Asymptotic analysis and simulation of mean first passage time for active Brownian particles in 1-D

TL;DR

This paper derives analytical expressions for the mean first passage time of active Brownian particles in a 1-D domain, revealing how swimming behavior and initial conditions influence escape times, validated by numerical simulations.

Contribution

It provides the first asymptotic analytical expressions for MFPT of ABPs in 1-D, considering weak swimming and initial orientation effects.

Findings

MFPT depends on initial position and orientation.

Bias in initial orientation causes asymmetry in MFPT.

Analytical results agree with numerical PDE solutions.

Abstract

Active Brownian particles (ABPs) are a model for nonequilibrium systems in which the constituent particles are self-propelled in addition to their Brownian motion. Compared to the well-studied mean first passage time (MFPT) of passive Brownian particles, the MFPT of ABPs is much less developed. In this paper, we study the MFPT for ABPs in a 1-D domain with absorbing boundary conditions at both ends of the domain. To reveal the effect of swimming on the MFPT, we consider an asymptotic analysis in the weak-swimming or small P\'eclet (Pe) number limit. In particular, analytical expressions for the survival probability and the MFPT are developed up to O(Pe$^2$). We explore the effects of the starting positions and starting orientations on the MFPT. Our analysis shows that if the starting orientations are biased towards one side of the domain, the MFPT as a function of the starting position becomes asymmetric about the center of the domain. The analytical results were confirmed by the numerical solutions of the full PDE model.