Continuous Time Random Walk with time-dependent jump probability : A Direct Probabilistic Approach

TL;DR

This paper presents a rigorous probabilistic approach to analyze continuous time random walks with time-dependent jump probabilities, revealing limitations of existing fractional Fokker-Planck models and exploring effects of extended jumps.

Contribution

It introduces a direct probabilistic method for deriving moments of CTRW with time-dependent jumps, bypassing fractional equations, and highlights the impact of higher moments and extended jumps.

Findings

Derived general moments for CTRW with arbitrary waiting times and jump probabilities.

Confirmed phenomena of 'death of linear response' and 'field-induced dispersion' in sub-diffusion.

Identified additional terms in higher moments, indicating limitations of FFPE in capturing full dynamics.

Abstract

We investigate the dynamics of a particle executing a general Continuous Time Random Walk (CTRW) in three dimensions under the influence of arbitrary time-varying external fields. Contrary to the general approach in recent works, our method invokes neither the Fractional Fokker-Planck equation (FFPE) nor the Stochastic Langevin Equation (SLE). Rather, we use rigorous probability arguments to derive the general expression for moments of all orders of the position probability density of the random walker for arbitrary waiting time density and jump probability density. Closed form expression for the position probability density is derived for the memoryless condition. For the special case of CTRW on a one-dimensional lattice with nearest neighbour jumps, our equations confirm the phenomena of "death of linear response" and "field-induced dispersion" for sub-diffusion pointed out in [I. M. Sokolov and J. Klafter, Phys. Rev. Lett. {\bf 97}, 140602 (2006)]. However, our analysis produces additional terms in the expressions for higher moments, which have non-trivial consequences. We show that the disappearance of these terms result from the approximation involved in taking the continuum limit to derive the generalized Fokker-Planck equation. This establishes the incompleteness of the FFPE formulation, especially in predicting the higher moments. We also discuss how different predictions of the model alter if we allow jumps beyond nearest neighbours and possible circumstances where this becomes relevant.