Transient anomalous diffusion in periodic systems: ergodicity, symmetry breaking and velocity relaxation.

Jakub Spiechowicz,Jerzy Łuczka,Peter Hänggi

doi:10.1038/srep30948

Abstract

We study far from equilibrium transport of a periodically driven inertial Brownian particle moving in a periodic potential. As detected for a SQUID ratchet dynamics, the mean square deviation of the particle position from its average may involve three distinct intermediate, although extended diffusive regimes: initially as superdiffusion, followed by subdiffusion and finally, normal diffusion in the asymptotic long time limit. Even though these anomalies are transient effects, their lifetime can be many, many orders of magnitude longer than the characteristic time scale of the setup and turns out to be extraordinarily sensitive to the system parameters like temperature or the potential asymmetry. In the paper we reveal mechanisms of diffusion anomalies related to ergodicity of the system, symmetry breaking of the periodic potential and ultraslow relaxation of the particle velocity towards its steady state. Similar sequences of the diffusive behaviours could be detected in various systems including, among others, colloidal particles in random potentials, glass forming liquids and granular gases.

Highlights

Experiments[40,41]
There, an archetypal model of the Brownian ratchet was studied revealing that the mean square deviation of the particle position from its average proceeds within three intermediate stages: initially starting as superdiffusion, followed as subdiffusion and as normal diffusion in the asymptotic long time limit
Its time significantly influences the duration of superdiffusion τ1 as we demonstrate in Anomalous Diffusion: Impact of Potential Asymmetry

Summary

Introduction

Experiments[40,41]. This kind of dynamics may not survive until the asymptotic long time regime, lately even its transient nature has been predicted theoretically and observed experimentally[42,43,44]. There, an archetypal model of the Brownian ratchet was studied revealing that the mean square deviation of the particle position from its average proceeds within three intermediate stages: initially starting as superdiffusion, followed as subdiffusion and as normal diffusion in the asymptotic long time limit. These transient diffusion anomalies may last many orders longer than the characteristic time scales of the setup and their lifetime can be controlled by variation of the system temperature and its asymmetry. The spatially periodic potential U(x) is assumed to be in a double-sine form[48] of period 2πL and a barrier height ΔU, namely

Methods

Discussion

Conclusion