The Fokker-Planck Equation: Methods of Solution and Applications (Springer Series in Synergetics) 2nd Edition by Hannes Risken (PDF)

Description

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.

User’s Reviews

Editorial Reviews: From the Back Cover 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.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐The book contents is OK. It is a classic reference in the field of stochastic processes and SDEs for natural scientists (engineers and financiers may find other titles more appealing), and it generally pays to have the title at hand.The caveat in this review is the following: the printing and binding quality of the book being sold really suck. I mean, really. It comes with flimsy covers (of magazine-like quality) filmed with chep plastic that is going to detach soon, and the binding will crack for sure as the book starts to get used.This Chinese quality standard comes from the fact that this kind of book is not printed in the usual sense anymore. It is produced “on demand:” when a customer orders it, it is printed (it seems that on a regular, $99 deskjet printer) and then bound by Amazon (by a third-party, of course) on agreement with Springer.More and more, after having been scammed a few times by now (see my other review on Ma’s “Statistical Mechanics” book), I will not buy technical books on paperback on Amazon anymore (except by Dover).

⭐For stoachstic processes with a physical perspective that avoids all the measure theoretic/filtration/submartingale hand wringing there are three books: Gardiner, van Kampen and this one, that I have found helpful. Of these I found this one to be the most intuitive, the least encumbered with hot-air proofs of adaptation and convergence and a host of concepts that have no use. Gardiner is undoubtably more complete, but it was only on reading the lucid description here of non-linear Langevin equations that I actually came to understand the meaning of the Stratanovich/Ito divide.. they correspond to different limiting procedures for approaching the delta function of Wiener noise correlation function.To those who want to build some intuition about stochastic processes, it’s an excellent first or second text I found it quite welcome especially for shoring up the confusing connections between the FP, and the propagator/symbol/generator/cumulant functions (which I find intuitive) and the Langevin-equation / SDE / random path approach (which I find annoying). The structure of the book is a little disorganized but the bottom line is that the content and chosen equations are tasteful and useful.Can you tell I’ve been really really enjoying this book?

⭐This book with Chapt. 12 on Statistical Properties of Laser Light is an example where the wish for general schemes creates selfmomentum that defeats the objective to describe real physics. The difficulties become serious when authors engage deeply in mathematical tools, here the Fokker-Planck equation, and impose methods onto systems for convenience, or to demonstrate an application – while compromising much on actual physics and the foundations. As a consequence, the missing integrity leaves readers behind in big uncertainty if obtained (formal) results represent, or do no longer represent (laser) physics.The approach in the book to “treat the electrical field classically, i.e., neglect its operator character[istic], provided that a proper classical noise source is added [with a] strength … so that it leads to the correct spontaneous emission rate” remains an ad hoc attempt (“semiclassical”) without systematic investigation of the involved approximations and their validity that depends upon the scope of the intended laser investigation. A test for evidence would have been to indicate how higher approximations should be obtained and how they affect the results obtained from the “simple laser model”.Chapt. 12 starts from macroscopic considerations, i.e., an equation for the (complex) amplitude “b” of the field: db/dt = beta*d*b – beta*|b|^2 * b (on the right side the first and second parts represent the pumping and (nonlinear) collision effects, respectively). To introduce fluctuations of the field, a formal Langevin term n(t) is then added heuristically, which implies the “simple laser model”. However, the delta-correlated Gaussian noise n(t) does not represent the real interactions; there is no true justification for simply adding this term. Finding a correct description of laser fluctuations requires studying the real physical mechanisms. One important source is the fluctuation in the pumping light itself. Therefore, the associated term (“beta*d”) represents actually a random fluctuation. Consequently, we face a problem and a stochastic differential equation very much different from the artificial application of the Langevin equation, or the equivalent Fokker-Planck equation. Regrettably, the book does not attempt corresponding answers although crucial for integrity that would be a responsibility of the author, not the reader.The book is not reader-friendly due to its difficult style of writing, apparently without a professional editing for clarity through better grammar. – The list of References is shockingly outdated (w/o exception all entries are older than 20 years). Note, the shown year of publication (1996) refers to a re-print, not to any revision of the 2nd edition which had been published years before, in 1989. It is very misleading, when the Synopsis to this book claims: “A supplement is included, containing a short review of new [!] material together with some recent [!] references.”

⭐This book is a classical reference in the subject of stochastic dynamics. It is a graduate level book written in clear and concise language. It covers all the basics about Langevin and Fokker-Planck equations (Chapters 3 and 4). In these chapters, Moyal expansion, Ito and Stratonovich interpretation of stochastic processes is presented carefully. Then they move on to study various methods of solving FP equation in the next 7 chapters. In the final chapter, FP equation and its application to Laser is discussed.I recommend reading this book along with Gardiner’s book (Handbook of Stochastic Methods) to anyone who wants to learn about stochastic dynamics seriously.

⭐I got the impression that there are very few good textbooks on the subject of random processes in continuous time and the Fokker-Planck equation, which are accessible for physicists. In this book the subject presented in a manner that I thought to be a good compromise between mathematical rigor and physical intuition. For example to the spirit of the book, white noise is introduced both from the point of view of a physicist (it has a very short correlation time etc) and from the point of view of a mathematician (as the “derivative” of a Wiener process). While I found the book not very friendly or easy to read, it was one of my main sources for self-learning this subject during my Ph. D. work. I found the book three years ago, own it for two years and keep learning from it until today. I recommend the book very much.

⭐You may also want to look at Van Kampen, stochastic processes in physics and chemistry.

⭐On August 1, 2011, Congress amended the CPSIA to exclude ordinary books and ordinary paper-based printed materials from the third party testing and certification requirements of the CPSIA. An “ordinary book” as defined by the amendment, means: “a book printed on paper or cardboard, printed with inks or toners, and bound and finished using a conventional method, and that is intended to be read or has educational value.” H.R.2715.The manufacturer of the book containing this web address only manufactures ordinary books bound and finished in a conventional manner; thus, the books produced by this manufacturer are not subject to the testing and certification requirements of the CPSIA.

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