Statistical Physics I: Equilibrium Statistical Mechanics

ISBN-13: 9783540536628 / Angielski / Miękka / 1992 / 256 str.

Morikazu Toda; Ryogo Kubo; Nobuhiko Saito

Opis

Statistical Physics I discusses the fundamentals of equilibrium statistical mechanics, focussing on basic physical aspects. No previous knowledge of thermodynamics or the molecular theory of gases is assumed. Illustrative examples based on simple materials and photon systems elucidate the central ideas and methods.

Spis treści

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) An Ideal 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 Ideal Gases.- 3.2.1 Level Density of a Free Particle.- 3.2.2 Ideal 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 Matrix Method.- 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, Weiss 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.- 4.7.3 Coherent Anomaly Method.- 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 (I).- 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 Ergodic Theorems (II).- 5.8.1 Weak Convergence.- 5.8.2 Ergodicity.- 5.8.3 Entropy and Irreversibility.- 5.9 Quantum Mechanical Systems.- 5.9.1 Theorems in Quantum Mechanical Systems.- 5.9.2 Chaotic Behavior in Quantum Systems.- 5.9.3 Correspondence Between Classical and Quantum Chaos.- 5.9.4 Quantum Mechanical Distribution Function.- General Bibliography.- References.- Subject Index,.

Blurb

Statistical Physics I discusses the fundamentals of equilibrium statistical mechanics, focusing on basic physical aspects. No previous knowledge of thermodynamics or the molecular theory of gases is assumed. Illustrative examples based on simple materials and photon systems elucidate the central ideas and methods.