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

Dan Shafir, Stanislav Burov

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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 ≤ F0ln(F0)-2/α where α = T/T g, for temperatures T < T g. We support our analytic results by comparison to numerical simulations.

Original languageEnglish
Article number033301
JournalJournal of Statistical Mechanics: Theory and Experiment
Issue number3
StatePublished - Mar 2022

Bibliographical note

Funding Information:
This work was supported by the Israel Science Foundation Grant No. 2796/20.

Publisher Copyright:
© 2022 IOP Publishing Ltd and SISSA Medialab srl.


  • diffusion in random media
  • ergodicity breaking
  • numerical simulations


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