Steady and transient states in low-energy swarms: Stability and first-passage times.

Abstract

We investigate a class of agent-based models of self-propelled particles (SPP) that interact according to a Morse potential in the presence of friction, a class which was able to reproduce many of the intriguing patterns of collective motion observed in nature. Specifically, we compare two closely related SPP models in the literature that differ by their prescription of particle drag and self-propulsion. Writing both models in terms of nondimensional parameters allows us to show that the dynamics in the highly viscous regime is independent of the precise forms of drag and propulsion. In contrast to what is indicated in the literature both models yield the same low-energy self-organized states: the coherent flock and the rigid rotation states which are highly ordered in both the coordinate and the velocity spaces and a velocity-disordered droplet state where particles are confined to rings which pass through the lattice points of the underlying Lagrange configuration. In contrast to the first two states which are stable, the third state is found to be a long-lived transient. In the regime studied, relaxing to one of the ordered steady states is inevitable, but how and when the transition occurs and what is the probability of ending in one state rather than the other are functions of the model parameters. Two types of transitions are characterized and first passage times are computed. Eventually, the evolution of the order parameter is explored in the framework of a Langevin-type equation, and the possible metastability of the random droplet state is discussed.