Emergent Dynamics of Active Systems on Curved Environments

Abstract

Curvature plays a central role in the proper function of many biological processes. With active matter being a standard framework for understanding many aspects of the physics of life, it is natural to ask what effect curvature has on the collective behaviour of active matter. In this paper, we use the classical theory of surfaces to explore the active motion of self-propelled agents confined to move on a smooth curved two-dimensional surface embedded in Euclidean space. Even without interactions and alignment, the motion is non-trivially affected by the presence of curvature, leading to effects akin, e.g.\ to gravitational lensing and tidal forces. Such effects can lead to intermittent trapping of particles and profoundly affect their flocking behaviour. We show that these effects are governed by a geometric torque that, in the absence of noise and interactions, compels particles to move along geodesics.