The Fokker-Planck Equation: Methods of Solution and Applications (Springer Series in Synergetics)
H. Risken | 1900-01-01 00:00:00 | Springer-Verlag | 472 | Mathematical Physics
The Fokker-Planck equation deals with those fluctuations of systems which stem from many tiny disturbances, each of which changes the variable of the system in an unpredictable way. In this book, 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 electrical circuit theory. In this second edition, some misprints have been corrected and a supplement is included, containing a short review of new material together with some recent references.
Reviews
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the missing integrity leaves readers behind in big uncertainty if obtained (formal) results represent, or do no longer represent (laser) physics.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."
Reviews
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.
Reviews
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.
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