Statistical Mechanics

1. Statistical Ensembles
   1. Definition and Purpose
      1. Role of Ensembles in Statistical Mechanics
      2. Representation of Macroscopic Systems
      3. Connection to Experimental Measurements
      4. Statistical Descriptions of Systems: Microstates vs. Macrostates
   2. Microcanonical Ensemble
      1. Definition and Characteristics
         1. Isolated Systems
         2. Fixed Energy, Volume, and Number of Particles
         3. Application to Simple Systems
      2. Phase Space and Liouville's Theorem
         1. Conservation of Phase Space Volume
         2. Invariance under Hamiltonian Dynamics
      3. Calculation of Thermodynamic Quantities
         1. Density of States
         2. Entropy and Temperature Relationships
         3. Energy Fluctuations
   3. Canonical Ensemble
      1. Definition and Environment
         1. Systems in Thermal Equilibrium with a Heat Bath
         2. Fixed Temperature, Volume, and Number of Particles
      2. Boltzmann Distribution
         1. Probability of a State
         2. Derivation from Maximum Entropy Principle
      3. Partition Function
         1. Definition and Calculation
         2. Connection to Thermodynamic Properties
         3. Helmholtz Free Energy
      4. Energy Fluctuations
         1. Variance and Standard Deviation of Energy
         2. Thermodynamic Stability and Response Functions
      5. Photon Gas Application
   4. Grand Canonical Ensemble
      1. Definition and Features
         1. Systems in Contact with a Particle Reservoir
         2. Fixed Temperature, Volume, and Chemical Potential
      2. Chemical Potential
         1. Definition and Role in Particle Exchange
         2. Relationships to Other Thermodynamic Quantities
      3. Grand Partition Function
         1. Definition and Importance
         2. Calculation of Average Quantities
         3. Pressure and Grand Canonical Potential
      4. Applications in Quantum Gases
         1. Bose-Einstein Condensation
         2. Fermi-Dirac Distribution
         3. Light-Matter Interactions
   5. Comparison and Suitability
      1. Appropriate Ensemble Selection for Different Physical Situations
      2. Thermodynamic Limit and Equivalence of Ensembles
      3. Limitations and Assumptions in Ensemble Theory
   6. Statistical Mechanics Beyond Equilibrium
      1. Non-equilibrium Extensions of Ensembles
      2. Perturbative Approaches
      3. Nonequilibrium Statistical Operators
   7. Advanced Topics in Ensembles
      1. Fluctuation Theorems
         1. Jarzynski Equality
         2. Crooks Fluctuation Theorem
      2. Bayesian Approaches to Statistical Mechanics
         1. Ensemble Inference
         2. Updating Ensembles with New Information
      3. Quantum Ensembles
         1. Density Matrices and Ensembles
         2. Quantum Entropy and Information
   8. Historical Development and Applications
      1. Contributions of Pioneers like Gibbs and Boltzmann
      2. Impact on Modern Computational Techniques
      3. Multidisciplinary Applications