A Modern Course in Statistical Physics, 3rd Revised and Updated Edition

Preface to the Third Edition XIII

Preface to the First Edition XV

1 Introduction 1

2 Complexity and Entropy 5

2.1 Introduction 5

2.2 CountingMicroscopic States 5

2.3 Multiplicity and Entropy of Macroscopic Physical States 9

2.4 Multiplicity and Entropy of a Spin System 10

2.4.1 Multiplicity of a Spin System 10

2.4.2 Entropy of Spin System 11

2.5 Multiplicity and Entropy of an Einstein Solid 14

2.5.1 Multiplicity of an Einstein Solid 15

2.5.2 Entropy of the Einstein Solid 15

2.6 Multiplicity and Entropy of an Ideal Gas 16

2.6.1 Multiplicity of an Ideal Gas 17

2.6.2 Entropy of an Ideal Gas 18

2.7 Problems 19

3 Thermodynamics 21

3.1 Introduction 21

3.2 Energy Conservation 23

3.3 Entropy 24

3.3.1 Carnot Engine 24

3.3.2 The Third Law 28

3.4 Fundamental Equation of Thermodynamics 29

3.5 Thermodynamic Potentials 32

3.5.1 Internal Energy 33

3.5.2 Enthalpy 34

3.5.3 Helmholtz Free Energy 35

3.5.4 Gibbs Free Energy 37

3.5.5 Grand Potential 38

3.6 Response Functions 40

3.6.1 Thermal Response Functions (Heat Capacity) 40

3.6.2 Mechanical Response Functions 42

3.7 Stability of the Equilibrium State 45

3.7.1 Conditions for Local Equilibrium in a PVT System 45

3.7.2 Conditions for Local Stability in a PVT System 46

3.7.3 Implications of the Stability Requirements for the Free Energies 50

3.7.4 Correlations Between Fluctuations 52

3.8 Cooling and Liquefaction of Gases 55

3.9 Osmotic Pressure in Dilute Solutions 58

3.10 The Thermodynamics of Chemical Reactions 61

3.10.1 The Affinity 62

3.11 Problems 67

4 The Thermodynamics of Phase Transitions 75

4.1 Introduction 75

4.2 Coexistence of Phases: Gibbs Phase Rule 76

4.3 Classification of Phase Transitions 77

4.4 Classical Pure PVT Systems 79

4.4.1 Phase Diagrams 79

4.4.2 Coexistence Curves: Clausius–Clapeyron Equation 80

4.4.3 Liquid–Vapor Coexistence Region 83

4.4.4 The van der Waals Equation 87

4.4.5 Steam Engines – The Rankine Cycle 90

4.5 Binary Mixtures 93

4.5.1 Equilibrium Conditions 94

4.6 The Helium Liquids 96

4.6.1 Liquid He4 97

4.6.2 Liquid He3 99

4.6.3 Liquid He3–He4 Mixtures 100

4.7 Superconductors 101

4.8 Ginzburg–Landau Theory 104

4.8.1 Theoretical Background 105

4.8.2 Applications of Ginzburg–Landau Theory 108

4.9 Critical Exponents 110

4.9.1 Definition of Critical Exponents 110

4.9.2 The Critical Exponents for Pure PVT Systems 111

4.9.3 The Critical Exponents for the Curie Point 113

4.9.4 The Critical Exponents for Mean Field Theories 114

4.10 Problems 116

5 Equilibrium Statistical Mechanics i – Canonical Ensemble 121

5.1 Introduction 121

5.2 Probability Density Operator-Canonical Ensemble 123

5.2.1 Energy Fluctuations 124

5.3 Semiclassical Ideal Gas of Indistinguishable Particles 125

5.3.1 Approximations to the Partition Function for Semiclassical Ideal Gases 126

5.3.2 Maxwell–Boltzmann Distribution 129

5.4 Interacting Classical Fluids 131

5.4.1 Density Correlations and the Radial Distribution Function 132

5.4.2 Magnetization Density Correlations 134

5.5 Heat Capacity of a Debye Solid 135

5.6 Order–Disorder Transitions on Spin Lattices 139

5.6.1 Exact Solution for a One-Dimensional Lattice 140

5.6.2 Mean Field Theory for a d-Dimensional Lattice 142

5.6.3 Mean Field Theory of Spatial Correlation Functions 145

5.6.4 Exact Solution to Ising Lattice for d = 2 146

5.7 Scaling 148

5.7.1 Homogeneous Functions 148

5.7.2 Widom Scaling 149

5.7.3 Kadanoff Scaling 152

5.8 Microscopic Calculation of Critical Exponents 155

5.8.1 General Theory 156

5.8.2 Application to Triangular Lattice 158

5.8.3 The S4 Model 161

5.9 Problems 163

6 Equilibrium Statistical Mechanics ii – Grand Canonical Ensemble 167

6.1 Introduction 167

6.2 The Grand Canonical Ensemble 168

6.2.1 Particle Number Fluctuations 169

6.2.2 Ideal Classical Gas 170

6.3 Virial Expansion for Interacting Classical Fluids 172

6.3.1 Virial Expansion and Cluster Functions 172

6.3.2 The Second Virial Coefficient, B2(T) 174

6.3.2.1 Square-Well Potential 175

6.4 Black Body Radiation 178

6.5 Ideal Quantum Gases 181

6.6 Ideal Bose–Einstein Gas 183

6.6.1 Bose–Einstein Condensation 187

6.6.2 Experimental Observation of Bose–Einstein Condensation 189

6.7 Ideal Fermi–Dirac Gas 191

6.8 Momentum Condensation in an Interacting Fermi Fluid 197

6.9 Problems 204

