# Statistical Mechanics

## 1st Edition

### International Series of Monographs in Natural Philosophy

**Author:**R.K. Pathria

**Editor:**D. ter Haar

**eBook ISBN:**9781483186887

**Imprint:**Pergamon

**Published Date:**25th September 1972

**Page Count:**342

## Description

Statistical Mechanics discusses the fundamental concepts involved in understanding the physical properties of matter in bulk on the basis of the dynamical behavior of its microscopic constituents. The book emphasizes the equilibrium states of physical systems. The text first details the statistical basis of thermodynamics, and then proceeds to discussing the elements of ensemble theory. The next two chapters cover the canonical and grand canonical ensemble. Chapter 5 deals with the formulation of quantum statistics, while Chapter 6 talks about the theory of simple gases. Chapters 7 and 8 examine the ideal Bose and Fermi systems. In the next three chapters, the book covers the statistical mechanics of interacting systems, which includes the method of cluster expansions, pseudopotentials, and quantized fields. Chapter 12 discusses the theory of phase transitions, while Chapter 13 discusses fluctuations. The book will be of great use to researchers and practitioners from wide array of disciplines, such as physics, chemistry, and engineering.

## Table of Contents

Preface

Acknowledgements

Historical Introduction

Chapter 1. The Statistical Basis of Thermodynamics

1.1 The Macroscopic and the Microscopic States

1.2 Contact Between Statistics and Thermodynamics: Physical Significance of Q(N,V,E)

1.3 Further Contact Between Statistics and Thermodynamics

1.4 The Classical Ideal Gas

1.5 The Entropy of Mixing and the Gibbs Paradox

1.6 The "Correct" Enumeration of The Microstates

Problems

Chapter 2. Elements of Ensemble Theory

2.1 Phase Space of a Classical System

2.2 Liouville's Theorem and its Consequences

2.3 The Microcanonical Ensemble

2.4 Examples

2.5 Quantum States and the Phase Space

2.6 Two Important Theorems—The "Equipartition" and the "Virial"

Problems

Chapter 3. The Canonical Ensemble

3.1 Equilibrium Between a System and a Heat Reservoir

3.2 A System in the Canonical Ensemble

3.3 Physical Significance of the Various Statistical Quantities

3.4 Alternative Expressions for the Partition Function

3.5 The Classical Systems

3.6 Energy Fluctuations in the Canonical Ensemble: Correspondence with the Microcanonical Ensemble

3.7 A System of Harmonic Oscillators

3.8 The Statistics of Paramagnetism

3.9 Thermodynamics of Magnetic Systems: Negative Temperatures

Problems

Chapter 4. The Grand Canonical Ensemble

4.1 Equilibrium Between a System and a Particle-Energy Reservoir

4.2 A System in the Grand Canonical Ensemble

4.3 Physical Significance of the Statistical Quantities

4.4 Examples

4.5 Density and Energy Fluctuations in the Grand Canonical Ensemble: Correspondence with Other Ensembles

Problems

Chapter 5. Formulation of Quantum Statistics

5.1 Quantum-Mechanical Ensemble Theory: The Density Matrix

5.2 Statistics of the Various Ensembles

5.3 Examples

5.4 Systems Composed of Indistinguishable Particles

5.5 The Density Matrix and the Partition Function of a System of Free Particles

Problems

Chapter 6. The Theory of Simple Gases

6.1 An Ideal Gas in a Quantum-Mechanical Microcanonical Ensemble

6.2 An Ideal Gas in Other Quantum-Mechanical Ensembles

6.3 Statistics of the Occupation Numbers

6.4 Kinetic Considerations

6.5 A Gaseous System in Mass Motion

6.6 Gaseous Systems Composed of Molecules with Internal Motion

A. Monatomic Molecules

B. Diatomic Molecules

C. Polyatomic Molecules

Problems

Chapter 7. Ideal Bose Systems

7.1 Thermodynamic Behavior of an Ideal Bose Gas

7.2 Thermodynamics of the Black-Body Radiation

7.3 The Field of Sound Waves

7.4 Inertial Density of the Sound Field

7.5 Elementary Excitations in Liquid Helium II

Problems

Chapter 8. Ideal Fermi Systems

8.1 Thermodynamic Behavior of an Ideal Fermi Gas

8.2 Magnetic Behavior of an Ideal Fermi Gas

A. Pauli Paramagnetism

B. Landau Diamagnetism and De Haas-Van Alphen Effect

8.3 The Electron Gas in Metals

A. Thermionic Emission

B. Photoelectric Emission

8.4 Statistical Equilibrium of White Dwarf Stars

8.5 Statistical Model of the Atom

Problems

Chapter 9. Statistical Mechanics of Interacting Systems: The Method Of Cluster

Expansions

9.1 Cluster Expansion for a Classical Gas

9.2 Virial Expansion of the Equation Of State

9.3 Evaluation of the Virial Coefficients

9.4 General Remarks on Cluster Expansions

9.5 Exact Treatment of the Second Virial Coefficient

9.6 Cluster Expansion for a Quantum-Mechanical System

9.7 The Binary Collision Method of Lee and Yang

9.8 Applications of he Binary Collision Method

A. A Gas of Noninteracting Particles

B. A Gas of Hard Spheres

Problems

Chapter 10. Statistical Mechanics of Interacting Systems: The Method of Pseudopotentials

