Instabilities and the emergence of self-organization in active matter: General considerations and an example

In nature, active particles can exhibit surprisingly complex self-coordinated movements. According to self-organization theory, the instability of the asynchronous state may induce these collective motions. Here, we present a general framework to model this using coupled Langevin equations. Then, based on the nonlinear Fokker-Planck equation formalism, we developed a comprehensive stability analysis of the asynchronous state. Furthermore, as an example, we applied these results to study the confluence of anti-aligning interactions with cohesive forces. Specifically, we performed numerical simulations of the active particles' system in a parameter region where the theory predicts that a spatially extended distribution of particles is unstable. As a result, the active particles form a cluster. We also detected two distinct types of such localized states, depending on the initial condition. In one of these states, the particles move disorderly, with a velocity dissociated from the one imposed by the self-propulsion drift. The other state exhibits two counterpropagating groups rotating with opposite angular velocities, where the particles primarily move with the velocity dictated by the self-propulsion force.