Universal dynamics of a passive particle driven by Brownian motion

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 exhibit 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 logt in two spatial dimensions. We compute the exact scaling functions for the position distribution in d=1 and 2. These functions are universal when the interaction is short-ranged. For long-ranged interactions, the MSD of the driven particle grows as t^{ϕ} with exponent ϕ depending on the tail of the potential.