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Statistical Physics

Advanced Texts in Physics

This program of advanced texts covers a broad spectrum of topics that are of current and emerging interest in physics. Each book provides a comprehensive and yet accessible introduction to a field at the forefront of modern research. As such, these texts are intended for senior undergraduate and graduate students at the M.S. and Ph.D. levels; however, research scientists seeking an introduction to particular areas of physics will also benefit from the titles in this collection.

Claudine Hermann

Statistical Physics

Including Applications to Condensed Matter

With 63 Figures

Claudine Hermann

Laboratoire de Physique de la Matie`re Condense´e Ecole Polytechnique

91128 Palaiseau France

claudine.hermann@polytechnique.edu

Library of Congress Cataloging-in-Publication Data is available.

ISBN 0-387-22660-5 Printed on acid-free paper.

2005 Springer Science+Business Media, Inc.

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.

The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.

Printed in the United States of America.

(MPY)

9 8 7 6 5 4 3 2 1

SPIN 10955284

 

springeronline.com

 

 

Table of Contents

Introduction

xi

Glossary

xv

1 Statistical Description of Large Systems. Postulates

1

1.1Classical or Quantum Evolution of a Particle; Phase Space . . 2 1.1.1 Classical Evolution . . . . . . . . . . . . . . . . . . . . . 2

1.1.2 Quantum Evolution . . . . . . . . . . . . . . . . . . . . 4

1.1.3Uncertainty Principle and Phase Space . . . . . . . . . . 4

1.1.4Other Degrees of Freedom . . . . . . . . . . . . . . . . . 5

1.2Classical Probability Density; Quantum Density Operator . . . 5

1.2.1Statistical Approach for Macroscopic Systems . . . . . . 6

1.2.2Classical Probability Density . . . . . . . . . . . . . . . 8

1.2.3Density Operator in Quantum Mechanics . . . . . . . . 9

1.3Statistical Postulates; Equiprobability . . . . . . . . . . . . . . 10

1.3.1

Microstate, Macrostate . . . . . . . . . . . . . . . . . .

10

1.3.2

Time Average and Ensemble Average . . . . . . . . . .

11

1.3.3Equiprobability . . . . . . . . . . . . . . . . . . . . . . . 12

1.4 General Properties of the Statistical Entropy . . . . . . . . . .

15

1.4.1 The Boltzmann Definition . . . . . . . . . . . . . . . .

15

1.4.2The Gibbs Definition . . . . . . . . . . . . . . . . . . . . 15

1.4.3 The Shannon Definition of Information . . . . . . . . .

18

Summary of Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . .

19

Appendix 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

Appendix 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

Appendix 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

2 The Di erent Statistical Ensembles. General Methods

31

2.1Energy States of an N -Particle System . . . . . . . . . . . . . . 32

2.2Isolated System in Equilibrium : “Microcanonical Ensemble” . 35

2.3Equilibrium Conditions for Two Systems in Contact . . . . . . 36

2.3.1Equilibrium Condition : Equal β Parameters . . . . . . 36

2.3.2Fluctuations of the Energy Around its Most Likely Value 37

v

vi

Table of Contents

2.4Contact with a Heat Reservoir, “Canonical Ensemble” . . . . . 40

2.4.1The Boltzmann Factor . . . . . . . . . . . . . . . . . . . 41

2.4.2Energy, with Fixed Average Value . . . . . . . . . . . . 42

2.4.3Partition Function Z . . . . . . . . . . . . . . . . . . . . 44

2.4.4 Entropy in the Canonical Ensemble . . . . . . . . . . . 45

2.4.5Partition Function of a Set of Two Independent Systems 46

2.5 Grand Canonical Ensemble . . . . . . . . . . . . . . . . . . . .

48

2.5.1 Equilibrium Condition : Equality of Both T ’s and µ’s .

49

2.5.2Heat Reservoir and Particles Reservoir . . . . . . . . . . 50

2.5.3Grand Canonical Probability and Partition Function . . 51

2.5.4Average Values . . . . . . . . . . . . . . . . . . . . . . . 52

2.5.5Grand Canonical Entropy . . . . . . . . . . . . . . . . . 53

2.6Other Statistical Ensembles . . . . . . . . . . . . . . . . . . . . 53

Summary of Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . .

55

Appendix 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

3 Thermodynamics and Statistical Physics

59

3.1Zeroth Law of Thermodynamics . . . . . . . . . . . . . . . . . . 60

3.2First Law of Thermodynamics . . . . . . . . . . . . . . . . . . . 60

3.2.1 Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.2.2Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.2.3Quasi-Static General Process . . . . . . . . . . . . . . . 63

3.3 Second Law of Thermodynamics . . . . . . . . . . . . . . . . . 64

3.4Third Law of Thermodynamics . . . . . . . . . . . . . . . . . . 66

3.5The Thermodynamical Potentials; the Legendre Transformation 67

3.5.1Isolated System . . . . . . . . . . . . . . . . . . . . . . . 67

3.5.2Fixed N , Contact with a Heat Reservoir at T . . . . . . 69

3.5.3Contact with a Heat and Particle Reservoir at T . . . . 70

3.5.4Transformation of Legendre; Other Potentials . . . . . . 72 Summary of Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4 The Ideal Gas

