1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Deterministic Differential Equation ..................... 1
1.1.2 Stochastic Differential Equation ........................ 2
1.1.3 Equation of Motion for the Distribution Function ......... 3
1.2 Fokker-Planck Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Fokker-Planck Equation for One Variable ............... 4
1.2.2 Fokker-Planck Equation for N Variables . . . . . . . . . . . . . . . . . 5
1.2.3 How Does a Fokker-Planck Equation Arise? ............. 5
1.2.4 Purpose of the Fokker-Planck Equation ................. 6
1.2.5 Solutions of the Fokker-Planck Equation ................ 7
1.2.6 Kramers and Smoluchowski Equations .................. 7
1.2.7 Generalizations of the Fokker-Planck Equation ........... 8
1.3 Boltzmann Equation ....................................... 9
1.4 Master Equation .......................................... 11
2. Probability Theory ............................................ 13
2.1 Random Variable and Probability Density ....... " .. " . .. . . . . . 13
2.2 Characteristic Function and Cumulants ....................... 16
2.3 Generalization to Several Random Variables ......... " . .. . . .. . 19
2.3.1 Conditional Probability Density ........................ 21
2.3.2 Cross Correlation .................................... 21
2.3.3 Gaussian Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Time-Dependent Random Variables .......................... 25
2.4.1 Classification of Stochastic Processes. . . . . . . . . . . . . . . . . . . . 26
2.4.2 Chapman-Kolmogorov Equation ....................... 28
2.4.3 Wiener-Khintchine Theorem ........................... 29
2.5 Several Time-Dependent Random Variables ................... 30
3. Langevin Equations ........................................... 32
3.1 Langevin Equation for Brownian Motion. .. . . . . . ... . . . . .. . .. . . 32
3.1.1 Mean-Squared Displacement ......... " .. " . . .. . .. . .. .. 34
3.1.2 Three-Dimensional Case .............................. 36
3.1.3 Calculation of the Stationary Velocity Distribution Function 36
3.2 Ornstein-Uhlenbeck Process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2.1 Calculation of Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2.2 Correlation Function ................................ 41
3.2.3 Solution by Fourier Transformation. . . . . . . . . . . . . . . . . . . . 42
3.3 Nonlinear Langevin Equation, One Variable ............. " .. . 44
3.3.1 Example ........................................... 45
3.3.2 Kramers-Moyal Expansion Coefficients. . . . . . . . . . . . . . . . . 48
3.3.3 Ito's and Stratonovich's Definitions. . . . . . . . . . . . . . . . . . . . 50
3.4 Nonlinear Langevin Equations, Several Variables ............ , . 54
3.4.1 Determination of the Langevin Equation from Drift and
Diffusion Coefficients ............................... 56
3.4.2 Transformation of Variables .......................... 57
3.4.3 How to Obtain Drift and Diffusion Coefficients for Systems 58
3.5 Markov Property ................. " ................. " .. . 59
3.6 Solutions of the Langevin Equation by Computer Simulation.. . . 60
4. Fokker-Planck Equation ....................................... 63
4.1 Kramers-Moyal Forward Expansion. . . . . . .. . .. . . ... . .. . .. ... 63
4.1.1 Formal Solution .................................... 66
4.2 Kramers-Moyal Backward Expansion . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2.1 Formal Solution .................................... 69
4.2.2 Equivalence of the Solutions of the Forward and Backward
Equations .......................................... 69
4.3 Pawula Theorem ......................................... 70
4.4 Fokker-Planck Equation for One Variable. . . . . . . . . . . . . . . . . . . . 72
4.4.1 Transition Probability Density for Small Times .......... 73
4.4.2 Path Integral Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.5 Generation and Recombination Processes .................... 76
4.6 Application of Truncated Kramers-Moyal Expansions . . . . . . . . . . 77
4.7 Fokker-Planck Equation for N Variables ..................... 81
