Authored by a well-known expert in the field of nonequilibrium statistical physics, this book is a coherent presentation of the subject suitable for masters and Ph D students, as well as postdocs in physics and related disciplines.
Starting from a general discussion of irreversibility and entropy, the method of nonequilibrium statistical operator is presented as a general concept. Stochastic processes are introduced as a necessary prerequisite to describe the evolution of a nonequilibrium state. Different standard approaches such as master equations, kinetic equations and linear response theory, are derived after special assumptions. This allows for an insight into the problems of nonequilibrium physics, a discussion of the limits of the approaches, and suggestions for improvements. The method of thermodynamic Green’s function is outlined that allows for the systematic quantum statistical treatment of many-body systems. Applications and typical examples are given, as well as fully worked problems.
Зміст
Preface XI
1 Introduction 1
1.1 Irreversibility: The Arrow of Time 2
1.1.1 Dynamical Systems 3
1.1.2 Thermodynamics 7
1.1.3 Ensembles and Probability Distribution 9
1.1.4 Entropy in Equilibrium Systems 11
1.1.5 Fundamental Time Arrows, Units 14
1.1.6 Example: Ideal Quantum Gases 17
1.2 Thermodynamics of Irreversible Processes 19
1.2.1 Quasiequilibrium 19
1.2.2 Statistical Thermodynamics with Relevant Observables 22
1.2.3 Phenomenological Description of Irreversible Processes 25
1.2.4 Example: Reaction Rates 29
1.2.5 Principle of Weakening of Initial Correlations and the Method of Nonequilibrium Statistical Operator 31
Exercises 38
2 Stochastic Processes 41
2.1 Stochastic Processes with Discrete Event Times 42
2.1.1 Potentiality and Options, Chance and Probabilities 43
2.1.2 Stochastic Processes 46
2.1.3 Reduced Probabilities 50
2.1.4 Properties of Probability Distributions: Examples 54
2.1.5 Example: One-Step Process on a Discrete Space–Time Lattice and Random Walk 58
2.2 Birth-and-Death Processes and Master Equation 61
2.2.1 Continuous Time Limit and Master Equation 63
2.2.2 Example: Radioactive Decay 67
2.2.3 Spectral Density and Autocorrelation Functions 69
2.2.4 Example: Continuum Limit of Random Walk and Wiener Process 76
2.2.5 Further Examples for Stochastic One-Step Processes 78
2.2.6 Advanced Example: Telegraph Equation and Poisson Process 84
2.3 Brownian Motion and Langevin Equation 89
2.3.1 Langevin Equation 89
2.3.2 Solution of the Langevin Equation by Fourier Transformation 94
2.3.3 Example Calculations for a Langevin Process on Discrete Time 95
2.3.4 Fokker–Planck Equation 96
2.3.5 Application to Brownian Motion 105
2.3.6 Important Continuous Markov Processes 107
2.3.7 Stochastic Differential Equations and White Noise 109
2.3.8 Applications of Continuous Stochastic Processes 110
Exercises 113
3 Quantum Master Equation 117
3.1 Derivation of the Quantum Master Equation 119
3.1.1 Open Systems Interacting with a Bath 119
3.1.2 Derivation of the Quantum Master Equation 124
3.1.3 Born–Markov and Rotating Wave Approximations 127
3.1.4 Example: Harmonic Oscillator in a Bath 132
3.1.5 Example: Atom Coupled to the Electromagnetic Field 135
3.2 Properties of the Quantum Master Equation and Examples 138
3.2.1 Pauli Equation 138
3.2.2 Properties of the Pauli Equation, Examples 143
3.2.3 Discussion of the Pauli Equation 146
3.2.4 Example: Linear Coupling to the Bath 148
3.2.5 Quantum Fokker–Planck Equation 151
3.2.6 Quantum Brownian Motion and the Classical Limit 154
Exercises 156
4 Kinetic Theory 157
4.1 The Boltzmann Equation 158
4.1.1 Distribution Function 159
4.1.2 Classical Reduced Distribution Functions 163
4.1.3 Quantum Statistical Reduced Distribution Functions 166
4.1.4 The Stoßzahlansatz 169
4.1.5 Derivation of the Boltzmann Equation from the Nonequilibrium Statistical Operator 173
4.1.6 Properties of the Boltzmann Equation 180
4.1.7 Example: Hard Spheres 181
