Vicsek Model — Collective Flocking

Each particle moves at constant speed and aligns its direction with neighbors within radius r, plus random noise η. Below a critical noise η_c the flock spontaneously orders — a non-equilibrium phase transition with order parameter φ (mean polarization) → 1.

Noise η: 0.4 Interaction radius r: 1.0 Speed v: 0.03 N particles: 200

Order φ: 0.00

Phase: —

Steps: 0

θᵢ(t+1) = ⟨θⱼ⟩_{|rᵢⱼ|<r} + η·ξᵢ
xᵢ(t+1) = xᵢ(t) + v·(cosθᵢ, sinθᵢ)

Critical noise: η_c ≈ 0.9 at this density.
Low noise: coherent flock (φ≈1); high noise: disordered (φ≈0).
The transition is discontinuous (first-order) in 2D (Grégoire & Chaté 2004). Color encodes heading angle.