Weak ergodicity breaking induced by global memory effects

TL;DR

This paper investigates how global memory effects in finite-state random walks can cause weak ergodicity breaking, leading to differences between time and ensemble averages even with finite mean waiting times.

Contribution

It introduces a class of globally correlated random walk models demonstrating ergodicity breaking driven by memory effects, independent of diverging waiting times.

Findings

Global memory induces ergodicity breaking without infinite mean waiting times.

Analytical and numerical results confirm the impact of memory on ergodic properties.

Time-averaged observables differ from ensemble averages due to memory effects.

Abstract

We study the phenomenon of weak ergodicity breaking for a class of globally correlated random walk dynamics defined over a finite set of states. The persistence in a given state or the transition to another one depends on the whole previous temporal history of the system. A set of waiting time distributions, associated to each state, set the random times between consecutive steps. Their mean value is finite for all states. The probability density of time-averaged observables is obtained for different memory mechanisms. This statistical object explicitly shows departures between time and ensemble averages. While the mean residence time in each state may result divergent, we demonstrate that this condition is in general not necessary for breaking ergodicity. Hence, global memory effects are an alternative mechanism able to induce this property. Analytical and numerical calculations support these results.