1 Basic concepts ............................................... 1
1.1 Velocity distribution function .......................... 1
1.2 The Maxwell-Boltzmann distribution function ............. 2
1.3 Densities and fluxes .................................... 3
1.3.1 Stress tensor and energy flux .................... 5
1.3.2 Stress tensor and heat flux in equilibrium ....... 6
1.3.3 Flux distribution ................................ 7
1.4 Collision frequency ..................................... 7
1.5 Mean free path .......................................... 8
1.6 Transport properties in the mean free path
approximation ........................................... 8
1.6.1 Thermal conductivity ............................. 9
1.6.2 Viscosity ....................................... 10
1.6.3 Wall slip ....................................... 11
1.6.4 Self-diffusion .................................. 11
1.7 Drude model for electric transport ..................... 12
Exercises ................................................... 13
2 Distribution functions ...................................... 15
2.1 Introduction ........................................... 15
2.2 Hamiltonian dynamics ................................... 16
2.3 Statistical description of the phase space ............. 16
2.4 Equilibrium distribution ............................... 19
2.5 Reduced distributions .................................. 21
2.6 Microscopic and average observables .................... 23
2.6.1 Global observables .............................. 23
2.6.2 Densities ....................................... 24
2.6.3 Fluxes .......................................... 24
2.6.4 Conservation equations .......................... 27
2.7 BBGKY hierarchy ........................................ 28
2.7.1 Equation for the one-particle distribution ...... 30
2.8 Generalisation to mixtures ............................. 31
2.9 Reduced distributions in equiUbrium and the pair
distribution function .................................. 32
2.10 Master equations ....................................... 33
2.11 Application: systems with overdamped dynamics .......... 34
Further reading ............................................. 36
Exercises ................................................... 37
3 The Lorentz model for the classical transport of charges .... 39
3.1 Hypothesis of the model ................................ 39
3.2 Lorentz kinetic equation ............................... 41
3.3 Ion distribution function .............................. 42
3.4 Equilibrium solution ................................... 43
3.5 Conservation laws and the collisional invariants ....... 43
3.6 Kinetic collision models ............................... 44
3.6.1 Rigid hard spheres .............................. 45
3.6.2 Thermalising ions: the BGK model ................ 46
3.7 Electrical conduction .................................. 46
3.7.1 Conservation equation ........................... 46
3.7.2 Linear response ................................. 47
3.7.3 Ohm's law ....................................... 47
3.7.4 Electrical conductivity ......................... 47
3.7.5 Frequency response .............................. 50
3.8 Relaxation dynamics .................................... 50
3.8.1 Properties of the linear operator ............... 51
3.8.2 Kinetic gap ..................................... 52
3.8.3 Spectrum of the linear operator ................. 53
3.8.4 Diffusive behaviour ............................. 54
3.8.5 Rigid hard spheres .............................. 54
3.8.6 Time scales ..................................... 55
3.9 The Chapman-Enskog method .............................. 56
3.10 Application: bacterial suspensions, run-and-tumble
motion ................................................. 58
Further reading ............................................. 60
Exercises ................................................... 61
4 The Boltzmann equation for dilute gases ..................... 63
4.1 Formulation of the Boltzmann model ..................... 63
4.1.1 Hypothesis ...................................... 63
4.1.2 Kinematics of binary colhsions .................. 64
4.2 Boltzmann kinetic equation ............................. 66
4.2.1 General case .................................... 66
4.2.2 Hard sphere model ............................... 67
4.3 General properties ..................................... 68
4.3.1 Balance equations and collisional invariants .... 68
4.3.2 H-theorem ....................................... 70
4.3.3 On the irreversibility problem .................. 73
4.4 Dynamics close to equilibrium .......................... 74
4.4.1 Linear Boltzmann operator ....................... 74
4.4.2 Spectrum of the linear Boltzmann equation ....... 75
4.4.3 Time scales ..................................... 77
4.5 BGK model .............................................. 77
4.6 Boundary conditions .................................... 79
4.7 Hydrodynamic regime .................................... 79
4.7.1 The hydrodynamic equations ...................... 79
4.7.2 Linear response ................................. 81
4.7.3 Variational principle ........................... 82
4.7.4 The Chapman-Enskog method ....................... 82
4.8 Dense gases ............................................ 86
4.8.1 The Enskog model for hard sphere gases .......... 86
4.8.2 Virial expansion ................................ 88
4.9 Application: granular gases ............................ 89
4.10 Application: the expanding universe .................... 91
Further reading ............................................. 92
Exercises ................................................... 92
5 Brownian motion ............................................. 95
5.1 The Brownian phenomenon ................................ 95
5.2 Derivation of the Foldcer-Planck equation .............. 96
5.3 Equilibrium solutions .................................. 98
5.3.1 Homogeneous equilibrium solution and the
fluctuation-dissipation relation ................ 98
5.3.2 Equilibrium solution under external potentials .. 99
5.4 Mobility under external fields ........................ 101
5.5 Long-time dynamics: diffusion ......................... 102
5.5.1 Solution of the diffusion equation ............. 102
5.5.2 Green-Kubo expression .......................... 104
