Inhomogeneous diffusion and ergodicity breaking induced by global memory effects

TL;DR

This paper introduces a discrete random walk model with global memory effects, revealing inhomogeneous, ballistic diffusion and ergodicity breaking, with trajectories exhibiting strong trajectory-to-trajectory variability and non-commuting limits.

Contribution

The paper presents an exact analytical framework for a memory-driven random walk model, demonstrating inhomogeneous diffusion and ergodicity breaking phenomena.

Findings

Ensemble of trajectories exhibits ballistic behavior.

Time-averaged moments become random variables in the long-time limit.

Discrepancy between time and ensemble averages due to memory effects.

Abstract

We introduce a class of discrete random walk model driven by global memory effects. At any time the right-left transitions depend on the whole previous history of the walker, being defined by an urn-like memory mechanism. The characteristic function is calculated in an exact way, which allows us to demonstrate that the ensemble of realizations is ballistic. Asymptotically each realization is equivalent to that of a biased Markovian diffusion process with transition rates that strongly differs from one trajectory to another. Using this "inhomogeneous diffusion" feature the ergodic properties of the dynamics are analytically studied through the time-averaged moments. In the long time regime they becomes random objects. While their mean values recover the corresponding ensemble averages, departure between time and ensemble averages is explicitly shown through their probability densities. For the density of the second time-averaged moment an ergodic limit and the limit of infinite lag times do not commutate. All these effects are induced by the memory effects.