2and setU= (u,p,v,q). We equip Hwith the scalar product\n/a\\}b∇acketle{tU,/tildewideU/a\\}b∇acket∇i}ht=/integraldisplay\nΩ/parenleftBig\n∇u·∇/tildewideu+p/tildewidep+∇v·∇/tildewidev+q/tildewideq/parenrightBig\ndx\nTheorem 3.1. Assume (HFg)and thatα∈L∞(Ω). Then for all initial data in energy\nspace, there exists a unique solution (u,v)∈ C([0,+∞);(H1\n0)2)∩C1([0,+∞);(L2)2)of(1.1).\nMoreover for any smooth initial data, the solution satisfies (u,v)∈L∞([0,+∞);(H2∩H1\n0)2)∩\nW1,∞([0,+∞);(H1\n0)2)∩W2,∞([0,+∞);(L2)2). Moreover, in this latter case the energy of\norder one, defined by\nE1(t) =1\n2/integraldisplay\nΩ(|utt|2+|∇ut|2+|vtt|2+|∇vt|2)dx, (3.1)\nis non increasing, i.e.,\nE1(t)≤E1(0). (3.2)\n3.2 One-step quasi-optimal energy decay rate for the wave-w ave system\nFirst case: polynomiaily growing dampings close 0\nFor the sake of clarity, we first provide the results in the cas e of a polynomially growing\nfeedbacks/ma√sto→ρ(·,s).\n8Theorem 3.2. Assume that p >1,(HFp)and(HC)hold. Assume also that ωdandωc\nsatisfy the PMGC. Then there exists α∗>0such that for any α+∈(0,α∗]and any non\nvanishing initial data in the energy space, the total energy of(1.1)defined in (1.2)decays as\nE(t)/lessorequalslantCE(0)t2\n1−p,∀t∈[TE(0),+∞), (3.3)\nwhereCE(0),TE(0)>0are constants depending on E(0).\nRemark 3.1. Ifp= 1, we can follow the proof of Theorem 3.2 and obtain, by using Lemma\n4.2 instead of Lemma 4.1, the exponential stability for syst em (1.1)\nE(t)/lessorequalslantCE(0)e−κt,∀t∈[0,+∞), (3.4)\nwhereC >0,κ>0 are constants independent of the initial data.\nSecond case: arbitrarily growing dampings close to the origin\nIf we consider a more general nonlinear damping ρ, we provide below a quasi-optimal one-\nstep explicit energy decay formula following the optimal-w eight convexity method together\nwith general weighted nonlinear integral inequalities dev eloped in [5, 8].\nTheorem 3.3. Assume(HFg)and(HC)hold. Assume that the function Hdefined by (2.3)is\nstrictly convex in [0,r2\n0]for some sufficiently small r0∈(0,1]and satisfies H(0) =H′(0) = 0.\nWe define the maps LandΛHrespectively as in (2.4)and(2.6). Assume also that ωdand\nωcsatisfy the PMGC. Then there exists α∗>0such that for any α+∈(0,α∗]and any non\nvanishing initial data in the energy space, the total energy of(1.1)defined in (1.2)decays as\nE(t)≤2βE(0)L/parenleftBig1\nψ−1\n0(t\nM)/parenrightBig\n,∀t≥M\nH′(r2\n0), (3.5)\nwhereβE(0)is defined by (5.30),Mis defined by (5.34)and independent of E(0). Further-\nmore, if limsup\nx→0+ΛH(x)<1, thenEsatisfies the following simplified decay rate\nE(t)≤2βE(0)/parenleftBig\nH′/parenrightBig−1/parenleftBigκM\nt/parenrightBig\n, (3.6)\nfortsufficiently large, and where κ>0is a constant independent of E(0).\nRemark 3.2. Note that when g(x) =x1/pwihp >1, thenH(x) =x(p+1)/2so that\nlimsup\nx→0+ΛH(x) =2\np+1<1. Then the formula (3.6) gives back the energy decay rate of\nt−2/(p−1)given in Theorem 3.2.\nRemark 3.3. The smallness of αcan be reduced if we assume additionally supp{α} ⊂ωd\nin Theorem 3.3. Actually, we can choose δ= 1 in the proof of (5.10), and in that case, the\nlast term of (5.10) can be replaced by\nC1/integraldisplayT\nSφ(t)/integraldisplay\nωd|u′|2dxdt\nThen (5.3) follows easily by choosing ε1>0 sufficiently small in (5.10). The proof afterwards\nis the same.\n93.3 Lower energy estimates\nThe optimality of the above estimates are open questions. He re we use the comparison\nmethod developed in [10] to establish the lower estimate of t he energy of the one-dimensional\ncoupled wave system. Actually, we consider thefollowing wa ve-wave system in Ω = (0 ,1)⊂R\nandρ(x,s) =a(x)g(s) for allx∈Ω and alls∈R.\n\n\nutt−uxx+α(x)vt+a(x)g(ut) = 0,in Ω×(0,+∞),\nvtt−vxx−α(x)ut= 0,in Ω×(0,+∞),\nu(t,0) =u(t,1) =v(t,0) =v(t,1) = 0,fort∈(0,+∞),\n(u,ut)(0,x) = (u0(x),u1(x)),(v,vt)(0,x) = (v0(x),v1(x)),forx∈Ω.(3.7)\nWe defineHby (2.3), Λ Hby (2.6) and consider the following assumptions\n\n\n∃r0>0 such that the function H: [0,r2\n0]/ma√sto→Rdefined by (2.3)\nis strictly convex on [0 ,r2\n0] and\neither 0<liminf\nx→0+ΛH(x)≤limsup\nx→0+ΛH(x)<1\nor there exist µ>0 andz1∈(0,z0]with (5.39) such that\n0<liminf\nx→0+/parenleftBig\nH(µx)\nµx/integraltextz1\nx1\nH(y)dy/parenrightBig\nand limsup\nx→0+ΛH(x)<1.(HFl)\nTheorem 3.4. Assume that (HFg),(HFl), and(HC)hold. Assume also that ωdandωc\nsatisfy the PMGC, and that α+∈(0,α∗]whereα∗>0is as in Theorem 3.3. Then for all\nnon vanishing smooth initial data, there exist T0>0andT1>0such that the energy of (3.7)\nsatisfies the lower estimate\nE(t)≥1\nγ2sC2s/parenleftBig/parenleftbig\nH′/parenrightbig−1/parenleftBig1\nt−T0/parenrightbig/parenrightBig2\n,∀t≥T1+T0, (3.8)\nwhereγs= 4/radicalbig\nE1(0),Csis as in Lemma 5.2.\n3.4 Some examples of decay rates\nFor the sake of completeness, we give some significative exam ples (taken from [5, 8]) of\nfeedback growths together with the resulting energy decay r ate when applying our results. In\nthe sequel, CE(0)>0 stands for a constant depending on E(0), whileC′\nE1(0)>0 is a constant\ndepending on E1(0).\nExample 3.1. [The Polynomial Case] Let g(x) =xpon (0,r0] withp>1. Then the energy\nof (1.1) satisfies the estimate\nE(t)≤CE(0)t−2\np−1,\nfortsufficiently large and for all non vanishing initial data in th e energy space. Moreover,\nthe energy of (3.7) satisfies the estimate\nE(t)/greaterorequalslantC′\nE1(0)t−4\np−1,\n10fortsufficiently large and for all non vanishing smooth initial da ta.\nExample 3.2. [Exponential growth of the feedback] Let g(x) =e−1\nx2on (0,r0].Then the\nenergy of (1.1) decays as\nE(t)≤CE(0)(ln(t))−1,\nfortsufficiently large and for all non vanishing initial data in th e energy space. Moreover,\nthe energy of (3.7) satisfies the estimate\nE(t)/greaterorequalslantC′\nE1(0)(ln(t))−2,\nfortsufficiently large and for all non vanishing smooth initial da ta.\nExample 3.3. [Polynomial-logarithmic growth] Let g(x) =xp(ln(1/x))qon (0,r0] with\np>2 andq>1. Then the energy of (1.1) decays as\nE(t)≤CE(0)t−2/(p−1)(ln(t))−2q/(p−1)\nfortsufficiently large and for all non vanishing initial data in th e energy space. Moreover\nthe energy of (3.7) satisfies the estimate\nE(t)/greaterorequalslantC′\nE1(0)t−4/(p−1)(ln(t))−4q/(p−1),\nfortsufficiently large and for all non vanishing smooth initial da ta.\nExample 3.4. [Faster than Polynomials, Less than Exponential] Let g(x) =e−(ln(1/x))pon\n(0,r0] withp>2. Then the energy of (1.1) decays as\nE(t)≤CE(0)e−2(ln(t))1/p,\nfortsufficiently large and for all non vanishing initial data in th e energy space. Moreover\nthe energy of (3.7) satisfies the estimate\nE(t)/greaterorequalslantC′\nE1(0)e−4(ln(t))1/p,\nfortsufficiently large and for all non vanishing smooth initial da ta.\nRemark 3.4. For these above four examples, one can show that lim\nx→0+ΛH(x)<1. Moreover,\nit is proved in [8], that the above decay rates are optimal in t he finite dimensional case.\n4 Weighted nonlinear integral inequalities and decay rates\nDefinition 4.1. Wesaythat anonnegative function Edefinedon[0 ,+∞) satisfies aweighted\nnonlinear integral inequality if there exists a nonnegativ e functionwdefined on [0 ,η) with\n0<η≤+∞and a constant M >0 such that E([0,+∞))⊂[0,η) and\n/integraldisplay∞\ntw(E(s))E(s)ds≤ME(t)∀t/greaterorequalslant0. (4.1)\n11Definition 4.2. We say that such a weight function wis of ”polynomial” type, if there exists\nα>0 such that\nw(s) =sα,∀s/greaterorequalslant0,withη= +∞.\nIt is well-known that when Eis a nonnegative, nonincreasing absolutely continuous fun ction\nsatisfying (4.1) with a polynomial weight function, then Esatisfies an optimal decay rate at\ninfinity, proved in [21, Theorem 9.1] (see also herein for oth er references), that we recall in\nthe next subsection.\n4.1 Polynomial weights\nLemma 4.1. [21, Theorem 9.1] Assume that E: [0,+∞)/ma√sto→[0,+∞)is a non-increasing\nfunction and that there are two constants α>0andT >0such that\n/integraldisplay∞\ntEα+1(s)ds≤TE(0)αE(t),∀t∈[0,+∞).\nThen we have\nE(t)≤E(0)/parenleftbiggT+αt\nT+αT/parenrightbigg−1\nα\n,∀t∈[T,+∞).\nRemark 4.1. Note that when applied to decay estimates for dissipative sy stems, the above\nLemma has to be used with a constant Twhich blows up as E(0) = 0, so that the minimal\ntime for which the above decay estimate is valid, blows up as E(0) = 0. This result can be\nreformulated as below, to give an estimate which is valid for E(0)>0 as well as for E(0) = 0\nand for any t/greaterorequalslant0 as explained below.\nCorollary 4.1. Assume that E: [0,+∞)/ma√sto→[0,+∞)is a non-increasing function and that\nthere are two constants α>0andM >0such that\n/integraldisplay∞\ntEα+1(s)ds≤ME(t),∀t∈[0,+∞).\nThen we have\nE(t)≤E(0)min/parenleftBig/parenleftBigM(α+1)\nM+αE(0)αt/parenrightBig1/α\n,1/parenrightBig\n,∀t/greaterorequalslant0.\nIn particular for E(0)>0, we deduce that\nE(t)≤E(0)/parenleftBigM(α+1)\nM+αE(0)αt/parenrightBig1/α\n,∀t/greaterorequalslantME(0)−1/α.\nLemma 4.2. [21, Theorem 8.1] Assume that E: [0,+∞)/ma√sto→[0,+∞)is a non-increasing\nfunction and that there is a constant T >0such that\n/integraldisplay∞\ntE(s)ds≤TE(t),∀t∈[0,+∞).\nThen we have\nE(t)≤E(0)e1−t/T,∀t∈[T,+∞).\n124.2 General weights\nFor general weight functions, semi-explicit optimal decay rates have been derived for the\nfirst time in [5], and later on a simplified form of the rates in [ 8].\nLetη >0 andM >0 be fixed and wbe a strictly increasing function from [0 ,η) onto\n[0,+∞). For anyr∈(0,η), we define a function Krfrom (0,r] on [0,+∞) by:\nKr(τ) =/integraldisplayr\nτdy\nyw(y), (4.2)\nand a function ψrwhich is a strictly increasing onto function defined from [1\nw(r),+∞) on\n[1\nw(r),+∞) by:\nψr(z) =z+Kr(w−1(1\nz))≥z ,∀z≥1\nw(r). (4.3)\nWe can now formulate our weighted integral inequality:\nTheorem 4.1. [5, Theorem 2.1] We assume that Eis a nonincreasing, absolutely continuous\nfunction from [0,+∞)on[0,η), satisfying the inequality\n/integraldisplayT\nSw(E(t))E(t)dt≤ME(S),∀0≤S≤T. (4.4)\nThenEsatisfies the following estimate:\nE(t)≤w−1/parenleftBig1\nψ−1r(t\nM)/parenrightBig\n,∀t≥M\nw(r), (4.5)\nwherer>0is such that\n1\nM/integraldisplay+∞\n0E(τ)w(E(τ))dτ≤r≤η.\nIn particular, we have lim\nt→+∞E(t) = 0with the decay rate given by (4.5).\nTheorem 4.2. [8, Theorem 2.3] Let Hbe a strictly convex function on [0,r2\n0]such the\nH(0) =H′(0) = 0. We define LandΛHas above. Let Ebe a given nonincreasing, absolutely\ncontinuous function from [0,+∞)on[0,+∞)withE(0)>0,M >0andβis a given\nparameter such that\nE(0)\n2L(H′(r2\n0))≤β.\nIn addition, Esatisfies the following weighted nonlinear inequality\n/integraldisplayT\nSL−1(E(t)\n2β)E(t)dt≤ME(S),∀0≤S≤T. (4.6)\nThenEsatisfies the following estimate:\nE(t)≤2βL/parenleftBig1\nψ−1\n0(t\nM)/parenrightBig\n,∀t≥M\nH′(r2\n0). (4.7)\nFurthermore, if limsup\nx→0+ΛH(x)<1, thenEsatisfies the following simplified decay rate\nE(t)≤2β/parenleftBig\nH′/parenrightBig−1/parenleftBigκM\nt/parenrightBig\n, (4.8)\nfortsufficiently large, and where κ>0is a constant independent of E(0).\n135 Proof of main results\nIn this section, we prove the main results including Theorem s 3.1-3.4 and the decay rates\nin Examples 3.1-3.4.\n5.1 Proof of Theorem 3.1\nWe define the following unbounded nonlinear operator AinHby\nAU= (p,∆u−αq−ρ(.,p),q,∆v+αp),\nwith the domain\nD(A) ={U∈ H;AU∈ H}.\nIt is easy to check that D(A) =H2∩(H1\n0)×H1\n0)2. Moreover, since ρis nondecreasing with\nrespect the second variable, we have for all U,˜U∈D(A),\n/a\\}b∇acketle{tAU−A/tildewideU,U−/tildewideU/a\\}b∇acket∇i}ht=−/integraldisplay\nΩ(ρ(x,p)−ρ(x,/tildewidep))(p−/tildewidep)dx≤0,\nThus−Ais a monotone operator. We now claim that −Ais a maximal operator. We\nproceed as follows. We denote by Athe unbounded operator in L2defined byA=−∆ and\nD(A) =H2∩H1\n0. ThenI−Ais invertible as an operator acting from H1\n0inH−1(Ω), so that\nthe operator ( I−A)−1is a well-defined, self-adjoint and if w∈L2then (I−A)−1w∈H2∩H1\n0.\nThen for any F= (f,g,h,r)∈ H, the equation\n(I−A)U=F\nwithU= (u,p,v,q)∈D(A) is equivalent to\n\n\nu−∆u+α(I−A)−1(αu)+ρ(.,u−f) =G,\nv= (I−A)−1(H2+αu),\np=u−f,q=v−h,(5.1)\nwhere \n\nH1=g+f+αh∈L2,\nH2=r+h−αf∈L2,\nG=H1−α(I−A)−1H2∈L2.(5.2)\nWe define for θ∈R\nR(x,θ) =/integraldisplayθ\n0ρ(x,s)ds.\nLet us define the functional J:H1\n0/ma√sto→Rdefined by\nJ(u) =/integraldisplay\nΩ/parenleftBig1\n2(u2+|∇u|2+/vextendsingle/vextendsingle/vextendsingle(I−A)−1/2(αu)/vextendsingle/vextendsingle/vextendsingle2\n)+R(x,u−f)−Gu/parenrightBig\ndx.\n14Note that thanks to our hypotheses, |ρ(x,s)| ≤C(1+|s|) for all (x,s)∈Ω×R, so thatJis\nwell-defined and continuously differentiable on H1\n0. Moreover, we have\nJ′(u).ϕ=/integraldisplay\nΩ(uϕ+∇u·∇ϕ+(I−A)−1/2(αu)(I−A)−1/2(αϕ)+ρ(x,u−f)ϕ−Gϕ)dx.\nWe denote by ||·||theL2norm. Since ρis nondecreasing with respect to the second variable,\nJis a convex function and we also have\nJ(u)/greaterorequalslant/parenleftBig||u||\n2−||G||/parenrightBig\n||u||+||∇u||2\n2,\nso thatJ(u)−→+∞as||∇u|| −→+∞. HenceJis coercive. Therefore Jattains a minimum\nat some point u∈H1\n0, which satisfies the Euler equation\nJ′(u) = 0.\nThe usual elliptic theory implies that the weak solution uof the variational problem\n\n\nu∈H1\n0,\nJ′(u).ϕ= 0,∀ϕ∈H1\n0\nis inH2. Henceu∈H2∩H1\n0. By defining vas in (5.1), and p,qas in (5.1), it follows that\nU= (u,p,v,q)∈D(A) and (I−A)U=F. Hence−Ais a maximal monotone operator. We\nconclude Theorem 3.1 using the classical theory of maximal m onotone operator (see e.g. [21]\nand the references therein).\n5.2 Proof of Theorems 3.2 and 3.3\nThe proof will be divided in three steps, following those des cribed in [14] (see also [5, 8]).\nStep 1: we first prove that the energy Esatisfies a suitable dominant energy estimate.\nThis is the step in which the geometric assumptions PMGC on both the damping and\nthe coupling regions are used, together with suitable multi pliers adapted to the coupled\nstructure of the wave-wave system. The proof is valid withou t specifying the growth\nassumptions on the feedback ρ.\nStep 2: we then prove that nonnegative and nonincreasing functions Esatisfying a suitable\ndominant energy estimate, satisfies a general weighted nonl inear inequality. In the\ncase of polynomially growing feedbacks ρ, the proof is easier since the weight function\nfor integral inequalities is known. The general growing cas e relies on the optimality-\nconvexity method of the first author [5].\nStep 3: we deduce energy decay rates, applying Corollary 4.1 for pol ynomially growing\nfeedbacks, whereas applying Theorem 4.2 for general growin g feedbacks.\n15Let usstart with Step 1.We usethe dominant energy methodas developed andexplained\nbythefirstauthorin[5,14]. Thismethodconsistsinestimat ingtimeintegralsofthenonlinear\nweighted energy of the system by corresponding dominant wei ghted energies, here in the\nfrictional case, it means by respectively the nonlinear kin etic energy and the localized linear\nkinetic energy. Note that this step is valid for feedbacks wi th polynomial as well as arbitrary\ngrowth close to the origin.\nTheorem 5.1. [Weighted dominant energy method] We assume that (HC)holds where ωc\nsatisfies the PMGC and thatρ∈C(Ω×R)is nondecreasing with respect to the second\nvariable, and ρ(·,0)≡0onΩ. Letωdbe a given subset of Ωsatisfying the PMGC. Let\nφ: [0,+∞)/ma√sto→[0,+∞)be a non-increasing and absolutely continuous function. The n, there\nexist constants δi>0(i= 1,2,3)andα∗>0depending only on Ω,ωdbut independent of φ\nsuch that for any initial data in the energy space, for all α+∈(0,α∗], the total energy of the\nsystem(1.1)satisfies the following nonlinear weighted estimate\n/integraldisplayT\nSφ(t)E(t)dt≤δ1φ(S)E(S)+\nδ2/integraldisplayT\nSφ(t)/integraldisplay\nΩρ2(x,u′)dxdt+δ3/integraldisplayT\nSφ(t)/integraldisplay\nωd|u′|2dxdt,(5.3)\nBefore proving Theorem 5.1, we give a Lemma on a weighted ener gy estimate for a non-\nhomogeneous wave equation. The proof of Lemma 5.1 will be giv en in Section 6.\nLemma 5.1. Letωbe a nonempty open subset of Ωsatisfying the PMGC. Let φ: [0,+∞)/ma√sto→\n[0,+∞)be a non-increasing and absolutely continuous function. The n, there exist con-\nstantsηi>0 (i= 1,2,3,4)independent on φsuch that for all (u0,u1)∈H1\n0×L2, all\nf∈L2((0,∞);L2(Ω))and all0≤S≤T, the solution uof\n\n\nu′′−∆u=finΩ×(0,+∞),\nu= 0onΓ×(0,+∞),\n(u,u′)(0) = (u0,u1)inΩ(5.4)\nsatisfies the estimate\n/integraldisplayT\nSφ(t)e(t)dt\n≤η1φ(S)[e(S) +e(T)]+η2/integraldisplayT\nS−φ′(t)e(t)dt\n+η3/integraldisplayT\nSφ(t)/integraldisplay\nΩ|f|2dxdt+η4/integraldisplayT\nSφ(t)/integraldisplay\nω|u′|2dxdt.(5.5)\nwheree(t) :=1\n2/integraltext\nΩ(|u′|2+|∇u|2)dx.\nProof of Theorem 5.1.\nWe first consider smooth initial data, then system (1.1) admi ts a unique solution ( u,v)∈\nC([0,+∞),(H2∩H1\n0)2)∩W1,∞([0,+∞),(H1\n0)2).\n16Let the weight function φbe a non-increasing absolutely continuous function. Let th en\ne1(t) :=1\n2/integraldisplay\nΩ(|u′|2+|∇u|2)dx.\nWe now apply Lemma 5.1 to the first equation of (1.1) with ω=ωd,f=−ρ(.,u′)−α(x)v′,\nwithEgiven by (1.2) and e(t) =e1(t). Usinge1(t)≤E(t) and the property that Eis\nnonincreasing, we obtain for all 0 ≤S≤Tand some constants ηi>0 (i= 1,2,3,4) that\n/integraldisplayT\nSφ(t)e1(t)dt\n≤η1φ(S)[e1(S)+e1(T)]+η2/integraldisplayT\nS(−φ′(t))e1(t)dt\n+2η3/integraldisplayT\nSφ(t)/integraldisplay\nΩα2(x)|v′|2dxdt+2η3/integraldisplayT\nSφ(t)/integraldisplay\nΩρ2(x,u′)dxdt\n+η4/integraldisplayT\nSφ(t)/integraldisplay\nωd|u′|2dxdt\n≤C1φ(S)E(S)+2η3/integraldisplayT\nSφ(t)/integraldisplay\nΩρ2(x,u′)dxdt\n+η4/integraldisplayT\nSφ(t)/integraldisplay\nωd|u′|2dxdt+2η3/integraldisplayT\nSφ(t)/integraldisplay\nΩα2(x)|v′|2dxdt.(5.6)\nWe now set\ne2(t) :=1\n2/integraldisplay\nΩ(|v′|2+|∇v|2)dx,\nand secondly, we apply Lemma 5.1 to the second equation of (1. 1) withω=ωc,f=α(x)u′,\nwithEgiven by (1.2), e(t) =e2(t). Again, using the inequality e2(t)≤E(t) and the property\nthatEis nonincreasing, we obtain for all 0 ≤S≤Tand some constants γi>0 (i= 1,2,3,4),\n/integraldisplayT\nSφ(t)e2(t)dt\n≤γ1φ(S)[e2(S)+e2(T)]+γ2/integraldisplayT\nS(−φ′(t))e2(t)dt\n+γ3/integraldisplayT\nSφ(t)/integraldisplay\nΩα2(x)|u′|2dxdt+γ4/integraldisplayT\nSφ(t)/integraldisplay\nωc|v′|2dt\n≤C2φ(S)E(S)+γ4/integraldisplayT\nSφ(t)/integraldisplay\nωc|v′|2dxdt+γ3/integraldisplayT\nSφ(t)/integraldisplay\nΩα2(x)|u′|2dxdt.(5.7)\nLetδ >0 be a real parameter to be chosen later on. Adding (5.6) to δ·(5.7), we obtain\nthat for all 0 ≤S≤Tand allδ>0\n/integraldisplayT\nSφ(t)/parenleftbig\ne1(t)+δe2(t)/parenrightbig\ndt\n≤C(1+δ)φ(S)E(S) +C/integraldisplayT\nSφ(t)/integraldisplay\nΩρ2(x,u′)dxdt\n+C/integraldisplayT\nSφ(t)/integraldisplay\nωd|u′|2dxdt+C/parenleftbig\nα++δ\nα−/parenrightbig/integraldisplayT\nSφ(t)/integraldisplay\nΩα|v′|2dxdt\n+Cδα+/integraldisplayT\nSφ(t)/integraldisplay\nΩα|u′|2dxdt,(5.8)\n17whereCdenotes generic positive constants which may vary from one l ine to another. Next,\nwe estimate the term/integraltextT\nSφ(t)/integraltext\nΩα(x)|v′|2dxdtthrough the coupling relation. Obviously, the\nfollowing identity holds for the solution ( u,v) of system (1.1):\n/integraldisplayT\nSφ(t)/integraldisplay\nΩ[v′(u′′−∆u+α(x)v′+ρ(x,u′))+u′(v′′−∆v−α(x)u′)]dxdt= 0.\nAfter integration by parts, we obtain by Cauchy-Schwartz in equality that for all ε1>0\n/integraldisplayT\nSφ(t)/integraldisplay\nΩα(x)|v′|2dxdt\n=/integraldisplayT\nSφ(t)/integraldisplay\nΩα(x)|u′|2dxdt−/bracketleftbigg\nφ(t)/integraldisplay\nΩ(u′v′+∇u·∇v)dx/bracketrightbiggT\nS\n+/integraldisplayT\nSφ′(t)/integraldisplay\nΩ(u′v′+∇u·∇v)dxdt\n−/integraldisplayT\nSφ(t)/integraldisplay\nΩρ(x,u′)v′dxdt\n≤/integraldisplayT\nSφ(t)/integraldisplay\nΩα|u′|2dxdt+Cφ(S)E(S)\n+ε1/integraldisplayT\nSφ(t)/integraldisplay\nΩ|v′|2dxdt+C\nε1/integraldisplayT\nSφ(t)/integraldisplay\nΩρ2(x,u′)dxdt.(5.9)\nUsing (5.9) in (5.8), we obtain for all 0 <δ≤1 and allε1>0 that\n/integraldisplayT\nSφ(t)/parenleftbig\ne1(t)+δe2(t)/parenrightbig\ndt\n≤C/parenleftbig\n1+δ+α++δ\nα−/parenrightbig\nφ(S)E(S)+C/parenleftBig\n1+/parenleftbig\nα++δ\nα−/parenrightbig1\nε1/parenrightBig/integraldisplayT\nSφ(t)/integraldisplay\nΩρ2(x,u′)dxdt\n+C/integraldisplayT\nSφ(t)/integraldisplay\nωd|u′|2dxdt+C2/parenleftbig\nα++δ\nα−/parenrightbig\nε1/integraldisplayT\nSφ(t)/integraldisplay\nΩ|v′|2dxdt\n+C1α+/parenleftbig\nα++δ\nα−/parenrightbig/integraldisplayT\nSφ(t)/integraldisplay\nΩ|u′|2dxdt,\n(5.10)\nwhereC,C1andC2are generic positive constants. Thus,\n/parenleftbig\n1−2C1α+/parenleftbig\nα++δ\nα−/parenrightbig/parenrightbig/integraldisplayT\nSφ(t)e1(t)dt+/parenleftbig\nδ−2C2ε1/parenleftbig\nα++δ\nα−/parenrightbig/parenrightbig/integraldisplayT\nSφ(t)e2(t)dt\n≤C/parenleftbig\n1+δ+α++δ\nα−/parenrightbig\nφ(S)E(S)+C/parenleftBig\n1+/parenleftbig\nα++δ\nα−/parenrightbig1\nε1/parenrightBig/integraldisplayT\nSφ(t)/integraldisplay\nΩρ2(x,u′)dxdt\n+C/integraldisplayT\nSφ(t)/integraldisplay\nωd|u′|2dxdt.\n(5.11)\nLetα+be small so that\n0<α+≤α∗:=/radicalbigg\n1\n4C1.\nWe then fix δ >0 so that\n/parenleftbig\n1−2C1α+/parenleftbig\nα++δ\nα−/parenrightbig/parenrightbig\n/greaterorequalslant1\n2\nand choose next ε1>0 so that\nδ−2C2ε1/parenleftbig\nα++δ\nα−/parenrightbig/parenrightbig\n/greaterorequalslantδ\n2.\n18With these successive choices of α+,δandε1, we deduce that\n/integraldisplayT\nSφ(t)E(t)dt≤C/parenleftbig\n1+δ+α++δ\nα−/parenrightbig\nφ(S)E(S)\n+C/parenleftBig\n1+/parenleftbig\nα++δ\nα−/parenrightbig1\nε1/parenrightBig/integraldisplayT\nSφ(t)/integraldisplay\nΩρ2(x,u′)dxdt\n+C/integraldisplayT\nSφ(t)/integraldisplay\nωd|u′|2dxdt.(5.12)\nFor initial data in the energy space, we conclude by a standar d argument using density of\n(H2∩H1\n0)×H1\n0inH1\n0×L2, together with the dissipativity of the underlying nonline ar\nsemigroup. This ends the proof of Theorem 5.1.\nRemark 5.1. The constant C1, and therefore the constant α∗depends only on Ω ,ωd.\nProof of Theorem 3.2. [Case of polynomially growing feedbacks]\nStep 2.Assume that ( HFp) holds.\nWe first consider smooth initial data, then system (1.1) admi ts a unique solution ( u,v)∈\nC([0,+∞),(H2∩H1\n0)2)∩W1,∞([0,+∞),(H1\n0)2). Moreover, the total energy Edefined by\n(1.2) is absolutely continuous and is non-increasing due to the monotonicity of ρ, that is\nE′(t) =−/integraldisplay\nΩρ(x,u′)u′dx≤0. (5.13)\nLett/greaterorequalslant0 be fixed and ω0,t\nd:={x∈ωd:|u′(t,x)|/greaterorequalslant1}andω1,t\nd:={x∈ωd:|u′(t,x)| ≤\n1}. In short, we just write ω0\nd,ω1\ndin the sequel. Then it follows from (HF p) and (5.13) that\n/integraldisplayT\nSφ(t)/integraldisplay\nω0\nd|u′|2dxdt≤ −C/integraldisplayT\nSφ(t)E′(t)dt≤Cφ(S)E(S) (5.14)\nSimilarly we obtain from (HF p), (5.13) and Young inequality that for every ε2>0,\n/integraldisplayT\nSφ(t)/integraldisplay\nω1\nd|u′|2dxdt≤C/integraldisplayT\nSφ(t)/integraldisplay\nΩ|ρ(x,u′)u′|2\np+1dxdt\n≤C/integraldisplayT\nSφ(t)/parenleftbigg/integraldisplay\nΩ|ρ(x,u′)u′|dx/parenrightbigg2\np+1\ndt\n≤C/integraldisplayT\nSφ(t)(−E′(t))2\np+1dt\n≤/integraldisplayT\nS[ε2(φ(t))p+1\np−1−C(ε2)E′(t)]dt\n≤ε2/integraldisplayT\nS(φ(t))p+1\np−1dt+C(ε2)E(S).(5.15)\nwhereC(ε2)>0 stands for a constant depending on ε2(going to + ∞asε2goes to zero).\nSumming (5.14) and (5.15) gives\n/integraldisplayT\nSφ(t)/integraldisplay\nωd|u′|2dxdt≤max/parenleftBig\nC(ε2),Cφ(S)/parenrightBig\nE(S)+ε2/integraldisplayT\nS(φ(t))p+1\np−1dt, (5.16)\n19Similarly, let Ω0:={x∈Ω :|u′|/greaterorequalslant1}and Ω1:={x∈Ω :|u′| ≤1}. Note that these\nsubsets depend as above on t. We get from (HF p), (5.13) and Young inequality that for every\nε3>0\n/integraldisplayT\nSφ(t)/integraldisplay\nΩρ2(x,u′)dxdt=/integraldisplayT\nSφ(t)/parenleftbigg/integraldisplay\nΩ0ρ2(x,u′)dx+/integraldisplay\nΩ1ρ2(x,u′)dx/parenrightbigg\ndt\n≤Cφ(S)E(S)+/integraldisplayT\nSφ(t)/integraldisplay\nΩ|u′ρ(x,u′)|2\np+1dxdt\n≤max/parenleftBig\nC(ε3),Cφ(S)/parenrightBig\nE(S)+ε3/integraldisplayT\nS(φ(t))p+1\np−1dt.(5.17)\nWe now choose the weight function φas follows\nφ(t) =Ep−1\n2(t),∀t/greaterorequalslant0. (5.18)\nCombining (5.3) , (5.16), (5.17), together with this choice forφ, and letting ε2,ε3small\nenough, we obtain that for all 0 ≤S≤T,\n/integraldisplayT\nSEp+1\n2(t)dt≤max/parenleftBig\nC1,C2Ep−1\n2(0)/parenrightBig\nE(S),\nwhereC1,C2are positive constants independent of E(0).\nWe finish the proof by Step 3, which says that nonnegative, nonincreasing functions E\nsatisfying a polynomial nonlinear integral inequality, th en satisfy a polynomial decay rate.\nApplying Corollary 4.1 with α=p−1\n2>0, andM=C(1 +Ep−1\n2(0)), we end the proof of\nTheorem 3.2 and obtain when E(0)>0\nE(t)≤CE(0)t−2/(p−1),∀t/greaterorequalslantC1E−(p−1)\n2(0)+C2, (5.19)\nwhere\nCE(0)=/parenleftBig\nmax/parenleftBig\nC1,C2Ep−1\n2(0)/parenrightBig\n(1+1\nα)/parenrightBig1/α\n.\nProof of Theorem 3.3. [Case of general growing feedbacks]\nInStep 2, We shall prove the following theorem\nTheorem 5.2. [Optimal-weight convexity method] Assume the hypotheses o f Theorem 5.1.\nAssume furthermore that (HFg)holds where gis such that the function Hdefined in (2.3)is\nstrictly convex on [0,r2\n0], andg′(0) = 0. We define Lby(2.4). Let the initial data be in the\nenergy space and be non vanishing, (u,v)be the solution of (1.1)andEbe its energy. Then\nEsatisfies the following nonlinear weighted integral inequa lity\n/integraldisplayT\nSL−1/parenleftBigE(t)\n2β/parenrightBig\nE(t)dt≤ME(S),∀0≤S≤T, (5.20)\nwhereβandMare respectively given by\nβ= max/parenleftBig\nC2,E(0)\n2L(H′(r2\n0))/parenrightBig\n,\n20and\nM= 2C1(1+H′(r2\n0)),\nwithC1>0,C2>0depending on δi(i= 1,2,3),Ω,ωdbut independent of E(0).\nRemark 5.2. This method is called the optimal-weight convexity method a ccording to the\nproperty that the weight function φis chosen in an optimal way by setting\nφ(.) =L−1/parenleftBigE(.)\n2β/parenrightBig\nthanks to suitable convexity arguments relying both on Jens en and Young’s inequalities for\nan appropriate convex function.\nProof.We consider as before smooth initial data, then the solution (u,v) of (1.1) is in\nC([0,+∞),(H2∩H1\n0)2)∩W1,∞([0,+∞),(H1\n0)2). Moreover, the total energy Esatisfies the\ndissipation relation (5.13). Thanks to Theorem 5.1, we know thatEsatisfies the weighted\ndominant energy estimate (5.3). We shall now use the optimal -weight convexity method of\nthe first author [5] to build an optimal weight function φto prove that the two terms\n/integraldisplayT\nSφ(t)/integraldisplay\nΩρ2(x,u′)dxdtand/integraldisplayT\nSφ(t)/integraldisplay\nωd|u′|2dxdt\nin (5.3), are bounded above by the term\nCE(S)(1+φ(S))+C/integraldisplayT\nSE(t)φ(t)dt∀0≤S≤T.\nWe proceed as in [5]. Choose a parameter ε0sufficiently small, e.g. ε0=min(1,g(r0)).\nFor fixedt/greaterorequalslant0, we define the subset Ωt\n1={x∈Ω,|u′(t,x)| ≤ε0}. Now thanks to (HF g),\nwe know that (2.2) holds. Hence, since gis increasing, we have\ng/parenleftbig|ρ(x,u′(t,x))|\nK/parenrightbig\n≤ |u′(t,x)|,for a.ex∈Ωt\n1, (5.21)\nwhereK=c2||a||∞, with||·||∞standing for the L∞norm. Now, we can note that parameter\nε0has been chosen to guarantee the following two properties\n1\n|Ωt\n1|/integraldisplay\nΩt\n1|ρ(x,u′(t,x))|2\nK2dx∈[0,r2\n0], (5.22)\nand\n1\n|Ωt\n1|K/integraldisplay\nΩt\n1ρ(x,u′(t,x))u′(t,x)dx∈[0,H(r2\n0)], (5.23)\nhold. Since Hhas been assumed to be convex on [0 ,r2\n0] and thanks to (5.22), the Jensen’s\ninequality, and (5.21), we obtain\nH/parenleftBig1\n|Ωt\n1|/integraldisplay\nΩt\n1|ρ(x,u′(t,x))|2\nK2dx/parenrightBig\n≤1\n|Ωt\n1|/integraldisplay\nΩt\n1H/parenleftBig|ρ(x,u′(t,x))|2\nK2/parenrightBig\ndx=\n1\n|Ωt\n1|K/integraldisplay\nΩt\n1|ρ(u′(t,x))|g/parenleftbig|ρ(x,u′(t,x))|\nK/parenrightbig\ndx≤1\n|Ωt\n1|K/integraldisplay\nΩt\n1u′(t,x)ρ(x,u′(t,x))dx.\n21By (5.23), we deduce that\n/integraldisplayT\nSφ(t)/integraldisplay\nΩt\n1|ρ(x,u′(t,x))|2dxdt≤/integraldisplayT\nSK2|Ωt\n1|φ(t)H−1/parenleftBig1\n|Ωt\n1|K/integraldisplay\nΩt\n1u′(t,x)ρ(x,u′(t,x))dx/parenrightBig\ndt,\nand using further Young’s inequality, the dissipation rela tion (5.13), we obtain\n/integraldisplayT\nSφ(t)/integraldisplay\nΩt\n1|ρ(x,u′(t,x))|2dxdt≤K2|Ω|/integraldisplayT\nS/hatwideH∗(φ(t))dt+KE(S),∀0≤S≤T.\nOn the other hand, we prove easily as in the proof of Theorem 3. 2 that\n/integraldisplayT\nSφ(t)/integraldisplay\nΩ\\Ωt\n1|ρ(x,u′(t,x))|2dxdt≤KE(S)φ(S),∀0≤S≤T.\nAdding these two inequalities, we obtain\n/integraldisplayT\nSφ(t)/integraldisplay\nΩ|ρ(x,u′(t,x))|2dxdt≤K2|Ω|/integraldisplayT\nS/hatwideH∗(φ(t))dt+KE(S)(1+φ(S)),∀0≤S≤T.\n(5.24)\nWe now turnto the estimate of thelocalized weighted linear k inetic energy. Thanksto (HF g),\nwe know that (2.2) holds. Choose a parameter ε1sufficiently small, e.g. ε1= min{r0,g(r1)}\nwherer1is defined by\nr2\n1=H−1/parenleftbiggk\nKH(r2\n0)/parenrightbigg\n,\nwherek=c1a−. For fixed t/greaterorequalslant0, we define the subset ωt\nd={x∈ωd,|u′(t,x)| ≤ε1}. Thanks\nto (2.2) and since (HF g) holds, we have\ng(|u′(t,x)|)≤|ρ(x,u′(t,x))|\nk,for a.ex∈ωt\nd. (5.25)\nNow, wecan notethat parameter ε1hasbeenchosentoguarantee thefollowing twoproperties\n1\n|ωt\nd|/integraldisplay\nωt\nd|u′(t,x)|2dx∈[0,r2\n0], (5.26)\nand\n1\n|ωt\nd|k/integraldisplay\nωt\ndρ(x,u′(t,x))u′(t,x)dx∈[0,H(r2\n0)], (5.27)\nhold. Since Hhas been assumed to be convex on [0 ,r2\n0] and thanks to (5.26), the Jensen’s\ninequality, and (5.25), we obtain\nH/parenleftBig1\n|ωt\nd|/integraldisplay\nωt\nd|u′(t,x)|2dx/parenrightBig\n≤1\n|ωt\nd|/integraldisplay\nωt\ndH(|u′(t,x)|2)dx\n=1\n|ωt\nd|/integraldisplay\nωt\nd|u′(t,x)|g(|u′(t,x)|)dx\n≤1\n|ωt\nd|k/integraldisplay\nωt\ndu′(t,x)ρ(x,u′(t,x))dx.\nThanks to (5.27), we deduce that\n/integraldisplayT\nSφ(t)/integraldisplay\nωt\nd|u′(t,x)|2dxdt≤/integraldisplayT\nS|ωt\nd|φ(t)H−1/parenleftBig1\n|ωt\nd|k/integraldisplay\nΩt\n1u′(t,x)ρ(x,u′(t,x))dx/parenrightBig\ndt,\n22and using further Young’s inequality, the dissipation rela tion (5.13), we obtain\n/integraldisplayT\nSφ(t)/integraldisplay\nωt\nd|u′(t,x)|2dxdt≤ |ωd|/integraldisplayT\nS/hatwideH∗(φ(t))dt+1\nkE(S),∀0≤S≤T.\nOn the other hand, we prove easily as in the proof of Theorem 3. 2 that\n/integraldisplayT\nSφ(t)/integraldisplay\nωd\\ωt\nd|u′(t,x)|2dxdt≤1\nkE(S)φ(S),∀0≤S≤T.\nAdding these two inequalities, we obtain\n/integraldisplayT\nSφ(t)/integraldisplay\nωd|u′(t,x)|2dxdt≤ |ωd|/integraldisplayT\nS/hatwideH∗(φ(t))dt+1\nkE(S)(1+φ(S)),∀0≤S≤T.(5.28)\nUsing (5.24) and (5.28) in the weighted dominant energy esti mate (5.3), we obtain\n/integraldisplayT\nSφ(t)E(t)dt≤C1E(S)(1+φ(S))+C2/integraldisplayT\nS/hatwideH∗(φ(t))dt,∀0≤S≤T, (5.29)\nwhere the constants C1,C2depend only on the δifori= 1,2,3 and on |Ω|and|ωd|in an\nexplicit way. In particular, they do not depend on φ.\nLet\nβ= max/parenleftBig\nC2,E(0)\n2L(H′(r2\n0))/parenrightBig\n. (5.30)\nwhereLis defined in (2.4). Since Eis a nonincreasing function, and thanks to (2.5), we have\nE(t)\n2β≤E(0)\n2β≤L(H′(r2\n0))<r2\n0.\nHence, since L−1is defined from [0 ,r2\n0) onto [0,+∞), we can define φby\nφ(t) =L−1/parenleftBigE(t)\n2β/parenrightBig\n,∀t/greaterorequalslant0. (5.31)\nBy definition of L,φis a nonnegative, non increasing and absolutely continuous function on\n[0,+∞). We first note that\nφ(S)≤H′(r2\n0),∀S/greaterorequalslant0. (5.32)\nThen, thanks to our ”optimal” choice of the weight function φand to the definition of L, we\nhave\nL(φ(t)) =E(t)\n2β=/hatwideH∗(φ(t))\nφ(t),∀t/greaterorequalslant0.\nThis implies\nC2/hatwideH∗(φ(t))≤β/hatwideH∗(φ(t)) =1\n2φ(t)E(t),∀t/greaterorequalslant0.\nCombining this estimate together with (5.32) in (5.29), we o btain\n/integraldisplayT\nSL−1/parenleftBigE(t)\n2β/parenrightBig\nE(t)dt≤ME(S),∀0≤S≤T, (5.33)\nwhere\nM= 2C1(1+H′(r2\n0)). (5.34)\n23We finish the proof of Theorem 3.3 by Step 3. Thanks to Step 2,Esatisfies the weighted\nnonlinear integral inequality (5.33), where βis defined by (5.30), Mis defined by (5.34).\nHence applying Theorem 4.2 with this βandMwe deduce that Esatisfies the decay rate\n(4.7) in the general case, and the simplified decay rate (4.8) if limsup\nx→0+ΛH(x)<1. This\nconcludes the proof of Theorem 3.3.\n5.3 Proof of Theorem 3.4\nThanks to our hypotheses, the simplified upper energy estima te (4.8) of Theorem 4.2\nholds, so that E(t) converges to 0 as tgoes to infinity. Hence, there exists T0/greaterorequalslant0 such that\nE(t)≤/parenleftBigr2\n0\nγs/parenrightBig2\n,∀t≥T0, (5.35)\nwhereγs= 4/radicalbig\nE1(0). Hence\nγs/radicalbig\nE(t)∈[0,r2\n0] for allt/greaterorequalslantT0. (5.36)\nOn the other hand, thanks to the regularity of u(see [10] for details), we have\n||u′(t,.)||L∞(Ω)≤γs/radicalbig\nE(t)\nUsing this inequality in the dissipation relation\n−E′(t) =/integraldisplay\nΩa(x)u′g(u′)dx, t/greaterorequalslant0,\ntogether with (5.36), we deduce that for all t/greaterorequalslantT0\n−E′(t)≤ ||a||L∞(Ω)2\nγs/radicalbig\nE(t)H(γs/radicalbig\nE(t)).\nTherefore, we have\nE(t)≥/parenleftBig1\nγsK−1(||a||L∞(Ω)(t−T0))/parenrightBig2\n,∀t/greaterorequalslantT0, (5.37)\nwhereK−1denotes the inverse function of Kdefined by\nK(τ) =/integraldisplayz0\nτdy\nH(y),∀τ∈(0,/radicalbig\nE(T0)), (5.38)\nwhere\nz0=γs/radicalbig\nE(T0). (5.39)\nWe denote by zthe solution of the following ordinary differential equation\nz′(t)+||a||L∞(Ω)H(z(t)) = 0,z(0) =z0. (5.40)\nThen we have the relation\nz(t−T0) =K−1(||a||L∞(Ω)(t−T0)),∀t/greaterorequalslantT0. (5.41)\nWe now use the following comparison Lemma, that we recall for the sake of completeness.\n24Lemma 5.2. [8, Lemma 2.4] Let Hbe a given strictly convex C1function from [0,r2\n0]toR\nsuch thatH(0) =H′(0) = 0, wherer0>0is sufficiently small and define ΛHon(0,r2\n0]by\n(2.6).\nLetzbe the solution of the ordinary differential equation:\nz′(t)+κH(z(t)) = 0, z(0) =z0t≥0, (5.42)\nwherez0>0andκ >0are given. Then z(t)is defined for every t≥0and decays to 0at\ninfinity. Moreover assume that (HFl)holds. Then there exists T1>0such that for all R>0\nthere exists a constant C >0such that\nz(t)≥C(H′)−1/parenleftBigR\nt/parenrightBig\n,∀t≥T1, (5.43)\nwhereT1is a positive constant.\nWe apply this Lemma to the solution zof (5.40) with R= 1 andκ=||a||L∞(Ω). Thus,\nthere exist two constants T1>0 andCs>0 such that\nz(t)≥Cs(H′)−1/parenleftBig1\nt/parenrightBig\n,∀t≥T1, (5.44)\nCombining (5.44) together with (5.37) and (5.41), we obtain the lower estimate (3.8).\nRemark 5.3. The constant Cof the above Lemma 5.2 depends explicitly on κ,R(and in\naddition of µif the second alternative of ( HFl) holds). This dependence is given in the proof\nof Lemma 2.4 in [8]. Moreover, one may assume that r0=∞in Lemma 5.2. In this case the\ninterval [0,r2\n0] becomes [0 ,+∞).\n5.4 Proof of the decay rates given in Examples 3.1-3.4\nExample 3.1 We haveH(x) =x(p+1)/2forx∈[0,r2\n0]. ThusH′(x) =p+1\n2x(p−1)/2andH\nis strictly convex on a right neighborhood of 0. Moreover, Λ H(x) =2\np+1<1 for allx∈[0,r2\n0].\nWe easily conclude applying (4.8) of Theorem 4.2 and Theorem 3.4 for the lower estimate in\nthe one-dimensional case.\nExample 3.2 We haveH(x) =√xe−1\nxforx∈[0,r2\n0]. ThusH′(x) =e−1/x√x(1\n2+1\nx), and\nHis strictly convex on a right neighborhood of 0. Moreover, we have Λ H(x) =1\n(1\n2+1\nx)for all\nx>0 sufficiently close to 0, so that lim\nx→0+ΛH(x) = 0. We apply (4.8) of Theorem 4.2. So we\nsetx(t) = (H′)−1/parenleftBig\nκM\nt/parenrightBig\n. Then one can prove that x(t) is equivalent to1\nln(t)−ln(κM)astgoes\nto +∞. We therefore obtain the desired upper bound, using this equ ivalence. One can show\nthat the second alternative of ( HFl) holds for any µ>1 (see subsection 7.10 in [8]). Thus,\nwe obtain in the same way by Theorem 3.4 the lower estimates in the one-dimensional case.\nExample 3.3 We haveH(x) =x(p+1)/2(ln(1/√x))qforx∈[0,r2\n0]. ThusH′(x) =\n1\n2x(p−1)/2(ln(1/√x))q/parenleftBig\np+1−q(ln(1/√x))−1/parenrightBig\nandHisstrictlyconvex onarightneighborhood\nof 0. Moreover, Λ H(x) =2\np+1−q(ln(1/√x))−1for allx >0 sufficiently close to 0, so that\n25lim\nx→0+ΛH(x) =2\np+1. We apply (4.8) of Theorem 4.2. So we set x(t) = (H′)−1/parenleftBig\nκM\nt/parenrightBig\nand\ny(t) =/parenleftBig\n2κM\nt/parenrightBig2/(p−1)\n. Then one can prove that/parenleftBig\nx(t)\ny(t)/parenrightBig(p−1)/2\n(ln(1/√x))qis equivalent to1\np+1\nastgoes to + ∞. On the other hand, computing ln( x(t)) and ln(y(t)), we find that ln( x(t))\nis equivalent to ln( y(t)) astgoes to + ∞. Using this relation in the previous one, we find\nthatx(t) is equivalent to Dt−2/(p−1)/parenleftBig\nln(t)/parenrightBig−2q/(p−1)\n, whereDis an explicit positive constant\nwhich depends on κ,M,pandq. We therefore obtain the desired upper estimate, using this\nequivalence. We obtain by Theorem 3.4 the lower estimates in the one-dimensional case.\nExample 3.4 We haveH(x) =√xe−(ln(1/√x))p\nforx∈[0,r2\n0]. Thus, we have H′(x) =\n1\n2√xe−(ln(1/√x) )p/parenleftBig\n1 +p/parenleftBig\nln/parenleftBig\n1√x/parenrightBig/parenrightBigp−1/parenrightBig\n, andHis strictly convex on a right neighborhood\nof 0. Moreover, we have Λ H(x) =2\n1+p/parenleftbig\nln(1/√x)/parenrightbigp−1for allx >0 sufficiently close to 0, so\nthat lim\nx→0+ΛH(x) = 0. We apply (4.8) of Theorem 4.2. So we set x(t) = (H′)−1/parenleftBig\nκM\nt/parenrightBig\nand\ny(t) =e−2(ln(t\nκM))1/p. Then one can prove that ln( x(t)) is equivalent to ln( y(t)) astgoes to\n+∞. We further set z(t) = ln(1//radicalbig\nx(t)) so thatz(t) goes to + ∞astgoes to + ∞, then\nwe havezp(t)(1−θ(t)) = ln(t\n2κM), whereθ(t) =z1−p(t) + ln(1 +pzp−1(t))z−p(t), so that\nθ(t) goes to 0 as tgoes to + ∞. Hence we have x(t) =e−2(ln(t\nκM))1/p 1\n(1−θ(t))1/p. We can check\nthat ln(t\n2κM)1/p(1−(1−θ(t))1/p) goes to 0 as tgoes to + ∞. Hence,x(t) is equivalent to\ne−2(ln(t))1/pastgoes to + ∞. We therefore obtain the desired upper estimate. One can sho w\nthat the second alternative of ( HFl) holds for any µ>1 (see subsection 7.10 in [8]). Thus,\nwe obtain by Theorem 3.4 the lower estimates in the one-dimen sional case.\n6 Proof of Lemma 5.1\nInthissection, weproveLemma5.1 bythepiecewise multipli ermethodwhichrelies onthe\ngeometric assumptions PMGC on the subset ω⊂Ω. Denoting by Ω jandxj(j= 1,···,J)\nthe sets and the points given by PMGC, we have ω⊃Nε(∪J\nj=1γj(xj)∪(Ω\\ ∪J\nj=1Ωj))∩Ω.\nHere,Nε(U) ={x∈Rn,d(x,U)≤ε}withd(·,U) the usual euclidian distance to the subset\nUofRn, andγj(xj) ={x∈Γj,(x−xj)·νj>0},whereνjdenotes the outward unit normal\nof the boundary Γ j=∂Ωj.\nLet 0< ε0< ε1< ε2< εand define Qi:=Nεi[∪J\nj=1γj(xj)∪(Ω\\ ∪J\nj=1Ωj)](i= 0,1,2).\nSince (Ωj\\Q1)∩Q0=∅, we introduce a cut-off function ψj∈C∞\n0(RN) satisfying\n0≤ψj≤1, ψ j= 1 on Ωj\\Q1;ψj= 0 onQ0. (6.1)\nFormj(x) =x−xj, we define the C1vector field on Ω:\nh(x) =/braceleftBigg\nψj(x)mj(x) ifx∈Ωj,j= 1,···,J\n0 ifx∈Ω\\∪J\nj=1Ωj(6.2)\nUsing the multiplier φ(t)h(x)·∇uto equation (5.4):\n/integraldisplayT\nSφ(t)/integraldisplay\nΩjh(x)·∇u(u′′−∆u−f)dxdt= 0\n26leads to/integraldisplayT\nSφ(t)/integraldisplay\nΓj∂u\n∂νjh·∇u+1\n2(h·νj)(|u′|2−|∇u|2)dΓdt\n=/bracketleftBigg\nφ(t)/integraldisplay\nΩju′h·∇udx/bracketrightBiggT\nS−/integraldisplayT\nSφ′(t)/integraldisplay\nΩju′h·∇udxdt\n+/integraldisplayT\nSφ(t)/integraldisplay\nΩj1\n2divh(|u′|2−|∇u|2)dxdt\n+/integraldisplayT\nSφ(t)/integraldisplay\nΩj\n/summationdisplay\ni,k∂u\n∂xi∂u\n∂xk∂hk\n∂xi−h·∇uf\ndxdt.(6.3)\nThanks to the choice of ψj, the terms in the left hand side of (6.3) vanish except on the\nboundary (Γ j\\γj(xj))∩Γ. Sinceu= 0 on this part of boundary, then u′= 0,∇u=∂u\n∂νjνj.\nHence, the left side of (6.3) becomes\n1\n2/integraldisplayT\nSφ(t)/integraldisplay\n(Γj\\γj(xj))∩Γ/vextendsingle/vextendsingle/vextendsingle∂u\n∂νj/vextendsingle/vextendsingle/vextendsingle2\nψjmj·νjdΓdt≤0.\nTherefore (6.3) implies that\n/integraldisplayT\nSφ(t)/integraldisplay\nΩj1\n2divh(|u′|2−|∇u|2)dxdt+/integraldisplayT\nSφ(t)/integraldisplay\nΩj/summationdisplay\ni,k∂u\n∂xi∂u\n∂xk∂hk\n∂xidxdt\n≤ −/bracketleftBigg\nφ(t)/integraldisplay\nΩju′h·∇udx/bracketrightBiggT\nS+/integraldisplayT\nSφ′(t)/integraldisplay\nΩju′h·∇udxdt\n+/integraldisplayT\nSφ(t)/integraldisplay\nΩjh·∇ufdxdt.\nSince, moreover h(x) =mj(x) on¯Ωj\\Q1, we obtain that\n/integraldisplayT\nSφ(t)/integraldisplay\nΩj\\Q1N\n2(|u′|2−|∇u|2)+|∇u|2dxdt\n≤ −/bracketleftBigg\nφ(t)/integraldisplay\nΩju′h·∇udx/bracketrightBiggT\nS+/integraldisplayT\nSφ′(t)/integraldisplay\nΩju′h·∇udxdt\n+/integraldisplayT\nSφ(t)/integraldisplay\nΩjh·∇ufdxdt\n−/integraldisplayT\nSφ/integraldisplay\nΩj∩Q1\n1\n2divh(|u′|2−|∇u|2)+/summationdisplay\ni,k∂u\n∂xi∂u\n∂xk∂hk\n∂xi\ndxdt.\nSumming the above inequality on jand using the facts that Ω \\Q1=∪J\nj=1(Ωj)\\Q1and\nh(x) = 0 on Ω \\∪J\nj=1Ωj, we obtain\n/integraldisplayT\nSφ(t)/integraldisplay\nΩ\\Q1N\n2(|u′|2−|∇u|2)+|∇u|2dxdt\n≤ −/bracketleftbigg\nφ(t)/integraldisplay\nΩu′h·∇udx/bracketrightbiggT\nS+/integraldisplayT\nSφ′(t)/integraldisplay\nΩu′h·∇udxdt\n+/integraldisplayT\nSφ(t)/integraldisplay\nΩh·∇ufdxdt\n−/integraldisplayT\nSφ(t)/integraldisplay\nΩ∩Q1\n1\n2divh(|u′|2−|∇u|2)+/summationdisplay\ni,k∂u\n∂xi∂u\n∂xk∂hk\n∂xi\ndxdt,(6.4)\n27Using the second multiplierN−1\n2φ(t)ufor (5.4):\nN−1\n2/integraldisplayT\nSφ(t)/integraldisplay\nΩu(u′′−∆u−f)dxdt= 0,\nyields that\nN−1\n2/integraldisplayT\nSφ(t)/integraldisplay\nΩ(|∇u|2−|u′|2)dxdt\n=−N−1\n2/bracketleftbigg\nφ(t)/integraldisplay\nΩu′udx/bracketrightbiggT\nS\n+(N−1)\n2/integraldisplayT\nSφ′(t)/integraldisplay\nΩu′udxdt\n+N−1\n2/integraldisplayT\nSφ(t)/integraldisplay\nΩufdxdt.(6.5)\nAdding (6.5) to (6.4) and using Cauchy-Schwarz and Poincar´ e’s inequalities, we obtain for\nallδ1>0 that/integraldisplayT\nSφ(t)e(t)dt\n=/integraldisplayT\nSφ(t)/integraldisplay\nΩ|∇u|2+|u′|2\n2dxdt\n≤Cφ(S)[e(S)+e(T)]+C/integraldisplayT\nS−φ′(t)e(t)dt\n+δ1/integraldisplayT\nSφ(t)e(t)dt+C\nδ1/integraldisplayT\nSφ(t)/integraldisplay\nΩ|f|2dxdt\n+C/integraldisplayT\nSφ(t)/integraldisplay\nΩ∩Q1(|u′|2+|∇u|2)dxdt. (6.6)\nCompared to the desired estimate (5.5), the term concerning |∇u|2on the right hand of\n(6.6) is crucial. We just follow the techniques developed in [27] to deal with this term.\nSinceRN\\Q2∩Q1=∅, there exists a cut-off function ξ∈C∞\n0(R) such that\n0≤ξ≤1, ξ= 1 onQ1, ξ= 0 on RN\\Q2. (6.7)\nApplying now the multiplier φ(t)ξ(x)uto (5.4) gives, after integration by parts, that\n/integraldisplayT\nSφ(t)/integraldisplay\nΩξ|∇u|2dxdt\n=/integraldisplayT\nSφ(t)/integraldisplay\nΩ(ξ|u′|2+1\n2u2∆ξ)dxdt\n+/integraldisplayT\nSφ′(t)/integraldisplay\nΩξuu′dxdt−/bracketleftbigg\nφ(t)/integraldisplay\nΩξuu′dx/bracketrightbiggT\nS\n+/integraldisplayT\nSφ(t)/integraldisplay\nΩξufdxdt.\n28Then it follows from the definition of ξthat\n/integraldisplayT\nSφ(t)/integraldisplay\nΩ∩Q1|∇u|2dxdt≤/integraldisplayT\nSφ(t)/integraldisplay\nΩξ|∇u|2dxdt\n≤Cφ(S)[e(S)+e(T)]+C/integraldisplayT\nS−φ′(t)e(t)dt\n+C/integraldisplayT\nSφ(t)/integraldisplay\nΩ|f|2dxdt\n+C/integraldisplayT\nSφ(t)/integraldisplay\nΩ∩Q2(|u′|2+|u|2)dxdt.(6.8)\nNow it remains to estimate the term concerning |u|2in (6.8). Since RN\\ω∩Q2=∅, there\nexists a function β∈C∞\n0(R) such that\n0≤β≤1, β= 1 onQ2, β= 0 onRN\\ω. (6.9)\nFix thetvariable and consider the solution zof the following elliptic problem in space:\n/braceleftBigg\n∆z=β(x)u,in Ω,\nz= 0,on Γ.\nHence,zandz′satisfy the following estimates\n/ba∇dblz/ba∇dblL2≤C/integraldisplay\nΩβ(x)|u|2dx. (6.10)\n/ba∇dblz′/ba∇dbl2\nL2≤C/integraldisplay\nΩβ(x)|u′|2dx. (6.11)\nApplying the multiplier φ(t)zto (5.4) gives, after integration by parts, that\n/integraldisplayT\nSφ(t)/integraldisplay\nΩβ(x)|u|2dxdt\n=/bracketleftbigg\nφ(t)/integraldisplay\nΩzu′dx/bracketrightbiggT\nS−/integraldisplayT\nSφ′(t)/integraldisplay\nΩzu′dxdt\n+/integraldisplayT\nSφ(t)/integraldisplay\nΩ(−z′u′−zf)dxdt.\nHence, using the estimates (6.10)-(6.11) in the above relat ion, and noting the definition of β,\nwe obtain for all δ2>0\n/integraldisplayT\nSφ(t)/integraldisplay\nΩ∩Q2|u|2dxdt≤/integraldisplayT\nSφ(t)/integraldisplay\nΩβ(x)|u|2dxdt\n≤Cφ(S)[e(S)+e(T)]+C/integraldisplayT\nS−φ′(t)e(t)dt\n+δ2/integraldisplayT\nSφ(t)e(t)dt+C\nδ2/integraldisplayT\nSφ(t)/integraldisplay\nω|u′|2dxdt\n+C\nδ2/integraldisplayT\nSφ(t)/integraldisplay\nΩ|f|2dxdt.(6.12)\nInserting (6.8) and (6.12) in (6.6), then choosing finally δ1andδ2sufficiently small, we obtain\n(5.5) which ends the proof of Lemma 5.1.\n29Acknowledgments\nThe authors are thankful to the support of the ERC advanced gr ant 266907 (CPDENL) and\nthe hospitality of the Laboratoire Jacques-Louis Lions of Universit ´ e Pierre et Marie Curie. 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Control\nOptimization, 28: 265-268, 1989.\n32"    },    {        "title": "1608.08326v3.Optimal_damping_ratios_of_multi_axial_perfectly_matched_layers_for_elastic_wave_modeling_in_general_anisotropic_media.pdf",        "content": "Optimal damping ratios of multi-axial perfectly matched layers for\nelastic-wave modeling in general anisotropic media\nKai Gaoa, Lianjie Huanga\naGeophysics Group, Los Alamos National Laboratory, Los Alamos, NM 87545\nAbstract\nThe conventional Perfectly Matched Layer (PML) is unstable for certain kinds of anisotropic media. This in-\nstability is intrinsic and independent of PML formulation or implementation. The Multi-axial PML (MPML)\nremoves such instability using a nonzero damping coe\u000ecient in the direction parallel with the interface be-\ntween a PML and the investigated domain. The damping ratio of MPML is the ratio between the damping\ncoe\u000ecients along the directions parallel with and perpendicular to the interface between a PML and the\ninvestigated domain. No quantitative approach is available for obtaining these damping ratios for general\nanisotropic media. We develop a quantitative approach to determining optimal damping ratios to not only\nstabilize PMLs, but also minimize the arti\fcial re\rections from MPMLs. Numerical tests based on \fnite-\ndi\u000berence method show that our new method can e\u000bectively provide a set of optimal MPML damping ratios\nfor elastic-wave propagation in 2D and 3D general anisotropic media.\nKey words: Anisotropic medium, elastic-wave propagation, Multi-axial Perfectly Matched Layers\n(MPML), damping ratio.\n1. Introduction\nElastic-wave modeling usually needs to absorb outgoing wave\felds at boundaries of an investigated\ndomain. Two main categories of boundary absorbers have been developed: one is called the Absorbing\nBoundary Condition (ABC) (e.g., Clayton and Engquist, 1977; Reynolds, 1978; Liao et al., 1984; Cerjan\net al., 1985; Higdon, 1986, 1987; Long and Liow, 1990; Peng and Toks oz, 1994), and the other is termed\nthe Perfectly Matched Layer (PML) (e.g., Berenger, 1994; Hastings et al., 1996; Collino and Tsogka, 2001).\nIn some literature, PML is considered as one of ABCs. However, there are fundamental di\u000berences in the\nconstruction of PML and its variants compared with traditional ABCs, we therefore di\u000berentiate them in\nnames. For a brief summary, please refer to Hastings et al. (1996) and other relevant references.\nThe PML approach was \frst introduced by Berenger (1994) for electromagnetic-wave modeling, and\nhas been widely used in elastic-wave modeling because of its simplicity and superior absorbing capability\n(e.g., Collino and Tsogka, 2001; Komatitsch and Tromp, 2003; Drossaert and Giannopoulos, 2007). Various\nimproved PML methods for elastic-wave modeling have been developed, such as non-splitting convolutional\nPML (CPML) to enahce absorbing capability for grazing incident waves (Komatitsch and Martin, 2007;\nPreprint submitted to Geophysical Journal International May 24, 2022arXiv:1608.08326v3  [physics.geo-ph]  21 Dec 2016Martin and Komatitsch, 2009), and CPML with auxiliary di\u000berential equation (ADE-PML) for modeling\nwith a high-order time accuracy formulation (Zhang and Shen, 2010; Martin et al., 2010). However, a well-\nknown problem of PML and its variants/improvements is that numerical modeling with PML is unstable in\ncertain kinds of anisotropic media for long-time wave propagation.\nTo address the instability problem of PML, B\u0013 ecache et al. (2003) analyzed the PML for 2D anisotropic\nmedia and found that, if there exists points where the ithcomponent of group velocity vhas an opposite\ndirection relative to the ithcomponent of wavenumber k, i.e.,viki<0 (no summation rules applied), then\nthexi-direction PML is unstable. The original version of this aforementioned condition was expressed with\nso-called \\slowness vector\" de\fned by B\u0013 ecache et al. (2003), but it can be recast in such form according\nto the de\fnition of the \\slowness vector\" in eq. (45) of (B\u0013 ecache et al., 2003). This PML instability is\nintrinsic and independent of PML/CPML formulations adopted for wave\feld modelings. To make PML\nstable, elasticity parameters of an anisotropic medium need to satisfy certain inequality relations (B\u0013 ecache\net al., 2003). These restrictions limit the applicability of PML for arbitrary anisotropic media.\nMeza-Fajardo and Papageorgiou (2008) presented an explanation for the instability of conventional PML.\nThey recast the elastic wave equations in PML to an autonomous system and found that the PML instability\nis caused by the fact that the PML coe\u000ecient matrix having one or more eigenvalues with positive imaginary\nparts. They showed that PML becomes stable when adding appropriate nonzero damping coe\u000ecients to PML\nin the direction parallel with the PML/non-PML interface. The ratio between the PML damping coe\u000ecients\nalong the directions parallel with and perpendicular to the PML/non-PML interface is called the damping\nratio. The resulting PML with nonzero damping ratios is termed the Multi-axial PML (MPML).\nA key step in the stability analysis of PML is to derive eigenvalue derivatives of the damped system\ncoe\u000ecient matrix. Meza-Fajardo and Papageorgiou (2008) derived expressions of the eigenvalue derivatives\nfor anisotropic media. However, these expressions are valid only for two-dimensional isotropic media and\nanisotropic media with up to hexagonal/orthotropic anisotropy, that is, C116= 0,C336= 0,C556= 0,\nC15=C35= 0, andC13can either be zero or nonzero depending on medium properties. Furthermore,\nalthough Meza-Fajardo and Papageorgiou (2008) showed that the nonzero damping ratios can stabilize PML,\nthey did not present a method to select the appropriate dampoing ratios. Adding these nonzero damping\ncoe\u000ecients makes the PML no longer \\perfectly matched\", and the larger are the ratios, the stronger the\narti\fcial re\rections become (Dmitriev and Lisitsa, 2011). This increase is linear. Therefore, it is necessary\nto \fnd a set of \\optimal\" damping ratios to not only ensure the stability of MPML, but to also eliminate\narti\fcial re\rections as much as possible.\nWe develop a new method to determine the optimal MPML damping ratios for general anisotropic media.\nWe show that, even for a two-dimensional anisotropic medium with nonzero C15andC35, the MPML stability\nanalysis is complicated, and new equations must be derived to calculate both the eigenvalues of the undamped\nsystem and the eigenvalue derivatives of the damped system. The resulting expressions are functions of all\nnonzeroCijcomponents as well as the wavenumber k. For 3D general anisotropic media, we \fnd that\n2such an analytic procedure becomes practically impossible because it requires de\fnite analytic expressions\nof eigenvalues and eigenvalue derivatives. In the 3D case, the dimension of the asymmetric system coe\u000ecient\nmatrix is up to 27 \u000227, and therefore a purely numerical approach should be employed. We present two\nalgorithms with slightly di\u000berent forms but essentially the same logic, to determine the optimal damping\nratios for 2D and 3D MPMLs. With these algorithms, it is possible to stabilize PML for any kind of\nanisotropic media without using a trial-and-error method. Our new algorithms enable us to use MPML\nfor \fnite-di\u000berence modeling of elastic-wave propagation in 2D and 3D general anisotropic media where all\nelastic parameters Cijmay be nonzero. These algorithms are also applicable to other elastic-wave modeling\nmethods such as spectral-element method (e.g., Komatitsch et al., 2000) and discontinuous Galerkin \fnite-\nelement method (e.g., de la Puente et al., 2007).\nOur paper is organized as follows. In the Methodology section, we derive the equations for the eigen-\nvalue derivatives for 2D and 3D general anisotropic media. We also present two algorithms to obtain the\noptimal MPML damping ratios. To validate our algorithms, we give six numerical examples in the Results\nsection, including three 2D anisotropic elastic-wave modeling examples and three 3D anisotropic elastic-wave\nmodeling examples, and show that our algorithms can give appropriate damping ratios for both 2D and 3D\nmodeling in general anisotropic media.\n2. Methodology\n2.1. Optimal damping ratios of 2D MPML\nIn this section, we concentrate our analysis on the x1x3-plane. This analysis is also valid for the x1x2-\nandx2x3-planes. We assume that C15andC35are generally nonzero for an anisotropic medium. The 2D\nelastic-wave equations in the stees-velocity form are given by (e.g., Carcione, 2007),\n\u001a@v\n@t=\u0003\u001b+f; (1)\n@\u001b\n@t=C\u0003Tv; (2)\nwhere \u001b= (\u001b11;\u001b33;\u001b13)Tis the stress wave\feld, v= (v1;v3)Tis the particle velocity wave\feld, fis the\nexternal force, \u001ais the mass density, Cis the elasticity tensor in Voigt notation de\fned as\nC=0\nBBB@C11C13C15\nC13C33C35\nC15C35C551\nCCCA; (3)\nand\u0003is the di\u000berential operator matrix de\fned as\n\u0003=0\n@@\n@x10@\n@x3\n0@\n@x3@\n@x11\nA: (4)\nIn the following analysis, we ignore the external force term fwithout loss of generality.\n3Using the convention in Meza-Fajardo and Papageorgiou (2008) for isotropic and VTI/HTI/othotropic\nmedia, the undamped system of eqs. (1){(2) can be also written as\n\u001a@\n@t2\n4v1\nv33\n5=2\n4@\n@x1\u001b11+@\n@x3\u001b13\n@\n@x1\u001b13+@\n@x3\u001b333\n5; (5)\n@\n@t2\n6664\u001b11\n\u001b33\n\u001b133\n7775=2\n6664C11C13C15C15\nC13C33C35C35\nC15C35C55C553\n77752\n6666664@\n@x1v1\n@\n@x3v3\n@\n@x1v3\n@\n@x3v13\n7777775: (6)\nEquivalently, the system of the above two equations can be written as\n@\n@tv=D1@\n@x1\u001b+D3@\n@x3\u001b; (7)\n@\n@t\u001b=C1@\n@x1v+C3@\n@x3v; (8)\nwhere\nv= (v1;v3)T; (9)\n\u001b= (\u001b11;\u001b33;\u001b13)T; (10)\nD1=\u001a\u000012\n41 0 0\n0 0 13\n5; (11)\nD3=\u001a\u000012\n40 0 1\n0 1 03\n5; (12)\nC1=2\n6664C11C15\nC13C35\nC15C553\n7775; (13)\nC3=2\n6664C15C13\nC35C33\nC55C553\n7775: (14)\nIn the conventional 2D PML, each \feld variable is split into two orthogonal components that are per-\npendicular to and parallel with the interface between the PML and the investigated domain, and the system\nof wave equations in PML can be written as (Meza-Fajardo and Papageorgiou, 2008)\n@\t\n@t=A\t; (15)\nwithA=A0+B, and\nA0=2\n6666664033 033@\n@x1C1@\n@x1C1\n033 033@\n@x3C3@\n@x3C3\n@\n@x1D1@\n@x1D1022 022\n@\n@x3D3@\n@x3D3022 0223\n7777775; (16)\n4B=2\n6666664\u0000d1I3033 032 032\n033\u0000d3I3032 032\n023 023\u0000d1I2022\n023 023 022\u0000d3I23\n7777775; (17)\nwhere 0mnis them\u0002nzero matrix, Imis them\u0002midentity matrix, and\n\t= (\u001b(1)\n11;\u001b(1)\n33;\u001b(1)\n13;\u001b(3)\n11;\u001b(3)\n33;\u001b(3)\n13;v(1)\n1;v(1)\n3;v(3)\n1;v(3)\n3)T(18)\nrepresents the split wave\feld variables in PML. In the damping matrix B,d1andd3represent the PML\ndamping coe\u000ecients along the x1- andx3-axis, respectively. The PML damping coe\u000ecients depend on the\nthickness of PML, the desired re\rection coe\u000ecient and the P-wave velocity at the PML/non-PML interface.\nUsually, they vary with the distance from a location inside the PML to the PML/non-PML interface according\nto the power of two or the power of three (e.g., Collino and Tsogka, 2001).\nTransforming system (15) into the wavenumber domain leads to\n@U\n@t=~AU; (19)\nwhere U=F[\t] is the Fourier transform of the split \fled variables \t,~A=~A0+B, and ~A0now is\n~A0=2\n6666664033 033ik1C1ik1C1\n033 033ik3C3ik3C3\nik1D1ik1D1022 022\nik3D3ik3D3022 0223\n7777775; (20)\nwherek1andk3are respectively the x1- andx3-components of wavenumber vector k.\nAs demonstrated by the stability theory of autonomous system in Meza-Fajardo and Papageorgiou (2008),\nfor a stable PML in an elastic medium, either isotropic or anisotropic, all the eigenvalues of the system matrix\n~Ashould have non-positive imaginary parts. In conventional PML, the outgoing wave\feld is damped only\nalong the direction perpendicular to the PML/non-PML interface, and the damping matrix Bfor PML in\nthex1andx3-directions can be respectively written as\nB1=2\n6666664\u0000d1I3033 032 032\n033 033 032 032\n023 023\u0000d1I2022\n023 023 022 0223\n7777775; (21)\nB3=2\n6666664033 033 032 032\n033\u0000d3I3032 032\n023 023 022 022\n023 023 022\u0000d3I23\n7777775; (22)\n5resulting in an unstable PML. Meza-Fajardo and Papageorgiou (2008) analyzed the derivatives of eigenvalues\nof~Awith respect to the damping parameters d1andd3, and showed that if an appropriate damping ratio\n\u00181or\u00183is added along the direction parallel with the PML/non-PML interface, i.e.,\nB1(\u00181) =2\n6666664\u0000d1I3 033 032 032\n033\u0000\u00181d1I33 032 032\n023 023\u0000d1I2 022\n023 023 022\u0000\u00181d1I223\n7777775; (23)\nB3(\u00183) =2\n6666664\u0000d3\u00183I33 033 032 032\n033\u0000d3I3 032 032\n023 023\u0000d3\u00183I22 022\n023 023 022\u0000d3I23\n7777775; (24)\nthen the PML becomes stable. The underlying principle of such stability comes from the fact that, after\nadding nonzero damping ratios \u00181and\u00183, the eigenvalue derivatives of ~Ahave negative values along all\nkdirections and consequently, all the relevant eigenvalues of ~A0have negative imaginary parts (other\neigenvalues are a pure zero), making autonomous system (15) stable.\nA key step for determing such damping ratios \u00181and\u00183is to compute the eigenvalue derivatives of ~A.\nMeza-Fajardo and Papageorgiou (2008) adopted the following procedure:\n1. Calculate eigenvalues eof undamped system coe\u000ecient matrix ~A0, of which six are a pure zero, and\nthe rest four are pure imaginary numbers as functions of elasticity coe\u000ecient Cand wavenumber k;\n2. Calculate the eigenvalue derivative of ~A=~A0+~B1(\u00181) or ~A=~A0+~B3(\u00183) with respect d1ord3at\nd1= 0 ord3= 0. These eigenvalue derivatives are functions of elasticity coe\u000ecient C, wavenumber k\nand damping ratio \u00181or\u00183;\n3. Choose an appropriate value to g ensure that the values of the eigenvalue derivatives are negative in the\nrange of (0;\u0019=2] (the direction of wavenumber k).\nBoth eigenvalues of ~A0and eigenvalue derivatives of ~Aare calculated analytically in the above procedure.\nSpecially, Step 2 involves implicit di\u000berentiation operation and solving roots for high-order polynomials, and\ncould not be accomplished numerically.\nWe adopt the above procedure for obtaining optimal damping ratios for 2D general anisotropic media\nwhereC15orC35may be nonzero. We \frst derive relevant expressions for the eigenvalues of ~A0and the\neigenvalue derivatives of ~A. Because our procedure is the same as that in Meza-Fajardo and Papageorgiou\n(2008), we only show the resulting equations. The four nonzero eigenvalues of ~A0(C;k) are\ne1(C;k) =\u0006ip\n2\u001aq\nP\u0000p\nQ; (25)\ne2(C;k) =\u0006ip\n2\u001aq\nP+p\nQ; (26)\n6P=\u001a[(C11+C55)k2\n1+ 2(C15+C35)k1k3+ (C33+C55)k2\n3]; (27)\nQ=\u001a2f[(C11+C55)k2\n1+ 2(C15+C35)k1k3+ (C33+C55)k2\n3]2\n+ 4[(C2\n15\u0000C11C55)k4\n1+ 2(C13C15\u0000C11C35)k3\n1k3\n+ (C2\n13\u0000C11C33\u00002C15C35+ 2C13C55)k2\n1k2\n3\n+ 2(\u0000C15C33+C13C35)k1k3\n3+ (C2\n35\u0000C33C55)]k4\n3g; (28)\nwhereCIJare components of the elasticity matrix and \u001ais the mass density. The two eigenvalues in e1or\ne2have the same length with di\u000berent signs, and we take only the negative ones, i.e.,\ne1(C;k) =\u0000ip\n2\u001aq\nP\u0000p\nQ; (29)\ne2(C;k) =\u0000ip\n2\u001aq\nP+p\nQ: (30)\nThe choice of the signs of e1ande2does not a\u000bect the following stability analysis and optimal damping\nratios.\nThe eigenvalue derivatives of ~Awith respect to the damping coe\u000ecient d1atd1= 0 can be written as\n\u001f(l)\n1(C;k;\u00181;el) = (2C15C35k2\n1k2\n3+ 3C15C33k1k3\n3\u00002C2\n35k4\n3\n+ 2C33C55k4\n3\u00002C2\n15k4\n1\u0018+ 2C15C35k2\n1k2\n3\u00181\n+C15C33k1k3\n3\u00181\u0000C2\n13k2\n1k2\n3(1 +\u00181)\n\u0000C13k1k3(C15k2\n1(1 + 3\u00181) +k3(2C55k1(1 +\u00181)\n+C35k3(3 +\u00181))) +e2\nl(k3(3C15k1(1 +\u00181)\n+ 3C35k1(1 +\u00181) +C33k3(2 +\u00181))\n+C55(k2\n3(2 +\u00181) +k2\n1(1 + 2\u00181)))\u001a+ 2e4\nl(1 +\u00181)\u001a2\n+C11k2\n1(2C55k2\n1\u00181+C33k2\n3(1 +\u00181) +C35k1k3(1 + 3\u00181) +e2\nl(1 + 2\u00181)\u001a))\n=(2((C2\n15\u0000C11C55)k4\n1+ 2(C13C15\u0000C11C35)k3\n1k3+ (C2\n13\u0000C11C33\u00002C15C35\n+ 2C13C55)k2\n1k2\n3+ 2(\u0000C15C33+C13C35)k1k3\n3+ (C2\n35\u0000C33C55)k4\n3)\n\u00003e2\nl((C11+C55)k2\n1+ 2(C15+C35)k1k3+ (C33+C55)k2\n3)\u001a\u00004e4\nl\u001a2); (31)\nwhere subscript \\ l\" is for the ltheigvenvalue, and elstands for the ltheigenvalue of ~A0. The eigenvalue\nderivatives of ~Awith respect to the damping coe\u000ecient d3atd3= 0 is given by\n\u001f(l)\n3(C;k;\u00183;el) = (\u00002C2\n15k4\n1+k2\n3(\u00002(C2\n35\u0000C33C55)k2\n3\u00183\n\u0000C2\n13k2\n1(1 +\u00183)\u0000C13k1(2C55k1(1 +\u00183)\n+C35k3(1 + 3\u00183))) +e2\nl(k3(3C35k1(1 +\u00183)\n+C33k3(1 + 2\u00183)) +C55(k2\n1(2 +\u00183) +k2\n3(1 + 2\u00183)))\u001a+ 2e4\nl(1 +\u00183)\u001a2\n+C15k1k3(\u0000C13k2\n1(3 +\u00183) +k3(2C35k1(1 +\u00183)\n7+C33k3(1 + 3\u00183)) + 3e2\nl(1 +\u00183)\u001a) +C11k2\n1(2C55k2\n1\n+k3(C33k3(1 +\u00183) +C35k1(3 +\u00183)) +e2\nl(2 +\u00183)\u001a))\n=(2((C2\n15\u0000C11C55)k4\n1+ 2(C13C15\u0000C11C35)k3\n1k3\n+ (C2\n13\u0000C11C33\u00002C15C35+ 2C13C55)k2\n1k2\n3+ 2(\u0000C15C33+C13C35)k1k3\n3\n+ (C2\n35\u0000C33C55)k4\n3)\u00003e2\nl((C11+C55)k2\n1\n+ 2(C15+C35)k1k3+ (C33+C55)k2\n3)\u001a\u00004e4\nl\u001a2): (32)\nThere exists a subtle trade-o\u000b between the PML stability and arti\fcail boundary re\rections for anisotropic\nmedia. On one hand, it is necessary to introduce nonzero damping ratios \u00181and\u00183to stabilize PML. On the\nother hand, adding these nonzero damping ratios to PML makes PML no longer perfectly matched, and the\nlarger are the damping ratios, the stronger the arti\fcial boundary re\rections become (Dmitriev and Lisitsa,\n2011). The original analysis of Meza-Fajardo and Papageorgiou (2008) only showed that a certain value of\n\u00181or\u00183can ensure that \u001f(l)\n1and\u001f(l)\n3are negative in all kdirections. However, it did not provide a method\nto determine how large the damping ratios \u00181and\u00183are adequate for an arbitrary anisotropic medium. We\ntherefore develop a procedure to determine the optimal damping ratios \u00181and\u00183to not only stabilize PMLs,\nbut to also minimize resulting arti\fcial boundary re\rections.\nWe employ the following procedure described in Algorithm 1 to obtain the optimal damping ratios \u00181\nand\u00183of MPML for 2D general anisotropic media.\nAlgorithm 1: Determine the optimal damping ratio \u0018i(i= 1;3) of MPML for 2D general anisotropic\nmedia\ninput :\u0018i= 0,\u000f=\u00000:005, \u0001\u0018= 0:001.\nfor\u00122(0;\u0019]do\n1) Calculate wavenumber k= (sin\u0012;cos\u0012)\n2) Calculate eigenvalues el(l= 1;2) of ~A0(C;k) using eqs. (25) and (26)\n3) Calculate eigenvalue derivatives \u001f(l)\ni(C;k;\u0018i;el)\n4)\u001fi;max= max(\u001f(1)\ni;\u001f(2)\ni) using eq. (31) or (32)\nif\u001fi;max>\u000fthen\n\u0018i=\u0018i+ \u0001\u0018\ngo to step 3\nend\nend\noutput:\u0018i\nWe apply the above procedure to both the x1- andx3-directions to obtain the optimal values of \u00181and\n\u00183.\n8Note that the eigenvalue derivatives along these two directions have di\u000berent expressions, although the\nexpressions for eigenvalues elare the same for both the x1- andx3-directions. Therefore, the optimal damping\nratios in the x1- andx3-directions might be di\u000berent from one another. In addition, the searching range for\nthe eigenvalue derivative should be (0 ;\u0019] instead of (0 ;\u0019=2]. We verify these \fndings in numerical examples\nin the next section.\nWe call the aforementioned procedure based on analytic expressions of eigenvalues and eigenvalue deriva-\ntives the analytic approach.\n2.2. Optimal damping ratios of 3D MPML\nElastic-wave equations (1){(2) are also valid for 3D general anisotropic media, but with\nv= (v1;v2;v3)T; (33)\n\u001b= (\u001b11;\u001b22;\u001b33;\u001b23;\u001b13;\u001b12)T; (34)\nC=2\n6666666666664C11C12C13C14C15C16\nC12C22C23C24C25C26\nC13C23C33C34C35C36\nC14C24C34C44C45C46\nC15C25C35C54C55C56\nC16C26C36C64C56C663\n7777777777775; (35)\n\u0003=2\n6664@\n@x10 0 0@\n@x3@\n@x2\n0@\n@x20@\n@x30@\n@x1\n0 0@\n@x3@\n@x2@\n@x103\n7775: (36)\nAnalogous to the 2D case, the 3D elastic-wave equations can be written using decomposed coe\u000ecient\nmatrices CiandDi(i= 1;2;3) as\n@\n@tv=D1@\n@x1\u001b+D2@\n@x2\u001b+D3@\n@x3\u001b; (37)\n@\n@t\u001b=C1@\n@x1v+C2@\n@x2v+C3@\n@x3v; (38)\nwhere\nC1=2\n6666666666664C11C16C15\nC12C26C25\nC13C36C35\nC14C46C45\nC15C56C55\nC16C66C563\n7777777777775; (39)\n9C2=2\n6666666666664C16C12C14\nC26C22C24\nC36C23C34\nC46C24C44\nC56C25C54\nC66C26C643\n7777777777775; (40)\nC3=2\n6666666666664C15C14C13\nC25C24C23\nC35C34C33\nC45C44C34\nC55C45C35\nC65C46C363\n7777777777775; (41)\nD1=2\n66641 0 0 0 0 0\n0 0 0 0 0 1\n0 0 0 0 1 03\n7775; (42)\nD2=2\n66640 0 0 0 0 1\n0 1 0 0 0 0\n0 0 0 1 0 03\n7775; (43)\nD3=2\n66640 0 0 0 1 0\n0 0 0 1 0 1\n0 0 1 0 0 03\n7775: (44)\nThe system of wave equations in PMLs for the 3D case can be expressed in the form of an autonomous\nsystem in the wavenumber domain as\n@U\n@t=~AU; (45)\nwhere\nU=F[\t]; (46)\n\t= (\u001b(1)\n11;\u001b(1)\n22;\u001b(1)\n33;\u001b(1)\n23;\u001b(1)\n13;\u001b(1)\n12;\u001b(2)\n11;\u001b(2)\n22;\u001b(2)\n33;\u001b(2)\n23;\u001b(2)\n13;\u001b(2)\n12;\n\u001b(3)\n11;\u001b(3)\n22;\u001b(3)\n33;\u001b(3)\n23;\u001b(3)\n13;\u001b(3)\n12;v(1)\n1;v(1)\n2;v(1)\n3;v(2)\n1;v(2)\n2;v(2)\n3;v(3)\n1;v(3)\n2;v(3)\n3)T; (47)\n~A=~A0+B; (48)\n10~A0=2\n6666666666664066 066 066ik1C1ik1C1ik1C1\n066 066 066ik2C2ik2C2ik2C2\n066 066 066ik3C3ik3C3ik3C3\nik1\u001a\u00001D1ik1\u001a\u00001D1ik1\u001a\u00001D1033 033 033\nik2\u001a\u00001D2ik2\u001a\u00001D2ik2\u001a\u00001D2033 033 033\nik3\u001a\u00001D3ik3\u001a\u00001D3ik3\u001a\u00001D3033 033 0333\n7777777777775; (49)\nB=2\n6666666666664\u0000d1I6066 066 063 063 063\n066\u0000d2I6066 063 063 063\n066 066\u0000d3I6063 063 063\n036 036 036\u0000d1I3033 033\n036 036 036 033\u0000d2I3033\n036 036 036 033 033\u0000d3I33\n7777777777775; (50)\nanddiis the damping coe\u000ecient along the xi-direction ( i= 1;2;3).\nTo stabilize the PML in the xi-direction, we need to employ nonzero damping ratios along the other two\ndirections perpendicular to xi. Therefore, for the x1-,x2- andx3-directions, we respectively set the damping\nmatrix to be\nB1(\u00181) =2\n6666666666664\u0000d1I6 066 066 063 063 063\n066\u0000d1\u00181I6 066 063 063 063\n066 066\u0000d1\u00181I6063 063 063\n036 036 036\u0000d1I3 033 033\n036 036 036 033\u0000d1\u00181I3 033\n036 036 036 033 033\u0000d1\u00181I33\n7777777777775; (51)\nB2(\u00182) =2\n6666666666664\u0000d2\u00182I6066 066 063 063 063\n066\u0000d2I6 066 063 063 063\n066 066\u0000d2\u00182I6 063 063 063\n036 036 036\u0000d2\u00182I3033 033\n036 036 036 033\u0000d2I3 033\n036 036 036 033 033\u0000d2\u00182I33\n7777777777775; (52)\nB3(\u00183) =2\n6666666666664\u0000d3\u00183I6 066 066 063 063 063\n066\u0000d3\u00183I6066 063 063 063\n066 066\u0000d3I6 063 063 063\n036 036 036\u0000d3\u00183I3 033 033\n036 036 036 033\u0000d3\u00183I3033\n036 036 036 033 033\u0000d3I33\n7777777777775: (53)\nIn the above 3D MPML damped matrices, we employ the same damping ratio along the two directions\n11parallel with the PML/non-PML interface. For instance, for damping along the x1-direction, we use the\nsame nonzero damping ratio \u00181for thex2- andx3-directions, so both the damping coe\u000ecients along the x2-\nandx3-directions in 3D MPML are \u00181d1. Using di\u000berent damping ratios along di\u000berent directions may also\nstabilize PML, but searching optimal values of damping ratios becomes even more complicated.\nWe develop a new approach to computing the eigenvalue derivatives of damped matrix ~Ausing eqs (51)-\n(53) for 3D general anisotropic media. Matrix ~A(as well as ~A0) has a dimension of 27 \u000227, resulting in\nan order of 27 of the characteristic polynomial of ~Aor~A0. Therefore, it is very di\u000ecult, if not impossible,\nto derive analytic expressions for the eigenvalues and eigenvalue derivatives, particularly for media with all\nCij6= 0.\nWe therefore adopt a numerical approach to solving for the eigenvalues and eigenvalue derivatives. Using\nthe de\fnitions of eigenvalues and eigenvectors of matrix ~A, we have\n(~A\u0000\u0015I27)P=0; (54)\nwhere\u0015is the eigenvalue of ~A, and the columns of Pare the eigenvectors of ~A. In addition, we have\nQT(~A\u0000\u0015I27) =0; (55)\nwhere the columns of QTare the left eigenvectors of ~A.\nDi\u000berentiating equation (54) with respect to damping parameter digives\n \n@~A\n@di\u0000@\u0015\n@diI27!\nP+ (~A\u0000\u0015I27)@P\n@di= 0: (56)\nMultiplying both sides of equation (56) with QTleads to\nQT \n@~A\n@di\u0000@\u0015\n@diI27!\nP+QT(~A\u0000\u0015I27)@P\n@di= 0; (57)\nwhich implies\nQT@~A\n@diP=QT@\u0015\n@diI27P: (58)\nTherefore,\n@\u0015\n@di=QT@~A\n@diP\nQTI27P: (59)\nBecause ~A=~A0+Bi(\u0018i) and ~A0is irrelevant to di, the eigenvalue derivative along xi-axis can be written\nas\n\u001fi(\u0018i) =QTRi(\u0018i)P\nQTP; (60)\n12where\nR1(\u00181) =2\n6666666666664\u0000I6066 066 063 063 063\n066\u0000\u00181I6066 063 063 063\n066 066\u0000\u00181I6063 063 063\n036 036 036\u0000I3033 033\n036 036 036 033\u0000\u00181I3033\n036 036 036 033 033\u0000\u00181I33\n7777777777775; (61)\nR2(\u00182) =2\n6666666666664\u0000\u00182I6066 066 063 063 063\n066\u0000I6066 063 063 063\n066 066\u0000\u00182I6063 063 063\n036 036 036\u0000\u00182I3033 033\n036 036 036 033\u0000I3033\n036 036 036 033 033\u0000\u00182I33\n7777777777775; (62)\nR3(\u00183) =2\n6666666666664\u0000\u00183I6066 066 063 063 063\n066\u0000\u00183I6066 063 063 063\n066 066\u0000I6063 063 063\n036 036 036\u0000\u00183I3033 033\n036 036 036 033\u0000\u00183I3033\n036 036 036 033 033\u0000I33\n7777777777775: (63)\nThese equations indicate that we only need to obtain the eigenvalues and the left and right eigenvectors\nof matrix ~A(C;k;\u0018i) for obtaining the optimal damping ratios of MPML in 3D general anisotropic media.\nThis can be achieved using a linear algebra library such as LAPACK, and the procedure is summarized in\nAlgorithm 2.\n13Algorithm 2: Determine the optimal damping ratio \u0018i(i= 1;2;3) for MPML in 3D general anisotropic\nmedia\ninput :\u0018i= 0,\u000f=\u00000:005, \u0001\u0018= 0:001.\nfor\u00122(0;\u0019]do\nfor\u001e2(0;\u0019]do\n1) Calculate wavenumber k= (cos\u001esin\u0012;sin\u001esin\u0012;cos\u0012)\n2) Calculate the left and right eigenvectors of ~A(C;k;\u0018i) using a numerical eigensolver\n3) Calculate the eigenvalue derivatives \u001f(l)\ni(l= 1;2;3) according to eq. (60)\n4)\u001fi;max= max(\u001f(1)\ni;\u001f(2)\ni;\u001f(3)\ni)\nif\u001fi;max>\u000fthen\n\u0018i=\u0018i+ \u0001\u0018\ngo to step 2\nend\nend\nend\noutput:\u0018i\nIn the above algorithm, it is not necessary to seek analytic forms of the left/right eigenvectors, which is\ngenerally impossible for matrix ~A. In our following numerical tests, we calculate the left/right eigenvectors\nwith the Intel Math Kernel Library wrapper for LAPACK. The above numerical approach is obviously\napplicable to the 2D case with trivial modi\fcations. Therefore, for 2D MPML, one can use either the\nanalytic approach or the numerical approach, yet for 3D MPML, one can use only the numerical approach.\n3. Results\nWe use three examples of 2D anisotropic media and three examples of 3D anisotropic media to validate\nthe e\u000bectiveness of our new algorithms for calculating optimal damping ratios in 2D and 3D MPMLs. In\nthe following, when presenting an elasticity matrix, we write only the upper triangle part of this matrix, but\nit should be clear that the elasticity matrix is essentially symmetric. We also assume that all the elasticity\nmatrices have units of GPa, and all the media have mass density values of 1000 kg/m3for convenience.\n3.1. MPML for 2D anisotropic media\nTo validate our new algorithm for determining the optimal damping ratios in MPML for 2D general\nanisotropic media, we consider a transversely isotropic medium with a horizontal symmetry axis (HTI\nmedium), a transversely isotropic medium with a tilted symmetry axis (TTI medium), and a transversely\nisotropic medium with a vertical symmetry axis (VTI medium) with serious qS triplication in both x1- and\nx3-directions.\n14-6000 -4000 -2000 0 2000 4000 6000\nx1component (m/s)\n-6000-4000-20000200040006000x3component (m/s)\nqP-wave\nqS-waveFigure 1: Wavefront curves in the 2D HTI medium with elasticity matrix (64).\nFor the HTI medium example, we use a well-known example with elasticity matrix (B\u0013 ecache et al., 2003;\nMeza-Fajardo and Papageorgiou, 2008):\nC=2\n66644 7:5 0\n20 0\n23\n7775: (64)\nNote that this medium is considered as an orthotropic medium in B\u0013 ecache et al. (2003) and Meza-Fajardo\nand Papageorgiou (2008). However, it could also be considered as an HTI medium on the x1x3-plane. The\nonly di\u000berences are that C11=C22andC44=C55for a 3D HTI medium, while there exists no such equality\nrestrictions for an orthotropic medium.\nThe wavefront curves of qP- and qS-waves in Fig. 1 show the anisotropy characteristics of this HTI\nmedium. We employ Algorithm 1 to determine the optimal damping ratios in MPML along the x1- and\nx3-directions, leading to\n\u00181= 0:108; \u0018 3= 0:259: (65)\nIn Meza-Fajardo and Papageorgiou (2008), the suggested values of damping ratios are \u00181= 0:30 and\n\u00183= 0:25 for this HTI medium. Their suggested value for damping ratio \u00181is much larger than the optimal\ndamping ratio given in eq. (65), while their suggested value for \u00183is similar to the optimal damping ratio.\nFigure 2 plots the values of eigenvalue derivatives of ~Aunder the optimal damping ratios in the x1-\nandx3-directions. In both panels, the blue curves represent the qP-wave eigenvalue derivatives, and the\nred curves are for the qS-wave eigenvalue derivatives. Clearly, the qS-wave gives rise to the large damping\nratios along both axes. Note that we set the threshold \u000f=\u00000:005, therefore, in both panels of Fig. 2, the\nmaximum values of the eigenvalue derivatives are \u00000:005.\nWe validate the e\u000bectiveness of our new MPML in numerical modeling of anisotropic elastic-wave pro-\n150 30 60 90 120 150\nWavenumber polar angle  (deg.)\n-1.5-1.2-0.9-0.6-0.30\nqPwave 1 with 1=0.108\nqSwave 1 with 1=0.108\n(a)\n0 30 60 90 120 150\nWavenumber polar angle  (deg.)\n-1.5-1.2-0.9-0.6-0.30\nqPwave 3 with 3=0.259\nqSwave 3 with 3=0.259\n (b)\nFigure 2: Eigenvalue derivatives of ~Aof MPML in the (a) x1- and (b)x3-directions under calculated optimal damping ratios\nin eq. (65) for the 2D HTI medium with elasticity matrix (64).\ngation. We use the rotated-staggered grid (RSG) \fnite-di\u000berence method (Saenger et al., 2000) to solve\nthe stress-velocity form elastic-wave equations (1){(2). The RSG \fnite-di\u000berence method has 16th-order\naccuracy in space with optimal \fnite-di\u000berence coe\u000ecients (Liu, 2014). We compute the wave\feld energy\ndecay curves of our wave\feld modelings to validate the e\u000bectiveness of MPML.\nIn our numerical modeling, the model is de\fned in a 400 \u0002400 grid, and a PML of 30-node thickness are\npadded around the model domain. The grid size is 10 m in both the x1- andx3-directions. A vertical force\nvector source is located at the center of the computational domain, and a Ricker wavelet with a 10 Hz central\nfrequency is used as the source time function. We simulate wave propagation for 20 s with a time interval of\n1 ms, which is smaller than what is required to satisfy the stability condition (about 1.54 ms). Figure 3 shows\nthe resulting wave\feld energy curve under the optimal damping ratios in eq. (65) , together with three others\nunder di\u000berent eigenvalue derivative threshold \u000fvalues, or equivalently, di\u000berent damping ratios. Figure 3\nshows that within the 20 s of wave propagation, the MPML with our calculated damping ratios \u00181= 0:108\nand\u00183= 0:259 is stable. These damping ratios are obtained under threshold value \u000f=\u00000:005, meaning\nthat the damping ratios have to ensure the eigenvalues derivatives \u001f1and\u001f3are not larger than \u00000:005 in\nthe entire range of wavenumber direction.\nWe test the behavior of MPML under threshold \u000f= 0:01, or equivalently, \u00181= 0:095 and\u00183= 0:248, and\nshow in Fig. 3 that the numerical modeling is stable. We further increase the threshold \u000fto be 0.05, and\nthe MPML becomes unstable quickly after about 2 s. Finally, the conventional PML, which is equivalent\nto MPML under \u00181=\u00183= 0, becomes unstable even earlier (before 1 s). These tests indicate that a small\npositive threshold \u000fmay still result in a stable MPML. However, there is no simple method to determine how\nlarge this positive \u000fto ensure the numerical stability. In this HTI medium case, \u000f= 0:01 results in stability\nwhile\u000f= 0:05 results in instability. Because a negative threshold resulting in stable MPML is consistent\nwith the stability theory presented by Meza-Fajardo and Papageorgiou (2008), we therefore should choose a\n160 5 10 15 20\nTime (s)1011\n109\n107\n105\n103\nWavefield energy |v|2=0.005,1=0.108,3=0.259\n=0.01,1=0.095,3=0.248\n=0.05,1=0.059,3=0.218\nConventional PMLFigure 3: Wave\feld energy decay curves under di\u000berent eigenvalue derivative thresholds for the 2D HTI medium with elasticity\nmatrix (64) within 20 s. Conventional PML can be considered as a special case of MPML with \u000f\u001d0 or equivalently \u00181=\u00183= 0.\nnegative threshold for the calculation of the optimal damping ratios to ensure that the resulting MPML is\nstable. This is also veri\fed in the hereinafter numerical examples.\nNext, we rotate the aforementioned HTI medium with respect to the x2-axis clockwise by \u0019=6 to obtain\na TTI medium represented by the following elasticity matrix:\nC=2\n66647:8125 7:6875 3:35585\n15:8125 3:57235\n2:18753\n7775; (66)\nwith unit GPa. The rotation can be accomplished by rotation matrix (e.g., Slawinski, 2010). The wavefront\ncurves in this TTI medium is shown in Fig. 4. Although this TTI medium is the rotation result of the HTI\nmedium in the previous numerical example, it is not obvious how to change the damping ratios accordingly.\nWe obtain the following optimal damping ratios of MPML under \u000f=\u00000:005 using Algorithm 1:\n\u00181= 0:157; \u0018 3= 0:226: (67)\nThe eigenvalue derivatives under this set of damping ratios are shown in Fig. 5. The eigenvalue derivative\ncurves are no longer symmetric with respect to \u0012=\u0019=2 (or has a period of \u0019=2) as those for the 2D HTI\nmedium (Fig. 2). Instead, they are periodic every \u0019angle, corresponding to the fact that there is always at\nleast one symmetric axis for whatever kind of 2D anisotropic medium in the axis plane. These curves also\nindicate that, for 2D general anisotropic medium (TTI medium in this example), it is necessary to determine\nthe values of eigenvalue derivatives within the range of (0 ;\u0019] instead of (0 ;\u0019=2]. Using only the range (0 ;\u0019=2]\ncan lead to a totally incorrect optimal value of \u00181, since the maximum value of \u001f1in the range (0 ;\u0019=2] is\nsmaller than that in the range ( \u0019=2;\u0019] for this TTI medium. In other words, even though \u001f1in (0;\u0019=2]\n17-6000 -4000 -2000 0 2000 4000 6000\nx1component (m/s)\n-6000-4000-20000200040006000x3component (m/s)\nqP-wave\nqS-waveFigure 4: Wavefront curves in the 2D HTI medium with elasticity matrix (66).\n0 30 60 90 120 150\nWavenumber polar angle  (deg.)\n-1.5-1.2-0.9-0.6-0.30\nqPwave 1 with 1=0.157\nqSwave 1 with 1=0.157\n0 30 60 90 120 150\nWavenumber polar angle  (deg.)\n-1.5-1.2-0.9-0.6-0.30\nqPwave 3 with 3=0.226\nqSwave 3 with 3=0.226\nFigure 5: Eigenvalue derivatives of ~Aof MPML in the (a) x1- and (b)x3-directions under calculated optimal damping ratios\nin eq. (67) for the 2D TTI medium with elasticity matrix (66).\nindicates a stable MPML, the MPML may still be unstable since \u001f1may be larger than zero in ( \u0019=2;\u0019].\nTherefore, for anisotropic media with symmetric axis not aligned with a coordinate axis, it is necessary to\nconsider the values of \u001fiin wavenumber direction \u00122(0;\u0019]. This statement is also true for 3D anisotropic\nmedia as shown in the hereinafter 3D numerical examples.\nFigure 6 displays the wave\feld energy decay curves for this TTI medium under the optimal damping\nratios, as well as under damping ratios calculated with positive eigenvalue derivative thresholds. In this\nexample, the wave\feld energy decays gradually within 20 s for both cases with \u000f=\u00000:005 and\u000f= 0:05.\nThe numerical modeling with \u000f= 0:15 becomes unstable. For comparison, in the previous HTI case,\n\u000f= 0:05 results in an unstable MPML. These results further demonstrate that a positive eigenvalue derivative\nthreshold should not be chosen to calculate the damping ratios, although a small positive \u000fmight result in\nstable MPML. In contrast, a negative \u000fcan always ensure the stability of MPML.\n180 5 10 15 20\nTime (s)1013\n1011\n109\n107\n105\n103\nWavefield energy |v|2=0.005,1=0.157,3=0.226\n=0.05,1=0.110,3=0.183\n=0.15,1=0.025,3=0.105\nConventional PMLFigure 6: Wave\feld energy decay curves under di\u000berent eigenvalue derivative thresholds in the 2D TTI medium with elasticity\nmatrix (66) within 20 s.\nOur next numerical example uses a VTI medium de\fned by\nC=2\n666410:4508 4:2623 0\n7:5410 0\n11:39343\n7775: (68)\nThe wavefront curves for this VTI medium are depicted in Fig. 7. The special feature of this VTI medium\nis that, in both the x1- andx3-directions, there exists serious qS-wave triplication phenomena. We obtain\nthe following optimal damping ratios using Algorithm 1:\n\u00181= 0:215; \u0018 3= 0:225: (69)\nThe corresponding eigenvalue derivatives in the x1- andx3-directions are displayed in Fig. 8. Again, it is the\nqS-wave that causes the damping ratios to be large to stabilize PML. Figure 9 depicts the wave\feld energy\ndecay curves under di\u000berent eigenvalue derivative thresholds. Similar with that of 2D TTI medium example,\na positive threshold 0.05 can still stabilize PML, yet a value of 0.15 makes the MPML unstable. We choose\na negative value of \u000fto ensure a stable MPML.\n3.2. MPML for 3D anisotropic media\nFor 3D anisotropic media, we need to determine the optimal MPML damping ratios along all three\ncoordinate directions. We use three di\u000berent anisotropic media (a quasi-VTI medium, a quasi-TTI medium\nand a triclinic medium) to demonstrate the determination of optimal MPML damping ratios.\n19-5000 -2500 0 2500 5000\nx1component (m/s)\n-5000-2500025005000x3component (m/s)\nqP-wave\nqS-waveFigure 7: Wavefront curves in the 2D VTI medium with elasticity matrix (68).\n0 30 60 90 120 150\nWavenumber polar angle  (deg.)\n-1.5-1.2-0.9-0.6-0.30\nqPwave 1 with 1=0.215\nqSwave 1 with 1=0.215\n0 30 60 90 120 150\nWavenumber polar angle  (deg.)\n-1.5-1.2-0.9-0.6-0.30\nqPwave 3 with 3=0.225\nqSwave 3 with 3=0.225\nFigure 8: Eigenvalue derivatives of ~Aof MPML in the (a) x1- and (b)x3-directions under calculated optimal damping ratios\nin eq. (69) for the 2D VTI medium with elasticity matrix (68).\n200 5 10 15 20\nTime (s)1020\n1015\n1010\n105\n100Wavefield energy |v|2=0.005,1=0.215,3=0.225\n=0.05,1=0.172,3=0.182\n=0.15,1=0.093,3=0.104\nConventional PMLFigure 9: Wave\feld energy decay curves under di\u000berent eigenvalue derivative thresholds in the 2D VTI medium with elasticity\nmatrix (68) within 20 s.\nWe \frst use a 3D anisotropic medium represented by the elasticity matrix\nC=2\n666666666666416:5 5 5 0 0 0\n16:5 5 0 0 0\n6:2 0 0 0\n4:96 0 0\n3:96 0\n5:963\n7777777777775: (70)\nThis elasticity matrix is modi\fed from the elasticity matrix of zinc (a VTI medium, or hexagonal anisotropic\nmedium) to increase the complexity of the resulting wavefronts and the characteristics of the eigenvalue\nderivatives along all three directions. This modi\fed elastic matrix still represents a physically feasible\nmedium since it is easy to verify that it satis\fes the following stability condition for anisotropic media\n(Slawinski, 2010):\ndet2\n6664C11\u0001\u0001\u0001C1n\n.........\nC1n\u0001\u0001\u0001Cnn3\n7775>0; (71)\nwheren= 1;2;\u0001\u0001\u0001;6. We call this anisotropic medium the quasi-VTI medium. Figure 10 shows the\nwavefront curves of this quasi-VTI medium on three axis planes.\nFor comparison, a standard 3D VTI medium can be expressed by its \fve independent elasticity constants\n21as (e.g., Slawinski, 2010)\nC=2\n6666666666664C11C12C13 0 0 0\nC11C13 0 0 0\nC33 0 0 0\nC44 0 0\nC44 0\nC11\u0000C12\n23\n7777777777775: (72)\nWe calculate the optimal damping ratios for MPML using Algorithm 2, and obtain\n\u00181= 0:088; \u0018 2= 0:131; \u0018 3= 0:041: (73)\nWe plot the eigenvalue derivatives in the polar angle range (0 ;\u0019] and azimuth angle range (0 ;\u0019] for three axis\ndirections in Fig. 11. Since the three symmetric axes of this VTI medium are aligned with three coordinate\naxes, the three eigenvalue derivatives are symmetric with respect to both \u0012=\u0019=2 and\u001e=\u0019=2 lines.\nWe conduct numerical wave\feld modeling to verify the stability of MPML under these optimal damping\nratios. The model is de\fned on a 400 \u0002400\u0002400 grid with a grid size of 10 m in all three directions. The\nthickness of PML layer is 25 grids. A vertical force vector is located at the center of the computational\ndomain, and the source time function is a Ricker wavelet with a central frequency of 10 Hz. The time step\nsize is 1 ms, which is smaller than the stability-required time step of 1.69 ms. A total of 15,000 time steps,\ni.e., 15 s, are simulated, and the wave\feld energy curve is shown in Fig. 12. The blue curve in Fig. 12 is\nfor the case with the optimal damping ratios in eq. (73). We also carry out a wave\feld modeling with the\neigenvalue derivative threshold \u000f= 0:015 and\u000f= 0:025. The MPMLs under these thresholds are unstable\naccording to the corresponding wave\feld energy variation curves in Fig. 12. As in the 2D MPML case, we\nshould always choose a negative \u000fto stabilize MPML for 3D anisotropic media.\nOur next numerical example uses a rotation version of the aforementioned quasi-VTI medium. We rotate\nthe quasi-VTI medium (70) with respect to the x1-axis by 30 degrees, the x2-axis by 50 degrees, and the\nx3-axis by 25 degrees, and the resulting elasticity matrix for this quasi-TTI medium is given by\nC=2\n666666666666415:7930 4:1757 4:9651 0:1582 0:6529\u00001:0343\n12:5979 4:1844 2:0903 0:8186\u00001:7513\n14:1587 1:9573 0:7643\u00000:1606\n3:7879\u00000:8909 0:8065\n5:0750 0:7668\n4:34233\n7777777777775: (74)\nThe corresponding wavefront curves are shown in Fig. 13.\nSimilar to the 3D quasi-VTI case, we obtain the following optimal damping ratios using Algorithm 2:\n\u00181= 0:089; \u0018 2= 0:051; \u0018 3= 0:080: (75)\n22-5000 -2500 0 2500 5000\nx1component (m/s)\n-5000-2500025005000x2component (m/s)\nqP-wave\nqS1-wave\nqS2-wave(a)\n-5000 -2500 0 2500 5000\nx1component (m/s)\n-5000-2500025005000x3component (m/s)\nqP-wave\nqS1-wave\nqS2-wave (b)\n-5000 -2500 0 2500 5000\nx2component (m/s)\n-5000-2500025005000x3component (m/s)\nqP-wave\nqS1-wave\nqS2-wave\n(c)\nFigure 10: Wavefront curves in the 3D quasi-VTI medium with elasticity matrix (70) on the (a) x1x2(b)x1x3and (c)x2x3\naxis plane. qS1 and qS2 represents the two qS-waves.\n2330 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1(a)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 (b)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 (c)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2\n(d)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 (e)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 (f)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1\n(g)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 (h)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 (i)\nFigure 11: Eigenvalue derivatives of ~Aof MPML in the (a)-(c) x1-, (d)-(f)x2-, and (g)-(i) x3-directions under calculated optimal\ndamping ratios in eq. (73) for the 3D quasi-VTI medium with elasticity matrix (70). (a), (d) and (g) represent qP-wave, (b),\n(e) and (h) represent qS1-wave, (c), (f) and (i) represent qS2-wave.\n240 5 10 15\nTime (s)1013\n1011\n109\n107\n105\nWavefield energy |v|2\n=0.005,1=0.089,2=0.131,3=0.041\n=0.015,1=0.068,2=0.112,3=0.020\n=0.025,1=0.058,2=0.102,3=0.009\nConventional PMLFigure 12: Wave\feld energy decay curve in the 3D VTI medium with elasticity matrix (70).\nThe eigenvalue derivatives under this set of damping ratios for three axis directions are shown in Fig. 14.\nObviously, for the quasi-TTI medium where the symmetric axes are not aligned with coordinate axes, the\neigenvalue derivatives of all three wave modes along any coordinate axis is no longer symmetric about any \u0012\nor\u001elines. Therefore, it is necessary to use the entire range of wavenumber polar angle \u0012and azimuth angle\n\u001e, i.e., (0;\u0019]\u0002(0;\u0019], to determine the optimal damping ratios.\nFigure 15 depicts the wave\feld energy decays under the optimal damping ratios in eq. (75) and damping\nratios with thresholds \u000f= 0:05 and\u000f= 0:075. For the case where \u000f= 0:05, the wave\feld energy does\nnot diverge immediately after maximum energy value occurred. Instead, the curve indicates a very slow\nenergy decay after about 1 s. In contrast, the optimal MPML with threshold \u000f=\u00000:005 shows a \\normal\"\nenergy decay. Therefore, although the MPML with \u000f= 0:05 does not show energy divergence within 15 s, it\nfails to e\u000bectively absorb the outgoing wave\feld, and we consider this as a \\quasi-divergence.\" Meanwhile,\nthe MPML with \u000f= 0:075 shows an energy divergence after about 4 s. These results further demonstrate\nthat the behavior of MPML with a positive eigenvalue derivative threshold is di\u000berent and unpredictable\nfor di\u000berent kinds of anisotropic media. Figure 15 also shows that the conventional PML gives an unstable\nresult.\nOur last 3D numerical example is based on a triclinic anisotropic medium represented by\nC=2\n666666666666410 3:5 2:5\u00005 0:1 0:3\n8 1:5 0:2\u00000:1\u00000:15\n6 1 0 :4 0:24\n5 0:35 0:525\n4\u00001\n33\n7777777777775; (76)\n25-5000 -2500 0 2500 5000\nx1component (m/s)\n-5000-2500025005000x2component (m/s)\nqP-wave\nqS1-wave\nqS2-wave(a)\n-5000 -2500 0 2500 5000\nx1component (m/s)\n-5000-2500025005000x3component (m/s)\nqP-wave\nqS1-wave\nqS2-wave (b)\n-5000 -2500 0 2500 5000\nx2component (m/s)\n-5000-2500025005000x3component (m/s)\nqP-wave\nqS1-wave\nqS2-wave\n(c)\nFigure 13: Wavefront curves in the 3D quasi-TTI medium with elasticity matrix (74) on the (a) x1x2(b)x1x3and (c)x2x3\naxis plane. qS1 and qS2 represents the two qS-waves. qS-wave wavefronts seem to be less complicated compared with those of\nthe 3D quasi-VTI medium (70) only because the qS-wave triplications are now out of axis planes after rotation.\n2630 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1(a)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 (b)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 (c)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1\n(d)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 (e)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 (f)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1\n(g)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 (h)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 (i)\nFigure 14: Eigenvalue derivatives of ~Aof MPML in the (a)-(c) x1-, (d)-(f)x2-, and (g)-(i) x3-directions under calculated optimal\ndamping ratios in eq. (75) for the 3D quasi-TTI medium with elasticity matrix (74). (a), (d) and (g) represent qP-wave, (b),\n(e) and (h) represent qS1-wave, (c), (f) and (i) represent qS2-wave.\n270 5 10 15\nTime (s)1024\n1019\n1014\n109\n104\nWavefield energy |v|2\n=0.005,1=0.089,2=0.051,3=0.080\n=0.05,1=0.034,2=0,3=0.024\n=0.075,1=0.009,2=0,3=0\nConventional PMLFigure 15: Wave\feld energy decay curve in the 3D quasi-TTI medium with elasticity matrix (74).\nwith unit GPa. The wavefront curves on three axis planes are shown in Fig. 16.\nWe solve for the optimal damping ratios for this triclinic anisotropic medium using Algorithm 2, and\nobtain the following optimal damping ratios with \u000f=\u00000:005:\n\u00181= 0:487; \u0018 2= 0:345; \u0018 3= 0:374: (77)\nThe damping ratios for this anisotropic medium are unexpectedly very large compared with those for the\nheretofore 2D and 3D examples. We seek the reasons of these large damping ratios from the eigenvalue\nderivatives shown in Fig. 17, and \fnd that it is the qS2-wave that leads to such large damping ratios to\nachieve a stable MPML. In fact, for the damping ratios in eq. (77), the corresponding eigenvalue derivatives\nof qP- and qS1-waves are far smaller than zero, yet the eigenvalue derivative of qS2-wave merely smaller\nthan zero (\u00000:005 under our threshold setting), leading to a set of relatively large damping ratios for this\n3D anisotropic medium.\nOur calculated optimal damping ratios result in a stable MPML, as indicated by the corresponding energy\ndecay curve shown in Fig. 18. The wave\feld energy decay curve with a threshold of \u000f= 0:1 displayed in\nFig. 18 is surprisingly almost identical with that of \u000f=\u00000:005. When using a threshold \u000f= 0:4, MPML\nbecome unstable, indicating that the positive threshold 0.4 is too large to make MPML stable. This veri\fes\nagain that, although a positive threshold might result in stable MPML, we should use a negative threshold\nto ensure a stable MPML for general anisotropic media. This is consistent with the stability condition\ndescribed in Meza-Fajardo and Papageorgiou (2008), and is perhaps the only practical method to stabilize\nPML using nonzero damping ratios.\n28-4000 -2000 0 2000 4000\nx1component (m/s)\n-3000-1500015003000x2component (m/s)\nqP-wave\nqS1-wave\nqS2-wave(a)\n-4000 -2000 0 2000 4000\nx1component (m/s)\n-3000-1500015003000x3component (m/s)\nqP-wave\nqS1-wave\nqS2-wave (b)\n-4000 -2000 0 2000 4000\nx2component (m/s)\n-3000-1500015003000x3component (m/s)\nqP-wave\nqS1-wave\nqS2-wave\n(c)\nFigure 16: Wavefront curves in the 3D triclinic anisotropic medium with elasticity matrix (74) on the (a) x1x2(b)x1x3and\n(c)x2x3axis plane. qS1 and qS2 represents the two qS-waves.\n2930 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5(a)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1.1-1-0.9-0.8-0.7-0.6-0.5-0.4 (b)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1.3-1.1-0.9-0.7-0.5-0.3-0.1 (c)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3\n(d)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3 (e)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1.3-1.1-0.9-0.7-0.5-0.3-0.1 (f)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3\n(g)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3 (h)\n30 60 90 120 150\nWavenumber polar angle  (deg.)\n30\n60\n90\n120\n150Wavenumber azimuth angle  (deg.)\n-1.3-1.1-0.9-0.7-0.5-0.3-0.1 (i)\nFigure 17: Eigenvalue derivatives of ~Aof MPML in the (a)-(c) x1-, (d)-(f)x2-, and (g)-(i) x3-directions under calculated\noptimal damping ratios in eq. (77) for the 3D triclinic anisotropic medium with elasticity matrix (76). (a), (d) and (g) represent\nqP-wave, (b), (e) and (h) represent qS1-wave, (c), (f) and (i) represent qS2-wave.\n300 5 10 15\nTime (s)1015\n1012\n109\n106\n103\nWavefield energy |v|2\n=0.005,1=0.487,2=0.345,3=0.374\n=0.1,1=0.420,2=0.258,3=0.291\n=0.4,1=0.230,2=0.010,3=0.054\nConventional PMLFigure 18: Wave\feld energy decay curve in the 3D triclinic anisotropic medium with elasticity matrix (76). The blue curve\n(\u000f=\u00000:005) and red curve ( \u000f= 0:1) are almost identical.\n4. Conclusions\nA de\fnite analytic method for determining the optimal damping ratios of multi-axis perfectly matched\nlayers (MPML) is generally impossible for 3D general anisotropic media with possible all nonzero elasticity\nparameters. We have developed a new method to e\u000eciently determine the optimal damping ratios of MPML\nfor absorbing unwanted, outgoing propagating waves in 2D and 3D general anisotropic media. 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Bulletin of the Seismological\nSociety of America 98 (4), 1811{1836.\nURL http://www.bssaonline.org/content/98/4/1811.abstract\n33Peng, C., Toks oz, M. N., 1994. An optimal absorbing boundary condition for \fnite di\u000berence modeling of\nacoustic and elastic wave propagation. The Journal of the Acoustical Society of America 95 (2), 733{745.\nURL http://scitation.aip.org/content/asa/journal/jasa/95/2/10.1121/1.408384\nReynolds, A. C., 1978. Boundary conditions for the numerical solution of wave propagation problems. Geo-\nphysics 43 (6), 1099{1110.\nURL http://geophysics.geoscienceworld.org/content/43/6/1099\nSaenger, E. H., Gold, N., Shapiro, S. A., 2000. Modeling the propagation of elastic waves using a modi\fed\n\fnite-di\u000berence grid. Wave Motion 31 (1), 77 { 92.\nURL http://www.sciencedirect.com/science/article/pii/S0165212599000232\nSlawinski, M. A., 2010. Waves and Rays in Elastic Continua. World Scienti\fc.\nURL http://www.worldscientific.com/worldscibooks/10.1142/7486#t=aboutBook\nZhang, W., Shen, Y., 2010. Unsplit complex frequency-shifted PML implementation using auxiliary di\u000ber-\nential equations for seismic wave modeling. Geophysics 75 (4), T141{T154.\nURL http://dx.doi.org/10.1190/1.3463431\n34"    },    {        "title": "1904.12678v6.Making_light_of_gravitational_waves.pdf",        "content": "Making light of gravitational-waves\nJustine Tarrant\u0003, Geo\u000b Beck, Sergio Colafrancesco\nSchool of Physics, University of the Witwatersrand,\nPrivate Bag 3, 2050, Johannesburg, South Africa\nAbstract\nMixing between photons and low-mass bosons is well considered in the litera-\nture. The particular case of interest here is with hypothetical gravitons, as we\nare concerned with the direct conversion of gravitons into photons in the pres-\nence of an external magnetic \feld. We examine whether such a process could\nproduce direct low-frequency radio counterparts to gravitational-wave events.\nOur work di\u000bers from previous work in the literature in that we use the results\nof numerical simulations to demonstrate that, although a single such event may\nbe undetectable without at least 105dipoles, an unresolved gravitational wave\nbackground from neutron star mergers could be potentially detectable with a lu-\nnar telescope composed of 103elements. This is provided the gravitational wave\nspectrum only experiences exponential damping above 80 kHz, a full order of\nmagnitude above the limit achieved by present simulation results. In addition,\nthe extrapolation cannot have a power-law slope .\u00002 (for 100 hours of ob-\nservation time) and background and foregrounds must be e\u000bectively subtracted\nto obtain the signal. This does not make detection impossible, but suggests it\nmay be unlikely. Furthermore, assuming a potentially detectable spectral sce-\nnario we show that, for the case when no detection is made by a lunar array, a\nlower bound, competitive with those from Lorentz-invariance violation, may be\nplaced on the energy-scale of quantum gravitational e\u000bects. The SKA is shown\nto have very limited prospects for the detection of either a single merger or a\nbackground.\nKeywords: lunar telescope, photon-graviton conversion, SKA-Low, quantum\ngravity\n1. Introduction\nElectromagnetic counterparts to gravitational-waves were well considered in\nthe literature [1, 2, 3, 4, 5, 6] prior to the \frst detection of compact mergers\n\u0003Corresponding author\nEmail addresses: justine.tarrant@wits.ac.za (Justine Tarrant),\ngeoffrey.beck@wits.ac.za (Geo\u000b Beck)\nThis paper is dedicated to the memory of Prof. Sergio Colafrancesco.\nPreprint submitted to Astroparticle Physics February 5, 2021arXiv:1904.12678v6  [astro-ph.HE]  4 Feb 2021by the Laser Interferometer Gravitational-wave Observatory (LIGO), made in\nSeptember 2015, and produced by the merger of two black holes. Furthermore,\nthis correlation was considered by [7, 8] following these events. It wasn't until\nthe discovery of counterparts associated with the \ffth LIGO event [9] detected\nin August 2017, that the \feld was revived as scientists poured over the new data\nobtained. The successful detection was only possible due to an immense, well\ncoordinated, global collaboration. The GW170817 event was the \frst detection\nof a binary neutron star merger, and was long hoped for since such events were\nexpected to form a kilonova and collimated out\rows, called jets, which result in\nelectromagnetic counterparts.\nEmissions across the entire electromagnetic spectrum were detected [10],\nthus heralding the dawn of multi-messenger astronomy. This includes the ob-\nservation of radio emissions in the broad band afterglow occurring when the\ngamma-ray burst interacts with the interstellar medium [11]. In this work, we\napproach this problem from a di\u000berent angle using the idea that gravitational-\nwaves can themselves directly generate plausibly detectable low-frequency radio\ncounterparts while propagating from their source to Earth.\nThe idea that gravitational degrees of freedom may be converted into elec-\ntromagnetic degrees of freedom is not a new one, see for example: [12, 13, 14, 15]\nand [16]. Interest in this subject resulted in the need to \fnd an indirect means of\nmeasuring gravitational-waves and also arose out of studies considering axion-\nphoton mixing [17, 18]. For example, early studies involved the scattering of\nelectromagnetic \felds o\u000b time-dependent gravitational \felds, showing that there\nwas a possible coherent interaction between linear gravitational-waves and elec-\ntromagnetic waves in which energy could be transformed from one degree of\nfreedom to the other [19]. Such conversions taking place inside an external\nmagnetic \feld could be extremely bright, even if a small percentage of the\ngravitational-wave pulse is transformed [16, 20]. Modern calculations include\nconsiderations about plasma frequency and QED corrections [7], see also [8, 12].\nAs mentioned in [7], these articles dealt with the graviton-photon conversion at\nhigh frequencies !=(2\u0019) which exceeded the plasma frequency, !p=(2\u0019), of the\nsurrounding and intervening medium. It makes sense that higher frequencies\nwere considered since low-frequency waves do not propagate in plasma. How-\never, for the case of LIGO events concerning the merger of two black holes of\nroughly 20 - 30 M\f, the realistic gravitational-wave frequencies were approxi-\nmately 100 - 200 Hz. This is problematic since typically the plasma frequency\nis approximately equal to 10 kHzpne\n1 cm\u00003, whereneis the electron number\ndensity.\nIn [7], they consider the possibility that the energy transition from gravitational-\nwaves into electromagnetic waves may still be possible despite their low fre-\nquency. In particular, they show that gravitational-waves travelling through a\nhigh-frequency plasma and non-zero magnetic \feld continue to transform some\nof their energy into non-propagating plasma waves which heat up the surround-\ning plasma, thus leading to a noticeable release of electromagnetic radiation. In\nthis work, they use an asymmetric conversion regime for which conversion from\nphotons into gravitons is less probable given that the wave vector for this solu-\n2tion is purely imaginary, corresponding to the damping of the electromagnetic\nwave travelling in the plasma with frequency higher than that of itself. The\nauthors conclude that the graviton-photon conversion mechanism studied can\nhardly account for plasma heating in LIGO black hole merger events.\nA further interesting paper is that of [8] where they considered the ef-\nfects generating dispersion and coherence braking of the electromagnetic waves.\nTherein they obtained the energy power and energy power \ruxes for quasi-\nperpendicular external magnetic \felds in the gravitational-wave propagation\ndirection. The authors found that the energy power was large, but that the\n\ruxes seen on Earth would remain faint. They considered waves with ! >!p,\nmaking the graviton-photon mixing for gravitational-waves of less than a few\nhundred Hertz less appealing. They noted, however, that the calculated plasma\nfrequency through which the waves travel depends on the line of sight, produc-\ning varying frequency cut-o\u000b's. Finally, they conclude that the detection of this\ngraviton to photon mechanism is unlikely to be made on Earth or in interplan-\netary space due to the large cut-o\u000b frequencies. They suggest that detection is\nonly probable outside of the solar system. We show that this is not necessarily\ntrue.\nOur proposal uses the idea that gravitational-waves may be converted into\nelectromagnetic waves following the mechanism laid out in [12]. Extending this\nmechanism, beyond the treatment of axions, to gravitons implies that the grav-\nitational \feld is quantized. Therefore, this model also provides a potential way\nto probe quantum gravity and its associated energy scales. Using the char-\nacteristics of relatively well studied environments, such as the known external\nmagnetic \feld, plasma density, and a phenomenological energy scale M, we can\ncompute the transition probability that gravitons are converted to photons by\nnumerically solving the equations of motion laid out in [8]. We then make use\nof data on the magnetic \feld and plasma environment within the nearby galaxy\nM31 to compute transition probabilities between photon and graviton when a\ngraviton \rux passes through the M31 galaxy. In particular, as the conversion\nspectrum from a single merger event will be shown to be likely unobservable,\nwe will also study the observability of conversion from an isotropic background\nformed by neutron star mergers out to redshift 6 by using population synthesis\nestimates of the merger rate for binary neutron stars [21].\nSupposing that it may be possible to \fnd the parameter Mthrough obser-\nvation of the electromagnetic spectrum corresponding to the gravitational-wave\nevent, we may then compare its value to the phenomenological MPl=p\n~c=G=\n1:22\u00021019GeV/c2, the Planck mass, as is widely used in the literature as an\nexpected energy scale for the appearance of quantum gravitational e\u000bects. In\nthis way, a bound may be placed on any deviation of the energy scale of quan-\ntum gravitational e\u000bects Mfrom that of the Planck mass. Furthermore, such\nan observation could determine the existence of the graviton and therefore that\nthe gravitational \feld is quantized.\nOur means of analysis is to make a phenomenological extrapolation of a\nnumerically determined gravitational-wave spectrum produced by a binary neu-\ntron star merger. We then determine how much the detectability of the resulting\n3counterpart photon spectrum depends upon the parameters of the extrapola-\ntion, which is justi\fed in that it accounts for both scenarios encountered in the\nliterature: mode decay and the contribution of higher order modes extending\nthe spectrum to higher frequencies. If the dependence is strong we conclude\nthat detection is improbable without signi\fcant \fne-tuning of the extrapolated\nspectrum. On the other hand, if the counterpart's detectability is largely in-\ndependent of the choice of parameters, we take this to indicate that detection\nwould be a strong possibility: in that, the extrapolation to detectable frequen-\ncies can be made far more naturally. To make the observations mentioned above,\nwe consider a radio array on the far side of the moon and consider the cases: 100\nand 1000 antennas. The far side of the moon is shielded from the interference\nfrom Earth and is thus preferable for conducting very low-frequency radio exper-\niments. Considerations for building telescopes on the moon are already under\nway and have been considered in the literature [22, 23, 24, 25, 26, 27, 28, 29].\nFurthermore, we consider the transition probabilities required for a detection\nwith the low-frequency Square Kilometre Array (SKA-LOW). Although the\n\frst attempts to search for these counterpart signals can be performed with the\nSKA-LOW, we \fnd that a detection of single merger events is probably impossi-\nble and, for an unresolved background of mergers, would require a considerably\n\fne tuned behaviour with regards to the extrapolation of the gravitational-wave\nspectrum from neutron stars merger. This is largely due to the fact that the\ntransition probabilities required for detection with the SKA-LOW are too high\nto be supported by any well known environments. Very large magnetic \felds, in\naddition to large intervening structures would be required. However, reasonable\ntransition probabilities result in a more achievable measurement using a lunar\narray. In this work, we demonstrate that, for an isotropic background formed\nby binary neutron star mergers, graviton-photon conversion emissions in the\nradio band are potentially detectable for the transition probabilities induced\nwithin the M31 environment provided exponential mode damping dominates\nonly for frequencies above 80 kHz, an order of magnitude above the highest\nfrequencies calculated in [30, 31, 32]. This suggests that detection of an unre-\nsolved background may be highly challenging but is not impossible. We note\nthat [16] discuss the idea of the observability of a single merging event as well,\nbut this is con\frmed here to be improbable. If the converted EM spectrum is\nnot detected by the hypothetical lunar array, and simulations do not indicate a\ncut-o\u000b before 80 kHz, we may also place a lower limit on the size of the scale M\nby considering the largest Mattainable when detection does not require signif-\nicant \fne-tuning. We also compare our results to Lorentz invariance violation\nconstraints and \fnd that we produce competitive limits with respect to the re-\nsults in [33]. These limits, however, are contingent on the spectral extrapolation\nmeeting the requirement of a minimum cut-o\u000b frequency value at 80 kHz and\na slope>\u00002. The 80 kHz cut-o\u000b receives some potential justi\fcation from\nthe fact that higher-order harmonics in binary mergers can result in power-law\nspectral tails and even, in some circumstances, MHz frequency waves [40, 41].\nA vital issue will be that of background radio emissions [34] and of the\nforeground emissions from the conversion environment itself. This issue, as\n4discussed in [35], in the context of reionisation science, will mean that it is vital\nto characterise the background/foreground emissions in detail in order to extract\nthe signal due to gravitational-wave conversion. One approach that we discuss\nis that of using a distant galaxy as a conversion environment, which allows the\nexpected signals to be only lightly impacted while minimising the issue of the\nforeground.\nThis paper is structured in the following manner: After the introduction,\nsection 2 discusses the model we use to formulate our study. Section 3 considers\nthe construction of a lunar array in an optimal setting. Section 4 presents our\nresults and discussion with the conclusion presented in section 5.\n2. Model\nOur model uses the spectrum for gravitational-waves resulting from a merg-\ning binary neutron star system found in [30, 31]. In this model the stars have\nequal masses of 1.5 M\fat a separation of 42.6 km. The resulting remnant is\na low-mass black hole of about 2.8 M\f. The total energy radiated away as\ngravitational-waves, in less than 3 ms, is \u0001 Egw\u00183\u00021051erg.\nNow let us suppose that part of the radiation emitted as gravitational-waves,\nis converted into electromagnetic radiation via the conversion of gravitons into\nphotons, as described below in Section 2.1. We denote this fraction by p=\nEEM=Etotal, called the conversion probability, or conversion fraction. Then\nEtotal=Egw+EEM= (1\u0000p)\u00001EgwwhereEgwis the energy that remains in the\nform of gravitational waves after conversion and Etotalis the initial gravitational\nwave energy of the event.\n2.1. The mechanism\nHere we summarize some of the details of the conversion mechanism de-\nscribed in [12], which illustrates the possibility for a low-mass (or zero-mass)\nparticle to be created from a photon (spin-1) passing through an external mag-\nnetic \feld, and vice versa. This formalism is applicable to the case of gravitons,\nwhich have spin-2. Conversion requires the presence of an external magnetic\n\feld supplying one virtual photon in order to satisfy symmetry constraints,\nthus conversion of real gravitons to real photons is one-to-one, as the second\nphoton in this interaction is virtual.\nIn [8] the authors demonstrate an equation of motion for propagating photon\nand graviton mixtures, when we choose our coordinate system such that the\npropagation direction aligns with the zaxis, of the form\n(!+@z) +M = 0; (1)\nwhere@z=@\n@z, = (hx;h+;Ax;Ay)Twithhx,h+,Ax, andAybeing graviton\nand photon polarisation states respectively. The mixing matrix Mis de\fned as\n5M=0\nBB@0 0\u0000iMx\ng\riMy\ng\r\n0 0 iMy\ng\riMx\ng\r\niMx\ng\r\u0000iMy\ng\rMxMCF\n\u0000iMy\ng\r\u0000iMx\ng\rM\u0003\nCFMy1\nCCA; (2)\nwhereMq\ng\r=gg\rkBq=(!+k) represents the photon-graviton coupling in the\ndirectionqwithgg\rbeing the coupling strength, Bqbeing the magnetic \feld\ncomponent in the qdirection,!being the total particle energy, and kbeing the\nmomentum. MCFcontains the Faraday and Cotton-Mouton e\u000bects, and Mq\nterms represent the `e\u000bective mass' picked up by the photon while propagating\nin a plasma. Note that we follow [8] in the de\fnition of all the matrix elements;\nso the reader is referred there for further details. It is possible to express the\ncoupling in terms of some energy scale Msuch thatgg\r=1\nM. Unless it is noted\nthat we are treating Mas a free parameter we will assume M=Mpl. Note\nthat the conversion of graviton to photon is one-to-one, as such they have equal\nenergies and frequencies.\nWe solve Eq. (1) numerically to obtain the state  after propagating a dis-\ntancedin the environment of the M31 galaxy. The state  will then be used\nto construct a probability p(\u0017) that a particle that starts as a graviton (with\nenergy corresponding to h\u0017) remains one after a distance d. Our initial state\nis taken to be composed entirely of gravitons so that  0=\u0000\n1=p\n2;1=p\n2;0;0\u0001\nwith the graviton terms chosen for simplicity. The solution method used will\nbe to divide the magnetic \feld up into domains with a length sampled from a\nKolmogorov turbulence distribution for coherence lengths between 50 and 100\npc [36]. Each domain is assigned a value Band plasma density neaccording\nto the models below (using the coordinates of the centre of the domain) and is\nrandomly allocated a magnetic \feld orientation. Each such allocation of \feld\norientations to each domain is termed a realisation, with the observable solu-\ntion taken to be the average over the ensemble of realisations. This approach is\nnecessary as the actual con\fguration of the magnetic \feld is unknown, and the\nrelative orientation in\ruences the mixing matrix in Eq. (2). The averaging over\nthe turbulent realisations will also remove any resonant spectral features.\nThe M31 environment is characterised by magnetic \feld and the gas distri-\nbutions for photon-graviton mixing, we will obtain these by following the model\nof [37]. This means that our magnetic \feld strength is B= 4:6\u00061:2\u0016G at\nr= 14 kpc from the centre of the galaxy, with a more general pro\fle following\nB(r) =4:6r1+ 64\nr1+r\u0016G; (3)\nwhereris the distance from the centre of M31 and r1= 200 kpc is taken to\nfollow the more conservative value from \ftting in [37]. This was found to be\nvalid when r\u001440 kpc in [37]. When extrapolating the magnetic \feld beyond\n40 kpc we will use the following conservative choice of pro\fle\nB(r) =4:6r1+ 64\nr1+ 40 kpcexp(\u0000(r\u000040 kpc)=(3:8rd))\u0016G; (4)\n6where the scale radius 3 :8rdis chosen motivated by arguments around the scale\nof spiral galaxy magnetic \felds in [38]. We take the gas density to be given by\nan exponential pro\fle\nne(r) =n0exp\u0012\n\u0000r\nrd\u0013\n; (5)\nwheren0= 0:06 cm\u00003is the central density [36], and rd\u00195 kpc is the disk\nscale radius \ftted by [37].\nWe \ft approximate power-law functions to the results of solving Eq. (1),\none is \ftted to p(\u0017) in the core region of M31 (when r\u001440 kpc as studied\nby [37]), the second is \ftted to an integration over an extended region of M31\nout tor= 200 kpc. The power-law takes the form p(\u0017) =a\u0000\u0017\n1 MHz\u0001band\nthe parameters are listed in Table 1. An example of the power-law \ft for\nthe extended case is displayed in Fig. 1, the deviation at low frequencies is\na result of more complex shapes for p(\u0017) in the outer regions of M31 where the\nelectron density drops faster than magnetic \feld strength, which can enhance\nconversion rates. In simple scenarios without multiple magnetic \feld domains\nor turbulence it is expected that, in the regime of weak mixing between graviton\nand photon and at low frequencies, the conversion probability scales as a steep\npower-law with index = \u00002 in a manner similar to the axion case [39]. However,\nthis mitigated by the radial variance of magnetic \felds and plasma densities\nconsidered here, as well as the turbulence and domain structured magnetic \felds.\nM31 region ab\nCore 1:2\u000210\u0000200.15\nExtended 9:3\u000210\u0000210.13\nTable 1: Fitting functions for conversion probability pbetween 1 kHz and 1 GHz.\n2.2. The gravitational wave spectrum from a neutron star merger\nThe neutron star merger gravitational wave spectrum found in [30, 31] is cal-\nculated by simulating the contribution of up to 7 modes and reaches a frequency\n\u0017f\u00197 kHz. We label this spectrum as Sgw(\u0017). To reach potentially detectable\nfrequencies we perform a power-law with exponential cut-o\u000b extrapolation for\nfrequencies \u0017 >\u0017f\nSex(\u0017) =A\u0017\u000bexp\u0012\n\u0000\u0017\n\u0017c\u0013\n; (6)\nso that the complete extrapolated spectrum is given by\nSgw;ex(\u0017) =(\nSgw(\u0017)\u0017\u0014\u0017f\nSex(\u0017)\u0017 >\u0017f; (7)\nwhere\u000bis the power law index and \u0017cis the cut-o\u000b frequency. The choice of\nusing a power law with a cut-o\u000b may be explained as follows: a cut-o\u000b is ap-\nplied because we know that the quasi-normal modes decay exponentially [40].\n7103\n102\n101\n100101102103\n (MHz)\n1020\np()\nPower-law fit\nExtended M31 averageFigure 1: Power-law \ft to the conversion probability pin the extended scenario from Table 1.\nFurthermore, we require a power law to account for the fact that higher order\nmodes may extend the spectrum seen in [30, 31] as is done in [19] in the presence\nof plasma and studied for MHz frequencies in [41]. Therefore, this extrapolation\nprovides a parameter space of \u000band\u0017cthat e\u000bectively considers a large vari-\nety of scenarios for the extension of the gravitational wave spectrum to higher\nfrequencies. The multi-order of magnitude extrapolation is justi\fed in that it\ndeals e\u000bectively with the cases of complete gravitational-wave mode decay (i.e.\nwhere\u0017c'\u0017f) as well as the potential contribution of additional modes where\n\u0017c>\u0017f. It is well known that higher order modes contribute gravitational-wave\npower at higher frequencies but with shorter lifetimes [40]. How much power\nis contributed by these modes is determined by the magnitude of their excita-\ntion, which is not yet well-understood for complex environments like neutron\nstars where many e\u000bects may amplify higher order mode magnitudes [40, 41].\nHowever, it is possible for even 1 MHz waves to be generated in eccentric sys-\ntems as explored in [41], this at least indicates that high-frequency waves are\npossible when higher harmonics are excited. Note that we do not require the\nsame level of excitation as explored in [41] as we are not considering direct\ngravitational-wave detection at high frequencies, all we require is the potential\nfor relatively weak excitation of higher order modes (as embodied by the power-\nlaw extrapolation). Additional justi\fcation for the power-law extrapolation can\nfound in the spectral modelling performed in [32], where the authors determine\nthe gravitational wave strain spectrum, from binary neutron star merger, out\nto 8 kHz. The aforementioned spectrum (\fgure 8 of [32]) shows a power-law\ntype decline at the highest displayed frequencies and no evidence of exponential\ndecay below 8 kHz. Although a strain spectrum is not an identical quantity to\nthe energy spectrum of [30, 31] it is su\u000ecient to establish a lack of exponen-\ntial decay in the gravitational-wave amplitude at frequencies studied so far in\n8the literature. In this regard our extrapolation is reasonable as it assumes the\nsmallest possible \u0017cis 8 kHz, individual modes decay exponentially [40] and it\nmay be possible that higher order mode contributions cut o\u000b via an exponential\ndecay. We use a power-law before this cut-o\u000b point as we note an overall trend\nof power-law like decrease in the amplitude of the gravitational-wave spectrum\nwith frequency in [31] and the tendency of gravitational wave amplitudes from\nmergers to have power-law tails after the ring-down [40]. By \ftting to peaks of\nspectrum from [31] we determine that the amplitude of the spectrum declines\nas a broken power-law with frequency. At frequencies below 5 kHz the slope is\nbetween\u00003:5 and\u00003 while it is &\u00001:8 above 5 kHz. This means that the vital\nrange of extrapolation for the power-law spectral index can be taken as being\nbetween\u00002 and\u00001:5.\nWe normalise Sgw;ex(\u0017) as follows: we require that Egwis the gravitational\nwave energy emitted from the source at a luminosity distance dLover a time\n\u0001t\u00192 ms and use this to determine the graviton \rux at Earth.\nThis power law is applied to the tail of the spectrum using the matching\ncondition: Sgw(\u0017f) =Sex(\u0017f). Furthermore, p(\u0017)S(\u0017) will then represent the\nspectrum obtained from the conversion of gravitational-waves into photons. The\nvalues\u000band\u0017cprovide us with the parameter space which allows us to deter-\nmine the detectability of the electromagnetic counterparts with a radio telescope\non the moon or with SKA-LOW. Note that this is a phenomenological extrapo-\nlation because it is di\u000ecult and beyond the scope of this work to calculate these\nspectra directly at frequencies high enough for radio-band detection. In order\nthat the exact details of the extrapolation don't bias the results, we will draw\nconclusions based on how insensitive the detection potential is to the choice of\nextrapolation parameters (the di\u000eculties of even producing Sgw(\u0017) are strongly\nnoted in [30, 31]).\n2.3. Isotropic gravitational wave background\nWe construct a gravitational wave background due to neutron star mergers\nas follows:\n\bgw(\u0017) =p(\u0017)\nobsZ6\n0:06dzdV\ndz\u001c(z)Rmerger (z)Sgw;ex(\u0017(1 +z)); (8)\nwhere \n obsis the solid angle being observed (in this case the sky-area of M31),\ndV\ndzis the co-moving volume element, \u001c(z) is the look-back time, and Rmerger (z)\nis the rate of neutron star mergers per unit volume as found in [21], the integral\nlimits \ft the plots presented in [21]. We note that black hole merger rates from\n[21] were found by the authors to be consistent with those of LIGO/VIRGO\nfor the local universe, this has remained the case with further analysis and also\ndemonstrates agreement for binary neutron stars [42, 43]. It is vital to note that\nuncertainties in \b gw(\u0017) will be in direct proportion with those of Rmerger (z=\n0) (as we could express R(z) =R0r(z)) and that these are substantial, as\ndemonstrated by the fact that LIGO/VIRGO \fnds R02[250;2810] Gpc\u00003\nyr\u00001[43] while [21] \fnd R02[20;600] Gpc\u00003yr\u00001. This implies, of course,\n9that the \rux from \b gw(\u0017) is uncertain by roughly a factor of 10. Notably\nthe overlap between LIGO/VIRGO and [21] suggest the higher end of the range\nR02[20;600] Gpc\u00003yr\u00001is more probable, which is promising for the detection\nprospects presented here. The cosmological quantities are calculated assuming\nthe Planck results from [44].\nWe could, in principle, consider the isotropic background of gravitational\nwaves passing through a succession of environments on the line of sight, thus\nleading to greater conversion fractions. However, to make a conservative esti-\nmate we consider only a single environment with some information as to its gas\nand magnetic \feld distributions. This is important as the evolution of these\nquantities with the host-environment redshift is largely uncertain.\nNote that, in our construction of \b gw, we have assumed that all the binary\nneutron star pairs and mergers are similar to that studied in [30, 31]. However,\nthis assumption shouldn't be overly problematic as [21] indicate a strong con-\ncentration of binary neutron stars towards lower masses, like those studied in\n[30, 31].\nIn the M31 core region \n obs= 4\u0019\u00026:73\u000210\u00004sr and in the extended case\n\nobs= 4\u0019\u00021:55\u000210\u00002sr.\n3. Lunar telescopes: Are they just science \fction?\nPresently, several large Earth-based radio telescopes have been built which\noperate at low-frequencies [45]. These include the Low Frequency Array (LO-\nFAR) [46] situated in Europe, the Long Wavelength Array (LWA) [47] situated\nin New Mexico and the Murchison Wide-Field Array (MWA) [48] situated in\nWestern Australia. All of these only work above 10 MHz due to the opaqueness\nof Earth's Ionosphere [45]. Furthermore, below frequences of 30 MHz, radio\nobservatories are severely limited due to Radio Frequency Interference (RFI)\ncaused by GPS, cellphones, radio broadcasts, etc.\nThis means that there is a whole range of frequencies below 30 MHz in\nthe electromagnetic spectrum to which Earth-based experiments are blind. To\nexplore this Ultra Long Wavelength (ULW) band, a space-based detector is re-\nquired, placed above the Earth's Ionosphere. In a report by the European Space\nAgency [49], they discuss the far side of the moon as a viable option for placing\na very low frequency array. Published in 1997, this is the most comprehensive\nstudy on ULW radio observations to date [45]. This idea is backed by the space-\nbased instrument called the Radio Astronomy Explorer (RAE), launched in the\n1970's. It discovered that the moon can act as a good shield against the RFI\nfrom Earth, thus providing an ideal radio-frequency observation environment.\nThe development of space-based detectors is still largely in the conceptual\nphase. Here we list some of the many concepts that have been presented for ex-\nploring ULW. These concepts include the single satellite element projects such as\nthe Dark Ages Explorer (DARE) [22] and the Lunar Radio Astronomy Explorer\n(LRX) [23]. Furthermore, there are many small satellites forming multi-element\ninterferometers: Formation-\rying sub-Ionospheric Radio astronomy science and\n10Technology (FIRST) [24], the Space-based Ultra-long wavelength Radio Ob-\nservatory (SURO-LC) [25] and the Orbiting Low Frequency Array (OLFAR)\n[26, 27]. Additionally, we mention Discovering the Sky at the Longest wave-\nlengths (DSL) [28], which would conduct single antenna measurements as well\nas form part of a ULW radio interferometer [29]. Finally, studies using dipole\nantennas have been considered [24, 25, 22, 26, 23]. Whilst there are currently\nno telescopes on the moon, it is evident there is much activity in this \feld with\nmajor international institutions and agencies participating and collaborating to\nput a detector on the moon or in orbit. We consider here some optimal speci\fca-\ntions for a lunar array, that may in principle, detect graviton-photon conversion\ncounterparts to gravitational-waves.\n3.1. Building a telescope on the moon\nIn this section we provide some details for designing an optimal lunar radio\ntelescope. As a test case, we consider two set-ups: \frstly, a con\fguration using\nN= 103log-periodic dual-polarized dipole antennas with bandwidth \u0001 \u0017= 30\nMHz, and secondly, the case with N= 100 antennas and the same bandwidth\n\u0001\u0017. The bandwidth is motivated by design considerations discussed in [35, 45].\nWe calculate the minimum observable \rux for the array as follows2\nSmin=2kBTsky\nNp\n\u0001\u0017\u001cAe; (9)\nwhereAe=\u00152=4\u0019is the e\u000bective collecting area and \u0015is the incoming wave-\nlength. Here kBis the Boltzmann constant, \u001cis the integration time. The sky\ntemperature is given by [50]\nTsky=8\n><\n>:16:3\u0002106K\u0010\n\u0017\n2MHz\u0011\u00002:53\n; \u0017 > 2MHz\n16:3\u0002106K\u0010\n\u0017\n2MHz\u0011\u00000:3\n; \u0017 < 2MHz:(10)\nWe can see, in \fgure 2, that a lunar array would less sensitive than the SKA\nat 50 MHz achieving around 10 \u0016Jy compared to\u00181 for the SKA (note we\nhave extrapolated the lunar array operating band for purposes of comparison\nwith SKA). The curve, for a lunar array, has an expected break or kink in its\nuniformity due to the behaviour of equation (10). In the aforementioned \fgure,\nwe have computed both telescope sensitivities for a total integration time of 100\nhours, to make comparison with standard \fgures quoted for the SKA [51]. For\na single merger event observation we would be limited to around a milli-second\nof integration time only, this being the event duration. For the lunar telescope\nwe consider two cases, one for N= 100 dipoles and the other for N= 1000\ndipoles. The SKA-LOW itself will possess N\u0018105dipoles. The orange dashed\n2http://sci.esa.int/science-e/www/object/doc.cfm?fobjectid=53829\n11line considers the conservative case of only N= 100 dipole antennas, and as\ncan seen we achieve better sensitivity in the case of N= 1000 antennas given\nby the solid blue line.\n100101102\n Frequency (MHz)1023\n1022\n1021\n1020\n1019\nSensitivity (erg cm2 s1)\n1 Jy\nN=1000\nN=100\nSKA\nFigure 2: Minimum \rux of photons that may be observed by a telescope on the moon within\n100 hours when there are N= 1\u0002102antennas (orange/dashed), and for N= 1\u0002103antennas\n(solid/blue) and \fnally we plot the sensitivity of the SKA-LOW (green/dashed-dotted) which\nhasN\u00181\u0002105antennas. The dotted line displays a 1 \u0016Jy threshold. We extrapolate the\nlunar array sensitivity to frequencies above 30 MHz for comparison purposes.\n3.2. Detectability of produced photons\nIn order for these counterparts to be detected by an observer on Earth or\non the moon, the frequency of these photons must be larger than the plasma\nfrequency,!p, of the environments through which they travel. This is to ensure\nthat the photons are not absorbed by the intervening medium. For a galaxy like\nM31 (ne= 0:06\u000210\u00003cm\u00003) we have!p\u00181:54\u000210\u00002MHz. For the same\nreasons, the photons must have frequencies higher than the plasma frequency of\nthe atmosphere through which they travel to arrive at the detector. Whilst the\nmoon has no atmosphere it does possess and Ionosphere with ne\u0018102cm\u00003\n[52] and therefore has !p\u00180:62 MHz and the Earth (n e\u0018105cm\u00003) [53]3has\n!p\u001819:9 MHz.\nThe sensitivity pro\fle for SKA-LOW may be found in [51], whilst for the\nmoon radio telescope we use the setup established in this section. The plasma\nfrequency for the moon, as calculated above, is roughly 0 :62 MHz. Therefore,\nthe telescope operating frequency should start just above this value, and we\nchose to start at 1 MHz. Hence, the operating frequency range is 1 \u000030 MHz.\nNote that the photons we are considering are those that dostrike the anten-\nnas after having passed through the moon's Ionosphere. Therefore, they have\nalso survived travel through the intervening space between the moon and the\ngravitational-wave event.\n3http://solar-center.stanford.edu/SID/science/Ionosphere.pdf\n124. Results and discussion\nWe provide the parameter space which indicates values of \u000band\u0017cfor which\ndetections may be possible within the bandwidths of SKA-LOW or a lunar array\nas described in section 3. That is, we study how the \rux varies compared to the\nsensitivity of the detector. When the \rux of the incoming photon is higher than\nthe sensitivity of the detector, we have made a detection, the shaded regions\nof the parameter space are those in which detections are possible. SKA-LOW\nis represented by blue/dark shading and the lunar array by the green/light\nshading. Plots showing parameter space detection coverage were computed at a\n5\u001bcon\fdence level and we will assume the energy scale of quantum gravitational\ne\u000bects is that of the Planck mass, making these results potentially conservative.\nWith regards to our choice of the parameter space, we require \u0017cto be larger\nthan the \fnal point in the spectrum in [30, 31], but smaller than 100 MHz, above\nwhich\u0017cbecomes irrelevant for SKA-LOW. Ultra-steep radio-band power-law\nindexes have been shown [54] to extend up until -2. In keeping with this, but\nallowing some lee-way for more extreme spectral fall-o\u000b, we chose our parameter\nspace to lie within \u000b2[\u00003;0].\nAs we make use of a phenomenological extrapolation of the gravitational-\nwave spectrum from [30, 31], we will discuss the implications of the results\nqualitatively. We are interested in the breadth of the detectable parameter\nspace, rather than the speci\fc model values for which signals will be detectable.\nThis is to ensure that our conclusions are not strongly dependent on the choice\nof extrapolation. So we will examine how strongly detection of a signal depends\non the choice of model parameters. A weak dependence will be taken to in-\ndicate that the signal is highly detectable, as it can be naturally extrapolated\nto detectable frequencies with little or no \fne-tuning. Whereas, a requirement\nof very particular values for \u000band\u0017cwill imply that detection is improbable,\nas signi\fcant \fne tuning of the spectral extrapolation is needed to reach de-\ntectable frequencies. We make a reasonable apriori assumption that a detection\nrequiring \fne-tuning is less plausible the more \fne-tuning that is required.\nFigure 3 displays the case of a single neutron star merger event occurring\nwithin the M31 galaxy. As is evident, the observation is only possible for more\nthan 100 dipole elements in the lunar array and the parameter space coverage\nis weak with 1000 dipoles and struggles to reach the cut-o\u000b frequency of 80\nkHz even with 105antennae. This suggests the detectability of the event is\nimprobable, as it requires a very large array even if the gravitational wave\nspectrum only begins exponential decay more than an order of magnitude above\nthe end of the spectrum from [30, 31, 32]. In addition to this, the power-law\nextension cannot have \u000b <\u00000:75 for 1000 antennae. In an attempt to test a\nvery close event we considered one taking place in the Milky-Way galactic centre\nassuming a magnetic \feld with an exponential pro\fle and central strength of 60\n\u0016G. This resulted in p\u001910\u000032, meaning that this cannot compensate for the\n\rux boost of\u0018104due to the smaller distance. We conclude that, barring some\nunconsidered nearby conversion environment, this makes it unlikely that single\nbinary neutron star merger events can be observable via photon-graviton mixing\n13without a very large array indeed. This result motivates our consideration of\nan isotropic background formed from a merging population of binary neutron\nstars instead.\n2.00\n 1.75\n 1.50\n 1.25\n 1.00\n 0.75\n 0.50\n 0.25\n 0.00\n102\n101\n100101102c (MHz)\nSKA\nMoon\n2.00\n 1.75\n 1.50\n 1.25\n 1.00\n 0.75\n 0.50\n 0.25\n 0.00\n102\n101\n100101102c (MHz)\nSKA\nMoon\n2.00\n 1.75\n 1.50\n 1.25\n 1.00\n 0.75\n 0.50\n 0.25\n 0.00\n102\n101\n100101102c (MHz)\nSKA\nMoon\nFigure 3: Detectability parameter space for the conversion of gravitons into photons in the\nM31 extended scenario with a single merger event. SKA-LOW is represented by the blue/dark\nregion and the lunar array by the green/light region. Top: 105lunar antennae. Bottom left:\n1000 lunar array antennae case. Bottom right: 100 antennae.\nFigure 4 displays the detectable parameter spaces using the M31 core prop-\nagation scenario for an isotropic gravitational wave background due to binary\nneutron star mergers. It demonstrates the e\u000bect of both the integration time, at\n1 s (left column) and 100 (right column), as well as the number of lunar array\nantennae, 1000 (top row) and 100 (bottom row). These plots demonstrate the\nimportance of the array size, as steeper power-law extrapolations are not so\neasily observable with the 100 element array and the extended integration time\nis important but only allows for approaching an order of magnitude above the\nhighest frequency studied in [32]. The coverage is not complete, even within an\norder of magnitude of [32], with cases where \u000b<\u00002 being unobservable if the\ncut-o\u000b starts at 80 kHz. The SKA, on the other hand, shows a somewhat limited\ndetection potential, requiring the cut-o\u000b to be at least two orders of magnitude\nabove the end of the spectra from [30, 31]. Although increased integration time\nis a considerable advantage over the single event results the real bene\ft comes\nfrom the combination with the \rux increase from the large number of unresolved\nevents.\n143.0\n 2.5\n 2.0\n 1.5\n 1.0\n 0.5\n 0.0\n102\n101\n100101102c/(1 MHz)\nSKA\nMoon\n3.0\n 2.5\n 2.0\n 1.5\n 1.0\n 0.5\n 0.0\n102\n101\n100101102c/(1 MHz)\nSKA\nMoon\n3.0\n 2.5\n 2.0\n 1.5\n 1.0\n 0.5\n 0.0\n102\n101\n100101102c/(1 MHz)\nSKA\nMoon\n3.0\n 2.5\n 2.0\n 1.5\n 1.0\n 0.5\n 0.0\n102\n101\n100101102c/(1 MHz)\nSKA\nMoonFigure 4: Detectability parameter space for the conversion of gravitons into photons in the\nM31 core scenario with an unresolved merger background. SKA-LOW is represented by the\nblue/dark region and the lunar array by the green/light region. Left: integration time of 1 s.\nRight: 100 hours of integration time. The upper row displays the 1000 lunar array antennae\ncase and the lower row displays that with 100 antennae.\nFigure 5 displays a conservative choice for the extended M31 propagation\nscenario again for an isotropic gravitational wave background due to binary\nneutron star mergers. For cases with 100 hours integration and/or 1000 dipoles\nthe results are very marginally superior to those from Fig. 4 as p(\u0017) is simi-\nlar but the observed sky area is an order of magnitude larger. However, for\nsmaller arrays and shorter integration times the detection prospects are im-\nproved substantially. The coverage of the parameter space is again incomplete,\nthe best achievable cut-o\u000b frequency required for observation being an order\nof magnitude above the highest frequency explored in [32]. This suggests that\nthe controlling variable over the detectability is the cut-o\u000b frequency as it most\nstrong dictates the parameter space coverage. Thus, the observability of the un-\nresolved spectrum hinges how far the power-law trend at higher frequencies [32]\nin the merger spectrum continues. This means that detection is possible but\nthere a large area of the extrapolation parameter space that could result in the\nsignal being undetectable. We will supplement this by exploring how close to the\nPlanck scale we can place stringent exclusion limits from the non-observation of\nsuch a GW-induced signal if the cut-o\u000b frequency can indeed be extended out\nto 80 kHz.\n153.0\n 2.5\n 2.0\n 1.5\n 1.0\n 0.5\n 0.0\n102\n101\n100101102c/(1 MHz)\nSKA\nMoon\n3.0\n 2.5\n 2.0\n 1.5\n 1.0\n 0.5\n 0.0\n102\n101\n100101102c/(1 MHz)\nSKA\nMoon\n3.0\n 2.5\n 2.0\n 1.5\n 1.0\n 0.5\n 0.0\n102\n101\n100101102c/(1 MHz)\nSKA\nMoon\n3.0\n 2.5\n 2.0\n 1.5\n 1.0\n 0.5\n 0.0\n102\n101\n100101102c/(1 MHz)\nSKA\nMoonFigure 5: Detectability parameter space for the conversion of gravitons into photons in the\nM31 extended scenario with an unresolved merger background. SKA-LOW is represented by\nthe blue/dark region and the lunar array by the green/light region. Left: integration time\nof 1 s. Right: 100 hours of integration time. The upper row displays the 1000 lunar array\nantennae case and the lower row displays that with 100 antennae.\nFigure 6 conveys the most optimistic detection case: an array for 105el-\nements (similar to that needed for 21 cm power spectrum observations [35])\nobserving for 1000 hours. In this case we are no longer dependent on the power-\nlaw spectral index to reach \u0017c.80 kHz.\nFigure 7 shows the lower limit that can be placed on the scale Mvia non-\nobservation with the lunar radio array for an isotropic gravitational wave back-\nground due to binary neutron star mergers. We derive the limits by \fnding\nthe largestMobservable at 2 \u001bcon\fdence interval in the most conservative, yet\ndetectable, scenario where \u0017c\u001980 kHz. The largest observable value will be\na lower limit in the case of non-observation. An interesting comparison may\nbe made with the Lorentz invariance violation limits for gamma-ray bursts.\nThe Lorentz invariance violations can be parameterised by En, whereEis the\ngamma-ray photon energy and nis an unknown parameter. When n= 1, it\nis shown that a maximum energy-scale of M1= 9:23\u00021019GeV [33] can be\nprobed, this can be achieved by the lunar array for \u000b\u0015\u00003 within 100 hours\nof integration time. In the case of the extended propagation our results exceed\nthe Lorentz invariance case for all \u000b\u0015\u00002 with 10 hours of integration time.\nForn= 2, the limits from Lorentz invariance violation are much weaker: that\n163.0\n 2.5\n 2.0\n 1.5\n 1.0\n 0.5\n 0.0\n102\n101\n100101102c/(1 MHz)\nSKA\nMoonFigure 6: Detectability parameter space for the conversion of gravitons into photons in the\nM31 extended scenario with an unresolved merger background. SKA-LOW is represented by\nthe blue/dark region and the lunar array by the green/light region. The integration time is\n1000 hours and the array contains 105dipoles.\nisM2= 1:3\u00021011GeV which is bettered by the lunar array within 1 second.\nTherefore, in both cases we are able to produce competitive non-observational\nlimits on the energy-scale of quantum gravitational e\u000bects. We also note that\nLorentz invariance violation limits strongly depend on the model-dependent\nparameter n, whereas our results don't have any such unknown dependence.\nAdditionally, even when \u000b=\u00003 and with only 100 dipoles, we can reach within\n2 orders of magnitude of Mplwith 1 second of integration time on the lunar\narray. The only caveat being that these limits require that \u0017c\u001580 kHz.\nOne point worth noting is that the need to align the line of sight with a\nconversion environment, like M31, implies that there will be strong foregrounds\nfrom this galaxy itself. These are not immediately quanti\fable at relevant fre-\nquencies, indeed the lowest frequency M31 radio \rux measurement in the NASA\nExtra-galactic Database [55] is at 151 MHz and produces 0 :75 Jy whereas the\nconversion signals require sensitivity of O(10\u0016Jy) (note that it is necessary to\ncompare to di\u000buse \ruxes only but the aforementioned is a point source observa-\ntion). We note that more extended surveys, as reported in [56, 57, 58] indicate\nfar higher \ruxes at low frequencies in M31 (around 200 Jy), however, to make\nthese comparable to our background all point sources would \frst need to be\nsubtracted from consideration. We therefore benchmark against the dominant\npoint source emission, as it is available, and it is unlikely the actual di\u000buse com-\nponent will be as bright, as even the di\u000buse emission in the Coma galaxy cluster\nradio halo falls below 100 Jy at frequencies below 100 MHz [59]. In summary,\nit is very possible that the signal we predict from graviton-photon conversion\n17103\n102\n101\n100101102\nintegration time [hr]102\n101\n100M/Mpl\nN=1000\nN=100\n103\n102\n101\n100101102\nintegration time [hr]101\n100M/Mpl\nN=1000\nN=100Figure 7: Maximum detectable energy-scale M, as a fraction of the Planck scale Mpl, versus\ntelescope integration time when \u0017c= 80 kHz for an unresolved merger background. This\nplot illustrates the lower bounds on Musing a lunar array with 1000 dipoles (solid lines) or\n100 dipoles (dotted lines) and apply for \u000b\u0015 \u00003. Left: we assume the M31 core propagation\nscenario. Right: M31 extended case\nwould be up to 5 orders of magnitude below the foreground. In addition, we\nnote that there may be other isotropic radio background contributions at low\nfrequencies, as argued in [34] but further understanding of these backgrounds\nwill make their subtraction more feasible. As a point of mitigation we note that\n[35] consider a similar foreground problem from the perspective of observing\nepoch of reionisation signals and conclude it to be necessary for a lunar array\nto consist of 103to 105elements to observe a non-isotropic signal like the 21 cm\npower-spectrum. The lower limit of this array size requirement \fts conveniently\nwith the number of antennae needed to detect graviton-photon conversion, with\na minimum cut-o\u000b frequency of 80 kHz. In addition, synchrotron spectra that\ndominate low-frequency galaxy emissions [60] will have a positive power-law\nexponent with frequency. Whereas, since it is cutting o\u000b, our graviton-photon\nsignal will have a negative slope. This potentially provides a valuable character-\nistic that can be leveraged to extract the signal despite large foregrounds. This\nmitigating factor might be cancelled out by free-free absorption modifying the\nspectral character of the background when viewed through a galactic conversion\nenvironment [60]. Additionally, we note that extraction of the signal in the face\nof the foregrounds and backgrounds remains the largest uncertainty here, as\ndiscussed in [35]. Thus, this would require precise calibration of the foreground\nand background \ruxes in order to extract the desired signal. We stress, however,\nthat for the foreground and background \ruxes any solution to similar challenges\nin global 21 cm observations can be leveraged. This can be illustrated by the\nextensive work that has gone into \fnding methods of extracting the 21 cm global\nsignal in the face of similarly daunting foregrounds [61, 62, 63, 64, 65]. These\nworks have illustrated that robust techniques exist by which low amplitude sig-\nnals can be extracted in the face of foregrounds several orders of magnitude\nbrighter and should be applicable beyond just the global 21 cm case [64]. In\nterms of backgrounds we note that it may be possible to mitigate these via di\u000ber-\n18ential observations, as the conversion signal will only be visible when looking at\nthe conversion environment target, whereas other isotropic radio backgrounds\nwill be visible o\u000b target as well. This is somewhat similar to the techniques\nused to measure the Sunyaev-Zel'dovich e\u000bect which is typically several orders\nof magnitude weaker than the cosmic microwave background that it perturbs.\nHowever, the e\u000bect on potential graviton-photon conversion cannot be estimated\nwithout precise knowledge of the conversion environment di\u000buse foreground and\ninstrumental systematics [63].\nOne possible approach to limit the foreground impact is the use of a distant\ngalaxy as a conversion environment. This is because the background gravita-\ntional wave \rux will be less diminished by the adjustment of the lower limit of\nthe integral in Eq. (8) than the conversion environment foreground \rux will be\nwhen moved out to a larger redshift. To illustrate this we present \fg. 8 which\ndisplays how the detectability varies as we adjust the conversion environment\nredshift. For simplicity we assume that it is possible to \fnd an M31-like galaxy\nat the given redshift. The results suggest that the minimum cut-o\u000b frequency\nrequired for detection is mildly sensitive to the conversion environment redshift.\nHowever, the \rux from the conversion environment galaxy will be reduced by a\nfactor of 107from that of M31. This approach could be a promising one but\nrequires the knowledge of the presence of a distant galaxy as well some con\f-\ndence in the assertion that the conversion fraction would be similar to that of\nwell-studied galaxies in the more local universe.\n5. Conclusion\nIn the presented work, we have demonstrated that it is possible to detect\nlow-frequency radio counterparts to gravitational-waves using a lunar dipole an-\ntenna array. While this was far more challenging for a single neutron star merger\nevent in the local universe (without a lunar array with at least 105elements).\nAs long as exponential mode damping does not dominate for frequencies below\n80 kHz (an order of magnitude above existing spectra [32] which show no expo-\nnential damping below 8 kHz), we have shown that the detection prospects for\nan unresolved background from binary neutron star mergers are largely robust\nto considerations of array size, power-law slopes >\u00002, and integration time.\nHowever, the best prospects can be obtained for 1000 hours of integration time\nand 105antennae (as recommended for reionisation science in [35]). We have\nalso demonstrated that the galaxy M31 provides a conversion environment in the\nlocal universe that allows for some prospect of detection, albeit quite unlikely,\nby numerically solving the equations of motion for photon-graviton mixing [8]\nand using magnetic \feld and plasma density information drawn from the liter-\nature [36, 38, 37]. Our modelling of the population of binary neutron stars was\ntaken from [21] and allowed us to compute the expected background signal with\nthe Planck 2015 cosmological parameters [44]. Crucially, our results reveal that\nthe detectability of a graviton-photon background from neutron star mergers\ndepends upon whether the power-law behaviour of the gravitational wave spec-\ntra [30, 31, 32] extends until 80 kHz with a power-law slope &\u00003 (\u00002 for 103\n193.0\n 2.5\n 2.0\n 1.5\n 1.0\n 0.5\n 0.0\n102\n101\n100101102c/(1 MHz)\nSKA\nMoon\n3.0\n 2.5\n 2.0\n 1.5\n 1.0\n 0.5\n 0.0\n102\n101\n100101102c/(1 MHz)\nSKA\nMoon\n3.0\n 2.5\n 2.0\n 1.5\n 1.0\n 0.5\n 0.0\n102\n101\n100101102c/(1 MHz)\nSKA\nMoon\n3.0\n 2.5\n 2.0\n 1.5\n 1.0\n 0.5\n 0.0\n102\n101\n100101102c/(1 MHz)\nSKA\nMoonFigure 8: Detectability parameter space for the conversion of gravitons, from an unresolved\nbackground, into photons as it varies with conversion environment redshift. All plots assume\n100 hours of integration time and 1000 antennae in the lunar array. The left column assumes\na conversion fraction like the extended M31 scenario whereas the core case is the right-hand\ncolumn. Top row: z= 0:5. Bottom row: z= 1:0.\nantennae and 100 hours observation). If exponential damping dominates below\nthis threshold, or the slope is steeper, then even a lunar array of 105elements\nwill not detect the resultant background. We note that the 80 kHz is an order\nof magnitude above the end of the spectrum modelled in [32, 30, 31] and this\nwould therefore seem to make detection highly improbable. However, it is pos-\nsible to obtain power-law extensions to gravitational wave spectra from mergers\nfrom higher-order harmonic contributions [40] and binary mergers can generate\nsignals out to 1 MHz in frequency with the aid of higher-order harmonics [41].\nAn important caveat is the issue of backgrounds and foregrounds. Both\nof which are expected to be an obstacle due the possibility of strong radio\nbackgrounds [34] and galaxy foregrounds from the conversion environment it-\nself. Thus, as in the case of epoch of reionisation science discussed in [35],\nthese provide the most signi\fcant uncertainties and the success of observa-\ntions discussed here will depend strongly upon characterisation of these back-\ngrounds/foregrounds. We have shown, however, that it may be possible to\nmitigate the foreground issue with the use of distant galaxies as conversion\nenvironments. We note that the background and foreground issues should be\naddressable following methods being devised to extract global 21 cm signals [64]\n20combined with potential di\u000berential observation techniques. A second caveat is\nthe uncertainty surrounding the neutron star merger rate, with [21] placing the\nrate in the local universe in the range [20 ;600] Gpc\u00003yr\u00001thus proving an order\nof magnitude uncertainty in the \rux from graviton-photon conversion of the un-\nresolved merger background. However, the overlap with the local universe range\nfrom LIGO/VIRGO [42, 43] of [250 ;2810] Gpc\u00003yr\u00001suggests that reducing\nuncertainties could improve the detection prospects presented here.\nWe note that several works (such as [8, 16]) have examined the question\nof whether photon-graviton conversion could be used to detect such merger\nevents. However, our work has presented novel results by both considering an\nunresolved gravitational wave background rather than just single events as well\nas using higher frequency numerical calculations from [30, 31, 32] and consid-\nering phenomenological extrapolations thereof. Crucially, we do not base our\nconclusions on the details of the extrapolation but rather on the relative in-\ndependence of the detectability on said details. In particular, our lower limits\non the energy scale of interaction assume both a power-law fall-o\u000b and an 80\nkHz exponential cut-o\u000b. It is worth noting that this requires the power-law\nextrapolation to hold for an order of magnitude beyond current knowledge of\nthe merger physics. Our use of a parameter space of potential extrapolations\nallows us to determine the required behaviour in order for a converted signal to\nbecome observable. Whether the requirements found for observability might be\nrealised in nature can, of course, be tested by future numerical simulations that\nextend the frequency range of gravitational wave spectrum.\nVital to this endeavour would be the prospect of placing a telescope on the\nlunar surface. Landing an array on the moon or placing one is orbit is promising,\nas can be see by the continuing activity in this \feld of radio astronomy [24,\n25, 22, 26, 23, 28]. The results presented here demonstrate that such a lunar\ninstrument may have the potential to probe the realm of quantum gravity at\nradio frequencies, using this mechanism of mixing between photons and low-\nmass bosons. It may even be possible to rule out or con\frm the existence of\nthe graviton. In particular, a non-observation of a neutron star merger induced\nradio background will place lower limits on the energy scale of putative quantum\ngravitational e\u000bects, subject to merger rate uncertainties. These lower limits\nare shown here to be competitive with Lorentz invariance violation experiments,\nas well as easily reaching within one or two orders of magnitude of the Planck\nscale. All of this, however, is contingent upon the lack of exponential mode\ndamping prior to 80 kHz and the ability to extract the signal in the presence of\nhighly dominant backgrounds and foregrounds.\n6. Acknowledgements\nJT and GB honour the memory of Prof. Sergio Colafrancesco, who con-\ntributed to this paper, and helped shape our academic lives, before his passing\nin September 2018. We thank several referees for their comments and criticism\nthat have profoundly shaped this work over a long period of time. We also thank\nDr Dmitry Prokhorov for his invaluable comments. This work is supported\n21by the South African Research Chairs Initiative of the Department of Science\nand Technology and National Research Foundation of South Africa (Grant No\n77948). J.T. acknowledges support from the DST/NRF SKA post-graduate\nbursary initiative. G.B acknowledges support from a National Research Foun-\ndation of South Africa Thuthuka grant no. 117969. This research has made use\nof the NASA/IPAC Extragalactic Database (NED), which is operated by the\nJet Propulsion Laboratory, California Institute of Technology, under contract\nwith the National Aeronautics and Space Administration. This work also made\nuse of the WebPlotDigitizer4.\nReferences\n[1] J. D. Schnittman, Coordinated Observations with Pulsar Timing Arrays\nand ISS-Lobster (2014). arXiv:1411.3994 .\n[2] R. 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Tauscher, Robust extraction of the cosmological global 21-cm signal\nfrom foreground and instrumental systematic e\u000bects, Bulletin of the AAS\n52 (3), https://baas.aas.org/pub/aas236-321p02-tauscher (6 2020).\nURL https://baas.aas.org/pub/aas236-321p02-tauscher\n28"    },    {        "title": "1004.4912v1.Radiation_Damping_in_a_Non_Abelian_Strongly_Coupled_Gauge_Theory.pdf",        "content": "arXiv:1004.4912v1  [hep-th]  27 Apr 2010Radiation Damping in a Non-Abelian\nStrongly-Coupled Gauge Theory\nMariano Chernicoff†, J. Antonio Garc´ ıa⋆and Alberto G¨ uijosa⋆\n†Departament de F´ ısica Fonamental, Universitat de Barcelo na, Marti i Franqu` es 1, E-08028\nBarcelona, Spain, Email: mchernicoff@ub.edu\n⋆Departamento de F´ ısica de Altas Energ´ ıas, Instituto de Ci encias Nucleares\nUniversidad Nacional Aut´ onoma de M´ exico, Apdo. Postal 70 -543, M´ exico D.F. 04510, M´ exico\nEmail:garcia, alberto@nucleares.unam.mx\nAbstract. We study a ‘dressed’ or ‘composite’ quark in strongly-coupl edN= 4 super-Yang-\nMills (SYM), making use of the AdS/CFT correspondence. We sh ow that the standard string\ndynamicsnicelycapturesthephysicsofthequarkanditssur roundingquantumnon-Abelianfield\nconfiguration, making it possible to derive a relativistic e quation of motion that incorporates\nthe effects of radiation damping. From this equation one can d educe a non-standard dispersion\nrelation for the composite quark, as well as a Lorentz covari ant formula for its rate of radiation.\n1. Motivation\nA considerable effort has been invested in the study of strongl y-coupled thermal non-Abelian\nplasmas by means of the AdS/CFT correspondence [1, 2, 3]. The jet quenching observed at\nRHIC indicates that partons crossing the quark-gluon plasm a experience significant energy loss.\nThis question has been studied in a strongly-coupled N= 4 SYM plasma for a variety of probes,\nincluding quarks [4, 5, 6], mesons [7], baryons [8, 9], gluon s [8, 10] and k-quarks [8]. Here we will\nanalyze quark energy loss in vacuum , which for a strongly-coupled non-Abelian gauge theory is\ninteresting already in itself, and is in addition helpful fo r developing intuition that might extend\nto the finite temperature case [11, 12].\nWe expect an accelerating quark to radiate, and to experienc e a damping force due to the\nemitted radiation. In the context of classical electrodyna mics, and for a non-relativistic electron\nmodeled as a vanishingly small charge distribution, this effe ct is incorporated in the classic\nAbraham-Lorentz equation [13, 14]\nm/parenleftigg\nd2/vector x\ndt2−ted3/vector x\ndt3/parenrightigg\n=/vectorF . (1)\nThe second term in the left-hand side is the damping term, whi ch is seen to be associated with\na characteristic timescale te≡2e2/3mc3, set by the classical electron radius. The search for a\nLorentz-covariant version of (1) led to the (Abraham-)Lore ntz-Dirac equation [15],\nm/parenleftigg\nd2xµ\ndτ2−te/bracketleftigg\nd3xµ\ndτ3−1\nc2d2xν\ndτ2d2xν\ndτ2dxµ\ndτ/bracketrightigg/parenrightigg\n=Fµ, (2)withτthe proper time and Fµ≡γ(/vectorF·/vector v/c,/vectorF) the 4-force. The second term within the square\nbrackets is the negative of the rate at which 4-momentum is ca rried away from the charge by\nradiation (as given by the covariant Lienard formula), so it is only this term that can properly\nbe called radiation reaction. The first term within the squar e brackets, usually called the Schott\nterm, and whose spatial part yields the damping force of (1) i n the non-relativistic limit, is\nknown to arise from the effect of the charge’s ‘near’ (as oppose d to radiation) field [16, 17].\nThe appearance of a third-order term in (1) and (2) leads to un physical behavior, including\npre-accelerating and self-accelerating (or ‘runaway’) so lutions. These deficiencies are known\nto originate from the assumption that the charge is pointlik e. For a non-relativistic charge\ndistribution of small but finite size ℓ, (1) is corrected by an infinite number of higher-derivative\nterms,\nm/parenleftigg\nd2/vector x\ndt2−te/bracketleftigg\nd3/vector x\ndt3−∞/summationdisplay\nn=1bnℓn\ncnd3+n/vector x\ndt3+n/bracketrightigg/parenrightigg\n=/vectorF (3)\n(withbnsome numerical coefficients), and is physically sound as long asℓ > ct e[17]. Upon\nshifting attention to the quantum case, one intuitively exp ects the pointlike ‘bare’ electron to\nacquire an effective size of order the Compton wavelength λC≡¯h/mc, due to its surrounding\ncloud of virtual particles. Given that λC≫cte, there should then be no room for unphysical\nbehavior. Indeed, in [18] it was shown that nonrelativistic QED leads to\nm/parenleftigg\nd2/vector x\ndt2−te/bracketleftigg\nd3/vector x\ndt3−∞/summationdisplay\nn=1dnλn\nC\ncnd3+n/vector x\ndt3+n/bracketrightigg/parenrightigg\n=/vectorF , (4)\nwhich has no runaway solutions, and shows that the charge dev elops a characteristic size ℓ=λC.\nGoing further to the quantum non-Abelian case is a serious ch allenge. Nevertheless, we\nwill show here that the AdS/CFT correspondence [1] allows us to address this question rather\neasily in certain strongly-coupled non-Abelian gauge theo ries [19, 20]. We expect the basic story\nwe will uncover to apply generally to all instances of the gau ge/string duality (including cases\nwith finite temperature or chemical potentials), but for sim plicity we will concentrate on the\nspecific example of N= 4 SYM. Besides the gauge field, this maximally supersymmetr ic and\nconformally invariant theory (CFT) contains 6 real scalar fi elds and 4 Weyl fermions, all in\ntheadjointrepresentation of the gauge group. We will be able to derive a Lorentz covariant\nequation summarizing the dynamics of a quark in this strongl y-coupled theory, which will turn\nout to be a nonlinear generalization of the Lorentz-Dirac eq uation (2), with a higher-derivative\nstructure somewhat similar to (4).\n2. Basic Setup\nFrom this point on we work in natural units, c= ¯h= 1. It is by now well-known that N= 4\nSU(Nc) SYM with coupling gYMis, despite appearances, completely equivalent [1] to Type\nIIB string theory on a background that asymptotically appro aches the five-dimensional anti-de\nSitter (AdS) geometry\nds2=GMNdxMdxN=R2\nz2/parenleftig\n−dt2+d/vector x2+dz2/parenrightig\n, (5)\nwhereR4/l4\ns=g2\nYMNc≡λdenotes the ’t Hooft coupling, and lsis the string length. The\nradial direction zis mapped holographically into a variable length scale in th e gauge theory,\nin such a way that z→0 andz→ ∞are respectively the ultraviolet and infrared limits.\nThe directions xµ≡(t,/vector x) are parallel to the AdS boundary z= 0 and are directly identified\nwith the gauge theory directions. The state of IIB string the ory described by the unperturbed\nmetric (5) corresponds to the vacuum of the N= 4 SYM theory, and the closed string sectordescribing (small or large) fluctuations on top of it fully ca ptures the gluonic (+ adjoint scalar\nand fermionic) physics. The string theory description is un der calculational control only for\nsmall string coupling and low curvatures, which translates intoNc≫1,λ≫1.\nIt is also known that one can add to SYM Nfflavors of matter in the fundamental\nrepresentation of the SU(Nc) gauge group by introducing in the string theory setup an ope n\nstring sector associated with a stack of NfD7-branes [21]. We will refer to these degrees of\nfreedom as ‘quarks,’ even though, being N= 2 supersymmetric, they include both spin 1 /2 and\nspin 0 fields. The D7-brane embedding is chosen to be translat ionally invariant along the gauge\ntheory directions xµ, and extend in the radial direction from the boundary of AdS a tz= 0 up\nto a location\nzm=√\nλ\n2πm(6)\ndeterminedbythemass mofthequarks. For Nf≪Nc, thebackreaction oftheD7-branes onthe\ngeometry can be neglected; in the field theory this correspon ds to a ‘quenched’ approximation.\nAn isolated quark of mass mis dual to an open string that has one endpoint on the D7-brane s\natz=zm, and extends radially all the way to the AdS horizon at z→ ∞. The string dynamics\nis governed by the Nambu-Goto action\nSNG=−1\n2πl2s/integraldisplay\nd2σ/radicalbig\n−detgab≡/integraldisplay\nd2σLNG, (7)\nwheregab≡∂aXM∂bXNGMN(X) (a,b= 0,1) denotes the induced metric on the worldsheet.\nWe can exert an external force /vectorFon the string endpoint by turning on an electric field F0i=Fi\non the D7-branes. This amounts to adding to (7) the usual mini mal coupling, which in terms\nof the endpoint/quark worldline xµ(τ)≡Xµ(τ,zm) reads\nSF=/integraldisplay\ndτ Aµ(x(τ))dxµ(τ)\ndτ. (8)\nVariation of SNG+SFimplies the standard Nambu-Goto equation of motion for all i nterior\npoints of the string, plus the boundary condition\nΠz\nµ(τ)|z=zm=Fµ(τ)∀τ , (9)\nwhere Πz\nµ≡∂LNG/∂(∂zXµ) is the worldsheet (Noether) current associated with space time\nmomentum, and Fµ=−Fνµ∂τxν= (−γ/vectorF·/vector v,γ/vectorF) the Lorentz four-force.\nNotice that the string is being described (as is customary) i n first-quantized language, and,\nas long as it is sufficiently heavy, we are allowed to treat it se miclassically. In gauge theory\nlanguage, then, we are coupling a first-quantized quark to th e gluonic (+ other SYM) field(s),\nand then carrying out the full path integral over the strongl y-coupled field(s)(the resultof which\nis codified by the AdS spacetime), but treating the path integ ral over the quark trajectory xµ(τ)\nin a saddle-point approximation.1\nIn more detail, it is really the endpoint of the string that co rresponds to the quark, while\nthe body of the string codifies the profile of the (near and radi ation) gluonic (+ other SYM)\nfield(s) set up by the quark.2The latter can be mapped out explicitly by computing one-poi nt\nfunctions of local operators ( ∝angbracketlefttrF2∝angbracketright,∝angbracketleftTµν∝angbracketright,...) in the presence of the quark, which, via the\nstandard GKPW recipe [2], requires a determination of the ne ar-boundary profile of the closed\nstring fields ( φ,hµν,...) generated by the macroscopic string.\n1For a study of quantum fluctuations about this classical stri ng configuration, see the very recent work [22].\n2In other words, the string here is the N= 4 SYM analog of the ‘QCD string’, with the surprising twist t hat it\nlives not in 4 but in 5 (+5) dimensions.For theinterpretation ofourresults, it willbecrucial tok eep inmindthat thequarkdescribed\nby this string is not ‘bare’ but ‘composite’ or ‘dressed’. Th is can be seen most clearly by working\nout the expectation value of the gluonic field surrounding a s tatic quark located at the origin\n[23],\n1\n4g2\nYM∝angbracketlefttrF2(x)∝angbracketright=√\nλ\n16π2|/vector x|4\n1−1+5\n2/parenleftig2πm|/vector x|√\nλ/parenrightig2\n/parenleftbigg\n1+/parenleftig2πm|/vector x|√\nλ/parenrightig2/parenrightbigg5/2\n. (10)\nForm→ ∞(zm→0), this is just the Coulombic field expected (by conformal in variance) for\na pointlike charge. For finite m, the profile is still Coulombic far away from the origin, but i n\nfact becomes non-singular at the location of the quark,\n1\n4g2\nY M∝angbracketlefttrF2(x)∝angbracketright=√\nλ\n128π2/bracketleftigg\n15/parenleftbigg2πm√\nλ/parenrightbigg4\n−35\n|/vector x|4/parenleftbigg2πm|/vector x|√\nλ/parenrightbigg6\n+.../bracketrightigg\nfor|/vector x|<√\nλ\n2πm.(11)\nAs seen in these equations, the characteristic thickness of this non-Abelian charge distribution\nis precisely the length scale zmdefined in (6). This is then the size of the gluonic cloud that\nsurrounds the quark, or in other words, the analog of the Comp ton wavelength for our non-\nAbelian source.\n3. Generalized Lorentz-Dirac Equation for the Quark\nBy Lorentz invariance, given the description of the static q uark, we know that a quark moving at\nconstant velocity corresponds to a purely radial string, mo ving as a rigid vertical rod. When the\nquark accelerates, the body of the string will trail behind t he endpoint, and will therefore exert\na force on the latter. Remembering that the body of the string codifies the SYM fields sourced\nby the quark, we know that in the gauge theory this force is int erpreted as the backreaction of\nthe gluonic field on the quark. In other words, in the AdS/CFT c ontext the quark has a ‘tail’,\nand it is this tail that is responsible for the damping effect we are after. This mechanism had\nbeen previously established in the computations of the drag force exerted on the quark by a\nthermal plasma, which is described in dual language in terms of a string living on a black hole\ngeometry [4, 5]. Our analysis here will make it clear that the damping effect is equally present\nin the gauge theory vacuum [19, 20].\nTo flesh out this story, we need to determine the string profile corresponding to a given\naccelerated quark/endpoint trajectory, by solving the non linear equation of motion following\nfrom the Nambu-Goto action (7). Fortunately, for this task w e can make use of the results\nobtained in a remarkable paper by Mikhailov [24], which we no w briefly review (a more detailed\nexplanation can be found in [11]). This author considered an infinitely massive quark ( zm= 0),\nand was able to solve the equation of motion for the dual strin g on AdS, for an arbitrary timelike\ntrajectory of the string endpoint. In terms of the coordinat es used in (5), his solution is\nXµ(τ,z) =zdxµ(τ)\ndτ+xµ(τ), (12)\nwithxµ(τ) the worldline of the string endpoint at the AdS boundary— or , equivalently, the\nworldline of the dual, infinitely massive, quark— parametri zed by its proper time τ. It is easy\nto check that the lines at constant τare null with respect to the induced worldsheet metric, a\nfact that plays an important role in Mikhailov’s constructi on.\nThe solution (12) is retarded. To see this, note that, parame trizing the quark worldline by\nx0(τ) instead of τ, and using dτ=√\n1−/vector v2dx0, where/vector v≡d/vector x/dx0, theµ= 0 component of (12)\ntakes the form\nt=z1√\n1−/vector v2+tret, (13)wheretretdenotes the value of the quark/endpoint coordinate time x0(τ) at the proper time\nτrelevant to the ( t,z) segment of the string under consideration, and the endpoin t velocity /vector v\nis meant to be evaluated at tret. In these same terms, the spatial components of (12) can be\nformulated as\n/vectorX(t,z) =z/vector v√\n1−/vector v2+/vector x(tret) = (t−tret)/vector v+/vector x(tret). (14)\nWe see here that the behavior at time t=X0(τ,z) of the string segment located at radial\nposition z(which, roughly speaking, codifies the gluonic field profile a t length scale zand\nposition /vector x=/vectorX(t,z) in the gauge theory) is completely determined by the behavi or of the string\nendpoint at the earliertimetret(t,z) determined from (13), i.e., by projecting back toward\nthe boundary along the null line at fixed τ. (Note the analogy with the construction of the\nLienard-Wiechert fields in classical electromagnetism.)\nAn analogous advanced solution built upon the same endpoint /quark trajectory can be\nobtained by reversing the sign of the first term in the right-h and side of (12). In gauge\ntheory language, this choice of sign corresponds to the choi ce between a purely outgoing or\npurely ingoing boundary condition for the waves in the gluon ic field at spatial infinity. Both\non the string and the gauge theory sides, more general configu rations should of course exist,\nbut obtaining them explicitly is difficult due to the highly no nlinear character of the system.\nHenceforth we will focus solely on the retarded solutions, w hich are the ones that capture the\nphysics of present interest, with influences propagating ou tward from the quark to infinity.\nUsing (13) and (14), Mikhailov was able to rewrite the total s tring energy in the form\nE(t) =√\nλ\n2π/integraldisplayt\n−∞dtret/vector a2−[/vector v×/vector a]2\n(1−/vector v2)3+Eq(/vector v(t)), (15)\nwhereofcourse /vector a≡d/vector v/dx0. Thesecond term intheabove equation arises fromatotal der ivative\non the string worldsheet, and gives the expected Lorentz-co variant expression for the energy\nintrinsic to the quark [11],\nEq(/vector v) =√\nλ\n2π/parenleftbigg1√\n1−/vector v21\nz/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsinglezm=0\n∞=γm . (16)\nThe first term in (15) must then represent the accumulated ene rgylostby the quark over all\ntimes prior to t. Surprisingly, the rate of energy loss is seen to have precis ely the same form\nas the standard Lienard formula from classical electrodyna mics. We therefore learn that in this\nnon-Abelian, strongly-coupled theory, the energy loss of a n infinitely massive (pointlike) quark\ndepends locallyon the quark worldline. For the spatial momentum, [24, 11] si milarly find\n/vectorP(t) =√\nλ\n2π/integraldisplayt\n−∞dtret/vector a2−[/vector v×/vector a]2\n(1−/vector v2)3/vector v+/vector pq(/vector v(t)), (17)\nwith\n/vector pq=√\nλ\n2π/parenleftbigg/vector v√\n1−/vector v21\nz/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsinglezm=0\n∞=γm/vector v . (18)\nWe see then that, in spite of the non-linear nature of the syst em, Mikhailov’s procedure leads to\na clean separation between the tip and the tail of the string, i.e., between the quark (including\nits near field) and its gluonic radiation field.\nIn fact, using (12), the standard string dynamics reduces to standard particle dynamics at\nthe level of the action: plugging (12) back into (7)+(8), we c an explicitly carry out the integraloverzto obtain [20]\nSNG+SF=−R2\n2πl2s/integraldisplay\ndτ/integraldisplay∞\nzm→0dz\nz2+/integraldisplay\ndτ Aµ(x(τ))dxµ\ndτ(τ) (19)\n=−m/integraldisplay\ndτ+/integraldisplay\ndτ Aµ(x(τ))dxµ\ndτ(τ),\nwhich is evidently the standard action for a pointlike exter nally forced relativistic particle (with\nmassm→ ∞). Notice that the associated equation of motion does notinclude a damping\nforce, which is just as one would expect for an infinitely mass ive charge, because the coefficient\nte∝1/mof the damping terms in (1) and (2) approaches zero as m→ ∞.\nLet us now consider the more interesting case of a quark with fi nite mass, zm>0, where\nthere should be a noticeable damping effect. As we emphasized a t the end of the previous\nsection, in this case our non-Abelian source is no longer poi ntlike but has size zm. On the string\ntheory side, the string endpoint is now at z=zm, and we must again require it to follow the\ngiven quark trajectory, xµ(τ). As before, this condition by itself does not pick out a uniq ue\nstring embedding. Just like we discussed for the infinitely m assive case, we additionally require\nthe solution to be retarded, in order to focus on the gluonic fi eld causally set up by the quark.\nAs in [11], we can inherit this structure by truncating a suit ably selected retarded Mikhailov\nsolution. The embeddings of interest to us can thus be regard ed as the z≥zmportions of\nthe solutions (12). These are parametrized by data at the AdS boundary z= 0, which are\nnow merely auxiliary and will henceforth be denoted with til des, to distinguish them from the\nactual physical data associated with the endpoint/quark at z=zm. In this notation, the string\nembedding reads\nXµ(˜τ,z) =zd˜xµ(˜τ)\nd˜τ+ ˜xµ(˜τ). (20)\nDifferentiation with respect to ˜ τand evaluation at z=zm(where we can read off the quark\ntrajectory xµ(˜τ)≡Xµ(˜τ,zm)) leads to\ndxµ\nd˜τ=zmd2˜xµ\nd˜τ2+d˜xµ\nd˜τ, (21)\nwhich in turn implies\ndτ2≡ −dxµdxµ=d˜τ2\n1−z2\nm/parenleftigg\nd2˜x\nd˜τ2/parenrightigg2\n (22)\nand\nd2xµ\nd˜τ2=zmd3˜xµ\nd˜τ3+d2˜xµ\nd˜τ2. (23)\nWe are now finally ready to derive the desired equation of moti on for the quark. This must\nsimply be dual to the equation of motion satisfied by the strin g endpoint, which we know to be\ngiven by the standard boundary condition (9). For the embedd ings (20), this condition reads\nΠz\nµ(τ) =√\nλ\n2πd˜τ\ndτ\n1\nzmd2˜xµ\nd˜τ2+/parenleftigg\nd2˜xµ\nd˜τ2/parenrightigg2d˜xµ\nd˜τ\n=Fµ. (24)\nUsing (22), (23) and carrying out some additional algebra (s ee [20] for details), this can be\nrewritten in the form\nd\ndτ\nmdxµ\ndτ−√\nλ\n2πmFµ\n/radicalig\n1−λ\n4π2m4F2\n=Fµ−√\nλ\n2πm2F2dxµ\ndτ\n1−λ\n4π2m4F2, (25)which is our main result [19, 20].\nNotice that the characteristic length scale appearing in (2 5) is precisely zm=√\nλ/2πm,\nwhich as discussed below (11), is the quark Compton waveleng th. Let us now examine the\nbehavior of a quark that is sufficiently heavy, or is forced suffi ciently softly, that the condition/radicalbig\nλ|F2|/2πm2≪1 holds. It is then natural to expand the equation of motion in a power series\nin this small parameter. To zeroth order in this expansion, w e correctly reproduce the pointlike\nresultm∂2\nτxµ=Fµ. If we instead keep terms up to first order, we find\nmd\ndτ/parenleftigg\ndxµ\ndτ−√\nλ\n2πm2Fµ/parenrightigg\n≃ Fµ−√\nλ\n2πm2F2dxµ\ndτ.\nIn theO(√\nλ) terms it is consistent, to this order, to replace Fµwith its zeroth order value,\nthereby obtaining\nm/parenleftigg\nd2xµ\ndτ2−√\nλ\n2πmd3xµ\ndτ3/parenrightigg\n≃ Fµ−√\nλ\n2πd2xν\ndτ2d2xν\ndτ2dxµ\ndτ. (26)\nInterestingly, this coincides exactlywith the Lorentz-Dirac equation (2), with the Compton\nwavelength (6) playing the role of characteristic size tefor the composite quark. This is indeed\nthe natural quantum scale of the problem. The radiation reac tion force in (26) is correctly\ngiven by the covariant Lienard formula, as expected from the result (15) [24], which we see\nthen arising as the pointlike limit of the full radiation rat e encoded in the right-hand side of\n(25). The Schott term in (26) (associated with the near field o f the quark), originated from the\nterms inside the τ-derivative in the left-hand side of (25), which we understa nd then to codify\na modified dispersion relation for our composite quark.\nTo second order in/radicalbig\nλ|F2|/2πm2, we similarly obtain\nmd2xµ\ndτ2−√\nλ\n2π\n/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright\nd3xµ\ndτ3−d2xν\ndτ2d2xν\ndτ2dxµ\ndτ/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n\n+λ\n4π2m\n/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright\nd4xµ\ndτ4−(1+2)d2xν\ndτ2d3xν\ndτ3dxµ\ndτ/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n (27)\n−λ\n4π2m\n/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright\n1\n2+ 1/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nd2xν\ndτ2d2xν\ndτ2d2xµ\ndτ2≃ Fµ.\nFor compactness, we have grouped together all terms arising at the same order in the expansion,\nand used underbraces to mark the radiation reaction terms or iginating from the right-hand side\nof (25) (which now include corrections beyond the standard L ienard formula), and overbraces to\nindicatethenear-fieldtermsarisingfromtheleft-handsid eof(25) (whichincorporatecorrections\nto the standard Schott term).\nWe can continue this expansion procedureto arbitrarily hig h order in/radicalbig\nλ|F2|/2πm2. Ourfull\nequation (25) is thus recognized as a compact (reduced-orde r) rewriting of an infinite-derivative\nextension of the Lorentz-Dirac equation that automaticall y incorporates the size zmof our non-\npointlike non-Abelian source. It is curious to note that (25 ), which clearly incorporates the\neffect of radiation damping on the quark, has been obtained fro m (12), which does notinclude\nsuch damping for the string itself. The latter arises from th e backreaction of the closed string\nfields set up by our macroscopic string, but these are of order 1/N2\nc, and therefore subleading\nat largeNc.The full physical content of (25) can be made transparent by r ewriting it in the form\ndPµ\ndτ≡dpµ\nq\ndτ+dPµ\nrad\ndτ=Fµ, (28)\nrecognizing Pµas the total string (= quark + radiation) four-momentum,\npµ\nq=mdxµ\ndτ−√\nλ\n2πmFµ\n/radicalig\n1−λ\n4π2m4F2(29)\nas the intrisic momentum of the quark including the contribu tion of the near-field sourced by it\n(or, in quantum mechanical language, of the gluonic cloud su rrounding the quark), and\ndPµ\nrad\ndτ=√\nλF2\n2πm2\ndxµ\ndτ−√\nλ\n2πm2Fµ\n1−λ\n4π2m4F2\n (30)\nas the rate at which momentum is carried away from the quark by chromo-electromagnetic\nradiation. Both (29) and (30) diverge when F2→ F2\ncrit, where\nF2\ncrit=4π2m4\nλ(31)\nis the critical value at which the force becomes strong enoug h to nucleate quark-antiquark pairs\n(or, in dual language, to create open strings) [25].\nUnlike its classical electrodynamic counterpart (2), our d ressed quark equation of motion has\nno self-accelerating (runaway) solutions: in the (continu ous) absence of an external force, (25)\nuniquely predicts that the 4-acceleration of the quark must vanish. Interestingly, the converse\nto this last statement is not true: constant 4-velocity does not uniquely imply a vanishing force.\nE.g., for motion purely along one dimension, (25) reads\nma=F(1−v2)3/2\n/radicalig\n1−λ\n4π2m4F2+√\nλ\n2πmdF\ndt(1−v2)\n1−λ\n4π2m4F2. (32)\nFora= 0, we obtain a differential equation with general solution\nF(t) =±2πm2\n√\nλsech/bracketleftigg\n2π\nγ√\nλm(t−t0)/bracketrightigg\n,\nwitht0an integration constant [20]. This is clearly non-vanishin g for all finite values of t0.\nWe conclude then that the energy provided to the system by thi s particular F(t) does not\ntranslate into an increase of the quark/endpoint velocity, but into a continuous modification of\nthe string tail, or, in gauge theory language, a change of the gluonic field profile. This is again\na consequence of the extended, and hence deformable, nature of the quark.\n4. Conclusions\nWe have shown how a straightforward analysis of the standard dynamics of a string in AdS\nneatly captures the physics of a ‘dressed’ or ‘composite’ qu ark that accelerates in the vacuum\nof strongly-coupled N= 4 SYM, providing a rather beautiful illustration of the pow er of the\nAdS/CFT correspondence. Specifically, we have learned that [24, 11, 19, 20]:•The total 4-momentum of the string (which is conserved at lea ding order in 1 /Nc) includes\nnot only the intrinsic 4-momentum of the dual quark, but also the accumulated 4-\nmomentum radiated by the quark.\n•The usual Nambu-Goto equation and forced boundary conditio n for the string imply a\ngeneralized Lorentz-Dirac equation for the dressed quark, which is nonlinear and\nphysically sensible.\n•Radiation damping is naturally incorporated in this equation of motion for the quark,\neven though string damping is negligible (at leading order i n 1/Nc).\n•A quark with finite mass is automatically non-pointlike , and has a non-trivial and non-\nlocaldispersion relation and radiation rate.\n•We expect an analogous story for other strongly-coupled theories (conformal or not,\nand with or without temperature or chemical potentials).\nAcknowledgements\nWe are grateful to the organizers of the Quantum Theory and Sy mmetries 6 Conference at the\nUniversity of Kentucky and the XII Mexican Workshop of Parti cles and Fields in Mazatl´ an,\nM´ exico, for putting together very useful meetings, and for the opportunity to present this work,\nwhich was partially supported by CONACyT grants 50-155I and CB-2008-01-104649, as well as\nDGAPA-UNAM grant IN116408.\n[1] J. 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Shaw1 \n \n1Quantum E lectromagnetics Division, National Institute of Standards and Technology, Boulder, CO \n80305 , USA  \n2Institute of Experimental and Applied Physics, University of Regensburg, 93053 Regensburg, \nGermany  \n3Department of Applied P hysics, Eindhoven University of Technology, 5600 MB Eindhoven, The \nNetherlands  \n \n \nDated: 01/05/2017  \n \n *Corresponding author: martin1.schoen@physik.uni -regensburg.de  \n \n \n \n \nAbstract  \nA systematic experimental study of Gilbert damping is performed via ferromagnet ic \nresonance for the disordered  crystalline  binary 3 d transition metal alloys Ni -Co, Ni -Fe and \nCo-Fe over the full range of alloy compositions. After accounting for inhomogeneous \nlinewidth  broadening , the damping shows clear evidence of both  interfacial da mping \nenhancement (by spin pumping) and radiative damping. We quantify these two extrinsic  \ncontributions  and thereby determine the intrinsic damping. The comparison of the intrinsic \ndamping to multiple theoretical calculation s yields good qualitative  and q uantitative  \nagreement  in most cases . Furthermore, the values of the damping obtained in this study are \nin good agreement with  a wide range of published experimental and theoretical values . \nAdditionally, we find a compositional dependence of the spin mixing  conductance.  \n \n \n \n \n 2  \n \n \n \n1  Introduction  \nThe magnetization dynamics in f erromagnetic films are phen omenologically well described \nby the Landau -Lifshitz -Gilbert formalism (LLG)  where  the damping  is described by a \nphenomenological damping parameter α.4,5 Over the past four decades, there have been \nconsiderable efforts to derive the phenomenological d amping parameter from first principles \ncalculations and to do so in a quantitative manner. One of the early promising  theories was that of \nKamberský, who introduced the so -called breathing Fermi surface model6–8. The name “breathing \nFermi  surface ” stems from the picture that the precessing magnetization, due to spin -orbit coupling, \ndistorts the Fermi surface. Re -populating the Fermi surface is delayed by the scattering time, \nresulting in a phase lag between the precession  and the Fermi sur face distortion. This  lag leads to  a \ndamping that is proportional to the scattering time. Although this approach describes the so-called \nconductivity -like behavior of the damping at low temperatures, it fails to describe  the high \ntemperature behavior of so me materials . The high temperature or resistivity -like behavior is \ndescribed by the so-called  “bubbling Fermi surface ” model. In the case of energetically shifted  \nbands , thermal  broaden ing can lead to a significant  overlap  of the spin-split bands  in 3d \nferromagnets.  A precessing magnetization can induce elec tronic transitions between such \noverlapping bands , leading to spin-flip process es. This process scales with the amount of band \noverlap. Since s uch overlap is further increased with the band broadening th at results from the finite \ntemperature of the sample, this contribution  is expected to increase as the temperature is increased. \nThis model for interband transition mediated damping describes the resistivity -like behavior of the \ndamping at higher temperatu res (shorter scattering times).  These two damping processes are  \ncombined in a torque correlation model by Gilmore , et al.9, as well as Thonig , et al.10, that  describes \nboth the low -temperature  (intra band transitio ns) and high -temperature  (inter band transitions) \nbehavior  of the damping . Another app roach via  scattering theory was successfully implemented by \nBrataas , et al.11 to describe damping in transition metals. Starikov , et al. ,2 applied the scattering \nmatrix approach to calculate the damping  of NixFe1-x alloys and  Liu, et al. ,12 expanded the formalism \nto include the influence of electron -phonon interaction s.  \nA numerical realization of the torque correlation model was performed  by Mankovsky , et \nal., for NixCo1-x, Ni xFe1-x, CoxFe1-x, and FexV1-x1. More recently , Turek , et al. ,3 calculated the \ndamping for NixFe1-x and CoxFe1-x alloys with the torque -correlation model, u tilizing non -local \ntorque correlators. It is important to stress that a ll of these  approaches  consider  only the intrinsic \ndamping. This complicates the quantitative comparison of calculated values for t he damping to \nexperimental data  since there are many extrinsic contributions to the damping that result from \nsample  structure , measurement geometr y, and/or sample properties . While some extrinsic \ncontributions to the damping  and linewidth  were discovered in the 1960 ’s and 1970 ’s, and are well \ndescribed  by the ory, e.g. eddy -current damping13,14, two -magnon scattering15–17, the slow rel axer \nmechanism18,19, or radiative damping20,21, interest in these mechanisms has been re -ignited \nrecently22,23. Further contributions, such as  spin-pumping, both extrinsic24,25 and intrinsic24,26, have 3 been discovered more recently and a re subject to extensive research27–31 for spintronics application.  \nTherefore , in order to allow a quantitative comparison to theoretical calculations  for intrins ic \ndamping, both the measurement and sample geometry must  be designed  to allow both the \ndetermination and possibl e minimization  of all additional contributions to the measured damping.  \nIn this study , we demonstrate methods to determine significant extrins ic contributions to the \ndamping , which  includ es a measurement of the effective spin mixing conductance for both the pure \nelements and select alloys. By precisely accounting for all of these extrinsic contributions, we \ndetermine the intrinsic damping  parame ters of the binary alloys Ni xCo1-x, Ni xFe1-x and Co xFe1-x and \ncompare them  to the calculation s by Mankovsky , et al. ,1, Turek , et al. , and Starikov , et al.2. \nFurthermore, we present the concentration -dependence of the  inhomogeneous linewidth \nbroadening, which for most alloys shows exceptionally small values, indicative  of the high \nhomogeneity of our samples.  \n2 Samples and method  \nWe deposited  NixCo1-x, Ni xFe1-x and Co xFe1-x alloys of varying composition (all compositio ns \ngiven in atomic percent)  with a thickness of 10 nm  on an oxidized (001)  Si substrate with  a Ta(3 \nnm)/Cu(3 nm) seed layer and  a Cu(3 nm) /Ta(3 nm) cap layer. In order to investigate interface \neffects , we also deposited multiple thickness series at 10 nm, 7 nm, 4 nm, 3 nm , and 2 nm of  both \nthe pure elements and select alloy s.  Structural characterization  was performed using X-ray \ndiffraction (XRD).  Field swept vector -network -analyzer  ferromagnetic resonance spectroscopy \n(VNA -FMR) was used in the out -of-plane geometry  to determine the total damping parameter αtot. \nFurther d etails of the deposition conditions, XRD, FMR measurement and fitting of the complex \nsusceptibility to the measured S21 parameter are reported in  Ref [66]. \nAn example of susceptibility fits t o the complex S21 data is shown in Fig.  1 (a) and (b). All \nfits were constrained  to a 3×  linewidth ΔH field window around the resonance field in order to \nminimize the influence of measurement drifts on the error in the susceptibility fits. The total \ndampin g parameter αtot and the inhomogeneous linewidth broadening Δ H0 are then determined  from \na fit to  the linewidth Δ H vs. frequency f plot22, as shown in Fig.  1 (c). \n ∆𝐻= 4𝜋𝛼𝑡𝑜𝑡𝑓\n𝛾𝜇0+ ∆𝐻0, (1) \nwhere γ=gμB/ħ is the gyro -magnetic ratio, μ0 is the vacuum permeability, μB is the Bohr -magneton , \nħ is the reduced Planck constant,  and g is the spectroscopic g-factor reported in Ref [ 66].  \n  4  \nFigure 1: (a) and (b) show respectively the real and imaginary part of the S21 transmission \nparameter (black squares)  measured at 20 GHz  with the complex susceptibility fit (red lines)  for \nthe Ni 90Fe10 sample . (c) The linewidths  from the suscept ibility fits  (symbols) and linear fits (solid \nlines) are plotted against frequency for different Ni -Fe compositions. Concentrations are denoted \non the right -hand  axis. The damping α and the inhomogeneous linewidth broadening Δ H0 for each \nalloy can be extracted from the fits via Eq.  (1). \n3 Results  \nThe first contribution t o the linewidth we discuss  is the  inhomogeneous linewidth broadening \nΔH0, which  is presumably  indicative of  sample inhomogeneity32,33. We plot Δ H0 for all the alloy \nsystem s against the respective concentrations in Fig.  2. For all alloys , ΔH0 is in the range of a few \nmT to 10  mT . There are only a limited  number of reports  for ΔH0 in the literature with which to \ncompare . For Permalloy (Ni 80Fe20) we measure Δ H0 = 0.35 mT, which is close to other reported \nvalues .34 For the  other  NixFe1-x alloys , ΔH0 exhibits a significant peak  near the fcc-to-bcc (face -\ncentered -cubic to body -centered -cubic) phase transition  at 30  % Ni , (see Fig.  2 (b)) which is easily \nseen in the raw data  in Fig. 1 (c). We speculate  that this  increase of  inhomogeneous broadening  in \nthe NixFe1-x is caused by the coexistence of the bcc and fcc  phases at the phase transition.  However,  \nthe CoxFe1-x alloys do not exhibit an increase in ΔH0 at the equivalent phase transition  at 70  % Co . \nThis suggests that the bcc and fcc phases of NixFe1-x tend to segregate near the phase transition, \nwhereas the same phases for CoxFe1-x remain intermixed throughout the transition.  \n5 One possible explanation for inhomogeneous broadening is magnetic anisotropy , as originally \nproposed  in Ref. [35]. However, this explanation does not account for our measured dependence of \nΔH0 on alloy concentration, since the perpendicular magnetic anisotropy , described in Ref [ 66] \neffectively  exhibits opposite behavior with alloy concentration.  For our alloys Δ H0 seems to roughly \ncorrelate to the inverse exchange constant36,37, which co uld be a starting point for future \ninvestigation of a quantitative theory of inhomogeneous broadening.  \n \n \n \n  \nFigure 2: The inhomogeneous linewidth -broadening Δ H0 is plotted vs. alloy composition for (a) \nNi-Co, (b) Ni -Fe and (c) Co -Fe. The alloy phases are denoted by color code described in Ref [ 66] \n \nWe plot the total measured damping αtot vs. composition for NixCo1-x, NixFe1-x and CoxFe1-x in \nFig. 3 (red crosses). The total damping of the NixCo1-x system increases monotonically  with \nincreased Ni content . Such smooth  behavior in the damping is  not surprising  owing to the absence \nof a phase transition for this alloy . In the NixFe1-x system , αtot changes very little from pure Fe to \napproximately 25 % Ni  where the bcc to fcc phase transition occurs . At the phase  transition,  αtot \nexhibits a step, increasing  sharply by approximately 30  %. For higher Ni concentrations , αtot \nincreases monotonically  with increasing Ni concentration . On the other hand, t he CoxFe1-x system \nshows a different behavior in the damping and d isplays a sharp minimum of (2.3 ± 0.1)×10-3 at 25 \n6 % Co as previously reported38. As the system changes to an fcc phase ( ≈ 70 % Co), αtot become \nalmost  constant.  \nWe compare our data to previously published values in Table I. However, direct c omparison \nof our data to previous report s is not trivial , owing to the variation in measurement conditions  and \nsample characteristic s for all the reported measurements . For example , the damping  can depend  on \nthe temperature .9,39 In addition, multiple intrinsic and intrinsic contributions to the total damping  \nare not always accounted for in the literature . This can be seen in the fact that the reported  damping \nin Ni80Fe20 (Permalloy) varies  from α=0.0 055 to α=0.04  at room tem perature  among studies . The \nlarge variation  for these reported data  is possibly the result of  different uncontrolled contributions  \nto the extrinsic damping  that add  to the total damping in the different experiments , e.g. spin-\npumping40–42, or roughness41. Therefore , the value for the intrinsic damping of Ni 20Fe80 is expected \nto be at the low end of this scatter . Our measured value of α=0.007 2 lies within the range of reported \nvalues. Similarly , many of our measured damping values for different alloy compositions lie within \nthe range  of reported values22,43 –48. Our measured  damping of the pure elements and the Ni 80Fe20 \nand Co 90Fe10 alloys is compared to room temperature values found in literature in Table 1, Col umns  \n2 and 3 . Column 5 contains theoretically calculated values .   \n \nTable 1: The total measured damping α tot (Col. 2)  and the intrinsic damping (C ol. 4) f or Ni 80Fe20, \nCo90Fe10, and the pure elements are compared to both experimental (Col. 3) and theoretical (Col. \n5) values from the literature . All values of the damping  are at room temperature if not noted \notherwise . \nMaterial  αtot (this study)  \n Liter ature values  αint (this study)  Calculated literature \nvalues  \nNi 0.029 (fcc)  0.06444 \n0.04549 0.024 (fcc)  0.0179 (fcc) at 0K  \n0.02212 (fcc) at 0K  \n0.0131 (fcc)  \nFe 0.0036 (bcc)  0.001944 \n0.002746 0.0025 (bcc)  0.00139 (bcc) at 0K  \n0.001012 (bcc)  at 0K  \n0.00121 (bcc) at 0K  \nCo 0.0047 (fcc)  0.01144  \n 0.0029 (fcc)  0.00119 (hcp)  at 0K  \n0.0007312 (hcp)  at 0K  \n0.0011 (hcp)  \nNi80Fe20 0.0073 (fcc)  0.00844 \n0.008 -0.0450 \n0.007848 \n0.00751 \n0.00652 \n0.00647 \n0.005553 0.0050 (fcc)  0.00462,54 (fcc) at 0K  \n0.0039 -0.00493 (fcc)  at 0K  \nCo90Fe10 0.0048 (fcc)  0.004344 \n0.004855 0.0030 (fcc)   7  \n \n \n \n  \nFigure 3: (color online) The measured damping αtot of all the alloys is plotted against the alloy \ncompositi on (red crosses) for (a) Ni -Co, (b) Ni -Fe and (c) Co -Fe (the data in (c) are taken from \nRef.[38]). The black squares  are the intrinsic damping  αint after correction for spin pumping and \nradiative contributions to the measured damping. The blue line is the intr insic damping calculated \nfrom the Ebert -Mankovsky theory ,1 where the blue circles are the values for the pure elements  at \n300K . The green  line is the calculated damping for the Ni -Fe alloys  by Starikov , et al.2 The inset \nin (b) depicts the damping in a smaller concentration window in order to better depict the small \nfeatures in the damping around the ph ase transition. The damping  for the Co -Fe alloys, calculated \nby Turek et al.3 is plotted as the orange  line. For the Ni -Co alloys the damp ing calculated by th e \nspin density of the respective alloy weighted bulk damping55 (purple dashed line).  \n8  \n \nThis scatter in the experimental data reported in the literature and its divergence from calculated \nvalues of the damping shows th e necessity to determine  the intrinsic damping  αint by quantification \nof all extrinsic contributions to the measured total damping α tot.  \nThe first extrinsic  contribution to the damping  that we consider  is the radiative damping α rad, \nwhich is caused by ind uctive coupling between sample and  waveguide , which results in energy \nflow from the sample back into the microwave circuit.23 αrad depends directly  on the measurement \nmethod and geometry. The effect is easily understood , since the strength of the inductive coupling \ndepends on the inductance of the FMR mode itself , which is in turn determined by the saturation \nmagnetization, sampl e thickness, sample  length, and waveguide width.  Assuming a homogeneous \nexcitation field , a uniform magnetization profile throughout the sample , and negligible spacing \nbetween the waveguide and sample , αrad is well approximated by23  \n 𝛼rad=𝛾𝑀𝑠𝜇02𝛿𝑙\n16 𝑍0𝑤𝑐𝑐, (2) \nwhere l (= 10 mm  in our case) is the sample length on the waveguide, wcc (= 100µm) is the width \nof the co -planar wave guide center conductor and Z0 (= 50 Ω), the impedance of the waveguide. \nThough inh erently small for most thin films , αrad can become  significant  for alloys with \nexceptionally small intrinsic damping and /or high saturation magnetization.  For example, it plays \na significant role (values of αrad ≈ 5x10-4) for the whole composition range of  the Co -Fe alloy system \nand the Fe -rich side of the Ni -Fe system. On the other hand, for pure Ni and Permalloy (Ni 80Fe20) \nαrad comprises only 3 % and 5  % of αtot, respectively.  \nThe second non -negligible contribution to the damping  that we consider  is the interfacial \ncontribution to the measured  damping , such as  spin-pumping into the adjacent Ta/Cu bilayers . Spin \npumping  is proportional to the reciprocal sample thickness as described in24 \n 𝛼sp= 2𝑔eff↑↓𝜇𝐵𝑔\n4𝜋𝑀s𝑡. (3) \nThe spectroscopic  g-factor and the saturation magnetization Ms of the alloys were reported  in \nRef [66] and the factor of 2 accounts for the presence of two nominally identical interfaces of the \nalloys in the cap and seed layers.  In Fig.  4 (a)-(c) we plot the damping dependence on reciprocal \nthickness  1/t for select alloy concentrations, which allows us to determine  the effective spin mixing \nconductance 𝑔eff↑↓  through fits to Eq. (3) . The effective spin m ixing conductance contains details of \nthe spin transport in the adjacent non -magnetic layers, such as the interfacial spin mixing \nconductance, both the conductivity and spin diffusion for all the non -magnetic layers with a non -\nnegligible spin accumulation,  as well as the details of the spatial profile for the net spin \naccumulation .56,57 The values of 𝑔eff↑↓, are plotted versus the alloy concentration in Fig. 4 (d), and are \nin the range of previously reported values  for samples prepare d under similar growth conditions55–\n59. Intermediate values of 𝑔eff↑↓ are determined by a guide to the eye interpolation [ grey lines, Fig.  4 \n(d)] and αsp is calculated for all alloy concentrations utilizing  those interpolated values.  \nThe data for 𝑔eff↑↓  in the NixFe1-x alloys shows approximately a factor two  increase of 𝑔eff↑↓ between \nNi concentrations of 30  % Ni and 50  % Ni, which we speculate to occur at the fcc to bcc phase \ntransition around 30  % Ni. According to this line of speculation , the previously mentioned step  \nincrease in the measured total damping at the NixFe1-x phase transition can be fully attributed to the \nincrease in spin pumping at the phase transition.  In CoxFe1-x, the presence of a step in  𝑔eff↑↓   at the \nphase transition is not confirmed, given the measurement precision, although  we do observe an \nincrease in the effective spin mixing conductance when transitioning from the bcc to fcc phase.  The 9 concentration dependence of 𝑔eff↑↓  requires further thorough investigation and we therefore restrict \nourselves to reporting the expe rimental findings.  \n \n \n           \n \nFigure 4: The damping for the thickness series at select alloy compositions vs. 1/ t for (a) Ni -Co, \n(b) Ni -Fe and (c) Co -Fe (data points, concentrations denoted in the plots), with linear fits to Eq. \n(3) (solid lines). (d) The extracted effective spin mixing conductance 𝑔eff↑↓  for the measured alloy \nsystems, where the gray lines show the  linear  interpolations for intermediate alloy concentrations. \nThe data for the Co -Fe system are taken from Ref.[38].  \n \n \n Eddy -current damping13,14 is estimated  by use of  the equations given in Ref.  [23]  for films  \n10 nm thick or less . Eddy currents are neglected because they are found to be less than 5  % of the \ntotal damping. Two -magnon scattering  is disregarded  because the mechanism is largely e xcluded \nin the out -of-plane measurement geometry15–17. The total measured damping  is therefore well \napproximated as the sum   \n 𝛼tot≅𝛼int+𝛼rad+𝛼sp, (4) \nWe determine  the intrinsic damping of the material by subtracting α sp and α rad from the measured \ntotal damping , as shown in Fig. 3 .  \n10 The intrinsic damping increases monotonically with Ni concentration  for the NixCo1-x alloys . \nIndicative of the importance of extrinsic sources of damping, αint is approximately 40 % smaller \nthan αtot for the Fe -rich alloy, though the difference decreases to only 15 % for pure Ni. This \nbehavior is expected, given that both αrad and αsp are proportional to  Ms. A comparison of αint to the \ncalculations by Mankovsky , et al. ,1 shows excellent quantitative agreement  to within 30 %.  \nFurthermore, w e compare αint of the NixCo1-x alloys to the spin density weighted average of the \nintrinsic damping of Ni and C o [purple dashed line in Fig. 3 (a)] , which gives good agreement with \nour data, as previously reported .55  \nαint for NixFe1-x (Fig. 3 (b)) also increases with Ni concentration after a small initial decrease \nfrom pure Fe to the first NixFe1-x alloys. The step increase found in αtot at the bcc to fcc phase \ntransition is fully attributed to αsp, as detailed in the previous section, and therefore does not occur \nin αint. Similar to the NixCo1-x system αint is significantly lower than αtot for Fe -rich alloys. With in \nerror bars, a  comparison to the calculations by Mankovsky , et al.1 (blue line) and Starikov , et al.2 \n(green line) exhibit excellent agreement in the fcc phase, with marginally larger deviations in the \nNi rich regime. Starikov , et al.2 calculated the damping over the ful l range of compositions, under \nthe assumption of continuous  fcc phase. This calcu lation deviates further from  our measured αint in \nthe bcc phase exhibiting qualitatively different behavior.  \nAs previously reported, t he dependence of αint on alloy compositio n in the CoxFe1-x alloys \nexhibits strongly non -monotonic behavior, differing from the two previously discussed alloys.38  \nαint displays a minimum at 25 % Co concentration with a, for conducting ferromagnets \nunprecedented, low value of int (5±1.8)  × 10-4. With increasing Co concentration , αint grows up \nto the phase transition, at which point  it increases by 10  % to 20 % unt il it reaches  the value for \npure Co.  It was shown  that αint scales with the density of states  (DOS)  at the Fermi energy n( EF) in \nthe bcc phase38, and the DOS also exhibits a sharp  minimum for Co 25Fe75. This scaling is \nexpected60,61 if the damping is dominated by the breathing Fermi surface process. With the \nbreathing surface model,  the intraband scattering  that leads to damping  directly scales with n( EF). \nThis scaling is particularly pronounc ed in the Co -Fe alloy system due to the small concentration  \ndependence of the spin -orbit coupling on alloy composition. The special properties of the CoxFe1-x \nalloy system are discussed  in greater detail in Ref.[38]. \nComparing αint to the calculations by Mankovsky  et al.1, we find good quantitative \nagreement with the value of the minimum. However, t he concentration of the minimum is \ncalculated to occur  at approximately 10 % to  20% Co, a slightly lower value than 25 % Co t hat we \nfind in this study. Furthermore , the strong concentration dependence around the minimum is not \nreflected in the calculations. More recent calculations by Turek et al.3, for the bcc CoxFe1-x alloys \n[orange line in Fig. 3 (c)] find the a minimum of the damping of 4x10-4 at 25 % Co concentration \nin good agreement with our experiment, but there is some deviation in concentration dependenc e \nof the damping around the minimum. Turek et al.3 also reported on the damping in the NixFe1-x \nalloy system, with similar qualitative and  quantitative results as the other two presented quantitative \ntheories1,2 and the results are therefore not plotted  in Fig.  3 (b) for the sake of comprehensibility of \nthe figure.  For both NixFe1-x and the CoxFe1-x alloys , the calculated spin density weighte d intrinsic \ndamping of the pure elements (not plotted) deviates significantly from the determined intrinsic \ndamping of the alloys, in contrary to the  good agreement  archived  for the  CoxNi1-x alloys. We \nspeculate that this difference between the alloy syste ms is caused by the non -monotonous \ndependence of the density of states at the Fermi Energy in the CoxFe1-x and NixFe1-x systems.  \nOther  calculated damping values for the pure elements and the Ni80Fe20 and Co 90Fe10 alloys \nare compared to the determined intr insic damping in Table 1. Generally , the calculations \nunderestimate the damping significantly, but our data are in good agreement with more recent \ncalculations for Permalloy ( Ni80Fe20).   11 It is important to point out that n one of the theories  considered he re include thermal \nfluctuations . Regardless, we find exceptional agreement with the calculations to αint at intermediate \nalloy concentrations . We speculate that the modeling of atomic disorder in the alloys in the \ncalculations, by the coherent potential approximation (CPA)  could be responsible for this \nexceptional agreement. The effect of disorder on the  electronic band structure possibly dominates \nany effect s due to nonzero temperature. Indeed, both effects cause a broadening of the bands due \nto enhanced  momentum scattering rates. This directly correlates to a change of the damping \nparameter  according to  the theory of Gilmore and Stiles9. Therefore , the inclusion of the inherent  \ndisorder  of solid -solution alloys  in the calculations  by Mankovsky et al1 mimic s the effects of \ntemperature on damping  to some extent . This argument is corroborated by the fact that the \ncalculations  by Mankovsky et al1 diverge for diluted alloys and pure elements  (as shown in Fig. 2 \n(c) for pure Fe) , where no  or to little disorder is introduced to account for temperature effects. \nMankovsky et al.1 performed temperature dependent calculations of the damping for pure bcc Fe, \nfcc Ni and hcp Co  and the values for 300 K are shown in Table 1 and Fig. 3. These calculations  for \nαint at a temperature of 300 K  are approximately a factor of two  less than our measured values , but \nthe agreement is  significantly improved relative to those  obtained by calculations that neglect \nthermal fluctuations . \n \n \n Figure 5: The intrinsic damping α int is plotted against ( g-2)2 for \nall alloys. We do not observe a proportionality between α int and \n(g-2)2. \n12 Finally, i t has been  reported45,64 that there is a  general proportionality between αint and (g-\n2)2 , as contained in the original microscopic BFS model proposed by  Kambersky .62 To examine \nthis relationship, w e plot αint versus (g-2)2 (determined in Ref [66]) for all samples measured here  \nin Figure 5 . While some samples with large values for ( g-2)2 also exhib it large αint, this is not a \ngeneral trend for all the measured samples . Given  that the damping is not purely a function of the \nspin-orbit strength, but also depends on the details of the band structure , the result  in Fig.  5 is \nexpected .  For example , the amount of  band overlap  will determine the amount of interband \ntransition leading to that damping channel.  Furthermore, the density of states at the Fermi energy \nwill affect the intraband contribution to the damping9,10.   Finally , the ratio of inter - to intra -band \nscattering that mediate s damping contributions at a fixed temperature (RT for our measurements) \nchanges  for different elements9,10 and therefore with alloy concentration. None of these f actors are \nnecessarily proportional to the spin -orbit coupling .  Therefore , we conclude that this simple \nrelation, which originally traces to an order of magnitude estimate  for the case of spin relaxation \nin semiconductors65, does not hold for  all magnetic systems  in general.   \n \n4  Summary  \nWe determined  the damping for the full compositi on range of the binary  3d transition  metal all oys \nNi-Co, Ni -Fe, and Co -Fe and showed that the measured damping can be explained by three \ncontributions to the damping: Intrinsic damping, radiative damping and damping due to spin \npumping. By quantifying all  extrinsic contributions to the measured damping, we determine  the \nintrinsic damping over the whole range of alloy compositions .  These values are compared  to \nmultiple theoretical calculations and yield excellent qualitative and good quantitative agreement for \nintermediate alloy concentrations. 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Schoen, Martin A. W. , Lucassen, Juriaan , Nembach, Hans T. , Silva, T. J. , Koopmans, Bert , Back, Christian H. , \nShaw, Justin M.  Magnetic properties of ultra -thin 3d transition -metal binary alloys I: spin and orbital mo ments, \nanisotropy, and confirmation of Slater -Pauling behavior . arXiv:1701.02177  (2017 ).  \n \n \n "    },    {        "title": "1304.0977v2.Damping_the_zero_point_energy_of_a_harmonic_oscillator.pdf",        "content": "arXiv:1304.0977v2  [quant-ph]  31 Jul 2013Damping the zero-point energy of a harmonic\noscillator\nT G Philbin and S A R Horsley\nPhysics and Astronomy Department, University of Exeter, Stock er Road, Exeter EX4\n4QL, UK.\nE-mail:t.g.philbin@exeter.ac.uk ,s.horsley@exeter.ac.uk\nAbstract. Thephysicsofquantumelectromagnetismin anabsorbingmedium isth at\nof a field of damped harmonic oscillators. Yet until recently the damp ed harmonic\noscillator was not treated with the same kind of formalism used to des cribe quantum\nelectrodynamics in a arbitrary medium. Here we use the techniques o f macroscopic\nQED, based on the Huttner–Barnett reservoir, to describe the q uantum mechanics of\na damped oscillator. We calculate the thermal and zero-point energ y of the oscillator\nfor a range of damping values from zero to infinity. While both the the rmal and\nzero-point energies decrease with damping, the energy stored in t he oscillator at fixed\ntemperature increases with damping, an effect that may be experimentally observable.\nAs the results follow from canonical quantization, the uncertainty principle is valid for\nall damping levels.\nPACS numbers: 03.65.-w, 03.65.Yz, 03.70.+kDamping the zero-point energy of a harmonic oscillator 2\n1. Introduction\nThe damped harmonic oscillator has a central place in physics, due to the prevalance\nof dissipation and linear response in our description of the physical w orld. Yet while\nthe classical treatment of the damped oscillator is elementary and w ell understood, the\nopposite is true in quantum physics. The dissipation of energy leads t o difficulties in\napplying the standard quantization rules to the damped oscillator [1, 2, 3, 4, 5, 6, 7,\n8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. Moreover, cu rrent experimental\nwork is probing the degree to which quantum mechanics may describe the macroscopic\nworld [22, 23, 24, 25, 26]. In these experiments the regime is often o ne where linear\nresponse is applicable and the degree of dissipation is of interest. It is not yet clear\nto what extent existing approaches to quantum damped systems a ccurately describe\nthe results [27]. Here we explore the consequences of a recent, an d particularly\ngeneral treatment of the damped harmonic oscillator [21], where a measured value\nof the dissipative response of the system as a function of frequen cy may be input\nas a parameter directly into the Hamiltonian, something which is not po ssible in\nprevious approaches. The thought is that, in analogy to electroma gnetism, where\nthe susceptibilities of a medium are routinely measured and then used as parameters\nwithin the quantum theory [28, 29], one might extend this approach t o describe the\naforementioned experiments. Through choosing a model suscept ibility where the system\nis exactly solvable, we find the interesting result that the ground st ate energy of a\ndamped oscillator—as calculated using the Hamiltonian of mean force— decreases as\nthe damping is increased, and propose a means whereby this phenom enon might be\nmeasured.\nThe technical difficulties in the canonical quantization of the damped harmonic\noscillator can be overcome through the inclusion of reservoir degre es of freedom that\ntake up the dissipated energy. If the reservoir has a finite, or cou ntably infinite, number\nof degrees of freedom, then a delicate limiting procedure must be em ployed to try to\ncapture genuine damping behaviour [3, 10, 11, 12, 13, 14, 15, 16, 1 7, 18, 19, 20]. This\nlimiting procedure is not usually performed in detail, but the subtleties involved are\nlucidly demonstrated by Tatarskii [13]. The limiting procedure amount s to a transition\nfrom a finite to an uncountably infinite number of dynamical degrees of freedom in the\nreservoir. The fact that this limit is imposed after the field equations have already\nbeen solved under the assumption of a finitereservoir obscures any connection with the\noriginal Hamiltonian. Indeed, it has been shown [13] that the limiting pr ocedure must\nbe separately constructed for each specific damping behaviour if it is to be rigorously\nperformed, something that is generally dispensed with in practice. R ather than follow\nthis well-trodden path, here we apply the technique used to quantiz e electromagnetism\nin absorbing media [30, 29], where a reservoir with an uncountably infin ite number of\ndegrees of freedom is used from the outset. This method was exte nded to the damped\noscillator in [21] and has the advantage that the imaginary part of th e susceptibility\nthat governs the dynamics of the oscillator appears explicitly as a pa rameter in theDamping the zero-point energy of a harmonic oscillator 3\nLagrangian or Hamiltonian. There is no limiting procedure to be perfor med and\nmoreover the quantum dynamics of the damped oscillator is then plac ed on the same\nfooting as macroscopic quantum electrodynamics. In what follows w e use this approach\nto explore the behaviour of the thermal and zero-point energy of the quantum damped\noscillator. We applythe previously derived results [21] for ageneral damped oscillator in\nthermal equilibrium to a simple model susceptibility that allows exact an alytic solution\nfor all quantities of interest. This example is used as a guide to the ty pe of effects that\nmay be measurable with current macroscopic quantum oscillators. W e find that the\nzero-point energy of the oscillator is less than /planckover2pi1ω0/2, where ω0is the free oscillation\nfrequency in the absence of damping. The energy removed in cooling the oscillator from\ntemperature T >0 to its quantum ground state is found to increase with damping,\nwhich offers one possibility of experimentally demonstrating the effec t derived here. It\nis also plausible that the damping might be engineered, as we discuss be low.\n2. Thermal equilibrium and a choice of susceptibility\nThe form of the Lagrangian for a single damped degree of freedom, q(t) (unit mass and\nfree oscillation frequency ω0) was previously given as [21]\nL=1\n2/parenleftbig\n˙q2−ω2\n0q2/parenrightbig\n+q/integraldisplay∞\n0dωα(ω)Xω+1\n2/integraldisplay∞\n0dω/parenleftBig\n˙X2\nω−ω2X2\nω/parenrightBig\n,(1)\nwhere the Xωconstitute the reservoir and are labelled by a continuum ‘index’ ω. The\ncoupling function α(ω) between reservoir and oscillator is related to the imaginary part\nof the susceptibility by\nα(ω) =ω0/radicalbigg\n2ωIm[χ(ω)]\nπ, (2)\nwhereχ(ω) is the linear susceptibility that quantifies the effect of the environm ent on\nthe motion of the oscillator (the factor of ω0inα(ω) is present so that the susceptibility\nis dimensionless). The susceptibility χ(ω) obeys the Kramer-Kronig relations [21], so\nthe imaginary part of χ(ω) that appears in the Lagrangian (1) determines the real part.\nIn fact the Kramers-Kronig relations give the full susceptibility in te rms of the coupling\nfunction α(ω) as [21]\nω2\n0χ(ω) = P/integraldisplay∞\n0dξα2(ξ)\nξ2−ω2+iπα2(ω)\n2ω. (3)\nTheresultingtheoryismoretransparentthanCaldeira–Leggettt ypeapproaches, inthat\nthe reservoir stipulated in the Lagrangian is sufficient for the task a nd does not have\nto be modified later. In fact, the approach in (1) is essentially identic al to that used in\nquantum electromagnetism within an arbitrary absorbing dielectric m edium, which can\nof course be viewed as a field theory of damped oscillators. In partic ular, the dynamics\nof a general damped harmonic oscillator is governed by an arbitrary susceptibility\nthat obeys the Kramers-Kronig relations [21], just as for the dyna mics of light in a\nmaterial medium. Our point of view is that canonical quantization of m acroscopicDamping the zero-point energy of a harmonic oscillator 4\nelectromagnetismforarbitrarydielectrics[29,31]andcanonicalq uantizationofageneral\ndamped harmonic oscillator [21] are both most effectively carried out using the powerful\nreservoir formalism originally introduced by Huttner and Barnett [30 ].\nFrom (1) we can derive a Hamiltonian and apply the standard quantiza tion rules.\nThe position operator for the oscillator is given by ˆ q, the canonical momentum by ˆΠq(t),\nand [ˆq(t),ˆΠq(t)] =i/planckover2pi1is always satisfied [21]. In thermal equilibrium the expectation\nvalues of the squares of the position and momentum operators wer e previously shown\nto be [21]\n/angbracketleftbig\nˆq2(t)/angbracketrightbig\n=/planckover2pi1\nπ/integraldisplay∞\n0dωcoth/parenleftbigg/planckover2pi1ω\n2kBT/parenrightbigg\nImG(ω), (4)\n/angbracketleftBig\nˆΠ2\nq(t)/angbracketrightBig\n=/planckover2pi1\nπ/integraldisplay∞\n0dωω2coth/parenleftbigg/planckover2pi1ω\n2kBT/parenrightbigg\nImG(ω), (5)\nwhereG(ω) is the Green function for the motion of the oscillator, containing th e\nsusceptibility χ(ω):\nG(ω) =−1\nω2−ω2\n0[1−χ(ω)]. (6)\nWhen the damping is zero the susceptibility χ(ω) vanishes and the Green function (6)\nis that of a free oscillator of frequency ω0. The results (4) and (5) could have been\nanticipated from the fluctuation–dissipation theorem [32], but here they are derived [21]\nfromthe Lagrangian(1), illustrating the consistency of our appro ach with known results\nfrom statistical physics.\nThe energy of the oscillator in thermal equilibrium cannot be anticipat ed from the\nfluctuation–dissipation theorem, but must be obtained from the to tal thermal energy\nof the coupled oscillator/reservoir system. This was calculated in [2 1] by subtracting,\nfrom the total thermal energy, the thermal energy of the rese rvoir in the absence of any\ncoupling to the oscillator, giving the result\n/angbracketleftˆH/angbracketrightq=/planckover2pi1\n2π/integraldisplay∞\n0dωcoth/parenleftbigg/planckover2pi1ω\n2kBT/parenrightbigg\nIm/braceleftbigg/bracketleftbigg\nω2\n0/parenleftbigg\nωdχ(ω)\ndω−χ(ω)+1/parenrightbigg\n+ω2/bracketrightbigg\nG(ω)/bracerightbigg\n.(7)\nA similar procedure in the electromagnetic case gives the Casimir (zer o-point plus\nthermal) stress-energy of the electromagnetic field in a material [2 9]. In the Appendix\nwe show that the prescription giving (7) is equivalent to the thermal average of the\nHamiltonian of mean force [33], consistent with earlier work on the the rmodynamics\nof strongly coupled systems [34]. In light of this, the results of [29] s how that the\nCasimir energy density is the thermal average of the Hamiltonian of m ean force for the\nelectromagnetic field in a macroscopic medium, a connection that was not recognised\nin [29] and which does not appear to be widely appreciated.\nIn accordance with our viewpoint that damping ought to be generally treated in\nthe same way as electromagnetic dissipation, the susceptibility χ(ω) of a real damped\noscillator should be measured rather than postulated. Recall that the electromagnetic\nsusceptibility of an individual material sample must be measured, and will vary even for\nsamples of the same material. The position and momentum correlation functions [21]Damping the zero-point energy of a harmonic oscillator 5\nof the damped oscillator in thermal equilibrium provide one method of e xperimentally\nextracting the quantities χ(ω) andω0that appear in (6).\nIn the absence of tabulated values, it is instructive to consider simp le formulae for\nthe susceptibility of a damped oscillator. In [21] the example of dampin g proportional\nto velocity was treated in detail, but this gives some problems with the zero-damping\nlimit at T >0 if the corresponding susceptibility χ(ω) is taken to hold strictly at\nall frequencies up to infinity. In addition, the case of damping propo rtional to velocity\nwould not be expected to be experimentally relevant [21]. We theref ore consider another\nexample, chosen to be simple enough to allow exact analytical solution while being well-\nbehaved in the limit of zero damping of the oscillator. Our particular su sceptibility\ntakes the form\nχ(ω) =2γ2(γ2\n1+ω2\n0)\nω2\n0(γ1+2γ2−iω), (8)\nwhereγ1andγ2are positive real constants. Being analytic in the upper-half comple x-\nfrequency plane, the real and imaginary parts of (8) exhibit Krame r-Kronig relations\nand from (2) and (3) we find that the coupling function α(ω) in (1) corresponding to\nthe susceptibility (8) is\nα(ω) =/radicalBigg\n4γ2ω2(γ2\n1+ω2\n0)\nπ[(γ1+2γ2)2+ω2]. (9)\nIt is important to note that the thermal results (4), (5) and (7) a re only valid for cases\nwhere the total Hamiltoniancan bediagonalized into normal modes, w hich isnot always\npossible [21]. A sufficient condition for the Hamiltonian to be diagonalizab le was found\nin [21] to be\nω2\n0>/integraldisplay∞\n0dξα2(ξ)\nξ2, (10)\na restriction that would not be transparent from the fluctuation– dissipation theorem.\nFor the coupling function (9), the condition (10) yields\nω2\n0>2γ1γ2, (11)\nwhich we assume to hold throughout. When γ2→0,γ1remaining fixed, the\nsusceptibility (8) vanishes and the case of an undamped oscillator is r ecovered.\nThe Green function (6) for the susceptibility (8) has poles\nω=−iγ1, ω=−iγ2±ω1, (12)\nω1=/radicalBig\nω2\n0−γ2(2γ1+γ2), (13)\nwhich correspond to the eigenfrequencies of the oscillator coupled to the reservoir. The\nconstants γ1andγ2thus serve as damping constants of the oscillator, while ω1is a\nmodified oscillation frequency when it is real. Note that (11) implies tha t the poles (12)\nare all in the lower-half complex-frequency plane so the Green func tion has retarded\nboundary conditions. Note also that the over-damped case with ima ginaryω1can occur\nwhile still satisfying (11).Damping the zero-point energy of a harmonic oscillator 6\n3. Thermal and zero-point results\nThe thermal expectation values (4), (5) and (7) can all be evaluat ed exactly for the\nsusceptibility (8). The integrands in each case are even functions o fωforT >0 and so\ncan be rewritten with lower integration limit of −∞; the integrals are then evaluated by\nclosing the integration contour in the upper (or lower) half-plane. T he infinite sum over\nthe residues of the poles of the hyperbolic cotangent function can be evaluated exactly,\nbut the resulting expressions are rather lengthy and we do not give them here. In the\nlimitγ2→0 we find the expectation values (4), (5) and (7) reduce to the fre e-oscillator\nvalues (/planckover2pi1/2ω0)coth(/planckover2pi1ω0/2kBT), (/planckover2pi1ω0/2)coth(/planckover2pi1ω0/2kBT) and (/planckover2pi1ω0/2)coth(/planckover2pi1ω0/2kBT),\nrespectively (by the virial theorem, the momentum-squared expe ctation value is equal\nto the thermal energy for a free oscillator with unit mass). The the rmal energy as a\nfunction of the damping γ2is plotted for temperature T=/planckover2pi1ω0/kBin Figure 2.\nThe zero-point ( T= 0) values of (4), (5) and (7) take a simpler form than the\nthermal results. They are best evaluated separately rather as t heT→0 limit of the\nthermal case, and we find them to be\n/angbracketleftbig\nˆq2(t)/angbracketrightbig\n=/planckover2pi1\nπω1(ω2\n0+γ2\n1−4γ1γ2)\n×/bracketleftbigg\n(ω2\n1+γ2\n1−γ2\n2)arctan/parenleftbiggω1\nγ2/parenrightbigg\n+γ2ω1ln/parenleftbiggω2\n0−2γ1γ2\nγ2\n1/parenrightbigg/bracketrightbigg\n, (14)\n/angbracketleftBig\nˆΠ2\nq(t)/angbracketrightBig\n=/planckover2pi1\nπω1(ω2\n0+γ2\n1−4γ1γ2)\n×/braceleftbigg/bracketleftbig\n(ω2\n1+γ2\n2)2+γ2\n1(ω2\n1−γ2\n2)/bracketrightbig\narctan/parenleftbiggω1\nγ2/parenrightbigg\n−γ2\n1γ2ω1ln/parenleftbiggω2\n0−2γ1γ2\nγ2\n1/parenrightbigg/bracerightbigg\n,(15)\n/angbracketleftˆH/angbracketrightq=/planckover2pi1\n2π/braceleftbigg\n2ω1arctan/parenleftbiggω1\nγ2/parenrightbigg\n+γ1ln/parenleftbigg\n1+2γ2\nγ1/parenrightbigg\n+γ2ln/bracketleftbigg(γ1+2γ2)2\nω2\n0−2γ1γ2/bracketrightbigg/bracerightbigg\n. (16)\nRecall that these expressions presuppose the inequality (11) and note that they are real\nin the over-damped case where ω1is imaginary. We now consider the dependence of\n(14)–(16) on the parameters within the susceptibility.\nFigure1showstheuncertainties inpositionandmomentum, ∆ qand∆p(thesquare\nroots of (14) and (15)), as functions of γ2forω0= 1010s−1andγ1=ω0/4. As damping\n(γ2) increases the product ∆ q∆pincreases from the minimum allowed value /planckover2pi1/2. The\nposition uncertainty ∆ qincreases with damping while ∆ pdecreases. The observed\nadherencetotheposition/momentumuncertaintyrelationisunsur prising, giventhatthe\nresults are based on canonical quantization, but violations of the u ncertainty principle\noccur in other approaches to the damped oscillator [1, 2].\nThe zero-point energy (16) of the oscillator is plotted in Figure 2 as a function of γ2\nforthesame valuesof ω0andγ1used inFigure1; theenergyattemperature T=/planckover2pi1ω0/kB\nis also plotted. As γ2increases the zero-point energy is damped below the free-oscillato r\nvalue/planckover2pi1ω0/2. In our example, the oscillation frequency for γ2>0 isω1, given by (13),\nprovided ω1is real. We emphasise, as is clear from (16), that the zero-point ene rgy of\nthe damped oscillator is not /planckover2pi1ω1/2. In fact the range of γ2in Figure 2 passes throughDamping the zero-point energy of a harmonic oscillator 7\n/CapDeltaq2Ω0\n/CapDeltap2/Slash1Ω0/CapDeltaq/CapDeltap\nΩ0/Slash12Ω0/Slash14 3Ω0/Slash140/HBar\n2/HBar\nΓ2\nFigure 1. Plots of the zero-point position uncertainty ∆ q(square root of (14)) and\nmomentum uncertainty ∆ p(square root of (15)) versus damping γ2withω0= 1010s−1\nandγ1=ω0/4. The squares of the uncertainties are scaled with an appropriate power\nofω0to havethe same units asthe product ∆ q∆p. The uncertaintyrelationis satisfied\nfor all parameters obeying (11).\nT/Equal0kBT/Equal/HBarΩ0\nΩ0/Slash12Ω0/Slash14 3Ω0/Slash140/HBarΩ0\n2/HBarΩ0\nΓ2/LAngleBracket1H/Hat/RAngleBracket1q\nFigure 2. Plots of the energy of the harmonic oscillator versus damping γ2, with\nω0= 1010s−1andγ1=ω0/4, forT= 0 and T=/planckover2pi1ω0/kB. The zero-point energy\n(T= 0) is damped below the free-oscillator value /planckover2pi1ω0/2. The energy for T >0 is\nalso damped below the free-oscillator value, though this damping is no t very apparent\nexcept for very low T. The energy that can be extracted from the oscillator at T >0\n(i.e. theT >0 energy minus the zero-point energy) increases with damping.\nω1= 0 and into the over-damped case where ω1is imaginary. The rather complicated\nzero-point energy (16) thus cannot be simply related to the oscillat ion behaviour, given\nby (12) and (13), although both are affected by the damping. The e nergy at T >0 is\nalso damped below the free-oscillator value but the difference betwe en theT >0 energy\nand the zero-point energy increases with damping. This shows that the energy stored\nin the oscillator at fixed Tincreases with damping, so that an increasing quantity ofDamping the zero-point energy of a harmonic oscillator 8\nenergy must be removed to bring the oscillator to its ground state.\nWe can also consider the limit of infinite damping, which occurs when γ2→ ∞.\nThe condition (11) then requires γ1→0, which we can satisfy by setting γ1=ω2\n0/(4γ2)\nfor example. With this value for γ1, the zero-point energy (16) goes to zero for infinite\ndamping γ2→ ∞, with a leading term of\n/angbracketleftˆH/angbracketrightq∼/planckover2pi1ω2\n0\n4πγ2/bracketleftbigg\n1+2ln/parenleftbigg23/2γ2\nω0/parenrightbigg/bracketrightbigg\n. (17)\nThe asymptotic approach of the zero-point energy to zero as γ2→ ∞is thus very slow.\nWe must note also that the position uncertainty diverges as ∆ q∼2/radicalbig\n/planckover2pi1γ2/π/ω0in this\ninfinite-damping limit (with γ1=ω2\n0/(4γ2)). A large displacement of the oscillator can\nbe expected to lead to nonlinear behaviour, so our assumption of a lin ear oscillator is\nnot realistic for extremely large damping with susceptibility (8).\nAs noted at the outset, the susceptibility is a quantity that must be measured,\nand in addition the “free-oscillation” frequency ω0is a parameter that must also be\nfitted to experimental data [21]. The theoretical ideal is an oscillato r whose damping\ncan be tuned from zero to a desired level, but this is a heavy demand in practice. A\nmore realistic scenario is a set of macroscopic oscillators prepared w ith slightly different\nmaterial geometries so that the damping varies slowly across the se t. In the absence\nof data for the susceptibilities of such a set of oscillators, the resu lts for the simple\nsusceptibility (8) give some qualitative indications. The results illustra ted in figure 2\nsuggest that if all oscillators in the set are brought to a fixed tempe rature, then the\nenergy removed from the oscillators in reaching their ground state s will increase with\ndamping, where the damping level is determined from the measured s usceptibilities.\n4. Conclusions\nThe quantum damped oscillator can be described using the technique s of macroscopic\nquantumelectrodynamics. Usingthisapproachwehavecalculatedt hethermalandzero-\npoint energy of a damped oscillator for a simple model susceptibility. E xperimental\nquantum oscillators will be characterized by susceptibilities that mus t be measured,\njust as the electromagnetic susceptibilities of materials must be mea sured to quantify\neffects such as Casimir forces. Our analytical results for a model s usceptibility show an\ninteresting effect that may also be present in experimental system s. The energy stored\nin the oscillator, which must be removed to reach the quantum groun d state, increases\nwith temperature due to damping of the zero-point energy. By eng ineering the damping\nof a set of oscillators, it may be possible to observe this effect in curr ent experimental\nsystems [22, 23, 24, 25, 26, 27].\nAcknowledgements\nWe are indebted to J. Anders for informing us of the Hamiltonian of me an force. We\nalso thank N. Kiesel and M. Aspelmeyer for interesting discussions.Damping the zero-point energy of a harmonic oscillator 9\nAppendix\nConsider a system composed of two interacting parts; the system of interest ( S), and a\nreservoir ( R). The total Hamiltonian of this system is of the form, ˆH=ˆHS+ˆHI+ˆHR,\nwhereˆHIcharacterises the coupling (of arbitrary strength) between Sand the reservoir.\nWe ask the question, what is the energy of Sin thermal equilibrium ?\nA choice of Hamiltonian, ˆH⋆that gives the correct equilibrium properties for S\nwithout reference to Ris the Hamiltonian of mean force [33]\nˆH⋆=−β−1log/parenleftBig\nZ−1\nRTrR/bracketleftBig\ne−βˆH/bracketrightBig/parenrightBig\n(A.1)\nwhereβ= 1/kT, andZR= TrR[exp(−βˆHR)]. The partition function Z⋆associated\nwithˆH⋆is then\nZ⋆= TrS/bracketleftBig\ne−βˆH⋆/bracketrightBig\n=Z\nZR, (A.2)\nwhereZ= Tr[exp( −βˆH)] is the total partition function. It is evident that equilibrium\naverages of quantities pertaining to Salone, computed using ˆH⋆will be identical to\nthose calculated from the full Hamiltonian ˆH. The factor of Z−1\nRwithin the logarithm\nplays no role in such a calculation, but is determined by the requiremen ts that (a) when\nˆHI→0 thenˆH⋆→ˆHS; and (b) the free energy F⋆associated with Sis [34]\nF⋆=−β−1log(Z⋆) =F−FR, (A.3)\nwhich is the amount of energy available to do work in a reversible, isoth ermal change\nof state of S, including that obtained through decoupling SandR[34]. In the case\nconsidered in the main text, where the q-oscillator plays the role of S, (A.3) will give\nthe correct generalized force (and therefore work done during a ny isothermal change of\nstate) when F⋆is differentiated with respect to the “free-oscillation” frequency ω0, or\nthe quantities γ1,2within the coupling of the oscillator to the reservoir. Furthermore,\nwhenγ1,2→0 thenF⋆→FS.\nGiven the above properties, (A.1) is interpreted as the effective Ha miltonian of S\nin thermal equilibrium. In answer to our initial question, the equilibrium average of the\nassociated energy is, using (A.2),\n/angbracketleftˆH⋆/angbracketright=−∂log(Z/ZR)\n∂β=/angbracketleftˆH/angbracketright−/angbracketleftˆHR/angbracketright, (A.4)\nwhere/angbracketleftˆHR/angbracketright=−∂log(ZR)/∂βis the equilibrium average of the energy of Rin the\nabsence of any coupling to S. This is the prescription that was previously used to\ncalculate the Casimir energy density [29] and the thermal energy of a damped harmonic\noscillator [21], the latter of which is given by (7).\nReferences\n[1] Dekker H 1981 Phys. Rep. 801\n[2] Um C I, Yeon K H and George T F 2002 Phys. Rep. 36263Damping the zero-point energy of a harmonic oscillator 10\n[3] Weiss U 2008 Quantum Dissipative Systems 3rd ed (Singapore: World Scientific)\n[4] Bateman H 1931 Phys. Rev. 38815\n[5] Grabert H and Weiss U 1984 Z. Phys. B5587\n[6] Blasone M and Jizba P 2002 Can. J. Phys. 80645; 2004 Ann. Phys. 312354\n[7] Latimer D C 2005 J. Phys. 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Phys. 14, 083043\n[22] O’Connell A D et al.2010Nature464697\n[23] Teufel J D, Donner T, Li D, Harlow J W, Allman M S, Cicak K, Sirois A J , Whittaker J D,\nLehnert K W and Simmonds R W 2011 Nature475359\n[24] Chan J, Alegre T P M, Safavi-Naeini A H, Hill J T, Krause A, Gr¨ obla cher S, Aspelmeyer M and\nPainter O 2011 Nature47889\n[25] Aspelmeyer M, Gr¨ oblacher S, Hammerer K and Kiesel N 2010 J. Opt. Soc. Am. B27A189\n[26] Poot M and Zant H S J 2012 Phys. Rep. 511273\n[27] Gr¨ oblacher S, Trubarov A, Prigge N, Aspelmeyer M and Eisert J , arXiv:1305.6942.\n[28] Scheel S and Buhmann S Y 2008 Acta Phys. Solv. 58675\n[29] Philbin T G 2010 New J. Phys. 12123008; 2011 New J. Phys. 13063026\n[30] Huttner B and Barnett S M 1992 Phys. Rev. A464306\n[31] Horsley S A R 2011 Phys. Rev. A84063822; 2012 Phys. Rev. A86023830; arXiv:1301.4178\n[quant-ph].\n[32] Landau L D and Lifshitz E M 1980 Statistical Physics, Part 1 3rd ed (Oxford: Butterworth-\nHeinemann)\n[33] Campisi M, Talkner P and H¨ anggi P 2009 Phys. Rev. Lett. 102210401; 2009 J. Phys. A42392002\n[34] Ford G W, Lewis J T and O’Connell R F 1985 Phys. Rev. Lett. 552273; 1988 J. Stat. Phys. 53\n439"    },    {        "title": "2211.04711v2.Gravitational_wave_constraints_on_spatial_covariant_gravities.pdf",        "content": "Gravitational-wave constraints on spatial covariant gravities\nTao Zhua;b,\u0003Wen Zhaoc;d,†and Anzhong Wange‡\naInstitute for theoretical physics and cosmology,\nZhejiang University of Technology, Hangzhou, 310032, China\nbUnited Center for Gravitational Wave Physics (UCGWP),\nZhejiang University of Technology, Hangzhou, 310032, China\ncCAS Key Laboratory for Research in Galaxies and Cosmology, Department of Astronomy,\nUniversity of Science and Technology of China, Hefei 230026, China\ndSchool of Astronomy and Space Sciences, University of Science and Technology of China, Hefei, 230026, China\neGCAP-CASPER, Physics Department, Baylor University, Waco, Texas 76798-7316, USA\n(Dated: February 22, 2023)\nThe direct discovery of gravitational waves (GWs) from the coalescence of compact binary com-\nponents by the LIGO/Virgo/KAGRA Collaboration provides an unprecedented opportunity for\nexploring the underlying theory of gravity that drives the coalescence process in the strong and\nhighly dynamical \feld regime of gravity. In this paper, we consider the observational e\u000bects of\nspatial covariant gravities on the propagation of GWs in the cosmological background and obtain\nthe observational constraints on coupling coe\u000ecients in the action of spatial covariant gravities from\nGW observations. We \frst decompose the GWs into the left- and right-hand circular polarization\nmodes and derive the e\u000bects of the spatial covariant gravities on the propagation equation of GWs.\nWe \fnd that these e\u000bects can be divided into three classes: 1) frequency-independent e\u000bects on\nGW speed and friction, 2) parity-violating amplitude and velocity birefringences, and 3) a Lorentz-\nviolating damping rate and dispersion of GWs. With these e\u000bects, we calculate the corresponding\nmodi\fed waveform of GWs generated by the coalescence of compact binaries. By comparing these\nnew e\u000bects with the publicly available posterior samples or results from various tests of gravities\nwith LIGO/Virgo/KAGRA data in the literature, we derive the observational constraints on cou-\npling coe\u000ecients of the spatial covariant gravities. These results represent the most comprehensive\nconstraints on the spatial covariant gravities in the literature.\nI. INTRODUCTION\nThe direct detection of gravitational waves (GWs) by\nLIGO/Virgo Collaboration has ushered in an entirely\nnew era of gravitational-wave astronomy [1{7]. To date,\nthe LIGO/Virgo/KAGRA Collaboration has announced\nthe detection of more than 90 con\fdent GW events in\nthe Gravitational-Wave Transient Catalog (GWTC) [5{\n7]. These signals are produced by the coalescence of com-\npact binaries, including binary black holes (BBH), binary\nneutron stars, and black hole-neutron star binaries. The\nGWs of these events, carrying valuable information about\nlocal spacetime properties of the compact binaries, allow\nus to explore the extreme gravity regime of spacetime,\nwhere the \felds are strong, nonlinear, and highly dynam-\nical. This has enabled a lot of model-independent tests\nof general relativity (GR) by the LIGO/Virgo/KAGRA\nCollaboration [8{13]. All of the tests to date have con-\n\frmed that GW data is consistent with the predictions\nof GR.\nWith these substantial successes of GR, the in-\ncreasing number of detected GW events from the\nLIGO/Virgo/KAGRA Collaboration also provides a\nvaluable window to explore, distinguish, or constrain any\nmodi\fed theories that exhibit deviations from GR. This\n\u0003zhut05@zjut.edu.cn; Corresponding author\n†wzhao7@ustc.edu.cn\n‡anzhong wang@baylor.eduhas stimulated a lot of work on constraining di\u000berent\nmodi\fed theories of gravity with GW data. In this pa-\nper, we consider a speci\fc type of modi\fed theory of\ngravity, the spatial covariant gravities [14{17], and test\nthem with the current population of GW events.\nThe spatial covariant gravities represent a series of\nalternative modi\fed theories of GR, which break time\ndi\u000beomorphism invariance but preserve the spatial one\n[14, 17]. This is very similar to the case of the Ho\u0014 rava-\nLifshitz (HL) theory of quantum gravity [18{25], in which\nthe symmetry of the theory is broken from the gen-\neral covariance down to the foliation-preserving di\u000beo-\nmorphisms. Such spatial covariance allows one to con-\nstruct the action of the theory only in terms of spatial\ndi\u000beomorphism invariants and study the e\u000bects of dif-\nferent terms. These new terms, which are absent in the\nEinstein-Hilbert action of GR, provide an e\u000ecient way\nto parametrize unknown high-energy physics e\u000bects on\nthe low-energy scale. On the other hand, the spatial co-\nvariant gravities can also represent a very general frame-\nwork for describing a lot of scalar-tensor theories in uni-\ntary gauge [14, 15, 26, 27]. To our knowledge, a lot of\nscalar-tensor theories can be mapped to the spatial co-\nvariant framework by imposing the unitary gauge, includ-\ning Horndeski theory, Chern-Simons modi\fed gravity,\nWeyl gravity, ghost-free parity-violating gravities, D!4\nGauss-Bonnet gravity, Ho\u0014 rava-Lifshitz gravities, etc. (see\ndetails in Ref. [28]).\nOne natural question now is whether the new terms\nbeyond GR introduced in the spatial covariant gravitiesarXiv:2211.04711v2  [gr-qc]  21 Feb 20232\ncan lead to any observational e\u000bects in the current and/or\nforthcoming experiments and observations, so the spatial\ncovariant gravities can be tested or constrained directly\nby observations. Such considerations have attracted a\ngreat deal of attention lately and several phenomenolog-\nical implications of the spatial covariant gravities have\nalready been investigated [28{34]. The phenomenologi-\ncal implications for other theories that can also be de-\nscribed in the spatial covariant framework under certain\nconditions have also been extensively studied, see for ex-\nample, Refs. [27, 35{40] and reference therein. In par-\nticular, the e\u000bects of the spatial covariant gravities on\nthe propagation of GWs in the cosmological background\nwere previously explored in [28]. In addition, the im-\nprints of the spatial covariant gravities on the primordial\nGWs was also calculated in detail recently in Ref. [29]. It\nwas shown that the parity-violating terms in the gravita-\ntional action of the spatial covariant gravities can induce\na nonzero circular polarization in the primordial GWs.\nThe possible signatures of these parity-violating e\u000bects\non the cosmic microwave background and the statistics\nof galaxy surveys were also brie\ry explored in Ref. [29].\nIn this paper, we focus on the imprints of spatial co-\nvariant gravities on the propagation of GWs, produced\nby the coalescence of compact binaries, and their ob-\nservational constraints with observational data of GW\nevents from LIGO/Virgo/KAGRA Collaboration. De-\ncomposing the GWs into the left- and right-hand cir-\ncular polarization modes, we \fnd that the equations of\nmotion of GWs in spatial covariant gravities can be ex-\nactly mapped to the parametrized propagation equation\nproposed in Refs. [41, 42]. Depending on di\u000berent terms\nin the gravitational action of spatial covariant gravities,\nthe new e\u000bects beyond GR can be fully characterized by\nfour parameters: \u0016 \u0017, \u0016\u0016,\u0017A, and\u0016A. The parameters \u0017A\nand\u0016Alabel the e\u000bects of the parity-violating terms in\nthe spatial covariant gravities, and \u0016 \u0017and \u0016\u0016describe the\ne\u000bects of other possible modi\fcations that are not rele-\nvant to parity violation. The correspondences between\ndi\u000berent terms in the spatial covariant gravities and the\nfour parameters are summarized in Table. II. In addition,\nwe present the expressions for these four parameters for\na number of speci\fc theories in Table III.\nDi\u000berent parameters correspond to di\u000berent e\u000bects on\nthe propagation of GWs. These e\u000bects can be divided\ninto three classes: 1) the frequency-independent e\u000bects\nwhich include the modi\fcation to the speed of GWs\nand GW friction, 2) the parity-violating e\u000bects which in-\nclude the amplitude and velocity birefringences of GWs,\nand 3) the Lorentz-violating e\u000bects which include the\nmodi\fed damping rate and dispersion relation of GWs.\nWe compare these new e\u000bects with existing observa-\ntional samples or results of various tests of gravities with\nLIGO/Virgo/KAGRA data in the literature and derive\nthe observational constraints on di\u000berent terms in the\nspatial covariant gravities. Using these constraints, we\nalso derive the corresponding bounds on the coupling\ncoe\u000ecients of a number of speci\fc theories in the Ap-pendix. Our results are summarized in Table V. With the\nfuture ground- and space-based detector network, more\nand more GW events in a wider frequency range will be\ndetected in the future, and we expect that the constraints\non the spatial covariant gravities will be improved dra-\nmatically and a deeper understanding of the nature of\ngravity will be achieved.\nThis paper is organized as follows. In the next section,\nwe present a brief review of the spatial covariant gravities,\nand in Sec. III we discuss the associated propagation of\nGWs in a homogeneous and isotropic cosmological back-\nground. In that section, we also map the di\u000berent new\ne\u000bects on the propagation of GWs arising from the spa-\ntial covariant gravities into four parameters. In Sec. IV,\nwe calculate the e\u000bects of the spatial covariant gravities\non the speed, frictions, and waveform of GWs produced\nby the coalescence of compact binaries, and derive their\nobservational constraints with observational data of GW\nevents from the LIGO/Virgo/KAGRA Collaboration. A\nbrief summary of our main results and some discussions\nare presented in Sec. V. We also present the detailed cor-\nrespondences between the coupling constants in a num-\nber of speci\fc theories and the coe\u000ecients in the spatial\ncovariant gravities and derive the corresponding observa-\ntional bounds on these speci\fc theories in the Appendix.\nThroughout this paper, the metric convention is cho-\nsen as (\u0000;+;+;+), and greek indices ( \u0016;\u0017;\u0001\u0001\u0001) run over\n0;1;2;3 and latin indices ( i; j; k ) run over 1 ;2;3. We\nset the units to c=~= 1.\nII. SPATIAL COVARIANT GRAVITIES\nIn this section, we present a brief introduction of the\nconstruction of the spatial covariant gravities. Most of\nthe expressions and results used here can be found in\nRefs. [14, 17, 28, 29] and references therein.\nThe spatial covariant gravity is only invariant un-\nder the three-dimensional spatial di\u000beomorphism, which\nbreaks the time di\u000beomorphism. Therefore, the gravita-\ntional action of this type of theory can be constructed\nonly in terms of spatial di\u000beomorphism invariants. In\norder to write down the gravitational action, it is con-\nvenient to write the metric of the spacetime in the\nArnowitt-Deser-Misner (ADM) form [43],\nds2=\u0000N2dt2+gij(dxi+Nidt)(dxj+Njdt);(2.1)\nwhereNis the lapse function, gijis the three-dimensional\nspatial metric, and Niis the shift vector. With these\nADM variables, the general action of the spatial covariant\ngravities can be written in the form,\nS=Z\ndtd3xNpgL(N;gij;Kij;Rij;ri;\"ijk);(2.2)\nwhereKijis the extrinsic curvature of t=const hyper-\nsurfaces,\nKij=1\n2N(@tgij\u0000riNj\u0000rjNi); (2.3)3\nRijis the intrinsic curvature tensor, riis the spatial co-\nvariant derivative with respect to gij, and\"ijk=pg\u000fijk\nis the spatial Levi-Civita tensor with \u000fijkbeing the total\nantisymmetric tensor.\nNormally, with the breaking of the time di\u000beomor-\nphism, extra degrees of freedom are often added on top of\nthe two tensorial degrees of freedom in GR. In particular,\nthe spatial covariant gravity which has three dynamical\ndegrees of freedom has been explored extensively [17].\nIt was also shown that under two necessary and su\u000e-\ncient conditions, the spatial covariant gravities can have\njust two tensorial degrees of freedom and no propagating\nscalar mode [44]. One condition is the degenerate condi-\ntion, which requires the lapse-extrinsic curvature sector\nof the Dirac matrix must be degenerate. Another condi-\ntion is the consistent condition, which requires that the\ndimension of the phase space at each spacetime point\nmust be even. In Refs. [16, 25], the above action has\nalso been extended by introducing _Nin the Lagrangian\nthrough1\nN(_N\u0000NiriN). This term can only contribute\nto the scalar-type modes of GWs which are expected to\nbe small, compared to the observed tensorial modes. For\nthis reason, we will not consider it in this paper.\nIn order to construct a concrete gravitational action\nwith spatial covariance, one \frst needs to specify the\nbuilding blocks which are invariant under the spatial dif-\nfeomorphisms. These building blocks consist of linear\ncombinations of the extrinsic curvature Kij, intrinsic cur-\nvatureRij, and their spatial derivatives and derivatives\nof the spatial metric itself. Up to the fourth order in\nderivatives of spatial metric variables gij, we have the\nbuilding blocks as shown in Table I which are all scalars\nunder the spatial di\u000beomorphisms. Then, the general\naction of the gravitational part of the spatial covariant\ngravities is given by [28]\nSg=Z\ndtd3xpgN\u0010\nL(0)+L(1)+L(2)+L(3)+L(4)\n+~L(3)+~L(4)\u0011\n; (2.4)\nwhereL(0);L(1);L(2);L(3), andL(4)are the parity-\npreserving terms, which are given by\nL(0)=c(0;0)\n1; (2.5)\nL(1)=c(1;0)\n1K; (2.6)\nL(2)=c(2;0)\n1KijKij+c(2;0)\n2K2+c(0;2)\n1R; (2.7)\nL(3)=c(3;0)\n1KijKjkKi\nk+c(3;0)\n2KijKijK+c(3;0)\n3K3\n+c(1;2)\n1rirjKij+c(1;2)\n2r2K+c(1;2)\n3RijKij\n+c(1;2)\n4RK; (2.8)\nL(4)=c(4;0)\n1KijKjkKi\nkK+c(4;0)\n2\u0000\nKijKij\u00012\n+c(4;0)\n3KijKijK2+c(4;0)\n4K4+c(2;2)\n1rkKijrkKij+c(2;2)\n2riKjkrkKij\n+c(2;2)\n3riKijrkKk\nj+c(2;2)\n4riKijrjK\n+c(2;2)\n5riKriK+c(2;2)\n6RijKi\nkKjk\n+c(2;2)\n7RKijKij+c(2;2)\n8RijKijK+c(2;2)\n9RK2\n+c(0;4)\n1rirjRij+c(0;4)\n2r2R+c(0;4)\n3RijRij\n+c(0;4)\n4R2; (2.9)\nand ~L(3)and ~L(4)are parity-violating terms which are\ngiven by\n~L(3)=c(2;1)\n1\"ijkKi\nlrjKkl+c(0;3)\n1!3(\u0000); (2.10)\n~L(4)=c(3;1)\n1\"ijkrmKi\nnKjmKkn+c(3;1)\n2\"ijkriKj\nmKk\nnKmn\n+c(3;1)\n3\"ijkriKj\nlKklK+c(1;3)\n1\"ijkRilrjKk\nl\n+c(1;3)\n2\"ijkriRj\nlKkl+c(1;3)\n3!3(\u0000)K: (2.11)\nAll of the coe\u000ecients like c(dt;ds)\ni are functions of tand\nN. Note that in Table I and Eqs. (2.10, 2.11), we add\nthe spatial Chern-Simons term !3(\u0000) and its coupling to\nK, which are absent in the original action in Ref. [28].\nIt is interesting to note that the above action reduces to\nGR if one imposes\nc(2;0)\n1=c(0;2)\n1=\u0000c(2;0)\n2=M2\nPl\n2(2.12)\nwhere all other coe\u000ecients c(dt;ds)\ni are set to zero and MPl\nis the reduced Planck mass.\nIII. GWS IN SPATIAL COVARIANT\nGRAVITIES\nIn this section, we consider the propagation of GWs of\nspatial covariant gravities in a homogeneous and isotropic\nbackground. The spatial covariant gravities can have\nthree degrees of freedom propagating in the theory, of\nwhich two are tensorial and one is of the scalar type.\nThe extra scalar mode, which is absent in GR, is in gen-\neral expected to be small compared to the two observed\ntensorial modes. For this reason, hereafter we only con-\nsider the two tensorial modes of GWs (the transverse\nand traceless modes). In the \rat Friedmann-Robertson-\nWalker spacetime, GW is described by the tensor pertur-\nbations of the metric, i.e.,\nds2=a2(\u001c)[\u0000d\u001c2+ (\u000eij+hij)dxidxj]; (3.1)\nwherea(\u001c) is the scale factor of the expanding Universe\nand hereafter we set a0= 1 as its present value. \u001cdenotes\nthe conformal time, which is related to the cosmic time\ntbydt=ad\u001c.hijdenotes the GWs, which we take to\nbe transverse and traceless, @ihij= 0 =hi\ni. Then, the\naction of GWs up to the quadratic action can be written\nin the form [28],4\nTABLE I. Building blocks of spatial covariant gravities up to fourth order in derivatives of hij, wheredtanddsare the\nnumbers of time and spatial derivatives respectively, and d=dt+dsdenotes the total number of time and spatial derivatives.\nHere!3(\u0000) denotes the spatial gravitational Chern-Simons term, and !3(\u0000) =\"ijk(\u0000m\njl@j\u0000l\nkm+2\n3\u0000n\nil\u0000l\njm\u0000m\nkn) with \u0000k\nij=\n1\n2gkm(@jgmj+@jgij\u0000@mgij) are the spatial Christo\u000bel symbols. The terms in this table are the same as those in Table I of\nRef. [28] except for the two new terms !3(\u0000) and!3(\u0000)K.\nd (dt;ds) Operators\n0 (0;0) 1\n1(1,0) K\n(0, 1) -\n2(2, 0) Kij,K2\n(1, 1) -\n(0, 2) R\n3(3, 0) KijKjkKi\nk,KijKijK,K3\n(2, 1) \"ijkKi\nlrjKkl\n(1, 2) rirjKij,r2K,RijKij,RK\n(0, 3) !3(\u0000)\n4(4, 0) KijKjkKi\nkK,\u0000\nKijKij\u00012,KijKijK2,K4\n(3, 1) \"ijkrmKi\nnKjmKkn; \"ijkriKj\nmKk\nnKmn; \"ijkriKj\nlKklK\n(2, 2)rkKijrkKij,riKjkrkKij,riKijrkKk\nj,riKijrjK,riKriK,RijKi\nkKjk,RKijKij,RijKijK,RK2\n(1, 3) \"ijkRilrjKk\nl; \"ijkriRj\nlKkl,!3(\u0000)K\n(0, 4) rirjRij;r2R; RijRij; R2\nS(2)=Z\ndtd3xa3\n2h\nG0(t)_hij_hij+G1(t)\u000fijk_hli1\na@j_hl\nk\u0000G2(t)_hij\u0001\na2_hij\n+W0(t)hij\u0001\na2hij+W1(t)\u000fijkhli1\na\u0001\na2@jhl\nk\u0000W 2(t)hij\u00012\na4hiji\n;\n(3.2)\nwhereGnandWnare given by [28]1\nG0=1\n2h\nc(2;0)\n1+ 3(c(3;0)\n1+c(3;0)\n2)H+ 3(3c(4;0)\n1+ 2c(4;0)\n2+ 3c(4;0)\n3)H2i\n; (3.3)\nG1=1\n2h\nc(2;1)\n1\u0000(c(3;1)\n1\u00002c(3;1)\n2\u00003c(3;1)\n3)Hi\n; (3.4)\nG2=1\n2c(2;2)\n1; (3.5)\nW0=1\n2h\nc(0;2)\n1+1\n2_c(1;2)\n3+1\n2\u0010\n3c(1;2)\n3+ 6c(1;2)\n4+ 2_c(2;2)\n6+ 3_c(2;2)\n8\u0011\nH\n+1\n2\u0010\n4c(2;2)\n6+ 6c(2;2)\n7+ 9c(2;2)\n8+ 18c(2;2)\n9\u0011\nH2+1\n2\u0010\n2c(2;2)\n6+ 3c(2;2)\n8\u0011\n_Hi\n; (3.6)\nW1=1\n4\u0010\n_c(1;3)\n1+ _c(1;3)\n2\u0011\n+c(0;3)\n1\u00003c(1;3)\n3H; (3.7)\nW2=\u00001\n2c(0;4)\n3: (3.8)\nHere a dot denotes a derivative with respect to the cosmic\ntimet,H= _a=ais the Hubble parameter, and \u0001 \u0011\n\u000eij@i@jwith\u000eijbeing the Kronecker delta. We consider\nthe GWs propagating in the homogeneous and isotropic\n1InW1we add the contributions from the two new terms !3(\u0000)\nand!3(\u0000)K.background , and ignore the source term. With the above\naction, one can obtain the equation of motion for hijas\n\u0012\nG0\u0000G2@2\na2\u0013\nh00\nij+h\n2HG0+G0\n0\u0000G0\n2@2\na2i\nh0\nij\n\u0000\u0014\nW0\u0000W 2@2\na2\u0015\n@2hij5\n+\u000filk@l\na\u0002\nG1@2\n\u001c+ (2HG1+G0\n1)@\u001c\u0000W 1@2\u0003\nhk\nj= 0;\n(3.9)\nwhereH\u0011a0=aand a prime denotes a derivative with\nrespect to the conformal time \u001c.\nIn order to study the propagation of GWs in the spa-\ntial covariant gravities, it is convenient to decompose the\nGWs into the circular polarization modes. To study the\nevolution of hij, we expand it over spatial Fourier har-\nmonics,\nhij(\u001c;xi) =X\nA=R;LZd3k\n(2\u0019)3hA(\u001c;ki)eikixieA\nij(ki);\n(3.10)\nwhereeA\nijdenotes the circular polarization tensors and\nsatis\fes the relation\n\u000fijknieA\nkl=i\u001aAejA\nl; (3.11)\nwith\u001aR= 1 and\u001aL=\u00001. We \fnd that the propagation\nequations of these two modes are decoupled, which can\nbe cast in the form [28]\nh00\nA+ (2 + \u0000A)Hh0\nA+!2\nAhA= 0; (3.12)\nwhere\nH\u0000A=\u0014\nln\u0012\nG0+\u001aAG1k\na+G2k2\na2\u0013\u00150\n;(3.13)\n!2\nA\nk2=W0+\u001aAW1k\na+W2k2\na2\nG0+\u001aAG1k\na+G2k2\na2: (3.14)\nThe properties of the propagation of GWs with nonzero\n\u0000Aand a modi\fed dispersion relation !2\nkwere discussed\nin Ref. [28]. Several speci\fc forms of the spatial covari-\nant gravities in which the GWs propagate at the speed\nof light were also explored [28]. The derivations of the\nspatial covariant gravities from GR are fully character-\nized by the quantities \u0000 Aand!2\nA. The former represents\nthe corrections to the damping rate which modi\fes the\namplitude damping rate of the GWs during their propa-\ngation in the cosmological background, and the latter is\nthe modi\fed dispersion relation of GWs which leads to a\nphase shifting of GWs from distant sources.\nWe expect that the derivations from GR are small,\nsuch that\n\u0000A\u001c1;\f\f\f\f!2\nA\nk2\u00001\f\f\f\f\u001c1: (3.15)\nThus, we can consider all of the new e\u000bects on GWs be-\nyond GR as small corrections to the standard GR result.\nIn this way, we are able to expand H\u0000Aand!Aas\nH\u0000A'(lnG0)0+\u001aA\nG0\u0012\nG1k\na\u00130\n+1\nG0\u0012\nG2k2\na2\u00130\n;(3.16)\n!2\nA\nk2'W0\nG0+\u001aAW1\u0000G1\nG0k\na+W2\u0000G2\nG0k2\na2:(3.17)\nNote that in the above expansion we only consider the\n\frst-order terms of each coe\u000ecient, i.e., 1 \u0000W 0=G0,W1,\nG1,W2, andG2.\nWith these considerations, the equation of motion\n(3.12) can be further simpli\fed into the standard\nparametrized form [41]\nh00\nA+ (2 + \u0016\u0017+\u0017A)Hh0\nA+ (1 + \u0016\u0016+\u0016A)k2hA= 0;\n(3.18)\nwith\nH\u0016\u0017= (lnG0)0+1\nG0\u0012\nG2k2\na2\u00130\n; (3.19)\nH\u0017A=\u001aA\nG0\u0012\nG1k\na\u00130\n; (3.20)\n\u0016\u0016=W0\nG0\u00001 +W2\u0000G2\nG0k2\na2; (3.21)\n\u0016A=\u001aAW1\u0000G1\nG0k\na: (3.22)\nIn such a parametrization, the new e\u000bects arising from\ntheories beyond GR are characterized by four parame-\nters: \u0016\u0017, \u0016\u0016,\u0017Aand\u0016A. The parameters \u0017A, and\u0016Alabel\nthe e\u000bects of the parity-violating terms in the spatial\ncovariant gravities, and \u0016 \u0017and \u0016\u0016describe the e\u000bects of\nother possible modi\fcations that are not relevant to par-\nity violation. Among these four parameters, \u0016Aand \u0016\u0016\ndetermine the speed of GWs, while \u0017Aand \u0016\u0017determine\nthe damping rate of GWs during their propagation.\nThe coe\u000ecients 1 \u0000W 0=G0,W1,G1,W2, andG2aris-\ning from di\u000berent terms in the spatial covariant gravity\ninduce distinct e\u000bects on GW propagation. In order to\nstudy the e\u000bects of the spatial covariant gravity on GW\npropagation, we separately consider each term in Eqs.\n(3.16) and (3.17) by setting the others to zero. In this\nway, the four parameters \u0016 \u0017, \u0016\u0016,\u0017A, and\u0016Acan be further\nparametrized in the following form [41]:\nH\u0016\u0017=h\n\u000b\u0016\u0017(\u001c) (k=aM LV)\f\u0016\u0017i0\n; (3.23)\n\u0016\u0016=\u000b\u0016\u0016(\u001c) (k=aM LV)\f\u0016\u0016; (3.24)\nH\u0017A=h\n\u001aA\u000b\u0017(\u001c) (k=aM PV)\f\u0017i0\n; (3.25)\n\u0016A=\u001aA\u000b\u0016(\u001c) (k=aM PV)\f\u0016; (3.26)\nwhere\f\u0016\u0017and\f\u0016\u0016are arbitrary even numbers and \f\u0017and\n\f\u0016are arbitrary odd numbers. \u000b\u0016\u0017,\u000b\u0016\u0016,\u000b\u0017, and\u000b\u0016are\nthe arbitrary functions of time. To write this form, we\nseparately consider each term in Eqs. (3.16) and (3.17)\nand set the others to zero. The di\u000berent terms in Eqs.\n(3.16) and (3.17) correspond to di\u000berent values of the\nabove parameters. The corresponding values of the pa-\nrameters (\u000b\u0016\u0017;\f\u0016\u0017;\u000b\u0016\u0016;\f\u0016\u0016;\u000b\u0017;\f\u0017;\u000b\u0016;\f\u0016) for the di\u000berent\nterms de\fned in Eqs. (3.16) and (3.17) are listed in Ta-\nble II. In Table III we present the corresponding values6\nof the parameters ( \u000b\u0016\u0017;\f\u0016\u0017;\u000b\u0016\u0016;\f\u0016\u0016;\u000b\u0017;\f\u0017;\u000b\u0016;\f\u0016) for sev-\neral speci\fc scalar-tensor theories that can be related to\nspatial covariant gravities in the unitary gauge.\nIV. EFFECTS OF THE SPATIAL COVARIANT\nGRAVITIES ON GWS AND THEIR\nCONSTRAINTS\nA. Frequency-independent e\u000bects from G0andW0\nThe coe\u000ecientsG0andW0induce two distinct and\nfrequency-independent e\u000bects on the propagation of\nGWs. One is the modi\fcation of the speed of the GWs\nifG06=W0, and another is the modi\fed fraction term of\nthe GWs if (lnG0)0is nonzero. In the following subsub-\nsections, we discuss these individually.\n1. Modi\fcation of speed of GWs\nWhenW06=G0, the speed of the GWs is modi\fed in\na frequency-independent manner,\ncgw=r\nW0\nG0: (4.1)For a GW event with an electromagnetic counterpart,\ncgwcan be constrained by comparison with the arrival\ntime of the photons. For the binary neutron star merger\nGW170817 and its associated electromagnetic counter-\npart GRB170817A, the almost coincident observation\nof both the electromagnetic wave and GW places an\nexquisite bound on the GW speed [45, 46],\n\u00003\u000210\u000015\u0014cgw\u00001\u00147\u000210\u000016: (4.2)\nNote that here we set the speed of light c= 1. This\nbound then leads to a constraint on W0=G0as\n\u00003\u000210\u000015\u0014r\nW0\nG0\u00001\u00147\u000210\u000016: (4.3)\nFrom this constraint, one has\n\u00003\u000210\u000015\u00141\nM2\nPl\"\n\u000ec(0;2)\n1\u0000\u000ec(2;0)\n1+1\n2_c(1;2)\n3\u00003(c(3;0)\n1+c(3;0)\n2)H+\u00103\n2c(1;2)\n3+ 3c(1;2)\n4+ _c(2;2)\n6+3\n2_c(2;2)\n8\u0011\nH\n+\u0010\n2c(2;2)\n6+ 3c(2;2)\n7+9\n2c(2;2)\n8+ 9c(2;2)\n9\u0011\nH2\u0000(9c(4;0)\n1+ 6c(4;0)\n2+ 9c(4;0)\n3)H2+\u0010\nc(2;2)\n6+3\n2c(2;2)\n8\u0011\n_H#\n\u00147\u000210\u000016:\n(4.4)\nHere\u000ec(0;2)\n1\u0011c(0;2)\n1\u00001\n2M2\nPland\u000ec(2;0)\n1\u0011c(2;0)\n1\u00001\n2M2\nPl.\nIn deriving the above bound, we have expanded all of the\ncoe\u000ecients c(dt;ds)\ni beyond GR in the modi\fed speed of\nGWs to the \frst order.\n2. Modi\fed GW friction from lnG0\nThe term (lnG0)0in Eq. (3.19) also induces an addi-\ntional friction term in the propagation equation of GWs.\nIn GR,G0is related to the Planck mass M2\nPlthrough\nG0=M2\nPl=4 and thus (lnG0)0= 0. In the spatial covari-\nant gravities,G0is time dependent and one can introduce\nan e\u000bective and time-dependent Planck mass M\u0003(t) by\nwritingG0=M2\n\u0003(t)=4. Then the modi\fed friction term\n(lnG0)0can be written in terms of the running of the\ne\u000bective Planck mass in the form\n(lnG0)0=HdlnM2\n\u0003\nlna: (4.5)Such an additional friction term also changes the damp-\ning rate of GWs during propagation. This leads to a\nGW luminosity distance dgw\nLwhich is related to the stan-\ndard luminosity distance of electromagnetic signals dem\nL\nas [47, 48]\ndgw\nL(z) =dem\nL(z) expn1\n2Zz\n0dz0\n1 +z0(lnG0)0\nHo\n=dem\nL(z) expn1\n2Z\nd(lnG0)o\n: (4.6)\nThus, it is possible to probe GW friction (ln G0)0using\nthe multimessenger measurements of dgw\nLanddem\nL.\nHowever, such a probe relies sensitively on the time\nevolution of (lnG0)0, which is in general unknown. In\norder to probe GW friction, there are two approaches\nto parametrize the time evolution of (ln G0)0. One is\ncM-parametrization [49], which is based on the evo-\nlution of the dark energy in the Universe, and an-\nother is \u0004-parametrization [47], which is a theory-based\nparametrization that can \ft a lot of modi\fed gravities.7\nTABLE II. Corresponding values of the parameters ( \u000b\u0016\u0017;\f\u0016\u0017;\u000b\u0016\u0016;\f\u0016\u0016;\u000b\u0017;\f\u0017;\u000b\u0016;\f\u0016) for the di\u000berent terms in de\fned in Eqs.\n(3.16) and (3.17).\nH\u0016\u0017 \u0016\u0016 H\u0017A \u0016A\n\u000b\u0016\u0017\f\u0016\u0017\u000b\u0016\u0016\f\u0016\u0016\u000b\u0017\f\u0017\u000b\u0016\f\u0016Related coe\u000ecients\nG0 lnG0 0\u00001 +W0=G00 | | | | c(2;0)\n1;c(3;0)\n1;c(3;0)\n2;c(4;0)\n1;c(4;0)\n2;c(4;0)\n3\nG1 | | | | G1MPV=G01\u0000G1MPV=G01c(2;1)\n1;c(3;1)\n1;c(3;1)\n2;c(3;1)\n3\nG2G2M2\nLV=G02\u0000G2M2\nLV=G02 | | | | c(2;2)\n1\nW0 | |\u00001 +W0=G00 | | | | c(0;2)\n1;c(1;2)\n3;c(1;2)\n4;c(2;2)\n6;c(2;2)\n7;c(2;2)\n8;c(2;2)\n9\nW1 | | | | | | W1MPV=G0 1c(0;3)\n1;c(1;3)\n1;c(1;3)\n2;c(1;3)\n3\nW2 | |G2M2\nLV=G0 2 | | | | c(0;4)\n3\nTABLE III. Corresponding values of the parameters ( \u000b\u0016\u0017;\f\u0016\u0017;\u000b\u0016\u0016;\f\u0016\u0016;\u000b\u0017;\f\u0017;\u000b\u0016;\f\u0016) for several speci\fc scalar-tensor theories\nthat can be related to spatial covariant gravities in the unitary gauge.\nH\u0016\u0017 \u0016\u0016 H\u0017A \u0016A\n\u000b\u0016\u0017\f\u0016\u0017\u000b\u0016\u0016\f\u0016\u0016 \u000b\u0017 \f\u0017 \u000b\u0016 \f\u0016\nHorndeski lnb0\u00003c0H\n20q\nd1\u0000_a1\nb0\u00003c0H\u00001 0 | | | |\nScalar-Gauss-Bonnet ln1+8_\u0018H\nM2\nPl0 8\u0018H+_\u0018_H\nM2\nPlH0 | | | |\nChern-Simons | | | | \u0000_#MPV 1 | |\nLorentz-violating Weyl | | 4 \rM2\nLV 2 | | | |\nChiral scalar-tensor | | | |MPV\u0010\n\u0000_#+ 2(a3+ 2a1)_\u001e2H\n\u00002b1_\u001e3+ 2(b4+b5\u0000b3)_\u001e4H\u00111MPV\u0010\n2a2@t\u0002\n(2a1+a3)__\u001e2a\u00002\u0003\n\u00002b1_\u001e3+ 2(b4+b5\u0000b3)_\u001e4H\u00111\nForcM-parametrization, the GW friction is written as\n[49]\n(lnG0)0\nH=cM\n\u0003(z)\n\n\u0003(0); (4.7)\nwherezis the redshift of the GW source and \n \u0003is the\nfractional dark energy density. If one considers the dark\nenergy density as a constant, then one has [50]\n\n\u0003(z) =\n\u0003(0)\n\n\u0003(0) + (1 +z)3\nm(0); (4.8)\nwhere \nm(0) is the value of the fractional energy density\nof matter. Several constraints on cMhave been derived\nusing both dgw\nLanddem\nLfrom GW events or populations\n[50{52]. Here we adopt a constraint from Ref. [50] from\na jointed parameter estimation of the mass distribution,\nredshift evolution, and GW friction with GWTC-3 for\ndi\u000berent BBH population models, which gives\ncM=\u00000:6+2:2\n\u00001:2: (4.9)\nThis corresponds to\n(lnG0)0\nH\f\f\f\f\nz=0=\u00000:6+2:2\n\u00001:2: (4.10)For \u0004 parametrization, the full redshift dependence of\nGW friction is described by two parameters (\u0004 0;n), with\nwhich the ratio between the GW and electromagnetic\nluminosity distances can be written as [47]\ndgw\nL(z)\ndem\nL(z)\u0011\u0004(z) = \u0004 0+1\u0000\u00040\n(1 +z)n: (4.11)\nSuch a parametrization corresponds to\n(lnG0)0\nH=2n(1\u0000\u00040)\n1\u0000\u00040+ \u00040(1 +z)n: (4.12)\nThe relation between the \u0004-parametrization and cM-\nparametrization was explored in Ref. [53]. Several con-\nstraints on (\u0004 ;n) were obtained using GW events with\nredshift information inferred from the corresponding elec-\ntromagnetic counterparts [51] or host galaxies [52], or the\nbinary black hole mass function [53]. A recent constraint\non (\u0004 0;n) was derived from an analysis of GW data in\nGWTC-3 with a BBH mass function, which gives [53]\n\u00040= 1:0+0:6\n\u00000:5; n= 2:5+1:7\n\u00001:1 (4.13)\nwith a prior uniform in ln \u0004 0. This bound leads to a\nconstraint on (lnG)0in the form\n\u00003:0<(lnG0)0\nH\f\f\f\f\nz=0<2:5: (4.14)\nThis leads to8\n\u00003:0<1\nH@tlnh\nc(2;0)\n1+ 3(c(3;0)\n1+c(3;0)\n2)H+ 3(3c(4;0)\n1+ 2c(4;0)\n2+ 3c(4;0)\n3)H2i\n<2:5: (4.15)\nB. Parity-violating e\u000bects from W1andG1\nAdding the parity-violating terms introduced in Eqs.\n(2.10) and (2.11) to the action of the spatial covariant\ngravities leads to nonzero coe\u000ecients W1andG1. Due to\nthe parity violation, these two coe\u000ecients induce two dis-\ntinct birefringent e\u000bects on the propagation of GWs: the\namplitude birefringences and the velocity birefringences.\n1. Amplitude birefringences of GWs from (G1k=a)0\nThe e\u000bects of the coe\u000ecient ( G1k=a)0are fully char-\nacterized by the parameter \u0017A, which leads to di\u000berent\ndamping rates for the left- and right-hand circular polar-\nizations of GWs, so the amplitude of the left-hand circu-\nlar polarization of GWs will increase (or decrease) during\nthe propagation, while the amplitude of the right-hand\nmodes will decrease (or increase). This e\u000bect induces\nmodi\fcations in the amplitude of the GW waveform in\nthe form [41]\nhA=hGR\nAexp\u0012\n\u00001\n2Z\u001c0\n\u001ceH\u0017A\u0013\nd\u001c=hGR\nAe\u001aA\u000eh1;\n(4.16)\nwith\n\u000eh1=\u00001\n2\"\na\u0017\u0012k\naMPV\u0013\f\u0017#\f\f\f\f\fa0\nae=\u00001\n2G1k\nG0a\f\f\f\fa0\nae;\n(4.17)\nwherehGRdenotes the waveform of GWs in GR, a0=\na(t0) witht0denoting the arrival time of GWs and ae=\na(te) withtebeing the emitted time. We can convert\nthe left- and right-hand GW polarization modes into the\nplus and cross modes which are used more often in GW\ndetections. Using the relation\nh+=hL+hRp\n2; h\u0002=hL\u0000hRp\n2i; (4.18)\nwe obtain\nh+(f) =hGR\n+cosh(\u000eh1)\u0000ihGR\n\u0002sinh(\u000eh1);(4.19)\nh\u0002(f) =hGR\n\u0002cosh(\u000eh1) +ihGR\n+sinh(\u000eh1):(4.20)\nWith the modi\fed waveform in the above, one is\nable to test the amplitude birefringent e\u000bect induced by\n(G1k=a)0=G0by comparing the modi\fed waveform with\nthe GW strain data from the GW detectors. By perform-\ning a Bayesian parameter estimation on the 12 LIGO-\nVirgo O1/O2 events with the modi\fed waveform, the\namplitude birefringent e\u000bect including parity violationhas been constrained, which leads to a combined lower\nbound on the corresponding energy scale MPVof [54, 55]\nMPV&10\u000022GeV; (4.21)\nwhich is a rather loose result. This is because GW de-\ntection is less sensitive to amplitude modi\fcation than\nphase.\nHere we would like to derive the constraint on the co-\ne\u000ecientG1by directly using the posterior samples ob-\ntained in Ref.[54] to test the amplitude birefringent e\u000bect\nwith 12 LIGO-Virgo O1/O2 events. The data for all 12\nGW events are available in Ref. [56]. In Ref. [54], the\namplitude birefringent e\u000bect due to parity violation was\ndescribed by a parameter A\u0017, which can be related to G1\nin the spatial covariant gravity via\n\u000eh1=\u0000A\u0017\u0019f=\u00001\n2G1k\nG0a\f\f\f\fa0\nae: (4.22)\nThus, one has\nA\u0017=G1\nG0\f\f\f\f\nz=0\u0000G1(z)\nG0(z)(1 +z): (4.23)\nHerezis the redshift of the GW source. In principle,\nthe coe\u000ecientG1is an arbitrary function of time which\ncan only be determined given a speci\fc model of spatial\ncovariant gravity. Considering that the redshifts of all 12\nGW sources are not large, we can approximately treat G1\nas constant, i.e., ignore its time dependence. Then, one\ncan relateA\u0017toG1by\nG1\nG0=\u0000A\u0017\nz: (4.24)\nThen, from the posterior distributions of A\u0017and the red-\nshiftzobtained in Ref. [54] for each GW event, one can\ncalculate the posterior distribution of G1for each GW\nevent. We plot the posterior probability distributions of\njG1jin Fig. 1. From this \fgure, we \fnd that the poste-\nrior probability distributions of G1with 90% con\fdence\nintervals are consistent with the GR value G1= 0 for all\n12 GW events.\nIn the above analysis, we have treated the quantity G1\nas a constant. In this sense, this quantity is also a uni-\nversal quantity for all GW events. Thus, one can com-\nbine all 12 individual posteriors of G1to get the overall\nconstraint. This can be done by multiplying the poste-\nrior distributions of the 12 GW events in LOGO-Virgo\nO1/O2 and then we \fnd that the coe\u000ecient G1can be\nconstrained to be\njG1=G0j<2065 km (4.25)9\n01000200030004000500060007000\n|1/0|[km]\n012345678Probability Densitiy×104\nGW151226\nGW170809\nGW170818\nGW190425\nGW170608\nGW170814\nGW170817\nGW170729\nGW170823\nGW170104\nGW150914\nGW151012\n90% upper limit\nFIG. 1. The posterior distributions for jG1jfrom twelve LIGO-\nVirgo O1/O2 GW events. The vertical dash line denotes the\n90% upper limits of jG1jfrom the combined result.\nat the 90% con\fdence level. This constraint can be con-\nverted into the constraint on the combination of coe\u000e-\ncientsc(2;1)\n1,c(3;1)\n1,c(3;1)\n2, andc(3;1)\n3as\njc(2;1)\n1\u0000(c(3;1)\n1\u00002c(3;1)\n2\u00003c(3;1)\n3)Hj\nM2\nPl<1033 km:\n(4.26)\nThe above results are obtained from the Bayesian pa-\nrameter estimation in Ref. [54] by comparing the mod-\ni\fed waveform with the GW strain data of the 12 GW\nevents in the LIGO-Virgo O1/O2 catalog. Here we would\nlike to mention that the amplitude birefringence also af-\nfects the statistical distribution of cos \u0013over the popula-\ntion of binary black hole mergers [57] with \u0013being the in-\nclination angle of the binary black hole system. One can\nconsider that the Universe is homogeneous and isotropic\non cosmological scales, and that gravitational physics\ndoes not have any preferred direction. This implies that\nthe underlying distribution for cos \u0013is \rat, meaning that\nits distribution is symmetric about zero when the ampli-\ntude birefringence is absent. When the amplitude bire-\nfringence induced by G1is included, the distribution of\ncos\u0013will preferentially have cos \u0013>0(<0) ifG1<0(>0)\n[57]. By checking the posterior distribution of cos \u0013for\nGW events in the GWTC-2 catalog, we can impose a\nconstraint onG1[57], i.e.,\njG1=G0j.1000 km: (4.27)\nThis bound corresponds to\nM\u00002\nPl\f\f\fc(2;1)\n1\u0000(c(3;1)\n1\u00002c(3;1)\n2\u00003c(3;1)\n3)H\f\f\f<500 km:\n(4.28)\nNote that this constraint improves on that in Eq. (4.26)\nby a factor of 2.\nIt is worth mentioning here that in some speci\fc mod-\nels, the coe\u000ecient G1could oscillate periodically. Oneexample is the axion-Chern-Simons theory studied in\nRefs. [58{60]. In this scenario, the axion oscillation can\ninduce parametric resonance in GWs at a certain fre-\nquency, which can produce resonance peaks in GW sig-\nnals. Searching for these peaks in GW signals can thus\nplace stringent constraints on both the axion-gravity cou-\npling and axion mass, which can in principle place a more\nstringent constraint on G1, depending sensitively on the\nspeci\fc coupling form of G1.\n2. Velocity birefringence of GWs from W1\u0000G1\nWhenW16=G1, the coe\u000ecient W1\u0000G 1induces a\nnonzero parameter \u0016A=\u001aA(W1\u0000G 1)k=(G0a), which\ndetermines the speed of the GWs. In particular, due\nto parity violation, the parameter \u0016A[or, equivalently,\n\u001aA(W1\u0000G 1)=G0] has the opposite signs for left- and\nright-hand circular polarizations of GWs. This leads to\ndi\u000berent velocities for left- and right-hand circular polar-\nizations of GWs, and therefore the arrival times of the\ntwo circular polarization modes could be di\u000berent. This\nphenomenon is known as velocity birefringence.\nAs shown in Ref. [41], with velocity birefringence, the\ndi\u000berent circular polarization modes will have di\u000berent\nphase velocities\nvA'1\u00001\n2\u001aA\u000b\u0016(\u001c)\u0012k\naMPV\u0013\n= 1\u00001\n2\u001aAW1\u0000G1\nG0k\na: (4.29)\nConsider GWs emitted at two di\u000berent times teandt0\ne,\nwith wave number kandk0, and received at correspond-\ning arrival times t0andt0\n0; then, the di\u000berent velocities of\ndi\u000berent circular polarization modes lead to a di\u000berence\nof their arrival times,\n\u0001t0= (1 +z)\u0001te+\u001aA\n2 \nk\f\u0016\nM\f\u0016\nPV\u0000k`\f\u0016\nM\f\u0016\nPV!Zt0\nte\u000b\u0016\na\f\u0016+1dt\n= (1 +z)\u0001te+\u001aA(k\u0000k0)\n2Zt0\nteW1\u0000G1\nG0a2dt:(4.30)\nHere \u0001te=te\u0000t0\ne. This velocity di\u000berence induces a\nmodi\fcation of the phase of the GW signal emitted from\na binary compact star system. The modi\fed GW wave-\nform in the Fourier domain reads\nhA(f) =hGR\nAei\u001aA\u000e\t1; (4.31)\nwhere the phase correction \u0001\t 1induced by the velocity\ndi\u000berence is expressed as\n\u000e\t1=A\u0016(\u0019f)2(4.32)\nwith\nA\u0016=Zt0\nteW1\u0000G1\nG0a2dt10\n=1\nH0Zz\n0(W1\u0000G1)(1 +z0)dz0\nG0p\n\nm(1 +z0)3+ \n \u0003:(4.33)\nHere we adopt H0= 67:8 km=s=Mpc, \nm= 0:308, and\n\n\u0003= 0:692.\nThere are several di\u000berent ways to test the velocity\nbirefringence induced by ( W1\u0000G1)=G0in spatial covari-\nant gravity. In Ref. [61], the velocity birefringence was\nconstrained by comparing the arrival times of GW170817\nand GRB170817a, which gives\nj(W1\u0000G1)=G0j.10\u000011km: (4.34)\nWith the arrival time di\u000berence of left- and right-hand\nGWs induced by velocity birefringence, Ref. [62] pro-\nposed a new method for constraining velocity birefrin-\ngence in a model-independent way by measuring the dif-\nference in arrival times of two GW polarizations. With\nthis method, it is expect to constrain j(W1\u0000G 1)=G0j\nto be.10\u000014km. This constraint is better than Eq.\n(4.34) by 3 orders of magnitude. The velocity birefrin-\ngence could also slightly widen or split the peak of the\nGW waveform [55]. By checking for waveform peak split-\nting in the \frst ever detected GW event, GW150914, Ref.\n[63] placed the \frst constraint on velocity birefringence,\nwhich corresponds to j(W1\u0000G1)=G0j.10\u000011km. Re-\ncently, the constraints from considering the width of the\npeak at the maximal amplitude of GW events have been\nimproved signi\fcantly with an analysis of 50 GW events\nin GWTC-1 [64] and GWTC-2 [65].\nSimilar to the case of amplitude birefringence, the ve-\nlocity birefringence due to parity violation can also be\ntested by comparing the modi\fed waveform (4.31) with\nthe GW strain data from the GW detectors; see Ref. [66]\nfor a review. Based on this, the tests of velocity bire-\nfringence have been carried out through full Bayesian\nparameter estimations on the GW events observed by\nthe LIGO/Virgo/KAGRA detectors in a series of papers\n[30, 54, 67{72]. Here we would like to derive the con-\nstraint on the coe\u000ecient ( W1\u0000G1)=G0by directly using\nthe posterior samples obtained in [68] for testing the ve-\nlocity birefringent e\u000bect with 94 GW events reported in\nthe 4th-Open Gravitational Wave Catalog (4-OGC) [73].\nThe data for these posterior samples were downloaded\nfrom [74]. With these data, one can derive the poste-\nrior distributions of ( W1\u0000G1)=G0from the posterior dis-\ntributions of the sampled parameter M\u00001\nPVfor each GW\nevent. In Fig. 2, we show the posterior distributions of\nj(W1\u0000G1)=G0jfor the 92 analyzed GW events2. Note\nthat here we treat the coe\u000ecient W1\u0000G1as a constant as\nwell. From this \fgure, we \fnd that the posterior distri-\nbutions ofj(W1\u0000G1)=G0jwith 90% con\fdence intervals\nare consistent with the GR value W1\u0000G1= 0 for all\n2Here we exclude the two events GW190521 and GW191109 since\ntheir posterior samples show intriguing non-zero results for ve-\nlocity birefringence. Some possible reasons that produce such\nsignatures were also explored in Ref. [68].\n0.02.55.07.510.012.515.017.520.0\n|(1 1)/0|[1018km]\n0.000.050.100.150.20Probability DensitiyGW191204_171526\nGW190708_232457\nGW190707_093326\nGW190512_180714\nGW200202_154313\nGW190720_000836\nGW190412_053044\nGW191215_223052\nGW190408_181802\nGW190814_211039\nother 4-OGC events\n90% upper limitFIG. 2. The posterior distributions for jW1\u0000G1jfrom 92 GW\nevents in the 4-OGC. The legend indicates the events that\ngive the tightest constraints. The vertical dash line denotes\nthe 90% upper limits from the combined result.\n92 GW events. By considering j(W1\u0000G1)=G0jas a uni-\nversal parameter, we also present its upper bound from\nthe combined posterior probability distributions in Fig. 2\n(the vertical dashed line), from which one is able to place\na constraint onj(W1\u0000G1)=G0jas\nj(W1\u0000G1)=G0j<4:4\u000210\u000018km; (4.35)\nat the 90% con\fdence level. This bound corresponds to\nM\u00002\nPl\f\f\f\f1\n4\u0010\n_c(1;3)\n1+ _c(1;3)\n2\u0011\n+c(0;3)\n1\u00003c(1;3)\n3H\n\u00001\n2h\nc(2;1)\n1\u0000(c(3;1)\n1\u00002c(3;1)\n2\u00003c(3;1)\n3)Hi\f\f\f\f\n<1:1\u000210\u000018km:(4.36)\nIt is worth mentioning here that a slightly stronger bound\non the velocity birefringence parameter was derived in\nRef. [69] by performing a full Bayesian analysis on GW\nevents in the LIGO-Virgo catalog GWTC-3.