Table of Contents
Preface v
Acknowledgments vii
1 Statistical Mechanics: The Basics 1
1.1 From Micro to Macro 1
1.2 Equilibrium Distribution 8
1.3 Boltzmann's Hypothesis 9
1.4 Dynamics, Dynamical Systems and Chaos 12
1.5 The Prevalence of Chaos 20
1.6 Quantum Dynamics 23
1.7 Problems for Chapter 1 26
2 The Micro canonical Ensemble 27
2.1 Examples 27
2.1.1 The Ideal Gas 27
2.1.2 An ideal solid 29
2.1.3 General oscillators 30
2.1.4 Magnetic moments in an external field 32
2.1.5 The quantum solid 34
2.1.6 The quantum free particle 37
2.2 Additivity of Entropy and Microcanonical Ensemble 38
2.3 Shannon Entropy 40
2.4 Problems for Chapter 2 41
3 Other Ensembles 43
3.1 Canonical Ensemble 43
3.2 Alternative Derivation 48
3.3 Varying Number of Particles 51
3.4 Grand Canonical Ensemble 53
3.5 Energy Fluctuation in Grand Canonical Ensemble 55
3.6 Physical Interpretation of Q 57
3.7 Problems for Chapter 3 58
4 Non-interacting Systems 61
4.1 Calculational Procedure for Canonical Ensemble 61
4.2 Ideal Gas 62
4.3 Ideal Gas in Grand Canonical Ensemble 65
4.4 Space-dependent Distribution 66
4.4.1 In a Gravitational Potential 66
4.4.2 In a Centrifuge 67
4.5 Oscillators in Contact with Heat Bath 68
4.5.1 Classical Oscillators 68
4.5.2 Quantum Oscillators 68
4.6 Specific Heat of Solids 69
4.7 Model for Paramagnetism 73
4.8 Electronic Energy in a Monoatomic Gas 76
4.9 Diatomic Molecules 77
4.9.1 Heteronuclear Molecules 77
4.9.2 Homonuclear Molecules 78
4.30 Solid-vapour Equilibrium 80
4.11 Saha Ionization Formula 81
4.12 Statistical Mechanics of Powders (Edwards' Model) 83
4.13 Problems for Chapter 4 87
5 Interacting Classical Systems 91
5.1 The Non-ideal Gas 91
5.2 Linked Cluster Expansion 95
5.3 Van der Waals Equation of State 100
5.4 Existence of Phase Transitions 105
5.5 Correlation Functions 106
5.6 Ising Model 110
5.6.1 Exact Solution in One Dimension 111
5.6.2 Approximate Method for Any Dimensions 113
5.7 Problems for Chapter 5 116
6 Quantum Statistics 119
6.1 Introduction 119
6.2 Quantum Distribution Functions 120
6.2.1 Fermions 121
6.2.2 Bosons 121
6.3 Equation of State 123
6.4 Entropy of the Quantum Ideal Gas 126
6.5 Chemical Potential in Two Dimensions 132
6.6 Problems for Chapter 6 133
7 Fermi Distribution: Examples 135
7.1 Degenerate Fermi Gas 135
7.2 White Dwarfs 136
7.3 Specific Heat of an Electron Gas 140
7.4 One Dimensional Metal: Effect of Periodic Lattice Structure 143
7.4.1 Energy Levels for a Periodic Potential 143
7.4.2 Fermi Level and Density of States 146
7.4.3 Specific Heat 147
7.4.4 Peierls' Instability 148
7.5 Emission of Electrons from a Metal Surface 150
7.6 Correlations in a Fermi Gas 152
7.7 Electrons in Graphene 153
7.7.1 The Tight-binding Approximation 154
7.7.2 Application to Grapliene 157
7.8 Problems for Chapter 7 161
8 Electrons in a Magnetic Field 163
8.1 Introduction 163
8.2 Magnetic Properties at T = 0 165
8.2.1 Intrinsic Magnetic Moment 166
8.2.2 Orbital Magnetic Moment 168
8.3 Magnetic Properties at T >> TF 170
8.4 De Haas - Van Alphen Effect: T = 0 171
8.5 Quantum Hall Effect 172
8.5.1 Integer Hall Effect 172
8.5.2 Fractional Hall effect 179
9 Bose-Einstein Distribution: Examples 181
9.1 Bose-Einstein Condensation 181
9.1.1 The Condensation Point 181
9.1.2 Order of the Transition 183
9.1.3 Near the Transition Point 186
9.1.4 Experimental Observation of the Transition: Bosons in a Trap 190
