Emergent Metric-Like States of Active Particles with Metric-Free Polar Alignment

We study a model of self-propelled particles interacting with their k nearest neighbors through polar alignment. By exploring its phase space as a function of two nondimensional parameters (alignment strength g and Péclet number Pe), we identify two distinct order-disorder transitions. One occurs at a low critical g value independent of Pe, has no significant density-order coupling, and is consistent with the transition previously predicted by the mean-field approach. Up to the system sizes studied, it appears continuous. The other is discontinuous, depends on a combined control parameter involving Pe and g that controls the alignment strength, and results from the formation of small, dense, highly persistent clusters of particles that follow metric-like dynamics. These dense clusters form at a critical value of the combined control parameter Pe/gα, with α≈1.5, which appears to be valid for different alignment-based models. Our study shows that models of active particles with metric-free interactions can produce characteristic length scales and self-organize into metric-like collective states that undergo metric-like transitions. Published by the American Physical Society 2025