Statistical Mechanics

1. Concepts of Probability in Statistical Mechanics
   1. Probability Distributions
      1. Definition and Importance
         1. Role in Predicting System Behavior
         2. Connection with Measurement Outcomes
      2. Types of Distributions
         1. Discrete vs. Continuous
         2. Probability Density Function (PDF)
         3. Cumulative Distribution Function (CDF)
         4. Joint, Marginal, and Conditional Distributions
      3. Common Distributions
         1. Gaussian Distribution
            1. Characteristics and Central Limit Theorem
            2. Applications in Thermodynamic Limits
         2. Poisson Distribution
            1. Use in Counting Statistics
         3. Binomial Distribution
            1. Applications in Spin Systems
   2. Entropy in Statistical Mechanics
      1. Boltzmann's Entropy Formula
         1. S = k_B ln W
         2. Relation to Microstates
         3. Applications to Ideal Gases
      2. Gibbs Entropy
         1. S = -k_B Σ_i p_i ln p_i
         2. Applicability to Ensembles
         3. Comparison to Boltzmann's Entropy
      3. Shannon Entropy
         1. Conceptual Overlap with Statistical Mechanics
         2. Use in Information Theory
      4. Entropy and Disorder
         1. Relationship to Thermodynamic Entropy
         2. Interpretation in Terms of Uncertainty
   3. The Principle of Equal A Priori Probabilities
      1. Definition and Justification
         1. Postulate of Equal Probabilities for Accessible Microstates
         2. Underlying Assumptions and Limitations
         3. Role in Deriving Statistical Distributions
      2. Applications
         1. Microcanonical Ensemble
            1. Application to Isolated Systems
            2. Connection to Energy Conservation
         2. Estimation of Density of States
      3. Challenges and Criticisms
         1. Justification in Quantum and Classical Systems
         2. Attempts to Derive from First Principles
   4. Connection between Probability and Physical Quantities
      1. Expectation Values
         1. Definition and Computation
         2. Use in Predicting Macroscopic Observables
      2. Variance and Standard Deviation
         1. Role in Understanding Fluctuations
         2. Importance in Response Functions
      3. Statistical Averages in Different Ensembles
         1. Canonical Averaging
         2. Grand Canonical Averaging
   5. Fluctuations and Correlations
      1. Nature of Fluctuations
         1. Spontaneous vs. Induced Fluctuations
         2. Importance in Phase Transitions
      2. Correlation Functions
         1. Definition and Calculation
         2. Use in Understanding Interactions
      3. Fluctuation-Dissipation Relations
         1. Connection to Transport Phenomena
         2. Role in Non-equilibrium Systems