Vicsek Model Explorer — interactive explainer

The Vicsek model (Vicsek et al., 1995) describes point particles that move at a fixed speed and update their heading to match the average heading of their neighbors, plus a small noise perturbation. Despite its simplicity, the model exhibits a sharp order-disorder phase transition as noise increases — below a critical η, particles spontaneously align into a coherent flock; above it, motion is disordered. The animation cycles through the heading update for a single focus particle; the right panel gives the equations and pseudocode. For an interactive simulation showing the phase regimes, see Tab ② Simulation.

Heading update Current state

▶ Focal particle ▶ Neighbor ▶ Non-neighbor → ⟨θ⟩_r ↺ Δθ → θ(t+1)

Update Equations

① Heading Update

⟨θ⟩_r = atan2(Σ_{j∈N_i} sin θ_j, Σ_{j∈N_i} cos θ_j) θ_i(t+1) = ⟨θ⟩_r + Δθ_i

Average heading of all particles within radius r (including i), plus noise. Kennedy & Eberhart (1995) borrowed this social-averaging idea for PSO.

② Noise

Δθ_i ~ U(−η/2, η/2) // uniform (default) Δθ_i ~ 𝒩(0, σ²) // Gaussian (optional)

Uniform noise is standard (Vicsek 1995). Gaussian noise has equivalent RMS magnitude at σ = η/√12.

③ Position Update

x_i(t+1) = x_i(t) + v₀(cos θ_i(t+1), sin θ_i(t+1))Δt

Each particle moves at constant speed v₀ in its new heading direction. Speed is fixed — only direction changes.

System Properties

Order Parameter φ

φ = (1/N)|Σ_j e^iθ_j| = (1/N)√((Σcosθ)²+(Σsinθ)²)

φ = 0: all headings random. φ = 1: perfect alignment. A global statistic — not part of the particle dynamics. Near the critical η_c, φ fluctuates strongly.

φ ≈ 0

φ ≈ 0.5

φ ≈ 1

Algorithm Sketch

initialize N particles in [0,L]²: x_i ∼ U([0,L]²); θ_i ∼ U(−π, π) repeat each timestep: for each particle i: N_i = {j : dist(x_i, x_j) < r} ⟨θ⟩_r = atan2(Σ_{j∈N_i}sinθ_j, Σ_{j∈N_i}cosθ_j) Δθ_i ∼ U(−η/2, η/2) θ_i ← ⟨θ⟩_r + Δθ_i x_i ← x_i + v₀(cosθ_i, sinθ_i) apply boundary conditions φ ← (1/N)|Σ_ie^iθ_i|

Phase Transition

The transition from disordered to ordered motion is driven by the competition between alignment (neighbor averaging) and randomness (η). Near the critical η_c, the order parameter φ shows large fluctuations and anomalous scaling. Whether the transition is continuous or weakly first-order remains debated (Grégoire & Chaté, 2004).