From quenched disorder to continuous time random walk

This work focuses on quantitative representation of transport in systems with quenched disorder. Explicit mapping of the quenched trap model to continuous time random walk is presented. Linear temporal transformation, $t\to t/{\mathrm{\Lambda }}^{1/\alpha }$, for a transient process in the subdiffusive regime is sufficient for asymptotic mapping. An exact form of the constant ${\mathrm{\Lambda }}^{1/\alpha }$ is established. A disorder averaged position probability density function for a quenched trap model is obtained, and analytic expressions for the diffusion coefficient and drift are provided.

Figure 1

Simulated behavior of fraction of moments of $S_{\alpha }$, $\overline{S_{\alpha }^2}/{\overline{S_{\alpha }^2}}^2-1$, as a function of the number of jumps ($N$) of a random walk. $◯$ are the results for a biased one-dimensional RW on a lattice, the transitions are allowed only to nearest neighbors with probability $q=0.7$ to the right and $1-q$ to the left. $▽$ presents the results for three-dimensional unbiased RW on a cubic lattice where the transitions are allowed only to nearest neighbors.

Figure 2

Moments and prefactors for one-dimensional biased QTM as presented by Eq. (11) (lines) and numerical simulations (symbols). (a) The first moment behavior as a function of time for three different $\alpha :♦$ is $\alpha =0.7,◯$ is $\alpha =0.5$, and $▽$ is $\alpha =0.3$. (b) The behavior of prefactor $\langle X\rangle /t^{\alpha }=V/\left(A\mathrm{\Gamma }[1+\alpha ]\right)$ for $\alpha =0.5$ and various $q>0.5$. Simulations were performed up to $t={10}^7$. (c) The growth of the second moment with time, the parameters similar to (a). ${10}^4$ realizations of disorder were used for averaging and $A=1$.

Figure 3

Second moment of position as function of time for unbiased QTM on a cubic lattice. Lines are analytical predictions provided by Eq. (12), and symbols are numerical simulations. $♦$ is $\alpha =0.7,◯$ is $\alpha =0.5$, and $▽$ is $\alpha =0.3$. ${10}^4$ realizations of disorder were used for averaging and $A=1$.