Non-Gaussian diffusion in static disordered media

Non-Gaussian diffusion is commonly considered as a result of fluctuating diffusivity, which is correlated in time or in space or both. In this work, we investigate the non-Gaussian diffusion in static disordered media via a quenched trap model, where the diffusivity is spatially correlated. Several unique effects due to quenched disorder are reported. We analytically estimate the diffusion coefficient $D_{\text{dis}}$ and its fluctuation over samples of finite size. We show a mechanism of population splitting in the non-Gaussian diffusion. It results in a sharp peak in the distribution of displacement $P(x,t)$ around $x=0$, that has frequently been observed in experiments. We examine the fidelity of the coarse-grained diffusion map, which is reconstructed from particle trajectories. Finally, we propose a procedure to estimate the correlation length in static disordered environments, where the information stored in the sample-to-sample fluctuation has been utilized.

Figure 1

The color map of a typical sample of correlated random energies with $L=1024$ and $r_c=16$. The color is assigned to each site according to $V_i=ln{D}_i^{\left(l\right)}$, where $D_0=0.25$ is chosen.

Figure 2

The distribution of mean waiting time ${\overline{\tau }}_N$ for samples with $r_c=16$, $D_0=0.25$ and various sizes $N=L^2$. 8000 landscapes are sampled for each size. The solid lines are the probability density function of one-sided Lèvy stable distribution with $\mu =1$, $\gamma =6.35$, and ${\delta }_N={\delta }_0+\frac{2}{\pi }\gamma lnN$ with ${\delta }_0=-26.1$.

Figure 3

${\langle D_{\text{dis}}\rangle }^{-1}$ (left) and ${\langle D_{\text{dis}}^2\rangle }^{-1/2}$ (right) are plotted against ${log}_2(N/r_c^2)$. The averaging is performed over 8000 samples, for various sample sizes $N=L^2$ and various extremal basin radii $r_c$. The dash lines are added for guidance.

Figure 4

The distribution of displacement $P(x,t)$ for a typical sample and various $t$. The leading term of Eq. (18) is plotted by the dash line. The inset enlarges the peak of $P(x,t=20)$ for a clearer view.

Figure 5

A scheme for the trajectory-based data analysis, discussed in Sec.4. The sample is divided into small grains of size $s$. The local diffusivity ${\delta }_q^{\left(s\right)}$ is evaluated for the grain $q$ from the segments of trajectories centered therein [see Eq. (20)]. The mean local diffusivity ${\overline{\delta }}^{\left(s\right)}$ and the typical time ${\overline{\tau }}^{\left(s\right)}$ can be evaluated for each sample at various resolution $s$. The sample-to-sample fluctuation of ${\overline{\tau }}^{\left(s\right)}$ is expected vanishing while $s$ approaches the typical correlation length $\xi$.

Figure 6

(Left) The inverse of mean local diffusivity of coarse-grained diffusion map ${\left({\overline{\delta }}^{\left(s\right)}\right)}^{-1}$ for various $s$. The colors denote different disordered samples. The orange cross mark shows $1/D_0$, which is expected for the genuine map. (Inset) ${\overline{\delta }}^{\left(s\right)}$ for various resolutions and different samples. (Right) The distribution of ${\left(D_{\text{dis}}\right)}^{-1}$ obtained by sampling 8000 disordered realizations (circles). It follows one-sided Lévy stable distribution with $\mu =1$ (solid line).