Self-propelled particles align with neighbors — order parameter jumps discontinuously at noise threshold
Order parameter φ = 0.00
Vicsek 1995: N particles move at constant speed v₀. Each particle's new direction = average direction of neighbors within radius r, plus uniform noise η. Order parameter φ = |⟨e^{iθ}⟩|: 0=disordered, 1=fully ordered. At low noise: flocking bands emerge. At high noise: disordered gas. The transition is first-order (discontinuous jump in φ) — a surprise for a simple alignment rule. This was a landmark result showing flocking as a phase transition, analogous to ferromagnetism but far from equilibrium.