Fluid limit of the continuous-time random walk with general Lévy jump distribution functions

Abstract

The continuous time random walk (CTRW) is a natural generalization of the Brownian random walk that allows the incorporation of waiting time distributions ψ(t) and general jump distribution functions η(x). There are two well-known fluid limits of this model in the uncoupled case. For exponential decaying waiting times and Gaussian jump distribution functions the fluid limit leads to the diffusion equation. On the other hand, for algebraic decaying waiting times, and algebraic decaying jump distributions, corresponding to Lévy stable processes, the fluid limit leads to the fractional diffusion equation of order α in space and order β in time. However, these are two special cases of a wider class of models. Here we consider the CTRW for the most general Lévy stochastic processes in the Lévy-Khintchine representation for the jump distribution function and obtain an integro-differential equation describing the dynamics in the fluid limit. The resulting equation contains as special cases the regular and the fractional diffusion equations. As an application we consider the case of CTRWs with exponentially truncated Lévy jump distribution functions. In this case the fluid limit leads to a transport equation with exponentially truncated fractional derivatives which describes the interplay between memory, long jumps, and truncation effects in the intermediate asymptotic regime. The dynamics exhibits a transition from superdiffusion to subdiffusion with the crossover time scaling as τc ∼ λ −α/β where 1/λ is the truncation length scale. The asymptotic behavior of the propagator (Green’s function) of the truncated fractional equation exhibits a transition from algebraic decay for t << τc to stretched Gaussian decay for t >> τc