Stochastic Processes Crossing from Ballistic to Fractional Diffusion with Memory: Exact Results

TL;DR

This paper derives an exact solution for a non-Markovian diffusion process that transitions from ballistic to fractional diffusion, accurately capturing short- and long-time behaviors within a unified framework.

Contribution

It introduces a method to select the memory kernel in the diffusion equation to exactly match both ballistic and fractional diffusion asymptotics, providing a comprehensive solution.

Findings

Exact probability distribution function for the process

Unified description of ballistic and fractional diffusion regimes

Improved short-time accuracy over previous models

Abstract

We address the now classical problem of a diffusion process that crosses over from a ballistic behavior at short times to a fractional diffusion (sub- or super-diffusion) at longer times. Using the standard non-Markovian diffusion equation we demonstrate how to choose the memory kernel to exactly respect the two different asymptotics of the diffusion process. Having done so we solve for the probability distribution function (pdf) as a continuous function which evolves inside a ballistically expanding domain. This general solution agrees for long times with the pdf obtained within the continuous random walk approach but it is much superior to this solution at shorter times where the effect of the ballistic regime is crucial.