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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Journal ArticleDOI
Autonomous robot photographer with KL divergence optimization of image composition and human facial direction
Kai Lan,Kosuke Sekiyama +1 more
TL;DR: Two technical issues of scene composition evaluation and viewpoint selection are solved by this robot photography system and the fact that better composed photos can be autonomously photographed by the proposed system is validated via experiments and human evaluations.
Journal ArticleDOI
Multiscale modelling and splitting approaches for fluids composed of Coulomb-interacting particles
TL;DR: In this article, the Fokker-Planck equation with a collision operator was considered for two-and multi-particle systems with different approximations for the Coulomb interaction and two different numerical schemes, one deterministic and the other stochastic, were proposed for coupling the interaction and transport equations.
Journal ArticleDOI
Numerical method for coupling the macro and meso scales in stochastic chemical kinetics
Lars Ferm,Per Lötstedt +1 more
TL;DR: In this article, a numerical method for simulation of stochastic chemical reactions is developed for the Fokker-Planck equation for the probability density of the molecular state, which preserves properties of the analytical solution such as non-negativity and constant total probability.
Journal ArticleDOI
Reconstructing Aggregate Dynamics in Heterogeneous Agents Models
TL;DR: In this paper, a statistical mechanics aggregation method is applied to a financial fragility model, which considers the heterogeneity of firms, their interactive behaviour and the feedback effects between micro, meso and macro level.
Book ChapterDOI
Upscaling Flow and Transport Processes
TL;DR: In this article, a detailed analysis of dispersion theories for pore-scale and continuum-scale solute transport is carried out, presenting the classical results as well as the most recent research in the field.
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.