Vicsek Flocking Transition

N self-propelled particles move at constant speed, aligning with neighbors within radius r plus noise η. A noise-driven order-disorder phase transition emerges: below critical noise, polar order φ → 1 (collective motion); above it, φ → 0 (disorder).

0.50
2.0
0.10
0.030
200
Order φ
—
Mean angle
—
Step
0
Vicsek update: θᵢ(t+1) = ⟨θⱼ⟩_{|rⱼ-rᵢ|<r} + ξᵢ  |  Order: φ = |N⁻¹ Σ eⁱθⱼ|