Trapping of an active Brownian particle at a partially absorbing wall

TL;DR

This paper studies the behavior of active Brownian particles near a partially absorbing wall, deriving explicit formulas for the mean first passage time using probabilistic methods, extending understanding of boundary interactions in active matter.

Contribution

It introduces a novel probabilistic approach to compute the mean first passage time for ABPs at partially absorbing boundaries, including a general encounter-based absorption model.

Findings

Derived explicit MFPT formulas for ABPs at partially absorbing walls.

Extended the model to include encounter-based absorption mechanisms.

Provided analytical tools for understanding boundary interactions in active matter.

Abstract

Active matter concerns the self-organization of energy consuming elements such as motile bacteria or self-propelled colloids. A canonical example is an active Brownian particle (ABP) that moves at constant speed while its direction of motion undergoes rotational diffusion. When ABPs are confined within a channel, they tend to accumulate at the channel walls, even when inter-particle interactions are ignored. Each particle pushes on the boundary until a tumble event reverses its direction. The wall thus acts as a sticky boundary. In this paper we consider a natural extension of sticky boundaries that allows for a particle to be permanently killed (absorbed) whilst attached to a wall. In particular, we investigate the first passage time (FPT) problem for an ABP in a two-dimensional channel where one of the walls is partially absorbing. Calculating the exact FPT statistics requires solving a non-trivial two-way diffusion boundary value problem (BVP). We follow a different approach by separating out the dynamics away from the absorbing wall from the dynamics of absorption and escape whilst attached to the wall. Using probabilistic methods, we derive an explicit expression for the MFPT of absorption, assuming that the arrival statistics of particles at the wall are known. Our method also allows us to incorporate a more general encounter-based model of absorption.