Epic Effort to Ground Physics in Math Opens Up the Secrets of Time
Wei-An Jin/Quanta Magazine
At the turn of the 20th century, the renowned mathematician David Hilbert had a grand ambition to bring a more rigorous, mathematical way of thinking into the world of physics. At the time, physicists were still plagued by debates about basic definitions — what is heat? how are molecules structured? — and Hilbert hoped that the formal logic of mathematics could provide guidance.
On the morning of August 8, 1900, he delivered a list of 23 key math problems to the International Congress of Mathematicians. Number six: Produce airtight proofs of the laws of physics.
The scope of Hilbert’s sixth problem was enormous. He asked “to treat in the same manner [as geometry], by means of axioms, those physical sciences in which mathematics plays an important part.”
His challenge to axiomatize physics was “really a program,” said Dave Levermore, a mathematician at the University of Maryland. “The way the sixth problem is actually stated, it’s never going to be solved.”
But Hilbert provided a starting point. To study different properties of a gas — say, the speed of its molecules, or its average temperature — physicists use different equations. In particular, they use one set of equations to describe how individual molecules in a gas move, and another to describe the behavior of the gas as a whole. Was it possible, Hilbert wondered, to show that one set of equations implied the other — that these equations were, as physicists had assumed but hadn’t rigorously proved, simply different ways of modeling the same reality?
For 125 years, even axiomatizing this small corner of physics seemed impossible. Mathematicians made partial progress, coming up with proofs that only worked when they considered the behavior of gases on extremely short timescales or in other contrived situations. But these fell short of the kind of result that Hilbert had imagined.
In 1900, David Hilbert came up with a list of 23 problems to guide the next century of mathematical research. His sixth problem challenged mathematicians to axiomatize physics.
University of Gottingen
Now, three mathematicians have finally provided such a result. Their work not only represents a major advance in Hilbert’s program, but also taps into questions about the irreversible nature of time.
“It’s a beautiful work,” said Gregory Falkovich, a physicist at the Weizmann Institute of Science. “A tour de force.”
Under the Mesoscope
Consider a gas whose particles are very spread out. There are many ways a physicist might model it.
At a microscopic level, the gas is composed of individual molecules that act like billiard balls, moving through space according to Isaac Newton’s 350-year-old laws of motion. This model of the gas’s behavior is called the hard-sphere particle system.
Now zoom out a bit. At this new “mesoscopic” scale, your field of vision encompasses too many molecules to individually track. Instead, you’ll model the gas using an equation that the physicists James Clerk Maxwell and Ludwig Boltzmann developed in the late 19th century. Called the Boltzmann equation, it describes the likely behavior of the gas’s molecules, telling you how many particles you can expect to find at different locations moving at different speeds. This model of the gas lets physicists study how air moves at small scales — for instance, how it might flow around a space shuttle.
Zoom out again, and you can no longer tell that the gas is made up of individual particles. It acts like one continuous substance. To model this macroscopic behavior — how dense the gas is and how fast it’s moving at any point in space — you’ll need yet another set of equations, called the Navier-Stokes equations.
Physicists view these three different models of the gas’s behavior as compatible; they’re simply different lenses for understanding the same thing. But mathematicians hoping to contribute to Hilbert’s sixth problem wanted to prove that rigorously. They needed to show that Newton’s model of individual particles gives rise to Boltzmann’s statistical description, and that Boltzmann’s equation in turn gives rise to the Navier-Stokes equations.
Mathematicians have had some success with the second step, proving that it’s possible to derive a macroscopic model of a gas from a mesoscopic one in various settings. But they couldn’t resolve the first step, leaving the chain of logic incomplete.
Now that’s changed. In a series of papers, the mathematicians Yu Deng, Zaher Hani and Xiao Ma proved the harder microscopic-to-mesoscopic step for a gas in one of these settings, completing the chain for the first time. The result and the techniques that made it possible are “paradigm-shifting,” said Yan Guo of Brown University.
Yu Deng usually studies the behavior of systems of waves. But by applying his expertise to the realm of particles, he has now resolved a major open problem in mathematical physics.
