Order–disorder transition driven by alignment noise
The Vicsek model: N self-propelled particles move at constant speed v. Each particle aligns its heading with the average heading of neighbors within radius r, plus a random noise term η. At low noise the system flocks coherently (φ≈1); at high noise order breaks down (φ≈0), constituting a kinetic phase transition. The order parameter φ = |⟨e^{iθ}⟩| measures global alignment.