Damian Almeida Baray

Ryerson Physical Laboratory.

Imagine filming two billiard balls colliding on a smooth pool table. If you were to play the video in reverse, nothing would appear out of place. The balls would approach, collide, and ricochet along physically permissible paths. In principle, the same should hold for the enormous numbers of particles in a gas. Reverse their velocities, and Newton’s laws suggest they should retrace their positions through space.

But at larger scales, gases do not behave reversibly. Plumes of smoke disperse throughout a room, but do not spontaneously coalesce. Gases mix together but do not separate themselves. A clear direction of time emerges, despite being governed by microscopic laws which work equally well forwards and backwards.

Explaining how complex behavior like this emerges from simple interactions is a recurring theme in much of University of Chicago mathematics professor Yu Deng’s work. In July, Deng became one of four mathematicians awarded the 2026 Fields Medal, one of the highest honors in mathematics. The International Mathematical Union, which awards the medal every four years, recognized Deng’s research in “partial differential equations, including the rigorous derivation of the Boltzmann equation from hard-sphere dynamics for rarefied gases, the derivation of wave kinetic equations from nonlinear dispersive systems, and probabilistic approaches to nonlinear Schrödinger dynamics.”

The first of those results —a “rigorous derivation of the Boltzmann equation”—proven alongside mathematicians Zaher Hani and Xiao Ma in 2024, is an extension of a result proven in 1975 by Oscar Lanford. Lanford’s theorem proved that the Boltzmann equation, which describes the larger-scale, time-irreversible behavior of gases, could be derived from time-reversible Newtonian laws—but only when restricted to very short, very specific windows of time. Deng, Hani, and Ma were able to generalize that result to the entire timeline.

The Maroon spoke to Deng about the mathematics behind the result, the process of working on 50-year-old problems, and how irreversible behavior emerges from reversible laws.

 Note: This interview has been edited and reordered for clarity and brevity.

Your Fields Medal citation references a few areas of your work: nonlinear wave equations, the derivation of the Boltzmann equation from Newtonian dynamics, and wave turbulence. From your perspective, what connects these three areas?

What we have here are these microscopic systems containing a huge number of objects that interact with each other. Individual interactions are usually simple, but the whole system becomes very complicated because there are so many objects.

So, you take a statistical point of view. Because there are so many objects, you don’t really care about the behavior of each individual object. You’re more interested in the global distribution—thinking about questions like “how many particles exist in this region, or in that region?” By looking at those statistical quantities, you obtain equations that describe the distribution of these objects. That is a common theme in wave turbulence, in the Boltzmann problem, in other questions like the random data problem: they are all interaction problems.

Much of your work connects different ways of describing a gas. Would you explain what those descriptions are, and how they relate?

At the microscopic level, you have tiny particles described by Newton’s laws, and those particles form the gas. The Boltzmann equation describes the density of particles. We regard the particles as indistinguishable, so we don’t care where each particle is. Instead, we ask how many particles occupy a given region with a given velocity. Our work concerns this step: obtaining a density function that describes the particle distribution and satisfies the Boltzmann equation.

Physicists had already shown that both equations held experimentally. Why was a mathematical derivation of the statistical Boltzmann picture from the Newtonian picture important?

Mathematics is our way of understanding this world. Something we can use is good, but we still need to understand why it works.

Of course, if we understand them, it might also help us understand other things, like the study of the Boltzmann equation, fluid equations, and other problems. But the first step is to have a good understanding of why these things work.

How would you explain the central ideas of your research to an undergraduate without much mathematics background?

An important part of understanding these systems in a statistical or probabilistic way is the so-called chaos assumption. There might be many objects, but these objects behave independently: the location or behavior of the first object does not affect, or is not related to, the behavior of the second or third object.

One of the main themes of our research is how this chaos assumption is generated and, most importantly, how it is propagated. Initially, the objects are independent, but due to their interactions, they are no longer exactly independent. We want to understand the extent to which independence persists or is recovered despite all the interactions.

[Suppose you already know the probabilities of finding a particle in a certain area and within a certain range of velocities: this information is called the statistical distribution. Boltzmann’s equation relies on the “molecular chaos assumption”: before two particles are about to collide, their velocities are statistically independent—knowing one particle’s velocity should give you essentially no additional information about the other beyond what the distribution tells you. A collision links their outgoing velocities: knowing how one ricocheted provides information about how the other did. Even if a system began in a “chaotic,” independent state, after many interactions, it’s not obvious that it should remain in one.]

Can you give a concrete example of the independence underpinning the chaos assumption, and how it might be lost?

Suppose you have a very large number of particles, but you are interested in only three of them: A, B and C. Initially, they’re independent: one’s movement does not affect another, but once particles A and B collide, their trajectories after the collision will not be independent. But C hasn’t collided, so A and C are independent, and so are B and C. But suppose B collides with C: then, with this collision, all three become related.

