Dynamical Renormalization Group Approach to the Collective Behavior of Swarms

Abstract

We study the critical behavior of a model with nondissipative couplings aimed at describing the collective behavior of natural swarms, using the dynamical renormalization group under a fixed-network approximation. At one loop, we find a crossover between an unstable fixed point, characterized by a dynamical critical exponent z ¼ d=2, and a stable fixed point with z ¼ 2, a result we confirm through numerical simulations. The crossover is regulated by a length scale given by the ratio between the transport coefficient and the effective friction, so that in finite-size biological systems with low dissipation, dynamics is ruled by the unstable fixed point. In three dimensions this mechanism gives z ¼ 3=2, a value significantly closer to the experimental window, 1.0 ≤ z ≤ 1.3, than the value z ≈ 2 numerically found in fully dissipative models, either at or off equilibrium. This result indicates that nondissipative dynamical couplings are necessary to develop a theory of natural swarms fully consistent with experiments.