4.1.8 Beyond the Boltzmann Kinetic Equation 183
4.2 Solutions of the Boltzmann Equation 186
4.2.1 The Linearized Boltzmann Equation 187
4.2.2 Relaxation Time Method 189
4.2.3 The Kohler Variational Principle 194
4.2.4 Example: Thermal Conductivity in Gases 196
4.3 The Vlasov–Landau Equation and Hydrodynamic Equations 199
4.3.1 Derivation of the Vlasov Equation 199
4.3.2 The Landau Collision Term 201
4.3.3 Example for the Vlasov Equation: The RPA Dielectric Function 203
4.3.4 Equations of Hydrodynamics 206
4.3.5 General Remarks to Kinetic Equations 213
Exercises 214
5 Linear Response Theory 217
5.1 Linear Response Theory and Generalized Fluctuation–Dissipation Theorem (FDT) 218
5.1.1 External Fields and Relevant Statistical Operator 219
5.1.2 Nonequilibrium Statistical Operator for Linear Response Theory 222
5.1.3 Response Equations and Elimination of Lagrange Multipliers 225
5.1.4 Example: Ziman Formula for the Conductivity and Force–Force Correlation Function 226
5.1.5 The Choice of Relevant Observables and the Kubo Formula 230
5.2 Generalized Linear Response Approaches 235
5.2.1 Thermal Perturbations 236
5.2.2 Example: Thermoelectric Effects in Plasmas 239
5.2.3 Example: Hopping Conductivity of Localized Electrons 243
5.2.4 Time-Dependent Perturbations 246
5.2.5 Generalized Linear Boltzmann Equation 249
5.2.6 Variational Approach to Transport Coefficients 251
5.2.7 Further Results of Linear Response Theory 254
Exercises 259
6 Quantum Statistical Methods 261
6.1 Perturbation Theory for Many-Particle Systems 262
6.1.1 Equilibrium Statistics of Quantum Gases 262
6.1.2 Three Relations for Elementary Perturbation Expansions 267
6.1.3 Example: Equilibrium Correlation Functions in Hartree–Fock Approximation 274
6.2 Thermodynamic Green’s Functions 279
6.2.1 Thermodynamic Green’s Functions: Definitions and Properties 280
6.2.2 Green’s Function and Spectral Function 285
6.2.3 Example: Thermodynamic Green’s Function for the Ideal Fermi Gas 289
6.2.4 Perturbation Theory for Thermodynamic Green’s Functions 291
6.2.5 Application of the Diagram Rules: Hartree–Fock Approximation 297
6.3 Partial Summation and Many-Particle Phenomena 300
6.3.1 Mean-Field Approximation and Quasiparticle Concept 301
6.3.2 Dyson Equation and Self-Energy 304
6.3.3 Screening Equation and Polarization Function 307
6.3.4 Lowest Order Approximation for the Polarization Function: RPA 312
6.3.5 Bound States 314
6.3.6 Excursus: Solution to the Two-Particle Schr€odinger Equation with a Separable Potential 318
6.3.7 Cluster Decomposition and the Chemical Picture 324
6.4 Path Integrals 329
6.4.1 The Onsager–Machlup Function 329
6.4.2 Dirac Equation in 1 þ 1 Dimensions 332
Exercises 335
7 Outlook: Nonequilibrium Evolution and Stochastic Processes 337
7.1 Stochastic Models for Quantum Evolution 338
7.1.1 Measuring Process and Localization 339
7.1.2 The Caldeira–Leggett Model and Quantum Brownian Motion 342
7.1.3 Dynamical Reduction Models 345
7.1.4 Stochastic Quantum Electrodynamics 347
7.1.5 Quantum Dynamics and Quantum Evolution 349
7.2 Examples 353
7.2.1 Scattering Theory 353
7.2.2 Bremsstrahlung Emission 355
7.2.3 Radiation Damping 359
7.2.4 The 1/f (Flicker) Noise 360
7.2.5 The Hydrogen Atom in the Radiation Field 362
7.2.6 Comments on Nonequilibrium Statistical Physics 365
References 371
Index 375
Про автора
Gerd Röpke is professor of Theoretical Physics at the University of Rostock, Germany. Having obtained his academic degrees from the University of Leipzig, he spent most of his career working at the Technical University Dresden before the appointment at Rostock. Professor Röpke has authored over 400 scientific publications on quantum statistics, nonequilibrium statistical mechanics, plasma physics and nuclear theory, including several monographs, and he received different awards. He is a member of the Saxonian Academy of Sciences and external member of the Max-Planck Society.
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