10.1 The Two-Body Pseudopotential

10.2 The Λγ-Body Pseudopotential and its Eigenvalues

10.3 Low-Temperature Behavior of an Imperfect Fermi Gas

10.4 Low-Temperature Behavior of an Imperfect Bose Gas

10.5 The Ground State Wave Function of Bose Fluid

10.6 States with Quantized Circulation

10.7 "Rotation" of the Superfluid

10.8 Quantized Vortex Rings and the Breakdown of Superfluidity

Problems

Chapter 11. Statistical Mechanics of Interacting Systems: The Method of Quantized Fields

11.1 The Formalism of Second Quantization

11.2 Low-Lying States of an Imperfect Bose Gas

11.3 Energy Spectrum of a Bose Liquid

11.4 Low-Lying States of an Imperfect Fermi Gas

11.5 Energy Spectrum of a Fermi Liquid: Landau's Phenomenological Theory

Problems

Chapter 12. Theory of Phase Transitions

12.1 General Remarks on the Problem of Condensation

12.2 Mayer's Theory of Condensation

12.3 The Theory of Yang and Lee

12.4 Further Comments on the Theory of Yang and Lee

A. The Gaseous Phase and the Cluster Integrals

B. An Electrostatic Analogue

12.5 A Dynamical Model for Phase Transitions

12.6 The Lattice Gas and The Binary Alloy

12.7 Ising Model in the Zeroth Approximation

12.8 Ising Model in the First Approximation

12.9 Exact Treatments of the One-Dimensional Lattice

A. The Combinatorial Method

B. The Matrix Method

C. The Zeros Of The Grand Partition Function

12.10 Study of the Two- and Three-Dimensional Lattices

12.11 The Critical Indices

12.12 The Law of Corresponding States

Problems

Chapter 13. Fluctuations

13.1 Thermodynamic Fluctuations

13.2 Spatial Correlations in a Fluid

13.3 Einstein-Smoluchowski Theory of the Brownian Motion

13.4 Langevin Theory of the Brownian Motion

13.5 Approach to Equilibrium: The Fokker-Planck Equation

13.6 Spectral Analysis of Fluctuations: The Wiener-Khintchine Theorem

13.7 The Fluctuation-Dissipation Theorem

13.8 The Onsager Relations

Problems

Appendixes

A. Influence of Boundary Conditions on the Distribution of Quantum States

B. Certain Mathematical Functions

C. "Volume" and "Surface Area" of an W-Dimensional Sphere of Radius R

D. On the Bose-Einstein Integrals

E. On the Fermi-Dirac Integrals

F. General Physical Constants

G. Defined Values and Equivalents

H. General Mathematical Constants

Bibliography

Index

## Details

- No. of pages:
- 342

- Language:
- English

- Copyright:
- © Pergamon 1972

- Published:
- 25th September 1972

- Imprint:
- Pergamon

- eBook ISBN:
- 9781483186887

## About the Author

### R.K. Pathria

Professor Raj Kumar Pathria is a theoretical physicist and an Urdu poet. He is known for his work on superfluidity in liquid helium, Lorentz transformation of thermodynamic quantities, a rigorous evaluation of lattice sums, and finite-size effects in phase transitions. Pathria obtained his BSc Honours degree in 1953 and MSc Honours degree in 1954 from Panjab University, Hoshiarpur, and earned a Ph.D. in Physics from University of Delhi in 1957. He has served on the physics faculties of the University of Delhi (1958-1964), McMaster University (1964-1965), University of Alberta (1965-1967), Panjab University, Chandigarh (1967-1969), and the University of Waterloo (1969-1998). After retirement, he moved to California to be close to family, and joined the faculty of the University of California San Diego as an adjunct professor of physics in 2000. The University of Waterloo honored him with both the Distinguished Teacher Award and the title Distinguished Professor Emeritus, and he is a Fellow of the American Physical Society. Raj is the author of over one-hundred published papers and two textbooks. The Theory of Relativity was first published in 1963 by Hindustan Publishing, with the second edition published by Pergamon Press, Oxford. This latter edition was republished by Dover Press in 2003. His widely used graduate physics textbook Statistical Mechanics is now in its fourth edition. It was originally published by Pergamon Press in 1974, with the second through fourth editions published by Elsevier in 1996, 2011, and 2021. Paul Beale joined Raj as coauthor of the third edition and fourth editions. Raj was nurtured in an atmosphere of Urdu language, idiom, and poetry. He published a compilation of his Urdu poems in Saihraa Saihraa, published under the pseudonym Raj Kumar Qais. The title means ‘desert after desert’ after the unending wanderings of Qais, the fabled lover of the Arabian damsel Laila. Raj is married to Raj Kumari Pathria. They have three children and five grandchildren.

### Affiliations and Expertise

Theoretical Physicist, University of California, San Diego, USA