Noise-driven order-disorder transition in self-propelled particles
The Vicsek model (1995) describes self-propelled particles that align their velocity with neighbors within radius r, plus angular noise η. Above a critical noise ηc, the system disorders; below ηc it exhibits long-range polar order — a spontaneous symmetry-breaking transition forbidden by the Mermin-Wagner theorem for equilibrium systems. The order parameter φ = |Σ e^(iθ)|/N measures collective alignment (0=disordered, 1=perfect flock).