Like meridian lines on a globe, two lines on a Gaussian-curved surface cannot be simultaneously straight and parallel everywhere. We find that this inescapable property of Gaussian curvature has important consequences for the clustering and swarming behavior of active matter. Focusing on the case of self-propelled particles confined to the surface of a sphere, we find that for high curvature, particles converge to a common orbit to form symmetry-breaking microswarms. We prove that this microswarm flocking behavior is distinct from other known examples in that it is a result of the curvature, and not incorporated through Vicsek-like alignment rules or collision-induced torques. Additionally, we find that clustering can be either enhanced or hindered as a consequence of both the microswarming behavior and curvature-induced changes to the shape of a cluster's boundary. Furthermore, we demonstrate how surfaces of non-constant curvature lead to behaviors that are not explained by the simple averaging of the total curvature. These observations demonstrate a promising method for engineering the emergent behavior of active matter via the geometry of the environment.

Self-propelled particles phase-separate into coexisting dense and dilute regions above a critical density. The statistical nature of their stochastic motion lends itself to various theories that predict the onset of phase separation. However, these theories are ill-equipped to describe such behavior when noise becomes negligible. To overcome this limitation, we present a predictive model that relies on two density-dependent timescales: τ_{F}, the mean time particles spend between collisions; and τ_{C}, the mean lifetime of a collision. We show that only when τ_{F}<τ_{C} do collisions last long enough to develop a growing cluster and initiate phase separation. Using both analytical calculations and active particle simulations, we measure these timescales and determine the critical density for phase separation in both two and three dimensions.

We simulate an Active Inertial Particle (AIP) model and find that inertia reduces particle motility, suppresses phase separation and results in interesting oscillatory behavior between a phase separated steady-state and a homogeneous fluid state.