Statistical Mechanics

1. Mathematical Tools
   1. Probability and Statistics
      1. Elementary Probability Concepts
         1. Definitions of Probability
         2. Conditional Probability and Bayes' Theorem
         3. Random Variables and Expectation Values
      2. Distributions
         1. Normal Distribution
         2. Binomial and Poisson Distributions
         3. Exponential and Gamma Distributions
      3. Statistical Inference
         1. Parameter Estimation
         2. Hypothesis Testing
         3. Confidence Intervals
      4. Applications in Statistical Mechanics
         1. Use in Ensemble Averages
         2. Fluctuation Theorems
   2. Complex Analysis
      1. Complex Numbers and Their Properties
         1. Real and Imaginary Components
         2. Polar and Exponential Forms
      2. Functions of a Complex Variable
         1. Analytic Functions
         2. Cauchy-Riemann Equations
      3. Integral Theorems
         1. Cauchy's Theorem
         2. Residue Theorem and Contour Integration
      4. Applications
         1. Evaluation of Partition Functions
         2. Analytic Continuation in Physical Problems
   3. Differential Equations
      1. Ordinary Differential Equations (ODEs)
         1. First and Second Order ODEs
         2. Solutions Techniques: Separation of Variables, Integrating Factor
         3. Boundary Value Problems and Eigenvalue Problems
      2. Partial Differential Equations (PDEs)
         1. Heat Equation
         2. Wave Equation
         3. Laplace's Equation
      3. Solving Techniques
         1. Series Solutions
         2. Fourier Transform and Laplace Transform Methods
      4. Applications to Statistical Mechanics
         1. Time Evolution of Probabilities
         2. Fokker-Planck and Langevin Equations
   4. Group Theory in Statistical Mechanics
      1. Basics of Group Theory
         1. Definition of Groups
         2. Subgroups and Cosets
         3. Homomorphisms and Isomorphisms
      2. Representation Theory
         1. Matrix Representations of Groups
         2. Character Tables
      3. Symmetry in Physical Systems
         1. Crystallographic Groups
         2. Symmetries in Phase Transitions
      4. Applications
         1. Classification of Phases
         2. Symmetry Breaking and Higgs Mechanism