Search results for: the-fokker-planck-equation

The Fokker Planck Equation

Author : Hannes Risken
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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.

Langevin And Fokker planck Equations And Their Generalizations Descriptions And Solutions

Author : Kwok Sau Fa
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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. Contents: Introduction Langevin and Fokker–Planck Equations Fokker–Planck Equation for One Variable and its Solution Fokker–Planck Equation for Several Variables Generalized Langevin Equations Continuous Time Random Walk Model Uncoupled Continuous Time Random Walk Model andits Solution Readership: Advanced undergraduate and graduate students in mathematical physics and statistical physics; biologists and chemists who are interested in nonequilibrium statistical physics. Keywords: Langevin Equation;Fokker-Planck Equation;Klein-Kramers Equation;Continuous Time Random Walk Model;Colored Noise;Tsallis Entropy;Population Growth Models;Wright Functions;Mittag-Leffler Function;Method of Similarity Solution;First Passage Time;Relativistic Brownian Motion;Fractional Derivatives;Integro-Differential Fokker-Planck EquationsReview: Key Features: This book complements Risken's book on the Langevin and Fokker-Planck equations. Some topics and methods of solutions are presented and discussed in details which are not described in Risken's book Several generalized Langevin equations are presented and discussed with some detail Integro-differential Fokker–Planck equation is derived from the uncoupled continuous time random walk model for generic waiting time probability distribution function which can be used to distinguish the differences for the initial and intermediate times with the same behavior in the long-time limit. Moreover, generalized Klein–Kramers equations are also described and discussed. To our knowledge these approaches are not found in other textbooks

Nonlinear Fokker Planck Equations

Author : T.D. Frank
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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.

Asymptotic Methods for the Fokker Planck Equation and the Exit Problem in Applications

Author : Johan Grasman
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Asymptotic methods are of great importance for practical applications, especially in dealing with boundary value problems for small stochastic perturbations. This book deals with nonlinear dynamical systems perturbed by noise. It addresses problems in which noise leads to qualitative changes, escape from the attraction domain, or extinction in population dynamics. The most likely exit point and expected escape time are determined with singular perturbation methods for the corresponding Fokker-Planck equation. The authors indicate how their techniques relate to the Itô calculus applied to the Langevin equation. The book will be useful to researchers and graduate students.

The Fokker Planck Equation for Stochastic Dynamical Systems and Its Explicit Steady State Solutions

Author : Christian Soize
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This is an analysis of multidimensional nonlinear dissipative Hamiltonian dynamical systems subjected to parametric and external stochastic excitations by the Fokker-Planck equation method.The author answers three types of questions concerning this area. First, what probabilistic tools are necessary for constructing a stochastic model and deriving the FKP equation for nonlinear stochastic dynamical systems? Secondly, what are the main results concerning the existence and uniqueness of an invariant measure and its associated stationary response? Finally, what is the class of multidimensional dynamical systems that have an explicit invariant measure and what are the fundamental examples for applications?

The Fokker Planck Equation

Author : Hannes Risken
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The Variational Formulation of the Fokker Planck Equation

Author : Richard Jordan
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Abstract: "The Fokker-Planck equation, or forward Kolmogorov equation, describes the evolution of the probability density for a stochastic process associated with an Ito stochastic differential equation. It pertains to a wide variety of time-dependent systems in which randomness plays a role. In this paper, we are concerned with Fokker-Planck equations for which the drift term is given by the gradient of a potential. For a broad class of potentials, we construct a time-discrete, iterative variational scheme whose solutions converge to the solution of the Fokker-Planck equation. The major novelty of this iterative scheme is that the time step is governed by the Wasserstein metric on probability measures. This formulation enables us to reveal an appealing, and previously unexplored, relationship between the Fokker-Planck equation and the associated free energy functional. Namely we demonstrate that the dynamics may be regarded as a gradient flux, or a steepest descent, for the free energy with respect to the Wasserstein metric."

Simulation of the Fokker Planck Equation by Random Walks of Test Particles in Velocity Space with Application to Magnetic Mirror Systems

Author : Gerald W. Englert
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Analysis of the Langevin Equation with Random Damping by the Fokker Planck Equation

Author : J. A. Young
File Size : 79.13 MB
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Model Predictive Control for the Fokker Planck Equation

Author : Arthur Fleig
File Size : 77.48 MB
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Expansion of the Fokker Planck Equation in Spherical Harmonics

Author : Frederic A. Lyman
File Size : 78.26 MB
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Bilinear Optimal Control of the Fokker Planck Equation

Author : Arthur Fleig
File Size : 51.82 MB
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Optimal Control of the Fokker Planck Equation with Space Dependent Controls

Author : Arthur Fleig
File Size : 77.36 MB
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Hypocoercivity Methods for the Fokker Planck Equation

Author : Raphael Winter
File Size : 49.47 MB
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Beyond The Triangle Brownian Motion Ito Calculus And Fokker planck Equation Fractional Generalizations

Author : Sabir Umarov
File Size : 51.5 MB
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The book is devoted to the fundamental relationship between three objects: a stochastic process, stochastic differential equations driven by that process and their associated Fokker-Planck-Kolmogorov equations. This book discusses wide fractional generalizations of this fundamental triple relationship, where the driving process represents a time-changed stochastic process; the Fokker-Planck-Kolmogorov equation involves time-fractional order derivatives and spatial pseudo-differential operators; and the associated stochastic differential equation describes the stochastic behavior of the solution process. It contains recent results obtained in this direction.This book is important since the latest developments in the field, including the role of driving processes and their scaling limits, the forms of corresponding stochastic differential equations, and associated FPK equations, are systematically presented. Examples and important applications to various scientific, engineering, and economics problems make the book attractive for all interested researchers, educators, and graduate students.

On the Solution of the Fokker Planck Equation for Multi dimensional Nonlinear Mechanical Systems

Author : Wolfram Martens
File Size : 23.16 MB
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Optimal Control of the Fokker Planck Equation with State Dependent Controls

Author : Arthur Fleig
File Size : 87.11 MB
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On Dissipativity of the Fokker Planck Equation for the Ornstein Uhlenbeck Process

Author : Arthur Fleig
File Size : 61.94 MB
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Extension of Dougherty s Model Fokker Planck Equation for a Plasma

Author : Robert J. Papa
File Size : 45.12 MB
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Electromagnetic waves can be severely attenuated and suffer distortion as they propagate through partially ionized gases. These facts must be considered in the design of any communication system in which waves must propagate through an intervening plasma medium, such as in reentry communications and ionospheric propagation. In this report, formulas are given that can predict such wave attenuation characteristics more accurately and for a much wider range of plasma conditions than previous theories. The conventional Appleton-Hartree equation used in ionospheric propagation studies gives the index of refraction of a wave traveling through a plasma in a magnetic field in terms of the properties of the plasma. This conventional Appleton-Hartree formula neglects important effects such as the random thermal motion of the particles, which can produce nonlocal effects. Also, the energy dependence of the electron-neutral collision frequency can alter the nature of the wave attenuation process. A generalization of the Appleton-Hartree equation is made to include these effects and to account for the Coulomb forces between charged particles. A kinetic equation is solved which includes the effects of energy-dependent electron-neutral collisions, Coulomb encounters and spatial dispersion. The perturbation method used in solving the kinetic equation assumes that the effects of Coulomb encounters and spatial dispersion are dominant, and electron-neutral collisions are relatively infrequent.

Fei Xian Xing Xi Tong Di FukePulangke Fang Cheng Zhi Tan Tao

Author : Firman So
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