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

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.

Figure 1

$D\left(t\right)$ estimated from the MSD in a numerical simulation of $5\times {10}^5$ Langevin particles, in one dimension, with local diffusion coefficient $\kappa \left(x\right)$ given by Eq. (45) with $\alpha =0.8$ and $l=4\pi l_0$ (continuous) black line (shown in units such that ${\kappa }_0=1$ and $l_0=1$). Shown as thick-dashed (red online), late-time prediction of $D\left(t\right)$ of the form predicted by Eq. (25) $D\left(t\right)=D_e+C/t$, where theory predicts that $D_e=1$ and $C=1.28$.

Figure 2

$D\left(t\right)-D_e$ (here $D_e={\kappa }_0$) estimated from the MSD in the same numerical simulation as Fig. (1) (open circles) compared at all times with a numerical evaluation of the Kubo formula Eq. (22) (solid line, red online, again shown in units such that ${\kappa }_0=1$ and $l_0=1$). Also shown is the asymptotic analytical late-time correction $C/t$ to $D\left(t\right)-D_e$ (dashed line, blue online) as well as the short time value $D\left(0\right)-D_e=\overline{\kappa }-D_e$ (dotted line, green online).