Foundations of Statistical Mechanics

International Series of Monographs in Natural Philosophy, Volume 22: Foundations of Statistical Mechanics: A Deductive Treatment presents the main approaches to the basic problems of statistical mechanics. This book examines the theory that provides explicit recognition to the limitations on one's powers of observation. Organized into six chapters, this volume begins with an overview of the main physical assumptions and their idealization in the form of postulates. This text then examines the consequences of these postulates that culminate in a derivation of the fundamental formula for calculating probabilities in terms of dynamic quantities. Other chapters provide a careful analysis of the significant notion of entropy, which shows the links between thermodynamics and statistical mechanics and also between communication theory and statistical mechanics. The final chapter deals with the thermodynamic concept of entropy. This book is intended to be suitable for students of theoretical physics. Probability theorists, statisticians, and philosophers will also find this book useful.

Preface

The Main Postulates of this Theory

Chapter I Basic Assumptions

1. Introduction

2. Dynamics

2.1. Exercises

3. Observation

3.1. Exercises

4. Probability

5. The Markovian Postulate

5.1. Exercises

6. Two Alternative Approaches

Chapter II Probability Theory

1. Events

1.1. Exercises

2. Random Variables

2.1. Exercise

3. Statistical Independence

3.1. Exercises

4. Markov Chains

4.1. Exercises

5. Classification of Observational States

5.1. Exercises

6. Statistical Equilibrium

6.1. Exercises

7. The Approach to Equilibrium

7.1. Exercises

8. Periodic Ergodic Sets

8.1. Exercises

9. The Weak Law of Large Numbers

9.1. Exercises

Chapter III The Gibbs Ensemble

1. Introduction

2. The Phase-Space Density

2.1. Exercise

3. The Classical Liouville Theorem

3.1. Exercises

4. The Density Matrix

4.1. Exercises

5. The Quantum Liouville Theorem

5.1. Exercises

Chapter IV Probabilities from Dynamics

1. Dynamical Images of Events

1.1. Exercise

2. Observational Equivalence

2.1. Exercise

3. The Classical Accessibility Postulate

3.1. Exercises

4. The Quantum Accessibility Postulates

4.1. Exercises

5. The Equilibrium Ensemble

5.1. Exercises

6. Coarse-Grained Ensembles

6.1. Exercises

7. The Consistency Condition

7.1. Exercises

8. Transient States

8.1. Exercise

Chapter V Boltzmann Entropy

1. Two Fundamental Properties of Entropy

2. Composite Systems

2.1. Exercise

3. The Additivity of Entropy

3.1. Exercises

4. Large Systems and the Connection with Thermodynamics

4.1. Exercises

5. Equilibrium Fluctuations

5.1. Exercises

6. Equilibrium Fluctuations in a Classical Gas

6.1. Exercises

7. The Kinetic Equation for a Classical Gas

8. Boltzmann's H Theorem

8.1. Exercise

Chapter VI Statistical Entropy

1. The Definition of Statistical Entropy

1.1. Exercises

2. Additivity Properties of Statistical Entropy

2.1. Exercises

3. Perpetual Motion

3.1. Exercise

4. Entropy and Information

5. Entropy Changes in the Observer

5.1. Exercises

Solutions to Exercises

Index

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