Vicsek Model: Active Matter

N self-propelled particles move at constant speed. Each aligns its direction with neighbors within radius r, plus random noise η. A phase transition from disorder to collective motion occurs as noise decreases or density increases.

θi(t+1) = ⟨θj|rij|<r + ξ  |  Order: φ = |⟨e⟩|  |  φ≈0 (disordered), φ≈1 (ordered)