Create 60_PHY_632_Statistical_Mechanics.py

pages/60_PHY_632_Statistical_Mechanics.py

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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
+""")