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

Abstract

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 2D, 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 → ∞). The first and second moments are calculated, and a formula for a general μth 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 ≳ F 0 ln ( F 0 ) − 2 / α where α = T/T g, for temperatures T < T g. We support our analytic results by comparison to numerical simulations.