[A and C never directly collided, but by altering the trajectory of B, A has indirectly affected C.]

These are tiny particles: for two to collide, their trajectories have to intersect, which happens only with very small probability. Making all three nonindependent requires another collision, so it has even smaller probability. There is a balance between the loss of independence and the small probability of the required collisions. For a small collection of particles, most remain independent; if they become dependent, it is only with a very small probability. That’s what we expect.

One of the classical results by Lanford is that the required independence holds for short periods of time. Our proof extends short times to arbitrarily long times. [In mathematics, “arbitrarily long” means the result holds for any finite duration, however long—there is no fixed cutoff. It does not refer to a literal infinite time.] We have to keep track of the independence relations over these longer intervals, which leads to the difficulty in the theorem.

Lanford’s result held only for a short time. In a physical setting, what does “short time” mean? Are we talking about seconds, milliseconds, nanoseconds?

It’s better to understand “short-time” in terms of the number of collisions. Consider the average number experienced by an individual particle. By short-time, we mean that this number is tiny—say one in a hundred. Then only one out of every hundred particles has experienced a collision, while the other 99 have experienced none at all.

A long-time result means that, on average, each particle may have experienced perhaps a hundred collisions. The difference between a small, expected number of collisions and a large one is what translates into the distinction between short- and long-time.

How did you, Hani, and Ma break this short-time barrier?

The key is to propagate the chaos assumption in time. A short-time argument can treat a very small time interval; to reach the final time, you need to iterate that short-time interval analysis many times. In the beginning, you have independence. After an interval, you no longer have exact independence. You have approximate independence.

The chaos assumption is not propagated exactly. So, from here, you have to be very careful, because if you do not treat the correlations carefully, you’d mess up the later argument.

We therefore keep track of objects called cumulants, which measure the dependence among different particles—the deviation from the chaos assumption. To control them, we expand them into very complicated diagrams. At short times, a diagram has only one layer and can be analyzed relatively easily. At longer times it has multiple layers, and even two layers are already sufficiently complicated.

One way to view Lanford’s argument is as the first layer. We found a way to reduce the multilayer case, in some sense, to the two-layer case. Even with only two layers, though, this is a completely different level of complexity from Lanford’s one-layer argument. There is a fundamental difference between one layer and two layers.

[As in Deng’s three-particle example, after one time interval, some particles have collided, so there is some dependence between them. Lanford proved that this dependence after a single interval is weak enough not to affect the Boltzmann equation’s independence requirements. But the “approximate independence” after a single interval is not the same as the true independence we started with, so you cannot just repeat the same argument going from interval one to interval two. This is the gap that had to be closed.]

How would you state the result of your paper with Hani and Ma for a general audience?

If you begin with a particle system satisfying the chaos assumption, then as it evolves through time, the chaos assumption is propagated to leading order, and the one-particle density evolves according to the Boltzmann equation. The limit remains valid throughout the interval on which the Boltzmann equation has a “nice” solution.

[Deng, Hani, and Ma proved that a gas whose particles begin in an independent state continues to behave, statistically, as though it were independent—any correlations stay small enough not to disturb the overall picture. The Boltzmann equation describes the statistical picture of an independent gas for as long as its own solution remains mathematically well-behaved.]

What are the result’s most important limitations?

There are a few. Obviously, you need to start with initial chaos. But I think the most important condition is that you have a “nice” solution to the Boltzmann equation. That solution is something the system converges to, but you do still need to have a solution. What happens if the Boltzmann solution is singular [in other words, a poorly-behaved edge case] is an important question that we have not answered.

Another important assumption is the Boltzmann-Grad scaling, which describes a low-density gas—closer to the gas in the outer atmosphere. Gas near the Earth’s surface is denser than the scenario we’ve been looking at, and strictly speaking it is not contained in the theorem. At higher densities, you have more collisions, while each interaction remains just as strong, and we do not have a good way to control all those effects.

Even considering these restrictions, why hadn’t anyone been able to unify these two descriptions before? What were the conceptual gaps?

One of the main difficulties is that you need to go from time-reversible Newtonian dynamics to the time-irreversible Boltzmann equation.

In a short-time derivation, like Lanford’s, you don’t really see this distinction because the entire evolution is treated in a single interval. But when dividing into multiple time-step intervals, you have to distinguish between moving from forward from one interval to the next and moving backward to the previous one. The forward evolution works one way; the backward evolution does not work in the same way. That reflects the irreversible nature of the Boltzmann equation.

We had heard talks discussing this fundamental difference, but previously people didn’t know how to perform a derivation that respected this fact. In fact, it really emerged from our work on wave turbulence.

