In a certain sense this book has been twenty-five years in the writing, since I first struggled with the foundations of the subject as a graduate student. It has taken that long to develop a deep appreciation of what Gibbs was attempting to convey to us near the end of his life and to understand fully the same ideas as resurrected by E.T. Jaynes much later. Many classes of students were destined to help me sharpen these thoughts before I finally felt confident that, for me at least, the foundations of the subject had been clarified sufficiently. More than anything, this work strives to address the following questions: What is statistical mechanics? Why is this approach so extraordinarily effective in describing bulk matter in terms of its constituents? The response given here is in the form of a very definite point of view-the principle of maximum entropy (PME). There have been earlier attempts to approach the subject in this way, to be sure, reflected in the books by Tribus [Thermostat­ ics and Thermodynamics, Van Nostrand, 1961], Baierlein [Atoms and Information Theory, Freeman, 1971], and Hobson [Concepts in Statistical Mechanics, Gordon and Breach, 1971].

Inhalt

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 Liquid State.- 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.