4.7.1 Probability Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.7.2 Joint Probability Distribution ......................... 85
4.7.3 Transition Probability Density for Small Times .......... 85
4.8 Examples for Fokker-Planck Equations with Several Variables. . . 86
4.8.1 Three-Dimensional Brownian Motion without Position
Variable ........................................... 86
4.8.2 One-Dimensional Brownian Motion in a Potential. . . . . . . . 87
4.8.3 Three-Dimensional Brownian Motion in an External Force 87
4.8.4 Brownian Motion of Two Interacting Particles in an External
Potential .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.9 Transformation of Variables ............................... 88
4.10 Covariant Form of the Fokker-Planck Equation. . . . . . . . . . . . . . . 91
5. Fokker-Planck Equation for One Variable; Methods of Solution. . . . . . 96
5.1 Normalization ........................................... 96
5.2 Stationary Solution ....................................... 98
5.3 Ornstein-Uhlenbeck Process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.4 Eigenfunction Expansion .................................. 101
5.5 Examples................................................ 108
5.5.1 Parabolic Potential ................................. 108
5.5.2 Inverted Parabolic Potential ......................... 109
5.5.3 Infinite Square Well for the Schrodinger Potential. . . . . . . 110
5.5.4 V-Shaped Potential for the Fokker-Planck Equation. . . .. 111
5.6 Jump Conditions.. ... ...... . ... ... ... . ... ... ... . .. . . ... .. 112
5.7 A Bistable Model Potential ................................. 114
5.8 Eigenfunctions and Eigenvalues of Inverted Potentials ......... 117
5.9 Approximate and Numerical Methods for Determining
Eigenvalues and Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.9.1 Variational Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 120
5.9.2 Numerical Integration .............................. 120
5.9.3 Expansion into a Complete Set ....................... 121
5.10 Diffusion Over a Barrier ................................... 122
5.10.1 Kramers' Escape Rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 123
5.10.2 Bistable and Metastable Potential. . . . . . . . . . . . . . . . . . . .. 125
6. Fokker-Planck Equation for Several Variables; Methods of Solution .. 133
6.1 Approach of the Solutions to a Limit Solution. . . . . . . . . . . . . . . .. 134
6.2 Expansion into a Biorthogonal Set .......................... 137
6.3 Transformation of the Fokker-Planck Operator, Eigenfunction
Expansions .............................................. 139
6.4 Detailed Balance ......................................... 145
6.5 Ornstein-Uhlenbeck Process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 153
6.6 Further Methods for Solving the Fokker-Planck Equation ...... 158
6.6.1 Transformation of Variables ......................... 158
6.6.2 Variational Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
6.6.3 Reduction to an Hermitian Problem. . . . . . . . . . . . . . . . . .. 159
6.6.4 Numerical Integration .............................. 159
6.6.5 Expansion into Complete Sets. . . . . . . . . . . . . . . . . . . . . . .. 159
6.6.6 Matrix Continued-Fraction Method. . . . . . . . . . . . . . . . . . . 160
6.6.7 WKB Method. . ..... . ... . ... ... ... .... . .. . .. . . ..... 162
7. Linear Response and Correlation Functions ....................... 163
7.1 Linear Response Function ................................. 164
7.2 Correlation Functions ..................................... 166
7.3 Susceptibility ............................................ 172
8. Reduction of the Number of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . .. 179
8.1 First-Passage Time Problems ............................... 179
8.2 Drift and Diffusion Coefficients Independent of Some Variables 183
8.2.1 Time Integrals of Markovian Variables ................ 184
8.3 Adiabatic Elimination of Fast Variables. . . . . . . . . . . . . . . . . . . . . 188
8.3.1 Linear Process with Respect to the Fast Variable ....... 192
