Statistical physics (2005)
.pdfStatistical 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.
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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
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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 . . . . . . . . . . . . . . . . . . |
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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 . . . . . . . . |
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116 |
5.3 Pauli Principle; Spin-Statistics Connection |
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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 |
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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 |
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149 |
6.7.2 Grand Canonical Partition Function of the Ideal Gas . . |
151 |
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Summary of Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . |
153 |
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7 Free Fermions Properties |
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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 . . . . |
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177 |
8.2.2 |
Electron on an Infinite and Periodic Chain |
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180 |
8.2.3Energy Bands and Bloch Functions . . . . . . . . . . . . 182
8.3 The Electron States in a Crystal . . |
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184 |
8.3.1 Wave Packet of Bloch Waves |
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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 |
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189 |
8.4.1 Filling of the Levels |
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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 |
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8.5.3 |
CD Readers : the Semiconductor Quantum Wells . . . . 197 |
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Summary of Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . 199 |
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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 . |
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. 218 |
Summary of Chapter 9 . . . . . . |
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. 221 |
Solving Exercises and Problems of Statistical Physics |
223 |
Units and Physical Constants |
225 |
Table of Contents |
ix |
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A few useful formulae |
229 |
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Exercises and Problems |
231 |
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Ex. 2000 : Electrostatic Screening . . . . . . . . . . . . . . . . . . . . 233 |
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Ex. 2001 : Magnetic Susceptibility of a “Quasi-1D” Conductor |
. . . 234 |
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Ex. 2002 : Entropies of the HC Molecule . . . . . . . . . . . |
. . . . 236 |
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Pr. |
2001 |
: Quantum Boxes and Optoelectronics . . . . . . . . |
. . . . 237 |
Pr. |
2002 |
: Physical Foundations of Spintronics . . . . . . . . |
. . . . 245 |
Solution of the Exercises and Problems |
251 |
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Ex. 2000 : Electrostatic Screening . . . . . . . . . . . . . . . . . . . . 253 |
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Ex. 2001 : Magnetic Susceptibility of a “Quasi-1D” Conductor |
. . . 255 |
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Ex. 2002 : Entropies of the HC Molecule . . . . . . . . . . . |
. . . . 258 |
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Pr. |
2001 |
: Quantum Boxes and Optoelectronics . . . . . . . . |
. . . . 259 |
Pr. |
2002 |
: Physical Foundations of Spintronics . . . . . . . . |
. . . . 266 |
Index |
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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