The authors' note ............................................... v
1 Phenomenological thermodynamics of irreversible processes .. 1
1.1 Main postulates of non-equilibrium thermodynamics .......... 1
1.1.1 Thermodynamic description of equilibrium and non-
equilibrium systems ................................. 1
1.1.2 Local equilibrium principle ......................... 3
1.1.3 Entropy balance equation and conservation laws ...... 4
1.1.4 Generalized flows and generalized thermodynamic
forces .............................................. 7
1.1.5 Generalized transport coefficients and the Onsager
symmetry relations .................................. 9
1.1.6 Variational principles in linear non-equilibrium
thermodynamics ..................................... 10
1.1.7 Minimum entropy production principle for weakly
non-equilibrium steady states ...................... 12
1.2 On the application of the Onsager theory .................. 13
1.2.1 Thermoelectric phenomena. The Peltier, Seebeck,
Thomson effects and their relationship ............. 13
1.2.2 Effects in an external magnetic field .............. 19
1.3 Self-organization in highly non-equilibrium systems ....... 28
1.3.1 Non-equilibrium dissipative structures ............. 28
1.3.2 The Glansdorff-Prigogine universal evolution
criterion .......................................... 29
1.3.3 Ways of describing strongly non-equilibrium
systems ............................................ 31
1.3.4 Stability of states of highly non-equilibrium
systems ............................................ 34
1.3.5 The Lyapunov global stability criterion ............ 36
1.3.6 Dynamical systems with one degree of freedom ....... 38
1.3.7 Dynamical systems with two degrees freedom ......... 39
1.3.8 Dynamic chaos ...................................... 45
1.3.9 Dynamic chaos in one-dimensional mappings .......... 51
Problems to Chapter 1 ..................................... 57
2 Brownian Motion ........................................... 59
2.1 The Langevin equation for a Brownian particle ............. 59
2.1.1 Nature of motion of a Brownian particle. Random
forces ............................................. 59
2.1.2 Displacement of a Brownian particle ................ 64
2.2 The Fokker-Planck equation for a Brownian particle ........ 67
2.2.1 Derivation of the Fokker-Planck equation ........... 67
2.2.2 The solution of the Fokker-Planck equation ......... 69
Problems to Chapter 2 ..................................... 73
3 Kinetic equations in non-equilibrium statistical
mechanics ................................................. 74
3.1 Description of non-equilibrium systems in statistical
mechanics ................................................. 74
3.1.1 Integrableand nonintegrable dynamical systems ...... 74
3.1.2 The evolution of dynamical systems in phase space .. 76
3.2 Substantiation of quasiclassical kinetic equations ........ 82
3.2.1 The Liouville equation for the distribution
function ........................................... 82
3.2.2 The chain of the Bogoliubov equations .............. 84
3.2.3 Equation for the one-particle distribution. The
relaxation time approximation ...................... 88
3.2.4 The Vlasov kinetic equation for a collisionless
plasma ............................................. 89
3.2.5 The Boltzmann equation for a low-density gas ....... 96
3.2.6 Qualitative derivation of the Boltzmann equation ... 97
3.2.7 Derivation of the Boltzmann equation from the
Bogoliubov equations chain ........................ 103
3.2.8 The Fokker-Planck equation ........................ 106
3.3 Solving for kinetic equations ............................ 110
3.3.1 The solution of the Boltzmann equation for the
equilibrium state ................................. 110
3.3.2 The Boltzmann H-theorem ........................... 112
3.3.3 The Hilbert expansion ............................. 114
3.3.4 The Enskog-Chapman method. Derivation of
hydrodynamic equations ............................ 120
3.3.5 The method of moments ............................. 125
Problems to Chapter 3 .................................... 127
4 Kinetic equation for electrons and phonons in
conducting crystals ...................................... 129
4.1 Kinetic coefficients in the relaxation time
approximation ............................................ 129
4.1.1 Kinetic equation for electrons and its solution
to the relaxation time approximation .............. 129
4.1.2 Conditions of applicability for the quasi-
classical description of electrons in conducting
crystals .......................................... 134
4.1.3 How to determine charge and heat fluxes and
calculate kinetic coefficients when H = 0 ......... 137
4.1.4 Scattering of electrons by lattice vibrations ..... 147
4.1.5 The Hamiltonian of interaction between electrons
and charged impurity centers ...................... 151
4.1.6 The collision integral for the electron-phonon
interaction ....................................... 153
4.1.7 Phenomenon of phonon drag ......................... 158
4.1.8 Expressions for charge and heat fluxes in
a magnetic field. Tensor structure of kinetic
