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+[    {        "title": "1305.3190v1.Dissipation_flow_frames__particle__energy__thermometer.pdf",        "content": "arXiv:1305.3190v1  [gr-qc]  14 May 201312th Joint European Thermodynamics Conference\nBrescia, July1-5, 2013\nDISSIPATIONFLOW-FRAMES: PARTICLE, ENERGY, THERMOMETER\nP. V´ an1,2,3andT.S.Bir´ o1\n1Dept. ofTheoreticalPhysics,WignerResearchCentre forPh ysics,InstituteforParticleandNuclearPhysics,\nH-1525Budapest,KonkolyThegeMikl´ os ´ ut29-33,Hungary;\n2Dept. ofEnergyEngineering,BudapestUniv. ofTechnologya ndEconomics,\nH-1111,Budapest,M˝ uegyetemrkp. 3-9,Hungary;\n3MontavidThermodynamicResearchGroup\nEmail: van.peter@wigner.mta.hu\nABSTRACT\nWe associate the following physical co-mover conditions of to different frame choices: i) Eckart: particle flow, ii) Lan dau-\nLifshitz: energy flow, iii) J¨ uttner: moving thermometer fr ame. The role of fixing a flow-frame is analysed with respect to local\nequilibrium concentratingondissipative currentsandfor ces insinglecomponent relativisticfluids. The specialrol eofa”J¨ uttner\nframe” is explored andcontrasted tothe more common Eckart a nd Landau-Lifshitz choices.\n1 Introduction\nIn dissipative theoriesof relativistic fluids we deal with f our\nfundamentalquestions.\nThe first considers causality. Only divergence type theories\nare, in general, causal because there the symmetric hyperbo lic-\nityofthesystemofnonlinearevolutionequationsisestabl ished\nbyconstruction[1;2; 3;4;5;6]. Theweakerversionofcausa l-\nity requires for the symmetric hyperbolicity only for the li n-\nearizedequations,andallowsforcharacteristicspeedsle ssthan\nthe speed of light [7]. This weak causality was studied in the\nIsrael-Stewarttheory;numerousresultinginequalitiesa regiven\nin [8]. From a physical point of view the causality of theorie s\nwithparabolicdifferentialequationsshouldalsobepossi ble. In\nthis case the validity of the continuum description is restr icted\nby the characteristic maximal speeds [9; 10; 11; 12]. A neces -\nsaryconditionforthistype ofrestrictionsrequirestheda mping\nof the perturbations, equivalent to the the linear stabilit y of the\ntheory[13].\nThe second question deals with generic stability. Generic\nstability is the linear stability of the homogeneous equili brium\nsolutions. Thesimplestrelativisticgeneralizationofth enonrel-\nativistic Fourier-Navier-Stokes equations was proved to b e un-\nstablebyHiscockandLindblom[14]. Inthesequeltheyformu -\nlated mathematical conditionsof generic stability of the I srael-\nStewart theory [8]specified to the Eckart frame. However, du e\nto the overwhelming complexity of these conditions they are\nnot connected to reasonable properties of equations of stat e or\ntransport coefficients. Since then several different propo sitions\narose suggesting a first order theory, mostly motivated by th e\nrestoration of the generic stability [15; 16; 12; 17; 18; 19; 20;\n21].\nThe third question is the correct distinction between ideal\nand dissipative fluids , especially in a relativistic context. It is\ncustomarytoassumethatperfect,nondissipativefluidsare char-\nacterizedbyaspecialformoftheenergy-momentumtensoran d\nthe particle current density vector [22; 23]. On the other ha nd\nphysicaldissipationisaccompaniedbynonzeroentropypro duc-tion. Fromthispointofviewthereisamoreextendedfamilyo f\nperfect fluids beyond the customarily treated ones [24]. The se\ndistinctionsaretechnicallyaddressedbythesocalledmat ching\nconditions[25;20;21;26].