The Fokker-Planck Equation Methods of Solution and Applications
One of the central problems synergetics is concerned with consists in the study of macroscopic qualitative changes of systems belonging to various disciplines such as physics, chemistry, or electrical engineering. When such transitions from one state to another take place, fluctuations, i.e., random...
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| Format: | eBook |
| Language: | English |
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Berlin, Heidelberg
Springer Berlin Heidelberg
1984, 1984
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| Edition: | 1st ed. 1984 |
| Series: | Springer Series in Synergetics
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| Subjects: | |
| Online Access: | |
| Collection: | Springer Book Archives -2004 - Collection details see MPG.ReNa |
Table of Contents:
- 11.2 Normalization of the Langevin and Fokker-Planck Equations
- 11.3 High-Friction Limit
- 11.4 Low-Friction Limit
- 11.5 Stationary Solutions for Arbitrary Friction
- 11.6 Bistability between Running and Locked Solution
- 11.7 Instationary Solutions
- 11.8 Susceptibilities
- 11.9 Eigenvalues and Eigenfunctions
- 12. Statistical Properties of Laser Light
- 12.1 Semiclassical Laser Equations
- 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
- 12.5 Transient Solution
- 12.6 Photoelectron Counting Distribution
- Appendices
- References
- 4.8 Examples for Fokker-Planck Equations with Several Variables
- 4.9 Transformation of Variables
- 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
- 5.6 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
- 5.10 Diffusion Over a Barrier
- 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
- 7. Linear Response and Correlation Functions
- 7.1 Linear Response function
- 1. Introduction
- 1.1 Brownian Motion
- 1.2 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
- 2.4 Time-Dependent Random Variables
- 2.5 Several Time-Dependent Random Variables
- 3. Langevin Equations
- 3.1 Langevin Equation for Brownian Motion
- 3.2 Ornstein-Uhlenbeck Process
- 3.3 Nonlinear Langevin Equation, One Variable
- 3.4 Nonlinear Langevin Equations, Several Variables
- 3.5 Markov Property
- 3.6 Solutions of the Langevin Equation by Computer Simulation
- 4. Fokker-Planck Equation
- 4.1 Kramers-Moyal Forward Expansion
- 4.2 Kramers-Moyal Backward Expansion
- 4.3 Pawula Theorem
- 4.4 Fokker-Planck Equation for One Variable
- 4.5 Generation and Recombination Processes
- 4.6 Application of Truncated Kramers-Moyal Expansions
- 4.7 Fokker-Planck Equation for N Variables
- 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
- 8.3 Adiabatic Elimination of Fast Variables
- 9. Solutions of Tridiagonal Recurrence Relations, Application to Ordinary and Partial Differential Equations
- 9.1 Applications and Forms of Tridiagonal Recurrence Relations
- 9.2 Solutions of Scalar Recurrence Relations
- 9.3 Solutions of Vector Recurrence Relations
- 9.4 Ordinary and Partial Differential Equations with Multiplicative Harmonic Time-Dependent Parameters
- 9.5 Methods for Calculating Continued Fractions
- 10. Solutions of the Kramers Equation
- 10.1 Forms of the Kramers Equation
- 10.2 Solutions for a Linear Force
- 10.3 Matrix Continued-Fraction Solutions of the Kramers Equation
- 10.4 Inverse Friction Expansion
- 11. Brownian Motion in Periodic Potentials
- 11.1 Applications