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The Fokker-Planck equation : methods of solution and applications

Author: Hannes Risken
Publisher: Berlin ; Paris : Springer, cop. 1989.
Series: Springer series in synergetics, 18; Springer series in synergetics, 18.
Edition/Format:   Print book : English : 2nd edView all editions and formats
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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  Read more...

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Document Type: Book
All Authors / Contributors: Hannes Risken
ISBN: 0387504982 9780387504988 354061530X 9783540615309 3540504982 9783540504986 0387130985 9780387130989
OCLC Number: 708362984
Description: 1 v. (XIV- 472 p.) : ill. ; 25 cm.
Contents: 1. Introduction.- 1.1 Brownian Motion.- 1.1.1 Deterministic Differential Equation.- 1.1.2 Stochastic Differential Equation.- 1.1.3 Equation of Motion for the Distribution Function.- 1.2 Fokker-Planck Equation.- 1.2.1 Fokker-Planck Equation for One Variable.- 1.2.2 Fokker-Planck Equation for N Variables.- 1.2.3 How Does a Fokker-Planck Equation Arise?.- 1.2.4 Purpose of the Fokker-Planck Equation.- 1.2.5 Solutions of the Fokker-Planck Equation.- 1.2.6 Kramers and Smoluchowski Equations.- 1.2.7 Generalizations of the Fokker-Planck Equation.- 1.3 Boltzmann Equation.- 1.4 Master Equation.- 2. Probability Theory.- 2.1 Random Variable and Probability Density.- 2.2 Characteristic Function and Cumulants.- 2.3 Generalization to Several Random Variables.- 2.3.1 Conditional Probability Density.- 2.3.2 Cross Correlation.- 2.3.3 Gaussian Distribution.- 2.4 Time-Dependent Random Variables.- 2.4.1 Classification of Stochastic Processes.- 2.4.2 Chapman-Kolmogorov Equation.- 2.4.3 Wiener-Khintchine Theorem.- 2.5 Several Time-Dependent Random Variables.- 3. Langevin Equations.- 3.1 Langevin Equation for Brownian Motion.- 3.1.1 Mean-Squared Displacement.- 3.1.2 Three-Dimensional Case.- 3.1.3 Calculation of the Stationary Velocity Distribution Function.- 3.2 Ornstein-Uhlenbeck Process.- 3.2.1 Calculation of Moments.- 3.2.2 Correlation Function.- 3.2.3 Solution by Fourier Transformation.- 3.3 Nonlinear Langevin Equation, One Variable.- 3.3.1 Example.- 3.3.2 Kramers-Moyal Expansion Coefficients.- 3.3.3 Ito's and Stratonovich's Definitions.- 3.4 Nonlinear Langevin Equations, Several Variables.- 3.4.1 Determination of the Langevin Equation from Drift and Diffusion Coefficients.- 3.4.2 Transformation of Variables.- 3.4.3 How to Obtain Drift and Diffusion Coefficients for Systems.- 3.5 Markov Property.- 3.6 Solutions of the Langevin Equation by Computer Simulation.- 4. Fokker-Planck Equation.- 4.1 Kramers-Moyal Forward Expansion.- 4.1.1 Formal Solution.- 4.2 Kramers-Moyal Backward Expansion.- 4.2.1 Formal Solution.- 4.2.2 Equivalence of the Solutions of the Forward and Backward Equations.- 4.3 Pawula Theorem.- 4.4 Fokker-Planck Equation for One Variable.- 4.4.1 Transition Probability Density for Small Times.- 4.4.2 Path Integral Solutions.- 4.5 Generation and Recombination Processes.- 4.6 Application of Truncated Kramers-Moyal Expansions.- 4.7 Fokker-Planck Equation for N Variables.- 4.7.1 Probability Current.- 4.7.2 Joint Probability Distribution.- 4.7.3 Transition Probability Density for Small Times.- 4.8 Examples for Fokker-Planck Equations with Several Variables.- 4.8.1 Three-Dimensional Brownian Motion without Position Variable.- 4.8.2 One-Dimensional Brownian Motion in a Potential.