Statistical Mechanics

1. Applications to Physical Systems
   1. Ideal Gas
      1. Maxwell-Boltzmann Statistics
         1. Distribution of molecular speeds
         2. Derivation from first principles
         3. Applicability to classical systems
         4. Role in determining macroscopic properties such as pressure and temperature
      2. Quantum Ideal Gases
         1. Bose-Einstein Statistics
            1. Characteristics of bosons
            2. Bose-Einstein condensation
            3. Applications and examples: superfluidity, laser cooling
         2. Fermi-Dirac Statistics
            1. Characteristics of fermions
            2. Fermi energy and temperature
            3. Applications and examples: electron gas in metals, white dwarfs
   2. Real Gases
      1. Virial Expansion
         1. Mathematical basis and series expansion
         2. Virial coefficients and their physical significance
         3. Practical applications in gas analysis
      2. Van der Waals Equation
         1. Correction terms for pressure and volume
         2. Critical temperature and pressure
         3. Phase behavior predictions for real gases
   3. Solids
      1. Einstein Model of Heat Capacity
         1. Assumptions and simplifications
         2. Derivation of specific heat formula
         3. Limitations and historical significance
      2. Debye Model of Heat Capacity
         1. Phonon theory and vibrational states
         2. Debye temperature
         3. Comparison to Einstein model and real solids
         4. Low-temperature behavior predictions
      3. Thermal Expansion
         1. Microscopic basis and statistical mechanisms
         2. Dependencies on temperature and pressure
         3. Practical implications in materials science
      4. Conductivity and Electrical Properties
         1. Role of charge carriers in solids
         2. Temperature dependence
         3. Applications in semiconductor theory
   4. Liquids and Liquid Mixtures
      1. Statistical Mechanics of Liquids
         1. Radial distribution function
         2. Structure and pair correlation
         3. Thermodynamic properties via statistical approach
      2. Phase Equilibria
         1. Liquid-vapor equilibrium
         2. Modeling phase transitions in mixtures
         3. Applications to distillation and material separation processes
   5. Magnetic Systems
      1. Paramagnetism and Diamagnetism
         1. Statistical basis for magnetic response
         2. Temperature and field dependence
      2. Ferromagnetism
         1. Domain theory and statistical models
         2. Curie-Weiss Law
         3. Role of quantum statistics
   6. Superconductivity
      1. Microscopic models and theories
         1. BCS theory and Cooper pairs
         2. Energy gap and superconducting state
      2. Thermodynamic properties of superconductors
         1. Entropy and energy considerations in transition
         2. Magnetic and electrical properties in low temperatures
   7. Polymers and Soft Matter
      1. Statistical models of polymer chains
         1. Random walk and ideal chain models
         2. Excluded volume effects
      2. Applications in colloids and gels
         1. Thermodynamic properties and phase transitions
         2. Structural dynamics and rheological properties
   8. Biological Systems
      1. Statistical mechanics of biomolecules
         1. Folding and stability of proteins
         2. Dynamics of nucleic acids
         3. Thermodynamic properties in biological environments
      2. Cellular Transport and Dynamics
         1. Mechanisms of diffusion and osmosis
         2. Role of statistical mechanics in understanding cellular functions
   9. Advanced Topics in Physical Applications
      1. Nanostructures and Nanoparticles
         1. Quantum size effects and statistical descriptions
         2. Applications in catalysis and material properties
      2. Statistical Mechanics in Astrophysics
         1. Stellar structure and thermodynamic stability
         2. Statistical models of matter under extreme conditions
         3. Applications in black hole thermodynamics