The Fokker-Planck Equation pdf epub mobi txt 电子书 下载 2026
出版者:Springer Verlag
作者:Risken, Hannes
出品人:
页数:486
译者:
出版时间:1996-9
价格:$ 111.87
装帧:Pap
isbn号码:9783540615309
丛书系列:Springer Series in Synergetics
图书标签:
- 数学
- 物理
- 英文原版
- The
- Physics
- Fokker-Planck equation
- Diffusion processes
- Stochastic differential equations
- Mathematical physics
- Probability theory
- Partial differential equations
- Non-equilibrium statistical mechanics
- Financial modeling
- Physics
- Engineering
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具体描述
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.
好的，这是一份关于一本名为《随机过程中的路径积分》的图书简介，该书内容与《The Fokker-Planck Equation》无关，并且力求内容详实，不露痕迹。 --- 图书名称：《随机过程中的路径积分：从理论基础到应用实践》 作者： [此处留空，模拟真实书籍信息] 出版社： [此处留空，模拟真实书籍信息] 出版年份： [此处留空，模拟真实书籍信息] --- 简介：跨越时空的桥梁——随机过程的路径积分视角 在现代物理学、金融数学以及复杂系统科学的广阔图景中，描述时间演化和不确定性行为的方法论始终是核心难题。《随机过程中的路径积分：从理论基础到应用实践》正是这样一本旨在系统性梳理和深入探讨随机过程理论中“路径积分”这一强大工具的专著。本书并非聚焦于某一特定微分方程的解析解法，而是将视角提升到更宏观、更具操作性的层面——如何通过对所有可能路径的累积求和来理解和预测系统的动态演变。 本书的撰写，旨在填补现有教材中对路径积分概念在随机动力学语境下阐述不足的空白。我们深知，路径积分，作为费曼的伟大遗产，在量子力学中取得了无可匹敌的成就，但其思想精髓——将系统从初始态到终态的演化视为所有可能轨迹的叠加——在处理热力学涨落、布朗运动、以及金融市场建模等经典随机系统中，同样展现出惊人的威力。 全书结构清晰，从基础概念的奠定开始，逐步深入到复杂的应用领域，确保读者，无论是来自数学、物理还是工程背景，都能构建起坚实的知识体系。 第一部分：随机过程的基石与积分的酝酿 本部分首先回顾了概率论和随机过程的必要背景知识，重点在于对马尔可夫过程、伊藤积分以及随机微分方程（SDEs）的深入理解。我们强调，SDEs虽然描述了瞬时演化，但其解本质上是概率测度在函数空间上的分布。 随后，我们将引入路径积分的核心思想——概率振幅的类比。在经典随机系统中，我们不讨论量子力学中的复值振幅，而是关注概率密度函数本身。路径积分的构建，是通过将时间离散化，然后对所有可能的中间路径进行加权平均来实现的。这一构建过程，深刻地揭示了路径积分作为演化算符的表示的本质。我们将详细讨论时间分裂（Time-Slicing）技术的数学细节，以及如何通过控制离散化步长的极限来恢复连续时间下的描述。 第二部分：核心理论：路径积分的构造与变换 在奠定了基础之后，本书的核心部分将集中于路径积分在不同随机框架下的具体构造方法。我们不会局限于单一的随机场模型，而是将讨论拓展到更一般的随机动力学系统。 变分原理与作用量： 路径积分的核心在于一个“作用量”（Action）泛函。在随机系统中，这一“作用量”往往与欧拉-拉格朗日方程或最小作用量原理有着深刻的联系，尽管这联系可能隐藏在概率的对数形式中。我们将探讨如何从SDE的扩散项和漂移项出发，构造出相应的欧拉作用量（Eulerian Action）或随机作用量。重点分析“噪声项”如何影响作用量泛函的结构，以及如何利用Onsager-Machlup函数来精确刻画路径的“阻力”或“能量耗散”。 概率的生成函数： 本书将路径积分与概率的生成函数（如矩函数生成函数、累积量生成函数）紧密联系起来。我们展示了路径积分如何作为计算这些生成函数的强大工具，尤其是在处理高阶矩和罕见事件概率时。 路径积分的“量子化”类比： 我们将详细探讨如何将路径积分应用于统计力学中的配分函数（Partition Function）计算。在统计物理中，配分函数与时间演化算符的迹（Trace）之间存在直接的映射关系。通过在有限温度下的路径积分，我们可以绕过复杂的格点求和，直接求解系统的宏观热力学量。 第三部分：应用前沿：从金融衍生品到复杂网络 路径积分方法的优越性在于其处理多维、非线性和高频扰动环境的能力。本书的第三部分旨在展示其在实际领域中的强大应用。 金融衍生品定价的非线性修正： 在不完全市场模型中，衍生品定价往往依赖于涉及随机波动率或随机利率的模型（例如Heston模型的一般化）。我们展示了如何利用路径积分来计算风险中性定价下欧式或奇异期权的贴现期望值。路径积分方法在这里的优势在于，它能够自然地处理由波动率变化带来的路径依赖性，提供比传统偏微分方程（PDE）方法更直观的路径分析。 有效势与平均场理论的扩展： 对于具有强相互作用的粒子系统或复杂网络，传统的平均场方法往往失效。我们将路径积分应用于噪声驱动的平均场动力学，构建有效作用量。通过对作用量进行鞍点近似或半经典近似（在随机背景下的类比），我们可以有效地提取出系统的宏观有效势，从而理解系统在扰动下的相变行为和稳态分布。 极端事件的概率估计： 在可靠性工程和风险管理中，计算系统发生极端故障的概率至关重要。本书介绍了大偏差理论（Large Deviation Theory）与路径积分的紧密结合。通过寻找“最小作用量路径”（即最有可能导致极端事件的路径），我们可以精确估计指数衰减项，这是传统线性方法难以企及的。 结论与展望 《随机过程中的路径积分：从理论基础到应用实践》并非一本简单的计算手册，它旨在培养读者用“路径集合”的思维方式来审视随机现象的直觉。本书的最终目标是使读者掌握如何将复杂的时间演化问题转化为对某一特定泛函（作用量）的积分问题，从而在理论探索和实际建模中获得极大的灵活性和洞察力。我们相信，路径积分这一跨越经典与量子、确定性与随机性的强大工具，将是未来处理复杂动态系统的必备利器。 ---
作者简介
Professor Dr. Hannes Risken
Abteilung fur Theoretische Physik, Universitat Ulm, Oberer Eselsberg, D-89081 Ulm, Germany
目录信息
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
· · · · · · (收起)
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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