Courtesy of Yu Deng
Declaration of Independence
Boltzmann could already show that Newton’s laws of motion give rise to his mesoscopic equation, so long as one crucial assumption holds true: that the particles in the gas move more or less independently of each other. That is, it must be very rare for a particular pair of molecules to collide with each other multiple times.
But Boltzmann could not definitively demonstrate that this assumption was true. “What he could not do, of course, is prove theorems about this,” said Sergio Simonella of Sapienza University in Rome. “There was no structure, there were no tools at the time.”
After all, there are infinitely many ways a collection of particles might collide and recollide. “You just get this huge explosion of possible directions that they can go,” Levermore said — making it a “nightmare” to actually prove that scenarios involving many recollisions are as rare as Boltzmann needed them to be.
In 1975, a mathematician named Oscar Lanford managed to prove this, but only for extremely short time periods. (The exact amount of time depends on the initial state of the gas, but it’s less than the blink of an eye, according to Simonella.) Then the proof broke down; before most of the particles got the chance to collide even once, Lanford could no longer guarantee that recollisions would remain a rare occurrence.
In the decades since, many mathematicians tried to extend his result, to no avail.
Then, in November of 2023, Deng, now at the University of Chicago, and Hani, of the University of Michigan, posted a preprint that teased the desired proof. A forthcoming paper, they wrote, would build off their latest result to investigate “the long-time extension of Lanford’s theorem.”
Other mathematicians didn’t know what to make of the announcement. “I didn’t think it was possible,” said Pierre Germain of Imperial College London. Deng and Hani didn’t even usually work with particle systems; until that point, they’d mainly been studying systems made up of waves (like rays of light).
So mathematicians eagerly awaited the promised proof.
When Particles Collide
Deng and Hani’s 2023 result involved an analysis of the transition from the microscopic scale to the mesoscopic scale in the context of waves. About a year before the mathematicians posted their paper online, Deng was at a conference, where he met with a graduate student at Princeton University named Xiao Ma. They ended up discussing Deng and Hani’s work, and how they might adapt the methods to particles. Doing so would allow them to extend Lanford’s result — to show that particle recollisions are rare even on longer timescales.
It was an idea that Deng and Hani had already been considering. Impressed by Ma’s insights on the topic, Deng invited him to help them turn their intuition into a proof.
The trio hoped to focus on a much-studied scenario where mathematicians had already proved the second, meso-to-macro step in Hilbert’s sixth problem. In this scenario, a dilute gas of spherical particles is trapped in a box. If a particle hits one of the box’s walls, it reappears on the opposite wall.
But to prove the harder micro-to-meso step for this setting — thereby resolving Hilbert’s sixth problem — Deng, Hani and Ma had to port their wave-based techniques over to particles. So they started in a setting where that task would be a little bit easier. They worked with a gas whose particles are distributed randomly in an infinite amount of space; unlike the particles in the boxed gas, which keep bouncing off each other forever, these particles eventually disperse and stop colliding. “In the whole-space case, there is a shortcut,” Deng said.
The three mathematicians first needed to tabulate the different patterns of collisions that might occur in their gas, and how likely each of those patterns was. They could easily rule out scenarios with particularly high rates of recollisions. This left them with a finite, though still massive, number of patterns to analyze — each involving a certain subset of particles colliding, in a certain order. Once they knew exactly what each pattern entailed, they could use that information to estimate its likelihood of occurring.
But that often felt like an impossible task, because many of the patterns involved huge numbers of particles and intricate, indirect interactions between them. “The structure of these sets [of colliding particles] gets exceedingly complicated,” Deng said. In principle, the mathematicians would need to keep track of every one of these particles simultaneously to compute the probability estimates they needed.
That’s where Deng and Hani’s previous work on waves gave them an important insight. In that result, they’d figured out ways to break up complicated patterns of interacting waves into simpler ones. They’d carefully crafted their technique so that, by working with only a few waves at a time, they could still get a good estimate for the likelihood of the more complicated complete wave pattern.