\nC. Lorentz-violating e\u000bects from W2andG2\nThe Lorentz-violating high-derivative terms intro-\nduced in Eq. (2.9) into the action of the spatial co-\nvariant gravities lead to nonzero coe\u000ecients W2andG2.\nThe coe\u000ecientW2arises from a term with four spatial\nderivatives, and thus it modi\fes the usual dispersion re-\nlation of GWs in GR. The coe\u000ecient G2, which arises\nfrom a term contained two time derivatives and two spa-\ntial derivatives, not only modi\fes the dispersion relation\nbut also leads to a modi\fed damping rate of GWs during\npropagation3. In the following, we discuss the e\u000bects of\n3Note that in some Lorentz-violating theories, vector modes can\nappear and propagate in the spacetime. This kind of vector mode11\nLorentz-violating high derivatives in the action of spatial\ncovariant gravity on the damping rate and dispersion of\nGWs respectively.\n1. Lorentz-violating damping rate\nThe coe\u000ecientG2induces a frequency-dependent fric-\ntion term in the propagation equation of GWs. In the\nparametrization of Eq. (3.18), this friction term leads\nto a nonzero parameter \u0016 \u0017, which provides a frequency-\ndependent damping of GW amplitudes during propaga-\ntion. This implies that at di\u000berent frequencies, GWs can\nexperience di\u000berent damping rates. This e\u000bect provides\nan amplitude modulation to the gravitational waveform\n[41],\nhA=hGR\nAexp\u0012\n\u00001\n2Zt0\nteH\u0016\u0017\u0013\n=hGR\nAe\u000eh2(4.37)\nwith\n\u000eh2=\u00001\n2\"\na\u0016\u0017\u0012k\naMLV\u0013\f\u0016\u0017#\f\f\f\f\fa0\nae=\u00001\n2G2k2\nG0a2\f\f\f\fa0\nae;\n(4.38)\nwherehGRdenotes the waveform of GWs in GR. We can\nconvert the left- and right-hand GW polarization modes\ninto the plus and cross modes, i.e.,\nh+(f) =hGR\n+e\u000eh2; (4.39)\nh\u0002(f) =hGR\n\u0002e\u000eh2: (4.40)\nWith this modi\fed waveform, it is possible to derive the\nconstraint onG2=G0by comparing the modi\fed wave-\nform with the GW strain data from the GW detectors.\nHowever, no test on the frequency-dependent damping\ne\u000bects has not been carried out yet in the literature, and\nwe expect to consider this in our future works.\n2. Lorentz-violating dispersion relation\nLorentz violation of gravity in general modi\fes the con-\nventional linear dispersion relation to a nonlinear one.\nDue to the existence of the coe\u000ecients W2andG2in\nspatial covariant gravity, the dispersion of GWs becomes\n!2\nk=k2(1 + \u0016\u0016) (4.41)\nis generated, for example in Einstein-\u001dther theory, by a time-like\n\u001ather \feld introduced in the theory. Such a time-like \u001ather \feld\nprovides a preferred time direction thus breaking the Lorentz\nsymmetry. This property is di\u000berent from the spatial covariant\ngravities we considered here, in which only an extra scalar mode\ncould be generated due to the breaking of the time di\u000beomor-\nphism and there is no new fundamental \feld is introduced in the\ntheory.with\n\u0016\u0016=W2\u0000G2\nG0k2\na2: (4.42)\nWith this modi\fed dispersion relation, the phase velocity\nof GWs reads\nv'1\u00001\n2\u000b\u0016\u0016\u0012k\naMLV\u0013\f\u0016\u0016\n= 1\u00001\n2W2\u0000G2\nG0k2\na2: (4.43)\nConsider GWs emitted at two di\u000berent times teandt0\ne,\nwith wave number kandk0, and received at correspond-\ning arrival times t0andt0\n0; then, the di\u000berent velocities of\nmodes lead to a di\u000berence in their arrival times [41, 75],\n\u0001t0= (1 +z)\u0001te+1\n2 \nk\f\u0016\u0016\nM\f\u0016\u0016\nLV\u0000k0\f\u0016\u0016\nM\f\u0016\u0016\nLV!Zt0\nte\u000b\u0016\u0016\na\f\u0016\u0016+1dt\n= (1 +z)\u0001te+k2\u0000k02\n2Zt0\nteW2\u0000G2\nG0a3dt: (4.44)\nHere \u0001te=te\u0000t0\ne. This velocity di\u000berence induces a\nmodi\fcation of the phase of the GW signal emitted from\na binary compact star system. The modi\fed GW wave-\nform in the Fourier domain reads [41, 75]\nhA(f) =hGR\nAei\u000e\t2; (4.45)\nwhere the phase correction \u000e\t2induced by the velocity\ndi\u000berence is expressed as\n\u000e\t2=A\u0016(\u0019f)3(4.46)\nwith\nA\u0016\u0016=4\n3Zt0\nteW2\u0000G2\nG0a3dt\n=4\n3H0Zz\n0(W2\u0000G2)(1 +z0)2dz0\nG0p\n\nm(1 +z0)3+ \n \u0003:(4.47)\nWe analyze the GW constraints on the Lorentz-\nviolating dispersion relation by comparing the modi\fed\nwaveform (4.45) with the GW strain data in GWTC-1\n[11], GWTC-2 [12], and GWTC-3 [13]. The gravitational\nconstraint on (W2\u0000G2)=G0can be obtained from the pos-\nterior samples of the Lorentz-violating parameter A4in\nRefs. [11{13]. In Refs. [11{13], the Lorentz-violating\nparameter A4was sampled separately for A4>0 and\nA4<0 in Refs. [11{13]. Here we consider positive and\nnegativeA4separately as well and derive the correspond-\ning bounds on (W2\u0000G2)=G0, which are presented in Table\nIV. From this table, we see that the most stringent con-\nstraint is from the combined posterior of GW events in\nGWTC-3, which gives\nj(W2\u0000G2)=G0j<1:2\u000210\u000010m2\n(4.48)\nat 90% C.L. This bound corresponds to\nM\u00002\nPl\f\f\fc(0;4)\n3+c(2;2)\n1\f\f\f<6\u000210\u000011m2: (4.49)12\nTABLE IV. 90% con\fdence level upper bounds on j(W2\u0000\nG2)=G0jfor positive and negative W2\u0000G2respectively from\nthe Bayesian inference by analyzing GW events in the\nLIGO/Virgo/KAGRA catalogs GWTC-1 [11], GWTC-2 [12],\nand GWTC-3 [13]. Note that the bounds on j(W2\u0000G2)=G0j\nis in the unit of 10\u000010m2withmbeing the meter.\ncatalogssampled with\nnegativeW2\u0000G2sampled with\npositiveW2\u0000G2\nGWTC-1 9.0 5.5\nGWTC-2 3.4 2.5\nGWTC-3 2.9 1.2\nV. SUMMARY AND DISCUSSIONS\nThe spatial covariant gravity is only invariant un-\nder the three-dimensional spatial di\u000beomorphism, which\nbreaks the time di\u000beomorphism. Therefore the gravita-\ntional action of this type of theory can be constructed\nin terms of spatial di\u000beomorphism invariants. A lot of\nscalar-tensor theories can be mapped to the spatial co-\nvariant framework by imposing the unitary gauge on the\ncoupling scalar \feld. This provides us with a general\nframework for exploring the e\u000bects of unknown high-\nenergy physics on the propagation of GWs.\nIn this paper, we studied the e\u000bects of the spa-\ntial covariant gravities on the propagation of GWs,\nproduced by the coalescence of compact binaries, and\ntheir observational constraints with GW events from the\nLIGO/Virgo/KAGRA Collaboration. For this purpose,\nwe calculated the e\u000bects of the spatial covariant gravities\non the friction, speed, amplitude, and velocity birefrin-\ngences of GWs, as well as the modi\fed dispersion rela-\ntion during GW propagation in the cosmological back-\nground. These e\u000bects can be described by the univer-\nsal parametrization proposed in Refs. [41, 42]. Di\u000ber-\nent e\u000bects correspond to di\u000berent parameters, as dis-\ncussed in detail in Sec. III and summarized in Ta-\nble II. These e\u000bects can be divided into three classes:\n1) frequency-independent e\u000bects which include modi\f-\ncations to GW speed and friction; (2) parity-violating\ne\u000bects which include the amplitude and velocity birefrin-\ngences of GWs; and 3) Lorentz-violating e\u000bects which in-\nclude the modi\fed damping rate and dispersion relation\nof GWs. Among these e\u000bects, the parity-violating and\nLorentz-violating e\u000bects are frequency dependent. De-\npending on di\u000berent coe\u000ecients in the spatial covariant\ngravities, these frequency-dependent e\u000bects can produce\namplitude modulation and phase corrections in the wave-\nform of GWs produced by the coalescence of compact bi-\nnaries. The calculation of these modi\fed GW waveforms\nwas presented in Sec. III. These modi\fed waveforms pro-\nvide important tools for constraining parity-violating or\nLorentz-violating e\u000bects in the spatial covariant gravities\nwith current available GW events or future GW detec-\ntions.\nWe compared these new e\u000bects with the publiclyavailable posterior samples or results from various tests\nof gravities using LIGO/Virgo/KAGRA data to ob-\ntain constraints on coupling coe\u000ecients in the action\nof spatial covariant gravities. Di\u000berent e\u000bects can\nbe tested with di\u000berent phenomena in GW observa-\ntions. For the frequency-independent e\u000bects, GW speed\nis constrained by the multimessenger observation of\nGW170817/GRB170817A, while GW friction is con-\nstrained by dark sirens in GWTC-1 with BBH population\nmodels. For the parity-violating e\u000bects, we constrained\nthe amplitude and velocity birefringent parameters from\nfull Bayesian parameter estimations of the GW events ob-\nserved by the LIGO/Virgo/KAGRA detectors. We also\nreported the constraint on the amplitude birefringent pa-\nrameter from the statistic analysis of the posterior dis-\ntributions of cos \u0013for GW events in GWTC-2. For the\nLorentz-violating e\u000bects, we reported the constraint on\nthe parameter in the modi\fed dispersion relation from a\nBayesian analysis of GW events in GWTC-3. Our results\nare summarized in Table V. Using these constraints, we\nalso derived the corresponding bounds on the coupling\ncoe\u000ecients of a number of speci\fc theories in the Ap-\npendix.\nIt is remarkable that the constraints on the e\u000bects\nthat modify the speed of GWs (including the e\u000bects of\nfrequency-independent modi\fcation to GW speed, am-\nplitude birefringence, and Lorentz-violating modi\fed dis-\npersion) are more stringent than those that a\u000bect the\namplitude of GWs. For example, for parity-violating ef-\nfects, the parameter jG1j(withW1= 0) can be con-\nstrained from tests of amplitude or velocity birefringence.\nIt is evident from Table V that the constraint on jG1j\nfrom the tests of velocity birefringence is stronger than\nthat from tests of amplitude birefringence by 20 orders\nof magnitude. This is because ground-based detectors\nare more sensitive to phase corrections than amplitude\nmodulations for the tests involving GW signals from the\ncoalescence of compact binaries.\nHere we would like to mention that the study per-\nformed in this paper can be extended in a few directions\nin future works. First, in order to derive the bounds\non the coupling coe\u000ecients, we considered the di\u000ber-\nent e\u000bects of spatial covariant gravities on the propa-\ngation of GWs separately. Thus, it would be interest-\ning to consider all of the new e\u000bects that arise from\nthe di\u000berent coe\u000ecients together. To do this, one would\nneed to simulate the modi\fed waveform with GW data\nby sampling all of the relevant coe\u000ecients. Second, it\nwould be interesting to see if future GW detectors such\nas the third-generation ground-based detectors, space-\nbased detectors, and pulsar-timing arrays, can improve\nthe bounds obtained in this paper. For the spatial co-\nvariant gravities, most of the e\u000bects on the propaga-\ntion of GWs are very sensitive to the higher frequency\nof GWs. This is because the amplitude and phase cor-\nrections to the waveform are proportional to f\f\u0016\u0017;\u0017Aand\nf1+\f\u0016\u0016;\u0016A, respectively. For this reason, it is not likely\nthat future space-based detectors or pulsar-timing arrays13\nTABLE V. Summary of estimations for bounds of the coupling coe\u000ecients in spatial covariant gravities. Note that all the\ncoe\u000ecients are estimated approximately at present time, i..e, z= 0. Here [amin;bmax] represents constraints with abinandbmax\nbeing the lower the upper bounds, respectively.\nCoe\u000ecients bounds related coe\u000ecients datasets used\nq\nW0\nG0\u00001 [\u000030;7]\u000210\u000016 c(2;0)\n1;c(3;0)\n1;c(3;0)\n2;c(4;0)\n1;c(4;0)\n2;c(4;0)\n3;\nc(0;2)\n1;c(1;2)\n3;c(1;2)\n4;c(2;2)\n6;c(2;2)\n7;c(2;2)\n8;c(2;2)\n9mult-messenge observations\nof GW170817 [45, 46]\n(lnG0)0\nH[\u00003:0;2:5] c(2;0)\n1;c(3;0)\n1;c(3;0)\n2;c(4;0)\n1;c(4;0)\n2;c(4;0)\n3dark sirens in GWTC-3 with\nBBH mass distributions [53]\njG1=G0j.2065 km\n.1000 kmc(2;1)\n1;c(3;1)\n1;c(3;1)\n2;c(3;1)\n3tests of amplitude birefringence\nwith LIGO-Virgo O1/O2 [54]\nfrom statistic distribution of\ncos\u0013in GWTC-2 [57]\n\f\f\fW1\u0000G1\nG0\f\f\f .4:4\u000210\u000018kmc(0;3)\n1;c(1;3)\n1;c(1;3)\n2;c(1;3)\n3;\nc(2;1)\n1;c(3;1)\n1;c(3;1)\n2;c(3;1)\n3tests of velocity birefringence\nwith 4-OGC [68]\n\f\f\fW2\u0000G2\nG0\f\f\f .1:2\u000210\u000010m2c(2;2)\n1; c(0;4)\n3tests of Lorentz-violating\ndispersion with GWTC-3 [13]\nwill be able to improve the bounds given in this paper\nsince the sensitive frequency of these detectors is much\nlower than those of the ground-based detectors. For the\nvelocity birefringence e\u000bect, it was shown [54] that the\nthird-generation gravitational-wave detectors are able to\nimprove the constraint on the energy scale of parity vi-\nolation toO(102) GeV. This implies that the bound\nonj(W2\u0000G2)=G0jcan be improved by about 3 orders\nof magnitude, i.e., improved to be j(W2\u0000G 2)=G0j.\nO(10\u000021) km. Similarly, it is expected that the third-\ngeneration gravitational-wave detectors could be able to\nsigni\fcantly improve the constraints on the other e\u000bects,\nsuch as the amplitude birefringence, Lorentz-violating\ndamping rate, and modi\fed dispersion relations. We ex-\npect to come back to these issues soon in future work.\nACKNOWLEDGEMENTS\nT.Z. and A.W. are supported in part by the Na-\ntional Key Research and Development Program of China\nGrant No.2020YFC2201503, and the Zhejiang Provincial\nNatural Science Foundation of China under Grant No.\nLR21A050001 and LY20A050002, the National Natural\nScience Foundation of China under Grant No. 12275238,\nNo. 11975203, No. 11675143, and the Fundamental\nResearch Funds for the Provincial Universities of Zhe-\njiang in China under Grant No. RF-A2019015. W.Z.\nis supported by the National Key Research and Devel-\nopment Program of China Grant No.2021YFC2203102\nand 2022YFC2200100, NSFC Grants No. 12273035 and\n11903030, the Fundamental Research Funds for the Cen-\ntral Universities.APPENDIX A: SEVERAL SPECIFIC SPATIAL\nCOVARIANT GRAVITIES\nThe spatial covariant gravities provide a unifying\nframework for describing scalar-tensor theories in the uni-\ntary gauge. In this appendix, we present several speci\fc\nscalar-tensor theories in the unitary gauge and Lorentz-\nviolating gravity by writing their gravitational actions in\nthe form of Eq. (2.4). We also provide relations between\nthe coupling coe\u000ecients in each theory and the corre-\nsponding coe\u000ecients in the spatial covariant gravities.\nIn addition, the observational constraints on these theo-\nries are derived from the constraints presented in Table\nV.\n1. Horndeski theory\nThe Horndeski theory is a general scalar-tensor theory\nconstructed from the metric tensor g\u0016\u0017and a scalar \feld\n\u001eand can have a second-order \feld equation [76]. The\nLagrangian of the Horndeski theory in the unitary gauge\nwith\u001e=\u001e(t) can be found in Refs. [28, 77, 78], and is\nwritten as [28]\nLunitary\nH =a0K\u00002a1(Rij\u00001\n2Rgij)Kij\n+b0(KijKij\u0000K2)\n+c0(K3\u00003KKijKij+ 2Ki\njKi\nkKk\ni)\n+d0+d1R; (A.1)\nwhere the six coe\u000ecients a0;a1;b0;c0;d0;d1are functions\noftandN, which can be related to the coe\u000ecients of\nthe Horndeski theory through Eqs. (8)-(13) in Ref. [28].\nComparing the above Lagrangian with Eq. (2.4), one\n\fnds\nc(0;0)\n1=d0; c(1;0)\n1=a0; c(0;2)\n1=d1; c(2;0)\n1=b0;\nc(2;0)\n2=\u0000b0; c(3;0)\n1= 2c0; c(3;0)\n2=\u00003c0; c(3;0)\n3=c0;14\nc(1;2)\n3=\u00002a1; c(1;2)\n4=a1; (A.2)\nwith all other coe\u000ecients c(t;s)\ni= 0. The corresponding\ncoe\u000ecientsG0;G1;G2andW0;W1;W2in the GW propa-\ngation equation (3.12) are\nG0=1\n2b0\u00003\n2c0H;G1= 0 =G2; (A.3)\nW0=1\n2d1\u00001\n2_a1;W1= 0 =G2: (A.4)\nThen using the constraints in Table. V, it is easy to infer\nthat\n\u00003\u000210\u000015<r\nd1\u0000_a1\nb0\u00003c0H\u00001<7\u000210\u000016(A.5)\nfrom the multimessenger observations of\nGW170817/GRB170817A [45, 46], and\n\u00003<_b0\u00003_c0H\u00003c0_H\nb0H\u00003c0H2<2:5 (A.6)\nby using the dark sirens in GWTC-3 with BBH mass\ndistributions [53].\n2. Scalar-Gauss-Bonnet gravity\nIt is also interesting to note that scalar-Gauss-Bonnet\ngravity with the Lagrangian LGB=\u0018(\u001e)R2\nGBcan be\nrecast in the form of the Horndeski theory [79], where\nR2\nGB\u00111\n4\"\u0016\u0017\u000b\f\"\u0015\u001c\n\u001a\u001b4R\u000b\f\u0015\u001c4R\u0016\u0017\u001a\u001b. Here\"\u0016\u0017\u001a\u001bis the\nLevi-Civita tensor and4R\u001a\u001b\u000b\f is the Riemann tensor\nde\fned in four-dimensional spacetime. In the unitary\ngauge, the propagation equation of GWs in scalar-Gauss-\nBonnet gravity corresponds to [28]\nG0=1\n4M2\nPl+ 2_\u0018H;G1= 0 =G2; (A.7)\nW0=1\n4M2\nPl+ 2\u0018;W1= 0 =G2: (A.8)\nUsing the constraints in Table V, one obtains\n\u00003\u000210\u000015.8\u0018\u0000_\u0018H\nM2\nPl.7\u000210\u000016(A.9)\nfrom the multimessenger observations of\nGW170817/GRB170817A [45, 46], and\n\u00003.8\u0018H+_\u0018_H\nM2\nPlH.2:5 (A.10)\nusing the dark sirens in GWTC-3 with BBH mass distri-\nbutions [53].\n3. Chern-Simons gravity\nChern-Simons gravity is an e\u000bective extension of GR\nthat includes a coupling between a scalar \feld \u001eand theChern-Pontryagin term [80{82]. The Lagrangian of the\nnew term beyond GR is\nLCS=M2\nPl\n21\n8#(\u001e)\"\u0016\u0017\u001a\u001b 4R\u001a\u001b\u000b\f4R\u000b\f\n\u0016\u0017;(A.11)\nIn the unitary gauge, one can \fnd that [28]\nL(u.g.)\nCS =M2\nPl\n2\"ijk#\u0000\nKilKlmrjKkm+Kl\niKm\njrmKkl\n\u0000KKl\nirjKkl\u00002Rl\nirjKkl\u00001\nN_#\n#Kl\nirjKkl\n\u00002\nNKl\niKjlKkmrmN\u00002\nNriKjlrkrlN\u0013\n:\n(A.12)\nIgnoring the terms containing spatial derivatives of N\nand comparingL(u.g.)\nCS with Eq. (2.4), one has\nc(2;0)\n1=c(0;2)\n1=\u0000c(2;0)\n2=M2\nPl\n2;\nc(3;1)\n2=\u0000c(3;1)\n1=\u0000c(3;1)\n3=\u00001\n2c(1;3)\n1=M2\nPl\n2#;\nc(2;1)\n1=\u0000M2\nPl\n2_#\nN: (A.13)\nThen the coe\u000ecients in the propagation equation (3.12)\nread\nG0=1\n4M2\nPl;G1=\u0000M2\nPl\n2_#\n2;G2= 0;(A.14)\nW0=1\n4M2\nPl;W1=\u0000M2\nPl\n2_#\n2;G2= 0:(A.15)\nThe nonzero coe\u000ecient G1induces amplitude birefrin-\ngence in the propagation of GWs. Using the constraints\nin Table. V, one gets\nj_#j.2065 km (A.16)\nfrom tests of amplitude birefringence with LIGO-Virgo\nO1/O2 events [54] and\nj_#j.1000 km (A.17)\nfrom the analysis of statistic distribution of cos \u0013in\nGWTC-2 [57].\n4. Lorentz-violating Weyl gravity\nWeyl gravity modi\fes GR by adding a Weyl-squared\nterm to the gravitational Lagrangian [83],\nLWeyl=\u0000M2\nPl\n2\r\n24C\u0016\u0017\u001a\u00154C\u0016\u0017\u001a\u0015; (A.18)\nwhere4C\u0016\u0017\u001a\u0015 is the Weyl tensor in four-dimensional\nspacetime. In the unitary gauge, the Lagrangian LWeyl\nbecomes [28, 83]\nL(u.g.)\nDSSY =M2\nPl\r(\u0000riKriK\u0000riKikrjKj\nk+ 2riKrjKj\ni15\n\u00002rkKijrjKik+ 2rkKijrkKij): (A.19)\nComparing this with Eq. (2.4), we have\nc(2;0)\n1=c(0;2)\n1=\u0000c(2;0)\n2=M2\nPl\n2;\nc(2;2)\n1=\u0000c(2;2)\n2=c(2;2)\n4=\u00001\n2c(2;2)\n3=\u00001\n2c(2;2)\n5= 2M2\nPl\r:\n(A.20)\nThen, the coe\u000ecients in the propagation equation (3.12)\nread\nG0=1\n4M2\nPl;G1= 0;G2=M2\nPl\r; (A.21)\nW0=1\n4M2\nPl;W1= 0;G2= 0: (A.22)\nUsing the constraints in Table. V, one gets\nj4\rj.1:2\u000210\u000010m2(A.23)\nfrom tests of Lorentz-violating dispersion with GWTC-3\n[13].\n5. Chiral scalar-tensor theory\nChern-Simons gravity can be extended to include\nparity-violating curvature terms with couplings between\nthe Riemann curvature and derivatives of the scalar \feld.\nOne theory of this type is the chiral scalar-tensor theory\nconsidered in Ref. [84], which includes couplings between\nthe Riemann curvature and the \frst and second deriva-\ntives of the scalar \feld.\nThe coupling of the \frst derivatives of the scalar\n\feld contains four terms with coupling coe\u000ecients\n(a1;a2;a3;a4) in the Lagrangian [84], which can be writ-\nten in the unitary gauge as\nLu:g:1\nchiral=M2\nPl_\u001e2\nN2h\n(4a1+a3)\"ijkKliKmjrmKk\nl\n\u0000(4a1+ 2a3)\"ijkRlirkKj\nl\n+a3\"ijk\u0000\nKi\nmKlm\u0000KKli\u0001\nrkKj\nli\n:(A.24)\nIn the above, we have used the condition 4 a1+ 2a2+\na3+8a4= 0 which makes the theory to be healthy in the\nunitary gauge.\nFor the coupling between the Riemann curvature and\nthe second derivatives of the scalar \feld, there are seven\nterms with coupling coe\u000ecients ( b1;b2;b3;b4;b5;b6;b7) in\nthe Lagrangian [84], which can be written in the unitary\ngauge as\nLu:g:2\nchiral=M2\nPl_\u001e3\nN3b1\"ijkKlirkKj\nl\n+M2\nPl_\u001e4\nN4(b4+b5\u0000b3)\"ijkKliKmjrmKk\nl:\n(A.25)In the above, we dropped all of the terms containing spa-\ntial derivatives of the lapse function, since they do not\ncontribute to the propagation of the tensorial GWs, and\nwe used the conditions b7= 0,b6= 2(b4+b5), and\nb2=\u0000_\u001e2\n2N2(b3\u0000b4) to make the theory to be healthy\nwhen the unitary gauge is imposed [84].\nNow the parity-violating Lagrangian of the chiral\nscalar-tensor theory reads Lu:g:\nchiral=Lu:g:\nCS+Lu:g:1\nchiral+Lu:g:2\nchiral.\nConsidering it with Eq. (2.4), one obtains\nc(2;0)\n1=c(0;2)\n1=\u0000c(2;0)\n2=M2\nPl\n2;\nc(3;1)\n1=M2\nPl\n2h\n\u0000#\u00002_\u001e2\nN2(4a1+a3)\n\u00002_\u001e4\nN4(b4+b5\u0000b3)i\n;\nc(1;3)\n1=M2\nPl\n2h\n\u00002#+2_\u001e2\nN2(4a1+ 2a3)i\n;\nc(2;1)\n1=M2\nPl\n2h\n\u0000\u0000_#\nN\u00002_\u001e3\nN3b1i\n;\nc(3;1)\n2=M2\nPl\n2h\n\u0000#\u00002_\u001e2\nN2a3i\n;\nc(3;1)\n3=M2\nPl\n2h\n\u0000#+2_\u001e2\nN2a3i\n: (A.26)\nThen the coe\u000ecients in the propagation equation (3.12)\nread\nG0=1\n4M2\nPl;G2= 0;\nG1=M2\nPl\n2h\n\u0000_#\n2\u0000b1_\u001e3+ 2(a3+ 2a1)_\u001e2H\n+(b4+b5\u0000b3)_\u001e4Hi\n;\nW0=1\n4M2\nPl;G2= 0;\nW1=M2\nPl\n2h\n\u0000_#\n2+ (4a1+ 2a3)_\u001e\u001e+ (2_a1+ _a3)_\u001e2i\n:\n(A.27)\nThe nonzero coe\u000ecient G1induces amplitude birefrin-\ngence in the propagation of GWs and the nonzero G1\u0000W 1\nleads to velocity birefringence. For jG1=G0j, using the\nconstraints in Table. 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D 68, 104012 (2003)\n[arXiv:gr-qc/0308071 [gr-qc]].\n[83] N. Deruelle, M. Sasaki, Y. Sendouda and A. Youssef,\nLorentz-violating vs ghost gravitons: the example of\nWeyl gravity, JHEP 09, 009 (2012) [arXiv:1202.3131 [gr-\nqc]].\n[84] M. Crisostomi, K. Noui, C. Charmousis and D. Langlois,\nBeyond Lovelock gravity: Higher derivative metric theo-\nries, Phys. Rev. D 97, 044034 (2018) [arXiv:1710.04531\n[hep-th]]."    },    {        "title": "1811.04821v1.Choking_non_local_magnetic_damping_in_exchange_biased_ferromagnets.pdf",        "content": "1 \n Choking non -local magnetic damping in exchange biased ferromagnets  \n \nTakahiro Moriy amaa), Kent  Oda, and Teruo Ono  \nInstitute for Chemical Research, Kyoto University, Gokasho, Uji, Kyoto, 611-0011, Japan.  \na ) corresponding  to :mtaka@scl.kyoto -u.ac.jp  \n \n \nAbstract  \nWe investigated the temperature dependence of the magnetic damping in the \nexchange biased Pt/ Fe 50Mn 50 /Fe 20Ni80 /SiO x multilayer s. In sample s having a stro ng \nexchange bias, we observed a  drastic decrease of the magnetic damping  of the FeNi  with \nincreasing temperature up to  the blocking temperature . The results essentially indicate  \nthat the non -local enhancement of the magnetic damping can be  choked by the  adjacent \nantiferromagne t and its temperature dependent  exchange bias. We also  point ed out that \nsuch a strong temperature dependent damping may be very beneficial for spintronic \napplications.   \n  2 \n The Gilbert damping constant , α, appearing in the Landau -Lifshitz -Gilbert equation  is \none of the important parameters characteri zing the magnetization dynamics [1 .. It \ninfluences  the switching speed of the magnetization  [2 , 3 . and also determines  the \nthreshold current for various spin-torque -related  phenomena [4, 5, 6, 7, 8.. With  those  \nnumerous  examples, i t is probably not an exagger ation to say that the performance  of \nevery spintronic device relies  on α. It is, howev er, unfortunate that the magnetic  damping \nis one of the least con trollable  magnetic  parameters among  those including  magnetic \nanisotropy, saturation magnetization, and gyromagnetic ratio which  a material \nengineering as well as  some exogenous  controls (e.g. temperature, electric field bias , \nstructural strain , etc. ) [9, 10, 11, 12, 13, 14, 15. can easily tune over a wide range . \nAlthough  some  efforts have been made to minimize α by carefully engineering the band \nstructure [ 16., an exogenous  control  of it, which is very much desirable for spintronic \napplications, has rarely been explored.  \n The magnetic damping is rooted in various relaxation processes of the spin \nangular momentum [17 .. The majority of such process es locally occur  within  the \nmagnetic material via, e.g., spin -phonon and spin -spin relaxations  [18.. On the other hand, \nan interaction between the magnetization  dynamics  and the itinerant electron spin , i.e. the \nspin pumping effect [ 19., results in a non -local magnetic damp ing of the spin angular \nmomentum.  In other words, when the spin angular momentum carried by the itinerant \nelectron diffuses into  an adjacent non -magnetic material, an additional damping \nenhancement can occur  depending on the spin relaxation  properties of the non -magnetic \nmaterial . For instance,  a strong damping enhancement  due to the spin pumping effect  has \nbeen observed  Pt/ ferrom agnet  (FM)  multilayers in which the Pt works as a good spin \ndissipative material [ 20 .. This non -local dampin g enhancement has been recently 3 \n revisited  for the system of the exchange biased FM/antiferromagnet (AFM) multialyers  \n[21.. In Ref.  21, it was shown  that the non-local dam ping was modified  by the N éel order \nin the AFM . In particular,  the linear relation between the strength of the exchange bias \n(EB)  and the en hanced damping strongly indicates  that an  additional spin relaxation  takes \neffect due to th e orientation of the N éel order . Furthermore, t he relatively gradual AFM \nthickness dependence of the damping enhancement , suggesting a spin current \ntransmission through the AFM  [22 , 23 , 24 , 25 , 26 , 27 , 28 ., was also  certainly \nintriguing .  \nFrom another point of view, t hese re sults indeed offer a  control  of the magnetic  \ndamping by the magnetic order in the adjacent AFM.  It is therefore worthwhile  to further \ninvestigate and understand  the magnetic  damping in exchange biased FM/AFM systems  \nand to gain a control of it for the sake of advancement in  spintronic application s. In this \nwork, we extend ed our investigations  to the temperature dependence of the non -local \ndamping in the exchange biased AFM/FM multilayer . We particularly paid attention to \nthe damping variation s toward  the blocking temperature TB of the EB. It is found  that the \nnon-local magnetic damping drastically decreases with increasing temperature  when EB \nis strong . Moreover,  the linear relation between the strength of the EB and the enhanced \ndamping, previously found in terms of  the AFM thickness dependence of EB [ 21., is \nfound to also hold  in terms of  the temperature dependence.   \nWe investigated Pt 5 nm/ Fe 50Mn 50 tFeMn nm/Fe 20Ni80 4 nm/SiO x 2nm ( tFeMn = 0, \n3, 20, and 60 nm) multilayers grown on a thermally oxidized Si substrate as illustrated in  \nFig. 1 (a) . The samples were photolithographically patterned into a 10 μm wide strip \nattached to a coplanar waveguide made of Ti/Au layer  facilitating a high frequency \nmeasurement . EB was set by a field cooling  process  with a n annealing  temperature of TFC 4 \n = 200 Cº and a field of  2 kOe.  No irreversible degradations of the film, e.g. due to atom \ndiffusions, were found in this field cooling process  [21.. High sensitivity f erromagnetic \nresonance (FMR)  measurements  were carried out by  a homodyne detection technique  [29, \n30 , 31 . with an identical circuitry  and methodology  used in Ref. 21. The homodyne \nvoltage Vdc were measured with a fixed frequency  f of the r.f. current and with a swept  \nexternal magnetic field applied either along or against t he direction of the exchange bias \nfield Heb which  are denoted by “along EB ” and “agains t EB”, respectively . The FMR  \nmeas urements were at first performed  at 298K and the measurement temperature was \nstepped up to 393 K.  \nFigure 1 (b) sh ows representative  FMR spectra obtained  at 298 K and 393 K for \ntFeMn = 20 nm with f = 8 GHz and the external field along and against EB . The \ndisplacement between  the two spectra at 298 K with the fields  along EB and against EB \nmanifests a unidirectional magnetic anisotropy, i.e. the exchange bias field. One can also \nsee that the exchange bias is diminished at 393 K.  FMR  spectra are fitted by the \ncombination of symmetric and anti -symmetric Lorentzians by which the resonant \nparame ters, such as the resonant field Hres and the spectral linewidth  ΔH, are extracted  \n[29..  \nFigure 2 shows  f vs. Hres and ΔH vs. f at 298 K and 393 K for tFeMn = 0, 3, 20, \nand 60 nm. In order to extract the effective demagnetizing field 4𝜋𝑀eff and the exchange \nbias field Heb, the f vs. Hres curves are fitted by the Kittel ’s equation 𝑓=\n(𝛾2𝜋⁄ )√(𝐻res+𝐻𝑢)(𝐻res+4𝜋𝑀eff), where 𝛾 is the gyromagnetic ratio and  Hu is the \nanisotropy field from which both uniaxial anisotropy field and the exchange bias field Heb \nare derived . The Gilbert  damping 𝛼 is extracted  from the  ΔH vs. f plot by using  ∆𝐻=\n∆𝐻0+2𝜋𝛼𝑓 𝛾⁄ [32., where ∆𝐻0 is a frequency independent -linewidth known as the 5 \n inhomogeneous broadening [ 33.. We note that our linewidth analyses explicitly separate \nthe frequency dependent damping , or the Gilbert damping  𝛼, from the independent on e, \nor inhomogeneous broadening which may be related to the two -magnon scattering  at the \nFM/AFM interface [ 34,35.. As shown in Fig. 2, a ll the samples exhibit a good  Kittel ’s \nfitting as well as a  linear fitting for  the α extraction . We note that, for tFeMn = 3 nm at 298 \nK, there was an exceptionally large uniaxial magnetic anisotropy field of 390 Oe, which \nmay lead to significantly large  ∆𝐻0  compared  to samples with  other tFeMn which \nessentially read ∆𝐻0 = 0. All the discussion s about the magnetic damping thereafter will \nbe referring to  α which reflects the intrinsic property of the system [33..  \nFigure s 3 (a), (b), and (c) summarize  4𝜋𝑀eff , Heb, and α, respectively, as a \nfunction of temperature. For all the samples, 4𝜋𝑀eff is ranging from 0.88 to 0.96 Tesla \nwhich is considered reasonable for the Fe 20Ni80 with a possible  interfacial  perpendicular \nanisotropy [ 36.. 4𝜋𝑀eff overall decreases by ~10 % with increasing temperature  from \n298 to 393 K, which can be attributed to  the thermal decay of the magnetization  as well \nas a change in the interfacial anisotropy . The exchange bias field i s found to be largest \nHeb = 79 Oe with tFeMn = 20 nm  and it monotonically decreases with  increasing \ntemperature.  The blocking temperature for the exchange bias is  found to be  TB = 393 K \nand 373 K for tFeMn = 20 and 60 nm, respectively.  No exchange bias field was observed \nfor tFeMn = 0 and 3 nm in the measurement temperature range.  It is remarkable that , for \ndifferent tFeMn , α behaves quite differently with respect to temperature . Namely, the \nsamples with tFeMn = 0 and 60 nm show a slight upward  trend of α with increasing \ntemperature but one  with tFeMn = 3 shows  almost constant with respect to temperature . It \nis noticeable that  α for tFeMn = 20 nm  undergoes a dra stic decrease, almost by a half , with \nincreasing temperature up to TB. We also emphasize that α measured with the field  along 6 \n EB was found to be always smaller than that measured with the field  against EB.  \nFigure 3 (d) plot s Δα as a function of Heb wher e Δα is the difference of α \nmeasured with the field  along EB and against EB. It is found that Δα increases  lineally \nwith respect to Heb. Although Heb varies due to temperature in this present case,  the \nobservation of the linear relation between Heb and Δα is essentially same as what was \nobserved previously with respect to Heb which varies with respect to tFeMn  [21.. The inset \nof Fig. 3 (d)  shows  α as a function of the field angle φ with respect to the exchange bias \ndirection ( φ = 0 corresponds to the direction along EB.) measured for tFeMn = 20 nm at \n298 and 343 K , which depicts the φ dependence of α similar to that observed in  the \nprevious report [ 21.. \nNow, o ne find s two peculiar point s in our results , especially when  focus ing on \ntFeMn = 20 nm . One is the strong temperature dependence of  α with the exchange bias . The \nother is the temperature dependence of  Δα which results in the linear relation between  Δα \nand Heb. While the latter is consistent with our previous report  and originates from the \nNéel order twisting [ 21., the former is quite intriguing in both physics and application \npoints of views.  Looking at Figs. 3 (c) and (d), one can presume  that the reduction of  α \nmay be correlated with the strength of the exchange bias.  \nIt has been  shown  that the intrinsic damping of  a single layer of FeNi  only \nslightly varies in this  temperature  range  [37.. A non -local damping enhancement by an \nadjacent Pt  due to the spin pumping effect should not significantly vary since the relevant \nparameters such  as the mixing conductance and the spin diffusion length in Pt do not vary  \nmuch  in this temperature range.  Very  little temperature  variation  of α in our control \nsample (tFeMn = 0 nm) is therefore consistent with those previous observation s as well as \nthe expectations . It is now very  clear  that the observed  temperature dependence of  α with 7 \n tFeMn =20 nm is solely associated  with the FeMn insertion and the exchange bias between \nthe FeMn and FeNi  [38. (Also, see Supplementary Information) . \nThe mechanism of the exchange bias is generally explained by two major factors \n[39.; i.e. the exchange coupling at the FM/AFM interface and the magnetic anisotropy \nenergy of t he AFM . Because a thermal agitation effectively reduces the magnetic \nanisotropy, the exchange bias field diminishes toward TB at which the effective magnetic \nanisotropy essentially becomes zero.  Considering the spin transmission in \nantiferromagnets by the form of the N éel order dynamics, there is a negative correlation \nbetween the magnetic anisotropy and  the spin diffusion length  (or the  so-called  healing \nlength for the spin transmission) λ𝐹𝑒𝑀𝑛  [24,40]. Ther efore, our observations shown in \nFigs. 3(b) and (c)  can be considered representing  a negative correlation between  α and \nλ𝐹𝑒𝑀𝑛 .  \nAccording to Ref. 40, α as a function of λ𝐹𝑒𝑀𝑛  behaves differently depending \non either the strong or weak damping limit  of the antiferromagnet . Our experimental \nresults are more consistent with the strong damping limit in which α increases with \ndecreasing λ𝐹𝑒𝑀𝑛 . We should note  here that the present  analysis neglects  a direct  thermal \neffect on the N éel order dynamics , such as thermal magnons,  which could  be more \nimportant  than considering the effective magnetic anisotropy  but is difficult to be taken \ninto account  analytically at this moment.  \nFinally , we highlight our results in the engineering point of view. As we pointed \nout above, t he magnetic  damping  of FeNi  should  generally  exhibit only a slight \ntemperature dependence , at around room temperature,  regardless of the local or non -local \nmechanisms. Our results suggest  that, by making use of the exchange biased bilayer,  one \ncan drastically “choke ” the non-local enhancement of the  magnetic damping in a 8 \n relatively narrow  temperature range . For  those spintronic applications i nvolving  a Joule \nheating up on the operation ( e.g. the spin torque magnetic memory  [6. requires quite a bit \nof current density to write information bits which heats up the ferromagnetic bits.), this \ntemperature control  of α may be beneficial since  it is reduced only when they are in \noperation.  \nIn summary, we investigated the temperature dependence of the magnetic \ndamping in the exchange biased Pt/ Fe 50Mn 50 /Fe 20Ni80 /SiO x multilayer in the \ntemperature range of 298 ~ 393 K. We reconfirmed the linear relation between Heb and \nΔα, which  have been seen in our previous report  [21., by the present  experimental \napproach  which is different f rom that in  the previous report . In sample s having a strong \nexchange bias, we observed the drastic decrease of the magnetic dampin g with increasing \ntemperature up to  the blocking temperature. The results lead us to the negative correlation \nbetween the magnetic anisotropy of the FeMn and the damping enhancement, implying \nthe mechanism of spin current transmission through the FeMn with a strong damping \nlimit [ 40.. Furthermore, w e pointed out that this strong temperature dependent damping \nmay be  very beneficial for spintronic applications.   \n \n \nAcknowledgements  \nThis work was supported in part by JSPS KAKENHI  Grant Numbers 26870300,  \n17H04924,  15H05702 , and  17H05181  (“Nano Spin Conversion Science ”). We also  \nacknowledge the support from Center for Spintronics Research Network (CSRN) .  \n  9 \n Figure captions:  \n \nFig. 1  (a) Schematic illustration of the layer structure  under  investigat ion by the \nhomodyne FMR  measurement . (b) Representative FMR spectra  (the homodyne voltage \nVdc as a function of the applied field either along (red  markers ) or against EB (blue  \nmarkers )) for tFeMn = 20 nm at f = 8 GHz measured at 298 K (upp er panel) and 393 K \n(lower panel).  The continuous lines are the fitting  by the combination of symmetric and \nanti-symmetric Lorentzians.  \n \nFig. 2  Resonant field Hres and the spectral linewidth ΔH as a function of frequency f \nmeasured at 298 K and 393 K for (a, e) tFeMn = 0 nm, (b, f) tFeMn = 3 nm, (c, g) tFeMn = 20 \nnm, and (d, h) tFeMn = 60 nm. The data with the applied field along and against EB are \nplotted.  The continuous lines are the fitting by the Kittel ’s equation and ∆𝐻=∆𝐻0+\n2𝜋𝛼𝑓 𝛾⁄ described in the main text.  \n \nFig. 3  (a) 4𝜋𝑀𝑒𝑓𝑓 as a function  of temperature , (b) Heb as a function of temperature, (c) \nα as a function of temperature, and (d) Δα as a function of Heb for tFeMn = 0 nm (dark blue), \n3 nm, (light blue), 20 nm (green), and 60 nm (red) . In panel (c), α obtained with the \napplied field along EB (upward triangle) and against EB (downward triangle) are \nseparately shown.  The inset of (d) shows  α as a function of the field  angle φ with respect \nto the exchange bias direction  for tFeMn = 20 nm.  (note that φ = 0° and 180° correspond to \n“along EB ” and “against EB ”, respectively.).  \n \n  10 \n  \nFigure 1  Moriyama et al.  \n \n  \n11 \n  \nFigure 2  Moriyama et al.  \n12 \n  \nFigure 3  Moriyama et al.  \n \n  \n13 \n References:  \n[1. T. L. Gilbert, Phys. Rev.  100, 1243 (1955).  \n[2. C.H. Back et al., Phys. Rev. Lett.  81, 3251 (1998).  \n[3. M. 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Th e damping term is assumed\nto be non-linear and localized to an open subset of the domain . We handle the disturbed case\nas an extension of [16], where stability results are given wi th a damping term active on the\nfull domain with no disturbances considered. We provide inp ut-to-state types of results.\n1 Introduction\nConsider the damped wave equation with localized damping, w ith Dirichlet boundary conditions\ngiven by\n(Pdis)\n\nutt−∆u=−a(x)g(ut+d)−e,inR+×Ω,\nu= 0, onR+×∂Ω,\nu(0,.) =u0, ut(0,.) =u1,(1)\nwhereΩis aC2bounded domain of R2,dandestand for a damping disturbance and a globally\ndistributed disturbance for the wave dynamics respectivel y. The term −a(x)g(ut+d)stands for the\n(perturbed) damping term, where g:R−→Ris aC1non-decreasing function verifying ξg(ξ)>0\nforξ/\\e}atio\\slash= 0whilea:Ω→Ris a continuous non negative function which is bounded below by a\npositive constant a0on some non-empty open subset ωofΩ. Here,ωis the region of the domain\nwhere the damping term is active, more precisely, the region where the localization function ais\nbounded from below by a0. As for the initial condition (u0,u1), it belongs to the standard Hilbert\nspaceH1\n0(Ω)×L2(Ω).\nIn this paper, our aim is to obtain input-to-state (ISS) type of results for (Pdis), i.e., estimates\nof the norm of the state uwhich, at once, show that trajectories tend to zero in the abs ence of\ndisturbances and remain bounded by a function of the norms of the disturbances otherwise.\n1UMA, ENSTA, Palaiseau and L2S, Université Paris Saclay, Fra nce.\n2Département de Mathématiques, Université Abou Bekr Belkai d, Tlemcen, Algeria.\n3L2S, Université Paris Saclay, France.\n∗This research was partially supported by the iCODE Institut e, research project of the IDEX Paris-Saclay,\nand by the Hadamard Mathematics LabEx (LMH) through the gran t number ANR-11-LABX-0056-LMH in the\n“Programme des Investissements d’Avenir”.\n1One can refer to [20] for a thorough review of ISS results and t echniques for finite dimension\nsystems and to the recent survey [19] for infinite dimensiona l dynamical systems. In the case of\nthe undisturbed dynamics, i.e., (1) with (d,e)≡(0,0), there is a vast literature regarding the\nstability of the corresponding system with respect to the or igin, which is the unique equilibrium\nstate of the problem. This in turn amounts to have appropriat e assumptions on aandg, cf. [3] for\nextensive references. We will however point out the main one s that we need in order to provide\nthe context of our work. To do so, we start by defining the energ y of the system by\nE(t) =1\n2/integraldisplay\nΩ/parenleftbig\n|∇u|2(t,·)+u2\nt(t,·)/parenrightbig\ndx, (2)\nwhich defines a natural norm on the space H1\n0(Ω)×L2(Ω). Strong stabilization has been established\nin the early works [8] and [10], i.e., it is proved with an argu ment based on the Lasalle invariance\nprinciple that limt→+∞E(t) = 0 for every initial condition in H1\n0(Ω)×L2(Ω). However, no decay\nrate of convergence for Eis established since it requires in particular extra assump tions ongand\nω.\nAs a first working hypothesis, we will assume that g′(0)>0, classifying the present work\nin those that aimed at establishing results of exponential c onvergence for strong solutions. We\nrefer to [3] for the line of work where gis assumed to be super-linear in a neighborhood of the\norigin (typically of polynomial type). Note that, in most of these works (except for the linear\ncase) the rate of exponential decay of Edepends on the initial conditions. That latter fact in\nturn relies on growth conditions of gat infinity. Regarding the assumptions on ω, they have\nbeen first put forward in the pioneering work [21] on semi-lin ear wave equations and its extension\nin [13], where the multiplier geometric conditions (MGC) ha ve been characterized for ωin order to\nachieve exponential stability. For linear equations, the s harpest geometrical results are obtained\nby microlocal techniques using the method of geometrical op tics, cf [4] and [7].\nIn this paper, our objective is to obtain results for non-lin ear damping terms and one should\nthink of the nonlinearity gnot only as a mean to provide more general asymptotic behavio r at\ninfinity than a linear one but also as modeling an uncertainty of the shape of the damping term.\nDealing with nonlinearities justifies why microlocal techn iques are not suited here and we will be\nusing the multiplier method as presented e.g. in [12]. Many r esults have been established in the\ncase where g′(0) = 0 , for instance, decay rates for the energy are provided in [15 ] in the localized\ncase but the non-linearity is to have a linear growth for larg e values of its arguments. Note that the\nestimates as presented in [15] are not optimal in general, as for instance in the case of a power-like\ngrowth. For general optimal energy decay estimates and for g eneral abstract PDEs, we refer the\nreader to [1] for a general formula for explicit energy decay estimates and to [2] for an equivalent\nsimplified energy decay estimate with optimality results in the finite dimensional case. However,\nwhen it comes to working under the hypothesis g′(0)>0, few general results are available. One\ncan find a rather complete presentation of the available resu lts in [16]. In particular, the proof\nof exponential stability along strong solutions has only be en given for general nonlinearities g, in\ndimension two and in the special case of a non-localized damp ing with no disturbances requiring\nonly one multiplier coupled with a judicious use of Gagliard o-Nirenberg’s inequality. Our results\ngeneralize this finding in the absence of disturbances (even though it has been mentioned in [16]\nwith no proof that this is the case). It has also to be noted tha t similar results are provided in [15]\nin the localized case but the nonlinearity is lower bounded b y a linear function for large values\nof its arguments. That simplifies considerably some computa tions. Recall also that the purpose\nof [15] is instead to address issues when g′(0) = 0 and to obtain accurate decay rates for E.\n2Hence a possible interest of the present paper is the fact tha t it handles nonlinearities gso that\ng(v)/vtends to zero as |v|tends to infinity with a linear behavior in a neighborhood of t he origin.\nAs for ISS purposes, this paper can be seen as an extension to t he infinite dimensional context\nof [14] where the nonlinearity is of the saturation type. Mor eover, the present work extends to the\ndimension two the works [17] and [18], where this type of issu es have been addressed by building\nappropriate Lyapunov functions and by providing results in dimension one. Here, we are not able\nto construct Lyapunov functions and we rely instead on energ y estimates based on the multiplier\nmethod, showing how these estimates change when adding the t wo disturbances dande. To develop\nthat strategy, we must impose additional assumptions on g′, still handling saturation functions.\nAs a final remark, we must recall that [16] contains other stab ility results in two directions. On\none hand,g′can simply admit a (possibly) negative lower bound and on the other hand, the space\ndimensionNcan be larger than 2, at the price of more restrictive assumptions on g, in particular,\nby assuming quasi-linear lower bounds for its asymptotic be havior at infinity. One can readily\nextend the results of the present paper in both directions by eventually adding growth conditions\nong.\n2 Statement of the problem and main result\nIn this section, we provide assumptions on the data needed to precisely define (1). We henceforth\nrefer to (1) as the disturbed problem (Pdis). Next, we state and comment the main results of this\nwork and discuss possible extensions.\nThroughout the paper, the domain Ωis a bounded open subset of R2of classC2, the assumptions\nongare the following.\n(H1): The function g:R−→Ris aC1non-decreasing function such that\ng(0) = 0, g′(0)>0, g(x)x>0forx/\\e}atio\\slash= 0, (3)\n∃C >0,∃1<q <5,∀ |x| ≥1,|g(x)| ≤C|x|q, (4)\n∃C >0,∃0<m<4,∀|x|>1,|g′(x)| ≤C|x|m. (5)\n(H2): The localization function a:Ω→Ris a continuous function such that\na≥0onΩand∃a0>0, a≥a0on ω. (6)\nIn order to prove the stability of solutions, we impose a mult iplier geometrical condition (MGC)\nonω. It is given by the following hypothesis.\n(H3): There exists an observation point x0∈R2for whichωcontains the intersection of Ω\nwith anǫ-neighborhood of\nΓ(x0) ={x∈∂Ω,(x−x0).ν(x)≥0}, (7)\nwhereνis the unit outward normal vector for ∂Ωand anǫ-neighborhood of Γ(x0)is defined by\nNǫ(Γ(x0)) ={x∈R2:dist(x,Γ(x0))≤ǫ}. (8)\n3Regarding the disturbances dande, we make the following assumptions.\n(H4): the disturbance function d:R+×Ω−→Rbelongs to L1(R+,L2(Ω))and satisfies the\nfollowing:\nd(t,·)∈H1\n0(Ω)∩L2q(Ω),∀t∈R+, t/ma√sto→/integraldisplayt\n0∆d(s,·)ds−dt(t,·)∈Lip/parenleftbig\nR+,H1\n0(Ω)/parenrightbig\n,(9)\nwhereLipdenotes the space of Lipschitz continuous functions. We als o impose that the following\nquantities\nC1(d) =/integraldisplay∞\n0/integraldisplay\nΩ(|d|2+|d|2q)dxdt, C 2(d) =/integraldisplay∞\n0/integraldisplay\nΩ|d|m(dt)2dxdt,\nC3(d) =/integraldisplay∞\n0/integraldisplay\nΩ(dt)2dxdt, C 4(d) =/integraldisplay∞\n0/parenleftbigg/integraldisplay\nΩ|dt|2(p\np−1)dx/parenrightbigg(p−1\np)\ndt, (10)\nare all finite, where pis a fixed real number so that, if 0<m≤2, thenp>2\nmand if2<m<4,\nthenp∈(1,m\nm−2).\nRemark 2.1 The fact that dbelongs toL1(R+,L2(Ω))means that the following quantity is finite\nC5(d) =/integraldisplay∞\n0||d||L2(Ω)dt, (11)\nwhich implies that the following quantity is also finite\nC6(d) =/integraldisplay∞\n0/integraldisplay\nΩ|d|dxdt. (12)\n(H5): The disturbance function e:R+×Ω−→Rbelongs to W1,1(R+,L2(Ω))and satisfies the\nfollowing\ne∈Lip/parenleftbig\nR+,H1\n0(Ω)/parenrightbig\n, e(0,.)∈L2(Ω),¯C1(e) =/integraldisplay∞\n0/integraldisplay\nΩe2dxdt<∞. (13)\nRemark 2.2 The fact that ebelongs toW1,1(R+,L2(Ω))means that the following quantities are\nfinite\n¯C2(e) =/integraldisplay∞\n0||e(t,·)||L2(Ω)dt,¯C3(e) =/integraldisplay∞\n0||et(t,·)||L2(Ω)dt. (14)\nRemark 2.3 In the rest of the paper, we will use various symbols C,CuandCd,ewhich are\nconstants independent of the time t. However, it is important to stress that these symbols have\nspecific dependence on other parameters of the problem. More precisely, the symbol Cwill be used\nto denote positive constants independent of initial condit ions and disturbances, i.e., only depending\non the domains Ω,ωand the functions aandg. The symbol Cudenotes a generic K-function of the\nnorms of the initial condition (u0,u1)and similarly the symbol Cd,edenotes a generic K-function of\nthe several quantities Ci(d)and¯Ci(e). HereKdenotes the set of continuous increasing functions\nγ:R+→R+withγ(0) = 0 , cf. [19].\nMoreover, in the course of intermediate computations, we wi ll try to keep all the previous constants\nas explicit as possible in terms of the norms of the initial co ndition and the Ci(d)and¯Ci(e)in\norder to keep track of the nature of generic constants. We wil l use the latter generic mainly in the\nstatements of the results.\n4Before we state the main results, we define the notion of a stro ng solution of (Pdis). To do so, we\nstart by giving an equivalent form of (Pdis):\nDefine for every (t,x)∈R+×Ω,¯d(t,x) =/integraltextt\n0d(s,x)ds. We translate uin(Pdis)asv=u+¯d, it is\nimmediate to see that (Pdis)is equivalent to the following problem:\n\n\nvtt−∆v+a(x)g(vt) = ˜e,inR+×Ω,\nv= 0, onR+×∂Ω,\nv(0,.) =v0, vt(0,.) = ¯u1,(15)\nwhere˜e=dt−∆¯d−e,v0=u0andv1=u1+d(0,.).\nDefine the unbounded operator\nA:H=H1\n0(Ω)×L2(Ω)−→H,\n(x1,x2)/ma√sto−→(x2,−∆x1+ag(x2)), (16)\nwith domain\nD(A) =/parenleftbig\nH2(Ω)∩H1\n0(Ω)/parenrightbig\n×H1\n0(Ω).\nFort≥0, set\nU(t) =/parenleftbiggu(t,·)\nut(t,·)/parenrightbigg\n, V(t) =/parenleftbiggv(t,·)\nvt(t,·)/parenrightbigg\n, D(t) =/parenleftbigg¯d(t,·)\nd(t,·)/parenrightbigg\n, G(t) =/parenleftbigg0\n˜e/parenrightbigg\n.\nNotice that G∈Lip(R+,L2(Ω)×H1\n0(Ω)). Then Problem (15) can be written as\nVt(t) =AV(t)+G(t), V(0) =V0=/parenleftbiggv0\nv1/parenrightbigg\n. (17)\nA strong solution of (17) in the sens of [5] is a function V∈C(R+,H), absolutely continuous in\nevery compact of R+, satisfying V(t)∈D(A),∀t∈R+and satisfying (17) almost everywhere in\nR+. On the other hand, the hypotheses satisfied by dimply that D(t)∈D(A)for everyt∈R+.\nSinceU=V−D, we can now give the following definition for a strong solutio n of(Pdis).\nDefinition 2.1 (Strong solution of (Pdis).)\nA strong solution uof(Pdis)is a function u∈C1(R+,L2(Ω))∩C(R+,H1\n0(Ω))such thatt/ma√sto→ut(t,·)\nis absolutely continous in every compact of R+. For allt∈R+,(u(t,·),ut(t,·))∈D(A)andu(t,·)\nsatisfies(Pdis)for almost all t∈R+.\nWe gather our findings in the following theorem regarding the disturbed system (Pdis).\nTheorem 2.1 Suppose that Hypotheses (H1)to(H5)are satisfied. Then, given (u0,u1)∈(H2(Ω)∩\nH1\n0(Ω))×H1\n0(Ω), Problem (Pdis)has a unique strong solution u. Furthermore, the following energy\nestimate holds:\nE(t)≤(C+Cu)E(0)e−t−1\nCu+C+Cd,e(Cu+1), (18)\nwhere the positive constant Cudepends only on the initial conditions and the positive cons tant\nCd,edepends only on the disturbances dande.\n5Remark 2.4 (Comments and extensions)\n•Theorem 2.1 holds true if the Lipschitz assumptions in (9)and(13)are replaced by bounded\nvariation ones.\n•In the case where the disturbances are both zero ( d≡0ande≡0), Theorem 2.1 holds without\nthe hypothesis on g′given by (5)(i.e. no restriction on qin(4)) and the hypothesis given by\n(4)can be then weakened to the following one\n∃C >0,∃q >1,∀ |x| ≥1,|g(x)| ≤C|x|q.\nIt is clear that if gsatisfies the last part of the condition above for 0≤q≤1, it would still\nsatisfy it for any q>1.\n•The geometrical condition MGC imposed in (H3)can be readily reduced to the weaker and\nmore general MGC introduced in [13] and called piecewise MGC in [3].\nRemark 2.5 Note that (18)is an ISS-type estimate but it fails to be a strict one (let say in the\nsense of Definition 1.6in [19]) for two facts. First of all, the estimated quantity Eis the norm of\na trajectory in the space H1\n0(Ω)×L2(Ω)while the constant Cudepends on the initial condition by\nits norm in the smaller space (H2(Ω)∩H1\n0(Ω))×H1\n0(Ω). This difference seems unavoidable since\nin the undisturbed case exponential decay can be proved only for strong solutions as soon as the\nnonlinearity gis not assumed to be bounded below at infinity by a linear funct ion. As a matter of\nfact, it would be interesting to prove that strong stability is the best convergence result one could get\nfor weak solutions, let say with damping functions gof saturation type functions and in dimension\nat least two.\nThe second difference lies in the second term in (18), namely it is not just a K-function of the\nnorms of the disturbances. We can get such a result if we have a n extra assumption on g, typically\ngof growth at most linear at infinity (i.e., q= 1) with bounded derivative (i.e., m= 0). In\nparticular, this covers the case of regular saturation func tions (increasing bounded functions gwith\nbounded derivatives).\nWe give now the proof of the well-posedness part of Theorem (2 .1).\nProof of the well-posedness: The argument is standard since −A, whereAis defined in (16), is\na maximal monotone operator on H1\n0(Ω)×L2(Ω)(cf. for instance [11] for a proof). We can apply\nTheorem 3.4 combined with Propositions 3.2 and Proposition s 3.3 in [5] to (17), which immediately\nproves the results of the well- posedness part.\n/squaresolid\nRemark 2.6 In [16], the domain of the operator has been chosen as\n{(u,v)∈H1\n0(Ω)×H1\n0(Ω) :−∆u+g(v)∈L2(Ω)}.\nHowever, in dimension two, taking the domain of Ain the case where d=e= 0asZ={(u,v)∈\nH1\n0(Ω)×H1\n0(Ω) :−∆u+a(x)g(v)∈L2(Ω)}or as(H2(Ω)∩H1\n0(Ω))×H1\n0(Ω)is equivalent. Indeed,\nusing the hypothesis given by (4), we have that |g(v)| ≤C|v|qfor|v|<1, which means when\n6combining it with the fact that g(0) = 0 that|g(v)| ≤C|v|q+C|v|for allv. From Gagliardo-\nNirenberg theorem (see in Appendix) we have for v∈H1\n0(Ω)that\n/ba∇dblv/ba∇dbl2q\nL2q(Ω)≤C/ba∇dblv/ba∇dbl2q−2\nH1\n0(Ω)/ba∇dblv/ba∇dbl2\nL2(Ω),\nwhich means that\n/ba∇dblg(v)/ba∇dbl2\nL2(Ω)=/integraldisplay\nΩ|g(v)|2dx≤C/integraldisplay\nΩ(|v|q+|v|)2dx≤C/ba∇dblv/ba∇dbl2q\nL2q(Ω)+C/ba∇dblv/ba∇dbl2\nL2(Ω)\n≤ /ba∇dblv/ba∇dbl2q−2\nH1\n0(Ω)/ba∇dblv/ba∇dbl2\nL2(Ω)+C/ba∇dblv/ba∇dbl2\nL2(Ω)<+∞(sincev∈H1\n0(Ω)),\ni.e.,g(v)∈L2(Ω). Then, by using Lemma 3.2 (with (d,e)≡(0,0)), we have that −∆u+ag(v)∈\nL2(Ω), which means that ∆u∈L2(Ω). On the other hand, /ba∇dbl∆u/ba∇dblL2(Ω)is an equivalent norm to the\nnorm ofH2(Ω)∩H1\n0(Ω)andΩis of classC2(the proof is a direct result of Theorem 4 of Section\n6.3 in [9]). We can finally conclude that Zis nothing else but (H2(Ω)∩H1\n0(Ω))×H1\n0(Ω).\n3 Proof of the energy estimate (18)\nTo prove the energy estimate given by (18), we are going to use the multiplier method combined\nwith a Gronwall lemma and other technical lemmas given in thi s section. We will be referring\nto [15] and [16] in several computations since our problem is a generalization of their strategy to\nthe case where the disturbances (d,e)are present.\nWe start with the following lemma stating that the energy Eis bounded along trajectories of\n(Pdis).\nLemma 3.1 Under the hypotheses of Theorem (2.1), the energy of a strong solution of Problem\n(Pdis), satisfies\nE′(t) =−/integraldisplay\nΩautg(ut+d)dx−/integraldisplay\nΩutedx,∀t≥0. (19)\nFurthermore, there exist positive constants CandCd,esuch that\nE(T)≤CE(S)+Cd,e,∀0≤S≤T. (20)\nProof of Lemma 3.1: Equation (19) follows after multiplying the first equation o f (1) byut\nand performing standard computations. Notice that we do not have the dissipation of Esince the\nsign ofE′is not necessarily constant. To achieve (20), we first write\n−/integraldisplay\nΩautg(ut+d)dx=−/integraldisplay\n|ut|≤|d|autg(ut+d)dx−/integraldisplay\n|ut|>|d|autg(ut+d)dx. (21)\nOn one hand, from (3) and the fact that (ut+d)anduthave the same sign if |ut|>|d|, we deduce\nthat\n−/integraldisplay\n|ut|>|d|autg(ut+d)dx≤0. (22)\n7On the other hand, since gis non-decreasing, has linear growth in a neighborhood of ze ro by (3),\nand satisfies (4), it follows that\n−/integraldisplay\n|ut|≤|d|autg(ut+d)dx≤C/integraldisplay\n|ut|≤|d||d||g(|2d|)|dx≤C/integraldisplay\nΩ|d||g(|2d|)|dx\n≤C/integraldisplay\n|d|<1|d||g(2d)|dx+C/integraldisplay\n|d|≥1|d||g(2d)|dx\n≤C/integraldisplay\n|d|<1|d|2dx+C/integraldisplay\n|d|≥1|d|q+1dx\n≤C/integraldisplay\nΩ(|d|2+|d|2q)dx. (23)\nCombining (21), (22), (23) and (19), we obtain that\nE′≤C/integraldisplay\nΩ(|d|2+|d|2q)dx−/integraldisplay\nΩutedxdt. (24)\nUsing Cauchy-Schwarz inequality,\nE′≤C/integraldisplay\nΩ(|d|2+|d|2q)dx+/parenleftbigg/integraldisplay\nΩ|e|2dxdt/parenrightbigg1\n2/parenleftbigg/integraldisplay\nΩ|ut|2dx/parenrightbigg1\n2\n≤C/integraldisplay\nΩ(|d|2+|d|2q)dx+C/ba∇dble/ba∇dblL2(Ω)√\nE,\nthen integrating between two arbitrary non negative times S≤T, we get\nE(T)≤E(S)+CC1(d)+C/integraldisplayT\nS/ba∇dble/ba∇dblL2(Ω)√\nEdt,\nwhich allows us to apply Theorem A.2 and conclude that\nE(T)≤CE(S)+CC1(d)+C¯C2(e)2=CE(S)+Cd,e.\nHence, the proof of Lemma 3.1 is completed.\n/squaresolid\nRemark 3.1 In the absence of disturbances, in other words when d=e= 0we have that:\nE′(t) =−/integraldisplay\nΩautg(ut)dx,∀t≥0, (25)\nand thus the energy Eis non increasing by using (3). That latter fact simplifies the proof of\nexponential decrease in this case.\nWe provide now an extension of Lemma 2 in [16] to the context of (Pdis).\nLemma 3.2 Under the hypotheses of Theorem 2.1, for every solution of Pr oblem(Pdis)with initial\nconditions (u0,u1)∈(H2(Ω)∩H1\n0(Ω))×H1\n0(Ω), there exist explicit positive constants CuandCd,e\nsuch that\n∀t≥0,/ba∇dbl−∆u(t,·)+a(·)g(ut(t,·)+d(t,·))+e(t,·)/ba∇dbl2\nL2(Ω)+/ba∇dblut(t,·)/ba∇dbl2\nH1\n0(Ω)≤Cu+Cd,e.(26)\n8Proof of Lemma 3.2: We setw:=ut, whereuis the strong solution of (Pdis). We know\nthatw(t)∈H1\n0(Ω)for everyt≥0. Moreover, it is standard to show that w(t)satisfies in the\ndistributional sense the following problem:\n\n\nwtt−∆w+ag′(w+d)(wt+dt)+et= 0, inΩ×R+,\nw= 0, on∂Ω×R+,\nw(0) =u1, wt(0) = ∆u0−g(u1+d(0))−e(0).(27)\nSetEw(t)to be the energy of wfor allt≥0. It is given by\nEw(t) =1\n2/integraldisplay\nΩ(w2\nt(t,x)+|∇w(t,x)|2)dx.\nUsingwtas a test function in (27), then performing standard computa tions, we derive\nEw(t)−Ew(0) =−/integraldisplayt\n0/integraldisplay\nΩ(ag′(w+d)(dt+wt)wt+etwt)dxdτ. (28)\nLetI:=/integraltextt\n0/integraltext\nΩa(.)g′(w+d)(dt+wt)wtdxdτ. We split the domain ΩinIaccording to whether\n|dt| ≤ |wt|or not. Clearly the part corresponding to |dt| ≤ |wt|is non negative since g′≥0,a≥0\nand(dt+wt)andwthave the same sign. From (5), one has the immediate estimate\ng′(a+b)≤C(1+|a+b|m)≤C(1+|a|m+|b|m),∀a,b∈R.\nUsing the above, we can rewrite (28) as\nEw(t)−Ew(0)≤/integraldisplayt\n0/integraldisplay\n|dt|>|wt|ag′(w+d)(dt+wt)wtdxdτ+/integraldisplayt\n0/integraldisplay\nΩ|et||wt|dxdτ\n≤C/integraldisplayt\n0/integraldisplay\nΩg′(w+d)d2\ntdxdτ+C/integraldisplayt\n0||et||L2(Ω)/radicalbig\nEwdτ\n≤C/integraldisplayt\n0/integraldisplay\nΩ(1+|w|m+|d|m)d2\ntdxdτ+C/integraldisplayt\n0/ba∇dblet/ba∇dblL2(Ω)/radicalbig\nEwdτ. (29)\nUsing Hölder’s inequality,\n/integraldisplayt\n0/integraldisplay\nΩ|w|md2\ntdxdτ≤/integraldisplayt\n0/parenleftbigg/integraldisplay\nΩ|w|pmdx/parenrightbigg1\np/parenleftbigg/integraldisplay\nΩ|dt|2p′dx/parenrightbigg1\np′\ndτ, (30)\nwithpdefined in (10) and p′>1is its conjugate exponent given by1\np+1\np′= 1. Thanks to the\nassumptions on p, one can use Gagliardo-Nirenberg’s inequality for wto get\n/parenleftbigg/integraldisplay\nΩ|w(t,x)|pmdx/parenrightbigg1\np\n≤CEw(t)mθ\n2E(t)(1−θ)m\n2, t≥0, (31)\nwhereθ= 1−2\nmp. Combining (31), (30) and (29), it follows that\nEw(t)−Ew(0)≤C/integraldisplayt\n0Emθ\n2wE(1−θ)m\n2/integraldisplay\nΩ/parenleftBig\n|dt|2p′dx/parenrightBig1\np′dτ\n+/integraldisplayt\n0/integraldisplay\nΩ(1+|d|m)d2\ntdxdτ+C/integraldisplayt\n0||et||L2(Ω)/radicalbig\nEwdτ. (32)\n9Note thatmθ\n2<1. Settingh1(t) =/integraltext\nΩ/parenleftbig\n|dt|2p′dx/parenrightbig1\np′,h2(t) =||et||L2(Ω)and using (20), (32) becomes\nEw(t)≤Ew(0)+C2(d)+C3(d)+(Cu+Cd,e)/integraldisplayt\n0Emθ\n2wh1(s)ds+C/integraldisplayt\n0h2(s)/radicalbig\nEwds. (33)\nWe know that\n/integraldisplay∞\n0h1(t)dt=C4(d)<∞,/integraldisplay∞\n0h2(t)dt=¯C3(e)<∞. (34)\nWe can now apply Theorem A.2 on (33) with\nS= 0, T=t, α1=mθ\n2, α1=1\n2, F(·) =Ew(·), C3=C2(d)+C3(d), C1=Cu+Cd,e, C2=C.\nWe obtain the following bound for Ew(·):\nEw(t)≤max/parenleftBig\n2(Ew(0)+C2(d)+C3(d)),(2˜C)1\n1−α/parenrightBig\n, (35)\nwhere˜C:=C1/ba∇dblh1/ba∇dbl1+C2/ba∇dblh2/ba∇dbl1andα:= max(α1,α2)if2˜C≥1orα:= min(α1,α2)if2˜C <1.\nIt is clear that ˜C= (Cu+Cd,e)C4(d)+C¯C3(e)≤Cu+Cd,e. One then rewrites (35) as\nEw(t)≤2(Ew(0)+C2(d)+C3(d))+(Cu+Cd,e)1\n1−α. (36)\nNote that for t≥0one obviously has that\nEw(t) =1\n2/integraldisplay\nΩ(w2\nt(t,x)+|���w(t,x)|2)dx\n=1\n2/parenleftBig\n||utt(t,·)||2\nL2(Ω)+/ba∇dblut(t,·)/ba∇dbl2\nH1\n0(Ω)/parenrightBig\n=/ba∇dbl−∆u(t,·)+a(·)g(ut(t,·)+d(t,·))+e(t,·)/ba∇dbl2\nL2(Ω)+/ba∇dblut(t,·)/ba∇dbl2\nH1\n0(Ω).\nThe conclusion of the lemma follows since, by taking into acc ount (4), it is clear that Ew(0)≤\nCu+Cd,e.\n/squaresolid\nWe next provide the following important estimate based on Ga gliardo-Nirenberg theorem:\nLemma 3.3 For allq>2, a strong solution uof(Pdis)satisfies\n/ba∇dblut(t,·)/ba∇dblq\nLq(Ω)≤(Cu+Cd,e)E(t), t≥0. (37)\nProof of Lemma 3.3: We derive immediately from (26) that /ba∇dblut/ba∇dblH1\n0(Ω)≤Cu+Cd,e. Then,\nusing Gagliardo-Nirenberg’s theorem, it follows that, for everyt≥0,\n/ba∇dblut(t,·)/ba∇dblq\nLq(Ω)≤C/ba∇dblut(t,·)/ba∇dblq−2\nH1\n0(Ω)/ba∇dblut(t,·)/ba∇dbl2\nL2(Ω)≤(Cu+Cd,e)E(t). (38)\n/squaresolid\n10We have all the tools now to start the proof of the second part o f Theorem 2.1. The stability result\nwill be achieved as a direct consequence of the following pro position:\nProposition 3.1 Suppose that the hypotheses of Theorem (2.1)are satisfied, then the energy E\nof the strong solution uof(Pdis)with(u0,u1)∈(H2(Ω)∩H1\n0(Ω))×H1\n0(Ω)), satisfies the following\nestimate: /integraldisplayT\nSE(t)dt≤(Cu+C)E(S)+(1+Cu)Cd,e, (39)\nwhere the positive constant Cudepends only on the initial condition, the positive constan tCd,e\ndepends only on the disturbances danderespectively and Cis a positive real constant.\n3.0.1 Proof of Proposition 3.1\nWe now embark on an argument for Proposition 3.1. It is based o n the use of several multipliers\nthat we will apply to the partial differential equation of (1) . For that purpose, we need to define\nseveral functions associated with Ω.\nLet(u0,u1)∈(H2(Ω)∩H1\n0(Ω))×H1\n0(Ω),S≤Ttwo non negative times and x0∈R2an ob-\nservation point. Define ǫ0,ǫ1andǫ2three positive real constants such that ǫ0<ǫ1<ǫ2<ǫwhere\nǫis the same defined in 8. Using ǫi, we define Qifori= 0,1,2asQi=Nǫi[Γ(x0)].\nSince(Ω\\Q1)∩Q0=∅, we are allowed to define a function ψ∈C∞\n0(R2)such that\n\n\n0≤ψ≤1,\nψ= 1on¯Ω\\Q1,\nψ= 0onQ0.\nWe also define the C1vector field honΩby\nh(x) :=ψ(x)(x−x0). (40)\nWhen the context is clear, we will omit the arguments of h.\nWe use the multiplier M(u) :=h∇u+u\n2to deduce the following first estimate:\nLemma 3.4 Under the hypotheses of Proposition 3.1, we have the followi ng inequality:\n/integraldisplayT\nSE dt≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketleftbigg/integraldisplay\nΩutM(u)dx/bracketrightbiggT\nS/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft /bracehtipupright\nT1+C/integraldisplayT\nS/integraldisplay\nΩ∩Q1|∇u|2dxdt\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nT2+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayT\nS/integraldisplay\nΩag(ut+d)M(u)dxdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nT3\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayT\nS/integraldisplay\nΩeM(u)dxdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nT4+C/integraldisplayT\nS/integraldisplay\nωu2\ntdxdt\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nT5, (41)\nwherehis defined in (40)andM(u)is the multiplier given by h.∇u+u\n2.\nProof of Lemma 3.4. The proof is based on multiplying (Pdis)by the multiplier M(u)and\nintegrating on [S,T]×Ω. Then, we follow the steps that led to the proof of equation (3.15)in [15]\nexcept that we take σ= 0andφ(t) =tin the beginning and we replace ρ(x,ut)bya(x)g(ut+d)+e.\n11/squaresolid\nRemark 3.2 From now on, whenever we refer to a proof in [15], we refer to th e steps of the proof\nwith the change of σ= 0andφ(t) =tas well as replacing ρ(x,ut)bya(x)g(ut+d)+e.\nThe goal now is to estimate the terms T1toT5.\nLemma 3.5 Under the hypotheses of Proposition 3.1, there exists a posi tive constant Csuch that\nT1≤CE(S)+Cd,e. (42)\nProof of Lemma 3.5: Exactly as the proof of equation (5.14)in [15] except that we use (20) in\nthe very last step since we do not have the non-increasing of t he energy here. We obtain (42).\n/squaresolid\nThe estimation of T2requires more work and it is given in the following lemma:\nLemma 3.6 Under the hypotheses of Proposition 3.1, T2is estimated by\nT2≤Cη0/integraldisplayT\nSE dt+C\nη0/integraldisplayT\nS/integraldisplay\nωu2\ntdxdt+1\nη0(C+Cu+Cd,e)E(S)+1\nη5\n0(Cd,eCu+Cd,e),(43)\nwhere0<η0<1is an arbitrary real positive number to be chosen later.\nProof of Lemma 3.6: The argument requires a new multiplier, namely ξu, where the function\nξ∈C∞\n0(R2)is defined by\n\n0≤ξ≤1,\nξ= 1onQ1,\nξ= 0onR2\\Q2.(44)\nSuch a function ξexists since R2\\Q2∩Q1=∅. Using the multiplier ξuand following the steps\nin the proof of Lemma 9in [15], yields the following identity:\n/integraldisplayT\nS/integraldisplay\nΩξ|∇u|2dxdt=/integraldisplayT\nS/integraldisplay\nΩξ|ut|2dxdt+1\n2/integraldisplayT\nS/integraldisplay\nΩ∆ξu2dxdt−/bracketleftbigg/integraldisplay\nΩξuutdx/bracketrightbiggT\nS\n−/integraldisplayT\nS/integraldisplay\nΩξu[a(x)g(ut+d)+e]dxdt. (45)\nCombining the fact that ∆ξis bounded and the definition of ξ, we derive from (45) that\nT2≤/integraldisplayT\nS/integraldisplay\nΩ∩Q2|ut|2dxdt+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketleftbigg/integraldisplay\nΩ∩Q2uutdx/bracketrightbiggT\nS/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft /bracehtipupright\nS1+C/integraldisplayT\nS/integraldisplay\nΩ∩Q2u2dxdt\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft /bracehtipupright\nS2\n+/integraldisplayT\nS/integraldisplay\nΩ|uag(ut+d)|dxdt\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nS3+/integraldisplayT\nS/integraldisplay\nΩ|ue|dxdt\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nS4. (46)\n12First, note that the first term of (46) is upper bounded by/integraltextT\nS/integraltext\nω|ut|2dxdtsinceΩ∩Q2⊂ω. Left\nto estimate the other terms in the right-hand side of (46). We start by treating S1. We easily get\nthe following estimate by using Young and Poincaré inequali ties:\n/integraldisplay\nΩ∩Q2|uut|dx≤1\n2/integraldisplay\nΩ∩Q2|u|2dx+1\n2/integraldisplay\nΩ∩Q2|ut|2dx≤CE. (47)\nUsing (20) with (47) we obtain the estimation of S1given by\nS1≤CE(S)+Cd,e. (48)\nTo estimate S2, we introduce the last multiplier in what follows:\nSince(Ω\\ω)∩(Q2∩Ω) =∅, there exists a function β∈C∞\n0(R2)such that\n\n\n0≤β≤1,\nβ= 1onQ2∩Ω,\nβ= 0onΩ\\ω.(49)\nFor everyt≥0, letzbe the solution of the following elliptic problem:\n/braceleftbigg∆z=βuinΩ,\nz= 0 on∂Ω.(50)\nOne can prove the following lemma:\nLemma 3.7 Under the hypotheses of Proposition 3.1 with zas defined in (50), it holds that\n||z||L2(Ω)≤C||u||L2(Ω),||zt||2\nL2(Ω)≤C/integraldisplay\nΩβ|ut|2dx,/ba∇dbl∇z/ba∇dblL2(Ω)≤C||∇u||L2(Ω), (51)\n∀S≤T∈R+,/integraldisplayT\nS/integraldisplay\nΩβu2dxdt=/bracketleftbigg/integraldisplay\nΩzutdx/bracketrightbiggT\nS+/integraldisplayT\nS/integraldisplay\nΩ(−ztut+z[ag(ut+d)+e])dxdt.\n(52)\nProof of Lemma 3.7: Equation 51 gathers standard elliptic estimates from the de finition ofz\nas a solution of (50) while (52) is obtained by using zas a multiplier for (Pdis). Steps of the proof\nare similar to the ones that led to equations (5.22),(5.25)and(5.26)in [15].\n/squaresolid\nSince the non negative βis equal to 1onQ2and0onR2\\ω, it follows from (52) that\nS2≤/bracketleftbigg/integraldisplay\nΩzutdx/bracketrightbiggT\nS/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nU1−/integraldisplayT\nS/integraldisplay\nΩztutdxdt\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nU2+/integraldisplayT\nS/integraldisplay\nΩz(ag(ut+d)+e)dxdt\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nU3. (53)\nWe estimate U1,U2andU3. We start by handling U1. One has from Cauchy-Schwarz inequality,\nthen (51) and Poincaré inequality that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nΩzutdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤ ||z||L2(Ω)||ut||L2(Ω)≤C||∇u||L2(Ω)||ut||L2(Ω)≤CE(t). (54)\n13Using (54) and the fact that Eis non-increasing, it is then immediate to derive that\n|U1|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg/integraldisplay\nΩzutdx/parenrightbigg\n(T)−/parenleftbigg/integraldisplay\nΩzutdx/parenrightbigg\n(S)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(E(T)+E(S)). (55)\nFinally, using (20) in (55), we obtain that\nU1≤CE(S)+Cd,e. (56)\nAs forU2, the use of Young inequality with an arbitrary real number 0<η0<1yields\n|U2| ≤/integraldisplayT\nS/integraldisplay\nΩ1\n2η0|zt|2dxdt+/integraldisplayT\nS/integraldisplay\nΩη0\n2|ut|2dxdt.\nThen, we use (51) and the fact that 0≤β≤1to conclude the following estimate:\nU2≤C\nη0/integraldisplayT\nS/integraldisplay\nωu2\ntdxdt+Cη0/integraldisplayT\nSEdxdt, (57)\nwhereη0is a positive real number to be chosen later.\nLeft to estimate U3. We can rewrite it as the following:\nU3=/integraldisplayT\nS/integraldisplay\n|ut+d|≤1a(x)zg(ut+d)dxdt\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nV1+/integraldisplayT\nS/integraldisplay\n|ut+d|>1a(x)zg(ut+d)dxdt\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nV2+/integraldisplayT\nS/integraldisplay\nΩa(x)zedxdt\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft /bracehtipupright\nV3.\n(58)\nWe estimate the three terms V1,V2andV3. We start by estimating V1. We have using Young\ninequality that\nV1≤Cη0/integraldisplayT\nSE dt+1\nη0/integraldisplayT\nS/integraldisplay\n|ut+d|≤1|ag(ut+d)|2dxdt. (59)\nThe fact that g(0) = 0 implies the existence of a constant C >0such that |g(x)| ≤C|x|for all\n|x| ≤1. Combining it with the fact that g(x)x≥0,∀x∈R, it follows that\n/integraldisplayT\nS/integraldisplay\n|ut+d|≤1|ag(ut+d)|2dxdt≤/integraldisplayT\nS/integraldisplay\n|ut+d|≤1a(.)(ut+d)g(ut+d)dxdt\n≤/integraldisplayT\nS/integraldisplay\nΩa(.)(ut+d)g(ut+d)dxdt. (60)\nUsing (19) and Young inequality with 0<η1<1,\n/integraldisplayT\nS/integraldisplay\nΩa(.)(ut+d)g(ut+d)dxdt=/integraldisplayT\nS/integraldisplay\nΩautg(ut+d)dxdt+/integraldisplayT\nS/integraldisplay\nΩadg(ut+d)dxdt\n≤/integraldisplayT\nS/integraldisplay\nΩautg(ut+d)dxdt+/integraldisplayT\nS/integraldisplay\nΩutedxdt−/integraldisplayT\nS/integraldisplay\nΩutedxdt+C/integraldisplayT\nS/integraldisplay\nΩ|d||g(ut+d)|dxdt\n≤/integraldisplayT\nS(−E′)dt+/integraldisplayT\nS/integraldisplay\nΩ|ut||e|dxdt+C/integraldisplayT\nS/integraldisplay\nΩ|d||g(ut+d)|dxdt\n≤E(S)+Cη1/integraldisplayT\nSE dt+C\nη1/integraldisplayT\nS/integraldisplay\nΩ|e|2dxdt+C/integraldisplayT\nS/integraldisplay\nΩ|d||g(ut+d)|dxdt\n≤E(S)+Cη1/integraldisplayT\nSE dt+C\nη1¯C1(e)+C/integraldisplayT\nS/integraldisplay\nΩ|d||g(ut+d)|dxdt. (61)\n14Left to estimate/integraltextT\nS/integraltext\nΩ|d||g(ut+d)|dxdt, we proceed as the following:\n/integraldisplayT\nS/integraldisplay\nΩ|d||g(ut+d)|dxdt=/integraldisplayT\nS/integraldisplay\n|ut+d|≤1|d||g(ut+d)|dxdt+/integraldisplayT\nS/integraldisplay\n|ut+d|>1|d||g(ut+d)|dxdt\n≤C/integraldisplayT\nS/integraldisplay\n|ut+d|≤1|d|dxdt+C\nη′\n1/integraldisplayT\nS/integraldisplay\n|ut+d|>1|d|2dxdt+η′\n1/integraldisplayT\nS/integraldisplay\n|ut+d|>1|g(ut+d)|2dxdt\n≤CC6(d)+C\nη′\n1C1(d)+Cη′\n1/integraldisplayT\nS/integraldisplay\n|ut+d|>1|ut+d|2qdxdt\n≤CC6(d)+C\nη′\n1C1(d)+Cη′\n1/integraldisplayT\nS/integraldisplay\nΩ|ut|2q+Cη′\n1/integraldisplayT\nS/integraldisplay\nΩ|d|2qdxdt, (62)\nwhere0<η′\n1<1. Then, using (37),\n/integraldisplayT\nS/integraldisplay\nΩ|d||g(ut+d)|dxdt≤CC6(d)+C\nη′\n1C1(d)+η′\n1(Cu+Cd,e)/integraldisplayT\nSE(t)dt+Cη′\n1C1(d)\n≤1\nη′\n1Cd,e+η′\n1(Cu+Cd,e)/integraldisplayT\nSE(t)dt. (63)\nCombining (61) and (63),\n/integraldisplayT\nS/integraldisplay\nΩa(.)(ut+d)g(ut+d)dxdt≤E(S)+(η1+η′\n1(Cu+Cd,e))/integraldisplayT\nSE dt+1\nη1η′\n1Cd,e.(64)\nCombining now (64), (61) and (59), we obtain that\nV1≤C/parenleftbigg\nη0+η′\n1\nη0(Cu+Cd,e)+η1\nη0/parenrightbigg/integraldisplayT\nSE dt+C\nη0E(S)+1\nη1η0η′\n1Cd,e.\nWe takeη1=η2\n0andη′\n1=η2\n0\nCu+Cd,eifCu+Cd,e>0. In that case, V1would be estimated by\nV1≤Cη0/integraldisplayT\nSE dt+C\nη0E(S)+1\nη5\n0Cd,e(Cu+Cd,e). (65)\nIfCu=Cd,e= 0, the above equation holds true trivially.\nRemark 3.3 With such a choice of η1andη′\n1, we have the following useful estimate obtained from\n(64):\n/integraldisplayT\nS/integraldisplay\nΩa(.)(ut+d)g(ut+d)dxdt≤E(S)+Cη2\n0/integraldisplayT\nSE dt+1\nη4\n0(Cd,eCu+Cd,e). (66)\nTo estimate V2, first notice that from Rellich-Kondrachov’s theorem in dim ension two (cf. [6]) that\nH1(Ω)⊂Lq+1(Ω),which means that ∃C >0such that /ba∇dblz/ba∇dblLq+1(Ω)≤C/ba∇dblz/ba∇dblH1(Ω), adding to that\nthe fact that z∈H1\n0(Ω)and (51), it holds that\n/ba∇dblz/ba∇dblLq+1(Ω)≤C√\nE. (67)\n15Then, using Hölder inequality yields\nV2≤/integraldisplayT\nS/parenleftbigg/integraldisplay\n|ut+d|>1(a|g(ut+d)|)q+1\nqdx/parenrightbiggq\nq+1/parenleftbigg/integraldisplay\n|ut+d|>1|z|q+1dx/parenrightbigg1\nq+1\ndt. (68)\nCombining (68) with the hypothesis given by (4), we get that\nV2≤C/integraldisplayT\nS/parenleftbigg/integraldisplay\n|ut+d|>1a|ut+d||g(ut+d)|dx/parenrightbiggq\nq+1/parenleftbigg/integraldisplay\n|ut+d|>1|z|q+1dx/parenrightbigg1\nq+1\ndt.\nUsing Young inequality for an arbitrary 0<η2<1,\nV2≤C/integraldisplayT\nS\n1\nηq+1\nq\n2/integraldisplay\n|ut+d|>1a(x)(ut+d)g(ut+d)dx+ηq+1\n2/integraldisplay\nΩ|z|q+1dx\ndt\n≤C/integraldisplayT\nS\n1\nηq+1\nq\n2/integraldisplay\nΩa(x)(ut+d)g(ut+d)dx+ηq+1\n2/integraldisplay\nΩ|z|q+1dx\ndt\n≤C/integraldisplayT\nS\n1\nηq+1\nq\n2/integraldisplay\nΩa(x)utg(ut+d)dx+C\nηq+1\nq\n2/integraldisplay\nΩ|d||g(ut+d)|dx+ηq+1\n2/integraldisplay\nΩ|z|q+1dx\ndt.(69)\nThe previous inequality combined with (19) and (67) implies that\nV2≤C/integraldisplayT\nS\n1\nηq+1\nq\n2(−E′)−1\nηq+1\nq\n2/integraldisplay\nΩutedx+C\nηq+1\nq\n2/integraldisplay\nΩ|d||g(ut+d)|dx+ηq+1\n2Eq+1\n2\ndt.\nThen, using (20), Esatisfies\n/integraldisplayT\nSEq+1\n2dt=/integraldisplayT\nSEq−1\n2Edt≤(CE(0)+Cd,e)q−1\n2/integraldisplayT\nSEdt≤(Cu+Cd,e)/integraldisplayT\nSEdt, (70)\nwhich gives that\nV2≤C\nηq+1\nq\n2E(S)+ηq+1\n2(Cu+Cd,e)/integraldisplayT\nSEdt+C\nηq+1\nq\n2/integraldisplayT\nS/parenleftbigg\n−/integraldisplay\nΩutedx+/integraldisplay\nΩ|d||g(ut+d)|dx/parenrightbigg\ndt.\nWe fixη2=/parenleftBig\nη0\n(Cu+Cd,e)/parenrightBig1\nq+1. It follows that\nηq+1\n2(Cu+Cd,e) =η0,\nC\nηq+1\nq\n2=C(Cu+Cd,e)1\nq\nη1\nq\n0≤C\nη0(C1\nq\nu+C1\nq\nd,e) =1\nη1\nq\n0(Cu+Cd,e),\nwhich leads to\nV2≤1\nη1\nq\n0(Cu+Cd,e)E(S)+η0/integraldisplayT\nSEdt+1\nη1\nq\n0(Cu+Cd,e)/integraldisplayT\nS/parenleftbigg\n−/integraldisplay\nΩutedx+/integraldisplay\nΩ|d||g(ut+d)|dx/parenrightbigg\ndt.\n(71)\n16To finish the estimation of V2, we still have to handle the last two integral terms in (71).\nOn one hand, we have already estimated the term/integraltextT\nS/integraltext\nΩ|d||g(ut+d)|dxdt in (63). We have\nimmediately for some 0<η3<1that\n(Cu+Cd,e)/integraldisplayT\nS/integraldisplay\nΩ|d||g(ut+d)|dxdt≤η3(Cu+Cd,e)/integraldisplayT\nSE dt+1\nη3(Cd,eCu+Cd,e). (72)\nChoosingη3to be equal toηq+1\nq\n0\n(Cu+Cd,e)implies that\nη3(Cu+Cd,e) =ηq+1\nq\n0,\n1\nη3(Cd,eCu+Cd,e)≤1\nηq+1\nq\n0(Cd,eCu+Cd,e),\nwhich gives that\n(Cu+Cd,e)/integraldisplayT\nS/integraldisplay\nΩ|d||g(ut+d)|dxdt≤ηq+1\nq\n0/integraldisplayT\nSE dt+1\nηq+1\nq\n0(Cd,eCu+Cd,e). (73)\nOn the other hand, we have for 0<η4<1that\n(Cu+Cd,e)/integraldisplayT\nS/integraldisplay\nΩutedxdt≤η4(Cu+Cd,e)/integraldisplayT\nSE dt+1\nη4(Cd,eCu+Cd,e).\nUsing the same concept as before, we fix η4=ηq+1\nq\n0\nCu+Cd,e, we obtain that\n(Cu+Cd,e)/integraldisplayT\nS/integraldisplay\nΩutedxdt≤ηq+1\nq\n0/integraldisplayT\nSE dt+1\nηq+1\nq\n0(Cd,eCu+Cd,e), (74)\nCombining (71), (73) and (74), we conclude that the estimati on ofV2is given by\nV2≤1\nη1\nq\n0(Cu+Cd,e)E(S)+η0/integraldisplayT\nSEdt+1\nηq+2\nq\n0(Cd,eCu+Cd,e). (75)\nAs forV3, we simply have when using (51) and Young inequality with η0that\nV3≤Cη0/integraldisplayT\nSE dt+C\nη0¯C1(e),\nwhich means that\nV3≤Cη0/integraldisplayT\nSE dt+1\nη0Cd,e. (76)\nTo achieve an estimation of S2, we just combine (56),(57) (65), (75) and (76) to get\nS2≤Cη0/integraldisplayT\nSE dt+C\nη0/integraldisplayT\nS/integraldisplay\nωu2\ntdxdt+\nC+C\nη0+1\nη1\nq\n0(Cu+Cd,e)\nE(S)\n+\n1\nη5\n0+1\nηq+2\nq\n0\n(Cd,eCu+Cd,e)+/parenleftbigg1\nη0+1/parenrightbigg\nCd,e. (77)\n17We can simplify the previous estimate by using the fact that 0<η0<1. As a result, (77) becomes\nS2≤Cη0/integraldisplayT\nSE dt+C\nη0/integraldisplayT\nS/integraldisplay\nωu2\ntdxdt+1\nη0(C+Cu+Cd,e)E(S)+1\nη5\n0(Cd,eCu+Cd,e).(78)\nRegardingS3, we follow the same steps we followed to get V1+V2. It is possible because usatisfies\nthe same result (67) as zfrom before. Hence, we obtain that\nS3≤Cη0/integraldisplayT\nSE dt+1\nη0(C+Cu+Cd,e)E(S)+1\nη5\n0(Cd,eCu+Cd,e). (79)\nFinally, to estimate S4, we simply have when using young inequality that\nS4≤η0/integraldisplayT\nSE dt+1\nη0Cd,e. (80)\nWe complete the estimate of T2in (46) by combining the estimations of S1,S2,S3andS4. Hence\nthe proof of Lemma 3.6 is completed.\n/squaresolid\nAn estimate of T3is provided in the next lemma:\nLemma 3.8 Under the hypotheses of Proposition 3.1, we have the followi ng estimate:\nT3≤Cη0/integraldisplayT\nSE dt+1\nη0[C+(1+Cη0)(Cu+Cd,e)]E(S)+1\nη5\n0/parenleftbig\nC3\nη0+1/parenrightbig\n(Cd,eCu+Cd,e),(81)\nwhere0< η0<1is a positive arbitrary real number to be chosen later and Cη0is an implicit\npositive constant that depends on η0only.\nProof of Lemma 3.8: First, note that\nT3≤1\n2S3+/integraldisplayT\nS/integraldisplay\nΩ|ag(ut+d)∇u.h|dxdt\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nX. (82)\nWe have already estimated S3in (79). It remains to deal with X. Using Young inequality implies\nthat\nX≤C\nη0/integraldisplayT\nS/integraldisplay\nΩ(a|g(ut+d)|)2dxdt+Cη0/integraldisplayT\nS/integraldisplay\nΩ|∇u|2dxdt\n≤C\nη0/integraldisplayT\nS/integraldisplay\nΩa|g(ut+d)|2dxdt+Cη0/integraldisplayT\nSE dt. (83)\nNow, setR1>1to be chosen later. We can rewrite the term/integraltextT\nS/integraltext\nΩa|g(ut+d)|2dxdtas\n/integraldisplayT\nS/integraldisplay\nΩa|g(ut+d)|2dxdt=/integraldisplayT\nS/integraldisplay\n|ut+d|≤R1a|g(ut+d)|2dxdt\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nY1+/integraldisplayT\nS/integraldisplay\n|ut+d|>R1a|g(ut+d)|2dxdt\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nY2.\n(84)\n18Sinceg(0) = 0 , it holds that |g(x)| ≤CR1|x|for some constant CR1and for|x|<R1. Combine it\nwith (66), it follows that Y1satisfies for some 0<η5<1\nY1≤CR1/integraldisplayT\nS/integraldisplay\n|ut+d|≤R1|ag(ut+d)||ut+d|dxdt\n≤CR1/integraldisplayT\nS/integraldisplay\nΩ|ag(ut+d)||ut+d|dxdt\n≤CR1E(S)+CCR1η2\n5/integraldisplayT\nSE dt+CR1\nη4\n5(Cd,eCu+Cd,e). (85)\nTakingη5=η0√\nCCR1leads to\nY1≤CR1E(S)+η2\n0/integraldisplayT\nSE dt+C3\nR1\nη4\n0(Cd,eCu+Cd,e) (86)\nAs forY2, we use (4) to obtain that\nY2≤C/integraldisplayT\nS/integraldisplay\n|ut+d|>R1|ut+d|2qdxdt\n≤C/integraldisplayT\nS/integraldisplay\n|ut+d|>R1|ut|2qdxdt+C/integraldisplayT\nS/integraldisplay\n|ut+d|>R1|d|2qdx\n≤C/integraldisplayT\nS/integraldisplay\n|ut+d|>R1|ut+d|\nR1|ut|2qdxdt+C/integraldisplayT\nS/integraldisplay\nΩ|d|2qdxdt\n≤C/integraldisplayT\nS/integraldisplay\nΩ|ut|\nR1|ut|2qdxdt+C/integraldisplayT\nS/integraldisplay\nΩ|d|\nR1|ut|2qdxdt+CC1(d)\n≤C\nR1/integraldisplayT\nS/integraldisplay\nΩ|ut|2q+1dxdt+C\nR2\n1/integraldisplayT\nS/integraldisplay\nΩ|ut|4qdxdt+Cd,e.\nThen, we use Lemma 3.3 as well as the fact that R1>1to conclude that Y2satisfies\nY2≤1\nR1(Cu+Cd,e)/integraldisplayT\nSE dt+Cd,e.\nWe takeR1=(Cu+Cd,e)\nη2\n0, we get the simplified estimate\nY2≤η2\n0/integraldisplayT\nSE dt+Cd,e. (87)\nRemark 3.4 For such a choice of R1, and based on how CR1is defined, we can assume that CR1\nin(86)is a constant of the type Cη0(Cu+Cd,e), whereCη0is a positive constant that depends on\nη0only.\nCombining (83), (84), (86) and (87) implies that\nX≤Cη0/integraldisplayT\nSE dt+Cη0\nη0(Cd,e+Cu)E(S)+C3\nη0\nη5\n0(Cd,eCu+Cd,e)+Cd,e\nη0. (88)\nFinally, we combine (82) and (88) with the estimation of S3, we obtain (81).\n19/squaresolid\nWe next seek to prove the upper bound of T4that is given by the following lemma\nLemma 3.9 Under the hypotheses of Proposition 3.1, the following esti mate holds:\nT4≤Cη0/integraldisplayT\nSE dt+C\nη0Cd,e, (89)\nwhere0<η0<1is a positive constant to be chosen later.\nProof of Lemma 3.9 :We have that\nT4≤1\n2/integraldisplayT\nS/integraldisplay\nΩ|eu|dxdt+/integraldisplayT\nS/integraldisplay\nΩ|e∇u.h|dxdt. (90)\nOn one hand, using Young inequality gives that\n/integraldisplayT\nS/integraldisplay\nΩ|eu|dxdt≤η0/integraldisplayT\nSE dt+C\nη0Cd,e. (91)\nOn the other hand, it gives that\n/integraldisplayT\nS/integraldisplay\nΩ|e∇u.h|dxdt≤η0/integraldisplayT\nSE dt+C\nη0Cd,e (92)\nCombining (90), (91) and (92), we prove (89).\n/squaresolid\nIt remains to handle the last term T5.\nLemma 3.10 Under the hypotheses of Proposition 3.1, we have the followi ng estimation:\nT5≤η0/integraldisplayT\nSE dt+¯Cη0(Cu+Cd,e)E(S)+¯C3\nη0\nη2\n0(Cd,eCu+Cd,e)+Cd,e, (93)\nwhere0< η0<1is a positive constant to be chosen later and and ¯Cη0is an implicit positive\nconstant that depends on η0only.\nProof of Lemma 3.10 :For everyR2>1, we have that\nT5≤1\na0/integraldisplayT\nS/integraldisplay\nωa(x)u2\ntdxdt≤C/integraldisplayT\nS/integraldisplay\nΩa(x)(ut+d)2dxdt+C/integraldisplayT\nS/integraldisplay\nΩa(x)d2dxdt\n≤C/integraldisplayT\nS/integraldisplay\n|ut+d|≤R2a(x)(ut+d)2dxdt\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nZ1+C/integraldisplayT\nS/integraldisplay\n|ut+d|>R2a(x)(ut+d)2dxdt\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nZ2+CC1(d).(94)\n20On one hand, since g′(0)>0, there exists αR2>0such that |g(v)| ≥αR2|v|for|v| ≤R2.\nCombining that with (66) yields for some 0<η6<1\nZ1≤/integraldisplayT\nS/integraldisplay\n|ut+d|≤R2a(x)(ut+d)g(ut+d)(ut+d)\ng(ut+d)dxdt\n≤1\nαR2/integraldisplayT\nS/integraldisplay\n|ut+d|≤R2a(x)(ut+d)g(ut+d)dxdt\n≤1\nαR2/integraldisplayT\nS/integraldisplay\nΩa(x)(ut+d)g(ut+d)dxdt\n≤1\nαR2E(S)+C1\nαR2η2\n6/integraldisplayT\nSE dt+1\nαR21\nη4\n6(Cd,eCu+Cd,e).\nWe choose η6=/radicalBig\nαR2\nCη0, we obtain that\nZ1≤1\nαR2E(S)+η0/integraldisplayT\nSE dt+1\nα3\nR2η2\n0(Cd,eCu+Cd,e).\nAs forZ2, we have that\nZ2≤C/integraldisplayT\nS/integraldisplay\n|ut+d|>R2|ut|2dxdt+C/integraldisplayT\nS/integraldisplay\n|ut+d|>R2|d|2dxdt\n≤C/integraldisplayT\nS/integraldisplay\n|ut+d|>R2|ut+d|\nR2|ut|2dxdt+CC1(d)\n≤C/integraldisplayT\nS/integraldisplay\n|ut+d|>R2|ut|3\nR2dxdt+C/integraldisplayT\nS/integraldisplay\n|ut+d|>R2|ut|2|d|\nR2dxdt+CC1(d)\n≤C\nR2/integraldisplayT\nS/integraldisplay\n|ut+d|>R2|ut|3dxdt+C\nR2\n2/integraldisplayT\nS/integraldisplay\n|ut+d|>R2|ut|4dxdt+CC1(d). (95)\nWe use Lemma 3.3 and the fact that R2>1, we derive the following:\nC\nR2/integraldisplayT\nS/integraldisplay\n|ut+d|>R2|ut|3dxdt+C\nR2\n2/integraldisplayT\nS/integraldisplay\n|ut+d|>R2|ut|4dxdt≤/parenleftbiggCu+Cd,e\nR2/parenrightbigg/integraldisplayT\nSE dt. (96)\nWe choose R2=(Cu+Cd,e)\nη0and we combine (95) and (96) we have that\nZ2≤η0/integraldisplayT\nSE dt+Cd,e. (97)\nRemark 3.5 For such a choice of R2, and based on how αR2is defined, we can assume that1\nαR2\nis also a constant of the type ¯Cη0(Cu+Cd,e), where¯Cη0is a constant that depends on η0only. As\na result,Z1is estimated by\nZ1≤η0/integraldisplayT\nSE dt+¯Cη0(Cu+Cd,e)E(S)+¯C3\nη0\nη2\n0(Cd,eCu+Cd,e). (98)\n21Combining (94), (97) and (98) and using (10) and (13), it foll ows that\nT5≤η0/integraldisplayT\nSE dt+¯Cη0(Cu+Cd,e)E(S)+¯C3\nη0\nη2\n0(Cd,eCu+Cd,e)+Cd,e,\nwhich proves Lemma 3.10.\n/squaresolid\nThe estimation of T5gives a direct estimation of the termC\nη0/integraltextT\nS/integraltext\nωu2\ntdxdt left in the estimation\nofT2. We can easily manage to have that\n1\nη0/integraldisplayT\nS/integraldisplay\nωu2\ntdxdt≤η0/integraldisplayT\nSE dt+1\nη0¯Cη2\n0(Cu+Cd,e)E(S)+1\nη0¯C3\nη2\n0\nη4\n0(Cd,eCu+Cd,e)+Cd,e.(99)\nIt is obtained by following the same steps that led to the esti mation ofT5with replacing η0byη2\n0.\nWe can finally finish the proof of Proposition 3.1: we combine t he estimations of Ti, i= 1,2,3,4,5,\nwhich are given by (42), (43), (81), (89) and (93) with (41), t hen we choose η0such thatCη0<1,\nwhich means that the term Cη0/integraltextT\nSE(t)dtgets absorbed by/integraltextT\nSE(t)dt. Then we use the fact that\nCd,eE(S)≤Cd,e(E(0)+Cd,e) =Cd,eCu+Cd,eand the fact that the choice of η0will be a constant\nC, we obtain (39).\n/squaresolid\nProof of the energy estimate of Theorem 2.1: Using the key result given by (39), we get at\nonce from Theorem A.1 that (101) holds true with T=C+CuandC0= (1+Cu)Cd,e. Using (20)\nfort≥1withT=tandS∈[t−1,t]and integrating it over [t−1,t], one gets that\nE(t)≤C/integraldisplayt\nt−1E(s)ds+Cd,e≤C/integraldisplay∞\nt−1E(s)ds+Cd,e.\nCombining the above with (100) yields (18) for t≥1. In turn, (20) with T∈[0,1]andS= 0\nprovides (18) for t≤1. The proof of Theorem 2.1 is then completed.\n/squaresolid\nA Appendix\nWe list in what follows, technical results used in the core of the paper.\nTheorem A.1 Gronwall integral lemma\nLetE:R+→R+satisfy for some C0,T >0:\n/integraldisplay+∞\ntE(s)ds≤TE(t)+C0,∀t≥0. (100)\nThen, the following estimate hold true\n/integraldisplay+∞\ntE(s)ds≤TE(0)e−t\nT+C0,∀t≥0. (101)\nIf in addition, t/ma√sto→E(t)is non-increasing, one has\nE(t)≤E(0)e1−t\nT+C0\nT,∀t≥0. (102)\n22The proof is classical, cf. for instance [3].\nTheorem A.2 Generalized Gronwall lemma\nLetF,h1andh2non negative functions defined on R+satisfying\n/ba∇dblh1/ba∇dbl1:=/integraldisplay∞\n0h1(t)dt<∞,/ba∇dblh2/ba∇dbl1:=/integraldisplay∞\n0h2(t)dt<∞,\nand\nF(T)≤F(S)+C3+C1/integraldisplayT\nSh1(s)Fα1(s)ds+C2/integraldisplayT\nSh2(s)Fα2(s)ds,∀S≤T, (103)\nwhereC1,C2,C3are positive constants and 0≤α1,α2<1. Then,Fsatisfies the following bound\nsup\nt∈[S,T]F(t)≤max/parenleftBig\n2(F(S)+C3),(2˜C)1\n1−α/parenrightBig\n,with˜C:=C1/ba∇dblh1/ba∇dbl1+C2/ba∇dblh2/ba∇dbl1, (104)\nwhereα:= max(α1,α2)if2˜C≥1orα:= min(α1,α2)if2˜C <1.\nProof of Theorem A.2: FixT≥S≥0. Fort∈[S,T]setY(t)for the right-hand side of (103)\napplied at the pair of times S≤t. It defines a non decreasing absolutely continuous function .\nSinceF(t)≤Y(t)≤Y(T)fort∈[S,T], one deduces that FS,T:= supt∈[S,T]F(t)is finite for every\nt∈[S,T]. One gets from (103) that\nFS,T≤F(S)+C3+˜Cmax(Fα1\nS,T,Fα2\nS,T),\nwith the notations of (104). The latter follows at once by con sidering whether F(S) +C3>\n˜Cmax(Fα1\nS,T,Fα2\nS,T)or not.\nWe recall the following useful result, cf. for instance [16] .\nTheorem A.3 Gagliardo–Nirenberg interpolation inequality\nLetΩ⊂RNbe a bounded Lipschitz domain, N≥1,1≤r<p≤ ∞,1≤q≤pandm≥0. Then\nthe inequality\n/ba∇dblv/ba∇dblp≤C/ba∇dblv/ba∇dblθ\nm,q/ba∇dblv/ba∇dbl1−θ\nrfor v∈Wm,q(Ω)∩Lr(Ω) (105)\nholds for some constant C >0and\nθ=/parenleftbigg1\nr−1\np/parenrightbigg/parenleftbiggm\nN+1\nr−1\nq/parenrightbigg−1\n, (106)\nwhere0<θ≤1 (0<θ <1ifp=∞andmq=N)and/ba∇dbl./ba∇dblpdenotes the usual Lp(Ω)norm and\n/ba∇dbl./ba∇dblm,qthe norm in Wm,q(Ω).\nReferences\n[1] F. Alabau-Boussouira. 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Exponential decay for the semilinear wave eq uation with locally distributed damp-\ning.Communications in Partial Differential Equations , 15(2):205–235, 1990.\n25"    },    {        "title": "1208.1761v3.Relaxation_of_Blazar_Induced_Pair_Beams_in_Cosmic_Voids.pdf",        "content": "arXiv:1208.1761v3  [astro-ph.CO]  28 May 2013Draft version March 9, 2022\nPreprint typeset using L ATEX style emulateapj v. 5/2/11\nRELAXATION OF BLAZAR INDUCED PAIR BEAMS IN COSMIC VOIDS\nFrancesco Miniati\nPhysics Department, Wolfgang-Pauli-Strasse 27, ETH-Z¨ ur ich, CH-8093, Z¨ urich, Switzerland; fm@phys.ethz.ch\nAndrii Elyiv\nInstitut d’Astrophysique et de G´ eophysique, Universit´ e de Li` ege, 4000 Li` ege, Belgium and\nMain Astronomical Observatory, Academy of Sciences of Ukra ine, 27 Akademika Zabolotnoho St., 03680 Kyiv, Ukraine\nDraft version March 9, 2022\nABSTRACT\nThe stability properties of a low density ultra relativistic pair beam pro duced in the intergalactic\nmedium by multi-TeV gamma-ray photons from blazars are analyzed. The problem is relevant for\nprobes of magnetic field in cosmic voids through gamma-ray observa tions. In addition, dissipation of\nsuch beams could affect considerably the thermal history of the int ergalactic medium and structure\nformation. We use a Monte Carlo method to quantify the properties of the blazar induced elec-\ntromagnetic shower, in particular the bulk Lorentz factor and the angular spread of the pair beam\ngenerated by the shower, as a function of distance from the blaza r itself. We then use linear and non-\nlinear kinetic theory to study the stability of the pair beam against th e growth of electrostatic plasma\nwaves, employing the Monte Carlo results for our quantitative estim ates. We find that the fastest\ngrowing mode, like any perturbation mode with even a very modest co mponent perpendicular to the\nbeam direction, cannot be described in the reactive regime. Due to t he effect of non-linear Landau\ndamping, which suppresses the growth of plasma oscillations, the be am relaxation timescale is found\nsignificantly longer than the inverse Compton loss time. Finally, densit y inhomogeneities associated\nwith cosmic structure induce loss of resonance between the beam p articles and plasma oscillations,\nstrongly inhibiting their growth. We conclude that relativistic pair bea ms produced by blazars in the\nintergalactic medium are stable on timescales long compared to the ele ctromagnetic cascade’s. There\nappears to be little or no effect of pair-beams on the intergalactic me dium.\nSubject headings: gamma rays: general – instabilities – intergalactic medium – plasmas – r adiation\nmechanisms: non-thermal – relativistic processes\n1.INTRODUCTION\nStreaming relativistic particles are common in ten-\nuous astrophysical plasma and their propagation and\nstability properties a recurrent theme. Examples in-\nclude type III solar radio burst (Benz 1993), quasars’\njets (Lesch & Schlickeiser 1987), cosmic-rays stream-\ning out of star forming galaxies (Miniati & Bell\n2011) and cosmic-ray transport in the intracluster\nmedium (Ensslin et al. 2011). Propagation and stabil-\nity properties, related in particular to the exitation of\nplasmawaves,issubjectofattentiveinvestigationasthey\ncan play a crucial role in the interpretation of observa-\ntional data.\nUltra-relativistic beams of e+e−pairs are also gener-\nated in the intergalactic medium (IGM) by very high en-\nergygamma-raysfrom distantblazars, bywayofphoton-\nphoton interactions with the extragalactic background\nlight (EBL, Gould & Schr´ eder 1967; Schlickeiser et al.\n2012). While blazars’ spectra, and in particular their\nmulti-TeV cut-off features, have been studied in detail\nto constraint the EBL (e.g. Aharonian et al. 2006), re-\ncently multi-GeVand TeVblazarsobservationshavealso\nbeen used to constrain magnetic field in cosmic voids for\nthe first time (Neronov & Vovk 2010; Tavecchio et al.\n2010). In fact, for flat enough blazar’s spectra, the\nelectromagnetic cascade should produce an observable\nspectral bump at multi-GeV energies. The absence of\nsuch a bump in a number of observed blazars is as-cribed to the presence of a sufficiently strong magnetic\nfield,Bv/greaterorsimilar10−16G, to deflect the pairs in less then an\ninverse Compton length, ℓIC≃Mpc(E±/TeV)−1(1 +\nz)−4, wherezis the cosmological redshift (Plaga 1995;\nNeronov & Semikoz 2009). When time variability of\nthe blazars is taken into account the above lower\nlimit is relaxed to a more conservative value of Bv/greaterorsimilar\n10−18G (Dermer et al. 2011; Taylor et al. 2011). The\nrequired filling factor of the magnetic field is about\n60% (Dolag et al. 2011). Other potential effects of a\nmagnetic field in voids on the electromagnetic cascades\nhave also been investigated, including extended emis-\nsion around gamma-ray point-like sources (Aharonian et\nal. 1994; Neronov & Semikoz 2007; Dolag et al. 2009;\nElyiv et al. 2009; Neronov et al. 2010a) and the de-\nlayed echoes of multi-TeV gamma-ray flares or gamma-\nray bursts (Plaga 1995; Takahashi et al. 2008; Murase\net al. 2008, 2009).\nHowever, in principle the pair-beam is subject to\nvarious instabilities, in particular microscopic plasma\ninstabilities of the two-stream family. On this ac-\ncount, Broderick et al. (2012) conclude that transverse\nmodes of the two-stream instability act on much shorter\ntimescales than inverse Compton scattering, effectively\ninhibiting the cascade and invalidating the above mag-\nnetic field measurements. In addition, as a result of\nthe beam’s relaxation, substantial amount of energy\nwould be deposited into the IGM, with dramatic con-2 Miniati & Elyiv\nsequences for its thermal history (Chang et al. 2012;\nPfrommer et al. 2012).\nIn this paper, we reanalyze the stability of blazars in-\nduced ultra-relativistic pair beams. In particular, we\nuse a Monte Carlo model of the electromagnetic shower\nto quantify the beam properties at various distances\nfrom the blazar, and analyze the stability of the pro-\nduced beam following the work of Breizman, Rytov and\ncollaborators (reviewed in Breizman & Ryutov 1974;\nBreizman 1990). We find that even for very modest\nperpendicular components of the wave-vector, the anal-\nysis of the instability requires a kinetic treatment. We\nthus estimate the max growth rate of the instability and\nfind that for bright blazars (with equivalent isotropic\ngamma-rayluminosityof1045ergs−1)itissuppressedby\nCoulomb collisions at distances D/greaterorsimilar50 and 20 physical\nMpc at redshift 0 and 3, respectively. Importantly, the\ngrowth rate of plasma oscillations is found to be severely\nsuppressed by non-linear Landau damping, so that even\nat closer distances to the blazar the beam relaxation\ntimescale remains considerably longer than the inverse\nCompton cooling time. Finally, the resonance condition\ncannot be maintained in the presence of density inhomo-\ngeneities associated to cosmological structure formation,\nwhich also act to dramatically suppress the instability.\nThus our findings support the magnetic field based in-\nterpretation of the gamma-ray observational results and\nrule out effects of blazars’ beam on the thermal history\nof the IGM. Broderick et al. did not consider the role of\ndensity inhomogeneities and concluded that non-linear\nLandau damping is unimportant, although they did not\npresent a quantitative analysis of the process.\nThe rest of this paper is organized as follows. Sec. 2\nsummarizes the physical properties of pair beams pro-\nduced by blazars and present the results of the Monte\nCarlo model. The two-stream instability in both the re-\nactive and kinetic regimes is discussed in Sec. 3, where\nthe max growth rate of the instability is also given and\ncompared to the collisional rate. Nonlinear effects are\nconsidered in Sec. 4, where the timescales for the beam\nrelaxation is derived. Finally, Sec. 5 briefly summarizes\nthe results.\n2.PAIR BEAMS IN VOIDS\nIn order to carry out the analysis of the stability of\nultra-relativistic pair beams produced by blazars, we\nneed to esimate characteristic quantities of the beam,\nincluding its density contrast to the IGM, the Lorentz\nfactor, and angular and velocity spread. These quanti-\nties derive from the energy and number density of the\npair producing photons, i.e. the blazar’s spectral flux,\nFγ, and the EBL model. Pair production has been stud-\nied extensively in the literature (e.g. Gould & Schr´ eder\n1967; Bonometto & Rees 1971; Schlickeiser et al. 2012)\nand in the following we briefly summarize its qualitative\nfeatures, which we then use to describe the results of our\nMonte Carlo model of a blazar induced cascade.\n2.1.Basic Qualitative Features\nPairs are most efficiently created just above the energy\nthreshold for production, i.e. where\ns≡EγEEBL(1−cosφ)/2m2\nec4≥1,(1)withφtheanglebetweentheinteractingphotons, Eγand\nEEBLthe energy of the incident and target EBL photon,\nrespectively, and the relativistic invariant, s, the center\nof mass energy square in units m2\nec4. The mean free\npath for the process depends on the details of the EBL\nmodel (Kneiske et al. 2004; Franceschini et al. 2008)\nbut is approximately\nℓγγ≃0.8/parenleftbiggEγ\nTeV/parenrightbigg−1\n(1+z)−ζGpc,(2)\nwithζ= 4.5 forz≤1,ζ= 0 other-\nwise (Neronov & Semikoz 2009; Broderick et al. 2012).\nThe particle number density of the beam, nb, is set by\nthe balance of pair production rate, 2 Fγ/ℓγγ, evaluated\nclose to production threshold, and energy loss rate. If\ninverse Compton losses dominate then, at a distance D\nfrom the blazar such that Fγ=Lγ/4πD2,\nnb≃2Fγ\ncℓIC\nℓγγ≃3×10−25cm−3/parenleftbiggEγLγ\n1045erg/s/parenrightbigg\n×/parenleftbiggD\nGpc/parenrightbigg−2/parenleftbiggEγ\nTeV/parenrightbigg\n(1+z)ζ−4. (3)\nwhere,EγLγ, is an estimate of the blazar’s equivalent\nisotropic gamma-ray luminosity for a source at distance\nD(see,Broderick et al. 2012). Eachpairparticlecarries\nabout half the energy of the incident gamma-ray, so the\nbeam Lorentz factor is\nΓ =Eγ\n2mec2∼106/parenleftbiggEγ\nTeV/parenrightbigg\n. (4)\nAnother important characteristic quantity of the beam\nis its angular spread, ∆ θ, determined by the distribution\nof angles θbetween the pair produced particles and the\nparentphotonsdirection. Thiscanbefoundtoberelated\nto Γ and the relativistic invariant as\n∆θ≤s\nΓ/parenleftbigg\n1−1\ns/parenrightbigg1\n2\n. (5)\n2.2.Monte Carlo Model\nIn this section we compute the characteristic quanti-\nties of a blazar induced pair beam, using a Monte Carlo\nmodel of the electromagnetic cascade, fully described\ninElyiv et al. (2009)andappliedalsoinNeronov et al.\n(2010). For the purpose, the blazar’s spectral emission\nis a typical power law distribution of primary gamma-\nray photons, dnγ/dEγ∝E−q\nγ, in the range 103≤\nEγ/mec2≤108, and with q= 1.8 (Abdo et al. 2010).\nIn addition, we use the nominal model of Aharonian\n(2001) for the EBL in the range from 0.1 to 1000 µm.\nFor the cosmic background radiations beyond 0.1 µm\nwe used data from Hauser et al. (2001). Contrary\nto Elyiv et al. (2009) we did not consider the deflection\nofe+e−pairs in the extragalactic magnetic field as well\nas the inverse Compton interactions with CMB photons.\nHere we took into account pair distributions resulting\nfrom just the first double photon collisions. Energy dis-\ntribution and cross section of the relevant reactions were\ntaken from (Aharonian 2003).Pair Beams in Cosmic Voids 3\nFig. 1.— Top: Energy distribution of the beam pairs generated\nat distances, from top to bottom, of 3.5 (black), 9.1 (red), 2 3.3\n(green), 60 (blue), 153 (cyan), 390 (magenta), and 1000 (yel low)\nMpc from a blazar with equivalent isotropic gamma-ray lumin osity\nof 1045erg s−1.Bottom: Peak (dash) and mean (solid) pair beam\nenergy as a function of distance from the blazar, for an equiv alent\nisotropic gamma-ray luminosity of 1045erg s−1. Vertical bars cor-\nrespond to 68% percentile of the energy spread about the medi am.\nGiven the primary gamma-ray photon spectrum, we\ngenerated random gamma-photon interaction events.\nThese are characterizedby the random distance from the\nblazar at which the double photon collision occurs based\non the EBL dependent mean free path of the high energy\nphotons. Next we randomly generated the energies E±\nofthe produced e+e−pairs, andevaluatedthe properan-\ngleθbetween each pair and the direction of the incident\ngamma-ray photon. For these quantities we used the an-\nalytical expressions for µe= cos(θ) in Schlickeiser et al.\n(2012).\nThe results of the calculation are shown in Fig. 1, for\na blazar with equivalent isotropic gamma-ray luminosity\nof 1045erg s−1. The top panel shows the spectral energy\ndistribution of the production rate of the pairs at several\ndistances from the blazar, ranging from 3.5 Mpc (top,\nblack line), to a 1 Gpc (bottom, yellow line). The bot-\ntompanelshowsthepeak(dash)andmean(solid)energy\nof the generated pairs, in units of mec2, as a function of\ndistance from the blazar. The Lorentz factor of the pairs\nis a monotonically decreasing function of distance, in the\nrange Γ=105−106up to a Gpc away from the blazar.\nThis shows that pair production typically peaks at near\ninfrared( EEBL≃0.1eV)EBLtargetphotonsinteracting\nwithgamma-rayswithenergy Eγ∼(0.1–1)TeV.Thever-\ntical barsindicate the energyrange encompassing68% of\nthe particles. In fact, there is considerable energy spread\nabout the mean value, which decreases towards larger\ndistances as the energy range of gamma-ray photons in-\nteracting with the EBL is also reduced. However, the\nbeam particles remain always ultra-relativistic, a detailFig. 2.— Angular distribution of beam pairs generated at 244\nMpc (solid), from the blazar. The colored curves indicate th e con-\ntribution to the angular distribution frompairsin the ener gy range:\n103–2.1×103(dot green), 2 .1×103–4.6×103(dot cyan), 4 .6×103–\n104(dot red), 104–2.15×104(dot blue), which dominate the large\nangle end, and 106–2.1×106(dash blue), 2 .1×106–4.6×107(dash\nred), 4.6×107–108(dash cyan), 108–2.15×108(dash green) which\ndominate the small angle end.\nrelevant in the analysis below.\nFig. 2 shows the angular distribution of beam pairs at\ndistances of 244 Mpc (black dash) from the blazar. The\nbeam angular spread is of order ∆ θ≃10−5, consistent\nwith Eq. (5) and the value of Γ estimated above. The\ncolored (dash and dot) curves show the angular distri-\nbution of pairs in eigth different energy bins, indicating\nthat the beam angularspread is primarily determined by\npairs with Γ ∼104–106.\nThedistribution that weuseforthe analysisofthe pair\nbeam stability below is the steady state one, obtained by\nbalancingthe production rategivenin Fig. 1with inverse\nCompton losses, i.e.\nf(Γ) =/integraldisplayΓ\n0τIC(ε)\nε/parenleftbiggdn(ε)\ndε/parenrightbigg\nγγdε (6)\nwhereτIC(Γ) =ℓIC/cis the energy dependent time scale\nfor inverse Compton losses.\nIn Table 1, we report as a function of distant, D, from\nthe blazar, the main properties of the beam, which are\nrelevant to the analysis below. These include, the num-\nber density of the beam pairs, nb; the mean value of\nthe inverse of the pairs Lorentz factor, ∝angb∇acketleftΓ−1∝angb∇acket∇ight−1, which\nenters the estimate of the max growth rate of the in-\nstability; the mean value of the pairs Lorentz factor, ∝angb∇acketleftΓ∝angb∇acket∇ight,\nwhichdeterminesthemeanenergyofthebeam; themean\nsquare value of the pairs Lorentz factor divided by the\nmean value of the same, ∝angb∇acketleftΓ2∝angb∇acket∇ight/∝angb∇acketleftΓ∝angb∇acket∇ight, which enters the es-\ntimate of the inverse Compton timescale; and the rms\nof the opening angle of the pairs, ∆ θ, which also enters\nthe growth rate of the instability. These Lorentz gamma\nfactors differ considerably, and they all monotonically\ndecrease with distance as in the bottom panel of Fig. 1.\nAlthough the results in this and the next sections as-\nsume a blazar equivalent isotropic gamma-ray luminos-\nity of 1045erg s−1. they can be generalized to other\nluminosities by rescaling the pair beam nubmer density\naccording to nb→nb(EγLγ/1045erg s−1).\n2.3.IGM in Voids4 Miniati & Elyiv\nTABLE 1\nBeam basic properties from Monte Carlo model\nD nb/angbracketleftΓ−1/angbracketright−1/angbracketleftΓ/angbracketright /angbracketleftΓ2/angbracketright//angbracketleftΓ/angbracketright∆θ\n(Mpc) cm−3(104) (105) (106) (10−5)\n0.87 2.81e-18 1.52 1.56 5.65 6.43\n1.39 1.17e-18 1.28 1.53 5.50 8.30\n2.22 4.73e-19 1.48 1.52 5.13 6.06\n3.55 1.79e-19 1.39 1.48 4.65 6.32\n5.68 7.48e-20 1.32 1.39 4.01 7.07\n9.09 2.93e-20 1.33 1.35 3.28 7.60\n14.55 1.14e-20 1.35 1.28 2.55 7.75\n23.28 4.48e-21 1.28 1.20 1.94 7.50\n37.25 1.65e-21 1.30 1.13 1.50 7.29\n59.60 5.25e-22 1.44 1.11 1.26 8.76\n95.37 1.86e-22 1.39 1.00 0.99 9.14\n152.59 6.31e-23 1.34 0.86 0.75 8.88\n244.14 2.03e-23 1.27 0.72 0.52 9.67\n390.63 6.13e-24 1.19 0.58 0.32 11.0\n625.00 1.75e-24 1.12 0.47 0.18 11.0\n1000.00 4.71e-25 1.04 0.39 0.12 11.8\nThe analysis of the pair beam instability depends also\non the thermodynamic properties of the plasma in cos-\nmic voids, namely the number density of free electrons\nand their temperature. The number density of free elec-\ntrons can be expressed as nv≃2×10−7(1+δ)(1+z)3\ncm−3. The typical overdensity δis taken to be the value\nat which the cumulative distribution of the IGM gas is\n0.5. Using the simulations results presented in Sec. 4.2,\nand in particular the insets in Fig. 9, we estimate as rep-\nresentative value for the voids δv=−0.9(1+z), where\nthe redshift dependence is approximate but sufficient for\nour purposes. Note that this implies a redshift evolution\nofthe bulk IGMdensity invoids nv∝(1+z)4. As forthe\ngastemperature we assume Tv≃a few×103K (1+z)1.5,\nwhich reproduces the IGM temperature at mean density\nof a few ×104at redshift 3. This redshift dependence,\nwhile again a rough approximation, is acceptable for our\npurposes.\n3.BEAM INSTABILITY: REACTIVE VS KINETIC\nTheblazarinduced pairbeam is subjectto microscopic\ninstabilities, in particular two-stream like instabilities,\nof both electrostatic and electromagnetic nature. The\nbeam is neutrally charged, so no return current is in-\nduced. In the following we assume a sufficiently weak\nmagnetic field, such that ωH≪ωp, whereωHis the cy-\nclotron frequency, ωp= (4πnve2/me)1/2the plasma fre-\nquency of the IGM in voids and ethe electron’s charge.\nIn this case, the instability is predominantly associated\nto Cherenkov emission of Langmuir waves, which oper-\nates under the resonant condition\nω−k·v= 0, (7)\nwherekis the wave-vectorof the perturbation mode and\nvthe beam particles velocity. The pair particles con-\ntributeequallytothedielectricfunction, astheyhavethe\nsame mass, number density, velocity distribution, and\nplasma frequency, ωp,b= (4πnbe2/me)1/2. After sep-\narating the contributions from the background plasma\nand the beam particles, the dispersion relation for Lang-\nmuir waves, valid in the relativistic case, can be written\nas (Breizman 1990)\n1−ω2\np\nω2−4πe2\nk2/integraldisplayk·∂f/∂p\nk·v−ωdp= 0,(8)−1•10−4−5•10−505•10−51•10−4\nk||c/ωp−110−510−410−310−210−1k⊥c/ωp\n  −6•104−4•104−2•10402•1044•1046•104\nFig. 3.— Normalized growth rate, γk/πωp(nb/nv), from Eq. (12)\nin the plane k/bardblc/ωp−k⊥c/ωp. Bright is positive, dark is negative\nand the uniformly colored region is zero, as it lies outside t he res-\nonant region.\nwhere,f(p), is the distribution function ofthe beam par-\nticles. There are two important regimes that character-\nize the unstable behavior of the beam, namely reactive\nand kinetic. In the reactive case, the beam’s velocity\nspread, ∆ v, is negligible so all particles can participate\nto the unstable behavior and the growth rate of the in-\nstability is therefore fastest. In the kinetic regime, on\nthe other hand, the velocity spread is considerable and\nonly the resonant particles contribute to the growth of\nLangmuir waves, so the growth rate is slower than in the\nreactive case. Formally, the reactive regime is applicable\nwhen (Breizman & Ryutov 1971)\n|k·∆v| ≪γr, (9)\nwhereγris the reactive growth rate. In this case, the\nintegral in Eq. (8) can be solve in a simplified way, which\ninvolves neglect of the velocity spread around the mean\nvalue. This leads to the estimate of the reactive growth\nratewhich, maximized alongthe longitudinal component\nof the wave-vector reads (Fainberg et al. 1970)\nγr≃ωp/parenleftbiggnb\nΓnv/parenrightbigg1\n3/parenleftBigg\nk2\n/bardbl\nk2Γ2+k2\n⊥\nk2/parenrightBigg1\n3\n,(10)\nwithk/bardbl=ωp/v, andk/bardbl, k⊥the components of the\nwave-vector parallel and perpendicular to the beam di-\nrection, respectively. It is well known that, for an ultra-\nrelativistic beam (Γ ≫1), the fastest growing modes\nin the reactive regime are those quasi-perpendicular\nto the beam. This is due to the large suppression\ncaused by relativistic inertia along the longitudinal di-\nrection (Fainberg et al. 1970). However, as shown later,\nfor quasi-perpendiculardirections ofthe wavevector, the\nreactive regime is not applicable.\nWhen the approximation (9) is not valid, the growth\nrate is evaluated from a pole of the integrand in the dis-Pair Beams in Cosmic Voids 5\nFig. 4.— Normalized growth rate as a function of k/bardblc/ωpin-\nside the the resonant region, for values of k⊥whereγkreaches its\nmaximum values.\npersion relation (8), namely\nγk=ωp2πe2\nk2/integraldisplay\nk·∂f\n∂pδ(ωp−k·v)dp.(11)\nIf, as is the case here, despite the energy spread\nthe particles remain ultra-relativistic and, |v|=c,\ncan be assumed, the above integral can be simplified\nto (Breizman & Mirnov 1970)\nγk=−ωpπnb\nn/parenleftBigωp\nkc/parenrightBig3/integraldisplayµ+\nµ−dµ2g+(µ−k/bardblc\nωp)∂g\n∂µ\n[(µ+−µ)(µ−µ−)]1\n2,(12)\nwhere the integration variable µis the angle between the\nparticles and the beam direction, and\nµ±= (ωp/kc)(k/bardbl/k±k⊥/k/radicalBig\nk2c2/ω2p−1),(13)\ng(θ) =mec\nnb/integraldisplay\npf(p,θ)dp≃ ∝angb∇acketleftΓ−1∝angb∇acket∇ight1\n∆θ2e−θ2\n∆θ2.(14)\nThe second equality for g(θ) in Eq. (14) is found to be a\ngood approximation based on results of the Monte Carlo\nmodel of the cascade. The integral for the growth rate in\nEq. (12) can be evaluated numerically. The qualitative\nbehavior of the growth rate, γk, on the plane k/bardbl−k⊥\nis summarized in Fig. 3 (Breizman & Ryutov 1971) for\na beam at a Gpc from the blazar. Outside the narrow\nresonant region of k-space, corresponding in the plot to\nthe uniform color, the growth rate is effectively null. In\nthenarrowresonantregionaround k/bardbl=ωp/c, thegrowth\ncanbepositive(bright), negative(dark)andnull, andfor\nlarge enough values of k⊥, it carries the sign of, ωp/k/bardbl−c\n(see below). Within the resonant region, the growth rate\nasafunctionof k/bardblhastypicallytwoextrema,amaximum\nand a minimum. This is shown in Fig. 4 for values of\nk⊥of interest, i.e. where γkreaches its maximum values.Fig. 5.— Normalized growth rate as a function of k/bardblc/ωpinside\nthe the resonant region, for large values of k⊥whereγkstarts to\ndrop compared to its maximum value.\nThe growthrate has its largestvalues where k⊥c/ωp/lessorsimilar1,\nanddecaysrapidlyintheoppositelimit. Thiscanbeseen\nfrom Fig. 5 where γkis plotted for values of k⊥close to\nand much larger than ωp/c(cf. scale of y-axis). Finally,\nwe find that the growth rate, maximized with respect to\nk/bardbland as a function of k⊥, can be well approximated by\nthe following expression by Breizman & Ryutov (1971)\nγk≃ωp∝angb∇acketleftΓ−1∝angb∇acket∇ightnb\nnv1\n∆θ2ω2\np\nω2p+k2\n⊥c2,(15)\nwhich we will be using in the following.\n3.1.Fastest Growing Modes vs Coulomb Collisions\nFor particles of an ultra-relativistic beam with modest\nangular spread, ∆ θ≪1, we can assume v/bardbl≃cand\nv⊥≃c∆θ. If the energy spread, ∆ E/E/lessorsimilar1, then the\nlongitudinal velocity spread of the beam is\n∆v/bardbl≃c∆E\n∝angb∇acketleftΓ∝angb∇acket∇ight2E+c∆θ2. (16)\nThus, for the angular spread and Lorentz factor char-\nacteristic of the blazars induced beam, the longitudinal\nvelocity spread is negligible with respect to the perpen-\ndicular velocity spread. It turns out that, the first and\nsecond terms in Eq. (16) are comparable to within a fac-\ntor of a few, so for the sake of simplicity in the following\nwe retain the second term only, neglecting any fudge fac-\ntor. If we then use Eq. (9) and (16), with the estimates\nforthebeamangularspreadandbulkLorentzfactorfrom\nthe previous Section, we find that virtually all modes re-\nquire a kinetic description, unless\nk⊥\nk/lessorsimilar×10−5/parenleftbiggnb/nv\n10−15/parenrightbigg/parenleftbigg∝angb∇acketleftΓ∝angb∇acket∇ight\n105/parenrightbigg−1/parenleftbigg∆θ\n10−5/parenrightbigg−3\n.(17)6 Miniati & Elyiv\nFig. 6.— Ratio of instability max growth rate, γmax, to Coulomb\ncollision rate, νc, for redshiftz=0 (solid), z=1 (short dash) and z=3\n(long dash). The shaded area corresponds to the absolutely s table\nregion where γmax≤νc.\nThe max growth rate occurs at k⊥provided by the\nabove estimate. For smaller values we enter the reac-\ntive regimes and relativistic inertia increases. For larger\nvalues we are in the kinetic regime where the growth\nrate decreasesdue to the increasingvelocity spreadalong\nk, although the decrease becomes significant only for\nk⊥≥ωp/c. The fastest growth rate for modes with\nk⊥≤ωp/cis therefore given by\nγmax≃ωp∝angb∇acketleftΓ−1∝angb∇acket∇ightnb\nnv1\n∆θ2= 4×10−12s−1\n×/parenleftbiggnv\n2×10−8cm−3/parenrightbigg−1\n2/parenleftbigg∝angb∇acketleftΓ−1∝angb∇acket∇ight\n10−4/parenrightbigg−1/parenleftbigg∆θ\n10−4/parenrightbigg−2/parenleftbiggD\nGpc/parenrightbigg−2\n,\n(18)\nwhere we have taken nb≃10−24cm−3at a Gpc from a\nblazar of luminosity EγLγ= 1045erg s−1. A basic con-\ndition for the growth of an instability is that its growth\nrateexceedsthe collisionaldamping rate, i.e. γmax≫νc,\nwhere (Huba 2009)\nνc≃10−11s−1/parenleftbiggnv\n2×10−8cm−3/parenrightbigg/parenleftbiggTv\n3×103K/parenrightbigg−3\n2\n,(19)\nand for the Coulomb logarithm we have used Λ c=\n27.4 (Huba 2009). In Fig. 6, we plot the ratio γmax/νc\nas a function of distance from the blazar, using the val-\nues reported in Table 1, which again apply for a blazar\nof equivalent isotropic gamma-ray luminosity of 1045erg\ns−1. The solid, dash and long-dash curves correspond\nto redshift z=0,z=1 and z=3, respectively. The red-\nshift dependence is obtained by using the void average\ndensity and temperature redshift dependences discussed\nin Sec. 2.3, together with the redshift dependence of nb\ngiven in Sec. 2.1. The shaded area corresponds to the re-\ngion where the instability is inhibited by collisions. The\nplot shows that the instability can only develop at dis-\ntances of less than a 50 Mpc at redshift z= 0 and about\n20 physical Mpc z=3.\n4.BEAM STABILIZATION\nAs shown in the previous section, pair beams within\na certain distance of the parent blazar may be unstable\ndue to the excitation of Langmuir waves. In this section\nwe further analyse these unstable conditions. In particu-\nlar we consider nonlinear effects on plasma waves due toscattering off thermal ions and density inhomogeneities.\nWe begin, however, with a brief outline of the main fea-\ntures of the relaxation process (for a detailed description\nsee, e.g., Melrose 1989; Breizman & Ryutov 1974). An\nimportant assumption in what follows is that the level of\nplasma turbulence remains low compared to the plasma\nthermal energy, so that a perturbative approach is valid.\nThis, will be verified at the end of the analysis.\nThe presence of excited plasma waves causes the beam\nparticles to diffuse in momentum space. This contin-\nues until the particle momentum distribution has flat-\ntened, and Cherenkov emission ( ∝∂f/∂p) is suppressed.\nAccording to the calculations of Grognard (1975), this\nprocess of quasilinear relaxation takes about 50-100 in-\nstability growth timescales to complete. In general, how-\never, other processes occur that reduce the energy of res-\nonant waves the particles interact with, thus stabilising\nthe beam. Spatial transport effects may contribute in\ntwo ways. On the one hand, waves drift along the energy\ndensity gradient at the group velocity, vg≃3v2\nt/c. For\nthe case of interest here, this process is negligible, due\nto the smallness of the group velocity and spatial gra-\ndients of the wave energy. In addition, however, if the\nplasma frequency is not constant in space due to plasma\ninhomogeities, the wave-vector will change in time, de-\nstroying the particle-wave resonant conditions. This ef-\nfect turns out to be important and will be considered\nfurther below.\nIn the limit of weak turbulence, second order ef-\nfects can also play an important role (Melrose 1989;\nBreizman & Ryutov 1974). Inshort, thesearedescribed\nin terms of three-wave interactions and particle-wave\nscattering. Three waves interactions involve, in addition\nto Langmuir waves, at least one electromagnetic wave,\nbecause the frequency resonance condition cannot be ful-\nfilled with three Langmuir waves alone. Compared to\nother processes discussed below, however, they are of or-\nderkBT/mec2, so they turn out to be negligible for the\nconditions of interest here. As for particle-wave scatter-\ning, Langmuir waves can undergo induce scattering ei-\nther by electrons or ions, into either Langmuir waves or\nelectromagnetic waves. The latter process is suppressed\nin presence of inhomogeneities, so it will be neglected in\nthe following. Furthermore, as we are considering waves\nwith wavelength larger then the Debye length, the scat-\ntering by thermal ions is considerably more important\nthan thermal electrons. This is because for thermal ions\nonly, the superposed effects of the bare and shielding\ncharge (basically an electron of opposite charge as the\nbare charge) do not cancel out, due to the much larger\nmass of the ion compared to the electron. Therefore,\nwith regard to second order nonlinear effects in the fol-\nlowing we only consider induced scattering off thermal\nions.\n4.1.Nonlinear Landau damping\nIn this section we consider in some detail the main pro-\ncessthat webelieve compensatesthe growthofLangmuir\nwaves,i.e. inducedscatteringoffplasmaions, alsoknown\nas non-linear Landau damping (Tsytovich & Shapiro\n1965;Breizman et al. 1971;Lesch & Schlickeiser 1987).\nIn this process, a thermal ion, with characteristic veloc-\nity,vti, interacts with the beat wave produced by twoPair Beams in Cosmic Voids 7\nFig. 7.— Ratio of beam relaxation timescale, τbeam, to invese\nCompton loss time, τIC, for redshift z=0 (solid), z=1 (short dash)\nand z=3 (long dash).\nLangmuir oscillations, ω(/vectork), ω(/vectork′), under the condition\nfor Cherenkov interaction, i.e.\nω(k)−ω(k′) = (k−k′)·vti. (20)\nThe rate of induced scattering of Langmuir waves off\nthermal ions in a Maxwellian plasma with number den-\nsitynand ion/electron temperature Ti/Terespectively,\nis (e.g., Melrose 1989)\nγnl(k) =3(2π)1\n2\n2TiTe\n(Ti+Te)2/integraldisplayd3k′\n(2π)3ˆW(k′)\nnmevti(21)\n×/parenleftbiggk·k′\nkk′/parenrightbigg2k′2−k2\n|k′−k|exp/bracketleftBigg\n−1\n2/parenleftbigg3\n2v2\nte\nωpvtik′2−k2\n|k′−k|/parenrightbigg2/bracketrightBigg\n,\nwhereˆW(k) indicates the spectral energy density of\nLangmuir waves. The growth rate, γnl(k), bears the sign\nof (k′−k). This indicates that as a result of induced\nscattering, Langmuir waves cascade towards regions of\nphase space of lower wave-vectors, i.e. lower energies,\nthe energy difference being absobed by the thermal ions.\nEventually, the wave energy is transferred to modes with\nwavenumber, k, smallenough that the wavephase-speed,\nω/k > c, exceeds the speed of light, and resonance with\nthe beam particles is lost. The wavenumbers allowed\nin the scattering process are constrained by the integral\nexpression in Eq. (21). In particular, the following con-\ndition must be fulfilled:\n|k′2−k2|\n|k′−k|≤ωpvti\nv2\nte≃35×ωp\nc/parenleftbiggTv\n3×103K/parenrightbigg−1\n2\n.(22)\nThe aboveconstrainis satisfiedforthe caseofdifferential\nscattering, i.e. ∆ k/k≪1, whereby k∼k′andk′∼ −k.\nIn this case Langmuir waves, generated with wavenum-\nber,k∼ωp/c, parallel to the beam, are isotropized.\nThis reduces the level of resonant energy density by a\nfactor∼∆θ2/4π, increasing somewhat the lifetime of\nthe beam. However, given the low temperature of the\nIGM in voids, the more efficient integral scattering, with\nk′≪k, is also allowed. In this case Langmuir waves\nare mostly kicked out of resonance in a single scattering\nevent, reducing dramatically the level of resonant energy\ndensity and suppressing the instability.\nThe general solution for the evolution of the energy\ndensity in plasma waves is a non trivial task, as it re-quires solving for integro-differential equations that de-\nscribe the detailed energy transfer of the wave energy\nacross different modes. However, for an conservative es-\ntimate, we can neglect differential scattering, and evalu-\nate the rate of induced integral scattering from Eq. (21)\nusing the condition k′≪k. We thus obtain\nγnl≃ωpWnr\nnvkBTvv2\nte\nvtic(23)\nwhereWnris the total energy density in Langmuir waves\natk≪ωp/c, i.e. non-resonant with the beam. This en-\nergy density is excited by the non-linear scattering pro-\ncess and for the most part is dissipated by Coulomb col-\nlisions at a rate νc(see further discussion below). Thus\nit evolves according to\n∂Wnr\n∂t= 2˜γnlWnrWr−νcWnr, (24)\nwhere we have used, ˜ γnl≡γnl/Wnr=\nωp(1/nvkBTv)(v2\nte/vtic) andWris the total energy\ndensity in Langmuir waves at k∼ωp/c, i.e. resonant\nwith the beam. The latter obviously evolves according\nto∂Wr\n∂t= 2γmaxWr−2˜γnlWnrWr,(25)\nwhere we have neglected the role of collisions (i.e. we\nassume, γmax≫νc, as required for the existence of\nthe instability). Eq. (25) and (24) form a well-known\nLotka-Volterra system of coupled non-linear differential\nequations, which has stable periodic solutions, with the\nfollowing average values for the energy densities:\nWnr=γmax\n˜γnl,Wr=νc\n2˜γnl. (26)\nIn this regime, the transfer rate of Langmuir waves out\nof resonance by non-linear Landau damping equals on\naverage their production rate, i.e. γnl≃γmax. Thus,\nthe beam emission of Langmuir waves is only linear in\ntime, with an average power P(Wr) = 2γmaxWr, and\nthe beam relaxation timescale at redshift z= 0 is:\nτbeam≃nb∝angb∇acketleftΓ∝angb∇acket∇ightmec2\n2γmaxWr= 1.5×109yr/parenleftbiggnv\n2×10−8cm−3/parenrightbigg−1\n×/parenleftbigg∝angb∇acketleftΓ−1∝angb∇acket∇ight\n10−4/parenrightbigg/parenleftbigg∝angb∇acketleftΓ∝angb∇acket∇ight\n105/parenrightbigg/parenleftbigg∆θ\n10−4/parenrightbigg2/parenleftbiggTv\n3×103K/parenrightbigg\n.(27)\nThe above timescale should be compared with the pairs\ncooling time on the Cosmic Microwave Background,\nτIC=ℓIC/c≃3×106(E±/TeV)−1(1+z)−4yr. The ra-\ntioofthese timescales isplotted in Fig. 7using the values\nreported in Table 1, as a function of distance from our\nreference blazar with isotropic gamma-ray luminosity of\n1045erg s−1. At redshift z= 0 (solid line) the beam ap-\npears to be stable on significantly longer timescales than\nthe inverse Compton emission energy loss timescale, par-\nticularly within 100 Mpc from the blazar, where the av-\nerage value of Γ of the pairs tends to be higher. This\nconclusion is reinforced at higher redshifts (dash line for\nz= 3), where the redshift dependence is inferred as in\nSec. 3.1.\nThe above analysis works in the weak turbulence\nregimes, which requires that the energy density of res-\nonant Langmuir waves be a small fraction of the beam8 Miniati & Elyiv\nFig. 8.— Ratio of nonresonant waves to thermal energy as a\nfunction of distance from our reference blazar, for redshif t z=0\n(solid), z=1 (short dash) and z=3 (long dash). The horizonta l\nline correspond to the thrshold for the onset of the modulati on\ninstability.\nenergy density. For the typical values of IGM gas and\nbeam parameters used above this requirement is readily\nfulfilled as, Wnr/nbΓmec2≃3×10−6, warranting our\napproach limited to second order processes.\nAnother consistency check to be performed concerns\ntheassumptionofcollisionaldissipationofthe longwave-\nlenght Langmuir waves. In fact, accumulation of energy\nin these non-resonant waves can generate modulation in-\nstability of the background plasma if (Breizman 1990),\nWnr\nnvkBTv≥k2λ2\nD∼kBTv/mc2. (28)\nIn Fig. 8 the ratio on the LHS of the above equation\nis plotted as a function of distance from our reference\nblazar using the values in Tab. 1. The solid line corre-\nsponds to redshift z= 0 and the horizontal line is the\nnominal threshold value for the onset of the modulation\ninstability. For redhisfts higher than z= 0, we plot for\ncomparison with the same threshold line the same ratio\non the LHS of Eq. (28) but divided by a factor (1+ z)3to\naccount for the IGM temperature redshift denpendence\n(see Sec. 2.3).\nThe plot shows that the assumption of collisional dissi-\npation of the long wavelength Langmuir waves is always\nvalid except at short distances from low redshift blazars.\nThe modulation instability deserves more attention that\nthe scope of the current paper can afford. Here we no-\ntice that, while the modulation instability could stabilize\nthe beam (Nishikawa & Ryutov 1976), it could also pro-\nvides an effective dissipation rate that is more efficient\nthan collisions. In this case the level of energy density\nof resonant waves, Wr, will increase with consequent re-\nduction of the beam lifetime (see Eq. 26-4.1). However,\nbecause the threshold condition for triggering the modu-\nlation instability depends quadratically on the tempera-\nture (see Eq. 28), should the background plasma suffers\neven modest heating caused by the beam relaxation, the\nmodulation instabilitywill quickly stabilize(at below the\nMpc scale), restoring the conditions for collisional dissi-\npation of the nonresonant waves.\nTherefore, in conclusion, from the above analysis to-\ngether with the findings in in Sec. 3.1 it appears that the\nbeam is stable at basically all relevant distances from\nthe blazar. As a result, the beam instability plays only asecondary role on the electromagnetic shower, the beam\ndynamics and the thermal history of the IGM.\n4.2.Plasma Inhomogeneities\nIn addition to the kinetic effects described above, the\nenergy density of the plasma waves evolves in time due\nto spatial gradients effects according to\nd\ndtˆW(x,t,k) =∂ˆW\n∂t+vg∇xˆW−∇xω·∇kˆW,(29)\nwhere for the rate of change of the wave-vector we have\nused the equation of geometric optics\ndk\ndt=−∇xω. (30)\nThefirstandsecondtermsontheRHSof(29)describeas\nusual explicit time dependence and the effects of spatial\ngradients discussed at the beginning of Sec. 4. The last\nterm describes the change in ˆWassociated with modifi-\ncations of the waves wave-vector as a result of inhomo-\ngeneities. This term is important because, just like in-\nduced scattering by thermal ions, it transfers the excited\nLangmuir waves to wavemodes that are out of resonance\nwith the beam particles, therefore suppressing the insta-\nbility (Breizman & Ryutov 1971; Nishikawa & Ryutov\n1976).\nFor Langmuir waves the most important contribution\nto,∇xω, comes from density inhomogeneities. In addi-\ntion, the beam stabilization mainly results from changes\ninthe longitudinal componentofthe wavevector. There-\nfore, we restrict our analysis to this case only, and write\ndk/bardbl\ndt≃1\n2ωp\nλ/bardbl, (31)\nwithλ/bardbl=nv/(/vector∇nv)/bardbl, the length scale of the density\ngradient along the beam.\nIn order to estimate the scale lengths of IGM den-\nsity gradients, λ, we have carried out a cosmologi-\ncal simulation of structure formation including hydro-\ndynamics, dark matter, and self-gravity as described\nin Miniati & Colella (2007). For the cosmological model\nwe adopted a flat ΛCDM universe with the following pa-\nrameters: total mass density, normalized to the criti-\ncal value for closure, Ω m= 0.2792; normalized bary-\nonic mass density, Ω b= 0.0462; normalized vacuum en-\nergy density, Ω Λ= 1−Ωm= 0.7208; Hubble constant\nH0= 70.1 km s−1Mpc−1; spectral index of primordial\nperturbation, ns= 0.96; and rms linear density fluctua-\ntion within a sphere of comoving radius of 8 h−1Mpc,\nσ8= 0.817, where h≡H0/100 (Komatsu et al. 2009).\nThe computational box has a comoving size L= 50h−1\nMpc, is discretized with 5123comovingcells, correspond-\ning to a nominal spatial resolution of 100 h−1comoving\nkpc. The collisionless dark matter component is repre-\nsented with 5123particles with mass 6 ×105h−1M⊙.\nFig. 9 shows the range of scale lengths of IGM den-\nsity gradients as a function of IGM gas over-density, for\nthree different cosmological redshifts, z= 0 (top), z= 1\n(middle) and z= 3 (bottom). Accordingly, the distance\ncovered by the beam particles during the fastest growth\ntime,∼cγ−1\nmax≃1 kpc, is much shorter than the typicalPair Beams in Cosmic Voids 9\nFig. 9.— Characteristic length scale of density gradient in the\nIGM as a function of IGM gas over-density. The shaded area cov ers\n±one root-mean-squared value about the average. The inset sh ows\nthe gas cumulative distribution as a function of overdensit y.\nscale-length of density gradients. This correspond to the\ncase of regular, as opposed to random, inhomogeneities.\nIt is clear that in order for the excited waves to have\nan effect on the beam, the beam-waves interaction un-\nder resonant conditions must continue for a sufficiently\nlong time. Therefore, the condition for wave excitation\nis expressed as (Breizman & Ryutov 1971)\nγmax∆k/bardbl\n|dk/bardbl/dt|=γmax2λ/bardbl∆k/bardbl\nωp>Λc,(32)\nwhere Λ cis the Coulomb logarithm and ∆ k/bardblthe change\nin longitudinal component of the wave-vector allowed by\nthe resonantcondition(7). Using Eq.(7) (and neglecting\nthe term ∆ E/EΓ2) to obtain, ∆ k/bardbl/lessorsimilarωp\nc∆θ2+k⊥∆θ,\nEq. (32) can be solved to express the condition for wave\nexcitation in terms of λ/bardbl, i.e.\nλ/bardbl≥c\n2ωp∝angb∇acketleftΓ−1∝angb∇acket∇ightnv\nnbΛc/parenleftbigg\n1+k⊥\nk/bardbl∆θ/parenrightbigg−1\n>106kpc\n×/parenleftbiggD\nGpc/parenrightbigg2/parenleftbigg∝angb∇acketleftΓ−1∝angb∇acket∇ight\n10−4/parenrightbigg/parenleftbigg∆θ\n10−4/parenrightbigg/parenleftbiggΛc\n30/parenrightbigg\n(1+z)2(33)\nwhere in the second inequality we have used, κ⊥≃κ/bardbl,\nwhich correspondsto the most favourablecase for the in-\nstability growth in the presence of inhomogeneities, and\nagain the refdshift dependence is derived as described\nin Sec. 3.1. We can again plot the minimal values of\nλ/bardblallowed for the growth of the beam instability us-\ning the parameter values for the beam from Table 1.\nThis is shown by the oblique lines in Fig. 10, for redshift\nz= 0 (solid), z= 1 (dash) and z= 3 (long dash). The\nthree horizontal thin lines (with the same line style as\nthe oblique lines at the same redshift), correspond to the\nmean scale-length of density inhomogeneities at typicalFig. 10.— Oblique line correspond to minimal values of λ/bardblal-\nlowed for the growth of the beam instability as a function of d is-\ntance from the blazar obtained using values in Table 1. Horiz ontal\nthin lines correspond to the mean scale-length of density in homo-\ngeneities at typical void overdensity (i.e., where the cumu lative gas\ndistribution function is 0.5), extracted from Fig. 9. Solid , dash and\nlong dash correspond to redshift z= 0,z= 1, and z= 3.\nvoid overdensity (i.e., where the cumulative gas distribu-\ntion function is 0.5), extracted from Fig. 9. The figure\nshowsthat the growthofLangmuirwavesis severelycon-\nstrainedbythepresenceofinhomoteneities,exceptforre-\ngions close to the blazars, i.e. at distances D <30,6,1\nMpc, for z= 0,1,3, respectively. Inhomogeneities pro-\nvide another independent argument against the growth\nof Langmuir waves and the unstable behavior of the pair\nbeam. While non-linear Landau damping weakens with\ndistance from the blazar (see Fig. 6), the impact of inho-\nmogeneities becomes stronger (see Fig. 10), so that the\nstabilizationeffects ofthe twoprocessescompensateeach\nother at different distances.\n5.CONCLUSION\nWe considered the stability properties of a low density\nultra relativistic pair beam produced in the intergalactic\nmedium by multi-TeV gamma-ray photons from blazars.\nThe physical properties of the pair beam are determined\nthrougha Monte Carlomodel ofthe electromagneticcas-\ncade. Insummarywefindthatthecombinationofkinetic\neffects, non-linear Landau damping and density inhomo-\ngeneitiesappearto considerablystabilizeblazarsinduced\nultra-relativistic beams over the inverse Compton loss\ntimescale, so that the electromagnetic cascade remains\nmostly unaffected by the beam instability. This implies\nthat the lack of a bumpy feature at multi-GeV energies\nin the gamma-ray spectrum of distant blazars cannot be\nattributed such instabilities and can in principle be re-\nlated to the presence of an intergalactic magnetic field.\nFinally, heating of the IGM by pair beams appears neg-\nligible.\nF.M. acknowledges very useful discussions with B. N.\nBreizman and A. Benz, and comments from D. D. Ryu-\ntov, and R. Schlickeiser. We are grateful to D. Potter for\nmaking available is grafic++ package for cosmological\ninitial conditions. The numerical calculations were per\nperformed at the Swiss National Supercomputing Cen-\nter.10 Miniati & Elyiv\nREFERENCES\nAbdo, A., Ackermann, M., Ajello, M. 2010, A&A, 710, 1271-128 5\nAjello, M. and Shaw, M. S. and Romani, R. W. and Dermer, C.\nD. and Costamante, L. and King, O. G. and Max-Moerbeck,\nW. and Readhead, A. and Reimer, A. and Richards, J. L. and\nStevenson, M. 2012, ApJ, 751, 108\nAharonian, A. 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D. 1965, Nuclear Fusion, 5 228"    },    {        "title": "1810.09857v3.Quantum_dissipation_of_planar_harmonic_systems__Maxwell_Chern_Simons_theory.pdf",        "content": "Quantum dissipation of planar harmonic systems: Maxwell-Chern-Simons theory\nAntonio A. Valido\u0003\nInstituto de F\u0013 \u0010sica Fundamental IFF-CSIC, Calle Serrano 113b, 28006 Madrid, Spain\nQOLS, Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom\n(Dated: January 7, 2019)\nConventional Brownian motion in harmonic systems has provided a deep understanding of a great\ndiversity of dissipative phenomena. We address a rather fundamental microscopic description for\nthe (linear) dissipative dynamics of two-dimensional harmonic oscillators that contains the conven-\ntional Brownian motion as a particular instance. This description is derived from \frst principles\nin the framework of the so-called Maxwell-Chern-Simons electrodynamics, or also known, Abelian\ntopological massive gauge theory. Disregarding backreaction e\u000bects and endowing the system Hamil-\ntonian with a suitable renormalized potential interaction, the conceived description is equivalent to\na minimal-coupling theory with a gauge \feld giving rise to a \ructuating force that mimics the\nLorentz force induced by a particle-attached magnetic \rux. We show that the underlying symme-\ntry structure of the theory (i.e. time-reverse asymmetry and parity violation) yields an interacting\nvortex-like Brownian dynamics for the system particles. An explicit comparison to the conventional\nBrownian motion in the quantum Markovian limit reveals that the proposed description represents\na second-order correction to the well-known damped harmonic oscillator, which manifests that there\nmay be dissipative phenomena intrinsic to the dimensionality of the interesting system.\nPACS numbers: 03.65.Yz, 11.10.Kk\nI. INTRODUCTION\nThe study of the physical process whereby an inter-\nesting system reaches asymptotically a stationary state\nfollowing a dissipative dynamics is ubiquitous in sev-\neral areas of physics such as quantum thermodynamics\n[1], condensed matter physics [2, 3], or cosmology [4{8].\nUnfortunately, this constitutes an intricate open-system\ntheory problem [1, 9{13] for which there is no a \"uni-\nversal\" recipe that could successfully provide a rigorous\nsolution. Indeed only a few speci\fc instances can be ex-\nactly solved, among which, the (linear) quantum Brown-\nian motion in harmonic systems (generically known as\nthe damped harmonic oscillator [14{21]) represents a\nprominent example [22, 23]. One of the most fruitful\napproaches to the latter rest on assuming that the mi-\ncroscopic Hamiltonians describing both the environment\nand system-environment interaction basically consist of\na large set of non-interacting harmonic oscillators lin-\nearly coupled to the system. This is commonly refereed\nto as the Feynman-Vernon [24], Caldeira-Leggett [25] or\nindependent-oscillator model [11, 21, 26], and recently, it\nhas been employed to investigate quantum thermometry\n[27{29], and non-equilibrium quantum thermodynamics\nor information properties [30{35].\nAlthough this standard model may look somehow ar-\nti\fcial, it resemblances to the Pauli-Fierz Hamiltonian\n[36, 37] in the strict dipole-approximation when the lat-\nter describres simple charged particles interacting with\nMaxwell electromagnetic \felds [21, 38{42]. That is, the\nindependent-oscillator model is essentially a particular\ninstance of the non-relativistic Maxwell electrodynam-\n\u0003a.valido@i\u000b.csic.esics [43, 44]. Remarkably, along with the usual Maxwell\ncontribution, the action of the two-dimensional Abelian\nelectrodynamics admits a Chern-Simons kinetic term [45]\nwhich preserves the essential ingredients demanded for a\nsensible (Abelian) gauge theory [46]: space-time local-\nity, as well as local U(1)-gauge and Lorentz invariance.\nThis is the so-called Maxwell-Chern-Simons electrody-\nnamics [47] or Abelain topological massive gauge the-\nory [46, 48], and has been successfully applied to study\nnew forms of gauge \feld mass generation [46, 47], the\ndynamics of vortices [49, 50], or the statistics transmu-\ntation [48] which have recently found appealing applica-\ntions in quantum computation theory [51]. Then two im-\nmediate question arises as to which kind of microscopic\ndescription is brought by this more fundamental gauge\ntheory, and further, whether it could shed new light on\ntwo-dimensional dissipative dynamics.\nMotivated by these natural questions, the present work\nis devoted to extensively examine the Abelian topological\nmassive gauge theory from the perspective of the quan-\ntum open-system theory. More concretely, we address\nthe dissipative dynamics in the low-energy regime of a\ntwo-dimensional system composed of charged harmonic\noscillators minimally interacting with a Maxwell-Chern-\nSimons electromagnetic \feld acting as a heat bath. Start-\ning from \frst principles, we derive a low-lying Hamilto-\nnian that provides a reliable and (numerically) solvable\ndissipative microscopic description within the Langevin-\nequation framework [5, 19, 22]. Interestingly, the Chern-\nSimons e\u000bects give rise to a Lorentz-like \ructuating force\nwhich represents an alternative to the geometric mag-\nnetism [52] in the context of recently extended environ-\nments [53, 54]. Unlike previous treatments, we show that\nthe components of such Chern-Simons (electric) force are\nnon-commutative owing to the \"topological\" nature of\nthe underlying theory, and cause an (ordinary) Hall re-arXiv:1810.09857v3  [quant-ph]  4 Jan 20192\nsponse of the system particles that recalls the dissipative\nHofstadter model [55, 56]. We also show that this re-\nsponse is enable to generate stationary correlations be-\ntween the transversal degrees of freedom in the quan-\ntum regimen, which may eventually induce new kinds of\nenvironmental-mediated entanglement between the sys-\ntem particles di\u000berent from the standard dissipative mod-\nels [39]. Moreover, the Chern-Simons kinetic term en-\ndows the Brownian motion with unusual statistical fea-\ntures that enrich the dissipative dynamics, for instance\nit prompts an anti-symmetric 1/f noise in the classical\nMarkovian Langevin equation that closely resemblances\nto the low-frequency magnetic \rux noise in superconduct-\ning circuits [57, 58]. Our main concern is to analyze\nthe main characteristics of the novel dissipative dynam-\nics provided by the Chern-Simons action as compared to\nthe conventional Brownian motion. Let us stress that our\nmotivation as well as approach is signi\fcantly distinct to\nmost previous treatments within quantum open-system\ntheory to the best of our knowledge [1, 9{13], in particu-\nlar, those related to the Brownian motion of charged par-\nticles moving in the presence of external magnetic \felds\n[59{62]. The present work is in the line to explore the\nintriguing interplay between dissipation and the latent\nsymmetry structure of the interesting problem (e.g. the\nin\ruence of time-reverse symmetry or parity conserva-\ntion on the spectral density), in much the same fashion\nas Refs.[63{65].\nThe present paper is organized as follows. In Sec.II\nthe quantum canonical Hamiltonian governing the whole\ndynamics is obtained via a Coulomb gauge quantiza-\ntion procedure by starting from the action characteris-\ntic of the Maxwell-Chern-Simons electrodynamics of a\nharmonicn-particle system. From this, in Sec.III we de-\nrive the dissipative microscopic Hamiltonian which is the\nbasis of the present work, and extensively discussed its\nproperties. The reduced dynamics of the system particles\nis addressed in terms of the Langevin equation formal-\nism in Secs. III A, III B, and III C, as well as we study\nthe asymptotic properties of the \ructuation-dissipation\nrelation and the conditions under which the system re-\nlax towards a thermal equilibrium state. The Secs. IV\nand IV A provide an explicit example of the proposed\ndissipative description applied to study the Markovian\ndynamics. Finally, we summarize and draw the main\nconclusions in Sec.V.\nII. GAUGE INVARIANT DESCRIPTION\nAs stated in the introduction, we consider the most\ngeneral action in Euclidean planar geometry of a U(1)\ngauge-invariant system composed of nharmonic oscilla-\ntors coupled to a homogeneous and isotropic gauge \feld\nA\u0016= (A0;A). This is given by\nS=SHO+Z\nd2xdtLMCS(x;t); (1)whereSHOis the usual action for the reduced harmonic\nsystem andLMCS represents the Lagrangian density of\nthe Maxwell-Chern-Simons electrodynamics [45{47, 51],\ni.e.\nLMCS =1\n2\u0000\nE2\u0000B2\u0001\n+\u0014\n2\u000f\u0016\u0017\u0015A\u0016@\u0017A\u0015+A\u0001J+A0\u001a;\nwith\u000f\u0016\u0017\u0015being the completely antisymmetric tensor\n(i.e.\u000f012= 1 and\u000fij=\u000f0ij). HereBandEare\nthe magnetic and electric \felds (i.e. B=\u000f\u000b\f@\u000bA\fand\nE\u000b=\u0000_A\u000b\u0000@\u000bA0), whereas\u001aandJare respectively the\ncharge and current densities of the harmonic n-particle\nsystem. The second term in the Lagrangian density\n(1) describes the Chern-Simons action whose strength is\ngiven by the coupling constant \u0014, while the \frst one is the\nusual Maxwell kinetic term. Importantly, we shall show\nthat well-known results for the damped harmonic oscilla-\ntor [15, 16, 21, 22, 25, 39] are recovered in any step of the\ntreatment by taking the limit \u0014!0. Throughout this\nwork, Latin indices (running from 1 to n) are reserved to\nthe system harmonic oscillators, and unless stated other-\nwise, we use Greek letters as well as Einstein convention\nof repeated indices for the two spatial dimensions. We\nuse the natural units c=~=kB= 1.\nWe shall consider that the total density matrix for the\nharmonicn-particle system and gauge \feld decouples at\nthe initial time t0, and further, the \feld is in a canoni-\ncal equilibrium state ^ \u001a\f/e\u0000\f^HMSC with ^HMSC being\nthe free Hamiltonian of the Maxwell-Chern-Simons gauge\n\feld (de\fned below), whilst the system may be in an arbi-\ntrary state. The restriction to free-correlation initial con-\nditions is not crucial for the subsequent treatment, rather\nit is an extensively used assumption [10, 11, 16, 22] that\nprovides a better exposition. Intuitively, this in agree-\nment with preparing the system separately and brought\ninto contact with the gauge \feld su\u000eciently fast such that\nthe subsequent dynamics is governed by the Maxwell-\nChern-Simons action (1). As a result, ^ \u001a\fwill completely\ncharacterize the statistical properties of the gauge \feld\noperators, and, as we shall see, the system particle oper-\nators in the asymptotic time limit as well.\nWithout loss of generality we assume that all harmonic\noscillators possess identical mass m, while distinct (bare)\nfrequencies !ifori2f1;ng. Moreover, the charge and\ncurrent densities can be expressed in terms of a function\n'that models the charge distribution of each harmonic\noscillator [66],\n\u001a(x) =enX\ni=1'(x\u0000qi(t));\nJ(x) =enX\ni=1'(x\u0000qi(t))_qi(t); (2)\nwhere 0< e determining the coupling strength to the\ngauge \feldA, andqi(t) denotes the spatial coordinate\nof thei-th harmonic oscillator. For seek of simplicity, we\nshall assume an identical 'for all particles.3\nFrom the Lagrangian density described by the general\naction (1), one obtains the following expressions for the\ncanonical momentum of the i-th harmonic oscillator pi\nand gauge \feld \u0005, [44]\n\u00050= 0; (3)\n\u0005\u000b=_A\u000b+@\u000bA0+\u0014\n2\u000f\u000b\fA\f;\npi=m_qi+eZ\nd2x'(x\u0000qi)A(x); (4)\nas well as the Gauss law [45, 47, 48, 66]\nr\u0005+\u0014\n2r\u0002A\u0000\u001a= 0; (5)\nwhich upon surface integration unveils that the har-\nmonicn-particle system possesses a magnetic-like \rux of\nstrength proportional to ne=\u0014 [46, 47]. The latter may be\nseen by realizing that the contribution from r\u0005vanishes\nsince this represents the longitudinal electric \feld which\nhere exponentially decays owing to the photon mass [51].\nOne may show that the classical Hamiltonian obtained\nvia canonical procedure from the action (1) reads [67]\nH=nX\ni=1\u00121\n2m\u0012\npi\u0000eZ\nd2x'(x\u0000qi)A(x)\u00132\n+V(qi)\u0013\n+1\n2Z\nd2x\u0012\n\u0005\u0001\u0005\u0000\u0014\u0005\u0002A+ (r\u0002A)2+\u00142\n4A\u0001A\n+A0\u0010\nr\u0001\u0005+\u0014\n2r\u0002A\u0000\u001a\u0011\u0013\n: (6)\nwhereVstands for the isotropic con\fning harmonic po-\ntential of the oscillators, which we shall take as V(qi) =\n1\n2m!2\ni(qi\u0000\u0016qi)2for simplicity, with \u0016qibeing the equilib-\nrium position of the ith harmonic oscillator.\nWe now quantize the Hamiltonian (6) preserving the\ngauge invariance by following the conventional Coulomb\ngauge quantization procedure [47, 67]. By writing the\ngauge \feld variables (6) in terms of the longitudinal\n(Ajj;\u0005jj) and transversal components ( A?;\u0005?), this\nmeans to set Ajjequal to zero, whilst \u0005jjis evaluated\nby demanding the Gauss law (5) as a non-dynamical con-\nstraint [47, 48, 68, 69], i.e.\n\u0005k(x) =rxZ\ndyGc(x\u0000y)\u0010\n\u001a(y)\u0000\u0014\n2r\u0002A?(y)\u0011\n;\n(7)\nwhereGc(x) is the two-dimensional Coulomb Green's\nfunction that satisfy r2Gc(x) =\u000e2(x), i.e.Gc(x\u0000y) =\n(2\u0019)\u00001logjx\u0000yj[46, 66, 69]. The quantization of the\nHamiltonian is achieved by \frstly imposing the Coulomb\ngauge equal-time commutation relations [47, 68],\nh\n^A?\n\f(y);^\u0005?\n\u000b(x)i\n=i\u000e?\n\u000b\f(x\u0000y);\n[^qi;^pi] =i\u000eij: (8)\nwhere\u000e?\n\u000b\f(x\u0000y) denotes the transverse delta function[48],\n\u000e?\n\u000b\f(x\u0000y) = \n\u000e\u000b\f\u0000@(x)\n\u000b@(x)\n\f\nr2x!\n\u000e(2)(x\u0000y)\n=P\u000b\f(x)\u000e(2)(x\u0000y); (9)\nandP\u000b\f(x) is called the transverse projective opera-\ntor. Notice that all other commutators vanish identi-\ncally. The quantum canonical Hamiltonian governing all\nthe system-\feld dynamics is then obtained from (6) after\nthe separation of the transverse and longitudinal compo-\nnents of the quantum gauge \feld and the replacement of\nthe gauge-\fxing constraints (see AppendixA for further\ndeatils). By substituting the expressions for the charge\ndensities (2) and the Coulomb Green's function in the ob-\ntained Hamiltonian (given by A1), it is simply to verify\nthat the latter can be cast as follows\n^H=nX\ni=1\u00121\n2m\u0012\n^pi\u0000eZ\nd2x'(x\u0000^qi)^A?(x)\u00132\n(10)\n+^V(^qi) +nX\nj=1^Vc(^qi\u0000^qj)\u0013\n+^HCS+^HMCS;\nwith\n^HMCS =1\n2Z\nd2x\u0010\n^\u0005?\u0001^\u0005?+^A?\n\u000b\u0000\n\u0000r2+\u00142\u0001^A?\n\u000b\u0011\n;\n(11)\nand where we have de\fned the system-\feld interaction\nterm characteristic of the Chern-Simons action,\n^HCS=e\u0014\n2\u0019nX\ni=1Z\nd2xd2y'(x\u0000^qi) logjx\u0000yjr\u0002 ^A?(y);\n(12)\nand the commonly known (two-dimensional) Coulomb\npotential [70],\n^Vc(^qi\u0000^qj) =\u0000e2\n4\u0019nX\ni;j=1Z\nd2xd2ylogjx\u0000yj'(x\u0000^qi)'(y\u0000^qj):\n(13)\nNote that (12) and (13) emerge from the interaction\nbetween the system particles and the longitudinal part\nof the MSC gauge \feld via the Gauss law, so they\nare fundamental for avoiding a gauge-invariance break-\ning. Here, ^HMCS models the Maxwell-Chern-Simons\n(MSC) environmental Hamiltonian, whilst all the system-\nenvironment interaction is mediated by (12) and the min-\nimal coupling to the gauge \feld appearing in (10). In\nthis way, the proposed description has two main char-\nacteristic that distinguish it from previous treatments\n[52, 54, 59]: (i) the MSC environmental spectrum is\ngaped by\u0014due to the Chern-Simons action endows the\nenvironmental quasiparticle excitations with a \"toplogi-\ncal\" mass\u0014, and additionally, (ii) the Chern-Simons ac-\ntion attaches a magnetic-like \rux to each system particle\n[46, 47] (see Eq.(5)) that give rise to an e\u000bective charge-\n\rux coupling between the harmonic oscillator mediated4\nby^HCS. In the next section, we build the dissipative\nmicroscopic description upon the canonical Hamiltonian\n(10).\nFor latter purposes it is convenient to express the envi-\nronmental Hamiltonian (11) in terms of the quasiparticle\nexcitations of the MCS gauge \feld [47],\n^HMCS =X\nk!(k)ay(k)a(k) +E0; (14)\nwhereay(k) (a(k)) stands for the creation (annihilation)\noperator for the gauge \feld mode k2(2\u0019=L)Z2and\nexcitation frequency\n!(k) =p\njkj2+\u00142; (15)\nandE0=P\nk!(k)\n2is the usual vacuum expectation value\nof the gauge \feld. Since E0does not play a crucial role\nin the dissipative dynamics, this can be disregarded in\nthe future treatment by rede\fning the quasiparticle op-\nerators. Furthermore, it is advantageous to express the\ncanonical variables of the MCS gauge \feld in terms of\nthe complete set of polarized plane waves,\n^A?\n\u000b(x) =X\nkL\u00001eik\u0001x\n2\u0019p\n2!(k)(\"\u000b(k)^a(k) +\"y\n\u000b(\u0000k)^ay(\u0000k));\n^\u0005?\n\u000b(k) =X\nk(iL)\u00001eik\u0001x\n2\u0019p\n2!\u00001(k)(\"\u000b(k)^a(k)\u0000\"y\n\u000b(\u0000k)^ay(\u0000k));\n(16)\nwhere we have introduced the spatial Fourier transform\nof the generalized polarization vector \"(k), whose com-\nponents satisfy [47],\n\"\u000b(k) =i\u000f\u000b\fk\f\njkjei\u0014\nj\u0014j\u0012(k);\n\"\u000b(k)\"y\n\f(k) =P\u000b\f(k);\n\u0012(k) = tan\u00001\u0012k2\nk1\u0013\n: (17)\nThe phase term e\u0006i\u0014\nj\u0014j\u0012re\rects the spin-1 property of\nthe quasiparticle excitations of the free MCS electrody-\nnamics which guarantees the Poincare algebra is satis\fed\n[47, 67]. Although such phase must be taken account in\norder to provide an appropriate creation-destruction al-\ngebra (endowed with the usual equal-time commutation\nrelations), we shall see that this has no apparent e\u000bect in\nthe asymptotic dissipative dynamics of the harmonic n-\nparticle system. This evidences that the gauge \feld can\nbe treated as a scalar electromagnetic \feld for practical\npurposes [70].\nThe physical results of the dissipative model (10)\nshould not depend upon details of the particle charged\ndistribution, which is ideally model by the Dirac delta\nfunction for point-like particles. Since we are considering\na harmonic con\fnement of the particles, it is advisableto assume a Gaussian distribution for the charged distri-\nbution of the i-th particle, i.e.\n'(x\u0000^qi) =1\n4\u0019\u001be\u0000jx\u0000^qij2\n4\u001b; (18)\nwhere 0< \u001b determines the width of the distribution.\nFor seek of simplicity we shall assume the same width\nfor all the harmonic oscillators. With this choice we may\nrecover the point-particle situation by taking the limit\n\u001b!0+, i.e.'(x\u0000^qi)!\u000e(2)(x\u0000^qi).\nFor completeness we would like now to brie\ry discuss\nthe case when we consider the Chern-Simons electrody-\nnamics alone. By rescaling ^A?!^A?=\u0015, and keeping\n\fxed\u0014=\u00152ande=\u0015after taking the limit \u00152!1 , the\nMaxwell term disappears from the action (1), leaving us\nwith the pure Chern-Simons electrodynamics coupled to\nthe harmonic n-particle system. Going further to the\ncanonical Hamiltonian governing the whole system-\feld\ndynamics, one may verify that ^HMCS and ^HCS(describ-\ning the MSC environment and Chern-Simons system-\feld\ninteraction, respectively) also disappear from (10). This\nis not surprising, it is a consequence of the fact that the\nChern-Simons action does not modify the energy because\nit is \frst-order in time derivatives [45{47]. So there would\nbe no environmental dynamics supporting an irreversible\ntransference of energy coming from the reduced system.\nIndeed ^A?takes the form of a statistical gauge \feld that\ncan be properly absorbed in the matter \feld in order\nto produce the desired statistics transmutation [48], for\ninstance the anyonic statistics [46]. Consequently, this\nissue prevents us to consider the Chern-Simons electro-\ndynamics alone as a legitimate microscopic model to de-\nscribe dissipative dynamics.\nIII. QUANTUM DISSIPATION\nAs similarly occurs in the case of a charged harmonic\noscillator coupled to the classical Maxwell electromag-\nnetic \feld, obtaining an analytical, exact treatment of\nthe open-system dynamics of the reduced system (10) is\nlikely out of reach [39, 71]. Yet, a reliable and rich dissi-\npative description may be provided by doing two approx-\nimations well understood and motivated in the theory\nof quantum open-systems and classical electrodynamics,\nthat eventually turns the Hamiltonian (10) (governing\nall the quantum dynamics) into a quadratic operator in\nthe canonical variables ^qiand ^a(k), fori2f1;ngand\nk2R2.\nConcretely, since we are dealing with con\fned parti-\ncles, it proves convenient to consider both approxima-\ntions: the small displacement of harmonic oscillators in\ncombination with the usual dipole approximation of the\ngauge \feld. Let us emphasize that the long-wavelength\nlimit is ubiquitous in most investigations in quantum op-\ntics, atomic physics, and quantum chemistry [36]. In\nthis way, the i-th harmonic oscillator is assumed to move5\naround the equilibrium position \u0016qiof the con\fning po-\ntentialV(^qi) previously de\fned, such that we may take\nthe small displacement approximation up to \frst order\n^qi!\u0016qi+^qiin the Chern-Simons interaction Hamilto-\nnian (12), i.e.\n^HCS'e\u0014\n8\u00192\u001bnX\ni=1Z\nd2x0e\u0000jx0j2\n4\u001bZ\nd2yr\u0002 ^A?(y)\n\u0002\u0012\nlogjy\u0000x0\u0000\u0016qij+(y\u0000x0\u0000\u0016qi)\u0001^qi\njy\u0000x0\u0000\u0016qij2\u0013\n(19)\n=X\nknX\ni=1g\f(k;\u0016qi)\u0010\n\"\f(k)^a(k) +\"y\n\f(\u0000k)^ay(\u0000k)\u0011\n+X\nknX\ni=1f\u000b\f(k;\u0016qi)^q\u000b\ni\u0010\n\"\f(k)^a(k) +\"y\n\f(\u0000k)^ay(\u0000k)\u0011\n;\nwhere we have replaced the Fourier decomposition of the\ngauge \feld (16) and de\fned the complex coe\u000ecients,\ng\f(k;\u0016qi) =iL\u00001e\u0014\u000f\r\fk\r\n4\u00193p\n2!(k)Zd2x\n4\u001b\n\u0002Z\nd2ye\u0000jxj2\n4\u001beik\u0001ylogjy\u0000x\u0000\u0016qij\n=ie\u0014\u000f\r\fk\r\n2\u0019Lp\n2!(k)eik\u0001\u0016qie\u0000\u001bjkj2\njkj2; (20)\nf\u000b\f(k;\u0016qi) =@\u0016q\u000b\nig\f(k;\u0016qi)\n=\u0000e\u0014\u000f\r\fk\rk\u000b\n2\u0019Lp\n2!(k)eik\u0001\u0016qie\u0000\u001bjkj2\njkj2: (21)\nWe must understand the spatial Fourier transform of the\nlogarithmic function as the solution of the homogeneous\ntwo-dimensional Poisson equation [69]. Paying attention\nto the \frst term on the right-hand side in (19), this can\nbe recognized as a displacement e\u000bect upon the envi-\nronmental quasiparticle operators, denoted by ^DBR, as\na consequence of the backreaction of the harmonic n-\nparticle system on the MSC environment. Importantly,\nthe second term is closely similar in structure to the vec-\ntor potential associated to certain magnetic \rux \b(\u0016qi)\nattached to the particle charge distribution '(x\u0000^qi)\n[46, 70], which shall be called Chern-Simons \rux. This\nresult is in complete agreement with the previous discus-\nsions in Sec.II. Speci\fcally, the Chern-Simons interaction\n(12) can be rewritten as a combination of these two con-\ntributions,\n^HCS'^DBR\u0000enX\ni=1^qi\u0001^ECS(\u0016qi); (22)\nwhere ^ECS(\u0016qi) is interpreted as the electric \feld self-\nconsistently induced by an electric charge \u0014\b(\u0016qi) accord-\ning to the induction Faraday's law [47, 66, 72]. This shall\nbe refereed to as the Chern-Simons electric \feld and pro-\nvides the desired interaction Hamiltonian in dipole ap-\nproximation. It is interesting to note that the Levi-Civitasymbol appearing in the de\fnition of the coe\u000ecients (20)\nand (21) signals that the time-reversal symmetry breaks\ndown, as well as the backreaction term and Chern-Simons\nelectric \feld inherit the axial symmetry from the Chern-\nSimons kinetic term [47]. It is important to note as well\nthat the bilinear structure of (19) and (22) is indepen-\ndent of the particular choice of ', wherein the speci\fc\nform of the coe\u000ecients g\fandf\u000b\fonly depends on this.\nFollowing the same procedure we may obtain an small-\ndisplacement expression of the Coulomb potential (13),\ni.e.^Vc'Pn\ni;j=1\u0010\n^V0\nc(\u0016qi\u0000\u0016qj) +^V0\nc(\u0016qi\u0000\u0016qj)\u0001(^qi\u0000^qj)\u0011\n,\nwhere ^V0\nc(\u0016qi\u0000\u0016qj) is a constant operator for given val-\nues of the particle central positions that may be removed\nfrom the Hamiltonian without perturbing the dissipative\ndynamics.\nOn the other side, the dipole approximation in the\nFourier decomposition of the gauge \feld (16) yields,\nZ\nd2x'(x\u0000^qi)^A?(x)' (23)\n'Z\nd2y'(y)^A?(y+\u0016qi)\n=X\nkeik\u0001\u0016qie\u0000\u001bjkj2\n2\u0019Lp\n2!(k)\u0000\n\"(k)^a(k) +\"y(\u0000k)^ay(\u0000k)\u0001\n=^Adip(\u0016qi);\nassuming that eik\u0001^qi\u00191 fori2f1;ng[21, 43](i.e. the\nsystem particles mainly interact with the \feld low-energy\nmodes in comparison with the oscillator bare frequen-\ncies).\nSubstituting these results (19) and (23) in (10), we ob-\ntain a quadratic Hamiltonian which constitutes a \frst-\norder approximation to the low-lying dissipative dynam-\nics (10). At this point, the latter Hamiltonian can be\nbrought into a suitable form by performing the Goeppert-\nMayer transformation [43],\n^UGM=e\u0000iePn\ni=1^qi\u0001^Adip(\u0016qi); (24)\nwhich basically consists of replacing (according to the\nBaker-Hausdor\u000b-Campbell formula)\n^pi!^pi+e^Adip(\u0016qi);\n^a(k)!^a(k) +ie \"y\n\u000b(k)e\u0000\u001bjkj2\n2\u0019Lp\n2!(k)nX\ni=1e\u0000ik\u0001\u0016qi^q\u000b\ni;\nwhereas the other terms remain invariant as they com-\nmute with ^UGM. Now, by grouping together the interac-\ntion terms we rede\fne the system-environment coupling\ncoe\u000ecient as follows,\nh\u000b(k;\u0016qi) =\u0000\"\f(k)\n!(k)f\u000b\f(k;\u0016qi) +ie \"\u000b(k)e\u0000\u001bjkj2eik\u0001\u0016qi\n2\u0019Lp\n2!(k)\n=e eik\u0001\u0016qie\u0000\u001bjkj2\n2\u0019Lp\n2!(k)\u0012\u0014\u000f\r\fk\rk\u000b\"\f(k)\njkj2!(k)+i\"\u000b(k)\u0013\n; (25)6\nand introduce the renormalized potential interaction be-\ntween system particles,\nV\u000b\f\nij=\u000eij\u000e\u000b\fm!2\ni (26)\n+e2\n4\u00192L2X\nkP\u000b\f(k)e\u00002\u001bjkj2cos(k\u0001(\u0016qi\u0000\u0016qj));\nV\u000b\ni=nX\nj=1\u0012\n2(V0\nc(\u0016qi\u0000\u0016qj))\u000b\n+X\nk\u0010\nh\u000b(k;\u0016qi)gy\n\f(k;\u0016qj)\"y\n\f(k)\n+hy\n\u000b(k;\u0016qi)g\f(k;\u0016qj)\"\f(k)\u0011\u0013\n; (27)\nwhere the second line of (26) corresponds to the so-called\ndipole self-energy associated to ^Adip[37], and (27) con-\ntains the Coulomb potential along with a backreaction\ncontribution. Taking a closer look to (25), one may see\nthat the particle-charge-distribution width (2 \u001b)\u00001\n2plays\nthe role of a frequency cut-o\u000b on the system-environment\ninteraction. Consequently, the Chern-Simons in\ruence\non the dissipative dynamics becomes weak (or strong)\nwhenp\n2\u001b\u0014\u001c1 (or 1\u001cp\n2\u001b\u0014), this will be seen\nmore clearly in the de\fnition of the spectral density in\nSec.III A. Furthermore, the breaking of time-reversal and\nparity symmetries are now hidden in the coupling coe\u000e-\ncients (25) and the linear potential V\u000b\ni.\nFinally, after some manipulation once Eqs. (25), (26),\nand (27) are substituted, we arrive to the desired mi-\ncroscopic Hamiltonian which is the basis of the present\nwork,\n^H0=nX\ni=1^p2\ni\n2m+nX\ni=1V\u000b\ni^q\u000b\ni+1\n2nX\nj;i=1\u0010\nV\u000b\f\nij\n\u0000X\nk!(k)(h\u000b(k;\u0016qi)hy\n\f(k;\u0016qj) +c:c:)\u0011\n^q\u000b\ni^q\f\nj\n+X\nk!(k)\f\f\f\f^ay(k)\u0000nX\ni=1\u0012\nh\u000b(k;\u0016qi)^q\u000b\ni\u0000g\f(k;\u0016qi)\"\f(k)\n!(k)\u0013\f\f\f\f2\n\u0000nX\ni;j=1X\nkP\u000b\f(k)\n!(k)gy\n\u000b(k;\u0016qi)g\f(k;\u0016qj): (28)\nwhere the subscript \"c.c.\" stands for Hermitian conjuga-\ntion. According to the thermodynamic limit, the gauge\n\feld is considered to be composed of an in\fnite number of\nmodes, then we may take the limit of a dense spectrum of\n\feld frequencies in (28) whenever convenient, and replace\nthe discrete momentum sum by an integral following the\nprescriptionP\nk!L2R1\n\u00001d2k. Doing this, we provide\nexplicit expressions for (26) and (27) in the AppendixC.Let us now discuss some important properties of the\ndissipative Hamiltonian (28). First, it is immediate to\nsee that disregarding the Chern-Simons e\u000bects (that is,\n^DBR!0 and ^E\u000b\nCS(\u0016qi)!0 for arbitrary i2f1;ng) in\n(28) returns the independent-oscillator model [21, 22, 39].\nConversely, (28) exhibits the symmetry structure char-\nacteristic of the underlying Chern-Simons action: parity\nbreaking and time-reversal asymmetry, as similarly oc-\ncurs in systems subjected to external magnetic \felds or\nrecent extended environments [54]. We shall see that\nsuch symmetry a\u000bords the appearance of a dissipative\nvortex-like dynamics driven by a Lorentz force arising\nfrom the aforementioned particle-attached Chern-Simons\n\rux. Concretely, by direct comparison to the so-called\nblackbody radiation bath [21], we identify a pseudo-\nelectric \feld ^E\u000b\ni(t) responsible for the environmental force\nacting upon the harmonic ith oscillator,\n^E\u000b\ni(t) =e\u0012\n\u0000@\n@t^A\u000b\ndip(\u0016qi;t) +^E\u000b\nCS(\u0016qi;t)\u0013\n+^F\u000b\nBR(\u0016qi;t);\n(29)\nwhere ^E\u000b\nCS(\u0016qi;t) and ^A\u000b\ndip(\u0016qi;t) are respectively obtained\nby replacing a(k)!a(k)e\u0000i!(k)(t\u0000t0)in the de\fni-\ntions (22) and (23), whilst ^FBR(\u0016qi;t) identi\fes with a\nbackreaction force obtained from substituting a(k)!\nh\u000b(k;\u0016qi)e\u0000i!(k)(t\u0000t0)in the expression of ^DBR.\nThe \frst term of the right-hand side of (29) bears the\ndissipation mechanism in the conventional Brownian mo-\ntion, and it coincides identically with the vector poten-\ntial contribution to the electric \feld corresponding to\nthe Maxwell electrodynamics action alone (i.e. \u0014= 0)\nin dipole approximation. Hence, the second term may\nbe interpreted as the Lorentz force due to the Chern-\nSimons electric \feld ^ECS(\u0016qi), whereas the backreaction\nforce ^FBR(\u0016qi;t) contains the previously mentioned e\u000bec-\ntive shift in the environmental quasiparticle operators.\nContrary to the so-called \"initial slip\" term in the stan-\ndard microscopic model that just depends on the initial\nharmonic oscillator positions [22], this backreaction dis-\nplacement is independent of the initial state of the har-\nmonic system. In Sec.III B, we shall see that the latter\ngives rise to non-stochastic \ructuations which eventually\ncancel out in the time asymptotic limit.\nUnlike the dipole approximation taken in (23), the va-\nlidity of the small displacement approximation described\nin (19) is intricate to elucidate just by looking at (22).\nThis can be better assessed by requiring the Hamiltonian\n(28) to be a positive de\fnite operator [16, 21, 39] in order\nit has a lower-bounded spectrum preventing \"runaway\"\nsolutions [73], and thus, it gives rise to simple dissipative\ndynamics (which preserves !ias the bare frequencies for\nthe system particles). As shown in detail in AppendixB,\nsuch condition is found to be equivalent to the following\ninequality,7\n1\n2nX\nj;i=1 \nV\u000b\f\nij\u0000\u000e2\u000b\u000e2\fe\u0000j\u0016qi\u0000\u0016qjj2\n8\u001b\n8\u0019\u001b+e2\n8\u0012\n(\u000e\u000b\f\u00002\u000e1\f\u000e1\u000b)\u0014H1(i\u0014j\u0001\u0016qijj)\nj\u0001\u0016qijj+\u000e1\u000b\u000e1\f\u00142H0(i\u0014j\u0001\u0016qijj)\u0013!\n^q\u000b\ni^q\f\nj\n+nX\ni;j=1 \n2 (V0\nc(\u0001\u0016qij))\u000b\u0000\u000e1\u000be2\u00142\n2\u0019J1(j\u0001\u0016qijj)!\n^q\u000b\ni\u0015e2\u00142\n4\u0019nX\ni;j=1J0(j\u0001\u0016qijj) (30)\nwhere\u0001\u0016qij=\u0016qi\u0000\u0016qjand we have introduced the auxil-\niary functions\nJl(j\u0001\u0016qijj) =Z1\n0dkkl+1e\u00002\u001bk2\nk2(k2+\u00142)Jl(kj\u0016qi\u0000\u0016qjj);\nwithJi(x) denoting the i-order Bessel function of\nthe \frst kind in the variable x, andHj(x) =\nij+1e2\u001b\u00142\u0010\nH(2)\nj(\u0000x) + (\u00001)j+1H(1)\nj(x)\u0011\n, withH(j)\ni(x)\nbeing thei-order Hankel function of the j-th kind [74].\nIt is worthwhile to note that the integral involved in the\nde\fnition of Jlmay present an infrared divergence (i.e.\nk!0) owing to the two-dimensional Coulomb Green\nfunction blows up at the origin (see Eq.(20)). This is a\nfeature characteristic of the Maxwell-Chern-Simons elec-\ntric and magnetic \felds that requires adequate regular-\nization schemes [47, 70].\nAlthough the positive condition may be looked rather\ncomplicated for supporting an intuitive interpretation at\n\frst sight, the right-hand side of (30), which emerges\nexclusively as a consequence of the backreaction on the\nMSC environment, re\rects a repulsive e\u000bect between the\nsystem particles that challenges with the con\fning har-\nmonic potential. To see this more clearly, let us focus in\nthe single harmonic oscillator system (i.e. n= 1). Hence,\nit can be shown that the formidable inequality (30) boils\ndown to\nX\n\u000b=1;2\u0012\nm!2\n1\u0000e2\u00142\n8\u0019\u0000\u0000\n0;2\u001b\u00142\u0001\ne2\u001b\u00142\u0013\n^q\u000b\n1^q\u000b\n1\u0015e2\u00142\n2\u0019R2\n0;\nwhereR2\n0=Pn\ni;j=1J0(0), and\u0000(0;x) denotes the in-\ncomplete Euler Gamma function [74]. Clearly, the above\ncondition may be interpreted as the single harmonic os-\ncillator is enforced to follow a \ructuating motion around\na circular area of radius larger than certain Rgiven by\nR2\n2\u001b=R2\n0\n4\u0019m\u001b!2\n1\ne2\u00142\u0010\n1\u00001\n8\u0019e2\u00142\u0000(0;2\u001b\u00142)e2\u001b\u00142\nm!2\n1\u0011: (31)\nTo be this result consistent with the small displace-\nment approximation considered previously, we demand\nthe length of Rto be su\u000eciently small in comparison to\nthe width of the particle charged distribution 2 \u001b, which\nimplies\ne2\n2m\u001b2!2\n1\u001c1\n\u001b\u00142: (32)Expression (32) entails that there must exist a trade-\no\u000b between the system-environment interaction strength\nand system particle bare frequencies. The physical in-\ntuition behind the latter is that the environment could\ndrive the particle to reach highly excited states for an\narbitrary large coupling, which would eventually lead to\nbreak down the small displacement approximation as-\nsumed in (19). For instance, for a strong Chern-Simons\naction 1\u001c\u001b\u00142, we may approximate \u0000\u0000\n0;2\u001b\u00142\u0001\ne2\u001b\u00142\u0018\n(2\u001b\u00142)\u00001, and then, the positive condition (31) holds for\n1\u001cm\u001b!2\n1=e2, which is equivalent to (32). In this way,\nthe small displacement approximation again requires that\nthe system-environment coupling must pay o\u000b the re-\npulsive counteract of a strong backreaction e\u000bect. From\nthis point onward we work within the parameter domain\nwhere expression (30) holds, and therefore, the subsidiary\ncondition (32) is always satis\fed for the bare frequencies\nof thenharmonic oscillators. In particular, this result\nis a manifestation of the issue that the Maxwell-Chern-\nSimons theory works better for developing models of con-\n\fned particle systems [46, 75].\nOur \fnal remark is that the Hamiltonian (28) can be\nregarded as a gauge-invariant microscopic description by\nconstruction, since it was derived from a gauge-invariant\nHamiltonian (10). This is a major di\u000berence with previ-\nous treatments [54], where it is not guaranteed the gauge-\ninvariance for a given choice of the system-environment\ncoupling coe\u000ecients. Remarkably, (28) looks very sim-\nilar to an environmental minimal-coupling Hamiltonian\n(in dipole approximation) [38] with gauge \feld\n^A\u000b\nMSC(\u0016qi) =i\neX\nk(h\u000b(k;\u0016qi)^a(k) +c:c:);\nand associated electric \feld,\n^E\u000b\nMCS(\u0016qi;t) =\u0000@\n@t^A\u000b\ndip(\u0016qi;t) +^E\u000b\nCS(\u0016qi;t): (33)\nConcretely, the Hamiltonian (28) is equivalent to a\nminimal-coupling theory of nharmonic oscillators with\nthe gauge \feld ^AMSC(\u0016qi) provided we disregard the\nbackreaction e\u000bects and endow the system Hamiltonian\nwith a renormalized potential interaction which cancels\nthe environmental in\ruence on the conservative dynam-\nics, i.e.\n^HRN=\u00002nX\nj=1(V0\nc(\u0016qi\u0000\u0016qj))\u000b^q\u000b\ni+1\n2nX\ni;j=1\u0010\n\u000eij\u000e\u000b\fm!2\ni\n\u0000V\u000b\f\nij+X\nk!(k)(h\u000b(k;\u0016qi)hy\n\f(k;\u0016qj) +c:c:)\u0011\n^q\u000b\ni^q\f\nj;8\nwhere the \frst term is the familiar Coulomb contri-\nbution. This statement can be explicitly veri\fed by\nabsorbing the gauge \feld in the canonical conjugate\nmomentum by means of a gauge transformation ^U=\neiePn\ni=1^qi\u0001^AMCS (\u0016qi)upon the Hamiltonian (28), once we\nhave dropped the backreaction terms (i.e. g\f(k;\u0016qi)!\n0) and introduced the renormalization ^HRN. The as-\nsociated Maxwell-Chern-Simons electric \feld (33) fea-\ntures non-commutative components (see Eq.(C7) in Ap-\npendixC for further details), i.e.\nh\n^E\u000b\nMCS(\u0016qi);^E\f\nMCS(\u0016qj)i\n/\u0000i\u0014\u000f\u000b\f: (34)\nInterestingly, this property is shared with the electric\n\feld of the free Maxwell-Chern-Simons electrodynamics\n(i.e. without matter-\feld interaction) [46], and has sev-\neral consequences in the dissipative dynamics illustrated\nin Sec.III B.\nWe spend the following sections to justify that the\nHamiltonian (28) regards a legitimate microscopic de-\nscription to simulate the relaxation process towards a\nthermal equilibrium state (see III A, III B and III C) de-\nspite the approximations taken to derive it, as well as we\nprovide an explicit comparison with the popular damped\nharmonic oscillator [4, 11, 18, 21, 22, 25] in the Markovian\nLangevin dynamics limit (see IV and IV A). Before pro-\nceeding with our treatment, it is convenient to recall the\n!-variable Fourier transform ~ r(!) of a time-dependent\nfunctionr(t),\n~r(!) =1\n2\u0019Z1\n\u00001dt ei!tr(t);\nand its corresponding real and imaginary parts,\nRe ~r(!) =~r(!) + ~ry(!)\n2;Im ~r(!) =~r(!)\u0000~ry(!)\n2i;\nwhere ~ry(!) represents the complex conjugate of ~ r(!).\nA. Generalized Lanvegin equation\nHaving determined the dissipative Hamiltonian (28)\nalong with the quasiparticle excitations of the gauge \feld,\nwe may turn the attention to the non-equilibrium dy-\nnamics of the harmonic n-particle system. Starting from\nthe Hamiltonian (28) we derive the following Heisenberg\nequations for the ith-oscillator position and momentum\noperators,\n_^q\u000b\ni=^p\u000b\ni\nm; (35)\n_^p\u000b\ni=\u00002nX\nj=1(V0\nc(\u0016qi\u0000\u0016qj))\u000b\u0000nX\nj=1V\u000b\f\nij^q\f\nj (36)\n+X\nk!(k)\u0000\nh\u000b(k;\u0016qi)^a(k) +hy\n\u000b(k;\u0016qi)^ay(k)\u0001\n;as well as for the quasiparticle creation operator of the\ngauge \feld,\n_^ay(k) =i!(k)^ay(k)\n+inX\ni=1\u0010\ng\f(k;\u0016qi)\"\f(k)\u0000!(k)h\f(k;\u0016qi)^q\f\ni\u0011\n:\nIt is straightforward to obtain the formal solution of the\nlatter equation by using the standard Green's function\nmethod, since it constitutes an inhomogeneous linear sys-\ntem of di\u000berential equations. First, we obtain for the\nquasiparticle operators of the gauge \feld\n^ay(k;t) = \n^ay(k;t0) +nX\ni=1\"\f(k)\n!(k)g\f(k;\u0016qi)!\nei!(k)(t\u0000t0)\n\u0000i!(k)nX\ni=1h\f(k;\u0016qi)Zt\nt0ei!(k)(t\u0000\u001c)^q\f\ni(\u001c)d\u001c\n\u0000nX\ni=1\"\f(k)\n!(k)g\f(k;\u0016qi); (37)\nwheret+\n0\u0014tin order to be physically consistent with\nthe considered initial preparation. Inserting the solution\n(37) into equation (36) and manipulating the subsequent\nresult, one gets to the desired generalized Langevin equa-\ntion,\nmd2^q\u000b\ni\ndt2+nX\nj=1V\u000b\f\nij^q\f\nj+V\u000b\ni (38)\n\u0000nX\nj=1Zt\nt0\u0006\u000b\f\nij(t\u0000\u001c)^q\f\nj(\u001c)d\u001c=^E\u000b\ni(t);\nwhere we have identi\fed the pseudo-electric \feld ^E\u000b\nide-\n\fned in (29) as the \ructuating force, and the generalized\nsusceptibility or self-energy as the retarded Green's func-\ntion,\n\u0006\u000b\f\nij(t\u0000t0) =i\u0002(t\u0000t0\u0000j\u0001\u0016qijj)Dh\n^E\u000b\ni(t);^E\fy\nj(t0)iE\n^\u001a\f;\n(39)\nwhere\u0002(t) denotes the Heaviside step function. Clearly,\nthe linear potential Virepresents an non-stochastic force\na\u000becting mainly the mean average position of the system\nparticles, so it could be neglected from the future discus-\nsion by doing a suitable renormalization of the harmonic\noscillators.\nAlthough equation (38) may look similar at \frst sight\nto the quantum Langevin equation in presence of mag-\nnetic \felds [59{61], both equations signi\fcantly di\u000ber\nin the statistical and analytical properties of the corre-\nsponding \ructuating force and retarded self-energy. On\none side, the backreaction e\u000bects in the pseudo-electric\nforce (29) prevents the dissipative dynamics to ful\fll the\n\ructuation-dissipation theorem at all times, in contrast\nto the conventional Brownian motion. On the other side,\nthe breaking of time-reversal and parity symmetry in the9\npresent context induces an imaginary contribution to the\n(\feld) spectral density that has no counterpart in the\nindependent-oscillator model [21, 39]. This deeply mod-\ni\fes the analytical structure of the Fourier transform of\nthe retarded self-energy, which can be compactly written\nas follows\n~\u0006\u000b\f\nij(!) =R~\u0006\u000b\f\nij(!) +iI~\u0006\u000b\f\nij(!); (40)\nwhereR~\u0006\u000b\f\nij(!) (I~\u0006\u000b\f\nij(!)) must not be confused with\nthe real (imaginary) part previously de\fned. Despite of\nthis, we would like to remark that the self-energy ~\u0006\u000b\f\nij(!)\nexhibits the general properties required to produce a reli-\nable dissipative dynamics: (causality condition) it is an-\nalytic in the upper-half !-complex plane, and further,\n(reality condition [21, 61]) it holds\n\u0010\n~\u0006\f\u000b\nij\u0011y\n(!) =~\u0006\u000b\f\nji(!): (41)\nBasically, these properties are encoded by the (\feld)\nspectral density arising from the Maxwell-Chern-Simons\nelectrodynamics, denoted by J\u000b\f, and which reduces\nto the well-known spectral density of the independent-\noscillator model for zero Chern-Simons constant.\nLet us draw more attention to the properties of the\nretarded self-energy. The Heaviside step function in theexpression of the retarded self-energy (39) guaranties the\ndissipative dynamics to be consistent with the initial de-\ncoupling of the harmonic particle system and gauge \feld,\nand further, it gives the usual pole prescription in the fre-\nquency domain mentioned above: \u0006\u000b\f\nij(t) is an analytic\nfunction in the upper-half complex plane. Moreover, it is\nin agreement with the fact that the pseudo-electric \felds\n^E\u000b\ni(t) and ^E\f\nj(t) must commute for space-like separations,\ni.e. [ ^E\u000b\ni(t0);^E\f\nj(t)] = 0 ifj\u0016qi\u0000\u0016qjj>jt\u0000t0j. This is usually\nknown as microscopic causality, and for instance, it is ful-\n\flled for the free Maxwell electromagnetic \feld [40]. As a\nconsequence, we show in Appendix C that the self-energy\nsatis\fes a generalized Kramers-Kronig identity [23, 39],\ni.e.\nR~\u0006\u000b\f\nij(!) =H\u0002\nI~\u0006\u000b\f\nij(!0)\u0003\n(!)\n=1\n\u0019PZ+1\n\u00001I~\u0006\u000b\f\nij(!0)\n!0\u0000!d!0; (42)\nwhereH[f(x)](!) denotes the Hilbert transform of the\nfunctionf(x) in the variable !, andPis the Cauchy\nprincipal value.\nBy replacing the pseudo-electric \feld (29) in (39) and\ntaking the dense spectrum limit, the retarded self-energy\ncan be cast in terms of the environmental spectral density\nas follows (see ApendixC for further details),\n\u0006\u000b\f\nij(t\u0000t0) =2\n\u0019\u0002(t\u0000t0\u0000j\u0001\u0016qijj)Z1\n0d!\u0010\nRefJ\u000b\f(!;\u0001\u0016qij)gsin(!(t\u0000t0)) + ImfJ\u000b\f(!;\u0001\u0016qij)gcos(!(t\u0000t0))\u0011\n;(43)\nwhereas the spectral density takes the form,\nJ\u000b\f(!;\u0001\u0016qij) =\u0010e\n2\u00112\ne\u00002\u001b(!2\u0000\u00142)Y\u000b\f\u0010\nj\u0001\u0016qijj;p\n!2\u0000\u00142\u0011\n;with\u0014\u0014! (44)\nand\nY(a;b) =\u0012\n\u00142J0(ab) +b\naJ1(ab)i\u0014!J 0(ab)\n\u0000i\u0014!J 0(ab)!2J0(ab)\u0000b\naJ1(ab)\u0013\n;\nwhere we may clearly observe that the o\u000b-diagonal elements arise exclusively from the Chern-Simons action. This\nspeci\fc form of the spectral density deserves some attention. Expression (44) shares some features with the usual\n(bath) spectral density of the damped harmonic oscillator model [19, 20]: (i) it features a broad gaped spectrum\n(\u0014 < ! <1) that may span the harmonic oscillator frequencies, and further, (ii) the strength of the system-\feld\ncoupling decays (exponentially) for su\u000eciently large frequencies compared to the aforementioned frequency cut-o\u000b\ngiven by (2 \u001b)\u00001\n2. Although the latter eventually prevents from ultraviolet divergence issues in the non-equilibrium\nparticle dynamics for most interesting cases, it is worthwhile to notice that the speci\fc case of point particles (i.e.\n\u001b!0+) is not free from this divergence. This may be seen as a consequence of the well-known self-energy problems\nthat su\u000bers the point-particle electrodynamics [45]. Furthermore, this shows that the cut-o\u000b factor of the spectral\ndensity is mainly determined by the choice of the particle charged distribution '(^q). Accordingly, the spectral density\nconstitutes a 2 n\u00022nHermitian matrix (i.e. J\u000b\f(!;\u0001\u0016qij) = (J\f\u000b)y(!;\u0001\u0016qji)), which immediately implies (41), and\ntherein, the retarded self-energy is a 2 n\u00022nreal matrix in the time domain.\nThe speci\fc form of the spectral density (44) also reveals interesting properties related to the dissipative dynamics.\nFor instance, the fact that the spectral density is highly oscillatory in the frequency domain unveils that the harmonic\nsystem may undergo a strong non-Markovian dissipative dynamics [9], rendering a richer quantum dissipative evolution\nthan the conventional Brownian motion. Furthermore, the diagonal elements of the spectral density manifest an\nanisotropic in\ruence to the transversal spatial degrees of freedom of distant harmonic oscillators. Nevertheless, this\ne\u000bect cancels out when the particles are very closed or localized in identical positions, which can be seen by taking the10\nasymptotic limit j\u0001\u0016qijj!0 in (44). By virtue of the o\u000b-diagonal shape of the spectral density, we may also realize\nthat the new dissipative Chern-Simons e\u000bects are mainly encoded in the Fourier cosine transform appearing in the\nretarded self-energy de\fnition (43). This result is consistent with previous treatments [54, 59] about Brownian motion\nin the presence of magnetic \felds, where it was shown that either a Berry's geometric magnetic or uniform magnetic\n\feld produces a \"transversal\" contribution to the memory kernel. Interestingly, we shall show in Sec.IV that in the\nMarkovian Langevin limit the o\u000b-diagonal elements of the retarded self-energy turn into an e\u000bective interaction which\nis akin to applying a non-conservative rotating force upon the harmonic oscillators, which is in agreement with the\nfact that such contribution is promoted by the time-reversal asymmetry and parity violation.\nA further simpli\fed expression between the retarded self-energy and spectral density is obtained by performing\nthe Fourier transform in (43) (the details of the derivation can be found in AppendixC). Doing this we arrive at the\nfollowing identity\nI~\u0006\u000b\f\nij(!) =Dh\n~E\u000b\ni(!);~E\fy\nj(!)iE\n=1\n2\u0019\u0010\n\u0002(!)Jy\n\u000b\f(!;\u0001\u0016qij)\u0000\u0002(\u0000!)J\u000b\f(\u0000!;\u0001\u0016qij)\u0011\n; (45)\nwhich completely characterizes the dissipative e\u000bects. It is well-known that a system, whose open-system dynamics is\ngoverned by a given quantum Langevin equation, will reach an asymptotic thermal equilibrium state if the dissipative\ne\u000bects are related to the \ructuations of the environmental noise via the so-called \ructuation-dissipation theorem\n[30, 39, 76], e.g.\nDn\n~\u0018\u000b\ni(!);~\u0018\fy\nj(!0)oE\n^\u001a\f\n2I~\u0006\u000b\f\nij(!)=\u000e(!\u0000!0)\u0000\n1 + 2n(!;\f\u00001)\u0001\n; (46)\nwhere ^\u0018\u000b\ni(t) represents a mean zero stationary Gaussian noise (i.e. a quantum Brownian noise) and 1 + 2 n(!;\f\u00001) =\ncoth (\f!=2). This relation manifests that the \ructuating force is only due to thermal \ructuations, which is the case\nfor the independent-oscillator model [21, 22, 25] (e.g. see the case of the electromagnetic \feld [41]). Going back to\nthe expression (29), the aforementioned backreaction contribution ^FBR(\u0016qi;t) to the pseudo-electric \feld constitutes\na non-stochastic force that breaks down the \ructuation-dissipation theorem (46) for the MSC environment initially\nin a canonical equilibrium state ^ \u001a\f. As shown in AppendixC, the statistics of the pseudo-electric force is related to\nthe dissipative e\u000bects via the following formidable equation,\n1\n2Dn\n~E\u000b\ni(!);~E\fy\nj(!0)oE\n^\u001a\f=\u000e(!\u0000!0)\u0000\n1 + 2n\u0000\n!;\f\u00001\u0001\u0001\nI~\u0006\u000b\f\nij(!) (47)\n+\u0002(!)\u0002(!0)\u0010\n~G\u000b\f\nij\u0011y\n(!;!0;t0) +\u0002(\u0000!)\u0002(\u0000!0)~G\u000b\f\nij(\u0000!;\u0000!0;t0)\n+\u0002(!)\u0002(\u0000!0)\u0010\n~F\u000b\f\nij\u0011y\n(!;\u0000!0;t0) +\u0002(\u0000!)\u0002(!0)~F\u000b\f\nij(\u0000!;!0;t0);\nwhere we have de\fned the following non-stochastic spectral functions for \u0014\u0014!;!0\n~G\u000b\f\nij(!;!0;t0) =e4\u00142\n16!!0ei(!\u0000!0)t0e\u00002\u001b(!2+!02\u00002\u00142)nX\nl;m=1R\u000b\f\u0010p\n!2\u0000\u00142;p\n!02\u0000\u00142;j\u0016qi+\u0016qlj;j\u0016qj+\u0016qmj\u0011\n;(48)\nand\n~F\u000b\f\nij(!;!0;t0) =\u0000e4\u00142\n16!!0ei(!+!0)t0e\u00002\u001b(!2+!02\u00002\u00142)nX\nl;m=1T\u000b\f\u0010p\n!2\u0000\u00142;p\n!02\u0000\u00142;j\u0016qi+\u0016qlj;j\u0016qj+\u0016qmj\u0011\n;(49)\nas well as we have introduced the matrices RandT, e.g.\nR11(a;b;c;d ) =1\nacbdu\u0012\n\u0014ac;2\u0014!\nj\u0014j\u00002\u0014;ac\u0013\nu\u0012\n\u0014bd;2\u0014!\nj\u0014j\u00002\u0014;bd\u0013\n;\nR12(a;b;c;d ) =i\nacbdu\u0012\n\u0000\u0014ac;\u00002\u0014!\nj\u0014j\u00002\u0014;ac\u0013\nu\u0012\n\u0000bd!0;\u0000\u00142\nj\u0014j+ 2!0;bd\u0013\n;\nR21(a;b;c;d ) =i\nacbdu\u0012\nac!0;\u0000\u00142\nj\u0014j\u00002!0;ac\u0013\nu\u0012\n\u0014b2d;\u00002\u0014!\nj\u0014j\u00002\u0014;bd\u0013\n;\nR22(a;b;c;d ) =1\nacbdu\u0012\nac!;\u00142\nj\u0014j\u00002!;ac\u0013\nu\u0012\nbd!0;\u00142\nj\u0014j\u00002!0;bd\u0013\n; (50)\nand the auxiliary function u(x;y;z ) =xJ1(z) +yJ2(z). Due to the detailed representation of Tis lengthy and11\nnot crucial for the future discussion, we move it to the\nAppendix D (see equation (D8), as well as the derivation\nof the \ructuation-dissipation relation (47). At this point,\nit is important to realize that both non-stochastic spec-\ntral functions, (48) and (49), are integrable functions in\nthe frequency domain, and further, they decay as fast as\nan exponential function for large arguments of !and!0.\nFor instance, it is readily to see that the matrix elements\nofRreduce to a \fnite and continuous algebraic function\nfor small arguments j\u0016qi=j+\u0016qljp\n!2\u0000\u00142\u001c1, by using\nthe asymptotic expressions of the Bessel functions, i.e.\nJ\u000b(z)\u0018\u0000(\u000b+ 1)\u00001(z=2)\u000bforz\u001c1. The matrix Tis\nfound to feature this property as well.\nFrom the derivation of (47) follows that the \frst line\nis just due to the Maxwell and Chern-Simons electric\ncontributions to the \ructuating force. That is, we can\nidentify the Lorentz force rendered by these electric \felds\nwith a stochastic thermal noise, i.e.\n^\u0018\u000b\ni(t) =e^E\u000b\nMCS(\u0016qi;t); (51)\nwhere ^E\u000b\nMCS(\u0016qi;t) was de\fned in (33). Recall that ^\u0018\u000b\ni(t)\nis an unbiased random operator (i.e.D\n^\u0018\u000b\ni(t)E\n= 0) that\nsatis\fes the \ructuation-dissipation relation illustrated in\n(46) even though the Chern-Simons electric \feld exhibits\ntime-reversal asymmetry [3]. On the other hand, the sec-\nond and third line represents the non-stochastic \ructua-\ntions owing to the backreaction force in (29). Concretely,\nthey come from non-stationary terms involving ^ a(k)^a(k)\nand ^ay(k)^ay(k), which represent non-conservative energy\nprocesses taking place in the gauge \feld at the initial time\nt0. Interestingly, paying attention to (48) and (49), we\nmay observe that such \ructuations become highly oscil-\nlatory in the long time limit t\u0000t0!1 , which could\nmake their contribution to the stationary dynamics ne-\nglectable. Indeed, we illustrate in the next section how\nthe non-stochastic \ructuations asymptotically cancel out\nin the strict limit by appealing to the fact that the afore-\nmentioned spectral functions (48) and (49) have both a\nbroad bandwidth and rapid decayment at large frequen-\ncies compared to the particle frequencies !i, recovering in\nturn the \ructuation-dissipation theorem (46). Before to\ncontinue, it is worthwhile to mention that these \ructua-\ntions would e\u000bectively disappear from the noise statistics\n(and thus, the theorem (46) would be valid for the \ruc-\ntuating force ^E\u000b\ni(t) during the whole time evolution) if\nwe would have access to the initial preparation of the\nMaxwell-Chern-Simons electromagnetic \feld and its ini-\ntial state could be tuned to\n^\u001a\f/exp \n\u0000\fX\nk!(k)^ay\nd(k)^ad(k)!\n;\ninstead of the canonical equilibrium state ^ \u001a\f/e\u0000\f^HMCS.\nThat is, the environmental annihilation (creation) op-\nerator would be initially replaced by a shifted operator\n^ad(^ay\nd) which e\u000bectively counteracts the backreaction ef-\nfects. As this is not the case for most interesting physicalsituations, in the next sections we \fnd useful to brie\ry\npresent the concrete arguments that justify the micro-\nscopic model (28) may reproduce a relaxation process\ntowards a thermal equilibrium state despite of this issue.\nB. Properties of the \ructuating force and retarded\nself-energy\nLet us draw attention to the retarded self-energy and\nforce \ructuations governing the quantum Langevin dy-\nnamics in the asymptotic time limit. From equation (47),\nwe may envisage the two-point autocorrelation function\nof the \ructuating force in the time domain for t0\u0014t, i.e.\nDn\n^E\u000b\ni(t);^E\fy\nj(t0)oE\n^\u001a\f=Dn\n^\u0018\u000b\ni(t);^\u0018\fy\nj(t0)oE\n^\u001a\f(52)\n+\u0007\u000b\f\nij(t;t0;t0) +\u0004\u000b\f\nij(t;t0;t0);\nwhere ^\u0018idescribes the previously mentioned thermal\nnoise upon the i-th harmonic oscillator given by (51),\nwhereas the non-stochastic \ructuations \u0007\u000b\f\nij(t;t0;t0) and\n\u0004\u000b\f\nij(t;t0;t0) represent the inverse Fourier transform of\nthe second and third line of the right-hand side of equa-\ntion (47), respectively.\nOwning to the functions (48) and (49) are continu-\nously di\u000berentiable for \u0014\u0014x <1as well as they\nexponentially decay for values xlarger than (2 \u001b)\u00001\n2,\n\u0007\u000b\f\nij(t;t0;t0) and\u0004\u000b\f\nij(t;t0;t0) can be computed for any\n\fnite value t0, though we may need to resort to numeri-\ncal computation methods in most interesting cases. In\nparticular, these \ructuations can be evaluated in the\nasymptotic time limit t\u0000t0!1 by appealing to the\nso-called Riemann-Lebesgue lemma [77], which is illus-\ntrated in Appendix D, and has been employed in the\nstudy of the stationary properties of the damped har-\nmonic oscillator [30, 76]. Essentially, this lemma states\nthat the factor e\u0006i(!\u0006!0)t0appearing in (48) and (49)\nbecomes so highly oscillatory that the integral of the\ncorresponding inverse Fourier transform averages out to\nzero over the bandwidth of the MSC environment. As\na result, it follows from the Riemann-Lebesgue lemma\nthat both\u0007\u000b\f\nij(t;t0;t0) and\u0004\u000b\f\nij(t;t0;t0) asymptotically\nvanishes in the long time limit t\u0000t0!1 . We elab-\norate on this discussion in AppendixD. In this way, the\nasymptotic dynamics of the harmonic n-particle system\nwill be dominated mainly by the thermal \ructuations,\ni.e. ^E\u000b\ni(t)!^\u0018\u000b\ni(t) fort\u0000t0!1 , and thus, the time\nasymptotic dissipative dynamics will follow a \ructuation-\ndissipation relation (46), as we wanted to show. We\nwould like to emphasize that this result is general and\njust rests on the basic properties of the spectral function:\nit exhibits a broad bandwidth, and a \fnite and continu-\nous coupling strength between the MSC environment and\nsystem particles.\nWe pay attention to the properties of the thermal noise\n^\u0018\u000b\ni(t) in what follows. By performing the inverse Fourier12\nFigure 1. (color online). Left: The elements of the retarded self-energy as a function of time. The solid blue and dashed\norange lines correspond respectively to \u000611\nij(t) and\u000622\nij(t), whereas\u000612\nij(t) is represented by the solid black line in the inset.\nRigth: The plot illustrates the elements of the thermal \ructuations in the zero-temperature regime as a function of time. The\ndiagonal correlations for \u000b= 1;2 are given by the solid blue and dashed orange lines, respectively. In the inset, the solid\nblack line depicts the o\u000b-diagonal correlationDn\n^E1\ni(t);^E2y\nj(t0)oE\n. In both pictures, we have \fxed e= 1,p\n2\u001b\u0014'0:01, and\nj\u0001\u0016qijj=p\n2\u001b'0:01.\ntransform in equation (46) after substituting the expres- sion of the retarded self-energy in terms of the spectral\ndensity (45), we obtain\nDn\n^\u0018\u000b\ni(t0+\u001c);^\u0018\fy\nj(t0)oE\n=1\n\u0019Z1\n0d!\u0000\n1 + 2n\u0000\n!;\f\u00001\u0001\u0001\u0010\nRefJ\u000b\f(!;\u0001\u0016qij)gcos(\u001c!) + ImfJ\u000b\f(!;\u0001\u0016qij)gsin(\u001c!)\u0011\n:\n(53)\nObserve that the Chern-Simons e\u000bects give rise to the o\u000b-diagonal contribution contained by the Fourier sine transform\nterm appearing in (53), as similarly occurs for extended Caldeira-Legget environments [54]. This term encodes all the\nthermal \ructuations emerging from the Chern-Simons electric \feld ^ECSacting upon the transversal spatial degrees\nof freedom. In Secs. IV and IV A, it is shown that such contribution may be interpreted as an ordinary Hall response\nassociated to ^ECS, and interestingly, it may generate long-time correlations between the transversal degrees of freedom\nof the system particles in the asymptotic equilibrium state.\nA profound analysis of the self-energy (43) and thermal \ructuations (53) inferred from the MSC environment is\nbeyond the scope of the present work. Rather we will focus the attention to the realistic physical situation when\nthe Chern-Simons action strength is weak, i.e.p\n2\u001b\u0014\u001c1, and all harmonic oscillators are very close to each other\ncompared with the particle charge distribution, i.e.j\u0001\u0016qijjp\n2\u001b\u001c1 for arbitrary iandj. Under these assumptions the\nde\fnition for the spectral density (44) may be brought into the simpli\fed expression,\nJ(!;\u0001\u0016qij)'\u0010e\n4\u00112\ne\u00002\u001b!2 \u00121\n4\u0000\n8\u00142+ 2\u0000\n4\u0000\u00142\u0000\nj\u0001\u0016qijj2\u00008\u001b\u0001\u0001\n!2\u0001\ni4\u0014!\n\u0000i4\u0014!1\n4(8\u00142+ 2(4 +\u00142(j\u0001\u0016qijj2+ 8\u001b))!2)\u0013\n\u0000\u0012\n(1 + 2\u00142\u001b)4i\u0014\n!\u0014\n\u00004i\u0014\n!\u0014 3(1 + 2\u00142\u001b)\u0013\u0012j\u0001\u0016qijj!2\n2\u00132!\n;with 0\u0014!: (54)\nEq.(54) is obtained by using the asymptotic form of the Bessel functions of the \frst kind for small arguments, and\nthen, performed a Taylor series expansion around j\u0001\u0016qijj= 0. Now, replacing (54) in (43) and using the standard\ntables of integration [74], one may obtain closed-form formulas for the retarded self energy with 0 \u0014t. For instance,\nthe o\u000b-diagonal elements takes the following form\n\u000612\nij(t) =\u0000\u000621\nij(t)'e2\u0014\n64\u0019\u001b2\u0010\nj\u0001\u0016qijj2(\u00001 +x2) + 8\u001b+xp\u0019\n2(j\u0001\u0016qijj2(3\u00002x2)\u000016\u001b)e\u0000x2er\f(x)\u0011\n; (55)\nwherex=tp\n8\u001band er\f(x) =i\u00001erf(ix), with erf( x) being the error function in the variable x[74]. The13\ncomputed form for the other terms can be found in the\nappendixE, see equations (E6) and (E7). The compo-\nnents of the retarded self-energy as functions of time\nare depicted in \fgure1. Paying attention to (55), we\nmay observe that the quantum Langevin dynamics of\nthe harmonic n-particle system presents an intricate non-\nMarkovian memory kernel, which exhibits an algebraic\nbehavior at small times, whilst it is dominated by an\nexponential decayment with vanishing time (2 \u001b)\u00001\n2in\nthe long time. Such non-Markovianity is a clear sig-\nnature of a rich dissipative dynamics [9]. Interestingly,\nthe expression for the o\u000b-diagonal component (55) re-\nveals that the \ructuating forces acting upon transversal\nspatial components of the harmonic oscillators do not\ncommute at t= 0, rather it takes a \fnite value pro-\nportional to the so-called topological mass \u0014[46]. Tak-\ning into account Eq.(29), this feature can be traced back\nto the fact that the Maxwell-Chern-Simons electric \feld\n^EMCS(\u0016qi) responsible for the dissipative dynamics has\nnon-commutative components (see Eq.(34)), as pointed\nout in Sec.III. It is well-known in the free Maxwell-\nChern-Simons electrodynamics [46, 47] that such non-\ncommutative property for the electric \felds arises from\nthe latent topological features of the microscopic theory,\nthus this feature of the memory kernel can be thought\nof as a topological trademark in the present dissipative\ndynamics [63].\nFigure 1 also illustrates the thermal \ructuations (53)\nobtained in the zero-temperature limit after replacing the\nspectral density by (54). The exact representation of\nthe diagonal and o\u000b-diagonal elements can be found in\nthe AppendixE (see (E8),(E9) and (E10)). Observe that\nthe time-dependent thermal \ructuations share a similar\nbehavior with the retarded self-energy. Moreover, both\ndiagonal components takes almost on the same values\ndue to the apparent anisotropy of the spectral density\nvanishes for closed particles, as was just discuss in the\nprevious section. Interestingly, we shall see in Sec.IV\nthat the transversal contribution (see the insest) at high\ntemperatures may be identi\fed with the \ructuations of\nan antisymmetric 1 =fnoise in the Markovian Langevin\ndynamics limit.\nIn summary, on one hand we have shown that the\nstationary dissipative e\u000bects characterized by (45) and\nthe stochastic electric force (51) follow a generalized\n\ructuation-dissipation relation (46). Remarkably, the\nretarded self-energy exhibits a non-commutative feature\ncharacteristic of the topologically massive gauge theory.\nOn the other hand, we have seen that the Maxwell-\nChern-Simons \feld induces non-stochastic \ructuations\nwhich may dominate the initial dissipative dynamics, but\nnevertheless their in\ruence on the long-time dynamics\ncan be disregarded, and thus, the system could eventu-\nally reach a thermal equilibrium state determined by the\ninitial gauge-\feld temperature \f\u00001. We shall discuss in\nthe following section under which conditions any initially\nprepared con\fguration of the harmonic n-particle system\ndecays into a thermal state at temperature \f\u00001due tothe interaction with the MCS environment.\nC. Equilibrium structure of propagators\nWe now turn the attention to the equilibrium proper-\nties of the harmonic n-particle system at late times. It\nproves convenient to analyze this by means of the (one-\nparticle) Green's function G\u000b\f\nij(t;t0) =D\nTC^Q\u000b\ni(t)^Q\f\nj(t0)E\nde\fned for a pair of particle position operators ^Qiand\n^Qj, whereTCdenotes the closed time path or Schwinger-\nKeldysh contour [3]. This is also known as the contour-\nordered propagator and may be conveniently expressed\nas follows,\nG\u000b\f\nij(t;t0) =\u0001\u000b\f\nij(t;t0)\u0000i\n2sgnTC(t\u0000t0)\u0003\u000b\f\nij(t;t0);\nin terms of the spectral and statistical correlators, re-\nspectively\n\u0003\u000b\f\nij(t;t0) =iDh\n^Q\u000b\ni(t);^Q\f\nj(t0)iE\n; (56)\n\u0001\u000b\f\nij(t;t0) =1\n2Dn\n^Q\u000b\ni(t);^Q\f\nj(t0)oE\n: (57)\nInterestingly, in a thermal equilibrium state the contour-\nordered propagator becomes time-translational invariant,\ni.e.G\u000b\f\nij(t;t0) =G\u000b\f\nij(t\u0000t0), and more importantly, the\nabove correlators are related such that it is satis\fed the\nKubo-Martin-Schwinger (KMS) boundary condition [3,\n7],\nG\u000b\f\nij(t\u0000t0+i\f)jt<t0=G\u000b\f\nij(t\u0000t0)jt0<t: (58)\nBy means of arguments analogous to derive the long-\ntime behavior of the environmental \ructuations, we shall\nshow that the harmonic n-particle system asymptotically\napproaches to a stationary state retrieving the KMS re-\nlation (58) with \f\u00001being the initial gauge-\feld tem-\nperature, under certain conditions consistent with the\nspectral density (44).\nIn order to evaluate the asymptotic time solution of the\ncontour-ordered propagator we \frstly solve the Cauchy\nproblem for the quantum Langevin equation (38). Since\nthe latter constitutes a linear integral-di\u000berential equa-\ntion, its solution can be straightforwardly obtained via\neither the Laplace or Fourier transform methods [19,\n20, 30]. In this context, the solution can be conve-\nniently expressed in terms of the (Kadano\u000b-Baym) re-\ntarded Green's function matrix GR(t) of the harmonic\nn-particle system [5, 20], whose entries are completely\ndetermined by the time Fourier transform\n\u0010\n~G\u00001\nR\u0011\u000b\f\nij(!) =\u0000\u000eij\u000e\u000b\fm(!+i0+)2+V\u000b\f\nij\u0000~\u0006\u000b\f\nij(!+i0+);\n(59)\nwhere ~G\u00001\nRdenotes the 2 n\u00022nmatrix inverse. Addition-\nally, the homogeneous solution ^ q\u000b\ni;h(t) of equation (38) is14\nobtained from,\nnX\nj=1Zt\nt0d\u001c\u0000\nG\u00001\nR\u0001\u000b\f\nij(t\u0000\u001c)^q\f\nj;h(\u001c) = 0; (60)\nby setting the initial conditions ^ q\u000b\ni;h(t0) = ^q\u000b\ni(t0) and\n_^q\u000b\nih(t0) = _^q\u000b\ni(t0) (which imply GR(t0) =I2nand\n_GR(t0) =m\u00001I2n, withI2nbeing the 2 n\u00022nidentity\nmatrix). Notice that ~GR(\u0000!) = ~Gy\nR(!) thanks to the\nproperties of the retarded self-energy (see Eqs. (42) and\n(45)). Then the solution of equation (38) formally reads,\n^q\u000b\ni(t) = ^q\u000b\ni;h(t) (61)\n+nX\nj=1Zt\nt0(GR)\u000b\f\nij(t\u0000\u001c)\u0010\n^\u0018\f\nj(\u001c)\u0000V\f\nj\u0011\nd\u001c;\nwhere all the stationary dynamics is completely con-\ntained by the second line expression on the right-hand\nside. For a better exposition, we shall disregard the non-\nstochastic force V\f\njfrom the following discussion by do-\ning a local unitary transformation,\n^Q\u000b\ni(t) = ^q\u000b\ni(t) +nX\nj=1Zt\nt0(GR)\u000b\f\nij(t\u0000\u001c)V\f\njd\u001c:\nFrom the expressions (60) and (61) one may see that,\nfor the system particles reach a stationary state inde-\npendent of an initially prepared con\fguration, the re-\ntarded Green function GR(t) must completely decay (ei-\nther algebraically or exponentially) at long time. As pre-\nviously show in Sec.III B, this information is completely\nencoded by its (time) Fourier transform (59). It is known\nfrom previous studies related to the generalized Langevin\nequation for the conventional damped harmonic oscilla-\ntor [19, 20, 76, 78, 79] that the existence of a well-de\fned\nstationary solution entails that ~GR(!) has no an isolated\npole!blying outside of the environment dense spectrum,\nor equivalently, it does not blow up at some value !b\nwith 0< !2\nb< \u00142(one way to see this is by evaluating\nits Fourier transform by means of the Riemann-Lebesgue\nlemma). This immediately implies that the determinant\nof~G\u00001\nR(!) can only have roots !robeying\u00142<!2\nr. By\nsubstituting the frequency-domain expression of the re-\ntarded self-energy in (59), the search of these roots may\nbe rephrased in terms of the eigenvalue problem of the\nreal part of ~G\u00001\nR(!), i.e. Re ~G\u00001\nR(\u0015) =m\u00152I2nwith\nRe\u0010\n~G\u00001\nR\u0011\u000b\f\nij(\u0015) =V\u000b\f\nij\u0000sgn (\u0015)\n2\u0019ImJ\u000b\f\u0000\f\f\u0015\f\f;\u0001\u0016qij\u0001\n\u00001\n2\u0019Hh\nReJ\u000b\f\u0000\nj!j;\u0001\u0016qij\u0001i\n(\u0015);(62)\nwhereI2ndenotes the 2 n\u00022nidentity matrix and sgn\nstands for the usual sign function. Using results borrowed\nfrom the matrix theory [80], the condition under which\nGR(t) asymptotically vanishes could be then expressed\nin the following compact form\nm\u00142I2n<Re~G\u00001\nR(!r); (63)that is, Re ~G\u00001\nR(!r)\u0000m\u00142I2nis a positive-de\fnite matrix\nfor!r. In the particular case of vanishing Chern-Simons\naction, the condition (63) returns the known result for\nthe conventional damped harmonic oscillator [19, 20, 79].\nWhen (63) is obeyed, the dissipative Hamiltonian (28)\nis prevented to have a bound, estable normal mode with\nfrequency!b. Intuitively speaking, the condition (63) en-\nsures that the spectrum of the gauge \feld will well accom-\nmodate the bare frequencies of the harmonic n-particle\nsystem (i.e. \u0014<!i<1=p\n2\u001bfori2f1;ng), and, conse-\nquently, it makes possible an irreversible energy transfer\nfrom the system to the environment, at least in a \fnite\ntime su\u000eciently larger than the natural time scale of the\nsystem particles.\nAs similarly occurs for an intricate non-Markovian\nLangevin dynamics in the conventional Brownian motion,\nit may be rather di\u000ecult to obtain the analytic structure\nof~G\u00001\nR(!) for most cases as manifested by the sophis-\nticated spectral density (44) and expression (45) for the\nretarded self-energy. Concretely, the real part of the re-\ntarded self-energy, which is obtained via the Kramers-\nKronig relation (42), regards a Bessel Hilbert transform\nthat has no analytic expression in general, and thus, one\nhas to resort to numerical methods [81]. Nonetheless,\nthis treatment substantially simpli\fes for the previously\ndiscussed situation when all the system particles are very\nclose to each other and the Chern-Simons action is weak\n(i.e.\u001b\u00142\u001c1 andj\u0001\u0016qijjp\n2\u001b\u001c1 for arbitrary iandj).\nRecall that the positive constraint upon the microscopic\nHamiltonian (30) discussed in Sec.III imposes the consis-\ntency condition (32) as well. As shown previously, the\ndissipation produced by the MCS environment is then\ncharacterized by a sort of super-Ohmic spectral density\n(54). Replacing this in (62) and computing the corre-\nsponding Hilbert transform, it may be veri\fed that there\nis no (non-trivial) solution !rwhich violates (63) pro-\nvided the particle frequencies lies in the MCS environ-\nment spectrum (i.e. \u0014 < !ifori2f1;ng), and further,\nthe subsidiary condition (32) is obeyed. To see this we\nmust realize that the o\u000b-diagonal elements of ~G\u00001\nR(!r)\nrepresent a perturbative contribution to det ~G\u00001\nR(!) by\nvirtue of (32). Hence, we expect that the harmonic os-\ncillators relax towards a thermal equilibrium state in\nthe closed-particle and weak-Chern-Simons-action pic-\nture. In more general scenarios, the condition (63) will be\nsatis\fed depending mainly on the values of the oscillator\nbare frequencies !i, width of the particle charged distri-\nbution\u001b, Chern-Simons constant \u0014, and environmental\ncouplinge.\nOnce the stationary solution is guaranteed (and the\ncondition (63) is fully satis\fed), it is convenient to take\nthe limitt\u0000t0!1 in Eq.(61) and rewrite the stationary\nsolution of the position operator in terms of its Fourier\ntransform, i.e. ^Q\u000b\ni;s(!) =Pn\nl=1\u0010\n~GR\u0011\u000b\r\nil(!)~\u0018\r\nl(!). After\nreplacing this into both the spectral (56) and statisti-\ncal (57) correlators, and averaging over the initial state\n^\u001a\f, we arrive at their Fourier transform ~\u0003\u000b\f\nij(!;!0) and15\n~\u0001\u000b\f\nij(!;!0), respectively. These are directly obtained by\ntaking into account the anti-commutator (45) and com-\nmutator (47) expressions of the \ructuating force in the\nfrequency domain. As a result, we \fnd for the spectral\ncorrelator\n~\u0003\u000b\f\nij(!;!0) = (64)\n= 2i\u000e(!\u0000!0)nX\nl;l0=1\u0014\u0010\n~GR\u0011\u000b\r\nilI~\u0006\r\r0\nll0\u0010\n~Gy\nR\u0011\r0\f\nl0j\u0015\n(!);\nwhich is absent of the non-stochastic \ructuations\n\u0007\u000b\f\nij(t;t0;t0) and\u0004\u000b\f\nij(t;t0;t0). In contrast, the Fourier\ntransform of the latter functions are involved in the ex-\npression for ~\u0001\u000b\f\nij(!;!0). Once again we may evaluate the\ntime asymptotic limit of the statistical correlator by us-\ning the aforementioned Riemann-Lebesgue lemma after\ninterchanging it by the integral in the frequency domain\n(as similarly discussed in the previous section). Recall\nthat ~\u0007\u000b\f\nij(!;!0;t0) and ~\u0004\u000b\f\nij(!;!0;t0) explicitly inherit\nthe fast oscillatory behavior exhibited by (48) and (49),\nsuch that we may drop the non-stochastic \ructuation\ncontribution in the asymptotic limit t\u0000t0!1 once\nwe appeal to the fact that the retarded Green's function\nis integrable according to the condition (63). Thus it can\nbe shown that this yields\n~\u0001\u000b\f\nij(!;!0) =\u0000i\n2\u0000\n1 + 2n\u0000\n!;\f\u00001\u0001\u0001~\u0003\u000b\f\nij(!;!0):(65)\nThis result is in agreement with the previous \fnding\nabout the asymptotic vanishing of the non-stochastic\n\ructuations in SecIII B. Now, it is simple to verify that\nthe KMS relation (58) is recovered from (64) and (65) via\ntheir inverse Fourier transform. Consequently, this certi-\n\fes that the harmonic n-particle system asymptotically\nreaches a thermal equilibrium state regardless its initial\ncon\fguration, as we wanted to show. On the other hand,\nthe KMS relation (58) reveals that the system particle\nundergoes a stationary Gaussian process in the long-time\nlimit completely determined by the statistical propaga-\ntor\u0001\u000b\f\nij(t0+\u001c;t0) =\u0001\u000b\f\nij(\u001c) and its time derivatives.\nThe latter probes that the microscopic Hamiltonian (28)\nin the asymptotic limit provides us with a (numerically)\nsolvable model for the dissipative dynamics of harmonic\nplanar systems.\nIn the following section, we shall focus the atten-\ntion in a non-equilibrium situation of physical interest\nwhich arises when the dissipative dynamics is well de-\nscribed by time-local damping, or equivalently, the re-\ntarded Green's function in the time domain is exclusively\ndominated by an exponential decaying behavior at long\ntime. This is commonly recognized as the Markovian\nregime [22], and has been extensively studied for the\ndamped harmonic oscillator based on the independent-\noscillator model [1, 16, 18, 21, 25], or alternatively, by\nphenomenological stochastic models [14]. We shall illus-\ntrate how this description emerges in the present treat-\nment and brie\ry discuss its validity.IV. EXAMPLE: MARKOVIAN LANGEVIN\nDYNAMICS\nSince the speci\fc form of the spectral density (56) con-\nsists of an algebraic combination of Bessel functions eval-\nuated in square roots and weighted by a Gaussian func-\ntion, we may expect that the retarded Green's function\n(endowed with the condition (63)) may display an intri-\ncate mixture of \"particle\" poles and brunch cut singu-\nlarities in the complex !-plane [5, 19, 20, 79]. It is well\nknown that the latter singularities contribute to the dy-\nnamics with a power-law decaying evolution, whilst the\nformer render the previously mentioned exponential be-\nhavior characteristic of the Markovian dynamics. Now\nwe shall assume that the dissipative dynamics of the har-\nmonicn-particle system is mainly dominated by particle\npoles. Since the latter mathematically represent (sim-\nple) complex poles, the Markovian case corresponds to\nfocus the attention when ( ~GR)\u000b\f\nij(!) takes the form of\na rational function [82]. Formally, this is equivalent to\napproximate ~GR(!) by a Breit-Wigner resonance shape\n[5, 7, 19, 20, 79], i.e. ~GR(!)'~GBW(!+i0+) with\n\u0010\n~G\u00001\nBW\u0011\u000b\f\nij(!) =\u0000\u000eij\u000e\u000b\f!2\u0000i2!\u0000\u000b\f\nij+\u0000\n\n\u000e2\u0001\u000b\f\nij\n(2\u0019m)\u00001Z\u000b\f\nij;\n(66)\nwhere the matrix entries\u0000\n\n\u000e2\u0001\u000b\f\nijand\u0000\u000b\f\nijare given by,\nm\u0000\n\n\u000e2\u0001\u000b\f\nij=V\u000b\f\nij (67)\n\u00001\n2\u0019sgn\u0010\n\n\u000b\f\nij\u0011\nImJ\u000b\f\u0010\f\f\n\u000b\f\nij\f\f;\u0001\u0016qij\u0011\n\u00001\n2\u0019Hh\nReJ\u000b\f\u0000\njxj;\u0001\u0016qij\u0001i\u0010\n\n\u000b\f\nij\u0011\n;\n\u0000\u000b\f\nij=m\u00001Z\u000b\f\nij\n2(2\u0019)\n\u000b\f\nij\u00121\n2\u0019ReJ\u000b\f\u0010\f\f\n\u000b\f\nij\f\f;\u0001\u0016qij\u0011\n(68)\n\u00001\n2\u0019Hh\nsgn(x)ImJ\u000b\f\u0000\njxj;\u0001\u0016qij\u0001i\u0010\n\n\u000b\f\nij\u0011\u0013\n;\nwithZ\u000b\f\nijbeing a renormalization factor,\nZ\u000b\f\nij= 2\u0019\"\n1 (69)\n+1\n2\u0019m@\n@!2\u0012\nHh\nReJ\u000b\f\u0000\njxj;\u0001\u0016qij\u0001i\n(\n\u000b\f\nij)\n+ sgn\u0010\n\n\u000b\f\nij\u0011\nImJ\u000b\f\u0010\f\f\n\u000b\f\nij\f\f;\u0001\u0016qij\u0011\u0013#\u00001\n:\nThe matrix elements \n\u000b\f\nijare obtained from the\nHadamard (entrywise) power\u0000\n\n\u000e2\u0001\u000b\f\nij=\n\u000b\f\nij\n\u000b\f\nij, and\nfurther, they satisfy \u000f\u000b\f(\n\u000e2)\u000b\f\nij=\u000f\f\u000b(\n\u000e2)\f\u000b\nijin order\n(66) preserves the Hermitian property of the equations.\nAdditionally, to be the Breit-Wigner approximation (66)16\nphysically consistent with a Markovian treatment of dis-\nsipative harmonic systems, we demand\n\n\u000b\f\nij\n\n\u000b\u000b\nij<\u0000\u000b\u000b\nij\n\n\u000b\u000b\nij\u001c1 with\u000b6=\f; (70)\nas well as 0 < \u0000\u000b\u000b\nij;\n\u000b\u000b\nijfori;j2 f1;ngand\u000b;\f =\n1;2. Expression (70) re\rects nothing else but the fact\nthat the Makovian dynamics emerges when the system-\nenvironment coupling is weak in comparison with the\nparticle bare frequencies [15, 16]. On the other hand,\nthe left-hand side of (70) is motivated by the subsidiary\ncondition (32) discussed in Sec.III. This point will be\ncleared in the next section when we deal with a concrete\nexample, e.g. the single harmonic oscillator case.\nThe ansatz (66) is inspired by the Breit-Wigner reso-\nnance shape for the scenario of a two-dimensional system\ncomposed of nindependent damped harmonic oscillators.\nFollowing the prescription from Refs.[5, 7], this may be\nobtained by doing a Taylor expansion of the real and\nimaginary parts of the retarded self-energy around the\nparticle pole. Recalling that Im ~\u0006\u000b\f\nij(!) and Re ~\u0006\u000b\f\nij(!)\nare respectively odd and even functions in the frequency\ndomain (according to the Kramers-Kronig relation (42)),\nsuch prescription may be extended to our treatment by\nconsidering the following Taylor expansions near \n\u000b\f\nij,\nIm~\u0006\u000b\f\nij(!)'Im~\u0006\u000b\f\nij\u0010\n\n\u000b\f\nij\u0011\n+4\u0019m\u0000\u000b\f\nij\nZ\u000b\f\nij\u0010\n!\u0000\n\u000b\f\nij\u0011\n;(71)\nRe~\u0006\u000b\f\nij(!)'Re~\u0006\u000b\f\nij\u0010\n\n\u000b\f\nij\u0011\n(72)\n\u0000m \n1\u00002\u0019\nZ\u000b\f\nij!\u0010\n\u000eij\u000e\u000b\f!2\u0000(\n\u000e2)\u000b\f\nij\u0011\n;\nwhere we have de\fned without loss of generality,\n@Im~\u0006\u000b\f\nij(!)\n@!\f\f\f\f\f\n!=\n\u000b\f\nij=4\u0019m\u0000\u000b\f\nij\nZ\u000b\f\nij;\n@Re~\u0006\u000b\f\nij(!)\n@!2\f\f\f\f\f\n!=\n\u000b\f\nij=\u0000m \n1\u00002\u0019\nZ\u000b\f\nij!\n:\nBy replacing equations (71) and (72) in the inverse of the\nretarded Green's function (59), we are lead to the con-\nstraints (67) and (68) by requiring this takes the form of\nthe ansatz (66). Notice that the \fnal expression of these\nconstraints is obtained after substituting the retarded\nself-energy by the spectral density, where the factor 2 \u0019\nessentially appears because the de\fnition of the latter\nconsider here. Similarly, (69) follows directly from (72),\nand corresponds to the so-called wave function renor-\nmalization in the particular case of independent damped\nharmonic oscillators. One may verify that Eqs. (71)\nand (72) retrieve the Breit-Wigner resonance shape for\nntwo-dimensional oscillator following the conventional\nBrownian motion, as desired.By considering the ansatz (66), we obtain a Markovian\ndynamics in agreement with the dissipative properties\nencoded by the spectral density. Indeed, it can be shown\nthat the inverse Fourier transform of (66) returns the\nfollowing Markovian Langevin equation for our harmonic\nn-particle system,\n2\u0019m\nZ\u000b\u000b\nij^Q\u000b\ni(t) + 2\u0019mnX\nj=1X\n\f=1;22\u0000\u000b\f\nij\nZ\u000b\f\nij_^Q\f\nj(t) (73)\n+ 2\u0019mnX\nj=1X\n\f=1;2\u0000\n\n\u000e2\u0001\u000b\f\nij\nZ\u000b\f\nij^Q\f\nj(t) =^\u0018\u000b\ni;BW(t);\nwhere ^\u0018\u000b\ni;BW(t) is the associated thermal noise which is\ndetermined from (53). Aside the contribution from the\nquadratic potential V\u000b\u000b\nijcontained in \n\u000b\u000b\nij(notice that\nV\u000b\f\nij= 0 for\u000b6=\f), one may realize from (73) that\nthe MSC environment in the Markovian limit may me-\ndiate an e\u000bective interaction between transversal spatial\ncomponents that is characteristic of a linear drift force\n[62]. More concretely, the latter is equivalent to an ar-\nray of non-conservative rotational forces acting on the\nharmonic oscillators\n^FR;i=\u00002\u0019mnX\nj=1\n2\nij(^Qj\u0002^e3);withi2f1;ng(74)\nwhere\u0000\n\n\u000e2\u0001\u000b\f\nij=\u000f\u000b\fZ\u000b\f\nij\n2\nijand ^e3denotes the (unit)\nnormal vector to the plane de\fned by the system. Upon\nclose inspection of equation (67), one may see that ^F\u000b\nR;i\nmust exclusively arise from the imaginary part of the\nspectral density (see Eqs. (43) and (44)). The latter\npinpoints the Chern-Simons electric \feld ^ECSas the\nresponsible for (74), which is primarily induced by the\nparticle-attached Chern-Simons \rux discussed in Secs. II\nand III. Interesting enough, this situation is common in\nstatic MCS electrodynamics [70], and invokes a physical\npicture for our dissipative system that recalls an intri-\ncate ensemble of dissipative coupled magnetic-like vorte-\nces: each oscillator follows a rotational symmetric motion\ncarrying certain Chern-Simons \rux that simultaneously\ninteracts between each other with strength determined\nby\nij. A similar vortex-like dynamics was also found\nin a disipative microscopic description of type-II super-\nconductors [83]. To be in agreement with an asymptotic\nstationary picture, there must exist a complex interplay\nbetween these driving forces and the energy dissipated\nby the corresponding environmental noise. Indeed, we\nshow below that the noise associated to (74) via the\n\ructuation-dissipation relation (46) resemblances to a\nmagnetic \rux noise, which reinforces the aforementioned\nvision of \rux-carrying particles. Accordingly, these rota-\ntional forces constitute a perturvative repulsive interac-\ntion (which may globally drift away the system particles)\nthat counteracts the particle con\fning potential.\nAlternatively, the ansatz (66) can be though of as the\nretarded Green's function resulting from the generalized17\nLangevin equation after assuming certain form for the\nspectral density, namely JBW\n\u000b\f, that ful\flls Eqs. (67)\nand (68). In other words, we should be able to deduce\na Markovian Langevin equation identical to the one as-\nsociated to the Breit-Wigner ansatz (73) if we replace\nsuch spectral density JBW\n\u000b\f in the expression (43) for\nthe retarded self-energy. In this way, we could follow\nan inverse line of thinking to \fgure out the speci\fc form\nofJBW\n\u000b\f by requiring the expression (43) reproduces a\ntime-local memory kernel which agrees with (73), i.e.\n\u0006\u000b\f\nij(t\u0000t0)/\u000e(t\u0000t0). On one hand, since the o\u000b-\ndiagonal elements of the spectral density are just con-\ntained in the Fourier cosine transform of (43), the trans-\nformation rules for the latter entails (for arbitrary i;j):\n\u0000\u000b\f\nij= 0 andJBW\n\u000b\f(!;\u0001\u0016qij)/i\n2\nijfor\u000b6=\f. On the\nother hand,JBW\n\u000b\f must be Hermitian to provide a sen-\nsible description as explained in Sec.III A. Introducing\ntogether these results in (67) and (68), we are led to the\nfollowing spectral density associated to the Breit-Wigner\napproximation (66),\nJBW\n\u000b\f(!;\u0001\u0016qij) =\u000e\u000b\f2m\u0019\u0000\u000b\u000b\nij\nZ\u000b\u000b\nij!\u0000i\u000f\u000b\f2\u0019m\n2\nij\n2:(75)\nUsing the Fourier sine and cosine transformation rules,\nit is simple to verify that expression (75) matches with\nthe Markovian Langevin equation (73). Interestingly,\nthe diagonal elements of JBW\n\u000b\f takes the form of an\nOhmic spectral density (which is characteristic of the\nstrict Markovian limit [1]), whilst the o\u000b-diagonal con-\ntribution exhibits a \rat or white spectrum.\nAdditionally, we may obtain the \ructuation-\ndissipation relation determining the corresponding\n\ructuating force ^\u0018\u000b\ni;BW(t) by replacing (75) in the ex-\npression (53). In the high temperature limit \u0000\u000b\u000b\nij\f\u001c1,\nthis takes the particular form\nDn\n^\u0018\u000b\ni;BW(t0+t);^\u0018\fy\nj;BW (t0)oE\n=\u000e\u000b\f2m(2\u0019)\u0000\u000b\u000b\nij\nZ\u000b\u000b\nij\f\u000e(t\u0000t0)\n+\u000f\u000b\f2\u0019m\n2\nij\n2\fsgn(t\u0000t0); (76)\nwhere again we have made use of the well-known proper-\nties of the Dirac delta function and the standard tables\nof Fourier cosine and sine transforms [74].\nThe statistical correlator (76) reveals intriguing fea-\ntures of the \ructuating force acting on the harmonic\nn-particle system at high temperatures, specially for\nthe o\u000b-diagonal correlations arising from the Chern-\nSimons e\u000bects. We may clearly recognize the \frst line\nin (76) as the well-known \ructuation-dissipation rela-\ntion due to the white noise presented in the conven-\ntional Brownian motion [22]. Remarkably, the second\nline closely resemblances to the ordinary Hall response of\ntwo-dimensional particles found in the dissipative Hofs-\ntadter model [55, 56], such that the \ructuations of the\ntransversal spatial components may be interpreted as theHall e\u000bect counterpart in the present context, occurring\nhere because the Chern-Simons \rux. Moreover, the sec-\nond line in the frequency domain identically coincides\nwith the \ructuations of an antisymmetric 1/f noise of\npower spectrum S(!) = (i!)\u00001. The latter may be real-\nized by paying attention to (53): the hyperbolic cotan-\ngent renders an e\u000bective inverse scaling for the power\nspectrum of the transversal correlations, whereas the\nspectral density contributes with a constant factor own-\ning to the form (75). Interestingly, 1 =fpower law is\ncharacteristic of the low-frequency magnetic \rux noise in\nsuperconducting circuits [57, 58], which suggests that the\nenvironmental noise acting upon transversal spatial de-\ngrees of freedom originates from the \ructuations of the\nChern-Simons \rux carried by the harmonic oscillators.\nThe time-reversal asymmetry and parity violation of the\npresent microscopic description can be clearly appreci-\nated from Eqs. (73) and (76). In principle, the latter can\nbe thought of as an extension to the classical Einstein\nrelation [22].\nA \fnal remark in this section is that the spectral den-\nsity (75) cannot provide a fully physical description of the\ndissipative dynamics (and thus neither (73)), as similarly\noccurs for the Ohmic spectral density in the standard mi-\ncroscopic model (i.e. it gives rise to the so-called ultra-\nviolet catastrophe). Here, it is important to emphasize\nthat the Markovian Langevin equation (73) will be valid\nwhen the coupling between the harmonic n-particle sys-\ntem and gauge \feld is weak according to (70). Physically,\nthis could well correspond to the scenario of closed parti-\ncles and weak Chern-Simons action in the weak-damping\nregime, i.e. e=!i\u001c1 and!ip\n2\u001b\u001c1.\nA. Single harmonic oscillator\nSo far we have followed a very general treatment\nof the harmonic n-particle system. Aiming to pro-\nvide a clear comparison with the conventional (two-\ndimensional) isotropic damped harmonic oscillator [22],\nwe now address the asymptotic dissipative dynamics of\nthe single harmonic oscillator case (i.e. n= 1) within the\nprevious Markovian framework. For a better exposition,\nwe take the parameters involved in Eqs. (66) and (75)\nas follows: \n12=\u0000\n21= (2\u0019)\u00001\n2\nCS, while for the\ndiagonal elements \n11=\n22=\n0and\u000011=\u000022=\u00000.\nFurthermore, substituting (76) in (69) yields Z\u000b\u000b= 2\u0019\nfor\u000b;\f = 1;2. Note that we have chosen an isotropic\ndamping rate since the apparent anisotropy of the spec-\ntral density cancels out for the single oscillator case as\ndiscussed in Sec.III A, and further, \nCSdetermines the\nstrength of the rotational forces discussed in the previous\nsection.\nOwing to the rational form of the Breit-Wigner approx-\nimation, the retarded Green's function ~GSO(!) is found\nto decay algebraically as faster as \u0018!\u00003and has no\nbrunch cut. Instead it possesses four complex-conjugate18\nsimple poles, denoted by \u0000i\u0015\u0006, in the!-plane, i.e.\n\u0015\u0006=\u00000\u0006\u0011with\u0011=\u0000\n\u00002\n0\u0000\n2\n0+i\n2\nCS\u00011\n2;(77)\naside its complex conjugate. Hence we may use contour\nintegration methods [82] to obtain its inverse Fourier\ntransform. Note that the contours at in\fnity do not\ncontribute to the \fnal integral due to ~GSO(!) vanishes\nrapidly at large frequency. Hence, it is simple to verify\nthat,\nGSO(\u001c) =\u0002(\u001c)\u0012\nf+(\u0011\u001c)\u0000if\u0000(\u0011\u001c)\nif\u0000(\u0011\u001c)f+(\u0011\u001c)\u0013\n;(78)\nwhere\u001c=t\u0000t0and we have introduced the following\nauxiliary functions\nf\u0006(tz) =e\u0000\u00000t\n2m\u0012sinh(tzy)\nzy\u0006sinh(tz)\nz\u0013\n;\nwhich returns the familiar solution of the Markovian\ndamped harmonic oscillator when \nCS!0. At \frst\nsight, the retarded Green's function (78) does not seem\nto decay at late times, preventing the single harmonic\noscillator to relax towards an equilibrium state. How-\never, one may show that the consistency condition (70)\nguarantees (78) asymptotically vanishes. Concretely, it\nis immediate to see that (78) will exhibit an exponential\ndecayment if the following inequality holds,\nRe\u001aq\n\u00002\n0\u0000\n2\n0+i\n2\nCS\u001b\n<\u00000:\nBy means of identities for the complex square root, this\ninequality is equivalent to \n4\nCS<4\u00002\n0\n2\n0, which is clearly\nsatis\fed by recalling (70), as we wanted to show. Note\nthis result is in complete agreement with the subsidiary\ncondition (32) derived in Sec.III.\nNow we study the position autocorrelation function\ndenoted by \u0001\u000b\u000b\nSO(t) and de\fned in (57). As before, this\nmay be obtained via equations (64) and (65) by using\nstandard contour integration techniques to compute the\ncorresponding inverse Fourier transform, once we have re-\nplaced the single-oscillator expressions for both the spec-\ntral density (76) and retarded Green's function ~GSO(!).\nSpeci\fcally, we arrive at the following identities\n\u000111\nSO(t) =\u000122\nSO(t)\n= RefS0(t) +SCS(t)g+4\u00000\nm\f_S(q)\n0(t);(79)\nwhere we have introduced,\nS0(t) =i\u0016+\n2m\u0016\u0000(\u0015+\u0000\u0015\u0000)\u0012\ncoth\u0012i\f\u0015\u0000\n2\u0013\ne\u0000\u0015\u0000t\n\u0000coth\u0012i\f\u0015+\n2\u0013\ne\u0000\u0015+t\u0013\n;\nSCS(t) =2\u00000\n2\nCS\nm\u0016\u0000(\u0015+\u0000\u0015\u0000)\u0012\n\u0015+coth\u0012i\f\u0015\u0000\n2\u0013\ne\u0000\u0015\u0000t\n\u0000\u0015\u0000coth\u0012i\f\u0015+\n2\u0013\ne\u0000\u0015+t\u0013\n; (80)with\u0016\u0006= 4\u00002\n0\n2\n0\u0006\n4\nCS, and the quantum corrections,\nS(q)\n0(t) =1X\nn=1e\u0000\u0017nt\u0010\u0000\n\u00172\nn+\n2\n0\u00012+ 3\n4\nCS\u00004\u00002\n0\u00172\nn\u0011\n(\u00172n\u0000\u00152\n+)(\u00172n\u0000\u00152\n\u0000)(\u00172n\u0000\u0015y2\n+)(\u00172n\u0000\u0015y2\n\u0000);\nS(q)\nCS(t) =1X\nn=1e\u0000\u0017nt\u0010\u0000\n\u00172\nn+\n2\n0\u00012+\n4\nCS\u000012\u00002\n0\u00172\nn\u0011\n(\u00172n\u0000\u00152\n+)(\u00172n\u0000\u00152\n\u0000)(\u00172n\u0000\u0015y2\n+)(\u00172n\u0000\u0015y2\n\u0000);\n(81)\nwith\u0017nbeing the positive bosonic Matsubara frequen-\ncies, i.e.\u0017n= 2\u0019n=\f . It is straightforward to show\nthat Eq.(79) returns the well-known results of the Marko-\nvian damped harmonic oscillator [14] for zero Chern-\nSimons action (i.e. \nCS!0). From expression (79)\nis immediate to get the mean-square dispersion of the\nharmonic oscillator position along one dimension, i.e.D\n^Q2E\n\f=\u000111\nSO(0) =\u000122\nSO(0). To evaluate the classi-\ncal and quantum limit, it is convenient to rewrite (79) in\nterms of the psi function  (x), which is the logarithmic\nderivative of the gamma function. In particular the quan-\ntum corrections _S(q)\n0(0) could be written as a combination\nof (x) by following a similar procedure to Ref.[14].\nIn the high temperature limit ( \u00000\f\u001c1), the quantum\ncorrections vanishes as can be clearly seen from equation\n(81). After some straightforward manipulation we then\narrive at,\nD\n^Q2E\n\f=\n4\nCS\n2\n0+ 4\u00002\n0\u0000\n\n4\n0+ 2\n4\nCS\u0001\nm\f(4\u00002\n0\n2\n0\u0000\n4\nCS) (\n4\n0+\n4\nCS)(82)\n=1\nm\f\n2\n0 \n1 +\u0012\n1 +\n2\n0\n2\u00002\n0\u0013\u0012\nCS\n\n0\u00134!\n+O \u0012\nCS\n\n0\u00138!\n;\nwhere the \frst term coincides identically with the clas-\nsical correlation function of the damped harmonic oscil-\nlator [14]. Clearly, the above expression shows that the\nChern-Simons action in the Markovian limit induces an\ne\u000bective broadening and shifting of the harmonic oscil-\nlator spectrum, so that the two-dimensional particle is\nexpected to be in a thermal equilibrium distribution in-\ncorporating these e\u000bects.\nAlthough (82) manifests that the Chern-Simons e\u000bects\nmay be eventually negligible in the classical dynamics,\nthis result may signi\fcantly diverse from the quantum\nscenario. In the zero temperature limit \f\u00001= 0, we\nconsider the asymptotic expression of the psi function\nfor large arguments, i.e.  (1 +z)\u0019logzfor 1\u001cz. It\ncan be shown that this leads to\nD\n^Q2E\n\f=log\u0012\n\u00000+p\n\u00002\n0\u0000\n2\n0\n\u00000\u0000p\n\u00002\n0\u0000\n2\n0\u0013\n2\u0019mp\n\u00002\n0\u0000\n2\n0+1\n2m\u00000\u0012\nCS\n\n0\u00132\n+O \u0012\nCS\n\n0\u00134!\n; (83)19\nFigure 2. (color online). The position cross-correlation as a\nfunction of time. The solid balck, dashed blue and dot-dashed\nred lines correspond to the temperatures \f\u00001= 0:01\n0,\n\f\u00001=\n0=2, and\f\u00001=\n0, respectively. We have \fxed\nthe rest of parameters as follows: \n0= 10,\u00000= 0:1\n0,\n\nCS=\u00000=2, andm= 1.\nwhere again the \frst term of the right-hand side identi\fes\nwith the quantum result for the mean-square dispersion\nfound in the damped harmonic oscillator [1, 13, 14, 25].\nUnlike the previous classical limit, this result underlines\nthat the Chern-Simons e\u000bects may substantially in\ru-\nence the Brownian motion, and could be interpreted as\nan \"hyper\fne\" structure of the two-dimensional dissipa-\ntive dynamics.\nFinally, we evaluate the position cross-correlation\n\u000112\nSO(t) between the transversal spatial degrees of free-\ndom. We can follow an identical procedure as to compute\n(79). We \fnd for 0 <t\n\u000112\nSO(t) = ImfS0(t) +SCS(t)g+2\n2\nCS\nm\fS(q)\nCS(t)\n+2\n2\nCS\n\u0019m\fZ1\n0d!sin(!t)\u0010\u0000\n!2\u0000\n2\n0\u00012+\n4\nCS+ 12\u00002\n0!2\u0011\n!(!2+\u00152\n+)(!2+\u00152\n\u0000)(!2+\u0015y2\n+)(!2+\u0015y2\n\u0000);\n(84)\nwhich vanishes for zero Chern-Simons action. Figure2\nillustrates the short-time behavior of the position cross-\ncorrelation at di\u000berent temperatures.\nFrom Eq.(84) we may also evaluate the long-time be-\nhavior of\u000112\nSO(t). Paying attention to Eqs. (80) and\n(81), it is readily to see that in the high temperature\nlimit\u00000\f\u001c1 the position cross-correlation (84) will de-\ncrease exponentially with a ratio given by \u00000at long time,\nsince the quantum corrections decay as e\u0000\f\u00001t. However,\nthis scenario drastically changes in the low temperature\nregime where the quantum corrections dominate the long\ntime behavior. In particular, in the zero temperature\nlimit the quantum corrections sum up to an algebraiclong-time behavior, i.e\n\u000112\nSO(t)\u0018\n2\nCS\nm\u0019(\n4\n0+\n4\nCS)1\nt=1\nm\u0019\n2\n01\nt\u0012\nCS\n\n0\u00132\n+O \u0012\nCS\n\n0\u00136!\n;fort!1: (85)\nTo obtain the above result we evaluate the in\fnite sum\ninvolved in the quantum corrections S(q)\nCS(t) after expand-\ning the rational part of its arguments about the origin\n\u0017n!0, and then taking the strict zero-temperature\nlimit. Interestingly, expression (85) shows that the\ntransversal spatial components exhibit long-time corre-\nlations in the quantum regimen. Let us stress that this\nfeature is a consequence of the quantum Hall response\npreviously discussed in Sec.IV and has no counterpart\nin the conventional Brownian motion, rather the latter\ngenerates a similar time algebraic decayment just for the\nposition autocorrelation functions [14] (i.e. \u0001\u000b\u000b\nSO(t)vt\u00002\nfor\u000b= 1;2). Thus we see from Eqs. (83) and (85) that\nthe dissipative dynamics arising from the Chern-Simons\naction in the Markovian limit constitutes a second-order\ncorrection to the damped harmonic oscillator in the quan-\ntum regime.\nV. OUTLOOK AND CONCLUDING REMARKS\nAdopting a combined gauge \feld and open-system the-\noretical view, we have shown that the non-relativistic\nMaxwell-Chern-Simons electrodynamics leads to a novel\nmicroscopic model for the non-equilibrium dynamics of\nplanar harmonic systems that ful\flls the essential ingre-\ndients of a (linear) non-anomalous dissipative descrip-\ntion: local U(1) gauge invariance and relaxation towards\na thermal equilibrium state, as well as it encompasses\nthe conventional Brownian motion as a particular in-\nstance (i.e. the deduced microscopic Hamiltonian ex-\nactly maps on the independent-oscillator model in the\nlimit of zero Chern-Simons constant). Speci\fcally, we\nhave shown that the symmetry structure provided by the\nChern-Simons action has two main e\u000bects in the long-\nwavelength regime: a gap environmental spectrum, and\na particle-attached magnetic-like \rux which yields an or-\ndinary Hall response of the harmonic oscillators. As a\ncounterpart, these come along with a backreaction on\nthe environment and renormalized potential interaction\nof the system particles. Importantly, if we disregard the\nlatter e\u000bects, the conceived dissipative model provides a\nlegitimate description for the Brownian motion of free\nparticles as well, owing to the fact that the microscopic\nHamiltonian turns identically into a minimal-coupling\ntheory with a gauge \feld bearing the basic dissipative\nmechanism. It is also worthwhile to remark that, though\nwe have considered a Gaussian particle charge distribu-\ntion, this condition could be substantially relaxed to con-\ntain a broad class of form factors (e.g. we could chose\na well-behaved function which falls to zero at su\u000eciently20\nlarge distances) without changing the overall properties\nof the dissipative dynamics, since such choice mainly af-\nfects the cuto\u000b factor of the spectral density.\nIn the Markovian regime, we have found that the ex-\nplicit in\ruence of the Chern-Simons action at the level\nof the Langevin equation is twofold: an additional ro-\ntational force and magnetic-like \rux noise acting upon\nthe harmonic oscillators; such that the system could be\nregarded as an ensemble of interacting dissipative vor-\ntex particles, unlike the conventional Brownian motion.\nMoreover, for the single harmonic oscillator case we have\nprovided concise expressions for the mean-square disper-\nsion in the high- and zero- temperature limit. Interest-\ningly, from the latter follows that the Chern-Simons ac-\ntion can be though of as a second-order correction to the\nwell-known damped harmonic oscillator model, from an\nopen-system theory perspective.\nFrom an experimental point of view, many results\nof planar condensed matter systems predicted by the\nMaxwell-Chern-Simons electrodynamics are still chal-\nlenging to be experimentally tested. Nonetheless, there\nexist several examples, for instance in the context of cold\nRydberg atoms [75], where this theory reproduces the\ntrue electromagnetic interaction. In particular, it was\nprobed in the Ref.[84] that an e\u000bective Maxwell-Chern-\nSimons description naturally emerges in the study of sys-\ntems composed of charged particles constrained to move\non an in\fnite plane and subjected to an ordinary elec-\ntromagnetic interaction, like in the quantum Hall e\u000bect.\nConcretely, it was shown that the Chern-Simons kinetic\nterm may arise from the underlying topological struc-\nture of the (3+1)-dimensional electrodynamics by dimen-\nsional reduction [84], which emphasize the genuine topo-\nlogical nature of the Chern-Simons action [45]. Together\nwith the previous discussion, we could draw the con-\nclusion that the proposed description could be tested in\nnear-future two-dimensional experiments addressing the\ndissipative dynamics induced by electric and magnetic\n\felds.\nIn the last decade the quantum physics of two-\ndimensional systems have attracted a renewed interest\n(e.g. in quantum computation theory), for which the\nMaxwell-Chern-Simons theory proved to be useful in thedescription of a great diversity of phenomena connected\nto condensed matter systems. Similarly, the proposed\nmicroscopic description could provide a theoretical test-\ning ground for new ideas in two-dimensional quantum\nthermodynamics, optics or information theory (e.g. as\noccurs for the independent-oscillator model). In partic-\nular, it is appealing to further investigate which are the\nnonequilibrium thermodynamic properties of the rising\n\rux-carrying Brownian particles.\nACKNOWLEDGMENTS\nThe author is grateful to L.A. Correa, S. Kohler, I. de\nVega and D. Alonso for fruitful discussions. The author\nwarmly thanks J.J. Garc\u0013 \u0010a-Ripoll and M.S. Kim for their\nsupport through the development of the present work.\nThe author acknowledges \fnancial support from Phys-\nical Science Research Council (project EP/K034480/1)\nand the Air Force O\u000ece of Scienti\fc Research (project\nFA2386-18-1-4019).\nAppendix A: Canonical Hamiltonian in the Coulomb\ngauge\nIn this Appendix we brie\ry illustrate the derivation\nof the canonical Hamiltonian (10) presented in Sec.II.\nFirst, let us indicate that we enforce A0as a Lagrange\nmultiplier in 6 [46, 66], in order to satisfy the Coulomb\nlaw constraint (5). It is easy to verify that equation (7)\nis a solution of the latter. Now, we replace this in (6)\nonce we have rewritten the \feld canonical variables in\nterms of their longitudinal and transversal parts. Doing\nthis, we obtain two terms coming from the dot and cross\nproducts, respectively,\nI1=Z\ndx\u0005k\u0001\u0005k; I 2=\u0000\u0014Z\ndx\u0005k\u0002A?:\nOwing to the boundary properties of the Coulomb\nGreen's function, gauge \feld, and charged distribution\nfunction (i.e.rGc(x)!0,@\u000bA?\n\f(x)!0, and\u001a(x)!0\nforjxj!1 ), both terms may be manipulated as follows,\nI1=Z\ndyZ\ndy0\u0010\n\u001a(y)\u0000\u0014\n2r\u0002A?(y)\u0011\u0010\n\u001a(y0)\u0000\u0014\n2r\u0002A?(y0)\u0011Z\ndx@\u000bGc(x\u0000y)@\u000bGc(x\u0000y0)\n=\u0000Z\ndxZ\ndy\u0010\n\u001a(y)\u0000\u0014\n2r\u0002A?(y)\u0011\nGc(x\u0000y)Z\ndy0r2\nxGc(x\u0000y0)\u0010\n\u001a(y0)\u0000\u0014\n2r\u0002A?(y0)\u0011\n=\u0000Z\ndxZ\ndy\u0000\n\u001a(y)\u0000\u0014r\u0002A?(y)\u0001\nGc(x\u0000y)\u001a(x)\u0000\u00142\n4Z\ndxZ\ndyr\u0002A?(x)Gc(x\u0000y)r\u0002A?(y)\n=\u0000Z\ndxZ\ndy\u0000\n\u001a(y)\u0000\u0014r\u0002A?(y)\u0001\nGc(x\u0000y)\u001a(x) +\u00142\n4Z\ndxZ\ndy0G(x0)\u000f\u000b\f\u000f\u000b0\f0A?\n\f0(x)@x\n\u000b@x\n\u000b0A?\n\f(x+y0);21\nas well as\nI2=\u0000\u0014Z\ndxZ\ndy0Gc(y0)\u000f\u000b\fA?\n\f(x)@x\n\u000b\u0010\n\u001a(x+y0)\u0000\u0014\n2r\u0002A?(x+y0)\u0011\n=\u0014Z\ndxZ\ndyr\u0002A?(x)G(x\u0000y)\u001a(y) +\u00142\n2Z\ndxZ\ndy0Gc(y0)\u000f\u000b\f\u000f\u000b0\f0A?\n\f0(x)@x\n\u000b@x\n\u000b0A?\n\f(x+y0):\nNow we may go further by gathering together the above results,\nI1+I2=\u0000Z\ndxZ\ndy\u0000\n\u001a(y)\u00002\u0014r\u0002A?(y)\u0001\nGc(x\u0000y)\u001a(x)\n+3\u00142\n4Z\ndxZ\ndyGc(x\u0000y)(\u000e\u000b\u000b0\u000e\f\f0\u0000\u000e\u000b\f0\u000e\u000b0\f)A?\n\f(x)@y\n\u000b0@y\n\u000bA?\n\f0(y)\n=\u0000Z\ndxZ\ndy\u0000\n\u001a(y)\u00002\u0014r\u0002A?(y)\u0001\nGc(x\u0000y)\u001a(x) +3\u00142\n4Z\ndxZ\ndyr2\nyGc(x\u0000y)A?\n\u000b(x)A?\n\u000b(y)\n=\u0000Z\ndxZ\ndy\u0000\n\u001a(y)\u00002\u0014r\u0002A?(y)\u0001\nGc(x\u0000y)\u001a(x) +3\u00142\n4Z\ndxA?\u0001A?;\nwhere we have made use of the fact @y\n\u000b0A?\n\u000b0(y) = 0 ac-\ncording to the Coulomb gauge \fxing. By substituting\nthis result in the Hamiltonian expressed in terms of the\nlongitudinal and transversal parts, we arrive at\n^H=nX\ni=1\u00121\n2m\u0012\n^pi\u0000eZ\nd2x'(x\u0000^qi)^A?(x)\u00132\n+V(^qi)\u0013\n\u00001\n2Z\nd2xd2y\u001a(x)Gc(x\u0000y)\u0010\n\u001a(y)\u00002\u0014r\u0002 ^A?(y)\u0011\n+^HMCS; (A1)\nwhere ^HMCS is de\fned in Sec.II. It is immediate to see\nthat the canonical Hamiltonian (10) is obtained after re-\nplacing the Coulomb Green's function and the charge\ndensity in (A1).\nAppendix B: Derivation of the positive condition\nHere we compute the positive constraint (30) presented\nin Sec.III, as well as we provide explicit expressions for\nthe potentials (26) and (27). First, we illustrate the basic\nprocedure by starting to compute the quadratic potential\nV\u000b\f\nij. To carry out the discrete sum in k, it is convenient\nto take the dense spectrum limit after switching the in-\ntegral to polar-coordinate variables,\nV\u000b\f\nij=\u000eij\u000e\u000b\fm!2\ni+e2\n8\u00192Z1\n0dk k e\u00002\u001bk2\n\u0002\u0012Z\u0019\n\u0000\u0019d\u0012eikj\u0016qi\u0000\u0016qjjcos\u0012\u0010\n\u000e1\u000b\u000e1\fsin2\u0012+\u000e2\u000b\u000e2\fcos2\u0012\n\u0000(1\u0000\u000e\u000b\f) cos\u0012sin\u0012\u0011\n+c:c:\u0013\n;\nwhere we have rewritten jkj=k, and\u0012is chosen to be\nthe azimuthal angle of the vector kand the axis de\fned\nby\u0001\u0016qij. The above equation is further simpli\fed byintroducing the de\fnition of the Bessel function of \frst\nkind, i.e.\nV\u000b\f\nij=\u000eij\u000e\u000b\fm!2\ni+e2\n2\u0019Z1\n0dk k e\u00002\u001bk2(B1)\n\u0002\u0012\n\u000e\u000b\fJ1(kj\u0016qi\u0000\u0016qjj)\nkj\u0016qi\u0000\u0016qjj\u0000\u000e2\u000b\u000e2\fJ2(kj\u0016qi\u0000\u0016qjj)\u0013\n:\nFortunately, the integral involving the Bessel functions\ncan be exactly obtained by making use of the integration\ntables (seex6.618 andx6.631 in Ref[74]),\nV\u000b\f\nij=\u000eij\u000e\u000b\fm!2\ni\n+e2e\u0000j\u0016qi\u0000\u0016qjj2\n16\u001b\n4\u0019p\n2\u001bj\u0016qi\u0000\u0016qjj\u0012\n\u000e\u000b\fp\u0019I1\n2\u0012j\u0016qi\u0000\u0016qjj2\n16\u001b\u0013\n\u0000\u000e2\u001b\u000e2\fM1\n2;1\u0012j\u0016qi\u0000\u0016qjj2\n8\u001b\u0013\u0013\n; (B2)\nwhereIndenotes the n-order modi\fed Bessel function of\nthe \frst kind and M1\n2;1is the Whitakker function [74].\nNote that the above expression is valid for 0 <j\u0016qi\u0000\u0016qjj\nand 0<\u001b.\nNow, paying attention to the Hamiltonian (28), it is\nclear that the latter will be a positive-de\fnite operator\nas long as the following quadratic expression is satis\fed,\n1\n2nX\nj;i=1\u0010\nV\u000b\f\nij\u0000X\nk!(k)(h\u000b(k;\u0016qi)hy\n\f(k;\u0016qj) +c:c:)\u0011\n^q\u000b\ni^q\f\nj\n+nX\ni=1V\u000b\ni^q\u000b\ni\u0000nX\ni;j=1X\nkP\u000b\f(k)\n!(k)gy\n\u000b(k;\u0016qi)g\f(k;\u0016qj)\u00150:\n(B3)\nLet us address the second term in the \frst line of equa-\ntion (B3). Firstly, by replacing the expression of the\nsystem-environment coupling coe\u000ecient (25), we obtain22\nafter some algebra\nh\u000b(k;\u0016qi)hy\n\f(k;\u0016qj) =e2eik\u0001(\u0016qi\u0000\u0016qj)e\u00002\u001bjkj2\n8\u00192L2!(k)\u0012\nP\u000b\f(k)\n+\u00142k\u000bk\f\njkj4!2(k)\u0000\njkj2\u0000k\rk\u000eP\r\u000e(k)\u0001\n+i\u0014\u000f\r\u000ek\r\njkj2!(k)(k\fP\u000b\u000e(k)\u0000k\u000bP\f\u000e(k))\u0013\n:\nInserting this result and the dispersion relation (15)\nyields\nX\nk!(k)h\u000b(k;\u0016qi)hy\n\f(k;\u0016qj) =e2\n4\u0019Z1\n0dk ke\u00002\u001bk2\n\u0002\u0014\u0012\u0012\n\u000e\u000b\f\u0000\u000e1\u000b\u000e1\f\u0012\n1\u0000\u00142\n(k2+\u00142)\u0013\u0013\nJ0(kj\u0016qi\u0000\u0016qjj)\n+ (\u000e\u000b\f\u00002\u000e2\u000b\u000e2\f)k\n(k2+\u00142)J1(kj\u0016qi\u0000\u0016qjj)\nj\u0016qi\u0000\u0016qjj\u0013\n\u0000i\u000f\u000b\f\u0014J0(kj\u0016qi\u0000\u0016qjj)p\nk2+\u00142\u0015\n:\nWe can analytically computed the above integral via\nstandard contour integration techniques by writing the\nBessel function as a combination of the Hankel functions\nand noting that the functions within the integral has sim-\nple poles\u0006i\u0014[85], i.e.\nX\nk!(k)(h\u000b(k;\u0016qi)hy\n\f(k;\u0016qj) +c:c:) = (B4)\n=\u000e2\u000b\u000e2\fe2\n8\u0019\u001be\u0000j\u0016qi\u0000\u0016qjj2\n8\u001b\n+e2e2\u001b\u00142\n8\u0014\u0014(\u000e\u000b\f\u00002\u000e1\u000b\u000e1\f)\nj\u0016qi\u0000\u0016qjj\u0012\nH(2)\n1(\u0000j\u0016qi\u0000\u0016qjji\u0014)\n+H(1)\n1(j\u0016qi\u0000\u0016qjji\u0014)\u0013\n+\u000e1\u000b\u000e1\fi\u00142\u0012\nH(1)\n0(j\u0016qi\u0000\u0016qjji\u0014)\n\u0000H(2)\n0(\u0000j\u0016qi\u0000\u0016qjji\u0014)\u0013\u0015\n:\nSimilarly, the previous procedure can be replicated to\nobtain the potential (56). That is,\nX\nkh\u000b(k;\u0016qi)gy\n\f(k;\u0016qj)\"y\n\f(k) =e2\u0014\n8\u00192Z1\n0dk ke\u00002\u001bk2\n\u0002Z\u0019\n\u0000\u0019d\u0012eikj\u0016qi\u0000\u0016qjjcos\u0012\nk2p\nk2+\u00142\u0012\n\u000f\r\u000bk\r\u0000i\u0014k\u000bp\nk2+\u00142\u0013\n=e2\u0014\n4\u0019Z1\n0dke\u00002\u001bk2J1(kj\u0016qi\u0000\u0016qjj)p\nk2+\u00142 \n\u0000\u0014\u000e1\u000bp\nk2+\u00142+i\u000e2\u000b!\n:\nand thus,\nV\u000b\ni=nX\nj=1 \n2 (V0\nc(\u0016qi\u0000\u0016qj))\u000b\n\u0000\u000e1\u000be2\u00142\n2\u0019Z1\n0dke\u00002\u001bk2\nk2+\u00142J1(kj\u0016qi\u0000\u0016qjj)!\n(B5)On the other hand, we \fnd for the independent term of\nthe quadratic form (B3),\nX\nkP\u000b\f(k)\n!(k)g\u000b(k;\u0016qi)g\f(k;\u0016qj) =\n=e2\u00142\n4\u0019Z1\n0dke\u00002\u001bk2\nk(k2+\u00142)J0(kj\u0016qi\u0000\u0016qjj) (B6)\nFinally, the positive constraint (30) is directly obtained\nby replacing (B4), (B5), and (B6) in (B3).\nAppendix C: Retarded self-energy and spectral\ndensity\nIn this appendix we illustrate the derivation of the re-\ntarded self-energy (43) and (\feld) spectral density (56),\nand the Kramers-Kronig relation (42) and (45) presented\nin Sec.III A.\nWe start from computing the anti-commutator appear-\ning in the de\fnition of the retarded self-energy (39) by\nusing the following expression of the pseudo-electric \feld\nwhich follows from (29),\n^E\u000b\ni(t) =X\nk!(k)\u0010\nh\u000b(k;\u0016qi)e\u0000i!(k)(t\u0000t0)^ad(k;t0) +c:c:\u0011\n;\n(C1)\nwhere ^ad(k;t0) are the environmental quasiparticle op-\nerators at the initial time after introducing the displace-\nment produced by the backreaction e\u000bects discussed in\nSec.III. By considering an equilibrium canonical initial\nstate ^\u001a\f, we obtain the following statistics for the dis-\nplaced quasiparticle operators\nDn\n^ad(k;t0);^ay\nd(k0;t0)oE\n^\u001a\f=\n=\u000e(k\u0000k0)(1 + 2n(!(k);\f\u00001))\n+ 2nX\ni;j=1\"\u000b(k)\"y\n\f(k0)\n!(k)!(k0)g\u000b(k;\u0016qi)gy\n\f(k0;\u0016qj);\nhf^ad(k;t0);^ad(k0;t0)gi^\u001a\f=\n= 2nX\ni;j=1\"\u000b(k)\"\f(k0)\n!(k)!(k0)g\u000b(k;\u0016qi)g\f(k0;\u0016qj);(C2)\nwithn(!(k);\f\u00001) denoting the average quasiparticle\nnumber of the free MCS \feld at temperature \f\u00001,\nn(!(k);\f\u00001) =1\ne\f!(k)\u00001: (C3)\nNow from (C1) we obtain23\n\u001ch\n^E\u000b\ni(t);^E\fy\nj(t0)i\n\u0006\u001d\n^\u001a\f=X\nk;k0!(k)!(k0) \nh\u000b(k;\u0016qi)hy\n\f(k0;\u0016qj)e\u0000i(!(k)t\u0000!(k0)t0\u0000(!(k)\u0000!(k0))t0)\u001ch\nad(k;t0);ay\nd(k0;t0)i\n\u0006\u001d\n^\u001a0\n+h\u000b(k;\u0016qi)h\f(k0;\u0016qj)e\u0000i(!(k)t+!(k0)t0\u0000(!(k)+!(k0))t0)\n[ad(k;t0);ad(k0;t0)]\u0006\u000b\n^\u001a0\n+hy\n\u000b(k;\u0016qi)hy\n\f(k0;\u0016qj)ei(!(k)t+!(k0)t0\u0000(!(k)+!(k0))t0)\u001ch\nay\nd(k;t0);ay\nd(k0;t0)i\n\u0006\u001d\n^\u001a0\n+hy\n\u000b(k;\u0016qi)h\f(k0;\u0016qj)ei(!(k)t\u0000!(k0)t0\u0000(!(k)\u0000!(k0))t0)\u001ch\nay\nd(k;t0);ad(k0;t0)i\n\u0006\u001d\n^\u001a0!\n: (C4)\nwhere [\u000f;\u000f]\u0006represents the anti-commutator and commutator in a short-hand notation, respectively. By taking the\ndense spectrum limit after some straightforward manipulation in the anti-commutator expression (C4), we arrive at\nDh\n^E\u000b\ni(t);^E\fy\nj(t0)iE\n^\u001a\f=X\nk;k0!(k)!(k0)\u0010\ne\u0000i(!(k)t\u0000!(k0)t0\u0000t0(!(k)\u0000!(k0)))h\u000b(k;\u0016qi)hy\n\f(k0;\u0016qj)h\nad(k;t0);ay\nd(k0;t0)i\n\u0000\n+ei(!(k)t\u0000!(k0)t0\u0000t0(!(k)\u0000!(k0)))hy\n\u000b(k;\u0016qi)h\f(k0;\u0016qj)h\nay\nd(k;t0);ad(k0;t0)i\n\u0000\u0011\n=\u00002iX\nk!2(k)\u0012\nRen\nh\u000b(k;\u0016qi)hy\n\f(k;\u0016qj)o\nsin(!(k)(t\u0000t0))\u0000Imn\nh\u000b(k;\u0016qi)hy\n\f(k;\u0016qj)o\ncos(!(k)(t\u0000t0))\u0013\n=\u00002i\n\u0019Z1\n0d!\u0012\nRe\b\nJ\u000b\f(\u0001\u0016qij;!)\t\nsin(!(t\u0000t0)) + Im\b\nJ\u000b\f(\u0001\u0016qij;!)\t\ncos(!(t\u0000t0))\u0013\n; (C5)\nwhereJ\u000b\frepresents the spectral density. It is immediate to see that we get expression (43) after substituting (C5)\nin the de\fnition (39). In the above equation, we introduced the following expression for the spectral density,\nJ\u000b\f(\u0001\u0016qij;!) =\u0019X\nk!2(k)\u0010\nRen\nh\u000b(k;\u0016qi)hy\n\f(k;\u0016qj)o\n\u0000iImn\nh\u000b(k;\u0016qi)hy\n\f(k;\u0016qj)o\u0011\n\u000e(!(k)\u0000!)\n=e2!\n8\u0019L2X\nkeik\u0001\u0001\u0016qije\u00002\u001bjkj2\u0012\n\u000e\u000b\f\u0000k\u000bk\f\njkj2+\u00142k\u000bk\f\njkj2!2\u0000i\u0014\njkj2!(\u000f\r\u000bk\rk\f\u0000\u000f\r\fk\rk\u000b)\u0013\n\u000e(!(k)\u0000!)\n=e2!2\n8\u0019Z1\n0dk ke\u00002\u001bk2\u000e\u0000\nk\u0000p\n!2\u0000\u00142\u0001\np\n!2\u0000\u00142Z\u0019\n\u0000\u0019d\u0012 eikj\u0001\u0016qijjcos\u0012W\u000b\f(k;\u0012); (C6)\nwhere we have replaced the expression for the system-environment coupling coe\u000ecients (25). In the last line of (C6),\nthe integral is expressed in terms of the polar-coordinate variables ( k=jkj;\u0012) (where\u0012is the azimuthal angle as\nbefore), and the following matrix,\nW\u000b\f(k;\u0012) =0\n@1 +\u0010\n\u00001 +\u00142\n!2\u0011\ncos2\u0012\u0010\n\u00001 +\u00142\n!2\u0011\ncos\u0012sin\u0012+i\u0014\n!\u0010\n\u00001 +\u00142\n!2\u0011\ncos\u0012sin\u0012\u0000i\u0014\n!1 +\u0010\n\u00001 +\u00142\n!2\u0011\nsin2\u00121\nA:\nTo get (C6) (after taking the dense spectrum limit) we employed the dispersion relation (15) of the Maxwell-Chern-\nSimons gauge \feld combined with the properties from the Dirac delta function[44], i.e. \u000e(!(k)\u0000!0) =!0(!02\u0000\n\u00142)\u00001\n2\u000e\u0000\nk\u0000p\n!02\u0000\u00142\u0001\n. Now, by paying attention to the trigonometric integral in (C6), we may identify the de\fnition\nof the zero- and \frst- order Bessel functions of \frst kind [86]. Once we have rewritten the integral in terms of the\nlatter, one may readily verify that the integral in the variable kdirectly leads to the desired expression for the spectral\ndensity (44).\nOn the other hand, from expression (C5) we may compute the commutator of the Maxwell-Chern-Simons electric\n\feld de\fned by Eq.(33) in Sec.III. Since the backreaction displacement on the environmental quasiparticle operators\nmust leave invariant the commutator of the pseudo-electric \feld force, from (C5) follows that\nh\n^E\u000b\nMCS(\u0016qi);^E\f\nMCS(\u0016qj)i\n=1\ne2h\n^E\u000b\ni(t);^E\fy\nj(t)i\n=\u0000i\u0014\u000f\u000b\f\n2\u0019Z1\n0dk k e\u00002\u001bk2J0(kj\u0001\u0016qijj); (C7)\nwhich clearly manifests that the Maxwell-Chern-Simons electric \feld has non-commutative components.24\nLet us turn the attention to the Kramers-Kronig relation (42) and expression (45). This may be obtained directly\nby carrying out the inverse Fourier transform on the previously deduced Eq.(43), i.e.\n~\u0006\u000b\f\nij(!) =1\n2\u0019Z1\n\u00001dt ei!t\u0006\u000b\f\nij(t) =\n=1\n\u0019X\nk!2(k)\u0012\nRen\nh\u000b(k;\u0016qi)hy\n\f(k;\u0016qj)o\nei!j\u0001\u0016qijjZ1\n\u00001d\u001c ei!\u001c\u0002(\u001c) sin(!(k)(\u001c+j\u0001\u0016qijj))\n\u0000Imn\nh\u000b(k;\u0016qi)hy\n\f(k;\u0016qj)o\nei!j\u0001\u0016qijjZ1\n\u00001d\u001c ei!\u001c\u0002(\u001c) cos(!(k)(\u001c+j\u0001\u0016qijj))\u0013\n=i\n2\u0019X\nk!2(k)\u0012\nRen\nh\u000b(k;\u0016qi)hy\n\f(k;\u0016qj)o\nlim\n\u000f!0+Z1\n0d\u001c\u0010\ne\u0000i(!(k)\u0000!)(\u001c\u0000\u000f\u001c+j\u0001\u0016qijj)\u0000ei(!(k)+!)(\u001c\u0000\u000f\u001c+j\u0001\u0016qijj)\u0011\n+iImn\nh\u000b(k;\u0016qi)hy\n\f(k;\u0016qj)o\nlim\n\u000f!0+Z1\n0d\u001c\u0010\ne\u0000i(!(k)\u0000!)(\u001c\u0000\u000f\u001c+j\u0001\u0016qijj)+ei(!(k)+!)(\u001c\u0000\u000f\u001c+j\u0001\u0016qijj)\u0011\u0013\n=i\n2X\nk!2(k)\u0012\u0010\n\u0000Ren\nh\u000b(k;\u0016qi)hy\n\f(k;\u0016qj)o\n+iImn\nh\u000b(k;\u0016qi)hy\n\f(k;\u0016qj)o\u0011\ne\u0000i(!(k)+!)j\u0001\u0016qijj\u000e(!(k) +!)\n+\u0010\nRen\nh\u000b(k;\u0016qi)hy\n\f(k;\u0016qj)o\n+iImn\nh\u000b(k;\u0016qi)hy\n\f(k;\u0016qj)o\u0011\ne\u0000i(!(k)\u0000!)j\u0001\u0016qijj\u000e(!(k)\u0000!)\u0013\n+1\n2\u0019X\nk!2(k)\u0012\nPZ1\n\u00001d!0\n!0\u0000!\u0010\n\u0000Ren\nh\u000b(k;\u0016qi)hy\n\f(k;\u0016qj)o\n+iImn\nh\u000b(k;\u0016qi)hy\n\f(k;\u0016qj)o\u0011\ne\u0000i(!(k)+!)j\u0001\u0016qijj\u000e(!(k) +!0)\n+PZ1\n\u00001d!0\n!0\u0000!\u0010\nRen\nh\u000b(k;\u0016qi)hy\n\f(k;\u0016qj)o\n+iImn\nh\u000b(k;\u0016qi)hy\n\f(k;\u0016qj)o\u0011\ne\u0000i(!(k)\u0000!)j\u0001\u0016qijj\u000e(!(k)\u0000!0)\u0013\n: (C8)\nRecalling that the self-energy must be analytic in the upper-half complex !plane (which is feature by the Heaviside\nstep function), in the above Fourier transform we have introduced a positive in\fnitesimal quantity 0+that provides\nthe correct pole prescriptions in the frequency domain. The latter gives rise to the principal-value terms. Paying\nattention to the real and imaginary parts in the last few lines of the right-hand side of Eq.(C8), we may easily identify\nthe spectral density de\fnition (C6). While the imaginary terms directly lead to (45), the principal-value terms only\ncontribute to the real part of the self-energy Fourier transform, and thus, they retrieve the extended Kramers-Kronig\nrelation (42).\nAppendix D: Derivation of the \ructuation-dissipation relation\nThis appendix is devoted to the derivation of the \ructuation-dissipation relation (47) of Sec.III A, and further, the\nnon-equilibrium spectral functions (48) and (49). We start from the commutator expression computed from Eq.(C4)\nin the AppendixC. By inserting the average values for the environmental displaced quasiparticle operators (C2), we\n\fnd\n1\n2Dn\n^E\u000b\ni(t);^E\fy\nj(t0)oE\n^\u001a0=X\nk;k0\u0010\n\u000e(k\u0000k0)!2(k)(1 + 2n(!(k);\f\u00001))Ren\nh\u000b(k;\u0016qi)hy\n\f(k;\u0016qj)e\u0000i!(k)(t\u0000t0)o\n+ 2nX\nl;m=1Ren\ne\u0000i(!(k)t\u0000!(k0)t0\u0000(!(k)\u0000!(k0))t0)\"\r(k)\"y\n\u000e(k0)g\r(k;\u0016ql)gy\n\u000e(k0;\u0016qm)h\u000b(k;\u0016qi)hy\n\f(k0;\u0016qj)o\n+ 2nX\nl;m=1Ren\ne\u0000i(!(k)t+!(k0)t0\u0000(!(k)+!(k0))t0)\"\r(k)\"\u000e(k0)g\r(k;\u0016ql)g\u000e(k0;\u0016qm)h\u000b(k;\u0016qi)h\f(k0;\u0016qj)o\u0011\n:25\nNow we compute the Fourier transform of the above anti-commutator by following a similar procedure as to compute\nthe commutator expression (C5) in the appendixC. Speci\fcally, considering t0<tone obtains\n1\n2Z1\n\u00001Z1\n\u00001dtdt0ei!te\u0000i!0t0Dn\n^E\u000b\ni(t);^E\fy\nj(t0)oE\n=\n=X\nk;k0\u0014\n\u000e(k\u0000k0)!2(k)(1 + 2n(!(k);\f\u00001))(2\u0019)2\n2\u0012\nRen\nh\u000b(k;\u0016qi)hy\n\f(k;\u0016qj)o\n(\u000e(!\u0000!0)(\u000e(!(k)\u0000!) +\u000e(!(k) +!)))\n\u0000iImn\nh\u000b(k;\u0016qi)hy\n\f(k;\u0016qj)o\n(\u000e(!\u0000!0)(\u000e(!(k) +!)\u0000\u000e(!(k)\u0000!)))\u0013\n+(2\u0019)2\n2nX\nl;m=1\u0012\nRen\nei(!(k)\u0000!(k0))t0\"\r(k)\"y\n\u000e(k0)g\r(k;\u0016ql)gy\n\u000e(k0;\u0016qm)h\u000b(k;\u0016qi)hy\n\f(k0;\u0016qj)o\u0010\n\u000e(!(k)\u0000!)\u000e(!(k0)\u0000!0)\n+\u000e(!(k) +!)\u000e(!(k0) +!0)\u0011\n\u0000iImn\nei(!(k)\u0000!(k0))t0\"\r(k)\"y\n\u000e(k0)g\r(k;\u0016ql)gy\n\u000e(k0;\u0016qm)h\u000b(k;\u0016qi)hy\n\f(k0;\u0016qj)o\u0010\n\u000e(!(k)\u0000!)\u000e(!(k0)\u0000!0)\n\u0000\u000e(!(k) +!)\u000e(!(k0) +!0)\u0013\n+(2\u0019)2\n2nX\nl;m=1\u0012\nRen\nei(!(k)+!(k0))t0\"\r(k)\"\u000e(k0)g\r(k;\u0016ql)g\u000e(k0;\u0016qm)h\u000b(k;\u0016qi)h\f(k0;\u0016qj)o\u0010\n\u000e(!(k)\u0000!)\u000e(!(k)0+!0)\n+\u000e(!(k) +!)\u000e(!(k0)\u0000!0)\u0011\n\u0000iImn\nei(!(k)+!(k0))t0\"\r(k)\"\u000e(k0)g\r(k;\u0016ql)g\u000e(k0;\u0016qm)h\u000b(k;\u0016qi)h\f(k0;\u0016qj)o\u0010\n\u000e(!(k)\u0000!)\u000e(!(k0) +!0)\n\u0000\u000e(!(k) +!)\u000e(!(k0)\u0000!0)\u0013\u0015\n: (D1)\nOne may bring the bracket appearing in the \frst and second lines of (D1) into the form of the stationary \ructuations\nin (47) after using the properties of the Dirac delta function (e.g. f(!(k))\u000e(!\u0000!(k)) =f(!)\u000e(!\u0000!(k)) for a given\nfunctionf(!)) as well as 1 + 2 n(!;\f\u00001) is an odd function in the variable !, and then, substituting the de\fnition\nof the spectral density given by (C6). Similarly, by directly comparing the \frst and second brackets on the variables\nl;min (D1) with the second and third lines of equation (47), it is easy to recognize the following expressions for the\nnon-stationary spectral functions,\n~G\u000b\f\nij(!;!0;t0) =(2\u0019)2\n2ei(!\u0000!0)t0nX\nl;m=1X\nk;k0\"\r(k)\"y\n\u000e(k0)g\r(k;\u0016ql)gy\n\u000e(k0;\u0016qm)h\u000b(k;\u0016qi)hy\n\f(k0;\u0016qj)\u000e(!(k)\u0000!)\u000e(!(k0)\u0000!0);\n(D2)\nas well as\n~F\u000b\f\nij(!;!0;t0) =(2\u0019)2\n2ei(!+!0)t0nX\nl;m=1X\nk;k0\"\r(k)\"\u000e(k0)g\r(k;\u0016ql)g\u000e(k0;\u0016qm)h\u000b(k;\u0016qi)h\f(k0;\u0016qj)\u000e(!(k)\u0000!)\u000e(!(k0)\u0000!0):\n(D3)\nLet us next to obtain equations (48) and (49) starting from (D2) and (D3), respectively. We show \frst the derivation\nfor~G\u000b\f\nij, and an identical procedure can be carried out to obtain ~F\u000b\f\nij. Substituting the corresponding expressions26\n(17), (20) and (25) in (D2), we \fnd\n2~G\u000b\f\nij(!;!0;t0)\n(2\u0019)2eit0(!\u0000!0)=e4\u00142\n4(2\u0019)4L4!!0nX\nl;m=1X\nk;k0e\u00002\u001b(jkj2+jk0j2)ei(k0\u0001(\u0016qj+\u0016qm)\u0000k\u0001(\u0016qi+\u0016ql))\n\u0002  \n\u00142k\u000bk0\n\f\u000f\u0013\u0017\u000f\u0017\r\u000f\u00130\u00170\u000f\u00170\r0k\u0013k0\n\u00130k\rk0\n\r0ei\u0014\nj\u0014j(\u0012(k0)\u0000\u0012(k))\njk0j2jkj2!(k)!(k0)\n\u0000i\u0014e\u0000i\u0014\nj\u0014j(\u0012(k0)\u0000\u0012(k))\u0012\u000f\u0013\u0017\u000f\u0017\r\u000f\f\r0k\u000bk\u0013k\rk0\n\r0\njkj2!(k)\u0000\u000f\u00130\u00170\u000f\u00170\r\u000f\u000b\r0k0\n\fk0\n\u00130k0\n\rk\r0\njk0j2!(k0)\u0013\n+\u000f\u000b\r\u000f\f\r0k\rk0\n\r0e\u0000i\u0014\nj\u0014j(\u0012(k0)\u0000\u0012(k))!\n(k2\n1+k2\n2)(k02\n1+k02\n2)\njkj3jk0j3e\u0000i\u0014\nj\u0014j(\u0012(k0)\u0000\u0012(k))!\n\u000e(!(k)\u0000!)\u000e(!(k0)\u0000!0)\n=e4\u00142\n4(2\u0019)4L4!!0nX\nl;m=1X\nk;k0e\u00002\u001b(jkj2+jk0j2)e\u0000i(k0\u0001(\u0016qj+\u0016qm)\u0000k\u0001(\u0016qi+\u0016ql))e\u0000i2\u0014\nj\u0014j(\u0012(k0)\u0000\u0012(k))\n\u0002 \n\u00142k\u000bk0\n\f\n!(k)!(k0)+i\u0014\u0010\u000f\f\r0k\u000bk0\n\r0\n!(k)\u0000\u000f\u000b\r0k0\n\fk\r0\n!(k0)\u0011\n+\u000e\u000b\f(k1k0\n1+k2k0\n2)\u0000k\fk0\n\u000b!\n1\njkjjk0j\u000e(!(k)\u0000!)\u000e(!(k0)\u0000!0)\n=e4\u00142\n4(2\u0019)4nX\nl;m=1Z1\n0dkdk0kk0e\u00002\u001b(k2+k02)\u000e\u0000\nk\u0000p\n!2\u0000\u00142\u0001\np\n!2\u0000\u00142\u000e\u0000\nk0\u0000p\n!02\u0000\u00142\u0001\np\n!02\u0000\u00142\n\u0002R\u000b\f(k;k0;j\u0016qi+\u0016qlj;j\u0016qj+\u0016qmj); (D4)\nwhere we have rewrite \u0012(k0) =\u00120and\u0012(k) =\u0012, and introduced the following matrix\nR\u000b\f(k;k0;j\u0016qi+\u0016qlj;j\u0016qj+\u0016qmj) =Z\u0019\n\u0000\u0019d\u0012d\u00120e\u0000i2\u0014\nj\u0014j(\u00120\u0000\u0012)e\u0000i(k0j\u0016qj+\u0016qmjcos\u0012\u0000kj\u0016qi+\u0016qljcos\u00120)D\u000b\f; (D5)\nwith\nD11=\u00142\n!!0cos\u0012cos\u00120+i\u0014\u00121\n!cos\u0012sin\u00120\u00001\n!0cos\u00120sin\u0012\u0013\n+ sin\u0012sin\u00120;\nD12=\u00142\n!!0cos\u0012sin\u00120\u0000cos\u00120sin\u0012\u0000i\u0014\u00121\n!cos\u0012cos\u00120+1\n!0sin\u0012sin\u00120\u0013\n;\nD21=\u00142\n!!0cos\u00120sin\u0012\u0000cos\u0012sin\u00120+i\u0014\u00121\n!0cos\u0012cos\u00120+1\n!sin\u0012sin\u00120\u0013\n;\nD22=\u00142\n!!0sin\u0012sin\u00120+i\u0014\u00121\n!0sin\u00120cos\u0012\u00001\n!sin\u0012cos\u00120\u0013\n+ cos\u0012cos\u00120:\nAs we have previously done in appendixC to obtain the expression for the spectral density, we now replace the trigono-\nmetric integrals appearing in (D5) by the corresponding de\fnition of the \frst- and second- order Bessel functions of\n\frst kind, i.e.\nR11(k;k0;a;b) =4\u00192\nabkk0!!0\u0012\n\u0014(akJ1(ak)\u00002J2(ak)) + 2\u0014!\nj\u0014jJ2(ak)\u0013\u0012\n\u0014(bk0J1(bk0)\u00002J2(bk0)) +!02\u0014\nj\u0014jJ2(bk0)\u0013\n;\nR12(k;k0;a;b) =\u00004i\u00192\nabkk0!!0\u0012\n\u0014(akJ1(ak)\u00002J2(ak))\u00002\u0014!\nj\u0014jJ2(ak)\u0013\u0012\nbk0!0J1(bk0)\u0000\u0012\n2!0\u0000\u00142k0\nj\u0014j\u0013\nJ2(bk0)\u0013\n;\nR21(k;k0;a;b) =4i\u00192\nabkk0!!0\u0012\n\u0014(bk0J1(bk0)\u00002J2(bk0))\u00002\u0014!0\nj\u0014jJ2(bk0)\u0013\u0012\nak!J 1(ak)\u0000\u0012\n2!\u0000\u00142k\nj\u0014j\u0013\nJ2(ak)\u0013\n;\nR22(k;k0;a;b) =4\u00192\nabkk0!!0\u0012\nak!J 1(ak)\u0000\u0012\n2!\u0000\u00142k\nj\u0014j\u0013\nJ2(ak)\u0013\u0012\nbk0!0J1(bk0)\u0000\u0012\n2!0\u0000\u00142k0\nj\u0014j\u0013\nJ2(bk0)\u0013\n: (D6)\nFinally, Eq.(48) is obtained by \frstly inserting (D6) into (D4), and then, carrying out the integral on the real variables\nkandk0, which is immediately obtained by using the properties of the Dirac delta function. Although we obtain a27\nformidable expression, the particular shape of (D6) permits to cast the non-equilibrium spectral density in the form\n(48) after some simple manipulation.\nApplying a similar procedure to (D3) as before yields\n2~F\u000b\f\nij(!;!0;t0)\n(2\u0019)2e\u0000it0(!+!0)=e4\u00142\n4(2\u0019)4L4!!0nX\nl;m=1X\nk;k0e\u00002\u001b(jkj2+jk0j2)ei(k0\u0001(\u0016qj+\u0016qm)+k\u0001(\u0016qi+\u0016ql))\u0012\u0012\u0014\u000f\u0013\u0017k\u0013k\u000b\"\u0017(k)\njkj2!(k)+i\"\u000b(k)\u0013\n(D7)\n\u0002\u0012\u0014\u000f\u00130\u00170k0\n\u00130k0\n\f\"\u00170(k0)\njk0j2!(k0)+i\"\f(k0)\u0013\u000f\u0015\r\u000f\u00150\u000ek\u0015k0\n\u00150\njkj2jk0j2\u000f\r\u0016\u000f\u000e\u00160k\u0016k0\n\u00160\njkjjk0jei\u0014\nj\u0014j(\u0012(k0)+\u0012(k))\u0013\n\u000e(!(k)\u0000!)\u000e(!(k0)\u0000!0)\n=e4\u00142\n4(2\u0019)4L4!!0nX\nl;m=1X\nk;k0e\u00002\u001b(jkj2+jk0j2)ei(k0\u0001(\u0016qj+\u0016qm)+k\u0001(\u0016qi+\u0016ql))ei2\u0014\nj\u0014j(\u0012(k0)+\u0012(k))\n\u0002 \n\u0000\u00142k\u000bk0\n\f\n!(k)!(k0)\u0000i\u0014\u0010\u000f\f\r0k\u000bk0\n\r0\n!(k)+\u000f\u000b\r0k0\n\fk\r0\n!(k0)\u0011\n\u0000(\u000e\u000b\f(k1k0\n1+k2k0\n2)\u0000k\fk0\n\u000b)!\n1\njkjjk0j\u000e(!(k)\u0000!)\u000e(!(k0)\u0000!0)\n=\u0000e4\u00142\n4(2\u0019)4nX\nl;m=1Z1\n0dkdk0kk0e\u00002\u001b(jkj2+jk0j2)\u000e\u0000\nk\u0000p\n!2\u0000\u00142\u0001\np\n!2\u0000\u00142\u000e\u0000\nk0\u0000p\n!02\u0000\u00142\u0001\np\n!02\u0000\u00142\n\u0002T\u000b\f(k;k0;j\u0016qj+\u0016qmj;j\u0016qi+\u0016qlj);\nwhere we have introduced the following matrix after doing some tedious algebra as before,\nT11(k;k0;a;b) =4\u00192\nabkk0!!0\u0012\nak\u0014J 1(ak)\u0012\n\u0000bk0\u0014J1(bk0) + 2\u0012\n\u0014\u0000\u0014!0\nj\u0014j\u0013\nJ2(bk0)\u0013\n(D8)\n+ 2J2(ak)\u0012\nbk0\u0014\u0012\n\u0014\u0000\u0014!\nj\u0014j\u0013\nJ1(bk0) + 2\u0012\u00142\nj\u0014j(!0+!) +!!0\u0000\u00142\u0013\nJ2(bk0)\u0013\u0013\n;\nT12(k;k0;a;b) =\u00004i\u00192\nabkk0!!0\u0012\na2k2\u0014J1(ak)\u0012\nbk0!0J1(bk0)\u00002\u0012\n!0\u0000\u00142\nj\u0014j\u0013\nJ2(bk0)\u0013\n+ 2akJ2(ak)\u0012\nbk0!0\u0012\n\u0014+\u0014!\nj\u0014j\u0013\nJ1(bk0) + 2\u0012\n(!0+!)\u0014+\u00142+\u0014\nj\u0014j!!0\u0013\nJ2(bk0)\u0013\u0013\n;\nT21(k;k0;a;b) =\u00004i\u00192\nabkk0!!0\u0012\na2k2!J1(ak)\u0012\nbk02\u0014J1(bk0)\u00002\u0012\n\u0014+\u0014!0\nj\u0014j\u0013\nJ2(bk0)\u0013\n\u00002akJ2(ak)\u0012\nbk0\u0014\u0012\n\u0014+\u0014!\nj\u0014j\u0013\nJ1(bk0)\u00002\u0012\n\u0014(!0+!)\u0000\u00142+\u0014\nj\u0014j!!0\u0013\nJ2(bk0)\u0013\u0013\n;\nT22(k;k0;a;b) =4\u00192\nabkk0!!0\u0012\n\u0000ak!J 1(ak)\u0012\nbk0!0\u0014J1(bk0) + 2\u0012\n!0+\u00142\nj\u0014jk0\u0013\nJ2(bk0)\u0013\n+ 2J2(ak)\u0012\nbk0!0\u0014\u0012\n2!+\u00142\nj\u0014jk\u0013\nJ1(bk0)\u00002\u0012\n\u00142+\u00142\nj\u0014j(!0+!)\u0000!!0)\u0013\nJ2(bk0)\u0013\u0013\n:\nOnce again, the integrals appearing in the last line of (D7) are directly obtained by making use of the Dirac delta\nproperties, once we have replaced D8. In this way, we \fnally arrive to the expression (49) for the non-stochastic\nspectral function presented in Sec.III A.\nAppendix E: Non-stochastic \ructuations, thermal noise and retarded self-energy for closed particles and\nweak Chern-Simons action\nIn this section, we extend the discussion, presented in Sec.III B, about the evaluation of the non-stochastic \ructu-\nations appearing in (52) when it is taking the strict limit t\u0000t0!1 , and we provide explicit expressions for the\nself-energy and the thermal correlations (53) in the zero-temperature limit as well.\nBy paying attention to the expressions obtained for (48) and (49), it is readily to see that both are integrable\nfunctions for !;!0into the domain of the MSC environment bandwidth (i.e. \u0014\u0014!\u0014(2\u001b)\u00001\n2), as well as they decay\nas fast as an exponential function for large frequencies since the Bessel functions decrease algebraically. As mentioned28\nin Sec.III A, these properties permit to use the Riemann-Lebesgue lemma to evaluate the non-equilibrium \ructuations\n\u0007\u000b\f\nij(t;t0;t0) and\u0004\u000b\f\nij(t;t0;t0) involved in expression (52), which are respectively obtained from the second and third\nline of (47) via the inverse Fourier transform. Speci\fcally, these non-stochastic \ructuations takes the following form\nin the time domain,\n\u0007\u000b\f\nij(t;t0;t0) =4\n\u00192Z\nd!d!0\u0010\nRen\n~G\u000b\f\nij(!;!0;t0)o\ncos(!t\u0000!0t0) + Imn\n~G\u000b\f\nij(!;!0;t0)o\nsin(!t\u0000!0t0)\u0011\n; (E1)\nand\n\u0004\u000b\f\nij(t;t0;t0) =4\n\u00192Z\nd!d!0\u0010\nRen\n~F\u000b\f\nij(!;!0;t0)o\ncos(!t+!t0) + Imn\n~F\u000b\f\nij(!;!0;t0)o\nsin(!t+!t0)\u0011\n: (E2)\nwhere we recall that ~G\u000b\f\nij(!;!0;t0) and ~F\u000b\f\nij(!;!0;t0) are given by (48) and (49), respectively. Formally, the afore-\nmentioned lemma can be enunciated as follows: Let f(!) be an complex-valued function that is absolutely integrable\nonR. Then the Riemann-Lebesgue lemma states that [77],\nlim\njtj!1Z1\n\u00001f(!)ei!td!!0: (E3)\nOne saysf(!) is an absolutely integrable function on Rif it is ful\fllZ1\n\u00001jf(!)jd!< 0, or equivalently, if it belongs\nto the class of L1(R)-functions. Clearly, the Riemann-Lebesgue lemma has an intuitive interpretation: the integral\nbecomes so highly oscillatory that everything cancels out.\nTo see more clearly how such lemma applied to (E1) and (E2), it is convenient to rewrite them as the integrals of two\nindependent dispersion functions in the variables !and!0. Let \frst pay attention to the non-stochastic \ructuation\nencoded by \u0007\u000b\f\nij(t;t0;t0). By using trigonometric identities once replaced equation (48) into (E1), we can bring this\ninto the following form\n\u0007\u000b\f\nij(t;t0;t0) =Z1\n0d!d!0Ren\n\u001a\u000b\f\ni(!)o\nRen\n\u001a\u000b\f\nj(!0)o\nc(!;!0;t\u0000t0)\n+Z1\n0d!d!0Imn\n\u001a\u000b\f\ni(!)o\nImn\n\u001a\u000b\f\nj(!0)o\ns(!;!0;t\u0000t0) (E4)\nwhere\nc(!;!0;\u001c) = cos (!\u001c) cos (!0\u001c) + sin (!\u001c) sin (!0\u001c)\ns(!;!0;\u001c) = sin (!\u001c) cos (!0\u001c)\u0000sin (!\u001c) cos (!0\u001c)\nand with the dispersion function given by,\n\u001a\u000b\f\ni=j(!) =e2\u0014e\u00002\u001b(!2\u0000\u00142)\np\n128!nX\nl=1r\u000b\f\ni=j\u0010p\n!2\u0000\u00142;j\u0016qi=j+\u0016qlj\u0011\n; \u0014\u0014! (E5)\nwhere the elements of the matrix function ri\u0000p\n!2\u0000\u00142;j\u0016qi+\u0016qlj\u0001\n(rj\u0000p\n!02\u0000\u00142;j\u0016qj+\u0016qlj\u0001\n) are directly obtained\nfrom the elements R\u000b\f, determined by (50), after removing the dependence on the variables !0(!) and \u0016qj(\u0016qi). It is\nfound that the elements r\u000b\f\ni=jare basically convergent combinations of \frst- and second- order Bessel functions of the\n\frst kind.\nFrom (E5) it is now clear that both dispersion functions \u001a\u000b\f\ni=j(x) (for arbitrary iandj) are continuously di\u000berentiable\nfor\u0014\u0014x<1, as well as they exponentially decay for values xlarger than (2 \u001b)\u00001\n2. As a result, it follows from the\nabove Riemann-Lebesgue lemma that all integrals in (E4) asymptotically vanishes in the long time limit t\u0000t0!1 .\nOn the other hand, starting from the expression for ~F\u000b\f\nij(!;!0;t0) (see equation (D8) in AppendixD), we can follow\nthe same procedure to show that the corresponding dispersion functions characterizing the non-stochastic \ructuation\n\u0004\u000b\f\nij(t;t0;t0) share these features with \u001a\u000b\f\ni=j(x), so we could conclude an identical statement for (E2) after applying the\nRiemann-Lebesgue lemma.\nNow we address the retarded self-energy and thermal \ructuations in the realistic situation when the system particles\nare su\u000eciently closed (i.ej\u0001\u0016qijjp\n2\u001b\u001c1) and the Chern-Simons action is considered to be weak (i.e.p\n2\u001b\u0014\u001c1). As a\nresult of these approximations, the spectral density substantially simpli\fes to (C6), which is illustrated in Sec.III B.29\nReplacing this in the expression for the retarded self-energy (43) yields upon integration,\n\u000611\nij(t)'e2\n16\u0019p\n2(4\u001b)5\u0012\u0010\n32\u001b(1 + 2\u00142\u001b)\u0000j\u0001\u0016qijj2(5 + 18\u00142\u001b)\u0011\nx+ 2j\u0001\u0016qijj2(1 + 2\u00142\u001b)x3(E6)\n\u0000p\u0019\n2e\u0000x2er\f(x)\u0010\n3j\u0001\u0016qijj2\u000032\u001b+ 14j\u0001\u0016qijj2\u00142\u001b\u0000192\u00142\u001b2\n+\u0010\n\u000012j\u0001\u0016qijj2+ 64\u001b\u000040j\u0001\u0016qijj2\u00142\u001b+ 128\u00142\u001b2\u0011\nx2+\u0010\n4j\u0001\u0016qijj2+ 8j\u0001\u0016qijj2\u00142\u001b\u0011\nx4\u0011\u0013\n;\nand\n\u000622\nij(t)'e2\n16\u0019p\n2(4\u001b)5\u0012\u0010\n32\u001b(1 + 2\u00142\u001b)\u0000j\u0001\u0016qijj2(15 + 22\u00142\u001b)\u0011\nx+ 6j\u0001\u0016qijj2(1 + 2\u00142\u001b)x3(E7)\n\u0000p\u0019\n2e\u0000x2er\f(x)\u0010\n9j\u0001\u0016qijj2\u000032\u001b+ 10j\u0001\u0016qijj2\u00142\u001b\u0000192\u00142\u001b2\n+\u0010\n\u000032j\u0001\u0016qijj2+ 64\u001b\u000056j\u0001\u0016qijj2\u00142\u001b+ 128\u00142\u001b2\u0011\nx2+\u0010\n12j\u0001\u0016qijj2+ 24j\u0001\u0016qijj2\u00142\u001b\u0011\nx4\u0011\u0013\n;\nas well as the o\u000b-diagonal element given by (55) in Sec.III B. The behavior of (E6) and (E7) in the time domain is\nillustrated in the \fgure1. Moreover, we may follow a similar procedure to get the zero-temperature limit of the thermal\n\ructuations (53) once we have replaced the spectral density by (C6). In this way, we consider the zero-temperature\nlimit where n\u0000\n!;\f\u00001\u0001\n!0. Using the standard tables of integration, we \fnd after some tedious manipulation\nDn\n^\u00181\ni(t0+t);^\u00181y\nj(t0)oE\n'e2e\u0000x2\n2p\n2\u0019(16\u001b)5\u0012\n\u00003j\u0001\u0016qijj2+ 32\u001b\u000014j\u0001\u0016qijj2\u00142\u001b+ 192\u00142\u001b2\n+\u0010\n12j\u0001\u0016qijj2\u000064\u001b+ 40j\u0001\u0016qijj2\u00142\u001b\u0000128\u00142\u001b2\u0011\nx2\n\u00004\u0010\nj\u0001\u0016qijj2+ 2j\u0001\u0016qijj2\u00142\u001b\u0011\nx4\u0013\n; (E8)\nand\nDn\n^\u00182\ni(t0+t);^\u00182y\nj(t0)oE\n'e2e\u0000x2\n2p\n2\u0019(16\u001b)5\u0012\n\u00009j\u0001\u0016qijj2+ 32\u001b\u000010j\u0001\u0016qijj2\u00142\u001b+ 192\u00142\u001b2\n+\u0010\n36j\u0001\u0016qijj2\u000064\u001b+ 56j\u0001\u0016qijj2\u00142\u001b\u0000128\u00142\u001b2\u0011\nx2\n\u000012\u0010\nj\u0001\u0016qijj2+ 2j\u0001\u0016qijj2\u00142\u001b\u0011\nx4\u0013\n; (E9)\nwhereas for the o\u000b-diagonal term,\nDn\n^\u00181\ni(t0+t);^\u00182y\nj(t0)oE\n'e2\u0014\n256p\u0019\u001b2xe\u0000x2\u0012\n16\u001b+j\u0001\u0016qijj2(\u00003 +x2)\u0013\n; (E10)\nwithx=tp\n8\u001b.The above expressions are illustrated in \fgure1 as functions of time.\n[1] U. 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We study the asymptotic\nprofile and optimal decay rates of solutions as t→ ∞inL2-sense. The damping terms\nconsidered in this paper is not studied so far, and in the low frequenc y parameters the\ndamping is rather weakly effective than that of well-studied power ty pe one such as ( −∆)θut\nwithθ∈(0,1). Inordertogettheoptimalrateofdecaywemeettheso-called hypergeometric\nfunctions, so the analysis seems to be more difficult and attractive.\n1 Introduction\nWe present and consider a new type of wave equation with a loga rithmic damping term:\nutt+Au+Lut= 0,(t,x)∈(0,∞)×Rn, (1.1)\nu(0,x) =u0(x), ut(0,x) =u1(x), x∈Rn, (1.2)\nwhere (u0,u1) are initial data chosen as\nu0∈H1(Rn), u1∈L2(Rn),\nand the operator Au:=−∆uforu∈H2(Rn), and a new operator\nL:D(L)⊂L2(Rn)→L2(Rn)\nis defined as follows:\nD(L) :=/braceleftbigg\nf∈L2(Rn)/vextendsingle/vextendsingle/integraldisplay\nRn(log(1+ |ξ|2))2|ˆf(ξ)|2dξ <+∞/bracerightbigg\n,\n∗ruy.charao@ufsc.br\n†Corresponding author: ikehatar@hiroshima-u.ac.jp\nKeywords and Phrases: Wave equation; Logarithmic damping; L2-decay; asymptotic profile, optimal decay.\n2010 Mathematics Subject Classification. Primary 35L05; Se condary 35B40, 35C20, 35S05.\n1forf∈D(L),\n(Lf)(x) :=F−1\nξ→x/parenleftBig\nlog(1+|ξ|2)ˆf(ξ)/parenrightBig\n(x),\nand symbolically writing, one can see\nL= log(I+A).\nHere, we denote the Fourier transform Fx→ξ(f)(ξ) off(x) by\nFx→ξ(f)(ξ) =ˆf(ξ) :=/integraldisplay\nRne−ix·ξf(x)dx\nas usual with i:=√−1, andF−1\nξ→xexpresses its inverse Fourier transform. Since the new\noperator Lis constructed by a nonnegative-valued multiplication one , it is nonnegative and\nself-adjoint in L2(Rn).\nThen, by a similar argument to [18, Proposition 2.1] based on Lumer-Phillips Theorem one\ncan find that the problem (1.1)-(1.2) has a unique mild soluti on\nu∈C([0,∞);H1(Rn))∩C1([0,∞);L2(Rn))\nsatisfying the energy inequality\nEu(t)≤Eu(0), (1.3)\nwhere\nEu(t) :=1\n2/parenleftbig\n/ba∇dblut(t,·)/ba∇dbl2\nL2+/ba∇dbl∇u(t,·)/ba∇dbl2\nL2/parenrightbig\n.\n(1.3) implies the decreasing property of the total energy be cause of the existence of some kind\nof dissipative term Lut. For details, see Appendix in this paper.\nA main topic of this paper is to find an asymptotic profile of sol utions in the L2topology as\nt→ ∞to problem (1.1)-(1.2), and to apply it to get the optimal rat e of decay of solutions in\nterms of the L2-norm.\nNow let us recall several previous works related to linear da mped wave equations with con-\nstant coefficients. We mention them from the viewpoint of the F ourier transformed equations.\n(1)In the case of weak damping for the equation (1.1) with L=I:\nˆutt+|ξ|2ˆu+ ˆut= 0, (1.4)\nas is usually observed, for small |ξ|, the solution to the equation (1.4) behaves like a constant\nmultiple of the Gauss kernel, while for large |ξ|the solution includes an oscillation property,\nwhich vanishes very fast. This property can be pointed out in the celebrated paper due to\nMatsumura [20] from the precise decay estimates point of vie w, and from the viewpoint of\nasymptotic expansions one should mention the so-called Nis hihara decomposition [25]. As a\nwork in a similar philosophy one can cite the paper due to Nara zaki [23], which deals with the\nhigher dimensional case. Higher order asymptotic expansio ns intof theL2-norm of solutions to\n(1.4) has been studied in Volkmer [32], and Said-Houari [29] restudies the diffusion phenomena\nfrom the viewpoint of the weighted L1-initial data. As an abstract theory in the case when\nthe operator Ais nonnegative and self-adjoint in Hilbert spaces one can ci te several papers due\nto Chill-Haraux [6], Ikehata-Nishihara [15], Radu-Todoro va-Yordanov [28], and Sobajima [31],\nwhere they also investigate the diffusion phenomenon of solut ions (ast→ ∞) more precisely.\n2(2)After the equation (1.4) one should give some comments to the following Fourier trans-\nformed equation of (1.1) with L=A:\nˆutt+|ξ|2ˆu+|ξ|2ˆut= 0. (1.5)\nThis corresponds to the so-called strongly damped wave equa tion case, which was set up by\nPonce [27] and Shibata [30] at the first stage of the research. On the contrary to (1.4) the\nsolution to the equation (1.5) behaves like a diffusion wave wi th complex-valued characteristic\nroots for small |ξ|, while for large |ξ|the solution does not include any oscillation property\nbecause the characteristic roots are real-valued. These ob servations are recently pointed out in\nthe papers Ikehata [14], Ikehata-Onodera [16] and Ikehata- Todorova-Yordanov [18] by capturing\nthe leading term as t→ ∞of the solution, and they have derived optimal estimates of s olutions\nin terms of L2-norm. In particular, it should be noted that optimal estima tes in [14, 16] do not\nnecessarily imply the decay estimates, and in fact, in space dimension 1 and 2 an infinite time\nblowup property occurs. Furthermore, it should be emphasiz ed that higher order asymptotic\nexpansions of the squared L2-norm of solutions as t→ ∞have been very precisely studied\nrecently by Barrera [1] and Barrera-Volkmer [2, 3], and in [2 1] Michihisa studies the higher or-\nder asymptotic expansion of the solution itself, and applie d it to investigate the lower bound of\ndecay rate of the difference between the leading term and the so lution itself in terms of L2norm.\n(3)As for the generalization of the study (1) and (2) one can c ite several papers due to\nD’Abbicco-Ebert [7], D’Abbicco-Ebert-Picon [8], D’Abbic co-Reissig [9], Char˜ ao-da Luz-Ikehata\n[5], Ikehata-Takeda [17], Karch [19] and Narazaki-Reissig [24], which deal with the equation\n(1.1) with the so-called structural damping L=Aθ(0< θ <1):\nˆutt+|ξ|2ˆu+|ξ|2θˆut= 0. (1.6)\nBy those works one has already known that θ= 1/2 is critical in the sense that for θ∈(0,1/2)\nthe solution to the equation (1.6) is parabolic like, and in t he case of θ∈[1/2,1) the solution to\ntheequation (1.6) behaves like adiffusion wave as t→ ∞, and in particular, θ= 1/2 corresponds\nto the scale invariant case.\nOn reconsidering our problem in the Fourier space our equati on (1.1) becomes\nˆutt+|ξ|2ˆu+log(1+ |ξ|2)ˆut= 0. (1.7)\nAn influence of the damping coefficient log(1 + |ξ|2) on the dissipative nature of the solution\nseems to be rather weak in both regions for large |ξ|and small |ξ|when we compare it with the\nfractional damping |ξ|2θwithθ∈(1/2,1). Furthermore, as is easily seen that the characteristic\nrootsλ±for the characteristic polynomial of (1.7) such that\nλ2+log(1+ |ξ|2)λ+|ξ|2= 0\nare all complex-valued for all ξ∈Rn, which implies that the corresponding solution has an\noscillating property for all frequency parameter ξ∈Rn. This property produces a big difference\nas compared with previously considered cases in the referen ces. So, the problem (1.1) may in-\nclude several difficulties which has never experienced so far , and in fact, we have to analyze the\nso-called hypergeometric functions naturally (see sectio n 2) when one captures the leading term\nand obtains the optimal rate of decay of the solution to probl em (1.1)-(1.2). In this sense, we\ndo present much new type of problems in the wave equation field through the analysis for the\nwave equation with log-damping.\nOur two main results read as follows.\n3Theorem 1.1 Letn≥1, and let [u0,u1]∈/parenleftbig\nH1(Rn)∩L1(Rn)/parenrightbig\n×/parenleftbig\nL2(Rn)∩L1,1(Rn)/parenrightbig\n. Then,\nthe unique solution u(t,x)to problem (1.1)-(1.2)satisfies\n/vextenddouble/vextenddouble/vextenddouble/vextenddoubleu(t,·)−/parenleftbigg/integraldisplay\nRnu1(x)dx/parenrightbigg\nF−1\nξ→x/parenleftbigg\n(1+|ξ|2)−t\n2sin(|ξ|t)\n|ξ|/parenrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2≤I0t−n\n4,(t≫1),\nwhere\nI0:=/ba∇dblu0/ba∇dblL2+/ba∇dblu1/ba∇dblL2+/ba∇dblu0/ba∇dblL1+/ba∇dbl(1+|x|)u1/ba∇dblL1.\nRemark 1.1 I0inTheorem1.1doesnotdependonanynorms /ba∇dblu0/ba∇dblH1. Thisisoneofdifferences\nas compared with the case for fractional damping L:= (−∆)θ(see [14]).\nAs a consequence of Theorem 1.1 one can get the optimal estima tes intof solutions in terms\nofL2-norm. We set\nP1:=/integraldisplay\nRnu1(x)dx.\nTheorem 1.2 Letn≥1, and let [u0,u1]∈/parenleftbig\nH1(Rn)∩L1(Rn)/parenrightbig\n×/parenleftbig\nL2(Rn)∩L1,1(Rn)/parenrightbig\n. Then,\nthe unique solution u(t,x)to problem (1.1)-(1.2)satisfies\n(i)n≥3⇒Cn|P1|t−n−2\n4≤ /ba∇dblu(t,·)/ba∇dblL2≤C−1\nnI0t−n−2\n4(t≫1),\n(ii)n= 2⇒C2|P1|√logt≤ /ba∇dblu(t,·)/ba∇dblL2≤C−1\n2I0√logt(t≫1),\n(iii)n= 1⇒C1|P1|√\nt≤ /ba∇dblu(t,·)/ba∇dblL2≤C−1\n1I0√\nt(t≫1),\nwhereI0is a constant defined in Theorem 1.1, andCn(n∈N)are constants independent from\nanytand initial data.\nRemark 1.2 In the case of |ξ| ≈0 one has\nlog(1+|ξ|2)≤ |ξ|2θ\nfor allθ∈(0,1). This implies, in the case of θ∈(0,1], the effect of damping log(1+ |ξ|2)ˆutis\nmuch weaker than all types of fractional damping |ξ|2θˆut. While, if we compare two types of\nfractional damping ( −∆)θwtwithθ∈(0,1/2) (parabolic like) and ( −∆)θwtwithθ∈(1/2,1]\n(diffusion wave like), in the case of small |ξ|(this is essential part of both solutions), the effect\nof (−∆)θwtwithθ∈(1/2,1) is weaker than ( −∆)θwtwithθ∈(0,1/2). In some sense, since\nthe effect of damping log(1+ |ξ|2)ˆutis weaker than |ξ|2θˆutwithθ∈(1/2,1] (wave like case), it\nseems to be natural that the results obtained in Theorem 1.2 c oincide with the case for strong\ndamping ( −∆)ut(see [14, 16]), however, its analysis is much more difficult, i n particular, when\nwe deal with the upper bound for several quantities. This is b ecause the ingredient (1+ |ξ|2)−t\n2\nof the leading term obtained in Theorem 1.1 behaves slower th an usual diffusion wave case such\nthate−t|ξ|2sin(|ξ|t)\n|ξ|in the Fourier space Rn\nξfor each t >0. As will be observed in section 2,\nthe function (1 + |ξ|2)−t\n2=e−t\n2log(1+|ξ|2)has a close relation to the so-called hypergeometric\nfunction with special parameters. In this connection, it wo uld be interesting to study a kind of\ndiffusion equation such that\nvt+1\n2log(I−∆)v= 0,\nin order to know more about deeper properties of the asymptot ic profile obtained in Theorem\n1.1.