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

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 -> infinity). The first and second moments are calculated, and a for mula 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 greater than or similar to (F-0 vertical bar In(F-0)vertical bar)(-2/alpha) where alpha = T/T-g, for temperatures T < T-g. We support our analytic results by comparison to numerical simulations.

Automated summary

In this study, we investigate the biased quenched trap model on a two-dimensional lattice with diverging expected dwell times. By using the double-subordination approach and calculating the return probability in 2D, we 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 -> infinity). We calculate the first and second moments and find a formula for a general mu th moment. The behavior of the first moment, < x(t)>, shows non-linear response in both time and the applied external force F-0.