9.1.5 Interaction Effects in BEC 193
9.2 Two Important Phenomena 195
9.2.1 Black Body Radiation and Stefan-Boltzmann law 195
9.2.2 Casimir Effect 199
9.3 Superfluidity of Helium-4 200
9.3.1 General Characteristics 200
9.3.2 The Energy Spectrum 203
9.3.3 Occurrence of Superfluidity 206
9.4 Problems for Chapter 9 208
10 Superconductivity 211
10.1 Introduction 211
10.2 Pairing Theory: Qualitative 213
10.3 Origin of the Attractive Interaction 221
10.4 Cooper Instability 223
10.5 A Different Pairing: High Temperature Superconductivity in Cuprates 225
11 Phase Transitions 231
11.1 Introduction 231
11.2 Transfer Matrix for the 2D Ising Model 234
11.2.1 The Conversion to a Quantum Problem 234
11.2.2 The Transition Temperature 238
11.2.3 The Specific Heat 241
11.3 Planar Model and Heisenberg Model 244
11.4 Statistical Mechanics and Field Theory 246
11.5 Kac-Hubbard-Stratonovich Transformation 250
11.6 Critical Phenomena: Scaling Laws 252
12 Landau Theory and Related Models 259
12.1 Landau Theory 259
12.2 The Gaussian Model 263
12.3 The Spherical Limit 264
13 The Renormalization Group 269
13.1 Renormalization Group in Real Space 269
13.1.1 Introduction 269
13.1.2 One-dimensional Ising Model 271
13.1.3 Two-dimensional Ising Model 273
13.2 Renormalization Group in Momentum Space 276
13.3 Fixed Points and Scaling Laws 277
13.4 Application to the Gaussian Model 282
13.5 Application to the Landau-Ginzburg Model 283
14 Disordered Systems 291
14.1 Introduction 291
14.2 Models with Quenched Disorder 293
14.3 Electron Localization in Disordered Systems 298
15 Transport Equation - I 305
15.1 Introduction 305
15.2 Distribution Function 306
15.3 In the Absence of Collisions 307
15.4 In the Presence of Collisions 309
16 Transport Equation - II 315
16.1 Introduction 315
16.2 Interaction without Collisions 316
16.3 Collisions with Fixed Scatterers 319
16.4 Binary Collisions 322
16.5 The H-theorem 326
17 Transport Equation - III 327
17.1 A Theorem 327
17.2 Transport Equation for Conserved Quantities 328
17.3 Equations of Hydrodynamics 331
18 Transport Equation - IV 335
18.1 Form of the Transport Equation for Quantum Particles 335
18.1.1 Fermions 336
18.1.2 Bosons 336
18.2 Equilibrium Distribution 337
18.3 Approach to Equilibrium (F≠0 but no Collisions) 339
18.4 Effect of Collisions in a Fermion Gas 340
18.5 Collision Term for Bosons 344
19 Metastable States 347
19.1 Introduction 347
19.2 A Phenomenological Approach 348
19.3 Decay of Metastable States 350
19.4 Beyond Metastability: Glasses 352
20 Langevin Equations 357
20.1 Introduction 357
20.2 An Example with Oscillators 358
20.3 Gaussian Distribution and the Central Limit Theorem 360
20.4 Langevin Equation 365
20.5 Langevin Equation and Equilibrium 367
20.6 Brownian Motion 369
20.7 Quantum Langevin Equation 375
21 Fokker-Planck Equations 379
21.1 Introduction 379
21.2 Fokker-Planck Equation: Derivation 380
21.3 The General Solution 383
21.4 Metastable State and the Lowest Eigenvalue 387
21.5 Passage to Equilibrium of a Non-equilibrium State 389
21.5.1 Regime - I: Diffusion 391
21.5.2 Regime - II: "Scaling" or "Sliding" 392
21.5.3 Regime-III: Kramers 393
21.6 Diffusion in a Periodic Potential 394
22 Fluctuation Dissipation Theorem 397
22.1 Introduction 397
22.2 The General Case 398
22.3 Jarzynski Equality 403
References 405
Index 413