\n4This paper is organized as follows. In section 2 we prepare se veral important propositions\nand lemmas, which will be used later, and in particular, in su bsection 2.1 we shall mention the\nso-called hypergeometric functions. Theorem 1.1 is proved in section 3. In section 4, we shall\nstudy the optimality of the L2-norm of solutions to problem (1.1)-(1.2) in the case of spac e\ndimension 1 and 2, and Theorem 1.2 will be proved at a stroke. A ppendix is prepared to check\nthe unique existence of the weak solution to problem (1.1)-( 1.2).\nNotation. Throughout this paper, /ba∇dbl · /ba∇dblqstands for the usual Lq(Rn)-norm. For simplicity of\nnotation, in particular, we use /ba∇dbl·/ba∇dblinstead of /ba∇dbl·/ba∇dbl2. Furthermore, we denote /ba∇dbl·/ba∇dblHlas the usual Hl-norm.\nFurthermore, we define a relation f(t)∼g(t) ast→ ∞by: there exist constant Cj>0 (j= 1,2) such\nthat\nC1g(t)≤f(t)≤C2g(t) (t≫1).\nWe also introduce the following weighted functional spaces.\nL1,γ(Rn) :=/braceleftbigg\nf∈L1(Rn)/vextendsingle/vextendsingle/ba∇dblf/ba∇dbl1,γ:=/integraldisplay\nRn(1+|x|γ)|f(x)|dx <+∞/bracerightbigg\n.\nFinally, we denote the surface area of the n-dimensional unit ball by ωn:=/integraldisplay\n|ω|=1dω.\n2 Hypergeometric functions\nOur interest refers to the historically most important hype rgeometric function 2F1(a,b;c;z)\ncalled Gauss’s hypergeometric function which may be defined by\n2F1(a,b;c;z) =∞/summationdisplay\nn=0(a)n(b)n\n(c)nzn\nn!, (2.1)\nwhere (a)n=a(a+1)···(a+n−1) is the Pochhammer symbol (upward factorial). The series\n(2.1)converges absolutelyin |z|<1(z∈C)forparameters a,b,c∈Cwithc/ne}ationslash= 0,−1,−2,−3,···\nThegeneralized hypergeometric functions pFq(a1,···,ap,b1,···,bp;c1,···,cq;z) are also de-\nfined similarly by hypergeometric power series, that includ e many other special functions as, for\nexample, Beta function.\nThese functions appear in many problems in statistics, prob ability, quantum mechanics\namong other areas. For a list of some of the many thousands of p ublished identities, symmetries,\nlimits, involving the hypergeometric functions we can refe r to the works by Erd´ elyi et al. (1955),\nGasper- Rahman (2004), Miller-Paris (2011). There is no any known systems for organizing all\nof the identities. In fact, there is no known algorithm that c an generate all identities. Moreover,\nthere are known a number of different algorithms that generate different series of identities. The\ntheory about the algorithmic remains an active research top ic.\nHypergeometric function is also given as a solution of the sp ecial Euler second-order linear\nordinary differential equation\nz(1−z)d2w\ndz2+/bracketleftbig\nc−(a+b+1)z/bracketrightbigdw\ndz−abw= 0.\nAround the singular point z= 0, there are two independent solutions. One of them, if cis not\na non-positive integer, is\nw=w(a,b;c;z) =2F1(a,b;c;z) =1\nB(b,c−b)/integraldisplay1\n0xb−1(1−x)c−b−1(1−zx)−adx,(2.2)\n5whereB(x,y) is the Beta function defined by\nB(x,y) =/integraldisplay1\n0tx−1(1−t)y−1dt\nwherexandyare complex numbers with positive real part.\nIn fact, (2.2) converges for z∈Csatisfying |z|<1, and for z=−1 the definition is formally,\nhowever, it should be mentioned that for z=−1 and special b,c, anda=t >0 the convergence\nof (2.2) makes sense.\nSeveral properties, as for example symmetries and some asym ptotic behavior on the param-\neterz, appear on the literature about this function (2.1)–(2.2) f or particular cases of a,b,c. For\nexample, Bessel functions can be expressed as a limit of hype rgeometric functions. However,\nthere seems to be no any results on asymptotic behavior for th e parameter awhena=trepre-\nsents the time. On hypergeometric functions we can mention t he works [10], [11], [12], [22] and\ntheir references.\nIn this work we get some results for some fixed a,b,c,z. Our result seems new and very\nimportant. In our Theorem 2.2 we prove that\nIp(t) =/integraldisplay1\n0(1+r2)−trpdr∼t−p+1\n2, t≫1\nfor each p≥0. But we can note that by a change of variable\nIp(t) =1\n2/integraldisplay1\n0(1+s)−tsp−1\n2ds=1\n2B/parenleftBigp+1\n2,1/parenrightBig\n2F1/parenleftBig\nt,p+1\n2;p+3\n2;−1/parenrightBig\n,\nsince we choose in (2.2): b−1 =p−1\n2,c=b+1,a=tandz=−1. In this case, one has\n2F1/parenleftBig\nt,p+1\n2;p+3\n2;−1/parenrightBig\n= (p+1)Ip(t),\nbecauseB/parenleftBigp+1\n2,1/parenrightBig\n=2\np+1.\nThen we obtain the following asymptotic behavior for a parti cular class of hypergeometric\nfunctions.\nProposition 2.1 Letp≥0. Then\n2F1/parenleftBig\nt,p+1\n2;p+3\n2;−1/parenrightBig\n∼t−p+1\n2, t≫1.\nFrom this proposition we have in particular\n2F1(t,1\n2;3\n2;−1)∼t−1\n2, t≫1,\nand\n2F1(t,3\n2;5\n2;−1)∼t−3\n2, t≫1.\nAs mentioned above, in the next subsection we show optimal as ymptotic behavior of the\nhypergeometric functions given by the integral\nIp(t) =/integraldisplay1\n0(1+r2)−trpdr, t > 1/2,\n6for each fixed p≥0, and to the case for each p∈R\nJp(t) =/integraldisplay∞\n1(1+r2)−trpdr, t > 1/2.\nIn particular, it is known that\nH0(t) :=/integraldisplay∞\n0(1+r2)−tdr=√π\n2Γ(t−1/2)\nΓ(t), t >1/2,\nwhere Γ = Γ( t) is the gamma function.\nThen, by combining our decay estimates in the next section wi thIp(t) andJp(t) one can\nobtain the following asymptotic behavior of the function Γ( t−1/2)/Γ(t)\nΓ(t−1/2)\nΓ(t)∼t−1/2, t≫1,\nwhose result does not seem to be well-known, although it is si mple to see that Γ( t−1)/Γ(t) =\n1\nt−1,t >1.\nFinally, it is important to observe that the behavior of hype rgeometric functions of the\ntype2F1(t,b;c;−1) appears when we study the asymptotic behavior of solution s for the wave\nequation under effects of a special dissipative term of logari thm type.\n2.1 General case\nLetp≥0 be a real number and Ip(t) be the function defined by\nIp(t) =/integraldisplay1\n0(1+r2)−trpdr,fort > p.\nThe following theorem gives the optimal asymptotic behavio r ofIp(t) for large t.\nTheorem 2.1 Assume that 0≤p≤3. Then\nIp(t)∼t−p+1\n2, t≫1.\nProof.Letf(r) be the function given by f(r) = (1+ r2)−trp,r≥0. Then β=/radicalbiggp\n2t−p,\nt > p, is a global maximum of fand 0< β <1 . Moreover f(r) is a decreasing function for\nr > β, increasing for 0 < r < β whenp >0,f(0) = 1 in case p= 0 and f(0) = 0 if p >0.\nCasep= 0:To prove this case we split the interval of integration in two parts as follows.\nI0(t) =/integraldisplayt−1/2\n0(1+r2)−tdr+/integraldisplay1\nt−1/2(1+r2)−tdr.\nNow we note that\n/integraldisplayt−1/2\n0(1+r2)−tdr≤t−1/2, (2.3)\nbecausef(0) = 1 is maximum global of f(r) on the interval (0 ,∞).\n7On the other hand, by using a change of variable u= log(1+ r2) we have\n/integraldisplay1\nt−1/2(1+r2)−tdr=/integraldisplay1\nt−1/2e−tlog(1+r2)dr\n=1\n2/integraldisplaylog2\nlog(1+1\nt)e−(t−1)u(eu−1)−1/2du\n≤1\n2/integraldisplaylog2\nlog(1+1\nt)e−(t−1)u(elog(1+1\nt)−1)−1/2du (2.4)\n≤1\n2/integraldisplaylog2\nlog(1+1\nt)e−(t−1)ut1/2du\n≤t1/2\n2e−(t−1)u\n(t−1)/vextendsingle/vextendsingle/vextendsingle\nlog(1+1\nt)≤Ct1/2\nt−1, t≥2\nwithC >0 a constant because e−(t−1)log(1+1\nt)is a time-bounded function on [2 ,∞).\nThe above estimates give an optimal upper bound to I0(t).\nThe estimate to I0(t) from below is very easy. Indeed, it is obvious that for t >1\nI0(t)≥/integraldisplayt−1/2\n0(1+r2)−tdr≥f(t−1/2)(t−1/2−0) = (1+1 /t)−tt−1/2.\nThen, from the fact that lim\nt→+∞(1 +1/t)−t=e−1, we may fix arbitrary positive C0< e−1and\nchooset0>1 depending on C0such that\nI0(t)≥C0t−1/2, t≥t0. (2.5)\nThe estimates (2.3), (2.4) and (2.5) prove the theorem to the casep= 0.\nCasep >0:To prove the theorem for 0 < p≤3 we split the interval of integration in\nthree parts, that is, we may write\nIp(t) =/integraldisplayβ\n0(1+r2)−trpdr+/integraldisplayt−1/4\nβ(1+r2)−trpdr+/integraldisplay1\nt−1/4(1+r2)−trpdr. (2.6)\nNote that β < t−1/4fort > p2.\nThe next step is to estimate each one of these integrals. Base d on the properties of f(r) and\nthe definition of βwe have\n/integraldisplayβ\n0(1+r2)−trpdr≤f(β)(β−0) = (1+ β2)−tβp+1(2.7)\n= (1+p\n2t−p)−t(p\n2t−p)p+1\n2≤Cp/parenleftBig1\n2t−p/parenrightBigp+1\n2, t > p,\nwhereCp>0 is a constant depending on pand we have used the fact that (1+p\n2t−p)−tis a\ntime-bounded function on the interval [ p,∞).\nNow we want to get an upper bound to the second integral on the r ight hand side of (2.6).\nTo do that we perform the following estimates using the defini tion ofβand integration by parts.\n8/integraldisplayt−1/4\nβ(1+r2)−trpdr=1\n2/integraldisplayt−1/4\nβ(1+r2)−t2rrp−1dr\n=(1+r2)−t+1rp−1\n2(−t+1)/vextendsingle/vextendsingle/vextendsinglet−1/4\nβ−1\n2/integraldisplayt−1/4\nβ(1+r2)−t+1\n−t+1(p−1)rp−2dr\n≤(1+p\n2t−p)−t+1(p\n2t−p)p−1\n2\n2(t−1)+p−1\n2/integraldisplayt−1/4\nβ(1+r2)−t+1\nt−1rp−2dr\n≤Cp\n(t−1)(2t−p)p−1\n2+p−1\n2(t−1)/integraldisplayt−1/4\nβ(1+r2)−t+1rp−2dr, t > max{1,p,p2}.\nAt this point we apply a change of variable u= log(1+ r2) to obtain\n/integraldisplayt−1/4\nβ(1+r2)−trpdr≤Cp\n(t−1)(2t−p)p−1\n2+p−1\n4(t−1)/integraldisplaylog(1+t−1/2)\nlog(1+β2)e−(t−1)u(eu−1)p−3\n2du\n(2.8)\nfor allt > max{1,p}.\nNow we also need to get an upper bound to the integral on the rig ht hand side of the above\nestimate for 0 ≤p≤3.\nFor 0< p≤3 we may estimate for t > max{1,p,p2}\np−1\n4(t−1)/integraldisplaylog(1+t−1/2)\nlog(1+β2)e−(t−1)u(eu−1)p−3\n2du\n≤p−1\n4(t−1)/integraldisplaylog(1+t−1/2)\nlog(1+β2)e−(t−1)u(elog(1+β2)−1)p−3\n2du\n=(p−1)βp−3\n4(t−1)/integraldisplaylog(1+t−1/2)\nlog(1+β2)e−(t−1)udu (2.9)\n≤(p−1)(p\n2t−p)p−3\n2\n4(t−1)e−(t−1)u\nt−1/vextendsingle/vextendsingle/vextendsingle\nlog(1+β2)\n≤Cp\n(t−1)2(2t−p)p−3\n2≤Cp\n(t−1)p+1\n2.\nThe last above inequality with Cp>0 is due to the fact that the function\ne−(t−1)u/vextendsingle/vextendsingle/vextendsingle\nlog(1+β2)= (1+p\n2t−p)−t+1\nis a bound function for t > p.\nNext we need to estimate the third integral on the right hand s ide of (2.6). From the\ndecreasing property of f(r) on the interval of integration, we have\n/integraldisplay1\nt−1/4(1+r2)−trpdr≤f(t−1/4)(1−t−1/4)≤f(t−1/4) = (1+1\nt1/2)−tt−p\n4. (2.10)\n9Now we observe that lim\nt→∞(1+1√\nt)−√\nt=e−1. Then, there exists t0>0 such that\n(1+1√\nt)−√\nt≤2\ne, t≥t0.\nIn particular\n(1+1√\nt)−t≤/parenleftbigg2\ne/parenrightbigg√\nt\n=/parenleftBige\n2/parenrightBig−√\nt\n, t≥t0.\nThen combining this inequality with (2.10) we may conclude t hat\n/integraldisplay1\nt−1/4(1+r2)−trpdr≤t−p\n4/parenleftBige\n2/parenrightBig−√\nt\n, t≥t0. (2.11)\nBy substituting the estimates (2.7), (2.8) combined with (2 .9) and (2.11) in (2.6), we obtain\nthe following optimal upper bound to Ip(t) to the case 0 < p≤3.\nIp(t)≤C t−p+1\n2, t≫1. (2.12)\nFinally we have to prove the lower estimate for the case 0 < p≤3. To this case the function\nf(r) = (1+ r2)−trpis increasing on the interval (0 ,β) withβ=/radicalbiggp\n2t−p, t > p.\nThe next estimate give us the conclusion of the proof of the op timality of the decay rate for\nIp(t), p >0. In fact, the limit\nlim\nt→∞(1+p\n4(2t−p))−t=e−p\n8\nimplies the existence of t0>0 such that\nIp(t)≥f/parenleftBigβ\n2/parenrightBig/parenleftBig\nβ−β\n2/parenrightBig\n=/parenleftbigg\n1+β2\n4/parenrightbigg−tβp+1\n2p+1≥1\n2e−p\n8βp+1\n2p+1≥e−p\n8\n2p+2/parenleftBigp\n2t−p/parenrightBigp+1\n2, t > p,\nbecauseβ=/radicalbiggp\n2t−pis the global maximum of f(r).\n/square\nThe proof of Theorem 2.1 is now established. However, our mai n aim in this section is to\nextend the result of this theorem for all p≥0. In order to do this we need the next important\nproperty of the hypergeometric function Ip(t) forp≥2.\nLemma 2.1 (Recurrence formula) Letp≥2be a real number. Then\nIp(t) =2−t+1\np+1−2t+p−1\n2t−p−1Ip−2(t), t >p+1\n2.\n10Proof.Letp≥2, andt >p+1\n2. It follows from integration by parts that\nIp(t) =1\n2/integraldisplay1\n0(1+r2)−t2rrp−1dr\n=2−t\n1−t+p−1\n2t−2/integraldisplay1\n0(1+r2)−t(1+r2)rp−2dr\n=2−t\n1−t+p−1\n2t−2/integraldisplay1\n0(1+r2)−trp−2dr+p−1\n2t−2/integraldisplay1\n0(1+r2)−trpdr\n=2−t\n1−t+p−1\n2t−2Ip−2(t)+p−1\n2t−2Ip(t),\nwhich implies the identity\nIp(t) =2−t+1\np+1−2t+p−1\n2t−p−1Ip−2(t).\nThis yields the desired equality for p≥2. /square\nCombining the recurrence formula with Theorem 2.1 we may pro ve the general result for\nIp(t).\nTheorem 2.2 Letp≥0be a real number. Then\nIp(t)∼t−p+1\n2, t≫1.\nProof.Applying Lemma 2.1 for 3 ≤p≤4 and using the result of Theorem 2.1, which holds\nfor 1≤p−2≤2 we get the proof for 3 ≤p≤4. By a similar argument to the case for 4 ≤p≤5\nand 2≤p−2≤3 we obtain the statement for 4 ≤p≤5. The general result follows using the\nprinciple of induction.\n/square\nRemark 2.1 It follows from Theorem 2.2 that the optimal rate of decay of t he function In−1(t)\nis the same as that of the Gauss kernel in L2-sense:/ba∇dblG(t,·)/ba∇dbl2∼t−n\n2ast→ ∞, where\nG(t,x) :=1\n(√\n4πt)ne−|x|2\n4t.\nIn order to deal with the high frequency part of estimates, on e defines a function\nJp(t) =/integraldisplay∞\n1(1+r2)−trpdr\nforp∈R.\nThen the next lemma is important to get estimates on the zone o f high frequency to problem\n(1.1)–(1.2).\nLemma 2.2 Letp∈R. Then it holds that\nJp(t)∼2−t\nt−1, t≫1.\n11Proof.We first note that\nJp(t) =/integraldisplay∞\n1e−tlog(1+r2)rpdr, t > 1.\nApplying a change of variable u= log(1+ r2) we get\nJp(t) =1\n2/integraldisplay∞\nlog2e−(t−1)u(eu−1)p−1\n2du.\nForp <1 andu≥log2 we have\nep−1\n2u≤(eu−1)p−1\n2≤1.\nThen using this inequality we obtain for t >1 the double below-above estimate\n2p−1\n22−t\nt−p+1\n2=1\n2/integraldisplay∞\nlog2e−(t−p+1\n2)udu≤Jp(t)≤1\n2/integraldisplay∞\nlog2e−(t−1)udu=2−t\nt−1\nForp≥1 andu≥log2 we have the inequality\n1≤(eu−1)p−1\n2≤ep−1\n2u.\nThus we also obtain for this case and t >1\n2−t\nt−1≤Jp(t)≤2p−1\n22−t\nt−p+1\n2.\nThese estimates imply the lemma. /square\nFor later use we prepare the following simple lemma, which im plies the exponential decay\nestimates of the middle frequency part.\nLemma 2.3 Letp∈R, andη∈(0,1]. Then there is a constant C >0such that\n/integraldisplay1\nη(1+r2)−trpdr≤C(1+η2)−t, t≥0.\n2.2 Inequalities and asymptotics\nLemma 2.4 Leta(ξ)andb(ξ)be the functions given by\na(ξ) =log(1+|ξ|2)\n2andb(ξ) =1\n2/radicalBig\n4|ξ|2−log2(1+|ξ|2) (2.13)\nforξ∈Rn. Then, the following estimates hold.\n(i)|a(ξ)|2\n|b(ξ)|2≤1\n3, ξ/ne}ationslash= 0;\n(ii)(b(ξ)−|ξ|)2\nb(ξ)2≤28\n3, ξ/ne}ationslash= 0.\nProof.To prove the lemma we use the elementary inequality\n|ξ|−log(1+|ξ|2)≥0,for allξ∈Rn.\n12Then\n(ii)|a(ξ)|2\n|b(ξ)|2=log2(1+|ξ|2)\n4|ξ|2−log2(1+|ξ|2)≤|ξ|2\n4|ξ|2−|ξ|2≤1\n3, ξ/ne}ationslash= 0;\n(ii)(b(ξ)−|ξ|)2\nb(ξ)2≤4(1+4|ξ|2\nb(ξ)2)≤4/parenleftbigg\n1+4|ξ|2\n4|ξ|2−log2(1+|ξ|2)/parenrightbigg\n≤4(1+4|ξ|2\n3|ξ|2)≤28\n3, ξ/ne}ationslash= 0.\n/square\nIn the next section, to study an asymptotic profile of the solu tion to problem (1.1)–(1.2) we\nconsider a decomposition of the Fourier transformed initia l data.\nRemark 2.2 Using the Fourier transform we can get a decomposition of the initial data ˆ u1as\nfollows\nˆu1(ξ) =A1(ξ)−iB1(ξ)+P1, ξ∈Rn,\nwhereP1,A1,B1are defined by\nP1=/integraldisplay\nRnu1(x)dx, A 1(ξ) =/integraldisplay\nRnu1(x)/parenleftbig\n1−cos(ξx)/parenrightbig\ndx, B 1(ξ) =/integraldisplay\nRnu1(x)sin(ξx)dx.\nAccording to the above decomposition we can derive the follo wing lemma (see Ikehata [13]).\nLemma 2.5 Letκ∈[0,1]. Foru1∈L1,κ(Rn)andξ∈Rnit holds that\n|A1(ξ)| ≤K|ξ|κ/ba∇dblu1/ba∇dblL1,κand|B1(ξ)| ≤M|ξ|κ/ba∇dblu1/ba∇dblL1,κ,\nwith positive constants KandMdepending only on n.\nIn order to show the optimality of the decay rates we need next two lemmas.\nLemma 2.6 Letn >2. Then there exists t0>0such that for t≥t0it holds that\nC−1t−n−2\n2≥/integraldisplay\nRne−tlog(1+|ξ|2)|sin(|ξ|t)|2\n|ξ|2dξ≥Ct−n−2\n2,\nwithCa positive constant depending only on n.\nProof.First, we may note that\nM(t) : =/integraldisplay\nRne−tlog(1+|ξ|2)|sin(|ξ|t)|2\n|ξ|2dξ\n=ωn/integraldisplay∞\n0e−tlog(1+r2)rn−3|sin(rt)|2dr\n≥ωn/integraldisplay∞\n0e−t r2rn−3|sin(rt)|2dr.\nConsidering a change of variable s=r√\nt, for fixt >0 we arrive at\nM(t)≥ωn\ntn−2\n2/integraldisplay∞\n0e−s2sn−3sin2/parenleftbig\ns√\nt)ds.\n13Using the identity\n2sin2x= (1−cos2x),\nwe obtain\nM(t)≥1\n2ωnt−n−2\n2/integraldisplay∞\n0e−s2sn−3/parenleftBig\n1−cos(2s√\nt)/parenrightBig\nds\n=1\n2ωnt−n−2\n2/parenleftbig\nA−Fn(t)/parenrightbig\n,\nwhere\nA=/integraldisplay∞\n0e−s2sn−3ds, F n(t) =/integraldisplay∞\n0e−s2sn−3cos/parenleftbig\n2s√\nt/parenrightbig\nds.\nDue to the fact e−s2sn−3∈L1(R) (n >2), we can apply the Riemann-Lebesgue theorem to\nget\nFn(t)→0, t→ ∞.\nThen we conclude the existence of t0>0 such that Fn(t)≤A\n2for allt≥t0. Thus, the half part\nof lemma is proved with C=ωnA\n4.\nNext, let us prove upper bound of decay estimates. Indeed,\nM(t)≤/integraldisplay\nRne−tlog(1+|ξ|2)|ξ|−2dξ\n=ωn/integraldisplay∞\n0e−tlog(1+r2)rn−3dr\n=ωn/integraldisplay∞\n0(1+r2)−trn−3dr=ωnIn−3(t)+ωnJn−3(t)\n≤C1,nt−n−2\n2+C2,n2−t\nt,\nwhere one has just used Theorem 2.2 and Lemma 2.2. These imply the desired estimates. /square\nFollowing the same ideas of Lemma 2.6 one can prove the follow ing result, however, this is\nnot used in the paper.\nLemma 2.7 Letn≥1. Then there exists t0>0such that vale\nC−1\nnt−n\n2≥/integraldisplay\nRne−tlog(1+|ξ|2)cos2(|ξ|t)dξ≥Cnt−n\n2, t≥t0,\nwhereCnis a positive constant depending only on n.\n3 Asymptotic profiles of solutions\nThe associated Cauchy problem to (1.1)-(1.2) in the Fourier space is given by\nˆutt(t,ξ)+|ξ|2ˆu(t,ξ)+log(1+ |ξ|2)ˆu= 0, (3.1)\nˆu(0,ξ) =u0(ξ),ˆut(0,ξ) =u1(ξ).\nThe characteristics roots λ+andλ−of the characteristic polynomial\nλ2+log(1+ |ξ|2)λ+|ξ|2= 0, ξ∈Rn\n14associated to the equation (3.1) are given by\nλ±=−log(1+|ξ|2)±/radicalBig\nlog2(1+|ξ|2)−4|ξ|2\n2. (3.2)\nIt should be mentioned that log(1+ |ξ|2)−4|ξ|2<0 for allξ∈Rn,ξ/ne}ationslash= 0, and the characteristics\nroots are complex and the real part is negative, for all ξ∈Rn,ξ/ne}ationslash= 0. Then we can write down\nλ±in the following form\nλ±=−a(ξ)±ib(ξ),\nwherea(ξ) andb(ξ) are defined by (2.13) in Lemma 2.4. In this case the solution o f the problem\n(3.1) is given explicitly by\nˆu(t,ξ) =/parenleftbigg\nˆu0(ξ)cos(b(ξ)t)+ˆu1(ξ)+ ˆu0(ξ)a(ξ)\nb(ξ)sin(b(ξ)t)/parenrightbigg\ne−a(ξ)t\nforξ∈Rn,ξ/ne}ationslash= 0 and t≥0.\nNext, in order to find a better expression for ˆ u(t,ξ) we apply the mean value theorem to get\nsin(b(ξ)t) = sin(|ξ|t)+t(b(ξ)−|ξ|)cos(µ(ξ)t), (3.3)\nwith\nµ(ξ) :=θ1b(ξ)+(1−θ1)|ξ|\nfor some θ1∈(0,1), and\n1/radicalbig\n1−g(r)= 1+log2(1+r2)\n8r21/radicalbig\n(1−θ2g(r))3(3.4)\nwith some θ2∈(0,1), where r:=|ξ|, and\ng(r) :=log2(1+r2)\n4r2.\nThe identity (3.4) was obtained applying the mean value theo rem to the function\nG(s) =1/radicalbig\n(1−sg(r))3,0≤s≤1.\nThen by using Remark 2.2, (3.3) and (3.4) ˆ u(t,ξ) can be re-written as\nˆu(t,ξ) =P1e−a(ξ)tsin(tr)\nr+P1log2(1+r2)\n8r31/radicalbig\n(1−θ2g(r))3e−a(ξ)tsin(tr)\n+e−a(ξ)tcos(b(ξ)t)ˆu0(ξ)+/parenleftbigga(ξ)\nb(ξ)/parenrightbigg\ne−a(ξ)tsin(b(ξ)t)ˆu0(ξ) (3.5)\n+/parenleftbiggA1(ξ)−iB1(ξ)\nb(ξ)/parenrightbigg\ne−a(ξ)tsin(b(ξ)t)+P1te−a(ξ)t/parenleftbiggb(ξ)−r\nb(ξ)/parenrightbigg\ncos(µ(ξ)t).\nWe want to introduce an asymptotic profile as t→ ∞in a simple form:\nP1e−a(ξ)tsin(|ξ|t)\n|ξ|, (3.6)\nwherea(ξ) =log(1+|ξ|2)\n2.\nOur goal in this section is to get decay estimates in time to th e remainder therms defined in\n(3.5). To proceed with that we define the next 5 functions whic h imply remainders with respect\nto the leading term (3.6).\n15•K1(t,ξ) =/parenleftBigA1(ξ)−iB1(ξ)\nb(ξ)/parenrightBig\ne−a(ξ)tsin(b(ξ)t);\n•K2(t,ξ) = ˆu0(ξ)a(ξ)\nb(ξ)e−a(ξ)tsin/parenleftbig\nb(ξ)t/parenrightbig\n;\n•K3(t,ξ) = ˆu0(ξ)e−a(ξ)tcos/parenleftbig\nb(ξ)t/parenrightbig\n;\n•K4(t,ξ) =P1e−a(ξ)tsin(rt)log2(1+r2)\n8r31/radicalbig\n(1−θ2g(r))3, r=|ξ|>0;\n•K5(t,ξ) =P1e−a(ξ)tt/parenleftbiggb(ξ)−|ξ|\nb(ξ)/parenrightbigg\ncos(µ(ξ)t),\nwherea(ξ) andb(ξ) are defined in Lemma 2.4. Note that using these Kj(t,ξ) (j= 1,2,3,4,5)\nthe solution ˆ u(t,ξ) to problem (3.1) can be expressed as\nˆu(t,ξ)−P1e−a(ξ)tsin(tr)\nr=5/summationdisplay\nj=1Kj(t,ξ). (3.7)\nLet us check, in fact, that {Kj(t,ξ)}become error terms by using previous lemmas studied\nin Section 2.\nFirst we obtain decay rates for each one of these functions on the zone of low frequency\n|ξ| ≪1.\nWe begin with K1(t,ξ).\nFor this function we prepare the following expression for 1 /b(ξ) based on (3.4):\n1\nb(ξ)=1\nr+log2(1+r2)\n8r31/radicalbig\n(1−θ2g(r))3, r=|ξ|>0. (3.8)\nThen,\nK1(t,ξ) :A1(ξ)−iB1(ξ)\n|ξ|e−a(ξ)tsin(b(ξ)t)\n+/parenleftbig\nA1(ξ)−iB1(ξ)/parenrightbiglog2(1+r2)\n8r3e−a(ξ)tsin(b(ξ)t)/radicalbig\n(1−θ2g(r))3=:K1,1(t,ξ)+K1,2(t,ξ).\nIt is easy to check the following estimate based on Lemma 2.5 w ithk= 1 and Theorem 2.2:\n/integraldisplay\n|ξ|≤1|K1,1(t,ξ)|2dξ≤(M+K)2/ba∇dblu1/ba∇dbl2\n1,1/integraldisplay\n|ξ|≤1e−tlog(1+|ξ|2)dξ\n≤ωn(M+K)2/ba∇dblu1/ba∇dbl2\n1,1/integraldisplay1\n0(1+r2)−trn−1dr\n≤Cωn(M+K)2t−n\n2/ba∇dblu1/ba∇dbl2\n1,1,(t≫1). (3.9)\nOn the other hand, since\nlim\nr→+0log2(1+r2)\nr2= 0,\nthere is a constant δ >0 such that for all 0 < r≤δit holds that\nlog2(1+r2)\n4r2≤1\n2. (3.10)\n16Then, the definition of g(r) implies\n1/radicalbig\n(1−θ2g(r))3≤2√\n2. (3.11)\nThus, from (3.10), (3.11) and Theorem 2.2 together with Lemm a 2.5 for k= 1, one has\n/integraldisplay\n|ξ|≤δ|K1,2(t,ξ)|2dξ≤8−1(M+K)2/ba∇dblu1/ba∇dbl2\n1,1/integraldisplay\n|ξ|≤δ/parenleftbigglog2(1+r2)\nr2/parenrightbigg2\ne−2ta(ξ)dξ\n≤1\n2(M+K)2/ba∇dblu1/ba∇dbl2\n1,1/integraldisplay\n|ξ|≤δe−2ta(ξ)dξ\n≤1\n2(M+K)2/ba∇dblu1/ba∇dbl2\n1,1ωn/integraldisplay1\n0(1+r2)−trn−1dξ\n≤Cωn(M+K)2/ba∇dblu1/ba∇dbl2\n1,1t−n\n2,(t≫1). (3.12)\nBy combining (3.9) and (3.12) we have the following estimate forK1(t,ξ),\n/integraldisplay\n|ξ|≤δ|K1(t,ξ)|2dξ≤C1,n/ba∇dblu1/ba∇dbl2\n1,1t−n\n2,(t≫1). (3.13)\nSimilarly to the computation for (3.13), one can obtain the e stimate for K4(t,ξ)\n/integraldisplay\n|ξ|≤δ|K4(t,ξ)|2dξ≤C1,n|P1|t−n\n2,(t≫1), (3.14)\nbecause\nlim\nr→+0log2(1+r2)\nr3= 0. (3.15)\nForK2(t,ξ) andK3(t,ξ), by using (i) of Lemma 2.4 one can easily obtain the estimate :\n/integraldisplay\n|ξ|≤1|Kj(t,ξ)|2dξ≤C1,n/ba∇dblu0/ba∇dbl1t−n\n2,(t≫1), (3.16)\nfor each j= 2,3. So, it suffices to deal with the case for K5(t,ξ). For this we remark that\nb(ξ)−r=r\n−log2(1+r2)\n4r2\n1+/radicalBig\n1−log2(1+r2)\n4r2\n.\nThis implies\n|b(ξ)−r| ≤r3log2(1+r2)\n4r4=:r3h(r),\nwhere we see\nh(r) =log2(1+r2)\n4r4→1\n4(r→+0).\nSo, there exists a constant δ0>0 such that for all r∈(0,δ0] it holds that\n0<log2(1+r2)\n4r4≤1.\nThus, one can estimate K5(t,ξ) as follows:\n/integraldisplay\n|ξ|≤δ0|K5(t,ξ)|2dξ≤ |P1|2t2/integraldisplay\n|ξ|≤δ0r6e−2ta(ξ)\nb(ξ)2dξ. (3.17)\n17On the other hand, since log2(1+r2)≤2r2for allr≥0, it follows that\n1\nb(ξ)2=4\n4r2−log2(1+r2)≤2\nr2,|ξ|=r >0.\nTherefore, one can estimate (3.17) for r∈(0,δ1] with sufficiently small δ1≤δ0as follows\n/integraldisplay\n|ξ|≤δ1|K5(t,ξ)|2dξ≤2|P1|2t2/integraldisplay\n|ξ|≤δ1r4e−2ta(ξ)\nb(ξ)2dξ≤C|P1|2t−n\n2, (3.18)\nwhere one has just used Theorem 2.2 and the definition of a(ξ) in Lemma 2.4.\nNow, by summarizing above discussion one can arrived at the f ollowing crucial lemma based\non (3.7), (3.13), (3.14), (3.16), and (3.18).\nProposition 3.1 Letn≥1. Then, there exists a small constant δ1∈(0,1]such that\n/integraldisplay\n|ξ|≤δ1/vextendsingle/vextendsingleˆu(t,ξ)−P1e−a(ξ)tsin(tr)\nr/vextendsingle/vextendsingle2dξ≤C/parenleftbig\n|P1|2+/ba∇dblu0/ba∇dbl2\n1+/ba∇dblu1/ba∇dbl2\n1,1/parenrightbig\nt−n\n2,(t≫1),\nwith some generous constant C=Cn>0depending only on the dimension n.\nNext, let us prepare the so-called high frequency estimates for such error terms Ki(t,ξ).\nThese terms decay very fast, as usual.\nLemma 3.1 Letn≥1. Then, it holds that\n/integraldisplay\n|ξ|≥δ1|Ki(t,ξ)|2dξ≤C||uj||2\n1o(t−n\n2),(t→ ∞),\nwherej= 1fori= 1,4,5andj= 0fori= 2,3, andδ1>0is a number defined in Proposition\n3.1.\nProof.We give the proof only for K5(t,ξ). The other cases are similar. Indeed, it follows from\n(ii) of Lemma 2.4, Lemmas 2.2 and 2.3 that\n/integraldisplay\n|ξ|≥δ1|K5(t,ξ)|2dξ≤16\n3|P1|2t2/integraldisplay\n|ξ|≥δ1e−log(1+ξ2)tdξ\n≤16\n3|P1|2t2/parenleftBigg/integraldisplay\n1≥|ξ|≥δ1e−log(1+ξ2)tdξ+/integraldisplay\n|ξ|≥δ1e−log(1+ξ2)tdξ/parenrightBigg\n≤16\n3|P1|2t2ωn/parenleftbigg/integraldisplay1\nδ1(1+r2)−trn−1dr+/integraldisplay∞\n1(1+r2)−trn−1dr/parenrightbigg\n≤16\n3|P1|2t2ωn/parenleftbigg/integraldisplay1\nδ1(1+r2)−trn−1dr+/integraldisplay∞\n1(1+r2)−trn−1dr/parenrightbigg\n≤16\n3|P1|2t2ωn/parenleftbigg\nC(1+δ2\n1)−t+2−t\nt−1/parenrightbigg\n,\nwhich implies the desired estimate for K5(t,ξ). /square\nNow, asadirectconsequenceofLemma3.1and(3.7)onecanget thehighfrequencyestimates\nfor the error terms.\n18Proposition 3.2 Letn≥1. Then, there exists a small constant δ1>0such that\n/integraldisplay\n|ξ|≥δ1|ˆu(t,ξ)−P1e−a(ξ)tsin(t|ξ|)\n|ξ||2dξ≤C(/ba∇dblu1/ba∇dbl2\n1+/ba∇dblu0/ba∇dbl2\n1)o(t−n\n2),(t→ ∞),\nwith some generous constant C >0.\nFinally, Theorem 1.1 is a direct consequence of Proposition s 3.1 and 3.2\nRemark 3.1 The decay rate stated in Proposition 3.2 can be drawn with a mo re precise fast\ndecay rate, however, since the decay rate in Proposition 3.1 is essential, and the rate of decay in\nProposition 3.2 can be absorbed into that of Proposition 3.1 , we have employed such style for\nsimplicity.\n4 Optimal rate of decay of solutions.\nIn this section we study the optimal decay rate in the sense of L2-norm of the solutions to\nproblem (1.1)-(1.2).\nWe first prepare the following proposition in the one dimensi onal case.\nProposition 4.1 It is true that\n/integraldisplay\nR(1+|ξ|2)−tsin2(t|ξ|)\n|ξ|2dξ∼t,(t≫1).\nProof.We set\nQ(t) :=/integraldisplay∞\n0(1+r2)−tsin2(tr)\nr2dr,\nand it suffices to obtain the estimate stated in Proposition fo rQ(t). Then, Q(t) can be divided\ninto two parts:\nQ(t) =Ql(t)+Qh(t),\nwith\nQl(t) :=/integraldisplay1/t\n0(1+r2)−tsin2(tr)\nr2dr,\nQh(t) :=/integraldisplay∞\n1/t(1+r2)−tsin2(tr)\nr2dr.\n(i)upper bound for Qj(t) withj=l,h.\nIndeed, let 1 /t <1. Then,\nQl(t)≤/integraldisplay1/t\n0(1+r2)−t(tr)2\nr2dr=t2/integraldisplay1/t\n0(1+r2)−tdr≤t2/integraldisplay1/t\n0dr=t,\nwhere one has just used the fact that if 0 ≤tr≤1 then 0 ≤sin(tr)≤tr. This implies\nQl(t)≤Ct(t >1). (4.1)\nOn the other hand, it follows from integration by parts one ca n get\nQh(t)≤/bracketleftbig\n−r−1(1+r2)−t/bracketrightbigr:=∞\nr:=1/t−2t/integraldisplay∞\n1/t(1+r2)−t−1dr≤t(1+1\nt2)−t.\n19Now, since\nlim\nt→∞(1+1\nt2)−t= 1, (4.2)\nthere is a constant t0≫1 such that for all t≥t0\nQh(t)≤2t. (4.3)\nEstimates (4.1) and (4.3) imply\nQ(t)≤Ct(t≫1). (4.4)\n(ii)lower bound for Qj(t) withj=l,h.\nIndeed, let 1 /t <1 again. Then, since 2sin( tr)≥trif 0≤tr≤1, the following estimate holds\nfort >1\nQl(t)≥t2\n4/integraldisplay1/t\n0(1+r2)−tdr≥t2\n4(1+1\nt2)−t1\nt=t\n4(1+1\nt2)−t.\nThus, because of (4.2) we can get\nQl(t)≥Ct(t≫1). (4.5)\nTo treat Qh(t) we set\nν:=5π\n4t, ν′:=7π\n4t, (4.6)\nand\nρ:=49π2\n16.\nIfν≤r≤ν′, then one has\n|sin(tr)| ≥1√\n2. (4.7)\nSo, one can get a series of estimates from below because of1\nt<5π\n4t(1< t):\nQh(t)≥1\n2/integraldisplayν′\nν(1+r2)−tr−2dr≥1\n2(7π\n4t)−2/integraldisplayν′\nν(1+r2)−tdr,\n≥1\n216t2\n49π2(1+ρ2\nt2)−t(ν′−ν)\n=4\n49π(1+ρ\nt2)−tt.\nSince\nlim\nt→∞(1+ρ\nt2)−t= 1,\none can arrive at the crucial estimate:\nQh(t)≥Ct(t≫1). (4.8)\nBy combining (4.5) and (4.8) it results that\nQ(t)≥Ct(t≫1). (4.9)\nFinally, the desired estimate can be accomplished by (4.4) a nd (4.9). /square\nNext we deal with the two dimensional case, which is rather di fficult.\n20Proposition 4.2 It is true that\n/integraldisplay\nR2(1+|ξ|2)−tsin2(t|ξ|)\n|ξ|2dξ∼logt,(t≫1).\nProof.It suffices to get the result to the following function after po lar coordinate transform:\nR(t) :=/integraldisplay∞\n0(1+r2)−tsin2(tr)\nrdr.\nThen,R(t) can be divided into two parts:\nR(t) =Rl(t)+Rh(t),\nwith\nRl(t) :=/integraldisplay1/t\n0(1+r2)−tsin2(tr)\nrdr,\nRh(t) :=/integraldisplay∞\n1/t(1+r2)−tsin2(tr)\nrdr.\n(i)upper bound for Rj(t) withj=l,h.\nIndeed, let 1 /t <1. Then,\nRl(t)≤/integraldisplay1/t\n0(1+r2)−t(tr)2\nrdr=t2/integraldisplay1/t\n0(1+r2)−trdr\n≤t/integraldisplay1/t\n0(1+r2)−tdr≤t/integraldisplay1/t\n0dr= 1,\nwhere one has just used the fact that if 0 ≤tr≤1 then 0 ≤sin(tr)≤tr. This implies\nRl(t)≤1 (t >1). (4.10)\nOn the other hand, it follows from the integration by parts on e can get\nRh(t)≤/integraldisplay∞\n1/t(1+r2)−tr−1dr\n=/bracketleftbig\n(logr)(1+r2)−t/bracketrightbigr:=∞\nr:=1/t+2t/integraldisplay∞\n1/t(rlogr)(1+r2)−t−1dr\n= (logt)(1+t−2)−t+2t/integraldisplay1\n1/t(rlogr)(1+r2)−t−1dr+2t/integraldisplay∞\n1(rlogr)(1+r2)−t−1dr\n≤(logt)(1+t−2)−t+2t/integraldisplay∞\n1(rlogr)(1+r2)−t−1dr\n≤(logt)(1+t−2)−t+2t/integraldisplay∞\n1r2(1+r2)−t−1dr (4.11)\n≤(logt)(1+t−2)−t+2t/integraldisplay∞\n1(1+r2)(1+r2)−t−1dr\n= (logt)(1+t−2)−t+2t/integraldisplay∞\n1(1+r2)−tdr.\nNow, because of Lemma 2.2 one can see that\n/integraldisplay∞\n1(1+r2)−tdr≤C2−t\nt−1,(t≫1), (4.12)\n21and since lim\nt→∞(1+t−2)−t= 1, one has (1+ t−2)−t≤2 fort≫1. Thus, it follows from (4.11)\nand (4.12), one can arrive at the estimate:\nRh(t)≤2logt+C2−t,(t≫1). (4.13)\n(4.10) and (4.13) implies the upper bound for tof the quantity R(t):\nR(t)≤Clogt,(t≫1). (4.14)\n(ii)lower bound for R(t). The lower bound for R(t) we do not need to separate R(t) into\nRj(t) withj=l,h, and prove at a stroke. The following property is essential.\nlog(1+r2)≤r2(r∈R). (4.15)\nOncewenotice (4.15), thederivation of thelower boundof in finitetimeblowuprateissimilar\nto [16]. Indeed, by using (4.15) and a change of variable one c an estimate R(t) as follows.\nR(t) =/integraldisplay∞\n0e−tlog(1+r2)r−1sin2(tr)dr\n≥/integraldisplay∞\n0e−tr2r−1sin2(tr)dr=/integraldisplay∞\n0e−σ2σ−1sin2(√\ntσ)dσ.\nBy setting\nνj:= (1\n4+j)π√\nt, ν′\nj:= (3\n4+j)π√\nt(j= 1,2,3,···)\nand using integration by parts one has\nR(t)≥1\n2∞/summationdisplay\nj=1/integraldisplayν′\nj\nνje−σ2σ−1dσ≥1\n4/integraldisplay∞\n5π\n4√\nte−σ2σ−1dσ\n=−1\n4log/parenleftbigg5π\n4√\nt/parenrightbigg\ne−25π2\n16t+1\n2/integraldisplay∞\n5π\n4√\nt(σlogσ)e−σ2dσ (4.16)\n≥1\n8e−25π2\n16tlogt−1\n4e−25π2\n16tlog5π\n4−1\n2/integraldisplay∞\n0σ|logσ|e−σ2dσ.\nSince\nlim\nt→∞e−25π2\n16t= 1,\nand /integraldisplay∞\n0σ|logσ|e−σ2dσ <+∞,\n(4.16) implies the desired estimate\nR(t)≥Clogt(t≫1). (4.17)\nWe may note that (4.14) and (4.17) imply the desired statemen t forR(t). /square\nFinally, let us now prove Theorem 1.2 at a stroke.\nProof of Theorem 1.2 completed. It follows from the Plancherel theorem and triangle inequal -\nity, with some constant Cn>0 one can get\nCn/ba∇dblu(t,·)/ba∇dbl ≥ |P1|/ba∇dbl(1+|ξ|2)−t\n2sin(t|ξ|)\n|ξ|/ba∇dbl−/ba∇dblˆu(t,·)−P1(1+|ξ|2)−t\n2sin(t|ξ|)\n|ξ|/ba∇dbl.\n22and\nCn/ba∇dblu(t,·)/ba∇dbl ≤ |P1|/ba∇dbl(1+|ξ|2)−t\n2sin(t|ξ|)\n|ξ|/ba∇dbl+/ba∇dblˆu(t,·)−P1(1+|ξ|2)−t\n2sin(t|ξ|)\n|ξ|/ba∇dbl.\nThese inequalities together with Theorem 1.1, Lemma 2.6 and Propositions 4.1 and 4.2 imply\nthe desired estimates. This part is, nowadays, well-known ( see [14, 16]). /square\nAcknowledgement. The work of the first author (R. C. CHAR ˜AO) was partially supported\nby PRINT/CAPES - Process 88881.310536/2018-00 and the work of the second author (R.\nIKEHATA) was supportedin part by Grant-in-Aid for Scientifi c Research (C)15K04958 of JSPS.\nReferences\n[1] J. 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Sobajima, Higher order asymptotic expansion of solu tions to abstract linear hyperbolic\nequations, Math. Ann. (2020). https://doi.org/10.1007/s 00208-020-01959-w\n[32] H. Volkmer, Asymptotic expansion of L2-norms of solutions to the heat and dissipative\nwave equations, Asymptotic Anal. 67(2010), 85-100. doi:10.3233/ASY-2010-0980.\n5 Appendix\nIn this appendix, let us describe the outline of proof of the unique ex istence of a mild solution to\nproblem (1.1)-(1.2) more in detail by applying the Lumer-Phillips Theor em (cf. Pazy [26, Theorem 4.3]).\nConcerning a relation between two nonnegative self-adjoint opera torsAandL, it holds that D(A)⊂\nD(A1/2)⊂D(L)⊂H:=L2(Rn) withD(A1/2) =H1(Rn). We first prepare the Kato-Rellich Theorem.\nTheorem 5.1 (Kato-Rellich) LetXbe a Hilbert space with its norm /ba∇dbl·/ba∇dbl, and let T:D(T)⊂X→X\nbe a self-adjoint operator in X. Furthermore, let V:D(V)⊂X→Xbe a symmetric operator in X.\nAssume that\n(1)D(T)⊂D(V),\n(2)there exist constants δ∈[0,1)andC >0such that /ba∇dblVu/ba∇dbl ≤δ/ba∇dblTu/ba∇dbl+C/ba∇dblu/ba∇dblforu∈D(T).\nThen, the operator T+Vis also self-adjoint in Xwith its domain D(T+V) =D(T).\nNow, let H0:=H1(Rn)×L2(Rn) be the Hilbert space with its inner product defined by\n</bracketleftbiggu\nv/bracketrightbigg\n,/bracketleftbiggw\nz/bracketrightbigg\n>:= (u,w)+(A1/2u,A1/2w)+(v,z),\nwhere (·,·) implies the usual inner product in L2(Rn). Furthermore, let us define a operator\nA:H0→ H0\nbyD(A) :=H2(R)×H1(R), and for U:=/bracketleftbiggu\nv/bracketrightbigg\n∈D(A);\nAU:=/bracketleftbiggv\n−Au−Lv/bracketrightbigg\n.\nNote that v∈H1(Rn) implies v∈D(L). Under these preparations we first show that\n(i)The operator A−1\n2Iis dissipative in H0.\nIndeed, let U:=/bracketleftbiggu\nv/bracketrightbigg\n∈D(A). Then\n<AU,U>=</bracketleftbiggv\n−Au−Lv/bracketrightbigg\n,/bracketleftbiggu\nv/bracketrightbigg\n>= (v,u)−(Lv,v)≤(u,v)≤1\n2(/ba∇dblu/ba∇dbl2+/ba∇dblv/ba∇dbl2)≤1\n2<U,U>,\nwhich implies the desired estimate. Here, one has just used the non- negativity of the self-adjoint operator\nLinL2(Rn) such that\n(Lv,v) =cn(log(1+ |·|2)ˆv,ˆv)≥0\nwith some constant cn>0.\n(ii)ForB:=A−1\n2I, we have to check R(1\n2I −B) =H0.\nOnce (i) and (ii) can be proved, it follows from the Lumer-Phillips Theor em that the operator B\ngenerates a C0semigroup etBof contractionson H0, and soetA=et\n2etBcan be a generated C0semigroup\nonH0(cf. [18, Proposition 2.1]).\n25It suffices to check that R(1\n2I −B) =R(I −A) =H0, that is, we have to solve the problem that for\neach/bracketleftbiggf\ng/bracketrightbigg\n∈ H0, there exists a solution/bracketleftbiggu\nv/bracketrightbigg\n∈D(A) such that\nu−v=f∈H1(Rn), (5.1)\nv+Au+Lv=g∈L2(Rn). (5.2)\nWe can find a pair of solution [ u,v] to problems (5.1) and (5.2) by\nu:= (I+A+L)−1(f+Lf+g), (5.3)\nand\nv:=u−f.\nWe easily see that u∈H2(Rn), andv∈H1(Rn). In order to check the well-posedness of the solution\n(5.3) it is enough to make sure that the operator A+L:D(A)→L2(R) is self-adjoint in L2(Rn).\nLet us apply the Kato-Rellich Theorem to check that A+Lis self-adjoint in H:=L2(Rn) with its\ndomainD(A+L) =D(A).\nSetφ(x) := log(1+ x)(1+x)−1. Then, since\nmax\nx≥0φ(x) =φ(e−1) =e−1,\nit holds that for v∈D(A) =H2(Rn),\n/ba∇dblLv/ba∇dbl2≤1\ne2/integraldisplay\nRn|ˆv(ξ)|2(1+|ξ|2)2dξ≤4\ne2(/ba∇dblv/ba∇dbl2+/ba∇dblAv/ba∇dbl2),\nwhich implies\n/ba∇dblLv/ba∇dbl ≤2\ne(/ba∇dblv/ba∇dbl+/ba∇dblAv/ba∇dbl) (5.4)\nwith 2/e∈(0,1). Therefore, by the Kato-Rellich theorem the operator A+Lbecomes self-adjoint, and\nnon-negative in H.\nFinally,\nU(t) = [u(t),u′(t)] :=etA[u0,u1]\nbecomes a unique mild solution to problem\ndU\ndt=AU(t), U(0) =/bracketleftbiggu0\nu1/bracketrightbigg\n.\nThis implies that the problem (1.1)-(1.2) has a desired unique weak solu tion\nu∈C([0,∞);H1(Rn))∩C1([0,∞);L2(Rn)).\nFinally, by density argument and the multiplier method one can get the energy inequality\nEu(t)≤Eu(s),(0≤s≤t).\n26"    },    {        "title": "1109.0930v1.Spectral_theory_of_damped_quantum_chaotic_systems.pdf",        "content": "arXiv:1109.0930v1  [math-ph]  5 Sep 2011SPECTRAL THEORY OF DAMPED QUANTUM CHAOTIC\nSYSTEMS\nST´EPHANE NONNENMACHER\nAbstract. We investigatethe spectral distribution of the damped waveequat ion\non a compact Riemannian manifold, especially in the case of a metric of n egative\ncurvature, for which the geodesic flow is Anosov. The main applicatio n is to\nobtain conditions (in terms of the geodesic flow on Xand the damping function)\nfor which the energy of the waves decays exponentially fast, at lea st for smooth\nenough initial data. We review various estimates for the high freque ncy spectrum\nin terms of dynamically defined quantities, like the value distribution of the time-\naveraged damping. We also present a new condition for a spectral g ap, depending\non the set of minimally damped trajectories.\n1.Introduction\n1.1.Spectrum of the damped wave equation. Given a Riemannian manifold\n(X,g) and adamping function a∈C∞(X,R+), we are interested in the solutions of\nthe damped wave equation (DWE)\n(1.1) ( ∂2\nt−∆+2a(x)∂t)v(x,t) = 0, v(x,0) =v0, ∂tv(x,0) =v1.\nThe natural setting is to take intial data ( v0,v1) in the space Hdef=H1(X)×L2(X).\nThis equation is equivalent with the system\n(1.2) ( i∂t+A)v(t) = 0,Adef=/parenleftbigg\n0I\n−∆−2ia/parenrightbigg\n,v(0)∈ H,\nwith the correspondence v(t) = (v(t),i∂v(t)).Agenerates a strongly continuous\nsemigroup on H, so the solution to (1.1,1.2) reads\n(1.3) v(t) =e−itAv(0).\nWe will always assume that the damping is nontrivial, a/\\e}atio\\slash≡0. Apart from the\nconstant solution v(t) = (1,0), all solutions then decay. Physically, this is expressed\nby the fact that the energyof the waves,\n(1.4) E(v(t)) =1\n2/parenleftbig\n/ba∇dbl∇v(t)/ba∇dbl2+/ba∇dbl∂tv(t)/ba∇dbl2/parenrightbig\n,\nwill decay to zero when t→ ∞for anyv(0)∈ H. In some sense, the waves are\nstabilized by the damping.\nTo analyze this decay, it it natural to try to expand the solution in te rms of the\nspectrum of A.This spectrum is discrete, consisting of countably many complex\neigenvalues {τn}with Reτn→ ±∞. It can be obtained by solving the generalized\neigenvalue equation\n(1.5)/parenleftbig\n−∆−τ2−2iaτ/parenrightbig\nu= 0.\nThis work has been partially supported by the grant ANR-09-JCJC- 0099-01 of the Agence\nNationale de la Recherche.\n1Xa(x)>0\nFigure 1.1. A damped geodesic.\nThe following properties are easily shown ( amindef= min x∈Xa(x), similarly for amax).\nProposition 1. [Leb93]All eigenvalues except τ0= 0satisfyImτn<0.\nIfReτn/\\e}atio\\slash= 0then−Imτn∈[amin,amax].\nThe spectrum is symmetric w.r.to the imaginary axis.\nTo each eigenvalue τncorresponds a quasi-stationary mode un(x), an eigenstate\nun= (un,τnun) ofAreads, and a solution vn(t,x) =e−itτnun(x) of the DWE. Hence\nImτnrepresents the quantum decay rate of the mode un.\nIfamin>0, then the energy decays exponentially, uniformly for initial data v(0)∈\nH. Precisely, there exists γ >0,C >0 such that\n(1.6) E(v(t))≤Ce−2γtE(v(0)),∀v(0)∈ H,∀t≥0.\n1.2.The Geometric Control Condition. The condition amin>0 is not neces-\nsary to ensure such a uniform exponential decay. Since all eigenva lues except τ0= 0\nsatisfy Im τn<0, for any C >0 the subspace HC⊂ Hspanned by the eigenstates\n{un,|Reτn| ≤C}is finite dimensional, and for any initial data v(0)⊂ HCthe en-\nergy will decay exponentially. Hence, the failure of exponential dec ay can only come\nfrom the behaviour of waves at high frequency. In this high freque ncy limit, a natu-\nral connection can be made with the classical ray dynamics on X, equivalently the\ngeodesicflow Φtontheunit cotangent bundle S∗X(see§1.4). Using this connection,\nRauch and Taylor [RauTay75] showed that the uniform exponential decay (1.6) is\nequivalent with the Geometric Control Condition (GCC), which state s that every\ngeodesic meets the damping region {x∈X, a(x)>0}(due to the compactness of\nS∗Xeach geodesic does it within some time T0>0).\nThis condition can be expressed in terms of the time averages of the damping,\nnamely the functions\n(1.7) /a\\}b∇acketle{ta/a\\}b∇acket∇i}htt(ρ) =1\nt/integraldisplayt\n0a◦Φs(ρ)ds, ρ∈T∗X, t >0.\nGCCisequivalent to thefact that, for t >0largeenough(say, t >2T0), thefunction\n/a\\}b∇acketle{ta/a\\}b∇acket∇i}httis strictly positive on S∗X. Lebeau [Leb93] generalized this result to the case of\nmanifolds with boundaries. He also showed that the optimal decay ra teγis given\nby min(G,a−), where G= inf{−Imτn, τn/\\e}atio\\slash= 0}is the spectral gap, while\n(1.8) a−def= lim\nt→∞min\nS∗X/a\\}b∇acketle{ta/a\\}b∇acket∇i}htt\nis the minimal asymptotic damping. We will explain the relevance of a−in§2.2.\nKoch and Tataru [KoTa94] studied the same question in a more gener al context\n(case of manifolds with boundaries, and of a damping taking place bot h in the\n2“bulk” and on the boundary). They showed that the averages /a\\}b∇acketle{ta/a\\}b∇acket∇i}httgovern the decay\nof the semigroup, up to a compact subspace.\nTheorem 2. [KoTa94, Thm 2] For each ǫ >0and each t >0there exists a subspace\nHǫ,t⊂ Hof finite codimension such that\n/vextenddouble/vextenddoublee−itA/vextenddouble/vextenddouble\nHǫ,t→H≤exp/braceleftBig\n−tmin\nS∗X/a\\}b∇acketle{ta/a\\}b∇acket∇i}htt/bracerightBig\n+ǫ.\nFor any subspace H1⊂ Hof finite codimension,\nexp/braceleftBig\n−tmin\nS∗X/a\\}b∇acketle{ta/a\\}b∇acket∇i}htt/bracerightBig\n≤/vextenddouble/vextenddoublee−itA/vextenddouble/vextenddouble\nH1→H.\nA consequence of this result is the characterization of the Fredholm spectrum of\nthe semigroup. In the present situation of damped waves on a comp actXwithough\nboundary, their result states1that this Fredholm spectrum is given by the annulus\n{z∈C, e−ta+≤ |z| ≤e−ta−}.\nNoticethat itispossible tohave apositivegap G >0andat thesame time a−= 0\n(failure of the GCC), see e.g. an example in [Ren94]. This reflects the f act that the\nspectrum of the semigroup e−itAis not controlled by the spectrum of generator A,\na frequent problem for nonnormal generators like A.\n1.3.Beyond Geometric control: a few cases of uniform energy deca y for\nregular data. If GCC fails, one can construct initial data v(0)∈ Hsuch that\nE(v(t)) decays arbitrarily slowly [Leb93]. Yet, it is possible to show that mor e\nregular data, say in some Sobolev space Hs=Hs+1×Hs,s >0, decay in some\nuniform way. In the most general situation, Lebeau showed a logar ithmic decay of\nthe energy [Leb93, Thm 1],\nE(v(t))1/2≤Cslog[3+log(3+ t)]\nlog(3+t)s/ba∇dblv/ba∇dblHs,∀v=v(0)∈ Hs,∀t≥0.\nThe proof uses Carleman estimates, which imply some control of the resolvent\n(τ−A)−1in a strip {Reτ≥C,|Imτ| ≤f(Reτ)}for some function f(r) which\ndecays exponentially as r→ ∞. By interpolation, proving such an estimate for\nsomes0>0 implies a similar estimate for all s >0.\nInspecific dynamical situations, onecan show resolvent estimates ina largerstrip,\ne.g. for some function f(r) decaying algebraically: one then gets an algebraic decay\nO(t−γs/ba∇dblv/ba∇dblHs) for the energy of regular data. Burq and Hitrik showed that this is\nthe case for damped waves in the (chaotic) stadium billiard , if the damping function\nvanishes only in some part of the central rectangle, so that the set of undamped\ntrajectories ,\n(1.9) Kdef=/braceleftbig\nρ∈S∗X,∀t∈R, a(Φt(ρ)) = 0/bracerightbig\n,\nconsists in a collection of (neutrally stable) “bouncing ball” trajecto ries [BuHi07].\nChristianson studied thecase where theset Kconsists inasingle hyperbolic closed\ngeodesic: the energy decay is bounded by a stretched exponentia lO(e−γst1/2/ba∇dblv/ba∇dblHs)\n[Chris07, Thm 5’][Chris11]. The hyperbolicity of the geodesic induces a s trong\ndispersion of the waves, responsible for this fast decay.\n1under the condition that, for any T >0, the set of T-periodic geodesics has measure zero.\n3In§3.3 and 3.4 we will present situations, with ( X,g) a manifold of negative\ncurvature, for which the energy of regular data decays exponen tially:\n(1.10) E(v(t))1/2≤Cse−γst/ba∇dblv/ba∇dblHs,∀v∈ Hs,∀t≥0.\nTheproofproceedsbycontrollingtheresolvent( τ−A)−1inastrip {Reτ > C,|Imτ| ≤c},\nand then uses standard arguments [Chris09]. Here as well, the reso lvent bounds are\nbased on certain hyperbolic dispersion estimates. See the Corollaries 16 and 19 for\nthe precise dynamical conditions, which involve the interplay betwee n the flow and\nthe damping function.\n1.4.High frequency limit – semiclassical formulation and gener alization.\nAs explained above, the decay of the energy is mainly governed by th e high fre-\nquencies Re τ≫1. A semiclassical formulation of the problem was used in [Sjo00],\nallowing to use /planckover2pi1-pseudodifferential techniques. Take an effective Planck’s “con-\nstant” 0</planckover2pi1≪1. In order to study the frequencies Re τ=/planckover2pi1−1+O(1), we perform\nthe rescaling\n(1.11) τ=√\n2z\n/planckover2pi1withzin the disk D(1/2,C/planckover2pi1).\nThe equation (1.5) becomes\n(1.12) (P(/planckover2pi1,z)−z)u= 0, P(/planckover2pi1,z) =−/planckover2pi12∆\n2−i/planckover2pi1√\n2za=−/planckover2pi12∆\n2−i/planckover2pi1a+O(/planckover2pi12).\nMore generally, we consider operators of the type\n(1.13) P(/planckover2pi1,z) =−/planckover2pi12∆\n2+i/planckover2pi1Op/planckover2pi1(qz).\nwhere the symbol2qz∈S0(T∗X) is now a function on the phase space T∗X, and\ndepends holomorphically on z∈D(1/2,C/planckover2pi1), withqz=q1/2+O(/planckover2pi1) [Sjo00]. With a\nslight abuse, will call q=q1/2(x,ξ) “the damping function” (a damping effectively\noccurs only at phase space points where q(x,ξ)<0, while positive values of qrather\ncorrespond to a “gain” or “pumping”).\nIn this framework, we are interested in the semiclassical distributio n of the gen-\neralized eigenvalues zn(/planckover2pi1) of the operator P(/planckover2pi1,z) in the disk D(1/2,C/planckover2pi1). Because\nwe will specify this spectrum at the scale /planckover2pi1, we may for convenience drop the term\nO(/planckover2pi12) in (1.13), and rather study the spectrum of the z-independent operator\n(1.14) P(/planckover2pi1) =P0(/planckover2pi1)+i/planckover2pi1Op/planckover2pi1(q), P 0(/planckover2pi1)def=−/planckover2pi12∆\n2.\nThe principal symbol of this operator is p0(x,ξ) =|ξ|2\n2, it generates the geodesic flow\nΦt= exp(tHp0) onT∗X, with unit speed on the energy shell p−1\n0(1/2) =S∗X.\n2For anyk∈R, the symbol class Sk(T∗X) denotes the functions a(x,ξ;/planckover2pi1) satisfying the follow-\ning bounds: for any multi-indices α,β∈Nd,\n/vextendsingle/vextendsingle/vextendsingle∂α\nx∂β\nξa(x,ξ;/planckover2pi1)/vextendsingle/vextendsingle/vextendsingle≤Cα,β(1+|ξ|)k−|β|uniformly for /planckover2pi1∈(0,1),\nBy Op/planckover2pi1(q) we refer to a semiclassical quantization on X, with the property that Op/planckover2pi1(q) is sym-\nmetric if qis real (see e.g [EvZw, App. E]). The operators Op/planckover2pi1(q) form the class Ψk(X) of\npseudodifferential operators (PDO).\n42.Semiclassical spectral distribution of P(/planckover2pi1). General results\nThe results we present in this section hold for arbitrary Riemannian m anifolds\n(X,g). We skip the proofs.\n2.1.Weyl law for the real parts. In the semiclassical limit, the number of\neigenvalues of the selfadjoint operator P0(/planckover2pi1) =−/planckover2pi12∆\n2in the interval/bracketleftbig1\n2±ǫ/bracketrightbigdef=/bracketleftbig1\n2−ǫ,1\n2+ǫ/bracketrightbig\nis approximately given by Weyl’s law. When applying the perturba-\ntioni/planckover2pi1Op/planckover2pi1(q) =O(/planckover2pi1), it seems natural to expect that most eigenvalues of P0(/planckover2pi1)\nin this interval will remain inside a (complex) neighbourhood/bracketleftbig1\n2±ǫ/bracketrightbig\n+O(/planckover2pi1). One\ncan indeed show that the “macroscopic” spectral distribution is giv en by the same\nWeyl’s law as in the undamped case.\nTheorem 3. [MarMat84, Sjo00] For anyǫ >0small enough (possibly depening on\n/planckover2pi1), the number of eigenvalues of P(/planckover2pi1)in the strip Sǫ=/braceleftbig1\n2−ǫ≤Rez≤1\n2+ǫ/bracerightbig\nis\nsemiclassically given by\n#{SpecP(/planckover2pi1)∩Sǫ}= (2π/planckover2pi1)−dVol{ρ∈T∗X,|p0(ρ)−1/2| ≤ǫ}+O/parenleftbig\n/planckover2pi1−d+1/parenrightbig\n.\nHereVolcorresponds to the symplectic volume in T∗X. This estimate implies that,\nforC >0large enough,\n(2.1) # {SpecP(/planckover2pi1)∩D(1/2,C/planckover2pi1)} ≍/planckover2pi1−d+1.\nThis result is actually not so easy to prove. It uses some subtle analy tic Fredholm\ntheory in order to relate the spectrum of P(/planckover2pi1) with that of a simpler “comparison\noperator”.\n2.2.Restrictions for the imaginary parts (quantum decay rates) .Even-\nthough the operator P(/planckover2pi1) was inspired by the wave equation, it is convenient for us\nto consider it instead as the generator of a Schr¨ odinger equation\n(2.2) i/planckover2pi1∂tv−P(/planckover2pi1)v= 0.\nAny eigenvalue/vector ( zn,un) ofP(/planckover2pi1) is thus associated with a solution vn(x,t) =\ne−iznt//planckover2pi1un(x) of this equation, and we will also call Im zn//planckover2pi1thequantum decay rate3\nassociated with un.\nFirstweremarkthataneigenmode unassociatedtoaneigenvalue zn∈D(1/2,C/planckover2pi1)\nissemiclassically microlocalized onp−1\n0(1/2) =S∗X. To state this property we need\nto introduce energy cutoffs.\nDefinition 4. An energy cutoff will be function χ∈C∞\nc([1/2±δ],[0,1]) for some\nδ >0, and satisfies χ(s) = 1 near s= 1/2. The cutoff χ0is said to be embedded in\nχ1, and we note χ1≻χ0, iffχ1≡1 near supp χ0. From the cutoff χwe construct\nthe (pseudodifferential) cutoff operator χ(P0(/planckover2pi1))∈Ψ−∞(X).\nProposition 5. Take an energy cutoff χ. Letu=u(/planckover2pi1)be an eigenstate of P(/planckover2pi1)\nwith eigenvalue z(/planckover2pi1)∈D(1/2,C/planckover2pi1). Then,\n/ba∇dbl(I−χ(P0))u/ba∇dbl=O(/planckover2pi1∞)/ba∇dblu/ba∇dbl.\nThis localization property explains why only the neighbourhood of S∗Xwill be\nimportant for us.\n3From(1.11)wehaveIm zn//planckover2pi1= Imτn+O(/planckover2pi1), sothisdenominationis(approximately)consistent\nwith the one of §1.1.\n52.2.1.A factorization of the Schr¨ odinger propagator. An easy way to realize how\nthe skew-adjoint subprincipal term i/planckover2pi1Op/planckover2pi1(q) implies a “damping” is to analyze the\npropagator for the Schr¨ odinger equation (2.2). The following Pro position adapts a\nmore general result due to Rauch and Taylor [RauTay75].\nProposition 6. Assumeq∈C∞\nc(T∗X), and take P(/planckover2pi1)as in (1.14). For any fixed\nt∈R, decompose the Schr¨ odinger propagator Vt=e−itP//planckover2pi1into\n(2.3) Vt=UtB(t),whereUt=e−itP0//planckover2pi1is the undamped propagator.\nThe operator B(t)is a PDO in Ψ0(X)of principal symbol\nb(t,ρ) = exp/braceleftbigg/integraldisplayt\n0q◦Φs(ρ)ds/bracerightbigg\n= exp{t/a\\}b∇acketle{tq/a\\}b∇acket∇i}htt(ρ)},\nwhere the time averaged damping /a\\}b∇acketle{tq/a\\}b∇acket∇i}httgeneralizes (1.7).\nThe factor b(t,ρ) is theaccumulated damping along the orbit {Φs(ρ),0≤s≤t}.\nIf one starts from a Gaussian wavepacket u0microlocalized at a point ρ, then the\nstateVtu0is a wavepacket microlocalized at Φt(ρ), and the above statement shows\nthat the L2norms are related through this factor:\n/ba∇dblut/ba∇dbl=/ba∇dblu0/ba∇dbl(b(t,ρ)+O(/planckover2pi1)).\nApplying anenergy cutoff allows toextend thefactorization(2.3)da mping functions\nq∈S0(T∗X) of noncompact support.\nLemma 7. Consider P(/planckover2pi1)with a damping function q∈S0(T∗X). Take two embed-\nded cutoffs χ1≻χ0supported in [1/2±δ], and the truncated damping ˜qdef=χ1(p0)q.\nThen, for any t≥0fixed, one has\n(2.4) Vtχ0(P0) =˜Vtχ0(P0)+OL2→L2(/planckover2pi1∞),\nwhere˜Vtis the propagator corresponding to ˜P=P0+i/planckover2pi1Op/planckover2pi1(˜q).\nThis identity uses the fact that the propagation Vtdoes not modify the energy\nlocalization properties.\n2.2.2.From propagator factorization to a resolvent bound. We can now easily obtain\nafirstconstraintonthequantumdecayrates. ApplyingPropositio n6tothefunction\n˜q=χ1(p0)qof Lemma 7, we get for any fixed t≥0:\n/vextenddouble/vextenddoubleVtχ0(P0)/vextenddouble/vextenddouble\nL2→L2=/vextenddouble/vextenddouble/vextenddouble˜B(t)χ0(P0)/vextenddouble/vextenddouble/vextenddouble\nL2→L2+O(/planckover2pi1∞)\n= max\nT∗X˜b(t)χ0(p0)+Ot(/planckover2pi1) (2.5)\n≤exp/braceleftbigg\ntmax\nsuppχ0◦p0/a\\}b∇acketle{t˜q/a\\}b∇acket∇i}htt/bracerightbigg\n+Ot(/planckover2pi1).\nOn the second line we used the sharp G˚ arding (in)equality for the op erator˜B(t)∈\nΨ0(X). The maximum maxp−1\n0([1/2±δ])/a\\}b∇acketle{t˜q/a\\}b∇acket∇i}httdecreases when t→ ∞or when δ→0,\nand converges to the asymptotic maximum on S∗X,\n(2.6) q+def= lim\nt→∞max\nρ∈S∗X/a\\}b∇acketle{tq/a\\}b∇acket∇i}htt(ρ).\nOne similarly defines an asymptotic minimum q−. We then get the following norm\nbounds for the propagator.\n6Proposition 8. Fixǫ >0. If the energy cutoff χ0has a small enough support, and\nTǫ>0is large enough, then for any (fixed) t≥Tǫand any small enough /planckover2pi1>0, the\nfollowing bounds holds:\n(2.7)/vextenddouble/vextenddoubleVtχ0(P0)/vextenddouble/vextenddouble\nL2→L2≤e(q++ǫ)t,/vextenddouble/vextenddoubleV−tχ0(P0)/vextenddouble/vextenddouble\nL2→L2≤e(−q−+ǫ)t.\nFrom there one easily obtains the following resolvent and spectral b ounds.\nTheorem 9. [Leb93, Sjo00] Take any ǫ >0. Then, there exists /planckover2pi1ǫ,Cǫ>0such that\nfor/planckover2pi1</planckover2pi1ǫ, the following resolvent estimate holds:\n(2.8)∀z∈D(1/2,C/planckover2pi1)\\{z,Imz//planckover2pi1∈[q−−ǫ,q++ǫ]},/vextenddouble/vextenddouble(P(/planckover2pi1)−z)−1/vextenddouble/vextenddouble≤Cǫ\n/planckover2pi1.\nAs a consequence, for /planckover2pi1</planckover2pi1ǫall eigenvalues zn∈D(1/2,C/planckover2pi1)ofP(/planckover2pi1)satisfy\nImzn\n/planckover2pi1∈[q−−ǫ,q++ǫ].\nIn the case of the damped wave equation ( q(x,ξ) =−a(x)) and in a situation of\ngeometric control ( a−=−q+>0), the resolvent estimate (2.8) can be used to show\nthe uniform exponential energy decay (1.6) [Hit03].\n2.3.Questions on the spectral distribution. So far we showed that the quan-\ntumdecay rates areboundedby the asymptotic extrema ofthe tim e-averaged damp-\ning. The following questions were raised in [Sjo00, AschLeb, Anan10] concerning\ntheir semiclassical distribution.\n(1) What are the possible accumulation points of the quantum decay rates when\n/planckover2pi1→0? In particular, are there sequences of decay rates (Im z(/planckover2pi1)//planckover2pi1)/planckover2pi1→0\nconverging to the extremal values q±?\n(2) Do the quantum decay rates admit an asymptotic distribution wh en/planckover2pi1→0?\nNamely, for a given interval I⊂[q−,q+] and 1≫ǫ(/planckover2pi1)≫/planckover2pi1, does the ratio\n#/braceleftbig\nz∈SpecP(/planckover2pi1),|Rez−1/2| ≤ǫ,Imz\n/planckover2pi1∈I/bracerightbig\n#{z∈SpecP(/planckover2pi1),|Rez−1/2| ≤ǫ}\nhave a limit when /planckover2pi1→0? Is this limit distribution related with the value\ndistributions of the averages /a\\}b∇acketle{tq/a\\}b∇acket∇i}htt?\n3.Spectral estimates on Anosov manifolds\nTheabove questions areopeningeneral. Inordertogetmoreprec ise informations\non the spectrum of P(/planckover2pi1), one needs to make specific assumptions on the geodesic\nflow onX. For instance, the case of a completely integrable dynamics has bee n con-\nsidered by Hitrik-Sj¨ ostrand in a sequence of papers (see e.g. [HitS jo08]and reference\ntherein). The case of nearly-integrable dynamics including KAM invar iant tori has\nbeen studied by Hitrik-Sj¨ ostrand-V˜ u Ngo .c [HSVN07]. In these cases, one can trans-\nformthe Hamiltonian flow into a normal form near each invariant torus, which leads\nto a precise description of the spectrum “generated” by this toru s. A Weyl law for\nthe quantum decay rates was recently obtained in [HitSjo11] (for s kew-adjoint per-\nturbations iθ(/planckover2pi1)Op/planckover2pi1(q), withθ(/planckover2pi1)≪/planckover2pi1). Ontheotherhand, Asch-Lebeau[AschLeb]\naddressed Question 1 for the case of the 2-dimensional standard sphere. They show\nthat, if the damping function qhas real analytic real and imaginary parts, there is\n7−\nE+E\nρt\nρt+ρ\nΦ(  )\nJ  (  )\nFigure 3.1. Structure of the Anosov flow near an orbit Φt(ρ)\ngenerically a spectral gap γ >0: for/planckover2pi1small enough, all eigenvalues zn∈D(1/2,C/planckover2pi1)\nhave quantum decay rates\nImzn//planckover2pi1∈[q−+γ,q+−γ].\nThe proof proceeds by applying a complex canonical transformatio n, such that the\nreal part of the pulled-back damping function takes values in the ab ove interval. In\nthis case, the range of the quantum decay rates is strictly smaller t han the range of\nclassical decay rates.\nWe will see below that such a spectral gap may also occur in the case o f Anosov\ngeodesic flows, thanks to a different mechanism, namely a hyperbolic dispersion\nproperty due to the instability of the classical flow (see Thms 15 and 18 below).\nIn the next section we recall the definition and properties of Anoso v manifolds.\n3.1.A short reminder on Anosov manifolds [KatHas95] .At the “antipode”\nof the completely integrable case, one finds the “strongly chaotic” flows, namely the\nAnosov (or uniformly hyperbolic) flows. Uniform hyperbolicity means that at each\npointρ∈S∗Xthere exists a splitting of the tangent space,\nTρS∗X=RHp(ρ)⊕E+(ρ)⊕E−(ρ),\nwhereHp(ρ) is the Hamiltonian vector field, E±(ρ) are the unstable and stable\nsubspaces at the point ρ; they both have dimension d−1, and are uniformly trans-\nverse toe.o.. The families {E±(ρ), ρ∈S∗X}formtheunstable/stable distributions,\nthey are invariant w.r.to the flow, H¨ older continuous, and are char acterized by the\nfollowing property: there exists C,λ >0 such that\n(3.1) ∀ρ∈S∗X,∀v∈E∓(ρ),∀t >0,/vextenddouble/vextenddoubledΦ±t\nρv/vextenddouble/vextenddouble≤Ce−λt/ba∇dblv/ba∇dbl.\nAs was shown by Hadamard, such a geodesic flow is obtained if the man ifold (X,g)\nhas a negative sectional curvature. The long time properties of su ch a strongly\nchaotic flow are well understood. In particular, Hopf proved that this geodesic flow\nisergodicw.r.to the Liouville measure on S∗X.\nDefinition 10. CallµLthe normalized Liouville measure on S∗X. Then, the ge-\nodesic flow ΦtonS∗Xis ergodic (w.r.to µL) iff, for any continuous function qon\nS∗X,\nforµL−almost every ρ∈S∗X,lim\nt→∞/a\\}b∇acketle{tq/a\\}b∇acket∇i}htt(ρ) = ¯qdef=µL(q).\n8The following quantities provide quantitative measures of the hyper bolicity:\nthe maximal expansion rate λmaxdef= lim\nt→∞sup\nρ∈S∗X1\ntlog/vextenddouble/vextenddoubledΦt\nρ/vextenddouble/vextenddouble, (3.2)\nthe unstable Jacobian J+(ρ,t) = det/parenleftbig\ndΦt↾E+(ρ)/parenrightbig\n, t >0,\nand its infinitesimal version ϕ+(ρ)def=d\ndtlogJ+(ρ,t)↾t=0, (3.3)\nthe minimal expansion rate νmindef=1\nd−1lim\nt→∞min\nρ∈S∗X1\ntlogJ+(ρ,t). (3.4)\nThe Jacobian depends on the choice of norms on the spaces TρS∗X. However, its\nlong time asymptotics is independent of this choice. In variable curva ture these\nrates are related by λmax≥νmin≥λ >0. In particular, if the positive Lyapunov\nexponents are not all equal, one has νmin< λmax.\nA particular class of Anosov manifolds consists in quotients of the d-dimensional\nhyperbolicspace Hdbyco-compactsubgroupsofitsisometrygroup. Thesemanifolds\nhave a constant curvature4−Λ2. Using the natural norms on TρS∗X, one then has\nJ+(ρ,t) =et(d−1)Λ, ϕ+(ρ) = Λ(d−1), λmax=νmin= Λ.\nIn this case the contraction/expansion are both homogeneous (in dependent of the\npointρ) and isotropic (independent in the direction).\nLet us finally notice that the study of the operator P(/planckover2pi1) on an Anosov manifold\nbelongs to the field of “quantum chaos”. The methods we will use belo w occur in\nvarious problems of this field, e.g. the study of eigenstates of the L aplacian P0(/planckover2pi1)\non such manifolds [Zel09].\n3.2.Fractal Weyl upper bounds for the quantum decay rates. Inthissection\nwe address Question 2, that is the asymptotic distribution of the qu antum decay\nrates, for the case of Anosov manifolds.\n3.2.1.Typical quantum decay rates for ergodic flows. A basic property of the geo-\ndesic flow on any Anosov manifold is ergodicity. Sj¨ ostrand has stud ied the asymp-\ntotic disribution of the quantum decay rates for any ergodic geode sic flow [Sjo00],\nand obtained the following result.\nTheorem 11. [Sjo00]Assume the geodesic flow on S∗Xis ergodic w.r.to the Liou-\nville measure. Then, For any C >0, ǫ >0, one has as /planckover2pi1→0,\n#/braceleftbigg\nz∈SpecP(/planckover2pi1)∩D(1/2,C/planckover2pi1),Imz\n/planckover2pi1/\\e}atio\\slash∈[¯q−ǫ,¯q+ǫ]/bracerightbigg\n=o(/planckover2pi1−d+1).\nComparing this bound with the Weyl law (2.1) shows that, in the semicla ssical\nlimit,almost all the quantum decay rates are close to the phase space average ¯ q.\nThe latter could be called the typical value for quantum decay rates, while values at\nfinite distance from ¯ qareatypical. The asymptotic distribution is simply the delta\nmeasure δ¯q.\nRemark 12.As was noticed in [Anan10], this concentration result could be seen\nas a “nonperturbative version” of quantum ergodicity . The latter property states\n4Usually one normalizes the metrics on Hdso that the curvature is −1. We prefer to keep track\nof the curvature in our notations.\n9thatalmost all the eigenstates u0\nnofP0(/planckover2pi1) with eigenvalues z0\nn∈D(1/2,C/planckover2pi1) sat-\nisfy/a\\}b∇acketle{tu0\nn,Op/planckover2pi1(q)u0\nn/a\\}b∇acket∇i}ht= ¯q+o/planckover2pi1→0(1). A naive perturbation theory argument would\npredict that, when switching on i/planckover2pi1Op/planckover2pi1(q), the eigenvalues z0\nnmove to zn=z0\nn+\ni/planckover2pi1/a\\}b∇acketle{tu0\nn,Op/planckover2pi1(q)u0\nn/a\\}b∇acket∇i}ht+O(/planckover2pi12),sothatthequantumdecayratesIm zn//planckover2pi1=/a\\}b∇acketle{tu0\nn,Op/planckover2pi1(q)u0\nn/a\\}b∇acket∇i}ht+\nO(/planckover2pi1), the RHS being equal to ¯ q+o(1) for almost all n. Of course, this argument\ndoes not apply because the perturbation i/planckover2pi1Op/planckover2pi1(q) is much stronger than the mean\nspacing between successive eigenvalues.\nTo prove the above theorem one relates the counting of quantum d ecay rates with\nthe value distribution of the quantum averages /a\\}b∇acketle{tq/a\\}b∇acket∇i}httonS∗X, that is the volumes\n(3.5) µL{ρ∈S∗X,/a\\}b∇acketle{tq/a\\}b∇acket∇i}htt(ρ)≥α}, α∈R, t >0.\nIndeed, the main intermediate result in the proof is the bound\n(3.6)\n#/braceleftbigg\nz∈SpecP(/planckover2pi1)∩D(1/2,C/planckover2pi1),Imz\n/planckover2pi1≥α/bracerightbigg\n≤Ct,δ/planckover2pi1−d+1µL{ρ∈S∗X,/a\\}b∇acketle{tq/a\\}b∇acket∇i}htt(ρ)≥α−δ},\nwhich holds for any fixed t >0 andδ >0 (this bound holds independently of the\nergodicity assumption). Ergodicity then implies that the value distrib ution of /a\\}b∇acketle{tq/a\\}b∇acket∇i}htt\nconverges to δ¯qwhent→ ∞; in particular, for any α >¯qthe volume (3.5) decays\nto zero when t→ ∞. We have then obtained the bound o(/planckover2pi1−d+1) for the quantum\ndecay rates ≥α >¯q. The case of values ≤α <¯qis treated analogously. /square\n3.2.2.Large deviation estimates for Anosov flows. ForanAnosovflow, onehasmore\nprecise estimates for the volumes (3.5), which can then induce shar per bounds on\nthe number of the atypical quantum decay rates. These volume es timate take the\nform oflarge deviation estimates. Let us introduce some notations. We call M\nthe set of Φt-invariant probability measures on S∗X. For each such measure µ, we\ndenoteby hKS(µ)itsKolmogorov-Sinaientropy: thisisanonnegativenumber, which\nmeasures the complexity of a µ-typical trajectory [KatHas95]. Then, we define the\nrate function ˜H:R→Ras follows:\n(3.7) ∀s∈R,˜H(s)def= sup/braceleftbig\nhKS(µ)−µ(ϕ+), µ∈ M, µ(q) =s/bracerightbig\n,\nwhere we recall that ϕ+is the infinitesimal unstable Jacobian (3.3). We are now\nready to state our large deviation result.\nTheorem 13. [Kif90]Assume the geodesic flow on S∗Xis Anosov. Then, for any\nclosed interval I⊂Rand for any q∈C∞(S∗X), the time averages /a\\}b∇acketle{tq/a\\}b∇acket∇i}httsatisfy\n(3.8) limsup\nt→∞1\ntlogµL{ρ∈S∗X,/a\\}b∇acketle{tq/a\\}b∇acket∇i}htt(ρ)∈I} ≤sup\ns∈I˜H(s).\nThe rate function ˜His continuous on [ q−,q+], smooth and strictly concave on\n(q−,q+), negative except at the point ˜H(¯q) = 0, satisfies ˜H(s)≥ −supµ∈Mµ(ϕ+)\nfors∈[q−,q+], and is equal to −∞outside [q−,q+]. As a consequence of (3.8), for\nanyα≥¯qandǫ >0 arbitrary small, there exists Tα,ǫ>0 such that\n(3.9) ∀t≥Tα,ǫ, µL{ρ∈S∗X,/a\\}b∇acketle{tq/a\\}b∇acket∇i}htt(ρ)≥α} ≤et(˜H(α)+ǫ).\nDue to the negativity of ˜H(α) forα/\\e}atio\\slash= ¯q, we see that the probability of /a\\}b∇acketle{tq/a\\}b∇acket∇i}htttaking\natypical values decays exponentially when t→ ∞.\n103.2.3.Fractal Weyl upper bounds on Anosov manifolds. Using these large deviation\nestimates, Anantharaman [Anan10] improved Thm 11 by letting the a veraging time\ngrow with /planckover2pi1in a controlled way. The optimal time is the Ehrenfest time\n(3.10) T=TEhr= (1−2ǫ)log1//planckover2pi1\nλmax,\nwhereλmaxis the largest expansion rate (3.2) and ǫ >0 arbitrary small. What is the\nsignification of this time? For any f∈S−∞(T∗X) supported in an ǫ-neighbourhood\nofS∗X, the classically evolved observable f◦Φtremains in the “good” symbol class5\nS−∞\n1/2−ǫ(T∗X) uniformly for times |t| ≤TEhr/2. In turn, the symmetric averages\n(3.11) /a\\}b∇acketle{tf/a\\}b∇acket∇i}htt,sym=1\nt/integraldisplayt/2\n−t/2f◦Φsds\nbelong to S−∞\n1/2−ǫ(T∗X) for|t| ≤TEhr, and this time is sharp.\nLet us insist on the fact that controlling the time evolution up to times t≍\nlog1//planckover2pi1is a crucial ingredient in order to obtainrefined spectral estimates on Anosov\nmanifolds. This will also bethe case in Sections 3.3 and3.4 when proving h yperbolic\ndispersion estimates and spectral gaps.\nThe pseudodifferential calculus on Ψ2\n1/2−ǫ(X) allows to extend the validity of the\nbound (3.6) up to the Ehrenfest time. The large deviation estimate ( 3.9) then leads\nto the following fractal Weyl upper bound for the number of atypical quantum decay\nrates.\nTheorem 14. [Anan10] Assume the geodesic flow on S∗Xis Anosov. Then, for any\nα≥¯qandǫ >0, one has for /planckover2pi1small enough\n(3.12) #/braceleftbigg\nz∈SpecP(/planckover2pi1)∩D(1/2,C/planckover2pi1),Imz\n/planckover2pi1≥α/bracerightbigg\n≤/planckover2pi1−˜H(α)\nλmax+(1−d)−ǫ,\nwhere˜His the rate function (3.7). A similar expression holds when c ounting quan-\ntum decay rates smaller than α′≤¯q.\nSince−˜H(α)>0 for allα >¯q, this upper bound improves the bound o(/planckover2pi1−d+1) of\nThm. 11 by a fractional power of /planckover2pi1(the name “fractal Weyl upper bound” takes its\norigin in the counting of resonances of chaotic scattering systems [SjoZwo07]). For\nanyα∈[¯q,q+] one has ˜H(α)≥ −supµµ(ϕ+)≥ −λmax(d−1), so the exponent of /planckover2pi1\nin (3.12) is negative, allowing the presence of (many) quantum decay rates arbitrary\nclose toq+.\nThese fractal upper bounds are not expected to be sharp in gene ral [Anan10]. In\nthe next sections we will be able, under certain conditions, to exclud e the possibility\nof quantum decay rates near q+(or near q−).\n5For any k∈R,δ∈[0,1/2), the symbol class Sk\nδ(T∗X) consists of functions g(x,ξ;/planckover2pi1) which\nmay become more and more singular when /planckover2pi1→0, but in a controlled way:\n∀α,β∈Nd,∀ρ∈T∗X,/vextendsingle/vextendsingle/vextendsingle∂α\nx∂β\nξg(ρ)/vextendsingle/vextendsingle/vextendsingle≤Cα/planckover2pi1−δ(|α|+|β|)(1+|ξ|)k−|β|.\nIn this class one can still use pseudodifferential calculus, and the ex pansions in powers of /planckover2pi1make\nsense [EvZw, Sec. 4.3].\n113.3.A pressure criterium for a spectral gap. The result of Thm 9, namely\nthe fact that all quantum decay rates belong to the interval [ q−−o(1),q++o(1)],\nwas obtained by studying the norm of the propagator Vt=e−itP//planckover2pi1for times |t| ≫1\nindependent of /planckover2pi1. Indeed, the sequence of equalities (2.5) directly leads to the uppe r\nbound\n(3.13) Im zn//planckover2pi1≤q++ǫ.\nThe crucial semiclassical ingredient in (2.5) is the second equality con necting the L2\nnorm of the PDO ˜B(t)χ(P0) with the supremum of its (principal) symbol. Hence,\none cannot hope to improve the bound (3.13) as long as ˜B(t)χ(P0) remains a “good”\nPDO, which is the case if the time t≤TEhr/2. To improve on (3.13) we will\ninvestigate the propagator Vtfor “large logarithmic times”, namely\n(3.14) t∼ Klog1//planckover2pi1,withK>0 large (but independent of /planckover2pi1).\nFor such times, the function ˜b(t)χ(p0) oscillates on scales much smaller than /planckover2pi1, so\nit cannot belong to any decent symbol class. As a result, the norm o f˜B(t)χ(P0) is\na priori unrelated with that function. However, using the hyperbo licity of the flow\none can prove an upper bound for the propagator Vtin terms of a certain topolog-\nical pressure depending on the damping and the hyperbolicity, and then obtain a\nconstraint on the quantum decay rates.\nBefore stating our bound, let us first introduce the notion of topo logical pressure\nassociated with the flow ΦtonS∗X. For any observable f∈C(S∗X), the pressure\nP(f) =P(f,Φt↾S∗X) can be defined by\n(3.15) P(f)def= sup{hKS(µ)+µ(f), µ∈ M}.\nNoticethatthedefinition(3.7)oftheratefunction ˜H(s)isverysimilar. Thepressure\nis also given by the growth rate of weighted sums over long closed geo desics6:\nP(f) = lim\nT→∞1\nTlog/summationdisplay\nT≤|γ|≤T+1e/integraltext\nγf.\nThe pressure appearing in the next theorem is associated with the f unctionf=\nq−ϕ+/2, where ϕ+is the infinitesimal unstable Jacobian (3.3), it thus mixes the\ndamping and hyperbolicity.\nTheorem 15. [Sche10]LetXbe an Anosov manifold, and q∈S0(T∗X)a damping\nfunction.\nAssume the following (purely classical) inequality holds:\n(3.16) P(q−ϕ+/2)< q+.\nThen, for any ǫ >0, there exists N >0such that, for /planckover2pi1small enough, the following\nresolvent bounds:\n(3.17)\n∀z∈D(1/2,C/planckover2pi1)∩/braceleftbig\nImz//planckover2pi1≥ P(q−ϕ+/2)+ǫ/bracerightbig\n,/vextenddouble/vextenddouble(P(/planckover2pi1)−z)−1/vextenddouble/vextenddouble≤/planckover2pi1−N.\n6The sum over long closed geodesics is analogousto a partition function in statistical mechanics,\nwith the time Tcorresponding to the volume of the system. This explains why the gr owth rate is\ncalled a “pressure”.\n12−ε ε\nhqhq1/2 1/2+ 1/2−\nPh\nhq−+\nFigure 3.2. Spectral gap in case the pressure P(q−ϕ+/2)< q+.\nAs a consequence, all quantum decay rates for zn(/planckover2pi1)∈D(1/2,C/planckover2pi1)satisfy\n(3.18)Imzn(/planckover2pi1)\n/planckover2pi1≤ P(q−ϕ+/2)+ǫ.\nin particular we have a spectral gap.\nIn the next subsection we present situations for which the inequalit y (3.16) is\nsatisfied. The proof of this theorem was inspired by a similar result in t he case of\nresonances in chaotic scattering [NZ2, NZ3]. We give some hints of t he proof in\n§3.3.2.\nLet us now apply this bound to the damped wave equation, that is tak eq(x,ξ) =\n−a(x).\nCorollary 16. [Sche10] Consider the DWE on Xwith nontrivial damping a≥0.\nAssume that the GCC fails ( a−= 0), but that the condition P(−a−ϕ+/2)<0holds\ntrue.\nThen the DWE has a spectral gap, and the energy decays exponen tially for regular\ninitial data, as in (1.10).\n3.3.1.Conditions for a spectral gap. Theinequality(3.16)canholdonlyifthedamp-\ning function qvaries “sufficiently” around its average value ¯ q. Indeed, one always\nhasP(−ϕ+/2)>0 for an Anosov flow, so by adding a constant dampint q≡¯qone\ngets\nP(¯q−ϕ+/2) = ¯q+P(−ϕ+/2)>¯q=q+.\nThe pressure depends continuously of q(in theC0topology), so if q= ¯q+δqwith\n/ba∇dblδq/ba∇dblC0small, the inequality (3.16) will not be satisfied.\nLet us consider the case of the DWE as in Corollary 16. The failure of t he GCC\nis equivalent to the fact that the set of undamped trajectories K(1.9) is nonempty.\nThis set is flow-invariant andclosed, so one candefine thepressure P(−ϕ+/2,Φt↾K)\nassociated with the flow Φtrestricted on K. It was shown in [Sche11] that if we\nmultiply the damping a(x) by a large constant C >0 (this does not modify the set\nK), then\nlim\nC→∞P(−Ca−ϕ+/2) =P(−ϕ+/2,Φt↾K).\nIf theset Kisthin enough, the pressure P(−ϕ+/2,Φt↾K) is negative (forinstance, if\nKconsists in a single closed geodesic, the pressure is equal to the ave rage of−ϕ+/2\nalong the geodesic, which is negative). In that case, the above limit s hows that the\npressure P(−Ca−ϕ+/2) is also negative if C≫1.\n133.3.2.Proof of the pressure bound: decomposing into “symbolic pat hs”.In this sec-\ntion we explain the strategy of proof of Thm 15. As mentioned befor e, we want to\nbound the norm of the propagator for large logarithmic times. We will prove the\nfollowing\nLemma 17. Chooseǫ >0. Then, for an energy cutoff χ0of small enough support,\nand a large enough constant K>0, for/planckover2pi1</planckover2pi1ǫone has the bound\n(3.19)/vextenddouble/vextenddoubleVtχ0(P0)/vextenddouble/vextenddouble≤et(P(q−ϕ+/2)+ǫ), t∼ Klog1//planckover2pi1.\nFrom there, obtaining the resolvent estimate (3.17) is relatively str aightforward.\nProof.Ouraimistoboundthenorm /ba∇dblVtu0/ba∇dblforanormalized stateu0microlocalized\ninp−1\n0([1/2±δ]), andatime t∼ Klog1//planckover2pi1. Wemayassumethat u0ismicrolocalized\nin an open subset Wof diameter ≤δ. We will first decompose u0in an adapted\nbasis of local “momentum” modes/braceleftbig\neη, η∈Rd,|η| ≤δ/bracerightbig\n. Here the parameter ηis\nthe“momentumcoordinate”inaDarbouxcoordinateframe {(y,η),|y| ≤δ,|η| ≤δ}\ninW, chosen such that η1=p0−1/2 represents the energy, y1the time, and the\n“momentum” Lagrangianleaves Λ η0={(y,η0)}areclose theunstable foliation. The\nmomentum states eηare Lagrangian states supported by the leaves Λ η, of norms\nO(1). The decomposition of u0in this “basis” reads\n(3.20) u0=/integraldisplay\n|η|≤δˆu0(η)eηdη\n(2π/planckover2pi1)d/2+O(/planckover2pi1∞),\nwhere the factor (2 π/planckover2pi1)−d/2ensures that /ba∇dblˆu0/ba∇dblL1=O(1).The reason for this de-\ncomposition is that we are able to precisely control the evolution of e ach individ-\nual state eη, up to times t∼ Klog1//planckover2pi1. The state Vteηis a Lagrangian state\nsupported on the Lagrangian leaf Φt(Λη). Due to the hyperbolicity of the flow,\nwhentgrows the leaf Φt(Λη) expands exponentially fast along the unstable direc-\ntions, and converges to the unstable foliation. To analyze the stat eVteη, we can\nview as the combination of many local Lagrangian states associated with bounded\npieces of Φt(Λη). A convenient bookkeeping consists in using a partition of unity\ninp−1\n0([1/2±δ]). One first covers this region by a finite family of open subsets\n{Wj, j= 1,...,J}of diameters ≤δ, and then constructs a smooth partition of\nunity{πj∈C∞\nc(Wj,[0,1]), j= 1,...,J}adapted to this cover:/summationtextJ\nj=1πj= 1 near\np−1\n0([1/2±δ]). This partitionisquantized into/braceleftBig\nΠjdef= Op/planckover2pi1(πj), j= 1,...,J/bracerightBig\n, such\nthat, for any energy cutoff χany supported in [1 /2±δ],\n(3.21)J/summationdisplay\nj=1Πjχ(P0) =χ(P0)+OL2→L2(/planckover2pi1∞).\nWe use this quantum partition to split the propagator Vninto “symbolic paths”:\n(3.22) Vneη=J/summationdisplay\nα1,...,αn=1Vα1···αneη+O(/planckover2pi1∞), Vα1···αn= ΠαnVΠαn−1···VΠα1V.\nFor each symbolic sequence α=α1···αn, the state Vαeηis a Lagrangian state sup-\nported on the Lagrangian leaf Λα\nηobtained from Λ ηby a sequence of nevolutions by\nΦ1and truncations on Wαk. The full evolved leaf is given by Φt(Λη) =/uniontext\n|α|=nΛα\nη.\nIf Λα\nηis empty Vαeη=O(/planckover2pi1∞). Otherwise ( α“admissible”), Λα\nηis a leaf of di-\nameter≤δ, close to the unstable foliation. One can compute the amplitude of\n14the Lagrangian state Vαeηto any order in /planckover2pi1. It is governed by both the accumu-\nlated damping and accumulated instability along the “path” α, leading to the norm\nestimate\n(3.23) /ba∇dblVαeη/ba∇dbl ≤Cb(α)J+(α)−1/2,where\n(3.24)b(α) =n/productdisplay\nk=1b(αk), b(j) = max\nρ∈Wjb(1,ρ),and similarly for/parenleftbig\nJ+/parenrightbig−1/2.\nApplying the triangular inequality to (3.22), we obtain a bound on the n orm ofVneη\nwhich can be related with the topological pressure:\n(3.25) /ba∇dblVneη/ba∇dbl ≤CJ/summationdisplay\nαadmis.b(α)J+(α)−1/2≤exp/braceleftbig\nn/parenleftbig\nP(q−ϕ+/2)+O(δ)/parenrightbig/bracerightbig\n.\nInserting thisboundintheexpansion (3.20) weget thesameboundf orVnu0withan\nextra factor /planckover2pi1−d/2; however, this factor is smaller than enδif we take n∼ Klog1//planckover2pi1\nandKlarge enough, in which case the bound (3.25) also applies to /ba∇dblVnu0/ba∇dbl. This\nends the proof of the norm bound (3.19). /square\nHad we inserted (3.23) in the expansion (3.20) before summing over α, we would\nhave obtained the following hyperbolic dispersion estimate :\n/ba∇dblVα/ba∇dbl ≤min/parenleftbig\nb(α), C/planckover2pi1−d/2b(α)J+(α)−1/2/parenrightbig\n,|α| ≤ Klog1//planckover2pi1.\nThis type of estimate was first proved in the case of the undamped o peratorP0(/planckover2pi1)\non an Anosov manifold [Anan08, AN1]. The adaptation to the case of t he damped\nwave equation was written in [Sche10]. A similar estimate was also shown in [NZ2]\nin the case of chaotic scattering.\n3.4.A “thickness” condition for a spectral gap. In this section we present a\nspectral gap condition expressed only in terms of the set of undam ped trajectories\n(1.9), but independently of the strength of the variations of the d amping (as op-\nposed to the “pressure condition” (3.16)). A proof of this result w ill appear in a\nforthcoming publication.\nTo state our result in the context of a damping function q(x,ξ), we introduce the\nset of “least damped trajectories”\n(3.26) K=/uniondisplay\n{suppµ, µ∈ M, µ(q) =q+},\nwhich generalizes7the set (1.9). Our condition for a spectral gap depends on a\ntopological pressure associated with the flow Φt↾K. We use quantities defined in\n§3.1.\nTheorem 18. LetXbe an Anosov manifold, and q∈S0(T∗X)a damping function.\nAssume the following condition holds:\n(3.27) P(−ϕ+,Φt↾K)<(d−1)/parenleftBigνmin\n2−λmax/parenrightBig\n.\nThen, there exists γ >0(“the gap”), C >0andN >0such that, for /planckover2pi1>0small\nenough, the following resolvent bound holds:\n(3.28) ∀z∈D(1/2,C/planckover2pi1)∩{Imz//planckover2pi1≥q+−γ},/vextenddouble/vextenddouble(P(/planckover2pi1)−z)−1/vextenddouble/vextenddouble≤C/planckover2pi1−N.\n7The two definitions are not strictly equal [Sche11], but this subtlet y will be irrelevant here.\n15As a consequence, all eigenvalues of P(/planckover2pi1)inD(1/2,C/planckover2pi1)satisfy\nImzn//planckover2pi1≤q+−γ,\nso we have a spectral gap.\nIntheframeworkofthedampedwaveequationwith(nontrivial)dam pingfunction\na(x)≥0, this gives the following\nCorollary 19. Assume that the set of undamped trajectories (1.9) is such th at the\ncondition (3.27) holds. Then, there exists γ >0,C >0andN >0such that\n/vextenddouble/vextenddouble(τ−A)−1/vextenddouble/vextenddouble\nL2→L2≤CτN,∀τ∈ {|Reτ| ≥C,Imτ≥ −γ}.\nAs a consequence, the energy decays exponentially for regul ar initial data, as in\n(1.10).\nOn a manifold of constant negative curvature −Λ2, the condition (3.27) takes the\nform\n(3.29) htop(Φt↾K)<(d−1)Λ/2,\nwherehtopis the topological entropy of the flow on K. It is usually interpreted as a\nmeasure of the “complexity” of the flow on K; we rather see it as a measure of the\n“thickness” of K. This bound reminds us of the lower bound obtained by Anan-\ntharamanwhen describing the localizationof eigenstates of theLap lacian onAnosov\nmanifolds [Anan08]: in constant curvature, she shows the semiclass ical measures as-\nsociated with any sequence of eigenstates cannot be supported o n a set of entropy\nsmaller than ( d−1)Λ/2, thus forbidding the eigenstates from being too localized.\nFollowing the thread of Remark 12, the spectral gap condition (3.29 ) could be seen\nas a “nonperturbative analogue” of this delocalization result.\nRemark20.For certain Anosov manifolds of dimension d≥3, the condition (3.27)\ncan never be satisfied. Indeed, the variational formula (3.15) sho ws that, for any\nclosed invariant set K′⊂S∗X, the pressure\nP(−ϕ+,Φt↾K′)≥ −sup\nµ∈Mµ(ϕ+)≥ −(d−1)λmax.\nIf the instability along E+isvery anistropic (meaning that the largest and smallest\npositive Lyapunov exponents are very different), then ( d−1)λmax−supµ∈Mµ(ϕ+)\ncan be strictly larger than(d−1)νmin\n2, in which case\nP(−ϕ+,Φt↾K′)>(d−1)/parenleftBigνmin\n2−λmax/parenrightBig\nfor any closed invariant set K′.\nThis remark hints at the fact that the condition (3.27) is probably no t sharp\n(even in 2 dimensions), except maybe on manifolds of constant curv ature. Rivi` ere\nhas recently improved Anantharaman’s lower bound on the support of semiclassical\nmeasures in variable curvature [Riv11]: he shows that any such supp ortSmust\nsatisfyP(−ϕ+/2,Φt↾S)≥0. If we follow the above analogy (and also following the\nresults of §3.3.1), it seems natural to expect the following\nConjecture 21. If the set Kof least damped trajectories satisfies\n(3.30) P(−ϕ+/2,Φt↾K)<0,\nthen there is a spectral gap in the spectrum of P(/planckover2pi1), in the sense of Thm 18.\n16In constant curvature the condition (3.30) is equivalent with (3.29) . In variable\ncurvature, it is weaker than (3.27). A Proof of (3.30) should make u se oflocal\nexpansion rates, instead of the globally defined rates λmaxandνmin, like in Rivi` ere’s\nwork on 2-dimensional Anosov manifolds [Riv10].\nIn the next subsection we sketch the proof of Thm 18.\n3.4.1.Sketch of proof for Thm 18. AswasthecaseforThm15, theproofproceedsby\nbounding the norm of the propagator Vtχ0(P0) for some logarithmic time (now the\ndouble of the Ehrenfest time, see (3.38)). In the course of the pr oof we will also need\nto control the evolution of a certain type of “microscopic” Lagran gian states. The\nnovelty comparedtotheproofofThm15isthatwewill nowdistinguish between two\ntypes of phase space points, the “weakly damped” vs. “strongly d amped” points.\nWithout loss of generality8, we may assume that the damping function q(x,ξ) is\ncompactly supported inside p−1\n0([1/2±ǫ]). From now on we denote the Ehrenfest\ntime byTdef=TEhr. We fix some level α∈(¯q,q+), and consider the set of “weakly\ndamped points”\nΩ+,αdef={ρ∈T∗X,/a\\}b∇acketle{tq/a\\}b∇acket∇i}htT,sym(q)≥α}\n(remember that /a\\}b∇acketle{tq/a\\}b∇acket∇i}htt,symis the symmetric time average (3.11)). The large deviation\nestimate (3.9) provides a bound on the volume of Ω +,α∩S∗X:\n(3.31) µL(Ω+,α∩S∗X)≤/planckover2pi1−˜H(α)\nλmax−O(ǫ).\nFrom the definition (3.7), ˜H(q+) is equal to the pressure P(−ϕ+,Φt↾K) appearing\nin(3.27). If we assume (3.27), then, bycontinuity of ˜H(s) on[q−,q+], wemay choose\nα∈(¯q,q+) large enough such that\n(3.32) β(α)def= (d−1)/parenleftBigνmin\n2−λmax/parenrightBig\n−˜H(α)>0.\nWe will now associate a quantum projector to the set Ω +,α. We first symmetrize the\nfactorization (2.3), by writing\n(3.33)\nVt=Ut/2Bs(t)Ut/2,whereBs(t) has principal symbol bs(t) =et/angbracketleftq/angbracketrightt,sym.\nAlthough this symbol is positive, the operator Bs(t) may not be selfadjoint; we then\ntake its polar decomposition\nBs(t) =W(t)A(t),where/braceleftBigg\nA(t) = (Bs(t)∗Bs(t))1/2is definite positive,\nW(t) is unitary.\nFor the same reasons as in §3.2.3 (and due to the support assumption on q),\nthe operators Bs(T),A(T),W(T) remain “good” PDO up to the Ehrenfest time:\nBs(T), A(T)∈/planckover2pi1−CΨ0\n1/2−ǫ(X), andW(T)∈Ψ0\n1/2−ǫ(X).\nThe operators A(T) andBs(T) have the same leading symbol bs(T) =eT/angbracketleftq/angbracketrightT,sym.\nHence, to the set Ω +,α=/braceleftbig\nbs(T,ρ)≥eαT/bracerightbig\nwe associate the spectral projector\nΠ+= Π+,αdef= 1lA(T)≥eαT,and call Π −=I−Π+.\n8Lemma 7 can be extended to logarithmic times.\n17We use these projectors to decompose the propagator at time 2 T: for some energy\ncutoffχ0we write\nV2Tχ0(P0) =UT/2W(T)A(T)UTW(T)A(T)UT/2χ0(P0) (3.34)\n=UT/2W(T)A(T)(Π++Π−)UTW(T)(Π++Π−)A(T)UT/2χ0(P0).\nThe RHS splits into four terms. Three terms contain at least one fac tor Π−: for\nthem we use the obvious bound\n(3.35) /ba∇dblA(T)Π−/ba∇dbl=/ba∇dblΠ−A(T)/ba∇dbl ≤eαT.\nThe remaining term contains the factor\nUT\n++def= Π+UTW(T)Π+.\nThe norm of this operator will be bounded by the following hyperbolic dispersion\nestimate, the proof of which is sketched in §3.4.2.\nProposition 22. Assume that for some α∈(¯q,q+)the condition (3.32) holds. Fix\nsomeǫ >0. Then, if the energy cutoff χ1has small enough support, for /planckover2pi1>0small\nenough one has\n(3.36)/vextenddouble/vextenddoubleUT\n++χ1(P0)/vextenddouble/vextenddouble\nL2→L2≤/planckover2pi1β(α)\nλmax−O(ǫ).\nSince propagation does not modify the energy localization, if choose in (3.34) a\ncutoffχ0≺χ1, we then have\nA(T)UT/2χ0(P0) =χ1(P0)A(T)UT/2χ0(P0)+O(/planckover2pi1∞).\nInserting this identity and the bounds (2.7), (3.35) and (3.36) in the identity (3.34),\nwe get\n(3.37)/vextenddouble/vextenddoubleV2Tχ0(P0)/vextenddouble/vextenddouble≤e2T(q++O(ǫ))/parenleftbig\ne−Tβ(α)+eT(α−q+)+e2T(α−q+)/parenrightbig\n.\nWe may optimize this upper bound over the level α: the optimal value of the\nexponent is reached for the (unique) parameter αc∈(¯q,q+) solving\nβ(α) =q+−α.\nFor any γ >0 satisfying q+−γ >q++αc\n2, we get (for /planckover2pi1>0 small enough) the\nfollowing norm bound for the propagator:\n(3.38)/vextenddouble/vextenddoubleV2Tχ0(P0)/vextenddouble/vextenddouble≤e2T(q+−γ).\nThe proof of the resolvent estimate (3.28) is then rather straight forward.\n3.4.2.Proof of the norm bound for UT\n++.In this last subsection we sketch the proof\nof Proposition 22, that is obtain an upper bound for\n(3.39)/vextenddouble/vextenddoubleUT\n++χ1(P0)/vextenddouble/vextenddouble\nL2→L2=/vextenddouble/vextenddoubleΠ+UTW(T)Π+χ1(P0)/vextenddouble/vextenddouble, T=TEhr.\nWe recall that Π +is the projector associated with the region Ω +,α. Let us indicate\nthat a similar type of dispersion estimate was used by S. Brooks, whe n studying\nthe delocalization of the eigenstates of quantized hyperbolic autom orphisms of the\n2-dimensional torus (the so-called “quantum cat maps”) [Bro10].\nTo estimate this norm, it will be useful to replace this projector by a smoothed\nmicrolocal projector obtained by quantizing a symbol χ+=χ+,α∈S−∞\n1/2−ǫ(T∗X),\nsuch that Op/planckover2pi1(χ+) “dominates” Π +:\nΠ+χ1(P0) = Op/planckover2pi1(χ+)Π+χ1(P0)+O(/planckover2pi1∞).\n18The norm (3.39) can then be bounded by:/vextenddouble/vextenddoubleUT\n++χ1(P0)/vextenddouble/vextenddouble≤sup\n/bardblu1/bardbl=/bardblu2/bardbl=1/vextendsingle/vextendsingle/a\\}b∇acketle{tOp/planckover2pi1(χ+)u2,UTW(T)Op/planckover2pi1(χ+)u1/a\\}b∇acket∇i}ht/vextendsingle/vextendsingle+O(/planckover2pi1∞).\nThe symbol χ+canbe chosen supported inside a set of theformΩ +,α−Cǫ∩p−1\n0([1/2±\nǫ]). This set is quite irregular, and the main information we have on it is an estimate\non its volume (using large deviation estimates like (3.31)). It is then co nvenient to\nuse ananti-Wick quantization scheme for Op/planckover2pi1(χ+), that is use a family of coherent\nstates (Gaussian wavepackets) {eρ, ρ∈T∗X}9, to define\nOp/planckover2pi1(χ+)def=/integraldisplaydρ\n(2π/planckover2pi1)dχ+(ρ)/a\\}b∇acketle{teρ,•/a\\}b∇acket∇i}hteρ.\nEach coherent state eρis normalized, and is microlocalized in a “microscopic ellipse”\naroundρ. This ellipse is chosen to be “adapted” to the flow. Let us describe it\nusing the local Darboux coordinates {(y,η)}as in§3.3.2. The ellipse is “short” in\nthe energy direction, ∆ η1∼/planckover2pi11−ǫ/2, and “long” in the time direction, ∆ y1∼/planckover2pi1ǫ/2.\nThe spread along the transverse directions is chosen isotropic: ∆ yj= ∆ηj∼/planckover2pi11/2,\nj= 2,...,d.\nThe scalar product /a\\}b∇acketle{tOp+(χ+)u2,UTWs(T)Op+(χ+)u1/a\\}b∇acket∇i}htcan now be expressed as\na double phase space integral/integraldisplay /integraldisplay\np−1\n0([1/2±ǫ])dρ1dρ2\n(2π/planckover2pi1)2d/a\\}b∇acketle{tu2,eρ2/a\\}b∇acket∇i}ht/a\\}b∇acketle{teρ1,u1/a\\}b∇acket∇i}htχ+(ρ2)χ+(ρ1)/a\\}b∇acketle{tU−T/2eρ2,UT/2W(T)eρ1/a\\}b∇acket∇i}ht.\nThe state W(T)eρ1is approximately identical to eρ1. We can then precisely describe\nthe (undamped) evolutions of the coherent states eρiup to the times ±T/2. The\nstateUT/2eρ1(resp.U−T/2eρ2) is a (microscopic) Lagrangian state along a leaf of\nthe weak unstable manifold of volume ∼/planckover2pi1(d−1+ǫ)/2J+\nT/2(ρ1) centered at ΦT/2(ρ1),\n(resp. a leaf of the weak stable manifold of volume ∼/planckover2pi1(d−1+ǫ)/2J+\nT/2(Φ−T/2(ρ2))\ncentered at Φ−T/2(ρ2)). Using the sharp energy localization of these states and the\nfact that stable and unstable manifolds intersect transversely to each other, one gets\nthe following bound:\n/vextendsingle/vextendsingle/a\\}b∇acketle{tU−T/2eρ2,UT/2W(T)eρ1/a\\}b∇acket∇i}ht/vextendsingle/vextendsingle≤θ/parenleftBig\nη1(ρ1)−η1(ρ2)\n/planckover2pi11−ǫ/parenrightBig\n/radicalBig\nJ+\nT/2(ρ1)J+\nT/2(Φ−T/2(ρ2))+O(/planckover2pi1∞),\nfor some θ∈C∞\nc([−1,1]). The denominator is bounded below by eνmin(d−1)T/2.\nInserting thisbound intheabove double integral andusing largedev iation estimates\nsimilar with (3.31) for the support of χ+, one finally gets (after some manipulations)\nthe bound (3.36). /square\nReferences\n[Anan08] N. Anantharaman, Entropy and the localization of eigenfunctions , Ann. Math. (2) 168,\n435–475 (2008)\n[Anan10] N. Anantharaman, Spectral deviations for the damped wave equation , GAFA 20 (2010)\n593–626\n[AN1] N. Anantharaman and S. Nonnenmacher, Half-delocalization of eigenfunctions for the\nLaplacian on an Anosov manifold , Ann. Inst. Fourier 57(7), 2465–2523 (2007)\n9Because χ+is supported inside p−1\n0([1/2−δ]),it is sufficient to construct a family of coherent\nstates with ρin this (compact) region.\n19[AschLeb] M. Asch and G. Lebeau, The Spectrum of the Damped Wave Operator for a Bounded\nDomain in R2, Exper. Math. 12 (2003) 227–241\n[Bro10] S. 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Zelditch, Recent developments in mathematical quantum chaos , inCurrent Developments\nin Mathematics, 2009 , D.Jerison, B.Mazur, T.Mrowka, W.Schmid, R.Stanley, S-T Yau\n(eds.), International Press 2009\n20Institut de Physique th ´eorique, CEA-Saclay, unit ´e de recherche associ ´ee au\nCNRS, 91191 Gif-sur-Yvette, France\n21"    },    {        "title": "1812.01798v1.Quasi_normal_modes_of_bumblebee_wormhole.pdf",        "content": "Quasi-normal modes of bumblebee wormhole\nR. Oliveiraa, D. M. Dantasa, Victor Santosa, C. A. S. Almeidaa\naUniversidade Federal do Cear´ a (UFC), Departamento de F´ ısica, Campus do Pici, Caixa\nPostal 6030, 60455-760, Fortaleza, Cear´ a, Brazil\nAbstract\nIn this work, we calculate the quasi-normal frequencies from a bumble-\nbee traversable wormhole. The bumblebee wormhole model is based on the\nbumblebee gravity, which exhibits a spontaneous Lorentz symmetry break-\ning. Supporting by the Lorentz violation parameter λ, this model allows the\nfulfillment of the flare-out and energy conditions, granted non-exotic matter\nto the wormhole. We analyze the parameters of bumblebee wormhole in or-\nder to obtain a Reege-Wheeler’s equation with a bell-shaped potential. We\nobtain the quasi-normal modes (QNMs) via the WKB approximation method\nfor both scalar and gravitational perturbations. All frequencies obtained are\nstable and the time domain profiles have decreasing oscillation (damping)\nprofiles for the bumblebee wormhole.\nKeywords: Gravitational Waves; Wormholes; Quasi-normal Modes;\nReege-Wheeler’s Potential.\n1. Introduction\nWormholes are solutions of Einstein equation denoted by tunnels that\nconnect two different regions of spacetime [1, 2]. The first conception of such\nstructures comes from 1935 by the Einstein-Rosen bridge [1], being the term\nwormhole later adopted by Misner and Wheeler in 1957 [2]. Unlike black\nholes, wormholes have no horizon, and it can be traversable depending on\nsome conditions at the wormhole throat [3–6].\nEmail addresses: rondinelly@fisica.ufc.br (R. Oliveiraa),davi@fisica.ufc.br\n(D. M. Dantasa),victor_santos@fisica.ufc.br (Victor Santosa),\ncarlos@fisica.ufc.br (C. A. S. Almeidaa)\nPreprint submitted to Elsevier December 6, 2018arXiv:1812.01798v1  [gr-qc]  5 Dec 2018In a very recently work, Bueno et al. [8] present wormholes as a model\nfor a class of exotic compact objects (ECOs). As a matter of fact, in the\ncontext of gravitational waves and its detections [7], wormholes would be dis-\ntinguished from black holes due to its features of ECOs, which can exhibit\ntwo bumps, instead of the usual single bump characteristic of black holes\n[8]. On the other hand, traversable wormholes violate the null convergence\ncondition (NCC) close at the throat, which leads to a violation of the null en-\nergy condition (NEC) [9, 10]. Hence, in the context of the usual formulation\nof Einstein-Hilbert General Relativity, traversable wormholes need an exotic\nmatter source [10–13]. However, recently some approaches were proposed in\norder to avoid such exotic matter by the modification of the gravity, namely,\nwith the wormhole in a Born-Infeld gravity [10], in the f(R) theories [11],\nin the framework of Gauss-Bonnet [12] and considering the wormhole in the\nbumblebee gravity scenario [14].\nThe so-called bumblebee model was developed from the string theory,\nwhich a spontaneous Lorentz symmetry breaking (LSB) was verified [15].\nSuch violations imply in the conception of the so-called Standard Model\nExtension (SME) [16, 17]. Recently, a Schwarzschild-like solution on a bum-\nblebee gravity was proposed in Ref. [18], where the Lorentz violation pa-\nrameter is upper-bounded by the Shapiro’s time delay of light. Moreover,\na traversable wormhole solution in the framework of the bumblebee gravity\nwas proposed by ¨Ovg¨ un et al. [14]. In this bumblebee wormhole, the Lorentz\nviolation parameter can support a normal matter source. At the limit of van-\nishing LSB, the deflection of light in this model becomes the same of the Ellis\nwormhole [19].\nIn this letter, we calculate the quasi-normal modes (QNMs) and time do-\nmain profiles for scalar and gravitational perturbations in the ¨Ovg¨ un bumble-\nbee wormhole [14]. We obtain a range over the Lorentz violation parameter\nλthat satisfies to all flare-out and energy conditions. Furthermore, we also\nchoose suitable LSB parameter that performs stable quasi-normal modes with\na decreasing time-domain profile.\nThe paper is organized as follows. In Sec 2, the ¨Ovg¨ un bumblebee worm-\nhole is reviewed. In Sec 3, the energy conditions and the flare-out conditions\nare analyzed by the LSB parameter. As a matter of fact, here we obtain a\ntranversable non-exotic wormhole. In subsection 3.1, we introduce the change\nof variable to obtain the Regge-Wheeler equation [20]. In Sec 4, we describe\nthe Regge-Wheeler equation for the scalar and tensorial perturbations in the\nbumblebee wormhole, where we made a suitable choice of parameters to ob-\n2tain an exact bell-shaped Regge-Wheeler potential. In subsection 4.1, the\nquasi-normal frequencies were computed by the third-order WKB method\n[21] and the damping time-domain profile was evaluated for both scalar and\ntensorial perturbations. In Sec 5, we present our last discussions and sum-\nmarize our results.\n2. Bumblebee wormhole\nIn this section, we review the exact solution of bumblebee wormhole pre-\nsented in Ref. [14]. Let us start with the following bumblebee action\nSB=/integraldisplay\ndx4√−g/bracketleftbiggR\n2κ+1\n2κξBµBνRµν−1\n4BµνBµν−V(BµBµ±b2)/bracketrightbigg\n+/integraldisplay\ndx4Lm,\n(1)\nwhereBµrepresents the bumblebee vector field, the Bµν=∂µBν−∂νBµis the\nbumblebee field strength, and the ξis the non-minimal curvature coupling\nconstant. For the vacuum solutions V(BµBµ∓b2) = 0, we have that b2=\n±BµBµ=±bµbµis the non-null vector norm associated to the vacuum\nexpectation value /angbracketleftBµ/angbracketright= bµ[14, 17]. The scalar curvature is denoted by R,\ngis the metric determinant and κthe gravitational constant.\nThe energy-momentum tensor is modified by the bumblebee field in the\nfollowing form [14, 18]:\nRµν−κG/bracketleftbigg\nTM\nµν+TB\nµν−1\n2gµν/parenleftBig\nTM+TB/parenrightBig/bracketrightbigg\n= 0, (2)\nwhereTM=gµνTM\nµνand the bumblebee energy-momentum tensor TB\nµνreads\nTB\nµν=−BµαBα\nν−1\n4BαβBαβgµν−Vgµν+ 2V/primeBµBν+\n+ξ\nκ/bracketleftbigg1\n2BαBβRαβgµν−BµBαRαν−BνBαRαµ+\n+1\n2∇α∇µ(BαBν)−1\n2∇2(BµBν)+\n−1\n2gµν∇α∇β(BαBβ)/bracketrightbigg\n. (3)\nThe modified Einstein equation in Eq. (2) with the energy-momentum\n3tensor in Eq. (3) can be explicited as\nEinstein\nµν =Rµν−κ/parenleftbigg\nTM\nµν−1\n2gµνTM/parenrightbigg\n−κTB\nµν−2κgµνV+\n+κBαBαgµνV/prime−ξ\n4gµν∇2(BαBα)+\n−ξ\n2gµν∇α∇β(BαBβ) = 0. (4)\nAt this point, the authors of Ref. [14] choose a static and spherically\nsymmetric traversable wormhole solution in the following form [5, 14]\nds2=e2Λdt2−dr2\n1−b(r)\nr−r2dθ2−r2sin2θdφ2, (5)\nwhere the red-shift function is made null (Λ = 0) and the bumblebee vector\nbµis set to be correlated to the wormhole shape function b(r) as following\n[14]\nbµ=\n0,/radicalBigga\n1−b(r)\nr,0,0\n, (6)\nwhereais a positive constant associated with the Lorentz violation term.\nBesides, following the reference [14], the isotropic energy–momentum ten-\nsor can be decomposed as a perfect fluid ( Tµ\nν)M= (ρ,−P,−P,−P) where\nP=wρ, (7)\nassumingρ≥0. The−1\n3<w≤1 is a dimensionless constant responsible to\nhold the energy conditions.\nSubstituting the wormhole metric Eq. (5) into Einstein equation (4), we\nhave the following new Einstein equations with the b(r) andwdependence\nEinstein\ntt =−κρr3(1 + 3w) +λr˙b(r)−λb(r) = 0, (8)\nEinstein\nrr =κρr3(w−1) + (2 + 3λ)r˙b−(2 + 3λ)b(r) = 0, (9)\nEinstein\nθθ =κρr3(w−1) +r˙b(r) + (2λ+ 1)b(r)−2λr= 0, (10)\nwhereλ=aξis defined as the Lorentz symmetry breaking (LSB) parameter\nand the dot denotes derivative with respect to the coordinate r.\n4The energy density ρcan be obtained from the system of equations\n(8−10). By solving directly ρfrom Eq. (8) we obtain:\nρ=λ/parenleftBig\nr˙b(r)−b(r)/parenrightBig\nκr3(1 + 3w). (11)\nMoreover,b(r) can be found by multiplying ( w−1) by Eq. (8) and\nsumming with (1 + 3 w) multiplied by Eq. (10). Additionally, the condition\nat wormhole throat b(r0) =r0leads to the solutions\nb(r) =λr\nλ+ 1+r0\nλ+ 1/parenleftbiggr0\nr/parenrightbiggγ\n, (12)\nwhereγ(w,λ) =λ(5w+3)+3w+1\nλ(w−1)+3w+1.\nThe derivative ˙b(r) can explicitly be obtained as\n˙b(r) =λ\nλ+ 1/bracketleftBigg\n1−γ/parenleftbiggr0\nr/parenrightbiggγ+1/bracketrightBigg\n. (13)\nFinally, from Eq. (12) and Eq. (13), the energy-density of Eq. (11) can\nbe found in the form\nρ(r) =−2λrγ+1\n0r−(γ+3)\nκ(λ(w−1) + 3w+ 1)(14)\nIn next section, we analyze the parameters λandwin order to obtain a\nspecial case of the ¨Ovg¨ un wormhole, where all energy conditions and flare-\nout holds. Moreover, we will also set these parameters in order to obtain a\nRegge-Wheeler potential with a bell-shaped curve.\n3. Energy conditions and flare-out\nIn this section we analyze the conditions imposed in the wormhole. We\nsummarize the conditions in Table 1. The NEC is the Null energy condition,\nWEC is the Weak energy conditions, SEC is the Strong energy condition,\nDEC is the Dominant energy conditions, and FOC is the flare-out condition.\nIn order to bound the wparameter, lets us assume that ρ≥0. Once that\nP=wρthe DEC holds for ρ≥|wρ|so−1≤w≤1. On the other hand,\n5NEC WEC SEC DEC FOC\nP+ρ≥0ρ≥0 and NEC ρ+ 3P≥0 and NEC ρ≥|P|˙b(r)<1\nTable 1: Energy and flare-out condition.\nthe SEC is verified for (1 + 3 w)ρ≥0 sow≤−1\n3. The NEC is expressed as\n(1 +w)ρ≥0, sow≥−1. Hence in order to obey NEC, SEC and DEC, the\nwparameter is must be such that\n−1\n3<w≤1. (15)\nFrom now on, we set κ=r0= 1. The energy (14) is positive when\nλ >3w+1\nw−1and it vanishes when r→∞ forγ > 3. The region where the\nconditions are valid is showed in Fig. 1, being all energy condition satisfied\nin the dark region.\n-1/3\nFigure 1: Representation of the region where the energy conditions are satisfied. In the\ndark region, all energy conditions hold. In the gray region, only the WEC is secured.\nOn the other hand, the flare-out condition (FOC) is necessary to maintain\nthe structure of the wormhole traversable [3, 5, 14, 22]. The FOC can be\nwritten as\nb(r)−r≤0 andr˙b(r)−b(r)<0⇒ ˙b(r)<1. (16)\nConsidering the equation (13), the FOC is satisfied for γ >0 ifλ > 0 and\nr > r 0. For the vanishing of energy (14) when r→∞ , it is necessary that\nγ >−3. So, the FOC is always obeyed when the energy (14) goes to zero at\ninfinity.\n63.1. Tortoise coordinates and the energy and flare-out conditions\nIn section 4, we will obtain the Regge-Wheeler equation for the ¨Ovg¨ un\nbumblebee wormhole. For this goal, a transformation to the radial variable r\ninto a new variable x(tortoise coordinate) is required. This transformation,\nfor Λ = 0, is given by following the integral[23, 24]\nx=/integraldisplay\ndr1/radicalBig\n1−b(r)\nr. (17)\nTo obtain an analytical function into xcoordinate, we choose w=−1\nandr0= 1, which transform the shape-function of Eq. (12) into\nb(r) =αr+β\nr, (18)\nwhereα=λ\nλ+1andβ=1\nλ+1= (1−α).\nHence, by solving Eq. (17) with the of Eq. (18), the tortoise coordinate\ncan be expressed as\nx=β−1/radicalBig\nβ(r2−1)⇒r=/radicalBig\nβx2+ 1. (19)\nFor this particular choice of w=−1, the SEC is violated. However, the\nenergy of Eq. (14) becomes ρ(r,λ) =λ2\nλ+1r−4which is positive when λ>−1,\nas can be seen in Fig. 2. So the DEC, NEC and WEC still hold. The FOC\nobtained by ˙b(r) of Eq. (18) readsλ\n(λ+1)−r−2\n(λ+1)<1, which it is still valid for\nλ>−1, as also can be seen in Fig. 2.\nFurthermore, in the section 4 we need to set a value for the λto compute\nthe quasi-normal modes. For a qualitative analysis, let us choose λ= 1.\nThe Fig. 3 shows the energy density ρ(r), the shape-function b(r) and its\nderivative ˙b(r) withw=−1 andλ= 1 fixed. Therefore, we note that, except\nthe SEC, all other conditions above mentioned are satisfied.\n4. Regge-Wheeler equation and quasi-normal modes\nIn this section, we obtain the wave equation for the bumblebee worm-\nhole. Let us consider an external perturbation, ignoring the back-reaction,\nand following all detailed linearization procedure for the scalar and the axial\n7Figure 2: The energy density ρ(r) is always positive for w=−1 andλ >1 (left panel).\nThe flare-out conditions is always verified for w=−1 andλ>0 (right panel).\nb(r)\nb(r)\nρ(r)\n0.0 0.5 1.0 1.5 2.0 2.5 3.0-20246\nr\nFigure 3: Plot of the shape-function b(r), its derivative ˙b(r) and the energy density ρ(r),\nwithw=−1 andλ= 1. The gray straight lines y= 0 andy= 1 help us to verify the\nWEC and the FOC, respectively.\ngravitational perturbations of the references [23–27]. By considering a sta-\ntionary solutions in the form Ψ( x,t) =ϕ(x) e−iωt, a simplified version of the\nso-called Regge-Wheeler equation can be represented by [20, 28, 29]\n/parenleftBiggd2\ndx2+ω2−V(r,l)/parenrightBigg\nϕ(x) = 0, (20)\nwhereϕ(x) is the wave function (for the scalar or gravitational perturbations)\nwithxtortoise coordinate (19), ωis the frequency and V(r,l) is the Regge-\nWheeler potential where lis the azimuthal quantum number, related to the\nangular momentum.\nFor the scalar perturbations, the Regge-Wheeler potential V(r,l) reads\n8[23, 26]\nVs(r,l) =e2Λ/bracketleftBiggl(l+ 1)\nr2−˙br−b\n2r3+1\nr/parenleftBigg\n1−b\nr/parenrightBigg\n˙Λ/bracketrightBigg\n. (21)\nSince that Λ = 0, by substituting the b(r) (18) into potential (21), the\nVs(r,l) =l(l+1)\nr2+β\nr4intoxcoordinate (19) becomes\nVs(x,l) =l(l+ 1)/parenleftBig\nx2\nλ+1+ 1/parenrightBig+/parenleftbigg1\nλ+ 1/parenrightbigg/parenleftBiggx2\nλ+ 1+ 1/parenrightBigg−2\n. (22)\nSimilarly, for the axial gravitational perturbations, the Regge-Wheeler\npotentialV(r,l) reads [25, 26, 29–31]\nVg(r,l) =e2Λ/bracketleftBiggl(l+ 1)\nr2+˙br−5b\n2r3+1\nr/parenleftBigg\n1−b\nr/parenrightBigg\n˙Λ/bracketrightBigg\n. (23)\nWithb(r) given by Eq. (18), the Vg(r,l) =l(l+1)−2α\nr2−3β\nr4intoxcoordinate\n(19) becomes\nVg(x,l) =/parenleftBigg\nl(l+ 1)−2λ\nλ+ 1/parenrightBigg/parenleftBiggx2\nλ+ 1+ 1/parenrightBigg−1\n−3\nλ+ 1/parenleftBiggx2\nλ+ 1+ 1/parenrightBigg−2\n.\n(24)\nThe Fig. 4 shows the plots of the scalar potential of Eq. (22) and grav-\nitational potential of Eq. (24). Note that both potentials are symmetric\nbell-shaped potentials centered at the origin. The increasing of the angular\nmomentum increases the peaks of potentials. For the scalar perturbation all\npotentials are repulsive. However, for the tensorial perturbation the first two\nvalues ofllead to attractive potentials. The height of peaks changes the\nbehavior of quasi-normal modes, as will be discussed in the next subsection.\n4.1. Quasi-normal modes and the time-domain\nTo compute the quasi-normal modes (QNMs) of Regge-Wheeler equation\n(20) we apply the semianalytic method of the third-order WKB approxi-\nmation presented in Ref. [21]. This method requires a positive bell-shaped\npotential [21]. Hence we impose l≥2 for the tensorial perturbations.\n9l=0\nl=1\nl=2\nl=3\n-6 -4 -2 0 2 4 6024681012\nxVS(x,l)\nl=0\nl=1\nl=2\nl=3\n-6 -4 -2 0 2 4 6-20246810\nxVG(x,l)Figure 4: The scalar (left) and gravitational (right) potentials of Regge-Wheeler equation\nfor somelparameters and λ= 1.\nBriefly, the QNMs can be found from the formula [21, 27, 32]\ni(ω2\nn−V0)/radicalBig\n−2¨V0−6/summationdisplay\ni=2Λi=n+1\n2, (25)\nwhere ¨V0is the second derivative of the potential on the maximum x0, Λiare\nconstant coefficients, and ndenotes the number of modes.\nAs a result, the quasi-normal modes for the scalar perturbations are writ-\nten in Table 2. Note that for lower l, largerncan be unstable (denoted by\n—, which were excluded due the positive imaginary part). This fact can be\ncorrelated to the small peaks of the scalar potential in Fig. 4.\nωn l= 0 l= 1 l= 2 l= 3\nn= 0±0.428623−0.496200i±1.481593−0.364167i±2.495248−0.357844i±3.498181−0.355942i\nn= 1±0.335244−1.557205i±1.283774−1.155671i±2.382690−1.089103i±3.421610−1.074500i\nn= 2±0.184733−2.616606i±0.990627−2.041774i±2.175852−1.859450i±3.274020−1.811550i\nn= 3 —±0.634513−2.973230i±1.898997−2.674104i±3.064840−2.574650i\nn= 4 —±0.204691���3.93439i±1.56697−3.52542i±2.80408−3.36589i\nn= 5 — —±1.18406−4.40578i±2.49918−4.18339i\nTable 2: QNMs for the scalar perturbations (22) with w=−1,κ=r0=λ= 1 and some\nl.\n.\nSimilarly, the quasi-normal modes for the gravitational perturbations are\nwritten in Table 3. We start with l= 3, to guarantee positive potentials.\nNote that all quasi-normal modes are stable presenting no positive imaginary\npart.\n10ωn l= 3 l= 4 l= 5\nn= 0±3.04939−0.321674i±4.156050−0.335541i±5.221450−0.341988i\nn= 1±2.987810−0.974895i±4.101510−1.010760i±5.175990−1.028450i\nn= 2±2.876140−1.653210i±3.995510−1.697660i±5.086450−1.722130i\nn= 3±2.730760−2.360980i±3.843440−2.401700i±4.955440−2.427030i\nn= 4±2.563960−3.092900i±3.65168−3.12544i±4.78639−3.146020i\nn= 5±2.380520−3.840940i±3.425930−3.868780i±4.583050−3.880670i\nTable 3: QNMs for the gravitational perturbations (24) with w=−1,κ=r0=λ= 1 and\nsomel.\n.\nMoreover, Fig. 5 shows the points of QNMs Tables 2 and 3. Note that\nall modes have smooth curves where the increase of lincreases the real part\nof each mode. The increasing of the number of modes ncan leads to QNMs\ninstabilities.\n●l=1 ▲l=2 ■l=3 ○l=4\n●\n●\n●▲\n▲\n▲\n▲■\n■\n■\n■○\n○\n○\n○●\n●\n●▲\n▲\n▲\n▲■\n■\n■\n■○\n○\n○\n○\n-3 -2 -1 0 1 2 3-3.0-2.5-2.0-1.5-1.0-0.50.0\nReωnImωn\n●l=3 ▲l=4 ■l=5\n●\n●\n●\n●\n●\n●▲\n▲\n▲\n▲\n▲\n▲■\n■\n■\n■\n■\n■●\n●\n●\n●\n●\n●▲\n▲\n▲\n▲\n▲\n▲■\n■\n■\n■\n■\n■\n-4 -2 0 2 4-4-3-2-10\nReωnImωn\nFigure 5: Plot of QNMs for scalar (left) and gravitational (right) perturbations with λ= 1.\nBesides, in Fig. 6, the time-domain of bumblebee wormhole is evaluated\nby the Gundlach’s method [33]. Note that both perturbation exhibits damp-\ning profiles, being the decreasing of scalar modes slower than the gravitational\nmodes. The lparameter faster the decay of these solutions.\n5. Conclusions\nIn this work, we study the bumblebee wormhole. This scenario has a\nLorentz violation parameter λ, which allows the preservation of energy con-\nditions, leading to a wormhole generated by non-exotic matter [14]. We study\nthe possible choices of the parameters λandw(associated to the relation\n11l=1\nl=2\nl=3\n20 30 40 50 60-8-6-4-20\ntlog(φ(t))\nl=3\nl=4\nl=5\n25 30 35 40 45 50 55 60-12-10-8-6-4-2\ntlog(φ(t))Figure 6: Time-domain log10(ϕ(t)) for the scalar (left) and the gravitational (right) per-\nturbations with λ= 1.\nP=wρ) that satisfies the flare-out and the energy conditions, as shown in\nFig. 1. In order to achieve an analytical and simplified tortoise coordinates\ntransformation of Eq. 19, we renounce the SEC condition. However, all the\nother ones, namely, NEC, WEC, DEC and FOC remain valid.\nMoreover, the scalar and tensorial perturbation of bumblebee wormhole\nwere obtained. We use the general expressions for the scalar and gravitational\npotential. Both the potentials admit positive bell-shaped (see Fig. 4). Hence\nwe evaluate the quasi-normal frequencies for both perturbations. The QNMs\nare stable (as denoted in Tables 2 and 3), and exhibits smooth curves (see\nFig. 5). Besides, we compute the time-domain profile for both perturbations\nin Fig. 6, from where we note that all QNMs studied perform damping\noscillation profiles.\nAcknowledgments\nThe authors thank the Funda¸ c˜ ao Cearense de apoio ao Desenvolvimento\nCient´ ıfico e Tecnol´ ogico (FUNCAP), the Coordena¸ c˜ ao de Aperfei¸ coamento\nde Pessoal de N´ ıvel Superior (CAPES), and the Conselho Nacional de De-\nsenvolvimento Cient´ ıfico e Tecnol´ ogico (CNPq) for financial support.\nReferences\n[1] A. Einstein and N. Rosen, Phys. Rev. 48, 73 (1935).\n[2] C. W. Misner and J. A. Wheeler, Annals Phys. 2, 525 (1957).\n[3] D. Hochberg and M. Visser, Phys. Rev. D 56, 4745 (1997).\n12[4] M. S. Morris, K. S. Thorne and U. Yurtsever, Phys. Rev. Lett. 61, 1446\n(1988).\n[5] M. S. Morris and K. S. Thorne, Am. J. Phys. 56, 395 (1988).\n[6] M. Visser, S. Kar and N. Dadhich, Phys. Rev. Lett. 90, 201102 (2003).\n[7] B. P. Abbott et al. [LIGO Scientific and Virgo Collaborations], Phys.\nRev. Lett. 116, no. 6, 061102 (2016) .\n[8] P. Bueno, P. A. Cano, F. Goelen, T. Hertog and B. Vercnocke, Phys.\nRev. D 97, no. 2, 024040 (2018).\n[9] S. Kar, N. Dadhich and M. Visser, Pramana 63, 859 (2004).\n[10] R. Shaikh, Phys. Rev. D 98, no. 6, 064033 (2018).\n[11] N. Godani and G. C. Samanta, “Traversable Wormholes and Energy\nConditions with Two Different Shape Functions in f(R) Gravity,”\narXiv:1809.00341 [gr-qc].\n[12] M. R. Mehdizadeh, M. K. Zangeneh and F. S. N. Lobo, Phys. Rev. D\n91, no. 8, 084004 (2015).\n[13] S. H. V¨ olkel and K. D. Kokkotas, Class. Quant. Grav. 35, no. 10, 105018\n(2018).\n[14] A. ¨Ovg¨ un, K. Jusufi and ˙I. Sakallı, “Exact traversable wormhole solution\nin bumblebee gravity,” arXiv:1804.09911 [gr-qc].\n[15] V. A. Kostelecky and S. Samuel, Phys. Rev. D 39, 683 (1989).\n[16] D. Colladay and V. A. Kostelecky, Phys. Rev. D 58, 116002 (1998).\n[17] V. A. Kostelecky, Phys. Rev. D 69, 105009 (2004).\n[18] R. Casana, A. Cavalcante, F. P. Poulis and E. B. Santos, Phys. Rev. D\n97, no. 10, 104001 (2018).\n[19] K. Jusufi, Int. J. Geom. Methods Mod. Phys. 14, 1750179 (2017).\n[20] T. Regge and J. A. Wheeler, Phys. Rev. 108, 1063 (1957).\n13[21] S. Iyer and C. M. Will, Phys. Rev. D 35, 3621 (1987).\n[22] M. Kord Zangeneh, F. S. N. Lobo and M. H. Dehghani, Phys. Rev. D\n92, no. 12, 124049 (2015).\n[23] S. W. Kim, Nuovo Cim. B 120, 1235 (2005).\n[24] S. E. Perez Bergliaffa and K. E. Hibberd, Phys. Rev. D 62, 044045\n(2000).\n[25] S. W. Kim, “Gravitational perturbation of traversable wormhole,” gr-\nqc/0401007.\n[26] S. W. Kim, Prog. Theor. Phys. Suppl. 172, 21 (2008).\n[27] V. Santos, R. V. Maluf and C. A. S. Almeida, Phys. Rev. D 93, no. 8,\n084047 (2016)\n[28] F. J. Zerilli, Phys. Rev. Lett. 24, 737 (1970).\n[29] P. Boonserm, T. Ngampitipan and M. Visser, Phys. Rev. D 88, 041502\n(2013)\n[30] S. Chandrasekhar, “The mathematical theory of black holes,” Oxford,\nUK: Clarendon (1985) 646 P.\n[31] K. A. Bronnikov, R. A. Konoplya and A. Zhidenko, Phys. Rev. D 86,\n024028 (2012)\n[32] R. A. Konoplya, Phys. Rev. D 68, 024018 (2003)\n[33] C. Gundlach, R. H. Price and J. Pullin, Phys. Rev. D 49, 883 (1994)\n14"    },    {        "title": "2004.04471v2.Unusual_stationary_state_in_Brownian_systems_with_Lorentz_force.pdf",        "content": "Unusual stationary state in Brownian systems with Lorentz force\nI. Abdoli,1H.D. Vuijk,1R. Wittmann,2J.U. Sommer,1, 3J.M. Brader,4and A. Sharma1, 3,\u0003\n1Leibniz-Institut f ur Polymerforschung Dresden, Institut Theorie der Polymere, 01069 Dresden, Germany\n2Institut f ur Theoretische Physik II, Weiche Materie,\nHeinrich-Heine-Universit at D usseldorf, 40225 D usseldorf, Germany\n3Technische Universit at Dresden, Institut f ur Theoretische Physik, 01069 Dresden, Germany\n4Department de Physique, Universit\u0013 e de Fribourg, CH-1700 Fribourg, Switzerland\nIn systems with overdamped dynamics, the Lorentz force reduces the di\u000busivity of a Brownian\nparticle in the plane perpendicular to the magnetic \feld. The anisotropy in di\u000busion implies that the\nFokker-Planck equation for the probabiliy distribution of the particle acquires a tensorial coe\u000ecient.\nThe tensor, however, is not a typical di\u000busion tensor due to the antisymmetric elements which\naccount for the fact that Lorentz force curves the trajectory of a moving charged particle. This gives\nrise to unusual dynamics with features such as additional Lorentz \ruxes and a nontrivial density\ndistribution, unlike a di\u000busive system. The equilibrium properties are, however, una\u000bected by the\nLorentz force. Here we show that by stochastically resetting the Brownian particle, a nonequilibrium\nsteady state can be created which preserves the hallmark features of dynamics under Lorentz force.\nWe then consider a minimalistic example of spatially inhomogeneous magnetic \feld, which shows\nhow Lorentz \ruxes fundamentally alter the boundary conditions giving rise to an unusual stationary\nstate.\nI. INTRODUCTION\nThe Lorentz force due to an external magnetic \feld\nmodi\fes the trajectory of a charged, moving particle\nwithout performing work on it. This results in charac-\nteristic helical trajectories in case of a constant magnetic\n\feld. Such motion is an idealization which compeletely\nignores dissipative e\u000bects that are highly relevant in, for\ninstance, plasma physics [1]. In fact, dissipative e\u000bects\nare dominant in colloidal systems where the dynamics\nare overdamped. Whereas the e\u000bect of Lorentz force in\nthe context of solid-state physics and plasma physics has\nbeen throughly studied, much less is known about its\ne\u000bect on di\u000busion systems subjected to an external mag-\nnetic \feld.\nA known consequence of the Lorentz force is a reduc-\ntion of the di\u000busion coe\u000ecient in the plane perpendic-\nular to the magnetic \feld, whereas the di\u000busion along\nthe \feld is una\u000bected [2, 3]. The anisotropy in di\u000busion\nimplies that the corresponding Fokker-Planck equation\nfor the probability distribution acquires a tensorial coef-\n\fcient, the components of which are determined by the\napplied magnetic \feld, the temperature, and the friction\ncoe\u000ecient. The tensor, however, is not a typical di\u000bu-\nsion tensor due to the antisymmetric elements which ac-\ncount for the fact that Lorentz force curves the trajectory\nof a charged, di\u000busing particle, giving rise to additional\nLorentz \ruxes [4, 5]. We have recently shown that the\ndynamics under this tensor are fundamentally di\u000berent\nfrom purely di\u000busive [6]. In particular, the nonequilib-\nrium dynamics are characerized by features such as ad-\nditional Lorentz \ruxes and a nontrivial density distribu-\ntion [see Fig. 1]. These have implications for dynamical\n\u0003sharma@ipfdd.deproperties of the system such as the mean \frst-passage\ntime, escape probability, and phase transition dynamics\nin \ruids [3, 6].\nSince the Lorentz force arising from an external mag-\nnetic \feld does no work on the system, the equilibrium\nproperties of the system are una\u000bected. This implies that\nto observe the nontrivial e\u000bects of Lorentz force, the sys-\ntem must be maintained out of equilibrium, possibly in a\nnonequilibrium steady state. This can be done by driving\nthe system out of equilibrium, for instance, via a time-\ndependent external potential or shear. Alternatively, one\nmay consider internally driven systems, a particularly\ninteresting example of which is active matter which is\nubiquitous in biology [7{9]. We recently demonstrated\nthat a system of active Brownian particles subjected to a\nspatially inhomogeneous Lorentz force relaxes towards a\nnonequilibrium steady state with inhomogeneous density\ndistribution and macroscopic \ruxes [10]. The distinctive\ndynamics of a charged, passive, di\u000busing particle under\nLorentz force may be appreciated by noting that if the\ntensor entering the Fokker-Planck equation was positive\nsymmetric, i.e., a di\u000busion-like tensor, there would be no\n\ruxes in the steady state.\nWe take a di\u000berent approach to drive the system into\na nonequilibrium steady state: the particle, while di\u000bus-\ning under the in\ruence of Lorentz force, is stochastically\nreset to a prescribed location at a constant rate. The\nconcept of stochastic resetting was introduced by Evans\nand Majumdar [11]. In their model, a Brownian particle\ndi\u000buses freely until it is reset to its initial location. The\nwaiting time between two consecutive resetting events\nis a random variable for which the Poissonian distribu-\ntion has been widely used. Evans and Majumdar showed\nthat di\u000busion under stochastic resetting gives rise to a\nnonequilibrium stationary state with a non-Gaussian po-\nsition distribution and particle \rux. They also demon-\nstrated that the mean \frst-passage time for this model isarXiv:2004.04471v2  [cond-mat.stat-mech]  18 Jun 20202\nFIG. 1. The nonequilibrium dynamics of a Brownian particle under Lorentz force are di\u000berent from purely di\u000busive. A hallmark\nsignature is the appearance of additional Lorentz \ruxes which result from the de\rection of di\u000busive \ruxes [6]. The particles\nare initially distributed in a disc of radius 0 :5 and evolve under Lorentz force due to a constant magnetic \feld. The \fgures\nshow the \ruxes after 1 :0 Brownian time unit. The total \rux, shown in (a), is decomposed in (b) the di\u000busive \rux and (c) the\nLorentz \rux. The direction of the \ruxes is shown by the arrows; the magnitude is color coded. Note that there is no \rux in\nthe steady state of a closed system where density is uniformly distributed [6].\n\fnite and has a minimum value at an optimal resetting\nrate. Over the last few years, stochastic resetting has\nbeen applied to a wide variety of random processes [12{\n16] and generalized to include non-Markovian resetting\nand dependence of resetting on internal dynamics [17{\n20]. It has been shown that it gives rise to intriguing\nphenomena such as dynamical phase transitions [21, 22],\nuniversal properties which are insensitive to details of\nunderlying random process [23{25] and optimal search\nstrategies [26].\nIn this paper, we show that under stochastic resetting a\nBrownian system settles into an unusual stationary state\nwhich preserves the hallmark features of dynamics un-\nder Lorentz force. In the case of a constant magnetic\n\feld, the nonequilibrium steady state is characterized by\na non-Gaussian probability density, di\u000busive and Lorentz\n\ruxes. These Lorentz \ruxes re\rect the behavior shown in\nFig. 1 and are reminiscent of Brownian vortices in a sys-\ntem of colloidal particles di\u000busing in an optical trap [27{\n29]. Due to the Lorentz force, the \rux is not along the\ndensity gradient. This holds even for a constant tensorial\ncoe\u000ecient. As a consequence, the boundary conditions\nfor di\u000busion in \fnite or semi-\fnite domains take a form\ndi\u000berent from the typical Neumann or Dirichlet condi-\ntions. By considering a minimalistic example, we show\nhow the modi\fed boundary condition gives rise to un\nunusual stationary state with no counterpart in purely\ndi\u000busive systems.\nThe paper is organized as follows. In Sec. II, we pro-\nvide a brief theoretical description of di\u000busion under\nLorentz force and stochastic resetting. In Sec. III, we\nderive the steady-state solution to the governing Fokker-\nPlanck equation for constant and inhomogeneous mag-\nnetic \felds. Finally, we discuss our results and present\nan outlook in section IV.II. THEORY AND SIMULATION\nWe consider a single di\u000busing particle which is stochas-\ntically reset to its initial position r0at a constant rate\n\u0016. The particle is subjected to Lorentz force arising from\nan external magnetic \feld B(r) =B(r)nwherenin-\ndicates the direction of the magnetic \feld and B(r) is\nthe magnitude. Our theoretical approach is based on\nthe Fokker-Planck equation for the position distribution\nof the particle. For a spatially inhomogeneous magnetic\n\feld, the probability for \fnding the particle at position\nrat timet, given that it started at r0,p(r;tjr0) obeys\nthe following Fokker-Planck equation [4, 5, 11]\n@tp(r;tjr0) =r\u0001[D(r)rp(r;tjr0)] (1)\n\u0000\u0016p(r;tjr0) +\u0016\u000e(r\u0000r0);\nwhere@tstands for derivative with respect to tand the\ntensorDis\nD(r) =D\u0014\u0012\n1+\u00142(r)\n1 +\u00142(r)M2\u0013\n\u0000\u0014(r)\n1 +\u00142(r)M\u0015\n=Ds(r) +Da(r); (2)\nwhereD=kBT=\r is the di\u000busion coe\u000ecient of a freely\ndi\u000busing particle and \u0014(r) =qB(r)=\rquanti\fes the\nstrength of Lorentz force relative to frictional force [10].\nHere\ris the friction coe\u000ecient, kBis the Boltzmann con-\nstant,Tis the temperature and qis the charge of the par-\nticle. The matrix Mis de\fned by B(r)\u0002v=B(r)Mv.\nDsandDaare the symmetric and antisymmetric parts\nof the tensor D.\nNote that Eq.(1) is not of the form of a continuity\nequation. The \frst term on the right hand side of Eq. (1)3\nFIG. 2. The stationary probability density of the particle's\nposition from Eq. (8) for a system with \u0014= 3:0 is shown in\nthe surface plot on top of the contour plot. The particle is\nstochastically reset to the origin r0=0with\u0016= 0:1. The\nsteady state is characterized by the symmetric, non-Gaussian\nprobability density, the di\u000busive and Lorentz \ruxes. Lorentz\n\ruxes are shown by white arrows.\nrepresents the contribution from overdamped motion un-\nder Lorentz force. The second and third terms stand for\nthe contribution due to the resetting of the particle: the\nsecond term represents the loss of the probability from\nthe positionrowing to resetting to the initial position\nr0while the third term stands for the gain of probability\natr0due to resetting from all other positions. The \rux\nin the system is given as\nJ(r;t) =\u0000D(r)rp(r;tjr0); (3)\nwhich can be decomposed into the di\u000busive \rux\nJs(r;t) =\u0000Ds(r)rp(r;tjr0); (4)\nand the Lorentz \rux\nJa(r;t) =\u0000Da(r)rp(r;tjr0): (5)\nNote that the di\u000busive \rux does not depend on the sign\nof the magnetic \feld. In contrast, the Lorentz \rux can\nbe reversed by reversing the magnetic \feld. Moreover, it\nis always perpendicular to the di\u000busive \rux. These prop-\nerties of Lorentz \rux, which are straightforward conse-\nquences of how the Lorentz force a\u000bects a particle's tra-\njectory, constitute the main rationale behind the above\ndecomposition. Although the dynamics are overdamped\nit is the presence of these Lorentz \ruxes which makes\nthe dynamics under Lorentz force distinct from a purely\ndi\u000busive system in which only di\u000busive \ruxes exist.\nIn order to con\frm our analytical predictions, Brow-\nnian dynamics simulations are performed using the\nLangevin equation of motion [30]. It has been shownthat the overdamped Langevin equation for a Brown-\nian motion in a magnetic \feld can yield unphysical val-\nues for velocity-dependent variables like \rux [5]. There-\nfore, we use the underdamped Langevin equation with\na su\u000eciently small mass. Omitting hydrodynamics, the\ndynamics of the particle are described by the following\nLangevin equations [5, 6]:\n_r(t) =v(t);\nm_v(t) =\u0000\rv+qv\u0002B(r) +p\n2\rkBT\u0011(t);(6)\nwheremis the mass of the particle and \u0011(t) is Gaus-\nsian white noise with zero mean and time correlation\nh\u0011(t)\u0011T(t0)i=1\u000e(t\u0000t0). The waiting time between two\nconsecutive resetting events is a random variable with\nPoisson distribution: in a small time interval \u0001 tthe par-\nticle is either reset to its initial position with probability\n\u0016\u0001tor continues to di\u000buse with probability 1 \u0000\u0016\u0001t.\nThroughout the paper we \fx the mass to m= 0:005\nand the integration time step to dt= 1\u000210\u00006\u001cwhere\n\u001c=\r=kBTis the time for di\u000busion over one unit dis-\ntance. In fact, it has been shown that even with a mass\nm= 0:02 the trajectory of the particle from Eq. (6) con-\nverges on the trajectory from the small-mass limit of this\nequation [5]. However, to ensure that the dynamics are\noverdamped, we have performed simulations with even a\nsmaller mass. The simulation results did not show any\nsigni\fcant change. Since the magnetic \feld is applied in\nthezdirection, the Lorentz force has no e\u000bect on the\nmotion in this direction. As a consequence, we restrict\nour analysis to the motion in the xyplane. Accordingly,\nthe vectorrdenotes the coordinates ( x;y) of the particle\nand the tensorial coe\u000ecient Dis a 2\u00022 matrix.\nIII. NONEQUILIBRIUM STEADY STATE\nIn this section, we determine the steady-state solution\nto the Eq. (1), \frst for a constant magnetic \feld and then\nfor a special choice of spatially inhomogeneous \feld.\nA. Constant magnetic \feld\nIn the case of a constant magnetic \feld \u0014(r) =\u0014, it\ncan be easily shown that r\u0001[Darp(r;tjr0)] = 0. This\nimplies that the tensor Din Eq. (1) can be replaced by\nDs=D=(1 +\u00142)1which yields the following Fokker-\nPlanck equation:\n@tp(r;tjr0) =D\n1 +\u00142r\u0001[rp(r;tjr0)] (7)\n\u0000\u0016p(r;tjr0) +\u0016\u000e(r\u0000r0);\nThe steady-state solution pss(rjr0) of this equation\nis obtained by setting @tp(r;tjr0) = 0 which, in two4\nFIG. 3. The stationary probability density distribution of the particle's position, the di\u000busive, and Lorentz \ruxes in the system\nobtained from the underdamped Langevin equation (6) with a mass m= 0:005 are shown in (a-c), respectively. The applied\nmagnetic \feld is constant such that \u0014= 3:0. The particle is stochastically reset to its initial position r0=0at a constant rate\n\u0016= 0:1. The direction of the \ruxes is shown by the arrows; the magnitude is color coded.\ndimensions, can be written as [11]\npss(rjr0) =\u000b2\n2\u0019K0(\u000bjr\u0000r0j); (8)\nwhereK0is the modi\fed Bessel function of the second\nkind of order zero and \u000b=p\n(1 +\u00142)\u0016=D. Using Eqs (3)\nand (8) the di\u000busive \rux can be written as\nJs(r) =\u000b3D\n2\u0019(1 +\u00142)K1(\u000br)^r; (9)\nwhereK1is the modi\fed Bessel function of the second\nkind of order one, r=jr\u0000r0jis the distance from the\nstarting point of the particle and ^ris a unit vector in the\nradial direction.\nThe steady-state solution in case of a constant mag-\nnetic \feld is the same as obtained in Ref. [11, 31] with\ntrivial rescaling of the di\u000busion coe\u000ecient wherein Dfor\na freely di\u000busing particle is replaced by D=(1+\u00142) for dif-\nfusion under Lorentz force. The distinctive new feature\nof the steady state is the presence of additional Lorentz\n\ruxes, which can be written as\nJa(r) =\u0000\u000b3D\u0014\n2\u0019(1 +\u00142)K1(\u000br)^\u0012; (10)\nwhere ^\u0012is a unit vector in the azimuthal direction.\nOn comparing Eqs. (10) and (9), it is evident that the\nLorentz \rux is nothing else but di\u000busive \rux de\rected by\nthe applied magnetic \feld.\nIn Fig. 2 we show a surface plot together with a con-\ntour plot of the probability density in the stationary state\nof the system from Eq. (8). The applied magnetic \feld\nis such that \u0014= 3:0. The particle is stochastically reset\nto its initial position r0=0at a constant rate \u0016= 0:1.\nLorentz \ruxes (Eq. (10)) are shown as white arrows on\ntop of the contour plot. These \ruxes resemble Brownianvortex observed in a system of colloidal particle di\u000busing\nin an optical trap [27{29]. Figures 3(a) to 3(c) show, re-\nspectively, the results for the probability density, di\u000busive\n\ruxes, and the Brownian vortices in the stationary state\nof the system, obtained from Brownian dynamics simula-\ntions. These results are in excellent agreement with the\ntheoretical results shown in Fig. 2.\nFIG. 4. Schematic showing how the motion of a charged\nBrownian particle is curved by Lorentz force. Two di\u000berent\ntrajectories are shown. The particle is subjected to magnetic\n\felds with the same magnitude and opposite directions in\neach half-plane. The red arrows (straight arrows) depict the\ndirection that the particle follows in the absence of a mag-\nnetic \feld, whereas in the presence of a magnetic \feld this\ndirection is curved, which is shown in blue.5\nB. Spatially inhomogeneous magnetic \feld: a\nminimal example\nAs shown above, Lorentz \ruxes in the steady state re-\nsult from de\rection of the radial \ruxes. In fact, for a\nconstant magnetic \feld, they do not a\u000bect the relaxation\ndynamics [6]. This is no longer the case when the mag-\nnetic \feld is inhomogeneous; the steady-state solution,\nas we show below, is determined by the di\u000busive and\nLorentz \ruxes.\nWe consider a minimalistic example of a spatially in-\nhomogeneous magnetic \feld to highlight how the Lorentz\n\ruxes fundamentally alter the boundary conditions giv-\ning rise to an unusual stationary state. The system is\ndivided into two half-planes by the line x= 0 [see Fig. 4].\nEach half plane is subjected to a constant magnetic \feld\nwith the same magnitude, but opposite direction such\nthat\n\u0014(r) =(\n\u0000\u00140; x\u00150;\n+\u00140; x< 0:(11)\nwhere\u00140is a (constant) parameter. In Fig. 4, two di\u000ber-\nent trajectories of the di\u000busing particle are shown. The\nred arrows depict the motion of the particle at a given\nposition without Lorentz force, whereas a similar motion\nin the presence of Lorentz force is shown by blue arrows.\nAs the particle moves away from the origin, the Lorentz\nforce makes the particle undergo a bias toward counter-\nclockwise motion if x >0 and clockwise if x <0. This\nimplies that there is no \rux across the line x= 0.\nThis particular choice of the magnetic \feld ensures\nthat the symmetric part of the tensor, Ds, is a constant\ntensor in the entire plane, whereas the antisymmetric\npart,Da, changes sign at x= 0. It thus follows that\nthe governing Fokker-Planck equation for the position\ndistribution of the particle is the same as in Sec. III A\n(Eq. (7)) with the boundary condition that the xcompo-\nnent of the \rux (Eq. (3)) is zero at x= 0. Since the \rux\nis composed of both di\u000busive and Lorentz components,\nthe boundary condition reads as\ns\u0001rp= 0 at x= 0; (12)\nwheres=(1; \u0014)is an oblique vector, the direction of\nwhich is determined by the magnetic \feld. This bound-\nary condition is known as the oblique boundary condition\nand is often employed in theory of wave propagation in\npresence of obstacles [32, 33]. Note that for \u0014= 0, this\nreduces to the ordinary Neumann boundary condition.\nThe Fokker-Planck equation (7) with the boundary\ncondition in Eq. (12) can be solved using the method\nof partial Fourier transforms [34] [see the appendix for\ndetails]. The steady-state solution, obtained for r0=0,\nis given as\npss(x;y) =\u000b2\n2\u0019Z1\n0d\u0018e\u0000\fjxj\n\f2+\u00142\u00182[\fcos(\u0018y)\u0000\u0014\u0018sin(\u0018y)]:\n(13)\nFIG. 5. The stationary probability density of the particle's\nposition from Eq. (13) for a system with \u0014=\u00002:0 ifx > 0\nand\u0014= 2:0 otherwise is shown in the surface plot on top of\nthe contour plot. The particle is stochastically reset to the\norigin r0=0with\u0016= 0:5. The steady state is characterized\nby the non-Gaussian probability density, which is symmetric\nwith respect to the line x= 0, the di\u000busive and Lorentz \ruxes.\nLorentz \ruxes are shown by white arrows.\nwhere\f=p\n\u00182+\u000b2. One can show that for a system\nwithout Lorentz force this expression correctly reduces to\nthe (analytical) results obtained by Evans and Majum-\ndar [11].\nIn Fig. 5 we show a surface plot on top of a contour plot\nof the probability density in the stationary state from\nEq. (13). The Lorentz \ruxes are shown by white arrows.\nThat an inhomogeneous magnetic \feld induces an un-\nusual stationary state in the system can be observed by\na comparison with Fig. 2.\nFigure 6 shows the results from Brownian dynamics\nsimulations with \u00140= 2:0 and\u0016= 0:5. The total, di\u000bu-\nsive, and Lorentz \ruxes in the system are shown in (a-c),\nrespectively. As can be seen in Fig. 6, the xcomponent\nof the total \rux is zero at x= 0.\nFigure 7 shows the steady-state distribution of the par-\nticle's position, obtained from simulations for di\u000berent\nvalues of\u00140. The particle is stochastically reset to its\ninitial position r0=0at a constant rate \u0016= 0:5 for all\nvalues of\u00140. As can be seen in Fig. 7, the distribution\nhas a candle-\rame-like form which is not symmetric with\nrespect toyaxis. This can be understood as the accu-\nmulation resulting from the equal and opposite Lorentz\n\ruxes atx= 0. The distribution becomes increasingly\nstretched along the ydirection with increasing magnetic\n\feld. A comparison of numerical solutions of Eq. (13)\n(not shown) with the simulations con\frms our analytical\npredictions.\nIn Fig. 8 we show the steady-state distribution of the\nposition of the particle from simulations for di\u000berent val-\nues of\u0016= 0:1;0:4;0:8 with\u00140= 2:0. The distribution is\nstretched along the ydirection. The width of the distri-\nbution along xdirection decreases with increasing \u0016.6\nFIG. 6. The total, di\u000busive, and Lorentz \ruxes in the stationary state of a di\u000busion system subjected to a magnetic \feld with\n\u0014=\u00002:0 ifx>0 and\u0014= 2:0 otherwise are shown in (a-c), respectively. The particle is stochatically reset to the origin r0=0\nat a constant rate \u0016= 0:5. The results are computed by Brownian dynamics simulations from Eq. (6) with a mass m= 0:005.\nThe direction of the \ruxes is shown by the arrows; the magnitude is color coded.\nThis minimalistic example shows how Lorentz \rux fun-\ndamentally alters the probability density and induces an\nunusual stationary state. Experimentally realizable mag-\nnetic \felds are likely to have more complicated shapes;\nhowever, this does not change the conclusions of this\nstudy.\nIV. DISCUSSION AND CONCLUSION\nLorentz force has the unique property that it depends\non the velocity of the particle and is always perpendicu-\nlar to it. Although this force generates particle currents,\nthey are purely rotational and do no work on the system.\nAs a consequence, the equilibrium properties of a Brow-\nnian system, for instance the steady-state density dis-\ntribution, are independent of the applied magnetic \feld.\nThe dynamics, however, are a\u000bected by Lorentz force:\nthe Fokker-Planck equation picks up a tensorial coe\u000e-\ncient, which re\rects the anisotropy of the particle's mo-\ntion. The di\u000busion rate perpendicular to the direction of\nthe magnetic \feld decreases with increasing \feld whereas\nthe rate along the \feld remains una\u000bected. In addition\nto this e\u000bect, Lorentz force gives rise to Lorentz \ruxes\nwhich result from the de\rection of di\u000busive \ruxes [5, 6].\nThe e\u000bects caused by the Lorentz force, however, oc-\ncur only in nonequilibrium and cease to exist when the\ndistribution of particles reaches equilibrium. A system\nsubjected to stochastic resetting, in contrast, is continu-\nously driven out of equilibrium. In this paper, we showed\nthat by stochastically resetting the Brownian particle to\na prescribed location, a nonequilibrium steady state can\nbe created which preserves the hallmark features of dy-namics under Lorentz force : a nontrivial density distri-\nbution and Lorentz \ruxes. We considered a minimalis-\ntic example of spatially inhomogeneous magnetic \feld,\nwhich shows how Lorentz \ruxes fundamentally alter the\nboundary conditions giving rise to an unusual stationary\nstate with no counterpart in (purely) di\u000busive systems.\nOne may wonder about the choice of stochastic reset-\nting in this study. Although there are several methods\nto drive a system into a nonequilibrium steady state,\nstochastic resetting is unique in the sense that it simply\nrenews the underlying (random) process and therefore, in\nsome sense, preserves the dynamics of the underlying pro-\ncess in the steady state. Contrast this with a system of\nactive Brownian particles subjected to Lorentz force [10]\nin which Lorentz force couples with the nonequilibrium\ndynamics of an active particle via its self-propulsion. Al-\nthough most of the research in stochastic resetting is the-\noretical, stochastic resetting has been realized experimen-\ntally in a system of a colloidal particle which is reset using\nholographic optical tweezers [35]. Resetting also features\nnaturally in the measurement of position-dependent dif-\nfusion of a particle di\u000busing near a wall which experi-\nences inhomogeneous drag due to hydrodynamics. The\nposition-dependent di\u000busion coe\u000ecient is measured by\nletting the particle di\u000buse freely from a given initial lo-\ncation for a certain period of time before resetting it,\nusing optical tweezers, to the initial location [36]. From\nthe `\fnite-time' ensemble of measurements, the di\u000busion\ncoe\u000ecient is obtained from the mean squared displace-\nment.\nIn this work we focused only on the steady-state prop-\nerties of the system. The investigation of how Lorentz\nforce a\u000bects the mean \frst-passage time and escape prob-7\nFIG. 7. The stationary probability density distribution of the particle's position. The applied magnetic \feld is such that\n\u0014=\u0000\u00140ifx >0 and\u0014=\u00140otherwise. \u00140is 0.1, 0.5, 1.0, 2.0, 3.0, 5.0 for systems (a) to (f), respectively. The distribution\nbecomes increasingly stretched along the ydirection with increasing magnetic \feld. The results are computed by Brownian\ndynamics simulations from Eq. (6) with a mass m= 0:005. The particle is stochastically reset to the origin r0=0at a constant\nrate\u0016= 0:5 in all systems.\nability in such systems is left for a future study.\nACKNOWLEDGMENTS\nWe would like to acknowledge Holger Merlitz for fruit-\nful discussions and suggestions.\nAppendix A: Oblique-Derivative Half-Plane Master\nEquation\nPartial di\u000berential equations with oblique derivative\nboundary conditions often arise in the theory of waves,\nfor instance, waves on the ocean or in a rotating plane [32,\n33]. There is a vast amount of mathematical literature on\nthis subject. Here we use the method of partial Fourier\ntransforms adopted from Ref. [34]. We consider x= 0 as\na re\recting boundary, for which the zero \rux condition\ncan be written as\ns\u0001rp= 0 at x= 0; (A1)wheres=(1; \u0014)is the oblique vector. We consider a\ndi\u000busing particle that is stochastically reset to ( x0;0) at\na constant rate \u0016. Later we will set x0= 0 to obtain the\nsolution for our particular case.\nThe master equation for the stationary probability\ndensitypss(x;y) is\nD\n1 +\u00142r2pss(x;y)\u0000\u0016pss(x;y)+\u0016\u000e(x\u0000x0)\u000e(y) = 0 (A2)\nWe de\fne the partial Fourier transform as\n^p(x;\u0018) =1p\n2\u0019Z1\n\u00001dypss(x;y)ei\u0018y: (A3)\nand its inverse as\npss(x;y) =1p\n2\u0019Z1\n\u00001d\u0018^p(x;\u0018)e\u0000i\u0018y: (A4)\nThe transformed Fokker-Planck equation [Eq. (A2)] be-\ncomes\n@2^p(x;\u0018)\n@x2\u0000\f2^p(x;\u0018) =\u0000\u000b2\u000e(x\u0000x0)p\n2\u0019; (A5)8\nFIG. 8. The stationary probability density distribution of the particle's position for di\u000berent values of \u0016= 0:1;0:4;0:8 with\n\u00140= 2:0 for systems (a) to (c). As in Fig. 7 the distribution is stretched along the ydirection. The width of the distribution\nalongxdecreases with increasing \u0016. The results are obtained by Brownian dynamics simulations from Eq. (6) with a mass\nm= 0:005.\nwhere\f=p\n\u00182+\u000b2and\u000b=p\n\u0016(1 +\u00142)=D. The\ntransformed boundary condition reads as\n@^p(x;\u0018)\n@x\u0000i\u0014\u0018^p(x;\u0018) = 0 at x= 0; (A6)\nwhereiis the imaginary unit. The general solution to\nEq. (A5) is:\n^p(x;\u0018) =\u001aAe\fx+Be\u0000\fx;0<x<x 0, (A7a)\nCe\fx+De\u0000\fx; x>x 0. (A7b)\nThe boundary condition that ^ p(x;\u0018) is zero as x!1\nimpliesC= 0. That the probability density is continuous\nonx=x0implies\nD=Ae2\fx0+B: (A8)\nSubstituting Eq. (A7a) into Eq. (A6) gives a relationship\nbetweenAandB:\nA(\f\u0000i\u0014\u0018) =B(\f+i\u0014\u0018) (A9)\nNow one can rewrite Eqs. (A7a) and (A7b) as\n^p(x;\u0018) =Ae\fx0\u0012\ne\u0000\fjx\u0000x0j+ \u0004e\u0000\fjx+x0j\u0013\n; (A10)\nwhere \u0004 = ( \f\u0000i\u0014\u0018)=(\f+i\u0014\u0018). Using this expression one\ngets\n@2^p(x;\u0018)\n@x2=Ae\fx0\u0002\n\f2e\u0000\fjx\u0000x0j\u00002\f\u000e(x\u0000x0)e\u0000\fjx\u0000x0j\n+ \u0004\f2e\u0000\fjx+x0j\u0003\n: (A11)\nThe second derivative of ^ p(x;\u0018) in Eq. (A5) can\nbe replaced by Eq. (A11), which results in A=\n(\u000b2e\u0000\fx0)=(2\fp\n2\u0019). After some simpli\fcations one gets\n^p(x;\u0018) =\u000b2\np\n2\u0019\u0012e\u0000\fjx\u0000x0j\u0000e\u0000\fjx+x0j\n2\f+e\u0000\fjx+x0j\n\f+i\u0014\u0018\u0013\n:\n(A12)For the system studied in this paper, we set x0= 0. Thus\n^p(x;\u0018) =\u000b2\np\n2\u0019e\u0000\fx\n\f+i\u0014\u0018: (A13)\nWe could not \fnd a closed analytical form for the inverse\nFourier transform of Eq. (A13). Nevertheless, the follow-\ning intergal can be evaluated numerically to obtain the\nsteady-state solution:\npss(x;y) =\u000b2\n2\u0019Z1\n0d\u0018e\u0000\fjxj\n\f2+\u00142\u00182[\fcos(\u0018y)\u0000\u0014\u0018sin(\u0018y)]:\n(A14)\nNote the factor 1 =2 on the right hand side of the\nEq. (A14), which accounts for the (symmetric) exten-\nsion of the solution to the x <0 half-plane. For special\ncase of\u0014= 0, it is easy to show that the above integral\nreduces to\npss(r) =\u000b2\n0\n2\u0019K0(\u000b0jrj); (A15)\nwhere\u000b0=p\n\u0016=D wherejrjis the distance from the\norigin, same as reported in Ref. [11] for a two dimensional\n(symmetric) di\u000busion under stochastic resetting.9\n[1] Robert J Goldston and Paul Harding Rutherford, Intro-\nduction to plasma physics (CRC Press, 1995).\n[2] V. Balakrishnan, Elements of Nonequilibrium Statistical\nMechanics (Ane Books, 2008).\n[3] H. D. Vuijk, J. M. Brader, and A. 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M. Silva\nInstituto de Física Teórica, Universidade Estadual Paulista,\nRua Dr. Bento Teobaldo Ferraz, 271, Bloco II, São Paulo, 01140-070, SP, Brazil\nThisarticlepresentsacomprehensivestudyoftheimpactofdecoherenceontheaveragecorrelation\nfor pure quantum states. We explore two primary mechanisms of decoherence: phase damping\nand amplitude damping, each having distinct effects on quantum systems. Phase damping, which\ndescribesthelossofquantumcoherencewithoutenergyloss, primarilyaffectsthephaserelationships\nbetween the components of a quantum system while amplitude damping involves energy dissipation\nand also affects the state’s occupation probabilities. We show that the average correlation follows\na predictable decaying pattern in both scenarios. Our analysis can be understood in the context of\nquantum computing, by focusing on how phase damping influences the entanglement and correlation\nbetween qubits, key factors in quantum computational efficiency and error correction protocols.\nI. INTRODUCTION\nThe study of quantum mechanics and its applications\nin quantum computing and quantum information theory\nhas highlighted the pivotal role of coherence in quantum\nsystems. Decoherence, the process by which quantum\nsystems interact with their environment leading to the\nloss of quantum coherence, poses significant challenges in\nmaintaining the integrity of quantum states [1–3]. Addi-\ntionally, measures of coherence and nonclassicality are\ncritical in understanding the transition from quantum\nto classical behavior in physical systems [4–7]. Coher-\nence, indicative of the superposition states in quantum\nmechanics, is a key feature that differentiates quantum\nsystems from classical ones. It allows for phenomena like\ninterference, which are absent in classical systems. Non-\nclassicality on the other hand extends beyond coherence,\nencompassing aspects like quantum entanglement and vi-\nolations of Bell inequalities, which defy classical explana-\ntions based on local realism and deterministic probabili-\nties [8, 9].\nTraditionally, nonclassicality has been quantified us-\ning Bell inequalities, which compare quantum mechan-\nical predictions against classical physics predictions for\ncertain physical systems. However, this approach, while\neffective, often demands precise control over experimen-\ntal parameters, posing practical challenges in laboratory\nsettings [10].\nA recent approach suggested the use of average cor-\nrelation as an alternative measure [11]. This innovative\napproach does not necessitate stringent control over mea-\nsurement directions and offers ease of computation, mak-\ning it highly suitable for practical applications. The pa-\npermeticulouslyarticulatesthederivationofaveragecor-\nrelation for various quantum states, including pure states\nand Werner states, providing a thorough understanding\nof how these states exhibit nonclassicality under specific\nconditions.\nMoreover, the study delves into the relationship be-\ntween Bell inequalities and correlation functions of di-\nchotomic observables, laying a solid theoretical founda-\ntionfortheproposedmethodology. Theauthors’analysisincludes an extensive mapping of two-qubit states based\nonaveragecorrelationandtheBellparameter. Thismap-\nping is instrumental in distinguishing between classical\nand nonclassical states and offers profound insights into\nthe behavior of mixed states in quantum systems [11].\nWithin this context we aim to analyze the impact of de-\ncoherence on average correlation, by initially focusing on\npure quantum states.\nDecoherence is not only a central theme in the study of\nquantum mechanics but also a critical barrier in the de-\nvelopment of quantum computing and information pro-\ncessing technologies [12–16]. Our study aims to exam-\nine how decoherence affects average correlation, thereby\ninfluencing the nonclassical characteristics of quantum\nstates. This analysis is particularly relevant in the con-\ntext of quantum computing and information processing,\nwheremaintainingnonclassicalityisessentialforharness-\ning the full potential of quantum technologies.\nII. AVERAGE CORRELATION FOR PURE\nSTATES\nStarting with the Schmidt decomposition for pure\nquantum states:\n|ψ⟩=c|0⟩A|1⟩B−p\n1−c2|1⟩A|0⟩B (1)\nWe wish to evaluate the average correlation given by:\nΣ =1\n4π2Z\ndΩaZ\ndΩb|E(a,b)| (2)\nwhere E(a,b)denotes the correlation function measured\nby two observers A and B and aandbare unit vectors\nwhich denote the axis in which the measurement of each\nobserver is made:\nE(a, b) =aTKb (3)\nthe correlation matrix Kin the formula above is defined\nas:\nKij=tr(ˆρˆσiA⊗ˆσjB) (4)arXiv:2403.10551v1  [quant-ph]  13 Mar 20242\nwith the corresponding Pauli matrices for each axis\ndirection given by ˆσA= (ˆσ1A,ˆσ2A,ˆσ3A)and ˆσB=\n(ˆσ1B,ˆσ2B,ˆσ3B). For a pure state of the form at Eq. (1),\nthe density matrix can be evaluated and shown to equal:\nρ=\n0 0 0 0\n0 c2−c√\n1−c20\n0−c√\n1−c21−c20\n0 0 0 0\n(5)\nThis leads to a correlation matrix given by:\nK=\n−2c√\n1−c2 0 0\n0 −2c√\n1−c20\n0 0 −1\n(6)\nIt is clear that there exits a singular value decomposition\nof the form:\nK=\n0 0 −1\n0−1 0\n−1 0 0\n\nα0 0\n0β0\n0 0 γ\n\n0 0 1\n0 1 0\n1 0 0\n(7)\nwhere α= 1, and β=γ= 2c√\n1−c2. Therefore defin-\ning:\n˜a=UTa,˜b=VTb (8)\nallows us to write the average correlation as:\nΣ =1\n16π2Z\ndΩaZ\ndΩb|(˜a)Tκ˜b| (9)\nand by redefining ˜bas˜bκ=κ˜b, we may write the scalar\nproduct in terms of the length of the vectors and theangle between them such that:\nΣ =1\n16π2Z\ndΩaZ\ndΩa|˜bκ||cosθB|(10)\nIf we now choose the z-axis of the coordinate system for\nvector ˜ato be aligned with that of vector ˜b, the integral\nbecomesindependentoftheazimuthalangleandwereach\nthe simple expression below:\nΣ =1\n8πZ\ndΩb|˜bκ| (11)\nNext, introducing spherical coordinates:\n˜bκ= (γcosϕsinθ, βsinϕsinθ, αcosθ)T(12)\nand the function:\nf(ϕ) =\u0012β\nα\u00132\nsin2ϕ+\u0010γ\nα\u00112\ncos2ϕ(13)\nwe may write:\nΣ =α\n8πZ2π\n0dϕZπ\n0dθsinθq\nf(ϕ) sin2θ+ cos2θ(14)\nPerforming the variable substitution u= cos θleads to:\nΣ =α\n8πZ2π\n0dϕp\nf(ϕ)Z1\n−1dus\n1 +1−f(ϕ)\nf(ϕ)u2(15)\nand solving the integral over uyields:\nΣ =α\n4\"\n1 +1\n2πZ2π\n0dϕf(ϕ)p\n1−f(ϕ)Arsinh s\n1−f(ϕ)\nf(ϕ)!#\n(16)\nSince for a pure state f(ϕ) =β2/α2, the integral over ϕ\ncan be performed immediately leading to:\nΣ =α\n4\"\n1 +1\nαβ2\np\nα2−β2Arsinh s\nα2−β2\nβ2!#\n(17)\nAs shown at [11] the average correlation in this case has\na maximum of Σ = 1 /2forc= 1/√\n2and a minimum of\nΣ = 1 /4forc= 1. Theauthorsalsoshowthatanaverage\ncorrelation value of Σ≤1\n4, is an indication of classical\nstates, whereas Σ>1\n2√\n2is associated exclusively with\nnonclassical states.III. IMPACT OF DECOHERENCE\nA. Phase Damping\nWe start our analysis by considering the action of the\nfollowing Kraus operators on the density matrix [17]:\nK0=\u0012\n1 0\n0√1−p\u0013\nK1=\u0012\n0 0\n0√p\u0013\n(18)\nHere, prepresents the probability of phase damping oc-\ncurring within the quantum system. When applying this\ntothedensitymatrixforthetwoquantumstateswehave:\nρ′=X\ni,j(Ki⊗Kj) ˆρ(Ki⊗Kj) (19)3\nLeading to:\nρ′=\n0 0 0 0\n0 c2−c√\n1−c2(1−p) 0\n0−c√\n1−c2(1−p) 1 −c20\n0 0 0 0\n\n(20)\nEvaluating the correlation matrix this time leads to:\nK=\n−2c√\n1−c2(1−p) 0 0\n0 −2c√\n1−c2(1−p) 0\n0 0 −1\n\n(21)\nThis matrix still posses a singular value decomposition,\nbut with a different parameter β=γ= 2c√\n1−c2(1−p).\nThe function f(ϕ)will therefore be given by:\nf(ϕ) =β2\nα2= 4c2(1−c2)(1−p)2(22)\nAnd knowing that p= 1−eΓt, where Γdenotes the\ndecoherence rate, we may substitute f(ϕ)on equation\n(16) for the average correlation and analyze how it varies\nwith time. The result is displayed below:\nFigure 1. Decay of average correlation over time due to\nphase damping for various decoherence rates, illustrating that\nhigher rates lead to faster decay in correlation.\nWe choose to set c= 1/√\n2such that the average cor-\nrelation starts at the maximum value of Σ = 1 /2and\ndecays with time to the minimum value Σ = 1 /4. It is\nevident that varying the decoherence rates, represented\nby different values of Γ, influences the rate at which aver-\nage correlation diminishes. Higher values of decoherence\nrate lead to a faster decay of the average correlation as\nexpected. The plot’s curves do not fall below 1/4, which\naligns with the observation that pure states, even when\nentangled, remain nonclassical and do not transition into\nclassical states as they maintain an average correlation\nabove the 1/4threshold. This suggests that even as thesystem experiences phase damping over time, it remains\nwithin the regime of nonclassical states.\nThe analysis reveals that the decay of Σhalts at the\nthreshold value of 1/4, irrespective of the decoherence\nrateapplied. Thisobservationissignificantasitindicates\nthat, despite the environmental perturbations modeled\nby different values of Γ, the system retains a minimum\nlevel of quantum correlation. This behavior underscores\nthe inherent resilience of quantum states to decoherence,\nsuggesting that entangled states, even under adverse con-\nditions, maintain a degree of nonclassicality.\nFurthermore, the consistent maintenance of Σabove\nthe1/4threshold across different decoherence scenarios\nposits that the system, subjected to phase damping, per-\nsistentlyresideswithinthenonclassicalregime. Thisout-\ncome is particularly relevant for quantum information\nprocessing, where the preservation of quantum correla-\ntions amidst environmental decoherence is crucial for the\nfunctionality of quantum technologies.\nB. Amplitude Damping\nIn the case of amplitude damping, the Kraus operators\nare given by [17]:\nK0=\u0012\n1 0\n0√1−p\u0013\nK1=\u0012\n0√p\n0 0\u0013\n(23)\nWe may apply these operators to the density matrix for\npure state as done in Eq. (19) leading to:\nρ′=\np 0 0 0\n0 c2(1−p) −c√\n1−c2(1−p) 0\n0−c√\n1−c2(1−p) (1 −c2)(1−p) 0\n0 0 0 0\n\n(24)\nAnd the correlation matrix will be:\nK=\n−2c√\n1−c2(1−p) 0 0\n0 −2c√\n1−c2(1−p) 0\n0 0 1 −2p\n\n(25)\nThe beta parameter is the same as in the phase damping\ncase, but the alpha parameter will be given by α=|1−\n2p|. Wetakethemodulusinordertoensurethepositivity\nof the singular values. The function f(ϕ)will be given\nthis time by:\nf(ϕ) =β2\nα2= 4c2(1−c2)(1−p)2(|1−2p|)−2(26)\nThe curve describing the effects of amplitude damping is\npresented below:4\nFigure 2. Decay of average correlation over time due to am-\nplitude damping for various decoherence rates. The curve\ncrosses the Σ = 1 /4threshold, indicating non-classical states.\nThe graph illustrates the effects of energy dissipation\non the time evolution of the average correlation. We\nnote that initially all curves start with Σvalues above\nthe nonclassicality threshold of 1/4, indicating that the\nsystem is in a nonclassical state. As time progresses, the\nvarious rates of amplitude damping cause Σto decrease,\ndemonstrating the loss of quantum correlations. Unlike\nphase damping, amplitude damping encompasses energy\ndissipation, which can lead to a complete loss of nonclas-\nsicality, asevidencedby Σpotentiallydroppingbelowthe\n1/4threshold.\nThe decay trajectories of Σunder different amplitude\ndamping rates show that higher values of Γcorrespond to\na more rapid decline in nonclassicality. This is a quantifi-\nable demonstration of how the system’s interaction with\nits environment accelerates the transition from quantum\ntoclassicalstates. Thedistinctratesofdecaynotonlyre-\nflect the sensitivity of nonclassical states to environmen-\ntal interactions but also highlight the temporal aspect of\nquantum coherence. We also point out that the effect\nof amplitude damping is more pronounced than that of\nphase damping presented earlier. This quantum channel\ninvolves energy dissipation which can significantly im-\npact the coherence properties of quantum systems[18–\n20] more than phase damping does. In the study [21] for\nexample, the authors explored the dynamics of a Jaynes-\nCummings model under phase-damping conditions and\nconcluded that, while phase damping does lead to de-\ncoherence, the absence of energy exchange between the\nsystem and its environment results in a comparatively\nslower progression towards classical states.\nIt is evident from Fig(2) that some states may retain\ntheir nonclassical character longer than others, depen-\ndent on the magnitude of Γ, which is crucial for quan-\ntum computing and communication tasks that rely on\nnonclassical states. The long-term behavior of the sys-tem, as depicted by the convergence of the curves to-\nwards a lower bound of Σ, suggests a stabilization of the\naverage correlation in the presence of amplitude damp-\ning. This equilibrium state, potentially indicative of a\nresidual quantum correlation or a complete transition to\nclassicality, emphasizes the nuanced nature of quantum\ndecoherence. Understanding the duration for which non-\nclassicality is preserved is of paramount importance for\nthe practical implementation of quantum technologies,\nas it dictates the window of opportunity for harnessing\nquantum mechanical advantages.\nIV. CONCLUSION\nIn this work, we systematically investigated the im-\npact of decoherence mechanisms, specifically phase and\namplitude damping, on the average correlation of quan-\ntum states. Our analysis revealed distinct effects of\nthese decoherence processes on quantum coherence and\ncorrelation, highlighting the differential impact of phase\nand amplitude damping. This distinction is crucial for\nunderstanding the resilience and vulnerability of quan-\ntum states under various decoherence scenarios, thereby\ninforming strategies for preserving quantum informa-\ntion integrity in quantum computing and communication\ntechnologies. Furthermore, our findings underscore the\nsignificance of coherence and correlation as fundamental\nresources in quantum technology applications, suggest-\ning that maintaining these properties is essential for the\npractical realization of quantum computational and in-\nformational tasks. The exploration of decoherence effects\npresented in this paper contributes to the broader field of\nquantum information science by providing a clearer un-\nderstanding of how quantum systems interact with their\nenvironment and the implications for quantum technol-\nogy development. Effects of phase damping and ampli-\ntude damping have been widely studied in the literature\n[22–24] and future research in this area may focus on de-\nveloping advanced quantum error correction techniques\nanddecoherence-resistantdesignstomitigatetheadverse\neffects observed, thereby enhancing the performance and\nreliability of quantum-based technologies.\nFurthermore, more work in this area can expand upon\nthe foundational insights garnered from this study by ex-\nploring generalizations of phase and amplitude damping\nchannels [25, 26]. This direction could yield a deeper un-\nderstanding of how quantum systems interact with com-\nplex environments over time. 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Otxoa de Zuazola3;4\n1RIKEN Center for Emergent Matter Science (CEMS) and RIKEN Cluster for\nPioneering Research (CPR), 2-1 Hirosawa, Wako, Saitama, 351-0198 Japan\n2Department of Theoretical and Applied Physics,\nAfrican University of Science and Technology (AUST),\nKm 10 Airport Road, Galadimawa, Abuja F.C.T, Nigeria\n3Hitachi Cambridge Laboratory, J. J. Thomson Avenue, CB3 OHE, Cambridge, United Kingdom and\n4Donostia International Physics Center, 20018 San Sebasti\u0013 an, Spain\n(Dated: May 2, 2022)\nMagnon emission and excitation by a relativistic domain wall at a constant velocity in antiferro-\nmagnet is theoretically studied. A pair emission due to a quadratic magnon coupling is shown to be\ndominant. The emission corresponds in the comoving frame to a vacuum polarization induced by\na zero-energy instability of the Lorentz-boosted anomalous response function. The emission rate is\nsensitive to the magnon dispersion and wall pro\fle, and is signi\fcantly enhanced for a thin wall with\nvelocity close to the e\u000bective light velocity. The Ohmic damping constant due to magnon excitation\nat low velocity is calculated.\nEmission from a relativistic moving object is a gen-\neral intriguing issue that has analogy to blackbody ra-\ndiation, blackhole physics1and can be applied for wave\nampli\fcation2. Solid-state systems are particularly in-\nteresting from the viewpoints of quantum e\u000bects and ex-\nperimental feasibility due to low 'light velocity'. Antifer-\nromagnets at low energy have been known to be typical\nrelativistic system3, and dynamic properties of domain\nwall has been explained in terms of Lorentz contraction4.\nIn this paper, we study the emission from moving do-\nmain wall, a relativistic soliton, in an antiferromagnet.\nWe discuss the low energy regime using a continuum\nmodel, valid when the wall thickness \u0015is larger than the\nlattice constant a. The system is described by a relativis-\ntic Lagrangian, and thus there are domain wall solutions\nmoving with a constant velocity smaller than the e\u000bective\nlight velocity c. The wall width \u0015is a\u000bected by Lorentz\ncontraction; \u0015=\r\u00150, where\u00150is the thickness at rest,\n\r(vw)\u0011p\n1\u0000(vw=c)2is a contraction factor, vwis the\nvelocity of the wall.\nEmission from a moving object is generally dominated\nby a linear process, where the object couples to its \ruc-\ntuation linearly. In the case of soliton solutions, such\nlinear coupling, absent at rest, arise from acceleration\nand deformation as argued for ferromagnetic domain\nwall5{8. The antiferroamgnetic case turns out to be qual-\nitatively distinct from the ferromagnetic case because of\nthe Lorentz invariance. The linear coupling, inducing\nsuper Ohmic dissipation, is negligible at low energy, and\nthe dominant emission arises from the second-order cou-\npling to the moving wall. The momentum is transferred\nfrom the wall to magnons, while the energy comes from\nDoppler shift. In the rest frame of the wall, the wall\npotential generates a localized magnon excitation. The\nexcitation is described by the normal (particle-hole) com-\nponent of magnon response function, which we call \u0005 q(q\nis the wave vector transferred). In the moving frame,\nthis excitation corresponds to a scattering of magnon,\nresulting in an Ohmic friction force at low velocity. Thescattering property of the normal response function \u0005 qis\nessentially the same as in the ferromagnetic case studied\nin Ref.8; Although the magnon dispersion in ferromag-\nnet, quadratic in the wave vector k, is di\u000berent from the\nantiferromagnetic linear behavior (in the absence of gap),\nit does not lead to qualitative di\u000berence in magnon scat-\ntering.\nA signi\fcant feature antiferromagnets have is the ex-\nistence of an anomalous particle-particle (or hole-hole)\npropagation, \u0000 q, like in superconductivity contributing to\nthe response function9. This is due to the quadratic time-\nderivative term of the relativistic Lagrangian, which al-\nlows positive and negative energy (or frequency) equally.\nThe anomalous response function thus can be regarded\nas a scattering of particles with a positive and negative\nenergies. The negative frequency mode exists generally in\nany relativistic excitations. In optics, for example, a scat-\ntering of negative frequency mode was argued to cause\nan ampli\fcation of photon current10. In the context of\nmagnons, the scattering of negative frequency mode cor-\nresponds to an emission/absorption of two magnons. The\nanomalous response function \u0000 qdescribing such process\nis shown to be sensitive to the magnon dispersion as well\nas the wall velocity. Its low energy weight is much smaller\ncompared to the normal response function \u0005 qfor the ide-\nally relativistic dispersion of k-linear dependence, while it\nis signi\fcantly enhanced if it deviates from linear to have\na \ratter dispersion. The anomalous response function in\nthis case has a sharp and large peak at \fnite wave vector\nfor the wall velocity close to the e\u000bective light velocity c,\nresulting in a strong forward emission of two magnons.\nOur results indicates that relativistic domain wall is use-\nful as a magnon emitter, and the e\u000eciency is tunable by\ndesigning magnon dispersion.\nThe pair emission here is analogous to the vac-\nuum polarization (Schwinger pair production) in\nelectromagnetism11, with the role of electric \feld played\nby the moving wall. In fact, in the laboratory frame, the\nmagnon creation gap of 2\u0001 is overcome by the energyarXiv:2007.13939v1  [cond-mat.mes-hall]  28 Jul 20202\nshift by the Doppler's e\u000bect, while in the moving frame\nwith the wall, a spontaneous vacuum polarization is in-\nduced by a zero-energy instability of the Lorentz-boosted\nanomalous magnon response function.\nMagnetic properties of anti\u000beromagnets are described\nby the staggered (N\u0013 eel) order parameter nof the unit\nlength. Its low energy Lagrangian is relativistic, namely,\ninvariant under the Lorentz transformation as for the ki-\nnetic parts3. We consider the case with an easy axis\nanisotropy energy along the zaxis, described by the con-\ntinuum Lagrangian of\nL=J\n2aZ\ndx\u00141\nc2_n2\u0000(rn)2+1\n\u00152\n0(nz)2\u0015\n;(1)\nwhereJis the exchange energy, J=\u00152\n0(=K) is the easy\naxis anisotropy energy. Our results are valid in the pres-\nence of hard-axis anisotropy simply by including the ef-\nfect in the gap of magnons. The e\u000bective light velocity\nisc=pgJ,gbeing a coupling constant9. The lattice\nconstant is included to simplify the dimensions of mate-\nrial constants12. We consider the one-dimensional case,\nalthough the the e\u000bects we discuss are general and ap-\nply to higher-dimensional walls. The Lagrangian is rela-\ntivistic, i.e., a Lorentz transformation to a moving frame\nwith a constant velocity v,t0= (t\u0000v\nc2x)=\r(v) andx0=\n(x\u0000vt)=\r(v) does not modify the form. The system has\na soliton (domain wall) solution, nz(x) = tanhx\n\u00150. The\nLorentz invariance indicates moving walls nz((x\u0000vt)=\r)\nare classical solutions for a constant v < c , with a con-\ntracted thickness \u0015=\u00150\r(v).\nThese constant velocity solutions are stable, meaning\nthat they have no linear coupling to magnons and there\nis no linear emission. Linear emission may occur during\nacceleration or by deformation. The emission is studied\nby introducing collective coordinates13. In the case of a\ndomain wall, of most interest is the wall position X14.\nThe coupling between the coordinate and \ructuation is\ngoverned by the kinetic part of the Lagrangian. In anti-\nferromagnets, it is second order in time derivative, and\nthus linear \ructuation, ', couples to the acceleration X\nas'X(See Ref.8). The emitted magnon amplitude h'i\nis thus proportional to X, and the recoil force on the\nwall is@2\n@t2h'i/@4\n@t4X. Hence the linear coupling does\nnot induce Ohmic friction and is negligibly small at low\nenergy. The result is the same for other collective vari-\nables like thickness oscillation. The motion of an an-\ntiferromagnetic domain wall is therefore protected from\nthe damping due to a linear coupling, in contrast to the\nferromagnetic case, where Ohmic dissipation arises from\nthickness oscillation8.\nInstead, emission due to the second-order coupling\ndominates in antiferromagnets. At low energy, contribu-\ntion containing less time derivative of the wall collective\ncoordinates dominates. The issue then reduces to a sim-\nple and general problem of the emission from a moving\npotential of a constant velocity15. Our domain wall so-\nlution of tanh-pro\fle induces an attractive potential of\nvw−k+qk/angbracketlefta†\nka†\n−k+q/angbracketright/angbracketleftaka−k−q/angbracketright\n/angbracketlefta†\nk+qak/angbracketrightFIG. 1. Schematic \fgure showing processes due to moving\ndomain wall, scattering\naya\u000b\n, pair emission\nayay\u000b\nand pair\nannihilation,haai. Momentum qis transferred from the wall\nwith a velocity vwto the magnons.\ncosh\u00002form. Taking account of the two magnon modes\nalong thexandy-directions, 'xand'y, respectively\n(n'('x;'y;1)), the potential reads13\nV=\u0000KZdx\na1\ncosh2x\u0000X(t)\n\u0015('2\nx+'2\ny); (2)\nwhereX(t) is the wall position and \u0015=\u00150\ris the thick-\nness of a moving wall16. We consider the case of a con-\nstant velocity, X(t) =vwt. A moving potential transfers\nmomentum qto \ructuations and an angular frequency \nas a result of the Doppler shift. Although the form of the\npotential, Eq. (2), is common for ferro and antiferromag-\nnetic cases, its e\u000bect is di\u000berent, due to di\u000berent nature of\nmagnon excitations. In ferromagnets, 'xand'yare rep-\nresented as linear combination of magnon \feld bandby\n(Holstein-Primakov boson). The potential in this case is\nproportional to magnon density as '2\nx+'2\ny= 4byb, induc-\ning scattering of magnons without changing total magnon\nnumber. (The feature is unchanged in the presence of\na hard-axis anisotropy.) This is due to the kinetic term\nlinear in the time-derivative for ferromagnetic magnon13,\niby_b, which allows a positive energy for the ferromagnetic\nmagnon boson. In contrast, a magnon boson in antifer-\nromagnets is described by a relativistic Lagrangian with\na kinetic term second-order of time derivative,1\nc2( _'i)2\n(i=x;y), which allows 'negative frequency' modes , and\nprocesses changing the total magnon number are allowed.\nIn fact, canonical magnon boson aiis de\fned for each\nmodei=x;yas'i(k;t) =qg\n!(i)\nk(a(i)\nk(t)+a(i)y\n\u0000k(t)), where\n!(i)\nk\u0011p\nc2k2+ \u00012\niis the energy with a gap \u0001 iof mode\ni9. The potential, Eq. (2), then reads\nV=\u0000Kg\u0015\naX\ni=x;yX\nk;qWqq\n!(i)\nk!(i)\nk+qe\u0000iqX(t)\n\u0002\u0010\na(i)\nka(i)\n\u0000k\u0000q+a(i)y\n\u0000ka(i)y\nk+q+ 2a(i)y\nk+qa(i)\nk\u0011\n;(3)\nwhereWq=\u0019q\u0015\nsinh\u0019\n2q\u0015is the Fourier transform of the\npotential pro\fle and !\u0000k=!kis assumed. The emission\nand absorption of two magnons, represented by terms aa\nandayay, are thus possible in antiferromagnet (Fig. 1).\nLet us evaluate the amplitudes of scattering and emis-\nsion/absorption as a linear response to the dynamic po-\ntential. We suppress the index ifor magnon branch. The3\nscattering amplitude,D\nay\nk+qakE\n(t) =iG<\nk;k+q(t;t), is a\nlesser Green's function for magnon. The amplitude after\nsummation over kis represented in terms of the normal\n(particle-hole) response function (including the form fac-\ntorWq)8,9,\n\u0005q;\n\u0011\u0000X\nkWqp!k!k+qnk+q\u0000nk\n!k+q\u0000!k\u0000\n + 2i\u0011;(4)\nasP\nkD\nay\nk+qakE\n=Kg\na\u0015eiqvwt\u0005q, where \u0005 q\u0011\u0005q;qv w.\nHerenk\u0011[e\f!k\u00001]\u00001is the Bose distribution function,\n\f\u0011(kBT)\u00001being the inverse temperature ( kBis the\nBoltzmann constant), \u0011is the damping coe\u000ecient of the\nmagnon Green's function. The angular frequency of \n =\nqvwin \u0005q;qv wis the one transferred to magnons as a\nresult of the Doppler shift. The emission amplitude of\ntwo magnons isP\nkD\nay\n\u0000k+qay\nkE\n=Kg\na\u0015eiqvwt\u0000q, where\n\u0000q\u0011\u0000q;qv wand\n\u0000q;\n\u0011X\nkWqp!k!\u0000k+q1 +n\u0000k+n\u0000k+q\n!\u0000k+q+!k\u0000\n + 2i\u0011;(5)\nis the anomalous (particle-particle) response function.\nThe absorption amplitude is given by this function asP\nkha\u0000k+qaki=Kg\na\u0015eiqvwt\u0000\u0003\n\u0000q(\u0003denotes the complex\nconjugate). The normal response function has symmetry\nof \u0005\u0000q;\n= \u0005q;\n, which leads in the case of \n = qvwto\n\u0005\u0000q;\u0000qvw= \u0005\u0003\nq;qv w, i.e., the real (imaginary) part of \u0005 q\nis even (odd) in q. The normal response has low energy\ncontribution around q= 0 and \n = 0. The asymmetric\nand localized character near q= 0 of Im[\u0005 q]17indicates\nan asymmetric real-space magnon distribution with re-\nspect to the wall center similarly to the ferromagnetic\ncase8. The anomalous response satis\fes \u0000 \u0000q;\n= \u0000q;\n.\nIt has a gap of 2\u0001 for \n, suppressing the low energy\ncontribution in the rest frame (Fig. 2). In the moving\nframe, the Lorentz boost, which transforms qand \n to\nq0= (q+vw\n=c2)=\rand \n0= (\n +vwq)=\r, distorts the\nresponse function, enhancing signi\fcantly the low energy\nweights at \fnite q. This induces spontaneous vacuum\npolarization, which corresponds to a 2 magnon emission\nin the laboratory frame.\nThere are two key factors governing the response func-\ntions, the form factor Wqand magnon dispersion. The\nform factor constrains the wave vector transfer qto\njqj.\u0015\u00001= (\u00150\r)\u00001. Because of this factor, magnon\ne\u000bects are signi\fcantly enhanced for thin walls at high\nvelocity (small \r). As the emission is dominated by the\nlargeqbehavior, it is sensitive to the wall pro\fle as we\nshall see below.\nThe role of the dispersion is clearly seen focusing on the\nimaginary part in the limit of \u0011!0, where the response\naries from the processes satisfying the energy and mo-\nmentum conservation. We consider the case of the disper-\nsion with a small gap and saturation around kmax=\u0019=a\n(See Ref.17), like the one in MnF 218. We choose vwas\npositive. The imaginary part of the normal response\nqΩv=0\nboost(v=0.8)\n 0\n 0.5\n 1 -0.2 0 0.2 0 2 4\nqΩv=0\nboost(v=0.8)\n 0 0.5 1-0.2 0 0.2 0.4 0.6 0 2 4 6FIG. 2. E\u000bect of Lorentz boost on the anomalous response\nfunction Im[\u0000 q;\n] atvw=c= 0:8 for a hyperbolic dispersion\n(\u0016= 5), ~\u00150= 2, ~\u0001 = 0:1, ~\u0011= 0:01 and ~T= 0:2 in dimension-\nless unit (See Ref.17). Blue is the amplitude at rest frame,\nwhich is localized at \n &2\u0001 with negligibly small weight at\n\n = 0. In the boosted frame, shown in red, the amplitude ex-\ntends to zero energy regime at \fnite q, inducing spontaneous\nvacuum polarization, which corresponds to a pair emission in\nthe laboratory frame.\nωk\nk k +qωk+q\nqωk+q−ωk\n=qVwω(a)\nωk\nk−k\n−k+q\n−ωkω−k+q\nqω−k+q+ωk\n=qVwω(b)k >0\nFIG. 3. Energy conservation conditions in (a) scattering\nand (b) emission/absorption of two magnons. The emission\nis regarded as a scattering from a hole state with energy \u0000!k\nto a particle state with energy !\u0000k+q. The slope of dotted\nstraight lines is vw.\narises from the process satisfying !k+q\u0000!k=qvw(Fig.\n3(a)), which leads to an asymmetric weight around q= 0.\nThe imaginary part of the anomalous response \u0000 qarises\nwhen\n!\u0000k+q+!k=qvw: (6)\nThis amplitude is much smaller than the normal response\nfor the relativistic dispersion, !k=p\n(ck)2+ \u00012, due\nto the following reason (Fig. 3(b)). The process sat-\nisfying Eq. (6) is regarded as a scattering process of\na particle and a hole having positive and negative en-\nergy,!\u0000k+qand\u0000!k, respectively. The condition re-\nquires that the average slope of the line connecting the\ntwo energies !\u0000k+qand\u0000!kisvw. However, the slope is\nlarger than cfor the relativistic dispersion, while vwhas\nan upper limit of c, which is the maximum group veloc-\nity. The condition cannot therefore be satis\fed by the\npurely relativistic dispersion, and the imaginary part of\nanomalous response thus arises only if the dispersion has\nan in\rection point like in Fig. 3(b). (In reality, a damp-\ning\u0011leads to a \fnite imaginary part, but it remains to\nbe negligibly small.) Those features are consistent with\na theory of spin transport in antiferromagnet9showing\nthat the anomalous correlation function is negligible.\nAs Fig. 3(b) suggests, the anomalous emission is en-\nhanced for a band with smaller average slope keeping the\nmaximum slope (the maximum group velocity) as c. We4\n 0 0.2 0.4 0.6 0.8 1\n-0.4 -0.2  0  0.2  0.4  0.6  0.8  1q/k maxImΓ=4\nλ=4\nT=0.8hyp v=0.6\nv=0.7\nv=0.8\nv=0.9\nv=0.96\nrel\n 0 1 2 3 4 5\n 0.6  0.7  0.8  0.9  1 0 0.2 0.4 0.6 0.8 1\nv/cI* q*\ntanhlinearq*\nλ~\n0=2\n4\n6\n8\nFIG. 4. (a) Im\u0000 qfor relativistic (dotted line) and hyperbolic\n(solid line) dispersion with \u0016= 5 for\u00150=a= 4. (b) The peak\namplitudeI\u0003and position q\u0003of Im\u0000 qfor di\u000berent \u00150=a. Wall\npro\fles are linear (solid line) and tanh (dashed line).\ntake here as an example a hyperbolic form of\n!(h)\nk= \u0001 +2ckmax\n\u0016\u0012\n1\u00001\ncosh\u0016k=k max\u0013\n; (7)\nwherekmax=\u0019=a and\u0016is a parameter de\fning the\naverage slope.19The dispersion does not bring quali-\ntative change in the normal response function \u0005 q(See\nRef.17), while the imaginary part of the anomalous re-\nsponse Im[\u0000 q] is signi\fcantly altered (Fig. 4(a)); A sharp\npeak appears for velocity vw=c&0:7 atq=q\u0003in high\nq-regime (0:5.q\u0003=kmax.1), indicating strong forward\nemission of two magnons. The minimum velocity neces-\nsary is determined by the dispersion; It is obviously larger\nthan!kmax=kmaxfor a monotonically increasing disper-\nsion, which is\u00182\n\u0016cfor hyperbolic dispersion with a small\ngap. The peak position q\u0003is independent on \u00150. The in-\ntensityI\u0003of the peak and q\u0003are plotted as function of\nvelocity in Fig. 4(b).\nThe anomalous emission, determined by large qbehav-\nior of the response function, is sensitive to the wall pro-\n\fle. In the case of very thin wall, linear pro\fle of nx(or\nny) inside the wall may appear instead of the ideal tanh\nwall, as argued in nanocontacts20. For the linear pro\fle,\nnx= (1\u0000jxj=\u0015(l))\u0012(\u0015(l)\u0000jxj), where\u0015(l)=\u0015\u0019=2, the\nform factor is Wq=2\u0019\n(q\u0015(l))2\u0010\n1\u0000sinq\u0015(l)\nq\u0015(l)\u0011\n(See Ref.17).\nThe anomalous response amplitude I\u0003is signi\fcantly en-\nhanced due to a slower decay at large q(Fig. 4(b)).\nThe emitted current amplitude is estimated by j\u0011P\nkqD\nay\n\u0000k+qay\nkE\n\u0018q\u0003I\u0003. For\u00150=a= 4,j&0:8 for\nvw=c > 0:8 andj= 0:2 for\u00150=a= 8 atvw=c= 0:9\natT= 0:8 and for linear wall pro\fle. Let us compare\nthe emitted spin wave current with the current due to\nthe wall motion. The spin wave current is de\fned as\nj=\u0000i\n2ay$\nra=\u00001\n4g( _'$\nr') in terms of real spin\nwave \feld'. For a domain wall, 'w= [coshx\u0000X(t)\n\u0015]\u00001.\nThe current at the wall center is thus jw\u0011Vw\n4g\u00152. Using\nJ=a2'c=a, we have\njw'kmax\n4\u0019~\u00152\n0~v\n1\u0000~v2(8)\nwhere ~v=vw=c. For ~\u0015= 4,jw=kmax'0:01(0:02) at ~v=\n0:8(0:9). The current due to the emission is thus by 1-2\n-0.4-0.2 0\n 0  0.2  0.4  0.6  0.8  1-0.1-0.05 0\nvFλ=4 Γ\nΠ\ntanh\nlinear\n 0 2 4 6\n 2  4  6  8 0 0.5 1 1.5 2\nλααGtanh\nlinear\nαG\n 0 0.2 0.4\n 4  6  8 0 0.02 0.04 0.06 0.08αGFIG. 5. (a) Plot of the force Ffor tanh and linear wall with\n\u00150=a= 4. The normal (\u0005) and anomalous (\u0000) contributions\nare shown by dashed and solid lines with axis at the left and\nright, respectively. Shaded region (~ v >0:97) shows a break-\ndown of continuum description for \u00150=a= 4. (b) Friction\nconstant\u000b(left axis) and contribution to the Gilbert damp-\ning constant (right axis).\norders of magnitude larger than the current of the wall\nitself in the relativistic regime. A thin and relativistic\nwall is therefore an extremely e\u000ecient magnon emitter.\nAs reaction to the scattering and emission/absorption,\na frictional force,\nF= 2Kg\naImX\nk;qWqe\u0000iqX(t)\np!k!k+qq\u0015D\nay\nk+qak+ay\n\u0000kay\nk+qE\n(9)\narises. As seen in the plot of Fig. 5, the emission con-\ntribution has a narrow peak at high velocity close to\nvw=c= 1, while the normal response (\u0005) contribution\nshows a broad peak starting from low velocity regime.\nThe normal contribution is larger than the emission con-\ntribution as the excitated magnon pro\fle is mostly local-\nized near the wall, resuting in a large overlap. The force\nat small velocity, dominated by the normal response, is\nan Ohmic friction, F=\u0000\u000bvw=\u00152, whose dimensionless\ncoe\u000ecient\u000bis plotted in Fig. 5. As the force arises\nfrom transfer of \fnite q, the friction constant \u000bdepends\nstrongly on the wall thickness. The friction coe\u000ecient\n\u000bcorresponds to a contribution to the Gilbert damp-\ning constant of \u000bG=a\n2\u0015\u000b, which is plotted by dashed\nlines. For linear wall pro\fle, the contribution \u000bGis 0.007\n(0.002) for \u0015=a= 6 (8) atT= 0:8, which is signi\fcantly\nlarge compared to the intrinsic Gilbert damping constant\nof most antiferromagnets. The damping due to magnon\nexcitation has clear temperature dependence, exponen-\ntially suppressed for T.\u0001 and increases linearly at high\ntemperature below the N\u0013 eel transition temperature17.\nFor quantitative study, the temperature-dependence of\n\u0011and the \ructuation near the N\u0013 eel temperature need to\nbe taken into account9.\nThe direction of the emitted magnons are determined\nby the sign of the wave vector k, while whether it is\nforward or behind the wall is determined by the group\nvelocity relative to the wall velocity. In the case of rel-\nativistic dispersion with a gap of ~\u0001 = 0:1, most part of\nthe normal response function at vw= 0:8 turns out to be\nthe magnon excitation behind the wall17. This is consis-\ntent with the observation based on the Landau-Lifshitz-\nGilbert (LLG) equation analysis in Ref.4that the moving5\nwall emits magnons mostly backward. The LLG study\n\fxes the magnon dispersion to be relativistic, and thus\ntheir results are due to the normal response function of\nthe present analysis.\nAs the amplitudeD\nay\n\u0000k+qay\nkE\nindicates, the two\nmagnons pair created by the mechanism proposed here\nare entangled quantum mechanically like in the case ofelectromagnetism21, suggesting interesting possibilities\nfor quantum magnonics.\nACKNOWLEDGMENTS\nThis work was supported by a Grant-in-Aid for Scien-\nti\fc Research (B) (No. 17H02929) from the Japan Soci-\nety for the Promotion of Science.\n1M. Petev, N. Westerberg, D. Moss, E. Rubino, C. Rimoldi,\nS. L. Cacciatori, F. Belgiorno, and D. Faccio, Phys. Rev.\nLett.111, 043902 (2013).\n2L. Ostrovskii, JETP, Vol. 34, No. 2, p. 293 (February 1972)\n(Russian original - ZhETF, Vol. 61, No. 2, p. 551, February\n1972 ) (1972).\n3F. D. M. Haldane, Phys. Rev. Lett. 50, 1153 (1983).\n4T. Shiino, S.-H. Oh, P. M. Haney, S.-W. Lee, G. Go, B.-G.\nPark, and K.-J. Lee, Phys. Rev. Lett. 117, 087203 (2016).\n5D. Bouzidi and H. Suhl, Phys. Rev. Lett. 65, 2587 (1990).\n6Y. L. Maho, J.-V. Kim, and G. Tatara, Phys. Rev. B 79,\n174404 (2009).\n7S. K. Kim, O. Tchernyshyov, V. Galitski, and\nY. Tserkovnyak, Phys. Rev. B 97, 174433 (2018).\n8G. Tatara and R. M. Otxoa de Zuazola, Phys. Rev. B 101,\n224425 (2020).\n9G. Tatara and C. O. Pauyac, Phys. Rev. B 99, 180405\n(2019).\n10E. Rubino, A. Lotti, F. Belgiorno, S. L. Cacciatori,\nA. Couairon, U. Leonhardt, and D. Faccio, Scienti\fc Re-\nports2, 932 (2012).\n11J. Schwinger, Phys. Rev. 82, 664 (1951).\n12E\u000bect of lattice beaking the relativistic nature of the wall\nwas discussed in Ref.22.\n13G. Tatara, H. Kohno, and J. Shibata, Physics Reports\n468, 213 (2008).\n14One could introduce other coordinates such as the an-\ngle of the wall plane and thickness, as in the case of aferromagnet8, but they are essentially decoupled from the\nXdynamics, except for the case where the driving \feld\nmixes them as discussed in Ref.4.\n15A second-order coupling of magnons to a wall velocity was\nstudied in Ref.7, and it was shown to give rise to a su-\nper Ohmic dissipation. Magnon-driven domain wall motion\nwas theoretically studied in Ref.23.\n16The potential was considered in Ref.23to study magnon-\ndriven motion of antiferroamgnetic domain wall.\n17See supplementary material.\n18S. M. Rezende, A. Azevedo, and R. L. Rodrguez-\nSurez, Journal of Applied Physics 126, 151101 (2019),\nhttps://doi.org/10.1063/1.5109132.\n19To be mathematically consistent, magnon dispersion is de-\ntermined by the Lagrangian (1) and is correlated with the\nequation of motion of domain wall. In reality, on the other\nhand, the dispersion is sensitive to details of lattice struc-\nture, while a collective behavior of a macroscopic wall is\nnot. We thus argue the case of a dispersion away from the\nrelativistic limit.\n20G. Tatara, Y.-W. Zhao, M. Mu~ noz, and N. Garc\u0013 \u0010a, Phys.\nRev. Lett. 83, 2030 (1999).\n21Z. Ebadi and B. Mirza, Annals of Physics 351, 363 (2014).\n22H. Yang, H. Y. Yuan, M. Yan, H. W. Zhang, and P. Yan,\nPhys. Rev. B 100, 024407 (2019).\n23E. G. Tveten, A. Qaiumzadeh, and A. Brataas, Phys. Rev.\nLett.112, 147204 (2014)."    },    {        "title": "2109.06034v1.Fourth_order_dynamics_of_the_damped_harmonic_oscillator.pdf",        "content": "Fourth-order dynamics of the damped harmonic oscillator\nJohn W. Sanders\nDepartment of Mechanical Engineering\nCalifornia State University, Fullerton\nFullerton, CA 92831, United States\nElectronic mail: jwsanders@fullerton.edu\nORCiD: 0000-0003-3059-3815\n(Dated: Received: date / Accepted: date)\n1arXiv:2109.06034v1  [physics.class-ph]  10 Sep 2021Abstract\nIt is shown that the classical damped harmonic oscillator belongs to the family of fourth-\norder Pais-Uhlenbeck oscillators. It follows that the solutions to the damped harmonic\noscillator equation make the Pais-Uhlenbeck action stationary. Two systematic approaches\nare given for deriving the Pais-Uhlenbeck action from the damped harmonic oscillator equa-\ntion, and it may be possible to use these methods to identify stationary action principles\nfor other dissipative systems which do not conform to Hamilton's principle. It is also shown\nthat for every damped harmonic oscillator x, there exists a two-parameter family of dual\noscillatorsysatisfying the Pais-Uhlenbeck equation. The damped harmonic oscillator and\nany of its duals can be interpreted as a system of two coupled oscillators with atypical spring\nsti\u000bnesses (not necessarily positive and real-valued). For overdamped systems, the resulting\ncoupled oscillators should be physically achievable and may have engineering applications.\nFinally, a new physical interpretation is given for the optimal damping ratio \u0010= 1=p\n2 in\ncontrol theory.\nI. Introduction and literature review\nEver since Newton1\frst laid the foundation for mechanics as a formal system, the de-\nvelopment of analytical mechanics has progressed via the principle of virtual work and its\ngeneralization, d'Alembert's principle,2reaching its pinnacle in the principle of stationary\naction.3{5During any time interval t2(t1;t2), the actual motion of a conservative mechanical\nsystem is given by a critical point of the action functional\nS=Zt2\nt1Ldt; (1)\nwhere the Lagrangian L=T\u0000Vis the di\u000berence between the system's total kinetic energy\nTand total potential energy V. Put simply, the action is stationary for the actual motion\nof a conservative system:\n\u000eS= 0: (2)\n2While sometimes attributed to Hamilton,4,5the principle of stationary action was already\nwell known by Lagrange3and contemporaries.6,7It has been demonstrated that the action\nalways achieves either a local minimum or a saddle point, but never a local maximum.7{9\nWhen dissipative forces are present, Hamilton's principle (2) may be exchanged for the\nd'Alembert-Lagrange principle3,10{12\n\u000eS+Zt2\nt1Qi\u000eqidt= 0; (3)\nwhere theqiare the generalized coordinates and the Qiare the generalized dissipative forces.\nLike d'Alembert's principle,2Hamilton's principle (2) and the d'Alembert-Lagrange principle\n(3) can be used as the basis for numerical solution methods.13,14Indeed, much modern work\nhas been devoted to the development of accurate and e\u000ecient variational time integrators\nbased on these principles.10,15{27Nevertheless, the fact remains that the d'Alembert-Lagrange\nprinciple is not a true variational principle, because in general the virtual work terms Qi\u000eqi\nare not the exact variations of a work function.6As a result, those terms must be inserted\ninto the equation manually after the variation operator \u000ehas been applied to the action.\nThe ad hoc insertion of the dissipative terms into an otherwise variational principle is ex-\ntremely dissatisfying on physical, mathematical, and aesthetic grounds, and over the last\ntwo centuries many attempts have been made to extend the Lagrangian framework and the\nstationary action principle to dissipative systems.\nOne of the most ubiquitous models of a dissipative system is given by the damped har-\nmonic oscillator, as shown in Figure 1. It consists of a mass mattached to a rigid support\nFIG. 1. The damped harmonic oscillator.\n3by a linear spring of sti\u000bness kand a linear damper of damping coe\u000ecient c, withx(t) the\ngeneralized coordinate along the single degree of freedom. The governing equation of motion\nis given by\nmx+c_x+kx= 0; (4)\nor equivalently,\nx+ 2!\u0010_x+!2x= 0; (5)\nwhere we have introduced the natural frequency !=p\nk=m and damping ratio \u0010=c=2m!.\nHenceforth, we shall regard ! > 0 and\u0010\u00150 as real-valued model parameters. Depending\non the damping ratio, the damped harmonic oscillator exhibits di\u000berent behaviors, including\nthe absence of damping ( \u0010= 0), underdamping (0 <\u0010 < 1), critical damping ( \u0010= 1), and\noverdamping ( \u0010 >1). While the damped harmonic oscillator does not conform to Hamilton's\nprinciple (2), there are known variational principles that reproduce the governing equation\n(5). As discussed in detail by Limebeer et al. ,26one class of stationary action principles\nis based on non-standard Lagrangians. For example, the dual system method of Bateman,\nMorse, Feshbach, and Tikochinsky28{30employs a Lagrangian of the form\nL= _x_y+!\u0010(x_y\u0000y_x)\u0000!2xy; (6)\nwherexandyare both to be varied. The Euler-Lagrange equations yield two oscillators:\nthe damped harmonic oscillator (5) and a dual anti-damped oscillator\ny\u00002!\u0010_y+!2y= 0: (7)\nAnother example is given by the Kanai-Caldirola oscillator,31,32which uses a Lagrangian of\nthe form\nL=1\n2e2!\u0010t( _x2\u0000!2x2); (8)\nwith Euler-Lagrange equation\ne2!\u0010t(x+ 2!\u0010_x+!2x) = 0: (9)\nA \fnal approach that will be mentioned here is the Caldeira-Leggett model,33{35which\ntreats a system as embedded within a heat bath, so that the combination of the system\n4and its surroundings is conservative.26An example is provided by Fukagawa and Fujitani,36\nwho used Hamilton's principle coupled with a nonholonomic constraint on the entropy to\nincorporate the damping force into the damped harmonic oscillator equation as well as the\nviscous terms into the momentum balance equation for a viscous \ruid, one special case being\nthe Navier-Stokes equations.37\nQuite independent of the damped harmonic oscillator is the Pais-Uhlenbeck oscillator,38\nwhich is used as a representative \\toy model\" to study higher-derivative theories for quantum\ngravity.39,40In general, the Pais-Uhlenbeck oscillator of order 2 nis given by\nnY\nj=1\u0012d2\ndt2+!2\nj\u0013\nx= 0; (10)\nwhere the!jare real-valued parameters. For n= 2, this yields the fourth-order equation\n....x+ (!2\n1+!2\n2)x+!2\n1!2\n2x= 0: (11)\nIn the present work, we will establish that the damped harmonic oscillator belongs to the\nfamily of fourth-order Pais-Uhlenbeck oscillators. To date, it seems that the connection be-\ntween the damped harmonic oscillator and the Pais-Uhlenbeck oscillator has gone unnoticed.\nThis connection leads to two interesting conclusions: (i) that the damped harmonic oscilla-\ntor is a critical point of the Pais-Uhlenbeck action, and (ii) that for every damped harmonic\noscillatorx, there exists a two-parameter family of coupled dual oscillators y.\nThe remainder of this paper is organized as follows. In Section II, we show that every\nsolution to the damped harmonic oscillator equation (5) is also a solution to a fourth-order\nequation of the Pais-Uhlenbeck type. In Section III, we investigate the two-parameter family\nof coupled-oscillator systems, leading to new interpretations for the various damping regimes.\nIn Section IV, we examine the special case of \u0010= 1=p\n2, which leads to new insights into this\n\\optimal control value.\" In Section V, we discuss the corresponding variational formulation\nbased on the Pais-Uhlenbeck action. We also present two systematic methods for arriving\nat the Pais-Uhlenbeck action from the original damped harmonic oscillator equation, which\nmay help identify variational principles for other dissipative systems. Finally, in Section VI,\nwe conclude with a brief summary of the present results.\n5II. The damped harmonic oscillator as a Pais-Uhlenbeck oscillator\nWe begin by observing that every solution to (5) is also a solution to the following fourth-\norder equation:\n....x+ 4!2\u00121\n2\u0000\u00102\u0013\nx+!4x= 0; (12)\nwhich is a kind of Pais-Uhlenbeck equation in which the coe\u000ecient of  xmay be positive,\nnegative, or zero, depending on the damping ratio. This can be demonstrated as follows.\nDi\u000berentiating (5) twice, we obtain\n....x+ 2!\u0010...x+!2x= 0: (13)\nSubstituting\n...x=\u00002!\u0010x\u0000!2_x (14)\nand\n_x=\u00001\n2!\u0010(x+!2x) (15)\ninto (13) yields (12). We conclude that the damped harmonic oscillator belongs to the family\nof Pais-Uhlenbeck oscillators. Indeed, with the same initial conditions\nx(0) =x0; (16)\n_x(0) =v0; (17)\nx(0) =a0\u0011\u00002!\u0010v 0\u0000!2x0; (18)\n...x(0) =j0\u0011\u00002!\u0010a 0\u0000!2v0; (19)\nthe uniqueness theorem guarantees that the Pais-Uhlenbeck oscillator (12) will yield the\nsame solution as the damped harmonic oscillator (5).\nUsing a trial solution of the form\nx(t) =Ae\u0015t; (20)\nwe solve the characteristic polynomial\n\u00154+ 4!2\u00121\n2\u0000\u00102\u0013\n\u00152+!4= 0; (21)\n6yielding four eigenvalues:\n\u00151;2=\u0006!vuut\u00002\u00121\n2\u0000\u00102\u0013\n+s\n4\u00121\n2\u0000\u00102\u00132\n\u00001 =\u0006!(\u0010+p\n\u00102\u00001); (22)\n\u00153;4=\u0006!vuut\u00002\u00121\n2\u0000\u00102\u0013\n\u0000s\n4\u00121\n2\u0000\u00102\u00132\n\u00001 =\u0006!(\u0010\u0000p\n\u00102\u00001): (23)\nThe behavior of the solution is determined by the damping coe\u000ecient. Just as in the classical\ntheory of the damped harmonic oscillator, there are four cases:\n1.No damping, \u0010= 0:For undamped oscillators, \u00151=\u00154= +i!and\u00152=\u00153=\u0000i!.\nIn this case, we have\nx(t) =Ae+i!t+Bte+i!t+Ce\u0000i!t+Dte\u0000i!t: (24)\nThis reduces to the classical simple harmonic oscillator solution\nx(t) =Ecos (!t) +Fsin (!t) (25)\nupon setting B=D= 0,A+C=E, andi(A\u0000C) =F.\n2.Underdamping, 0< \u0010 < 1:In this case, the eigenvalues are complex-valued and\ndistinct. Writing\n\u00151;2=\u0006!(\u0010+ip\n1\u0000\u00102); (26)\n\u00153;4=\u0006!(\u0010\u0000ip\n1\u0000\u00102); (27)\nwhereiis the imaginary unit, we have\nx(t) =Ae!\u0010tei!p\n1\u0000\u00102t+Be\u0000!\u0010te\u0000i!p\n1\u0000\u00102t+Ce!\u0010te\u0000i!p\n1\u0000\u00102t+De\u0000!\u0010tei!p\n1\u0000\u00102t:(28)\nThis reduces to the classical damped solution\nx(t) =e\u0000!\u0010th\nEcos\u0010\n!p\n1\u0000\u00102t\u0011\n+Fsin\u0010\n!p\n1\u0000\u00102t\u0011i\n: (29)\nupon setting A=C= 0,B+D=E, and\u0000i(B\u0000D) =F.\n73.Critical damping, \u0010= 1:For critically damped systems, \u00151=\u00153= +!and\n\u00152=\u00154=\u0000!. In this case, we have\nx(t) =Ae+!t+Bte+!t+Ce\u0000!t+Dte\u0000!t: (30)\nThis reduces to the classical damped harmonic solution\nx(t) = (C+Dt)e\u0000!t(31)\nupon setting A=B= 0.\n4.Overdamping, \u0010 >1:For overdamped systems, all eigenvalues are real-valued and\ndistinct, leading to a linear combination of exponential growth and decay:\nx(t) =Ae\u00151t+Be\u00152t+Ce\u00153t+De\u00154t: (32)\nThis reduces to the classical damped harmonic solution\nx(t) =Be\u0000!(\u0010+p\n\u00102\u00001)t+De\u0000!(\u0010\u0000p\n\u00102\u00001)t(33)\nupon setting A=C= 0.\nIII. Equivalent coupled-oscillator system\nIt is well-known that the fourth-order Pais-Uhlenbeck equation (12) is mathematically\nequivalent to the following system of two second-order equations:\nx+\u00161x\u0000\u001a1y=0; (34)\ny+\u00162y\u0000\u001a2x=0; (35)\nwhere\n\u00161+\u00162= 4!2\u00121\n2\u0000\u00102\u0013\n; (36)\n\u00161\u00162\u0000\u001a1\u001a2=!4: (37)\nand bothx(t) andy(t) satisfy the Pais-Uhlenbeck equation (12) individually. This can be\nthought of as a system of two coupled oscillators, as shown schematically in Figure 2. The\n8FIG. 2. Coupled oscillator system equivalent to the fourth-order Pais-Uhlenbeck oscillator.\nsprings in Figure 2 are not meant to suggest that a damped oscillator is equivalent to two\nconservative oscillators: as we will see, the sti\u000bnesses of the supporting springs ( k1,k2) and\ncoupling spring ( k\u0003) are generally not real-valued and positive, and so the dynamics are not\nconservative. In general, the masses and sti\u000bnesses of the coupled system are related to the\nmodel parameters as follows:\n\u00161=k1+k\u0003\nm1; \u001a 1=k\u0003\nm1; \u0016 2=k\u0003+k2\nm2; \u001a 2=k\u0003\nm2: (38)\nwhence we obtain\nm2\nm1=\u001a1\n\u001a2;k1\nk\u0003=\u00161\n\u001a1\u00001;k2\nk\u0003=\u00162\n\u001a2\u00001: (39)\nWith initial conditions\nx(0) =x0; y (0) = (a0+\u00161x0)=\u001a1; (40)\n_x(0) =v0; _y(0) = (j0+\u00161v0)=\u001a1; (41)\nxwill coincide with the damped harmonic oscillator. Now we observe that there are only\ntwo damped harmonic oscillator parameters ( !and\u0010), but there are four coupled oscillator\nparameters ( \u00161,\u00162,\u001a1, and\u001a2). It follows that for every damped harmonic oscillator xthere\nexists a two-parameter family of dual oscillators ysatisfying the Pais-Uhlenbeck equation.\nFor example, provided \u00106= 1=p\n2 (this is a unique case that will be treated in Section IV),\nwe may set \u00162= (1=\u000b\u00001)\u00161(\u000b6= 0,\u000b6= 1) and\u001a2=\f\u00002\u001a1(\f6= 0), leading to\n\u00161= 4\u000b!2\u00121\n2\u0000\u00102\u0013\n; \u0016 2= 4(1\u0000\u000b)!2\u00121\n2\u0000\u00102\u0013\n; (42)\n\u001a1=\f!2s\n16\u000b(1\u0000\u000b)\u00121\n2\u0000\u00102\u00132\n\u00001; \u001a 2=1\n\f!2s\n16\u000b(1\u0000\u000b)\u00121\n2\u0000\u00102\u00132\n\u00001;(43)\n9where we have chosen the positive radical for \u001a1in order to make the coupling sti\u000bness k\u0003\npositive when k\u0003is real-valued and \fis positive.\nA. Symmetric coupled oscillator\nHenceforth, we will restrict attention to the symmetric coupled oscillator, in which the\ntwo masses are identical and real-valued ( m1=m2=m2R) and the two supporting springs\nhave identical sti\u000bnesses ( k1=k2=k, not necessarily real or positive), giving \u001a1=\u001a2=\u001a\nand\u00161=\u00162=\u0016. This corresponds to the particular dual oscillator with \u000b= 1=2 and\f= 1,\ngiving\n\u0016= 2!2\u00121\n2\u0000\u00102\u0013\n; (44)\n\u001a=!2s\n4\u00121\n2\u0000\u00102\u00132\n\u00001 = 2!2\u0010p\n\u00102\u00001: (45)\nThe coupling sti\u000bness is given by\nk\u0003=m\u001a= 2m!2\u0010p\n\u00102\u00001; (46)\nTABLE I. Summary of all possible scenarios for a symmetric coupled oscillator\n(m1=m2=m2R,k1=k2=k). The sign of \u001ahas been chosen so as to make the\ncoupling sti\u000bness k\u0003positive when it is real-valued.\nCase \u0016 \u001a k\u0003=m\u001a k =m\u0016\u0000k\u0003r=k\u0003=k\n\u0010= 0 + !20 0 + m!20\n\u00102(0;1=p\n2) real, positive imaginary imaginary complex complex\n\u0010= 1=p\n2 0 + i!2+im!2\u0000im!2-1\n\u00102(1=p\n2;1) real, negative imaginary imaginary complex complex\n\u0010= 1\u0000!20 0 \u0000m!20\n\u0010 >1 real, negative real, positive real, positive real, negative real, negative\n\u0010!+1 \u00001 +1 +1 \u00001 \u0000 1=2\n10FIG. 3. Real and imaginary parts of the sti\u000bness ratio rversus damping ratio \u0010.\nand the supporting sti\u000bnesses are given by\nk=m\u0016\u0000k\u0003=m(\u0016\u0000\u001a) = 2m!2\u00121\n2\u0000\u00102\u0000\u0010p\n\u00102\u00001\u0013\n: (47)\nWe de\fne the sti\u000bness ratio as\nr=k\u0003\nk=\u0012\u0016\n\u001a\u00001\u0013\u00001\n=\u0000\"\n\u00102\u00001\n2\n\u0010p\n\u00102\u00001+ 1#\u00001\n: (48)\nThe real and imaginary parts of the sti\u000bness ratio rare plotted versus \u0010in Figure 3. Below\nwe consider the various damping cases in turn, which are summarized in Table I.\n1.No damping, \u0010= 0:In the absence of damping, \u0016= +!2,\u001a= 0,k\u0003= 0,k= +m!2,\nandr= 0. This corresponds to a case of two uncoupled simple harmonic oscillators\nwith positive sti\u000bnesses and identical natural frequencies !.\n2.Underdamping, 0<\u0010 < 1:For underdamping, the sti\u000bnesses kandk\u0003are non-real.\nTo be more precise, we must make a distinction between damping ratios less than,\nequal to, and greater than \u0010= 1=p\n2:\n11FIG. 4. Representative coupled oscillator dynamics for an underdamped system with\n\u0010= 0:2,!= 1,x0= 1, andv0= 0. The damped harmonic oscillator x(t) is real-valued,\nwhile the dual oscillator y(t) is imaginary.\n(a) 0<\u0010 < 1=p\n2: We have that \u0016is real and positive, \u001ais imaginary, k\u0003is imaginary,\nkis complex, and the sti\u000bness ratio ris complex.\n(b)\u0010= 1=p\n2: We have that \u0016= 0,\u001a= +i!2,k\u0003= +im!2,k=\u0000im!2, andr=\u00001.\nAgain, this case will be discussed in more detail in Section IV.\n(c) 1=p\n2<\u0010 < 1: We have that \u0016is real and negative, \u001ais imaginary, k\u0003is imaginary,\nkis complex, and the sti\u000bness ratio ris complex.\nFigure 4 shows representative coupled oscillator dynamics for an underdamped case\nwith\u0010= 0:2,!= 1,x0= 1, andv0= 0. In this case, the damped harmonic oscillator\nx(t) is real-valued, while the dual oscillator y(t) is imaginary.\n3.Critical damping, \u0010= 1:For critical damping, \u0016=\u0000!2,\u001a= 0,k\u0003= 0,k=\u0000m!2,\nandr= 0. As in the absence of damping, this corresponds to another case in which\nthe coupling spring sti\u000bness is zero. The di\u000berence is that here the supporting spring\n12sti\u000bnesses are negative .\n4.Overdamping, \u0010 >1:All quantities are real-valued. We have that \u0016is negative, \u001a\nis positive, k\u0003is positive, kis negative, and ris negative. In the limit as \u0010!+1,\nk\u0003!+1andk!\u00001 , andr!\u0000 1=2. This leads to an interesting interpretation\nof overdamping. As the system becomes more and more overdamped, the coupling\nspring becomes ever more positively sti\u000b, and the supporting springs become ever\nmore negatively sti\u000b, with their ratio approaching \u00001=2 asymptotically. Figure 5 shows\nrepresentative coupled oscillator dynamics for an overdamped case with \u0010= 1:5,!= 1,\nx0= 1, andv0= 0. In this case, the damped harmonic oscillator x(t) and the dual\noscillatory(t) are both real-valued.\nIt should be noted that spring-like elements with e\u000bectively negative sti\u000bnesses are physically\nachievable.41Thus, it should be possible to test the results presented here for overdamped os-\ncillators experimentally. Coupled oscillators mimicking the overdamped harmonic oscillator\nFIG. 5. Representative coupled oscillator dynamics for an overdamped system with\n\u0010= 1:5,!= 1,x0= 1, andv0= 0. The damped harmonic oscillator x(t) and the dual\noscillatory(t) are both real-valued.\n13may even \fnd applications to vibration suppression systems.\nIV. Optimal control: \u0010= 1=p\n2\nConsider a forced damped harmonic oscillator of the form\nx+ 2!\u0010_x+!2x=f0sin (\nt); (49)\nwheref0is the forcing magnitude and \nis the forcing frequency. At steady state, the\nsolution has normalized magnitude\nX(\u0017) =1p\n(1\u0000\u00172)2+ (2\u0010\u0017)2; (50)\nwhere\u0017=\n=!is the frequency ratio. A plot of X(\u0017) versus\u0017for various values of \u0010\nis shown in Figure 6. It is straightforward to show that X(\u0017) has a local maximum at\n\u0017=p\n1\u00002\u00102, provided\u0010 <1=p\n2. For\u0010 >1=p\n2, there is no longer a local maximum. The\ncritical value \u0010= 1=p\n2 at which the local maximum disappears happens to be the \\optimal\nFIG. 6. Plot of normalized response amplitude X(\u0017) versus frequency ratio \u0017for various\ndamping ratios \u0010. The local maximum does not exist for \u0010 >1=p\n2\u00190:7071.\n14control value\" in terms of the tradeo\u000b between rise time and percent overshoot in response\nto a step input.42\nLooking at the second-order damped harmonic oscillator equation (5), \u0010= 1=p\n2 does\nnot appear to have a special role; it is simply one case of underdamping. However, it is\nimmediately clear from the Pais-Uhlenbeck equation (12) that \u0010= 1=p\n2 is signi\fcant. The\ninertial term,\n4!2\u00121\n2\u0000\u00102\u0013\nx; (51)\nis the only term in the Pais-Uhlenbeck equation (12) that involves the damping ratio. We\nmight call this the \\inertial damping\" term. Interestingly, the inertial damping term vanishes\nfor\u0010= 1=p\n2, yielding\n....x+!4x= 0: (52)\nThis leads to a physical (rather than a purely control-theoretical) interpretation for the\noptimal control value. When \u0010= 1=p\n2, the oscillator's \\jounce\" (time rate-of-change of jerk)\nis not a\u000bected by its current acceleration, only by its current displacement from equilibrium.\nIn other words, there is no inertial damping for the optimal control value.\nAdditionally, \u0010= 1=p\n2 leads to a degenerate case in which the coupled oscillator param-\neters\u00161=\u0000\u00162, according to (36). Setting \u00161=\rand\u001a2=\f\u00002\u001a1(\f6= 0), we have\n\u00161=\r; \u0016 2=\u0000\r; (53)\n\u001a1=\fp\n\u0000\r2\u0000!4; \u001a 2=1\n\fp\n\u0000\r2\u0000!4: (54)\nIn Section III A, when we considered the case of \u0010= 1=p\n2, we were assuming identical\nmasses and identical supporting springs, which amounts to setting \r= 0 and\f= 1 here.\nThat was the only underdamped value for which the sti\u000bness ratio (48) was real-valued\n(r=\u00001), giving\nk\u0003= +im!2; k =\u0000im!2: (55)\nThus, optimal control is the only case in which the symmetric coupled oscillator with real\nmasses has purely imaginary spring sti\u000bnesses. Figure 7 shows representative coupled oscil-\nlator dynamics for an optimal control case with \u0010= 1=p\n2,!= 1,x0= 1, andv0= 0. In\n15FIG. 7. Representative coupled oscillator dynamics for an optimal control case with\n\u0010= 1=p\n2,!= 1,x0= 1, andv0= 0. The damped harmonic oscillator x(t) is real-valued,\nwhile the dual oscillator y(t) is imaginary.\nthis case, the damped harmonic oscillator x(t) is real-valued, while the dual oscillator y(t)\nis imaginary.\nV. Variational formulation\nThe fact that every solution to the damped harmonic oscillator equation (5) also satis\fes\nthe fourth-order Pais-Uhlenbeck equation (12) also leads to a new variational formulation\nfor the damped harmonic oscillator. The Pais-Uhlenbeck action is given by\nS=Zt2\nt1\u0014\nx2\u00004!2\u00121\n2\u0000\u00102\u0013\n_x2+!4x2\u0015\ndt; (56)\nand the Pais-Uhlenbeck equation (12) can be recovered by taking \u000eS= 0. It follows that\nevery solution to the damped harmonic oscillator equation (5) constitutes a critical point of\nthe Pais-Uhlenbeck action. It is instructive to compare this result to previous variational\nformulations of the damped harmonic oscillator. Like the dual system approach of Bateman,\n16Morse, Feshbach, and Tikochinsky,28{30the Pais-Uhlenbeck action introduces dual oscilla-\ntors. The di\u000berence here is that there is actually a two-parameter family of dual oscillators\ny; moreover, the dual oscillators do not appear explicitly in the action and are not varied\nindependently of x. Unlike the Kanai-Caldirola Lagrangian,31,32the Pais-Uhlenbeck La-\ngrangian does not depend explicitly on time. And unlike the Caldeira-Leggett approach33{35\ntaken by Fukagawa and Fujitani,36the Pais-Uhlenbeck action does not require the use of\nnon-holonomic constraints to include the dissipative terms.\nIt may be that variational formulations can be found for other dissipative systems by con-\nsidering higher-order equations. The problem becomes how to identify those equations. One\npossible method is to manipulate the second-order equation directly, as we did in Section II.\nHaving arrived at (12), we may multiply by \u000ex,\n....x\u000ex+ 4!2\u00121\n2\u0000\u00102\u0013\nx\u000ex+!4x\u000ex= 0; (57)\nintegrate by parts judiciously,\nZt2\nt1\u0014\nx\u000ex\u00004!2\u00121\n2\u0000\u00102\u0013\n_x\u000e_x+!4x\u000ex\u0015\ndt= 0; (58)\nthus completing the variation and arriving at the Pais-Uhlenbeck action principle\n\u000eZt2\nt1\u0014\nx2\u00004!2\u00121\n2\u0000\u00102\u0013\n_x2+!4x2\u0015\ndt= 0: (59)\nAnother method is to start from the d'Alembert-Lagrange principle (3) and look for a\nparticular variation that puts it in variational form. For the damped harmonic oscillator,\nthe d'Alembert-Lagrange principle gives\n\u0000Zt2\nt1(mx+c_x+kx)\u000exdt = 0: (60)\nBoth the inertial term mx\u000exand the spring term kx\u000ex can be put into variational form,\nbecause they each contain an even number of time derivatives. However, the damping term\nc_x\u000ex has an odd number of derivatives and therefore cannot be expressed as the exact\nvariation of a work function. If we had an additional term proportional to x\u000e_x, it could\ncancel out the damping term. Thus, we might look for a variational principle of the form\nZt2\nt1(mx+c_x+kx)(\u000ex+\u000b\u000e_x)dt= 0: (61)\n17However, this will give a term proportional to  x\u000e_x, which also has an odd number of deriva-\ntives. Thus, in general we must look for a variational principle of the form\nZt2\nt1(mx+c_x+kx)(\u000ex+\u000b\u000e_x+\f\u000ex)dt= 0: (62)\nExpanding, we have\nZt2\nt1(mx\u000ex+\u000bmx\u000e_x+\fmx\u000ex+c_x\u000ex+\u000bc_x\u000e_x+\fc_x\u000ex+kx\u000ex +\u000bkx\u000e _x+\fkx\u000e x)dt= 0:(63)\nThe terms with odd numbers of derivatives are \u000bmx\u000e_x,c_x\u000ex,\fc_x\u000ex, and\u000bkx\u000e _x. Upon\nintegration by parts, we want \u000bmx\u000e_xto cancel\fc_x\u000ex, and forc_x\u000exto cancel\u000bkx\u000e _x. This\nrequires\n\u000bm=\fcandc=\u000bk; (64)\nwhich gives \u000b=c=kand\f=m=k. Hence, we are looking for a variational principle of the\nformZt2\nt1(mx+c_x+kx)\u0010\n\u000ex+c\nk\u000e_x+m\nk\u000ex\u0011\ndt= 0: (65)\nWe observe that \u000ex+ (c=k)\u000e_x+ (m=k)\u000exis actually a variation of the governing equation,\nup to a factor of k. Expanding, we have\nZt2\nt1\u0012\nmx\u000ex+mc\nkx\u000e_x+m2\nkx\u000ex+c_x\u000ex+c2\nk_x\u000e_x+mc\nk_x\u000ex+kx\u000ex +cx\u000e_x+mx\u000ex\u0013\ndt= 0:\n(66)\nJudicious integration by parts yields\nZt2\nt1\u0012\n\u0000m_x\u000e_x+mc\nkx\u000e_x+m2\nkx\u000ex+c_x\u000ex+c2\nk_x\u000e_x\u0000mc\nkx\u000e_x+kx\u000ex\u0000c_x\u000ex\u0000m_x\u000e_x\u0013\ndt= 0;\n(67)\nwhich simpli\fes to\nZt2\nt1\u0014m2\nkx\u000ex+\u0012c2\nk\u00002m\u0013\n_x\u000e_x+kx\u000ex\u0015\ndt= 0: (68)\nEach term can now be put into variational form, yielding\n\u000eZt2\nt1\u0014m2\n2kx2+1\n2\u0012c2\nk\u00002m\u0013\n_x2+1\n2kx2\u0015\ndt= 0; (69)\nor equivalently,\n\u000eZt2\nt1\u0014\nx2\u00004!2\u00121\n2\u0000\u00102\u0013\n_x2+!4x2\u0015\ndt= 0: (70)\nFuture work may consider whether either of the above approaches generalizes to other kinds\nof dissipative systems.\n18VI. Summary and conclusion\nThe main result of this paper is that the classical damped harmonic oscillator belongs to\nthe family of fourth-order Pais-Uhlenbeck oscillators. Three immediate corollaries follow:\n(I) For every damped harmonic oscillator x, there exists a two-parameter family of dual\noscillatorsysatisfying the Pais-Uhlenbeck equation. The damped harmonic oscillator\nand any of its duals constitute a system of two coupled oscillators with non-standard\nspring sti\u000bnesses. The various cases are summarized in Table I for coupled oscillators\nwith identical masses and identical supporting springs. Of particular interest is the case\nof overdamping \u0010 >1, for which the sti\u000bness of the coupling spring is positive while the\nsti\u000bnesses of the supporting springs are negative. This scenario should be physically\nachievable41and may even \fnd applications to vibration suppression systems.\n(II) The optimal control value \u0010= 1=p\n2, which gives the best tradeo\u000b between rise time\nand percent overshoot in response to a step input,42appears prominently in the fourth-\norder formulation and has a clear physical interpretation. For this value of the damping\nratio, there is no inertial damping.\n(III) The solutions of the damped harmonic oscillator make the Pais-Uhlenbeck action sta-\ntionary, yielding a new variational formulation for the damped harmonic oscillator.\nTwo systematic methods have been given for arriving at the Pais-Uhlenbeck action\nstarting from the damped harmonic oscillator equation. It may be possible to gen-\neralize these approaches and identify stationary actions for other kinds of dissipative\nsystems which do not conform to Hamilton's principle.\nDeclarations\nCon\ricts of interest\nThe author declares that he has no con\ricts of interest or competing interests.\n19Data availability\nData sharing is not applicable to this article as no new data were created or analyzed in\nthis study.\nReferences\n1I. Newton, Philosophiae Naturalis Principia Mathematica (Royal Society of London, 1687).\n2J. L. R. d'Alembert, Trait\u0013 e de dynamique (J. B. Coignard, 1743).\n3J. L. Lagrange, M\u0013 ecanique Analytique (Gauthier-Villars, 1811).\n4W. R. Hamilton, \\On a general method in dynamics,\" Philosophical Transactions of the\nRoyal Society 124, 247{308 (1834).\n5W. R. Hamilton, \\Second essay on a general method in dynamics,\" Philosophical Trans-\nactions of the Royal Society 125, 95{144 (1835).\n6C. 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Jacobs, \\The damping ratio of an optimal control system,\" IEEE Transactions\non Automatic Control 10, 473{476 (1965).\n23"    },    {        "title": "1902.03199v1.Quantitative_analysis_of_the_interaction_between_a_dc_SQUID_and_an_integrated_micromechanical_doubly_clamped_cantilever.pdf",        "content": "Quantitative analysis of the interaction between a dc SQUID and an\nintegrated micromechanical doubly clamped cantilever\nMajdi Salman,1,a)Georgina M Klemencic,1Soumen Mandal,1Scott Manifold,1Luqman Mustafa,2Oliver A\nWilliams,1and Sean R Giblin1\n1)School of Physics and Astronomy, Cardi\u000b University, Queen's Building, The Parade,\nCardi\u000b, CF24 3AA, United Kingdom\n2)Centre for Innovation Competence SiLi-nano, Martin-Luther-University Halle-Wittenberg,\nKarl-Freiherr-von-Fritsch-Strasse 3, 06120 Halle (Saale), Germany\n(Dated: 11 February 2019)\nBased on the superconducting quantum interference device (SQUID) equations described by\nthe resistively- and capacitively-shunted junction model coupled to the equation of motion of\na damped harmonic oscillator, we provide simulations to quantitatively describe the interac-\ntion between a dc SQUID and an integrated doubly clamped cantilever. We have chosen to\ninvestigate an existing experimental con\fguration and have explored the motion of the can-\ntilever and the reaction of the SQUID as a function of the voltage-\rux V(\b) characteristics.\nWe clearly observe the Lorentz force back-action interaction and demonstrate how a sharp\ntransition state drives the system into a nonlinear-like regime, and modulates the cantilever\ndisplacement amplitude, simply by tuning the SQUID parameters.\nPACS numbers: Valid PACS appear here\nI. INTRODUCTION\nTheoretical and experimental studies1{7of linear and\nnonlinear micro and nanomechanical resonators are of\ngreat interest as they can be used for sensitive force and\ndisplacement measurements. The physical parameters of\nthe resonators can also be tuned to observe the transi-\ntion from the classical to quantum regimes with relative\nexperimental ease, enabling observations of macroscopic\nquantum systems.8Signi\fcant experimental progress in\nthe detection of resonators as they enter the quantum\nground state has been achieved by capactiave coupling\nto superconducting \rux qubits,9and quantum state con-\ntrol of a mechanical drum resonator in a superconduct-\ning resonant circuit has been achieved by phonon-photon\ncoupling.10,11The state detection is an integral part of\nany coupled resonator system as the coupling mechanism\nis implicit in any experimental endeavour.\nConsidering a doubly clamped cantilever, it is obvious\nthat as the cantilever oscillates the displacement changes,\nand the transduction technique will cause a back-action\nthat in\ruences the cantilever position.12The impact\nof back-action can be positive in terms of cooling3and\nsqueezing the resonator motion,13{15and coupling and\nsynchronising multiple resonators.16,17Depending on the\nspeci\fc transduction technique, back-action can be due\nto radiation pressure,18electron tunnelling,19or pho-\ntothermal e\u000bects.20\nPrevious experiments have used a dc SQUID to detect\nthe motion of a suspended doubly clamped cantilever in-\ntegrated directly into a SQUID loop,1,21and for a tor-\nsional SQUID cantilever.22For this SQUID-based trans-\nduction scheme, the back-action has a simple inductive\ncomponent caused by the Lorentz force due to the cir-\nculating current.21{23Experimentally, the Lorentz back-\na)Electronic mail: Salmanm2@cardi\u000b.ac.uk\nDoubly clampedcantileverI\"J1J2Φ$%&𝑩(a)\n0.0 0.5 1.0 1.5 2.005101520(b)Ib=2.0I0Ib=1.3I0SQUID Voltage (PV)\n)ext/)0FIG. 1. (a) scheme for the dc SQUID displacement detector in\nwhich the two Josephson junctions are labelled by J 1and J 2.\nThe cantilever displacement is out-of-plane, and the applied\nmagnetic \feld, B, is in-plane. (b) V(\b) characteristics for\na dc SQUID with \fL= 0:115 and\fC= 1:61. Four regimes\nare identi\fed: ( i) the simple oscillatory regime where the bias\ncurrent,Ib= 2:0I0. The other regimes are: ( ii) the rapidly\nchanging regime (red), ( iii) the zero voltage response regime\n(blue), and ( iv) the intermediate regime (green).\naction was shown to shift the mechanical cantilever reso-\nnant frequency and quality factor by \u0001 fand \u0001Qrespec-\ntively. To understand the e\u000bect of back-action on \u0001 fand\n\u0001Q, two transfer functions were obtained,21which are\ncoe\u000ecients for the average circulating current expanded\nin the terms of the cantilever displacement, u, and veloc-\nity, _u.\nIn previous work, however, it was not possible to obtain\nthe velocity-dependent transfer function in the frame of\nthe SQUID equations coupled to the equation of motion\nof the doubly clamped cantilever. To simplify this issue,\nPoot et al21modulated the \rux change in the SQUID\nloop caused by the cantilever oscillation. Subsequently,\nthe total \rux in the SQUID loop was assumed to be\na function of the externally applied \rux, \b ext, and the\nmodulation of the \rux due to the changing area of the\nloop, \b!\bext+ \b modcos(!modt). Such a modulationarXiv:1902.03199v1  [physics.app-ph]  8 Feb 20192\ncan describe the in\ruence of the back-action on \u0001 fand\n\u0001Qof the cantilever when the SQUID displacement de-\ntector is tuned within limited regions of the V(\b) curve.21\nHowever, a full description of the SQUID-cantilever in-\nteraction requires a comprehensive model to provide in-\nformation not only about the in\ruence of back-action\nin all regions of V(\b), but also about the amplitude,\nwidth, line shape, and responsivity,dV\ndu, which must be\ncalculated by linking the cantilever displacement to the\nSQUID voltage. Thus, the need for quantitative treat-\nments of the unscaled SQUID equations coupled explic-\nitly to the equation of motion for the integrated beam\nbecomes important. Though such treatments are com-\nplicated and challenging,24they can be performed nu-\nmerically with improving computational capabilities.\nIn this paper, we simulate the interaction between a\ndc SQUID and an embedded micromechanical doubly\nclamped cantilever as experimentally demonstrated by\nEtaki et al1and shown schematically in Fig. 1(a). The\nSQUID-cantilever interaction is analysed in di\u000berent re-\ngions of the V(\b) curve, as shown in Fig. 1(b). Within\nthis framework, we have explored some regions of the\nV(\b) curve, where the SQUID-cantilever response is ap-\nparently strongly nonlinear. Futhermore, the back-action\nand the subsequent response of the SQUID is linked to\nthe cantilever displacement. The e\u000bect of changing the\nSQUID operating point is discussed in depth, and it is\ndemonstrated that the SQUID itself can be used to con-\ntrol the cantilever response by simple modi\fcation of the\ncontrollable SQUID parameters.\nII. THE MODEL\nThe model we present is based on the experimental\nparameters of Etaki et al1to allow for experimental ver-\ni\fcation of the results. As such the inductive screening\nparameter, \fL, and Stewart-McCumber parameter, \fC,\nare selected to be 0 :115 and 1:61 respectively. With these\nvalues for\fLand\fC,V(\b) characteristics of an over-\ndamped dc SQUID are shown in Fig. 1(b) to demonstrate\nthe possible operating points of a SQUID displacement\ndetector. The V(\b) curves are calculated using the time-\nscaled SQUID equations described by the resistively-\nand capacitively-shunted junction (RCSJ) model25. In\nFig. 1(b), four di\u000berent regimes in the SQUID V(\b)\nresponse are de\fned: ( i) the simple oscillatory regime\nwhere the bias current, Ib= 2:0I0. The other regimes\nare (ii) the rapidly changing regime (red), ( iii) the zero\nvoltage response regime (blue), and ( iv) the intermediate\nregime (green). Our analysis covers the interaction be-\ntween a dc SQUID and an integrated cantilever when the\nsystem is tuned to operating points within these de\fned\nregimes, and the resulting e\u000bect on the cantilever-SQUID\ndynamics.\nWe use the equation of motion of a damped harmonic\noscillator given in21to describe the displacement, u(t),\nof the mechanical cantilever:\nmu+m!0\nQ0_u+m!2\n0u=Fd(t) +FL(t); (1)\n2.0005 2.0010 2.0015 2.0020 2.0025 2.00300.00.20.40.60.81.0\nFrequency (MHz)SQUID Voltage (PV)(a) )ext/)0= 0.1 0.25 0.40 0.55\n2.0005 2.0010 2.0015 2.0020 2.0025 2.003005101520Displacement (pm)Frequency (MHz)(c) )ext/)0= 0.1 0.25 0.40 0.55 )ext/)0= 0.10 0.25 0.40 0.555 10 15 200.00.20.40.60.81.0SQUID Voltage (PV)\nDisplacement (pm)(e)(f)\n0.1 0.3 0.5 0.7 0.90.2 0.4 0.6 0.80.010.020.030.040.05Ib=2.0I0\n)ext/)0dV/du (nV.fm-1)2.001 2.002 2.0030.130.380.630.88\n0.250.500.75(b) )ext/)0\nFrequency (MHz)0.0000.11810.23620.35440.47250.59060.70870.82690.9450SQUID Voltage (PV)\n2.001 2.002 2.0030.130.380.630.88\n0.250.500.75(d)\nFrequency (MHz) )ext/)0\n0.45002.8065.1637.5199.87512.2314.5916.9419.30Displacement (pm)FIG. 2. Line shapes for (a) SQUID voltage and (c) cantilever\ndisplacement calculated as a function of \b extforIb= 2:0I0.\n(b) and (d) density plots for SQUID voltages and cantilever\ndisplacement respectively. (e) the linear displacement-voltage\ntrace as extracted by linking (b) and (d) via the frequency.\n(f) the responsivity (dV\ndu) as calculated from the slopes of the\ndisplacement-voltage lines.\nwheremis the beam mass, !0= 2\u0019f0is the intrinsic\nfrequency,Q0is the quality factor, Fd=F0cos(!0t) is\nthe driving force, and FL(t) is the Lorentz force FL(t) =\naB`(Ib=2+J). Here,Bis the in-plane magnetic \feld, `is\nthe length of the cantilever, Jis the circulating current,\nanda= 0:911is a geometrical factor that depends on\nthe mode shape. Eq. (1) is coupled to the dc SQUID\nequations given by the RCSJ model:\n\b0\n2\u0019C\u000e1+\b0\n2\u0019C1\nR_\u000e1+I0sin(\u000e1) =1\n2(Ib+J);(2)\n\b0\n2\u0019C\u000e2+\b0\n2\u0019C1\nR_\u000e2+I0sin(\u000e2) =1\n2(Ib\u0000J);(3)\n\u000e1\u0000\u000e2= 2\u0019\u0001\btot=\b0; (4)\nwhere\u000e1;2are the phase di\u000berences of the junctions, \b 0is\nthe \rux quantum, Ibis the bias current, I0is the critical\ncurrent. The total \rux, \b tot, has three contributions: ( i)\nthe external \rux \b ext, (ii) the \rux due to the circulating\ncurrent,J, \rowing through the inductance of the loop,\nL, and (iii) the change in \rux through the loop due to\nthe cantilever displacement, aB`u . Therefore, \b tot=\n\bext+LJ+aB`u (t), and Eqs. (1-3) are coupled via the\ncirculating current as J=1\nL(\u000e1\u0000\u000e2\n2\u0019\b0\u0000\bext\u0000aB`u ).3\nThese coupled di\u000berential equations are numerically\nsolved without averaging the SQUID voltage and cir-\nculating currents, or scaling the time. Therefore, the\ntime span Tmaxmust be large enough to be suitable\nfor the cantilever, while the time step d tmust be small\nenough to resolve the impact of the fast changes dom-\ninated by the relatively high SQUID characteristic fre-\nquency!c=2\u0019RI 0\n\b0. Although this can be computation-\nally expensive for cantilevers with very low frequencies\nrelative to !c, the experimental results of Etaki et al1\nallow their experiment to be modelled within a relatively\nsmall time window.\nHere, we solve a system for identical experimental con-\nditions demonstrated by Etaki et al1withf0'2 MHz.\nTo calculate the time dependent voltage, V= \b 0_\u000e1+_\u000e2\n2\u0019,\nthe Runge-Kutta method (RK4) was used to numerically\nintegrate the equations presented above. The SQUID re-\nsponse was then obtained in the frequency domain by\nevaluating the Fourier transform of the SQUID voltage\nand the cantilever displacement. Our calculations were\nperformed for I0= 0:7\u0016A,R= 29:5 \n,B= 111 mT,\nC= 0:91 pF, and L= 170 pH. These values give a\nMcCumber-Stewart parameter \fC=2\u0019I0R2C\n\b0= 1.61 and\na screening parameter \fL=2I0L\n\b0= 0:115. The cantilever\nhas a length `= 50\u0016m, a massm= 6\u000210\u000013kg, and was\nassumed to have a resonant frequency f0= 2:0018 MHz\nand a quality factor Q0= 25000. The piezo drive which\ncontrolsFdis used only to locate the eigenmodes and is\nturned o\u000b during measurement22. Thus, at t= 0 the\ninitial velocity v0=du\ndt\f\f\nu=u0= 0, where u0is the initial\ndisplacement amplitude. Here, u0= 20 pm.\nThe time span chosen for these calculations was\nTmax = 25 ms, i.e. more than six times the life-\ntime of the cantilever, and the optimised time step cho-\nsen was dt= 0:0125 ns. The calculations were re-\npeated at di\u000berent values of normalised \rux in the range\n0:90\b 0\u0014\bext\u00140:05\b 0, and bias currents in the range\n2:0I0\u0014Ib\u00141:10I0. In the frequency domain, we se-\nlected frequency steps of d f= 12:5 Hz. The units of the\nresponse which were calculated directly from a Fourier\ntransform are V\u0001s for the unnormalised SQUID voltage\nand m:s for the unnormalised cantilever displacement. To\nconvert the units of the voltage-response from V \u0001s to V,\nthe response was multiplied by1\n\u001c, where\u001cis the lifetime\nof the cantilever, which is related to the full width at half\nmaximum (FWHM) as1\n\u0019\u001c=fFWHM . A similar proce-\ndure was used to convert the units of the displacement-\nresponse from m\u0001s to m.\nIII. RESULTS\nA. The simple oscillatory regime behaviour\nTo ensure our calculations are based in physical re-\nality, we contextualized the calculations with the exist-\ning experimental parameters.1As experimentally demon-\nstrated, the voltage responses exhibit Lorentzian distri-\nbutions and for \b ext= 0:25\b 0, i.e. the highest SQUID\nsensitivity for Ib= 2:0I0shown in Fig. 1(b), there was\n144145146147148149150(c)FWHM (Hz) Ib/I0= 1.60 1.70 1.80 1.90 2.00\n0.3 0.4 0.5 0.6 0.70.00.51.01.52.02.5(e)\n)ext/)0Max. SQUID Voltage (PV) Ib/I0= 1.60 1.70 1.80 1.90 2.00020406080100(a)'  Ib/I0= 1.60 1.70 1.80 1.90 2.00(d)\n1.61.71.81.92.0b0\n145.1145.7146.4147.0147.7148.3149.0149.6150.3FWHM (Hz)(d)1.61.71.81.92.0(b)b00.00013.7527.5041.2555.0068.7582.5096.25110.0'f (Hz)\n0.3 0.5 0.70.4 0.61.61.71.81.92.0b0(f)\n)ext/)00.080.420.771.111.461.802.142.492.83Max.SQUID Voltage (PV)FIG. 3. Calculations for the range 0 :75\b 0\u0014\bext\u00140:25\b 0\nand 2:0I0\u0014Ib\u00141:55I0for (a) the frequency shift, \u0001 f, (c)\nFWHM and (e) the maximum SQUID voltage, Vmax. The\ncorresponding density plots are shown in (b), (d), and (f)\nrespectively.\nno relative experimental shift in \u0001 fof the cantilever.\nChanging the operating point of the SQUID by changing\n\bextwithin the simple oscillatory region shown in Fig.\n1(b) a\u000bects \u0001 f, and the operating point clearly a\u000bects\nthe sensitivity to the SQUID voltage as clearly shown in\nFig. 2(a) and (b). Moreover the subsequent cantilever\ndisplacement is also a\u000bected (Fig. 2(c) and (d)). These\nresults clearly demonstrate the in\ruence of the Lorentz\nback-action on the resonator from the SQUID displace-\nment detector, and the expected magnitude of change in\nthe experimental variables.\nThe cantilever displacement and SQUID voltage are\nexplicitly linked in the frequency domain, i.e., the dis-\nplacementu(f) is parametrically linked to the voltage\nV(f). The subsequent analysis was performed at Ib=\n2:0I0and 0:05\b 0\u0014\bext\u00140:95\b 0, and the displacement-\nvoltage trace is plotted in Fig. 2(e). The traces show a\nlinear dependence of voltage on displacement, which al-\nlows determination of the cantilever position in a respon-\nsivity speci\fed by the slope of the displacement-voltage\nlines. Consequently, the responsivity (dV\ndu) was calcu-\nlated atIb= 2:0I0for di\u000berent \b extvalues, with the\nresult shown in Fig. 2(f). The \fgure shows a sinusoidal\nbehaviour fordV\nduwhich varies from 4 :7\u000210\u00002nV:fm\u00001at\n\bext= 0:25\b 0to 0:5\u000210\u00002nV:fm\u00001at \b ext= 0:50\b 0.\nImportantly, Fig. 2 shows an appropriate representa-\ntion of the experimental results by Etaki et al1, thereby\ndemonstrating a good computational model.4\nB. The intermediate regime behaviour\nFurther calculations were performed through the V(\b)\ncurve identi\fed in Fig. 1(b) to examine the SQUID-\ncantilever coupling and explore the system response as\nthe coupling/back-action is modi\fed. Fig. 3(a)-(f) shows\n\u0001f, the FWHM, and the SQUID voltage as the bias\ncurrent and \b extare tuned. The largest frequency shift\ncorresponds to the smallest gradient (dV\nd\b) of the work-\ning point. This can clearly be understood by Eq. (1),\nwhere the frequency of the cantilever is controlled by the\ndisplacement coe\u000ecient. As the cantilever frequency is\nshifted by changing \b extandIb, a modi\fcation in this\ncoe\u000ecient emerges due to the circulating current depen-\ndence onu. Such a dependence was previously anal-\nysed by expanding the circulating current in terms of\nthe displacement, u21. In this way, the new displace-\nment coe\u000ecient, which arises from the back-action of the\nSQUID current on the cantilever, modi\fes the frequency\nand causes a slight or signi\fcant shift depending on \b ext\nandIb.\nThe Lorentz back-action also a\u000bects the cantilever\nquality factor; FWHMs of simulated line shapes are ex-\ntracted and presented as a function of \b extfor various\nvalues ofIbin Fig. 3(c) and (d). The variation of the\nFWHM can be interpreted in an identical way to that of\n\u0001f, where the only di\u000berence being that FWHM =!0\n2\u0019Q0\nis given in terms of velocity coe\u000ecient in Eq. 1. Thus,\nthe FWHM is modi\fed if Jis assumed to have a de-\npendence on the velocity in addition to the displacement\nwhich modi\fes the frequency21. The corresponding peak\nvoltage,Vmax, dependence on \b extandIbis shown in\nFig. 3(e) and (f). The behaviour of Vmaxas a function of\n\bextis consistent with d V=d\bextof the SQUID V(\bext)\ncurve shown in Fig. 1(b).\nC. The rapidly changing regime behaviour\nNow we turn to a di\u000berent regime from Fig 1(b), where\nthe largest e\u000bect of back-action on the cantilever is ob-\nserved, and the SQUID response is apparently nonlinear.\nTo examine the e\u000bect of back-action on the cantilever\nmotion, a point in such region was selected as shown in\nthe inset of Fig. 4(a). Subsequently, at \b ext= 0:30\b 0\nandIb= 1:20I0, the unnormalised cantilever displace-\nment and corresponding unnormalised SQUID voltage\nresponse for various displacement amplitudes u0, are ob-\ntained and plotted in Fig. 4(a) and Fig. 4(b). When\nu0= 20 pm, the cantilever appears to have a nonlinear\nbehaviour as demonstrated by the modi\fed line shape\nof the cantilever and the SQUID response. Speci\fcally,\nas the displacement is reduced from u0= 20 pm to\nu0= 10 pm to u0= 5pm, the change in the \rux through\nthe loop is 0 :05, 0:025 and 0:0125\b 0respectively. This\nchange of \rux in the SQUID loops drives the cantilever\nto experience two V(\b) regions of di\u000berent responsivity,\nwhich results in the nonlinear-like behaviour. It should\nbe noted, however, that as the cantilever returns to its\ndynamical equilibrium position, the response becomes\nmore Lorentzian as expected.\n2.000 2.001 2.002 2.003024681012\n0.0 0.1 0.2 0.3 0.4 0.502468101214Ib= 1.20I0)ext/)0 \u0016\u0013SQUID Voltage (PV)SQUID\u0003)-V curve)ext/)0Unnorm. Disp. Amp (fm.s)\nFrequency (MHz)(a) u0=5pm u0=10pm u0=20pm\n2.000 2.001 2.002 2.0030.00050.00100.00150.0020Unnorm.SQUID Voltage nV.s(b)\nFrequency (MHz) u0=5pm u0=10pm u0=20pm\n0.300.350.400.450.500.55\n0.00.51.01.52.02.53.0FIG. 4. (a) unnormalised displacement and (b) corresponding\nunnormalised SQUID voltage when the SQUID displacement\ndetector is tuned (\b ext= 0:30\b 0andIb= 1:20I0) to the\nworking point shown in the inset of (a). The initial cantilever\namplitudes are u0= 20 pm (blue), u0= 10 pm (red), and\nu0= 5 pm (black), which correspond to a change of \rux in the\nSQUID loop of 0 :05\b 0, 0:025\b 0, and 0:0125\b 0respectively.\n1.15 1.20 1.25 1.30 1.35 1.402468101214161820(c) )ext/)0  0.28 0.30 0.32NIb/I00.2 0.4 0.6 0.81.251.501.752.00(d)b0)ext/)02.0004.2506.5008.75011.0013.2515.5017.7520.00uN(pm)\nFIG. 5. (a) Snapshot for the time array of cantilever displace-\nments at (a) \b ext= 0:25\b 0andIb= 2:0I0(a point in the sim-\nple oscillatory regime) versus (b) a point \b ext= 0:30\b 0and\nIb= 1:20I0in the rapidly changing regime in which a sharp\ntransition state emerges until the cantilever enters the normal\nstate atu=uN. (c) calculations for the normal state posi-\ntions of speci\fc lines around \b ext= 0:30\b 0andIb= 1:20I0,\nand (d) the yellow-blue islands in the density plot indicate\na shift in the normal state positions that starts emerging at\nt=tNandu=uN.5\nThe e\u000bect of the SQUID-cantilever interaction on the\ncantilever motion can be more clearly observed by com-\nparing the time evolution of the cantilever displacement\nfor two di\u000berent bias and \rux values. The time depen-\ndent displacement for \b ext= 0:25\b 0andIb= 2:0I0\nis plotted in Fig. 5(a), and for \b ext= 0:30\b 0and\nIb= 1:20I0in Fig. 5(b). Clearly if the SQUID operating\npoint is in the rapidly changing regime (Fig. 5b) there is a\nsharp transition state as the cantilever returns to its equi-\nlibrium position. Naively if the SQUID bias is switched\nwhen the cantilever motion is large, there is an instanta-\nneous damping which can be used to modify the motion\nof the cantilever. Normal state positions, uN, for speci\fc\nlines around \b ext= 0:30\b 0andIb= 1:20I0are shown\nin Fig. 5(c). These positions are extracted when the\ncantilever enters the normal state that accounts for the\nLorentzian pro\fle in the frequency domain, and when the\namplitude starts decaying exponentially at time t=tN,\nas shown in Fig. 5(b). A more comprehensive analy-\nsis is presented in the density plot shown in Fig. 5(d).\nThe plot given in Fig. 5(c) exhibits details for one of the\nyellow-blue islands in the density plot. The islands cor-\nrespond to the intermediate regimes in the V(\b) curves.\nIts anticipated that such e\u000bect could be employed to pre-\ncisely and rapidly control the amplitude of the cantilever\ndisplacement below its initial amplitude which can be\nset by a piezo drive used to locate the eigenmodes of the\ncantilever. In other words, putting the system in such re-\ngions enables modulating the cantilever amplitude after\nisolating the system from the external actuator.\nIV. CONCLUSION\nIn conclusion, we have shown how the tuning of\nthe SQUID device a\u000bects the back-action between the\nSQUID and the doubly clamped cantilever. Speci\fcally,\nwe have quanti\fed the line shapes expected from the\nSQUID response and the corresponding cantilever dis-\nplacement. The e\u000bect can be quantitatively analysed via\nthe shift in the cantilever frequency, the line width, inten-\nsity, and shift in the position of the normal state. Direct\nsolutions for the unscaled dc SQUID equations coupled\nto the equations of motion of an integrated cantilever\nallow determination of voltage-displacement traces of a\ndisplacement detector. For a SQUID displacement de-\ntector tuned to a working point in the rapidly changing\nregion, a sharp transition state emerges and a nonlinear-\nlike response due to the emergence of such state is ob-\nserved. This state could be used to employ the system as\na self modulator for the displacement amplitude of the\ncantilever. These results should allow a clearer under-standing and manipulation of future experimental work.\nACKNOWLEDGMENTS\nThe authors would like to acknowledge Andrew Ar-\nmour for numerous conversations and bene\fcial com-\nments and N. Peretto, R. M. Smith, and R. A. Frewin for\nthe computational facilities, and E. Riordan for reading\nand discussions. 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Science 325,\n1103 (2004).\n20G. Jourdan, F. Comin, J. Chevrier, Phys. Rev. Lett., 101, 133904\n(2008).\n21M. Poot, S. Etaki, I. Mahboob, K. Onomitsu, H. Yamaguchi,\nYa. M. Blanter, and H. S. J. van der Zant, Phys. Rev. Lett. 105,\n207203 (2010).\n22S. Etaki, F. Konschelle, Ya. M. Blanter, H. Yamaguchi, and H.\nS. J. van der Zant, Nat. Commun. 41803 (2013).\n23O. Shevchuk, R. Fazio, Y. M Blanter , Phys. Rev. B 90, 205411\n(2014).\n24The supplementary Information: S. Etaki, F. Konschelle, Ya. M.\nBlanter, H. Yamaguchi, and H. S. J. van der Zant, Nat. Commun.\n41803 (2013).\n25J. Clarke and A. I. Braginski. The SQUID Handbook Vol. 1 (Wi-\nley VCH, GmbH and Co. KGaA, Weinheim, 2004)."    },    {        "title": "2107.02340v2.Theory_of_vibrators_with_variable_order_fractional_forces.pdf",        "content": "Theory of vibrators with variable-order fractional forces \nMin\ng Li 1, 2 \n1 Ocean College, Zhejiang University, Zhejiang 310012, P. R. China \n2 Village 1, East China Normal University, Shanghai 200062, PR. China \nEmails: mli@ee.ecnu.edu.cn, ming_lihk@yahoo.com, mli15@zju.edu.cn \n(Correspondence: Ocean College, Zhejiang University, China) \nURL: http://orcid.org/0000-0002-2725-353X  \nAb\nstract: In this paper, we present a theory of six classes of vibrators with variable-order fractional \nforces of inertia, damping, and restoration. The novelty and contributions of the present theory are reflected \nin six aspects. 1) Equivalent motion equations of those variable-order fractional vibrators are proposed. 2) The analytical expressions of the effective mass, damping, and stiffness of those variable-order fractional vibrators are presented. 3) The asymptotic properties of the effective mass, damping, and stiffness of a class of variable-order fractional vibrators are given. 4) The restricted effective parameters (damping ratio, damping free natural frequency, damped natural frequency, frequency ratio) of the variable-order fractional vibrators are put forward. 5) We bring forward the analytical representations of the free responses, the impulse responses, and the frequency transfer functions of those variable-order fractional vibrators. 6) We \npropose a solution to an open problem of how to mathematically explain the Rayleigh damping assumption \nbased on the present theory of variable-order fractional vibrations.  \nKeywords: Fractional vibrations, effective mass, effective damping, effective stiffness, Rayleigh \ndamping assumption. \n1.Introduction\nFractional vibrations attract the interests of researchers in various fields, ranging from physics to \nmathematics, see e.g., [1-20]. The literature for a class of fractional vibrators below is rich, \n()() () ,     1 2 ,dx tmk x t f tdt\n     (1.1) \nsee e.g., Uchaikin [1, Chap. 7], Duan [2, Eq. (3)], Duan et al. [3, Eq. (4.2)], Al-rabtah et al. [4, Eq. (3.1)], \nZurigat [5, Eq. (16)], Blaszczyk and Ciesielski [6, Eq. (1)], Blaszczyk et al. [7, Eq. (10)], Drozdov [8, Eq. (9)], Stanislavsky [9], Achar et al. ([10, Eq. (1)], [11, Eq. (9)], [12, Eq. (2)]), Tofighi [13, Eq. (2)], Ryabov \nand Puzenko [14, Eq. (1)], Ahmad and Elwakil [15, Eq. (1)], Blaszczyk [16], Tavazoei [17], Sandev and \nTomovski [18, Eq. (36)], Singh et al. [19], Eab and Lim [20], Li et al. [21, 22], Li [23, 24].  \nFollowing the terms used in the author’s recent work [23, 24], a vibrator in (1.1) is called a class I \nfractional one. A class I fractional vibrator is only with the fractional inertia force\n()() . mx tGenerally, 1 <  \n2 [1-22].arXiv:2107.02340v2 [math.DS] 7 Jul 20212\nIf a vibrator's motion equation takes the following form, it is called a class II fractional vibrator in [23, \n24]  \n2\n2() ()() () ,     0 1 .dx t dx tmc k x t f tdt dt\n      (1.2) \nA class II fractional vibrator is simply with the fractional damping force()() . cx tThe reports regarding the \nresearch of (1.2) are affluent, see e.g., Li [23, 24], Lin et al. [25, Eq.(2)], Duan [26, Eq. (31)], Alkhaldi et \nal. [27, Eq. (1a)], Dai et al. [28, Eq. (1)], Ren et al. [29, Eq. (1)], Xu et al. [30, Eq. (1)], He et al. [31, Eq. (4)], Leung et al. [32, Eq. (2)], Chen et al. [33, Eq. (1)], Deü and Matignon [34, Eq. (1)], Dră gănescu et al. \n[35, Eq. (4)], Rossikhin and Shitikova [36, Eq. (3)], Xie and Lin [37, Eq. (1)], Ren et al. [38, Eq. (1)], Yuan et al. [39, Eq. (8)], Dai et al. [40, Eq. (1)], Lin et al. [41, Eq. (1)], Naranjani et al. [42, Eq. (1)], Lim et al. [43], Matteo et al. [44], Tomovski and Sandev [45, Eq. (44)], Kumar et al. [46], Tian et al. [47, 48], He et al. \n[49, Eq. (1)], Varanis et al. [50], Pang et al. [51, Eq. (1.1)], Yu et al. [52, Eq. (2.1)], He\n et al. [53, Eq. (7)], \nSpanos and Malara [54, 55], Golmankhaneh [56], Duan et al [57, Eq. (8)].  \nWhen the motion equation of a fractional vibrator is expressed by \n33\n3() ()() () ,     1 2 , 0 1 ,dxt dxtmck x t f tdt dt\n       (1.3) \nit is called a class III fractional vibrator [23, 24]. A class III fractional vibrator is with both the fractional \ninertia force()() mx tand the fractional damping one()() . cx tFor the reports about (1.3), we refer to Gomez-\nAguilar et al. [58, Eq. (10)], Tian et al. [59], Berman and Cederbaum [60], Coronel-Escamilla et al. [61, Eq. \n(12)], Sene et al. [62, Eq. (8)], Vishwamittar et al. [63, Eq. (1)], Ismail et al. [64, Eq. (3)], Li, et al. [65]. \nThere are other three classes of fractional vibrators that are expressed by (1.4), (1.5), and (1.6), \nrespectively. \n() ()( ) ,    1 3 , 0 1 ,dx t dx tmk f tdt dt\n      (1.4) \n2\n2() ()() ,     0 1 ,dx t dx tmk f tdt dt\n     (1.5) \nand  \n() () ()( ) ,    1 3 , 0 2 , 0 1 .dx t dx t dx tmck f tdt dt dt\n        (1.6) \nThe literature regarding (1.4), (1.5),  and (1.6) is rare, except the recent work [24], to the best of my \nknowledge.  \nFollowing [24], we call (1.4) the motion equation of a class IV fractional vibrator, (1.5) for a class V \nfractional vibrator, and (1.6) a class VI fractional one. A class IV fractional vibrator is with both the arXiv:2107.02340v2 [math.DS] 7 Jul 20213\nfractional inertia force()() mx tand the fractional restoration force()() . kx tA class V fractional vibrator is \nonly with the fractional restoration force()() .kx tA class VI fractional vibrator contains the fractional \ninertia force()() , mx tthe fractional damping force()() , cx tand the fractional restoration one()()kx tfor 1 < \n < 3, 0 <  < 2, and 0   < 1, see [24].  \nConsidering variable fractional orders ():[0, ) (1, 3), ():[0, ) (1, 2), and \n():[0, ) [0,1) for (1.1) – (1.6)1, we have six classes of variable-order fractional vibrators respectively \nexpressed by (1.7) – (1.12) below. \n()\n()()() () ,dx tmk x t f tdt\n (1.7 ) \n2( )\n2( )() ()() () ,dx t d x tmc k x t f tdt dt\n  (1.8 ) \n() ()\n() ()() ()() () ,dx t dx tmck x t f tdt dt \n   (1.9 ) \n() ()\n() ()() ()() ,dx t dx tmk f tdt dt \n  (1. 10) \n2( )\n2( )() ()() ,dx t d x tmk f tdt dt\n (1. 11) \nand  \n() () ()\n() () ()() () ()() .dx t dx t dx tmck f tdt dt dt  \n   (1. 12) \nIn (1.7) – (1.12),()\n()(),dx tmdt\n()\n()(),dx tcdt\nand()\n()() dx tkdt\ndesignate variable-order fractional forces of \ninertia, damping, and restoration, respectively. The literature regarding (1.7) – (1.12) is rarely seen. \nDifferent classes of vibrators have their specific application areas. For instance, (1.7) implies a vibrator \nthat is damping free in form but m moves at()\n()() dx t\ndt\ninstead of x(t) and its displacement is x(t).  \nIn this paper, we aim at establishing a theory with respect to the variable-order fractional vibrators (1.7) \n–(1.12). Note that (1.7) – (1.11) are the special cases of (1.12). For example, when c = 0 and () = 0,\n(1.12) reduces to (1.7). Therefore, we detail the analysis of (1.12) in Sections 2-5. The results regarding\n(1.7) – (1.11) are given in Section 7 as the consequences of the results from (1.12).\n1 In structrual vibrations, effective mass or damping or stiffness is frequency-dependent [87, 90, 109].  arXiv:2107.02340v2 [math.DS] 7 Jul 20214\nThe present theory consists of a set of results in six aspects as follows. 1) The equivalent equations of \n(1.7) – (1.12) are proposed in Sections 2 and 7. 2) The analytical expressions of the effective mass, \ndamping, and stiffness of (1.7) – (1.12) are presented in Sections 2 and 7. 3) The asymptotic properties of \nits effective mass, damping, and stiffness of (1.12) are put forward in Theorems 3.1-3.3 and Corollaries 3.2 \nand 3.3. 4) The analytic expressions of the restricted effective damping ratio, natural frequencies, and \nfrequency ratio regarding (1.7) – (1.12) are brought forward in Sections 4 and 7. 5) The close form \nsolutions of free responses, impulse ones, and frequency transfer functions of (1.7) – (1.12) are given in \nSections 5 and 7. 6) A solution to the open problem of how to mathematically explaining the Rayleigh damping assumption is given in Theorem 6.1.  \nThe rest of the paper is organized as follows. In Section 2, we propose an equivalent equation of (1.12) \nand the analytic expressions of the effective mass, damping, and stiffness of (1.12). In Section 3, we present the asymptotic properties of the effective mass, damping, and stiffness of (1.12). In Section 4, we bring forward the analytic expressions of the restricted effective damping ratio, restricted effective natural \nfrequencies, and restricted effective frequency ratio of (1.12). In Section 5, the expressions of the free \nresponse, impulse one, and frequency transfer function regarding (1.12) are presented. In Section 6, we propose a mathematical explanation of the Rayleigh damping assumption. The corresponding results of (1.7) \n– (1.11) are given in Section 7, which is followed by conclusion s.\n2. Equivalent motion equation and effective mass, damping, and stiffness of fractional vibrator (1.12)\nIn this section, we first brief the preliminaries. Then, we present the equivalent motion equation of \n(1.12). Finally, we propose the analytical expressions of the effective mass, damping, and stiffness of (1.12). \n2.1. Preliminaries \nIn this research, we use the Weyl fractional derivative [66-83]. Let X() be the Fourier transform of \nx(t). Using the Weyl fractional derivative, the Fourier transform of()()vxt for v  0 is given by \n()F( ) ( ) ( ) ,vvxt i X  (2.1 ) \nwhere F the Fourier transform operator, see Uchaikin [1], Miller and Ross [82], Lavoie et al, [83]. \nLemma 2.1. Let F1() = F[ f1(t)] and F2() = F[ f2(t)]. If F[ f1(t)  f2(t)] = 0, f1(t)  f2(t) is a null function. \nIf f1(t)  f2(t) is a null function, f1(t) = f2(t) in the sense of F1() = F2() (Papoulis [84], Gelfand and \nVilenkin [85], Bracewell [86]).  \n2.2. Equivalent motion equation of (1.12) \nThe theorem proposed below gives an equivalent motion equation of (1.12). \nTheorem 2.1.  The motion equation of the fractional vibrator (1.12) is equivalently given by arXiv:2107.02340v2 [math.DS] 7 Jul 20215\n2\n()2 ()2\n2\n() 1 () 1 () 1\n()() () ( )cos cos22\n() () () ( )sin sin sin222\n()cos ( ) ( ),   1 ( ) 3,  0 ( ) 2,  0 ( ) 1.2dx tmcdt\ndx tmckdt\nkx t ft \n  \n \n  \n   \n\n \n (2.2 ) \nProof . Let X() be the Fourier transform of x(t). Rewrite (1.12) by \n() () ()\n1 () () ()() () ()() .dx t dx t dx tft m c kdt dt dt  \n    (2.3 ) \nOn one hand, (2.2) is rewritten by \n2\n() 2 () 2\n2 2\n() 1 () 1 () 1 ()() () ( )() c o s c o s22\n() () () ( ) ()sin sin sin cos ( ).222 2dx tft m cdt\ndx tmck k x tdt \n     \n    \n \n  (2.4 ) \nDoing F[ f1(t)] yields \n() () ()\n1F[ ( )] ( ) ( ) ( ) ( ).ft m i c i k i X       (2.5) \nOn the other side, doing F[ f2(t)] produces \n() ()\n2\n() () () ()() ()F[ ( )] cos cos ( )22\n() () () ()sin sin sin ( ) cos ( ),222 2ft m c X\nim c k X k X \n     \n     \n  (2.6 ) \nwhere 1. i Rewriting the above yields \n() ()\n2\n() () () ()() ()F[ ( )] cos sin ( )22\n() () () ()cos sin ( ) cos sin ( ).22 22ft m i m X\nci c X ki k X \n    \n     \n    (2.7 ) \nSince the principal values of(),i (),iand()iare given by \n()\n()\n()() ()cos sin ,22\n() ()cos sin ,22\n() ()cos sin ,22ii\niiii\n\n \n \n \n\n(2.8 ) \nwe can write (2.7) by arXiv:2107.02340v2 [math.DS] 7 Jul 20216\n() () ()\n2F[ ( )] ( ) ( ) ( ) ( ).ft m i c i k i X       (2.9) \nFrom (2.5) and (2.9), we see that F[ f1(t)] = F[ f2(t)]. Thus, F[ f1(t)  f2(t)] = 0. Hence, f1(t) = f2(t) as f1(t)  f2(t) \nis a null function.  \n2.3. Effective mass, damping, and stiffness of (1.12) \nNote that, in (1.12), the unit of()() mx tis not Newton. Thus, the primary mass m is not an effective \nquantity to be an inertia measure unless  = 2. Also, the primary damping coefficient c does not effectively \nquantify a damping measure in (1.12) since the unit of()() cx tis not Newton if   1. Similarly, the primary \nstiffness k in (1.12) is not an effective quantity to be a restoration measure as the unit of()()kx tis non-\nNewton when   0. In vibrations, the effective mass of a vibrator is the coefficient of x(t), the effective \ndamping is the coefficient of x(t), and the effective stiffness is that of x(t) (Harris [87], Den Hartog [88], \nTimoshenko [89], Palley et al. [90], Nakagawa and Ringo [91]). Now, we present the effective mass, \ndamping, and stiffness of (1.12) by the theorems below. \nTheorem 2.2.  Let meff be the effective mass of the fractional vibrator (1.12). Then, \n()2 ()2\neff() ()cos cos .22mm c     (2.10) \nProof . The above is the coefficient of x(t) in (2.2). Thus, meff is an inertia measure of the vibrator (2.2) \nand equivalently (1.12).  \nThough the unit of meff is not kg unless () = 2 and () = 1, it measures the inertia of a vibrator (2.2) \nand equivalently (1.12). The quantity meff reduces to the primary m when () = 2 and () = 1. \nTheorem 2.3.  Denote by ceff the effective damping coefficient for the fractional vibrator (1.12). Then, \n() 1 () 1 () 1\neff() () ()sin sin sin .222cm c k        (2.11) \nProof . The above is the coefficient of  x(t) in (2.2) that is equivalently to (1.12).  \nAlthough the unit of ceff is not that of the standard damping, that is, N  m1  s, it measures the \ndamping coefficient of a vibrator (2.2) and equivalently (1.12). The quantity ceff degenerates to c if () = \n2, () = 1, and () = 0. \nTheorem 2.4.  Let keff be the effective stiffness of the fractional vibrator (1.12). Then, \n()\neff()cos .2kk (2.1 2) \nProof . In (2.2), the above is the coefficient of x(t) in (2.2) which is equivalently to (1.12).  \nThe unit of keff is not N  m1, but it measures the restoration of the vibrator (2.2) and equivalently \n(1.12). The quantity keff reduces to the primary k if () = 0. arXiv:2107.02340v2 [math.DS] 7 Jul 20217\n3. Asymptotic properties of effective mass, damping, and stiffness of variable-order fractional\nvibrator (1.12) \nAs meff, ceff, and keff are the functions of vibration frequency , we may write them by meff(), ceff(), \nand keff(). \nTheorem 3.1  (Asymptotic property for   ). If 2 < () < 3, meff()   when   . On the \nother side,  meff()  0 when 1 < () < 2 if   . That is, \neff,2 ( ) 3lim ( ) .0,1 ( ) 2m\n  (3.1) \nProof . Because() 2lim\n\nand()cos2< 0 if 2 < () < 3. On the other hand,() 2lim 0\n\n \nfor 1 < () < 2. Besides,() 2lim 0\n\nfor 0 < () < 2. Thus, the above is valid. \nTheorem 3.2 (Negative mass).  If 0 < () < 1 and 2 < () < 3, meff   if   0. \nProof . Note that()cos2> 0 and2\n0lim\n\nfor 0 < () < 1. Besides,() 2\n0lim 0\n\nfor 2 < ()\n< 3. Thus, for 0 < () < 1 and 2 < () < 3, we have \neff0lim ( ) .m\n\n (3.2 ) \nThe proof finishes2.  \nThe above implies that the range of meff() is ( , ) in general. A few of plots of meff() are shown \nin Fig. 3.1. \n0 2468 1 00.751.52.253Effective mass\n0 2468 1 00.751.52.253Effective mass\n\n(a)    (b\n) \n2 Refer [24, Chap. 15] and references therein about negative mass. arXiv:2107.02340v2 [math.DS] 7 Jul 20218\n0 2468 1 01.534.56Effective mass\n0 2468 1 00.751.52.253Effective mass\n\n(c)    (d\n) \n0 0.005\n4000320024001600800Effective mass\n01 02 051015Effective mass\n\n(e)   \n                                           (f) \nFig. 3.1. Plots of meff() with m = 1, c = 1, and k = 1. (a).  = 1. Solid:  = 1.3. Dot:  = 1.6. Dash:  = \n1.9. (b).   = 1. Solid:  = 2.3. Dot:  = 2.6. Dash:  = 2.9. (c).  = 1.3. Solid:  = 1.3. Dot:  = 1.6. Dash: \n = 1.9. (d).  = 1.3. Solid:  = 2.3. Dot:  = 2.6. Dash:  = 2.9. (e). Observing negative meff at small  \nwhen  = 0.9. Solid:  = 1.3. Dot:  = 1.6. Dash:  = 1.9. (f). () = 1.10 + 1.89|sin( )|, () = 1 +\n0.99|cos()|.\nWe now consider the asymptotic property of ceff for   . \nTheorem 3.3.  If   ,  \neff,1 ( ) 2,0 ( ) 2.,2 ( ) 3 ,0 ( ) 1c \n        (3.3 ) \nProof . Since 0  () < 1, we have() 1lim 0.\n\nIn addition, because()sin 02when 0 < () <\n2 and ()sin 02if 1 < () < 2,  ceff()   if    for 1 < () < 2 and 0 < () < 2. Additionally,arXiv:2107.02340v2 [math.DS] 7 Jul 20219\nbecause() 1lim 0\n\n when 0 < () < 1 but()sin 02if 2 < () < 3, we have ceff()   for 2 < \n() < 3 and 0 < () < 1 when   .  \nThe above exhibits that  < ceff < 3. \nCorollary 3.1.  The fractional vibrator (1.12) may be self-vibrated and accordingly non-stable if  2 < \n() < 3 and 0 < () < 1. \nProof . As ceff may be negative if  2 < () < 3 and 0 < () < 1, a vibrator (1.12) may be self-vibrated \nand non-stable in that case.  \nCorollary 3.2.  If 1 < () < 3, 0 < () < 1 and  is large enough, we have \n() 1\neff()sin .2cm (3.4 ) \nIn addition, for 1 < () < 3 and 0 < () < 1, if  is small enough, \n() 1 () 1\neff() ()sin sin .22cc k     (3.5) \nProof . If 1 < () < 3, 0 < () < 1 and  is sufficiently large,() 10and() 10.Thus, (3.4) \nholds. On the other hand, if  is small enough,() 10.Thus, (3.5) results.  \nFig. 3.2 gives a few illustrations of ceff. \n01 02 03 051015Effective damping\n01 02 03 0\n800600400200200Effective damping\n\n(a)\n  (b) \n3 For several particular cases of negative damping in engineering, refer Den Harton [88] and Nakagawa and Ringo [91]. For  \ngeneral cases of negative damping in fractional vibrations with constant fractional orders, refer to [24]. arXiv:2107.02340v2 [math.DS] 7 Jul 202110\n01 02 03 0\n800600400200200Effective damping\n01 02 03 0\n402020Effective damping\n\n(c) \n                                                  (d) \nFig. 3.2. Plots of ceff with m = 1, c = 1, and k = 1. (a).  = 0.3,  = 0.3. Solid:  = 1.3. Dot:  = 1.6. \nDash:  = 1.9. (b). Negative damping with constant fractional orders,  = 0.3,  = 0.3. Solid:  = 2.3. Dot: \n = 2.6. Dash:  = 2.9. (c). Negative damping with variable fractional orders, () = 1.10 + 1.89|cos(0.5 )|,\n() = 1 + 0.99|sin( )|, () = 0.99|sin( )|. (d). () = 1 + 0.99|sin( )|, () = 0.99|cos( )|. Solid:  = 1.3.\nDot (negative damping):  = 2.3.\nCorollary 3.3.  For keff, we have keff()  0.Besides,()lim\n\nand()\n0lim 0.\n\n  \nProof . The proof is straightforward as 0  () < 1 in (2.12).  \n02 55 07 5 1 0 02.557.510Effective stiffness\n0 2.5 5 7.5 100.751.52.253Effective stiffness\n\n(a)\n  (b) \nFig. 3.3. Plots of keff with k = 1. (a). Solid:  = 0.3. Dot:  = 0.6. Dash:  = 0.9. (b). () = 0.99|cos( )|.  \nWe show its plots in Fig. 3.3. arXiv:2107.02340v2 [math.DS] 7 Jul 202111\n4. Restricted effective damping ratio, natural frequencies, and frequency ratio of variable-order\nfractional vibrator (1.12) \nThe effective parameters meff, ceff, and keff are directly obtained from (2.2). In this section, we address \nthe restricted effective damping ratio, natural frequencies, and frequency ratio. By restricted, we mean that \nadditional conditions are considered for expressing those parameters.  \n4.1. Restricted effective damping ratio of (1.12) \nAs can be seen from Section 3, meff  (, ), ceff  (, ), and k eff  (0, ). Let eff be the restricted \neffective damping ratio (effective damping ratio for short) of the fractional vibrator (1.12). Here, we only \nconsider eff in the case of meff > 0 from a view of engineering. With that restriction, we define eff by \neff\neff\neff eff.\n2c\nmk  (4.1) \nTheorem 4.1.  The quantity eff is in the form \n() 1 () 1 () 1\neff\n() 2 () 2 ()() () ()sin sin sin222.\n() () ()2c os c os c os22 2mck\nmc k  \n    \n\n  \n\n\n(4.2) \nProof . Substituting meff, ceff, and keff into (4.1) results in (4.2).  \nAlthough eff is not dimensionless in general, it effectively reflects the damping ratio of (2.2) and \nequivalently (1.12) in the case of meff > 0. It is dimensionless if () = 2, () = 1, and () = 0. In that \ncase, it equals to the standard damping ratio sinceeff () 2 ,() 1 , () 0.\n2c\nmk     \nA few of plots of eff are indicated in Fig. 4.1. \n0 2.5 5 7.5 107.51522.530Effective damping ratio\n0 2.5 5 7.5 10\n30201010Effective damping ratio\n\n(a)\n  (b) arXiv:2107.02340v2 [math.DS] 7 Jul 202112\n0 2.5 5 7.5 10\n2001255025100Effective damping ratio\n0 2.5 5 7.5 10\n10062.52512.550Effective damping ratio\n\n(c\n)                                                                   (d) \nFig. 4.1. Plots of eff for m = 1 = k  = c = 1. (a). Damping ratio for some range of frequency with \nconstant fractional orders.   = 0.3,  = 0.3. Solid:  = 1.3. Dot:  = 1.6. Dash:  = 1.9. (b). Negative \ndamping ratio for some range of frequency with constant fractional orders.   = 1.9,  = 0.3. Solid:  = 2.3. \nDot:  = 2.6. Dash:  = 2.9. (c). Negative damping ratio for some range of frequency with variable \nfractional orders. () = 1.10 + 1.89|cos(0.1 )|, () = 1 + 0.99|sin( )|, () = 0.99|cos( )|. (d). Negative \ndamping ratio for some range of frequency with variable fractional orders. () = 1.10 + 1.89|cos(0.1 )|, \n() = 1 + 0.99|sin( )|, and () = 0.99|exp( )|. \n4.2. Restricted effective damping free natural frequency of (1.12) \nFor the fractional vibrator (1.12), with the restriction meff > 0 from the point of view of engineering, we \ncoin a term restricted effective damping free natural frequency (effective damping free natural frequency in \nshort) to the quantity defined by \neff\neffn\neff.k\nm (4.3 ) \nTheorem 4.2.  The quantity effn is in the form \n()\neffn\n() 2 () 2()cos2.() ()cos cos22k\nmc\n \n \n (4.4) \nProof . Substituting meff and keff into (4.3) yields (4.4).  \nThough the unit of effn is not rad/s, but it in functional takes the form of the conventional damping \nfree natural frequency. Its unit reduces to rad/s if () = 2, () = 1, and () = 0 as \neffn n () 2 ,() 1 , () 0.k\nm    (4.5 ) arXiv:2107.02340v2 [math.DS] 7 Jul 202113\nIn general, effn is a function of . Thus,  effn = effn(). Fig. 4.2 illustrates a few plots of effn. For variable \nfractional orders, effn() is complicated, see Figs. 4.2 (e) and (f). \n01 02 03 02040Effetive damping free natur fre\n01 02 03 0510Effective damping free natur fre\n\n(a\n)    (b) \n01 02 03 020406080Effective damping free natur fre\n01 02 03 012Effective damping free natur fre\n\n(c\n)    (d) \n01 02 03 02468Effective damping free natur fre\n01 02 03 0510Effective damping free natur fre\n\n(e)\n   (f) \nFig. 4.2. Plots of effn with m = 1 and k = 1, c  = 0.2. (a). effn with constant fractional orders.  = 0.3,  \n= 0.3. Solid:  = 1.3. Dot:  = 1.6. Dash:  = 1.9. (b).  effn with constant fractional orders.  = 1.3,  = 0.3. \nSolid:  = 1.3. Dot:  = 1.6. Dash:  = 1.9. (c). effn with constant fractional orders.  = 0.3,  = 0.3. Solid: \n = 2.3. Dot:  = 2.6. Dash:  = 2.9. (d).  effn with constant fractional orders.  = 1.3,  = 0.3. Solid:  =\n2.3. Dot:  = 2.6. Dash:  = 2.9. (e).  effn with variable fractional orders. () = 1.10 + 1.89|cos(0.1 )|,arXiv:2107.02340v2 [math.DS] 7 Jul 202114\n() = 1 + 0.99|sin( )|, () = 0.99|cos( )|. (f). effn with variable fractional orders. () = 1.10 + \n1.89|cos(0.1 )|, () = 1 + 0.99|sin( )|, () = 0.99|exp( )|. \n4.3. Restricted effective damped natural frequency of (1.12) \nNow we introduce a term restricted effective damped natural frequency (effective damped natural \nfrequency in short) for (1.12). Denote it by effd. With the restriction | eff|  14, we define it by \n2\neffd effn eff eff 1 ,     1.     (4.6 ) \nTheorem 4.3.  The representation of effd is in the form \n()\neffd\n() 2 () 2\n2\n() 1 () 1 () 1\n() 2 () 2 ()()cos2\n() ()cos cos22\n() () ()sin sin sin2221.\n() () ()2 cos cos cos22 2k\nmc\nmck\nmc k\n \n  \n  \n \n  \n  \n\n\n\n\n (4.7 ) \nProof . Substituting eff and effn into (4.6) yields (4.7).  \nNote that if () = 2, () = 1, and () = 0, effd degenerates the conventional d with the unit of \nrad/s. It is generally a function of . Hence, effd = effd(). \n4.4. Restricted effective frequency ratio of (1.12) \nIntroduce a term restricted effective frequency ratio (effective frequency ratio in short) for the \nfractional vibrator (1.12). Denote it by eff. Define it by \neff\neffn. (4.8 ) \nBoth effn and eff are restricted by meff > 05.  \nTheorem 4.4.  The representation of eff is given by \n() 2 () 2\nn\neff\n()() ()cos 2 cos22,()cos2 \n  \n  (4.9) \nwhere\nn.  \n4 Small damping is commonly assumed in vibration engineering, see [24, 87-92, 109]. The quantity effd is restricted in the  \nsense of small damping. \n5 That restriction of meff > 0 is taken from the point of view of vibration engineering. arXiv:2107.02340v2 [math.DS] 7 Jul 202115\nProof . Substituting effn into (4.8) results in (4.9).  \nThe quantity eff reduces to  when () = 2, () = 1, and () = 0. Its plots are indicated in Fig. 4.3. \n01 02 03 05101520Effective frequency ratio\n01 02 03 0204060Effective frequency ratio\n\n(a\n)   (b) \n01 02 03 0255075100Effective frequency ratio\n\n(c) \nFi\ng. 4.3. Plots of eff with m = 1, k  = 1, c = 0.2. (a). Effective frequency ratio with constant fractional \norders.  = 0.3,  = 0.3. Solid:  = 1.3. Dot:  = 1.6. Dash:  = 1.9. (b). Effective frequency ratio with \nconstant fractional orders.  = 0.3,  = 0.3. Solid:  = 2.3. Dot:  = 2.6. Dash:  = 2.9. (c). Effective \nfrequency ratio with variable fractional orders. () = 1.10 + 1.89|cos(0.1 )|, () = 1 + 0.99|sin( )|, () \n= 0.99|cos( )|. \n5.Restricted responses (free, impulse) of the variable-order fractional vibrator (1.12)\nHere and below, we use two restrictions, namely, meff > 0 and |eff|  1, unless otherwise stated.\n5.1. Equivalent representation of (2.2) \nBased on m eff(), ceff(), and k eff(), we rewrite (2.2) by \n2\neff eff eff 2() ()() () .dx t d x tmc k x t f tdt dt  (5.1 ) \nThe above can be further written by arXiv:2107.02340v2 [math.DS] 7 Jul 202116\n2\n2\neff effn effn 2\neff() () ()2( ) .dx t d x t ftxtdt dt m   (5.2 ) \n5.2. Restricted free response \nLet x(t) be the restricted free response (free response for short) of the fractional vibrator (1.12). It is a \nsolution to the differential equation given by \n2\n2\neff effn effn 2\n00() ()2( )0,    1 ( ) 3,0 ( ) 2,0 ( ) 1.\n(0) , (0) .dx t d x txtdt dt\nxx x v       \n  (5.3) \nThe theorem proposed below provides its solution.  \nTheorem 5.1.  The free response x(t) to (1.12) is given by \neff effn 0 eff effn 0\n0 effd effd\neffd( ) cos sin ,    0.t vxxt e x t t t   \n (5.4) \nProof . According to Theorem 2.1, (2.2) is equivalent to (1.12). On the other hand, (5.2) is equivalent to \n(2.2). Since (5.3) is a standard vibration equation in form, (5.4) holds.  \nFig. 5.1 shows a few plots of x(t).  \n04 81 2 1 6 2 0\n10.40.20.81.42\ntx(t)04 81 2 1 6 2 0\n10622610\ntx(t)\n(a)    (b\n) \n0 3 6 9 12 15\n104281420\ntx(t)\n0 3 6 9 12 15\n104281420\ntx(t)\n(c)  (\nd) arXiv:2107.02340v2 [math.DS] 7 Jul 202117\nFig. 5.1. Plots of x(t) for x 0 = 1, v 0 = 1, m  = 1 and k = 1. (a). For ω = 1.1 and c = 0.2,  = 0.3,  = 0.3. \nSolid:  = 1.3. Dot.  = 1.6. Dash:  = 1.9. (b). Self-vibration. x(t) with constant fractional orders for ω = \n1.1 and c  = 0.2,  = 1.9,   = 0.3.  = 2.4. (c). x (t) with variable fractional orders for   (0, 1) and c = 0.2, \n() = 1.10 + 1.89|cos(0.1 )|, () = 1 + 0.99|sin( )|, () = 0.99|cos( )|. (d). x(t) with variable fractional \norders for   (0, 1) and c = 1.2, () = 1.10 + 1.89|cos(0.1 )|, () = 1 + 0.99|sin( )|, () = \n0.99|cos()|. \nIn time-frequency plane, we have \nx(t) = x(t, ). (5.5) \n5.3. Restricted impulse response \nThe present theorem below gives the expression of the restricted impulse response (impulse response \nfor short) to the fractional vibrator (1.12). \nTheorem 5.2.  Let h(t) be the impulse response to the fractional vibrator (1.12). It is a solution to the \nfollowing fractional differential equation: \n() () ()\n() () ()() () ()() ,     1 ( ) 3 , 0 ( ) 2 , 0 ( ) 1dh t dh t dh tmck tdt dt dt  \n              (5.6) \nwith zero initial conditions. The  expression of h(t) is given by \neff effn\neffd\neff effd1() s i n ,     0 .tht e t tm (5.7 ) \nProof . According to Theorem 2.1, (5.6) is equivalently expressed by \n2\neff eff eff 2() ()() () .dht d htmc k h t tdt dt  \nThe\n above is a standard vibration equation in form. Its solution is (5.7).  \nFig. 5.2 shows a few plots of h(t).  \n0 3 6 9 12 15\n0.50.20.10.40.71\nth(t)0 3 6 9 12 15\n10622610\nth(t)\n(a)  (\nb) arXiv:2107.02340v2 [math.DS] 7 Jul 202118\n0 3 6 9 12 15\n20.80.41.62.84\nth(t)0 3 6 9 12 15\n10.60.20.20.61\nth(t)\n(c)  \n                                                              (d) \nFig. 5.2. Plots of h(t) for m  = 1 and k = 1. (a). h (t) with constant fractional orders for ω = 1.1 \nand c = 0.2,  = 0.3,  = 0.3. Solid:  = 1.3. Dot:  = 1.6. Dash:  = 1.9. (b). Self-vibration.  h(t) \nwith constant fractional orders for ω = 1.1 and c = 0.2,  = 2.5,  = 1.5,  = 0.3. (c). h (t) with \nvariable fractional orders for   (0, 1) and c = 0.2, () = 1.10 + 1.89|cos(0.1 )|, () = 1 + \n0.99|sin()|, () = 0.99|cos( )|. (d). h(t) with variable fractional orders for   (0, 1) and c = \n1.2, () = 1.10 + 1.89|cos(0.1 )|, () = 1 + 0.99|sin( )|, () = 0.99|cos( )|. \nIn time-frequency plan, h(t) may be written as \nh(t) = h(t, ω). (5.8)\n5.4. Restricted frequency transfer function \nTheorem 5.3.  Let H () be the restricted frequency transfer function (frequency transfer function in \nshort) of the fractional vibrator (1.12). It is given by \n2\neff eff eff eff1() .\n12H\nki\n \n(5.9) \nProof . Doing the Fourier transform of (5.7) produces the above.   \nUsing polar coordinates, we have \n()() () ,iHH e  (5.10) \nwhere the amplitude-frequency response | H()| is in the form \n2 2 2eff\neff eff eff11() ,\n12Hk\n \n(5.11) \nand the phase-frequency response function () is expressed by arXiv:2107.02340v2 [math.DS] 7 Jul 202119\n2\n1 eff\n2 2 2\neff eff eff1() c o s .\n12\n  \n (5.12) \nFigs. 5.3 and 5.4 illustrate several plots of | H()| and (), respectively. \n0 0.4 0.8 1.2 1.6 2510Effective amp fre transfer function\n0 0.4 0.8 1.2 1.6 2510Effective amp fre transfer function\n\n(a)  (\nb) \n0 0.4 0.8 1.2 1.6 2510Effective amp fre transfer function\n0 0.4 0.8 1.2 1.6 22468Effective amp fre transfer function\n\n(c)  (d\n) \nFig. 5.3. Plots of | H()| when m = 1 and k  = 1. (a). | H()| with constant fractional orders for ω = 1.1 \nand c = 0.2,  = 1.3,  = 0.3. Solid:  = 1.3. Dot:  = 1.6. Dash:  = 1.9. (b). | H()| with constant fractional \norders for ω = 1.1 and c = 0.2,  = 1.3,  = 0.3. Solid:  = 2.3. Dot:  = 2.6. Dash:  = 2.9. (c). | H()| with \nvariable fractional orders for   (0, 2) and c = 0.2, () = 1.10 + 1.89|cos( )|, () = 1 + 0.99|sin( )|, \n() = 0.99|cos( )|. (d). | H()| with variable fractional orders for   (0, 2) and c = 1.2, () = 1.10 + \n1.89|cos()|, () = 1 + 0.99|sin( )|, () = 0.99|cos( )|. arXiv:2107.02340v2 [math.DS] 7 Jul 202120\n0 0.4 0.8 1.2 1.6 2123Effective phase frequency function\n0 0.4 0.8 1.2 1.6 224Effective phase frequency function\n\n(a)   (\nb) \n0 0.4 0.8 1.2 1.6 224Effective phase frequency function\n0 0.4 0.8 1.2 1.6 2123Effective phase frequency function\n\n(c\n)    (d) \nFig. 5.4. Plots of () when m = 1, k  = 1. (a). () with constant fractional orders for ω = 1.1 and c = \n0.2,  = 1.3,  = 0.3. Solid:  = 1.3. Dot:  = 1.6. Dash:  = 1.9. (b). () with constant fractional orders \nfor ω = 1.1 and c  = 0.2,  = 1.3,  = 0.3. Solid:  = 2.3. Dot:  = 2.6. Dash:  = 2.9. (c). () with variable \nfractional orders for   (0, 2) and c = 0.2, () = 1.10 + 1.89|cos( )|, () = 1 + 0.99|sin( )|, () = \n0.99|cos()|. (d). () with variable fractional orders for   (0, 2) and c = 1.2, () = 1.10 + 1.89|cos( )|, \n() = 1 + 0.99|sin( )|, () = 0.99|cos( )|. \n6.Mathematical explanation of Rayleigh damping assumption\nRayleigh introduced his damping assumption in [92]. It has been widely used in structural mechanics,\nsee\n e.g., Palley et al. [90], Trombetti and Silvestri [93], Poul and Zerva [94], Nåvik et al. [95], Park et al. \n[96], Hussein et al. [97], Sigmund and Jensen [98], Chen et al. [99], Battisti et al. [100], Cox et al. [101], \nTisseur and Meerbergen [102], Chu et al. [103], Fay et al. [104], Naderian et al. [105], Tian et al. [106], \nIovane et al. [107], Kouris et al. [108], Jin and Xia [109], just mentioning a few. In the field, it is well arXiv:2107.02340v2 [math.DS] 7 Jul 202121\nknown that how to give a mathematical explanation of the Rayleigh damping assumption is an open \nproblem.  \nRecently, the author presented a mathematical explanation of the Rayleigh damping assumption when \n, , and  are constants [24, Chap. 14]. In this paper, we address the further explanation by taking into \naccount the functions (), (), and () instead of , , and  being constants.  \nLet cr be the Rayleigh damping. Its standard form is expressed by \ncray = am + bk,  (6.1) \nwhere a and b are frequency-dependent parameters. Precisely, a is proportional to  while b is inversely \nproportional to . With the variable-order fractional vibrator (1.12), we use its effective damping to give a \nmore general explanation of the Rayleigh damping assumption.  \nTheorem 6.1.  Let c = 0 in (2.11). Then, ceff in (2.11) reduces to the generalized Rayleigh damping, \ndenoted by c gray, in the form \n() 1 () 1\ngray() ()sin sin .22cm k     (6.2) \nProof.  Let c  = 0 in (2.11). Then, c eff becomes the above. The above exhibits that cgray is proportional to \nm with the coefficient a  in the form \n() 1 ()() s i n .2aa (6.3 ) \nBesides, cgray is proportional to k with the coefficient b in the form \n() 1 ()() s i n .2bb (6.4 ) \nThus, we have \ngray () () . ca m b k   (6.5) \nBecause a() is proportional to  while b() is inversely proportional to . Eq. (6.2) is a mathematical \nexpression that is consistent with the Rayleigh damping assumption.   \nNote that a()  0 and b()  0 for 1 < () < 2 and 0  () < 1. Accordingly, cgray  0 in that cases \nof 1 < () < 2 and 0  () < 1. On the other hand, if 2 < () < 3 and 0  () < 1, a() may be \nnegative and b ()  0. In that case, cgray may be negative. Because () and () are varying in terms of , \n(6.2) is a general form of the Rayleigh damping. In fact, when both () and () are constants  and , \ncgray reduces to a specific form of the standard Rayleigh damping in the form  \n11\nr sin sin .22cm k   (6.6) \nFig. 6.1 shows the plots of a() and b() while Fig. 6.2 demonstrates the plots of cgray. arXiv:2107.02340v2 [math.DS] 7 Jul 202122\n0 2.5 5 7.5 102010010203040a\n0 2.5 5 7.5 1000.170.330.50.670.831b   \n\n(a\n)                                                             (b) \nFig. 6.1. Plots of a() and b () with m = k = 1. (a). a () with variable fractional order. () = 1.80 + \n1.19|sin()|. (b). b() with variable fractional order. () = 0.99|cos( )|. \n0 2.5 5 7.5 102010010203040Generalized Rayleigh damping   \n0 2.5 5 7.5 1010.500.511.52Generalized Rayleigh damping\n\n(a) \n                                                           (b)  \nFig. 6.2. Plots of c gray with m = k  = 1. (a). c gray with variable fractional orders.  () = 1.80 + \n1.19|sin()| and () = 0.99|cos( )|. (b).  cgray with variable fractional orders. () = 2.99exp( ) and () \n= 0.99|cos( )|. \n7. Results for the variable-order fractional vibrators (1.7) – (1.11)\nWe now propose the results regarding the variable-order fractional vibrators expressed by (1.7) –\n(1.11). Mathematically, (1.7) – (1.11) are the special cases of (1.12). However, from a view of engineering, \nstudying each class from (1.7) to (1.11) is meaningful since each has its specific application area. For instance, (1.7) is for the case of damping free structures with variable-order fractional inertia force and \nconventional restoration one. The class (1.10) stands for the case of damping free structures with variable-\norder fractional inertia force and variable-order fractional restoration one. The importance of the results in \nSection 7 lies in representing the results for each class from (1.7) – (1.11) based on the results with respect \nto (1.12) addressed in Sections 2 – 5. In what follows, x\n0 and v0 are initial conditions. \n7.1. Results regarding variable-order fractional vibrator (1.7) \nConsider (1.7). Letting c = 0 and () = 0 in (2.2) yields the equivalent equation of (1.7) in the form arXiv:2107.02340v2 [math.DS] 7 Jul 202123\n2\n()2 () 1\n2() ( ) () ( )cos sin ( ) ( ),   1 ( ) 3.22dx t d x tmm k xt ftdt dt       (7.1) \nLet meff1 be the effective mass of the fractional vibrator (1.7). Then, letting c = 0 in (2.10) produces \n() 2\neff1()cos .2mm (7.2 ) \nDenote by ceff1 the effective damping of the fractional vibrator (1.7). Then, letting c = () = 0 in (2.11) \nresults in  \n() 1\nff1()sin .2ecm (7.3 ) \nLet effn1 be the restricted effective damping free natural frequency of (1.7) for meff1 > 0. It is given by \neffn1\n()2eff1.()cos2kk\nmm\n(7.4) \nLet eff1 be the restricted effective damping ratio of (1.7) under the condition of meff1 > 0. Then, from (4.1), \nwe have \n() 1\neff1\neff1\n() 2 eff1()sin2.\n2( )2c os2mc\nmkmk\n\n\n\n\n (7.5) \nConsider the restricted effective damped natural frequency for (1.7) when meff1 > 0. Denote it by effd1. \nWith the restriction | eff1|  1, we have \n2\neffd1 effn1 eff1 1.   (7.6) \nLet eff1 be the restricted effective frequency ratio of (1.7). It is, according to (4.8), given by \neff1\neffn1.  (7.7) \nBoth effn and eff are restricted by meff > 0.  \nDenote by x1(t) the free response to (1.7). According to Theorem 5.1, we have \neff1 effn1 0 eff1 effn1 0\n1 0 effd1 effd1\neffd1( ) cos sin ,    0.t vxxt e x t t t    \n (7.8) \nLet h1(t) be the impulse response to (1.7). Based on Theorem 5.2, we have arXiv:2107.02340v2 [math.DS] 7 Jul 202124\neff1 effn1\n1e ffd1\neff1 effd11() s i n ,     0 .tht e t tm  (7.9) \nLet H1() be the restricted frequency transfer function of the fractional vibrator (1.7). Following Theorem \n5.3, we have \n1 2\neff1 eff1 eff11() .\n12H\nki\n \n (7.10) \n7.2. Results regarding variable-order fractional vibrator (1.8) \nTaking into account (1.8), we substitute () by 2 and () with 0 in (2.2). Then, we have the \nequivalent equation of (1.8) given by \n2\n() 2 () 1\n2() ( ) () ( )cos sin ( ) ( ),   0 ( ) 2.22dx t d x tmc c k x t f tdt dt         (7.11) \nWrite the above by \n2\neff2 eff2 2() ()() () ,dx t d x tmc k x t f tdt dt  (7. 12) \nwhere meff2 and c eff2 are the effective mass and damping of the fractional vibrator (1.8), respectively. From \n(7.11) and (7.12), we see that \n()2\neff2()cos ,2mm c (7. 13) \n() 1\neff 2()sin .2cc (7. 14) \nDenote by effn2 the restricted effective damping free natural frequency of (1.8) when meff2 > 0. It is \neffn2\neff2.k\nm (7. 15) \nLet eff2 be the restricted effective damping ratio of (1.8) in the case of meff2 > 0. Then,  \neff2\neff2\neff2.\n2c\nmk  (7.16) \nDenote by effd2 the restricted effective damped natural frequency of (1.8) for meff2 > 0. Using the \nrestriction | eff2|  1, we have \n2\neffd2 effn2 eff2 1.    (7.17) arXiv:2107.02340v2 [math.DS] 7 Jul 202125\nLet eff2 be the restricted effective frequency ratio of (1.8). It is given by \neff2\neffn2.  (7.18) \nLet x2(t) be the free response to (1.8). With Theorem 5.1 or from (7.12), we have \neff2 effn2 0 eff2 effn2 0\n2 0 effd2 effd2\neffd2( ) cos sin ,    0.t vxxt e x t t t    \n (7.19) \nDenote by h2(t) the impulse response to (1.8). Following Theorem 5.2 or from (7.12), we have \neff2 effn2\n2 effd2\neff2 effd21() s i n ,     0 .tht e t tm  (7.20) \nLet H2() be the restricted frequency transfer function of the fractional vibrator (1.8). With Theorem 5.3 or \nby doing the Fourier transform of h2(t), we have \n2 2\neff2 eff2 eff21() .\n12H\nki\n \n (7.21) \n7.3. Results regarding variable-order fractional vibrator (1.9) \nConsider (1.9). Replacing () with 0 in (2.2) results in \n2\n()2 ()2\n2\n() 1 () 1() () ( )cos cos22\n() () ( )sin sin22\n( ) ( ) ,  1 ( ) 3 , 0 ( ) 2 .dx tmcdt\ndx tmcdt\nkx t f t \n  \n \n \n\n\n   (7.22) \nThe above is the equivalent equation of (1.9). Let meff3 and c eff3 be the effective mass and damping of the \nfractional vibrator (1.9), respectively. Write the above by \n2\neff3 eff3 2() ()() () ,dx t d x tmc k x t f tdt dt  (7. 23) \nBy comparing (7.22) to (7.23), we have \n()2 ()2\neff3() ()cos cos ,22mm c     (7.24) \n() 1 () 1\nef f3() ()sin sin .22cm c     (7.25) \nLet effn3 be the restricted effective damping free natural frequency of (1.9) for meff3 > 0. It is given by arXiv:2107.02340v2 [math.DS] 7 Jul 202126\neffn3\neff3.k\nm (7. 26) \nLet eff3 be the restricted effective damping ratio of (1.9) when meff3 > 0. Then, \neff3\neff3\neff3.\n2c\nmk  (7.27) \nDenote by effd3 the restricted effective damped natural frequency of (1.9) when meff3 > 0. With | eff3|  1, \nwe have \n2\neffd3 effn3 eff3 1.   (7.28) \nDenote by eff3 the restricted effective frequency ratio of (1.9). It is in the form \neff3\neffn3.  (7.29) \nDenote x3(t) the free response to (1.9). Since (7.23) is a standard vibration equation, x3(t) is given by \neff3 effn3 0 eff3 effn3 0\n3 0 effd3 effd3\neffd3( ) cos sin ,    0.t vxxt e x t t t    \n (7.30) \nDenote by h3(t) the impulse response to (1.9). Following Theorem 5.2 or from (7.23), we have \neff3 effn3\n3 effd3\neff3 effd31() s i n ,     0 .tht e t tm  (7.31) \nLet H3() be the restricted frequency transfer function of the fractional vibrator (1.9). Doing the Fourier \ntransform of h3(t) yields \n3 2\neff3 eff3 eff31() .\n12H\nki\n \n (7.32) \n7.4. Results regarding variable-order fractional vibrator (1.10) \nLet the equivalent equation of (1.10) be \n2\neff4 eff4 eff4 2() ()() () ,dx t d x tmc k x t f tdt dt  (7. 33) \nwhere meff4, ceff4, and keff4 are the effective mass, damping, and stiffness of the fractional vibrator (1.10), \nrespectively. Taking into account (1.10), we replace c by 0 in (2.2) and have the equivalent equation of \n(1.10) expressed by arXiv:2107.02340v2 [math.DS] 7 Jul 202127\n2\n()2 () 1 () 1\n2\n()() ( ) () () ( )cos sin sin22 2\n()cos ( ) ( ),   1 ( ) 3,  0 ( ) 1.2dx t d x tmm kdt dt\nkx t ft  \n   \n     \n (7. 34) \nBy comparing (7.33) to (7.34), we see that \n()2\neff4()cos ,2mm (7. 35) \n() 1 () 1\nef f4() ()sin sin ,22cm k     (7.36) \n()\ne ff4()cos .2kk (7. 37) \nBecause (7.33) is a standard vibration equation in form, we write its free response, denoted by x4(t), by \neff4 effn4 0 eff4 effn4 0\n4 0 effd4 effd4\neffd4( ) cos sin ,    0,t vxxt e x t t t    \n (7.38) \nwhereeff4\neff4\neff4 eff4 2c\nmk is the restricted effective damping ratio of (1.10),eff4\neffn4\neff4k\nm is the restricted \neffective damping free natural frequency of (1.10),2\neffd4 effn4 eff4 1  for |eff4|  1 is the restricted\neffective damped natural frequency of (1.10). By restricted, we mean that they are in the sense of meff4 > 0.  \nDenote by h4(t) the impulse response to (1.10). From Theorem 5.2 or (7.33), we at once write it by \neff4 effn4\n4 effd4\neff4 effd41() s i n ,     0 .tht e t tm  (7.39) \nAccording to Theorem 5.3, the restricted frequency transfer function of the fractional vibrator (1.10), which \nis denoted by H4(), is given by \n4 2\neff4 eff4 eff4 eff41() ,\n12H\nki\n \n(7. 40) \nwhereeff4\neffn4.   \n7.5. Results regarding variable-order fractional vibrator (1.11) \nThe equivalent equation of (1.11) is expressed by arXiv:2107.02340v2 [math.DS] 7 Jul 202128\n2\n() 1 ()\n2() ( ) () ( )sin cos ( ) ( ),   0 ( ) 1.22dx t d x tmk k x t f tdt dt        (7.41) \nAs a matter of fact, in (2.2), letting () = 2 and c = 0 produces the above.  \nLet ceff5 and k eff5 be the effective damping and stiffness of the fractional vibrator (1.11), respectively. \nFrom the above, we have \n() 1\neff 5()sin ,2ck (7. 42) \n()\ne ff5()cos .2kk (7. 43) \nThus, (7.41) can be rewritten by \n2\neff5 eff5 2() ()() () .dx t d x tmc k x t f tdt dt   (7.44) \nThe above in form is a standard vibration equation. Let x5(t) be the free response of (7.44). Then,  \neff5 effn5 0 eff5 effn5 0\n5 0 effd5 effd5\neffd5() c o s s i n ,     0 ,t vxxt e x t t t    \n (7.45) \nwhereeff5\neff5\neff5,\n2c\nmkeff5\neffn5 ,k\nm and2\neffd5 effn5 eff5 1  for |eff5|  1.\nDenote by h5(t) the impulse response to (1.11). Then, \neff5 effn5\n5e ffd5\neffd51( ) sin ,     0.tht e t tm  (7.46) \nLet H5() be the frequency transfer function of the fractional vibrator (1.11). Then, \n5 2\neff5 eff5 eff5 eff51() ,\n12H\nki\n \n (7.47) \nwhereeff5\neffn5.  \n8. Conclusions\nWe have derived out an equivalent vibration equation of a class of variable-order fractional vibrators\n(1.12), which is expressed by (2.2). Base on it, we have presented the analytical expressions of its effective \nmass, and damping, stiffness in Section 2. We have brought forward the analytical expressions of its \ndamping ratio, damping free natural frequency, damped natural frequency, and frequency ratio in Section 4. arXiv:2107.02340v2 [math.DS] 7 Jul 202129\nIn Section 5, we have put forward the close form expressions of free response, impulse response, and \nfrequency transfer function. The asymptotic properties of its effective mass, damping, and stiffness have \nbeen given in Section 3. In addition, we have given a mathematical explanation of the Rayleigh damping \nassumption when () and () are varying in terms of  in Section 6 Besides, based on (2.2), we have \nproposed equivalent equations, effective vibration parameters (mass, damping, stiffness, damping ratio, \ndamping free natural frequency, damped natural frequency, frequency ratio), as well as free responses, \nimpulse responses, and frequency transfer functions of five classes of variable-order fractional vibrators \n(1.7) – (1.11) in Section 7. When (), () and () are constants, the present th eory is applicable to the \nfractional vibrators expressed by (1.1) – (1.6)6. \nDeclaration of competing interest \nThe author declares that there are no known competing financial interests or personal relationships that \ncould have appeared to influence the work reported in this paper. \nAcknowledgements \nThis work was supported in part by the National Natural Science Foundation of China (NSFC) under \nthe project grant number 61672238. The views and conclusions contained in this document are those of the \nauthor and should not be interpreted as representing the official policies, either expressed or implied, of \nNSFC or the Chinese government. \nReferences \n[1] V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers , Vol. II, Springer, 2013.\n[2] J.-S. Duan, The periodic solution of fractional oscillation equation with periodic input, Advances in\nMathematical Physics , vol. 2013, 2013 .\n[3] J.-S. Duan, Z. 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Nasedkin, Finite element dynamic analysis of anisotropic elastic solids with voids,\nComputers & Structures , 87(15-16) 2009, 981-989.\n[108] L. A. S. Kouris, A. Penna, and G. Magenes, Seismic damage diagnosis of a masonry building using\nshort-term damping measurements, Journal of Sound and Vibration , vol. 394, 2017, 366-391.