Statistical Physics I
Equilibrium Statistical Mechanics
Published on 1. June 1983
Book
Hardback
978-3-540-11460-4 (ISBN)
Description
This first volume of Statistical Physics is an introduction to the theories of equilibrium statistical mechanics, whereas the second volume (Springer Ser. Solid-State Sci., Vol. 31) is devoted to non equilibrium theories. Particular emphasis is placed on fundamental principles and basic con cepts and ideas. We start with physical examples of probability and kinetics, and then describe the general principles of statistical mechanics, with appli cations to quantum statistics, imperfect gases, electrolytes, and phase tran sitions, including critical phenomena. Finally, ergodic problems, the me chanical basis of statistical mechanics, are presented. The original text was written in Japanese as a volume of the Iwanami Series in Fundamental Physics, supervised by Professor H. Yukawa. The first edition was published in 1973 and the second in 1978. The English edition has been divided into two volumes at the request of the publisher, and the chapter on ergodic problems, which was at the end of the original book, is included here as Chapter 5. Chapters 1,2,3 and part of Chapter 4 were written by M. Toda, and Chapters 4 and 5 by N. Saito. More extensive references have been added for further reading, and some parts of the final chapters have been revised to bring the text up to date. It is a pleasure to express my gratitude to Professor P. Fulde for his detailed improvements in the manuscript, and to Dr. H. Lotsch of Springer Verlag for his continued cooperation.
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Content
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.