Natural swarms in 3.99 dimensions

The dynamical critical exponent z of natural swarms of insects is calculated using the Renormalization Group (RG) to order ε=4-d. The hydrodynamic theory of swarms combines the effects of activity with the presence of inertial behavior, which has been found to be a key ingredient in the description of the dynamics of natural swarms in the field. A novel RG fixed point emerges, where both off-equilibrium activity and inertial mode-coupling inertial interactions are relevant. In three dimensions we predict a critical exponent z = 1.35, in excellent agreement with the experimental value, z_exp = 1.37 ± 0.11.

To account for the presence of inertial behavior, the velocity field is coupled to the generator of its rotations, namely the spin, through a mode-coupling interaction. Because of the rotational symmetry, the spin is assumed to be exactly conserved. However, this might not always be true in biological systems. Hence, we show also that in the presence of weak spin dissipation our predictions might still hold for finite-size systems such as swarms. This result gives additional evidence for the fact that inertial mode-coupling dynamics plays a crucial role, together with activity, in characterizing the collective behavior in biological systems.