Ergodic transitions in continuous-time random walks

TL;DR

This paper derives a general framework for understanding ergodic and nonergodic behavior in continuous-time random walks with arbitrary waiting time distributions, applicable across various lattice structures.

Contribution

It provides a unified expression for the distribution of time-averaged observables, extending recent theoretical results to systems with diverse trapping time distributions.

Findings

Transitions between ergodic and weakly nonergodic regimes identified.

Systems with non-identical trapping times are typically nonergodic.

Results applicable regardless of lattice topology and dimensionality.

Abstract

We consider continuous-time random walk models described by arbitrary sojourn time probability density functions. We find a general expression for the distribution of time-averaged observables for such systems, generalizing some recent results presented in the literature. For the case where sojourn times are identically distributed independent random variables, our results shed some light on the recently proposed transitions between ergodic and weakly nonergodic regimes. On the other hand, for the case of non-identical trapping time densities over the lattice points, the distribution of time-averaged observables reveals that such systems are typically nonergodic, in agreement with some recent experimental evidences on the statistics of blinking quantum dots. Some explicit examples are considered in detail. Our results are independent of the lattice topology and dimensionality.