Theory of Fluids Enters the 21st Century

With this machinery, Wilson solved the mystery of why many magnets magnetize at the same rate. The zooming-out step passes through a point at which all magnets look the same, whether their atoms are arranged in cubes or tetrahedra. All that matters is that the atoms have a symmetry that lets them point in any direction. (Magnets whose atoms are pinned down to spin in a plane, for instance, magnetize at a different rate.)

Wilson’s work also clarified why a few properties, like the temperature at which the magnetization takes place, vary wildly from magnet to magnet, even when they have the same spin symmetry. These properties are related to the size of the few surviving terms.

The symmetries tell you the overall shape of the terms that stay large, but not exactly how large they are. The sizes of these terms act like threads lightly tethering the macroscopic world to the microscopic one.

Over the following decades, Wilson’s machinery seeped into many areas of physics. Physicists used Wilson’s calculations to justify the previously murky mechanics of quantum field theory, which treats each particle as a wave while washing out the less significant effects of the smallest vibrations. Wilson’s contributions also shaped modern theories of certain phases of matter, like solids.

But when it came to fluids, scientists remained stuck. Their defining symmetries weren’t yet clear.

How To Define a Fluid

The first break in the case came in the 2000s, when a group of cosmologists was using Wilsonian thinking to develop an effective field theory for the universe as a whole. As they did so, they stumbled upon a key insight: The universe’s expansion breaks a crucial symmetry in space-time.

In general, space and time have no reference point against which you can measure speed. If you’re in a windowless spaceship, you can’t tell whether you’re moving quickly, slowly, or not at all. Space-time has a symmetry with respect to speed.

But in the expanding universe, there is a special reference point against which you can discern a motion: It’s the one in which the expansion of space itself moves galaxies uniformly away from you. If you were to leave your galaxy in a spaceship and travel against this cosmic recession in a particular direction, you would see the galaxies in front of you recede more slowly than the galaxies behind you.

Fluids, the group noted, break the same symmetry. If you’re immersed in a resting liquid, you can tell you’re at rest. And if you start to swim, you’ll feel the drag of the fluid as it moves past you. The resting fluid, like the expanding universe, lacks the underlying speed symmetry of space-time. Meanwhile, it has other standard space-time symmetries; rotations and translations don’t change the fluid.

“At the level of the symmetries, they are the same,” said Alberto Nicolis, a physicist now at Columbia University, who worked on the effective field theory of the cosmos.

Alberto Nicolis, a physicist at Columbia University, took inspiration from cosmology to help devise a new theory of fluids. Anna DeBeer

The resemblance got the group thinking, but they needed one more ingredient. They found it by considering what changes they could make to a fluid without changing its energy — another set of symmetries. They realized you could always swap two parcels of a fluid for free. You could also shuffle three parcels, or four, or any number. In contrast, you can’t exchange any regions of a solid without paying an energy toll, through rupture or serious internal stress, so solids lack these swapping symmetries.

The group had identified an unlimited number of fluid symmetries. They considered what terms these symmetries would require in the theory and then, channeling the spirit of Wilson, zoomed out and watched the microscopic details wash away. They landed right on the Euler equations, perfect for perfect fluids, derived from fundamental symmetry principles.

The resulting theory — initially buried in a cosmology paper in 2005 and highlighted in a hydrodynamics paper in 2012 — was the first to apply the full effective field theory treatment to fluids. “It’s the foundational text of all of this,” Landry said.

The next step would be to include the imperfect fluids, too.

A Black Hole Lead

The new effective field theory accelerated an effort within a community of scientists studying black holes, which have their own strange connection to fluids.

A theoretical breakthrough from the late 1990s had established that, under special conditions, you could view a spherical black hole as a flat quantum soup. Theorists had connected the viscosity of the soup, which enabled it to dissipate energy, to the black hole’s ability to gobble up energy and hide it. “Things can fall into the black hole,” Nicolis said. “That’s a form of dissipation.”

Physicists already had an effective field theory of black holes: Einstein’s theory of gravity. If a black hole had a secret identity as an imperfect, energy-dissipating, viscous fluid, then an effective field theory for imperfect, energy-dissipating, viscous fluids should exist, too. Multiple teams raced to find it.

At the Massachusetts Institute of Technology, Hong Liu led a team that used black holes as a jumping-off point to understand fluids at a deeper level. Courtesy of Hong Liu

Hong Liu, a physicist at MIT, led one of the groups. The group noticed that the surface of a black hole had a swapping symmetry akin to the one Nicolis and company had used to define a fluid; you could exchange patches of a black hole with each other without disturbing the black hole’s structure.

