Response to a small external force and fluctuations of a passive particle in a one-dimensional diffusive environment

TL;DR

This paper studies the long-term behavior of a passive particle in a one-dimensional diffusive environment, revealing trapping phenomena, scaling laws, and transient effects under small external forces and in the large diffusion limit.

Contribution

It provides new insights into the asymptotic trapping and fluctuation scaling of particles in random environments, including transient regimes and crossover behaviors.

Findings

Particle asymptotic speed scales quadratically with small external force.

Fluctuations are diffusive with possible logarithmic corrections.

Transient regimes and crossover behaviors are characterized in the large diffusion limit.

Abstract

We investigate the long time behavior of a passive particle evolving in a one-dimensional diffusive random environment, with diffusion constant $D$. We consider two cases: (a) The particle is pulled forward by a small external constant force, and (b) there is no systematic bias. Theoretical arguments and numerical simulations provide evidence that the particle is eventually trapped by the environment. This is diagnosed in two ways: The asymptotic speed of the particle scales quadratically with the external force as it goes to zero, and the fluctuations scale diffusively in the unbiased environment, up to possible logarithmic corrections in both cases. Moreover, in the large $D$ limit (homogenized regime), we find an important transient region giving rise to other, finite-size scalings, and we describe the cross-over to the true asymptotic behavior.