Vicsek Model — Active Matter Flocking

The Vicsek model (1995) is the canonical model of collective motion in active matter — flocks of birds, schools of fish, bacterial swarms. Each particle moves at constant speed v₀ and aligns its direction with neighbors within radius r, plus random noise η. Update rule: θᵢ(t+1) = ⟨θⱼ⟩_{|rᵢ-rⱼ|<r} + ξ, where ξ ∈ [−ηπ, +ηπ]. The order parameter Φ = |N⁻¹Σ e^{iθⱼ}| measures global alignment: Φ→1 (ordered flock) at low noise/high density, Φ→0 (disordered) at high noise. Vicsek et al. showed this is a genuine phase transition in 2D, violating the Mermin-Wagner theorem because the system is out of equilibrium. The transition is first-order (discontinuous) with coexisting bands of high-density ordered and low-density disordered regions.