Why birds ignore Newton: New theory could sharpen models of flocks, crowds and cells

Birds in flocks, bacteria and cells: In many collective systems, individual elements respond to only part of their surroundings, seemingly defying Newton’s third law of motion—action equals reaction. These exceptions are known as nonreciprocal interactions. A Dresden physics team working with Roderich Moessner, a founding member of the Würzburg–Dresden Cluster of Excellence ctd.qmat, has now developed a theory that makes it possible to describe these interactions efficiently and simulate them far more precisely.

This could help researchers study motion in swarms and flocks, as well as biological processes, in much greater detail. The findings have been published in Nature Physics.

Newton and fields of vision

Birds have a wide field of vision. Yet when they fly in a flock, they orient themselves only toward the birds in front of them or alongside them. Because a bird never aligns itself with a bird behind it, the flock seemingly defies Newton’s third law—the principle of action and reaction, often summed up as “For every action, there is an equal and opposite reaction.”

When we run, for example, our feet push against the ground and the ground pushes back with an equal but opposite force. The same principle is at work when we drive a car, jump, row or let air escape from a balloon: When the air is expelled backward, the balloon flies forward.

Everyday life is full of movements that obey Newton’s third law, which is more than 300 years old and forms a cornerstone of classical mechanics. “Whatever we normally teach our students in theoretical mechanics, it ultimately rests on the action-reaction principle,” explains research group leader Marín Bukov.

Flocks of birds, schools of fish, swarms of bacteria, people in crowds and tissue cells, by contrast, do not obey Newton’s third law because the components of these systems respond to only part of their surroundings. This makes the interaction unidirectional, meaning that “action equals reaction” no longer applies. These exceptions are known as nonreciprocal interactions. Until now, they could not be fully described with the classical theories developed for reciprocal interactions, and therefore these systems could not be simulated efficiently.

Efficient simulation, however, is essential for studying processes in the human body or the motion of flocks and swarms. This research gap has now been closed by the findings of a Dresden physics team working with Moessner. Moessner is a principal investigator of the Würzburg–Dresden Cluster of Excellence ctd.qmat—Complexity, Topology and Dynamics in Quantum Matter—and director of the Max Planck Institute for the Physics of Complex Systems in Dresden.

Newton reloaded: Physicists in Dresden find an elegant solution

“The research team has developed and proven a theory that makes much of what we teach our students applicable to nonreciprocal systems as well. These systems, where Newton’s third law does not apply, can now finally be described exactly and simulated precisely—even using established methods. This is exactly the kind of tool that has been missing in recent years,” says Bukov.

To achieve this, the team of physicists expanded the original action-reaction framework. To describe nonreciprocal systems using the tools developed for reciprocal systems, all that is needed are additional artificial variables. Here is how it works: Theoretical physicists usually model nature in equations. Each variable describes a degree of freedom that actually exists—such as the position or speed of a bird, the position of a fish in a school or the position of a car in traffic.

“The trick behind the new theory is that it constructs a partner for each component of the system—a fictitious partner that doesn’t exist in nature. The original nonreciprocal interactions are replaced by reciprocal interactions with these auxiliary degrees of freedom,” explains Bukov’s colleague Ricard Alert, a biophysicist.

What does that mean for a flock of birds? “To simulate the birds’ movements precisely, we describe the dynamic system ‘flock of birds’ using established methods—as if it were a reciprocal system, even though it is not. The elegant solution is to artificially place a fictitious bird in front of each real bird, aligned in exactly the opposite direction,” says Alert.

Putting the results in context

Introducing auxiliary degrees of freedom is nothing new in physics. What is new, however, is that these auxiliary degrees of freedom now make it easier to study systems with nonreciprocal interactions. On the one hand, this allows researchers to use the established theoretical framework of many-body physics. On the other, it enables nonreciprocal systems to be simulated with much greater accuracy. Above all, the findings deepen physicists’ fundamental understanding of these processes—and such understanding is always the basis for future discoveries.

“In Würzburg and Dresden, we study quantum matter whose particles interact under certain conditions in ways that give rise to new phenomena such as magnetism or lossless current transport. The exciting question now is whether these exceptions to Newton’s law lead to entirely new forms of collective quantum behavior. We still know very little about this—and that is precisely what makes this so fascinating,” says Moessner.

Who’s behind this story?

Lisa Lock

BA art history, MA material culture. Former museum editor, paramedic, and transplant coordinator. Editing for Science X since 2021.

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Robert Egan

Bachelor’s in mathematical biology, Master’s in creative writing. Well-traveled with unique perspectives on science and language.

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