Introductory Statistical Mechanics for

This concise introduction to statistical mechanics is geared toward students and professionals concerned with any branch of solid state physics or low temperature physics. It presents the principles, along with examples showing their generality, with simplicity and clarity. It also reviews issues of critical interest to physicists. 1963 edition.

1 BASIC PRINCIPLES

1

(36)

1.1 Introduction

1

(4)

1.1.1 The Role of Thermodynamics

1

(3)

1.1.2 The Aim of Statistical Mechanics

4

(1)

1.2 The Boltzmann-Planck Equation

5

(6)

1.2.1 The Foundations by Boltzmann

5

(1)

1.2.2 The Planck Generalization

6

(1)

1.2.3 Planck's Analysis

7

(2)

1.2.4 The "Ergodic Theorem" and "Equally Probable States"

9

(2)

1.3 A Simple Model of a Solid

11

(6)

1.3.1 Paramagnetic Salt Model

12

(1)

1.3.2 The Entropy

12

(2)

1.3.3 Alteration of Entropy by Magnetic Fields

14

(1)

1.3.4 Introduction of Degeneracy

14

(3)

1.4 Distribution of Microscopic Complexions

17

(4)

1.5 Basic Results

21

(6)

1.5.1 The "Partition Function," or "Zustandsumme"

25

(1)

1.5.2 The "Internal Distribution" and Degeneracy

25

(2)

1.6 "Internal Excitation" of Solids—Schottky Anomaly

27

(7)

1.6.1 Dependence on Temperature and Volume

27

(2)

1.6.2 Agreement with the Third Law

29

(1)

1.6.3 Energy and Specific Heat (the "Schottky Anomaly")

29

(5)

1.7 Magnetic Susceptibility

34

(3)

2 THE VIBRATIONS OF SOLIDS

37

(32)

2.1 Normal Coordinates and Modes of Vibration

37

(21)

2.2 The Einstein Model for a Crystalline Solid

39

(6)

2.3 The Debye Model for the Lattice Vibrations of a Solid

45

(11)

2.4 Comment on the Analysis

56

(2)

2.5 Some Remarks on Anharmonicity

58

(3)

2.5.1 The Quasi-harmonic Model

58

(1)

2.5.2 The Gruneisen Parameter and Grüneisen's Equation

58

(3)

2.6 Electrical Resistivity of Metals

61

(8)

2.6.1 Dependence on Temperature and Volume

61

(4)

2.6.2 Change of Resistivity on Melting

65

(2)

2.6.3 Resistivity at Very Low Temperatures

67

(2)

3 GASLIKE ASSEMBLIES

69

(45)

3.1 Introduction

69

(2)

3.2 The Analysis of the Gaseous Model

71

(2)

3.3 The Real "Chemical" Gas

73

(5)

3.3.1 Partition Function for "Ideal Gases"

76

(2)

3.4 Bose-Einstein Statistics

78

(4)

3.4.1 Summary of Analytical Situation

80

(1)

3.4.2 The Classical Limit

81

(1)

3.5 Pauli Exclusion Principle and Spin

82

(2)

3.6 Fermi-Dirac Statistics

84

(14)

3.6.1 General Behavior of a Fermi-Dirac Assembly

84

(5)

3.6.2 Behavior of Idealized Fermi-Dirac Gas

89

(4)

3.6.3 General Remarks about Conduction Electrons

93

(5)

3.7 The Liquid Helium Problem

98

(10)

3.7.1 Liquid Helium as an Assembly of Localized Particles

98

(1)

3.7.2 Liquid Helium as a "Gaslike" Assembly

99

(1)

3.7.3 Liquid He4 as a Bose-Einstein "Gaslike" Model

99

(6)

3.7.4 Specific Comparison with Liquid He4

105

(2)

3.7.5 The Problem of Liquid He³

107

(1)

3.8 Electromagnetic Radiation

108

(6)

3.8.1 The "Oscillator Model"

110

(2)

3.8.2 The "Photon Model"

112

(2)

4 CLASSICAL STATISTICAL MECHANICS

114

(23)

4.1 The General Transition to Classical Mechanics

114

(6)

4.1.1 The Ideal Gas

117

(1)

4.1.2 Assembly of Classical Harmonic Oscillators

117

(2)

4.1.3 Physical Interpretation of Classical Statistical Mechanics

119

(1)

4.2 The Maxwell-Boltzmann Distribution

120

(2)

4.3 The Isothermal Equilibrium Atmosphere

122

(1)

4.4 Assembly of Anharmonic Oscillators

123

(4)

4.4.1 Anharmonic Correction to Lattice Specific Heat

126

(1)

4.5 The Problem of Irreversible Processes

127

(10)

4.5.1 The Basic Problem in Outline

127

(6)

4.5.2 Fluctuations and the "Ergodic Theorem"

133

(4)

Appendix I Basic Expressions for "Localizable" Elements

137

(4)

Appendix II Assembly of Localized Elements

141

(8)

Appendix III Bose-Einstein Statistics

149

(5)

Appendix IV Fermi-Dirac Statistics

154

(3)

Appendix V "Intermediate" Statistics

157

(5)

REFERENCES

162

(3)

A GLOSSARY OF SYMBOLS WITH THEIR SIGNIFICANCE

165

(6)

INDEX

171

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Introductory Statistical Mechanics for Physicists

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