Weak subordination breaking for the quenched trap model

Abstract

We map the problem of diffusion in the quenched trap model onto a new stochastic process: Brownian motion which is terminated at the coverage "time" ${\mathcal{S}}_{\alpha }={\sum }_{x=-\infty }^{\infty }(n_x{)}^{\alpha }$ with $n_x$ being the number of visits to site $x$. Here $0<\alpha =T/T_g<1$ is a measure of the disorder in the original model. This mapping allows us to treat the intricate correlations in the underlying random walk in the random environment. The operational "time" ${\mathcal{S}}_{\alpha }$ is changed to laboratory time $t$ with a Lévy time transformation. Investigation of Brownian motion stopped at "time" ${\mathcal{S}}_{\alpha }$ yields the diffusion front of the quenched trap model which is favorably compared with numerical simulations. In the zero temperature limit of $\alpha \to 0$ we recover the renormalization group solution obtained by C. Monthus. Our theory surmounts critical slowing down which is found when $\alpha \to 1$. Above the critical dimension two mapping the problem to a continuous time random walk becomes feasible though still not trivial.

15 pages. 7 figures

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