Dynamics of a passive sliding particle on a randomly fluctuating surface

TL;DR

This study investigates how a particle slides on a fluctuating surface, revealing different scaling behaviors depending on the relative speeds of surface fluctuations and particle motion, with numerical and theoretical analysis supporting the findings.

Contribution

The paper introduces a detailed analysis of the particle's dynamics on a fluctuating surface, identifying distinct scaling regimes and providing a self-consistent approximation for the anomalous diffusion exponent.

Findings

Anomalous diffusion with exponent ~0.67 when surface fluctuates faster

Normal diffusion with exponent 0.5 when particle moves faster

Gaussian displacement distribution in both regimes

Abstract

We study the motion of a particle sliding under the action of an external field on a stochastically fluctuating one-dimensional Edwards-Wilkinson surface. Numerical simulations using the single-step model shows that the mean-square displacement of the sliding particle shows distinct dynamic scaling behavior, depending on whether the surface fluctuates faster or slower than the motion of the particle. When the surface fluctuations occur on a time scale much smaller than the particle motion, we find that the characteristic length scale shows anomalous diffusion with $\xi(t)\sim t^{2\phi}$, where $\phi\approx 0.67$ from numerical data. On the other hand, when the particle moves faster than the surface, its dynamics is controlled by the surface fluctuations and $\xi(t)\sim t^{{1/2}}$. A self-consistent approximation predicts that the anomalous diffusion exponent is $\phi={2/3}$, in good agreement with simulation results. We also discuss the possibility of a slow cross-over towards asymptotic diffusive behavior. The probability distribution of the displacement has a Gaussian form in both the cases.