coefficients ...................................... 162
4.1.9 Galvanomagnetic and thermomagnetic effects in
semiconductors with a parabolic dispersion law .... 164
4.2 Hydrodynamic description of a hot electrons .............. 169
4.2.1 Transition to a hydrodynamic description .......... 169
4.2.2 The momentum balance equation ..................... 172
4.2.3 Balance equations of energy and particle number ... 178
4.2.4 Solving a set of balance equations. Applications
of hydrodynamic approach .......................... 180
4.2.5 Negative differential resistance .................. 182
Problems to Chapter 4 .................................... 186
5 Theory of linear response to an external mechanical
perturbation ............................................. 189
5.1 Electrical conductivity of an electron gas. The Kubo
method ................................................... 189
5.1.1 The Liouville equation and its solution ........... 189
5.1.2 Linear response of a dynamical system to an
external field .................................... 193
5.1.3 Calculation of electrical conductivity ............ 197
5.1.4 High-frequency magnetic susceptibility ............ 207
5.2 Electrical conductivity in a quantizing magnetic field ... 214
5.2.1 Charge and heat fluxes in a quantizing magnetic
field ............................................. 214
5.2.2 Dynamics of electron motion in a quantizing
magnetic field .................................... 216
5.2.3 The conductivity tensor in a quantizing magnetic
field ............................................. 218
5.2.4 The conductivity in the case quasielastic
scattering by phonons ............................. 221
5.3 Symmetry properties of correlation functions ............. 227
5.3.1 Additive conservation laws and selection rules
for averages ...................................... 227
5.3.2 Symmetry properties of correlation functions for
operations of spatial rotation, complex
conjugation and time reversal ..................... 230
Problems to Chapter 5 .................................... 235
6 Non-equilibrium statistical operator method .............. 238
6.1 Non-equilibrium and quasi-equilibrium statistical
operators ................................................ 238
6.1.1 Quasi-equilibrium distribution .................... 238
6.1.2 Extremal properties of a quasi-equilibrium
distribution. Thermodynamics of a quasi-
equilibrium ensemble .............................. 241
6.1.3 Boundary conditions and the Liouville equation
for the NSO ....................................... 248
6.1.4 Linear relaxation equations in the NSO-method ..... 251
6.2 The projection operators method in non-equilibrium
statistical mechanics .................................... 256
6.2.1 Why is it necessary to introduce projection
operators? ........................................ 256
6.2.2 The Mori projection operator method ............... 260
6.2.3 Using the Mori projection operators to calculate
conductivity ...................................... 267
6.2.4 Relationship between a linear variant of the
NSO-method and Mori's method ...................... 274
6.2.5 High-frequency susceptibility ..................... 277
6.2.6 Determination of non-equilibrium parameters by
the NSO-method .................................... 279
6.3 Hydrodynamic modes and singularity of dynamic
correlation functions .................................... 285
6.3.1 Spin diffusion .................................... 285
6.3.2 The fluctuation-dissipation theorem ............... 289
6.3.3 Long-range correlations and slow modes ............ 294
6.3.4 Bogoliubov inequality and 1/k2 divergence
theorem ........................................... 298
Problems to Chapter 6 .................................... 304
7 Response of a highly non-equilibrium system to a weak
measuring field .......................................... 307
7.1 NSO for highly non-equilibrium systems ................... 307
7.1.1 Set up of the problem. A boundary condition for
the NSO ........................................... 307
7.1.2 Generalized susceptibility of a non-equilibrium
system ............................................ 313
7.2 Projection operator for non-equilibrium systems .......... 317
7.2.1 Magnetic susceptibility ........................... 317
7.2.2 Electrical conductivity of highly non-
equilibrium systems ............................... 319
Problems to Chapter 7 .................................... 337
8 Master equation approach ................................. 338
8.1 The basic idea of the method ............................. 338
8.1.1 Problem statement ................................. 338
8.1.2 The Zwanzig kinetic equation ...................... 339
8.2 Master equation for the quasi-equilibrium distribution ... 342
8.2.1 Robertson projection operator ..................... 342
8.2.2 Use of the master equation to calculate kinetic
coefficients ...................................... 345
Problems to Chapter 8 .................................... 350
Bibliography .................................................. 353
Index ......................................................... 355
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