Fokker Planck Kolmogorov Equations
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| Author | : Vladimir I. Bogachev |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 495 |
| Release | : 2015-12-17 |
| Genre | : Mathematics |
| ISBN | : 1470425580 |
This book gives an exposition of the principal concepts and results related to second order elliptic and parabolic equations for measures, the main examples of which are Fokker-Planck-Kolmogorov equations for stationary and transition probabilities of diffusion processes. Existence and uniqueness of solutions are studied along with existence and Sobolev regularity of their densities and upper and lower bounds for the latter. The target readership includes mathematicians and physicists whose research is related to diffusion processes as well as elliptic and parabolic equations.
| Author | : Hannes Risken |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 486 |
| Release | : 2012-12-06 |
| Genre | : Mathematics |
| ISBN | : 3642615449 |
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.
| Author | : Joseph L. McCauley |
| Publisher | : Cambridge University Press |
| Total Pages | : 219 |
| Release | : 2013-02-21 |
| Genre | : Business & Economics |
| ISBN | : 0521763401 |
Provides graduate students and practitioners in physics and economics with a better understanding of stochastic processes.
| Author | : Vladimir I. Bogachev |
| Publisher | : American Mathematical Society |
| Total Pages | : 495 |
| Release | : 2022-02-10 |
| Genre | : Mathematics |
| ISBN | : 1470470098 |
This book gives an exposition of the principal concepts and results related to second order elliptic and parabolic equations for measures, the main examples of which are Fokker–Planck–Kolmogorov equations for stationary and transition probabilities of diffusion processes. Existence and uniqueness of solutions are studied along with existence and Sobolev regularity of their densities and upper and lower bounds for the latter. The target readership includes mathematicians and physicists whose research is related to diffusion processes as well as elliptic and parabolic equations.
| Author | : Grigorios A. Pavliotis |
| Publisher | : Springer |
| Total Pages | : 345 |
| Release | : 2014-11-19 |
| Genre | : Mathematics |
| ISBN | : 1493913239 |
This book presents various results and techniques from the theory of stochastic processes that are useful in the study of stochastic problems in the natural sciences. The main focus is analytical methods, although numerical methods and statistical inference methodologies for studying diffusion processes are also presented. The goal is the development of techniques that are applicable to a wide variety of stochastic models that appear in physics, chemistry and other natural sciences. Applications such as stochastic resonance, Brownian motion in periodic potentials and Brownian motors are studied and the connection between diffusion processes and time-dependent statistical mechanics is elucidated. The book contains a large number of illustrations, examples, and exercises. It will be useful for graduate-level courses on stochastic processes for students in applied mathematics, physics and engineering. Many of the topics covered in this book (reversible diffusions, convergence to equilibrium for diffusion processes, inference methods for stochastic differential equations, derivation of the generalized Langevin equation, exit time problems) cannot be easily found in textbook form and will be useful to both researchers and students interested in the applications of stochastic processes.
| Author | : T.D. Frank |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 414 |
| Release | : 2005-01-07 |
| Genre | : Science |
| ISBN | : 3540212647 |
Centered around the natural phenomena of relaxations and fluctuations, this monograph provides readers with a solid foundation in the linear and nonlinear Fokker-Planck equations that describe the evolution of distribution functions. It emphasizes principles and notions of the theory (e.g. self-organization, stochastic feedback, free energy, and Markov processes), while also illustrating the wide applicability (e.g. collective behavior, multistability, front dynamics, and quantum particle distribution). The focus is on relaxation processes in homogeneous many-body systems describable by nonlinear Fokker-Planck equations. Also treated are Langevin equations and correlation functions. Since these phenomena are exhibited by a diverse spectrum of systems, examples and applications span the fields of physics, biology and neurophysics, mathematics, psychology, and biomechanics.
| Author | : Sau Fa Kwok |
| Publisher | : World Scientific |
| Total Pages | : 208 |
| Release | : 2018-03-07 |
| Genre | : Science |
| ISBN | : 9813228423 |
This invaluable book provides a broad introduction to a rapidly growing area of nonequilibrium statistical physics. The first part of the book complements the classical book on the Langevin and Fokker-Planck equations (H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications (Springer, 1996)). Some topics and methods of solutions are presented and discussed in details which are not described in Risken's book, such as the method of similarity solution, the method of characteristics, transformation of diffusion processes into the Wiener process in different prescriptions, harmonic noise and relativistic Brownian motion. Connection between the Langevin equation and Tsallis distribution is also discussed.Due to the growing interest in the research on the generalized Langevin equations, several of them are presented. They are described with some details.Recent research on the integro-differential Fokker-Planck equation derived from the continuous time random walk model shows that the topic has several aspects to be explored. This equation is worked analytically for the linear force and the generic waiting time probability distribution function. Moreover, generalized Klein-Kramers equations are also presented and discussed. They have the potential to be applied to natural systems, such as biological systems.
| Author | : William Coffey |
| Publisher | : World Scientific |
| Total Pages | : 436 |
| Release | : 1996 |
| Genre | : Mathematics |
| ISBN | : 9789810216511 |
The book is suitable for a lecture course on the theory of Brownian motion, being based on final year undergraduate lectures given at Trinity College, Dublin. Topics that are discussed include: white noise; the Chapman-Kolmogorov equation ? Kramers-Moyal expansion; the Langevin equation; the Fokker-Planck equation; Brownian motion of a free particle; spectral density and the Wiener-Khintchin theorem ? Brownian motion in a potential application to the Josephson effect, ring laser gyro; Brownian motion in two dimensions; harmonic oscillators; itinerant oscillators; linear response theory; rotational Brownian motion; application to loss processes in dielectric and ferrofluids; superparamagnetism and nonlinear relaxation processes.As the first elementary book on the Langevin equation approach to Brownian motion, this volume attempts to fill in all the missing details which students find particularly hard to comprehend from the fundamental papers contained in the Dover reprint ? Selected Papers on Noise and Stochastic Processes, ed. N Wax (1954) ? together with modern applications particularly to relaxation in ferrofluids and polar dielectrics.
| Author | : Luigi Ambrosio |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 333 |
| Release | : 2008-10-29 |
| Genre | : Mathematics |
| ISBN | : 376438722X |
The book is devoted to the theory of gradient flows in the general framework of metric spaces, and in the more specific setting of the space of probability measures, which provide a surprising link between optimal transportation theory and many evolutionary PDE's related to (non)linear diffusion. Particular emphasis is given to the convergence of the implicit time discretization method and to the error estimates for this discretization, extending the well established theory in Hilbert spaces. The book is split in two main parts that can be read independently of each other.
| Author | : Simo Särkkä |
| Publisher | : Cambridge University Press |
| Total Pages | : 327 |
| Release | : 2019-05-02 |
| Genre | : Business & Economics |
| ISBN | : 1316510085 |
With this hands-on introduction readers will learn what SDEs are all about and how they should use them in practice.
