Vicsek Flocking Model

Alignment-based collective motion and order-disorder transition

Noise η η = 0.20
Interaction Radius r = 25
Density ρ N = 200
Speed v = 3

Vicsek (1995): each particle aligns its velocity with neighbors within radius r, plus angular noise η. Order parameter φ = |⟨e^{iθ}⟩| transitions from 0 (disordered) to 1 (ordered flock) as η decreases. Color encodes heading direction. Panel shows real-time order parameter φ.