\nFinally the proper choice of flow-frames continues to be an\nunsettledquestion[16]. Onegenerallybelievesthatinrel ativis-\ntic fluids the flow field uacan be chosen arbitrarily, since it is\nasomewhatvaguelydefinedphysicalproperty,belongingtot he\nflow of volatile quantities, once the energy,once the conser ved\ncharge. In this situation it is customary to fix the flow either to\nthemotionofparticles(Eckartframe)[27],orthatoftheen ergy\ndensity (Landau-Lifshitz frame) [28]. The flow fixing deter-\nminesacontinuousset oflocalrestframesinthefluid:wesha ll\nrefertothedifferentchoicesoffixingtheflowas flow-framesor\nframes. Contrarytothebeliefinafreechoiceoftheflow-frame\nwe point out that this may not be completely arbitrary, as one\nassociates a given physical content of the dissipation to ea ch.\nFurther choices than the two classical ones should be prefer red\nbydemandingagivenformoflocalGibbsrelations.\nIn this paper we present the general flow-frame, the separa-\ntion of perfect and dissipative parts of energy-momentum an d\nparticlenumbercurrentdensityandtheirrelationtogener icsta-\nbility. The key theoretical aspect connecting these proble ms is\nrelativistic thermodynamics. Our most important observat ion\nis that the usual assumption of kinetic equilibriumby intro duc-\ning the velocity field parallel to the local thermometer and L a-\ngrangemultiplierfield βaalso appearingin the collisioninvari-\nantψ=α+βaka, already acts as a flow-frame fixing. This\nchoice we tag as thermometer frame or J¨ uttner frame, distin -\nguishing from the Eckart, Landau-Lifshitz and other conven -\ntions.\n2 Generalone componentdissipative relativisticfluids\nIn this paper the Lorentz form is given as gab=\ndiag(1,1,1,1)and all indexes a,b,c,...run over 0 ,1,2,3. We\nusenaturalunits, h=k=c=1.\nA single component fluid is characterized by the particlenumber four-vector Naand the symmetric energy-momentum\ndensity tensor Tab. The velocity field of the fluid, the flow-\nframeua, introduces a local rest frame and the basic fields Na\nandTabcan be expanded by their local rest frame components\nparallelandperpendicularto theflow:\nNa=nua+ja, (1)\nTab=euaub+qaub+uaqb+Pab. (2)\nHerenis the flow-frame particle number density, jais in this\nframe the non-convective particle number current density, eis\nthe energy density, qais the momentum density and Pabis the\npressure tensor. These components are flow-frame dependent ,\nin particular jaua=0,qaua=0 andPabub=0. Introducing\nthe substantial time derivatived\ndt:=ua∂adenoted by and over-\ndot, the balances of the particle current density and energy -\nmomentumareexpressedbythe localrestframequantities:\n∂aNa=˙n+n∂aua+∂aja=0, (3)\n∂bTab=˙eua+e˙ua+eua∂bub+˙qa+qa∂bub+\nua∂bqb+qb∂bua+∂bPab=0a. (4)\nTheenergyandmomentumbalancesarethe timeandspace-\nlike componentsof the energy-momentumbalanceprojectedi n\ntheflow-frame:\nua∂bTab=˙e+e∂bub+ua˙qa+∂bqb−Pab∂bua=0,(5)\nΔa\nc∂bTcb=e˙ua+Δa\nb˙qb+qa∂bub+qb∂bua+Δa\nc∂bPcb=0a.