Generalized master equations and fractional Fokker–Planck equations from continuous time random walks with arbitrary initial conditions

Abstract

In the standard continuous time random walk the initial state is taken as a non-equilibrium state, in which the random walking particle has just arrived at a given site. Here we consider generalizations of the continuous time random walk to accommodate arbitrary initial states. One such generalization provides information about the initial state through the introduction of a first waiting time density that is taken to be different from subsequent waiting time densities. Another generalization provides information about the initial state through the prior history of the arrival flux density. The master equations have been derived for each of these generalizations. They are different in general but they are shown to limit to the same master equation in the case of an equilibrium initial state. Under appropriate conditions they also reduce to the master equation for the standard continuous time random walk with the non-equilibrium initial state.The diffusion limit of the generalized master equations is also considered, with Mittag-Leffler waiting time densities, resulting in the same fractional Fokker–Planck equation for different initial conditions.