Statistical Mechanics

1. Non-equilibrium Statistical Mechanics
   1. Time-Dependent Processes
      1. Overview of Non-equilibrium Systems
         1. Definition of Non-equilibrium
         2. Differences from Equilibrium Systems
         3. Relevance to Real-world Systems
      2. Approach to Study Time Evolution
         1. Liouville's Theorem in Non-equilibrium
         2. Time-dependent Schrödinger Equation
         3. Evolution of Probability Densities
   2. Fluctuation-Dissipation Theorem
      1. Statement of the Theorem
      2. Physical Interpretation
      3. Derivation from Linear Response Theory
         1. Perturbative Analysis
         2. Stochastic Differential Equations
      4. Applications
         1. Noise in Electronic Circuits
         2. Statistical Models of Complex Fluids
   3. Linear Response Theory
      1. Fundamental Concepts
         1. Perturbation and Response
         2. Susceptibility and Response Functions
      2. Kubo Relations
         1. Derivation from Statistical Mechanics
         2. Connection to Green's Functions
      3. Applications in Different Areas
         1. Electrical Conductivity
         2. Magnetic Susceptibility
         3. Thermal Transport
   4. Transport Processes
      1. Diffusion
         1. Fick's Laws of Diffusion
         2. Brownian Motion
         3. Applications in Physics and Chemistry
      2. Conductivity
         1. Electrical Transport in Solids
         2. Role in Semiconductor Physics
         3. Conductivity in Ionic and Molecular Systems
      3. Thermal Transport
         1. Heat Conduction Mechanisms
         2. Phonons and Their Role
         3. Experimental Techniques for Measurement
   5. Stochastic Processes
      1. Langevin Equation
         1. Introduction and Historical Context
         2. Formulation and Solutions
         3. Applications in Simulating Physical Systems
      2. Fokker-Planck Equation
         1. Derivation and Relation to the Langevin Equation
         2. Solution Techniques
         3. Role in Describing Brownian Motion
      3. Master Equation Approach
         1. Discrete State Systems
         2. Transition Probabilities and Rates
         3. Application in Reaction Kinetics
      4. Markov Processes and Their Importance
         1. Definition and Characteristics
         2. Continuous vs. Discrete Time Markov Chains
         3. Use in Modeling Complex Systems
   6. Irreversibility and the Arrow of Time
      1. Microscopic Reversibility
         1. Time-reversal Symmetry in Microscopic Laws
      2. Emergence of Macroscopic Irreversibility
         1. Entropy Production and the Second Law of Thermodynamics
         2. Role of Coarse-graining and Statistical Ensembles
      3. The H-Theorem and its Limitations
         1. Boltzmann's Hypothesis
         2. Critiques and Extensions
      4. Extension into Thermodynamic Processes
         1. Work, Heat, and Efficiency in Non-equilibrium Systems
         2. Experimental Observations and Theoretical Predictions