# Transport in disordered systems: the single big jump approach

## TL;DR

This paper investigates super-diffusive behavior in disordered systems using the single big jump principle, highlighting how rare, large trapping times dominate fluctuations and influence overall dynamics.

## Contribution

It introduces a modified extreme value statistics approach to account for correlated large trapping times affecting super-diffusive transport.

## Key findings

- Large trapping times govern super-diffusion behavior.
- Initial conditions influence long-term fluctuation dynamics.
- Modified Fréchet law describes extreme trapping time fluctuations.

## Abstract

In a growing number of strongly disordered and dense systems, the dynamics of a particle pulled by an external force field exhibits super-diffusion. In the context of glass forming systems, super cooled glasses and contamination spreading in porous medium it was suggested to model this behavior with a biased continuous time random walk. Here we analyze the plume of particles far lagging 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 non typical 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 stationary state or non-equilibrium case, persist for ever in the sense that the rare fluctuations are sensitive to the initial preparation. To describe the fluctuations of the largest trapping time, we modify Fr\'{e}chet'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.