(6)\nThe frame related quantities are important in the separatio n\noftheidealanddissipativepartsofthebasicfields. Thisse para-\ntion is best performed by analyzing the thermodynamical rel a-\ntions. Inorderto achievethisonepostulatesthe existence ofan\nadditional vector field, the entropy current as a function of the\nbasic fields Sa(Na,Tab). It must not decrease by fulfilling the\nconditionof the balances(3)and (4). That conditionalineq ual-\nity can be best representedby introducingthe Lagrange-Far kas\nmultipliers1αandβa, respectively:\nΣ:=∂aSa+α∂aNa−βb∂aTba≥0. (7)\nThe left hand side of this inequality shows, that the definiti on\nof the entropy production is done before specifying the flow-\nframe. However, the separation of ideal and dissipative par ts\nof basic physical quantities, is a consequence to the choice of\nthatflow-frame. Citing theauthorsof[32],whenarguingabo ut\nthe uniqueness of the Landau-Lifshitz frame “The uniquenes s\noftheenergyframecomesfrom... thephysicalassumptionth at\nthe dissipative effect comes from only the spatial inhomoge ne-\nity.“. However,whatis spacelikeisaframedependentquest ion\nand one hopes only that physical systems may reveal by their\ninternal dynamics a physical ground for such a separation. A\npossible candidate for this separation can be the thermomet er\nvector,βa, reconstructablefrom observationsof a multiparticle\nspectrastemmingfroma relativisticfluid.\n1Lagrange multipliers are introduced for conditional extre ma. For condi-\ntional inequalities Gyula Farkas suggested analogous quan tities and proved the\ncorresponding theorem oflinear algebra, called Farkas’ le mma[29; 30;31].3 Thermodynamicsofrelativisticfluids– equilibrium\nTheconceptofperfectfluidsdealswiththeabsenceofdissi-\npation,theentropyproductionvanishes:\nΣ0=∂aSa\n0+α∂aNa\n0−βb∂aTba\n0=0. (8)\nTheequilibriumentropydensity Sa\n0is connectedto the equi-\nlibrium particle number density Na\n0and equilibrium energy-\nmomentum density Tab\n0by the following definition of the\nisotropicpressure:\np0βa=Sa\n0+αNa\n0−βbTab\n0. (9)\nStandard kinetic theory definitions and calculations satis fy\nthe above expressions. Then αandβaare coefficients in the\ncollisioninvariantof the equilibriumdistributionfunct ion,ψ=\nα+βaka,andthepressureisthat ofanidealgas p0=n0T.\nKinetic theory describes a perfect fluid by the detailed bal-\nance requirement. Out of equilibrium dissipation can occur .\nIn a dissipative fluid all physical quantities in principle d evi-\nate from their local equilibrium values. There also may exis t\nnon-dissipativecurrents(presumablydrivenby non-dissi pating\nforces, like the Lorentz force in magnetic fields). The therm o-\ndynamicapproachaimsattheseparationofdissipativeandn on-\ndissipativelocalcurrents,inordertoensurethepositivi tyofthe\nexpression (7). Physical freedom in the choice of a flow-fram e\nshould be restricted to different handlings of non-dissipa tive\ncurrents.\nIt is natural to introduce the J¨uttner frame ua\nJdefined by the\ndirectionof βa(thermometermotion):\nua\nJ=βa\n/radicalbig\n/bardblβaβa/bardbl. (10)\nInthat frametheequilibriumfieldsaredecomposedas:\nNa\n0=nJua\nJ, (11)\nTab\n0=eJua\nJub\nJ−pΔab\nJ, (12)\nSa\n0= (βJhJ−αnJ)ua\nJ, (13)\nwherehJ=eJ+p0is the equilibrium enthalpy density in the\nJ¨ uttner frame, and βJ=βaua\nJ=1/TJis the reciprocal J¨ uttner\ntemperature. α,βaandp0do not carry a frame index, because\nthey are introduced before specifying the flow-frame. On the\notherhandtherepresentations(11)-(13)areframedepende nt. In\ncaseofageneral���ow-frame ua,thatisnotparallelto βa,onecan\ncharacterizethisdifferencebyintroducing wa=βa/(βbub)−ua.\nThenwais orthogonalto ua(waua=0) and spacelike ( wawa=\n−w2). The Lagrange multiplier four-vector, βa, can be splitted\nas\nβa=βJua\nJ=β(ua+wa), (14)\nwhereβ=βauaisthereciprocaltemperatureinageneralframedefinedby ua. Theequilibriumfieldsin thisframearegivenas\nNa\n0=n0ua+ja\n0, (15)\nTab\n0=e0uaub+qa\n0ub+qb\n0ua−pΔab+qa\n0qb\n0\nh0,(16)\nSa\n0= (βh0+βwbqb\n0−αn0)ua+βqa\n0−αja\n0.