Dmitrii Zubarev Vladimir Morozov Gerd Röpke Statistical Mechanics of Nonequilibrium Processes Volume 1: Basic Concepts, Kinetic Theory

Akademie Verlag Contents

1 Basic concepts of Statistical mechanics 11 1.1 Classical distribution functions 11 1.1.1 Distribution functions in phase space 11 1.1.2 Liouville's theorem 15 1.1.3 Classical Liouville equation 17 1.1.4 Time reversal in classical Statistical mechanics 20 1.2 Statistical Operators for quantum Systems 23 1.2.1 Pure quantum ensembles 24 1.2.2 Mixed quantum ensembles 27 1.2.3 Passage to the classical limit of the density matrix 29 1.2.4 Second quantization 34 1.2.5 Quantum Liouville equation 40 1.2.6 Time reversal in quantum Statistical mechanics 43 1.3 Entropy 48 1.3.1 Gibbs entropy 48 1.3.2 The Information entropy 54 1.3.3 Equilibrium Statistical ensembles 57 1.3.4 Extremal property of the microcanonical ensemble 59 1.3.5 Extremal property of the canonical ensemble 63 1.3.6 Extremal property of the grand canonical ensemble 66 1.3.7 Entropy and thermodynamic relations 68 1.3.8 Nernst's theorem 72 1.3.9 Equilibrium fluctuations of thermodynamic quantities 76 1.3.10 Equilibrium fluctuations of dynamical variables 79 Appendices to Chapter 1 84 1A Time-ordered evolution Operators 84 1B The maximum entropy for quantum ensembles 86 Problems to Chapter 1 87 8 Contents

2 Nonequilibrium Statistical ensembles 89 2.1 Relevant Statistical ensembles 90 2.1.1 Reduced description of nonequilibrium Systems 90 2.1.2 Relevant Statistical distributions 96 2.1.3 Entropy and thermodynamic relations in relevant ensembles . . 98 2.2 Examples of relevant distributions 100 2.2.1 Local equilibrium in classical fluids 100 2.2.2 Relevant distribution for classical gases 104 2.2.3 Relevant distribution for quantum gases 107 2.2.4 Diagonal relevant distribution for quantum Systems 114 2.2.5 Relevant distribution for weakly interacting Subsystems .... 115 2.3 The method of the nonequilibrium Statistical Operator 117 2.3.1 Retarded Solutions of the Liouville equation 117 2.3.2 Generalized transport equations for observables 123 2.3.3 Entropy production in nonequilibrium states 127 2.3.4 Perturbation expansions of Statistical distributions 129 2.3.5 Exponential form of the nonequilibrium distribution 132 2.3.6 Boundary conditions for the Liouville equation and the method of quasi-averages 136 2.4 Other approaches to the theory of nonequilibrium processes • 142 2.4.1 Zwanzig's projection method 143 2.4.2 Robertson's projection method 146 2.4.3 The method of ergodic conditions 149 2.5 Simple examples of nonequilibrium processes 154 2.5.1 Relaxation of the momentum of foreign particles in a medium . 154 2.5.2 The Pauli equation 160 2.5.3 Chemical reactions 164 Appendices to Chapter 2 172 2A Wick's theorem for nonequilibrium quantum gases 172 2B Some useful Operator identities 174 2C Properties of the projection Operators 176 2D Boundary conditions in the quantum scattering theory 179 2E Equivalence of nonequilibrium distributions 183 Problems to Chapter 2 186

3 The classical kinetic theory 188 3.1 Cluster expansions in classical kinetic theory 189 3.1.1 The generalized kinetic equation 189 3.1.2 Reduced distribution functions 192 3.1.3 Hierarchy of equations for reduced distribution functions . . . . 193 3.1.4 The Boltzmann kinetic equation 194 3.1.5 Cluster expansion of the collision integral 201 3.2 Diagram methods in kinetic theory 209 3.2.1 The diagram technique 209 3.2.2 Diagram representation of correlation functions 217 3.2.3 Diagram representation of the collision integral 220 3.2.4 Simple examples 223 3.3 Kinetic theory of nonideal gases 227 3.3.1 Boltzmann collision integral for nonideal gases 227 3.3.2 Three-particle events 229 3.3.3 Many-particle events 232 3.3.4 The relevant distribution for dense gases 238 3.3.5 The Enskog kinetic equation 244 3.4 Kinetic equations for plasmas 248 3.4.1 The simplest kinetic equations: Vlasov and Landau 248 3.4.2 The binary correlation function for plasmas 256 3.4.3 The Balescu-Lenard collision integral 262 3.4.4 Generalized collision integrals 264 Appendices to Chapter 3 268 3A Normal Solutions of the Boltzmann equation 268 3B Cluster expansions of distribution functions 276 3C The three-particle resolvent 277 3D Binary correlation functions for low-density gases 278 3E Evaluation of binary correlation functions for plasmas 280 Problems to Chapter 3 281

The quantum kinetic theory 285 4.1 Quantum Systems with weak interactions 286 4.1.1 The generalized quantum kinetic equations 286 4.1.2 The Markovian form of the collision term 290 4.1.3 The kinetic equation for the single-particle density matrix . . . 291 4.1.4 The quantum Vlasov equation 294 4.1.5 The dielectric function for collisionless plasmas 297 4.1.6 The collision integral in the Born approximation 302 4.1.7 Electron-phonon interactions in metals 303 4.2 Cluster expansions in quantum kinetic theory 305 4.2.1 Quantum hierarchy for reduced density matrices 306 4.2.2 The quantum Boltzmann equation 309 4.2.3 Scattering of electrons by impurities in crystals 315 4.3 Kinetic theory of quantum Systems beyond the Boltzmann equation . 324 4.3.1 The approximation of binary correlations 325 4.3.2 The collision integral for quantum plasmas 328 4.3.3 Relevant Statistical Operators for dense quantum Systems . . . 331 4.3.4 The quantum Enskog equation 335 10 Contents

4.4 Quantum kinetic processes in strong external nelds 341 4.4.1 The kinetic equation for weakly interacting Systems in an alter­ nating field 341 4.4.2 The gauge-invariant Wigner function 344 4.4.3 Kinetics of electrons in an electromagnetic field 347 4.4.4 The field corrections to electric conductivity 351 Appendices to Chapter 4 354 4A Rearrangement of the quantum collision integral 354 4B Conductivity in the electron-impurity System 357 4C The Markovian form of kinetic equations in an alternating field 362 Problems to Chapter 4 363

Bibliography 367

Index 372