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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Approximate Moment Methods for Population Balance Equations in Particulate and Bioengineering Processes
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Cluster approximations for infection dynamics on random networks.
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TL;DR: This paper constructs an uncorrelated triplet approximation that captures the behavior of the system in a region of parameter space where the pair approximation fails to give a good quantitative or even qualitative agreement.
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Active and reactive behaviour in human mobility: the influence of attraction points on pedestrians
Mario Gutiérrez-Roig,Oleguer Sagarra,Aitana Oltra,John R. B. Palmer,Frederic Bartumeus,Albert Díaz-Guilera,Josep Perelló +6 more
TL;DR: A general modelling framework based on Langevin dynamics is developed, which allows us to test the influence of two distinct types of ingredients on mobility: reactive or context-dependent factors, modelled by means of a force field generated by attraction points in a given spatial configuration and active or inherent factors,modelled from intrinsic movement patterns of the subjects.
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.