5.5.3 Coarse-grained master equation ................. 105
5.5.4 Eigenvalue analysis ............................ 106
5.5.5 Chapman-Enskog method .......................... 107
5.5.6 Boundary conditions ............................ 108
5.6 Early relaxation ...................................... 108
5.7 Rotational diffusion .................................. 109
5.8 Application: light diffusion .......................... 110
5.9 Application: bacterial alignment ...................... 111
Further reading ............................................ 112
Exercises .................................................. 113
6 Plasmas and self-gravitating systems ....................... 115
6.1 Long-range interactions ............................... 115
6.2 Neutral plasmas ....................................... 116
6.2.1 Introduction ................................... 116
6.2.2 Debye screening ................................ 117
6.2.3 Vlasov equation ................................ 119
6.2.4 Stationary solutions ........................... 122
6.2.5 Dynamical response ............................. 122
6.3 Waves and instabilities in plasmas .................... 125
6.3.1 Plasma waves ................................... 125
6.3.2 Landau damping ................................. 126
6.3.3 Instabilities .................................. 130
6.4 Electromagnetic effects ............................... 131
6.4.1 Magnetic fields ................................ 131
6.4.2 Hydrodynamic equations ......................... 131
6.5 Self-gravitating systems .............................. 132
6.5.1 Kinetic equation ............................... 132
6.5.2 Self-consistent equilibrium solutions .......... 133
6.5.3 Jeans instability .............................. 135
6.6 Beyond mean field ..................................... 135
6.6.1 Velocity relaxation and dynamical friction ..... 135
6.6.2 Slow relaxation ................................ 136
6.6.3 Kinetic equations .............................. 137
6.7 Application: point vortices in two dimensions ......... 138
Further reading ............................................ 140
Exercises .................................................. 140
7 Quantum gases .............................................. 143
7.1 Boson and fermion ideal gases at equilibrium .......... 143
7.1.1 Description of the quantum state ............... 143
7.1.2 Equilibrium distributions ...................... 145
7.2 Einstein coefficients ................................. 146
7.3 Scattering transition rates ........................... 148
7.4 Master kinetic equation ............................... 149
7.5 Equilibrium solutions ................................. 151
7.6 Where is the molecular chaos hypothesis? .............. 152
7.7 Phonons ............................................... 153
7.7.1 Ideal gas of phonons ........................... 153
7.7.2 Phonon-phonon interactions ..................... 155
7.7.3 Phonon-electron interactions ................... 160
7.8 Application: lasers ................................... 161
7.9 Application: quark-gluon plasma ....................... 164
Further reading ............................................ 166
Exercises .................................................. 166
8 Quantum electronic transport in solids ..................... 169
8.1 Electronic structure .................................. 169
8.2 Fermi-Dirac distribution, conductors, and insulators .. 169
8.3 Boltzmann-Lorentz equation ............................ 171
8.3.1 Distribution function .......................... 171
8.3.2 Scattering processes ........................... 172
8.3.3 Semiclassical kinetic equation ................. 173
8.3.4 Linear collision operator ...................... 174
8.4 Time-independent point defects ........................ 175
8.4.1 Transition rates ............................... 175
8.4.2 Spherical models ............................... 176
8.5 Relaxation time approximation ......................... 177
8.6 Electrical conductivity ............................... 177
8.6.1 Qualitative description: metals and
insulators ..................................... 177
8.6.2 Conductivity of metals ......................... 179
8.6.3 Finite-temperature effects ..................... 181
8.6.4 Electron-phonon interactions ................... 182
8.6.5 Multiple scattering mechanisms and the
Matthiessen rule ............................... 184
8.7 Thermal conductivity and Onsager relations ............ 185
В Tensor analysis ............................................ 230
B.l Basic definitions ..................................... 230
8.2 Isotropie tensors ..................................... 232
8.3 Tensor products, contractions, and Einstein notation .. 233
B.4 Differential operators ................................ 234
B.5 Physical laws ......................................... 234
Exercises .................................................. 235
С Scattering processes ....................................... 236
C.l Classical mechanics ................................... 236
C.1.1 Kinematics of binary colhsions ................. 236
C.1.2 Geometrical parameterisation ................... 237
C.1.3 Scattering for hard sphere. Coulomb, and
gravitational potentials ....................... 237
C.2 Quantum mechanics ..................................... 238
C.2.1 Time-dependent perturbation theory ............. 238
C.2.2 Fermi golden rule .............................. 239
Exercises .................................................. 240
D Electronic structure in crystalline solids ................. 242
D.l Crystalline solids .................................... 242
D.2 Band structure ........................................ 242
D.2.1 Bloch theorem .................................. 243
D.2.2 Energy bands ................................... 245
D.2.3 Bloch velocity and crystal momentum equation ... 245
D.2.4 Self-consistent potential ...................... 246
D.3 Density of states ..................................... 247
D.3.1 Free electron gas .............................. 247
D.3.2 General case in three dimensions ............... 248
Exercises .................................................. 249
References .................................................... 250
Index ......................................................... 255
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