Inhomogeneous diffusion and ergodicity breaking induced by global 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 previoushistory of the walker, being defined by an urn-like memory mechanism. The characteristic function is calculated in an exact way, which allows us todemonstrate that the ensemble of realizations is ballistic. Asymptoticallyeach 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. Even in the long time regime they remain random objects. While their average over realizations 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. A generalized Einstein fluctuation-dissipation relation is also obtained for the time-averaged moments.