Active matter: self-propelled particles align with neighbors — continuous phase transition to collective motion
Update rule: θᵢ(t+Δt) = ⟨θⱼ⟩_{|rᵢⱼ|<r} + η·ξᵢ (Vicsek et al 1995)
Order parameter: φ = (1/N)|Σᵢ e^{iθᵢ}| ∈ [0,1]
Phase transition: disordered (φ≈0) → polar flock (φ→1) as η decreases
Giant number fluctuations: ΔN ~ N^α, α≈0.8 (not ½) — hallmark of active matter