# Statistical Mechanics

**By**

- R K Pathria, University of California at San Diego
- Paul D. Beale, University of Colorado at Boulder

Statistical Mechanics explores the physical properties of matter based on the dynamic behavior of its microscopic constituents. After a historical introduction, this book presents chapters about thermodynamics, ensemble theory, simple gases theory, Ideal Bose and Fermi systems, statistical mechanics of interacting systems, phase transitions, and computer simulations. This edition includes new topics such as BoseEinstein condensation and degenerate Fermi gas behavior in ultracold atomic gases and chemical equilibrium. It also explains the correlation functions and scattering; fluctuationdissipation theorem and the dynamical structure factor; phase equilibrium and the Clausius-Clapeyron equation; and exact solutions of one-dimensional fluid models and two-dimensional Ising model on a finite lattice. New topics can be found in the appendices, including finite-size scaling behavior of Bose-Einstein condensates, a summary of thermodynamic assemblies and associated statistical ensembles, and pseudorandom number generators. Other chapters are dedicated to two new topics, the thermodynamics of the early universe and the Monte Carlo and molecular dynamics simulations. This book is invaluable to students and practitioners interested in statistical mechanics and physics.

View full description### Audience

Graduate and Advanced Undergraduate Students in Physics. Researchers in the field of Statisical Physics.

### Book information

- Published: February 2011
- Imprint: ACADEMIC PRESS
- ISBN: 978-0-12-382188-1

### Reviews

"An excellent graduate-level text. The selection of topics is very complete and gives to the student a wide view of the applications of statistical mechanics. The set problems reinforce the theory exposed in the text, helping the student to master the material"--**Francisco Cevantes**

**"**Making sense out of the world around us in one of the most appealing facets of physics. One may start by putting together seemingly isolated observations and as the different pieces start to fall into place, more complicated arrangements and more fundamental explanations are sought. This is indeed the case for instance when trying to understand the behaviour of a collection of particles. On the one hand, thermo- dynamics provides us with a satisfactory explanation of the macroscopic phenomena observed, however, in order to get to the core of the physical system it becomes necessary to take into account the microscopic constituents of the system as well as the fact that quantum mechanical effects are at play. This is the realm of statistical mechanics and the subject of one of the most widely recognised textbooks around the globe: Pathria’s Statistical Mechanics.**…**The original style of the book is kept, and the clarity of explanations and derivations is still there. I am convinced that this third edition of Statistical Mechanics will enable a number of new generations of physicists to gain a solid background of statistical physics and that can only be a good thing."--**Contemporary Physics, pages 619-620**

### Table of Contents

Preface to the Third Edition

Preface to the Second Edition

Preface to the First Edition

Historical Introduction

1. The Statistical Basis of Thermodynamics

1.1. The macroscopic and the microscopic states

1.2. Contact between statistics and thermodynamics: physical significance of the number Ω(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

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

Problems

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 in the canonical ensemble

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. Two theorems-the “equipartition” and the “virial”

3.8. A system of harmonic oscillators

3.9. The statistics of paramagnetism

3.10. Thermodynamics of magnetic systems: negative temperatures

Problems

4. The Grand Canonical Ensemble 91

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 various statistical quantities

4.4. Examples

4.5. Density and energy fluctuations in the grand canonical ensemble: correspondence with other ensembles

4.6. Thermodynamic phase diagrams

4.7. Phase equilibrium and the Clausius-Clapeyron equation

Problems

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

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. Gaseous systems composed of molecules with internal motion

