Vicsek Model — Flocking Transition
Self-propelled particles align with neighbors; noise-driven order-disorder transition
Order φ = 0.000
The Vicsek model (1995): each particle moves at speed v₀ and aligns its heading
with the average direction of neighbors within radius r, plus angular noise η.
Despite having no explicit ferromagnetic coupling, a genuine phase transition occurs:
below critical noise η_c, the system develops long-range polar order (φ→1, all aligned).
Above η_c, motion is disordered (φ→0). In 2D, this violates the Mermin-Wagner theorem —
the transition is discontinuous (first-order) and involves traveling density bands.
Color encodes direction of motion.