Enhancement, slow relaxation, ergodicity and rejuvenation of diffusion in biased continuous-time random walks

Abstract

Bias plays an important role in the enhancement of diffusion in periodic potentials. Using the continuous-time random walk in the presence of a bias, we provide a novel mechanism for the enhancement of diffusion in a random energy landscape. When the variance of the waiting time diverges, in contrast to the bias-free case the dynamics with bias becomes superdiffusive. In the superdiffusive regime, we find a distinct initial ensemble dependence of the diffusivity. We show that the time-averaged variance converges to the corresponding ensemble-averaged variance, i.e., ergodicity is preserved. However, trajectory-to-trajectory fluctuations of the time-averaged variance decay slowly. Our finding suggests that in the superdiffusive regime the diffusivity for a non-equilibrium initial ensemble gradually increases to that for an equilibrium ensemble when the start of the measurement is delayed, corresponding to a rejuvenation of diffusivity.