From the Ehrenfest model to time-fractional stochastic processes
type: Item
instance of: scholarly article
title: From the Ehrenfest model to time-fractional stochastic processes
zbMATH Open document ID: 1185.60048
published in: Journal of Computational and Applied Mathematics
publication date: 2009-10-09T00:00:00Z
review text: The author shows that the time-honoured Ehrenfest model is equivalent to an explicit difference scheme (with spatial boundaries at \(-R\) and \(+R\), \(R\) being proportional to \(h^{(-2)}\) where \(h\) is the spatial grid-length) for the one-dimensional diffusion equation with central linear drift. Then the diffusion equation is modified to one with fractional derivative in time for which the solution to the Cauchy problem (the fundamental solution) is given for the \(\delta\)-initial condition at an arbitrary point \(x_0\). (Unfortunately, there is an easily detectable misprint in the formula given for the stationary solution that is identical to the same origin-centered Gaussian as for the equation with time derivative of order \(1\).) By discretizing the time-fractional differential equation in space as usual, in time by the Gruenwald-Letnikov scheme, a time-dependent transition matrix for a time-fractional Ehrenfest model is obtained and discussed with respect to its properties. The time-fractional model has the same stationary solution as the non-fractional model Finally the question of reversibilty of the stationary behaviour is touched, and numerical numerical calculations suggest an affirmative answer. The author corroborates her theory by illustrations of results of numerical case studies.
reviewed by: Rudolf Gorenflo
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 26A33
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 65N06
Mathematics Subject Classification ID: 82C41
Mathematics Subject Classification ID: 82C80
zbMATH DE Number: 5612592
zbMATH Keywords: Ehrenfest urn model
zbMATH Keywords: diffusion processes
zbMATH Keywords: central drift
zbMATH Keywords: difference schemes
zbMATH Keywords: time-fractional derivative
zbMATH Keywords: stochastic processes
zbMATH Keywords: reversibility
author: Entsar A. Abdel-Rehim
MaRDI profile type: Publication
full work available at URL: j.cam.2009.07.010
OpenAlex ID: W2000002722
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DOI: 10.1016/J.CAM.2009.07.010
description: scientific article; zbMATH DE number 5612592