Coarsening dynamics in the Vicsek model

We numerically study the flocking model introduced by Vicsek et al. (1995) in the coarsening regime. At standard self-propulsion speeds, we find two distinct growth laws for the coupled density and velocity fields. The characteristic length scale of the density domains grows as $L_{\rho}(t) \sim t^{1/4}$, while the velocity length scale grows much faster, $viz.$, $L_{v}(t) \sim t^{5/6}$. The spatial fluctuations in the density and velocity ordering are studied by calculating the two-point correlation function and the structure factor, which show deviations from the well-known Porod's law. This is a natural consequence of scattering from irregular morphologies that dynamically arise in the system. In contrast, at lower self-propulsion speeds, the morphology is distinct, and as a result a new set of scaling exponents emerge. Most strikingly, the velocity order follows the density order with $L_{\rho}(t) \sim L_v(t) \sim t^{1/4}$.