Steady states of active Brownian particles interacting with boundaries

TL;DR

This paper develops an approximation method for analyzing steady states of non-interacting active Brownian particles near boundaries, connecting boundary conditions to phenomena like accumulation and depletion interactions.

Contribution

It introduces a novel approximation strategy linking asymptotic solutions of the Smoluchowski equation to boundary conditions for active particles.

Findings

Good agreement with exact and numerical solutions when boundary conditions vary slowly.

Framework elucidates how boundary-induced breaking of detailed balance causes observed phenomena.

Applicable to characterizing long-range flows and depletion interactions.

Abstract

An active Brownian particle is a minimal model for a self-propelled colloid in a dissipative environment. Experiments and simulations show that, in the presence of boundaries and obstacles, active Brownian particle systems approach nontrivial nonequilibrium steady states with intriguing phenomenology, such as accumulation at boundaries, ratchet effects, and long-range depletion interactions. Nevertheless, theoretical analysis of these phenomena has proven difficult. Here we address this theoretical challenge in the context of non-interacting particles in two dimensions, basing our analysis on the steady-state Smoluchowski equation for the 1-particle distribution function. Our primary result is an approximation strategy that explicitly connects asymptotic solutions of the Smoluchowski equation to boundary conditions. We test this approximation against the exact analytic solution in a 2d planar geometry as well as numerical solutions in circular and elliptic geometries. We find good agreement so long as the boundary conditions do not vary too rapidly with respect to the persistence length of particle trajectories. Our results are relevant for characterizing long-range flows and depletion interactions in such systems. In particular, our framework shows how such behaviors are connected to the breaking of detailed balance at the boundaries.