A Finite Volume Method for Continuum Limit Equations of Nonlocally Interacting Active Chiral Particles

TL;DR

This paper develops a finite volume numerical method to solve continuum equations describing nonlocally interacting active chiral particles, enabling analysis of pattern formation and phase transitions in such systems.

Contribution

It introduces a general finite volume framework for numerically solving nonlinear PDEs of active particle systems with nonlocal interactions, extending analytical approaches.

Findings

The method accurately reproduces analytical results for homogeneous problems.

It predicts pattern formation in inhomogeneous systems.

Different types of phase transitions are identified in the models.

Abstract

The continuum description of active particle systems is an efficient instrument to analyze a finite size particle dynamics in the limit of a large number of particles. However, it is often the case that such equations appear as nonlinear integro-differential equations and purely analytical treatment becomes quite limited. We propose a general framework of finite volume methods (FVMs) to numerically solve partial differential equations (PDEs) of the continuum limit of nonlocally interacting chiral active particle systems confined to two dimensions. We demonstrate the performance of the method on spatially homogeneous problems, where the comparison to analytical results is available, and on general spatially inhomogeneous equations, where pattern formation is predicted by kinetic theory. We numerically investigate phase transitions of particular problems in both spatially homogeneous and inhomogeneous regimes and report the existence of different first and second order transitions.