\n[109] X. D. Jin and L. J. Xia, Ship Hull Vibration , Shanghai Jiaotong University Press, 2011. In Chinese.arXiv:2107.02340v2 [math.DS] 7 Jul 2021"    },    {        "title": "2108.01193v1.Wide_Area_Damping_Control_for_Interarea_Oscillations_in_Power_Grids_Based_on_PMU_Measurements.pdf",        "content": "Wide-Area Damping Control for Interarea\nOscillations in Power Grids Based on PMU\nMeasurements\nIlias Zenelis, Xiaozhe Wang, Member, IEEE\nAbstract —In this paper, a phasor measurement unit (PMU)-\nbased wide-area damping control method is proposed to damp\nthe interarea oscillations that threaten the modern power system\nstability and security. Utilizing the synchronized PMU data, the\nproposed almost model-free approach can achieve an effective\ndamping for the selected modes using a minimum number of\nsynchronous generators. Simulations are performed to show the\nvalidity of the proposed wide-area damping control scheme.\nIndex Terms —Power systems, estimation, control applications\nI. INTRODUCTION\nLOW-FREQUENCY interarea oscillations, involving two\ncoherent generator groups swinging against each other\nat a frequency typically less than 2Hz, lead to a small-\nsignal stability concern for the modern inter-connected power\nsystems. The undesirable existence of interarea oscillations\ndue to weakly-tied transmission lines may limit the power\ntransmission capability between different areas and damage\nthe power grid elements, and therefore needs to be constantly\nmonitored and controlled. Conventionally, power system sta-\nbilizers (PSSs) have been employed to damp the interarea\noscillations. However, classical PSSs are incapable of damping\nthe iterarea modes, the frequencies of which lie beyond their\nlimited bandwidth. Although a vast amount of techniques\naiming to improve the traditional PSSs have been proposed,\nincluding the multiple-input PSSs (e.g., [1]), multi-band PSSs\n(e.g., [2]) and supervisory level PSSs (e.g., [3]), PSS tech-\nniques may not effectively damp the interarea modes involving\ndifferent areas and may strongly depend on the assumed\nnetwork model [4].\nThe implementation of a synchrophasor-based wide-area\nmeasurement system (WAMS) in power grids greatly enhances\nthe observability of power system dynamics, providing a\nunique opportunity to observe, identify and damp the inter-\narea oscillations. Multiple control methodologies have been\ndeveloped for damping the interarea oscillations deploying\na WAMS. A comprehensive discussion about the formula-\ntion of wide-area control problem in power systems was\npresented in [5]. In [6], a decentralized/hierarchical archi-\ntecture for wide-area damping control using PMU remote\nThis work is supported by Natural Sciences and Engineering Research\nCouncil (NSERC) Discovery Grant (NSERC RGPIN-2016-04570) and Fonds\nde Recherche du Qu ´ebec — Nature et technologies (FRQ-NT NC-253053).\nIlias Zenelis and Xiaozhe Wang are with the Department of Electrical and\nComputer Engineering, McGill University, Montr ´eal, QC H3A 0G4, Canada.\nemail: ilias.zenelis@mail.mcgill.ca, xiaozhe.wang2@mcgill.cafeedback signals was discussed. In [7], a sparsity-promoting\noptimal wide-area control was employed to damp the interarea\noscillations in bulk power systems. References [4] [8] pro-\nposed the design of wide-area damping controllers (WADCs)\nthat provide supplementary damping control to synchronous\ngenerators. The authors of [9] applied a networked control\nsystem model for wide-area closed-loop power systems. The\nauthors of [10] introduced a power oscillation damping (POD)\ncontroller based on a WAMS using a modal linear quadratic\nGaussian (MLGC) methodology. However, these approaches\nrequire the detailed and accurate knowledge of the complete\nnetwork model (both topology and parameter values), that is\nunavailable or corrupted in practice as a result of communica-\ntion failures, bad data in state estimation etc. In addition, the\nimpact of disturbances on the interarea oscillations cannot be\nwell captured by these methods.\nIn this paper, we attempt to develop a wide-area damping\ncontrol strategy for interarea modes utilizing PMU measure-\nments, which does not rely on the assumed network model (the\nonly required knowledge is the damping and inertia constants\nof generators which are not subject to constant changes). To\nthe knowledge of authors, the proposed wide-area damping\nmethod for interarea oscillations seems to be the first method\nthat is completely independent of the network model and its\nparameter values. The main contributions of the paper are as\nbelow:\n\u000fA measurement-based (almost model-free) technique is\napplied to accurately estimate the system state matrix\nin ambient conditions, which is completely independent\nof the system network model and is computationally\nefficient.\n\u000fAn effective wide-area damping control scheme for in-\nterarea modes is proposed using the participation factors\nof the estimated system state matrix, which can damp\na target mode by a desired coefficient using the least\npossible number of generators while maintaining the\nother modes unaffected.\n\u000fNumerical studies are conducted in the IEEE 39-bus 10-\ngenerator New England system to show that the proposed\nwide-area damping control method is fast, effective, and\nrobust again measurement noise.\nII. THE STOCHASTIC POWER SYSTEM MODEL\nIn this paper, we investigate the power system dynamic\noperation in quasi steady-state, i.e., in ambient conditions.arXiv:2108.01193v1  [eess.SY]  2 Aug 2021Since interarea modes are predominantly determined by the\nmachine rotor angles and speeds, classical swing equations\nare used to model generator dynamics:\n_\u000ei=!i\u0000!S\nMi_!i=Pmi\u0000Pei\u0000Di(!i\u0000!S)i= 1;:::;n(1)\nwhere\u000eiis the generator rotor angle, !iis the rotor angular\nvelocity,!Sis the synchronous speed, Miis the inertia\nconstant,Diis the damping coefficient, Pmiis the generator’s\nmechanical power input from the prime mover, Peiis the\ngenerator’s electrical power output, and nis the number of\ngenerators in the system. Peiis defined as\nPei=EinX\nj=1EjYijcos(\u000ei\u0000\u000ej\u0000\u001eij)i= 1;:::;n (2)\nwhereEiis the constant voltage behind the transient reactance\nX0\nd, andYij\\\u001eijis the (i,j)thentry of the reduced admittance\nmatrix containing generators’ impedances. It should be pointed\nout that in (1) each generator represents the equivalent aggre-\ngation of thousands of actual generators.\nIn the power system model of (1)-(2), the loads are modeled\nas constant impedances. However, other types of loads such\nas ZIP loads can be incorporated in this formulation It is\ncommon and reasonable to assume that the load power varies\nstochastically following a Gaussian distribution [11]. The load\nfluctuation manifests itself in the diagonal elements of the\nreduced admittance matrix as proposed in [12], [13]:\nY(i;i) =Yii(1 +\u001bidWt;i)\\\u001eii (3)\nwhereWtis a Wiener process and \u001biis the standard deviation\nof the variation describing load fluctuations. Therefore, the\npower system equations become:\n_\u000ei=!i\u0000!S\nMi_!i=Pmi\u0000Pei\u0000Di(!i\u0000!S)\u0000E2\niGii\u001bi\u0018i(4)\nwhereGii=Yiicos(\u001eii), and\u0018i=dWt;i\ndt;i= 1;:::;n , are\nindependent Gaussian random variables.\nNote that (4) represents a set of stochastic differential\nequations. To conduct the small-signal stability analysis, we\nlinearize (4) around the steady-state operating point as shown\nbelow:\n\u0014_\u000e\n_!\u0015\n=\u00140 In\n\u0000M\u00001@Pe\n@\u000e\u0000M\u00001D\u0015\u0014\u000e\n!\u0015\n+\u00140\n\u0000M\u00001E2G\u0006\u0015\n\u0018\n(5)\nwhere\u000e= [\u000e1;:::;\u000en]T,!= [!1\u0000!S;:::;!n\u0000\n!S]T,M=diag([M1;:::;Mn]),D=diag(D1;:::;Dn),\nPe= [Pe1;:::;Pen]T,E=diag([E1;:::;En]),G=\ndiag([G11;:::;Gnn]),\u0006 = diag([\u001b1;:::;\u001bn]), and\u0018=\n[\u00181;:::;\u0018n]T.\nLetx= [\u000e;!]T,A=\u00140 In\n\u0000M\u00001@Pe\n@\u000e\u0000M\u00001D\u0015\n,B=\n[0;\u0000M\u00001E2G\u0006]T, then (5) takes the following compact\nform:\n_x=Ax+B\u0018 (6)In short, the stochastic power system dynamic model in\nambient conditions can be represented as a vector Ornstein-\nUnlenbeck process that is Gaussian and Markovian. It will be\ndiscussed in Section III that the dynamic system state matrix\nAcan be estimated from the statistical properties of the PMU\nmeasurements, based on which a measurement-based wide-\narea damping control scheme is developed.\nIII. PMU-BASED WIDE-AREA DAMPING CONTROL\nA. An (Almost) Model-Free Approach of Estimating A\nAssuming that the state matrix Ais stable (satisfied in\nambient conditions), the stationary covariance matrix Cxx\nsatisfies the following Lyapunov equation [14]:\nACxx+CxxAT=\u0000BBT(7)\nwhereCxx=\u0014\nC\u000e\u000eC\u000e!\nC!\u000eC!!\u0015\n. Equation (7) integrates the\nstatistical properties of states that can be extracted from PMUs\nand the model knowledge, providing an ingenious way to\nestimate the model information from measurements.\nSupposing that PMUs are installed at all the generator\nterminal buses (optimistic currently, yet not unreasonable in\nthe near future), we can use the PMU measurements to\ncalculate the values of rotor angle \u000eand rotor speed !in\nambient conditions as discussed in many previous works (e.g.,\n[15]). We, therefore, can further estimate the covariance matrix\nCxxof\u000eand!(see Appendix). If the damping Dand inertia\nconstantsMare known, it has been shown in [12] that the\ndynamic state Jacobian matrix@Pe\n@\u000ecan be estimated by the\nfollowing equation derived from (7):\n(@Pe\n@\u000e) =MC!!C\u00001\n\u000e\u000e\u0000DC!\u000eC\u00001\n\u000e\u000e(8)\nImportantly, we do not require any information about the\nnetwork model (topology and parameter values) that is usually\nsubject to inaccuracy due to, for instance, communication\nerrors. Therefore, this method for estimating the dynamic state\nJacobian matrix and the system state matrix is almost model-\nfree (except the knowledge of DandM). A brief overview\nof the detailed derivation of@Pe\n@\u000eis presented in Appendix.\nNote that the conventional model-based method calculates\nthe matrix@Pe\n@\u000eby differentiating (2) with respect to \u000ethat\nheavily depends on the network topology and parameter values\nembedded in the admittance matrix Y.\nOnce the dynamic state Jacobian matrix@Pe\n@\u000eis estimated,\nthe system state matrix Acan be easily computed by:\nA=\u00140 In\n\u0000M\u00001@Pe\n@\u000e\u0000M\u00001D\u0015\n(9)\nB. Modal Analysis and Linear Feedback Control\nThe eigenvalues \u0003 = diag([\u00151;:::;\u0015 2n])ofAappearing in\ncomplex conjugate pairs \u0015i=\u0011i\u0006!i;i= 1;:::;n , the right\neigenvectors \b = [\u001e1;:::;\u001e 2n]and the left eigenvectors \t =\n[ T\n1;:::; T\n2n]TofAcan be readily extracted from the esti-\nmated matrix A. Therefore, the mode frequencies fi=!i\n2\u0019;i=\n1;:::;n and the damping ratios \u0010i=\u0000\u0011ip\n\u0011i2+!i2;i= 1;:::;n arestraightforwardly obtained. Moreover, the participation factor\nPiof\u0015idefined as:\nPi= [P1;i;:::P 2n;i]T= [\u001e1;i i;1;:::;\u001e 2n;i i;2n]T(10)\ncan be estimated from the right and left eigenvectors.\nIn addition, the matrix \u0003with the eigenvalues of Aas\ndiagonal elements, can be written as:\n\u0003 = \tA\b (11)\nThe left and right eigenvectors corresponding to \u0015iand\u0015j\nsatisfy the following relation:\n j\u001ei=(\n1;ifi=j\n0;ifi6=j(12)\nwhere a vector normalization has been applied.\nC. The Proposed Wide-Area Damping Control Scheme\nIn this paper, we intend to develop a wide-area damping\ncontrol scheme using PMU measurements. Actually, we add a\nstate feedback control loop to the original linear time-invariant\nopen-loop system described by (5) as shown below:\n_x=Ax+Bcu\nu=Kx(13)\nwherex= [\u000e;!]Tis obtained from PMU measurements.\nThe gain matrix Kis designed to damp the targeted interarea\noscillation modes. The control center sends the input control\nsignalsu=Kxto the generators that participate in the\nWAMS-based central control as indicated by Bc.\nThe matrixBcis defined as: Bc=\u0014\nBc\u000e 0\n0Bc!\u0015\n, whereBc\u000e\nandBc!refer to\u000eand!respectively. Ideally, the remedial\ncontrol scheme is applied to all ngenerators and thus, Bc=I.\nHowever, it is rather impractical and expensive to apply a\ncontrol measure to every synchronous machine. In this paper,\nthe generators with the largest participation factors in regard\nto the mode of interest, are selected to conduct the damping\ncontrol. Mathematically speaking, the diagonal entries of Bc\ncorresponding to the generators that no controls are carried\nout, are substituted by 0.\nThe closed-loop plant matrix Aclis given by:\nAcl=A+ \u0001A (14)\nwhere \u0001A=BcK, according to the state feedback loop\ndefined in (13). Representing \u0001Ain diagonal canonical form\nby applying the similarity transformation described by (11),\n\u0001\u0003 = \t\u0001A\b (15)\nHence, substituting \u0001A=BcKin the above relation,\n\u0001\u0003 = \tBcK\b (16)\nInspired by the model-based damping technique introduced\nin [16], we propose a 2n\u00022ndamping matrix to damp\nthe particular interarea oscillation mode k. In contrast to the\nmodel-based method in [16], the proposed wide-area damping\ncontrol releases the dependence of the method on the accuratenetwork model, topology, and parameter values, which are\nsubject to frequent changes. The subscript ( k) is attached\nto the mathematical symbols thereafter to denote their refer-\nence to mode k. For instance, K(k)symbolizes the feedback\nmatrix devoted to mode k. The two open-loop eigenvalues\nassociated with mode kare denoted as \u0015k1=\u0011k+j!kand\n\u0015k2=\u0011k\u0000j!k.\n1) Ideal Case: As we have seen before, ideally Bc=Iif\nall generators receive the damping control signals. Assuming\nthat we want to move the eigenvalues of mode k(the conjugate\npair\u0015k1and\u0015k2) by a coefficient \u001bk<0, we propose to use\nthe following damping matrix:\nK(k)=\u001bk[\u001ek1;\u001ek2][ T\nk1; T\nk2]T(17)\nSubstituting (17) to (16) with Bc=I, we have:\n\u0001\u0003 (k)=\u001bk\t[\u001ek1;\u001ek2][ T\nk1; T\nk2]T\b (18)\nthe(i;j)thentry of which is:\n\u0001\u0003ij(k)=\u001bk i\u001ek1 k1\u001ej+\u001bk i\u001ek2 k2\u001ej (19)\nApplying the orthogonality principle illustrated by (12), we\nhave:\n\u0001\u0003ij(k)=(\n\u001bk;if(i;j)2f(k1;k1);(k2;k2)g\n0;otherwise(20)\nTherefore,\u0015k1=\u001bk+\u0011k+j!kand\u0015k2=\u001bk+\u0011k\u0000j!k, the\neigenvalues of the closed-loop state matrix Aclcorresponding\nto the targeted mode k, migrate to the left by a coefficient\n\u001bk, leading to an improved damping. The rest of the eigenval-\nues remain unaffected under the proposed feedback damping\ncontrol.\n2) Practical Case: In reality,Bcmay not be equal to I\nas mentioned previously considering the cost of conducting\ncontrol for all generators. If we still let the damping matrix\nKto be:\nK(k)=\u001bk[\u001ek1;\u001ek2][ T\nk1; T\nk2]T(21)\nthen\n\u0001\u0003 (k)=\u001bk\tBc[\u001ek1;\u001ek2][ T\nk1; T\nk2]T\b\n=\u001bk\tBc\u001ek1 k1\b +\u001bk\tBc\u001ek2 k2\b\n=\u001bk\t^\u001ek1 k1\b +\u001bk\t^\u001ek2 k2\b (22)\nwhere ^\u001eki=Bc\u001ekii= 1;2:\nAs a result, ^\u001ekiwill have nonzero entries only if the\ncorresponding generators carry out the WAMS-based control.\nFor instance, presuming that only Generators 4-6 receive\ndamping control signals, then we have ^\u001eki= [^\u001e\u000e\nki;^\u001e!\nki] =\n[0;0;0;\u001e\u000e4\nki;\u001e\u000e5\nki;\u001e\u000e6\nki;0;:::;0;\u001e!4\nki;\u001e!5\nki;\u001e!6\nki;0;:::;0]T.\nBy the eigenvalue perturbation theory [17], the eigenvalues\n[^\u00151;:::;^\u00152n]of\u0003 + \u0001\u0003 (k), and thus of A+ \u0001A(k)satisfy:\n^\u0015i=\u0015i+eT\ni\u001bk(\t^\u001ek1 k1\b + \t ^\u001ek2 k2\b)ei\n=(\n\u0015i+\u001bkeT\ni\t^\u001ei;ifi2fk1;k2g\n\u0015i;otherwise(23)\nwhereeidenotes a unit vector that has 1in theithposition and\n0 elsewhere. It is observed that although the damping effect tothe targeted mode kis slightly affected compared to the ideal\ncase sinceeT\ni\t^\u001eiis typically different from 1, the other modes\nstill remain unaffected. To ensure that an effective damping is\nacted to mode kwhile minimizing the number of generators,\nwe choose the generators with the largest participation factors\nin modekto carry out the control signals. The proposed\nWAMS-based damping control algorithm is presented below\nand is illustrated in Fig. 1.\nStep 1. Estimate the system state matrix Ausing the PMU\nmeasurements by (8)-(9).\nStep 2. Calculate the eigenvalues \u0003, the right eigenvectors\n\b, the left eigenvectors \tof the estimated A, and the\nparticipation factor Pifor each mode \u0015i. Select the interarea\noscillation mode kto damp.\nStep 3. Compute the damping matrix Kby (17) for the\ntargeted mode k.\nStep 4. Select the generators with the largest participation\nfactors, find the corresponding Bc, and send the damping\ncontrol input signals u=BcKxto the selected generators.\nIn practice, the damping control signals are transmitted to\nthe remote terminal units (RTUs) of the favored generators\nwhere they can serve either as ancillary control inputs to the\ngenerators’ PSSs or as direct inputs to the generators’ exciters.\nFig. 1: Schematic diagram of the closed-loop system.\nIV. NUMERICAL RESULTS\nThe IEEE 39-bus 10-generator New England system, is\nused to demonstrate the effect of the proposed control tech-\nnique The topology of the system can be found in [12].\nFor validation purposes, two case studies are presented. The\nfirst study intends to test the proposed control method on\nthe classical generator models under which the method is\ndeveloped. The second study is employed to demonstrate\nthe validity of the suggested method in the real-world case\nwhere the generators are modelled as higher-order models and\nare controlled by exciters and PSSs. In addition, the PMU\nmeasurement noise is also considered. All parameters for the\ntwo studies are available in: https://github.com/zenili/Mode-\nParticipation-Estimation-2017. PSAT-2.1.9 [18] is used for all\nsimulations.\nA. Study I: Classical Generator Model\nThe 10 generators are modelled as the classical model\ndescribed by (4). The angle of Generator 1 (G1) serves as the\nreference. The load fluctuations are characterized by a standard\ndeviation\u001bi= 5 in (3) for all generators. We assume that thesampling rate is 20 Hz, lying within the typical range of PMU\nsampling rate: 6-60 Hz [6]. By executing the system state\nmatrix estimation and modal analysis described in Section III,\nall the eigenvalues can be estimated with a very good accuracy.\nIndeed, the estimation error is less than 2% for frequencies and\nbelow 6% for damping ratios for all modes. Specifically, Mode\n7 that is characterized by the estimated values f7= 1:662Hz\n(0.54% estimation error) and \u00107= 1:03% (5.56% estimation\nerror) is considered to be weakly damped as \u00107<10%, which\nis a widely accepted criterion for satisfactory damping.\nThe estimated mode shapes and participation factors for\nMode 7 are presented in Fig. 2. It is clear from Fig. 2a that\nMode 7 is an interarea oscillation mode, in which Generator\n4 (G4) and 5 (G5) are swinging against Generator 6 (G6) and\n7 (G7). The influence of the rest of the generators in Mode 7\nis negligible as their participation factors are close to zero.\n  0.2  0.4  0.6  0.8\n30\n21060\n24090\n270120\n300150\n330180 0\n  Gen 5\nGen 6\nGen 7\nGen 8\nGen 9\nGen 10\nGen 2\nGen 3\nGen 4\n(a) Mode shapes for Mode 7.\n234567891000.10.20.30.4\nGenerator number  \nthe estimated\nparticipation factor (b) Participation factors for\nMode 7.\nFig. 2: Study I: The estimated mode shapes and participation\nfactors for Mode 7.\nIt is worth noting that the total CPU time needed for the\ncalculation of Ais 9.642 ms using a computer of 2.50GHz\nand 8.00GB memory, indicating that the real-time estimation\nof the system state matrix of the reduced network model is\nfeasible in practical applications.\nBy the proposed WAMS-based damping control algorithm,\nthe most significant participants in Mode 7, Generator 5-\n7, are chosen to conduct the control. To illustrate how the\nnumber of controlled generators influences the damping effect,\nwe perform the following numerical experiments. In the 1st\nexperiment, the damping control signal is adopted only at\nthe generator with the largest participation factor—G5. In the\n2ndexperiment, both G5 and G7 receive the damping control\nsignals. In the 3rdexperiment, we include G6 together with\nG5 and G7 to apply the control signals. In the 4thexperiment,\nall generators participate in the damping feedback loop. The\ndamping coefficient is set to be \u001b7= 2in all experiments. The\ncomparison between the open-loop eigenvalues and the closed-\nloop eigenvalues is illustrated in Fig. 3. It can be observed\nthat the selected interarea mode gains more damping as the\nnumber of connected stations increases. Also, it seems that\nthe exclusion of the generators with negligible participation\nfactors does not have a notable impact on the effectiveness\nand efficiency of the damping control scheme. Moreover,\nTable I shows that the threshold 10% is met in the last three\nexperiments, indicating that the proposed technique requires\nonly two generators (G5 and G7) to achieve a desirabledamping performance, although more controlled generators\nwill provide an even enhanced damping effect. Note that the\nrest of the modes are not affected by the method.\n−2.5 −2−1.5 −1−0.5 0−20−1001020\nRealImaginary\n  open−loop eigenvalues\nclosed−loop eigenvalues\n(a) Control at G5\n−2.5 −2−1.5 −1−0.5 0−20−1001020\nRealImaginary\n  open−loop eigenvalues\nclosed−loop eigenvalues (b) Control at G5 and G7\n−2.5 −2−1.5 −1−0.5 0−20−1001020\nReal Imaginary\n  open−loop eigenvalues\nclosed−loop eigenvalues\n(c) Control at G5,G7 and G6\n−2.5 −2−1.5 −1−0.5 0−20−1001020\nReal Imaginary\n  open−loop eigenvalues\nclosed−loop eigenvalues (d) Control at all Generators\nFig. 3: Study I: A comparison between the open-loop and the\nclosed-loop eigenvalues.\nTABLE I: Study I: Closed-loop damping ratio for Mode 7.\nGenerators Closed-loop\ndamping ratio (%)\nG5 8.55\nG5, G7 13.11\nG5, G7, G6 17.15\nAll Generators 19.77\nB. Study II: Detailed Generator Model with PMU Measure-\nment Noise\nIn this study, all the 10generators in the IEEE 39-bus system\nare modelled by the fourth-order models, which are controlled\nby field exciters and PSSs. Besides this, a Gaussian-distributed\nmeasurement noise with standard deviation of 10\u00003for angles\nand10\u00006for rotor speeds is added to the emulated PMU\nmeasurements according to the IEEE standard [19].\nThe eigenvalues of the system state matrix Aare accurately\nestimated with an error lower than 2% for mode frequencies\nand less than 8% for damping ratios. Particularly, Mode 6\nthat is described by the estimated values f6= 1:800 Hz\n(0.67% estimation error) and \u00106= 2:66%<10% (3.42%\nestimation error) is obviously underdamped. Fig. 4 presents\nthe estimated mode shapes and participation factors for Mode\n6 that is apparently an interarea mode. Indeed, Generator 10\n(G10) and 8 (G8) oscillate against Generator 2 (G2) and 9\n(G9). These generators take the most responsibility for the\nexcitation of Mode 6.\nIt should be noted that in this case the total CPU time\nneeded for the calculation of Ais 9.731 ms.\nThe developed damping control technique is implemented\nby G10, G2 and G8 , the most important participants in Mode\n6, utilizing four different experiments. The experiments with\n  0.2  0.4  0.6  0.8\n30\n21060\n24090\n270120\n300150\n330180 0\n  Gen 8\nGen 9\nGen 10\nGen 2\nGen 3\nGen 4\nGen 5\nGen 6\nGen 7(a) Mode shapes for Mode 6.\n234567891000.10.20.30.4\nGenerator number  \nthe estimated\nparticipation factor (b) Participation factors for Mode\n6.\nFig. 4: Study II: The estimated mode shapes and participation\nfactors for Mode 6.\nan increasing number of controlled generators are designed\nbased on the participation factor ranking while the damping\nfactor is selected to be \u001b6= 2. The relationship between the\nopen-loop and the closed-loop eigenvalues is shown in Fig. 5.\nIt can be seen that the damping effect increases as the number\nof the generators participating in the central control grows,\nwhich is also corroborated by the damping ratios presented\nat Table II. Furthermore, the 10% damping ratio requirement\nis satisfied by all experiments, implying that the proposed\ntechnique can achieve an effective damping impact with only\none generator (G10) under control.\n−2.5 −2−1.5 −1−0.5 0−15−10−5051015\nRealImaginary\n  open−loop eigenvalues\nclosed−loop eigenvalues\n(a) Control at G10\n−2.5 −2−1.5 −1−0.5 0−15−10−5051015\nRealImaginary\n  open−loop eigenvalues\nclosed−loop eigenvalues (b) Control at G10 and G2\n−2.5 −2−1.5 −1−0.5 0−15−10−5051015\nRealImaginary\n  open−loop eigenvalues\nclosed−loop eigenvalues\n(c) Control at G10,G2 and G8\n−2.5 −2−1.5 −1−0.5 0−15−10−5051015\nRealImaginary\n  open−loop eigenvalues\nclosed−loop eigenvalues (d) Control at all Generators\nFig. 5: Study II: Comparison between the open-loop and the\nclosed-loop eigenvalues.\nTABLE II: Study II: Closed-loop damping ratio for Mode 6.\nGenerators Closed-loop\ndamping ratio (%)\nG10 10.39\nG10, G2 17.29\nG10, G2, G8 18.86\nAll Generators 19.93\nThe last but not the least, the effectiveness of the pro-\nposed approach is demonstrated by comparing it with the\nconventional PSS technique. Particularly, the interarea Mode7 (f7= 1:662Hz and\u00107= 1:03%) was excited and the time-\ndomain response of \u000e4(Fig. 6) was simulated for the following\ncases:\n\u000fCase A: No PSS control;\n\u000fCase B: PSS control at all generators;\n\u000fCase C: WAMS-based control at all generators;\n\u000fCase D: WAMS-based control at G5 (biggest participa-\ntion factor);\nIt can be seen that the proposed method (Case C and Case\nD) achieves an improved damping performance compared to\nCase A and Case B. Even when the control is conducted at\nonly one generator (G5), the damping performance of the\nproposed wide-area damping method is better than the PSS\nlocal control. As known, the conventional PSS is only effective\nin a typically narrow frequency range. Although multi-band\nPSS [2] may enhance the performance, a complicated tuning\nprocess is required and may affect the rest of the modes. In\naddition, these approaches may not work well if the assumed\nnetwork model is subject to constant changes. In contrast, the\nproposed wide-area damping control method can effectively\ndamp the target mode by any selected damping coefficient\nusing a small number of generators while maintaining the other\nmodes unaffected. More importantly, the network model and\nparameter values are not assumed to be known.\n0246810−0.1−0.0500.050.1\ntime (s)δ4 (rad)\n  no PSS control\nonly PSS control \nWAMS−based control (all)\nWAMS−based control (G5)\nFig. 6: Study II: Time-domain response of \u000e4to the excitation\nof Mode 5.\nV. CONCLUSIONS AND PERSPECTIVES\nThis paper proposes a wide-area damping control method\nusing PMU data to damp the undesirable interarea oscillations\nin the modern power grid. The proposed method does not\ndepend on the network model and can be integrated into\nonline dynamic security assessment (DSA) for continuous\nmonitoring and controlling the interarea oscillations. It has\nbeen shown analytically and numerically that the targeted\nmode can be adequately damped using a small number of\nsynchronous machines. In the future, our efforts will focus\non simultaneously damping multiple interarea modes in larger\npower systems exploiting the estimated participation factors.\nAPPENDIX\nThe stationary covariance matrix is given by:\nCxx=\u0014C\u000e\u000eC\u000e!\nC!\u000eC!!\u0015\nwhere, for instance, C\u000ei\u000ej=E[(\u000ei\u0000\u0016i)(\u000ej\u0000\u0016j)], and\u0016iis the\nmean of\u000ei. In practice, C\u000e\u000eis usually unknown due to insuf-\nficient data. Thus, C\u000e\u000eis estimated by the sample covariancematrixQ\u000e\u000e, the(i;j)thelement of which is computed as [14]:\nQ\u000ei\u000ej=1\nN\u00001NP\nk=1(\u000eki\u0000\u0016\u000ei)(\u000ekj\u0000\u0016\u000ej), where \u0016\u000eisymbolizes the\nsample mean of \u000ei, andNis the sample size. Similarly, Q!!\nandQ!\u000eare used to estimate C!!andC!\u000erespectively. A\nwindow size of 450s is used in the examples of this paper,\nwhich shows good accuracy.\nIt should be noted that the proposed technique of estimating\nCxxis fast and efficient as shown in the simulation study.\nMoreover,Cxxcan be estimated recursively using a fast\niterative approach, which will further reduce the computational\neffort.\nREFERENCES\n[1] I. Kamwa, R. Grondin, D. Asber, J. P. Gingras, and G. Trudel, Active-\npower stabilizers for multi-machine power systems: Challenges and\nprospects , IEEE Trans. Power Syst., vol. 13, no. 4, pp. 1352–1358, Nov.\n1998.\n[2] I. Kamwa, L. Gerin-Lajoie and G. Trudel, Multi-loop power system\nstabilizers using wide-area synchronous phasor measurements, Pro-\nceedings of the 1998 American Control Conference. ACC (IEEE Cat.\nNo.98CH36207), Philadelphia, PA, USA, 1998, pp. 2963-2967 vol.5.\n[3] H. Ni, G. T. Heydt, and L. Mili, Power system stability agents using\nrobust wide area control, IEEE Trans. Power Syst., vol. 17, no. 4, pp.\n1123–1131, Nov. 2002.\n[4] Y . Zhang and A. Bose, Design of wide-area damping controllers for\ninterarea oscillations , IEEE Trans. Power Syst., vol. 23, no. 3, pp.\n1136–1143, Aug. 2008.\n[5] A. Chakrabortty and P. P. Khargonekar, Introduction to wide-area control\nof power systems , in Proc. Amer. Control Conf. , 2013, pp. 6758-6770.\n[6] I. Kamwa, R. Grondin, and Y . Hebert, Wide-area measurement\nbased stabilizing control of large power systems—–A decentral-\nized/hierarchical approach , IEEE Trans. Power Syst., vol. 16, no. 1,\npp. 136–153, Feb. 2001.\n[7] F. D ¨orfler, M. R. Jovanovic, M. Chertkov, and F. Bullo, Sparsity-\npromoting optimal wide-area control of power networks , IEEE Trans.\nPower Syst., vol. 29, no. 5, pp. 2281-2291, 2014.\n[8] M. E. Raoufat, K. Tomsovic, and S. M. Djouadi, Virtual actuators for\nwide-area damping control of power systems , IEEE Trans. on Power\nSystems, vol. 31, pp. 1-9, 2016.\n[9] S. Wang, X. Meng, and T. Chen, Wide-area control of power systems\nthrough delayed network communication, IEEE Trans. Contol Syst.\nTechnol., vol. 20, no. 2, pp. 495–503, Mar. 2012.\n[10] R. Preece, J. Milanovic, A. Almutairi, and O. Marjanovic, Damping of\ninter-area oscillations in mixed AC/DC networks using WAMS based\nsupplementary controller , IEEE Trans. Power Syst., vol. 28, no. 2, pp.\n1160-1169, May 2013.\n[11] R. Singh, B. C. Pal, and R. A. Jabr, Statistical representation of distri-\nbution system loads using Gaussian mixture model . IEEE Transactions\non Power Systems, vol. 25, no. 1, pp. 29-37, 2010.\n[12] X. Wang, J. Bialek, and K. Turitsyn, PMU-based estimation of dynamic\nstate Jacobian matrix and dynamic system state matrix in ambient\nconditions. IEEE Transactions on Power Systems, June 2017.\n[13] X. Wang, I. Zenelis, Estimating Participation Factors and Mode Shapes\nfor Electromechanical Oscillations in Ambient Conditions . 2018 IEEE\nCanadian Conference on Electrical & Computer Engineering.\n[14] C. Gardiner, Stochastic Methods: A Handbook for the Natural and Social\nSciences . Springer Series in Synergetics. Springer, Berlin, Germany,\n2009.\n[15] N. Zhou, S. Lu, R. Singh, and M. A. Elizondo, Calibration of reduced\ndynamic models of power systems using phasor measurement unit\n(PMU) data . North American Power Symposium (NAPS), 2011.\n[16] H. G. Far, H. Banakar, P. Li, C. Luo, and B.-T. Ooi, Damping interarea\noscillations by multiple modal selectivity method , IEEE Transactions on\nPower Systems, vol. 24, no. 2, pp. 766–775, 2009.\n[17] L. N. Trefethen and D. Bau, III , Numerical Linear Algebra , SIAM,\nPhiladelphia, 1997.\n[18] F. Milano, An open source power system analysis toolbox. IEEE Trans-\nactions on Power Systems, vol. 20, no. 3, pp. 1199-1206.\n[19] IEEE Standard for Synchrophasor Measurements for Power Systems-\nAmendment 1: Modification of Selected Performance Requirements .\nIEEE Std C37.118.1a-2014 (Amendment to IEEE Std C37.118.1-2011),\npp. 1-25, April 2014."    },    {        "title": "1407.8369v2.Plasmons_in_finite_spherical_ionic_systems.pdf",        "content": "arXiv:1407.8369v2  [cond-mat.mes-hall]  19 Nov 2014Plasmons in finite spherical ionic systems\nW. Jacak\nInstitute of Physics, Wroc/suppress law University of Technology,\nWyb. Wyspia´ nskiego 27, 50-370 Wroc/suppress law, Poland,\n(Dated: Received: date / Accepted: date)\nThe challenging question on possible plasmon type excitati ons in finite ionic systems is discussed.\nRelated theoretical model is formulated and developed in or der to describe surface and volume\nplasmons of ion liquid in finite electrolyte systems. Irradi ation of ionic surface plasmon fluctuations\nis studied in terms of the Lorentz friction of oscillating ch arges. Attenuation of surface plasmons\nin the ionic sphere is calculated and minimized with respect to the sphere size. Various regimes\nof approximation for description of size effect for damping o f ionic plasmons are determined and a\ncross-over in damping size-dependence is demonstrated. Th e most convenient, optimal dimension\nof finite electrolyte system for energy and information tran sfer by usage of ionic dipole plasmons is\ndetermined. The overall shift of size effect to micrometer sc ale for ions in comparison to nanometer\nscale for electrons in metals is found, as well as thered shif t byseveral orders ofplasmonic resonances\nin ion systems predicted in a wide range of variation dependi ng on ion system parameters. This\nconvenient opportunity of tuning resonances differs proper ties of ionic plasmons from plasmons in\nmetals where electron concentration is firmly fixed.2\nI. INTRODUCTION\nRecent experimental and theoretical investigationof plasmon osc illationsin metallic nanoparticlesfocused attention\non the fundamental character of this phenomenon and also on gre at prospects of applications. In particular, the so-\ncalled plasmon effect in solar cells modified in nano-scale with on surface deposited metallic particles leads to the\nsignificant growth of their efficiency [1–6]. The mediating role in collectin g of sun-light energy is played by surface\nplasmon oscillations in metallic nanoparticles due to their radiative prop erties. Irradiation of energy of plasmon\noscillations is preferable for energy transport applications. As it wa s observed experimentally and later predicted\ntheoretically, irradiation losses of plasmon energy are strongly sen sitive to the metallic nanoparticles size [7].\nStrong irradiation of plasmon oscillations in metallic nanoparticles plays also a major role in construction of plas-\nmonic wave-guides with high transference efficiency. Many experime ntal studies [8, 9] indicated that periodic linear\nstructuresofmetallicnanoparticlesserveasefficientplasmonwave -guideswithlowdamping[10–12]. The wave-lenghts\nof propagating in such structures plasmon-polaritons typically are by one order shorter in comparison to light with\nthe same frequency, which allows for avoiding diffraction limits in optica l circuits [13–15]. This is perceived as a way\nto forthcoming constructions of plasmon opto-electronic nano-d evices, not attainable by using only light wave-guides\nlimited by diffraction constraints. An efficient energy transfer in plas mon wave-guides is also conditioned by radiative\nproperties of surface plasmons in metallic nano-components.\nRadiative losses of plasmon oscillations can be described by the so-ca lled Lorentz friction [16, 17]. Accelerating\ncharges irradiate electro-magnetic wave and the related energy lo ss can be accounted for as an effective electric field\nwhich hampers electron movement. For the case of the oscillating dip ole as for the dipole-type surface plasmons in\na metallic nanosphere, the Lorentz friction force is proportional t o the third order time-derivative of this dipole [16].\nLet us emphasize here that the strong irradiation of surface plasm ons in metallic nanospheres, linked to the Lorentz\nfriction, is exclusively present in sufficiently large metallic nano-partic les. Small metallic nano-paricles in form of the\nclusters of size 1 −5 nm do not exhibit irradiation efficiency so high as nanospheres with ra diia >10 nm. Especially\nmuch attention was focused on large nanoparticles of noble metals ( gold, silver and copper) because of location of\nplasmon resonances in particles of these metals within the visible light s pectrum.\nAn interesting question which we try to discuss in the present paper is the possibility for occurrence of similar\nplasmon effects with ionic carriers instead of electrons. Many finite io nic systems in a form of enclosed by membranes\nelectrolyte systems can be encountered in biological structures a nd the question arises regarding possible significance\nof such ion plasmonic phenomena, the role it would take in such struct ures and whether the radiative properties of\nplasmon fluctuation would also be so pronounced in ionic system as the y were in metals. It is quite reasonable that\nionic plasmon effect would be located in other regions of energy and wa ve lengths in comparison to metallic systems.\nThis, let’s call it ’soft’ plasmonics could be linked with functionality of bio logical systems where electricity is rather\nof ionic than electronic character. For instance the cell signaling, m embrane transfer or nerve cell conductivity would\nserve as examples.\nThe theoretical plasmonic model we will adopt for ions, as far as pos sible, upon analogy to the metallic nanospheres\nwith plasmon excitations theory. The ionic systems are much more co mplicated in comparison to a metal crystal\nstructure with free electrons. Therefore, an identification of an appropriate simplifications of the approach to ionic\nsystem is of a primary significance. The model must be capable of rep etition for ions in electrolyte the plasmonic\nscenario known from electrons in metals.\nIn the present paper we will consider the finite spherical ionic syste m (e.g., liquid electrolyte artificially confined\nwith a membrane) and identify the plasmon excitations of ions in this sy stem. We will determine their energies for\nvarious parameters of the ionic system with special attention paid t o irradiation properties of ionic plasmons.\nIn the subsequent paper [18] we will analyze an ionic plasmon-polarito n propagation in electrolyte sphere chains\nwith prospective relation to signaling in biological systems. Taking in min d that metallic nano-chains serve as very\nefficient wave-guides for electro-magnetic signals in the form of colle ctive surface plasmon excitation of wave-type\ncalled plasmon-polaritons, we will try to model the similar phenomenon in the ionic spheres chains.\nFor the initial crude model we will study the spherical or prolate sph eroidal ionic conducting system with balanced\ncharges in analogy to the jellium model in metals when local fluctuation s of ion density, negative and positive beyond\nthe equilibrium level, can form plasmons in ionic finite system [19].\nII. FLUCTUATIONS OF THE CHARGE DENSITY IN THE SINGLE CONDUCT ING IONIC\nSPHERICAL SYSTEM\nThe problem of how to establish an adequate model for multi-ionic sys tem to grasp essential properties of ionic\nplasmons is a main issue. For simple two-component ionic system we dea l with both sign ion solutions creating an\nelectrolyte with balanced negative and positive total charge. In eq uilibrium these charge cancellation is also local. As3\n+++++\n--\n--- A+B-\nl=3,m=2 \nl=1,m=0,1 \nl=2,m=0,1,2 \nl=3,m=0,1,2,3 D(t) \nFIG. 1. Dipole D(t) creation in a single sphere by the simplest surface plasmon oscillations (left); examples of surface plasmon\ncharge distribution with various multiplicity l, m—different colors indicate distinct values of local charge d ensity from negative\nto positive ones (right)\nwe deal with two kinds of carriers they both would form density fluct uations resulting in violation of the local electric\nequilibrium. The total compensation of both sign charges requires, however, that any density fluctuation of negative\ncharges must be accompanied by distant, in general, but ideally equiv alent positive ions fluctuation, and conversely.\nThis means that effectively we deal with density fluctuation of ions, e ither positive or negative in charge value (always\nmutually compensated) with respect to uniform charge distribution assumed as ideally cancelled by the opposite sign\nuniform background—fictitious jellium in analogy to metal. In this way w e can model a two component ionic system\nby two single component ion systems with jellium of the opposite sign. F or simplicity we assume that the charge and\nmass of the opposite ions are the same, but generalization is straigh tforward.\nTo simplify the model according to the above described lines let us con sider the spherical shape system with the\nradiusaand with balanced total charge of both sign ions with uniform equilibriu m density distributions n+(−)(r) =\nnΘ(a−r) (Θ(r) is the Heaviside step function, ais the sphere radius). For simplicity, we assume the same absolute\nvalue of charges of plus- and minus-charged ions. By m+(−)one can denote the mass of positive (negative) ions. They\nboth are of the order of104−5me, whereme= 3.1×10−31kg, is the mass of the electron. Reducing the two component\nsystem to two systems with the jellium, is an approximation but may se rve for a recognition of ionic dynamics and of\nscales for its quantitative characteristics, at least. The advanta ge of such an approach is a close analogy to description\nof plasmon in metals including direct definition of the rigid shape of the s ystem by the explicit jellium form.\nUpon the above model assumptions we will consider the ionic carries w ith density oscillating around the zero\nvalued equilibrium density as screened by the fictitious positive charg ed background (the effect of the opposite sign\nionspresence). Hence, the descriptionoffluctuationsoflocalde nsityofelectronsin metallicnanospherecanbedirectly\nused to model fluctuations of effective ions density, substituting e lectron mass by the ion mass and the electron charge\nby the ion charge. The dynamical equation for the charged fluid in ion system can be thus repeated from the case of\nmetal nanosphere with electrons [19]. The equilibrium density of the e ffective charged liquid, denoted by n, will be\ntreated as a parameter and assumed equal to n=ηN0whereηis the molarity of the electrolyte in the sphere and N0\nis the one-molar electrolyte concentration of ions. The equilibrium de nsity determines the bulk plasmon frequency for\nthe ion system according to the formula, ω2\np=4πnq2\nm, wherenandmare the equilibrium uniform concentration and\nthe mass of ions with the charge q. Due to larger mthan the electron mass, me, and usually smaller concentration\nof effective ions than the one for electrons in metals, ωpcan be considerably reduced, even by several orders of\nmagnitude. Note, that for electrons in metals ¯ hωp≃10 eV and typically falls into ultra-violet region of radiation\nwith corresponding energy of photons. In the ionic system, the pla smon frequency ¯ hωpcan be much lower and placed\nin the range of infra-red or even in lower energy part of the electro -magnetic wave spectrum.\nA. Definition of the model\nThe Hamiltonian for the two type ion system has the form,\nˆHion=−N−/summationdisplay\ni=1¯h2∇2\ni\n2m−−N+/summationdisplay\nj=1¯h2∇2\nj\n2m+−N−,N+/summationdisplay\ni,jq−q+\nε|ri−rj|+1\n2N−/summationdisplay\ni,i′,i/negationslash=i′(q−)2\nε|ri−ri′|+1\n2N+/summationdisplay\nj,j′j/negationslash=j′(q+)2\nε|rj−rj′|, (1)\nwhereq−(+),M−(+),N−(+)are the charge, the mass and the number of the −(+) ions, respectively. To analyze this\ncomplicated system we propose the following approximation, assumin g, for simplicity, q−=−q+=q,N−=N+=N,4\nm−=m+=mand let us add and subtract the same terms as written below,\nˆHion=−N/summationtext\ni=1¯h2∇2\ni\n2m−N/summationtext\nj=1¯h2∇2\nj\n2m−/summationtext\ni,jq2\nε|ri−rj|+1\n2N/summationtext\ni,i′,i/negationslash=i′q2\nε|ri−ri′|+1\n2N/summationtext\nj,j′j/negationslash=j′q2\nε|rj−rj′|\n−q2/summationtext\nj/integraltextn(r)d3r\nε|rj−r|−q2/summationtext\ni/integraltextn(r)d3r\nε|ri−r|+q2/summationtext\nj/integraltextn(r)d3r\nε|rj−r|+q2/summationtext\ni/integraltextn(r)d3r\nε|ri−r|,(2)\nwhere we have introduced formally the jellium of the spherical shape for both types of ions, with the density n\nideally compensating opposite charges of uniformly distributed ions, n(r) =nΘ(a−r),ais the sphere radius (the\npositivejellium with the negativejellium mutually cancelthemselves). A ssuming that q2/summationtext\nj/integraltextn(r)d3r\nε|rj−r|+q2/summationtext\ni/integraltextn(r)d3r\nε|ri−r|−\n/summationtext\ni,jq2\nε|ri−rj|≃0, which is fulfilled for not too strong ion concentration fluctuations beyond the uniform distribution,\nwe can separate the Hamiltonian into the sum, ˆHions=ˆH−+ˆH+, where,\nˆH−(+)=/summationdisplay\nj/bracketleftBigg\n−¯h2∇2\nj\n2m−q2/integraldisplayn(r)d3r\nε|rj−r|/bracketrightBigg\n+1\n2/summationdisplay\nj/negationslash=j′q2\nε|rj−rj′|. (3)\nThe latter term in r.h.s. of Eq. (3) corresponds to interaction betw een ions of the same sign, whereas the second\nterm in the first sum describes interaction of these ions with the effe ctive jellium (of opposite sign), εis the dielectric\nconstant of the electrolyte medium. The confined electrolyte syst em could be, in particular, a electrolyte medium\nshaped by appropriately formed membrane, as frequently occurs in biological systems. Because of separation of the\nHamiltonian (1) one can consider single Hamiltonian (3). The ion wave fu nction corresponding to Hamiltonian (3) is\ndenoted by Ψ ion(t).\nThe form of Hamiltonian (3) allows for repetition of its further discus sing along the scheme applied to electrons\nin metals [19], which we will recall below, for the sake of completeness. A local density of chosen type ions can be\nwritten, in analogy to semiclassical Pines-Bhom random phase appro ximation (RPA) approach to electrons in metal\n[20, 21], in the following form:\nρ(r,t) =<Ψion(t)|/summationdisplay\njδ(r−rj)|Ψion(t)>, (4)\nwhererjdenotes coordinate of j−thion and the Dirac delta quasiclassically fixes j−thion position. The Fourier\npicture of the above density has the form:\n˜ρ(k,t) =/integraldisplay\nρ(r,t)e−ik·rd3r=<Ψion(t)|ˆρ(k)|Ψion(t)>, (5)\nwhere the ’operator’ ˆ ρ(k) =/summationtext\nje−ik·rj.\nUsing the above notation one can rewrite ˆHionin the following form, in an analogy to the bulk case for metallic\nplasmon description [19–21]:\nˆHion=N/summationdisplay\nj=1/bracketleftBigg\n−¯h2∇2\nj\n2m/bracketrightBigg\n−q′2\n(2π)3/integraldisplay\nd3k˜n(k)2π\nk2/parenleftBig\nˆρ+(k)+ ˆρ(k)/parenrightBig\n+q′2\n(2π)3/integraldisplay\nd3k2π\nk2/bracketleftBig\nˆρ+(k)ˆρ(k)−N/bracketrightBig\n,(6)\nwhere: ˜n(k) =/integraltext\nd3rn(r)e−ik·ris the Fourier picture of jellium distribution (in the derivation of Eq. (6 ) we have taken\ninto account that4π\nk2=/integraltext\nd3r1\nre−ik·r),q′2=q2\nε.\nUtilizing this form of the effective ion Hamiltonian one can write out the d ynamic equation in Heisenberg repre-\nsentation for ion density fluctuations,\nd2ˆρ(k)\ndt2=1\n(i¯h)2/bracketleftBig/bracketleftBig\nˆρ(k),ˆHion/bracketrightBig\n,ˆHion/bracketrightBig\n, (7)\nwhich attains the following form,\nd2ˆρ(k)\ndt2=−/summationtext\nje−ik·rj/braceleftBig\n−¯h2\nm2(k·∇j)2+¯h2k2\nm2ik·∇j+¯h2k4\n4m2/bracerightBig\n−4πq′2\nm(2π)3/integraltext\nd3p˜n(p)k·p\np2ˆρ(k−p)−4πq′2\nm(2π)3/integraltext\nd3pˆρ(k−p)k·p\np2ˆρ(p).(8)5\nOne can notice that ˆ ρ(k−p)ˆρ(p) =δˆρ(k−p)δˆρ(p)+˜n(k−p)δˆρ(p)+δˆρ(k−p)˜n(p)+˜n(k−p)˜n(p) and ˜n(p)ˆρ(k−p) =\n˜n(p)δˆρ(k−p) + ˜n(p)˜n(k−p), where δˆρ(k) = ˆρ(k)−˜n(k) describes the ’operator’ of local ion density fluctuation\nabove the uniform distribution. Therefore, one can rewrite Eq. (8 ) as follows,\nd2δˆρ(k)\ndt2=−/summationtext\nje−ik·rj/braceleftBig\n−¯h2\nm2(k·∇j)2+¯h2k2\nm2ik·∇j+¯h2k4\n4m2/bracerightBig\n−4πq′2\nm(2π)3/integraltextd3p˜n(k−p)k·p\np2δˆρ(p)−4πq′2\nm(2π)3/integraltextd3pδˆρ(k−p)k·p\np2δˆρ(p).(9)\nTaking averaging over the quantum states |Ψion>, for the ion density fluctuation δ˜ρ(k,t) =<Ψion|δˆρ(k,t)|Ψion>=\n˜ρ(k,t)−˜n(k), we obtain the following equation,\n∂2δ˜ρ(k,t)\n∂t2=−<Ψion|/summationtext\nje−ik·rj/braceleftBig\n−¯h2\nm2(k·∇j)2+¯h2k2\nm2ik·∇j+¯h2k4\n4m2/bracerightBig\n|Ψion>\n−4πq′2\nm(2π)3/integraltext\nd3p˜n(k−p)k·p\np2δ˜ρ(p,t)−4πq′2\nm(2π)3/integraltext\nd3pk·p\np2<Ψion|δˆρ(k−p)δˆρ(p)|Ψion> .(10)\nFor small k, in analogy to the semiclassical approximation for electrons [19, 21 ], the contributions of the second and\nthirdcomponentstothefirsttermontheright-hand-sideofEq. ( 10)canbeneglectedassmallincomparisontothefirst\ncomponent. Small and thus negligible is also the third term in the right- hand-side of Eq.(10), as involving a product\nof twoδ˜ρ(which we assumed small, δ˜ρ/n << 1). This approach corresponds to the random-phase-approxima tion\n(RPA) formulated for bulk metal [20, 21]. Within the RPA, Eq. (10) a ttains the following shape,\n∂2δ˜ρ(k,t)\n∂t2=2k2\n3m<Ψion|/summationdisplay\nje−ik·rj¯h2∇2\nj\n2m|Ψvion>−4πq′2\nm(2π)3/integraldisplay\nd3p˜n(k−p)k·p\np2δ˜ρ(p,t), (11)\nand due to the spherical symmetry,\n<Ψion|/summationdisplay\nje−ik·rj¯h2\nm2(k·∇j)2|Ψion>≃2k2\n3m<Ψion|/summationdisplay\nje−ik·rj¯h2∇2\nj\n2m|Ψion> .\nOne can rewrite Eq. (11) in the position representation,\n∂2δ˜ρ(r,t)\n∂t2=−2\n3m∇2<Ψion|/summationtext\njδ(r−rj)¯h2∇2\nj\n2m|Ψion>\n+ω2\np\n4π∇/braceleftBig\nΘ(a−r)∇/integraltext\nd3r11\n|r−r1|δ˜ρ(r1,t)/bracerightBig\n.(12)\nIn the case of metals it was next used the Thomas-Fermi formula to assess the averaged kinetic energy [20]:\n<Ψion|−/summationtext\njδ(r−rj)¯h2∇2\nj\n2m|Ψion>≃3\n5(3π2)2/3¯h2\n2m(ρ(r,t))5/3\n=3\n5(3π2)2/3¯h2\n2mn5/3Θ(a−r)/bracketleftBig\n1+5\n3δ˜ρ(r,t)\nn+.../bracketrightBig\n.(13)\nThe above Thomas-Fermi formula is addressed, however, to ferm ionic and degenerate quantum systems, as electrons\nin metals. Forionicsystems suchestimation ofkinetic energyisinappr opriate, because the ion concentrationis usually\nmuch lower than that one of electrons in metals and the system is not degenerated even if ions are fermions. The\nMaxwell-Boltzmann distribution should be applied instead of the Fermi- Dirac or Bose-Einstein ones. Independently\nof fermionic of bosonic statistics of ions, the Maxwell-Boltzmann dist ribution allows for estimation of the averaged\nkinetic energy of ions located inside the sphere with the radius a, in the following form,\n<Ψion|−/summationdisplay\njδ(r−rj)¯h2∇2\nj\n2m|Ψion>≃(n+δρ(r,t))Θ(a−r)3kT\n2, (14)\nwherekis the Boltzmann constant and Tis the temperature. For ions of the 3D shape or of the linear shape, the\ninclusion of rotational degrees of freedom would result in the facto r6kT\n2or5kT\n2, respectively, instead of3kT\n2for point\nlike ion model.\nUsing the formula (14) and taking into account that ∇Θ(a−r) =−r\nrδ(a−r), one can rewrite Eq. (12) in the\nfollowing manner,\n∂2δ˜ρ(r,t)\n∂t2=/bracketleftbigkT\nm∇2δ˜ρ(r,t)−ω2\npδ˜ρ(r,t)/bracketrightbig\nΘ(a−r)\n−kT\nm∇/braceleftbig\n[n+δ˜ρ(r,t)]r\nrδ(a−r)/bracerightbig\n−/bracketleftBig\nkT\nmr\nr∇δ˜ρ(r,t)+ω2\np\n4πr\nr∇/integraltext\nd3r11\n|r−r1|δ˜ρ(r1,t)/bracketrightBig\nδ(a−r).(15)6\nIn the above formula ωpis the bulk ion-plasmon frequency, ω2\np=4πnq′2\nm. The solution of Eq. (15) can be decomposed\ninto two parts related to the distinct domains—inside the sphere and on the sphere surface,\nδ˜ρ(r,t) =/braceleftbigg\nδ˜ρ1(r,t), for r < a,\nδ˜ρ2(r,t), for r≥a,(r→a+),(16)\ncorresponding to the volume and surface excitations, respective ly. These two parts of ion local density fluctuations\nsatisfy the equations (according to Eq. (15)),\n∂2δ˜ρ1(r,t)\n∂t2=kT\nm∇2δ˜ρ1(r,t)−ω2\npδ˜ρ1(r,t), (17)\nand (here ǫ= 0+)\n∂2δ˜ρ2(r,t)\n∂t2=−kT\nm∇/braceleftbig\n[n+δ˜ρ2(r,t)]r\nrδ(a+ǫ−r)/bracerightbig\n−/bracketleftBig\nkT\nmǫF\nmr\nr∇δ˜ρ2(r,t)+ω2\np\n4πr\nr∇/integraltext\nd3r11\n|r−r1|(δ˜ρ1(r1,t)Θ(a−r1)+δ˜ρ2(r1,t)Θ(r1−a))/bracketrightBig\nδ(a+ǫ−r).(18)\nThe Dirac delta in Eq. (18) results due to the derivative of the Heavis ide step function—ideal jellium charge distri-\nbution. In Eq. (18) an infinitesimal shift, ǫ= 0+, is introduced to fulfill requirements of the Dirac delta definition\n(its singular point must be an inner point of an open subset of the dom ain). This shift is only of a formal character\nand does not reflect any asymmetry.\nThe electric field due to surface charges is zero inside the sphere, a nd therefore cannot influence the volume excita-\ntions. Oppositely, the volume charge fluctuation-induced-electric -field can excite the surface fluctuations. Therefore,\nthe equation for volume plasmons is independent of surface plasmon s, whereas the volume plasmons contribute the\nequation for the surface plasmons.\nThe problem of separation between surface and volume plasmons ha s been thoroughly analyzed for metal clusters\nand was identified as significant for very small clusters. In the size- scale of 1 −3 nm for metallic clusters, the effect of\nso-calledspill-outofelectronsbeyondthejellium edgewasimportant andcausedthesurfacefuzzy resultingin coupling\nofvolumeand surfaceplasmonoscillations. Manydirectnumericalsim ulations(TDLDA, i.e., the time dependent local\ndensity approximation) [22, 23] have been verified that the volume– surface excitation miss-mass gradually disappears\nin larger clusters [22, 23], which supports accuracy of semiclassical RPA description, within which volume plasmons\ncan be separated from the surface ones (even though the latter can be excited by the former ones, due to the last\nterm in Eq. (18)). The quantum spill-out effect disappears gradually with growing sphere dimension and in the\nrange of several nanometers for the metallic sphere radius is comp letely negligible. In the present paper we consider\nthe radius range of ionic systems of micrometer order, when quant um effects are negligible. Such an opportunity\nallows us to formulate an analytical RPA semiclassical description in th e form of an oscillator equation, allowing for\nphenomenological inclusion of the damping effects. The energy dissip ation effects turned out to be overwhelming\nphysical property in the case of larger metallic nanospheres [7, 19] (witha >10 nm for Au or Ag) and also for much\nlarger ionic systems, as we will demonstrate it below.\nB. Solution of RPA equations: volume and surface plasmon fre quencies\nEqs (17) and (18) are solved for metallic nanospheres [19] and thes e solutions can be directly applied to ionic\nsystems. To summarize briefly this analysis, both parts of the plasm a fluctuation can represented as follows,\nδ˜ρ1(r,t) =nF(r,t), for r < a,\nδ˜ρ2(r,t) =σ(Ω,t)δ(r+ǫ−a), ǫ= 0+, for r≥a,(r→a+),(19)\nwith initial conditions, F(r,t)|t=0= 0, σ(Ω,t)|t=0= 0, (Ω is the spherical angle), F(r,t)|r=a= 0,/integraltext\nρ(r,t)d3r=N\n(neutrality condition). For the aboveinitial and boundary condition s and taking advantageof the spherical symmetry,\none can write out the time-dependent parts of the ion concentrat ion fluctuations in the form [19] (cf. Appendix),\nF(r,t) =∞/summationdisplay\nl=1l/summationdisplay\nm=−l∞/summationdisplay\ni=1Almnjl(knlr)Ylm(Ω)sin(ωlit), (20)\nand\nσ(Ω,t) =∞/summationtext\nl=1l/summationtext\nm=−lBlm\na2Ylm(Ω)sin(ω0lt)\n+∞/summationtext\nl=1l/summationtext\nm=−l∞/summationtext\ni=1Almn(l+1)ω2\np\nlω2\np−(2l+1)ω2\nliYlm(Ω)nea/integraltext\n0dr1rl+2\n1\nal+2jl(klir1)sin(ωlit),(21)7\nwherejl(ξ) =/radicalBig\nπ\n2ξIl+1/2(ξ) is the spherical Bessel function, Ylm(Ω) is the spherical function, ωli=ωp/radicalbigg\n1+kTx2\nli\nω2\npa2m\nare the frequencies of the ion volume self-oscillations (volume plasmo n frequencies), xliare the nodes of the Bessel\nfunction jl(ξ) numerated with i= 1,2,3...(cf. Fig. 2), kli=xli/a,ωl0=ωp/radicalBig\nl\n2l+1are the frequencies of the\nion surface self-oscillations (surface plasmon frequencies). The d erivation of the self-frequencies for ionic plasmon\noscillations is presented with all the details in Appendix. Amplitudes AlmiandBlmare arbitrary in the homogeneous\nproblem and can be adjusted to the initial conditions for the first de rivatives.\nThe function F(r,t) describes volume plasmon oscillations, whereas σ(Ω,t) describes the surface plasmon oscilla-\ntions. Let us emphasize that the first term in the Eq. (21) corresp onds to the surface self-oscillations, while the\nsecond one term describes the surface oscillations induced by the v olume plasmons. The frequencies of the surface\nself-oscillations are equal to,\nω0l=ωp/radicalbigg\nl\n2l+1, (22)\nwhich, for l= 1, is the dipole type surface oscillation frequency, described for m etallic nanosphere by Mie [24],\nω01=ωp/√\n3.\nl/c610\nl/c611\nl/c612\n0 5 10 15 20/c450.20.00.20.40.60.81.0\nrlspherical Bessel function Jl\nl/c611,i/c611\nl/c611,i/c612\nl/c612,i/c611\n200 220 240 260 280 3002.0 /c1801074.0 /c1801076.0 /c1801078.0 /c1801071.0 /c1801081.2 /c180108\ntemperature Klia 50 mionic sphere\nFIG. 2. The spherical Bessel functions Jl(r) forl= 0,1,2 displaying possible charge density fluctuations in the sph ere along\nthe sphere radius r(arbitrary units) for volume plasmon modes; the angular dis tribution for these modes is governed by the\nreal spherical functions Ylm(Ω) similarly as for the surface plasmon modes (cf. Fig. 1 rig ht). The exemplary temperature\ndependence of self-frequencies of volume plasmon modes ωli,li= 11,12,21, for diluted electrolyte n≃10141/m3and ion\nmass∼104me,a∼50µm—right\nC. Ionic surface plasmon frequencies for a nanosphere embed ded in a dielectric medium, with ε1>1\nOne can now include the influence of a dielectric surroundings (in gene ral, distinct from the inner one of considered\nionic system) on plasmons in this system. In order to do it let us assum e that ions on the surface ( r=a+, i.e.,\nr≥a, r→a) interact with Coulomb forces renormalized by the relative dielectric constant ε1>1 (distinct from ε\nfor inner medium). Thus a small modification of Eq. (18) is of order,\n∂2δ˜ρ2(r)\n∂t2=−2\n3m∇/braceleftbig/bracketleftbig3\n5ǫFn+ǫFδ˜ρ2(r,t)/bracketrightbigr\nrδ(a+ǫ−r)/bracerightbig\n−/bracketleftBig\n2\n3ǫF\nmr\nr∇δ˜ρ2(r,t)+ω2\np\n4πr\nr∇/integraltext\nd3r11\n|r−r1|/parenleftBig\nδ˜ρ1(r1,t)Θ(a−r1)+1\nε1δ˜ρ2(r1,t)Θ(r1−a)/parenrightBig/bracketrightBig\nδ(a+ǫ−r),(23)\n(note that Eq. (17) is not affected by the outer medium). The solut ion of the above equation is of the same form as\nthat one for the Eq. (18) case, but with the renormalized surface plasmon frequencies,\nω0l=ωp/radicalbigg\nl\n2l+11\nε1. (24)8\nIII. DAMPING OF PLASMON OSCILLATIONS IN IONIC SYSTEMS\nThe semiclassical RPA treatment of plasmon excitations in finite ion sy stems as presented above, does not account\nfor plasmon damping. The damping of plasmon oscillations can be, howe ver, included in a phenomenological manner,\nby addition of an attenuation term to plasmon dynamic equations, i.e., the term, −2\nτ0∂δρ(r,t)\n∂t, added to the right\nhand sides of both Eqs (17) and (18), taking into account their osc illatory form. The introduced damping ratio1\nτ0accounts for ion scattering losses and can be approximated in analo gy to metallic systems, by inclusion of energy\ndissipation caused by irreversible its transformation into heat via va rious microscopic channels similar to those for\nOhmic resistivity [25],\n1\nτ0≃v\n2λb+Cv\n2a, (25)\nwhereais the sphere radius, vis the mean velocity of ions, v=/radicalBig\n3kT\nm,λbis the ion mean free path in bulk electrolyte\nmaterial the same as the sphere is made of (including scattering of io ns on other ions, and on solvent particles and\nadmixtures). The second term in Eq. (25) accounts for scatterin g of ions on the boundary of the finite ionic system,\nthe sphere with the radius a, the constant Cis of order of unity [25].\nIn order to explicitly express a forcing field which moves ions in the sys tem, the inhomogeneous time dependent\nterm should be added to the homogeneous equations (17) and (18) . The forcing field may be the time dependent\nelectric field. Ifone considersit asthe electric component ofthe inc ident e-m wavethen the comparisonofthe resonant\nwave-length with the system size is of order. Similarly as for metallic na no-spheres also for finite ionic systems the\nsurface plasmon resonant wave-length highly exceeds the system dimension and the forcing field is practically uniform\nalong whole the system. Such a perturbation could excite only surfa ce dipole plasmons, i.e., the mode with l= 1,\nwhich can be described by the function Q1m(t) (l= 1 and mare angular momentum numbers related to the assumed\nspherical symmetry of the system). The corresponding dynamica l equation for surface plasmons reduced to only mode\nQ1m(t) has the following form,\n∂2Q1m(t)\n∂t2+2\nτ0∂Q1m(t)\n∂t+ω2\n1Q1m(t)\n=/radicalBig\n4π\n3qn\nm/bracketleftbig\nEz(t)δm,0+√\n2(Ex(t)δm,1+Ey(t)δm,−1)/bracketrightbig\n,(26)\nwhereω1=ωp√3ε1(it is a dipole surface plasmon frequency, i.e., the Mie frequency [24], ε1is the dielectric susceptibility\nof the system surroundings). Because only Q1mcontribute to the plasmon response to the homogeneous electric fi eld,\nthus the effective ion density fluctuation has the form [19],\nδρ(r,t) =\n\n0, r < a,\n1/summationtext\nm=−1Q1m(t)Y1m(Ω)r≥a, r→a+,(27)\nwhereYlm(Ω) is the spherical function with l= 1. One can also explicitly calculate the dipol D(t) corresponding to\nsurface plasmon oscillations given by Eq. (27),\n\n\nDx(t) =q′/integraltext\nd3rxδρ(r,t) =√\n2π√\n3q′Q1,1(t)a3,\nDy(t) =q′/integraltext\nd3ryδρ(r,t) =√\n2π√\n3q′Q1,−1(t)a3,\nDz(t) =q′/integraltext\nd3rzδρ(r,t) =√\n4π√\n3q′Q1,0(t)a3.(28)\nThe dipole D(t) satisfies the equation (it is rewritten Eq. (26)),\n/bracketleftbigg∂2\n∂t2+2\nτ0∂\n∂t+ω2\n1/bracketrightbigg\nD(t) =a34πq′2n\n3mE(t) =εa3ω2\n1E(t). (29)\nOnecannoticethatthedipole(28)scalesasthesystemvolume, ∼a3, whichmaybeinterpretedthatallionsactually\ncontribute to the surface plasmon oscillations. This is connected wit h the fact that the surface modes correspond\nto uniform translation-type oscillations of ions in the system, when in side the sphere the charge of ions is exactly\ncompensated by oppositely signed ions, whereas the not balanced c harge density occurs only on the surface despite\nall ions oscillate. For the volume plasmons the non-compensated cha rge density fluctuations are present also inside9\n0204060801001201400.0000.0010.0020.0030.0040.0050.006\na10 nm0 100 200 300 400 5000.0000.0050.0100.015\na10 nm\n1 1m=10 m , q=3e, n=10 N , T=300K4\ne 0-2m=10 m , q=3e, n=10 N , T=300K4\ne 0-3\nFIG. 3. The cross-over in ionic system-size dependence of da mping rate for surface plasmons, for T= 300K, m= 104me,\nq= 3e,n= 10−2N0(N0is the concentration of one molar electrolyte) (left) and fo rn= 10−3N0(right); in the size-region\nclose to the cross-over the perturbative treatment for Lore ntz friction perfectly coincides with the exact approach\nthe sphere as volume plasmon modes have the compressional chara cter with not balanced charge fluctuations along\nthe system radius.\nThescatteringeffectsaccountedforbytheapproximateformula (25)causedampingofplasmonsespeciallystrongfor\nsmallsizeofthesystemduetothenanosphere-edgescatteringc ontributionproportionalto1\na. Thistermis, however,of\nlowering significance with the radius growth. We will show that radiatio n losses resulted due to accelerated movement\nof ions scales as a3, and for rising athese irradiative energy losses quickly dominate plasmon attenuatio n. Due to\nopposite size dependence of scattering and irradiation contributio ns to the plasmon damping one can thus observe\nthe cross-over in dumping with respect to its size dependence, as it is depicted in Fig. 3. One can also determine the\nradiusa∗for which the total attenuation rate for surface plasmons is minima l,a∗=/parenleftBig\n33/2Cc3v\n2ω1ω3p/parenrightBig1/4\n. The system sizes\na∗for two distinct ionic systems are listed in Tab. I.\nThe irradiation of energy of the oscillating dipole is expressed by the s o-called Lorentz friction [16], i.e., the effective\nelectric field slowing down the motion of charges,\nEL=2√ε\n3c3∂3D(t)\n∂t3. (30)\nHence, we can rewrite Eq. (29) including the Lorentz friction term,\n/bracketleftbigg∂2\n∂t2+2\nτ0∂\n∂t+ω2\n1/bracketrightbigg\nD(t) =εa3ω2\n1E(t)+εa3ω2\n1EL, (31)\nor forE= 0,\n/bracketleftbigg∂2\n∂t2+ω2\n1/bracketrightbigg\nD(t) =∂\n∂t/bracketleftBigg\n−2\nτ0D(t)+2\n3ω1/parenleftbiggωpa\nc√\n3/parenrightbigg3∂2\n∂t2D(t)/bracketrightBigg\n. (32)\nOne can apply the perturbation method for solution of Eq. (32) whe n the right hand side of this equation is\ntreated as a small perturbation. In the zeroth step of the pertu rbation we have/bracketleftBig\n∂2\n∂t2+ω2\n1/bracketrightBig\nD(t) = 0, from which\n∂2\n∂t2D(t) =−ω2\n1D(t). Hence, for the first step of the perturbation, we put the latte r formula to the right hand side of\nEq. (32), i.e.,\n/bracketleftbigg∂2\n∂t2+2\nτ∂\n∂t+ω2\n1/bracketrightbigg\nD(t) = 0, (33)\nwhere\n1\nτ=1\nτ0+ω1\n3/parenleftbiggωpa\nc√\n3/parenrightbigg3\n. (34)\nWithin the first step of perturbation, the Lorentz friction can be in cluded into the total attenuation rate1\nτ. Nev-\nertheless, this approximation is justified only for sufficiently small pe rturbations, i.e., when the second term in Eq.\n(34), proportional to a3, is small enough to fulfill the perturbation restrictions. The relate d limiting value, ˜ a, of the\nionic system size depends on the ion concentration, charge, mass, dielectric susceptibility, as is exemplified below in\nthe following subsection.10\nThe solution of Eq. (33) is of the form D(t) =Ae−t/τcos(ω′\n1t+φ), where ω′\n1=ω′\n1/radicalBig\n1−1\n(ω1τ)2, which gives the\nred shift of the plasmon resonance due to strong, ∼a3, growth of attenuation caused by the irradiation. The Lorentz\nfriction term in Eq. (34) dominates plasmon damping for a≤˜adue to this a3dependence—cf. Fig. 3. The plasmon\ndamping grows rapidly with aand this results in pronounced redshift of resonance frequency.\nA. Exact inclusion of the Lorentz damping to the attenuation of ionic dipole surface plasmons\nNow we will consider the dynamic equation for surface plasmons in the ionic spherical system (32) with the Lorentz\nfriction term, but without application of the perturbation method f or solution resulting in substitution of the Lorentz\nfriction term2\n3ω1/parenleftBig\nωpa\nv√\n3/parenrightBig3∂3D(t)\n∂t3with the approximate formula, −2ω1\n3/parenleftBig\nωpa\nv√\n3/parenrightBig3∂D(t)\n∂t, what was the result of taking\nin the right hand side of Eq. (32) the zeroth order its solution, for w hich∂2D(t)\n∂t2=−ω2\n1D(t). To compare various\ncontributions to Eq. (32) we change to dimensionless variable t→t′=ω1t. Then Eq. (32) attains the form,\n∂2D(t′)\n∂t′2+2\nτ0ω1∂D(t′)\n∂t′+D(t′) =2\n3/parenleftbiggωpa\nv√\n3/parenrightbigg3∂3D(t′)\n∂t′3. (35)\nIn the case of solution of Eq. (35) by perturbation we get renorma lized attenuation rate for effective damping\nterm,1\nω1τ0+1\n3/parenleftBig\nωpa\nv√\n3/parenrightBig3\n. This term quickly achieves the value 1, for which the oscillator falls int o the over-damped\nregime. For system parameters as assumed for Fig. 3, the achieve ment by the attenuation rate of the value equal 1\ntakes place at 25,5 µm and 8 µm forn= 10−3N0andn= 10−2N0, respectively. At these values of a, the frequency\nω′\n1=ω′\n1/radicalBig\n1−1\n(ω1τ)2goes to zero, which indicates an apparent artifact of the perturb ation method. To verify how\nbehaves the exact damped frequency in the considered system on e has to solve the dynamical equation without any\napproximations. As this equation is of third order linear differential e quation, one can find its solution in the form,\n∼eiΩt′, with the analytical expressions for three possible values of the ex ponent,\nΩ1=−i\n3g−i21/3(1+6gu)\n3g/parenleftbig\n2+27g2+18gu+√\n4(−1−6gu)3+(2+27 g2+18gu)2/parenrightbig1/3\n−i/parenleftbig\n2+27g2+18gu+√\n4(−1−6gu)3+(2+27 g2+18gu)2/parenrightbig1/3\n3×21/3g∈Im(=iα),\nΩ2=−i\n3g+i(1+i√\n3)(1+6gu)\n3×22/3g/parenleftbig\n2+27g2+18gu+√\n4(−1−6gu)3+(2+27 g2+18gu)2/parenrightbig1/3\n+i(1−i√\n3)/parenleftbig\n2+27g2+18gu+√\n4(−1−6gu)3+(2+27 g2+18gu)2/parenrightbig1/3\n6×21/3g=ω+i1\nτ,\nΩ3=−i\n3g+i(1−i√\n3)(1+6gu)\n3×22/3g/parenleftbig\n2+27g2+18gu+√\n4(−1−6gu)3+(2+27 g2+18gu)2/parenrightbig1/3\n+i(1+i√\n3)/parenleftbig\n2+27g2+18gu+√\n4(−1−6gu)3+(2+27 g2+18gu)2/parenrightbig1/3\n6×21/3g=−ω+i1\nτ,(36)\nwhereu=1\nτ0ω1andg= 2/3/parenleftBig\naωp\nc√3ε1/parenrightBig3\n.\nIn Fig. 4 we haveplotted the damping rate ( ImΩ) and the self-frequency( ReΩ) (also translatedfor resonancewave-\nlength—right pannels) with respect to the system radius a. For comparison, the approximate perturbative solutions\nare plotted also—the blue line, whereas the exact solution of Eq. (35 ) is plotted in red line. The blue line finishes at\nalimit, when the attenuation rate within the perturbation approach rea ches the critical value 1 (then λ→ ∞). For\nthe accurate solution of Eq. (35) this singular behavior disappears and the oscillating solution, eiΩt, exists for larger\naas well.\nWe notice that the red-shift of the plasmonresonanceis stronglyo verestimatedin the frameworkofthe perturbative\napproach to the Lorentz friction unless a <˜a, where ˜ais sensitive to ionic system parameters and especially to ion\nconcentration (as is demonstrated in Fig. 4).\nLet us emphasize that the equation (35) has in general two types o f particular solutions, eiΩt′, with complex self-\nfrequencies Ω. The solutions given by Ω 2and Ω 3are of oscillating type with damping ( iΩ2andiΩ3are mutually\nconjugated, thusΩ 2andΩ3havetherealpartsofoppositesign,whereasthesameimaginaryp arts,thelatterispositive\ndisplaying the damping rate) and the second one—given by Ω 1, which turns out to be an unstable exponentially\nrising solution (negative imaginary solution). This unstable solution is t he well known artifact in the Maxwell\nelectrodynamics (cf. e.g., $ 75 in [16]) and corresponds to the infinite self-acceleration of the free charge due to11\n0200 400 600 800 1000 12000.00.20.40.60.81.0\na10 nm0200 400 600 8001000 12000.000060.000080.000100.000120.00014\na10 nm0200 400 600 800 1000 12000.00.20.40.60.81.0\na10 nm0 1000 2000 3000 4000 50000.00.20.40.60.81.0\na10 nm0 1000 2000 3000 4000 50000.00.20.40.60.81.0\na10 nm01000 2000 3000 4000 50000.000150.000200.000250.000300.000350.000400.00045\na10 nmm=10 m , q=3e, n=10 N , T=300K4\ne 0-311\nm=10 m , q=3e, n=10 N , T=300K4\ne 0-21 1\nFIG. 4. Comparison of the damping rate and the resonance freq uency (transformed also into the resonance wave-length—\nin right panels), i.e., the damping rate and frequency (wave length) of oscillating solution of Eq.(35), exact (red line ) and\napproximate upon perturbation approach (blue line), both w ith respect to the ionic finite system radius a\nmaterial ionic system sample 1 sample 2\nion concentration n(N0is one-molar con-\ncentr.)10−2N0 10−3N0\neffective ion mass m(meelectron mass) 104me 104me\ncharge of effective ion q/√ε 3e 3e\ntemperature T 300 K 300 K\nmean velocity of ions v=/radicalbig3kT\nm1168 m/s 1168 m/s\nbulk plasmon frequency ωp 9.3×10131/s 2.93×10121/s\ndielectric constant of souroundings ε1 2 2\nMie frequency ω1=ωp/√3ε1 3.8×10131/s 1.2×10121/s\nconstant in Eq. (25) C 2 2\nbulk mean free path (room temp.) λb 0.5µm 0.1µm\nradius for minimal damping a∗=/parenleftBig\n33/2Cc3v\n2ω1ω3p/parenrightBig1/4\n2.7×10−7m 8.6×10−7m\nTABLE I. The ion-system parameters assumed for calculation of damping rates and self-frequency for dipole surface plas mons\nLorentz friction force (i.e., to the singular solution of the equation m˙v=const.רv, which is associated with a formal\nrenormalization of the field-mass of the charge—infinite for point-lik e charge and canceled in an artificial manner by\narbitrary assumed negative infinite non-field mass, resulting in ordin ary mass of e.g., an electron, which is, however,\nnot defined mathematically in a proper way). This unphysical singular particular solution should be thus discarded.\nThe other oscillatory type solution resembles the solution of the ord inary damped harmonic oscillator, though with\ndistinct attenuationrate andfrequency. They areexpressedby analyticalformulaefor Ω 2and Ω3by Eqs(36) and then\nare calculated for various aand compared with the corresponding quantities found within the pe rturbation approach.\nThis comparison is presented in Fig. 4. From this comparison it is clearly visible that application of the perturbation\napproach leads to high overestimation of the damping rate for a >˜a. Therefore, we can conclude that the usage of\nthe approximate formula for the Lorentz friction damping in the for m (34) is justified up to a≃˜a, while for a >˜a,\nthese approximate values strongly differ from the exact ones. The value ˜a < alimitsharply depends on ionic system\nparameters and approximately ˜ a≃alimit\n2.12\nIV. CONCLUSIONS\nConcluding, we can state that in ionic finite systems we may observe p lasmons similar as in the metallic nanopar-\nticles. The structure of surface and volume plasmons for ions is rep eated from the similar properties of electronic\nplasmons in metallic spherical systems, however, with the significant shift of resonant energy towards lower one cor-\nrespondingly to by few orders larger mass of ions in comparison to ele ctron mass and different concentration of ions\nin electrolyte. Thus corresponding to the resonant energy electr o-magnetic wave length is shifted to deep infra-red or\neven longer wave lengths depending on ion concentration. The typic al for metal clusters cross-overin size dependence\nof plasmon damping between the scattering, leading to Ohmic type en ergy dissipation, versus the irradiation losses\nalso is observable in ionic spherical system with similar to metal size-de pendence, though shifted toward the microm-\neter scale for ions instead on the nanometer scale for metals. Of pa rticular interest is the high irradiation regime\nfor dipole plasmons in ionic system with prospective application for sign aling and energy transfer in ionic systems.\nThe initial strong enhancement of efficiency of the Lorentz friction with the radius growth of electrolyte sphere is\nobserved on the micrometer scale with typical a3radius dependence above some threshold radius which value depend s\non electrolyte parameters. At certain value of the radius (variatin g in a wide range also depending on ion system\nparameters) this enhancement saturates and then the irradiatio n losses slowly diminish, which allows for definition of\nthe most convenient sizes of electrolyte finite system for optimizing radiation mediated transport efficiency preferring\nthe highest radiation losses.\nAcknowledgments Authors acknowledge the support of the present work upon the N CN project\nno. 2011/03/D/ST3/02643 and the NCN project no. 2011/02/A/ ST3/00116.\nAppendix A: Derivation of plasmon frequencies\n1. Volume plasmons\nInordertodetermineself-frequenciesofthe volumeionicplasmonin thespherewemustsolveEq. (17) withthe form\noftherelevantsolutiongivenbyEq. (19). Forinitialconditionslisted belowEq. (19)weassume F(r,t) =Fω(r)sin(ωt)\nand by substitution of this function into Eq. (17) we get,\n∆Fω(r)+k2Fω(r) = 0, (A1)\nwherek2=(ω2−ω2\np)m\nkT. This is a well known Helmholtz differential equation of which solutions ( finite in the origin)\ncan be expressed by the spherical Bessel functions (for the rad ial dependence of Fω(r)),\nFω(r) =Ajl(kr)Ylm(Ω), (A2)\nwherejl(x) =/radicalbigπ\n2xIl+1/2(x) is the the lth spherical Bessel function linked to the Bessel function of first kind. The\nboundary condition, F(a) = 0, gives quantization of k,kli=xli\na, wherexliis theith zero of the lth Bessel function\n(cf. Fig. 2 left). Following this quantization one arrives at the corre sponding self-frequency quantization,\nω2\nli=ω2\np/parenleftbigg\n1+kTxli\nω2pma2/parenrightbigg\n. (A3)\nThus the volume ionic plasmons in the sphere are described by functio ns,\nδρ1(r,t) =n∞/summationdisplay\nl=1m=l/summationdisplay\nm=−l∞/summationdisplay\ni=1Almijl(klir)Ylm(Ω)sin(ωlit), (A4)\nwhereAlmiarearbitraryconstants. Thecomponentwith l= 0vanishesbecauseofneutralitycondition,a/integraltext\n0r2drdΩF(r,t) =\n0 (as/integraltext\ndΩYlm(Ω) =√\n4πδl0δm0,dΩ =sinΘdΘdφ). Note that in ionic systems self-frequencies of volume plasmons in\nthe sphere are temperature dependent—cf Eq. (A3) and Fig. 2 rig ht.13\n2. Surface plasmons\nIn order to determine self-frequencies for surface plasmons, on e has to consider Eq. (18) and solution for it given\nby Eq. (19). The first term in the right hand side of Eq. (18) can be r ewritten to the form,\nkT\nm∇(n+δρ2)∇Θ(a−r)+kT\nm(n+δρ)∆Θ\n=−kT\nmδ(a−r)∂\n∂r(n+δρ) =kT\nm1\nr2∂\n∂r(r2δ(a−r)\n=−kT\nm1\nr2∂\n∂r/bracketleftbig\n(n+δρ2)r2δ(a−r)/bracketrightbig\n,(A5)\nwhere we used formulae ∇Θ(a−r) =−r\nrδ(a−r),r\nr∇=∂\n∂r. The next term in the right hand side of Eq. (18) can be\ntransformed into,\n−kT\nmδ(a−r)r\nr∇δρ2−ω2\np\n4πδ(a−r)r\nr∇/integraltextd3r1δρ(r1)\n|r−r1|\n=−kT\nmδ(a−r)∂\n∂rδρ−2−ω2\np\n4πδ(a−r)∂\n∂r/integraltextd3r1δρ(r1)\n|r−r1|.(A6)\nEq. (18) attains thus the form,\n∂2ρ2\n∂t2=−kT\nm1\nr2∂\n∂r/bracketleftbig\n(n+δρ2)r2δ(a−r)/bracketrightbig\n−kT\nmδ(a−r)∂\n∂rδρ−2−ω2\np\n4πδ(a−r)∂\n∂r/integraltextd3r1δρ(r1)\n|r−r1|.(A7)\nWe suppose the solution of the above equation in the form, δρ2=σ(Ω,t)δ(a+0+−r) and multiply both sides of this\nequation by r2and integrate with respect to rin arbitrary limits, i.e.,L/integraltext\nlr2dr..., such that a∈(l,L) (this integration\nremoves Dirac deltas), which leads to the equation,\na2∂2σ(Ω,t)\n∂t2=��kT\nmL/integraltext\nldr∂\n∂r/bracketleftbig\n(n+δρ2)r2δ(a−r)/bracketrightbig\n−kT\nmσ(Ω,t)L/integraltext\nlr2drδ(a−r)∂\n∂rδ(a−r)\n−ω2\np\n4πL/integraltext\nlr2drδ(a−r)∂\n∂r∞/integraltext\nar2\n1dr1/integraltext\ndΩδρ1(r2)\n|r−r1|\n−ω2\np\n4πL/integraltext\nlr2drδ(a−r)∂\n∂ra/integraltext\n0r2\n1dr1/integraltext\ndΩδρ1(r1)\n|r−r1|.(A8)\nTwo first terms in the right hand side of the above equation vanish, b ecause,\n−kT\nmL/integraldisplay\nldr∂\n∂r/bracketleftbig\n(n+δρ2)r2δ(a−r)/bracketrightbig\n=−kT\nm/bracketleftbig\n(n+δρ2)r2δ(r−a)/bracketrightbig\n|L\nl= 0 (A9)\nand\n−kT\nmσ(Ω,t)L/integraltext\nlr2drδ(a−r)∂\n∂rδ(a−r) =−kT\nma2L/integraltext\nldr1\n2∂\n∂rδ2(a−r)\n=−kT\nma2\n2δ2(a−r)|L\nl=−kT\nma2\n2limµ→01\nπµ\nµ2+(a−r)2δ(a−r)|L\nl= 0.(A10)\nTwolasttermsofr.h.s. ofEq. (A8)canbetransformedusingthefo rmula[26],1√\n1+z2−2zcosγ=∞/summationtext\nl=0Pl(cosγ)zl, for z <\n1, where Pl(cosγ) =4π\n2l+1l/summationtext\nm=−lYlm(Ω)Y∗\nlm(Ω) are Legendre polynomials. This formula leads to the following one,\n∂\n∂a1\n|a−r1|=\n\n∞/summationtext\nl=0lal−1\nrl+1\n1Pl(cosγ), for a < r 1,\n−∞/summationtext\nl=0(l+1)rl\n1\nal+2Pl(cosγ), for a > r 1,(A11)14\nwherea=ar\nr,cosγ=a·r1\nar1. Employing Eq. (A11), the last two terms in Eq. (A8) can be transfo rmed as follows,\n−ω2\np\n4πL/integraltext\nlr2drδ(a−r)∂\n∂r∞/integraltext\nar2\n1dr1/integraltext\ndΩ1δρ2(r1)\n|r−r1|\n=−ω2\np\n4πa2/integraltext\ndΩ1∞/integraltext\nar2\n1dr1δρ2(r1)∂\n∂a1√\na2+r2\n1−2ar1cosγ\n=−ω2\np\n4πa2/integraltext\ndΩ1∞/integraltext\nar2\n1dr1σ(Ω1)δ(a+0+−r1)∞/summationtext\nl=0lal−1\nrl+1\n1Pl(cosγ)\n=−ω2\np\n4πa2/integraltext\ndΩ1σ(Ω1)1\na2∞/summationtext\nl=04πl\n2l+1l/summationtext\nm=−lYlm(Ω)Y∗\nlm(Ω1)\n=−ω2\npa2∞/summationtext\nl=0l/summationtext\nm=−ll\n2l+1Ylm(Ω)/integraltext\ndΩ1σ(Ω1)Y∗\nlm(Ω1),(A12)\nand\n−ω2\np\n4πL/integraltext\nlr2drδ(a−r)∂\n∂ra/integraltext\n0r2\n1dr1/integraltext\ndΩ1δρ1(r1)\n|r−r1|\n=−ω2\np\n4πa2/integraltextdΩ1a/integraltext\n0r2\n1dr1nF(r1,t)(r1)∂\n∂a1√\na2+r2\n1−2ar1cosγ\n=ω2\np\n4πa2/integraltext\ndΩ1nF(r1,t)∞/summationtext\nl=0(l+1)rl\n1\nal+2Pl(cosγ)\n=ω2\npn∞/summationtext\nl=0l+1\n2l+1Ylm(Ω)a/integraltext\n0r2\n1dr1rl\n1\nal∞/summationtext\nl1=1l1/summationtext\nm1=−l1/summationtext\niAlmijl1(kl1ir1)sin(ωl1it)/integraltext\ndΩ1Y∗\nlm(Ω1)Yl1m1(Ω1)\n=ω2\npn∞/summationtext\nl=0l/summationtext\nm=−l/summationtext\nil+1\n2l+1Ylm(Ω)Almia/integraltext\n0rl+2\n1dr1\naljl(klir1)sin(ωlit).(A13)\nEquation (A8) attains thus the form,\n∂2σ(Ω,t)\n∂t2=−ω2\npa2∞/summationtext\nl=0l/summationtext\nm=−ll\n2l+1Ylm(Ω)/integraltext\ndΩ1σ(Ω1)Y∗\nlm(Ω1)\n+ω2\npn∞/summationtext\nl=0l/summationtext\nm=−l/summationtext\nil+1\n2l+1Ylm(Ω)Almia/integraltext\n0rl+2\n1dr1\naljl(klir1)sin(ωlit),(A14)\nAssuming now, σ(Ω,t) =∞/summationtext\nl=0l/summationtext\nm=−lqlm(t)Ylm(Ω) and putting it to the above equation, we obtain,\n∞/summationtext\nl=0l/summationtext\nm=−lYlm(Ω)∂2qlm(t)\n∂t2=−∞/summationtext\nl=0/summationtextl\nm=−lω2\npl\n2l+1Ylm(Ω)qlm(t)\n+ω2\np∞/summationtext\nl=1l/summationtext\nm=−l/summationtext\nil+1\n2l+1Ylm(Ω)Alma/integraltext\n0rl+2\n1dr1\nal+2jl(klir1)sin(ωlit).(A15)\nFrom the above equation we notice that for l= 0 we get∂2q00\n∂t2= 0 and thus q00(t) = 0 (as q(0) = 0 and lim t→∞q(t)<\n∞). Forl≥1 we get\n∂2qlm(t)\n∂t2=−ω2\npl\n2l+1qlm(t)+/summationdisplay\niω2\npl+1\n2l+1Almna/integraldisplay\n0rl+2\n1dr1\nal+2jl(klir1)sin(ωlit), (A16)\nwhich requires the solution form,\nqlm(t) =Blm/a2sin(ωp/radicalBig\nl\n2l+1t)\n+/summationtext\niAlm(l+1)ω2\np\nω2p−(2l+1)ω2\nlina/integraltext\n0rl+2\n1dr1\nal+2jl(klir1)sin(ωlit),(A17)\nandδρ2(r,t) =∞/summationtext\nl=1l/summationtext\nm=−lqlm(t)Ylm(Ω)δ(a−r). The first term in Eq. (A17) describes the self-oscillations of surf ace\nplasmons, whereas the second one displays the surface plasmon os cillations induced by the volume plasmons. This15\ninduced part of surface oscillations is nonzero only when the volume m odes are excited and their amplitudes, Almi,\nare nonzero. The frequencies of self-oscillations of the surface p lasmons are equal to ωl0=ωp/radicalBig\nl\n2l+1, corresponding to\nvarious multipole modes (numbered with l). 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B 31, p. 6360, 1985.\n[24] G. Mie, “Beitrige zur Optik tr¨ uber Medien, speziell ko lloidaler Metall¨ osungen,” Ann. Phys. 25, p. 376, 1908.\n[25] M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Elec tromagnetic energy transfer and switching in nanoparticle\nchain arrays below the diffraction limit,” Phys. Rev. B 62, p. R16356, 2000.\n[26] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals Series and Products , Academic Press, Inc., Boston, 1994."    },    {        "title": "1312.3100v1.Modelling_of_the_optical_properties_of_silver_with_use_of_six_fitting_parameters.pdf",        "content": "MODELING OF THE OPTI CAL PROPERTIES OF SI LVER WITH USE OF \nSIX FITTING PARAMETERS \n \n \nA.O.Melikyan \nInstitute of Mathematics and High Technologies, Russian-Armenian (Slavonic) State University, Yerevan \narmen_melikyan@hotmail.com   \nand  \nB.V.Kryzhanovsky \nDepartament of Nanotechnologies, Center of Optical Neural Technologies SRISA RAS, Moscow  \nkryzhanov@gmail.com   \n \n \nAbstract   We propose a realistic model of the optical properties of  of silver, in which inter-band transition with a \nthreshold energy of ~ 4 eV is described phenomenologically by an ensemble of oscillators with same  damping \nconstant  and a certain distribution of resonant frequencies in the interband transition threshold to infinity. The \ncontribution of the conduction electrons  in the dielectric  function  is de termined by the Drude formula. The \nproposed model actually contains the featur es of both the Drude-Lorentz model (Raki ć et al. 1998) and Tauc-\nLorentz model (Jian-Hong Qiu et al. 2005). However, unlike these works proposed model contains only six fitting parameters, with the square root of the mean square  deviation of the absorption coefficient and refractive index of silver from the experimental values in the range of  0.6 nm - 6.0 nm being of the order of  0.05. \n \n \n 1. INTRODUCTION \n Modeling of the optical properties of various \nmaterials and parallel comparison of results with \nexperimental data allows to identify the most important physical processes that determine the dependence of the \ndielectric function (DF) on frequency. Some progress in \nthis area has been made for the noble metals, which are \nintensively investigated in connection with applications in \nnanooptics and photonics [1-5]. In the low-frequency region of the spectrum where the photon energy lies below \nthe interband transitions threshold the behavior of the DF \nof silver is sufficiently well described by the Drude-like \nmodel in the range 1.8 ÷2.8 eV [4,5]. At the same time, at \nenergies above the threshold of interband transitions good \nagreement is achieved for gold with the use of two or four damped oscillator model in addition to the free electron \ncontribution. Another approach is based on the so called \ncritical point model [1-3]. A universal approach using 17 parameters for each metal was developed in [6]. \nIt is clear that the superposition of a sufficiently large number of Lorentz oscillators can describe properly the frequency dependence of the DF. However, the physical \nnature of the electron dispersion in the conduction band  \nand the d-band sufficiently differs from the Lorentz oscillator physics. In this connection it should be noted that \nin [7, 8] a better agreement is achieved  with use of the \nTauc-Lorentz model, which is a finite ensemble of Lorentz \noscillators with eigenfrequencies distributed continuously \nin the range from the threshold frequency to infinity. However in this model there is no Drude term describing \nthe contribution of the conduction electrons, and an \nadditional fitting parameter corresponding to the DF at \nasymptotically high photon energies is included.  \n In this paper we introduce a hybrid model, containing only six parameters in which the contribution of free electrons is described by the Drude formula with two adjustable parameters, and the contribution of the \ninterband transitions by an ensemble of Lorentz oscillators \nwith a continuous distribution of the eigenfrequencies and with the same damping constants. The density of states is \nchosen in such way that at electron energies close to the \nthreshold of interband absorption the dispersion law is \nquadratic, and at high energies nonparabolicity of the \nconduction band is taken into account. \n \n 2.  DESCRIPTION OF THE MODEL \n Let D  be the distribution function of the \neigenfrequencies of the oscillator, then DF of silver in the \nproposed model can be represented as ) (0ω\n∫∞\nγω − ω − ωω ω+γ + ω ωω− = ω ε\n02 2\n00 0\nD2\np\nid D\ni1) (\n) () (\np  (1) \nwhere  ω is the plasma frequency and Dγ Drude \nrelaxation constant. \n We consider the contributions to the dielectric \nconstant of silver ) (ωε  from the plasma oscillations of s-\nelectrons and the interband transition with the threshold \nenergy of ~ 4 eV of d-electrons. For the former we use the \nsimple Drude expression and the latter is the contribution \nof the interband  transitions.   \n We choose D ) (0ω in the following form \n⎩⎨⎧\nΔ < ωΔ ≥ ω ω ω= ωσ ω −\n00 0 p\n00e fD0\n,,) (/\nf    (2) \nΔ  a n d  γ, where , σ are fitting parameters of the \ninterband transition: Δ is the interband transition \nthreshold,   is oscillator strength, and f γ is the damping  \nconstant. Choice of the partition function in the form as in \nexpression (2) can be easily interpreted: if the partition \nfunction for the electron momentum is written as  \n, then taking into account the \ndependence of electron energy on momentum and \nsubstituting ,[] dp p /p p exp ~ W(p)dp2 2\n02−\n02pω →0 0 d dpω ω →/ ,   one can \nobtain σ →2\n0p\n0 0d D dp p Wωω →) ( ) ( . Finally we write down the \nexpression for the DF with six fitting parameters as follows \n ∫∞\nΔσ ω −\nγω − ω − ωω ω ω\n+γ + ω ωω− = ω εid e\nfi12 2\n00 0 p\nD2\np0/\n) () (  (3) \nObviously, the complex dielectric constant, defined by the \nexpression (3) satisfies Kramers-Kronig dispersion relations.      \n To determine the constants in the Eq. (3) one should compare the results of simulation  for the refractive index \n or extinction coefficient   \n) (ω =n n ) (ω =k k\n[]2 1\n12\n22\n121k/\n) ( ε − ε + ε = ω  \n  (4) \n[]2 1\n1222\n121\nn/\n) ( ε + ε + ε = ω  \nwhere  and , with the \nexperimental data. It is known (see e.g.  [9]) that the results \nof measuring  of the absorption coefficient are more \nreliable as compared with the da ta on the refractive index.  \nThe data from the different so urces such as [9-11] differ \neven in the long wavelength region, which is well described by the Drude term. At the same time, it turns out \nthat the extinction coefficient is more sensitive to values of \nthe parameters than the refrac tive index.  For these reasons \nwe carry out the fitting procedure using the data for the \nextinction coefficient  ) ( Reω ε = ε1 ) ( Imω ε = ε2\n) (ω=k k . The values obtained by \nthe least-squares method are given below: \n eV 022 0D .=γ , eV 042 9p .=ω ,  eV 050 4 .= Δ\n  (5) \n eV 260 0 .=γ ,    ,    V 935 9e.= σ 994 2 f.=\nIt is important to note that  the values of  Drude parameters \nDγ and   are determined from the best fit with the \nexperimental data in the range from 0.64 eV to 1 eV, \nwhere the contribution of  the interband transition can be neglected.  pω\nIt is found that the root-mean-square deviation of  ) (ωδk  \nvalue for the frequency range  0.64 ÷6.22eV is equal to 0.067. Calculations for the refractive index give close values. \n Figure 1 show respectively the results for real and imaginary parts of dielectric function, the refractive index \nand extinction coefficient according to the expressions (3)- \n(4) (solid lines) with use of the values of (5). The \ninterpolation curves for th e same quantities according to \nthe measured data are shown by dotted lines.    \n-8-4048\n0246\n eV energy Photon ,a  1ε 2ε\n 251/ε\n \n  \n0246\n0246-  Pali k\n eV energy Photon ,b ) (ωk\n ) (ω⋅n 4\n Fig.1. Real and imaginary parts of the optical \ndielectric function of Ag (a) and the refractive index \n) (ω=n n  and extinction coefficient ) (ω=k k  of \nAg (b). Solid lines correspond to the expressions (3) and (4); dotted lines are the experimental data points from Jonson and  Christy [9] and for \n] . , . [eV 5 4 eV 5 3 ∈ωh  from Palik [10]. The values \nof the fitting parameters are given in (5). \n \n 3. RESULTS AND DISCUSSION \n First, it can be seen that the results of calculations \nusing the expressions (3) and (4) manifest excellent \nagreement with the experimental data. In addition, the \nvalues of the Drude parameters  and pωDγ estimated by \nthe least square method are very close to those extracted \nfrom the experimental data (see [9]). It is interesting to \nnote that the value of Dγ for Ag determined in [9] from \nthe optical measurements agrees with that extracted from \nthe measurement of conductivity [12].  \n Note that (3) describes adequately the optical \nproperties of silver only at frequencies below 6 eV. At high \nfrequencies, the new terms should be added similar to the \nintegral in (3) describing the contribution of interband \ntransitions at higher frequencies. The analysis shows that \nthe contribution of the next interband transition (with a  \npeak at ~ 14 eV [13]) becomes significant (~ 10%) even at \nfrequencies close to 6 eV.  \nIn Fig. 2 we present the results of calculations \nbased on (3)-(5) for the transmission T and reflection R \ncoefficients of silver films of thickness 20 and 50 nm on a \nquartz substrate. Solid lin es correspond to the \ntransmittance and reflectance of films with thickness of 20 \nand 50 nm, respectively,  calculated with use of (3) – (5), \nwhile the squares and diamonds correspond to the data of \n[9].  \n \n \n \n It should be noted that the fitting procedure based on \nthe expression (3) with use of  the SOPRA data [11] leads \nto the worst match even at low frequencies.In particular it \ngives for the relaxation constant  Dγ the value of  \nwhich is four times more than the value of [9]. This discrepancy in the data is easily explained. The matter is that the measurement of th e absorption coefficient is \ncarried out indirectly - on the transmission of a thin film. A transmission (extinction) is determined by the absorption in the film and the scattering by inhomogeneities, which is difficult to measure and, moreover, to account for. For this \nreason deposition of thin silver film is recommended to \nperform at low (less than 150°C) temperature of the substrate in order to minimize a film crystallite size and \nresulting dispersion. (for Au see [9], for Ag - [14]). The \npresence of impurities should be accounted for as well. It is especially difficult to get rid of the impurities of copper, \nwhich, even in the case of ma king a film of high purity \nsilver (~ 99.99) can make a significant contribution to the \nmeasured value of the imaginary component of the \ndielectric constant. For exam ple on the dispersion curve \n) ( Im\n085 0.ωε=ε2  shown in [9], a flat maximum in the \nnge eV 2 \n0.000.250.500.751.00\n024 nm20d=\n eV energy Photon ,T frequency ra ~ω  is clearly visible which  \ncorresponds to the m  (with the value of ~ 6) of the \nimaginary part of the diel ectric function of copper. \n aximum\n 4. CONCLUSION \n Realistic phenomenological model of the permittivity \no ver as a function of frequency is proposed that takes f sil\ninto account qualitatively the band structure. The values of \nsix fitting parameters are determined using  the least-\nsquare method and the experimental data of [9].  The \nmodel operate with only six parameters for Drude term and \none interband transition with threshold energy of about 4 \neV, and provides excellent agreement with the data of [9] \nin the range of photon energy from 0.62 eV to 6.22 eV. At \nthe same time more complicated model [6] in which 21 \nfitting parameters are used gives very large deviations \nfrom the experimental data. From this we can conclude \nthat the description under the proposed model (3) - (5)  fits \nbetter with the physics of the process.  6R \n \n0.000.250.500.751.00\n024 nm50d=\n eV energy Photon , \n 5.  ACKNOWLEDGMENTS \n This paper is supported in part by the  Presidium of \nR (projects #1.8 and 2.1). \n AS \nReferences \ne Ru E C, and Meyer M.  Erratum: An \n [2] omparison of gold and silver \n [3] Description of dispersion \n [4] and Weber W H.   Electromagnetic \n [5] .  Theory of surface enhanced Raman \n [6]  D, Djurisic, A B et al . Optical properties of metal \n [7] u, Xiao-Yong Gao et al. \n [8] S., \nStchakovsky M.  Optical Characterization  of   HfO2 by  \n [1] Etchegoin  P G, L\n6R \nT analytic model for the optical properties of gold. J. Chem. \nPhys 125  164705 , 2006. \nVial A., and Laroche T. C\ndispersion laws suitable  for FDTD simulations. Applied \nPhysics B 93  139-143,  2008. \nVial A. and Laroche T. \nproperties of metals by means of  the critical points \nmethod and application to the st udy of resonant structures \nusing the FDTD method.  J. Phys. D: Appl. Phys. 40  7152-8, 2007. \nFord G W, Fig 2. Transmission T   and refraction R  coefficients \nof silver films of thickness 20 nm (a) and 50 nm (b) on quartz substrate. Solid lines correspond to the \ntransmittance and reflectance calculated with use of \n(3) –(5), squares and diamonds correspond to the \ndata \npoints of [9]. interactions of molecules with metal surfaces. Phys. Rep. 113 195-287, 1984. \nRojas R, and Claro F\nscattering in colloids . J.Chem. Phys. 98 (2)  998-1006, \n1993. \nRakic A\nfilms for vertical-cavity optoelectronic devices. Applied Optics 37 5271-5283, 1998. \nJian-Hong Qiu, Peng Zho\nEllipsometric Study of the Optical Properties of Silver \nOxide Prepared by Reactive Magnetron Sputtering. J. \nKorean Phys. Soc., Vol. 46,  pp. S269\u001fS275, 2005. \nSancho-Parramon J., Modreanu M., Bosch  \nspectroscopic ellipsometry: Di spersion models and direct \ndata inversion. Thin  Solid Films, v. 516, pp.7990-7995, \n2008. \n [9] \n Phys. Rev. B 6, pp.4370–9, 1972. \n \n [12] . D. Solid State Physics  \nUsing Surface Johnson P B, Christy R W. Op tical constants of the noble \nmetals.\n [10] Palik E D (ed) Handbook of Optical Constants of Solids, \n(New York:  Academic). 1985. \n [11] http://www.sspectra.com/files/misc/win/SOPRA.EXE  \nAshcroft N. W., and Mermin N\n(Saunders College P ublishing) p.10, 1976. \n [13] Ehrenreich H, and Philipp H R  Optical properties of Ag \nand Cu. Phys. Rev. 128   1622-9 , 1962. \n [14] Palagushkin A N, Prokopenko et al.   Measurement of \nMetal nanolayers Optical Parameters \nPlasmon  Resonance Method. Optical Memory and Neural \nNetworks (Information Optics) 16  288-294, 2007.  "    },    {        "title": "1902.08700v1.Strongly_Enhanced_Gilbert_Damping_in_3d_Transition_Metal_Ferromagnet_Monolayers_in_Contact_with_Topological_Insulator_Bi2Se3.pdf",        "content": "1 \n Strongly Enhanced Gilbert Damping in 3 d Transition Metal \nFerromagnet Monolayers in Contact with Topological Insulator Bi 2Se3 \nY. S. Hou1, and R. Q. Wu1 \n1 Department of Physics and Astronomy, University of California, Irvine, California \n92697 -4575, USA  \n \nAbstract  \nEngineering Gilbert damping of ferromagnetic metal films is of great importance to \nexploit and design spintronic devices that are operated with an ultrahigh speed. Based on \nscattering theory of Gilbert damping, we extend the torque method originally  used in \nstudies of magnetocrystalline anisotropy to theoretically determine Gilbert dampings of \nferromagnetic metals. This method is utilized to investigate Gilbert dampings of 3 d \ntransition metal ferromagnet iron, cobalt and nickel monolayers that are co ntacted by the \nprototypical topological insulator Bi 2Se3. Amazingly, we find that their Gilbert dampings \nare strongly enhanced by about one order in magnitude, compared with dampings of their \nbulks and free -standing monolayers, owing to the strong spin -orbit coupling of Bi 2Se3. \nOur work provides an attractive route to tailoring Gilbert damping of ferromagnetic \nmetallic films by putting them in contact with topological insulators.  \n  \n \n \n \n \nEmail: wur@uci.edu   \n \n \n \n \n \n 2 \n I. INTRODUCTION  \nIn ferromagnets, the time -evolution of their magnetization M can be described by the \nLandau -Lifshitz -Gilbert (LLG) equation [1-3]  \n1Meff\nSdd\ndt dt   MMM H M\n, \nwhere \n0B g  \n  is the gyromagnetic ratio, and \nMSM  is the saturation \nmagnetization. The first term describes the precession motion of magnetization M about \nthe effective magnetic field, Heff, which  includes contributions from external field, \nmagnetic anisotropy, exchange, dipole -dipole and Dzyaloshinskii -Moriya  interactions [3]. \nThe second term represents the decay of magnetization prece ssion with a dimensionless \nparameter \n , known as the Gilbert damping [4-8]. Gilbert  damping is known to be \nimportant for the performance of various spintronic devices such as hard drives, magnetic \nrandom access memories, spin filters, and magnetic sensors [3, 9, 10]. For example, \nGilbert damping in the free layer of reader head in a magnetic hard drive determines its \nresponse speed and signal -to-noise ratio [11, 12]. The bandwidth, insertion loss , and \nresponse time of a magnetic thin film microwave device also critically depend on the \nvalue of \n  in the film [13]. \n \nThe rapid developm ent of spintronic technologies calls for the ability of tuning Gilbert  \ndamping in a wide range. Several approaches have been proposed for the engineering of \nGilbert damping in ferromagnetic (FM) thin films, by using non -magnetic or rare earth \ndopants, addi ng differ ent seed layers for growth, or adjusting composition ratios in the \ncase of alloy films [9, 14-16]. In par ticular, tuning \n  via contact with other materials \nsuch as heavy metals, topological insulators (TIs), van der Waals monolayers or magnetic \ninsulators is promising as the selection of material combinations is essentially unlimited.  \nSome of these materials may have fundamentally different damping mechanism and offer \nopportunity for studies of new phenomena such as spin -orbit torque, spin -charge \nconversion, and thermal -spin-behavior [17, 18].  \n \nIn this work, we systematically investigate the effect of Bi 2Se3 (BS), a prototypical TI, on \nthe Gilbert damping of 3d transitio n metal (TM) Fe, Co and Ni monolayers (MLs) as they 3 \n are in contacted with each other. We find that the Gilbert dampings in the TM/TI \ncombinations are enhanced by about an order of magnitude than their counterparts in \nbulk Fe, Co and Ni as well as in the fr ee-standing TM MLs.  This drastic enhancement \ncan be attributed to the strong spin -orbit coupling (SOC) of the TI substrate and might \nalso be related to its topological nature . Our work introduces an appealing way to \nengineer Gilbert dampings of FM metal fi lms by using the peculiar physical properties of \nTIs.  \n \nII. COMPUTATIONAL DETAILS  \nOur density functional theory (DFT) calculations are carried out using the Vienna Ab-\ninitio  Simulation Package (VASP) at the level of the generalized gradien t approximation \n[19-22]. We treat Bi -6s6p, Se -4s4p, Fe -3d4s, Co -3d4s and Ni -3d4s as valence electrons \nand employ the projector -augmented wave pseudopotentials to d escribe core -valence \ninteractions [23, 24].  The energy cutoff of plane -wave expansion is 450 eV [22]. The BS \nsubstrate is simulated by five quintuple layers ( QLs), with an in -plane lattice constant of \naBS = 4.164 Å and a vacuum space of 13 Å between slabs along the normal axis. For the \ncomputational convenience, we put Fe, Co and Ni MLs on both sides of the BS slab. For \nthe structural optimization of the BS/TM slabs, a 6× 6× 1 Gamma -centered k -point grid is \nused, and the positions of all atoms except those of the three central BS QLs are fully \nrelaxed with a criterion that the force on each atom is less than 0.01 eV/Å. The van der \nWaals (vdW) correction in the form of the nonlocal vdW functional (optB86b -vdW) [25, \n26] is included in all calculations.  \n \nThe Gilbert dampings are determined by extending the torque method that we developed \nfor the study of magnetocrystalline anisotropy [27, 28]. To ensure the numerical \nconvergence, we use very dense Gamma -centered k -point grids and, furthermore, large \nnumbers of unoccupied bands. For example, the first Bri llouin zone of BS/Fe is sampled \nby a 37× 37× 1 Gamma -centered k -point grid, and the number of bands for the second -\nvariation step is set to 396, twice of the number (188)  of the total valence electrons. More \ncomputational details are given in Appendix A. Mag netocrystalline anisotropy energies \nare determined by computing total energies with different magnetic orientations [29].  4 \n  \nIII. TORQUE METHOD OF DETERMINING GILBERT DAMPING  \nAccording to the scattering theory of Gilbert damping [30, 31], the energy dissipation \nrate of the electroni c system with a Hamiltonian, H(t), is determined by  \n dis 2i j j i F i F j\nijE E E E Euu\n         \nHHuu\n. \nHere, EF is the Fermi level and \nu  is the deviation of  normalized magnetic moment away \nfrom its equilibrium, i.e., \n0m m u  with \n00 MsM m . On the other hand, the time \nderivative of the magnetic energy in the LLG equation is [32] \n mag 3S\neffM dEdt\n MH\n mm\n. \nBy taking \ndis magEE\n  , one obtains the Gilbert damping as:  \n4i j j i F i F j\nij SE E E EM u u\n        \nHH\n. \nNote that, to obtain Eq. (4), we use \n mu  since the eq uilibrium normalized \nmagnetization m0 is a constant. In practical numerical calculations, \nFEE  is \ntypically substituted by the Lorentzian function \n 22\n0 0.5 0.5 L          . \nThe half maximum parameter, \n1 , is adjusted to reflect different scattering rates of \nelectron -hole pairs created by the precession of magnetization M [10]. This procedure  \nhas been already used in several ab initio  calculations for Gilbert dampings of metallic \nsystems [8, 9, 32-35], where the electronic responses play the major role for energy \ndissipation .  \n \nIn this work, we focus on the primary Gilbert damping in FM metals that arises from \nSOC [10, 36-38]. There are two important effects in a uniform precession of \nmagnetization M, when SOC is taken into consideration. The first is the F ermi surface \nbreathing as M rotates, i.e., some occupied states shift to above the Fermi level and some \nunoccupied states shift to below the Fermi level. The second is the transition between \ndifferent states across the Fermi level as the precession can be viewed as a perturbation to 5 \n the system. These two effects generate electron -hole pairs near the Fermi level and their \nrelaxation through lattice scattering leads to the Gilbert damping.  \n \nNow we demonstrate how to obtain the Gilbert damping due to SOC thro ugh extending \nour previous torque method [27]. The general Hamiltonian in Eq. (4) can be replaced by \n SOC j j jr\n H l s\n [4, 27] where  the index j refers to atoms, and \njji lr and s \nare orbital and spin operators, respectively. This is in the same spirit for the determination \nof the magnetocrystal line anisotropy [27], for which  our torque meth od is recognized as a \npowerful tool in the framework of spin -density theory [27]. When m points at the \ndirection of \n , , ,x y zm m m n , the term \nls  in HSOC is written as follows:   \n22\n2211cos sin sin22\n1sin sin cos 52 2 2\n1sin cos sin2 2 2ii\nz\nii\nz\nii\nzs l l e l e\ns l l e l e\ns l l e l e\n\n  \n\n\n\n\n  \n\n     \n   \n  \nn ls\n \nTo obtain the derivatives of H in Eq. (4), we assume that the magnitude of M is constant \nas its direction changes [36]. The processes of getting angular derivatives of H are \nstraightforward and the results are given by Eq. (A1) -(A5) in Appendix B.  \n \nIV. RESULTS AND DISCUSSION  \nIn this section, we first show that our approach of determining  Gilbert damping works \nwell for  FM metals such as 3d TM Fe, Co and Ni bulks. Following that, we demonstrate \nthe strongly enhanced Gilbert dampings of Fe, Co and Ni MLs due to the contact with BS \nand then discuss the underlying physical mechanism of these enhancements.  \n \nA. Gilbe rt dampings of 3d TM Fe, Co and Ni bulks  \nGilbert dampings of 3d TM bcc Fe, hcp Co and fcc Ni bulks calculated by means of our \nextended torque method are consistent with previous theoretical results [10]. As shown in \nFig. 1, the intraband contributions decrease whereas the interband contributions increase \nas the scattering rate \n  increases. The minimum values of  \n have the same magnitude 6 \n as those in Ref. [10] for all three metals, showing the applicability of our approach for the \ndetermination of Gilbert dampings of FM metals.  \n \n \nFigure 1 (color online) Gilbert dampings of (a) bcc Fe, (b) hcp Co and (c) fcc Ni bulks. Black \ncurves give the total Gilbert damping. Red and blue curves give the intraband and interband \ncontributi ons to the total Gilbert damping, respectively.  \n \nB. Strongly enhanced Gilbert dampings of Fe, Co and Ni MLs in contact with BS  \nWe now investigate the magnetic properties of heterostructures of BS and Fe, Co and Ni \nMLs. BS/Fe is taken as an example and its atom arrangement is shown in Fig. 2a. From \nthe spatial distribution of charge density difference \nBS+Fe-ML BS Fe-ML        in Fig. \n2b, we see that there is fairly obvious charge transfer between Fe and the topmost Se \natoms. By taking  the average of \n  in the xy plane, we find that charge transfer mainly \ntakes place near the interface (Fig.2c). Furthermore, the charge transfer induces non -\nnegligible magnetization  in the topmost QL of BS (Fig. 2b). Similar charge  transfers and \ninduced magnetization are also found in BS/Co and BS/Ni (Fig. A1 and Fig. A2 in \n7 \n Appendix C). These suggest that interfacial interactions between BS and 3 d TMs are very \nstrong.  Note that BS/Fe and BS/Co have in -plane easy axes whereas the BS/ Ni has an \nout-of-plane one.     \n \n \nFigure 2 (color online) (a) Top view of atom arrangement  in BS/Fe. (b) Charge density difference \n\n near the interface in BS/Fe. Numbers give the induced magnetic moments (in units of \nB ) in \nthe top most  QL BS. Color bar indicates the weight of negative (blue) and positive (red) charge \ndensity differences. (c) Planer -averaged charge density difference \n in BS/Fe. In (a), (b), (c), \ndark green, light gra y and red balls represent Fe, Se and Bi atoms, respectively.  \n \nFig. 3a and 3b show the \n  dependent Gilbert dampings of BS/Fe, BS/Co and BS/Ni. It is \nstriking that Gilbert dampings of BS/Fe, BS/Co and BS/Ni are enhanced by about one  or \ntwo order s in magnitude  from the counterparts of Fe, Co and Ni bulks as well as their \nfree-standing MLs, depending on the choice of scattering rate in the range from 0. 001 to \n1.0 eV.  Similar to Fe, Co and Ni bulks, the intraband contributions monotonically  \ndecrease while the interband contributions increase as the scattering rate \n  gets larger \n(Fig. A3 in Appendix D).  Note that our calculations indicate that there is no obvious \ndifference between the Gilbert dampings of BS/Fe when f ive- and six -QL BS slabs are \nused (Fig. A4 in Appendix E). This is consistent with the experimental observation that \nthe interaction  between the top and bottom topological surface states is negligible  in BS \nthicker than five QLs [39].  \n \n8 \n  \nFigure 3 (color online) Scattering rate \n  dependent Gilbert dampings of (a) Fe ML, bcc Fe bulk, \nBS/Fe and PbSe/Fe, (b) Co ML, hcp Co bulk, BS/Co, Ni ML, fcc Ni bulk and BS/Ni. (c) \nDependence of the Gilbert dampin g of BS/Fe on the scaled SOC \nBS  of BS in the range from \nzero (\nBS0 ) to full strength (\nBS1 ). Solid lines show the fitting of Gilbert damping \nBS/Fe  \nto Eq. (6). The inse t shows Gilbert damping comparisons between BS/Fe at \nBS0 , bcc Fe bulk \nand Fe ML.  \n \nAs is well -known, TIs are characterized by their strong SOC and topologically nontrivial \nsurface states. An important issue is how they affect the Gilbert damping s in BS/TM \nsystems. Using BS/Fe as an example, we artificially tune the SOC parameter \nBS of BS \nfrom zero to full strength and fit the Gilbert damping \nBS/Fe  in powers of \nBS  as \n2\nBS/Fe 2 BS BS/Fe BS 0 (6)       \n. \nAs shown in Fig. 3c, we obtain two interesting results: (I) when \nBS  is zero, the \ncalculated residual Gilbert damping \nBS/Fe BS 0    is comparable to Gilbert dampings of \nbcc Fe bulk and Fe free -standin g ML (see the inset in Fig. 3c) ; (II) Gilbert damping \nBS/Fe\n increases almost linearly with \n2\nBS  , simi lar to previous results [36]. These reveal \nthat the strong SOC of BS is crucial for the enhancement of Gilbert damping.    \n  \nTo gain insight i nto how the strong SOC of BS affects the damping of BS/Fe, we explore \nthe k-dependent contributions to Gilbert damping, \nBS/Fe . As shown in Fig. 4a , many \nbands near the Fermi  level show strong intermixing between Fe and BS orbitals (mar ked \nby black arrows ). Accordingly, these k-points have large contributions to the Gilbert \n9 \n damping  (marked by red arrows in Fig. 4b). However, if the hybridized states are far \naway from the Fermi level, they make almost zero contribution to the Gilbert damp ing. \nTherefore, we conclude that only hybridizations at or close to Fermi level have dominant \ninfluence on the Gilbert damping. This is understandable, since energy differences EF-Ei \nand EF-Ej are important in the Lorentzian functions in Eq. (4).   \n \n \nFigu re 4 (color online) (a) DFT+SOC calculated band structure of BS/Fe. Color bar indicates \nthe weights of BS (red) and Fe ML (blue). Black dashed line indicates the Fermi level. (b) k -\ndependent contributions to Gilbert damping \nBS/Fe at sc attering rate \n26meV . Inset shows \nthe first Brillouin zone and high -symmetry k -points \n , \nK and \n .  \n \nIt appears that there is no direct link between the topologic al nature of BS and the strong \nenhancement of Gilbert damping. The main contributions to Gilbert damping are not \nfrom the vicinity around the \n -point,  where the topological nature of BS manifests.  \nBesides,  BS should undergo a topol ogical phase transition from trivial to topological as \nits SOC \nBS  increases [40]. If the topological nature of BS dictates the e nhancement of \nGilbert damping, one should expect a kink in the \nBS  curve at this phase transition \npoint but this is obviously absent i n Fig. 3c.  \n \n10 \n To dig deeper into this interesting issue, we replace the topologically nontrivial BS with a \ntopologically trivial insulator PbSe, because the latter has a nearly the same SOC as the \nformer. As shown in Fig. 3a, the Gilbert damping of PbSe/Fe is noticeably smaller than \nthat of BS/Fe, although both are significantly enhanced from the values of \n  of Fe bulk \nand Fe free -standing ML. Taking the similar SOC and surface geometry between BS and \nPbSe  (Fig. A5 in Appendix F) , the large difference between the Gilbert dampings of \nBS/Fe and PbSe/Fe suggests that the topological nature of BS still has an influence on \nGilbert damping. One possibility is that the BS surface is metallic with the presence of \nthe time -reversal protected t opological surface states and hence the interfacial \nhybridization is stronger.      \n \n \nFigure 5 (color online) Comparisons between Gilbert damping \n  of BS/Fe at \n26meV  and \n(a) total DOS, (b) Fe projected DOS and (c) BS projected DOS. Red arrows and light cyan \nrectangles highlight the energy windows where Gilbert damping \n  and the total DOS and Fe \nPDOS have a strong correlation . In (a), (b) and (c), all DOS are in units of state per eV and \nFermi level EF indicated by the vertical green lines is set to be zero.  \n \nA previous study of Fe, Co and Ni bulks suggested a strong correlation between Gilbert \ndamping and total density of states (DOS) around the Fermi level [36]. To attest if this is \n11 \n applicable here, we show the total DOS and Gilbert damping \nBS/Fe  of BS/Fe as a \nfunction of the Fermi level based on the rigid band a pproximation. As shown in Fig. 5a, \nGilbert damping \nBS/Fe and the total DOS behave rather differently in most energy \nregions. From the Fe projected DOS (PDOS) and BS projected PDOS (Fig. 5b and 5c), \nwe see a better  correlation between G ilbert damping \nBS/Fe and Fe -projected DOS, \nespecially in regions highlighted by the cyan rectangles . We perceive that although the \n\n-DOS correlation might work for simple systems, it doesn’t hold when hybridiza tion and \nSOC are complicated as the effective SOC strength may vary from band to band.  \n \nV. SUMMARY  \nIn summary, we extend our previous torque method from determining magnetocrystalline \nanisotropy energies [27, 28] to calculating Gilbert da mping of FM metals and apply this  \nnew approach to Fe, Co and Ni MLs in contact with TI  BS. Remarkably, the presence of \nthe TI BS substrate causes order of magnitude enhancements in their Gilbert dampings. \nOur studies demonstrate such strong enhancement is mainly due to the strong SOC of TI \nBS substrate . The topological nature of BS may also play a role by facilitati ng the \ninterfacial hybridiz ation and leaving more states around the Fermi level . Although \nalloying with heavy elements also enhances Gilbert dampings  [32], the use of  TIs pushes \nthe enhancement into a much wide r range. Our work thus establishes an attractive way \nfor tuning the Gilbert damping of FM metallic films, especially in the ultrathin reg ime.  \n \nACKNOWLEDGMENTS  \nWe thank Prof. A. H. MacDonald and Q. Niu at University Texas, Austin, for insightful \ndiscussions. We also thank Prof. M. Z. Wu at Colorado State University and Prof. J. Shi \nat University of California, Riverside for sharing their ex perimental data before \npublication. Work was supported by DOE -BES (Grant No. DE -FG02 -05ER46237). \nDensity functional theory calculations were performed on parallel computers at NERSC \nsupercomputer centers.   \n \n 12 \n Appendix A: Details of Gilbert damping calcula tions  \nTo compare Gilbert dampings of Fe, Co and Ni free -standing MLs with BS/Fe, BS/Co, \nand BS/Ni, we use \n33  supercells containing three atoms and set their lattice \nconstants to 4.164 Å, same as that of the BS substrate. This means that the lattice \nconstant of their primitive unit cell (containing one atom) is fixed at 2.40 Å. The relaxed \nlattice constants of Fe (2.42 Å), Co (2.35 Å) and Ni (2.36 Å) free -standing MLs are close \nto this value.   \nSystems  a (Å)  b (Å) c (Å) k-point grid            \nFe bulk  2.931  2.931  2.931  35× 35× 35  16 36 2.25 \n Co bulk  2.491  2.491  4.044  37× 37× 23  18 40 2.22 \nNi bulk  3.520  3.520  3.520  31× 31× 31  40 80 2.00 \nFe ML  4.164  4.164  -- 38× 38× 1  24 56 2.33 \nCo ML  4.164  4.164  -- 37× 37× 1  27 64 2.37    \nNi ML  4.164  4.164  -- 39× 39× 1  30 72 2.40 \nBS/Fe  4.164  4.164  -- 37× 37× 1  188 396 2.11 \nBS/Co  4.164  4.164  -- 37× 37× 1  194 408 2.10 \nBS/Ni  4.164  4.164  -- 37× 37× 1  200 432 2.16 \nPbSe/Fe  4.265  4.265  -- 37× 37× 1  174 376 2.16 \n \nTable A1. Here are details of Gilbert damping calculations of all systems that are studied \nin this work.     is abbreviated for the number of  valence electrons and     stands for the \nnumber of total bands.   is the ratio between     and    , namely,          . Note that \nfive QLs of BS are used in calculations for BS/Fe, B S/Co and BS/Ni.   \n \n \n \n \n \nAppendix B: Derivatives of SOC Hamiltonian HSOC with respect to the \nsmall deviation \nu  of magnetic moments  \nBased on the SOC Hamiltonian HSOC in Eq. (5) in the main text, derivatives of the term \nls\n against the polar angle \n  and azimuth angle \n  are    13 \n \n11sin cos cos22\n1 1 1cos sin sin A1 ,2 2 2\n1 1 1cos sin sin2 2 2ii\nnz\nii\nz\nii\nzs l l e l e\ns l l e l e\ns l l e l e\n\n  \n  \n  \n\n\n  \n\n       \n   \n  \nls \nand \n \n  \n 22\n22110 sin sin22\n10 sin cos A2 .2 2 2\n10 cos sin2 2 2ii\nn\nii\niis i l e i l e\ns i l e i l e\ns i l e i l e\n\n\n\n\n\n\n  \n\n         \n     \n    \nls\n \nNote that magnetization M is assumed to have a constant magnitude when it precesses , so \nwe have \n0SOC SOC    H M H m . When the normalized magnetization  m points \nalong the direction of \n , , ,x y zm m m n , we have: \nsin cosxm , \nsin sinym  \nand \ncoszm . Taking \n0m m u  and the chain rule together, we obtain derivatives of \nSOC Hamiltonian HSOC with respect to the small deviation  of magnetic moments as \nfollows:   \nsincos cos A3 ,sinSOC SOC SOC SOC SOC\nx x x x x\nSOC SOCu m m m m\n\n                   \nH H H H H m\nm\nHH\n \ncoscos sin A4 ,sinSOC SOC SOC SOC SOC\ny y y y y\nSOC SOCu m m m m\n\n                   \nH H H H H m\nm\nHH\n \nand  \nu14 \n \n sin A5 .SOC SOC SOC SOC SOC\nz z z z z\nSOCu m m m m\n\n                 \nH H H H H m\nm\nH \nCombining Eq. (5) and Eq. (A1 -A6), we can easily obtain the final formulas of \nderivatives of SOC Hamiltonian HSOC of magnetization m. \n \n \n \n \nAppendix C: Charge transfers and induced magnetic moments in BS/Fe, \nBS/Co and BS/Ni  \n \nFigure A1 (color online) Planar -averaged char ge difference \nBS TM ML BS TM-ML         \n(TM = Fe, Co and Ni) of  (a) BS/Fe, (b) BS/Co and (c) BS/Ni . The atoms positions are given along \nthe z axis.  \n \n15 \n  \nFigure A2 (Color online) Charge density difference \nBS TM ML BS TM ML          (TM = Fe, \nCo and Ni) nea r the interface betwee n the TM monolayer and the top most  QL BS of (a) BS/Fe, (b) \nBS/Co and (c) BS/Ni. The color bar shows the weights of the negative (blue) and positive (red) \ncharge density differences. Numbers give the induced magnetic moments (in units  of \nB ) in the \ntopmost  QL BS. Bi and Se atoms are shown by the purple and light green balls, respectively.  \n \n \nAppendix D: Contributions of intraband and interband to the Gilbert \ndampings of BS/Fe, BS/Co and BS/Ni  \n \nFigure A3 (color  online) Calculated Gilbert dampings of (a) BS/Fe, (b) BS/Co and (c) BS/Ni. \nBlack curves give the total damping. Red and blue curves give the intraband and interband \ncontributions, respectively.  \n16 \n Appendix E: Gilbert dampings of BS/Fe with five - and six -QLs of BS slabs  \n8  \nFigure A4 (color online). Gilbert dampings of BS/Fe with five (red) and six (black) QLs of BS \nslabs. In the calculations of the Gilbert damping of BS/Fe with six QLs of BS, we use a 39 ×39×1 \nGamma -centered k -point grid, and the number of the  total bands is 448 which is twice \nlarger than the number of the total valence electrons (216).    \n \nAppendix F: Structural c omparisons between BS/Fe and PbSe/Fe  \n \n17 \n Figure A5 (color online) (a) Top view and (c) side view of atom arrangement  in BS/Fe. (b) Top  \nview and (d) side view of atom arrangement  in PbSe/Fe. In (a) and (c), the xyz -coordinates are \nshown by the red arrows. In (b) and (d), the rectangles with blue dashed lines highlight the most \ntop QL BS in BS/Fe which is similar to the Pb and Se atom laye rs in PbSe/Fe. 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Ohmic damping of the oscilla tor can be\nexactly treated with a 1D scalar field reservoir, whereas general n on-Ohmic damping\nis conveniently treated with a continuum reservoir of harmonic oscilla tors. Using the\ndiagonalizedHamiltonianofthe totalsystem, wecalculateanumbero fthermodynamic\nquantities for the damped oscillator: the mean force internal ener gy, mean force free\nenergy, and another internal energy based on the free-oscillato r Hamiltonian. The\nclassicalmean force energyis equal to that ofa free oscillator, fo r both Ohmic and non-\nOhmic damping and no matter how strong the coupling to the reservo ir. In contrast,\nthe quantum mean force energy depends on the details of the damp ing and diverges\nfor strictly Ohmic damping. These results give additional insight into t he steady-\nstate thermodynamics of open systems with arbitrarily strong cou pling to a reservoir,\ncomplementing results for energies derived within dynamical approa ches (e.g. master\nequations) in the weak-coupling regime.\nPACS numbers: 03.65.-w, 03.65.Yz, 03.70.+k\n1. Introduction\nDamped oscillators are of importance in numerous experimental and natural settings\nand their dynamics has been extensively modelled [1, 2]. Various appro aches to the\nquantization of such oscillators have been explored. For example, o ne approach deals\nsolely with the oscillator as the total dynamical system in which case t he energy is not\nconserved [3, 4]. Other approaches add additional degrees of fre edom, i.e. different kinds\nof reservoirs, in addition to the system of interest. These reserv oir approaches allow the\nquantization of systems with dissipation to bedeveloped with a time-in dependent global\nHamiltonian, which offers advantages in applying the quantum formalis m. The usual\npractice to arrive at damped motion is to couple the oscillator to addit ional harmonic\noscillators, eitheradiscreteset[5,6,7,8,9,2]oracontinuum[10, 11,12,13]. Alternative\nmodels couple the oscillator to a scalar field [14]. The quantization tech niques\nemployed with reservoirs include path integrals [6, 7], canonical qua ntization [10], and\nphenomenological approaches [9].Thermal energies of classical and quantum damped oscillato rs coupled to reservoirs 2\nThe aim of this paper is to derive thermal-equilibrium energies and free energies of\ndamped oscillators in the classical and quantum regimes. We employ bo th a scalar field\nreservoir [14] and the continuum reservoir [10] , and our approach throughout is based\non canonical quantization and diagonalization of the total Hamiltonia n. Contrary to\nmany master equation approaches which assume an initial state of t he oscillator and\nreservoir that is a product state [7, 18, 14, 2, 21], we will here ass ume that the global\nsystem is in a thermal equilibrium state, implying that it is not in a produc t state\nbecause of the coupling. We consider two different definitions of the thermal energy\nof a damped oscillator: the mean-force energy, which includes a con tribution from the\noscillator-reservoir coupling term in the Hamiltonian [15, 16, 17], and a n energy based\nsolely on the oscillator part of the Hamiltonian. A case of particular int erest is Ohmic\ndamping of the oscillator, where the damping is proportional to veloc ity, but we also\nwish to provide expressions for thermal energies for non-Ohmic da mping. It is well\nknown that the energy of a quantum oscillator with Ohmic damping is div ergent [22], a\nproblem usually treated by the introduction of a cut-off frequency [7, 22, 2] that changes\nthe damping from strictly Ohmic damping. Interestingly, in the most w idely used\ntreatment of damped motion, which begins with a discrete set of har monic oscillators as\na reservoir [7, 2], even the classical case contains divergent quant ities in the equations\nof motion. Using the scalar-field reservoir [14] and the Huttner-Ba rnett reservoir (a\ncontinuum of harmonic oscillators) [10] to model Ohmic and non-Ohm ic damping, we\nwill obtain finite results for the energies of the oscillator in the classic al case, from\na dynamics that contains no divergent quantities. We will thereby ex hibit within a\nreservoir treatment that Ohmic damping is physically possible in classic al mechanics\nbut impossible in quantum mechanics. A comparison will also be made bet ween the\nOhmic-damping thermal energies and those for general damping, a nd between classical\nand quantum thermal energies.\nWhile we here consider the case of a global thermal state, the two r eservoir\napproaches we discuss can readily be applied to investigate non-equ ilibrium dynamics\n(see [14], for example). The use of reservoirs for non-equilibrium pr oblems is discussed\nby many authors, e.g. [7, 18, 19, 20, 21, 23].\n2. Ohmic damping with a scalar field reservoir\nOhmic damping corresponds to the following simple equation of motion o f a harmonic\noscillator:\n¨q(t)+γ˙q(t)+ω2\n0q(t) =f(t), (1)\nwhereqis the oscillator displacement, γis a damping constant, ω0is the frequency of\nan externally applied potential and fis an external force. The question now arises as\nto how a reservoir and coupling can be chosen to obtain (1) as the eff ective equation of\nmotion for the oscillator. Coupling the oscillator to a reservoir of har monic oscillators,\neither a discrete or continuous set, does not give (1) as the equat ion of motion for q,Thermal energies of classical and quantum damped oscillato rs coupled to reservoirs 3\nexcept as a limiting case [7, 8, 2] or with an accompanying zero-frequ ency solution [11].\nIn order to obtain (1) exactly, a reservoir [14] consisting of a 1D sc alar field φ(x,t) may\nbe used, linearly coupled to the oscillator. In detail, the Lagrangian is\nL=1\n2/parenleftbig\n˙q2(t)−ω2\n0q2(t)/parenrightbig\n+1\n2/integraldisplay∞\n−∞dx/bracketleftbigg1\nc2˙φ2(x,t)−(∂xφ(x,t))2/bracketrightbigg\n−α˙q(t)φ(0,t),(2)\nwhereαis the coupling constant. This gives the equations of motion\n¨q(t)+ω2\n0q(t) =α˙φ(0,t) and1\nc2¨φ(x,t)−∂2\nxφ(x,t) =−α˙q(t)δ(x).(3)\nAs was shown previously [14], the solution for the scalar field is\nφ(x,t) =−c\n2αq/parenleftbigg\nt−|x|\nc/parenrightbigg\n+φh(x,t), (4)\nwhereφh(x,t) is a solution of the homogeneous φ-equation ¨φ/c2−∂2\nxφ= 0. This yields\naq-equation of the desired form stated in Eq. (1):\n¨q(t)+γ˙q(t)+ω2\n0q(t) =α˙φh(0,t) with γ:=c\n2α2, (5)\nwithγproportional to α2. As a main result of this paper we will now quantize\nand diagonalise the Hamiltonian for this simple system, which exhibits ex act Ohmic\ndamping.\n3. Quantization and diagonalization of the scalar field mode l\nFrom Eq. (2) one obtains the canonical momenta\nΠq(t) = ˙q(t)−αφ(0,t) and Π φ(x,t) =1\nc2˙φ(x,t). (6)\nWequantizeintheHeisenbergpicturebyimposingtheequal-timecomm utationrelations\n[ˆq(t),ˆΠq(t)] = i/planckover2pi1and [ˆφ(x,t),ˆΠφ(x′,t)] = i/planckover2pi1δ(x−x′), (7)\nwhile all other commutators of these operators vanish. Here the q uantized operators\nare indicated with hats. Combining Eq. (2) and Eq. (6) the quantized Hamiltonian is\nˆH=1\n2/bracketleftBig\nˆΠ2\nq+ω2\n0ˆq2/bracketrightBig\n+1\n2/integraldisplay∞\n−∞dx/bracketleftBig\nc2ˆΠ2\nφ+(∂xˆφ)2/bracketrightBig\n+α\n2/bracketleftBig\nˆΠqˆφ(0,t)+h.c./bracketrightBig\n+α2\n2ˆφ2(0,t) (8)\nwhereh.c.stands for Hermitian conjugate and a Hermitian combination of oper ators\nhas been taken in the second-last term.\nThe diagonalization of a Hamiltonian of the general form (8) has been described\nin detail in [11] (see also [13]). The Hamiltonian (8) is diagonalized by tra nsforming it\ninto the normal form\nˆH=1\n2/integraldisplay∞\n−∞dk/planckover2pi1ω/bracketleftBig\nˆC†(k)ˆC(k)+ˆC(k)ˆC†(k)/bracketrightBig\nwithω:=c|k|,(9)\nwhereˆC†(k) andˆC(k) are the creation and annihilation operators for a free scalar field,\nˆΨ(x,t) =/integraldisplay∞\n−∞dk/radicalbigg\nc2/planckover2pi1\n4πω/bracketleftBig\nˆC(k)eikx−iωt+ˆC†(k)e−ikx+iωt/bracketrightBig\n. (10)Thermal energies of classical and quantum damped oscillato rs coupled to reservoirs 4\nThe ladder operators obey the standard bosonic field commutation relations\n[ˆC(k),ˆC†(k′)] =δ(k−k′) and [ ˆC(k),ˆC(k′)] = 0. (11)\nThe diagonalizationalso goesthroughclassically, withcommutators r eplaced by Poisson\nbrackets. We show in the Appendix that the relationship between th e dynamical\nvariables appearing in Eq. (8) and in Eq. (9) is given by\nˆq(t) =/integraldisplay∞\n−∞dk/radicalbigg\n/planckover2pi1ω\n4π/bracketleftbiggicα\nω2−ω2\n0+iγωˆC(k)e−iωt+h.c./bracketrightbigg\n, (12)\nˆφ(x,t) =/integraldisplay∞\n−∞dk/radicalbigg\nc2/planckover2pi1\n4πω/bracketleftbigg/parenleftbigg\neikx−iγω\nω2−ω2\n0+iγωeiω|x|/c/parenrightbigg\nˆC(k)e−iωt+h.c./bracketrightbigg\n,(13)\nˆΠq(t) =˙ˆq(t)−αˆφ(0,t), (14)\nˆΠφ(x,t) =1\nc2˙ˆφ(x,t). (15)\n4. Thermal equilibrium of the scalar-field model\nWearenowreadytocalculatethermalequilibrium expectationvalues ofthetotalsystem\nand deduce the internal energy and free energy of the open quan tum oscillator, as well\nas of its classical counterpart. A common assumption in dynamical a pproaches is that\nthe reservoir is always in a thermal state while the system evolves in t ime towards a\nsteady state [1, 2]. Incontrast, we here consider the case where theoscillator andsystem\nhave been interacting for so long that a global thermal state has b een reached.\nFor a non-equilibrium treatment of the scalar-field reservoir, in the context of the\nquantum-classical transition, see [14]. In global thermal equilibrium , i.e.ρtot(β) =e−βˆH\nZ,\nβ−1=kBT, the normal modes of the total system have the expectation valu es:\n/an}bracketle{tˆC†(k)ˆC(k′)/an}bracketri}httot=N(ω)δ(k−k′) with N(ω) =/bracketleftbigg\nexp/parenleftbigg/planckover2pi1ω\nkBT/parenrightbigg\n−1/bracketrightbigg−1\n(16)\n/an}bracketle{tˆC(k)ˆC(k′)/an}bracketri}httot= 0. (17)\nThese equations now determine the thermal correlation functions of theq-oscillator.\nUsing Eq. (12) it is straightforward to show that\n1\n2/an}bracketle{tˆq(t)ˆq(t′)+h.c./an}bracketri}httot=/planckover2pi1\nπ/integraldisplay∞\n0dωγω\n(ω2−ω2\n0)2+γ2ω2cos[ω(t−t′)]coth/parenleftbigg/planckover2pi1ω\n2kBT/parenrightbigg\n.(18)\nThis result can also be obtained by using Eq. (1) and appealing to the fl uctuation-\ndissipation theorem [22]. In this scalar field model the fluctuation-dis sipation theorem\ndoes not have to be imposed, rather it arises as a consequence of t hermal equilibrium of\nthe total system as formalised in Eqs. (16)–(17). The above auto correlation function is\nfinite and depends on the damping parameter γbut further results show the quantum\ncase to be unphysical (see [22] and below).\nTo calculate the thermodynamic quantities for the oscillator, such a s its internal\nenergy, one must quantify the contributions that come from its co upling to the\nreservoir which causes the damping [24]. The energy of an open oscilla tor, whichThermal energies of classical and quantum damped oscillato rs coupled to reservoirs 5\nis part of a total thermal state, can be accounted for by the Ham iltonian of mean\nforce [15]. The Hamiltonian of mean force appears naturally in non-eq uilibrium work\nrelations [16, 17, 25], in the thermodynamic analysis of the second law and Landauer’s\nprinciple for a damped oscillator [26], in the evolution to steady state of open systems\ncoupled to a thermal reservior [21], and also in the Casimir effect [12]. The Hamiltonian\nof mean force for the oscillator is defined as [17]\nˆH⋆(β) =−1\nβlntrφ[e−βˆH]\nZφ, Z φ= trφ[e−βˆHφ], β=1\nkBT,(19)\nwhere the traces are taken over the scalar field φonly and ˆHφ=\n1\n2/integraltext∞\n−∞dx/bracketleftBig\nc2ˆΠ2\nφ+(∂xˆφ)2/bracketrightBig\nis the free scalar-field Hamiltonian, a part of the total Hamil-\ntonianˆHgiven in (8). Thus, Zφin (19) is the partition function of a free scalar field in\nthermal equilibrium. The partition function Z⋆associated with ˆH⋆is\nZ⋆= trS/bracketleftBig\ne−βˆH⋆/bracketrightBig\n=Z\nZφ, Z= tr[e−βˆH], (20)\nwhereZis the partition function of the total system. The free energy F⋆associated\nwithZ⋆is, from (20),\nF⋆=−β−1ln(Z⋆) =F−Fφ, (21)\nwhereF=−β−1ln(Z) is the free energy of the total system and Fφ=−β−1ln(Zφ)\nis the free energy of a free scalar field in thermal equilibrium. From th e standard\nthermodynamic relation F=U−TSrelating the free energy to the internal energy\nUand entropy S, we can find the internal energy associated with the mean force fr ee\nenergyF⋆. From (21) we have\nF⋆=U−Uφ−T(S−Sφ), (22)\nwhereUis the total internal energy, Uφis the internal energy of a free scalar field, Sis\nthe total entropy and Sφis the entropy of a free scalar field. Equation (22) shows that\nthe mean force thermal energy U⋆associated with F⋆is\nU⋆=U−Uφ=/an}bracketle{tˆH/an}bracketri}httot−Z−1\nφtrφ[ˆHφe−βˆHφ], (23)\nwhich is the total thermal energy minus the thermal energy of a fr ee scalar field. Note\nthatU⋆isnotdefined as the expectation value of ˆH⋆in the global thermal state, i.e.\nU⋆:/ne}ationslash=/an}bracketle{tˆH⋆/an}bracketri}httot. The total Hamiltonian (8) can be rewritten with (14) in terms of ˆ qand\nˆφas\nˆH=1\n2˙ˆq2+1\n2ω2\n0ˆq2+1\n2/integraldisplay∞\n−∞dx/bracketleftbigg1\nc2˙ˆφ2\n+(∂xˆφ)2/bracketrightbigg\n. (24)\nNowU⋆can be obtained as the expectation value of (24) in the global therm al stateρtot\ncalculated using (12), (13), (16) and (17), and dropping all γ-independent terms in the\nfinal scalar-field contribution (this subtracts out the free scalar -field energy as required\nby (23)). Remarkably, the γ-dependent terms arising from the scalar-field part of (24)Thermal energies of classical and quantum damped oscillato rs coupled to reservoirs 6\ncancel out in the expectation value, and so U⋆is the same as would be obtained from\njust theq-terms in (24). The result for U⋆is\nU⋆=/planckover2pi1\n2π/integraldisplay∞\n0dωγω(ω2+ω2\n0)\n(ω2−ω2\n0)2+γ2ω2coth/parenleftbigg/planckover2pi1ω\n2kBT/parenrightbigg\n. (25)\nThis expression diverges for any temperature Twhich proves that Ohmic damping of a\nquantum oscillator is unphysical. Such divergences are often avoide d by introducing a\nhigh frequency cut-off at the outset [22] which in turn results in ap proximately Ohmic\ndamping for those frequencies that can be supported by the syst em.\n4.1. Classical Ohmic damping\nClassically the ˆC(k) in (10) become the complex amplitudes C(k) of the normal modes,\nand the thermal-equilibrium expectation value corresponding to (16 ) is/an}bracketle{tC∗(k)C(k′)/an}bracketri}ht=\nδ(k−k′)kBT/(/planckover2pi1ω). The occurrence of /planckover2pi1in this classical expression is due to the /planckover2pi1-\ndependent normalization of the complex amplitudes in (10). We will der ive classical\nthermal results as limits /planckover2pi1→0 of the quantum expressions, but they may of course be\ndirectly obtained from the classical normal-mode expectation value s.\nIn the classical limit of (25) the mean force internal energy is finite a nd neatly gives\nU⋆=kBT\nπ/integraldisplay∞\n0dωγ(ω2+ω2\n0)\n(ω2−ω2\n0)2+γ2ω2=kBT, (26)\nwhich is the same result as for an undamped oscillator. This result, U⋆=kBT, is\nactually true not only for Ohmic damping, but also for very general d amping, see next\nsection.\nAn alternative definition of the internal energy of the oscillator tha t is often\nconsidered [24] is\nU:=1\n2/an}bracketle{t˙q2+ω2\n0q2/an}bracketri}httot, (27)\nwhere the expectation value is again taken in the global thermal sta te. Note that this\nenergy could depend on the coupling to the reservoir through both the correlations in\nthe thermal state and the dependence of qand ˙qon the coupling α. As discussed after\nEq. (24), for Ohmic damping it turns out that the scalar field contrib ution toU⋆cancels\nout, i.e. the mean force energy is the sameas would be obtained using the definition\nofUin Eq. (27). Thus for the case of Ohmic damping one obtains U=U⋆=kBT.\nInterestingly, for general damping U⋆differs from U, see next section.\n5. Huttner-Barnett reservoir and general damping\nIn the previous sections the reservoir was taken to be a 1D scalar fi eld because this leads\nvery simply to Ohmic damping of the q-oscillator. For general damping, a reservoir of\nharmonic oscillators can be used. In [11, 12, 13] an oscillator coupled to a reservoir\nconsisting of a continuum of harmonic oscillators {Xω:ω∈[0,∞)}was considered.Thermal energies of classical and quantum damped oscillato rs coupled to reservoirs 7\nThis continuum reservoir was introduced by Huttner and Barnett [ 10]. The resulting\nHamiltonian is [11]\nˆH=1\n2ˆΠ2\nq+1\n2ω2\n0ˆq2+1\n2/integraldisplay∞\n0dω/parenleftBig\nˆΠ2\nXω+ω2ˆX2\nω/parenrightBig\n−1\n2/integraldisplay∞\n0dωα(ω)/bracketleftBig\nˆqˆXω+ˆXωˆq/bracketrightBig\n,(28)\nwhereα(ω) is a function describing the coupling between the oscillator and rese rvoir\nmodes. Theresultingdynamicsofthe q-oscillatorisgovernedbyacomplexsusceptibility\nχ(ω) =χ⋆(−ω) whose imaginary part is proportional to α2(ω). The susceptibility\nobeys Kramers-Kronig relations and so χ(ω) is analytic in the upper-half complex ω-\nplane [11]. In addition there is a sufficient condition on χ(ω) for the total Hamiltonian\nto be diagonalizable, namely [11]\n/integraldisplay∞\n0dωIm[χ(ω)]<π\n2. (29)\nThe damping term in the effective equation of motion for q, see (1) for the Ohmic\ndamping case, now features a damping kernel, ω2\n0/integraltext∞\n−∞dt′χ(t−t′)ˆq(t′). The position\noperator for the q-oscillator then becomes [11]\nˆq(t) =/integraldisplay∞\n0dω/radicalbigg\n/planckover2pi1\n2ω/bracketleftbigg−α(ω)\nω2−ω2\n0[1−χ(ω)]ˆC(ω)e−iωt+h.c./bracketrightbigg\n, (30)\nwhereˆC(ω) are the annihilation operators for the modes that diagonalize the f ull\nHamiltonian (28), analogous to Eq. (9). The explicit form of the Hamilt onian in (28)\nallows the calculation of the mean force internal energy [11]:\nU⋆=/planckover2pi1\n2π/integraldisplay∞\n0dωcoth/parenleftbigg/planckover2pi1ω\n2kBT/parenrightbigg\nIm\n\nω2\n0/bracketleftBig\nωdχ(ω)\ndω−χ(ω)+1/bracketrightBig\n+ω2\nω2\n0[1−χ(ω)]−ω2\n\n.(31)\nIt is easy to obtain from the results in [11] that the internal energy Udefined by (27)\ndiffers from (31) by not having the χ-dependent terms in the numerator inside the curly\nbrackets, i.e.\nU=/planckover2pi1\n2π/integraldisplay∞\n0dωcoth/parenleftbigg/planckover2pi1ω\n2kBT/parenrightbigg\nIm/braceleftbiggω2\n0+ω2\nω2\n0[1−χ(ω)]−ω2/bracerightbigg\n. (32)\nThis means that for general damping the two energy expressions ( 31) and (32) will differ\n(i.e.U∗/ne}ationslash=U) both in the quantum and classical cases. For the classical oscillato r, both\nU⋆andUcan be evaluated exactly for arbitrary χ, see below.\n5.1. Ohmic damping\nAs can be seen by comparing Eq. (31) with the Ohmic expression Eq. ( 25), Ohmic\ndamping corresponds to choosing a “susceptibility”\nχ(ω) =iγω\nω2\n0. (33)\nHowever, this choice is not physical as it obeys neither Kramers-Kr onig relations nor\ncondition (29), reflecting the well-known result that strictly Ohmic d amping cannot beThermal energies of classical and quantum damped oscillato rs coupled to reservoirs 8\ntreated by a reservoir of harmonic oscillators [8]. (We note, howev er, that the case of\nOhmic damping when accompanied by a zero-frequency solution for t heq-oscillator can\nbe properly treated by a valid susceptibility [11].)\nDespite the fact that the Ohmic damping “susceptibility” (33) is not a valid choice,\ninserting it into the general result (31) gives the Ohmic-damping res ult (25) derived\nwith the scalar field reservoir. Also, in (31) the χ-dependent terms in the numerator\ninside the curly brackets cancel out for this choice (33), which aga in givesU⋆=Uin\nthe Ohmic damping case, as found in the previous section.\n5.2. Classical limit\nWe now derive the important result that in the classical limit /planckover2pi1→0 one obtains\nU⋆=kBTfor almost any susceptibility, whereas Udepends on the susceptibility (i.e.\nthe damping). If we take the Im outside the integration in (31) then the real part of\nthe resulting integral does not converge. However, setting the lo wer integration limit\nto−∞and dividing the whole integral by 2 removes the diverging real part, because\nthe real part of the integrand is odd in ω(recall that χ(ω) =χ∗(−ω), so Re[χ(ω)] is\neven and Im[ χ(ω)] is odd). The new integral from −∞to +∞now requires the pole at\nω= 0 to be treated as a principal value so that the integration picks ou t the even part\nof the integrand. The resulting expression in the classical limit is\nU⋆=kBT\n2πIm\nP/integraldisplay∞\n−∞dωω2\n0/bracketleftBig\nωdχ(ω)\ndω−χ(ω)+1/bracketrightBig\n+ω2\nω(ω2\n0[1−χ(ω)]−ω2)\n=kBT,(34)\nwhere P denotes principal value. For very general χ(ω), the principal-value integral in\n(34) evaluates to 2 πi, as shown by the following analysis. If ωdχ(ω)\ndω|ω=0= 0, then the\nintegrand in (34) has a simple pole at ω= 0. Consider the same integrand integrated\nover a closed contour Cin the complex ω-plane that runs along the real line but goes\nbelow the pole at ω= 0, and then closes anti-clockwise in a large semicircle in the\nupper-half plane. This contour integral can be decomposed as/contintegraldisplay\nCdz= P/integraldisplay∞\n−∞dω+/integraldisplay\nCǫdz+/integraldisplay\nRdz, (35)\nwhereCǫis an infinitesimal semicircle running anti-clockwise around the pole at ω= 0\nandRisa largesemicircle of radius Rinthe upper-half plane taken inthe anti-clockwise\ndirection. As the integrand in (34) is analytic everywhere inside Cexcept at the simple\npole atω= 0 (recall that χ(ω) is analytic in the upper-half plane), the integral around\nCis 2πi. The integral along Cǫisπi, and it is easy to show that the integral along the\nsemicircle of radius R, asR→ ∞, is−πi. From (35) this shows that the principal-\nvalue integral in (34) is 2 πi so we obtain the classical result U⋆=kBTfor very general\nsusceptibility.\nThe classical limit of the energy U(32) is\nU=kBT\n2πIm/bracketleftbigg\nP/integraldisplay∞\n−∞dωω2\n0+ω2\nω(ω2\n0[1−χ(ω)]−ω2)/bracketrightbigg\n. (36)Thermal energies of classical and quantum damped oscillato rs coupled to reservoirs 9\nIn evaluating this integral the only changes to the above analysis of U⋆are that the\nintegral around the closed contour Cis now 2πi/[1−χ(0)], and the integral along Cǫis\nπi/[1−χ(0)], assuming χ(0) is finite and not equal to 1. Hence, the classical internal\nenergyUbecomes\nU=kBT/bracketleftbigg\n1+χ(0)\n2[1−χ(0)]/bracketrightbigg\n, χ(0)/ne}ationslash= 1. (37)\nThis depends on the damping unless χ(0) = 0. The general result (37) reproduces the\nclassical value U=kBTfor Ohmic damping obtained in the last section if we again\nsubstitute the “susceptibility” (33), because in this case χ(0) = 0. Thus, in the classical\nlimit considered here, the general damping dependence of the ener gyU(37) contrasts\nwith the damping independence of the mean force energy U⋆=kBT.\n5.3. Comparison of energies for different damping\nTable 1 summarises the classical and quantum results for UandU⋆with different types\nof damping. The calculations above showed that for Ohmic damping of a classical\noscillator, the energy can be taken to be either UorU⋆as both reduce to kBT. In\ncontrast, for a quantum oscillator, both UandU⋆diverge for Ohmic damping. For\ngeneral non-Ohmic damping, classically U⋆is always kBTwhereas Udepends on the\ndamping. Quantum mechanically both U⋆andUdepend on the damping, however they\nare not equal. Note that for a classical oscillator small deviations fr om Ohmic damping\nresult in small changes in U. In contrast, for a quantum oscillator with Ohmic damping\nthe energies diverge, while small deviations from Ohmic damping make UandU⋆finite.\nThus, even for oscillators that are classically well described by Ohmic damping, their\nquantum (including zero-point) energies are entirely determined by the deviations from\nOhmic damping.\ndamping classical quantum\nno damping U=U⋆=kBT U=U⋆=/planckover2pi1ω0\n2coth/planckover2pi1βω0\n2\nOhmic damping U=U⋆=kBT U,U⋆→ ∞\ngeneral damping U=kBT/bracketleftBig\n1+χ(0)\n2[1−χ(0)]/bracketrightBig\n/ne}ationslash=U⋆=kBTU/ne}ationslash=U⋆, see (32) and (31)\nTable 1. Table comparing energies for classical and quantum damped oscillato rs,\nfor no damping, Ohmic damping, and general non-Ohmic damping desc ribed by a\nsusceptibility χ(ω).\n5.4. Free energy and entropy\nFinally, itisalsointeresting toderivetheHelmholtzfreeenergy F⋆arisingfromthemean\nforce energy U⋆for general damped oscillators. Using the standard thermodynam ic\nrelations F⋆=U⋆−TS⋆andS⋆=−∂F⋆\n∂T, we obtain U⋆=−T2d\ndTF⋆\nT, which gives F⋆as\nF⋆=−T/integraldisplay\ndTU⋆\nT2+aT, (38)Thermal energies of classical and quantum damped oscillato rs coupled to reservoirs 10\nfor some constant a. The entropy S⋆=−∂F⋆\n∂Tand must vanish at T= 0 in line with the\nthird law of thermodynamics, and this allows the value of ain (38) to be determined.\nThe results for F⋆andS⋆are\nF∗=kBT\nπ/integraldisplay∞\n0dωln/bracketleftbigg\nsinh/parenleftbigg/planckover2pi1ω\n2kBT/parenrightbigg/bracketrightbigg\nIm\n\nω2\n0/bracketleftBig\nωdχ(ω)\ndω−χ(ω)+1/bracketrightBig\n+ω2\nω(ω2\n0[1−χ(ω)]−ω2)\n\n+kBTln2,\n(39)\nS⋆=/planckover2pi1\n2π/integraldisplay∞\n0dω/braceleftbigg1\nTcoth/parenleftbigg/planckover2pi1ω\n2kBT/parenrightbigg\n−2kB\n/planckover2pi1ωln/bracketleftbigg\nsinh/parenleftbigg/planckover2pi1ω\n2kBT/parenrightbigg/bracketrightbigg/bracerightbigg\n×Im\n\nω2\n0/bracketleftBig\nωdχ(ω)\ndω−χ(ω)+1/bracketrightBig\n+ω2\nω2\n0[1−χ(ω)]−ω2\n\n−kBln2. (40)\nTo verify that the entropy (40) vanishes at T= 0, first note that the T-dependent factor\nin the integral reduces to2kB\n/planckover2pi1ωln2 in the T→0 limit. The integral is then proportional\nto an integral evaluated above, see (34).\nThe classical limit of the free energy (39) is\nF⋆=kBT\nπ/integraldisplay∞\n0dωln/parenleftbigg/planckover2pi1ω\n2kBT/parenrightbigg\nIm\n\nω2\n0/bracketleftBig\nωdχ(ω)\ndω−χ(ω)+1/bracketrightBig\n+ω2\nω2\n0[1−χ(ω)]−ω2\n\n+kBTln2,(41)\nwhere/planckover2pi1still appears as the phase space volume element, which will cancel in f ree\nenergy differences [27]. In contrast to the damping independence o fU⋆, the classical\nfree energy F⋆does depend on the details of the damping in almost all cases. The\nnotableexceptionisOhmicdamping, forwhich F⋆canbefoundwiththe“susceptibility”\n(33). The resulting integral can be evaluated exactly and gives the free energy of an\nundamped oscillator, i.e. F⋆=kBTln/parenleftBig\n/planckover2pi1ω0\nkBT/parenrightBig\n, independent of γ. Alternatively, the\nclassical Ohmic-damping result can be evaluated using the scalar-fie ld reservoir of the\nlast section by the same analysis leading from (31) to (41).\n6. Conclusions\nTo model damped harmonic motion we considered two time-independe nt Hamiltonians\nfor an oscillator coupled to a reservoir, one with a scalar-field reser voir and one with a\nHuttner-Barnett reservoir. Using the diagonalised Hamiltonians we derived expressions\nfor thermodynamic quantities of a damped oscillator, for both Ohmic and general\ndamping, when the total system is in a global thermal state. These were evaluated\nfor both the quantum and classical regimes. We recovered the fac t that strictly Ohmic\ndamping of a quantum oscillator cannot physically occur due to divergences forcing one\ntoabandontheexact Ohmicregime. Incontrast, Ohmicdampingofa classical oscillator\ncan be treated exactly using the scalar-field reservoir, giving finite and physically\nmeaningful results. The diagonalized form of the Hamiltonians allowed the calculation\nof thermal energies of the oscillator, i.e. the mean force energy U∗and its correspondingThermal energies of classical and quantum damped oscillato rs coupled to reservoirs 11\nfree energy F⋆, and the commonly used internal energy U. We found that classically\nU⋆=kBTfor any damping type, no matter how strong the coupling to the res ervoir.\nThis demonstrates a remarkable and non-trivial property of the c lassical mean force\nenergy. In contrast to U⋆, the classical internal energy Uand the mean force free energy\nF⋆do depend on the coupling strength for general non-Ohmic damping .\nForquantum oscillatorsitissurprisingthatwhilestrictlyOhmicdampingisplagued\nwith divergences, infinitesimal changes to the damping result in finite expressions for\nbothU⋆andU. In addition, the quantum mean force energy U⋆does depend on the\ncoupling, as do UandF⋆. The treatment of classical and quantum open systems\noften assumes initial product states between the system of inter est and a reservoir\nthat are then evolved with a global Hamiltonian, and under a number o f assumptions,\nto a long-time steady-state [28, 1, 2]. In contrast we here conside red the stationary\nsituation of the total system being in a global thermal state. The r esults presented\nadd a new perspective on the thermodynamics of open systems with arbitrarily strong\ncouplingtoareservoir. Forexample, thedifferentenergymeasure smaybeofsignificance\nwhen calculating efficiencies of small scale and quantum engines that o perate between\nequilibrium configurations in the strong coupling limit [29]. Such engine c ycles may\nshow departures from standard thermodynamics which assumes w eak coupling. Finally,\nextending the thermal equilibrium analysis of the Huttner-Barnett reservoir presented\nhere to analyse the non-equilibrium dynamics of damped oscillators is a n interesting\ntopic for future investigation.\nAcknowledgements\nWe thank I. Hooper and S. Horsley for discussions that led to this wo rk. JA\nacknowledges support by EPSRC (EP/M009165/1).\nReferences\n[1] Breuer H-P and Petruccione F 2002 The Theory of Open Quantum Systems (Oxford; Oxford\nUniversity Press)\n[2] Weiss U 2012 Quantum Dissipative Systems 4th ed (Singapore: World Scientific)\n[3] Dekker H 1981 Phys. Rep. 801\n[4] Um C I, Yeon K H and George T F 2002 Phys. Rep. 36263\n[5] Magalinskii V B 1959 Sov. Phys. JETP 91381\n[6] Feynman R P and Vernon F L 1963 Ann. Phys. 24118\n[7] Caldeira A O and Leggett A J 1983 PhysicaA121587\n[8] Tatarskii V P 1987 Sov. Phys. Usp. 30134\n[9] Yu L H and Sun C P 1994 Phys. Rev. A 49592; Yu L H 1995 Phys. Lett. A202167; Yu L H and\nSun C P 1996 Phys. Rev. A 543779\n[10] Huttner B and Barnett S M 1992 Phys. Rev. A464306\n[11] Philbin T G 2012 New J. Phys. 14, 083043\n[12] Philbin T G and Horsley S A R 2013 arXiv:1304.0977[quant-ph]\n[13] Barnett S M, Cresser J D and Croke S 2015 arXiv:1508.02442[qua nt-ph]\n[14] Unruh W G and Zurek W H 1989 Phys. Rev. D401071Thermal energies of classical and quantum damped oscillato rs coupled to reservoirs 12\n[15] Kirkwood J G 1935 J. Chem. Phys. 3300\n[16] Jarzynski C 2004 J. Stat. Mech.: Theor. Exp. P09005\n[17] Campisi M, Talkner P and H¨ anggi P 2009 Phys. Rev. Lett. 102210401\n[18] Gardiner C W and Collett M J 1985 Phys. Rev. A313761\n[19] Paz J P and Roncaglia 2008 Phys. Rev. Lett. 100220401\n[20] Correa L A, Valido A A and Alonso D 2012 Phys. Rev. A86012110\n[21] Subasi Y, Fleming C H, Taylor J M and Hu B L 2012 Phys. Rev. E86061132\n[22] Grabert H and Weiss U 1984 Z. Phys. B5587\n[23] Valenti D, Magazz` u L, Caldara P and Spagnolo B 2015 Phys. Rev. B 91235412; Magazz` u L,\nValenti D, Spagnolo B and Grifoni M 2015 Phys. Rev. E 92032123\n[24] Gelin M and Thoss M 2009 Phys. Rev. E 79051121\n[25] Philbin T G and Anders A 2014 arXiv:1404.5181[cond-mat.stat-mec h]\n[26] Hilt S, Shabbir S, Anders J and Lutz E 2011 Phys. Rev. E 83030102(R)\n[27] Landau L D and Lifshitz E M 1980 Statistical Physics, Part 1 3rd ed (Oxford: Butterworth-\nHeinemann)\n[28] Hatano T and Sasa S 2001 Phys. Rev. Lett. 863463\n[29] Kosloff R and Levy A 2014 Ann. Rev. Phys. Chem. 65365\nAppendix\nHere we describe the diagonalization of the Hamiltonian (8). The proc edure is very\nsimilar to the diagonalization of the damped harmonic oscillator with a re servoir\ncomposed of a continuum of harmonic oscillators [11, 13] (which is in t urn similar to\npart of the Huttner-Barnett model [10]).\nWe seek a linear transformation between the dynamical variables in ( 8) and (9). It\nis more convenient to work with the time-dependent operators\nˆC(k,t) =ˆC(k)e−iωt,ˆC†(k,t) =ˆC†(k)eiωt, (A.1)\n[ˆC(k,t),ˆH] =/planckover2pi1ωˆC(k,t), (A.2)\nthe last relation following from (9) and (11). The required transfor mation must take\nthe form\nˆq(t) =/integraldisplay∞\n−∞dk/bracketleftBig\nfq(k)ˆC(k,t)+h.c./bracketrightBig\n,ˆΠq(t) =/integraldisplay∞\n−∞dk/bracketleftBig\nfΠq(k)ˆC(k,t)+h.c./bracketrightBig\n(A.3)\nˆφ(x,t) =/integraldisplay∞\n−∞dk/bracketleftBig\nfφ(x,k)ˆC(k,t)+h.c./bracketrightBig\n, (A.4)\nˆΠφ(x,t) =/integraldisplay∞\n−∞dk/bracketleftBig\nfΠφ(x,k)ˆC(k,t)+h.c./bracketrightBig\n, (A.5)\nfor some unknown functions fq(k), etc. The commutation relations (11) with (A.3)–\n(A.5) give:\nfq(k) = [ˆq(t),ˆC†(k,t)], f Πq(k) = [ˆΠq(t),ˆC†(k,t)], (A.6)\nfφ(x,k) = [ˆφ(x,t),ˆC†(k,t)], f Πφ(x,k) = [ˆΠφ(x,t),ˆC†(k,t)].(A.7)\nThe transformation (A.3)–(A.5) must be invertible, which together with (7), (A.6) and\n(A.7) implies\nˆC(k,t) =−i\n/planckover2pi1/braceleftbigg\nf∗\nΠq(k)ˆq(t)−f∗\nq(k)ˆΠq(t)Thermal energies of classical and quantum damped oscillato rs coupled to reservoirs 13\n+/integraldisplay∞\n−∞dx/bracketleftBig\nf∗\nΠφ(x,k)ˆφ(x,t)−f∗\nφ(x,k)ˆΠφ(x,t)/bracketrightBig/bracerightbigg\n. (A.8)\nWe find equations for the f-coefficients in (A.3)–(A.5) as follows. Insert (A.8) and (8)\ninto (A.2) and simplify using (7). This gives an expression for ˆC(k,t) which can be\ncompared with (A.8) to find\nfΠq(k)+αfφ(0,k) =−iωfq(k),iωfΠq(k) =ω2\n0fq(ω), c2fΠφ(x,k) =−iωfφ(x,k),(A.9)\niωfΠφ(x,k) =αfΠq(k)δ(x)−∂2\nxfφ(x,k)+α2fφ(0,k)δ(x), (A.10)\nwhich give\nω2\n0fq(k) =ω2fq(k)−iαωfφ(0,k),ω2\nc2fφ(x,k) =−iαωfq(k)δ(x)−∂2\nxfφ(x,k).(A.11)\nThese are the same as the classical equations (3) in the frequency domain and their\nsolution is\nfφ(x,k) =−1\n2cαeiω|x|/cfq(k)+hφ(k)eikx, fq(k) =iαωhφ(k)\nω2−ω2\n0+iγω, (A.12)\nwherehφ(k) is the amplitude of the solution to the homogeneous fφequation ( α= 0).\nThe value of hφ(k) is determined by the fact that (A.8) and its Hermitian conjugate\nhave commutator [ ˆC(k,t),ˆC†(k′,t)] =δ(k−k′) (see (11) and (A.1)). A tedious\ncalculation shows that this commutator holds with (A.8) expanded in t he solutions\nfor thef-coefficients if\nhφ(k) =/radicalbigg\nc2/planckover2pi1\n4πω. (A.13)\nThe commutator [ ˆC(k,t),ˆC(k′,t)] = 0 is identically satisfied by (A.8) with the solutions\nfor thef-coefficients. The expansions (A.3)–(A.5) have now been determine d and give\n(12)–(14). Consistency ofthediagonalizationisdemonstratedby showing that(12)–(14)\nobey the commutation relations (7) because of (11)."    },    {        "title": "1005.3791v1.Long_wavelength_unstable_modes_in_the_far_upstream_of_relativistic_collisionless_shocks.pdf",        "content": "arXiv:1005.3791v1  [astro-ph.HE]  20 May 2010Draft version April 25, 2022\nPreprint typeset using L ATEX style emulateapj v. 11/10/09\nLONG WAVELENGTH UNSTABLE MODES IN THE FAR UPSTREAM OF RELATI VISTIC COLLISIONLESS\nSHOCKS\nItay Rabinak, Boaz Katz and Eli Waxman\nDepartment of Particle Physics and Astrophysics, The Weizm ann Institute of Science, Rehovot 76100, Israel\n(Dated: April 25, 2022)\nDraft version April 25, 2022\nABSTRACT\nThe growth rate of long wavelength kinetic instabilities arising due to t he interaction of a collimated\nbeam of relativistic particles and a cold unmagnetized plasma are calcu lated in the ultra relativistic\nlimit. For sufficiently culminated beams, all long wave-lengthmodes are shown to be Weibel-unstable,\nand a simple analytic expression for their growth rate is derived. For large transversevelocity spreads,\nthese modes become stable. An analytic condition for stability is given . These analytic results, which\ngeneralize earlier ones given in the literature, are shown to be in agre ement with numerical solutions\nof the dispersion equation and with the results of novel PIC simulatio ns in which the electro-magnetic\nfields are restricted to a given k-mode. The results may describe th e interaction of energetic cosmic\nrays, propagating into the far upstream of a relativistic collisionless shock, with a cold unmagnetized\nupstream. The long wavelength modes considered may be efficient in d eflecting particles and could\nbe important for diffusive shock acceleration. It is shown that while t hese modes grow in relativistic\nshocks propagating into electron-positron pair plasmas, they are damped in relativistic shocks propa-\ngating into electron-proton plasmas with moderate Lorenz factor s Γsh/lessorsimilar(mp/me)1/2. If these modes\ndominate the deflection of energetic cosmic rays in electron-positr on shocks, it is argued that particle\naccelerationis suppressed at shock frame energies that are large rthan the downstreamthermal energy\nby a factor of /greaterorsimilarΓsh.\nSubject headings: shock waves acceleration of particles cosmic rays\n1.INTRODUCTION\nCurrent understanding of gamma-ray burst (GRB)\n”afterglows,” the delayed low energy emission follow-\ning the prompt γ-ray emission, suggests that the ra-\ndiation observed is the synchrotron emission of ener-\ngetic non-thermal electrons in the downstream of an\nultra-relativistic collisionless shock driven into the sur-\nrounding interstellar medium (ISM) or stellar wind\n(Zhang & M´ esz´ aros 2004; Piran 2004).\nThis model requires a strong magnetic field and a\nlarge population of energetic electrons to be present\nin the downstream. Observations suggest that the\nfraction of post-shock thermal energy density car-\nried by non-thermal electrons, ǫe, is large, ǫe≈\n0.1 (e.g. Zhang & M´ esz´ aros 2004; Frail et al. 2001;\nFreedman & Waxman 2001; Berger et al. 2003). The\nfractionofpost-shockthermalenergycarriedbythemag-\nnetic field, ǫB, is less well constrained by observations.\nHowever, in cases where ǫBcan be reliably constrained\nby multi waveband spectra, values close to equipartition,\nǫB��0.01 to 0.1, are inferred (e.g. Frail et al. 2000).\nThe non-thermal energetic electron (and proton)\npopulation is believed to be produced by the dif-\nfusive (Fermi) shock acceleration (DSA) mechanism\n(for reviews see Drury 1983; Blandford & Eichler 1987;\nMalkov & O’C Drury 2001).\nThe required magnetic fields in the shock frame in\nthe downstream (e.g. Frail et al. 2000) and upstream\n(Li & Waxman 2006) regions are much larger than the\nambient field, and thus require substantial amplification.\nThe accelerated particles are likely to have an important\nitay.rabinak@weizmann.ac.ilrole in generating and maintaining the inferred magnetic\nfields.\nThe main challenge associated with the downstream\nmagnetic field is that the field amplitude must remain\nclosetoequipartitiondeepintothedownstream,overdis-\ntances∼1010lsd(Gruzinov & Waxman 1999; Gruzinov\n2001a). While near equipartition fields on skin depth\nscalearelikelyto be produced in the vicinityofthe shock\nby electromagnetic (e.g. Weibel-like) instabilities (e.g.\nBlandford & Eichler 1987; Gruzinov & Waxman 1999;\nMedvedev & Loeb 1999; Wiersma & Achterberg 2004),\nthey are expected to decay within a few skin-depths\ndownstream (Gruzinov 2001a). This suggests that the\ncorrelation length of the magnetic field far downstream\nandpossiblyupstreammustbemuchlargerthan theskin\ndepth,L≫lsd, perhaps even of the order of the distance\nfrom the shock (Gruzinov & Waxman 1999; Gruzinov\n2001a; Katz et al. 2007).\nThe search for a self-consistent theory of collision-\nless shocks has led to extensive numerical studies\nusing the particle in cell (PIC) based algorithms (e.g.\nGruzinov 2001a,b; Medvedev et al. 2005; Silva et al.\n2003; Nishikawa et al. 2003; Frederiksen et al. 2004;\nJaroschek et al. 2004; Spitkovsky 2005, 2008a;\nMartins et al. 2009). Such simulations have pro-\nvided compelling evidence for acceleration of particles\nand generation of long lasting near-equipartition mag-\nnetic fields. However numerically simulating the long\nterm behavior is challenging and is currently restricted\nto pair ( e+e−) plasmas in 2D (e.g. Spitkovsky 2008b;\nKeshet et al. 2009).\nLarge scale magnetic fields may possibly be gen-\nerated in the upstream by the interaction of the2\nbeam of CRs propagating ahead of the shock and the\nupstream plasma (e.g. Katz et al. 2007; Keshet et al.\n2009). In particular high energy CRs naturally in-\ntroduce large scales due to their large Larmor ra-\ndius, and the large distances to which they prop-\nagate into the upstream. Instabilities arising from\nthe interaction of relativistic beams and cold plasmas\nhave long been studied (Akhiezer 1975, and references\ntherein) and are suspected of amplifying the magnetic\nfield in the shock transition layer (Gruzinov & Waxman\n1999; Medvedev & Loeb 1999; Wiersma & Achterberg\n2004; Bret et al. 2005; Lyubarsky & Eichler 2006;\nAchterberg et al. 2007; Achterberg & Wiersma 2007;\nBret 2009). In (Lemoine & Pelletier 2009) a systematic\nstudy of these instabilities for the lowest energy CRs,\nwith energies comparable to the thermal energy of the\nshocked plasma, and their application for Fermi acceler-\nation is given.\nIn this paperwe analyzelong wavelengthplasma insta-\nbilities resulting from the counter-streaming flow of high\nenergyCRs , γ≫Γsh, running far aheadofthe shockand\na non-magnetized upstream plasma. The analysis is re-\nstricted to long wavelength modes, k≪ω0/c, which are\nexpected to deflect particles efficiently. For simplicity it\nis assumed that the particle distribution is homogenous.\nThe paper is organized as follows. In §2 we calculate\nthe growth rate of long wavelength modes. We sepa-\nrately discuss highly collimated beams and beams with\na significant transversevelocity spread, and derive a con-\ndition for the stability of these modes. In §3 we discuss\nthe possible implications of these results to collisionless\nshocks. In §4wesummarizethemain resultsandconclu-\nsions. An estimate of the saturation level of the modes is\nbeyond the scope of this paper. Throughout this paper,\nunits with c= 1 are assumed ( cis retained in some of\nthe expressions).\n2.ANALYSIS\nConsider a homogenous, anisotropic distribution of\nparticles consisting of a cold plasma and an axi-\nsymmetric beam of ultra relativistic CRs. The analysis\nis carried out in the rest frame of the cold plasma which\ninitially has zero magnetic and electric fields.\nThe plasma frequencies, in this frame, of the cold\nplasma and the beam are denoted by ω0andωCR, re-\nspectively, where the plasma frequency of a plasma with\nspeciesiis defined by\nω2\np=/summationdisplay\ni4πq2\ni\nmi/integraldisplayd3p′\nγ(p′)fi(p′), (1)\nwhereqi,mi,fi(p) are the species’ charge, mass and mo-\nmentum distribution. It is assumed that ω0≫ωCR, and\nthattheCRshaveasmallbutfinitespreadinthevelocity\ndirections.\nIn this section we analyzethe linear growth ofunstable\nmodes with long wavelengths, k≪ω0. We start by\nconsidering a beam with no transverse velocity spread in\n§2.1. We show that the entire k-space regime considered\nis unstable and provide a simple analytic expression of\nthe instability growth rate. The effects of a spread in the\nvelocity directions of the CRs are discussed in §2.2.\n2.1.No spreadConsider the simplest case in which all the particles in\nthe beam propagate in the same direction and are ultra\nrelativistic (with an arbitrary energy distribution). It\nis straight forward to write the full dispersion equation\nwhich turns out to be a sixth order polynomial equation\nforωwith real coefficients (cf. §A and e.g. Akhiezer\n1975,§6.4.2). Four of the six solutions for ωare small\nperturbations, of order ω2\nCR/ω2\n0, of the four cold plasma\noscillating modes ω=±ω0,±(ω2\n0+k2)0.5, and are real\n(stable). The two remaining solutions have a non zero\nimaginary part and therefore are complex conjugates of\neach other. Hence for each kthere is one unstable mode.\nBelow we derive, directly from the Maxwell-Vlasovequa-\ntions, an approximate expression for the growth rate of\nthis mode, Eqs. (10), and (11). More details and a nu-\nmerical solution for the dispersion equation are given in\n§A.\nFor any axi-symmetric distribution of particles the lin-\nearmodescanbe separatedinto modeshavinganelectric\nfield in the x−kplane, where xis the axis of symme-\ntry (and magnetic field perpendicular to this plane), and\nmodes with an electric field perpendicular to xandk\n(cf.§A). The unstable mode has an electric field, E, in\nthex−kplane, and a magnetic field, B, perpendicular\nto this plane. The electrical currents carried by the cold\nplasma and the CRs, as derived by the Vlasov equations,\nare respectively given by\n4πJ0=ω2\n0\n−iωE, (2)\nand\n4πJCR,⊥=iω2\nCR(B−E⊥)/Ω; (3)\n4πJCR,||=−iω2\nCR(B−E⊥)k⊥/Ω2,(4)\nwheresubscripts ⊥and||correspondtocomponents that\nare perpendicular to the beam and parallel to the beam\nrespectively, Ω ≡k||−ω, and where we used the ultra\nrelativistic approximation β= 1 for the CRs.\nBy neglecting the displacement current ∂tE, compared\nto the current carried by the cold plasma [using ω≪ω0\nwhich is self consistently implied by the result, Eqs. (10)\nand (11)], the Maxwell equations read\nik⊥B= 4π(JCR,||+J0,||); (5)\n−ik||B= 4π(JCR,⊥+J0,⊥); (6)\niωB=−ik⊥E||+ik||E⊥. (7)\nEquation (7) can be written as\niω(B−E⊥) =−ik⊥E||, (8)\nbyselfconsistentlyneglectingΩ E⊥asfollows. Inregimes\nwhere Ω ≪k||, this term is negligible compared to the\ntermk||E⊥. Otherwise, where Ω /greaterorsimilark||, Eq. (11), implies\nthat Ω∼ωand Eqs. (6) and (3) imply that E⊥≪B,\nmaking the term Ω E⊥negligible compared to ωB.\nBy neglecting the term ik⊥Bin Eq. (5), compared to\nJCR,||[using Eqs. (4) and (10) ], and substituting for\nJCR,||,J0,||,E||,equations (4), (2), and (8) respectively,\nequation (5) becomes\nω2\nCRBk⊥/Ω2=−ω2\n0B/k⊥. (9)3\n10−410−210010−610−510−410−310−2\nk⊥/ω0η0/ω0\n  \nk||/ω0= 0.05\nk||/ω0= 0.5\nk||/ω0= 1\nAnalytic approx.\nFig. 1.— Growth rate of unstable modes for a delta function\nmomentum distribution of beam particles with plasma freque ncy\nωCR/ω0= 0.01 and Lorentz factor γ= 5000. The solid lines are\nthe exact solutions ofthe dispersion relation, evaluated n umerically\n(cf.§A.) The dashed line (covering the solid line with k||=\n0.05ω0)isthe analytical approximation given inequation (10). Th e\ncurvature at low values of k⊥is due to the electrostatic mode.\nThe solution of this equation for ωis\nIm{ω}=±ωCR\nω0k⊥≡ ±η0(k⊥); (10)\nRe{ω}=k||. (11)\nEq. (6) determines the ratio between E⊥andB. The\nresult, Eqs. (10) and (11), agrees with that of (Akhiezer\n1975,§6.4.2) and (Lemoine & Pelletier 2009) in the rel-\nevantkspace regimes.\nBy retaining the term ik⊥Bin Eq. (5), and assum-\ning finite Lorentz factors, the dispersion relation can be\nsimilarly solved resulting in\nIm{ω}=±ωCR/radicalBigg\nk2\n⊥\nω2\n0+k2\n⊥+k2\n||/angbracketleftbigg1\nγ2/angbracketrightbigg\n,(12)\nwhere averagingis over q2\nifi(p)/(miγ) [cf. Eq. (1)]. This\nsolution, which is valid for all kwithk||≪1, reduces to\nEq. (10) in the regime\nk||/angbracketleftbig\nγ−2/angbracketrightbig1/2≪k⊥≪ω0. (13)\nIn figure (1) the analytical approximation for the\ngrowth rate given in equation (10) is compared with a\nnumerical solution of equation (A2) for different modes.\nFor illustration modes with k||∼ω0that are not ana-\nlyzed in this paper are shown. As can be seen in the\nfigure, the analytic expression provides an excellent ap-\nproximation in the relevant k-space regime.\n2.2.With spread\nFor a velocity distribution of the beam particles that\ndiffers from a delta function, Eq. (9) should be replaced\nwith\nik⊥B=−iω2\nCRBk⊥/angbracketleftbigg1\n(Ω−k⊥β⊥)2/angbracketrightbigg\n−iω2\n0B/k⊥,(14)\nor /angbracketleftbigg1\n(Ω−k⊥β⊥)2/angbracketrightbigg\n=−1\nη2\n0, (15)where the same approximations leading to Eq. (9) were\nused, Ω≡k||β||−ω, andη0is the growth rate for a delta\nfunction momentum distribution Eq. (10). Averaging is\ncarried over the velocity distribution function,\n/angbracketleftbigg1\n(Ω−k⊥β⊥)2/angbracketrightbigg\n=1\nω2\nCR/integraldisplay∞\n−∞f(β⊥)\n(Ω−k⊥β⊥)2dβ⊥,(16)\nwhere\nf(β⊥) =/summationdisplay\ni4πq2\ni\nmi/integraldisplayd3p′\nγ(p′)fi(p′)δ(β′\n⊥−β⊥).(17)\nIt is to be understood that whenever a singularity is en-\ncountered, the expression should be evaluated at Ω →\nΩ−iǫin the limit ǫ→0+.\nConsider first the following 1D rectangular distribu-\ntion:βx≡β||=βcos(θ), βy≡β⊥=βsin(θ), βz= 0,\nwithθuniformly distributed between ±∆θ, andβ= /radicalbig\n1−γ−2(see Achterberg et al. 2007, for a discussion\nof a distribution with two identical counter streaming\nbeams). For small angular spread the growth rate can be\napproximatedanalyticallyby neglecting the variationsof\nβx(assuming βx=β). Under this assumption Eq. (16)\nreads/angbracketleftbigg1\n(Ω−k⊥β⊥)2/angbracketrightbigg\n=1\n(k⊥β∆θ)2−Ω2,(18)\nthe dispersion relation [Eq. (15)] reads\n1\n(k⊥β∆θ)2−Ω2=−1\nη2\n0, (19)\nand the solution for Ω2is\nΩ2= (k⊥β∆θ)2−η2\n0. (20)\nForillustration, thegrowthratesoftheunstablemodes\nare shown in figure (2) as a function of the spread ∆ θfor\nthe 1D rectangular distribution considered above. The\nsolidlinesinthisfigurearetheresultsofasemi-analytical\ncalculation in which the velocity integrals where evalu-\nated analytically, and a continuous solution for the dis-\npersion equation as a function of the spread was found\nnumerically, starting from the solution for a delta func-\ntion distribution. The obtained solution is verified to be\nthe fastest growing one at a given k, by comparing it\nto the results of a ’1-mode’ PIC simulation (shown as x\nsigns, cf. §B). These results are compared to the growth\nrate estimate of Eq. (20) (dots) and to the growth rate\nof the delta function η0distribution (dotted line). As\ncan be seen in the figure the estimate given in Eq. (20)\nis in very good agreement with both the simulation and\nthe semi-analytical calculation.\nNote the followingfeatures ofthe solution given by Eq.\n(20):\n1. For a sufficiently small spread, ∆ θ <∆θcritthere\nexists an unstable mode with Re{ω}=k||β, where\nthe condition for instability is\nk⊥β∆θ < η0, (21)\nor\nβ∆θ < ωCR/ω0 (22)\n2. The growth rate, Im(Ω) monotonically decreases\nfromη0to 0 as the spread ∆ θis increased.4\n10−310−210−100.511.522.533.5x 10−3\n∆ θ η/ω0\n  \nk⊥/ω0= 0.4\nk⊥/ω0= 0.2\nFig. 2.— Growth rate calculations in the presence of a spread in\nthe transverse velocities of the beam particles. Different m ethods,\nas described in the text in §2.2, were used to calculate the growth\nrates. Results are given for {k||/ω0= 0.05,k⊥/ω0= 0.4(0.2)}\nWeibel modes in red (blue) of a beam with γu= 5000, ωCR/ω0=\n10−2and different spreads, ∆ θ. Solid lines, dots and xsymbolsgive\nthe growth rates obtained using the semi-analytical calcul ation,\nEq. (20), and a ’1 mode PIC simulation’ (cf. §B) respectively.\nDotted lines show the growth rate for a delta function moment um\ndistribution (no spread, ∆ θ= 0).\n3. For ∆ θ >∆θcritthe mode becomes stable.\nThe criterion for instability given in (21) can be inter-\npreted asthe simple requirement(Akhiezer 1975, §6.4.2)\nthat the particles in the beam do not move in the direc-\ntion perpendicular to the beam a distance exceeding the\nwavelength of the mode during one e-folding time (of the\ndelta-function instability).\nWenextshowthatmostofthefeaturesofequation(20)\nare generic to a large class of axi-symmetricdistributions\nof CRs with a small spread.\nFor purely imaginary Ω = −iη, Eq. (16) reads\n/angbracketleftbigg1\n(Ω−k⊥β⊥)2/angbracketrightbigg\n= 2/integraldisplay∞\n0k2\n⊥β2\n⊥−η2\n(η2+k2\n⊥β2\n⊥)2f(β⊥)dβ⊥,\n(23)\nand has a zero imaginary part for all real η. In the\nlimitsη→0+,∞the term/angbracketleftbig\n(Ω−k⊥β⊥)−2/angbracketrightbig\ngoes to\n−k−2\n⊥/angbracketleftbig\nβ−2\n⊥/angbracketrightbig\nand 0 respectively, where\n/angbracketleftbigg1\nβ2\n⊥/angbracketrightbigg\n≡ −lim\nǫ→0/integraldisplay∞\n−∞f(β⊥)\n(β⊥+iǫ)2dβ⊥ (24)\n=/integraldisplay∞\n−∞f(β⊥)|β⊥=0−f(β⊥)\nβ2\n⊥dβ⊥.(25)\nThe last equality follows from the fact that/integraldisplay∞\n−∞1\n(x+iǫ)2dx= 0. (26)\nThe following features of Eq. (15) generalize the fea-\ntures of the 1D distribution considered above with the\ncondition for instability,\nk⊥/angbracketleftbig\nβ−2\n⊥/angbracketrightbig−1/2< η0, (27)or /angbracketleftbig\nβ−2\n⊥/angbracketrightbig−1/2< ωCR/ω0, (28)\ngeneralizing the condition (21).\n1. For a small spread,/angbracketleftbig\nβ−2\n⊥/angbracketrightbig−1/2< η0/k⊥(equiv-\nalent to k−2\n⊥/angbracketleftbig\nβ−2\n⊥/angbracketrightbig\n>1/η2\n0), there exists an un-\nstable mode, with Re{ω}=k||β. To see this,\nnote that the term/angbracketleftbig\n(Ω−k⊥β⊥)−2/angbracketrightbig\ncontinuously\nchanges from −k2\n⊥/angbracketleftbig\nβ−2/angbracketrightbig\nto 0 asηchanges from 0\nto∞, and must be equal to −η2\n0for some positive\nη.\n2. Assuming that the term/angbracketleftbig\n(Ω−k⊥β⊥)−2/angbracketrightbig\nis mono-\ntonically increasing with η, for a marginal spread /angbracketleftbig\nβ−2\n⊥/angbracketrightbig−1/2→η0/k⊥the mode becomes stable\nη→0. This suggests that for larger spreads the\nmode is stable.\nAs for the 1-D distribution considered above, the cri-\nterion for instability given in (27) can roughly be inter-\npreted as the requirement that the particles in the beam\ndo not move in the direction perpendicular to the beam\na distance exceeding the wavelength of the mode dur-\ning one e-folding time. Note however that the simple\nexpression, ∆ θβ, in the 1-D case, which is equal to the\nmaximal velocity of the particles in the direction per-\npendicular to the beam, is replaced by the non trivial\naverage,/angbracketleftbig\nβ−2\n⊥/angbracketrightbig−1/2[the velocity averagedefined in (24)],\nwhich has a less obvious meaning.\n3.APPLICATION TO COLLISIONLESS SHOCKS\nWe next discuss the possible application of the above\nresults to the study of long wavelength magnetic field\ngeneration in the far upstream of collisionless shocks.\nThe application of linear analysis of homogenous distri-\nbutions to the non-homogenous and non-linear problem\nof collisionless shocks is far from being trivial. Further-\nmore, the long wavelength modes studied above are not\nthe fastest growing modes, and can be affected by the\nfaster growing, short wavelength modes once the latter\nreach the non-linear stages. Nevertheless, the analysis of\nlinear growth of long-wave length modes is an important\nstep in the study of long-wave length magnetic field gen-\neration and can be used as a basis for comparison once\nmore accurate calculations are made (e.g. PIC simula-\ntions). In addition it is possible that some of the main\nfeaturesofthe linearmodes alsoappearin the more com-\nplicated shock scenario.\nConsider a shock with Lorentz factor Γ sh≫1 propa-\ngating into a cold plasma with particle density n0and\nplasma frequency ω0. We first assume that all parti-\ncles in the plasma have the same mass (e.g. electrons-\npositrons) and then generalize the results to an electron\nproton plasma.\nAssume that high energy, shock accelerated cosmic\nrays carry a fraction ∼ǫpof the post-shock energy.\nIn the shock frame, the cosmic rays are not highly\nbeamed and have an energy distribution ncr,s(> γs)∼\nǫpΓshn0(γs/Γsh)−p+1withp≈2, where γsis the Lorenz\nfactor of the cosmic rays in the shock frame.\nWe wish to analyze the generation of long wavelength\nmagnetic fields in a region surrounding a point, x, deep\nin the upstream due to the interaction of the CRs that\nreachthispointandtheincomingupstreamparticles. We5\nassumethatthispointisreachedbyasubstantialfraction\nof the CRs that have Lorenz factors larger than a space\ndependent minimum γs(x), and study the instabilities in\na simplified homogenous model of the upstream frame.\nThe cosmic rays in the upstream frame are beamed to\nan angular separation of ∼1/Γsh, and have a plasma\nfrequency\nωCR∼ǫ1/2\np(γs/Γsh)−p/2ω0, (29)\nwhereω0is the upstream plasma frequency and different\nvalues of γsrepresent different positions in the simplified\npicture1. Deep in the upstream, where γs(x)≫Γsh,\nthe plasma frequency of the cosmic rays is much smaller\nthan that of the upstream, and the analysis of section §2\nholds.\nEqs. (10) and (29) imply that long wavelength modes\nwill grow with a growth rate of approximately\nη0=k⊥ǫ1/2\np(γs/Γsh)−p/2. (30)\nIt is useful to compare the e-folding time η−1\n0of this instability to the time it takes the ambi-\nent magnetic field B0to deflect a cosmic ray par-\nticle by an angle of 1 /Γshback to the downstream,\nTR,0∼Γshγsmc/(eB0Γsh) =γsmc/(eB0). The ratio of\nthe two times is\nTR,uη0∼Γshk⊥\nω0ǫ1/2\npǫ−1/2\nB,0(γs/Γsh)1−p/2,(31)\nwhereǫB,0∼B2\n0/(8πn0mc2) is the ratio of the energy\ndensity in the ambient magnetic field to the upstream\nrest mass energy density. For small values of the mag-\nnetic field in the upstream, there will be a large range of\nkvectors\nk⊥/greaterorsimilarΓ−1\nshǫ−1/2\npǫ1/2\nB,0(32)\nfor which the instability will grow on time scales that are\nmuch shorter than the deflection time of the particles.\nFarawayfromthe shockin the upstream, where ωCRis\nsufficiently small, the instability will be suppressed due\nto the 1/Γshspread in the cosmic rays transverse veloc-\nities. Using Eqs. (27) [or (21)] and (30), the modes are\nunstable only at locations in the upstream where\nγs(x)/lessorsimilarΓsh/parenleftBig\nǫ1/2\npΓsh/parenrightBig2/p\n. (33)\nFor illustration, the growth rate of a specific unsta-\nble mode is shown in figure (3) as a function of γs. For\nsimplicity, the momentum distribution is assumed to be\nthe same as in §2.2 with parameters chosen in terms of\nshock parameters as: ±∆θ=±1/Γsh(with Γ sh= 25),\nupstream Lorentz factor γu= 2γsΓsh, and beam plasma\nfrequency, ωcras given in Eq. (29) with p= 2, and\nǫp= 0.1. In the figure we also show the results of the\nsemi-analytical calculation (cf. §2.2), the growth rate\nestimate of Eq. (20) (dots) and the results of the 1-\nmode PIC simulation (cf. §B, x signs). For compari-\nson the growth rate η0of the delta function distribution\n(cf.§2.1, dotted line), and the maximal γsfrom equa-\ntion (33) (red circle, y axis value arbitrarily set to 0)\nare also shown. As can be seen in the figure the delta\nfunction momentum distribution result is a good approx-\nimation at small angular spread, and the angular spread\n1Note that the energy carried by the cosmic rays in this frame\ngreatly exceeds that of the upstream particle rest mass ener gy den-\nsity for Γ s≫ǫ1/4\np,e0=n0mc2.10200.10.20.30.4\nγcr,s η/ωcr\n  \nFig. 3.— Growth rate for a specific Weibel mode, with k||/ω0=\n0.05 andk⊥/ω0= 0.5, shown as a function of the shock frame\nLorentz factor, γs, ofthe beam particles. Thebeam particleshave a\n1D rectangular phase space distribution (cf. §2.2) with parameters\nas discussed in the text in §3. The solid line, dots and x symbols\ngive the growth rates obtained using the semi-analyticalca lculation\n(cf.§2.2), Eq. (20), and a ’1 mode PIC simulation’ (cf. §B)\nrespectively. The dotted line is the growth rate obtained fo r a delta\nfunction distribution, η0. The red circle shows the value of γCR,s\ngiven by equation (33), beyond which the modes are predicted to\nbe stable.\nsuppresses the growth rate at high cosmic ray Lorentz\nfactors in accordance with the estimate in equation (33).\nNext, consider a shock propagating into a electron-\nproton plasma. The equation for the plasma frequency\nof the beam (29) should be replaced by\nωCR∼ǫ1/2\np(me/mp)1/2(γs,p/Γsh)−p/2ω0,(34)\nwhereγs,pistheminimal CRprotonshockframeLorentz\nfactor at the position considered and where the contri-\nbution of the electron CRs at a given particle energy was\nneglected. Given the transverse velocity spread, 1 /Γsh,\nof the CRs Eqs. (28) and (22) imply that the long wave-\nlength modes do not grow for moderate shock Lorentz\nfactors (see also Lyubarsky & Eichler 2006)\nΓsh/lessorsimilar100ǫ1/2\np,−1, (35)\nwhereǫp= 0.1ǫp,−1.For higher shock Lorentz factors,\nthe long wavelength modes grow at locations where the\nminimal shock frame CR energy, ǫs(x) (equal for CR\nelectrons and protons), is sufficiently low [cf. Eq. (33)]\nǫs(x)/lessorsimilarΓshmp/parenleftBig\nǫ1/2\npΓsh/parenrightBig2/p\n(me/mp)1/p.(36)\n4.DISCUSSION\nIn this paper the growth rates of the long wavelength\nunstable modes arising from the interaction of a beam of\nultra-relativistic CRs and an unmagnetized cold plasma\nwere calculated. We have shown that in the ultra-\nrelativistic limit all long wavelength modes are unstable\nwith a growth rate η0=k⊥ωCR/ω0[Eq. (10)], as long as\nthe spread in the transverse velocity distribution of the\nbeam is sufficiently small. An extension of this result,\nfor finite Lorentz factors and k⊥/greaterorsimilarω0, is given in Eq.6\n(12). For large transverse velocity spreads the instability\nis suppressed. The condition for instability was derived\nfor a large class of velocity distributions [Eqs. (28) and\n(24)].\nThe possible application of these results to the interac-\ntionofCRswiththeincomingplasmainthefarupstream\nofunmagnetizedcollisionlessshockswasaddressedin §3.\nIn a shock propagating into an electron positron plasma,\nthe instability grows in upstream regions where the min-\nimal shock frame CR Lorentz factor is sufficiently low,\nγs<Γ2\nshǫ1/2\np[cf. Eq. (33)]. Farther upstream of the\nshock the instability is suppressed due to the low density\nof the CRs. If these instabilities dominate the particle\ndeflectionsresponsibleforparticleacceleration,theaccel-\neration of CRs to shock frame Lorentz factors exceeding\nΓ2\nshǫ1/2\np[cf. Eq. (33)] may be obstructed.The long wavelength instabilities considered do not\ngrow in the upstream of a shock propagating into an\nelectron proton plasma having Lorentz factors /lessorsimilar100 (cf.\nEq. (35), see also Lyubarsky & Eichler 2006) implying\nthat: 1. The modes considered are not important\nin the relativistic shocks responsible for the observed\nGRBs afterglow emission (except possibly at the earliest\nstages, when Γ sh/greaterorsimilar100); 2. Particle acceleration in\nelectron-positron plasmas suggested by the results of\nPIC simulations Spitkovsky (2008b), where these modes\ncan grow, may differ from acceleration in electron-proton\nplasmas.\nWewouldliketothankAnatolySpitkovsky,UriKeshet\nand Avi Loeb for useful discussions. This research was\npartially supported by ISF, AEC and Minerva grants.\nAPPENDIX\nA.FULL SOLUTION OF THE DISPERSION EQUATION\nConsider the case, described in §2, where the velocity distribution of the particles is axi-symmetric (a roundx). In\nthis case averages over the velocity distributions of the type /an}bracketle{tβ⊥/an}bracketri}ht, and/an}bracketle{tβz/an}bracketri}ht, where⊥,zare directions perpendicular to\nx, are zero. As a result, terms in the beam susceptibility (c.f. Melros e 1986),χCRcontaining such averages cancel out,\nand the susceptibility will only have non-diagonal term in the plane defi ned by beam direction and the wavenumber\nvector,k(thex−kplane). In this plane the beam susceptibility, the electro-magnetic fi eld susceptibility, and the cold\nplasma susceptibility have matrix forms which are respectively\nχCR=ω2\nCR/parenleftBigg\n1−2βk||\nΩ+(k2−ω2)β2\nΩ2−βk⊥ω2\nΩ\n−βk⊥ω2\nΩ1/parenrightBigg\n;χEM=−/parenleftbiggω2−k2\n⊥k||k⊥\nk||k⊥ω2−k2\n||/parenrightbigg\n;χ0=ω2\n0,(A1)\nwhere Ω is defined after equation (4), and the magnetic field Bis perpendicular to the x−kaxis. In the remaining\naxis only the plasma frequency term remains and this axis gives rise on ly to the plasma oscillation modes.\nIn the absence of the beam the dispersion equation, det {χ0+χEM}= 0, is a forth order polynomial equation in\nωwith real coefficients which has the following 4 solutions, ω=±ω0,±(ω2\n0+k2)0.5, that represent respectively the\nplasma oscillation and the electro-magnetic mode. In the presence o f the beam the dispersion equation becomes\ndet{χ0+χCR+χEM}= 0, (A2)\nwhich is a 6’th order polynomial equation in ωwith real coefficients. This equation is a slight perturbation of the\noriginal dispersion equation with 6 distinct solutions. Four of these s olutions are slight deviation from the solutions\nof the original dispersion equation, while the other two solutions, fo r thekspace regime that is considered above, are\nthe solutions discussed in §2.1. The full dispersion equation can be solved numerically for any kand these solutions\nare shown in figure (1).\nB.1-MODE PIC SIMULATIONS\nThe solution described above was numerically verified to be the faste st growing one at several wavevectors, k. This\nwas done by performing an efficient PIC simulation in which only one kmode is treated while the rest of the modes are\nneglected. In this simulation, the electric field, magnetic field, and ele ctric current are sinusoidal with a given /vectorkvalue\nand a time dependent amplitude, while the cosmic rays are treated as particles with continuous position (1D along /vectork)\nand momenta (3D). The upstream, which is assumed to have a delta f unction momentum distribution, is written in\nterms of fluid quantities in the linear approximation and is likewise sinuso idal with the given /vectork. The time dependent\namplitude of the electric current carried by the cosmic rays is derive d from the distribution of the particles by\njCR\nk=/summationdisplay\njqjβjexp(k·xj), (B1)\nwhereqj,βj,andxjare the charge velocity and position of the particle j. As such, the simulation is only accurate for\nthe linear regime (the amplitude at which the mode becomes non-linear can in principle be identified). The growth rate\nis calculated by fitting the time evolution of the amplitude of the mode w ith an exponential function in time. With\nthis method only the fastest growing mode at the given /vectorkis accounted for. This is a direct, physically transparent\nmethod of studying the linear regime of a single wavevector, with no r estriction on the 3D velocity distribution, which\nis time and memory efficient.\nFor illustration, the results of the simulation used for figure (3) are presented in figure (B1). As an underlaying\nquantity for the fitting we use the amplitude of the electric current ,jk, which is normalized to the current j0=qn0c,7\n10 20 30 40 50 60−5−4−3−2−1012\ntω0ln[|jk|/(qn0c)]\nFig. B1.— The growth of a {k||/ω0= 0.05,k⊥/ω0= 0.5}Weible mode in a 1-mode PIC simulation described in the text i n§B. The\nparameters chosen for this simulation were taken to reprodu ce theγs= 25 (ω2\nCR= 0.1,γu= 1250) point given in figure (3). The green\ncurve is the logarithm of the amplitude of the current vector ,jk, obtained from the simulation, and the blue line is a linear fi t to this curve\nin the range 19 .6< tω0<44.5. Its slope, 0 .131, represents the growth rate.\nwheren0is the upstream density, and qis the electron charge. As can be seen in the figure, the current jkgrows\nexponentiallywithtime andthe growthrateoftheinstabilityiseasilyob tainedfromthefit. Theaccuracyofthegrowth\nrate obtained numerically is high despite the fact that only 106particles (cosmic rays) were used. The saturation level\nofjkcan also be obtained from this figure. However, since all quantities, except those related to the cosmic rays, are\ntreated in the linear regime, this saturation is representative only if in reality the saturation is governed by the cosmic\nrays.\nREFERENCES\nAchterberg, A. & Wiersma, J. 2007, A&A, 475, 1\nAchterberg, A., Wiersma, J., & Norman, C. A. 2007, A&A, 475,\n19\nAkhiezer, A. I. 1975, Plasma electrodynamics - Vol.1: Linea r\ntheory; Vol.2: Non-linear theory and fluctuations, ed. Akhi ezer,\nA. I.\nBerger, E., Kulkarni, S. R., & Frail, D. A. 2003, ApJ, 590, 379\nBlandford, R. & Eichler, D. 1987, Phy. Rep., 154, 1\nBret, A. 2009, ApJ, 699, 990\nBret, A., Firpo, M., & Deutsch, C. 2005, Physical Review Lett ers,\n94, 115002\nDrury, L. O. 1983, Reports on Progress in Physics, 46, 973\nFrail, D. A., Kulkarni, S. R., Sari, R., Djorgovski, S. G., Bl oom,\nJ. S., Galama, T. J., Reichart, D. E., Berger, E., Harrison,\nF. A., Price, P. A., Yost, S. 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Madejski, 345–350\nSpitkovsky, A. 2008a, ApJ, 673, L39\n—. 2008b, ApJ, 682, L5\nWiersma, J. & Achterberg, A. 2004, A&A, 428, 365\nZhang, B. & M´ esz´ aros, P. 2004, International Journal of Mo dern\nPhysics A, 19, 2385"    },    {        "title": "1412.0038v2.Non_equilibrium_thermodynamics_of_damped_Timoshenko_and_damped_Bresse_systems.pdf",        "content": "arXiv:1412.0038v2  [math-ph]  6 Mar 2015Non-equilibrium thermodynamics of damped Timoshenko\nand damped Bresse systems\nManh Hong Duong\nMathematics Institute,\nUniversity of Warwick,\nCoventry CV4 7AL, UK.\nEmail: m.h.duong@warwick.ac.uk\nAugust 4, 2018\nAbstract\nIn this paper, we cast damped Timoshenko and damped Bresse sy stems into a general\nframework for non-equilibrium thermodynamics, namely the GENERIC (General Equation\nfor Non-Equilibrium Reversible-Irreversible Coupling) f ramework. The main ingredients of\nGENERIC consist of five building blocks: a state space, a Pois son operator, a dissipative\noperator, an energy functional, and an entropy functional. The GENERIC formulation of\ndamped Timoshenko and damped Bresse systems brings several benefits. First, it provides\nalternativeways toderivethermodynamically consistent m odels ofthesesystems byconstruct-\ning building blocks instead of invoking conservation laws a nd constitutive relations. Second, it\nreveals clear physical and geometrical structures of these systems, e.g., the role of the energy\nand the entropy as the driving forces for the reversible and i rreversible dynamics respectively.\nThird, it allows us to introduce a new GENERIC model for dampe d Timoshenko systems\nthat is not existing in the literature.\nKey words. Non-equilibrium thermodynamics, GENERIC, damped Timoshe nko systems, damped\nBresse systems.\nPACS.05.70.Ln, 02.30.Jr\n1 Introduction\nGENERIC (General Equation for Non-Equilibrium Reversible-Irreve rsible Coupling [ ¨Ott05])\nis a formalism for non-equilibrium thermodynamics which unifies both re versible and irreversible\ndynamics. The main ingredients of GENERIC consist of five building bloc ks: a state space Z, a\nPoisson operator L, a dissipative operator M, an energy functional E, and an entropy functional\nS, which are required to satisfy certain conditions. A consequence o f these conditions is that the\nfirst and the second law of thermodynamics are fulfilled, i.e., the tota l energy is preserved and the\nentropy is increasing in time, see Section 2. To show a particular realiz ation of GENERIC one\nneeds to specify the five building blocks and verify the conditions impo sed on them.\nGENERIC has been proven to be a powerful framework for mathem atical modelling of com-\nplex systems. In the original papers [G ¨O97,¨OG97], it was originally introduced in the context of\ncomplex fluids with applications to the classical hydrodynamics and to non-isothermal kinetic\ntheory of polymeric fluids. Since then many models have been shown t o have a GENERIC\nstructure. Recently it has been applied further to anisotropic inela stic solids [HT08a], to vis-\ncoplastic solids [HT08b], to thermoelastic dissipative materials [Mie11], t o the soft glassy rheology\nmodel [FI13], and to turbulence [ ¨O14], just to name a few. More recently, the mathematically\n12\nrigorousstudy ofGENERIC hasreceiveda lotofattention. In [ADPZ 11, DLR13, DPZ13, MPR14,\nDuo14] the authors show that there is a deep connection between GENERIC structures of many\npartial differential equations including the diffusion equation, the Fo kker-Planck equation and\nthe Vlasov-Fokker-Planck equation and the large deviation principle of underlying stochastic pro-\ncesses. The connection provides microscopic interpretation for t he GENERIC structures of these\nequations [ADPZ11, DLR13, DPZ13, MPR14, Duo14], gives ideas to co nstruct approximation\nschemes for a generalized Kramers equation [DPZ14] and offers tec hniques to handle singular\nlimits of partial differential equations [AMP+12, Duo14, DPS13]. We refer to the original pa-\npers [G¨O97,¨OG97] and the book [ ¨Ott05] for an exposition of GENERIC and the mentioned\npapers as well as references therein for further information.\nThe aim of this paper is to show that damped Timoshenko and damped B resse systems\ncan be cast into the GENERIC framework. Both Timoshenko and Bre sse systems are described\nby wave equations. In damped Timoshenko and damped Bresse syst ems, irreversible behaviour\nis introduced via additional, frictional or heat conduction, damping m echanisms. This can be\naccomplished in various ways. The two systems play important roles in theory of elasticity and\nstrength of materials [Tim53]. They have been studied extensively in t he literature from various\nperspectives such as existence and uniqueness of solutions, their exponential stability and rate of\ndecay, see for instance [RR02, RFSC05, HK13, SR09, RR08, FRM14 ] for damped Timoshenko\nsystems and [SJ10, BRJ11, FM12, AFSM14, FR10, HS14, LR09] for damped Bresse systems.\nIn this paperwefocus onthe GENERICstructuresofthe twosyst ems. The formulationin the\nGENERIC framework has three benefits. Firstly, it provides altern ative ways to derive thermo-\ndynamically consistent models of these systems by constructing th e building blocks {Z,L,M,E,S}\ninstead of invoking conservation laws and constitutive relations. We demonstrate that all previ-\nous known models can be constructed in this way using a common proc edure. In addition, we\nalso illustrate how to use GENERIC to come up with a new model for the damped Timoshenko\nsystem. Secondly, thanks to its splitting formulation, the GENERIC framework provides clear\nphysical and geometrical structures for these systems, e.g., th e role of the energy and the entropy\nas the driving forces for the reversible and irreversible dynamics re spectively. Thirdly, this has the\npossibility of application of the recent developments for GENERIC as mentioned in the second\nparagraph to mathematically analyse the two systems.\nThe organization of the paper is as follows. In Section 2, we review th e GENERIC framework\nand its basic properties. In Section 3, we introduce a procedure fo r mathematically modelling\nof complex systems using GENERIC. In Section 4, we place damped Tim oshenko systems with\ndamping either by frictional mechanisms or by heat conduction of va rious types in the GENERIC\nframework. A similar analysis for damped Bresse systems is shown in S ection 5. Conclusion and\nfurther discussion are given in Section 6. Finally, Appendix A contains some detailed computa-\ntions, and Appendix B shows more models of the two systems.\n2 GENERIC\nIn this section, we summarize the GENERIC framework and its main pr operties. We refer to\nthe original papers [G ¨O97,¨OG97] and the book [ ¨Ott05] for an exposition of GENERIC.\n2.1 Definition of GENERIC\nAs mentioned in the introduction, GENERIC is a formulation for non-e quilibrium thermody-\nnamics that couples both reversible and irreversible dynamics in a cer tain way. Let z∈Zbe a set\nof variables which appropriately describe the system under conside ration, and Zdenotes a state\nspace. Let E,S:Z→Rbe two functionals, which are interpreted respectively as the tota l energy\nand the entropy. Finally, suppose that for each z∈Z,L(z) andM(z) are two operators, which\nare called Poisson operator and dissipative operator respectively, that map the cotangent space\natzto onto the tangent space at z. A GENERIC equation for zis then given by the following3\ndifferential equation\nzt=L(z)δE(z)\nδz+M(z)δS(z)\nδz, (1)\nwhere\n•ztis the time derivative of z.\n•δE\nδz,δS\nδzare appropriate derivatives, such as either the Fr´ echet derivat ive or a gradient with\nrespect to some inner product, of EandSrespectively.\n•L=L(z) is for each zan antisymmetric operator satisfying the Jacobi identity, i.e., for a ll\nfunctionals F,G,Fi:Z→R, i= 1,2,3\n{F,G}L=−{G,F}L, (2)\n{{F1,F2}L,F3}L+{{F2,F3}L,F1}L+{{F3,F1}L,F2}L= 0, (3)\nwhere the Poisson bracket {·,·}Lis defined via\n{F,G}L:=δF(z)\nδz·L(z)δG(z)\nδz. (4)\n•Similarly, M=M(z) is for each za symmetric and positive semidefinite operator, i.e., for all\nfunctionals F,G:Z→R,\n[F,G]M= [G,F]M, (5)\n[F,F]M≥0, (6)\nwhere the dissipative bracket [ ·,·]Mis defined by\n[F,G]M:=δF(z)\nδz·M(z)δG(z)\nδz. (7)\n•Moreover, {L,M,E,S}are required to fulfill the degeneracy conditions: for all z∈Z,\nL(z)δS(z)\nδz= 0,M(z)δE(z)\nδz= 0. (8)\n{Z,L,M,E,S}are called the building blocks and a GENERIC system is then fully charac terised\nby its building blocks.\n2.2 Basic properties of GENERIC\nWe recall two basis properties of GENERIC. The first one is that GEN ERIC automatically\nverifiesthe first and secondlaws ofthermodynamics. Moreprecise ly, the degeneracyconditions (8)\ntogetherwith the symmetries ofthe Poissonand dissipativebracke ts(2)-(6) ensurethat the energy\nis conserved along a solution\ndE(z(t))\ndt=δE(z)\nδz·dz\ndt(1)=δE(z)\nδz·/parenleftbigg\nL(z)δE(z)\nδz+M(z)δS(z)\nδz/parenrightbigg\n(5),(8)=δE(z)\nδz·L(z)δE(z)\nδz(4)={E,E}L(2)= 0,\nand that the entropy is a non-decreasing function of time\ndS(z(t))\ndt=δS(z)\nδz·dz\ndt(1)=δS(z)\nδz·/parenleftbigg\nL(z)δE(z)\nδz+M(z)δS(z)\nδz/parenrightbigg\n(2),(8)=δS(z)\nδz·M(z)δS(z)\nδz(7)= [S,S]M(6)\n≥0.4\nThe second property of GENERIC, which is useful when construct ing the building blocks, is\nthat it is invariant under coordinatetransformations[G ¨O97]. Let z/mapsto→zbe a one-to-onecoordinate\ntransformation. The new building blocks {L,M,E,S}are obtained via\nE(z) =E(z),S(z) =S(z),L(z) =/bracketleftbigg∂(z)\n∂(z)/bracketrightbigg−1\nL(z)/bracketleftbigg∂(z)\n∂(z)/bracketrightbigg−T\n,M(z) =/bracketleftbigg∂(z)\n∂(z)/bracketrightbigg−1\nM(z)/bracketleftbigg∂(z)\n∂(z)/bracketrightbigg−T\n,\n(9)\nwhere∂(z)\n∂(z)is the transformation matrix, [ ·]−1is the inverse of [ ·] and [·]−Tdenotes the transpose\nof [·]−1. Then the transformed GENERIC system {Z,L,M,E,S}is equivalent to the original one.\n3 Mathematical modelling of complex systems usingGENERIC\nAs explained in Section 2, the GENERIC framework provides a system atic method to derive\nthermodynamically consistent evolution equations. The modelling pro cedure of complex systems\nusing GENERIC can be summarized in the following procedure consistin g of three steps.\nStep 1. Choose state variable z∈Z;\nStep 2. Choose (construct) GENERIC building blocks that include\n•two functionals EandS\n•two operators LandM\nsuch that the GENERIC conditions (cf. Section 2) are fulfilled;\nStep 3. Derive the equation.\nWe emphasize again that the quintuple {Z,E,S,L,M}completely determine the model and that\nthe thermodynamics laws are automatically justified in the GENERIC f ramework.\nIn this paper, we apply this procedure to cast the existing damped T imoshenko and damped\nBresse systems into the GENERIC and to derive new models.\nAll the existing damped Timoshenko and damped Bresse models share a common feature:\nthey are described by wave equations coupled with damping mechanis ms, either by frictional or\nheat conduction. It is well-known that, without the damping effect, the energy of a wave equation\nis conserved. In other words, a wave equation is conservative. Ho wever, when the damping effect\nis present, the energy is no longer conserved. This implies that the d amped Timoshenko and\ndamped Bresse systems exhibit both conservative and dissipative e ffects represented in the wave\nequations and the damping mechanisms respectively. It is the reaso n why GENERIC would be a\nnatural framework to work with. To place them in the GENERIC sett ing, we need to specify the\nbuilding blocks and verify the conditions imposed on them, cf. Section 2. As also mentioned in\nSection 2, one might obtain the same GENERIC system from different building blocks provided\nthat they are related by a co-ordinate transformation (9). Ther efore, one can mathematically\nsimplify the construction of the building blocks by choosing the state space in such a way that\nthe entropy is multiple of one of its components. This technique has b een used in [DPZ14] for the\nVlasov-Fokker-Planck equation and will be used throughout the pr esent paper.\nWe arenowreadytointroduceacommonprocedureto placethe exis tingdampedTimoshenko\nanddamped Bressesystemsin the GENERICframework. Thisproce dureisaslightlymodification\nof the one described above where the first step is split into Step 1a a nd Step 1b and the second\nstep is decomposed to Step 2a and Step 2b.\nStep 1a. Re-write the system as a system of parabolic partial differ ential equations;\nStep 1b. Introduce a new variable e, depending only on time, to capture the lost of the energy of the\noriginal system. The time derivative of eis simply determined by the negative of that of the\nenergy of the original system.5\nStep 2a. Construct the building blocks of the GENERIC: the GENERI C-energy Eis defined to be\nthe summation of the original energy and e; the GENERIC-entropy Sis a multiple of e; the\nPoisson operator Lis easily deduced from the wave equations, the anti-symmetry prop erty\nand and the degeneracycondition L(z)δS\nδz(z) = 0; and the dissipativeoperator Mis built from\nthe damping effect, the symmetry property and the degeneracy c ondition M(z)δE\nδz(z) = 0;\nStep 2b. Verify the remaining conditions;\nStep 3. Derive the equation.\n4 GENERIC formulation of damped Timoshenko systems\nIn this section, we place damped Timoshenko systems of various typ es in the GENERIC\nframework. We perform the procedure described in Section 3 in det ails for two systems: the\nTimoshenko system with dual frictional damping and the Timoshenko system damped by heat\nconduction of type I. We also show how to derive a new GENERIC mode l. Some details of\ncomputation will be givenin Appendix A, and the Timoshenkosystem da mped by heatconduction\nof type II and type III will be presented in Appendix B.\n4.1 The Timoshenko system\nThe Timoshenko system, which describes the transverse vibration s of a beam, is a set of two\ncoupled wave equations of the form\nϕtt=k(ϕx+ψ)x,\nψtt=bψxx−k(ϕx+ψ).(10)\nHere,tis the time variable and xis the space coordinate along the beam. The function ϕis the\ntransverse displacement and ψis the rotation angle of a filament of the beam. Throughout this\npaper, subscripts denote derivatives of the functions with respe ct to the corresponding variables;\nfor instance, ϕxis the first order derivative of ϕwith respect to x. Finally,k,bare positive\nconstants.\nThe energy of the beam is\nE(t) =/integraldisplay\nΩ/parenleftbigg1\n2ϕ2\nt+1\n2ψ2\nt+k\n2(ϕx+ψ)2+b\n2ψ2\nx/parenrightbigg\ndx.\nSolutions of (10) do not decay and the system’s energy remains con stant at all times. This can\nbe seen easily by computing the derivative of E(t) with respect to time\nd\ndtE(t) =/integraldisplay\nΩ[ϕtϕtt+ψtψtt+k(ϕx+ψ)(ϕxt+ψt)+bψxψxt]dx\n=/integraldisplay\nΩ[kϕt(ϕx+ψ)x+ψt(bψxx−k(ϕx+ψ))+k(ϕx+ψ)(ϕxt+ψt)+bψxψxt]dx\n= 0.\nThere is a large amount of research in the literature devoted to find the damping necessary\nto add to the system (10) in order to stabilize its solutions [RR02, RFS C05, HK13, SR09, RR08,\nFRM14]. Two damping mechanisms have been mainly considered, namely damping by frictions\nand by heat conduction. The latter can be accomplished in various wa ys. By adding damping\neffects, we get damped Timoshenko systems, and the energy abov e is no longer preserved. The\nmain results of the mentioned papers are that damped Timoshenko s ystems are exponentially\nstable.\nIn the next sections, we will show that damped Timoshenko systems , either by frictions or by\nheat conduction, can be cast into the GENERIC framework.6\n4.2 The Timoshenko system with dual frictional damping\nWe start with the Timoshenko system (10) with two frictional terms added, i.e., the following\nsystem/braceleftigg\nϕtt=k(ϕx+ψ)x−δ1ϕt,\nψtt=bψxx−k(ϕx+ψ)−δ2ψt.(11)\nThe two terms δ1ϕtandδ2ψt, whereδ1,δ2≥0, represent the frictions. This system has been\nstudied, e.g., in[RFSC05], wheretheauthorsprovethatitisexponen tialstable. Set p=ϕt,q=ψt,\nthen Eq. (11) can be re-written as a system of parabolic differentia l equations.\n\n\nϕt=p,\nψt=q,\npt=−δ1p+k(ϕx+ψ)x,\nqt=−δ2q−k(ϕx+ψ)+bψxx.(12)\nWe introduce an auxiliary variable e,depending only on t, such that\nd\ndte(t)+d\ndt/integraldisplay\nΩ/parenleftbigg1\n2p2+k\n2(ϕx+ψ)2+1\n2q2+b\n2ψ2\nx/parenrightbigg\ndx= 0,\nwhich results in,\net=/integraldisplay\nΩ(δ1p2+δ2q2)dx. (13)\nWe give more detailed discussion on the introduction of ein Remark 4.1 below. We now show that\nthe system (12)-(13) can be cast into the GENERIC framework. T he building blocks are given by\nz= (ϕ ψ p q e )T,E(z) =e+/integraldisplay\nΩ/parenleftbigg1\n2p2+k\n2(ϕx+ψ)2+1\n2q2+b\n2ψ2\nx/parenrightbigg\ndx,S(z) =αe,\nL(z) =\n0 0 1 0 0\n0 0 0 1 0\n−1 0 0 0 0\n0−1 0 0 0\n0 0 0 0 0\n,M(z) =1\nα\n0 0 0 0 0\n0 0 0 0 0\n0 0 δ1 0 −δ1p\n0 0 0 δ2 −δ2q\n0 0−δ1/integraltext\nΩp·/squaredx−δ2/integraltext\nΩq·/squaredx/integraltext\nΩ(δ1p2+δ2q2)dx\n.\nIn the above expressions, α∈R\\{0}is some scaled parameter. The squares /squareinM(z) represent\nthe arguments that M(z) acts on.\nWe now derive the GENERIC equation associated to these building bloc ks by computing\nδE(z)\nδz=\n−k(ϕx+ψ)x\n−bψxx+k(ϕx+ψ)\np\nq\n1\n,δS(z)\nδz=\n0\n0\n0\n0\nα\n,\nL(z)δE(z)\nδz=\np\nq\nk(ϕx+ψ)x\nbψxx−k(ϕx+ψ)\n0\n,M(z)δS(z)\nδz=\n0\n0\n−δ1p\n−δ2q/integraltext\nΩ(δ1p2+δ2q2)dx\n.\nIt is now clear that the GENERIC equation\nzt=L(z)δE(z)\nδz+M(z)δS(z)\nδz,\nis equivalent to the system (12)-(13). Note how the GENERIC stru cture reveals/clarifies the\nconservative and dissipative parts. The verification that the buildin g blocks {Z,L,M,E,S}above\nsatisfy the conditions of GENERIC is given in the Appendix A.7\nRemark 4.1. We emphasize that the system (12)-(13) is coupled only in one direct ion: the newly\nadded equation for eis slaved to the original damped Timoshenko system. In other words , the\nintroductionof edoesnotchangethe originalsystem. Onecanthink ofitasapurelym athematical\ntechnique. Alternatively, from a physical point of view, one can also interpret this as embedding\nthe original system into a bigger reservoir/environmentand ecaptures the exchange of the energy\nbetween them.\n4.3 The Timoshenko system damped by heat conduction of type I\nIn this section, we consider the Timoshenko system (10) damped by heat conduction of the\nform (type I),\n\n\nϕtt=k(ϕx+ψ)x,\nψtt=bψxx−k(ϕx+ψ)−γθx,\nθt=κθxx−γψtx.(14)\nInthismodel, θisthetemperature. ItiscoupledtotheTimoshenskosystemviathe term−γθx\nin the equation for ψ. The system above, which is often known as the Timoshenko-Fourie r system\nsince the heat conduction is described by the classical Fourier law, h as been studied extensively\nin the literature, see for instance [RR02, HK13, SR09].\nSetp=ϕt,q=ψt, then Eq. (14) can be re-written as a system of parabolic different ial equations\n\n\nϕt=p,ψt=q,\npt=k(ϕx+ψ)x,\nqt=−k(ϕx+ψ)+bψxx−γθx,\nθt=κθxx−γqx.(15)\nSimilarly as in the previous section, we introduce an auxiliary variable e, depending only on\nt, such that\nd\ndte+d\ndt/integraldisplay\nΩ/parenleftbigg1\n2p2+1\n2q2+k\n2(ϕx+ψ)2+b\n2ψ2\nx+1\n2θ2/parenrightbigg\ndx= 0,\nwhich gives rise to\net=κ/integraldisplay\nΩθ2\nxdx. (16)\nWe now show that the coupled systems (15)-(16) can be cast into t he GENERIC framework. The\nbuilding blocks are as follows\nz= (ϕ ψ p q θ e )T,E(z) =/integraldisplay\nΩ/parenleftbigg1\n2p2+k\n2(ϕx+ψ)2+1\n2q2+b\n2ψ2\nx+1\n2θ2/parenrightbigg\ndx+e,S(z) =αe,\nL(z) =\n0 0 1 0 0 0\n0 0 0 1 0 0\n−1 0 0 0 0 0\n0−1 0 0 −γ∂x0\n0 0 0 −γ∂x0 0\n0 0 0 0 0 0\n,M(z) =1\nα\n0 0 0 0 0 0\n0 0 0 0 0 0\n0 0 0 0 0 0\n0 0 0 0 0 0\n0 0 0 0 −κ∂xx κθxx\n0 0 0 0 −κ/integraltext\nΩθx·∂x/squaredx κ/integraltext\nΩθ2\nxdx\n,\nHaving the building blocks, we can derive the GENERIC equation assoc iated to them. A direct8\ncalculation gives\nδE(z)\nδz=\n−k(ϕx+ψ)x\n−bψxx+k(ϕx+ψ)\np\nq\nθ\n1\n,δS(z)\nδz=\n0\n0\n0\n0\n0\nα\n,\nL(z)δE(z)\nδz=\np\nq\nk(ϕx+ψ)x\nbψxx−k(ϕx+ψ)−γθx\n−γqx\n0\n,M(z)δS(z)\nδz=\n0\n0\n0\n0\nκθxx\nκ/integraltext\nΩθ2\nxdx\n.\nIt follows that the GENERIC equation\nzt=L(z)δE(z)\nδz+M(z)δS(z)\nδz,\nis the same as the coupled systems (15)-(16). The verification tha t{L,M,E,S}satisfy the condi-\ntions of GENERIC is presented in the Appendix A.\nIn Appendix B, we list two more models: the Timoshenko system dampe d by heat conduction\nof type II and type III.\n4.4 A new model\nAswehaveshown, in ordertoplaceeachdampedTimoshenkosystem sinthe previoussections\nin the GENERIC framework, we need to introduce an extra auxiliary v ariablee. Motivated\nby [Mie11], we introduce the following system, which is GENERIC on its ow n, i.e., it is not\nnecessary to complement with an auxiliary variable. This example show s how GENERIC can be\nused to build up a new model that is thermodynamically consistent.\n\n\nϕtt=k(ϕx+ψ)x,\nψtt=bψxx−k(ϕx+ψ)+γθx,\nθt=δθxx+γθψtx.(17)\nThe difference between this model and the previousone lies in the sec ond term of the equation\nfor the temperature. The coupling to the mechanical part (i.e., the Timoshenko system) is now\nnon-linear. Set p=ϕt,q=ψt, then Eq. (17) can be re-written as\n\n\nϕt=p,\nψt=q,\npt=k(ϕx+ψ)x,\nqt=−k(ϕx+ψ)+bψxx+γθx,\nθt=δθxx+γθqx.(18)\nWe now show that this system is a GENERIC equation. The building block s are defined as9\nfollows\nz= (ϕ ψ p q θ )T\nE(z) =/integraldisplay\nΩ/parenleftbigg1\n2p2+k\n2(ϕx+ψ)2+1\n2q2+b\n2ψ2\nx+θ/parenrightbigg\ndx,S(z) =/integraldisplay\nΩlogθdx,\nL(z) =\n0 0 1 0 0\n0 0 0 1 0\n−1 0 0 0 0\n0−1 0 0 γ(θ/square)x\n0 0 0 γθ·/squarex0\n,M(z) =\n0 0 0 0 0\n0 0 0 0 0\n0 0 0 0 0\n0 0 0 0 0\n0 0 0 0 −δ(θ/squarex)x\n.\nWe have\nδE(z)\nδz) =\n−k(ϕx+ψ)x\n−bψxx+k(ϕx+ψ)\np\nq\n1\n,δS(z)\nδz=\n0\n0\n0\n0\n1\nθ\n,\nL(z)δE(z)\nδz) =\np\nq\nk(ϕx+ψ)x\nbψxx−k(ϕx+ψ)+γθx\nγθqx\n,M(z)δS(z)\nδz) =\n0\n0\n0\n0\nδθxx\n.\nSubstituing the above computation to the GENERIC equation\nzt=L(z)δE(z)\nδz)+M(z)δS(z)\nδz), (19)\nyields the system (18). It can be verified analogouslyas in the Appen dix A that {L,M,E,S}satisfy\nthe conditions of the GENERIC.\nWe now move to the second class of systems.\n5 Damped Bresse systems\nIn this section, we cast damped Bresse systems into the GENERIC f ramework. We recall the\nundamped Bresse system first.\n5.1 The Bresse system\nThe Bresse system, which is also knownas the circulararchproblem, is given by three coupled\nwave type equations as follows\n\n\nϕtt=k(ϕx+ψ+lφ)x+k0l(φx−lϕ),\nψtt=bψxx−k(ϕx+ψ+lφ),\nφtt=k0(φx−lϕ)x−kl(ϕx+ψ+lφ).(20)\nIn this equation, tandxare respectively the time and space variables. The functions φ,ϕand\nψrepresent the longitudinal, vertical, and shear angle displacements of elastic materials such\nas flexible beams. k,l,k0andbare positive constants. More information about mathematical\nmodeling of the Bresse system can be found in [LLS93]. Eq. (20) is con servative in the sense that\nit preserves the energy\nE(t) =/integraldisplay\nΩ/parenleftbigg1\n2ϕ2\nt+1\n2ψ2\nt+1\n2φ2\nt+k\n2(ϕx+ψ+lφ)2+b\n2ψ2\nx+k0\n2(φx−lϕ)2/parenrightbigg\ndx.10\nLike in the Timoshenko system (10), Eq. (20) is not stable and much r esearch have been done to\nfind damping effects that need to be included to stabilize (20). Again, two mechanisms are taken\ninto account: frictional damping and heat conduction.\nIn the next sections, we show that damped Bresse systems of var ious types can be placed in\nthe GENERIC framework.\n5.2 The Bresse system with frictional damping\nIn this section, we consider the Bresse system with frictional dissip ation,\n\n\nϕtt=k(ϕx+ψ+lφ)x+k0l(φx−lϕ)−γ1ϕt,\nψtt=bψxx−k(ϕx+ψ+lφ)−γ2ψt,\nφtt=k0(φx−lϕ)x−kl(ϕx+ψ+lφ)−γ3φt.(21)\nThis system is (20) with three frictional terms added γ1ϕt,γ2ψt,γ3φt, whereγ1,γ2andγ3are\nnon-negative constants. Stability property and rate of decay of solutions of this system have been\nstudied in, e.g., [SJ10, BRJ11, FM12, AFSM14].\nWe now cast this system into the GENERIC framework. Since the pro cedure is similar as in\nSection 4, we only list the computations here.\n•Re-write the system by introducing new variables p,qandw\n\n\nϕt=p,\nψt=q,\nφt=w,\npt=k(ϕx+ψ+lφ)x+k0l(φx−lϕ)−γ1p,\nqt=bψxx−k(ϕx+ψ+lφ)−γ2q,\nwt=k0(φx−lϕ)x−kl(ϕx+ψ+lφ)−γ3w.(22)\n•Equation for e\net=−d\ndt/integraldisplay/parenleftbigg1\n2p2+1\n2q2+1\n2w2+k\n2(ϕx+ψ+lφ)2+b\n2ψ2\nx+k0\n2(φx−lϕ)2/parenrightbigg\ndx=/integraldisplay\nΩ(γ1p2+γ2q2+γ3w2)dx.\n(23)\n•GENERIC building blocks\nz=/parenleftbigϕ ψ φ p q w e/parenrightbigT,\nE(z) =e+/integraldisplay/parenleftbigg1\n2p2+1\n2q2+1\n2w2+k\n2(ϕx+ψ+lφ)2+b\n2ψ2\nx+k0\n2(φx−lϕ)2/parenrightbigg\ndx,S(z) =αe,\nL(z) =\n0 0 0 1 0 0 0\n0 0 0 0 1 0 0\n0 0 0 0 0 1 0\n−1 0 0 0 0 0 0\n0−1 0 0 0 0 0\n0 0 −1 0 0 0 0\n0 0 0 0 0 0 0\n,\nM(z) =1\nα\n0 0 0 0 0 0 0\n0 0 0 0 0 0 0\n0 0 0 0 0 0 0\n0 0 0 γ1 0 0 −γ1p\n0 0 0 0 γ2 0 −γ2q\n0 0 0 0 0 γ3 −γ3w\n0 0 0 −γ1/integraltext\nΩp·/squaredx−γ2/integraltext\nΩq·/squaredx−γ3/integraltext\nΩw·/squaredx/integraltext\nΩ(γ1p2+γ2q2+γ3w2)dx\n.11\n•A direct calculations gives\nδE(z)\nδz=\n−k(ϕx+ψ+lφ)x−k0l(φx−lϕ)\n−bψxx+k(ϕx+ψ+lφ)\n−k0(φx−lϕ)x+kl(ϕx+ψ+lφ)\np\nq\nw\n1\n,δS(z)\nδz=\n0\n0\n0\n0\n0\n0\nα\n,\nL(z)δE(z)\nδz=\np\nq\nw\nk(ϕx+ψ+lφ)x+k0l(φx−lϕ)\nbψxx−k(ϕx+ψ+lφ)\nk0(φx−lϕ)x−kl(ϕx+ψ+lφ)\n0\n,M(z)δS(z)\nδz=\n0\n0\n0\n−γ1p\n−γ2q\n−γ3w/integraltext\nΩ(γ1p2+γ2q2+γ3w2)dx\n.\nThen the GENERIC\nzt=L(z)δE(z)\nδz+M(z)δS(z)\nδz,\nis the same as the system (22)-(23).\n5.3 The Bresse system damped by heat conduction: type I\nIn this section, we consider the Bresse system coupled to a heat co nduction of the form (type\nI)\n\nϕtt=k(ϕx+ψ+lφ)x+k0l(φx−lϕ),\nψtt=bψxx−k(ϕx+ψ+lφ)−γθx,\nφtt=k0(φx−lϕ)x−kl(ϕx+ψ+lφ),\nθt=κθxx−γψxt.(24)\nThis system is (20) coupled to the heat conduction given by the last e quation. Stability property\nand rate of decay of solutions of this system have been studied in, e .g., [FR10, HS14].\nWe now show the steps to place this system in the GENERIC setting.\n•Re-write the system\n\n\nϕt=p,\nψt=q,\nφt=w,\npt=k(ϕx+ψ+lφ)x+k0l(φx−lϕ),\nqt=bψxx−k(ϕx+ψ+lφ)−γθx,\nwt=k0(φx−lϕ)x−kl(ϕx+ψ+lφ),\nθt=κθxx−γqx.(25)\n•Equation for e\net=−d\ndt/integraldisplay/parenleftbigg1\n2p2+1\n2q2+1\n2w2+k\n2(ϕx+ψ+lφ)2+b\n2ψ2\nx+k0\n2(φx−lϕ)2+1\n2θ2/parenrightbigg\ndx=κ/integraldisplay\nΩθ2\nxdx.\n(26)12\n•GENERIC building blocks\nz=/parenleftbigϕ ψ φ p q w θ e/parenrightbigT,\nE(z) =e+/integraldisplay/parenleftbigg1\n2p2+1\n2q2+1\n2w2+k\n2(ϕx+ψ+lφ)2+b\n2ψ2\nx+k0\n2(φx−lϕ)2+1\n2θ2/parenrightbigg\ndx,S(z) =αe,\nL(z) =\n0 0 0 1 0 0 0 0\n0 0 0 0 1 0 0 0\n0 0 0 0 0 1 0 0\n−1 0 0 0 0 0 0 0\n0−1 0 0 0 0 −γ∂x0\n0 0 −1 0 0 0 0 0\n0 0 0 0 −γ∂x0 0 0\n0 0 0 0 0 0 0 0\n,\nM(z) =1\nα\n0 0 0 0 0 0 0\n0 0 0 0 0 0 0\n0 0 0 0 0 0 0\n0 0 0 0 0 0 0\n0 0 0 0 0 0 0\n0 0 0 0 0 0 0\n0 0 0 0 0 −κ∂xx κθxx\n0 0 0 0 0 −κ/integraltext\nΩθx·∂x/squaredx κ/integraltext\nΩθ2\nxdx\n.\n•A direct calculation gives\nδE(z)\nδz=\n−k(ϕx+ψ+lφ)x−k0l(φx−lϕ)\n−bψxx+k(ϕx+ψ+lφ)\n−k0(φx−lϕ)x+kl(ϕx+ψ+lφ)\np\nq\nw\nθ\n1\n,δS(z)\nδz=\n0\n0\n0\n0\n0\n0\n0\nα\n,\nL(z)δE(z)\nδz=\np\nq\nw\nk(ϕx+ψ+lφ)x+k0l(φx−lϕ)\nbψxx−k(ϕx+ψ+lφ)−γθx\nk0(φx−lϕ)x−kl(ϕx+ψ+lφ)\n−γqx\n0\n,M(z)δS(z)\nδz=\n0\n0\n0\n0\n0\n0\nκθxx\nκ/integraltext\nΩθ2\nxdx\n.\nThen it follows that the GENERIC\nzt=L(z)δE(z)\nδz+M(z)δS(z)\nδz,\nis the same as the system (25)-(26).\nIn Appendix B, we show one more model: the Bresse system damped b y heat conduction of type\nII.\n6 Conclusion and discussion\nIn this paper we have introduced a modelling procedure of complex sy stems using GENERIC.\nAs a concrete example, this procedure allows us to unify many existin g damped Timoshenko and13\ndamped Bresse systems into the GENERIC framework and derive a n ew model. This formulation\nnot only provides an alternative thermodynamically consistent deriv ation but also reveals geomet-\nrical structures, via the GENERIC building blocks, of these system s. An important question for\nfurther researchwould be on mathematical analysis, such as well-p osedness and asymptotic limits,\nof the damped Timoshenko and damped Bresse systems (and more g eneral thermodynamical sys-\ntems) using GENERIC structure. Recently it has become clear, see e.g., [SS04, Ser11, AMP+12]\nand very recent papers [Mie14, Mie15], that variational structure has important consequences for\nthe analysis of an evolution equation. This is because variational str ucture can provide many good\nconcepts of weak solution and many techniques in calculus of variatio ns, such as Gamma conver-\ngence, can be exploited. There is a large literature on mathematical analysis of gradient flows,\nwhich is an instance of the GENERIC where the reversible effect is abs ent, using variational for-\nmulation, see the papers mentioned above and references therein . However, that of for GENERIC\nis still lacking. A variational formulation for a full GENERIC system ha s been proposed recently\nin [DPZ13]. We expect that this variational can be used for the GENER IC systems studied in\nthis paper.\n7 Appendix A: Verification of the GENERIC conditions\nIn this Appendix, we present the verification of the GENERIC condit ions for the Timoshenko\nsystem with two frictional damping and the Timoshenko system damp ed by heat conduction of\ntype I. The verification for the other models are similar and hence om itted.\n7.1 The Timoshenko system with dual frictional damping\nWe now verify that {L,M,E,S}constructed in Section 4.2 satisfy the conditions of the\nGENERIC. Let F,G:Z→Rbe given. We have\n{F,G}L=δF(z)\nδz·L(z)δG(z)\nδz=\nδF(z)\nδϕ\nδF(z)\nδψ\nδF(z)\nδp\nδF(z)\nδq\nδF(z)\nδe\n·\nδG(z)\nδp\nδG(z)\nδq\n−δG(z)\nδϕ\n−δG(z)\nδψ\n0\n\n=/integraldisplay\nΩ/bracketleftbiggδF(z)\nδϕδG(z)\nδp+δF(z)\nδψδG(z)\nδq−δF(z)\nδpδG(z)\nδϕ−δF(z)\nδqδG(z)\nδψ/bracketrightbigg\ndx\n=−/integraldisplay\nΩ/bracketleftbiggδG(z)\nδϕδF(z)\nδp+δG(z)\nδψδF(z)\nδq−δG(z)\nδpδF(z)\nδϕ−δG(z)\nδqδF(z)\nδψ/bracketrightbigg\ndx=−{G,F}L,\ni.e.,Lis anti-symmetric.\nTheverificationthat LsatisfiestheJacobiidentitycanbedonebycomputing {{F1,F2}L,F3}L+\n{{F2,F3}L,F1}L+{{F3,F1}L,F2}Ldirectly similarly as above. The computation is lengthy and\ntedious, hence it is omitted. It seems that there is no general meth od to verify the Jacobi identity.\nAccording to [Gol01, Chapter 9] “there seems to be no simple way of p roving Jacobi’s identity\nfor the Poisson bracket without lengthy algebra.”. However, it sho uld be mentioned that Kr¨ oger\nand H¨ utter [KH10] have developed a Mathematica notebook which f acilitates verification of the\nJacobi identity. Next, we check conditions on M. Without loss of generality and for simplicity of14\nnotation, we set α= 1 throughout this Appendix. We have\n[F,G]M=δF(z)\nδz·M(z)δG(z)\nδz\n=\nδF(z)\nδϕ\nδF(z)\nδψ\nδF(z)\nδp\nδF(z)\nδq\nδF(z)\nδe\n·\n0\n0\nδ1δG(z)\nδp−δ1pδG(z)\nδe\nδ2δG(z)\nδq−δ2qδG(z)\nδe/integraltext\nΩ/bracketleftig\n−δ1pδG(z)\nδp−δ2qδG(z)\nδq+δG(z)\nδe(δ1p2+δ2q2)/bracketrightig\ndx\n\n=/integraldisplay\nΩ/bracketleftbigg\nδ1δF(z)\nδp/parenleftbiggδG(z)\nδp−pδG(z)\nδe/parenrightbigg\n+δ2δF(z)\nδq/parenleftbiggδG(z)\nδq−qδG(z)\nδe/parenrightbigg/bracketrightbigg\ndx\n+δF(z)\nδe/integraldisplay\nΩ/bracketleftbigg\n−δ1pδG(z)\nδp−δ2qδG(z)\nδq+δG(z)\nδe(δ1p2+δ2q2)/bracketrightbigg\ndx\n=/integraldisplay\nΩ/bracketleftbigg\nδ1δG(z)\nδp/parenleftbiggδF(z)\nδp−pδF(z)\nδe/parenrightbigg\n+δ2δG(z)\nδq/parenleftbiggδF(z)\nδq−qδF(z)\nδe/parenrightbigg/bracketrightbigg\ndx\n+δG(z)\nδe/integraldisplay\nΩ/bracketleftbigg\n−δ1pδF(z)\nδp−δ2qδF(z)\nδq+δF(z)\nδe(δ1p2+δ2q2)/bracketrightbigg\ndx\n= [G,F]M,\ni.e.,Mis symmetric. It also follows from the above computation that\n[F,F]M=/integraldisplay\nΩ/bracketleftigg\nδ1/parenleftbiggδF(z)\nδp−pδF(z)\nδe/parenrightbigg2\n+δ2/parenleftbiggδF(z)\nδq−qδF(z)\nδe/parenrightbigg2/bracketrightigg\ndx≥0,\ni.e.,Mis positive semidefinite. It remains to verify the degeneracy conditio ns. Indeed,\nM(z)δE(z)\nδz=\n0 0 0 0 0\n0 0 0 0 0\n0 0 δ1 0 −δ1p\n0 0 0 δ2 −δ2q\n0 0−δ1/integraltext\nΩp·/squaredx−δ2/integraltext\nΩq·/squaredx/integraltext\nΩ(δ1p2+δ2q2)dx\n·\n−k(ϕx+ψ)x\n−bψxx+k(ϕx+ψ)\np\nq\n1\n= 0,\nL(z)δS(z)\nδz=\n0 0 1 0 0\n0 0 0 1 0\n−1 0 0 0 0\n0−1 0 0 0\n0 0 0 0 0\n·\n0\n0\n0\n0\n1\n= 0.15\n7.2 The Timoshenko system damped by heat conduction of type I\nIn this section, we verify that {L,M,E,S}constructed in Section 4.3 satisfy the conditions of\nthe GENERIC. Let F,G:Z→Rbe given. We have\n{F,G}L=δF(z)\nδz·L(z)δG(z)\nδz=\nδF(z)\nδϕ\nδF(z)\nδψ\nδF(z)\nδp\nδF(z)\nδq\nδF(z)\nδθδF(z)\nδe\n·\nδG(z)\nδp\nδG(z)\nδq\n−δG(z)\nδϕ\n−δG(z)\nδψ−γ∂\n∂x/parenleftig\nδG(z)\nδθ/parenrightig\n−γ∂\n∂x/parenleftig\nδG(z)\nδq/parenrightig\n0\n\n=/integraldisplay\nΩ/bracketleftbiggδF(z)\nδϕδG(z)\nδp+δF(z)\nδψδG(z)\nδq−δF(z)\nδpδG(z)\nδϕ−δF(z)\nδqδG(z)\nδψ−γδF(z)\nδq∂xδG(z)\nδθ−γδF(z)\nδθ∂xδG(z)\nδq/bracketrightbigg\ndx\n=−/integraldisplay\nΩ/bracketleftbiggδG(z)\nδϕδF(z)\nδp+δG(z)\nδψδF(z)\nδq−δG(z)\nδpδF(z)\nδϕ−δG(z)\nδqδF(z)\nδψ−γδG(z)\nδq∂xδF(z)\nδθ−γδG(z)\nδθ∂xδF(z)\nδq/bracketrightbigg\ndx\n=−{G,F}L,\ni.e.,Lis anti-symmetric. The verification that Lsatisfies the Jacobi identity can be done by\ncomputing {{F1,F2}L,F3}L+{{F2,F3}L,F1}L+{{F3,F1}L,F2}Ldirectly similarly as above. The\ncomputation is lengthy and tedious, hence it is omitted.\n[F,G]M=δF(z)\nδz·M(z)δG(z)\nδz\n=\nδF(z)\nδϕ\nδF(z)\nδψ\nδF(z)\nδp\nδF(z)\nδq\nδF(z)\nδθδF(z)\nδe\n·\n0\n0\n0\n0\n−κ∂xxδG(z)\nδθ+κθxxδG(z)\nδe\nκ/integraltext\nΩ/bracketleftig\n−θx∂xδG(z)\nδθ+δG(z)\nδeθ2\nx/bracketrightig\n\n=κ/integraldisplay\nΩ/bracketleftbiggδF(z)\nδθ/parenleftbigg\n−∂xxδG(z)\nδθ+θxxδG(z)\nδe/parenrightbigg\n+δF(z)\nδe/parenleftbigg\n−θx∂xδG(z)\nδθ+δG(z)\nδeθ2\nx/parenrightbigg/bracketrightbigg\ndx\n=κ/integraldisplay\nΩ/bracketleftbiggδG(z)\nδθ/parenleftbigg\n−∂xxδF(z)\nδθ+θxxδF(z)\nδe/parenrightbigg\n+δG(z)\nδe/parenleftbigg\n−θx∂xδF(z)\nδθ+δF(z)\nδeθ2\nx/parenrightbigg/bracketrightbigg\ndx\n= [G,F]M,\ni.e.,Mis symmetric. It also follows from the above computation that\n[F,F]M=κ/integraldisplay\nΩ/parenleftbigg\n∂xδF(z)\nδθ−δF(z)\nδeθx/parenrightbigg2\ndx≥0,16\ni.e.,Mis positive semi-definite. The degeneracy condition also can be verifie d. Indeed,\nM(z)δE(z)\nδz=\n0 0 0 0 0 0\n0 0 0 0 0 0\n0 0 0 0 0 0\n0 0 0 0 0 0\n0 0 0 0 −κ∂xxκθxx\n0 0 0 0 −κ/integraltext\nΩθx·∂xκ/integraltext\nΩθ2\nxdx\n·\n−k(ϕx+ψ)x\n−bψxx+k(ϕx+ψ)\np\nq\nθ\n1\n= 0,\nL(z)δS(z)\nδz=\n0 0 1 0 0 0\n0 0 0 1 0 0\n−1 0 0 0 0 0\n0−1 0 0 −γ∂x0\n0 0 0 −γ∂x0 0\n0 0 0 0 0 0\n·\n0\n0\n0\n0\n0\n1\n= 0.\n8 Appendix B: Other models\n8.1 The Timoshenko system damped by heat conduction: type II\nIn this section, we cast the Timoshenko system damped by heat con duction of type II into\nthe GENERIC framework. The steps are summarized as follows.\n•The original system\n\n\nϕtt=k(ϕx+ψ)x,\nψtt=bψxx−k(ϕx+ψ)−γθx,\nθt=−sx−γψtx,\nst=−θx−βs.Re-write the system\n\n\nϕt=p,\nψt=q,\npt=k(ϕx+ψ)x,\nqt=−k(ϕx+ψ)+bψxx−γθx,\nθt=−sx−γqx,\nst=−θx−βs.(27)\nHereθandsare respectively the temperature difference and the heat flux. Th e heat conduc-\ntion in this model is described by the Cattaneo law, see for instance [R R02, HK13, SR09].\n•Equation for e\net=−d\ndt/integraldisplay\nΩ/parenleftbigg1\n2p2+1\n2q2+k\n2(ϕx+ψ)2+b\n2ψ2\nx+1\n2θ2+1\n2s2/parenrightbigg\ndx=β/integraldisplay\nΩs2dx.(28)17\n•GENERIC building blocks\nz= (ϕ ψ p q θ s e )T\nE(z) =/integraldisplay\nΩ/parenleftbigg1\n2p2+k\n2(ϕx+ψ)2+1\n2q2+b\n2ψ2\nx+1\n2θ2+1\n2s2/parenrightbigg\ndx+e,S(z) =αe,\nL(z) =\n0 0 1 0 0 0 0\n0 0 0 1 0 0 0\n−1 0 0 0 0 0 0\n0−1 0 0 −γ∂x0 0\n0 0 0 −γ∂x0−∂x0\n0 0 0 0 −∂x0 0\n0 0 0 0 0 0 0\n,\nM(z) =1\nα\n0 0 0 0 0 0 0\n0 0 0 0 0 0 0\n0 0 0 0 0 0 0\n0 0 0 0 0 0 0\n0 0 0 0 0 0 0\n0 0 0 0 0 β −βs\n0 0 0 0 0 −β/integraltext\nΩs·/squaredx β/integraltext\nΩs2dx,\n.\n•Computation\nδE(z)\nδz=\n−k(ϕx+ψ)x\n−bψxx+k(ϕx+ψ)\np\nq\nθ\ns\n1\n,δS(z)\nδz=\n0\n0\n0\n0\n0\n0\nα\n,\nL(z)δE(z)\nδz=\np\nq\nk(ϕx+ψ)x\nbψxx−k(ϕx+ψ)−γθx\n−γqx−sx\n−θx\n0\n,M(z)δS(z)\nδz=\n0\n0\n0\n0\n0\n−βs\nβ/integraltext\nΩs2dx\n.\nFromtheabovecomputation,weobtainthattheGENERICequation withbuilding {z,L,M,E,S}\nis equivalent to the system (27)-(28).\nRemark 8.1. The last two equations in (27) can be coupled to get one equation for θas follows\nθtt=θxx−βθt−βγψtx−γψttx.\nOne should compare and see the difference between the above equa tion and the equation for the\nheat conduction of the model in the next subsection.\n8.2 The Timoshenko system damped by heat conduction: type II I\nWe now consider the Timoshenko system damped by heat conduction of the form (type III),18\n•The original system\n\n\nϕtt=k(ϕx+ψ)x,\nψtt=bψxx−k(ϕx+ψ)−βθtx,\nθtt=δθxx−γψtx+Kθtxx.(29)Re-write the system\n\n\nϕt=p,\nψt=q,\npt=k(ϕx+ψ)x,\nqt=−k(ϕx+ψ)+bψxx−γwx,\nθt=w,\nwt=δθxx−γqx+Kwxx.(30)\nThis system has been investigated, e.g., in [FRM14].\n•Equation for e\net=−d\ndt/integraldisplay\nΩ/parenleftbigg1\n2p2+1\n2q2+1\n2w2+k\n2(ϕx+ψ)2+b\n2ψ2\nx+δ\n2θ2\nx/parenrightbigg\ndx=K/integraldisplay\nΩw2\nxdx.(31)\n•GENERIC building blocks\nz= (ϕ ψ p q θ w e )T\nE(z) =/integraldisplay\nΩ/parenleftbigg1\n2p2+k\n2(ϕx+ψ)2+1\n2q2+b\n2ψ2\nx+δ\n2θ2\nx/parenrightbigg\ndx+e,S(z) =αe,\nL(z) =\n0 0 1 0 0 0 0\n0 0 0 1 0 0 0\n−1 0 0 0 0 0 0\n0−1 0 0 0 −γ∂x0\n0 0 0 0 0 1 0\n0 0 0 −γ∂x−1 0 0\n0 0 0 0 0 0 0\n,\nM(z) =1\nα\n0 0 0 0 0 0 0\n0 0 0 0 0 0 0\n0 0 0 0 0 0 0\n0 0 0 0 0 0 0\n0 0 0 0 0 0 0\n0 0 0 0 0 −K∂xx Kwxx\n0 0 0 0 0 −K/integraltext\nΩwx·∂x/squaredx K/integraltext\nΩw2\nxdx\n,\n•Computation\nδE(z)\nδz=\n−k(ϕx+ψ)x\n−bψxx+k(ϕx+ψ)\np\nq\n−δθxx\nw\n1\n,δS(z)\nδz=\n0\n0\n0\n0\n0\n0\nα\n,\nL(z)δE(z)\nδz=\np\nq\nk(ϕx+ψ)x\nbψxx−k(ϕx+ψ)−γwx\nw\n−γqx+δθxx\n0\n,M(z)δS(z)\nδz=\n0\n0\n0\n0\n0\nKwxx\nK/integraltext\nΩw2\nxdx\n.\nIt is straightforward to verify that the GENERIC system is indeed t he system (30)-(31).19\n8.3 The Bresse system damped by heat conduction: type II\nIn this section, we investigate the Bresse system damped by two te mperature equations of\nthe following form (type II)\n\n\nϕtt=k(ϕx+ψ+lφ)x+k0l(φx−lϕ)−γlη,\nψtt=bϕxx−k(ϕx+ψ+lφ)−δθx,\nφtt=k0(φx−lϕ)x−kl(ϕx+ψ+lφ)−γηx,\nθt=κ1θxx−δψtx,\nηt=κ2ηxx−γ(φxt−lϕt).(32)\nThis system has one extra equation, which is the last one, compared to (24). Stability property\nand rate of decay of solutions of this system have been studied in, e .g., [LR09].\n•Re-write the system\n\n\nϕt=p,\nψt=q,\nφt=w,\npt=k(ϕx+ψ+lφ)x+k0l(φx−lϕ)−γlη,\nqt=bψxx−k(ϕx+ψ+lφ)−δθx,\nwt=k0(φx−lϕ)x−kl(ϕx+ψ+lφ)−γηx,\nθt=κ1θxx−δqx,\nηt=κ2ηxx−γ(wx−lp).(33)\n•Equation for e\net=d\ndt/integraldisplay/parenleftbigg1\n2p2+1\n2q2+1\n2w2+k\n2(ϕx+ψ+lφ)2+b\n2ψ2\nx+k0\n2(φx−lϕ)2+1\n2θ2+1\n2η2/parenrightbigg\ndx=/integraldisplay\nΩ(κ1θ2\nx+κ2η2\nx)dx.\n(34)\n•GENERIC building block\nz=/parenleftbigϕ ψ φ p q w θ η e/parenrightbigT,\nE(z) =e+/integraldisplay/parenleftbigg1\n2p2+1\n2q2+1\n2w2+k\n2(ϕx+ψ+lφ)2+b\n2ψ2\nx+k0\n2(φx−lϕ)2+1\n2θ2+1\n2η2/parenrightbigg\ndx,S(z) =αe,\nL(z) =\n0 0 0 1 0 0 0 0 0\n0 0 0 0 1 0 0 0 0\n0 0 0 0 0 1 0 0 0\n−1 0 0 0 0 0 0 −γl0\n0−1 0 0 0 0 −δ∂x0 0\n0 0 −1 0 0 0 0 −γ∂x0\n0 0 0 0 −δ∂x0 0 0 0\n0 0 0 γl0−γ∂x0 0 0\n0 0 0 0 0 0 0 0 0\n,\nM(z) =1\nα\n0 0 0 0 0 0 0 0 0\n0 0 0 0 0 0 0 0 0\n0 0 0 0 0 0 0 0 0\n0 0 0 0 0 0 0 0 0\n0 0 0 0 0 0 0 0 0\n0 0 0 0 0 0 0 0 0\n0 0 0 0 0 0 −κ1∂xx 0 κ1θxx\n0 0 0 0 0 0 0 −κ2∂xx κ2ηxx\n0 0 0 0 0 0 −κ1/integraltext\nΩθx·∂x/squaredx−κ1/integraltext\nΩθx·∂x/squaredx/integraltext\nΩ(κ1θ2\nx+κ2η2\nx)dx\n.20\nNext we place the coupled system (33)-(34) in the GENERIC framew ork with the building blocks,\nWe compute each term on the right hand side of the GENERIC equatio n\nδE(z)\nδz=\n−k(ϕx+ψ+lφ)x−k0l(φx−lϕ)\n−bψxx+k(ϕx+ψ+lφ)\n−k0(φx−lϕ)x+kl(ϕx+ψ+lφ)\np\nq\nw\nθ\nη\n1\n,δS(z)\nδz=\n0\n0\n0\n0\n0\n0\n0\n0\nα\n,\nL(z)δE(z)\nδz=\np\nq\nw\nk(ϕx+ψ+lφ)x+k0l(φx−lϕ)−γlη\nbψxx−k(ϕx+ψ+lφ)−δθx\nk0(φx−lϕ)x−kl(ϕx+ψ+lφ)−γηx\n−δqx\nγlp−γwx\n0\n,M(z)δS(z)\nδz=\n0\n0\n0\n0\n0\n0\nκ1θxx\nκ2ηxx /integraltext\nΩ(κ1θ2\nx+κ2η2\nx)dx\n.\nSubstituting these building blocks to the GENERIC equation\nzt=L(z)δE(z)\nδz+M(z)δS(z)\nδz,\nyields the coupled system (33)-(34).\nReferences\n[ADPZ11] S. 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History of Strength of Materials: With a Brief Account of the\nHistory of Theory of Elasticity and Theory of Structures . McGraw-Hill, 1953."    },    {        "title": "2309.15778v3.Exploring_antisymmetric_tensor_effects_on_black_hole_shadows_and_quasinormal_frequencies.pdf",        "content": "Exploring antisymmetric tensor effects on black hole shadows and\nquasinormal frequencies\nA. A. Ara´ ujo FilhoID∗\nDepartamento de F´ ısica, Universidade Federal da Para´ ıba,\nCaixa Postal 5008, 58051-970, Jo˜ ao Pessoa, Para´ ıba, Brazil\nJ. A. A. S. ReisID†\nUniversidade Estadual do Sudoeste da Bahia (UESB),\nDepartamento de Ciˆ encias Exatas e Naturais,\nCampus Juvino Oliveira, Itapetinga – BA, 45700-00, Brazil\nH. HassanabadiID‡\nPhysics Department, Shahrood University of Technology, Shahrood, Iran and\nDepartment of Physics, University of Hradec Kr ´alov´e,\nRokitansk ´eho 62, 500 03 Hradec Kr ´alov´e, Czechia.\nAbstract\nThis study explores the impact of antisymmetric tensor effects on spherically symmetric black holes,\ninvestigating photon spheres, shadows, emission rate and quasinormal frequencies in relation to a parameter\nwhich triggers the Lorentz symmetry breaking. We examine these configurations without and with the\npresence of a cosmological constant. In the first scenario, the Lorentz violation parameter, denoted as λ, plays\na pivotal role in reducing both the photon sphere and the shadow radius, while also leading to a damping\neffect on quasinormal frequencies. Conversely, in the second scenario, as the values of the cosmological\nconstant (Λ) increase, we observe an expansion in the shadow radius. Also, we provide the constraints of\nthe shadows based on the analysis observational data obtained from the Event Horizon Telescope (EHT)\nfocusing on Sagittarius A∗shadow images. Additionally, with the increasing Λ, the associated gravitational\nwave frequencies exhibit reduced damping modes.\n∗Electronic address: dilto@fisica.ufc.br\n†Electronic address: jalfieres@gmail.com\n‡Electronic address: hha1349@gmail.com\n1arXiv:2309.15778v3  [gr-qc]  26 Mar 2024I. INTRODUCTION\nLorentz symmetry, a fundamental pillar of modern physics, postulates the consistent applicabil-\nity of physical laws across all inertial reference frames. While firmly established as a fundamental\nprinciple, corroborated by extensive experimental and observational background, it has become\nevident that Lorentz symmetry may exhibit deviations under specific energy conditions within di-\nverse theoretical approaches, such as string theory [1], loop quantum gravity [2], Horava–Lifshitz\ngravity [3], non–commutative field theory [4], Einstein–aether theory [5], massive gravity [6], f(T)\ngravity [7], very special relativity [8], and more.\nLorentz symmetry breaking (LSB) can be manifested in two distinct manners: explicit and\nspontaneous [9]. The first one case occurs when the Lagrangian density lacks Lorentz invariance,\nresulting in the formulation of distinct physical laws in specific reference frames. Conversely, the\nsecond one arises when the Lagrangian density maintains Lorentz invariance, but the ground state\nof a physical system does not exhibit Lorentz symmetry [10].\nThe investigation of spontaneous Lorentz symmetry breaking [11–16] finds its foundation in the\nStandard Model Extension. Within this framework, the simplest field theories are encapsulated by\nbumblebee models [1, 12–15, 17–20]. In these models, a vector field, termed the bumblebee field,\nacquires a non–zero vacuum expectation value (VEV). This aspect establishes a distinct direction,\nresulting in the local Lorentz invariance violation for particles, which in turn leads to remarkable\nconsequences for instance to the thermodynamic properties [21–31].\nRef. [32] presents an exact solution for a static and spherically symmetric spacetime within\nthe framework of bumblebee gravity. Analogously, a Schwarzschild–like solution has undergone\nrigorous examination from multiple perspectives, including Hawking radiation [33], the accretion\nprocess [34, 35] gravitational lensing [36], and quasinormal modes [37].\nFollowing this, Maluf et al. expanded upon these discoveries by deriving an (A)dS–\nSchwarzschild–like solution, relaxing the vacuum conditions [38]. Additionally, Xu et al. introduced\ninnovative categories of static spherical bumblebee black holes by incorporating a background bum-\nblebee field with a non–zero temporal component, exploring their thermodynamic characteristics\nand observational consequences in Refs. [39–42].\nDing et al. investigated the domain of rotating bumblebee black holes in Ref. [43, 44], cov-\nering a wide range of topics including their accretion processes [45], shadows [46], quasi–periodic\noscillations [47], and quasinormal modes [48]. Furthermore, an exact rotating BTZ–like black hole\nsolution was derived in [49], and its quasinormal modes were subject to analytical investigation in\n2reference [50].\nA Schwarzschild–like black hole incorporating a global monopole was introduced in [51], and\nits quasinormal modes were analyzed [52, 53]. Moreover, various other black hole solutions were\nexplored within the framework of the Bumblebee gravity model in Refs. [54–56], as well as in the\ncontext of metric affine formalism [57–59]. Additionally, a traversable bumblebee wormhole solution\nwas recently proposed in the literature [60], with subserquents investigations of its corresponging\ngravitational waves [61, 62].\nBeyond the vector field theories, another approach for exploring LSB involves a rank–two an-\ntisymmetric tensor field referred to as the Kalb–Ramond field [63, 64]. It naturally arises in the\nspectrum of bosonic string theory [65]. When this field is non–minimally coupled to gravity and\nobtains a non–zero vacuum expectation value, it breaks the Lorentz symmetry spontaneously. In\nRef. [66], an exact solution for a static and spherically symmetric configuration within this context\nis demonstrated. This discovery was followed by an exploration of the behavior of both massive\nand massless particles near this static spherical Kalb–Ramond black hole in Reference [67]. Addi-\ntionally, Ref. [68] investigates the gravitational deflection of light and the shadows cast by rotating\nblack holes.\nOver the past few years, there has been a significant focus on the exploration of gravitational\nwaves and their spectra, as evidenced by recent studies [69–73]. This heightened interest can be\nattributed in large part to the remarkable advancements in gravitational wave detection technology,\nnotably exemplified by the VIRGO and LIGO detectors. These advanced instruments have played\na crucial role in providing profound understanding into the captivating domain of black hole physics\n[74–77]. In Ref. [78], presents novel exact solutions for static and spherically symmetric spacetime,\nboth in the presence and absence of the cosmological constant. These solutions are derived within\nthe context of a non–zero vacuum expectation value background of the Kalb–Ramond field.\nWithin the field of black hole research, a significant facet of investigation revolves around the\nexploration of quasinormal modes (QNMs). These QNMs represent intricate oscillation frequencies\nthat emerge as a consequence of black holes responding to initial perturbations. To derive these\nfrequencies, specific boundary conditions must be imposed [79, 80]. The study of gravitational\nwaves and their spectra over the last years [69–73], particularly, with the advancements in grav-\nitational wave detectors such as VIRGO and LIGO. These ones have provided valuable insights\ninto black hole physics [74–77]. One aspect of this research involves investigating the quasinormal\nmodes (QNMs), which are complex oscillation frequencies that arise in the response of black holes\nto initial perturbations. These frequencies can be obtained under specific boundary conditions\n3[79, 80].\nIn this work, we study the impact of anti–symmetric tensor effects, which triggers the Lorentz\nsymmetry breaking, on spherically symmetric black holes, analyzing photon spheres, shadows, and\nquasinormal frequencies. We explore these configurations both in the absence and presence of a\ncosmological constant.\nII. THE ANTISYMMETRIC BLACK HOLE SOLUTION\nWe commence by considering the Einstein–Hilbert action, which is non-minimally coupled to a\nself–interacting Kalb–Ramond (KR) field, expressed in the following form [81]:\nS=1\n2κ2Z\nd4x√−g\u0014\nR−2Λ−1\n6HµνρHµνρ−V(BµνBµν) +ξ2BρµBν\nµ+ξ3BµνBµνR\u0015\n+Z\nd4x√−gLm,(1)\nwhere ξ2andξ3are the coupling constants between the Ricci tensor and the Kalb–Ramond field,\nκ= 8πG, Λ is the cosmological constant, and the field strength Hµνρ≡∂[µBνρ]. The potential\nV(BµνBµν) is responsible for triggering the spontaneous Lorentz symmetry breaking. With such\nan aspect, we are able to maintain this theory invariant under local Lorentz transformations.\nIn order to obtain the gravitational field equations, we vary Eq. (1) with respect to the metric\ntensor gµν, which gives\nRµν−1\n2gµνR+ Λgµν=Tm\nµν+TKR\nµν=Tµν, (2)\nwhere Tµνis the total stress–energy tensor, Tm\nµνis the stress–energy of the matter fields and TKR\nµν\nbeing given by\nTKR\nµν=1\n2HµαβHαβ\nν−1\n12gµνHαβρHαβρ+ 2V′(X)BαµBα\nν−gµνV(X)\nξ2\u001a1\n2gµνBαγBβ\nγRαβ−Bα\nµBβ\nνRαβ−BαβBνβRµα−BαβBµβRνα\n1\n2∇α∇µ(BαβBνβ) +1\n2∇α∇ν(BαβBµβ)−1\n2∇α∇αBγ\nµBνγ\n−1\n2gµν∇α∇β(BαγBβ\nγ)\u001b\n.(3)\nIn this context, the prime symbol denotes the derivative with respect to the argument of the\nrespective functions. Utilizing the Bianchi identity, we can ascertain that the Tµνis conserved.\nTo induce a non-zero vacuum expectation value (VEV) for the Kalb–Ramond field, denoted as\n⟨Bµν⟩=bµν, we consider a potential of general form V=V(BµνBµν±b2). Here, the choice of the\n4sign±ensures that b2is a positive constant. As a result, the vacuum expectation value (VEV) is\ndefined by the condition bµνbµν=∓b2. The gauge invariance Bµν→Bµν+∂[µΓν]associated with\nthe Kalb–Ramond field is spontaneously broken. Since the non–minimal coupling of the KR field\nwith gravity is present, this symmetry–breaking process due to a VEV background gives rise to a\nviolation of local Lorentz invariance for particles. Additionally, the term ξ3BµνBµνRin Eq. (1)\nbecomes ∓ξ3b2Rin the vacuum state, and this can be integrated into the Einstein–Hilbert terms\nthrough a suitable redefinition of variables.\nFor convenience, the antisymmetric tensor Bµνcan be decomposed as Bµν=˜E[µvν]+ϵµναβvα˜Bβ,\nwhere vαis a timelike 4–vector. The pseudo-fields ˜Eµand ˜Bµare spacelike and satisfy the condi-\ntions ˜Eµvµ=˜Bµvµ= 0. Drawing a parallel to Maxwell’s electrodynamics, these pseudo-fields can\nbe thought of as pseudo–electric and pseudo–magnetic fields.\nIf we assume that the only non–zero terms in the VEV are b10=−b01=˜E(r), or similarly,\nb2=−˜E(r) dt∧dr, it becomes apparent that the vacuum field turns out to present a configuration\nof a pseudo–electric field. As a result, this specific configuration leads to the vanishing of the\nKalb–Ramond (KR) field strength, meaning Hλµν= 0 or H3= db2= 0.\nIn this study, we concentrate on exploring a static and spherically symmetric spacetime, set\nagainst a non–zero (VEV) for the KR field. The proposed metric form is described as follows: The\nmetric tensor for this spacetime is presented in terms of the following line element:\nds2=−A(r)dt2+B(r)dr2+r2dθ2+r2sin2θdϕ2. (4)\nHere, A(r) and B(r) are functions of the radial coordinate r, which will be determined by the\nunderlying dynamics of the system, including the KR field. Accordingly, the pseudo–electric field\n˜E(r) can be reformulated as ˜E(r) =|b|q\nA(r)B(r)\n2,ensuring that the constant norm condition\nbµνbµν=−b2is fulfilled.\nIt becomes suitable to rewrite the field equation in terms of bas follows:\nRµν= Λgµν+V′(bµαbα\nν+b2gµν) +ξ2h\ngµνbαγbβγRαβ−bα\nµbβ\nνRαβ−bαβbµβRνα\n−bαβbνβRµα+1\n2∇α∇µ(bαβbνβ) +1\n2∇α∇ν(bαβbµβ)−1\n2∇α∇α(bγ\nµbνγ)\u0015\n.(5)\nGiven the specified metric ansatz, the field equations, denoted as Eq. (5), can be explicitly refor-\n5mulated as follows [78]:\n2A′\nA−A′B′\nAB−A′2\nA2+4A′\nrA+4Λ\n1−λB= 0,\n2A′′\nA−A′B′\nAB−A′2\nA2−4B′\nrB+4Λ\n1−λB= 0,\n2A′′\nA−A′B′\nAB−A′2\nA2+1 +λ\nλr\u0012A′\nA−B′\nB\u0013\n−(1−Λr2−b2r2V′)2B\nλr2+2(1−λ)\nλr2= 0,(6)\nwhere λ≡ξ2b2\n2.\nA. For Λ = 0\nInitially, in the scenario where a cosmological constant is absent, our aim is to formulate a black\nhole solution resembling that of the Schwarzschild geometry within the framework of this theory.\nHere, we assume V′= 0, indicating that the vacuum expectation value (VEV) is situated at a local\nminimum of the potential. This condition can be conveniently verified, for example, by considering\na smooth quadratic potential V=1\n2σX2, where X≡BµνBµν+b2andσserves as a coupling\nconstant.\nAfter some algebraic manipulations, we obtain [78]:\nA(r) =1\nB(r)=1\n1−λ−2M\nr, (7)\nwhich yields\nds2=−\u00121\n1−λ−2M\nr\u0013\ndt2+dr2\n1\n1−λ−2M\nr+r2dθ2+r2sin2dφ2. (8)\nIt gives rise to the following event horizon\nrΛ=0= 2(1 −λ)M. (9)\nIn the next subsections, we shall calculate the photon spheres as well as the shadows of the black\nhole under consideration. In addition, the quasinormal frequencies are addressed in this context.\n1. Photon sphere and shadows\nFor a comprehensive understanding of particle dynamics and the patterns of light rays in proxim-\nity to black holes, it is fundamental to comprehend the aspects of photon spheres. These spherical\nregions play an integral role in deciphering the shadows projected by black holes, as well as in\nunderstanding the influence of the KR field on the spacetime under investigation.\n6In terms of black hole dynamics, the significance of the photon sphere, i.e., often designated as\nthe critical orbit, necessitates the utilization of the Lagrangian method for instance to compute\nthe null geodesics. The objective of this study is to scrutinize the influence of specific parameters,\nspecifically Mandλ, on it. To offer further clarification, we assert:\nL=1\n2gµν˙xµ˙xν. (10)\nUpon setting the angle to θ=π/2, the aforementioned equation simplifies as follows:\ng−1\n00E2+g−1\n11˙r2+g33L2= 0. (11)\nIn this context, Lsignifies the angular momentum while Erepresents the energy [82]. Consequently,\nEq. (11) can be expressed as:\n˙r2=E2−\u00121\n1−λ−2M\nr\u0013\u0012L2\nr2\u0013\n, (12)\nwhere the effective potential is denoted as V ≡\u0010\n1\n1−λ−2M\nr\u0011\u0010\nL2\nr2\u0011\n. To ascertain the critical radius,\none must solve the equation ∂V/∂r= 0. This solution reads:\nrc= 3(1 −λ)M. (13)\nHere, we denote rcis the radius of the critical orbits. Obviously, when λ→0, we recover the\nSchwarzschild case. Furthermore, the Lorentz symmetry breaking brings about a constraint to\nparameter λ, i.e., bounded from above. In other words, the physical limits of such a value is\nλ= 1. After that, rcwill not show physical values. Additionally, recent literature, including\nreferences [70, 83], has explored analogous studies in the realm of dark matter. It is also important\nto underscore that the appearance of dual photon spheres has been reported within the framework\nof the Simpson–Visser solution [84, 85], and others [86].\nThe left side of Table I presents various calculations of the critical orbit rc, each corresponding to\ndifferent values of λwhile maintaining M= 1. In this setting, as the Lorentz–violating parameter λ\nincreases, rcundergoes a corresponding decrease until it reaches a trivial contribution (when λ= 1).\nGiven that this parameter is intrinsically linked to the Kalb–Ramond field, our results indicate that\nLorentz–violating coefficients exert a considerable influence on this scenario by reducing the radius\nof the photon sphere. On the other hand, when the mass Mis varied, a significant increase in the\ncritical orbits is observed.\nThe scrutiny of shadows within the framework of the Kalb–Ramond field and black hole config-\nurations is critically important. Characterized by the unique silhouette of a black hole contrasted\n7TABLE I: The critical orbit, denoted by rc, is illustrated for a range of values pertaining to mass M, and\nparameter λ.\nM λ r cM λ r c\n1.00 0.00 3.00 —– —– —–\n1.00 0.10 2.70 1.00 0.10 2.70\n1.00 0.20 2.40 2.00 0.10 5.40\n1.00 0.30 2.10 3.00 0.10 8.10\n1.00 0.40 1.80 4.00 0.10 10.8\n1.00 0.50 1.50 5.00 0.10 13.5\n1.00 0.60 1.20 6.00 0.10 16.2\n1.00 0.70 0.90 7.00 0.10 18.9\n1.00 0.80 0.60 8.00 0.10 21.6\n1.00 0.90 0.30 9.00 0.10 24.3\n1.00 1.00 0.00 10.0 0.10 27.0\nagainst a luminous backdrop, these shadows serve as windows into the geometry of spacetime and\nnearby gravitational interactions. Analysis of these phenomena enables us to extract invaluable\ninformation for refining theoretical models, thereby providing a more robust validation of gravi-\ntational theories. To facilitate our investigation, we introduce two novel parameters, as outlined\nbelow [87, 88]:\nξ=L\nEandη=K\nE2, (14)\nwhere Kis commonly referred to as the Carter constant [89, 90]. Following a series of algebraic\nmanipulations, the resulting expression is:\nξ2+η=r2\nc\nf(rc). (15)\nIn our quest to determine the radius of the shadow, we will employ the celestial coordinates ˜ α\nandβ[72, 91–93] as: α=−ξandβ=±√η. Thereby, we are able to write the radius shadow as\nfollows\nRΛ=0=rcp\nf(rc)=3(1−λ)Mq\n1\n3−3λ. (16)\nThe illustration in Fig. 1 demonstrates the impact of parameters Mandλon the black hole\nshadows. Notably, from a broader perspective, an increase in the parameter λ, i.e., λ= 0.100,\n80500100015002000\n01020304050FIG. 1: The circles in the figure illustrate shadows formed by varying the parameters Mandλ. Specifically,\non the left section, the shadows are generated with Mvalues ranging from 1 (representing the innermost\nradius) to 4 (representing the outermost radius), focusing solely on integer increments. In contrast, on the\nright features shadows that vary with λvalues from 0.1 (outermost radius) to 0.175 (innermost radius),\nwhen Mis set to 1. The radius increments by 0.025 for each successive circle in this case. Color density\nrefers to the magnitude associated with the coordinates αandβ, as indicated in the plot legends situated\non the right side of each respective graph.\nλ= 0.125,λ= 0.150, and λ= 0.175, is associated with a decrease in the radius of the shadows,\nwhich is shown in the right side. This behavior can be directly attributed to the presence of Lorentz\nviolation in the theoretical framework under consideration.\nFurthermore, on the left section of Fig. 1 presents black hole shadows for mass values ranging\nfrom M= 1 (corresponding to the innermost radius) to M= 4 (corresponding to the outermost\nradius), considering only integer increments while keeping λfixed at λ= 0.1. It is evident that as\nthe mass Mincreases, the radius of the shadow expands correspondingly.\n2. Quasinormal modes\nIn the ringdown phase, the phenomenon of quasinormal modes [94–107] comes to the fore, ex-\nhibiting unique oscillation patterns that remain unaffected by the system’s initial state. These\nmodes serve as a fingerprint for it, arising from the inherent vibrations of spacetime and remaining\nconsistent regardless of specific initial conditions. In contrast to normal modes, which are associ-\nated with closed systems, quasinormal modes are relevant to open systems, leading to a gradual\nenergy loss through the emission of gravitational waves. From a mathematical standpoint, these\nmodes are identified as poles in the complex Green’s function.\nTo determine the frequencies of quasinormal modes, it is essential to solve the wave equation\nin a system governed by the background metric gµν. However, deriving analytical solutions for\n9these modes is frequently a complex task. A range of methods for solving for these modes has\nbeen proposed in the scientific literature, with the WKB (Wentzel–Kramers–Brillouin) method\nstanding out as one of the most widely used. This technique was significantly influenced by the\ngroundbreaking work of Will and Iyer [108, 109]. Subsequent refinements, extending the method\nto the sixth order, have been made by Konoplya [110]. For the computations in our study, we focus\non perturbations mediated through the scalar field, utilizing the Klein–Gordon equation within the\ncontext of curved spacetime\n1√−g∂µ(gµν√−g∂νΦ) = 0 . (17)\nWhile the exploration of backreaction effects within this context is undoubtedly important, the\nfocus of this manuscript is directed differently. We primarily center our investigation on the scalar\nfield, treating it as a minor perturbation. The inherent spherical symmetry of the scenario at hand\nallows for a specific decomposition of the scalar field, which we write as follows\nΦ(t, r, θ, φ ) =∞X\nl=0lX\nm=−lr−1Ψlm(t, r)Ylm(θ, φ), (18)\nHere, the spherical harmonics are represented by Ylm(θ, φ). When we incorporate the decomposi-\ntion of the scalar field, as outlined in Eq. (18), into Eq. (17), the equation assumes a Schr¨ odinger–\nlike form. This transformation endows the equation with wave–like characteristics, making it\nparticularly well–suited for working on it\n−∂2Ψ\n∂t2+∂2Ψ\n∂r∗2+Veff(r∗)Ψ = 0 . (19)\nThe potential Veffis often designated as the Regge–Wheeler potential or the effective potential,\ncapturing critical aspects of the black hole geometric structure. Additionally, we employ the\ntortoise coordinate r∗, which spans the entire extent of spacetime and approaches r∗→ ±∞ . This\ncoordinate is computed using the equation d r∗=p\n[1/f(r)2]dr, as elaborated below:\nr∗=−2(λ−1)2M+ 2(λ−1)2Mln[2(λ−1)M+r] +r(1−λ). (20)\nThrough algebraic manipulation, the effective potential can be reformulated as follows:\nVeff=f(r)\u0012l(l+ 1)\nr2+2M\nr3\u0013\n. (21)\nTo facilitate a more nuanced understanding of Eq. (21), we direct attention to Fig. 2. The\nfigure reveals that a barrier–like configuration takes shape when, for example, Mandλ= 2 are\n10-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0-1.5-1.0-0.50.00.51.0FIG. 2: The effective potential Veffis depicted as a function of the tortoise coordinate r∗, considering\ndifferent values of l.\nconsidered. Moreover, an increase in lis accompanied by a substantial elevation in the value of\nVeff.\nRecent literature has seen similar examinations of quasinormal modes within various frame-\nworks, including non–commutativity [93, 111], bumblebee gravity [72], regular black holes [70, 83],\nand more. For the computation of these quasinormal modes, our methodology centers on the WKB\napproach. Our principal objective is to derive stationary solutions for the system under study. To\nrealize this aim, we posit that the function Ψ( t, r) can be expressed as Ψ( t, r) =e−iωtψ(r), where ω\ndenotes the frequency. Employing this formulation allows us to readily isolate the time–independent\ncomponent of Eq. (19) through the procedure detailed subsequently:\n∂2ψ\n∂r∗2−\u0002\nω2−Veff(r∗)\u0003\nψ= 0. (22)\nThe WKB method, initially pioneered by Schutz and Will [112], has become an invaluable tool\nfor identifying quasinormal modes, particularly in the realm of particle scattering near black holes.\nThis methodology has undergone refinements over time, most notably due to substantial contri-\nbutions by Konoplya [110, 113]. It is important to acknowledge that the efficacy of this technique\nhinges on the potential adopting a barrier–like shape, which ultimately flattens to constant values\nasr∗→ ±∞ . By aligning the terms in the power of solution series with the peak potential turn-\ning points, quasinormal modes can be accurately computed. Given these conditions, the formula\ndeveloped by Konoplya is articulated as follows:\ni(ω2\nn−V0)q\n−2V′′\n0−6X\nj=2Λj=n+1\n2. (23)\nIn essence, Konoplya’s framework for determining quasinormal modes comprises multiple elements.\n11Specifically, the term V′′\n0represents the second derivative of the potential, evaluated at its apex\nr0. Furthermore, the constants Λ jare shaped by both the effective potential and its derivatives\nat this pinnacle. It is worth highlighting that recent advancements in the field have led to the\ndevelopment of a 13th–order WKB approximation, initiated by Matyjasek and Opala [114]. This\nprogress significantly enhances the accuracy of quasinormal frequency calculations.\nIt is imperative to note that the quasinormal frequencies pertinent to the scalar field exhibit a\nnegative imaginary component. This distinctive attribute signifies that these modes are subject to\nexponential decay over temporal intervals, thereby representing the mechanism of energy dissipa-\ntion via scalar waves. This characteristic aligns coherently with extant scholarly literature that has\ninvestigated scalar, electromagnetic, and gravitational perturbations within spherically symmetric\nsetup [79, 80, 115].\nIn Tables II, III, and IV, we exhibit the behavior of quasinormal modes, for different values\nofl, as they evolve with changes in two key parameters, namely, Mandλ, by employing the\nsixth–order WKB approximation. Remarkably, a consistent trend emerges across all cases: as λ\nincreases, these frequencies exhibit a notable increase in damping. In contrast, when we increment\nthe mass parameter, we observe a contrasting behavior: the modes exhibit reduced damping.\nB. With presence of cosmological constant\nIn the presence of a cosmological constant, it becomes evident that the initial assumption of\nV′= 0 does not yield a self–consistent solution that complies with all the equations of motion.\nAs a result, we adopt the methodology described in Ref. [116] to relax the vacuum conditions to\nV= 0 while allowing V′̸= 0. The linear potential, V=σX, is often the most frequently discussed\nform that meets this criteria, where σserves as a Lagrange multiplier field [10].\nTaking the derivative of this potential with respect to Xyields V′(X) =σ. The equation\nof motion for the Lagrange multiplier σconstrains the theory to the potential extrema, which\ncorrespond to X= 0. In this case, bµνbecomes the VEV of the KR field for the on–shell σ.\nIt should be noted that, to maintain the positivity of the potential V, the off–shell value of\nthe Lagrange multiplier σmust share the same sign as X. Although it is theoretically possible to\nexpand the Lagrange multiplier field σaround its vacuum value, i.e., expressed as σ=⟨σ⟩+ ˜σ, and\nallow⟨σ⟩to vary with spacetime coordinates, for the sake of simplicity we opt to set ˜ σ= 0 through\nappropriate initial conditions. We further assume that ⟨σ⟩is a real constant. This ensures that\nthe on–shell value of σ, denoted as σ≡ ⟨σ⟩, is uniquely determined by the field equations [10].\n12TABLE II: Utilizing the sixth–order WKB approximation, we demonstrate the quasinormal frequencies\ncorresponding to varying values of Mandλ, specifically for l= 0.\nM λ ω0 ω1 ω2\n1.0, 0.00 0.110464 - 0.100819 i0.089023 - 0.344552 i0.191783 - 0.476507 i\n1.0, 0.10 0.136377 - 0.124466 i0.109907 - 0.425363 i0.236779 - 0.588257 i\n1.0, 0.11 0.139431 - 0.127304 i0.112340 - 0.435169 i0.241795 - 0.602365 i\n1.0, 0.12 0.142685 - 0.130153 i0.115034 - 0.444630 i0.248184 - 0.614008 i\n1.0, 0.13 0.145994 - 0.133153 i0.117709 - 0.454849 i0.254012 - 0.627978 i\n1.0, 0.14 0.149360 - 0.136312 i0.120373 - 0.465837 i0.259346 - 0.644182 i\n1.0, 0.15 0.152878 - 0.139554 i0.12319 - 0.476984 i0.265269 - 0.659956 i\n1.0, 0.16 0.156544 - 0.142893 i0.126151 - 0.488373 i0.271701 - 0.675582 i\n1.0, 0.17 0.160371 - 0.146327 i0.129268 - 0.499981 i0.278682 - 0.690960 i\n1.0, 0.18 0.164315 - 0.149909 i0.132448 - 0.512204 i0.285565 - 0.707774 i\n1.0, 0.19 0.168411 - 0.153622 i0.135773 - 0.524814 i0.292906 - 0.724792 i\n1.0, 0.20 0.172607 - 0.157523 i0.139113 - 0.538309 i0.299758 - 0.744307 i\nM λ ω0 ω1 ω2\n1.0, 0.1 0.1363770 - 0.1244660 i0.1099070 - 0.4253630 i0.2367790 - 0.5882570 i\n2.0, 0.1 0.0681880 - 0.0622330 i0.0549530 - 0.2126820 i0.1183900 - 0.2941280 i\n3.0, 0.1 0.0454482 - 0.0414980 i0.0366170 - 0.1418610 i0.0788040 - 0.1963900 i\n4.0, 0.1 0.0340940 - 0.0311160 i0.0274760 - 0.1063410 i0.0591948 - 0.1470640 i\n5.0, 0.1 0.0272707 - 0.0248976 i0.0219733 - 0.0851046 i0.0473030 - 0.1177820 i\n6.0, 0.1 0.0227320 - 0.0207421 i0.0183228 - 0.0708752 i0.0394972 - 0.0979586 i\n7.0, 0.1 0.0194746 - 0.0177881 i0.0156873 - 0.0608201 i0.0337353 - 0.0842628 i\n8.0, 0.1 0.0170472 - 0.0155583 i0.0137384 - 0.0531704 i0.0295974 - 0.0735321 i\n9.0, 0.1 0.0151562 - 0.0138266 i0.0122178 - 0.0472394 i0.0263493 - 0.0652601 i\n10.0, 0.1 0.0136372 - 0.0124471 i0.0109901 - 0.042539 i0.0236742 - 0.0588351 i\nFurthermore, after some algebraic manipulations, we get [78]\nA(r) =1\nB(r)=1\n1−λ−2M\nr−Λr2\n3(1−λ), (24)\nwhich yields\nds2=−\u00121\n1−λ−2M\nr−Λr2\n3(1−λ)\u0013\ndt2+dr2\n1\n1−λ−2M\nr−Λr2\n3(1−λ)+r2dθ2+r2sin2dφ2. (25)\n13TABLE III: By applying the sixth–order WKB approximation, we elucidate the quasinormal frequencies\nassociated with diverse values of Mandλ, with a specific focus on cases where l= 1.\nM λ ω0 ω1 ω2\n1.0, 0.00 0.292910 - 0.097761 i0.264471 - 0.306518 i0.231014 - 0.542166 i\n1.0, 0.10 0.345670 - 0.120878 i0.309733 - 0.380212 i0.270155 - 0.673692 i\n1.0, 0.11 0.351810 - 0.123630 i0.314977 - 0.389000 i0.274720 - 0.689366 i\n1.0, 0.12 0.358132 - 0.126477 i0.320369 - 0.398100 i0.279412 - 0.705614 i\n1.0, 0.13 0.364646 - 0.129424 i0.325920 - 0.407519 i0.284254 - 0.722420 i\n1.0, 0.14 0.371359 - 0.132474 i0.331636 - 0.417272 i0.289249 - 0.739817 i\n1.0, 0.15 0.378280 - 0.135634 i0.337523 - 0.427378 i0.294400 - 0.757846 i\n1.0, 0.16 0.385418 - 0.138907 i0.343589 - 0.437854 i0.299715 - 0.776531 i\n1.0, 0.17 0.392782 - 0.142301 i0.349839 - 0.448720 i0.305199 - 0.795918 i\n1.0, 0.18 0.400383 - 0.145820 i0.356286 - 0.459989 i0.310869 - 0.816013 i\n1.0, 0.19 0.408231 - 0.149471 i0.362936 - 0.471686 i0.316729 - 0.836866 i\n1.0, 0.20 0.416338 - 0.153261 i0.369797 - 0.483834 i0.322785 - 0.858524 i\nM λ ω0 ω1 ω2\n1.0, 0.1 0.345670 - 0.120878 i0.309733 - 0.380212 i0.270155 - 0.673692 i\n2.0, 0.1 0.172835 - 0.060438 i0.154867 - 0.190106 i0.135077 - 0.336846 i\n3.0, 0.1 0.115223 - 0.040292 i0.103245 - 0.126737 i0.090052 - 0.224563 i\n4.0, 0.1 0.086417 - 0.030219 i0.077433 - 0.095053 i0.067538 - 0.168423 i\n5.0, 0.1 0.069134 - 0.024175 i0.061947 - 0.076042 i0.054031 - 0.134736 i\n6.0, 0.1 0.057611 - 0.020146 i0.051622 - 0.0633685 i0.045026 - 0.112281 i\n7.0, 0.1 0.049381 - 0.017268 i0.044247 - 0.054316 i0.038593 - 0.096241 i\n8.0, 0.1 0.043208 - 0.015109 i0.038716 - 0.047526 i0.033769 - 0.084211 i\n9.0, 0.1 0.038407 - 0.013430 i0.034414 - 0.042245 i0.030017 - 0.074854 i\n10.0, 0.1 0.034567 - 0.012087 i0.030973 - 0.038021 i0.027015 - 0.067368 i\nAbove expression gives rise to the following physical horizon\nrΛ̸=0=Λ +\u0010p\nΛ3(9(λ−1)2ΛM2−1) + 3( λ−1)Λ2M\u00112/3\nΛ3qp\nΛ3(9(λ−1)2ΛM2−1) + 3( λ−1)Λ2M. (26)\n14TABLE IV: Utilizing the sixth–order WKB approximation, we present the quasinormal frequencies corre-\nsponding to various values of Mand Λ, particularly when l= 2.\nM λ ω0 ω1 ω2\n1.0, 0.00 0.483642 - 0.096766 i0.463847 - 0.295627 i0.430386 - 0.508700 i\n1.0, 0.10 0.568021 - 0.119536 i0.542536 - 0.365844 i0.500283 - 0.631162 i\n1.0, 0.11 0.577801 - 0.122246 i0.551626 - 0.374211 i0.508323 - 0.645777 i\n1.0, 0.12 0.587865 - 0.125048 i0.560972 - 0.382868 i0.516583 - 0.660905 i\n1.0, 0.13 0.598224 - 0.127948 i0.570587 - 0.391828 i0.525071 - 0.676570 i\n1.0, 0.14 0.608890 - 0.130951 i0.580479 - 0.401107 i0.533799 - 0.692794 i\n1.0, 0.15 0.619877 - 0.134060 i0.590662 - 0.410718 i0.542775 - 0.709607 i\n1.0, 0.16 0.631199 - 0.137281 i0.601147 - 0.420679 i0.552009 - 0.727038 i\n1.0, 0.17 0.642869 - 0.140620 i0.611946 - 0.431007 i0.561511 - 0.745117 i\n1.0, 0.18 0.654903 - 0.144082 i0.623074 - 0.441719 i0.571295 - 0.763873 i\n1.0, 0.19 0.667317 - 0.147673 i0.634544 - 0.452835 i0.581368 - 0.783347 i\n1.0, 0.20 0.680128 - 0.151401 i0.646371 - 0.464375 i0.591747 - 0.803569 i\nM λ ω0 ω1 ω2\n1.0, 0.1 0.568021 - 0.119536 i0.542536 - 0.365844 i0.500283 - 0.631162 i\n2.0, 0.1 0.284011 - 0.059768 i0.271268 - 0.182922 i0.250142 - 0.315581 i\n3.0, 0.1 0.189340 - 0.039845 i0.180845 - 0.121948 i0.166761 - 0.210387 i\n4.0, 0.1 0.142005 - 0.029884 i0.135634 - 0.091461 i0.125071 - 0.157790 i\n5.0, 0.1 0.113604 - 0.023907 i0.108507 - 0.073168 i0.100057 - 0.126232 i\n6.0, 0.1 0.094670 - 0.019922 i0.090422 - 0.060974 i0.083380 - 0.105194 i\n7.0, 0.1 0.081145 - 0.017076 i0.077505 - 0.052263 i0.071469 - 0.090165 i\n8.0, 0.1 0.071002 - 0.014942 i0.067817 - 0.045730 i0.062535 - 0.078895 i\n9.0, 0.1 0.063113 - 0.013281 i0.060281 - 0.040649 i0.055587 - 0.070129 i\n10.0, 0.1 0.056802 - 0.011953 i0.054253 - 0.036584 i0.050028 - 0.063116 i\n1. Photon sphere and shadows\nRemarkably, it is important to mention that the cosmological constant does not affect the\nphoton sphere radius. The same facet, in the other hand, can not be inferred to the shadow radius.\nIn addition, in the case of substantial cosmic distances and the presence of a cosmological constant,\nit has been demonstrated that the size of a black hole’s shadow can explicitly vary with respect\nto the radial coordinate of a distant observer denoted as rO. This relationship is detailed in [117]\n1502468\n00.20.40.6\n00.20.40.6FIG. 3: The diagram showcases an array of circles, each representing the shadow formations resulting\nfrom variations in the parameters M,λ, and Λ. In the upper–left quadrant, these circles correspond to\nshadows generated by Mvalues ranging from 0 .1 (innermost circle) to 0 .4 (outermost circle), increasing\nincrementally by integers. In this context, λremains fixed at 0 .1, while Λ is set at −0.1. Conversely,\nin the upper-right quadrant, the circles portray shadows influenced by varying λvalues. These λvalues\nrange from 0 .1 (innermost circle) to 0 .175 (outermost circle), with Mmaintained at 0 .1 and Λ at −0.1. In\nthis arrangement, each successive circle exhibits a λvalue incremented by 0 .025, and the shadow radius\ndiminishes as λincreases. In the lower section of the illustration, the circles are determined by a decrement\nof−0.1 in Λ for each subsequent representation. The span commences with Λ = −0.1 for the innermost\ncircle and descends to Λ = −0.4 for the outermost circle.\nand can be represented as:\nRΛ̸=0=rcs\nf(rO)\nf(rc)=3(1−λ)Mq\n−6λM+6M+Λr3\nO−3rO\n(λ−1)rOq\n1\n1−λ+ 9(λ−1)ΛM2. (27)\nIn Fig. 3, it is exhibited the shadows for different values of varying the parameters M,λ, and Λ.\nIn the left section of the diagram, shadows (circles) are formed by varying the mass parameter M,\ni.e., 0 .1,0.2,0.3 and 0 .4. Here, the innermost circle corresponds to M= 0.1 and the outermost to\nM= 0.4. For these circles, λremains constant at 0 .1, while Λ is set to −0.1. The increment of M\nis responsible for making larger values of shadow radius.\nIn contrast, the middle section features circles whose properties are dictated by changes in the\ncoupling constant λ. In this setting, λvaries from 0 .125 (outermost circle) to 0 .2 (outermost circle),\nwith each consecutive circle incremented by 0 .025 in λ. Notably, the radius of the shadow shrinks\nasλincreases, while Mand Λ are fixed at 0 .1 and −0.1, respectively. Finally, the right section\nof the diagram, the circles are characterized by decrements of −0.1 in Λ for each subsequent plot.\nThe value of Λ starts at −0.1 for the innermost circle and decreases to −0.4 for the outermost\ncircle.\n162. Constraints on the parameters of antisymmetric black hole with the EHT observations Sgr A*\nIn this section, we derive constraints on the parameters of antisymmetric black holes by ana-\nlyzing observational data obtained from the Event Horizon Telescope (EHT) [118–121] focusing on\nSagittarius A∗shadow images. The shadow image’s radius variation for Sgr. A∗is quantified with\nuncertainties at 1 σand 2 σlevels.\nFig. 4 illustrates the relationship between the shadow and the parameters Λ and λ. Tables V\nand VI present the acceptable parameter constraints for λand Λ based on experimental limits at\n1σand 2 σlevels, respectively.\nTABLE V: The results of the constraints for λfor some values of Λ based on σ1 and σ2.\nΛ -0.01 -0.03 -0.05\nσ1Upper band for λ0.08 0.049 0.014\nLower band for λ- - -\nσ2Upper band for λ0.045 0.007 -\nLower band for λ- - -\nTABLE VI: The results of the constraints for Λ for some values of λbased on σ1 and σ2.\nλ 0.001 0.01 0.1\nσ1Upper band for Λ -0.057 -0.052 -\nLower band for Λ - - -\nσ2Upper band for Λ -0.033 -0.028 -\nLower band for Λ - - -\n3. Quasinormal modes\nFrom Eq. (25), the Regge–Wheeler potential reads\nVeff(Λ)=f(r) 2M\nr2−2Λr\n3(1−λ)\nr+l(l+ 1)\nr2!\n. (28)\nAnalogously what we have accomplished in the previous sections, here we present the analysis of\nthequasinormal modes, considering Λ ̸= 0 instead. The main features ascribed to this, will be\nshown as follows.\nIn Fig. 5, we observe the effective potential, denoted as Veff(Λ), plotted against the tortoise\ncoordinate r∗. This representation explores various values of lwhile maintaining constant values\nfor the parameters Λ = −0.01,λ= 0.1, and M= 1.\n17Λ= -0.01 Λ= -0.03 Λ= -0.05\n2σ\n1σ\n0.02 0.04 0.06 0.08 0.144.555.56\nλRShadow\nλ=0.001 λ=0.01 λ=0.1\n2σ\n1σ\n-0.07 -0.05 -0.03 -0.01 044.555.56\nΛRShadowFIG. 4: The shadow radius plots according to λare shown in different color lines for Λ = −0.01,−0.03,−0.05\nin left panel. On the right panel, color lines represent the shadow radius versus Λ for λ= 0.001,0.01,0.1.\nIn both panels two pairs of dashed lines for experimental constraints are shown.\n-10 -5 0 5 10 150.0000.0050.0100.0150.0200.0250.0300.035\nFIG. 5: The effective potential Veff(Λ)is depicted as a function of the tortoise coordinate r∗, considering\ndifferent values of land fixed values of Λ = −0.01,λ= 0.1, and M= 1.\nIn Tables VII, VIII, and IX, we present the quasinormal frequencies computed for various\nvalues of the quantum number l, utilizing the sixth–order WKB method. Additionally, we conduct\na comprehensive analysis of the behavior of these modes as they depend on the parameters M,\nλ, and Λ. It is noteworthy that across all considered values of l, i.e., specifically, l= 0, l= 1,\nandl= 2, the observed trends in the modes, with respect to variations in Mandλ, closely\nresemble those identified in the previous case, as one might logically anticipate. It is important to\nhighlight that as the cosmological constant Λ attains increasingly negative values, the quasinormal\nfrequencies demonstrate a damper aspect.\nIt is worth noting that a remarkable feature emerges when l= 0: certain modes exhibit insta-\nbility for the considered parameter values. However, as lvaries, these instabilities are no longer\nmaintained.\n18III. EMISSION RATE\nThe energy emission rate can be obtained according to [122]\nd2E\ndωdt=2π2σlimω3\neω\nTH−1, (29)\nwhere ωrepresents the frequency of photon, and Tis the Hawking temperature for the outer event\nhorizon, being defined as\nTH=f′(r)\n4π\f\f\f\f\nr=rh, (30)\nwhere f(r) is taken in accordance with Eq. (24) and rhrepresents the radius of the horizon and\nσlimis the limiting cross section intricately connected to the radius of the event horizon. The\nequation governing σlimindspacetime dimensions is provided in Reference [123], and additional\ninsights can be found in the works of [124, 125]\nσlim=πd−2\n2Rd−2\ns\nΓ\u0000d\n2\u0001, (31)\nwhere Rsrepresents the shadow radius and Γ\u0000d\n2\u0001\ndenotes the gamma function.\nTo ascertain the absorption cross–section, we have two methodologies at our disposal. The initial\napproach involves deriving the effective potential using Equation (28). Meanwhile, the second\nmethod entails incorporating the purely imaginary quantity represented as Γ, thereby facilitating\nthe analysis\nΓ =i\u0000\n˜ω2−V0\u0001\np\n−2V′′\n0−6X\nj=2Λj, (32)\nwhere V0is the potential calculated in its maximum point. The aforementioned equation stems\nfrom the WKB method. Within this framework, the coefficients corresponding to the complex\nfunctions defining the effective potential are denoted as Λ j, and ˜ ωsignifies a purely real frequency\nassociated with quasinormal modes. The reflection and transmission coefficients are articulated as\nfollows:\n|R|2=1\n1 +e−2iπγ,|T|2=1\n1 +e2iπγ, (33)\nThe partial absorption cross–section for ℓmodes is precisely defined as:\nσℓ=π(2ℓ+ 1)\n˜ω2|Tℓ(˜ω)|2. (34)\n19The total absorption cross–section is established through the summation of all partial absorption\ncross–sections, as expressed by:\nσabs=X\nℓσℓ. (35)\nMoreover, by using Eq. (27) for d= 2 it can be written as\nσlim=πR2\ns=π\n3(1−λ)Mq\n−6λM+6M+Λr3\nO−3rO\n(λ−1)rOq\n1\n1−λ+ 9(λ−1)ΛM2\n2\n. (36)\n0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.00.10.20.30.40.50.6\n0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.00.20.40.60.81.01.2\nFIG. 6: The emission rates are represented with respect to frequency for M= 1, in left plot different values\nofλ, fixed values of Λ = −0.02 and in right plot different values of Λ and fixed values of λ= 0.02 are\nconsidered.\nThe emission rate as a function of frequency ωis depicted in Fig. 6. In the left panel, the Λ\nvalue is held constant, while different values of λ(specifically, 0 .02,0.04,0.06,0.08) are considered.\nConversely, in the right panel, λis fixed, and varying Λ values of −0.02,−0.04,−0.06,−0.08 are\nexplored.\nIn the left panel, discernible peaks in the emission rate are observed, indicating that the max-\nimum energy emission increases with higher values of the parameter λ. Additionally, these peaks\nshift towards higher frequencies. Conversely, in the right panel, as the absolute value of Λ increases,\nthe emission rate’s peak shifts to higher frequencies, accompanied by an increase in the maximum\nemission value.\nIV. CONCLUSION\nThis study explored the impact of antisymmetric tensor effects on spherically symmetric black\nholes, examining photon spheres, shadows, emission rate, and quasinormal frequencies concerning\n20a parameter that triggered Lorentz symmetry breaking. Both scenarios, one without and one\nwith the presence of a cosmological constant, were investigated. In the first scenario, the Lorentz\nviolation parameter, denoted as λ, played a pivotal role in reducing both the photon sphere and\nthe shadow radius, while also inducing a damping effect on quasinormal frequencies. Conversely,\nin the second scenario, as the values of the cosmological constant (Λ) increased, an expansion in\nthe shadow radius was observed, accompanied by a reduction in damping modes associated with\ngravitational wave frequencies. Furthermore, we determined the shadow constraints by analyzing\nobservational data acquired from the Event Horizon Telescope (EHT), specifically concentrating\non the shadow images of Sagittarius A∗.\nAcknowledgments\nThe authors would like to thank the anonymous referee for a careful reading of the manuscript\nand for the remarkable suggestions given to us. A. A. 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Liu, “Observing the shadow of einstein-maxwell-dilaton-axion black hole,” Jour-\nnal of Cosmology and Astroparticle Physics , vol. 2013, no. 11, p. 063, 2013.\n29TABLE VII: Utilizing the sixth–order WKB approximation, we demonstrate the quasinormal frequencies\ncorresponding to varying values of M,λ, and Λ specifically for l= 0.\nM λ Λ ω0 ω1 ω2\n1.00, 0.10, -0.01 0.140914 - 0.124590 i0.0356274 - 0.589974 i Unstable\n1.01, 0.10, -0.01 0.139526 - 0.123374 i0.0321124 - 0.593042 i Unstable\n1.02, 0.10, -0.01 0.138166 - 0.122178 i0.0285944 - 0.596445 i Unstable\n1.03, 0.10, -0.01 0.136818 - 0.121013 i0.0250711 - 0.600259 i Unstable\n1.04, 0.10, -0.01 0.135483 - 0.119876 i0.0215458 - 0.604477 i Unstable\n1.05, 0.10, -0.01 0.134172 - 0.118756 i0.0180223 - 0.609043 i Unstable\n1.06, 0.10, -0.01 0.132868 - 0.117668 i0.0144995 - 0.614046 i Unstable\n1.07, 0.10, -0.01 0.131578 - 0.116601 i0.0109807 - 0.619443 i Unstable\n1.08, 0.10, -0.01 0.130301 - 0.115558 i0.0074673 - 0.625253 i Unstable\n1.09, 0.10, -0.01 0.129026 - 0.114544 i0.0039610 - 0.631505 i Unstable\n1.10, 0.10, -0.01 0.127765 - 0.113549 i0.0004630 - 0.638159 i Unstable\nM λ Λ ω0 ω1 ω2\n1.00, 0.10, -0.01 0.140914 - 0.124590 i0.0356274 - 0.589974 i Unstable\n1.00, 0.11, -0.01 0.144078 - 0.127389 i0.0399678 - 0.593609 i Unstable\n1.00, 0.12, -0.01 0.147355 - 0.130275 i0.0444021 - 0.597706 i Unstable\n1.00, 0.13, -0.01 0.150718 - 0.133280 i0.0489170 - 0.602466 i Unstable\n1.00, 0.14, -0.01 0.154207 - 0.136370 i0.0535301 - 0.607694 i Unstable\n1.00, 0.15, -0.01 0.157812 - 0.139589 i0.0582345 - 0.613502 i Unstable\n1.00, 0.16, -0.01 0.161528 - 0.142928 i0.0630258 - 0.619972 i Unstable\n1.00, 0.17, -0.01 0.165388 - 0.146375 i0.0679225 - 0.626949 i Unstable\n1.00, 0.18, -0.01 0.169366 - 0.149968 i0.0729055 - 0.634656 i Unstable\n1.00, 0.19, -0.01 0.173483 - 0.153690 i0.0779843 - 0.643031 i Unstable\n1.00, 0.20, -0.01 0.177773 - 0.157550 i0.0831850 - 0.651918 i Unstable\nM λ Λ ω0 ω1 ω2\n1.0, 0.1 -0.0010 0.136954 - 0.124470 i0.109389 - 0.428021 i0.222318 - 0.608012 i\n1.0, 0.1 -0.0011 0.137019 - 0.124463 i0.109298 - 0.428297 i0.220240 - 0.610849 i\n1.0, 0.1 -0.0012 0.137059 - 0.124479 i0.109150 - 0.428766 i0.217755 - 0.614682 i\n1.0, 0.1 -0.0013 0.137138 - 0.124459 i0.109060 - 0.428976 i0.215557 - 0.617572 i\n1.0, 0.1 -0.0014 0.137192 - 0.124463 i0.108914 - 0.429380 i0.212961 - 0.621467 i\n1.0, 0.1 -0.0015 0.137243 - 0.124469 i0.108751 - 0.429820 i0.210211 - 0.625691 i\n1.0, 0.1 -0.0016 0.137293 - 0.124475 i0.108577 - 0.430276 i0.207349 - 0.630143 i\n1.0, 0.1 -0.0017 0.137359 - 0.124467 i0.108418 - 0.430641 i0.204510 - 0.634402 i\n1.0, 0.1 -0.0018 0.137422 - 0.124462 i0.108242 - 0.431039 i0.201534 - 0.638973 i\n1.0, 0.1 -0.0019 0.137471 - 0.124470 i0.108027 - 0.431560 i0.198299 - 0.644263 i\n1.0, 0.1 -0.0020 0.137537 - 0.124461 i0.107832 - 0.431968 i0.195135 - 0.649239 i30TABLE VIII: Utilizing the sixth–order WKB approximation, we demonstrate the quasinormal frequencies\ncorresponding to varying values of M,λ, and Λ specifically for l= 1.\nM λ Λ ω0 ω1 ω2\n1.00, 0.10, -0.01 0.360556 - 0.123469 i0.317340 - 0.394362 i0.260939 - 0.717176 i\n1.01, 0.10, -0.01 0.357277 - 0.122294 i0.314274 - 0.390758 i0.257535 - 0.711608 i\n1.02, 0.10, -0.01 0.354065 - 0.121143 i0.311264 - 0.387230 i0.254151 - 0.706211 i\n1.03, 0.10, -0.01 0.350918 - 0.120015 i0.308308 - 0.383775 i0.250782 - 0.700980 i\n1.04, 0.10, -0.01 0.347835 - 0.118909 i0.305404 - 0.380392 i0.247428 - 0.695930 i\n1.05, 0.10, -0.01 0.344812 - 0.117824 i0.302549 - 0.377078 i0.244084 - 0.691043 i\n1.06, 0.10, -0.01 0.341849 - 0.116760 i0.299743 - 0.373832 i0.240748 - 0.686324 i\n1.07, 0.10, -0.01 0.338944 - 0.115716 i0.296984 - 0.370652 i0.237419 - 0.681770 i\n1.08, 0.10, -0.01 0.336095 - 0.114692 i0.294269 - 0.367536 i0.234093 - 0.677383 i\n1.09, 0.10, -0.01 0.333301 - 0.113687 i0.291597 - 0.364483 i0.230767 - 0.673162 i\n1.10, 0.10, -0.01 0.330560 - 0.112700 i0.288966 - 0.361491 i0.227439 - 0.669107 i\nM λ Λ ω0 ω1 ω2\n1.00, 0.10, -0.01 0.360556 - 0.123469 i0.317340 - 0.394362 i0.260939 - 0.717176 i\n1.00, 0.11, -0.01 0.366647 - 0.126212 i0.322546 - 0.403170 i0.265762 - 0.732657 i\n1.00, 0.12, -0.01 0.372920 - 0.129050 i0.327902 - 0.412285 i0.270718 - 0.748687 i\n1.00, 0.13, -0.01 0.379385 - 0.131988 i0.333415 - 0.421722 i0.275813 - 0.765296 i\n1.00, 0.14, -0.01 0.386048 - 0.135029 i0.339091 - 0.431496 i0.281054 - 0.782508 i\n1.00, 0.15, -0.01 0.392919 - 0.138178 i0.344938 - 0.441622 i0.286449 - 0.800349 i\n1.00, 0.16, -0.01 0.400006 - 0.141442 i0.350962 - 0.452119 i0.292004 - 0.818853 i\n1.00, 0.17, -0.01 0.407319 - 0.144826 i0.357172 - 0.463003 i0.297727 - 0.838048 i\n1.00, 0.18, -0.01 0.414869 - 0.148334 i0.363575 - 0.474295 i0.303627 - 0.857972 i\n1.00, 0.19, -0.01 0.422665 - 0.151975 i0.370180 - 0.486014 i0.309713 - 0.878656 i\n1.00, 0.20, -0.01 0.430720 - 0.155753 i0.376997 - 0.498183 i0.315994 - 0.900142 i\nM λ Λ ω0 ω1 ω2\n1.0, 0.1 -0.0010 0.347181 - 0.121152 i0.310679 - 0.381601 i0.270925 - 0.676553 i\n1.0, 0.1 -0.0011 0.347332 - 0.121179 i0.310772 - 0.381739 i0.270992 - 0.676845 i\n1.0, 0.1 -0.0012 0.347482 - 0.121206 i0.310866 - 0.381878 i0.271057 - 0.677139 i\n1.0, 0.1 -0.0013 0.347633 - 0.121233 i0.310958 - 0.382017 i0.271118 - 0.677437 i\n1.0, 0.1 -0.0014 0.347784 - 0.121260 i0.311051 - 0.382156 i0.271178 - 0.677734 i\n1.0, 0.1 -0.0015 0.347934 - 0.121288 i0.311144 - 0.382295 i0.271235 - 0.678034 i\n1.0, 0.1 -0.0016 0.348085 - 0.121315 i0.311236 - 0.382434 i0.271290 - 0.678336 i\n1.0, 0.1 -0.0017 0.348235 - 0.121342 i0.311328 - 0.382573 i0.271344 - 0.678638\n1.0, 0.1 -0.0018 0.348386 - 0.121369 i0.311420 - 0.382712 i0.271395 - 0.678941 i\n1.0, 0.1 -0.0019 0.348536 - 0.121396 i0.311511 - 0.382850 i0.271443 - 0.679247 i\n1.0, 0.1 -0.0020 0.348687 - 0.121422 i0.311603 - 0.382989 i0.271490 - 0.679552 i31TABLE IX: Utilizing the sixth–order WKB approximation, we demonstrate the quasinormal frequencies\ncorresponding to varying values of M,λ, and Λ specifically for l= 2.\nM λ Λ ω0 ω1 ω2\n1.00, 0.10, -0.01 0.590234 - 0.123075 i0.561064 - 0.378474 i0.513965 - 0.656422 i\n1.01, 0.10, -0.01 0.584825 - 0.121924 i0.555866 - 0.374973 i0.509103 - 0.650431 i\n1.02, 0.10, -0.01 0.579527 - 0.120797 i0.550774 - 0.371544 i0.504335 - 0.644563 i\n1.03, 0.10, -0.01 0.574335 - 0.119691 i0.545783 - 0.368183 i0.499658 - 0.638814 i\n1.04, 0.10, -0.01 0.569248 - 0.118608 i0.540891 - 0.364889 i0.495071 - 0.633182 i\n1.05, 0.10, -0.01 0.564261 - 0.117546 i0.536095 - 0.361659 i0.490571 - 0.627663 i\n1.06, 0.10, -0.01 0.559372 - 0.116504 i0.531393 - 0.358493 i0.486154 - 0.622253 i\n1.07, 0.10, -0.01 0.554578 - 0.115482 i0.526781 - 0.355388 i0.481818 - 0.616951 i\n1.08, 0.10, -0.01 0.549877 - 0.114480 i0.522258 - 0.352343 i0.477561 - 0.611752 i\n1.09, 0.10, -0.01 0.545265 - 0.113497 i0.517820 - 0.349356 i0.473380 - 0.606654 i\n1.10, 0.10, -0.01 0.540742 - 0.112532 i0.513465 - 0.346425 i0.469274 - 0.601655 i\nM λ Λ ω0 ω1 ω2\n1.00, 0.10, -0.01 0.590234 - 0.123075 i0.561064 - 0.378474 i0.513965 - 0.656422 i\n1.00, 0.11, -0.01 0.599921 - 0.125781 i0.570043 - 0.386850 i0.521905 - 0.671080 i\n1.00, 0.12, -0.01 0.609892 - 0.128579 i0.579278 - 0.395516 i0.530064 - 0.686250 i\n1.00, 0.13, -0.01 0.620157 - 0.131475 i0.588780 - 0.404485 i0.538451 - 0.701957 i\n1.00, 0.14, -0.01 0.630729 - 0.134473 i0.598559 - 0.413773 i0.547076 - 0.718226 i\n1.00, 0.15, -0.01 0.641622 - 0.137578 i0.608627 - 0.423394 i0.555949 - 0.735083 i\n1.00, 0.16, -0.01 0.652848 - 0.140794 i0.618996 - 0.433365 i0.565079 - 0.752559 i\n1.00, 0.17, -0.01 0.664421 - 0.144128 i0.629679 - 0.443702 i0.574477 - 0.770682 i\n1.00, 0.18, -0.01 0.676359 - 0.147585 i0.640689 - 0.454423 i0.584154 - 0.789487 i\n1.00, 0.19, -0.01 0.688675 - 0.151172 i0.652040 - 0.465549 i0.594123 - 0.809006 i\n1.00, 0.20, -0.01 0.701387 - 0.154894 i0.663748 - 0.477100 i0.604395 - 0.829277 i\nM λ Λ ω0 ω1 ω2\n1.0, 0.1 -0.0010 0.570273 - 0.119902 i0.544435 - 0.367128 i0.501786 - 0.633691 i\n1.0, 0.1 -0.0011 0.570498 - 0.119939 i0.544624 - 0.367256 i0.501935 - 0.633944 i\n1.0, 0.1 -0.0012 0.570722 - 0.119975 i0.544813 - 0.367384 i0.502085 - 0.634196 i\n1.0, 0.1 -0.0013 0.570947 - 0.120011 i0.545002 - 0.367512 i0.502233 - 0.634449 i\n1.0, 0.1 -0.0014 0.571172 - 0.120048 i0.545191 - 0.367640 i0.502382 - 0.634701 i\n1.0, 0.1 -0.0015 0.571396 - 0.120084 i0.545380 - 0.367768 i0.502530 - 0.634954 i\n1.0, 0.1 -0.0016 0.571621 - 0.120121 i0.545569 - 0.367896 i0.502679 - 0.635207 i\n1.0, 0.1 -0.0017 0.571845 - 0.120157 i0.545758 - 0.368024 i0.502827 - 0.635459 i\n1.0, 0.1 -0.0018 0.572069 - 0.120193 i0.545946 - 0.368152 i0.502975 - 0.635711 i\n1.0, 0.1 -0.0019 0.572293 - 0.120229 i0.546135 - 0.368280 i0.503122 - 0.635964 i\n1.0, 0.1 -0.0020 0.572518 - 0.120266 i0.546323 - 0.368407 i0.503269 - 0.636216 i32"    },    {        "title": "2211.05478v4.Connect_the_Lorentz_Violation_to_the_Glashow_Resonance_Event.pdf",        "content": "Connect the Lorentz Violation to the Glashow Resonance Event\nDing-Hui Xu1,∗and Shu-Jun Rong \u00001,†\n1College of Science, Guilin University of Technology, Guilin, Guangxi 541004, China\nThe recent reported Glashow resonance (GR) event shows a promising prospect in the test\nof the Lorentz violation (LV) by the high-energy astrophysical neutrinos (HANs) around the\nresonant energy. However, since the production source and the energy spectra of HANs are\nuncertain at present, moderate LV effects may be concealed at the TeV energy-scale. In this\npaper, we propose the LV Hamiltonian of a special texture which can lead to the decoupling\nofνµ(¯νµ). On the base of the decoupling, a noticeable damping of the GR event rate is shown\nfor the HANs from the source dominated by ¯ νµ, irrespective of the energy spectra of the\nHANs at Earth. Accordingly, the observation of GR events may bring stringent constraints\non the LV and the production mechanism of HANs.\n∗3036602895@qq.com\n†rongshj@glut.edu.cnarXiv:2211.05478v4  [hep-ph]  30 May 20232\nI. INTRODUCTION\nThe Glashow resonance (GR)[1] at Eν≃6.3 PeV in the rest frame of electrons is a probe\nof high-energy astrophysical antineutrinos ¯ νe[2, 3]. The observation of GR events can provide\nuseful information on the energy spectra, production source, and flavor transition of high-energy\nastrophysical neutrinos (HANs). In recent years, several PeV events of HANs have been reported\nby the IceCube neutrino observatory[4]. In particular, the first GR event was detected at the 2 .3σ\nlevel[5]. Although there is just one event, the source where no antineutrinos are produced may\nbe excluded as the sole origin of HANs. Furthermore, the detection of the GR event can also set\nconstraints on new physics effects which become significant at the PeV energy-scale. Many novel\neffects in the field of HANs, such as those from pseudo-Dirac neutrinos[6–9], neutrino-decay[10–\n15], neutrino secret interactions[16–18], etc[19–22], have been proposed. In this paper, we take into\naccount the impacts of the Lorentz violation (LV)[23–32] on the flux of ¯ νearound the resonance\nenergy and show an interesting observation that the GR events may be damped by the LV.