Self-propelled particles — order-disorder transition at critical noise η_c
Vicsek Model (1995): Each particle moves at constant speed v, aligning direction with neighbors within radius r, plus angular noise η.
Below critical noise η_c: ordered phase Φ→1 (all particles align).
Above η_c: disordered phase Φ→0 (random motion). Transition is first-order in 2D (Toner-Tu 1995).
Right panel shows order parameter Φ = |⟨e^{iθ}⟩| vs time and η sweep history.