Mean first passage time of active Brownian particles in two dimensions

TL;DR

This paper develops a PDE-based framework to analyze the mean first passage time of active Brownian particles in two-dimensional domains, revealing complex behaviors influenced by activity, geometry, and initial conditions.

Contribution

It introduces a PDE approach to characterize MFPT for ABPs in various geometries, including asymptotic analysis and validation against simulations, advancing understanding of active particle escape dynamics.

Findings

MFPT exhibits non-monotonic dependence on initial conditions.

Increasing swimming speed can either increase or decrease MFPT.

Rich behaviors differ from passive particles, influenced by geometry and activity.

Abstract

The mean first passage time (MFPT) is a key metric for understanding transport, search, and escape processes in stochastic systems. While well characterized for passive Brownian particles, its behavior in active systems-such as active Brownian particles (ABPs)-remains less understood due to their self-propelled, nonequilibrium dynamics. In this paper, we formulate and analyze an elliptic partial differential equation (PDE) to characterize the MFPT of ABPs in two-dimensional domains, including circular, annular, and elliptical regions. For annular regions, we analyze the MFPT of ABPs under various boundary conditions. Our results reveal rich behaviors in the MFPT of ABPs that differ fundamentally from those of passive particles. Notably, the MFPT exhibits non-monotonic dependence on the initial position and orientation of the particle, with maxima often occurring away from the domain center. We also find that increasing swimming speed can either increase or decrease the MFPT depending on the geometry and initial orientation. Asymptotic analysis of the PDE in the weak-activity regime provides insight into how activity modifies escape statistics of the particles in different geometries. Finally, our numerical solutions of the PDE are validated against Monte Carlo simulations.