Anomalous diffusion in correlated continuous time random walks

TL;DR

This paper investigates how correlations in waiting times and jump lengths in continuous time random walks lead to anomalous diffusion behaviors, including subdiffusion and superdiffusion, with implications for ergodicity breaking.

Contribution

It introduces a model linking correlated waiting times and jump lengths to anomalous diffusion, extending understanding of CTRW dynamics with correlated variables.

Findings

Correlated waiting times cause subdiffusion with <r^2(t)>~t^{alpha/(1+alpha)}.

Correlations in jump lengths induce superdiffusion.

Weak ergodicity breaking occurs in both correlated scenarios.

Abstract

We demonstrate that continuous time random walks in which successive waiting times are correlated by Gaussian statistics lead to anomalous diffusion with mean squared displacement <r^2(t)>~t^{2/3}. Long-ranged correlations of the waiting times with power-law exponent alpha (0<alpha<=2) give rise to subdiffusion of the form <r^2(t)>~t^{alpha/(1+alpha)}. In contrast correlations in the jump lengths are shown to produce superdiffusion. We show that in both cases weak ergodicity breaking occurs. Our results are in excellent agreement with simulations.