- 4.8.3 Three-Dimensional Brownian Motion in an External Force.- 4.8.4 Brownian Motion of Two Interacting Particles in an External Potential.- 4.9 Transformation of Variables.- 4.10 Covariant Form of the Fokker-Planck Equation.- 5. Fokker-Planck Equation for One Variable; Methods of Solution.- 5.1 Normalization.- 5.2 Stationary Solution.- 5.3 Ornstein-Uhlenbeck Process.- 5.4 Eigenfunction Expansion.- 5.5 Examples.- 5.5.1 Parabolic Potential.- 5.5.2 Inverted Parabolic Potential.- 5.5.3 Infinite Square Well for the Schrudinger Potential.- 5.5.4 V-Shaped Potential for the Fokker-Planck Equation.- 5.6 Jump Conditions.- 5.7 A Bistable Model Potential.- 5.8 Eigenfunctions and Eigenvalues of Inverted Potentials.- 5.9 Approximate and Numerical Methods for Determining Eigenvalues and Eigenfunctions.- 5.9.1 Variational Method.- 5.9.2 Numerical Integration.- 5.9.3 Expansion into a Complete Set.- 5.10 Diffusion Over a Barrier.- 5.10.1 Kramers' Escape Rate.- 5.10.2 Bistable and Metastable Potential.- 6. Fokker-Planck Equation for Several Variables; Methods of Solution.- 6.1 Approach of the Solutions to a Limit Solution.- 6.2 Expansion into a Biorthogonal Set.- 6.3 Transformation of the Fokker-Planck Operator, Eigenfunction Expansions.- 6.4 Detailed Balance.- 6.5 Ornstein-Uhlenbeck Process.- 6.6 Further Methods for Solving the Fokker-Planck Equation.- 6.6.1 Transformation of Variables.- 6.6.2 Variational Method.- 6.6.3 Reduction to an Hermitian Problem.- 6.6.4 Numerical Integration.- 6.6.5 Expansion into Complete Sets.- 6.6.6 Matrix Continued-Fraction Method.- 6.6.7 WKB Method.- 7. Linear Response and Correlation Functions.- 7.1 Linear Response Function.- 7.2 Correlation Functions.- 7.3 Susceptibility.- 8. Reduction of the Number of Variables.- 8.1 First-Passage Time Problems.- 8.2 Drift and Diffusion Coefficients Independent of Some Variables.- 8.2.1 Time Integrals of Markovian Variables.- 8.3 Adiabatic Elimination of Fast Variables.- 8.3.1 Linear Process with Respect to the Fast Variable.- 8.3.2 Connection to the Nakajima-Zwanzig Projector Formalism.- 9. Solutions of Tridiagonal Recurrence Relations, Application to Ordinary and Partial Differential Equations.- 9.1 Applications and Forms of Tridiagonal Recurrence Relations.- 9.1.1 Scalar Recurrence Relation.- 9.1.2 Vector Recurrence Relation.- 9.2 Solutions of Scalar Recurrence Relations.- 9.2.1 Stationary Solution.- 9.2.2 Initial Value Problem.- 9.2.3 Eigenvalue Problem.- 9.3 Solutions of Vector Recurrence Relations.- 9.3.1 Initial Value Problem.- 9.3.2 Eigenvalue Problem.- 9.4 Ordinary and Partial Differential Equations with Multiplicative Harmonic Time-Dependent Parameters.- 9.4.1 Ordinary Differential Equations.- 9.4.2 Partial Differential Equations.- 9.5 Methods for Calculating Continued Fractions.- 9.5.1 Ordinary Continued Fractions.- 9.5.2 Matrix Continued Fractions.- 10. Solutions of the Kramers Equation.- 10.1 Forms of the Kramers Equation.- 10.1.1 Normalization of Variables.- 10.1.2 Reversible and Irreversible Operators.- 10.1.3 Transformation of the Operators.- 10.1.4 Expansion into Hermite Functions.- 10.2 Solutions for a Linear Force.- 10.2.1 Transition Probability.- 10.2.2 Eigenvalues and Eigenfunctions.- 10.3 Matrix Continued-Fraction Solutions of the Kramers Equation.- 10.3.1 Initial Value Problem.- 10.3.2 Eigenvalue Problem.- 10.4 Inverse Friction Expansion.- 10.4.1 Inverse Friction Expansion for K0(t), G0,0(t) and L0(t).- 10.4.2 Determination of Eigenvalues and Eigenvectors.- 10.4.3 Expansion for the Green's Function Gn,m(t).