Self-propelled hard disks: implicit alignment and transition to collective motion

We formulate amodel of self-propelled hard disks whose dynamics is governed by mutually coupled vectors for velocity and body orientation. Numerical integration at low densities reveals that the expected transition from isotropic to aligned collective motion is present. However, the transition at the Landau mean-field level is strongly first-order, while it is continuous in the Vicsek model. We show that this difference is rooted in the completely opposite effect that individual scattering events have on alignment. We argue that such differences will generically hold for systems of self-propelled particles with repulsive short-ranged interactions. We obtain these results by matching the numerical results to the framework of Boltzmann theory, based on the statistics of binary scattering properties, always assuming that the system is small enough to stay homogeneous. We further show that the presence of noise in the dynamics can change the nature of the transition from discontinuous to continuous.