Ergodicity, rejuvenation, enhancement, and slow relaxation of diffusion in biased continuous-time random walks.

Takuma Akimoto,Andrey G Cherstvy,Ralf Metzler

doi:10.1103/physreve.98.022105

Takuma Akimoto, Andrey G Cherstvy + Show 1 more

Open Access

https://doi.org/10.1103/physreve.98.022105

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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 report on an interesting phenomenon for the enhancement of diffusion by the start of the measurement 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. Moreover, the diffusivity can be increased by the aging time when the initial ensemble is not in equilibrium. 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 unexpectedly slowly. Our findings provide a rejuvenation phenomenon in the superdiffusive regime, that is, the diffusivity for a nonequilibrium initial ensemble gradually increases to that for an equilibrium ensemble when the start of the measurement is delayed.

Highlights

Mixing of more than two fluids is a key operation of microfluidic devices in chemistry, biology, and industry, in which diffusion is an essential mechanism for mixing [1,2,3]
We investigate an initial-ensemble dependence of the variance of the displacement and ergodic properties of the time-averaged variance of the displacement in the continuous-time random walk (CTRW) with drift
In the paradigmatic CTRW model, a bias plays a significant role in the enhancement of diffusion, which is supported by an increased variance of waiting times

Summary

INTRODUCTION

Mixing of more than two fluids is a key operation of microfluidic devices in chemistry, biology, and industry, in which diffusion is an essential mechanism for mixing [1,2,3]. The diffusivity is always suppressed by the bias for discrete-time random walks. This trend may be reversed when the time steps are continuous random variables. When the second moment of the waiting time diverges, the variance becomes superdiffusive In this regime there is a clear initial-ensemble dependence of statistical quantities such as the correlation function and the mean-squared displacement [39,40], which will give rise to a nonequivalence of time and ensemble averages [40,41]. In CTRW, waiting times are independent and identically distributed (IID) random variables, which do not depend on the position of a particle.

ENSEMBLE-AVERAGED VARIANCE

TIME-AVERAGED VARIANCE

CONCLUSION