79

4.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.2Kinetic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.2.1Scattering Cross Section, Mean Free Path . . . . . . . . 80

4.2.2Kinetic Calculation of the Pressure . . . . . . . . . . . . 81

4.3Classical or Quantum Statistics? . . . . . . . . . . . . . . . . . 83

4.4 Classical Statistics Treatment of the Ideal Gas . . . . . . . . . 85

4.4.1Calculation of the Canonical Partition Function . . . . . 85

4.4.2Average Energy; Equipartition Theorem . . . . . . . . . 87

4.4.3 Free Energy; Physical Parameters (P, S, µ) . . . . . . . 89

4.4.4Gibbs Paradox . . . . . . . . . . . . . . . . . . . . . . . 90

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

Summary of Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . .

93

Appendix 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

Table of Contents

vii

Appendix 4.2 . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . 103

5 Indistinguishability, the Pauli Principle

113

5.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.2States of Two Indistinguishable Particles . . . . . . . . . . . . . 116 5.2.1 General Case . . . . . . . . . . . . . . . . . . . . . . . . 116

5.2.2 Independent Particles . . . . . . . .

. . . .

. . . . .

.

.

116

5.3 Pauli Principle; Spin-Statistics Connection

. . . .

. . . . .

.

.

119

5.3.1Pauli Principle; Pauli Exclusion Principle . . . . . . . . 119

5.3.2Theorem of Spin-Statistics Connection . . . . . . . . . . 120

5.4Case of Two Particles of Spin 1/2 . . . . . . . . . . . . . . . . . 121

5.4.1Triplet and Singlet Spin States . . . . . . . . . . . . . . 121

5.4.2Wave Function of Two Spin 1/2 Particles . . . . . . . . 123

5.5Special Case of N Independent Particles . . . . . . . . . . . . . 124

5.5.1Wave Function . . . . . . . . . . . . . . . . . . . . . . . 124

5.5.2Occupation Numbers . . . . . . . . . . . . . . . . . . . . 125

5.6 Return to the Introduction Examples . . . . . . . . . . . . . . 126

5.6.1

Fermions Properties . . . . . . . . . . . . . . . . . . . .

126

5.6.2

Bosons Properties . . . . . . . . . . . . . . . . . . . . .

126

Summary of Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6 General Properties of the Quantum Statistics

131

6.1Use of the Grand Canonical Ensemble . . . . . . . . . . . . . . 132

6.1.12 Indistinguishable Particles at T , Canonical Ensemble . 132

6.1.2Description in the Grand Canonical Ensemble . . . . . . 133

6.2Factorization of the Grand PartitionFunction . . . . . . . . . . 134

6.2.1Fermions and Bosons . . . . . . . . . . . . . . . . . . . . 134

6.2.2 Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.2.3Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.2.4Chemical Potential and Number of Particles . . . . . . . 136

6.3Average Occupation Number;Grand Potential . . . . . . . . . . 136

6.4Free Particle in a Box; Density of States . . . . . . . . . . . . . 138

6.4.1Quantum States of a Free Particle in a Box . . . . . . . 138

6.4.2Density of States . . . . . . . . . . . . . . . . . . . . . . 143

6.5Fermi-Dirac Distribution; Bose-Einstein Distribution . . . . . . 147

6.6Average Values of Physical Parameters at T . . . . . . . . . . . 148

6.7Common Limit of the Quantum Statistics . . . . . . . . . . . . 149

6.7.1 Chemical Potential of the Ideal Gas

. . . . . . . . . . .

149

6.7.2 Grand Canonical Partition Function of the Ideal Gas . .

151

Summary of Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . .

153

7 Free Fermions Properties

 

155

7.1Properties of Fermions at Zero Temperature . . . . . . . . . . . 155

7.1.1Fermi Distribution, Fermi Energy . . . . . . . . . . . . . 155

7.1.2Internal Energy and Pressure at Zero Temperature . . . 158

viii

Table of Contents

7.1.3Magnetic Properties. Pauli Paramagnetism . . . . . . . 159

7.2Properties of Fermions at Non-Zero Temperature . . . . . . . . 160

7.2.1Temperature Ranges and Chemical Potential Variation . 160

7.2.2Specific Heat of Fermions . . . . . . . . . . . . . . . . . 163

7.2.3Thermionic Emission . . . . . . . . . . . . . . . . . . . . 165 Summary of Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Appendix 7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

8 Elements of Bands Theory and Crystal Conductivity

175

8.1What is a Solid, a Crystal? . . . . . . . . . . . . . . . . . . . . 176

8.2The Eigenstates for the Chosen Model . . . . . . . . . . . . . . 177

8.2.1

Recall : the Double Potential Well . . . .

. .

. .

. . .

.

177

8.2.2

Electron on an Infinite and Periodic Chain

.

. .

. . .

.

180

8.2.3Energy Bands and Bloch Functions . . . . . . . . . . . . 182

8.3 The Electron States in a Crystal . .

. . . .

.

. .

. .

