Brownian particles driven by spatially periodic noise

TL;DR

This paper investigates the dynamics of a Brownian particle influenced by spatially periodic noise, revealing slow mean displacement, effects of external forces, and phase shifts on long-term diffusion and drift.

Contribution

It provides analytical and simulation insights into how spatially periodic noise affects Brownian motion, including effects of external forces and phase shifts.

Findings

Mean displacement scales as t^{-1/2} at long times

Stationary current exhibits an essential singularity near zero noise strength

Phase shift significantly influences long-time drift and diffusion

Abstract

We discuss the dynamics of a Brownian particle under the influence of a spatially periodic noise strength in one dimension using analytical theory and computer simulations. In the absence of a deterministic force, the Langevin equation can be integrated formally exactly. We determine the short- and long-time behaviour of the mean displacement (MD) and mean-squared displacement (MSD). In particular we find a very slow dynamics for the mean displacement, scaling as $t^{-1/2}$ with time $t$. Placed under an additional external periodic force near the critical tilt value we compute the stationary current obtained from the corresponding Fokker-Planck equation and identify an essential singularity if the minimum of the noise strength is zero. Finally, in order to further elucidate the effect of the random periodic driving on the diffusion process, we introduce a phase factor in the spatial noise with respect to the external periodic force and identify the value of the phase shift for which the random force exerts its strongest effect on the long-time drift velocity and diffusion coefficient.