Weak subordination breaking for the quenched trap model

TL;DR

This paper introduces a novel stochastic process mapping for diffusion in the quenched trap model, enabling analysis of complex correlations and providing accurate diffusion front predictions validated by simulations.

Contribution

It presents a new mapping of the quenched trap model onto a Brownian motion stopped at a coverage-dependent time, facilitating analysis of correlations and diffusion behavior.

Findings

Accurately predicts the diffusion front of the quenched trap model.

Recovers known solutions in the zero-temperature limit.

Overcomes critical slowing down near lpha=1.

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" ${\cal S}_\alpha=\sum_{x=-\infty} ^\infty (n_x)^\alpha$ 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" ${\cal S}_\alpha$ is changed to laboratory time $t$ with a L\'evy time transformation. Investigation of Brownian motion stopped at "time" ${\cal S}_\alpha$ 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.