\nAs is known, the ¯ νeflux controlling the GR event rate is affected by neutrino oscillations\nover cosmological distances. The effects of the LV on neutrino oscillations are determined by the\nHamiltonian H=H0+HLV, where the first and the second term are expressed respectively as\nfollowing[33]:\nH0=U\n0 0 0\n0∆m2\n21\n2Eν0\n0 0∆m2\n31\n2Eν\nU+, (1)\nHLV=±\n0 aT\neµaT\neτ\n(aT\neµ)∗0aT\nµτ\n(aT\neτ)∗(aT\nµτ)∗0\n−4Eν\n3\n0 cTT\neµ cTT\neτ\n(cTT\neµ)∗0 cTT\nµτ\n(cTT\neτ)∗(cTT\nµτ)∗0\n. (2)\nHere Uis the leptonic mixing matrix in vacuum [34] and matter effects are not considered. For\nantineutrinos, all terms should be replaced by their complex conjugates and the signal of the a\nmatrix is minus. According to the recent constraints on the LV parameters from neutrino oscillation\nexperiments[33, 35], the energy-scale of the amatrix can take 10−24GeV at the 95% confidence\nlimit. Hence, employing the global fit data on the squared mass differences of neutrinos [36], we\ncan see that HLVbecomes dominant in Hwhen the neutrino energy is of order TeV. Moreover,\neven if the a(c) parameters are as tiny as 10−28GeV (10−31),HLVis still significant when Eν3\nreaches PeV. Accordingly, the ¯ νeflux leading to the GR events is sensitive to the LV effect which\ncannot be detected by the neutrino oscillation experiments at low energies.\nIn the light of the realistic situation of neutrino astronomy, to obtain information on LV from\nthe GR events, two troublesome issues should be addressed. First, although the first promising\npoint-source of HANs, namely the blazar TXS0506+056, was identified at the 3 σlevel[37], the\ngenuine discovery of the origin of HANs is still in lack at present epoch. Thus, the original\nflavor composition of the HANs at the source is totally unknown. For the sake of undermining\nthis problem, two typical sources, namely the pion ( π±) decay source, the muon ( µ±) damped\nsource, are considered in this paper. Second, the fitted energy spectra of HANs by now have large\nuncertainties. Several models, such as the single power-law, the double power-law, the power-law\nwith cut-off, etc, can coordinate the observed events of HANs[4, 38]. Furthermore, the constraints\non the parameters of the spectra are loose. Thus, moderate LV effects on GR events may be\nconcealed by the uncertainties of the energy spectra. To overcome this issue, we propose the\nmatrix HLVof a special texture which can bring the decoupling of νµ(¯νµ). The GR events from\nthe typical sources are damped by the decoupling effect. Consequently, stringent constraints on\nthe LV Hamiltonian and the production source of HANs may be obtained.\nIn the following section, the specific impacts of the LV on the flavor transition of HANs, the\nflux of ¯ νe, and the strength of the GR event rate are shown. Finally, conclusions are given.\nII. GLASHOW RESONANCE IMPACTED BY LORENTZ VIOLATION\nA. The flavor transition of HANs and the texture of HLV\nThe coherence of neutrino mass eigenstates has disappeared after HANs travelled a cosmological\ndistance. Hence, the averaged neutrino oscillation probability is written as\nPs\nαβ=X\ni|Uαi|2|Uβi|2, (3)\nwhere α, β=e, µ, τ ,i= 1,2,3. This is the so-called standard flavor transition probability. When\nthe LV is taken into account, Ps\nαβis changed into Pαβwhich reads\nPαβ=X\ni|UN\nαi|2|UN\nβi|2. (4)\nThe new mixing matrix UNis obtained by the diagonalization of the total Hamiltonian H=\nH0+HLV. The eigenvalues of Hare written as[33]\nEi=−2p\nQcos(θi\n3)−a\n3, i= 1,2,3. (5)4\nThe components are expressed as follows[33]:\nQ=a2−3b\n9, (6)\nθ1= arccos( RQ−3\n2), θ2=θ1+ 2π, θ 3=θ1−2π, (7)\na=−Tr(H), b=Tr(H)2−Tr(H2)\n2, c=−det(H), (8)\nR=2a3−9ab+ 27c\n54, (9)\nwhere the notations Tr and det denote the trace and determinant respectively. Accordingly, the\nelements of UNare written as [33]\nUN\nei=B∗\niCi\nNi, UN\nµi=AiCi\nNi, UN\nτi=AiBi\nNi, (10)\nin which\nAi=Hµτ(Hee−Ei)−HµeHeτ, (11)\nBi=Hτe(Hµµ−Ei)−HτµHµe, (12)\nCi=Hµe(Hττ−Ei)−HµτHτe, (13)\nN2\ni=|AiBi|2+|AiCi|2+|BiCi|2. (14)\nGiven the analytical expression of UNand the specific LV parameters, we can predict how the\nflavor ratio at the source is changed into that at Earth. However, the issue of the flavor transition\nincluding the LV effects is more subtle. On the one hand, since the probability Pαβis dependent\non both the magnitude and the texture of the matrix HLV, too many patterns of Pcan be obtained\nfrom the LV parameters in the allowed ranges shown in Tab.I. On the other hand, the experimental\nconstraints on the flavor ratio of HANs at the PeV energy-scale are loose at present. In order to\ngive robust observations on the impacts of the LV on the GR events, we propose a special pattern\nwith the decoupling of νµ(¯νµ), i.e.,\nP∼\n1\n201\n2\n0 1 0\n1\n201\n2\n. (15)5\nTABLE I. Constraints on the Lorentz violation parameters [33].\nLV Parameters 95% confidence limit Best fit\nRe(aT\neµ) 1 .8×10−23GeV 1 .0×10−23GeV\nIm(aT\neµ) 1 .8×10−23GeV 4 .6×10−24GeV\nRe(cTT\neµ) 8 .0×10−271.0×10−28\nIm(cTT\neµ) 8 .0×10−271.0×10−28\nRe(aT\neτ) 4 .1×10−23GeV 2 .2×10−24GeV\nIm(aT\neτ) 2 .8×10−23GeV 1 .0×10−28GeV\nRe(cTT\neτ) 9 .3×10−251.0×10−28\nIm(cTT\neτ) 1 .0×10−243.5×10−25\nRe(aT\nµτ) 6 .5×10−24GeV 3 .2×10−24GeV\nIm(aT\nµτ) 5 .1×10−24GeV 1 .0×10−28GeV\nRe(cTT\nµτ) 4 .4×10−271.0×10−28\nIm(cTT\nµτ) 4 .2×10−277.5×10−28\nThe decoupling pattern can be obtained from the Hamiltonian HLVdominated by the parameter\n(HLV)eτ, namely |HLV|eτ≫ |HLV|αβwith α(β)̸=e, τ. According to the expression of HLV\nshown in Eq.2, we know that the texture of HLVis dependent on the neutrino energy Eν. For\nthe antineutrinos (neutrinos) with Eν≥1TeV, the decoupling texture can be realised with the LV\nparameters in the allowed ranges in Tab.I. The specific decoupling energy-scale Edis determined by\nthe magnitudes of the LV parameters. Here we consider two representative decoupling energy-scales\nfor antineutrinos (neutrinos). For the LV parameters aT\nαβ= 10−3(aT\nαβ)bf,CTT\nαβ= 10−2(cTT\nαβ)bf,\nwe have Ed1∼5TeV. For the parameters aT\nαβ= 10−7(aT\nαβ)bf,CTT\nαβ= 10−7(cTT\nαβ)bf, we obtain\nEd2∼1PeV. The dependence of PαβonEνis shown in Fig.1. In the first case, the behavior of\nPαβis simple. When Eν> E d1,Pαβshows the decoupling pattern. In the energy range [1TeV,\n5TeV], a small variation is added to the pattern. In the second case, the energy Eνcan be classified\ninto 3 areas. In the range [1TeV, 10TeV], H0is dominant in H, namely Pαβ∼Psαβ. In the range\n[10TeV, 1PeV], the impact of HLVis noticeable. The variation of Pαβwith Eνis steep, especially in\nthe range [100TeV, 1PeV]. In the range Eν> E d2,Pαβis also converged to the decoupling pattern.\nGeneral speaking, we can see that tiny LV parameters may bring an interesting phenomenology of\nthe flavor transition of HANs, which hence impacts the flux of ¯ νe.6\n1000 1041051061071080.400.450.500.55\nEν[GeV]Pee\n1000 1041051061071080.40.60.81.0\nEν[GeV]Pμμ\nFIG. 1. The flavor transition probability of antineutrinos. Pαβis obtained from the leptonic mixing param-\neters at the 3 σlevel of the global fit data with normal mass ordering(NO)[36] and the LV parameters. For\nblue lines, aT\nαβ= 10−7(aT\nαβ)bf,cTT\nαβ= 10−7(cTT\nαβ)bf. For red lines, aT\nαβ= 10−3(aT\nαβ)bf,cTT\nαβ= 10−2(cTT\nαβ)bf.\n(aT\nαβ)bf, (cTT\nαβ)bfare the best fit values listed in Tab.I. Black line: Psαβfrom the best fit values of the global\nfit data with NO[36].\nB. The energy spectrum of ¯νemodified by the LV\nAs has been shown, the LV resulting in the νµ(¯νµ) decoupling has significant impacts on the\nflavor transition of HANs. Thus, the energy spectrum of ¯ νemay be modified by the LV effect. In\naddition to HLV, the flux of ¯ νeat Earth is also dependent on the production source of HANS. Since\nthe source of ¯ νeis uncertain at present, here we consider two typical sources to set the original\nflavor composition. They are listed as follows: the pion ( π±) decay source with the original flavor\nratio\nRs(π±)= (1/3,2/3,0), (16)\nand the muon ( µ±) damped source with the ratio\nRs(µ±)= (0,1,0). (17)\nFor the both sources, the flux of neutrinos is equal to that of antineutrinos. Employing the flavor\nratio at the source and the transition matrix P, we can derive the ratio of ¯ νeat Earth, namely\nre(Eν) =ϕ¯νe(Eν)\nϕν+¯ν(Eν)=1\n2X\nα=e,µ,τPeα(Eν)Rs\nα, (18)\nin which ϕ¯νeandϕν+¯νare the differentia flux of ¯ νeand that of total HANs respectively, Rs\nαis\nthe component of the flavor ratio vector at the source. Accordingly, the differentia flux of ¯ νeis7\nTABLE II. Energy-spectrum parameters of HANs at the 68.3% confidence level[38].\nModel normalization spectral index\nsingle power-law ϕastro γastro\n5.68+1.56\n−1.55 2.89+0.23\n−0.20\ndouble power-law ϕhard ϕsoft γhard γsoft\n3.26+1.88\n−2.44 0.08+3.64\n−0.08 2.78+0.24\n−0.32 3.12+1.01\n−0.30\npredicted to be\nϕ¯νe=1\n2ϕν+¯ν(Eν)×(X\nα=e,µ,τPeα(Eν)Rs\nα). (19)\nAt present, the uncertainty of ϕν+¯νis large. Several models of the energy spectra can work,\ne.g., the single power-law, double power-law, single power law with spectral cutoff, log-parabola,\nsegmented power-law. For the sake of illustration, we consider the single power-law and the double\npower-law spectrum in this paper. They are expressed respectively as[38]\nϕspl\nν+¯ν=ϕastro×(Eν\n100TeV)−γastro·10−18GeV−1cm−2s−1sr−1, (20)\nϕdpl\nν+¯ν= (ϕhard×(Eν\n100TeV)−γhard+ϕsoft×(Eν\n100TeV)−γsoft)·10−18GeV−1cm−2s−1sr−1.(21)\nThe ranges of the normalization and the spectral index at the 68.3% confidence level are listed in\nTab.II. Using these parameters, we show the LV effect on the energy spectrum of ¯ νein Fig.2 and\nFig.3.\nFor the muon ( µ±) damped source (see Fig.3), the LV can bring a noticeable damping of the\nflux of ¯ νewhen Eν≥Ed. The damping rate ϕLV\n¯νe/ϕS\n¯νeis of order [10−6, 10−4] at the resonance\nenergy. Here ϕLV\n¯νeandϕS\n¯νedenote the flux including the LV and that without the LV respectively.\nThe specific damping value is dependent on the form of ϕν+¯ν. For the pion ( π±) decay source (see\nFig.2), the damping effect is moderate and concealed by the large uncertainty of ϕν+¯ν, irrespective\nof the magnitude of the decoupling energy. In general, we can expect that the LV effect can lead\nto a noticeable decrease of the ¯ νeflux around the resonance energy for a source dominated by ¯ νµ\n(νµ). Thus, the detection of GR events may set stringent constraints on the texture of HLVand\nthe original flavor ratio.8\n1000 10410510610710810-1210-1110-1010-910-810-710-6\nEν[GeV]Eν2ϕνe_[GeVcm-2sr-1s-1]\n1000 10410510610710810-1110-910-710-5\nEν[GeV]Eν2ϕνe_[GeVcm-2sr-1s-1]\n1000 10410510610710810-1210-1110-1010-910-810-710-6\nEν[GeV]Eν2ϕνe_[GeVcm-2sr-1s-1]\n1000 10410510610710810-1110-910-710-5\nEν[GeV]Eν2ϕνe_[GeVcm-2sr-1s-1]\nFIG. 2. The energy spectrum of ¯ νefrom the pion ( π±) decay source. Left plots: from the total flux of the\nsingle power-law. Right plots: from the total flux of the double power-law. The ranges of the parameters of\nthe energy spectrum ϕν+¯νare taken as those listed in Tab.II. Blue lines: arising from Pswithout the LV\neffects. Red lines: from Pwith the LV effects. For PsandP, the leptonic mixing parameters are in the\n3σallowed ranges of the global fit data with NO[36]. The plots in the first row: with the LV parameters\ntaken as aT\nαβ= 10−3(aT\nαβ)bf,cTT\nαβ= 10−2(cTT\nαβ)bf. The plots in the second row: aT\nαβ= 10−7(aT\nαβ)bf,\ncTT\nαβ= 10−7(cTT\nαβ)bf. The black line: arising from Pwith the best fit values of the energy spectrum ϕν+¯ν\nand leptonic mixing parameters. The dashed black line: from Pswith the best fit values of the leptonic\nmixing parameters.\nC. The strength of the GR event rate damped by the LV\nThe GR events provide a window to explore the texture of HLVand the flavor composition of\nthe HANs at the PeV energy-scale. Now let us examine the impacts of the decoupling of νµ(¯νµ)\non the GR event rate. To weaken the influence of the uncertain energy spectrum ϕν+¯ν, for down-\ncoming events, the ratio of the resonance events to the nonresonant continuum events is proposed9\n1000 10410510610710810-1410-1110-8\nEν[GeV]Eν2ϕνe_[GeVcm-2sr-1s-1]\n1000 10410510610710810-1510-1310-1110-910-710-5\nEν[GeV]Eν2ϕνe_[GeVcm-2sr-1s-1]\n1000 10410510610710810-1410-1110-8\nEν[GeV]Eν2ϕνe_[GeVcm-2sr-1s-1]\n1000 10410510610710810-1510-1210-910-6\nEν[GeV]Eν2ϕνe_[GeVcm-2sr-1s-1]\nFIG. 3. The energy spectrum of ¯ νefrom the muon ( µ±) damped source. The conventions of the parameters,\ncolors and the lines are the same as those in Fig.2.\nto describe the strength of the GR event rate, i.e.[2],\nNRes\nNnon−Res(Eν> Eminν)=10π\n18(ΓW\nMW)(σpeak\nRes\nσCC\nνN(Eν= 6.3PeV))(α−1.4)(Emin\nν\n6.3PeV)α−1.4\n[1−(Eminν\nEmaxν)(α−1.4)]×[re]Eν=6.3PeV\n= 11×(α−1.4)(Emin\nν\n6.3PeV)α−1.4\n[1−(Eminν\nEmaxν)(α−1.4)]×[re]Eν=6.3PeV.\n(22)\nHere the spectral index αis dependent on the acceleration mechanism of neutrinos. In absence of\nGR events, α≥2.3[39, 40]. In this paper, we take α= 2. We note that a moderate variation of α\nwill not lead to a noticeable change of the observation on the LV effect. Employing the best fit values\nof the leptonic mixing parameters with NO[36] and the LV parameters aT\nαβ= 10−3(aT\nαβ)bf,cTT\nαβ=\n10−2(cTT\nαβ)bfwith ( aT\nαβ)bf, (cTT\nαβ)bflisted in Tab.I, we calculate the ratio of resonance event rate\naround the resonance energy. The specific values are listed in Tab.III. For the sake of comparison,\nthe parenthetic values obtained from the standard flavor transition matrix Psare also shown.\nFrom Tab.III, we can see that the ratio of resonance event rate to the nonresonant event rate is\nmainly determined by the ratio of ¯ νeat the resonance energy-scale. The damping rate of the GR10\nTABLE III. The ratio of resonance event rate to nonresonant event rate with and without the LV effect for\nthe pion ( π±) decay source, muon ( µ±) damped source, and a source rich in νµ(¯νµ) with ϕν=ϕ¯ν. Here\nα= 2, Emin\nν=1, 2, 3, 4, 5 PeV, Emax\nν=∞,retakes the value [ re]Eν=6.3PeV. The parenthetic values are\nobtained from the standard flavor transition matrix Ps.\nEmin\nν(PeV) 1 2 3 4 5\nRs(π±)= (1/3,2/3,0) 0.18 0.28 0.35 0.42 0.48 re=0.083\n(0.32) (0.50) (0.63) (0.75) (0.86) ( re=0.149)\nRs(µ±)= (0,1,0) 7 .8×10−61.2×10−51.5×10−51.8×10−52.0×10−5re= 3.5×10−6\n(0.19) (0.28) (0.36) (0.43) (0.49) ( re=0.086)\nRs(x)= (0.1,0.9,0) 0.05 0.08 0.10 0.12 0.14 re=0.025\n(0.23) (0.35) (0.44) (0.53) (0.60) ( re=0.105)\nevent rate approximates that of [ re]Eν=6.3PeV, namely\n(NRes\nNnon−Res(Eν> Eminν))LV/(NRes\nNnon−Res(Eν> Eminν))S∼([re]Eν=6.3PeV)LV/([re]Eν=6.3PeV)S,\n(23)\nin which the superscript ’LV’ (’S’) denotes the value with (without) the LV effect. For the original\nflavor ratios Rs(π±),Rs(µ±), and Rs(x), ([re]Eν=6.3PeV)LV/([re]Eν=6.3PeV)Stakes 0.557, 4 .07×10−5,\n0.238, respectively. The damping rate is sensitive to the increase of the original ratio of ¯ νµaround\nthe resonance energy when ϕ¯νµ/ϕ¯ν>0.9. Furthermore, when the decoupling energy is around\n1PeV, i.e., aT\nαβ= 10−7(aT\nαβ)bf,cTT\nαβ= 10−7(cTT\nαβ)bf, the values of ([ re]Eν=6.3PeV)LVare similar to\nthose listed in Tab.III. Thus, the damping rate of the GR event rate is mainly sensitive to the\ntexture of HLV, irrespective the magnitudes of the LV parameters. However, we should keep in\nmind that the magnitudes of the parameters determine the decoupling energy which impacts the\nevolution of Pwith Eν. In the range Eν< E d, the flavor ratio of HANs at Earth is dependent on\nthe magnitudes of the LV parameters.\nD. Comments on the realistic source and the ντevent at the PeV energy-scale\nAt present, the production source and the acceleration mechanism of HANs are unknown. How-\never, it is widely believed that the strong magnetic field plays an important role in the acceleration\nof HANs[41], e.g. acceleration by the magnetic reconnection[42–44]. As is known, the synchrotron\nradiation of the charged particle in the strong magnetic field becomes significant at high energies.\nWe can expect that the muon ( µ±) damped source may be dominant at the PeV energy-scale.11\nAccordingly, when Ed<6.3PeV, the decoupling pattern can show a noticeable impact on the GR\nevents. Furthermore, we note that the decoupling pattern can bring the following observation:\nϕνe∼ϕντ, ϕ ¯νe∼ϕ¯ντ. (24)\nHence, for the muon ( µ±) damped source, the proposed LV texture can lead to a notable decrease\nof both the GR event rate and the double bang events from ντat the PeV energy-scale. In other\nwords, we can obtain the double constraints on the texture of HLVfrom the GR event and the\ndouble bang event.\nIII. CONCLUSIONS\nThe Lorentz invariance is a fundamental symmetry in nature. Strong constraints on the LV\nparameters have been obtained at low energies. To connect the LV to the GR event at the PeV\nenergy-scale, we proposed the LV Hamiltonian of the special texture which can bring the νµ(¯νµ)\ndecoupling at high energies. On the base of the νµ(¯νµ) decoupling, we analysed the impacts of the\nLV parameters on the flavor transition of HANs. For the typical sources of HANs, the damping\neffect from the decoupling pattern on the energy spectrum of ¯ νewas shown. For the muon ( µ±)\ndamped source, the decrease of the ¯ νeflux at the PeV energy-scale is notable, irrespective of the\nform of the total flux ϕν+¯ν. The LV effect can lead to a noticeable damping of the GR event rate if\nthe HANs are mainly from the source dominated by ¯ νµ(νµ). 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The α-particle coefficients would lead to anisotropies in the α-decays\nof nuclei, and because the decay process involves quantum tunnelin g, the effects of any\nLorentz violations could be exponentially enhanced.\n1baltschu@physics.sc.edu1 Introduction\nAt present, there exists a great deal of interest in the possibility t hat Lorentz and CPT\ninvariance may not be exact in nature. If the fundamental physica l laws do not respect\nthese symmetries, then we could expect to see evidence of Lorent z and CPT violations\neven in the effective theory that governs conventional low-energ y phenomena. If small\nviolations of these symmetries were discovered, this would provide a critically important\npiece of information about the fundamental structure of physics and a clue to what other\nnew effects we could expect to see. There is a parameterization of L orentz and CPT\nviolations in low-energy effective field theory, known as the standar d model extension\n(SME), which contains possible Lorentz- and CPT-violating correct ions to the standard\nmodel [1, 2] and general relativity [3]. Both the renormalizability [4, 5] and stability [6]\nof the SME have been carefully examined.\nThe SME provides a useful framework for interpreting experiment al tests of Lorentz\nand CPT symmetry. Sensitive searches for Lorentz violation have in cluded studies of\nmatter-antimatter asymmetries for trapped charged particles [1 3, 14, 15] and bound\nstate systems [16, 17], determinations of muon properties [18, 19], analyses of the be-\nhavior of spin-polarized matter [20, 21], frequency standard comp arisons [22, 23, 24, 25],\nMichelson-Morley experiments with cryogenic resonators [26, 27, 2 8], Doppler effect mea-\nsurements [29, 30], measurements of neutral mesons [31, 32, 33, 34, 35, 36], polarization\nmeasurements on the light from distant galaxies [37, 38, 39, 40], high -energy astrophysical\ntests [41, 42, 43, 44] and others. The results of these experimen ts set bounds on various\nSME coefficients. Up-to-date information about bounds on the SME coefficients may\nbe found in [45]; at the present time, many of the SME coefficients are quite strongly\nconstrained, but many others are not.\nThere are many systems and reaction processes that could poten tially be used to\nset further bounds on the SME’s coefficients for Lorentz violation. We shall consider a\nparticular system—the helium nucleus, or α-particle—because of the insights it provides\ninto the general properties of Lorentz-violating physics.\nOne challenging problem in the study of Lorentz violation is understan ding Lorentz\nviolation for composite particles. The fundamental fields in the SME a re the gauge\nfields, leptons, quarks, and the Higgs. However, bounds on Loren tz violation are usually\nformulated in terms of bounds on SME coefficients for hadrons, rat her than the more\nfundamental quark and gluon coefficients. The reason for this is ob vious; hadrons are\nthe physical excitation of the strongly interacting fields. However , they have a nontrivial\nstructure, and the relationship between the quark, gluon, and ha dron coefficients is not\nentirely clear.\nThereareverypreciseboundsonmanyLorentzviolationcoefficient sfortheprotonand\nneutron. However, most measurements of these coefficients are made not on free nucleons\nbut on more complicated nuclei. Some model must be used to relate th e possible effects of\nLorentzviolationinthenuclear system tothecoefficients forindividu al nucleons. This isa\n1simplerproblemthanrelatinghadroncoefficientstothecoefficientsf orsubnuclear partons.\nIt is often reasonable to take a simplified model, such as the Schmidt m odel [46], which\nassigns all the angular momentum of nucleus to a single unpaired nucle on constituent.\nHowever, this model obviously entails a great deal of idealization, an d we would like to\nunderstand the nature of Lorentz violation in nuclei and other com posite particles more\nfully.\nIn this paper, we shall consider one of the simplest composite partic les: the helium\nnucleus, containing two protons and two neutrons. The simple close d shell structure of\nthis nucleus ensures that many spin-dependent coefficients must v anish. Yet there is still\nsome nontrivial structure which we shall uncover. We shall then ex amine the possible\nimpact of Lorentz violation on the α-decays of nuclei.\nThere are several motivations for this work. First, it presents a n ew calculation of the\neffective Lorentz violation coefficients for a composite particle, in te rms of the coefficients\nfor its constituents. Second, it examines how the effects of Loren tz violation can affect\nthe intrinsically quantum mechanical phenomenon of tunneling. Finally , it suggests a\nnew method for placing laboratory bounds on a number of Lorentz v iolation coefficients\nfor protons and neutrons. Although the likely constraints are not particularly tight,\nthis method offers a way to constrain several coefficients that hav e not previously been\nbounded by laboratory experiments.\nThispaperisorganizedasfollows. Section2discusses theLorentzv iolationcoefficients\nrelevant for α-particles and how these are related to the coefficients for the con stituent\nprotonsand neutrons. Then Section 3 shows how these coefficient s could impact the phys-\nical process of α-decay. In Section 4 we look quantitatively at how α-decay studies could\nbe used to place bounds on a number of Lorentz violation coefficients and in Section 5\npresent our conclusions.\n2 Lorentz Violation for α-Particles\nIn this section, we shall derive the effective Hamiltonian for an α-particle, including the\neffects of Lorentz symmetry violations in the proton and neutron s ectors. In both this\nsection and the next, some approximations will be necessary in orde r to calculate the\neffects of the Lorentz violation on α-decays. However, the results will be at least semi-\nquantitative and good enough to place order of magnitude bounds.\nThe minimal SME Lagrange density of a single species of fermion is\nLf=¯ψ(iΓµ∂µ−M)ψ, (1)\nwhere\nM=m+/negationslasha−/negationslashbγ5+1\n2Hµνσµν+im5γ5, (2)\nand\nΓµ=γµ+cνµγν−dνµγνγ5+eµ+ifµγ5+1\n2gλνµσλν. (3)\n2Electromagnetic interactions are introduced via the minimal coupling substitution pµ→\npµ−qAµ, whereqis the charge. A nonrelativistic Hamiltonian Hfmay be derived from\n(1) using a Foldy-Wouthuysen transformation [47]; this effective H amiltonian is [48]\nHf=p2\n2m+/bracketleftbigg\nm/parenleftbigg\n−cjk−1\n2c00δjk/parenrightbigg/bracketrightbiggpjpk\nm2\n+/bracketleftbigg/parenleftbigg\n−bj+mdj0−1\n2mǫjklgkl0+1\n2ǫjklHkl/parenrightbigg\nσj−aj−m(c0j+cj0)+mej/bracketrightbiggpj\nm\n−/bracketleftbigg\nb0δjk−m(dkj+d00δjk)−mǫklm/parenleftbigg1\n2gmlj+gm00δjl/parenrightbigg\n−ǫjklHl0/bracketrightbiggpjσk\nm\n+/braceleftbigg/bracketleftbigg\nm(d0j+dj0)−1\n2/parenleftbigg\nbj+mdj0+1\n2mǫjmngmn0+1\n2ǫjmnHmn/parenrightbigg/bracketrightbigg\nδkl\n+1\n2/parenleftbigg\nbl+1\n2mǫlmngmn0/parenrightbigg\nδjk−mǫjlm(gm0k+gmk0)/bracerightbiggpjpkσl\nm2. (4)\nThis is the free nonrelativistic SME Hamiltonian for a single proton or ne utron, to leading\norder in the SME coefficients.\nMany of the terms in (4) will not contribute when two protons and tw o neutrons are\ncombined to forman α-particle. The effective Hamiltonian for the composite particle, Hα,\ninvolves a sum of four HfHamiltonians—one for each of the nucleon constituents—plus\nadditional contributions due to nuclear binding effects. We shall not consider the binding\nindetail; it’srolewill simply betoensure thatthefournucleons followes sentially identical\nspacetime trajectories. Moreover, to an excellent approximation , the two protons and two\nneutrons in this nucleus are each separately in a spin singlet state.\nSo the operator to create an α-particle at /vector xisa†(/vector x) = 2b†\np↑(/vector x)b†\np↓(/vector x)b†\nn↑(/vector x)b†\np↓(/vector x), where\nb†\np↑(orb†\nn↓) is the creation operator for a spin up proton (or spin down neutro n). Because\nof the anticommutivity of the fermion operators,\nb†\n↑(/vector x)b†\n↓(/vector x) =1\n2/bracketleftBig\nb†\n↑(/vector x1)b†\n↓(/vector x2)−b†\n↑(/vector x2)b†\n↓(/vector x1)/bracketrightBig/vextendsingle/vextendsingle/vextendsingle\n/vector x1=/vector x2=/vector x, (5)\nanda†produces an excitation with the total proton spin and total neutr on spin both\nequal to zero.\nEach of the two protons carries the same mechanical momentum an d likewise for the\ntwo neutrons. (Since the proton and neutron masses are very sim ilar, each nucleon carries\napproximately one-fourth of the total momentum of the α-particle.) Consequently, when\nthe spin-dependent terms in each Hfare added together, they give contributions to Hα\nwhich depend only on /vector σp1+/vector σp2and/vector σn1+/vector σn2—the total proton and neutron spin vectors\nin the nucleus. But in the spin-0 singlet state, both of these operat ors vanish identically.\nTherefore, none of the spin dependent terms in Hfwill contribute to Hα.\n3So we may neglect all spin-dependent terms in Hf. Doing this, we are left with a\nHamiltonian equivalent to\nHf≃p2\n2m−/parenleftbigg\ncjk+1\n2c00δjk/parenrightbiggpjpk\nm−[aj+m(c0j+cj0)−mej]pj\nm. (6)\nHowever, it is known that the aµparameters are unobservable, except in interactions that\ninvolve gravitation or flavor changing interactions [49]. The field red efinition\nψ→e−ia·xψ,¯ψ→eia·x(7)\neliminates afrom the Lagrangian (1) entirely. This is equivalent to a translation in\nmomentum space, pµ→pµ−aµ. In a theory with aonly, the shifted /vector pis the correct me-\nchanical momentum γm/vector v; including ain the action merely corresponds to a poor choice\nof canonical momentum. Since neither gravity nor the weak interac tion are involved with\nα-decay,awill not contribute to any observable quantity in this kind of decay pr ocess.\nMoreover, any coefficients that enter in the same manner as amust also prove unob-\nservable; the whole expression [ aj−m(c0j+cj0)−mej]pj\nmcannot affect α-decay physics.\nNeglecting this term, we are left with the final fermion Hamiltonian rele vant toα-decay,\nHf≃p2\n2m−/bracketleftbig\nc(jk)+c00δjk/bracketrightbigpjpk\n2m, (8)\nwherec(jk)=cjk+ckj. Only the cterms in the fermion sector can affect α-decays, and\nthe remaining terms in the effective Hamiltonian are all separately inva riant under C, P,\nand T.\nThec00term is not observable in solely nonrelativistic experiments; it merely c hanges\nthe effective value of m. However, we shall retain it, and its effects can be observed by\ncomparing the kinetic energy of a nonrelativistically moving particle wit h the mass energy\nmobserved in particle creation or annihilation processes.\nTo find a useful effective Hamiltonian for the α-particle, we must take the sum of four\nHfterms and then perform a canonical transformation. The transf ormation will separate\nthe center of mass motion from the relative motions of the four nuc leons, and it is the\ncenter of mass motion that determines the motion of the α-particle.\nTo see how the center of mass Hamiltonian is modified by c-type Lorentz violation for\nthe constituents, it is simpler to first consider the case of a two-pa rticle bound state. The\nproblem can be further simplified by considering a situation in which the re is Lorentz\nviolation in the Hamiltonian for only one of the two constituents. Once this simple\nexample has been worked out, the generalization to more constitue nt particles and more\nsources of Lorentz violation is straightforward.\nFor the two-particle example, the Hamiltonian is\nH=p2\n1\n2m1+p2\n2\n2m2−/bracketleftbig\nc(jk)+c00δjk/bracketrightbigp2jp2k\n2m2. (9)\n4Since the term in brackets in (9) is symmetric in ( jk) and represents a small correction,\nwe may diagonalize it by a rotation, choosing spatial coordinates in wh ich the kinetic\nenergy for particle 2 has no off-diagonal pjpkterms. In these coordinates, the kinetic\nenergy splits, as it conventionally does, into three pieces, each of t he form\nHj=p2\n1j\n2m1+/bracketleftbig\n1−c(jj)−c00/bracketrightbigp2\n2j\n2m2. (10)\nIn (10) and for the remainder of this paragraph, jrepresents a specific coordinate and is\nnot to be summed over. Hjdescribes one-dimensional dynamics equivalent to those for\ntwo particles of masses m1andm′\n2=m2//bracketleftbig\n1−c(jj)−c00/bracketrightbig\n. The effective center of mass\ncoordinate is Rj=m1r1j+m′\n2r2j\nm1+m′\n2, and the conjugate momentum is the total momentum\nPj=p1j+p2j. The center of mass part of the Hamiltonian then becomes P2\nj/2(m1+m′\n2).\nIncluding the dynamics in all three directions, the total center of m ass Hamiltonian is\ntherefore\nHCM=1\n2(m1+m2)/braceleftbigg\nδjk−m2\nm1+m2/bracketleftbig\nc(jk)+c00δjk/bracketrightbig/bracerightbigg\nPjPk (11)\n(once again summing over j). The results of the previous paragraph dictate this this\nformula holds in coordinates for which c(jk)is diagonal; and since (11) is written in a\ntensor form, it must also hold in the original, unrotated coordinate s ystem.\nThegeneralizationtomoreLorentz-violatingparticlesisstraightfo rward. Thecenterof\nmass Hamiltonian will contain a mass-weighted sum of the Lorentz viola tion coefficients\nfor the constituents. The weights represent the fraction of the momentum carried by\nvarious constituents, and the sum is the effective Lorentz violation coefficient for the\ncomposite particle. For the α-particle, we find\nHα=1\n4(mp+mn)/braceleftbigg\nδjk−mp\nmp+mn/bracketleftBig\ncp\n(jk)+cp\n00δjk/bracketrightBig\n−mn\nmp+mn/bracketleftbig\ncn\n(jk)+cn\n00δjk/bracketrightbig/bracerightbigg\nPjPk.\n(12)\nThis does not yet include the effects of nuclear binding. As a first app roximation, we may\ntreat the binding energy −ǫ=mα−2mp−2mnas if it were divided up equally among\nthe four constituent hadrons. This means replacing mpandmninHαwithmp−ǫ/4 and\nmn−ǫ/4, respectively. If we also neglect the neutron-proton mass differ ence, the effective\nHamiltonian for a free α-particle becomes\nHα=1\n2mα/parenleftbigg\nδjk−1\n2kα\njk/parenrightbigg\nPjPk, (13)\nwithkα\njk=cp\njk+cn\njk+(cp\n00+cn\n00)δjk.\nThekα\njkcoefficients are nonrelativistic versions of the Lorentz violation coe fficientskµν\nfor a scalar field. In the presence of kµν, the Lagrange density for a free relativistic scalar\nis\nLφ=1\n2(∂µφ)(∂µφ)+1\n2kµν(∂νφ)(∂µφ)−m2\n2φ2. (14)\n5However, the coefficients k0jork00are intrinsically relativistic. They control violations\nof boost invariance, and they cannot be measured in a purely nonre lativistic experiment.\nYet the relationship between cn\njk,cp\njk, andkα\njkmust be the same in every Lorentz frame,\nsince we may always repeat the preceding analysis with an α-particle that is moving\nnonrelativistically in the desired frame. In order for this to hold, the full effective Lorentz\nviolation coefficient for the α-particle must be kα\nµν=cp\nνµ+cn\nνµ. (The change in the\nplacement of c00terms just corresponds to a change in the overall normalization of the\nfield and has no physical meaning.) For a more general spin-0 bound s tate, the coefficient\nkµνwillbeamass-weighted sumoftheconstituents’ cνµcoefficients, liketheoneappearing\nin (12).\nWhen each nucleon is minimally coupled to the electromagnetic field by th e replace-\nment/vector p→/vector p−q/vectorAandHf→Hf−qA0, the total momentum and α-particle Hamiltonian\ntransform as /vectorP→/vectorP+ 2e/vectorAandHα→Hα+ 2eA0, wheree=−|e|is the charge of\nthe electron. For nonrelativistic motion in the absence of an extern al magnetic field, the\ncontribution of the vector potential /vectorAmay be neglected. There can also be interactions\nbetween the α-particle and a strong interaction potential Vs; when included, this gives\nthe final interacting Hamiltonian\nHα=1\n2mα/parenleftbigg\nδjk−1\n2kα\njk/parenrightbigg\nPjPk+2eA0(/vector x)+Vs(/vector x). (15)\n3 Calculating α-Decay Rates\nWe shall now proceed to a calculation of how Lorentz violation could imp act theα-decay\nof a nucleus. There are three major places in which Lorentz violation can enter the decay\nprocess. These can be loosely related to three constituents of th e interaction: the attrac-\ntive interaction in the parent nucleus, the expelled α-particle, and the electromagnetic\ninteraction between the α-particle and the daughter nucleus.\nThe decay process can be understood as the α-particle tunneling from the deep attrac-\ntive potential well of the nucleus, through the Coulomb barrier, an d escaping to infinity.\nWhen theα-particle energy is small compared with the height of the barrier, th e details\nof the potential inside the nucleus are fairly unimportant. We shall t herefore neglect the\neffects of Lorentz violations on the nuclear potential. In fact, if th e Lorentz violations in\nvarious sectors of the SME are comparable in magnitude (as natura lness conditions would\nsuggest), the dominant Lorentz-violating contribution to the dec ay rate Γ will arise from\none particular factor in our calculation. The exponential part of th e barrier penetration\nfactor depends much more strongly on the Lorentz violation param eters than any other\nelement in the formula for Γ; and the stronger dependence of this f actor onkαis most\npronounced when the energy of the decay is smallest compared with the height and width\nof the Coulomb barrier.\nWe shall examine the various factors that together compose the d ecay rate below.\n6However, we must first discuss the question of Lorentz violation in t he electromagnetic\nsector, since electromagnetic Lorentz violation effects do not bec ome negligible compared\nwith the effects of kαin any kinematical limit. There is one form of spin-independent\nLorentz violation like corkαfor each particle species in the standard model. Consid-\neration only of spin-independent forms is justified by a combination o f factors. Since\ntheα-particle has no spin angular momentum, it is automatic that any Lore ntz viola-\ntions in its sector must be spin independent. This straightforward g eneral argument was\nconfirmed by the calculations in Section 2. In contrast, the electro magnetic sector of\nthe SME certainly contains terms that depend on photon polarizatio n. However, these\nterms are extremely tightly constrained by astrophysical polarime try [38, 39, 40]. Any\npolarization dependence in the phase speed of light will lead to birefrin gence—a change\nin the polarization of light waves as they propagate. Birefringence w ith the right energy\ndependence to indicate Lorentz violation is not observed even in rad iation that has tra-\nversed cosmological distances. Therefore it is reasonable to negle ct these terms in the\naction, leaving behind an electromagnetic sector that is spin indepen dent. With only the\nconventional terms and spin-independent Lorentz violation, the L agrange density for the\nfree electromagnetic sector is\nLA=−1\n4FµνFµν−1\n4(kF)β\nµβν(FρµFρν+FµρFν\nρ). (16)\nThe relevant type of Lorentz violation in each sector can be parame terized by a two-\nindex symmetric tensor. However, only the differences between th ese tensors are ob-\nservable physically. A coordinate transformation can be used to elim inate this kind of\nLorentz violation entirely from one sector, at the cost of changing the coefficients in\nall other sectors. Specifically, once the birefringent terms in the e lectromagnetic sector\nare set to zero, the entire sector may be made conventional by a c oordinate redefinition\nxµ→xµ−1\n2(kF)βµ\nβνxν. Under such a redefinition, the Lorentz violation in the α-\nparticle sector transforms kα\nµν→kα\nµν−(kF)β\nµβν. It is this difference of coefficients that\nis ultimately measurable; however, for notational simplicity, we shall henceforth assume\n(kF)β\nµβν= 0.\nNow we can discuss the details of the α-decay process in the presence of kα. There are\ntwo ways of treating the tunneling process by which the α-particle escapes through the\nelectrostatic potential of the nucleus. The tunneling rate may be d etermined using either\nthepropertiesof exact Coulomb wave functionsor theWentzel-Kr amers-Brillouin-Jeffreys\n(WKBJ) approximation. In generalizing the analysis of α-decay to cover the possibility\nof Lorentz violation, we shall opt for the latter approach.\nHowever, this approach immediately runs into a problem. In the pres ence ofkα, the\nmodified Schr¨ odinger is no longer separable (in either spherical or p arabolic coordinates).\nIn order to use the WKBJ approximation, it is generally necessary to separate the vari-\nables; the WKBJ technique can then be applied to one variable at a time . So in order to\nget around this difficulty, we must introduce another approximation .\n7The new approximation is a form of the eikonal approximation. The co nventional\neikonal approximation, when applied to scattering problems, proce eds as follows. One\nexamines the linear path that a particle of a given impact parameter w ould follow in the\nabsence of interactions. When the interaction is “turned on,” a par ticle moving along this\nline would acquire a phase shift—which is just the time integral of the s cattering potential\nalong the straight trajectory. The scattering amplitude can then be calculated from this\nphase shift.\nIn the case of α-decay, a different sort of eikonal approximation is required. To ca l-\nculate the barrier penetration factor, we shall treat the ejecte dα-particle as if it were\nescaping along a well-defined linear path in the direction of its ultimate v elocity. How-\never, since the region of interest along this path is part of the class ically forbidden region,\nwe shall not be calculating a phase accrued along the path; instead w e shall calculate the\nexponential suppression of the wave function along the straight lin e. This reduces the\nthree-dimensional problem to a one-dimensional problem for the mo tion in the direction\nˆvof the velocity. The kinetic term in the α-particle Hamiltonian for the motion along\nthis axis is\nKˆv=1\n2mα/parenleftbigg\n1−1\n2kα\njkˆvjˆvk/parenrightbigg\nP2, (17)\nwherePis now the one-dimensional momentum in the relevant direction. [Actu ally, in\nthe presence of the Lorentz violation, the directions of the momen tum and the velocity\ngenerally differ; the velocity is vk=1\nmα(pk−1\n2kα\njk)pj. However, the difference between\nthe two directions only contributes a higher order correction to th e penetration factor.]\nFor motion purely in the ˆ v-direction, the effect of the Lorentz violation is to modify the\ninertial mass of the α-particle to mα/parenleftbig\n1+1\n2kα\njkˆvjˆvk/parenrightbig\n.\nNote that this use of the eikonal approximation is only straightforw ard if the nucleus\nand theα-particle are in a relative S state. However, the barrier penetratio n factors for\nL= 0 angular momentum states are always greater than for higher an gular momentum\nstates at the same energy, because for L >0 states there is an additional centrifugal\nterm in the effective potential. Because of this (and because of the related fact that the\nwave function for two particles with relative angular momentum L>0 vanishes when the\nparticles’positionscoincide), Sstatesfrequentlydominateinnucle arinteractionprocesses,\nand we shall consider only S-wave α-decays in our calculations.\nApplying the standard WKBJ technique, the barrier penetration fa ctor is then\nT=β2exp/bracketleftbigg\n−2√\n2mα/parenleftbigg\n1+1\n4kα\njkˆvjˆvk/parenrightbigg/integraldisplayrE\nrNdr/radicalbig\n2(Z−2)e2/4πr−E/bracketrightbigg\n,(18)\nwhererNandrE= (Z−2)e2/2πEare the classical turning points. rNrepresents the\nnuclear radius, at which the attractive nuclear potential binding th eα-particle to the\nnucleus comes into play; for a nucleus containing Anucleons,rN≈1.4A1/3fm. The\nescape radius rEis the beginning of the classically allowed region outside the repulsive\nCoulomb potential. The prefactor β2would be negligible for a sufficiently slowly varying\n8potential; while the exponential arises from the lowest order term in an expansion in\npowers of /planckover2pi1,βcomes from a higher order term. However, the higher order corre ction\nbecomes relevant in the region where the dominant potential chang es rapidly from being\nthe repulsive Coulomb to the attractive nuclear interaction; taking it into account, we\nfindβ2=/radicalbig\n(rE/rN)−1.\nThe integral in the exponent gives\n/integraldisplay\ndr/radicalbigg\n1\nr−1\nrE=r/radicalbigg\n1\nr−1\nrE+1\n2√rEtan−1/bracketleftBigg\n2r−rE\n2/radicalbig\nr(rE−r)/bracketrightBigg\n. (19)\nWhenEissmallcomparedwiththecharacteristicpotentialatthenucleus, (Z−2)e2/2πrN,\nthe lower limit of integration becomes relatively unimportant. This limit is equivalent to\nhavingrE≫rN, so that the width of the barrier region is large compared with the siz e\nof the nucleus. The fact that the dependence on rNin this limit is weak is fortuitous,\nsince the relevant nuclear size can only be determined approximately (and might itself be\naffected by Lorentz violation). In the rE≫rNregime, it is a reasonable approximation to\nsetrN≈0. (However, the finite size of the nucleus is an important effect in re alα-decays.\nSome of the earliest measurements of the size of certain α-emitting nuclei were actually\nbased on analyses of the finite nuclear size corrections to the deca y rate.)\nIn the energy regime for which the penetration factor is relatively in sensitive to the\nprecise value of rN, any Lorentz violation effects in the strong force that holds the pa rent\nnucleus together are of lesser importance. The most crucial feat ure of the penetration\nfactorTis that it depends on kαas\nT=β2/parenleftbiggT0\nβ2/parenrightbigg(1+1\n4kα\njkˆvjˆvk)\n, (20)\nwhereβ2T0is the penetration factor in the absence of kαLorentz violation. Tdepends\nexponentially on kα; the smaller the penetration factor is, the larger the fractional d epen-\ndence on the Lorentz violation will be.\nTaking the rN≈0 limit in the exponent, the penetration factor becomes\nT=β2exp/bracketleftBigg\n−2/radicalbigg\nmα(Z−2)e2\nπ/parenleftbigg\n1+1\n4kα\njkˆvjˆvk/parenrightbigg/parenleftBigπ\n2√rE/parenrightBig/bracketrightBigg\n(21)\n=β2exp/bracketleftbigg\n−(Z−2)e2/radicalbiggmα\n2E/parenleftbigg\n1+1\n4kα\njkˆvjˆvk/parenrightbigg/bracketrightbigg\n, (22)\nwith the well-known dependence on exp( −bE−1/2). The leading corrections to the expo-\nnential for finite rNare\nT=β2exp/braceleftBigg\n−bE−1/2/bracketleftBigg\n1−4\nπ/parenleftbiggrN\nrE/parenrightbigg1/2\n+2\n3π/parenleftbiggrN\nrE/parenrightbigg3/2/bracketrightBigg/bracerightBigg\n. (23)\n9The full decay rate is Γ =ω\n2πT, whereω\n2πis the frequency for oscillations of the\nα-particle inside the attractive nuclear potential region. The frequ ency gives the rate at\nwhichtheα-particlestrikestheinsideoftheCoulombbarrier, and Tistheprobabilitythat\nit tunnels through and escapes during a single collision. An estimate of ωisω∼1\nrN/radicalBig\nE\nmα;\nthis is approximately the inverse of the time which a particle with kinetic energyEand\nmassmαtakes to traverse a distance rN.\nNeitherβnorωdepend exponentially on the Lorentz violation coefficients. We there -\nfore expect the dominant contribution of the Lorentz violation to b e made through its\neffects on the exponential in T. Neglecting the Lorentz violation in the prefactor β2ω, the\ndecay rate may be written in the simplified form\nΓ = Γ 0exp/parenleftbigg\n−(Z−2)e2\n4/radicalbiggmα\n2Ekα\njkˆvjˆvk/parenrightbigg\n, (24)\nwhere Γ 0is the rate in the absence of Lorentz violation. In evaluating Γ 0, many of the\napproximations (such as rN≈0) that were used in our determination of the Lorentz-\nviolating correction are not necessary. In fact, Γ 0does not need to be calculated at all; it\ncan be taken from experimental data.\n4 Comparison with Experiment\nWhat (24) predicts is an anisotropy in the emission of the α-particles. The decay rate\ndepends on the emission direction ˆ v. Presuming the anisotropy is a small correction,\nΓ = Γ 0/parenleftbigg\n1−(Z−2)e2\n4/radicalbiggmα\n2Ekα\njkˆvjˆvk/parenrightbigg\n. (25)\nThe effects of Lorentz violation are enhanced by the potentially larg e parameter\n(Z−2)e2\n4/radicalbiggmα\n2E= 0.99(Z−2)/parenleftbiggE\n1MeV/parenrightbigg−1/2\n. (26)\nTo detect an anisotropy, the fractional difference between the d ecay rates in different\ndirections must be greater than the fractional error in the measu red rate due to random\nerrors. IfNcounts are collected, the signal to noise ratio goes as√\nN, and repeated\nmeasurements made on the same isotope can produce bounds on kαthat scale as N−1/2.\nTo measure an anisotropy in Γ is, in principle, straightforward. A sam ple of decaying\nmaterial should be placed inside a detector that would identify the dir ections of any out-\ngoingα-particles. This directional datawouldbeusedtosearchforeviden ce ofanisotropy.\nOf course, in a laboratory located on the Earth, the planet’s rotat ion must be accounted\nforinanyabsolutedirectional measurement. Thedirection ofeach decayinthelaboratory\n10coordinates must be re-expressed in a non-rotating coordinate s ystem. There is a partic-\nular sun-centered celestial equatorial coordinate system of this nature, in which bounds\non Lorentz-violating coefficients are conventionally expressed. Th e conversion between\nlaboratory and sun-centered coordinates is obviously dependent on the sidereal time, and\nso the time associated with each decay event must be recorded alon g with its direction.\nDetails of the sun-centered coordinates are given in [50].\nIn practice, the kind of experiment described here could be tricky. For example, it is\ndesirable to have a small physical sample, so that rescattering of α-particles produced by\ndecays well inside the sample could be minimized. The ideal α-emitter for these purposes\nis one whose half-life is well known, which has only one major decay mod e, and which has\nnuclear spin I= 0 anddecays into another I= 0 nucleus. The spin condition is important\nfor two reasons. First, having both the parent and daughter nuc lei spinless (as well as\ntheα-particle) ensures that the decay proceeds via an S-wave tunnelin g state. Second, if\nthe decaying isotope had a nonzero spin, then a slight polarization of the population of\nparent nuclei could lead to a small anisotropy in the decay rate, whic h could depend on\nthe relative orientations of the net nuclear spin and the outgoing α-particles’ trajectories.\nA nuclide with the desired properties is222Rn [51]. Its half-life of 3.8235 days is\nknown with a fractional 1 σerror of less than 10−4.222Rn is anIP= 0+state, which\ndecays overwhelmingly to218Po, another 0+isotope. The decay energy is 5 .5903±0.0003\nMeV.\nIf any anisotropy in the α-decay of222Rn can be ruled out with the same /lessorsimilar10−4\naccuracy with which the total half-life is known, (25) and (26) dicta te that the Lorentz\nviolation coefficients kαmay be bounded at the level |kα\njk|/lessorsimilar2×10−6(although the trace\nkα\njjwould not be constrained, since it does not lead to any anisotropy). These bounds are\nnot very strong, especially compared with the results of atomic cloc k experiments, but\nthey could give new constraints on neutron and proton ccoefficients that have not been\nbounded in the laboratory.\nThe reason that an experiment like this would be sensitive to new coeffi cients is that\ntheα-decay experiment would involve the measurement of actual decay directions. The\nextremely precise clock comparison experiments instead measure e nergy shifts. The en-\nergyof certain transitions can varyas the anglebetween the labor atorymagnetic field and\na background vector field changes. However, not all possible magn etic field directions are\nsampled as the Earth rotates, which is why certain coefficients are n ot measured. Con-\nstraints on the various coefficients which are difficult to measure in clo ck tests can come\nfrom cosmic ray observations, because cosmic rays coming from all possible directions\ncan potentially be observed [52]. However, these astrophysical b ounds lack the certainty\nassociated with controlled laboratory measurements.\n115 Conclusion\nIn this paper, we have examined Lorentz violation for composite par ticles, specifically\nα-particles. In low-energy physics, Lorentz violation can enter the in theα-particle sector\nonly through the coefficients kα\nµν. These coefficients are linear combinations of the proton\nandneutroncoefficients cp\nνµandcn\nνµ. Inthelimitofexactisospinsymmetry, kα\nµν=cp\nνµ+cn\nνµ.\nMore generally, the coefficients for the Lorentz-violating, spin-ind ependent modifica-\ntion of a composite particle’s nonrelativistic kinetic energy are linear c ombinations of the\nanalogous coefficients for its constituent particles. The weight give n to each coefficient\nin this sum is determined by the fraction of the total momentum that is carried by the\ncorresponding constituent. This is the solution to a simple instance o f the more general\nproblem of determining the Lorentz violation coefficients for a compo site particle in terms\nof the coefficients for the constituents.\nThe example of the α-particle is especially straightforward, because it is a spin singlet\ncomposed of four extremely similar particles. It would be desirable to get an equivalent\nunderstanding of more complicated nuclei. This would be particularly u seful for improv-\ning the bounds on Lorentz violation that come from clock comparison experiments. Such\nbounds are currently evaluated using extremely crude models of th e nucleus. The scope\nandaccuracy of the resulting bounds would be significantly improved with a better under-\nstanding of the relationship between the SME coefficients for nucleo ns and the hyperfine\ntransition frequencies that can actually be observed in the laborat ory.\nEven trickier than the problem of relating the effective Lorentz viola tion coefficients\nfor nuclei to the coefficients for their hadronic constituents is tha t of relating the proton,\nneutron, and other hadron coefficients to the coefficients for the underlying quark and\ngluon fields. These constituents interact strongly through quant um chromodynamics, and\nthe interaction cannot be treated as a small correction. Renorma lization group effects are\nalso very important; the ultimate relationship between the Lorentz violation coefficients\nfor the physical hadrons and the coefficients for the parton fields will depend sensitively\non the renormalization scale. The whole problem is quite difficult; howev er, the subject\nis also extremely interesting, and it represents one of the most impo rtant open problems\nconcerning the structure of the SME.\nThis paper also examined another phenomenon in quantum mechanics whose interac-\ntion with Lorentz violation has been very little studied: tunneling. In r eactions, such as\nα-decay, whose rate is principally determined by a quantum barrier pe netration factor T,\nthe sensitivity to Lorentz-violating effects may be enhanced. This is a consequence of the\npenetration factor’s exponential dependence on various parame ters. If Lorentz violation\nleads to a small increase in the height of the potential barrier throu gh which a particle\nmust tunnel, the tunneling rate will be diminished by an exponentially gr eater amount.\nWhile an experiment searching for a possible anisotropy in the emission ofα-particles\nby a decaying isotope is interesting, the bounds that such an exper iment might place on\nphysical Lorentz violation coefficients are not very precise. Such a n experiment would be\n12sensitive to certain coefficients that have not previously been cons trained by laboratory\nexperiments; however, the coefficients in question have been boun ded at the 5 ×10−14\nlevel by observations of ultra-high-energy cosmic rays. Constra ints based on controlled\nlaboratoryexperiments are generally preferable to bounds based on inferences drawn from\nastrophysical data, but it is unfortunate that the laboratory te sts proposed here might\nonly give bounds at the 2 ×10−6level.\nHowever, this work opens the door to the possibility of constraining Lorentz violation\nwith other experiments involving tunneling. The ideal experiment for placing such con-\nstraints would have a high barrier, which would make the tunneling rat e more sensitive to\nany Lorentz violations. To compensate for the low tunneling rate, a large flux of particles\nagainst the barrier would be required. If a process with the desired characteristics is\nfound, it could be used to place new bounds on coefficients for Loren tz violation.\nReferences\n[1] D. Colladay, V. A. Kosteleck´ y, Phys. Rev. D 55, 6760 (1997).\n[2] D. Colladay, V. A. Kosteleck´ y, Phys. Rev. D 58, 116002 (1998).\n[3] V. A. Kosteleck´ y, Phys. Rev. D, 69105009 (2004).\n[4] V. 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D 78, 085018 (2008).\n15"    },    {        "title": "1805.06355v1.Sequence_Lorentz_spaces_and_their_geometric_structure.pdf",        "content": "arXiv:1805.06355v1  [math.FA]  16 May 2018SEQUENCE LORENTZ SPACES AND THEIR GEOMETRIC\nSTRUCTURE\nMACIEJ CIESIELSKI1∗AND GRZEGORZ LEWICKI\nAbstract. This article is dedicated to geometric structure of the Lore ntz and\nMarcinkiewicz spaces in case of the pure atomic measure. We s tudy complete\ncriteria for order continuity, the Fatou property, strict m onotonicity and strict\nconvexity in the sequence Lorentz spaces γp,w. Next, we present a full char-\nacterization of extreme points of the unit ball in the sequen ce Lorentz space\nγ1,w. We also establish a complete description with an isometry o f the dual\nand predual spaces of the sequence Lorentz spaces γ1,wwritten in terms of the\nMarcinkiewicz spaces. Finally, we show a fundamental appli cation of geometric\nstructure of γ1,wto one-complemented subspaces of γ1,w.\n2000 Mathematics Subjects Classification: 46E30, 46B20, 46 B28.\nKey Words and Phrases: Lorentz and Marcinkiewicz spaces, strict monotonicity,\nstrict convexity, order continuity, extreme point, existe nce set, one-complemented sub-\nspace.\n1. Introduction\nGeometricstructureswithapplicationoftheLorentzspacesΓ p,wandMarcinkiewicz\nspacesMφin case of the non-atomic measure have been investigated extensiv ely by\nmany authors [3, 5, 6, 12, 13]. In contrast to the non-atomic case there are only\nfew papers concerning geometric structure of sequence Lorent z and Marcinkiewicz\nspaces. The first crucial paper devoted to the Marcinkiewicz spac es appeared in\n2004 [9], where authors have studied the biduals and order continuo us ideals of\nthe Marcinkiewicz spaces for the pure atomic measure. The next sig nificant paper\nwas published in 2009 [10], in which there has been investigated, among others,\nstrict monotonicity, smooth points and extreme points with applicat ion to one-\ncomplemented subspaces. For other results concerning the issue devoted to one-\ncomplemented subspaces please see a.g. [7, 8, 11].\nThe purpose of this article is to explore geometric properties of the sequence\nLorentz spaces γp,wand its dual and predual spaces. It is worth mentioning that\nwe present an application of geometric properties to a characteriz ation of one-\ncomplemented subspaces in the Lorentz spaces γp,win case of the pure atomic\nmeasure. It is necessary to mention that a characterization of ge ometric structure\nof the sequence Lorentz and Marcinkiewicz spaces does not follow im mediately as a\nconsequence of well known results from the case of non-atomic me asure in general.\nThe paper is organized as follows. In section 2, we present the need ed terminol-\nogy. In section 3, we show an auxiliary result devoted to a relationsh ip between\n12 M. CIESIELSKI & G. LEWICKI\nthe global convergence in measure of a sequence ( xn)⊂ℓ0and the pointwise con-\nvergence of its sequence of decreasing rearrangements ( x∗\nn). In case of the pure\natomic measure, we also establish a correspondence between an ide ntity of signs\nof the values for two different sequences in ℓ0and an additivity of the decreasing\nrearrangement operation for these sequences. Section 4 is devo ted to an investi-\ngation of geometric structure of sequence Lorentz spaces γp,w. Namely, we focus\non complete criteria for order continuity and the Fatou property in Lorentz spaces\nfor the pure atomic measure. Next, we present a characterizatio n of strict mono-\ntonicity and strict convexity of γp,wwritten in terms of the weight sequence w.\nIn spirit of the previous result, we describe an equivalent condition f or extreme\npoints of the unit ball in the sequence Lorentz space γ1,w. In section 5, we solve\nthe essential problem showing a full description of the dual and pre dual spaces of\nthe sequence Lorentz space γ1,w. First, we answer a crucial question under which\ncondition does an isometric isomorphism exist between the dual spac e of the se-\nquence Lorentz space γ1,wand the sequence Marcinkiewicz space mφ. Next, we\ndiscuss complete criteria which guarantee that the predual space of the sequence\nLorentz space γ1,wcoincides with the sequence Marcinkiewicz space m0\nφ. Addi-\ntionally, we investigate necessary condition for the isometry betwe en the predual\nofγ1,wand the Marcinkiewicz space m0\nφ. In section 5, we present an application\nof geometric properties of the sequence Lorentz space γ1,wto a characterization\nof one-complemented subspaces. Namely, using an isometry betwe en the classical\nLorentz space d1,wand the Lorentz space γ1,w, we prove that there exists norm\none projection on any nontrivial existence subspace of γ1,w. Additionally, by the\nprevious investigation and in view of [10], we establish a full character ization of\nsmooth points in the sequence Lorentz space γ1,wand its predual and dual spaces.\nFinally, we study an equivalent condition for an extreme points in the d ual space\nof the sequence Lorentz space γ1,w.\n2. Preliminaries\nLetR,R+andNbe the sets of reals, nonnegative reals and positive integers,\nrespectively. A mapping φ:N→R+is said to be quasiconcave ifφ(t) is increasing\nandφ(t)/tis decreasing on Nand alsoφ(n)>0 for alln∈N. We denote by ℓ0the\nset of all real sequences, and by SX(resp.BX) the unit sphere (resp. the closed\nunit ball) in a Banach space ( X,/bardbl·/bardblX). Let us denote by ( ei)∞\ni=1a standard basis\ninR∞. A sequence quasi-Banach lattice ( E,/bardbl · /bardblE) is said to be a quasi-Banach\nsequence space (or aquasi-K¨ othe sequence space ) if it is a sequence sublattice of ℓ0\nand holds the following conditions\n(1) Ifx∈ℓ0,y∈Eand|x| ≤ |y|, thenx∈Eand/bardblx/bardblE≤ /bardbly/bardblE.\n(2) There exists a strictly positive x∈E.\nForsimplicity let us use the short symbol E+={x∈E:x≥0}. An element x∈E\nis called a point of order continuity , shortlyx∈Ea, if for any sequence ( xn)⊂E+\nsuch thatxn≤ |x|andxn→0 pointwise we have /bardblxn/bardblE→0.A quasi-Banach\nsequence space Eis said to be order continuous , shortlyE∈(OC), if any element\nx∈Eis a point of order continuity. A space Eis said to be reflexive ifEand\nits associate space E′are order continuous. Given a quasi-Banach sequence space\nEis said to have the Fatou property if for all (xn)⊂E+, supn∈N/bardblxn/bardblE<∞and\nxn↑x∈ℓ0, thenx∈Eand/bardblxn/bardblE↑ /bardblx/bardblE(see [16, 2]). We say that Eisstrictly\nmonotone if for anyx,y∈E+such thatx≤yandx/\\e}atio\\slash=ywe have /bardblx/bardblE</bardbly/bardblE.SEQUENCE LORENTZ SPACES AND THEIR GEOMETRIC STRUCTURE 3\nLet (X,/bardbl·/bardblX) be a Banach space. Recall that x∈SXis anextreme point ofBX\nif for anyy,z∈SXsuch thatx= (y+z)/2 we havex=y=z. A Banach space\nXis called rotundorstrictly convex if anyx∈SXis an extreme point of BX. An\nelementx∈Xis called a smooth point ofXif there exists a unique linear bounded\nfunctionalf∈SX∗such thatf(x) =/bardblx/bardblX.\nThedistribution for any sequence x∈ℓ0is defined by\ndx(λ) = card {k∈N:|x(k)|>λ}, λ≥0.\nFor any sequence x∈ℓ0itsdecreasing rearrangement is given by\nx∗(n) = inf{λ≥0 :dx(λ)≤n−1}, n∈N.\nIn this article we use the notation x∗(∞) = limn→∞x∗(n). For any sequence x∈ℓ0\nwe denote the maximal sequence ofx∗by\nx∗∗(n) =1\nnn/summationdisplay\ni=1x∗(i).\nIt is easy to notice that for any point x∈ℓ0,x∗≤x∗∗,x∗∗is decreasing, continuous\nand subadditive. For more details of dx,x∗andx∗∗see [2, 14].\nWe saythat twosequences x,y∈ℓ0areequimeasurable , shortlyx∼y, ifdx=dy.\nA quasi-Banach sequence space ( E,/bardbl·/bardblE) is called symmetric orrearrangement\ninvariant (r.i. for short) if whenever x∈ℓ0andy∈Esuch thatx∼y,thenx∈E\nand/bardblx/bardblE=/bardbly/bardblE. Thefundamental sequence φEof a symmetric space Ewe define\nas followsφE(n) =/bardblχ{i∈N:i≤n}/bardblEfor anyn∈N(see [2]). Let 0 < p <∞and\nw= (w(n))n∈Nbe a nonnegative real sequence and let for any n∈N\nW(n) =n/summationdisplay\ni=1w(i) andWp(n) =np∞/summationdisplay\ni=n+1w(i)\nip<∞.\nFor short notation the sequence wis called a nonnegative weight sequence. In\nthe whole paper, unless we say otherwise we suppose that wa nonnegative weight\nsequence is nontrivial, i.e. there is n∈Nsuch thatw(n)>0. Now, we recall the\nsequence Lorentz space d1,wwhich is a subspace of ℓ0such that for any sequence\nx= (x(n))n∈N∈d1,wwe have\n/bardblx/bardbld1,w=∞/summationdisplay\ni=1x∗(n)w(n)<∞.\nIt is well known that the Lorentz space d1,wis a symmetric space with the Fatou\nproperty (see [13]). The sequence Lorentz space γp,wis a collection of all real\nsequencesx= (x(n))n∈Nsuch that\n/bardblx/bardblγp,w=/parenleftBigg∞/summationdisplay\ni=1(x∗∗(n))pw(n)/parenrightBigg1/p\n<∞.\nLet us notice that for any nonnegative sequence w= (w(n))n∈Nthe sequence\nLorentzspace γp,wis ar.i. (quasi-)Banachsequencespaceequipped withthe (quasi-\n)norm/bardbl·/bardblγp,w. It is easy to observe that the fundamental sequence of the Lor entz\nspaceγp,wis given by φγp,w(n) =/vextenddouble/vextenddoubleχ{i≤n,i∈N}/vextenddouble/vextenddouble\nγp,w= (W(n)+Wp(n))1/pfor every4 M. CIESIELSKI & G. LEWICKI\nn∈N. Letφbe a quasiconcave sequence. The Marcinkiewicz space mφand (resp.\nm0\nφ) consists of all real sequences x= (x(n))n∈Nsuch that\n/bardblx/bardblmφ= sup\nn∈N{x∗∗(n)φ(n)}<∞/parenleftBig\nresp.m0\nφ⊂mφand lim\nn→∞x∗∗(n)φ(n) = 0/parenrightBig\n.\nRecall that mφandm0\nφare symmetric spaces equipped with the norm /bardbl·/bardblmφ(for\nmore details see [9]).\n3. properties of decreasing rearrangement for a pure atomic\nmeasure\nIn this section, first we present an auxiliary lemma devoted to a corr espondence\nbetween the global convergence in measure on Nof an arbitrary sequence of ele-\nments inℓ0to an element in ℓ0and the pointwise convergence of their decreasing\nrearrangements. Although the similar result emerges in case of the non-atomic\nmeasure space (see [14]), the proof of it is not valid in case of the pu re atomic\nmeasure space. It is worth mentioning that in the pure atomic measu re space the\nproof of the wanted result is quite long and requires new techniques .\nLemma 3.1. Letxm,x∈ℓ0for allm∈N. Ifxmconverges to xglobally in\nmeasure, then x∗\nmconverges to x∗onN.\nProof.Let (xm)⊂ℓ0,x∈ℓ0be such that xm→xglobally in measure. Since for\nanyǫ>0 andm∈Nwe have\ncard{n∈N:|xm(n)−x(n)|>ǫ} ≥card{n∈N:||xm(n)|−|x(n)||>ǫ},\nwithout loss of generality we may assume that x≥0 andxm≥0 for alln∈N.\nLetB={bi}be a set of all values for a function x:N→R+. Define for any\ni∈ {1,...,card(B)},\nNi={n∈N:x(n) =bi},andci=i/summationdisplay\nj=1card(Nj), c0= 0.\nWithout loss of generality we may assume that ( bi) is strictly decreasing. Now we\npresent the proof in three cases.\nCase1.Suppose that card( N1) =∞. Then, it is easy to see that x∗(n) =b1χN. If\nb1= 0 then for all m≥Mδ1we have\ndxm(δ1) = card{n∈N:|xm(n)|>δ1}<1.\nHence, since dx∗m(δ1) =dxm(δ1) for every m≥Mδ1, we getx∗\nm→0 globally in\nmeasure, whence we infer that x∗\nm→0 pointwise. In case when B={b1}then we\ntakeb2= 0. Denote δ1= (b1−b2)/4. Sincexm→xglobally in measure, there\nexistsMδ1∈Nsuch that for all m≥Mδ1,\n(1) card {n∈N:|xm(n)−x(n)|>δ1}<1.\nNow, we claim that for any n∈N,x∗\nm(n)→x∗(n). Indeed, by (1) we conclude\nthat for any m≥Mδ1andn∈N,\n|x(n)−xm(n)| ≤δ1.\nIf card(N\\N1) = 0, then we are done. Otherwise, for any n∈N1andk∈N\\N1\nwe observe that\nxm(n)≥x(n)−δ1=b1+3(b1−b2)\n4=b1+3δ1>x(k)+3δ1≥xm(k)+2δ1SEQUENCE LORENTZ SPACES AND THEIR GEOMETRIC STRUCTURE 5\nfor allm≥Mδ1. Consequently, for every m≥Mδ1we obtainx∗\nm= (xmχN1)∗and\nalso|b1−xm(n)| ≤δ1for eachn∈N1. Therefore, for all m≥Mδ1andn∈Nit is\neasy to notice that\nδ1≥ |b1−x∗\nm(n)|=|x∗(n)−x∗\nm(n)|.\nCase2.Assume that there exists bj0∈B\\ {0}such that card( Nj0) =∞and\n0<card(Nj)<∞for anyj∈ {1,...,j0−1}. Then, we have\n(2) x∗(n) =\nj0/summationdisplay\nj=1bjχNj\n∗\n(n) =j0/summationdisplay\nj=1bjχ{i∈N:cj−1+1≤i≤cj}(n).\nIn case when card( B) =j0then we assume that bj0+1= 0. Denote for any\ni∈ {1,...,card(B)},\nδi=bi−bi+1\n4andδ= min\n1≤i≤j0{δi}.\nSincexm→xglobally in measure, there exists Mδ∈Nsuch that for all m≥Mδ,\ncard{n∈N:|xm(n)−x(n)|>δ}<1.\nTherefore, for any m≥Mδandni∈Niwhere 1≤i≤j0we have\n(3) δ≥ |x(ni)−xm(ni)|=|bi−xm(ni)|.\nHence, for all m≥Mδandni∈Niwhere 1≤i≤j0−1 we easily observe\nxm(ni) =bi−δ≥bi+1+3δ≥xm(ni+1)+2δ.\nIn consequence, by (3) we get for every m≥Mδandn∈N,\n(4)x∗\nm(n) =\nj0/summationdisplay\nj=1xmχNj\n∗\n(n) =j0/summationdisplay\nj=1/parenleftbig\nxmχNj/parenrightbig∗(n−cj−1)χ{i∈N:cj−1+1≤i≤cj}(n).\nClearly, there exists σ:N→/uniontextj0\nj=1Nja permutation such that x∗(n) =x(σ(n)) for\nalln∈N. Thus, for any n∈Nthere exists j∈ {1,...,j 0}such thatσ(n)∈Nj\nand by (3) we obtain\nδ≥ |xm(σ(n))−x(σ(n))|=|xm(σ(n))−bj|=|(xmχNj)∗(n−cj−1)−bj|\nfor allm≥Mδ. Therefore, by (2) and (4) we infer that\nx∗\nm(n) =j0/summationdisplay\nj=1/parenleftbig\nxmχNj/parenrightbig∗(n−cj−1)χ{i∈N:cj−1+1≤i≤cj}(n)\n→j0/summationdisplay\nj=1bjχ{i∈N:cj−1+1≤i≤cj}(n) =x∗(n).\nCase3.Suppose that for any bj∈B\\{0}we have card( Nj)<∞. If card(B)<∞\nthen without loss of generality we may assume that j0= card(B) andbj0= 0.\nNext, letting for any i∈ {1,...,j 0−1},\nδi=bi−bi+1\n4andδ= min\n1≤i≤j0−1{δi},6 M. CIESIELSKI & G. LEWICKI\nand proceeding analogously as in case 2 we may show that x∗\nm→x∗onN, in case\nwhen card( B)<∞. Now, assume that card( B) =∞. Then, since ( bj) is strictly\ndecreasing and bounded we conclude\nlim\nj→∞bj=b≥0.\nFirst, let us consider that b= 0. Letǫ>0. Then, there exists j0∈Nsuch that for\nallj≥j0we have\n(5) 0 <bj<ǫ\n4andbj0−1≥ǫ\n4.\nDefine for any i∈ {1,...,j0},\nδi=bi−bi+1\n4andδ= min/braceleftbiggǫ/4−bj0\n4,min\n1≤i≤j0{δi}/bracerightbigg\n.\nSimilarly as in case 2 there is Mδ∈Nsuch that for all m≥Mδ,n∈Nand\nk∈/uniontext\nj≥j0Njwe get\n(6)|xm(n)−x(n)| ≤δandxm(k)≤δ+x(k)≤δ+bj0<δ+ǫ\n4<ǫ\n2.\nMoreover, we may observe that\nxm(ni)≥xm(ni+1)+2δ\nfor everym≥Mδandni∈Niwherei∈ {1,...,j0−1}. Next, assuming that\nσ:N→/uniontext∞\nj=1Njis a permutation such that x∗(n) =x(σ(n)) for alln∈N, then\nfor anyn∈Nwithn≤cj0−1there exists j∈ {1,...,j 0−1}such thatσ(n)∈Nj\nand by (6) we obtain\nǫ>δ≥ |xm(σ(n))−x(σ(n))|=|xm(σ(n))−bj|\n=|(xmχNj)∗(n−cj−1)−bj|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nj0−1/summationdisplay\nj=1xmχNj\n∗\n(n)−bj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nfor allm≥Mδ. On the other hand, if n > cj0−1then there is j≥j0such that\nσ(n)∈Njand by (5) and (6) it follows that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n∞/summationdisplay\nj=j0xmχNj\n∗\n(n−cj0−1)−x∗(n)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=|xm(σ(n))χNj(σ(n))−x(σ(n))|\n=|xm(σ(n))−bj|<ǫ\nfor allm≥Mδ. Now, let us notice that for every n∈N,\nx∗(n) =∞/summationdisplay\nj=1bjχ{i∈N:cj−1+1≤i≤cj}(n)\nand\nx∗\nm(n) =\n\n/parenleftBig/summationtextj0−1\nj=1xmχNj/parenrightBig∗\n(n) if n≤cj0−1,/parenleftBig/summationtext∞\nj=j0xmχNj/parenrightBig∗\n(n−cj0−1) ifn>cj0−1.\nHence, we infer that for any m≥Mδandn∈N,\nx∗\nm(n)→x∗(n).SEQUENCE LORENTZ SPACES AND THEIR GEOMETRIC STRUCTURE 7\nNow, we assume that b >0. Then, it is easy to see that x∗(∞) =b >0. Next,\ntaking\ny=xχsupp(x)+bχN\\supp(x)andym=xmχsupp(x)+bχN\\supp(x)\nfor allm∈N, we may show that x∗=y∗andx∗\nm=y∗\nmfor sufficiently large m∈N.\nNext, passing to subsequence and relabeling if necessary, it is enou gh to prove that\ny∗\nm→y∗onN. Clearly, by definition of yandymfor allm∈Nwe may observe\nthatym−bconvergesy−bglobally in measure and ( y−b)∗(∞) = 0. Finally, using\nanalogous technique as previously, in case 3 for b= 0, we finish the proof. /square\nRemark 3.2. Let us notice that using analogous techniques as in the proof of the\nproperty 90in[14]and by the property 70in[14](see pp. 64-65), in view of Theorem\n2.7 and Proposition 3.3 in [2]we are able to show the below assertion.\nFor any two sequences xandywithx∗(∞) = 0andy∗(∞) = 0the following\nconditions are equivalent.\n(i)For anyi∈N,\n(x+y)∗(i) =x∗(i)+y∗(i).\n(ii) sgn(x(i)) = sgn(y(i))for anyi∈Nand there exists (En)n∈Na countable\ncollection of sets such that for every n∈Nwe have card(En) =nand\nx∗∗(n) =1\nn/summationdisplay\ni∈En|x(i)|andy∗∗(n) =1\nn/summationdisplay\ni∈En|y(i)|.\n4. geometric structure of sequence lorentz spaces γp,w\nIn this section, we discuss complete criteria for order continuity, t he Fatou prop-\nerty, strict monotonicity and strict convexity and also extreme po ints of the unit\nball in the sequence Lorentz space γp,w.\nTheorem 4.1. Letwbe a nonnegative weight sequence and 0< p <∞. The\nLorentz space γp,wis order continuous if and only if W(∞) =∞.\nProof. Necessity. Suppose that γp,wis not order continuous. Then, there exists\n(xm)⊂γ+\np,w\\{0}such thatxm↓0 pointwise and d= infn∈N/bardblxm/bardblγp,w>0. Next,\npassing to subsequence and relabeling if necessary we may assume t hat/bardblxm/bardblγp,w↓\nd. SinceW(∞) =∞we claim that dx(λ)<∞for allλ>0 andx∈γp,w. Indeed,\nassumingforacontrarythatthereis x∈γp,wsuchthatx∗(∞) = limn→∞x∗(n)>0\nwe obtainℓ∞֒→γp,w. Definez=χN. Then, we have z∗∗=z∈γp,wand also\n/bardblz/bardblγp,w=W(∞) =∞, which gives us a contradiction and proves the claim. Let\nǫ>0. Define two sets\nE1={n∈N:x1(n)>ǫ}andE2=N\\E1.\nNow, since x∗\n1(∞) = 0 it is easy to notice card( E1) =dx1(ǫ)<∞andE1∩E2=∅.\nTherefore, since xm↓0 pointwise we have\ndxm(ǫ) = card{n∈N:xm(n)>ǫ} →0 asm→ ∞.\nHence, by Lemma 3.1 it follows that x∗\nm→0 pointwise on N. Consequently,\nsince/bardblx1/bardblγp,w<∞andx∗∗(n)<∞for alln∈N, applying twice the Lebesgue\nDominated Convergence Theorem we conclude /bardblxm/bardblγp,w→0.\nSufficiency. Assume for a contrary that W(∞)<∞. Then, it is easy to see that\nx=χN∈γp,w,x∗∗=xand/bardblx/bardblγp,w=W(∞). Definexm=χ{i∈N:i≥m}for any8 M. CIESIELSKI & G. LEWICKI\nm∈N. Clearly, we have xm↓0 andxm≤xpointwise for every m∈N. Moreover,\nwe observe that x∗∗\nm=x∗∗for anym∈N. Hence, we get /bardblxm/bardblγp,w=W(∞)>0\nfor alln∈N, which contradicts with assumption that γp,wis order continuous. /square\nRemark 4.2. First, let us observe that for any sequence symmetric space E, Propo-\nsition 5.9 in [2]is true. Namely, using analogous technique as in [2]we clearly get\nthe embedding E ֒→mφholds with constant 1, i.e. for all x∈E,\n/bardblx/bardblmφ= sup{x∗∗(n)φE(n)} ≤ /bardblx/bardblE,\nwhereφEis the fundamental sequence of a symmetric space EonN. Next, in view\nof Remark 3.2 in [4]and assuming that Ehas the Fatou property, we may show\nthatφE(∞) =∞if and only if x∗(∞) = 0for anyx∈E.\nLemma 4.3. Letwbe a nonnegative weight sequence and 0<p<∞. The Lorentz\nspaceγp,whas the Fatou Property.\nProof.Let (xm)⊂γ+\np,w,x∈ℓ0andxm↑xpointwise and supm∈N/bardblxm/bardblγp,w<∞.\nImmediately, by Proposition 1.7 in [2] it follows that x∗\nm↑x∗. Next, applying twice\nLebesgue Monotone Convergence Theorem [18] we get /bardblxm/bardblγp,w↑ /bardblx/bardblγp,w. Finally,\nsince supm∈N/bardblxm/bardblγp,w<∞it follows that x∈γp,w. /square\nTheorem 4.4. Letwbe a nonnegative weight sequence and 0< p <∞. The\nLorentz space γp,wis strictly monotone if and only if W(∞) =∞.\nProof. Necessity. Assume for a contrary that W(∞)<∞. Then, we may show\nthatℓ∞֒→γp,w. Next, defining two sequences\nx=χ{i∈N:i>1}andy=χN\nwe easily observe that x≤y,x/\\e}atio\\slash=yandx∗∗=y∗∗=y. Consequently, /bardblx/bardblγp,w=\n/bardbly/bardblγp,w, which contradicts with assumption that the Lorentz space γp,wis strictly\nmonotone.\nSufficiency. Letx,y∈γ+\np,w,x≤yandx/\\e}atio\\slash=y. Sincex/\\e}atio\\slash=ythere exists n0∈Nsuch\nthatx(n0)<y(n0). Define\nδ0= max{y(n0)/2,x(n0)}andN0={n∈N:y(n)>δ0}.\nSinceW(∞) =∞, by the proof of Theorem 4.1 it follows that y∗(∞) =x∗(∞) = 0.\nHence, since n0∈N0we get\n0<card(N0) =dy(δ0)<∞.\nNow,weclaimthatthereexists m0∈ {1,...,card(N0)}suchthatx∗(m0)<y∗(m0).\nIndeed, if it is not true then for all n∈ {1,...,card(N0)}we havex∗(n) =y∗(n).\nMoreover, there is a permutation σ:N→Nsuch thatσ(n)∈N0andy∗(n) =\ny(σ(n)) for every n∈ {1,...,card(N0)}. So, we have\nx∗(n) =y∗(n) =y(σ(n))≥x(σ(n))\nfor anyn∈ {1,...,card(N0)}. Letm0∈ {1,...,card(N0)}be such that σ(m0) =\nn0. Then, we observe that\nx∗(m0) =y∗(m0) =y(σ(m0)) =y(n0)>x(n0).\nTherefore, we obtain\nx∗(m0)>x(n0) =x(σ(m0)),SEQUENCE LORENTZ SPACES AND THEIR GEOMETRIC STRUCTURE 9\nwhich implies that there exists k0∈N\\N0such thatx(k0) =y(n0). On the other\nhand, it is well known that x(k0)≤y(k0), whence\ny(k0)≥x(k0) =y(n0) =y∗(m0).\nIn consequence, by definition of N0this yields that k0∈N0and gives us a contra-\ndiction. Now, since x∗(n)≤y∗(n) for anyn∈Nandx∗(n0)< y∗(n0) for some\nn0∈Nit follows that\nx∗∗(n)≤y∗∗(n) andk/summationdisplay\ni=1x∗(i)<k/summationdisplay\ni=1y∗(i)\nfor alln∈Nandk≥n0. Finally, by assumption that W(∞) =∞there exists\n(nk)⊂Nsuch thatw(nk)>0 for every k∈N. Hence, we infer that /bardblx/bardblγp,w<\n/bardbly/bardblγp,w. /square\nThe immediate consequence of the previous theorem and Propositio n 2.1 in [10]\nis the following result.\nCorollary 4.5. Letw≥0be a weight sequence such that W(∞) =∞and let\n0<p<∞. An element x∈Sγp,wis an extreme point of Bγp,wif and only if x∗is\nan extreme point of Bγp,w.\nNext, we show that the Lorentz space γp,wis strictly convex for 1 <p<∞and\nwa positive weight sequence such that W(∞) =∞. In some parts of the proof\nof the following theorem we use the similar techniques to Theorem 3.1 in [6] (see\nalso Theorem 2.3 in [5]). For the sake of completeness and reader’s co nvenience we\nshow all details of the proof.\nTheorem 4.6. Letwbe a nonnegative weight sequence. The Lorentz space γp,w\nis strictly convex if and only if 1< p <∞andw(n)>0for anyn∈Nand also\nW(∞) =∞.\nProof. Necessity. Assume that γp,wis strictly convex. For a contrary we suppose\nthatp= 1. Letx,y∈Sγp,wand/bardblx+y/bardblγp,w= 2. Without loss of generality we\nmay assume that x=x∗andy=y∗. Then, we have ( x+y)∗∗=x∗∗+y∗∗and also\n/bardblx+y/bardblγp,w=/bardblx/bardblγp,w+/bardbly/bardblγp,w= 2.\nConsequently, since xandyare arbitrary and γp,wis strictly convex we conclude a\ncontradiction. Now, assume that W(∞)<∞. Define\nx=1\nW(∞)1/pχ{2n:n∈N}andy=1\nW(∞)1/pχN.\nClearly, we have for any n∈N,\nx∗∗(n) =y∗∗(n) =1\nW(∞)1/p.\nMoreover, we observe that\n(x+y)∗∗(n) =1\nW(∞)1/p/parenleftbig\n2χ{2n:n∈N}+χ{2n−1:n∈N}/parenrightbig∗∗(n) =2\nW(∞)1/p\nfor anyn∈N. Hence, we get\n/bardblx/bardblγp,w=/bardbly/bardblγp,w=/bardblx+y/bardblγp,w\n2= 1.10 M. CIESIELSKI & G. LEWICKI\nTherefore, by assumption that γp,wis strictly convex we obtain a contradiction.\nNow, let us suppose for a contrary that there is n0∈Nsuch thatw(n0) = 0. If\nn0= 1, then take ǫ∈(0,1/φγp,w(2)) and define\nx=1\nφγp,w(2)χ{1,2}andy=/parenleftbigg1\nφγp,w(2)+ǫ/parenrightbigg\nχ{1}+/parenleftbigg1\nφγp,w(2)−ǫ/parenrightbigg\nχ{2}.\nIt is easy to see that x/\\e}atio\\slash=yand\nx∗∗(n) =1\nφγp,w(2)χ{1,2}(n)+2\nnφγp,w(2)χN\\{1,2}(n)\nand also\ny∗∗(n) =/parenleftbigg1\nφγp,w(2)+ǫ/parenrightbigg\nχ{1}(n)+1\nφγp,w(2)χ{2}(n)+2\nnφγp,w(2)χN\\{1,2}(n).\nTherefore, since w(1) = 0, we have\n/bardblx/bardblγp,w=/bardbly/bardblγp,w=/parenleftBigg\n1\n(φγp,w(2))pw(2)+2p\n(φγp,w(2))p∞/summationdisplay\nn=3w(n)\nnp/parenrightBigg1/p\n=1\nφγp,w(2)(W(2)+Wp(2))1/p= 1.\nFurthermore, we observe that\n(x+y)∗∗(n) =/parenleftbigg/parenleftbigg2\nφγp,w(2)+ǫ/parenrightbigg\nχ{1}+/parenleftbigg2\nφγp,w(2)−ǫ/parenrightbigg\nχ{2}/parenrightbigg∗∗\n(n)\n=/parenleftbigg2\nφγp,w(2)+ǫ/parenrightbigg\nχ{1}(n)+4\nnφγp,w(2)χN\\{1}(n).\nHence, since w(1) = 0, we get\n/bardblx+y/bardblγp,w=/parenleftBigg\n4p\n(φγp,w(2))p∞/summationdisplay\nn=2w(n)\nnp/parenrightBigg1/p\n=2\nφγp,w(2)/parenleftBigg\nw(2)+2p∞/summationdisplay\nn=3w(n)\nnp/parenrightBigg1/p\n=2\nφγp,w(2)(W(2)+Wp(2))1/p= 2.\nSo, in case when w(1) = 0, it follows that γp,wis not strictly convex. Assume that\nn0>1. Define\nx=1\nφγp,w(n0)χ[1,n0]andy=1\nφγp,w(n0)/parenleftbigg\nχ[1,n0−1]+1\n2χ{n0,n0+1}/parenrightbigg\n.\nThen, we easily observe that x/\\e}atio\\slash=yand/bardblx/bardblγp,w= 1. Moreover, we have\ny∗∗(n) =1\nφγp,w(n0)\n\n1 ifn<n0,\nn0−1/2\nn0ifn=n0,\nn0\nnifn>n0,SEQUENCE LORENTZ SPACES AND THEIR GEOMETRIC STRUCTURE 11\nand\n(x+y)∗∗(n) =1\nφγp,w(n0)/parenleftbigg\n2χ[1,n0−1]+3\n2χ{n0}+1\n2χ{n0+1}/parenrightbigg∗∗\n(n)\n=2\nφγp,w(n0)\n\n1 ifn<n0,\nn0−1/4\nn0ifn=n0,\nn0\nnifn>n0.\nHence, since w(n0) = 0, we conclude that\n/bardbly/bardblγp,w=/bardblx+y/bardblγp,w\n2=1\nφγp,w(n0)(W(n0−1)+Wp(n0))1/p= 1.\nIn consequence, by assumption that γp,wis strictly convex we get a contradiction.\nSufficiency. Letx,y∈Sγp,wandx/\\e}atio\\slash=y. We consider the proof in two cases.\nCase1.Assume that there exists n0∈Nsuch thatx∗∗(n0)/\\e}atio\\slash=y∗∗(n0). Then, by\nstrict convexity of the power function upfor 1<p<∞we have\n/parenleftbigg1\n2x∗∗(n0)+1\n2y∗∗(n0)/parenrightbiggp\n<1\n2x∗∗p(n0)+1\n2y∗∗p(n0).\nTherefore, since for any n∈N,\n/parenleftbigg1\n2x∗∗(n)+1\n2y∗∗(n)/parenrightbiggp\n≤1\n2x∗∗p(n)+1\n2y∗∗p(n)\nby assumption that w(n)>0 for alln∈Nwe infer that /bardblx+y/bardblγp,w<2.\nCase2.Suppose that x∗∗(n) =y∗∗(n) for every n∈N. Thus, we have x∗(n) =\ny∗(n) for anyn∈N. We claim that there exists n0∈Nsuch that\n(x+y)∗∗(n0)<x∗∗(n0)+y∗∗(n0).\nIndeed, assuming that it is not true it follows that ( x+y)∗(n) =x∗(n)+y∗(n) for\nalln∈N. Consequently, since W(∞) =∞, by Remark 3.2 we obtain |x+y|(n) =\n|x(n)|+|y(n)|for alln∈Nand there exists ( En) an increasing sequence of sets\nsuch that card( En) =nfor everyn∈Nand also\n/summationdisplay\ni∈En|x(i)|=n/summationdisplay\ni=1x∗=n/summationdisplay\ni=1y∗=/summationdisplay\ni∈En|y(i)|.\nIn consequence, |x(n)|=|y(n)|for anyn∈Nand sox(n) =y(n) for every n∈N.\nTherefore, in view of assumption x/\\e}atio\\slash=ywe get a contradiction. Finally, applying\nthe triangle inequality for the maximal function we infer that\n/vextenddouble/vextenddouble/vextenddouble/vextenddoublex+y\n2/vextenddouble/vextenddouble/vextenddouble/vextenddoublep\nγp,w<1\n2/bardblx/bardblp\nγp,w+1\n2/bardbly/bardblp\nγp,w= 1.\n/square\nFinally, we present a complete criteria for an extreme point in the ball of the\nLorentz space γ1,w. It is worth mentioning that in some parts of the proof we use\nsimilar technique to the proof of Theorem 2.6 in [10]. For the sake of co mpleteness\nandreader’sconveniencewepresent alldetails ofthe proofofthe followingtheorem.12 M. CIESIELSKI & G. LEWICKI\nTheorem 4.7. Letw≥0be a weight sequence such that W(∞) =∞. An element\nx∈Sγ1,wis an extreme point of Bγ1,wif and only if there exists n0∈Nsuch that\n(7) x∗=1\nφγ1,w(n0)χ{i∈N:i≤n0}\nand in case when n0>1,W(n0−1)>0.\nProof.Lettingx∈Sγ1,w, by Corollary 4.5 we may consider that x=x∗is an\nextreme point of Bγ1,w. Denote\nn0= sup{n∈N:x∗(n) =x∗(1)}.\nSinceW(∞) =∞andφγ1,w(n) =W(n) +W1(n) for anyn∈N, by Lemma 4.3\nand by Remark 4.2 it follows that x∗(∞) = 0 and so n0∈N. We claim that\nx∗(n0+1) = 0. Suppose on the contrary that x∗(n0+1)>0 and denote\nn1= card{n∈N:x∗(n) =x∗(n0+1)}\nand\nd= min{x∗(1)−x∗(n0+1),x∗(n0+1)−x∗(n0+n1+1)}.\nFirst, notice that φγ1,w(n+ 1)> φγ1,w(n)>0 for anyn∈N. Indeed, since\nW(∞) =∞we infer that φγ1,w(n)>0 for alln∈N. Now, assuming for a contrary\nthat there is n∈Nsuch thatφγ1,w(n+1) =φγ1,w(n), we easily obtain\nw(n+1) =−(n+1)∞/summationdisplay\ni=n+2w(i)\ni<0.\nHence, since w(n+ 1)≥0 we get a contradiction. Now, we are able to find\na,b∈(0,d) such that\n(8) b=aφγ1,w(n0+n1)−φγ1,w(n0)\nφγ1,w(n0).\nDefine\ny=x∗−bχ{i∈N:i≤n0}+aχ{i∈N:n0<i≤n0+n1}\nand\nz=x∗+bχ{i∈N:i≤n0}−aχ{i∈N:n0<i≤n0+n1}.\nClearly,y/\\e}atio\\slash=zandx= (y+z)/2. Sincey=y∗andz=z∗, by (8) we have\n/bardbly/bardblγ1,w=∞/summationdisplay\nn=1y∗∗(n)w(n)\n=∞/summationdisplay\nn=1w(n)\nnn/summationdisplay\nj=1/parenleftbig\nx∗(j)−bχ{i∈N:i≤n0}(j)+aχ{i∈N:n0<i≤n0+n1}(j)/parenrightbig\n=∞/summationdisplay\nn=1x∗∗(n)w(n)−b/parenleftBiggn0/summationdisplay\nn=1w(n)+n0∞/summationdisplay\nn=n0+1w(n)\nn/parenrightBigg\n+a/parenleftBiggn0+n1/summationdisplay\nn=n0+1w(n)\nn(n−n0)+n1∞/summationdisplay\nn=n0+n1+1w(n)\nn/parenrightBigg\n=/bardblx/bardblγ1,w−bφγ1,w(n0)+a/parenleftbig\nφγ1,w(n0+n1)−φγ1,w(n0)/parenrightbig\n=/bardblx/bardblγ1,w= 1.SEQUENCE LORENTZ SPACES AND THEIR GEOMETRIC STRUCTURE 13\nSimilarly, we may show that /bardblz/bardblγ1,w= 1. Therefore, in view of assumption that x\nis an extreme point of Bγ1,wwe conclude a contradiction, which proves our claim.\nIn case when n0>1 we assume that w(n) = 0 for all n∈ {1,...,n 0−1}. Then,\nfora∈(0,x∗(n0)) we define\ny=x∗+aχ{1}−aχ{n0}andz=x∗−aχ{1}+aχ{n0}.\nNext, it is clearly observe that y/\\e}atio\\slash=z,x= (y+z)/2,y∗=y=z∗and\n/bardblz/bardblγ1,w=/bardbly/bardblγ1,w=∞/summationdisplay\nn=n0w(n)\nnn/summationdisplay\nj=1/parenleftbig\nx∗(j)+aχ{1}(j)−aχ{n0}(j)/parenrightbig\n= 1.\nConsequently, by assumption that xis an extreme point of Bγ1,wwe have a con-\ntradiction. So, this implies that if n0>1 then it is needed W(n0−1)>0. Now,\nassume that x∈γ1,wand satisfies (7). For simplicity of our notation we denote\nc= 1/γ1,w(n0). Ifn0= 1, then by Theorem 4.4 we conclude that xis an ex-\ntreme point of Bγ1,w. Consider that n0>1. suppose that y,z∈Sγ1,w,y/\\e}atio\\slash=zand\nx= (y+z)/2. We claim that y(i) =z(i) = 0 for all i > n0. Indeed, if y(i)>0\nfor somei > n0, then it is obvious that z(i) =−y(i)<0 for some i > n0. Next,\ndefining two elements\nu=yχ{i∈N:i≤n0}andv=zχ{i∈N:i≤n0}\nwe havex= (u+v)/2. On the other hand, by Theorem 4.4 we infer that /bardblu/bardblγ1,w<\n/bardbly/bardblγ1,w= 1 and /bardblv/bardblγ1,w</bardblz/bardblγ1,w= 1. In consequence, we get\n1 =/bardblx/bardblγ1,w=1\n2/bardblu+v/bardblγ1,w≤/bardblu/bardblγ1,w+/bardblv/bardblγ1,w\n2<1,\nwhich yields a contradiction and proves our claim. Now, define\nI1={i∈N,i≤n0;y(i)>c},\nI2={i∈N,i≤n0;y(i) =c},\nI3={i∈N,i≤n0;y(i)<c}.\nWe can easily notice that y,z∈γ+\n1,w. Indeed, if it is not true then we may define\nu,v∈γ+\n1,wsuch thatu≤ |y|,u/\\e}atio\\slash=|y|andv≤ |z|,v/\\e}atio\\slash=|z|and alsox= (u+v)/2.\nTherefore, by Theorem 4.4 we obtain a contradiction. Next, since γ1,wis strictly\nmonotone and y∈Sγ1,w,y/\\e}atio\\slash=xwe observe that card( I1)>0 and card( I3)>0,\nwhencey(1)> y(n0). Without loss of generality we may assume that y=y∗.\nThen, we have\n1 =n0−1/summationdisplay\nn=1y∗∗(n)w(n)+n0/summationdisplay\ni=1y(i)∞/summationdisplay\nn=n0w(n)\nn(9)\n=n0−1/summationdisplay\nn=1n/summationdisplay\ni=1y(i)w(n)\nn+n0/summationdisplay\ni=1y(i)∞/summationdisplay\nn=n0w(n)\nn.\nMoreover, by assumption that z∈Sγ1,wandx= (y+z)/2 it follows that z(i) =\n2c−y(i) for anyi∈ {1,...,n 0}andz(i) = 0 for all i>n0. Thus, we obtain\nz∗(n) = (2c−y(n0+1−n))χ{i∈N:i≤n0}(n)14 M. CIESIELSKI & G. LEWICKI\nfor everyn∈N. Consequently, we have\n1 =n0/summationdisplay\nn=1z∗∗(n)w(n)+n0/summationdisplay\ni=1z(i)∞/summationdisplay\nn=n0+1w(n)\nn\n=n0/summationdisplay\nn=1/parenleftBigg\n2cn−n/summationdisplay\ni=1y(n0+1−i)/parenrightBigg\nw(n)\nn+/parenleftBigg\n2cn0−n0/summationdisplay\ni=1y(i)/parenrightBigg∞/summationdisplay\nn=n0+1w(n)\nn\n=2cφγ1,w(n0)−n0/summationdisplay\nn=1n/summationdisplay\ni=1y(n0+1−i)w(n)\nn−n0/summationdisplay\ni=1y(i)∞/summationdisplay\nn=n0+1w(n)\nn.\nHence, by definition of cwe obtain that\n(10) 1 =n0−1/summationdisplay\nn=1n/summationdisplay\ni=1y(n0+1−i)w(n)\nn+n0/summationdisplay\ni=1y(i)∞/summationdisplay\nn=n0w(n)\nn.\nFurthermore, since y=y∗andy(1)>y(n0), we infer that for every n<n0,\nn/summationdisplay\ni=1y(i)>n/summationdisplay\ni=1y(n0+1−i).\nIn consequence, since W(n0−1)>0, by (9) and (10) we conclude\n1 =n0−1/summationdisplay\nn=1n/summationdisplay\ni=1y(i)w(n)\nn+n0/summationdisplay\ni=1y(i)∞/summationdisplay\nn=n0w(n)\nn\n>n0−1/summationdisplay\nn=1n/summationdisplay\ni=1y(n0+1−i)w(n)\nn+n0/summationdisplay\ni=1y(i)∞/summationdisplay\nn=n0w(n)\nn= 1,\nwhich gives us a contradiction and finishes the proof. /square\n5. dual and predual spaces of sequence lorentz spaces γ1,w\nNow, wepresentacharacterizationofthedualandpredualspac esofthesequence\nLorentz space γ1,w.\nTheorem 5.1. Letw= (w(n))n∈Nbe a nonnegative weight sequence and let φγ1,w\nbe the fundamental sequence of the sequence Lorentz space γ1,w. ThenW(∞) =∞\nif and only if every linear bounded functional fonγ1,whas the form\nf(x) =∞/summationdisplay\nn=1x(n)y(n) for any x∈γ1,w,and/bardblf/bardblγ∗\n1,w=/bardbly/bardblmψ\nwherey∈mψandψ(n) =n/φγ1,w(n)for everyn∈N.\nProof. Sufficiency. Suppose that W(∞)<∞. We claim that ℓ∞֒→γ1,w. Indeed,\ntakingx=χNit is easy to see that x∗∗=xand/bardblx/bardblγ1,w=W(∞)<∞, which\nimplies our claim. Let f∈γ∗\n1,w. Then, by assumption there exists y∈mψsuch\nthat\n/bardblf/bardblγ∗\n1,w=/bardbly/bardblmψ≥1\nφγ1,w(n)n/summationdisplay\ni=1y∗(k)\nfor alln∈N. Next, in view of the inequality\nW(n)≤φγ1,w(n)≤W(∞)<∞SEQUENCE LORENTZ SPACES AND THEIR GEOMETRIC STRUCTURE 15\nfor everyn∈N, it follows that φγ1,w(∞) =W(∞)<∞. Thus , we have\nφγ1,w(∞)/bardblf/bardblγ∗\n1,w≥∞/summationdisplay\ni=1y∗(k),\nwhencey∈ℓ1. Therefore, we observe that mψ֒→ℓ1. Moreover, since γ1,wandmψ\nare symmetric by Corollary 6.8 in [2] we conclude that γ1,w֒→ℓ∞andℓ1֒→mψ.\nHence, since ℓ∞is the dual space of ℓ1(see [16]) we have a contradiction.\nNecessity. Sinceγ1,wis a symmetric space, by Corollary 4.4 and Theorem 2.7 in\n[2] we get the associate space γ′\n1,wofγ1,wis a symmetric space and\nγ′\n1,w=/braceleftBigg\ny= (y(n)) : sup\nx∈Bγ1,w/braceleftBigg∞/summationdisplay\nn=1y∗(n)x∗(n)/bracerightBigg\n<∞/bracerightBigg\n.\nNext, in view of Theorem 4.1 it follows that γ1,wis order continuous if and only\nifW(∞) =∞. Hence, by Theorem 4.1 in [2] we have W(∞) =∞if and only\nif the dual space γ∗\n1,wand the associate space γ′\n1,wof the sequence Lorentz space\nγ1,wcoincide, i.e. γ∗\n1,w=γ′\n1,w. Consequently, assuming that ψis the fundamental\nsequence of the dual space γ∗\n1,w, by Remark 4.2 we conclude that γ∗\n1,w֒→mψ. Now,\nwe prove the reverse embedding, i.e. mψ֒→γ∗\n1,w. First, by Theorem 5.2 in [2] we\nobtainψ(n)φγ1,w(n) =nfor alln∈N. Therefore, we have\nmψ=/braceleftbigg\nx= (x(n))n∈N: sup\nn∈N{x∗∗(n)ψ(n)}<∞/bracerightbigg\n(11)\n=/braceleftbigg\nx= (x(n))n∈N: sup\nn∈N/braceleftbigg/summationtextn\ni=1x∗(i)\nφγ1,w(n)/bracerightbigg\n<∞/bracerightbigg\n.\nWe claim that for any y∈mψthe mapping fygiven by\nfy(x) =∞/summationdisplay\nn=1x(n)y(n) for all x∈γ1,w\nis a linear bounded functional on γ1,w. Indeed, taking y∈mψ, by the Hardy-\nLittlewood inequality we obtain that for any x∈γ1,w,\n|fy(x)| ≤∞/summationdisplay\nn=1|x(n)y(n)| ≤∞/summationdisplay\nn=1x∗(n)y∗(n).\nNow, for simplicity of our notation let us denote [1 ,i] ={1,2,...,i}for anyi∈N.\nNext, picking x∈γ1,wwith a finite measure support, without loss of generality we\nmay assume that\nx∗(n) =N/summationdisplay\nk=1akχ[1,ik](n),16 M. CIESIELSKI & G. LEWICKI\nfor everyn∈N, where (ik)N\nk=1⊂Nis strictly increasing and ak>0 for any\nk∈ {1,...,N}. Then, we get\n|fy(x)| ≤∞/summationdisplay\nn=1x∗(n)y∗(n) =N/summationdisplay\nk=1ak∞/summationdisplay\nn=1χ[1,ik](n)y∗(n) (12)\n=N/summationdisplay\nk=1akik/summationdisplay\nn=1y∗(n)≤N/summationdisplay\nk=1akφγ1,w(ik) sup\n1≤k≤N/braceleftBigg/summationtextik\nn=1y∗(n)\nφγ1,w(ik)/bracerightBigg\n≤ /bardbly/bardblmψN/summationdisplay\nk=1akφγ1,w(ik).\nFurthermore, we observe that\n/bardblx/bardblγ1,w=∞/summationdisplay\nn=1x∗∗(n)w(n) =∞/summationdisplay\nn=1w(n)\nnn/summationdisplay\nj=1N/summationdisplay\nk=1akχ[1,ik](j)\n=∞/summationdisplay\nn=1w(n)\nnN/summationdisplay\nk=1akmin{n,ik}\n=N/summationdisplay\nk=1ak∞/summationdisplay\nn=1w(n)\nnmin{n,ik}\nand\nφγ1,w(ik) =W(ik)+W1(ik) =∞/summationdisplay\nn=1w(n)\nnmin{n,ik}\nfor everyk∈ {1,...,N}. Hence, by (12) it follows that\n(13) |fy(x)| ≤ /bardbly/bardblmψ/bardblx/bardblγ1,w\nfor anyx∈γ1,wwith a finite measure support. Finally, since W(∞) =∞, by\nTheorem 4.1 we get γ1,wis order continuous, and so every element x∈γ1,wcan be\nexpressed as a limit of a sequence of elements in γ1,wwith finite measure support.\nThus, we conclude that (13) holds for any x∈γ1,w. Now, we show that there\nexists an isometry between mψandγ∗\n1,w. Next, it is easy to see that there exists\na permutation σ:N→Nsuch thaty∗(n) =|y◦σ(n)|for anyn∈N. We present\nthe proof in two cases.\nCase1.Assume that there is n0∈Nsuch that\n(14) /bardbly/bardblmψ=1\nφγ1,w(n0)n0/summationdisplay\ni=1y∗(i)\nWithout loss of generality we may assume that y∗(i)>0 for alli∈[1,n0]. Define\nx(n) =/braceleftBiggsgn(y(n))\nφγ1,w(n0)ifn∈σ([1,n0]),\n0 otherwise.\nThen, we have for any n∈N,\nx∗∗(n) =1\nnn/summationdisplay\ni=11\nφγ1,w(n0)χ[1,n0](i) =1\nφγ1,w(n0)/parenleftBig\nχ[1,n0](n)+n0\nnχN\\[1,n0](n)/parenrightBig\n.SEQUENCE LORENTZ SPACES AND THEIR GEOMETRIC STRUCTURE 17\nConsequently, we get\n/bardblx/bardblγ1,w=∞/summationdisplay\nn=1x∗∗(n)w(n) =1\nφγ1,w(n0)/parenleftBiggn0/summationdisplay\nn=1w(n)+n0∞/summationdisplay\nn=n0+1w(n)\nn/parenrightBigg\n= 1.\nNext, by (14) we observe that\nfy(x) =∞/summationdisplay\nn=1x(n)y(n) =1\nφγ1,w(n0)/summationdisplay\nn∈σ([1,n0])|y(n)| (15)\n=1\nφγ1,w(n0)/summationdisplay\nn∈[1,n0]|y◦σ(n)|\n=1\nφγ1,w(n0)n0/summationdisplay\nn=1y∗(n) =/bardbly/bardblmψ.\nCase2.Suppose that\n(16) /bardbly/bardblmψ= limsup\nn→∞1\nφγ1,w(n)n/summationdisplay\ni=1y∗(i).\nThen, defining a sequence ( xm) by\nxm(n) =/braceleftBiggsgn(y(n))\nφγ1,w(m)ifn∈σ([1,m]),\n0 otherwise.\nand proceeding analogously as in case 1 we obtain /bardblxm/bardblγ1,w= 1 for all m∈N.\nMoreover, by (16), passing to subsequence and relabeling if neces sary we have\n(17)fy(xm) =1\nφγ1,w(m)/summationdisplay\nn∈σ([1,m])|y(n)|=1\nφγ1,w(m)m/summationdisplay\nn=1y∗(n)→ /bardbly/bardblmψ.\nFinally, combining both cases and according to (13), (15) and (17) w e finish the\nproof. /square\nTheorem 5.2. Letwbe a nonnegative weight sequence. The Marcinkiewicz space\nm0\nψis the predual of the sequence Lorentz space γ1,wif and only if W(∞) =∞,\nwhere\nψ(n) =n\nφγ1,w(n)for anyn∈N.\nAdditionally, if W(∞) =∞then there exists an isometry between the sequence\nLorentz space γ1,wand the dual space (m0\nψ)∗of the Marcinkiewicz space m0\nψ.\nProof.First, we define for any i∈N,\nv(i) =∞/summationdisplay\nk=iw(k)\nk.\nClearly,(v(i))i∈Nisadecreasingsequenceand0 ≤v(i)<∞foralli∈N. Moreover,\nwe easily observe that for every i∈N,\nv(i) =∞/summationdisplay\nk=iw(k)\nk=w(i)+∞/summationdisplay\nk=i+1w(k)\nk−(i−1)w(i)\ni.18 M. CIESIELSKI & G. LEWICKI\nHence, we evaluate\nφγ1,w(n) =n/summationdisplay\ni=1w(i)+n∞/summationdisplay\ni=n+1w(i)\ni\n=n/summationdisplay\ni=1/parenleftBigg\nw(i)+∞/summationdisplay\nk=i+1w(k)\nk−(i−1)w(i)\ni/parenrightBigg\n=n/summationdisplay\ni=1v(i)\nfor alln∈N. Next, since ψ(n) =n/φγ1,w(n) for eachn��N, by (11) and by\nTheorem 3.4 in [9] it follows that mψis the bidual of m0\nψif and only if φγ1,w(∞) =\n∞. Now, we claim that W(∞) =∞if and only if φγ1,w(∞) =∞. Indeed, it is easy\nto see that for any n∈N,\nW(n)≤φγ1,w(n)≤n/summationdisplay\ni=1w(i)+n∞/summationdisplay\ni=n+1w(i)\nn=W(∞),\nwhich implies our claim. Therefore, according to Theorem 5.1 we obtain that the\nMarcinkiewicz space m0\nψis the predual of the sequence Lorentz space γ1,wif and\nonly ifW(∞) =∞. Now, we show that there exists an isometry between ( m0\nψ)∗\nandγ1,w. First, since φγ1,w(∞) =∞, in view of Theorem 3.2 in [9] it follows that\nm0\nψis a non-trivial subspace of all order continuous elements of mψ. Then, defining\nfor anyx∈γ1,wthe linear mapping fxby\nfx(y) =∞/summationdisplay\nn=1x(n)y(n) for any y∈m0\nψ,\nand proceeding analogously as in Theorem 5.1 we are able to show that\n(18) |fx(y)| ≤ /bardbly/bardblmψ/bardblx/bardblγ1,w\nfor anyy∈m0\nψ. On the other hand, it is well known that there exists σ:N→Na\npermutation such that x∗(n) =|x◦σ(n)|for alln∈N. Define\ny(n) =/braceleftBigg\nsgn(x(n))v(σ−1(n)) ifn∈σ(N),\n0 otherwise,\nfor anyn∈N. Then, we have\nfx(y) =∞/summationdisplay\nn=1x(n)y(n) =/summationdisplay\nn∈σ(N)|x(n)|v◦σ−1(n) =∞/summationdisplay\nn=1x∗(n)v(n) =/bardblx/bardblγ1,w,\nwhence, according to (18) we finish the proof. /square\n6. Application\nThis section is devoted to a relationship between the existence set a nd one-\ncomplemented subspaces of the sequence Lorentz space γ1,w. Moreover, we present\na complete characterization of smooth points in the sequence Lore ntz spaceγ1,w\nand its dual space and predual space. Finally we show full criteria fo r extreme\npoints in the dual space of the sequence Lorentz space γ1,w.\nFirst, let us recall some basic definitions and notations that corres ponds to the\nbest approximation. Let Xbe a Banach space and C⊂Xbe a nonempty set. A\ncontinuous surjective mapping P:X→Cis called a projection onto C, whenever\nP|C= Id, i.e.P2=P. Given a subspace Vof a Banach space X, byP(X,V) weSEQUENCE LORENTZ SPACES AND THEIR GEOMETRIC STRUCTURE 19\ndenote the set of all linear bounded projections from XontoV. Let us recall that\na closed subspace Vof a Banach space Xis said to be one-complemented if there\nexists a norm one projection P∈P(X,V). A setC⊂Xis said to be an existence\nsetof the best approximation if for any x∈Xwe have\nRC(x) =/braceleftbigg\ny∈C:/bardblx−y/bardblX= inf\nc∈C/bardblx−c/bardblX/bracerightbigg\n/\\e}atio\\slash=∅.\nIt is obvious that any one-complemented subspace is an existence s et. The converse\nin general is not true. By a deep result of Lindenstrauss [15] ther e exists a Banach\nspaceXand a linear subspace VofXsuch thatVis an existence set in Xand\nVis not one-complemented in X. However, if Xis a smooth Banach space both\nnotions are equivalent (see Proposition 5 in [1]). We will show that both notions\nare equivalent in γ1,w, which is obviously not a smooth space.\nFirst, we establish an isometry between the sequence Lorentz spa cesγ1,wand\nd1,vfor some nonnegative sequences wandv.\nRemark 6.1. Assuming that w= (w(n))n∈Nis a nonnegative weight sequence, we\nmay easily show that there exits a linear surjective isometr yTform the sequence\nLorentz space γ1,wonto the sequence Lorentz space d1,v, wherev= (v(n))n∈Nis\ngiven by\n(19) v(i) =∞/summationdisplay\nn=iw(n)\nnfor anyi∈N.\nIndeed, taking x∈γ1,wwe observe that\n/bardblx/bardblγ1,w=∞/summationdisplay\nn=1w(n)\nnn/summationdisplay\ni=1x∗(i) =∞/summationdisplay\ni=1x∗(i)∞/summationdisplay\nn=iw(n)\nn=∞/summationdisplay\nn=1x∗(n)v(n) =/bardblx/bardbld1,v.\nTheorem 6.2. Letwbe a nonnegative weight sequence and let V⊂γ1,w,V/\\e}atio\\slash={0}\nbe a linear subspace. If Vis an existence set, then Vis one-complemented.\nProof.Letvbe a nonnegative sequence given by (19). Then, by Remark 6.1 ther e\nexistsalinearsurjectiveisometry T:γ1,w→d1,v. Hence, since Vis anexistenceset\ninγ1,w, by Lemma 3.4 in [10] it follows that T(V)/\\e}atio\\slash={0}is an existence set in d1,v.\nIn consequence, by Theorem 3.10 in [10] we infer that T(V) is one complemented in\nd1,v. Finally, applying again Lemma 3.4 in [10] we get that Vis one-complemented\ninγ1,w. /square\nWe present a full criteria for smooth points in the sequence Lorent z spaceγ1,w\nand its dual and predual spaces. First, let us notice that by Theor em 1.10 in [10]\nand by Remark 6.1, the next theorem follows immediately.\nTheorem 6.3. Letwbe a nonnegative weight sequence and let x∈Sγ1,w. Then,\nan element xis a smooth point in γ1,wif and only if the following conditions are\nsatisfied\n(i) card(supp( x)) =∞.\n(ii)If there isn∈Nsuch thatw(n)>0, thenx∗(n)>x∗(n+1).\nTheorem 6.4. Letwbe a nonnegative weight sequence and ψ(n) =n/φγ1,w(n)for\nanyn∈Nandx∈Sm0\nψ. Then, an element xis a smooth point in m0\nψif and only\nif\ncard{n∈N:x∗∗(n)ψ(n) = 1}= 1.20 M. CIESIELSKI & G. LEWICKI\nProof.Letvbe a sequence given by (19) and let V(n) =/summationtextn\ni=1v(i). Then, by\nRemark 6.1 we easily observe that V(n) =φγ1,w(n) =n\nψ(n)for everyn∈Nand\nm0\nψ=/braceleftBigg\nx∈mψ: lim\nn→∞1\nV(n)n/summationdisplay\ni=1x∗(i) = 0/bracerightBigg\n.\nHence, in view of Theorem 1.5 in [10] we complete the proof. /square\nDirectly, by Theorem 1.9 in [10] and Remark 6.1 and also Theorem 5.1 we in fer\nthe following theorem.\nTheorem 6.5. Letwbe a nonnegative weight sequence and ψ(n) =n/φγ1,w(n)for\nanyn∈Nandx∈Sγ∗\n1,w. Then, an element xis a smooth point in Bγ∗\n1,wif and\nonly if there exists n0∈Nsuch that\nx∗∗(n0)ψ(n0) = 1>sup\nn/\\e}atio\\slash=n0{x∗∗(n)ψ(n)}.\nThelastessentialapplicationofTheorem5.1andRemark6.1,inviewof Theorem\n2.2 in [10], is the next result which presents an equivalent condition for extreme\npoints in the dual space γ∗\n1,wof the sequence Lorentz space γ1,w.\nTheorem 6.6. Letwbe a nonnegative weight sequence and ψ(n) =n/φγ1,w(n)\nfor anyn∈Nandx∈Sγ∗\n1,w. Then,xis an extreme point of Bγ∗\n1,wif and only if\nx∗(n) =/summationtext∞\ni=nw(i)\nifor alln∈N.\nRemark 6.7. Although applying Theorem 2.6 in [10]and Remark 6.1 we are able\nto find successfully an equivalent condition for an extreme p oint in the sequence\nLorentz space γ1,w, withwa nonnegative weight sequence, we present the proof of\nthis problem with all details (see Theorem 4.7). It is worth m entioning that the\ntechniques, that was presented in the proof of Theorem 4.7, m ight be interesting for\nreaders and applicable to search a complete characteristic of an extreme point in\nγp,wwith1<p<∞.\nAcknowledgement.∗Thefirstauthor(MaciejCiesielski)issupportedbytheMin-\nistry of Science and Higher Education of Poland, grant number 04/4 3/DSPB/0094.\nReferences\n[1] B. Beauzamy, and B. Maurey, Points minimaux et ensembles optimaux dans les espaces de\nBanach , J. Functional Analysis 24(1977), no. 2, 107139.\n[2] C. Bennett and R. Sharpley, Interpolation of operators , Pure and Applied Mathematics, 129.\nAcademic Press, Inc., Boston, MA, 1988.\n[3] M. Ciesielski, On geometric structure of symmetric spaces , J. Math. Anal. Appl. 430(2015),\nno. 1, 98-125.\n[4] M. Ciesielski, Relationships between K-monotonicity and rotundity properties with applica-\ntion, J. Math. Anal. Appl. (2018), https://doi.org/10.1016/j. jmaa.2018.05.008.\n[5] M. Ciesielski, A. Kami´ nska, P. Kolwicz and R. P/suppress lucienn ik,Monotonicity and rotundity of\nLorentz spaces Γp,w, Nonlinear Anal. 75(2012), no. 5, 2713-2723.\n[6] M. Ciesielski, A. Kami´ nska and R. P/suppress luciennik, Gˆ ateaux derivatives and their applications to\napproximation in Lorentz spaces Γp,w, Math. Nachr. 282(2009) no. 9, 1242-1264.\n[7] W.J. Davis and P. Enflo, Contractive projections on lpspaces , Analysis at Urbana, Vol. I\n(Urbana, IL, 1986-1987), 151-161, London Math. Soc. Lectur e Note Ser., 137, Cambridge\nUniv. Press, Cambridge, 1989.\n[8] J.E. Jamison, A. Kami´ nska, and G. Lewicki, One-complemented subspaces of Musielak-Orlicz\nsequence spaces , J. Approx. Theory 130(2004), no. 1, 1-37.SEQUENCE LORENTZ SPACES AND THEIR GEOMETRIC STRUCTURE 21\n[9] A. Kami´ nska and H. J. Lee, M-ideal properties in Marcinkiewicz spaces , Comment. Math.\n(Prace Mat.) 2004, Tomus specialis in Honorem Juliani Musie lak, 123-144.\n[10] A. Kami´ nska, H. J. Lee and G. Lewicki, Extreme and smooth points in Lorentz and\nMarcinkiewicz spaces with applications to contractive pro jections , Rocky Mountain J. Math.\n39(2009), no. 5, 1533-1572.\n[11] A. Kami´ nska and G. Lewicki, Contractive and optimal sets in modular spaces , Math. Nachr.\n268(2004), 74-95.\n[12] A. Kami´ nska and L. Maligranda, On Lorentz spaces Γp,w, Israel J. Math. 140 (2004), 285-\n318.\n[13] A. Kami´ nska and L. Maligranda, Order convexity and concavity of Lorentz spaces Λp,w,\n0< p <∞, Studia Math. 160(2004), no. 3, 267-286.\n[14] S. G. Krein, Yu. I. Petunin and E. M. Semenov, Interpolation of linear operators , Trans-\nlated from the Russian by J. Sz˝ ucs. Translations of Mathema tical Monographs, 54. American\nMathematical Society, Providence, R.I., 1982.\n[15] J. Lindenstrauss, On projections with norm 1-an example , Proc. Amer. Math. Soc. 151964\n403-406.\n[16] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces. II. Function spaces , Ergebnisse\nder Mathematik und ihrer Grenzgebiete [Results in Mathemat ics and Related Areas], 97.\nSpringer-Verlag, Berlin-New York, 1979.\n[17] B. Randrianantoanina, Norm-one projections in Banach spaces. International Conf erence\non Mathematical Analysis and its Applications (Kaohsiung, 2000) , Taiwanese J. Math. 5\n(2001), no. 1, 35-95.\n[18] H. L. Royden, Real analysis , Third edition, Macmillan Publishing Company, New York, 19 88.\nMaciej Ciesielski Grzegorz Lewicki\nInstitute of Mathematics Department of Mathematics\nFaculty of Electrical Engineering and Computer Science\nPozna´ n University of Technology Jagiellonian University\nPiotrowo 3A, 60-965 Pozna´ n, Poland /suppress Lojasiewicza 6, 30-34 8 Krak´ ow, Poland\nemail: maciej.ciesielski@put.poznan.pl email: grzegorz .lewicki@im.uj.edu.pl"    },    {        "title": "2305.02674v1.Vibrational_resonance_in_a_damped_and_two_frequency_driven_system_of_particle_on_a_rotating_parabola.pdf",        "content": "arXiv:2305.02674v1  [nlin.CD]  4 May 2023Springer Nature 2021 LATEX template\nVibrational resonance in a damped and\ntwo-frequency driven system of particle on a\nrotating parabola\nR Kabilan1†, M Sathish Aravindh1,2†, A Venkatesan1*\nand M Lakshmanan2\n1*PG & Research Department of Physics, Nehru Memorial\nCollege (Autonomous), Affiliated to Bharathidasan Universi ty,\nPuthanampatti, Tiruchirappalli, 621007, Tamil Nadu, Indi a.\n2*Department of Nonlinear Dynamics, School of Physics,\nBharathidasan University, Tiruchirappalli, 620024, Tami l Nadu,\nIndia.\n*Corresponding author(s). E-mail(s): av.phys@gmail.com ;\nContributing authors: kabilanrajagopal82@gmail.com ;\nsathisharavindhm@gmail.com ;lakshman.cnld@gmail.com ;\n†These authors contributed equally to this work.\nAbstract\nIn the present work, we examine the role of nonlinearity in vi brational\nresonance (VR) of a forced and damped form of a velocity-depe ndent\npotential system. Many studies have focused on studying the vibra-\ntional resonance in different potentials, like bistable pot ential, asym-\nmetrically deformed potential, and rough potential. In thi s connection,\nvelocity-dependent potential systems are very important f rom a physical\npoint of view (Ex: pion-pion interaction, cyclotrons and ot her electro-\nmagnetic devices influenced by the Lorentz force, magnetron s, mass\nspectrometers). They also appear in several mechanical con texts. In\nthis paper, we consider a nonlinear dynamical system with ve locity-\ndependent potential along with additional damping and driv en forces,\nnamely a particle moving on a rotating-parabola system, and study\nthe effect of two-frequency forcing with a wide difference in t he fre-\nquencies. We report that the system exhibits vibrational re sonance in\na certain range of nonlinear strength. Using the method of se paration\n1Springer Nature 2021 L ATEX template\n2Vibrational resonance in a system of particle on a rotating p arabola\nof motions (MSM), an analytical equation for the slow oscill ations of\nthe system is obtained in terms of the parameters of the fast s ignal.\nThe analytical computations and the numerical studies conc ur well.\nKeywords: Nonlinear systems, Vibrational Resonance, nonpolynomial\noscillator, velocity-dependent potential system\n1 Introduction\nSince its observation by Landa and McClintock [ 1], the phenomenon of vibra-\ntional resonance (VR) has been studied analytically, computationa lly, and\nempirically in several models of nonlinear systems, spanning many field s of\nscience and engineering. Vibrational resonance primarily relies on th e cooper-\nation oftwo input signalswith radicallydifferent frequency values - a w eak low\nfrequency and a fast high frequency - to improve the quality of the response\nof the weak frequency signal. Theoretical studies and experiment al realization\nof VR have been reported in diverse fields of research.\nIn particular, the VR phenomenon has been studied in bistable syste ms\n[2–4], multistable systems [ 5], ratchet devices [ 6], excitable systems [ 7], quintic\noscillators [ 8], coupled oscillators [ 9], overdamped systems, delayed dynami-\ncal systems [ 8], asymmetric potential in Duffing oscillators, fractional order\npotential oscillators[ 10], FHN neuralmodels [ 11], oscillatorynetworks[ 12,13],\nbiological nonlinear systems [ 14], parametrically excited systems [ 15], systems\nwith nonlinear damping [ 16], deformed potentials [ 17] as well as harmonically\ntrapped and roughed potentials [ 16,18].\nMore significantly, experimental realizations of VR have been made, par-\nticularly in optical system, bistable surface emitting lasers [ 19], multistable\nsystems, arrays of hard limiters [ 20], and Chua’s circuits [ 21]. The potential\napplication of this novel phenomenon has been found in many areas o f science,\nnamely energy harvester from bistable oscillator, energy detecto rs [22], signal\ndetection, signal transmission and amplification [ 23,24], and the detection of\nfaults in bearings [ 25], also in the design of logic gates and memory devices\n[26–28].\nAt the vibrational resonance, it is observed that the effective pot ential is\nbifurcated. As a result many studies have focused on the effect of potentials.\nFurther the roleplayedbythe systemparameters,delayterms, f ractionalorder\nterms, and potential deformation have been also considered on th e induction\nand control of VR [ 8,17].\nDespite the fact that VR is realized and observed in many systems, o ne\ncan realize that they all belong to systems whose potentials are ass umed to\nbe polynomial functions. It will be interesting to investigate the pos sibility\nand nature of VR in physically interesting nonpolynomial systems. Fr om this\nconsideration, in the present work a mechanical system which corr esponds to\na nonpolynomial velocity-dependent potential is considered for su ch a study.Springer Nature 2021 L ATEX template\nVibrational resonance in a system of particle on a rotating p arabola 3\nThe model under consideration is a mechanical model with a rich unde rly-\ning nonlinear dynamics [ 29–31]. This model often depicts a particle’s motion\non a rotating parabola. This mechanical model also describes centr ifugation\nequipment, centrifugal forces, industrial events, and a motorb ike being driven\nin a revolving parabolic well in a circus.\nAs mentioned above, many of the works on VR are concerned with po ly-\nnomial oscillators and very few works have focused on VR in nonpolyn omial\noscillators. In particular, VR in driven oscillators with position-depen dent\nmass [32], VR in the Morse oscillator [ 3], VR in a dual-frequency-driven\nTietz-Hua quantum well [ 33], identification and parameter estimation of non-\npolynomial forms of damping nonlinearity in dynamic systems [ 34] are few of\nthe works on VR in nonpolynomial oscillators. More detailed investigat ions on\nVRinnonpolynomialoscillatorsarerequiredtounravelthenatureo fresonance\nbehavior in these oscillators depending upon the various system par ameters.\nIn particular, it will be important to explore the role of nonlinearity pa ram-\neter in contrast to the damping and forcing parameters in the onse t of VR\nphenomenon.\nIn this connection, there are numerous methods available to appro ximate\nthe solution of nonlinear equations. Methods such as the Lindstedt –Poincar´ e\nperturbation method, the multiple scale perturbation method, the harmonic\nbalance method, and the averaging method are quite useful for th is purpose.\nEach one of the above procedures has its own set of advantages a nd disad-\nvantages, as well as ranges of validity and applicability in specific situa tions.\nHowever, the method of separation of motion (MSM) [ 17,32] features sev-\neral significant advantages over the above methods specifically in c onnection\nwith the study of VR analytically, e.g., the simplicity in application and the\ntransparency of the physical interpretation. Hence, in order to investigate VR\ntheoretically, we used the MSM to divide the equation of motion into a s low\nmotion and a fast motion and obtain a set of integro-differential equ ations.\nThe composite system is completely solved by this pair of integro-diffe rential\nequations, which describe the equations of slow oscillations and fast vibra-\ntions, respectively, and their superposition. The method of direct separationof\nmotion appears to be a convenient and simple tool for obtaining the e quation\nof a system’s slow motion. It is also applicable to solving many mechanica l\nproblems. This MSM method is employed to study vibrational resonan ce in\nmany nonlinear systems with polynomial potential functions. In the present\nwork,weextend the generalframeworkofthe MSM method followin gthe work\nof Roy-Layinde et al.[32] to study the vibrational resonance in the nonpoly-\nnomial oscillator with velocity-dependent potential. Also, in the pres ent work,\nthe role of nonlinearity on VR in a nonpolynomial system of damped, dr iven\nparticle moving on a rotating parabola is examined. Most of the studie s on\nVR in the literature are concerned with role of damping in either suppr essing\nor increasing the resonance behaviour [ 3]. But in the present system, it is also\nshown that the VR occurs in an optimal range of nonlinear strength therebySpringer Nature 2021 L ATEX template\n4Vibrational resonance in a system of particle on a rotating p arabola\nemphasizingtheroleofnonlinearity.Itisquiteinterestingandentire lydifferent\nfrom the VR studies on other polynomial and nonpolynomial oscillator s.\nIt is an established fact that resonance occurs in a nonlinear oscillat or sys-\ntem if the matching of the natural frequency of an underdamped o scillation of\nthe system with an external periodic driving force causes the resp onse ampli-\ntude of the system to be enhanced. In recent times, it has been fo und that the\nsystem amplification of response, or the term “resonance,” is being used by\nadjusting other system parameters, including different kinds of fo rces.\nAsaresult,manytermsarebeingintroduced,namely,stochasticr esonance,\nchaotic resonance, ghost resonance, auto resonance, anti re sonance, and so on,\ndepending upon the manifestation of forces or system parameter s [details are\ngiven in ref.[ 3,35]].\nIn the present work, we essentially consider a nonpolynomial nonline ar\noscillator model, namely the motion of a particle on a rotating parabola\nwith additional interactions, which is quite different from the genera lized\nMathews-Lakshmanan oscillator model with additional interactions studied in\nref.[32]. The present oscillator is though analogous to a position-dep endent\nmass (PDM) system, the mass is associated with the strength of no nlinearity.\nIn the present paper, we mainly focus on a specific form of vibration al res-\nonance, which is discussed in terms of a slowly driven system’s respon se to\nvariations in the parameters associated with a high-frequency per iodic force.\nIn the absenceofnonlinearity,the model consideredbecomesaline arharmonic\noscillator. We have shown that no VR occurs in this system without no nlinear-\nity. On the other hand for a lowlevel of nonlinearitymaximum respons eoccurs\nin this model, which is the most novel feature. Thus, it is essential an d impor-\ntant to study the effect of nonlinearity in the system. We have show n that the\nenhancement or suppression of vibrational resonance is possible t hrough vari-\nation of the nonlinear system parameter in the presence of low-fre quency and\nhigh-frequency forces within appropriate parameter regimes.\nOur results, while confirming the basic features of VR as identified in t he\ncase of PDM Duffing and Morse oscillators, specifically brings out nove l fea-\ntures on the effect of nonlinearity. The main contribution of this pap er is that\nthe mere presence of nonlinearity alone does not guarantee the em ergence of\nVR, as in the caseof polynomial nonlinear oscillators.An appropriate strength\nof nonlinearity region is solely responsible for the induction and contr ol of VR.\nThus, in the present study, we have investigated the role of nonpo lynomial\nnonlinearity in oscillator systems for inducing vibrational resonance . In par-\nticular, we have shown that the vibrational resonance is realized an d observed\nwithin an optimal range of nonlinear strength. From our study, it is in ferred\nthat resonance cannot be observed at all parameter values of no nlinearity.\nThe article is organized as follows : In Section 2, the dynamical model is\ndiscussed.Analyticalstudiesandcomparisonbetweennumericala ndanalytical\nstudies based on method of separation of motions are given in Sectio ns3&4,\nrespectively. Finally, the paper is concluded in Section 5.Springer Nature 2021 L ATEX template\nVibrational resonance in a system of particle on a rotating p arabola 5\nFig. 1 The red circle shows how a freely sliding particle moves down a parabolic wire that\nrotates at a constant angular velocity Ω [ 29].\n2 The dynamical model\nIn this article, we consider the nonpolynomial oscillator governed by the\nequation of motion [ 29–31]\n(1+µx2)¨x+µx˙x2+ω2\n0x= 0, (1)\nand subjected to two additional periodic forcings and extra linear d amping.\nFor the analysis of fundamentally nonlinear processes in a non-gene ric\nmechanical system that correspond to a well-known model [ 30,31] is Eq.(1)\nwith the Lagrangian,\nL=1\n2/bracketleftbig\n(1+µx2)˙x2−ω2\n0x2/bracketrightbig\n. (2)\nThe corresponding Hamiltonian is\nH=1\n2/bracketleftbig\np2(1+µx2)−1+ω2\n0x2/bracketrightbig\n, (3)\nwhere the canonical conjugate momentum is\np= ˙x(1+µx2). (4)\nWe take into consideration a mechanical model that depicts how a fr eely\nmoving particle of unit mass moves down a parabolic wire that rotates around\nthe particle’s axis z=√µx2with a constant angular velocity Ω (Ω2= Ω2\n0=\n−ω2\n0+g√µ), where µ >0 andω0>0 as shown in Fig. 1. Here, ‘g’ stands forSpringer Nature 2021 L ATEX template\n6Vibrational resonance in a system of particle on a rotating p arabola\nthe acceleration brought on by gravity, ‘1√µ’ stands for the rotating parabola’s\nsemi-lotus rectum, and ‘ ω0’ stands for the initial angular velocity [ 29,31].\nNext, we investigate the nonlinear dynamics associated with Eq.( 1) under\nthe influence of additional damping and two periodic external forcin gs, leading\nto the equation of motion\n(1+µx2)¨x+µx˙x2+α˙x+ω2\n0x=f1cosω1t+f2cosω2t. (5)\nIn the above f1andf2correspond to the strength of two periodic forces\nwith period 2 π/ω1and 2π/ω2respectively. Here ω1andω2are the strength of\nthe low frequency and high frequency forces, respectively.\nForasingleforce,Eq.( 5)hasbeenwellinvestigatedandit showsthe period-\ndoubling route to chaos, strange nonchaotic attractors, interm ittency, crisis\n[29] and prediction of extreme events using machine learning [ 36].\nWe will express Eq.( 5) in a way that makes our analytical process manage-\nable, in accordance with the work of Roy-Layinde et al.[32]. In particular, we\ndivide Eq.( 5) by (1+ µx2) throughout so that it can be expressed as\n¨x+(µx˙x2+α˙x+ω2\n0x)(1+µx2)−1\n= (1+µx2)−1(f1cosω1t+f2cosω2t). (6)\nNow apply the binomial expansion to the terms (1+ µx2)−1and restrict\nour consideration to the first three terms of the expansion only. E q.(6) can be\nexpressed as,\n¨x+(µx˙x2+α˙x+ω2\n0x)(1−µx2+µ2x4)\n= (1−µx2+µ2x4)(f1cosω1t+f2cosω2t). (7)\nThe potential V(x) associated with ( 7) may then be written\nV(x) =ω2\n0\n2x2−µω2\n0\n4x4+µ2ω2\n0\n6x6. (8)\nThe system potential shown in Fig. 2for given parameter values, ω2\n0andµ,\nis computed from Eq.( 8).\nIn connection with the analysis of the above nonlinear system, a cou -\nple of natural questions arise: In a linear system, is it possible to det ect\nvibrational resonance? What happens when a linear system is subje cted to a\nhigh-frequency force? To answer these questions we consider Eq .(5) without\nthe nonlinear term ( µ= 0.0). The linear equation can be easily integrated and\nthe exact analytical solution is given by\nx(t) =A1em1t+A2em2t+fω1cos(ω1t+φ1)+fω2cos(ω2t+φ2),(9)\nwhereSpringer Nature 2021 L ATEX template\nVibrational resonance in a system of particle on a rotating p arabola 7\n 0 0.25 0.5\n−1  0  1V(x)\nx\nFig. 2 The system potential ( 8) forµ= 0.5 andω2\n0= 0.25.\nfω1=f1/radicalbig\n(ω2\n0−ω2\n1)2+α2ω2\n1, fω2=f2/radicalbig\n(ω2\n0−ω2\n2)2+α2ω2\n2(10)\nFromthe above,itmaybeobservedthatthe structureof fω1isindependent\nof the amplitude f2and the frequency ω2. Therefore, the amplitude fω1is\nunaffected by the presence of the high frequency force in a linear s ystem [3].\nConsequently, no VR occurs in the linear system and so it is important to\nstudy the effect of nonlinearity via analytical and numerical method s. These\nare carried out in the upcoming sections.\n3 Theoretical analysis\nThe conventional method of separation of motions (MSM), which is a per-\nturbation method, is now used, in which it is assumed that the system\ndynamics consists of a slow component y(t) and a fast component z(t,τ). The\nmain equation of the system, Eq.( 7), is solved entirely by superimposing the\nsolutionsofthe twointegro-differentialequationsderivedfromea chofthecom-\nponents using the MSM approach. Following the factorization of like t erms\nand the definition x=y+z, the system equation ( 7) can be represented as\n¨y+ ¨z+µ[µ2y5+5µ2zy4+(10µ2z2−µ)y3+(10µ2z3−3µz)y2+(5µ2z4−3µz2+1)y\n+µ2z5−µz3+z](˙y2+2˙y˙z+ ˙z2)+α[1+µ2y4+4µ2zy3+(6µ2z2−µ)y2\n+(4µ2z3−2µz)y](˙y+ ˙z)+ω2\n0[µ2y5+5µ2zy4+(10µ2z2−µ)y3+(10µ2z3−3µz)y2\n+(5µ2z4−3µz4−3µz2+1)y+µ2z5−µz3+z] = [1+µ2y4+4µ2zy3\n+(6µ2z2−µ)y2+(4µ2z3−2µz)y+µ2z4−µz2](f1cosω1t+f2cosω2t). (11)\nNowtakingthetimeaverageinaperiodoverthefastvariable τandconsidering\nthe fact that the fast signal ’z’ is rapidly oscillating in the term with pe riodSpringer Nature 2021 L ATEX template\n8Vibrational resonance in a system of particle on a rotating p arabola\n2π\nω2, we have\n¨y+µ[µ2y5+5µ2zy4+(10µ2z2−µ)y3+(10µ2z3−3µz)y2+(5µ2z4−3µz2+1)y\n+µ2z5−µz3+z](˙y2+˙z2)+α[1+µ2y4+4µ2zy3+(6µ2z2−µ)y2\n+(4y2z3−2µz)y]˙y+ω2\n0[µ2y5+5µ2zy4+(10µ2z2−µ)y3+(10µ2z3−3µz)y2\n+(5µ2z4−3µz2+1)y+µ2z5−µz3+z] = [1+µ2y4+4µ2zy3\n+(6µ2z2−µ)y2+(4y2z3−2µz)y+µ2z4−µz2+1](f1cosω1t+f2cosω2t).(12)\nIn the above, the average value of ‘z’ in relation to the fast variable ‘τ=ω2t’\nis given by\nz=1\n2π/integraldisplay2π\n0zdτ= 0 (13)\nso that Eq.( 12) becomes,\n¨y+µ[µ2y5+(10µ2z2−µ)y3+(10µ2z3)y2+(5µ2z4−3µz2+1)y\n+µ2z5−µz3](˙y2+˙z2)+α[1+µ2y4+(6µ2z2−µ)y2+4µ2z3y]˙y\n+ω2\n0[µ2y5+(10µ2z2−µ)y3+10µ2z3y2+(5µ2z4−3µz2+1)y+µ2z5−µz3]\n= 1+µ2y4+(6µ2z2−µ)y2+(4µ2z3y+µ2z4−µz2+1)(f1cosω1t),(14)\nEq.(14) describes the slow motion of the system.\nThe averages in the slow motion equation can now be found using the\napproximation method. For the composite system ’ x’, this is done by first\nfinding the equation for the initial oscillations in ’ z’ by deducting Eq.( 14) from\nEq.(11), which is the equation for the slow component ‘ y’. Consequently, the\nequation governing the system’s fast oscillations can be expressed as\n¨z+2µ˙y[µ2y5+5µ2zy4+(10µ2z2−µ)y3+(10µ2z3−3µz)y2+(5µ2z4−3µz2+1)y\n+µ2z5−µz3+z]˙z+µ[5µ2zy4+10µ2y3(z2−z2)+10µ2y2(z3−z3)−3µzy2+\n5µ2y(z4−z4)−3µy(z2−z2)+µ2(z5−z5)−µ(z3−z3)+z]˙y2+[µ2y5(˙z2−˙z2)\n+5µ2z˙z2y4+10µ2y3(z2˙z2−z2˙z2)−µy3(˙z2+˙z2)+10µ2y2(z3˙z2−z3˙z2)−3µy2z˙z2\n+5µ2y(z4˙z2−z4˙z2)−3µy(z2˙z2−z2˙z2)+y(˙z2−˙z2)+µ2(z5˙z2−z5˙z2)\n−µ(z3˙z2−z3˙z2)+z˙z2]+α[4µ2zy3+6µ2y2(z2−z2)+4µ2y(z3−z3)\n−2µzy+µ2(z4−z4)−µ(z2−z2)]˙y+[µ2y4+4µ2zy3+(6µ2z2−µ)y2\n+(4µ2z3−2µz)y+µ2z4−µz2+1]˙z+ω2\n0[5µ2zy4+10µ2y3(z2−z2)\n+10µ2y2(z3−z3)−3µy2z+5µ2y(z4−z4)−3µy(z2−z2)\n+µ2(z5−z5)−µ(z3−z3)+z] = [4zy3+bµ2(z2−z2)y2+4µ2(z3−z3)y−2µzy\n+µ2(z4−z4)−µ(z2−z2)](f1cosω1t)+[µ2y4+4zy3+(6z2µ2−µ)y2\n+(4µ2z3−2µz)y+µ2z4−µz2+1](f2cosω2t). (15)Springer Nature 2021 L ATEX template\nVibrational resonance in a system of particle on a rotating p arabola 9\n 0 0.1 0.2\n-1  0  1Veff(x)\ny\nFig. 3 The effective potential ( 23) forµ= 0.5 andω0= 0.25.\nWe observe that Eqs.( 14) and (15) are a pair of integro-differential\nequations that, when combined, totally solve the composite system Eq.(7)\nin an average sense. They describe the equations for the slow oscilla tions ’y’\nand the fast vibrations z, respectively. Then, using the assumption that the\ncomponent ’ z’ oscillates considerably faster than the slow component ’ y’, we\napply the inertial approximation ¨ z >>˙z >> z >> z2and treat the variable\n’y’ as constant in Eq.( 15). Hence Eq.( 15) is reduced to\n¨z=f2cosω2t (16)\nwhich has a solution\nz=−f2\nω2\n2cosω2t, (17)\nleading to the mean values\nz=z3=z5= 0; z2=f2\n2\n2ω4\n2; (18a)\nz4=3f4\n2\n8ω8\n2;˙z2=f2\n2\n2ω2\n2(18b)\nUsing Eq.( 18) in Eq.( 14), after simplification it reads as the equation of\nmotion for the slow component as\n¨y+µ[µ2y5+(10µ2z2−µ)y3+(5µ2z4−3µz2+1)]˙y2+α[µ2y4+(6µ2z2−µ)y2\n+µ2z4−µz2+1]˙y+[µ(5µ2z4−3µz2+1)˙z2+ω2\n0(5µ2z4−3µz2+1)]y\n+[µ(10µ2z2−µ)˙z2+ω2\n0(10µ2z2−µ)]y3+(µ3˙z2+ω2\n0µ2)y5\n= (µ2y4+(6µ2z2−µ)y2+µ2z4−µz2+1)f1cosω1t (19)\nKeeping in mind the values of various mean values given in Eqs.( 18) and\nredefining the quantities by the following quantities,Springer Nature 2021 L ATEX template\n10Vibrational resonance in a system of particle on a rotating p arabola\n 0 2 4\n 0 5 10 15 20Q\nf2\nFig. 4The relationship between f2and the response amplitude Q. The solid line represents\nanalytically computed response amplitude and the dotted li ne represents the numerical\nresponse amplitude for the fixed value of µ= 0.5,f1= 0.05α= 0.2,ω1= 1.0 andω2= 4.5.\nC1= (5µ2z4−3µz2+1);\nC2= (6µ2z2−µ);\nC3= (10µ2z2−µ);\nC4= 1+µ2z4−µz2, (20)\nη1= (µ˙z2+ω2\n0)C1;\nη2= (µ˙z2+ω2\n0)C3;\nη3=µ2(µ˙z2+ω2\n0), (21)\nthe slow oscillations of the system described by Eq.( 14) can be written as\n¨y+µ(C1y+C3y3+µ2y5)˙y2+α[C4+C2y2+µ2y4]˙y+η1y+η2y3+η3y5\n= (C4+C2y2+µ2y4)f1cosω1t.(22)\nConsequently, the system’s effective potential is given by\nVeff(y) =η1\n2y2+η2\n4y4+η3\n6y6. (23)\nThe linearized equation of motion becomes Eq.( 22) by neglecting the\nnonlinear components and applying the approximation f1<<1 such that\n|y|<<1 in the long-term limit t→ ∞:\n¨y+λ˙y+ω2\nry=Fcosω1t, (24)\nwhere the resonant frequency is ωr=√η1,λ=αC4andF=C4f1. It is\nclear that the steady state solution of equation Eq.( 22) takes the form y(t) =Springer Nature 2021 L ATEX template\nVibrational resonance in a system of particle on a rotating p arabola 11\n-1 0 1\n 0  0.2  0.4  0.6(a)x.\n \nx-2-1 0 1 2\n-1.2 -0.9 -0.6 -0.3  0(b)x.\n \nx-2-1 0 1 2\n-1.2 -0.9 -0.6 -0.3  0(c)x.\n \nx\nFig. 5 Phase-space trajectories for different values of frequenci es of the second periodic\nforce. Panles (a)-(c) represent f2= 5.0,9.0 and 11 .0, respectively, with fixed value of µ= 0.5,\nf1= 0.05 andα= 0.2.\nALcos(ω1t+φ). We define the quantity AL/f1as the response amplitude ‘ Q’.\nThe oscillation amplitude AL=F√\n(ω2r−ω2\n1)2+λ2ω2\n1,φ= tan−1/parenleftbiggλω1\nω2\n1−ω2r/parenrightbigg\nand\nthe response amplitude can be computed as\nQana=AL\nf1=C4/radicalbig\n(ω2r−ω2\n1)2+λ2ω2\n1. (25)\nUsing the original definitions of the various quantities occurring in ( 25), the\nresponse amplitude can be computed as\nQana=1+3µ2f4\n2\n8ω28−µf2\n2\n2ω24\n[((ω2\n0+µf2\n2\n2ω2\n2)(15µ2f4\n2\n8ω8\n2−3µf2\n2\n2ω4\n2+1)−ω2\n1)2+α2(1+3µ2f4\n2\n8ω8\n2−µf2\n2\n2ω4\n2)2ω2\n1]1\n2.(26)\nNow we analyze the VR using Eq.( 26) and we verify numerically the above\ntheoretical results. In order to appreciate the nature of Qana, we consider the\nquantity S= ((ω2\n0+µf2\n2\n2ω2\n2)(15µ2f4\n2\n8ω8\n2−3µf2\n2\n2ω4\n2+1)−ω2\n1)2+α2(1+3µ2f4\n2\n8ω8\n2−µf2\n2\n2ω4\n2)2ω2\n1\nin the denominator of equation ( 26) and we observe that the qualitative fea-\nture ofQanacan be deduced from the nature of S. That is Qanareaches\nthe maximum value provided the value of S is minimum. It is clear from the\ndenominator of equation ( 26), the appearance of resonance depends on the\nsystem parameters ω2\n0,µ,α,f2,ω1, andω2. Now we consider the nonlinear\nparameterand study its effect. We compute numericallyby analyzing the cases\ndS/df2= 0 and d2S/df2\n2>0 at resonance. The results are depicted in Fig. 4.\n4 Numerical Analysis\nNext, the theoretical response amplitude Qanaprovided by Eq.( 26) has been\ncompared with the numerical Qnumobtained from the Fourier spectrum of the\nsolution of the velocity-dependent equation represented as coup led first-order\nautonomous ordinary differential equations (ODEs) of the following form:\ndx\ndt=y,Springer Nature 2021 L ATEX template\n12Vibrational resonance in a system of particle on a rotating p arabola\n 2 3 4\n 11  11.5  12(a)Q\nf2-0.2 0 0.2\n 11  11.5  12(b)x(t)\nf2\nFig. 6The corresponding response curve Qand the bifurcation diag ramofthe displacement\nx(t) forω1= 1.5. The remaining parameters are fixed as µ= 0.5,f1= 0.05 andα= 0.2.\n 0 2 4\n 0  10  20  30µ=0.1 µ=0.5µ=1.0\nµ=1.5Q\nf2\nFig. 7The relationship between f2and the response amplitude Q. The solid lines represent\nthe response amplitudes for different values of µand that curves for µ= 0.1,0.5,1.0 and 1.5\nare represented by different colors.\ndy\ndt= (−µx˙x2−α˙x−ω2\n0x+f1cosω1t+f2cosω2t)×\n(1−µx2+µ2x4). (27)\nTo substantiate our analytical study, we numerically integrate Eq.( 27)\nusing the Runge-Kutta Fourth order (RK4) method with step size ∆ t= 0.01t\nover a simulation of finite time interval Ts=nT, withT= 2π/ω1being\nthe period of oscillations, where the amplitude f1and frequency ω1belong to\nthe slow component and the amplitude f2and frequency ω2are considered\nas belonging to the fast component, and n=1,2,3... is the number of co m-\nplete oscillations. For our computation we fix the system parameter values as\nµ= 0.5,ω2\n0= 0.25,α= 0.2,f1= 0.05 andω1= 1.0. The remaining factors,\nf2andω2, are selected in a way that promotes the emergence of VR.\nAfter solving numerically Eqs.( 27) and using the results in the following\nexpression for the response function after discarding the trans ients, we obtainSpringer Nature 2021 L ATEX template\nVibrational resonance in a system of particle on a rotating p arabola 13\n 1 2 3 4 5\n 5  10  15α=0.2\nα=0.3\nα=0.5Q\nf2\nFig. 8 The dependence of response amplitude Qwithf2. The analytical and numerical\nresults are represented by the solid and dotted line curves, respectively, for different value\nofα.\n 0 2 4 6\n 5  7  9  11  13ω1=0.8\nω1=1.0ω1=1.5Q\nf2\nFig. 9The relationship between f2and the response amplitude Q. The solid line represents\nanalytically computed response amplitude and the dotted li ne represents the numerical\nresponse amplitude for different values of ω1and that curves for ω1= 0.8,1.0 and 1.5 are\nrepresented by different colors.\n[1]\nQnum=/radicalbig\nQ2s+Q2c\nf1, (28)\nover a range of values of the forcing strength of high frequency d riving force\n(f2). Here the values of the quantities QsandQcare computed from the\nFourier spectrum of the time series of the output signal x(t) as\nQc=2\nnT/integraldisplaynT\n0x(t)cosωtdt,\nQs=2\nnT/integraldisplaynT\n0x(t)sinωtdt. (29)\nThe response curves of Qversus selected parameters for a variety of system\nparameters are superposed to compare the numerically calculated Qwith theSpringer Nature 2021 L ATEX template\n14Vibrational resonance in a system of particle on a rotating p arabola\n 0 3 6 9 12\n 0  0.5  1(a)f2\nµ\n 0 5 10 15 20 25\n 0 3 6 9 12\n 0  0.5  1(b)f2\nµ\n 0 1 2 3 4 5 6 7 8 9 10\n 0 3 6 9 12\n 0  0.5  1(c)f2\nµ\n 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5\n 0 3 6 9 12\n 0  0.5  1(d)f2\nµ\n 0 2 4 6 8 10 12 14\nFig. 10 Two-parameter phase diagram for µvs second force f2for fixed ω1=\n0.1,0.5,1.0,1.5: the maximum response amplitude Qas indicated in the figure.\npreviously obtained analytical results. The maximum of each of the a nalytical\ncurveswellmatcheswiththenumericallycomputedresultsandonec anobserve\nthat the VR is clearly indicated by an adequate difference between th e low\nand high frequencies.\nWe begin by examining the occurrence of VR in the response curve of the\nsystem as given by Eq.( 28) above. We fix the parameters as below: µ= 0.5,\nω2\n0= 0.25,α= 0.2,f1= 0.05,ω1= 1.0,andω2= 4.5.Fig.4showstheresponse\namplitude Qof the system depending on the amplitude ofhigh-frequencydrive\nforcing strength f2.\nIt is obvious from Fig. 4that as the fast driving force f2increases, the\nresponseamplitude increasesandit reachesamaximum value around f2= 9.0.\nFurther,asweincreasethevalueof f2,theresponseamplitude Qstartsdecreas-\ning. The analytically calculated response amplitude from Eq.( 26), shown as a\nsolid line for a set of system parameter values, and Eq.( 29), which is computed\ndirectlyfromthesystem’smainequation,iscomparedwherethecor responding\nnumerical response amplitude shown by broken/dotted lines[see Fig .4].\nTo investigate further, we consider the phase trajectories of th e system.\nThese are illustrated in Figs. 5. In Fig.5(a) the phase space of the trajectories\ncorresponds to a one well structure. Further, on increasing the forcing param-\neter to the value f2= 9.0, the attractor of the system expands in the phase\nspace and consequently the resonance amplitude grows until it ach ieves its\nhighestQvalue [see Fig. 5(b)]. On further increase of f2values, the expansion\nof the attractor decreases and the resonance amplitude also get s decreased [seeSpringer Nature 2021 L ATEX template\nVibrational resonance in a system of particle on a rotating p arabola 15\nFig.5(c)]. Consequently, it is observed in Fig. 4that the results are consistent\nwith theoretical and numerical studies.\nBy investigating the underlying dynamics and, in particular, the bifur ca-\ntion structure, we can now attempt to comprehend the mechanism of VR in\nthe present system. Resonance curves in nonlinear systems are w ell known to\nbe closely related to the underlying bifurcation structure. In part icular, earlier\nstudies support the notion that symmetry-breaking (sb) and Hop f bifurcations\noccur between resonances [ 5,37]. In this article, we present a new dynamical\nmechanism linked to resonance. Figs. 6(a-b) show the response curve Q and\nbifurcation diagram, respectively, obtained by increasing the forc ing strength\nf2.Theresponsecurve Qattainsthemaximumvaluejustwhentheperioddou-\nbling bifurcation takes place at f2= 11.766 on increasing the value of f2. The\nresponse curve Qfirst increases exponentially, and then attains a maximum at\nthe period-doubling bifurcation point ( f2= 11.766). Thus, the maximum in Q\nsignalling VR clearly originates from the period-doubling bifurcation, t hereby\nwe confirm that VR is linked to the bifurcation of the attractors.\nTo investigate how nonlinearity affects the detected vibrational re sonance,\nthe response amplitude Qfor different values of f2on the value of the nonlin-\nearity (µ= 0.1,0.5,1.0,1.5)ofthe system is presented in Fig. 7. It is interesting\nto note that decreasing the strength of the nonlinearity significan tly enhances\nthe response amplitude Qand it reaches a maximum amplitude for µ <0.1.\nBelow this value of µ, the Q value remains unchanged. Enhancing the vibra-\ntional frequency in the nonlinear system requires an understandin g of the role\nof the damping coefficient α. In our study, on varying the damping term α\n(α= 0.2,0.3, and 0.5), our findings demonstrate that the response amplitude\nQdiminishes as the damping term αis increased (see Fig. 8).\nTo gain more insight on the VR phenomenon in the present mechanical\nmodel, the response amplitude Qfor different ‘ ω1’ values is investigated for\nvarious values of the strength of high-frequency force f2. It is observed that\non increase in the forcing strength of high-frequency causes a de crease in the\nresponse amplitude [see Fig. 9].\nFig.10is a 3-dimensional plot which depicts the numerically computed\nresponse as a function of the nonlinearity µand the forcing strength f2of high\nfrequency for different values of ω1(ω1= 0.1,0.5,1.0,1.5).It is clearly demon-\nstrated in Fig. 10that the response amplitude ‘ Q’ is high for small values of\nnonlinearity µ. Increasingnonlinearityvalue decreasesthe response amplitude.\nFurther increase in the value of ‘ ω1’ also suppresses the response amplitude.\n5 Conclusion\nIn this paper, we have examined the vibrational resonance (VR) ex hibited by\na particle moving on a rotating parabola with additional damping force and\nsubjected to combined low-frequency and high-frequency driving forces using\nanalytical and numerical studies. It is found that nonlinearity cont ributes sig-\nnificantly to the occurrence of VR. The origin and mechanism of vibra tionalSpringer Nature 2021 L ATEX template\n16Vibrational resonance in a system of particle on a rotating p arabola\nresonance in the present system are identified as corresponding t o period dou-\nbling bifurcation and it has been confirmed numerically. Usually, in many\nnonlinear systems, the VR stretches along the nonlinearity, provid ed the other\nparameters are appropriately chosen. But in the present model, n onlinearity\nplays a vital role in the occurrence of VR. It is an established fact th at the\npresence of nonlinearity induces secondary resonances apart fr om primary res-\nonances. In most of the cases, the nonlinearity is essential for th e occurrence\nof VR. However, the changes in the values of the nonlinear paramet er does not\nmake a big difference in the emergence of VR. In the present study, we have\nshown that the values of the nonlinear parameter in the present no npolyno-\nmial system significantly contributes to the emergence of VR. In pa rticular, we\nhave observed that VR occurs in an optimal range of the nonlinear p arameter.\nIt is established in our study that the induction or control of VR in a\nnonpolynomial system is due to an appropriate range of a nonlinear s ystem\nparameter. Our results, while confirming the basic features of VR a s identified\nin the case of PDM Duffing [ 32] and Morse oscillators [ 18], specifically brings\nout novel features on the effect of nonlinearity.\nAlso, the nonlinearity is represented by the semi-latus rectum of th e rotat-\ning parabola and is also part of the angular velocity of the parabola. T hus, the\nappropriate range of the nonlinear strength in the rotating parab ola mechan-\nical system will cause resonance in the system. From a practical po int of\nview, this will lead to the malfunctioning or destruction of parts of th e rotat-\ning parabola machines, like centrifugation equipment, industrial hop pers, etc.\nThus, the present analysis of VR in a particle moving on a rotating par abola\nis necessary from a technological point of view also. Further, the m odel is ana-\nlyzed as a potential-dependent mass (PDM) system, where the mas s is defined\nas the strength of nonlinearity. Thus, the study of nonlinearity in t he system\nreveals the role played by the PDM of the system.\nAcknowledgment. A.V.expresseshisgratitudetotheDST-SERBforfund-\ning a research project with Grant No.EMR/2017/002813.Sincere a ppreciation\nis extended by M.S. to the Council of Scientific & Industrial Researc h, India\nfor funding his fellowship via SRF Scheme No.08/711(0001)2K19-EMR -I.A.V.\nadditionally thanks the DST-FIST for funding research projects v ia Grant\nNo.SR/FST/College-2018-372(C). M.L. thanks the DST-SERB Natio nal Sci-\nence Chair program for funding under Grant No.NSC/2020/000029 .\nAuthor contributions\nRK contributed in setting general idea and implementation its analytic al anal-\nysis as well as in writing manuscript draft. MS contributed in numerica l\nsimulations as well as in writing manuscript draft. AV and ML participat ed in\nchoosingthemethodsforproblemtreatmentandpresentationof thesimulation\nresults as well as writing the paper draft.Springer Nature 2021 L ATEX template\nVibrational resonance in a system of particle on a rotating p arabola 17\nData availability. The datasets generated during and/or analyzed during\nthe current study are available from the corresponding author on reasonable\nrequest.\nReferences\n[1] Landa, P., McClintock, P.V.: Vibrational resonance. J. Phys. A: M ath.\nTheor33(45), 433 (2000)\n[2] Lakshmanan,M., Rajasekar,S.: NonlinearDynamics:Integrabilit y,Chaos\nand Patterns. Springer, Verlag Berlin Heidelberg (2003)\n[3] Rajasekar,S., Sanjuan, M.A.F.: Nonlinear Resonances. Springer , Switzer-\nland (2016)\n[4] Baltan´ as, J., Lopez, L., Blechman, I., Landa, P., Zaikin, A., Kurth s,\nJ., Sanju´ an, M.A.F.: Experimental evidence, numerics, and theory of\nvibrational resonance in bistable systems. Phys. Rev. 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Chaos 26(9), 093117\n(2016)"    },    {        "title": "1401.3396v1.Damping_of_Terahertz_Plasmons_in_Graphene_Coupled_with_Surface_Plasmons_in_Heavily_Doped_Substrate.pdf",        "content": "arXiv:1401.3396v1  [cond-mat.mes-hall]  15 Jan 2014Damping of Terahertz Plasmons in Graphene\nCoupled with Surface Plasmons in Heavily-Doped Substrate\nA. Satou1,2,∗Y. Koseki1, V. Ryzhii1,2, V. Vyurkov3, and T. Otsuji1,2\n1Research Institute of Electrical Communication, Tohoku Un iversity, Sendai 980-8577, Japan\n2CREST, Japan Science and Technology Agency, Tokyo 107-0075 , Japan\n3Institute of Physics and Technology, Russian Academy of Sci ences, Moscow 117218, Russia\nCoupling of plasmons in graphene at terahertz (THz) frequen cies with surface plasmons in a\nheavily-doped substrate is studied theoretically. We reve al that a huge scattering rate may com-\npletely damp out the plasmons, so that proper choices of mate rial and geometrical parameters are\nessential to suppress the coupling effect and to obtain the mi nimum damping rate in graphene.\nEven with the doping concentration 1019−1020cm−3and the thickness of the dielectric layer be-\ntween graphene and the substrate 100 nm, which are typical va lues in real graphene samples with\na heavily-doped substrate, the increase in the damping rate is not negligible in comparison with\nthe acoustic-phonon-limited damping rate. Dependence of t he damping rate on wavenumber, thick-\nnesses of graphene-to-substrate and gate-to-graphene sep aration, substrate doping concentration,\nand dielectric constants of surrounding materials are inve stigated. It is shown that the damping\nrate can be much reduced by the gate screening, which suppres ses the field spread of the graphene\nplasmons into the substrate.\nI. INTRODUCTION\nPlasmons in two-dimensional electron gases (2DEGs)\ncan be utilized for terahertz (THz) devices. THz\nsources and detectors based on compound semicon-\nductor heterostructures have been extensively investi-\ngated both experimentally and theoretically.1–8The two-\ndimensionality, which gives rise to the wavenumber-\ndependent frequency dispersion, and the high electron\nconcentration on the order of 1012cm−2allow us to have\ntheir frequency in the THz rangewith submicron channel\nlength. Most recently, a very high detector responsivity\noftheso-calledasymmetricdouble-grating-gatestructure\nbased on an InP-based high-electron-mobility transistor\nwas demonstrated.9However, resonant detection as well\nas single-frequency coherent emission have not been ac-\ncomplished so far at room temperature, mainly owning\nto the damping rate more than 1012s−1in compound\nsemiconductors.\nPlasmons in graphene have potential to surpass those\nin the heterostructures with 2DEGs based on the stan-\ndard semiconductors, due to its exceptional electronic\nproperties.10Massiveexperimental and theoretical works\nhave been done very recently on graphene plasmons in\nthe THz and infrared regions (see review papers Refs. 11\nand 12 and references therein). One of the most im-\nportant advantages of plasmons in graphene over those\nin heterostructure 2DEGs is much weaker damping rate\nclose to 1011s−1at room temperature in disorder-free\ngraphene suffered only from acoustic-phonon scatter-\ning.13That is very promising for the realization of the\nresonant THz detection14and also of plasma instabili-\nties, which can be utilized for the emission. In addition,\ninterbandpopulation inversioninthe THz rangewaspre-\ndicted,15,16andithasbeeninvestigatedfortheutilization\nnot only in THz lasers in the usual sense but also in THz\nactive plasmonic devices17,18and metamaterials.19Many experimental demonstrations of graphene-based\ndevices have been performed on graphene samples with\nheavily-doped substrates, in order to tune the carrier\nconcentration in graphene by the substrate as a back\ngate. Typically, either peeling or CVD graphene trans-\nferred onto a heavily-doped p+-Si substrate, with a SiO 2\ndielectric layer in between, is used (some experiments\non graphene plasmons have adapted undoped Si/SiO 2\nsubstrates20,21). Graphene-on-silicon, which is epitax-\nial graphene on doped Si substrates,22is also used.\nFor realization of THz plasmonic devices, properties of\nplasmons in such structures must be fully understood.\nAlthough the coupling of graphene plasmons to sur-\nface plasmons in perfectly conducting metallic substrates\nwith/without dielectric layers in between have been the-\noretically studied,23,24the influence of the carrier scat-\ntering in a heavily-doped semiconductor substrate (with\nfinite complex conductivity) has not been taken into ac-\ncount sofar. Since the scatteringratein the substratein-\ncreases as the doping concentration increases, it is antic-\nipated that the coupling of graphene plasmons to surface\nplasmonsin the heavily-dopedsubstrate causesundisired\nincrease in the damping rate.\nThe purpose of this paper is to study theoretically the\ncoupling between graphene plasmons and substrate sur-\nface plasmons in a structure with a heavily-doped sub-\nstrate and with/without a metallic top gate. The paper\nis organized as follows. In the Sec. II, we derive a dis-\npersion equation of the coupled modes of graphene plas-\nmons and substrate surface plasmons. In Sec. III, we\nstudy coupling effect in the ungated structure, especially\nthe increase in the plasmon damping rate due to the cou-\npling and its dependences on the doping concentration,\nthe thickness of graphene-to-substrate separation, and\nthe plasmon wavenumber. In Sec. IV, we show that the\ncoupling in the gated structures can be less effective due\nto the gate screening. We also compare the effect in\nstructures having different dielectric layers between the2\nFIG. 1. Schematic views of (a) an ungated graphene structure\nwith a heavily-doped Si substrate where the top surface is\nexposed on the air and (b) a gated graphene structure with a\nheavily-doped Si substrate and a metallic top gate.\ntop gate, graphene layer, and substrate, and reveal the\nimpact of values of their dielectric constants. In Sec. V,\nwe discuss and summarize the main results of this paper.\nII. EQUATIONS OF THE MODEL\nWe investigateplasmonsinanungatedgraphenestruc-\nture with a heavily-doped p+-Si substrate, where the\ngraphene layer is exposed on the air, as well as a gated\ngraphene structure with the substrate and a metallic top\ngate, whichareschematicallyshowninFigs.1(a)and(b),\nrespectively. The thickness of the substrate is assumed\nto be sufficiently larger than the skin depth of the sub-\nstrate surface plasmons. The top gate can be considered\nasperfectlyconducting metal, whereasthe heavily-doped\nSisubstrateischaracterizedbyitscomplexdielectriccon-\nstant.\nHere, we use the hydrodynamic equations to describe\nthe electronmotion in graphene,26while using the simple\nDrude model for the hole motion in the substrate (due\nto virtual independence of the effective mass in the sub-\nstrate on the electron density, in contrast to graphene).\nIn addition, these are accompanied by the self-consistent\n2D Poisson equation (The formulation used here almost\nfollows that for compound semiconductor high-electron-\nmobility transistors, see Ref. 25). Differences are the\nhydrodynamic equations accounting for the linear dis-\npersion of graphene and material parameters of the sub-\nstrate and dielectric layers. In general, the existence of\nboth electrons and holes in graphene results in various\nmodes such as electrically passive electron-hole sound\nwaves in intrinsic graphene as well as in huge damping\nof electrically active modes due to the electron-hole fric-\ntion, as discussed in Ref. 26. Here, we focus on the case\nwhere the electron concentration is much higher than thehole concentration and therefore the damping associated\nwith the friction can be negligibly small. Besides, for the\ngeneralization purpose, we formulate the plasmon dis-\npersion equation for the gated structure; that for the un-\ngated structure can be readily found by taking the limit\nWt→ ∞(see Fig. 1).\nThen, assuming the solutions of the form exp( ikx−\niωt), where k= 2π/λandωare the plasmon wavenum-\nber and frequency ( λdenotes the wavelength), the 2D\nPoisson equation coupled with the linearized hydrody-\nnamic equations can be expressed as follows:\n∂2ϕω\n∂z2−k2ϕω=−8πe2Σe\n3meǫk2\nω2+iνeω−1\n2(vFk)2ϕωδ(z),\n(1)\nwhereϕωis the ac (signal) component of the potential,\nΣe,me, andνeare the steady-state electron concentra-\ntion, the hydrodynamic “fictitious mass”, and the col-\nlision frequency in graphene, respectively, and ǫis the\ndielectric constant which is different in different layers.\nThe electron concentration and fictitious mass are re-\nlated to each other through the electron Fermi level, µe,\nand electron temperature, Te:\nΣe=/integraldisplay∞\n02ε\nπ/planckover2pi12v2\nF/bracketleftbigg\n1+exp/parenleftbiggε−µe\nkBTe/parenrightbigg/bracketrightbigg−1\ndε,(2)\nme=1\nv2\nFΣe/integraldisplay∞\n02ε2\nπ/planckover2pi12v2\nF/bracketleftbigg\n1+exp(ε−µe\nkBTe)/bracketrightbigg−1\n.(3)\nIn the following we fix Teand treat the fictitious mass as\na function of Σ e. The dielectric constant can be repre-\nsented as\nǫ=\n\nǫt, 0< z < W t,\nǫb, −Wb< z <0,\nǫs[1−Ω2\ns/ω(ω+iνs)], z <−Wb,(4)\nwhereǫt,ǫb, andǫsare thestaticdielectric constants of\nthe top and bottom dielectric layers and the substrate,\nrespectively, Ω s=/radicalbig\n4πe2Ns/mhǫsis the bulk plasma\nfrequency in the substrate with Nsandmhbeing the\ndoping concentration and hole effective mass, and νsis\nthe collision frequency in the substrate, which depends\non the doping concentration. The dielectric constant in\nthe substrate is a sum of the static dielectric constant of\nSi,ǫs= 11.7 and the contribution from the Drude con-\nductivity. The dependence of the collision frequency, νs,\non the doping concentration, Ns, is calculated from the\nexperimental data for the hole mobility at room temper-\nature in Ref. 27.\nWe use the following boundary conditions: vanish-\ning potential at the gate and far below the substrate,\nϕω|z=Wt= 0 and ϕω|z=−∞= 0; continuity conditions\nof the potential at interfaces between different layers,\nϕω|z=+0=ϕω|z=−0andϕω|z=−Wb+0=ϕω|z=−Wb−0;\na continuity condition of the electric flux density at\nthe interface between the bottom dielectric layer and3\nthe substrate in the z-direction, ǫb∂ϕω/∂z|z=−Wb+0=\nǫs∂ϕω/∂z|z=−Wb−0; and a jump of the electric flux den-\nsity at the graphene layer, which can be derived from\nEq. (1). Equation (1) together with these boundary con-\nditions yield the following dispersion equation\nFgr(ω)Fsub(ω) =Ac, (5)\nwhere\nFgr(ω) =ω2+iνeω−1\n2(vFk)2−Ω2\ngr,(6)\nFsub(ω) =ω(ω+iνs)−Ω2\nsub, (7)\nAc=ǫ2\nb(H2\nb−1)\n(ǫbHb+ǫtHt)(ǫs+ǫbHb)Ω2\ngrΩ2\nsub,(8)\nΩgr=/radicalBigg\n8πe2Σek\n3meǫgr(k), ǫgr(k) =ǫtHt+ǫbǫb+ǫsHb\nǫs+ǫbHb,(9)\nΩsub=/radicalBigg\n4πe2Ns\nmhǫsub(k), ǫsub(k) =ǫs+ǫbǫb+ǫtHtHb\nǫbHb+ǫtHt,\n(10)\nandHb,t= cothkWb,t. In Eq. (5), the term Acon the\nright-handsiderepresentsthecouplingbetweengraphene\nplasmons and substrate surface plasmons. If Acwere\nzero, the equations Fgr(ω) = 0 and Fsub(ω) = 0 would\ngive independent dispersion relations for the former and\nlatter, respectively. Qualitatively, Eq. (8) indicates that\nthe coupling occurs unless kWb≫1 orkWt≪1,\ni.e., unless the separation of the graphene channel and\nthe substrate is sufficiently large or the gate screening\nof graphene plasmons is effective. Note that the non-\nconstant frequency dispersion of the substrate surface\nplasmon in Eq. (10) is due to the gate screening, which\nis similar to that in the structure with two parallel metal\nelectrodes.28Equation (5) yields two modes which have\ndominantpotentialdistributionsnearthegraphenechan-\nnel and inside the substrate, respectively. Hereafter, we\nfocus on the oscillating mode primarily in the graphene\nchannel; we call it “channel mode”, whereas we call the\nother mode “substrate mode”.\nIII. UNGATED PLASMONS\nFirst, we study plasmons in the ungated structure.\nHere, the temperature, electron concentration, and col-\nlision frequency in graphene are fixed to Te= 300 K,\nΣe= 1012cm−2, andνe= 3×1011s−1. With these val-\nues of the temperature and concentration the fictitious\nmass is equal to 0 .0427m0, wherem0is the electron rest\nmass. The value of the collision frequency is typical to\nthe acoustic-phonon scattering at room temperature.13\nAs for the structural parameters, we set ǫt= 1 and\nWt→ ∞, and we assume an SiO 2bottom dielectric layer\nwithǫb= 4.5. Then Eq. (5) is solved numerically.(a)\n(b) 0 1 2 3 4 5 6 7\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-3 0 1 2 3 4 5 6 7\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-3 0 1 2 3 4 5 6 7\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-3 0 1 2 3 4 5 6 7\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-3 0 1 2 3 4 5 6 7\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-3 0 1 2 3 4 5 6 7\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.150.200.250.30\n10191020\n 0 0.2 0.4 0.6 0.8 1 1.2\n1017101810191020Frequency, THz\nDoping concentration, cm-3Wb = 50, 100, 200, \n \t \t \t \t 300, 400 nm\nFIG. 2. Dependencesof(a) theplasmon dampingrate and(b)\nfrequencyonthesubstrate dopingconcentration, Ns, withthe\nplasmon wavenumber k= 14×103cm−1(the wavelength λ=\n4.5µm)and with differentthicknesses of thebottom dielectric\nlayer,Wb, in the ungated graphene structure. The inset in (a)\nshows the damping rate in the range Ns= 1019−1020cm−3\n(in linear scale).\nFigures 2(a) and (b) show the dependences of the plas-\nmondamping rateandfrequencyon the substratedoping\nconcentrationwiththeplasmonwavenumber k= 14×103\ncm−1(i.e., the wavelength λ= 4.5µm) and with differ-\nent thicknesses of the bottom dielectric layer, Wb. The\nvalue of the plasmon wavelength is chosen so that it gives\nthe frequency around 1 THz in the limit Ns→0. They\nclearlydemonstratethatthereisahugeresonantincrease\nin the damping rate at around Ns= 3×1017cm−3as\nwell as a drop of the frequency. This is the manifestation\nofthe resonantcoupling ofthe grapheneplasmonand the\nsubstratesurfaceplasmon. The resonancecorrespondsto\nthe situation where the frequencies of graphene plasmons\nand substrate surface plasmons coincide, in other words,\nwhere the exponentially decaying tail of electric field of\ngraphene plasmons resonantly excite the substrate sur-\nface plasmons.\nAt the resonance, the damping rate becomes larger\nthan 1012s−1, over 10 times larger than the contribu-\ntion from the acoustic-phonon scattering in graphene,\nνe/2 = 1.5×1011s−1. For structures with Wb= 504\nand 100 nm, even the damping rate is so large that the\nfrequency is dropped down to zero; this corresponds to\nan overdamped mode. It is seen in Figs. 2(a) and (b)\nthat the coupling effect becomes weak as the thickness\nof the bottom dielectric layer increases. The coupling\nstrength at the resonance is determined by the ratio of\nthe electric fields at the graphene layer and at the inter-\nface between the bottom dielectric layer and substrate.\nIn the case of the ungated structure with a relatively low\ndoping concentration, it is roughly equal to exp( −kWb).\nSinceλ= 4.5µm is much larger than the thicknesses of\nthebottomdielectriclayerinthestructuresunderconsid-\neration, i.e., kWb≪1, the damping rate and frequency\nin Figs. 2(a) and (b) exhibit the rather slow dependences\non the thickness.\nAway from the resonance, we have several nontrivial\nfeatures in the concentration dependence of the damp-\ning rate. On the lower side of the doping concentra-\ntion, the damping rate increase does not vanish until\nNs= 1014−1015cm−3. This comes from the wider field\nspread of the channel mode into the substrate due to\nthe ineffective screening by the low-concentration holes.\nOn the higher side, one can also see a rather broad\nlinewidthoftheresonancewithrespecttothedopingcon-\ncentration, owning to the large, concentration-dependent\ndamping rate of the substrate surface plasmons, and a\ncontribution to the damping rate is not negligible even\nwhen the doping concentration is increased two-orders-\nof-magnitude higher. In fact, with Ns= 1019cm−3,\nthe damping rate is still twice larger than the contribu-\ntion from the acoustic-phonon scattering. The inset in\nFig. 2(a) indicates that the doping concentration must\nbe at least larger than Ns= 1020cm−3for the cou-\npling effect to be smaller than the contribution from\nthe acoustic-phonon scattering, although the latter is\nstill nonnegligible. It is also seen from the inset that,\nwith very high doping concentration, the damping rate\nis almost insensitive to Wb. This originates from the\nscreeningbythe substratethat stronglyexpandsthe field\nspread into the bottom dielectric layer.\nAs for the dependence of the frequency, it tends to a\nlower value in the limit Ns→ ∞than that in the limit\nNs→0, as seen in Figure 2(b), along with the larger de-\npendence on the thickness Wb. This corresponds to the\ntransition of the channel mode from an ungated plasmon\nmode to a gated plasmon mode, where the substrate ef-\nfectively acts as a back gate.\nTo illustrate the coupling effect with various frequen-\nciesinthe THzrange, dependencesoftheplasmondamp-\ning rate and frequency on the substrate doping concen-\ntration and plasmon wavenumber with Wb= 300 nm\nare plotted in Figs. 3(a) and (b). In Fig. 3(a), the peak\nof the damping rate shifts to the higher doping concen-\ntration as the wavenumber increases, whereas its value\ndecreases. The first feature can be understood from\nthe matching condition of the wavenumber-dependent\nfrequency of the ungated graphene plasmons and the\ndoping-concentration-dependent frequency of the sub-\n(a)\n(b) 0.2 0.4 0.6 0.8 1 1.2 1.4\n 5 10 15 20 25 30 35Ns = 2 x 1018 cm-3\n5 x 1018 cm-3\n1 x 1019 cm-3\nFIG. 3. Dependencesof(a) theplasmon dampingrate and(b)\nfrequency on the substrate doping concentration, Ns, and the\nplasmon wavenumber, k, with different the thickness of the\nbottom dielectric layer Wb= 300 nm in the ungated graphene\nstructure. The inset of(a) shows thewavenumberdependence\nof the damping rate with certain doping concentrations. The\nregion with the damping rate below 0 .2×1012s−1is filled\nwith white in (a).\nstrate surface plasmons, i.e., Ω gr∝k1/2, roughlly speak-\ning, and Ω sub∝N1/2\ns. The second feature originates\nfrom the exponential decay factor, exp( −kWb), of the\nelectricfield ofthe channelmode at the interfacebetween\nthe bottom dielectric layer and the substrate; since the\ndoping concentrationis /lessorsimilar1018cm−3at the resonancefor\nany wavevector in Fig. 3, the exponential decay is valid.\nAlso, with a fixed doping concentration, say Na>1019\ncm−3, the damping rate has a maximum at a certain\nwavenumber, resulting from the first feature (see the in-\nset in Fig. 3(a)).\nIV. GATED PLASMONS\nNext, we study plasmons in the gated structures.\nWe consider the same electron concentration, ficticious\nmass, and collision frequency, Σ e= 1012cm−2,me=\n0.0427m0, andνe= 3×1011s−1, as the previous sec-\ntion. As examples of materials for top/bottom dielectric5\n(a)\n(b)0.150.250.350.450.550.65\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.150.250.350.450.550.65\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.150.250.350.450.550.65\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.150.250.350.450.550.65\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.150.250.350.450.550.65\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.150.200.250.30\n10191020\n0.951.001.051.10\n1017101810191020Frequency, THz\nDoping concentration, cm-3Wb = 50, 100, 200, \n \t \t \t \t 300, 400 nm0.150.200.250.300.35\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.150.200.250.300.35\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.150.200.250.300.35\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.150.200.250.300.35\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.150.200.250.300.35\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.150.20\n10191020\n0.780.790.800.810.820.830.840.85\n1017101810191020Frequency, THz\nDoping concentration, cm-3Wb = 50, 100, 200, \n \t \t \t \t 300, 400 nm\nFIG. 4. Dependences of (a) the plasmon damping rate and\n(b)frequency on thesubstrate doping concentration, Ns, with\nthe plasmon wavelength λ= 1.7µm (the wavenumber k=\n37×103cm−1), with thicknesses of the Al 2O3top dielectric\nlayerWt= 20 and 40 nm (left and right panels, respectively),\nand with different thicknesses of the SiO 2bottom dielectric\nlayer,Wb, in the gated graphene structure. The insets in (a)\nshow the damping rate in the range Ns= 1019−1020cm−3\n(in linear scale).\nFIG. 5. Dependence of the plasmon damping rate on the sub-\nstrate doping concentration, Ns, and the plasmon wavenum-\nber,k, with different the thicknesses of the Al 2O3top dielec-\ntric layer Wt= 20 nm and the SiO 2bottom dielectric layer\nWb= 50 nm in thegated graphene structure. The region with\nthe damping rate below 0 .2×1012s−1is filled with white.\nlayers, we examine Al 2O3/SiO2and diamond-like carbon\n(DLC)/3C-SiC. These materials choices not only reflect\nthe realistic combination of dielectric materials available\ntoday, but also demonstrate two distinct situations for\nthe coupling effect under consideration, where ǫt> ǫbfor\nthe former and ǫt< ǫbfor the latter.Figures 4(a) and (b) show the dependences of the plas-\nmondamping rateandfrequencyon the substratedoping\nconcentration with the wavenumber k= 37×103cm−1\n(the plasmon wavelength λ= 1.7µm), with thicknesses\nof the Al 2O3top dielectric layer Wt= 20 and 40 nm, and\nwith different thicknesses of the SiO 2bottom dielectric\nlayer,Wb. As seen, the resonant peaks in the damping\nrate as well as the frequency drop due to the coupling\neffect appear, although the peak values are substantially\nsmaller than those in the ungated structure (cf. Fig. 2).\nThe peak value decreases rapidly as the thickness of the\nbottom dielectric layerincreases; it almostvanishes when\nWb≥300 nm. These reflect the fact that in the gated\nstructuretheelectricfieldofthechannelmodeisconfined\ndominantly in the top dielectric layer due to the gate\nscreening effect. The field only weakly spreads into the\nbottom dielectric layer, where its characteristic length is\nroughtly proportional to Wt, rather than the wavelength\nλas in the ungated structure. Thus, the coupling effect\non the damping rate together with on the frequency van-\nishesquicklyas Wbincreases, evenwhen the wavenumber\nis small and kWb≪1. More quantitatively, the effect is\nnegligible when the first factor of Acgiven in Eq. (8) in\nthe limit kWb≪1 andkWt≪1,\nǫ2\nb(H2\nb−1)\n(ǫbHb+ǫtHt)(ǫs+ǫbHb)≃1\n1+(Wb/ǫb)/(Wt/ǫt)(11)\nis small, i.e., when the factor ( Wb/ǫb)/(Wt/ǫt) is much\nlarger than unity. A rather strong dependence of the\ndamping rate on Wbcan be also seen with high doping\nconcentration, in the insets of Fig. (4)(a).\nFigure 5 shows the dependence of the plasmon damp-\ning rate on the substrate doping concentration and plas-\nmon wavenumber, with dielectric layer thicknesses Wt=\n20 andWb= 50 nm. As compared with the case of\nthe ungated structure (Fig. 3(a)), the peak of the damp-\ning rate exhibits a different wavenumber dependence;\nit shows a broad maximum at a certain wavenumber\n(around 150 ×103cm−1in Fig. 5) unlike the case of\nthe ungatedstructure, where the resonantpeakdecreases\nmonotonically as increasing the wavenumber. This can\nbe explained by the screening effect of the substrate\nagainst that of the top gate. When the wavenumber is\nsmall and the doping concentration corresponding to the\nresonance is low, the field created by the channel mode\nis mainly screened by the gate and the field is weakly\nspread into the bottom direction. As the doping concen-\ntration increases (with increase in the wavevector which\ngivesthe resonance),thesubstratebeginstoactasaback\ngateand the field spreadsmoreinto the bottom dielectric\nlayer,sothatthecouplingeffectbecomesstronger. When\nthe wavenumberbecomessolargethat kWb≪1doesnot\nhold, the field spread is no longer governed dominantly\nby the substrate or gate screening, i.e., the channel mode\nbegins to be “ungated” by the substrate. Eventually,\nthe coupling effect on the damping rate again becomes\nweak, with the decay of the field being proportional to\nexp(−kWb).6\n0.00.51.01.52.02.5\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.00.51.01.52.02.5\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.00.51.01.52.02.5\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.00.51.01.52.02.5\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.00.51.01.52.02.5\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.150.200.250.30\n10191020\n0.600.700.800.901.001.10\n1017101810191020Frequency, THz\nDoping concentration, cm-3Wb = 50, 100, 200, \n \t \t \t \t 300, 400 nm0.00.20.40.60.81.01.2\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.00.20.40.60.81.01.2\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.00.20.40.60.81.01.2\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.00.20.40.60.81.01.2\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.00.20.40.60.81.01.2\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.150.20\n10191020\n0.500.550.600.650.700.750.80\n1017101810191020Frequency, THz\nDoping concentration, cm-3Wb = 50, 100, 200, \n \t \t \t \t 300, 400 nm(a)\n(b)\nFIG. 6. The same as Figs. 4(a) and (b) but with the DLC\ntop and 3C-SiC bottom dielectric layer.\nAs illustrated in Eq. (11), the coupling effect in\nthe gated strcture is characterized by the factor\n(Wb/ǫb)/(Wt/ǫt) when the conditions kWb≪1 and\nkWt≪1 are met. This means that not only the thick-\nnesses ofthe dielectric layersbut alsotheir dielectric con-\nstants are very important parameters to determine the\ncoupling strength. For example, if we adapt the high-k\nmaterial,e.g.,HfO 2inthetopdielectriclayer,itresultsin\nthe more effective gate screening than in the gated struc-\nture with the Al 2O3top dielectric layer, so that the cou-\npling effect can be suppressed even with the same layer\nthicknesses. The structure with the DLC top and 3C-SiC\nbottom dielectric layers (with ǫt= 3.129andǫb= 9.7)\ncorresponds to the quite opposite situation, where the\ngate screening becomes weak and the substrate screening\nbecomes more effective, so that stronger coupling effect\nis anticipated. Figures 6(a), (b), and 7 show the same\ndependences as in Figs 4(a), (b), and 5, respectively, for\nthe structure with the DLC top and 3C-SiC bottom di-\nelectric layers. Comparing with those for the structure\nwith the Al 2O3top and SiO 2bottom dielectric layers,\nthe damping rate as well as the frequency are more in-\nfluenced by the coupling effect in the entire ranges of the\ndoping concentration and wavevector. In particular, the\nincrease in the damping rate with high doping concen-\ntrationNs= 1019−1020cm−3and the thickness of the\nbottom layer Wb= 50−100 nm, which are typical val-\nues in real graphene samples, is much larger. However,\nthis increase can be avoided by adapting thicker bottom\nlayer, say, Wb/greaterorsimilar200 nm or by increasing the doping\nconcentration.\nFIG. 7. The same as Fig. 5 but with the DLC top and 3C-SiC\nbottom dielectric layer.\nV. CONCLUSIONS\nIn summary, we studied theoretically the coupling of\nplasmons in graphene at THz frequencies with surface\nplasmonsinaheavily-dopedsubstrate. Wedemonstrated\nthat in the ungated graphene structure there is a huge\nresonant increase in the damping rate of the “channel\nmode” at a certain doping concentration of the substrate\n(∼1017cm−2) and the increase can be more than 1012\ns−1, due to the resonant coupling of the graphene plas-\nmon and the substrate surface plasmon. The depen-\ndences of the damping rate on the doping concentration,\nthe thickness of the bottom dielectric layer, and the plas-\nmon wavenumber are associated with the field spread of\nthe channel mode into the bottom dielectric layer and\ninto the substrate. We revealed that even with very high\ndoping concentration (1019−1020cm−2), away from the\nresonance, the coupling effect causes nonnegligible in-\ncrease in the damping rate compared with the acoustic-\nphonon-limited damping rate. In the gated graphene\nstructure, the coupling effect can be much reduced com-\npared with that in the ungated structure, reflecting the\nfact that the field is confined dominantly in the top di-\nelectric layer due to the gate screening. However, with\nvery high doping concentration, it was shown that the\nscreening by the substrate effectively spreads the field\ninto the bottom dielectric layer and the increase in the\ndamping rate can be nonnegligible. These results suggest\nthat the structural parameters such as the thicknesses\nand dielectric constants of the top and bottom dielectric\nlayers must be properly chosen for the THz plasmonic\ndevices in order to reduce the coupling effect.\nACKNOWLEDGMENTS\nAuthorsthankM.SuemitsuandS.Sanbonsugeforpro-\nviding information about the graphene-on-silicon struc-\nture and Y. Takakuwa, M. Yang, H. Hayashi, and T. Eto7\nfor providing information about the diamond-like-carbon\ndielectric layer. This work was supported by JSPSGrant-in-Aid for Young Scientists (B) (#23760300), by\nJSPS Grant-in-Aid for Specially Promoted Research\n(#23000008), and by JST-CREST.\n∗a-satou@riec.tohoku.ac.jp\n1M. Dyakonov and M. Shur, Phys. Rev. Lett. 71, 2465\n(1993).\n2M. Dyakonov and M. Shur, IEEE Trans. Electron Devices\n43, 380 (1996).\n3M. S. Shur and J.-Q. L¨ u, IEEE Trans. Micro. Th. Tech.\n48, 750 (2000).\n4M. S. Shur and V. Ryzhii, Int. J. High Speed Electron.\nSys.13, 575 (2003).\n5F. Teppe, D. Veksler, A. P. Dmitriev, X. Xie, S. Rumyant-\nsev, W. Knap, and M. S. Shur, Appl. Phys. Lett. 87,\n022102 (2005).\n6T. Otsuji, Y. M. Meziani, T. Nishimura, T. Suemitsu, W.\nKnap, E. Sano, T. Asano, V. V. and Popov, J. Phys.:\nCondens. Matter 20, 384206 (2008).\n7V. Ryzhii, A. Satou, M. Ryzhii, T. Otsuji, and M. S. Shur,\nJ. Phys.: Condens. Matter 20, 384207 (2008).\n8V. V. Popov, D. V. Fateev, T. Otsuji, Y. M. Meziani, D.\nCoquillat, and W. Knap, Appl. Phys. Lett. 99, 243504\n(2011).\n9T. Watanabe, S. Boubanga Tombet, Y. Tanimoto, Y.\nWang, H. Minamide, H. Ito, D. Fateev, V. Popov, D. Co-\nquillat, W. Knap, Y. Meziani, and T. Otsuji, Solid-State\nElectron. 78, 109 (2012).\n10V. Ryzhii, Jpn. J. Appl. Phys. 45, L923 (2006); V. Ryzhii,\nA. Satou, and T. Otsuji, J. Appl. Phys. 101, 024509\n(2007).\n11T.Otsuji, S.A.BoubangaTombet, A.Satou, H.Fukidome,\nM. Suemitsu, E. Sano, V.Popov, M. Ryzhii, andV.Ryzhii,\nJ. Phys. D: Appl. Phys. 45, 303001 (2012).\n12A. N. Grigorenko, M. Polini, and K. S. Novoselov, Nat.\nPhoton. 6, 749 (2012).\n13E. Hwang and S. Das Sarma, Phys. Rev. B 77, 115449\n(2008).\n14V. Ryzhii, T. Otsuji, M. Ryzhii, and M. S. Shur, J. Phys.\nD: Appl. Phys. 45, 302001 (2012).15V. Ryzhii, M. Ryzhii, and T. Otsuji, J. Appl. Phys. 101,\n083114 (2007).\n16M. Ryzhii and V. Ryzhii, Jpn. J. Appl. Phys. 46, L151\n(2007).\n17A. A. Dubinov, V. Ya Aleshkin, V. Mitin, T. Otsuji, and\nV. Ryzhii, J. Phys.: Condens. Matter 23, 145302 (2011).\n18V. Popov, O. Polischuk, A. Davoyan, V. Ryzhii, T. Otsuji,\nand M. Shur, Phys. Rev. B 86, 195437 (2012).\n19Y. Takatsuka, K. Takahagi, E. Sano, V. Ryzhii, and T.\nOtsuji, J. Appl. Phys. 112, 033103 (2012).\n20L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H.\nBechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, Nat.\nNanotechnol. 6, 630 (2011).\n21J. H. Strait, P. Nene, W.-M. Chan, C. Manolatou, S. Ti-\nwari, F. Rana, J. W. Kevek, and P. L. Mceuen, Phys. Rev.\nB87, 241410 (2013).\n22M. Suemitsu, Y. Miyamoto, H. Handa, and A. Konno, e-J.\nSurf. Sci. Nanotech. 7, 311 (2009).\n23N. Horing, Phys. Rev. B 80, 193401 (2009).\n24J. Yan, K. Thygesen, and K. Jacobsen, Phys. Rev. Lett.\n106, 146803 (2011).\n25A. Satou, V. Vyurkov, and I. Khmyrova, Jpn. J. Appl.\nPhys.43, 566 (2004).\n26D. Svintsov, V. Vyurkov, S. Yurchenko, T.Otsuji, and V.\nRyzhii, J. Appl. Phys. 111, 083715 (2012).\n27C. Bulucea, Solid-State Electron. 36, 489 (1993).\n28S. A. Maier, Plasmonics: Fundamentals and Applications\n(Springer Science, NY, 2007).\n29The dielectric constant of DLC varies in the range between\n3.1 and 7.8, depending on its growth condition.30Here, we\nchoose thelowest valuefor demonstration ofthecase where\nǫt≪ǫb.\n30H. Hayashi, S. Takabayashi, M. Yang, R. Jeˇ sko, S. Ogawa,\nT. Otsuji, and Y. Takakuwa, “Tuning of the dielec-\ntric constant of diamond-like carbon films synthesized by\nphotoemission-assisted plasma-enhanced CVD,” in 2013\nInternational Workshop on Dielectric Thin Films for Fu-\nture Electron Devices -Science and Technology- (2013\nIWDTF), Tokyo, Japan, November 7-9, 2013."    },    {        "title": "1302.6301v1.Modelling_Fast_Alfvén_Mode_Conversion_Using_SPARC.pdf",        "content": "Modelling Fast-Alfv\u0013 en Mode Conversion Using\nSPARC\nH Moradi, P S Cally\nMonash Centre for Astrophysics, School of Mathematical Sciences, Monash University,\nVictoria 3800, Australia\nE-mail: hamed.moradi@monash.edu\nAbstract. We successfully utilise the SPARC code to model fast-Alfv\u0013 en mode conversion in\nthe region cA\u001dcSvia 3-D MHD numerical simulations of helioseismic waves within constant\ninclined magnetic \feld con\fgurations. This was achieved only after empirically modifying the\nbackground density and gravitational strati\fcations in the upper layers of our computational\nbox, as opposed to imposing a traditional Lorentz Force limiter, to ensure a manageable\ntimestep. We found that the latter approach inhibits the fast-Alfv\u0013 en mode conversion process\nby severely damping the magnetic \rux above the surface.\n1. Introduction\nA series of recent studies (e.g., [1, 2, 3, 4, 5, 6, 7, 8]) have shown that an important feature not\nwidely accounted for in sunspot seismology is fast-Alfv\u0013 en wave mode conversion - a process which\noccurs at, and beyond, the fast wave re\rection height (where cA\u0019!=kh;cAdenotes the Alfv\u0013 en\nwave speed, !the wave frequency and khthe horizontal wavenumber) - being spread over many\nscale heights for wavenumbers typical of local helioseismology. This process is most e\u000ecient\nfor\u0012(\feld inclination from vertical) between 30\u000e\u000040\u000e, and\u001e(angle between the magnetic\n\feld and wave propagation planes) between 60\u000e\u000080\u000e, and appears to have the potential to\nmodify the seismic wave-path through the solar atmosphere, thereby a\u000becting the wave travel\ntimes that are the basis of our inferences about the subsurface (see [9, 10, 11, 12] for recent\nreviews). Motivated by these studies, our aim is to use the Seismic Propagation through Active\nRegions and Convection (SPARC) code, a 3-D magnetohydrodynamic (MHD) wave-propagation\ncode developed by [13] for computational heliosiesmology, to numerically simulate this process,\nand investigate the implications of fast-Alfv\u0013 en wave mode conversion on the seismology of the\nphotosphere. In this paper we describe our attempts at forward modelling this process with\nSPARC using a quiet-Sun background model permeated by homogenous inclined magnetic \felds.\n2. Numerical Setup\nThe SPARC code solves the 3-D linearized Euler and induction equations of magneto\ruid motion\nin Cartesian geometries to investigate wave interactions with local perturbations (e.g., sound\nspeed, pressure, density, \rows, magnetic \feld etc.). Over the past few years, a number of various\nsolar phenomena have been studied using this code (e.g., [14, 15, 16, 17, 18]). The computational\nbox we employ for our simulations using SPARC spans 186 :6 Mm in the horizontal direction\n(128 evenly-spaced grid points in the horizontal directions xandy; \u0001x= \u0001y= 1:46 Mm/pixel),arXiv:1302.6301v1  [astro-ph.SR]  26 Feb 2013and from 2:5 Mm above the surface ( z= 0; where zdenotes height in Mm) to 25 Mm below in\nthe vertical direction (300 non-uniformly spaced grid points in z; \u0001zvaries from several hundred\nkilometres at depth, to tens of kilometres in the near-surface layers). The vertical boundaries\nof the box are absorbent, with perfectly matched layer (PML) boundary layers spanning the\ntop 10 and bottom 7 grid points in z, while periodic boundary conditions are imposed on the\nhorizontal sides. In a similar manner to [5, 6, 7], we use a monochromatic ( \u0017=!=2\u0019= 5 mHz)\nplane-parallel wave driver, which only excites waves propagating in the ( x,z) planes. We do\nthis by imposing a perturbation of the form:\nsin(!t) e\u0000(x\u0000x0)2=2\u000e2\nxe\u0000(z\u0000z0)2=2\u000e2\nz (1)\nin a few grid points near z0=\u00005 Mm and centred at around x0= 0 Mm, in pressure, density\nand velocity ( xandzcomponents only).\n3. Model Atmosphere\nThe background model used is a convectively stabilised solar model (CSM B) from [19]. On top\nof this background model we employ a constant, inclined magnetic \feld con\fguration using the\nprescription from [1]:\nB0=B0(sin\u0012cos\u001e;sin\u0012sin\u001e;cos\u001e); (2)\nWe choose B0= 1500 G and a number of di\u000berent angle orientations in the range 0 <\u0012 < 90\u000e\nand 0<\u001e< 90\u000e.\nIn 3-D MHD simulations, the timestep (\u0001 t\u0018\u0001z=cA) is often highly constrained by\nthe Courant-Friedrichs-Lewy (CFL) condition, due to the exponentially increasing value of\ncA=B0=(\u00160\u001a)1=2(where\u001adenotes density and \u00160the magnetic permeability) above the surface.\nThis causes the wavelengths of both the fast and Alfv\u0013 en waves to become quite large, resulting\nin an extremely sti\u000b numerical problem. The most common way of dealing with this issue, in\ncomputational helioseismology, has been to simply apply a cA\\limiter\" to moderate the action\nof the Lorentz Force when the ratio cA/cS(wherecSdenotes sound speed) becomes exceedingly\nlarge. Some choices for these limiters have been discussed before (e.g., [20, 21, 15, 22, 23]).\nGenerally, they tend to pre\fx the Lorentz-Force terms in the momentum equations, with cA\ntypically being capped at \u001820\u000060 km/s. The physical implications of arti\fcially limiting\nthe Lorentz Force in such a manner (particularly on the seismology) have not been explored.\nRecently though [24] has shown that signi\fcant internal re\rection of Alfv\u0013 en waves can occur if\nthecApro\fle is not adequately treated above the surface.\nOur method of ensuring a reasonable \u0001 tfor our calculations involves empirically modifying\nthe background density ( \u001a0(z)) in the upper layers ( \u00180:5< z < 2:5 Mm) of CSM B, in\nconjunction with a commensurate modi\fcation of the gravitational acceleration ( g0(z)) pro\fle\nover the same zrange (in order to o\u000bset the change in the density scale height), to obtain\na maximum global fast ( cF) andcAof\u001980 km/s. All other background variables remain\nunaltered. The modi\fed pro\fles of \u001a0(z) andg0(z) are shown in Figure 1. The value of 80 km/s\nwas chosen because it provides us with a reasonable time step (\u0001 t= 0:5 s), and is safely higher\nthan the largest horizontal phase-speeds ( !=kh) typically sampled in sunspot seismology (e.g.,\n[25]). This is important since !=kh\u0019cAalso denotes the location of the fast mode re\rection\nheight in the solar atmosphere [6]. Figure 2 a) shows the resulting cAandcFpro\fles in our\nmodel as a function of height. For comparison purposes, the cAandcFpro\fles which would\nresult from imposing a Lorentz Force limiter, instead of modifying \u001a0(z) andg0(z) above the\nsurface, are shown in Figure 2 b).\nThe obvious downside of empirically modifying \u001a0(z) andg0(z) in order to satisfy the\nCFL condition is that background model will no longer be as `solar-like' (i.e., in terms of\neigenfrequencies, eigenfunctions and power spectrum) as CSM B. The larger \u001a0(z) pro\fle which−2−101210−1510−1010−5\nz (Mm)log ρ0 (g/cm3)\n  \nCSM_B ρ0\nModified ρ0\ncA≈ cS\n−2−1012050100150200250300\nz (Mm)g0 (m/s2)\n  \nCSM_B g0\nModified g0\ncA≈ cSFigure 1. Plots of the original CSM B ([19]; dot-dashed line) and modi\fed (solid line)\nbackground density (left) and gravitational acceleration (right) pro\fles as a function of height.\nThe solid vertical line represents the cA\u0019cSheight.\nnow results above the surface also modi\fes acoustic cut-o\u000b frequency ( !ac=cS=2H\u001a; whereH\u001a\ndenotes the density scale height), which is reduced from \u0017= 5:2 mHz to 3 :6 mHz. We also \fnd\nthat the modi\fed atmosphere produces large-amplitude convective ( g-) modes at \u0017\u00191:5\u00001:7\nmHz (it is worth noting though that this is a frequency range which is typically associated with\nsupergrannulation noise and is generally \fltered out/ignored in sunspot seismology).\n−2−1 0120102030405060708090\nz (Mm)cS, cF, cA (km/s)a)\n  \ncA\ncS\ncF\ncA≈ cS\n−2−1 0120102030405060708090\nz (Mm)cS, cF, cA (km/s)b)\n  \ncA\ncS\ncF\ncA≈ cS\nFigure 2. a): Plots of various wave speeds as a function of height resulting from the modi\fed\nCSM B background model. b): Same as a) but for an unmodi\fed CSM B background in\nconjunction with a Lorentz Force limiter (i.e, the Lorentz Force is multiplied by a factor of\n200c2\nS=(200c2\nS+c2\nA), leading to cAandcFbeing capped at\u001980 km/s). The solid vertical line\nrepresents the cA\u0019cSheight.\nHowever, more importantly for our concerns, this method ensures that we satisfy our CFL\ncondition without any direct modi\fcation of the Lorentz Force via the introduction of anarti\fcial term in Maxwell's equations, which as we shall show in the proceeding section, results\nin unphysical damping of the magnetic \rux above the surface and inhibits the fast-Alfv\u0013 en mode\nconversion process.\n4. Results\nFollowing [5, 6, 7], we use velocity projections onto three orthogonal directions ( ^ elong, which\nselects the longitudinal component of wave propagation, i.e, the slow mode; ^ etrans, which selects\nthe transversal component of wave propagation, i.e., the fast mode; ^ eperp, which selects the\nperpendicular component of wave motion, i.e., the Alfv\u0013 en mode; see equations 1-3 in [5] for\nde\fnitions) to separate the Alfv\u0013 en mode from the fast and slow magneto-acoustic modes in\nthe regioncA\u001dcS. We also calculate the temporally averaged acoustic ( Fac=hp1v1i; where\nprepresents pressure and vrepresents the 3-D velocity respectively, subscript \\1\" represents\nperturbations), magnetic ( Fmag=hB1\u0002(v1\u0002B0)=\u00160i) and total ( Ftot=Fac+Fmag) energy\n\ruxes in order to measure the e\u000eciency of conversion to Alfv\u0013 en waves around the cA\u0019cS\nequipartition height.\nt (min)z (Mm)Vlong, θ = 0 °, φ = 80°\n51015202530−1012\nVtrans, θ = 0 °, φ = 80°\nt (min)z (Mm)\n51015202530−1012\nVperp, θ = 0 °, φ = 80°\nt (min)z (Mm)\n51015202530−1012\nt (min)z (Mm)Vlong, θ = 30 °, φ = 80°\n51015202530−1012\nVtrans, θ = 30 °, φ = 80°\nt (min)z (Mm)\n51015202530−1012\nVperp, θ = 30 °, φ = 80°\nt (min)z (Mm)\n51015202530−1012\nt (min)z (Mm)Vlong, θ = 60 °, φ = 80°\n51015202530−1012\nVtrans, θ = 60 °, φ = 80°\nt (min)z (Mm)\n51015202530−1012\nVperp, θ = 60 °, φ = 80°\nt (min)z (Mm)\n51015202530−1012\nFigure 3. Figures above represent the velocity projections derived from simulations using\nconstant inclined magnetic \feld con\fgurations (left column: \u0012= 0\u000e;\u001e= 80\u000e; middle column:\n\u0012= 30\u000e;\u001e= 80\u000e; right column: \u0012= 60\u000e;\u001e= 80\u000e) as a function of height (Mm) and time\n(minutes). VperpandVtrans amplitudes have been scaled by a factor of ( \u001a0cA)1=2, whileVlong\nhas been scaled by a factor of ( \u001a0cS)1=2. The grayscale is the same in all panels. The horizontal\nsolid and dotted lines represent the cA\u0019cSand fast-mode re\rection heights respectively.\nThe results of the projected velocities derived from simulations where \u001eis \fxed at 80\u000eand\n\u0012varies from 0\u000eto 30\u000eto 60\u000eare shown in Figure 3. The inclination of the ridges in these\nprojections indicates the wave propagation speed: the more inclined the ridges, the lower the\npropagation speed and vice versa. The presence of the (primarily acoustic) slow mode ( Vlong) is\nclearly visible above the cA\u0019cSlevel (solid line) in all three cases (with the reduced !acresulting\nin a signi\fcant amount of acoustic waves propagating along the \feld both above and below the\nequipartition height), while the rapidly propagating Alfv\u0013 en mode ( Vperp) only appears when the\nmagnetic \feld is su\u000eciently inclined and oriented out of the plane, i.e, \u0012= 30\u000e;\u001e= 80\u000eand\n\u0012= 60\u000e;\u001e= 80\u000e, with the latter con\fguration appearing to be the more e\u000ecient in producing\nAlfv\u0013 en waves. For these two cases, a faint presence of the magnetically dominated fast mode(Vtrans) above the re\rection height (dotted line) can still be made out. This is a result of\nour plane-parallel driver, which excites all wavenumbers, leading to a proportion of fast modes\nbeing transmitted through to the PML, rather than being re\rected at cA\u0019!=kh. In the\n\u0012= 60\u000e;\u001e= 80\u000eprojection, there appears to be a wavefront with an opposite inclination of\nridges close to the top boundary of the domain. This could either be an artefact of the colour\nscheme/scaling, or a numerical artefact (i.e, re\rection) from the upper boundary condition\nof the simulations. While we do not observe any re\rection from the upper boundary in the\ncorresponding acoustic \rux (see Figure 4), given that the wavefornt appears to arrive at the\nupper boundary prior to the arrival of the slow waves, it could also be possible that the signature\nis a yet unmodelled product of the fast-Alfven conversion process. This is something which we\nhope investigate in a future work.\nFigure 4 shows the vertical component of the averaged \ruxes as a function of height. We\nobserve that the \rux variations are strongest near the conversion layer (solid vertical line), with\nthe magnetic \rux exceeding the acoustic \rux when \u0012= 60\u000e;\u001e= 80\u000eforz>1 Mm. These results\nare in good agreement with previous numerical simulations of fast-Alfv\u0013 en mode conversion using\nhomogenous inclined magnetic \felds (e.g., see Figure 3 from [5]).\n−2−1012−4−202468x 10−9\nz (Mm)<flux>θ = 0°, φ = 80°\n−2−1012−4−202468x 10−9\nz (Mm)<flux>θ = 30°, φ = 80°\n−2−1012−4−202468x 10−9\nz (Mm)<flux>θ = 60°, φ = 80°\nFigure 4. Time-averaged magnetic (bold sold line), acoustic (dashed line) and total (dotted\nline) \ruxes (in non-dimensional units) calculated as a function of height. The solid vertical line\nrepresents the cA\u0019cSheight.\nFor comparison purposes, we also conducted simulations where, instead of modifying \u001a0(z)\nandg0(z) above the surface, we employ a Lorentz Force limiter (with cAandcFcapped at\u001980\nkm/s, i.e, as shown in in Figure 2 b)). The resulting averaged \ruxes are shown in Figure 5.\nWhile di\u000berences in the magnitude and height variations of the acoustic \ruxes between these\nresults, and those contained in Figure 4, can be explained by the change in !ac, the di\u000berences\nin the magnetic \ruxes, particularly when considering the \u0012= 60\u000e;\u001e= 80\u000ecases, are almost\nentirely due to the Lorentz Force limiter. With the Lorentz Force limiter in place, the magnetic\n\rux above z > 1 Mm appears to just be able to creep above the acoustic \rux for a couple\nof hundred kilometres, before being completely damped prior to reaching PML. We observed\nthis phenomenon regardless of the value of the cAcap that was used with the Lorentz Force\nlimiter. As expected, the resulting velocity projections for these cases (\fgures not included) also\ncon\frmed the absence of any signi\fcant Alfv\u0013 en modes above cA\u0019cS.\n5. Summary\nUnderstanding the physics of propagating waves within regions of strong magnetic \felds, of\nwhich fast-Alfv\u0013 en wave mode conversion has recently been shown to be a critical component, is\nessential for helioseismic studies of sunspots and active regions. We used the 3-D linear MHD\nsolver SPARC to simulate this process in a convectively stabilised solar model (CSM B with−2−1012−4−202468x 10−9\nz (Mm)<flux>θ = 0°, φ = 80°\n−2−1012−4−202468x 10−9\nz (Mm)<flux>θ = 30°, φ = 80°\n−2−1012−4−202468x 10−9\nz (Mm)<flux>θ = 60°, φ = 80°Figure 5. Time-averaged magnetic (bold sold line), acoustic (dashed line) and total (dotted\nline) \ruxes (in non-dimensional units) calculated as a function of height for simulations where a\nLorentz Force limiter is used with cAandcFcapped as shown in Figure 2 b). The solid vertical\nline represents the cA\u0019cSheight.\nempirically modi\fed \u001a0(z) andg0(z) pro\fles in the upper layers to ensure a reasonable \u0001 t)\npermeated by homogenous inclined magnetic \felds. We found that employing a traditional\nLorentz Force limiter to arti\fcially cap cAabove the surface tends to inhibit the fast-Alfv\u0013 en\nmode conversion process by signi\fcantly damping the magnetic \rux above the surface.\nThe next steps in our forward modelling process will include the introduction of random\nstochastic sources and more realistic (i.e., sunspot-like) background atmospheres, in order to\nsimulate arti\fcial helioseismology data sets. With the aid of local helioseismic diagnostic tools,\nsuch as time-distance helioseismology and helioseismic holography, we will then be able to\nattempt to quantify the e\u000bects of fast-Alfv\u0013 en mode conversion on the wave travel times.\nAcknowledgments\nThis work was supported by an award under the Merit Allocation Scheme on the NCI National\nFacility at the ANU.\n[1] Cally P S and Goossens M 2008 Sol. Phys. 251251{265\n[2] Cally P S and Andries J 2010 Sol. Phys. 26617{38 ( Preprint 1007.1808 )\n[3] Cally P S and Hansen S C 2011 ApJ 738119 ( Preprint 1105.5754 )\n[4] Hanson C S and Cally P S 2011 Sol. 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Phys. 268293{308 ( Preprint 1003.0528 )\n[22] Rempel M, Sch ussler M and Kn olker M 2009 ApJ 691640{649 ( Preprint 0808.3294 )\n[23] Braun D C, Birch A C, Rempel M and Duvall T L 2012 ApJ 74477\n[24] Cally P S 2012 Sol. Phys. 28033{50 ( Preprint 1206.2114 )\n[25] Couvidat S, Gizon L, Birch A C, Larsen R M and Kosovichev A G 2005 ApJS 158217{229"    },    {        "title": "1207.5639v2.Doppler_Signatures_of_the_Atmospheric_Circulation_on_Hot_Jupiters.pdf",        "content": "ApJ, accepted and published\nPreprint typeset using L ATEX style emulateapj v. 5/2/11\nDOPPLER SIGNATURES OF THE ATMOSPHERIC CIRCULATION ON HOT JUPITERS\nAdam P. Showman1, Jonathan J. Fortney2, Nikole K. Lewis1,3, and Megan Shabram4\nApJ, accepted and published\nABSTRACT\nThe meteorology of hot Jupiters has been characterized primarily with thermal measurements, but\nrecent observations suggest the possibility of directly detecting the winds by observing the Doppler\nshift of spectral lines seen during transit. Motivated by these observations, we show how Doppler\nmeasurements can place powerful constraints on the meteorology. We show that the atmospheric\ncirculation|and Doppler signature|of hot Jupiters splits into two regimes. Under weak stellar\ninsolation, the day-night thermal forcing generates fast zonal jet streams from the interaction of\natmospheric waves with the mean \row. In this regime, air along the terminator (as seen during transit)\n\rows toward Earth in some regions and away from Earth in others, leading to a Doppler signature\nexhibiting superposed blue- and redshifted components. Under intense stellar insolation, however,\nthe strong thermal forcing damps these planetary-scale waves, inhibiting their ability to generate\njets. Strong frictional drag likewise damps these waves and inhibits jet formation. As a result, this\nsecond regime exhibits a circulation dominated by high-altitude, day-to-night air\row, leading to a\npredominantly blueshifted Doppler signature during transit. We present state-of-the-art circulation\nmodels including nongray radiative transfer to quantify this regime shift and the resulting Doppler\nsignatures; these models suggest that cool planets like GJ 436b lie in the \frst regime, HD 189733b is\ntransitional, while planets hotter than HD 209458b lie in the second regime. Moreover, we show how\nthe amplitude of the Doppler shifts constrains the strength of frictional drag in the upper atmospheres\nof hot Jupiters. If due to winds, the \u00182 km s\u00001blueshift inferred on HD 209458b may require drag\ntime constants as short as 104{106seconds, possibly the result of Lorentz-force braking on this planet's\nhot dayside.\nSubject headings: planets and satellites: general, planets and satellites: individual: HD 209458b,\nmethods: numerical, atmospheric e\u000bects\n1.INTRODUCTION\nTo date, the exotic meteorology of hot Jupiters has\nbeen characterized primarily with thermal emission ob-\nservations, particularly infrared light curves (e.g. Knut-\nson et al. 2007, 2009, 2012; Cowan et al. 2007; Cross-\n\feld et al. 2010) and secondary eclipse measurements\n(e.g., Charbonneau et al. 2005; Deming et al. 2005). To-\ngether, these observations place important constraints on\nthe vertical temperature pro\fles, day-night temperature\ndi\u000berences, and magnitude of day-night heat transport\ndue to the atmospheric circulation. Moreover, in the\ncase of HD 189733b and Ups And b, infrared lightcurves\nindicate an eastward displacement of the hottest region\nfrom the substellar longitude (Knutson et al. 2007, 2009;\nCross\feld et al. 2010). This feature is a common out-\ncome of atmospheric circulation models, which gener-\nally exhibit fast eastward wind\row at the equator that\ndisplaces the thermal maxima to the east (Showman &\nGuillot 2002; Cooper & Showman 2005; Showman et al.\n2008, 2009; Dobbs-Dixon & Lin 2008; Dobbs-Dixon et al.\n2010; Menou & Rauscher 2009, 2010; Rauscher & Menou\n1Department of Planetary Sciences and Lunar and Planetary\nLaboratory, The University of Arizona, 1629 University Blvd.,\nTucson, AZ 85721 USA; showman@lpl.arizona.edu\n2Department of Astronomy and Astrophysics, University of\nCalifornia, Santa Cruz, CA 95064, USA\n3Currently a Sagan Fellow at the Department of Earth, At-\nmospheric, and Planetary Sciences, Massachusetts Institute of\nTechnology, Cambridge, MA 02139\n4Department of Astronomy, University of Florida, 211 Bryant\nSpace Science Center, Gainesville, FL 32611-2055, USA2010, 2012; Burrows et al. 2010; Thrastarson & Cho\n2010; Lewis et al. 2010; Heng et al. 2011b,a; Showman\n& Polvani 2011; Perna et al. 2012). In this way, the\nlight curves provide information|albeit indirectly|on\nthe atmospheric wind regime.\nRecent developments, however, open the possibility\nof direct observational measurement of the atmospheric\nwinds on hot Jupiters. Snellen et al. (2010) presented\nhigh-resolution groundbased, 2- \u0016m spectra obtained dur-\ning the transit of HD 209458b in front of its host star.\nFrom an analysis of 56 spectral lines of carbon monoxide,\nthey reported an overall blueshift of 2 \u00061 km s\u00001relative\nto the expected planetary motion, which they interpreted\nas a signature of atmospheric winds \rowing from dayside\nto nightside toward Earth along the planet's terminator.\nIn a similar vein, Hedelt et al. (2011) presented trans-\nmission spectra of Venus from its 2004 transit, in which\nthey detected Doppler shifted spectral lines in the upper\natmosphere, again seemingly the result of atmospheric\nwinds. These observations pave the way for an entirely\nnew approach to characterizing hot Jupiter meteorology.\nThe possibility of characterizing hot Jupiter meteorol-\nogy via Doppler provides a strong motivation for deter-\nmining the types of Doppler signatures generated by the\natmospheric circulation. Seager & Sasselov (2000) \frst\nmentioned the possible in\ruence of exoplanet winds on\ntheir transit spectra, and Brown (2001) considered the\ne\u000bect in more detail. More recently, Miller-Ricci Kemp-\nton & Rauscher (2012) took a detailed look at the abil-\nity of the atmospheric circulation to a\u000bect the transmis-arXiv:1207.5639v2  [astro-ph.EP]  11 Jun 20132 Showman et al.\n(a) (b) \nFig. 1.| Schematic implications of atmospheric circulation\nfor Doppler measurements of a hot Jupiter observed in transit.\nPlanet's host star is depicted at right, planet (viewed looking down\nover north pole) is at the center, and Earth is at the left. (a):\nIn the presence of zonal jets, air \rows along latitude circles (col-\nored arrows), leading to air\row toward Earth along one terminator\n(blue arrow) and away from Earth along the other (red arrow). A\nDoppler signature that is broadened, or in extreme cases may be\nbimodally split into blue- and redshifted components, results. (b):\nWhen zonal jets are damped, air \rows primarily from day to night\nat low pressure, leading to air\row toward Earth along both ter-\nminators (blue arrows). A primarily blueshifted Doppler signature\nresults.\nsion spectrum. Here, we continue this line of inquire\nto show how Doppler measurements can place powerful\nconstraints on the meteorology of hot Jupiters. We show\nthat the atmospheric circulation of hot Jupiters splits\ninto two regimes|one with strong zonal jets and super-\nposed eddies, and the other comprising predominant day-\nto-night \row at high-altitudes, with weaker jets|which\nexhibit distinct Doppler signatures.\nIn Section 2, we present theoretical considerations\ndemonstrating why two regimes should occur and the\nconditions for transition between them. In Section 3,\nwe test these ideas with an idealized dynamical model.\nSection 4 presents state-of-the-art three-dimensional dy-\nnamical models of three planets|GJ 436b, HD 189733b,\nand HD 209458b|that bracket a wide range of stellar\nirradiation and plausibly span the transition from jet to\neddy-dominated5at the low pressures sensed by Doppler\nmeasurements. Section 5 presents the expected Doppler\nsignatures from these models, and Section 6 concludes.\n2.TWO REGIMES OF ATMOSPHERIC\nCIRCULATION: THEORY\nWe expect the Doppler signature of the atmospheric\ncirculation on hot Jupiters to fall into two regimes, illus-\ntrated in Figure 1.\n2.1. Jet-dominated regime\nOn rotating planets, the interaction of atmospheric\nturbulence with the anisotropy introduced by the merid-\nional gradient of the Coriolis parameter (known as the\n\fe\u000bect) leads to the emergence of zonal jets, which of-\nten dominate the circulation (e.g., Rhines (1975, 1994),\n5Eddies refer to the deviation of the winds from their zonal\naverage.Williams (1978, 1979), Vallis & Maltrud (1993), Cho &\nPolvani (1996) Dritschel & McIntyre (2008); for recent\nreviews in the planetary context, see Vasavada & Show-\nman (2005) and Showman et al. (2010)). When radiative\nforcing and friction are weak, the heating of air parcels\nas they cross the dayside or nightside will be too small to\ninduce signi\fcant day-night temperature variations; the\ndominant driver of the \row will then be the meridional\n(latitudinal) gradient in the zonal-mean radiative heat-\ning. Such a \row will exhibit signi\fcant zonal symmetry\nin temperature and winds with the primary horizontal\ntemperature variations occurring between the equator\nand the poles. For this regime to occur, the radiative\ntimescale must be signi\fcantly longer than the timescale\nfor air parcels to cross a hemisphere; the rotation rate\nmust also be su\u000eciently fast, and the friction su\u000eciently\nweak. The speed and number of the zonal jets will de-\npend on a zonal momentum balance between Coriolis\naccelerations acting on the mean-meridional circulation\nand eddy accelerations resulting from baroclinic and/or\nbarotropic instabilities, if any.\nWhen the radiative forcing is su\u000eciently strong, as\nexpected for typical hot Jupiters, large day-night heat-\ning contrasts will occur. As shown by Showman &\nPolvani (2011), such heating contrasts induce standing,\nplanetary-scale Rossby and Kelvin waves. For typical\nhot Jupiter parameters, these waves cause an equator-\nward \rux of eddy angular momentum that drives a su-\nperrotating (eastward) jet at the equator (Showman &\nPolvani 2011). This provides a theoretical explanation\nfor the near-ubiquitous emergence of eastward equatorial\njets in atmospheric circulation models of hot Jupiters.\nIn these jet-dominated regimes6(Fig. 1a), air along the\nterminator|as seen during transit|\rows toward Earth\nin some regions and away from Earth in others. This\nleads to a Doppler signature where spectral lines are\nbroadened, with minimal overall shift in the central wave-\nlength. In extreme cases the Doppler signature may be\nsplit into distinct, superposed blue- and redshifted veloc-\nity peaks.\n2.2. Suppression of jets by damping\nThe presence of su\u000eciently strong radiative or fric-\ntional damping can suppress the formation of zonal jets,\nleading to a circulation that at high altitudes is dom-\ninated by day-to-night \row rather than jets that are\nquasi-symmetric in longitude (Fig. 1b). Here, we demon-\nstrate the conditions under which the mechanisms of\nShowman & Polvani (2011) are suppressed.\nShowman & Polvani (2011) identi\fed two speci\fc\nmechanisms for the emergence of equatorial superrota-\ntion in models of synchronously rotating hot Jupiters.\nWe consider each in turn.\n2.2.1. Di\u000berential zonal wave propagation\nAs described above, the day-night thermal forcing on\na highly irradiated, synchronously rotating planet gener-\nates standing, planetary-scale Rossby and Kelvin waves.\n6The interaction of eddies with the mean \row is generally re-\nsponsible for driving zonal jets, so eddies are almost never negli-\ngible to the dynamics, even when zonal jets are strong. Here, by\n\\jet dominated\" we do not mean that eddies are unimportant but\nrather simply that the resulting jets have velocity amplitudes that\nsigni\fcantly exceed the amplitude of the eddies.Atmospheric circulation of hot Jupiters 3\nThe Kelvin waves straddle the equator while the Rossby\nwaves exhibit pressure perturbations peaking in the mid-\nlatitudes for typical hot Jupiter parameters. The (group)\npropagation of Kelvin waves is to the east while that\nof long Rossby waves is to the west; this di\u000berential\nzonal propagation induces an eastward phase shift of\nthe standing wave pattern near the equator and a west-\nward phase shift at high latitudes. The result is a pat-\ntern of eddy velocities (northwest-southeast in the north-\nern hemisphere and southwest-northeast in the southern\nhemisphere) that causes an equatorward \rux of eddy an-\ngular momentum.\nIf the radiative or frictional timescales are signi\fcantly\nshorter than the time required for Kelvin and Rossby\nwaves to propagate over a planetary radius, the waves are\ndamped, inhibiting their zonal propagation and prevent-\ning the latitude-dependent phase shift necessary for the\nmeridional angular momentum \ruxes. Therefore, this\nmechanism for generating zonal jets is suppressed when\nradiative or frictional damping timescales are su\u000eciently\nshort. The Kelvin-wave dispersion relation in the primi-\ntive equations7\n!=Nk\n\u0000\nm2+1\n4H2\u00011=2(1)\nwhere!is wave frequency, k > 0 andmare zonal\nand vertical wavenumbers, respectively, Nis the Brunt-\nVaisala frequency, and His the scale height. The fastest\npropagation speeds occur in the limit of long vertical\nwavelength ( m!0), which yields != 2NHk and thus\nphase and group propagation velocities of 2 NH. The\npropagation time across a hemisphere is thus roughly\na=NH , whereais the planetary radius. We thus expect\nthis jet-driving mechanism to be inhibited when\n\u001crad\u001ca\nNHor\u001cdrag\u001ca\nNH: (2)\nFor typical hot-Jupiter parameters ( a= 108m,H\u0019\n400 km, and N\u00193\u000210\u00003s\u00001appropriate to a ver-\ntically isothermal temperature pro\fle for a gravity of\n10 m s\u00002and speci\fc heat at constant pressure of 1 :3\u0002\n104J kg\u00001K\u00001), we obtain a=NH\u0018105sec. Thus, this\nmechanism should be inhibited when the radiative or\ndrag time scales are much shorter than \u0018105s.\n2.2.2. Multi-way force balance\nEven when the radiative timescale is extremely short\nand zonal propagation of Rossby and Kelvin waves is\ninhibited, an eddy-velocity pattern that promotes equa-\ntorial superrotation can occur under some conditions. As\npointed out by Showman & Polvani (2011) in the con-\ntext of linear solutions, a three-way horizontal force bal-\nance between pressure-gradient, Coriolis, and frictional\n7The primitive equations are the standard equations for large-\nscale atmospheric \rows in stably strati\fed atmospheres. They are\na simpli\fcation of the Navier-Stokes equations wherein the verti-\ncal momentum equation is replaced with local hydrostatic balance,\nand are valid when N2\u001d\n2(where \n is the planetary rotation\nrate) and horizontal length scales greatly exceed the vertical length\nscales. These conditions are generally satisi\fed for the large-scale\n\row in planetary atmospheres, including that on hot Jupiters. See\nShowman et al. (2010) or Vallis (2006, Chapter 2) for a more de-\ntailed discussion.drag forces can lead to eddy velocities tilted northwest-\nsoutheast in the northern hemisphere and southwest-\nnortheast in the southern hemisphere if the drag and\nCoriolis forces are comparable. This occurs because drag\ngenerally points opposite to the wind direction, whereas\nthe Coriolis force points to the right (left) of the wind\nin the northern (southern) hemisphere. When these two\nforces are comparable, balancing them with the pressure-\ngradient force requires that the horizontal wind rotates\nclockwise of the day-night pressure-gradient force in the\nnorthern hemisphere and counterclockwise of it in the\nsouthern hemisphere (Figure 2). In the limit of short\n\u001cradwhen the horizontal pressure-gradient force points\nfrom day to night, these arguments imply that, at low\nlatitudes, the eddy velocities tilt northwest-southeast in\nthe northern hemisphere and southwest-northeast in the\nsouthern hemisphere (Figure 2; see Showman & Polvani\n2011, Appendix D, for an analytic demonstration).\nEven when frictional drag is too weak to play an im-\nportant role in the force balance, a similar three-way bal-\nance between pressure-gradient, Coriolis, and advection\nforces can under appropriate conditions lead to veloc-\nity tilts that promote equatorial superrotation. As air\n\rows from day to night, the Coriolis force will de\rect\nthe trajectory of the air\row to the right of the pressure-\ngradient force in the northern hemisphere and to the left\nof it in the southern hemisphere. When the pressure-\ngradient force per mass, Coriolis force per mass, and ad-\nvective acceleration are all comparable, as expected un-\nder the Rossby number Ro\u00181 conditions typical of hot\nJupiters, then the de\rection will be substantial. In the\nlimit of short \u001cradwhen the horizontal pressure-gradient\nforce points from day to night, these arguments again\nimply that the eddy velocities tilt northwest-southeast\nin the northern hemisphere and southwest-northeast in\nthe southern hemisphere.\nNow consider the e\u000bect of damping on this mecha-\nnism. To the degree that the radiative timescale is short\nenough for temperatures to be close to radiative equi-\nlibrium, radiative damping will not inhibit this mech-\nanism; however, strong frictional damping can prevent\nit from occurring. When the frictional force is much\nstronger than the Coriolis and advective forces, the hor-\nizontal force balance is no longer a multi-way force bal-\nance but rather becomes essentially a two-way balance\nbetween the pressure-gradient force and drag. In this\ncase, winds simply \row down the pressure gradient from\nday to night. There is thus no overall tendency for pro-\ngrade eddy-velocity tilts to develop, so the jet-pumping\nReynolds stress, and the jets themselves, are weak.\nTo quantify the amplitude of drag needed for this tran-\nsition to occur, consider a drag force per mass parame-\nterized by\u0000v=\u001cdrag, where vis horizontal velocity and\n\u001cdragis the drag time constant. The drag force dom-\ninates over the Coriolis force when \u001cdrag\u001cf\u00001, where\nf= 2\n sin\u001eis the Coriolis parameter, \n is the planetary\nrotation rate (2 \u0019over the rotation period), and \u001eis lat-\nitude. Models of hot Jupiters predict \rows whose domi-\nnant length scales are global, in which case the advective\nacceleration should scale as U2=a, whereUis the char-\nacteristic horizontal wind speed. Drag will then domi-\nnate over the advection force when \u001cdrag\u001c(a=jr\bj)1=2,\nwherejr\bjis the characteristic amplitude of the horizon-4 Showman et al.\n(a)\n (b)\nFig. 2.| Schematic illustrating two mechanisms for driving equatorial superrotation on a hot Jupiter from Showman &\nPolvani (2011). (a) The day-night thermal forcing generates standing, planetary-scale Kelvin and Rossby waves. Di\u000berential\nzonal (east-west) propagation of these waves|Kelvin wave to the east and Rossby waves to the west|leads to an eastward\ndisplacement of the thermal structure at the equator and a westward displacement at midlatitudes and, in turn, eddy velocities\nthat tilt northwest-southeast in the northern hemisphere and southwest-northeast in the southern hemisphere. This pattern\nleads to an equatorward \rux of eddy momentum and the emergence of equatorial superrotation. (b). Even when radiative\nforcing is su\u000eciently strong to suppress the di\u000berential zonal thermal o\u000bsets just mentioned, a three-way force balance between\npressure-gradient, Coriolis, and drag forces can lead to equatorward-eastward and poleward-westward velocity tilts, thereby\ndriving equatorial superrotation. Light regions indicate the dayside (with the substellar point marked by an \u0002), and darker\nregions indicate the nightside. When the radiative time constant is short over a broad range of pressures, the pressure-gradient\nforce points from day to night (long gray open arrows). In the linear limit, the pressure-gradient force is balanced by the sum\nof drag (short gray open arrows) and Coriolis forces (short white open arrows). The fact that the Coriolis force points to the\nright (left) of the wind vector in the northern (southern) hemisphere, and that drag typically points in the opposite direction\nof the wind itself, implies that drag and Coriolis forces will exhibit orientations qualitatively similar to those drawn in the\n\fgure when their amplitudes are comparable. This three-way force balance therefore implies that the wind vectors themselves\nexhibit an orientation which is rotated clockwise relative to \u0000rpin the northern hemisphere and counterclockwise relative to\n\u0000rpin the southern hemisphere. The \fgure makes clear that, at low latitudes, these eddy wind-vector orientations correspond\nto northwest-southeast tilts in the northern hemisphere and southwest-northeast tilts in the southern hemisphere. The result\nwould be the equatorward transport of eddy angular momentum and the development of equatorial superrotation.\ntal day-night pressure-gradient force on isobars8, given to\norder of magnitude by jr\bj\u0018R\u0001Thoriz\u0001 lnp=a, where\nRis the speci\fc gas constant, \u0001 Thoriz is the character-\nistic day-night temperature di\u000berence, and \u0001 ln pis the\nrange of ln pover which this temperature di\u000berence ex-\ntends. For typical hot Jupiter parameters, both condi-\ntions imply drag dominance for \u001cdrag\u001c105sec. When\nthis condition is satis\fed, the horizontal force balance\nis between the pressure-gradient and drag forces. As\nmentioned above, the resulting circulation at low pres-\nsure involves day-to-night \row with minimal zonal-mean\neddy-momentum \rux convergences in the meridional di-\nrection and weak zonal jets.\n2.2.3. Direct damping of jets by friction\nFrictional drag can also directly damp the zonal jets.\nA robust understanding of how drag in\ruences the equi-\n8The condition for dominance of drag over advection can be\nmotivated as follows. When advection and drag are comparable,\nand both together balance the pressure-gradient force, it implies\nto order-of-magnitude that\nU2\na\u0018U\n\u001cdrag\u0018jr \bj: (3)\nThese two relations yield \u001cdrag\u0018a=U andU\u0018\u001cdragjr\bj, which\ntogether imply \u001cdrag\u0018(a=jr\bj)1=2. For drag time constants sig-\nni\fcantly shorter than this value, the drag force exceeds the ad-\nvection force.librated jet speed|and hence a rigorous theoretical\nprediction of the amplitudes of drag needed to damp\nthe jet|requires a detailed theory for the full, three-\ndimensional interactions of the global-scale planetary\nwaves with the background \row, which is currently lack-\ning. It is therefore not possible at present to provide\na robust theoretical estimate of the amplitude of drag\nnecessary to damp the zonal jets. Still, because the\njets are fundamentally driven by global-scale waves that\nresult from the day-night heating gradients (Showman\n& Polvani 2011), and because the radiative time con-\nstant increases rapidly with depth, we expect that the\nmagnitude of zonal-mean acceleration of the zonal-mean\nzonal wind varies strongly with depth. These arguments\nheuristically suggest that the necessary frictional damp-\ning times are less than a value ranging from 104sec at low\npressures of say .0.1 bar to 106sec or more at pressures\nof several bars, below the infrared photosphere.\n2.2.4. Recap\nWhen the jets and the waves that generate them are\nsuppressed, the planet will tend to exhibit a large day-\nnight temperature di\u000berence at low pressure, resulting\nin a large horizontal pressure gradient force between day\nto night that will drive a day-night \row (modi\fed by\nthe Coriolis e\u000bect) at low pressure. In this regime, air\n\rows toward Earth along most of the terminator, leading\nto a predominantly blueshifted Doppler signature dur-Atmospheric circulation of hot Jupiters 5\ning transit. Mass continuity requires the existence of a\nreturn \row from night to day in the deep atmosphere\n(below the regions sensed by Doppler transit measure-\nments). Because the density at depth is much larger\nthan that aloft, the velocities of this return \row can be\nsmall.\n3.TEST OF THE TWO REGIMES WITH AN\nIDEALIZED MODEL\nWe now demonstrate this transition from jet- to eddy-\ndominated circulation regimes in an idealized dynami-\ncal model. As in Showman & Polvani (2011), we con-\nsider a two-layer model, with constant densities in each\nlayer; the upper layer represents the strati\fed, meteoro-\nlogically active atmosphere and the lower layer represents\nthe denser, quiescent deep interior. When the lower layer\nis taken to be in\fnitely deep and the lower-layer winds\nand pressure \feld are steady in time, the governing equa-\ntions are the shallow-water equations for the \row in the\nupper layer:\ndv\ndt+grh+fk\u0002v=R\u0000v\n\u001cdrag(4)\n@h\n@t+r\u0001(vh) =heq(\u0015;\u001e)\u0000h\n\u001crad\u0011Q (5)\nwhere v(\u0015;\u001e;t ) is horizontal velocity, h(\u0015;\u001e;t ) is the up-\nper layer thickness, \u0015is longitude, ttime,gis the (re-\nduced) gravity,9andd=dt\u0011@=@t +v\u0001ris the material\n(total) derivative. The term Rin Eq. (4) represents mo-\nmentum advection between the layers; it is \u0000vQ=h in\nregions of heating ( Q > 0) and zero in regions of cool-\ning (Q< 0). See Showman & Polvani (2011) for further\ndiscussion and interpretation of the equations.\nIn the context of a three-dimensional (3D) atmosphere,\nthe boundary between the layers represents an atmo-\nspheric isentrope, and radiative heating/cooling, which\ntransports mass between layers, is therefore represented\nas a mass source/sink, Q, in the upper-layer equations.\nWe parameterize this as a Newtonian cooling that relaxes\nthe thickness toward a radiative-equilibrium thickness,\nheq(\u0015;\u001e), over a prescribed radiative time constant \u001crad.\nHere, we set\nheq(\u0015;\u001e) =\u001aH on the nightside;\nH+ \u0001heqcos\u0015cos\u001eon the dayside\n(6)\nwhere the substellar point is at ( \u0015;\u001e) = (0\u000e;0\u000e). This\nexpression incorporates the fact that, on the nightside,\nthe radiative equilibrium temperature pro\fle of a syn-\nchronously rotating hot Jupiter is constant (e.g., Show-\nman et al. 2008), whereas on the dayside the radiative-\nequilibrium temperature increases from the terminator to\nthe substellar point. An important property of Eq. (6) is\nthat the zonal-mean radiative-equilibrium thickness, heq,\nis greater at the equator than the poles, re\recting the fact\nthat a planet with zero obliquity (whether tidally locked\nor not) absorbs more sunlight at low latitudes than high\n9The reduced gravity is the gravity times the fractional density\ndi\u000berence between the two layers. For a hot Jupiter with a strongly\nstrati\fed thermal pro\fle, where entropy increases sign\fcantly over\na scale height, the reduced gravity is comparable to the actual\ngravity.latitudes. Note that Eq. (6) di\u000bers from the formulation\nofheqadopted by Showman & Polvani (2011), where heq\nwas set toH+ \u0001heqcos\u0015cos\u001eacross the entire planet\n(dayside and nightside).\nIn addition to radiation, we include frictional drag\nparameterized with Rayleigh friction, \u0000v=\u001cdrag, where\n\u001cdragis a speci\fed drag timescale. The drag could result\nfrom vertical turbulent mixing (Li & Goodman 2010),\nLorentz-force braking (Perna et al. 2010), or other pro-\ncesses.\nOur model formulation is identical to that described\nin Showman & Polvani (2011, Section 3.2) in all ways\nexcept for the prescription of heq(\u0015;\u001e).\nParameters are chosen to be appropriate for hot\nJupiters. We take gH = 4\u0002106m2sec\u00002and set\n\u0001heq=H= 1, implying that the radiative-equilibrium\ntemperatures vary by order-unity from nightside to day-\nside. We also take \n = 3 :2\u000210\u00005sec\u00001anda=\n8:2\u0002107m, implying a rotation period of 2.2 Earth days\nand radius of 1.15 Jupiter radii, similar to the values for\nHD 189733b. The radiative and frictional timescales are\nvaried over a wide range to characterize the dynamical\nregime.\nWe solved Equations (4){(5) in full spherical geome-\ntry using the Spectral Transform Shallow Water Model\n(STSWM) of Hack & Jakob (1992). The equations are in-\ntegrated using a spectral truncation of T170, correspond-\ning to a resolution of 0 :7\u000ein longitude and latitude (i.e.,\na global grid of 512 \u0002256 in longitude and latitude). All\nmodels were integrated until a steady state is reached.\nThe solutions con\frm our theoretical predictions of a\nregime transition. Figure 3 illustrates the equilibrated\nsolutions for radiative time constants, \u001crad, of 10, 1, 0.1,\nand 0.01 days10for the case where drag is turned o\u000b\n(i.e.,\u001cdrag! 1 ).11As expected, when the radiative\ntime constant is long (10 days, panel (a)), jets dominate\nthe circulation, with relatively weak eddies in compar-\nison to the zonal-mean zonal winds. At intermediate\nvalues of the radiative time constant (1 and 0.1 days,\npanels (b) and (c)), the \row consists of strong jets and\nsuperposed eddies. At short values of the radiative time\nconstant (0.01 days, panel (d)), the jets are relatively\nweak|though not absent|and day-to-night eddy \row\ndominates the circulation.\nThe dynamical behavior of this sequence can be un-\nderstood as follows. When the radiative time constant\nis long (Figure 3(a)), the day-night thermal forcing is\nweak|air parcels experience only weak heating/cooling\nas they circulate from day to night|and the circula-\ntion is instead dominated by the equator-to-pole vari-\nation in the zonal-mean heq(i.e., by the zonal-mean ra-\ndiative heating at low latitudes and cooling at high lati-\ntudes). At intermediate values of the radiative time con-\nstant (panels (b) and (c)), the day-night thermal forc-\ning becomes su\u000eciently strong to generate a signi\fcant\nplanetary wave response, and the eddy-momentum con-\nvergence induced by these waves generates equatorial su-\nperrotation via the mechanisms identi\fed by Showman &\n10In this paper, 1 day is de\fned as 86400 sec.\n11As described by Showman & Polvani (2011), the coupling be-\ntween layers|speci\fcally, mass, momentum, and energy exchange\nin the presence of heating/cooling|ensures that even cases with-\nout drag in the upper layer readily equilibrate to a steady state.\nAll the models shown here are equilibrated.6 Showman et al.\n(a)\u001crad= 10 days\n(b)\u001crad= 1 day\n(c)\u001crad= 0:1 days\n(d)\u001crad= 0:01 days\nFig. 3.| Geopotential gh(orange scale, units m2s\u00002) and winds\n(arrows) for the equilibrated (steady-state) solutions to the shallow-\nwater equations (Equations 4{5) in full spherical geometry assuming\nno upper-level drag ( \u001cdrag!1 ) and\u001crad=10, 1, 0.1, and 0.01 Earth\ndays from top to bottom, respectively. Note the transition from a\ncirculation dominated by zonally symmetric jets at long \u001cradto one\ndominated by day-to-night \row at short \u001crad.\nPolvani (2011), particularly the di\u000berential zonal propa-\ngation of the standing Kelvin and Rossby waves. At short\nradiative time constant (panel (d)), such zonal propa-\ngation is inhibited but, as predicted by the theory in\nSection 2, the three-way force balance between pressure-\ngradient, Coriolis, and advection forces still generates\nprograde phase tilts in the velocities. Although, visually,\nthe \row appears dominated primarily by day-to-night\n\row (Figure 3(d)), these phase tilts still drive superrota-\ntion near the equator, and even at higher latitudes thezonal-mean zonal wind remains a signi\fcant fraction of\nthe eddy wind amplitude.\nThe transition from a regime dominated by jets to\na regime dominated by day-to-night \row is even more\nstriking when drag is included. Figure 4 shows a sequence\nof models with radiative time constants, \u001crad, of 10, 1,\n0.1, and 0.01 days (as in Figure 3) but with \u001cdrag= 10\u001crad\nin all cases. Overall, the trend resembles that in Figure 3:\nat long radiative time constants (panel (a)), the \row is\ndominated by high-latitude, highly zonal jets; at inter-\nmediate radiative time constants (panel (b) and (c)), the\n\row is transitional, exhibiting strong eddies associated\nwith the standing planetary-scale wave response to the\nday-night thermal forcing, and zonal jets driven by those\neddies (cf Showman & Polvani 2011); and at short radia-\ntive time constants (panel (d)), the Kelvin and Rossby\nwaves are damped and the circulation consists almost\nentirely of day-to-night \row. As predicted by the the-\nory in Section 2, drag in this case is strong enough to\noverwhelm the advection and Coriolis forces, leading to\na two-way horizontal force balance between the pressure-\ngradient force and drag. As a result, there is no overall\nprograde phase tilt of the velocity pattern. The eddy\nforcing of the zonal-mean \row, and the jets themselves,\nare therefore weak.\nTo better characterize the dominance of jets versus\nday-night \row, we performed integrations over a com-\nplete grid including all possible combinations of 0.01, 0.1,\n1, 10, and 100 days in \u001cradand 0.1, 1, 10, 100, and 1\ndays in\u001cdrag. The typical amplitude of the jets can be\ncharacterized by the root-mean-square of the zonal-mean\nzonal wind variation in latitude:\nurms=\"\n1\n\u0019Z\u0019=2\n\u0000\u0019=2u(\u001e)2d\u001e#1=2\n(7)\nwhere the overbar denotes a zonal average. To char-\nacterize the amplitude of the eddies, we adopt a metric\nrepresenting the variation of the zonal wind in longitude:\nueddy(\u001e) =\u00141\n2\u0019Z\u0019\n\u0000\u0019(u\u0000u)2d\u0015\u00151=2\n(8)\nand then determine the root-mean-square variations of\nthis quantity in latitude:\nueddy;rms=\"\n1\n\u0019Z\u0019=2\n\u0000\u0019=2ueddy(\u001e)2d\u001e#1=2\n: (9)\nThe ratio of urmstoueddy;rmsthen provies a measure of\nthe relative dominance of jets versus day-night \row.\nThese calculations demonstrate that jets dominate\nwhen both the radiative and drag time constant are long,\nand that day-to-night eddy \row dominates when either\ntime constant is very short. This is shown in Figure 5,\nwhich presents the ratio urms=ueddy;rmsversus\u001cradand\n\u001cdrag. The dashed curve in panel (b), corresponding to\nurms=ueddy;rms= 1, demarcates the approximate transi-\ntion between regimes (jets dominate above and to the\nright of the curve, while day-night \row dominates below\nand to the left of the curve). Although extremely short\nvalues of either\u001crador\u001cdragare su\u000ecient to ensure eddy-\ndominated \row, the trend of the transition di\u000bers for the\ntwo time constants. When drag is weak or absent, \u001cradAtmospheric circulation of hot Jupiters 7\n(a)\u001crad= 10 days \u001cdrag= 100 days\n(b)\u001crad= 1 day \u001cdrag= 10 days\n(c)\u001crad= 0:1 day\u001cdrag= 1 day\n(d)\u001crad= 0:01 day \u001cdrag= 0:1 day\nFig. 4.| Geopotential gh(orange scale, units m2s\u00002) and winds\n(arrows) for the equilibrated (steady-state) solutions to the shallow-\nwater equations (Equations 4{5) in full spherical geometry assuming\n\u001cdrag= 10\u001cradand\u001crad=10, 1, 0.1, and 0.01 days from top to bottom,\nrespectively. Note the transition from a circulation dominated by\nzonally symmetric jets at long \u001cradto one dominated by day-to-night\n\row at short \u001crad.\nmust be extremely short|less than 0.1 day|to ensure\neddy- rather than jet-dominated \row (Figure 5). On the\nother hand, over a wide range of \u001cradvalues,\u001cdragneed\nonly be less than\u00183 days to ensure eddy-dominated \row.\nThe transition between jet and eddy-dominated regimes\nas a function of drag occurs more sharply when \u001cradis\nlarge than when it is small.\n4.THREE-DIMENSIONAL MODELS\nFig. 5.| Ratiourms=ueddy;rms, characterizing the ratio of jets\nto eddies, versus \u001cradand\u001cdrag.Top: triple-dashed-dotted,\ndashed-dotted, dashed, dotted, and solid curves depict results\nfrom models with \u001cdrag values of 0.1, 1, 10, 100 days, and in-\n\fnite (i.e., no drag in the upper layer), respectively. Bottom:\nTwo-dimensional representation of the same data. Colorscale\ndepicts log10(urms=ueddy;rms) with even log-spacing of the con-\ntour intervals. Values of urms=ueddy;rmsrange from 0.06 (i.e.,\n10\u00001:193) to 31 (i.e., 101:496). The dashed black contour denotes\nurms=ueddy;rms= 1 (i.e., log10(urms=ueddy;rms) = 0), giving an\napproximate demarcation between the jet- and eddy-dominated\nregimes. Jets dominate when both \u001cradand\u001cdrag are long, and\neddies (essentially day-night \row) dominate when either time con-\nstant becomes very short.\nWe now demonstrate this regime shift in three-\ndimensional atmospheric circulation models including re-\nalistic radiative transfer. GJ 436b, HD 189733b, and HD\n209458b are selected as examples that bracket a large\nrange in stellar irradiation and yet are relatively easily\nobservable, hence representing good targets for Doppler\ncharacterization.\n4.1. Model setup\nWe solve the radiation hydrodynamics equations us-\ning the Substellar and Planetary Atmospheric Circula-\ntion and Radiation (SPARC) model of Showman et al.\n(2009). This model couples the dynamical core of the\nMITgcm (Adcroft et al. 2004), which solves the primitive\nequations of meteorology in global, spherical geometry,\nusing pressure as a vertical coordinate, to the state-of-\nthe-art, nongray radiative transfer scheme of Marley &\nMcKay (1999), which solves the multi-stream radiative\ntransfer equations using the correlated- kmethod to treat\nthe wavelength dependence of the opacities. Here, we use\na two-stream implementation of this model. To date, this\nis the only circulation model of hot Jupiters to include a8 Showman et al.\nTABLE 1\nModel Parameters\nGJ 436b HD 189733b HD 209458b\nradius (m) 2 :69\u00021078:2396\u00021079:437\u0002107\nrotation period (days) 2.3285 2.2 3.5\ngravity (m s\u00002) 12.79 21.4 9.36\npbase(bars) 200 200 200\nptop(bars) 2\u000210\u000052\u000210\u000062\u000210\u00006\nNr 47 53 53\nmetallicity (solar) 1 and 50 1 1\nrealistic radiative transfer solver, which is necessary for\naccurate determination of heating rates, temperatures,\nand \row\feld. The composition and therefore opacities\nin hot-Jupiter atmospheres are uncertain. Here, gaseous\nopacities are calculated assuming local chemical equilib-\nrium for a speci\fed atmospheric metallicity, allowing for\nrainout of any condensates. We neglect any opacity due\nto clouds or hazes.\nModel parameters are summarized in Table 1. Al-\nthough the atmosphere of GJ 436b is likely enriched\nin heavy elements (Spiegel et al. 2010; Lewis et al.\n2010; Madhusudhan & Seager 2011), solar metallicity\nrepresents a reasonable baseline for HD 189733b and\nHD 209458b, and to establish the e\u000bect of di\u000bering stel-\nlar \rux at constant metallicity we therefore adopt solar\nmetallicity for the gas opacities in our nominal models of\nall three planets. To bracket the range of plausible metal-\nlicities, we also explore a model of GJ 436b with 50 times\nsolar metallicity. For HD 189733b and GJ 436b, we ne-\nglect opacity due to strong visible-wavelength absorbers\nsuch as TiO and VO, as TiO and VO are not expected\nfor these cooler planets. For HD 209458b, secondary-\neclipse measurements suggest the presence of a strato-\nsphere (Knutson et al. 2008), and we therefore include\nTiO and VO opacity for this planet, which allows a ther-\nmal inversion due to the large visible-wavelength opac-\nity of these species (Hubeny et al. 2003; Fortney et al.\n2008; Showman et al. 2009). While debate exists about\nthe ability of TiO to remain in the atmosphere (Show-\nman et al. 2009; Spiegel et al. 2009), our present pur-\npose is simply to use TiO as a proxy for any chemical\nspecies that strongly absorbs in the visible wavelengths\nand hence allows a stratosphere to exist; other strong\nvisible-wavelength absorbers would exert qualitatively\nsimilar e\u000bects. Synchronous rotation is assumed for HD\n189733b and HD 209458b (with a substellar longitude\nperpetually at 0\u000e); the rotation of GJ 436b, however,\nis assumed to be pseudosychronized with its slightly ec-\ncentric orbit (Lewis et al. 2010).12The obliquity of all\nmodels is zero so that the substellar point lies along the\nequator.\nOur nominal models do not include explicit frictional\ndrag in the upper levels.13However, several frictional\n12This has only a modest e\u000bect on the results; synchronously\nrotating models of GJ 436b on circular orbits exhibit similar cir-\nculation patterns.\n13All of the models include a Shapiro \flter to maintain numer-\nical stability. In some of the models, particularly those for HD\n209458b, we also include a drag term in the deep atmosphere be-\nlow 10 bars; this allows the total kinetic energy of the model to\nequilibrate while minimally a\u000becting the circulation in the upperprocesses may be important for hot Jupiters, including\nvertical turbulent mixing (Li & Goodman 2010), break-\ning small-scale gravity waves (Watkins & Cho 2010), and\nmagnetohydrodynamic drag (Perna et al. 2010). The lat-\nter may be particularly important for hot planets such as\nHD 209458b. Accordingly, we additionally present a se-\nquence of HD 209458b integrations that include frictional\ndrag, which we crudely parameterize as a linear relax-\nation of the zonal and meridional velocities toward zero14\nover a prescribed drag time constant, \u001cdrag. Within any\ngiven model, we treat \u001cdrag as a constant everywhere\nwithin the domain. This is not a rigorous representa-\ntion of drag (for example, Lorentz forces will vary greatly\nfrom dayside to nightside and may act anisotropically on\nthe zonal and meridional winds); still, the approach al-\nlows a straightforward evaluation of how drag of a given\nstrength alters the circulation.\nFor all three planets, we solve the equations on the\ncubed-sphere grid using a horizontal resolution of C32,\ncorresponding to an approximate global resolution of\n128\u000264 in longitude and latitude. The lowermost Nr\u00001\nvertical levels are evenly spaced in log-pressure from an\naverage basal pressure pbaseof 200 bars to a top pres-\nsure,ptop, of 20\u0016bars for GJ 436b and 2 \u0016bars for HD\n189733b and HD 209458b. The uppermost model level\nextends from a pressure of ptopto zero. Our models of GJ\n436b and HD 189733b were originally presented in Lewis\net al. (2010) and Fortney et al. (2010), respectively, while\nfor HD 204958b we present new models here. These new\nintegrations adopt 11 opacity bins in our correlated- k\nscheme; detailed tests show that this 11-bin scheme pro-\nduces net radiative \ruxes, heating rates, and atmospheric\ncirculations very similar to those of the 30-bin models\n(see Kataria et al. 2012). We integrate these models\nuntil they reach an essentially steady \row con\fguration\nat pressures <1 bar, corresponding to integration times\ntypically ranging from one to four thousand Earth days,\ndepending on the model.\n4.2. Results: nominal models\nOur nominal, three-dimensional models exhibit a fun-\ndamental transition in the upper-atmospheric behavior|\nat pressures where Doppler measurements are likely to\nsense|as stellar insolation increases from modest (for\nGJ 436b) to intermediate (for HD 189733b) to large (for\nHD 209458b). This is illustrated in Figures 6 and 7.\nFigure 6 shows temperature and winds over the globe\nfor models of GJ 436b, HD 189733b, and HD 209458b.\nFigure 7 presents histograms of the fraction of termi-\nnator arc length versus terminator zonal-wind speed for\natmosphere. Note that, for computationally feasible integration\ntimes (thousands of Earth days), models that include drag in the\ndeep layers (but not at pressures less than \u001810 bars) exhibit \row\npatterns and wind speeds in the observable atmosphere that are\nextremely similar to those in models that entirely lack large-scale\ndrag. This is due to the fact that, even in such drag-free models,\nthe wind speeds at pressures &10 bars remain weak. For brevity,\nin this paper, we use the term \\drag free\" to refer to models lack-\ning an explicit large-scale drag term, \u0000v=\u001cdrag, in the observable\natmosphere; nevertheless, it should be borne in mind that some of\nthose models do contain drag in the bottommost model layers, and\nall of them include the Shapiro \flter.\n14In other words, we add a term \u0000v=\u001cdragto the horizontal mo-\nmentum equations, where vis the horizontal velocity. This simple\nscheme is called \\Rayleigh drag\" in the atmospheric dynamics lit-\nerature.Atmospheric circulation of hot Jupiters 9\n(a)\n(b)\n(c)\n(d)\nFig. 6.| Global temperature (orange scale) and winds (arrows) for\na sequence of 3D SPARC/MITgcm models at a pressure of 0.1 mbar,\nwhere Doppler measurements are likely to sense. Panels (a), (b), (c),\nand (d) show solar-metallicity GJ 436b, 50 \u0002solar GJ 436b, solar-\nmetallicity HD 189733b, and solar-metallicity HD 209458b models,\nrespectively. The vertical blue solid and dashed lines show the loca-\ntion of the terminators 90\u000ewest and east of the substellar longitude,\nrespectively. The substellar point is at longitude 0\u000ein all panels. The\nmodels show a gradual transition from a circulation dominated by\nzonal jets (top) to one dominated by day-night \row at low pressure\n(bottom).\nthese same models,15which gives an approximate sense\nof how a discrete spectral line would be split, shifted, or\nsmeared in frequency due to the Doppler shift of zonal\nwinds along the terminator. To isolate the e\u000bect of dy-\n15For each model, we de\fne a one-dimensional array uicorre-\nsponding to the terminator velocity at 0.1 mbar versus terminator\nangle\u0012ifrom 0 to 360\u000e. We de\fne 20 velocity bins, equally spaced\nbetween the minimum and maximum terminator velocities from\nthe arrayui. We then determine the fraction of the points in the\nuiarray that fall into each velocity bin. This is what is plotted in\nFigure 7. The qualitative results are similar when di\u000berent choices\nare made for the number of velocity bins.\n(a)\n(b)\n(c)\n(d)\nFig. 7.| Histograms showing the fraction of the full, 360\u000etermina-\ntor, at a pressure of 0.1 mbar, \rowing at various wind speeds toward\nor away from Earth for the four models presented in Figure 6. (To\nisolate the dynamical contribution, this does not include the contri-\nbution of planetary rotation to the inertial-frame velocity.) Negatives\nvalues are toward Earth and positive values are away from Earth.\nPanels (a), (b), (c), and (d) show solar-metallicity GJ 436b, 50 \u0002so-\nlar GJ 436b, solar-metallicity HD 189733b, and solar-metallicity HD\n209458b models, respectively. This gives a crude sense of the dynam-\nical contribution to the Doppler splitting of a discrete spectral line.\nThe models exhibit a transition from \rows that exhibit both blue-\nand red-shifted components (top) to a \row whose Doppler signature\nwould be predominantly blueshifted (bottom).\nnamics, the contribution of planetary rotation to the ve-\nlocity is not included in Figure 7, although we will return\nto its e\u000bects subsequently.\nIn the case of solar-metallicity GJ 436b (Figure 6a),\nthe predominant dynamical feature is a broad superro-\ntating (eastward) jet that extends over all longitudes and\nin latitude almost from pole to pole. The jet exhibits sig-\nni\fcant wave activity, manifesting as small-scale \ructua-\ntions in temperature and zonal wind, particularly at the\nhigh latitudes of both hemispheres where the zonal-mean10 Showman et al.\nzonal winds peak. Nevertheless, the model exhibits little\ntendency toward a zonal-wavenumber-one pattern that\nwould be associated with a predominant day-to-night\n\row. Save for small regions near the poles, the zonal\nwinds at low pressure are everywhere eastward, imply-\ning that, during transit, the zonal winds \row away from\nEarth along the leading limb and toward Earth along the\ntrailing limb. This would lead to almost equal blueshifted\nand redshifted Doppler components, with a relative min-\nimum near zero Doppler shift (Figure 7a).\nNext consider HD 189733b (Figure 6c). The model\nagain exhibits a superrotating equatorial jet, which is\nfast across most of the nightside|achieving eastward\nspeeds of 4 km s\u00001|but slows down considerably over\nthe dayside, reaching zero speed near the substellar\npoint. Despite this variation, the zonal winds within\nthe jet (latitudes equatorward of 60\u000e) are eastward along\nboth terminators. In contrast, the high-latitude zonal\nwind (poleward of 60\u000elatitude) is westward along the\nwestern terminator and eastward along the eastern ter-\nminator16, as expected for day-to-night \row. As seen\nduring transit, along the trailing limb, the zonal winds\n\row toward Earth. Along the leading limb, they \row\ntoward Earth poleward of 60\u000elatitude and away from\nEarth equatorward of 60\u000elatitude. Spectral lines would\nthus exhibit a broadened or bimodal character, with the\nblueshifted component considerably stronger than the\nredshifted component (Figure 7c). HD 189733b is thus\na transitional case between the two regimes discussed in\nSection 2.\nIn the case of HD 209458b (Figure 6d), the strong su-\nperrotating jet continues to exist, but at and west of the\nwestern terminator it is con\fned substantially closer to\nthe equator than in our GJ 436b or HD 189733b mod-\nels. Poleward of\u001830\u000elatitude on the western terminator,\nand everywhere along the eastern terminator, the air\row\ndirection is from day to night. This implies that, as seen\nduring transit, the trailing limb exhibits zonal winds to-\nward Earth. The leading limb exhibits zonal winds that\nare toward Earth poleward of \u001830\u000elatitude and away\nfrom Earth equatorward of \u001830\u000elatitude. This would\nlead to Doppler shifts that are almost entirely blueshifted\n(Figure 7d).\nTo summarize, these models exhibit a transition from\na circulation dominated by zonal jets at modest insola-\ntion (GJ 436b) to one dominated by day-night \row at\nhigh insolation (HD 209458b). Qualitatively, this transi-\ntion matches well the predictions from our theory in Sec-\ntion 2|as the stellar insolation increases, the e\u000bective\nradiative timescale decreases, and this damps the stand-\ning planetary-scale Rossby and Kelvin waves, limiting\ntheir ability to drive a dominant zonal \row and lead-\ning to a circulation comprised primarily of day-to-night\n\row at low pressure. We emphasize that the models in\nFigures 6{7 do not contain frictional drag at the low pres-\nsures sensed by remote measurements, and so the only\nsource of damping is the radiation (as well as the Shapiro\n\flter, which exerts minimal e\u000bect at large scales). The\nmodels show that the regime transition occurs very grad-\nually as stellar insolation is varied (Figures 6a{d). This\nis also consistent with theoretical expectations; as shown\n16Eastern and western terminators refer here to the terminators\n90\u000eof longitude east and west, respectively, of the substellar point.\nFig. 8.| Winds toward or away from Earth (colorscale, m s\u00001)\nalong the full, 360\u000eterminator in a sequence of models as viewed\nduring the center of transit. Colorscale is such that red (posi-\ntive) represents redshifted velocities while blue (negative) repre-\nsents blueshifted velocities. The radial coordinate represents log\npressure, and the plotted range is from 200 bars at the inside to\n2\u0016bar at the outside. The \frst, second, and third rows show our\nsolar-metallicity nominal models of GJ 436b, HD 189733b, and\nHD 209458b, respectively. The fourth row shows our model of HD\n209458b where frictional drag is imposed with a drag time con-\nstant of 104s. For each model, the left panel shows the winds\nalone, and the right panel shows the sum of the winds and the\nplanet's rotation. From top to bottom, the transition from high-\naltitude velocities that have both blue and redshifted components\nto velocities that are entirely blueshifted is clearly evident.\nin Figure 5, when large-scale drag is absent, the radia-\ntive time constant must be decreased by over a factor\nof\u001830 (from\u00183 days to less than 0.1 day) to force the\n\row from the jet-dominated to eddy-dominated regime.\nMoreover, as discussed in Section 2, damping through\nradiation alone can inhibit di\u000berential zonal propagationAtmospheric circulation of hot Jupiters 11\nFig. 9.| Temperature and winds at 0.1 mbar pressure in models of HD 209458b with frictional drag. Left: a model with a drag time\nconstant of 3\u0002104s. Right: a model with a drag time constant of 104s. Top panels show temperature in K (orange scale) and winds\n(arrows). Bottom panels show zonal wind in m s\u00001. In the case with weaker drag, an equatorial jet extends partway across the nightside,\nbut the jet is damped out in the case with stronger drag. Vertical solid and dashed lines show the terminators.\nof the planetary-scale waves, but the multi-way horizon-\ntal force balance between pressure-gradient, Coriolis, and\nadvective forces can still produce prograde phase tilts\nnear the equator. Thus, we still expect a narrow equato-\nrial jet over at least some longitudes. This can be seen in\nthe nonlinear shallow-water solutions (see Figure 3(d))\nand also explains the continuing existence of a narrow\nequatorial jet even for extreme radiative forcing in the\n3D models (Figure 6(d)).\nThis regime transition manifests clearly in plots of ter-\nminator winds. Figure 8 shows the wind component pro-\njected along the line of sight to Earth at the termina-\ntor for a sequence of models. Red represents velocities\naway from Earth (hence redshifted) while blue repre-\nsents velocities toward Earth (hence blueshifted). The\n\frst, second, and third rows of Figure 8 show our nom-\ninal models of GJ 436b, HD 189733b, and HD 209458b,\nwhile the fourth row depicts our model of HD 209458b\nadopting frictional drag with \u001cdrag= 104s. For GJ 436b,\nthe leading limb is redshifted while the trailing limb is\nblueshifted. For HD 189733b, the redshifted portion|\ncorresponding to the equatorial jet|is con\fned to the\nlow and midlatitudes on the leading limb. As a result,\nat high altitudes, only about one-quarter of the limb\nis redshifted, while about three-quarters is blueshifted.\nFor our nominal HD 209458b model, the con\fnement of\nthe equatorial jet to low latitudes on the leading limb is\neven stronger, such that only about \u001810% of the high-\naltitude limb is redshifted while \u001890% is blueshifted. In\nthe HD 209458b model with frictional drag, the high-\naltitude winds are blueshifted over the entire terminator,\ncompleting the transition from a circulation dominated\nby jets to one dominated by high-altitude day-to-night\n\row.\nThe regime transition discussed here is a\u000bected not\nonly by the stellar insolation but also the atmospheric\nmetallicity. Larger metallicities imply larger gaseousopacities (due to the increased abundance of H 2O, CO,\nand CH 4), and this moves the photosphere to lower pres-\nsures (cf Spiegel et al. 2010; Lewis et al. 2010), imply-\ning that the bulk of the starlight is then absorbed in a\nregion with very little atmospheric mass. As a result,\nincreasing the atmospheric metallicity enhances the day-\nside heating and nightside cooling per mass at the pho-\ntosphere even when the stellar insolation remains un-\nchanged. The e\u000bects of this are illustrated in Figure 6b,\nwhich shows a GJ 436b model identical to that in Fig-\nure 6a except that the metallicity is 50 times solar (Lewis\net al. 2010). Because of the greater absorption of stellar\nradiation at high levels, the atmosphere exhibits a large\nday-night temperature di\u000berence and signi\fcant zonal-\nwavenumber-one structure in the zonal wind, with strong\nlongitudinal variations in the equatorial jet reminiscent\nof that in our HD 189733b model (compare Figures 6(b)\nand (c)). Although eastward \row still dominates along\nmost of the terminator, as in the solar-metallicity GJ\n436b model, the western terminator exhibits westward\n\row within\u001830\u000elatitude of the pole. Spectral lines as\nseen during transit still exhibit bimodel blue and red-\nshifts, but the blueshifts are now slightly more dominant\n(Figure 7b).\nAlthough we have focused on the existence of a regime\ntransition in models with di\u000bering stellar \ruxes, it is\nworth emphasizing that the same transition often occurs\nwithin a given model from low pressure to high pres-\nsure. Generally speaking, the radiative time constants\nare short at low pressure and long at high pressure (Iro\net al. 2005; Showman et al. 2008). The theory presented\nhere therefore predicts that, as long as the incident stellar\n\rux is su\u000eciently high and frictional drag is su\u000eciently\nweak, the air should transition from a day-to-night \row\npattern at low pressure to a jet-dominated zonal \row\nat high pressures. Just such a pattern is seen in many\npublished 3D hot Jupiter models (e.g., Cooper & Show-12 Showman et al.\nman 2005, 2006; Showman et al. 2008, 2009; Rauscher &\nMenou 2010; Heng et al. 2011b). Note, however, that\nif the incident stellar \rux is su\u000eciently low (and the\ndrag is very weak), the atmosphere may be in a regime\nof jet-dominated \row throughout; on the other hand, if\nfrictional drag is su\u000eciently strong, jets may be unable\nto form at all, and the atmosphere may be in a regime\nof day-night \row aloft with a very weak return \row at\ndepth.\n4.3. Results: in\ruence of drag\nWe now consider the e\u000bect of frictional drag in 3D\nmodels. As discussed in Section 2, su\u000eciently strong fric-\ntional drag (i) damps the standing planetary-scale waves\nthat are the natural response to the day-night heating\ngradient, and (ii) drives the horizontal force balance into\na two-way balance between pressure-gradient and drag\nforces, both of which inhibit the development of prograde\nphase tilts in the eddy velocities and in turn the pumping\nof zonal jets, and, \fnally (iii) directly damps the zonal\njets. Thus, we expect that an atmosphere with su\u000e-\nciently strong frictional drag will lack zonal jets and that\nits circulation will instead consist primarily of day-to-\nnight \row at high altitude, with return \rows at depth.\nFigure 9 illustrates this for solar-metallicity models of\nHD 209458b where Rayleigh drag is implemented with\ntime constants of 3 \u0002104s (left column) and 104s (right\ncolumn). As predicted, the air at 0.1 mbar \rows directly\nfrom dayside to nightside over both terminators. The\nmodel with \u001cdrag= 3\u0002104s exhibits a remnant equa-\ntorial jet on the nightside that extends from the eastern\nterminator to the antistellar point. Because the stronger\nfrictional drag damps it out, the model with \u001cdrag= 104s\nlacks such a jet, and the \row exhibits only modest asym-\nmetry (due to the \fe\u000bect) between the western and\neastern terminators. Doppler lines would be entirely\nblueshifted in both cases.\nFriction a\u000bects not only the qualitative circulation\nregime (e.g., existence or lack of zonal jets) but also the\nspeed of the high-altitude \row between day and night.\nFigure 10 shows the root-mean-square zonal wind speeds\nat the terminator for a sequence of HD 209458b mod-\nels with di\u000bering drag time constants. All of these runs\nare in the same regime as the model in Figure 9, where\nday-to-night \row dominates at low pressure. When \u001cdrag\nis su\u000eciently long, the \row speeds are independent of\nthe drag time constant, but they start to decrease when\n\u001cdragis su\u000eciently short. When drag is absent in the up-\nper atmosphere, our HD 209458b model equilibrates to\nan rms terminator wind speed of 5 :2 km s\u00001at 0.1 mbar,\ndecreasing with depth to 3 :8, 2:6, and 1:9 km s\u00001at 1,\n10, and 100 mbar, respectively. As shown in Figure 10,\nthe addition of weak drag ( \u001cdrag= 3\u0002105s) exerts only\na modest e\u000bect on the day-night \row speeds at pressures\n.10 mbar. Drag time constants \u001cdrag.105s start to\nmatter signi\fcantly in the upper atmosphere, however;\nfor example, for \u001cdrag= 104s, the rms terminator speeds\nare 2:1, 1:1, 0:6, and 0:2 km s\u00001at 0.1, 1, 10, and 100\nmbar|signi\fcantly less than the equilibrated speeds in\nthe absence of upper-level drag.\nThe above results suggest that the amplitude of the\nobserved Doppler shift can place constraints on the\nstrength of frictional drag in the upper atmospheres of\nhot Jupiters. Snellen et al. (2010)'s inference of winds on\nFig. 10.| Steady-state root-mean-square wind speeds at the\nterminator versus frictional drag time constant from a sequence\nof HD 209458b models including drag. For each 3D model, per-\nformed for a given drag time constant, the root-mean-square wind\nspeeds|calculated along the terminator|are shown at pressures\nof 0.1, 1, 10, and 100 mbar. For these models the speeds gener-\nally represent day-night \row. The dashed lines to the right of the\nrightmost points are connecting to the model with no drag in the\nupper atmosphere ( \u001cdrag!1 ), where the rms terminator wind\nspeed is 5200, 3800, 2600, and 1900 m s\u00001at 0.1, 1, 10, and 100\nmbar, respectively. The equilibrated speeds depend signi\fcantly\non the drag time constant and, within a given model, on pressure.\nHD 209458b is tentative but, at face value, suggests wind\nspeeds toward Earth of 2 \u00061 km s\u00001. Snellen et al. (2010)\nsuggest that their measurements are sensing pressure lev-\nels of 0.01{0.1 mbar. At these levels, the winds in our\nmodels equilibrate to \u00184{6 km s\u00001when drag is weak,\nand Figure 10 shows that reducing the wind speed to\n2 km s\u00001requires drag time constants potentially as short\nas\u0018104s. This hints that strong frictional drag processes\nmay operate in the atmosphere of HD 209458b. But\ncaution is warranted: Figure 10 also demonstrates that\nthe rms terminator wind speeds also depend strongly\non pressure within any given model; therefore, making\nrobust inferences about drag amplitudes from observed\nDoppler shifts requires extremely careful and accurate\nestimates of the pressure levels being probed. This may\nbe a challenge, at least until the composition and hence\nwavelength-dependent opacity of hot Jupiters are better\nunderstood. If the Snellen et al. (2010) measurements\nare actually sensing deeper pressures of \u001810 mbar, say,\nthen explaining their 2-km s\u00001signal would require little\nif any drag in the observable atmosphere.\nIn light of Figure 10, it is interesting to brie\ry comment\non the pressures being probed in transmission spectra\ncomputed from our models. In Section 5, we will present\ntransmission spectra for our 3D models computed self-\nconsistently from high-spectral-resolution versions of the\nsame opacities used to integrate the GCM. These cal-\nculations indicate that, in the K-band region consid-\nered by Snellen et al., our synthetic transmission spectra\nprobe pressures ranging from \u001810 mbar in the contin-\nuum between spectral lines to less than \u00180.1 mbar at line\ncores. It is the Doppler shifts of the spectral lines that\nare observable|the Doppler shift of the continuum, ifAtmospheric circulation of hot Jupiters 13\nany, is almost undetectable since the absorption depends\nonly weakly on wavelength there. As a result, the overall\nDoppler signal detected in a spectral cross-correlation is\nheavily weighted toward the Doppler shift of the spectral\nlines. We \fnd that, when cross-correlating our synthetic\ntransmission spectra with template spectra, our models\nof HD 209458b primarily probe the atmospheric winds\nat pressures of 0.1 to 1 mbar.\nThe qualitative dependence of terminator wind speed\non the drag time constant|illustrated in Figure 10|can\nbe understood analytically. To order of magnitude, the\nhorizontal pressure gradient force in pressure coordinates\nbetween day and night can be written R\u0001Thoriz\u0001 lnp=a.\nThis is balanced by some combination of advection, of\nmagnitude U2=a, Coriolis force, of magnitude fU, and\ndrag, of magnitude U=\u001c drag. Drag will dominate when\n\u001cdrag.f\u00001and whenU=\u001c drag&U2=a, which requires\n\u001cdrag\u001c(a=jr\bj)1=2, equivalent to the requirement that\n\u001cdrag\u001ca=(R\u0001Thoriz\u0001 lnp)1=2. As long as these con-\nditions are satis\fed, we can balance drag against the\npressure-gradient force. Solving for \u001cdrag then implies\nthat the amplitude of drag necessary to obtain a wind\nspeedUis\n\u001cdrag\u0018Ua\nR\u0001Thoriz\u0001 lnp(10)\nInserting parameters appropriate to the 0.1 mbar level\non HD 209458b ( a\u0018108m,R\u00183700 J kg\u00001K\u00001,\n\u0001Thoriz\u00181000 K, and \u0001 ln p\u00185), and adopting U\u0018\n2 km s\u00001motivated by the Snellen et al. measurement of\nHD 209458b, we obtain \u001cdrag\u0018104s. This value agrees\nwell with the strength of drag needed in our 3D model\nintregrations to achieve a speed of 2 km s\u00001at the 0.1\nmbar level (leftmost black triangle in Figure 10).\n5.TRANSMISSION SPECTRUM CALCULATIONS\nTo quantify the implications for observations, in this\nsection we present theoretical transmission spectra from\nour 3D models demonstrating the in\ruence of Doppler\nshifts due to atmospheric winds. These spectra illus-\ntrate how the dynamical regime shifts described in the\npreceding sections manifest in transit spectra.\n5.1. Methods\nWe have previously developed a code to compute the\ntransmission spectrum of transiting planet atmospheres,\nwhich we extend here to include Doppler shifts due to at-\nmospheric winds. The \frst generation of the code, which\nused one-dimensional atmospheric pressure-temperature\n(p-T) pro\fles, is described in Hubbard et al. (2001) and\nFortney et al. (2003). In Shabram et al. (2011) the one-\ndimensional (1D) version of the code was well-validated\nagainst the analytic transmission atmosphere model of\nLecavelier Des Etangs et al. (2008). In Fortney et al.\n(2010) we implemented a method to calculate the trans-\nmission spectrum of fully 3D models.\nThe calculation of the absorption of light passing\nthrough the planet's atmosphere is based on a sim-\nple physical picture. One can imagine a straight path\nthrough the planet's atmosphere, parallel to the star-\nplanet-observer axis, at an impact parameter rfrom this\naxis. The gaseous optical depth \u001cG, starting at the ter-\nminator and moving outward in one direction along thispath, can be calculated via the equation:\n\u001cG=Z1\nrr0\u001b(r0)n(r0)\n(r02\u0000r2)1=2dr0; (11)\nwherer0is the distance between the local location in the\natmosphere and the planetary center, nis the local num-\nber density of molecules in the atmosphere, and \u001bis the\nwavelength-dependent cross-section per molecule. Later\nwe will discuss the role of winds leading to a Doppler\nshifted\u001baway from rest wavelengths. We assume hydro-\nstatic equilibrium with a gravitational acceleration that\nfalls o\u000b with the inverse of the distance squared. The\nbase radius is taken at a pressure of 10 bars, where the\natmosphere is opaque, and this radius level is adjusted\nto yield the best \ft to observations, where applicable.\nHere we de\fne the wavelength-dependent transit radius\nas the radius where the total slant optical depth reaches\n0.56, following Lecavelier Des Etangs et al. (2008). Ad-\nditional detail and description can be found in Fortney\net al. (2010), as the 3D setup here is the same as de-\nscribed in that paper.\nFor any particular column of atmosphere, hydrostatic\nequilibrium is assumed, and we use the given local p-T\npro\fle to interpolate in a pre-tabulated chemical equi-\nlibrium and opacity grid that extends out to 1 \u0016bar.\nThe equilibrium chemistry mixing ratios (Lodders 1999;\nLodders & Fegley 2002, 2006) are paired with the opac-\nity database (Freedman et al. 2008) to generate pres-\nsure - , temperature-, and wavelength-dependent absorp-\ntion cross-sections that are used for that particular col-\numn. For a di\u000berent column of atmosphere, with a dif-\nferent p-Tpro\fle, local pressures and temperatures will\nyield di\u000berent mixing ratios and wavelength-dependent\ncross-sections.\nWe include the Doppler shifts due to the local atmo-\nspheric winds and planetary rotation when evaluating the\nopacity at any given region of the 3D grid. At high spec-\ntral resolution, rotation tends to cause a broadening of\nspectral lines (Spiegel et al. 2007), while the atmospheric\nwind speeds lead to absorption features that are Doppler\nshifted from their rest wavelengths (Snellen et al. 2010;\nMiller-Ricci Kempton & Rauscher 2012). The cross sec-\ntion\u001bis not evaluated at the rest wavelength, \u00150, but\nrather at the Doppler shifted value, \u0015, found via\n\u0015=\u00150(1\u0000vlos\nc); (12)\nwherevlosis the line-of-sight velocity|including both\nrotation and atmospheric winds|and cis the speed of\nlight. The Snellen et al. (2010) observations were per-\nformed at a resolving power of R\u0018105. For additional\nclarity in presentation, we have computed opacities and\ntransmission spectra at R= 106. In practice we inter-\npolate within our R= 106opacity database to yield the\ncorrect\u001bfor every height in the atmosphere, on every\ncolumn, given the calculated velocities at every location\nin our grid. This is done at 128 locations around the ter-\nminator. The contribution to the transmission spectrum\nis strongly weighted toward regions near the terminator,\nand falls essentially to zero more than \u001820\u000efrom the ter-\nminator (where the transit chord reaches extremely low\npressures). Therefore, we only include in the calculation\nregions within\u000620\u000eof the terminator (i.e., a total swath14 Showman et al.\n40\u000ewide centered on the terminator). Note that for sim-\nplicity we do not include the Doppler shift due to orbital\nmotion, and we are therefore essentially evaluating the\ntransmission spectrum at the center of the transit for a\nplanet with zero orbital eccentricity. The e\u000bect of or-\nbital motion was considered by Miller-Ricci Kempton &\nRauscher (2012).\n5.2. HD 209458b Cases and the Role of Drag\nWe now turn to a detailed analysis of the HD 209458b\nmodels in the vicinity of the Snellen et al. (2010) obser-\nvations of HD 209458b near 2.2 \u0016m. The Doppler shift\nwas not measured across all wavelengths, but only within\nthe narrow CO lines, since \rat transmission spectra (cor-\nresponding to the continuum between the spectral lines)\nyield no leverage on the Doppler shifts. These peaks are\nall of nearly the same strength (see, e.g. Snellen et al.\n2010, supplemental online material), so they probe very\nsimilar heights in the atmosphere. In Figure 11 we have\ncomputed the transmitted spectrum at R= 106, 1300\nwavelengths, from 2.3080 to 2.3011 \u0016m, for three mod-\nels of HD 209458b: a model with no drag in the upper\natmosphere (top left panel), a weak-drag model with a\ndrag time constant of 3 \u0002105s (middle left panel), and a\nstrong-drag model with a drag time constant of 1 \u0002104s\n(bottom left panel).\nFor each of the three drag cases we calculated the\nmodel planet radius vs. wavelength for four variants of\nthe dynamical model. One uses rest-wavelength values\n(black, this corresponds to a reference case where winds\nare assumed to be zero), one includes atmospheric dy-\nnamics only (magenta, ignoring rotation but using the\nfull 3D winds), one includes only rotation (green, ig-\nnoring dynamics), and in orange is the full model, in-\ncluding both dynamics and rotation. The transmission\nspectra in Figure 11 are somewhat di\u000ecult to interpret.\nTherefore we have also calculated the cross-correlation,\ncompared to the rest wavelength model, across the 1300\nwavelengths in our calculation.\nThere are several aspects of note for these plots. Start-\ning in the upper right of Figure 11, the drag free HD\n209458b model, we see that the self cross-correlation is\nstrongly peaked at 0 m s\u00001, as expected. The dynamics-\nonly model shows winds that peaked at \u00002500 m s\u00001\n(meaning a blueshift), with strong winds (generally at\nthe equator) reaching beyond \u00005000 m s\u00001. As seen in\nthe dynamics output, there is also a component of red-\nshift winds, taking up a relatively small fraction of the\nterminator, which range from \u00180 to 5000 m s\u00001. Interest-\ningly, for this drag-free case, the cross-correlation curve\nof the rotation-only model is not symmetric about the\nzero-velocity point. This asymmetry only appears in\nmodels with a strong leading/trailing hemispheric tem-\nperature contrast. It appears to be due the trailing (hot-\nter) hemisphere having a larger scale height, and there-\nfore more prominent absorption features. The full model,\nincluding dynamics and rotation, has a broader peak\nthan the dynamics-only model due to rotational broad-\nening. The peak \u00005000 m s\u00001velocities from dynam-\nics and\u00002000 m s\u00001, from rotation, lead to velocities on\nthe trailing hemisphere's equator of \u00007000 m s\u00001. The\nfull model is not merely just a broadened dynamics-only\nmodel due to the asymmetric rotational component.The weak-drag case (middle right of Figure 11) has\na more constrained atmospheric \row, which is gener-\nally day-to-night, with a much reduced super-rotating\njet. Velocities from dynamics are peaked more narrowly\naround\u00002200 m s\u00001(red curve). The rotational com-\nponent is close to symmetric, due to the small lead-\ning/trailing temperature di\u000berence. The full model looks\nmuch like the dynamics-only model, broadened due to\nplanetary rotation.\nThe strong-drag case (lower right of Figure 11) has\na very constrained circulation, with a relatively uni-\nform day-to-night \row around the entire planet, with\nlittle sign of an equatorial jet. The dynamics velocities\nare strongly peaked at \u00001000 m s\u00001and the small lead-\ning/trailing temperature contrast leads to a symmetric\nrotational component. The full model looks very much\nlike a broadened version of the dynamics-only model.\nThe slightly higher peak on the right side of the full\nmodel is due to a slight asymmetry in the velocities\naround\u00001000 m s\u00001in the dynamics-only model.\nOverall, a comparison of the models in Figure 11 high-\nlights the possibility that the amplitude of drag in the\natmosphere of HD 209458b can be inferred from observa-\ntions. The no-drag, weak-drag, and strong-drag models\n(Figure 11) exhibit peaks in the cross-correlation cen-\ntered at\u00004,\u00002:5, and\u00001 km s\u00001, respectively. We em-\nphasize that these values are the quantitative result of\nour rigorously calculated transmission spectra from our\nfully coupled 3D models (and are not, for example, sim-\nply the velocity at some assumed pressure of the 3D mod-\nels). At face value, the results in Figure 11 indicate that\nmodels with a drag time constant of 104{106s provide a\nbetter \ft to the Snellen et al. (2010) observations than\nmodels with no drag in the upper atmosphere. This is\nconsistent with the inferences drawn in Section 4.3.\nThe possibility of drag in the atmosphere of HD\n209458b is particularly interesting in light of recent sug-\ngestions that thermal ionization of alkali metals at high\ntemperature can lead to Lorentz forces that act to brake\nthe atmospheric winds (Perna et al. 2010; Menou 2012).\nIn the regime of day-to-night \row, air at the termina-\ntor has just crossed much of the dayside, and so the\nwind speed at the terminator is predominantly deter-\nmined by drag on the dayside rather than the nightside.\nSecondary-eclipse observations indicate that HD 209458b\nexhibits a dayside stratosphere with temperatures poten-\ntially reaching\u00182000 K (Knutson et al. (2008); see also\nour Figure 6). Based on the scaling relations in Perna\net al. (2010), such high temperatures should lead to very\nshort drag times, potentially consistent with the infer-\nences on drag drawn here.\n5.3. Comparing Three Di\u000berent Planets\nFigure 12 allows us to diagnose the di\u000berent atmo-\nspheric dynamics and Doppler shift signatures of HD\n209458b, HD 189733b, and GJ 436b. All cases are drag-\nfree. The top row is the same HD 209458b model de-\nscribed in the top row of Figure 11. The dynamical wind\nvelocities for HD 189733b (middle panels of Figure 12)\nare relatively similar to those of HD 209458b, but lack-\ning a very high velocity component. The planet's rota-\ntion period of 2.2 days is only 63% of the period of HD\n209458b, meaning HD 189733b has a signi\fcantly largerAtmospheric circulation of hot Jupiters 15\nFig. 11.| Left panels : High spectral resolution ( R\u0018106) planet radius versus wavelength for three models of HD 209458b with various\ndrag strengths. Solid black is the \\Rest\" model calculated without a Doppler shift. Magenta includes Doppler shifts due to atmospheric\ndynamics only. Green includes Doppler shifts due to planetary rotation only. In orange, the \\Full Model\" includes both dynamics and\nrotation. Right panels : Cross-correlations for the planetary transmission spectra shown at left. As drag becomes stronger one generally\n\fnds slower and more peaked wind speeds. See text for additional details.\nrotational velocity component. The asymmetry shown\nis also due to a relatively large leading/trailing tempera-\nture contrast. The full model has a smaller, high velocity\npeak at -5000 m s\u00001due to the strong blue-shifted peak\nof rotational velocity.\nThe GJ 436b spectrum clearly shows a transition to\na di\u000berent regime of atmospheric dynamics. The \row\nis dominated by a wide super-rotating jet, with little\n\row purely being day to night. This manifests itself in\nthe dynamics-only model as being somewhat symmetric,\nwith a slightly higher cross-correlation peak on the blue-\nshifted side. The rotational component is nearly sym-\nmetric, owning to relative temperature homogenization\nof the planet. The magnitude of the rotational velocities\nare quite small because the planet has a small radius.\nThe full model shows little Doppler shift. The spectrum\nplot at left shows very little di\u000berence between all of the\nmodels, other than that they have been Doppler broad-\nened compared to the rest model.6.DISCUSSION\nRecent observations suggest that atmospheric winds\non hot Jupiters can be directly inferred via the Doppler\nshift of spectral lines seen during transit (Snellen et al.\n2010). Motivated by these observations, we have shown\nthat the atmospheric circulation of hot Jupiters divides\ninto two regimes depending on the strength of the radia-\ntive forcing and frictional drag, with implications for the\nDoppler signature:\n\u000fUnder moderate stellar \ruxes and weak to mod-\nerate drag, atmospheric waves generated by the\nday-night thermal forcing interact with the mean\n\row to produce fast east-west (zonal) jets. In this\nregime, air along the terminator \rows toward Earth\nin some regions and away from Earth in others,\nleading to blue-shifted and red-shifted contribu-\ntions to the Doppler signature seen during transit.\nDepending on the speed of the winds relative to16 Showman et al.\nFig. 12.| Left panels : High spectral resolution ( R\u0018106) planet radius versus wavelength for drag-free models of HD 209458b, HD\n189733b, and GJ 436b. Solid black is the \\Rest\" model calculated without a Doppler shift. Magenta includes Doppler shifts due to\natmospheric dynamics only. Green includes Doppler shifts due to planetary rotation only. In orange, the \\Full Model\" includes both\ndynamics and rotation. Right panels : Cross-correlations for the planetary transmission spectra shown at left. As the incident \rux becomes\nsmaller, from HD 209458b, to HD 189733b, to GJ 436b, peak wind speeds become smaller, and the planet's dynamics become more\ndominated by a super-rotating jet. See text for additional details.\nthe planetary rotation, as well as the variation of\nzonal winds in latitude and height, this will cause\nDoppler lines observed during transit to be broad-\nened or, in extreme cases, split into distinct, super-\nposed blue-shifted and red-shifted velocity peaks.\n\u000fUnder extreme stellar \ruxes and/or strong fric-\ntional drag, however, the radiative and/or fric-\ntional damping is so strong that it damps these\nwaves and inhibits jet formation. At the low\npressures sensed by transit measurements, the at-\nmospheric circulation then involves a day-to-night\n\row, with a return \row at deeper levels. In this\nregime, the air\row at levels sensed by transit mea-\nsurements is toward Earth along most or all of the\nterminator, leading to a predominantly blue-shifted\nDoppler signature of spectral lines observed during\ntransit.We presented a theory predicting this regime transition,\nand we con\frmed its existence and explored its prop-\nerties in one-layer shallow-water models and in three-\ndimensional models coupling the dynamics to realistic\nnon-gray radiative transfer. We then presented detailed\nradiative transfer calculations of the transit spectra ex-\npected from our 3D models in the 2- \u0016m spectral region\nobserved by Snellen et al. (2010); these calculations can\nhelp to guide future observational e\u000borts.\nWe also showed that, in the second regime described\nabove, the speed of the day-night wind\row depends on\nthe amplitude of the drag at the low pressures sensed\nby transit measurements. Under relatively weak drag,\nthe wind speeds at the terminator of our HD 209458b\nmodels reach\u00184{6 km s\u00001depending on altitude and\nforcing conditions. Under strong drag, the wind speeds\nare slower. Interestingly, at the low pressures sensed by\ntransit observations, the drag must be relatively strong|Atmospheric circulation of hot Jupiters 17\nwith e\u000bective drag time constants of \u0018106s or less|to\nreduce the speeds by a signi\fcant fraction. Our mod-\nels of HD 209458b without signi\fcant drag in the upper\natmosphere produce peak cross-correlations of the tran-\nsit spectrum corresponding to blueshifts of \u00183{7 km s\u00001;\nthis exceeds, albeit marginally, the \u00182 km s\u00001blueshift\ninferred by Snellen et al. (2010). On the other hand,\nour models that agree best with the Snellen et al.\n(2010) inference|where the peak cross-correlations of\nthe transit spectrum lie at \u00181{3 km s\u00001|exhibit drag\ntimescales of 104{106s. This suggests, tentatively, that\nfrictional drag may be important on the dayside of HD\n209458b. An attractive possibility is that the dayside\nof HD 209458b is su\u000eciently hot for partial ionization\nto occur, leading to Lorentz-force braking of the winds\n(Perna et al. 2010; Menou 2012). Regardless, these mod-\nels demonstrate that, in principle, measurements of the\nDoppler shift of spectral lines can place constraints on the\namplitude of drag in the atmospheres of hot Jupiters.\nIn light of this issue, we note that our results di\u000ber\nfrom those of Miller-Ricci Kempton & Rauscher (2012),\nwho obtained peak cross-correlations in the transmission\nspectra corresponding to blueshifted velocities of about\n\u00002:5 km s\u00001and\u00001 km s\u00001in models without and with\ndrag, respectively. Their velocity shifts for their drag-\nfree case are signi\fcantly slower than the velocity shifts\nwe obtain of\u00004 km s\u00001for our drag-free HD 209458b\nmodel. A signi\fcant di\u000berence in the two studies is\nthat, in Miller-Ricci Kempton & Rauscher (2012), the\nheating/cooling in the thermodynamic energy equation\nwas determined using a simpli\fed Newtonian relaxation\nscheme based on that presented in Cooper & Showman\n(2005); in contrast, our dynamical models are fully cou-\npled to non-gray radiative transfer, from which the ra-\ndiative heating/cooling rates are calculated. This may\nlead to a quantitative di\u000berence in the radiative heating\nrates and hence three-dimensional wind structure. Fu-\nture work may shed light on the discrepancies between\nthe results.\nIt is worth mentioning that, within each of the two\nbroad dynamical regimes studied in this paper, there\nmay lie additional subregimes involving important tran-\nsitions between dynamical mechanisms. For example, in\nthe regime of zonal jets, we have emphasized the develop-\nment of equatorial superrotation by standing, planetary-\nscale Rossby and Kelvin waves induced by the day-night\nthermal forcing (Showman & Polvani 2011). However,\nwhen the stellar \rux is lower than considered in most hot-\nJupiter models, the importance of the day-night thermal\nforcing decreases and the equator-to-pole heating gradi-\nent becomes dominant. Baroclinic instabilities can then\noccur, particularly when the rotation rate is fast, and\nthese may lead to multiple mid-latitude east-west jets,\nwith an equatorial jet that may be of either sign de-\npending on the details. A transition analogous to this is\nevident in models of GJ 436b presented by Lewis et al.\n(2010). We will explore such dynamical transitions fur-\nther in future work.\nIt is also worth discussing the proposal of Mon-\ntalto et al. (2011), who suggested that the \u00182 km s\u00001blueshift inferred by Snellen et al. (2010) results not\nfrom atmospheric winds but from planetary orbital mo-\ntion due to an eccentric orbit. The radial velocity\nat the time the planet crosses the line of sight to\nEarth is RV 0=~Kecos!=p\n1\u0000e2whereeis the ec-\ncentricity,!is the argument of periastron, and ~K\u0011\n[2\u0019G=P ]1=3M?sini=(M?+mp)2=3= 1:47\u0002105m sec\u00001\nis a constant, which we have evaluated for HD 209458\nparameters. Here, Pis the orbital period, Gis the grav-\nitational constant, iis the orbital inclination, and M?\nandmpare the mass of the star and planet, respectively.\nExplaining a 2-km sec\u00001blueshift would thus require\nthatecos!\u00190:014. Montalto et al. (2011) point out\nthat the eccentricity itself is rather poorly constrained;\nhowever, what matters is not eccentricity alone but the\ncombination ecos!. A key point, apparently not ap-\npreciated by Montalto et al. (2011), is that ecos!is\ntightly constrained by observations of transit and sec-\nondary eclipse. Observations of the relative timing of\ntransit and secondary eclipse from Deming et al. (2005)\nshow that ecos! < 0:002 at 1-\u001b. Observations from\nKnutson et al. (2008) and Cross\feld et al. (2012) place\neven tighter upper limits on ecos!; the latter study\nyieldsecos!= 0:00004\u00060:00033, corresponding to a\n3-\u001bupper limit of the orbit-induced Doppler shift of\n140 m s\u00001at the center of transit. This appears to rule\nout any orbital explanation for the Doppler shift inferred\nby Snellen et al. (2010).\nOne also might wonder whether the Snellen et al.\n(2010) measurements could be explained by a greater\nabundance of CO on the eastern terminator, where tem-\nperatures are warm and wind preferentially \rows from\nday to night, and a reduced abundance of CO (and en-\nhancement of CH 4) on the western terminator, where\ntemperatures are generally cooler. This is unlikely, how-\never, because the timescales for chemical interconversion\nbetween CO and CH 4in the observable atmosphere are\norders of magnitude longer than dynamical timescales, so\nCO and CH 4should be chemically quenched (Cooper &\nShowman 2006). Therefore, the abundance of CO should\nbe essentially the same everywhere along the terminator\nat pressures low enough to be sensed remotely.\nFinally, while we have emphasized the wind patterns\nand implications for transit Doppler measurements, the\ndynamics described here also predict a regime transition\nin the temperature structure that may be important in\nexplaining thermal observations from light curves and\nsecondary eclipses. The shallow-water models in Fig-\nures 3 and 4, and the 3D models in Figure 6, show that\nthe \row tends to a state with small longitudinal temper-\nature variations when radiation and friction are weak,\nwhereas the day-night temperature di\u000berences become\nlarge when either radiation or friction become strong.\nOur models therefore predict a transition from small to\nlarge fractional day-night temperature di\u000berences at the\ninfrared photosphere as stellar \rux increases from small\nto large. We will explore this issue further in future work.\nThis research was supported by NASA Origins and\nPlanetary Atmospheres grants to APS.\nREFERENCES\nAdcroft, A., Campin, J.-M., Hill, C., & Marshall, J. 2004,\nMonthly Weather Review, 132, 2845Brown, T. M. 2001, ApJ, 553, 100618 Showman et al.\nBurrows, A., Rauscher, E., Spiegel, D. S., & Menou, K. 2010,\nApJ, 719, 341\nCharbonneau, D., Allen, L. E., Megeath, S. T., Torres, G.,\nAlonso, R., Brown, T. M., Gilliland, R. L., Latham, D. 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In addition, by allowing the damping to vary in a fas hion we describe, one obtains a fixed-\npoint quantum search algorithm in which ignorance of the num ber of targets increases the number\nof oracle queries only by a factor of 1.5.\nDissipation has generally been regarded as a destruc-\ntivefoeinthearenaofquantummechanics,ruiningquan-\ntum effects and greatly complicating theory. This is par-\nticularly true in the case of quantum information science,\nwhich depends on maintaining delicate quantum super-\npositions and entanglement. On the other hand, dissipa-\ntion has many productive uses in the classical regime,\nfrom automobile shocks to toaster ovens. One there-\nforewonderswhether dissipationmight be employedcon-\nstructively in the quantum information context [1].\nWe propose here a natural application of dissipation\nin the quantum search algorithm. This algorithm, due to\nGrover[2], is a mainstay ofquantum information science.\nGiven an unsorted database of Nitems of which Mare\ntargetitems, thequantumsearchlocatesoneofthetarget\nitems with O(/radicalbig\nN/M) queries of an oracle. A classical\nsearch would require O(N/M) queries. A well known\nvulnerability of the quantum search is the need for prior\nknowledgeof the value of M[3, 4]. In the absenceof such\nknowledge, the quantum search is not robust, produc-\ning results that oscillate between target and non-target\nitems. In this Letter, we show that introducing dissipa-\ntion into the search algorithm can damp out these oscil-\nlations. Strikingly, we find that a critical damping value\nemerges from the theory that divides between the quan-\ntum regime (low dissipation, , O(/radicalbig\nN/M) queries) and\ntheclassicalregime(highdissipation, , O(N/M)queries).\nAlthough recently other fixed-point quantum searches\nhave been developed [5, 6, 7], they are not designed to\npreserve the signature O(/radicalbig\nN/M) behavior of the quan-\ntum search. When the damping is chosen appropriately,\nthe dissipative approachmaintains the O(/radicalbig\nN/M) quan-\ntum behavior. Furthermore, when the damping is al-\nlowed to vary in a manner we describe, ignorance of M\ncosts only a factor of 1 .5 in the number of oracle calls,\nwhich is a very low overhead compared to other ways\nof handling ignorance of M. Overall, our results con-\nvincingly demonstrate the productive use of dissipation\nin quantum algorithms and provide an example where\nthe appropriate amount of dissipation emerges explicitly\nfrom the theory.\nWe briefly review Grover’s quantum search [8]. Given\nan unsorted database of Nitems, one is charged withthe task of finding any one of Mtarget items dispersed\nthroughoutthe database. The tool toprobe the database\nis an oracle that indicates whether a given item is a tar-\nget or not. The algorithm begins by placing the state |ψ/an}brack⌉tri}ht\nof the system into an equal superposition of all database\nstates/summationtext\ns=1,...,N|s > /√\nN= cosθ/2|α/an}brack⌉tri}ht+ sinθ/2|β/an}brack⌉tri}ht.\nHere, we have defined |α/an}brack⌉tri}htto be an equal superposition of\nallM−Nnon-target states, |β/an}brack⌉tri}htto be an equal superpo-\nsition of all Mtarget states, and sin θ/2 =/radicalbig\nM/N. It is\nconvenient to define Pauli operators in the 2-dimensional\nHilbert space of |α/an}brack⌉tri}htand|β/an}brack⌉tri}htbyX=|β/an}brack⌉tri}ht/an}brack⌉tl⌉{tα|+|α/an}brack⌉tri}ht/an}brack⌉tl⌉{tβ|,\nY=i|β/an}brack⌉tri}ht/an}brack⌉tl⌉{tα| −i|α/an}brack⌉tri}ht/an}brack⌉tl⌉{tβ|, andZ=|α/an}brack⌉tri}ht/an}brack⌉tl⌉{tα| − |β/an}brack⌉tri}ht/an}brack⌉tl⌉{tβ|. In\nterms of these operators, the oracle is just Z. Grover de-\nveloped a series of gates, consisting of one call to the\noracleZfollowed by one ”inversion about the mean”\nE, that produces the rotation G=EZ= exp(−iθY).\nAfterRapplications of G, the system has rotated to\ncos(2R+ 1)θ/2|α/an}brack⌉tri}ht+ sin(2R+ 1)θ/2|β/an}brack⌉tri}ht. Ifθis not\ntoo big (M≪N), and one chooses Rto be near\nR(M)≡ ⌈arccos(/radicalbig\nM/N)/θ⌉, then the final state will\nbe close to the target |β/an}brack⌉tri}ht. Clearly, if one is ignorant of\nMand keeps applying GpastR∼R(M), the system\nwill rotate past |β/an}brack⌉tri}ht.