Ageing first passage time density in continuous time random walks and quenched energy landscapes

Abstract

We study the first passage dynamics of an ageing stochastic process in the continuous time random walk (CTRW) framework. In such CTRW processes the test particle performs a random walk, in which successive steps are separated by random waiting times distributed in terms of the waiting time probability density function $\psi(t) \simeq t^{-1-\alpha}$) ($0\le \alpha \le 2$). An ageing stochastic process is defined by the explicit dependence of its dynamic quantities on the ageing time $t_a$, the time elapsed between its preparation and the start of the observation. Subdiffusive ageing CTRWs describe systems such as charge carriers in amorphous semiconductors, tracer dispersion in geological and biological systems, or the dynamics of blinking quantum dots. We derive the exact forms of the first passage time density for an ageing subdiffusive CTRW in the semi-infinite, confined, and biased case, finding different scaling regimes for weakly, intermediately, and strongly aged systems: these regimes, with different scaling laws, are also found when the scaling exponent is in the range $1<\alpha<2$, for sufficiently long $t_a$. We compare our results with the ageing motion of a test particle in a quenched energy landscape. We test our theoretical results in the quenched landscape against simulations: only when the bias is strong enough, the correlations from returning to previously visited sites become insignificant and the results approach the aging CTRW results. With small or without bias, the ageing effects disappear and a change in the exponent compared to the case of a completely annealed landscape can be found, reflecting the build-up of correlations in the quenched landscape.