Approach to asymptotically diffusive behavior for Brownian particles in media with periodic diffusivities

TL;DR

This paper investigates the long-term diffusive behavior of Brownian particles in media with periodic spatial diffusivity, providing formulas for effective diffusion constants and the approach to diffusion.

Contribution

It introduces a Kubo type formula for the mean squared displacement in media with periodic diffusivity, elucidating the asymptotic approach to diffusion and calculating key constants.

Findings

Derived a Kubo type formula for effective diffusion constant

Provided a method to understand the approach to asymptotic diffusion

Calculated the intercept of the mean squared displacement at late times

Abstract

We analyze the mean squared displacement of a Brownian particle in a medium with a spatially varying local diffusivity which is assumed to be periodic. When the system is asymptotically diffusive the mean squared displacement, characterizing the dispersion in the system, is, at late times, a linear function of time. A Kubo type formula is given for the mean squared displacement which allows the recovery of some known results for the effective diffusion constant $D_e$ in a direct way, but also allows an understanding of the asymptotic approach to the diffusive limit. In particular, as well as as computing the slope of a linear fit to the late time mean squared displacement, we find a formula for the constant where the fit intersects the y axis.