\nIn terms of the system density matrix, the evolution\nρ′=GρG†implies\n\nTr(ρ′X)\nTr(ρ′Z)\nTr(ρ′)\n=\ncos2θsin2θ0\n−sin2θcos2θ0\n0 0 1\n\nTr(ρX)\nTr(ρZ)\nTr(ρ)\n(1)\nwhere the initial density matrix ρ=|ψ/an}brack⌉tri}ht/an}brack⌉tl⌉{tψ|satisfies\n[Tr(ρX)Tr(ρZ)Tr(ρ)]T= [sinθcosθ1]T.\nSo far, we have simply reviewed Grover’s algorithm,\npointing out how it assumes prior knowledge of the num-\nber of targets M. Suppose that we do not know M. How\ncan we find a target reliably? We propose introducing\ndamping to slow the rotation in the |α/an}brack⌉tri}ht,|β/an}brack⌉tri}htplane as the\nsystem approaches the target state.\nWe append an external spin to the system, taking\n|ψ/an}brack⌉tri}htto|ψ/an}brack⌉tri}ht|↓/an}brack⌉tri}ht. Let the external spin Pauli operators be\nSx,Sy,Sz. This external spin will serve to indicate the\nproximity of the system to the target state. We replace2\nthe Grover rotation Gwith\nU=/bracketleftbigg\nG1−Sz\n2+1+Sz\n2/bracketrightbigg/bracketleftbigg\ne−iφSy1−Z\n2+1+Z\n2/bracketrightbigg\n.\n(2)\nFirst,/bracketleftbig\ne−iφSy(1−Z)/2+(1+Z)/2/bracketrightbig\ncalls the oracle and\nflips the external spin if |ψ/an}brack⌉tri}hthas reached the target state\n(Z=−1). Second, [ G(1−Sz)/2+(1+Sz)/2] applies a\nGrover rotation only if the external spin has not flipped\n(Sz=−1). Thus, the external spin stifles the Grover\nrotation as the target is approached, effectively damping\nthe system. Although it appears at first that Urequires2 controlled oracle calls, in fact it can be written using\none controlled oracle call as\nU=/bracketleftbigg\nE1−Sz\n2+1+Sz\n2/bracketrightbigg\neiφSy/2/bracketleftbigg\nZ1−Sz\n2+1+Sz\n2/bracketrightbigg\ne−iφSy/2.\nAssume we follow application of Uwith measurement\nof the external spin, and repeat these two steps until\nthe external spin has flipped. The density matrix of the\nsystem will have the form ρ|↓/an}brack⌉tri}ht/an}brack⌉tl⌉{t↓|until the external spin\nflips. The iteration produces the following effect on ρ:\n\nTr(ρ′X)\nTr(ρ′Z)\nTr(ρ′)\n=\ncos2θcosφsin2θ1+cos2φ\n2sin2θ1−cos2φ\n2\n−sin2θcosφcos2θ1+cos2φ\n2cos2θ1−cos2φ\n2\n01−cos2φ\n21+cos2φ\n2\n\nTr(ρX)\nTr(ρZ)\nTr(ρ)\n (3)\nNote that, since there is some probability at each itera-\ntion that the external spin will flip, Tr(ρ) can decrease.\nThe value of φdetermines the amount of damping of\nthe iteration. When φ= 0, there is no damping, and\nwe recover Grover’s quantum search. When φ=π/2,\nthe damping is strongest – the external spin essentially\nacts as a pointer for a full measurement of Z, the system\nundergoes collapse at each oracle call, and the search is\nnearly classical. This is the limit discussed in [6, 7]. To\nchoose the optimal value of φfor our purposes, we con-\nsider the eigenvalues of the matrix in (3). As Fig. 1\nshows, for small φ, whenUis nearlyG= exp(−iθY) and\n(3) is nearly (1), there are two complex conjugate eigen-\nvalues∼exp(±i2θ) associated with the rotation and one\neigenvalue ∼1 that keeps Tr(ρ) nearly constant. When\nφislarge,there arethreerealeigenvalues. Inthe extreme\ncaseφ=π/2 (not shown in Fig. 1), two eigenvalues are\n0 and one equals cos2θ. The eigenvalue associated with\nTr(ρX) is 0 since the measurement destroys the coher-\nence. The eigenvalue associated with Tr(ρ(1−Z)/2) is\n0 sinceρis associated with the non-target part of the\nsystem for which Z= 1. The remaining eigenvalue\nis cos2θbecause a projection to the non-target state\nTr(ρ(1 +Z)/2) followed by a Grover rotation through\nθreturns a non-target state with probability cos2θ.\nRemarkably, there is a critical damping defined by\ncosφ= (1−sinθ)/(1 + sinθ) at which all three eigen-\nvalues are cos φ. At this critical damping, all three com-\nponents Tr(ρX),Tr(ρZ), andTr(ρ) appearing in the\nmap (3) tend to be suppressed since all the eigenvalues\nhave magnitude under 1. Thus, 1 −Tr(ρ), the probabil-\nity that the target state has been found and the external\nspin has flipped, tends to increase. Fig. 2 exhibits the\nspecial character of the critical damping value. It de-00.20.40.60.81\nφEigenvalues\n  \nCritical\ndamping\nImaginary PartReal Part\nFIG. 1: Real and imaginary parts of the 3 eigenvalues of\nthe matrix (3) as a function of the damping φnear the crit-\nical damping. Since the qualitative appearance of graph is\nindependent of MforM≪N, we have avoided putting nu-\nmerical values on the φaxis specific to a particular M. The\ncritical damping occurs at cos φ= (1−sinθ)/(1 + sinθ) =\n(N−2/radicalbig\nM(N−M))/(N+2/radicalbig\nM(N−M)).\npicts the average number of oracle calls to find the tar-\nget as a function of Nandφin the case M= 1. As\nthe damping increases, the Grover rotation shrinks, so\nthe number of oracle queries tends to increase. For small\ndamping, the averagenumberoforaclequeriesto find the\ntarget has the quantum behavior O(√\nN), while for large\ndamping, the number of queries goes like the classical\nexpectation O(N). However, for the critical value of the\ndamping, there is a distinct valley that separates these\ntwo regimes. (The number of oracle calls on the z-axis\nof Fig. 2 is computed assuming the following reasonable\nstrategy. Knowing Nandφ, one plans to execute Rcalls\ntoU. If the external spin flips before the Rcalls are\ncomplete, a target has been found with certainty and no3\n1500010000\n00.10.20100200300\nN φ/πOracle Calls\nFIG. 2: Average number of oracle calls to find the target for\nM= 1 as a function of Nandφ. Note the linear dependence\nonNfor large φ, the√\nNdependence for small φ, and the\nvalley separating these two regions at the critical damping .\nmore calls are needed. Otherwise, the Rcalls toUare\nfollowed with one query to the oracle to determine if a\ntarget has been found. If not, one starts the procedure\nover. By fixing a judicious choice of R, one minimizes\nthe expected number of oracle queries. This minimum is\nplotted on the z-axis in Fig. 2.)\nHaving pointed out the existance of a critical damping\nvalue and noted its special character, we now consider\nhow the damping can be used in the case of unknown M.\nThe qualitative appearance of Fig. 1 is maintained for\ngeneralM≪N, but the value of the critical damping\ndepends upon Mthroughθ. In the absence of knowl-\nedge ofM, suppose we assume the worst-case scenario of\nfewest targets, M= 1, in which the damping cos φ=\n(1−sinθ)/(1 + sinθ) = (N−2/radicalbig\nM(N−M))/(N+\n2/radicalbig\nM(N−M)) = (√\nN−1−1)2/(√\nN−1+1)2is weak-\nest. Fig. 3 shows the progress of the algorithm given\nthis choice of φ, forN= 10,000 withM= 1 and with\nM= 40. The effect of the dissipation is evident; the os-\ncillations into and out of the target state are effectively\ndamped even for Msubstantially greater than 1. The\nvalue ofTr(ρZ) tends to zero, and with high probability\nthe external spin flips. Since the external spin signals\nsuccess, it is not necessary to know Mto run the search\neffectively.\nAlthough the choice cos φ= (√\nN−1−1)2/(√\nN−1+\n1)2is effective for a broadrangeof Mvalues near M= 1,\nit is not effective when the unknown value of Mhappens\nto be very large, on the order of N/2. For such large\nvalues ofM, the angleθis much greater than this choice\nofφ, and the system approaches the target state long be-\nfore the external spin flips to indicate success. To adapt\nour algorithm so that the spin can flip early when ap-\npropriate, we allow φto change from iteration to itera-\ntion. In the first application of U, there is large damping\nφ=π/2 in caseMis large. If the external spin has not\nflipped after this iteration of U, this is taken as evidence\nthatMmust be somewhat smaller, so the damping is\ndecreased. For concreteness, for iteration n >1, we set0 50 100 150−1−0.500.51\nNumber of queriesTr(ρ Z)\n  \nM = 1\nM = 40\nFIG. 3: Damping of oscillations of Tr(ρZ). They approach 0\nrather than -1 since ρis only the part of the density matrix\nfor which the external since has not flipped.\ncosφn= (1−sin(π/2n))/(1 +sin(π/2n)), which should\nbe somewhat near the critical damping for the Msatisfy-\ningR(M)∼n. Although this choice of φnhas not been\noptimized, we find that it yields good behavior.\nThe results are shown in Fig. 4, which compares the\naveragenumberofiterationsbeforetheexternalspinflips\nto the average number of iterations of the undamped\nGrover’salgorithmassumingpriorknowledgeof M. (The\naverage number of iterations of the undamped Grover’s\nalgorithm is determined as follows. First, one computes\np(R), the probability of finding a target after Rcalls\nto the oracle given NandM. One then minimizes\noverRthe average number of iterations ( R+1)/p(R) =\n(R+1)p(R)+2(R+1)p(R)(1−p(R))+3(R+1)p(R)(1−\np(R))2+...assuming that one performs a verification call\nto the oracle after the Riterations and then repeats the\nwhole procedure if the verification turns out negative.)\nIn the worst case shown, ignorance of Mleads to an ex-\ntra factor of roughly 1 .5 oracle calls, although occasion-\nally damping can actually decrease the number of oracle\ncalls. This damping method of coping with ignorance of\nMcompares very favorably with other methods such as\nsuccessively applying 2nGrover iterations and measur-\ning, which behaves quite erractically as a function of M\nsometimes costing an extra factor of 30 or more oracle\ncalls, or quantum counting [3], which yields the value of\nMbut at a factor far in excess of 1 .5 ifMis determined\nwith reasonable accuracy.\nSo far, the damping has been effected ”artificially” us-\ning a single ancilla spin that is subjected to unitary evo-\nlution with the system and then measured. This could be\nthe best way to proceed in an actual quantum computer,\nbut it is also possible to produce the damping in a more\ntraditional fashion by coupling the system to a bath. To\nshow this, we first append a flag qubit to the system to\nsignal when the target has been reached, taking |ψ/an}brack⌉tri}htto\n|ψ/an}brack⌉tri}ht|0/an}brack⌉tri}ht. The flag qubit operators are denoted σx,σy, and\nσz. Next, we introduce a low-temperature bath, modeled\nas a collection of spins with operators {Sx,i,Sy,i,Sz,i}.4\n0 2000 4000 6000 8000 100000.60.70.80.911.11.21.31.41.5\nMRatio of Oracle Calls\nFIG. 4: Average number of calls to oracle for damped\nsearch with varying damping cos φn= (1−sin(π/2n))/(1 +\nsin(π/2n)) as a function of M. We take N= 10,000 and as-\nsume ignorance of M. For each M, the damped search result\nis divided by the average number of calls to the oracle for the\nundamped quantum search assuming that Mis known. Note\nthat ignorance of Mcosts at most a factor of 1 .5 in oracle\ncalls.\nWe can achieve the same results as (2) by revising it to\nU= [G1−σz\n2+1+σz\n2] (4)\n/productdisplay\ni[σx1+Sz,i\n2+1−Sz,i\n2]\n[(e−iφSy,i−1)1−Z\n21−σz\n2+1]\nThe similarity to (2) is evident, but there is an ex-\ntra operator [ σx(1 +Sz,i)/2 + (1−Sz,i)/2] that flips\nthe flag qubit if the bath spin has flipped. We can\nwrite (4) as U= exp(−iθY(1−σz)/2)/producttext\ni[σx1+Sz,i\n2+\n1−Sz,i\n2]exp(−iφSy,i(1−Z)(1−σz)/4) = exp( −iθY(1−\nσz)/2)/producttext\niexp(φ(1−Z)/2(σ+S+,i−σ−S−,i))[σx1+Sz,i\n2+\n1−Sz,i\n2]. Assuming that the bath is cold so that Sz,i=−1\ninitially, the last factor can be removed. Now, one can\nthink of Grover’s algorithm as evolution under a Hamil-\ntonian [9], and since the Grover rotation has the form\nG= exp(−iθY) it follows that H=Y. To add damping\nto the Hamiltonian, we are motivated by the form of U\nto writeH=Y(1−σz)/2+i(φ/θ)(1−Z)/2(σ+/summationtext\niS+,i+\nσ−/summationtext\niS−,i) +Hbath. Assuming that Hbathallows us\nto make a Markovian approximation, we write a Lind-\nbladt equation [10] for the portion ρof system den-sity matrix for which the flag qubit has not flipped:\n˙ρ=−i[Y,ρ]−C(1−Z)ρ/2−Cρ(1−Z)/2. Here,C\ndepends on φ/θand on spin correlation functions of the\nbath. This Lindbladt equation leads to dynamics analo-\ngous to those resulting from (2).\nIn summary, we have introduced damping into\nGrover’s search to mitigate over-rotation past the tar-\nget states when the number of target states is unknown.\nA critical damping value has emerged that divides be-\ntween the classical and quantum regimes. Tuning the\ndamping appropriately permits quantum search without\nknowledge of Mwith only a factor of 1 .5 overhead. We\nhave presented one promising application of dissipation\nin quantum information, but others can be identified,\nsuggesting that this is an exciting avenue for further\nstudy.\nThe author acknowledges helpful discussions with An-\ndrew Cross, Daniel Gottesman, Patrick Hayden, and\nKevin Obenland.\n∗Electronic address: ari.m.mizel@saic.com\n[1] The idea that decoherence can sometimes play a con-\nstructive role has surfaced sporadically. Some examples\ninclude: M. B. Plenio, S. F. Huelga, A. Beige, and\nP. L. Knight, Phys. Rev. A 59, 2468 (1999); M. S.\nKim, J. Lee, D. Ahn, and P. L. Knight, Phys. Rev. A\n65, 040101(R):1-4 (2002); V. Kendon and B. Tregenna,\nquant-ph/0209005; S.DasguptaandD.A.Lidar, J.Phys.\nB40, S127 (2007); R. J. C. Spreeuw and T. W. Hijmans,\nPhys. Rev. A 76, 022306 (2007); F. Verstraete, M. M.\nWolf, J. I. Cirac, quant-ph/0803.1447.\n[2] L. K. Grover, Phys. Rev. Lett. 78, 325 (1997).\n[3] M. Boyer, G. Brassard, P. Hoyer, and A. Tapp, Fortschr.\nPhys.46, 493 (1998).\n[4] L. K. Grover, Phys. Rev. Lett. 80, 4329 (1998).\n[5] L. K. Grover, Phys. Rev. Lett. 95, 150501 (2005).\n[6] T. Tulsi, L. K. Grover, and A. Patel, Quant. Inform. and\nComput. 6, 482 (2006).\n[7] L. K. Grover, A. Patel, and T. Tulsi, quant-ph/0603132.\n[8] A good pedagogical presentation can be found in M.\nA. Nielsen and I. L. Chuang, Quantum Computation\nand Quantum Information, Cambridge University Press\n(2000).\n[9] E. Farhi and S. Gutmann, Phys. Rev. A 57, 2403 (1998).\n[10] H.-P. Breuer and F. Petruccione, The Theory of Open\nQuantum Systems, Oxford University (2002)."    },    {        "title": "2307.07591v2.A_new_radiation_reaction_approximation_for_particle_dynamics_in_the_strong_field_regime.pdf",        "content": "Astronomy &Astrophysics manuscript no. 46515corr ©ESO 2023\nAugust 3, 2023\nA new radiation reaction approximation for particle dynamics in the\nstrong field regime\nJ. Pétri\nUniversité de Strasbourg, CNRS, Observatoire astronomique de Strasbourg, UMR 7550, F-67000 Strasbourg, France.\ne-mail: jerome.petri@astro.unistra.fr\nReceived ; accepted\nABSTRACT\nContext. Following particle trajectories in the intense electromagnetic field of a neutron star is prohibited by the large ratio between\nthe cyclotron frequency ωBand the stellar rotation frequency Ω. No fully kinetic simulations on a macroscopic scale and with realistic\nfield strengths have been performed so far due to the huge computational cost implied by this enormous scale of separation.\nAims. In this paper, we derive new expressions for the particle velocity subject to strong radiation reaction that are intended to be\nmore accurate than the current state-of-the-art expression in the radiation reaction limit regime, the so-called Aristotelian regime.\nMethods. We shortened the timescale hierarchy by solving the particle equation of motion in the radiation reaction regime, where\nthe Lorentz force is always and immediately balanced by the radiative drag, and including a friction not necessarily opposite to the\nvelocity vector, as derived in the Landau-Lifshitz approximation.\nResults. Starting from the reduced Landau-Lifshitz equation (i.e. neglecting the field time derivatives), we found expressions for the\nvelocity depending only on the local electromagnetic field configuration and on a new parameter related to the field strength that\ncontrols the strength of the radiative damping. As an example, we imposed a constant Lorentz factor γduring the particle motion. We\nfound that for ultra-relativistic velocities satisfying γ≳10, the di fference between strong radiation reaction and the radiation reaction\nlimit becomes negligible.\nConclusions. The new velocity expressions produce results similar in accuracy to the radiation reaction limit approximation. We\ntherefore do not expect this new method to improve the accuracy of neutron star magnetosphere simulations. The radiation reaction\nlimit is a simple but accurate, robust, and e fficient way to follow ultra-relativistic particles in a strong electromagnetic field.\nKey words. magnetic fields – relativistic processes – methods: numerical – stars: neutron – acceleration of particles – radiation:\ndynamics\n1. Introduction\nWith the recent increase in computational power, performing full\nkinetic simulations of neutron star magnetospheres can now be\nenvisaged. However, the di fference in timescales between gyro\nfrequency and stellar frequency prevents realistic values from\nbeing applied to these parameters. So far the only way to circum-\nvent this scaling problem is to downsize these frequencies while\nstill keeping them well separated by respecting the ordering of\nthese frequencies. Although this is helpful for understanding the\ndynamics of charged particles in extreme environments, it does\nnot permit estimations of the true e fficiency of particle accelera-\ntion and radiation reaction to be made because the Lorentz fac-\ntors reached are several orders of magnitude lower than the ones\npredicted from observations of very high energy photons (see\nthe review by Philippov & Kramer (2022)). Recently some at-\ntempts have been made to simulate realistic parameters by Tom-\nczak & Pétri (2020) and Pétri (2022), but the computational time\nremains prohibitive, and only test particles have been investi-\ngated while neglecting their back reaction to the field.\nIt is highly desirable to overcome this limitation by em-\nploying an approximation known as the radiation reaction limit\n(RRL) regime, sometimes also called Aristotelian dynamics, for\nwhich the equation of motion with radiative friction is shortened\nby use of an algebraic expression for the particle velocity de-\npending only on the local value of the electric and magnetic field.This idea was applied by Mestel et al. (1985) and Finkbeiner\net al. (1989). Spectra and light curves in this regime were exten-\nsively studied by Pétri (2019) in a vacuum field. He found realis-\ntic Lorentz factors and photon energies in reasonable agreement\nwith the spectra observed by Fermi /LAT (Abdo et al. 2013).\nRecently, Chang et al. (2022) generalised the RRL velocity\nby including the Landau-Lifshitz term proportional to the veloc-\nity (Landau & Lifchitz 1989) and by computing the associated\nradiation spectra. They found a complicated formula that unfor-\ntunately does not apply to any electromagnetic field configura-\ntions. Moreover they introduced some hypotheses that are not\nwell justified to derive an expression for the velocity. Following\na different approach, Cai et al. (2022) studied the validity of the\nRRL equilibrium by describing the particle motion in a Frenet\nframe with a finite Lorentz factor. They introduced the principal\nnull directions, which are the eigenvectors of the electromag-\nnetic field tensors. The spatial part is equal to the Aristotelian\nspatial velocity, or stated di fferently, it is equal to the RRL ve-\nlocity. Although their analysis is based on the Landau-Lifshitz\nequations, including the time evolution of the Lorentz factor, at\nthe end of their derivation they had to resort to the computation\nof the curvature radius in order to estimate this aspect of the\nLorentz factor. In this work, we attempt to estimate the Lorentz\nfactor by evolving it in time from the initial conditions, but as\nwe show, the curvature radiation interpretation leads to more ac-\ncurate estimates of the Lorentz factor. Cai et al. (2022) applied\nArticle number, page 1 of 8arXiv:2307.07591v2  [astro-ph.HE]  2 Aug 2023A&A proofs: manuscript no. 46515corr\ntheir idea to a rather artificial magnetic field configuration. Our\naim is to apply such techniques to realistic fields, such as a ro-\ntating magnetic dipole.\nIn this paper, we derive formulas for the velocity in an ar-\nbitrary electromagnetic field configuration starting from the re-\nduced Landau-Lifshitz equation (LLR; i.e. neglecting the field\ntime derivatives). In section 2, we derive the algorithm for the\nnew velocity field according to the LLRs and that we call im-\nproved radiation reaction (IRR). Some explicit expressions of\nthis velocity are given in Section 3. In Section 4, we then quan-\ntify the improvement brought by the inclusion of the radiative\nfriction term proportional to the velocity compared to the stan-\ndard RRL. Conclusions and perspectives are touched on in Sec-\ntion 5.\n2. Strong radiation reaction regime\nRadiation reaction can be thought of as a friction drag opposing\nsome resistance to the Lorentz force. It acts as a brake and is\nappropriately depicted by a force opposite to the velocity vector.\nHowever, in the Landau-Lifshitz approximation, the radiation re-\naction force is opposite to the velocity only in the limit of ultra-\nrelativistic particles. For an arbitrary particle speed, there are ad-\nditional components along the electric field E, the magnetic field\nB, and the electric drift motion E∧B. We aim to quantify the\neffect of these additional forces in the particle trajectory by first\nderiving a new expression for the velocity.\n2.1. Equation of motion\nAs an approximation of the Lorentz-Abraham-Dirac equation,\nwe employ the Landau-Lifshitz expression according to Landau\n& Lifchitz (1989) such that\ndui\ndτ=q\nmFikuk+qτm\nmgi, (1a)\ngi=uℓ∂ℓFikuk+q\nm \nFikFkℓuℓ+(Fℓmum) (Fℓkuk)ui\nc2!\n,(1b)\nwith the typical timescale related to the particle classical radius\ncrossing time\nτm=q2\n6πε0m c3, (2)\nwithτbeing the proper time, ui=γ(c,v) as the 4-velocity, qbe-\ning the particle charge, mas the mass, cas the speed of light, E\nandBas the electric and magnetic field, ε0as the vacuum permit-\ntivity, vas the particle velocity, and Fikas the electromagnetic\ntensor.\nTo derive the velocity vector vand the Lorentz factor γ, it\nis judicious to switch to the 3 +1 formalism by introducing the\nobserver time dt=γdτ. Therefore,\ndp\ndt=qFL+γqτm\"dE\ndt+v∧dB\ndt#\n(3a)\n+q2τm\nm\u0002FL∧B+(β·E)E/c\u0003+q2τm\nm c2γ2[(β·E)2−F2\nL]v\ndγ\ndt=q\nmc\"\nβ·E+τmγβ·dE\ndt+qτm\nm c\u0010\nFL·E+γ2[(β·E)2−F2\nL]\u0011#\n,\n(3b)\nwhere we define the vector field\nFL=E+v∧B, (4)the normalised velocity β=v/c, and the momentum by p=\nγmv. In the constant field approximation, we drop the time\nderivatives and obtain the fundamental equation of motion for\na particle as follows\ndp\ndt=qFL+q2τm\nm\u0002FL∧B+(β·E)E/c\u0003(5a)\n+q2τm\nm c2γ2[(β·E)2−F2\nL]v\ndγ\ndt=q\nmc\u0014\nβ·E+qτm\nm c\u0010\nFL·E+γ2[(β·E)2−F2\nL]\u0011\u0015\n. (5b)\n2.2. Derivation of the velocity: First approach\nThe derivation of the particle velocity follows the procedure out-\nlined by Mestel (1999). Nevertheless, instead of using a friction\nof the form−Kvwith K>0, we use the three-dimensional ver-\nsion of the radiation reaction force, neglecting the space-time\ndependence of the electromagnetic field such that the radiative\nforce reduces to the second and third term in the right-hand side\nof Eq. (5a). Writing the radiation reaction force as\nFrad=K2\u0002FL∧B+(β·E)E/c\u0003−K1v (6)\nand balancing the Lorentz force\nFext=qFL (7)\nwith this radiation reaction Fext=Frad, we arrive at\nqFL=(K1+K2B2)v−K2(E∧B+(B·v)B+(β·E)E/c).(8)\nWe note that there is no assumption about particles moving at\nthe speed of light, their Lorentz factor is arbitrary, and v<c.\nThis represents a novelty compared to all other radiation reaction\nexpressions, which are always enforcing v=c. The coe fficients\nK1andK2are deduced from Eq. (5a) and given by\nK1=q2τm\nm c2γ2h\nF2\nL−(β·E)2i\n(9a)\nK2=q2τm\nm. (9b)\nWe note that these coe fficients are algebraic, being positive what-\never the sign of the charge q. However, K1depends on the\nLorentz factor γ, and we leave it unconstrained. In order to solve\nEq. (8), the velocity is advantageously decomposed into three\ncomponents ( σ,δ,η ) such that\nv=σE+δB+ηE∧B. (10)\nThese components must satisfy a linear system of three equa-\ntions of unknowns ( σ,δ,η ) according to\nq(1−ηB2)=[K1+K2B2]σ−K2[σE2+δ(E·B)]/c2(11a)\nqη(E·B)=K1δ−K2(E·B)σ (11b)\nqσ=[K1+K2B2]η−K2. (11c)\nWe recall that K1is unconstrained. Therefore, in order to fully\nsolve the system, an additional condition is required for K1.\nTo this end, we could enforce v=c, but as a generalisation,\nwe impose an user-defined Lorentz factor γ=(1−v2/c2)−1/2.\nEquations (10) and (11), supplemented with the condition on the\nspeed v, fully determine the velocity vector v. Solving for δ, we\nget\nδ=E·B\nK1[qη+K2σ], (12)\nArticle number, page 2 of 8J. Pétri: A new radiation reaction approximation for particle dynamics in the strong field regime\nreducing the system to a 2x2 size. Indeed, the smaller linear sys-\ntem to be solved reads\nK1+K2\u0010\nB2−E2\nc2\u0011\n−K2\n2\nK1\u0010\nE·B\nc\u00112q\u0014\nB2−K2\nK1\u0010\nE·B\nc\u00112\u0015\n−q K 1+K2B2\u0012σ\nη\u0013\n=\u0012q\nK2\u0013\n.\n(13)\nDeviation from the standard RRL arises because of the terms\ncontaining K2, which are usually neglected in the ultra-\nrelativistic regime. In order to deduce the number of relevant free\nparameters in the problem, it is preferable to employ quantities\nwithout dimensions, as explained in the next section.\n2.3. Dimensionless system\nAs generally required for numerical simulations, we introduce\nseveral useful quantities without dimensions relevant for the\ncomputation of the velocity. Following our previous work in\nPétri (2022), the primary fundamental variables are: the speed\nof light c; a typical frequency ωinvolved in the problem; the\nparticle electric charge q; and the particle rest mass m. From\nthese quantities we derive a typical time and length scale as\nwell as electromagnetic field strengths such that the length\nscale L=c/ω; the time scale T=1/ω; the magnetic field\nstrength Bn=mω/|q|; the electric field strength En=c Bn;\nand the typical electromagnetic force strength Fn=|q|En. The\ntwo important parameters defining the family of solutions are\nthe field strength parameters aBandaEand the radiation reac-\ntion e fficiency k2=ωτ maccording to the following definitions\naB=B\nBn=ωB\nω(14a)\naE=E\nEn=ωE\nω. (14b)\nThe external force becomes\nFext\nFn=sign( q) (e+β∧b) (15)\nwith sign( q)=q/|q|and the radiative force\nFrad\nFn=k2\u0002e∧b+b∧(b∧β)+(β·e)e\u0003−k1β (16)\nwith the normalised fields e=E/En,b=B/Bnand\nk1=K1\n|q|Bn; k2=K2Bn\n|q|=ωτ m. (17)\nThe velocity expansion coe fficients are also normalised accord-\ning to\n˜σ=σBn ; ˜η=ηB2\nn ; ˜δ=δBn/c. (18)\nThe normalised system to be solved then reads with ζ=sign(q)\n \nk1+k2\u0010\nb2−e2\u0011\n−k2\n2\nk1(e·b)2ζh\nb2−k2\nk1(e·b)2i\n−ζ k1+k2b2!\u0012˜σ\n˜η\u0013\n=\u0012ζ\nk2\u0013\n.(19)\nIn the above system, k2is fixed by the nature of the charged\nparticle ( q,m) and the typical frequency ω. There is no freedom\nto choose it arbitrarily. However k1is undetermined and needs to\nbe fixed by an additional constraint on the velocity. Choosing theLorentz factor γ, the coe fficient k1is found from the condition\n∥v∥/c=1−γ−2.\nActually k1is related to the velocity vby Eq. (9) in the LLR\napproximation. But solving for the coe fficientsσ,δ,η , and vis\nonly a function of k1. Thus Eq. (9) is a non-linear equation for k1\nsolely, which connects back to the fact that Eq. (8) is a non-linear\nequation for vinvolving the Lorentz factor. This first approach\nhas the drawback of implicitly including the Lorentz factor in\nthe linear system via the parameter K1. In the next sub-section,\nwe develop a second approach that is quadratic in the velocity\nand does not contain the Lorentz factor.\n2.4. Derivation of the velocity: Second approach\nIn a second approach, called velocity radiation reaction (VRR),\ninstead of cancelling the relativistic momentum time derivative\ndp\ndt, we decided to cancel the velocity time derivativedv\ndtgiven in\nthe Landau-Lifshitz approximation by\nγmdv\ndt=q[E+v∧B−(β·E)β]+\nq2τm\nm\u0002E∧B+(v∧B)∧B+(β∧E)∧E/c+β· (E∧B)β\u0003.\n(20)\nThe advantage of this approach is that it sticks closer to the Aris-\ntotelian regime. Indeed, if the term involving τmis removed, we\nretrieve the RRL and the associated Aristotelian velocity expres-\nsion that exactly satisfies\nE+v∧B−(β·E)β=0. (21)\nTranslated into normalised units, we get\nζk2\u0002e∧b+(β∧b)∧b+(β∧e)∧e+β· (e∧b)β\u0003\n+e+β∧b−(β·e)β=0.(22)\nThis expression is quadratic in β,and unlike the previous ap-\nproach, it does not involve the Lorentz factor. It can thus be\nsolved by standard root finding techniques for a fixed Lorentz\nfactor (or equivalently a fixed velocity norm). In a simple pre-\nscription, we set the velocity norm to ∥v∥=cagain, but any\nLorentz factor can be imposed. Departure from the RRL arises\ndue to the term involving ζk2. In the next section, we discuss the\ndifferent approximations to the particle Lorentz factor.\n3. Approximations of the particle Lorentz factor\nIn this section, we explore several approximations to estimate the\nparticle velocity without resorting to a full time integration of the\nequation of motion. We first remind the standard expression for\nthe velocity, how it compares to our new expression and then\ndiscuss the Lorentz factor estimation.\n3.1. Friction opposite to velocity\nStarting from the radiation reaction description of Mestel (1999)\nwhere the radiative friction is opposite to the particle velocity\nvector v, we write\nq(E+v∧B)=Kv, (23)\nwhere Kis a positive parameter related to the power radiated by\nthe particle. Solving for the velocity, we find\n \nB2+K2\nq2!\nv=K\nqE+E∧B+q\nK(E·B)B. (24)\nArticle number, page 3 of 8A&A proofs: manuscript no. 46515corr\nMoreover Kis the only positive solution of the bi-quadratic\nequation\nK4v2−q2(E2−v2B2)K2−q4(E·B)2=0. (25)\nThus, it satisfies\nK\n|q|=s\nE2−v2B2+p\n(E2−v2B2)2+4v2(E·B)2\n2v2. (26)\nSo far there are no constraints on the particle speed v<c. In the\nlimit of v=c, we retrieve the velocity expression used in the\nliterature, namely,\nv±=E∧B±(E0E/c+c B0B)\nE2\n0/c2+B2, (27)\nassuming particles moving at the speed of light ∥v±∥=c. In\nthe equation, v+represents positively charged particles, whereas\nv−represents negatively charged particles. The electromagnetic\nfield strength E0andB0are deduced from the electromagnetic\ninvariants E2−c2B2=E2\n0−c2B2\n0andE·B=E0B0with the con-\nstraint E0≥0. Therefore, the radiated power is PR=qFL·v=\n|q|c E0andK=|q|E0/c≥0.\nExcept for this ultra-relativistic limit for which we assume\nv=c, another relation is required to set the particle Lorentz\nfactorγ. To this end, we equate the radiated power according to\nthe local curvature radius ρcof the particle trajectory as\nPR=q2\n6πε0γ4c\nρ2c=γ4τmm c4\nρ2c=|q|c E0 (28)\nfrom which the Lorentz factor becomes\nγ= |q|E0\nτmm c3ρ2\nc!1/4\n=β·e\nk2ρ2\nc\nr2\nL1/4\n. (29)\nMoreover, the curvature is found from the acceleration by\nκc=1\nρc≈\r\r\r\r\rdβ\nc dt\r\r\r\r\r. (30)\nAs long as γ≫1, the RRL Eq. (27) remains a very good ap-\nproximation.\nIn Eq. (23), the strength of the damping Kis undetermined\nbut usually set by the particle velocity vor equivalently by its\nLorentz factor γ. Looking at LLR, we observed that the radia-\ntion reaction force term proportional to γ2is also opposite to the\nvelocity v. We could therefore identify K1with Kto get\nKτmv2=m c2. (31)\nHence, K/|q|≥m/|q|τm=9×1011T for electrons and positrons.\nThis value is much too high. As a consequence, there are two\nequations, Eqs. (26) and (31), for the two unknowns Kandv.\nIn the ultra-relativistic limit K≈|q|E0/cand\nβ2≈m c\n|q|E0τm=1\nωE0τm. (32)\nKeeping the velocity less than the speed of light leads to E0≥\n2.7×1020V/m, which is even larger than the critical value of\nEcrit≈1.3×1018V/m. Therefore, this idea fails and gives\nthe same value as before for the magnetic equivalent of c E0=\n9×1011T. We must conclude that the only reasonable way to\ncompute the Lorentz factor is via the curvature radiation power\nEq. (28). Before switching back to the LLR equation, we check\nhow the new velocity approximation compares to the simple pre-\nscription presented in this paragraph.3.2. Comparison to the radiation reaction limit\nIn the literature about approximated radiation reaction formulas,\nonly the force opposite to the velocity is considered. Translated\ninto our more general approach dealing with the full set of terms\nin the LLR equation, we enforce k2=0. The system (19) then\nsimplifies into\n \nk1ζb2\n−ζk1! \n˜σ\n˜η!\n= \nζ\n0!\n. (33)\nThe solution is readily found with\n˜η=1\nk2\n1+b2(34a)\n˜σ=ζk1˜η (34b)\n˜δ=ζe·b\nk1˜η. (34c)\nThis solution is exactly the same as the one presented in Eq. (24)\nexcept that all quantities are now normalised. The speed is not\nexplicitly imposed to be equal to the speed of light. Therefore,\nk1needs to be deduced, for instance, from the Lorentz factor, as\nin the previous discussion about curvature radiation power.\nWhen k2,0,k1is the root of a polynomial of high degree\nwith no analytical expression. We compute this coe fficient by\napplying a root finding algorithm via Newton-Raphson. A good\ninitial guess for k1in the system (19) is\nk1≈E0\nEn=|q|E0\nm cω=aE0. (35)\nWe checked that very few iterations are required to converge to\na highly accurate solution with several digits of precision. Some\nsimulations are shown in the next section. Finally, in the last ap-\nproach, we use the full terms in the original LLR equation and\nsolve for the velocity while taking into account the term inde-\npendent ofγ.\n3.3. Lorentz factor from LLR\nThe system of equations (19) solves the particle velocity vec-\ntor by assuming the coe fficient k1is freely adjustable. Actually,\nfrom the LLR equation, it is not tuneable and must be determined\nself-consistently with the expression (9) in which there are no\nfree parameters once the velocity vis fixed. This approach would\nlead to a first algorithm for finding the particle Lorentz factor γ.\nWe need to solve for γsuch that Eq. (9) is verified. However, as\nwe show in the next section, the IRR algorithm finds very simi-\nlar velocities compared to the ‘standard’ algorithm. In this case\nEq. (23) also holds, approximately. Then it can be shown that\nγ2q2h\n(β·E)2−F2\nLi\n≈−K2\n1v2, (36)\nwhich gives values of K1very di fferent from the expectation in\nEq. (9).\nIn a second alternative algorithm using the LLR equation, we\ncan try to set dγ/dt=0 and solve for the value of K1such that it\nsatisfies\nβ·E+qτm\nm c\u0010\nFL·E+γ2[(β·E)2−F2\nL]\u0011\n=0. (37)\nHere again there are no free parameters once the velocity vis\nfixed. This equation constrains the Lorentz factor because it de-\npends on v,which is fully solved once K1is fixed. Therefore,\nArticle number, page 4 of 8J. Pétri: A new radiation reaction approximation for particle dynamics in the strong field regime\nthe procedure consists of finding the root of Eq. (37) depend-\ning only on γ. However, here also, as the solution is close to the\nexpression (27), we found instead that\nFL·E+γ2[(β·E)2−F2\nL]≈0, (38)\nwhich is not compatible with Eq. (37).\nActually, the second term in Eq. (37) corresponds to the op-\nposite of the curvature radiation power PR. It is related to the\ncurvatureκcin the case of an ultra-relativistic particle such that\nκc=|q|\nγ2m c2q\nγ2[F2\nL−(β·E)2]−FL·E. (39)\nIf the term FL·Eis negligible, we retrieve the result (Kelner\net al. 2015)\nκc≈|q|\nγm c2q\nF2\nL−(β·E)2. (40)\nThe curvature would vanish in the limiting case investigated in\nthis section because, by construction, dv/dt=0. Consequently,\nas in the previous section, the best procedure to compute the\nLorentz factor is through the curvature radiation power PR, again\nby replacing E0byβ·Ein Eq. (28) and Eq. (29).\nA final trial consisted of integrating the Lorentz factor di ffer-\nential equation (5b) in time from the initial conditions. Because\nthe regime is close to the RRL, the second term expressing the\npower radiated almost always vanishes. Contrary to the magnetic\nfield, only the electric field produces work and is able to acceler-\nate particles. The results are less good compared to the curvature\nradius approach. Actually, the curvature radius represents only\nan auxiliary variable to compute the Lorentz factor. It could be\nderived straightforwardly from the definition of Eq. (30) but at\nthe expense of computing the Lagrangian time derivatives of the\nelectric and magnetic fields as\ndE\ndt=∂E\n∂t+v·∇E (41)\nand with a similar expression for B. These expressions are, how-\never, unwieldy to implement because they require the comput-\ning of partial time and space derivatives ∂tand∂r. We prefer to\ncompute the curvature from a finite di fference approximation of\nEq. (30), which is an equivalent description but much simpler\nto implement numerically. In the next section, we explore the\nefficiency and accuracy of the above mentioned methods for a\nrotating magnetic dipole with an electric quadrupole component.\n4. Simulations around a rotating dipole\nAs a typical macroscopic frequency, we used the neutron star\nrotation frequency Ωand setω= Ω. The numerical setup, elec-\ntromagnetic field configuration, and initial conditions for parti-\ncle position and velocity are exactly the same as in Pétri (2022).\nWe simulated a sample of test particles evolving in the Deutsch\n(1955) electromagnetic field.\n4.1. Radiation reaction limit accuracy\nThe improved version of the radiation reaction regime in the\nfirst approach di ffers significantly from the standard version only\nwhenever the ratio k2b2/k1becomes comparable or greater than\none. This means that the braking force no longer aligns with\nthe particle velocity vector and that it also involves friction in\nFig. 1. Ratio of the coe fficient k1andk2expressed as k2b2/k1in log\nscale for a sample of eight trajectories starting at several distances from\nthe surface given by r0/rL≈{1,0.37,0.14,0.05}.\nFig. 2. Time evolution of the ratio c K 1/|q|E0in log scale for a sample\nof eight representative trajectories. Both numbers are identical to eight\ndigits of precision.\ntheE,B, and E∧Bdirections. To check if this situation hap-\npens for electrons in the rotating magnetic dipole, we plotted\nthis ratio in a log scale (see Fig.1) for a sample of eight tra-\njectories starting at di fferent locations r0within the light cylin-\nder. The radial distances are given in the legend of the figure\nand were normalised to the light cylinder radius rLsuch that\nr0/rL≈ {1,0.37,0.14,0.05}. As can be seen in this plot, this\nratio is mostly much lower than one, meaning that the improved\nversion of radiation reaction does not significantly di ffer from\nthe straightforward RRL, except for sparse events of very few\ntrajectories. Moreover, as we later show, even in these cases, the\ntrajectories are not drastically a ffected by the corrections brought\nthrough the IRR expression.\nFigure 2 shows the deviation of K1from|q|E0/cin the IRR\nversion for the same sample shown in Fig. 1. The ratio equals one\nto very high accuracy. Both parameters are identical up to eight\ndigits of precision. This supports the fact that the improvement is\nmarginal. Finally, in Fig. 3, we compare the parallel electric field\nE∥=β·Eto the value E0corresponding to the parallel electric\nfield in the strict radiation reaction regime. Because E∥<0 for\nnegatively charged particles, we plotted ζE∥/E0to keep positive\nnumbers. Both values of the parallel electric field are identical to\nmore that 12 digits of precision.\nArticle number, page 5 of 8A&A proofs: manuscript no. 46515corr\nFig. 3. Time evolution of the true parallel electric field E∥compared to\nthe estimated parallel electric field E0for a sample of eight trajectories.\nBecause E∥<0 is negative for electrons, we plot ζE∥/E0.\nBased on all the above observations, we did not expect to ob-\nserve a drastic change in the particle dynamics between the RRL\nand IRR regime. For a more quantitative analysis, we plotted the\nLorentz factor evolution in time for the same sample of electrons.\nNo di fference in the Lorentz factor evaluation was observed be-\ntween both approximations. We therefore concluded that there is\nno advantage in including the correction brought by the Landau-\nLifshitz equation with a friction term not anti-aligned with the\nvelocity vector.\n4.2. Comparison between IRR, VRR, and LLR\nWe have checked that the IRR does not bring significant im-\nprovements compared to radiation reaction. To complete the\nanalysis of accuracy and e fficiency of the IRR approximation,\nwe compared it to the more reliable LLR equation of motion.\nFigure 4 shows the evolution of the Lorentz factor for the\nthree descriptions of the particle motion, RRL, IRR, and LLR.\nThe curves only di ffer by their initial condition. The RRL and its\nimproved version show Lorentz factor estimates agreeing with\nthe LLR computations to reasonably good accuracy. We note that\nthe time evolution of the Lorentz factor is reproduced with the\nassociated fluctuations for one of the trajectories. For the RRL\nand IRR case, the Lorentz factors were computed according to\nexpression (29). The curvature κcin Eq. (30) was estimated with\na finite di fference approximation\nκc=\r\r\r\r\r\rvn+1/2−vn−1/2\nc2dt\r\r\r\r\r\r. (42)\nIt only involved the value of the electromagnetic field at two\nneighbouring times: tn+1/2andtn−1/2.\nIf we integrate the time evolution of the Lorentz factor in-\nstead, we get less accurate estimates of the Lorentz factor, as\nshown in Fig. 5 in green dashed lines for the VRR approach and\nindicated as VRR2. The blue dashed lines show the Lorentz fac-\ntor computed via the curvature and give similar results to IRR in\nFig. 4, indicated as VRR1.\nRepresenting another check of the e fficiency of the IRR and\nradiation reaction approximation, Fig. 6 overlaps the LLR trajec-\ntories shown in black solid lines onto the IRR trajectories shown\nin coloured, thick solid lines for a sample of particles starting\natr0/rL≈{1,0.37,0.14,0.05}from the top to the bottom row,\nrespectively see the legend in the right-column panels). For the\nFig. 4. Evolution of the Lorentz factor for the three approximations of\nthe equation of motion of an electron. Colours are as follows: RRL in\ndashed blue, IRR in dashed green, and LLR in solid red lines.\nFig. 5. Evolution of the Lorentz factor for the VRR approximation in\ndashed green and dashed blue lines compared to LLR in solid red lines.\nVRR2 stands for evaluation by integration in time of the Lorentz factor,\nwhereas VRR1 stands for evaluation of the Lorentz factor by the curva-\nture radius.\ntrajectories starting well above the stellar surface, correspond-\ning to the first row with r0/rL≈1 and to the second row with\nr0/rL≈0.37, all trajectories agree and overlap. Only some tra-\njectories starting from the stellar surface, corresponding to the\nfourth row with r0/rL≈0.05,do not match, though the be-\nhaviour remains the same. For particles starting at r0/rL≈0.14,\nthird row, the agreement is also excellent.\nFinally, we stress that all the above results rely on the as-\nsumption that dE/dt=dB/dt=0are valid. This is correct\nas long as the terms involving dE/dtanddB/dtremain small\ncompared to the other terms in Eq. (3). We checked this a poste-\nriori by computing the time derivatives dlnE/dtanddlnB/dt,\nin normalised units, during a full simulation span. The time evo-\nlution of these derivatives is shown in Fig. 7. They remain of\norder unity, with a maximal value of about ten. If multiplied by\nthe correct factors given in the Landau-Lifshitz equation, we no-\nticed that these terms in the radiation reaction force indeed stay\nat a negligible level, even when multiplied by a factor γ. A sim-\nple criterion for dropping these terms is γk2≪1. Thus, we can\nconfidently ignore these time derivatives even outside the light\ncylinder.\nArticle number, page 6 of 8J. Pétri: A new radiation reaction approximation for particle dynamics in the strong field regime\nFig. 6. Comparison of IRR and LLR trajectories of a relevant sample of electrons. The solid black lines depict the LLR trajectories, and the coloured\nsymbols show the IRR approximation. In each horizontal panel, particles start at a fixed spherical radius given by r0/rL≈{1,0.37,0.14,0.05},\nfrom top to bottom.\n5. Conclusions\nTracking a charged particle motion in an ultra-strong electro-\nmagnetic field is computationally a very demanding task. How-\never, finding accurate approximations able to follow these ultra-relativistic trajectories with radiative friction is a central prob-\nlem in modelling realistic neutron star magnetospheres. In this\npaper, we extended the velocity vector expression in the RRL\nby including a radiative force linear in velocity as derived from\nArticle number, page 7 of 8A&A proofs: manuscript no. 46515corr\nFig. 7. Time evolution of the electric field derivative dlnE/dt(blue)\nand magnetic field derivative dlnB/dt(red), in normalised units, along\na sample of three particle trajectories depicted by solid lines, dashed\nlines, and dotted lines.\nthe Landau-Lifshitz equation. We showed that integrating the\nparticle trajectories with this new expression gives very simi-\nlar results to the ‘standard’ radiation reaction expression of the\nAristotelian dynamics. The Lorentz factors are identical in both\ncases. A new parameter was introduced to control the strength\nof this force linear in velocity compared to the ultra-relativistic\nterm proportional to γ2. It almost always remains negligible\ncompared to the γ2term anti-aligned with the velocity vector.\nIncluding such a refinement in the radiation reaction regime to\nobtain more accurate solutions is therefore not recommended be-\ncause it also requires more computational time for no benefit.\nNevertheless, we observed some discrepancy between the\nLandau-Lifshitz solution and the IRR solution for some parti-\ncles starting from regions close to the surface where the field\nstrength is maximal. In such cases, the Landau-Lifshitz integra-\ntion scheme is recommended if accuracy becomes an issue in ob-\ntaining reliable results. An alternative approach therefore would\nbe to evolve the velocity vector in time and the Lorentz factor\nusing the ultra-relativistic equation of motion approximation for\na charged particle while assuming that the speed is and remains\nvery close to the speed of light.\nAnother possible application beyond neutron stars but not\nexplored in this work is using lasers in the extreme light regime\nto investigate high energy physics in ultra-strong electromag-\nnetic fields in the laboratory. Indeed current technology pushes\nthe laser nominal intensity above I0≳1022W/cm2(Gonoskov\net al. 2022), corresponding to magnetic field strengths on the or-\nder of B≳107T, which are similar to field strengths met around\ncompact objects in high energy astrophysics. At such laser inten-\nsities, the field strength is expected to reach the radiation domi-\nnated regime and even the strong field quantum electrodynamics\ndomain where electron-positron pair cascades are triggered.\nAcknowledgements. I am grateful to the referee for helpful comments and sug-\ngestions. This work has been supported by the CEFIPRA grant IFC /F5904-\nB/2018 and ANR-20-CE31-0010.\nReferences\nAbdo, A. A., Ajello, M., Allafort, A., et al. 2013, ApJS, 208, 17\nCai, Y ., Gralla, S. E., & Paschalidis, V . 2022, arXiv:2209.07469\nChang, S., Zhang, L., Jiang, Z., & Li, X. 2022, MNRAS, 513, 925\nDeutsch, A. J. 1955, Annales d’Astrophysique, 18, 1\nFinkbeiner, B., Herold, H., Ertl, T., & Ruder, H. 1989, A&A, 225, 479Gonoskov, A., Blackburn, T., Marklund, M., & Bulanov, S. 2022, Rev. Mod.\nPhys., 94, 045001\nKelner, S. R., Prosekin, A. Y ., & Aharonian, F. A. 2015, AJ, 149, 33\nLandau, L. & Lifchitz, E. 1989, Physique théorique : Tome 2, Théorie des\nchamps (Moscou: Mir)\nMestel, L. 1999, Stellar magnetism (Oxford : Clarendon, 1999. (International\nseries of monographs on physics ; 99))\nMestel, L., Robertson, J. A., Wang, Y . M., & Westfold, K. C. 1985, MNRAS,\n217, 443\nPhilippov, A. & Kramer, M. 2022, ARA&A, 60, 495\nPétri, J. 2019, MNRAS, 484, 5669\nPétri, J. 2022, A&A, 666, A5\nTomczak, I. & Pétri, J. 2020, J. Plasma Phys., 86, 825860401\nArticle number, page 8 of 8"    },    {        "title": "1604.05688v1.Lorentz_atom_revisited_by_solving_Abraham_Lorentz_equation_of_motion.pdf",        "content": "arXiv:1604.05688v1  [quant-ph]  19 Apr 2016Lorentz atom revisited by solving Abraham–Lorentz equatio n of motion\nJ. Bosse\nFachbereich Physik, Freie Universität Berlin, 14195 Berli n, Germany\n(Dated: April 5, 2018)\nBy solving the non–relativistic Abraham-Lorentz (AL) equa tion, I demonstrate that AL equation\nof motion is notsuited for treating the Lorentz atom, because a steady–stat e solution does not exist.\nThe AL equation serves as a tool, however, for deducing appro priate parameters Ω,Γto be used\nwith the equation of forced oscillations in modelling the Lorentz atom. The electric polarizability ,\nwhich many authors “derived” from AL equation in recent year s, is found to violate Kramers–Kronig\nrelations rendering obsolete the extracted photon–absorp tion rate, for example. Fortunately, errors\nturn out to be small quantitatively, as long as light frequen cyωis neither too close to nortoo far\nfromresonance frequency Ω. Polarizability and absorption cross section are derived f or the Lorentz\natom by purely classical reasoning and shown to agree with qu antum–mechanical calculations of\nthe same quantities. In particular, oscillator parameters Ω,Γdeduced by treating the atom as a\nquantum oscillator are found equivalent to those derived fr om classical AL equation. The instructive\ncomparison provides a deep insight into understanding the g reat success of Lorentz’s model which\nwas suggested long before the advent of quantum theory.\nPACS numbers: 02.30Hq, 03.50.De, 31.15xp, 37.10.-x\nI. INTRODUCTION\nIn recent years, the classical Lorentz –oscillator model\nserving as an intuitive description of an atom under\nthe influence of the ac–electric field associated with a\nstanding wave of visible light has celebrated a revival\nin quantum–optics literature [1], [2]. While the classi-\ncalequation of forced oscillations , with a friction force\nproportional to the 1sttime–derivative of elongation,\nhas many applications (e. g., the ac–current circuit with\nimpedance and capacity [3] or simplified models of den-\nsity fluctuations in liquids [4]) besides the Lorentz\natom, the latter has played a special role. When deriv-\ning the so–called ‘radiative reaction force’ from classica l\nelectrodynamics to account for energy loss of an acceler-\nated charge by radiation, Abraham andLorentz (AL)\narrived at a modified equation of motion with a friction\nforce proportional to the 3rdtime–derivative of the oscil-\nlator elongation. The radiative reaction force has been\ndiscussed extensively for more than 100 years, both for\nrelativistic and non–relativistic velocities of the charg ed\nparticle, because its implications have raised fundamen-\ntal problems such as “pre–acceleration” or “run–aways in\nthe absence of external forces” which are still open for\ndiscussion (see, e. g., [5, Ch. 17], [6], [7], [8, Ch. 11]). In\ncase of the Lorentz atom, the oscillating electron may\nbe described non–relativistically. So the non–relativist ic\nAL equation, only, will be studied here.\nIn Sec. II, a concise review is presented of the unique\nsolution of the forced–oscillations equation , inclusive of\nits steady–state limit, by introducing classical elongation\nresponse and relaxation functions.\nIn Sec. III, the unique solution of the AL equation\nfor given initial values (x0, v0, b0)is determined. The\nunique solution is shown to be a “run–away” implying\nnon–existence of a steady–state solution and spotting the\nAL equation of motion as an inappropriate tool for de-scribing the Lorentz atom. The forced–oscillation equa-\ntionis suggested, instead, to do the job together with os-\ncillator parameters (Ω,Γ)derived from AL equation. In\nthis context, a widely spread error is pointed out regard-\ning the “complex polarizability” of the Lorentz atom\n(see, e. g., [5, Sec. 17.9], [1, Sec. II.A], [2, Sec. 2A]).\nIn Sec. IV, for a quantum–mechanical system per-\nturbed by an oscillatory external field, a representation–\nfree perturbation expansion in the field strength is pre-\nsented for an expectation value. With its help, the ‘ab-\nsorbed power’ (dipole moment to 1storder) and the ‘ ac–\nStark shift’ (energy to 2ndorder) are derived in terms of\nthe dipole–dipole response function, or rather the com-\nplex polarizability. The quantum–mechanical response\nfunction is evaluated for a charged quantum oscillator\nand compared to the classical dipole–response function\nof the Lorentz atom derived in Sec. III. Perfect agreement\nbetween classical and quantum–mechanical calculation is\nfound, which renders an explanation for the great success\nof the classical Lorentz atom.\nIn Sec. V, the reader finds a Summary and Conclu-\nsions. In Sec. VI, Appendix, the unique solution of the\nAL equation of motion for given initial values ( x0, v0, b0)\nis presented in terms of classical response and relaxation\nfunctions. In addition, the Appendix contains a short\ncompendium on integral transforms used in this work.\nII. CLASSICAL OSCILLATOR\nA. Response and relaxation functions\nThe ordinary second–order differential equation\n¨x(t)+Γ˙x(t)+Ω2x(t) =f(t)/m (1)2\nwith positive constants ( m,Ω,Γ) and external force f(t)\nhas for given initial values,\nx0=x(t0), v 0= ˙x(t0), (2)\na unique solution. Finding this solution belongs to the\nfirst exercises in every math course on ordinary differen-\ntial equations. For physical applications, it is useful to\ncast the unique solution into the intuitive form ( t≥t0),\nx(t;t0) =φ(t−t0)x0+iχ(t−t0)mv0 (3)\n+/integraldisplayt\nt0dt′iχ(t−t′)f(t′),\nwith abbreviations\nχ(t) =eζ1t−eζ2t\nim(ζ1−ζ2), φ(t) =ζ1eζ2t−ζ2eζ1t\nζ1−ζ2(t≥0)\n(4)\ndenoting, resp., classical elongation–response and (nor-\nmalized) –relaxation function defined, at this stage, for\nnon–negative arguments, only. Here ζ1andζ2denote\nroots of the characteristic polynomial associated with\nEq. (1),ζ1,2=−Γ/2±/radicalbig\n(Γ/2)2−Ω2, which obey\nℜ[ζ1,2]<0, ζ 1ζ2= Ω2, ζ1+ζ2=−Γ.(5)\nDue to negative real parts of both roots ζ1andζ2, the\nfunctions φ(t)andχ(t)will decay to zero if time argu-\nments grow large.\nIt is convenient to extend the definitions of response\nand relaxation functions to negative time arguments.\nIn accordance with quantum–mechanical linear–response\ntheory (see Sec. IV below), I postulate\nχ(−t) =−χ(t) =χ(t)∗, φ(−t) =φ(t) =φ(t)∗.(6)\nAfter introducing phase angle ϑand frequency ˜Ωby\nϑ= arctan/bracketleftbiggΓ\n2˜Ω/bracketrightbigg\n,˜Ω =/braceleftbigg/radicalbig\nΩ2−(Γ/2)2,Ω>Γ/2\ni/radicalbig\n(Γ/2)2−Ω2,Ω≤Γ/2,\n(7)\nwhich will be real valued, if Γ<2Ω(low–damping\nregime), the response and relaxation functions may be\nexpressed in the more descriptive way,\nχ(t) = e−Γ\n2|t|sin(˜Ωt)\nim˜Ω, φ(t) = e−Γ\n2|t|cos(˜Ω|t|−ϑ)\ncosϑ.\n(8)\nHere occurrence of |t|reflects the symmetry introduced\nin Eq. (6).\nB. Steady–state elongation\nThe initial time t0in Eqs. (2),(3) is properly inter-\npreted as the instant, when the external force f(t)is\nswitched on. After switch–on, the elongation x(t,t0)will\nat first depend on t0and initial values ( x0, v0) until ‘tran-\nsients’ have died off due to relaxation processes, and thesystem described by Eq. (1) acquires a steady state. The\ncorresponding steady–state elongation ξ(t)is found by\nswitching on the force f(t)adiabatically and choosing\nt0=−∞in Eq. (3),\nξ(t) = lim\nt0→−∞x(t,t0) =/integraldisplay∞\n0dt′e−ot′iχ(t′)f(t−t′).(9)\nAdiabatical switch–on is described by replacing under the\nintegral in Eq. (3): f(t′)→f(t′)e−o(t−t′)(o >0). Sub-\nsequent substitution t−t′→t′results in Eq. (9). It is\nunderstood from here on that o→0is taken aftertime\nintegrations have been performed —without repeatedly\nemploying the explicit notation limo→0. This convention\nregarding treatment of the small positive frequency owill\nbe used throughout.\nIt is to be noted that the steady–state elongation\nEq. (9) is independent of initial values (x0, v0), because\nthe general solution of the homogeneous equation (Eq. (1)\nforf(t)≡0),xh(t,t0) =φ(t−t0)x0+ iχ(t−t0)mv0\nwhich doesdepend on initial conditions, will vanish in\nthe steady–state limit. This independence of initial val-\nues is a physical requirement on a steady–state solution,\nbecause initial values x0andv0are not (and cannot usu-\nally be) measured.\nC. Dynamical susceptibility\nIf the force entering the integrand in Eq. (9) is repre-\nsented by its Fourier integral, the steady–state solution\nwill also appear in Fourier –expanded form,\nξ(t) =1\n2π/integraldisplay∞\n−∞dω ξωexp(−itω), ξ ω= ˜χ(ω+io)fω,\n(10)\nwith the dynamical susceptibility ˜χ(ω+io) =ξω/fωdeter-\nmining the ratio between Fourier–transformed elongation\nand force. Here ˜χ(z), theFourier–Laplace transform\n(FLT, see Sec.VI B for details) of the response function\nχ(t), has been introduced. For the classical response and\nrelaxation functions given in Eq. (8), FLTs are readily\ncalculated ( s= signℑ[z]),\n˜χ(z) =1/m\nΩ2−z(z+siΓ),˜φ(z) =z+siΓ\nΩ2−z(z+siΓ).\n(11)\nFor the dynamical susceptibility, one has\n˜χ(ω+io) =Ω2\nΩ2−ω2−iωΓ˜χ0≡χ′(ω)+iχ′′(ω)(12)\nwith real and imaginary parts\nχ′(ω) =(Ω2−ω2)Ω2\n(Ω2−ω2)2+(ωΓ)2˜χ0, (13)\nχ′′(ω) =ωΓΩ2\n(Ω2−ω2)2+(ωΓ)2˜χ0, (14)3\nand the static susceptibility\n˜χ0=1\nmΩ2= ˜χ(io) =χ′(0). (15)\nIt is worth pointing out the close relationship between\nresponse and relaxation function known as Kubo iden-\ntity,\nχ(z)≡z˜φ(z)+1\nmΩ2⇐⇒χ(t)≡i˙φ(t)\nmΩ2,(16)\nwhich is reflected by Eqs. (8), (11), and also mentioning\nthe exact rewriting,\n˜χ(z) =Ω\n˜Ω/bracketleftBigg\nΓ\n2\n˜Ω−/parenleftbig\nz+siΓ\n2/parenrightbig+Γ\n2\n˜Ω+/parenleftbig\nz+siΓ\n2/parenrightbig/bracketrightBigg\nΩ˜χ0\nΓ,\n(17)\nwhich highlights the resonance patterns emerging near\nω=±˜Ω(cf. Eq. (7)) in case of Γ/Ω≪1.\nFinally, it is important to notice that the two in-\ngredients χ′(ω) =χ′(−ω)andχ′′(ω) =−χ′��(−ω)of\nthe dynamical susceptibility are intimately connected via\nKramers–Kronig relations, cf. Eq. (83) below. These\ndispersion relations are an immediate consequence of\nthe generalized susceptibility ˜χ(z)appearing as the\nFourier–Laplace transform of the response function\nχ(t). Violation of Kramers–Kronig relations is an in-\ndicator for a faulty determination of ˜χ(ω+io). Similarly,\nexperimental results on χ′(ω)andχ′′(ω)would not be\ntrustworthy, if available measured data permitted some-\none to demonstrate violation of Kramers–Kronig re-\nlations.\nD. Oscillatory force\nThe steady–state solution Eq. (9) acquires a specially\nsimple form, if one assumes a sinusoidal t–dependence of\nfrequency ωfor the force,\nf(t) =f0cos(ωt)≡f0ℜ/bracketleftbig\ne∓itω/bracketrightbig\n. (18)\nwith real f0andω. Inserting this force into Eq. (9), re-\nsults in the steady–state elongation\nξ(t) =ℜ/bracketleftbig\n˜χ(ω+io)f0e−itω/bracketrightbig\n= [χ′(ω)cos(ωt)+χ′′(ω)sin(ωt)]f0(19)\nwhich may be cast into the clearly arranged form\nξ(t) =Acos(ωt−ϕ) (20)\nwithω–dependent amplitude and phase shift,\nA=Ω2˜χ0f0/radicalbig\n(Ω2−ω2)2+(ωΓ)2≤Am=Ω2˜χ0f0\n˜ΩΓ,(21)\nϕ= arctan/parenleftbiggωΓ\nΩ2−ω2/parenrightbigg\n+π1−sign/parenleftbig\nΩ2−ω2/parenrightbig\n2.The oscillator picks up energy from the oscillatory force\nanddissipates this energy via friction ( Γ>0). The work\ndone by the external force during time interval (t, t+dt)\namounts to [ξ(t+dt)−ξ(t)]f(t) =dt˙ξ(t)f(t). Integrat-\ning this energy over an oscillation period T= 2π/ωand\ndividing by Tresults in the average absorbed power\nP(ω) =1\nT/integraldisplayT\n0dt˙ξ(t)f(t) =1\n2ωχ′′(ω)f2\n0≥0.(22)\nA glance at Eq. (14) shows that no power will be ab-\nsorbed, if a constant external force is applied ( ω= 0),\nwhile maximum power absorption will be achieved, if\nthe ‘resonance value’ ωr=±Ωis chosen for the ap-\nplied oscillatory–force frequency ω. Finally, it is to be\nnoted that the driven elongation ξ(t)develops its ampli-\ntude maximum Amfor a different driving–field frequency\nωm=±Ω/radicalbig\n1−Γ2/(2Ω2). Moreover, both ωrandωm\ndiffer from ˜Ωin Eq. (8). For Γ≪Ω, however, the three\nfrequencies|ωm|<˜Ω<|ωr|will differ only slightly, and\nmerge for Γ→0.\nIII. LORENTZ ATOM\nA. Abraham–Lorentz equation\nThe forced–oscillations Eq. (1) presents an ingenious\nmodel first suggested by Lorentz for describing an\natom under the influence of visible light. Lorentz as-\nsumed an electron (charge q=−e, massm=me)\nwhich is bound to the atomic nucleus by a restoring\nforcefΩ(t) =−mΩ2x(t)and subject to a friction force\nfΓ(t) =−mΓ˙x(t). If light is shining on the atom this\nelectron will, in addition, be exposed to an oscillating\nforcef(t) =qE0cos(ωt)exerted on a charge by the elec-\ntric field associated with a standing light wave of fre-\nquencyω(neglecting much smaller magnetic–field con-\ntributions). While the value of the restoring–force pa-\nrameter Ω2was roughly known, because Ω≈1015s−1\ncould be detected by finding the light–wave frequency ωr\n‘in resonance’ with the atom, there was little experimen-\ntal information on the extremely small but finite damping\nconstant ( Γ≪Ω) at the end of the nineteenth century.\nIn summary, the Lorentz –model parameters ΩandΓ\nhad to be determined from theoretical reasoning.\nIn the Abraham–Lorentz (AL) equation of motion\n[5, Eq. (17.9)],\n¨x(t)−τ...x(t)+ω2\n0x(t) =f(t)/m , (23)\nthe radiation–reaction force fRR(t) =mτ...x(t)replaces\nthe unknown friction force fΓ(t) =−mΓ˙x(t)of the\nforced–oscillator equation of motion (Eq. (1)), while the\nresonance frequency which determines the restoring force\nhas been denoted by ω0here, for clarity reasons. The\nradiation–reaction force has been derived from classical\nelectrodynamics. It accounts for the energy loss which4\ntheaccelerated electron will suffer due to Hertz radia-\ntion of electromagnetic waves. The parameter\nτ= 2R/(3c) =q2/(6πǫ0mc3) (24)\nwith classical charge radius R=q2/(4πǫ0mc2), permit-\ntivityǫ0, and light velocity cin vacuum denotes a very\nshort characteristic time. For the time it “takes light to\npass by an electron”, one finds τ≈10−23sresulting in\nthe small parameter τω0≈10−8for an atomic electron.\nIn view of the smallness of the characteristic time τ\nand the dimensionless parameter (τω0), it is tempting to\nrewrite the AL equation\n¨x=−ω2\n0x+f\nm+τd\ndt¨x\n=/parenleftbigg\n1−τd\ndt/parenrightbigg−1/bracketleftbigg\n−ω2\n0x+f\nm/bracketrightbigg\n=/bracketleftbigg\n1+τd\ndt+O(τ2)/bracketrightbigg/bracketleftbigg\n−ω2\n0x+f\nm/bracketrightbigg\n=−ω2\n0x−τω2\n0˙x+f(t+τ)\nm+O(τ2).(25)\nIn the representation Eq. (25), one of AL equation’s\nstrange properties shows up: the acceleration at (present\ntime)t,¨x(t), is induced by a force f(t+τ)to be applied at\n(future time) t+τ. Leaving aside philosophical questions\narising from the ‘pre–acceleration’ problem (see, e.g., [5 ,\nSec. 17.7], [8, Sec. 11.2.2]) and, in view of τω0≪1, sim-\nply replacing f(t+τ)→f(t)by assuming a sufficiently\nslowly varying force, the expansion Eq. (25) shows that\nthe widely used Abraham–Lorentz equation (Eq. (23))\ncould be replaced to a very good approximation with the\nequation of forced oscillations (Eq. (1)), if damping con-\nstant and restoring–force constant were chosen as follows:\nΓ→τω2\n0andΩ2→ω2\n0.\nInstead of further endeavours to find an approximate\nequation by exploiting the smallness of τ, let us solve the\nAL equation of motion itself.\nB. Roots of AL characteristic polynomial\nThe inhomogeneous third–order ordinary differential\nequation with constant coefficients may be solved by\n‘brute force’. It is straight–forward to find the unique\nsolution of Eq. (23) for given initial conditions\nx0=x(t0), v0= ˙x(t0), b0= ¨x(t0). (26)\nThe unique solution xAL(t,t0) =xh\nAL(t,t0) +xp\nAL(t,t0),\nwhich is the sum of the general solution of the homoge-\nneous and one particular solution of the inhomogeneous\nequation, will then be used to derive the steady–state\nelongation ξAL(t) =xAL(t,t0→−∞)following the pro-\ncedure applied in Sec. II B.\nDenoting by ζ1, ζ2, ζ3the roots of the characteristic\npolynomial associated with Eq. (23),\nζ2−τζ3+ω2\n0= 0, (27)FIG. 1: Roots of characteristic polynomial. Solid red:\nζ2= 1/τ−2u=ω0[(τω0)−1+(τω0)−2(τω0)3+...]>0. Solid\ngreen:ℜ[ζ1,3] =−ω0[(τω0)/2−(τω0)3+...]<0. Dashed\ngreen:ℑ[ζ1] =ω0[1−5(τω0)2/8 +...]>0,ℑ[ζ3] =−ℑ[ζ1].\nDashed gray: small– (τω0)asymptotes. Grid lines mark\nℑ[ζ1,2]extrema at (τω0= 0,v/ω0=±1), andℜ[ζ1,3]min-\nimum at (τω0= 2, u/ω0=−1/4).\none finds, as expected for the roots of a 3rdorder poly-\nnomial, a pair of complex–conjugate besides a real root,\nζ1=u+iv , ζ 2=1\nτ−2u , ζ 3=u−iv(28)\nwith real and imaginary parts of ζ1given by\nu=−(w−1)2\n6τw≤0, v =1−w2\n2τw√\n3≥0,\nw=/bracketleftbigg\n1+3\n2τω0/parenleftbigg\n9τω0−/radicalBig\n12+81τ2ω2\n0/parenrightbigg/bracketrightbigg1\n3\n,(29)\nwhere0≤w≤1. Real and imaginary parts of the\ncharacteristic–polynomial roots in Eq. (28) are displayed\nin Fig. 1 as function of the parameter τω0.\nC. Absence of AL steady–state solution\nIt is important to realize that the positive rootζ2(red\nline in Fig. 1) implies a unique solution xAL(t,t0)which\nwill diverge in the steady–state limit,\nξAL(t) = lim\nt0→−∞xAL(t,t0) =∞, (30)\nfor generic initial conditions (see Appendix for details).\nEvidently, a steady–state solution of the AL equation\ndoesnot, in general, exist. Facing this staggering fact,\none must admit that Eq. (23) is notsuited to model the\nsteady–state elongation of a bounded atomic electron.\nThis conclusion is corroborated by closer inspection of\nthe special case, f(t)≡0:\n¨x(t)−τ...x(t)+ω2\n0x(t) = 0. (31)5\nIntroducing abbreviations\nΓ =−2u ,Ω2=u2+v2, (32)\nthe general solution of the homogeneous AL equation,\nEq. (31), may be cast into the form ( t≥t0)\nxh\nAL(t,t0) =φ(t−t0)x0+iχ(t−t0)mv0 (33)\n+e(t−t0)ζ2/bracketleftbig\nb0+Γv0+Ω2x0/bracketrightbig\nt2\n1(t,t0),\nwhich explicitly shows the contribution that will diverge,\ndue toζ2>0, when(t−t0)grows large. That holds, be-\ncauset2\n1(t,t0)abbreviates an expression, which reduces\nto the positive constant t2\n1(t,−∞) =τ2/(1 + 4v2τ2)\nin the steady–state limit, while φandχdenote relax-\nation and response function, resp., defined in Eq. (4) (or\nEq. (8), equivalently) —for parameters ( Γ,Ω2) provided\nin Eq. (32). Hence, the first line on r.h.s. of Eq. (33),\nwhich evidently represents the general homogeneous–\nequation solution of Eq. (1), will vanish in the steady–\nstate limit.\nAn interesting aspect of the ‘run–away solution’\nEq. (33) is the observation that the diverging contribu-\ntion would have been absent, if one assumed initial val-\nuesnotchosen independently as in Eq. (26) but in such\na way that the pre–factor of e(t−t0)ζ2in brackets on the\nr.h.s. of Eq. (33) will vanish,\n¨x(t0)+Γ˙x(t0)+Ω2x(t0) = 0, (34)\nwithΓ = Γ(τ,ω0)andΩ2= Ω2(τ,ω0)given in Eq. (32).\nThis condition will prevent the solution of Eq. (31) from\nrunning away, —a noteworthy observation, because any\ninstant of time could have been chosen to play the role\nof initial time t0. I conclude:\n1. Equation (34) must hold at any time t, ifbounded\nelongations of the Lorentz –atom electron are to\nbe guaranteed.\n2. The equation of forced oscillations (Eq. (1) and\nproperties discussed in Sec. II) with parameters\nΓ =−2u\n=τω2\n0/bracketleftbig\n1−2(τω0)2+O/parenleftbig\n(τω0)4/parenrightbig/bracketrightbig\n,\nΩ =/radicalbig\nu2+v2\n=ω0/bracketleftbig\n1−(τω0)2/2+O/parenleftbig\n(τω0)4/parenrightbig/bracketrightbig\n(35)\nshould be used for treating the Lorentz atom.\n3. My conclusions are corroborated by the fact that\nthe suggested procedure is consistent with the\nsmall–τexpansion given in Eq. (25).\nD. Lorentz–atom polarizability\nFollowing the conclusion of Sec. III C - item 2, the\natomic dipole moment d(t) = (−e)ξ(t), which is inducedby the electric field of a standing light wave exerting\nthe force f(t) = (−e)E0cos(ωt)on the electron (within\ndipole approximation), can be read from Eq. (19),\nd(t) =ℜ[˜χ(ω+io)e2E0e−itω] (36)\nwith oscillator parameters\nΩ→ω0,Γ→τω2\n0 (37)\ntaken from Eq. (35) —with perfectly sufficient precision\nin consideration of τω0≈10−8. The Lorentz model\nalso allows to account for additional damping processes\nbesides radiative loss by replacing Γin Eq. (36) with a\ntotaldamping constant Γt,\nΓ→Γt= Γ+Γ′, (38)\nwhich, as opposed to [5, Eq. (17.61)], does notdepend on\nfrequency.\nIn view of the constant dipole moment d(t) =d0in-\nduced by a static field E0,\nd0= (˜χ0e2)E0=αE0(ω= 0) (39)\nwithα= ˜χ0e2denoting the (static) polarizability , it has\nbecome common to name the dynamical dipole suscep-\ntibility,˜α(ω+ io) = ˜χ(ω+ io)e2, the “complex polariz-\nability” [1, Sec. II.A]. Its real part, the (generalized ω–\ndependent) polarizability\nα(ω) =ℜ[˜α(ω+io)] =χ′(ω)e2, (40)\ndetermines a force F=−/vector∇Udipacting on the atom in the\nlight field, where\nUdip(r) =−1\n2α(ω)|E(r,t)|2=−χ′(ω)e2E2\n0\n4(41)\ndenotes the ‘optical dipole potential’ which will be iden-\ntified as the average atomic energy shift, known as\n‘ac–Stark effect’ in Sec. IVE below. Within classical\nelectrodynamics, the optical dipole potential can only\nbe made plausible to within a factor 2, because one\nhas−d(t)·E0cos(ωt) =−χ′(ω)e2E2\n0/2for the time–\naveraged potential energy of an electric dipole moment\nin an external electric field.\nE. Absorption and scattering of radiation by\nLorentz atom\nThe imaginary part of the dynamical dipole suscepti-\nbility,ℑ[˜α(ω+ io)] =χ′′(ω)e2, via Eq. (22) determines\nthe average power P(ω)absorbed by the atom from the\nelectric field, which implies the absorption cross–section\n(λ0=c/ω0, wavelength at resonance, divided by 2π),\nσabs(ω) =P(ω)\nǫ0cE2\n0/2= 6πλ2\n0ΓΓtω2\n(ω2\n0−ω2)2+(ωΓt)2(42)6\nwhich obeys the famous f–sum rule,\n/integraldisplay∞\n0dω σabs(ω) =π\n2ǫ0ce2\nm, (43)\nalso known as ‘dipole sum rule’ in the present context.\nIt must be emphasized that the f-sum rule, valid for\nboth classical and quantum mechanical systems, states\nthe following interesting fact. The integrated absorption\ncross section on the r.h.s. of Eq. (43) is determined by the\nratioe2/malone. It does not depend on further details\nof the system, here represented by oscillator frequency\nand damping constants.\nA photon-absorption rate Γabs(ω)has been considered\nin [1, Sec. II.A] which is determined by χ′′(ω), too. From\nquantum–mechanical scattering theory, one finds ( Θ(x)\ndenoting unit–step function)\nΓabs(ω) =χ′′(ω)\n2/planckover2pi1e2E2\n0Θ(ω) =P(ω)\n/planckover2pi1ωΘ(ω),(44)\nif the atom is assumed in its electronic ground state (i. e.,\nat zero temperature) when hit by photons. In Eq. (44),\nΓabs(−ω) = 0 forω >0expresses the fact that an atom\nin its ground state cannot loose energy by stimulated (or\nspontaneous) emission of a photon of energy /planckover2pi1ω. It can\nonly win energy by absorbing a photon of energy /planckover2pi1ω.\nFinally, the time–dependent dipole moment induced\nby the oscillatory external field (Eq. (36)) will produce\nan electromagnetic field which in the far–field dipole–\napproximation (E,B)rad=¨d(t−r/c)\n4πǫ0c2r/bracketleftbig\ne×r\nr/bracketrightbig/parenleftbig\n×r\nr,1\nc/parenrightbig\nmay\nbe interpreted in terms of a radiation–scattering cross\nsection [2, Eq. (2A.48)], [5, Eq. (17.63)]\nσsc(ω) =8π\n3R2 ω4\n(Ω2−ω2)2+(ωΓt)2(45)\n→6πλ2\n0/parenleftbiggΓ\nω0/parenrightbigg2\n\nω4/ω4\n0, ω≪ω0\n(ω0/2)2\n(ω−ω0)2+(Γt/2)2, ω≈ω0\n1 , ω≫ω0\nfrom which well–known scattering regimes are easily\nidentified as limiting cases: Rayleigh scattering for\nω≪Ω,Thomson scattering for ω≫Ω, and, for ω≈Ω,\nthe resonant Lorentz scattering exhibiting the char-\nacteristic line shape with ‘full width at half maximum\n(FWHM)’, Γt, and ‘peak cross section’,\nσsc(Ω) =8π\n3R2Ω2\nΓ2\nt= 6πλ2\n0/parenleftbiggΓ\nΓt/parenrightbigg2\n. (46)\nHereR= 3cτ/2denotes the classical electron radius,\nand oscillator parameters are given by Ω =ω0and total\ndecay constant Γt= Γ+Γ′withΓ =τω2\n0.\nAs opposed to the statement in [5, Eq. (17.72)] which\nrefers to allω, Eqs. (42) and (45) imply σL\nabs(ω) =\nσL\nsc(ω)+σL\nr(ω)for frequencies|ω−ω0|≪ω0only, i. e., for\nthe resonant Lorentz –absorption (or total) cross sec-\ntion. The total cross section is composed of a scatteringcontribution σL\nsc(ω), spelled out in Eq. (45) ( ω≈ω0), and\na ‘reaction cross section’ σL\nr(ω)with the same Lorentz\nresonance denominator, but Γreplaced with√\nΓΓ′in\nthe numerator. Consequently, σL\nabs(ω)must be given by\nσL\nsc(ω)withΓreplaced by√ΓΓtin the numerator, which\nis easily verified from Eq. (42). The reason for the dis-\ncrepancy with Ref. [5] will become clear in Sec. III F.\nF. Pitfalls\nRegarding the classical model of the atomic complex\npolarizability, much confusion has been created in litera-\nture by erroneous conclusions drawn from the AL equa-\ntion of motion, Eq. (23), with oscillatory external force\nf(t) =f0cos(ωt). In Refs. [5], [8], [1], [2], e. g., and in\nnumerous other publications, authors search for a partic-\nularsolution of Eq. (23) which oscillates with frequency\nωof the driving force. Indeed, there is one such solution,\nxosc(t) =(ω2\n0−ω2)cos(ωt)+τω3sin(ωt)\n(ω2\n0−ω2)2+(τω3)2f0\nm,(47)\nwhich can be checked easily by inserting xosc(t)into\nEq. (23). But xosc(t)isnotthe steady–state solution of\nthe inhomogeneous AL equation. According to Eq. (30),\nsuch a steady–state solution does not exist which brought\nme to rule out the AL equation of motion as a candidate\nfor describing the Lorentz –atom elongation.\nIt must be emphasized that, in contrast to my findings,\nEq. (47) is frequently claimed to present the steady–state\nsolution to the AL equation with oscillatory force, which\nis not true as I demonstrated in Sec. III C. Since xosc(t)\nisnotthe steady–state solution, we are notallowed to\ninterpret Eq. (47) as if it were the analog of Eq. (19). Ex-\ntracting from Eq. (47) a “susceptibility”,\nX(ω) =ω2\n0\nω2\n0−ω2−iτω3X0, X 0=1\nmω2\n0(48)\nis a frequently repeated mistake which\n•is found already in the high–impact monograph\n[5], where in [5, Eqs. (17.60-61)] a non–radiative\ndecay constant Γ′was assumed in addition to\nthe radiative decay constant Γ =τω2\n0, both of\nwhich were combined into a total decay constant\nΓt(ω) = Γ′+(ω/ω0)2Γ, which is evidently not con-\nstant! Moreover, Γt(ω)violates the f–sum rule and\nsuppresses the high–frequency Thomson scattering\nin [5, Eq. (17.63)]. According to my findings in\nEqs. (35–36), the total decay rate must here read\nΓt= Γ′+Γas arrived at in Eq. (38) above, which\nwill repair the mentioned deficiencies.\n•was made in the monograph [8, Ex. 11.4, p. 468]\nimplicitly, when claiming Γ =τω2instead of the\ncorrect result Eq. (35),7\nFIG. 2: Kramers–Kronig check. Full green: ωχ′′(ω)/˜χ0\n(Eq.(14)); dashed green: χ′(ω)/˜χ0(Eq.(13)); full red:\nωℑ[X(ω)]/X0(Eq.(48)); dashed red: ℜ[X(ω)]/X0(Eq.(48));\nfull gray: integrals on r.h.s. (Eq.(50) & first Eq.(83)). Gri d\nlines indicate positions of Ω,ωX\nm,ω0(from left to right).\n•has been carried on into the optical–dipole poten-\ntial community by the very informative and often\ncited review article [1, Sec. II. A],\n•is even found in the more recent monograph [2],\nwhere it shows up in [2, Eq. (2A.53)] and again, as\na nasty suppressor of Thomson scattering, in [2,\nEq. (2A.48)].\n•would also result, if one erroneously applied to\nEq. (23) the mnemonic trick which is so helpful in\nremembering ˜χ(ω+io).\nNamely, Fourier transforming Eq. (1) which is, of\ncourse, obeyed by the steady–state elongation ξ(t),\nm/bracketleftbig\n(−iω)2+(−iω)Γ+Ω2/bracketrightbig\nξω=fω\n= [˜χ(ω+io)]−1ξω, (49)\nand reading the result Eq. (12) for ˜χ(ω+ io)from\ntheFourier –transformed equation of motion.\nNote that the “short–cut” Eq. (49) works out al-\nright only, because I proved in Sec. II B above that\nEq. (1) does indeed have a unique steady–state so-\nlution. The same, however, does not hold true for\nthe AL equation (Eq. (23)) as I demonstrated in\nSec. III C above.\nBut why can X(ω)not serve as a proper dynamical sus-\nceptibility, anyway? Answer: Because it does notobey\nKramers–Kronig relations,\nℜ[X(ω)]/negationslash=1\nπ−/integraldisplay∞\n−∞d¯ωℑ[X(¯ω)]\n¯ω−ω, (50)\nwhich is a consequence of xosc(t)not being the steady–\nstate solution of the AL equation. Inequality Eq. (50)\nis clearly demonstrated in Fig. 2, where the full gray\nline (cutting the ordinate at ≈0.5) displays the numer-\nically evaluated principal–value integral from r.h.s. ofEq. (50) which has been divided by the constant X0. This\nshould be compared with ℜ[X(ω)]/X0depicted as dashed\nred line. Both curves differ markedly indicating viola-\ntion of the Kramers–Kronig relation. As pointed out\nabove, however, a proper susceptibility must obey this\nrelation. As opposed to X(ω), the real and imaginary\nparts of the dynamical susceptibility in Eq. (12) do form\naKramers–Kronig pair. This has also been demon-\nstrated in Fig. 2: the gray line representing the numeri-\ncally evaluated principal–value integral (first Eq. (83) fo r\nf′′(ω)→χ′′(ω), after dividing by ˜χ0) cannot be distin-\nguished from the dashed green line displaying χ′(ω)/˜χ0\nfrom Eq. (13). The very large parameter value chosen\nin Fig. 2 for demonstration purposes, τω0= 2, requires\nexact evaluation of ΓandΩ2using Eq. (32) with Eq. (29).\nThe difference between correct (green) and faulty (red)\nmodel polarizability curves will diminish for decreasing\nvalues of τω0. This observation is substantiated by the\nrelations\nℜ[˜X(ω)]\nχ′(ω)= 1−(τω0)2/braceleftBigg\n1−ω4/bracketleftbig\n1+O/parenleftbig\n(τω0)2/parenrightbig/bracketrightbig\n(ω2\n0−ω2)ω2\n0/bracerightBigg\n(51)\nℑ[˜X(ω)]\nχ′′(ω)=ω2\nω2\n0/braceleftBigg\n1−(τω0)2ω2/bracketleftbig\n1+O/parenleftbig\n(τω0)2/parenrightbig/bracketrightbig\nω2\n0/bracerightBigg\n(52)\nfound from comparison of Eq. (48) with Eqs. (13–14),\nwhereΓandΩare given in Eq. (35). For electron pa-\nrameters ( τω0≈10−8), it seems that use of the incorrect\n˜X(ω)will produce quantitatively acceptable polarizabil-\nity results, if one restricts to the frequency range\n1≫/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−ω2\nω2\n0/vextendsingle/vextendsingle/vextendsingle/vextendsingle≫(τω0)2(53)\nwhich allows to set ω2/ω2\n0≈1in Eq. (52) and, at the\nsame time, keep sufficient distance to the pole in Eq. (51).\nThe limitations imposed on the range of frequencies by\nEq. (53) are best appreciated by throwing a glance at\nFig. 3, where relative errors derived from Eqs. (51–52) are\ndisplayed together with χ′(ω)andχ′′(ω)for the realistic\nvalueτω0= 10−8referring to the oscillating electron\nof the Lorentz atom. From Fig. 3 it is clear that big\nquantitative errors ( ±20% for|ω/ω0−1| ≈0.1) only\noccur in the imaginary part and in a detuning range,\nwhereχ′′(ω)is very small, while the quantitative error in\nthe real part is negligibly small in magnitude ( <3.×10−5\n% for|ω/ω0−1|>0.005).\nOn the one hand, this observation may possibly ex-\nplain, why the analytically incorrect polarizability X(ω)\n(Eq. (48)) could survive in literature for so long without\nbeing debunked.\nOn the other hand, Fig. 3 clearly demonstrates that the\nphoton–absorption rate Γabs(ω)deduced from Eq. (48)\nis systematically under– (over–)estimating the true\nphoton–absorption rate in the red– (blue–)detuned fre-\nquency regime (see, e.g., in [1, Eqs. (11),(41)]). For the8\nΤΩ0/Equal1./MulΤiply10/Minus8/MulΤiply106\n0.95 1.00 1.05 1.10Ω\nΩ0\n/Minus30/Minus20/Minus1010203040Χ'/LParen1Ω/RParen1\nΧ/OverTilde\n0,Χ''/LParen1Ω/RParen1\nΧ/OverTilde\n0\nFIG. 3: Polarizability: reactive (blue) and dissipative\n(red,106–fold amplified) parts. Errors from Eqs.(51),(52):\n{ℜ[X(ω)]/χ′(ω)−1}×102(blue–dashed, 106–fold amplified)\nand{ℑ[X(ω)]/χ′′(ω)−1}×102(red–dashed) reflecting factor\nω2/ω2\n0in Eq.(52).\nexperimentally relevant ratio of absorption rate and op-\ntical dipole potential (cf. [1, Eq. (14)]), one finds from\nEqs. (44),(22) the simple result,\n/planckover2pi1Γabs\nUdip=2ωΓ\nω2−Ω2=/braceleftbigg\n−2ωΓ/Ω2/bracketleftbig\n1+O/parenleftbig\nω2/Ω2/parenrightbig/bracketrightbig\nΓ/∆ [1+O(∆/Ω)]\n(54)\nwhere∆ =ω−Ωandω≥0. For a “quasi–electrostatic\ntrap” (QUEST), this ratio is under–estimated by roughly\n100% (forω≪Ω), because Γabswill be larger by the\nfactor(Ω/ω)2if deduced from Eq. (48). For a “far off–\nresonance trap” (FORT), which is understood to be de-\ntuned sufficiently slightly to still obey |∆|≪Ω, the incor-\nrect polarizability happens to produce the correct ratio\nΓ/∆given in Eq. (54), because ω≈Ω.\nIV. QUANTUM–MECHANICAL REASONING\nA. Perturbation expansion for expectation value\nA physical system (Hamiltonian ˆH) will take on the\nexplicitly time–dependent Hamiltonian\nˆHt=ˆH−ˆDE0cos(ωt) (55)\nunder the influence of an external, linearly polarized, os-\ncillatory field eE0cos(ωt)which couples to the dynamical\nvariable ˆD=ˆD·e. As a consequence, the average value\nat timetof an arbitrary operator ˆOtwill deviate from\nits stationary–state value,\n/angbracketleftBig\nδˆOt/angbracketrightBig\nt≡/angbracketleftBig\nˆOt/angbracketrightBig\nt−/angbracketleftBig\nˆOt/angbracketrightBig\n=/angbracketleftBig\nˆOt/angbracketrightBig(1)\nt+/angbracketleftBig\nˆOt/angbracketrightBig(2)\nt+... . (56)\nAssuming the perturbing field switched on adiabati-\ncally at time t0=−∞(which amounts to replacingE0cos(ωt′)→e−o(t−t′)E0cos(ωt′)witho >0for all\nt′≤t), then–th order in E0contribution to/angbracketleftBig\nδˆOt/angbracketrightBig\nt\nreads explicitly ( τ0= 0),\n/angbracketleftBig\nˆOt/angbracketrightBig(n)\nt= in/integraldisplay∞\nτ0dτ1.../integraldisplay∞\nτn−1dτnχˆO†\nt;ˆD(τ1,...,τ n)\n×n/productdisplay\nj=1e−oτjE0cos[ω(t−τj)](57)\nwith the n–th order response function,\nχˆA;ˆB(τ1,...,τ n) = (58)\n1\n/planckover2pi1n/angbracketleftBig/bracketleftBig\n.../bracketleftBig/bracketleftBig\nˆA†,ˆB(−τ1)/bracketrightBig\n,ˆB(−τ2)/bracketrightBig\n... ,ˆB(−τn)/bracketrightBig/angbracketrightBig\n.\nHereˆB(t) = exp(i tˆH//planckover2pi1)ˆBexp(−itˆH//planckover2pi1)denotes a\nHeisenberg operator referring to the unperturbed sys-\ntem, and the stationary–state average is defined as\n/angbracketleftBig\nˆA/angbracketrightBig\n= Tr/braceleftBig\nˆAˆW/bracerightBig\n,/bracketleftBig\nˆH ,ˆW/bracketrightBig\n= 0 (59)\nwith statistical operator ˆWdescribing the initial station-\nary state of the unperturbed system.\nIt must be emphasized that/angbracketleftBig\nδˆOt/angbracketrightBig\ntin Eqs. (56–58)\ndescribes the steady–state deviation from the unper-\nturbed expectation value, which is induced by the ex-\nternal field. Interestingly enough, the first–order result/angbracketleftBig\nˆOt/angbracketrightBig(1)\nt(written out explicitly in Eq. (60) for the in-\nduced dipole moment) has the same structure found for\nthe steady–state solution in Eq. (9) for the classical os-\ncillator elongation.\nB. Induced dipole moment\nFor the atomic dipole moment induced by a linearly–\npolarized standing light wave of frequency ω, one reads\nfor the first–order result d(t) =/angbracketleftBig\nδˆD/angbracketrightBig(1)\ntfrom Eq. (57),\nd(t) = i/integraldisplay∞\n0dτe−oτχˆD;ˆD(τ)E0cos[ω(t−τ)]\n=ℜ/bracketleftBig\n˜χˆD;ˆD(ω+io)E0e−iωt/bracketrightBig\n, (60)\nwith the dipole–dipole response function\nχˆD;ˆD(t) =1\n/planckover2pi1/angbracketleftBig/bracketleftBig\nˆD(t),ˆD/bracketrightBig/angbracketrightBig\n=−χˆD;ˆD(−t),(61)\nwhereˆDis identified with the component in field direc-\ntion of the atomic dipole–moment operator ˆD.\nThe corresponding dynamical dipole–susceptibility\n(“complex polarizability”) resulting from Fourier–\nLaplace transforming χˆD;ˆD(t)according to Eq. (81),\nmay quite generally be cast into the form [4]\n˜χˆD;ˆD(z) =Ω2\nD\nΩ2\nD−z2−z˜KD(z)˜χˆD;ˆD(io).(62)9\nThis formally exact expression is cited here only to point\nout the following facts.\n•The relaxation kernel ˜KD(z) =/integraltext∞\n−∞dω\nπK′′\nD(ω)\nω−zis determined by an even, non–negative, and\nbounded spectral function K′′\nD(ω). This gener-\nally frequency–dependent ‘total damping constant’\nK′′\nD(ω) =ℑ[˜KD(ω+ io)]will inevitably be asso-\nciated with a resonance–frequency renormalization\nviaℜ[˜KD(ω+io)]. Such a real contribution is miss-\ning in [5, Eq. (17.60)] resulting in the violation of\nKramers–Kronig relations and f–sum rule dis-\ncussed in Sec. III F above. Moreover, Γt(ω)in [5,\nEq. (17.61)] is not bounded.\n•The relaxation kernel ˜KD(z)is the FLT of the\ndipole memory function KD(t)governing the gen-\neralized oscillator equation,\n¨φˆD;ˆD(t)+Ω2\nDφˆD;ˆD(t)+/integraldisplayt\n0dt′KD(t′)˙φˆD;ˆD(t−t′) = 0\n(63)\nwith initial conditions φˆD;ˆD(0) = 1 ,˙φˆD;ˆD(0) = 0 ,\nwhich is obeyed by the (normalized) dipole relax-\nation function. Both, Eq. (62) and Eq. (63), are\nformally exact and, in view of Kubo ’s identity\nEq. (16), equivalent statements.\nTo conclude these general remarks, I emphasize that\nmemory effects may be neglected in some applications,\nrendering\nKD(t)≈2Γδ(t) =⇒K′′\nD(ω)≈Γ =⇒˜KD(z)≈siΓ(64)\na reasonable approximation —as is the case for the quan-\ntum oscillator in Sec. IVC. Under these circumstances,\nEq. (63) reduces to a free–oscillations equation of the\nsame type obeyed by the classical relaxation function\nφ(t)introduced in Sec. II. These remarks on very gen-\neral quantum mechanical (and quantum statistical) re-\nsults may illuminate the great success of models such\nas the Lorentz atom, which are based on the classical\nforced–oscillations equation of motion.\nC. Quantum oscillator\nAssuming the eigenvalue problem of the unperturbed\nHamiltonian solved ( ˆH|n/angb∇acket∇ight=|n/angb∇acket∇ightεn,n= 0,1,2,...)\nand the atom in its ground state initially ( ˆW=|0/angb∇acket∇ight/angb∇acketleft0|),\nthe dipole–response function defined in Eq. (61) is easily\nevaluated,\nχˆD;ˆD(t) =2\ni/planckover2pi1/summationdisplay\nn/negationslash=0|Dn0|2sin(ωn0t) e−Γn\n2|t|,(65)\nwith the dynamical susceptibility,\n˜χˆD;ˆD(z) =/summationdisplay\nn/negationslash=02mωn0\n/planckover2pi1e2|Dn0|2Ω2\nn˜χn(io)\nΩ2n−z2−siΓnz,(66)given by Fourier–Laplace transformation. Here\nDn0=/angb∇acketleftn|ˆD·e|0/angb∇acket∇ightare dipole–moment matrix elements\nandωn0= (εn−ε0)//planckover2pi1denote atomic excitation frequen-\ncies (n=1,2,...). Abbreviations have been introduced\nfor partial static polarizabilities and resonance frequen -\ncies,˜χn(io) =e2/(mΩ2\nn)andΩ2\nn=ω2\nn0+ (Γn/2)2, re-\nspectively.\nIn Eq. (65), ad–hoc damping factors have been inserted\nwhich account approximately for the natural lifetimes\nof excited atomic states while preserving the symmetry\nspelled out in Eq. (61). Excited atomic states are well\nknown to have a finite natural lifetime τn= 1/Γneven\nif no electromagnetic field is applied, because there is\n“spontaneous emission” due to the atom interacting with\nvacuum fluctuations, interactions which have not been in-\ncluded into the unperturbed Hamiltonian H. In leading\norder (electric dipole transitions), spontaneous emissio n\nwill occur at a rate [9, Chap. V]\nΓn=4α\n3c2εn′<εn/summationdisplay\nn′ω3\nnn′|/angb∇acketleftn′|ˆ r|n/angb∇acket∇ight|2, (67)\nwhereα=e2/(4πǫ0/planckover2pi1c)≈1/137denotes the Sommer-\nfeld fine–structure constant.\nIt is very instructive to evaluate the dipole–response\nfunction in detail for a simple model of an atom. Within\nthe quantum–oscillator model for the atomic electron,\nˆH=/planckover2pi1ω10(ˆa†ˆa+ 1/2), one has for the electric dipole–\nmoment operator ˆD=−ex0(ˆa†+ ˆa)with oscillator\nlengthx0=/radicalbig\n/planckover2pi1/(2meω10)resulting in matrix elements\n|Dn0|2=e2x2\n0δn,1, which leave only a single term in the\nsum on r.h.s of Eq. (65),\nχˆD;ˆD(t) =e2sin(ω10t)\nimeω10e−Γ1\n2|t|. (68)\nEvaluation of the damping constant Γ1using the transi-\ntion rate of Eq. (67) results in\nω10= (E1−E0)//planckover2pi1←→˜Ω\nΓ1= ¯τω2\n10←→Γ, (69)\nwhere the characteristic time ¯τturns out to be identical\nto the time constant τintroduced in Eq. (24),\n¯τ≡τ=2α/planckover2pi1\n3mec2≈6.3×10−24s. (70)\nThe quantum–mechanical results derived above are note-\nworthy in several respects, as they demonstrate why the\nclassical oscillator model discussed in Sec. II has been so\nextremely successful in describing an atom irradiated by\nlight.\n1. The general result Eq. (60) for the induced dipole\nmoment of any physical system in a weak electric\nfield has the same formal structure as one finds for\nthe steady–state elongation of a classical oscillator\nsubjected to an external field, see Eq. (9).10\n2. Dipole–dipole response function of a quantum os-\ncillator (Eq. (68)) and elongation–response function\nof a classical oscillator (Eq. (8)) become identical\n—after multiplying the latter by (−e)2and iden-\ntifying induced moment (−e)x(t)→/angbracketleftBig\nδˆD/angbracketrightBig\nt, force\nf(t)→(−e)E(t), and fixing oscillation frequency\nand damping constant of the classical Lorentz\natom according to Eq. (69).\nNote that the latter identification solves, by quantum–\nmechanical arguments, the problem of finding appropri-\nate parameters Ω,Γto be used for the classical Lorentz–\natom:Ω =ω10/bracketleftbig\n1+O/parenleftbig\n(τω10)2/parenrightbig/bracketrightbig\n,Γ =τω2\n10, which to\nleading order in the small parameter (τω10)agree with\nthe classical solution, provided one also identifies /planckover2pi1times\nthe classical resonance frequency ω0with the energy dif-\nference(E1−E0), i. e.,ω0≡ω10.\nLorentz andAbraham at the end of 19thcentury,\nof course, did not have recourse to results from quan-\ntum theory (cf. Eqs. (68–69)). They had to specify their\nmodel parameters by using classical electrodynamics,\nonly. While Ωin Eq. (1) could naturally be associated\nwith the frequency of resonantly absorbed light, deter-\nmination of Γrequired introduction of a radiative reac-\ntion force which lead to the strange new AL equation of\nmotion (Eq. (23)) for the oscillator elongation.\nIt is therefore noteworthy and comforting to see that\ntheclassical radiation–damping constant Γderived from\nAL equation (Eq. (35)) is perfectly reproduced by the\nquantum–mechanical result in Eq. (69).\nD. Average absorbed power\nThe average power absorbed from the external oscilla-\ntory field by the physical system described in Eq. (55) is\ngiven by\nP(ω) =d\ndt/angb∇acketleftHt/angb∇acket∇ightt=/angbracketleftbigg∂Ht\n∂t/angbracketrightbigg\nt=/angbracketleftBig\nδˆD/angbracketrightBig\ntsin(ωt)E0\n=1\n2ωχ′′\nˆD;ˆD(ω)E2\n0+O(E3\n0), (71)\nwhereF(t) =1\nT/integraltextT\n0dt F(t),T= 2π/ωand, in the last\nline, use has been made of Eq. (60). The quantum–\nmechanical result in lowest non–vanishing order of per-\nturbation theory, Eq. (71), should be compared to the\nclassical expression Eq. (22). As in case of the induced\ndipole moment, the formal structures of both, quantum\nand classical, results for P(ω)are identical. The average\npower absorbed from the ac-electric field by a charged\nquantum oscillator in its ground state will coincide with\nthe power absorbed by the classical oscillator, because of\nequivalent response functions, cf. Sec. IVC - item 2.E.ac–Stark effect and optical dipole potential\nThe energy/angb∇acketleftH/angb∇acket∇ightof an atom is expected to change upon\napplying an electric field E0cos(ωt). Such a phenomenon\nis well known as Stark shift in case of a constant electric\nfield (ω= 0). Since an atom in its ground state has no\npermanent dipole moment, the Stark shift is typically\nof2ndorder inE0. The rapidly oscillating electric field of\nvisible light will also induce a shift of the atomic energy,\nwhich is rapidly oscillating with frequency ωand known\nasac–Stark effect. Due to the high frequency of light,\nthe induced shift cannot be detected by time–resolved\nmeasurements. Therefore, only the time–averaged shift\nis of interest here (averaging over period T= 2π/ω).\nApplying the perturbation expansion Eq. (56) to the\noperator of total energy, ˆOt→ˆHt, the averaged induced\nenergy shift is\n/angbracketleftBig\nδˆHt/angbracketrightBig\nt=/angbracketleftBig\nˆHt/angbracketrightBig\nt−/angb∇acketleftH/angb∇acket∇ight= ∆ε+O(E3\n0),\n∆ε=−E0/angbracketleftBig\nˆD/angbracketrightBig(1)\ntcos(ωt)+/angbracketleftBig\nˆH/angbracketrightBig(2)\nt.(72)\nSince one may replace under the time average/angbracketleftBig\nˆD/angbracketrightBig(1)\nt→\n/angbracketleftBig\nδˆD/angbracketrightBig(1)\nt, the first contribution to ∆εis easily evaluated\nwith the help of the induced dipole moment in Eq. (60),\n−E0/angbracketleftBig\nˆD/angbracketrightBig(1)\ntcos(ωt) =−χ′\nˆD;ˆD(ω)E2\n0\n2.(73)\nHereχ′\nˆD;ˆD(ω) =α(ω)is the electric polarizability defined\nquantum–mechanically, which should be compared to its\nclassical pendent in Eq. (40). The second contribution to\n∆ε(Eq. (72)) is read from Eq. (57). Noting the relation\nχˆH;ˆD(τ1,τ2) =1\ni∂\n∂τ2χˆD;ˆD(τ2−τ1) (74)\nbetween quadratic and linear dipole–dipole response\nfunction, and employing sin(ωt)/t→πδ(t)for large ω\nunder the final integral, one finds\n/angbracketleftBig\nˆH/angbracketrightBig(2)\nt= i2/integraldisplay∞\n0dτ1/integraldisplay∞\nτ1dτ2χˆH;ˆD(τ1,τ2)\n×e−o(τ1+τ2)cos[ω(τ2−τ1)]E2\n0\n2\n=i\n2/integraldisplay∞\n0dte−otχˆD;ˆD(t)\n×/bracketleftBig\ncos(ωt)+ω\nosin(ωt)/bracketrightBigE2\n0\n2\n=1\n2χ′\nˆD;ˆD(ω)E2\n0\n2, (75)\nwhich is just (−1/2)times the energy of the induced\ndipole moment in the external field. Summing both con-\ntributions, Eq. (73) and Eq. (75), there will be a non–zero11\naverage shift of the system energy induced by the electric\nfield (“ac–Stark effect”),\n∆ε=−1\n4χ′\nˆD;ˆD(ω)E2\n0. (76)\nAs expected, the conventional quadratic Stark shift fol-\nlows from Eq. (76) for ω= 0. If the external electric field\nis produced, e. g., by the standing wave of linearly (in\nz–direction) polarized light created by two laser beams\ncounter–propagating along x–axis,\nE0cos(ωt) =˜E0cos(k·r−ωt)+˜E0cos(−k·r−ωt),\nthe field strength E0→2˜E0cos(2πx/λ)will acquire a\nspatial dependence, E0=E0(r). As long as field varia-\ntions over distances of the order of system diameter are\nnegligible, which is the case for an atom in visible light\n(λ≫a0), Eq. (76) applies. The energy shift —and thus\nthe energy of the atom itself, too— will be a function of\nthe atomic position rviaE0(r)2resulting in a force act-\ning on the atom (“dipole force”, −/vector∇∆ε(r)). Hence one\ndefines an “optical–dipole potential”,\nUdip(r) = ∆ε(r), (77)\nwhich crucially depends on the frequency of the laser light\nused to produce the potential via the electric polarizabil-\nityα(ω) =χ′\nˆD;ˆD(ω).\nV. CONCLUSIONS\nBy determining the unique solution of the non–\nrelativistic AL equation (Eq. (23)), which turns out to be\na ‘run–away’ for generic initial conditions, I showed that\nthere is no steady–state solution that will describe the\ndriven oscillations of an atomic dipole moment induced\nby the electric field of light. Due to its run–away solu-\ntion, Eq. (23) does not qualify for modelling the bounded\nelectron of Lorentz ’s atom.\nTherefore, an attempt to determine the complex\natomic polarizability by employing any one particular so-\nlution of the AL equation, which is notthe steady state,\nwill be a misleading effort. The erroneous “polarizabil-\nity” Eq. (48) which, besides other deficiencies, violates\nKramers–Kronig relations and f–sum rule, has spread\nwidely in literature. The error is obviously invoked by\n(and has been traced back to) authors’ unjustified as-\nsumption of having found the steady–state solution of\nthe AL equation which, as I proved by finding the unique\nsolution Eq. (78), does not exist.\nHowever, according to the discussion in Sec. III C, there\nis also a positive aspect of the AL equation. In an endeav-\nour to account for radiative dissipation processes within\nclassical electrodynamics, the AL equation allows to de-\ntermine the appropriate oscillator parameters ΩandΓto\nbe used with Eq. (1), when implementing radiative dissi-\npation in the Lorentz atom.Finally, in Sec. IV the steady–state induced dipole mo-\nment of a system placed into an external electric field\nis studied by quantum mechanical perturbation theory\nin a ‘semi–classical approach’. The quantum mechanical\ndipole–dipole response function, which determines elec-\ntric polarizability, average power absorbed from the field,\nand optical dipole potential, is identified as a quantum\nanalog of the classical elongation–response function in-\ntroduced in Sec. II. By the formally exact Eqs. (62)–(63),\nit is demonstrated that, in case of negligible system mem-\nory, the dipole–dipole response function will acquire the\nsame functional form as the classical response function\n(Eq. (8)). If, moreover, a quantum oscillator is chosen as\na simple atomic model, the quantum–mechanically deter-\nmined values for ( Ω,Γ) turn out to be in perfect agree-\nment with the classical oscillator parameters determined\nfrom AL equation.\nThe intimate relations between quantum–mechanical\nand classical response and relaxation functions carved\nout in Sec. IV above raise well–founded expectations that\ntheLorentz atom, modelled by Eq. (1), will have inter-\nesting future applications, in which oscillator parameter s\nare nowadays determined in quantum–mechanical calcu-\nlations.\nAcknowledgement:\nIt was my pleasure to discuss with A. Pelster and J.\nAkram many aspects of this work. Special thanks go\nto V. Bagnato and E. dos Santos for their warm hospi-\ntality and fruitful discussions during a visit to USP Sao\nCarlos, Brazil, where part of this work developed.\nVI. APPENDIX\nA. Unique solution of AL equation of motion\nThe unique solution of Eq. (23), which is the general\nsolution of the homogeneous equation plus a particular\nsolution of the inhomogeneous equation, may be cast into\nthe following form ( t≥t0),\nxAL(t,t0) =xh\nAL(t,t0)+xp\nAL(t,t0), (78)\nxh\nAL(t,t0) =φ(t−t0)x0+iχ(t−t0)mv0\n−(b0+Γv0+Ω2x0)τ2\n1+4˜Ω2τ2/bracketleftBig\n−e(t−t0)(Γ+1/τ)\n+φ(t−t0)+(Γ+1 /τ)iχ(t−t0)m/bracketrightBig\n(79)\nxp\nAL(t,t0) =/integraldisplayt−t0\n0dt′/bracketleftBig\n−et′(Γ+1/τ)+φ(t′)\n+(Γ+1/τ)iχ(t′)m/bracketrightBigτf(t−t′)\n(1+4˜Ω2τ2)m.(80)\nHere oscillator relaxation and response functions, φ(t)\nandχ(t), and frequency ˜Ω, are defined in terms of (Ω,Γ)12\nand are given, resp., in Eq. (4) and Eq. (7). The oscilla-\ntor parameters Ω=Ω(τ,ω0)andΓ=Γ(τ,ω0)are given in\nterms of the AL parameters (τ,ω0)in Eq. (35).\nAs discussed in Sec. III C above, Eq. (79) implies\nthat the unique solution of Eq. (23) for initial values\n(x0, v0, b0)will diverge, if (t−t0)→∞, because the\ncharacteristic polynomial of Eq. (23) has a positive root,\nz2= Γ + 1/τ >0. From limt0→−∞xAL(t,t0) =∞, I\nconclude that a steady–state solution of the AL equation\ndoes not exist. A steady–state solution would require\nthatxh\nAL(t,−∞) = 0 for generic (x0, v0, b0).\nB. Fourier–Laplace transform (FLT)\nIn Eq. (10), the Fourier–Laplace transform (FLT)\nof a bounded function f(t)(|f(t)|≤M <∞) has been\nintroduced,\n˜f(z) =/integraldisplay∞\n−∞dteitzisΘ(st)f(t), s= signℑ[z]/negationslash= 0,\n(81)\nwhich is an analytical function for all complex zoutside\nthe real axis. The FLT of f(t)has as a Cauchy –integral\nrepresentation\n˜f(z) =/integraldisplay∞\n−∞dω\nπf′′(ω)\nω−zz=ω±io−→f′(ω)±if′′(ω)(82)\nwithf′′(ω) =1\n2i/bracketleftBig\n˜f(ω+io)−˜f(ω−io)/bracketrightBig\ndenoting the\nspectral function , or dissipative part of ˜f(ω+ io), and\nf′(ω) =1\n2/bracketleftBig\n˜f(ω+io)+˜f(ω−io)/bracketrightBig\ndenoting the reactive\npartof˜f(ω+ io). Dissipative and reactive parts obeydispersion relations,\nf′(ω) =−/integraldisplay∞\n−∞d¯ω\nπf′′(¯ω)\n¯ω−ω, f′′(ω) =−−/integraldisplay∞\n−∞d¯ω\nπf′(¯ω)\n¯ω−ω,\n(83)\nknown as Kramers–Kronig relations in physics litera-\nture.\nIn general, f′(ω)andf′′(ω)will be complex functions\nof the real variable ω. Functions f(t), which vanish for\nlarge|t|(as is the case for response and relaxation func-\ntions discussed above), are related to their spectral func-\ntion by conventional Fourier transform,\nf′′(ω) =/integraldisplay∞\n−∞dt\n2eitωf(t), f(t) =/integraldisplay∞\n−∞dω\nπe−itωf′′(ω),\n(84)\nand one easily verifies for the response function χ(t)\n(Eq. (4)), which is purely imaginary, odd in t, and van-\nishing for|t|→∞ ,\nχ′′(ω) =±ℑ[˜χ(ω±io)] =−χ′′(−ω) =χ′′(ω)∗,(85)\na spectral function which is real, odd in ω, and1/2of the\nconventional Fourier transform of χ(t). Similarly, the\nrelaxation function φ(t)(Eq. 4), which is real, even in t,\nand vanishing for |t|→∞ , will have a spectral function,\nφ′′(ω) =±ℑ[˜φ(ω±io)] =φ′′(−ω) =φ′′(ω)∗(86)\nwhich is real, even in ω, and just 1/2of the conven-\ntional Fourier transform of φ(t). For response and\nrelaxation spectrum, Kubo ’s identity takes the simple\nform:χ′′(ω) =ωφ′′(ω)/(mΩ2).\n[1] R. Grimm, M. Weidemüller, and Y. Ovchinnikov. Opti-\ncal dipole traps for neutral atoms. Advances in Atomic,\nMolecular, and Optical Physics , 42:95 – 170, 2000.\n[2] G. Gilbert, A. Aspect, and C. Fabre. Introduction to\nQuantum Optics . Campridge University Press, New York,\n2010.\n[3] L. Bergmann and Cl. Schaefer. Lehrbuch der Experi-\nmentalphysik , volume II (Elektrizitätslehre). Walter de\nGruyter & Co., Berlin, 1961.\n[4] P. C. Martin. In C. de Witt and R. Balian, editors, Prob-\nlème à N Corps , page 37 ff, New York, 1968. Gordon and\nBreach.[5] John David Jackson. Classical Electrodynamics . John Wi-\nley & Sons, New York, 1962.\n[6] George L. Murphy. A Note on the Abraham-Lorentz Equa-\ntion.Aust. J. Phys. , 30:675–6, 1977.\n[7] J.L. Jiménez and I. Campos. A critical examination of the\nAbraham-Lorentz equation for a radiating charged parti-\ncle.Am. J. Phys. , 55(11):1017, 1987.\n[8] David J. Griffiths. Introduction to Electrodynamics .\nPrentice-Hall, Inc., 3rdedition, 1999.\n[9] W. Heitler. The Quantum Theory of Radiation . Zürich,\n3rdedition, 1953."    },    {        "title": "1908.11198v2.Observations_of_Electromagnetic_Electron_Holes_and_Evidence_of_Cherenkov_Whistler_Emission.pdf",        "content": "Observations of Electromagnetic Electron Holes and Evidence of Cherenkov Whistler\nEmission\nKonrad Steinvall,1, 2,\u0003Yuri V. Khotyaintsev,1Daniel B. Graham,1\nAndris Vaivads,3Olivier Le Contel,4and Christopher T. Russell5\n1Swedish Institute of Space Physics, Uppsala, 75121, Sweden\n2Space and Plasma Physics, Department of Physics and Astronomy, Uppsala University, Uppsala, 75120, Sweden.\n3Division of Space and Plasma Physics, School of Electrical Engineering and Computer Science,\nKTH Royal Institute of Technology, Stockholm, 11428, Sweden\n4Laboratoire de Physique des Plasmas, CNRS/Ecole Polytechnique/Sorbonne\nUniversit\u0013 e/Univ. Paris Sud/Obs. de Paris, Paris, F-75252 Paris Cedex 05, France\n5Department of Earth and Space Sciences, University of California, Los Angeles, California, 90095, USA\n(Dated: January 8, 2020)\nWe report observations of electromagnetic electron holes (EHs). We use multi-spacecraft analysis\nto quantify the magnetic \feld contributions of three mechanisms: the Lorentz transform, electron\ndrift within the EH, and Cherenkov emission of whistler waves. The \frst two mechanisms account for\nthe observed magnetic \felds for slower EHs, while for EHs with speeds approaching half the electron\nAlfv\u0013 en speed, whistler waves excited via the Cherenkov mechanism dominate the perpendicular\nmagnetic \feld. The excited whistlers are kinetically damped and typically con\fned within the EHs.\nElectron holes (EHs) are localized nonlinear plasma\nstructures in which electrons are self-consistently trapped\nby a positive potential [1{3]. By scattering and heating\nelectrons, EHs play an important part in plasma dynam-\nics [4, 5]. EHs are frequently observed in space [6{10]\nand laboratory [11{13] plasmas. They are typically man-\nifested in data as diverging, bipolar, electric \felds par-\nallel to the ambient magnetic \feld. EHs are formed by\nvarious instabilities [14, 15], and are thus indicators of\nprior instability and turbulence. Their connection with\nstreaming instabilities leads them to frequently appear\nduring magnetic reconnection [16{19]. Furthermore, sim-\nulations of magnetic reconnection have shown EHs can\nCherenkov radiate whistler waves which in turn a\u000bect the\nreconnection rate [20]. Studying EHs can thus prove im-\nportant for understanding key plasma phenomena such\nas magnetic reconnection.\nThough EHs are usually considered electrostatic, ob-\nservations of electromagnetic EHs have been made in\nEarth's magnetotail [21, 22]. The observed magnetic\n\felds (\u000eB) were argued to be the sum of two independent\n\felds. First, \u000eBLgenerated by the Lorentz transform, of\nthe electrostatic \feld, and second, \u000eBdgenerated by the\n\u000eE\u0002B0drift of electrons associated with the EH electric\n\feld and ambient magnetic \feld [21, 23]. These studies\nwere limited either by the fact that the EHs were only\nobserved at one point in space [21], or provided only es-\ntimates of\u000eBdkat the EH center [22]. With the Magne-\ntospheric Multiscale (MMS) [24] mission, it is possible to\nuse four-spacecraft measurements to obtain a complete\nthree-dimensional description of EHs [25{27], enabling\n\u000eBto be investigated in greater detail [25].\nIn this letter we use data from MMS to investigate\nelectromagnetic EHs frequently observed during bound-\nary layer crossings in the magnetotail. We use multi-\nspacecraft methods to quantify di\u000berent contributions to\u000eB. Our results show that \u000eBd;kwell explains the ob-\nserved\u000eBk, and that \u000eBd;?is in good agreement with\nobservations for EHs that are much slower than the elec-\ntron Alfv\u0013 en speed. For increasing EH speeds we show,\nfor the \frst time, that localized whistler waves are ex-\ncited from the EHs via the Cherenkov mechanism and\ncontribute signi\fcantly to \u000eB?.\nFig. 1 shows an example of a plasma sheet boundary\nlayer crossing containing signatures of magnetic recon-\nnection and EHs with magnetic \felds. At 2017-07-26\n07:00 UT, MMS was in the plasma sheet and detected\na fast reconnection jet moving tailward (Fig. 1c). At\n07:01:30, the ion \row reversed, and MMS entered the\nboundary layer between the plasma sheet and the tail\nlobes (Fig. 1d) where strong wave activity was observed\n(Fig. 1e). First as low-frequency E?oscillations con-\nsistent with lower hybrid drift waves [33], and later as\nsolitaryEkwaves marked by the vertical dashed line\nin Fig. 1e, and exempli\fed in Figs. 1g,h. The solitary\nwaves were accompanied by a high-energy electron beam\n(Fig. 1f) parallel to B0. By timing Ekbetween the space-\ncraft we \fnd the structures to be EHs moving together\nwith the beam. Notably the EHs have magnetic \feld\n\ructuations \u000eBassociated with them. We show two EH\nexamples in Figs. 1g-j. While both EHs have positive\nand monopolar \u000eBk(distorted in the \fgure by high-pass\n\fltering) con\fned within the EH, there are signi\fcant\ndi\u000berences in \u000eB?. For the \frst EH (Figs. 1g,i), \u000eB?\nis localized within the EH, whereas for the second EH,\n\u000eB?oscillates multiple times and forms a trailing tail\n(Fig. 1h,j). Note that of the roughly 40 EHs that were\nobserved during this time, only two EHs had the tail-like\nfeature in Fig. 1j, the others resembled Fig. 1i. The po-\nlarization of \u000eB?is right handed for all cases (Figs. 1k,l)\nwith dominant frequency !\u00190:7\nce< !pe, where \n ce\nand!peare the electron cyclotron and plasma frequen-arXiv:1908.11198v2  [physics.space-ph]  7 Jan 20202\nFIG. 1. Left: Event overview. (a) Magnetic \feld from FGM [28] in geocentric solar magnetospheric (GSM) coordinates, (b)\nplasma density from FPI [29], (c) ion velocity from FPI in GSM, (d) electron energy spectrogram from FPI, (e) electric \feld\nfrom EDP [30, 31] in \feld-aligned coordinates, (f) spectrogram of the ratio of the parallel and anti-parallel electron phase-space\ndensity from FPI. The vertical dashed line shows where EHs are observed. Right: Examples of electromagnetic EHs. The data\nis high-pass \fltered at 100 Hz. (g,h) Electric \feld from EDP, (i,j) magnetic \feld from SCM [32], (k,l) hodograms of \u000eB?.\ncies.\nWe perform a statistical study to investigate how \u000eB\ndepends on EH properties. To accurately estimate the\nelectron hole speed, vEH, and parallel length scale, lk,\nthe EHs should be detected by as many spacecraft as\npossible, and all four spacecraft are needed to accurately\nestimate the EH center potential, \b 0, and perpendicular\nlength scale, l?[26, 27]. We therefore limit the study\nto June-August 2017, when MMS was probing the mag-\nnetotail with electron scale spacecraft separation. We\ntake 9 data intervals where one or more groups of elec-\ntromagnetic EHs are observed, resulting in a data-set of\n336 EHs, all observed in connection to boundary layers\nsimilar to that in Fig. 1.\nWe use the multi-spacecraft timing method discussed\nin Ref. 27, cross-correlating \u000eEkbetween the space-\ncraft, to determine vEH,lk, and the measured potential\n\bm=R\n\u000eEkvEHdtof the 336 EHs. The median propa-\ngation angle of the EHs with respect to B0is 12\u000ewhich\nis within the uncertainty of the four-spacecraft timing,\nsovEHis assumed to be \feld aligned. In Fig. 2 we\nplot \bmagainstvEH=vAe(vAe=c\nce=!peis the elec-\ntron Alfv\u0013 en speed), with the peak value of \u000eB?color-\ncoded. The \fgure shows that \u000eB?increases with poten-\n10-210-1100\nvEH/vAe100101102103104m (V)\n2017-06-11T17:22:01\n2017-06-11T17:25:16\n2017-07-03T21:55:11\n2017-07-06T00:54:15\n2017-07-06T14:07:28\n2017-07-18T01:40:31\n2017-07-23T19:49:00\n2017-07-26T07:01:37\n2017-08-23T15:01:17\n00.050.10.150.20.25\nB (nT)FIG. 2. Measured EH potential \b magainstvEH=vAefor 336\nEHs, with the peak value of \u000eB?color-coded. EHs from the\nsame burst-data interval have the same symbol.\ntial and velocity. A dependence on \b mis expected since\n\u000eBL;\u000eBd/\u000eE?/\b0and thevEH=vAedependence is\nqualitatively consistent with \u000eBL/vEHsince the EHs\nwere observed in the same plasma region with, for the\nmost part, similar vAe.\nNext, we investigate the di\u000berent mechanisms that can\ngenerate\u000eB. For weakly relativistic EHs (i.e. \r\u00191)3\n\u000eBL;f?1;?2g=\u0007vEH\u000eEf?2;?1g=c2[34]. By assuming the\nEH potential\n\b(r;\u0012;z ) = \b 0e\u0000r2=2l2\n?e\u0000z2=2l2\nk; (1)\n\u000eBdis given by the Biot-Savart law of the \u000eE\u0002B0current\nJ\u0012=en0r\b(r;z)=(B0l2\n?) [23] as\n\u000eBd(x) =en0\u00160\n4\u0019B0Zr0\nl2\n?\b(r0;z0)^\u0012\u0002x\u0000x0\njx\u0000x0j3d3x0;(2)\nwheren0is the electron density, and eis the elemen-\ntary charge. In Fig. 3 we show two examples of EHs\nwhere we calculate and compare \u000eBLand\u000eBdwith ob-\nservations. The \frst EH (Figs. 3a-d) is small amplitude\n(\bm= 680 V), slow ( vEH=vAe= 1=9) and has a weak\n\u000eB\u00180:01 nT. We use the method of Ref. 26 (using, in-\nstead of the maximum value, \u000eE?evaluated at \u000eEk= 0)\nto \ft the\u000eEdata of the four spacecraft to the electro-\nstatic \feld corresponding to Eq. (1), giving l?= 26 km\n= 0:6de= 1:6lk, wherede=c=!peis the electron inertial\nlength; \b 0= 915 V = 1 :4Te=e, whereTeis the electron\ntemperature; and the position of the EH. A represen-\ntation of the \ft is shown in Fig. 3a, where we plot the\nspacecraft (colored dots) and the EH (grey cross) position\nin the perpendicular plane. The arrows are the measured\n(colored) and predicted (grey) \u000eE?evaluated at \u000eEk= 0,\nshowing that the EH \ft well describes \u000eE?for all four\nspacecraft. A time series representation of the \ft is shown\nin Fig. 3b for MMS4, where the measured and \ftted \u000eE\nare the solid and dashed lines respectively, a\u000erming that\nthe \ft is in good agreement with observations. With \b 0,\nlkandl?known, we solve Eq. (2) numerically to obtain\n\u000eBd.\u000eBLis small,j\u000eBLj\u00190:004 nT. We plot MMS4\ndata of\u000eB(solid) together with \u000eBL+\u000eBd(dashed)\nin Fig. 3c, and the residual \u000eBRes=\u000eB\u0000\u000eBL\u0000\u000eBd\nin Fig. 3d. We \fnd that \u000eB\u0019\u000eBd, the only discrep-\nancy being thatj\u000eBd;?1jis overestimated initially. This\nmight be due to the fact that the EH has a steeper in-\ncrease of\u000eEkthan the model (Fig. 3b). The second EH\n(Fig. 3e-h) has larger amplitude (\b m= 3:5 kV), is faster\n(vEH=vAe= 1=4) and has a stronger \u000eB\u00180:1 nT. We\nperform the same analysis and present analogous plots\nin Fig. 3e-h. As before, the EH \ft of \u000eE(Fig. 3e,f)\nagrees well with observations (\b 0= 4:2 kV = 1:9Te=e\nandl?= 40 km = 1 :1de= 1:6lk),j\u000eBLj \u0019 0:02 nT\nis small compared to j\u000eB?j, and\u000eBkis well traced by\n\u000eBd;k. However, when it comes to \u000eB?there is signi\f-\ncant\u000eBRes;?implying an additional mechanism is con-\ntributing to \u000eB?. We note that \u000eBRes;?is right hand\npolarized and its dominant frequency f\u0019400 Hz is be-\nlowfce\u0019650 Hz. We estimate the wave normal angle of\n\u000eBRes;?bykk=k?=\u000eB?=\u000eBk= 2:6, corresponding to a\nwave normal angle 21\u000e. We thus \fnd that while \u000eBof\nthe slower EH can be fully explained by \u000eBd, the faster\nEH has an additional \u000eBRes;?with features consistent\nwith whistler waves.\n            \n2017-07-06 UTC-50050100E (mV/m)MMS1\n||21(f)\n            \n2017-07-06 UTC-0.100.10.2B (nT)\n||21(g)\n  00:54:15.990      00:54:15.993    \n2017-07-06 UTC-0.100.10.2BRes (nT)\n||21(h)             \n2017-07-06 UTC-1001020E (mV/m)MMS4\n||21(b)\n             \n2017-07-06 UTC-0.0200.020.04B (nT)\n||21(c)\n 00:55:40.275      00:55:40.278     \n2017-07-06 UTC-0.0200.020.04BRes (nT)\n||21(d)-50 0 50\nx (km)-50050y (km)Fast EH: vEH/vAe = 0.25\n1000200030004000(e)\n20 mV/mMMS1\nMMS2\nMMS3\nMMS4\n-40 -20 0 20 40\nx (km)-40-2002040y (km)Slow EH: vEH/vAe = 0.11\n900\n700\n500\n300\n100(a)\n20 mV/mFIG. 3. Two examples of EH \fts and induced magnetic\n\felds. (a) The position of MMS (colored dots) and the EH\n(grey cross) in the perpendicular plane. The measured and\n\ftted\u000eE?are illustrated by the colored and grey arrows, re-\nspectively, where the arrow length is proportional to j\u000eE?j.\nThe grey contours are EH equipotential lines in Volts, and the\nmagenta circle corresponds to r=l?. (b) Measured (solid)\nand \ftted (dashed) \u000eE. (c) Measured \u000eB(solid) and calcu-\nlated\u000eBL+\u000eBd(dashed). (d) \u000eB\u0000\u000eBL\u0000\u000eBd. (e)-(h) Same\nformat as (a)-(d) for a di\u000berent EH. All \felds are high-pass\n\fltered at 50 Hz.\nWe are able to apply this method and calculate \u000eBd\nfor a total of 19 EHs. The remaining EHs were either\nnot observed by all four spacecraft ( \u001850%), had\u000eEthat\nwas qualitatively inconsistent with the assumed poten-\ntial model, e.g. bipolar \u000eE?(\u001825%), or gave \ftting re-\nsults deemed too di\u000berent from observations to be useful\n(\u001815%). For these 19 EHs, \u000eBkis consistently well de-\nscribed by\u000eBd;k, andj\u000eBLj\u001cj\u000eBd;?j, meaning\u000eBdis\nmore important for generating \u000eBin the observed param-\neter range of Fig. 2. For all 19 EHs, when \u000eBRes;?6= 0,\nit is right hand polarized with ! < \nce< !pewhich we\ninterpret as being related to the whistler mode.\nBecause\u000eBRes;?is localized to the EHs, we believe\nthe EHs to be the source of the whistlers, rather than for\nexample temperature anisotropy or Landau resonance.\nIn fact, for most observations Te?=Tek<1, so whistlers\nshould not grow from temperature anisotropy. In this\nsection we consider the generation of whistler waves from\nEHs via the Cherenkov mechanism, and show that this\nis consistent with our observations.\nThe theory of whistler waves Cherenkov emitted by\nEHs is developed and discussed in Ref. 20. In sum-\nmary, the Cherenkov resonance condition is !=kk=vEH4\nwhich speci\fes !andkkof the excited wave. Further,\nthe ratio of the whistler electric \feld to that of the EH\ngrows secularly (linearly in time) at a rate proportional\nto (vEH=vAe)4, subject to vEH\u0014vAe=2.\nTo put our EH observations into the context of the\nCherenkov mechanism, we plot the kinetic (orange and\npink from WHAMP [35]) and cold (blue) whistler dis-\npersion relation ( k?= 0) for one group of slow EHs\n(vEH\u0019vAe=16) withT?=Tk= 1:0 in Fig. 4a, and for\none group of fast EHs ( vEH\u0019vAe=4) withT?=Tk= 0:3\nin Fig. 4b. We de\fne and plot !EH=\u0019=tpp, wheretpp\nis the peak-to-peak time of \u000eEk, andkEH=!EH=vEH,\ncolor-coding \u000eB?. The Cherenkov resonance condition is\nfor a given EH manifested in the plots as the intersection\nof!r(kk) with the straight line passing through the origin\nand the point ( kEH;!EH). The slope of this line corre-\nsponds tovEH, meaning faster EHs excite whistlers with\nsmallerkk. The shaded regions contain EH velocities\nbetween max( vEH) and min(vEH) for the two groups.\nFor the slow EHs (Fig. 4a), these intersections occur\natkkde\u001d1. However, for the fast EHs (Fig. 4b) we\n\fnd that the EHs can excite whistlers in the wavenum-\nber range 2 :3\u0014kkde\u00144:7. This interval is marked by\nthe blue vertical lines at the intersection for the fastest\nand slowest EHs. We note that there is an additional\npermitted region for small kkde\u001c1, which was observed\nin Ref. [20]. For the observed EHs however, kk\u0019kEH,\nwhich is consistent with waves in the larger kkinterval.\nFor the permitted waves in the larger kkinterval,\r\nis large and negative. The resonant whistlers are thus\nstrongly damped, providing a possible explanation to\nwhy\u000eBRes;?is typically con\fned within the EHs. Note\nthat we are investigating the classic Cherenkov mecha-\nnism, where waves are excited by a propagating charge\nacting as an antenna [36, 37], not by kinetic Landau reso-\nnance. This is why the growth from the Cherenkov mech-\nanism does not appear in Fig. 4.\nExtending the dispersion relation in Fig. 4b to include\nk?>0 yields the surface in Fig. 4c and 4d, showing the\nrelative damping \r=!rand ellipticity respectively. By in-\ncludingk?>0, the resonant waves go from being points\non a curve, to contours on a surface. The blue contours\nin Figs. 4c,d show the waves that can be excited by the\nfastest and slowest EHs in Fig. 4b, meaning the other\nEHs in Fig. 4b can excite whistlers between these con-\ntours. From observations we have ellipticity values close\nto 1, consistent with the permitted k?.kkregion in\nFig. 4d.\nAdditionally, the fact that we observe a strong\nvEH=vAedependence of \u000eB?(Fig. 2) is explained by the\n(vEH=vAe)4dependence of the secular whistler growth.\nvEH=vAeis 4 times larger for the EHs in Fig. 4b than for\nthose in Fig. 4a, meaning they grow \u0018250 times faster.\nThis explains why signi\fcant \u000eBRes;?is observed only\nfor the fast EHs as was found in Fig. 3.\nAs an example we consider the EH with the tail-like\n/r\n/ce\nkde(c) (d)\nEllipticity\nk||de/ce\nk||de kde 0510201\n0.5\n00\n-0.1\n-0.2-0.3\n-0.4\n-0.50510201\n0.5\n01\n0.5\n0\n-0.5-102468 1 0\nk||de-0.500.511.52/cer(cold)\nr(1200 eV)\n(1200 eV)Fast EHs:\nvEHvAe/4(b)\n00.050.10.150.20.250.3\nBB(nT)02468 1 0\nk||de-0.500.511.52/cer(cold)\nr(1700 eV)\n(1700 eV)\nforbiddenCherenkov radiation permittedSlow EHs:\nvEHvAe/16(a)\n00.050.10.150.20.250.3\nB(nT)FIG. 4. (a)-(b) Cold (blue) and kinetic (orange and pink)\nwhistler dispersion relation ( k?= 0) for two di\u000berent groups\nof EHs. The dotted lines are extrapolations (based on the\ncold plasma dispersion relation) of the kinetic results, and\nare not exact. EH data is plotted with symbols and colorbar\nconsistent with Fig. 2. The average vEHisvAe=16 in (a),\nandvAe=4 in (b). The shaded intervals show min( vEH)\u0014\nv\u0014max(vEH), and the corresponding kkintervals satisfying\n!=kk=vEHare marked in blue. The black line and cross\nin (b) show the EH speed and observed properties of \u000eB?in\nFig. 1j. (c) Whistler dispersion relation for k?\u00150, color-\ncoding the relative damping \r=!r. The blue contours show\nthe boundaries of the Cherenkov-permitted regions, and the\nblack contour corresponds to the resonant waves of the EH\nin Fig. 1h. (d) Same as (c), but with ellipticity of \u000eBcolor\ncoded, 1 and\u00001 meaning right and left handed, respectively.\n\u000eB?shown in Figs. 1g,j. This EH is located at the point\nkEHde= 2:0,!EH=!ce= 0:55 in Fig. 4b, and its veloc-\nityvEH= 0:28vAecorresponds to the black line. From\nthe Cherenkov resonance condition we expect the emit-\nted whistler to have !=\nce= 0:73 andkkde= 2:7. The\nEH is observed by all four MMS spacecraft and we apply\na generalized four-spacecraft version of the method dis-\ncussed in Ref. 10 on \u000eB?to determine !=\nce= 0:76 and\nkkde= 3:2. This point is marked in Fig. 4b with a black\ncross. The predicted damping for the observed wave is\n\r\u0019\u00000:25\nce, qualitatively consistent with the strong5\ndecay seen in Fig. 1j. Taking the observed k?de= 0:53\ninto account in Figs. 4c,d, the black contour corresponds\nto the Cherenkov resonant waves, and we see that the ob-\nserved wave (black cross) is still close to the modes pre-\ndicted by the Cherenkov mechanism. We thus conclude\nthat the Cherenkov mechanism is in good agreement with\nobservations, and is likely the source of \u000eBRes;?.\nConclusions. In summary, we report MMS observa-\ntions of electron holes (EHs) with magnetic \feld signa-\ntures consisting of monopolar \u000eBkand right hand po-\nlarized\u000eB?. Typically, \u000eB?is con\fned within the EH\nand only one wave period is observed. In rare cases how-\never, multiple periods can be observed extending outside\nthe EH while rapidly decaying. The frequency of \u000eB?\nis below \n ce. Using spacecraft timing we calculate vEH\nand \bm, \fnding\u000eB?to correlate with both parameters.\nWe are able to calculate the magnetic \feld generated by\n\u000eE\u0002B0drifting electrons, \u000eBd, in a few cases, concluding\nthat this mechanism is responsible for the observed \u000eBk,\nand that\u000eBL\u001c\u000eBd, where\u000eBLis the Lorentz trans-\nform of the EHs electric \feld, in the observed parameter\nrange. For slow EHs ( vEH=vAe.0:1)\u000eB?\u0019\u000eBd?,\nwhereas an additional \u000eB?source is required for faster\nEHs. We show that this additional \feld is consistent\nwith whistler waves generated by EHs via the classic\nCherenkov mechanism (not Landau resonance). This is\nsupported by the right-hand polarization and ! < \nce,\nand the fact that signi\fcant \u000eB?is observed for EHs with\nspeeds approaching vAe=2. The kinetic whistler disper-\nsion relation shows that there is signi\fcant damping for\nthe wavenumbers predicted from the Cherenkov mecha-\nnism, which suggests that mainly a near-\feld signal will\nbe excited. This is consistent with our observation of\n\u000eB?being localized to the EH itself.\nUsing multi-spacecraft MMS observations we can for\nthe \frst time quantify individual contributions to \u000eBof\nEHs. We report the \frst observational evidence of EHs\nCherenkov radiating whistler waves, though the waves\ntend to be localized within the EHs rather than freely\npropagating.\nAcknowledgements. We thank the entire\nMMS team and instrument PIs for data ac-\ncess and support. MMS data are available at\nhttps://lasp.colorado.edu/mms/sdc/public. This\nwork is supported by the Swedish National Space Board,\ngrant 128/17, the Swedish Research Council, grant\n2016-05507. 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Gekelman,Cherenkov radiation of shear alfv\u0013 en waves, Physics of\nPlasmas 15, 082101 (2008)."    },    {        "title": "1609.07019v1.Lorentz_Covariant_Canonical_Symplectic_Algorithms_for_Dynamics_of_Charged_Particles.pdf",        "content": "arXiv:1609.07019v1  [physics.plasm-ph]  22 Sep 2016Lorentz Covariant Canonical Symplectic Algorithms for\nDynamics of Charged Particles\nYulei Wang,1, 2Jian Liu,1, 2, ∗and Hong Qin1, 3\n1School of Nuclear Science and Technology and Department of M odern Physics,\nUniversity of Science and Technology of China, Hefei, Anhui 230026, China\n2Key Laboratory of Geospace Environment, CAS, Hefei, Anhui 2 30026, China\n3Plasma Physics Laboratory, Princeton University, Princet on, NJ 08543, USA\nAbstract\nIn this paper, the Lorentz covariance of algorithms is intro duced. Under Lorentz transformation,\nboth the form and performance of a Lorentz covariant algorit hm are invariant. To acquire the\nadvantages of symplectic algorithms and Lorentz covarianc e, a general procedure for constructing\nLorentz covariant canonical symplectic algorithms (LCCSA ) is provided, based on which an explicit\nLCCSA for dynamics of relativistic charged particles is bui lt. LCCSA possesses Lorentz invariance\nas well as long-term numerical accuracy and stability, due t o the preservation of discrete symplectic\nstructure and Lorentz symmetry of the system. For situation s with time-dependent electromagnetic\nfields, which is difficult to handle in traditional constructi on procedures of symplectic algorithms,\nLCCSA provides a perfect explicit canonical symplectic sol ution by implementing the discretization\nin 4-spacetime. We also show that LCCSA has built-in energy- based adaptive time steps, which\ncan optimize the computation performance when the Lorentz f actor varies.\n∗corresponding author: jliuphy@ustc.edu.cn\n1I. INTRODUCTION\nThe advanced structure-preserving geometric algorithms h ave stepped into the field of\nplasma physics and attracted more and more attentions in rec ent years [1–11]. Through\npreserving different geometric structures, such as the phas e-space volume, the symplectic\nstructure, and the Poisson structure, geometric algorithm s possess long-term numerical ac-\ncuracy and stability and have shown powerful capabilities i n dealing with multi-scale and\nnonlinear problems. Volume-preserving algorithms (VPA) o f different orders for both non-\nrelativistic and relativistic full-orbit dynamics of char ged particles have been constructed in\nseveral publications [12–14]. A Poisson-preserving algor ithm for solving the Vlasov-Maxwell\nsystem is built through splitting the Hamiltonian and using the Morrison-Marsden-Weistein\nbracket [15]. As an important aspect of geometric algorithm s, symplectic methods have\nproduced fruitful results. For gyro-center dynamics of cha rged particles, variational sym-\nplectic methods have been studied and applied to plasma simu lations [1–3]. It is also feasible\nto canonicalize the gyro-center equations to construct can onical symplectic algorithms for\ntime-independent magnetic fields [16]. The Particle-in-Ce ll (PIC) method, known as the\nfirst principle simulation method for plasma systems, has be en reconstructed by the use of\ndifferent symplectic methods, including variational sympl ectic method, canonical symplec-\ntic method, and non-canonical symplectic method [6–9]. The oretically, symplectic methods\nimpose numerical results with a set of constrains, the numbe r of which is determined by the\nfreedom degrees of the systems [9, 17], by preserving the glo bal symplectic structure of the\nsystem. Correspondingly, the global relative errors of mot ion constants can be restricted to\nbounded small values, which enable symplectic algorithms t o retain many key properties of\nthe origin continuous systems. However, another essential geometric property of physical\nsystems has long been ignored in structure-preserving algo rithms, i.e., the Lorentz covari-\nance. The lack of Lorentz covariance leads to inconsistent n umerical solutions in different\ninertial frames. In this paper, we equip the symplectic algo rithm with the Lorentz covariance\nto obtain better performances.\nAs an intrinsic property of continuous physical systems, th e Lorentz covariance has be-\ncome a common sense in modern physics, which states that the p hysical rules and events keep\ninvariant under Lorentz transformation [18]. It is also imp ortant for algorithms to satisfy the\nLorentz covariance. Similar to continuous covariant syste m, Lorentz covariant algorithms\n2have invariant forms and describe invariant processes unde r Lorentz transformation. The\nLorentz invariance of each one-step map ensures the referen ce-independence of numerical\nresults, which leads to that the numerical properties, such as stability, convergence, and\nconsistency are also independent with the choice of referen ce frames. In applications, the\nLorentz covariant algorithms make it convenient and safe to adopt the same set of discretized\nequations in all inertial frames.\nThe combination of Lorentz covariance and symplectic metho d can generate algorithms\npossessing benefits from the both. If the long-term numerica l accuracy and stability are\nunavailable, Lorentz covariant algorithms cannot guarant ee the long-term correctness of\nsimulations, even though the results are reference-indepe ndent. On contrary, although sym-\nplectic methods without Lorentz covariance have long-term conservativeness and stability,\nthey break the Lorentz symmetry of the original continuous s ystems and produce incon-\nsistent numerical solutions in different inertial frames. O n the other hand, it is difficult\nto construct conventional symplectic algorithms for time- dependent Hamiltonian systems.\nMeanwhile, it is not straightforward to develop convention al symplectic algorithms with\noptimized adaptive time steps. These two problems can be sol ved automatically by the con-\nstruction of Lorentz covariant symplectic algorithms. Cov ariant algorithms directly iterate\ngeometric objects in 4-spacetime and discretize the worldl ines with respect to the discrete\nproper time τ. Consequently, the time, t, as a component of the 4-spacetime, plays the\nsame role as spatial coordinates in time-dependent Hamilto nians. Taking the place of t, the\nproper time is employed as the dynamical parameter and leads to proper-time-independent\nHamiltonians for time-dependent systems. Because covaria nt algorithms directly discretize\nthe worldline, one can obtain energy-based adaptive-time- step symplectic schemes given the\nfixed proper-time step ∆ τ= ∆t/γ. The adaptive time step can improve the performance of\nsymplectic algorithms when the Lorentz factor varies.\nTo endue symplectic algorithms with Lorentz covariance, a s traightforward way is to start\nfrom the view point of geometry. The Lorentz covariant syste ms reside in the 4-dimentional\nspacetime. Considering the reference-independence, the L orentz covariant discretized equa-\ntions can be regarded as the one-step maps of geometric objec ts in spacetime. As a result,\nif one starts from covariant continuous geometric equation s, and discretizes these equa-\ntions without breaking the integrity of all the geometric ob jects, the Lorentz covariance can\nbe naturally inherited. The canonical symplectic methods d irectly deal with the Hamil-\n3tonian equations of physical systems. During discretizati on, the symplectic structure of\nHamiltonian equations is retained, and each of the physical quantities is treated as an in-\nseparable discretized geometric object, updated at differe nt proper-time steps [17]. It is\nreadily to see that the canonical symplectic method provide s a convenient way to combine\nthe symplectic method and the Lorentz covariance. Here we su mmarize a general procedure\nfor constructing Lorentz covariant canonical symplectic a lgorithms (LCCSA), namely, 1)\nto write down the covariant geometric Hamiltonian equation for a target physical system\nin 4-dimentional spacetime, 2) to discretize the Hamiltoni an equations by using a canoni-\ncal symplectic scheme, such as Euler-symplectic scheme and implicit mid-point symplectic\nscheme, described by geometric objects in 4-spacetime.\nFollowing this procedure, we construct an explicit LCCSA fo r the simulation of relativis-\ntic dynamics of charged particles. Compared with a non-cova riant algorithm, LCCSA ex-\nhibits the reference-independent form and good long-term p erformances in different Lorentz\nframes. As a symplectic algorithm, LCCSA shows outstanding long-term numerical accu-\nracy than a covariant fourth-order Runge-Kutta algorithm ( RK4). Meanwhile, LCCSA can\nautomatically adjust the time step-length according to the energy of a particle and guarantee\nthe approximate constant time-sampling number in one gyro- period. The performance in\nsimulating energy-changing processes can be improved. As e xamples, both the computation\nefficiency for simulating acceleration and braking processe s of charged particle by use of\nLCCSA are optimized compared with those fixed-time-step alg orithms.\nThe rest part of this paper is organized as follows. The defini tion and properties of\nLorentz covariant symplectic algorithms are introduced in Sec. II. The detailed procedure\nof constructing an explicit LCCSA is explained in Sec. III. I n Sec. IV, the performances of\nLCCSA are exhibited through several typical numerical case s. We summarize this article in\nSec. V.\nII. LORENTZ COVARIANT SYMPLECTIC ALGORITHMS\nBefore introducing Lorentz covariant symplectic algorith ms, we first provide the rigorous\ndefinition of Lorentz covariant algorithm. For a given conti nuous Lorentz covariant system\nF, an algorithm Ais called Lorentz covariant if and only if it satisfies\nDA◦ TLF=TL◦ D AF, (1)\n4where TLdenotes the Lorentz transformation operator, DAdenotes the discretization oper-\nator determined by the algorithm A, the operation “ ◦” means composite mapping, and L\ndenotes the Lorentz transformation matrix satisfying LTgL=g, where gis the Lorentz met-\nric tensor of the 4-dimentional spacetime [18–20]. General ly speaking, Lcan be both proper\nLorentz transformation (det L= +1) and improper Lorentz transformation (det L=−1).\nA Lorentz transformation includes the rotation and the Lore ntz boost of inertial frames [18].\nSuppose that Ais applied to system Fin the inertial frame O, the first operation on the\nright-hand side of Eq. 1, φA=DAF, gives a realization of algorithm A, i.e., a set of discrete\nequations in this frame. In another inertial frame O′moving with speed βrelative to O, this\ndiscrete system are described by discrete equations φ′\nA=TLφAfollowing the Lorentz trans-\nformation of φA. On the left-hand side of Eq. 1, because the original system Fis Lorentz\ncovariant, F′=TLFtakes the same form as F. Consequently, the realization of algorithm A\nonF′, i.e., ξA=DAF′, also takes the same form as φAexcept that the physical quantities in\nξAare observed in the frame O′. So Eq. 1 concludes that the discrete equations generated by\na Lorentz covariant algorithm Ahave the invariant form and provides the same discretized\nsystem in different Lorentz inertial frames.\nTo make the picture of covariant algorithms clearer, for com parison, we investigate an ex-\nample of a non-covariant algorithm, i.e., the VPA for relati vistic charged particles dynamics\nas constructed in [13]. This algorithm has been applied to th e study of long-term dynamics\nof runaway electrons in tokamaks and shown its outstanding l ong-term numerical accuracy\n[10, 11]. However, its non-Lorentz-covariant property can be proved according to Eq. 1 as\nfollows. The target continuous system is the relativistic L orentz force equations FL3\ndx\ndt=p\nγ, (2)\ndp\ndt=E+p×B\nγ, (3)\nwhere xis the position, pis the mechanical momentum, γ=√1 +p2is the Lorentz factor,\nandEandBare respectively electric and magnetic fields. Notice that a ll the physical\nquantities in this paper are normalized according to Tab. I u nless noted otherwise. As a\ncommon wisdom, FL3is Lorentz covariant. We will show that DV P A◦TLFL3/negationslash=TL◦DV P AFL3.\nFirstly, we derive the discrete system φ′\nV P A=TL◦ D V P AFL3. In the reference frame O,\nby applying the discrete operator to FL3we obtain φV P A =DV P AFL3, which is a set of\n5Names Symbols Units\nTime, Proper Time, Gyro-period t,τ,Tcem0/eB0\nPosition x m0c/eB0\nMechanical/Canonical Momentum p,P m0c\nVelocity v,β c\nElectric field E B0c\nMagnetic field B B0\nVecter field A e/m0c\nScalar field φ e/m0c2\nHamiltonian H m0c2\nTable I. Units of all the physical quantities used in this pap er. m 0is the rest mass of a particle, e\nis the elementary charge, c is the speed of light, and B0is the given reference magnetic field.\ndifference equations in Oand can be written explicitly as [13],\ntk+1=tk+ ∆t , (4)\nxk+1=xk+ ∆tpk\nγk, (5)\npk+1=pk+W, (6)\nwhere γk=/radicalig\n1 +p2\nk, and W(tk+1,xk+1,pk,Ek+1,Bk+1,∆t) is a function given by\nW= ∆tEk+1+/parenleftig\nDˆBk+1+dDˆB2\nk+1/parenrightig/parenleftigg\npk+∆t\n2Ek+1/parenrightigg\n, (7)\nwhere Ek+1=E(tk+1,xk+1),Bk+1=B(tk+1,xk+1),d= ∆t//bracketleftbigg\n2/radicalig\n1 + (pk+ ∆tEk+1/2)2/bracketrightbigg\n,\nandD= 2d//parenleftig\n1 +d2B2\nk+1/parenrightig\n, and in Cartesian coordinate system ˆBis defined as\nˆB=\n0 Bz−By\n−Bz0Bx\nBy−Bx0\n. (8)\nThen, we transform φV P Ainto another frame O′. We suppose that O′moves with a fixed\nspeed β= (β1, β2, β3) relative to O. In this case, the Lorentz matrix Ldenotes the Lorentz\n6boost matrix which can be written explicitly in the Cartesia n coordinate system as,\nL=\nΓ −Γβ1 −Γβ2 −Γβ3\n−Γβ11 +(Γ−1)β2\n1\nβ2(Γ−1)β1β2\nβ2(Γ−1)β1β3\nβ2\n−Γβ2(Γ−1)β1β2\nβ2 1 +(Γ−1)β2\n2\nβ2(Γ−1)β2β3\nβ2\n−Γβ3(Γ−1)β1β3\nβ2(Γ−1)β2β3\nβ2 1 +(Γ−1)β2\n3\nβ2\n, (9)\nwhere β=|β|, and Γ = 1 /√1−β2is the Lorentz factor of frame O′. Without loss of\ngenerality, we set βas (β,0,0). Substituting xk,pk,γk, ∆t,EkandBkin Eq. 5 and Eq. 6\nby\ntk= Γ ( t′\nk+βx′\nk), (10)\nxk= Γ ( βt′\nk+x′\nk), (11)\nyk=y′\nk, (12)\nzk=z′\nk, (13)\nγk= Γ/parenleftig\nγ′\nk+βp′\nx,k/parenrightig\n, (14)\npx,k= Γ/parenleftig\nβγ′\nk+p′\nx,k/parenrightig\n, (15)\npy,k=p′\ny,k, (16)\npz,k=p′\nz,k, (17)\n∆t=γk\nγ′\nk∆t′=Γ/parenleftig\nγ′\nk+βp′\nx,k/parenrightig\nγ′\nk∆t′, (18)\nEk=fE(E′\nk,B′\nk), (19)\nBk=fB(E′\nk,B′\nk), (20)\nwhere fEandfBare the Lorentz transformation functions for electric and m agnetic fields\n[18]. After simplification, the difference equations in O′,φ′\nV P A=TLφV P A, becomes\nt′\nk+1=t′\nk+ ∆t′, (21)\nx′\nk+1=x′\nk+ ∆t′p′k\nγ′k, (22)\np′\nx,k+1=p′\nx,k+β/parenleftigg/radicalig\n1 + (p′\nk)2−/radicalbigg\n1 +/parenleftig\np′\nk+1/parenrightig2/parenrightigg\n+W′\nx\nΓ, (23)\np′\ny,k+1=p′\ny,k+W′\ny, (24)\n7p′\nz,k+1=p′\nz,k+W′\nz, (25)\nwhere W′\nx,W′\ny, and W′\nzare three components of W′/parenleftig\nt′\nk+1,x′\nk+1, γ′\nk,p′\nk,E′\nk+1,B′\nk+1,∆t′/parenrightig\nwhich is given by\nW′=W/bracketleftig\ntk+1/parenleftig\nt′\nk+1,x′\nk+1/parenrightig\n,xk+1/parenleftig\nt′\nk+1,x′\nk+1/parenrightig\n,pk(γ′\nk,p′\nk), fE, fB,∆t/parenleftig\nγ′\nk, p′\nx,k,∆t′/parenrightig/bracketrightig\n.(26)\nAccording to Eq. 23, φ′\nV P Ais an implicit scheme.\nNext, we derive the difference equations determined by ξV P A=DV P A◦ TLFL3. Because\nEqs. 2 and 3 are covariant equations, the target continuous s ystem in frame O′takes the\nform F′\nL3=TLFL3, i.e.,\ndx′\ndt′=p′\nγ′, (27)\ndp′\ndt′=E′+p′×B′\nγ′. (28)\nDiscretizing F′\nL3by VPA, the difference equation ξV P A=DV P AF′\nL3is given by\nx′\nk+1=x′\nk+ ∆t′p′k\nγ′k, (29)\np′\nk+1=p′\nk+V′, (30)\nwhere V′/parenleftig\nt′\nk+1,x′\nk+1,p′\nk,E′\nk+1,B′\nk+1,∆t′/parenrightig\n=W/parenleftig\nt′\nk+1,x′\nk+1,p′\nk,E′\nk+1,B′\nk+1,∆t′/parenrightig\n. It is obvi-\nous that ξV P A/negationslash=φ′\nV P A. That the VPA in [13] is not Lorentz covariant is therefore pr oved.\nBeing both Lorentz covariant and symplectic, an algorithm i s of significance in two as-\npects. In the first place, the preservation of the symplectic structure guarantees that the nu-\nmerical solutions are good enough to approximate the contin uous solutions in arbitrary long\ntime. Secondly, the Lorentz covariance of algorithm makes t he numerical results reference-\nindependent, which preserves the geometric nature of origi nal systems. Figure 1 depicts the\nschematic diagram for the relation between a covariant cont inuous system and the corre-\nsponding discrete systems generated by the Lorentz covaria nt symplectic algorithm A. The\n4-spacetime is denoted by MST. The continuous evolution of the original system forms a\nworldline, marked by Cwl, starting from the initial condition p0. The reference frames Oand\nO′are two chosen Lorentz inertial frames. For a covariant cont inuous system, the master\nequations F′inO′has identical form as FinO. According to the Lorentz covariance, the\nsolutions of FandF′, i.e., zτandz′τ, express the same worldline in MST. Given Ais a co-\nvariant algorithm, the corresponding discretized equatio ns in O′is expressed as ξA=DAF′,\n8Figure 1. Schematic diagram for the covariance of continuou s systems and the Lorentz covariant\nsymplectic algorithms. MSTis the configuration space of 4-spacetime. The worldline, Cwl, is\ndenoted by the black solid curve. The sequence of purple poin ts,pk, denote the discrete approx-\nimation of Cwl. Two Lorentz frames, OandO′, are chosen to express the Lorentz tranformation\nrelations. For covariant continuous system, FandF′have the same form, and their solutions in\ndifferent reference frames give the same worldline on MSTwith the initial condition p0. Similarly,\nfor Lorentz covariant algorithm A, the discrete systems φA=DAFandξA=DAF′have the\nsame form, and their results zkandz′kdescribe the same sequence pkonMST, if the Lorentz\ntransformation can be calculated exactly.\nandφA=DAFdenotes the discretized equations in frame O. The sequence determined\nbyφAinOis denoted by zk, and the sequence determined by ξAinO′in denoted by z′k.\nAccording to Eq. 1, we have the relation ξA=TLφAand thus z′k=TLzkfor each proper-time\nstep k. Consequently, as the analogy with the continuous case, the numerical results of A\nin different Lorentz frames provide different numerical solu tions zkandz′kbut the same\n4-worldpoint sequence pkinMST. On the other hand, because Ais a symplectic algorithm,\nthe conservation of discrete symplectic structure ensures pklocates adjacent to the exact\n9solution of the original continuous system CwlinMST, see the purple curve in Fig. 1. We\ncan conclude that the Lorentz covariant symplectic algorit hms have long-term numerical\nconservativeness, accuracy, and stability, which are inde pendent of the choice of reference\nframes.\nIII. CONSTRUCTION OF LCCSA\nIn this section, we introduce a convenient procedure for the construction of LCCSA.\nThe construction of an explicit LCCSA for relativistic dyna mics of charged particles is in-\ntroduced step by step for demonstration. This procedure can be generally applied for the\nconstruction of Lorentz covariant symplectic algorithms f or any other Lorentz covariant con-\ntinuous Hamiltonian systems. Since the Lorentz covariance should be preserved during the\ndiscretization, the geometric properties in 4-spacetime s hould be preserved. It is convenient\nto employ the Lorentz-covariant forms of the continuous sys tem to construct LCCSA.\nFirstly, write explicitly down the covariant Hamiltonian e quations for charged particles\nin 4-spacetime. The covariant Hamiltonian describing char ged particle dynamics in electro-\nmagnetic fields is [21]\nH=gαβ(Pα−Aα) (Pβ−Aβ)\n2, (31)\nwhere Xαis the 4-position vector, Pαis the canonical momentum 1-form, and Aαdenotes\nthe 4-vector-potential 1-form. In Cartesian coordinate sy stem, we have Xα= (t,x),Pα=\n(γ+φ,−P),Aα= (φ,−A), and\ngαβ=gαβ=\n1 0 0 0\n0−1 0 0\n0 0 −1 0\n0 0 0 −1\n,\nwhere Pis the canonical momentum, and φandAare respectively the scalar and vector\npotentials of electromagnetic fields. Before deriving the H amiltonian equations, one should\nnotice that the evolution parameters should be Lorentz scal ars, which is vital to keep the\nLorentz invariance of step-length after discretization. A s a direct consideration, we choose\nthe proper time τas the evolution parameter. Correspondingly, according to the Hamilto-\n10nian given in Eq. 31, we obtain the covariant Hamiltonian equ ations FL4[18, 21],\ndPα\ndτ=−∂H\n∂Xα=/parenleftig\nPβ−Aβ/parenrightig\n∂αAβ, (32)\ndXα\ndτ=∂H\n∂Pα=Pα−Aα, (33)\nwhere ∂α=∂/∂Xα= (∂/∂X0,∇),Pα=gαβPβ, and Aα=gαβAβ. It is readily to see that\nEqs. 32 and 33 are geometric equations and have reference inv ariant forms in all Lorentz\ninertial frames.\nSecondly, discretize the Hamiltonian equations by using a c anonical symplectic method.\nTo obtain an explicit scheme with high efficiency, here we choo se the Euler-symplectic\nmethod, which can be expressed by [9, 17]\nPk+1=Pk−h∂H\n∂X/parenleftig\nPk+1, Xk/parenrightig\n, (34)\nXk+1=Xk+h∂H\n∂P/parenleftig\nPk+1, Xk/parenrightig\n, (35)\nwhere his the step-length. The Euler-symplectic method does not br eak the geometric\nobject or the form of continuous equations. Combining Eqs. 3 2-35, we can obtain the discrete\nequations of the LCCSA φLCCSA =DLCCSA FL4as\nPk+1\nα=Pk\nα+ ∆τ/parenleftig\nPβ,k+1−Aβ,k/parenrightig∂Ak\nβ\n∂Xα, (36)\nXα,k+1=Xα,k+ ∆τ/parenleftig\nPα,k+1−Aα,k/parenrightig\n, (37)\nwhere ∆ τis the step-length of proper time. The difference equations, Eqs. 36-37, act as one-\nstep maps of geometric objects/parenleftig\nXα,k, Pk\nα/parenrightig\n/ma√sto→/parenleftig\nXα,k+1, Pk+1\nα/parenrightig\n. As a property of geometric\nequations, Eqs. 36-37 naturally inherit the reference-ind ependence of Eqs. 32-33. The Lorentz\ncovariance of the LCCSA can also be verified directly through the definition Eq. 1. The\nLorentz transformation of φLCCSA ,φ′\nLCCSA =TLφLCCSA , can be given by left-multiplying\nthe Lorentz matrix on both sides of Eqs. 36 and 37. Considerin g the linear relations of all\nthe terms in Eqs. 36-37, it is obvious to see that φ′\nLCCSA has the same form with ξLCCSA =\nDLCCSA ◦ TLFL4. Therefore, LCCSA satisfies the definition of Lorentz covari ant algorithms.\nDuring discretization, the Lorentz covariance cannot be in herited without keeping geo-\nmetric objects in 4-spacetime, even though the 4-dimention al covariant Hamiltonian equa-\ntions are used. To explain this, we provide a counter-exampl e, a non-covariant algorithm\n11(NCOVA) of Eqs. 32-33, namely, φNCOV A ,\nPk+1\nα=Pk\nα+ ∆τ/parenleftig\nPβ,k−Aβ,k/parenrightig\n∂αAk\nβ, (38)\nX0,k+1=X0,k+ ∆τ/parenleftig\nP0,k+1−A0,k/parenrightig\n, (39)\nxk+1=xk+ ∆τ/parenleftig\nPk−Ak/parenrightig\n, (40)\nwhere X0,P0, and A0denote the 0-components of Xα,Pα, and Aα, respectively. The one-\nstep map of Pαdetermined by Eq. 38 is the Euler method. In Eqs. 39-40, the 4- canonical-\nmomentum for pushing Xαis treated in different ways. When calculating X0,k+1,P0,k+1is\nused. And Pkis used to calculate xk+1. The integrity of 4-dimentional 1-form Pαin Eq. 33 is\nthus broken, which lead to different forms of Eqs. 39-40 after Lorentz transformations. The\nbad performance of this NCOVA under Lorentz transformation is presented in numerical\nexamples in Sec. IV, which shows numerically that TLφNCOV A /negationslash=ξNCOV A , where ξNCOV A =\nDNCOV A ◦ TLFL4.\nIV. NUMERICAL EXPERIMENTS\nIn this section, we analyze and test the performances of LCCS A through several numerical\nexperiments.\nA. The Lorentz covariance\nTo test the Lorentz covariance of algorithms, the motion of a n electron is simulated in\ndifferent Lorentz frames. The background magnetic field is gi ven by\nB=B0R\nR0ez, (41)\nwhich has the vector potential\nA=B0R2\n3R0eθ, (42)\nwhere R=√x2+y2,ezandeθare the unit vectors of cylindrical coordinates. The pa-\nrameters of field are set as B0= 1 T and R0= m 0c/eB0≈1.69×10−3m. We mark\nthe lab reference frame as O, where the initial condition of the charged particle is set a s\nx0= (0,2R0,0) and p0= (0,m0c,0). We then find another frame O′moves with velocity\n12βcor= (0.5,0,0) relative to O. Initially, the local time of the OandO′are both set to be\n0, and the origin points of OandO′coincide in 4-spacetime.\nIn the case of LCCSA, we first apply ξLCCSA inO′. Once obtained the numerical solution\nz′k\nξinO′, we transform it back to the frame Oto get the result of zk\nξ=TL−1◦ξLCCSA z′0,\nwhere z= (x,p). On the other hand, by using φLCCSA , we can get discrete solution\nzk\nφ=φCCSA z0inOdirectly. The orbits of the electron in the x-y plane are plot ted in Fig. 2,\nand the difference between the x-components of zk\nφandzk\nξis denoted by Dk\nx=xk\nφ−xk\nξ. It\ncan be observed that the numerical difference comes from calc ulations in different Lorentz\nframes is about 10−15m, which is in the order of machine precision. Meanwhile, Dk\nxis nearly\nindependent with the step-length, see Figs. 2c and Figs. 2f. It is shown in Fig. 2 that the\ndifference equations of LCCSA in OandO′, namely, φLCCSA andξLCCSA , can produce the\nsame results if the numerical error caused by the calculatio n of Lorentz transformation is\nneglected. As a result, the stability, convergence, and con sistency of LCCSA are reference\nindependent, which makes it safe to use LCCSA directly in diff erent frames.\nFor comparison, the relativistic VPA and NCOVA are also used to calculate the same\ncase. Because VPA is not a covariant algorithm as discussed i n Sec. II, if we calculate\nthe dynamics of a charged particle in O′by use of ξV P A, its results z′k\nξcannot be simply\ntransformed back to the results zk\nφgiven by φV P A inO, namely, TL−1z′k\nξ/negationslash=zk\nφ. In other\nwords, if observing in O, the VPA carried out in different reference frames TL−1◦ξV P Az′0\nandφV P Az0are actually two different algorithms with different propert ies and outputs.\nWith the same field configuration and initial conditions, the results from ξV P AandφV P A\nare shown in Fig. 3. If ∆ t= 0.1, see Fig. 3c, the position difference of orbits in Fig. 3a and b\nis in the order of R0∼10−3m. If ∆ t= 0.628, see Fig. 3e, TL−1◦ξV P Az′0becomes unstable\nand gives wrong numerical results. Similarly, the non-cova riant property of NCOVA is shown\nin Fig. 4. When applied in different frames, NCOVA also become s different algorithms and\nhence has different performances, see numerical results φNCOV A z0andTL−1◦ξNCOV A z′0in\nFig. 4a, b, d, and e. The Dxis also comparable to the value of R0and dependent with the\nstep-length, see Fig. 4e, f. According to Figs. 3 and 4, the no n-covariant problem results\nfrom the non-covariant algorithms in different Lorentz fram e are well exhibited.\n13-5 0 5\nx(m)×10-3-505y(m)×10-3\n-5 0 5\nx(m)×10-3-505y(m)×10-3\n0 0.5 1\nt(s)×10-9-101Dx(m)×10-14\n-5 0 5\nx(m)×10-3-505y(m)×10-3\n-5 0 5\nx(m)×10-3-505y(m)×10-3\n0 0.5 1\nt(s)×10-9-2-101234Dx(m)×10-15(b)\n(d) (e) (f)(a)\n∆τ= 0.628 ∆τ= 0.628 TL−1◦ξLCCSAz′0TL−1◦ξLCCSAz′0 φLCCSAz0\nφLCCSAz0∆τ= 0.1\n∆τ= 0.628∆τ= 0.1 ∆τ= 0.1\n(c)\nFigure 2. The comparison of simulation results are given by L CCSA in different Lorentz frames.\nSubfigures a, b and c are simulated with the step-length ∆ τ= 0.1, while subfigures d, e and f are\ncalculated by ∆ τ= 0.628. The difference between the results calculated in two fra mes is in the\norder of machine precision, which caused by the imprecision of Lorentz transformation instead of\nthe algorithm itself.\nB. The secular stability\nAll LCCSAs possess good long-term properties belonging to s tandard symplectic algo-\nrithms. The covariant Hamiltonian in Eq. 31, known as the mas s-shell, is a constant of\nmotion. Through conserving the symplectic structure, LCCS A can restrict the global error\nof the mass-shell under a small value [17]. For comparison, w e develop a Lorentz covariant\nbut non-symplectic algorithm, i.e., a fourth-order Runge- Kutta method (RK4), to solve the\n4-dimentional covariant Lorentz equations,\ndXα\ndτ=Uα, (43)\ndpα\ndτ=FαβUβ, (44)\nwhere pαis the 4-mechanical-momentum, Uαis the 4-velocity, and Fαβis the electromagnetic\ntensor [18]. We can see that RK4 is a Lorentz covariant algori thm because its discretization\n14-5 0 5\nx(m)×10-3-505y(m)×10-3\n-5 0 5\nx(m)×10-3-505y(m)×10-3\n0 0.5 1\nt(s)×10-9-1-0.500.511.5Dx(m)×10-3\n-5 0 5\nx(m)×10-3-505y(m)×10-3\n-0.03 -0.02 -0.01 0\nx(m)-505y(m)×10-3\n0 0.5 1\nt(s)×10-9-0.04-0.03-0.02-0.0100.01Dx(m)(b) (c)\n(d) (e) (f)(a)∆t= 0.1 ∆t= 0.1 ∆t= 0.1\n∆t= 0.628 ∆t= 0.628 ∆t= 0.628φV PAz0\nφV PAz0TL−1◦ξV PAz′0\nTL−1◦ξV PAz′0\nFigure 3. The comparison of simulation results are given by V PA in different Lorentz frames.\nSubfigures a, b and c are simulated with the step-length ∆ t= 0.1, while subfigures d, e and f are\ncalculated with ∆ t= 0.628. The position difference between the numerical results i n two frames\nis comparable to R0with ∆ t= 0.1. The VPA applied in the frame O′turns out unstable with\n∆t= 0.628.\ndoes not break the geometric structure of Eqs. 43 and 44.\nFigure 5 compares the evolutions of relative numerical erro r of mass-shell calculated by\nRK4 and the LCCSA. The electromagnetic field and initial cond itions are set the same\nas in Fig. 2, and the step-length is set to be ∆ τ= 0.1. After 2 ×106proper-time steps,\nthe relative mass-shell error of RK4 accumulates to a signifi cant value, which results in\nunreliable numerical results. However, the relative error of LCCSA keeps bounded in a\nsmall region due to its symplectic nature. According to this numerical experiment, Lorentz\ncovariant algorithms without secular conservativeness su ffer from coherent accumulation of\nnumerical errors, which implies the necessity to combine th e Lorentz covariance and the\nstructure-preserving methods.\n15-5 0 5\nx(m)×10-3-505y(m)×10-3\n-5 0 5\nx(m)×10-3-505y(m)×10-3\n0 0.5 1\nt(s)×10-9-6-4-202Dx(m)×10-5\n-5 0 5\nx(m)×10-3-505y(m)×10-3\n-5 0 5\nx(m)×10-3-505y(m)×10-3\n0 0.5 1\nt(s)×10-9-50510Dx(m)×10-4(b)\n(d) (e) (f)(a) (c)∆τ= 1e−4 ∆τ= 1e−4 ∆τ= 1e−4 TL−1◦ξNCOV Az′0φNCOV Az0\nφNCOV Az0∆τ= 1e−3 TL−1◦ξNCOV Az′0∆τ= 1e−3 ∆τ= 1e−3\nFigure 4. The comparison of simulation results are given by N COVA in different Lorentz frames.\nSubfigures a, b and c are simulated with the step-length ∆ τ= 0.0001, while subfigures d, e and\nf are calculated with ∆ τ= 0.001. The NCOVA is a 1st-order non-covariant algorithm and do es\nnot preserve symplecitc structure. The time step-length re quired for stability is much smaller than\nLCCSA. The inconsistence between the results calculated in different frames is comparable to R0\nthough with small time steps.\nC. The energy-based adaptive time step\nThrough the discretization of the proper time τ, LCCSA also possesses built-in energy-\nbased adaptive-time-step property. The discrete relation between ∆ τand ∆ tcan be reflected\nby the 0th component of Eq. 37 as\n∆t=tk+1−tk= ∆τ/parenleftig\nPk+1\n0−φk/parenrightig\n, (45)\nwhere P0is the 0th component of canonical momentum, and φis the electric potential.\nConsidering that the expression in the bracket of Eq. 45 can b e rewritten as Pk+1\n0−φk=\nγk+1+φk+1−φkand ∆ φ=φk+1−φkgenerally is a small value, the time step ∆ tis\napproximately proportional to γk+1. Equation 45 is actually a discrete version of the relation\ndt=γdτ. For constant ∆ τ, the time step ∆ tcan be self-adapted according to the energy\n160 0.5 1 1.5 2 2.5\nSteps×106-0.200.20.40.60.81∆H/H0RK4\nLCCSA\nFigure 5. The evolutions of relative errors of mass-shell by RK4 and LCCSA. The step-length is\n∆τ= 0.1. The error of mass-shell given by RK4 becomes comparable to H0after 2 ×106steps,\nwhile the relative error is limited under a small value in the case of LCCSA.\nof particles. When a charged particle moves in an extern magn etic field, the gyro-period\nTce= 2πγm0/eBdetermines the smallest time-scale of the particle dynamic s. In simulations,\nto resolved the dynamical behaviors smaller the time-scale of gyro-period, the time step\nshould be restricted smaller than Tce. When considering the efficiency of computation, too\nsmall time step brings heavy computation consuming. One sho uld choose a suitable ∆ tto\nbalance the accuracy and the efficiency. Because Tceis proportional to γ, for algorithms\nwith fixed time step ∆ t, time steps lie in one gyro-period grows as the increase of γ, which\ncause the waste of calculation resources in problems with in creasing γ. On the other hand, if\nthe particle loses energy quickly in some processes, Tcemay drop to smaller than ∆ t, which\nresults in numerical instabilities for algorithms. Howeve r, the time-step problems can be\navoided easily by using the LCCSA.\nTo show the advantages of the energy-based adaptive time ste ps, the acceleration and\nbraking process of an electron is simulated in a uniform magn etic field. We compare the\n170 20 40 60 80 100\n∆K/K000.511.522.53Steps×106\nVPA\nLCCSA\nFigure 6. The numbers of simulation steps in terms of the rela tive increase of energy required by\nVPA and LCCSA to simulate the same acceleration process of an electron.\nperformance of LCCSA with VPA which has a fixed time step [13]. Both the electric and\nmagnetic fields have only z-component, namely, B=B0ezandE=E0ez. In the acceleration\nprocess, the particle is released at x= 1.8 m,y=z= 0, the magnetic field is set as B0= 2 T,\nand the electric field is E0= 10000 V /m. The initial momentum of the electron is given by\np0= (0,1 m 0c,0.1 m 0c). Figure 6 shows the number of steps iterated by VPA and LCCS A\nin terms of different relative increments of kinetic energy. As the increase of the energy, the\nslope of red curve keeps unchanged, while the slope of blue cu rve decreases significantly, see\nFig. 6. Therefore, to reach the same energy, the computation efficiency of LCCSA is much\nbetter than VPA. In the case of braking process, the initial p osition and the magnetic field\nare the same as before, the electric field is set as E0= 1 MV /m, and the initial momentum\nis given by p0= (0 ,1 m 0c,−10 m 0c). Figure 7 depicts the number of time samplings\nduring each gyro-period. The sampling number of VPA in one gy ro-period decreases as the\ndecrease of energy due to the fixed time step, while the time sa mpling number of LCCSA\nkeeps unchanged. In this case, through adjusting the time st ep automatically, LCCSA can\nprovide higher accuracy than VPA and avoid numerical instab ilities in the simulation of\nenergy decrease processes.\n180 0.5 1 1.5 2\nt(s)×10-8010203040506070Samples Per Gyro-period\nVPA\nLCCSA\nFigure 7. Numbers of time steps in one gyro-period when emplo ying VPA and LCCSA to simulate\nthe same decelerate process of an electron. As the decrease o f the energy, the number of time-\nsamplings in one gyro-period for LCCSA keeps unchanged, whi le the number of time-samplings for\nVPA decreases.\nV. CONCLUSIONS\nIn this paper, we provide the definition of Lorentz covariant algorithms and introduce\nLorentz covariant symplectic algorithms in detail. Lorent z covariant algorithms can generate\ndiscretized equations, which inherits the Lorentz covaria nt nature of original continuous\nsystems. Symplectic algorithms without Lorentz covarianc e only performs well in one specific\ninertial frame. While covariant symplectic algorithms are reference-independent and possess\nlong-term conservativeness, which make it convenient and s afe to employ the same algorithm\nin any Lorentz frame. Because of the essentiality of Lorentz covariance, the Lorentz covariant\nsymplectic algorithms have wide applications.\nOn the other hand, because the time-variable becomes a compo nent of coordinate for\n4-spacetime in the construction of LCCSA, the time-depende nt Hamiltonian system is no\nlonger a problem for the construction of required symplecti c algorithms. Taking the proper\ntime τas the dynamical parameter, all time-dependent Hamiltonia n system becomes proper-\ntime-independent. The explicit symplectic algorithm, lik e the LCCSA in Eqs. 36-37, for time\ndependent systems can be easily constructed. According to t he idea and procedure in this\n19paper, many other Lorentz covariant symplectic algorithms as well as other kinds of Lorentz\ncovariant structure-preserving algorithms can be readily constructed. In the future work,\nwe will further investigate the Lorentz covariant structur e-preserving algorithms and apply\nthe LCCSAs to study key physical problems.\nACKNOWLEDGMENTS\nThis research is supported by National Magnetic Confinement Fusion Energy Research\nProject (2015GB111003, 2014GB124005), National Natural S cience Foundation of China\n(NSFC-11575185, 11575186, 11305171), JSPS-NRF-NSFC A3 Fo resight Program (NSFC-\n11261140328), Key Research Program of Frontier Sciences CA S (QYZDB-SSW-SYS004),\nand the GeoAlgorithmic Plasma Simulator (GAPS) Project.\n[1] H. Qin and X. Guan, Phys. Rev. Lett. 100, 035006 (2008).\n[2] H. Qin, X. Guan, and W. M. Tang, Phys. Plasmas 16, 042510 (2009).\n[3] J. Li, H. Qin, Z. Pu, L. Xie, and S. Fu, Phys. Plasmas 18, 052902 (2011).\n[4] X. Guan, H. Qin, and N. J. Fisch, Phys. Plasmas 17, 092502 (2010).\n[5] J. Liu, H. Qin, N. J. Fisch, Q. Teng, and X. Wang, Phys. Plas mas21, 064503 (2014).\n[6] J. Xiao, J. Liu, H. Qin, and Z. Yu, Phys. Plasmas 20, 102517 (2013).\n[7] J. Xiao, J. Liu, H. Qin, Z. Yu, and N. Xiang, Phys. Plasmas 22, 092305 (2015).\n[8] J. Xiao, H. Qin, J. Liu, Y. He, R. Zhang, and Y. Sun, Phys. Pl asmas 22, 112504 (2015).\n[9] H. Qin, J. Liu, J. Xiao, R. Zhang, Y. He, Y. Wang, Y. Sun, J. W . Burby, L. Ellison, and\nY. Zhou, Nucl. Fusion 56, 014001 (2015).\n[10] J. Liu, Y. Wang, and H. Qin, Nucl. Fusion 56, 064002 (2016).\n[11] Y. Wang, H. Qin, and J. Liu, Physics of Plasmas 23, 062505 (2016).\n[12] Y. He, Y. Sun, J. Liu, and H. Qin, J. Comput. Phys. 281, 135 (2015).\n[13] R. Zhang, J. Liu, H. Qin, Y. Wang, Y. He, and Y. Sun, Phys. P lasmas 22, 044501 (2015).\n[14] Y. He, Y. Sun, J. Liu, and H. Qin, J. Comput. Phys. 305, 172 (2016).\n[15] Y. He, H. Qin, Y. Sun, J. Xiao, R. Zhang, and J. Liu, Phys. P lasmas 22, 124503 (2015).\n[16] R. Zhang, J. Liu, Y. Tang, H. Qin, J. Xiao, and B. Zhu, Phys . Plasmas 21, 032504 (2014).\n20[17] E. Hairer, C. Lubich, and G. Wanner, Geometric numerical integration: structure-preserving\nalgorithms for ordinary differential equations , vol. 31 (Springer Science & Business Media,\n2006), ISBN 3540306668.\n[18] J. D. Jackson, Classical electrodynamics , vol. 3 (Wiley New York etc., 1962).\n[19] H. Qin, Report, Princeton Plasma Physics Lab., Princet on, NJ (US) (2005).\n[20] H. Qin, R. Cohen, W. Nevins, and X. Xu, Phys. Plasmas 14, 056110 (2007).\n[21] H. Goldstein, Classical mechanics (Pearson Education India, 1965), ISBN 8131758915.\n21"    },    {        "title": "1106.1135v2.Frictional_damping_in_radiative_electrodynamics_and_its_scaling_to_macroscopic_systems.pdf",        "content": "Frictional damping in radiative electrodynamics  \nand its scaling to macroscopic systems  \n \nD. Das* \n(Bhabha Atomic Research Centre  \nTrombay, Mumbai 400 085.  \n \n (e-mail: dasd 1951 @gmail.com ) \n \nAbstract  \n                Radiation force in Abraham -Lorentz -Dirac equati on is revisited for possible \nsignature of irreversible action in the dynamics. The analysis shows that the classical \nelectron can ‘dissipate’ out a certain fraction of field energy that distinguishes itself from \nthe well known Larmor radiation loss. The thermal power loss is shown to follow bi -\nquadratic acceleration functionality , which is akin to the characteristic s of Hawking -\nUnruh radiation emission from warm surrounding fi eld of a non -inertial observer.  \nReversibility in nonstationary evolution is possi ble at the expense of power from \nconcerned external field . By revealing  nonlocal mitigation characteristics in \nnonstationary evolutions , a measure of dissipative relaxation  in the radiative \nelectrodynamics is worked out to compare the two distinctly differ ent modes of energy \nlosses. The measure is shown to be applicable uniquely in all scales of externally \nperturbed systems undergoing nonstationary dynamics , and is used as a common \nthread to explain frictional contributions of phonons and electrons reported  for metals in \nsuperconductive phase transitions .  \n \nKey words: Abraham -Lorentz -Dirac equation, radiation force, Hawking -Unruh \nradiation, frictional dissipation.  \nPACS: 45.20. -d, 03.30.+p  \n---------------------------------------------------------------------- -----------------------------  \n*(Presently, superannuated from the official position)  \n \n \n \n \n \n \n  \nFrictional damping in radiative electrodynamics  and its scaling to \nmacroscopic systems  \n \n(D. Das ) \n \nIn macroscopic systems dynamic friction is a common phenomenon wh ere one body \nsliding over other experiences resistance. The dynamic resistance due to friction is expressed \nempirically with time asymmetric term proportional to the sliding velocity (\nv ). The motion is then \ndescribed by \n0 externalv = F - κv m\n , where \n0m , \nexternal F  and \nκ  are the particle’s mass, external force \nand frictional constant respectively; \nv\n  being observable acceleration. The quanti ty, \n2κv / 2  \ncorroborates to frictional power loss. Current approach in understanding the frictional property is \nbased on case specific modeling and numerical simulation analyses. Recently, there is \nsignificant progress in measuring the friction property using nano -tip electromechanical probes  \n[1,2,3 ]. In this study, the frictional power loss is generally described by considering decoherence \neffect in the sliding operation of one solid surface (probe) over the other (specimen). The drivin g \nforce in sliding torsionally perturbs the surface phonon modes to result in decoherence, where \nthe transversely driven modes are significantly perturbed and undergo critical damping. \nConsidering the limiting counteraction characteristics of omnipresent n onlocal defense against \ncritical perturbation, it has been possible to generally express the driving force in terms of the \nbasic properties of phonon modes. This helps defining the damping power loss as well as the \neffective frictional constant.  \nBackgroun d \nQuantum mechanics has established that dynamic interaction of field and particle in \ncoherent evolution is nonlocally mediated. It further points out that the nonlocally mitigated \ncounteraction to external perturbation has a limit beyond which the coheren tly evolved system \nundergoes decoherence. Considering the case of electrodynamics this study shows that at the \ncriticality of decoherence the omnipresent nonlocal field in its incessant effort to reinstate \ncoherent evolution counteracts perturbation and re laxation occurs as a result of stress yield at \nthe supersession of perturbation over the nonlocal defense ( details given in the appendix).  The \nmitigation limit in critical counteraction is registered in the radiation recoil effect to perturbation. \nConsider ing universality of nonlocal mitigation in decoherence, it is envisaged that the obtained \nmeasure of critical reaction to perturbation can lead the way of its generalized description in \nmacroscopic cases such as in the dynamic friction  where the coherent p honon oscillations are \nperturbed leading to dissipation loss under damped oscillations , besides the regular loss of \nsonic power . As elaborated below, the r elaxation features  in dynamic friction is quite similar to \nthose in electrodynamics for which the req uired formulation is comparatively easy  because of its \nrich classical and quantum mechanical bases. Before describing the general formulation, the \ndecoherence characteristics in radiative motion of electrodynamics is briefly mentioned here.  \nJerk force influencing upon radiative dynamics of an accelerated charged particle \n(charge, \nq  and mass, \n0m ) involves the two mutually orthogonal components, 2 3 2 3(2 / 3 )v (2 / 3 )[ve + ve]q c q c\n; \nv d(ve)/dt\n , \ne being unit acceleration vector, \ne v / v=\n , \nand \ne\n  being angular rate arising from torsion of the normal,\ne . The direct component in \nundergoing displacement results in a ccelerated growth rate of the kinetic energy (\n2\n0 k  v / 2 m ) \nas \n23(2 / 3 )(ve v) = qc •\n2 3 2 3 2\n0 [(2 / 3 )k - [(2 / 3 )v ve•v]]q m c q c +\n . The equation also indicate s \nthat the accelerated growth is impeded by two power loss  terms:  one due to the well recognized  \nLarmor radiation loss while the other from  reversed displacement rate (\n-v ) of the torsion based \njerk component, \n23(2 / 3 )veqc\n . However, when displacement operation is carried out on direct \nas well as transverse compo nents of jerk force, the two torsion based power terms  cancel  out: \n2 3 2 3 2\n0 (2 / 3 )v v (2 / 3 )[(k / ) v ]q c q c m•  −\n. This is anyway expected as  the transverse force \ncomponent of jerk contributes to disordered power loss which cannot figure in  the displacement \nbased  energy cons ervation analysis .  \nHowever, i t is to be recognized that the reverse displacement of the charged particle \narises from the radiation recoil effect due to the rate of momentum loss \n2 2 4\nrad (2 v / 3 )nqc\n  of \nLarmor radiation; \nradn ( ) / E H EH , is the radiation vector, (\nradn  being unit radiation vector, \n2\nradn1=\n),  \nE  and \nH  (with \nEH= ) being transversely oriented electric and magne tic field \ncomponents in the emitted radiation. Impact due to radiation recoil on mass \n0m  of the charged \nparticle generates a velocity component given by, \nrecoilv=\n2 2 4\n0 0 rad 0 0/ (2 v n / 3 ) /m q c m− −\n , \nwhere \n23\n00(2 / 3 )q m c  is time involved in  recoil -impact, and \n\n  is the rate of momentum loss \nIn Larmor radiation. It is interesting to note that the implied power loss, \n2\n00/m\n , is having \n4v\n  \ndepe ndence, which is different in nature as compared to the \n2v\n  depende nce of Larmor radiation \nloss. The recoil based power is functionally distinct from th e one involved  in the power balance \nequation so far revealed in electrodynamics.  The conventional approach of electrodynamics as \nsuch cannot sight the possibility for  thermal power loss. This could be rooted to the absence of \nany consideration on dynamic balance in between the two mutually opposing transverse forces \narising from jerk and radiation recoil. The unexplored consequence  of radiation recoil effect \nneeds further attention as it might have underlying connection with Hawking -Unruh type \nradiation emission  [4,5]. It is important to mention here that in the special case of uniform ly \naccelerated charged particle although jerk has null expectation value (\nv = 0\n ), the transverse \nforce component, \n23\n00 (2q /3 ) ve ve cm\n , has finite magnitu de in the dynamics  which is of \ncourse nonstationary ; finite torsion (\ne\n )  results from evolution of unit acceleration vector in the  \ncritically perturbed motion with \nd e /dt 0= , \nde/dt 0 . In general, \ne\n  evolves as \nk k k\nke c (-i ω )exp(-iω t)=\n, \n*\nkk\nkc c =1 . Electrodynamics  with constant  expectation value of \nacceleration allows its evolution entail ing internal frequency characteristics related to \n1\n0− . And \nthe relaxation losses remain unabated in the nonstatio nary evolution with the observable \nuniform linear acceleration.  Torsion , \ne\n, the pertinent property in disordered relaxation  gets established on the polarized vacuum field inherently existing around the point -like charged \nparticle.  The spherically polarized field having dynamically relevant radius of \n22\n002 / 3r q m c=  \nexperiences torque stress of \n23(e/ ) (2q /3 )ve F c c= \n , (\nr0[ n v]orm \n , \nrn being unit radial \nvector)  in the accelerat ed dynamics .  \nAs elaborated in the appendix, the dynamic evolution aspect of transverse jerk \ncomponent is incessantly mitigated by the omnipresent nonlocal field and the mitigation \ncharacteristics at critically perturbed dynamics holds the key for the regu lar (Larmor) and \ndisordered relaxation losses. While the recoil from regular relaxation signifies the manifested \nnonlocal reaction effect to critical torsional perturbation , the disorder ed one endorses \nspontaneity in decoherence.  These aspects of nonlocal mitigation characteristics could be \nrevealed by reformulating the electrodynamics purely from classical framework but with the \nadditional measure of optimization of radiation loss/gain in a dynamic course, which is missing \nin the conventional analysis. When the analysis is based solely on energy -momentum \nconservation in the radiative interaction of field and (charged ) particle, the problem is not closely \ndefined because of no measure on  the power in or out of the interaction volume . In absence of \nthe measur e, power  expense of external source in a dynamic passage is not optimized and left \nopen.  The incomplete definition leads to issues like a cceleration runaway or generally, in \nacausality. Classical formalism  being blind to nonlocal interplay of nature, restr icts its analysis \nfor the field -particle interaction  at the radiating boundary  (event horizon)  at a finite radial \ndistance (\n0r>>r ) from the accelerated charge . As consequence, the analysis could not sight the \nnonlocal criterion for disordered relaxation . Quantum field theory does recognize the disordered \nfeature  using  accelerated  frame  commoving with the charge. According to the commoving \nobserver the surrounding vacuum field of the accelerated charge is thermal ly populated over \nground  and excited states ; population corroborat ing to the thermal state predicted by Hawking \nand Unruh. QFT has not attempted to reveal  criticality aspect of the nonlocal mediation in field -\nparticle interaction.  \nAs worked out in the appendix, r adiative relaxat ion occurs in the presence of critical \nperturbation by transverse force, whe re nonlocal field attains its limiting defense failing to g uard \nthe coherent dynamic state. Relaxation in fact registers the resulting loss of coherence . The \ncriticality is express ible as \nrad e = v/c ncr cr\n ,  (\nlim\nrad / 1 wn  nwc→= ,  \nwn  being phase velociy \nvector) . It gives a measure of critical torsion that is intrinsically involved in radiation reaction. \nThe manifested torsion leads to  therm al relaxation loss in electrodynamics, which occurs along \nwith the regular (Larmor) relaxation at the expense of power from external supply. Thermal \nrelaxation is the result of critically damped oscillations of internal modes of the polarized sphere \nassoci ated with  accelerated charge as the  modes  are driven by the transverse force. Unlike the \nregular relaxation, the thermal effect occurs without any work signature. Using standard \nanalytical results of critically damped oscillation [6] driven by a transverse  force \nF  one finds \nthat energy relaxation occurs at the rate of \n-1 2\n0 0 relax (2 ω ) E (say) F  =\n ; \n0  and \n0ω  are \nrespectively the oscillator’s effective mass and frequency, \n  the damping parameter which is \nunity in this case of driven oscillation, and \n21\n0 [(2 / ) 1]  −=  +  is the off-resonance coupling \nefficiency of the driving force, (\n0 rad    −  being deviation of the driver’s frequency from \n0ω ; ~\n1 for near resonance cases (\n0  ), and \n ~\n1/ 5  for far off resonance cases \n(\n0~ ). For electrodynamics, since the rate of change of radiation momentum gives the \nmeasure of transverse force one writes \n2 4 2\nrad (2 / 3 )v nF q c= \n , the thermal power loss \nrelaxE\n  \ncan be rewritten as \nrelax 0 [( / ) / ] Em   = • \n , where, \n0 0 0 recoil/ / vmm = \n  is the \nmagnitude of recoil displacement rate , and \n2\nLarmor rad 0 0 max rad( / )n ( / )(v/v ) nE c m c  = \n . It can \nbe seen that the thermal power  in torsionally driven damping is expressible as  \n4 2 6\nrelax 0 [ v / 6 ] E S c  =\n, where \n2/ ~ 1/137qc\n  is the fine structure constant, and \n0S\n(\n22\n0 4c= ) is the surface area of the dynamically significant sphere of radius, \n0c . \nrelaxE\n has \nthe bi-quadratic functionality in a cceleration  indeed . Considering that the relaxation results in \nthermal radiation emission uniformly over the entire \n4  solid angle of the polarized sphere, the \npower density expression can be rearranged as \n4\n0 ( / ) [ (160 )][ ( v / 2 ) ]relax BE S c     =\n , \nwhere \n2 4 3 2 = / 60B c   \n , is the Stefan -Boltzmann constant. The power density corroborates to \nblack body emission temperatre of  \nradT ( )K= \n1/4\nB (160 ) [ v / 2 ] c   \n1/4\nB 1.04 [ v / 2 ] c   \n, \nwhere, \nB  is Boltzmann constant. \n  assumes unity at the highest possible nonlocal defense, \nwhich is at the critical acceleration of \nmaxv\n .  With lowering of acceleration, \n  reduces and \nconv erges to the lowest limit of 1/5, (\n1/ 5 1 ). Nonetheless, \nradT  has semblance  to \nHawking -Unruh radiation temperature [4,5]. \nNow, t he general form of transverse force \nF  can be obtained by co nsidering the \noscillation characteristics of the relevant internal modes. As for electrodynamics the dynamically \nrelevant time, \n00 /rc= , corroborates to the internal oscillation frequency as \n00ω 2 / 4= , \nwhere \n04  expresses time period of one full oscillation that covers sweeping across the \ndiametric stretch of the polarized sphere. As described already, the dynamic evolution involves \nthe transverse force, \ne/ Fc=\n , \nr0[ n v]eorm \n .  \nF  can be rewritten now using the internal \noscillation frequency (\n0ω )  as  \n                                          \n00 ( / 2)( / )vecr Fm=\n                       (1).  \nThe force expressi on (1) now involves the characteristic mass, internal mode frequency and the \ndynamic feature of torsionally stressed polarized sphere. As described below, the expression \nhelps formulating the transverse force in torsionally perturbed phonon modes involved in \nfrictional phenomenon.  \nBasic formulation  of damping power loss in dynamic friction  \nNoting universality of incessantly mitigating nature of nonlocal field to optimize energy \nlosses in critically perturbed dynamics, the relaxation loss in frictional phe nomenon is \nformulated by considering decoherence characteristics under the critical mitigation. Regular \nrelaxation in frictional dynamics across a pair of interacting surfaces occurs through \nmanifestation of sonic energy. In addition to the power loss thro ugh sonic propagation there is finite thermal relaxation through the driven damped oscillations of the surface phonon modes. \nTo formulate the frictional damping property in the sliding course of one solid over the surface of \nanother one, it is necessary to  define the variables involved in defining decoherence associated \nwith the oscillating phonon modes. Perturbations in the sliding process involve direct driving of \nthe surface phonon modes and also, indirectly through electron plasma oscillation of conduct ion \nelectrons of the surfaces concerned; the latter one is significant in the cases of electronic \nconductors like metals/alloys. Driving forces for the direct and indirect ways of mode \nperturbations are to be defined by considering the gen eral form of tran sverse force in \nexpression( 1) involv ed in decoherence. Thus using ( 1), the damping relaxation can be \nexpressed: \n-1 2\nrelax 0 0E =(2 ) F \n , where, \n00 ( / 2)( / ) vecr F  =\n , \n0  and \n0  respectively \nbeing t he reduced mass and frequency of the mode concerned. The acceleration (\nv\n ), and the \ntorsion rate (\necr\n ) of phonon modes as involved in \nF  are to be specifically defined for the \npertur bations. In frictional measurements, the off -resonant coupling efficiency (\n ) of torsional \nperturbation in surface sliding of solid objects falls under the far off resonance cases, \n~ 1 / 5 . \nThis is because the mode oscillation involves significantly high frequency (\n0 ) as compared to \nthose associated with perturbation in the surface sliding operation. Thus, \n-1 2\nrelax 0 0E = (10 ) F\n .  \nIn the sliding course of friction monitoring probe o n the surface of a solid specimen, \ndriving force imparts torsional oscillations of phonon modes of the contact area (in x -y plane, \nsay). In surface oscillation, two orthogonal modes respectively along x and y directions of its ith \nlattice point are involve d according to their coupling angles with the probe movement.  \nOscillation of surface mode along z direction being not as active as the two other modes of the \nhalf space, its effect is generally not considered. But in the description of transversely driven  \noscillations, it cannot be neglected and it is to be separately considered. The sliding direction \nmaking \nx  and  \ny  angles with the respective modes leads to torsional oscillations of the two \nmodes in x -y plane; \n/2xy  += . With sliding velocity of \nV , the overall torsional \ndisplacement rate of the x and y modes is expressible as \n,ˆˆ e  = (V/Q )[i cos( ) i cos( )]cr i i x x y y +\n , \n(\ni i iz x y= ; \n2 2 2i i i 1x y z= = = ), where \nQi  represents oscillation amplitude (\niz  being unit vector \nrepresenting the x -y plane). Therefore, \n22\n,ie (V/Q )cr i=\n . The above analysis suggests that \neffectively one of the two surface modes  (x- and y -mode) contribute to torsional oscillation. \nSuperscript ‘\ncr ’ indicates criticality in torsional oscillations of the phonon modes in the surface \nsliding process, where the acceleration magnitude involved in the harmonic os cillators is \nrepresented by \n2v  Q =- Q\n . Thus, the product \n(ve )cr\n  has the magnitude of \n2V . Modes are \nconsidered as degenerate with fundamental frequency, \n , and all lat tice points are treated to \nbe equivalent. Recalling the expression of thermal loss in damped oscillation which is critically \ndriven by the force \n0 ( / 2)( / ) vecr F  =\n  one writes the relaxation rate in an active mode as \n-1 2\n0 E = (10 ) F\n22\n0 ( / 40) V  =\n. As indicated, \nE \n  effectively represents thermal power loss \nfrom x and y modes of a lattice point. Besides this contribution of the surface modes, sliding \naction on contact plane of a specimen of finite thick ness can torsionally perturb the mode along z direction as indicated already. The damping loss to this additional mode need not double the \nvalue expressed above. For this presentation, the half -space contribution from z -mode will be \nconsidered later as 50%  of \nE\n  as an approximation.  \nUnlike the motion of an accelerated charge, the thermal relaxation in sliding motion of \nmacroscopic objects can be significant even at nominal sliding speed (\nV ). This is due to  the \nfact that oscillating phonon modes of the two surfaces in dynamic contact are huge in numbers. \nAssuming the simpler case of equivalent lattice points, each point contributing one x -y mode, \nthe rate of power loss per unit area over a sliding surface is  generally expressed by considering \ndamping loss over the lattice points having surface density \nsN  as \n()sNE\n , \nE\n being damping \npower loss from a lattice point. being Considering the reduc ed mass \n0  of a lattice point, and \ncoupling efficiency \n  of the two surfaces in contact in the sliding process one writes, \n22\n0 E =[( / 40) V ] \n. Surface roughness together with external load applied  normal to the \nsurface decides the coupling. Torsional displacement rate is very much dependent on \n . \nStiction and wear are common problems in nano -tip electromechanical probes used in the \nfriction measurement [2]. Considering tha t each of the modes can vibrate with a frequency \ndistribution from zero to a maximum value, that is, \n0m , the above expression of \ndamping loss per unit area needs to be modified by using phonon density of states on surface \nwith freq uency of \n  as \n2( ) (1/ ) / vs g  = , (\nvs  being the speed of stress wave ) [7]. Since out \nof the two distinguishable surfaces modes (longitudinal and transverse), one is effectively \nperturbed by torsional ly driven force, one evaluates the maximum value of actively involved \nmode frequency,\nm , by normalizing the total surface modes per unit area as, \n0()m\ns g d N\n= . \nOn integration of RHS, the normalization leads to \n1/2 1/22vm s s N = .  \nThe power loss per unit area is thus expressible as \nfriction\nunit areaE=\n  \n2 2 2\n0\n0[( / 40) V ] ( )m\ngd\n    \n, where the reduced mass (\n0 ) and surface coupling efficiency \n(\n) are taken to be identical irrespective of the active modes. The integral expression on \nsubstitution of \n()g leads to, \nfriction 2 3\nunit area 0 ( /120) ( V/v )sm E   =\n , \n being the coupling efficiency \nof the sliding surface, which is the measur ement probe of friction property. Using the \nm  value \nin this result one gets the power loss as \nfriction 2 3/2\nunit area 0 0.412 ( V) vss EN =\n . The frictional power loss \nper unit area can be used now to evaluate the average damping constant in the oscil lation of a \nmode as \n2 1/2\n0 0.82[ v ]ssN    , where the sliding speed of probe is taken as 1cm s-1. As \ndiscussed an additional 50% over this result is to be considered because of contribution from z -\nmode in the half -space (sample thickness). Thus, the da mping const per lattice point is \n2 1/2\nlatticepoint 1.23[ v ]ssN  \n Thus, for Nb lattice (bulk density, 8.57 gcm-3, and atomic weight, \n92.9064 a.m.u.), the \nsN  value is 1.4759x1015 cm-2, and sound speed is 348000 cms-1; the frictional coeffi cient in phonon mode in the Nb lattice works out to be \nlattice pointNb ~\n12 22.5 10− kgs-1, \n1\n. The measured value, if not corrected for the surface -coupling efficiency of measuring \nprobe, will be lower tha n \n122.5 10−  kgs-1. For silicon lattice (\nμ  28.0855 a.m.u.\n , \nsv= 843310 \ncm s-1, and   \nsN =1.35x1015 cm-2) which is generally used as a probe in the measurement of \nthe friction, the f riction constant can be estimated. The frictional coefficient per phonon mode in \nthe Si lattice works out as \nlattice pointSi ~\n12 21.8 10− kgs-1.  \nIn metallic lattice, there are additional components of dissipative losses in the sh ear \naction of sliding the friction monitoring probe over specimen surface. The shear force perturbs \nthe oscillating plasma of conduction electron, which in turn can torsionally perturb active modes \nof positively charged ion cores representing the lattice p oints. The charged cores undergo \noscillations in the plasma field of conduction electrons to be described as \n2\n0 [4 / ]c plasma q c plasmaQ n zq Q −−=−\n \n2\n00 [ ( / )]p c plasmaz m Q− =− , where \nqn , \nzq, \n0, \n0m , and \np  \nare respectively the number density of conduction electrons, core charge, reduced mass of the \ncore, electron mass, and plasma frequency, \n2 1/2\n0 (4 / )pq n q m= . Noting that each core is \nundergoin g harmonic oscillation with the unique frequency of \n1/2\n00[ ( / )]c plasma p zm   −= , the \naction of a shear force in orthogonal direction to the linear oscillations will lead to the frictional \ndissipation per lattice core atom as, \n22\nc-  0 = 1.5[( / 40) ( V) ]plasma c plasmaE   −\n , \n being the \ncoupling efficiency of the sliding surface with the friction monitoring probe. In \nc-  plasmaE\n , the factor \n1.5 accounts for the number of torsionally active modes per core atom; it is a approximation. \nThus,  the damping const per core atom is given by \n22\nc-  0 = 3( / 40)plasma c plasma   − . For the \ncase of Nb metal lattice \nqn  = 4.33x1022 cm-3 and \n0.79 0.03z= (at low temperatures) [8], and \none thus finds the value of \np  as  \n161.17 10p= s-1 and \n132.52 10c plasma−= s-1. Thus, for Nb \nmetal lattice the value of \n- c plasmaE\n corroborates to frictional coefficient of \nNb\nc plasma−\n~\n2 123.0 10− kgs-1. The f rictional contribution from electron plasma oscillation alone is \nsignificantly low (~8.77x10-15 kgs-1) and is not considered. The analysis on the whole provides \nestimates of frictional coefficients using basic solid state properties of matter. Following gi ves \ncomparison of a meticulously measured data with the results of this analysis.  \nThe estimated value of frictional constant due to electron plasma induced core oscillation \nin Nb surface, namely, \nc plasma\nNb− ~\n2 123.0 10−  kgs-1, \n1 , can be compared with the recently \nreported value of electronic contribution in Nb lattice [3]. In the superconducting phase \ntransition, the observed decrease in the friction property by about 3.7x10-12 kgs-1 is quite \ncompar able with the estimated value. Following the transition, the noted residual value of the \ncoefficient of about 1.75x10-12 kgs-1 is also quite close to the present estimate of the phonon \ncontribution in Nb lattice as \nphonon\nNb ~\n12 22.5 10− kgs-1. The disparity is evidently resulting from \nlow surface to probe coupling value, (\n~ 0.7 ). Appendix: Electrodynamics –in abridged version of classical and quantum  mechanics   \nIn the accelerated motion of a charged pa rticle in external electromagnetic field , the self \nfield loss that occurs  at the radiation boundary (\n classical radius of the charge) is the net \nresult of critical negotiation in  nonlocal ly mediat ed field-particle interaction leading to the  \noptimum relaxat ion as radiation . This negotiation aspect  is not consider ed in the conventional \nanalysis, because of which  the causally interconnected radiation relaxation events remain \nincompletely specified  in the classical description.  The incomplete ness  is reflected by the well \nmarked presence of unwanted acausal consequence like acceleration runaway under \nwithdrawal of external power  source  abruptly. The nonlocal mitigation involved in optimizing \nradiation relaxations remaining unaccounted , the power loss ultimately borne by the external \nfield is left open without a measure. Radiation related force designed solely on the basis of \nenergy -momentum conservation falls short of the measure required. The presented analysis \nproves that adoption of th e additional measure of optimizing radiation loss (or, gain) in the \ndynamic course  can describe electrodynamics in a modified form over the well known \ndescription by Abrah am-Lorentz -Dirac (ALD) equation . The modification of radiation related \nforce involves  a nonlocally mediated feature  governing relaxation loss in  the nonstationary \ndynamics under  external field. R adiation loss (gain) could be generally optimized in the dynamic \ncourse variationally defined in between arbitrarily selected pair of space -like s urfaces with their \ntime-like separation, only if the course is characteristically endowed with features of delocalized \nevolution instead of the classically defined course of a world path (line). Using the classical \nLagrangian formalism, the delocalized cou rse involving minimum radiation loss/gain in the \ndynamic passage could be described by linear combination of a set of infinitesimally differed \nstationary action paths \n()px  with nonlocally governed optimal displacement.  This is so as the \nminimum radiation loss (relaxation loss) within a dynamic course can be accomplished under \nthe criterion of optimized path displacement , which inherently involves entanglement of the \nvariational elements, \np xx ,  with instant 4-accelerations of the infinitesimally differed \npaths at any instant as   \n    \n2\n12 1/2()p p p\npc dx dx g\n\n\n    \n2\n2\n1\n12[( ) ( ) ] 0p p p p p\npc g v x g v x d\n   \n \n  −=  \n           (A1).  \nIn the displacement optimization equation ( A1), the integrated term as defined by the scalars, \n2\n,()p p p\npc v x\n\n at \n1=  and \n2= , are in fact variations of proper time  in the respective \ncases, which are anyway null by definition of the two time -like boundaries. The remaining last \nterm involves the entanglemen t of 4 -accelerations with the nonzero variation elements.  (For the \nflat space, entanglement is satisfied by a classical geodesic path expressible as \n0pv=\n ; \ngeodesic course supports free motion of particles). Thus f or general dynamic c ourses, the \nmeasure of minimized relaxation loss implied by e q.(A1), suggests that nonlocal mediation \ncanonically governs the weight factors \n2\npc  of the linearly combined paths such that the \ndynamically relevant  4-acceleration , \n( ) ( )pp\npv c v\n , (\np vv\n , \n21p\npc= ) corrob orates to the 4 -orthogoal connection  with the variation , \np xx , \npp\npx c x   as \n0 vx\n=\n , \n(\np p ppcc= , \npp  being Kronecker delta ). Replacing the set of variational elements with the \ncorresponding set of space -like unit 4 -vectors, \np ii , \npp\npi c i , one rewrites the 4 -\northogonal connecti on as \n0 vi\n=\n , (\n1 ii\n=− ). Since the nonlocal quantities, \ni  are \nindependent of displacement, one writes \n0 v i v i\n==\n . Furthermore, like 4 -acceleration and \n4-jerk one finds that  the canonically averaged 4 -velocity,  \np vv , \n( ) ( )pp\npv c v  is 4-\northogonal to a reduced form of the space -like unit 4 -vector,  \n() i i v v  \n− , (\nv i v\n  ). In \nessence, for meeting the minim um relaxation loss, the nonlocal mediation with dynamical \nquantities are governed by the following set of connectivities:  \n                       \n0 vi\n=\n ,       \n0 vi\n=\n , and      \n0 vi\n= ,  (\ni i v v  \n− )             ( A1a). \nThe orthogonal connection, \n0 iv\n= , suggests that the conventionally used radiation 4 -force, \n22(2 / 3 )[ ]q c v v v+\n needs modification by additional term involving  the nonlocal mediation \nfeature \ni . The modified 4 -force will endorse minimum radiation relaxation in the dynamic \npassage. The na ture of nonlocal mediation and dynamic feature of the envisaged radiation 4 -\nforce will be explored now.  \nConsidering  now the Langra ngian formalism,  the stationary path action is represented in \nthe following form:  \n        \n2\n2\n1\n122\n, , , [ ( / / ) ] [ ( )]p p p p p p p p p\nppc L x d d f x d c x\n  \n  \n        + − −\n0S= , \nwhere, \nppvx\n , \n( / )p p p Lv−   , the canonical 4 -momentum on path \np . Overall Lagrangian \nin the linear combination is \n( , ) ( , )p p p p\npL x v c L x v   = , \np LL . The 4 -vector, \npf  \nrepresents the radiation related 4 -force involved in path s \np  of the family.  Now,  with the \nconsideration of supplementary condition, namely, \n22\n, 1 , 2( ),  at  = ( ),  at p p p p p p\nppc x c x\n     \n, the delocalized dynamic course of the infinite set \nof minimally displacing paths \n()px  of stationary action is describable irrespe ctive of the \narbitrarily selected variational displacements, \nx  and boundaries (\n1 ,\n2) involved in the \nformalism as  \n                                     \n/ / 0L x d d f\n   + − =                                 (A2), \nwhere, \np ff . \nf  like the conventional 4 -forces being  space -like as \n0 fv\n= , eq.(A2) \nwill conserve the rest mass of the charged particle in dynamics. Fu rthermore, noting that  eq.(A2) endorses  that the 4 -projection, \nfx\n  is governed by that of the conventional terms as ,  \n( / / ) f x L x d d x  \n   =   +\n. Since,  the conventional terms,  \n( / / )L x d d\n   +  are local \nquantities havin g no components along \nx , it leads to   \n                                                                  \n0 fi\n=                                       (A2a).  \nFunctionality of \nf  require s detailed consideration of the nonlocal mediator’s \ncharacteristics, which is ingrained within the supplementary condition, which can be rewritten as  \n                                                       \n12( ) ( )ii\n   = = \nC (say)                    ( A2b).  \nThis equality is valid irrespective of the arbitrarily selected variation boundaries. The parameter \nC\n can be evaluated by considering that at the initial boundary, \n1  the external field \nasymptotically disappears and the particle assumes free motion. It is to be noted that for free \nparticle motion with constant 4 -velocity components, \npp\npv c v , (\n221p\npvc== ), the \nevolution equation , (eq.(A1a)), \n0 vi\n=\n  corroborates to invariant value of \nvi\n  irrespective of \nthe nonlocally evolved quantities,  \np ii . For free motion, the canonically averaged 4 -\nvelocity remaining unaltered, t he invariancy of \nvi\n  is accountable  only when the nonlocal \nevolution in the motion is expressed by the 4 -orthogonality, \n0 v i v x\n== . Using \n0[1, w / ] w p c\n, \n0w / i/ic= , \n0[i , i ]i , (\n0p  being a multiplier), the evolution can be rewritten \nas \n0 v i v w\n== ,  where, \nw  is nonlocal velocity, or, phase velocity that has magnitudes \nwithin, \nwc  . Since for free motion of a particle of mass \n0m , the canonical 4 -momentum is \nexpressed as \n0m cv , the mediator’s evolution can be equivalently represented by \n0i\n=\n. Now recalling back the supplementary co ndition, ( A2b), it is evident that the selected \nfree motion of concerned particle with asymptotic disappearance of the external field at the \ninitial boundar y \n1  is mediated by the evolution \n0i\n= . This makes  the parameter \nC  \ncharacterizing th e supplementary condition (1b) a s a null in general and this implies that \ndynamic evolution with minimum radiation loss corroborates to  the 4 -orthogonality   \n                                                         \n0 iw\n==                           (A2c).  \nEq.(A2c) speaks for an organized (coherent) evolution of the infinitesimally differed path family \n()px\n mediated by the nonlocally defined f eature \ni .  \nNow, as per equations in ( A1a) the mediating feature \ni  is having its own identity \nindependent of the local hyperplane of 4 -tangent, 4 -normal \ne , (\n2/ e v v−\n ), and with 4 -jerk \nv\n, which has 4 -binormal connection as \n2() v e v e e v   \n −  +\n , wherein the quantity, \ne\n  \ncorroborates to 4 -binormal, \n2() e e v v   − −\n , \n0 ee\n=\n , \n0 e v e v\n==\n . Besides 4 -\northogonalities with the 4 -tangent and 4 -normal, (\n0 iv\n= , \n0 ei\n= ), the feature \ni  is having finite projection on the 4 -binormal as \n2i e v v\n =− −\n ; nonzero projection on the local feature \ne\n confirms involvement of the nonlocal mediator in the dynamics.  It is now worthwhile to note \nthat the ALD 4 -force, \n22(2 / 3 )[ ]q c v v v+\n , (\n2v v v\n\n ), is desc ribed in the hyperplane of 4 -\nnormal and 4 -binormal as  \n22 (2 / 3 )[ ( )]q c v e e v e  \n−−\n . Considering this and recall ing the 4 -\northogonal connection, \n0 iv\n= , one can express radiation related 4 -force \nf  as \n2 2 2(2 / 3 )[{ ( )} ]f q c v e e v e v i    \n   − − +\n. The space -like 4 -force is  lying in the local \nhyperplane  of the 4 -normal, and 4 -binormal (\ne\n ; \n0 ee\n=\n ). The parameter, \n  gets normalized \nto \n2/ (1 )vv=+  by considering equation ( A2a), and also, the equalities: \n2e i e i v v\n   = =− −\n and \n2(1 ) i i i i v\n   = =− + . Thus, electrodynamics that follows from \neq.(1) is expressible as  \n  \n2 2 2\n0 ( / ) (2 / 3 )[{ ( )} ] 0 m cv q c F v q c v e e v e v i   \n      − − − − + =\n           (A2d). \nIn the delocalized represe ntation of electrodynamics eq.( A2d), the dynamic variables \ninvolved are to be recalled with their canonically averaged characteristics as was realized over a \nfamily of minimally displacing paths of optimum action in a dynamic course. The averaged \nrepresentat ion of a quantity \nX  will be generally denoted by \nX . The canonically averaged \nrepresentation is necessary to express the nonstationary dynamics which is critically mitigated \nby the nonlocal field  in attain ing minimum relaxation loss ; mitigating 4 -force term being \n22(2 / 3 )q c v i\n. The Lorentz 4 -force, \n( / )q c F v\n , being 4 -orthogonal to 4 -tangent (\nv ) and \nalso, to the nonlocal mediator \ni , it can be generally described in the hyperplane of 4 -normal \nand 4 -binormal. But its canonically averaged projection on 4-binormal ,\nF v e\n\n  when \nexpressed in an instant commoving inertial frame as \nEe−•\n  resul ts in null averaged value \nbecause of the freely evolving binormal vector \ne\n  under a given external field . In the averaged \nrepresentation, Lorentz 4 -force is oriented solely along 4 -normal ; \n( / ) ( ) ( / ) ( )q c F v q c F v e e  \n  \n. Thus , by cons idering this aspect and n oting that \nf  is \nmade of two mutually 4 -orthogonal components,  \n2 2 2(2 / 3 )[{ } ]q c v e v i−+\n  and \n2(2 / 3 )( )q c v e e\n −\n, the dynamics can be expressed  in the following two components:  \n                        \n2\n0 (2 / 3 )( ) ( / )( ) 0 m cv q c v e e q c F v\n   + − =\n,                 (A3), \n          and,                                            \n2 2 2(2 / 3 ) 0q c v e v i− + =\n                (A4). \nDynamic variable s in the component equation (A3) are oriented along 4 -normal. Eq.(A3) \nrepresent s dynamics in the presence of direct component of jerk 4 -force, which in the referred \nlocal frame is given by   \n23(2 / 3 ) veqc−\n . Its presence results in impediment of the inertial force, \n0vm\n  expressible as  \n0 0 maxv[1- dln(v/v ) /dt]m\n , where,  \n23\n0 0 max(2 / 3 ) / vq m c c=\n . \nElectrodynamics thereby involves modified inertial force, wherein the proper mass of the \ncharged particle ge ts renormalized to a lowe r value as \n0 0 max 0[1 d ln (v/v )/dt] m m m = − \n . The \nequality limit for the proper  mass holds for the special case of uniformly accelerated dynamics . \nIn this case, one can a lways find an instantly commoving inertial  frame to ensure that there is no \nchange of proper mass in the dynamics (\n0 mm= ). \nComponent equation (A4) represents dynamic balance of two 4 -forces respectively from \ntorsion based 4-jerk component and the nonlocally mitigating counteraction , \n22(2 / 3 )q c v i\n . \nThe feature of nonlocal counteraction to  external perturbation is missing in the conve ntional \ndescription. As mentioned  before the torsion al perturbation  ingrained in jerk evolution is the root \nof thermal relaxation (similar to  Hawking -Unruh effect ) in the nonstationary dynamics. I n the  \ninstant commoving inertial frame, the mitigating force  can be further simplified and the dynamic \nbalance (eq.( A4)) can be expressed as  \n2 3 2\nw (2 / 3 )[ ve v (n /w)] = 0qc −\n ,  (\n1 /w 0c ),  \nwhere \nwn  is the unit vector along the phase velocity, \nw . At cri ticality, the nonlocal reaction to \nthe torsional perturbation attains its highest value as \n/wc  approaches unity. Thus the critical \nbalance is represented by \nrad e = v/c ncr cr\n ,  (\nlim\nrad / 1 wn  nwc→= ,  \nradn  being radiation vector). \nThe result gives a measure of critical torsion that is intrinsically involved in radiation reaction. In \nthe changeover of coherent to incoherent evolution occurring at critical perturbation, the \nmanifested torsion lea ds to  dissipative relaxation loss in electrodynamics, which occurs along \nwith the regular (Larmor) relaxation  at the expense of power from external supply. Recoil \nmomentum from regular relaxation is now identifiable with the nonlocal reaction, \n2 3 2 limit 2\nw (2 / 3 ) v ( w/w )c qc→ −\n. For the special case of uniformly accelerated linear motion of a \ncharge (\nv0=\n ), the torsion component is inherently present as \nv  d(ve)/dt  = v e  ve+\n. Therefore, the relaxation losses remain unabated in the \nnons tationary evolution with the observable uniform linear acceleration. It is important to note \nthat t he presence of relaxation s abhors acceleration runaway as an alternative solution to the \ndynamics under abrupt withdrawal of external field, prohibiting ther eby acausality issue.  \nThe delocalized description of dynamics arrived in ( A2) could be rationalized after \nunderstanding the canonical governance of the nonlocal mediator \ni  expressed by the \nsupplementary condition, \n0 iw\n== . Now, recognizing  that the velocity -like 4 -vector , \n[1,w / ] wc\n, \n1 /w 0c , corroborates to the phase velocity, its 4 -components in the \nreciprocal space  coordinates  is expressible as \n0k[1,ω / k]kw p c , where \nk[ω / , k] kc , \n(\n0\nkω / kc , and \n0p  is a multiplier).  \nkw  being  perennially 4 -orthogonal to the spectral  4-\ncordinates as  \n,0k kw\n= , one finds that the s pectrally represented supplementary condition, \nk ,k 0 w\n=\n, has the following characteristics : \nk, evolves in the reciprocal space as \nkk . \nSince this spectral connection of the nonlocal media tor stands irrespective of the particle, external field, and their interactions, the proportionality constant is versatile in nature. The \nproportionality constant is having the unit of action, which can be identified with the Planck’s \nconstant, \n . The canonical 4 -momentum thus evolves in quantized form as \nk k=\n . Noting \nthat in electrodynamic evolution, \nk  can be expressed as \n0 /kkm cv qA c  =+ , (\nA  being 4-\npotential components of electromagnetic field), it is seen that the scalar representation of \ncanonical 4 -momentum evolution is having the form \n22\n0, [( / )( / ) - ] 0k k k k\nkka a k qA c k qA c m c v v  \n   \n − − = \n, with \nk k kkaa= . Noting that the \nsummation \n, ()k k k k\nkka a v v\n\n  in the evolution equation represents the scalar \n()vv\n , which is \nunity, the result corroborates to the dispersion property  of the nonlocal mediator (\nw )  as \n2 2 2\n0 ( / )k qA c m c−=\n. Phase velocity and group velocity of the mediator are obtainable from \nthe dispersion equation. Dynamic equation ( A2) along with the dispersion relation governs the \ncanonical rule of the delocalized passage of the (scalar) charged particle in the abridged \n(classico -quantu m) version of the arrived electrodynamics.  Dispersion characteristics of the \nscalar evolution under non -relativistic approximation can be similarly represented. Dispersion \ncharacteristics for spinor evolution case can also be obtained by involving Dirac ma trices in \nexpressing the quantized canonical 4 -momentum of the spinor particle.  \nThe canonical rules in the declocalized description of dynamics though  arrived purely \nfrom the consideration of nonlocal mitigation feature for minimized relaxation characteris tics do \nconform to the well established quantum mechanical prescriptions. (It is worthwhile to mention \nhere that the deductively obtained description of delocalized dynamic has similarity with the well \nknown Feynman’s  approach of summed over phase histori es of all possible paths). For \ndynamic passage that involves no relaxation loss, the canonical rule  of coherent evolution \nshows null expectation values of acceleration and jerk, and thus ensures the absence of the \nradiation related force. However, with the  loss of coherence under critically high external field, \nthe expectation values of acceleration and jerk are finite to result in radiation relaxations of the \nordered and disordered kinds respectively as Larmor radiation loss and thermal radiation loss \nfrom the Hawking -Unruh effect.  \n(Feynman R P and Hibbs A R (1965) Quantum Mechanics and Path Integrals ,  \nNew York:  McGraw -Hill) \n \n \n \n \n \n \n \n \n \n \n References  \n \n1.  A. Da yo, W. Alnasrallah, and J. Krim, Superconductivity -Dependent Sliding Friction, Phys  \n      Rev Lett  80 (8), 1998,1690 -1693.  \n2.  Anisoara Socoliuc, Enrico Gnecco, Sabine Maier, Oliver Pfeiffer, Alexis Baratoff, Roland  \n      Bennewitz, Ernst Meyer, 2010 Atomic -Scale Control of Friction by Actuation of Nanometer - \n      Sized Contacts, SCIENCE VOL 313, 207 -210.  \n3.  Marcin Kisiel, Enrico Gnecco , Urs Gysin , Laurent Marot , Simon Rast and Ernst Meyer  \n      2011 Suppression of electronic friction on Nb films in the superconducting state Nature  \n      Materials (Letters) vol.10, p119 -122.  \n4.  Hawking Stephe n W (1975) Particle creation by black holes, Commun. Math. Phys. 43,  \n     199-220.   \n5.  Unruh W. G. (1976) Notes on black -hole evaporation.  Physical Review  D. 14 (4):  \n     p870 -892.  \n6.  Richard Fitzpatrick, Driven Damped Harmonic Oscillation,  farside.ph.utexas.edu › teaching             \n    › Waves › node13   \n7.  C. Kittel 2005 Introduction t o solid state physics (Density of phonon states in Chapter 5 ). \n8.  A. I. Golovashkin, I. E. Leksina, G. P. Motulevich, and A. A. Shubin 1969 The Optical  \n     Properties of Niobium,  Soviet Physics JETP Vol. 29, p27 -34. \n \n  \n "    },    {        "title": "2307.05932v2.Lorentz_covariant_spinor_wave_packet.pdf",        "content": "Lorentz-covariant spinor wave packet\nKin-ya Oda∗and Juntaro Wada†\nApril 8, 2024\n∗Department of Mathematics, Tokyo Woman’s Christian University, Tokyo 167-8585, Japan\n†Department of Physics, University of Tokyo, Tokyo 113-0033, Japan\nAbstract\nWe propose a novel formulation for a manifestly Lorentz-covariant spinor wave-packet\nbasis. The traditional definition of the spinor wave packet is problematic due to its un-\navoidable mixing with other wave packets under Lorentz transformations. Our approach\nresolves this inherent mixing issue. The wave packet we develop constitutes a complete\nset, enabling the expansion of a free spinor field while maintaining Lorentz covariance.\nAdditionally, we present a Lorentz-invariant expression for zero-point energy.\n∗E-mail: odakin@lab.twcu.ac.jp\n†E-mail: wada-juntaro@g.ecc.u-tokyo.ac.jp\n1arXiv:2307.05932v2  [hep-th]  5 Apr 2024Contents\n1 Introduction 3\n2 Lorentz-covariant spinor wave packet 4\n2.1 Spinor plane waves, revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . 4\n2.2 Lorentz-covariant spinor wave packet . . . . . . . . . . . . . . . . . . . . . . . 5\n2.2.1 Brief review of Lorentz-invariant scalar wave packet . . . . . . . . . . 5\n2.2.2 Difficulty in spin-diagonal representation . . . . . . . . . . . . . . . . . 7\n2.2.3 Phase-space-diagonal representation . . . . . . . . . . . . . . . . . . . 8\n2.3 Momentum expectation value . . . . . . . . . . . . . . . . . . . . . . . . . . . 10\n2.4 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11\n3 Spinor field expanded by wave packets 12\n4 Energy, momentum, and charge 13\n5 Summary and discussion 16\nA “Slanted” foliation 16\nB Wigner representation 18\nC Energy, momentum, and number operator in scalar case 19\n21 Introduction\nIn quantum mechanics, wave packets serve as a crucial conceptual and mathematical foun-\ndation. In real observations, we never encounter idealized plane-wave states, characterized\nby zero uncertainty in momentum and infinite uncertainty in position. They do not belong\nto the Hilbert space because of its non-normalizability. In quantum field theory (QFT), the\nplane-wave S-matrix is traditionally used, but this approach results in divergences due to the\nsquared energy-momentum delta function. This makes the plane-wave S-matrix computation\nmore of a mnemonic than a rigorous derivation for observables; see e.g. Ref. [1].\nWave packet states have been extensively discussed in various contexts of particle physics\nphenomenology, including anomalies in vector meson decay [2], corrections to Fermi’s golden\nrule [3, 4], searches for dark photons [5], applications in quantum computation [6, 7, 8], and\nstudies of neutrino oscillation [9, 10, 11, 12, 13, 14, 15, 16, 17, 18].\nDespite these applications, theoretical efforts to construct QFT based on wave-packet\nstates have been limited. Previous work has employed Gaussian formalism, utilizing Gaussian\nwave functions as a complete basis for expanding the free one-particle Hilbert space. This\napproach has revealed phenomena like time-boundary effects, which are unobservable in plane-\nwave formalism [19]. However, the Gaussian formalism breaks manifest Lorentz covariance,\nnecessitating a refinement for both aesthetic and pragmatic reasons.\nThe absence of manifest Lorentz covariance is unsatisfying, as historically, physics has\nadvanced through symmetry-based formalisms, such as the Dirac [20] and Becchi-Rouet-\nStora-Tyutin (BRST) [21, 22, 23] formalisms. Therefore, it is desirable to develop a manifestly\nLorentz-covariant wave packet formalism.\nPractically, the Gaussian formalism complicates calculations due to the lack of Lorentz\ncovariance, evident in the difficulty of deriving explicit analytic formulas for Gaussian wave\nfunctions in position space. In contrast, our previous study demonstrated the expansion of\nscalar fields in QFT using a complete basis of Lorentz-invariant wave packets, facilitating the\nderivation of analytic formulas for Lorentz-invariant wave functions in scalar fields [24]. Also,\nthe Gaussian wave packet is shown to be a non-relativistic limit of the Lorentz-invariant wave\npacket for the scalar fields [24].\nIn this study, we investigate the wave packet basis for spinors within a Lorentz-covariant\nframework. Traditional methodologies, including Gaussian formalism (refer to Appendix A\nin Ref. [25] for an overview) and other approaches towards Lorentz-covariant spinor wave\npackets [26, 27, 28, 29, 30], assume that the spin dependence of spinor wave packets aligns with\nthat of plane waves. However, this assumption introduces a significant complication: it leads\nto a complex Lorentz transformation law, which unexpectedly intertwines wave-packet states\nwith different central momenta and positions. Such a phenomenon is physically paradoxical.\nImagine a scenario where a single particle is depicted by a wave packet with a distinct central\nmomentum and position. The blending of this wave packet with others having varying central\nmomenta and positions effectively results in an unphysical merging of distinct particles under\nLorentz transformation, which is a challenging notion for conventional theories.\nIn response to this issue, this paper introduces a novel definition of the spinor wave packet\nthat circumvents this problematic mixing. Our definition ensures that wave packets remain\nseparate and distinct under Lorentz transformations. We then establish the completeness\nof this newly defined spinor wave packet within the free one-particle subspace. Expanding\non this concept, we demonstrate that the spinor field can be effectively expanded using our\nspinor wave packet. Additionally, we explore the application of this approach to several well-\n3established operators in the wave packet basis, providing a comprehensive understanding of\nits implications in quantum field theory.\nThe paper is organized as follows: Section 2 introduces the new Lorentz-covariant spinor\nwave-packet basis in the one-particle subspace. Section 3 extends this to the creation and\nannihilation operators, showing how the free fermion field can be expanded using this basis.\nFinally, Section 4 presents the expression of several QFT operators in terms of wave packets.\n2 Lorentz-covariant spinor wave packet\nIn this section, we point out that the known representation of a spinor wave packet suffers\nfrom mixing with other wave packets under Lorentz transformations, and propose a complete\nset of Lorentz-covariant spinor wave-packet basis without the difficulty of mixing.\nWe work in the ( d+ 1)-dimensional Minkowski space Md+1spanned by coordinate system\nx=\u0000\nx0,x\u0001\n=\u0000\nx0, x1, . . . , xd\u0001\n∈R1,d, with d= 3 spatial dimensions. We take the almost-plus\nmetric signature diag( −,+, . . . , +). We only consider a massive field, m > 0, and always take\n(d+ 1)-momenta on-shell, p0=p\nm2+p2, throughout this paper unless otherwise stated.\nWhen an on-shell momentum appears in an argument of a function such as f(p), we use both\ndand ( d+ 1)-dimensional notations interchangeably: f(p) =f(p).\n2.1 Spinor plane waves, revisited\nTo spell out our notation, we summarize basic known facts on the spinor plane waves. A free\nDirac field bψ(x) can be expanded by plane wave as follows,\nbψ(x) =X\nsZddp\n2p0 \nu(p, s)eip·x\n(2π)d/2bα(p, s) +v(p, s)e−ip·x\n(2π)d/2bβ†(p, s)!\n, (1)\nwhere u(p, s) and v(p, s) are plane-wave solutions of the Dirac equation\n(i/p+m)u(p, s) = 0 ,\n(i/p−m)v(p, s) = 0 , (2)\nwith s=±1/2 being the spin in the rest frame of each solution. Throughout this paper, we\nsuppress the spinor indices a= 1, . . . , 2⌊(d+1)/2⌋forbψ,u,v, etc. when unnecessary.\nThese solutions satisfy the following completeness relations,\nX\nsu(p, s)u(p, s) =−i/p+m,\nX\nsv(p, s)v(p, s) =−i/p−m, (3)\nand their normalization is\nu(p, s)u\u0000\np, s′\u0001\n= 2mδss′,\nv(p, s)v\u0000\np, s′\u0001\n=−2mδss′, (4)\n4where ψ:=ψ†βis the Dirac adjoint.1The coefficients bα(p, s) andbβ†(p, s) in Eq. (1) are the\nannihilation and creation operators for particle and anti-particle, respectively, that satisfy the\nfollowing anticommutation relations:\nn\nbα(p, s),bα†\u0000\np′, s′\u0001o\n=δss′2p0δd\u0000\np−p′\u0001b1,\nn\nbβ(p, s),bβ†\u0000\np′, s′\u0001o\n=δss′2p0δd\u0000\np−p′\u0001b1,\nothers = 0 . (5)\nFree one-particle subspaces of particle and antiparticle are spanned by the following plane-\nwave bases:\nbα†(p, s)|0⟩=:|p, s, n⟩⟩, bβ†(p, s)|0⟩=:|p, s, nc⟩⟩, (6)\nwhere nandncdenote the particle and antiparticle of nth species, respectively. The anti-\ncommutator (5) leads to the inner product:\n\n\np, s, η\f\fp′, s′, η′\u000b\u000b\n:= 2p0δd\u0000\np−p′\u0001\nδss′δηη′, (7)\nwhere η=n, nclabels the particle and anti-particle. The normalization (7) leads to the\ncompleteness relation (resolution of identity) in the free one-particle subspace of each η=\nn, nc:\nX\nsZddp\n2p0|p, s, η⟩⟩⟨⟨p, s, η|=ˆ1. (8)\nThe Lorentz transformation law of the plane wave reads\nbU(Λ)|p, s⟩⟩=X\ns′\f\fΛp, s′\u000b\u000b\nDs′s\u0000\nW(Λ, p)\u0001\n, (9)\nwhere Dis the spin- srepresentation of the Winger rotation SO(d); see Appendix B for details.\n2.2 Lorentz-covariant spinor wave packet\nIn this subsection, we first briefly review basic facts on Lorentz-invariant scalar wave pack-\nets [31, 32], which is discussed in our previous work [24]. Next, we point out the difficulty\nin the conventional treatment of the spinor wave packet [26, 27, 28, 29]. Then, we propose\na new definition of the spinor wave packet and show that we can avoid this difficulty in our\nexpression.\n2.2.1 Brief review of Lorentz-invariant scalar wave packet\nFor central position Xand momentum Pin (d+ 1)-dimensions, a Lorentz-invariant scalar\nwave packet |Π⟩⟩is defined by [31, 32]:2\n⟨⟨p|Π⟩⟩:=Nϕe−ip·(X+iσP), (10)\n1We adopt the spinor notation in Ref. [1]: {γµ, γν}= 2ηµνI, where η:= diag( −1,1, . . . , 1) and Iis the\nunit matrix in the spinor space. Here, β:=iγ0is distinguished from the operator bβ.\n2See e.g. Refs. [28, 24] for reviews.\n5where Π denotes the phase space3\nΠ := ( X, P), (11)\nand the normalization factor\nNϕ:=\u0000σ\nπ\u0001d−1\n4\nq\nKd−1\n2(2σm2)(12)\nprovides ⟨⟨Π|Π⟩⟩= 1, in which Kn(z) is the modified Bessel function of the second kind. Here\nand hereafter, we fix σunless otherwise stated.\nThe wave function and the inner product are obtained as [31, 24]\n⟨⟨x|Π⟩⟩=Nϕmd−1\n√\n2πKd−1\n2(∥ξ∥)\n∥ξ∥d−1\n2, (13)\n\n\nΠ\f\fΠ′\u000b\u000b\n=N2\nϕ\u0000\n2πm2\u0001d−1\n2Kd−1\n2(∥Ξ∥)\n∥Ξ∥d−1\n2, (14)\nwhere for any complex vector Vµ, we write ∥V∥:=√\n−V2, namely,\n∥ξ∥=mq\nσ2m2+ (x−X)2−2iσP·(x−X), (15)\n∥Ξ∥=mq\u0000\n(X−X′)−iσ(P+P′)\u00012, (16)\nwith ξµ:=m[σPµ+i(x−X)µ] and Ξµ:=m[σ(Pµ+P′µ) +i(X−X′)µ].4We note that\nthere is no branch-cut ambiguity for the square root as long as m > 0 [24].\nWith this state, the momentum expectation value and its (co)variance become [31]\n⟨⟨ˆpµ⟩⟩ϕ:=Zddp\n2p0⟨⟨Π|p⟩⟩pµ⟨⟨p|Π⟩⟩=MϕPµ, (17)\n⟨⟨ˆpµˆpν⟩⟩ϕ=Kd+3\n2\u0000\n2σm2\u0001\nKd−1\n2(2σm2)PµPν+Mϕ\n2σηµν, (18)\nwhere\nMϕ:=Kd+2\n2\u0000\n2σm2\u0001\nKd−1\n2(2σm2). (19)\nIn general, a matrix element of ˆ pbecomes\n⟨⟨ˆpµ⟩⟩Π,Π′:=Zddp\n2p0⟨⟨Π|p⟩⟩pµ\n\np\f\fΠ′\u000b\u000b\n=\u0000\n2πm2\u0001d−1\n2N2\nϕmΞµKd+1\n2(∥Ξ∥)\n∥Ξ∥d+1\n2. (20)\n3Here, Π includes the wave-packet central time X0. Though P0=p\nm2+P2is not an independent\nvariable, we also include it for the convenience of writing its Lorentz transformation below.\n4The abuse of notation is understood such that a vector-squared V2:=−\u0000\nV0\u00012+V2is distinguished from\nthe second component of Vby the context.\n6Let us consider a spacelike hyperplane Σ N,T={X|N·X+T= 0}in the space of\ncentral position X; see Appendix A. One can write the completeness relation in the position-\nmomentum phase space in a manifestly Lorentz- invariant fashion [24] (see also Ref. [31]):\nZ\nd2dΠϕ|Π⟩⟩⟨⟨Π|=ˆ1, (21)\nwhere ˆ1 denotes the identity operator in the one-particle subspace and the Lorentz- invariant\nphase-space volume element is given by\nZ\nd2dΠϕ:=1\nMϕZddΣµ\nX\n(2π)d(−2Pµ)ddP\n2P0, (22)\nin which\nddΣµ\nX:= dd+1X δ(N·X+T)Nµ(23)\nis the Lorentz-covariant volume element. We stress that σis not summed nor integrated in\nthe identity (21) and that the identity holds for any fixed σ.\nLet us consider a “time-slice frame” ˇXof the central-position space in which ˇΣˇN,Tbecomes\nan equal-time hyperplane ˇX0=T,\nˇX:=L−1(N)X, (24)\nwhere the “standard” Lorentz transformation L(N) is defined by N=:L(N)ℓ, with ℓ\ndenoting ℓ:= (1 ,0) in any frame; note that ˇN=L−1(N)N=ℓby definition; see Appendix A\nfor details. On the constant- ˇX0hyperplane ˇΣˇN,T=\bˇX\f\fˇX0=T\t\n, the Lorentz-invariant\nphase-space volume element reduces to the familiar form:\nZ\nd2dˇΠϕ=1\nMϕZ\nˇX0=TddˇXddˇP\n(2π)d. (25)\nNote that Mϕ→1 in the non-relativistic limit σm2≫1.\n2.2.2 Difficulty in spin-diagonal representation\nIn the literature [26, 27, 28, 29] a so to say spin-diagonal one-particle wave-packet state\n|Π, S⟩⟩Dwith a spin Shas been defined as\n⟨⟨p, s|Π, S⟩⟩D:=⟨⟨p|Π⟩⟩δsS (26)\nwhere ⟨⟨p|Π⟩⟩is nothing but the scalar Lorentz-invariant wave packet (10).5Its normalization\nbecomes\n\n\nΠ, S\f\fΠ′, S′\u000b\u000b\nD=X\nsZddp\n2p0⟨⟨Π, S|p, s⟩⟩D\n\np, s\f\fΠ′, S′\u000b\u000b\nD\n=X\nsZddp\n2p0⟨⟨Π|p⟩⟩\n\np\f\fΠ′\u000b\u000b\n=\n\nΠ\f\fΠ′\u000b\u000b\nδSS′, (27)\n5In the literature, the normalization and X-dependence [24] have been omitted.\n7where ⟨⟨Π|Π′⟩⟩is given in Eq. (14).6This leads to the following completeness relation in the\none-particle subspace\nX\nSZ\nd2dΠϕ|Π, S⟩⟩D⟨⟨Π, S|D=ˆ1, (28)\ngeneralizing the completeness relation of the scalar wave packet (21).\nOnce the wave-packet state is defined, its Lorentz transformation law is obtained as\nbU(Λ)|Π, S⟩⟩D=X\nsZddp\n2p0bU(Λ)|p, s⟩⟩⟨⟨p, s|Π, S⟩⟩D\n=X\ns,s′Zddp\n2p0X\nS′Z\nd2dΠ′\nϕ\f\fΠ′, S′\u000b\u000b\nD\n\nΠ′, S′\f\fΛp, s′\u000b\u000b\nDDs′s\u0000\nW(Λ, p)\u0001\n⟨⟨p, s|Π, S⟩⟩D\n=X\nS′Z\nd2dΠ′\nϕ\f\fΛΠ′, S′\u000b\u000b\nD\n\nDS′S\u0000\nW(Λ,ˆp)\u0001\u000b\u000b\nΠ′,Π, (29)\nwhere\nΛΠ := (Λ X,ΛP). (30)\nand\n\n\nDS′S\u0000\nW(Λ,ˆp)\u0001\u000b\u000b\nΠ′,Π:=\n\nΠ′\f\f Zddp\n2p0DS′S\u0000\nW(Λ, p)\u0001\n|p⟩⟩⟨⟨p|!\n|Π⟩⟩. (31)\nWe see that the spin-diagonal choice (26) leads to the complicated transformation law (29)\nmixing the wave-packet state with the others having various centers of momentum and posi-\ntion.\nBelow, we will show that we can indeed realize a physically reasonable transformation\nlaw, so to say the phase-space-diagonal representation, which evades the mixing with other\nstates (29):\nbU(Λ)|Π, S⟩⟩=X\nS′\f\fΛΠ, S′\u000b\u000b\nCS′S(Λ,Π), (32)\nwhere CS′S(Λ,Π) is a yet unspecified representation function.\n2.2.3 Phase-space-diagonal representation\nInstead of the conventional choice (26), we propose to define\n\n\np, s, η\f\fΠ, S, η′\u000b\u000b\n:=Nψe−ip·(X+iσP)MsS(p, P, η )δηη′, (33)\n6An inner product of the spin-diagonal wave-packet state and another state |ψ⟩is understood as\n⟨⟨Π, S|ψ⟩D:=\u0000\n|Π, S⟩⟩D\u0001†|ψ⟩=:⟨⟨Π, S|D|ψ⟩. We will never consider an inner product of the spin-diagonal\nwave-packet state and a phase-space-diagonal wave-packet state that appears below so that this notation will\nnot cause confusion.\n8where the key element is\nMsS(p, P, η ) :=\n\nu(p, s)u(P, S)\n2mforη=n,\n−v(P, S)v(p, s)\n2mforη=nc,(34)\nandNψis a normalization factor to be fixed below. Note that MsS(p, p, η ) =δsSand that\nMsS(p, P, n ) =MsS(p, P, nc) from u(p, s) =Cv∗(p, s) (with the charge conjugation matrix\nC=−γ2in our notation). The definition (34) leads to7\nMsS(p, P, η ) =X\ns′,S′D∗\ns′s\u0000\nW(Λ, p)\u0001\nMs′S′(Λp,ΛP, η)DS′S\u0000\nW(Λ, P)\u0001\n. (35)\nThen it follows that\n⟨⟨p, s, η|Π, S, η⟩⟩=X\ns′,S′D∗\ns′s\u0000\nW(Λ, p)\u0001\n⟨⟨Λp, s, η|ΛΠ, S, η⟩⟩DS′S\u0000\nW(Λ, P)\u0001\n. (36)\nHere and hereafter, for notational simplicity, we omit the label ηand concentrate on the case\nof the particle when the distinction is irrelevant. The identity (36) results in8\nbU(Λ)|Π, S⟩⟩=X\nS′\f\fΛΠ, S′\u000b\u000b\nDS′S\u0000\nW(Λ, P)\u0001\n. (37)\nAs promised, we have realized the phase-space-diagonal representation (32).\nNow we show that the normalization ⟨⟨Π, S|Π, S⟩⟩= 1 is realized by the choice\nNψ=s\n2\n1 +MϕNϕ, (38)\n7One can show it as\nMsS(p, P, n ) =1\n2mu(p, s)S−1(Λ)S(Λ)u(P, S) =1\n2mX\ns′,S′D∗\ns′s\u0000\nW(Λ, p)\u0001\nu\u0000\nΛp, s′\u0001\nu\u0000\nΛP, S′\u0001\nDS′S\u0000\nW(Λ, P)\u0001\n=X\ns′,S′D∗\ns′s\u0000\nW(Λ, p)\u0001\nMs′S′(Λp,ΛP, n)DS′S\u0000\nW(Λ, P)\u0001\n,\nand similarly for MsS(p, P, nc).\n8This can be shown as\nbU(Λ)|Π, S⟩⟩=X\nsZddp\n2p0bU(Λ)|p, s⟩⟩⟨⟨p, s|Π, S⟩⟩=X\ns′,S′Zddp\n2p0\f\fΛp, s′\u000b\u000b\n\nΛp, s′\f\fΛΠ, S′\u000b\u000b\nDS′S\u0000\nW(Λ, P)\u0001\n=X\nS′\f\fΛΠ, S′\u000b\u000b\nDS′S\u0000\nW(Λ, P)\u0001\n,\nwhere we have used Eqs. (9) and (36) and then the unitarity (95) in the second equality.\n9where Nϕis given in Eq. (12). Let us first compute\n⟨⟨Π, S|Π, S⟩⟩=X\nsZddp\n2p0⟨⟨Π, S|p, s⟩⟩⟨⟨p, s|Π, S⟩⟩\n=N2\nψ\n(2m)2u(P, S) Zddp\n2p0⟨⟨Π|p⟩⟩⟨⟨p|Π⟩⟩\nN2\nϕ\u0000\n−i/p+m\u0001!\nu(P, S)\n=N2\nψ\n(2m)2u(P, S)⟨⟨−i/ˆp+m⟩⟩ϕ\nN2\nϕu(P, S), (39)\nwhere we used Eq. (3) in the second line. The expectation value ⟨⟨ˆpµ⟩⟩ϕis presented in Eq (17),\nfrom which we get\n⟨⟨−i/ˆp+m⟩⟩ϕ=−iMϕ/P+m. (40)\nTherefore, using the Dirac equation (2) and then the normalization (4), we see that the\nchoice (38) provides the normalized state.\nFinally, the inner product is given by\n\n\nΠ, S, η\f\fΠ′, S′, η′\u000b\u000b\n=X\ns,η′′Zddp\n2p0\n\nΠ, S, η\f\fp, s, η′′\u000b\u000b\n\np, s, η′′\f\fΠ′, S′, η′\u000b\u000b\n=N2\nψ\n(2m)2u(P, S)⟨⟨−i/ˆp+m⟩⟩Π,Π′\nN2\nϕu\u0000\nP′, S′\u0001\nδηη′, (41)\nwhere, ⟨⟨−i/ˆp+m⟩⟩Π,Π′=−i⟨⟨/ˆp⟩⟩Π,Π′+mwith⟨⟨ˆpµ⟩⟩Π,Π′being given in Eq. (20). Hereafter,\nwe adopt this representation for the Lorentz-covariant spinor wave packet.\n2.3 Momentum expectation value\nIn this subsection, we compute the momentum expectation value of the Lorentz covariant\nspinor wave packet:\n⟨⟨ˆpµ⟩⟩ψ:=X\nsZddp\n2p0⟨⟨Π, S|p, s⟩⟩pµ⟨⟨p, s|Π, S⟩⟩. (42)\nThis will be an important parameter in the following. Putting Eq. (33), we obtain\n⟨⟨ˆpµ⟩⟩ψ=N2\nψ\n4m2X\nsZddp\n2p0⟨⟨Π|p⟩⟩pµ⟨⟨p|Π⟩⟩\nN2\nϕu(P, S)u(p, s)u(p, s)u(P, S)\n=Nψ\n4m2u(P, S)⟨⟨ˆpµ(−i/ˆp+m)⟩⟩ϕ\nN2\nϕu(P, S), (43)\nwhere we used Eq. (3). The expectation value and its covariance ⟨⟨ˆpµ⟩⟩ϕ,⟨⟨ˆpµˆpν⟩⟩ϕare shown\nin Eqs (17) and (18). Thus,\n⟨⟨ˆpµ(−i/ˆp+m)⟩⟩ϕ=−i Kd+3\n2\u0000\n2σm2\u0001\nKd−1\n2(2σm2)Pµ/P+Mϕ\n2σγµ!\n+MϕmPµ. (44)\n10Hence, using the Dirac equation (2) and then the normalization (4), we get\n1\n4m2u(P, S)⟨⟨ˆpµ(−i/ˆp+m)⟩⟩ϕu(P, S) =1\n2 \nMϕ+Mϕ\n2σm2+Kd+3\n2\u0000\n2σm2\u0001\nKd−1\n2(2σm2)!\nPµ. (45)\nTherefore, the momentum expectation value is given by\n⟨⟨ˆpµ⟩⟩ψ=MψPµ, (46)\nwhere\nMψ:=1\n1 +Mϕ\"Kd+3\n2\u0000\n2σm2\u0001\nKd−1\n2(2σm2)+Mϕ\u0012\n1 +1\n2σm2\u0013#\n. (47)\nNote that Mψ→1 in the non-relativistic limit σm2≫1.\n2.4 Completeness\nIn this subsection, we will prove the following completeness relation for Lorentz-covariant\nspinor wave packet,\nX\nSZ\nd2dΠψ|Π, S⟩⟩⟨⟨Π, S|=ˆ1, (48)\nwhere\nZ\nd2dΠψ:=1\nMψZ\nΣN,TddΣµ\nX\n(2π)d(−2Pµ)ddP\n2P0\n=Mϕ\nMψZ\nd2dΠϕ, (49)\nin which d2dΠϕ,Mϕ, andMψare given in Eqs. (22), (19), and (47) respectivity.\nTo prove Eq. (48), we rewrite it as a matrix element for both-hand sides, sandwiched by\nthe plane-wave bases (6):\nN2\nψ\nMψZ\nΣN,Tdd+1X\n(2π)dδ(N·X+T) (−2P·N)ddP\n2P0⟨⟨p|Π⟩⟩⟨⟨Π|q⟩⟩\nN2\nϕ\n×1\n4m2X\nSu(p, s)u(P, S)u(P, S)u\u0000\nq, s′\u0001\n= 2p0δd(p−q)δss′, (50)\nwhere we used Eq. (7) on the right-hand side. On the left-hand side, we integrate Xover\nΣN,Tby exploiting its Lorentz invariance, choosing a coordinate system where it becomes a\nconstant- ˇX0hyperplane ˇΣˇN,Twith ˇX0=T. Then left-hand side in Eq. (50) becomes\n(l.h.s.) =N2\nψ\nM2\nψδd(p−q)ZddP\n2P02P0e2σP·p\nN2\nϕu(p, s)\u0000\n−i/P+m\u0001\nu(p, s′)\n4m2\n=N2\nψ\nM2\nψδd(p−q)u(p, s)\n\n2ˆp0(−i/ˆp+m)\u000b\u000b\nϕ\nN2\nϕu\u0000\nq, s′\u0001\n= 2p0δd(p−q), (51)\n11where we used Eq. (3) in the first line, and Eqs. (45) and (47) in the last line. Thus, Eq. (50),\nand hence the completeness (48), is proven.\n3 Spinor field expanded by wave packets\nNow we define the creation and annihilation operators of the Lorentz-covariant wave packet.\nWe write a free spin-1 /2 one-particle state of nth spinor particle |Π, S;n⟩⟩and of its anti-\nparticle |Π, S;nc⟩⟩. Similarly to the plane wave case, we define wave-packet creation operators\nby\nbA†(Π, S)|0⟩:=|Π, S, n⟩⟩, (52)\nbB†(Π, S)|0⟩:=|Π, S, nc⟩⟩, (53)\nand annihilation operators bA(Π, S),bB(Π, S) by their Hermitian conjugate, with mass di-\nmensionsh\nbA†(Π, S)i\n=\u0002\n|Π, S;n⟩⟩\u0003\n= 0, etc. Then, the completeness relation (48) on the\none-particle subspace reads\n⟨0|bα(p, s;n) =X\nSZ\nd2dΠψ⟨⟨p, s;n|Π, S;n⟩⟩⟨0|bA(Π, S), (54)\nand similarly for the anti-particles. Then, we can naturally generalize it to an operator\nrelation that is valid on the whole Fock space:\nbα(p, s) =X\nSZ\nd2dΠψ⟨⟨p, s;n|Π, S;n⟩⟩bA(Π, S), (55)\nbβ(p, s) =X\nSZ\nd2dΠψ⟨⟨p, s;nc|Π, S;nc⟩⟩bB(Π, S). (56)\nSimilarly, the completeness of the plane wave (8) leads to the expansion of these creation and\nannihilation operators:\nbA(Π, S) =X\nsZddp\n2p0⟨⟨Π, S;n|p, s;n⟩⟩bα(p, s), (57)\nbB(Π, S) =X\nsZddp\n2p0⟨⟨Π, S;nc|p, s;nc⟩⟩bβ(p, s). (58)\nFrom the above equations, we can derive the anti-commutation relation of the creation and\nannihilation operators:\nn\nbA(Π, S),bA†\u0000\nΠ′, S′\u0001o\n=\n\nΠ, S, n\f\fΠ′, S′, n\u000b\u000bb1, (59)\nn\nbB(Π, S),bB†\u0000\nΠ′, S′\u0001o\n=\n\nΠ, S, nc\f\fΠ′, S′, nc\u000b\u000bb1, (60)\nothers = 0 , (61)\nwhereb1 denotes the identity operator in the whole Fock space, and ⟨⟨Π, S|Π′, S′⟩⟩is the inner\nproduct of the Lorentz covariant wave packets, given in Eq. (41).\n12Finally, the free spinor field can be expanded as\nbψ(x) =X\nSZ\nd2dΠψh\nU(x,Π, S)bA(Π, S) +V(x,Π, S)bB†(Π, S)i\n, (62)\nwhere the Dirac spinor wave functions are given by\nU(x,Π, S) =X\nsZddp\n2p0u(p, s)eip·x\n(2π)d\n2⟨⟨p, s;n|Π, S;n⟩⟩\n=1\n2mNψ\nNϕ⟨⟨x|(−i/ˆp+m)|Π⟩⟩u(P, S)\n=1\n2Nψmd−1\n√\n2π \n−i/ ξKd+1\n2(∥ξ∥)\n∥ξ∥d+1\n2+Kd−1\n2(∥ξ∥)\n∥ξ∥d−1\n2!\nu(P, S), (63)\nV(x,Π, S) =X\nsZddp\n2p0v(p, s)e−ip·x\n(2π)d\n2⟨⟨Π, S;nc|p, s;nc⟩⟩\n=CU∗(x,Π, S), (64)\nwhere we have used the scalar wave function (13). Here, ∥ξ∥andξµare given in Eq. (15) and\nbelow it, respectively.\nThe normalization conditions of these Dirac spinors are\nZdd+1X\n(2π)dδ(N·X+T)U(x,Π, S)U(x,Π, S′) = 2 mδSS′,\nZdd+1X\n(2π)dδ(N·X+T)V(x,Π, S)V(x,Π, S′) =−2mδSS′ (65)\nwhere we used Eqs. (4) and (8). The normalization is as same as the case of plane waves (4),\nexcept for the integration of X.\nNext, the completeness relations can be computed by\nX\nSZdd+1X\n(2π)dδ(N·X+T)U(x,Π, S)U(x,Π, S) =−i/PMψ+m,\nX\nSZdd+1X\n(2π)dδ(N·X+T)V(x,Π, S)V(x,Π, S) =−i/PMψ−m, (66)\nwhere we have used Eqs. (3), (40) and (44). These relations are similar to that of plane\nwaves (3), except for the integration of Xand factor Mψon the right-hand side.\n4 Energy, momentum, and charge\nIn this section, we rewrite well-known operators in QFT, i.e. the total Hamiltonian, momen-\ntum, and charge operators, in the language of the spinor wave packet. Since the wave packet\nis not the momentum eigenstate, the total Hamiltonian and momentum operators cannot be\ndiagonalized in the wave packet basis. However, the zero-point energy can be described in a\nfully Lorentz invariant manner using this basis. In Appendix C, we also show the correspond-\ning expressions for the scalar wave packet.\n13First, let us consider the convergent part of the total Hamiltonian and momentum oper-\nators. In the momentum space, these operators are given by\nbPµ:=Zddp\n2p0X\nspµ\u0010\nbα†(p, s)bα(p, s) +bβ†(p, s)bβ(p, s)\u0011\n. (67)\nPutting Eqs. (55) and (56) into the above expression, we get\nbPµ=X\nS,S′Z\nd2dΠψZ\nd2dΠ′\nψ\u0010\nbA†(Π)bA\u0000\nΠ′\u0001\n+bB†(Π)bB\u0000\nΠ′\u0001\u0011\n⟨⟨ˆpµ⟩⟩(Π,S),(Π′,S′), (68)\nwhere\n⟨⟨ˆpµ⟩⟩(Π,S),(Π′,S′):=X\nsZddp\n2p0⟨⟨Π, S|p, s⟩⟩pµ\n\np, s\f\fΠ′, S′\u000b\u000b\n. (69)\nWe see that the total Hamiltonian and momentum operators are not diagonal on the wave\npacket basis, unlike the plane-wave eigenbasis.\nLet us discuss the divergent part of this operator, coming from the zero-point energy:\nbPµ\nzero:=X\nsZddp\n2p0(−pµ)n\nbβ(p, s),bβ†(p, s)o\n. (70)\nSimilarly as above, putting Eq. (55) into this commutator, we obtain\nbPµ\nzero=X\nS,S′Z\nd2dΠψZ\nd2dΠ′\nψ⟨⟨−2ˆpµ⟩⟩(Π,S),(Π′,S′)\n\nΠ′, S′\f\fΠ, S\u000b\u000bb1\n=X\nSZ\nd2dΠψ⟨⟨−2ˆpµ⟩⟩ψb1\n=X\nSZdd+1X\n(2π)dddP\n2P0(2P·N)δ(N·X+T)Pµb1. (71)\nwhere we have used the completeness relation (48) in the second line and, in the last line, the\nexpectation value (46) and the Lorentz-invariant phase-space volume element (49).\nLet the time-like normal vector Nµanddspace-like vectors N⊥i(i= 1, . . . , d ) compose\nan orthonormal basis: N·N⊥i= 0 and N⊥i·N⊥j=δijsuch that we can decompose Pµinto\nthe components parallel and perpendicular to Nµ,\nPµ=−(P·N)Nµ+X\ni(P·N⊥i)Nµ\n⊥i. (72)\nWhen we put this into Eq. (71), the perpendicular components vanish in a regularization\nscheme that makes Lorentz covariance manifest, namely in the dimensional regularization:\nZddP\n2P0(P·N⊥i) (P·N) =ZddP\n2P0PµPνNµN⊥i ν\n=ZddP\n2P01\nd+ 1(P·P) (N·N⊥i)\n= 0. (73)\n14Therefore, the divergent part Pµ\nzerohas only one independent component, which can be inter-\npreted as zero-point energy, defined in a manifestly Lorentz-invariant fashion:\nEzero:=−NµPµ\nzero\n=X\nSZdd+1X\n(2π)dddP\n2P0(−2) (P·N)2δ(N·X+T), (74)\nwhere Pµ\nzerois the coefficient in front of b1 in the right-hand side of Eq. (71). We note that\nthe zero-point energy should be a scalar as we have shown, otherwise, an infinite momentum\nwould appear from a Lorentz transformation.\nPhysically, we would expect that the zero-point energy is independent of the choice of\nspace-like hyperplane Σ N,T. We can show it by exploiting the Lorentz invariance of the\nexpression (74) by choosing Nµ=ℓµ(= (1 ,0)), without loss of generality. Then, the zero-\npoint energy reduces to the well-known form:\nEzero=X\nSZ\nX0=TddXddP\n(2π)d(−P0). (75)\nIt is remarkable that this zero point energy of the Dirac spinor is exactly −4 times that\nof a real scalar, shown in Eq. (112) in Appendix, although the expression of momentum\nexpectation values in Eqs. (46) and (17) are completely different between the spinor and\nscalar. The factor 4 is the number of degrees of freedom, and the negative sign cancels the\nbosonic contribution in a supersymmetric theory.\nNext, we consider the following charge operator\nbQ:=Zddp\n2p0X\ns[−bα†(p, s)bα(p, s) +bβ†(p, s)bβ(p, s)] (76)\nSubstituting Eq. (55) into the above expression, we obtain\nbQ=X\nSZ\nd2dΠψ[−bA†(Π, S)bA(Π, S) +bB†(Π, S)bB(Π, S)], (77)\nwhere we have used\nbA(Π, S) =X\nSZ\nd2dΠ′\nψ\n\nΠ, S\f\fΠ′, S′\u000b\u000bbA\u0000\nΠ′, S\u0001\n,\nbB(Π, S) =X\nSZ\nd2dΠ′\nψ\n\nΠ′, S′\f\fΠ, S\u000b\u000bbB\u0000\nΠ′, S\u0001\n, (78)\nwhich follows from Eq. (48). This expression Eq. (77) means that the creation operators\nA†(Π, S) and B†(Π, S) create the wave packet with charge −1, and +1 respectively. In fact,\nbQbA†(Π, S)|0⟩=n\nbQ ,bA†(Π, S)o\n|0⟩\n=−X\nS′Z\nd2dΠ′\nψbA†\u0000\nΠ′, S′\u0001\n\nΠ′, S′\f\fΠ, S\u000b\u000b\n|0⟩\n=−bA†(Π, S)|0⟩, (79)\nis valid. Here we have used Eq. (78) in the last line.\n155 Summary and discussion\nIn this paper, we have proposed fully Lorentz-covariant wave packets with spin. In the conven-\ntional definition of the wave packet, spin dependence of the wave function in the momentum\nspace is just given by Kronecker delta, δsS, and such a wave packet with spin transforms under\nLorentz transformation mixing wave-packet states that have different centers of momentum\nand position. Our proposal overcomes this difficulty.\nWe have also proven that these wave packets form a complete basis that spans the spinor\none-particle subspace in the manifestly Lorentz-invariant fashion. Generalizing this com-\npleteness relation to the whole Fock space, we have shown that the creation and annihilation\noperators of plane waves can be expanded by that of these wave packets. This relation leads\nto the expansion of the spinor field in a Lorentz covariant manner. In addition to this, we\nhave expressed the well-known operators in a wave packet basis: the total Hamiltonian, mo-\nmentum, and charge operators. In particular, we have given the Lorentz covariant expression\nof zero point energy, in terms of centers of momentum and position of this wave packet.\nSince neutrino oscillation requires wave-packet formulation, our new definition of wave\npacket may have an impact on this context. The novel Lorentz covariant basis that we\npropose will be useful in handling the wave packet quantum field theory [25, 33, 19, 34] more\ntransparently. It may also be interesting that consider Bell’s inequality of this wave packet\nstate.\nAcknowledgement\nWe thank Ryusuke Jinno for a useful comment. This work is supported in part by the JSPS\nKAKENHI Grant Nos. 19H01899, 21H01107 (K.O.), and 22J21260 (J.W.).\nAppendix\nA “Slanted” foliation\nIn this appendix, we briefly introduce “slanted” foliation which is necessary to write down the\ncompleteness relation of Lorentz-invariant wave packets in the fully Lorentz-invariant manner.\nLet us consider the following spacelike hyperplane:\nΣn,τ:=n\nx∈R1,d\f\f\fn·x+τ= 0o\n, (80)\nwhere nis an arbitrary fixed vector that is timelike-normal n2=−1 and is future-oriented\nn0>0, namely n0=√\n1 +n2, and τ∈Rparametrizes the foliation. Physically, nis the\nnormal vector to the hyperplanes and τis the proper time for this foliation. A schematic\nfigure is given in the left panel in Fig. 1. 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1: Σ n,τinxframe (left) and ˇΣˇn,τ(= Σ n,τ) in ˇxframe (right)\nwhere the “standard vector” ℓis defined to be\n(ℓµ)µ=0,...,d= (1,0) (82)\nin any frame9andL(n) is the “standard boost to the foliation.” Concretely, for the vector\nwith n0=√\n1 +n2,\nL(n) =\u0014n0nt\nnI +\u0000\nn0−1\u0001nnt\nn2\u0015\n, (83)\nwhere t denotes a transpose, I is the identity matrix in ddimensions, nis given in the n×1\nmatrix representation, and L(n) =\u0002\nLµν(n)\u0003\nµ,ν=0,...,d. Note that L−1(n) =L(−n). Now\nan equal-time hyperplane x0=τin the arbitrary reference frame is written as Σ ℓ,τbecause\nℓ·x+τ=−x0+τ= 0 on it.\nFor any given foliation Fn, we may Lorentz-transform from the reference frame xto the\n“time-slice” frame ˇ xthat gives ˇ nµ=ℓµ:\nxµ→ˇxµ=Lνµ(n)xν, (84)\nnµ→ˇnµ=Lνµ(n)nν(=ℓµ), (85)\nwhere we used\u0000\nL−1\u0001µν=Lνµas usual. In the time-slice coordinate system ˇ x, the same\nplane is written as\nˇΣˇn,τ:=n\nˇx=L−1(n)x, x∈R1,d\f\f\fˇn·ˇx+τ= 0o\n(= Σ n,τ). (86)\nAs said above, since ˇ n·ˇx+τ=−ˇx0+τ= 0 on ˇΣˇn,τ, they are equal-time hyperplanes\nparametrized by τ∈Rin the ˇ xcoordinate system. A schematic figure is given in the right\npanel in Fig. 1.\n9In the language of differential geometry, the basis-independent vector is written as ˇ n:= ˇnµ∂µwith∂µ=∂\n∂xµ\nbeing the basis vectors in the reference coordinates. Under the change of basis ∂µ→∂′\nµ= Λ µν∂ν, where\n∂′\nµ:=∂\n∂x′µ, ˇnshould remain the same ˇ n→ˇn′µ∂′\nµ= ˇn′µΛµν∂ν!= ˇnν∂ν, that is, ˇ n′µ= Λµ\nνˇnν.\n17B Wigner representation\nIn this appendix, we briefly review the Wigner representation in the case of massive one-\nparticle state |p, s⟩⟩to spell out our notation; see e.g. Ref. [1] for more details. Here and\nhereafter, we neglect the label for the particle and antiparticle since it is irrelevant for the\ncurrent discussion.\nThe Poincar´ e transformation on a plane-wave state can be written as\n|p, s⟩⟩ →bU(Λ, a)|p, s⟩⟩ (87)\nwhere\nbU(Λ, a) =e−ia·bPbU(Λ) = eia0bH−ia·bPbU(Λ), (88)\nin which bP=\u0010\nbP0,bP\u0011\nis the generator of the spacetime translation. Since the translational\npart is the same as the scalar case, we concentrate on the Lorentz transformation.\nWithout loss of generality, we can choose sto be the of the particle in its rest frame:\n|p, s⟩⟩=bU\u0000\nL(p)\u0001\n|0, s⟩⟩, (89)\nconsistently with the definition (7) as we will see below. Here, sis the spin eigenvalue for the\nrotation in, say, x1-x2plane in the rest frame and the standard boost L(p) is defined by\npµ=:Lµν(p)mℓν, (90)\nin which ℓµis given in Eq. (82). Concretely, the standard boost to pcan be written in terms\nof the “standard boost to a foliation” (83) as10\nL(p) =L(p/m). (91)\nSince |p, s⟩⟩has an internal degree of freedom s, the Lorentz group representation for this\nstate could be nontrivial. To deal with this, we introduce the well-known procedure, Wigner\nrepresentation. First, under the Lorentz transformation, the plane-wave basis transforms as\nbU(Λ)|p, s⟩⟩=bU(Λ)bU\u0000\nL(p)\u0001\n|0, s⟩⟩=bU\u0000\nL(Λp)\u0001bU\u0000\nL−1(Λp)\u0001bU(Λ)bU\u0000\nL(p)\u0001\n|0, s⟩⟩\n=bU\u0000\nL(Λp)\u0001bU\u0000\nW(Λ, p)\u0001\n|0, s⟩⟩, (92)\nwhere\nW(Λ, p) :=L−1(Λp) ΛL(p). (93)\nHere, W(Λ, p) is corresponding to the rotation because this transformation does not change\nthe momentum p. We call this Wigner rotation in SO(d).11Next, we may always write\nbU\u0000\nW(Λ, p)\u0001\n|0, s⟩⟩=X\ns′\f\f0, s′\u000b\u000b\nDs′s\u0000\nW(Λ, p)\u0001\n, (94)\nwhere Dis a finite-dimensional unitary representation of SO(d):\nX\nsDs′′s\u0000\nW(Λ, p)\u0001\nD∗\ns′s\u0000\nW(Λ, p)\u0001\n=δs′s′′. (95)\nPutting this into Eq. (92), we obtain Eq. (9).\n10In general, these two are different concepts, L(p)̸=L(n), since pandnare different.\n11We sloppily write SO(d) when it is to be understood as Spin( d).\n18C Energy, momentum, and number operator in scalar case\nWe give the energy, momentum, and number operators for a real scalar field in terms of the\nLorentz-invariant wave-packet basis. First, we briefly review the scalar wave packet in QFT,\nand then we will show newly-found expressions of these operators in terms of the Lorentz-\ninvariant scalar wave packets.\nThe free field is usually expressed in the plane wave basis:\nbϕ(x) =Zddp\n2p0 \nbα(p)eip·x\n(2π)d/2+bα†(p)e−ip·x\n(2π)d/2!\n, (96)\nwhere bα(p) andbα†(p) are the creation and annihilation operators of the plane waves, which\nsatisfy\u0002\nbα(p),bα†(p′)\u0003\n= 2p0δd(p−p′), etc.\nNow, we define a wave-packet creation operator by [24]\nbA†(Π)|0⟩:=|Π⟩⟩, (97)\nand an annihilation operator bA(Π) by its Hermitian conjugate. The completeness of the scalar\nwave packet (21) leads to the following expansion of the creation and annihilation operators\nof the plane waves:\nbα(p) =Z\nd2dΠ⟨⟨p|Π⟩⟩bA(Π), bα†(p) =Z\nd2dΠbA†(Π)⟨⟨Π|p⟩⟩. (98)\nThus, the free scalar field can be expanded as [24]\nbϕ(x) =Z\nd2dΠh\n⟨⟨x|Π⟩⟩bA(Π) + bA†(Π)⟨⟨Π|x⟩⟩i\n, (99)\nwhere the wave function is given in Eq. (13).\nNow let us rewrite well-known operators in QFT, i.e. the total Hamiltonian, momentum,\nand number operators, into the language of the scalar wave packet.\nFirst, in momentum space, the convergent part of the number operator is described by\nbN:=Zddp\n2p0bα†(p)bα(p). (100)\nSubstituting Eq. (98) into the above expression, we obtain\nbN=Z\nd2dΠZ\nd2dΠ′bA†(Π)\n\nΠ\f\fΠ′\u000b\u000bbA\u0000\nΠ′\u0001\n=Z\nd2dΠbA†(Π)bA(Π). (101)\nOn the second line, we have used\nbA(Π) =Z\nd2dΠ′\n\nΠ\f\fΠ′\u000b\u000bbA\u0000\nΠ′\u0001\n, bA†(Π) =Z\nd2dΠ′bA†\u0000\nΠ′\u0001\n\nΠ′\f\fΠ\u000b\u000b\n, (102)\nwhich follows from Eq. (21). From Eq. (101), we can read off a Lorentz-covariant number-\ndensity operator in the 2 d-dimensional phase space:\nbN=bA†(Π)bA(Π). (103)\n19We now consider the divergent part of the plane-wave number operator, coming from the\nzero-point oscillation:\nbNzero:=Zddp\n2p01\n2h\nbα(p),bα†(p)i\n. (104)\nPutting Eq. (98) into the above expression, we obtain\nbNzero=Z\nd2dΠZ\nd2dΠ′1\n2\n\nΠ\f\fΠ′\u000b\u000b\n\nΠ′\f\fΠ\u000b\u000bb1\n=1\n2Z\nd2dΠb1. (105)\nTherefore, including the divergent part, the number-density operator can be described by\nbN+bNzero:=bA†(Π)bA(Π) +1\n2(106)\nFrom this expression, it can be interpreted that there is one zero-point oscillation per 2 d-\ndimensional phase space volume.\nNext, we consider the convergent part of the total Hamiltonian and momentum operators.\nIn the momentum space, these operators are given by\nbPµ:=Zddp\n2p0pµbα†(p)bα(p). (107)\nPutting Eq. (98) into the above expression, we get\nbPµ=Z\nd2dΠZ\nd2dΠ′bA†(Π)bA\u0000\nΠ′\u0001\n⟨⟨ˆpµ⟩⟩Π,Π′, (108)\nwhere ⟨⟨ˆpµ⟩⟩Π,Π′is given in Eq.(20). We see that the total Hamiltonian and momentum\noperators are not diagonal on the wave packet basis, unlike the plane-wave eigenbasis.\nNow, let us discuss the divergent part of this operator, coming from the zero-point energy:\nbPµ\nzero:=Zddp\n2p0pµ1\n2h\nbα(p),bα†(p)i\n. (109)\nPutting Eq. (98) into the above commutator, we obtain\nbPµ\nzero=1\n2Z\nd2dΠ (Π)Z\nd2dΠ′⟨⟨ˆpµ⟩⟩Π,Π′\n\nΠ′\f\fΠ\u000b\u000bb1\n=1\n2Z\nd2dΠ⟨⟨ˆpµ⟩⟩ϕb1,\n=Zdd+1X\n(2π)dddP\n2P0(−P·N)δ(N·X+T)Pµb1\n=Zdd+1X\n(2π)dddP\n2P0(P·N)2δ(N·X+T)Nµb1. (110)\nwhere we have used the completeness relation (21) in the second line, the formula (17) in the\nthird line, and the same argument as in Eq. (73) in the last line. It is noteworthy that the\nresult becomes the same as in the spinor case (71) up to the factor −4.\n20We may define the zero-point energy in a manifestly Lorentz-invariant fashion:\nEzero:=−NµPµ\nzero\n=Zdd+1X\n(2π)dddP\n2P0(P·N)2δ(N·X+T), (111)\nwhere Pµ\nzerois the coefficient of b1 in the right-hand side of Eq. (110).\nPhysically, we expect that the zero-point energy should be independent of the choice of\nthe space-like hyperplane Σ N,T. We can show it by exploiting the Lorentz invariance of the\nexpression (111) to choose Nµ=ℓµ(= (1 ,0)), without loss of generality. Then, the zero-point\nenergy reduces to the well-known form:\nEzero=Z\nX0=TddXddP\n(2π)d1\n2P0. (112)\nReferences\n[1] S. Weinberg, The Quantum theory of fields. Vol. 1: Foundations . Cambridge University\nPress, 2005.\n[2] K. Ishikawa, O. Jinnouchi, K. Nishiwaki, and K.-y. Oda, Wave-packet effects: a\nsolution for isospin anomalies in vector-meson decay ,Eur. Phys. J. C 83(2023), no. 10\n978, [ arXiv:2308.09933 ].\n[3] K. Ishikawa, O. Jinnouchi, A. Kubota, T. Sloan, T. H. Tatsuishi, and R. Ushioda, On\nexperimental confirmation of the corrections to Fermi’s golden rule ,PTEP 2019\n(2019), no. 3 033B02, [ arXiv:1901.03019 ].\n[4] R. Ushioda, O. Jinnouchi, K. Ishikawa, and T. Sloan, Search for the correction term to\nthe Fermi’s golden rule in positron annihilation ,PTEP 2020 (2020), no. 4 043C01,\n[arXiv:1907.01264 ].\n[5] S. Demidov, S. Gninenko, and D. Gorbunov, Light hidden photon production in high\nenergy collisions ,JHEP 07(2019) 162, [ arXiv:1812.02719 ].\n[6] S. P. Jordan, K. S. M. Lee, and J. Preskill, Quantum Computation of Scattering in\nScalar Quantum Field Theories ,Quant. Inf. Comput. 14(2014) 1014–1080,\n[arXiv:1112.4833 ].\n[7] S. P. Jordan, K. S. M. Lee, and J. Preskill, Quantum Algorithms for Quantum Field\nTheories ,Science 336(2012) 1130–1133, [ arXiv:1111.3633 ].\n[8] Z. Davoudi, C.-C. Hsieh, and S. V. Kadam, Scattering wave packets of hadrons in gauge\ntheories: Preparation on a quantum computer ,arXiv:2402.00840 .\n[9] C. Y. Cardall, Coherence of neutrino flavor mixing in quantum field theory ,Phys. Rev.\nD61(2000) 073006, [ hep-ph/9909332 ].\n[10] E. K. Akhmedov, J. Kopp, and M. Lindner, Oscillations of Mossbauer neutrinos ,JHEP\n05(2008) 005, [ arXiv:0802.2513 ].\n21[11] E. K. Akhmedov and A. Y. Smirnov, Paradoxes of neutrino oscillations ,Phys. Atom.\nNucl. 72(2009) 1363–1381, [ arXiv:0905.1903 ].\n[12] J. Kopp, Mossbauer neutrinos in quantum mechanics and quantum field theory ,JHEP\n06(2009) 049, [ arXiv:0904.4346 ].\n[13] E. K. Akhmedov and A. Y. Smirnov, Neutrino oscillations: Entanglement,\nenergy-momentum conservation and QFT ,Found. Phys. 41(2011) 1279–1306,\n[arXiv:1008.2077 ].\n[14] E. K. Akhmedov and J. Kopp, Neutrino Oscillations: Quantum Mechanics vs.\nQuantum Field Theory ,JHEP 04(2010) 008, [ arXiv:1001.4815 ]. [Erratum: JHEP 10,\n052 (2013)].\n[15] J. Wu, J. A. Hutasoit, D. Boyanovsky, and R. Holman, Neutrino Oscillations,\nEntanglement and Coherence: A Quantum Field theory Study in Real Time ,Int. J.\nMod. Phys. A 26(2011) 5261–5297, [ arXiv:1002.2649 ].\n[16] E. Akhmedov, D. Hernandez, and A. Smirnov, Neutrino production coherence and\noscillation experiments ,JHEP 04(2012) 052, [ arXiv:1201.4128 ].\n[17] M. Blasone, S. De Siena, and C. Matrella, Wave packet approach to quantum\ncorrelations in neutrino oscillations ,Eur. Phys. J. C 81(2021), no. 7 660,\n[arXiv:2104.03166 ].\n[18] H. Mitani and K.-y. Oda, Decoherence in neutrino oscillation between 3D Gaussian\nwave packets ,Phys. Lett. B 846(2023) 138218, [ arXiv:2307.12230 ].\n[19] K. Ishikawa, K. Nishiwaki, and K.-y. Oda, New effect in wave-packet scattering of\nquantum fields ,Phys. Rev. D 108(2023), no. 9 096013, [ arXiv:2102.12032 ].\n[20] P. A. M. Dirac, The quantum theory of the electron ,Proc. Roy. Soc. Lond. A 117\n(1928) 610–624.\n[21] C. Becchi, A. Rouet, and R. Stora, The Abelian Higgs-Kibble Model. Unitarity of the S\nOperator ,Phys. Lett. B 52(1974) 344–346.\n[22] C. Becchi, A. Rouet, and R. Stora, Renormalization of Gauge Theories ,Annals Phys.\n98(1976) 287–321.\n[23] I. V. Tyutin, Gauge Invariance in Field Theory and Statistical Physics in Operator\nFormalism ,arXiv:0812.0580 . Lebedev Institute preprint No. 39 (1975).\n[24] K.-y. Oda and J. Wada, A complete set of Lorentz-invariant wave packets and modified\nuncertainty relation ,Eur. Phys. J. C 81(2021), no. 8 751, [ arXiv:2104.01798 ].\n[25] K. Ishikawa and K.-y. Oda, Particle decay in Gaussian wave-packet formalism revisited ,\nPTEP 2018 (2018), no. 12 123B01, [ arXiv:1809.04285 ].\n[26] D. V. Naumov and V. A. Naumov, Relativistic wave packets in a field theoretical\napproach to neutrino oscillations ,Russ. Phys. J. 53(2010) 549–574.\n22[27] D. V. Naumov and V. A. Naumov, A Diagrammatic treatment of neutrino oscillations ,\nJ. Phys. G37 (2010) 105014, [ arXiv:1008.0306 ].\n[28] D. V. Naumov, On the Theory of Wave Packets ,Phys. Part. Nucl. Lett. 10(2013)\n642–650, [ arXiv:1309.1717 ].\n[29] D. Naumov and V. Naumov, Quantum Field Theory of Neutrino Oscillations ,Phys.\nPart. Nucl. 51(2020), no. 1 1–106.\n[30] V. A. Naumov and D. S. Shkirmanov, Virtual neutrino propagation at short baselines ,\nEur. Phys. J. C 82(2022), no. 8 736, [ arXiv:2208.02621 ].\n[31] G. Kaiser, Phase Space Approach to Relativistic Quantum Mechanics. 1. Coherent State\nRepresentation for Massive Scalar Particles ,J. Math. Phys. 18(1977) 952–959.\n[32] G. R. Kaiser, Phase Space Approach to Relativistic Quantum Mechanics. 2.\nGeometrical Aspects ,J. Math. Phys. 19(1978) 502–507.\n[33] K. Ishikawa, K. Nishiwaki, and K.-y. Oda, Scalar scattering amplitude in Gaussian\nwave-packet formalism ,arXiv:2006.14159 .\n[34] A. Edery, Wave packets in QFT: Leading order width corrections to decay rates and\nclock behavior under Lorentz boosts ,Phys. Rev. D 104(2021), no. 12 125015,\n[arXiv:2106.13768 ].\n23"    },    {        "title": "2301.12705v1.Measuring_Lorentz_Violation_in_Weak_Gravity_Fields.pdf",        "content": "arXiv:2301.12705v1  [gr-qc]  30 Jan 2023Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022\n1\nMeasuring Lorentz Violation in Weak Gravity Fields\nZonghao Li\nDepartment of Physics, Indiana University, Bloomington, I N 47405, USA\nMany new linearized coefficients for Lorentz violation are di scovered in our\nrecent work on the construction of a generic Lorentz-violat ing effective field\ntheory in curved spacetime. The new coefficients can be constr ained by ex-\nperiments in weak gravity fields. In this work, we compare exp eriments in dif-\nferent gravitational potentials and study three types of gr avity-related exper-\niments: free-fall, gravitational interferometer, and gra vitational bound-state\nexperiments. First constraints on the new coefficients for Lo rentz violation are\nextracted from those experiments.\n1. Lorentz violation in gravity\nIn recent years, Lorentz violation has been a popular topic in the se arch\nfor physics beyond the Standard Model (SM) and General Relativit y (GR).\nThe Standard-Model Extension (SME)1,2has been widely used as a com-\nprehensive framework to study Lorentz violation in the context of effective\nfield theory. The minimal terms in the Lagrange density of the SME in\ncurved spacetime were constructed by Kosteleck´ y in 2004,2and the non-\nminimal terms were systematically constructed in our recent work.3,4The\nlinearizations of those terms in weak gravity fields were also obtained .5\nThe present contribution to the proceedings of CPT’22 studies the experi-\nmental implications of the linearized terms with a focus on matter–gr avity\ncouplingsinweakgravityfields. ThisworkisbasedontheresultsinRef .[5].\n2. Potential-dependent experiments\nAn interesting implication ofthe linearized Lagrangedensity constru cted in\nourrecentwork5isthatthemeasuredSMEcoefficientsforLorentzviolation\ncan depend on the gravitationalpotential of the laboratory. Coe fficients for\nLorentz violation have been measured in many experiments under th e as-\nsumption that spacetime is flat.6However, these experiments are typically\nperformed at different elevations and hence at different gravitatio nal po-\ntentials, so the SME coefficients can depend on the potentials.Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022\n2\nTaking the b-type coefficients as an example, we know that the term in\nthe Lagrange density containing the minimal bκcoefficient is L ⊃bκψγκψ\nin flat spacetime.1Adding couplings with the gravitational field, we can\nwrite the generalization of the term in a weak gravity field as5\nL ⊃(bκ\nasy+(bL)κµνhµν+···)ψγκψ≡bκ\nexptψγκψ, (1)\nwherehµν≡gµν−ηµνis the linearized gravitational field. In a nonrela-\ntivistic weak gravityfield, hµνcan be approximatedby h00≈ −2φ,h0j≈0,\nandhjk≈ −2φδjk, whereφis the gravitational potential. The actual bκ\ncoefficients measured in experiments should be the effective value\nbκ\nexpt=bκ\nasy+(bL)κµνhµν+··· ≈bκ\nasy−2(bL)κΣΣφ, (2)\nwhere ΣΣ in the index means a summation over space and time indices in\nthe Sun-centered frame,6i.e., (bL)κΣΣ= (bL)κTT+ (bL)κXX+(bL)κY Y+\n(bL)κZZ. We see that this effective coefficient depends on the gravitational\npotential. Moreover, the combination ( bL)κΣΣcan be constrained by com-\nparing experiments measuring bκat different elevations.\nAs an example, an experiment in Seattle constrained the /tildewidebX\necoefficient,\na combination of bκand other SME coefficients in the electron sector,6\nas|/tildewidebX\ne|<3.7×10−31GeV.7Another experiment in Taiwan measured\nthe same combination and got |/tildewidebX\ne|<3.1×10−29GeV.8The results are\nobtained atdifferent elevationswith different gravitationalpotent ials, sowe\ncan compare them to get a constraint on the linearized coefficient /tildewidebXΣΣ\neas\n|/tildewidebXΣΣ\ne|<3.2×10−15GeV.Similaranalysescanbedoneforothercoefficients\nusing experimental data summarized in the data tables6for the SME.5\nMore experiments can be done to measure SME coefficients in differen t\ngravitational potentials, and those can be used to constrain the lin earized\ncoefficients.\n3. Free-fall experiments\nAsidefromcomparingresultsindifferentexperiments,wecanalsoco nstrain\nthe linearized coefficients by single gravity-related experiments. To better\nanalyze those experiments, we derived the nonrelativisticHamiltonia n from\nthe linearized Lagrange density. The Hamiltonian can be written as\nH=H0+Hφ+Hσφ+Hg+Hσg+..., (3)\nwhereH0is the Hamiltonian without background fields, and other compo-\nnents are the corrections from background fields. Components w ith sub-\nscriptσare spin dependent, and those without are spin independent. The\nexact terms in the components can be found in Ref. [5].Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022\n3\nThe Hamiltonian can modify the gravitational acceleration experienc ed\nby a dynamical system on the Earth’s surface. The spin-dependen t com-\nponents permits us to study spin–gravity couplings in the SME frame work\nfor the first time. In this section, we use free-fall experiments to test the\nspin–gravity couplings.\nOne experiment9compares the effective gravitational accelerations of\ntwo isotopes of strontium atoms, the spin-zero bosonic88Sr and the spin-\n9/2 fermionic87Sr. Unpolarized87Sr atoms were used there, so if effective\ngravitational accelerations depend on spin orientations, the meas ured grav-\nitationalaccelerationsof87Sr atomsshouldspanabroaderrangethan those\nof88Sr atoms. They found no such effect to a sensitivity of 10−7. Ana-\nlyzing the result in our framework, we get bounds on nonrelativistic S ME\ncoefficients as\n/vextendsingle/vextendsingle/vextendsingle(kNR\nσφ)Z\nn/vextendsingle/vextendsingle/vextendsingle<1×10−4GeV,\n/vextendsingle/vextendsingle/vextendsingle(kNR\nσφpp)ZJJ\nn−0.4(kNR\nσφpp)ZZZ\nn/vextendsingle/vextendsingle/vextendsingle<5×10−2GeV−1, (4)\nwhere the subscript nmeans the coefficients are for neutrons, and repeated\nJindices mean a summation over special coordinates J=X,Y,Zin the\nSun-centered frame.\nAnother experiment10compares the gravitational acceleration experi-\nenced by87Rb atoms with different spin orientations. They found no dif-\nference to a sensitivity of 10−7. This can be translated to constraints on\nnonrelativistic coefficients as\n/vextendsingle/vextendsingle/vextendsingle(kNR\nσφ)Z\np−0.6(kNR\nσφ)Z\ne/vextendsingle/vextendsingle/vextendsingle<2×10−5GeV,\n/vextendsingle/vextendsingle/vextendsingle(kNR\nσφpp)ZJJ\np+0.3(kNR\nσφpp)JJZ\np/vextendsingle/vextendsingle/vextendsingle<7×10−3GeV−1, (5)\nwhere the subscripts pandemean proton and electron flavor, respectively.\nAnother type of interesting free-fall experiment is to compare th e falls\nof hydrogen H and antihydrogen H. This can provide insights on CPT\nsymmetry. Several groups have been designing experiments for t hat.11A\ndetailed theoretical analysis of the falls in our framework can be fou nd in\nRef. [5]. We expect new results from those experiments in the near f uture.\n4. Gravitational interferometer experiments\nOur nonrelativistic Hamiltonian can also modify the gravity-induced ph ase\nshiftingravitationalinterferometerexperiments. Inthissection , weanalyzeProceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022\n4\nseveral interferometer experiments with neutrons and use them to extract\nbounds on the nonrelativistic coefficients.\nThe first gravitational interferometer experiment was performe d by\nColella, Overhauser, and Werner (COW).12They used Bragg diffraction\nto split a coherent neutron beam into two paths at different heights and\nmeasured the relative gravity-induced phase shift between the tw o paths.\nUnpolarized neutron beams were used in the experiment, so it is mainly\nsensitive to the spin-independent terms in our Hamiltonian.\nThe effective gravitational acceleration measured in the original CO W\nexperiment attains an accuracy of 10%. From this, we deduce a con straint\non a nonrelativistic coefficient as\n(kNR\nφ)n<1×10−1GeV, (6)\nwhere (kNR\nφ)nis a spin-independent coefficient in the neutron sector. More\nrecent versions of the COW experiment can improve this result.5\nThe next type of interferometer experiments we consider is the Off Spec\nexperiment, which uses polarized nonrelativistic neutron beams and splits\nthe beams by magnetic fields.13This is sensitive to spin–gravity couplings.\nThe experiment measured the effective gravitational acceleration to an ac-\ncuracy of 2.5%. After some analysis,5we get the constraint/vextendsingle/vextendsingle/vextendsingle(kNR\nφ)n+(kNR\nσφ)j\nnˆsj/vextendsingle/vextendsingle/vextendsingle<2.5×10−2GeV, (7)\nwhere (kNR\nφ)nand (kNR\nσφ)j\nnare coefficients in the neutron sector, and ˆ sj\nis the initial polarization direction of the neutron beams. We expect a\nmore precise result to be obtained from a more detailed analysis of th e ex-\nperiment. Also, our understanding of spin–gravity couplings and Lo rentz\nviolation can be further improved by future experiments using similar se-\ntups with the OffSpec experiment. For example, the coefficients ( kNR\nσg)jk\nnin\nour Hamiltonian can be constrained by comparing the phase shifts be tween\nhorizontally split neutron beams with different spin orientations.\n5. Gravitational bound-state experiments\nAnother application of the nonrelativistic Hamiltonian concerns grav ita-\ntional bound-state experiments,15,16where the bounds states of neutrons\nin the Earth’s gravitational field are measured. In those experimen ts, our\nnonrelativistic Hamiltonian can modify the energy states by changing the\npotential experienced by the neutrons. Specifically, the spin-inde pendent\nterms in the Hamiltonian shift the energy levels, and the spin-depend ent\nterms split the energy levels.Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022\n5\nThe first gravitation bound-state experiment15measured the critical\nheights of the bound states, which are related to the energy levels . The\nprecision of the measurement is around 10%. A later experiment16im-\nproved the precision to around 0.3% by measuring the transition fre quen-\ncies between different energy levels. From those results, constra ints on\nnonrelativistic SME coefficients are found to be5,17\n/vextendsingle/vextendsingle(kNR\nφ)n/vextendsingle/vextendsingle<1×10−3GeV,\n/radicalBig/bracketleftbig\n(kNR\nσφ)Jn/bracketrightbig2<8×10−3GeV, (8)\nwhere (kNR\nφ)nand (kNR\nσφ)J\nnare nonrelativistic coefficients in the neutron\nsector, and the square implies a summation over J=X,Y,Zin the Sun-\ncentered frame. We expect these results to be improved by more p recise\nfuture measurements.\nAcknowledgments\nThis work was supported in part by the U.S. Department of Energy a nd\nby the Indiana University Center for Spacetime Symmetries (IUCSS ).\nReferences\n1. D. Colladay and V.A. Kosteleck´ y, Phys. Rev. D 55, 6760 (1997); Phys. Rev.\nD58, 116002 (1998).\n2. V.A. Kosteleck´ y, Phys. Rev. D 69, 105009 (2004).\n3. V.A. Kosteleck´ y and Z. Li, Phys. Rev. D 99, 056016 (2019).\n4. V.A. Kosteleck´ y and Z. Li, Phys. Rev. D 103, 024059 (2021).\n5. V.A. Kosteleck´ y and Z. Li, Phys. Rev. D 104, 044054 (2021).\n6.Data Tables for Lorentz and CPT Violation, V.A. Kosteleck´ y and N. Russell,\n2023 edition, arXiv:0801.0287v16.\n7. B.R. Heckel et al., Phys. Rev. D 78, 092006 (2008).\n8. L.-S. Hou et al., Phys. Rev. Lett. 90, 201101 (2003).\n9. M.G. Tarallo et al., Phys. Rev. Lett. 113, 023005 (2014).\n10. X.C. Duan et al.., Phys. Rev. Lett. 117, 023001 (2016).\n11. S. Aghion et al., Nat. Commun. 5, 4538 (2014); C. Amole et al., Phys. Rev.\nLett. 112, 121102 (2014); P. Indelicato et al., Nat. Commun. 4, 1787 (2013).\n12. R. Colella, A.W. Overhauser, and S.A. Werner, Phys. Rev. Lett.34, 23\n(1975).\n13. V.-O. de Haan et al., Phys. Rev. A 89, 063611 (2014).\n14. V. de Haan et al., Phys. Rev. A 89, 063661 (2014).\n15. V.V. Nesvizhevsky et al., Nature 415, 297 (2002).\n16. G. Cronenberg et al., Nat. Phys. 14, 1022 (2018).\n17. A.N.Ivanov, M. Wellenzohn, and H. Abele, Phys. Lett. B 822136640 (2021)."    },    {        "title": "1207.4390v1.New_complete_orthonormal_sets_of_exponential_type_orbitals_in_standard_convention_and_their_origin.pdf",        "content": "New  complete or thonormal se ts of exponential type orbitals in stand ard convention and \ntheir origin  \nI.I. Guseino v \nDepartment of Physics, Faculty of Arts and Sciences,  Onsekiz Mart University, Çanakkale, Turkey   \nAbstract \n      In standard convention, the new com plete orthonrm al sets of ()*αψ -exponential type \norbitals (()*αψ -ETOs) are introduced as functions of th e com plex or real spherical harm onics \nand ()*αL -modifi ed and ()*pL\u0000-generalized L aguerre polynom ials (()*αL -MPLs and ()*pL-\nGLPs), \n()()()()()** 32 2,( 2) ,x\nnlm nl lm re xSααψζ ζ θϕ−=G\n\u0000L  \n()()()1\n2**\n1 *(1 )!()(2)(*1)pl\nnl nlnlx xL xnqα\nα −−⎡⎤−−=⎢⎥Γ+⎣⎦\u0000L , \nwhere, 0ζ<<∞, 2xrζ= ,*2 2 * pl α=+− ,* 1 qn l *α =++−  and *αis the noninteger or \nintege r (for*αα=) frictional quantum  number ( *3α−∞<<). It is shown that the origin of \nthe ()*αψ -ETOs, ()*\nnlα\n\u0000L -MLPs and ()*pL-GLPs is the self -frictional quan tum forces whic h are  \nanalog of radiation damping or self-frictiona l forces in troduced by Lorentz in class ical \nelectrodynam ics. The relations for the quantum  self-frictional poten tials in term s of ()*αψ -\nETOs, ()*\nnlα\n\u0000L -MLPs and ()*pL\u0000-GLPs, respectively, are establishe d. We note that,  in the c ase of \ndisappearing frictional forces, the ()*αψ -ETOs are reduces to the Schöringer’s wave functions \nfor the hydrogen-like atom s in standard convent ion and, therefore, becom e the noncom plete. \nKeyw ords: Exponential type orbitals, Generalized Laguerre polynom ials, Modified Laguerre \nPolynom ials, Frictional quantum  number \n \n1. Introduction  \nIn a previous paper [1], we  have suggest ed in nonstandard convention the complete \northonorm al sets of αψ-ETOs with integer α, where 2α−∞<≤. It was shown that the  \nLambda and Coulom b-Sturm ian ETOs introduced in Refs. [2-5] are the special cases of the \nαψ-ETOs for 0α=and 1α=, respectively, i.e., 0\nnlm nlm ψ≡Λ and 1\nnlm nlm ψψ≡   (see als o Refs. \n[6-9]). The purpose o f this work is to cons truct the a nalytical re lations f or com plete \northonorm al sets of ETOs and qua ntum  self-frictional potentials  using genera lized Laguerre \npolynom ials defined in standard convention.  \n 2. Definition and basic formulas \n   The com plete orthonorm al sets of ()*αψ -ETOs with in teger () *αα=  and noninteger \nfriction al quantum  num ber *α in this w ork are defined as  \n()()()()(**,,nlm nl lm rR rSαα), ψζ ζθϕ =G                                                                                             (1) \n()()()()()*3\n2 ,2nl nl*R r Rαζζ= xα                                                                                                      (2) \n()()()()**2x\nnl nl Rx e xαα−=\u0000L ,                                                                                                            (3) \nwhere \n()()()1\n2*\n1 *(1 )!()(2)(*1)pl\nnl nlnl *x xL xnqα\nα −−⎡⎤−−=⎢ ⎥Γ+⎣⎦\u0000L                                                                                 (4) \n()[](*\n11 1(* 1)() 1;*1;.(1 )!(*1)p\nnlq) L xF nlnl p−−Γ+= −−−−−Γ+p x+                                                       (5) \n Here, 2xrζ= , *2 2 * pl α=+−  , * 1 qn l *α =++−  and *3α−∞<<; (,lmS)θϕ are the \ncomplex or real spherical harm onics; ()*\n1p\nnlL−− and ()*\nnlαL  are th e gen eralized and  modified \nLaguerre polynom ials, respectively. The ()11 ;; Fηγ xis the confluent hypergeom etric function \ndeterm ined as f ollows: \n()()\n()11\n0;; ,!k\nk\nkkxFxkηηγγ∞\n==∑                                                                                                        (6) \nwhere \n()()() 1... 1kk ηηηη =+ +− with  ()01η=                                                                              (7) \nis the Pochh ammer symbol. In this work, the mathem atical notation for ()*pL\u0000-GLPs (see, for \nexam ple, Re f. [11]) is used. \nWe notice that the Eqs. (1)-(3) are  introduced by the use of m ethod set out in previous works \n[1,10]. \n    The ortho norm ality relations of ()*pL-GLPs, ()*αL -MLPs and ()*αψ -ETOs are defined as \nfollows: \n() () ** *\n1' 1 '\n0(*1 )() ()(1 )!pp xp\nnl nl nnqex L xL xdxnlδ∞\n−\n−− −−Γ+=−− ∫                                                             (8) ()()()()** 2\n'\n0x\nnl n'l nn ex x xdxααδ∞\n−= ∫ \u0000\u0000LL                                                                                             (9) \n()()()()** *3\n''' '' '\n0,,nlm nlm nnllmm rr drααψζ ψζ δδδ∞\n= ∫GG G,                                                                             (10) \nwhere \n()()()()*\n* 2\nn'l n'ln *x xxα\nα⎛⎞=⎜⎟⎝⎠\u0000Lα\n\u0000L                                                                                                    (11) \n()()()(**2,nlm nlmnrxαα\n)*,rαψζ ψζ⎛⎞=⎜⎟⎝⎠G G.                                                                                            (12) \n    The ()*pL\u0000-GLPs satisf y the f ollowing di fferential equations [12] \n() () *\n1() ()p p\nnl nldLx L xdx−− −=−*                                                                                                           (13) \n() () ()2\n** 2\n11 2() (*1) () ( 1) ()0pp\nnl nl nlddxL x p x L x nl L xdx dx−− −− −− ++− +−− =*\n1p.                                         (14) \n   By the use of Eqs. (3), (4), (13) and (14)  it is easy to derive  the f ollowin g formulae: \nfor ()()*\nnlxα\n\u0000L  \n()()()()\n()()()**\n[* ]nl nl nl-1dl x *x xdx x nl qαα α=−\n−\u0000\u0000 \u0000 LL L x                                                                 (15) \n()()()()()()()2\n**\n2*(* 12 ) (1 ) 0nl nl nllp l ddxx p lx x n xdx dx xαα −++−− +−− =\u0000\u0000LL*α\n\u0000L  ,                    (16) \nfor ()()*\nnlR xα\n\u0000  \n()()()()\n()()()** 1\n2 [* ]nl nl nl-1dl x *R xR x Rdx x nl qαα⎛⎞=−+ − ⎜⎟⎝⎠ −\u0000\u0000 xα\n\u0000                                                      (17) \n()()()()() ()()2\n**\n21 1(3 *) (1 *) 024nl nl nlll dd l xxR x R x n R xdx dx x xαααα+ ⎛⎞ ⎛⎞+− ++−−−− = ⎜⎟ ⎜⎟⎝⎠ ⎝⎠\u0000\u0000 \u0000*α.     (18) \nIn section 3, we need also to use th e following re lations: \n()() ()()()()()()* *\n* 1/2nl nl\nnl nldx dR x*/ R xdx dxα α\nα=−+\u0000 \u0000\n\u0000LL xα\n\u0000                                                                  (19) ()() ()()()()()()* *\n* 1/2p\nnl-1 p nl\nnl nl-1dL x dR x l */ R xdx x dxα\nα −\n− =−++\u0000 \u0000\n\u0000 L x\u0000 .                                                         (20) \n    As we see that the definition of the ()*αψ -ETOs and ()*\nnlαL -MLPs, which are the radial p arts \nof ()*αψ -ETOs, and derivation of their differe ntial equations are based on the use of ()*pL-\nGLPs. \n \n3. Expressions for quantum frictional potentials \n    As we pointed ou t that the o rigin of  the ()*αψ -ETOs is the quantum  friction al forces \nproduced b y the particles its elf. According ly, the qua ntum s elf-frictional potential has to \ndepend on the radial parts of ()*αψ -ETOs the Schrödinger equation of whic h is defined by \n(see Ref.[10]) \n() ()()()()* * 2\n22 21 11 1042nl nlll ddx V xxdx dx xαα\nζ+ ⎡⎤⎛⎞−+ ++ ⎢⎜⎟⎝⎠⎣⎦R x= ⎥ .                                                (21) \nThis equatio n can be rew ritten as the f ollows: \n()()\n()()() ()()()()2\n**\n2 2 *1* 13* 042nl nl\nnlll dd d x xxVdx dx dx x Rxαα\nαααζ⎡⎤ −++− − −−− = ⎢⎥\n⎢⎥⎣⎦. xR x                      (22) \nThus, we obtain for the ()()*\nnlR xα\n\u0000  two kinds of  independent equa tions, Eqs. (18) and (22), one \nof which, Eq.(22), contains th e quantum  frictio nal poten tials ()()*\nnlV xα. The comparison of \nEqs. (18) and (22) gives  \n()() ()()() ()()* 22\n** 22 11* /2nl\nnl nldR x lVx n R x .x xd xα\nα ζζα⎡⎤\n=− −− +− ⎢\n⎢⎥⎣⎦xα⎥                                        (23) \n   Now we t ake into account Eqs. (17), (19) and (20). Then, we obtain for the quantum  self-\nfriction al potentials the f ollowing r elations : \n()() ()()()*,,nl n nl Vr U rUαζζ ζ =+*,,rα                                                                                      (24) \nwhere  \n(),nnUrrζζ=−                                                                                                                      (25) ()()()()( )()()()()\n()()()()()()\n()()()()()** 2\n1\n** * 2\n1\n**\n12/ * /   (26)\n,1 *2/ * /   (27)\n// .   (nl nl\nnl nl nl\npp\nnl nlnl qR xR x\nUr nlq x x\nrL xL xαα\nαα αζ\nζα ζ\nζ−\n−\n−− −⎧−⎪\n⎪=− − ⎨\n⎪\n⎪\n⎩LL\n28)\nHere, the function (),nUζr is th e cor e attraction s elf-frictiona l potential.  As we s ee th at the \ntotal self -frictiona l pote ntial ()()*,nlVαζr is a function of the self-frictional constant ζand \nquantum  numbers and ,nl *α. The self -frictiona l parameter ζ can be chosen properly \naccord ing to  the natu re of the particle and corresp onding field under consideration.\n     It should be noted that the se lf-frictional proper ties disappear for Z\nnζ= and *1αα==, \ni.e., \n()()*,nlZVrrαζ=−                 for Z\nnζ= and *1α=.                                                               (29) \nIn th is case,  the ()*αψ -ETOs are reduced to the Schrödi nger’s eigenfunction for the hydrogen-\nlike atom s and becom e the noncomplete, i.e., ()1\nnlm nlm ψψ≡ , where nlmψ is the Schrödinger’s \nwave function in standard convention. \n \n4. Conclusion  \n   In this paper, by the use of explicit e xpression for the generalized L aguerre polynom ials \nthrough the confluent hypergeom etric series, the com plete ort honorm al sets of ETOs and self-\nfrictional potentials are construc ted. The relations for com plete orthonorm al sets of modified \nLaguerre polynom ials through the GLPs are suggest ed. It is shown that the origin of the \nGLPs, therefore, of the ETOs and MLPs, is the self-frictional potentials of the field produced \nby the particle itse lf. The f ormulas f or the ()*αψ -ETOs, ()*\nnlαL -MLPs and ()*pG\u0000-GLP s \npresented in this work can be useful tool  in the num erous physical and m athem atical  \napplication s. They can also be u sed for th e wide applications in  electronic structure \ncalcu lations of atom s, molecules and  solids. \n \nReferences  \n1. I. I. Guseinov, Int. J. Quantum  Chem ., 90 (2002) 114. \n2. E. A. Hylleraas, Z. Phys. 48 (1928) 469. \n3. E. A. Hylleraas, Z. Phys. 54 (1929) 347. \n4. H. Shull, P.O. Löwdi n, J.Che m. Phys., 23 (1955) 1362. 5. P.O. Löwdin, H. Shull, Phys. Rev., 101 (1956) 1730. \n6. K. Rotenberg, Adv. At. Mol. Phys., 6 (1970) 233. \n7. E. J. Weniger, J. Math. Phys., 26 (1985) 276. \n8. E. J. Weniger, E. O. Steinbo rn, J. Math. Phys., 30 (1989) 774. \n9. H. H. H. Homeier, E. J. Weniger, E. O. Steinborn, Int. J. Quantum Chem., 44 (1992) 405. \n10. I. I. Guseinov, American Institute of  Physics Conference Proceeding, 899 (2007) 65. \n11. W. Magnus, F. Oberhettinger, R. P. Soni , Formulas and Theorems for the Specials \nFunctions of Mathematical Physics, New York, Springer, 1966. \n12. I. S. Gradsteyn, I. M. Ryzhik, Tables of Integrals, Sums, Series and Products, 4th ed., \nAcademic Press, New York, 1980. "    },    {        "title": "1412.8646v6.The_classical_Maxwell_Lorentz_electrodynamics_aspects_of_the_electron_inertia_problem_within_the_Feynman_proper_time_paradigm.pdf",        "content": "arXiv:1412.8646v6  [physics.class-ph]  30 Oct 2015THE CLASSICAL MAXWELL-LORENTZ ELECTRODYNAMICS ASPECTS OF\nTHE ELECTRON INERTIA PROBLEM WITHIN THE FEYNMAN PROPER\nTIME PARADIGM\nANATOLIJ K. PRYKARPATSKI AND NIKOLAI N. BOGOLUBOV (JR.)\nAbstract. The Maxwell electromagnetic and the Lorentz type force equa tions are derived in\nthe framework of the R. Feynman proper time paradigm and the r elated vacuum field theory\napproach. The electron inertia problem is analyzed within t he Lagrangian and Hamiltonian\nformalisms and the related pressure-energy compensation p rinciple. The modified Abraham-\nLorentz damping radiation force is derived, the electromag netic elctron mass origin is argued.\n1.Introduction\nTheelementarypointchargedparticle, likeelectron, massproblemw asinspiringmanyphysicists\n[32] from the past as J. J. Thompson, G.G. Stokes, H.A. Lorentz, E . Mach, M. Abraham, P.A. M.\nDirac, G.A. Schott and others. Nonetheless, their studies have no t given rise to a clear explanation\nof this phenomenon that stimulated new researchers to tackle it fr om different approaches based\non new ideas stemming both from the classical Maxwell-Lorentz elect romagnetic theory, as in\n[13, 22, 23, 50, 20, 21, 26, 27, 34, 35, 41, 42, 44, 45, 46, 48, 52, 57, 58, 59, 62], and modern quantum\nfield theories of Yang-Mills and Higgs type, as in [3, 28, 29, 61] and oth ers, whose recent and\nextensive review is done in [60].\nInthepresentworkIwillmostlyconcentrateondetailanalysisand consequencesoftheFeynman\nproper time paradigm [20, 21, 15, 16] subject to deriving the electr omagnetic Maxwell equations\nand the related Lorentz like force expression considered from the vacuum field theory approach,\ndevelopedinworks[11,10,12,8], andfurther, onitsapplicationsto theelectromagneticmassorigin\nproblem. Our treatment ofthis and related problems, based on the least action principle within the\nFeynman proper time paradigm [20], has allowed to construct the res pectively modified Lorentz\ntype equation for a moving in space and radiating energy charged po int particle. Our analysis\nalso elucidates, in particular, the computations of the self-interac ting electron mass term in [41],\nwhere there was proposed a not proper solution to the well known c lassical Abraham-Lorentz\n[1, 38, 39, 40] and Dirac [14] electron electromagnetic ”4/3-electr on mass” problem. As a result of\nour scrutinized studying the classical electromagnetic mass proble m we have stated that it can be\nsatisfactorysolved within the classicalH. Lorentz and M. Abraham reasoningsaugmented with the\nadditional electron stability condition, which was not taken before in to account yet appeared to be\nvery important for balancing the related electromagnetic field and m echanical electron momenta.\nThe latter, following recent enough works [52, 44], devoted to ana lyzing the electron charged shell\nmodel, can be realized within there suggested pressure-energy compensation principle, suitably\napplied to the ambient electromagnetic energy fluctuations and the own electrostatic Coulomb\nelectron energy.\n2.Feynman proper time paradigm geometric analysis\nIn this section, we will develop further the vacuum field theory appr oach within the Feynman\nproper time paradigm, devised before in [12, 10], to the electromagn etic J.C. Maxwell and H.\nLorentz electron theories and show that they should be suitably mo dified: namely, the basic\nLorentz force equations should be generalized following the Landau -Lifschitz least action recipe\n[37], taking also into account the pure electromagnetic field impact. W hen applied the devised\nDate : present.\n1991Mathematics Subject Classification. PACS: 11.10.Ef, 11.15.Kc, 11.10.-z; 11.15.-q, 11.10.Wx, 0 5.30.-d.\nKey words and phrases. classical Maxwell elctrodynamics, electron inertia probl em, Feynman proper time par-\nadigm, least action principle, Lagrangian and Hamiltonian formalisms, Lorentz type force derivation, modified\nAbraham-Lorentz damping radiation force.\n12 ANATOLIJ K. PRYKARPATSKI AND NIKOLAI N. BOGOLUBOV (JR.)\nvacuum field theory approach to the classical electron shell model, the resulting Lorentz force\nexpression appears to satisfactorily explaine the electron inertial mass term exactly coinciding\nwith the electron relativistic mass, thus confirming the well known as sumption [31, 53] by M.\nAbraham and H. Lorentz.\nAs was reported by F. Dyson [15, 16], the original Feynman approac h derivation of the electro-\nmagnetic Maxwell equations was based on an a priori general form of the classical Newton type\nforce, acting on a charged point particle moving in three-dimensiona l spaceR3endowed with the\ncanonical Poisson brackets on the phase variables, defined on the associated tangent space T(R3).\nAs a result of this approach there only the first part of the Maxwell equations were derived, as the\nsecond part, owing to F. Dyson [15], is related with the charged matt er nature, which appeared\nto be hidden. Trying to complete this Feynman approach to the deriv ation of Maxwell’s equa-\ntions more systematically we have observed [10] that the original Fe ynman’s calculations, based\non Poisson brackets analysis, were performed on the tangent space T(R3) which is, subject to the\nproblem posed, not physically proper. The true Poisson brackets c an be correctly defined only on\nthecoadjoint phase space T∗(R3),as seen from the classical Lagrangian equations and the related\nLegendre transformation [33, 4, 25, 7] from T(R3) toT∗(R3).Moreover, within this observation,\nthe corresponding dynamical Lorentz type equation for a charge d point particle should be written\nfor the particle momentum, not for the particle velocity, whose valu e is well defined only with\nrespect to the proper relativistic reference frame, associated w ith the charged point particle owing\nto the fact that the Maxwell equations are Lorentz invariant.\nThus, from the very beginning, we shall reanalyze the structure o f the Lorentz force exerted\non a moving charged point particle with a charge ξ∈Rby another point charged particle with a\nchargeξf∈R, making use of the classical Lagrangian approach, and rederive th e corresponding\nelectromagnetic Maxwell equations. The latter appears to be stro ngly related to the charged point\nmass structure of the electromagnetic origin as was suggested by R. Feynman and F. Dyson.\nConsider a charged point particle moving in an electromagnetic field. F or its description, it is\nconvenient to introduce a trivial fiber bundle structure π:M →R3,M=R3×G,with the abelian\nstructure group G:=R\\{0},equivariantly acting on the canonically symplectic coadjoint space\nT∗(M) endowed both with the canonical symplectic structure\nω(2)(p,y;r,g) :=d pr∗α(1)(r,g) =< dp,∧dr >+ (2.1)\n+< dy,∧g−1dg >G+< ydg−1,∧dg >G\nfor all (p,y;r,g)∈T∗(M),whereα(1)(r,g) :=< p,dr > +< y,g−1dg >G∈T∗(M) is the\ncorresponding Liouville form on M,and with a connection one-form A:M→T∗(M)×Gas\n(2.2) A(r,g) :=g−1< ξA(r),dr > g+g−1dg,\nwithξ∈ G∗,(r,g)∈R3×G,and<·,·>being the scalar product in E3.The corresponding\ncurvature 2-form Σ(2)∈Λ2(R3)⊗ Gis\n(2.3) Σ(2)(r) :=dA(r,g)+A(r,g)∧A(r,g) =ξ3/summationdisplay\ni,j=1Fij(r)dri∧drj,\nwhere\n(2.4) Fij(r) :=∂Aj\n∂ri−∂Ai\n∂rj\nfori,j=1,3 is the electromagnetic tensor with respect to the reference fra meKt,characterized\nby the phase space coordinates ( r,p)∈T∗(R3). As an element ξ∈ G∗is still not fixed, it is natural\nto apply the standard [33, 4, 7] invariant Marsden–Weinstein–Meye r reduction to the orbit factor\nspace ˜Pξ:=Pξ/Gξsubject to the related momentum mapping l:T∗(M)→ G∗,constructed\nwith respect to the canonical symplectic structure (2.1) on T∗(M),where, by definition, ξ∈ G∗\nis constant, Pξ:=l−1(ξ)⊂T∗(M) andGξ={g∈G:Ad∗\nGξ}is the isotropy group of the element\nξ∈ G∗.