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 → 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.
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BibTeX
@misc{tzella2026stationary,
title = {{Stationary distribution of active Brownian particles in a bounded domain}},
author = {Alexandra Tzella and Daniel Loghin and David J. Needham and Jacques Vanneste},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03865},
eprint = {2611.03865},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03865},
}
RIS
TY - UNPB
TI - Stationary distribution of active Brownian particles in a bounded domain
AU - Alexandra Tzella
AU - Daniel Loghin
AU - David J. Needham
AU - Jacques Vanneste
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03865
UR - https://arxiv.org/abs/2611.03865
AB - 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 → 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.
ER -