Foundations of Statistical Mechanics

1 Introduction.- A. Physical Foundations.- Many Degrees of Freedom.- B. Kinetic Theory.- C. The Notion of Ensembles.- D. Ergodic Theory.- E. Critique.- Problems.- References.- 2 Theory of Probability.- A. Historical Background.- B. The Algebra of Probable Inference.- Axiomatic Formulation.- Extensions of the Theory.- Probabilities and Frequencies.- C. Calculus of Probable Inference.- Principle of Maximum Entropy.- Further Properties of SI.- Probabilities and Frequencies.- General Observations.- Problems.- References.- 3 Equilibrium Thermodynamics.- A. Canonical Ensemble.- B. Fluctuations.- Measured Values.- Measurable Fluctuations.- Stability of the Equilibrium State.- C. The Efficacy of Statistical Mechanics.- Macroscopic Uniformity.- Generalized Inverse Problems.- Infinite Volume Limit.- Problems.- References.- 4 Quantum Statistical Mechanics.- A. Review of the Principles of Quantum Mechanics.- B. Principle of Maximum Entropy.- The Entropy.- The PME.- C. Grand Canonical Ensemble.- Single-Component Systems.- Many-Body Quantum Mechanics.- The Necessity of Quantum Statistics.- Pressure Ensemble.- Summary.- D. Physical Entropy and the Second Law of Thermodynamics.- Classical Background.- The Theoretical Connection.- Physical Interpretation.- Irreversibility.- E. Space-Time Transformations.- Rotations.- Galilean Transformations.- Lorentz Transformations.- Relativistic Statistical Mechanics.- Problems.- References.- 5 Noninteracting Particles.- A. Free-Particle Models.- Historical Observations.- B. Boltzmann Statistics.- Weak Degeneracy.- C. The Degenerate Fermi Gas.- D. The Degenerate Bose Gas.- The Photon Gas.- E. Relativistic Statistics.- Weak Degeneracy.- Degenerate Fermions.- Bose-Einstein Condensation.- The Function f(x).- Problems.- References.- 6 External Fields.- A. Inhomogeneous Systems in Equilibrium.- Uniformly Rotating Bucket.- Uniform Gravitational Field.- Harmonic Confinement.- Bose-Einstein Condensation in a Gravitational Field.- B. ‘Classical Magnetism’.- Paramagnetism.- Diamagnetism.- The Importance of Quantum Mechanics.- C. Quantum Theory of Magnetism.- Spinless Bosons.- Degenerate Electron Gas.- High-Field Pauli Paramagnetism.- D. Relativistic Paramagnetism.- Degenerate Equation of State.- Ground-State Magnetization.- Evaluation of the Integrals J1 and J2.- Problems.- References.- 7 Interacting Particles I: Classical and Quantum Clustering.- A. Cluster Integrals and the Method of Ursell.- The Symmetry Problem.- B. Virial Expansion of the Equation of State.- Inversion of the Fugacity Expansion.- Ideal Quantum Gases.- The Virial Coefficients.- C. Classical Virial Coefficients.- Hard Spheres.- Point Centers of Repulsion-Soft Spheres.- Repulsive Exponential.- Hard Core Plus Square Well.- Sutherland Potential.- Triangle Well.- Trapezoidal Well.- Lennard-Jones Potential.- Miscellaneous Models.- Experimental Survey.- D. Quantum Corrections to the Classical Virial Coefficients.- Hard Spheres.- Other Models.- Higher Virial Coefficients and General Results.- E. Quantum Virial Coefficients.- Higher Virial Coeficients.- F. Paramagnetic Susceptibility.- Problems.- References.- 8 Interacting Particles, II: Fock-Space Formulation.- A. Particle Creation and Annihilation.- B. Ground State of the Hard-Sphere Bose Gas.- C. The Phonon Field.- Gas of Noninteracting Phonons.- D. Completely Degenerate Electron Gas.- E. Digression: A Perturbation Expansion of f(?,µV).- F. Long-Range Forces.- Coulomb Interactions and Screening.- Gravitational Interactions.- Problems.- References.- 9 The Phases of Matter.- A. Correlations and the LiquidState.- Radial Distribution Function.- Ideal Quantum Fluids.- Ornstein-Zernike Theory.- Theory of Liquids.- B. Crystalline Solids.- Free-Electron Model.- Electrons and Phonons.- C. Phase Transitions.- Phenomenological Theory.- Modern Developments.- D. Superconductivity.- The BCS Theory.- Problems.- References.- Appendix A.- Highpoints in the History of Statistical Mechanics.- Appendix B.- The Law of Succession.- Appendix C.- Method of Jacobians.- Appendix D.- Convex Functions and Inequalities.- Appendix E.- Euler-Maclaurin Summation Formula.- Appendix F.- The First Four Ursell Functions and Their Inverses.- Appendix G.- Thermodynamic Form of Wick’s Theorem.