Nonequilibrium Statistical Physics

Linear Irreversible Processes

Noelle Pottier Laboratoire Matiere et Systemes Complexes, CNRS, and

Universite Paris Diderot - Paris 7

OXPORD UNIVERSITY PRESS

Contents

Chapter 1 Random variables and random processes 1

1 Random variables, moments, and characteristic function 2 2 Multivariate distributions 4 3 Addition of random variables 6 4 Gaussian distributions 7 5 The central limit theorem 9 6 Random processes 12 7 Stationarity and ergodicity 14 8 Random processes in physics: the example of Brownian motion 16 9 Harmonic analysis of stationary random processes 17

10 The Wiener-Khintchine theorem 19 Appendix

1A An alternative derivation of the Wiener-Khintchine theorem 23 Bibliography 25 References 25

Chapter 2 Linear thermodynamics of irreversible processes 27

1 A few reminders of equilibrium thermodynamics 28 2 Description of irreversible processes: affinities and fluxes 29 3 The local equilibrium hypothesis 32 4 Affinities and fluxes in a continuous medium in local equilibrium 34 5 Linear response 37 6 A few simple examples of transport coefficients 38 7 Curie's principle 42 8 The reciprocity relations 43 9 Justification of the reciprocity relations 45

10 The minimum entropy production theorem 48 Bibliography 50 References 50

xii Contents

Supplement 2A Thermodynamic fluctuations 51

1 The fluctuations 51 2 Consequences of the maximum entropy principle 52 3 Probability of a fluctuation: the Einstein formula 53 4 Equilibrium fluctuations in a fluid of N molecules 54

Bibliography 58 References 58

Supplement 2B Thermoelectric effects 59

1 Introduction 59 2 The entropy source 60 3 Isothermal electrical conduction 61 4 Open-circuit thermal conduction 62 5 The Seebeck effect 62 6 The Peltier effect 63 7 The Thomson effect 65 8 An illustration of the minimum entropy production theorem 66

Bibliography 67

Supplement 2C Thermodiffusion in a fluid mixture 68

1 Introduction 68 2 Diffusive fluxes in a binary mixture 68 3 The entropy source 69 4 Linear relations between fluxes and affinities 70 5 The Soret and Dufour effects 72

Bibliography 73 References 73

Chapter 3 Statistical description of out-of-equilibrium systems 75

1 The phase space distribution function 76 2 The density operator 80 3 Systems at equilibrium 83 4 Evolution of the macroscopic variables: classical case 84 5 Evolution of the macroscopic variables: quantum case 86

Bibliography 88

Chapter 4 Classical systems: reduced distribution functions 89

1 Systems of classical particles with pair interactions 90 2 The Liouville equation 91 3 Reduced distribution functions: the BBGKY hierarchy 93 4 The Vlasov equation 96 5 Gauge invariance 97

Contents xiii

Appendices 4A Pair interaction potentials 99 4B Hamilton's equations for a charged particle 100 4C Gauge invariance of the Liouville equation 102

Bibliography 104

Chapter 5 The Boltzmann equation 105

1 Statistical description of dilute classical gases 106 2 Time and length scales 107 3 Notations and definitions 108 4 Evolution of the distribution function 109 5 Binary collisions 110 6 The Boltzmann equation 113 7 Irreversibility 116 8 The Я-theorem 117 9 Equilibrium distributions 120

10 Global equilibrium 121 11 Local equilibrium 123

Bibliography 125 References 125

Supplement 5A The Lorentz gas 126

1 Gas in the presence of fixed scattering centers 126 2 Time scales 126 3 Collisions with the fixed scatterers 127 4 Kinetic equation of the Lorentz gas 127

Bibliography 130 References 130

Supplement 5B The irreversibility paradoxes 131

1 The paradoxes 131 2 The time-reversal paradox 131 3 The recurrence paradox 132

Bibliography 133 References 133

Chapter 6 Transport coefficients 135

1 The relaxation time approximation 136 2 Linearization with respect to the external perturbations 138 3 Kinetic coefficients of a Lorentz gas 138 4 Electrical conductivity 142 5 Diffusion coefficient 144

Bibliography 147 References 147

xiv Contents

Supplement 6A Landau damping 148

1 Weakly coupled plasma 148 2 The Vlasov equations for a collisionless plasma 148 3 Conductivity and electrical permittivity of a collisionless plasma 151 4 Longitudinal waves in a Maxwellian plasma 154

Bibliography 157

Chapter 7 From the Boltzmann equation to the hydrodynamic equations 159

1 The hydrodynamic regime 160 2 Local balance equations 161 3 The Chapman-Enskog expansion 165 4 The zeroth-order approximation 168 5 The first-order approximation 169

Appendices 7A A property of the collision integral 175 7B Newton's law and viscosity coefficient 176

Bibliography 180

Chapter 8 The Bloch—Boltzmann theory of electronic transport 181

1 The Boltzmann equation for the electron gas 182 2 The Boltzmann equation's collision integral 184 3 Detailed balance 187 4 The linearized Boltzmann equation 188 5 Electrical conductivity 189 6 Semiclassical transport in the presence of a magnetic field 192 7 Validity limits of the Bloch-Boltzmann theory 198

Bibliography 200 References 200

Supplement 8A Collision processes 201

1 Introduction 201 2 Electron-impurity scattering 201 3 Electron-phonon scattering 207

Bibliography 211 References 211

Supplement 8B Thermoelectric coefficients 212

1 Particle and heat fluxes 212 2 General expression for the kinetic coefficients 213 3 Thermal conductivity 213 4 The Seebeck and Peltier coefficients 215

