Collective ring formation in active matter

We study the formation of ring-like structures in interacting active particle systems in two dimensions. The emergent structure shows signatures of both spatial and orientational organization. The spatial organization is characterized by the radial distance of a tagged particle from the centroid of the assembly, while orientational organization is characterized by the radial alignment of its self-propulsion direction. We derive exact analytical expressions for the radial and polarization distributions for systems of active Brownian particles and run-and-tumble particles. While both models exhibit annular steady states, we show that their spatial and orientational organization differ qualitatively in the strongly active regime. A direct comparison of the two models reveals how the nature of the propulsion mechanism leads to the distinction in both the structure of the annulus and the statistics of particle orientations. Our results provide a unified analytical framework for characterizing emergent annular states in active matter and identify robust signatures that distinguish persistent active dynamics with continuous and discrete reorientation.