Stochastic Ergodicity Breaking: a Random Walk Approach

TL;DR

This paper investigates the non-ergodic behavior of continuous time random walks (CTRWs), revealing how divergence in waiting times leads to deviations from classical statistical mechanics and deriving a generalized law for occupation times.

Contribution

It provides an analytical and numerical study of non-ergodic properties in CTRWs, including derivation of occupation time distributions in different phases.

Findings

Non-ergodic phase occurs when average waiting time diverges.

Occupation time distribution differs from Boltzmann-Gibbs in non-ergodic phase.

Derived a generalized non-ergodic statistical law.

Abstract

The continuous time random walk (CTRW) model exhibits a non-ergodic phase when the average waiting time diverges. Using an analytical approach for the non-biased and the uniformly biased CTRWs, and numerical simulations for the CTRW in a potential field, we obtain the non-ergodic properties of the random walk which show strong deviations from Boltzmann--Gibbs theory. We derive the distribution function of occupation times in a bounded region of space which, in the ergodic phase recovers the Boltzmann--Gibbs theory, while in the non-ergodic phase yields a generalized non-ergodic statistical law.