Curvature Effects on Boundary Accumulation of Self-Propelled Particles: A Fokker-Planck Approach
Abstract
It is well known that self-propelled particles tend to accumulate near boundaries. Although the influence of boundary geometry on this accumulation has been extensively studied using individual-based models, a population-level PDE description becomes desirable when a large number of particles are involved. In this work, we investigate boundary accumulation of self-propelled particles confined in two-dimensional circular and annular domains. We develop Fokker--Planck models that qualitatively reproduce the behavior observed in individual-based simulations. The main contribution is the formulation of appropriate evolution equations for particles accumulated at the boundary, incorporating transitions between the interior free-swimming state and the boundary-contact state. Starting from a general spatial two-dimensional model, we simplify the associated boundary conditions by assuming radial symmetry and then design an exponentially weighted difference scheme that preserves the total mass of the coupled system. Because transitions between different particle states induce singular behavior near the boundary, establishing discrete mass conservation is nontrivial. In particular, the velocity space must be divided into two subdomains, with the resulting discontinuity treated explicitly. Stochastic simulations and numerical experiments are performed both to validate the proposed models and to investigate the effects of geometric and dynamical parameters on boundary accumulation.
Related
- Long-time asymptotics of nonlinear Fokker-Planck equations with inhomogeneous diffusion and natural boundary conditions
- Constant mean curvature disks with circular boundary
- Constant mean curvature disks with circular boundary in the $3$-sphere and in hyperbolic space
- A remark on the compactness of manifolds with non-negative Ricci curvature and uniformly convex boundary
- Obata theorem of σk-flat and constant associated kth boundary curvature equation
BibTeX
@misc{fu2026curvature,
title = {{Curvature Effects on Boundary Accumulation of Self-Propelled Particles: A Fokker-Planck Approach}},
author = {Jingyi Fu and Jinqi Liu and Benoit Perthame and Min Tang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24513},
eprint = {2610.24513},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24513},
}
RIS
TY - UNPB
TI - Curvature Effects on Boundary Accumulation of Self-Propelled Particles: A Fokker-Planck Approach
AU - Jingyi Fu
AU - Jinqi Liu
AU - Benoit Perthame
AU - Min Tang
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24513
UR - https://arxiv.org/abs/2610.24513
AB - It is well known that self-propelled particles tend to accumulate near boundaries. Although the influence of boundary geometry on this accumulation has been extensively studied using individual-based models, a population-level PDE description becomes desirable when a large number of particles are involved. In this work, we investigate boundary accumulation of self-propelled particles confined in two-dimensional circular and annular domains. We develop Fokker--Planck models that qualitatively reproduce the behavior observed in individual-based simulations. The main contribution is the formulation of appropriate evolution equations for particles accumulated at the boundary, incorporating transitions between the interior free-swimming state and the boundary-contact state. Starting from a general spatial two-dimensional model, we simplify the associated boundary conditions by assuming radial symmetry and then design an exponentially weighted difference scheme that preserves the total mass of the coupled system. Because transitions between different particle states induce singular behavior near the boundary, establishing discrete mass conservation is nontrivial. In particular, the velocity space must be divided into two subdomains, with the resulting discontinuity treated explicitly. Stochastic simulations and numerical experiments are performed both to validate the proposed models and to investigate the effects of geometric and dynamical parameters on boundary accumulation.
ER -