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 $2$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 ($t \rightarrow \infty$). 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. $\langle x(t)\rangle$, 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 $t\gtrsim \big(F_0 \left| \ln (F_0)\right|\big)^{-2 / \alpha}$ where $\alpha=T/T_g$, for temperatures $T

Keywords:

• Limit (mathematics)
• Mathematical physics
• Position (vector)
• Physics
• Probability density function
• Lattice (group)
• Function (mathematics)
• Moment (mathematics)

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