Self-propelled particles align with neighbors within radius r, subject to angular noise η. At low noise: long-range polar order emerges spontaneously.
Update rule: θᵢ(t+1) = ⟨θⱼ⟩_{|rᵢ−rⱼ|<r} + ξ
where ξ ∈ [−ηπ, ηπ] is random noise.
The flocking transition at η_c(ρ) is a genuine non-equilibrium phase transition: the ordered phase breaks a continuous symmetry (rotation), which would be impossible in equilibrium 2D by Mermin-Wagner, but is allowed here because the system is far from equilibrium.
The transition is first-order (discontinuous), with density bands (smectic-like) in the coexistence region — a striking non-equilibrium pattern.