How birds fly together: Long-range order in a two-dimensional dynamical XY model

We propose a non-equilibrium 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 non-trivial, and different from the equilibrium model, for $d<d_c=4$; nonetheless, we are able to determine its scaling exponents {\it 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 Viscek et. al.