\nAs a result of the Marsden–Weinstein–Meyer reduction, one finds t hatGξ≃G,the factor-space\n˜Pξ≃T∗(R3) is endowed with a suitably reduced symplectic structure ¯ ω(2)\nξ∈T∗(˜Pξ) and theTHE CLASSICAL MAXWELL ELECTRODYNAMICS 3\ncorresponding Poisson brackets on the reduced manifold ˜Pξare\n{ri,rj}ξ= 0,{pj,ri}ξ=δi\nj, (2.5)\n{pi,pj}ξ=ξFij(r)\nfori,j=1,3,considered with respect to the reference frame Kt.Introducing a new momentum\nvariable\n(2.6) ˜ π:=p+ξA(r)\non˜Pξ,it is easy to verify that ¯ ω(2)\nξ→˜ω(2)\nξ:=< d˜π,∧dr >,giving rise to the following “minimal\ninteraction” canonical Poisson brackets:\n(2.7) {ri,rj}˜ω(2)\nξ= 0,{˜πj,ri}˜ω(2)\nξ=δi\nj,{˜πi,˜πj}˜ω(2)\nξ= 0\nfori,j=1,3 with respect to some new reference frame ˜Kt′,characterized by the phase space\ncoordinates ( r,˜π)∈˜Pξand a new evolution parameter t′∈Rif and only if the Maxwell field\ncompatibility equations\n(2.8) ∂Fij/∂rk+∂Fjk/∂ri+∂Fki/∂rj= 0\nare satisfied on R3for alli,j,k=1,3 with the curvature tensor (2.4).\nNow we proceed to a dynamic description of the interaction between two moving charged point\nparticles ξandξf,moving respectively, with the velocities u:=dr/dtanduf:=drf/dtsubject\nto the reference frame Kt.Unfortunately, there is a fundamental problem in correctly formu lating\na physically suitable action functional and the related least action co ndition. There are clearly\npossibilities such as\n(2.9) S(t)\np:=/integraldisplayt2\nt1dtL(t)\np[r;dr/dt]\non a temporal interval [ t1,t2]⊂Rwith respect to the laboratory reference frame Kt,\n(2.10) S(t′)\np:=/integraldisplayt′\n2\nt′\n1dt′L(t′)\np[r;dr/dt′]\non a temporal interval [ t′\n1,t′\n2]⊂Rwith respect to the moving reference frame Kt′and\n(2.11) S(τ)\np:=/integraldisplayτ2\nτ1dτL(τ)\np[r;dr/dτ]\non a temporal interval [ τ1,τ2]⊂Rwith respect to the proper time reference frame Kτ,naturally\nrelated to the moving charged point particle ξ.\nIt was first observed by Poincar´ e and Minkowski [47] that the te mporal differential dτis not\na closed differential one-form, which physically means that a particle can traverse many different\npaths in space R3with respect to the reference frame Ktduring any given proper time interval dτ,\nnaturally related to its motion. This fact was stressed [17, 18, 43, 47, 49] b y Einstein, Minkowski\nand Poincar´ e,and later exhaustively analyzed by R. Feynman, who argued [20] that the dynamical\nequation of a moving point charged particle is physically sensible only wit h respect to its proper\ntime reference frame. This is Feynman’s proper time reference fra me paradigm, which was recently\nfurther elaborated and applied both to the electromagnetic Maxwe ll equations in [22, 23, 24] and\nto the Lorentz type equation for a moving charged point particle un der external electromagnetic\nfield in [10, 12, 11, 7]. As it was there argued from a physical point of v iew, the least action\nprinciple should be applied only to the expression (2.11) written with re spect to the proper time\nreference frame Kτ,whose temporal parameter τ∈Ris independent of an observer and is a closed\ndifferential one-form. Consequently, this action functional is also mathematically sensible, which\nin part reflects the Poincar´ e’sand Minkowski’s observation that th e infinitesimal quadratic interval\n(2.12) dτ2= (dt′)2−|dr−drf|2,\nrelating the reference frames Kt′andKτ,can be invariantly used for the four-dimensional rel-\nativistic geometry. The most natural way to contend with this prob lem is to first consider the\nquasi-relativistic dynamics of the charged point particle ξwith respect to the moving reference\nframeKt′subject to which the charged point particle ξfis at rest. Therefore, it possible to write4 ANATOLIJ K. PRYKARPATSKI AND NIKOLAI N. BOGOLUBOV (JR.)\ndown a suitable action functional (2.10), up to O(1/c4),as the light velocity c→ ∞, where the\nquasi-classical Lagrangian function L(t′)\np[r;dr/dt′] can be naturally chosen as\n(2.13) L(t′)\np[r;dr/dt′] :=m′(r)|dr/dt′−drf/dt′|2/2−ξϕ′(r).\nwherem′(r)∈R+is the charged particle ξineryial mass parameter and ϕ′(r) is the potential\nfunction generated by the charged particle ξfat a point r∈R3with respect to the reference\nframeKt′.Since the standard temporal relationships between reference fr amesKtandKt′:\n(2.14) dt′=dt(1−|drf/dt′|2)1/2,\nas well as between the reference frames Kt′andKτ:\n(2.15) dτ=dt′(1−|dr/dt′−drf/dt′|2)1/2,\ngive rise, up to O(1/c2),asc→ ∞,todt′≃dtanddτ≃dt′,respectively, it is easy to verify that\nthe least action condition δS(t′)\np= 0 is equivalent to the dynamical equation\n(2.16) dπ/dt=∇L(t′)\np[r;dr/dt] = (1\n2|dr/dt−drf/dt|2)∇m−ξ∇ϕ(r),\nwhere we have defined the generalized canonical momentum as\n(2.17) π:=∂L(t′)\np[r;dr/dt]/∂(dr/dt) =m(dr/dt−drf/dt),\nwith the dash signs dropped and denoted by “ ∇” the usual gradient operator in E3.Equating the\ncanonical momentum expression (2.17) with respect to the refere nce frame Kt′to that of (2.6)\nwith respect to the canonical reference frame ˜Kt′,and identifying the reference frame ˜Kt′with\nKt′,one obtains that\n(2.18) m(dr/dt−drf/dt) =mdr/dt−ξA(r),\ngiving rise to the important inertial particle mass determining expres sion\n(2.19) m=−ξϕ(r),\nwhich right away follows from the relationship\n(2.20) ϕ(r)drf/dt=A(r).\nThe latter is well known in the classical electromagnetic theory [31, 3 7] for potentials ( ϕ,A)∈\nT∗(M4) satisfying the Lorentz condition\n(2.21) ∂ϕ(r)/∂t+<∇,A(r)>= 0,\nyet the expression (2.19) looks very nontrivial in relating the “inertial” mass of the charged point\nparticleξto the electric potential, being both generated by the ambient char ged point particles\nξf.As was argued in articles [11, 10, 51], the above mass phenomenon is c losely related and from\na physical perspective shows its deep relationship to the classical e lectromagnetic mass problem.\nBefore further analysis of the completely relativistic the charge ξmotion under consideration,\nwe substitute the mass expression (2.19) into the quasi-relativistic action functional (2.10) with\nthe Lagrangian (2.13). As a result, we obtain two possible action fun ctional expressions, taking\ninto account two main temporal parameters choices:\n(2.22) S(t′)\np=−/integraldisplayt′\n2\nt′\n1ξϕ′(r)(1+1\n2|dr/dt′−drf/dt′|2)dt′\non an interval [ t′\n1,t′\n2]⊂R,or\n(2.23) S(τ)\np=−/integraldisplayτ2\nτ1ξϕ′(r)(1+1\n2|dr/dτ−drf/dτ|2)dτTHE CLASSICAL MAXWELL ELECTRODYNAMICS 5\non an [τ1,τ2]⊂R. The direct relativistic transformations of (2.23) entail that\nS(τ)\np=−/integraldisplayτ2\nτ1ξϕ′(r)(1+1\n2|dr/dτ−drf/dτ|2)dτ≃ (2.24)\n≃ −/integraldisplayτ2\nτ1ξϕ′(r)(1+|dr/dτ−drf/dτ|2)1/2dτ=\n=−/integraldisplayτ2\nτ1ξϕ′(r)(1−|dr/dt′−drf/dt′|)−1/2dτ=−/integraldisplayt′\n2\nt′\n1ξϕ′(r)dt′,\ngiving rise to the correct, from the physical point of view, relativist ic action functional form (2.10),\nsuitably transformed to the proper time reference frame repres entation (2.11) via the Feynman\nproper time paradigm. Thus, we have shown that the true action fu nctional procedure consists\nin a physically motivated choice of either the action functional expre ssion form (2.9) or (2.10).\nThen, it is transformed to the proper time action functional repre sentation form (2.11) within the\nFeynman paradigm, and the least action principle is applied.\nConcerning the above discussed problem of describing the motion of a charged point particle\nξin the electromagnetic field generated by another moving charged p oint particle ξf,it must\nbe mentioned that we have chosen the quasi-relativistic functional expression (2.13) in the form\n(2.10) with respect to the moving reference frame Kt′,because its form is physically reasonable\nand acceptable, since the charged point particle ξfis then at rest, generating no magnetic field.\nBased on the above relativistic action functional expression\n(2.25) S(τ)\np:=−/integraldisplayτ2\nτ1ξϕ′(r)(1+|dr/dτ−drf/dτ|2)1/2dτ\nwritten with respect to the proper reference from Kτ,one finds the following evolution equation:\n(2.26) dπp/dτ=−ξ∇ϕ′(r)(1+|dr/dτ−drf/dτ|2)1/2,\nwhere the generalized momentum is given exactly by the relationship ( 2.17):\n(2.27) πp=m(dr/dt−drf/dt).\nMaking use of the relativistic transformation (2.14) and the next on e (2.15), the equation (2.26)\nis easily transformed to\n(2.28)d\ndt(p+ξA) =−∇ϕ(r)(1−|uf|2),\nwhere we took into account the related definitions: (2.19) for the c harged particle ξmass, (2.20)\nfor the magnetic vector potential and ϕ(r) =ϕ′(r)/(1−|uf|2)1/2for the scalar electric potential\nwith respect to the laboratory reference frame Kt.Equation (2.28) can be further transformed,\nusing elementary vector algebra, to the classical Lorentz type fo rm:\n(2.29) dp/dt=ξE+ξu×B−ξ∇< u−uf,A >,\nwhere\n(2.30) E:=−∂A/∂t−∇ϕ\nis the related electric field and\n(2.31) B:=∇×A\nis the related magnetic field, exerted by the moving charged point pa rticleξfon the charged point\nparticleξwithrespecttothelaboratoryreferenceframe Kt.TheLorentztypeforceequation (2.29)\nwas obtained in [10, 11] in terms of the moving reference frame Kt′,and recently reanalyzed in\n[11, 50]. The obtained results follow in part [54, 55] from Amp` ere’s cla ssical works on constructing\nthe magnetic force between two neutral conductors with station ary currents.6 ANATOLIJ K. PRYKARPATSKI AND NIKOLAI N. BOGOLUBOV (JR.)\n3.Analysis of the Maxwell and Lorentz force equations: the ele ctron inertial\nmass problem\nAs a moving charged particle ξfgenerates the suitable electric field (2.30) and magnetic field\n(2.31) via their electromagnetic potential ( ϕ,A)∈T∗(M4) with respect to a laboratory reference\nframeKt,wewillsupplementthemnaturallybymeansoftheexternalmaterial equationsdescribing\nthe relativistic charge conservation law:\n(3.1) ∂ρ/∂t+<∇,J >= 0,\nwhere (ρ,J)∈T∗(M4) is a related four-vector for the charge and current distribution in the space\nR3.Moreover, one can augment the equation (3.1) with the experiment ally well established Gauss\nlaw\n(3.2) <∇,E >=ρ\nto calculate the quantity ∆ ϕ:=<∇,∇ϕ >from the expression (2.30):\n(3.3) ∆ ϕ=−∂\n∂t<∇,A >−<∇,E > .\nHaving taken into account the relativistic Lorentz condition (2.21) a nd the expression (3.2) one\neasily finds that the wave equation\n(3.4) ∂2ϕ/∂t2−∆ϕ=ρ\nholds with respect to the laboratory reference frame Kt.Applying the rot-operation “ ∇×” to the\nexpression (2.30) we obtain, owing to the expression (2.31), the eq uation\n(3.5) ∇×E+∂B/∂t= 0,\ngiving rise, together with (3.2), to the first pair of the classical Max well equations. To obtain the\nsecond pair of the Maxwell equations, it is first necessary to apply t he rot-operation “ ∇×”to the\nexpression (2.31):\n(3.6) ∇×B=∂E/∂t+(∂2A/∂t2−∆A)\nand then apply −∂/∂tto the wave equation (3.4) to obtain\n(3.7)−∂2\n∂t2(∂ϕ\n∂t)+<∇,∇∂ϕ\n∂t>=∂2\n∂t2<∇,A >−\n−<∇,∇<∇,A >>=<∇,∂2A\n∂t2−∇×(∇×A)−∆A >=\n=<∇,∂2A\n∂t2−∆A >=<∇,J > .\nThe result (3.7) leads to the relationship\n(3.8) ∂2A/∂t2−∆A=J,\nif we take into account that both the vector potential A∈E3and the vector of current J∈E3are\ndetermined up to a rot-vector expression ∇ ×Sfor some smooth vector-function S:M4→E3.\nInserting the relationship (3.8) into (3.6), we obtain (3.5) and the se cond pair of the Maxwell\nequations:\n(3.9) ∇×B=∂E/∂t+J,∇×E=∂B/∂t.\nIt is important that the system of equations (3.9) can be represen ted by means of the least action\nprinciple δS(t)\nf−p= 0,where the action functional\n(3.10) S(t)\nf−p:=/integraldisplayt2\nt1dtL(t)\nf−p\nis defined on an interval [ t1,t2]⊂Rby the Landau-Lifschitz type [37] Lagrangian function\n(3.11) L(t)\nf−p=/integraldisplay\nR3d3r((|E|2−|B|2)/2+< J,A > −ρϕ)\nwith respect to the laboratory reference frame Kt,which is subject to the electromagnetic field a\nunique and physically reasonable. From (3.11) we deduce that the ge neralized field momentum\n(3.12) πf:=∂L(t)\nf−p/∂(∂A/∂t) =−ETHE CLASSICAL MAXWELL ELECTRODYNAMICS 7\nand its evolution is given as\n(3.13) ∂πf/∂t:=δL(t)\nf−p/δA=J−∇×B,\nwhich is equivalent to the first Maxwell equation of (3.9). As the Maxw ell equations allow the\nleast action representation, it is easy to derive [33, 4, 7, 11, 51] th eir dual Hamiltonian formulation\nwith the Hamiltonian function\n(3.14) Hf−p:=/integraldisplay\nR3d3r < πf,∂A/∂t > −L(t)\nf−p=/integraldisplay\nR3d3r((|E|2−|B|2)/2−< J,A > ),\nsatisfying the invariant condition\n(3.15) dHf−p/dt= 0\nfor allt∈R.\nIt is worth noting here that the Maxwell equations were derived und er the important condition\nthat the charged system ( ρ,J)∈T(M4) exerts no influence on the ambient electromagnetic field\npotentials ( ϕ,A)∈T∗(M4).As this is not actually the case owing to the damping radiation\nreaction on accelerated charged particles, one can try to describ e this self-interacting influence\nby means of the modified least action principle, making use of the Lagr angian expression (3.11)\nrecalculated with respect to the separately chosen charged part icleξendowed with the uniform\nshell model geometric structure and generating this electromagn etic field.\nFollowing the slightly modified well-known approach from [37] and reaso nings from [6, 44] this\nLandau-Lifschitz type Lagrangian (3.11) can be recast (further in the Gauss units) as\n(3.16)L(t)\nf−p=/integraltext\nR3d3r((|E|2−|B|2)/2+/integraltext\nR3d3r(1\nc< J,A > −ρϕ)−< k(t),dr/dt > =\n=/integraltext\nR3d3r(1\n2<−∇ϕ−1\nc∂A/∂t,−∇ϕ−1\nc∂A/∂t > −\n−1\n2<∇×(∇×A),A >)+/integraltext\nR3d3r(1\nc< J,A > −ρϕ)−< k(t),dr/dt > =\n=/integraltext\nR3d3r(1\n2<−∇ϕ,E >−1\n2c< ∂A/∂t,E > −1\n2< A,∇×B >)+\n+/integraltext\nR3(1\nc< J,A > −ρϕ)−< k(t),dr/dt > =\n=/integraltext\nR3d3r(1\n2ϕ <∇,E >+1\n2c< A,∂E/∂t > −1\n2c< A, J+∂E/∂t > )+\n+/integraltext\nR3(1\nc< J,A > −ρϕ)−1\n2cd\ndt/integraltext\nR3d3r < A,E > −\n−1\n2limr→∞/integraltext\nS2r< ϕE+A×B,dS2\nr>−< k(t),dr/dt > =\n=−1\n2/integraltext\n/notlessorslnteql+(ξ)d3r(1\nc< J,A > −ρϕ)+/integraltext\n/notlessorslnteql+(ξ)∪Ω−(ξ)(1\nc< J,A > −ρϕ)−< k(t),dr/dt > −\n−1\n2cd\ndt/integraltext\nR3d3r < A,E > −1\n2limr→∞/integraltext\nS2r< ϕE+A×B,dS2\nr>=8 ANATOLIJ K. PRYKARPATSKI AND NIKOLAI N. BOGOLUBOV (JR.)\n=−1\n2/integraltext\n/notlessorslnteql+(ξ)d3r(1\nc< J,A > −ρϕ)−1\n2/integraltext\n/notlessorslnteql−(ξ)d3r(1\nc< J,A > −ρϕ)+\n+1\n2/integraltext\n/notlessorslnteql−(ξ)d3r(1\nc< J,A > −ρϕ)+/integraltext\n/notlessorslnteql+(ξ)∪Ω−(ξ)(1\nc< J,A > −ρϕ)−< k(t),dr/dt > −\n−1\n2cd\ndt/integraltext\nR3d3r < A,E > −1\n2limr→∞/integraltext\nS2r< ϕE+A×B,dS2\nr>=\n=1\n2/integraltext\n/notlessorslnteql−(ξ)d3r(1\nc< J,A > −ρϕ)−1\n2/integraltext\n/notlessorslnteql+(ξ)∪Ω−(ξ)d3r(1\nc< J,A > −ρϕ)+\n+/integraltext\n/notlessorslnteql+(ξ)∪Ω−(ξ)(1\nc< J,A > −ρϕ)−< k(t),dr/dt > −\n−1\n2cd\ndt/integraltext\nR3d3r < A,E > −1\n2limr→∞/integraltext\nS2r< ϕE+A×B,dS2\nr>=\n=1\n2/integraltext\n/notlessorslnteql−(ξ)d3r(1\nc< J,A > −ρϕ)+1\n2/integraltext\n/notlessorslnteql+(ξ)∪Ω−(ξ)d3r(1\nc< J,A > −ρϕ)−\n−1\n2cd\ndt/integraltext\nR3d3r < A,E > −1\n2limr→∞/integraltext\nS2r< ϕE+A×B,dS2\nr>,\nwherewehaveintroducedstillnotdeterminedaradiationdampingfo rcek(t)∈E3,havedenotedby\n/notlessorslnteql+(ξ) :=supp ξ+⊂R3and/notlessorslnteql−(ξ) :=supp ξ−⊂R3the corresponding charge ξsupports, located\non the electromagnetic field shadowed rear and electromagnetic fie ld exerted front semispheres (see\nFig.1) of the electron shell, respectively to its motion with the fixed ve locityu(t)∈E3,as well as\nwe denoted by S2\nra two-dimensional sphere of radius r→ ∞.\n[The courtesy picture from [44]]\nHaving naturally assumed that the radiated charged particle energ y at infinity is negligible, the\nLagrangian function (3.16) becomes equivalent to\n(3.17)\nL(t)\nf−p=1\n2/integraltext\n/notlessorslnteql−(ξ)d3r(1\nc< J,A > −ρϕ)+1\n2c/integraltext\n/notlessorslnteql+(ξ)∪Ω−(ξ)(< J,A > −ρϕ)−< k(t),dr/dt >,\nwhich we now need to additionally recalculate taking into account that the electromagnetic po-\ntentials ( ϕ,A)∈T∗(M4) are retarded, generated by only the front part of the electron shell and\ngiven as 1 /c2→0 in the following expanded into Lienard-Wiechert series form:\n(3.18)ϕ=/integraltext\nR3d3r′ρ(t′,r′)\n|r−r′|/vextendsingle/vextendsingle/vextendsingle\nt′=t−|r−r′|/c= limε↓0/integraltext\nR3d3r′ρ(t−ε,r′)\n|r−r′|+\n+limε↓01\n2c2/integraltext\nR3d3r′|r−r′|∂2ρ(t−ε,r′)/∂t2+\n+limε↓01\n6c3/integraltext\nR3d3r′|r−r′|2∂ρ(t−ε,r′)/∂t+O(1/c4) =\n=/integraltext\n/notlessorslnteql+(ξ)d3r′ρ(t,r′)\n|r−r′|+1\n2c2/integraltext\n/notlessorslnteql+(ξ)d3r′|r−r′|∂2ρ(t,r′)/∂t2+\n+1\n6c3/integraltext\n/notlessorslnteql+(ξ)d3r′|r−r′|2∂ρ(t,r′)/∂t+O(1/c4),\nA=1\nc/integraltext\nR3d3r′J(t′,r′)\n|r−r′|/vextendsingle/vextendsingle/vextendsingle\nt′=t−|r−r′|/c= limε↓01\nc/integraltext\nR3d3r′J(t−ε,r′)\n|r−r′|−\n−limε↓01\nc2/integraltext\nR3d3r′∂J(t−ε,r′)/∂t+\n+limε↓01\n2c3/integraltext\nR3d3r′|r−r′|∂2J(t−ε,r′)/∂t2+O(1/c4) =\n=1\nc/integraltext\n/notlessorslnteql+(ξ)d3r′J(t,r′)\n|r−r′|−1\nc2/integraltext\n/notlessorslnteql+(ξ)d3r′∂J(t,r′)/∂t+\n+1\n2c3/integraltext\n/notlessorslnteql+(ξ)d3r′|r−r′|∂2J(t,r′)/∂t2+O(1/c4),\nwhere thecurrentdensity J(t,r) =ρ(t,r)dr/dtforallt∈Randr∈Ω(ξ) :=/notlessorslnteql+(ξ)∪/notlessorslnteql+(ξ)≃S2:=\nsupp ρ(t;r)⊂R3,being the spherical compact support of the charged particle dens ity distribution,\nand the limit lim ε↓0was treated physically, that is taking into account the assu med shell modell\nof the charged particle ξand its corresponding charge density self interaction . Moreover, theTHE CLASSICAL MAXWELL ELECTRODYNAMICS 9\npotentials (3.18) are both considered to be retarded and non singu lar, moving in space with the\nvelocity u∈T(R3) subject to the laboratory reference frame Kt.As a result of simple enough\ncalculations like in [31], making use of the expressions (3.18) one obtain s that the Lagranfian\nfunction (3.17) brings about\n(3.19) L(t)\nf−p=Ees\n2c2|u|2−< k(t),dr/dt >,\nwhere we took into account that owing to the reasonings from [6, 44 ] the only front half the electric\ncharge interacts with the whole virtually identical charge charge ξ,as well as made use of the\nfollowing up to O(1/c4) limiting integral expressions:\n(3.20)/integraltext\n/notlessorslnteql+(ξ)∪/notlessorslnteql−(ξ)d3r/integraltext\n/notlessorslnteql+(ξ)∪/notlessorslnteql−(ξ)d3r′ρ(t,r′)ρ(t,r′) :=ξ2,\n1\n2/integraltext\n/notlessorslnteql+(ξ)∪/notlessorslnteql−(ξ)d3r/integraltext\n/notlessorslnteql+(ξ)∪/notlessorslnteql−(ξ)d3r′ρ(t,r′)ρ(t,r′)\n|r−r′|:=Ees,\n/integraltext\n/notlessorslnteql+(ξ)d3rρ(t,r)/integraltext\n/notlessorslnteql+(ξ)d3r′ρ(t;r′)\n|r′−r|=1\n2Ees,\n/integraltext\n/notlessorslnteql−(ξ)d3rρ(t,r)/integraltext\n/notlessorslnteql−(ξ)d3r′ρ(t;r′)\n|r′−r|=1\n2Ees,\n/integraltext\n/notlessorslnteql−(ξ)d3rρ(t,r)/integraltext\n/notlessorslnteql+(ξ)d3r′ρ(t;r′)\n|r−r′||<r′−r,u>\n|r′−r||2>:=Ees\n6|u|2,\n/integraltext\n/notlessorslnteql+(ξ)d3rρ(t,r)/integraltext\n/notlessorslnteql+(ξ)d3r′ρ(t;r′)\n|r−r′||<r′−r,u>\n|r′−r||2>:=Ees\n6|u|2.\nTo obtain the corresponding evolution equation for our charged pa rticleξwe need, within the\nFeynman proper time paradigm, to transform the Lagrangian func tion (3.19) to the one with\nrespect to the proper time reference frame Kτ:\n(3.21) L(τ)\nf−p= (mes/2)|˙r|2(1+|˙r|2/c2)−1/2−< k(t),˙r >,\nwhere, for brevity, we have denoted by ˙ r:=dr/dτthe charged particle velocity with respect to the\nproper reference frame Kτand by, definition, mes:=Ees/c2its so called electrostatic mass with\nrespect to the laboratory refrence frame Kt.\nThus, the generalized charged particle ξmomentum (up to O(1/c4)) equals\n(3.22)πp:=∂L(τ)\nf−p/∂˙r=mes˙r\n(1+|˙r|2/c2)1/2−mes|˙r|2˙r\n2c2(1+|˙r|2/c2)3/2−k(t) =\n=mesu(1−|u|2\n2c2)−k(t)≃mesu(1−|u|2/c2)1/2−k(t) = ¯mesu−k(t),\nwhere we denoted, as before, by u:=dr/dtthe charged particle ξvelocity with respect to the\nlaboratory reference frame Ktand put, by definition,\n(3.23) ¯ mes:=mes(1−|u|2)1/2\nits mass parameter ¯ mes∈R+with respect to the proper reference frame Kτ.\nThe generalized momentum (3.23) satisfies with respect to the prop er reference frame Kτthe\nevolution equation\n(3.24) dπp/dτ:=∂L(τ)\nf−p/∂r= 0,\nbeing equivalent, with respect to the laboratoryreference frame Kt,to the Lorentz type equation\n(3.25)d\ndt(¯mesu) =dk(t)/dt.\nThe evolution equation (3.25) allows the corresponding canonical Ha miltonian formulation on the\nphase space T∗(R3) with the Hamiltonian function\n(3.26)Hf−p:=< πp,r >−L(τ)\nf−p≃<mes˙r\n(1+|˙r|2/c2)1/2−mes|˙r|2˙r\n2c2(1+|˙r|2/c2)3/2−k(t),˙r >−\n−(mes/2)|˙r|2(1+|˙r|2/c2)−1/2+< k(t),˙r >= ¯mes|u|2/2,10 ANATOLIJ K. PRYKARPATSKI AND NIKOLAI N. BOGOLUBOV (JR.)\nnaturally looking and satisfying up to O(1/c4) for allτandt∈Rthe conservation conditions\n(3.27)d\ndτHf−p= 0 =d\ndtHf−p.\nLooking at the equation (3.25) and (3.26), one can state that the p hysically observable inertial\ncharged particle ξmass parameter\n(3.28) mphys:= ¯mes,\nbeing exactly equal to the relativistic charged particle ξelectromagnetic mass, as it was assumed\nby H. Lorentz and Abraham.\nTo determine the damping radiation force k(t)∈E3,we can make use of the Lorentz type force\nexpression (3.17)andobtain, similarlyto[31], upto O(1/c4) accuracy,the resultingself-interecting\nAbraham-Lorentz type force expression. Thus, owing to the zer o net foirce condition, we have that\n(3.29) dπp/dt+Fs= 0,\nwhere the Lorentz force\nFs=−1\n2c/integraldisplay\n/notlessorslnteql−(ξ)d3rρ(t,r)d\ndtA(t,r)−1\n2c/integraldisplay\n/notlessorslnteql+(ξ)∪/notlessorslnteql−(ξ)d3rρ(t,r)d\ndtA(t,r)− (3.30)\n−1\n2/integraldisplay\n/notlessorslnteql−(ξ)d3rρ(t,r)∇ϕ(t,r) (1−|u/c|2)−1\n2/integraldisplay\n/notlessorslnteql+(ξ)∪/notlessorslnteql−(ξ)d3rρ(t,r)∇ϕ(t,r) (1−|u/c|2).\nThis expression easily follows from the least action condition δS(t)= 0,whereS(t):=/integraltextt2\nt1L(t)\nf−pdt\nwith the Lagrangian function given by the derived above Landau-Lif schitz type expression (3.20),\nandthe potentials( ϕ,A)∈T∗(M4) givenbythe Lienard-Wiechertexpressions (3.18). Followedby\ncalculations similar to those of [31, 5], from (3.30) and (3.18) one can o btain, within the assumed\nbefore uniform shell electron model, for small |u/c| ≪1 and slow enough acceleration that\n(3.31)Fs=/summationtext\nn∈Z+(−1)n+1\n2n!cn(1−|u/c|2)[/integraltext\n/notlessorslnteql−(ξ)ρ(t,r)d3r(·)+\n+/integraltext\n/notlessorslnteql+(ξ)∪/notlessorslnteql−(ξ)ρ(t,r)d3r(·)]/integraltext\n/notlessorslnteql+(ξ)d3r′∂n\n∂tnρ(t,r′)∇|r−r′|n−1+\n+/summationtext\nn∈Z+(−1)n+1\n2n!cn+2[/integraltext\n/notlessorslnteql−(ξ)ρ(t,r)d3r(·)+\n+/integraltext\n/notlessorslnteql+(ξ)∪/notlessorslnteql−(ξ)ρ(t,r)d3r(·)]/integraltext\n/notlessorslnteql+(ξ)d3r′|r−r′|n−1∂n+1\n∂tn+1J(t,r′] =\n=/summationtext\nn∈Z+(−1)n+1\n2n!cn+2(1−|u/c|2)[/integraltext\n/notlessorslnteql−(ξ)ρ(t,r)d3r(·)+\n+/integraltext\n/notlessorslnteql+(ξ)∪/notlessorslnteql−(ξ)ρ(t,r)d3r(·)]/integraltext\n/notlessorslnteql+(ξ)d3r′∂n=2\n∂tn+2ρ(t,r′)∇|r−r′|n+1+\n+/summationtext\nn∈Z+(−1)n+1\n2n!cn+2[/integraltext\n/notlessorslnteql−(ξ)ρ(t,r)d3r(·)+\n+/integraltext\n/notlessorslnteql+(ξ)∪/notlessorslnteql−(ξ)ρ(t,r)d3r(·)]/integraltext\n/notlessorslnteql+(ξ)d3r′|r−r′|n−1∂n+1\n∂tn+1J(t,r′).THE CLASSICAL MAXWELL ELECTRODYNAMICS 11\nThe relationship above can be rewritten, owing to the charge contin uity equation (3.1), giving rise\nto the radiation force expression\n(3.32)\nFs=/summationtext\nn∈Z+(−1)n\n2n!cn+2(1−|u/c|2)[/integraltext\n/notlessorslnteql−(ξ)ρ(t,r)d3r(·)+/integraltext\n/notlessorslnteql+(ξ)∪/notlessorslnteql−(ξ)ρ(t,r)d3r(·)]×\n×/integraltext\n/notlessorslnteql+(ξ)d3r′|r−r′|n−1∂n+1\n∂tn+1/parenleftBig\nJ(t,r′)\nn+2+n−1\nn+2<r−r′,J(t,r′)>(r−r′)\n|r−r′|2/parenrightBig\n+\n+/summationtext\nn∈Z+(−1)n+1\n2n!cn+2[/integraltext\n/notlessorslnteql−(ξ)ρ(t,r)d3r(·)+/integraltext\n/notlessorslnteql+(ξ)∪/notlessorslnteql−(ξ)ρ(t,r)d3r(·)]/integraltext\n/notlessorslnteql+(ξ)d3r′|r−r′|n−1∂n+1\n∂tn+1J(t,r′) =\n=/summationtext\nn∈Z+(−1)n+1\n2n!cn+2(1−|u/c|2)[/integraltext\n/notlessorslnteql−(ξ)ρ(t,r)d3r(·)+/integraltext\n/notlessorslnteql+(ξ)∪/notlessorslnteql−(ξ)ρ(t,r)d3r(·)]×\n×/integraltext\n/notlessorslnteql+(ξ)d3r′|r−r′|n−1∂n+1\n∂tn+1/parenleftBig\nJ(t,r′)\nn+2+n−1\nn+2|r−r′,u|2J(t,r′)\n|r−r′|2|u|2/parenrightBig\n+\n+/summationtext\nn∈Z+(−1)n+1\n2n!cn+2[/integraltext\n/notlessorslnteql−(ξ)ρ(t,r)d3r(·)+/integraltext\n/notlessorslnteql+(ξ)∪/notlessorslnteql−(ξ)ρ(t,r)d3r(·)]/integraltext\n/notlessorslnteql+(ξ)d3r′|r−r′|n−1∂n+1\n∂tn+1J(t,r′).\nNow, having applied to (3.32) the rotational symmetry property fo r calculation of the internal\nintegrals, one easily obtains in the case of a charged particle ξuniform shell model that\nFs=/summationtext\nn∈Z+(−1)n\n2n!cn+2(1−|u/c|2)[/integraltext\n/notlessorslnteql−(ξ)ρ(t,r)d3r(·)+/integraltext\n/notlessorslnteql+(ξ)∪/notlessorslnteql−(ξ)ρ(t,r)d3r(·)]×\n×/integraltext\n/notlessorslnteql+(ξ)d3r′|r−r′|n−1∂n+1\n∂tn+1/parenleftBig\nJ(t,r′)\nn+2+(n−1)J(t,r′)\n3(n+2)/parenrightBig\n+\n+/summationtext\nn∈Z+(−1)n+1\n2n!cn[/integraltext\n/notlessorslnteql−(ξ)ρ(t,r)d3r(·)+/integraltext\n/notlessorslnteql+(ξ)∪/notlessorslnteql−(ξ)ρ(t,r)d3r(·)]/integraltext\n/notlessorslnteql+(ξ)d3r′|r−r′|n+1\nc2∂n+1\n∂tn+1J(t,r′) =\n(3.33)=d\ndt[/summationtext\nn∈Z+(−1)n+1\n6n!cn+2[/integraltext\n/notlessorslnteql−(ξ)ρ(t,r)d3r(·)+/integraltext\n/notlessorslnteql+(ξ)∪/notlessorslnteql−(ξ)ρ(t,r)d3r(·)]×\n×/integraltext\n/notlessorslnteql+(ξ)d3r′|r−r′|n−1∂n\n∂tnJ(t,r′)−/summationtext\nn∈Z+(−1)n|u|2\n6n!cn+4)[/integraltext\n/notlessorslnteql−(ξ)ρ(t,r)d3r(·)+\n+/integraltext\n/notlessorslnteql+(ξ)∪/notlessorslnteql−(ξ)ρ(t,r)d3r(·)]/integraltext\n/notlessorslnteql+(ξ)d3r′|r−r′|n−1∂n\n∂tnJ(t,r′)].\nNow, having took into account the integral expressions (3.20), on e finds from (3.33) that up to\ntheO(1/c4) accuracy the following radiation reaction force expression\nFs=−d\ndt/parenleftbiggEes\nc2u/parenrightbigg\n+d\ndt/parenleftbiggEes\n2c2|u/c|2u(t)/parenrightbigg\n+2ξ2\n3c3d2u\ndt2+O(1/c4) = (3.34)\n=−d\ndt/parenleftbigg\nmes(1−|u/c|2\n2)u/parenrightbigg\n+2ξ2\n3c3d2u\ndt2+O(1/c4) =\n=−d\ndt/parenleftBig\nmes(1−|u/c|2)1/2u/parenrightBig\n+2ξ2\n3c3d2u\ndt2+O(1/c4) =\n=−d\ndt(¯mesu−2ξ2\n3c3du\ndt)+O(1/c4)\nholds. We mention here that following the reasonings from [6, 44, 52], in the expressions above\nthere is taken into account an additional hidden and the velocity u∈T(R3) directed electrostatic\nCoulomb surface self-force, acting only on the front half part of the spherical electron shell. As a\nresult, from (3.29), (3.30) and the relationship (3.22) one obtains t hat the electron momentum\n(3.35) πp:= ¯mesu−2ξ2\n3c3du\ndt= ¯mesu−k(t),12 ANATOLIJ K. PRYKARPATSKI AND NIKOLAI N. BOGOLUBOV (JR.)\nthereby defyning both the radiation reaction momentum k(t) =2ξ2\n3c3du\ndtand the corresponding\nradiation reaction force\n(3.36) Fr=2ξ2\n3c3d2u\ndt2+O(1/c4),\ncoincides exactly with the classical Abraham–Lorentz–Dirac expre ssion. Moreover, it also follows\nthat the observable physical shell model electron inertial mass\n(3.37) mph=mes:=Ees/c2,\nbeing completely of the electromagnetic origin, giving rise to the final force expression\n(3.38)d\ndt(mphu) =2ξ2\n3c3d2u\ndt2+O(1/c4).\nThis means, in particular, that the real physically observed “inertia l” massmphof an electron\nwithin the uniform shell model is stronglydetermined by its electroma gneticself-interactionenergy\nEes. A similar statement there was recently demonstrated using comple tely different approaches\nin [52, 44], based on the vacuum Casimir effect considerations. Moreo ver, the assumed above\nboundednessoftheelectrostaticself-energy Eesappearstobe completelyequivalenttothe existence\nof so-called intrinsic Poincar´ e type “ tensions”, analyzed in [6, 44], and to the existence of a special\ncompensatingCoulomb“ pressure”,suggestedin[52], guaranteeingtheobservableelectronstability .\nRemark 3.1.Some years ago there was suggested in the work [41] a ”solution” to the mentioned\nbefore ”4 /3-electron mass” problem, expressed by the physical mass mass r elationship (3.37) and\nformulated more than one hundred years ago by H. Lorentz and M. Abraham. To the regret, the\nabove mentioned ”solution” appeared to be fake that one can easily observe from the main not\ncorrectassumptions on which the work[41] has been based: the fir st one is about the particle-field\nmomentum conservation, taken in the form\n(3.39)d\ndt(p+ξA) = 0,\nand the second one is a speculation about the 1 /2-coefficient imbedded into the calculation of the\nLorentz type self-interaction force\n(3.40) F:=−1\n2c/integraldisplay\nR3d3rρ(t;r)∂A(t;r)/∂t,\nbeing not correctly argued by the reasoning that the expression ( 3.40) represents ”... the inter-\naction of a given element of charge with all other parts, otherwise w e count twice that reciprocal\naction” (cited from [41], page 2710). This claim is fake as there was no t taken into account the\nimportant fact that the interaction in the integral (3.40) is, in realit y, retarded and its impact into\nit should be considered as that calculated for two virtually different c harged particles, as it has\nbeen done in the classical works of H. Lorentz and M. Abraham. Sub ject to the first assumption\n(3.39) it is enough to recall that a vector of the field momentum ξA∈E3is not independent\nand is, within the charged particle model considered, strongly relat ed with the local flow of the\nelectromagnetic potential energy in the Lorentz constraint form :\n(3.41) ∂ϕ/∂t+<∇,A >= 0,\nunder which there hold the exploited in the work [41] the Lienard-Wiec hert expressions (3.17)\npotentials for calculation of the integral (3.40). Thus, the equatio n (3.39), following the classical\nNewton second law, should be replaced by\n(3.42)d\ndt′(p′+ξA′) =−∇(ξϕ′),\nwritten with respect to the reference frame K(t′;r) subject to which the charged particle ξis at\nrest. Taking into account that with respect to the laboratory ref erence frame Ktthere hold the\nrelativistic relationships dt=dt′(1− |u|2/c2)1/2andϕ′=ϕ(1− |u|2/c2)1/2,from (3.42) one\neasily obtains that\n(3.43)d\ndt(p+ξA) =−ξ∇ϕ(1−|u|2/c2) =\n=−ξ∇ϕ+ξ\nc∇< u,uϕ/c > =−ξ∇ϕ+ξ\nc∇< u,A > .THE CLASSICAL MAXWELL ELECTRODYNAMICS 13\nHere we made use of the well-known relationship A=uϕ/cfor the vector potential generated by\nthis charged particle ξmoving in space with the velocity u∈T(R3) with respect to the laboratory\nreference frame Kt.Based now on the equation (3.43) one can derive the final expressio n for the\nevolution of the charged particle ξmomentum:\ndp/dt=−ξ∇ϕ−ξ\ncdA/dt+ξ\nc∇< u,A > = (3.44)\n=−ξ∇ϕ−ξ\nc∂A/∂t−ξ\nc< u,∇> A+ξ\nc∇< u,A > =\n=ξE+ξ\ncu×(∇×A) =ξE+ξ\ncu×B,\nthat is exactly the well known Lorentz force expression, used in th e works of H. Lorentz and M.\nAbraham.\nRecently enough there appeared other interesting works devote d to this ”4 /3-electron mass”\nproblem, amongst which we would like to mention [44, 52], whose argume ntations are close to\neach other and based on the charged shell electron model, within wh ich there is assumed a vir-\ntual interaction of the electron with the ambient ”dark” radiation e nergy. The latter was first\nclearly demonstrated in [52], where a suitable compensation mechanis m of the related singular\nelectrostatic Coulomb electron energy and the wide band vacuum ele ctromagnetic radiation energy\nfluctuations deficit inside the electron shell was shown to be harmon ically realized as the electron\nshell radius a→0.Moreover, this compensation happens exactly when the induced ou tward di-\nrected electrostatic Coulomb pressure on the whole electron coinc ides, up to the sign, with that\ninduced by the mentioned above vacuum electromagnetic energy flu ctuations outside the electron\nshell, since there was manifested their absence inside the electron s hell.\nReally, the outward directed electrostatic spatial Coulomb pressu re on the electron equals\n(3.45) ηcoul:= lim\na→0ε0|E|2\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=a= lim\na→0ξ2\n32ε0π2a4,\nwhereE=ξr\n4πε0|r|3∈E3is the electrostatic field at point r∈Rsubject to the electron center\nat the point r= 0∈R.The related inward directed vacuum electromagnetic fluctuations s patial\npressure equals\n(3.46) ηvac:= lim\nΩ→∞1\n3/integraldisplayΩ\n0dE(ω),\nwheredE(ω) is the electromagnetic energy fluctuations density for a frequen cyω∈R,and Ω∈R\nis the corresponding electromagnetic frequency cutoff. The integ ral (3.46) can be calculated if to\ntake into account the quantum statistical recipe [19, 30, 9] that\n(3.47) dE(ω) :=/planckover2pi1ωd3p(ω)\nh3,\nwhere the Plank constant h:= 2π/planckover2pi1and the electromagnetic wave momentum p(ω)∈E3satisfies\nthe relativistic relationship\n(3.48) |p(ω)|=/planckover2pi1ω/c.\nWhence by substituting (3.48) into (3.47) one obtains\n(3.49) dE(ω) =/planckover2pi1ω3\n2π2c3dω,\nwhich entails, owing to (3.46), the following vacuum electromagnetic e nergy fluctuations spatial\npressure\n(3.50) ηvac= lim\nΩ→∞/planckover2pi1Ω4\n24π2c3.\nFor the charged electron shell model to be stable at rest it is neces sary to equate the inward\n(3.50) and outward (3.45) spatial pressures:\n(3.51) lim\nΩ→∞/planckover2pi1Ω4\n24π2c3= lim\na→0ξ2\n32ε0π2a4,14 ANATOLIJ K. PRYKARPATSKI AND NIKOLAI N. BOGOLUBOV (JR.)\ngiving rise to the balance electron shell radius ab→0 limiting condition:\n(3.52) ab= lim\nΩ→∞/bracketleftBigg\nΩ−1/parenleftbigg3ξ2c2\n2/planckover2pi1/parenrightbigg1/4/bracketrightBigg\n.\nSimultaneously we can calculate the corresponding Coulomb and elect romagnetic fluctuations\nenergy deficit inside the electron shell:\n(3.53) ∆ Wb:=1\n2/integraldisplay∞\nabε0|E|2d3r−/integraldisplayab\n0d3r/integraldisplayΩ\n0dE(ω) =ξ2\n8πε0ab−/planckover2pi1Ω4a3\nb\n6πc3= 0,\nadditionally ensuring the electron shell model stability.\nAnother important consequence from this pressure-energy com pensation mechanism can be\nderived concerning the electron ienrtial mass mph∈R+,entering the momentum expression (3.35)\nin the case of the electron slow enough movement. Namely, following t he reasonings from [44], one\ncan observe that during the electton movement there arises an ad ditional hidden not compensated\nand velocity u∈T(R3) directed electrostatic Coulomb surface self-pressure acting on ly on the\nfront half part of the electron shell and equal to\n(3.54) ηsurf:=|Eξ|\n4πa2\nb1\n2=ξ2\n32πε0a4\nb,\ncoinciding, evidently, with the alreadycompensatedoutwarddirect ed electrostaticCoulombspatial\npressure (3.45). As, evidently,duringtheelectronmotioninspace itssurfaceelectriccurrentenergy\nflow is not vanishing [44], one ensues that the electron momentum ga ins an additional mechanical\nimpact, which can be expressed as\n(3.55) πξ:=−ηsurf4πa3\nb\n3c2u=−1\n3ξ2\n8πε0abc2u=−1\n3¯mesu,\nwhere we took into account that within this electron shell model the corresponding electrostatic\nelectron mass equals its electrostatic energy\n(3.56) ¯ mes=ξ2\n8πε0abc2.\nThus, one can claim that, owing to the structural stability of the ele ctron shell model, its\ngeneralized self-interaction momentum πp∈T∗(R3) gains during the movement with velocity\nu=dr/dt∈T(R3) the additional backwarddirected hidden impact (3.55), which can b e identified\nwith the back-directed momentum component\n(3.57) πξ=−1\n3¯mesu,\ncomplementing the classical [31, 5] momentum expression\n(3.58) πp=4\n3¯mesu,\nwhich can be easily obtained from the Lagrangian expression expres sion, if one not to take into\naccount the shading property of the moving uniform shell electron model. Then, owing to the\nadditional momentum (3.57), the full momentum becomes as\n(3.59) πp=πξ+4\n3¯mesu= (−1\n3¯mes+4\n3¯mes)u= ¯mesu,\ncoinciding with that of (3.22) modulo the radiation reaction momentum k(t) =2ξ2\n3c3du\ndt,strongly\nsupporting the electromagnetic energy origin of the electron inert ion mass for the first time con-\nceived by H. Lorentz and M. Abraham.\n4.Comments\nThe electromagnetic mass origin problem was reanalyzed in details with in the Feynman proper\ntime paradigm and related vacuum field theory approach by means of the fundamental least action\nprinciple and the Lagrangian and Hamiltonian formalisms. The resulting electron inertia appeared\nto coincide in part, in the quasi-relativistic limit, with the momentum exp ression obtained more\nthan one hundred years ago by M. Abraham and H. Lorentz [1, 38, 3 9, 40], yet it proved to\ncontain an additional hidden impact owing to the imposed electron sta bility constraint, which wasTHE CLASSICAL MAXWELL ELECTRODYNAMICS 15\ntaken into account in the original action functional as some prelimina rily undetermined constant\ncomponent. As it was demonstrated in [52, 44], this stability constr aint can be successfully realized\nwithin the charged shell model of electron at rest, if to take into ac count the existing ambient\nelectromagnetic “dark” energy fluctuations, whose inward direct ed spatial pressure on the electron\nshell is compensated by the related outward directed electrostat ic Coulomb spatial pressure as\nthe electron shell radius satisfies some limiting compatibility condition. The latter also allows to\ncompensate simultaneously the corresponding electromagnetic en ergy fluctuations deficit inside\nthe electron shell, thereby forbidding the external energy to flow into the electron. In contrary to\nthe lack of energy flow inside the electron shell, during the electron m ovement the corresponding\ninternal momentum flow is not vanishing owing to the nonvanishing hidd en electron momentum\nflow caused by the surface pressureflow and compensated by the suitably generated surface electric\ncurrent flow. As it was shown, this backward directed hidden momen tum flow makes it possible to\njustify the corresponding self-interaction electron mass expres sion and to state, within the electron\nshell model, the fully electromagnetic electron mass origin, as it has b een conceived by H. Lorentz\nand M. Abraham and strongly supported by R. Feynman in his Lectur es [20]. This consequence is\nalso independently supported by means of the least action approac h, based on the Feynman proper\ntime paradigm and the suitably calculated regularized retarded elect ric potential impact into the\ncharged particle Lagrangian function.\nThe charged particle radiation problem, revisited in this Section, allow ed to conceive the expla-\nnation of the charged particle mass as that of a compact and stable object which should be exerted\nby a vacuum field self-interaction energy. The latter can be satisfie d iff the expressions (3.20)\nhold, thereby imposing on the intrinsic charged particle structure [4 2] some nontrivial geometrical\nconstraints. Moreover, as follows from the physically observed pa rticle mass expressions (3.37),\nthe electrostatic potential energy being of the self-interaction o rigin, contributes into the inertial\nmass as its main relativistic mass component.\nThere exist different relativistic generalizations of the force expre ssion (3.38), which suffer the\ncommon physical inconsistency related to the no radiation effect of a charged particle in uniform\nmotion.\nAnother deeply related problem to the radiation reaction force ana lyzed above is the search for\nan explanation to the Wheeler and Feynman reaction radiation mecha nism, called the absorption\nradiation theory, strongly based on the Mach type interaction of a charged particle with the\nambient vacuum electromagnetic medium. Concerning this problem, o ne can also observe some\nof its relationships with the one devised here within the vacuum field th eory approach, but this\nquestion needs a more detailed and extended analysis.\n5.Acknowledgements\nA.P.would liketo conveyhis cordialthanks to Prof. Hal Puthoff(Ins titute forAdvanced Studies\nat Austin, Texas USA) for sending me his original works, and to Jerr old Zacharias Professor of\nPhysics Roman Jackiw (Department of Physics at the Massachuset t Institute of Technology, MT,\nUSA) for instrumental discussion during his collaborative research stay at the NJIT, NJ USA\nduring May 20-31, 2015, as well as for the related comments and us eful remarks. The authors\nalso very acknowledged to Prof. Denis Blackmore (NJIT, Newark NJ , USA) and Prof. Edward\nKapuscik (Institute for Nuclear Physics at PAS, Krak´ ow, Poland) for friendly cooperation and\nimportant discussions. The work of N.B. was supported by the RSF u nder a grant 14-50-00005.\nReferences\n[1] Abraham M. Dynamik des Electrons. Nachrichten von der Ge selschaft der Wissenschafften zu G¨ ottingen.\nMathematisch-Physikalische Klasse, 1902, S. 20\n[2] Abraham R. and Marsden J. Foundations of Mechanics, Seco nd Edition, Benjamin Cummings, NY,\n[3] Annila A. The Meaning of Mass. International Journal of T heoretical and Mathematical Physics, 2(4), 2012,\np. 67-78\n[4] Arnold V.I. Mathematical Methods of Classical Mechanic s., Springer, NY, 1978\n[5] Di Bartolo B. Classical theory of electromagnetism. The Second Edition, World Scientific, NJ, 2004\n[6] Becker R Theorie der Elektrizitat Bd. II Elektronentheo rie, Berlin: B.G. Teubner, 1933\n[7] Blackmore D., Prykarpatsky A.K. and Samoylenko V.Hr. No nlinear dynamical systems of mathematical physics:\nspectral and differential-geometrical integrability anal ysis. World Scientific Publ., NJ, USA, 201116 ANATOLIJ K. PRYKARPATSKI AND NIKOLAI N. 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QCD and natural phylosophy. Ann. Henry Poinc are, 4, 2003, 211-228\n[62] Yaremko Y. and Tretyak V. Radiation Reaction in Classic al Field Theory. LAP LAMBERT Academic Pub-\nlishing, Germany, 2012\nThe Ivan Franko State Pedagogical University of Drohobych, L viv region, Ukraine, and the Faculty\nof Applied Mathematics at AGH University of Science and Techno logy, Krakow 30059 Poland\nE-mail address :pryk.anat@ua.fm, prykanat@cybergal.com\nThe Abdus Salam InternationalCentre for Theoretical Physic s, Trieste, Italy, andthe V.A. Steklov\nMathematical Institute of RAS, Moscow, Russian Federation\nE-mail address :nikolai bogolubov@hotmail.com"    },    {        "title": "0903.2644v2.Gauge_Invariant_Summation_of_All_QCD_Virtual_Gluon_Exchanges.pdf",        "content": "arXiv:0903.2644v2  [hep-th]  8 Dec 2009Brown-HET-1574\nGauge Invariant Summation of All QCD Virtual Gluon Exchange s\nH. M. Fried§,‡, Y. Gabellini†, T. Grandou†, Y.-M. Sheu§,†,∗\n§Physics Department, Brown University, Providence, RI 0291 2, USA\n†Institut Non Lin ´eaire de Nice, UMR 6618 CNRS;\n1361, Route des Lucioles, 06560 Valbonne, France\n‡Perimeter Institute for Theoretical Physics, Waterloo, ON , Canada N2L 2Y5\n(Dated: November 14, 2018)\nAbstract\nThe interpretation of virtual gluons as ghosts in the non-li near gluonic structure of QCD permits\nthe formulation and realization of a manifestly gauge-inva riant and Lorentz covariant theory of in-\nteracting quarks/anti-quarks, for all values of coupling. The simplest example of quark/anti-quark\nscattering in a high-energy, quenched, eikonal model at lar ge coupling is shown to be expressible as\na set of finite, local integrals which may be evaluated numeri cally; and before evaluation, it is clear\nthat the result will be dependent only on, and is damped by inc reasing momentum transfer, while\ndisplaying physically-reasonable color dependence in a ma nner underlying the MIT Bag Model\nand an effective, asymptotic freedom. Similar but more compli cated integrals will result from all\npossible gluonic-radiative corrections to this simplest e ikonal model. Our results are compatible\nwith an earlier, field-strength analysis of Reinhardt et al.\nPACS numbers: 12.38.-t, 11.15.-q, 12.38.Lg\n∗Electronic address: ymsheu@mailaps.org\n1I. INTRODUCTION\nThere has long been a strong-coupling framework in Abelian QFT, who se lowest-order\napproximation is the Eikonal Model; and, with due attention to color in dices and their dis-\nruptive effects on the coherence of Abelian eikonalization, these te chniques can be extended\nto QCD [1]. In this spirit, we would like to call attention to a novel, manife stly gauge-\ninvariant (MGI) method of calculating the sum of all virtual-gluon exc hange graphs in\nQCD, including—and, in fact, made possible by—cubic and quartic gluon interactions. We\nillustrate this technique by its application to quark–quark ( QQ) or quark–anti-quark( Q¯Q)\nscattering in aneikonal-style, quenched approximation; in effect, w e concentrate onthe sum-\nmation of all possible gluon exchanges with color coupling constant gtreated as anaveraged,\nor constant quantity, neglecting its renormalization, along with qua rk mass and propagator\nrenormalizations.\nBy ’eikonal model’ we mean one specific restriction: that all virtual-g luon 4-momenta\nemitted or absorbed by the scattering quarks are small compared with the incident and final\n4-momenta of the quarks in their center of mass (CM). Correction s to such an eikonal model\nwere defined years ago [2], and in principle may be adjoined to the pres ent discussion; but\nthat is outside the present analysis. In Abelian Physics, this assump tion leads to coherent\nscalar or Neutral Vector Meson (NVM) exchanges; in QCD specific c olor fluctuations are\nintroducedwhichcandestroysuchcoherence. Thesecolortechn iqueswerefirstintroduced[3]\nas an intelligent, quasi-Abelian (QA) approximation to a theory of simp le non-Abelian\nexchanges; but, inthe present paper, with its emphasis onMGIand a concurrent summation\nover all cubic and quartic gluon interactions, such approximations b ecome exact.\nWe treat the quarks as effectively asymptotic particles, since it tak es only two or three\nin combination to make an asymptotic hadron. One can continue to re tain gluons in the\nformalism, and if needed, introduce Faddeev–Popov ghosts [4] to in sure that, when the such\ngauge-dependent gluon propagators are renormalized, expecte d properties are maintained.\nBut in this paper such considerations are suppressed, for we are c oncerned only with the\nenumeration and summation of all virtual gluon exchanges; and it is f or these exchanges\nthat the MGI properties hold. It should also be mentioned, and will be noted below at an\nappropriate point, that all possible gluon-exchange corrections t o the relatively simple forms\npresented below will possess the MGI property.\n2Perhaps the most frequently-used way of introducing gauge invar iance in QCD is by the\nuse of the functional integral (FI)\nZ[j,η,¯η] =N/integraldisplay\nd[A]δ[F(A)]det[δF/δω]·exp/bracketleftbigg\n−i\n4/integraldisplay\nF2/bracketrightbigg\n(1)\n·exp/bracketleftbigg\ni/integraldisplay\n¯η·Gc[A]·η+L[A]+i/integraldisplay\nj·A/bracketrightbigg\n,\nwhere\nGc[A] = [m+γ·(∂−igA·λ)]−1,\nL[A] = Trln[1 −igγ·A·λSc],\nand where ja\nµ,ηµ, and ¯ηµare gluon and quark sources, respectively, the delta-functional of\nF[A] defines the particular gauge adopted, and the det[ δF/δωa] guarantees the color-gauge\ninvariance of the FI when a change of gauge is made by the variation o f a relevant function\nωa(x).Nis a normalization constant which is chosen such that Z[0,0,0] = 1, and the FIs\nover quark coordinates have already been performed. The enume ration of gluonic degrees\nof freedom in this formalism is muted, but perturbative expansions o f (1) are equivalent to\nthose obtained immediately below.\nThere is another, independent method of arriving at the equivalent of (1) in which one\nstarts from Schwinger’s Action principle [5], where the enumeration of proper degrees of\nfreedom is paramount, while gauge invariance takes a secondary an d circuitous path. There,\nasinQED,oneimmediatelyfindsthattheequal-time-commutation-re lations(ETCRs)ofthe\ngluon field operators lead to proper quantization in the Coulomb gaug e; but because of the\nmicro-causality of thefields, [ Aa\ni,Ab\nj]/vextendsingle/vextendsingle\nx0=y0= 0, the morecomplicated, canonically-conjugate\nfield momentum operator πa\ni(x) may bereplaced by ∂0Aa\nifor purposes of calculating relevant\npropagators in a variety of gauges.\nSchwinger’s formalismistheonethatweshall initiallyadopt, beginning w iththechoice of\na relativistic gauge ( e.g., one of standard gauges used in QED, or an axial gauge) for the fr ee\ngluon generating functional (GF), expressing the full GF as a well-d efined Action operator\nacting upon the free GF for gluons and quarks, and then employing a convenient rearrange-\nment of the functional operations in terms of an equivalent but con ceptually-simpler linkage\noperator. For specific processes, in that selected gauge, with th e aid of Halpern and Frad-\nkin/eikonal representations, onesums over ALLvirtual, gluonicflu ctuations, including those\n3due to cubic and quartic gluon interactions; and one then trusts th at subsequent events pro-\nvide the necessary gauge invariance, at least for all physical proc esses, and renormalizations,\nas in QED\nIn this presentation, we begin as above; but before all gluonic fluct uations are performed,\nwe observe that, in QCD, unlike QED, there is one special way of insur ing MGI. This\nsimple step corresponds to treating virtual gluons as ghost gluons , with the result that all\nof the initial, gauge-dependent gluon propagators cancel away; t his is gauge invariance with\na vengeance! With one small exception—which will be discussed and ju stified in Section\nII—the form of this result has previously been found in field-streng th analysis, e.g., that of\nReinhardt et al.[6], with the difference that our result is gauge independent, while tha t of\nRef. [6] allows the choice of an arbitrary gauge; this difference is disc ussed following Eq. (29)\nof the present paper. It may also be of interest to note that the a nalysis of Ref. [6] is in\nEuclidean space, while ours is directly in Minkowski space. And in the co ntext of an eikonal\nmodel, the ’local’ simplifications obtained are sufficient to reduce all th e FIs (used in the\nFradkin representation of Gc[A] andL[A]) to ordinary integrals, susceptible to numerical\nintegration.\nThese ideas, and their application to eikonal models of hadronic scat tering (built out\nof eikonal models for the underlying quarks), should be of use to ph enomenologists and\nexperimentalists, who must translatepureQCD theoryinto practic al predictions andrequire\na separateanalysis of binding, andthattopic isnot covered in thispa per. Rather, we confine\nourselves to the basic properties of quark scattering by the multip le gluon exchanges noted\nabove, and observe qualitative results depending upon the impact p arameter, which are\nreminiscent of the MIT Bag Model, and of asymptotic freedom.\nOne of the common features of QED is that MGI is incompatible with man ifest Lorentz\ncovariance (MLC); that is, one must choose, and has always chose n, a gauge-dependent\nformulation as the price of MLC. The reasons are well known, stemm ing from the effective\nbalancing act of constraints vs. true degrees of photonic freedo m. Traditionally, it has been\nmost convenient to choose a gauge dependence for the covariant photon propagator, assure\noneself of the gauge independence of radiative corrections to tha t photon propagator (by\nmeansofrigorousfermion-chargeconservation), andaccept th enecessity ofgauge-dependent\nphoton propagators as long as all properly-defined S-matrix eleme nts of the theory can be\nshowntobeindependent ofgauge[7]. InQED,asSchwinger hasshow n[8], Green’sfunctions\n4of operators calculated in the Coulomb gauge can be transformed in to Green’s functions in\nconventional relativisticgaugesbyadjoininganoperatorgaugetr ansformationtotheoriginal\noperator, and so retain the basic quantum formalism without the ne ed for indefinite metric\nquantization.\nThe same, basic formalism may be followed in analytic treatments of pe rturbative QCD.\nAn additional, additive feature has been the apparent necessity of adjoining ”ghost” fields\nto the theory, in order to produce a conventional representatio n of the gluon propagator\nwith its proper degrees of freedom [4]. As shown in this note, there exists a simple, ’virtual-\ngluon–ghost’ interpretation which can be used to ’spark’ a MGI and MLC formulation of\nquark/anti-quark interactions; and in this formulation, the non-p erturbative, mathematical\nrepresentation of physical processes is expressed by an FI over the position and color co-\nordinates of a single, anti-symmetric color tensor, χa\nµν(x). This integral, which, long ago,\nwas suggested by Halpern [9, 10], begins life in the definition of an FI; b ut due to the ghost\nnature of the virtual gluons, is reduced to a single n-fold integral o ver ’local’ position and\ncolor coordinates. One advantage of the present method is that it is accessible to couplings\nof any size; and, in fact, the calculations appear to simplify consider ably in the limit of\nstrong coupling.\nAspossiblythesimplest, non-trivialillustration, wesetupthecalcula tionofahigh-energy\nQand/or¯Qeikonal scattering amplitude, in quenched approximation, and at lar ge coupling,\nusing recent eikonal techniques for non-Abelian interactions [2]. At the end of this model\ncalculation, one can see that Halpern’s integral describes an effect ive, ’almost-contact’ in-\nteraction between the Q’s and/or ¯Q’s, replacing the conventional, boson-propagator ’action-\nat-a-distance’ Abelian eikonal result. As in QED, or any Abelian theo ry [11, 12, 13, 14, 15],\na logarithmic growth of a total cross section will require at least a pa rtial lifting of the\nquenched approximation; in this simplest model invoking quenching, o ne finds a scattering\namplitude dependent only upon momentum transfer (or impact para meter), with reasonable\ncolor structure.\nOne simplification employed below should be stressed, for although th e model we present\nresembles an eikonal calculation, certain rather complicated norma lization factors have, for\nconvenience, been omitted, as noted at the appropriate place. Th e thrust of this presenta-\ntion is therefore limited to display the method of virtual-gluon exchan ges; and to show, in a\nscattering context, how dependence upon momentum transfer o r impact parameter controls\n5the disruptive effects of color fluctuations on otherwise-coheren t, eikonal-like exchanges. To\nput this calculation into a strict eikonal framework, as in Appendix B o f the QA refer-\nence [3], one need calculate the neglected normalization factors; an d hence when we refer\nto the ’scattering amplitude’ we mean an unrenormalized, MGI and ML C quantity whose\nmagnitude we can only compare for different impact parameters.\nA list of abbreviations of frequently-used phrases has been added as Appendix D.\nII. FORMULATION\nBegin with QED, and its free-photon Lagrangian,\nL0=−1\n4f2\nµν=−1\n4(∂µAν−∂νAµ)2. (2)\nIts action integral may be rewritten as\n/integraldisplay\nd4xL0=−1\n2/integraldisplay\n(∂νAµ)2+1\n2/integraldisplay\n(∂µAµ)2(3)\n=−1\n2/integraldisplay\nAµ/parenleftbig\n−∂2/parenrightbig\nAµ+1\n2/integraldisplay\n(∂µAµ)2,\nand the difficulty of maintaining both MGI and MLC appears at this stag e. What has\ntypically been done since the original days of Fermi, who simply neglect ed the inconvenient\n(∂µAµ)2term, is to use the latter to define a relativistic gauge in which all calcu lations\nretain MLC, while relying upon strict charge conservation to maintain an effective gauge\ninvariance of the theory.\nThe choice of relativistic gauge can be arranged in various ways; and what shall be done\nhere, in the context of the preceding paragraphs, is to multiply this inconvenient term by\nthe real parameter λ, and transfer it into an effective ’interaction’ term. For definitene ss,\nbegin with the free-field, ( λ= 0, Feynman) propagator D(0)\nc,µν=δµνDc, where ( −∂2)Dc= 1,\nand the free-field Generating Functional (GF)\nZ(0)\n0[j] =ei\n2Rj·D(0)\nc·j, (4)\nand operate upon it by the ’interaction’ λ-term, to produce a new, free-field GF\nZ(ζ)\n0[j] =ei\n2λR(∂µAµ)2/vextendsingle/vextendsingle/vextendsingle\nA→1\niδ\nδj·ei\n2Rj·D(0)\nc·j(5)\n=ei\n2R\nj·D(ζ)\nc·j·e−1\n2Trln[1−λ∂∂\n∂2],\n6whereD(ζ)\nc,µν= (δµν−ζ∂µ∂ν/∂2)Dc, withζ=λ/(λ−1). The functional operation of (5)\nis fully equivalent to a bosonic, gaussian, FI; and such ’linkage operat ion’ statements are\nfrequently more convenient than the standard FI representatio ns, since they do not require\nspecification of infinite normalization constants.\nThe Tr-Log term is an infinite phase factor, representing the sum o f the vacuum energies\ngenerated by longitudinal and time-like photons, with a weight λarbitrarily inserted; this\nquantity could have been removed by an appropriate version of nor mal ordering, but can\nmore simply be absorbed into an overall normalization constant.\nAgain starting from the D(0)\nc,µνof a Feynman propagator, and including the usual fermion\ninteraction Lint=ig¯ψγ·Aψ, and the gauge ’interaction’1\n2λ(∂µAµ)2, it is also easy to show\nthat one generates the standard, Schwinger functional solution in the gauge ζ,\nZ(ζ)\nQED[j,η,¯η] =NeiR¯η·Gc[A]·η+L[A]+i\n2λR(∂µAµ)2/vextendsingle/vextendsingle/vextendsingle\nA→1\niδ\nδj·ei\n2Rj·D(0)\nc·j, (6)\nwhere the phase factor of (5) has been absorbed into N. It will be convenient to rearrange\n(6) using the easily-proven identity\nF/bracketleftbigg1\niδ\nδj/bracketrightbigg\n·ei\n2R\nj·D(ζ)\nc·j≡ei\n2R\nj·D(ζ)\nc·j·eD(ζ)\nA·F[A]/vextendsingle/vextendsingle/vextendsingle\nA=R\nD(ζ)\nc·j, (7)\nwhereD(ζ)\nA=−i\n2/integraltextδ\nδA·D(ζ)\nc·δ\nδA, so that (6) now reads\nZ(ζ)\nQED[j,η,¯η] =Nei\n2R\nj·D(ζ)\nc·j·eD(ζ)\nA·eiR\n¯η·Gc[A]·η+L[A]/vextendsingle/vextendsingle/vextendsingle\nA=R\nD(ζ)\nc·j. (8)\nThis is the functional QED we know, and have used for a half-centur y.\nWe now come to QCD, with\nL=−1\n4F2\nµν−¯ψ[m+γ·∂−igγ·A·λ]ψ, (9)\nandFa\nµν=∂µAa\nν−∂νAa\nµ+gfabcAb\nµAc\nν≡fa\nµν+gfabcAb\nµAc\nν. Since ’proper’ quantization in\nthe Coulomb gauge, for the free and interacting theories yield the s ame ETCRs for QCD as\nfor QED (with an extra δabcolor factor appearing in all relevant equations); and since at\ng= 0, QCD is the same free-field theory as QED (except for additional color indices); and\nsince QED in any of the conventional relativistic gauges can be obtain ed by treating the\ni\n2λ/integraltext\n(∂µAµ)2as an ’interaction’ (as above); and therefore, rather than re-in vent the wheel,\nwe set up QCD in the form used above for QED.\n7As a final preliminary step, we write\n−1\n4/integraldisplay\nF2=−1\n4/integraldisplay\nf2−1\n4/integraldisplay\n(F2−f2) (10)\n≡ −1\n4/integraldisplay\nf2+/integraldisplay\nL′[A],\nwithfa\nµν=∂µAa\nν−∂νAa\nµand\nL′[A] =−1\n4(2fa\nµν+gfabcAb\nµAc\nν)(gfadeAd\nµAe\nν); and for subsequent usage, after an integration-\nby-parts, we note the exact relation\n−1\n4/integraldisplay\nF2=−1\n2/integraldisplay\nAa\nµ/parenleftbig\n−∂2/parenrightbig\nAa\nµ+1\n2/integraldisplay/parenleftbig\n∂µAa\nµ/parenrightbig2+/integraldisplay\nL′[A], (11)\n[In the next few paragraphs, for simplicity, we suppress the quark variables, which will be\nre-inserted at the end of this gluon argument.]\nTo choose a particular relativistic gauge, multiply the 2nd RHS term of (11) byλ, and\ninclude this termaspartofthe interaction, toobtainthefamiliarQCD generating functional\n(GF) in the relativistic gauge specified by ζ=λ/(λ−1)\nZ(ζ)\nQCD[j] =NeiRL′[1\niδ\nδj]·ei\n2λRδ\nδjµ∂µ∂νδ\nδjν·ei\n2Rj·D(0)\nc·j, (12)\nor after rearrangement\nZ(ζ)\nQCD[j] =NeiRL′[1\niδ\nδj]·ei\n2R\nj·D(ζ)\nc·j, (13)\nwith the determinantal phase factor of (5) included in the normaliza tionN, and aδab\nassociated with each free-gluon propagator D(ζ)ab\nc,µν.\nAfter re-inserting the quark variables, and after rearrangemen t, expansion of (13) in\npowers ofggenerates the conventional Feynman graphs of perturbation th eory in the gauge\nζ. If one wishes to have a conventional form for the renormalized glu on propagators, one\ncan insert Faddeev-Popov ghosts into the Lagrangian. But it is clea r that all choices of\nλare possible except λ= 1, for that choice leads to ζ→ ∞and an undefined gluon\npropagator. This is an unfortunate situation because the choice λ= 1 is precisely the one\nwhich corresponds to MGI in QCD, as is clear from (11).\nBut there is a very simple, alternate way of writing (13), by replacing the/integraltext\nL′of that\nequation by the relation given by the exact (11),\n/integraldisplay\nL′[A] =−1\n4/integraldisplay\nF2+1\n2/integraldisplay\nAa\nµ/parenleftbig\n−∂2/parenrightbig\nAa\nµ−1\n2/integraldisplay/parenleftbig\n∂µAa\nµ/parenrightbig2, (14)\n8which (continuing to suppress the quark variables) yields\nZ(ζ)\nQCD[j] =Ne−i\n4RF2−i\n2(1−λ)R(∂µAa\nµ)2+i\n2RAa\nµ(−∂2)Aa\nµ/vextendsingle/vextendsingle/vextendsingle\nA→1\niδ\nδj·ei\n2Rj·D(0)\nc·j.(15)\nIt is now obvious that the choice λ= 1 can be made. It will become clear below that, in\nthe form (15), these operations are exactly equivalent to the intr oduction of a gluonic ghost\nfield; and it is this ’ghost property’ for virtual gluonexchanges tha t generates an exceedingly\nsimple, MGI and MLC result for the present eikonal model—and for all subsequent radiative\ncorrections to this model that can be written. This ghost mechanis m occurs because the\nghost gluon has been introduced by the Feynman propagator assu mption which leads to\nthe factor exp/bracketleftBig\ni\n2/integraltext\nj·D(0)\nc·j/bracketrightBig\nof (15), while the term exp/bracketleftbigi\n2/integraltext\nAa\nµ(−∂2)Aa\nµ/bracketrightbig/vextendsingle/vextendsingle\nA→1\niδ\nδjis that\nfunctional operator which will remove every such propagator fro m the sum of all virtual\nprocesses of every n-point function of the theory, without exce ption. In effect, the gluon\nghost acts as a ’spark plug’ to generate the MGI and MLC interactio ns of the theory, which\nthen take on a remarkably simple form.\nIf one argues that because no color gluons can ever be asymptotic , it is then reasonable to\nsuppress the leading RHS factor exp/bracketleftBig\ni\n2/integraltext\nj·D(0)\nc·j/bracketrightBig\nof the rearranged GF; or, if one wishes\nto retain the specification of individual gluons, that factor may be r etained, and standard\nFaddeev–Popov ghosts inserted to guarantee its proper pertur bative renormalization. In the\nexample to be given shortly, this factor plays no role and will therefo re be omitted.\nAfter rearrangement, and after re-inserting the quark variable s, (15) becomes\nZ(ζ)\nQCD[j,η,¯η] =Ne−i\n2Rδ\nδA·D(0)\nc·δ\nδA·e−i\n4RF2+i\n2RAa\nµ(−∂2)Aa\nµ·eiR\n¯η·Gc[A]·η+L[A]/vextendsingle/vextendsingle/vextendsingle\nA=R\nD(0)\nc·j,(16)\nand we next invoke the representation suggested by\nHalpern [9],\ne−i\n4R\nF2=N′/integraldisplay\nd[χ]ei\n4R\n(χa\nµν)2+i\n2R\nχa\nµνFa\nµν, (17)\nwhere/integraltext\nd[χ] =/producttext\ni/producttext\na/producttext\nµ>ν/integraltext\ndχa\nµν(wi), so that (17) represents a functional integral over\nthe anti-symmetric tensor χa\nµν(w). Here, all space-time is broken up into small regions of\nsizeδ4about each point wiandN′is a normalization constant so chosen that the RHS\nof (17) becomes equal to unity as Fa\nµν→0. In this way, the GF may be rewritten as\n(N′·N=N′′→ N)\nZQCD[j,η,¯η] =N/integraldisplay\nd[χ]ei\n4R(χa\nµν)2·eD(0)\nA·ei\n2R\nχ·F+i\n2R\nAa\nµ(−∂2)Aa\nµ (18)\n·eiR\n¯η·Gc[A]·η+L[A]/vextendsingle/vextendsingle/vextendsingle\nA=RD(0)\nc·j,\n9whereD(0)\nA=−i\n2/integraltextδ\nδA·D(0)\nc·δ\nδA.\nAs noted above, we treat the quarks and anti-quarks as stable en tities during the scatter-\ning; and then must calculate functional derivatives with respect to the sources ¯ η(x1),η(y1),\n¯η(x2), andη(y2), which bring down factors of GI\nc(x1,y1|A) andGII\nc(x2,y2|A), where the su-\nperscripts I and II refer to the scattering fermions. With standa rd mass-shell amputation,\nwe pass to the small-momentum-transfer limit of the eikonal model [ 3], derived in detail in\nAppendix B of this reference for the specific case of QQscattering, and using the conven-\ntional, FI approach in an axial gauge. (The discussion of Appendix B o f ref. [3] contains the\nfull QCD, with cubic and quartic gluon interactions.)\nThe quark scattering amplitude is given by the familiar eikonal form [3 ],\nT(s,t) =is\n2m2/integraldisplay\nd2b ei/vector q·/vectorb/bracketleftbig\n1−eiX(s,b)/bracketrightbig\n, (19)\ns=−(p1+p2)2,\nt=−(p1−p′\n1)2=−q2CM− − → −/vector q2,\nwhile the exponential of the eikonal function, E = exp[ iX], is obtained in this quenched for-\nmalism by the appropriately normalized (as in (B.32) of this reference ) action of the linkage\noperator: exp/bracketleftBig\n−i\n2/integraltextδ\nδA·D(0)\nc·δ\nδA/bracketrightBig\nuponexp/bracketleftbigi\n2/integraltext\nχ·F+i\n2/integraltext\nAa\nµ(−∂2)Aa\nµ/bracketrightbig\n·OE{p1,p′\n1,p2,p′\n2}\nin the limit A→0, where the factors denoted by OE {···}are the ordered exponentials con-\ntributed by the Green’s functions corresponding to the incident an d outgoing particles, as\nnoted below. As remarked in the previous section, for simplicity of pr esentation, certain\nnormalization factors shall be suppressed; and therefore our re sult is only a qualitative ex-\npression of the scattering amplitude, or rather, of the eikonal ex ponential E = exp[ iX], in its\ndependence upon impact parameter. But from this qualitative resu lt it will be possible—by\nmeans of two numerical integrations—to obtain a qualitative picture of the effective inter-\naction potential between a pair of quarks or of a quark and an anti- quark.\nIn QED each such Green’s function Gc[A] would contribute an exponential factor\nexp/bracketleftbig\nig/integraltext\nd4wRµ(w)Aµ(w)/bracketrightbig\nwith\nRµ(w) =pµ/integraldisplay0\n−∞dsδ(w−y+sp)+p′\nµ/integraldisplay∞\n0dsδ(w−y+sp′) (20)\n≃pµ/integraldisplay+∞\n−∞dsδ(w−y+sp),\n10but in QCD it will generate an ordered exponential (OE) of form\n/parenleftbigg\nexp/bracketleftbigg\nigpµ/integraldisplay+∞\n−∞dsAa\nµ(y−sp)λa/bracketrightbigg/parenrightbigg\n+. (21)\nIn order to extract the A-dependence from such OE, we rewrite (21) as\n/integraldisplay\nd[α]δ/bracketleftbig\nαa(s)−gpµAa\nµ(y−sp)/bracketrightbig\n·/parenleftBig\neiR+∞\n−∞dsλaαa(s)/parenrightBig\n+, (22)\nwhere/integraltext\nd[α] is a functional integral defined over all values of the mesh coordin ates−∞ ≤\nsi≤+∞. One then writes a representation for the δ-functional of (22), so that the OE of\n(21) becomes\nN′/integraldisplay\nd[α]/integraldisplay\nd[Ω]e−iR+∞\n−∞dsΩa(s)[αa(s)−gpµAa\nµ(y−sp)]·/parenleftBig\neiR+∞\n−∞dsλaαa(s)/parenrightBig\n+,(23)\nwhereN′=/parenleftbig∆\n2π/parenrightbign/vextendsingle/vextendsingle\nn→∞, a normalizationconstant for thefunctional integral over the’pr oper\ntime’svalues, with the width of each mesh given by ∆ (of dimension (length)2), ∆≃δ2.\nThese operations have become routine in eikonal analysis [1, 3].\nIn our present QCD eikonal scattering amplitude, each Q, or¯Q, is described by a Green’s\nfunction Gc(xi,yi|A), and each has an OE of the form expressed by (23), with corresp onding\npi,yi, and Ωc\nµ(si) variables. In QCD eikonal models, only the interaction correspond ing\nto multiple gluon exchanges between the scattering Q¯Qare retained, and the functions\ncontributing to the eikonal amplitude will contain those pair-wise-int eraction variables in\nthe manner of (36), below.\nThere is another A-dependence contribution to Gc[A], the OE denoted by\n/parenleftbigg\nexp/bracketleftbigg\ng/integraldisplay+∞\n−∞dsσµνFa\nµν(y−sp)λa/bracketrightbigg/parenrightbigg\n+, (24)\nwhere this OE is again defined by its s-value. However, in the present virtual-ghost–gluon\ncalculation, a simple scaling argument shows that these spin-sensitiv e terms do not appear.\nWe retain the basic idea of an eikonal model, concerned with the inter action of a Q,\nor¯Q, each treated as a particle of renormalized mass mand color charge g; and to effect\nthis statement, we suppress all self-energy structure of each Q, or¯Q. We also suppress the\nL[A]-dependence, as in a quenched approximation, where the scatter ing is assumed to occur\nso quickly that charge renormalization effects and any change in the fundamental vacuum\nstructure have insufficient time to react. As shown in the original ca lculation of Cheng and\nWu [11, 12, 13, 15], contributions from L[A] are essential for the increase of total cross\n11sections with scattering energy; and such effects will be missing in th e simple model here\ndescribed.\nFrom (18) and (23), our eikonal exponential function E = eiXwill be proportional to\n/integraldisplay\nd[χ]ei\n4R\nχ2·e−i\n2Rδ\nδA·D(0)\nc·δ\nδA·ei\n2R\nAa\nµKab\nµνAb\nν+iR\nQa\nµAa\nµ/vextendsingle/vextendsingle/vextendsingle\nA→0, (25)\nwhere\nKab\nµν=gfabcχc\nµν+δµνδab(−∂2),\nQa\nµ=−∂νχa\nµν+g/bracketleftbig\nRa\nIµ+Ra\nIIµ/bracketrightbig\n,\n(f·χ)ab\nµν=fabcχc\nµν.\nThe linkage operation is again Gaussian, and yields\ne−1\n2Trln“\n1−K·D(0)\nc”\n·ei\n2R\nQ·»\nD(0)\nc1\n1−K·D(0)\nc–\n·Q, (26)\nand here is where the ’ghost-magic’ appears, since\n1−K·D(0)\nc= 1−[gf·χ+(−∂2)]D(0)\nc (27)\n=−g(f·χ)D(0)\nc.\nTheQ-dependence of (26) is then just\ni\n2/integraldisplay\nQ·D(0)\nc/bracketleftbig\n−g(f·χ)D(0)\nc/bracketrightbig−1·Q (28)\n=i\n2/integraldisplay\nQ·D(0)\nc/bracketleftbig\nD(0)\nc/bracketrightbig−1[−g(f·χ)]−1·Q\n=−i\n2g/integraldisplay\nQ·(f·χ)−1·Q,\nwithallD(0)\ncpropagatorscancelingaway, leavinganintegraloverasinglespace -timevariable,\nw,\n−i\n2g/integraldisplay\nd4wQa\nµ(w) [f·χ(w)]−1/vextendsingle/vextendsingleab\nµνQb\nν(w). (29)\nPrecisely this form of effective interaction was previously found in an instanton approxima-\ntion to a QCD field-strength formalism [6] with the difference that our (29) does not contain\na gauge-fixing term of form δ[ga(x)] of Eq. (17) of that paper (using the notation of that\npaper, where the original gauge-fixing dependence of the Aa\nµvariables was replaced by a\n12simpler gauge fixing of the Fa\nµν, and then transferred to the χa\nµνfield). We consider this an\nunphysical difference because of the following argument.\nDifferent gauges are traditionally chosen in order to simplify calculatio ns of different\nprocesses; for example, in the calculation of infrared effects in QED , the Yennie gauge,\nζ=−2, is to be preferred over that of any other ζvalue because it simplifies the analysis.\nGauge A is chosen to calculate process A because gauge B or C would e ntail a great deal of\nunnecessary work; but because the Physics is independent of gau ge, if no errors are made,\nthen the use of gauge B or C or any other gauge must give exactly th e same results as does\nthe use of gauge A. Instead of using gauge A to calculate any proce ss, we could use the\naverage of gauge A and of gauge B, or take the average over all po ssible gauges; the physical\nanswer must be the same. In the language of reference [6], the cho ice ofga(x) is arbitrary;\nand so we suppose that we may average over an arbitrary number o f suchga(x); and if there\nare a continuous number of such functions, as can surely be imagine d and constructed, then\ninstead of using a particular ga(x), we may simply calculate/integraltext\nd[g]δ[g(x)] over the complete\nfunctional space of such functions, and divide by the (infinite) volu me of such a space, which\nlatter quantity may be absorbed into an overall normalization const ant. The result of this\nlast summation, taken under the/integraltext\nd[χ] integrals, is just unity, which is the form of our (29).\nJustasanaverageoverallpathsispath-independent, soanaver ageover allpossiblegauge\nchoices is gauge independent; our result is gaugeinvariant [16] beca use the property of gauge\nindependence was forced by the ghost mechanism, which automatic ally removes the gluon\npropagators carrying any arbitrary choice of initial gauge. The re striction initially made for\nthe use of the Feynman gauge for D(0)\ncwas only for simplicity of presentation; the entire\ndiscussion could have been carried through by adding and subtract ing another gauge-fixing\nterm to the Lagrangian, adding it to L0and subtracting it from L′. Again expressing the\ninteraction in terms of Halpern’s integral, one finds exactly the same cancelations, except\nthat it is a propagator in an arbitrary gauge that is removed; the de tails are in Appendix C.\nAt this point it may be useful to digress into just how ghost enter QF T, especially in the\ncontext of a linkage operation. For immediate relevance, consider a bosonic ghost, which\nmight be introduced in order to have an intuitive representation of a determinantal factor\n13exp[−1\n2TrlnB], whereBis any desired quantity, or operator. Consider the operation\ne−i\n2Rδ\nδA·Dc·δ\nδA·ei\n2RA·K·A(30)\n=ei\n2R\nA·[K·1\n1−Dc·K]·A·e−1\n2Trln(1−Dc·K),\nwith the choice K=D−1\nc+B. Then, because Dc·D−1\nc= 1, 1−Dc· K=−Dc· B, and\nDc(−B ·Dc)−1=−B−1, so that (30) becomes\ne−i\n2RA·[(Dc·B·Dc)−1+D−1\nc]·A·e−1\n2TrlnB·e−1\n2Trln(−Dc). (31)\nSettingA= 0, and treating exp/bracketleftbig\n−1\n2Trln(−Dc)/bracketrightbig\nas an unimportant, if divergent, nor-\nmalization constant, the remainder is just the desired term. Of cou rse, (31) is completely\nequivalent to the Gaussian functional integral\nN/integraldisplay\nd[φ]ei\n2Rφ·B·φ, (32)\nwith an appropriate normalization N. But the ghost mechanism is not simply just the\nGaussian integral (32), for in the linkage operator formalism one se es the removal of all Dc\nfrom the internal, virtual structure of the theory. It is in this sen se that our virtual QCD\nformalism corresponds to the use of a gluon ghost, which has been f orced upon us by the\nrequirement of MGI; and that requirement is then rigorously satisfi ed by the removal of all\ngauge-dependent gluon propagators.\nIt should be emphasized that this ghost removal will occur automat ically for every cor-\nrection, quenched or unquenched, to the simplified limits of this exam ple. For example,\ntheL[A] terms neglected in this quenched calculation can be retained by a st raightforward\nexpansion of exp {L[A]}in powers of L’s; and every L[A] so included may be expressed in\nterms of an exact Fradkin representation [5, 17], which is itself not m ore complicated than\na sequence of operations upon an exponential of linear and quadra ticA-dependence. The\ntotality of such radiative corrections, exactly or in any form of app roximation, will always\nretain the same form as in (29) or (31) above, with KandQhaving added terms; but the\nremoval of all Dcpropagators must again occur, as MGI is maintained.\nThe question then arises: If the gluon propagators are to disappe ar, what is going to\nreplace them as the ’carriers’ of interactions from one Q, or¯Q, to another? And the an-\nswer is that the Halpern field χa\nµνtakes on a new and physical significance as the carrier of\nthe totality of virtual-gluon interactions, in the form (as seen below ) of an ’almost contact’\n14interaction. Anditisthisnovel interpretationwhichhastheability, in onesuccinct ifcompli-\ncated representation, to display the effective QCD interaction for all values of the coupling,\nlarge or small. To the associated question of a possible phase change for large values of the\ncoupling, we, in this paper, make no prediction, for the answer to th at question demands an\nevaluation of our results for small g, and comparison with simple QCD p erturbation theory.\nOur final Halpern integral is much simpler to evaluate for large couplin g, rather than small;\nand in the interests of simplicity of presentation, that question has been left unanswered.\nAnother untouched question is whether color-charge renormaliza tion in this formalism will\nrequire additional Faddeev–Popov ghosts, which could certainly be inserted, if desired.\nReturning to (25) and (26), now written in the form\nN/integraldisplay\nd[χ]ei\n4R\nχ2·[det(gf·χ)]−1\n2·e−i\n2gRQ·[f·χ]−1·Q. (33)\nWe first drop the self-energy parts of the Q-dependence, retaining for the exponential factor\n−ig2/integraldisplay\nRa\nIµ·(gf·χ)−1/vextendsingle/vextendsingleab\nµν·Rb\nIIν (34)\n−i\n2/parenleftbig\n∂λχa\nµλ/parenrightbig\n·(gf·χ)−1/vextendsingle/vextendsingleab\nµν·/parenleftbig\n∂σχb\n��σ/parenrightbig\n,\nand then, for simplicity, discard all but the largest g-dependence of (34),\n−ig2/integraldisplay\nRa\nIµ·(gf·χ)−1/vextendsingle/vextendsingleab\nµν·Rb\nIIν. (35)\nInserting the eikonal representations of Ra\nIµandRb\nIIν, in the CM of the scattering quarks,\nwe need to evaluate\n−ig/integraldisplay+∞\n−∞ds1/integraldisplay+∞\n−∞ds2p1µp2ν·Ωa\nI(s1)·/integraldisplay\nd4w[f·χ(w)]−1/vextendsingle/vextendsingleab\nµν·Ωb\nII(s2) (36)\n×δ(4)(w−y1+s1p1)·δ(4)(w−y2+s2p2).\nHere,p1,p2,y1andy2are the relevant 4-momenta and space-time coordinates appearin g in\neachQ/¯QGreen’s function, and they are evaluated in the CM of the scatterin gQ/¯Q, which\nare (initially) assumed to have zero relative transverse momentum. In this way,\np1,4=p2,4=iE, p 1,3=−p2,3=p, (37)\np1,1=p1,2=p2,1=p2,2= 0, z=y1−y2,\n(f·χ)−1/vextendsingle/vextendsingleab\n34=i(f·χ)−1/vextendsingle/vextendsingleab\n30, s±=1\n2/bracketleftbiggzL\np±z0\nE/bracketrightbigg\n,\n15so that the product of the two delta-functions of (36) becomes\nδ(2)(/vector y1,⊥−/vector y2,⊥)·δ(s1−s+)·δ(s2−s−) (38)\n·δ(2)(/vector w⊥−/vector y⊥)·δ/parenleftbigg\nwL−1\n2(y1,L+y2,L)/parenrightbigg\n·δ/parenleftbigg\nw0−y1,0+E\npy1,L/parenrightbigg\n·1\n2pE,\nwhere/vector y⊥=/vector y1,⊥=−/vector y2,⊥≡1\n2/vectorb, and the zero of CM time is chosen when both particles\nare at their distance of closest approach, when y1,0=y2,0= 0; then, for all times, z0=\ny1,0−y2,0= 0. Hence, s1=s2; and since y1,0=γms1,s1=y1,0/(γm), and for large γ\nand any reasonable duration of the scattering, s1≈0≈s2. Also,y1,0+y2,0≡2y0, and\ny1,L+y2,L= 0, and the entire (36) may be written as\nigδ(2)(/vectorb)Ωa\nI(0) [f·χ(w)]−1/vextendsingle/vextendsingleab\n30Ωb\nII(0), (39)\nwhere the expected anti-symmetry of the µ,νvariables of [ f·χ]−1has been used, together\nwith thep1,µ,p2,νvalues appropriate to the CM. Note that the wvariable of [ f·χ]−1is a\nfixed 4-vector, given by w(0)\nµ= (/vector y⊥,/vector0L;y0) forE/p≈1.\nThis last restriction immediately means that only this w(0)\nµ, of all the possible w-values of\nthe original Halpern representation, is relevant to this interaction ; and all of the other wµ-\nterms of that functional integral, with their normalization factors , are effectively removed\nfrom the computation in the form of an uninteresting, convergent , normalization factor,\nN′/integraldisplay\nd[χ]ei\n4R\nχ2/radicalBig\ndet(gf·χ)−1, (40)\nwhich separates itself from the b-dependent part of the calculation. The latter, in contrast,\nis now given by\n/productdisplay\na/productdisplay\nµ>ν/bracketleftbigg∆\n(2π)2/bracketrightbigg/integraldisplay+∞\n−∞dχa\nµν(w(0))/radicalBig\ndet(gf·χ)−1ei\n4∆2(χa\nµν(w(0)))2\n(41)\n×eigδ(2)(/vectorb)Ωa\nI(0) [f·χ(w)]−1|ab\n30Ωb\nII(0),\nwhere ∆ = δ2, andδrefers to the small distance, in each of four space-time directions\nsurrounding the point w(0). For a later purpose, we shall borrow a divergent factor from\none of the normalization terms, and so insert a ∆, multiplying χ, into the determinant of\n(41). Henceforth, we suppress the w(0)symbol, with the understanding that the integral of\n16(41) refers to the summation over all possible values of the quantit yχa\nµν(w(0)); and we write\nthe measure of (41) as/producttext\nµ>ν/integraltext\ndnχµν, wherenrefers to the number of independent color\ncontributions of SU(N).\nThe next step is to rescale the ∆-dependence of (41), defining ¯ χa\nµν= ∆χa\nµνand so obtain\n(2π)−2/productdisplay\nµ>ν/integraldisplay\ndn¯χµνei\n4¯χ2/radicalBig\ndet(gf·¯χ)−1(42)\n×eig[∆δ(2)(/vectorb)]Ωa\nI(0) [f·¯χ]−1|ab\n30Ωb\nII(0),\nand we must then interpret the quantity ∆ δ(2)(/vectorb). For this, first write a Fourier representa-\ntion\nδ(2)(/vectorb) = (2π)2/integraldisplay\nd2/vectork⊥ei/vectork⊥·/vectorb, (43)\nand realize that this integral requires a specification of all /vectork⊥. But is this reasonable in an\neikonal model of quarks, where we understand that such quarks can never be measured in\nisolation, with precise values of momenta? Rather, we must extend t his eikonal model to\nallow for unmeasurable transverse momenta exchanged between q uarks of the same hadron,\nbefore any quarks in different hadrons can be imagined to interact w ith each other, as\nis the conceptual situation of this calculation. That transverse mo menta, which can be\ntreated as an average quantity even though it can never be measu red with precision, will\ncertainly be smaller than the CM momenta, or the CM energy, of the h adrons which are\nactually scattering; andit will be onthe same order of magnitude ast he transverse momenta\ndefining the δ-function above. In other words, taking into account that we are talking about\nquark scattering, rather than particle scattering, the magnitud e of the transverse momenta\ninside/integraltext\nd2/vectork⊥must be limited; and the natural parameter which sets the scale for high-\nenergy scattering, in which eikonal models are most relevant, is the CM scattering energy\nof the hadrons. We therefore insert under the integral of/integraltext\nd2/vectork⊥a limiting factor for its\ntransverse momenta; this can be done in many, physically-equivalen t ways, but perhaps the\nsimplest is to use exp/bracketleftBig\n−/vectork2\n⊥/M2/bracketrightBig\n, withMon the order of the CM scattering energy. This\nreplacesδ(2)(/vectorb) by a more realistic Gaussian distribution ( M2/4π)·exp/bracketleftBig\n−M2/vectorb2/4/bracketrightBig\n, and has\nthe further advantage that the product ∆ δ(2)(/vectorb) is now proportional to the dimensionless\nquantity ∆ M2. Since it was our eikonal model that, in part, defined ∆, it is reasona ble to\nchoose the product ∆ M2≡ξas a number ∼O(1), thereby replacing the original ∆ δ(2)(/vectorb)\n17byϕ(b) =ξ\n4πexp/bracketleftBig\n−M2/vectorb2/4/bracketrightBig\n. Eq. (42) then becomes\n(2π)−2/productdisplay\nµ>ν/integraldisplay\ndn¯χµν/radicalBig\ndet(gf·¯χ)−1·ei\n4¯χ2+igϕ(b)Ωa\nI(0) [f·¯χ]−1|ab\n30Ωb\nII(0).(44)\nIII. EVALUATION\nWe next turn to the evaluation of (44), which is to be inserted under the normalized\nfunctional integrals\nN′′/integraldisplay\nd[ΩI]/integraldisplay\nd[ΩII] exp/bracketleftbigg\n−i/integraldisplay+∞\n−∞ds/parenleftbig\nαa\nI(s)Ωa\nI(s)+αb\nII(s)Ωb\nII(s)/parenrightbig/bracketrightbigg\n.(45)\nBut theb-dependence of (44) is associated with Ω I(0) and Ω II(0); and this means that all of\nthe othersivalues,si/ne}ationslash= 0, of the/integraltext\nd[ΩI] and/integraltext\nd[ΩII] may beintegrated immediately, withall\nyielding factors of δ(αI(si)) andδ(αII(sj)),si/ne}ationslash= 0/ne}ationslash=sj. Only the normalized contributions\nof each functional integral with s= 0 are relevant here. In the Quasi-Abelian model of\nreference [3], this was suggested as an intelligent approximation for the SU(2) eikonal model\nconsidered there; but here, for SU(3), it is an almost automatic co nsequence of the ’locality’\nof the gluon-ghost mechanism. And it has the extremely convenient effect of transforming\nthe remaining OE integrations over\n/parenleftbig\nexp/bracketleftbig\n−i/integraltext\nαI·λI/bracketrightbig/parenrightbig\n+and/parenleftbig\nexp/bracketleftbig\n−i/integraltext\nαII·λII/bracketrightbig/parenrightbig\n+into ordinary integrals over unordered quan-\ntities,\n/parenleftbigg∆\n2π/parenrightbiggn/integraldisplay\ndnαIe−i∆αI(0)·λI(46)\n×/parenleftbigg∆\n2π/parenrightbiggn/integraldisplay\ndnαIIe−i∆αII(0)·λII,\nwith the result that all that remains of the color dynamics are the te dious but straightfor-\nward, ordinary integrals\n(2π)−2n/integraldisplay\ndnαI/integraldisplay\ndnαII/integraldisplay\ndnΩI/integraldisplay\ndnΩII (47)\n×eigϕΩa\nI[f·χ]−1|ab\n30Ωb\nII·e−iαI·ΩI−iαII·ΩII\n×e−iαI·λI·e−iαII·λII,\nwhere we have rescaled the αI,IIvariables, and suppressed their now useless (0) notation, as\nwell as the notational change: ¯ χ→χ, for the remaining χintegration.\n18To evaluate (47) one needs a representation of the inverse of the doubly anti-symmetric\nmatrix, [f·χ]−1/vextendsingle/vextendsingleab\nµν. If there exist ncolor and 4 space-time coordinates, there are then\n3n(n−1) independent quantities comprising this quantity, and the simplest , compatible\nassumption is to write\n[f·χ]−1/vextendsingle/vextendsingleab\nµν=Gab·Hµν, (48)\nwhere we expect GabandHµνeach to be anti-symmetric. If (48) is true, there then follows\nthe necessary condition\nδabδµν=/summationdisplay\nc,λ(f·χ)|ac\nµλGcb·Hλν, (49)\nwhich can be used to provide implicit representations for both GandH, as follows. Set\na=bin (49) and sum over all color coordinates to obtain\nδµν=/summationdisplay\nλ/bracketleftBigg\n1\nn/summationdisplay\nb,c(f·χ)|bc\nµλGcb/bracketrightBigg\n·Hλν, (50)\nfrom which it follows that\n/parenleftbig\nH−1/parenrightbig\nµν=1\nn/summationdisplay\nb,c(f·χ)|bc\nµνGcb. (51)\nSimilarly, set µ=νand sum over space-time indices to obtain\n/parenleftbig\nG−1/parenrightbigab=1\n4/summationdisplay\nµ,λ(f·χ)|ab\nµλHλµ. (52)\nIt will be convenient to define\nQµν=1\nn/summationdisplay\nb,cfbceGcbχe\nµν≡/summationdisplay\neqeχe\nµν, (53)\nso that (51) and its inverse can be expressed as\n/parenleftbig\nH−1/parenrightbig\nµν=Qµν, Hµν=/parenleftbig\nQ−1/parenrightbig\nµν. (54)\nThe general statement of the inverse of an anti-symmetric, 4 ×4 matrix can be used to\nrepresentQ−1as\n/parenleftbig\nQ−1/parenrightbig\nµν=1\n2ǫµναβQαβ√detQ, (55)\nbutit will bemost useful tonotethatonlyoneofthesix, independen tHµν,H30, ismultiplied\nby the factor gϕin the exponential of (44); and for small band largeg, this contribution\n19will be large. Does this carry the implication that, for all color indices, theχc\n30will typically\nbe larger than the χc\nαβof the other Lorentz indices? Not necessarily, but in the interest o f\nsimplifying the computations we shall assume that only χ30andχ12are of interest. [In order\nto prevent det[ Q] from vanishing, it is necessary to retain one other χαβin addition to χ30.]\nWith this approximation, det[ Q]→Q2\n12Q2\n30, and\n/parenleftbig\nQ−1/parenrightbig\nµν=δµ3δν0Q12+δµ1δν2Q30\nQ12Q30, (56)\nso that\nH30=/parenleftbig\nQ−1/parenrightbig\n30=1\nQ30, (57)\nand\n/parenleftbig\nG−1/parenrightbigab=1\n2/summationdisplay\ndfabd/braceleftbiggχd\n30\nQ30+χd\n12\nQ12/bracerightbigg\n, (58)\nwithGabgiven by the inverse of (58).\nDoes the inverse of G−1exist? The inverse of an anti-symmetric matrix Mabof eight\nrows and columns is given by\n/parenleftbig\nM−1/parenrightbigab=1\n48[detM]−1\n2ǫabcdefghMcdMefMgh, (59)\nwhereǫabcdefghis the unit anti-symmetric tensor of eight dimensions. However, if Mab=\n/summationtext\ncfabcVc≡(f·V)ab, calculation shows [18] that det[ f·V] = 0, for any and every value of\nthe color vector Vc. In general, inverses of such Lie-valued sums do not exist, and it mig ht\nappear that this MGI calculation must grind to a halt. However, what is relevant is the\ncombination of GwithH, notGalone; and therefore let us give a physicist’s redefinition of\nthe problem. We shall define the determinant of a matrix Mas: det[M]+λ2, whereλ→0\nas a subsequent condition. It is also understood that the elements ofMare dimensionless.\nRewriting (52) in the form ( G−1)ab= (f·V)ab, whereVc=1\n4/summationtext\nµλχc\nµλHλµ, the corre-\nspondingGabmay be expressed as Gab=¯Gab/λ, where¯Gabis defined by (1 /48) multiplying\nthe corresponding numerator of (59), with Mab= (f·V)ab. From (53) the quantities Qµν\nandqemay be written as Qµν=¯Qµν/λ,qe= ¯qe/λ, where¯Qand ¯qare defined in terms of\nthe finite ¯G. Then, from (54) one may write Hµν=λ/parenleftbig¯Q−1/parenrightbig\nµν≡λ¯Hµν; and in this way the\nproductGab·Hµνof (48) becomes ¯Gab·¯Hµν, and is independent of λ. Only¯Gabquantities are\nneeded in the subsequent analysis of color dynamics; although one fi nds a factor of det[ G−1]\nrequired at one point in the calculation, it is immediately followed by a fac tor det[G]; and\n20the product of two such determinants is unity. But were there a div ergent contribution of\nany form associated with the original ( f·χ)−1, which we have represented by the product\nfromG·H, there exist separate arguments to show that such divergences have no effect\non the Physics; one of those arguments appears in the paper by Re inhardt, et al. [6], and\nan independent proof is given in Appendix A of the present paper. Pe rhaps the simplest\nargument is to observe that any singularity of ( f·χ)−1will cause the exponent of (44) to\noscillate infinitely rapidly, and make no contribution to the integral.\nFinally, in the special limit of small impact parameter, the eikonal expo nential E[χ] will\nreduce toafiniteset ofpossible terms—all considerably smaller than thatofthelargeimpact\nparameter result—involving the magnitude of the diagonalized compo nents ofG. In this\nway, because the color coordinates are coupled to space-time, th e procedure is well defined\nand yields qualitative results in agreement with QCD intuition. We shall fi nd that for\nlarge impact parameters the scattering is coherent, with the quar ks retaining their original\ncolor, while for smaller impact parameters, color fluctuations reduc e the magnitude of the\namplitude.\nIV. ESTIMATION\nIf the Halpern variable χa\nµνis written as za\nµνrµν, whereza\nµνrepresents the color-projection\nof a ’magnitude’ rµν, inspection of the original inter-relations of GandHshows that G\nis independent of the ’magnitudes’ r, and depends only upon the z; and we shall assume\nthe same dependence for ¯Hand¯G. In contrast, ¯H, while dependent upon the z, varies as\nthe inverse of the rvariables; and it is this latter r-dependence which appears to be most\nrelevant to the overall color properties of the amplitude. We shall t herefore treat the z-\ndependence as producing relatively unimportant averages which ar e to be relegated to later\nnumerical integrations, and concentrate in what follows on the out put of the r-integrals.\nSince thegϕ-dependence of (47) is associated with the dependence of H30, integration over\nχ12variables can be moved into a separate, uninteresting normalization constant; and we\nsuppress the (30)-subscripts of the remaining χ30variables.\nThegϕ-dependent exponential factor of (47) is then\nexp/bracketleftbig\nigϕΩa\nI¯GabΩb\nII/r/bracketrightbig\n(60)\n21and we first consider the integral\n(2π)−n/integraldisplay\ndnΩIe−iαa\nIΩa\nIeigϕΩa\nI¯GabΩb\nII/r(61)\n=/parenleftbiggr\ngϕ/parenrightbiggn\nδ(n)/parenleftbigg\n¯GabΩb\nII−/parenleftbiggr\ngϕ/parenrightbigg\nαa\nI/parenrightbigg\n.\nNow define Ωb\nII≡(¯G−1)bc¯Ωc\nII, so that (61) becomes\n/parenleftbiggr\ngϕ/parenrightbiggn\nδ(n)/parenleftbigg\n¯Ωa\nII−/parenleftbiggr\ngϕ/parenrightbigg\nαa\nI/parenrightbigg\n(62)\nand/integraltext\ndnΩIIyields\n/parenleftbiggr\ngϕ/parenrightbiggn\ndet/bracketleftbig¯G−1/bracketrightbig\ne−iαa\nII(¯G−1)abαb\nI(r\ngϕ)(63)\nSincei\n4¯χ2⇒i\n4/summationtext\nc[(χc\n12)2−(χc\n30)2], after removing the χc\n12-dependence, there remain the\ngϕ-dependent integrals\n(2π)−n/integraldisplay\ndnαIe−iλI·αI/integraldisplay\ndnαIIe−iλII·αII(64)\n×det/bracketleftbig¯G−1/bracketrightbig\n·/integraldisplay\ndnχe−ir2/4/parenleftbiggr\ngϕ/parenrightbiggn\n×e−i(r\ngϕ)αa\nII(¯G−1)abαb\nI,\nwhereχc\n30≡χc=rzc,\n/integraldisplay\ndnχ≡/productdisplay\nc/integraldisplay+∞\n−∞dχc(65)\n=/productdisplay\nc/integraldisplay+∞\n−∞dχc/integraldisplay∞\n0dr2δ(r2−/summationdisplay\na(χa)2);\nand with dχc=rdzc,\n/integraldisplay\ndnχ→2/productdisplay\nc/integraldisplay+1\n−1dzcδ(1−/summationdisplay\na(za)2)/integraldisplay∞\n0drrn−1. (66)\nThen, (64) may be rewritten as\n(2π)−n2/productdisplay\nc/integraldisplay+1\n−1dzcδ(1−/summationdisplay\na(za)2) (67)\n×det/bracketleftbig¯G−1/bracketrightbig/integraldisplay∞\n0drrn−1/parenleftbiggr\ngϕ/parenrightbiggn\ne−ir2/4\n×/integraldisplay\ndnαIe−iλI·αI/integraldisplay\ndnαIIe−iλII·αII\n×e−i(r\ngϕ)αa\nII(¯G−1)abαb\nI.\n22For clarity of presentation, in the passage from (44) and (45) to ( 62), we have suppressed\nthefactor of/radicalbig\ndet(gf·χ)−1of(44). Fromthediscussion of SectionIIIandthat ofAppendix\nA, this omitted term will contribute a factor of r−1/2to the integrand of (67), which will\nhave no bearing on the qualitative conclusions of Sections IV and V.\nIt will now be most convenient to isolate the αI,IIfactors from the λI,IIfactors, by writing\ne−iλI·αI= (2π)−n/integraldisplay\ndnv/integraldisplay\ndnΩeiΩ·(v−αI)·e−iv·λI(68)\ne−iλII·αII= (2π)−n/integraldisplay\ndnw/integraldisplay\ndn¯Ωei¯Ω·(w−αII)·e−iw·λII,\nso that integration over the αI,IImay be performed,\n/integraldisplay\ndnαI/integraldisplay\ndnαIIe−iαI·Ω−iαII·¯Ω·e−iαa\nI(¯G−1)abαb\nII/α(69)\n= (2π)nαn/integraldisplay\ndnαIIe−iαII·¯Ωδ(αΩa−(¯G−1)abαb\nII),\nwhereα=gϕ/r. With the variable change: αb\nII=¯Gbcβc, this becomes\n(2π)nαndet[¯G]/integraldisplay\ndnβe−i¯Ω·¯G·βδ(β−αΩ) = (2π)nαne−iα¯Ω·¯G·Ω, (70)\nand one notes that the determinantal factor of (70) combines wit h that of (67) to produce\na factor of unity.\nAt this point is useful to perform the remaining v,wintegrals written in the form\n/integraldisplay\ndnΩ/integraldisplay\ndn¯Ωe−iα¯Ω·¯G·ΩJI(Ω)JII(¯Ω), (71)\nwhere\nJI(Ω) = (2π)−n/integraldisplay\ndnve−iv·λI·eiv·Ω, (72)\nand\nJII(¯Ω) = (2π)−n/integraldisplay\ndnwe−iw·λII·eiw·¯Ω. (73)\nClearly, for gϕ(b)→0, (71) reduces to a constant, independent of color factors, so that in\nthis limit the initial and final quark colors must remain the same; but fo r largegϕ(b), there\nwill be oscillations involving changing color coordinates away from that constant, so that\nthe magnitude of the b-dependent amplitude will be reduced.\nWe have carried out a simple estimation of this effect for the simplest c ase of SU(2) in\nAppendix B, and find that the expectations described in the above p aragraph hold true:\n23Color fluctuations at small impact parameter diminish the coherent s cattering produced at\nlarger impact parameter. This non-perturbative and gauge-invar iant statement can formthe\nconceptual basis of quark scattering and binding, as well as asymp totic freedom. A more\nprecise statement must await a careful program of numerical inte gration, which we are not\nable to perform. But there can be little doubt of the qualitative natu re of the output of\nsuch a detailed calculation; and for this reason, we believe that the m ethods described in\nthis paper open a door to the realistic estimation and calculation of de tailed QCD processes,\nproperly gauge invariant, and containing all orders of coupling.\nV. SUMMARY AND EXPECTATIONS\nThe above Sections have described a new method of calculating a par ticular scattering\nprocess in QCD, to all orders of the coupling and with GI and LC assur ed. We have\nmade a number of approximations for ease of presentation, as well as for our inability of\nperforming certain relatively unimportant integrations which must b e left for subsequent\nnumerical integration. Our result is a qualitative expression of the e ikonal exponential\nfunction E = exp[ iX], given as a function of the square of the impact parameter betw een the\nscattering particles. And from this quantity, by a process requirin g numerical integrations,\nit is, in principle, possible to obtain a qualitative idea of the effective inte raction potential\nbetween a pair of quarks or of a quark and an anti-quark.\nTo see this, return for a moment to the simple, potential theory pr oblem of a particle\nscattering froman external potential V(|r|). There, the corresponding function E is given by\nthe exponential of a simple kinematical factor multiplying the two-dim ensional expression\nof that potential, obtained—as a result of a relevant, eikonal-calcu lation prescription—by\ncalculating the three-dimensional Fourier transform of that pote ntial,˜V(|k|), and setting\nthe longitudinal component of that 3-momentum equal to zero, to obtain˜V(|k⊥|).\nIn all previous field theory models, or approximate calculations of su bsets of Feynman\ngraphs, which yield eikonals, χ(b), dependent upon the square of the impact parameter, the\ntwo-dimensional Fourier transform of that eikonal generates an effective ˜V(|k⊥|); and the\nsimple ’extension’ of |k⊥|to the full, three-dimensional |k|, produces the Fourier transform\nof the original potential ˜V(|r|). The same process may be considered for the log of the\nfunction E we have obtained, with its built-in, qualitative approximatio ns. Because of\n24the relative complexity of our result, the Fourier transform over it sb-dependence must\nbe done numerically; but that is certainly possible, in principle; and it will generate a\nqualitative ˜V(|k⊥|). Then, the simple enlargement of that argument, from |k⊥|to the full\n|k|, produces ˜V(|k|); and a subsequent Fourier transform, again performed numeric ally, will\nyield a qualitative form for the effective potential V(|r|) between quarks and/or anti-quarks.\nOf course, the potential will, in SU(3), involve Gell-Mann color matrice s, as in SU(2) it\ninvolves Pauli matrices; but these can be included, in principle, in a per haps tedious but\nstraightforward way (as in Eq. (3.8) and the following paragraph of reference [3]). Improve-\nments to our qualitative E can surely be made, by numerical integrat ion over the r- and\nz-factors, at different stages. But here is a method of analytically p roducing a qualitative,\neffectiveV(|r|)—aswell asanassociatedscattering amplitude—which includes cont ributions\nfrom every single QCD Feynman graph relevant to the process.\nOf course, we have left out, again for simplicity of presentation, th ose parts of the Physics\ndealing with charge renormalization, and with the production of part icles in the scattering\nprocess, inelastic effects which have such a unitarity importance to a scattering amplitude.\nBut, as explained in the text, these effects can be systematically inc luded in the MGI/MLC\ncalculations. They may not be able to be calculated exactly, but it will s urely be possible\nto understand their qualitative features.\nThe qualitative results seen above for the scattering amplitude—co herent, multiple gluon\nexchange at larger impact parameters, with color fluctuations des troying that coherence at\nsmaller distances—are intuitively in agreement with the MIT Bag Model, where quarks are\n’free’ when close together but are subject to a confining potentia l, and tend to bind as they\nmove apart; for example, a pion as a bound state of a Qand¯Q, with the distance between\nthem continuously oscillating as they remain bound. Another expect ed example would be\nthe simple vertex function, where the impact parameter of the pre sent calculation becomes\nthe conjugate Fourier variable of momentum transfer, so that lar ger momentum transfers\ncorrespond to induced color fluctuations and a decrease of the eff ective coupling strength;\nthis is just what would be expected of a theory with ’true’ asymptot ic freedom, arising from\nthe exchanges of an infinite number of gluons.\nFinally, one must comment on the obvious fact that scattering expe riments are performed\nwith hadrons, and not with individual quarks; each hadron will involve integrals over the\ntransverse momentum or spatial distributions of individual quark w ave functions. What we\n25have estimated is the idealized case of two quark/anti-quark scatt ering, suppressing the fact\nthat each is bound within its own hadron, and it must be possible to tak e into account that\nbinding. The proper way is to carry out those integrations over the quark coordinates; but\na simple, physical argument can serve to modify our idealized calculat ion, as follows.\nBinding suggests that the scale of individual transverse distances , or of the difference\nbetween those distances of two interacting quarks is controlled by the wave functions, such\nthat/an}bracketle{tb/an}bracketri}htis never appreciably less than 1 /µ, whereµmay be taken as on the order of the\nhadron mass. But our /vectorb=/vectorB−/vectorb, where/vectorBrefers to the difference of transverse positions of\nthe two hadrons, while /vectorbdenotes the difference of transverse positions of each quark with in\nits hadron. Physically, one expects |/vectorB|/greaterorsimilar|/vectorb|, and the smallest b-values would be controlled\nby the largest k⊥values of/integraltext\nd2k⊥, which are surely limited, in any eikonal model, by\nthe requirement that all transverse momenta associated with, or arising from the exchange\nof gluons must be less than the corresponding longitudinal momenta of the quarks, i.e.,\n|k⊥|/lessorsimilarM∼O(E).\nBut there is another question, related to large transverse separ ations, when the hadron\namplitude is expected to vanish, because we are fundamentally dealin g with short range nu-\nclear forces. How large can the Bvalues become, or how small can the hadronic momentum\ntransfer become, before some form of screening sets in and redu ces the hadronic amplitude\nto zero? In this case, the needed screening must arise from an inte rplay of the integrals over\nquark wave functions such that for sufficiently large b, there is effectively no scattering, and\na ’short-range’ force has been achieved; the quark wave functio ns modify that form of the\noverall, hadronic, eikonal amplitude, such that screening sets in fo r distances larger than\n1/µ—giving a Yukawa effect between hadrons—while there remains an ove rall, non-zero and\ncoherent scattering for distances less than 1 /µ. But from a quark point of view, the essential\nand interesting aspect of our result is that when bbecomes so small that b <1/M, color\nfluctuations begin, and destroy that coherence.\nFinally, one may contrast the qualitative output of such MGI/MLC es timations with\nother, traditional methods of ’summing’ Feynman graphs, such as the use of a Bethe–\nSalpeter equation, whose kernel is only known in a low-order pertur bative approximation;\nor a renormalization group argument, set up to represent the sum of all perturbative effects,\nbut whose beta function is then estimated by a few orders of pertu rbation theory; or by\nthe sum of ’leading-order’ perturbative terms, which then omit who le classes of Feynman\n26graphs. In contrast, we believe that the present method holds gr eat hope for generating at\nleast qualitative descriptions of field-theory Physics which include, o r can be systematically\nmade to include, every virtual exchange.\nAPPENDIX A: ZERO EIGENVALUES OF [f·χ]−1\nIn this appendix, one wishes to get some insight into the role of the op erator [f·χ]−1’s\npossible zero eigenvalues. One then focuses on the expression (44 ), here rewritten as\nN2−1/productdisplay\na=1/integraldisplay\ndχa\n30det[gf·χ]−1\n2ei\n4χ2\n30+igϕ(b)Ωa\nI[f·χ]−1|ab\n30Ωb\nII (A1)\nwhich is a part of the larger expression\n(2π)−2n/integraldisplay\ndnαIe−iαa\nIλa/integraldisplay\ndnαIIe−iαa\nIIλa(A2)\n×/integraldisplay\ndnΩI/integraldisplay\ndnΩIIe−iαI·ΩIe−iαII·ΩII\n×(2π)−2/productdisplay\na/integraldisplay\ndχa\n30det[gf·χ]−1\n2\n×ei\n4χ2\n30+igϕ(b)Ωa\nI[f·χ]−1|ab\n30Ωb\nII\nand wheren, as in the main text, is a shortcut for N2−1. One has the relation\nχ2\n30=N2−1/summationdisplay\na=1(χa\n30)2=1\nNtr(χa\n30λa)2, (A3)\ntr(λaλb) =Nδab,\nwhere theλa’s are thentraceless generators of the SU(N) Lie algebra, taken in its n×n-\ndimensional adjoint representation with ( λa)bc=−ifabc.\nBeing symmetric under the combined exchange a↔b, 3↔0 the operator [ f·χ30] can be\ndiagonalized and has real eigenvalues. Note that this property app lies to [f·χµν] and can\nbe deduced from (33) with Qa\nµ, the current given after (25). In the form (A1), though, this\nproperty is not transparent. With the pi,µ, ati= 1,2, given after (36), this is because (A1)\nresults from a re-arrangement of an original expression\ngϕ(b)×···×(p1,3p2,0−p1,0p2,3)Ωa\nIΩb\nII[f·χ]−1/vextendsingle/vextendsingleab\n30, (A4)\non which that symmetry can be read off easily.\n27The O[χ30] orthogonal matrix that effects the diagonalization of [ f·χ]−1can be used to\nre-define the integrations on Ωa\nIand Ωb\nII. With this re-definition, the two Jacobians will\ncompensate one another, so that keeping the same symbol for th e re-defined Ω’s, under the\nintegration over Ωa\nIand Ωb\nII, one can proceed to the replacement\nigϕΩa\nI/parenleftbig\n[f·χ]−1/parenrightbigab\n30Ωb\nII−→igϕΩa\nIδab\nξaΩb\nII, (A5)\nwhere theξa’s are theN2−1 eigenvalues of the matrix [ f·χ30], some of them, possibly zero.\nNow, relying on Theorem 3.2 in Ref. [19], and taking (A3) and (A5) into a ccount, it is\npossible to rewrite (A1) as\n1\nN/integraldisplay+∞\n−∞dξ1···dξnδ(n/summationdisplay\n1ξi)/productdisplay\n1≤i<j≤n|ξi−ξj|ei\n4NPn\na=1ξ2\na\n√ξ1···ξneigϕΩa\nIδab\nξaΩb\nII,(A6)\nwhere the delta-function accounts for the traceless property o f any [f·χ] matrix, and where\nNis the normalization constant\nN=/integraldisplay+∞\n−∞dξ1···dξnδ(n/summationdisplay\n1ξi)/productdisplay\n1≤i<j≤n|ξi−ξj|ei\n4NPn\na=1ξ2\na. (A7)\nOf course, calculations can be continued further [20], but in (A6) it a lready appears that\nthe possible occurrence of vanishing eigenvalues is not a problem, an d that they should not\ncontribute significantly.\nAPPENDIX B: ESTIMATION IN SU(2)\nThe quantities/producttextn\nc=1/integraltext+1\n−1dzcδ(1−/summationtext\na(za)2) used repeatedly in the text are simply solid\nangle factors, as can be seen immediately for n= 3 of SU(2). There, one can make a variable\nchange from z1,z2,z3toλ,θ,φ, where−1≤λ≤+1,z1=λsinθcosφ,z2=λsinθsinφ,\nz3=λcosθ. The Jacobian of the transformation d z1dz2dz3=Jdλdθdφis simple to obtain,\nJ=λ2sinθ, so that\n3/productdisplay\nc=1/integraldisplay+1\n−1dzcδ(1−/summationdisplay\na(za)2) (B1)\n=/integraldisplay+1\n−1dλ/integraldisplayπ\n0dθ/integraldisplay2π\n0dφδ(1−λ2)λ2sinθ\n=/integraldisplayπ\n0dθ/integraldisplay2π\n0dφsinθ,\n28as expected.\nToillustratehowcolorfluctuationscanreduceacoherentamplitude , considerthesimplest\nSU(2) case of (71), where\nJ(Ω) = (2π)−3/integraldisplay\nd3ve−iv·σ·eiv·Ω, (B2)\nUpon performing the angular integrations, and then integration ov er the magnitude of v,\none obtains\nJ(Ω) =−1\n4π/bracketleftbigg1\nΩ∂\n∂Ω+/vector σ·∂\n∂/vectorΩ/bracketrightbiggδ(1−Ω)\nΩ. (B3)\nThe integral/integraltext\nd3ΩJI(Ω)eiα/vectorΩ·¯G·/vector¯Ωthen becomes\n1\n4π3/productdisplay\nc=1/integraldisplay+1\n−1dzcδ(1−(zc)2)/bracketleftbig\n1+iα/parenleftbig\nσI\ni+zi/parenrightbig¯Gij¯Ωj/bracketrightbig\neiαP\nabza¯Gab¯Ωb,(B4)\nand performing the final/integraltext\nd3¯ΩJII(¯Ω) on the result of (B4) produces for the SU(2) form of\n(71) the result\n/parenleftbigg1\n4π/parenrightbigg23/productdisplay\nc=1/integraldisplay+1\n−1dzcδ(1−/summationdisplay\na(za)2) (B5)\n·3/productdisplay\nd=1/integraldisplay+1\n−1d¯zdδ(1−/summationdisplay\nb(¯zb)2)·eiαz·¯G·¯z\n·/braceleftbig\n1+iαξ1/parenleftbig\nσI·¯G·σII/parenrightbig\n+iαξ2/parenleftbig\nσI·¯G·¯z+z·¯G·σII/parenrightbig\n+iαξ3/parenleftbig\nz·¯G·¯z/parenrightbig\n+(iα)2ξ4/parenleftbig\nσI·¯G·¯z/parenrightbig/parenleftbig\nz·¯G·σII/parenrightbig\n+(iα)2ξ5/parenleftbig\nz·¯G·¯z/parenrightbig2/bracerightBig\n,\nwhereξ1,...,ξ 5are numerical constants. The ¯Gabare gi-ven by that numerator function\nof (59), where the Mabdepend upon the zc-components of χc\n30, as given by (58); and those\nzc-components have been suppressed, for they require a separat e, numerical integration.\nIn SU(2), only the diagonal σI,II\n3spin matrices can contribute to matrix elements between\nunchanged isotropic ( i.e., color) states, whereas for SU(3) there would be two such matric es,\nλI,II\n3andλI,II\n8.\nLet us estimate the α/ne}ationslash= 0 effect by evaluating the ’1’ term of the curly bracket of (B5);\nand for this it is most convenient to consider an orthogonal transf ormation to diagonalize\nthe real, anti-symmetric ¯Gab, by simultaneously transforming to a new set of variables z′\na,\n¯z′\nb. Under such a transformation, the measures and form of the ’1’ t erms contributing to\n29(B5) are unchanged, but the exponential factor z′·¯G′·¯z′is simplified because ¯G′is diagonal.\nAfter converting to angular coordinates, let us simplify even furth er by suppressing the φ′-,\n¯φ′-dependence of that exponential, and merely calculate the θ′,¯θ′integrals, using z′= cosθ′\nand¯z′= cos¯θ′,\nI(a) =/integraldisplay+1\n−1dz′/integraldisplay+1\n−1d¯z′eiaz′¯z′, (B6)\nwherea=gϕ(b)¯G′\n33/r, and we assume that ¯G′\n33/ne}ationslash= 0. These integrals are elementary and\nyield\nI(a) =4\na/integraldisplaya\n0dxsinx\nx. (B7)\nFor large values of a, corresponding to small impact parameter, I(a)≃2π/a, and is damped\nin comparison to the corresponding integrals one finds for a→0,I(0) = 4. This sort\nof damping is naturally to be expected when color fluctuations destr oy the large impact\nparameter coherence, as stated in the text at the end of Section IV.\nAPPENDIX C: GAUGE INDEPENDENCE\nThe QCD Lagrangian can be expressed as\nLQCD=Lgluon+Lquark+Lint (C1)\n=−1\n4Fa\nµνFa\nµν−¯ψ[m+γµ(∂µ−igAa\nµλa)]ψ,\nwhereAa\nµare gauge fields, Fa\nµνis the field strength with\nFa\nµν=∂µAa\nν−∂νAa\nµ+gfabcAb\nµAc\nν, (C2)\nandλaare the color matrices of SU(3). Separate the gluon sector of Lag rangian into two\nparts as [21]\nLgluon=−1\n4Fa\nµνFa\nµν (C3)\n=−1\n4/bracketleftbig\nfa\nµνfa\nµν+(Fa\nµνFa\nµν−fa\nµνfa\nµν)/bracketrightbig\n,\nwherefa\nµν=∂µAa\nν−∂νAa\nµas defined in Eq. (10). Thus, Lgluon=L(0)\ngluon+L′\ngluonwith\nL(0)\ngluon=−1\n4fa\nµνfa\nµν, (C4)\nL′\ngluon=−1\n4(Fa\nµνFa\nµν−fa\nµνfa\nµν). (C5)\n30One can add and subtract a ’gauge-fixing’ term to the gluon Lagran gian, which does not\nchange its overall gauge invariance.\nL(0)\ngluon=−1\n4fa\nµνfa\nµν−1\n2ζ(∂µAa\nµ)2, (C6)\nL′\ngluon=Lgluon−L(0)\ngluon (C7)\n=−1\n4(Fa\nµνFa\nµν−fa\nµνfa\nµν)+1\n2ζ(∂µAa\nµ)2,\nThe QCD Lagrangian can then be written in terms of free L(0)\nQCDand interacting L′\nQCD\nparts asLQCD=L(0)\nQCD+L′\nQCD, where the free and interacting parts are\nL(0)\nQCD=Lquark+L(0)\ngluon (C8)\n=−¯ψ[m+γµ·∂µ]ψ−1\n4fa\nµνfa\nµν−1\n2ζ(∂µAa\nµ)2,\nL′\nQCD=Lint+L′\ngluon (C9)\n= +ig¯ψ(γµ·Aa\nµλa)ψ−1\n4(Fa\nµνFa\nµν−fa\nµ��fa\nµν)\n+1\n2ζ(∂µAa\nµ)2,\nrespectively.\nThe generating functional of QCD is\nZ{j,¯η,η}=1\n/an}bracketle{tS/an}bracketri}htexp/bracketleftbigg\ni/integraldisplay\nL′\nQCD/braceleftbigg1\niδ\nδj,1\niδ\nδ¯η,−1\niδ\nδη/bracerightbigg/bracketrightbigg\n·Z0{j,¯η,η},(C10)\nwhereja\nµ,ηµ, and ¯ηµare gluon, quark and anti-quark sources, respectively. Following t he\nconventional approach, either functional integral or Schwinger ’s Action principle [7], the free\ngenerating functional with L(0)\nQCD=L(0)\ngluon+Lquarkis\nZ0{j,¯η,η}= exp/braceleftbiggi\n2/integraldisplay\nj·D(ζ)\nc·j+i/integraldisplay\n¯η·Sc·η/bracerightbigg\n, (C11)\nwhere the gauge field propagator is now defined by the gauge condit ion with a gauge pa-\nrameterζas\ni/integraldisplay\nL(0)\ngluon=−i\n4/integraldisplay\nfa\nµνfa\nµν−i\n2ζ/integraldisplay\n(∂µAa\nµ)2(C12)\n= +i\n2/integraldisplay\nAa\nµδab/bracketleftbigg\nδµν∂2+/parenleftbigg1\nζ−1/parenrightbigg\n∂µ∂ν/bracketrightbigg\nAb\nν\n=−i\n2/integraldisplay\nAa\nµ/parenleftBig\nD(ζ)\nc−1/parenrightBigab\nµνAb\nν\n31and/parenleftBig\nD(ζ)\nc−1/parenrightBigab\nµν=−δab/bracketleftbigg\nδµν∂2+/parenleftbigg1\nζ−1/parenrightbigg\n∂µ∂ν/bracketrightbigg\n. (C13)\nAfter rearrangement, one finds\nZ{j,¯η,η}=1\n/an}bracketle{tS/an}bracketri}htei\n2Rj·D(ζ)\nc·j·e−i\n2Rδ\nδA·D(ζ)\nc·δ\nδA (C14)\n·eiRL′\nQCD[A,1\niδ\nδ¯η,−1\niδ\nδη]·eiR\n¯η·Sc·η\n=ei\n2R\nj·D(ζ)\nc·j·e−i\n2Rδ\nδA·D(ζ)\nc·δ\nδA\n·eiR\nL′\ngluon[A]·eiR¯η·Gc[A]·η·eL[A]\n/an}bracketle{tS/an}bracketri}ht,\nwhereAa\nµ(x) =/integraltext\ndyD(ζ)ab\ncµν(x−y)jb\nν(y). The exponential factor involving L′\ngluoncan be cast\ninto the form\neiR\nL′\ngluon[A](C15)\n= exp/braceleftbigg\n−i\n4/integraldisplay\n(Fa\nµνFa\nµν−fa\nµνfa\nµν)+i\n2ζ/integraldisplay\n(∂µAa\nµ)2/bracerightbigg\n=N′/integraldisplay\nd[χ]ei\n4R\nχa\nµνχa\nµν+i\n2R\nχa\nµνFa\nµν·e+i\n4R\nfa\nµνfa\nµν+i\n2ζR\n(∂µAa\nµ)2,\nwhere Eq. (17) is used. The χa\nµν-independent factor becomes\n+i\n4/integraldisplay\nfa\nµνfa\nµν+i\n2ζ/integraldisplay\n(∂µAa\nµ)2(C16)\n=−i\n2/integraldisplay\nAa\nµδab/bracketleftbigg\nδµν∂2+/parenleftbigg1\nζ−1/parenrightbigg\n∂µ∂ν/bracketrightbigg\nAb\nν\n= +i\n2/integraldisplay\nAa\nµ/parenleftBig\nD(ζ)\nc−1/parenrightBigab\nµνAb\nν.\nThe generating function of QCD becomes\nZ{j,¯η,η} (C17)\n=ei\n2Rj·D(ζ)\nc·j·e−i\n2Rδ\nδA·D(ζ)\nc·δ\nδA\n×N′/integraldisplay\nd[χ]ei\n4R\nχ2+i\n2R\nχ·[f+gfAA]\n·e+i\n2RA·D(ζ)\nc−1·A·eiR¯η·Gc[A]·η·eL[A]\n/an}bracketle{tS/an}bracketri}ht.\nExcept for the expansion of the closed-fermion-functional L[A], the gauge field dependence\nin the exponent is at most quadratic; however, an expansion in powe rs ofL[A], using a\nmodified Fradkin representation for each L[A] andGc[A], generates a totally quadratic\nA-dependence.\n32For theQQorQ¯Qscattering, one will encounter\ne−i\n2Rδ\nδA·D(ζ)\nc·δ\nδA·/bracketleftbigg\nGI\nc[A]GII\nc[A]eL[A]e+i\n2R\nχ·[f+gfAA]+i\n2R\nA·D(ζ)\nc−1·A/bracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nA→0.(C18)\nUnder the eikonal and quenched approximations, the coefficients o f linear and quadratic\nAa\nµ-dependent terms are ( cf.Eq. (25))\nQa\nµ=g(Ra\nIµ+Ra\nIIµ)−∂νχa\nµν, (C19)\nand\nKab\nµν=gfabcχc\nµν+/parenleftBig\nD(ζ)\nc−1/parenrightBigab\nµν, (C20)\nrespectively, and where Ra\nIνandRa\nIIνcome from the eikonal approximation of the Green’s\nfunction of the quarks or anti-quarks. The linkage operation can b e worked out as\ne−i\n2Rδ\nδA·D(ζ)\nc·δ\nδA·e+i\n2RA·K·A+iRA·Q/vextendsingle/vextendsingle/vextendsingle\nA→0(C21)\n=e−1\n2Trln“\n1−D(ζ)\nc·K”\n·ei\n2R\nQ·»\nD(ζ)\nc·“\n1−K·D(ζ)\nc”−1–\n·Q.\nThe kernel in the quadratic term of Qa\nµis\nDζ\nc·/parenleftbig\n1−K·Dζ\nc/parenrightbig−1(C22)\n=Dζ\nc·/parenleftBig\n1−/bracketleftBig\ngf·χ+Dζ\nc−1/bracketrightBig\n·Dζ\nc/parenrightBig−1\n=−(gf·χ)−1.\nThe result is independent of the gluon (gauge field) propagator. Th e derivation is valid for\narbitrary relativistic gauge conditions.\nAPPENDIX D: LIST OF ABBREVIATIONS\nThe following abbreviations have been used freely throughout the t ext.\n33CM Center of Mass\nETCRs Equal-time Commutation Relations\nFI Functional Integral\nGF Generating Functional\nGI Gauge-Invariant\nLC Lorentz Covariant\nMGI Manifestly Gauge Invariant\nMLC Manifestly Lorentz Covariant\nNVM Neutral Vector Meson\nOE Ordered Exponential\nQQuark\n¯QAnti-quark\nQA Quasi-Abelian\nQFT Quantum Field Theory\nRHS Right Hand Side\nACKNOWLEDGMENTS\nOne of us (H.M.F.) was supported in part by a Travel Grant from the J ulian Schwinger\nFoundation.\n[1] H. M. Fried and Y. Gabellini, Phys. Rev. D 55, 2430 (1997).\n[2] H.-T. Cho, H. M. Fried, and T. Grandou, Phys. Rev. D 37, 960 (1988).\n[3] H. M. Fried, Y. Gabellini, and J. Avan, Eur. Phys. J. C 13, 699 (2000).\n[4] L. D. Faddeev and V. N. Popov, Phys. Lett. B 25, 29 (1967).\n[5] H. M. Fried, Basics of Functional Methods and Eikonal Models (Editions Fronti` eres, Gif-sur-\nYvette Cedex, France, 1990).\n[6] H. Reinhardt, K. Langfeld, and L. v. Smekal, Phys. Lett. B 300, 111 (1993).\n[7] H. M. Fried, Functional Methods and Models in Quantum Field Theory (The MIT Press,\nCambridge, MA, 1972).\n[8] J. Schwinger, Stanford lectures (1956).\n34[9] M. B. Halpern, Phys. Rev. D 16, 1798 (1977).\n[10] M. B. Halpern, Phys. Rev. D 16, 3515 (1977).\n[11] H. Cheng and T. T. Wu, Phys. Rev. 182, 1852, 1868, 1873, 1899 (1969).\n[12] H. Cheng and T. T. Wu, Phys. Rev. D 1, 1069, 1083 (1970).\n[13] H. Cheng and T. T. Wu, Phys. Rev. Lett. 24, 1456 (1970).\n[14] L. N. Lipatov and G. V. Frolov, Yad. Fiz. 13, 588 (1971), [Sov. J. Nucl. Phys. 13, 333 (1971)].\n[15] H. Cheng and T. T. Wu, Expanding Protons: Scattering at High Energies (The MIT Press,\nCambridge, MA, 1987).\n[16] Oliver Rosten has pointed out to us that MGInvariance an d MGIndependence need not corre-\nspond to the same properties, at least in the context of an exa ct RG analysis; see, for example,\nT. R. Morris and O. J. Rosten, J. Phys. A 39, 11657 (2006). In the present case, as suggested\nby the above argument, they are the same thing.\n[17] E. S. Fradkin, Nucl. Phys. 76, 588 (1966).\n[18] We thank Marcus Spradlin for his kind and most efficient he lp with Mathematica.\n[19] M. L. Mehta, Random Matrices and the Statistical Theory of Energy Levels (Academic Press,\nNew York, 1967).\n[20] T. Grandou, H.-T. Cho, and H. M. Fried, Phys. Rev. D 37, 946 (1988).\n[21] H. M. Fried, Phys. Rev. D 46, 5574 (1992).\n35"    },    {        "title": "1608.01642v1.Extending_the_graviton_propagator_with_a_Lorentz_violating_vector_field.pdf",        "content": "arXiv:1608.01642v1  [gr-qc]  4 Aug 2016Proceedings of the Seventh Meeting on CPT and Lorentz Symmet ry (CPT’16), Indiana University, Bloomington, June 20-24, 20 16\n1\nExtending the Graviton Propagator with a Lorentz-Violatin g\nVector Field\nMichael D. Seifert\nDepartment of Physics, Astronomy, & Geophysics, Connectic ut College\n270 Mohegan Ave., New London, CT 06375\nE-mail: mseifer1@conncoll.edu\nI discuss progress towards “bootstrapping” a Lorentz-viol ating gravity the-\nory: namely, extending a linear Lorentz-violating theory o f a rank-2 tensor to\na non-linear theory by coupling this field to its own stress-e nergy tensor.\nThe gravitational sector of the Standard Model Extension (SME) has\nbecome of great interest in recent years, particularly with the rec ent detec-\ntion of gravitational waves by the LIGO Collaboration. The treatme nt of\ngravity by the SME differs in an important way from its treatment of o ther\nsectors. In the context of a gravitationallycurved spacetime, Lo rentz viola-\ntion cannot be thought of as due to a fixed background tensor; ins tead, the\nLorentz-violating tensor field (represented abstractly by Ψ...) must have\nits own dynamics.1The gravitational sector for the SME must therefore\nbe thought of as including the Einstein-Hilbert action, various dynam ical\nterms for Ψ..., and small coupling terms between Ψ...and the Riemann\ntensor.\nThe great majority of the work thus far in the gravitational secto r of\nthe SME2–4has focused on linearized perturbations about a solution where\nspacetime is flat ( gab=ηab) and the Lorentz-violating tensor field is con-\nstant (∇Ψ...= 0). In this limit, the dynamics of Ψ...do not greatly affect\nthe gravitational dynamics.2However, in strongly curved spacetimes, a\nconstant tensor field Ψ...will in general not exist. If the SME framework is\nto address such spacetimes (such as compact objects, black hole s, or cosmo-\nlogical spacetimes), we will have to address the dynamics of the und erlying\ntensor field Ψ....\nAn old idea in the context of gravitational physics is the idea of “boot -\nstrapping”gravityfromalineartheorytoanon-lineartheory.5Thismethod\nisbasedontheideathat “gravitygravitates”: the stress-energ yofthe grav-\nitational field should act as a source forthe gravitational field. This ideaProceedings of the Seventh Meeting on CPT and Lorentz Symmet ry (CPT’16), Indiana University, Bloomington, June 20-24, 20 16\n2\npredates the SME framework by a few decades, and it is instructive to ask\nwhether one can extend this idea to situations with violation of Loren tz\nsymmetry. In this process, three main questions arise:\n(1) At the level of a linear free-field gravity theory, what kinds of t heories\ncan I write down if I allow for violations of Lorentz symmetry?\n(2) Can such linear theories be bootstrapped to non-linear theorie s?\n(3) Does requiring that the linear theory be “bootstrappable” plac e con-\nstraints on the dynamics of the Lorentz-violating field Ψ...?\nLet us consider a general linear theory of a source-free symmetr ic rank-\ntwo tensor field habin flat spacetime, with action\nS=1\n2/integraldisplay\nd4x∂ahbcPabcdef∂dhef (1)\nwhere the propagator tensor Pabcdefis some constant tensor to be deter-\nmined. The resulting equations of motion are then\n/parenleftbig\nPabcdef+Paefdbc/parenrightbig\n∂a∂dhef= 0. (2)\nBy symmetry in (1), we can assume that Pabcdefis symmetric under the\nexchanges b↔c,e↔f, and{abc} ↔ {def}. From Eq. (2), we can also\ntake the propagator to be symmetric under the exchange {bc} ↔ {ef}.\nFinally, we will want to insert a conserved stress-energy tensor as a source\non the right-hand side of (2); this implies that the divergence of the left-\nhand side of (2) must also vanish. In Fourier space, this implies that t he\nquantity Pabcdefkakbkd= 0 for all choices of wave propagation vector ka.\nWe expect that in the end, Pabcdefwill not be a fundamental object\nbut rather a function of simpler tensors, such as a “fiducial” flat me tric\nηabor a Lorentz-violating tensor field of some kind. The strategy is the n\nto write down the most general Pabcdefthat can be constructed out of\nthese simpler tensors, subject to the above symmetry constrain ts. Using\nthe fiducial metric alone, for example, we find that the unique propa gator\nsatisfying the desired symmetry properties is the usual Lorentz- invariant\nlinearized gravity propagator, as expected:\n(PLI)abcdef=ηa(bηc)dηef+ηa(eηf)dηbc−ηa(bηc)(eηf)d\n−ηa(eηf)(bηc)d−ηadηbcηef+ηadηb(eηf)c.(3)\nThe simplest possible Lorentz-violating tensor field Ψ...would be a\nfour-vector field Aa. If we follow the above procedure, constructing the\npropagator out of ηabandAa, we find that the resulting Lorentz-violatingProceedings of the Seventh Meeting on CPT and Lorentz Symmet ry (CPT’16), Indiana University, Bloomington, June 20-24, 20 16\n3\npropagator ( PLV)abcdefonly has one free parameter ξ. What’s more, this\n(PLV)abcdefis equivalent to the Lorentz-invariant ( PLI)abcdefunder the\nsubstitution\nηab→˜ηab≡ηab+ξAaAb. (4)\nIn other words, the introduction of a Lorentz-violating vector fie ld only\nallows one to change the “effective metric” ˜ ηabfor linearized gravity. This\nwill have the effect of changing the “light cones” for gravitational w ave\npropagation; however, it will not allow for more exotic effects such a s dis-\npersion or birefringence of gravitational waves. This result is in agr eement\nwith the more general results of Ref. 4.\nHaving classified the ways in which the linearized theory can break\nLorentz symmetry, I now turn to the question of extending it to a n on-\nlinear theory. To do this, I follow the method of Deser,6,7and write down\nafirst-order linearmodelintermsofa densitized tensorhabandanauxiliary\ntensor Γabc. In the Lorentz-invariant case, the action for this model is\nS=/integraldisplay\nd4x/bracketleftbig\n2hab∂[cΓc\nb]a+2ηabΓc\nd[cΓd\na]b+Lmat(ηab,Aa,∂aAb)/bracketrightbig\n.(5)\nThe equations of motion for haband Γabc, combined, are equivalent to the\nlinearized vacuum Einstein equations ifwe interpret habasthe perturbation\nto the densitized inverse metric: gab=ηab+hab. Under this interpretation,\nhabmust be coupled to the trace-reversedstress-energy τab=δL/δηab. The\nsecond term in (5), along with the matter Lagrangian Lmat, contribute to\nthe stress-energy (note that the first term is independent of ηab.) We will\nthus need to add two new coupling terms to the action (5):\nS→S+/integraldisplay\nd4xhab/bracketleftbig\n2Γc\nd[cΓd\na]b+(τmat)ab/bracketrightbig\n. (6)\nImportantly, the new term in the gravitational sector (the first t erm in (6))\ndoes not depend on ηab, and so the bootstrap procedure terminates here for\nthe gravitational sector. In the matter sector, the term ( τmat)abmay itself\ndepend on the metric ηab, and so the contributions to the stress-energy\nfrom this coupling term must be added in as well. The iteration of this\nprocedure can, in principle, generate an infinite series of terms. Ho wever,\nassuming that various integrability conditions are satisfied,7the resulting\nseries can be summed up to yield a matter action that is minimally coupled\nto the densitized metric gab. The gravitational terms, meanwhile, combine\ninto the Palatini action for general relativity:\nS=/integraldisplay\nd4x/bracketleftbig\ngabRab[Γ]+Lmat(gab,Aa,∇aAb)/bracketrightbig\n, (7)Proceedings of the Seventh Meeting on CPT and Lorentz Symmet ry (CPT’16), Indiana University, Bloomington, June 20-24, 20 16\n4\nwhere the Ricci tensor Rabis viewed here as a function of the connection\ncoefficients Γ.\nPerhaps surprisingly, this scenario changes very little when we relax the\nassumption of Lorentz symmetry. As found above, the only possib le mod-\nification that can be made to the linearized gravity action in the prese nce\nof a Lorentz-violating vector field is to replace the matter metric ηabwith\nan effective metric ˜ ηab=ηab+ξAaAb. This will give rise to a new term\nξAaAbΓcd[cΓda]bin the action; but this term is independent of ηab, and so\ndoes not contribute to the stress-energy tensor. Thus, the en tire bootstrap\nprocedure carries through as before; the only difference is that t he densi-\ntized metric that appears in the Palatini action is not the same as tha t\nappearing in the matter action:\nS=/integraldisplay\nd4x/bracketleftbig˜gabRab[Γ]+Lmat(gab,Aa,∇aAb)/bracketrightbig\n(8)\nwhere˜gab≡gab+ξAaAb.\nThis action could then be rewritten using the (undensitized) gravita -\ntional metric as a fundamental variable; the result would be some kin d\nof exotic tensor-vector theory of gravity. It is important to not e, how-\never, that the construction of this theory required that the act ion for the\nLorentz-violating field Aaitself be amenable to “bootstrapping”; in partic-\nular, it must satisfy various integrability constraints at each stage of the\nbootstrap procedure. I conjecture that a symmetry-breaking potential and\na “Maxwell-type” kinetic term for Aawill satisfy these constraints, and\nthat more exotic kinetic terms will fail; but this has not yet been prov en.\nAcknowledgments\nI would like to thank Connecticut College for their financial support in\nattending this conference.\nReferences\n1. V. A. Kosteleck´ y, Phys. Rev. D 69, 105009 (2004).\n2. Q. Bailey & V. A. Kosteleck´ y, Phys. Rev. D 74, 045001 (2006).\n3. V. A. Kosteleck´ y & J. D. Tasson, Phys. Lett. B 749, 551–559 (2015).\n4. V. A. Kosteleck´ y & M. Mewes, Phys. Lett. B 757, 510–514 (2016).\n5. R. Kraichnan, Phys. Rev. 98, 1118–1122 (1955).\n6. S. Deser, Gen. Rel. Grav. 1, 9–18 (1970).\n7. V. A. Kosteleck´ y & R. Potting, Phys. Rev. D 79, 065018 (2009)."    },    {        "title": "1703.00708v1.Pinch_dynamics_in_a_low__β__plasma.pdf",        "content": "Pinch dynamics in a low- \fplasma\nH. K. Mo\u000batt1zand K. Mizerski2x\n1DAMTP Centre for Mathematical Sciences, University of Cambridge, Wilberforce\nRoad, Cambridge, CB3 0WA, UK\n2Department of Magnetism, Institute of Geophysics, Polish Academy of Sciences,\nKsiecia Janusza 64, 01-452 Warsaw, Poland\nE-mail: hkm2@cam.ac.uk\nAbstract. The relaxation of a helical magnetic \feld B(x;t) in a high-conductivity\nplasma contained in the annulus between two perfectly conducting coaxial cylinders\nis considered. The plasma is of low density and its pressure is negligible compared\nwith the magnetic pressure; the \row of the plasma is driven by the Lorentz force\nand and energy is dissipated primarily by the viscosity of the medium. The axial\nand toroidal \ruxes of magnetic \feld are conserved in the perfect-conductivity limit,\nas is the mass per unit axial length. The magnetic \feld relaxes during a rapid initial\nstage to a force-free state, and then decays slowly, due to the e\u000bect of weak resistivity\n\u0011, while constrained to remain approximately force-free. Interest centres on whether\nthe relaxed \feld may attain a Taylor state; but under the assumed conditions with\naxial and toroidal \rux conserved inside every cylindrical Lagrangian surface, this is\nnot possible. The e\u000bect of an additional \u000b-e\u000bect associated with instabilities and\nturbulence in the plasma is therefore investigated in exploratory manner. An assumed\npseudo-scalar form of \u000bproportional to q\u0011(j\u0001B) is adopted, where j=r\u0002Bandqis\nanO(1) dimensionless parameter. It is shown that, when qis less that a critical value\nqc, the evolution remains smooth and similar to that for q= 0; but that if q > qc,\nnegative-di\u000busivity e\u000bects act on the axial component of B, generating high-frequency\nrapidly damped oscillations and an associated transitory appearance of reversed axial\n\feld. However, the scalar quantity \r=j\u0001B=B2remains highly non-uniform, so that\nagain the \feld shows no sign of relaxing to a Taylor state for which \rwould have to\nbe constant.\nKeywords : Magnetic relaxation; Taylor conjecture; \u000b-e\u000bect; reversed-\feld pinch\nzCorresponding author: hkm2@cam.ac.uk\nxCorresponding author: krzysztof.mizerski@gmail.comarXiv:1703.00708v1  [physics.plasm-ph]  2 Mar 2017Pinch dynamics in a low- \fplasma 2\n1. Introduction\nMagnetic relaxation is the process by which a magnetic \feld in a highly conducting\n\ruid seeks a minimum energy state subject to pertinent topological constraints (Mo\u000batt\n1985). In the perfect conductivity limit, the lines of force (` B-lines') are frozen in the\n\ruid, a topological constraint represented by the family of magnetic helicity invariants\nHV=Z\nVA\u0001BdV; (1)\nwhere B=r\u0002A, andVis any Lagrangian volume on whose surface @V(with unit\nnormal n)n\u0001B= 0. These invariants represent the conserved degree of linkage of B-\nlines within V. Of particular importance is the global magnetic helicity H, integrated\nover the whole domain Dof \ruid. The magnetic energy\nM(t) =1\n2Z\nDB2dV (2)\nthen has a lower bound (Arnold 1974)\nM(t)>\u0015jHj; (3)\nwhere\u0015>0 is a constant that depends on the scale and geometry of D.\nThe particular problem addressed in this paper concerns relaxation in a plasma of\nextremely low-density \u001a, in which the \ruid pressure pis negligible compared with the\nmagnetic pressure pM=1\n2\u00160B2, i.e.\f=p=pM\u001c1. Flow of the plasma is then driven\nsolely by the Lorentz force j\u0002B. The low density implies further that inertia is negligible\ncompared with the viscous force in the Navier-Stokes equation, the viscosity \u0016being\nessentially independent of \u001ain the limit \u001a!0. These approximations have been adopted\nin a cartesian model by Bajer & Mo\u000batt (2013) who treated relaxation of a single-\ncomponent \feld, and by Mo\u000batt (2015) who considered the case of a two-component\n\feld with non-zero helicity. Here we shall consider the situation in a cylindrical geometry,\nfor which the Lorentz force includes the `hoop stress' associated with curvature of the\nB-lines.\n2. Relaxation in a low- \fplasma\nWe consider a two-component helical \feld in cylindrical polar coordinates fr;\u0012;zgof\nthe form\nB=B0(0;b\u0012(r;t);bz(r;t)): (4)\nThe associated current distribution is given by k\nj=r\u0002B=B0\u0012\n0;\u0000@bz\n@r;1\nr@\n@r(rb\u0012)\u0013\n: (5)\nThe \frst objective is to determine how the relaxation of this \feld is constrained by the\ninitial magnetic helicity distribution. The component b\u0012of the \feld is responsible for\nkFor simplicity of notation, we absorb the conventional constant \u00160in the de\fnition of j.Pinch dynamics in a low- \fplasma 3\nthe classic `pinch e\u000bect' (Bennet 1934). In this scenario, it is natural to suppose that the\ninitialz-component of \feld is uniform, and that the initial \u0012-component is concentrated\nnear the outer cylindrical boundary, such a \feld then providing a radial Lorentz force\nthat tends to drive the plasma inwards. We shall suppose that the resulting radial\nmotion\nu= (u(r;t);0;0) (6)\nis controlled by viscosity, which, as indicated above, dominates over inertia when the\nplasma density is su\u000eciently small. We further suppose that the \ruid pressure is\nnegligible compared with the magnetic pressure, i.e. this is a `low- \f' plasma. Of course,\nthe \ruid pressure increases in the inner region where the density increases; the e\u000bect of\nthis increase can be included without di\u000eculty in the numerical treatment.\nIn the perfect conductivity limit \u0011= 0, the magnetic \feld evolves according to the\n`frozen-\feld' equation\n@B=@t=r\u0002 (u\u0002B); (7)\nand, with the neglect of inertia and pressure gradient, the Navier-Stokes equation\ndegenerates to\n0=j\u0002B+\u0016sr2u+\u00121\n3\u0016s+\u0016b\u0013\nr(r\u0001u); (8)\nwhere\u0016sand\u0016bare the shear and bulk viscosities. From (7) and (8), an equation may\neasily be derived for the magnetic energy:\ndM=dt =\u0000Z\nV\u0014\n\u0016s(r\u0002u)2+\u00124\n3\u0016s+\u0016b\u0013\n(r\u0001u)2\u0015\ndV: (9)\nThusM(t) is monotonic decreasing and bounded below by (3). Equilibrium is attained\nonly when u\u00110, and then from (8), j\u0002B= 0, i.e. the \feld is `force-free'. Hence\nj=\rB; (10)\nfor some pseudo-scalar function \r(x) satisfying ( B\u0001r)\r= 0.\nFor the particular one-dimensional geometry considered in this paper, it is therefore\nto be expected that, when the magnetic di\u000busivity \u0011is su\u000eciently weak, the \feld will\nrelax rapidly during an initial stage to a force-free state (with here \r=\r(r)) that has\nminimum energy compatible with its initial (conserved) topology. Minimising energy\nsubject to the single topological constraint of conserved global helicity yields a force-\nfree \feld structure with \r=cst., a condition that provides reversed axial \feld near the\nouter boundary (Taylor 1974). However, the dynamical process through which such\na reversed \feld may spontaneously appear is not revealed by the simple process of\nseeking a minimum-energy state. Such a reversal cannot in fact appear for so long as\nthe pinching motion is purely radial. However, it seems possible that instabilities of the\nbasic relaxing \feld may lead to an \u000b-e\u000bect, which could conceivably achieve reversal.\nWe shall explore this possibility in x8; \frst however, we treat simple radial relaxation\nneglecting any instabilities that may be present.Pinch dynamics in a low- \fplasma 4\nWe suppose that the plasma is contained in the cylindrical annulus \u000e < r=a < 1,\nwhere 0< \u000e < 1; we shall scale all lengths so that, in e\u000bect, a= 1. The boundaries\nr=\u000eandr= 1 are assumed to be thin perfectly conducting cylinders, separating\nthe plasma from the internal and external regions; these boundaries can therefore\nsupport current sheets with both z- and\u0012-components. The electric \feld is given by\nE=\u0011j\u0000u\u0002B= (0;E\u0012;Ez), and we suppose that E= 0 in the internal and external\nregions, assumed insulating, i.e. for r < \u000e andr >1. Sinceu= 0 on both boundaries\nand both tangential components of Eare continuous across them, it follows that, with\nn= (1;0;0),\n\u0011n\u0002j=n\u0002E= 0 onr=\u000eand onr= 1; (11)\ni.e. that\n\u0011@bz=@r= 0; \u0011@ (rb\u0012=)@r= 0;onr=\u000eand onr= 1: (12)\nSome type of boundary-layer behaviour is to be expected in the limit \u0011!0.\n3. Field evolution and \rux conservation\nWhen the magnetic di\u000busivity \u0011is nonzero, the \feld evolution is described by the\ninduction equation,\n@B\n@t=\u0000r\u0002 E=r\u0002 (u\u0002B) +\u0011r2B: (13)\nand the \ruid density \u001asatis\fes the mass conservation equation\n@\u001a\n@t=\u0000r\u0001 (\u001au): (14)\nThese equations may be combined to give\nD\nDt\u0012B\n\u001a\u0013\n=\u0012B\n\u001a\u0001r\u0013\nu+\u0011\n\u001ar2B; (15)\nwhereD=Dt\u0011(@=@t +u\u0001r), the Lagrangian (or `material') derivative. Noting that,\nfrom (4) and (6),\n(B\u0001r)u=B0(0;b\u0012u=r;0); (16)\nwhen\u0011= 0;eqn.(15) gives\nD\nDt\u0012bz\n\u001a\u0013\n= 0 andD\nDt\u0012b\u0012\nr\u001a\u0013\n= 0; (17)\ni.e. when following any material element of \ruid, bz=\u001aand (b\u0012=r)=\u001aare constant.\nIn Eulerian form, eqn.(13) has components\n@b\u0012\n@t=\u0000@\n@r(ub\u0012) +\u0011@\n@r1\nr@\n@r(rb\u0012); (18)\nand\n@bz\n@t=\u00001\nr@\n@r(rubz) +\u00111\nr@\n@rr@bz\n@r: (19)Pinch dynamics in a low- \fplasma 5\nThe \rux of bzbetween the two cylinders is\n\bz=Z1\n\u000ebz(r;t) 2\u0019rdr; (20)\nand we note, using (19) and the conditions u= 0; \u0011@bz=@r= 0 on both boundaries,\nthat\n1\n2\u0019d\bz\ndt=Z1\n\u000e@bz\n@trdr=Z1\n\u000e@\n@r\u0012\n\u0000rubz+\u0011r@bz\n@r\u0013\ndr=\u0014\n\u0000rubz+\u0011r@bz\n@r\u00151\n\u000e= 0:(21)\nIt follows that \b z= cst.\nSimilarly, the \rux of b\u0012in the\u0012-direction, per unit axial length between the\ncylinders, is\n\b\u0012=Z1\n\u000eb\u0012(r;t) dr; (22)\nand it follows in the same way from (18), and the conditions u= 0; \u0011@(rb\u0012)=@r= 0 on\nboth boundaries, that \b \u0012= cst. also. The constraints\n\bz= cst.;\b\u0012= cst.; (23)\nprovide an important check on the numerical computations that follow (see Fig. 2(f)).\nThe mass per unit axial length Mis of course similarly constant:\nM=Z1\n\u000e\u001a(r;t) 2\u0019rdr= cst. (24)\nThe results (23) are clearly compatible with (17) when \u0011= 0.\n4. Initial conditions\nWe adopt as initial conditions for the magnetic \feld\nbz(r;0) =1\n\u0019(1\u0000\u000e2)(so \bz= 1); (25)\nand\nrb\u0012(r;0) =c\n3(1\u0000\u000e)3\u0002\n3(r\u0000\u000e)2(1\u0000\u000e)\u00002(r\u0000\u000e)3\u0003\ne\u0000k(1\u0000r)2: (26)\nThese are chosen to be compatible with the boundary conditions (12) for any values of\nthe parametersf\u000e;c;kg, and to satisfy b\u0012(1;0) =c. Fig. 1 shows the \feld rb\u0012(r;0) for\n\u000e=c= 0:5 and for three values k= 1;10;30, together with corresponding values of\nthe \rux \b \u0012. Increasing kleads to increasing concentration of rb\u0012(r;0) near the outer\nboundary. Rather arbitrarily we choose\n\u000e= 0:5; c = 0:5; k = 10; (27)\nin the computations that follow. We further adopt initial conditions for the velocity and\ndensity \felds,\nu(r;0) = 0; \u001a (r;0) =\u001a0; (28)\ni.e. the plasma is initially at rest with uniform density \u001a0.Pinch dynamics in a low- \fplasma 6\n0.50.60.70.80.91.00.10.20.30.40.5    k=1       10     30Φθ = 0.144     0.110     0.076rr bθ(r,0)\nFigure 1. Initial pro\fles of rb\u0012for\u000e= 0:5; c= 0:5, and three values of k, with\ncorresponding values of the \rux \b \u0012.\n5. Dynamics of the relaxation process\nThe Lorentz force in the cylindrical geometry considered here takes the form\nj\u0002B=B2\n0\u0012\n\u00001\n2@\n@r(b2\n\u0012+b2\nz)\u0000b2\n\u0012\nr;0;0\u0013\n: (29)\nThe Navier-Stokes equation, including this term, has only a radial component:\n\u001a\u0012@u\n@t+u@u\n@r\u0013\n=\u0000B2\n0\u00121\n2@\n@r(b2\n\u0012+b2\nz) +b2\n\u0012\nr\u0013\n+\u0016\u0012@2u\n@r2+1\nr@u\n@r\u0000u\nr2\u0013\n;(30)\nwhere\u0016= 4\u0016s=3+\u0016bis an e\u000bective viscosity. As in Mo\u000batt (2015), it is now convenient\nto introduce dimensionless variables\n^r=r=a; ^t=tB2\n0=\u0016; ^\u001a=\u001a=\u001a 0;^u=u\u0016=B2\n0a; (31)\nand dimensionless parameters\n\u0014=\u0011\u0016=B2\n0a2; \u000f =\u001a0B2\n0a2=\u00162; (32)\nwe assume that both these parameters are small: \u0014\u001c1 (i.e. small di\u000busivity); and\n\u000f\u001c1 (i.e. low density). With the variables (31), and immediately dropping the hats,\neqn. (14) is unchanged, while in eqns. (18) and (19), \u0011is simply replaced by \u0014:\n@b\u0012\n@t=\u0000@\n@r(ub\u0012) +\u0014@\n@r1\nr@\n@r(rb\u0012); (33)\nand\n@bz\n@t=\u00001\nr@\n@r(rubz) +\u00141\nr@\n@rr@bz\n@r: (34)\nThe momentum equation (30) becomes\n\u000f\u001a\u0012@u\n@t+u@u\n@r\u0013\n=\u00001\n2@\n@r(b2\n\u0012+b2\nz)\u0000b2\n\u0012\nr+\u0012@2u\n@r2+1\nr@u\n@r\u0000u\nr2\u0013\n: (35)\nHere, the term1\n2@(b2\n\u0012+b2\nz)=@ris the gradient of magnetic pressure, and the term \u0000b2\n\u0012=r\nis the additional `hoop stress' that arises due to curvature of the B-lines.Pinch dynamics in a low- \fplasma 7\n6. Numerical integration\nWe can now proceed to numerical integration of these equations, with boundary\nconditions as already stated. When \u00146= 0, these are\nu= 0; @bz=@r= 0; @(rb\u0012=)@r= 0;onr=\u000eand onr= 1; (36)\nand initial conditions as speci\fed in the previous section. Results obtained with\nMathematica are summarised in Fig. 2(a -h), for the particular choice of parameters\n(27), together with \u000f= 0:01;\u0014= 0:0001 (the behaviour for this choice is quite typical).\nThe panels of the \fgure show (a) the inward movement of the density \feld \u001a(r;t) in\nresponse to the negative radial Lorentz force (b), which collapses rapidly to near zero;\n(c,d) the corresponding evolution of the magnetic \feld components; (e) the rise of both\n\feld components at the inner boundary; (f) the decay of magnetic energy; (g) the\npseudo-scalar coe\u000ecient \r(r) when the nearly force-free state has been established; and\n\fnally (h) the \ruxes \b zand \b\u0012which, as expected, remain constant to within numerical\nerror throughout the whole computational period 0 <t< 500.\nThe following points are particularly worth noting. First, there are clearly two\nphases to the evolution: an initial phase (here 0 <t.7) during which magnetic di\u000busion\nis negligible and the magnetic energy decreases relatively rapidly on the (dimensional)\ntime-scale\u0016=B2\n0; and a slow di\u000busive stage t&7, during which bzslowly relaxes back\nto its initial uniform value (its \rux remaining constant); during this phase, rb\u0012also\nslowly decays to a constant C(= \b\u0012=log(1=\u000e)), implying ultimate concentration of\naxial current as a current sheet on the inner boundary r=\u000e.\nSecond, although not shown in the \fgure, the energy1\n2R\nb2\nzdVof thebz-\feld\nactually increases during the initial phase (reaching a maximum at t\u001918), but this\nincrease is more than compensated by the decrease of energy of the b\u0012-\feld. Later,\nduring the slow di\u000busive stage, both contributions to energy slowly decrease. Fig. 2(f)\nshows the decay of energy for \u0014= 0:00001 as well as for \u0014= 0:0001; as expected, the\ndi\u000busive e\u000bect is less apparent in the former case, but the initial (non-di\u000busive) stage is\nquite similar.\nThird, as previously noted, when the Lorentz force is e\u000bectively zero (i.e. during\nthe di\u000busive phase), j=\r(r;t)B; the coe\u000ecient \ris then given by\n\r= (j\u0001B)=B2: (37)\nThis coe\u000ecient, shown in Fig. 2(g), is far from uniform in r, so this is certainly not a\nTaylor state (for which \rwould necessarily be uniform). The weak time-dependence\nof\rresults from slow continuing evolution during the di\u000busive stage. This stage is\ninteresting because the \feld components continuously adjust themselves in such a way\nthat the force-free condition is maintained; in other words, this is not a pure di\u000busive\nprocess, but one that is still constrained to remain nearly force-free through the dynamics\nencapsulated in eqn. (35).Pinch dynamics in a low- \fplasma 8\n0.50.60.70.80.91.00.70.80.91.01.11.21.30.60.70.80.91.0\n-0.8-0.6-0.4-0.2t = 0t = 02.510100100102.5\n(a)                                                                                      (b)ρ(r, t)(j × B)r \n0.50.60.70.80.91.00.10.20.30.40.5\n0.50.60.70.80.91.00.350.400.450.50\n0204060801000.10.20.30.40.50.6\n01002003004005000.050.100.150.200.250.30   (c)                                                                                         (d)\n     (e)                                                                                           (f)\n       (g)                                                                                            (h)Φz/4Φθt = 02.510100500r bθ(r, t)bz(r, t)t = 02.510010t = 500rrbz(0.5, t)5 bθ(0.5, t)t051015200.3100.3150.3200.3250.330M(t)\ntκ = 0.00001  κ = 0.0001\n0.60.70.80.91.00.51.01.52.02.53.03.5γ(r)tt=200 ———   300 ———   400 ———   500 ———\nFigure 2. Pinch e\u000bect with \u000e=c= 0:5;k= 10;\u000f= 0:01;\u0014= 0:0001; (a) Evolution\nof density \feld \u001a(r;t); (b) collapse of Lorentz force ( j\u0002B)rduring early stage of\nrelaxation; (c) evolution of rb\u0012(r;t) and (d) of bz(r;t); (e) the increase of 5 b\u0012(\u000e;t) and\nbz(\u000e;t) during the initial phase; (f) the decay of magnetic energy; the dashed line\nseparates the early phase from the later di\u000busive phase of evolution; (g) the function\n\r(r) = (j\u0001B)=B2att= 200;300;400;500; and (h) the \ruxes \b zand \b\u0012, which remain\nsensibly constant.Pinch dynamics in a low- \fplasma 9\nFigure 3. Comparison of situation when \u0014= 0 and\u0014= 0:00001: (a) rb\u0012(r;t) at\nt= 200 for\u0014= 0 (blue) and \u0014= 0:00001 (red); (b) expanded view of the same curves\nin the region 0 :9< r < 1; (c)bz(r;t) att= 200 for\u0014= 0 (blue) and \u0014= 0:00001\n(red); (d) expanded view of the bz-curves in the region 0 :9<r< 1;\n7. Limiting behaviour as \u0011!0\nIn the limit \u0011= 0, i.e.\u0014= 0, we drop the di\u000busion terms in (33) and (34), and only the\nboundary condition u= 0 of (36) survives, (12) being then automatically satis\fed. The\ntwo \ruxes \b zand \b\u0012are still conserved, within numerical error, as in Fig. 2. Comparison\nwith the situation when \u0014= 0:00001 is interesting. The di\u000berence is in fact very slight\nup to about t= 100, but becomes visible, particularly near the boundary r= 1 by\nt= 200, as shown in Fig. 3(a,b). The expanded views in Figs. 3(b,d) show that indeed\nwhen\u0014= 0 the boundary condition @(rb\u0012)=@r= 0 is notmaintained at r= 1; when\n\u0014= 0:00001 a weak boundary layer is required, within which the solution adapts to this\nboundary condition.\nAs time advances, the curve for \u0014= 0:00001 continues to evolve due to weak\ndi\u000busion towards the situation rb\u0012=C, whereas the curve for \u0014= 0 remains static.\nIn this case of \u0014= 0, there is only the initial phase of non-di\u000busive relaxation to a\nforce-free minimum-energy \feld, which then remains essentially static.\n8. Inclusion of an \u000b-e\u000bect\nIt is obvious that purely radial \row cannot lead to local reversal of an axial \feld that is\ninitially uniform, even in conjunction with di\u000busion. However, the situation considered\nby Taylor (1974) in the context of the reversed-\feld pinch was that of a fully turbulent\nplasma in which more complex processes must be present. Such turbulence presumably\nresults from persistent instability of the evolving mean \feld B, which has non-zeroPinch dynamics in a low- \fplasma 10\nhelicity represented both by the integrals (1) and also by the non-zero pseudo-scalar\nj\u0001b. Such instabilities must inherit the helicity of this mean \feld, and may be expected\nto provide an \u000b-e\u000bect, such that the (non-dimensionalised) mean electric \feld becomes\nE=\u000bb\u0000u\u0002b+\u0014j; (38)\nwhere ustill represents the mean radial \row, and \u0014now includes turbulent as well as\nmolecular di\u000busivity. A stability analysis following the approach of Furth, Killeen &\nRosenbluth (1963) has been carried out by Mizerski (2017), and a resulting anisotropic\n\u000b-e\u000bect deduced. Here, we adopt the simpler isotropic prescription\n\u000b(r;t) =q\u0014j\u0001b; (39)\nwhereqis a pure scalar constant (positive or negative); the factor \u0014is included here in\nrecognition of the di\u000busive origin of the \u000b-e\u000bect. The choice (39) of course ensures that\n\u000bhas the same pseudo-scalar character as j\u0001b.\nWith this prescription for \u000b, the boundary condition n\u0002E= 0 becomes\n\u0014n\u0002(q(j\u0001b)b+j) = 0 onr=\u000e;1; (40)\nand, in the cylindrical geometry, this is still obviously satis\fed by j\u0012=jz= 0 onr=\u000e;1.\nThe only modi\fcation required is therefore the inclusion of a term\nr\u0002 (\u000bb) =\u0012\n0;\u0000@(\u000bbz)\n@r;1\nr@\n@r(\u000brb\u0012)\u0013\n(41)\nin the induction equation. In exploratory vein, we include only the z-component {, thus\nreplacing (34) by\n@bz\n@t=\u00001\nr@\n@r(rubz) +\u00141\nr@\n@rr@bz\n@r+q\u00141\nr@\n@r[(j\u0001b)rb\u0012]; (42)\nwhile leaving (33) unaltered. From (4) and (5), we then have\n@bz\n@t=\u00001\nr@\n@r(rubz)+\u00141\nr@\n@rr@bz\n@r+q\u00141\nr@\n@r\u0014\nbzb\u0012@\n@r(rb\u0012)\u0000rb2\n\u0012@bz\n@r\u0015\n;(43)\n@b\u0012\n@t=\u0000@\n@r(ub\u0012) +\u0014@\n@r1\nr@\n@r(rb\u0012); (44)\n\u000f\u001a\u0012@u\n@t+u@u\n@r\u0013\n=\u00001\n2@\n@r(b2\n\u0012+b2\nz)\u0000b2\n\u0012\nr+\u0012@2u\n@r2+1\nr@u\n@r\u0000u\nr2\u0013\n; (45)\n@\u001a\n@t=\u00001\nr@\n@r(r\u001au): (46)\nwhere, for convenience, we include the remaining unaltered evolution equations.\n{Justi\fcation is suggested by study of the interaction of tearing modes proportional to exp i(kzz+m\u0012)\n(Mizerski 2017, particularly eqn. (44)). It is found that the \u0012- andz- components of the curl of the\nresulting mean emf di\u000ber by a factor proportional to kz=m, and that, summing over all Fourier modes,\none of these components must vanish.Pinch dynamics in a low- \fplasma 11\n8.1. Field reversal and negative di\u000busivity\nConsider \frst whether and under what circumstances reversals of the axial \feld\ncomponent bzmay occur. Suppose that this does happen, and that the reversal is\ninitiated at a critical time t\u0003; thenbz(r;t\u0003) must equal zero for some r=r\u0003wherer\u0003\nmay be an internal point or an end-point of the closed interval [ \u000e;1]. Moreover, for\nt<t\u0003,bz(r;t)>0 for allr2[\u000e;1]. Then, at t=t\u0003, we have\nbz(r;t\u0003) = 0; @bz(r;t\u0003)=@r= 0;atr=r\u0003; (47)\nand\n@2bz(r;t\u0003)=@r2>0 atr=r\u0003; (48)\nsince the curvature is necessarily positive at this point. From eqn.(43), we then have\n\u0014@bz(r\u0003;t)]\n@t\u0015\nt=t\u0003=\u0014\u0000\n1\u0000q[b\u0012(r\u0003;t\u0003)]2\u0001\u0014@2bz(r;t\u0003)\n@r2\u0015\nr=r\u0003; (49)\nall other terms vanishing by virtue of (47). Hence it would appear that bz(r;t) will\nindeed become negative in a neighbourhood of r\u0003, provided\nq[b\u0012(r\u0003;t\u0003)]2>1: (50)\nThis can therefore occur only if q >0 andb\u0012is su\u000eciently strong; and it is likely to\noccur \frst in the region where jb\u0012jis maximal. More generally, (43) may be written\n@bz\n@t=\u00001\nr@\n@r(rubz) +\u0014\u0000\n1\u0000qb2\n\u0012\u00011\nr@\n@rr@bz\n@r\n+q\u0014\nr@\n@r\u0014\nbzb\u0012@\n@r(rb\u0012)\u0015\n\u0000q\u0014@bz\n@r@\n@r\u0000\nb2\n\u0012\u0001\n: (51)\nThe second term on the right-hand side has a di\u000busive character, but with negative\ndi\u000busivity in any region where qb2\n\u0012>1. (The remaining terms of the right-hand side\ninvolve only bzand@bz=@r.) It follows that bzcan become negative only if this is\n`triggered' by a period of negative di\u000busivity in some r-interval.\nWe continue to use the parameter values (27). With these values, b\u0012(r;0) is maximal\natr\u00190:9765, with maximum value 0 :5059; an interval of negative di\u000busivity therefore\noccurs forq&3:907.\n8.2. Results for q\u00144\nNumerical integration for q.3:5 were quite regular, and not greatly di\u000berent from\nthe situation when q= 0. However, as might be expected from the above discussion,\nnumerical instabilities that are di\u000ecult to control appear when q&4. This however is\nthe regime that must be investigated in seeking possible reversal of bz(r;t).\nThis led us to adopt a controllable numerical procedure, speci\fcally 4th-order \fnite-\ndi\u000berences in the radial direction and 2nd-order Adams-Bashforth time-stepping with\nCrank-Nicolson treatment of the di\u000busive terms. The results for q= 0 were as expected\nin complete agreement with those obtained using Mathematica . We focus \frst on thePinch dynamics in a low- \fplasma 12\nFigure 4. Computed di\u000berences \u0001 bz(r;t;q), \u0001rb\u0012(r;t;q), \u0001\u001a(r;t;q) and \u0001u(r;t;q)\nbetween the case q= 4 and the case q= 0 for\u0014= 0:01 at times t= 0;0:5;2;4;8. The\nleft column shows, for reference, the evolution of bz(r;t),rb\u0012(r;t),\u001a(r;t) andu(r;t)\nwhenq= 0.\nnear-critical situation when q= 4. Let\u001a(r;t;q) denote the density computed for any\nparticular value of q, and let\n\u0001\u001a(r;t;q)=\u001a(r;t;q)\u0000\u001a(r;t; 0); (52)\nsimilarly for \u0001 bz(r;t;q);\u0001rb\u0012(r;t;q);\u0001u(r;t;q);\u0001(j\u0002b)r(r;t;q) and \u0001\r(r;t;q).\nFigs. 4 and 5 show numerical results for \u0014= 0:01; the left-hand columns show curves\nforq= 0, while the right-hand columns show the di\u000berences \u0001 bz(r;t;q), etc.\nNote \frst from the last row of Fig. 4 that the negative (pinching) velocity is initially\ndecreased in magnitude by the \u000b-e\u000bect when q= 4; it actually becomes weakly positivePinch dynamics in a low- \fplasma 13\nFigure 5. Computed di\u000berences \u0001( j\u0002b)r(r;t;q) and \u0001\r(r;t;q) for same parameter\nvalues as in Fig. 4. The left column shows, for reference, the evolution of ( j\u0002b)r(r;t)\nand\r(r;t) whenq= 0.\nnearr= 1 fort&2 and is positive over the whole range ( \u000e;1) fort&2:5. This implies\na corresponding net decrease in the inward transport of mass; however, the decrease in\ntransport for the magnetic \feld component bz(r;t) is more than compensated by the\ndirect action of the \u000b-e\u000bect. When q= 4,bz(r;t) decreases much more rapidly than\nwhenq= 0 (by a factor of about 3) near r= 1 in the early stage of relaxation; rb\u0012(r;t)\nincreases more rapidly near r= 1, but by a more modest amount ( \u00186%). The function\n\r(r;t) de\fned by eqn. (37) is also changed by \u001825% when q= 4, but there is no\napparent tendency for \r(r;t) to become more uniform.\n8.3. Results for q= 5:5\nAs indicated above, we anticipated numerical problems for q&4, and we did indeed\nrun into these. Typically, in the range of qbetween 4 and 6, a packet of oscillations\ninbzof very short wavelength appears at t= 0+ in the region of negative di\u000busivity\nnearr= 1. These oscillations move inwards, in tandem with rb\u0012which decreases till\nthe local di\u000busivity \u0014(1\u0000qb2\n\u0012) becomes positive, at which stage the oscillations in bzare\ndamped out, the subsequent evolution being quite smooth. Fig. 6 shows this subsequent\nevolution for q= 5:5. Note that already at the early time t= 0:05,bzis negative near\nr= 1, presumably a consequence of the early negative di\u000busivity in this region. The\n\feld then relaxes back, becoming positive at r= 1 by time t= 0:15.\nThe short period of reversed bznearr= 1 is interesting in the context of the\nreversed-\feld pinch. However, we can't be certain that this is a genuine physical e\u000bect\nrather than just a consequence of adopting a unphysical model for the \u000b-e\u000bect yielding\na period of negative di\u000busion. The behaviour for t= 0+ is evidently non-analytic; anPinch dynamics in a low- \fplasma 14\nFigure 6. Early stage of evolution of bz(r;t) for q=5.5, \u0014= 0:01, and other parameter\nvalues as in Fig. 4.\nasymptotic treatment of the behaviour as t#0 is presented in Appendix A.\n9. Conclusions\nWe have investigated the relaxation of an axisymmetric magnetic \feld having both\naxial and toroidal components in a cylindrical geometry with perfectly conducting\nboundaries. The density is assumed very small and \ruid pressure is neglected compared\nwith magnetic pressure. A purely radial \row is driven by the Lorentz force and energy\nis dissipated by viscosity. In the zero-resistivity limit, the \feld rapidly relaxes to a\nforce-free state. When weak resistivity is taken into account, the initial rapid relaxation\nis followed by slow decay of the \feld which is constrained to remain nearly force-free\nwithj=\r(r;t)B. However\r(r;t) is quite strongly non-uniform, so this is not a Taylor\nstate.\nInx8, we have explored the possibility that an \u000b-e\u000bect, with \u000bproportional to\nj\u0001b, might be capable of causing axial \feld reversal near the outer boundary where\nthe toroidal \feld component b\u0012is initially strong. We have found that if this \u000b-e\u000bect\nis su\u000eciently strong, it can produce a region of negative e\u000bective di\u000busivity of the bz-\n\feld near the outer boundary; this can instantaneously generate high-frequency short-\nwavelength oscillations which are rapidly damped as they move into the interior region\nof positive di\u000busivity. A transitory reversal of Bzoccurs near the outer boundary during\nthis process.\nThere are serious numerical di\u000eculties in handling such a situation; nevertheless,\nthis work points to one possibility whereby a reversed axial \feld, as observed in the\nreversed \feld pinch (Taylor 1974), can be dynamically generated from an initially\nuniform axial \feld, through the combined action of pinching by the b\u0012-\feld and a\nsuitably contrived \u000b-e\u000bect. Of course, it would be desirable to derive a correct form of\nthis\u000b-e\u000bect, through investigation of the turbulence that results from instabilities ofPinch dynamics in a low- \fplasma 15\nthe relaxing \feld. Work is ongoing on this aspect of the problem.\nAcknowledgments . The partial funding of the Ministry of Science and Higher\nEducation of Poland within the grant no IP 2014 031373 and statutory activities No\n3841/E-41/S/2015 is gratefully acknowledged. This work was also partially funded from\nthe Leading National Research Centre (KNOW) received by the Centre for Polar Studies\nin Poland for the period 2014-2018.\nReferences\nArnold, V.I. (1974) The asymptotic Hopf invariant and its applications. Proc. Summer School in Di\u000b.\nEqs. Erevan, Armenia [In Russian]. 229-256.\nBajer, K. & Mo\u000batt, H. K. (2013) Magnetic relaxation, current sheets, and structure formation in an\nextremely tenuous \ruid medium. Astrophys. J. 779, 169-182.\nBender, C.M. & Orszag, S.A. (1978) Advanced Mathematical Methods for Scientists and Engineers.\nMcGraw?Hill, New York.\nBennett, W. H. (1934) Magnetically self-focussing streams. Phys. Rev. 45.12: 890.\nFurth, H. P., Killeen, J. & Rosenbluth, M. N. (1963) Finite-resistive instabilities of a sheet pinch Phys.\nFluids , 459-\nMizerski, K. (2017) Large scale EMF in current sheets induced by tearing modes, Fluid Dyn. Res. ,\nSubmitted.\nMo\u000batt, H. K. (1985) Magnetostatic equilibria and analogous Euler \rows of arbitrarily complex\ntopology. Part 1. Fundamentals. J. Fluid Mech. 159359-378.\nMo\u000batt, H. K. (2015) Magnetic relaxation and the Taylor conjecture. J.Plasma Phys. 81.06: 905810608.\nTaylor, J. B. (1974) Relaxation of toroidal plasma and generation of reverse magnetic \felds, Phys. Rev.\nLett.33, 1139-1141.\nAppendix A. Short-time asymptotics\nThe short-time behaviour of bz(r;t) is bound to be non-analytic, since even the linear\ndi\u000busion term introduces short-time dependence of the sort \u0018exp(\u0000r2=4\u0014t). Although\nthe problem is nonlinear, to get some insight into this short-time behaviour, we suppose\nthat the controlling factor of bz, i.e. its most rapidly changing component (Bender &\nOrszag 1978), has the asymptotic WKB form\nbz(r;t)\u0000bz(r;0)\u0018eT(t)R(r)ast!0; (A.1)\nwhereT(t) andR(r) are functions to be determined. We further assume, that since\nthe velocity is weak throughout the entire evolution, the terms involving u(r;t) in (43)\nand (44) can be neglected at the initial stage leaving only the e\u000bect of di\u000busion in the\nb\u0012component. The behaviour at small tofb\u0012can therefore be expressed by a simple\nformulab\u0012\u0019b\u0012(r;0) +\f(r;t) where\f(r;t)!0 ast!0. Substituting the asymptotic\nexpressions for the magnetic \feld components into equation (43) we obtain, for small t,\nR_T \u0019\u0014K(r)R02T2;whereK(r) = 1\u0000qb\u0012(r;0)2; (A.2)\nand where the dot denotes a time derivative and the prime a derivative with respect to\nr. It follows that,\n_T\nT2= cst:=\u0014K(r)R02\nR: (A.3)Pinch dynamics in a low- \fplasma 16\nThe solution of the above equations is\nT(t)R(r) =\u00001\n4\u0014t\u001aZr\n\u000e[K(r)]\u00001=2dr+ cst:\u001b2\n: (A.4)\nWe have seen that increasing the value of qleads to negative di\u000busion in the\nevolution of bz(r;t). As long as q.1=b\u0012(1;0)2= 4 the short-time asymptotics of\nbz(r;t) involve no signi\fcant irregularities. For qabove the critical value, the negative\ndi\u000busion introduces serious irregularities and the point r=rcat which the di\u000busion\ncoe\u000ecientK(r) changes sign becomes a singular (critical) point. In the vicinity of rc,\nthe WKB solution ceases to be valid since gradients become in\fnite and a critical layer+\nof widthO(\u0015)\u001c1 is required to match the bz-derivatives across rc. A continuous WKB\nsolution for t\u001c1 for the controlling factor, with a jump in the \frst derivative at r=rc,\nsatisfying the boundary conditions (12), can be found in the form\nbz(r;t)\u0000bz(r;0)\u0018exp8\n<\n:\u00001\n4\u0014t\"Zr\n\u000edrp\nK(r)#29\n=\n;; (A.5)\nforr<rc, and\nbz(r;t)\u0000bz(r;0)\u0018\nexp\u001a\n\u00001\n4\u0014t\u0002\nI2\n1\u0000I(r)2+ 2I2I(r)\u0003\u001b\ncos\"p\nI2\n1+I2\n2\n2\u0014t(I(r)\u0000I2)#\n;(A.6)\nforr>rc, where\nI(r) =Zr\nrcdrp\n\u0000K(r); I 1=Zrc\n\u000edrp\nK(r); I 2=Z1\nrcdrp\n\u0000K(r): (A.7)\n(Strictly, the upper limit in I1should berc\u0000\u0015, and the lower limit in I2andI(r) should\nberc+\u0015, due to the presence of the critical layer at rc, which matches the derivatives of\nthe WKB solutions on either side of the critical point rc, but this correction is negligibly\nsmall.) It is clear, therefore, that for r<rcthe solution is regular and controlled mainly\nby di\u000busion. However, for r > rcoscillations of very short O(t) wavelength appear,\nwhich is smallest near rcand increases with r; sinceI1> I 2>I(r) for anyq, the\namplitude of these oscillations is exponentially small and decreases with increasing r.\nThey appear instantaneously at t= 0+ and their wavelength increases with increasing\nt. At very short times they are strongly damped by the very small exponential term in\n(A.6). The situation is depicted (for q= 6) in Fig. A1.\nThese fast short-time oscillations are just as described in x8.3 for the run with\nq= 5:5. At the earliest stage of evolution we observed a small drop in the axial \rux \b z\nofbzfrom 1 to about 0 :94, which subsequently remained constant; the toroidal \rux \b \u0012\n+With\u0018= (r\u0000rc)=\u0015,\u0015\u001c1 being the critical layer thickness, one can expand 1 \u0000qb\u0012(rc+\u0018\u0015;0)2\u0019\n\u0000q\u0018\u0015[b\u0012(r;0)2]0\nr=rcand thebzcritical-layer equation is @t\u0003bz\u0019\u0000\u0018@2\n\u0018bzwith\u0015=q\u0014\u001c[b\u0012(r;0)2]0\nr=rcand\nt=t\u0003\u001c{ note that as the time scale \u001cincreases the critical layer thickens; this equation may be solved\nin terms of a Laplace transform in time with r-dependence of each Laplace mode \u0018e\u001b\u001cin the form\n\u0018\u00181=2J1(\u001b\u00181=2), and matched to the WKB solutions on both sides, i.e for r<rcandr>rc.Pinch dynamics in a low- \fplasma 17\nFigure A1. Plots of the asymptotic r-dependence of the controlling factor, i.e. the\nmost rapidly changing component of bz(r;t\u001c1) at times 4 \u0014t= 0:001 - top row, and\n4\u0014t= 0:01 - bottom row, for q= 6 (rc\u00190:8725) and other parameter values as in the\nnumerical simulations (Fig. 4). The regions r < rcandr > rcare plotted separately\nin the left and right columns respectively. Note that in reality the sharp change in\nderivative at r=rcis smoothed out by a boundary layer, thickening with time.\nand the mass Mexperienced much smaller jumps at the same moment as that of \b z, but\nthen also remained constant. These jumps are an indication of unavoidable numerical\ninaccuracy at this earliest stage when the extremely short wavelength oscillations cannot\nbe adequately resolved by numerical procedure, however much re\fned."    },    {        "title": "1410.7609v2.Equations_of_a_Moving_Mirror_and_the_Electromagnetic_Field.pdf",        "content": "arXiv:1410.7609v2  [physics.optics]  11 Jun 2015Equations of a Moving Mirror and the\nElectromagnetic Field\nLuis Octavio Casta˜ nos1and Ricardo Weder2\nDepartamento de F´ ısica Matem´ atica, Instituto de Investigacion es en Matem´ aticas\nAplicadas y en Sistemas, Universidad Nacional Aut´ onoma de M´ exico , Apartado\nPostal 20-126, M´ exico DF 01000, M´ exico\nE-mail:1. loccj@yahoo.com , 2. weder@unam.mx\nAbstract. We consider a system composed of a mobile slab and the electromagne tic\nfield. We assume that the slab is made of a material that has the follow ing properties\nwhen it is at rest: it is linear, isotropic, non-magnetizable, and ohmic w ith zero\nfree charge density. Using instantaneous Lorentz transformat ions, we deduce the\nset of self-consistent equations governing the dynamics of the sy stem and we obtain\napproximate equations to first order in the velocity and the acceler ation of the slab.\nAs a consequence of the motion of the slab, the field must satisfy a w ave equation with\ndamping and slowly varying coefficients plus terms that are small when the time-scale\nof the evolution of the mirror is much smaller than that of the field. Als o, the motion\nof the slab and its interaction with the field introduce two effects in th e slab’s equation\nof motion. The first one is a position- and time-dependent mass relat ed to the effective\nmasstaken in phenomenological treatments of this type of systems. Th e second one\nis a velocity-dependent force that can give rise to friction and that is related to the\nmuch sought coolingof mechanical objects.\nPACS numbers: 42.50.Wk, 03.50.De, 42.65.k, 05.45.a\nSubmitted to: Phys. Scr.Equations of a Moving Mirror and the Electromagnetic Field 2\n1. INTRODUCTION\nOptomechanics studies systems composed of light(the electromagnetic field) and\nmechanical objects such as movable membranes or mirrors [1]-[4]. Th e interaction\nbetween the two components is governed by two principal forces: radiation pressure\nand thermal forces. The latter are also called bolometric forces and consist in light\nabsorption deflecting the mechanical object [5]. In general, both forces are always\npresent, but, depending on the optomechanical system, one can be much larger than\nthe other and they can point in different directions [5]. Using certain experimental\nsetups and certain types of materials one can manage to have mech anical objects in\nwhich one force predominates over the other. In this way, one has systems in which\nradiation pressure dominates [6, 7, 8], others in which thermal forc es are much larger\n[9, 10], and some others in which both types of forces are comparab le [11]-[14].\nResearch in optomechanics is driven mainly by technological applicatio ns in areas\nsuch as lightwave communications [15] and studies in classical and qu antum physics.\nIn particular, optomechanical systems exhibit intricate classical n on-linear dynamics\n[7]-[14], [16, 17] and may be a setting where quantum physics can be st udied in the\nmacroscopic domain [1]-[4]. A first step that may open the door for su ch research is\nthat mechanical objects have been cooled to their ground state in some experimental\nset ups [18, 19].\nIn this work we study a system composed of the electromagnetic fie ld and a mobile\nmirror in the form of a slab. Using the transformation properties of electromagnetic\nquantities under Lorentz transformations and not considering th ermal effects, we\nestablish the classical equations that determine the evolution of th e system. Afterwards,\nwe deduce equations that are correct up to first order in both the velocity and\nacceleration of the slab. With these results at hand, we conclude th at the motion of\nthe mirror and its interaction with the field give rise to a position- and t ime-dependent\nmass related to the effective mass taken in phenomenological treatments of this type\nof systems [4, 10, 13] and to a velocity-dependent force that is re lated to the coolingof\nmechanical objects [4, 5]. Moreover, the field must satisfy a wave e quation that depends\non the slab’s position, velocity, and acceleration.\nThere are a fair number of works discussing the equations governin g these type\nof systems, see [20, 21] and references therein. In particular, [2 0] considers a one-\ndimensional cavity composed of one perfect, fixed mirror and one p erfect, mobile mirror\nwith empty space in between, while [21] considers a one-dimensional c avity composed of\ntwo perfect, fixed mirrors and one mobile, non-conducting mirror w ith constant electric\nsusceptibility in between. Both [20, 21] have the objective of deduc ing a Hamiltonian\nthat approximately describes the mobile-mirror + electromagnetic fi eld system and give\na justification of the Hamiltonian usually used in the area of quantum o ptomechanics.\nOur work is completely different in spirit, since our objective is to esta blish the exact\nclassical equations governing the system from first principles and t o obtain consistent\napproximations of them. Furthermore, we consider that the slab is an ohmic conductorEquations of a Moving Mirror and the Electromagnetic Field 3\nand we establish our equations for general electromagnetic fields a nd dielectric and\nconductivity functions. Moreover, we deduce the time-dependen t mass and velocity-\ndependent force affecting the motion of the mirror mentioned abov e, quantities that\ncannot be deduced correctly from the approximate Lagrangian us ed in [21] because it\ndoes not give the correct equation of motion for the mirror to first order in its velocity\nand acceleration, as it is mentioned in [21].\nFinally, we have strived in keeping the treatment simple enough, since only an\nelementary knowledge of electromagnetism and special relativity is r equired.\nThe material is organized as follows: In Sec. II we introduce the sys tem under\nstudy and we establish the model used to describe it. In Sec. III we determine the\nequations governing the evolution of the electromagnetic field, while Sec. IV considers\nthe special case of a linearly polarized electric field and obtains appro ximations to the\nequation governing the evolution of the field. In Sec. V we deduce th e force affecting\nthe motion of the mobile mirror and establish the equations governing the evolution of\nthe complete field-mobile mirror system. In Sec. VI we discuss the va lidity of the model\nand in Sec. VII we present a Lagrangian density for the electromag netic field. Finally,\nthe conclusions are given in Sec. VIII. Part of this work was presen ted, without giving\nany details of the deduction of the results, at the Latin America Optics and Photonics\nConference (LAOP) 2014 held at Canc´ un, M´ exico [22].\n2. THE SYSTEM\nThe system under study is composed of a mobile mirror and the electr omagnetic field.\nFor simplicity, we assume that the mirror has the form of a slab that h as infinite length\nand width and that has thickness δ0when it is at rest. Moreover, we assume that the\nmirror is made of a material that has the following properties when it is at rest (we\ncall them the rest properties of the mirror ): it is linear, isotropic, non-magnetizable,\nand ohmic with zero free charge density, electric susceptibility χ, and conductivity σ.\nIn all that follows (unless otherwise stated), we assume that the e lectric susceptibility\nand conductivity have continuous first derivatives in all of space an d thatχandσare\nequal to zero outside of the mirror. Also, we use the Minkowski met ric (that is, the\nict-system) for special relativity and Gaussian units for the electrom agnetic field.\nWe mention that it is important to distinguish which properties are sat isfied when\nthe mirror is at rest, since it will be shown explicitly in the following sectio ns that these\ndo not hold when the mirror is in motion.\nWe want to establish the equations governing the evolution of the sy stem in an\ninertial reference frame LS(theLaboratory System ) where\n(i) A right-handed Cartesian coordinate system is specified by unit v ectorsˆ x,ˆ y, and\nˆ zalong the positive directions of the coordinate axes.\n(ii) Thecoordinatesofanarbitraryevent aredenotedby ( x,y,z,ict ). Also,r≡(x,y,z)\nandcis the speed of light in vacuum.Equations of a Moving Mirror and the Electromagnetic Field 4\n(iii) The mirror can move only along the x-axis and it fills the region\nR(t) =/braceleftigg\nr∈R3:|x−q(t)| ≤δ(t)\n2/bracerightigg\n. (1)\nHereq(t) is the position of the midplane of the mirror along the x-axis, while δ(t)\nis the mirror’s thickness in the xdirection. Observe that δ(t) is time-dependent\nbecausethemirrorcanbeinmotionand,consequently, itcanbeLor entzcontracted.\nIn all that follows we call q(t) the midpoint and δ(t) the thickness of the mirror in\nLS. Also, ˙q(t) = (dq/dt)(t)and ¨q(t) = (d2q/dt2)(t)arethevelocity andacceleration\nof the midpoint q(t), respectively.\n(iv) Outside of R(t) there is vacuum.\n(v) We assume that all electromagnetic quantities are functions of onlyxandt.\nMoreover, E(x,t) denotes the electric field, B(x,t) the magnetic field, P(x,t)\nthe polarization, M(x,t) the magnetization, Jf(x,t) the free current density, and\nρf(x,t) the free charge density.\nNow the goal is to deduce the form of P(x,t),M(x,t),Jf(x,t), andρf(x,t). It\nis tempting to calculate these quantities by assuming that the rest p roperties of the\nmirror are valid even if the mirror is in motion, especially in the case of sm all velocities.\nNevertheless, it is shown explicitly in the following sections that one ne glects important\nphenomena if one decides to take that approach. Therefore, we t ake another path\nto determine the aforementioned quantities in LS. We now describe the idea behind\nour approach. At each instant of time in LSwe consider another inertial reference\nframeMSwhere the midpoint q(t) of the mirror is instantaneously at rest. If the\nmirror is not subject to very large accelerations, the mirror will be a pproximately at\nrest inMSduring a small time interval. Hence, one can use the rest properties of the\nmirror to calculate the polarization, magnetization, free current d ensity and free charge\ndensity in MSduring this small time interval. Afterwards, one can use their well kn own\ntransformation properties under Lorentz transformations to o btain these quantities in\nLS. A word of caution, this instantaneous Lorentz transformation works only if both\nthe coordinate origin and the origin of time in LSare first translated to be at q(t) and\nat the instant of time one is considering. We now formalize this idea. In particular,\nwe determine quantitatively the aforementioned small time interval and we establish\nconditions under which the mirror is subject to small accelerations.\nLett0∈Rbe fixed. Consider an inertial reference frame LS0obtained from LSby\na time and space translation where t0is the new origin of time and ( q(t0),0,0) is the\nnew origin of space. In LS0one has the following properties:\n(i) A Cartesian coordinate system is specified by unit vectors ˆ x′,ˆ y′, andˆ z′parallel\nrespectively to ˆ x,ˆ y, andˆ z.\n(ii) The coordinates of an arbitrary event are denoted by ( x′,y′,z′,ict′) and they are\nrelated to the corresponding coordinates ( x,y,z,ict ) inLSby\nx′=x−q(t0), y′=y ,Equations of a Moving Mirror and the Electromagnetic Field 5\nt′=t−t0, z′=z . (2)\nAlso,r′≡(x′,y′,z′).\n(iii)E′(x′,t′) denotes the electric field, B′(x′,t′) the magnetic field, P′(x′,t′) the\npolarization, M′(x′,t′) the magnetization, J′\nf(x′,t′) the free current density, and\nρ′\nf(x′,t′)thefreechargedensity. Noticethat aquantity f′(x′,t′) inLS0isconnected\nto the corresponding quantity f(x,t) inLSby\nf′(x′,t′) =f[x′+q(t0),t′+t0]. (3)\nObserve that at time t′the mirror fills the region\nR′(t′) =/braceleftigg\nr′∈R3:|x′−q′(t′)| ≤δ′(t′)\n2/bracerightigg\n, (4)\nwith\nq′(t′) =q(t′+t0)−q(t0), δ′(t′) =δ(t′+t0), (5)\nits midpoint and its thickness along the x′-axis. As in the case of q(t) andδ(t), we call\nq′(t′) the midpoint and δ′(t′) the thickness of the mirror in LS0. Also, there is vacuum\noutside of R′(t′).\nIn particular, at time t′= 0 the mirror fills the region\nR′(0) =/braceleftigg\nr′∈R3:|x′| ≤δ(t0)\n2/bracerightigg\n, (6)\nand its mid-point satisfies\nq′(0) = 0,dq′\ndt′(0) = ˙q(t0),d2q′\ndt′2(0) = ¨q(t0). (7)\nNow that we have established a reference frame in which the coordin ate origin is at\nthe midpoint of the mirror and the time origin has been redefined appr opriately, we\nintroduce another reference frame in which the mirror is instantan eously at rest.\nDefine\nv0= ˙q(t0), β0=v0\nc, γ0=1/radicalig\n1−β2\n0. (8)\nNotice that v0and, consequently, β0can be positive or negative.\nConsider an inertial reference frame MS0(forMirror System ) where\n(i) A Cartesian coordinate system is specified by unit vectors ˆ x′′,ˆ y′′, andˆ z′′parallel\nrespectively to ˆ x′,ˆ y′, andˆ z′.\n(ii)MS0moves with velocity v0ˆ x′with respect to LS0.\n(iii) The coordinates of an arbitrary event are denoted by ( x′′,y′′,z′′,ict′′) and they\nare related to the corresponding coordinates ( x′,y′,z′,ict′) inLS0by a Lorentz\ntransformation:\nct′′=γ0(ct′−β0x′), y′′=y′,\nx′′=γ0(x′−β0ct′), z′′=z′. (9)\nHere the space-time origin ( r′=0,ict′= 0) ofLS0coincides with the space-time\norigin (r′′=0,ict′′= 0) ofMS0. Also,r′′≡(x′′,y′′,z′′).Equations of a Moving Mirror and the Electromagnetic Field 6\n(iv)E′′(x′′,t′′) denotes the electric field, B′′(x′′,t′′) the magnetic field, P′′(x′′,t′′) the\npolarization, M′′(x′′,t′′) the magnetization, J′′\nf(x′′,t′′) the free current density, and\nρ′′\nf(x′′,t′′) the free charge density.\nUsing the Lorentz transformation in (9) one can relate the coordin ates of the midpoint\nq′(t′) inLS0with those of the midpoint q′′(t′′) of the mirror along the x′′-axis inMS0.\nConsider the event whose coordinates in LS0are given by\nb′= (q′(t′),y′,z′,ict′). (10)\nFrom (4) it is clear that b′is an event associated with a midpoint of the mirror. Using\n(9) it follows that the aforementioned event has coordinates in MS0given by\nb′′= (q′′(t′′),y′,z′,ict′′), (11)\nwith\nq′′(t′′) =γ0[q′(t′)−β0ct′],\nct′′=γ0[ct′−β0q′(t′)]. (12)\nUsing (5) and (9) one can also relate the velocities and accelerations inLS0with those\ninMS0. In particular, for the midpoint one has\ndq′′\ndt′′(t′′) =˙q(t′+t0)−v0\n1−β0\nc˙q(t′+t0),\nd2q′′\ndt′′2(t′′) =¨q(t′+t0)\nγ3\n0/bracketleftig\n1−β0\nc˙q(t′+t0)/bracketrightig3. (13)\nFrom (7), (8), (12), and (13) one obtains at time t′= 0 that\nt′′= 0,dq′′\ndt′′(0) = 0 ,\nq′′(0) = 0,d2q′′\ndt′′2(0) =γ3\n0¨q(t0). (14)\nTherefore, the midpoint q′′(t′′) of the mirror is at rest at the coordinate origin in MS0\nat timet′′= 0, although it can have a non-zero acceleration. Note that, in gen eral, the\nargument above does not imply that the other points of the mirror a re at rest in MS0at\ntimet′= 0 because the events with coordinates ( x′/ne}ationslash= 0,t′= 0) inLS0have coordinates\n(x′′/ne}ationslash= 0,t′′/ne}ationslash= 0) inMS0. This is related to the issue of rigid bodies in special relativity\n[28].\nIn the following we assume that MS0is an inertial reference frame in which all\nthe points of the mirror are instantaneously at rest at time t′′= 0. Notice that this\nassumption holds only approximately if the mirror doesnot move withc onstant velocity.\nIt follows that the mirror occupies the following region in MS0at timet′′= 0:\nR′′(0) =/braceleftigg\nr′′∈R3:|x′′| ≤δ0\n2/bracerightigg\n. (15)\nRecall that δ0is the thickness of the mirror along the x′′-axis when it is at rest.\nSince there is vacuum outside the mirror, we know that the polarizat ion,\nmagnetization, and free current and charge densities are zero in LSat timet0for allxEquations of a Moving Mirror and the Electromagnetic Field 7\noutside the mirror, that is, for all xsuch that |x−q(t0)|> δ(t0)/2, see (1). Therefore,\nwe only have to determine these quantities in LSat timet0for allxinside the mirror,\nthat is, for all xsuch that |x−q(t0)| ≤δ(t0)/2. Equivalently, we have to determine\nthem inLS0at timet′= 0 for all |x′| ≤δ(t0)/2, see (6). We want to take advantage of\nthe rest properties of the mirror. Hence, we can determine the va rious quantities first in\nMS0where the mirror is instantaneously at rest at time t′′= 0 andthen transformthem\nappropriately to LS0. Nevertheless, the condition that the mirror is instantaneously at\nrest at time t′′= 0 is not enough to be able to use the rest properties to calculate th e\nrequiredquantitiesin LS0attimet′= 0. Thereasonforthisisthat, asmentionedabove,\naccording to (9) the events with coordinates ( x′/ne}ationslash= 0,t′= 0) inLS0have coordinates\n(x′′/ne}ationslash= 0,t′′/ne}ationslash= 0) inMS0and all the points of the mirror may not be at rest in MS0for\nt′′/ne}ationslash= 0 (for example, if the mirror does not move with constant velocity in LS, then, in\ngeneral, not all of its points will be at rest in MS0fort′′/ne}ationslash= 0). Consequently, one has\nto assume that the mirror is approximately at rest in MS0during a small time interval\ncentred at t′′= 0 to be able to use the mirror’s rest properties. We now determine t his\ntime interval explicitly.\nFirst observe from (9) the following relation:\nx′∈/bracketleftigg\n−δ(t0)\n2,δ(t0)\n2/bracketrightigg\n, t′= 0\n⇒\n\nx′′∈/bracketleftig\n−γ0δ(t0)\n2,γ0δ(t0)\n2/bracketrightig\n,\nt′′∈/bracketleftig\n−γ0|β0|δ(t0)\n2c,γ0|β0|δ(t0)\n2c/bracketrightig\n.(16)\nIn other words, points x′in the mirror at time t′= 0 inLS0correspond in MS0to\npointsx′′in the mirror at some time t′′in the interval on the right-hand side of (16). If\nwe assume that the mirror is at rest in MS0for allt′′in the aforementioned interval,\nthen we will be able to use the rest properties of the mirror for ( x′′,t′′) corresponding\nto (x′,t′) such that x′is inside the mirror and t′= 0.\nWith the discussion of the two previous paragraphs in mind we make th e stronger\nassumption that the mirror is (approximately) at rest in MS0during the time interval\n[−t′′\n1, t′′\n1], t′′\n1=γ0|β0|δ(t0)\n2c. (17)\nThe mirror is approximately at rest in MS0during the time interval given in (17) if\nand only if it is subject to very small accelerations in MS0. This happens if and only\nif the magnitudes of the electric and magnetic fields are not very larg e, since the force\naffecting the mirror depends on the fields (see Sec. VI). In addition , the results of Sec.\nVI indicate that these requirements are satisfied in most experimen tal situations. Also,\nwe consider that the mirror is approximately at rest in MS0during the time interval\ngiven in (17) if the midpoint q′′(t′′) moves a distance much smaller than δ0/2 during this\ntime interval. Recall that δ0is the thickness of the mirror along the x′′-axis when it is\nat rest.\nSince(byassumption)themirroris(approximately)atrestin MS0fort′′∈[−t′′\n1,t′′\n1],Equations of a Moving Mirror and the Electromagnetic Field 8\nit follows that the region occupied by the mirror in MS0fort′′∈[−t′′\n1,t′′\n1] is given by\nR′′(t′′) =R′′(0) =/braceleftigg\nr′′∈R3:|x′′| ≤δ0\n2/bracerightigg\n. (18)\nNotice that the x′′interval in (18) must coincide with the x′′interval in (16) because\nthey both correspond to the region occupied by the mirror in MS0along the x′′-axis\nand the mirror is (approximately) at rest in MS0fort′′∈[−t′′\n1,t′′\n1]. Therefore,\nδ(t0) =δ0\nγ0. (19)\nNoticethat (19)states that themirror appearstobeLorentz co ntracted alongthe x-axis\nto an observer in LS.\nFinally recall that the electric susceptibility χand the conductivity σof the mirror\nare continuously differentiable functions that are zero outside of t he mirror when it is\nat rest. From (18) it follows that\nχ(x′′) = 0, σ(x′′) = 0 for x′′/ne}ationslash∈/parenleftigg\n−δ0\n2,δ0\n2/parenrightigg\n. (20)\n2.1. The electric and magnetic fields\nThe electric and magnetic fields can be accommodated into a second- rank anti-\nsymmetric tensor (called the electromagnetic tensor ) and, therefore, change accordingly\nunder Lorentz transformations [23]. One can view these transfor mation properties in\nmatrix form as follows:\n/parenleftigg\nE′(x′,t′)\nB′(x′,t′)/parenrightigg\n=M0/parenleftigg\nE′′(x′′,t′′)\nB′′(x′′,t′′)/parenrightigg\n, (21)\nwhere\nM0=\n1 0 0 0 0 0\n0γ00 0 0 γ0β0\n0 0 γ00−γ0β00\n0 0 0 1 0 0\n0 0 −γ0β00γ00\n0γ0β00 0 0 γ0\n. (22)\nHere (x′,t′) are coordinates in LS0and (x′′,t′′) are the corresponding coordinates in\nMS0with the connection given by (9). Also, the j-th component of the column vector\non the left of (21) is E′\nj(x′,t′) ifj= 1,2,3 orB′\nj−3(x′,t′) ifj= 4,5,6. A similar relation\nholds for the column vector on the right-hand side of (21).\n2.2. Polarization and magnetization\nWe want to determine the polarization P(x,t) and magnetization M(x,t) of the mirror\ninLSat timet0. Equivalently, we can determine P′(x′,t′) andM′(x′,t′) inLS0at time\nt′= 0, see (3).Equations of a Moving Mirror and the Electromagnetic Field 9\nThe polarization and magnetization can also be accommodated into a s econd-rank\nanti-symmetric tensor (called the moments tensor ) and, therefore, change accordingly\nunder Lorentz transformations [23]. The relationship between the polarization and\nmagnetization in LS0and inMS0can also be viewed in matrix form as follows:\n/parenleftigg\nP′(x′,t′)\nM′(x′,t′)/parenrightigg\n=M−1\n0/parenleftigg\nP′′(x′′,t′′)\nM′′(x′′,t′′)/parenrightigg\n. (23)\nAgain, (x′,t′) are coordinates in LS0and (x′′,t′′) are the corresponding coordinates in\nMS0with the connection given by (9). Also, the j-th component of the column vector\non the left of (23) is P′\nj(x′,t′) ifj= 1,2,3 orM′\nj−3(x′,t′) ifj= 4,5,6. A similar relation\nholds for the column vector on the right-hand side of (23). Notice t hat the electric\nand magnetic fields are connected by M0, while the polarization and magnetization are\nconnected by M−1\n0, see (21) and (23). This difference is simply due to how the moments\nand electromagnetic tensors are defined, see the Appendix for th e details.\nSince (by assumption) the mirror is (approximately) at rest in MS0during the time\ninterval (17) and the mirror is linear, isotropic, and non-magnetiza ble when it is at rest,\none has\nP′′(x′′,t′′) =χ(x′′)E′′(x′′,t′′),\nM′′(x′′,t′′) =0for allx′′andt′′∈[−t′′\n1,t′′\n1]. (24)\nRecall that the region occupied by the mirror in LS0at timet′= 0 is given in (6). From\n(6), (16), (18), and (19) one has that a point x′inside the mirror at time t′= 0 inLS0\ncorrespondstoapoint x′′insidethemirroratsometime t′′∈[−t′′\n1,t′′\n1]inMS0. Therefore,\n(24) is satisfied for the ( x′′,t′′) that corresponds to x′∈[−δ(t0)/2,δ(t0)/2] andt′= 0.\nHence, one can apply (21), (23), and (24) to obtain that for x′∈[−δ(t0)/2,δ(t0)/2] and\nt′= 0/parenleftigg\nP′(x′,0)\nM′(x′,0)/parenrightigg\n=M−1\n0/parenleftigg\nχ(γ0x′)I3O3×3\nO3×3O3×3/parenrightigg\nM−1\n0×\n×/parenleftigg\nE′(x′,0)\nB′(x′,0)/parenrightigg\n. (25)\nHere we used (9) with t′= 0 to obtain that χ(x′′) =χ(γ0x′). Also, here and in the\nfollowing I3is the identity 3 ×3 matrix and On×mis then×mzero matrix.\nAlso,P′(x′,0) =M′(x′,0) =0for|x′|> δ(t0)/2 because there is vacuum\noutside of the mirror, see (6). Using (19) and (20) in (25) one obta ins precisely that\nP′(x′,0) =M′(x′,0) =0for|x′|> δ(t0)/2. Therefore, (25) is actually valid for all\nx′. Expanding the product in (25) and using (3) and the fact that t0is arbitrary, one\nconcludes that\nP(x,t) =γ(t)2χLS(x,t)[E(x,t)+β(t)ˆ x×B(x,t)\n−β(t)2E1(x,t)ˆ x/bracketrightig\n,\nM(x,t) =−β(t)ˆ x×P(x,t), (26)\nwhere\nβ(t)≡˙q(t)\nc, γ(t)≡1/radicalig\n1−β(t)2, (27)Equations of a Moving Mirror and the Electromagnetic Field 10\nand\nχLS(x,t) =χ{γ(t)[x−q(t)]}. (28)\nNotice that although the mirror is linear, isotropic, and non-magnet izable when it is\nat rest, to an observer in LSit appears to have a magnetization and the polarization\ndepends not only on the electric field, but also on the velocity of the m irror and on the\nmagnetic field.\n2.3. Free current and charge\nSince the mirror has zero free charge density and satisfies Ohm’s law when it is at rest\nand the mirror is (approximately) at rest during the time interval (1 7), one has\nρ′′\nf(x′′,t′′) = 0,J′′\nf(x′′,t′′) =σ(x′′)E′′(x′′,t′′), (29)\nfor allx′′andt′′∈[−t′′\n1,t′′\n1]. From (29) it follows that the current four-vector in MS0is\ngiven by [23]\ns′′(x′′,t′′) =/parenleftig\nJ′′\nf(x′′,t′′), icρ′′\nf(x′′,t′′)/parenrightigT,\n= (σ(x′′)E′′(x′′,t′′),0)T, (30)\nfor allx′′andt′′∈[−t′′\n1,t′′\n1].\nOne can express the connection between (30) and the current fo ur-vector s′(x′,t′)\ninLS0[23] in matrix form as follows:\ns′(x′,t′) =/parenleftig\nJ′\nf(x′,t′), icρ′\nf(x′,t′)/parenrightigT,\n=M1s′′(x′′,t′′), (31)\nwith\nM1=\nγ00 0−iγ0β0\n0 1 0 0\n0 0 1 0\niγ0β00 0γ0\n. (32)\nAgain, (x′,t′) are coordinates in LS0and (x′′,t′′) are the corresponding coordinates in\nMS0with the connection given by (9).\nUsing the assumption that the mirror is at rest during the interval [ −t′′\n1,t′′\n1] in (17)\nand an argument similar to that used with the polarization and magnet ization in the\nprevious section (that is, establishing a formula valid for points inside the mirror and\nthenobservingthatitisalsovalidforpointsoutsidethemirrorbecau sethereisvacuum),\nit follows from (21), (30), and (31) that\ns′(x′,0) =M1/parenleftigg\nσ(γ0x′)I3O3×3\nO1×3O1×3/parenrightigg\nM−1\n0×\n×/parenleftigg\nE′(x′,0)\nB′(x′,0)/parenrightigg\n, (33)Equations of a Moving Mirror and the Electromagnetic Field 11\nfor allx′. Here we used (9) with t′= 0 to obtain that σ(x′′) =σ(γ0x′). Expanding the\nproduct in (33) and using (3) along with the fact that t0is arbitrary, one concludes that\nthe free current density Jf(x,t) and the free charge density ρf(x,t) inLSare given by\nJf(x,t) =γ(t)σLS(x,t)[E(x,t)+β(t)ˆ x×B(x,t) ],\nρf(x,t) =γ(t)\ncβ(t)σLS(x,t)E1(x,t), (34)\nwith\nσLS(x,t) =σ{γ(t)[x−q(t)]}. (35)\nNotice that, even if the mirror has zero free charge density when it is at rest, it appears\nto be charged to an observer in LSifE1(x,t)/ne}ationslash= 0. Also, observe that the mirror does\nnot satisfy Ohm’s law when it is in motion.\n3. MAXWELL’S EQUATIONS\nMaxwell’s equations in LScan be written as\n∇×B(x,t) =4π\nc[Jf(x,t)+Jb(x,t)]+1\nc∂E\n∂t(x,t),\n∇×E(x,t) =−1\nc∂B\n∂t(x,t),\n∇·E(x,t) = 4π[ρf(x,t)+ρb(x,t)],\n∇·B(x,t) = 0, (36)\nwhereρf(x,t) andJf(x,t) are the free charge and current densities given in (34) and\nρb(x,t) andJb(x,t) are the bound charge and current. Using (26) one has\nρb(x,t)≡ −∇· P(x,t),\n=−χLS(x,t)∇·E(x,t)−E1(x,t)∂\n∂xχLS(x,t)\n(37)\nand\nJb(x,t)≡∂P\n∂t(x,t)+c∇×M(x,t),\n=γ(t)2χLS(x,t)/braceleftigg∂E\n∂t(x,t)\n+β(t)/bracketleftigg\nc∂E\n∂x(x,t)+ˆ x×∂B\n∂t(x,t)−c∂E1\n∂x(x,t)ˆ x/bracketrightigg\n+β(t)2/bracketleftigg\nc∇×B(x,t)−∂E1\n∂t(x,t)ˆ x/bracketrightigg\n+dβ\ndt(t)[ˆ x×B(x,t)−2β(t)E1(x,t)ˆ x]/bracerightigg\n+f1(x,t)−β(t)f2(x,t). (38)\nHere\nf1(x,t) = [E(x,t)+β(t)ˆ x×B(x,t)]×Equations of a Moving Mirror and the Electromagnetic Field 12\n×/parenleftigg\nβ(t)c∂\n∂x+∂\n∂t/parenrightigg\nγ(t)2χLS(x,t),\nf2(x,t) =ˆ xE1(x,t)/parenleftigg\nc∂\n∂x+β(t)∂\n∂t/parenrightigg\nγ(t)2χLS(x,t).\n(39)\nNotice that, when the mirror is at rest (that is, ˙ q(t)/c=β(t) = 0 for all t), one has\nχLS(x,t) =χ(x−q) from (28) and the right-hand side of (38) correctly reduces to\n(∂P/∂t)(x,t) =χ(x−q)(∂E/∂t)(x,t) .\n3.1. The case of a piecewise constant susceptibility and con ductivity\nUp to now we have assumed that χ(x′′) andσ(x′′) are continuously differentiable\nfunctions. In this subsection we consider the case where they are given by the following\ntwo piecewise constant functions:\nχ(x′′) =/braceleftigg\nχ0if−δ0\n2≤x′′≤δ0\n2,\n0 elsewhere .\nσ(x′′) =/braceleftigg\nσ0if−δ0\n2≤x′′≤δ0\n2,\n0 elsewhere .(40)\nFrom (28), (35), and (40) it follows that\nχLS(x,t) =/braceleftigg\nχ0if|x−q(t)| ≤δ(t)\n2,\n0 elsewhere .\nσLS(x,t) =/braceleftigg\nσ0if|x−q(t)| ≤δ(t)\n2,\n0 elsewhere .(41)\nHere we have used (19) and the fact that t0∈Ris arbitrary to conclude that\nδ(t) =δ0\nγ(t). (42)\nAll the results we have derived up to now hold with (40). The differenc e consists\nin that one must pasteE(x,t) andB(x,t) correctly at the boundaries of the mirror. In\nother words, one first solves (36) inside the mirror, that is, in R(t) given in (1). Then,\none solves (36) outside the mirror, that is, outside of R(t), and, finally, one applies\nboundary conditions at x=q(t)±δ(t)/2. These are obtained by considering the usual\nboundary conditions for materials at rest in MS0during the time interval [ −t′′\n1,t′′\n1] and\nthen using the transformation equations for the field in (21) to obt ain the corresponding\nboundary conditions in LS0. Afterwards, one simply uses (3) to express the relations\nwith the quantities in LS. This is what we do now.\nSince the mirror is at rest in MS0during the time interval [ −t′′\n1,t′′\n1] given in (17),\nwe can use the usual boundary conditions [24]:\nB′′(x′′\n0+,t′′)·ˆ x′′=B′′(x′′\n0−,t′′)·ˆ x′′,\nE′′(x′′\n0+,t′′)׈ x′′=E′′(x′′\n0−,t′′)׈ x′′, (43)\nand\n[D′′(x′′\n0+,t′′)−D′′(x′′\n0−,t′′)]·ˆ x′′= 4πσ′′\nf(x′′\n0,t′′),Equations of a Moving Mirror and the Electromagnetic Field 13\nˆ x′′×[H′′(x′′\n0+,t′′)−H′′(x′′\n0−,t′′)] =4π\ncK′′\nf(x′′\n0,t′′),\n(44)\nforx′′\n0=±δ0/2 andt′′∈[−t′′\n1,t′′\n1]. Here\nf(a±) = lim x′′→a±f(x′′). (45)\nIn (44) the quantities σ′′\nf(x′′\n0,t′′) andK′′\nf(x′′\n0,t′′) are the free surface charge and current\ndensities induced at the boundaries of the mirror. Also, from (24) o ne finds that the\nelectric displacement vector D′′(x′′,t′′) and the H′′(x′′,t′′) field are given by\nD′′(x′′,t′′)≡E′′(x′′,t′′)+4πP′′(x′′,t′′),\n= [1+4πχ(x′′)]E′′(x′′,t′′),\nH′′(x′′,t′′)≡B′′(x′′,t′′)−4πM′′(x′′,t′′),\n=B′′(x′′,t′′), (46)\nfor allx′′andt′′∈[−t′′\n1,t′′\n1].\nSince the mirror satisfies Ohm’s law and has zero free charge density when it is at\nrest and the mirror is at rest in MS0during the time interval [ −t′′\n1,t′′\n1] given in (17), it\nfollows that [24]\nK′′\nf(x′′\n0,t′′) = 0, σ′′\nf(x′′\n0,t′′) = 0, (47)\nforx′′\n0=±δ0/2 andt′′∈[−t′′\n1,t′′\n1].\nFrom (43)-(47) it is straightforward to show that\nB′′(x′′\n0+,t′′) =B′′(x′′\n0−,t′′),\nE′′\nj(x′′\n0+,t′′) =E′′\nj(x′′\n0−,t′′) (j= 2,3),\nE′′\n1/parenleftiggδ0\n2+,t′′/parenrightigg\n= (1+4πχ0)E′′\n1/parenleftiggδ0\n2−,t′′/parenrightigg\n,\nE′′\n1/parenleftigg\n−δ0\n2−,t′′/parenrightigg\n= (1+4πχ0)E′′\n1/parenleftigg\n−δ0\n2+,t′′/parenrightigg\n, (48)\nforx′′\n0=±δ0/2 andt′′∈[−t′′\n1,t′′\n1].\nFrom (9) and (19) one has that the events with coordinates in LS0\n/parenleftigg\nx′\n0=±δ(t0)\n2, t′\n0= 0/parenrightigg\n, (49)\nhave coordinates in MS0/parenleftigg\nx′′\n0=±δ0/2, t′′\n0=∓γ0β0δ(t0)\n2c=∓β0δ0\n2c/parenrightigg\n. (50)\nHence, it follows from the boundary conditions in (48) and the relatio nship between the\nfields inLS0and inMS0given in (21) that\nB′(x′\n0+,0) = B′(x′\n0−,0),\nE′\nj(x′\n0+,0) = E′\nj(x′\n0−,0) (j= 2,3),\nE′\n1/parenleftiggδ(t0)\n2+,0/parenrightigg\n= (1+4πχ0)E′\n1/parenleftiggδ(t0)\n2−,0/parenrightigg\n,Equations of a Moving Mirror and the Electromagnetic Field 14\nE′\n1/parenleftigg\n−δ(t0)\n2−,0/parenrightigg\n= (1+4πχ0)E′\n1/parenleftigg\n−δ(t0)\n2+,0/parenrightigg\n, (51)\nwithx′\n0=±δ(t0)/2.\nUsing (3) to connect the quantities in (51) with those in LSand recalling that\nt0∈Ris arbitrary, one concludes that\nB(x0+,t) =B(x0−,t),\nEj(x0+,t) =Ej(x0−,t) (j= 2,3),\nE1(x1−,t) = (1+4 πχ0)E1(x1+,t),\nE1(x2+,t) = (1+4 πχ0)E1(x2−,t), (52)\nforx0=q(t)±δ(t)/2,x1=q(t)−δ(t)/2,x2=q(t)+δ(t)/2, andt∈R. Therefore, the\nboundaryconditionsstatethat themagnetic field B(x,t)andthetangential components\nEj(x,t) (j= 2,3) of the electric field must be continuous at the boundaries x=\nq(t)±δ(t)/2 of the mirror and that the normal component E1(x,t) has a discontinuity\nif it is different from zero.\n4. A SPECIAL CASE\nIn the rest of the article we assume that the electric field is linearly po larized along the\nz-axis. Then, there is no free and bound charge in LS, see (34) and(37), and the electric\nand magnetic fields can be derived from vector and scalar potentials of the following\nform:\nA(x,t) =A0(x,t)ˆ z, V(x,t) = 0. (53)\nExplicitly, one has\nB(x,t) =∇×A(x,t) =−∂A0\n∂x(x,t)ˆ y,\nE(x,t) =−1\nc∂A\n∂t(x,t) =−1\nc∂A0\n∂t(x,t)ˆ z. (54)\nNotice that we are working in the Coulomb gauge and that it coincides w ith the Lorentz\ngauge.\nWith this choice of the electromagnetic field, all the Maxwell equation s in (36)\nare automatically satisfied except for the Amp´ ere-Maxwell equat ion (that is, the first\nequation in (36)), which now takes the form\nα1(x,t)\nc2∂2A0\n∂t2(x,t)+α2(x,t)\nc∂2A0\n∂x∂t(x,t)\n+α3(x,t)\nc∂A0\n∂x(x,t)+α4(x,t)\nc2∂A0\n∂t(x,t)\n=α0(x,t)∂2A0\n∂x2(x,t). (55)\nHere\nα0(x,t) = 1−4πγ(t)2β(t)2χLS(x,t),\nα1(x,t) = 1+4 πγ(t)2χLS(x,t),Equations of a Moving Mirror and the Electromagnetic Field 15\nα2(x,t) = 8πγ(t)2β(t)χLS(x,t),\nα3(x,t) = 4πdβ\ndt(t)/bracketleftig\nγ(t)2χLS(x,t)+β(t)f0(x,t)/bracketrightig\n+4πγ(t)σLS(x,t)β(t),\nα4(x,t) = 4πf0(x,t)dβ\ndt(t)+4πγ(t)σLS(x,t), (56)\nand\nf0(x,t) =γ(t)4β(t)/braceleftigg\n2χ(x′′)+x′′dχ\ndx′′(x′′)/bracerightigg\n,\nx′′=γ(t)[x−q(t)]. (57)\nNotice that in (57) it is χthat appears and not χLS, see (28) for the definition of χLS.\nWe remark that (55) was presented at the LAOP 2014 conference [22] without giving\nany details of its derivation.\nObserve that the coefficients of the partial differential equation in (55) are position-\nand time-dependent. Therefore, we expect that, in general, the dynamics of the field\ncannot be restricted to a single-mode, that is, we expect that (55 ) has no solutions of\nthe form F(x)T(t).\nUp to now we have not taken advantage of two facts: a) one can ex amine the case\nwhere the velocity and acceleration of the mirror are small, and b) th e mirror normally\nevolves on a time-scale much larger than that in which the field evolves . In the following\nsubsection we use these facts to simplify (55).\n4.1. Introduction of non-dimensional quantities\nIn the rest of the article we assume that\n(i)λ0is the characteristic wavelength of the field.\n(ii)ν0=c/λ0is the characteristic frequency of the field.\n(iii)A00is the characteristic value of A0(x,t).\n(iv)ν−1\noscis the time scale in which q(t) changes appreciably.\nWe measure length in units of λ0and time in units of ν−1\n0, that is, we take x=λ0ξ\nandt=ν−1\n0τ=ν−1\noscτosc.\nDefine\nǫpert=νosc\nν0,˜˜q(τosc) =q(ν−1\noscτosc)\nλ0,\n˜χ(ξ) =χ(λ0ξ),˜q(τ) =q(ν−1\n0τ)\nλ0,\n˜σ(ξ) =σ(λ0ξ)\nν0,˜δ0=δ0\nλ0,\n˜A0(ξ,τ) =1\nA00A0(λ0ξ,ν−1\n0τ). (58)\nNotice that these are non-dimensional quantities and that ǫpertcompares the time scale\nν−1\n0in which the field changes appreciably with the time-scale ν−1\noscin which q(t) changesEquations of a Moving Mirror and the Electromagnetic Field 16\nappreciably. Since one normally has ν−1\n0≪ν−1\nosc(that is, the field evolves on a much\nsmaller time-scale than the mirror), one expects that ǫpert≪1. For example, using the\nexperimental values from [30] one has ν0= 2.82×1014Hz,νosc= 1.34×105Hz, and\nǫpert= 4.8×10−10. Moreover, observe that\n˜q(τ) =˜˜q(ǫpertτ). (59)\nA straightforward calculation using the quantities defined in (58) sh ows that the\nnon-dimensional form of (55) is given by\nα1(x,t)∂2˜A0\n∂τ2(ξ,τ) +α2(x,t)∂2˜A0\n∂ξ∂τ(ξ,τ)\n+α3(x,t)\nν0∂˜A0\n∂ξ(ξ,τ) +α4(x,t)\nν0∂˜A0\n∂τ(ξ,τ)\n=α0(x,t)∂2˜A0\n∂ξ2(ξ,τ), (60)\nwithx=λ0ξandt=ν−1\n0τ.\nIn the following we deduce an approximate equation for ˜A0(ξ,τ) for the case in\nwhich the velocity and the acceleration of the mirror are small. In ord er to do this,\nassume that\nd˜q\ndτ(τ) =ǫP˜Q(τ), (61)\nwhere 0< ǫP≪1 is a perturbation parameter. Notice that ǫP/ne}ationslash=ǫpert. Recall that ǫpert\ncompares the time-scale in which the field changes appreciably with th e time-scale in\nwhichq(t) changes appreciably, see (58). Equation (61) serves as a definit ion of˜Q(τ).\nWe are simply factoring out ǫPfrom (d˜q/dτ)(τ), so that the perturbation parameter\nappears explicitly.\nBefore proceeding we delve on the meaning of (61). From (27) and ( 58) one has\n˙q(ν−1\n0τ)\nc=β(ν−1\n0τ) =d˜q\ndτ(τ), (62)\nso thatǫPin (61) is a perturbation parameter that indicates that the velocity of the\nmirror is very small compared to the speed of light c. From (61) it follows that\nd2˜q\ndτ2(τ) =ǫPd˜Q\ndτ(τ). (63)\nNotice that (62) implies that\n¨q(ν−1\n0τ)\ncν0=1\nν0dβ\ndt(ν−1\n0τ) =d2˜q\ndτ2(τ), (64)\nso thatǫPin (63) is a perturbation parameter that also indicates that the acc eleration of\nthe mirror is very small compared to the speed of light cmultiplied by the characteristic\nfrequency ν0of the field. From the discussion above it follows that an approximatio n to\nfirst order in ǫPcorresponds to an approximation to first order in the velocity ˙ q(t) and\nthe acceleration ¨ q(t) of the mirror.Equations of a Moving Mirror and the Electromagnetic Field 17\nUsing (58), (61), and (63) and neglecting terms of order ǫn\nPwithn≥2 it follows\nfrom (60) that\n∂2˜A0\n∂ξ2(ξ,τ) ={1+4π˜χ[ξ−˜q(τ)]}∂2˜A0\n∂τ2(ξ,τ)\n+4π˜σ[ξ−˜q(τ)]∂˜A0\n∂τ(ξ,τ)\n+8π˜χ[ξ−˜q(τ)]d˜q\ndτ(τ)∂2˜A0\n∂ξ∂τ(ξ,τ),\n+4π˜σ[ξ−˜q(τ)]d˜q\ndτ(τ)∂˜A0\n∂ξ(ξ,τ)\n+4π˜χ[ξ−˜q(τ)]d2˜q\ndτ2(τ)∂˜A0\n∂ξ(ξ,τ).\n(65)\nWe emphasize that (65) is correct to first order in ǫP.\nNotice that ˜ q(τ) uses the characteristic time-scale of the field, since τ=ν0t. It is\nmuch better to express equation (65) in terms of the non-dimensio nal position of the\nmirror˜˜q(τosc) because it uses the time-scale of the mirror τosc=νosct=ǫpertτand this\nallows two of the different time-scales involved in the system to appea r explicitly in\nterms of the perturbation parameter ǫpert. Using the relationship ˜ q(τ) =˜˜q(ǫpertτ) given\nin (59), it follows from (65) that\n/braceleftig\n1+4π˜χ/bracketleftig\nξ−˜˜q(ǫpertτ)/bracketrightig/bracerightig∂2˜A0\n∂τ2(ξ,τ)\n+4π˜σ/bracketleftig\nξ−˜˜q(ǫpertτ)/bracketrightig∂˜A0\n∂τ(ξ,τ)\n+ǫpert8π˜χ/bracketleftig\nξ−˜˜q(ǫpertτ)/bracketrightigd˜˜q\ndτosc(ǫpertτ)∂2˜A0\n∂ξ∂τ(ξ,τ)\n+ǫpert4π˜σ/bracketleftig\nξ−˜˜q(ǫpertτ)/bracketrightigd˜˜q\ndτosc(ǫpertτ)∂˜A0\n∂ξ(ξ,τ)\n+ǫ2\npert4π˜χ/bracketleftig\nξ−˜˜q(ǫpertτ)/bracketrightigd2˜˜q\ndτ2\nosc(ǫpertτ)∂˜A0\n∂ξ(ξ,τ)\n=∂2˜A0\n∂ξ2(ξ,τ). (66)\nWe emphasize again that (66) is correct to first order in ǫP. Notice that (66) is a wave\nequation with damping and with slowly varying coefficients plus terms mu ltiplied by\nǫpertandǫ2\npertthat are a small perturbation when ǫpert≪1 (which is the usual case).\nTherefore, a multiple-scales approach [25] is an adequate perturb ation method to solve\nit approximately. This is the subject of future work [26].\nFor completeness we now express (65) with units. Using (58) it follow s that\n∂2A0\n∂x2(x,t) =1+4πχ[x−q(t)]\nc2∂2A0\n∂t2(x,t)\n+4πσ[x−q(t)]\nc2∂A0\n∂t(x,t)\n+8πχ[x−q(t)]\nc2˙q(t)∂2A0\n∂x∂t(x,t)Equations of a Moving Mirror and the Electromagnetic Field 18\n+4πσ[x−q(t)]\nc2˙q(t)∂A0\n∂x(x,t)\n+4πχ[x−q(t)]\nc2¨q(t)∂A0\n∂x(x,t). (67)\nNotice that this equation is correct to first order in ˙ q(t) and ¨q(t) because it comes from\na non-dimensional equation that is correct to first order in ǫP.\nWenotethat, taking σ= 0andχin(40)andusing appropriateunits, (67)coincides\nwith equation (7) of [21] (see also references cited therein). That reference obtained it\nusing an approximate Lagrangian density correct to first order in β(t) for the field, see\nalso Sec. VII below.\n4.2. The case of piecewise constant susceptibility and cond uctivity\nIf the electric susceptibility χ(x′′) and the conductivity σ(x′′) are given by (40), then one\nsolves(55), (60), (65), (66), and(67)insideandoutsideofthem irrorandonethenpastes\nthem using the boundary conditions in (52). With the choice of A(x,t) given in (53),\nthe latter simply amounts to asking that A0(x,t), (∂A0/∂t)(x,t), and (∂A0/∂x)(x,t) be\ncontinuousattheboundariesofthemirroror,equivalently, that ˜A0(ξ,τ),(∂˜A0/∂τ)(ξ,τ),\nand (∂˜A0/∂ξ)(ξ,τ) be continuous at the boundaries of the mirror. These are located at\nx=q(t)±δ(t)/2 if (55) or (60) are used. If (65), (66), or (67) are used, then t hey are\nlocated at x=q(t)±δ0/2 because one has to make an approximation to first order in\nǫP, see (42).\n5. FORCE ON THE MIRROR\nIn equations (55), (60), (65), (66), and (67) governing the dyn amics of the field, the\nmirror could have a position q(t) determined by an external agent. In this section we\ndetermine the force that the field exerts on the mirror and, conse quently, the equation\ngoverning its dynamics, namely, the time-derivative of the mechanic al momentum (per\nunit area) equals the force (per unit area). One of the aforement ioned equations\ncombined with one for the mirror constitutes a self-consistent set of equations governing\nthe dynamics of the field-mirror system.\nConsider the volume\nV(t) =/bracketleftigg\nq(t)−δ(t)\n2, q(t)+δ(t)\n2/bracketrightigg\n×[y0,y1]×[z0,z1],\n(68)\nwithy0< y1andz0< z1.\nRecall that there is no free and bound charge with the choice of A(x,t) in (53), see\n(34), (37), and (54). Therefore, the force on the mirror in V(t) at time tis given by\n[23, 24]\nF(t) =/integraldisplay\nV(t)d3r1\ncJ(r,t)×B(r,t). (69)Equations of a Moving Mirror and the Electromagnetic Field 19\nOne can rewrite F(t) using the density of electromagnetic momentum gem(x,t) and\nthe Maxwell stress tensor T(x,t) [23, 24]. Recall that they are given by\ngem(x,t) =1\n4πcE(x,t)×B(x,t),\nT(x,t) =1\n4π/bracketleftbigg\nE(x,t)E(x,t)T−1\n2E(x,t)2I3/bracketrightbigg\n+1\n4π/bracketleftbigg\nB(x,t)B(x,t)T−1\n2B(x,t)2I3/bracketrightbigg\n. (70)\nHereE(x,t) andB(x,t) are taken to be column vectors. Using (70) it follows that [24]\nF(t) =/contintegraldisplay\n∂V(t)T(x,t)n(t)da−/integraldisplay\nV(t)d3r∂\n∂tgem(x,t).\n(71)\nHere∂V(t) is the surface bounding V(t),n(t) is the unit vector normal to ∂V(t) and\nexterior to V(t), andT(x,t)n(t) is the product of a matrix times a column vector.\nNotice that one cannot take the partial derivative with respect to tofgem(x,t) out of\nthe integral in (71) as in the case where the volume of integration is t ime-independent.\nAfter a straightforward calculation it follows from (54), (68), (70 ), and (71) that\nF(t) =ˆ x\n\n−1\n8π/bracketleftigg∂A0\n∂x(x,t)/bracketrightigg2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglex=q(t)+δ(t)/2\nx=q(t)−δ(t)/2\n+1\n4πc2/integraldisplayq(t)+δ(t)/2\nq(t)−δ(t)/2dx∂2A0\n∂t2(x,t)∂A0\n∂x(x,t)/bracerightigg\n×\n×(y1−y0)(z1−z0). (72)\nHere and in the following\nh(x,t)/vextendsingle/vextendsingle/vextendsinglex=b\nx=a=h(b,t)−h(a,t). (73)\nWe now relate F(t) with the mechanical momentum of the mirror. We assume that\nthe mirror has a uniform mass per unit volume ρM0when it is at rest. Then its mass\ndensity in LSis given by ρM(t) =γ(t)ρM0. Note that this holds because we are assuming\nthat the mirror is at rest during the time-interval [ −t′′\n1,t′′\n1] given in (17). Using (42) it\nfollows that the amount of mirror mass in V(t) is given by\nM=/integraldisplay\nV(t)ρM(t)d3r=δ(t)(y1−y0)(z1−z0)γ(t)ρM0\n= (y1−y0)(z1−z0)M0, (74)\nwithM0the mirror’s mass per unit area, that is,\nM0=ρM0δ0. (75)\nWe now consider the mirror in V(t) to be a single point particle with mass Mgiven\nin (74). Then, its relativistic mechanical momentum is given by [23, 28]\nPmech(t) = (y1−y0)(z1−z0)M0γ(t)˙q(t)ˆ x, (76)\nand the equation of motion of the mirror is given by\nd\ndtPmech(t) =F(t). (77)Equations of a Moving Mirror and the Electromagnetic Field 20\nUsing (72) and (76) the equation above simplifies to\nd\ndtp(t) =f(t), (78)\nwithp(t) the mechanical momentum of the mirror (along the x-axis) per unit area\nperpendicular to the x-axis and f(t) the pressure exerted by the field along the x-axis,\nthat is,\np(t) =M0γ(t)˙q(t)\nf(t) =−1\n8π/bracketleftigg∂A0\n∂x(x,t)/bracketrightigg2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglex=q(t)+δ(t)/2\nx=q(t)−δ(t)/2\n+1\n4πc2/integraldisplayq(t)+δ(t)/2\nq(t)−δ(t)/2dx∂2A0\n∂t2(x,t)∂A0\n∂x(x,t).\n(79)\nObserve that (78) combined with (55) constitutes the self-c onsistent set of equations\ngoverning the dynamics of the mirror-field system.\nIn order to further simplify (78) one must use the equation for A0(x,t). We now\ndo this to first order in ǫPor, equivalently, to first order in ˙ q(t) and ¨q(t). Substituting\n∂2A0(x,t)/∂t2from (67) in f(t) given in (79) and simplifying one obtains that\nM[q(t),t]¨q(t) =F0[q(t),t]−F1[q(t),t]˙q(t)\nc, (80)\nwhere\nF0=−1\n2/integraldisplayq(t)+δ0\n2\nq(t)−δ0\n2dxχ[x−q(t)]\n1+4πχ[x−q(t)]×\n×∂\n∂x/bracketleftigg∂A0\n∂x(x,t)/bracketrightigg2\n,\n−1\nc2/integraldisplayq(t)+δ0\n2\nq(t)−δ0\n2dxσ[x−q(t)]\n1+4πχ[x−q(t)]×\n×∂A0\n∂x(x,t)∂A0\n∂t(x,t),\nF1=1\nc/integraldisplayq(t)+δ0\n2\nq(t)−δ0\n2dx/braceleftiggσ[x−q(t)]\n1+4πχ[x−q(t)]\n+χ[x−q(t)]\n1+4πχ[x−q(t)]∂\n∂t/bracerightigg/bracketleftigg∂A0\n∂x(x,t)/bracketrightigg2\n,\nM=M0+1\nc2/integraldisplayq(t)+δ0\n2\nq(t)−δ0\n2dxχ[x−q(t)]\n1+4πχ[x−q(t)]×\n×/bracketleftigg∂A0\n∂x(x,t)/bracketrightigg2\n. (81)\nFor simplicity we omitted the point [ q(t),t] whereF0,F1, andMare evaluated in (81).\nWe remark that (80) was presented at the LAOP 2014 conference [22] without giving\nany details of its derivation.Equations of a Moving Mirror and the Electromagnetic Field 21\nWe emphasize that (80) is correct to first order in ǫPor, equivalently, to first\norder in ˙q(t) and ¨q(t).Observe that (67) combined with (80) gives a self-consisten t set\nof equations correct to first order in ǫPthat governs the dynamics of the mirror-field\nsystem.\nNotice that M[q(t),t] reduces to M0and the right-hand side of (80) reduces to\nF0[q(t),t] when the mirror is at rest (recall that the time dependent term in M[q(t),t]\narises from a term linear in the acceleration of the mirror appearing in the force).\nTherefore, the motion of the mirror and the coupling to the field give rise to two effects.\nThe first one is a position- and time-dependent mass related to the effective mass taken\nin phenomenological treatments of this type of systems [4, 10, 13]. The second one is a\nvelocity-dependent force that can give rise to friction and that is r elated to the cooling\nof mechanical objects [4, 5]. We note that a friction force has also been obtained using\na scattering matrix approach and a dispersive dielectric constant f or a non-conducting\ndelta-function mirror in [29]. We will investigate the dynamics given by (80) in [27].\n5.1. Piecewise constant susceptibility and conductivity\nThe results of Sec. V are valid for arbitrary continuously differentia bleχ(x′′) andσ(x′′)\nthat are zero outside of the mirror, see (20). One can also conside r the case where they\nare given in (40). In this case, χ[x−q(t)] =χ0andσ[x−q(t)] =σ0for|x−q(t)| ≤δ0/2\nto first order in ǫP, so that factors involving these quantities can be taken out of the\nintegrals. In particular, the first term in F0[q(t),t] given in (81) can be integrated\nexplicitly.\nThe simplest case for the dynamics of the mirror+field system is to co nsider a non-\nconducting material (that is, σ0= 0) and to make an approximation to order zero in\n˙q(t) and ¨q(t) in both the equation for the field and in the force affecting the mirro r. In\nthis case equations (67) and (80) reduce to\nM0¨q(t) =−1\n2/parenleftiggχ0\n1+4πχ0/parenrightigg/bracketleftigg∂A0\n∂x(x,t)/bracketrightigg2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglex=q(t)+δ0\n2\nx=q(t)−δ0\n2,\n∂2A0\n∂x2(x,t) =1+4πχ[x−q(t)]\nc2∂2A0\n∂t2(x,t).\n(82)\nNotice that the time-dependent term modifying M0in the last equation of (81)\ndisappears in (82) because it comes from corrections to the force of first order in ˙ q(t)\nand ¨q(t). We remark that (82) was presented at the LAOP 2014 conferen ce [22] without\ngiving any details of its derivation.\nThe dynamics of (82) have already been studied in [16, 17]. In these w orks we\nassumed that a perfect mirror is fixed at x= 0 and that the mobile mirror is free to\nmove for x >0. Moreover, we assumed that the mobile mirror is very thin so that\nthe electric susceptibility can be approximated by a delta-function: χ(x′′) =χ00δ(x′′).\nAmong several results, it was found that, within the rotating-wav e-approximation, theEquations of a Moving Mirror and the Electromagnetic Field 22\nforce on the mirror can be deduced from a periodic potential with pe riod half the\nwavelength of the field. Finally, we note that the delta-function cas eχ(x′′) =χ00δ(x′′)\nis obtained by taking the following limits in (82): δ0↓0,ρM0→+∞, andχ0→+∞so\nthatρM0δ0=M0= constant and δ0χ0=χ00= constant.\n6. VALIDITY OF THE ASSUMPTION THAT THE MIRROR IS\nAPPROXIMATELY AT REST\nWe have assumed that the mirror is approximately at rest in MS0during the time\ninterval [ −t′′\n1,t′′\n1] given in (17). Moreover, we stated that we consider that the mirr or\nis approximately at rest in MS0during the time interval given in (17) if the midpoint\nq′′(t′′) inMS0moves a distance much smaller than half the mirror’s thickness (when it is\nat rest)δ0/2 during the time interval. In this section we establish necessary con ditions\nfor this to be true in the case of the electric susceptibility in (40), ze ro conductivity\nσ(x′′) = 0, and for a field of the form given in (53) and (54).\nAssume that the mirror is at rest in MS0during the time interval [ −t′′\n1,t′′\n1] in (17).\nWe now deduce the force per unit area (pressure) acting on the mir ror inMS0.\nFirst recall that the potentials can be accommodated into a four-v ector as\n(A(x,t),iV(x,t)) [23]. If one denotes de potentials in LS0byA′(x′,t′) andV′(x′,t′)\nand the potentials in MS0byA′′(x′′,t���′) andV′′(x′′,t′′), then it follows that they are\nconnected by the following relation:\n/parenleftigg\nA(x,t)\niV(x,t)/parenrightigg\n=/parenleftigg\nA′(x′,t′)\niV′(x′,t′)/parenrightigg\n,\n=M1/parenleftigg\nA′′(x′′,t′′)\niV′′(x′′,t′′)/parenrightigg\n,\n(83)\nHere (x,t) are coordinates in LS, while ( x′,t′) and (x′′,t′′) are the corresponding\ncoordinates in LS0andMS0, respectively. The connection between the coordinates\nis given in (2) and (9). Also, M1is given in (32) and notice that (3) holds with Aand\nVinstead of f. Using (53) in (83) it immediately follows that\n\nA′′(x′′,t′′)\niV′′(x′′,t′′)\n=\n0\n0\nA0(x,t)\n0\n. (84)\nTherefore, potentialsof theform(53)in LSimply that thefields in MS0canbededuced\nfrom potentials A′′(x′′,t′′) =A′′\n0(x′′,t′′)ˆ z′′andV′′(x′′,t′′) = 0 with formulae for the fields\nsimilar to those in (54).\nNow observe that the right-hand side of the first equation in (82) g ives the pressure\naffecting the mirror when it is at rest, since the right-hand side of th at equation gives\nthe correct pressure on mirror to order zero in β(t) and˙β(t). Therefore, to determine\nthe force acting on the mirror in MS0during the time interval [ −t′′\n1,t′′\n1] we simplyEquations of a Moving Mirror and the Electromagnetic Field 23\nhave to take the right-hand side of the first equation in (82) and re place (∂A0/∂x)(x,t)\nby (∂A′′\n0/∂x′′)(x′′,t′′) andq(t) byq′′(t′′) and use that B′′(x′′,t′′) =∇′′×A′′(x′′,t′′) =\n−(∂A′′\n0/∂x′′)(x′′,t′′)ˆ y′′. Then, it follows from (82) that the pressure affecting the mirror\nat timet′′= 0 inMS0is given by\nf0=−χ0/2\n1+4πχ0B′′(x′′,0)2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglex′′=δ0/2\nx′′=−δ0/2. (85)\nNotice that we used q′′(0) = 0, see (14). We emphasize that the pressure in (85) was\nobtained assuming that the mirror is at rest in MS0during the time interval [ −t′′\n1,t′′\n1].\nWe now determine if the mirror really moves a negligible distance if this pr essure is\nintroduced in its equation of motion.\nAccording to (78) the equation of motion of the mirror in MS0is\nd\ndt′′/bracketleftigg\nM0γ1(t′′)dq′′\ndt′′(t′′)/bracketrightigg\n≃f0, (86)\nwhere\nγ1(t′′) =\n\n1−/bracketleftigg1\ncdq′′\ndt′′(t′′)/bracketrightigg2\n\n−1/2\n, (87)\nand we have approximated the pressure acting on the mirror in MS0at timet′′by the\npressure acting on it at time t′′= 0.\nIt is straightforward to show that the solution of (86) with q′′(0) = 0 and\n(dq′′/dt′′)(0) = 0 is given by\nq′′(t′′)≃f0\n2M0(t′′)2fort′′∈[0,t′′\n1], (88)\nif\n/parenleftiggf0t′′\n1\nM0c/parenrightigg2\n≪1. (89)\nIt follows from(88) that the mirror will beapproximately at rest dur ing the interval\n[0,t′′\n1] if\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglef0\n2M0(t′′\n1)2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≪δ0\n2,/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglef0\ncM0t′′\n1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n≪1. (90)\nThe first condition in (90) simply states that the midpoint of the mirro r must have\na displacement much smaller than half the thickness of the mirror whe n it is at rest.\nMeanwhile, the second condition in (90) is identical to (89) and state s that the square\nof the velocity of the midpoint of the mirror must be much smaller than c2.\nWe now obtain a sufficient condition for (90). Assume that\n|E(x,t)| ≤Emax,|B(x,t)| ≤Bmax. (91)\nUsing (21) in combination with (54) it follows that\n|E′′(x′′,0)| ≤γ0(Emax+|β0|Bmax),\n|B′′(x′′,0)| ≤γ0(Bmax+|β0|Emax). (92)Equations of a Moving Mirror and the Electromagnetic Field 24\nUsing (92) in (90) it is straightforward to show that (90) will hold whe never\nBmax,|β0|Emax≪/radicaltp/radicalvertex/radicalvertex/radicalbt2πρM0c2\nγ2\n0|β0|. (93)\nWe now illustrate (93) with an example. Reference [30] introduces an experimental set\nup that can be analyzed with the model of this article. They have a mir ror with the\nfollowing properties:\nL=W= 10−3(m),\nδ0= 50×10−9(m),\nM= 4×10−11(kg). (94)\nwhereL= (y1−y0) andW= (z1−z0) denote the length and the width of the mirror,\nrespectively.\nThe quantity on the right-hand side of (93) is a strictly decreasing f unction of β0\nand is minimized for β0= 1, that is, when the speed of the mirror equals the speed of\nlight. With this in mind we take β0= 1/2, since smaller values of β0give larger bounds.\nTo relate better to quantities used in a laboratory, we now transfo rm the electric\nand magnetic fields from Gaussian to MKS units. It can be shown that [24]\nE(MKS)=E(gauss)\n√4πǫ0,B(MKS)=B(gauss)\nc√4πǫ0, (95)\nwhereE(MKS)(B(MKS)) is the electric (magnetic) field in MKS units, E(gauss)(B(gauss)) is\nthe electric (magnetic) field in Gaussian units, and ǫ0is the permittivity of vacuum.\nFrom (93), (95), the parameters in (94), and the value β0= 1/2, it follows that the\nmaximum electric E(MKS)\nmaxand magnetic B(MKS)\nmaxfields in MKS units must satisfy\nE(MKS)\nmax≪1\n|β0|√4πǫ0/radicaltp/radicalvertex/radicalvertex/radicalbt2πρM0c2\nγ2\n0|β0|,\n≃5.2×1015/parenleftbiggV\nm/parenrightbigg\n,\nB(MKS)\nmax≪1\nc√4πǫ0/radicaltp/radicalvertex/radicalvertex/radicalbt2πρM0c2\nγ2\n0|β0|≃107(T). (96)\nTo have an idea of the order of magnitude of E(MKS)\nmaxandB(MKS)\nmaxwe now consider\na plane wave describing the electromagnetic field of a laser. It can be shown that the\nnorm of (the average in one cycle of) the Poynting vector of a plane wave in vacuum\nand in MKS units is given by [28]\n|/an}b∇acketle{tSMKS/an}b∇acket∇i}ht|=cǫ0\n2(E(MKS)\nmax)2. (97)\nHereE(MKS)\nmaxis the magnitude of the electric field of the plane wave in MKS units.\nTherefore, the incident power (in Watts) on the mirror is\nP=LW|/an}b∇acketle{tSMKS/an}b∇acket∇i}ht|. (98)\nFactoring out E(MKS)\nmax, substituting the parameters in (94), and using the well-known\nrelationship between the magnitudes of the electric and magnetic fie lds of a plane waveEquations of a Moving Mirror and the Electromagnetic Field 25\nin vacuum [28] one gets\nE(MKS)\nmax= 2.7√\nP×104/parenleftbiggV\nm/parenrightbigg\n,\nB(MKS)\nmax=1\ncE(MKS)\nmax=√\nP×10−4(T). (99)\nFor a laser power of P= 1 (Watt) one observes that both quantities in (99) are\nmuch smaller than the bounds in (96). Therefore, we conclude that it is a reasonable\nassumption to consider that the mirror is approximately at rest dur ing the time interval\n[−t′′\n1,t′′\n1].\n7. A LAGRANGIAN DENSITY FOR THE FIELD\nIn this section we derive a Lagrangian density for the field in the case where the\nconductivity is zero, that is, σ(x′′) = 0.\nSince (24) holds during the time interval [ −t′′\n1,t′′\n1], it follows that the Lagrangian\ndensity for the electromagnetic field in MS0is given by [24]\nL′′=1\n8π/bracketleftig\nE′′(x′′,t′′)·D′′(x′′,t′′)\n−B′′(x′′,t′′)·H′′(x′′,t′′)/bracketrightig\n,\n=1\n8π/bracketleftig\nǫ(x′′)E′′(x′′,t′′)2−B′′(x′′,t′′)2/bracketrightig\n, (100)\nfort′′∈[−t′′\n1,t′′\n1] and the dielectric function\nǫ(x′′) = 1+4 πχ(x′′). (101)\nOne can write (100) in matrix form as follows:\nL′′=1\n8π/parenleftigg\nE′′(x′′,t′′)\nB′′(x′′,t′′)/parenrightiggT/parenleftigg\nǫ(x′′)I3O3×3\nO3×3−I3/parenrightigg\n×\n×/parenleftigg\nE′′(x′′,t′′)\nB′′(x′′,t′′)/parenrightigg\n. (102)\nRecall that the mirror is at rest in MS0during the time interval [ −t′′\n1,t′′\n1] in (17).\nThen one can use (21) to connect the fields in MS0with those in LS0to obtain the\nLagrangian density in LS0at timet′= 0 (here one uses an argument similar to that\nused to determine the polarization and magnetization where one obt ains a formula\nvalid inside the mirror and then observes that it also gives the correc t value outside the\nmirror). If one then uses (3) to connect the fields in LS0with those in LSand recalls\nthatt0is arbitrary, one concludes that the Lagrangian density in LSis given by\nL=1\n8π/parenleftigg\nE(x,t)\nB(x,t)/parenrightiggT/parenleftig\nM−1\n0/parenrightigT/parenleftigg\nǫ(x′′)I3O3×3\nO3×3−I3/parenrightigg\n×\n×M−1\n0/parenleftigg\nE(x,t)\nB(x,t)/parenrightigg\n, (103)\nwithx′′=γ(t)[x−q(t)] andATthe transpose of matrix A.Equations of a Moving Mirror and the Electromagnetic Field 26\nEquation (103) is valid for an arbitrary electromagnetic field. For th e special case\ngiven in (53) and (54) one obtains from (103) the following Lagrangia n density:\nL=1\n8π\n\n/bracketleftigg1\nc∂A0\n∂t(x,t)/bracketrightigg2\n−/bracketleftigg∂A0\n∂x(x,t)/bracketrightigg2\n\n\n+γ(t)2χLS(x,t)\n2/bracketleftigg1\nc∂A0\n∂t(x,t)+β(t)∂A0\n∂x(x,t)/bracketrightigg2\n.\n(104)\nObserve that (104) expresses the Lagrangian density as the sum of a part corresponding\nto the free field plus a part associated with the presence of the mirr or. We note that\n(104) is identical to the Lagrangian density given in [21] (see also re ferences therein) for\nthe case of the piecewise constant electric susceptibility χgiven in (40).\nThe Euler-Lagrange equation associated with Lis then given by\n∂\n∂t/braceleftigg∂L\n∂[∂tA0(x,t)]/bracerightigg\n+∂\n∂x/braceleftigg∂L\n∂[∂xA0(x,t)]/bracerightigg\n= 0,\n(105)\nwhere∂tand∂xdenote partial derivatives with respect to tandx, respectively. After\na lengthy calculation one can show from (104) that the Euler-Lagra nge equation (105)\ndoes indeed give (55) with σLS(x,t) = 0.\nOne can be interested in using a simpler Lagrangian density from which\napproximate equations for the field can be obtained (for example, t o first order in the\nvelocity and the acceleration of the mirror). One way to achieve this is to expand the\nLagrangian density in (104) in powers of β(t) as follows:\nL=L0+β(t)L1+1\n2β(t)2L2+... , (106)\nwhere the first three terms are given by\nL0=1\n8π\n\nǫ[x−q(t)]\nc2/bracketleftigg∂A0\n∂t(x,t)/bracketrightigg2\n−/bracketleftigg∂A0\n∂x(x,t)/bracketrightigg2\n\n,\nL1=χ[x−q(t)]1\nc∂A0\n∂t(x,t)∂A0\n∂x(x,t),\nL2=χ[x−q(t)]\n\n/bracketleftigg1\nc∂A0\n∂t(x,t)/bracketrightigg2\n+/bracketleftigg∂A0\n∂x(x,t)/bracketrightigg2\n\n\n+x−q(t)\n2dχ\ndx′′[x−q(t)]/bracketleftigg1\nc∂A0\n∂t(x,t)/bracketrightigg2\n. (107)\nNow we make a few comments on (106). First, the term β(t)nLn/n! introduces terms of\norderǫn\nPandǫn+1\nPin theEuler-Lagrangeequations forthe field, since onehasto calcu late\nderivatives with respect to t(see (105)) and factors χ[x−q(t)] are present. Therefore,\nthe equations of order nin ˙q(t) and ¨q(t) for the field cannot be deduced exactly with\n(106) if one neglects Lmform > n, since terms of order n+1 would have to be neglected\nin the Euler-Lagrange equations to obtain the correct equations o f motion. This shows\nthat it appears not to be possible to have an approximate Lagrangia n density of a givenEquations of a Moving Mirror and the Electromagnetic Field 27\norder for the field that yields the correct equations without having to discard terms.\nThe argument above holds for a continuously differentiable electric s usceptibility χ(x′′).\nFor the piecewise constant χ(x′′) in (40), the derivatives of χ[x−q(t)] are zero inside\nand outside of the mirror, so that one obtains the correct equatio ns of order nin ˙q(t)\nand ¨q(t) for the field if one neglects Lmform > n. Also, in this case one must paste\nthe solution at the boundaries of the mirror so as to have a continuo usly differentiable\nfunction, see (52).\nFor an approximate Lagrangian of order one in β(t) for the complete mirror+field\nsystem we refer the reader to [21], where the following Lagrangian is proposed:\nL(1)=1\n2M0˙q(t)2+V[q(t)]+/integraldisplayL\n0dx[L0+β(t)L1].\n(108)\nHereV[q(t)] is a potential affecting the mirror and [0 ,L] is the region where the mirror\ncan move. Moreover, two perfect, fixed mirrors are located at x= 0 and x=L. This\nLagrangian gives the correct force on the mirror only to order zer o in ˙q(t) and ¨q(t) and\nhas been used to quantize the mirror+field system [21]. Since one doe s not recover the\ncorrect time-dependent mass and velocity-dependent force affe cting the mirror, physical\nphenomena associated with these terms have to be included by othe r means, such as\nmaster equationmethods, inasimilarway inwhich dampingisintroduced inaharmonic\noscillator and spontaneous emission is introduced in the interaction o f a two level atom\nwith a single-mode electromagnetic field [4].\n8. CONCLUSIONS\nIn this article we established the equations that govern the dynamic s of a system\ncomposed of a mobile slab interacting with the electromagnetic field us ing a relativistic\ntreatment and not considering thermal effects. The slab is made of a material\nthat satisfies the following properties when it is at rest: it is linear, iso tropic, non-\nmagnetizable, and ohmic with zero free charge density. Moreover, we obtained\napproximate equations for the slab-field system correct to first o rder in the velocity\nand the acceleration of the slab. On one hand, we showed that the e lectromagnetic field\nsatisfies a wave equation with damping and slowly varying coefficients p lus terms that\nare small when the slab evolves on a time-scale much larger than that of the field. These\npropertiesarisefromthefactthatthemobileslabappearstohave amagnetizationanda\npolarizationthatdepends onthe magneticfield when it isinmotion. Ont he other hand,\ntheslabsatisfies adynamical equationwithtwotermsthatariseasa result ofthemotion\nof the slab and the coupling to the electromagnetic field. The first on e is a position- and\ntime-dependent massrelatedtothe effective mass takeninphenomenological treatments\nof this type of systems. The second is a velocity-dependent force that can give rise to\nfriction and that is related to the coolingof mechanical objects. Also, the fact that\nthere are two separate time-scales, one associated with the fast evolution of the field\nand another associated with the slower evolution of the slab, allows t he use of theEquations of a Moving Mirror and the Electromagnetic Field 28\nmultiple scales method [25] to study the dynamics of the system. This is the topic of\nwork in preparation [26].\nOne may inquire why a relativistic treatment is needed, especially in the case\nwhere the slab does not move at relativistic speeds. Well, one is confr onted with the\nfollowing problem: one knows the properties of the slab (such as the polarization and\nthe magnetization) only when the slab is at rest and one needs to det ermine them in the\nLaboratory reference frame LSwhere it canbein motion. Oneway to solve thisproblem\nis discussed in this article and consists in first calculating the propert ies of the slab in\nan inertial reference frame where the slab is approximately at rest during a small time\nintervalandthenusingtheirtransformationpropertiesunderLo rentztransformationsto\ndetermine theirformin LS.Theadvantageofusing Lorentztransformationsisthreefold.\nFirst, all electromagnetic quantities (such as the polarization and t he magnetization)\nhave simple, well-defined transformation properties under Lorent z transformations.\nSecond, the use of Lorentz transformations illuminates the result s, enables a better\nunderstanding of the physics of the system, and allows one to obta in correction terms\nin the case of small velocities and accelerations. For example, one fin ds that, although\nthe slab is made up of a linear, isotropic, and non-magnetizable mater ial when it is at\nrest, when the slab is in motion it has a magnetization and a polarization that depend\non the electric and magnetic fields and on its velocity. Moreover, one also finds that\nthese modified magnetization and polarization lead to a time dependen t mass and to\na velocity-dependent force affecting the motion of the slab. Althou gh these terms are\nsmall for small velocities of the slab, they give rise to important phen omena related to\nthe effective mass and coolingof mechanical objects. Third, the relativistic treatment\nallows one to obtain consistent approximations for both the field and the slab at a given\norder of the velocity and acceleration.\nIt is also important to note that in our treatment the dynamics of th e slab are\ngeneralandarenotrestricted, forexample, tooscillatorymotion s. Infact, ourtreatment\nallows the slab to accelerate slowly up to relativistic velocities. Moreov er, all of our\nexpressions reduce to the correct value when the slab has consta nt velocity.\nFinally, deducing the equations for both the field and the mobile slab co nstitutes a\nproblem of fundamental physics and these equations can be import ant in other physical\ncontexts.\nACKNOWLEDGEMENTS\nR. Weder is a fellow of the Sistema Nacional de Investigadores. L. O. Casta˜ nos\nthanks the Universidad Nacional Aut´ onoma de M´ exico for suppor t. Research partially\nsupported by project PAPIIT-UNAM IN102215.Equations of a Moving Mirror and the Electromagnetic Field 29\nAPPENDIX\nIn theict-system or Minkowski metric (which is the one used in this article) with\ncoordinates ( x1=x,x2=y,x3=z,x4=ict), one first introduces the electromagnetic\ntensor[23]:\nFνµ=\n0B3−B2−iE1\n−B30B1−iE2\nB2−B10−iE3\niE1iE2iE30\n. (109)\nHereEjandBj(j= 1,2,3) are the (Cartesian) components of the electric Eand\nmagnetic Bfields, respectively.\nIn order to write Maxwell’s equations in four-dimensional form in term s of free\ncharges and free currents only, one has to incorporate the boun d charges and bound\ncurrents into the fields. This can be accomplished by first introducin g themoments\ntensorMνµgiven by [23]:\nMνµ=\n0M3−M2iP1\n−M30M1iP2\nM2−M10iP3\n−iP1−iP2−iP30\n. (110)\nHereMjandPj(j= 1,2,3) are the (Cartesian) components of the magnetization M\nand polarization P, respectively. One then defines the tensor Hνµ= (Fνµ−4πMνµ) and\nmakes use of the usual definitions D=E+4πPandH=B−4πMto express Hνµin\nterms of the electric displacement vector Dand the field H.\nUsing the tensors introduced above one can write Maxwell’s equation s in four-\ndimensional form as follows:\n4/summationdisplay\nµ=1∂Hνµ\n∂xµ=4π\ncsν,\n∂Fνµ\n∂xλ+∂Fµλ\n∂xν+∂Fλν\n∂xµ= 0. (111)\nHeresν= (Jf,icρf) is the free-current four-vector.\nTo obtain (111) one uses that\n1\ncgν=4/summationdisplay\nµ=1∂\n∂xµMνµ, (112)\nwheregνis the bound-current four-vector given by\ngν≡(Jb, icρb),\n=/parenleftigg\nc∇×M+∂P\n∂t,−ic∇·P/parenrightigg\n. (113)\nNotice that Mνµis obtained from Fνµby replacing BjbyMjandEjby−Pj,\nrespectively. This difference in signs is responsible for the fact that the fields EandB\ntransform with matrix M0, while the polarization Pand magnetization Mtransform\nwith the inverse matrix M−1\n0, see equations (21) and (23).Equations of a Moving Mirror and the Electromagnetic Field 30\nREFERENCES\n[1] Jin-Jin L and Ka-Di Z 2013 Generalized Optomechanics and Its Applications (World Scientific)\n[2] Hunger D et al2011 Coupling ultracold atoms to mechanical oscillators C. R. Physique 12\n871-887\n[3] Rogers B, Lo Gullo N, De Chiara G, Palma G M, and PaternostroM 201 4 Hybrid optomechanics\nfor Quantum Technologies Quantum Meas. 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A 84, 023812\n[22] Casta˜ nos L O and Weder R 2014 Classical Dynamics of a Mobile Mirr or and the Electromagnetic\nFieldLatin America Optics and Photonics Conference, OSA Technic al Digest (online) paper\nLM3A.3\n[23] Becker R 1964 Electromagnetic Fields and Interactions (Dover)\n[24] Jackson J D 1975 Classical Electrodynamics 2nd edn (Wiley)\n[25] Holmes M H 1995 Introduction to Perturbation Methods (Springer)\n[26] Casta˜ nos L O and Weder R How the electromagnetic field evolves in the presence of a mobileEquations of a Moving Mirror and the Electromagnetic Field 31\nmirror,in preparation\n[27] Casta˜ nos L O and Weder R Multiple Scales Solutions in Classical Opt omechanics: the oscillator,\nin preparation\n[28] Griffiths D J 1999 Introduction to Electrodynamics 3rd edn (Addison Wesley)\n[29] Xuereb A, Domokos P, Asb´ oth J, Horak P, and Freegarde T 20 09 Scattering theory of cooling\nand heating in optomechanical systems Phys. Rev. A 79053810\n[30] Thompson J D et al2008Strong dispersive coupling of a high-finesse cavity to a microme chanical\nmembrane Nature45206715"    },    {        "title": "2106.04948v1.Grammage_of_cosmic_rays_in_the_proximity_of_supernova_remnants_embedded_in_a_partially_ionized_medium.pdf",        "content": "MNRAS 000, 000{000 (0000) Preprint 10 June 2021 Compiled using MNRAS L ATEX style \fle v3.0\nGrammage of cosmic rays in the proximity of supernova remnants\nembedded in a partially ionized medium\nS. Recchia1;2?, D. Galli3, L. Nava4, M. Padovani3, S. Gabici5, A. Marcowith6,\nV. Ptuskin7, G. Morlino3\n1Dipartimento di Fisica, Universit\u0013 a di Torino, via P. Giuria 1, 10125 Torino, Italy\n2Istituto Nazionale di Fisica Nucleare, Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy\n3INAF{Osservatorio Astro\fsico di Arcetri, Largo E. Fermi 5, 50125 Firenze, Italy\n4INAF-Osservatorio Astronomico di Brera, Via Bianchi 46, I-23807 Merate, Italy\n5Universit\u0013 e de Paris, CNRS, Astroparticule et Cosmologie, F-75006 Paris, France\n6Laboratoire Univers et particules de Montpellier, Universit\u0013 e Montpellier/CNRS, F-34095 Montpellier, France\n7Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radiowave Propagation, 108840, Troitsk, Moscow, Russia\nAccepted XXX. Received YYY; in original form ZZZ\nABSTRACT\nWe investigate the damping of Alfv\u0013 en waves generated by the cosmic ray resonant\nstreaming instability in the context of the cosmic ray escape and propagation in the\nproximity of supernova remnants. We consider ion-neutral damping, turbulent damp-\ning and non linear Landau damping in the warm ionized and warm neutral phases of\nthe interstellar medium. For the ion-neutral damping, up-to-date damping coe\u000ecients\nare used. We investigate in particular whether the self-con\fnement of cosmic rays\nnearby sources can appreciably a\u000bect the grammage. We show that the ion-neutral\ndamping and the turbulent damping e\u000bectively limit the residence time of cosmic rays\nin the source proximity, so that the grammage accumulated near sources is found to\nbe negligible. Contrary to previous results, this also happens in the most extreme\nscenario where ion-neutral damping is less e\u000bective, namely in a medium with only\nneutral helium and fully ionized hydrogen. Therefore, the standard picture, in which\nCR secondaries are produced during the whole time spent by cosmic rays throughout\nthe Galactic disk, need not to be deeply revisited.\nKey words:\n1 INTRODUCTION\nThe most popular hypothesis for the origin of Galactic cos-\nmic rays (CRs) invokes supernova remnants (SNRs) as the\nmain sources of such particles (see e.g. Blasi 2013; Gabici\net al. 2019). In this scenario, which in the last decades had\nbecome a paradigm, CR di\u000busion plays a central role. Di\u000bu-\nsion is the key ingredient at the base of the di\u000busive shock\nacceleration of particles at SNRs (e.g. Drury 1983). Di\u000bu-\nsion also a\u000bects the escape of CRs from the acceleration site\nand the subsequent propagation in the source region, with\nprominent implications for \r-ray observations (Aharonian\n& Atoyan 1996; Gabici et al. 2009; Casanova et al. 2010;\nOhira et al. 2011; Nava & Gabici 2013). Finally, di\u000busion\ndetermines the con\fnement time of CRs in the Galaxy, thus\na\u000becting the observed spectrum and the abundances of sec-\nondary spallation nuclei and of unstable isotopes (Ptuskin\n& Soutoul 1998; Wiedenbeck et al. 2007).\n?E-mail: sarah.recchia@unito.itThe di\u000busion of CRs is thought to be mostly due to the\nresonant scattering o\u000b plasma waves whose wavelength is\ncomparable to the particle's Larmor radius rL=\rmpc2=eB,\nwherempis the proton mass, Bis the magnetic \feld\nstrength and \rthe Lorentz factor (see e.g. Skilling 1975a).\nThe magneto-hydrodynamic (MHD) turbulence relevant for\nCR propagation is composed of incompressible Alfv\u0013 enic and\ncompressible (fast and slow) magnetosonic \ructuations (Cho\n& Lazarian 2002; Fornieri et al. 2021). MHD turbulence is\nubiquitous in the interstellar space and may be injected by\nastrophysical sources (see e.g. Mac Low & Klessen (2004))\nbut also by CRs themselves. The active role of CRs in pro-\nducing the waves responsible for their scattering has been\nwidely recognized (see e.g. Wentzel 1974; Skilling 1975b; Ce-\nsarsky 1980; Amato 2011). In fact, spatial gradients in the\nCR density, as those found in the source vicinity, lead to the\nexcitation of Alfv\u0013 en waves at the resonant scale (Ptuskin\net al. 2008). This process, called resonant streaming insta-\nbility , produces waves that propagate along magnetic \feld\nlines in the direction of decreasing CR density.\n©0000 The AuthorsarXiv:2106.04948v1  [astro-ph.HE]  9 Jun 20212S. Recchia et al.\nThe density of Alfv\u0013 en waves that scatter CRs is lim-\nited by several damping processes. The most relevant are:\n(i) ion-neutral damping in a partially ionized medium (Kul-\nsrud & Pearce 1969; Kulsrud & Cesarsky 1971; Zweibel &\nShull 1982); ( ii) turbulent damping, due to the interaction of\na wave with counter-propagating Alfv\u0013 en wave packets. Such\nwaves may be the result of a background turbulence injected\non large scales and cascading to the small scales (we indicate\nthis damping as FG, after Farmer & Goldreich 2004); ( iii)\nnon-linear Landau (NLL) damping, due to the interaction\nof background thermal ions with the beat of two interfering\nAlfv\u0013 en waves (see e.g. Felice & Kulsrud 2001; Wiener et al.\n2013). The relative importance of these e\u000bects depends sig-\nni\fcantly on the physical conditions and chemical composi-\ntion of the ambient medium. A few other collisionless and\ncollisional damping processes can impact magnetohydrody-\nnamical wave propagation in a partially ionized gas but they\nmostly a\u000bect high-wavenumber perturbations (Yan & Lazar-\nian 2004). Recently, it has been suggested that dust grains\nmay also contribute to the damping of Alfv\u0013 en waves (Squire\net al. 2021).\nIn this paper we investigate the escape of CRs from\nSNRs, and their subsequent self con\fnement in the source\nregion, as due to the interplay between the generation of\nAlfv\u0013 en waves by CR streaming instability, and the damp-\ning process mentioned above. Our main goal is to establish\nwhether the self-con\fnement of CRs nearby sources can ap-\npreciably a\u000bect the grammage accumulated by these parti-\ncles. In fact, if this is the case, a signi\fcant fraction of CR\nsecondaries would be produced in the vicinity of CR sources,\nand not during the time spent by CRs in the Galactic disk,\nas commonly assumed. This would constitute a profound\nmodi\fcation of the standard view of CR transport in the\nGalaxy (see, e.g. D'Angelo et al. 2016). In particular, we\nfocus on the CR propagation in partially ionized phases of\nthe interstellar medium (ISM), showing that the ion-neutral\nand FG damping can signi\fcantly a\u000bect the residence time\nof CRs nearby their sources. We \fnd that, for typical condi-\ntions, the grammage accumulated by CRs in the vicinity of\nsources is negligible compared to that accumulated during\nthe time spent in the Galaxy. Even in the case of a medium\nmade of fully ionized H and neutral He, the combination of\nion-neutral and turbulent damping can substantially a\u000bect\nthe con\fnement time1.\nThis paper is organized as follows: in Sec. 2 we describe\nthe damping of Alfv\u0013 en waves by ion-neutral collisions in vari-\nous partially ionized phases of the ISM, and by other damp-\ning mechanisms; in Sec. 3 we illustrate the equations and\nthe setup of our model of CR escape and propagation in the\nproximity of SNRs, the time dependent CR spectrum and\ndi\u000busion coe\u000ecient, the residence time of CRs in the source\nproximity and the implications on the grammage; in Sec. 4\nwe describe our results; and \fnally in Sec. 5 we draw our\nconclusions.\n1The case of a fully neutral (atomic or partially molecular)\nmedium and of a di\u000buse molecular medium (see, e.g. Brahimi\net al. 2020) are not treated here, since the \flling factor of such\nphases is small, but we report the ion-neutral damping rate for\nsuch media for the sake of completeness. The case of a fully ion-\nized medium has been extensively treated by Nava et al. (2019).2 DAMPING OF ALFV \u0013EN WAVES\n2.1 Ion-neutral damping\nThe Galaxy is composed, for most of its volume, by three\nISM phases, namely the warm neutral medium (WNM, \fll-\ning factor\u001825%), warm ionized medium (WIM, \flling fac-\ntor\u001825%) and hot ionized medium (HIM, \flling factor\n\u001850%, see e.g. Ferri\u0012 ere 2001; Ferri\u0012 ere 2019). The physi-\ncal characteristics of these phases are summarised in Ta-\nble 1 (from Jean et al. 2009, see also Ferri\u0012 ere 2001; Fer-\nri\u0012 ere 2019). The physical characteristics of the cold neu-\ntral medium (CNM) and the di\u000buse medium (DiM) are also\nlisted for completeness, while their \flling factor is .1% (Fer-\nri\u0012 ere 2001; Ferri\u0012 ere 2019). In the regions where neutrals are\npresent, like the WNM and the WIM, the rate of ion-neutral\ndamping depends on the amount and chemical species of the\ncolliding particles. In the WNM and WIM the ions are H+,\nwhile neutrals are He atoms (with a H/He ratio of \u001810%)\nand H atoms with a fraction that varies from phase to phase.\nThe main processes of momentum transfer (mt) be-\ntween ions and neutrals are elastic scattering by induced\ndipole, and charge exchange (ce). In the former case, domi-\nnant at low collision energies, the incoming ion is de\rected\nby the dipole electric \feld induced in the neutral species,\naccording to its polarizability (Langevin scattering); in the\nlatter case the incoming ion takes one or more electrons from\nthe neutral species, which becomes an ambient ion. The fric-\ntion force per unit volume Fiexerted on an ion iis thus the\nsum of Fi;mt+Fi;ce.\nWith the exception of collisions between an ion and a\nneutral of the same species, as in the important case of col-\nlisions of H+ions with H atoms (see Sec. A1), the two pro-\ncesses are well separated in energy. At low collision energies\nelastic scattering dominates, and the friction force is\nFi;mt=ninn\u0016inh\u001bmtviin(un\u0000ui); (1)\nwhereniandnnare the ion and neutral densities, uiand\nunare the ion and neutral velocities, \u0016inis the reduced\nmass of the colliding particles, \u001bmtis the momentum trans-\nfer (hereafter m.t.) cross section, and the brackets denote\nan average over the relative velocity of the colliding particles.\nAt high collision energies (above \u0018102eV), the dom-\ninant contribution to the transfer of momentum is charge\nexchange\nA++ B!A + B+: (2)\nIf the charge exchange rate coe\u000ecient is approximately in-\ndependent of temperature, and there is no net backward-\nforward asymmetry in the scattering process (two conditions\ngenerally well satis\fed), Draine (1986) has shown that the\nfriction force on the ions takes the form\nFi;ce=ninnh\u001bceviinm2\nnun\u0000m2\niui\nmn+mi; (3)\nwhere\u001bceis the charge exchange (hereafter c.e.) cross sec-\ntion, andmn(i)the mass of the neutral (ion).\nThe collisional rate coe\u000ecients h\u001bmtviinandh\u001bceviin\nare often estimated from the values given by Kulsrud & Ce-\nsarsky (1971) or Zweibel & Shull (1982) for H+{ H collisions\n(e.g. D'Angelo et al. 2016; Nava et al. 2016; Brahimi et al.\n2020). The rate coe\u000ecients for collisions between various\nMNRAS 000, 000{000 (0000)3\nTable 1. ISM phases and parameters adopted in this work. Tis the gas temperature, Bthe interstellar magnetic \feld, nthe total gas\ndensity,fthe ionisation fraction, \u001fthe helium fraction and Linjthe injection scale of the background magnetic turbulence.\nT(K)B(\u0016G)n(cm\u00003) neutral ion f \u001f L inj(pc)\nWIM 8000 5 0.35 H, He H+0.6\u00000.9 0\u00000.1\nHe H+1 0.1 50\nWNM 8000 5 0.35 H, He H+7\u000210\u00003\u00005\u000210\u000020\u00000.1 50\nCNM 80 5 35 H, He C+4\u000210\u00004\u000010\u000030.1 1-50\nDiM 50 5 300 H 2, He C+10\u000040.1 1-50\nHIM 1065\u00180:01 - H+1.0 0.0 100\nspecies of ions and neutrals adopted in this study are de-\nscribed in detail in Sec. A1. For elastic collisions, they have\nbeen taken from the compilation by Pinto & Galli (2008); for\ncharge exchange, they have been calculated from the most\nupdated available cross sections.\nIon-neutral collisions are one of the dominant damping\nprocesses for Alfv\u0013 en waves propagating in a partially ionized\nmedium (see Piddington 1956; Kulsrud & Pearce 1969). In\nthe case of elastic ion-neutral collisions, (Eq. 1), the disper-\nsion relation for Alfv\u0013 en waves in this case is\n!(!2\u0000!2\nk) +i\u0017in[(1 +\u000f)!2\u0000\u000f!2\nk] = 0; (4)\nwhere!is the frequency of the wave, !k=kvA;iis the\nwavevector in units of the Alfv\u0013 en speed of the ions\nvA;i=Bp4\u0019mini; (5)\n\u0017inis the ion-neutral collision frequency\n\u0017in=mn\nmi+mnh\u001bmtviinnn; (6)\nand\u000fis the ion-to-neutral mass density ratio\n\u000f=mini\nmnnn: (7)\nNotice that \u000fis a small quantity in the WNM and CNM but\nnot in the WIM2.\nThe dispersion relation Eq. (4) is a cubic equation for\nthe wave frequency !(with real and imaginary parts) as a\nfunction of the real wavenumber !k. Writing!=<(!)\u0000\ni\u0000in\nd, where \u0000in\nd>0 is the ion-neutral damping rate, and\nsubstituting in Eq. (4), one obtains (Zweibel & Shull 1982)\n!2\nk=2\u0000in\nd\n\u0017in\u00002\u0000in\nd[(1 +\u000f)\u0017in\u00002\u0000in\nd]2; (8)\nwhich implies 0 <\u0000in\nd<\u0017in=2. If\u000f\u001c1, then\n\u0000in\nd\u0019!2\nk\u0017in\n2[!2\nk+ (1 +\u000f)2\u00172\nin]: (9)\nAlfv\u0013 en waves resonantly excited by CR protons have fre-\nquency!k\u0019vA;i=rLThus, the frequency is related to the\n2To be precise, the dispersion relation Eq. (4) is valid only if\nthe friction force is proportional to the ion-neutral relative speed\nun\u0000ui, as in the case of momentum transfer by elastic collisions.\nHowever, we use the same relation also in the case of charge ex-\nchange, simply replacing h\u001bmtviinwithh\u001bceviin.kinetic energy of the CR proton E=\rmpc2as\n!k\u0019eBv A;i\nE: (10)\nThe e\u000bective Alfv\u0013 en velocity, vA=<(!)=k, felt by CRs de-\npends on the coupling between ions and neutrals. In general,\nthe following asymptotic behavior can be identi\fed:\n\u000fLow wavenumber, !k\u001c\u0017in:\nat large CR energy ions and neutrals are well coupled; the\ntotal density is n=nH+nHe+niand the Alfv\u0013 en speed\nrelevant for CRs resonant with the waves is\nvA;n=Bp4\u0019\u0016m pn; (11)\nwhere\u0016\u00181:4 is the mean molecular weight, and \u0000in\nd/E\u00002;\n\u000fHigh wavenumber, !k\u001d\u0017in:\nat small CR energy ions and neutrals are weakly coupled and\nion-neutral damping is most e\u000bective. The Alfv\u0013 en speed is\nthe one in the ions, vA;i, and \u0000in\nd\u0018const.\nNotice that if \u000f<1=8 there is a range of wavenumbers\nfor which the waves do not propagate in a partially ionized\nmedium (Zweibel & Shull 1982). This is marked as a shaded\nregion in Fig. 1-2. On the other hand, such non-propagation\nband is found in the absence of CRs propagating in the par-\ntially ionized medium. Recently it has been suggested (Re-\nville et al. 2021) that taking into account the presence of\nCRs may allow for the propagation of waves in that band.\nIntroducing the fraction of ionized gas fand the helium-\nto-hydrogen ratio \u001f,\nf=ni\nnH+ni; \u001f =nHe\nnH+ni; (12)\nEq. (6) becomes\n\u0017in=\u00141\u0000f\n1 + ~mih\u001bmtvii;H+4\u001f\n4 + ~mih\u001bmtvii;He\u0015n\n1 +\u001f:(13)\nwhere ~mi=mi=mp. In the following, the standard value\n\u001f= 0:1 is assumed, but the case \u001f= 0 is also considered for\nillustrative purposes and for a comparison with the results\nof D'Angelo et al. (2016), who neglect the contribution of\nhelium to ion-neutral damping.\n2.1.1 WIM and WNM\nIn this case H is partially ionized and the dominant ion is H+\n( ~mi= 1). Therefore \u000f=nH+=(nH+4nHe) =f=(1\u0000f+4\u001f),\nMNRAS 000, 000{000 (0000)4S. Recchia et al.\ni.e.\u000f= 0:005{0:05 and 0.75{9 for the WNM and the WIM,\nrespectively. The ion-neutral collision frequency is\n\u0017in=\u00141\u0000f\n2h\u001bmtviH+;H+4\u001f\n5h\u001bmtviH+;He\u0015n\n1 +\u001f:(14)\nFig. 1 shows the damping rate for waves resonant with CRs\nof energyE, as a function of the CR energy E. Notice the\nnon-propagation band found in the WNM ( \u000f<1=8).\n2.1.2 CNM and DiM\nIn this case H is neutral and the dominant ion is C+( ~mi=\n12), with fractional abundance nC+=nH\u0019(0:4{1)\u000210\u00003.\nTherefore\u000f= 12nC+=(nH+ 4nHe)\u0019(3{9)\u000210\u00003and\n\u0017in=\u00141\n13h\u001bmtviC+;H+\u001f\n4h\u001bmtviC+;He\u0015n\n1 +\u001f: (15)\nFig. 2 shows the damping rate for waves resonant with CRs\nof energyE, as a function of the CR energy E. Also in this\ncase non-propagation regions are found.\n2.2 Wave cascade and turbulent damping\nThe turbulent damping (FG) of self-generated Alfv\u0013 en waves\nis due to their interaction with a pre-existing background\nturbulence. Such turbulence may be injected by astrophys-\nical sources (see e.g. Mac Low & Klessen 2004) with a tur-\nbulent velocity vturband on scales, Linj, much larger than\nthe CR Larmor radius. For waves in resonance with particles\nwith a given energy E, the damping rate, that accounts for\nthe anisotropy of the turbulent cascade, has been derived\nby Farmer & Goldreich (2004); Yan & Lazarian (2004) and\nreads\n\u0000FG\nd=\u0012v3\nturb=Linj\nrLvA\u00131=2\n; (16)\nwherevAis the e\u000bective Alfv\u0013 en speed felt by CRs, as de\fned\nin Sec. 2.1. We take the turbulence as trans-Alfv\u0013 enic at the\ninjection scale, namely vturb=vA;n(at large scales waves\nare in the low wavenumber regime, where ions and neutrals\nare well coupled, as illustrated in Sec. 2.1). This is the\nlikely situation if the turbulence is mainly injected by old\nSNRs, with a forward shock becoming trans-sonic and trans-\nAlfv\u0013 enic. The FG damping rate is shown in Fig. 1 for the\nWIM and WNM.\nIn highly neutral media, such as the the WNM, CNM\nand DiM, the background turbulence responsible for the FG\ndamping can be damped by ion-neutral friction at a scale,\nlmin= 1=kmin(Xu et al. 2015, 2016; Lazarian 2016; Brahimi\net al. 2020). Correspondingly, there is a minimum particle\nenergy,Emin, such that rL(Emin) =lmin, below which the\nFG damping cannot a\u000bect the self-generated Alfv\u0013 en waves\n(Brahimi et al. 2020):\n1\nlmin=L1=2\ninj\u00122\u000f\u0017in\nvA;n\u00133=2r\n1 +vA;n\n2\u000f\u0017inLinj: (17)\nIn Fig. 1 the FG damping rate for the WNM is truncated\natEmin. In the WIM the cascade rate is found to be always\nlarger than the ion-neutral damping rate and there is no\nEmin.2.3 Non-linear Landau damping\nThe non-linear Landau (NLL) damping is caused by the in-\nteraction between the beat of two Alfv\u0013 en waves and the ther-\nmal (at temperature T) ions in the background medium. The\ndamping rate for resonant waves is given by (Kulsrud 1978;\nWiener et al. 2013)\n\u0000NLL\nd=1\n2s\n\u0019\n2\u0012kBT\nmp\u0013I(kres)\nrL; (18)\nwherekBis the Boltzmann constant and I(kres) is the wave\nenergy density (see Sec. 3 below for the de\fnition) at the\nresonant wavenumber kres= 1=rL.\n3 COSMIC RAY PROPAGATION IN THE PROXIMITY\nOF SNRS\nWe consider the escape of CRs from a SNR and the sub-\nsequent propagation in the source proximity. The propaga-\ntion region is assumed to be embedded in a turbulent mag-\nnetic \feld, with a large scale ordered component of strength\nB0. CRs are scattered by Alfv\u0013 en waves, which constitute a\nturbulent magnetic \feld background of relative amplitude\n\u000eB=B 0, wherekis the wavenumber. We only consider waves\nthat propagate along the uniform background \feld B0. In\nthe limit of \u000eB=B 0\u001c1, which is the one relevant for the\ncases treated in this paper, the CR di\u000busion along \feld lines\ncan be treated in the quasi-linear regime, with a di\u000busion\ncoe\u000ecient given by Berezinskii et al. (1990) and Kulsrud\n(2005)\nD(E) =4\u0019cr L(E)\n3I(kres)\f\f\f\f\nkres=1=rL=DB(E)\nI(kres); (19)\nwherecis the speed of light, I(kres) =\u000eB(kres)2=B2\n0is the\nwave energy density calculated at the resonant wavenumber\nkres= 1=rL, andDB(E) = (4\u0019=3)crL(E) is the Bohm di\u000bu-\nsion coe\u000ecient. We also assume that the dominant source of\nAlfv\u0013 enic turbulence is produced by the CR resonant stream-\ning instability.\nIn our model we adopt the \rux tube approximation for\nthe CR transport along B0(see, e.g., Ptuskin et al. 2008),\nand we neglect the di\u000busion across \feld lines, which is sup-\npressed in the \u000eB=B 0\u001c1 regime (see e.g. Drury 1983;\nCasse et al. 2002). Thus, we do not address the perpen-\ndicular evolution of the \rux tube (see, e.g., Nava & Gabici\n2013) and any possible CR feedback on it and, in general,\non the ISM dynamics (see, e.g., Schroer et al. 2020). Such\none-dimensional model for the CR propagation is applicable\nfor distances from the source below the coherence length, Lc,\nof the background magnetic turbulence, i.e. the scale below\nwhich the magnetic \rux tube is roughly preserved (see, e.g.,\nCasse et al. 2002).\nWhen particles di\u000buse away from the source at distances\nlarger than Lc, di\u000busion becomes 3-D and the CR density\ndrops quickly. In the Galactic disk, Lcis estimated observa-\ntionally and may range from few pc to \u0019100 pc, depending\non the ISM phase (see, e.g., Nava & Gabici 2013, and refer-\nences therein).\nWe follow the approach proposed by Nava et al. (2016,\n2019), and we determine: ( i) the escape time of CRs of a\nMNRAS 000, 000{000 (0000)5\nFigure 1. Damping rates \u0000in\ndand \u0000FG\nd(ion-neutral and turbulent) of Alfv\u0013 en waves in the WNM ( left-hand panel ) and WIM ( right-hand\npanel ) vs. CR energy E. Di\u000berent colors are used for di\u000berent values of the hydrogen ionization fraction f. Unless stated otherwise, a\nstandard\u001f= 0:1 He abundance is assumed. The considered parameters for the WNM and WIM are given in Table 1. In the left-hand\npanel the dotted lines refers to ion-neutral damping, while the dashed lines to the FG damping. The last are truncated to the minimum\nenergy,Emin, below which the background turbulence is damped by ion-neutral friction before reaching the scale relevant for damping\nself-generated waves resonant with particle of energy <E min(see Sec. 2.2). The shaded regions represent the range of non-propagation\nof Alfv\u0013 en waves (see Sec. 2.1). In the right-hand panel, the solid lines refer to ion-neutral damping, while the thin dotted lines refer to\nthe case where the contribution to damping from He is neglected ( \u001f= 0). The dashed lines refer to the FG damping, which in the case\nof WIM is found to depend little on the ionization fraction f.\nFigure 2. Ion-neutral damping rate \u0000in\ndof Alfv\u0013 en waves in the CNM (left-hand panel) and DiM (right-hand panel) as a function of the\nCR energy E. Di\u000berent colors are used for di\u000berent values of the C+abundance. The parameters adopted for the CNM and DiM are\ngiven in Table 1. The shaded regions represent the range of non-propagation of Alfv\u0013 en waves (see Sec. 2.1).\ngiven energy from the remnant; ( ii) the time dependent evo-\nlution of the CR cloud and of the self-generated di\u000busion\ncoe\u000ecient after escape; ( iii) the time spent by CRs in the\nsource vicinity and the corresponding grammage. We focus\non the warm ionized and warm neutral phases of the ISM.\nAs shown in the following sections, the results depend sig-\nni\fcantly on the considered ISM phase, and in particular on\nthe amount and type of neutral and ionized atoms in the\nbackground medium.3.1 Transport equations and initial/boundary conditions\nThe coupled CR and wave transport equations read (Nava\net al. 2016, 2019)\n@PCR\n@t+vA@PCR\n@z=@\n@z\u0012DB\nI@PCR\n@z\u0013\n(20)\nand\n@I\n@t+vA@I\n@z= 2(\u0000 CR\u0000\u0000d)I+Q: (21)\nMNRAS 000, 000{000 (0000)6S. Recchia et al.\nHerevAis an e\u000bective Alfv\u0013 en velocity that takes into ac-\ncount the coupling between ions and neutrals (see Sec. 2.1),\nwhilePCRis the partial pressure of CRs of momentum p\nnormalized to the magnetic \feld pressure\nPCR=4\u0019\n3cp4f(p)\nB2\n0=8\u0019: (22)\nThe term 2 \u0000 CRIgives the wave growth due to the CR\nstreaming instability and can be expressed as\n2\u0000CRI=\u0000vA@PCR\n@z: (23)\nThe term \u0000 dencompasses the relevant wave damping rates,\nwhich are described in detail in Sec. 2. The term Qrepre-\nsents the possible injection of turbulence from an external\nsource, other than the CR streaming. Here it is taken as\nQ= 2\u0000 dI0, whereI0is a parameter such that typical val-\nues of the Galactic CR di\u000busion coe\u000ecient are recovered at\nlarge distances from the source (see, e.g., Strong et al. 2007).\nIn this way, when the streaming instability is not relevant,\ndi\u000busion is regulated by the background turbulence. Notice\nthat in these equations we neglected adiabatic losses that\narises when Alfv\u0013 en speed varies in space (see, e.g., Brahimi\net al. 2020), since we are considering a homogeneous ISM\naround the SNR.\nThe coordinate zis taken along B0andz= 0 refers\nto the centre of the CR source. The equations are solved\nnumerically with a \fnite di\u000berence explicit method, using\nthe initial conditions\nPCR=\u001aP0\nCR ifz<R esc(E);\n0 ifz>R esc(E);\nand\nI=I0 everywhere: (24)\nHereResc(E) is the size of the region \flled by CRs at the\ntime of escape, and P0\nCRis the initial CR pressure inside this\nregion. The method used to determine the escape radius and\ntime for particles of energy Eis described in detail in Nava\net al. (2016, 2019). As for the initial condition for the waves,\nit is also possible to choose I\u001dI0forz<R esc(E) in order\nto mimic Bohm di\u000busion inside the source, as proposed by\nMalkov et al. (2013). However, as discussed by Nava et al.\n(2019), di\u000berent choices of Iinside the source have little im-\npact on the \fnal solution. As boundary conditions we impose\na symmetric CR distribution at z= 0, and\nPCR= 0; I =I0 atz&Lc: (25)\nThe 1-D model used in this paper is valid only up to a dis-\ntance from the source given by the coherence length Lcof\nthe magnetic \feld. At larger distances the di\u000busion becomes\n3-D and the CR density quickly drops, a behavior that can\nbe described in terms of a free escape boundary at Lc. The\nvalue ofLcis constrained from observations to be .100 pc\nbut its value is matter of debate and is likely phase- and\nposition-dependent in our galaxy. We take the free escape\nboundary at z&Lc, and we check that this assumption is\nnot signi\fcantly a\u000becting the results, for instance changing\nLcto 10Lc.4 RESULTS\nIn what follows we discuss the release of CRs of energy E\nfrom a SNR, giving an estimate of the age and radius of the\nsource at the moment of escape. Moreover, we investigate the\npropagation of runaway CRs in the source proximity, with\nparticular focus on the CR spectrum and self-generated dif-\nfusion coe\u000ecient. Finally, we estimate the residence time of\nCRs in the source proximity and we infer general implica-\ntions for the grammage accumulated in that region.\nWe treat these aspects in the cases of a WIM or a WNM\nsurrounding the remnant, with special emphasis on the role\nplayed by the ion-neutral damping of Alfv\u0013 en waves.\nIn our calculations we assume that: ( i) runaway CRs\nhave a total energy ECR= 1050erg, a power-law spectrum\nin energy between 1 GeV{5 PeV with spectral index \r=\n\u00002:0; and ( ii) the typical ISM di\u000busion coe\u000ecient is D(E) =\nD0(E=10 GeV)0:5withD0= 1028cm2s\u00001(see, e.g., Strong\net al. 2007).\n4.1 Escape of cosmic rays: source age and radius\nThe age and radius of escape of CRs of energy Eis estimated\nas follows (Nava et al. 2016, 2019): we de\fne the half-time,\nt1=2(E;R) of a CR cloud as the time after which half of the\nCRs of a given energy, initially con\fned within a region of\nsizeR, have escaped that region. The evolution of such CR\ncloud is studied by solving Eq. (20) and Eq. (21) with initial\nconditions given by Eq. (24) and boundary conditions given\nby Eq. (25).\nThe radius of a SNR expanding in a homogeneous\nmedium of density ncan be estimated as (Truelove & McKee\n1999)\nRSNR(t) = 0:5\u0012E51\nn\u00131=5\"\n1\u00000:09M5=6\nej\f\nE1=2\n51n1=3tkyr#2=5\nt2=5\nkyrpc;\n(26)\nwhereE51is the supernova explosion energy in units of\n1051erg,nis the total density of the ambient ISM in cm\u00003,\nMej\fis the mass of the supernova ejecta in solar masses,\nandtkyris the SNR age in kyr. This equation is valid\nin the adiabatic phase of the expansion, which starts af-\nter\u001986M5=6\nej\fE\u00001=2\n51n\u00001=3yr. Here we take E51= 1 and\nMej\f= 1:4, while the gas density depends on the medium\nand is reported in Table 1. At earlier times an approx-\nimate expression for the free expansion phase has to be\nused (Chevalier 1982). The adiabatic phase stops at roughly\n\u00191:4\u0002104E3=14\n51n\u00004=7yr, when the radiative phase starts\n(Cio\u000e et al. 1988). At this epoch we also assume that the\nacceleration of CRs becomes ine\u000bective and that all CRs are\ninstantly released (the validity of the assumption of instant\nrelease may depend on the conditions in the shock precursor,\nas discussed by Brahimi et al. 2020).\nThe escape time, tesc(E), of particles with energy Eis\nestimated as the time such that t1=2(E;R SNR) equals the\nage of a SNR of radius RSNR. This is the timescale over\nwhich waves can grow. The dependence of t1=2(E;R) on the\ninitial radius Rat \fxed energy E, and varying the back-\nground di\u000busion coe\u000ecient and the CR spectral index, has\nbeen extensively explored by Nava et al. (2016, 2019), and\nwe refer the reader to these works for a detailed discussion.\nHere we brie\ry summarize the most relevant points. At small\nMNRAS 000, 000{000 (0000)7\nRthe CR gradient is large and the ampli\fcation of waves is\nvery e\u000bective. In this regime, the NLL damping, that scales\nwith the wave energy density I(k), can play an important\nrole and may dominate over ion-neutral and FG damping\nat small enough radii. At intermediate R, ion-neutral damp-\ning dominates in most cases. In fact, as shown in Fig. 1,\nthe ion-neutral damping rate is larger than the FG rate at\nleast up to particle energies of \u001810 TeV, with the only ex-\nception of a WIM with fully ionized hydrogen and \u001810%\nof neutral helium. At large R, the CR gradient is reduced\nand the streaming instability is less e\u000bective. In this case\nthe expansion of the CR bubble is determined by the back-\nground turbulence and the test particle limit is recovered,\nwitht1=2/R2=D0.\nFig. 3 and Fig. 4 show the SNR age and radius at the\ntime of escape of CRs, as a function of the CR energy,in the\ncase of a WNM and WIM respectively. We con\frm the \fnd-\nings by Nava et al. (2016, 2019) and the qualitative results\nthat particles with higher energy escape earlier than the low\nenergy particles (see, e.g., Gabici 2011).\nIn the case of the WIM, Rescandtesctend to decrease\nwith a larger neutral density. The e\u000bect of ion-neutral damp-\ning is especially visible in the energy range \u00181{10 TeV,\nwhere the ion-neutral damping rate starts to decrease from\na roughly constant value at di\u000berent particle energies, de-\npending on the value of the ionized hydrogen fraction fand\non the helium-to-hydrogen ratio \u001f, as shown in Fig. 1. The\ndrop of \u0000in\ndcorresponds to an increase of tesc, and therefore\nto a better CR con\fnement.\nThe situation is more subtle in the case of the WNM.\nIn fact, here the con\fnement appears to be slightly more\ne\u000bective for a smaller value of f, a result opposite to what\nhappens in the WIM. This can be explained taking into ac-\ncount that, while the \u0000in\ndis practically the same at energies\nbelow\u001810 TeV for f\u00187\u000210\u00003\u00005\u000210\u00002, the e\u000bective\nAlfv\u0013 en speed felt by CRs, which at these energies is roughly\nthat of ions, as illustrated in Sec. 2, is a factor of \u00182{3 larger\nforf= 7\u000210\u00003. This re\rects on an increase by the same\namount of the FG damping rate (above the cut-o\u000b energy,\nsee Sec. 2.2), as shown in Fig. 1, but also of the growth rate\nterm, which is proportional to vA. On the other hand, at\nenergies in the range \u001810 GeV{1 TeV, the FG damping is\nalways subdominant compared to the ion-neutral damping,\nand the net e\u000bect of this change of vAis a slightly better\ncon\fnement at smaller fas a result of an enhanced wave\ngrowth rate.\nAlso in the case of a WIM the e\u000bective vAis a bit larger\nat smallerfbelow\u00181 TeV, but here the increase is only\nby a factor of\u00181:2 and cannot overcome the e\u000bect of ion-\nneutral damping.\n4.2 Cosmic ray spectra and di\u000busion coe\u000ecient\nThe spectrum of runaway CRs and the corresponding self-\ngenerated turbulence depend signi\fcantly on the distance\nfrom the source and on the time. In Fig. 5 and Fig. 6 we show\nthe spectrum of CRs that have already escaped to a distance\nof 50 pc from the centre of the remnant at di\u000berent times,\nfromt= 6\u0002103yr tot= 105yr, and for the WNM ( f=\n0:05,\u001f= 0:1) and the WIM ( f= 0:9,\u001f= 0:1), respectively.\nTo remove the e\u000bect of the simple 1-D geometry that we are\nadopting we multiply PCRbyR2\nesc(E)=R2\nSNR(t). For eachcase we also show the ratio between the self-generated and\nbackground di\u000busion coe\u000ecients.\nIn the same \fgures we also show the spectrum and di\u000bu-\nsion coe\u000ecient at the position of the shock of CRs that have\nescaped at earlier times (and are considered as decoupled\nfrom the accelerator) as well as the spectrum of particle still\ncon\fned in the accelerator, at di\u000berent ages of the remnant\nfromt= 6\u0002103yr tot= 3\u0002104yr. The spectrum of the\ncon\fned particles is estimated by assuming that the remnant\nprovides\u00191050erg in CRs with a /E\u00002spectrum that ex-\ntends from\u00181 GeV to\u00185 PeV, with an exponential cuto\u000b\nat the energy of the particles that escape at the considered\nage. The SNR is assumed to stop accelerating particles at\nthe onset of the radiative phase, which for the typical den-\nsity of the WIM and WNM takes place at trad\u00193\u0002104yr.\nThe corresponding shock radius is Rrad\u001922 pc. In all cases,\nat each time and location, the spectrum exhibit a sharp rise\nat low energies and a peak followed by a nearly power-law\nlike behaviour at higher energies.\nSince high energy particles escape earlier and di\u000buse\nfaster, they reach a given distance at earlier times, as shown\nfor the spectra at 50 pc, where the peak moves at lower\nenergies with time. At energies lower than the peak, particles\nhave not yet reached that position, giving the sharp rise. At\nlarger energies, particles have di\u000bused over a bigger distance,\ngiving a spectrum steeper than the spectrum released from\nthe SNR, here assumed to be /E\u00002.\nA similar behaviour is observed with the spectra at the\nshock. Here the position at which the spectra are shown\nvaries with time, since it is given by the shock radius at\na given age. This radius is always <50 pc for the cases\nillustrated here. Comparing the spectra at the shock and at\n50 pc for a given age, it is evident that the peak is at lower\nenergy in the former case. This again is the result of an\nearlier escape and faster di\u000busion of high energy particles.\nAs for theD=D 0ratio as a function of energy, it is possi-\nble to infer from the plots that, at a given time, the di\u000busion\ncoe\u000ecient is equal to its background value at low energies,\nwhere particles have not yet escaped or not yet reached the\nposition where Dis calculated, and at high energies, where\nthe turbulence produced by such particle is being damped.\nAt intermediate energies, and roughly corresponding to the\npeak in the spectrum, the di\u000busion coe\u000ecient is suppressed\ncompared to D0due to CR streaming instability. The ener-\ngies at which the suppression is evident move to lower values\nwith time.\nA more complex situation can be observed in some\ncases, for instance at t= 104yr at the shock and in the\nTeV range, for the WIM. Here Dstarts to deviate from D0\nat\u001850 GeV, then, above \u0018500 GeV,Dbegins to ap-\nproach again D0, but above\u00183 TeV the level of turbulence\nincreases again, with the result of a suppression of D. This\nis due to the energy dependence of the escape radius, which\nbecomes steeper above few TeV. This results in the fact that,\nabove that energy, the escape radius becomes small enough\nas to e\u000bectively excite the streaming instability. In fact the\ndensity of runaway CRs is /1=R2\nesc. The more e\u000ecient self-\ncon\fnement at this high energies also re\rects in a hardening\nof the spectrum (see also Nava et al. 2019 for a discussion).\nMNRAS 000, 000{000 (0000)8S. Recchia et al.\n102103104105\n101102103104 1 10 100\n-- Resc- tesctesc [yr]\nResc [pc]\nE [GeV]WNM\nf=0.007\nf=0.05\n104105106\n10110210310-210-1tres [yr]\nX [g/cm2]\nE [GeV]WNM\nf=0.007\nf=0.05 \nFigure 3. Case of WNM. Left panel: SNR age and radius at the time of escape of CRs, as a function of the CR energy. Right panel:\nResidence time of CRs in a region of \u0018100 pc around the source, as a function of the CR energy, and the corresponding grammage.\nDi\u000berent colors are used for di\u000berent values of the hydrogen ionization fraction f, while the helium-to-hydrogen ratio \u001f= 0:1. The\nparameters of the WNM are listed in Table 1.\n103104105\n101102103104 1 10 100\n-- Resc\n- tesctesc [yr]\nResc [pc]\nE [GeV]WIM\nf=0.6\nf=0.9\nf=0.9, χ =0.0\n104105106\n10110210310-210-1100tres [yr]\nX [g/cm2]\nE [GeV]WIM\nf=0.6\nf=0.9\nf=1\nf=1, χ=0\nf=1, χ=0, 20 pc\nFigure 4. Case of WIM. Left panel: SNR age and radius at the time of escape of CRs, as a function of the CR energy. Right panel:\nResidence time of CRs in a region of \u0018100 pc around the source, as a function of the CR energy, and the corresponding grammage. The\ncyan curve is computed by assuming that particles of all energies escape at SNR radius of 20 pc, as assumed by D'Angelo et al. (2016).\nDi\u000berent colors are used for di\u000berent values of the hydrogen ionization fraction fand helium-to-hydrogen ratio \u001f. The parameters of the\nWIM are listed in Table 1.\n4.3 Residence time and grammage\nThe grammage accumulated by CRs close to their sources,\nand its relevance compared to the grammage accumulated\nwhile di\u000busing in the whole Galaxy, is related to the resi-\ndence time in the proximity of the source. A formal deter-\nmination of the grammage should be done by solving the CR\ntransport equation for nuclei with the inclusion of spallation\ncontributions (see, e.g., Berezinskii et al. 1990; Ptuskin &\nSoutoul 1990).\nHere we adopt the following approach: we assume that\nN0particles of energy Eare instantly injected by a source\nlocated in a region of constant gas density \u001a=\u0016mpnand are\nsubject to free escape at a boundary located at a distance\nLcfrom the source. At a given time t, the total number\nof particles that are still in the region is Nin(t) while the\nnumber of escaped particles is Nesc=N0\u0000Nin(t). Particles\nthat cross the boundary at time thave accumulated the\ngrammageX(t) =\u001actand the average grammage gained byall escaped particles from t= 0 tot=1is given by\nhXi=1\nN0Z1\n0\u001actdNesc\ndtdt (27)\n=\u001ac\nN0\u0014Z1\n0Nin(t)dt\u0000lim\nt\u0000!1(tNin(t))\u0015\n=\u001ac\u001c res:\nThe residence time \u001cresis given by\n\u001cres=1\nN0\u0014Z1\n0Nin(t)dt\u0000lim\nt\u0000!1(tNin(t))\u0015\n: (28)\nThe number of particles with a given energy contained in\nthe region at a given time can be calculated from the CR\npressure (see Sec. 3):\nNin(t) =N0\nRescP0\nCRZLc\n0PCR(z;t;E )dz; (29)\nwhere we took into account that CRs on energy Eare ini-\ntially released from a region of size Resc(E).\nMNRAS 000, 000{000 (0000)9\nWNM \n10-210-1100\n10110210310450 pc6e+3\n1e+4 3e+4\n6e+4\n1e+5f(E)E2 x (Resc/Rsh)2 [eV/cm3]\nE [GeV]10-310-210-1100101102\n101102103104shock\n6e+3\n1e+43e+4f(E)E2 x (Resc/Rsh)2 [eV/cm3]\nE [GeV]\n 1\n101102103104D/D0\nE [GeV] 1\n101102103104D/D0\nE [GeV]\nFigure 5. Case of WNM ( f= 0:05,\u001f= 0:1): CR spectrum and ratio D=D 0as a function of energy, where D0(E) =\n1028(E=10 GeV)0:5cm2s\u00001. The di\u000berent colors refer to di\u000berent ages in yr (as marked). The left panels refer to a location at 50 pc\nfrom the center of the SNR, while the right panels refer to the shock position. In this case also the spectrum of CRs still con\fned in the\naccelerator is shown (dashed lines).\nFig. 3 and 4 show the residence time of CRs in a re-\ngion of\u0018100 pc around the remnant, for the WNM and\nthe WIM, respectively, and for di\u000berent values of the neu-\ntral fraction, and the corresponding grammage. Even in the\n(not very plausible) case of a fully ionized WIM ( f= 1:0,\n\u001f= 0:0), the grammage is nearly two orders of magnitudes\nsmaller than that accumulated in the disk (see e.g. Jones\net al. 2001; Gabici et al. 2019). This is due to the action of\nthe ion-neutral and FG damping. The presence of a \u001810%\nof neutral helium (which is less e\u000bective at wave damping\ncompared to hydrogen) noticeably reduces the CR residence\ntime compared to the case of a fully ionized background\nmedium, above few tens GeV.\nThese results are at odds with what previously sug-\ngested by D'Angelo et al. (2016), namely that ion-neutral\ndamping is totally negligible in the case of only neutral he-\nlium, since the c.e. cross section is very small. However, as\nshown in Sec. 2, not only c.e. but also m.t. has to be taken\ninto account in the ion-neutral damping. In fact, D'Angelo\net al. (2016) \fnd a grammage nearly a factor of ten larger\nthan what we \fnd. This is in part due to their assumption\nof ion-neutral damping of waves with neutral helium. In ad-dition, they assume a 20% acceleration e\u000eciency, while we\nassume 10%, which enhances the e\u000bect of streaming insta-\nbility, and they do not include the FG damping. Finally,\nD'Angelo et al. (2016) assume that CRs at all energies are\nreleased when the SNR radius is 20 pc, a scenario implying\na smaller release radius at low energies compared to our cal-\nculation, which translates in an enhancement of the stream-\ning instability (and the CR con\fnement) at energies below\n\u00191 TeV, as shown in Fig. 4.\nAny addition of neutral hydrogen compared to the\ncase of only neutral helium further reduces the con\fnement\ntime. Correspondingly, the CR grammage accumulated in\nthe source proximity results to be well below that inferred\nfrom observations. A similar result was found by Nava et al.\n(2019) for the HIM, where the residence time tends to be\nlarger than in a partially ionized medium, but the ISM den-\nsity is lower (\u00180:01 cm\u00003).\n5 CONCLUSIONS\nWe followed the method and the setup proposed by Nava\net al. (2016, 2019) to investigated the escape of CRs from\nMNRAS 000, 000{000 (0000)10 S. Recchia et al.\nWIM \n10-310-210-1100\n10110210310450 pc\n6e+31e+4\n3e+4\n6e+4\n1e+5f(E)E2 x (Resc/Rsh)2 [eV/cm3]\nE [GeV]10-310-210-1100101102\n101102103104shock\n6e+31e+43e+4f(E)E2 x (Resc/Rsh)2 [eV/cm3]\nE [GeV]\n 1\n101102103104D/D0\nE [GeV] 1\n101102103104D/D0\nE [GeV]\nFigure 6. Case of WIM ( f= 0:9,\u001f= 0:1): CR spectrum and ratio D=D 0as a function of energy, where D0(E) =\n1028(E=10 GeV)0:5cm2s\u00001. The di\u000berent colors refer to di\u000berent ages in yr (as marked). The left panels refer to a location at 50 pc\nfrom the center of the SNR, while the right panels refer to the shock position. In this case also the spectrum of CRs still con\fned in the\naccelerator is shown (dashed lines).\nSNRs embedded in a WNM and WIM, and the CR self con-\n\fnement in the source proximity. Our main objective was\nto determine whether, in realistic situations, the grammage\naccumulated by CRs in the source region could become com-\nparable to that inferred from observations. Ifthiswasthe\ncase, In this case, the standard picture in which CR secon-\ndaries are produced during the whole time spent by cosmic\nrays throughout the disk of the Galaxy, should be profoundly\nrevisited.\nWe con\frm the results found by Nava et al. (2016, 2019)\nthat CRs escaping from SNRs drive the excitation of Alfv\u0013 en\nwaves through the resonant CR streaming instability, which\nresults in a suppression of the di\u000busion coe\u000ecient and in the\nCR self-con\fnement in the source region. The SNR radius\nat which CRs of a given energy leave the source, Resc(E),\nresults to be a decreasing function of the energy, and regu-\nlates the CR streaming instability through a dilution factor\n/1=R2\nesc.\nWe found that the growth of self-generated Alfv\u0013 en\nwaves, and consequently the residence time of CRs in the\nsource region, is signi\fcantly limited by several damping\nprocesses, especially by the FG and ion-neutral damping.In particular, for the ion-neutral damping of Alfv\u0013 en waves\nwe have used up-to-date damping coe\u000ecients, based on ac-\ncurate experimental or theoretical determinations of the mo-\nmentum transfer and charge exchange cross sections between\nions and neutrals of di\u000berent species.\nIn the 1-D geometry adopted in our calculation a sup-\npression of the di\u000busion coe\u000ecient is found within a distance\nform the source of the order of the magnetic \feld coherence\nlengthLc\u001950\u0000100 pc, for a timescale that can be as large\nas\u0018105yr at\u001810 GeV. A smaller value for Lcwould make\nthe CR propagation to become 3-D closer to the remnant,\nthus reducing the residence time compared to our results.\nWe conclude that ion-neutral damping strongly limits\nthe CR grammage that can be accumulated in the source\nregion. Under the conditions typically met in the WIM and\nWNM of the Galactic disk, the CR source grammage is\nfound to be negligible compared to that inferred form ob-\nservation. A similar result was found for the HIM by Nava\net al. (2019), where the CR residence time is typically larger\nthan in a partially ionized medium but the ISM density is\nmore than a factor ten smaller. This makes alternative sce-\nnarios for the interpretation of quantities such as the B/C\nMNRAS 000, 000{000 (0000)11\nratio, in which an important contribution to the production\nof secondaries comes from the source region, less attractive.\nOur results on the residence time (and grammage) may\nchange with the inclusion of the contribution of other sources\nof turbulence to the CR con\fnement. In particular, it has\nrecently been suggested that another another CR-induced\ninstability, the so called non-resonant streaming instability ,\nwhich is mediated by the CR current and plays a crucial role\nin the acceleration of CRs (Bell 2004), may signi\fcantly en-\nhance the CR self-con\fnement in the source region above\n\u0018TeV energies (Schroer et al. 2020). Further investigations\nare thus needed in order to \frmly establish the importance\namount of CR grammage accumulated in the vicinity of\nsources by very-high energy particles.\nREFERENCES\nAharonian F. A., Atoyan A. 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B, 22,\n3037\nSkilling J., 1975a, MNRAS, 172, 557\nSkilling J., 1975b, MNRAS, 173, 255\nSquire J., Hopkins P. F., Quataert E., Kempski P., 2021, MNRAS,\n502, 2630\nStancil P. C., et al., 1998, J. Phys. B, 31, 3647\nStrong A. W., Moskalenko I. V., Ptuskin V. S., 2007, Ann. Rev.\nNucl. Part. Sci., 57, 285{327\nTruelove J. K., McKee C. F., 1999, ApJS, 120, 299\nWentzel D. G., 1974, ARA&A, 12, 71\nWiedenbeck M. E., et al., 2007, Space Sci. Rev., 130, 415\nWiener J., Zweibel E. G., Oh S. P., 2013, ApJ, 767, 87\nXu S., Lazarian A., Yan H., 2015, ApJ, 810, 44\nXu S., Yan H., Lazarian A., 2016, ApJ, 826, 166\nYan H., Lazarian A., 2004, ApJ, 614, 757\nZweibel E. G., Shull J. M., 1982, ApJ, 259, 859\nAPPENDIX A: COLLISIONAL COEFFICIENTS\nA1 Collisions of H+with H and He atoms\nIn the WNM and the WIM the relevant damping processes\nfor Alfv\u0013 en waves are collisions of H+with either H or He\natoms. The case of H+{ H collisions is special because the\nproton elastically scattered by the H atom is indistinguish-\nable from the recoiling proton produced by charge exchange\n(Krsti\u0013 c & Schultz 1999; Glassgold et al. 2005; Schultz et al.\n2008, 2016). Therefore, the elastic scattering and the charge\nexchange channels for collisions of H+with H, as for any col-\nlision between ions and their parent gas, cannot in general\nbe separated. Only at collision energies Ecm&1 eV can the\nMNRAS 000, 000{000 (0000)12 S. Recchia et al.\nforward elastic and the backward charge exchange be ap-\nproximately separated. In this limit, \u001bmt\u00192\u001bce(Dalgarno\n1958), a frequently used approximation (see e.g. Kulsrud &\nCesarsky 1971).\nFig. A1 shows the m.t. cross section for H+{ H colli-\nsions (Krsti\u0013 c & Schultz 1999; Glassgold et al. 2005), and\nthe corresponding rate coe\u000ecient. The \fgure also shows the\nm.t. cross sections and rate coe\u000ecients derived from either\ncharge exchange and elastic scattering in the distinguishable\nparticle approach (Schultz et al. 2008, 2016), and the exper-\nimental determination of the m. t. cross section obtained\nby Brennan & Morrow (1971) by measuring the attenuation\nof Alfv\u0013 en waves propagating in a partially-ionized hydrogen\nplasma atEcm\u00195 eV. Also shown in Fig. A1 are the rate co-\ne\u000ecients used by Kulsrud & Cesarsky (1971) and Zweibel &\nShull (1982), and frequently adopted also for other species\nby scaling the collision rate with the ratio mn=miof the\nneutral and ion masses (see e.g. (Zweibel & Shull 1982)). As\nshown in the following, this approximation is not generally\ncorrect, since the rate coe\u000ecients in the case of other species\nexhibit a di\u000berent dependence on the temperature compared\nto the H+{ H collisions case.\nFig. A2 shows the m.t. (Krsti\u0013 c & Schultz 1999, 2006)\nand c.e. (Loreau et al. 2018) cross sections for H+{ He col-\nlisions, and the corresponding rate coe\u000ecients. Charge ex-\nchange contributes to the transfer of momentum above col-\nlision energies\u0018103eV, corresponding to relative velocities\nof\u0018500 km s\u00001, much larger than typical thermal or Alfv\u0013 en\nspeeds in the WNM and WIM.