You don’t have to know the position and momentum of every molecule in the North Atlantic Ocean to study the Gulf Stream current, but small- and large-scale phenomena are still connected. That’s especially true in the vicinity of a phase transition. If you want to understand what’s going on in water near its critical point, where the distinction between liquid and gas fades away, or in a magnet near its Curie temperature, at which permanent magnetization vanishes, you can’t confine your investigation to just one length scale. They’re all important.

Nowadays it may be tempting to approach such a problem by brute force: Set up a computer simulation sufficiently intricate to capture a wide range of scales, and let it run. But half a century ago, when that was a less realistic option, physicists were driven by necessity to seek a more analytically tractable solution. The result, the renormalization group (RG), is the art of mathematically zooming out—of blurring over a system’s finest details and compressing everything else to a smaller scale—in search of scaling laws and self-similar phenomena. When applied across diverse areas of physics, spanning both condensed matter and high energy, RG calculations have identified deep connections among ostensibly dissimilar systems.^{1}

The RG’s reach now includes biology, thanks to new work by researchers led by Andrea Cavagna, of the Institute for Complex Systems of the National Research Council in Rome, who applied the method to a dynamical field theory of insect swarms.^{2} Their theoretical results agree extremely 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.

## Cloud computing

Cavagna and colleagues have long been interested in collective biological behaviors. They were initially inspired by the flocking starlings that 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, despite having no one leader. The collective synchrony emerges from each bird’s tendency to fly in the same direction as its neighbors.

Swarming insects, in contrast, don’t all fly in the same direction: The cartoon depiction of a horde of angry wasps determinedly chasing down their fleeing victim is a myth. Rather, a real swarm tends to hover in place like a cloud, with its constituent insects buzzing to and fro, as shown in the photo in figure 1a and the composite trajectory image in figure 1b.

Nevertheless, swarming insects do imitate their neighbors—just not strongly enough to create any swarm-wide order. In physics parlance, swarms and flocks are the disordered and ordered phases of the same system, much like a magnet above and below its Curie temperature.

Whatever the temperature, a magnet’s spins are governed by the competition between an energetic preference to align with their neighbors and an entropic preference to orient randomly. Below the Curie temperature, energy wins, and the spins align. Above the Curie temperature, entropy wins, and there’s no net magnetization.

But something special happens in a magnet that’s only barely above its Curie temperature. Clumps of aligned spins emerge that, although smaller than the whole magnet, can be quite large. In fact, they span all length scales—no matter how large the magnet, there’s a good chance that it contains a clump that’s a significant fraction of its size—the very problem that necessitated the invention of the RG to explain systems near phase transitions.

From their initial observations of insect swarms in the wild, Cavagna and colleagues found a marked resemblance to barely demagnetized magnets.^{3} The swarms lacked a single preferred direction, but plenty of clumps of insects momentarily aligned their motion. The larger the swarm, the larger the clumps—the hallmark of a scale-free, near-critical system.

A magnet’s Hamiltonian is easy to understand, even if its near-critical behavior is not. Could an insect swarm—a collection of living animals, each of which can sense and respond to all the complexities of its environment—be governed by a similarly simple set of rules? “The space and time correlation functions were horrible beasts,” says Cavagna, “but in physics, you can make a big simplification and describe everything with one exponent.” The dynamical critical exponent *z* quantifies how correlations in space scale with correlations in time. In 2017 the researchers tried applying the dynamical scaling hypothesis to their swarm data. It worked.^{4}

## Order of life

But the value of *z* for the swarms was extremely low: 1.37 ± 0.11, as shown by the orange bell curve in figure 2. For magnets, in contrast, *z* is exactly 2, shown by the black square. Intuitively, *z* can be thought of as a measure of how fast fluctuations spread across a system, with smaller values representing faster spread and a fundamental speed limit at *z* = 1. Insect swarms were not only faster than magnets at propagating fluctuations, they were also faster than any existing theory could explain.

Two key ingredients, the researchers found, distinguish a swarm from a magnet and cause fluctuations to propagate faster in the former. The first is activity: Whereas a spin in a magnet always interacts with the same set of neighbors, insects move under their own power, so their nearest neighbors are constantly changing. The second is inertia: Insects don’t react immediately to what their neighbors are doing, so small-scale heterogeneities in a swarm can persist for some time. In other words, if a swarm is viewed as a fluid, it’s one without a lot of viscous drag.