Bibliography 217

Contents xv

Chapter 9 Master equations 219

1 Markov processes: the Chapman-Kolmogorov equation 220 2 Master equation for a Markovian random process 223 3 The Pauli master equation 226 4 The generalized master equation 228 5 From the generalized master equation to the Pauli master equation 229 6 Discussion 231

Bibliography 233 References 233

Chapter 10 Brownian motion: the Langevin model 235

1 The Langevin model 236 2 Response and relaxation 238 3 Equilibrium velocity fluctuations 243 4 Harmonic analysis of the Langevin model 247 5 Time scales 249

Bibliography 251 References 251

Supplement 10A The generalized Langevin model 253

1 The generalized Langevin equation 253 2 Complex admittance 255 3 Harmonic analysis of the generalized Langevin model 255 4 An analytical model 257

Bibliography 259 References 259

Supplement 10B Brownian motion in a bath of oscillators 260

1 The Caldeira-Leggett model 260 2 Dynamics of the Ohmic free particle 265 3 The quantum Langevin equation 267

Bibliography 269 References 269

Supplement IOC The Nyquist theorem 270

1 Thermal noise in an electrical circuit 270 2 The Nyquist theorem 270

Bibliography 275 References 275

xvi Contents

Chapter 11 Brownian motion: the Fokker-Planck equation 277

1 Evolution of the velocity distribution function 278 2 The Kramers-Moyal expansion 279 3 The Fokker-Planck equation 282 4 Brownian motion and Markov processes 285

Bibliography 288 References 288

Supplement I I A Random walk 290

1 The drunken walker 290 2 Diffusion of a drunken walker on a lattice 291 3 The diffusion equation 292

Bibliography 293 References 293

Supplement I I B Brownian motion: Gaussian processes 294

1 Harmonic analysis of stationary Gaussian processes 294 2 Gaussian Markov stationary processes 295 3 Application to Brownian motion 297

Bibliography 300 References 300

Chapter 12 Linear responses and equilibrium correlations 301

1 Linear response functions 302 2 Generalized susceptibilities 303 3 The Kramers-Kronig relations 306 4 Dissipation 307 5 Non-uniform phenomena 308 6 Equilibrium correlation functions 310 7 Properties of the equilibrium autocorrelation functions 314

Appendix 12A An alternative derivation of the Kramers-Kronig relations 319

Bibliography 321 References 321

Supplement 12A Linear response of a damped oscillator 322

1 General interest of the study 322 2 The undamped oscillator 322 3 Oscillator damped by viscous friction 323 4 Generalized susceptibility 324 5 The displacement response function 327

Bibliography 328

Contents xvii

Supplement 12B Electronic polarization 329

1 Semiclassical model 329 2 Polarization response function 330 3 Generalized susceptibility 331 4 Comparison with the Lorentz model 331

Bibliography 334

Supplement 12C Some examples of dynamical structure factors 335

1 The examples 335 2 Free atom 335 3 Atom in a harmonic potential 337

Bibliography 340

Chapter 13 General linear response theory 341

1 The object of linear response theory 342 2 First-order evolution of the density operator 342 3 The linear response function 345 4 Relation with the canonical correlation function 347 5 Generalized susceptibility 348 6 Spectral function 350 7 Relaxation 352 8 Symmetries of the response and correlation functions 357 9 Non-uniform phenomena 359

Appendices 13A Classical linear response 361 13B Static susceptibility of an isolated system and isothermal susceptibility 363

Bibliography 367 References 367

Supplement 13A Dielectric relaxation 368

1 Dielectric permittivity and polarizability 368 2 Microscopic polarization mechanisms 371 3 The Debye theory of dielectric relaxation 371 4 A microscopic model of orientational polarization 374

Bibliography 378 References 378

Supplement 13B Magnetic resonance 379

1 Formulation of the problem 379 2 Phenomenological theory 380 3 A microscopic model 383

Bibliography 388

xviii Contents

Chapter 14 The fluctuation-dissipation theorem 389

1 Dissipation 390 2 Equilibrium fluctuations 393 3 The fluctuation-dissipation theorem 395 4 Positivity ofwxAAH 398 5 Static susceptibility 398 6 Sum rules 400

Bibliography 403 References 403

Supplement 14A Dissipative dynamics of a harmonic oscillator 404

1 Oscillator coupled with a thermal bath 404 2 Dynamics of the uncoupled oscillator 404 3 Response functions and susceptibilities of the coupled oscillator 407 4 Analysis of \xx (J) 409 5 Dynamics of the weakly coupled oscillator 415

Bibliography 417 References 417

Chapter 15 Quantum theory of electronic transport 419

1 The Kubo-Nakano formula 420 2 The Kubo-Greenwood formula 423 3 Conductivity of an electron gas in the presence of impurities 427

Bibliography 431 References 431

Supplement 15 A Conductivity of a weakly disordered metal 433

1 Introduction 433 2 The Kubo-Greenwood formula 433 3 Conductivity of a macroscopic system 436 4 Conductance of a mesoscopic system: Landauer's approach 438 5 Addition of quantum resistances in series: localization 440

Bibliography 445 References 445

Chapter 16 Thermal transport coefficients 447

1 The indirect Kubo method 448 2 The source of entropy and the equivalent 'Hamiltonian' 452

Bibliography 457 References 457

Contents xix

Supplement 16A Diffusive light waves 458

1 Diffusive light transport 458 2 Diffusion coefficient of light intensity 459 3 Diffusive wave spectroscopy 462

Bibliography 467 References 467

Supplement 16B Light scattering by a fluid 468

1 Introduction 468 2 Linearized hydrodynamic equations 468 3 Transverse fluctuations 470 4 Longitudinal fluctuations 472 5 Dynamical structure factor 478

Bibliography 480 References 480

Index 481