Long-Range Order in a Two-Dimensional DynamicalXYModel: How Birds Fly Together

We propose a nonequilibrium continuum dynamical model for the collective motion of large groups of biological organisms (e. g. , flocks of birds, slime molds, etc. ) Our model becomes highly nontrivial, and different from the equilibrium model, for d<d₂=4; nonetheless, we are able to determine its scaling exponents exactly in d=2 and show that, unlike equilibrium systems, our model exhibits a broken continuous symmetry even in d=2. Our model describes a large universality class of microscopic rules, including those recently simulated by Vicsek et al.