Statistical Mechanics For Beginners: A Textbook For Undergraduates

by Lucien Gilles Benguigui

Overview

This textbook is for undergraduate students on a basic course in Statistical Mechanics. The prerequisite is thermodynamics. It begins with a study of three situations — the closed system and the systems in thermal contact with a reservoir — in order to formulate the important fundamentals: entropy from Boltzmann formula, partition function and grand partition function. Through the presentation of quantum statistics, Bose statistics and Fermi-Dirac statistics are established, including as a special case the classical situation of Maxell-Boltzmann statistics. A series of examples ensue it: the harmonic oscillator, the polymer chain, the two level system, bosons (photons, phonons, and the Bose-Einstein condensation) and fermions (electrons in metals and in semiconductors). A compact historical note on influential scientists forms the concluding chapter.The unique presentation starts off with the principles, elucidating the well-developed theory, and only thereafter the application of theory. Calculations on the main steps are detailed, leaving behind minimal gap. The author emphasizes with theory the link between the macroscopic world (thermodynamics) and the microscopic world.

Product Details

ISBN-13:	9789814299121
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	08/20/2010
Pages:	180
Product dimensions:	6.00(w) x 8.90(h) x 0.50(d)

Table of Contents

Dedication v

Preface vii

List of Figures xiii

Physical Constants xv

Introduction 1

I Fundamentals 7

1 The Closed System or the Microcanonical Ensemble 9

1.1 The Microcanonical Ensemble 9

1.2 Properties of the Entropy 12

1.3 An Example 17

2 The System in Thermal Contact with a Reservoir: The Canonical and Grand Canonical Ensembles 22

2.1 The Canonical Ensemble 23

2.1.1 The partition function 23

2.1.2 Energy, entropy and thermodynamic potential 26

2.1.3 A two-level system 29

2.1.4 The ideal gas; equipartition of energy in classical mechanics 30

2.2 The Grand Canonical Ensemble 33

2.2.1 The grand partition function 34

2.2.2 The number of particles, energy, entropy and the grand potential 36

2.2.3 An example 38

2.3 Summary 41

2.3.1 Fluctuations 42

2.3.2 Final remark 43

3 Quantum Statistics 44

3.1 The Partition Function and the Free Energy 45

3.1.1 For variable N and for fermions 46

3.1.2 For variable N and for bosons 46

3.1.3 For fixed N, and for both fermions and bosons 47

3.2 The Energy and the Entropy 49

3.3 The Classical Ideal Gas: Maxwell-Boltzmann Statistics 50

3.4 Qualitative Behavior of the Chemical Potential and the Derivation of (∂μ/∂T)_V,N <0 53

3.4.1 Bosons 53

3.4.2 Fermions 54

4 The Density of States 55

4.1 The Wave Vector 56

4.2 The Density of States 57

4.3 The Monatomic Ideal Gas 58

4.3.1 The partition function 58

4.3.2 The internal energy, entropy and equation of state 59

4.3.3 The classical limit 60

5 Some Problems 61

5.1 The Quantum Harmonic Oscillator 61

5.1.1 Low temperature limit 64

5.1.2 High temperature limit 65

5.2 The Polyatomic Ideal Gas 66

5.3 Bosons and Fermions in a Two-Level System 68

5.3.1 The particles are bosons 69

5.3.2 The particles are fermions 72

5.3.3 Classical particles 73

5.4 The Magnetic Chain 75

II Applications 79

6 The Gas of Photons: The Black Body Radiation 81

6.1 The Energy and the Energy Spectrum 83

6.2 The Free Energy and the Entropy 87

6.2.1 The relation with the wave picture 88

6.3 Light Emission and Absorption of Solids; Kirchhoff's Law 88

6.4 The Black Body Emission 90

6.5 The Properties of Photon Gas are Independent of the Shape and the Material of the Cavity 90

7 Atomic Vibration in Solids: Phonons 92

7.1 Atomic Vibration in Solids 92

7.2 The Properties of Phonons 97

7.3 The Low Temperature Case 97

7.4 The High Temperature Case 99

7.5 The Debye Formula 100

7.6 Resolution of the Differential Equation (7.1) by Means of Trigonometric Functions 104

7.7 Derivation of the Expression (7.30) Giving C_v in the Debye Model 105

8 The Boson Gas at Low Temperature: The Bose-Einstein Condensation 107

8.1 The Chemical Potential 108

8.2 The Energy, Specific Heat, Free Energy and Entropy 112

8.3 Experimental Verfication 114

9 The Gas of Fermions: Electrons in Metals and in Semiconductors 116

9.1 Free Electrons in a Box 116

9.1.1 The Fermi-Dirac function 116

9.1.2 The chemical potential or the Fermi level 120

9.1.3 The energy 123

9.1.4 The specific heat 127

9.1.5 Applications to metals 129

9.2 Electrons in Semiconductors 131

10 A History of Statistical Mechanics 136

10.1 Thermodynamics and Statistical Mechanics Before Maxwell and Bolztmann 136

10.2 The Kinetic Theory of Maxwell 138

10.3 Boltzmann and Irreversibility 139

10.4 Gibbs, the Father of Statistical Mechanics 141

10.5 Planck and Einstein: Quantum Theory and Statistics 142

10.6 The Method of Bose and the Bose-Einstein Condensation 145

10.7 The Principle of Pauli and the Statistics of Fermi and Dirac 146

10.8 Modern Developments 147

Exercises 148

Index 161