From the Ehrenfest model to time-fractional stochastic processes
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.
scientific article; zbMATH DE number 5612592
scientific article