Fractional diffusion in periodic potentials

TL;DR

This paper derives an analytical expression for fractional diffusion in periodic potentials, compares it with normal diffusion through simulations, and explores their long-term similarities after rescaling time.

Contribution

It provides a closed-form analytical solution for the effective fractional diffusion coefficient in periodic potentials, supported by numerical validation.

Findings

Analytical solution for fractional diffusion coefficient derived.

Numerical simulations confirm theoretical predictions.

Long-time behaviors of normal and fractional diffusion can be matched by time rescaling.

Abstract

Fractional, anomalous diffusion in space-periodic potentials is investigated. The analytical solution for the effective, fractional diffusion coefficient in an arbitrary periodic potential is obtained in closed form in terms of two quadratures. This theoretical result is corroborated by numerical simulations for different shapes of the periodic potential. Normal and fractional spreading processes are contrasted via their time evolution of the corresponding probability densities in state space. While there are distinct differences occurring at small evolution times, a re-scaling of time yields a mutual matching between the long-time behaviors of normal and fractional diffusion.