Active fluids consist of self-propelled particles (bacteria, birds, cytoskeletal motors) that consume energy to maintain directed motion.
The Vicsek model is the canonical polar active matter model: particles align with neighbors within radius r, subject to noise η. Above a critical density or below a noise threshold, spontaneous polar order (flocking) emerges — a non-equilibrium phase transition.
The Toner-Tu field theory shows the ordered phase has anomalous long-range order even in 2D, violating Mermin-Wagner theorem (activity saves the day).
Rich patterns emerge: bands, lanes, vortices, asters — depending on symmetry and boundary conditions.