Active swarms on a sphere

TL;DR

This paper investigates how curvature influences collective motion in active systems, revealing unique patterns like polar vortices and circulating bands on a sphere, driven by topological constraints and elastic distortions.

Contribution

It introduces a model of self-propelled particles on a sphere, demonstrating how curvature induces novel stable motion patterns and elastic energy storage, advancing understanding of active matter in curved environments.

Findings

Curvature causes stable polar vortex and circulating band patterns.

Topological constraints lead to elastic distortions and energy storage.

Analytical results connect curvature-induced frustration to pattern formation.

Abstract

Here we show that coupling to curvature has profound effects on collective motion in active systems, leading to patterns not observed in flat space. Biological examples of such active motion in curved environments are numerous: curvature and tissue folding are crucial during gastrulation, epithelial and endothelial cells move on constantly growing, curved crypts and vili in the gut, and the mammalian corneal epithelium grows in a steady-state vortex pattern. On the physics side, droplets coated with actively driven microtubule bundles show active nematic patterns. We study a model of self-propelled particles with polar alignment on a sphere. Hallmarks of these motion patterns are a polar vortex and a circulating band arising due to the incompatibility between spherical topology and uniform motion - a consequence of the hairy ball theorem. We present analytical results showing that frustration due to curvature leads to stable elastic distortions storing energy in the band.