Activity and inertia had both been treated separately in RG calculations before. The equilibrium (nonactive) inertial model was one of the classic RG successes^{5} from the 1970s: It describes the behavior of superfluids, among other things, and has a *z* of 1.5, shown by the blue diamond in figure 2. The active noninertial model didn’t emerge until later, once interest in the physics of active matter—that is, living things—had begun to gain traction as a field of study.^{6} It gives *z* = 1.73, shown by the green triangle.

Encouraged by how activity and inertia each push *z* in the right direction, the researchers began to think about applying the RG to a model that incorporates both. When they embarked on the project in the summer of 2019, it was a two-person effort of Cavagna and PhD student Luca Di Carlo. But the complexity quickly spiraled out of control.

In simple terms, an RG calculation entails renormalizing an equation in two steps—coarse-graining and rescaling—and hoping to get a new equation of the same mathematical form. When that happens, the calculation is finished: The equation describes how the system behaves self-similarly across all length scales.

Often, however, renormalizing an equation gives a more complicated equation. “When that happens, the RG is telling you, ‘Hey dummy, you forgot to put a term in your equation,’” says Cavagna. Renormalizing the more complicated equation may complete the process, or it may yield an even more complicated equation. And so on.

The active inertial model has more ways than usual for its equations to become complicated. Roughly speaking, in the viscous regime, the social force—how individuals influence their neighbors—is proportional to the first time derivative of velocity. But in the inertial regime, it’s proportional to the second time derivative. The additional derivative introduces more ways to make nonlinear terms, which complicate the equations.

The RG calculation was soon yielding more terms than could feasibly be calculated by hand. “It was a nightmare,” says Cavagna. Only after Di Carlo and Mattia Scandolo, another student in the group, developed a specialized Mathematica code to help with the calculation did they start to get a handle on the renormalization. “I was very close to giving up,” says Cavagna, “but they made the final push to get it done. I could never have done it without them.”

The result: *z* = 1.35, shown by the red circle in figure 2, and in excellent agreement with the data. The researchers wrote up their work that summer and submitted the paper for publication. When the reviews came in, says Cavagna, “they said, ‘This looks great, but it could just be an accident. You need to do numerical simulations for us to believe this is real.’” That task fell to Giulia Pisegna, another student who’d worked on the project. Even though she’d already left the group for a postdoctoral position, she set up the large, intricate simulations, let them run, and found *z* = 1.35 ± 0.04, shown by the pink stripe.

Even with the convergence of theory, simulation, and experiment, *z* is still just one quantity. The active inertial model successfully predicts other properties of insect swarms, says Cavagna, “but we’d really be happier if we had more exponents.” More critical exponents exist than just *z*, and the RG is capable of calculating them. 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 in an applied magnetic field, and it’s not yet clear what the equivalent experiment would even be on an insect swarm. “Until we have data, I’m not sure we want to invest all the time to do the calculations,” says Cavagna. “As we say in our group, ‘No data, no party.’”

## Not dead yet

The applications to biology have the potential to breathe new life into the RG. A major theme in the story of RG calculations has been about finding the same critical exponents and behaviors across disparate physical systems. Nevertheless, the *z* = 1.35 regime is completely new. (So, for that matter, was the *z* = 1.73 regime of the active noninertial model.) Finding new critical phenomena in active systems may be a way that new physics lurks in living matter (see the article by Paul Davies, *Physics Today*, August 2020, page 34).

“But fewer and fewer people remember how to do these calculations,” says Cavagna. The sophisticated math of the RG builds on the methods of quantum field theory, which is increasingly seen as a merely optional part of the physics curriculum, especially for would-be biophysicists. “Biophysics is so rich in amazing and fascinating problems,” says Cavagna. “But we have so many students coming to us wanting to study the birds, and we have to tell them that they can’t because they don’t know field theory. And you really have to learn it when you’re young—trying to pick it up when you’re 35 or 40 is too late.”

## References

*Sci. Am.*

*Phys. Rev. Lett.*

*Nat. Phys.*

*Rev. Mod. Phys.*

*New J. Phys.*

*Physics Today*

*Physics Today*