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To understand the collective behaviors of biological swarms, flocks, and colonies, we investigated the non-equilibrium dynamic patterns of self-propelled particle systems using statistical mechanics methods and H-stability analysis of Hamiltonian systems. By varying the individual vision range, we observed phase transitions between four phases, i.e., gas, crystal, liquid, and mill-liquid coexistence patterns. In addition, by varying the inter-particle force, we detected three distinct milling sub-phases, i.e., ring, annulus, and disk. Based on the coherent analysis for collective motions, one may predict the stability and adjust the morphology of the phases of self-propelled particles, which has promising potential applications in natural self-propelled particles and artificial multi-agent systems

Pattern phase transitions of self-propelled particles: gases, crystals, liquids, and mills