Dynamical Renormalization Group Approach to the Collective Behavior of Swarms

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 $z=d/2$, and a stable fixed point with $z=2$, 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 $z=3/2$, a value significantly closer to the experimental window, $1.0\le z\le 1.3$, than the value $z\approx 2$ 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.

Figure 1

The renormalization group flow and crossover. Top: Flow diagram on the $(X_l,f_l)$ plane for $d=3$. When the initial friction ${\eta }_0$ is small, i.e., $X_0\sim 1$ (red dots), the flow converges towards the unstable fixed point, $z=d/2$, and remains in its neighborhood for many iterations, before crossing over to the $z=2$ fixed point. Bottom: When plotting the running parameters and the critical exponent as a function of the iteration step along a flow line with small ${\eta }_0$, the RG crossover clearly emerges. The initial values of the parameters are $f_0=2$, $X_0=0.9999$, $w_0=3$.

Figure 2

Numerical simulations. Top: Relaxation time vs correlation length in $d=3$, at various values of the friction $\widehat{\eta }$. Lines are best fit to $z=3/2$ (blue, low $\widehat{\eta }$) and $z=2$ (orange, large $\widehat{\eta }$). Bottom, left: The normalized dynamical correlation function, $C(k,t)/C(k,0)$, at $k=1/\xi$. Right: The relaxation form factor, $h(t/\tau )\equiv \dot{C}(t/\tau )/C(t/\tau )$, goes to 1 for exponential relaxation, while it goes to 0 for inertial relaxation. Experimental data on swarms and data on the $3d$ Vicsek model close to the ordering transition are from [5].