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| import streamlit as st | |
| # Set the page title | |
| st.title("PHY 632: Statistical Mechanics") | |
| # Course Details | |
| st.markdown(""" | |
| ## Course Details | |
| - **Course Title**: Statistical Mechanics | |
| - **Credits**: 3 | |
| - **Prerequisites**: PHY 504, PHY 520, PHY 522 | |
| - **Instructor**: [Instructor Name] | |
| - **Office Hours**: [Office Hours] | |
| ## Course Description | |
| This course provides an in-depth exploration of the principles of statistical mechanics and their applications to thermodynamics and various physical systems. Topics include the thermodynamic description of matter, perfect gases, and quantum statistics. | |
| ## Course Objectives | |
| Upon completing this course, students will: | |
| - Understand the connection between statistical mechanics and thermodynamics. | |
| - Solve problems involving classical and quantum statistical systems. | |
| - Apply statistical mechanics to real-world physical systems. | |
| --- | |
| """) | |
| # Weekly Outline with Problems | |
| # Week 1: Introduction to Statistical Mechanics and Thermodynamics | |
| with st.expander("**Week 1: Introduction to Statistical Mechanics and Thermodynamics**"): | |
| st.markdown(""" | |
| ### Topics Covered | |
| - Review of thermodynamic principles. | |
| - Introduction to statistical mechanics: microstates and macrostates. | |
| - Connection between thermodynamics and statistical mechanics. | |
| ### Problems | |
| 1. Derive the first law of thermodynamics from a microscopic statistical perspective. | |
| 2. Calculate the change in entropy for an ideal gas in an adiabatic process. | |
| 3. Use statistical mechanics to explain the concept of temperature in terms of microstates and macrostates. | |
| 4. Compute the thermodynamic variables (energy, temperature, pressure) for a system of non-interacting particles. | |
| 5. Explain the connection between the partition function and thermodynamic quantities. | |
| 6. Derive the thermodynamic identity and apply it to a simple thermodynamic process. | |
| 7. Calculate the entropy change in a reversible isothermal expansion of an ideal gas. | |
| 8. Use statistical mechanics to describe the fluctuation of energy in a small subsystem. | |
| """) | |
| # Week 2-3: Microcanonical Ensemble and Classical Thermodynamics | |
| with st.expander("**Week 2-3: Microcanonical Ensemble and Classical Thermodynamics**"): | |
| st.markdown(""" | |
| ### Topics Covered | |
| - Microcanonical ensemble and the definition of entropy. | |
| - Energy, entropy, and temperature in the microcanonical ensemble. | |
| - Applications to classical thermodynamics. | |
| ### Problems | |
| 9. Solve for the entropy of a system of N non-interacting particles in the microcanonical ensemble. | |
| 10. Derive the thermodynamic quantities (energy, temperature, entropy) from the microcanonical partition function. | |
| 11. Calculate the probability distribution of energy in a microcanonical ensemble for a small system. | |
| 12. Apply the microcanonical ensemble to a system of harmonic oscillators and derive the thermodynamic quantities. | |
| 13. Explain the role of the density of states in determining the entropy of an isolated system. | |
| 14. Calculate the change in entropy for a system of non-interacting spins in an external magnetic field. | |
| 15. Solve for the temperature of a system in terms of the multiplicity of microstates. | |
| 16. Use the microcanonical ensemble to derive the ideal gas law. | |
| 17. Compute the entropy of an ideal gas in a microcanonical ensemble and compare with classical thermodynamics. | |
| 18. Explain the concept of ergodicity and its significance in statistical mechanics. | |
| """) | |
| # Week 4-5: Canonical Ensemble: Partition Functions, Thermodynamic Quantities | |
| with st.expander("**Week 4-5: Canonical Ensemble: Partition Functions, Thermodynamic Quantities**"): | |
| st.markdown(""" | |
| ### Topics Covered | |
| - Canonical ensemble for systems in thermal contact with a heat bath. | |
| - Partition function as the central quantity in the canonical ensemble. | |
| - Calculation of thermodynamic quantities from the partition function. | |
| ### Problems | |
| 19. Derive the canonical partition function for a system of non-interacting particles in a potential well. | |
| 20. Use the canonical partition function to calculate the Helmholtz free energy for an ideal gas. | |
| 21. Calculate the specific heat of a system from the partition function and discuss its temperature dependence. | |
| 22. Solve for the energy fluctuations in a canonical ensemble and derive the corresponding heat capacity. | |
| 23. Calculate the partition function for a system of N harmonic oscillators. | |
| 24. Derive the relationship between the canonical partition function and the free energy. | |
| 25. Apply the canonical ensemble to a system of particles in a gravitational potential and compute the thermodynamic quantities. | |
| 26. Use the partition function to calculate the entropy and internal energy of an ideal gas. | |
| 27. Derive the expression for the pressure of an ideal gas using the canonical ensemble. | |