When we first worked on the wave turbulence problem, we began with a short-time theorem, which was already very complicated, and then we found a way to extend it to longer times. We realized that our scheme for solving this problem—dividing the time interval, propagating the information between intervals, and expressing that information through multilayer diagrams—was very general. It applied to the wave picture, but it could also apply to the particle picture. We had to produce new algorithms to account for the differences between particles and waves, which took several months, perhaps half a year, but the general scheme worked perfectly when applied to the Boltzmann problem.

Many have framed this paper as a solution to Hilbert’s sixth problem. Do you agree with this characterization?

[In 1900, German mathematician David Hilbert posed 23 then-unsolved problems that would become a roadmap for 20th-century mathematics. The sixth called for an axiomatic treatment of the theory of probabilities and mechanics.]

Hilbert’s sixth problem has a very broad interpretation: axiomatizing all of physics. Certainly, that’s too broad. There is a narrower description: deriving macroscopic limits from microscopic systems. What we did was solve that specific problem under the Boltzmann-Grad scaling, which corresponds to the dilute gas case.

There are other natural scalings, including the dense gas case. If we could solve both the dilute and dense gas scenarios, we could probably say that we had addressed essentially the main part of the microscopic-derivation program within Hilbert’s sixth problem. For now, we can treat the low-density gas case. The dense gas case I have been discussing with Zaher [Hani]. We’re trying to start with simpler models, but we haven’t had significant progress at this time.

How do you assess the significance of the result yourself?

It answers one of the main questions about how you break down continuous descriptions of fluids all the way to the fundamental particles. Subject to some assumptions, this is the first time we can really understand fluid equations from basic microscopic particles.

I also believe the method we developed is probably more important than the result itself. We developed a way of analyzing diagrams of arbitrary size, and there are many situations in mathematical physics where you have to handle huge diagrams. A theorem is a theorem, but the method we developed should provide a general philosophy for dealing with them.

You’ve mentioned that the method emerged from work studying waves. What brought you to that work, and how did you connect it to the Boltzmann derivation?

It was during a 2018 conference here in Chicago that Zaher came to me. He was working on wave turbulence and asked me how I felt about that work and how they should proceed. At the time, I did not take it too seriously. I thought it might be a very difficult problem.

But, at the same time, I was making progress on another project involving Gibbs measures. In that project, we had expansions with arbitrarily large diagrams. I realized that the wave turbulence problem also had expansions with arbitrarily large diagrams. The two things were probably related, and in the end, we realized that they were almost essentially the same thing. Around 2019 or 2020, we finished a paper on wave turbulence covering vanishing time scales.

Then, Zaher asked what would happen if we tried to reach a larger time scale. I was not confident because this was a critical problem, and I knew critical problems were very hard. But he insisted that we should work on it because it was a physically important problem.

[In the study of partial differential equations, critical is a technical term describing problems whose difficulty is the same at every scale—zooming in or out does not make anything easier.]

We spent almost all of 2020 developing the tools we used ourselves. In the end, we had the number theory results, the expansions, diagrams, and algorithms we needed to extend the result from a vanishing time scale to a small but constant time scale. This was the wave turbulence analogue of Lanford’s result.

In 2021, we started on the longer-time problem. At some point, we realized it might be solved by dividing the long interval into short intervals. This did not happen in one shot. I’ve been saying in interviews that we suddenly realized something worked, but strictly speaking, it wasn’t a single step. We gradually realized there was the method of dividing time, the method of propagating the answers, and the whole scheme gradually developed. But it was a very smooth process, once we had the idea, the argument grew naturally, and we were able to close the long-time derivation of wave turbulence in 2023.

We then began the Boltzmann problem and invited Xiao [Ma] to join us. We started in the spring of 2023. The first few months involved setting things up. From the end of 2023 through spring 2024, we spent roughly a year working on the main object—the algorithm. By the end of spring, we essentially knew how to solve it, and the paper was ready in August.

The first talk with Zaher was in the usual room [UChicago math faculty] use for seminars. That was where I became involved in these discussions. It really is the place where everything started.

What’s next for you?

There are many critical problems in mathematical physics and quantum field theory. We are trying to apply similar methods to more problems, including the famous Yang-Mills problem. We want to start with simpler scenarios and see what the limits of the method are.

[The Yang-Mills existence and mass gap problem was one of the seven open questions named a Millennium Prize Problem by the Clay Mathematics Institute in 2000. They are among the most famous open problems in mathematics; each carries a $1 million prize for its solution. At the time of the interview, only one, the Poincaré conjecture, had been solved. ChatGPT creator OpenAI released what it said was a solution to a second Millennium Prize problem—Navier-Stokes existence and smoothness—after the interview but before publication, and has since claimed that its artificial intelligence models have made “substantial progress” on a third problem.]

What do you consider the most beautiful takeaway from your work?

First, you can characterize and derive macroscopic, global phenomena from basic, microscopic interactions.

Second, you can see the emergence of time-irreversibility by keeping track of the interactions between different objects. I think this is interesting on a philosophical level: How do you understand the emergence of irreversibility?