Universal Dynamics of a Passive Particle Driven by Brownian Motion

TL;DR

This paper studies the universal long-time behavior of a passive particle driven by Brownian motion under nonreciprocal interactions, revealing dimension-dependent MSD growth and universal distribution functions for short-ranged potentials.

Contribution

It demonstrates the universality of the driven particle's long-time dynamics and provides exact scaling functions for position distributions in one and two dimensions.

Findings

MSD grows as t^{1/2} in 1D and log t in 2D for short-range interactions.

Exact universal scaling functions for position distributions are derived.

Long-range interactions lead to MSD growth as t^{} with potential-dependent exponent.

Abstract

We investigate the overdamped dynamics of a `passive' particle driven by nonreciprocal interaction with a `driver' Brownian particle. When the interaction between them is short-ranged, the long-time behavior of the driven particle is remarkably universal -- the mean-squared displacement (MSD) and the typical position of the driven particle exhibits the same qualitative behaviors independent of the specific form of the potential. In particular, the MSD grows as $t^{1/2}$ in one dimension and $\log t$ in two spatial dimensions. We compute the exact scaling functions for the position distribution in $d=1$ and $d=2$. These functions are universal when the interaction is short-ranged. For long-ranged interactions, the MSD of the driven particle grows as $t^{\phi}$ with exponent $\phi$ depending on the tail of the potential.