A theory of insect swarms, courtesy of the renormalization group

You don’t have to know the position and momentum of every molecule in the Atlantic Ocean to meaningfully study the Gulf Stream current, but the small- and large-scale behaviors of water are still connected. Such connections can be brought to light by the renormalization group (RG), the art of mathematically blurring over irrelevant details of a small-scale model to distill out its measurable large-scale properties. The portfolio of RG successes includes such diverse areas as condensed matter, fluid dynamics, and particle physics.

Its reach now includes biology, thanks to new work by Andrea Cavagna (of the Institute for Complex Systems of the National Research Council in Rome) and colleagues, who applied the RG to a dynamical field theory of insect swarms. Their theoretical results agree stunningly well with both numerical simulations and experimental observations. It’s not the first attempt to perform an RG calculation on a living active system. But the successful comparison with experiment is new.

Cavagna and colleagues have long been interested in collective biological behaviors. They were initially inspired by flocking starlings, which put on especially impressive displays in their home city of Rome. (See Physics Today, October 2007, page 28 .) A flock of thousands of birds can undulate and swirl in near unison, its synchrony driven not by any one leader but by each bird’s tendency to fly in the same direction as its neighbors.

Swarming insects, too, tend to imitate their neighbors, but not strongly enough to cause the whole group to fly in the same direction. In physics parlance, swarms and flocks are the disordered and ordered phases of the same system, akin to a ferromagnet above and below its critical temperature. Like a demagnetized magnet, a swarm lacks long-range order. But it still has plenty of correlations and collective behavior.

The macroscopic quantity in question is called the dynamic critical exponent z, a measure of how a system’s correlations in space are related to its correlations in time. It can also be thought of as a measure of how quickly fluctuations spread across the system, with smaller exponents representing swifter propagation. For a standard ferromagnet, z = 2. For observed swarms, Cavagna and colleagues found a value of 1.37 ± 0.11, so low as to be unexplained by any previously existing theory.

Two key ingredients, Cavagna and colleagues found, distinguish a magnet from a swarm. The first is activity: The insects are constantly moving, so their set of nearest neighbors is constantly changing. The second is inertia: Insects take some time to react to what their neighbors are doing.

Incorporating activity into an RG calculation had been done already by other researchers in 2015 , and it yielded a z of 1.73. Adding in inertia was another story. The complexity of the calculation exploded to the point where it could no longer be done by hand. Only after two of Cavagna’s students, Luca Di Carlo and Mattia Scandolo, developed a Mathematica code to help with the calculation did everything fall into place. The result: z = 1.35. The equally challenging numerical simulations were tamed by Giulia Pisegna, another student in the group, who found z = 1.35 ± 0.04.

The agreement among theory, experiment, and simulations is impressive, but could it be an accident? The activity–inertia model already does a good job of explaining some other behaviors of insect swarms, but predicting and testing other critical exponents would strengthen the case for its correctness. The RG is capable of calculating more critical exponents than just z; the limitation now is in the experimental data. Some critical exponents, for example, manifest themselves only in a system’s response to an external stimulus—such as a magnet’s response to an applied magnetic field. It’s not yet clear how to conduct the equivalent experiment on a swarm of insects. (A. Cavagna et al., Nat. Phys., 2023, doi:10.1038/s41567-023-02028-0 .)