Aging Continuous Time Random Walks

TL;DR

This paper analyzes aging effects in continuous time random walks, deriving exact formulas for their behavior depending on the age of the process, and explores implications for anomalous diffusion and transport in disordered systems.

Contribution

It introduces a generalized framework for aging CTRWs, providing exact solutions and detailed asymptotic analysis of their Green functions, including aging effects and biased transport.

Findings

Exact expression for the Green function in aging CTRWs.

Identification of regimes with significant aging effects.

Slow convergence to standard CTRW behavior for small diffusion exponents.

Abstract

We investigate aging continuous time random walks (ACTRW), introduced by Monthus and Bouchaud [{\em J. Phys. A} {\bf 29}, 3847 (1996)]. Statistical behaviors of the displacement of the random walker ${\bf r}={\bf r}(t) - {\bf r}(0)$ in the time interval $(0,t)$ are obtained, after aging the random walk in the time interval $(-t_a,0)$. In ACTRW formalism, the Green function $P({\bf r}, t_a, t)$ depends on the age of the random walk $t_a$ and the forward time $t$. We derive a generalized Montroll--Weiss equation, which yields an exact expression for the Fourier double Laplace transform of the ACTRW Green function. Asymptotic long times $t_a$ and $t$ behaviors of the Green function are investigated in detail. In the limit of $t\gg t_a$, we recover the standard non-equilibrium CTRW behaviors, while the important regimes $t\ll t_a$ and $t \simeq t_a$ exhibit interesting aging effects. Convergence of the ACTRW results towards CTRW behavior, becomes extremely slow when the diffusion exponent becomes small. In the context of biased ACTRW, we investigate an aging Einstein relation. We briefly discuss aging in Scher-Montroll type of transport in disordered materials.