Geometric control of motility-induced phase separation.

Curvature fundamentally alters the collective properties of soft, active, and biological materials. Here we study motility-induced phase separation (MIPS), a canonical non-equilibrium transition, and demonstrate that even weak and slowly varying curvature provides robust geometric control over the dense MIPS phase. This includes dictating both the location and morphology of the MIPS cluster, even in regimes where curvature has minimal effect on the overall phase boundaries. Focusing on active Brownian particles confined to the surface of a torus, we show that varying the aspect ratio drives a structural transition of the dense cluster from a disk localized at the outer equator to a band wrapping the minor circumference. We then discuss how the curved geometry provides a platform for comparing different theoretical frameworks for the MIPS phase: by analyzing the geometries of the cluster boundaries, we show that the dense phase shape is more consistent with a boundary-length-minimizing, thermodynamic picture than with the simplest kinetic picture in the large-particle-number limit. Our results establish curved space not only as a tool to shape and guide non-equilibrium dynamics, but as a uniquely sensitive arena for probing the fundamental mechanisms of active matter.