Statistical Physics I

1. General Preliminaries.- 1.1 Overview.- 1.1.1 Subjects of Statistical Mechanics.- 1.1.2 Approach to Equilibrium.- 1.2 Averages.- 1.2.1 Probability Distribution.- 1.2.2 Averages and Thermodynamic Fluctuation.- 1.2.3 Averages of a Mechanical System - Virial Theorem.- 1.3 The Liouville Theorem.- 1.3.1 Density Matrix.- 1.3.2 Classical Liouville's Theorem.- 1.3.3 Wigner's Distribution Function.- 1.3.4 The Correspondence Between Classical and Quantum Mechanics.- 2. Outlines of Statistical Mechanics.- 2.1 The Principles of Statistical Mechanics.- 2.1.1 The Principle of Equal Probability.- 2.1.2 Microcanonical Ensemble.- 2.1.3 Boltzmann's Principle.- 2.1.4 The Number of Microscopic States, Thermodynamic Limit.- a) A Free Particle.- b) A Perfect Gas.- c) Spin System.- d) The Thermodynamic Limit.- 2.2 Temperature.- 2.2.1 Temperature Equilibrium.- 2.2.2 Temperature.- 2.3 External Forces.- 2.3.1 Pressure Equilibrium.- 2.3.2 Adiabatic Theorem.- a) Adiabatic Change.- b) Adiabatic Theorem in Statistical Mechanics.- c) Adiabatic Theorem in Classical Mechanics.- 2.3.3 Thermodynamic Relations.- 2.4 Subsystems with a Given Temperature.- 2.4.1 Canonical Ensemble.- 2.4.2 Boltzmann-Planck's Method.- 2.4.3 Sum Over States.- 2.4.4 Density Matrix and the Bloch Equation.- 2.5 Subsystems with a Given Pressure.- 2.6 Subsystems with a Given Chemical Potential.- 2.6.1 Chemical Potential.- 2.6.2 Grand Partition Function.- 2.7 Fluctuation and Correlation.- 2.8 The Third Law of Thermodynamics, Nernst's Theorem.- 2.8.1 Method of Lowering the Temperature.- 3. Applications.- 3.1 Quantum Statistics.- 3.1.1 Many-Particle System.- 3.1.2 Oscillator Systems (Photons and Phonons).- 3.1.3 Bose Distribution and Fermi Distribution.- a) Difference in the Degeneracy of Systems.- b) A Special Case.- 3.1.4 Detailed Balancing and the Equilibrium Distribution.- 3.1.5 Entropy and Fluctuations.- 3.2 Perfect Gases.- 3.2.1 Level Density of a Free Particle.- 3.2.2 Perfect Gas.- a) Adiabatic Change.- b) High Temperature Expansion.- c) Density Fluctuation.- 3.2.3 Bose Gas.- 3.2.4 Fermi Gas.- 3.2.5 Relativistic Gas.- a) Photon Gas.- b) Fermi Gas.- c) Classical Gas.- 3.3 Classical Systems.- 3.3.1 Quantum Effects and Classical Statistics.- a) Classical Statistics.- b) Law of Equipartition of Energy.- 3.3.2 Pressure.- 3.3.3 Surface Tension.- 3.3.4 Imperfect Gas.- 3.3.5 Electron Gas.- 3.3.6 Electrolytes.- 4. Phase Transitions.- 4.1 Models.- 4.1.1 Models for Ferromagnetism.- 4.1.2 Lattice Gases.- 4.1.3 Correspondence Between the Lattice Gas and the Ising Magnet.- 4.1.4 Symmetric Properties in Lattice Gases.- 4.2 Analyticity of the Partition Function and Thermodynamic Limit.- 4.2.1 Thermodynamic Limit.- 4.2.2 Cluster Expansion.- 4.2.3 Zeros of the Grand Partition Function.- 4.3 One-Dimensional Systems.- 4.3.1 A System with Nearest-Neighbor Interaction.- 4.3.2 Lattice Gases.- 4.3.3 Long-Range Interactions.- 4.3.4 Other Models.- 4.4 Ising Systems.- 4.4.1 Nearest-Neighbor Interaction.- a) One-Dimensional Systems.- b) Many-Dimensional Systems.- c) Two-Dimensional Systems.- d) Curie Point.- 4.4.2 Method of Matrix.- a) One-Dimensional Ising System.- b) Two-Dimensional Ising Systems.- 4.4.3 Zeros on the Temperature Plane.- 4.4.4 Spherical Model.- 4.4.5 Eight-Vertex Model.- 4.5 Approximate Theories.- 4.5.1 Molecular Field Approximation.- 4.5.2 Bethe Approximation.- 4.5.3 Low and High Temperature Expansions.- 4.6 Critical Phenomena.- 4.6.1 Critical Exponents.- 4.6.2 Phenomenological Theory.- 4.6.3 Scaling.- 4.7 Renormalization Group Method.- 4.7.1 Renormalization Group.- 4.7.2 Fixed Point.- 5. Ergodic Problems.- 5.1 Some Results from Classical Mechanics.- 5.1.1 The Liouville Theorem.- 5.1.2 The Canonical Transformation.- 5.1.3 Action and Angle Variables.- 5.1.4 Integrable Systems.- 5.1.5 Geodesics.- 5.2 Ergodic Theorems.- 5.2.1 Birkhoff s Theorem.- 5.2.2 Mean Ergodic Theorem.- 5.2.3 Hopf's Theorem.- 5.2.4 Metrical Transitivity.- 5.2.5 Mixing.- 5.2.6 Khinchin's Theorem.- 5.3 Abstract Dynamical Systems.- 5.3.1 Bernoulli Schemes and Baker's Transformation.- 5.3.2 Ergodicity on the Torus.- 5.3.3 K-Systems (Kolmogorov Transformation).- 5.3.4 C-Systems.- 5.4 The Poincaré and Fermi Theorems.- 5.4.1 Bruns' Theorem.- 5.4.2 Poincaré-Fermi's Theorem.- 5.5 Fermi-Pasta-Ulam's Problem.- 5.5.1 Nonlinear Lattice Vibration.- 5.5.2 Resonance Conditions.- 5.5.3 Induction Phenomenon.- 5.6 Third Integrals.- 5.7 The Kolmogorov, Arnol'd and Moser Theorem.- 5.8 Quantum Mechanical Systems.- 5.8.1 Theorems in Quantum Mechanical Systems.- 5.8.2 Chaotic Behavior in Quantum Systems.- 5.8.3 Adiabatic Processes and Susceptibility.- General Bibliography.- References.