INTERMEDIATE STATISTICAL MECHANICS

by Jayanta K Bhattacharjee, Dhruba Banerjee

Overview

In this new textbook, a number of unusual applications are discussed in addition to the usual topics covered in a course on Statistical Physics. Examples are: statistical mechanics of powders, Peierls instability, graphene, Bose-Einstein condensates in a trap, Casimir effect and the quantum Hall effect. Superfluidity and super-conductivity (including the physics of high-temperature superconductors) have also been discussed extensively.The emphasis on the treatment of these topics is pedagogic, introducing the basic tenets of statistical mechanics, with extensive and thorough discussion of the postulates, ensembles, and the relevant statistics. Many standard examples illustrate the microcanonical, canonical and grand canonical ensembles, as well as the Bose-Einstein and Fermi-Dirac statistics.A special feature of this text is the detailed presentation of the theory of second-order phase transitions and the renormalization group, emphasizing the role of disorder. Non-equilibrium statistical physics is introduced via the Boltzmann transport equation. Additional topics covered here include metastability, glassy systems, the Langevin equation, Brownian motion, and the Fokker-Planck equation.Graduate students will find the presentation readily accessible, since the topics have been treated with great deal of care and attention to detail.

Product Details

ISBN-13:	9789813201163
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	12/15/2016
Sold by:	Barnes & Noble
Format:	eBook
Pages:	440
File size:	29 MB
Note:	This product may take a few minutes to download.

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 >> T_F 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