Active Matter: Vicsek Model + Run-and-Tumble

How does self-propulsion give rise to collective order and to anomalous transport? Two canonical models bracket the question from opposite ends. The Vicsek model takes the collective view — N aligning self-propelled particles on a periodic domain, swept over noise η to trace the order parameter φ = |⟨e^iθ⟩| through its disorder-to-order nonequilibrium phase transition. Run-and-tumble dynamics take the single-particle view — one active walker whose Poisson reorientations set a ballistic-to-diffusive crossover in the mean-squared displacement, with the effective diffusion constant checked analytically and under harmonic confinement. Built in Julia as an interactive Pluto notebook.