Hydrodynamic theory of flocking in the presence of quenched disorder

The effect of quenched (frozen) orientational disorder on the collective motion of active particles is analyzed. We find that, as with annealed disorder (Langevin noise), active polar systems are far more robust against quenched disorder than their equilibrium counterparts. In particular, long-ranged order (i. e. , the existence of a nonzero average velocity) persists in the presence of quenched disorder even in spatial dimensions d=3, while it is destroyed even by arbitrarily weak disorder in d4 in equilibrium systems. Furthermore, in d=2, quasi-long-ranged order (i. e. , spatial velocity correlations that decay as a power law with distance) occurs when quenched disorder is present, in contrast to the short-ranged order that is all that can survive in equilibrium. These predictions are borne out by simulations in both two and three dimensions.