The cohesive motion of autonomous agents is ubiquitous in natural, social and technological settings. Its current models are often discretized in time and include one or more of the following components: explicit velocity alignment (also called neighbor following), attraction/adhesion, inelastic collisions and friction. However, real moving agents (animals, humans, robots, etc.) are usually asynchronous and perceive the coordinates of others with higher precision than their velocities. Therefore, here we work with a minimal model that applies none of the listed components and is continuous in both time and space. The model contains (i) radial repulsion among the particles, (ii) self-propelling parallel to each particle's velocity and (iii) noise. First, we show that in this model two particles colliding symmetrically in 2 dimensions at a large angle leave at a smaller angle, i.e., their total momentum grows. For many particles we find that such local gains of momentum can lead to stable global ordering. As a function of noise amplitudes we observe a critical slowing down at the order-disorder boundary, indicating a dynamical phase transition. Our current numerical results -- limited by the system's slowing down -- show that the transition is discontinuous.
Collective motion in a minimal continuous model
Illes J. Farkas, Jeromos Kun, Yi Jin, Gaoqi He, Mingliang Xu