Zaitsev R.O. Introduction to Modern Kinetic Theory:
A Set of Lectures

CONTENTS

CONTENTS 3

INTRODUCTION 11

Lecture I. THE BOLTZMANN THEORY 14
§ 1. The Boltzmann equation 16
Problem 1. Molecule collision number 20
§ 2. The Boltzmann iï-theorem 21
§ 3. Conservation laws 25
P r o b 1 e m 2. Locally equilibrium distributions 26
§ 4. The r-approximation 27
P r o b 1 e m 3. The thermal conductivity in the
r-approximation 28
P r o b 1 e m 4. The coefficient of viscosity in the
r-approximation 30
§ 5. An approximate solution of the
Boltzmann equation 31
§5.1. The thermal conductivity 35
§ 5.2. Coefficients of viscosity 37
§ 6. The Boltzmann equation for a binary mixture 41
§ 7. The Chapman—Enskog equation for a binary
mixture 43
§7.1. The system of algebraic equations 47
§ 7.2. Diffusion and thermal diffusion 51
§ 7.3. The thermal conductivity of a gas mixture 53
§ 7.4. The transversal viscosity of a gas mixture 55
P r o b 1 e m 5. The diffusion coefficient 56
P r o b 1 e m 6. The thermal diffusion coefficient 59
Appendix. The Bogolubov dynamical theory .... 62 References 66

Lecture II. HYDRODYNAMIC EQUATIONS 67
§ 1. General relations 68
§ 1.1. Mass conservation law 68 § 1.2. Momentum conservation law 69
§ 1.3. Energy conservation law 70
§ 2. The closed system of equations 72
§ 3. Linearized hydrodynamic equations 75
§ 4. Hydrodynamic excitations 77
§ 5. Excitations in a charged system 79
§ 6. Meteorologist's equations 80
References 82

Lecture III. THE ENSKOG THEORY 83
§ 1. The Enskog equation 84
§ 2. Conservation laws 88
§ 3. Linearization of the collision integral 91
§ 4. The equations of hydrodynamics 95
§ 5. Solution of the transport equation 96
§ 6. Calculation of the transport coefficients 97
§6.1. transport corrections 97
§ 6.2. Corrections caused by momentum transfer 99
§ 6.3. The shear and volume viscosity factors 101
§ 6.4. Corrections caused by energy transfer 102
§ 6.5. The thermal conductivity 104
§ 7. Conclusions 104
Appendix. Calculation of angular integrals 105
References 110

Lecture IV. TRANSPORT EQUATIONS
FOR CHARGED PARTICL Ill
§ 1. Small momentum transfer 112
§ 2. Landau's collision integral 114
Problem 1. Transport coefficients
in the r-approximation 119
P r o b 1 e m 2. The cooling rate of hot electrons 122
§ 3 The H theorem for Landau's collision integral .... 124
§ 4. The Fokker—Planck collision integral 126
§ 4.1. The Fokker-Planck equation for charged particles 127 § 4.2. The Fokker-Planck equation for heavy particles 130
P r o b 1 e m 3. The mobility of heavy particles 131
P r o b 1 e m 4. The diffusion and thermodiffusion coefficients 132 References 134

Lecture V. ELECTRONS IN A METAL (T < 6) 135
§ 1. The equation for the distribution function 136
Problem 1. The residual resistance 141
P r o b 1 e m 2. The electron thermal conductivity 142
P r o b 1 e m 3. The thermoelectric effect 143
P r o b 1 e m 4. The law of increasing entropy 144
§ 2. The Kondo effect 146
§ 2.1. The perturbation theory 146
§ 2.2. The temperature correction 153
§ 2.3. Reducing to a one-dimensional problem 156
§ 3. Non-stationary phenomena 157
§ 3.1. Longitudinal fields 157
§3.2. Transversal fields 161
References 166

Lecture VI. ELECTRONS AND PHONONS IN
A METAL 167
§ 1. The electron—phonon interaction 168
§ 2. The transport equation in metals 170
§ 2.1. The if-theorem on entropy increase 171
§ 3. The transport equation at equilibrium phonons ...174
§ 3.1. Integration over virtual electron momenta 180
§ 3.2. Integration over scattering angles 180
§ 3.3. A trial function of the first approximation 181
§ 3.4. The Bloch-GrMneisen equation 182
§ 4. The resistance in dependence of temperature .... 183 § 5. The thermal conductivity in dependence of
temperature 186
§ 5.1. Low temperatures T < O 186
§ 5.2. The kernel of the integral equation 190 § 5.3. High temperatures T ^> 0 191
Problem 1. Qualitative considerations 192
P r o b 1 e m 2. The rate of phonon relaxation 195
P r o b 1 e m 3. The rate of electron relaxation 196
§ 6. The conductivity of semiconductors 198
§ 7. Umpklapp processes and their role 200
References 205

Lecture VII. NON-STATIONARY PERTURBATION
THEORY 206
§ 1. The Schwinger—Mills—Keldysh theory 207
§ 1.1. The transition to the interaction representation 208
§ 1.2. Averaging over the states of an ideal gas 209
§ 1.3. The diargammatic technique 212
§ 2. Tunneling through a fiat interface 214
§ 3. The Kubo formula 217
§ 4. A longitudinal external field 219
§4.1. The longitudinal non-stationary correction 220
§ 4.2. The longitudinal dielectric permittivity 222
§ 5. A transversal external field 224
§5.1. The transversal non-stationary correction 226
§ 5.2. The transversal dielectric permittivity 226
§ 5.3. The calculation of the penetration depth 231
§ 6. The rate of relaxation of impurity spins 232
References 237

