Book ChapterDOI
Fokker-Planck Equation
Hannes Risken
- pp 63-95
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TLDR
In this paper, an equation for the distribution function describing Brownian motion was first derived by Fokker [11] and Planck [12] and it is shown that expectation values for nonlinear Langevin equations (367, 110) are much more difficult to obtain.Abstract:
As shown in Sects 31, 2 we can immediately obtain expectation values for processes described by the linear Langevin equations (31, 31) For nonlinear Langevin equations (367, 110) expectation values are much more difficult to obtain, so here we first try to derive an equation for the distribution function As mentioned already in the introduction, a differential equation for the distribution function describing Brownian motion was first derived by Fokker [11] and Planck [12]: many review articles and books on the Fokker-Planck equation now exist [15 – 15]read more
Citations
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Langevin model for real-time Brownian dynamics of interacting nanodefects in irradiated metals
Sergei L. Dudarev,Mark R. Gilbert,Mark R. Gilbert,Kazuto Arakawa,Kazuto Arakawa,Hideki Mori,Zhongwen Yao,Zhongwen Yao,Michael L. Jenkins,Peter M. Derlet +9 more
TL;DR: In this article, the dynamics of microstructural evolution in these materials are strongly influenced by long-range elastic interactions between mobile nanoscale radiation defects, and simulations, performed using an algorithm where nanoscales radiation defects are treated as interacting Langevin particles, are able to match and explain the real-time dynamics of nanodefects observed in in situ electron microscope experiments.
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TL;DR: The stability and bifurcations in oscillator models describing electric power grids are analyzed and it is demonstrated that these networks exhibit instabilities without overloads.
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Diffusion and trapping in heterogeneous media: An inhomogeneous continuous time random walk approach
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References
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Journal ArticleDOI
Fluctuations and Irreversible Processes
Lars Onsager,S. Machlup +1 more
TL;DR: In this paper, the probability of a given succession of (nonequilibrium) states of a spontaneously fluctuating thermodynamic system is calculated, on the assumption that the macroscopic variables defining a state are Gaussian random variables whose average behavior is given by the laws governing irreversible processes.
Journal ArticleDOI
The Radiation Theories of Tomonaga, Schwinger, and Feynman
TL;DR: In this article, a unified development of the subject of quantum electrodynamics is outlined, embodying the main features both of the Tomonaga-Schwinger and of the Feynman radiation theory.
Journal ArticleDOI
Covariant formulation of non-equilibrium statistical thermodynamics
TL;DR: In this paper, the diffusion matrix of the Fokker Planck equation is used as a contravariant metric tensor in phase space, and the covariance of the Langevin-equations and the fokker equation is demonstrated.
Journal ArticleDOI
Approximation of the Linear Boltzmann Equation by the Fokker-Planck Equation
TL;DR: In this article, the first two terms of the Kramers-Moyal expansion of the Fokker-Planck equation were used to approximate the linear Boltzmann integral operator.