(17)\nHereβ=βJ/√\n1−w2,n0=nJ/√\n1−w2,e0=\n(eJ+pw2)/(1−w2),αandp0does not change, ja\n0=n0wa,\nqa\n0=h0wa[24]. (15)-(17) and (11)-(13) are the forms of the\nsame equilibriumfields in the J¨ uttnerand in the generalfra mes\nrespectively. In the specific equilibriumthe J¨ uttner, Eck art and\nLandau-Lifshitz frames coincide, the different choices le ad to\nthesamecondition: wa=0.\n4 Thermodynamics of relativistic fluids – out of equilib-\nrium\nIn classical non-equilibriumthermodynamics,without int er-\nnal variables, oneassumes that the gradientsof the equilib rium\nfields characterizethe deviationfrom local kinetic equili brium.\nIn that case the concept of local equilibrium is not modified.\nTheinternalvariabletheories,liketheIsrael-Stewartth eory[33;\n34; 25; 35; 36] or GENERIC [37; 38], choose a different char-\nacterization: local equilibrium is modified, some formerly dis-\nsipative currents appear among the state variables and as a\nconseqence their contribution may reduce the entropy produ c-\ntion.Therelativistictheoriesrevealedthattheflow-framefixin g\nplays a special role in the specification of local equilibriu m. It\nhas been an observation of Planck and Einstein, that the mo-\nmentum density (energy current density) is not purely dissi pa-\ntive and therefore in relativistic theories it has to be take n into\naccounteveninlocalequilibrium[39;40].\nOurstartingpointisthefundamentalinequalityoftheseco nd\nlaw (7). We introducethe followingrelationof the fieldsout of\nequilibrium,asageneralizationof(9):\nSa+αNa−βaTab=Φa. (18)\nWith a general Φathis relation is valid without any restriction.\nInageneralflow-frame, ua,wedefinethethermostaticpressure\nas:\np=uaΦa\nβ. (19)\nThereforethegeneralformofthepotential Φacanbewrittenas\nΦa=βp(ua+ga),whereuaga=0.(20)\nThe parallel and perpendicular components of (18) to the flow\nuaare\ns+αn−β(h+wbqb) =0,(21)\nJa+αja−β(qa+wbΠab)+βp(wa−ga) =0a,(22)\nwhereh=e+pandΠab=Pab+pΔabis the viscous pressure.\nThen we rewrite the entropy production (7) with flow relatedquantities:\nΣ=∂aSa+α∂aNa−βa∂bTab=−Na∂aα+Tab∂aβb+∂aΦa=\n−n˙α+h˙β+qa(βwa˙)+β˙p+Πab∂aβb−ja∂aα+qa∂aβ+\nβ˙ub(qb−hwb)+βqaub∂awb+p(ga−wa)∂aβ+\npβ∂a(ga−wa)+gaβ∂ap. (23)\nThermodynamics is taken into account by the following two\npostulates.\n1)The underlined part in the above expression with proper\ntime derivatives(totaldifferentials)iszero.\nβd\ndtp=nd\ndtα−hd\ndttβ−qad\ndt(βwa). (24)\nThis is the relativistic Gibbs-Duhem relation. Considerin g this\ntogether with the vanishing differential of (21), we obtain the\nGibbsrelation[41]:\nβ(de+wadqa)=ds+αdn. (25)\nBasedonthisresultweconcludethattheentropyhastobegiv en\nbyafunctionalrelationshipbetweenthelocaldensities(b utcer-\ntainlyincludingthemomentumdensity qa),i.e. theproperrela-\ntivisticandlocalequationofstateisafunction s(e,qa,n). Ithas\nthefollowingpartialderivatives:\n∂s\n∂e/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nqa,n=β,∂s\n∂n/vextendsingle/vextendsingle/vextendsingle/vextendsingle\ne,qa=−α,∂s\n∂qa/vextendsingle/vextendsingle/vextendsingle/vextendsingle\ne,n=βwa,(26)\nidentifying the thermodynamical entropic intensive param eters\nas beingβ,α=βµandβwa. The four-vector wais constrained\nbyitsorthogonalitytothelocalflow,soit containsindepen dent\ninformation on a spatial three-vector only. In isotropic me dia\nthisdegreeoffreedomisreducedtothelengthofthisvector ,w2.