From continuous time random walks to the fractional Fokker-Planck equation

TLDR

The fractional Fokker‑Planck equation describes anomalous diffusion in an external force field near thermal equilibrium. The study generalizes the continuous time random walk to incorporate space‑dependent jump probabilities. The authors extend the CTRW with spatially varying jump probabilities, derive the fractional Fokker‑Planck equation when the mean waiting time diverges, discuss its validity domain, and compare CTRW and FFPE solutions in the force‑free case. They derive the fractional Fokker‑Planck equation under diverging waiting times and demonstrate that, in the force‑free case, the CTRW solution agrees with the FFPE solution.

Abstract

We generalize the continuous time random walk (CTRW) to include the effect of space dependent jump probabilities. When the mean waiting time diverges we derive a fractional Fokker-Planck equation (FFPE). This equation describes anomalous diffusion in an external force field and close to thermal equilibrium. We discuss the domain of validity of the fractional kinetic equation. For the force free case we compare between the CTRW solution and that of the FFPE.