# Methods of Statistical Physics

### International Series in Natural Philosophy

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Methods of Statistical Physics is an exposition of the tools of statistical mechanics, which evaluates the kinetic equations of classical and quantized systems. The book also analyzes the equations of macroscopic physics, such as the equations of hydrodynamics for normal and superfluid liquids and macroscopic electrodynamics. The text gives particular attention to the study of quantum systems. This study begins with a discussion of problems of quantum statistics with a detailed description of the basics of quantum mechanics along with the theory of measurement. An analysis of the asymptotic behavior of universal quantities is also explained. Strong consideration is given to the systems with spontaneously broken system. Theories such as the kinetic theory of gases, the theory of Brownian motion, the theory of the slowing down of neutrons, and the theory of transport phenomena in crystals are discussed. The book will be a useful tool for physicists, mathematicians, students, and researchers in the field of statistical mechanics.

## Table of Contents

Chapter 1 - Kinetic Equations for Classical Systems

1.1. Many-Particle Distribution Functions

1.1.1. Boltzmann's Kinetic Equation

1.1.2. Probability Density of Phase Points

1.1.3. Equations for Many - Particle Distribution Functions

1.2. Integral Equations for Many - Particle Distribution Functions

1.2.1. Integral Equations for Distribution Functions at the Kinetic Stage of Evolution

1.2.2. Construction of a Perturbation Theory for Systems with a Low Particle Density

1.3. Kinetic Equations and Transport Phenomena in Gases

1.3.1. Kinetic Equation in the Case of Weak Interactions

1.3.2. Kinetic Equation in the Low Density Case

1.3.3. Theory of Transport Phenomena in Gases

1.4. Kinetic Equations for Particles Interacting with a Medium

1.4.1. The Fokker-Planck Equation for Slow Processes

1.4.2. The Theory of Brownian Movement

1.4.3. The Theory of Neutron Moderation

1.5. Statistical Mechanics of a System of Charged Particles

1.5.1. A Kinetic Equation for Electrons in a Plasma

1.5.2. Theory of Screening

1.5.3. Dispersion Equation for Waves in Plasmas

1.6. Irreversibility of Macroscopic Processes and the Ergodic Hypothesis

1.6.1. Reversibility of Mechanical Motions and Irreversibility of Macroscopic Processes

1.6.2. The Ergodic Hypothesis

Chapter 2 - General Principles of the Statistical Mechanics of Quantum Systems

2.1. Principles of Quantum Mechanics

2.1.1. Pure States and Mixed States

2.1.2. The Dynamic Law of Quantum Mechanics

2.1.3. The Measuring Process

2.2. Second Quantization

2.2.1. Particle Creation and Annihilation Operators

2.2.2. Operators of Physical Quantities

2.3. Symmetry of Equations of Quantum Mechanics

2.3.1. Invariance of Equations of Quantum Mechanics Relative to Continuous Transformations

2.3.2. Invariance of Equations of Quantum Mechanics under Spatial Reflection and Time Reversal

2.4. The Principle of Attenuation of Correlations and Ergodic Relations for Quantum Systems

2.4.1. The Principle of Attenuation of Correlations

2.4.2. Equations of Motion

2.4.3. Ergodic Relations for Quantum Systems

Chapter 3 - Theory of Equilibrium States of Quantum Systems

3.1.Theory of Weakly Non-Ideal Quantum Gases

3.1.1. The Bose-Einstein and Fermi-Dirac Distributions

3.1.2. Thermodynamic Perturbation Theory

3.1.3. Quantum Virial Expansions

3.2. Superfluidity of a Gas of Bosons or Fermions

3.2.1. Quasi-Averages

3.2.2. Theory of Superfluidity of a Bose Gas

3.2.3. Theory of Superfluidity of a Fermi Gas and the Phenomenon of Superconductivity

Chapter 4 - Methods of Investigating Non-Equilibrium States of Quantum Systems