\nIn cases containingradiation it appears as a velocity param eter\noftheDopplershift[41].\nByutilizingtheabovefunctionalformoftheequationofsta te\nonederivesthatthepressure,theintensiveparameterasso ciated\ntomechanicalwork,satisfiesthefollowingfour-vectorgen eral-\nized Gibbs-Duhem relation, now written by the traditional d if-\nferentials:\nβ∂ap=n∂aα−h∂aβ−qb∂a(βwb). (27)\n2)Our second postulate is ga=wa.By doing so we spell\noutthefundamentalcompatibilityofnon-equilibrium(18) with\nthe equilibrium(9) definitionsof pressure. In this way we tr eat\nthe non-dissipative part of the thermodynamical potential , and\nwith that the influence of the pressure gradient on the entrop y\nproductionratepossiblyclosesttotheidealgasbehavior. Thisis\na special matching condition known from kinetic theory ( δn=\n0,δe=0): inthiscasethepressurefour-vector Φaisparallelto\nthethermometervector βa.\nNow a short calculation reduces (23) to a form collecting\nterms according to the gradients of intensives. A chemical d if-\nfusion part is associated to ∂aα, a heat diffusion (Fourier-)partto the gradient of β, and finally a viscosity term with the sym-\nmetricgradienttensorofthefullfour-vector ∂aβb. Wealsogain\none further term containing the gradient of the difference b e-\ntweenuaandwa. The antisymmetry of the multiplier enforces\ntheantisymmetryofthisvelocityrelatedgradient,theref orethis\nterm we tag as ”vorticity”. We arrive at the following expres -\nsion:\nΣ= (nwa−ja)∂aα+(qa−hwa)(∂aβ+β˙ua)\n+ (Πab−q(awb))∂aβb+q[bwa]∂a(β(ub−wb))≥0.(28)\nHereq(awb)andq[awb]denotesthesymmetricandantisymmet-\nric parts of qawbrespectively. (28) is the entropy production\nrate without fixing the flow-frame. For a perfect fluid, charac -\nterizedby(15)-(16),thelocalentropyproductioniszero. Nowit\nis straightforward to identify thermodynamic fluxes and for ces\nand establish functional relationships, that are strictly linear in\nthefirst approximation2:\nDiffusive Thermal Viscous Vortical\nFluxes nwa−jaqa−hwaΠab−q(awb)q[bwa]\nForces ∇aα∇aβ+β˙uaΔ(bc∇a)βcΔ[bc∇a](β(uc−wc))\nTable1. Thermodynamicfluxesandforcesin ageneralflow\nframe\nHere∇a=Δb\na∂b. Thecorrespondinglinearresponserelations\nforisotropiccontinuaare:\nnwa−ja=D∇aα+σ(∇aβ+β˙ua),(29)\nqa−hwa=σ∇aα+λ(∇aβ+β˙ua), (30)\nΠab−q(awb)=ζΔab∂cβc+2ηΔ/angbracketleftbc∇a/angbracketrightβc,(31)\nq[bwa]=χΔ[bc∇a](β(uc−wc)). (32)\nHere/angbracketleft/angbracketrightdenotes the symmetric traceless part in the bracketed\nindices,λis the heat conduction coefficient, Dis the diffusion\ncoefficient, σis the Soret-Dufour coefficient of thermal diffu-\nsion.ζis the bulk viscosity, ηis the shear viscosity, and χis\nthe vortical viscosity coefficient. Because of the nonnegat ive\nentropy production (28) the linear transport coefficients m ust\nfulfillthefollowinginequalities:\nD≥0,λ≥0,λD−σ2≥0,ζ≥0,η≥0,χ>0.(33)\nHere the first three inequalities are coupled channel condit ions\nforstability,whilethelast threeareindependentones.\nThe proceduredescribed here ensures the existence of a ho-\nmogeneous flow field as a time independent solution of the\nequations of motion of the fluid. That is why deviation from\nlocalequilibriumisbestcharacterizedbygradientsofthe basic\nfieldsinthefirst approximation .\nIn the followingwe studysome importantparticularchoices\nfortheflow-frame.