We discuss the relationship between kinetic equations of the Fokker-Planck type (two linear and one non-linear) and the Kolmogorov (a.k.a. master) equations of certain N -body diffusion processes, in the context of Kac’s propagation-of-chaos limit. The linear Fokker-Planck equations are well-known, but here they are derived as a limit N → ∞ of a simple linear diffusion equation on 3N − C-dimens...

We study the behavior of skyrmions in thin films under the action of stochastic torques arising from thermal fluctuations. We find that the Brownian motion of skyrmions is described by a stochastic Thiele’s equation and its corresponding Fokker-Planck equation. The resulting Fokker-Planck equation is recognized as the one for a high-friction Brownian particle which has been studied extensively ...

We derive the generalized Fokker-Planck equation associated with a Langevin equation driven by arbitrary additive white noise. We apply our result to study the distribution of symmetric and asymmetric Lévy flights in an infinitely deep potential well. The fractional Fokker-Planck equation for Lévy flights is derived and solved analytically in the steady state. It is shown that Lévy flights are ...

We consider an evolving network of a fixed number of nodes. The allocation of edges is a dynamical stochastic process inspired by biological reproduction dynamics, namely by deleting and duplicating existing nodes and their edges. The properties of the degree distribution in the stationary state is analysed by use of the Fokker-Planck equation. For a broad range of parameters exponential degree...

Time evolution of the expectation values of various dynamical operators of the harmonic oscillator with dissipation is analitically obtained within the framework of the Lindblad theory for open quantum systems. We deduce the density matrix of the damped harmonic oscillator from the solution of the Fokker-Planck equation for the coherent state representation, obtained from the master equation fo...

We extend a recently developed relation between the master equation describing the Parrondo’s games and the formalism of the Fokker–Planck equation to the case in which the games are modified with the introduction of “self–transition probabilities”. This accounts for the possibility that the capital can neither increase nor decrease during a game. Using this exact relation, we obtain expression...

Analysis and Implementation of Numerical Methods for Simulating Dilute Polymeric Fluids David Knezevic Doctor of Philosophy Balliol College Michaelmas Term 2008 In this thesis we develop, analyse and implement a number of numerical methods for simulating dilute polymeric fluids. We use a well-known model in which the polymeric fluid is represented by a suspension of dumbbells in a Newtonian sol...

A new algorithm for nonconvex optimization by means of a so-called Fokker-Planck Learning Machine is proposed in this paper. This is done by considering the Fokker-Planck (FP) equation related to continuous simulated annealing, which has been proven to convergence to the global optimum under certain conditions. An approximate solution to the FP equation is sought by parametrizing the transition...

The a posteriori error estimates are studied for class of nonlinear stead-state Poisson–Nernst–Planck equations, which coupled system consisting the Nernst–Planck equation and Poisson equation. Both global upper bounds local lower estimators obtained by using averaging operator. Numerical experiments given to confirm reliability efficiency estimators.

The classical Fokker–Planck equation is a linear parabolic equation which describes the time evolution of the probability distribution of a stochastic process defined on a Euclidean space. Corresponding to a stochastic process, there often exists a free energy functional which is defined on the space of probability distributions and is a linear combination of a potential and an entropy. In rece...