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, sets 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 residence time in each state may have a divergent mean value, we demonstrate that this condition is in general not necessary for breaking ergodicity. Hence, we conclude that global memory effects are an alternative mechanism able to induce ergodicity breaking without involving power-law statistics. Analytical and numerical calculations support these results.
Highlights
Ergodicity plays a fundamental role in the formulation of statistical physics
The main goal of this paper is to demonstrate that systems whose dynamics involves global memory effects may develop ergodicity breaking (EB)
III, we study three different global memory mechanisms: the elephant random walk model, a random walk driven by an urn-like dynamics, and an imperfect case of the last one
Summary
Ergodicity plays a fundamental role in the formulation of statistical physics. This property is usually stated by saying that ensemble average and time average of observables are equals, the last one being taken in the long time (infinite) limit. Global memory (or correlation) effects refer to systems whose stochastic dynamics at a given time depends on its whole previous temporal history (trajectory). These kinds of dynamics has been studied previously [37,38,39,40,41,42,43,44,45], mainly as a mechanism that induces superdiffusion. Our main results rely on alternative memory mechanisms They are related to a Polya urn dynamics [46–50], which is one of the simplest models of contagion process, being of interest in various disciplines [47]. Analytical calculations that support the main results are presented in the Appendixes
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