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We analyze the mean squared displacement of a Brownian particle in a mediumwith 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 linearfunction of time. A Kubo type formula is given for the mean squareddisplacement which allows the recovery of some known results for the effectivediffusion constant $D_e$ in a direct way, but also allows an understanding ofthe asymptotic approach to the diffusive limit. In particular, as well as ascomputing 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.

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