The Fokker-Planck Equation: Methods of Solution and Applications (Springer Series in Synergetics, 18)

This book deals with the derivation of the Fokker-Planck equation, methods of solving it and some of its applications. Various methods such as the simulation method, the eigenfunction expansion, numerical integration, the variational method, and the matrix continued-fraction method are discussed. This is the first time that this last method, which is very effective in dealing with simple Fokker-Planck equations having two variables, appears in a textbook. The methods of solution are applied to the statistics of a simple laser model and to Brownian motion in potentials. Such Brownian motion is important in solid-state physics, chemical physics and electric circuit theory. This new study edition is meant as a text for graduate students in physics, chemical physics, and electrical engineering. Booknews Provides graduate students and researchers (physics, chemical physics, electrical engineering) with a beautifully organized and detailed account of the Fokker-Planck equation: how it arises, and how information is tickled out of it. Twelve chapters, plus appendices and a supplement outlining developments subsequent to the appearance (1984) of the 1st edition. Attractively printed on slick paper. (NW) Annotation c. Book News, Inc., Portland, OR (booknews.com) 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 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. Front Matter....Pages I-XIV Introduction....Pages 1-12 Probability Theory....Pages 13-31 Langevin Equations....Pages 32-62 Fokker-Planck Equation....Pages 63-95 Fokker-Planck Equation for One Variable; Methods of Solution....Pages 96-132 Fokker-Planck Equation for Several Variables; Methods of Solution....Pages 133-162 Linear Response and Correlation Functions....Pages 163-178 Reduction of the Number of Variables....Pages 179-195 Solutions of Tridiagonal Recurrence Relations, Application to Ordinary and Partial Differential Equations....Pages 196-228 Solutions of the Kramers Equation....Pages 229-275 Brownian Motion in Periodic Potentials....Pages 276-373 Statistical Properties of Laser Light....Pages 374-413 Back Matter....Pages 414-472 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.