Collective motion in polar active matter — ordered phase exists in 2D
The Vicsek model shows a flocking transition: below noise threshold η_c, particles spontaneously align into a moving flock — an ordered phase in 2D that violates the Mermin-Wagner theorem!
Toner & Tu showed this is possible because active matter breaks time-reversal symmetry. Their hydrodynamic theory predicts anomalous fluctuations with exponents χ=3/5, z=6/5 in 2D — confirmed numerically.
Polar order φ = |⟨e^{iθ}⟩| ranges 0 (disordered) to 1 (perfect flock).