Basics of Statistical Physics

271 Pages·2022·6.72 MB·English

by  Harald J .W. Müller-Kirsten

BASICS OF

STATISTICAL PHYSICS
Third Edition
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BASICS OF

STATISTICAL PHYSICS
Third Edition
Harald J. W. Müller-Kirsten
University of Kaiserslautern, Germany
World Scientific
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BASICS OF STATISTICAL PHYSICS
Third Edition
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Contents
Preface to Third Edition xi
Preface to Second Edition xiii
Preface to First Edition xv
1 Introduction 1
1.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Thermodynamic Potentials . . . . . . . . . . . . . . . . . . . 2
1.3 Capacity of Heat . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Frequently Used Terms . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Applications and Examples . . . . . . . . . . . . . . . . . . . 8
1.6 Problems without Worked Solutions . . . . . . . . . . . . . . 9
2 Statistical Mechanics of an Ideal Gas (Maxwell) 11
2.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Maxwell’s Treatment . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Lagrange’s Method of Multipliers . . . . . . . . . . . . . . . . 14
2.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4.1 Pressure exerted on the wall of a vessel . . . . . . . . 17
2.4.2 Effusion of gas through a hole . . . . . . . . . . . . . . 18
2.4.3 Thermionic emission . . . . . . . . . . . . . . . . . . . 19
2.5 Distribution Function for all Directions . . . . . . . . . . . . 20
2.6 Applications and Examples . . . . . . . . . . . . . . . . . . . 21
2.7 Problems without Worked Solutions . . . . . . . . . . . . . . 25
3 The a priori Probability 27
3.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . 27
3.2 The a priori Probability . . . . . . . . . . . . . . . . . . . . . 27
3.3 Examples Illustrating Liouville’s Theorem . . . . . . . . . . . 30
3.4 Insertion of Physical Conditions . . . . . . . . . . . . . . . . . 31
v
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3.5 Applications and Examples . . . . . . . . . . . . . . . . . . . 35
3.6 Problems without Worked Solutions . . . . . . . . . . . . . . 38
4 Classical Statistics (Maxwell–Boltzmann) 39
4.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . 39
4.2 The Number of Arrangements of Elements in
Maxwell–Boltzmann Statistics . . . . . . . . . . . . . . . . . . 39
4.3 Method of Maximum Probability . . . . . . . . . . . . . . . . 41
4.3.1 The case of nonconserved elements . . . . . . . . . . . 43
4.3.2 The case of conserved elements . . . . . . . . . . . . . 43
4.3.3 The meaning of μ . . . . . . . . . . . . . . . . . . . . 44
4.3.4 Identification of μ with 1/kT . . . . . . . . . . . . . . 45
4.3.5 Distribution of particles in the atmosphere . . . . . . 47
4.3.6 Law of equipartition of energy . . . . . . . . . . . . . 47
4.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.4.1 The monatomic gas . . . . . . . . . . . . . . . . . . . 48
4.4.2 A solid . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.5 Applications and Examples . . . . . . . . . . . . . . . . . . . 49
4.6 Problems without Worked Solutions . . . . . . . . . . . . . . 54
5 Entropy 57
5.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . 57
5.2 The Boltzmann Formula . . . . . . . . . . . . . . . . . . . . . 57
5.3 Applications and Examples . . . . . . . . . . . . . . . . . . . 61
5.4 Problems without Worked Solutions . . . . . . . . . . . . . . 66
6 Quantum Statistics 69
6.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . 69
6.2 A priori Weighting in Quantum Statistics . . . . . . . . . . . 70
6.2.1 Approximate calculation of number of states . . . . . 73
6.2.2 Accurate calculation of number of states . . . . . . . . 74
6.2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.3 The Allowed Number of Elements in Quantum States . . . . 80
6.3.1 One element . . . . . . . . . . . . . . . . . . . . . . . 80
6.3.2 Two non-interacting elements . . . . . . . . . . . . . . 80
6.3.3 More than two elements stuck together . . . . . . . . 82
6.4 Counting of Number of Arrangements . . . . . . . . . . . . . 82
6.4.1 Fermi–Dirac statistics . . . . . . . . . . . . . . . . . . 82
6.4.2 Bose–Einstein statistics . . . . . . . . . . . . . . . . . 84
6.5 Quantum Statistics at High Temperatures . . . . . . . . . . . 84
6.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
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Contents vii
6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.8 Applications and Examples . . . . . . . . . . . . . . . . . . . 88
6.9 Problems without Worked Solutions . . . . . . . . . . . . . . 95
7 Exact Form of Distribution Functions 97
7.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . 97
7.2 Fermi–Dirac Occupation Numbers . . . . . . . . . . . . . . . 97