. . . .

.

.

184

8.3.1 Wave Packet of Bloch Waves

. . . .

.

. .

. .

. . . .

.

.

184

8.3.2Resistance; Mean Free Path . . . . . . . . . . . . . . . . 184

8.3.3Finite Chain, Density of States, E ective Mass . . . . . 185

8.4 Statistical Physics of Solids

. . . . . . .

. .

. . .

. .

. .

. .

.

.

189

8.4.1 Filling of the Levels

. . . . . . .

. .

. . .

. .

. .

. .

.

.

189

8.4.2Variation of Metal Resistance versus T . . . . . . . . . . 190

8.4.3Insulators’ Conductivity Versus T ; Semiconductors . . . 191

8.5 Examples of Semiconductor Devices . . . . . . . . . . . . . . . 196

8.5.1The Photocopier : Photoconductivity Properties . . . . 196

8.5.2

The Solar Cell : an Illuminated p − n Junction

. . . . . 196

8.5.3

CD Readers : the Semiconductor Quantum Wells . . . . 197

Summary of Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . 199

9 Bosons : Helium 4, Photons,Thermal Radiation

201

9.1Material Particles . . . . . . . . . . . . . . . . . . . . . . . . . . 201

9.1.1Thermodynamics of the Boson Gas . . . . . . . . . . . . 202

9.1.2Bose-Einstein Condensation . . . . . . . . . . . . . . . . 203

9.2Bose-Einstein Distribution of Photons . . . . . . . . . . . . . . 206

9.2.1 Description of the Thermal Radiation; the Photons . . 206

9.2.2Statistics of Photons, Bosons in Non-Conserved Number 207

9.2.3Black Body Definition and Spectrum . . . . . . . . . . . 210

9.2.4Microscopic Interpretation . . . . . . . . . . . . . . . . . 212

9.2.5Photometric Measurements : Definitions . . . . . . . . . 214

9.2.6Radiative Balances . . . . . . . . . . . . . . . . . . . . . 217

9.2.7 Greenhouse E ect .

. .

. . . . . . . . . . . . . . . .

.

. 218

Summary of Chapter 9 . . . . . .

. .

. . . . . . . . . . . . . . . .

.

. 221

Solving Exercises and Problems of Statistical Physics

223

Units and Physical Constants

225

Table of Contents

ix

A few useful formulae

229

Exercises and Problems

231

Ex. 2000 : Electrostatic Screening . . . . . . . . . . . . . . . . . . . . 233

Ex. 2001 : Magnetic Susceptibility of a “Quasi-1D” Conductor

. . . 234

Ex. 2002 : Entropies of the HC Molecule . . . . . . . . . . .

. . . . 236

Pr.

2001

: Quantum Boxes and Optoelectronics . . . . . . . .

. . . . 237

Pr.

2002

: Physical Foundations of Spintronics . . . . . . . .

. . . . 245

Solution of the Exercises and Problems

251

Ex. 2000 : Electrostatic Screening . . . . . . . . . . . . . . . . . . . . 253

Ex. 2001 : Magnetic Susceptibility of a “Quasi-1D” Conductor

. . . 255

Ex. 2002 : Entropies of the HC Molecule . . . . . . . . . . .

. . . . 258

Pr.

2001

: Quantum Boxes and Optoelectronics . . . . . . . .

. . . . 259

Pr.

2002

: Physical Foundations of Spintronics . . . . . . . .

. . . . 266

Index

 

 

275

Introduction

A glosssary at the end of this introduction defines the terms specific to Statistical Physics. In the text, these terms are marked by an asterisk in exponent.

A course of Quantum Mechanics, like the one taught at Ecole Polytechnique, is devoted to the description of the state of an individual particle, or possibly of a few ones. Conversely, the topic of this book will be the study of systems containing very many particles, of the order of the Avogadro number N, for example the molecules in a gas, the components of a chemical reaction, the adsorption sites for a gas on a surface, the electrons of a solid. You certainly previously studied this type of system, using Thermodynamics which is ruled by “exact” laws, such as the ideal gas one. Its physical parameters, that can be measured in experiments, are macroscopic quantities like its pressure, volume, temperature, magnetization, etc.

It is now well-known that the correct microscopic description of the state of a system, or of its evolution, requires the Quantum Mechanics approach and the solution of the Schroedinger equation, but how can this equation be solved when such a huge number of particles comes into play? Printing on a listing the positions and velocities of the N molecules of a gas would take a time much longer that the one elapsed since the Big Bang! A statistical description is the only issue, which is the more justified as the studied system is larger, since the relative fluctuations are then very small (Ch. 1).

This course is restricted to systems in thermal equilibrium : to reach such an equilibrium it is mandatory that interaction terms should be present in the hamiltonian of the total system : even if they are weak, they allow energy exchanges between the system and its environment (for example one may be concerned by the electrons of a solid, the environment being the ions of the same solid). These interactions provide the way to equilibrium. This approach takes place during a characteristic time, the so-called “relaxation time”, with a range of values which depends on the considered system (the study of o - equilibrium phenomena, in particular transport phenomena, is another branch

xi