Vicsek Model — Flocking Phase Transition
PARAMETERS
Noise η
0.30
Interaction radius r
0.10
Num particles N
300
Speed v₀
0.03
Density ρ
—
ORDER PARAMETER
|φ| (alignment):
—
Avg velocity x:
—
Avg velocity y:
—
Phase:
—
VICSEK 1995
θᵢ(t+1) = ⟨θⱼ⟩_{|rᵢⱼ|<r} + ηξᵢ
rᵢ(t+1) = rᵢ(t) + v₀·êᵢ(t)
Order parameter: φ = |⟨e^{iθ}⟩|
Phase transition at η_c(ρ,v):
• Disordered: φ → 0 (η > η_c)
• Ordered: φ → 1 (η < η_c)
Transition is FIRST ORDER
(Grégoire & Chaté 2004)