Vicsek Flocking Model

Local alignment rule + noise drives an order-disorder transition — true long-range order in 2D, unlike equilibrium systems (Mermin-Wagner forbids it at equilibrium).

η=0.20
r=25
v=3
N=200
Order Φ: —
Phase: —
Vicsek (1995): each agent moves at fixed speed v and aligns its direction with neighbors within radius r, plus random noise η. Despite simplicity, the model shows a continuous phase transition at critical noise η_c. Below η_c: collective flocking (Φ≈1). Above: disordered motion (Φ≈0). Remarkably, 2D flocking violates Mermin-Wagner theorem because the system is out of equilibrium — activity itself stabilizes long-range order.