Stationary distribution of active Brownian particles in a bounded domain

Abstract

Self-propelled particles such as swimming micro-organisms often accumulate at boundaries. Active Brownian particles (ABPs), that is, particles moving at constant speed with an orientation undergoing rotational diffusion, provide a simple model. ABPs have a finite probability to be on the boundary. Their probability density therefore includes both regular bulk and singular boundary components. Exchanges between these represent capture at the boundary and re-entry into the bulk. We examine the distribution of ABPs in the stationary regime. We first obtain a similarity solution that describes the probability density near the turning points, where orientation is tangential to the boundary and particles re-enter the bulk. This shows that the bulk density is singular and the boundary density is non-differentiable. We then add spatial diffusion to the ABP model and investigate the limit $\eps \to 0$ of small spatial diffusivity. We use matched asymptotics to describe the regularising effect of a finite $\eps$, focusing on the vicinity of the turning points. We assess our predictions against numerical simulations using both lattice Markov chain and finite-element discretisations.