In a growing number of strongly disordered and dense systems, the dynamics of a particle pulled by an external force field exhibits superdiffusion. In the context of glass-forming systems, supercooled glasses, and contamination spreading in porous media, it was suggested that this behavior be modeled with a biased continuous-time random walk. Here we analyze the plume of particles lagging far behind the mean, with the single big jump principle. Revealing the mechanism of the anomaly, we show how a single trapping time, the largest one, is responsible for the rare fluctuations in the system. These nontypical fluctuations still control the behavior of the mean square displacement, which is the most basic quantifier of the dynamics in many experimental setups. We show how the initial conditions, describing either the stationary state or nonequilibrium case, persist forever in the sense that the rare fluctuations are sensitive to the initial preparation. To describe the fluctuations of the largest trapping time, we modify Fréchet's law from extreme value statistics, taking into consideration the fact that the large fluctuations are very different from those observed for independent and identically distributed random variables.

• Received 18 June 2019
• Revised 31 October 2019

DOI:https://doi.org/10.1103/PhysRevResearch.1.033172

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society