8.3.2 Connection to the Nakajima-Zwanzig Projector
Formalism ....................................... 194
9. Solutions of Tridiagonal Recurrence Relations, Application to Ordinary
and Partial Differential Equations .............................. 196
9.1 Applications and Forms of Tridiagonal Recurrence Relations. . . 197
9.1.1 Scalar Recurrence Relation ......................... 197
9.1.2 Vector Recurrence Relation. . . . . . . . . . . . . . . . . . . . . . . . . 199
9.2 Solutions of Scalar Recurrence Relations .................... 203
9.2.1 Stationary Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
9.2.2 Initial Value Problem .............................. 209
9.2.3 Eigenvalue Problem ............................... 214
9.3 Solutions of Vector Recurrence Relations. . . . . . . . . . . . . . . . . . . . 216
9.3.1 Initial Value Problem .............................. 217
9.3.2 Eigenvalue Problem ............................... 220
9.4 Ordinary and Partial Differential Equations with Multiplicative
Harmonic Time-Dependent Parameters ..................... 222
9.4.1 Ordinary Differential Equations. . . . . . . . . . . . . . . . . . . . . 222
9.4.2 Partial Differential Equations ....................... 225
9.5 Methods for Calculating Continued Fractions. . . . . . . . . . . . . . .. 226
9.5.1 Ordinary Continued Fractions. . . . . . . . . . . . . . . . . . . . . . . 226
9.5.2 Matrix Continued Fractions. . . . . . . . . . . . . . . . . . . . . . . .. 227
10. Solutions of the Kramers Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 229
10.1 Forms of the Kramers Equation . . . . . . . . . . . . . . . . . . . . . . . . . . .. 229
10.1.1 Normalization of Variables ......................... 230
10.1.2 Reversible and Irreversible Operators. . . . . . . . . . . . . . . . . 231
10.1.3 Transformation of the Operators .................... 233
10.1.4 Expansion into Hermite Functions ................... 234
10.2 Solutions for a Linear Force. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 237
10.2.1 Transition Probability ............................. 238
10.2.2 Eigenvalues and Eigenfunctions ..................... 241
10.3 Matrix Continued-Fraction Solutions of the Kramers Equation. 249
10.3.1 Initial Value Problem .............................. 251
10.3.2 Eigenvalue Problem ............................... 255
10.4 Inverse Friction Expansion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 257
10.4.1 Inverse Friction Expansion for Ko(t), Go,o(t) and Lo(t) . . 259
10.4.2 Determination of Eigenvalues and Eigenvectors. . . . . . . . 266
10.4.3 Expansion for the Green's Function Gn,m(t) ........... 268
10.4.4 Position-Dependent Friction ........................ 275
11. Brownian Motion in Periodic Potentials ......................... 276
11.1 Applications ............................................ 280
11.1.1 Pendulum........................................ 280
11.1.2 Superionic Conductor ....... , ................ " . '" 280
11.1.3 Josephson Tunneling Junction ...................... 281
11.1.4 Rotation of Dipoles in a Constant Field ............... 282
11.1.5 Phase-Locked Loop ............................... 283
11.1.6 Connection to the Sine-Gordon Equation ............. 285
11.2 Normalization of the Langevin and Fokker-Planck Equations .. 286
11.3 High-Friction Limit ...................................... 287
11.3.1 Stationary Solution ... '" ... , ...... , . . .. ... . . ... . .. 287
11.3.2 Time-Dependent Solution .......................... 294
11.4 Low-Friction Limit ...................................... 300
11.4.1 Transformation to E and x Variables ................. 301
11.4.2 'Ansatz' for the Stationary Distribution Functions . . . . . . 304
11.4.3 x-Independent Functions ........................... 306
11.4.4 x-Dependent Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
11.4.5 Corrected x-Independent Functions and Mobility. . . . . . . 310
11.5 Stationary Solutions for Arbitrary Friction .................. 314
11.5.1 Periodicity of the Stationary Distribution Function.. . .. 315
11.5.2 Matrix Continued-Fraction Method.. . . .. .. . .. . . . .. . . 317