\nA2 Collisions of C+with H and He atoms\nIn the CNM and DiM, ion-neutral damping is dominated by\nthe collisions between neutral hydrogen and ionized carbon.\nFig. A3 shows the cross sections and reaction rates for m.t.\nand c.e. in the case of collisions of C+ions with H atoms.\nFor collisions of C+ions with He atoms, no theoretical or\nexperimental are available. The m.t. transfer rate coe\u000ecient,\naccording to the Langevin theory, is h\u001bmtviC+;He= 1:33\u0002\n10\u00009cm3s\u00001(Pinto & Galli 2008). As in the case of H+{\nHe collisions, the large di\u000berence with the rate coe\u000ecients\nused in this work resides in the assumption that the rate\ncoe\u000ecient is the same for all species.\nFigure A1. Top panel: cross sections for collisions of H+with H\nvs. collision energy in the center-of-mass frame Ecm. Momentum\ntransfer cross section, from Krsti\u0013 c & Schultz (1999), Glassgold\net al. (2005), and Schultz et al. (2008); charge exchange cross\nsection, from Hodges & Breig (1991), and Schultz et al. (2008);\ndashed line: Langevin m.t. cross section. Experimental data for\nthe c.e. cross section: Newman et al. (1982) ( \flled triangles ). The\nempty circle shows the value of the m.t. cross section measured\nby Brennan & Morrow (1971) Bottom panel: Collisional rate co-\ne\u000ecient for momentum transfer and charge exchange. The \flled\nand empty circles show the rate coe\u000ecient adopted by Kulsrud\n& Cesarsky (1971) and Zweibel & Shull (1982), respectively.\nMNRAS 000, 000{000 (0000)13\nFigure A2. Top panel: cross sections for collisions of H+with\nHe vs. collision energy in the center-of-mass frame Ecm. Momen-\ntum transfer cross section, from Krsti\u0013 c & Schultz (1999, 2006);\nLangevin m.t. cross section ( dashed line ); charge exchange cross\nsection, from Loreau et al. (2018); and ionization cross section,\nfrom Shah & Gilbody (1985). Experimental data for the c.e. cross\nsection: Martin et al. (1981) ( empty triangles ), Shah & Gilbody\n(1985) ( \flled triangles ), Shah et al. (1989) ( empty squares ), and\nKusakabe et al. (2011) ( \flled squares ). Experimental data for\nthe ionization cross section: Shah & Gilbody (1985) ( empty cir-\ncles).Bottom panel: Collisional rate coe\u000ecient for Langevin m.t.\n(dashed line ), m.t. and c.e. ( solid lines ).\nFigure A3. Top panel: cross sections for collisions of C+with H\nvs. collision energy in the center-of-mass frame Ecm: m.t. cross\nsection computed by Flower & Pineau des Forets (1995) ( \flled\ntriangles ); \ft, from Pinto & Galli (2008) ( solid line ); Langevin\nm.t. cross section ( dashed line ). Experimental data for the c.e.\ncross section: Phaneuf et al. (1978) ( empty circles ), Go\u000be et al.\n(1979) ( empty squares ) and Stancil et al. (1998) ( \flled triangles );\nc.e. cross section recommended by Stancil et al. (1998) ( solid line ).\nBottom panel: Collisional rate coe\u000ecients: Langevin m.t. ( dashed\nline), m.t. and c.e.( solid lines ).\nMNRAS 000, 000{000 (0000)"    },    {        "title": "1201.4314v1.New_development_in_theory_of_Laguerre_polynomials.pdf",        "content": "                                                                                                                                                \nNew  development in th eory of Laguerre polyn omials  \nI. I. Guseino v \nDepartment of Physics, Faculty of Arts and Sciences,  Onsekiz Mart University, Çanakkale, Turkey   \nAbstract \n    The new com plete orthonorm al sets of αL-Laguerre type polynom ials (αL-LTP, \n2,1,0,1,2,... α=− − ) are suggested. Using Schrödinger equa tion for com plete orthonorm al sets \nof αψ-exponential type orbitals (αψ-ETO) introduced by the aut hor, it is shown that the \norigin  of these polyno mials is the  centrally s ymmetric p otential which conta ins the co re \nattraction potential and the quantum  frictional po tential of  the field pr oduced by the particle \nitself. The quantum  frictional f orces are the ana log of  radiation dam ping or f riction al forces \nsuggested by Lorentz in classical electrodynam ics. The new αL-LTP are com plete withou t \nthe inclusion of the continuum  states of hydrogen like atom s. It is shown that the nonstandard \nand standard conventions of αL-LTP and their weight functi ons are th e sam e.  As an  \napplic ation,  the s ets of infinite expansion form ulas in term s of αL-LTP and L-Generalized \nLaguerre polynom ials (L-GLP) for atom ic nuclear at traction integrals of Slater type orbitals \n(STO) and Coulom b-Yukawa like correlated inte raction p otentials (C IP) with integer and  \nnoninteger indices are obtained. The arrange and rearranged power  series of a general power \nfunction are also investigated. The convergence of these series is tested by calculating \nconcrete cases for arbitrary values of pa rameters of orbitals and power function.  \nKey Words: Centrally  symm etric potentials,  αψ- exponential type orbitals, Frictional \nquantum  numbers, Generalized  Laguerre polynom ials  \n1. Introduction \n    It is well known that the ei genfunctions of the Schrödinger equation f or the hydrogen-like \natom s the radial parts of which contain the L- GLP are not com plete unless the continuum  is \nincluded [1]. Because o f this, som e difficulties  arise in th e solution o f different physical \nproblem s when the L -GLP are employed (see Re f. [2] and references quoted therein). \nTherefore, the necessity  arises for the constru ction of the new αL-LTP using com plete \northonorm al sets of radial parts of αψ-ETO [3]. As  has been show n in a previous paper [4], \nthe eigenfunctions αψ-ETO correspo nd to the  total centrally symmetric poten tial which \ncontains the core attraction poten tial and the Lorentz frictional potential of the field produced \n1 \n                                                                                                                                                 \nby the p article its elf. We notice that the L ambda and Coulomb-Sturm ian ETO introduced in \nRefs. [5-8] are the sp ecial cases of th e αψ-ETO for 0α= and 1α=, respectively. \n2. Definition and basic formulas  \n    The new αL-LTP in non-standard conventions suggested in this work are defined as      \n()1\n3() !(1) ()(2)(!)n\nlp lk\nnl q nk\nklqpx xL x xnqαα\nα−\n=⎡⎤−=− =Π⎢⎥⎣⎦∑ \u0000Lα,                                                                    (1)    \nwhere \n()()\n()()\n()()12\n12\n2klpkl l\nnkqp F q\na\nkl nqα\nα−+−⎡⎤−⎢⎥ Π=−⎢⎥ −⎣⎦!\n! !\n \n()()()\n()02\n00mn\n2for m n b\nmn m Fn\nform andm n c⎧≤≤ ⎪− =⎨\n⎪<>⎩!\n!!\n.\nHere, α is the frictional quantum  num ber ( )2α−∞<≤  (see Ref. [4]), 22pl α =+−, \n1 qn lα =++−  and  are the non-standard L-GLP. The ()p\nqLxαL-LTP are orthonorm al with \nrespect to the weight function 2n\nxα⎛⎞\n⎜⎟⎝⎠: \n()()2\n0x\nnl nl nnex x xdxααδ∞\n−\n′= ∫LL′                                                                                                  (3) \n() ()2\nnl nlnx xxα\nα⎛⎞=⎜⎟⎝⎠.\u0000L Lα\n\u0000                                                                                                         (4) \n    Now we  take into account Eq. (1) in the for mulae for radial parts of αψ-ETO in \nnonstardard conventions [3] which are the comple te orthonorm al sets of eigenfunctions of  \nSchrödinger equation for hydrogen-like atom s. Then, we  obtain:                                                                        \n()()()3\n22nl nlR r Rαζζ= , xα                                                                                                      (5a) \n()()()3\n22nl nlR r Rαζζ= , xα ,                                                                                                    (5b) \nwhere 2x rζ=  and  \n()()2x\nnl nlRxe xα−=Lα                                                                                                                 (6a)                        \n2 \n                                                                                                                                                 \n()()2x\nnl nlRxe xα−=Lα .                                                                                                              (6b) \n     We note that th e sim ilar form ulas can al so be derived in standa rd conventions using the \nrelation  \n()()()()()() 1 1p p pp\nqq pLx qL x qL xα\n−=− =−()! !qp−,                                                                         (7) \nwhere ()()p\nqpL x−are the standard L-GLP. The  non-standard  and standard conventions for the  \nL-GLP were discussed in Refs. [9] and [10], re specti vely (see Ref. [11 ] for t he defi nition of \nαψ-ETO in standard conventions). Taking into account Eqs. (1) and (7) we obtain for the \nαL-LTP in standard conventi ons the following relations: \n()()()()() 1pl\nnl nl qpx qN xL xα α α\n−=− !\u0000L                                                                                             \n(8a) \n()()()()2\nnl nlnx xxα\nα⎛⎞=⎜⎟⎝⎠.\u0000L Lα\n\u0000                                                                                                    (8b) \nIt is easy to show  that th e ()α\n\u0000L are o rthogonal with respect  to the weight function 2n\nxα⎛⎞\n⎜⎟⎝⎠: \n()()()()2\n0x\nnl n'l nn ex x xdxααδ∞\n−\n′= ∫ \u0000\u0000LL .                                                                                            (9) \n    The radial parts of complete or tonorm al sets of  ()αψ -ETO in standard conventions through \nthe ()α\n\u0000L  are defined as  \n ()()()()()3\n22nl nlR r Rαζζ= , xα                                                                                                \n(10a) \n()()()()()3\n22nl nlR r Rαζζ= , xα ,                                                                                              (10b)   \nwhere   \n()()()()2x\nnl nlR xe xα−=Lα                                                                                                           (11a) \n()()()()2x\nnl nlR xe xα−=Lα .                                                                                                        (11b) \n    It is easy to show that the nonstandard αL and standard ()αL  and their weight functions \nα\n\u0000L and ()α\n\u0000L , respectively, are th e sam e, i.e., ()α α=\u0000LL  and ()α α=\u0000LL .   \n3 \n                                                                                                                                                 \n    The nonstandard αL- and standard ()αL -LTP for m the com plete orthonorm al sets on the \ninterval  with the w eight functions  (0,∞)αL and ()αL , respectiv ely. T he com pleteness \nrelations can be proved by the use of m ethod s et out in Ref. [10] (see also [12]). It is easy to \nshow that  \n()()() ()()()1\n2\n11xx\nnl nl nl nl\nnl nlex x RxRxαα α αδ∞∞′−+\n=+ =+′ ′ = = ∑∑ \u0000\u0000 \u0000 LL xx′−                                               (12)   \n()()()()()()()()()()1\n2\n11xx\nnl nl nl nl\nnl nlex x R xR xαα α αδ\u0000\u0000 \u0000 LL∞∞′−+\n=+ =+′ ′ = = ∑∑ xx′− .                                    (13) \n     It shou ld be noted that, in the special cases of the αL- and ()αL -LTP for 0α= and \n1α=, the Eqs. (12) and (13) de scribe the com pleteness propert ies of Lam bda and Coulomb-\nSturm ian functions with nonstandard and standard conve ntions, respectively (see Refs.[5-8]).       \n3. Differential equation of αL-LTP \n     In order to derive the differential equation for αL-LTP we use the Schrödinger equation \nfor the rad ial parts ()nlRxαin the  form \n()()()()22\n22 21 1122nl nlll ddx VxRx Rxxdx dx xα αζεζ+ ⎡⎛⎞−+ + = ⎜⎟⎢⎝⎠⎣⎦⎤\n⎥,                                          (14) \nwhere 22εζ=−  (see Ref. [4 ]). Here, (),V rζ  denotes the p otential of  the centrally \nsymmetric f ield which corres ponds to the eigenfunctions αψ-ETO . The substitu tion of Eq. \n(6a) in to (14 ) yields the f ollowing eq uation f or the αL-LTP: \n() ()()()2\n2 2131 12nl nl nl\nnlll dd d xxx Vxdx dx dx xαα α\nαααζ+⎛+−−−− −++ = ⎜⎝⎠LL LL 0⎞\n⎟ .                   (15) \nNext we use the form ula (1) in the equation  \n() ()2\n21pp\nqq p\nqdL dLxp x qpLdx dx++−+−=0                                                                             (16) \nfor non-standard L-GLP [13]  and the condition \n kp\nq pk\nq kdLLdx+= .                                                                                                                         (17) \nThen, a simple alg ebra leads to  \n() ()()()2\n21131nl nl\nnlll l ddxx ndx dx x xαα\nααα+− ⎛+−−−−++ =⎜⎝⎠LLL 0⎞\n⎟.                                    (18)  \n4 \n                                                                                                                                                 \nThus, w e obtained f or the αL-LTP two kinds of i ndependent equations one of which, Eq. \n(15), contains the potential ()(),nl Vx V rαζ≡ . The comparison of these equations gives  \n()()(nl n nlVr U rUαζ ζ=+ , , )rαζ,,                                                                                          (19)         \nwhere the first and secon d term s are the core attraction and f rictiona l pote ntials, re spectively, \n()22,nUr nxζζ=−                                                                                                                (20) \n()()()() ()\n()()() ()2\n2\n121\n21\n21\n21nl\nnl nl\npp\nqqdx lUr x adx x x\nLx Lx bxα\nααζα\nζ\nζα+⎛⎞−\n⎜⎟ =− −⎜⎟⎝⎠\n−\n=−,\n,L\nL\n       \nwhere 2. x rζ=                                            \nHere, the function nlUα is the Lorentz f riction al self-poten tial of the field  produced b y the \nparticle a t the point w here it is loca ted. See R ef. [4] for the description of properties of these \npotentials.  \n    Thus, we have established a large num ber ( 21012 α=−− ,,,, ,...) of independent com plete \northonorm al sets of relations for αL-LTP the o rigin of  which is the core attra ction and \nfriction al potentials of  the field produ ced by the p article its elf.  \n4. Use of αL-LTP  and L- GLP in study of pow er series of a general pow er function  \n    In this sec tion w e cons ider the ar range and rearranged power series of a function  \n()1 rnfr re reµµ r ξηξξ∗ ∗ ∗−+−== ,−,                                                                                  (22) \nwhere  is th e integer p art of n *1µ− and 0* 1η<<. For this purpose we utilize the following \nLaguerre series obtaine d with the help of αL-LTP and L-GLP: \nfor αL-LTP \n()()\n1rre A rηξ αν α\nµν ηµ\nµνξ∗\n∗∞\n−\n=+=∑ L ,                                                                                            (23) \nfor L- GLP  \n()()rre B Lrηξ ν ν\nµ ηµ\nµνξ∗\n∗∞\n−\n==∑ ,                                                                                                   (24)    \nwhere 0ξ≤<∞  , 2,1,0,1,2,..., 0,1,2,... α ν =−− = and \n5 \n                                                                                                                                                 \n()()()\n()1\n33\n2\n1ssss\nAµααν αν\nµ ηµ ηανηα\nξµ\nξ∗ ∗∗−\n−++=Γ−++\n=Π\n+∑                                                                            (25) \n()() ()\n()3 101\n1ssss\nBµν\nν ν\nµ ηµ ηνην µν\nξβ\nµξ∗ ∗∗−\n+++=Γ+++\n=\n+∑-!\n(! ) .                                                                        (26) \nHere, sν\nµβare th e Laguerre coefficients d etermined by \n0()s\ns\nsLx xµν\nν\nµβ−\n==∑ν\nµ                                                                                                                    (27) \n()()()() () 1 28s\nss ssF Fνν\nµνβµ µµ+\n+=− − !.\n \n       Now we obtain the power series. For this purpose, we use the m ethod set out in previous \npapers [3,14]. Then, taking into ac count the p roperties (2b ) and (2 c) in Eqs. (23) and (27) we \nobtain for the arrange, Eqs. (29a) and (30a), and rearranged, Eqs. ( 29b) and (30b), power \nseries the following formulae: \nfor αL-LTP \n() ()11\n11N\nrs\nsNssre A r lim A rµµ\nηξ αν αν αν αν\nµ ηµ ηµ\nµν ν µν νξ∗\n∗ ∗−− ∞\n−\n→∞=+ = =+ ==Π = ∑∑ ∑∑s\nsµξΠ                                              (29a) \n()1\nlim ,N\nNQN rαν µ\nηµ\nµνξ∗−\n→∞==∑ ,                                                                                                        (29b) \nfor L-GLP \n() ()\n00N\nrs\nsNssre B r lim B rµν µν\nηξ ν ν ν ν\nµ ηµ ηµ\nµν µνξβ ξβ∗\n∗ ∗−− ∞\n−\n→∞== ====∑∑ ∑∑s\nsµ                                                    (30a)                         \n()\n0N\nNlim D N rν\nν µ\nηµ\nµξ∗−\n→∞==∑ , ,                                                                                                     (30b)                         \nwhere \n() ()\n1s sN\nsQN Aαν αν αν\nµ ηµ η\nνξ ξ∗ ∗\n=+Π =∑,                                                                                                 (31) \n() () s sN\nsDN Bν ν ν\nµ ηµ ηνξ βξ∗ ∗\n==∑, .                                                                                              (32) \n     As an exam ple of application of Eqs. ( 29a), (29b), (30a) and (30b) we calculate the atom ic \nnuclea r attraction  integrals of STO  and C oulom b-Yukaw a like  CIP w ith in teger and  \nnoninteger indices defin ed by [15] \n6 \n                                                                                                                                                 \n() ()()()3 q q\npp p pI rr f r ζζξχζχζ ξ∗ ∗\n∗∗ ∗ ∗∗\n′ ′′ ′ =∫GG G,, , , ,drG\n≡ q,                                                           (33)             \nwher e , , pn lm∗∗≡ pn lm∗∗′′′′µνσ∗∗≡\n), and \n() ()( ,,lm nlm nrR rS χζ ζθϕ ∗∗=G                                                                                            (34)  \n()()()1212 122 1 ,n n\nnrR r n reζζζ∗ ∗\n∗−+ ∗ −⎡⎤ =Γ +⎣⎦−                                                                      (35) \n() ()(1\n24\n21fr f rSµν�� µ\nνσπ) ξ ξθν∗ ∗ ⎛⎞=⎜⎟+⎝⎠G, ,ϕ,                                                                        (36) \n()1 rfr reµ µξξ∗ ∗−−= , .                                                                                                              (37) \nIt is easy to derive for integral (33 ) the relation \n() ()()q\nmm pp nnIC lmlmAIνσ σµζζξ ζζξ∗ ∗\n∗∗ ∗∗′ ′′′ ′ = ,, , ,,′′,                                                                 (38) \nwhere () , Cl mlmνσ′′ are the generalized Gaunt coefficients [16] and  \n() ()()()2\n0nn n nI Rr R rf rrdµζζξ ζζ ξ∗ ∗\n∗∗ ∗ ∗∞\n′ ′′ ′ =∫,, , , , rµ.                                                           (39) \nThese integrals are determ ined from  the followin g analytical relation :  \n() ()()\n()11\nnn nn NN\nINµ\nµµ\nζζξ ζζ\nεξ∗\n∗∗ ∗∗ ∗∗∗∗\n++Γ+ +\n=\n+'',' , ,' ,                                                                      (40) \nwhere ',* *'*1 Nn n εζζ=+ =+− and  \n()()()\n()()12 12\n1\n222\n21 2 1nn\nnnN\nnnζζ\nζζ∗∗\n∗∗++\n∗∗=\n⎡⎤ ′ Γ+ Γ+⎣⎦/' /\n''\n,' .                                                                          (41) \n    Now we evaluate the integral (39) using relations (29a), (29b), (30a) and (30b). Then, it is \neasy to derive for the atom ic nuclear attractio n integrals th e followin g large nu mber of  \narrange, Eqs. (42a) and (43a), and rearranged,  Eqs. (42b) and (43b), power series expansion \nrelations : \n for αL-LTP \n \n() () () () ()()\n()() (11\n11142\n42nn ss Nss\nN\nn\nnn NNns ns\nnn nn\n)I AJ lim AJ\nlim Q N J bµµαν ανµ\nµµ ηη\nνν\nαν µ\nηµ\nµναν αν\nµµµν µνζζξ\nξζ ζξζ ζ ξ ζζ∗\n∗∗\n∗∗ ∗−−\n′→∞==\n−\n+\n′→∞=∗∗ ∗ ∗ ∗∗∞++\n′′=+ =+′==\n′ =′′ ∑∑ΠΠ\n∑∑∑,,\n,, ,,, a\n  \n7 \n                                                                                                                                                 \n \n for L-GLP \n() () () () ()()\n() (\n00043\n43nn N\nN\nn\nnn N)Nns ns\nnn nn ssssI BJ lim BJ\nlim D N J bµ\nν\nνµ\nηµ\nµaµνµ ννννν\nµµ µµµν µνζζξη η\nξζ ζξζ ζ ξ ζζ ββ∗\n∗∗\n∗∗ ∗′→∞\n−\n+\n′→∞=∗∗ ∗ ∗ ∗∗−− ∞++\n′′== ==′==\n′ =′′\n∑∑∑ ∑∑,,\n(, ) , ,,,\n \nwhere 2α−∞<≤ and 0ν≤<∞. \nHere, the qu antities ()nnJκζζ∗∗′′, are determ ined by \n() ()() ()()2\n2\n02\nnn n n nn NN\nJR rR rrdrNκκ\nκκ\nζζ ζζ ζζ\nε∗∗ ∗ ∗ ∗∗ ∗∗∞\n+\n′′ ++Γ++\n′′==∫ ',, , ,'  ,                               (44) \nwhere and nsκ=+ nκµ=+.   \n5. Numerical results and discussion  \nThe applicability of the arrange an d rearra nged power series obtained from  the use of \ncomplete orthonorm al sets of αL-LTP and L-GLP is  tested by ca lculating the atom ic nuclear \nattraction integrals determ ined by Eqs. (42a), (42b) and (43a), (43b), respectively. On the  \nbasis of these for mulae we constructed the program s which a re perform ed in the Mathem atica \n7.0 language packages. The converg ence properties of Coulomb (for 0ξ=) and Yukawa (for \n51ξ=.) like nuclear attraction integrals for 0ν= and 2 2α−≤≤ are shown in Figures 1 \nand 2, respectively.  \nThe Figures 1 and 2 show a good rate agreem ent of values obtained from the arrange and \nrearranged series expansion relations. Thus, the Eqs. (42a), (42b), (43a) and (43b) display the \nmost rapid convergence as a func tion of summation lim its for 40 N= . We notice that the  \ngreater accu racy is atta inable by the  use of  more terms in series expansion relations obtained \nin this w ork. \n6. Conclusion  \nWe have demonstrated that th e determ ination of arrange and rearranged  power series  of  a \nfunction () f rµξ∗, obtained by the use of αL-LTP and L-G LP is legitim ate for the integer a nd \nnoninteger values of indices *µ.  As we see from our tests th at the power series derived in \nthis w ork with the he lp of αL-LTP and L-GLP can be useful  tool for evalua tion o f the \nmulticen ter nuclea r attraction in tegrals w hen arrange and rearrange d one-range addition \ntheorem s for STO and Coulom b-Yukawa like CIP presented in our previous papers are \nemployed. \n8 \n                                                                                                                                                 \n \nReferences  \n1. I. N. Levine,  Quantum  Chem istry, 5th ed. (Prentice Hall, New Jersey, 2000). \n2. E. J. W eniger, J. Math. C hem., 50 (2012) 17. \n3. I. I. Guseinov, Int. J. Quantum  Chem., 90 (2002) 114. \n4. I. I. Guse inov, 6th International Co nference of the Balkan Physics Union, Am erican \nInstitute of Physics Conferen ce Proceedings, 899 (2007) 65. \n5. E.A. Hylleraas, Z. Phys., 48 (1928) 469. \n6.  E.A. Hylleraas, Z. Phys., 54 (1929) 347. \n7. H. Shull, P.O. Lowdin, J. Che m. Phys., 23 (1955) 1362. \n8. P.O. Lowdi n, H. Shull,  J. Che m. Phys., 101 (1956) 1730. \n9. P. Kaijser, V.H. Sm ith, Adv. Quantum  Chem., 10 (1977) 37. \n10. W. Magnus, F. Oberhetting er, R.P.  Soni, Formulas and Theorem s for the Special \nFunctions of  Mathem atical Physics (Springer, New York, 1966). \n11. I. I. Guseinov, J. Math. Che m., 43 (2008) 427. \n12. R. Sz mytkowski, J. Phys. B, 30 (1997) 825, Appendix E. \n13. J. C. Slater,  Quantum  theory of  atom ic structur e, Vol. 2, Mc G raw-Hill, N ew York, \n1960. \n14. I.I Guseinov, Phys. Rev. A, 31 (1985) 2851. \n15. I. I. Guseinov, J. Theor.  Com put. Chem ., 7 (2008) 257. \n16. I. I. Guseinov, J. Phys. B, 3 (1970) 1399.\n9 \n                                                                                                                                                 \næ æ\næààààààààààì\nì\nì\nì\nì\nì\nì\nìììììììì ììììììòòòòòòòòòòòòòòòòòòòòò\nôôôôôôôôôôôôôô ôô ôô ôô ô\nççççç çççççççççççççççç\n20 25 30 35 4093190932009321093220932309324093250\nNa0I2.3,41.1H3.56,4.65,0Lx10-5\nça=-2ôa=-1òa=0ìa=1àa=2æ analytical\nFig.1. Convergence of arrange  (42a) and rearranged (42b) power series for Coulomb like \ninteg rals\n ( )01.1\n2.3,43.56,4.65,0 Iα for 22α−≤≤\n \n \næ ææ\nààààààààààààààààààààà\nìììììììììììììììììììììò\nò\nòòòòòòòòòòòòòòòòòòòô\nô\nô\nô\nôôôôôôôôôôôôôôççççççççççççççççççççç\n20 25 30 35 4025002550260026502700\nNa0I2.3,41.1H3.56,4.65,5.1Lx10-5\nça=-2ôa=-1òa=0ìa=1àa=2æanalytical\nFig.2. Convergence of arrange (42a) and rear ranged (42b) power series for Yukawa  like \ninteg rals\n ( )01.1\n2.3,43.56,4.65,5.1 Iα for 22α−≤≤ \n \n10 \n "    },    {        "title": "1206.3984v1.Quantum_Breathing_of_an_Impurity_in_a_One_dimensional_Bath_of_Interacting_Bosons.pdf",        "content": "Quantum Breathing of an Impurity in a One-dimensional Bath of Interacting Bosons\nSebastiano Peotta,1,\u0003Davide Rossini,1Marco Polini,2Francesco Minardi,3, 4and Rosario Fazio1\n1NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, I-56126 Pisa, Italy\n2NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, I-56126 Pisa, Italy\n3LENS-European Laboratory for Non-Linear Spectroscopy and Dipartimento di Fisica,\nUniversit\u0012 a di Firenze, via N. Carrara 1, IT-50019 Sesto Fiorentino-Firenze, Italy\n4CNR-INO, via G. Sansone 1, IT-50019 Sesto Fiorentino-Firenze, Italy\nBy means of time-dependent density-matrix renormalization-group (TDMRG) we are able to\nfollow the real-time dynamics of a single impurity embedded in a one-dimensional bath of interacting\nbosons. We focus on the impurity breathing mode, which is found to be well-described by a single\noscillation frequency and a damping rate. If the impurity is very weakly coupled to the bath, a\nLuttinger-liquid description is valid and the impurity su\u000bers an Abraham-Lorentz radiation-reaction\nfriction. For a large portion of the explored parameter space, the TDMRG results fall well beyond\nthe Luttinger-liquid paradigm.\nPACS numbers: 71.38.-k, 05.60.Gg, 67.85.-d\nI. INTRODUCTION\nThe dynamics of impurities jiggling in classical and\nquantum liquids has tantalized many generations of\nphysicists since the early studies on Brownian motion.\nIn particular, the interaction of a quantum system with\nan external environment strongly a\u000bects its dynamics1{3.\nBecause of this coupling, the motion of a quantum parti-\ncle is characterized by a renormalized mass, decoherence,\nand damping. Polarons4, originally studied in the con-\ntext of slow-moving electrons in ionic crystals, and impu-\nrities in3He5are two prototypical examples in which the\nbath is bosonic and fermionic, respectively. These prob-\nlems have been at the center of great interest for many\ndecades in condensed matter physics.\nRecent advances in the \feld of cold atomic gases6have\nmade it possible to observe and study these phenomena\nfrom a di\u000berent perspective and hence to disclose new\naspects not addressed so far. It is indeed possible to\naccurately tune the coupling between a quantum particle\nand the bath and to modify the many-body nature of the\nbath itself. Furthermore, the dynamics of the dressed\nparticle can be studied in real time, thus giving direct\naccess to both mass renormalization and damping. This\nproblem becomes of particular relevance if the bath is\none-dimensional (1D). In this case interactions strongly\na\u000bect the excitation spectrum of the bath7and therefore\nthe e\u000bective dynamics of the coupled system.\nThe dynamics of impurities in cold atomic gases has\nattracted a great deal of experimental8and theoretical9\nattention in recent years. In particular, Catani et al.10\nhave recently studied experimentally the dynamics of K\natoms (the \\impurities\") coupled to a bath of Rb atoms\n(the \\environment\") con\fned in 1D \\atomic wires\". Mo-\ntivated by Ref. 10 we perform a time-dependent density-\nmatrix renormalization group (TDMRG)11study of the\ndynamics of the breathing mode in a 1D bath of inter-\nacting bosons.\nAt a \frst sight one might think that the problem un-\nder consideration reduces to the study of a particle cou-\n0 1 2 3 4 5 6\nt/(T/2)0.00.20.40.60.81.0σ(t)//lscripthou12= +0.6\nu12=−0.6FIG. 1. (Color online) The width of the impurity breathing\nmode\u001b(t), in units of `ho= (J2=V2)1=4\u000e(\u000eis the lattice\nspacing), is plotted as a function of time t(in units of T=2 =\n\u0019=! 2). Data labeled by empty symbols represent TDMRG\nresults corresponding to two opposite values of the impurity-\nbath coupling constant u12. The strength of interactions in\nthe bath has been \fxed to u1= 1. The data for u12=\u00000:6\n(squares) are shifted downward by 0 :37`ho. The black solid\nlines are \fts to the TDMRG data based on Eq. (3).\npled to a Luttinger liquid12. As we will discuss in the\nremainder of this work, it turns out that the nonequilib-\nrium impurity dynamics eludes this type of description.\nThis is the reason why we choose to tackle the problem\nwith an essentially exact numerical method. Several fea-\ntures of the experimental data in Ref. 10 are also seen\nin our simulations. As we will discuss in the conclusions,\nhowever, a detailed quantitative account of the data in\nRef. 10 may require additional ingredients and is outside\nthe scope of the present work. Here, we highlight a num-\nber of distinct signatures of the impact of interactions on\nthe breathing motion of an impurity, which are amenable\nto future experimental testing.arXiv:1206.3984v1  [cond-mat.quant-gas]  18 Jun 20122\nII. MODEL HAMILTONIAN AND IMPURITY\nBREATHING MODE\nWe consider a 1D bath of interacting bosons coupled to\na single impurity con\fned in a harmonic potential. The\nbath is modeled by a Bose-Hubbard Hamiltonian with\nhoppingJ1and on-site repulsion U1>0:\n^HB=\u0000J1X\ni(^by\ni^bi+1+H:c:)+U1X\ni^n2\ni+X\niWi^ni:(1)\nHere ^by\ni(^bi) is a standard bosonic creation (annihilation)\noperator on the i-th site. To avoid spurious e\u000bects due\nto quantum con\fnement along the 1D system, we con-\nsider a nearly-homogeneous bath: the external con\fning\npotentialWiis zero in a large region in the middle of the\nchain (1\u0014i\u0014L) and raises smoothly at the edges (see\nSec. A). The local density h^nii=h^by\ni^biiis thus essentially\nconstant in a region of length \u00182L=3. In (almost all) the\nresults shown below we \fx the number of particles in the\nbath toNbath= 22 and distribute them over L= 250\nsites, thus keeping the average density to a small value,\nh^nii.0:1. For this choice of parameters the lattice is\nirrelevant and the model (1) is ideally suited to describe\na continuum. Indeed, in the low-density limit, Eq. (1)\nreduces to the Lieb-Liniger model13(a mapping between\nthe coupling constants of the two models is summarized\nin Sec. A). The Hamiltonian describing the impurity dy-\nnamics is\n^HI(t) =\u0000J2X\ni\u0000\n^ay\ni^ai+1+ H:c:\u0001\n+V2(t)X\ni\u0016i2^Ni;(2)\nwith ^Ni= ^ay\ni^aithe impurity density operator and \u0016i=\ni\u0000i0. Eq. (2) includes a kinetic term and a harmonic\npotential centered at i0= (L+1)=2, whose strength V2(t)\ndepends on time, mimicking the quench performed in the\nexperimental study of Catani et al.10. Because the on-\nsite impurity density h^Nii.0:15 is low, also in this case\nthe model (2) well describes the corresponding contin-\nuum Hamiltonian (Sec. A). In this work we have \fxed\nJ2=J1= 2 to take into account the mass imbalance be-\ntween the impurity and bath atoms. For future purposes\nwe introduce !(t) = 2p\nV2(t)J2=~, with!1=!(t <0)\nand!2=!(t\u00150),i.e.the harmonic-potential frequen-\ncies before and after the quench, respectively. The full\ntime-dependent Hamiltonian ^H(t) =^HB+^HI(t)+^Hcoupl\ncontains a further density-density coupling between bath\nand impurity ^Hcoupl =U12P\ni^ni^Ni.\nThe quench in !(t) excites the impurity breathing\nmode (BM), i.e. a mode in which the width \u001b(t)\u0011\u0002P\ni\u0016i2h^Ni(t)i\u00031=2, associated with the impurity density\nh^Ni(t)i, oscillates in time14. This quantity is evaluated\nwith the TDMRG.III. NUMERICAL RESULTS\nIn Fig. 1 we illustrate the time evolution of the im-\npurity width \u001b(t) dictated by ^H(t)15. Di\u000berent sets of\ndata refer to two values of the impurity-bath interaction\nu12=U12=J2. Timetis measured in units of T=2, where\nT= 2\u0019=! 2is the period set by the harmonic-con\fnement\nfrequency!(t) after the quench. The TDMRG results\n(empty symbols) have been obtained by setting u1=\nU1=J2= 1 andV2(t)=J2= 10\u00003fort <0 and 10\u00004for\nt\u0015016.\nThe black solid lines are \fts to the TDMRG data based\non the following expression:\n\u001b2(t)\n\u001b2(+1)= 1 +e\u00002\u0000t\ncos2(\u001e)X\ni=x;p\u0001icos2\u0010\ntp\n\n2\u0000\u00002\u0000\u0012i\u0011\n;\n(3)\nwhere\u001e= arccos(p\n1\u0000\u00002=\n2),\u0012x=\u001e,\u0012p=\u0019=2,\n\u0001x= [\u001b(0)=\u001b(+1)]2\u00001, and \u0001p= [\u001b(+1)=\u001b(0)]2\u00001.\nEq. (3) is the prediction for the BM width obtained by\nsolving a quantum Langevin equation for the impurity\nposition operator ^X(t) in the presence of Ohmic damp-\ning and a random Gaussian force with colored spectrum\n(see Sec C):\n@2\nt^X(t) + 2\u0000@t^X(t) + \n2^X(t) =^\u0018(t): (4)\nThe three parameters \u001b(+1), \n, and \u0000 (respectively the\nasymptotic width at long times, the frequency of the\nbreathing oscillations, and the friction coe\u000ecient) have\nbeen used to \ft the data. The initial width \u001b(0) is ex-\ntracted from numerical data for the ground-state width\natt <0. Note that, in the limit in which the impurity-\nbath interaction is switched o\u000b ( u12= 0),\u001b(t) must os-\ncillate at the frequency 2 !2, since only even states (under\nexchangex$\u0000x) of the harmonic-oscillator potential\nare involved in the time evolution of a symmetric mode\n(like the BM). This implies \n !!2in the limit u12!0,\nwhere Eq. (3) reproduces the exact non-interacting dy-\nnamics.\nIn Fig. 2 we plot the values of the frequency \n as a\nfunction of u12and for di\u000berent values of u1. Several fea-\ntures of the data in this \fgure are worth highlighting: i)\nthe behavior of \n is dramatically di\u000berent when the sign\nof interactions is switched from attractive to repulsive,\nexcept at weak coupling, for a tiny region of small u12\nvalues; ii) the behavior becomes more symmetric with\nrespect to the sign of u12as the bath is driven deeper\ninto the Tonks-Girardeau (TG) regime, i.e., foru1!1\n(in passing, we notice that our results in this limit are\nrelevant in the context of the so-called \\Fermi polaron\"\nproblem17); iii) the renormalization of the frequency \n is\nreduced on increasing the strength of repulsive interac-\ntions in the bath u1. It becomes almost independent of\nu12foru1&0:5.\nFig. 3 illustrates the dependence of the damping rate\n\u0000 onu12, for di\u000berent values of u1. Three features of\nthe data are remarkable: i) \u0000 displays a strong asym-\nmetrical behavior with respect to u12= 0 away from3\n−0.6−0.4−0.20.0 0.2 0.4 0.6\nu120.40.60.81.01.2Ω/ω2\nu1\n0.1\n0.3\n0.5\n1.0\n∞\nFIG. 2. (Color online) The oscillation frequency of the breath-\ning mode (in units of !2) as a function of the impurity-bath\nLieb-Liniger parameter u12, for di\u000berent values of the bath\ndimensionless coupling constant u1. All the data in this \fg-\nure have been obtained by setting V2(t)=J2= 10\u00003fort<0\nand 10\u00004fort\u00150. Error bars refer to the \ftting procedure.\nSolid lines are guides to the eye.\nthe weak-coupling limit; ii) \u0000 decreases with increasing\nu1, saturating to a \fnite result in the TG limit; and, \f-\nnally, iii) \u0000 depends quadratically on u12forju12j\u001c1.\nThe non-monotonic behavior of \u0000 on the attractive side\ncan be explained as following. For u12\u00190 the damping\nrate must be small. Increasing ju12jthe damping rate\nincreases because the coupling of the impurity to the\nbath increases. However, upon further increasing ju12j\nanother e\u000bect kicks in. We have indeed discovered (data\nnot shown here) that \u0000 decreases monotonically with de-\ncreasing frequency. As shown in Fig. 2, on the attractive\nside \n decreases rapidly as ju12jincreases, thereby reduc-\ning the damping rate. The non-monotonic behavior of \u0000\ndoes not occur for u12>0 because on the repulsive side\n\n changes slightly with respect to u12.\nThe asymptotic width \u001b(+1), shown in Fig. 4, fairly\nagrees with the equilibrium value at the frequency !2\nthat one can calculate numerically. This \fnding seems\nto suggest that the impurity has nearly \\thermalized\"\nwith the bath over the time scale of our simulations.\nIV. LUTTINGER-LIQUID THEORY AND THE\nABRAHAM-LORENTZ FRICTION\nWe now discuss which features in Figs. 2-4 can (or can-\nnot) be explained by employing a low-energy Luttinger-\n−0.6−0.4−0.20.0 0.2 0.4 0.6\nu120.00.10.20.30.40.50.60.7Γ/ω2\n 0.0 0.02 0.04 0.06\nω2τ0.20.30.40.50.60.70.8Γ/(ω2\n2τ)FIG. 3. (Color online) Same as in Fig. 2, but for the fric-\ntion coe\u000ecient \u0000 (in units of !2). Note that \u0000 vanishes\nquadratically for weak impurity-bath couplings ( ju12j\u001c 1)\nand saturates to a \fnite value in the limit u1!1 . The\ninset illustrates the dependence of \u0000 [in units of !2\n2\u001c] on\n!2(in units of 1 =\u001c). For each value of !2, the tDMRG\ndata (green symbols) have been obtained by performing a\nquench corresponding to a value of !1=p\n10!2. The\nother parameters are: Nbath = 40,L= 600 (h^nii\u00190:07),\nu12= 0:1, andu1= 1. The solid line represents the predic-\ntion \u0000 AL(!2\u001c1=\u001c)=(!2\n2\u001c) = 1=2, based on the Abraham-\nLorentz model with the value of \u001ccorresponding to u1= 1.\nThe dashed line at \u0000 =!2\n2\u001c= 0:44 is the result of a best \ft to\nthe data.\nliquid description of the bath.\nThe Hamiltonian of a single impurity of mass M, de-\nscribed by the pair of conjugate variables ( ^X;^P), coupled\nto a bath of harmonic oscillators (the bosonic excitations\nof the Luttinger liquid) with dispersion !k=vsjkj, is1,10:\n^H(t) =^P2\n2M+V(^X;t) +X\nk6=0~!k^\ry\nk^\rk+g12^\u001a(^X);(5)\nwhereV(x;t) =M!2(t)x2=2 (the identi\fcation with the\nlattice model \fxes M=~2=(2J2\u000e2) with\u000ethe lattice\nspacing) and g12is a coupling constant playing the role\nofu12in the discrete model. In Eq. (5) ^ \ry\nk(^\rk) is the\ncreation (annihilation) operator for an acoustic-phonon\nmode with wave number kand ^\u001a(x) is the bath den-\nsity operator. The sound velocity vsis related to the\nLuttinger parameter Kof the Lieb-Liniger model by\nGalilean invariance18;Kis in turn de\fned by the rela-\ntion\u0014=K=(\u0019~vs), where\u0014is the compressibility of the\nbath7. The parameters Kandvs, which completely char-4\n−0.6−0.4−0.20.0 0.2 0.4 0.6\nu120.500.550.600.650.70σ(+∞)//lscriptho\nu1\n0.1\n0.3\n∞\nFIG. 4. (Color online) Same as in Fig. 2, but for the asymp-\ntotic value \u001b(+1) at long times of the width \u001b(t) (in units\nof`ho). Here the lines are notguides to the eye, but repre-\nsent the equilibrium value \u001beqfor the width in the harmonic\npotential with frequency !2(\u001beq=`ho=p\n2 foru12= 0).\nacterize the Luttinger liquid, can be expressed in terms\nof the coupling constants of the model in Eq. (1) (see\nSec. B). We now observe that the sign of the impurity-\nbath coupling g12can be gauged away from the Hamil-\ntonian (5) by the canonical transformation ^ \rk!\u0000 ^\rk.\nThis means that if the bath was truly a Luttinger liquid,\nimpurity-related observables such as \n and \u0000 should not\ndepend on g12being attractive or repulsive. This low-\nenergy description seems to apply only in a tiny region\naroundu12= 0. All the deviations from this prediction\nseen in Figs. 2-4 have to be attributed to physics beyond\nthe Luttinger-liquid paradigm.\nThe Heisenberg equation of motion induced by the\nHamiltonian (5) reads (see Sec. B):\nM@2\nt^X(t) +M!2\n2^X(t) +MZt\n0dt0^\u0000(t;t0)@t0^X(t0)\n=\u0000g12@x^\u001a(x;t)\f\f\nx=^X(t); (6)\nwhere\n^\u0000(t;t0) =X\nk6=0c2\nkk2\nM!2\nkeik^X(t)e\u0000ik^X(t0)cos [!k(t\u0000t0)];(7)\nwithck=\u0000g12[Kvs=(\u0019~L)]1=2jkje\u0000jkj=2kc, is the memory\nkernel1(kcis an ultraviolet cut-o\u000b).\nIf the dynamics of the impurity is slow with respect to\nthe speedvsof propagation of information in the bath,\nthen \\retardation e\u000bects\" can be neglected and we canapproximate the operator ^\u0000(t;t0) with the following c-\nnumber \u0000(t\u0000t0) =P\nk6=0c2\nkk2cos [!k(t\u0000t0)]=(M!2\nk).\nIn this limit it is possible to show (see Sec. B) that\nEq. (6) reduces to a quantum Langevin equation with an\nAbraham-Lorentz (AL) term, which describes the reactive\ne\u000bects of the emission of radiation from an oscillator19.\nThis is a term of the form \u0000M\u001c@3\nt^X(t) with a \\char-\nacteristic time\" \u001c=g2\n12K=(\u0019M~v4\ns). Remarkably, ne-\nglecting the well-known \\runaway\" solution19and keep-\ning only the damped solutions, we \fnd that the quan-\ntum Langevin equation with the AL term yields an ex-\npression for \u001b(t) after the quench which is identical to\nEq. (3) with \u0000 = \u0000 AL(!2)!2\u001c\u001c1=!2\n2\u001c=2. The full func-\ntional dependence of \u0000 ALon!2is reported in Sec. C.\nNote that \u0000 ALis proportional to g2\n12. This is in agree-\nment with the TDMRG results shown in Fig. 3 in the\nweak-couplingju12j!0 limit. Moreover, \u0000 ALis propor-\ntional toK=v4\ns/K5(in our case vs/K\u00001from Galilean\ninvariance18) and proportional to !2\n2. The former state-\nment implies a fast saturation of the friction coe\u000ecient\nto a \fnite value in the TG limit ( K= 1). This is in\nagreement with the TDMRG data shown in Fig. 3. The\nquadratic dependence of the damping rate on !2is also\nwell displayed by the TDMRG data, at least for !2\u001c\u001c1,\nas shown in the inset to Fig. 3.\nIt is instructive to compare our \fndings with the ex-\nperimental data of Ref. 10. The latter show that, in a\nsizable range of interaction strength u12, the frequency\nof the breathing mode does not vary appreciably while\nthe damping coe\u000ecient increases up to \u0000 \u00180:2!2. As\nshown in Figs. 2 and 3, we do observe the same behav-\nior foru1>0:2. Moreover, as in the experiment, we do\nsee that the width of the breathing mode reduces upon\nincreasingu12. A detailed quantitative comparison with\nthe experiment is, however, not possible at this stage: i)\none notable di\u000berence is that our calculations are car-\nried out at T= 0, while temperature e\u000bects seem to be\nimportant in Ref. 10; ii) furthermore, the impurity trap-\nping frequency in our calculations is considerably larger\nthan in the experiment16. Extending the current calcu-\nlations to take into account these di\u000berences lies beyond\nthe scope of this work.\nIn summary, we have shown that, in the dynamics of\nan impurity coupled to a 1D bosonic bath, a Luttinger-\nliquid description of the bath is applicable only in a very\nsmall region of parameter space, where the impurity suf-\nfers an AL radiation-reaction friction. Among the most\nstriking features we have found, we emphasize the non-\nmonotonic behavior of the damping rate and the large\nrenormalization of the oscillation frequency for attractive\nimpurity-bath interactions.\nAppendix A: Lattice to continuum mapping\nIn our simulations we consider a 1D bath of interact-\ning bosons coupled to a single impurity con\fned in a har-\nmonic potential. The bath is modeled by a Bose-Hubbard5\n0 50 100 150 200 250\ni0.000.020.040.060.080.10Wi/J1\nFIG. 5. External con\fning potential for the bath in a lattice\nwithL= 250 sites, as de\fned by Eq. (A1) (we set \u0001 = 50\nandW=J 1= 0:1).\nHamiltonian ^HB, Eq. (1) in the main text, with hopping\nJ1and on-site repulsion U1>0. The external con\fning\npotentialWihas the following explicit form:\nWi=8\n>>>>>>><\n>>>>>>>:W i = 1;\nW\n2h\n1\u0000tanh\u0010\n3(i\u0000(\u0001+1)=2)\n2p\n(i\u00001)(\u0001\u0000i)\u0011i\n1<i< \u0001;\n0 \u0001 \u0014i\u0014L\u0000\u0001;\nW\n2h\n1 + tanh\u0010\n3(i\u0000(2L\u0000\u0001)=2)\n2p\n(i\u0000L+\u0001)(L\u0000i)\u0011i\nL\u0000\u0001<i<L;\nW i =L:\n(A1)\nThe parameter \u0001 is the length in unit of lattice sites of\nthe left and right boundary regions, where the potential\ngoes from 0 to a \fnite value W > 0. The interpolation\nbetween the two values is as smooth as possible, since the\npotential, as a continuous function of x=i\u000e(\u000ebeing the\nlattice spacing), has zero derivatives of all orders at the\njoining points in Eq. (A1). We set \u0001 = 50 lattice sites,\nwhile we used W=J 1= 0:1 forL= 250 andN= 22 (see\nFig. 5),W=J 1= 0:06 forL= 600 and N= 40, where L\nis the length of the (two) lattices used in our simulations,\nandNthe number of particles in the bath. In the latter\ncaseWis smaller since the on-site density ni\u00190:07 in\nthe middle of the chain is smaller and a weaker potential\nis used.\nWith our choice of parameters, the local density h^nii=\nh^by\ni^biiis essentially constant in a region of length \u00182L=3,\nand kept to a low value h^nii.0:1 everywhere. At such\nlow densities and for not too strong repulsive interaction\n(U1=J1.10), the lattice is irrelevant and the model can\nbe mapped to a Lieb-Liniger Hamiltonian13describing\n1D bosons of mass m=~2=(2J1\u000e2) interacting through\na contact (repulsive) two-body potential. The upper\nbound for the parameter U1=J1can be understood as\nfollows: as the interaction between bosons is increased,\nthe healing length of the Lieb-Liniger gas gets smaller,\nand, when it is comparable with \u000e, lattice e\u000bects be-\ncomes relevant. An extensive discussion of this point can\nbe found in Refs. 20{22. Other relevant parameters of\nthe continuum model are the density n=h^nii=\u000e(where\nh^niiis the on-site density taken in the central region\nwhere the bath is homogeneous) and the dimensionless\nLieb-Liniger parameter \r1=mg1=(~2n) =U1=(2J1h^nii),whereg1=U1\u000e>0 is the strength of the contact repul-\nsion between bosons.\nThe Hamiltonian ^HI(t) describing the impurity is writ-\nten in Eq. (2) in the main text. Given the low impurity\ndensityh^Nii.0:15, also in this case the lattice model\nwell describes the continuum Hamiltonian of a particle\nof massM=~2=(2J2\u000e2) moving in a parabolic potential\nV(x;t) =V2(t)(x=\u000e)2=M!2(t)x2=2 (centered without\nloss of generality at x= 0). The impurity mass is \fxed\ntoM=m=2 (thus corresponding to a ratio between the\ntwo lattice hoppings J2=J1= 2), such to take into ac-\ncount the mass imbalance between Rb and K atoms, as\nexperimentally done in Ref. 10.\nThe total Hamiltonian ^H(t) =^HB+^HI(t)+^Hcoupl con-\ntains a further density-density coupling between bath and\nimpurity, ^Hcoupl =U12P\ni^ni^Ni, which in the continuum\nlimit corresponds to a \u000e-function of strength g12=U12\u000e.\nAppendix B: Quantum Langevin equation for a\nparticle in a Luttinger liquid\nIn this section we derive a quantum Langevin equa-\ntion1,2,23for a single particle, described by the conju-\ngate variables ( ^X;^P), that is coupled to a Luttinger\nliquid by a density-density interaction. The relevant\nHamiltonian10is given by Eq. (5) where the \frst two\nterms describe the impurity Hamiltonian, the potential\nV(^X)\u0011\u0000M!2^X2=2 representing an harmonic con\fning\ntrap of frequency !for the impurity, while the third one\ndenotes the quadratic Luttinger Hamiltonian, in which\n^\ry\nk(^\rk) is the creation (annihilation) operator for an\nacoustic-phonon mode with wave vector kand dispersion\n!k=vsjkj. Apart from the coupling constant g12, the last\nterm, which contains an ultraviolet cut-o\u000b ( kc), de\fnes\nthe bath density operator ^ \u001a(^X) of the Luttinger liquid7,\nand is controlled by the parameter K. This, in turn,\ncontrols the speed of \\sound\" vsby virtue of Galilean\ninvariance: vs=~\u0019n=(mK). Repulsive interactions in\nthe bath enter the problem through the dependence of vs\nandKon the Lieb-Liniger parameter \r113.\nIn Eq. (5) the phonon modes couple linearly to the\nparticle position, the latter entering through the ex-\nponential exp ( ik^X). As a consequence, the harmonic\nexcitations of the Luttinger liquid can be integrated\nout leaving an e\u000bective dissipative equation (quantum\nLangevin equation23) for the impurity degree of freedom.\nWe \frst switch from annihilation and creation opera-\ntors to (complex) position and momentum, by de\fning\n^xk=p\n~=2!k\u0000\n^\rk+ ^\ry\n\u0000k\u0001\nand ^pk=ip\n~!k=2\u0000\n^\ry\nk\u0000^\r\u0000k\u0001\n,\nwith\u0002\n^xk;^pk0\u0003\n=i~\u000ekk0. Notice that, unlike the usual\nposition and momentum, they are complex and their ad-\njoint are ^xy\nk= ^x\u0000kand ^py\nk=\u0000^p\u0000k. Using these new6\nvariables one gets\n^H=^P2\n2M+V(^X) +X\nk>0\u0012\nj^pkj2+!2\nk\f\f\f^xk\u0000ck\n!2\nke\u0000ik^X\f\f\f2\u0013\n+ const.; (B1)\nwhere the coe\u000ecients ck=\u0000g12q\nKvs\n\u0019~Ljkje\u0000jkj=2kchave\nbeen de\fned according to Ref. 1. Usually, after complet-\ning the square, some additional terms depending only on\n^Xare left out and their e\u000bect is to renormalize the po-\ntentialV(^X). Given the peculiar non-linear nature of\nthe coupling/exp(ik^X), this does not happen in our\ncase: the terms completing the square turn out to be\nindependent of ^X, therefore the potential V(^X) is not\nrenormalized.The Heisenberg equations of motion dictated by\nEq. (B1) are given by\nM@2\nt^X\u0000F(^X) =X\nk6=0ikck^xkeik^X; (B2)\n@2\nt^xk+!2\nk^xk=cke\u0000ik^X; (B3)\nwhereF(x)\u0011 \u0000@xV(x). The solution of the second\nequation can be immediately written down:\n^xk(t) = ^xk(0) cos!kt+^p\u0000k(0)\n!ksin!kt\n+ck\n!kZt\n0dt0sin[!k(t\u0000t0)]e\u0000ik^X(t0):(B4)\nIntegrating by parts and substituting into Eq. (B2), we\nget\nM@2\nt^X(t)\u0000F(^X(t))+MZt\n0dt0^\u0000(t;t0)@t0^X(t0) =X\nk6=0ikckeik^X(t)h\u0010\n^xk(0)\u0000ck\n!2\nke\u0000ik^X(0)\u0011\ncos!kt+^p\u0000k(0)\n!ksin!kti\n(B5)\nwith a memory kernel function\n^\u0000(t;t0) =X\nk6=0c2\nkk2\nM!2\nkeik^X(t)e\u0000ik^X(t0)cos!k(t\u0000t0):(B6)\nthat depends separately ontandt0(and not only on\nt\u0000t0), due to the presence of the two non-commuting\noperators ^X(t) and ^X(t0) evaluated at di\u000berent times.\nUpon the substitution ^ xk(0)\u0000cke\u0000ik^X(0)=!2\nk!^xk(0),\nthe right hand side is just \u0000g12@x^\u001a(x;t)jx=^X(t). In or-\nder to see this, simply rewrite ^ xk(0) and ^pk(0) using the\ncorresponding annihilation and creation operators and\ncompare to the density-density coupling in the Hamilto-\nnian (5). As discussed in Ref. 1, one can forget about the\nterm/eik^X(0)in the right hand side of Eq. (B5), since\nit has no e\u000bect when taking averages on the equilibrium\nstate of the bath coupled to the particle at t= 0 (as a con-\nsequence the stochastic term in the quantum Langevin\nequation (B5) has zero average: h@x^\u001a(^X(t);t)i= 0).\nWhen evaluating the noise correlator one runs into dif-\n\fculties since eik^X(t)e\u0000ik^X(t0)6=eik[^X(t)\u0000^X(t0)]. In the\nlimit when the particle has a small velocity with respect\ntovs, we can however perform the following approxima-\ntion:\neik^X(t)e\u0000ik^X(t0)e\u0006ivsjkj(t\u0000t0)\u0019e\u0006ivsjkj(t\u0000t0); (B7)\nsince in this case the product of the \frst two exponentials\nis assumed to be slowly varying with respect to the third\none. Therefore in this limit the noise correlator ^\u0004(t;t0)\u0011g2\n12h[@x^\u001a(^X(t);t)]y@x^\u001a(^X(t0);t0)ireads:\n^\u0004(t;t0)\u0019g2\n12K\n\u0019~v4sZ+1\n0d!\n2\u0019e\u0000!=! c~!3\u0002\n\u0002\u0012\ncoth\f~!\n2cos!(t\u0000t0)\u0000isin!(t\u0000t0)\u0013\n;(B8)\nwhere we passed in the continuum by substituting the\nseries overkwith an integral over the frequencies !. In\nthe same approximation, the memory kernel (B6) enter-\ning the third term in the left hand side of Eq. (B5) reads\n^\u0000(t;t0)\u00192g2\n12K\n\u0019~v4sZ+1\n0d!\n2\u0019e\u0000!=! c!2cos!(t\u0000t0):(B9)\nNotice that, when the cut-o\u000b !cgoes to in\fnity, this\nkernel basically reduces to the second derivative of a delta\nfunction:\n^\u0000(t;t0)!c!+1=\u0000M\u001c\u000e00(t\u0000t0); (B10)\nwhere\u001c=g2\n12K=(\u0019M~v4\ns) is the proper time scale.\nThe relation<\u0002e\u0004(!)\u0003\n=M~!coth(\f~!=2)<\u0002e\u0000(!)\u0003\nstates the \ructuation-dissipation theorem1,2,23, and holds\nbetween the real parts of the Fourier transforms of the\nnoise correlator, e\u0004(!), and of the memory kernel, e\u0000(!).\nThis ensures the consistency of the slow-particle approx-\nimation , see Eq. (B7). Within this approximation, the\nquantum Langevin equation takes the linear form\nM@2\nt^X(t)\u0000F(^X(t))\u0000M\u001c@3\nt^X(t) =\u0000g12@x^\u001a(^X(t);t):\n(B11)7\nThe fact that the position operator appears on the right\nhand side as an argument of the bath density does not\nspoil linearity, since the noise correlator is independent\nof the di\u000berence X(t)\u0000X(t0) in the slow-particle approx-\nimation.\nConsidering ^X(t) as a classical variable, the k-sum in\nthe third term on the left of Eq. (B5) can be easily per-\nformed, thus obtaining the following classical Langevin\nequation:\nM@2\ntX(t)\u0000F(X(t))\u0000g2\n12K\n2\u0019~v3sZt\n0dt0@3\ntX(t0)\u0002\n\u0002X\n\u000f=\u0006\u0002\n\u000e\u0000\nX(t)\u0000X(t0) +\u000fvs(t\u0000t0)\u0001\u0003\n=\u0000g12@x\u001a(X(t);t):\n(B12)\nFrom here it is apparent that the particle is interacting at\npointX(t) and time twith the phonons (density \ructu-\nations in the Luttinger liquid) emitted by itself at point\nX(t0) in a past time t0. This is similar to radiation damp-\ning of the motion of a charge particle, where the role of\nthe electromagnetic \feld is now played by the Luttinger\nliquid. The problem of damping due to the emission of\nradiation is a very old one and the classical version of\nEq. (B11) has been known in this context for a long time\nas the Abraham-Lorentz equation19.\nThe quantum version of Eq. (B12) is more compli-\ncated, due to its operatorial character (in this case in-\ndeedeik^X(t)e\u0000ik^X(t0)6=eik[^X(t)\u0000^X(t0)]). In the speci\fc,\nEq. (B11) has been obtained under the assumption (B7),\nthus it is expected to be valid only when the particle does\nnot move too fast with respect to the phonons. This can\nbe quanti\fed by considering a small value for the ratio \u0017\nbetween the maximum velocity of the impurity distribu-\ntion when it expands, !2`2\nho=\u001b(0) (here`ho=p\n~=M! 2is\nthe harmonic oscillator length), and the velocity of sound\nvs. Such ratio is independent of the trap frequency, and\nis \fxed only by the initial squeezing through the uncer-\ntainty principle { if the uncertainty in position is \u001b(0)\nthen the uncertainty in velocity is ~=(M\u001b(0)).\nIn our simulations this parameter is actually not so\nsmall. An estimate can be given as follows: the velocity\nof soundvsis a fraction of the Fermi velocity for a gas\nof free fermions at the same density, say vs\u0018~\u0019n=m ,\nthe initial squeezing is \u001b(0)=`ho\u00190:4 and the density\nn`ho\u00191, so\n\u0017=!2`2\nho\nvs\u001b(0)=~\nMvs\u001b(0)=m=M\n\u0019\u001b(0)n\u00191:6: (B13)\nSo we expect corrections to Eq. (B11) due to retardation\ne\u000bects embodied in Eq. (B5).\nAppendix C: Impurity breathing mode within the\nquantum Langevin equation\nIn this section we derive the function used to \ft the\nTDMRG data [see Eq. (3) in the main text]. We showhow it can be obtained starting both from the quantum\nLangevin equation for an ohmic bath1,2:\nM@2\nt^X(t) + 2M\u0000@t^X(t) +M!2\n2^X(t) =^\u0018(t);(C1)\nwith noise correlator\n\u0004(t) =h^\u0018(t)^\u0018(0)i (C2)\n=2M\u0000\n\u0019Z+1\n0d!~!\u0014\ncoth\f~!\n2cos!t\u0000isin!t\u0015\n;\nand from the Langevin equation (B11) in the previous\nsection, with a noise correlator given by Eq. (B8) that\ndescribes a superohmic bath1,2.\nThe predictions for the oscillation frequency and the\ndamping coe\u000ecient are di\u000berent in the two cases. How-\never it is possible to treat both equations in the same\nway, because their respective Green functions G(!) have\na similar pole structure in the frequency domain. Both\nhave two poles at complex frequencies !\u0006with negative\nimaginary part (\u0000 >0):\n!\u0006=\u0006p\n\n2\u0000\u00002\u0000i\u0000: (C3)\nThese are the physically relevant ones and correspond to\nexponentially decaying solutions. It is essential for our\npurposes that !+=\u0000!\u0003\n\u0000. We point out that Eq. (B11)\nis of third order and its Green function has a third pole in\nthe upper half of the complex plane that corresponds to\nan unphysical diverging solution, also called \\run-away\"\nsolution19, which will be discarded in the following. Note\nthat, while for the ohmic case \n is equal to the trap\nfrequency!2, and \u0000 is exactly the coe\u000ecient that appears\nin Eq. (C1), for the Langevin equation in (B11) one has a\nmore complex dependence on the trap frequency !2and\non the time scale \u001c. In the speci\fc, the three roots of the\ncubic characteristic equation \u0000i!3\u001c\u0000!2+!2\n2= 0 read\n!\u0006=\u00061\n2p\n3\u001c\u0000\nz\u0000z\u00001\u0001\n+i\n3\u001c\u0012\n1\u0000z+z\u00001\n2\u0013\n;(C4)\n!run\u0000away=i\n3\u001c\u0002\n1 +z+z\u00001\u0003\n; (C5)\nwith\nz\u00061=\u001227!2\n2\u001c2+ 2\u0006p\n(27!2\n2\u001c2+ 2)2\u00004\n2\u00131=3\n:(C6)\nNote that!\u0006have always negative imaginary part, while\n!run\u0000awayhas positive imaginary part and zero real part.\nThe symmetry of the physical roots allows us to write\nthe solution of both the two Langevin equations as\n^X(t) =^X(0)e\u0000\u0000tcosp\n\n2\u0000\u00002t\n+@t^X(0) + \u0000 ^X(0)p\n\n2\u0000\u00002e\u0000\u0000tsinp\n\n2\u0000\u00002t\n+Zt\n0dt0G(t\u0000t0)^\u0018(t0);(C7)8\n^\u0018(t) andG(t) denoting the noise term and the Green\nfunction respectively.\nUsing Eq. (C7), and the fact that h^\u0018(t)i= 0, we can\nevaluate some asymptotic averages which turn out to be\nuseful to calculate the impurity breathing mode. Some\nof them are null:\nh^X(+1)i=h@t^X(+1)i=hf^X;@t^Xgit!+1= 0;\n(C8)\nwhile other ones, such as h^X2(+1)iandh(@t^X(+1))2i,\ncan be implicitly written in terms of the spectral func-\ntion1, whose form depends on the nature of the bath.\nFinally, call ^E(t) =Rt\n0dt0G(t\u0000t0)^\u0018(t0) and note that\nh^X(0)^E(t)i=h@t^X(0)^E(t)i= 0: (C9)\nWith all these results at hand, the average of ^X2(t) can\nbe calculated at an arbitrary time t. Suppose that the\nparticle starts its motion at t=\u00001 in a squeezed har-\nmonic potential. At t= 0 it will have equilibrated with\nthe bath, and expectation values are taken on the equilib-\nrium state of the whole system, with the particle at rest\nin the squeezed harmonic potential. Then the frequency\nof the harmonic con\fnement is suddenly quenched to a\nnew value!2. The width of the impurity density distri-\nbution as a function of time, for t>0, can be explicitly\ncalculated by taking the square of Eq. (C7) on the global\nstate:\u001b2(t) =h^X2(t)i. Using Eqs. (C8)-(C9), one ends\nup with the following expression:\n\u001b2(t) =h^E2(t)i+e\u00002\u0000t\ncos2\u001e\u0014\nh^X2(0)icos2\u0000p\n\n2\u0000\u00002t\u0000\u001e\u0001\n+h(@t^X(0))2i\n\n2sin2p\n\n2\u0000\u00002t\u0015\n;(C10)\nwith\u001e= arcsin(\u0000=\n). It is not necessary to calculate ex-\nplicitly the average h^E2(t)i, since it has to cancel exactly\nthe other terms in Eq. (C10) when h^X2(0)i=h^X2(+1)i\nandh(@t^X(0))2i=h(@t^X(+1))2i. Therefore the \fnal\nresult is\n\u001b2(t) =\u001b2(+1)\u001a\n1 +e\u00002\u0000t\ncos2\u001eh\n\u0001xcos2\u0000p\n\n2\u0000\u00002t\u0000\u001e\u0001\n+ \u0001psin2p\n\n2\u0000\u00002ti\u001b\n;(C11)\nwhere the de\fnitions below have been used:\n\u0001x=h^X2(0)i\u0000h ^X2(+1)i\nh^X2(+1)i; (C12)\n\u0001p=h(@t^X(0))2i\u0000h(@t^X(+1))2i\n\n2h^X2(+1)i: (C13)We employ Eq. (C11) in our \ftting procedure as fol-\nlows. \u0001xand \u0001pare \fxed to their non-interacting values\n\u0001x=\u001b2(0)\n\u001b2(+1)\u00001 and \u0001 p=\u001b2(+1)\n\u001b2(0)\u00001;(C14)\nwhile\u001b(+1), \n, \u0000 are used as \ftting parameters. The\ninitial width \u001b(0) is provided directly by the numerical\ndata. This choice allows reliable \fts that smoothly inter-\npolate with the exact non-interacting result:\n\u001b(t) =s\n\u001b2(0) cos2!2t+`4\nho\n4\u001b2(0)sin2!2t: (C15)\nSurprisingly, the same functional form of Eq. (C11)\nholds for two quite di\u000berent models. This is due to\nthe form of the two relevant complex frequencies [see\nEq. (C3)] that applies to both of them. However the\ntwo models provide distinct predictions for the modulus\n\n =j!\u0006jand the damping coe\u000ecient \u0000, with some ap-\npreciable di\u000berences. For an ohmic bath one has \n = !2,\nso that the frequency is not renormalized in this case.\nMoreover \u0000 is a \fxed time constant independent of the\ntrap frequency. For the superohmic case of Eq. (B11),\nfor!2\u001c\u001c1 it can be shown that \n =!2is a function of\n!2\u001c, the same holds for \u0000 =!2(see Fig. 6). Notice also\nthat!\u0006always has a non-zero real part, therefore there\nis no overdamped solution for any value of \u001c.\nFinally we emphasize that the functional form given in\nEq. (C11) is essential to extract the parameters \n, \u0000 and\n\u001b(+1) from the numerical data. Indeed we have veri\fed\nthat the numerical \fts performed using the functional\nform10\n\u001b(t) =\u001b(+1) +Ae\u0000\u0000tcos(p\n\n2\u0000\u00002t\u0000\u001e);(C16)\nwithAbeing the oscillation amplitude (grey lines in\nFig. 7), are signi\fcantly di\u000berent from the ones that are\nobtained employing Eq. (C11) (black lines in Fig. 7). For\nexample, the \ft done with Eq. (C16) often overestimates\nthe asymptotic with \u001b(+1).\nACKNOWLEDGMENTS\nWe acknowledge \fnancial support by the EU FP7\nGrant No. 248629-SOLID.9\n0.0 0.5 1.0 1.5 2.0 2.5 3.0√ω2τ0.00.20.40.60.81.0Ω/ω2\n0.0 0.5 1.0 1.5 2.0 2.5 3.0√ω2τ0.00.10.20.30.40.5Γ/(ω2\n2τ)\nFIG. 6. (Color online) Plot of \n =!2and \u0000=!2as a function ofp!2\u001c/g12, to allow a comparison with the results in the main\ntext. In particular, after expanding Eq. (C4) for !2\u001c\u001c1, we get: \n =!2\u00191\u0000!2\n2\u001c2=2 and \u0000=(!2\n2\u001c)\u00191=2.\n\u0003s.peotta@sns.it\n1U. 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(Color online) Width of the impurity breathing mode \u001b(t) (in units of `ho= [~=(M! 2)]1=2) as a function of time t(in\nunits ofT=2 =\u0019=! 2). Data in di\u000berent panels corresponds to four values of the impurity-bath Lieb-Liniger parameter u12, for\na \fxed value of the bath Lieb-Liniger parameter u1= 1. Filled red circles label the tDMRG results. The black solid lines are\n\fts to the tDMRG data using Eq. (C11). The thin grey lines are obtained by employing the \ftting function in Eq. (C16)."    },    {        "title": "1509.01487v1.Damped_transverse_oscillations_of_interacting_coronal_loops.pdf",        "content": "arXiv:1509.01487v1  [astro-ph.SR]  4 Sep 2015Astronomy&Astrophysics manuscriptno.ms c/ci∇clecopy∇tESO2021\nJune27,2021\nDamped transverseoscillations ofinteracting coronal loo ps\nRoberto Soler1,2& Manuel Luna3,4\n1Departament de Física,Universitat de les IllesBalears, E- 07122 Palmade Mallorca, Spain.\n2Institut d’Aplicacions Computacionals de Codi Comunitari (IAC3), Universitat de les Illes Balears, E-07122 Palma de Mallor ca,\nSpain.\n3Institutode Astrofísicade Canarias,E-38200 La Laguna, Te nerife, Spain.\n4Departamento de Astrofísica,Universidad de La Laguna, E-3 8206 La Laguna, Tenerife,Spain.\ne-mail:roberto.soler@uib.es, mluna@iac.es\nReceived XXX;accepted XXX\nABSTRACT\nDamped transverse oscillations of magnetic loops are routi nely observed in the solar corona. This phenomenon is interp reted as\nstanding kink magnetohydrodynamic waves, which are damped by resonant absorption owing to plasma inhomogeneity acros s the\nmagnetic field. The periods and damping times of these oscill ations can be used to probe the physical conditions of the cor onal\nmedium. Some observations suggest that interaction betwee n neighboring oscillating loops in an active region may be im portant\nand can modify the properties of the oscillations compared t o those of an isolated loop. Here we theoretically investiga te resonantly\ndamped transverse oscillations of interacting non-unifor m coronal loops. We provide a semi-analytic method, based on the T-matrix\ntheory of scattering, to compute the frequencies and dampin g rates of collective oscillations of an arbitrary configura tion of parallel\ncylindrical loops. The e ffect of resonant damping is included in the T-matrix scheme in the thin boundary approximation. Analytic\nand numerical results inthe specific case of twointeracting loops are givenas anapplication.\nKey words. Magnetohydrodynamics (MHD) —Sun: atmosphere —Sun: corona —Sun: oscillations — Waves\n1. INTRODUCTION\nTransverse oscillations of magnetic loops in the solar coro na\nare under intense research since the first observational rep orts\nby theTransition Region And Coronal Explorer (TRACE) (see,\ne.g., Nakariakovetal. 1999; Aschwandenetal. 1999). Large -\namplitude coronal loop oscillations are usually excited af -\nter energetic events as, e.g., solar flares, coronal mass eje c-\ntions, or low coronal eruptions (see Zimovets&Nakariakov\n2015). Based on magnetohydrodynamic (MHD) wave theory\n(e.g., Nakariakov&Verwichte 2005), transverse loop oscil la-\ntions have been interpreted as standing kink MHD waves.\nKink MHD modes are nearly incompressible waves, mainly\ndriven by magnetic tension, and responsible for global tran s-\nverse motions of the flux tube (see, e.g., Edwin& Roberts\n1983; Goossensetal. 2009, 2012). A relevant feature of larg e-\namplitude loop oscillations is that they are strongly dampe d. It\nhas been shown that resonant absorption, caused by plasma in -\nhomogeneityinthedirectionperpendiculartothemagnetic field,\nisanefficientdampingmechanismofkinkMHDwavesincoro-\nnal loops (see, e.g., Ruderman&Roberts 2002; Goossensetal .\n2002). Due to resonant absorption, the energy from the globa l\nkink motion of the flux tube is transferred to small-scale, un re-\nsolved rotational motions around the nonuniform boundary o f\nthe tube (see, e.g., Terradasetal. 2006; Goossenset al. 201 4;\nSoler&Terradas 2015). As a result of this process, the globa l\nkink oscillation of the coronal loop is quickly damped in tim e.\nTheinterestedreaderisreferredtoGoossenset al.(2011), where\nthe theoryandapplicationsof resonantwavesin the solar at mo-\nspherearereviewed.\nObservations often show that neighboring oscillating loop s\ninanactiveregioninteractwitheachotherandexhibitcoll ectivebehaviour(e.g.,Schrijver&Brown2000;Verwichteetal.20 04;\nSchrijveret al. 2002; White et al. 2013). Interaction betwe en\nloops can modify the properties of their transverse oscilla tions\ncompared to those of the classic kink mode of an isolated loop .\nTherefore,advancedmodelsdescribingcoronallooposcill ations\nshould take into account interactions within loop systems. A\nnumber of works have studied collective transverse oscilla tions\nin Cartesian geometry (e.g., Díaz etal. 2005; Díaz &Roberts\n2006; Lunaetal. 2006; Arreguiet al. 2007, 2008). In cylindr i-\ncal geometry, Lunaetal. (2008) numerically investigated t rans-\nverse oscillations of two cylindrical loops, and Ofman (200 9)\nperformed numerical simulations in the case of four interac t-\ning loops. Concerning analytical works in cylindrical geom e-\ntry, Lunaet al. (2009, 2010) used the T-matrix theory of scat -\ntering (see, e.g., Twersky 1952; Waterman 1969; Bogdan 1987 ;\nKeppensetal.1994)toinvestigatetransverseoscillation softwo\nandthreeloops(Lunaetal. 2009)andofbundlesofmanyloops\n(Lunaetal. 2010) in the β=0 approximation,where βrefersto\ntheratioofthegaspressuretothemagneticpressure.Soler et al.\n(2009) later extended the method of Lunaet al. (2009, 2010)\nby incorporating gas pressure and longitudinal flows and stu d-\nied collective oscillationsof flowingprominencethreads. These\nworksshowedthatloopinteractiona ffectsthepropertiesoftheir\noscillations. Lunaet al. (2008, 2009) obtained that a syste m of\ntwo loops of arbitrary radii supports four kink-like collec tive\nmodes.Theshiftofthecollectivemodefrequencieswithres pect\ntotheindividualkinkfrequenciesoftheloopsissignifican twhen\nthe distance between loops is small (of the order of the loop r a-\ndius)andwhen loopshavesimilar densities. Conversely,th e os-\ncillatingloopsshowlittleinteractionwhentheyarefarfr omeach\notherandwhentheir densitiesaresubstantiallydi fferent.Onthe\nother hand, VanDoorsselaereetal. (2008) and Robertsonet a l.\nArticlenumber, page 1of 11A&Aproofs: manuscript no. ms\n(2010) used a different method based on bycilindrical coordi-\nnates to study transverse oscillations of two pressure-les s loops\ninthethintube(TT)approximation.Ofthefourkink-likeco llec-\ntivemodesobtainedbyLunaet al.(2009)intheT-matrixtheo ry,\nonly two different modes remain in the TT approximation con-\nsidered by VanDoorsselaereet al. (2008) and Robertsonetal .\n(2010).\nConcerning the damping of the oscillations, Arreguietal.\n(2007,2008)investigatedresonantlydampedoscillations oftwo\nslabs, while Terradaset al. (2008) numerically studied the res-\nonant damping of transverse oscillations of a multi-strand ed\nloop. Those works showed that the process of resonant damp-\ning is not compromised by the irregular geometry of a real-\nistic loop model and still produces the e fficient attenuation of\nglobal transverse oscillations. The damping of transverse oscil-\nlations of two cylindrical loops was analytically investig atedby\nRobertson&Ruderman (2011) and Gijsen& VanDoorsselaere\n(2014),whoconsideredtheTTapproximationandusedbycili n-\ndrical coordinates. Results obtained with bycilindrical c oordi-\nnates should be treated with caution when the distance betwe en\nloops is small. Geometrical e ffects intrinsically associated to\nthe bycilindrical coordinates may produce unphysical resu lts.\nOur purpose is to use the T-matrix method of Lunaet al. (2009,\n2010) to investigate resonantly damped oscillations of bun dles\nof loops. The present paper is partially based on unpublishe d\nresults included in Soler (2010)1. The effect of resonant ab-\nsortion in the Alfvén continuum is incorporated to the T-mat rix\nscheme by using the methodthat combinesthe jump conditions\nof the perturbationsat the resonance position with the so-c alled\nthinboundary(TB)approximation(see,e.g.,Sakuraietal. 1991;\nGoossenset al. 1992). A similar method has previously been\nused by Keppens (1995) to investigate absorption of acousti c\nwaves. We provide a general analytic theory, which is valid f or\nbundles of many transversely nonuniform parallel loops of a r-\nbitrary radii. Specific results in the case of two loops are ob -\ntained and compared to those given in Robertson&Ruderman\n(2011) and Gijsen &VanDoorsselaere (2014). Our results are\nalso compared to those of Arreguiet al. (2007, 2008) obtaine d\ninCartesian geometry.\nThispaperisorganizedasfollows.Section2containsthede -\nscription of the equilibrium configurationand the basic gov ern-\ning equations. The general analytic T-matrix theory of scat ter-\ning to compute the frequencies and damping rates of collecti ve\nlooposcillationsisgivenin Section3.Later,thespecificc ase of\ndamped oscillations of two loops is discussed both analytic ally\nandnumericallyin Section 4. Finally,someconcludingrema rks\naregiveninSection5.\n2. MODEL AND GOVERNINGEQUATIONS\nOurequilibriumconfigurationiscomposedof Nstraightandpar-\nallelmagneticcylindersoflength Lembeddedinauniformcoro-\nnalplasma.Theendsofthemagnetictubesarefixedattworigi d\nwalls representing the solar photosphere. We set the z-direction\nto be along the axes of the tubes. The magnetic field is straigh t\nalongthe z-direction,namely B=Bˆez,whereBisaconstantev-\nerywhere.Weusesubscripts‘i’and‘e’toreferto,ingenera l,the\ninternalregionofthetubesandtheexternalplasma,respec tively.\nThe subscript or superscript ‘j’ is used to refer to a particu lar\nloop.Wedenoteby Rjtheradiusofthejthtube.Thedistancebe-\ntweenthecentersofthejthandj’thloopsis djj′.Wedenotebyρj\n1The full text of Soler (2010) is available at\nhttp://www.uib.es/depart/dfs/Solar/thesis_robertoso ler.pdfFig. 1.Sketch of the equilibrium configuration in the specific case o f\ntwotransverselynon-uniform coronal loops.\ntheinternaldensityofthejth tube,while ρedenotestheexternal\ndensity,i.e.,thedensityofthecoronalenvironment.Inou rmodel\nρjandρeareconstants.Thereisa transverselynonuniformtran-\nsitional layer surroundingeach magnetictube in which the d en-\nsity continuously varies from the internal density, ρj, to the ex-\nternal density,ρe. The thicknessesof the non-uniformboundary\nlayer of the jth cylinter is lj. A sketch of the equilibriumconfig-\nuration in the case of two magnetic tubes ( N=2) is given in\nFigure1.\nWe adopt theβ=0 approximation,where βrefersto the ra-\ntio of the thermal pressure to the magnetic pressure. This is an\nappropriateapproximationtoinvestigatetransversewave sinthe\nsolar corona. In the β=0 approximation,the ideal MHD equa-\ntions governing linear perturbations superimposed on the s tatic\nequilibriumstate are\nρ∂2ξ\n∂t2=1\nµ(∇×b)×B, (1)\nb=∇×(ξ×B), (2)\nwhereξis the plasma Lagrangian displacement, bis the mag-\nnetic field Eulerian perturbation, ρis the density, and µis the\nmagneticpermittivity.\nWe assume the temporal dependence of perturbations as\nexp(−iωt), whereωis the oscillation frequency. In the case of\ntransverselynonuniformtubes,theglobaltransverseosci llations\narequasi-modeswhosefrequencyiscomplexowingtodamping\nby resonant absorption, i.e., ω=ωR+iωI, whereωRandωI\nare the real and imaginary parts of the frequency, respectiv ely.\nThereal partofωisrelatedtotheperiodandtheimaginarypart\ncorrespondstothedampingrateoftheoscillations.We cons ider\nthat the oscillating flux tubes are line-tied at the photosph ere,\nwhich acts as a perfectly reflecting wall in this model owing t o\nits largedensity comparedto the coronaldensity.Hence, we as-\nsume perturbations to be proportional to exp (ikzz), withkzthe\nlongitudinalwavenumber.Forstandingoscillations, kzgivenby\nkz=nπ\nL,withn=1,2,... (3)\nWeshallrestictourselvestothefundamentalmodeofoscill ation,\nsowe take n=1.\nArticlenumber, page 2of 11RobertoSoler &Manuel Luna: Damped collective looposcilla tions\nIn the regions with constant density, Equations (1) and (2)\ncanbereducedtofollowingequation,\n∇2\n⊥P′+k2\n⊥P′=0, (4)\nwhereP′=B·b/µis the total pressure Eulerian pertubation\nand the subscript⊥refers to the direction perpendicular to the\nmagnetic field. Thus, ∇2\n⊥denotes the perpendicular part of the\n∇2operator.Inturn,thequantity k⊥playstheroleoftheperpen-\ndicularwavenumberandisdefinedas\nk2\n⊥=ω2−ω2\nA\nv2\nA, (5)\nwhereω2\nA=k2\nzv2\nAis the square of the Alfvén frequency and\nv2\nA=B2/µρis the square of the Alfvén velocity. We stress that\nEquation(4)isonlyvalidintheregionswithconstantdensi ty,so\nitdoesnotapplywithinthenonuniformboundariesoftheloo ps.\n3. T-MATRIXTHEORY OF SCATTERING\nEquation (4) is the two-dimensional Helmholtz Equation. To\nsolve Equation (4) we use the scattering theory in its T-matr ix\nformalism (see, e.g., Waterman 1969). In the solar context, the\nT-matrixtheoryhaspreviouslybeenusedtoinvestigatethe scat-\ntering and absorption properties of bundles of magnetic flux\ntubes (e.g. Bogdan&Zweibel 1985; Bogdan&Cattaneo 1989;\nKeppenset al. 1994; Keppens 1995, among others). Lunaetal.\n(2009, 2010) used of this techniqueto compute the eigenmode s\nof systems of magnetic tubes. Because of the inhomogeneityo f\nthe tubes in the transverse direction, the modes with freque n-\ncies between the internal Alfvén frequencies of the loops an d\nthe external Alfvén frequency are resonant in the Alfvén con -\ntinuum.As a result, the oscillationsare dampedby resonant ab-\nsorption. The effect of resonant absorption was not considered\nbyLunaet al.(2009,2010).Herewe extendtheirtheoryto con -\nsiderresonantdamping.\n3.1. SolutionsintheInternaland ExternalPlasmas\nWe use local polar coordinates associated to the jth loop. We\ndenoteby rjandϕjtheradialandazimuthalcoordinates,respec-\ntively, of the coordinate system whose origin is located at t he\ncenter of the jth tube. We can define an equivalent coordinate\nsystem in each tube. In this local coordinate system, the sol u-\ntion to Equation (4) in the internal region of the jth tube can be\nexpressedas\nP′j\ni=∞/summationdisplay\nm=−∞Aj\nmJm/parenleftig\nk⊥jrj/parenrightig\nexp/parenleftig\nimϕj/parenrightig\n, (6)\nwheremis the azimuthal wavenumber, Jmis the usual Bessel\nfunctionofthefirstkindoforder m,andAj\nmareconstants.Unlike\nthecase ofisolatedtubes(see,e.g.,Edwin&Roberts1983), the\nsolutionisnotentirelydescribedbyasinglevalueof m.Because\nof interaction between tubes, the values of mare coupled. For\ntransverse, kink-like oscillations the dominant terms in t he ex-\npansion are those with m=±1, but the contribution from other\nm’s is not negligible unless the tubes are far from each other\n(Lunaet al.2009).\nThesolutionto Equation(4)in theexternalregioniswritte n\nusingtheprincipleofsuperposition,whichisapplicablet olinearwaves.Thetotalexternalsolutioniscomputedbyaddingthe net\ncontributionsofall fluxtubes,namely\nP′\ne=/summationdisplay\njP′j\ne, (7)\nwhereP′j\neis the net contribution of the jth tube to the external\nsolution.Thekeyideabehindthescatteringtheoryisthatt heso-\nlutionofEquation(4)intheexternalplasmacanbedecompos ed\ninto several fieldswith different physical meanings, namely the\ntotal,exciting,andscatteredfields .Herewegiveanoverviewof\nthe method. Interested readers are referred to Lunaetal. (2 009,\n2010)forextensiveexplanations.\nIn the external plasma, the total field associated to the jth\ncylindercanbeexpressedas\nP′j\ntotal=∞/summationdisplay\nm=−∞/bracketleftig\nαj\n1mH(1)\nm/parenleftig\nk⊥erj/parenrightig\n+αj\n2mH(2)\nm/parenleftig\nk⊥erj/parenrightig/bracketrightig\nexp/parenleftig\nimϕj/parenrightig\n,(8)\nwhereH(1)\nmandH(2)\nmare the usual Hankel functions of the first\nand second kind, respectively, and αj\n1mandαj\n2mconstants. The\nfirst term on the right-hand side of Equation (8) represents o ut-\ngoing waves from the jth tube, and the second term represents\nincomingwavestowardthejthtube.Importantly,we notetha t\nP′\ne/nequal/summationdisplay\njP′j\ntotal. (9)\nThe reason for this inequality is that the outgoing wave asso -\nciated to a particular tube contributes as an incoming wave f or\nall the other tubes. In other words,/summationtext\njP′j\ntotalis not the net exter-\nnalsolution.Toovercomethisproblem,the totalfield associated\nto the jth cylinder, P′j\ntotal, is decomposed into a scattered field ,\nP′j\nscat,andaexcitingfield ,P′j\nexcit.Conceptually,the scatteredfield\nrepresentsthe actual contributionof the varioustubes to t he net\nexternal solution, whereas the exciting field can be understood\nasthecross-talkmechanismresponsibleforinteractionbe tween\nfluxtubes(seedetailsin, e.g.,Bogdan&Cattaneo1989).\nThe full net external solution, P′\ne, is defined so that it corre-\nsponds to the sum of the scattered fields associated to all tubes,\nnamely\nP′\ne=/summationdisplay\njP′j\nscat. (10)\nConversely,the excitingfield associatedtothejthtubeisdefined\nas the difference between the full net contribution and the scat-\nteredfield ofthejthtube,namely\nP′j\nexcit=P′\ne−P′j\nscat=/summationdisplay\nn/nequaljP′n\nscat. (11)\nWaterman (1969) introduced the T-matrix operator, Tj, which\nlinearlyrelatesthe scatteredandexcitingfieldsas\nP′j\nscat=TjP′j\nexcit. (12)\nBogdan (1987) showed that for cylindrical scatterers the T-\nmatrixisdiagonal,andKeppenset al.(1994)gaveanexpress ion\nofits elements,namely\nTj\nmm=1\n21−αj\n1m\nαj\n2m, (13)\nArticlenumber, page 3of 11A&Aproofs: manuscript no. ms\nwhereαj\n1mandαj\n2mare the same constants the appear in Equa-\ntion(8).WecanuseEquation(13)toeliminate αj\n1mandwriteall\ntheexpressionsintermsof αj\n2malone.Withthehelpoftheselast\nformulae, and after some algebraic manipulations using wel l-\nknown properties of the Bessel functions, we can rewrite Equ a-\ntion(8) as\nP′j\ntotal=∞/summationdisplay\nm=−∞2αj\n2m/bracketleftig\nJm/parenleftig\nk⊥erj/parenrightig\n−Tj\nmmH(1)\nm/parenleftig\nk⊥erj/parenrightig/bracketrightig\nexp/parenleftig\nimϕj/parenrightig\n,\n(14)\nfrom where it is straightforward to identify both exciting a nd\nscatteredfields,namely\nP′j\nexcit=∞/summationdisplay\nm=−∞2αj\n2mJm/parenleftig\nk⊥erj/parenrightig\nexp/parenleftig\nimϕj/parenrightig\n, (15)\nP′j\nscat=−∞/summationdisplay\nm=−∞2αj\n2mTj\nmmH(1)\nm/parenleftig\nk⊥erj/parenrightig\nexp/parenleftig\nimϕj/parenrightig\n.(16)\nFinally,weusetheexpressionof P′j\nscatintoEquation(10)toarrive\nat thetotalnet solutionintheexternalplasma,namely\nP′\ne=−/summationdisplay\nj∞/summationdisplay\nm=−∞2αj\n2mTj\nmmH(1)\nm/parenleftig\nk⊥erj/parenrightig\nexp/parenleftig\nimϕj/parenrightig\n. (17)\nEquations (6) and (17) formally describe the total pressure\nperturbation in the interior and in the exterior of the tubes , re-\nspectively.However,we recall that these expressionsdo no t ap-\nply in the nonuniform boundary layers. The T-matrix element s,\nTj\nmm, contain the information about how the solutions are con-\nnectedacrossthenon-uniformboundariesofthe tubes.\n3.2. T-matrixElementsinthe ThinBoundaryApproximation\nAtthisstageweincorporatethee ffectofthenonuniformbound-\narylayers.Inthenonuniformlayerofthejthtubetheglobal wave\nmodesare resonantin the Alfvén continuumat the resonantpo -\nsition,rj=rA,j, where the global oscillation frequency matches\nthe local Alfvén frequency. The resonant position, rA,j, is de-\nfined through the resonant condition ω2=k2\nzv2\nA(rA,j). We use\nthe TB approximation and restrict ourselves to lj/Rj≪1. The\nTB approximation assumes that the jump of the perturbations\nacross the resonant layer is the same as their jump across the\nwhole nonuniform layer. Thus, the connection formulae of th e\nwave perturbations across the resonance are used as jump con -\nditions for the total pressure and the Lagrangian displacem ent\nat the boundariesof the tubes. This method and its applicati ons\nhave been reviewed by Goossenset al. (2011). The TB approx-\nimation within the formalism of the T-matrix theory has prev i-\nouslybeenusedbyKeppenset al.(1994)andKeppens(1995).\nGeneral expressions of the connection formulae for the\nperturbations across the resonant layer can be found in, e.g .,\nSakuraiet al.(1991).Inlocalcoordinates,theconnection formu-\nlae forthe total pressure Eulerianpertubation, P′, and the radial\ncomponentoftheLagrangiandisplacement, ξr,atrj=rA,jare\n/bracketleftbigP′/bracketrightbig=0,/bracketleftbigξr/bracketrightbig=−iπm2/r2\nA,j\n|ρ∆A|jP′, (18)\nwhere [X]=Xe−Xjdenotes the jump of the quantity Xacross\ntheresonantlayerand |ρ∆A|jisdefinedas\n|ρ∆A|j=ρ(rA,j)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingled\ndrj/parenleftig\nω2−k2\nzv2\nA/parenrightig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglerA,j=ω2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingledρ\ndrj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglerA,j, (19)wherewe have used the resonantcondition ω2=k2\nzv2\nA(rA,j). Af-\nterimposingthejumpconditionsgiveninEquation(18),weo b-\ntainboththeT-matrixelementsandthedispersionrelation ofthe\ncollectivemodes.\nOnthe onehand,theT-matrixelementsare\nTj\nmm=k⊥e\nρe(ω2−k2zv2\nAe)J′\nm(k⊥eRj)\nJm(k⊥eRj)−k⊥j\nρj/parenleftig\nω2−k2zv2\nAj/parenrightigJ′\nm(k⊥jRj)\nJm(k⊥jRj)+iπm2/r2\nAj\n|ρ∆A|\nk⊥e\nρe(ω2−k2zv2\nAe)H(1)\nm′(k⊥eRj)\nH(1)\nm(k⊥eRj)−k⊥j\nρj/parenleftig\nω2−k2zv2\nAj/parenrightigJ′m(k⊥jRj)\nJm(k⊥jRj)+iπm2/r2\nAj\n|ρ∆A|\n×Jm/parenleftig\nk⊥eRj/parenrightig\nH(1)\nm/parenleftig\nk⊥eRj/parenrightig, (20)\nwheretheprime′denotesthe derivativeoftheBessel orHankel\nfunctionwithrespecttoitsargument.Intheabsenceofreso nant\ndamping,Equation (20) consistently reverts to Equation (1 7) of\nLunaet al.(2009).\nOntheotherhand,theconstants αj\n2msatisfyanalgebraicsys-\ntemofequations,namely\nαj\n2m+/summationdisplay\nj′/nequalj∞/summationdisplay\nm′=−∞αj′\n2m′Tj′\nm′m′H(1)\nm′/parenleftig\nk⊥edjj′/parenrightig\nexp/parenleftig\ni(m′−m)ϕjj′/parenrightig\n=0,\n(21)\nfor−∞<m<∞, whereϕjj′is the angle formed by the vec-\ntor positions of the centers of the two tubes with respect to t he\nglobal reference frame. Equation (21) is system with an infin ite\nnumberofalgebraicequations.Theconditionthatthereisa non-\ntrivial solution, i.e., the determinant formed by the coe fficients\nset equal to zero, provides us with the dispersion relation ( see\ndetails in Lunaetal. 2009, 2010). The solution of the disper -\nsion relation is the complex frequencyof the damped collect ive\nquasi-mode. The imaginary part of the frequency is the reso-\nnant damping rate. In the case of Lunaet al. (2009, 2010), the\nsolution frequency was real because of the absence of resona nt\ndamping.\nThe dispersion relation is a complicated expression and has\nto be solved by numericalmethods.For practicalcomputatio nal\npurposes, in Equation (21) the indices mandm′must be trun-\ncated into a finite number. The truncation term must be large\nenough to avoid inaccuracy of the solutions. Then, the dispe r-\nsion relation can be solved using standardnumericalroutin esto\nfindthe rootsoftranscendentalequations.\n4. APPLICATION TODAMPEDOSCILLATIONS OF\nTWOLOOPS\nWerecallthattheT-matrixtheorydescribedintheprevious Sec-\ntionisvalidforanarbitrarynumberofloops.Hereweapplyt hat\nmethod to investigate damped transverse oscillations in a l oop\nsystem composed of two magnetic tubes alone. First we derive\napproximate expressions of the period and damping time in th e\ncase of two identical thin tubes. Later, we consider two tube s\nwith different properties and arbitrary radii and perform a nu-\nmericalstudy.\n4.1. Approximatesolutionsfortwo identicalthintubes\nWe consider the paradigmatic case of two tubes with identica l\nproperties. We denote by ρi,R, andlthe internal density, the\nradius, and the thickness of the boundary layer of both tubes ,\nArticlenumber, page 4of 11RobertoSoler &Manuel Luna: Damped collective looposcilla tions\nrespectively. The external density remains ρe. The distance be-\ntweenthecentersofthetwo tubesis d.\nTo obtain an approximate dispersion relation for kink-like\nmodes we assume that the contribution from the azimuthal\nwavenumbers m=±1 to the collectiveoscillation is muchmore\nimportant than the contributions from other values of m. Thus,\nwe takem,m′=±1 in Equation (21) and roughly neglect the\nother terms. As pointed out by Lunaet al. (2009), the couplin g\nbetweenthedifferentvaluesof mgetsstrongerasthe separation\nbetween the cylinders decreases. In terms of the parameters of\nthemodel,thismeansthatourapproximationappliestothec ase\noflargeseparations,i.e., R/d≪1.\nThe system defined in Equation (21) now becomes an al-\ngebraic system of four equations for the unknowns α1\n2,−1,α1\n2,1,\nα2\n2,−1, andα2\n2,1. We obtain the dispersion relation from the con-\nditionthatthereisanon-trivialsolutionofthesystem,na mely\n/parenleftig\nH(1)\n0(k⊥ed)2−H(1)\n2(k⊥ed)2/parenrightig2T4\n11\n−2/parenleftig\nH(1)\n0(k⊥ed)2+H(1)\n2(k⊥ed)2/parenrightig\nT2\n11+1=0, (22)\nwhere we used the property that T11=T−1−1according to\nthe symmetry relations of the Bessel and Hankel functions (s ee\nAbramowitz&Stegun 1972). Equation (22) is an equation for\nω, which is enclosed in the definitions of T11andk⊥e. To go\nfurther analytically, we perform a series expansion of the H an-\nkelfunctionsforsmall argumentsandretainthefirsttermal one.\nThis would be approximately valid for thin tubes. After some\nalgebraicmanipulations,Equation(22) canberecastas\nT11±iπ\n4(k⊥ed)2≈0. (23)\nNow, we look for a simplified expression of T11. We use the\nTT approximation,i.e., R/L≪1. In Equation (20) we perform\nan asymptoticexpansionofthe Bessel and Hankel functionsf or\nsmall arguments.Equation(20)becomes\nTmm≈ −1\nρe(ω2−k2zv2\nAe)−1\nρi(ω2−k2zv2\nAi)+iπm/R\nω2\nR|dρ/dr|R\n1\nρe(ω2−k2zv2\nAe)+1\nρi(ω2−k2zv2\nAi)−iπm/R\nω2\nR|dρ/dr|R\n×iπ\n22m(k⊥eR)2m\nm!(m−1)!, (24)\nwhere we also assumed rA≈Rfor simplicity. We take m=\n1, substitute Equation (24) into Equation (23), and arrive a t the\nfollowingexpression,\nρi/parenleftig\nω2−k2\nzv2\nAi/parenrightig\nδ+ρe/parenleftig\nω2−k2\nzv2\nAe/parenrightig\n−iπ\nRρiρe\n|dρ/dr|R/parenleftig\nω2−k2\nzv2\nAi/parenrightig/parenleftig\nω2−k2\nzv2\nAe/parenrightig\nω2=0, (25)\nwheretheparameter δisdefinedas\nδ=1∓/parenleftigR\nd/parenrightig2\n1±/parenleftigR\nd/parenrightig2. (26)\nEquation (25) is the dispersion relation for resonantly\ndampedkinkmodesoftwoidenticaltubesintheTTandTBap-\nproximationsandforlargeseparations.Inthelimitthatth etubes\nare far from each other, R/d→0 and soδ→1. Then, Equa-\ntion (25) consistently reverts to the dispersion relation o f reso-\nnant kink modes in an isolated thin tube (e.g., Goossensetal .\n1992).4.1.1. Frequencyof the Oscillations\nFirst we neglect the presence of the nonuniform boundary lay -\ners and study undamped oscillations. We drop the last term on\nthe left-hand side of Equation (25) and the dispersion relat ion\nsimplifiesto\nρi/parenleftig\nω2−k2\nzv2\nAi/parenrightig\nδ+ρe/parenleftig\nω2−k2\nzv2\nAe/parenrightig\n=0. (27)\nTheexactsolutionis\nω2=ω2\nk\n1∓ζ−1\nζ+1/parenleftigR\nd/parenrightig2, (28)\nwhereζ=ρi/ρeis the density contrast and ωkis the frequency\nof the kink mode in an individual thin tube (Edwin&Roberts\n1983),namely\nω2\nk=ρiv2\nAi+ρev2\nAe\nρi+ρek2\nz. (29)\nEquation (28) can be compared to Equation (51) of\nVanDoorsselaereetal.(2008).Bothexpressionsagreeifth epa-\nrameterEof VanDoorsselaereetal. (2008) is approximated as\nE=exp[−2arccosh(d\n2R)]≈/parenleftigR\nd/parenrightig2, which is valid for R/d≪1.\nIn agreementwith VanDoorsselaereetal. (2008), in the TT ap -\nproximationwe obtain two kink-likesolutionscorrespondi ngto\nthe+and−signs in Equation (28). As explained by Lunaet al.\n(2009), there are actually four kink-like modes beyond the T T\napproximation. For arbitrary radii, the low-frequency sol ution\n(that with the+sign in Equation (28)) splits in the SxandAy\nmodes of Lunaet al. (2009), while the high-frequency soluti on\n(that with the−sign in Equation(28)) becomestheir SyandAx\nsolutions.\nFrom Equation (28) we compute the period of the oscilla-\ntions,P=2π/ω,as\nP=Pk/radicaligg\n1∓ζ−1\nζ+1/parenleftbiggR\nd/parenrightbigg2\n, (30)\nwithPk=2π/ωkthe period of the kink mode of an isolated\ntube.Forlargeseparationsthetubesfeel little interacti on.Then,\nthe periods of the two solutions consistently tend to that of the\nkinkmodeofanisolatedtube.\n4.1.2. DampingRate\nHereweincorporatethee ffectofresonantdampingandconsider\nthefull expressionofthedispersionrelation(Equation(2 5)).To\nobtain an approximate expression of the damping rate, we sub -\nstituteω=ωR+iωIinEquation(25)andassumeweakdamping,\ni.e.,|ωI|≪|ωR|. Then,we are allowed to neglect terms of order\nω2\nIand higher orders. After some algebraic maniputations (not\ngivenhere for the sake of simplicity) we arrive at an express ion\nfortheratioωI/ωR, namely\nωI\nωR=π\n21\nRρiρe\nρiδ+ρe1\n|dρ/dr|R/parenleftig\nω2\nR−k2\nzv2\nAi/parenrightig/parenleftig\nω2\nR−k2\nzv2\nAe/parenrightig\nω2\nR.(31)\nTosimplifyEquation(31)weconsiderasmoothmonotonicpro -\nfile for the density in the nonuniformboundariesso that we ca n\nexpressthederivativeofthe densityprofileas\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingledρ\ndr/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleR=π2\n4Fρi−ρe\nl, (32)\nArticlenumber, page 5of 11A&Aproofs: manuscript no. ms\nwithFa factor that depends on the form of the profile. For in-\nstance,F=4/π2for a linear profile and F=2/πfor a sinu-\nsoidal profile. In addition, we approximate ωRby the value of\nthe frequencyin the undampedcase (Equation(28)) and subst i-\ntutethe expressionof δ(Equation(26))to arriveat\nωI\nωR≈−1\n2π1\nFl\nRζ−1\nζ+11±/parenleftigR\nd/parenrightig2\n1∓ζ−1\nζ+1/parenleftigR\nd/parenrightig2/bracketleftigg\n1−/parenleftbiggR\nd/parenrightbigg4/bracketrightigg\n. (33)\nFor a linear density profile, Equation (33) becomes Equa-\ntion(86)ofRobertson&Ruderman(2011)iftheapproximatio n\nexp[−2arccosh(d\n2R)]≈/parenleftigR\nd/parenrightig2valid for R/d≪1 is again per-\nformedintheirexpression.\nNow, we define the damping time as τD=1/|ωI|and use\nEquation (33) to obtain the expression for the ratio τD/P. By\nkeepingtermsupto( R/d)2,theexpressionforthedampingratio\nis\nτD\nP≈/parenleftbiggτD\nP/parenrightbigg\nk/bracketleftigg\n1∓2ζ\nζ+1/parenleftbiggR\nd/parenrightbigg2/bracketrightigg\n, (34)\nwhere\n/parenleftbiggτD\nP/parenrightbigg\nk=FR\nlζ+1\nζ−1, (35)\nis the damping ratio of the individual kink mode (see, e.g.,\nRuderman&Roberts 2002; Goossensetal. 2002). As for indi-\nvidual kink modes, the damping ratio of collective modes is i n-\nversely proportional to the thickness of the nonuniform lay er.\nEquation (34) shows that the solution with the +sign is more\nefficientlydampedbyresonantabsorptionthanthesolutionwit h\nthe−sign.Thisdifferenceinthedampingratesofthetwomodes\nqualitatively agrees with the results of Robertson&Ruderm an\n(2011).\n4.2. NumericalStudy\nHereweperformanumericalstudybeyondtheapproximatelim -\nitsstudiedbefore.Todoso,weconsiderthefulldispersion rela-\ntionobtainedfromEquation(21)byusingthegeneralexpres sion\nof the T-matrix elements (Equation (20)). The truncated dis per-\nsion relation is then numerically solved. Convergence test s of\ntheresultshavebeenperformedtomakesurethatthetruncat ion\nvalue of the azimuthal series is large enoughfor the errorin the\nsolutions to be negligible. In short, we found that for kink- like\nmodes the truncation value, namely mt, has no important e ffect\nunlessmt<5andthetwoloopsarenexttoeachother( d/R≈2).\nWhenmt>5and/ortheloopsarefarfromeachother,weessen-\ntiallyobtainthattheresultsareindependentof mt.Wehaveused\nmt=30 in all computationsgivenhere, which reducesthe error\nand assures the excellent converge of the solutions even whe n\nd/R≈2.\nThe numerical method follows a two-step procedure. First,\nwe solve the dispersion relation in the absence of nonunifor m\nboundary layers. In that case, the solution is a real frequen cy\nthat corresponds to an approximation to ωR. This approximate\nvalue ofωRis used to compute the resonance positions and the\nderivative of the density profile at the resonances. We assum e\nsinusoidal density profiles in the nonuniform boundary laye rs.\nThen, we use these parameters to solve the complete dispersi on\nrelation,whichnowincludesthee ffectofresonantdamping.The\nfrequency obtained from the second run is complex, so that it\nprovides us with a more accurate value of ωRand also gives us\nthevalueofωI.4.2.1. Identicalloops\nWe initially study the case of two identical tubes. We set the\nCartesiancoordinatessystemsothatthe xy-planeisperpendicu-\nlar to the axes of the loops. In that plane, the centers of the t wo\nloops are located on the x-axis. We use the notation introduced\nbyLunaetal. (2008)to denotethefourkink-likemodesprese nt\nin a two-loop configuration. The modes are labeled as Sx,Ax,\nSy, andAy, whereSandAdenotesymmetric or anti-symmetric\nmotions of the two loops, respectively, and the subscripts xand\nyindicate the main direction of polarization of the oscillat ions\nin the coordinates system defined above. The eigenfunctions of\nthesefourmodesinthecaseofloopswithoutnonuniformboun d-\narylayerscanbefoundinFigure2ofLunaet al.(2008).\nFigure 2 shows the dependenceof ωRand the ratio|ωI|/ωR\non the separation between loops, d/R, for a particular set of\nparameters given in the caption of the figure. We note that the\ncurves corresponding to the SxandAymodes, and those of the\nAxandSymodes,arealmostsuperimposedbecauseweareinthe\nTTregime(weused L/R=100).Concerningthebehaviorof ωR,\nthe frequencies of the four solutions tend to the kink freque ncy\nofanisolatedloopinthelimit d/R≫1.Conversely,thesmaller\nthe separation between loops, the more important the splitt ing\nof the collective frequencies with respect to the kink frequ ency\nof an isolated loop. Figure 2(a) can be compared to Figure 3 of\nLunaet al. (2008) and with Figure 4 of VanDoorsselaereet al.\n(2008).However,unlikeinthosepreviousworkswenotethat in\nourcasethefrequenciesofthehigh-frequencysolutions( Axand\nSy)donottendtotheexternalAlfvénfrequencywhen d/R→2.\nThe reason for this di fference is probably that the AxandSy\nmodes are strongly damped when d/R→2, and this fact has\nsomeimpactonthereal partofthefrequencyaswell.\nOntheotherhand,Figure2(b)showsthatthehigh-frequency\nmodes (AxandSy) are more efficiently damped by resonant\nabsorption than the low-frequency modes ( SxandAy). This\nresult agrees with that of Robertson&Ruderman (2011) for\nlarge separations. However, the behavior of |ωI|/ωRobtained\nhere for small separations is dramatically di fferent from that\nof Robertson& Ruderman (2011). They found that the oscil-\nlations become undamped in the limit d/R→2, while we\nfind that the modes remain damped. Although the analysis of\nRobertson&Ruderman (2011) is mathematically correct, we\nfind no physical reason for which the oscillations should be\nundamped when the loops are close to each other. Our com-\nputations show that the damping of the low-frequency modes\n(SxandAy) is roughly independent on d/R, whereas the damp-\ning of the high-frequency modes ( AxandSy) gets stronger as\nthe separation between loops is reduced. As pointed out by\nRobertson&Ruderman (2011) and Gijsen &VanDoorsselaere\n(2014),thephysicalsignificanceoftheresultsobtainedwi thby-\ncilindrical coordinates should be treated with caution whe n the\nseparationbetweenthetubesissmall.\nPanels(c)and(d)ofFigure2showthesameresultsaspanels\n(a) and (b), respectively, but now log10(l/R−2)is used in the\nhorizontal axes. These additional graphs are included to sh ow\nin more detail the behavior of the solutions obtained with th e\nT-matrixmethodforsmall separationsbetweentubes.\nWe haveoverplottedinFigure2theanalyticapproximations\nofωRand|ωI|/ωRgiven in Equations (28) and (33). These ap-\nproximationswere derivedin the limit d/R≫1 and reasonably\nagree with the numericalsolutions when d/R/greaterorsimilar3. As expected,\nthe approximationsdo not work well for small separations. T he\nanalytic approximationswere derived considering the cont ribu-\ntionsfrom m=±1 alone,but the contributionof high m’s to the\nArticlenumber, page 6of 11RobertoSoler &Manuel Luna: Damped collective looposcilla tions\nFig.2.Numerical resultsinthe case oftwoidentical coronal loops . (a)Dependence of ωR/ωA,eond/R,whereωA,e=kzvA,eistheexternal Alfvén\nfrequency. (b)Dependence of |ωI|/ωRond/R.The meaningof thevarious linesisindicatedwithinthepan els. Thedashed lines correspond tothe\nanalytic approximations in the limit d/R≫1 (Equations (28) and (33)). We have used ζ=5,L/R=100, andl/R=0.2. Panels (c) and (d) show\nthe same results as panels (a)and (b),respectively, but asf unction of log10(d/R−2).\nFig.3.Numericalresultsinthecase oftwoidentical coronal loops . (a)Dependence of ωR/ωA,eonL/R,whereωA,e=kzvA,eistheexternal Alfvén\nfrequency. (b)Dependence of |ωI|/ωRonL/R.Themeaning ofthevarious linesisindicatedwithinthefigu re.Wehave usedζ=5,l/R=0.2, and\nd/R=2.5. Wenote thatthe horizontal axes of both panels are inlogar ithmic scale.\nfullsolutionisimportantforsmallseparationbetweenloo ps(see\nLunaet al. 2009).\nFigure 3 displays the dependence of the solutions on L/R.\nThis figure is included to show that the almost degenerate cou -\nplesSx–AyandAx–Sysplit into four different solutions for\nsmall values of L/Rbeyondthe TT regime. We point out, how-ever, that the impact of the value of L/Ron the solutions is\nnot relevant when realistic values of this parameter are con sid-\nered. The TT limit used by Robertson&Ruderman (2011) and\nGijsen&VanDoorsselaere(2014) isthereforeadequate.\nNow we plot in Figure 4 the dependenceof ωRand|ωI|/ωR\non the nonuniform layer thickness, l/R. We consider a small\nArticlenumber, page 7of 11A&Aproofs: manuscript no. ms\nseparation, namely d/R=2.5, and the remaining parameters\nare the same as in Figure 2. Consistently, when l/R=0 the\nmodes are undamped. The real part of the frequency of the\nlow-frequency modes is almost independent of l/R, while their\n|ωI|/ωRis roughly linear with l/R. Conversely, the real part of\nthe frequency of the low-frequency modes decreases when l/R\nincreases,andtheir |ωI|/ωRisonlylinearwith l/Rforsmallval-\nues of this parameter. As discussed before, the behavior of t he\nlow-frequency modes ( SxandAy) is similar to that of the kink\nmode of an isolated loop. However, the high-frequency modes\n(AxandSy) seem to be more a ffected by the interaction be-\ntween loops and show a somewhat di fferent behavior when l/R\nincreases.We notethatbecauseoftheTBapproximationwear e\nrestrictedtoconsidersmall valuesof l/R.\nIt is useful to relate the present results with those of\nArreguietal. (2007, 2008), who studied the damping of trans -\nverse oscillations of two nonuniformslabs. They found that the\nratio|ωI|/ωRcorresponding to the symmetric kink mode of\nthe two slabs is weakly dependent of the separation between\nthe two slabs (see Arreguietal. 2007, their Figure 4). In tur n,\nArreguietal.(2008)foundthattheanti-symmetrickinkmod eof\nthetwoslabsismoree fficientlydampedthanthesymmetrickink\nmode (see their Figure 6). The symmetric and anti-symmetric\nkink modes of two slabs would be equivalent to the SxandAx\nmodesoftwocylinders.Thus,ourresultsincylindricalgeo metry\nareconsistentwithpreviousfindingsin Cartesiangeometry .\nIt is also convenient to consider the physical arguments of\nArreguietal. (2007, 2008) to explain why the high-frequenc y\nmodes damp more e fficiently than the low-frequency modes.\nArreguietal. (2007, 2008) related the e fficiency of the damp-\ning with the magnitude of the total pressure perturbationwi thin\nthe resonant layers. According to Andrieset al. (2000), the ef-\nficiency of the resonant coupling between the global transve rse\nmodeandtheAlfvéncontinuummodesisproportionaltotheto -\ntalpressureperturbationsquared.Soler(2010)plottedth esquare\nof the total pressure perturbation corresponding to the Sxand\nAxmodes (see his Figures 9.7 and 9.8) and found that, when\nthe quantities are normalized, the perturbation of the Axmode\nreaches a larger value in the resonant layers than that of the Sx\nsolution. This result qualitatively explains why resonant damp-\ning is more efficient for the Axmode than for the Sxmode.\nEquivalently,asimilarreasoninghelpusunderstandthedi fferent\nattenuationofthe SyandAymodes.Nevertheless,amorerobust\nstudy of the process of resonant absorption in two-dimensio nal\nconfigurationswouldbeneededforacompleteunderstanding of\nthedifferentdampingrates(see Russell &Wright2010).\n4.2.2. Non-identicalloops\nHere we consider two loops with di fferent properties and com-\npare our results to those of Gijsen&VanDoorsselaere (2014) .\nWe usesubscripts1and2torefertothetwodi fferentloops.For\nsimplicity,we take R1=R2=RandL/R=100in all following\ncomputations.\nFirst, we assume the same density contrast in the two loops,\nnamelyζ1=ζ2=5, and vary l2/Rwhilel1/Ris kept fixed\ntol1/R=0.2. These results are shown in Figure 5 and can\nbe compared to those already displayed in Figure 4 in the case\nofl1=l2. Importantly, we find that the collective oscillations\nremain damped when l2/R=0. Although this is a rather par-\nticular situation, the results have interestingimplicati ons. When\nl2/R=0 resonant absorption only occurs in the boundarylayer\nof loop #1. However, this is enough for the collective oscill a-\ntions of the two loops to be e fficiently damped. We note that inthe study by Gijsen &VanDoorsselaere (2014) the thicknesse s\nofthe nonuniformlayersare linkedtothe coordinatesystem .\nNow we consider the case of two loops with di fferent den-\nsity contrasts. We fix ζ2=3 and compute the solutions as func-\ntions ofζ1. These results are displayed in Figure 6, where the\nremaining parameters used in the computations are specified in\nthe caption. For consistency, we keep the same notation as be -\nfore to denote the various modes according to the ordering of\ntheirfrequencies,althoughtheydonotrepresenttrulycol lective\noscillations ifζ1/nequalζ2(Lunaet al. 2009). Figure 6 can be com-\nparedtoFigure6ofGijsen &VanDoorsselaere(2014).Tomake\na proper comparison, we note that Gijsen &VanDoorsselaere\n(2014) plotted the signed damping ratio, while here we plot t he\nabsolutevalue.\nConcerning the real part of the frequency (Figure 6(a)), we\nfind the same results as Lunaetal. (2009). When ζ1<ζ2, the\nhigh-frequencymodes are associated to loop #1 alone, where as\nthe low-frequency modes represent individual oscillation s of\nloop #2. The opposite happenswhen ζ1>ζ2. Conversely,when\nζ1≈ζ2the four modes approach and interact in the form of an\n‘avoided crossing’. Only in that case the modes represent tr uly\ncollectiveoscillations(Lunaetal. 2009).\nFigure 6(a) also shows that the high-frequency modes are\nalwayswithintheAlfvéncontinuaofthe twoloops,i.e.,the fre-\nquencies of the AxandSymodes are always larger than ωA,1\nandωA,2and smaller thanωA,e. This is true except in the limit\nζ1→1, where the frequencies of the AxandSymodes tend to\nωA,e. On the contrary, the low-frequency SxandAymodes are\nbelow the Alfvén continuumof loop #1 when ζ1/lessorsimilar2 and below\nthe Alfvén continuum of loop #2 when ζ1/greaterorsimilar5. This result may\nhaveimplicationsforthedampingbyresonantabsorption.\nFigure 6(b) displays the damping ratio of the modes as a\nfunction ofζ1. The result for the high-frequency modes can be\nunderstood as follows. When ζ1→1 the frequencies of the\nhigh-frequencymodestendto ωA,eand,asa consequence,these\nmodes become undamped in that limit. When 1 <ζ1<ζ2, the\ndampingratioincreaseswhen ζ1increases.When1<ζ1<ζ2the\nhigh-frequency modes represent individual oscillations o f loop\n#1. Then, the damping ratio reaches a maximum when ζ1≈ζ2.\nWhenζ1>ζ2the damping ratio saturates to a constant value\nbecause the high-frequencymodes representnow individual os-\ncillationsofloop#2,andthevalueof ζ2isfixedinthecomputa-\ntions.Thebehaviorofthedampingofthehigh-frequencymod es\nagreeswiththatplottedbyGijsen&VanDoorsselaere(2014) in\ntheirFigure6.\nThe overall behavior of the damping ratio of the low-\nfrequency modes displayed in Figure 6(b) also agrees with\nGijsen&VanDoorsselaere (2014). The damping ratio of the\nlow-frequencymodes is roughly constant when ζ1<ζ2and in-\ncreaseswhenζ1>ζ2.Thefactthatthelow-frequencymodesare\nbelow the Alfvén continuumof loop #1 when ζ1/lessorsimilar2 and below\ntheAlfvéncontinuumofloop#2when ζ1/greaterorsimilar5havenoimportant\nimpact on the damping. Again, these results can be understoo d\nby considering that the low-frequency modes are associated to\nloop#2whenζ1<ζ2,whiletheyareassociatedtoloop#1when\nζ1>ζ2.\nThe low-frequency modes computed here do not show the\npronounced minimum of the damping rate seen in the solu-\ntion plotted by Gijsen& VanDoorsselaere (2014) when ζ1≈\nζ2. There are several e ffects that may explain this di fference.\nThe most obvious one is the di fferent geometry considered\nin Gijsen &VanDoorsselaere (2014) and here. Another pos-\nsible explanation is that the density profile in the nonuni-\nform layers used by Gijsen& VanDoorsselaere (2014) is dif-\nArticlenumber, page 8of 11RobertoSoler &Manuel Luna: Damped collective looposcilla tions\nFig.4.Numerical results inthe case of two identical coronal loops . (a) Dependence of ωR/ωA,eonl/R, whereωA,e=kzvA,eis the external Alfvén\nfrequency. (b) Dependence of |ωI|/ωRonl/R. The meaning of the various lines is indicated within the figu re. We have usedζ=5,L/R=100,\nandd/R=2.5.\nFig.5.Numericalresultsinthecaseoftwonon-identicalcoronall oopswith R1=R2=R.(a)Dependence of ωR/ωA,eonl2/R,whereωA,e=kzvA,e\nistheexternalAlfvénfrequency.(b)Dependence of |ωI|/ωRonl2/R.Themeaningofthevariouslinesisindicatedwithinthefigu re.Wehaveused\nζ1=ζ2=5,L/R=100,l1/R=0.2andd/R=2.5.\nFig. 6.Numerical results in the case of two non-identical coronal l oops with R1=R2=R.(a) Dependence of ωR/ωA,eonζ1, whereωA,e=kzvA,e\nistheexternalAlfvénfrequency.(b)Dependence of |ωI|/ωRonζ1.Thedottedlinescorrespond totheAlfvénfrequenciesofth etwoloops,andthe\nmeaning of the remaining lines isindicated withinthe figure . Wehave usedζ2=3,L/R=100,l1/R=l2/R=0.2andd/R=3.\nferent from that used here. A linear density profile is used in\nGijsen &VanDoorsselaere(2014),sothatthederivativeofd en-sity at the resonancepositionis independentof thefrequen cyof\nthemode.Hereweuseasinusoidalprofileandtakeintoaccoun t\nArticlenumber, page 9of 11A&Aproofs: manuscript no. ms\nthat the positionof theresonanceandthe valueof the deriva tive\nof density at the resonance position are functions of the mod e\nfrequency.\nInFigure6(b),thedampingrateofthelow-frequencymodes\nshowsasmallbumparound ζ1≈2,whentherealpartofthefre-\nquency approximately crosses the internal Alfvén frequenc y of\nloop #1. The reason for this bump is that the the low-frequenc y\nmodes intersect with and ‘avoid cross’ the fluting modes that\ncluster toward the internal Alfvén frequency. This bump is a b-\nsent from Figure 6 of Gijsen& VanDoorsselaere (2014) prob-\nably because coupling between kink and fluting modes is not\ndescribedinthe TTapproximation.\n5. CONCLUDING REMARKS\nIn this paper we have extended the analytic T-matrix theory\nof scattering of Lunaet al. (2009, 2010) to investigate reso -\nnantly damped oscillations of an arbitrary configuration of par-\nallel cylindrical coronal loops. After presenting the gene ral the-\nory,we have performeda specific applicationin the case of tw o\nloops. This work is partially based on unpublished results i n-\ncluded in Soler (2010), where collective damped oscillatio ns of\nprominencethreadswerestudied.\nWe have compared our results to those of the papers by\nRobertson&Ruderman (2011) and Gijsen& VanDoorsselaere\n(2014). They investigated the damping of collective oscill a-\ntions of two loops in the TT approximation and used a method\nbased on bicylindrical coordinates. In general, the result s of\nRobertson&Ruderman (2011) and Gijsen& VanDoorsselaere\n(2014) are in good agreementwith the present results, speci ally\nwhen the separation between loops is large. However, when\nthe separation between the loops is small, i.e., for separat ions\nof few radii, the results of those previous works show impor-\ntant differences compared to the present findings. For instance,\nRobertson&Ruderman (2011) and Gijsen& VanDoorsselaere\n(2014) obtained that by decreasing the distance between loo ps,\nthe efficiency of resonant damping is reduced. In their compu-\ntations,bothlow-andhigh-frequencymodesbecomeundampe d\nwhen the loops are in contact. However, this result lacks of a\nphysical explanationand contradictspreviousfindingsin C arte-\nsian geometry (Arreguiet al. 2007, 2008). In our computatio ns,\nwe find that the damping of the high-frequency modes gets\nstronger by decreasing the separation between loops, while the\ndampingof the low-frequencymodesis roughlyindependento f\nthe separation. Our solutions do not become undamped when\nthe two tubes are in contact. Thus, the results obtained here are\nconsistentwithpreviousresultsbyArreguietal.(2007,20 08)of\ncollectiveoscillationsoftwo slabs.\nAlthough the mathematical analysis of\nRobertson&Ruderman (2011) and Gijsen& VanDoorsselaere\n(2014) is flawless, their results by may be a ffected by un-\navoidable geometrical problems related to the bicylindric al\ncoordinates when the loops are close to each other. In bicyli n-\ndricalcoordinatestheshapesofnonuniformboundarylayer sare\nnot symmetric and change when the separation between tubes\ndecreases.Thenonuniformlayersgetthickerintheouterpa rtsof\nthetubesandthinnerintheinnerparts.Asalreadymentione dby\nRobertson&Ruderman (2011) and Gijsen& VanDoorsselaere\n(2014), these geometrical limitations may lead to unphysi-\ncal results for small separations. The T-matrix method used\nhere is not constrained by the geometrical problems of the\nbicylindrical coordinates. Therefore, we may conclude tha t the\nresults given here are more generally applicable than those ofRobertson&Ruderman (2011) and Gijsen &VanDoorsselaere\n(2014) whentheloopsareclose toeachother.\nBecauseoftheTBapproximationwe wererestrictedtocon-\nsider small values of l/R, i.e.,l/R≪1. The effect of thick\nnonuniform layers could be included into the T-matrix forma l-\nism with the method of Frobeniusused by Soleretal. (2013) in\nthecaseofanisolatedloop.Thiswouldsubstantiallyincre asethe\nmathematicalcomplexityof the problembut, on the otherhan d,\nit would provide a more accurate description of the damping\nof largely nonuniform loops. It has been shown by Soleret al.\n(2014) that the error in the damping rate associated to the us e\nof the TB approximation can be important when the loops are\nlargely non-uniform. Apart from a numerical factor, the dam p-\ning rate in the TB approximation is independent of the specifi c\ndensity profile considered within the nonuniform boundary, but\nthe density profile can have a more important impact when the\nnon-uniformlayersarethick.Inaddition,therealpartoft hefre-\nquency depends on l/Rbeyond the limit l/R≪1. The effect\nof thick nonuniform boundaries on collective loop oscillat ions\ncouldbeexploredinthefuture.\nThe method given here to compute resonantly damped col-\nlective oscillations can have multiple applications in the future.\nFor instance, damped oscillations of a coronal arcade could be\nstudied by modeling the arcade as a long line of parallel loop s.\nAnother interesting application is the investigation of os cilla-\ntionsofloopsformedbymanystrands.AsshownbyLunaet al.\n(2010), the global oscillation of the whole loop would be de-\ntermined by the interaction of the oscillations of the indiv idual\nstrands.Theresonantabsorptionprocessworkingintheind ivid-\nual strands would a ffect the damping on the global loop motion\n(see Terradaset al. 2008). In principle, the presence of mul tiple\nresonances in the system may cause the transverse oscillati ons\nof a multi-strandedloop to damp more quickly than the oscill a-\ntions of an equivalent monolithic loop. This idea could be co n-\nfirmedusingtheT-matrixmethod.Also,asshownbySoleret al .\n(2009),theeffectsofgaspressureandmassflowalongtheloops\ncaneasilybeincludedintheT-matrixformalism,thusexten ding\ntheapplicabilityofthemethod.\nAcknowledgements. We thank Ramón Oliver for useful comments on a draft\nof this paper. R.S. acknowledges support from MINECO throug h a ‘Juan de\nla Cierva’ grant and through projects AYA2011-22846 and AYA 2014-54485-P,\nfromMECDthrough projectCEF11-0012,fromthe‘Vicerector at d’Investigació\ni Postgrau’ of the UIB, and from FEDER funds. M.L. acknowledg es support\nby MINECO through projects AYA2011-24808 and AYA2014-5507 8-P. M.L.is\nalso grateful to ERC-2011-StG 277829-SPIA.\nReferences\nAbramowitz, M.& Stegun, I. 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V.&Nakariakov, V.M. 2015, A&A,577, A4\nArticlenumber, page 11of 11"    },    {        "title": "2304.09025v1.Tests_of_modified_gravitational_wave_propagations_with_gravitational_waves.pdf",        "content": "Tests of modi\fed gravitational wave propagations with gravitational waves\nTao Zhua;b,\u0003Wen Zhaoc;d,†Jian-Ming Yana;b, Cheng Gonga;b;e, and Anzhong Wangf‡\naInstitute for theoretical physics & cosmology, Zhejiang University of Technology, Hangzhou, 310032, China\nbUnited Center for Gravitational Wave Physics, Zhejiang University of Technology, Hangzhou, 310032, China\ncCAS Key Laboratory for Research in Galaxies and Cosmology, Department of Astronomy,\nUniversity of Science and Technology of China, Hefei 230026, China\ndSchool of Astronomy and Space Sciences, University of Science and Technology of China, Hefei, 230026, China\neKey Laboratory of Cosmology and Astrophysics (Liaoning) & Department of Physics,\nCollege of Sciences, Northeastern University, Shenyang 110819, China\nfGCAP-CASPER, Physics Department, Baylor University, Waco, Texas 76798-7316, USA\n(Dated: April 19, 2023)\nAny violation of the fundamental principles of general relativity (GR), including the violations\nof equivalence principle and parity/Lorentz symmetries, could induce possible derivations in the\ngravitational wave (GW) propagations so they can be tested/constrained directly by the GW data.\nIn this letter, we present a universal parametrization for characterizing possible derivations from\nGW propagations in GR. This parametrization provides a general framework for exploring possible\nmodi\fed GW propagations arising from a large number of modi\fed theories of gravity. With this\nparameterization, we construct the modi\fed GW waveforms generated by the coalescence of compact\nbinaries with the e\u000bects of the gravitational parity/Lorentz violations, then analyze the open data\nof compact binary merging events detected by LIGO/Virgo/KAGRA collaboration. We do not \fnd\nany signatures of gravitational parity/Lorentz violations, thereby allowing us to place several of\nthe most stringent constraints on parity/Lorentz violations in gravity and a \frst constraint on the\nLorentz-violating damping e\u000bect in GW. This also represents the most comprehensive tests on the\nmodi\fed GW propagations\n1. Introduction | The direct detection of gravita-\ntional waves (GWs) from the coalescence of compact bi-\nnary systems by the LIGO/Virgo/KAGRA Collabora-\ntion has opened a new era in gravitational physics [1{5].\nThe GWs of these events, carrying valuable information\nabout local spacetime properties of the compact bina-\nries, allow us to test the fundamental building blocks of\nEinstein's general relativity (GR), including the equiva-\nlence principle, parity and Lorentz symmetries, the four-\ndimensional spacetime, etc.\nIn GR, GWs possess two independent polarization\nmodes, which propagate at the speed of light with an\namplitude damping rate as the inverse of the luminosity\ndistance of the GW sources. Any violation of the funda-\nmental principles of GR could induce possible derivations\nfrom the above standard propagation properties of GWs.\nWith speci\fc derivations in GW propagation from GR,\none is able to obtain the constraints on the e\u000bects from\nGW data. This has enabled a lot of tests of GW propa-\ngation by the LIGO/Virgo/KAGRA Collaboration [6{8].\nDi\u000berent modi\fcations to GR may induce di\u000berent\ne\u000bects in GW propagation. Given a large number of\nmodi\fed theories, one challenging task is to construct\na uni\fed framework for characterizing di\u000berent e\u000bects so\nthey could be directly tested with GW data in a model-\nindependent way. Several parametrized frameworks have\nbeen proposed for this purpose [9{13]. In this letter,\n\u0003Corresponding author: zhut05@zjut.edu.cn\n†wzhao7@ustc.edu.cn\n‡anzhong wang@baylor.eduwe introduce a universal parametrization for character-\nizing possible derivations from GW propagation in GR\n[9]. This parametrization provides a general framework\nfor exploring possible modi\fed GW propagations arising\nfrom a large number of modi\fed theories of gravity. With\nthis parameterization, we study the modi\fed GW wave-\nforms with the e\u000bects of the parity/Lorentz violations\nin gravity, which enables us to constrain the GR deriva-\ntions with compact binary merging events detected by\nLIGO/Virgo/KAGRA collaboration. The tests carried\nout in this letter represent the most comprehensive con-\nstraints on the modi\fed GW propagations in the litera-\nture.\n2. Modi\fed GW propagation | We consider GWs\npropagating on a homogeneous and isotropic background,\nand the spatial metric is written as gij=a(\u001c)(\u000eij+\nhij(\u001c;xi)), where\u001cdenotes the conformal time, which\nrelates to the cosmic time tbydt=ad\u001c, andais the\nexpansion factor of the universe. Throughout this letter,\nwe set the present expansion factor a0= 1.hijdenote\nGWs, which are transverse and traceless, @ihij= 0 =hi\ni.\nIt is convenient to expand hijover spatial Fourier har-\nmonics,\nhij(\u001c;xi) =X\nA=R;LZd3k\n(2\u0019)3hA(\u001c;ki)eikixieA\nij(ki);(1)\nwhereeA\nijdenote the circular polarization tensors and\nsatisfy the relation \u000fijknieA\nkl=i\u001aAejA\nlwith\u001aR= 1 and\n\u001aL=\u00001. To study the possible derivation of propagation\nof GWs from that of GR, we write the modi\fed propa-\ngation equations of the two GW modes in the followingarXiv:2304.09025v1  [gr-qc]  18 Apr 20232\nparametrized form [9]\nh00\nA+ (2 + \u0016\u0017+\u0017A)Hh0\nA+ (1 + \u0016\u0016+\u0016A)k2hA= 0;(2)\nwhere a prime denotes the derivative with respect to the\nconformal time \u001candH=a0=a. In such a parametriza-\ntion, the new e\u000bects arising from theories beyond GR\nare fully characterized by four parameters: \u0016 \u0017, \u0016\u0016,\u0017A, and\n\u0016A. Di\u000berent parameters correspond to di\u000berent e\u000bects\non the propagation of GWs. These e\u000bects can be divided\ninto three classes: 1) the frequency-independent e\u000bects\ninduced by \u0016 \u0016and \u0016\u0017, which include modi\fcations to the\nGW speed and friction; 2) the parity-violating e\u000bects in-\nduced by\u0017Aand\u0016A, which include the amplitude and\nvelocity birefringences of GWs respectively; and 3) the\nLorentz-violating e\u000bects induced by frequency-dependent\n\u0016\u0017and \u0016\u0016, which include the frequency-dependent damp-\ning and nonlinear dispersion relation of GWs respectively.\nThe corresponding modi\fed theories with speci\fc forms\nof the four parameters H\u0016\u0017, \u0016\u0016,H\u0017A, and\u0016Aare summa-\nrized in Table I.\n2.a. Frequency-independent e\u000bects |When the\nparameters \u0016 \u0016and \u0016\u0017are frequency-independent, they can\ninduce two distinct and frequency-independent e\u000bects on\nthe propagation of GWs. One is the modi\fcation to the\nspeed of GWs due to the nonzero of \u0016 \u0016, and the other is\nthe modi\fed friction term of the GWs if \u0016 \u0017is nonzero.\nThese frequency-independent e\u000bects can arise from sev-\neral modi\fed theories of gravity, for example, the scalar-\ntensor theory, extra dimensions, Einstein-\u001dther theory,\netc., as summarized in Table I.\nWith the parameter \u0016 \u0016, the speed of GWs are modi-\n\fed in a frequency-independent manner, cgw'1 +1\n2\u0016\u0016.\nFor a GW event with an electromagnetic counterpart,\ncgwcan be constrained by comparison with the arrival\ntime of the photons. For the binary neutron star merger\nGW170817 and its associated electromagnetic counter-\npart GRB170817A [14], the almost coincident observa-\ntion of the electromagnetic wave and the GW places an\nexquisite bound on \u0016 \u0016,\u00003\u000210\u000015<1\n2\u0016\u0016 < 7\u000210\u000016.\nNote that here we set the speed of light c= 1.\nThe parameter \u0016 \u0017induces an additional friction term on\nthe propagation equation of GWs, which can be written\nin terms of the running of the e\u000bective Planck mass as\nH\u0016\u0017=HdlnM2\n\u0003\ndlna, whereM\u0003is the running of the e\u000bective\nPlanck mass. Such a friction term changes the damp-\ning rate of the GWs during their propagation, leading\nto a GW luminosity distance dgw\nLwhich is related to the\nstandard luminosity distance dem\nLof electromagnetic sig-\nnals bydgw\nL=dem\nLexpn\n1\n2Rz\n0dz0\n1+z0\u0016\u0017(z))o\n[15]. Thus, it is\npossible to probe this GW friction H\u0016\u0017by using the mul-\ntimessenger measurements of dgw\nLanddem\nL. Such a probe\nrelies sensitively on the time evolution of H\u0016\u0017, and there\nare two approaches to parametrize its time evolution: the\ncM- [16] and \u0004-parametrizations [15] from di\u000berent mo-\ntivations. A recent constraint on the GW \frction with\nthe \u0004-parametrization was derived from an analysis of\nthe GW data of GWTC-3 with BBH mass function [17],\nwhich leads to\u00003:0<\u0016\u0017(0)<2:5.2.b. Parity-violating birefringences | The pa-\nrameters\u0017Aand\u0016Alabel the gravitational parity-\nviolating e\u000bects. The parameter \u0016Ainduces velocity\nbirefringence, leading to di\u000berent velocities of left- and\nright-hand circular polarizations of GWs, so the arrival\ntimes of them are di\u000berent. The parameter \u0017A, on the\nother hand, induces amplitude birefringence, leading to\ndi\u000berent damping rates of left- and right-hand circular\npolarizations of GWs, so the amplitude of the left-hand\nmode increases (or decreases) during its propagation,\nwhile the amplitude of the right-hand mode decreases (or\nincreases). For a large number of parity-violating theo-\nries,\u0017Aand\u0016Aare frequency-dependent. Thus, one can\nfurther parametrize \u0017Aand\u0016Aas [9]\nH\u0017A=h\n\u001aA\u000b\u0017(\u001c) (k=aM PV)\f\u0017i0\n; (3)\n\u0016A=\u001aA\u000b\u0016(\u001c) (k=aM PV)\f\u0016; (4)\nwhere\f\u0017,\f\u0016are arbitrary numbers, \u000b\u0017,\u000b\u0016are arbitrary\nfunctions of time, and MPVdenotes the energy scale of\nthe parity violation. For the GW events at the local Uni-\nverse, these two functions can be approximately treated\nas constant. The parity-violating theories with di\u000berent\nvalues of (H\u0017A,\u0016A) and (\f\u0017;\f\u0016) are summarized in Table\nI.\nWith the above parametrization, one can derive their\nexplicit GW waveforms by solving the equation of motion\n(2). It is shown [9] that the amplitude and phase mod-\ni\fcations to the GR-based waveform due to the parity-\nviolating e\u000bects can be written as\n~hA(f) =~hGR\nAe\u001aA\u000eh1ei(\u001aA\u000e\t1); (5)\nwhere ~hGR\nAis the corresponding GR-waveform, and its\nexplicit form can be found in the previous works [9].\nThe amplitude correction \u000eh1=A\u0017(\u0019f)\f\u0017is caused by\nthe parameters \u0017A, while the phase correction \u000e\t1=\nA\u0016(\u0019f)\f\u0016+1for\f\u00166=\u00001, and\u000e\t1=A\u0016lnufor\f\u0016=\n\u00001, is caused by the parameters \u0016Awith\nA\u0017=1\n2\u00122\nMPV\u0013\f\u0017h\n\u000b\u0017(\u001c0)\u0000\u000b\u0017(\u001ce)(1 +z)\f\u0017i\n;(6)\nA\u0016=(2=MPV)\f\u0016\n\u0002(\f\u0016+ 1)Zt0\nte\u000b\u0016\na\f\u0016+1dt; (7)\nwherete(t0) is the emitted (arrival) time for a GW event,\nz= 1=a(te)\u00001 is the redshift, fis the GW frequency at\nthe detector, and u=\u0019MfwithMbeing the measured\nchirp mass of the binary system, and the function \u0002(1 +\nx) = 1 +xforx6= 1 and \u0002(1 + x) = 1 forx=\u00001.\n2.c. Lorentz-violating damping and disper-\nsions |The violations of Lorentz symmetry or di\u000beo-\nmorphisms can lead to nonzero and frequency-dependent\n\u0016\u0017and \u0016\u0016. The parameter \u0016 \u0016induces frequency-dependent\nfriction in the propagation equation of GWs, while \u0016 \u0016\nmodi\fes the conventional linear dispersion relation of\nGWs to nonlinear ones. Considering both \u0016 \u0017and \u0016\u0016are3\nTABLE I. Corresponding parameters H\u0016\u0017, \u0016\u0016,H\u0017A, and\u0016Ain speci\fc modi\fed theories of gravity. The numbers in the brackets\nare the values of \f\u0016\u0017,\f\u0016\u0016,\f\u0017, and\f\u0016for each theory, which represent the frequency dependences of H\u0016\u0017, \u0016\u0016,H\u0017A, and\u0016A.\nFriction and speed Birefringences Damping and dispersion\nTheories of gravity H\u0016\u0017 \u0016\u0016H\u0017A(\f\u0017)\u0016A(\f\u0016)H\u0016\u0017(\f\u0016\u0017) \u0016\u0016(\f\u0016\u0016)\nNonlocal gravity [15, 18, 19] X | | | | |\nTime-dependent Planck mass gravity [20] X | | | | |\nExtra dimension (DGP) [21, 22] X | | | | |\nf(R) gravity [23] X | | | | |\nf(T) gravity [24] X | | | | |\nf(T;B) gravity [25] X | | | | |\nf(Q) gravity [28] X | | | | |\nGalileon Cosmology [29] X | | | | |\nHorndeski [30, 31] X X | | | |\nbeyond Horndeski GLPV [32] X X | | | |\nDHOST [33] X X | | | |\nSME gravity sector [34, 35] X X | | | |\ngeneralized scalar-torsion gravity [37] X X | | | |\nteleparallel Horndeski [25] | X | | | |\ngeneralized TeVeS theory [26, 27] | X | | | |\ne\u000bective \feld theory of in\ration [38] | X | | | |\nScalar-Gauss-Bonnet [36] | X | | | |\nEinstein-\u001dether [39, 40] | X | | | |\nbumblebee gravity [41] | X | | | |\nChern-Simons gravity [42, 49{51] | | X(1) | | |\nPalatini Chern-Simons [43] | | X(1) X(1) | |\nChiral-scalar-tensor [44{46] | | X(1) X(1) | |\nParity-violating scalar-nonmetricity [52{54] | | X(1) X(-1, 1) | |\nMetric-a\u000ene Chern-Simons [47, 48] | | | X(-1) | |\nNieh-Yan teleparallel [55{57] | | | X(-1) | |\nNew general relativity [58] | | | X(-1) | |\nChiral Weyl gravity [85] | | | X(1) | X(2)\nSpatial covariant gravities [61{63] X X X (1) X(1, 3) X(2) X(2, 4)\nHavara with parity violation [64{66] | X | X(1, 3) | X(2, 4)\nlinear gravity with Lorentz violation [77] | X | X(d\u00004\u00151) | X(d\u00004\u00152)\ndi\u000beomorphism/Lorentz violating linear gravity [78] | X | X(d\u00004\u0015\u00001) | X(d\u00004\u0015\u00002)\nHorava with mixed derivative coupling [67] | X | | X(2) X(2, 4)\nHorava gravity [68{72] | X | | | X(2, 4)\nmodi\fed dispersion in extra dimension [79] | | | | | X(2)\nNoncommutative Geometry [80, 81] | | | | | X(-2, 2)\nDouble special relativity theory [82{84] | | | | | X(-2, 1)\nconsistent 4D Einstein-Gauss-Bonnet [73{75] | | | | | X(2)\nLorentz violating Weyl gravity [76] | | | | | X(2)\nMassive gravity [59, 60] | | | | | X(-2)\nfrequency-dependent, one can parametrize them as\nH\u0016\u0017=h\n\u000b\u0016\u0017(\u001c) (k=aM LV)\f\u0016\u0017i0\n; (8)\n\u0016\u0016=\u000b\u0016\u0016(\u001c) (k=aM LV)\f\u0016\u0016; (9)\nwhere\f\u0016\u0017,\f\u0016\u0016are arbitrary numbers, \u000b\u0016\u0017,\u000b\u0016\u0016are arbitrary\nfunctions of time, and MLVdenotes the energy scale of\nLorentz violation. Similarly, we treat them as constants\nfor GW events in a local Universe. The Lorentz-violating\ntheories with di\u000berent values of ( H\u0016\u0017;\u0016\u0016) and (\f\u0016\u0017;\f\u0016\u0016) are\nsummarized in Table I.\nWith \u0016\u0017, the GWs at di\u000berent frequencies can experi-\nence di\u000berent damping rates that lead to an amplitude\nmodulation to the gravitational waveform, and with \u0016 \u0016\nthe GWs at di\u000berent frequencies can have di\u000berent phasevelocities, which lead to a phase correction to GW wave-\nforms. The modi\fed waveform with \u000b\u0016\u0017and\u000b\u0016\u0016read\n~hA(f) =~hGR\nA(f)e\u000eh2ei\u000e\t2; (10)\nwhere\u000eh2=\u0000A\u0016\u0017(\u0019f)\f\u0016\u0017and\u000e\t2=A\u0016\u0016(\u0019f)\f\u0016\u0016+1for\n\f\u0016\u00166=\u00001 and\u000e\t2=A\u0016\u0016lnufor\f\u0016\u0016=\u00001 with\nA\u0016\u0017=1\n2\u00122\nMLV\u0013\f\u0016\u0017h\n\u000b\u0016\u0017(\u001c0)\u0000\u000b\u0016\u0017(\u001ce)(1 +z)\f\u0016\u0017i\n;(11)\nA\u0016\u0016=(2=MLV)\f\u0016\u0016\n\u0002(\f\u0016\u0016+ 1)Zt0\nte\u000b\u0016\u0016\na\f\u0016\u0016+1dt: (12)\n3. Bayesian inferences on the Modi\fed wave-\nforms with GWTC-3 | With both the parity- and4\nTABLE II. The numbers of the GW events used in the\nBayesian analysis and the combined posteriors in each test. In\nseveral tests, we exclude a few events that have the strongest\nimpact in biasing the combined posterior.\nModels Number of analyzed events Combined\n\f\u0017= 1\n\f\u0016\u0017= 2\n\f\u0016\u0016= 288 in GWTC-387\n78\n87\n\f\u0016=\u00001\n\f\u0016= 3\n\f\u0016\u0016= 444 + 44 in [57] with di\u000berent template\n41 + 47 in [63] with di\u000berent template\n41 + 47 in [63] with di\u000berent template86\n84\n87\nLorentz-violating e\u000bects, the modi\fed GW waveform of\nthe circular polarization modes are given by [9]\n~hA(f) =~hGR\nA(f)e\u001aA\u000eh1+\u000eh2ei(\u001aA\u000e\t1+\u000e\t2):(13)\nThe circular polarization modes ~hRand~hLare related to\nthe modes ~h+andh\u0002via~h+=~hL+~hRp\n2and~h\u0002=~hL\u0000~hRp\n2i,\nfrom which one can obtain the waveforms for the plus and\ncross modes. Eq. (13) represents the modi\fed waveform\nwe use to compare with the GW data.\nThe tests are performed within the framework of\nBayesian inference by analyzing the open data of bi-\nnary black hole merger events in GWTC-3. We consider\nthe cases of parity- and Lorentz-violating waveforms in\nEq. (13) with di\u000berent values of \f\u0016\u0017,\f\u0016\u0016,\f\u0017, and\f\u0016sep-\narately. For parity-violating e\u000bects, we consider the am-\nplitude birefringence with \f\u0017= 1 and velocity birefrin-\ngence with \f\u0016=\u00001;3. The test of velocity birefrin-\ngence with \f\u0016= 1 was explored in [86, 87], which will\nnot be considered here. For Lorentz-violating e\u000bects, we\nconsider the frequency-dependent damping with \f\u0016\u0017= 2\nand nonlinear dispersion relations with \f\u0016\u0016= 2;4. The\nnumber of GW events analyzed in each test are summa-\nrized in Table II. Therefore we consider in total six sepa-\nrate tests, in which we employ template IMRPhenomXPHM\nfor the GR-based waveform ~hGR\n+;\u0002except GW170817 and\nIMRPhenomPv2 NRTidal for GW170817. For GR param-\neters in these waveforms, we use the prior distributions\nthat are consistent with those used in [3{5]. The priors\nfor parity- and Lorentz-violating parameters, A\u0017,A\u0016,A\u0016\u0017,\nandA\u0016\u0016are chosen to be uniformly distributed. We use\nthe open source package BILBY and a nested sampling\nmethod dynesty to perform parameter estimations with\nthe modi\fed waveform to [88]. The results of our analysis\nare presented in Fig. 1 and Table III.\nFrom the posterior distributions of A\u0017,A\u0016,A\u0016\u0017, and\nA\u0016\u0016and the redshift zof the analyzed GW events in\neach analysis, one can obtain posterior distributions of\nMPVandMLVthrough Eqs. (6), (7), (11), and (12), re-\nspectively. In Fig. 1, we display the marginalized pos-\nterior distributions of M\u0000\f\u0017\nPV with\f\u0017= 1,M\u0000\f\u0016\nPV with\n\f\u0016=\u00001;3,M\u0000\f\u0016\u0017\nLVwith\f\u0016\u0017= 2, andM\u0000\f\u0016\u0016\nLVwith\f\u0016\u0016= 2;4\nfrom selected GW events in the GWTC-3. For most GW\nevents we analyze in each test, we do not \fnd any sig-ni\fcant signatures of parity and Lorentz violations, so\nwe can obtain the combined constraints on M\u0000\f\u0017;\u0000\f\u0016\nPV\nandM\u0000\f\u0016\u0017;\u0000\f\u0016\u0016\nLV for each case by multiplying the posterior\ndistributions of the individual events together, which are\npresented as the 90% upper limits by the vertical dash\nline in each \fgure of Fig. 1. The corresponding combined\nconstraints on MPVandMPVfor each case are summa-\nrized in Table III.\nComparing with previous results, the bounds on MPV\nfor\f\u0017= 1,\f\u0016= 3, andMLVfor\f\u0016\u0016= 4 improves those\ngiven in [63, 89] by a factor of 4.1, 1.2, and 1.3, respec-\ntively. They represent the most stringent constraints for\nparity/Lorentz violations in these cases. The bounds on\nMPVfor\f\u0016=\u00001 andMLVfor\f\u0016\u0016= 2 are compati-\nble with those obtained in [57] and [6{8] from di\u000berent\nwaveform templates and methods. We also obtain the\n\frst bound on MLVfor\f\u0016\u0017= 2, which stands for the\nLorentz-violating damping e\u000bect in GWs. These con-\nstraints can be used to derive corresponding bounds on\nthe coupling coe\u000ecients of a large number of modi\fed\ntheories of gravity, which will be considered separately\nsoon.\n4. Summary | We have derived new constraints\non possible modi\fcations of GW propagation in various\ntheories of gravity, by using the GW data of the compact\nbinary merger events of GWTC-3. We began with a uni-\nversal parametrization to the modi\fed GW propagation\nwhich could arise from a large number of modi\fed theo-\nries of gravity, allowing us to analyze the GW data given\nin GWTC-3 with the parity- and Lorentz-violating GW\nwaveforms. Our new results provide several strongest\nconstraints on the gravitational parity and Lorentz viola-\ntions and the \frst constraint on Lorentz-violating damp-\ning e\u000bect. These constraints are essential in the study\nof both parity and Lorentz symmetries as fundamental\nproperties of GR, an endeavor that should provide deep\ninsight into the construction of the quantum theory of\ngravity.\nAcknowledgements | T.Z. and A.W. are supported\nin part by the National Key Research and Development\nProgram of China Grant No.2020YFC2201503, and the\nZhejiang Provincial Natural Science Foundation of China\nunder Grant No. LR21A050001 and LY20A050002, the\nNational Natural Science Foundation of China under\nGrant No. 12275238, No. 11675143, and No. 11975203.\nW.Z. is supported by the National Key Research and De-\nvelopment Program of China Grant No.2022YFC2204602\nand No. 2021YFC2203102, NSFC Grants No. 12273035,\nthe Fundamental Research Funds for the Central Uni-\nversities, and the 111 Project for \"Observational and\nTheoretical Research on Dark Matter and Dark Energy\"\n(B23042). We are grateful that this research has made\nuse of data or software obtained from the Gravitational\nWave Open Science Center (gw-openscience.org), a ser-\nvice of LIGO Laboratory, the LIGO Scienti\fc Collabora-\ntion, the Virgo Collaboration, and KAGRA.5\nTABLE III. Results from the Bayesian analysis of the parity- and Lorentz-violating waveforms with GW events in GWTC-3.\nThe table shows 90%-credible upper bounds on MPVfor\f\u0017=\u00001 (for velocity birefringence) and lower bounds on MPVand\nMLVfor other cases. We also include bounds for several cases derived from existing tests with GWTC-1/GWTC-2/GWTC-3\nin Refs. [6{8, 57, 63, 89] for comparison.\nMPV[GeV] MLV[GeV]\n\f\u0017= 1 \f\u0016=\u00001 \f\u0016= 3 \f\u0016\u0017= 2 \f\u0016\u0016= 2 \f\u0016\u0016= 4\nGWTC-1 1 :0\u000210\u000022[89] | | 0 :8\u000210\u000011[6] |\nGWTC-2 | 6 :5\u000210\u000042[57] 1:0\u000210\u000014[63] | 1 :3\u000210\u000011[7] 2:4\u000210\u000016[63]\nGWTC-3 | | | | 1 :8\u000210\u000011[8] |\nThis work 4:1\u000210\u0000227:8\u000210\u0000421:2\u000210\u0000141:4\u000210\u0000211:2\u000210\u0000113:2\u000210\u000016\n0 2 4 6 8 10 12 14\nM1\nPV[1021/GeV]\n0.000.050.100.150.200.250.300.350.40Probability DensitiyGW190727_060333\nGW200112_155838\nGW190513_205428\nGW190915_235702\nGW190803_022701\nGW200311_115853\nother GW events\n90% upper limit\n0 10 20 30 40 50\nMPV[1042GeV]\n0.000.010.020.030.040.050.06Probability DensitiyGW200128_022011\nGW190413_134308\nGW190828_063405\nGW190910_112807\nGW190915_235702\nGW190731_140936\nother GW events\n90% upper limit\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nM3\nPV[1042/GeV3]\n0.00.20.40.60.81.01.21.41.6Probability DensitiyGW191129_134029\nGW200316_215756\nGW190924_021846\nGW191204_171526\nGW191105_143521\nGW190725_174728\nother GW events\n90% upper limit\n0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0\nM2\nLV[1042/GeV2]\n0.000.250.500.751.001.251.501.752.00Probability DensitiyGW191204_171526\nGW190707_093326\nGW190512_180714\nGW200225_060421\nGW190708_232457\nGW191215_223052\nother GW events\n90% upper limit\n0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0\nM2\nLV[1022/GeV2]\n0.00.20.40.60.81.0Probability DensitiyGW190708_232457\nGW190512_180714\nGW190720_000836\nGW200225_060421\nGW190412\nGW190408_181802\nother GW events\n90% upper limit\n0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9\nM4\nLV[1063/GeV4]\n0.02.55.07.510.012.515.017.5Probability DensitiyGW200115_042309\nGW191129_134029\nGW190917_114630\nGW170817\nGW200316_215756\nGW190924_021846\nother GW events\n90% upper limit\nFIG. 1. 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J.\n908, 58 (2021)."    },    {        "title": "1406.2403v1.Wigner_s_Space_time_Symmetries_based_on_the_Two_by_two_Matrices_of_the_Damped_Harmonic_Oscillators_and_the_Poincaré_Sphere.pdf",        "content": "arXiv:1406.2403v1  [math-ph]  10 Jun 2014Wigner’s Space-time Symmetries based on the\nTwo-by-two Matrices of the Damped Harmonic\nOscillators and the Poincar´ e Sphere\nSibel Ba¸ skal\nDepartment of Physics, Middle East Technical University, 06800 An kara, Turkey\ne-mail: baskal@newton.physics.metu.edu.tr\nYoung S. Kim\nCenter for Fundamental Physics, University of Maryland,\nCollege Park, Maryland 20742, U.S.A.\ne-mail: yskim@umd.edu\nMarilyn E. Noz\nDepartment of Radiology, New York University\nNew York, New York, 10016, U.S.A.\ne-mail: marilyne.noz@gmail.com\nAbstract\nThesecond-order differential equation for a dampedharmonic oscillator can becon-\nverted to two coupled first-order equations, with two two-by -two matrices leading to\nthe groupSp(2). It is shown that this oscillator system contains the ess ential features\nof Wigner’s little groups dictating the internal space-tim e symmetries of particles in\nthe Lorentz-covariant world. The little groups are the subg roups of the Lorentz group\nwhose transformations leave the four-momentum of a given pa rticle invariant. It is\nshown that the dampingmodes of the oscillator correspond to the little groups for mas-\nsive and imaginary-mass particles respectively. When the s ystem makes the transition\nfrom the oscillation to damping mode, it corresponds to the l ittle group for massless\nparticles. Rotations around the momentum leave the four-mo mentum invariant. This\ndegree of freedom extends the Sp(2) symmetry to that of SL(2,c) corresponding to\nthe Lorentz group applicable to the four-dimensional Minko wski space. The Poincar´ e\nsphere contains the SL(2,c) symmetry. In addition, it has a non-Lorentzian parameter\nallowing us to reduce the mass continuously to zero. It is thu s possible to construct\nthe little group for massless particles from that of the mass ive particle by reducing its\nmass to zero. Spin-1/2 particles and spin-1 particles are di scussed in detail.1 Introduction\nWe are quite familiar with the second-order differential equa tion\nmd2y\ndt2+bdy\ndt+Ky= 0, (1)\nfor a damped harmonic oscillator. This equation has the same mathematical form as\nLd2Q\ndt2+RdQ\ndt+1\nCQ= 0, (2)\nforelectrical circuits, where L,R,andCaretheinductance, resistance, andcapacitance\nrespectively. These two equations play fundamental roles i n physical and engineering\nsciences. Sincethey start from the sameset of mathematical equations, one set of prob-\nlems can be studied in terms of the other. For instance, many m echanical phenomena\ncan be studied in terms of electrical circuits.\nIn Eq.(1), when b= 0, the equation is that of a simple harmonic oscillator with\nthe frequency ω=/radicalbig\nK/m. Asbincreases, the oscillation becomes damped. When bis\nlarger than 2√\nKm, the oscillation disappears, as the solution is a damping mo de.\nConsider that increasing bcontinuously, while difficult mechanically, can be done\nelectrically using Eq.(2) by adjusting the resistance R.The transition from the oscil-\nlation mode to the damping mode is a continuous physical proc ess.\nThisbterm leads to energy dissipation, but is not regarded as a fun damental force.\nIt is inconvenient in the Hamiltonian formulation of mechan ics and troublesome in\ntransition to quantum mechanics, yet, plays an important ro le in classical mechanics.\nIn this paper this term will help us understandthe fundament al space-time symmetries\nof elementary particles.\nWe are interested in constructing the fundamental symmetry group for particles in\nthe Lorentz-covariant world. For this purpose, we transfor m the second-order differen-\ntial equation of Eq.(1) to two coupled first-order equations using two-by-two matrices.\nOnly two linearly independent matrices are needed. They are the anti-symmetric and\nsymmetric matrices\nA=/parenleftbigg0−i\ni0/parenrightbigg\n,andS=/parenleftbigg0i\ni0/parenrightbigg\n, (3)\nrespectively. The anti-symmetric matrix Ais Hermitian and corresponds to the oscil-\nlation part, while the symmetric Smatrix corresponds to the damping.\nThese two matrices lead to the Sp(2) group consisting of two-by-two unimodu-\nlar matrices with real elements. This group is isomorphic to the three-dimensional\nLorentz group applicable to two space-like and one time-lik e coordinates. This group\nis commonly called the O(2,1) group.\nThisO(2,1) group can explain all the essential features of Wigner’s l ittle groups\ndictating internal space-time symmetries of particles [1] . Wigner defined his little\ngroups as the subgroups of the Lorentz group whose transform ations leave the four-\nmomentum of a given particle invariant. He observed that the little groups are different\nfor massive, massless, and imaginary-mass particles. It ha s been a challenge to design\na mathematical model which will combine those three into one formalism, but we show\nthat the damped harmonic oscillator provides the desired ma thematical framework.\n2For the two space-like coordinates, we can assign one of them to the direction of the\nmomentum, and the other to the direction perpendicular to th e momentum. Let the\ndirection of the momentum be along the zaxis, and let the perpendicular direction be\nalong thexaxis. We therefore study the kinematics of the group within t hezxplane,\nthen see what happens when we rotate the system around the zaxis without changing\nthe momentum [2].\nThe Poincar´ e sphere for polarization optics contains the SL(2,c) symmetry isomor-\nphic to the four-dimensional Lorentz group applicable to th e Minkowski space [3, 4, 5,\n6, 7]. Thus, the Poincar´ e sphere extends Wigner’s picture i nto the three space-like and\none time-like coordinates. Specifically, this extension ad ds rotations around the given\nmomentum which leaves the four-momentum invariant [2].\nWhile theparticle mass isaLorentz-invariant variable, th ePoincar´ e spherecontains\nan extra variable which allows the mass to change. This varia ble allows us to take the\nmass-limit of the symmetry operations. The transverse rota tional degrees of freedom\ncollapse into onegauge degree offreedom andpolarization o f neutrinosisaconsequence\nof the requirement of gauge invariance [8, 9].\nTheSL(2,c) group contains symmetries not seen in the three-dimension al rotation\ngroup. While we are familiar with two spinors for a spin-1/2 p article in nonrelativistic\nquantum mechanics, there are two additional spinors due to t he reflection properties of\nthe Lorentz group. There are thus sixteen bilinear combinat ions of those four spinors.\nThis leads to two scalars, two four-vectors, and one antisym metric four-by-four tensor.\nThe Maxwell-type electromagnetic field tensor can be obtain ed as a massless limit of\nthis tensor [10].\nInSec. 2, wereview thedampedharmonicoscillator inclassi cal mechanics, andnote\nthat the solution can be either in the oscillation mode or dam ping mode depending\non the magnitude of the damping parameter. The translation o f the second order\nequation into a first order differential equation with two-by- two matrices is possible.\nThis first-order equation is similar to the Schr¨ odinger equ ation for a spin-1/2 particle\nin a magnetic field.\nSection 3showsthat thetwo-by-two matrices of Sec. 2can bef ormulated interms of\ntheSp(2)group. ThesematricescanbedecomposedintotheBargman nandWignerde-\ncompositions. Furthermore, this group is isomorphic to the three-dimensional Lorentz\ngroup with two space and one time-like coordinates.\nIn Sec. 4, it is noted that this three-dimensional Lorentz gr oup has all the essential\nfeatures of Wigner’s little groups which dictate the intern al space-time symmetries of\ntheparticles intheLorentz-covariant world. Wigner’s lit tle groupsarethesubgroupsof\nthe Lorentz group whose transformations leave the four-mom entum of a given particle\ninvariant. The Bargmann Wigner decompositions are shown to be useful tools for\nstudying the little groups.\nIn Sec. 5, we note that the given momentum is invariant under r otations around\nit. The addition of this rotational degree of freedom extend s theSp(2) symmetry to\nthe six-parameter SL(2,c) symmetry. In the space-time language, this extends the\nthree dimensional group to the Lorentz group applicable to t hree space and one time\ndimensions.\nSection 6 shows that thePoincar´ e spherecontains the symme tries ofSL(2,c) group.\nIn addition, it contains an extra variable which allows us to change the mass of the\nparticle, which is not allowed in the Lorentz group.\n3In Sec. 7, the symmetries of massless particles are studied i n detail. In addition to\nrotation aroundthemomentum, Wigner’s little groupgenera tes gauge transformations.\nWhile gauge transformations on spin-1 photons are well know n, the gauge invariance\nleads to the polarization of massless spin-1/2 particles, a s observed in neutrino polar-\nizations.\nIn Sec. 8, it is noted that there are four spinors for spin-1/2 particles in the Lorentz-\ncovariant world. It is thus possible to construct sixteen bi linear forms, applicable to\ntwo scalars, and two vectors, and one antisymmetric second- rank tensor. The elec-\ntromagnetic field tensor is derived as the massless limit. Th is tensor is shown to be\ngauge-invariant.\n2 Classical Damped Oscillators\nFor convenience, we write Eq.(1) as\nd2y\ndt2+2µdy\ndt+ω2y= 0, (4)\nwith\nω=/radicaligg\nK\nm,andµ=b\n2m, (5)\nThe damping parameter µis positive when there are no external forces. When ωis\ngreater than µ, the solution takes the form\ny=e−µt/bracketleftbigC1cos(ω′t)+C2sin(ω′t)/bracketrightbig, (6)\nwhere\nω′=/radicalig\nω2−µ2, (7)\nandC1andC2are the constants to be determined by the initial conditions . This\nexpression is for a damped harmonic oscillator. Conversely , whenµis greater than ω,\nthe quantity inside the square-root sign is negative, then t he solution becomes\ny=e−µt/bracketleftbigC3cosh(µ′t)+C4sinh(µ′t)/bracketrightbig, (8)\nwith\nµ′=/radicalig\nµ2−ω2, (9)\nIfω=µ, both Eq.(6) and Eq.(8) collapse into one solution\ny(t) =e−µt[C5+C6t]. (10)\nThese three different cases are treated separately in textboo ks. Here we are inter-\nested in the transition from Eq.(6) to Eq.(8), via Eq.(10). F or convenience, we start\nfromµgreater than ωwithµ′given by Eq.(9).\nFor a given value of µ, the square root becomes zero when ωequalsµ. Ifωbecomes\nlarger, the square root becomes imaginary and divides into t wo branches.\n±i/radicalig\nω2−µ2. (11)\n4This is a continuous transition, but not an analytic continu ation. To study this in\ndetail, we translate the second order differential equation o f Eq.(4) into the first-order\nequation with two-by-two matrices.\nGiven the solutions of Eq.(6), and Eq.(10), it is convenient to useψ(t) defined as\nψ(t) =eµty(t),andy=e−µtψ(t). (12)\nThenψ(t) satisfies the differential equation\nd2ψ(t)\ndt2+(ω2−µ2)ψ(t) = 0. (13)\n2.1 Two-by-two Matrix Formulation\nIn order to convert this second order equation to a first order system we introduce\nψ1,2(t) = (ψ1(t),ψ2(t)). Then we have a system of two equations\ndψ1(t)\ndt= (µ−ω)ψ2(t),\ndψ2(t)\ndt= (µ+ω)ψ1(t) (14)\nwhich can be written in matrix form as\nd\ndt/parenleftbiggψ1\nψ2/parenrightbigg\n=/parenleftbigg0µ−ω\nµ+ω0/parenrightbigg/parenleftbiggψ1\nψ2/parenrightbigg\n. (15)\nUsing the Hermitian and anti-Hermitian matrices of Eq.(3) i n Sec. 1, we construct\nthe linear combination\nH=ω/parenleftbigg0−i\ni0/parenrightbigg\n+µ/parenleftbigg0i\ni0/parenrightbigg\n. (16)\nWe can then consider the first-order differential equation\ni∂\n∂tψ(t) =Hψ(t). (17)\nWhile this equation is like the Schr¨ odinger equation for an electron in a magnetic field,\nthe two-by-two matrix is not Hermitian. Its first matrix is He rmitian, but the second\nmatrix is anti-Hermitian. It is of course an interesting pro blem to give a physical inter-\npretation to this non-Hermitian matrix in connection with q uantum dissipation [11],\nbut this is beyond the scope of the present paper. The solutio n of Eq.(17) is\nψ(t) = exp/braceleftbigg/parenleftbigg0−ω+µ\nω+µ0/parenrightbigg\nt/bracerightbigg/parenleftbiggC7\nC8/parenrightbigg\n, (18)\nwhereC7=ψ1(0) andC8=ψ2(0) respectively.\n2.2 Transition from the Oscillation Mode to Damping\nMode\nIt appears straight-forward to compute this expression by a Taylor expansion, but it\nis not. This issue was extensively discussed in previous pap ers by two of us [12, 13].\nThe key idea is to write the matrix\n/parenleftbigg0−ω+µ\nω+µ0/parenrightbigg\n(19)\n5as a similarity transformation of\nω′/parenleftbigg0−1\n1 0/parenrightbigg\n(ω>µ), (20)\nand as that of\nµ′/parenleftbigg0 1\n1 0/parenrightbigg\n(µ>ω), (21)\nwithω′andµ′defined in Eq.(7) and Eq.(9), respectively.\nThen the Taylor expansion leads to\n/parenleftbiggcos(ω′t) −/radicalbig\n(ω−µ)/(ω+µ) sin(ω′t)/radicalbig\n(ω+µ)/(ω−µ) sin(ω′t) cos( ω′t)/parenrightbigg\n,(22)\nwhenωis greater than µ. The solution ψ(t) takes the form\n/parenleftbiggC7cos(ω′t)−C8/radicalbig\n(ω−µ)/(ω+µ) sin(ω′t)\nC7/radicalbig(ω+µ)/(ω−µ) sin(ω′t)+C8cos(ω′t)/parenrightbigg\n. (23)\nIfµis greater than ω, the Taylor expansion becomes\n/parenleftbiggcosh(µ′t)/radicalbig\n(µ−ω)/(µ+ω) sinh(µ′t)/radicalbig\n(µ+ω)/(µ−ω) sinh(µ′t) cosh( µ′t)/parenrightbigg\n.(24)\nWhenωis equal toµ, both Eqs.(22) and (24) become\n/parenleftbigg1 0\n2ωt1/parenrightbigg\n. (25)\nIfωis sufficiently close to but smaller than µ, the matrix of Eq.(24) becomes\n/parenleftbigg1+(ǫ/2)(2ωt)2+ǫ(2ωt)\n(2ωt) 1+( ǫ/2)(2ωt)2/parenrightbigg\n, (26)\nwith\nǫ=µ−ω\nµ+ω. (27)\nIfωis sufficiently close to µ, we can let\nµ+ω= 2ω,andµ−ω= 2µǫ. (28)\nIfωis greater than µ,ǫdefined in Eq.(27) becomes negative, the matrix of Eq.(22)\nbecomes /parenleftbigg1−(−ǫ/2)(2ωt)2−(−ǫ)(2ωt)\n2ωt 1−(−ǫ/2)(2ωt)2/parenrightbigg\n. (29)\nWe can rewrite this matrix as\n/parenleftigg\n1−(1/2)/bracketleftbig/parenleftbig2ω√−ǫ/parenrightbigt/bracketrightbig2−√−ǫ/bracketleftbig/parenleftbig2ω√−ǫ/parenrightbigt/bracketrightbig\n2ωt 1−(1/2)/bracketleftbig/parenleftbig2ω√−ǫ/parenrightbigt/bracketrightbig2/parenrightigg\n. (30)\nIfǫbecomes positive, Eq.(26) can be written as\n/parenleftbigg1+(1/2)[(2ω√ǫ)t]2√ǫ[(2ω√ǫ)t]\n2ωt 1+(1/2)[(2ω√ǫ)t]2/parenrightbigg\n. (31)\nThe transition from Eq.(30) to Eq.(31) is continuous as they become identical when\nǫ= 0.Asǫchanges its sign, the diagonal elements of above matrices te ll us how\ncos(ω′t) becomes cosh( µ′t). As for the upper-right element element, −sin(ω′t) becomes\nsinh(µ′t).This non-analytic continuity is illustrated in Fig. 1.\n6Figure1: Transitions fromsintosinh, andfromcostocosh .They arecontinuous transitions.\nTheir first derivatives are also continuous, but the second derivat ives are not. Thus, they\nare not analytically but only tangentially continuous.\n2.3 Mathematical Forms of the Solutions\nInthissection, weusetheHeisenbergapproachtotheproble m, andobtainthesolutions\nin the form of two-by-two matrices. We note that\n1. For the oscillation mode, the trace of the matrix is smalle r than 2. The solution\ntakes the form of /parenleftbiggcos(x)−e−ηsin(x)\neηsin(x) cos(x)/parenrightbigg\n, (32)\nwith trace 2cos( x). The trace is independent of η.\n2. For the damping mode, the trace of the matrix is greater tha n 2.\n/parenleftbiggcosh(x)e−ηsinh(x)\neηsinh(x) cosh(x)/parenrightbigg\n, (33)\nwith trace 2cosh( x). Again, the trace is independent of η.\n3. For the transition mode, the trace is equal to 2, and the mat rix is triangular and\ntakes the form of /parenleftbigg1 0\nγ1/parenrightbigg\n. (34)\nWhenxapproaches zero, the Eq.(32) and Eq.(33) take the form\n/parenleftbigg1−x2/2−xe−η\nxeη1−x2/2/parenrightbigg\n,and/parenleftbigg1+x2/2xe−η\nxeη1+x2/2/parenrightbigg\n, (35)\nrespectively. These two matrices have the same lower-left e lement. Let us fix this\nelement to be a positive number γ. Then\nx=γe−η. (36)\nThen the matrices of Eq.(35) become\n/parenleftbigg1−γ2e−2η/2−γe−2η\nγ 1−γ2e−2η/2/parenrightbigg\n,and/parenleftbigg1+γ2e−2η/2γe−2η\nγ 1+γ2e−2η/2/parenrightbigg\n,(37)\n7If we introduce a small number ǫdefined as\nǫ=√γe−η, (38)\nthe matrices of Eq.(37) become\n/parenleftbigge−η/20\n0eη/2/parenrightbigg/parenleftbigg1−γǫ2/2√γǫ√γǫ1−γǫ2/2/parenrightbigg/parenleftbiggeη/20\n0e−η/2/parenrightbigg\n,\n/parenleftbigge−η/20\n0eη/2/parenrightbigg/parenleftbigg1+γǫ2/2√γǫ√γǫ1+γǫ2/2/parenrightbigg/parenleftbiggeη/20\n0e−η/2/parenrightbigg(39)\nrespectively, with e−η=ǫ/√γ.\n3 Groups of Two-by-two Matrices\nIf atwo-by-two matrix has fourcomplex elements, it has eigh t independentparameters.\nIf the determinant of this matrix is one, it is known as an unim odular matrix and\nthe number of independent parameters is reduced to six. The g roup of two-by-two\nunimodular matrices is called SL(2,c). This six-parameter group is isomorphic to the\nLorentz group applicable to the Minkowski space of three spa ce-like and one time-like\ndimensions [15].\nWe can start with two subgroups of SL(2,c).\n1. While the matrices of SL(2,c) are not unitary, we can consider the subset con-\nsisting of unitary matrices. This subgroup is called SU(2), and is isomorphic to\nthe three-dimensional rotation group. This three-paramet er group is the basic\nscientific language for spin-1/2 particles.\n2. We can also consider the subset of matrices with real eleme nts. This three-\nparametergroupiscalled Sp(2)andisisomorphictothethree-dimensionalLorentz\ngroup applicable to two space-like and one time-like coordi nates.\nIn the Lorentz group, there are three space-like dimensions withx,y,andzco-\nordinates. However, for many physical problems, it is more c onvenient to study the\nproblem in the two-dimensional ( x,z) plane first and generalize it to three-dimensional\nspace by rotating the system around the zaxis. This process can be called Euler\ndecomposition and Euler generalization [2].\nFirst we study Sp(2) symmetry in detail, and achieve the generalization by au g-\nmenting the two-by-two matrix corresponding to the rotatio n around the zaxis. In\nthis section, we study in detail properties of Sp(2) matrices, then generalize them to\nSL(2,c) in Sec. 5.\nTherearethreeclassesof Sp(2)matrices. Theirtracescanbesmallerorgreaterthan\ntwo, or equal to two. While these subjects are already discus sed in the literature [16,\n17, 18] our main interest is what happensas thetrace goes fro mless thantwo to greater\nthan two. Here we are guided by the model we have discussed in S ec. 2, which accounts\nfor the transition from the oscillation mode to the damping m ode.\n83.1 Lie Algebra of Sp(2)\nThe two linearly independent matrices of Eq.(3) can be writt en as\nK1=1\n2/parenleftbigg0i\ni0/parenrightbigg\n,andJ2=1\n2/parenleftbigg0−i\ni0/parenrightbigg\n. (40)\nHowever, the Taylor series expansion of the exponential for m of Eq.(22) or Eq.(24)\nrequires an additional matrix\nK3=1\n2/parenleftbiggi0\n0−i/parenrightbigg\n. (41)\nThese matrices satisfy the following closed set of commutat ion relations.\n[K1,J2] =iK3,[J2,K3] =iK1,[K3,K1] =−iJ2. (42)\nThese commutation relations remain invariant under Hermit ian conjugation, even\nthoughK1andK3are anti-Hermitian. The algebra generated by these three ma -\ntrices is known in the literature as the group Sp(2) [18]. Furthermore, the closed set\nof commutation relations is commonly called the Lie algebra . Indeed, Eq.(42) is the\nLie algebra of the Sp(2) group.\nThe Hermitian matrix J2generates the rotation matrix\nR(θ) = exp( −iθJ2) =/parenleftbiggcos(θ/2)−sin(θ/2)\nsin(θ/2) cos(θ/2)/parenrightbigg\n, (43)\nand the anti-Hermitian matrices K1andK2, generate the following squeeze matrices.\nS(λ) = exp( −iλK1) =/parenleftbiggcosh(λ/2) sinh(λ/2)\nsinh(λ/2) cosh(λ/2)/parenrightbigg\n, (44)\nand\nB(η) = exp( −iηK3) =/parenleftbiggexp(η/2) 0\n0 exp( −η/2)/parenrightbigg\n, (45)\nrespectively.\nReturning to the Lie algebra of Eq.(42), since K1andK3are anti-Hermitian, and\nJ2is Hermitian, the set of commutation relation is invariant u nder the Hermitian\nconjugation. In other words, the commutation relations rem ain invariant, even if we\nchange the sign of K1andK3, while keeping that of J2invariant. Next, let us take\nthe complex conjugate of the entire system. Then both the JandKmatrices change\ntheir signs.\n3.2 Bargmann and Wigner Decompositions\nSince theSp(2) matrix has three independent parameters, it can be writt en as [16]\n/parenleftbiggcos(α1/2)−sin(α1/2)\nsin(α1/2) cos(α1/2)/parenrightbigg/parenleftbiggcoshχsinhχ\nsinhχcoshχ/parenrightbigg/parenleftbiggcos(α2/2)−sin(α2/2)\nsin(α2/2) cos(α2/2)/parenrightbigg\n(46)\nThis matrix can be written as\n/parenleftbiggcos(δ/2)−sin(δ/2)\nsin(δ/2) cos(δ/2)/parenrightbigg/parenleftbigga b\nc d/parenrightbigg/parenleftbiggcos(δ/2) sin(δ/2)\n−sin(δ/2) cos(δ/2)/parenrightbigg\n(47)\n9where\n/parenleftbigga b\nc d/parenrightbigg\n=/parenleftbiggcos(α/2)−sin(α/2)\nsin(α/2) cos(α/2)/parenrightbigg/parenleftbiggcoshχsinhχ\nsinhχcoshχ/parenrightbigg/parenleftbiggcos(α/2)−sin(α/2)\nsin(α/2) cos(α/2)/parenrightbigg\n(48)\nwith\nδ=1\n2(α1−α2),andα=1\n2(α1+α2). (49)\nIf we complete the matrix multiplication of Eq.(48), the res ult is\n/parenleftbigg(coshχ)cosαsinhχ−(coshχ)sinα\nsinhχ+(coshχ)sinα(coshχ)cosα/parenrightbigg\n. (50)\nWe shall call hereafter the decomposition of Eq.(48) the Bar gmann decomposition.\nThis means that every matrix in the Sp(2) group can be brought to the Bargmann\ndecomposition by a similarity transformation of rotation, as given in Eq.(47). This\ndecomposition leads to an equidiagonal matrix with two inde pendent parameters.\nFor the matrix of Eq.(48), we can now consider the following t hree cases. Let us\nassumethat χispositive, andtheangle θislessthan90o. Let uslookat theupper-right\nelement.\n1. If it is negative with [sinh χ <(coshχ)sinα], then the trace of the matrix is\nsmaller than 2, and the matrix can be written as\n/parenleftbiggcos(θ/2)−e−ηsin(θ/2)\neηsin(θ/2) cos(θ/2)/parenrightbigg\n, (51)\nwith\ncos(θ/2) = (coshχ)cosα,ande−2η=(coshχ)sinα−sinhχ\n(coshχ)sinα+sinhχ.(52)\n2. If it is positive with [sinh χ >(coshχ)sinα)], then the trace is greater than 2,\nand the matrix can be written as\n/parenleftbiggcosh(λ/2)e−ηsinh(λ/2)\neηsinh(λ/2) cosh( λ/2)/parenrightbigg\n, (53)\nwith\ncosh(λ/2) = (coshχ)cosα,ande−2η=sinhχ−(coshχ)sinα\n(coshχ)sinα+sinhχ.(54)\n3. If it is zero with [(sinh χ= (coshχ)sinα)], then the trace is equal to 2, and the\nmatrix takes the form /parenleftbigg1 0\n2sinhχ1/parenrightbigg\n, (55)\nThe above repeats the mathematics given in Subsec. 2.3.\nReturning to Eq.(51) and Eq.(52), they can be decomposed int o\nM(θ,η) =/parenleftbiggeη/20\n0e−η/2/parenrightbigg/parenleftbiggcos(θ/2)−sin(θ/2)\nsin(θ/2) cos(θ/2)/parenrightbigg/parenleftbigge−η/20\n0eη/2/parenrightbigg\n,(56)\n10and\nM(λ,η) =/parenleftbiggeη/20\n0e−η/2/parenrightbigg/parenleftbiggcosh(λ/2) sinh(λ/2)\nsinh(λ/2) cos(λ/2)/parenrightbigg/parenleftbigge−η/20\n0eη/2/parenrightbigg\n,(57)\nrespectively. In view of the physical examples given in Sec. 7, we shall call this the\n“Wigner decomposition.” Unlike the Bargmann decompositio n, the Wigner decompo-\nsition is in the form of a similarity transformation.\nWe note that both Eq.(56) and Eq.(57) are written as similari ty transformations.\nThus\n[M(θ,η)]n=/parenleftbiggcos(nθ/2)−e−ηsin(nθ/2)\neηsin(nθ/2) cos(nθ/2)/parenrightbigg\n,\n[M(λ,η)]n=/parenleftbiggcosh(nλ/2)eηsinh(nλ/2)\ne−ηsinh(nλ/2) cosh(nλ/2)/parenrightbigg\n,\n[M(γ)]n=/parenleftbigg1 0\nnγ1/parenrightbigg\n. (58)\nThese expressions are useful for studying periodic systems [14].\nThe question is what physics these decompositions describe in the real world. To\naddress this, we study what the Lorentz group does in the real world, and study\nisomorphism between the Sp(2) group and the Lorentz group applicable to the three-\ndimensional space consisting of one time and two space coord inates.\n3.3 Isomorphism with the Lorentz group\nThe purpose of this section is to give physical interpretati ons of the mathematical for-\nmulas given in Subsec. 3.2. We will interpret these formulae in terms of the Lorentz\ntransformations which are normally described by four-by-f our matrices. For this pur-\npose, it is necessary to establish a correspondence between the two-by-two representa-\ntion of Sec. 3.2 and the four-by-four representations of the Lorentz group.\nLet us consider the Minkowskian space-time four-vector\n(t,z,x,y) (59)\nwhere/parenleftbigt2−z2−x2−y2/parenrightbigremainsinvariantunderLorentztransformations. TheLore ntz\ngroup consists of four-by-four matrices performing Lorent z transformations in the\nMinkowski space.\nIn order to give physical interpretations to the three two-b y-two matrices given\nin Eq.(43), Eq.(44), and Eq.(45), we consider rotations aro und theyaxis, boosts\nalong thexaxis, and boosts along the zaxis. The transformation is restricted in the\nthree-dimensional subspace of ( t,z,x). It is then straight-forward to construct those\nfour-by-four transformation matrices where the ycoordinate remains invariant. They\nare given in Table 1. Their generators also given. Those four -by-four generators satisfy\nthe Lie algebra given in Eq.( 42).\n11Table 1: Matrices in the two-by-two representation, and their cor responding four-by-four\ngenerators and transformation matrices.\nMatrices Generators Four-by-four Transform Matrices\nR(θ) J2=1\n2/parenleftbigg0−i\ni0/parenrightbigg\n0 0 0 0\n0 0−i0\n0i0 0\n0 0 0 0\n\n1 0 0 0\n0 cosθ−sinθ0\n0 sinθcosθ0\n0 0 0 1\n\nB(η) K3=1\n2/parenleftbiggi0\n0−i/parenrightbigg\n0i0 0\ni0 0 0\n0 0 0 0\n0 0 0 0\n\ncoshηsinhη0 0\nsinhηcoshη0 0\n0 0 1 0\n0 0 0 1\n\nS(λ) K1=1\n2/parenleftbigg0i\ni0/parenrightbigg\n0 0i0\n0 0 0 0\ni0 0 0\n0 0 0 0\n\ncoshλ0 sinhλ0\n0 1 0 0\nsinhλ0 coshλ0\n0 0 0 1\n\n4 Internal Space-time Symmetries\nWe have seen that there corresponds a two-by-two matrix for e ach four-by-four Lorentz\ntransformation matrix. It is possible to give physical inte rpretations to those four-by-\nfour matrices. It must thus be possible to attach a physical i nterpretation to each\ntwo-by-two matrix.\nSince 1939 [1] when Wigner introduced the concept of the litt le groups many papers\nhave been published on this subject, but most of them were bas ed on the four-by-four\nrepresentation. In this section, we shall give the formalis m of little groups in the\nlanguage of two-by-two matrices. In so doing, we provide phy sical interpretations to\nthe Bargmann and Wigner decompositions introduced in Sec. 3 .2.\n4.1 Wigner’s Little Groups\nIn[1], Wignerstarted withafreerelativistic particle wit hmomentum, thenconstructed\nsubgroups of the Lorentz group whose transformations leave the four-momentum in-\nvariant. These subgroups thus define the internal space-tim e symmetry of the given\nparticle. Without loss of generality, we assume that the par ticle momentum is along\nthezdirection. Thus rotations around the momentum leave the mom entum invariant,\nand this degree of freedom defines the helicity, or the spin pa rallel to the momentum.\nWe shall use the word “Wigner transformation” for the transf ormation which leaves\nthe four-momentum invariant\n1. For a massive particle, it is possible to find a Lorentz fram e where it is at rest\nwith zero momentum. The four-momentum can be written as m(1,0,0,0), where\n12mis the mass. This four-momentum is invariant under rotation s in the three-\ndimensional ( z,x,y) space.\n2. For an imaginary-mass particle, there is the Lorentz fram e where the energy\ncomponent vanishes. The momentum four-vector can be writte n asp(0,1,0,0) ,\nwherepis the magnitude of the momentum.\n3. If the particle is massless, its four-momentum becomes p(1,1,0,0). Here the first\nand second components are equal in magnitude.\nThe constant factors in these four-momenta do not play any si gnificant roles. Thus we\nwrite them as (1 ,0,0,0),(0,1,0,0), and (1,1,0,0) respectively. Since Wigner worked\nwith these three specific four-momenta [1], we call them Wign er four-vectors.\nAll of these four-vectors are invariant under rotations aro und thezaxis. The\nrotation matrix is\nZ(φ) =\n1 0 0 0\n0 1 0 0\n0 0 cosφ−sinφ\n0 0 sinφcosφ\n. (60)\nIn addition, the four-momentum of a massive particle is inva riant under the rotation\naroundthe yaxis, whosefour-by-fourmatrixwasgiven inTable1. Thefou r-momentum\nof an imaginary particle is invariant under the boost matrix S(λ) given in Table 1. The\nproblem for the massless particle is more complicated, but w ill be discussed in detail\nin Sec. 7. See Table 2.\nTable 2: Wigner four-vectors and Wigner transformation matrices applicable to two space-\nlike and one time-like dimensions. Each Wigner four-vector remains inv ariant under the\napplication of its Wigner matrix.\nMass Wigner Four-vector Wigner Transformation\nMassive (1 ,0,0,0)\n1 0 0 0\n0 cosθ−sinθ0\n0 sinθcosθ0\n0 0 0 1\n\nMassless (1 ,1,0,0)\n1+γ2/2−γ2/2γ0\nγ2/2 1−γ2/2γ0\n−γ γ 1 0\n0 0 0 1\n\nImaginary mass (0 ,1,0,0)\ncoshλ0 sinhλ0\n0 1 0 0\nsinhλ0 coshλ0\n0 0 0 1\n\n134.2 Two-by-two Formulation of Lorentz Transformations\nThe Lorentz group is a group of four-by-four matrices perfor ming Lorentz transforma-\ntions on the Minkowskian vector space of ( t,z,x,y),leaving the quantity\nt2−z2−x2−y2(61)\ninvariant. It is possible to perform the same transformatio n using two-by-two matri-\nces [15, 7, 19].\nIn this two-by-two representation, the four-vector is writ ten as\nX=/parenleftbiggt+z x−iy\nx+iy t−z/parenrightbigg\n, (62)\nwhere its determinant is precisely the quantity given in Eq. (61) and the Lorentz trans-\nformation on this matrix is a determinant-preserving, or un imodular transformation.\nLet us consider the transformation matrix as [7, 19]\nG=/parenleftbiggα β\nγ δ/parenrightbigg\n,andG†=/parenleftbiggα∗γ∗\nβ∗δ∗/parenrightbigg\n, (63)\nwith\ndet(G) = 1, (64)\nand the transformation\nX′=GXG†. (65)\nSinceGis not a unitary matrix, Eq.(65) not a unitary transformatio n, but rather we\ncall this the “Hermitian transformation”. Eq.(65) can be wr itten as\n/parenleftbiggt′+z′x′−iy′\nx+iy t′−z′/parenrightbigg\n=/parenleftbiggα β\nγ δ/parenrightbigg/parenleftbiggt+z x−iy\nx+iy t−z/parenrightbigg/parenleftbiggα∗γ∗\nβ∗δ∗/parenrightbigg\n,(66)\nIt is still a determinant-preserving unimodular transform ation, thus it is possible\nto write this as a four-by-four transformation matrix appli cable to the four-vector\n(t,z,x,y) [15, 7].\nSince theGmatrix starts with four complex numbers and its determinant is one by\nEq.(64), it has six independent parameters. The group of the seGmatrices is known\nto be locally isomorphic to the group of four-by-four matric es performing Lorentz\ntransformations on the four-vector ( t,z,x,y). In other words, for each Gmatrix there\nis a corresponding four-by-four Lorentz-transform matrix [7].\nThematrix Gis not aunitary matrix, becauseits Hermitian conjugate is n ot always\nitsinverse. Thisgrouphasaunitarysubgroupcalled SU(2)andanotherconsistingonly\nof real matrices called Sp(2). For this later subgroup, it is sufficient to work with the\nthreematrices R(θ),S(λ), andB(η) given inEqs.(43), (44), and(45) respectively. Each\nof these matrices has its corresponding four-by-four matri x applicable to the ( t,z,x,y).\nThese matrices with their four-by-four counterparts are ta bulated in Table 1.\nThe energy-momentum four vector can also be written as a two- by-two matrix. It\ncan be written as\nP=/parenleftbiggp0+pzpx−ipy\npx+ipyp0−pz/parenrightbigg\n, (67)\nwith\ndet(P) =p2\n0−p2\nx−p2\ny−p2\nz, (68)\n14which means\ndet(P) =m2, (69)\nwheremis the particle mass.\nThe Lorentz transformation can be written explicitly as\nP′=GPG†, (70)\nor/parenleftbiggp′\n0+p′\nzp′\nx−ip′\ny\np′\nx+ip′\nyE′−p′\nz/parenrightbigg\n=/parenleftbiggα β\nγ δ/parenrightbigg/parenleftbiggp0+pzpx−ipy\npx+ipyp0−pz/parenrightbigg/parenleftbiggα∗γ∗\nβ∗δ∗/parenrightbigg\n.(71)\nThis is an unimodular transformation, and the mass is a Loren tz-invariant variable.\nFurthermore, it was shown in [7] that Wigner’s little groups for massive, massless, and\nimaginary-mass particles can be explicitly defined in terms of two-by-two matrices.\nWigner’s little group consists of two-by-two matrices sati sfying\nP=WPW†. (72)\nThe two-by-two Wmatrix is not an identity matrix, but tells about the interna l space-\ntime symmetry of a particle with a given energy-momentum fou r-vector. This aspect\nwas not known when Einstein formulated his special relativi ty in 1905, hence the\ninternal space-time symmetry was not an issue at that time. W e call the two-by-two\nmatrixWthe Wigner matrix, and call the condition of Eq.(72) the Wign er condition.\nIf determinant of Wis a positive number, then Pis proportional to\nP=/parenleftbigg1 0\n0 1/parenrightbigg\n, (73)\ncorresponding to a massive particle at rest, while if the det erminant is negative, it is\nproportional to\nP=/parenleftbigg1 0\n0−1/parenrightbigg\n, (74)\ncorresponding to an imaginary-mass particle moving faster than light along the zdi-\nrection, with a vanishing energy component. If the determin ant is zero, Pis\nP=/parenleftbigg1 0\n0 0/parenrightbigg\n, (75)\nwhich is proportional to the four-momentum matrix for a mass less particle moving\nalong thezdirection.\nFor all three cases, the matrix of the form\nZ(φ) =/parenleftbigge−iφ/20\n0eiφ/2/parenrightbigg\n(76)\nwill satisfy the Wigner condition of Eq.(72). This matrix co rresponds to rotations\naround the zaxis.\nFor the massive particle with the four-momentum of Eq.(73), the transformations\nwith the rotation matrix of Eq.(43) leave the Pmatrix of Eq.(73) invariant. Together\nwith theZ(φ) matrix, this rotation matrix leads to the subgroup consist ing of the\nunitary subset of the Gmatrices. The unitary subset of GisSU(2) corresponding to\nthe three-dimensional rotation group dictating the spin of the particle [15].\n15For the massless case, the transformations with the triangu lar matrix of the form\n/parenleftbigg1γ\n0 1/parenrightbigg\n(77)\nleave the momentum matrix of Eq.(75) invariant. The physics of this matrix has\na stormy history, and the variable γleads to a gauge transformation applicable to\nmassless particles [8, 9, 20, 21].\nFor a particle with an imaginary mass, a Wmatrix of the form of Eq.(44) leaves\nthe four-momentum of Eq.(74) invariant.\nTable 3 summarizes the transformation matrices for Wigner’ s little groups for mas-\nsive, massless, and imaginary-mass particles. Furthermor e, in terms of their traces, the\nmatrices given in this subsection can be compared with those given in Subsec. 2.3 for\nthe damped oscillator. The comparisons are given in Table 4.\nOf course, it is a challenging problem to have one expression for all three classes.\nThis problem has been discussed in the literature [12], and t he damped oscillator case\nof Sec. 2 addresses the continuity problem.\nTable 3: Wigner vectors and Wigner matrices in the two-by-two repr esentation. The trace\nof the matrix tells whether the particle m2is positive, zero, or negative.\nParticle mass Four-momentum Transform matrix Trace\nMassive/parenleftbigg1 0\n0 1/parenrightbigg /parenleftbiggcos(θ/2)−sin(θ/2)\nsin(θ/2) cos( θ/2)/parenrightbigg\nless than 2\nMassless/parenleftbigg1 0\n0 0/parenrightbigg /parenleftbigg1γ\n0 1/parenrightbigg\nequal to 2\nImaginary mass/parenleftbigg1 0\n0−1/parenrightbigg /parenleftbiggcosh(λ/2) sinh( λ/2)\nsinh(λ/2) cosh( λ/2)/parenrightbigg\ngreater than 2\n5 LorentzCompletionofWigner’sLittleGroups\nSo far we have considered transformations applicable only t o (t,z,x) space. In order to\nstudythe full symmetry, we have to consider rotations aroun d thezaxis. As previously\nstated, when a particle moves along this axis, this rotation defines the helicity of the\nparticle.\nIn [1], Wigner worked out the little group of a massive partic le at rest. When the\nparticle gains a momentum along the zdirection, the single particle can reverse the\ndirection of momentum, the spin, or both. What happens to the internal space-time\nsymmetries is discussed in this section.\n16Table 4: Damped Oscillators and Space-time Symmetries. Both share Sp(2) as their sym-\nmetry group.\nTrace Damped Oscillator Particle Symmetry\nSmaller than 2 Oscillation Mode Massive Particles\nEqual to 2 Transition Mode Massless Particles\nLarger than 2 Damping Mode Imaginary-mass Particles\n5.1 Rotation around the zaxis\nIn Sec. 3, our kinematics was restricted to the two-dimensio nal space of zandx, and\nthus includes rotations around the yaxis. We now introduce the four-by-four matrix of\nEq.(60) performing rotations around the zaxis. Its corresponding two-by-two matrix\nwas given in Eq.(76). Its generator is\nJ3=1\n2/parenleftbigg1 0\n0−1/parenrightbigg\n. (78)\nIf we introduce this additional matrix for the three generat ors we used in Secs. 3 and\n3.2, we end up the closed set of commutation relations\n[Ji,Jj] =iǫijkJk,[Ji,Kj] =iǫijkKk,[Ki,Kj] =−iǫijkJk,(79)\nwith\nJi=1\n2σi,andKi=i\n2σi, (80)\nwhereσiare the two-by-two Pauli spin matrices.\nFor each of these two-by-two matrices there is a correspondi ng four-by-four matrix\ngenerating Lorentz transformations on the four-dimension al Lorentz group. When\nthese two-by-two matrices are imaginary, the correspondin g four-by-four matrices were\ngiven in Table 1. If they are real, the corresponding four-by -four matrices were given\nin Table 5.\nThis set of commutation relations is known as the Lie algebra for the SL(2,c),\nnamely the group of two-by-two elements with unit determina nts. Their elements are\ncomplex. This set is also the Lorentz group performing Loren tz transformations on the\nfour-dimensional Minkowski space.\nThis set has many useful subgroups. For the group SL(2,c), there is a subgroup\nconsisting only of real matrices, generated by the two-by-t wo matrices given in Table 1.\nThis three-parameter subgroup is precisely the Sp(2) group we used in Secs. 3 and 3.2.\nTheir generators satisfy the Lie algebra given in Eq.(42).\n17Table 5: Two-by-two and four-by-four generators not included in Table 1. The generators\ngiven there and given here constitute the set of six generators fo rSL(2,c) or of the Lorentz\ngroup given in Eq.( 79).\nGenerator Two-by-two Four-by-four\nJ31\n2/parenleftbigg1 0\n0−1/parenrightbigg\n0 0 0 0\n0 0 0 0\n0 0 0 −i\n0 0i0\n\nJ11\n2/parenleftbigg0 1\n1 0/parenrightbigg\n0 0 0 0\n0 0 0 i\n0 0 0 0\n0−i0 0\n\nK21\n2/parenleftbigg0 1\n−1 0/parenrightbigg\n0 0 0 i\n0 0 0 0\n0 0 0 0\ni0 0 0\n\nIn addition, this group has the following Wigner subgroups g overning the internal\nspace-time symmetries of particles in the Lorentz-covaria nt world [1]:\n1. TheJimatrices form a closed set of commutation relations. The sub group gener-\nated by these Hermitian matrices is SU(2) for electron spins. The corresponding\nrotation group does not change the four-momentum of the part icle at rest. This\nis Wigner’s little group for massive particles.\nIftheparticleisat rest, thetwo-by-two formofthefour-ve ctor isgiven byEq.(73).\nThe Lorentz transformation generated by J3takes the form\n/parenleftbiggeiφ/20\n0e−iφ/2/parenrightbigg/parenleftbigg1 0\n0 1/parenrightbigg/parenleftbigge−iφ/20\n0eiφ/2/parenrightbigg\n=/parenleftbigg1 0\n0 1/parenrightbigg\n(81)\nSimilar computations can be carried out for J1andJ2.\n2. There is another Sp(2) subgroup, generated by K1,K2, andJ3. They satisfy the\ncommutation relations\n[K1,K2] =−iJ3,[J3,K1] =iK2,[K2,J3] =iK1.(82)\nThe Wigner transformation generated by these two-by-two ma trices leave the\nmomentum four-vector of Eq.(74) invariant. For instance, t he transformation\nmatrix generated by K2takes the form\nexp(−iξK2) =/parenleftbiggcosh(ξ/2)isinh(ξ/2)\nisinh(ξ/2) cosh(ξ/2)/parenrightbigg\n(83)\n18and the Wigner transformation takes the form\n/parenleftbiggcosh(ξ/2)isinh(ξ/2)\n−isinh(ξ/2) cosh(ξ/2)/parenrightbigg/parenleftbigg1 0\n0−1/parenrightbigg/parenleftbiggcosh(ξ/2)isinh(ξ/2)\n−isinh(ξ/2) cosh(ξ/2)/parenrightbigg\n=/parenleftbigg1 0\n0−1/parenrightbigg\n.\n(84)\nComputations with K2andJ3lead to the same result.\nSince the determinant of the four-momentum matrix is negati ve, the particle has\nanimaginarymass. Inthelanguageofthefour-by-fourmatri x, thetransformation\nmatrices leave the four-momentum of the form (0 ,1,0,0) invariant.\n3. Furthermore, we can consider the following combinations of the generators:\nN1=K1−J2=/parenleftbigg0i\n0 0/parenrightbigg\n,andN2=K2+J1=/parenleftbigg0 1\n0 0/parenrightbigg\n.(85)\nTogether with J3, they satisfy the the following commutation relations.\n[N1,N2] = 0,[N1,J3] =−iN2,[N2,J3] =iN1, (86)\nIn order to understand this set of commutation relations, we can consider an xy\ncoordinate system in a two-dimensional space. Then rotatio n around the origin\nis generated by\nJ3=−i/parenleftbigg\nx∂\n∂y−y∂\n∂x/parenrightbigg\n, (87)\nand the two translations are generated by\nN1=−i∂\n∂x,andN2=−i∂\n∂y, (88)\nfor thexandydirections respectively. These operators satisfy the comm utations\nrelations given in Eq.(86).\nThe two-by-two matrices of Eq.(85) generate the following t ransformation matrix.\nG(γ,φ) = exp[−iγ(N1cosφ+N2sinφ)] =/parenleftbigg1γe−iφ\n0 1/parenrightbigg\n. (89)\nThe two-by-two form for the four-momentum for the massless p article is given by\nEq.(75). The computation of the Hermitian transformation u sing this matrix is\n/parenleftbigg1γe−iφ\n0 1/parenrightbigg/parenleftbigg1 0\n0 0/parenrightbigg/parenleftbigg1 0\nγeiφ1/parenrightbigg\n=/parenleftbigg1 0\n0 0/parenrightbigg\n, (90)\nconfirming that N1andN2, together with J3, are the generators of the E(2)-like little\ngroup for massless particles in the two-by-two representat ion. The transformation that\ndoes this in the physical world is described in the following section.\n5.2 E(2)-like Symmetry of Massless Particles\nFrom the four-by-four generators of K1,2andJ1,2,we can write\nN1=\n0 0i0\n0 0i0\ni−i0 0\n0 0 0 0\n,andN2=\n0 0 0 i\n0 0 0 i\n0 0 0 0\ni−i0 0\n. (91)\n19These matrices lead to the transformation matrix of the form\nG(γ,φ) =\n1+γ2/2−γ2/2γcosφ γsinφ\nγ2/2 1−γ2/2γcosφ γsinφ\n−γcosφ γcosφ1 0\n−γsinφ γsinφ0 1\n(92)\nThis matrix leaves the four-momentum invariant, as we can se e from\nG(γ,φ)\n1\n1\n0\n0\n=\n1\n1\n0\n0\n. (93)\nWhen it is applied to the photon four-potential\nG(γ,φ)\nA0\nA3\nA1\nA2\n=\nA0\nA3\nA1\nA2\n+γ(A1cosφ+A2sinφ)\n1\n1\n0\n0\n, (94)\nwith the Lorentz condition which leads to A3=A0in the zero mass case. Gauge\ntransformations are well known for electromagnetic fields a nd photons. Thus Wigner’s\nlittle group leads to gauge transformations.\nIn the two-by-two representation, the electromagnetic fou r-potential takes the form\n/parenleftbigg2A0A1−iA2\nA1+iA20/parenrightbigg\n, (95)\nwith the Lorentz condition A3=A0. Then the two-by-two form of Eq.(94) is\n/parenleftbigg1γe−iφ\n0 1/parenrightbigg/parenleftbigg2A0A1−iA2\nA1+iA20/parenrightbigg/parenleftbigg1 0\nγeiφ1/parenrightbigg\n, (96)\nwhich becomes\n/parenleftbiggA0A1−iA2\nA1+iA20/parenrightbigg\n+/parenleftbigg2γ(A1cosφ−A2sinφ) 0\n0 0/parenrightbigg\n. (97)\nThis is the two-by-two equivalent of the gauge transformati on given in Eq.(94).\nFor massless spin-1/2 particles starting with the two-by-t wo expression of G(γ,φ)\ngiven in Eq.(89), and considering the spinors\nu=/parenleftbigg1\n0/parenrightbigg\n,andv=/parenleftbigg0\n1/parenrightbigg\n, (98)\nfor spin-up and spin-down states respectively,\nGu=u,andGv=v+γe−iφu, (99)\nThis means that the spinor ufor spin up is invariant under the gauge transformation\nwhilevis not. Thus, the polarization of massless spin-1/2 particl e, such as neutrinos,\nis a consequence of the gauge invariance. We shall continue t his discussion in Sec. 7.\n205.3 Boosts along the zaxis\nIn Subsec. 4.1 and Subsec. 5.1, we studied Wigner transforma tions for fixed values of\nthe four-momenta. The next question is what happens when the system is boosted\nalong thezdirection, with the transformation\n/parenleftbiggt′\nz′/parenrightbigg\n=/parenleftbiggcoshηsinhη\nsinhηcoshη/parenrightbigg/parenleftbiggt\nz/parenrightbigg\n. (100)\nThen the four-momenta become\n(coshη,sinhη,0,0),(sinhη,coshη,0,0), eη(1,1,0,0), (101)\nrespectively for massive, imaginary, and massless particl es cases. In the two-by-two\nrepresentation, the boost matrix is\n/parenleftbiggeη/20\n0e−η/2/parenrightbigg\n, (102)\nand the four-momenta of Eqs.(101) become\n/parenleftbiggeη0\n0e−η/parenrightbigg\n,/parenleftbiggeη0\n0−e−η/parenrightbigg\n,/parenleftbiggeη0\n0 0/parenrightbigg\n, (103)\nrespectively. These matrices become Eqs.(73), (74), and (7 5) respectively when η= 0.\nWe are interested in Lorentz transformations which leave a g iven non-zero momen-\ntum invariant. We can consider a Lorentz boost along the dire ction preceded and\nfollowed by identical rotation matrices, as described in Fi g.(2) and the transformation\nmatrix as\n/parenleftbiggcos(α/2)−sin(α/2)\nsin(α/2) cos(α/2)/parenrightbigg/parenleftbiggcoshχ−sinhχ\n−sinhχcoshχ/parenrightbigg/parenleftbiggcos(α/2)−sin(α/2)\nsin(α/2) cos(α/2)/parenrightbigg\n,(104)\nwhich becomes\n/parenleftbigg(cosα)coshχ −sinhχ−(sinα)coshχ\n−sinhχ+(sinα)coshχ (cosα)coshχ/parenrightbigg\n. (105)\nExcept thesign of χ, thetwo-by-two matrices of Eq.(104) and Eq.(105) areident ical\nwith those given in Sec. 3.2. The only difference is the sign of t he parameter χ. We\nare thus ready to interpret this expression in terms of physi cs.\n1. If the particle is massive, the off-diagonal elements of Eq. (105) have opposite\nsigns, and this matrix can be decomposed into\n/parenleftbiggeη/20\n0e−η/2/parenrightbigg/parenleftbiggcos(θ/2)−sin(θ/2)\nsin(θ/2) cos(θ/2)/parenrightbigg/parenleftbiggeη/20\n0e−η/2/parenrightbigg\n.(106)\nwith\ncos(θ/2) = (coshχ)cosα,ande2η=cosh(χ)sinα+sinhχ\ncosh(χ)sinα−sinhχ,(107)\nand\ne2η=p0+pz\np0−pz. (108)\nAccording to Eq.(106) the first matrix (far right) reduces th e particle momentum\nto zero. The second matrix rotates the particle without chan ging the momentum.\nThe third matrix boosts the particle to restore its original momentum. This is\nthe extension of Wigner’s original idea to moving particles .\n21fig.(a) fig.(b) \nFigure 2: Bargmann and Wigner Decompositions. The Bargmann deco mposition is illus-\ntrated in fig.(a). Starting from a momentum along the zdirection, we can rotate, boost, and\nrotate to bring the momentum to the original position. The resulting matrix is the prod-\nuct of one boost and two rotation matrices. In the Wigner decompo sition, the particle is\nboosted back to the frame where the Wigner transformation can b e applied. Make a Wigner\ntransformation there and come back to the original state of the m omentum as illustrated in\nfig.(b). This process also can also be written as the product of thre e simple matrices.\n2. If the particle has an imaginary mass, the off-diagonal elem ents of Eq.(105) have\nthe same sign,\n/parenleftbiggeη/20\n0e−η/2/parenrightbigg/parenleftbiggcosh(λ/2)−sinh(λ/2)\nsinh(λ/2) cosh(λ/2)/parenrightbigg/parenleftbiggeη/20\n0e−η/2/parenrightbigg\n,(109)\nwith\ncosh(λ/2) = (coshχ)cosα,ande2η=sinhχ+cosh(χ)sinα\ncosh(χ)sinα−sinhχ,(110)\nand\ne2η=p0+pz\npz−p0. (111)\nThisisalsoathree-stepoperation. Thefirstmatrixbringst heparticlemomentum\nto the zero-energy state with p0= 0. Boosts along the xorydirection do not\nchange the four-momentum. We can then boost the particle bac k to restore its\nmomentum. This operation is also an extension of the Wigner’ s original little\ngroup. Thus, it is quite appropriate to call the formulas of E q.(106) and Eq.(109)\nWigner decompositions.\n3. If the particle mass is zero with\nsinhχ= (coshχ)sinα, (112)\n22theηparameter becomes infinite, and the Wigner decomposition do es not appear\nto be useful. We can then go back to the Bargmann decompositio n of Eq.(104).\nWith the condition of Eq.(112), Eq.(105) becomes\n/parenleftbigg1−γ\n0 1/parenrightbigg\n, (113)\nwith\nγ= 2sinhχ. (114)\nThe decomposition ending with a triangular matrix is called the Iwasawa decom-\nposition [17, 22] and its physical interpretation was given in Subsec. 5.2. The γ\nparameter does not depend on η.\nThus, we have given physical interpretations to the Bargman n and Wigner de-\ncompositions given in Sec. (3.2). Consider what happens whe n the momentum be-\ncomes large. Then ηbecomes large for nonzero mass cases. All three four-moment a in\nEq.(103) become\neη/parenleftbigg1 0\n0 0/parenrightbigg\n. (115)\nAs for the Bargmann-Wigner matrices, they become the triang ular matrix of Eq.(113),\nwithγ= sin(θ/2)eηandγ= sinh(λ/2)eη,respectively for the massive and imaginary-\nmass cases.\nIn Subsec. 5.2, we concluded that the triangular matrix corr esponds to gauge trans-\nformations. However, particles with imaginary mass are not observed. For massive\nparticles, we can start with the three-dimensional rotatio n group. The rotation around\nthezaxis is called helicity, and remains invariant under the boo st along the zdirec-\ntion. As for the transverse rotations, they become gauge tra nsformation as illustrated\nin Table 6.\n5.4 Conjugate Transformations\nThe most general form of the SL(2,c) matrix is given in Eq.(63). Transformation\noperators for the Lorentz group are given in exponential for m as:\nD= exp/braceleftigg\n−i3/summationdisplay\ni=1(θiJi+ηiKi)/bracerightigg\n, (116)\nwhere theJiare the generators of rotations and the Kiare the generators of proper\nLorentz boosts. They satisfy the Lie algebra given in Eq.(42 ). This set of commutation\nrelations is invariant under the sign change of the boost gen eratorsKi. Thus, we can\nconsider “dot conjugation” defined as\n˙D= exp/braceleftigg\n−i3/summationdisplay\ni=1(θiJi−ηiKi)/bracerightigg\n, (117)\nSinceKiare anti-Hermitian while Jiare Hermitian, the Hermitian conjugate of the\nabove expression is\nD†= exp/braceleftigg\n−i3/summationdisplay\ni=1(−θiJi+ηiKi)/bracerightigg\n, (118)\n23Table 6: Covariance of the energy-momentum relation, and covaria nce of the internal space-\ntime symmetry. Under the Lorentz boost along the zdirection, J3remains invariant, and\nthis invariant component of the angular momentum is called the helicity . The transverse\ncomponent J1andJ2collapse into a gaugetransformation. The γparameter for the massless\ncase has been studied in earlier papers in the four-by-four matrix f ormulation [8, 21].\nMassive, Slow COVARIANCE Massless, Fast\nE=p2/2m Einstein’s E=mc2E=cp\nJ3 Helicity\nWigner’s Little Group\nJ1,J2 Gauge Transformation\nwhile the Hermitian conjugate of Gis\n˙D†= exp/braceleftigg\n−i3/summationdisplay\ni=1(−θiJi−ηiKi)/bracerightigg\n, (119)\nSince we understand the rotation around the zaxis, we can now restrict the kine-\nmatics to the ztplane, and work with the Sp(2) symmetry. Then the Dmatrices\ncan be considered as Bargmann decompositions. First, Dand˙D, and their Hermitian\nconjugates are\nD(α,χ) =/parenleftbigg(cosα)coshχsinhχ−(sinα)coshχ\nsinhχ+(sinα)coshχ(cosα)coshχ/parenrightbigg\n,\n˙D(α,χ) =/parenleftbigg(cosα)coshχ −sinhχ−(sinα)coshχ\n−sinhχ+(sinα)coshχ (cosα)coshχ/parenrightbigg\n.(120)\nThese matrices correspond to the “D loops” given in fig.(a) an d fig.(b) of Fig. 3 respec-\ntively. The “dot” conjugation changes the direction of boos ts. The dot conjugation\nleads to the inversion of the space which is called the parity operation.\nWe can also consider changing the direction of rotations. Th en they result in the\nHermitian conjugates. We can write their matrices as\nD†(α,χ) =/parenleftbigg(cosα)coshχsinhχ+(sinα)coshχ\nsinhχ−(sinα)coshχ(cosα)coshχ/parenrightbigg\n,\n˙D†(α,χ) =/parenleftbigg(cosα)coshχ −sinhχ+(sinα)coshχ\n−sinhχ−(sinα)coshχ (cosα)coshχ/parenrightbigg\n.(121)\n24fig.(a) fig.(b) \nfig.(c) fig.(d) \nFigure 3: Four D-loops resulting from the Bargmann decomposition. Let us go back to\nFig. 2. If we reverse of the direction of the boost, the result is fig.( a). From fig.(a), if we\ninvert the space, we come back to fig.(b). If we reverse the direct ion of rotation from fig.(a),\nthe result is fig.(c). If both the rotation and space are reversed, the result is the fig.(d).\nFrom the exponential expressions from Eq.(116) to Eq.(119) , it is clear that\nD†=˙D−1,and˙D†=D−1. (122)\nThe D loop given in Fig. 2 corresponds to ˙D. We shall return to these loops in Sec. 7.\n6 SymmetriesderivablefromthePoincar´ eSphere\nThe Poincar´ e sphere serves as the basic language for polari zation physics. Its underly-\ning language is the two-by-two coherency matrix. This coher ency matrix contains the\nsymmetryof SL(2,c) isomorphictothetheLorentz groupapplicable tothreespa ce-like\nand one time-like dimensions [4, 6, 7].\nFor polarized light propagating along the zdirection, the amplitude ratio and phase\ndifference of electric field xandycomponents traditionally determine the state of\npolarization. Hence, the polarization can be changed by adj usting the amplitude ratio\nor the phase difference or both. Usually, the optical device wh ich changes amplitude\nis called an “attenuator” (or “amplifier”) and the device whi ch changes the relative\nphase a “phase shifter.”\n25Let us start with the Jones vector:\n/parenleftbiggψ1(z,t)\nψ2(z,t)/parenrightbigg\n=/parenleftbiggaexp[i(kz−ωt)]\naexp[i(kz−ωt)]/parenrightbigg\n. (123)\nTo this matrix, we can apply the phase shift matrix of Eq.(76) which brings the Jones\nvector to /parenleftbiggψ1(z,t)\nψ2(z,t)/parenrightbigg\n=/parenleftbiggaexp[i(kz−ωt−iφ/2)]\naexp[i(kz−ωt+iφ/2)]/parenrightbigg\n. (124)\nThe generator of this phase-shifter is J3given Table 5.\nThe optical beam can be attenuated differently in the two direc tions. The resulting\nmatrix is\ne−µ/parenleftbiggeη/20\n0e−η/2/parenrightbigg\n(125)\nwith the attenuation factor of exp( −µ0+η/2) and exp( −µ−η/2) for thexandy\ndirections respectively. We are interested only the relati ve attenuation given in Eq.(45)\nwhich leads to different amplitudes for the xandycomponent, and the Jones vector\nbecomes /parenleftbiggψ1(z,t)\nψ2(z,t)/parenrightbigg\n=/parenleftbiggaeµ/2exp[i(kz−ωt−iφ/2)]\nae−µ/2exp[i(kz−ωt+iφ/2)]/parenrightbigg\n. (126)\nThe squeeze matrix of Eq.(45) is generated by K3given in Table 1.\nThe polarization is not always along the xandyaxes, but can be rotated around\nthezaxis using Eq.(76) generated by J2given in Table 1.\nAmong the rotation angles, the angle of 45oplays an important role in polarization\noptics. Indeed, if we rotate the squeeze matrix of Eq.(45) by 45o, we end up with the\nsqueeze matrix of Eq.(44) generated by K1given also in Table 1.\nEach of these four matrices plays an important role in specia l relativity, as we\ndiscussed in Secs. 3.2 and 7. Their respective roles in optic s and particle physics are\ngiven in Table 7.\nThe most general form for the two-by-two matrix applicable t o the Jones vector\nis theGmatrix of Eq.(63). This matrix is of course a representation of theSL(2,c)\ngroup. It brings the simplest Jones vector of Eq.(123) to its most general form.\n6.1 Coherency Matrix\nHowever, the Jones vector alone cannot tell us whether the tw o components are coher-\nent with each other. In order to address this important degre e of freedom, we use the\ncoherency matrix defined as [3, 23]\nC=/parenleftbiggS11S12\nS21S22/parenrightbigg\n, (127)\nwhere\n<ψ∗\niψj>=1\nT/integraldisplayT\n0ψ∗\ni(t+τ)ψj(t)dt, (128)\nwhereTis a sufficiently long time interval. Then, those four element s become [4]\nS11=<ψ∗\n1ψ1>=a2, S 12=<ψ∗\n1ψ2>=a2(cosξ)e−iφ,\nS21=<ψ∗\n2ψ1>=a2(cosξ)e+iφ, S 22=<ψ∗\n2ψ2>=a2.(129)\n26Table 7: Polarization optics and special relativity share the same mat hematics. Each matrix\nhas its clear role in both optics and relativity. The determinant of the Stokes or the four-\nmomentum matrix remains invariant under Lorentz transformation s. It is interesting to note\nthat the decoherence parameter (least fundamental) in optics co rresponds to the ( mass)2\n(most fundamental) in particle physics.\nPolarization Optics Transformation Matrix Particle Symmetry\nPhase shift by φ/parenleftbigge−iφ/20\n0eiφ/2/parenrightbigg\nRotation around z.\nRotation around z/parenleftbiggcos(θ/2)−sin(θ/2)\nsin(θ/2) cos( θ/2)/parenrightbigg\nRotation around y.\nSqueeze along xandy/parenleftbiggeη/20\n0e−η/2/parenrightbigg\nBoost along z.\nSqueeze along 45o/parenleftbiggcosh(λ/2) sinh( λ/2)\nsinh(λ/2) cosh( λ/2)/parenrightbigg\nBoost along x.\na4(sinξ)2Determinant (mass)2\n27The diagonal elements are the absolute values of ψ1andψ2respectively. The angle\nφcould be different from the value of the phase-shift angle give n in Eq.(76), but this\ndifference does not play any role in the reasoning. The off-diago nal elements could\nbe smaller than the product of ψ1andψ2, if the two polarizations are not completely\ncoherent.\nThe angle ξspecifies the degree of coherency. If it is zero, the system is fully\ncoherent, while the system is totally incoherent if ξis 90o. This can therefore be called\nthe “decoherence angle.”\nWhile the most general form of the transformation applicabl e to the Jones vector\nisGof Eq.(63), the transformation applicable to the coherency matrix is\nC′=G C G†. (130)\nThe determinant of the coherency matrix is invariant under t his transformation, and\nit is\ndet(C) =a4(sinξ)2. (131)\nThus, angle ξremains invariant. In the language of the Lorentz transform ation appli-\ncable to the four-vector, the determinant is equivalent to t he (mass)2and is therefore\na Lorentz-invariant quantity.\n6.2 Two Radii of the Poincar´ e Sphere\nLet us write explicitly the transformation of Eq.(130) as\n/parenleftbiggS′\n11S′\n12\nS′\n21S′\n22/parenrightbigg\n=/parenleftbiggα β\nγ δ/parenrightbigg/parenleftbiggS11S12\nS21S22/parenrightbigg/parenleftbiggα∗γ∗\nβ∗δ∗/parenrightbigg\n. (132)\nIt is then possible to construct the following quantities,\nS0=S11+S22\n2, S 3=S11−S22\n2,\nS1=S12+S21\n2, S 2=S12−S21\n2i. (133)\nThese are known as the Stokes parameters, and constitute a fo ur-vector (S0,S3,S1,S2)\nunder the Lorentz transformation.\nReturning to Eq.(76), the amplitudes of the two orthogonal c omponent are equal,\nthus, thetwodiagonal elements ofthecoherency matrixaree qual. Thisleadsto S3= 0,\nand the problem is reduced from the sphere to a circle.\nIn this two-dimensional subspace, we can introduce the pola r coordinate system\nwith\nR=/radicalig\nS2\n1+S2\n2\nS1=Rcosφ,\nS2=Rsinφ. (134)\nThe radius Ris the radius of this circle, and is\nR=a2cosξ. (135)\n28R\n0S\nFigure 4: Radius of the Poincar´ e sphere. The radius Rtakes its maximum value S0when\nthe decoherence angle ξis zero. It becomes smaller as ξincreases. It becomes zero when the\nangle reaches 90o.\nThe radius Rtakes its maximum value S0whenξ= 0o. It decreases as ξincreases\nand vanishes when ξ= 90o. This aspect of the radius R is illustrated in Fig. 4.\nIn order to see its implications in special relativity, let u s go back to the four-\nmomentum matrix of m(1,0,0,0). Its determinant is m2and remains invariant. Like-\nwise, thedeterminant of the coherency matrix of Eq.(127) sh ouldalso remain invariant.\nThe determinant in this case is\nS2\n0−R2=a4sin2ξ. (136)\nThisquantity remains invariant undertheHermitian transf ormation of Eq.(132), which\nis a Lorentz transformation as discussed in Secs. 3.2 and 7. T his aspect is shown on\nthe last row of Table 7.\nThe coherency matrix then becomes\nC=a2/parenleftbigg1 (cos ξ)e−iφ\n(cosξ)eiφ1/parenrightbigg\n. (137)\nSince the angle φdoes not play any essential role, we can let φ= 0, and write the\ncoherency matrix as\nC=a2/parenleftbigg1 cosξ\ncosξ1/parenrightbigg\n. (138)\nThe determinant of the above two-by-two matrix is\na4/parenleftig\n1−cos2ξ/parenrightig\n=a4sin2ξ. (139)\nSince the Lorentz transformation leaves the determinant in variant, the change in\nthisξvariable is not a Lorentz transformation. It is of course pos sible to construct a\nlarger group in which this variable plays a role in a group tra nsformation [6], but here\nwe are more interested in its role in a particle gaining a mass from zero or the mass\nbecoming zero.\n29fig.(a)fig.(b)\nFigure 5: Transition from the massive to massless case. Within the fr amework of the Lorentz\ngroup, it is not possible to go from the massive to massless case direc tly, because it requires\nthe change in the mass which is a Lorentz-invariant quantity. The on ly way is to move\nto infinite momentum and jump from the hyperbola to the light cone, a nd come back, as\nillustrated in fig.(a). The extra symmetry of the Poincar´ e sphere a llows a direct transition\nas shown in fig. (b).\n6.3 Extra-Lorentzian Symmetry\nThe coherency matrix of Eq.(137) can be diagonalized to\na2/parenleftbigg1+cosξ0\n0 1 −cosξ/parenrightbigg\n(140)\nby a rotation. Let us then go back to the four-momentum matrix of Eq.(67). If\npx=py= 0, andpz=p0cosξ, we can write this matrix as\np0/parenleftbigg1+cosξ0\n0 1 −cosξ/parenrightbigg\n. (141)\nThus, with this extra variable, it is possible to study the li ttle groups for variable\nmasses, including the small-mass limit and the zero-mass ca se.\nFor a fixed value of p0, the (mass)2becomes\n(mass)2= (p0sinξ)2,and (momentum )2= (p0cosξ)2, (142)\nresulting in\n(energy)2= (mass)2+(momentum )2. (143)\nThis transition is illustrated in Fig. 5. We are interested i n reaching a point on the\nlight conefrommasshyperbolawhilekeepingtheenergy fixed . Accordingtothisfigure,\nwe do not have to make an excursion to infinite-momentum limit . If the energy is fixed\nduring this process, Eq.(143) tells the mass and momentum re lation, and Figure 6\nillustrates this relation.\n30Momentum Mass Energy \nξangle \nFigure 6: Energy-momentum-mass relation. This circle illustrates th e case where the energy\nis fixed, while the mass and momentum are related according to the tr iangular rule. The\nvalue of the angle ξchanges from zero to 180o. The particle mass is negative for negative\nvalues of this angle. However, in the Lorentz group, only ( mass)2is a relevant variable, and\nnegative masses might play a role for theoretical purposes.\nWithin the framework of the Lorentz group, it is possible, by making an excursion\nto infinite momentum where the mass hyperbola coincides with the light cone, to then\ncome back to the desired point. On the other hand, the mass for mula of Eq.(142)\nallows us to go there directly. The decoherence mechanism of the coherency matrix\nmakes this possible.\n7 Small-mass and Massless Particles\nWe now have a mathematical tool to reduce the mass of a massive particle from its pos-\nitive value to zero. During this process, the Lorentz-boost ed rotation matrix becomes\na gauge transformation for the spin-1 particle, as discusse d Subsec. 5.2. For spin-1/2\nparticles, there are two issues.\n1. It was seen in Subsec. 5.2 that the requirement of gauge inv ariance lead to a\npolarization of massless spin-1/2 particle, such as neutri nos. What happens to\nanti-neutrinos?\n2. Therearestrongexperimentalindicationsthatneutrino shaveasmallmass. What\nhappens to the E(2) symmetry?\n7.1 Spin-1/2 Particles\nLet usgo back to the two-by-two matrices of Subsec. 5.4, and t he two-by-two Dmatrix.\nFor a massive particle, its Wigner decomposition leads to\nD=/parenleftbiggcos(θ/2)−e−ηsin(θ/2)\neηsin(θ/2) cos(θ/2)/parenrightbigg\n(144)\n31This matrix is applicable to the spinors uandvdefined in Eq.(98) respectively for the\nspin-up and spin-down states along the zdirection.\nSince the Lie algebra of SL(2,c) is invariant under the sign change of the Kima-\ntrices, we can consider the “dotted” representation, where the system is boosted in the\nopposite direction, while the direction of rotations remai n the same. Thus, the Wigner\ndecomposition leads to\n˙D=/parenleftbiggcos(θ/2)−eηsin(θ/2)\ne−ηsin(θ/2) cos(θ/2)/parenrightbigg\n(145)\nwith its spinors\n˙u=/parenleftbigg1\n0/parenrightbigg\n,and ˙v=/parenleftbigg0\n1/parenrightbigg\n. (146)\nFor anti-neutrinos, the helicity is reversed but the moment um is unchanged. Thus,\nD†is the appropriate matrix. However, D†=˙D−1as was noted in Subsec 5.4. Thus,\nwe shall use ˙Dfor anti-neutrinos.\nWhen the particle mass becomes very small,\ne−η=m\n2p, (147)\nbecomes small. Thus, if we let\neηsin(θ/2) =γ,ande−ηsin(θ/2) =ǫ2, (148)\nthen theDmatrix of Eq.(144) and the ˙Dof Eq.(145) become\n/parenleftbigg1−γǫ2/2−ǫ2\nγ1−γǫ2/parenrightbigg\n,and/parenleftbigg1−γǫ2/2−γ\nǫ21−γǫ2/parenrightbigg\n, (149)\nrespectively where γis an independent parameter and\nǫ2=γ/parenleftbiggm\n2p/parenrightbigg2\n. (150)\nWhen the particle mass becomes zero, they become\n/parenleftbigg1 0\nγ1/parenrightbigg\n,and/parenleftbigg1−γ\n0 1/parenrightbigg\n, (151)\nrespectively, applicable to the spinors ( u,v) and (˙u,˙v) respectively.\nFor neutrinos,\n/parenleftbigg1 0\nγ1/parenrightbigg/parenleftbigg1\n0/parenrightbigg\n=/parenleftbigg1\nγ/parenrightbigg\n,and/parenleftbigg1 0\nγ1/parenrightbigg/parenleftbigg0\n1/parenrightbigg\n=/parenleftbigg0\n1/parenrightbigg\n. (152)\nFor anti-neutrinos,\n/parenleftbigg1−γ\n0 1/parenrightbigg/parenleftbigg1\n0/parenrightbigg\n=/parenleftbigg1\n0/parenrightbigg\n,and/parenleftbigg1−γ\n0 1/parenrightbigg/parenleftbigg0\n1/parenrightbigg\n=/parenleftbigg−γ\n1/parenrightbigg\n.(153)\nIt was noted in Subsec. 5.2 that the triangular matrices of Eq .(151) perform gauge\ntransformations. Thus, for Eq.(152) and Eq.(153) the requi rement of gauge invariance\n32leads to the polarization of neutrinos. The neutrinos are le ft-handed while the anti-\nneutrinos are right-handed. Since, however, nature cannot tell the difference between\nthe dotted and undotted representations, the Lorentz group cannot tell which neutrino\nis right handed. It can say only that the neutrinos and anti-n eutrinos are oppositely\npolarized.\nIf the neutrino has a small mass, the gauge invariance is modi fied to\n/parenleftbigg1−γǫ2/2−ǫ2\nγ1−γǫ2/2/parenrightbigg/parenleftbigg0\n1/parenrightbigg\n=/parenleftbigg0\n1/parenrightbigg\n−ǫ2/parenleftbigg1\nγ/2/parenrightbigg\n, (154)\nand /parenleftbigg1−γǫ2/2−γ\nǫ21−γǫ2/parenrightbigg/parenleftbigg1\n0/parenrightbigg\n=/parenleftbigg1\n0/parenrightbigg\n+ǫ2/parenleftbigg−γ/2\n1/parenrightbigg\n, (155)\nrespectivelyforneutrinosandanti-neutrinos. Thusthevi olation ofthegaugeinvariance\nin both cases is proportional to ǫ2which ism2/4p2.\n7.2 Small-mass neutrinos in the Real World\nWhether neutrinos have mass or not and the consequences of th is relative to the Stan-\ndard Model and lepton number is the subject of much theoretic al speculation [24, 25],\nand of cosmology [26], nuclear reactors [27, 28], and high en ergy experimentations [29,\n30]. Neutrinos are fast becoming an important component of t he search for dark mat-\nter and dark radiation [31]. Their importance within the Sta ndard Model is reflected\nby the fact that they are the only particles which seem to exis t with only one direction\nof chirality, i.e. only left-handed neutrinos have been con firmed to exist so far.\nIt was speculated some time ago that neutrinos in constant el ectric and magnetic\nfields would acquire a small mass, and that right-handed neut rinos would be trapped\nwithin the interaction field [32]. Solving generalized elec troweak models using left- and\nright-handed neutrinoshas beendiscussedrecently [33]. T oday these right-handedneu-\ntrinos which do not participate in weak interactions are cal led “sterile” neutrinos [34].\nA comprehensive discussion of the place of neutrinos in the s cheme of physics has been\ngiven by Drewes [31].\n8 Scalars, Four-vectors, and Four-Tensors\nIn Secs. 5 and 7, our primary interest has been the two-by-two matrices applicable to\nspinors for spin-1/2 particles. Since we also used four-by- four matrices, we indirectly\nstudied the four-component particle consisting of spin-1 a nd spin-zero components.\nIf there are two spin 1/2 states, we are accustomed to constru ct one spin-zero state,\nand one spin-one state with three degeneracies.\nIn this paper, we are confronted with two spinors, but each sp inor can also be\ndotted. For this reason, there are sixteen orthogonal state s consisting of spin-one and\nspin-zero states. How many spin-zero states? How many spin- one states?\nFor particles at rest, it is known that the addition of two one -half spins result\nin spin-zero and spin-one states. In the this paper, we have t wo different spinors\nbehaving differently under the Lorentz boost. Around the zdirection, both spinors are\ntransformed by\nZ(φ) = exp( −iφJ3) =/parenleftbigge−iφ/20\n0eiφ/2/parenrightbigg\n. (156)\n33However, they are boosted by\nB(η) = exp( −iηK3) =/parenleftbiggeη/20\n0e−η/2/parenrightbigg\n,\n˙B(η) = exp(iηK3),=/parenleftbigge−η/20\n0eη/2/parenrightbigg\n(157)\napplicable to the undotted and dotted spinors respectively . These two matrices com-\nmute with each other, and also with the rotation matrix Z(φ) of Eq.(156). Since K3\nandJ3commute with each other, we can work with the matrix Q(η,φ) defined as\nQ(η,φ) =B(η)Z(φ) =/parenleftbigge(η−iφ)/20\n0e−(η−iφ)/2/parenrightbigg\n,\n˙Q(η,φ) =˙B(η)˙Z(φ) =/parenleftbigge−(η+iφ)/20\n0e(η+iφ)/2/parenrightbigg\n. (158)\nWhen this combined matrix is applied to the spinors,\nQ(η,φ)u=e(η−iφ)/2u, Q (η,φ)v=e−(η−iφ)/2v,\n˙Q(η,φ)˙u=e−(η+iφ)/2˙u,˙Q(η,φ)˙v=e(η+iφ)/2˙v. (159)\nIf the particle is at rest, we can construct the combinations\nuu,1√\n2(uv+vu), vv, (160)\nto construct the spin-1 state, and\n1√\n2(uv−vu), (161)\nfor the spin-zero state. There are four bilinear states. In t heSL(2,c) regime, there are\ntwo dotted spinors. If we include both dotted and undotted sp inors, there are sixteen\nindependent bilinear combinations. They are given in Table 8. This table also gives\nthe effect of the operation of Q(η,φ).\nAmong the bilinear combinations given in Table 8, the follow ing two are invariant\nunder rotations and also under boosts.\nS=1√\n2(uv−vu),and˙S=−1√\n2(˙u˙v−˙v˙u). (162)\nThey are thus scalars in the Lorentz-covariant world. Are th ey the same or different?\nLet us consider the following combinations\nS+=1√\n2/parenleftig\nS+˙S/parenrightig\n,andS−=1√\n2/parenleftig\nS−˙S/parenrightig\n. (163)\nUnder the dot conjugation, S+remains invariant, but S−changes its sign.\nUnder the dot conjugation, the boost is performed in the oppo site direction. There-\nfore it is the operation of space inversion, and S+is a scalar while S−is called the\npseudo-scalar.\n34Table 8: Sixteen combinations of the SL(2,c) spinors. In the SU(2) regime, there are two\nspinors leading to four bilinear forms. In the SL(2,c) world, there are two undotted and two\ndotted spinors. These four spinors lead to sixteen independent bilin ear combinations.\nSpin 1 Spin 0\nuu,1√\n2(uv+vu), vv,1√\n2(uv−vu)\n˙u˙u,1√\n2(˙u˙v+ ˙v˙u),˙v˙v,1√\n2(˙u˙v−˙v˙u)\nu˙u,1√\n2(u˙v+v˙u), v˙v,1√\n2(u˙v−v˙u)\n˙uu,1√\n2(˙uv+ ˙vu),˙vv,1√\n2(˙uv−˙vu)\nAfter the operation of Q(η,φ) and˙Q(η,φ)\ne−iφeηuu,1√\n2(uv+vu), eiφe−ηvv,1√\n2(uv−vu)\ne−iφe−η˙u˙u,1√\n2(˙u˙v+ ˙v˙u), eiφeη˙v˙v,1√\n2(˙u˙v−˙v˙u)\ne−iφu˙u,1√\n2(eηu˙v+e−ηv˙u), eiφv˙v,1√\n2(eηu˙v−e−ηv˙u)\ne−iφ˙uu,1√\n2(˙uv+ ˙vu), eiφ˙vv,1√\n2(e−η˙uv−eη˙vu)\n358.1 Four-vectors\nLetusconsiderthebilinearproductsofonedottedandoneun dottedspinoras u˙u, u˙v,˙uv, v˙v,\nand construct the matrix\nU=/parenleftbiggu˙v v˙v\nu˙u v˙u/parenrightbigg\n. (164)\nUnder the rotation Z(φ) and the boost B(η) they become\n/parenleftbiggeηu˙v e−iφv˙v\neiφu˙u e−ηv˙u/parenrightbigg\n. (165)\nIndeed, this matrix is consistent with the transformation p roperties given in Table 8,\nand transforms like the four-vector\n/parenleftbiggt+z x−iy\nx+iy t−z/parenrightbigg\n. (166)\nThis form was given in Eq.(62), and played the central role th roughout this paper.\nUnder the space inversion, this matrix becomes\n/parenleftbiggt−z−(x−iy)\n−(x+iy)t+z/parenrightbigg\n. (167)\nThis space inversion is known as the parity operation.\nThe form of Eq.(164) for a particle or field with four-compone nts, is given by\n(V0,Vz,Vx,Vy). The two-by-two form of this four-vector is\nU=/parenleftbiggV0+VzVx−iVy\nVx+iVyV0−Vz/parenrightbigg\n. (168)\nIf boosted along the zdirection, this matrix becomes\n/parenleftbiggeη(V0+Vz)Vx−iVy\nVx+iVye−η(V0−Vz)/parenrightbigg\n. (169)\nIn the mass-zero limit, the four-vector matrix of Eq.(169) b ecomes\n/parenleftbigg2A0Ax−iAy\nAx+iAy0/parenrightbigg\n, (170)\nwith the Lorentz condition A0=Az. The gauge transformation applicable to the\nphoton four-vector was discussed in detail in Subsec. 5.2.\nLet us go back to the matrix of Eq.(168), we can construct anot her matrix ˙U. Since\nthe dot conjugation leads to the space inversion,\n˙U=/parenleftbigg˙uv˙vv\n˙uu˙vu/parenrightbigg\n. (171)\nThen\n˙uv≃(t−z),˙vu≃(t+z)\n˙vv≃ −(x−iy),˙uu≃ −(x+iy), (172)\nwhere the symbol ≃means “transforms like.”\nThus,Uof Eq.(164) and ˙Uof Eq.(171) used up eight of the sixteen bilinear forms.\nSince there are two bilinear forms in the scalar and pseudo-s calar as given in Eq.(163),\nwe have to give interpretations to the six remaining bilinea r forms.\n368.2 Second-rank Tensor\nIn this subsection, we are studying bilinear forms with both spinors dotted and un-\ndotted. In Subsec. 8.1, each bilinear spinor consisted of on e dotted and one undotted\nspinor. There are also bilinear spinors which are both dotte d or both undotted. We are\ninterested in two sets of three quantities satisfying the O(3) symmetry. They should\ntherefore transform like\n(x+iy)/√\n2,(x−iy)/√\n2, z, (173)\nwhich are like\nuu, vv, (uv+vu)/√\n2, (174)\nrespectively in the O(3) regime. Since the dot conjugation is the parity operatio n, they\nare like\n−˙u˙u,−˙v˙v,−(˙u˙v+ ˙v˙u)/√\n2. (175)\nIn other words,\n(uu˙) =−˙u˙u,and (vv˙) =−˙v˙v. (176)\nWe noticed a similar sign change in Eq.(172).\nIn order to construct the zcomponent in this O(3) space, let us first consider\nfz=1\n2[(uv+vu)−(˙u˙v+ ˙v˙u)], g z=1\n2i[(uv+vu)+(˙u˙v+ ˙v˙u)],(177)\nwherefzandgzare respectively symmetric and anti-symmetric under the do t conju-\ngation or the parity operation. These quantities are invari ant under the boost along\nthezdirection. They are also invariant under rotations around t his axis, but they are\nnot invariant under boost along or rotations around the xoryaxis. They are different\nfrom the scalars given in Eq.(162).\nNext, in order to construct the xandycomponents, we start with g±as\nf+=1√\n2(uu−˙u˙u)g+=1√\n2i(uu+ ˙u˙u)\nf−=1√\n2(vv−˙v˙v)g−=1√\n2i(vv+ ˙v˙v). (178)\nThen\nfx=1√\n2(f++f−) =1\n2[(uu−˙u˙u)+(vv−˙v˙v)]\nfy=1√\n2i(f+−f−) =1\n2i[(uu−˙u˙u)−(vv−˙v˙v)]. (179)\nand\ngx=1��\n2(g++g−) =1\n2i[(uu+ ˙u˙u)+(vv+ ˙v˙v)]\ngy=1√\n2i(g+−g−) =−1\n2[(uu+ ˙u˙u)−(vv+ ˙v˙v)]. (180)\nHerefxandfyare symmetric under dot conjugation, while gxandgyare anti-\nsymmetric.\n37Furthermore, fz,fx,andfyofEqs.(177)and(179)transformlikeathree-dimensional\nvector. The same can be said for giof Eqs.(177) and (180). Thus, they can grouped\ninto the second-rank tensor\nT=\n0−gz−gx−gy\ngz0−fyfx\ngxfy0−fz\ngy−fxfz0\n, (181)\nwhose Lorentz-transformation properties are well known. T hegicomponents change\ntheir signs under space inversion, while the ficomponents remain invariant. They are\nlike the electric and magnetic fields respectively.\nIf the system is Lorentz-booted, fiandgican be computed from Table 8. We are\nnow interested in the symmetry of photons by taking the massl ess limit. According to\nthe procedure developed in Sec. 6, we can keep only the terms w hich become larger for\nlarger values of η. Thus,\nfx→1\n2(uu−˙v˙v), f y→1\n2i(uu+ ˙v˙v),\ngx→1\n2i(uu+ ˙v˙v), g y→ −1\n2(uu−˙v˙v), (182)\nin the massless limit.\nThen the tensor of Eq.(181) becomes\nF=\n0 0 −Ex−Ey\n0 0 −ByBx\nExBy0 0\nEy−Bx0 0\n, (183)\nwith\nBx≃1\n2(uu−˙v˙v), B y≃1\n2i(uu+ ˙v˙v),\nEx=1\n2i(uu+ ˙v˙v), E y=−1\n2(uu−˙v˙v). (184)\nThe electric and magnetic field components are perpendicula r to each other. Fur-\nthermore,\nEx=By, E y=−Bx. (185)\nIn order to address this question, let us go back to Eq.(178). In the massless limit,\nB+≃E+≃uu, B −≃E−≃˙v˙v (186)\nThe gauge transformation applicable to uand ˙vare the two-by-two matrices\n/parenleftbigg1−γ\n0 1/parenrightbigg\n,and/parenleftbigg1 0\n−γ1/parenrightbigg\n. (187)\nrespectively as noted in Subsecs. 5.2 and 7.1. Both uand ˙vare invariant under gauge\ntransformations, while ˙ uandvdo not.\nTheB+andE+are for the photon spin along the zdirection, while B−andE−\nare for the opposite direction. In 1964 [35], Weinberg const ructed gauge-invariant\nstate vectors for massless particles starting from Wigner’ s 1939 paper [1]. The bilinear\nspinorsuuand and ˙v˙vcorrespond to Weinberg’s state vectors.\n38fig.(a) \nfig.(b) \nFigure7: Contractionsofthethree-dimensional rotationgroup. Thisgroupcanbeillustrated\nby asphere. This groupcanbecome thetwo-dimensional Euclidean g roupona plane tangent\nat the north poleas illustrated in fig.(a). It was later noted that the re is a cylinder tangential\nto this sphere, and the up and down translations on this cylinder cor respond to the gauge\ntransformation for photons [20]. As illustrated in fig.(b), the four- dimensional representation\nof of the Lorentz group contains both elongation and contraction of of the zaxis, as the\nsystem is boosted along this direction. The elongation and the contr action become the\ncylindrical and Euclidean groups, respectively [21].\n8.3 Possible Symmetry of the Higgs Mechanism\nIn this section, we discussed how the two-by-two formalism o f the group SL(2,c) leads\nthescalar, four-vector, andtensorrepresentationsofthe Lorentzgroup. Wediscussedin\ndetail how the four-vector for a massive particle can be deco mposed into the symmetry\nof a two-component massless particle and one gauge degree of freedom. This aspect\nwas studied in detail by Kim and Wigner [20, 21], and their res ults are illustrated in\nFig. 7.\nThis subject was initiated by In¨ on¨ u and Wigner in 1953 as th e group contrac-\ntion [36]. In their paper, they discussed the contraction of the three-dimensional ro-\ntation group becoming contracted to the two-dimensional Eu clidean group with one\nrotational and two translational degrees of freedom. While theO(3) rotation group\ncan be illustrated by a three-dimensional sphere, the plane tangential at the north pole\nis for theE(2) Euclidean group. However, we can also consider a cylinde r tangential\nat the equatorial belt. The resulting cylindrical group is i somorphic to the Euclidean\ngroup [20]. While the rotational degree of freedom of this cy linder is for the photon\nspin, the up and down translations on the surface of the cylin der correspond to the\ngauge degree of freedom of the photon, as illustrated in Fig. 7.\nThe four-dimensional Lorentz group contains both the Eucli dean and cylindrical\ncontractions. These contraction processes transform a fou r-component massive vector\nmeson into a massless spin-one particle with two spin one-ha lf components, and one\ngauge degree of freedom.\n39Since this contraction procedure is spelled out detail in Re f. [21], as well as in the\npresent paper, its reverse process is also well understood. We start with one two-\ncomponent massless particle with one gauge degree of freedo m, and end up with a\nmassive vector meson with its four components.\nThe mathematics of this process is not unlike the Higgs mecha nism [37, 38], where\none massless field with two degrees of freedom absorbs one gau ge degree freedom to\nbecome a quartet of bosons, namely that of W,Z±plus the Higgs boson. As is well\nknown, thismechanismisthebasisforthetheoryofelectro- weak interaction formulated\nby Weinberg and Salam [39, 40].\nThe word ”spontaneous symmetry breaking” is used for the Hig gs mechanism. It\ncould be an interesting problem to see that this symmetry bre aking for the two Higgs\ndoublet model can be formulated in terms of the Lorentz group and its contractions.\nIn this connection, we note an interesting recent paper by D´ ee and Ivanov [41].\nConclusions\nIt was noted in this paper that the second-order differential e quation for damped\nharmonicoscillators canbeformulatedintermsoftwo-by-t wo matrices. Thesematrices\nproduce the algebra of the group Sp(2). While there are three trace classes of the two-\nby-two matrices of this group, the damped oscillator tells u s how to make transitions\nfrom one class to another.\nIt is shown that Wigner’s three little groups can be defined in terms of the trace\nclasses of the Sp(2) group. If the trace is smaller than two, the little group i s for\nmassive particles. If greater than two, the little group is f or imaginary-mass particles.\nIf the trace is equal to two, the little group is for massless p articles. Thus, the damped\nharmonicoscillator provides a procedurefor transition fr om one little groupto another.\nThe Poincar´ e sphere contains the symmetry of the six-param eterSL(2,c)group.\nThus, the sphere provides the procedure for extending the sy mmetry of the little group\ndefinedwithinthespaceofthree-dimensionalLorentzgroup tothefullfour-dimensional\nMinkowski space. In addition, the Poincar´ e sphere offers the variable which allows us\nto change the symmetry of massive particle to that of massles s particle by continuously\nchanging the mass.\nIn this paper, we extracted the mathematical properties of t he Lorentz group and\nWigner’s little groups from the damped harmonic oscillator and the Poincar´ e sphere.\nInaddition, it shouldbenotedthat thesymmetryoftheLoren tzgroupisalsocontained\nin thesqueezed state of light [15] and the ABCDmatrix for optical beam transfers[14].\nIn addition, we mentioned the possibility of understanding the the mathematics of\nthe Higgs mechanism in terms of the Lorentz group and its cont ractions.\nReferences\n[1] Wigner, E. On Unitary Representations of the Inhomogene ous Lorentz Group,\nAnn. Math. 1939,40, 149-204.\n[2] Han, D; Kim, Y. S.; Son, D. Eulerian parametrization of Wi gner little groups\nand gauge transformations in terms of rotations in 2-compon ent spinors, J. Math.\nPhys.1986,27, 2228-2235.\n40[3] Born, M.; Wolf, E. Principles of Optics. 6th Ed. (Pergamon, Oxford) 1980.\n[4] Han, D; Kim, Y. S.; Noz, M. E. Stokes parameters as a Minkow skian four-vector,\nPhys. Rev. E 1997,56, 6065-76.\n[5] Brosseau, C. Fundamentals of Polarized Light: A Statistical Optics Appro ach\n(John Wiley, New York) 1998.\n[6] Ba¸ skal, S; Kim, Y. S. de Sitter group as a symmetry for opt ical decoherence, J.\nPhys. A 2006,39, 7775-88.\n[7] Kim, Y. S.; Noz, M. E. Symmetries Shared by the Poincar´ e G roup and the\nPoincar´ e Sphere, Symmetry 2013,5, 233-252.\n[8] Han, D; Kim, Y. S; Son, D. E(2)-like little group for massl ess particles and polar-\nization of neutrinos, Phys. Rev. D 1982,26, 3717-3725.\n[9] Han, D.; Kim Y. S.; Son D. Photons, neutrinos and gauge tra nsformations, Am.\nJ. Phys. 1986,54, 818-821.\n[10] Ba¸ skal, S; Kim, Y. S. Little groups and Maxwell-type te nsors for massive and\nmassless particles, Europhys. 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Lett. 1967,19, 1265-1266.\n[40] Weinberg, S. Quantum Theory of Fields, Volume II, Modern Applications (Cam-\nbridge University Press) 1996.\n[41] D´ ee, A; Ivanov, I. P. Higgs boson masses of the general t wo-Higgs-doublet model\nin the Minkowski-space formalism Phys. Rev. D 2010,81, 015012-8.\n42"    },    {        "title": "0906.0086v2.Lorentz_Ricci_solitons_on_3_dimensional_Lie_groups.pdf",        "content": "arXiv:0906.0086v2  [math.DG]  3 Jul 2009Lorentz Ricci solitons on 3-dimensional Lie\ngroups\nKensuke Onda∗\nGraduate School of Mathematics, Nagoya University\nAbstract\nThe three-dimensional Heisenberg group H3has three left-invariant Lorentz\nmetricsg1,g2, andg3asin[R92] . They arenot isometric each other. Inthis\npaper, we characterize the left-invariant Lorentzian metric g1as a Lorentz\nRicci soliton. This Ricci soliton g1is a shrinking non-gradient Ricci soliton.\nLikewise we prove that the isometry group of flat Euclid plane E(2) and the\nisometry group of flat Lorentz plane E(1,1) have Lorentz Ricci solitons.\n1 Introduction\nIn this paper, we consider Lorentz metrics on 3-dimensional Lie gro ups. The\nHeisenberg group H3has three left-invariant Lorentzian metrics g1,g2, and\ng3.Rahmani [R92] proved that these metrics are non-isometric each o ther,\nand Turhan [T08] showed that these metrics are geodesically comple te. Rah-\nmani [R92 ,RR06] exhibited that the Lie algebra of infinitesimal isometries\nof (H3,g1) and (H3,g2) are four-dimensional and solvable but not nilpotent,\nand the Lie algebra of infinitesimal isometries of ( H3,g3) is six-dimensional.\nMoreover Rahmani [RR06] showed that the left-invariant Lorentz ian metric\ng2has a negative constant curvature −1/4 ,g3is flat, and g1is not Einstein.\nThere area difference ofa Riemannian caseand aLorentzian case, a ndoneof\n∗kensuke.onda@math.nagoya-u.ac.jp\n1interesting phenomenon of a Lorentzian case. In this paper, we ch aracterize\nthe left-invariant Lorentzian metric g1as a Lorentz Ricci soliton, that is one\nof the generalization of Einstein metrics.\nLetg0be a pseudo-Riemannian metric on manifold Mn. Ifg0satisfies\n2Ric[g0]+LXg0+αg0= 0,\nwhereXis some vector field and αis some constant, then ( Mn,g0,X,α) is\ncalled aRicci soliton structure andg0is called the Ricci soliton. Moreover\nwe say that the Ricci soliton g0is agradient Ricci soliton if the vector field\nXsatisfiesX=∇f, wherefis some function, and the Ricci soliton gis\nanon-gradient Ricci soliton if the vector field XsatisfiesX/ne}ationslash=∇ffor any\nfunction f. If a constant αis negative, zero, or positive, then gis called\na shrinking, steady, or expanding Ricci soliton, respectively. Acco rding to\n[CK04], Ricci solitons have a relation with the Ricci flow.\nProposition 1. A pseudo-Riemannian metric g0is a Ricci soliton if and\nonly ifg0is a solution of the Ricci flow equation,\n∂\n∂tg(t)ij=−2Ric[g(t)]ij,\nsatisfying g(t) =c(t)(ϕt)∗g0, wherec(t)is a scaling parameter, and ϕtis a\ndiffeomorphism.\nAn interesting example of Ricci solitons are ( R2,gst,X,α), where the\nmetricgstis the Euclidean metric on R2, the vector field Xis\nX=−1\n2(αx+βy)∂\n∂x+1\n2(βx−αy)∂\n∂y,\nandαandβare any constants. If β= 0, then ( R2,gst,X,α) are gradient\nRicci solitons, named Gaussian solitons, and if β/ne}ationslash= 0, then ( R2,gst,X,α)\nare non-gradient Ricci solitons. In closed Riemannian case, Perelma n [P02]\nproved that Ricci solitons become gradient Ricci solitons, and any s teady or\nexpanding Ricci solitonsbecomeEinstein metrics withEinstein consta nt zero\nor negative, respectively. But in non-compact Riemannian case, it is false. A\ncounterexample is that any left-invariant Riemannian metrics on the three-\ndimensional Heisenberg group are expanding non-gradient Ricci so litons but\nnot Einstein metrics ([BD07] ,[GIK06],[L07]).\n2Our goal is to characterize the left-invariant Lorentzian metric g1as a\nLorentz Ricci soliton, and to prove that the isometry group of flat Euclid\nplaneE(2) and the isometry group of flat Lorentz plane E(1,1) have Lorentz\nRicci solitons. This paper is organized as follows. Section 2 contains a brief\nreview of the Levi-Civita connection of a left-invariant metric to pre pare for\nSection3, 4, and5. InSection3, wecalculateRiccitensorofthelef t-invariant\nLorentzian metric g1onH3, and prove that g1satisfies Ricci soliton equation.\nIn Section 4, we calculate Ricci tensors of a left-invariant Lorentz ian metric\nonE(2), and prove that the metric satisfies a Ricci soliton equation. Fin ally,\nin Section 5 we consider a left-invariant Lorentzian metric on E(1,1), that is\ngeodesically incomplete and satisfies a Ricci soliton equation.\n2 Preliminaries\nLetGbe a 3-dimensional Lie group, and gthe Lie algebra of all left-invariant\nvector fields on G. Suppose that X,Y, andZare left-invariant vector fields\nbelonging to g. In [CK04], a map ad : g→gl(g) is defined by\n(adX)Y:= [X,Y],\nand the adjoint ad′:g→gl(g) with respect to a left-invariant metric gof\nthe map ad : g→gl(g) given by\ng/parenleftbig\n(adX)′Y,Z/parenrightbig\n=g/parenleftbig\nY,(adX)Z/parenrightbig\n=g/parenleftbig\nY,[X,Z]/parenrightbig\n.\nAs known, the Levi-Civita connection is determined as follow.\nProposition 2 ([CK04]) .The Levi-Civita connection of a left-invariant met-\nricgis determined by adandad′via the formula\n∇XY=1\n2/braceleftbig\n[X,Y]−(adX)′Y−(adY)′X/bracerightbig\n. (1)\nThe above proposition is proved for a pseudo-Riemannian metric. We use\nthe formula as later.\n33 The three-dimensional Heisenberg group\nThe Heisenberg group H3has three left-invariant Lorentzian metrics g1, g2,\nandg3,giving by\ng1=−dx2+dy2+(x dy+dz)2,\ng2=dx2+dy2−(x dy+dz)2,\ng3=dx2+(x dy+dz)2−/parenleftbig\n(1−x)dy−dz/parenrightbig2.\nRahmani [RR06] showed that the left-invariant Lorentzian metric g2has a\nnegative constant curvature −1/4 ,g3is flat, and g1is not Einstein. In this\nsection, we characterize the left-invariant Lorentzian metric g1as a Lorentz\nRicci soliton. Let our frame be defined by\nF1=∂\n∂z,F2=∂\n∂y−x∂\n∂z,F3=∂\n∂x,\nand coframe\nθ1=x dy+dz ,θ2=dy ,θ3=dx .\nIt is easy to check that the metric g1is represented by g1= (θ1)2+(θ2)2−\n(θ3)2, and all brackets [ Fi,Fj] vanish except [ F2,F3] =F1. Then the Levi-\nCivita connection of the metric g1is given by\n(∇FiFj) =1\n2\n0F3F2\nF30F1\nF2−F10\n, (2)\nand its Ricci tensor is expressed by\nR11=−1\n2,R22=1\n2,R33=−1\n2,\nand other components are 0. Since R11=−1\n2g11andR22=1\n2g22,g1is not\nEinstein. But g1satisfies a Ricci soliton equation as follows.\nTheorem 3. The left-invariant Lorentzian metric\ng1=−dx2+dy2+(x dy+dz)2\nsatisfies a Ricci soliton equation\n2Ric+LXg−3g= 0,\n4where the vector field Xis defined by\nX= (2z+xy)F1+yF2+xF3.\nMoreover the vector field XsatisfiesX/ne}ationslash=∇ffor any function f. Therefore\nthe left-invariant Lorentzian metric g1is a shrinking non-gradient Lorentz\nRicci soliton.\nProof.Using (2), we get\n(∇iXj) =\n21\n2x −1\n2y\n−1\n2x1−1\n2(2z+xy)\n1\n2y1\n2(2z+xy)−1\n. (3)\nHence the metric g1is a shrinking Ricci soliton. Since\n∇1X2−∇2X1=x/ne}ationslash= 0,\nwe obtain the statement.\nAs a consequence, Lorentz metrics on H3are negative constant curvature\nmetric, flat metric, or shrinking Ricci soliton, which is one of the gene raliza-\ntion of positive Einstein metrics.\nRemark 4. According to Theorem 3, the left-invariant Lorentzian metric\ng1is a “shrinking” Ricci soliton. But any left-invariant Riemannian metric s\nonH3are “expanding” Ricci solitons. This phenomenon is understood as\nbelow.\nWe consider left-invariant Riemannian metrics g(t) onH3, giving by\ng(t) =A(t)(θ1)2+B(t)(θ2)2+C(t)(θ3)2, where it’s frame {Fi}3\ni=1satis-\nfies [F2,F3] =F1, andA(t),B(t), andC(t) are positive constants. For any\nA,B, andC, the metric g(t) becomes an expanding Ricci soliton. The Ricci\nflow ofg(t) is given by\n\nd\ndtA=−A2\nBC,\nd\ndtB=A\nC,\nd\ndtC=A\nB,(4)\nand the Backward Ricci flow\n∂\n∂tg(t)ij= 2Ric[g(t)]ij\n5ofg(t) is expressed by\n\nd\ndtA=A2\nBC,\nd\ndtB=−A\nC,\nd\ndtC=−A\nB.(5)\nThe Ricci flow and Backward Ricci flow exist on −∞< t < T and−T <\nt <+∞, respectively, where Tis some constant depending only on A(0),\nB(0), and C(0) . In the Ricci flow case, if t→T, thenABC→+∞. This\nphenomenon means that the metric is expanding. In the Backward R icci flow\ncase, ift→+∞, thenABC→0. This phenomenon means that the metric\nis shrinking.\nOn the other hand, we consider left-invariant Lorentzian metrics g(t) on\nH3, giving by g(t) =A(t)(θ1)2+B(t)(θ2)2−C(t)(θ3)2, where it’s frame\n{Fi}3\ni=1satisfies [ F2,F3] =F1, andA(t),B(t), andC(t) are positive con-\nstants. If A=B=C= 1, the metric g(t) becomes g1. For any A,B, and\nC, the metric gbecomes a shrinking Lorentzian Ricci soliton. The Ricci flow\nequations of g(t) are\n\nd\ndtA=A2\nBC,\nd\ndtB=−A\nC,\nd\ndtC=−A\nB.(6)\nThis equations are same as Backward Ricci flow equations of Riemann ian\nmetricsg(t). So in Lorentz case, a parameter texists on ( −T,+∞), whereT\nis some constant depending only on A(0),B(0), and C(0) , and if t→+∞,\nthenABC→0, and this phenomenon means that the metric is shrinking.\n4 The isometry group of flat Euclid plane\nIn this section, we consider the isometry group of flat Euclid plane E(2).\nThe isometry group of flat Euclid plane has at least two kinds of Loren tzian\n6metrics as follow :\ng1= (siny dx+cosy dz)2+dy2−(cosy dx−siny dz)2,\ng2= (siny dx+cosy dz)2−dy2+(cosy dx−siny dz)2.\nAccording to Theorem 4 .1 of [G96], these metrics are geodesically complete.\nThe Lorentzian metric g2is expressed by\ng2=−dy2+dx2+dz2.\nThereforetheLorentzianmetric g2isaflatLorentzianmetric. Inthissection,\nwe characterize the left-invariant Lorentzian metric g1as a Lorentz Ricci\nsoliton.\nLet our frame be defined by\nF1= siny∂\n∂x+cosy∂\n∂z, F2=∂\n∂y, F3= cosy∂\n∂x−siny∂\n∂z,\nand coframe\nθ1= siny dx+cosy dz , θ2=dy , θ3= cosy dx−siny dz .\nIt iseasy tocheck thatthe metric g1is given by g1= (θ1)2+(θ2)2−(θ3)2, and\nthe bracket relations are [ F1,F2] =−F3, [F2,F3] =−F1, and [F3,F1] = 0 .\nThen the Levi-Civita connection of the metric g1is represented by\n(∇FiFj) =\n0 0F2\nF30−F1\nF20 0\n, (8)\nand its Ricci tensor vanish except R22= 2 . The Lorentzian metric g1is not\nEinstein obviously, but satisfies a Ricci soliton equation as follows.\nTheorem 5. The left-invariant Lorentzian metric\ng1= (siny dx+cosy dz)2+dy2−(cosy dx−siny dz)2\nsatisfies a Ricci soliton equation\n2Ric+LXg−4g= 0,\n7where the vector field Xis defined by\nX= 2(zcosy+xsiny)F1+2(xcosy−zsiny)F3.\nMoreover the vector field XsatisfiesX/ne}ationslash=∇ffor any function f. Thus the\nleft-invariant Lorentzian metric g1is a shrinking non-gradient Lorentz Ricci\nsoliton.\nProof.Using (8), we get\n(∇iXj) =\n2 −2(−zsiny+xcosy) 0\n2(−zsiny+xcosy) 0 2( xsiny+zcosy)\n0 −2(xsiny+zcosy) −2\n.\nIt follows that the metric g1is a shrinking Ricci soliton. Since\n∇1X2−∇2X1=−4(−zsiny+xcosy)/ne}ationslash= 0,\nwe obtain the statement.\n5 The isometry group of flat Lorentz plane\nIn this section, we consider the isometry group of flat Euclid plane E(1,1).\nThe isometry group of flat Lorentz plane has at least two kinds of Lo rentzian\nmetrics as follow :\ng1=−dz2+(ezdx+e−zdy)2+(ezdx−e−zdy)2,\ng2=dz2+(ezdx+e−zdy)2−(ezdx−e−zdy)2.\nAccording to Theorem 4 .2 of [G96], g1is geodesically incomplete, however\ng2is geodesically complete. Since g2is the same as [N79] , g2is flat. In this\nsection, we characterize the left-invariant Lorentzian metric g1as a Lorentz\nRicci soliton.\nLet our frame be defined by\nF1=1\n2/parenleftBig\ne−z∂\n∂x+ez∂\n∂y/parenrightBig\n, F2=1\n2/parenleftBig\ne−z∂\n∂x−ez∂\n∂y/parenrightBig\n, F3=∂\n∂z\n8and coframe\nθ1=ezdx+e−zdy , θ2=ezdx−e−zdy , θ3=dz .\nIt is easy to check that the metric g1is given by g1= (θ1)2+(θ2)2−(θ3)2,\nand the bracket relations are [ F1,F2] = 0 , [F2,F3] =F1, and [F3,F1] =−F2.\nThen the Levi-Civita connection of the metric g1is represented by\n(∇FiFj) =\n0F3F2\nF30F1\n0 0 0\n, (10)\nand its Ricci tensor vanish except R33=−2 . The Lorentzian metric g1is\nnot Einstein obviously, however satisfies a Ricci soliton equation as f ollows.\nTheorem 6. The left-invariant Lorentzian metric\ng1=−dz2+(ezdx+e−zdy)2+(ezdx−e−zdy)2\nsatisfies a Ricci soliton equation\n2Ric+LXg−4g= 0,\nwhere the vector field Xis defined by\nX= 4a(xezF1+xezF2−1\n2F3)+4(1−a)(ye−zF1−ye−zF2+1\n2F3)\n+b(ezF1+ezF2)+c(e−zF1−e−zF2),\nwherea,b, andcare any constants. Moreover the vector field Xsatisfies\nX/ne}ationslash=∇ffor any function f. Thus the left-invariant Lorentzian metric g1is\na shrinking non-gradient Lorentz Ricci soliton.\nProof.Using (10), we get the following :\n∇1X1= 2,\n∇1X2=∇2X1= 0,\n∇1X3=∇3X1=−4axez+4(1−a)ye−z−bez+ce−z,\n∇2X2= 2,\n∇2X3=∇3X2=−4axez−4(1−a)ye−z−bez−ce−z,\n∇3X3= 0.\n9It follows that the metric g1is a shrinking Ricci soliton. Since\n∇3X2−∇2X3= 8axez−8(1−a)ye−z+2bez−2ce−z/ne}ationslash= 0,\nwe obtain the statement.\nRemark 7. The vector field Xis described by\nX= 4ye−zF1−4ye−zF2+2F3\n+4aY1+bY2+cY3,\nwhereY1= (xez−ye−z)F1+(xez+ye−z)F2−F3,Y2=ezF1+ezF2,Y3=\ne−zF1−e−zF2.Vector fields {Yi}3\ni=1form the Lie algebra of infinitesimal\nisometries of ( E(1,1),g1), that is solvable.\nReferences\n[Be] A. Besse, Einstein Manifolds , Springer-Verlag, New York, 1987.\n[BD07] P. Baird and L. Danielo, Three-dimensional Ricci solitons which\nproject to surfaces , J. reine angew. Math. 608(2007), 65-91.\n[C07] B. Chow, S-C. Chu, D. Glickenstein, C. Guenther, J. Isenber g, T.\nIvey, D. Knopf, P. Lu, F. Luo, L. Ni, The Ricci flow: techniques and ap-\nplications. Part I. Geometric aspects, Mathematical Surveys and Mono-\ngraphs, 135. American Mathematical Society, Providence, RI, 20 07.\n[CK04] B. Chow and D. Knopf, The Ricci Flow : An Introduction , AMS\n(2004).\n[G96] M. Guediri, On completeness of left-invariant Lorentz metrics on solv-\nable Lie groups, Rev. Mat. Univ. Complut. Madrid 9 (1996), no. 2, 337–\n350.\n[GIK06] C. Guenther, J. Isenberg, D.Knopf, Linear stability of homogeneous\nRicci solitons, Int. Math. Res. Not. 2006, Art. ID 96253, 30 pp.\n[L07] J. Lott, On the long-time behavior of type-III Ricci flow solutions,\nMath. Ann. 339 (2007), no. 3, 627–666.\n10[M76] J. Milnor, Curvatures of left invariant metrics on Lie groups, Ad-\nvances in Math. 21 (1976), no. 3, 293–329.\n[N79] K. Nomizu, Left-invariant Lorentz metrics on Lie groups, Osaka J.\nMath. 16 (1979), no. 1, 143–150.\n[P02] G. Perelman, The entropy formula for the Ricci flow and its geometric\napplications, preprint: math. DG/0211159 (2002).\n[R92] S. Rahmani, Metriques de Lorentz sur les groupes de Lie unimodu-\nlaires, de dimensiontrois (French), [ Lorentz metrics on three-dimensional\nunimodular Lie groups ] J. Geom. Phys. 9 (1992), no. 3, 295–302.\n[RR06] N. Rahmani, and S. Rahmani, Lorentzian geometry of the Heisenberg\ngroup,Geom. Dedicata 118 (2006), 133–140.\n[T08] E. Turhan, Completeness of Lorentz metric on 3-dimensional Heisen-\nberg group, Int. Math. Forum 3 (2008), no. 13-16, 639–644.\n11"    },    {        "title": "2111.02502v2.Interplay_of_interactions_and_disorder_at_the_charge_density_wave_transition_of_two_dimensional_Dirac_semimetals.pdf",        "content": "Interplay of interactions and disorder at the charge-density wave transition of\ntwo-dimensional Dirac semimetals\nMikolaj D. Uryszek1and Frank Kr uger1, 2\n1London Centre for Nanotechnology, University College London,\nGordon St., London, WC1H 0AH, United Kingdom\n2ISIS Facility, Rutherford Appleton Laboratory, Chilton,\nDidcot, Oxfordshire OX11 0QX, United Kingdom\nWe consider the e\u000bects of weak quenched fermionic disorder on the quantum-phase transition\nbetween the Dirac semimetal and charge density wave (CDW) insulator in two spatial dimensions.\nThe symmetry breaking transition is described by the Gross-Neveu-Yukawa (GNY) theory of Dirac\nfermions coupled to an Ising order parameter \feld. Treating the disorder using the replica method,\nwe consider chemical potential, vector potential (gauge), and random mass disorders, which all\narise from non-magnetic charge impurities. We self-consistently account for the Landau damping of\nlong-wavelength order-parameter \ructuations by using the non-perturbative RPA re-summation of\nfermion loops, and compute the renormalization-group (RG) \row to leading order in the disorder\nstrength and 1 =N(Nthe number of Dirac fermion \ravors). We \fnd two \fxed points, the clean\nGNY critical point which is stable against weak disorder, and a dirty GNY multi-critical point, at\nwhich the chemical potential disorder is \fnite and the other forms of disorder are irrelevant. We\ninvestigate the scaling of physical observables at this \fnite-disorder multi-critical point which breaks\nLorentz invariance and gives rise to distinct non-Fermi liquid behavior.\nI. INTRODUCTION\nOver the past couple of decades, two and three dimen-\nsional topological semimetals have been at the forefront\nof research in condensed matter physics [1{3]. Within\nthis family of systems, Dirac fermions have been found to\nbe ubiquitous, where the most famous example of a host\nsystem is graphene - a two dimensional carbon mono-\nlayer which exhibits a four-fold degenerate band cross-\ning at charge neutrality. Due to the unique properties of\nDirac/Weyl semimetals, e.g. point-like Fermi surface and\nlinearly vanishing density of states, they are well suited\nto realizations of high energy phenomena in an accessible\ncondensed matter setting.\nIn any realistic condensed matter system disor-\nder is present, hence its understanding is paramount.\nQuenched, non-dynamical disorder has been widely stud-\nied in the non-interacting limit of systems that exhibit\ntwo-dimensional Dirac fermions, e.g. degenerate (or zero-\ngap) semiconductors [4, 5], graphene [6{12], and d-wave\nsuperconductors [13{15].\nA lot of interest was triggered by the \frst graphene\nexperiments [16{18] which showed a minimal conductiv-\nity of the order of the conductance quantum e2=hover\na wide range of temperatures. It was shown theoreti-\ncally that the transport properties depend crucially on\nthe type of disorder [7] but that for randomness which\npreserves one of the chiral symmetries of the clean Hamil-\ntonian the conductivity is equal to the minimal value\n[7, 8], suggesting that the transport is not a\u000bected by\nlocalization and remains ballistic. However, this univer-\nsal result is based on a self-consistent Born approxima-\ntion, which is not applicable to massless Dirac fermions in\ntwo spatial dimensions [6, 13, 14]. More recently, it was\nargued that over the experimentally accessible temper-\nature range, graphene is in the Drude{Boltzmann di\u000bu-sive transport regime and that density inhomogeneities\nfrom remote charge impurities render the Dirac points\ne\u000bectively inaccessible to experiments [9, 10]. Using a\nself consistent RPA-Boltzmann approach, the authors\nshowed that the conductivity is indeed of order e2=hbut\nwith a non-universal pre-factor that depends on the disor-\nder distribution. Remote charge impurities can be viewed\nas random chemical potential shifts that give rise to pud-\ndles of electron and hole-doped regions in the graphene\nlayer. Building on that picture, the scaling of the con-\nductivity was obtained within a random resistor network\nmodel that describes the percolation of p- andn-type\nregions [11].\nAs the minimal conductivity puzzle shows, there is a\nlot of rich physics already at the non-interacting level.\nHowever, an accurate description of a Dirac semimetal\nalso must include the e\u000bects of electron-electron inter-\nactions, on top of the disorder. For weak Coulomb in-\nteractions the clean two-dimensional Dirac \fxed point\nis unstable against generic disorder and the RG \row is\ndominated by the randomness in the chemical potential\n[19{21], similar to the non-interacting case [6] and con-\nsistent with the picture of local electron and hole \\pud-\ndles\". On the other hand, in the regime of moderate\nto strong Coulomb interactions, it was found that \ruc-\ntuations associated with such random potential disorder\nare parametrically cut o\u000b by screening and that instead\nthe runaway \row is dominated by vector potential disor-\nder [22]. Such disorder from elastic lattice deformations\n(\\ripples\") [7] and topological lattice defects [23{25].\nThe situation is very di\u000berent in three-dimensional\nDirac/Weyl semimetals with long-range Coulomb inter-\nactions. In these systems the semi-metallic phase is sta-\nble against short-range correlated disorder. Above a crit-\nical disorder strength, the semi-metallic phase undergoes\na quantum phase transition into a disorder controlled dif-arXiv:2111.02502v2  [cond-mat.str-el]  24 Feb 20222\nfusive metallic phase with a \fnite density of states at\nthe Fermi level [26{30]. It remains a controversial issue\nwhether the disorder transition is rounded out by non-\nperturbative, rare region e\u000bects [31{33] or not [34, 35].\nUnder a su\u000eciently strong short-ranged electron-\nelectron interaction a Dirac semimetal will undergo a\nquantum phase transition into a symmetry broken state\nwhere the fermionic spectrum is gapped. Such a transi-\ntion is best described using a composite fermion-boson\napproach, resulting from a Hubbard-Stratonovich decou-\npling of the fermionic interaction vertex in the relevant\nchannel with a dynamical order parameter \feld. In the\ncase of Dirac fermions the resulting \feld theory is known\nas the Gross-Neveu-Yukawa (GNY) model which de-\nscribes chiral symmetry breaking and spontaneous mass\ngeneration in high energy physics [36, 37]. The symmetry\nbroken phase is dependent on the nature of the micro-\nscopic interactions; for the half-\flled Hubbard model on\nthe honeycomb lattice with competing interactions a vast\narray of phases were found [38], including antiferromag-\nnetism, di\u000berent types of charge order, Kekule phases\nand topological Quantum Hall states.\nThe e\u000bects of weak quenched disorder on the\nsemimetal-to-superconductor transition, described by\nthe XY GNY model, were studied using \u000fexpansions\nbelow the upper critical dimension [39, 40]. It was found\nthat chemical potential disorder is strongly irrelevant at\nthe clean quantum-critical point in D= 4\u0000\u000fspace-\ntime dimensions but that disorder in the superconduct-\ning order parameter mass plays a crucial role. Such\nbosonic disorder would arise from randomness in the at-\ntractive fermion interaction after Hubbard-Stratonovich\ndecoupling. In the supersymmetric case of a single two-\ncomponent Dirac \feld coupled to the XY order parame-\nter, there is a marginal \row away from the clean critical\npoint to strong disorder [39, 40]. However, if degeneracies\nsuch as spin or valley pseudo-spin are included, the clean\n\fxed point becomes stable against weak bosonic mass\ndisorder and a \fnite-disorder multi-critical point with\nnon-integer dynamical exponent ( z > 1) can be identi-\n\fed within the double \u000fexpansion [40]. Similar \fnite\ndisorder \fxed points were established in the chiral Ising\nand Heisenberg GNY models with bosonic random-mass\ndisorder, using triple \u000fexpansion [41].\nIn this work we revisit the e\u000bects of disorder on\nthe quantum criticality of two dimensional Dirac/Weyl\nfermions. For simplicity, we focus on quantum phase\ntransitions that, in the absence of disorder, are described\nby the chiral Ising GNY theory. An example is the CDW\ntransition of electrons on the half-\flled honeycomb lat-\ntice that is driven by a repulsive nearest-neighbor inter-\naction and characterized by an imbalance of charge on\nthe two sublattices. Our work departs in two important\naspects from previous studies [39{41]. Firstly, we omit\nan\u000fexpansion and compute the quantum corrections\nin two spatial dimensions. Away from the upper criti-\ncal dimension, the Landau damping of long-wavelength\norder-parameter \ructuations is a non-perturbative e\u000bect.It renders the order parameter propagator non-analytic\nin the IR limit, thereby changing the universal critical\nbehavior [42, 43]. This physics is not captured by the \u000f\nexpansion since the boson propagator remains analytic\nat the upper critical dimension. Secondly, we consider\ndisorder on the level of the original fermionic theory, e.g.\nin the form of a random potential from point-like impuri-\nties. In the physical dimension, such fermionic potential\ndisorder is marginal at the clean GNY \fxed point.\nOur paper is organized as follows. In Sec. II we present\nthe Yukawa theory for Dirac fermions subject to a strong-\nshort ranged interaction and uncorrelated disorder. We\nutilize the RPA to account for the non-perturbative Lan-\ndau damping of long-wavelength order-parameter \ructu-\nations. The disorder is treated using the replica formal-\nism [44, 45]. In Sec. III we employ Wilson's momentum\nshell RG within the large- Nexpansion to derive the \row\nof the disorder couplings, and compute the critical expo-\nnents to leading order in 1 =Nand weak disorder. Lastly,\nin Sec. IV we present a summary of our main \fndings,\nand compare them with previous literature.\nII. MODEL\nA. Clean Dirac fermions in 2+1 dimensions\nWe consider Dirac Fermions with dispersion \u000f(k) =\n\u0006vjkjin two spatial dimensions, described by the imag-\ninary time action\nS =Z\nd2xZ\nd\u001c\ty(@\u001c+iv@\u0001\u001b) \t; (1)\nover fermionic Grassmann \felds \t( x;\u001c). Here@=\n(@x;@y) and\u001b= (\u001bx;\u001by) are the conventional 2 \u00022\nPauli matrices. This action describes non-interacting\nelectrons on the half-\flled honeycomb lattice in the long-\nwavelength, low-energy limit, where in this case the Pauli\nmatrices act on the fA;Bgsublattice pseudospin sub-\nspace. In addition, the fermionic Grassmann \felds carry\nthe electron spin \ravors and the valley indices from the\ntwo distinct Dirac points in the Brillouin zone.\nIn the following, we do not consider spontaneous sym-\nmetry breaking or disorder that lift the spin and valley\ndegeneracies. We further generalize to a total number of\nNcomponents of the fermion \felds, \t = (  1;:::; N),\nin order to gain analytic control through an expansion\nin 1=N. For brevity, we use the short-hand notation\nTr[\u001bi\u001bj] =N\u000eij.\nWe consider the case where strong short-range inter-\nactions drive an instability in the charge channel, which\ncorresponds to a quantum phase transition from a Dirac\nsemimetal to a CDW insulator where the sublattice sym-\nmetry is spontaneously broken. Generally this transition\nbelongs to the chiral Ising GNY universality class [36, 37],\nwhich is best studied within the Yukawa language where\nthe Dirac fermions couple to a real-valued, scalar dynam-\nical order parameter \u001e(x;\u001c). This results in the chiral3\nIsing GNY model,\nSGNY =S +Sg+S\u001e+S\u0015; (2)\nwhere\nSg=gp\nNZ\nd2xZ\nd\u001c \u001e\ty\u001bz\t; (3)\nS\u001e=1\n2Z\nd2xZ\nd\u001c \u001e(\u0000@2\n\u001c\u0000c2@2+m2)\u001e; (4)\nS\u0015=\u0015\nNZ\nd2xZ\nd\u001c \u001e4: (5)\nStarting from a lattice model on the honeycomb lattice,\nthe Yukawa coupling Sgarises naturally from a Hubbard\nStratonovich decoupling of a repulsive nearest-neighbor\ninteraction. The Yukawa coupling anti-commutes with\nthe non-interacting action, Eq.(1), and thereby fully\ngaps the fermionic spectrum in the ordered phase where\nh\u001ei6= 0. Here cis the bosonic velocity, and m2is the\ntuning parameter for the quantum phase transition. In\nthe context of the CDW transition on the honeycomb\nlattice,m2\u0018Vc\u0000V, whereVis the repulsion between\nelectrons on adjacent sites and Vcthe critical interaction\nstrength.\nB. Landau damping of order-parameter\n\ructuations in d= 2\nThe functional form of Eq. (4) is obtained naively from\nconsidering the most relevant analytical behavior that\nthe boson can exhibit. At the upper critical dimension,\nwhich for the GNY theory of Dirac fermions is d= 3,\nthis is su\u000ecient. However when considering systems in\nphysical dimensions like in d= 2, as done in this work,\nit is imperative to consider the phenomenon of Landau\ndamping of the order parameter \ructuations by gapless\nelectronic particle-hole excitations. To self-consistently\naccount for these damped dynamics, we use the non-\nperturbative RPA resummation of fermion loops, shown\ndiagrammatically in Fig. 1, to obtain the dressed inverse\nboson propagator. In d= 2 the bosonic self energy is\ngiven by,\n\u0005(~k) =g2\nNZd3q\n(2\u0019)3Trh\n\u001bzG (~k+~ q)\u001bzG (~ q)i\n(6)\nwhere~k= (k;!) and the fermion propagator is given by\nG (k;!) =i!+vk\u0001\u001b\n!2+v2k2: (7)\nSince the leading IR behavior of the integral for \u0005( k;!)\nis independent of the UV momentum cut-o\u000b we can send\nthe cut-o\u000b to in\fnity and perform the integration over\nthe entire three dimensional frequency-momentum space.\nThis leads to the result [42]\n\u0005(k;!) =g2\n16v2\u0000\n!2+v2k2\u00011=2: (8)\n+=FIG. 1: Random phase approximation to account for\nthe non-perturbative Landau damping of IR\norder-parameter \ructuations. The black wavy line\nrepresents the bare order parameter, while the fermionic\npropagator is denoted by the solid arrowed line.\nIn the long-wavelength limit of small frequency and\nmomenta the self energy dominates over the ( !2+c2k2)\nterms in the bare inverse propagator (4). In order to\nidentify the universal critical behavior we drop the sub-\nleading terms, which are irrelevant in an RG sense, and\nuse the inverse bosonic propagator\nG\u00001\n\u001e(k;!) = \u0005(k;!) +m2: (9)\nFormally, the RPA contribution dominates in the\nlarge-Nlimit, which is evident after making the rescal-\ningg2!g2N. The Landau damped dynamics a\u000bects the\nscaling of the e\u000bective order parameter \feld. Crucially,\nthe quartic \u001e4term of Eq. (5) is rendered irrelevant at\ntree level and so is neglected in the following. This is a\ncommon feature of the \\interaction driven scaling\" [46]\nof gapless fermionic systems.\nC. Coupling to Disorder: Replica Field Theory\nWe will consider di\u000berent forms of quenched disorder\n\feldsVi(x) that arise from non-magnetic charge impuri-\nties and are expected to a\u000bect the quantum phase tran-\nsition between the Dirac semimetal and CDW insulator.\nThese \felds couple to the fermions in the di\u000berent chan-\nnels of the 2\u00022 sublattice pseudospin space,\nSdis=X\ni=0;x;y;zZ\nd2xZ\nd\u001cVi(x)\ty(x;\u001c)\u001bi\t(x;\u001c);\n(10)\nwhere in addition to the three Pauli matrices we have\nde\fned the identity matrix as \u001b0. Other forms of disor-\nder which would break degeneracies of the other fermion\n\ravors, e.g. spin or valley degeneracies, are not consid-\nered here. Note that for simplicity, we do not consider\ndisorder that couples to the bosonic order parameter.\nV0andVzare random potentials that couple to the\nsymmetric (  y\nA A+ y\nB B) and anti-symmetric (  y\nA A\u0000\n y\nB B) combinations of the local electron densities on the\ntwo sites in the unit cell. The latter combination is re-\nquired as some charge impurities will a\u000bect the two sites\ndi\u000berently and locally break the symmetry between the\ntwo sub-lattices. In the following, we will refer to V0as\n\\chemical potential disorder\" since it can be viewed as\nspatial variations of the homogeneous chemical potential4\n\u0016= 0, and to Vzas \\random mass disorder\" since it cou-\nples in the same way as the electronic mass gap generated\nby the condensation of the CDW order parameter.\nThe components V?:=Vx=Vycorrespond to ran-\ndom gauge (vector) potential disorder. As discussed in\nthe context of graphene, the random gauge potential de-\nscribes elastic lattice deformations or ripples [22, 47, 48],\nwhich will be caused by impurity atoms. The di\u000berent\ndisorder \felds Viare present in any system with non-\nmagnetic impurities and, as we will show later, there ex-\nists a rich interplay between them.\nWe assume that the random potentials Vi(x) are un-\ncorrelated and that they follow Gaussian distributions\nwith zero mean and variances \u0001 i\u00150,\nhVi(x)idis= 0; (11)\nhVi(x1)Vj(x2)idis= \u0001i\u000eij\u000e(x1\u0000x2); (12)\nwhereh:::idisdenotes the average over the disor-\nder. The presence of disorder on the level of the\nquadratic fermion action, Eq. (10), does not a\u000bect the\nHubbard-Stratonovich decoupling of the fermion inter-\naction. The resulting \feld theory is therefore given by\nSGNY[\ty;\t;\u001e] +Sdis[\ty;\t]. It is important to stress\nthat disorder does not enter in the bosonic sector of the\ntheory, e.g. in the form of random-mass disorder of the\nCDW order parameter \feld \u001e. In order to average the\nfree energy over the quenched disorder, we use the replica\ntrick [44, 45],\nhFidis=\u0000ThlnZidis=\u0000Tlim\nn!0hZnidis\u00001\nn; (13)\nwhereZ=R\nD[\ty;\t;\u001e]e\u0000(SGNY+Sdis)denotes the parti-\ntion function. After taking nreplicas of the system and\nperforming the average over the uncorrelated Gaussian\ndisorder, using Eqs. (11) and (12), we obtain the e\u000bec-\ntive replica \feld theory\nS=nX\na=1Z\nd2xZ\nd\u001c\ty\na\u0012\n@\u001c+iv@\u0001\u001b+gp\nN\u001ea\u001bz\u0013\n\ta\n+1\n2nX\na=1Z\njkj\u0014\u0003d2k\n(2\u0019)2Z1\n\u00001d!\n2\u0019G\u00001\n\u001e(k;!)j\u001ea(k;!)j2\n\u00001\n2nX\na;b=1Z\nd2xZ\nd\u001cZ\nd\u001c0X\ni=0;x;y;z\u0001i\n\u0002\u0014\n\ty\na(x;\u001c)\u001bi\ta(x;\u001c)\u0015\u0014\n\ty\nb(x;\u001c0)\u001bi\tb(x;\u001c0)\u0015\n;(14)\nat zero temperature. Here G\u00001\n\u001e(!;k), de\fned in Eq.(9),\nis the inverse dressed bosonic propagator that is obtained\nby the RPA resummation as seen in Fig. 1. Unlike\nRefs. [39{41], we do not include a four-boson disorder\nvertex. Such a vertex would arise from a replica average\nof random-mass disorder of the CDW order parameter\n\feld\u001ewhich is not present in our theory. In Appendix\nA, we show that starting with the bare replica action\n(14), a four-boson disorder vertex is not generated under\nthe RG at two-loop order.III. RENORMALISATION GROUP ANALYSIS\nIn the following we perform a momentum-shell RG\nanalysis of the replica action (14). We integrate out\nfast modes with momenta from an in\fnitesimal shell\n\u0003e\u0000d`<jkj<\u0003 near the UV momentum cuto\u000b \u0003. This\nis followed by the conventional rescaling of momenta,\nfrequency and \felds. To restore the original cuto\u000b we\nrescale momenta as k=k0e\u0000d`while!=!0e\u0000zd`with\nzthe dynamical exponent. The \felds are rescaled as\n\t(k;!) = \t0(k0;!0)e\u0000\u000e\td`=2;\n\u001e(k;!) =\u001e0(k0;!0)e\u0000\u000e\u001ed`=2:(15)\nWe start with a simple tree-level scaling analysis. In\nthe absence of disorder and at the critical point m2= 0\nthe \feld theory remains invariant under the above rescal-\ning forz= 1,\u000e\t=\u00002\u00002z, and\u000e\u001e=\u00004. As the tuning\nparameter of the quantum phase transition, the order-\nparameter mass is a relevant perturbation with tree-level\nscaling dimension [ m2] = 2\u0000z. On the other hand, the\nbosonic\u001e4vertex (5) is irrelevant with scaling dimension\n[\u0015] =\u00006\u00003z\u00002\u000e\u001e=\u00001, justifying why we neglected\nit in our theory, Eq. (14). Under these scaling conven-\ntions the fermionic disorder is vertex is marginal at tree\nlevel which motivates a perturbative expansion in the\ncouplings \u0001 iof fermionic disorder.\nWe compute all diagrams, shown in Fig. 2, that con-\ntribute atO(\u0001i;1\nN) ind= 2. We \frst consider the\nfermionic self-energy corrections due to the Yukawa cou-\npling at second order and the disorder vertex at linear\norder, which are shown by the two diagrams in Fig. 2(a),\nrespectively. The \frst diagram leads to a renormalization\nof the overall prefactor of the inverse fermion propaga-\ntor, resulting in an anomalous dimension of the fermion\n\felds. The disorder induced self energy only a\u000bects the\nfrequency term and therefore breaks the symmetry be-\ntween momentum and frequency scaling, leading to a cor-\nrection to the dynamical exponent z. The inverse fermion\npropagator remains invariant under the RG for\n\u000e\t=\u00004 +8\n3\u00192N\u00001\n2\u0010\n~\u00010+~\u0001z+ 2~\u0001?\u0011\n; (16)\nz= 1 +1\n2\u0010\n~\u00010+~\u0001z+ 2~\u0001?\u0011\n; (17)\nwhere we have de\fned the rescaled disorder variances\n~\u0001i=\u0001i\n\u0019v2(18)\nand~\u0001?:=~\u0001x=~\u0001y. Hence the theory will no longer be\nLorentz invariant for any \fnite disorder \fxed point. The\nrenormalization of the Yukawa vertex is calculated from\nthe diagrams in Fig. 2(b),\ndg\nd`=\u0014\n\u00004\u00002z\u0000\u000e\t\u0000\u000e\u001e\n2(19)\n\u00008\n\u00192N+1\n4\u0010\n2~\u0001?\u0000~\u0001z\u0000~\u00010\u0011\u0015\ng:5\n(a)(b)(c)\n(d)(a)(b)(c)\n(d)(a)(b)(c)\n(d)(a)(b)(c)\n(d)(a)(b)(c)\n(d)(a)(b)(c)(d)\nFIG. 2: Feynman diagrams of O(\u0001i;1\nN) for the large- N\ntheory, Eq. (14). (a) Fermion self energy corrections\nthat renormalize the fermionic propagator. (b)\nRenormalization of the Yukawa vertex. (c)\nRenormalization of the bosonic mass. (d) Corrections to\nthe fermionic disorder vertex. The dashed line\nrepresents the replicated disorder interaction.\nSince the coupling gcan be scaled out of the large- N\nreplica theory, Eq. (14), using \u001e!\u001e=g,m2!g2m2, we\ndemand that it is scale invariant. This determines the\ncritical dimension of the order-parameter \feld \u001e,\n\u000e\u001e=\u00004\u000064\n3\u00192N\u00001\n2\u0010\n2~\u0001?+ 3~\u0001z+ 3~\u00010\u0011\n;(20)\nwhere we have eliminated \u000e\tandz, using Eqs. (16) and\n(17). The renormalization of the order parameter mass\nm2is given by the two-loop diagrams in Fig. 2(c). The\nresulting RG equation is given by\ndm2\nd`=\u0012\n1\u000032\n3\u00192N+~\u0001z+~\u00010\u0013\nm2; (21)\nwhere the dependence on disorder arrises through z(17)\nand\u000e\u001e(20). At any \fnite disorder \fxed point the order\nparameter correlation length exponent \u0017, de\fned through\nthe identi\fcation dm2=d`=\u0017\u00001m2, will therefore di\u000ber\nfrom the one in the clean system.\nFinally, the coupled RG equations for the di\u000berent\ntypes of disorder are obtained from the diagrams inFig. 2(d),\nd~\u00010\nd`=~\u00010\u0012\n~\u00010+~\u0001z+ 2~\u0001?\u000032\n9\u00192N\u0013\n+ 2~\u0001?~\u0001z;\nd~\u0001?\nd`=\u0000~\u0001? ~\u0001z\n6+32\n9\u00192N!\n+~\u00010~\u0001z; (22)\nd~\u0001z\nd`=~\u0001z \n5~\u0001?\n3\u0000~\u0001z\u0000~\u00010\u000032\n3\u00192N!\n+ 2~\u0001?~\u00010:\nIn the non-interacting limit, corresponding to diagrams\nin Fig. 2 that only include the disorder vertex, the RG\nequations for the disorder variances agree with previous\nresults [7, 11, 20, 49].\nA. RG \row and \fxed points\nWe start by summarizing the critical exponents for\nthe interaction-driven semimetal to CDW insulator tran-\nsition of the clean system at T= 0 ind= 2. The\ncritical exponents of order 1 =Nat the clean interacting\ncritical \fxed point, which we denote by Pclean, are ob-\ntained from Eqs. (16), (17), (20) and (21) by setting\n~\u00010=~\u0001?=~\u0001z= 0. In the absence of disorder the the-\nory satis\fes Lorentz invariance with dynamical exponent\nz= 1. The anomalous critical dimensions of the \felds,\nde\fned through \u000e\t=\u00004 +\u0011\t,\u000e\u001e=\u00004 +\u0011\u001e, and the\ncorrelation length exponent \u0017reduce to\n\u0011clean\n\t =8\n3\u00192N; \u0011clean\n\u001e =\u000064\n3\u00192N;\n\u0017clean = 1 +32\n3\u00192N:(23)\nThese exponents are in agreement with those obtained\nfrom soft cuto\u000b RG [42] and with previous results using\nthe largeNconformal bootstrap [50{52] and the critical\npoint large Nformalism [53{55]. At Pclean, the order\nparameter mass m2is the only relevant parameter, rep-\nresenting the tuning parameter of the quantum phase\ntransition.\nIn order to analyze whether the clean system CDW\ncritical point is stable against weak charge-impurity dis-\norder, we numerically integrate the coupled RG equations\nfor~\u00010,~\u0001?and ~\u0001z(22). The resulting RG \row of the\nthree disorder variances on the critical manifold m2= 0\nis shown in Fig. 3. For small disorder, in the regime\nbounded by the transparent purple surface, the \row is\ntowards the clean system critical point Pclean, demon-\nstrating that the CDW quantum critical point is stable\nagainst weak disorder. This is in line with the Harris\ncriterion which states that a non-disordered \fxed point\nis stable if\u0017clean\u00152=d, wheredis the dimensionality of\nthe system [56, 57].\nClose to the boundary surface, the RG \row is con-\ntrolled by the only \fnite disorder \fxed point in the ac-6\nFIG. 3: RG \row in the disorder subspace on the critical\nmanifoldm2= 0, as de\fned by Eqs. (22). Within the\nregion bounded by the transparent surface disorder\nrenormalizes to zero, showing that the CDW critical\npointPclean is stable against small disorder. Near this\nboundary surface the RG \row is towards a \fnite\ndisorder \fxed point P(c)\ndisat which only chemical\npotential disorder is relevant.\ncessible region of positive variances,\nP(c)\ndis:\u0010\n~\u0001(c)\n0;~\u0001(c)\n?;~\u0001(c)\nz\u0011\n=\u001232\n9\u00192N;0;0\u0013\n: (24)\nP(c)\ndisis unstable along the ~\u00010direction but stable\nagainst ~\u0001?and ~\u0001z. This is consistent with the RG \row\nfor initial values ~\u0001i(0) that are very close to the separat-\ning surface in Fig. 3. Shown are three pairs of trajectories\nwith initial values that are slightly inside (blue) and out-\nside (red) of the region bounded by the surface. The\ntrajectories closely track the surface and split very close\ntoP(c)\ndis, where the \row is either to the clean \fxed point,\n~\u00010!0 or strong chemical potential disorder, ~\u00010!1 .\nOur RG analysis shows that the transition to a glassy\nstate is always driven by potential disorder, even if the\nother forms of disorder initially dominate. Since the ran-\ndom gauge \feld and random mass disorders are irrelevant\natP(c)\ndiswe neglect them in the following. The RG equa-\ntions for the chemical potential disorder ~\u00010and the orderparameter mass m2then reduce to\nd~\u00010\nd`=~\u00010\u0012\n~\u00010\u000032\n9\u00192N\u0013\n; (25)\ndm2\nd`=\u0012\n1\u000032\n3\u00192N+~\u00010\u0013\nm2: (26)\nInserting the critical disorder strength ~\u0001(c)\n0=32\n9\u00192N\ninto Eqs. (16), (17), (20) and (26) we obtain the critical\nexponents at the \fnite disorder multi-critical point P(c)\ndis,\n\u0011dirty\n\t=8\n9\u00192N; \u0011dirty\n\u001e=\u000080\n3\u00192N;\n\u0017dirty= 1 +64\n9\u00192N; z dirty= 1 +16\n9\u00192N:(27)\nAt both the clean system semimetal to CDW insula-\ntor transition and at the \fnite disorder multicritical point\nthe fermion anomalous dimension \u0011\tis greater than zero.\nThis implies that at the quantum critical points (QCPs)\nthe fermion Green's function has branch cuts rather than\nquasiparticle poles, and the fermionic liquid is therefore\na non-Fermi liquid. Approaching the QCPs from the\nmetallic side, V <VcandVc\u0000V!0, the quasiparticle\nresidue has to vanish with some characteristic exponent.\nOn the CDW side, the condensation of the order param-\neter leads to the formation of a gap Min the fermion\nspectrum, which increases as a power of V\u0000Vc>0.\nIn order to extract these exponents we perform a scal-\ning analysis of the fermionic spectral function. Details\ncan be found in Ref. [58]. Here we only give the re-\nsults. Approaching the quantum phase transition from\nthe metallic side, the quasiparticle pole strength vanishes\nas\nZ\u0018(Vc\u0000V)(z\u00001+\u0011\t)\u0017= (Vc\u0000V)8\n3\u00192N;(28)\nwhere to order 1 =Nthe critical exponents are the same\nfor the clean and dirty \fxed points Pclean andPdirty. The\nFermi velocity behaves as\nv\u0018jVc\u0000Vj(z\u00001)\u0017=\u001aconst at Pclean\njVc\u0000Vj16\n9\u00192NatPdirty(29)\nFinally, on the CDW insulator side of the quantum\nphase transition the gap in the electron spectrum in-\ncreases as\nM\u0018(V\u0000Vc)z\u0017=(\n(V\u0000Vc)1+32\n3\u00192NatPclean\n(V\u0000Vc)1+80\n9\u00192NatPdirty(30)\nThe behavior of Z,vandMnear the clean and \fnite-\ndisorder QCPs is illustrated in Fig. 4.\nIn order to estimate the phase boundary between the\nCDW insulator and the disordered phase in the close\nproximity of PdirtyforV >Vcand~\u00010>~\u0001(c)\n0we compare\nthe CDW induced gap Min the electron spectrum with\nthe standard deviationp~\u00010of the chemical potential7\nDirac semimetal\n+⇢\u0000⇢CDWinsulator\nFIG. 4: Behavior of the the quasiparticle pole strength\nZ, the Fermi velocity vand the gap Min the fermion\nspectrum at the clean semimetal/CDW insulator\ntransition and at the \fnite disorder multicritical point,\nas a function of the nearest neighbor repulsion V\u0000Vc.\nHere we evaluated the critical exponents for N= 8,\ncorresponding to Dirac electrons on the honeycomb\nlattice with valley and spin degeneracies.\ndisorder. Close to Pdirty, the disorder increase exponen-\ntially under the RG,\n~\u00010(`)\u0000~\u0001(c)\n0'\u0010\n~\u00010\u0000~\u0001(c)\n0\u0011\ne\u0017\u00001\n\u0001`with\u0017\u00001\n\u0001=32\n9\u00192N:\nWe evaluate the disorder variance at the \\correlation\nlength\"\u0018\u0018e`\u0003\u0018(V\u0000Vc)\u0000\u0017, wherem2(`\u0003)' \u0000 1.\nEquating the resulting standard deviation with the gap\nMnearPdirty, Eq. (30), we obtain the phase boundary\n\u0010\n~\u00010\u0000~\u0001(c)\n0\u0011\n'(V\u0000Vc)(2zdirty+\u0017\u00001\n\u0001)\u0017dirty\n'(V\u0000Vc)2(1+32\n3\u00192N): (31)\nA schematic phase diagram as a function of the inter-\naction strength V\u0000Vc'\u0000m2and the chemical potential\ndisorder ~\u00010is shown in Fig. 5.\nIV. DISCUSSION\nWe have investigated the e\u000bects of quenched short-\nranged disorder on the quantum phase transition between\na two-dimensional Dirac semi-metal and a charge density\nwave (CDW) insulator. In the absence of disorder, the\nphase transition belongs to the chiral Ising Gross-Neveu-\nYukawa (GNY) universality class. In order to achieve\nanalytic control in d= 2, far below the upper critical di-\nmension, we have analyzed the problem in the limit of a\nlarge number Nof Dirac fermion \ravors. We have used\nthe RPA fermion loop resummation to self-consistently\naccount for the Landau damping of the boson dynamics\nCDW InsulatorDirac SemimetalDisordered Phase(s)?FIG. 5: Schematic Phase diagram as a function of the\ninteraction strength V\u0000Vcand the variance ~\u00010of the\nchemical potential disorder.\nby electronic particle-hole excitations. As pointed out in\nthe literature [42, 43], this is a non-perturbative e\u000bect in\ntwo spatial dimensions that changes the IR physics and\nhence the universal critical behavior. As we have demon-\nstrated in our work, Landau damping also plays a crucial\nrole in how the critical system responds to disorder.\nWe have considered three types of electronic disorder\nthat all arise from non-magnetic charge impurities. The\nrandom potential from the impurities is decomposed into\nrandom mass disorder, which locally breaks the symme-\ntry between the two sublattices, and symmetric random\nchemical potential disorder. The local lattice deforma-\ntions caused by impurity atoms is accounted for by ran-\ndom gauge potential disorder [7]. For simplicity, we have\nneglected correlations between the di\u000berent types of dis-\norder and assumed that disorder is uncorrelated between\ndi\u000berent positions in space.\nAfter averaging over disorder, using the replica formal-\nism, we have performed a perturbative RG calculation to\nleading order in the disorder strength and in 1 =N. Our\nanalysis shows that the clean GNY critical point is stable\nagainst weak disorder. This is in stark contrast to non-\ninteracting or weakly interacting two-dimensional Dirac\nfermions where disorder is a relevant perturbation, result-\ning in a run-away \row towards strong disorder [6, 19{21].\nMost importantly, we have identi\fed a dirty GNY crit-\nical point at a \fnite value of the chemical potential disor-\nder of order 1 =N. At this multicritcal point, the random\nmass and random gauge potential disorders are irrele-\nvant. This shows that the transition into a disordered\nstate is driven by chemical potential disorder, even if the\nother forms of disorder dominate on shorter length and\ntime scales.\nThe irrelevance of random mass disorder Vzat the\nclean and \fnite disorder GNY critical points might seem\nsurprising since this type of disorder breaks the AB sub-\nlattice symmetry, similar to a random \feld that cou-\nples to the Ising CDW order parameter \u001e. According\nto the scaling arguments by Imry/Ma [59, 60] and Aizen-8\nman/Wehr [61], such a random \feld would destroy any\nlong-range Ising order and associated quantum critical\npoint in two spatial dimensions. However, the GNY uni-\nversality class falls outside a pure order-parameter de-\nscription used in these arguments, and is formulated in\nterms of bosonic and fermionic degrees of freedom. The\ncoupling to gapless fermion excitations changes the IR\ndynamics of the bosonic order parameter \feld, resulting\nin unusual scaling properties in the bosonic sector, e.g.\nthe\u001e4vertex is rendered irrelevant in d= 2. Moreover,\nthe disorder Vzcouples to the fermion operator \ty\u001bz\t\nand not to the CDW order parameter \u001e. Although af-\nter integrating out the fermions, Vzwould translate into\na random \feld in the resulting order parameter theory,\nthis step is not allowed since the fermions are gapless\nat the QCP. Future quantum Monte-Carlo studies could\nshed some light on this subtle question.\nThe disorder driven phase transition along the line of\ncritical interaction in the two-dimensional system might\nbe similar to the transition in weakly interacting, three\ndimensional Weyl/Dirac semimetals [26{30]. In both\ncases, the transition is driven by chemical potential disor-\nder which is expected to induce a \fnite zero-energy den-\nsity of states in the disordered phase, giving rise to di\u000bu-\nsive metallic behavior. This would be consistent with our\nnaive picture for the transition between the CDW insula-\ntor, which forms above the critical interaction strength,\nand the disordered phase: if the standard deviation of the\nrandom chemical potential shifts exceeds the electronic\ngap induced by the symmetry breaking, the system will\ndevelop a \fnite density of states at the average chemical\npotential, leading to di\u000busive metallic behavior. How-\never, further calculations are required to ascertain the\nproperties of the disordered phase in the strongly inter-\nacting, two dimensional system. An investigation of the\ndependence on the form of the disorder distribution, e.g.\nwhether it is bounded, Gaussian or exhibits long tails, as\nwell as of any potential replica symmetry breaking [62],\nindicative of glassy behavior, would be very interesting.\nThe random-mass disorder might play an important role\nin stabilizing a \fnite disorder multi-critical point with\nbroken Replica symmetry.\nOur renormalization-group approach does not capture\nnon-perturbative, rare region e\u000bects, which have spurned\na lot of discussion in the context of three dimensional\nWeyl/Dirac semimetals. A study by Nandkishore et al.\n[31] \frst proposed that rare region e\u000bects induce a non-\nvanishing density of states at the Weyl/Dirac points,\nthereby turning the disorder-driven phase transition into\na crossover. This was substantiated by numerical calcula-\ntions [32, 33] but remains at odds with recent theoretical\nliterature [34, 35]. However, as chemical potential disor-\nder is marginal in two spatial dimensions, and irrelevant\nin three, it is expected that rare region resonances will\nhave a \\sub-leading e\u000bect\" on the physics of the transi-\ntion in two dimensions [63].\nWe have shown that the symmetry-breaking quantum\nphase transition at the dirty GNY does not belong tothe chiral-Ising GNY universality of the clean system.\nWe have computed the critical exponents at the \fnite-\ndisorder multi-critical point to order 1 =Nand found\nthat the anomalous dimensions of the boson and fermion\n\felds, the correlation length exponent of the CDW or-\nder parameter and the dynamical critical exponent di\u000ber\nfrom those at the clean GNY \fxed point. This leads\nto di\u000berent critical behavior of physical observables such\nas the electronic gap, the Fermi velocity, and the quasi-\nparticle residue near the transition and results in a novel\nnon-Fermi liquid state at the multicritical point.\nThe interplay between symmetry breaking and disor-\nder was previously studied for the XY GNY [39, 40] and\nthe chiral Ising and Heisenberg GNY models [41], us-\ning the replica formalism combined with \u000fexpansions.\nNear the upper critical dimension fermionic disorder is\nstrongly irrelevant at the clean system quantum critical\npoints. Instead, short-ranged disorder of the bosonic or-\nder parameter mass (sometimes referred to as random Tc\ndisorder) gives rise to a \fnite disorder multicritical point,\nregardless of the symmetry of the order parameter. At\nthis \fnite disorder critical point the Lorentz invariance is\nbroken with a dynamical exponent z >1, similar to our\ndirty GNY \fxed point, while the fermionic and bosonic\nanomalous dimensions remain unchanged, which is not\nthe case in our theory.\nThe irrelevance of the chemical potential disorder\nseems to be only valid near the upper critical dimen-\nsion, hence any extrapolation to the physical dimension\nofd= 2 without the inclusion of it is questionable.\nMoreover, the non-perturbative Landau damping which\nis crucial for the universal critical behavior of the two-\ndimensional system, is not captured by an \u000fexpansion\nbelow the upper critical dimension. On the other hand,\nwe have not included bosonic disorder in our e\u000bective\n\feld theory, for simplicity. Starting from an interact-\ning fermionic model with a random potential, bosonic\ndisorder would not arise from a Hubbard-Stratonovich\ndecoupling of the fermionic interaction vertex. However,\nas pointed out in Refs. [39] and [40], at two-loop order\nchemical potential disorder could generate a bosonic dis-\norder vertex in the replica theory. We have presented an\nexplicit calculation in Appendix A, demonstrating that\nthis is not the case.\nIt is important to stress, however, that bosonic ran-\ndom mass and random \feld disorders do not break sym-\nmetries that are not already broken by the fermionic dis-\norder potentials. An e\u000bective low energy \feld theory ob-\ntained from careful coarse-graining of a microscopic lat-\ntice Hamiltonian will therefore also contain the symme-\ntry allowed bosonic disorder. The presence of additional\nbosonic disorder could potentially a\u000bect our conclusions\nand should be considered in future work. Random \feld\ndisorder is known to have a detrimental e\u000bect on the\nCDW order and associated quantum phase transition.\nHowever, in the case of remote charge impurities that\ndo not break the symmetry between the two sublattices,\nsuch random \feld disorder would be suppressed. As the9\nfermionic random potentials, the bosonic random mass\ndisorder is marginal at the GNY interacting \fxed point\nand might therefore alter the multi-critical behavior.\nIt is also interesting to compare our results with re-\ncent work [64{66] on the role of generic types of fermionic\ndisorder in strongly coupled QED 3, which describes the\ninteraction of massless Dirac fermions with U(1) gauge\nbosons in 2+1 space-time dimensions. Similar to our\nwork, the problem was generalized to a large number N\nof fermion \ravors and analyzed within the Replica frame-\nwork. In QED 3, su\u000eciently strong gauge coupling leads\nto dynamical chiral symmetry breaking and spontaneous\nfermion mass generation. However, this quantum phase\ntransition is lost above a critical number Nc= 32=\u00192of\nfermion \ravors [67{70] and therefore no longer accessible\nin the large Nlimit [64]. This might explain why disorder\nis found to be a relevant perturbation, similar to the case\nof weakly interacting Dirac fermions in 2+1 dimensions\n[19{21]. However, unlike in the weakly interacting case,\nthe \row is towards a stable \fnite-disorder \fxed point\nwith a broken \ravour degeneracy and z > 1. This be-\nhaviour is very di\u000berent from that of chiral Ising GNY\ntheories at criticality, investigated in our work: the clean\nGNY \fxed point in 2+1 dimensions is stable against weak\nfermionic disorder and the transition to a di\u000busive metal-\nlic state is characterized by a multi-critical point at \fnite\nchemical potential disorder and z>1.\nFor simplicity, we have analyzed critical GNY theories\nwith an Ising order parameter. We believe that the be-\nhaviour is similar for GNY theories with continuous order\nparameter symmetries and that the stability of the clean\nGNY \fxed point against disorder is the consequence of\ngapless fermion excitations that completely change the\nlong-wavelength order-parameter dynamics in two spa-\ntial dimensions. The Wilson-Fisher critical \fxed point\nin conventional bosonic theories, e.g. for the super\ruid-\ninsulator transition d= 2, is indeed unstable towards the\nformation of a \fnite disorder \fxed point [71]. Although\nthe behaviour of the large N\feld theory in d= 2 is\nsimilar to that of a double \u000fexpansion near the upper\ncritical dimension, the latter shows a spiralling RG \row\ninto the \fnite disorder \fxed point [71]. Similar behaviour\nis found in a double \u000fexpansion of critical GNY theories\nwith bosonic random mass disorder [40]. This could ei-\nther point towards pathologies of the double \u000fexpansion\nor otherwise indicate important physical behaviour that\nis lost in the oversimpli\fed large Ntreatment.\nIn future extensions of our work it would be interest-\ning to investigate the e\u000bects of long-range correlations\nof disorder. It is often assumed that impurities and im-\nperfections are screened e\u000bectively and that disorder can\ntherefore be taken to be uncorrelated. However, it has\nbeen reported that in graphene the correlations between\ndisorder-induced puddles of electron- and hole-doped re-\ngions decay algebraically [72{74]. Such power-law cor-\nrelations are expected to change the long-wavelength\nphysics and hence the universal critical behavior. One\nmight also include other types of disorder, e.g. defectsthat lead to inter-valley scattering, magnetic impurities\nthat break the spin degeneracy, or topological lattice de-\nfects that are described by random non-Abelian gauge\n\felds. The interplay of the di\u000berent types of disorder is\nexpected to lead to rich phase behavior and novel critical\nphenomena, in particular if competing fermionic interac-\ntions are taken into account.\nACKNOWLEDGMENTS\nF.K. acknowledges \fnancial support from EPSRC un-\nder Grant No. EP/P013449/1.\nAppendix A: Two-loop fermion diagram that\ngenerates the boson disorder\nHere we address the question if the electronic disorder,\nwhich are de\fned on the level of the quadratic fermion\naction [see Eq. (10)], can generate random mass disorder\nof the bosonic order parameter \feld at two loop order,\nas suggested in Refs. [39] and [40]. In the disorder aver-\naged replica theory the electronic disorder is described by\na disorder vertex that is quartic in the fermionic Grass-\nmann \felds, couples di\u000berent replicas, and is non-local in\nimaginary time [see Eq. (14)]. Similarly, bosonic random\nmass disorder gives rise to a disorder vertex\nSdis\n\u001e=\u0000\u001b2\n2nX\na;b=1Z\nd2xZ\nd\u001cZ\nd\u001c0\u001e2\na(x;\u001c)\u001e2\nb(x;\u001c0)\n(A1)\nin the replica \feld theory, where \u001b2is the variance of the\nbosonic random mass disorder distribution. This vertex\nwould be generated by the two-loop diagram shown in\nFig. 6 where the external momenta in the loop integrals\nare set to zero. This results in\n\u001b2\u0018g4\nN2X\ni=0;x;y;zD2\ni\u0001i (A2)\nwith\nDi=Z\nk;!Tr\u0002\nG\t(k;!)\u001bzG\t(k;!)\u001bzG\t(k;!)\u001bi\u0003\n:\n(A3)\nIt is straightforward to see that electronic random mass\ndisorder \u0001 zdoes not contribute since the trace over the\nproduct of Pauli matrices vanishes in this case, Dz= 0.\nIn the other channels we obtain the integrals\nD0=\u0000NZ\nk;!i!\n(!2+v2k2)2;\nDx=\u0000NZ\nk;!vkx\n(!2+v2k2)2;\nDy=\u0000NZ\nk;!vky\n(!2+v2k2)2;10\nσz\nσzσiσi\nσzσz\nFIG. 6: The two-loop diagram that according to\nRefs.[39, 40] generates the bosonic disorder vertex.\nafter taking the trace. 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