- 10.4.4 Position-Dependent Friction.- 11. Brownian Motion in Periodic Potentials.- 11.1 Applications.- 11.1.1 Pendulum.- 11.1.2 Superionic Conductor.- 11.1.3 Josephson Tunneling Junction.- 11.1.4 Rotation of Dipoles in a Constant Field.- 11.1.5 Phase-Locked Loop.- 11.1.6 Connection to the Sine-Gordon Equation.- 11.2 Normalization of the Langevin and Fokker-Planck Equations.- 11.3 High-Friction Limit.- 11.3.1 Stationary Solution.- 11.3.2 Time-Dependent Solution.- 11.4 Low-Friction Limit.- 11.4.1 Transformation to E and x Variables.- 11.4.2 `Ansatz' for the Stationary Distribution Functions.- 11.4.3 x-Independent Functions.- 11.4.4 x-Dependent Functions.- 11.4.5 Corrected x-Independent Functions and Mobility.- 11.5 Stationary Solutions for Arbitrary Friction.- 11.5.1 Periodicity of the Stationary Distribution Function.- 11.5.2 Matrix Continued-Fraction Method.- 11.5.3 Calculation of the Stationary Distribution Function.- 11.5.4 Alternative Matrix Continued Fraction for the Cosine Potential.- 11.6 Bistability between Running and Locked Solution.- 11.6.1 Solutions Without Noise.- 11.6.2 Solutions With Noise.- 11.6.3 Low-Friction Mobility With Noise.- 11.7 Instationary Solutions.- 11.7.1 Diffusion Constant.- 11.7.2 Transition Probability for Large Times.- 11.8 Susceptibilities.- 11.8.1 Zero-Friction Limit.- 11.9 Eigenvalues and Eigenfunctions.- 11.9.1 Eigenvalues and Eigenfunctions in the Low-Friction Limit.- 12. Statistical Properties of Laser Light.- 12.1 Semiclassical Laser Equations.- 12.1.1 Equations Without Noise.- 12.1.2 Langevin Equation.- 12.1.3 Laser Fokker-Planck Equation.- 12.2 Stationary Solution and Its Expectation Values.- 12.3 Expansion in Eigenmodes.- 12.4 Expansion into a Complete Set; Solution by Matrix Continued Fractions.- 12.4.1 Determination of Eigenvalues.- 12.5 Transient Solution.- 12.5.1 Eigenfunction Method.- 12.5.2 Expansion into a Complete Set.- 12.5.3 Solution for Large Pump Parameters.- 12.6 Photoelectron Counting Distribution.- 12.6.1 Counting Distribution for Short Intervals.- 12.6.2 Expectation Values for Arbitrary Intervals.- Appendices.- A1 Stochastic Differential Equations with Colored Gaussian Noise.- A2 Boltzmann Equation with BGK and SW Collision Operators.- A3 Evaluation of a Matrix Continued Fraction for the Harmonic Oscillator.- A4 Damped Quantum-Mechanical Harmonic Oscillator.- A5 Alternative Derivation of the Fokker-Planck Equation.- A6 Fluctuating Control Parameter.- S. Supplement to the Second Edition.- S.1 Solutions of the Fokker-Planck Equation by Computer Simulation (Sect. 3.6).- S.2 Kramers-Moyal Expansion (Sect. 4.6).- S.3 Example for the Covariant Form of the Fokker-Planck Equation (Sect. 4.10).- S.4 Connection to Supersymmetry and Exact Solutions of the One Variable Fokker-Planck Equation (Chap. 5).- S.5 Nondifferentiability of the Potential for the Weak Noise Expansion (Sects. 6.6 and 6.7).- S.6 Further Applications of Matrix Continued-Fractions (Chap. 9).- S.7 Brownian Motion in a Double-Well Potential (Chaps. 10 and 11).- S.8 Boundary Layer Theory (Sect. 11.4).- S.9 Calculation of Correlation Times (Sect. 7.12).- S.10 Colored Noise (Appendix A1).- S.11 Fokker-Planck Equation with a Non-Positive-Definite Diffusion Matrix and Fokker-Planck Equation with Additional Third-Order-Derivative Terms.- References.
Series Title: Springer series in synergetics, 18; Springer series in synergetics, 18.
Responsibility: H. Risken.
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