| 28. Solve for the thermodynamic properties of a two-level system in thermal equilibrium with a heat bath. | |
| 29. Calculate the probability of finding a system in a particular energy state using the Boltzmann distribution. | |
| 30. Use the canonical ensemble to solve for the entropy of a paramagnetic system in an external magnetic field. | |
| """) | |
| # Week 6-7: Grand Canonical Ensemble: Applications to Gases | |
| with st.expander("**Week 6-7: Grand Canonical Ensemble: Applications to Gases**"): | |
| st.markdown(""" | |
| ### Topics Covered | |
| - Grand canonical ensemble for systems with varying particle numbers. | |
| - Chemical potential and applications to ideal and real gases. | |
| - Thermodynamic properties of systems in the grand canonical ensemble. | |
| ### Problems | |
| 31. Derive the grand partition function for an ideal gas and use it to calculate the average particle number. | |
| 32. Calculate the chemical potential of an ideal gas using the grand canonical ensemble. | |
| 33. Solve for the particle number fluctuations in a grand canonical ensemble and relate them to the compressibility. | |
| 34. Use the grand canonical ensemble to derive the equation of state for a van der Waals gas. | |
| 35. Calculate the thermodynamic properties of a photon gas using the grand canonical ensemble. | |
| 36. Derive the relationship between the grand partition function and the thermodynamic potential. | |
| 37. Apply the grand canonical ensemble to a system of fermions and calculate the Fermi energy at low temperatures. | |
| 38. Solve for the particle number distribution in a grand canonical ensemble of non-interacting particles. | |
| 39. Calculate the entropy of a Bose gas in the grand canonical ensemble and discuss Bose-Einstein condensation. | |
| 40. Use the grand partition function to calculate the pressure and chemical potential of a photon gas. | |
| """) | |
| # Week 8: Quantum Statistical Mechanics: Bose-Einstein and Fermi-Dirac Statistics | |
| with st.expander("**Week 8: Quantum Statistical Mechanics: Bose-Einstein and Fermi-Dirac Statistics**"): | |
| st.markdown(""" | |
| ### Topics Covered | |
| - Quantum statistics: Bose-Einstein and Fermi-Dirac distributions. | |
| - Application to real-world systems like photon gases and electron systems. | |
| - Understanding condensation phenomena and degenerate Fermi gases. | |
| ### Problems | |
| 41. Derive the Bose-Einstein distribution function and apply it to a system of non-interacting bosons. | |
| 42. Solve for the critical temperature of a Bose-Einstein condensate in three dimensions. | |
| 43. Calculate the Fermi energy for a system of non-interacting fermions at absolute zero. | |
| 44. Use the Fermi-Dirac distribution to compute the average energy of a system of electrons in a metal. | |
| 45. Derive the equation of state for an ideal Fermi gas at high temperatures. | |
| 46. Calculate the entropy and specific heat of a Fermi gas at low temperatures using the Fermi-Dirac distribution. | |
| 47. Apply Bose-Einstein statistics to a photon gas and derive Planck’s law of blackbody radiation. | |
| 48. Solve for the thermodynamic properties of a Bose gas near the critical temperature of condensation. | |
| 49. Derive the chemical potential of a Bose gas in terms of the particle number and temperature. | |
| 50. Calculate the thermodynamic properties of a degenerate Fermi gas and compare with classical ideal gases. | |
| """) | |
| # Week 9-10: Ideal Gases: Classical and Quantum Regimes | |
| with st.expander("**Week 9-10: Ideal Gases: Classical and Quantum Regimes**"): | |
| st.markdown(""" | |
| ### Topics Covered | |
| - Classical ideal gas in the microcanonical and canonical ensembles. | |
| - Quantum ideal gas and quantum corrections to classical behavior. | |
| - Comparing classical and quantum behavior of ideal gases. | |
| ### Problems | |
| 51. Solve for the thermodynamic properties of a classical ideal gas using the partition function. | |
| 52. Compare the behavior of a classical ideal gas and a quantum ideal gas at low temperatures. | |
| 53. Derive the equation of state for an ideal gas in both classical and quantum regimes. | |
| 54. Calculate the compressibility and specific heat of a quantum ideal gas. | |
| 55. Solve for the quantum corrections to the thermodynamic properties of an ideal gas at low temperatures. | |
| 56. Use quantum statistical mechanics to calculate the pressure and chemical potential of a photon gas. | |
| 57. Calculate the specific heat of a classical ideal gas and compare with the Dulong-Petit law. | |
| 58. Derive the expression for the entropy of a quantum ideal gas and compare with the classical result. | |
| 59. Apply quantum statistical mechanics to a system of non-interacting particles in a box and calculate the energy levels. | |
| 60. Solve for the energy density and pressure of a photon gas using quantum statistics. | |
| 61. Compare the behavior of a Bose gas and a Fermi gas at low temperatures and discuss their differences. | |
| 62. Calculate the thermodynamic quantities of an ideal gas in the classical limit using the canonical partition function. | |
| """) | |
| # Week 11: Blackbody Radiation and Photon Gas | |
