# Foundations of Classical and Quantum Statistical Mechanics

### International Series of Monographs in Natural Philosophy

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Foundations of Classical and Quantum Statistical Mechanics details the theoretical foundation the supports the concepts in classical and quantum statistical mechanics. The title discusses the various problems set by the theoretical justification of statistical mechanics methods. The text first covers the the ergodic theory in classical statistical mechanics, and then proceeds to tackling quantum mechanical ensembles. Next, the selection discusses the the ergodic theorem in quantum statistical mechanics and probability quantum ergodic theorems. The selection also details H-theorems and kinetic equations in classical and quantum statistical mechanics. The book will be of great interest to students, researchers, and practitioners of physics, chemistry, and engineering.

## Table of Contents

Preface

Preface to the English Edition

General Introduction

Part I. Ergodic Theory

Chapter I. The Ergodic Theory in Classical Statistical Mechanics

I. Statistical Ensembles of Classical Systems

II. Ergodic Theorems in Classical Mechanics

III. The Hypothesis of Metric Transitivity

Chapter Ii. Quantum Mechanical Ensembles. Macroscopic Operators

I . Statistical Ensembles of Quantum Systems

II. Macroscopic Operators

Chapter III. The Ergodic Theorem in Quantum Statistical Mechanics

I. The Ergodic Problem in Quantum Mechanics

II. The First Quantum Ergodic Theorem

III. The Second Quantum Ergodic Theorem

IV. The Proofs of Von Neumann and Pauli-Fierz

Chapter IV. Probability Quantum Ergodic Theorems

I. Comments on the Quantum Ergodic Theory

II. First Probability Ergodic Theorem

III. The Macroscopic Probability Ergodic Theorem

IV. Statistical Properties of Macroscopic Observables. Comparison with Classical Theory

V. Relations between Microcanonical, Canonical and Grand-Canonical Ensemble

Part II. H-Theorems

Introduction

Chapter V. H-Theorems and Kinetic Equations in Classical Statistical Mechanics

I. Mechanical Reversibility and Quasi-Periodicity

II. Coarse-Grained Densities and the Generalized H-Theorem

III. Transition Probabilities and Boltzmann's Equation

IV. Stochastic Processes and H-Theorems

V. Integration of the Liouville Equation

VI. Prigogine's Theory of Irreversible Processes

Chapter VI. H-Theorems and Kinetic Equations in Quantum Statistical Mechanics

I. Fine- and Coarse-Grained Densities in Quantum Mechanics

II. The H-Theorem for an Ensemble of Quantum Systems

III. The Kinetic Equation and Irreversible Processes

IV. Boltzmann's Equation and Stochastic Processes in Quantum Theory

V. Zwanzig's Method

Chapter Vii. General Conclusions. Macroscopic Observation and Quantum Measurement

I. Applications of Statistical Mechanics

II. Quantum Measurement and Macroscopic Observation

Appendix I

1. Historical Review of Ergodic Theory

2. Birkhoff's Theorem

3. Notes On the Metric Transitivity of Hypersurfaces

4. Structure Functions in Classical Statistical Mechanics

Appendix II. Probability Laws in Real n-Dimensional Euclidean Space

1. The Unit Hypersphere in n-Dimensional Space

2. The Unit Hypersphere in 2n-Dimensional Space 340

3. Probability Laws for the Quantities Dii(y) and µij(α)

Appendix III A. Ehrenfests' Model

1. The Function H(Z, t) and the "H-Curve"

2. Ehrenfests' Model

3. Transition Probabilities and the Fundamental Equation

4. Stationary Distribution

5. Properties of the "Δs-Curve"

6. Calculation of P(n│m, s)

Appendix III B. Notes on the Definition of Entropy

Appendix IV. Note on Recent Developments in Classical Ergodic Theory

I. The Concept of an Abstract Dynamic System

II. Asymptotic Properties of Abstract Dynamic Systems

1. Definitions

2. Asymptotic Properties

III. Entropy and K-Systems

1. Measurable Decompositions

2. Entropy

3. K-Systems

Bibliography

Index

## Product details

- No. of pages: 440
- Language: English
- Copyright: © Pergamon 1963
- Published: January 1, 1963
- Imprint: Pergamon
- eBook ISBN: 9781483186269

## About the Author

### R. Jancel

## About the Editor

### D. ter Haar

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