Self-propelled particles — order/disorder transition at critical noise η_c
Order φ: —
Order parameter: φ = |⟨v⃗⟩|/v₀. Above η_c: disordered (φ→0). Below η_c: flocking (φ→1). The transition is discontinuous (first-order) for finite-density systems — a surprise discovered in 2004.
Vicsek Model (1995): Each particle aligns its velocity with the average direction of neighbors within radius R, plus angular noise η. The order parameter φ = |mean velocity|/v₀ undergoes a phase transition at critical noise η_c. Originally thought to be a continuous (second-order) transition analogous to XY model ferromagnetism, Grégoire & Chaté (2004) showed it is discontinuous (first-order) for large systems. The model belongs to a universality class distinct from equilibrium magnets — it is an active matter system with genuine non-equilibrium dynamics. Giant number fluctuations (∝ N^{0.8} rather than √N) are a characteristic signature.