Inhomogeneous diffusion and ergodicity breaking induced by global memory effects
Abstract
We introduce a class of discrete randomwalk model driven by global memory effects. At any time, the rightleft transitions depend on the whole previous history of the walker, being defined by an urnlike 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 timeaveraged moments. Even in the longtime regime, they remain random objects. While their average over realizations recovers the corresponding ensemble averages, departure between time and ensemble averages is explicitly shown through their probability densities. For the density of the second timeaveraged 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 fluctuationdissipation relation is also obtained for the timeaveraged moments.
 Publication:

Physical Review E
 Pub Date:
 November 2016
 DOI:
 10.1103/PhysRevE.94.052142
 arXiv:
 arXiv:1609.00259
 Bibcode:
 2016PhRvE..94e2142B
 Keywords:

 Condensed Matter  Statistical Mechanics
 EPrint:
 9 pages, 5 figures