7 Brownian Motion and Fluctuation–Dissipation 207

7.1 Introduction 207

7.2 Brownian Motion 208

7.2.1 Langevin Equation 209

7.2.2 Correlation Function and Spectral Density 210

7.3 The Fokker–Planck Equation 212

7.3.1 Probability Flow in Phase Space 214

7.3.2 Probability Flow for Brownian Particle 214

7.3.3 The Strong Friction Limit 217

7.4 Dynamic Equilibrium Fluctuations 221

7.4.1 Regression of Fluctuations 223

7.4.2 Wiener–Khintchine Theorem 225

7.5 Linear Response Theory and the Fluctuation – Dissipation Theorem 226

7.5.1 Linear Response Theory and the Fluctuation – Dissipation Theorem 227

7.5.2 Causality 228

7.5.3 The Fluctuation–Dissipation Theorem 231

7.5.4 Power Absorption 233

7.6 Microscopic Linear Response Theory 235

7.6.1 The Perturbed Density Operator 235

7.6.2 The Electric Conductance 236

7.6.3 Power Absorption 241

7.6.4 Thermal Noise 242

7.7 Problems 243

8 Hydrodynamics 247

8.1 Introduction 247

8.2 Navier–Stokes Hydrodynamic Equations 248

8.2.1 Balance Equations 248

8.2.2 Entropy Source and Entropy Current 253

8.2.3 Transport Coefficients 257

8.3 Linearized Hydrodynamic Equations 260

8.3.1 Linearization of the Hydrodynamic Equations 260

8.3.2 Transverse Hydrodynamic Modes 264

8.3.3 Longitudinal Hydrodynamic Modes 265

8.3.4 Dynamic Correlation Function and Spectral Density 267

8.4 Light Scattering 268

8.4.1 Scattered Electric Field 270

8.4.2 Intensity of Scattered Light 272

8.5 Hydrodynamics of Mixtures 274

8.5.1 Entropy Production in Multicomponent Systems 275

8.5.2 Fick’s Law for Diffusion 277

8.5.3 Thermal Diffusion 279

8.6 Thermoelectricity 280

8.6.1 The Peltier Effect 280

8.6.2 The Seebeck Effect 282

8.6.3 Thomson Heat 284

8.7 Superfluid Hydrodynamics 284

8.7.1 Superfluid Hydrodynamic Equations 284

8.7.2 Sound Modes 288

8.8 Problems 291

9 Transport Coefficients 295

9.1 Introduction 295

9.2 Elementary Transport Theory 295

9.2.1 The Mean Free Path 296

9.2.2 The Collision Frequency 296

9.2.3 Tracer Particle Current 298

9.2.4 Transport of Molecular Properties 300

9.2.5 The Rate of Reaction 301

9.3 The Boltzmann Equation 303

9.3.1 Derivation of the Boltzmann Equation 304

9.4 Linearized Boltzmann Equation 304

9.4.1 Kinetic Equations for a Two-Component Gas 305

9.4.2 Collision Operators 306

9.5 Coefficient of Self-Diffusion 308

9.5.1 Derivation of the Diffusion Equation 308

9.5.2 Eigenfrequencies of the Lorentz–Boltzmann Equation 309

9.6 Coefficients of Viscosity and Thermal Conductivity 311

9.6.1 Derivation of the Hydrodynamic Equations 311

9.6.2 Eigenfrequencies of the Boltzmann Equation 315

9.6.3 Shear Viscosity and Thermal Conductivity 317

9.7 Computation of Transport Coefficients 318

9.7.1 Sonine Polynomials 319

9.7.2 Diffusion Coefficient 319

9.7.3 Thermal Conductivity 321

9.7.4 Shear Viscosity 323

9.8 Problems 324

10 Nonequilibrium Phase Transitions 327

10.1 Introduction 327

10.2 Near Equilibrium Stability Criteria 328

10.3 The Chemically Reacting Systems 330

10.3.1 The Brusselator – A Nonlinear Chemical Model 330

10.3.2 Boundary Conditions 332

10.3.3 Stability Analysis 333

10.3.4 Chemical Crystals 335

10.4 The Rayleigh–Bénard Instability 337

10.4.1 Hydrodynamic Equations and Boundary Conditions 337

10.4.2 Linear Stability Analysis 340

10.5 Problems 343

Appendix A Probability 345

A.1 Definition of Probability 345

A.2 Probability Distribution Functions 347

A.2.1 Discrete Stochastic Variables 348

A.2.2 Continuous Stochastic Variables 348

A.2.3 Characteristic Function 349

A.2.4 Jointly Distributed Stochastic Variables 350

A.3 Binomial Distributions 351

A.3.1 The Binomial Distribution 351

A.3.2 The Gaussian (or Normal) Distribution 353

A.3.3 The Poisson Distribution 356

A.4 Markov Chains 358

A.5 Probability Density for Classical Phase Space 359

A.6 Quantum Probability Density Operator 363

A.7 Problems 366

Appendix B Exact Differentials 369

Appendix C Ergodicity 373

Appendix D Number Representation 377

D.1 The Number Representation 383

D.1.1 The Number Representation for Bosons 383

D.1.2 The Number Representation for Fermions 385

D.1.3 Thermodynamic Averages of Quantum Operators 387

Appendix E Scattering Theory 389

Appendix F Useful Mathematics and Information 395

F.1 Series Expansions 395

F.2 Reversion of Series 395

F.3 Derivatives 395

F.4 Integrals 395

References 397

Index 403