But that symmetry wasn’t enough to describe an imperfect fluid. To achieve that, Liu’s group — which included the researchers Michael Crossley and Paolo Glorioso — resorted to an old trick from quantum mechanics. They duplicated the substance in their theory, effectively adding a second fluid with a clock that ticked backward while the first fluid’s clock ticked forward. Comparing the two fluids at any given moment let the researchers keep track of random variations. (The real physical fluid was essentially an average of the two mathematical fluids.)

The team had one last problem. They knew that a zoomed-out fluid must obey the laws of thermodynamics: It has to have a temperature that varies from place to place, and a tendency for any hot patches to blend with cold patches as time ticks along. At the same time, they knew that at the zoomed-in level, the buzzing of atoms makes no distinction between past and future. (You wouldn’t be able to tell if a video of atomic motion was playing forward or backward.) Was there a way to tie the thermodynamics of the fluid to the two ways that time worked?

After some trial and error, the group found a symmetry that did the trick. The transformation switched the two fluids, reversed their clocks, and fiddled with the temperature in a particular way. If it happened a second time, the fluid would return to its original state. The different notions of time, and their ultimate compatibility, flowed from this symmetry, which was unlike any that physicists had seen before. “It’s a funny symmetry,” said Kristan Jensen, a physicist now at the University of Victoria in Canada, who also contributed to the black-holes-to-fluids effort. “And it’s essential. Before that, there are a ton of terms. And then [everything] collapses down and you just get known phenomena, nothing more and nothing less.”

They were done. From these symmetries they could write down a theory and zoom out to derive the Navier-Stokes equations. In 2015, Liu and his group posted their new effective field theory. It was a magnum opus, spanning 110 pages. “It blew my mind,” Landry said.

Next, physicists would put the theory to work.

Fluid Follow-up

It wasn’t immediately obvious how to harness the highly technical theory for specific applications. But Liu and Glorioso rewrote the equations in a more accessible way in 2018, and the calculations started to trickle out.

The new work allowed theorists to work more efficiently and push their calculations further. In particular, they were able to ferret out tiny terms in Navier-Stokes equations that had previously been ignored, terms that capture some of the effects of random molecular motion. “The stuff that we’re doing is more than just sort of an alternate version [of fluid dynamics],” Landry said. “It is actually strictly more correct.”

Luca Delacrétaz at the University of Chicago used the new theory to calculate more precisely how heat spreads through a liquid — and found that it moves more slowly than expected during its first few moments, due to random jitters. The effect is far too small to measure in water, but it becomes more substantial in quantum systems made from a limited number of particles. “Doing precision physics in chaotic quantum many-body systems sounds crazy, but now it’s possible,” Delacrétaz said.

Other researchers used the theory to derive Navier-Stokes analogues for exotic states of matter, like fracton matter. Roughly speaking, fracton matter is composed of groups of particles that collectively act like one particle. Individual “fractons” stay trapped in place; only when fractons band together can they move about.

In 2020, Andrew Lucas, a physicist at the University of Colorado, Boulder, and his collaborators spotted new symmetries in the fracton phase and used the effective field theory framework to derive a fracton version of the Navier-Stokes equations. The work opened the door to understanding whole new classes of liquidlike phases of matter. “It’s just a few lines of algebra to kind of plug it into the formalism that [Liu] developed and ask what’s possible,” Lucas said.

In 2024, Lucas and his group built on Liu’s theory by making explicit the basis for some of its construction. They used a newly discovered “strong” symmetry, related to the fact that the total number of particles in a fluid is always the same. That symmetry is partially broken, becoming a “weak” symmetry, due to the fact that the exact number of particles in a specific region of a fluid can vary from moment to moment. The consequence of this broken symmetry, they found, is a long, slow diffusion. This helps explain why it takes many seconds for an ink drop to spread through a fluid made of particles furiously colliding from picosecond to picosecond.

Lucas and his collaborators integrated this symmetry into Liu’s effective field theory in 2024. “Our main contribution was to justify what Liu was doing,” Lucas said. “To me it’s a closure of a story.”

While the effective field theory framework has catalyzed new calculations, the real prize may be more conceptual: a new definition of what makes a fluid a fluid. A fluid is any material with a particular set of symmetries, no matter what it’s made of.

“There’s something magical about fluid dynamics,” Nicolis said. “There’s oil, water, mercury. These are all very different, microscopically, very different things that, in practice, behave very similarly when you look at how they flow.”