This is a minimal model of collective motion with asynchronous (sequential) updates. Each agent aligns its velocity with the average direction of neighbors within radius r, then adds angular noise η. The update is random-sequential (one agent per step), unlike the original Vicsek model's synchronous update.
The polar order parameter φ = |⟨v̂⟩| measures global alignment: φ≈1 is perfect flocking, φ≈0 is disordered. As noise increases past a critical threshold η_c, the system undergoes a phase transition. For finite systems this transition shows strong fluctuations — the susceptibility χ = N·Var(φ) peaks at η_c.
Watch cohesive flocks form at low noise (blue arrows align), and dissolve into chaos at high noise. The order bar tracks φ in real time.