Vicsek model — order-disorder transition at critical noise η_c, φ = |⟨v⟩|/v₀
The Vicsek model (1995) describes N self-propelled particles moving at constant speed v₀, each aligning with the average direction of neighbors within radius r plus angular noise η. The order parameter φ = |⟨e^(iθ)⟩| undergoes a continuous transition near η_c ≈ 0.45 (density-dependent). Unlike equilibrium systems, this is an active phase transition violating detailed balance, showing giant density fluctuations and propagating density waves.