Self-Propelled Particle Flocking (Vicsek Model)
Active matter phase transition: disorder → collective motion via local alignment
N: 300
Order φ: 0.00
Phase: disordered
Vicsek model (1995): each particle aligns its direction with neighbors within radius r, plus angular noise η.
Update: θᵢ(t+1) = ⟨θⱼ⟩_{|rᵢ-rⱼ|<r} + ξ, where ξ∈[-η/2, η/2].
Order parameter φ = |Σeⁱᶿ|/N measures collective alignment: 0=random, 1=perfect flock.
Phase transition: above critical noise η_c, the flock disorders. Below η_c, spontaneous symmetry breaking creates collective motion — an active matter analog of the XY model ferromagnet.
Colors encode heading direction (hue = angle). Watch coherent bands and clusters emerge.