Evolution of a passive particle in a one-dimensional diffusive environment

TL;DR

This paper investigates the dynamics of a passive particle in a one-dimensional fluctuating environment, revealing Gaussian sub-diffusive behavior at short times and diffusive trapping at long times.

Contribution

It provides a detailed analysis of the particle's transition from sub-diffusive to diffusive behavior in a fluctuating potential environment.

Findings

Gaussian sub-diffusive fluctuations with exponent 3/4 at short times

Particle becomes trapped by local minima at long times

Long-time behavior is diffusive with exponent 1/2

Abstract

We study the behavior of a tracer particle driven by a one-dimensional fluctuating potential, defined initially as a Brownian motion, and evolving in time according to the heat equation. We obtain two main results. First, in the short time limit, we show that the fluctuations of the particle become Gaussian and sub-diffusive, with dynamical exponent $3/4$. Second, in the long time limit, we show that the particle is trapped by the local minima of the potential and evolves diffusively i.e. with exponent $1/2$.