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Coarsening Dynamics in the Vicsek Model of Active Matter

We study the flocking model introduced by Vicsek 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 $L_ρ(t) \sim t^{θ_ρ}$ (with $θ_ρ\simeq 0.25$), while the velocity length scale grows much faster, $viz.$, $L_{v}(t) \sim t^{θ_v}$ (with $θ_v \simeq 0.83$). The spatial fluctuations in the density and velocity fields 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. At large values of the scaled wave-vector, the scaled structure factors for the density and velocity fields decay with powers $-2.6$ and $-1.52$, respectively.