\n5 Thermometerframe\nThethermometerorJ¨ uttnerframeisthenaturalchoiceinki -\nnetictheorycalculations. Inthiscasethedirectionof βadefines\nthe flow-frame similarly to the natural frame in perfect fluid s:\n2Since dissipative fluxes are orthogonal to ua, only the projected gradient\nterms,Δa\nb∂b,constitute thermodynamical forces.β=/radicalbig\n/bardblβbβa/bardblandua=βa/β. Inthissectionweapplythisdef-\ninition of the flow-frame. Then the local equilibrium relati ons\nare:\ns+αn−βh=0,ds+αdn−βde=0.(34)\nTheentropycurrentdensity, Jasatisfies\nJa+αja−βqa=0a, (35)\nandtheentropyproductionratefulfillstheinequality,\nΣ=−ja∂aα+qa(∂aβ+β˙ua)+βΠab∂aub≥0.(36)\nThis form of the entropy production was derived originally\nbyEckartrestrictingtothecase ga=wa=0.\nEckart identified the following thermodynamic fluxes and\nforces\nDiffusive Thermal Mechanical\nFluxes −jaqaβΠab\nForces ∇aα∇aβ+β˙uaΔ(bc∇a)uc\nTable2. ThermodynamicfluxesandforcesbyEckart.\nUnfortunatelyinthiscaseagenericinstabilityoccurs,th elin-\nearinstabilityofthehomogeneousequilibrium,asitwaspr oved\nbyHiscockandLindblomin[14]. Nonnegativeentropyproduc -\ntion is established only if considering the basic balance eq ua-\ntions (for energy, momentum and further conserved Noether-\ncharges) as constraints. However, by deriving (36) the bala nce\nof momentum does not enter the calculations. Therefore the\nlinearrelationbetweenthethermalpartofthefluxesandfor ces\nwiththeaccelerationterm, β˙ua,connectschangesinthesequan-\ntitiesirrespectivetothemomentumbalanceequation(6). A cor-\nrect treatmentof thermodynamicforcesand fluxeson the othe r\nhand should introduce the momentum balance into the above\nentropyproductionformula. A shortcalculationleadsto:\nΣ=/parenleftBign\nhqa−ja/parenrightBig\n∂aα−β\nhqa/parenleftBig\n˙qa+qa∂bub+qb∂bua+∂bΠb\na/parenrightBig\n+βΠab∂aub≥0. (37)\nThis step makes an important difference with respect to stab il-\nity properties of the homogeneous equilibrium of a fluid. The\ncorresponding thermodynamic fluxes and forces in the J¨ uttn er\nframeare\nDiffusive Thermal Mechanical\nFluxesn\nhqa−ja−β\nhqaβΠab\nForces ∇aαXa=Δab˙qb+qa∂bub+qb∂bua+Δac∂bΠb\ncΔ(bc∇a)uc\nTable3. ThermodynamicfluxesandforcesinJ¨ uttnerframe\nprovidinggenericstability\nHereXais a convenient abbreviation for the thermal force,\nthe thermodynamicalforce associated to the dissipative cu rrent\nof the heat. Linear transport relations for isotropic conti nua in\ntheJ¨ uttnerframecannowbeeasily established:\nn\nhqa−ja=D∇aα+σXa, (38)\n−β\nhqa=σ∇aα+λXa, (39)\nβΠab=ζΔab∂cuc+2ηΔ/angbracketleftb\nc∇a/angbracketrightuc. (40)With this modification the generic stability of the theory in\nJ¨ uttnerfframeisestablished: theheattransfervector qareceives\na positive relaxation factor, β/hλ>0. It is easy to realize that\nbyignoringviscosity,componentdiffusionandcrosseffec ts,in\nhomogeneousequilibrium,where all spacelike projectedgr adi-\nents of the velocity field vanish, the only surviving term in t he\nthermal force is that with the total time derivative of the he at\nvector:\nλXa=λΔab˙qb=−β\nhqa. (41)\nMultipliedby qathisleadstoarelaxationequationforthelength\nofthevector, Q=−qaqaasfollows\n˙Q=−2β\nhλQ. (42)\nThismeansa relaxationtowardsthe qa=0 valueof the energy\ncurrentdensity.