Occupation Time Statistics in the Quenched Trap Model

TL;DR

This paper studies how the time a particle spends in different states varies in a disordered system, revealing a transition from Boltzmann to Lamperti distribution depending on temperature relative to a disorder parameter.

Contribution

It introduces a detailed analysis of occupation time distributions in quenched trap models, highlighting a temperature-dependent transition and linking to Lévy statistics.

Findings

Occupation times follow Boltzmann statistics for T > T_g.

For T < T_g, occupation times follow a Lamperti distribution.

Distribution asymmetry is governed by the Boltzmann factor with T_g.

Abstract

We investigate the distribution of occupation times for a particle undergoing a random walk among random energy traps and in the presence of a deterministic potential field $U^{{\rm det}}(x)$. When the distribution of energy traps is exponential with a width $T_g$ we find that the occupation time statistics behaves according to (i) the canonical Boltzmann theory when $T>T_g$, (ii) while for $T<T_g$ they are distributed according to the Lamperti distribution with the asymmetry of the distribution determined by the Boltzmann factor $\exp(-U^{{\rm det}}(x)/T_g)$ with $T_g$ and not $T$ being the effective temperature. We explain how our results describe occupation times in other systems with quenched disorder, when the underlying partition function of the problem is a random variable distributed according to L\'evy statistics.