Basic Statistical Physics Book

Preface vii

Notations and Fundamental Constants xi

1 Basic Concepts 1

1.1 Introduction 1

1.2 Master Equation and Hypothesis of Equal á-priori Probability 3

1.2.1 Example of 3 Level Systems 4

1.3 Phase Space, Phase Point, Phase Trajectory 4

1.4 Statistical Distribution Function and Ergodic Hypothesis 5

1.5 Statistical Fluctuation and Statistical Independence 7

1.6 Statistical Fluctuation and Generalized Susceptibility 11

1.7 Generalized Ornstein-Zernicke Relation 13

1.8 Problems 15

2 Motion of Systems in Phase Space 17

2.1 Integral Invariants 17

2.2 Classical Liouville's Equation 20

2.3 Role of Energy 22

2.4 Quantum Mechanical Density Matrix 22

2.5 Quantum Liouville's Equation 24

2.6 Problems 26

3 States in Statistical Physics 27

3.1 Microscopic and Macroscopic States 27

3.2 Statistical Weight and Density of States 28

3.3 Examples: Non-interacting One- and N-Particle Systems and Spin-1/2 Particles 29

3.4 Entropy and Boltzmann's Principle 31

3.5 Boltzmann's H-Theorem 34

3.6 Problems 35

4 Statistical Ensembles 37

4.1 Introduction 37

4.2 Microcanonical Distribution Function 37

4.3 Canonical (Gibbs) Distribution Function 39

4.3.1 Thermodynamic Temperature and Distribution Function 39

4.3.2 Spin-1/2 Particles and Negative Temperature 40

4.3.3 Partition Function and Different Thermodynamic Functions 41

4.3.4 System of Linear Harmonic Oscillators in Canonical Ensemble 42

4.3.5 Energy Fluctuation in Canonical Ensemble and Equivalence of Canonical and Microcanonical Ensembles 43

4.4 Grandcanonical Distribution Function 44

4.4.1 Dependence of Thermodynamic Functions on Number of Particles 44

4.4.2 Chemical Potential and Distribution Function 47

4.4.3 Density Fluctuation in Grandcanonical Ensemble and Equivalence of Grandcanonical and Canonical Ensembles 49

4.5 Problems 50

5 Ideal Gas 51

5.1 Boltzmann Distribution 51

5.2 Partition Function, Free Energy and Equation of State 52

5.3 Specific Heat: Translational, Vibrational and Rotational Components 55

5.4 Degeneracy Temperature 59

5.5 Problems 61

6 Chemical Reaction Equilibrium 63

6.1 Conditions of Chemical Equilibrum 63

6.2 Law of Mass Action 65

6.3 Heat of Reaction and Direction of Reaction 66

6.4 Ionization Equilibrium 68

6.5 Saha Formula 70

6.6 Problems 71

7 Real Gas 73

7.1 Free Energy, Virial Equation of State 73

7.2 Second Virial Coefficient and Applicability of Virial Equation 76

7.3 Model Calculation and van-der-Waal's Equation of State 78

7.4 Joule-Thomson Expansion and Inversion Temperature 80

7.5 Problems 83

8 Strong Electrolytes 85

8.1 Debye-Hückel Approximation, Debye Length 85

8.2 Screened Coulomb Potential 88

8.3 Equation of State and Osmotic Pressure 89

8.4 Problems 90

9 Quantum Statistics 91

9.1 Bose and Fermi Distributions 91

9.2 Quantum Gases of Elementary Particles: Number Density and Chemical Potential, Energy Density, Equation of State 94

9.3 Black Body Radiation and Planck's Law 98

9.4 Lattice Specific Heat and Phonons 100

9.5 Degenerate Bose Gas, Bose Condensation 103

9.6 Liquid He and Superfluidity 106

9.6.1 Systematics of Liquid ⁴He 106

9.6.2 Landau's 2-Fluid Model 111

9.6.3 Systematics of Liquid ³He 117

9.7 Degenerate Fermi Gas, Degeneracy Pressure, Specific Heat 118

9.8 Magnetism of Free Fermions 124

9.8.1 Preamble 124

9.8.2 Landau Diamagnetism 124

9.8.3 Pauli Paramagnetism 127

9.9 Interacting Fermi System: Fermi Liquid Theory 129

9.10 Relativistic Degenerate Fermi Gas 133

9.11 Problems 136

10 Bose-Einstein Condensate 139

10.1 Introduction 139

10.2 Trapping of Atoms 140

10.3 Cooling of Atoms 142

10.4 Problems 145

11 Statistical Astrophysics 147

11.1 Introduction 147

11.2 Stars: Stability and Evolution 148

11.3 High Temperature Dense Matter 152

11.4 Neutron Stars and Black Holes 154

11.5 Problems 156

12 Phase Transitions 157

12.1 Systematics of Phase Transitions 157

12.2 Ehrenfest's Classification of Phase Transitions 159

12.3 Order Parameter, Continuous and Discontinuous Transitions 160

12.4 Landau's Theory of Continuous Phase Transitions 163

12.5 Continuity of Entropy and Discontinuity of Specific Heat 165

12.6 Generalized Susceptibility 167

12.7 Critical Exponents and Fluctuations of Order Parameter 170

12.8 Ising Model 173

12.8.1 Zero-Field 1-Dimensional Case 175

12.8.2 Non-Zero-Field 1-Dimensional Case 175

12.8.3 Multi-Dimensional Case 177

12.8.4 2-Dimensional Ising System 178

12.9 Problems 181

13 Irreversible Processes 183

13.1 Introduction 183

13.2 Linear Response Theory (Kubo Formalism) 184

13.2.1 Mechanical Process 184

13.2.2 Thermal Process 189

13.3 Symmetry Relations 191

13.4 Fluctuation-Dissipation Theorem 192

13.5 Problems 196

14 Mathematical Appendix 197

14.1 Beta and Gamma Functions 197

14.2 Dirac Delta Function (Distribution) 198

14.3 Functional Derivative 200

14.4 Mathematical Identities 201

14.5 Multiple Summation 202

14.6 Pauli Matrices 203

14.7 Probability Theory 203

14.7.1 Elementary Results of Probability Theory 203

14.7.2 Statistical Distributions 204

14.7.3 Central Limit Theorem 207

14.8 Quantum Mechanics, A Retrospect 209

14.9 Riemann, Bernoulli and Fourier 210

14.9.1 Riemann's ζ-Function 210

14.9.2 Bernoulli Numbers and Polynomials 211

14.9.3 Fourier Series 212

14.9.4 Integrals of Quantum Statistics 214

14.10 Sanskrit Transliteration 216

14.11 Stirling's Theorem 217

14.12 Summation and Integration 219

14.13 Volume of an N-Dimensional Sphere 220

Bibliography 223

Index 225