\nAn important property of these equations is the expected\ngenericstatiblity ofthe homogeneousequilibrium. Withou t the\ndetailedcalculations(tobeshownelswhere)wewanttoemph a-\nsizethattheconditionsofgenericstabilityarepurelythe rmody-\nnamic. Namely,itisfulfilledwheneverthetransportcoeffic ients\nλ,˜ηare nonnegativeand the following inequalities for thermo-\ndynamicstabilityi.e. theconcavityoftheentropy s(e,n,qa)are\nsatisfied:\n∂eT>0,∂nµ\nT>0∂eT∂nµ\nT−/parenleftbigg∂nT\nT/parenrightbigg2\n≥0.(43)\n6 Otherflow-frames\nThe other flow frames can be conveniently defined in our\ngeneralframework.\nThe Eckart frame is defined by the direction of the particle\ncurrentdensityvector ua=Na//radicalbig\n/bardblNbNb/bardbl. One realizesthatin\ncase of dissipative fluids the J¨ uttner and Eckart frames do n ot\ncoincide.\nIn case of a Landau-Lifshitz frame the flow field is de-\nfined by the direction of the momentum density vector ua=\nubTa\nb//bardblucTd\nc/bardbl, therefore qa=0a. In case of dissipative fluids\nthe J¨ uttner and Landau-Lifshitz frames also do not coincid e.\nHowever,intheabsenceof qa,thethermodynamicrelationsare\nsimilartothe onesina J¨ uttnerframe\ns+αn−βh=0,ds+αdn−βde=0,ndα−hdβ−βdp=0.\n(44)\nIn principle there are several further possibilities of fra me\nfixing. One of them introduces wa=βa/h. This choice fixes\nthevelocityfieldcompatibletosomekinetictheorycalcula tions\n[24;18].\nOnceachoiceofthelinearresponsehasbeenmade,onecan\ntransform the description in one frame to the other. The diff er-\nenttransportcoefficientsarenotequivalent,aconstantin variant\ncoefficient may become flow-frame dependent in other frames.\nWether the primaryflow-frameindependentchoiceis preferr ed\nornotrequiresfurtherinvestigations.7 Summary\nThermodynamicrelationsinrelativisticfluidsadhoretoflo w-\nframes, while dividing spacial homogeneouschanges from th e\nforces enforcing this homogeneity. It is made transparent i n\nthe train of thoughts from (18) to (23), where we calculated\nentropy production separating comoving time derivatives a nd\nspacial gradients. We have seen, that α,βaand p are flow-\nframe independent. Then local equilibrium was postulated by\nthethermodynamicrelation(25),containinghomogeneoust her-\nmodynamics. In[41],presentingadifferentreasoning,we h ave\nshownthatthedifferenttransformationformulasoftherel ativis-\ntictemperature,duetoPlanck-Einstein,Blanu˘ sa-Ott,La ndsberg\nandDoppler,canbeunifiedandreasonablyexplainedinexact ly\nthisthermodynamicframework.\nWe propose that the thermometer frame, defined in (10),\nshould be a preferredchoice. In general βacan be divided into\nparts orthogonal and parallel to the flow ua:βa=β(ua+wa),\nwhereuawa=0. We have revealed how far this choice differs\nfrom the Eckart and Landau-Lifshitz frames. There are argu-\nments, that the widely used Landau-Lifshitz frame should be\nperferred [10; 42]. However, these studies do not distingui sh\nthethermometerframe.\nThe entropyproductionin a general frame(28) helpsto rec-\nognize\n–thatviscouspressureisdampstheinhomogeneitiesin βa,\n– that there are perfect fluidswith zero entropyproduction\nbutja/n=qa/h=wa/negationslash=0andΠab=hwawb/negationslash=0,\n–thereisa vorticityrelateddissipativeterm.\nFurthermore we have mentioned, that generic stability is pr op-\nerlyderivedifthemomentumbalanceconstraintisalsocons id-\neredinthecalculationoftheentropyproduction(36).\nIn our previous works we have shown further examples of\nflow-frames. In [12; 17; 43; 41] the wa=qa/ecase was ex-\nploredandin[24]and[18]we haveanalyzedthekinetictheor y\ncompatibility and thermodynamics when wa=qa/h. 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