| with st.expander("**Week 11: Blackbody Radiation and Photon Gas**"): | |
| st.markdown(""" | |
| ### Topics Covered | |
| - Blackbody radiation and photon gas. | |
| - Application of Bose-Einstein statistics to photons. | |
| - Planck’s law and thermodynamic properties of photon gases. | |
| ### Problems | |
| 63. Derive Planck’s radiation law using Bose-Einstein statistics for photons. | |
| 64. Solve for the energy density of blackbody radiation as a function of temperature. | |
| 65. Calculate the entropy of a photon gas and compare with classical thermodynamics. | |
| 66. Use the Stefan-Boltzmann law to calculate the radiation pressure of blackbody radiation. | |
| 67. Solve for the spectral energy density of blackbody radiation at different temperatures. | |
| 68. Derive the relationship between the energy density and temperature for blackbody radiation. | |
| 69. Calculate the specific heat of a photon gas using the partition function. | |
| 70. Analyze the thermodynamic properties of blackbody radiation in terms of the photon gas model. | |
| """) | |
| # Week 12: Non-Ideal Systems: Interacting Gases and Phase Transitions | |
| with st.expander("**Week 12: Non-Ideal Systems: Interacting Gases and Phase Transitions**"): | |
| st.markdown(""" | |
| ### Topics Covered | |
| - Non-ideal gas behavior and interactions between particles. | |
| - Phase transitions: first-order and second-order. | |
| - Application of statistical mechanics to phase transitions. | |
| ### Problems | |
| 71. Solve the van der Waals equation for a real gas and calculate its thermodynamic properties. | |
| 72. Use statistical mechanics to derive the conditions for a first-order phase transition. | |
| 73. Calculate the critical temperature and pressure of a real gas using the van der Waals equation. | |
| 74. Solve for the coexistence curve of a phase transition in a two-phase system. | |
| 75. Derive the Gibbs free energy for a system undergoing a phase transition. | |
| 76. Apply the van der Waals equation to a liquid-gas phase transition and calculate the latent heat. | |
| 77. Calculate the specific heat and compressibility of a system near a critical point. | |
| 78. Solve for the entropy change during a first-order phase transition using statistical mechanics. | |
| 79. Use statistical mechanics to derive the Clausius-Clapeyron equation for phase transitions. | |
| 80. Analyze the behavior of a gas near the critical point using the van der Waals equation. | |
| """) | |
| # Week 13-14: Advanced Topics: Critical Phenomena, Renormalization | |
| with st.expander("**Week 13-14: Advanced Topics: Critical Phenomena, Renormalization**"): | |
| st.markdown(""" | |
| ### Topics Covered | |
| - Critical phenomena and second-order phase transitions. | |
| - Renormalization group theory and its application to phase transitions. | |
| - Scaling laws and critical exponents. | |
| ### Problems | |
| 81. Derive the critical exponents for a system undergoing a second-order phase transition. | |
| 82. Apply the renormalization group theory to calculate the scaling laws near the critical point. | |
| 83. Solve for the correlation length and susceptibility near the critical point of a phase transition. | |
| 84. Calculate the behavior of the specific heat near a second-order phase transition. | |
| 85. Use scaling arguments to derive the temperature dependence of thermodynamic quantities near a critical point. | |
| 86. Solve for the critical temperature of a system using renormalization group methods. | |
| 87. Analyze the behavior of a magnetic system near its critical temperature using statistical mechanics. | |
| 88. Apply renormalization techniques to derive the scaling behavior of thermodynamic quantities near a phase transition. | |
| 89. Calculate the critical exponents for a system undergoing a phase transition and compare with experimental data. | |
| 90. Use renormalization group theory to solve for the behavior of a system near a tricritical point. | |
| """) | |
| # Week 15: Review and Final Exam Preparation | |
| with st.expander("**Week 15: Review and Final Exam Preparation**"): | |
| st.markdown(""" | |
| ### Topics Covered | |
| - Comprehensive review of statistical mechanics concepts. | |
| - Focus on key topics such as quantum statistics, partition functions, and phase transitions. | |
| - Problem-solving sessions and final exam preparation. | |
| ### Problems | |
| 91. Mixed problems covering microcanonical, canonical, and grand canonical ensembles. | |
| 92. Problems integrating quantum statistical mechanics with real systems such as photon gases and Bose-Einstein condensation. | |
| 93. Calculate thermodynamic quantities for systems undergoing phase transitions. | |
| 94. Use quantum statistics to solve real-world problems in condensed matter physics. | |
| 95. Analyze thermodynamic properties of gases using classical and quantum statistics. | |
| 96. Derive critical exponents and scaling laws for systems near critical points. | |
| """) | |
| # Textbooks Section | |
| st.markdown(""" | |
| ## Textbooks | |
| - **"Statistical Mechanics"** by R.K. Pathria and Paul D. Beale | |
| - **"Statistical Physics: Volume 5 (Course of Theoretical Physics)"** by L.D. Landau and E.M. Lifshitz | |
| """) |