Active Brownian Motion in two-dimensions under Stochastic Resetting

TL;DR

This paper analyzes the effects of stochastic resetting on active Brownian particles in two dimensions, deriving exact stationary distributions for certain protocols and characterizing the transition from ballistic to diffusive motion.

Contribution

It provides the first exact calculations of stationary position distributions under different resetting protocols for active Brownian particles.

Findings

Stationary states exist when both position and orientation are reset.

Position distribution diverges near the reset point at high resetting rates.

Orientation resetting leads to a transition from ballistic to diffusive motion.

Abstract

We study the position distribution of an active Brownian particle (ABP) in the presence of stochastic resetting in two spatial dimensions. We consider three different resetting protocols : (I) where both position and orientation of the particle are reset, (II) where only the position is reset, and (III) where only the orientation is reset with a certain rate $r.$ We show that in the first two cases the ABP reaches a stationary state. Using a renewal approach, we calculate exactly the stationary marginal position distributions in the limiting cases when the resetting rate $r$ is much larger or much smaller than the rotational diffusion constant $D_R$ of the ABP. We find that, in some cases, for a large resetting rate, the position distribution diverges near the resetting point; the nature of the divergence depends on the specific protocol. For the orientation resetting, there is no stationary state, but the motion changes from a ballistic one at short-times to a diffusive one at late times. We characterize the short-time non-Gaussian marginal position distributions using a perturbative approach.