Self-propelled particles: polar order parameter φ vs noise η
Controls
0.30
1.0
0.3
200
φ (order): —
Phase: —
Vicsek Model:
Each particle moves at speed v₀ and aligns its velocity with neighbors within radius r, plus random noise η. At low η: ordered flock (φ→1). At high η: disordered (φ→0). Near the critical noise η_c: band formation, giant density fluctuations. The transition is discontinuous (first-order) — Vicsek 1995, Toner-Tu 1998 field theory.