Vicsek Model — Flocking & Active Matter

Self-propelled particles align with neighbors up to noise η. Above critical noise η_c, collective motion breaks down in a discontinuous (first-order) transition.

η_c ≈ 0.6–1.2 depending on density
Order parameter φ: —
Particles N: —
Phase: —
Step: 0

Vicsek (1995): update rule
θᵢ(t+1) = ⟨θⱼ⟩_{|rᵢ-rⱼ|<r} + η·ξ
ξ ∈ [−π,π] uniform

φ = |Σe^{iθⱼ}|/N
φ≈1: ordered, φ≈0: disordered.

Transition: 1st order for finite density, becomes continuous in the thermodynamic limit (debate ongoing — Toner-Tu field theory predicts long-range order in 2D, violating Mermin-Wagner).
Angular distribution P(θ) — uniform=disordered, peaked=ordered