They hoped the same idea would work in the particle setting.
But after a collision, particles behave very differently from waves. For instance, particles, unlike waves, bounce off each other, greatly affecting the resulting pattern of collisions and its probability of happening. Deng, Hani and Ma needed to rework the details of their strategy from the beginning.
Zaher Hani studies solutions to equations that arise in oceanography, plasma physics and quantum mechanics.
Courtesy of Zaher Hani
First, they tackled the simplest cases, in which each particle collides just a few times over a very short time span, with no recollisions. They then gradually moved on to harder and harder cases — longer amounts of time, with more collisions and recollisions.
It was as much an art as a science. “The intuition was developed gradually, starting with some unsuccessful attempts,” Deng said. They had to get a sense for how to slice up large, complicated patterns of particle collisions in a way that would simplify their calculations while keeping their estimates highly accurate.
“This is a process that takes months,” Hani said. “We would be stuck constantly.” Nearly every day, they jumped on a Zoom meeting to talk things through. “Much to the dismay of my wife, some of them happened very late at night, or very early in the morning,” Hani said. “I would put my daughter to sleep, and then we would have two or three hours of Zoom meetings.”
Finally, by the spring of 2024, the trio was sure they had covered everything. Their proof, which they posted online that summer, confirmed that recollisions had to be very, very uncommon. They’d shown, as they’d hoped to, that in their infinite-space setting, Boltzmann’s description of the gas could be derived from Newton’s. The microscopic and mesoscopic scales fell under a single rigorous mathematical framework.
“I think it’s outstanding work,” said Alexandru Ionescu, a mathematician at Princeton who was also Deng and Ma’s doctoral adviser. “These are some of the most significant advances in many, many years.”
They were now ready to return to the gas-in-a-box setting, where they could finally solve Hilbert’s sixth problem.
The Completed Chain
It didn’t take long for them to extend their result from the infinite-space setting to the boxed one. “Eighty percent of the proof is still the same in the whole-space case,” Deng said.
In March, they posted a new paper that combined their proof with the earlier results connecting the Boltzmann equation to the Navier-Stokes equations. The logical chain was complete: They’d shown that, for a realistic model of a gas, a microscopic description of individual particles does indeed ultimately give rise to a macroscopic description of the gas’s large-scale behavior.
The work didn’t just mark the resolution of a major case of Hilbert’s sixth problem. It also provided a rigorous mathematical resolution of an old paradox.
At the microscopic scale, where particles act like billiard balls, time is reversible. Newton’s equations predict both where a particle comes from and where it’s going. The future is not fundamentally different from the past.
But at the mesoscopic and macroscopic levels, there is no going back in time. “We know very well that, going forward in time, one ages but does not rejuvenate; heat does not spontaneously pass from a cold body to a warm body; a drop of ink in a glass of water spreads, darkening the liquid, but does not spontaneously return to the small, round shape it originally had,” Simonella wrote. Neither the Boltzmann equation nor the Navier-Stokes equations are time-reversible; if you try to run time backward, the results will be nonsensical.
To Boltzmann’s contemporaries, this was perplexing. How could a time-irreversible equation be derived from a time-reversible system?
But Boltzmann argued that there was no paradox: Even if each particle can be modeled in a time-reversible way, almost every collision pattern ends up with a gas dispersing. The chance of, say, a gas suddenly contracting is essentially zero.
Lanford had confirmed this intuition mathematically for his very short time frame. Now Deng, Hani and Ma’s result confirms it for more realistic situations.
Going forward, mathematicians — who are still poring over the details of the new proof — want to test whether similar techniques might be useful in other, even more realistic contexts. These might include gases made up of particles of different shapes, or particles that interact in more complicated ways.
Meanwhile, Falkovich said, these sorts of rigorous proofs can help physicists understand why a gas behaves a certain way at various scales, and why different models might be more or less effective in different scenarios. “What mathematicians do to physicists,” he said, “is they wake us up.”
Editor’s Note: Deng and Hani’s work on the system of waves was funded in part by the Simons Foundation, which also funds this editorially independent magazine.