4.1. The Reaction of a System to an External Perturbation

4.1.1. The Statistical Operator of a System Located in a Weak External Field

4.1.2. Properties of the Green Functions

4.2. General Theory of Relaxation Processes

4.2.1. An Integral Equation for the Statistical Operator in the Case of Weak Interactions

4.2.2. An Integral Equation for the Statistical Operator in the Case of Small Inhomogeneities

4.2.3. An Integral Equation for the Statistical Operator of In homogeneous Systems with Weak Interactions

4.3. Summation of Secular Terms

4.3.1. Asymptotic Operators

4.3.2. A Functional Equation for the Asymptotic Operators

4.3.3. Summation of Secular Terms and the Coarse-Grained Statistical Operators

4.4. The Low-Frequency Asymptotics of the Green Functions

4.4.1. Linearization of the Equations for the Course-Grained Statistical Operator

4.4.2. The Asymptotics of the Green Functions in the Region of Low Frequencies and Small Wavevectors

Chapter 5 - Kinetic Equations for Quantum Systems

5.1. Kinetic Equations in the Case of Weak Interactions

5.1.1. The Kinetic Stage of the Evolution

5.1.2. Kinetic Equations for Boson and Fermion Gases in the Second Approximation of Perturbation Theory

5.1.3. Zero Sound

5.1.4. The Collision Integral in the Third Approximation of Perturbation Theory

5.1.5. Entropy of a Weakly Non-Ideal Gas

5.2. Kinetic Equations Taking Pair Collisions into Account

5.2.1. The Statistical Operator in the Kinetic Stage of the Evolution

5.2.2. Expansion of the Collision Integral in a Power Series of the Single-Particle Distribution Function

5.2.3. Quantum Virial Expansion of the Collision Integral

5.3. Kinetic Equations for Particles and Radiation Interacting with an External Medium

5.3.1. A Kinetic Equation for Particles Interacting with a Medium

5.3.2. A Kinetic Equation for Photons in a Medium

5.4. Kinetic Equations for Particles in an External Field and Green Functions in the Kinetic Approximation

5.4.1. An Integral Equation for the Statistical Operator

5.4.2. Kinetic Equations for Charged Particles in an Electromagnetic Field

5.4.3. The Connection between the Low-Frequency Asymptotic Behavior of Green Functions and Kinetic Equations

5.4.4. Integral Equations for Determining the Low-Frequency Asymptotic Behavior of the Green Functions of Degenerate Bose-Systems

5.4.5. Green Functions in the Case of Weak Interactions between Quasi-Particles

5.4.6.Single-Particle Green Functions of Degenerate Bose-Systems in the Kinetic Approximation

5.5. The Collision Integral for Phonons and the Theory of the Thermal Conductivity of Dielectrics

Chapter 6 - Equations of Macroscopic Physics

6.1. Hydrodynamic Equations for a Normal Liquid

6.1.1. Hydrodynamic Equations for an Ideal Liquid

6.1.2. Equations of Hydrodynamics for a Non-Ideal Liquid

6.1.3. Low-Frequency Asymptotic Behavior of the Hydrodynamic Green Functions

6.2. Hydrodynamic Equations of a Superfluid Liquid

6.2.1. Fluxes of Hydrodynamic Quantities

6.2.2. Equations of Motion for the Statistical Operator of a Superfluid Liquid

6.2.3. Hydrodynamic Equations for an Ideal Superfluid Liquid

6.3. Macroscopic Electrodynamic Equations

6.3.1. The Maxwell-Lorentz Equations for the Electromagnetic Field Operators

6.3.2. Response of a System to an External Electromagnetic Perturbation

6.3.3. The Connection between External and Internal Susceptibilities

Bibliography

Index

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## Product details

- No. of pages: 446
- Language: English
- Copyright: © Pergamon 1981
- Published: January 1, 1981
- Imprint: Pergamon
- eBook ISBN: 9781483189376

## About the Authors

### A. I. Akhiezer

### S. V. Peletminskii

## About the Editor

### D. ter Haar

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