The Fokker-Planck Equation: Methods of Solution and ApplicationsOne 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 processes, may play an im portant role. Over the past decades it has turned out that the Fokker-Planck equation pro vides a powerful tool with which the effects of fluctuations close to transition points can be adequately treated and that the approaches based on the Fokker Planck equation are superior to other approaches, e.g., based on Langevin equa tions. Quite generally, the Fokker-Planck equation plays an important role in problems which involve noise, e.g., in electrical circuits. For these reasons I am sure that this book will find a broad audience. It pro vides the reader with a sound basis for the study of the Fokker-Planck equation and gives an excellent survey of the methods of its solution. The author of this book, Hannes Risken, has made substantial contributions to the development and application of such methods, e.g., to laser physics, diffusion in periodic potentials, and other problems. Therefore this book is written by an experienced practitioner, who has had in mind explicit applications to important problems in the natural sciences and electrical engineering. |
Contents
1 | |
60 | 24 |
FokkerPlanck Equation | 103 |
FokkerPlanck Equation for Several Variables Methods of Solution | 133 |
Linear Response and Correlation Functions | 163 |
Langevin Equations | 169 |
Reduction of the Number of Variables | 179 |
Solutions of Tridiagonal Recurrence Relations Application to Ordinary | 196 |
Brownian Motion in Periodic Potentials | 276 |
88 | 302 |
FokkerPlanck Equation for One Variable Methods of Solution | 313 |
Statistical Properties of Laser Light | 374 |
Appendices | 414 |
S Supplement to the Second Edition | 436 |
References | 448 |
463 | |
Other editions - View all
The Fokker-Planck Equation: Methods of Solution and Applications Hannes Risken,Till Frank Limited preview - 1996 |
The Fokker-Planck Equation: Methods of Solution and Applications Hannes Risken No preview available - 2012 |
Common terms and phrases
approximation bistable Brownian motion c₁ calculate continued fraction correlation function cosine potential damping constants depend derivative detailed balance differential equations diffusion coefficients diffusion matrix distribution function drift and diffusion drift coefficient eigenfunctions eigenvalues equation of motion expression əxi finite Fokker-Planck equation Fokker-Planck operator follows Fourier transform Gaussian given Green's function Hermitian initial condition inserting integral iteration Kramers equation Langevin equation Langevin force Laplace transform laser matrix continued fractions matrix continued-fraction method natural boundary conditions noise normalized obtain one-variable Ornstein-Uhlenbeck process oscillator parameter particles periodic potential probability current probability density problem random variable recurrence relation Schrödinger equation Sect Smoluchowski equation solved stationary distribution stationary solution stochastic variable t₁ tion transition probability vanish vector recurrence relation velocity W₁ x₁ Xmax Xmin zero дх