7.3 Bose–Einstein Occupation Numbers . . . . . . . . . . . . . . 99
7.4 Thermodynamical Functions . . . . . . . . . . . . . . . . . . . 102
7.5 Applications and Examples . . . . . . . . . . . . . . . . . . . 102
7.6 Problems without Worked Solutions . . . . . . . . . . . . . . 108
8 Application to Radiation (Light Quanta) 109
8.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . 109
8.2 Planck’s Radiation Law . . . . . . . . . . . . . . . . . . . . . 110
8.3 Black Body Thermal Radiation . . . . . . . . . . . . . . . . . 113
8.4 Applications and Examples . . . . . . . . . . . . . . . . . . . 114
8.5 Problems without Worked Solutions . . . . . . . . . . . . . . 119
9 Debye Theory of Specific Heat of Solids 123
9.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . 123
9.2 The Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 125
9.3 Applications and Examples . . . . . . . . . . . . . . . . . . . 128
9.4 Problems without Worked Solutions . . . . . . . . . . . . . . 133
10 Electrons in Metals 135
10.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . 135
10.2 Evaluation of the Distribution Function . . . . . . . . . . . . 136
10.2.1 First approximation . . . . . . . . . . . . . . . . . . . 137
10.2.2 Second degree of approximation . . . . . . . . . . . . . 138
10.3 Applications and Examples . . . . . . . . . . . . . . . . . . . 140
10.4 Problems without Worked Solutions . . . . . . . . . . . . . . 151
11 Limitations of the Preceding Theory — Improvement
with Ensemble Method 155
11.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . 155
11.2 Ensembles — Three Types . . . . . . . . . . . . . . . . . . . 158
11.2.1 Ensembles and ergodic hypothesis . . . . . . . . . . . 158
11.2.2 The ensemble distribution function . . . . . . . . . . . 160
11.3 The Canonical Ensemble of a Closed System . . . . . . . . . 162
11.3.1 Thermodynamics of a closed system in a heat bath . . 163
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viii Contents
11.4 The Grand Canonical Ensemble . . . . . . . . . . . . . . . . . 165
11.5 Ensemble Method of Maximum Probability . . . . . . . . . . 166
11.6 Comments on the Function ρ . . . . . . . . . . . . . . . . . . 167
11.7 Applications and Examples . . . . . . . . . . . . . . . . . . . 169
11.8 Problems without Worked Solutions . . . . . . . . . . . . . . 174
12 Averaging instead of Maximization, and Bose–Einstein
Condensation 177
12.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . 177
12.2 The Darwin–Fowler Method of Mean Values . . . . . . . . . . 178
12.2.1 Mean occupation number n . . . . . . . . . . . . . . . 179
j
12.2.2 Taking subsidiary condition into account. . . . . . . . 180
12.3 Classical Statistics . . . . . . . . . . . . . . . . . . . . . . . . 181
12.4 Quantum Statistics . . . . . . . . . . . . . . . . . . . . . . . . 183
12.4.1 Fermi–Dirac statistics . . . . . . . . . . . . . . . . . . 183
12.4.2 Bose–Einstein statistics . . . . . . . . . . . . . . . . . 184
12.4.3 Evaluation of the coefficient of ωN in Z . . . . . . . 184
ω
12.5 Bose–Einstein Condensation . . . . . . . . . . . . . . . . . . . 190
12.5.1 The phenomenon of Bose–Einstein condensation . . . 190
12.5.2 Derivation of the Bose–Einstein distribution function
under condensation conditions . . . . . . . . . . . . . 191
12.6 Applications and Examples . . . . . . . . . . . . . . . . . . . 201
12.7 Problems without Worked Solutions . . . . . . . . . . . . . . 212
13 The Boltzmann Transport Equation 215
13.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . 215
13.2 Distribution Functions . . . . . . . . . . . . . . . . . . . . . . 215
13.3 Solution of the Boltzmann Equation . . . . . . . . . . . . . . 217
13.3.1 Solving the Boltzmann equation for two typical cases . 218
13.3.2 Calculation of the current density . . . . . . . . . . . 219
13.3.3 Application to metals . . . . . . . . . . . . . . . . . . 220
13.3.4 Calculation of the relaxation time . . . . . . . . . . . 222
13.4 Applications and Examples . . . . . . . . . . . . . . . . . . . 228
13.5 Problems without Worked Solutions . . . . . . . . . . . . . . 228
14 Thermal Radiation of Black Holes 231
14.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . 231
14.2 Background Geometry . . . . . . . . . . . . . . . . . . . . . . 231
14.3 Rindler Coordinates . . . . . . . . . . . . . . . . . . . . . . . 232
14.4 Introduction of Fields . . . . . . . . . . . . . . . . . . . . . . 234
14.5 Thermalization . . . . . . . . . . . . . . . . . . . . . . . . . . 236
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Contents ix
14.6 Black Hole Evaporation . . . . . . . . . . . . . . . . . . . . . 238
14.7 Applications and Examples . . . . . . . . . . . . . . . . . . . 239
14.8 Problems without Worked Solutions . . . . . . . . . . . . . . 240
Bibliography 243
Index 249