Ergodicity Breaking and Localization

TL;DR

This paper investigates how combined non-Poisson renewal events with fractional Brownian motion or scaled Brownian motion lead to ergodicity breaking and localization, challenging existing views on determinism and randomness in complex systems.

Contribution

It introduces a novel analysis of ergodicity breaking and localization resulting from the joint action of NPR with FBM and SBM, revealing new phenomena in anomalous diffusion.

Findings

Localization occurs with NPR and SBM during sub-diffusion.

NPR and FBM can cause localization even with super-diffusive FBM.

The results suggest a need to refine theoretical perspectives on determinism and randomness.

Abstract

We study the joint action of the non-Poisson renewal events (NPR) yielding Continuous Time Random Walk (CTRW) with index alpha < 1 and two different generators of Hurst coefficient H not equal to 0.5, one generating fractional Brownian motion (FBM) and another scaled Brownian motion (SBM). We discuss the ergodicity breaking emerging from these joint actions and we find that in both cases the adoption of time averages leads to localization. In the case of the joint action of NPR and SBM, localization occurs when SBM would produce sub-diffusion. The joint action of NPR and FBM, on the contrary, may lead to localization when FBM would be a source of super-diffusion. We argue that the second effect might require a refinement of the theoretical perspective about determinism and randomness.