This is the first textbook to include the matrix continued-fraction method, which is very effective in dealing with simple Fokker-Planck equations having two variables. Other methods covered are the simulation method, the eigen-function expansion, numerical integration, and the variational method. Each solution is applied to the statistics of a simple laser model and to Brownian motion in potentials. The whole is rounded off with a supplement containing a short review of new material together with some recent references. This new study edition will prove to be very useful for graduate students in physics, chemical physics, and electrical engineering, as well as for research workers in these fields.
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Brownian Motion .......................................... 1
1.2 Fokker-Planck Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
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
2.4 Time-Dependent Random Variables. . . .. . ... . . ... . . .. ... . . . .. 25
2.5 Several Time-Dependent Random Variables ................... 30
3. Langevin Equations ........................................... 32
3.1 Langevin Equation for Brownian Motion ............. '" . . ... . 32
3.2 Ornstein-Uhlenbeck Process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3 Nonlinear Langevin Equation, One Variable .................. 44
3.4 Nonlinear Langevin Equations, Several Variables.. . . . . . . . . .. . . 54
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
4.3 Pawula Theorem ......................................... 70
4.4 Fokker-Planck Equation for One Variable. . . . . . . . . . . . . . . . . . . . 72
4.5 Generation and Recombination Processes .................... 76
4.6 Application of Truncated Kramers-Moyal Expansions . . . . . . . . . . 77
4.7 Fokker-Planck Equation for NVariables ..................... 81
4.8 Examples for Fokker-Planck Equations with Several Variables. . . 86
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-UWenbeck 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 Potentialfor 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 ofthe 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 a Complete Set ....................... 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 InitialValueProblem .............................. 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 ofthe 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 aConstant Field ............... 282
11.1.5 Phase-Locked Loop ............................... 283
11.1.6 Connection to the Sine-Gordon Equation ............. 285
11.2 Normalization ofthe 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 MatrixContinued-FractionMethod .................. 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 Langevin Equation ................................ 379
12.1.3 LaserFokker-PlanckEquation ...................... 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 EigenfunctionMethod ............................. 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 ofthe Fokker-Planck Equation ........ 429
A6 Fluctuating Control Parameter ............................ 431
References ...................................................... 436
Subject Index ................................................... 445
Fokker-Planck方程在物理学各个领域中有广泛的应用，如Fokker-Planck方程在核物理、粒子物理、量子物理、计算物理中的应用；在金融领域，Fokker-Planck方程常用于描述随机过程，在金融工程中有广泛的应用；著名的Black-Scholes-Merton模型即可以理解为Fokker-Planck方程的一个具体应用；此书之前论坛上有DJVU格式的，现上传pdf格式。
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