6.6. Chemical equilibrium

Problems

7. Ideal Bose Systems

7.1. Thermodynamic behavior of an ideal Bose gas

7.2. Bose-Einstein condensation in ultracold atomic gases

7.3. Thermodynamics of the blackbody radiation

7.4. The field of sound waves

7.5. Inertial density of the sound field

7.6. Elementary excitations in liquid helium II

Problems

8. Ideal Fermi Systems

8.1. Thermodynamic behavior of an ideal Fermi gas

8.2. Magnetic behavior of an ideal Fermi gas

8.3. The electron gas in metals

8.4. Ultracold atomic Fermi gases

8.5. Statistical equilibrium of white dwarf stars

8.6. Statistical model of the atom

Problems

9. Thermodynamics of the Early Universe

9.1. Observational evidence of the Big Bang

9.2. Evolution of the temperature of the universe

9.3. Relativistic electrons, positrons, and neutrinos

9.4. Neutron fraction

9.5. Annihilation of the positrons and electrons

9.6. Neutrino temperature

9.7. Primordial nucleosynthesis

9.8. Recombination

9.9. Epilogue

Problems

10. Statistical Mechanics of Interacting Systems: The Method of Cluster Expansions

10.1. Cluster expansion for a classical gas

10.2. Virial expansion of the equation of state

10.3. Evaluation of the virial coefficients

10.4. General remarks on cluster expansions

10.5. Exact treatment of the second virial coefficient

10.6. Cluster expansion for a quantum-mechanical system

10.7. Correlations and scattering

Problems

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

11.1. The formalism of second quantization

11.2. Low-temperature behavior of an imperfect Bose gas

11.3. Low-lying states of an imperfect Bose gas

11.4. Energy spectrum of a Bose liquid

11.5. States with quantized circulation

11.6. Quantized vortex rings and the breakdown of superfluidity

11.7. Low-lying states of an imperfect Fermi gas

11.8. Energy spectrum of a Fermi liquid: Landau’s phenomenological theory

11.9. Condensation in Fermi systems

Problems

12. Phase Transitions: Criticality, Universality, and Scaling

12.1. General remarks on the problem of condensation

12.2. Condensation of a van der Waals gas

12.3. A dynamical model of phase transitions

12.4. The lattice gas and the binary alloy

12.5. Ising model in the zeroth approximation

12.6. Ising model in the first approximation

12.7. The critical exponents

12.8. Thermodynamic inequalities

12.9. Landau’s phenomenological theory

12.10. Scaling hypothesis for thermodynamic functions

12.11. The role of correlations and fluctuations

12.12. The critical exponents ν and n

12.13. A final look at the mean field theory

Problems

13. Phase Transitions: Exact (or Almost Exact) Results for Various Models

13.1. One-dimensional fluid models

13.2. The Ising model in one dimension

13.3. The n-vector models in one dimension

13.4. The Ising model in two dimensions

13.5. The spherical model in arbitrary dimensions

13.6. The ideal Bose gas in arbitrary dimensions

13.7. Other models

Problems

14. Phase Transitions: The Renormalization Group Approach

14.1. The conceptual basis of scaling

14.2. Some simple examples of renormalization

14.3. The renormalization group: general formulation

14.4. Applications of the renormalization group

14.5. Finite-size scaling

Problems

15. Fluctuations and Nonequilibrium Statistical Mechanics

15.1. Equilibrium thermodynamic fluctuations

15.2. The Einstein-Smoluchowski theory of the Brownian motion

15.3. The Langevin theory of the Brownian motion

15.4. Approach to equilibrium: the Fokker-Planck equation

15.5. Spectral analysis of fluctuations: the Wiener-Khintchine theorem

15.6. The fluctuation-dissipation theorem

15.7. The Onsager relations

Problems

16. Computer Simulations

16.1. Introduction and statistics

16.2. Monte Carlo simulations

16.3. Molecular dynamics

16.4. Particle simulations

16.5. Computer simulation caveats

Problems

Appendices

A. Influence of boundary conditions on the distribution of quantum states

B. Certain mathematical functions

C. “Volume” and “surface area” of an n-dimensional sphere of radius R

D. On Bose-Einstein functions

E. On Fermi-Dirac functions

F. A rigorous analysis of the ideal Bose gas and the onset of Bose-Einstein condensation

G. On Watson functions

H. Thermodynamic relationships

I. Pseudorandom numbers

Bibliography

Index