Active Matter — Self-Propelled Particles (Vicsek Model)

Each particle moves at constant speed and aligns its direction with neighbors within radius r, plus noise η. At low noise / high density, flocking emerges — a non-equilibrium phase transition.

Parameters

Order Parameter

Polar order φ —
Mean speed —
Clusters —
Vicsek model (1995): continuous active matter with velocity alignment.

Order parameter: φ = |Σvᵢ|/(N·v₀)
φ≈1: ordered flock, φ≈0: disordered gas.

Phase transition at critical noise η_c depends on density ρ. The transition is first-order with giant number fluctuations — anomalous even for critical phenomena.