11.5.3 Calculation of the Stationary Distribution Function .... 320
11.5.4 Alternative Matrix Continued Fraction for the Cosine
Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
11.6 Bistability between Running and Locked Solution ............ 328
11.6.1 Solutions Without Noise ........................... 329
11.6.2 Solutions With Noise .............................. 334
11.6.3 Low-Friction Mobility With Noise ................... 335
11.7 Instationary Solutions .................................... 337
11.7.1 Diffusion Constant ................................ 342
11.7.2 Transition Probability for Large Times ............... 343
11.8 Susceptibilities .......................................... 347
11.8.1 Zero-Friction Limit.. . . . .. . .. . . . .. . .. . .. . .. . . . .. .. . 355
11.9 Eigenvalues and Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
11.9.1 Eigenvalues and Eigenfunctions in the Low-Friction Limit 365
12. Statistical Properties of Laser Light ............................. 374
12.1 Semiclassical Laser Equations ............................. 377
12.1.1 Equations Without Noise .... , ...... ,. . .. . .. . . ... ... 377
12.1.2 LangevinEquation ................................ 379
12.1.3 Laser Fokker-Planck Equation ...... ,. . .. . .. . . .. . .. . 382
12.2 Stationary Solution and Its Expectation Values. . . . . . . . . . . . . . . 384
12.3 Expansion in Eigenmodes ................................. 387
12.4 Expansion into a Complete Set; Solution by Matrix Continued
Fractions ............................................... 394
12.4.1 Determination of Eigenvalues ....................... 396
12.5 Transient Solution ....................................... 398
12.5.1 Eigenfunction Method ............................. 398
12.5.2 Expansion into a Complete Set ...................... 401
12.5.3 Solution for Large Pump Parameters. . . .. .. ... . . ... .. 404
12.6 Photoelectron Counting Distribution ....................... 408
12.6.1 Counting Distribution for Short Intervals ............. 409
12.6.2 Expectation Values for Arbitrary Intervals ............ 412
Appendices ..................................................... 414
A1 Stochastic Differential Equations with Colored Gaussian Noise 414
A2 Boltzmann Equation with BGK and SW Collision Operators .. , 420
A3 Evaluation of a Matrix Continued Fraction for the Harmonic
Oscillator .............................................. 422
A4 Damped Quantum-Mechanical Harmonic Oscillator .......... 425
A5 Alternative Derivation of the Fokker-Planck Equation ........ 429
A6 Fluctuating Control Parameter ............................ 431
S. Supplement to the Second Edition ............................... 436
S.1 Solutions of the Fokker-Planck Equation by Computer
Simulation (Sect. 3.6) .................................... 436
S.2 Kramers-Moyal Expansion (Sect. 4.6) . . . . . . . . . . . . . . . . . . . . . . . 436
S.3 Example for the Covariant Form of the Fokker-Planck Equation
(Sect. 4.10) ............................................. 437
S.4 Connection to Supersymmetry and Exact Solutions of the
One Variable Fokker-Planck Equation (Chap. 5) ............. 438
S.5 Nondifferentiability of the Potential for the Weak Noise
Expansion (Sects. 6.6 and 6.7) ............................. 438
S.6 Further Applications of Matrix Continued-Fractions
(Chap. 9) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 439
S.7 Brownian Motion in a Double-Well Potential
(Chaps. 10 and 11) ....................................... 439
S.8 Boundary Layer Theory (Sect. 11.4) ........................ 440
S.9 Calculation of Correlation Times (Sect. 7.12) ................ 441
S.10 Colored Noise (Appendix A1) ............................. 443
S.11 Fokker-Planck Equation with a Non-Positive-Definite Diffusion
Matrix and Fokker-Planck Equation with Additional Third-
Order-Derivative Terms .................................. 445
References ...................................................... 448
Subject Index ................................................... 463
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