Order parameter φ = 0.00
Critical noise η_c ≈ 0.9 (ρ=0.6)
Vicsek Model (1995)
Each particle moves at constant speed and aligns its direction with neighbors within radius r, plus noise η. The order parameter is the mean normalized velocity:
φ = |⟨e^iθ⟩|
At low noise (η ≪ η_c), particles form coherent flocks — φ→1. Above the critical noise, motion is disordered — φ→0. The transition is first-order in 2D (controversial since 1995; confirmed by Chate et al. 2004). Near criticality, giant density fluctuations arise: Δρ ∼ ρ^(0.8) — anomalous, unlike equilibrium.
active matter
flocking
symmetry breaking
non-equilibrium