Self-propelled particles • alignment interaction • polar order parameter φ
Flock Parameters
θᵢ(t+1) = ⟨θ⟩ᵣ + ξη
ξ ∈ [−π, π] uniform
Order param:
φ = |⟨eⁱθ⟩| ∈ [0,1]
φ→1: ordered flock
φ→0: disordered gas
Critical η* ~ ρ, r
Order φ = –
Phase: –
Vicsek model (1995): each agent aligns with average heading of neighbors, plus angular noise η. Above a critical noise, collective motion breaks down in a discontinuous (first-order in 2D) transition. A paradigm of non-equilibrium phase transitions in flocking birds, bacteria, and crowd dynamics.