Lecture VIII. THE TUNNELING CURRENT AND
THE JOSEPHSON EFFECT 238
§ 1. Tunneling at a given voltage applied 239
§ 1.1. The tunneling Hamiltonian and average current 239
§ 1.2. The (u-v) transformation with a phase factor 241
§ 1.3. The calculation of the volt ampere characteristic 243
§ 2. The stationary (d.c.) supercurrent 249
§2.1. The calculation of the Josephson current amplitude ... .250 § 3. The non-stationary Josephson effect 256 §3.1. The Josephson current. The temperature T <^ Tc 258
§ 3.2. The interference current. The temperature T < Tc .... 262
§ 4. Conclusion 264
Appendix The calculation of elliptic integrals 266
References 270

Lecture IX. QUANTUM TRANSPORT
PHENOMENA 271
§ 1. The equation for the density matrix 272
§ 2. The quantum transport equation 277
§ 3. The Dyson equation 280
§ 4. Collision integrals 285
§4.1. The scattering by impurities 288
§ 4.2. The scattering by equilibrium phonons 289
§ 4.3. The electron-electron interaction 292
§ 4.4. The Landau collision integral 295
§ 5. The screening and plarization 298
§5.1. The plasmon exchange 305
§ 6. The retarded and advanced Green functions 305
§ 7. Diffusons and Coopérons 310
§ 7.1. "Long tails" of correlation functions 319
References 321

Lecture X. NOISE IN ELECTRICAL CIRCUITS ...322
§ 1. The fluctuation-dissipation theorem 324
§ 1.1. The Kubo formula 325
§ 1.2. The white noise 326
§ 2. The 1//-noise in electrical circuits 330
§ 2.1. The l//-noise. Description of phenomenon 330
§ 2.2. The 1/f -noise. The setting of problem 330
§ 3. The equations at T = 0 331
§3.1. The equation for four-current correlator 333
§ 3.2. Equations for two-current vertices 336
§ 3.3. Equations for scalar vertices 339
§ 3.4. Calculation of the exponent a 342 § 4. The l//a-noise at a finite temperature 343
§4.1. Experimental data 343
§ 4.2. Scheme of calculations 344
§ 4.3. Formulation of problem 347
§ 5. Renormalization-group equations 351
§5.1. Equations for scalar vertices 351
§ 5.2. Equations for two-current verticies 354
§ 5.3. Equations for four-current vertices 356
§ 6. Calculation of the exponent a 357
§ 7. Calculation of ip values 360
§7.1. Low temperatures (T < O) 362
§ 7.2. High temperatures (T ^> 0) 363
§ 7.3. Qualitative comparisons with experiment 364
§ 7.4. Conclusions 365
References 366

Lecture XL TRANSPORT EQUATION FOR
PHONONS 367
§ 1. The transport equation 368
§ 2. The phonon—phonon interaction 368
§ 3. The three—phonon collision integral 370
§ 4. The H-theorem 373
§ 5. The thermal conductivity of dielectrics 377
§5.1. The equation at a given temperature gradient 377
§ 5.2. The thermal conductivity at high temperatures 379
§ 5.3. The thermal conductivity at low temperatures 380
§ 6. The phonon hydrodynamics equations 384
§ 7. Sound absorbtion. Short waves 387
§ 8. Sound absorbtion. Long waves 392
§ 8.1. Long waves. High temperatures T ^ 0 395
§ 8.2. Long waves. Low temperatures T < O 396
Problem. Sound absorbtion in the r approximation 397
References 402
Lecture XII. TRANSPORT PHENOMENA IN
FERMI-LIQUID 403
§ 1. The equation for the distribution function 404
§ 2. The relaxation time in dependence of tempera- ture 406
§ 3. The conservation laws 407
§3.1. The continuity equation for quasi-particles 407
§ 3.2. The momentum conservation law 408
§ 3.3. The energy conservation law 409
§ 4. Linearization of the transport equation 410
Problem 1. The thermal conductivity in dependence of
temperature 412
Problem 2. The viscosity in dependence of temperature 415
Problem 3. Electron's entrainment 417
§ 5. The exact solution of the transport equation 418
§ 5.1 Separation of variables 418
§ 5.2 Transformation of the kernel of the integral
equation 418
§ 6. Solution of integral equations 425
§ 6.1. The inhomogeneous equation. Even functions 426
§ 6.2. The inhomogeneous equation. Odd functions 429
§ 7. The shear viscosity. The exact solution 431
§ 8. The thermal conductivity. The exact solution 434
§ 9. The second viscosity 436
§ 10. The sound propagation in a Fermi-liquid 438
§ 10.1. The zero sound in a Fermi-liquid 438
§ 10.2. The velocity of hydrodynamic sound 438
§ 10.3. The sound absorbtion in the r-approximation 441
§ 11. The if-theorem 451
Appendix A. The effective mass 455
Appendix B. Calculation of integrals 457
Appendix C. Calculation of angular integrals 460
References 462
Lecture XIII. TRANSPORT PHENOMENA IN
SUPERCONDUCTORS 463
§ 1. The non-stationary Meissner effect 464
§ 1.1. The Pippard case 472
§ 1.2. The London case 477
§ 2. The absorbtion of ultrasound 480
§ 3. The rate of relaxation of the nuclear spins 485
§ 4. The thermal conductivity of a superconductor .... 489 § 5. The Ginzburg—Landau non-stationary
equations 495
Problem 1: The gauge invariant GLAG equation 501
P r o b 1 e m 2: Relaxation in an ideal superconductor 502
P r o b 1 e m 3: Relaxation in a non-ideal superconductor .... 503 References 510

BIBLIOGRAPHY 511

Manuals and monographs 515