Werner Ebeling

Igor M. Sokolov

Humboldt-Universität, Germany

STATISTICAL THERMODYNAMICS * AND STOCHASTIC THEORY OF • NONEQUILIBRIUM SYSTEMS

\jj^ World Scientific NEW J E R S E Y ' - L O N D O N • S I N G A P O R E • BEIJING • S H A N G H A I • H O N G K O N G • TAIPEI • C H E N N A I

Contents

Preface v

Chapter 1 Introduction 1 1.1 The task of statistical physics 1 1.2 On history of fundamentals of statistical thermodynamics 3 1.3 On history of the concept of Brownian motion 12

Chapter 2 Thermodynamic, Deterministic and Stochastic Levels of Description 19

2.1 Thermodynamic level: The fundamental laws 19 2.2 Lyapunov functions: Entropy and thermodynamic potentials 28 2.3 Energy, entropy, and work 32 2.4 Deterministic models on the mesoscopic level 39 2.5 Stochastic models on the mesoscopic level 49

Chapter 3 Reversibility and Irreversibility, Liouville and Markov Equations 57

3.1 Boltzmann's kinetic theory 57 3.2 Probability measures and ergodic theorems 64 3.3 Dynamics and probability for one-dimensional maps 69 3.4 Hamiltonian dynamics: The Liouville equation 73 3.5 Markov models 79

Chapter 4 Entropy and Equilibrium Distributions 83 4.1 The Boltzmann-Planck principle 83 4.2 Isolated systems: The microcanonical distribution 87 4.3 Gibbs distributions for closed and for open systems 89 4.4 The Gibbs-Jaynes maximum entropy principle 92

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Chapter 5 Fluctuations and Linear Irreversible Processes 97 5.1 Einstein's theory of fluctuations 97 5.2 Fluctuations of many variables 100 5.3 Onsager's theory of linear relaxation processes 102 5.4 Correlations and spectra of stationary processes near equilibrium . . . . 105 5.5 Symmetry relations and generalizations 109

Chapter 6 Nonequilibrium Distribution Functions 111 6.1 A simple example — driven Brownian particles I l l 6.2 Canonical-dissipative systems 115 6.3 Microcanonical non-equilibrium ensembles 118 6.4 Systems driven by energy depots 120 6.5 Systems of particles coupled to Nose-Hoover thermostats 126 6.6 Nonequilibrium distributions from information-theoretical

methods 131

Chapter 7 Brownian Motion 139 7.1 Einstein's concept of Brownian motion 139 7.2 The Langevin equation: Theme and variations 141

7.2.1 The Langevin equation 141 7.2.2 Thermalization of the velocity: The fluctuation-dissipation

theorem 142 7.2.3 The properties of the noise 144

7.3 Taylor-Kubo formula and velocity-velocity correlations 146 7.4 The overdamped limit 148 7.5 Example: The Ornstein-Uhlenbeck process 150 7.6 The path integral representation 153 7.7 A non-Markovian Langevin equation 155

Chapter 8 Fokker—Planck and Master Equations 161 8.1 Equations for the probability density 161 8.2 Special stochastic processes 165

8.2.1 Example 1: The Ornstein-Uhlenbeck process revisited 165 8.2.2 Example 2: The Klein-Kramers equation 167

8.3 The Fokker-Planck equation and the Liouville equation 169 8.4 Transition rates and master equations 172 8.5 Energy diffusion and detailed balance 173 8.6 System in contact with several heat baths 176

Chapter 9 Escape and First Passage Problems 183 9.1 General considerations 183 9.2 The renewal approach 185 9.3 Example: Free diffusion in presence of boundaries 188

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9.4 Mean life time in a potential well 189 9.4.1 The flow-over-population approach 189 9.4.2 The Arrhenius law 191 9.4.3 Diffusion in a double-well 192

9.5 Moments of the first passage time 194 9.6 An underdamped situation 196

Chapter 10 Reaction Kinetics 199 10.1 The mass action law 199 10.2 Classical kinetics 202 10.3 The Smoluchowski approximation 203 10.4 Fluctuation effects in chemical kinetics 207

10.4.1 The Vaks-Balagurov kinetics in trapping 208 10.4.1.1 The Vaks-Balagurov problem in one dimension . . . . 209 10.4.1.2 General asymptotic behavior in trapping 212

10.4.2 Fluctuation-dominated kinetics in A + В —> 0 Reaction 215 10.4.3 Autocatalytic reactions and front propagation phenomena . . . . 221

Chapter 11 Random Walk Approaches 225 11.1 The random walk approach to transport processes 226

11.1.1 Random walks on lattices 228 11.1.2 The continuous-time random walks (CTRW) 232 11.1.3 CTRW and the master equation 235

11.2 Power-law waiting-time distributions 236 11.3 Aging behavior of CTRW systems 241 11.4 CTRW and fractional Fokker-Planck equations 245 11.5 Superdiffusion: Levy flights and Levy walks 251

Chapter 12 Active Brownian Motion 261 12.1 The general model 261

12.1.1 Self-propelling of Brownian particles 261 12.1.2 Equations of motions 263

12.2 Force-free motion of active particles and mean square displacement . . . 268 12.2.1 Distribution functions 268 12.2.2 Mean-square displacement 270

12.3 Deterministic motion in external potentials 273 12.3.1 Parabolic confinement 273 12.3.2 Perturbed and transient limit cycles 276

12.4 Stochastic motion in symmetric external potentials 279 12.5 Collective dynamics of clusters and swarms 285

12.5.1 Dynamics of active clusters and swarms 285 12.5.2 Confined systems of active particles 287 12.5.3 Dynamics of self-confined driven particles 292 12.5.4 The influence of hydrodynamic Oseen-type interactions 300

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References 305

Index 327