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, 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 non-reciprocal 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 Roderich 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 non-reciprocal 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 non-reciprocal 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 non-reciprocal 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 an fictitious bird in front of each real bird, aligned in exactly the opposite direction," says Alert.
Putting the Results in Context, Outlook
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 non-reciprocal interactions. On the one hand, this allows researchers to use the established theoretical framework of many-body physics. On the other, it enables non-reciprocal 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.
The findings of the Dresden physics team have been published in the journal Nature Physics.
Publication
Hamiltonian description of non-reciprocal interactions; Yu-Bo Shi, Roderich Moessner, Ricard Alert & Marín Bukov, Nature Physics (2026), https://doi.org/10.1038/s41567-026-03317-0
ctd.qmat
The Cluster of Excellence ctd.qmat — Complexity, Topology and Dynamics in Quantum Matter — at Julius-Maximilians-Universität Würzburg and Technische Universität Dresden explores and develops novel quantum materials with tailored properties. Around 300 researchers from over 30 countries work at the interface of physics, chemistry, and materials science to lay the foundations for tomorrow's technologies. In 2026, the cluster entered the second funding period of the German Excellence Strategy of the Federal and State Governments — with an expanded focus on the dynamics of quantum processes.
