VICSEK MODEL FLOCKING

N particles align with neighbors in radius r plus noise η — order-disorder phase transition in collective motion
Vicsek et al., 1995
φ = |⟨e^(iθ)⟩| — magnitude of mean unit velocity vector. φ=1: perfect order (flock), φ≈0: disorder (gas).

Phase transition occurs at critical noise η_c depending on density ρ. Above η_c: disordered. Below: ordered flock emerges.