The case of the biased quenched trap model in two dimensions with diverging mean dwell times

Abstract

We investigate the biased quenched trap model on top of a two-dimensional lattice in the case of diverging expected dwell times. By utilizing the double-subordination approach and calculating the return probability in $2$d, we explicitly obtain the disorder averaged probability density function of the particle's position as a function of time (for any given bias) in the limit of large times ($t\to \infty$). The first and second moments are calculated, and a formula for a general $\mu$-th moment is found. The behavior of the first moment, i.e. $⟨x(t)⟩$, presents non-linear response both in time and in the applied external force $F_0$. While the non-linearity in time occurs for any measurement time $t$, the non-linearity in $F_0$ is expected only when $t\gtrsim (F_0\left|\ln (F_0)\right|{)}^{-2/\alpha }$ where $\alpha =T/T_g$, for temperatures $T<T_g$. We support our analytic results by comparison to numerical simulations.

arXiv comment34 pages, 6 figures

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