Stochastic Processes in Physics and Chemistry (eBook)

Front Cover 1
Stochastic Processes in Physics and Chemistry 4
Copyright Page 5
TABLE OF CONTENTS 14
PREFACE TO THE FIRST EDITION 8
PREFACE TO THE SECOND EDITION 10
ABBREVIATED REFERENCES 11
PREFACE TO THE THIRD EDITION 12
Chapter I. STOCHASTIC VARIABLES 18
1. Definition 18
2. Averages 22
3. Multivariate distributions 27
4. Addition of stochastic variables 31
5. Transformation of variables 34
6. The Gaussian distribution 40
7. The central limit theorem 43
Chapter II. RANDOM EVENTS 47
1. Definition 47
2. The Poisson distribution 50
3. Alternative description of random events 52
4. The inverse formula 57
5. The correlation functions 58
6. Waiting times 61
7. Factorial correlation functions 64
Chapter III. STOCHASTIC PROCESSES 69
1. Definition 69
2. Stochastic processes in physics 72
3. Fourier transformation of stationary processes 75
4. The hierarchy of distribution functions 78
5. The vibrating string and random fields 81
6. Branching processes 86
Chapter IV. MARKOV PROCESSES 90
1. The Markov property 90
2. The Chapman–Kolmogorov equation 95
3. Stationary Markov processes 98
4. The extraction of a subensemble 103
5. Markov chains 106
6. The decay process 110
Chapter V. THE MASTER EQUATION 113
1. Derivation 113
2. The class of W-matrices 117
3. The long-time limit 121
4. Closed, isolated, physical systems 125
5. The increase of entropy 128
6. Proof of detailed balance 131
7. Expansion in eigenfunctions 134
8. The macroscopic equation 139
9. The adjoint equation 144
10. Other equations related to the master equation 146
Chapter VI. ONE-STEP PROCESSES 151
1. Definition the Poisson process
2. Random walk with continuous time 153
3. General properties of one-step processes 156
4. Examples of linear one-step processes 160
5. Natural boundaries 164
6. Solution of linear one-step processes with natural boundaries 166
7. Artificial boundaries 170
8. Artificial boundaries and normal modes 174
9. Nonlinear one-step processes 178
Chapter VII. CHEMICAL REACTIONS 183
1. Kinematics of chemical reactions 183
2. Dynamics of chemical reactions 188
3. The stationary solution 190
4. Open systems 193
5. Unimolecular reactions 195
6. Collective systems 199
7. Composite Markov processes 203
Chapter VIII. THE FOKKER–PLANCK EQUATION 210
1. Introduction 210
2. Derivation of the Fokker–Planck equation 214
3. Brownian motion 217
4. The Rayleigh particle 221
5. Application to one-step processes 224
6. The multivariate Fokker–PIanck equation 227
7. Kramers' equation 232
Chapter IX. THE LANGEVIN APPROACH 236
1. Langevin treatment of Brownian motion 236
2. Applications 238
3. Relation to Fokker–Planck equation 241
4. The Langevin approach 244
5. Discussion of the Itô–Stratonovich dilemma 249
6. Non-Gaussian white noise 254
7. Colored noise 257
Chapter X. THE EXPANSION OF THE MASTER EQUATION 261
1. Introduction to the expansion 261
2. General formulation of the expansion method 265
3. The emergence of the macroscopic law 271
4. The linear noise approximation 275
5. Expansion of a multivariate master equation 280
6. Higher orders 284
Chapter XI. THE DIFFUSION TYPE 290
1. Master equations of diffusion type 290
2. Diffusion in an external field 293
3. Diffusion in an inhomogeneous medium 296
4. Muitivariate diffusion equation 299
5. The limit of zero fluctuations 304
Chapter XII. FIRST-PASSAGE PROBLEMS 309
1. The absorbing boundary approach 309
2. The approach through the adjoint equation – Discrete case 315
3. The approach through the adjoint equation – Continuous case 320
4. The renewal approach 324
5. Boundaries of the Smoluchowski equation 329
6. First passage of non-Markov processes 336
7. Markov processes with large jumps 339
Chapter XIII. UNSTABLE SYSTEMS 343
1. The bistable system 343
2. The escape time 350
3. Splitting probability 354
4. Diffusion in more dimensions 358
5. Critical fluctuations 361
6. Kramers' escape problem 364
7. Limit cycles and fluctuations 372
Chapter XIV. FLUCTUATIONS IN CONTINUOUS SYSTEMS 380
1. Introduction 380
2. Diffusion noise 382
3. The method of compounding moments 384
4. Fluctuations in phase space density 388
5. Fluctuations and the Boltzmann equation 391
Chapter XV. THE STATISTICS OF JUMP EVENTS 400
1. Basic formulae and a simple example 400
2. Jump events in nonlinear systems 403
3. Effect of incident photon statistics 405
4. Effect of incident photon statistics – continued 409
Chapter XVI. STOCHASTIC DIFFERENTIAL EQUATIONS 413
1. Definitions 413
2. Heuristic treatment of multiplicative equations 416
3. The cumulant expansion introduced 422
4. The general cumulant expansion 424
5. Nonlinear stochastic differential equations 427
6. Long correlation times 433
Chapter XVII. STOCHASTIC BEHAVIOR OF QUANTUM SYSTEMS 439
1. Quantum probability 439
2. The damped harmonic oscillator 445
3. The elimination of the bath 453
4. The elimination of the bath – continued 457
5. The Schrödinger–Langevin equation and the quantum master equation 461
6. A new approach to noise 466
7. Internal noise 468
SUBJECT INDEX 474