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

David S. Dean (1) , Thomas Guérin (1)

1 LOMA - Laboratoire Ondes et Matière d'Aquitaine

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.

Mots clés: Stochastic analysis methods; Stochastic processes

Domaines: Mécanique statistique [cond-mat.stat-mech]

David S. Dean, Thomas Guérin. Approach to asymptotically diffusive behavior for Brownian particles in media with periodic diffusivities. Physical Review E : Statistical, Nonlinear, and Soft Matter Physics, 2014, 90 (6), pp.062114 (1-6). ⟨10.1103/PhysRevE.90.062114⟩. ⟨hal-01119884⟩