Vicsek Model — Flocking & Polar Domain Coarsening

The Vicsek model (1995): each particle aligns its velocity with neighbors within radius r, plus angular noise η. Below the critical noise η_c, a flocking phase emerges — spontaneous polar order in 2D, violating Mermin-Wagner (active matter is out of equilibrium!). The order parameter ⟨v⟩/v₀ tracks polar domain coarsening as clusters align and merge over time.