When the greatest mathematician alive unveils a vision for the next century of research, the math world takes note. That’s exactly what happened in 1900 at the International Congress of Mathematicians at Sorbonne University in Paris. Legendary mathematician David Hilbert presented 10 unsolved problems as ambitious guideposts for the 20th century. He later expanded his list to include 23 problems, and their influence on mathematical thought over the past 125 years cannot be overstated.
Hilbert’s sixth problem was one of the loftiest. He called for “axiomatizing” physics, or determining the bare minimum of mathematical assumptions behind all its theories. Broadly construed, it’s not clear that mathematical physicists could ever know if they had resolved this challenge. Hilbert mentioned some specific subgoals, however, and researchers have since refined his vision into concrete steps toward its solution.
In March mathematicians Yu Deng of the University of Chicago and Zaher Hani and Xiao Ma of the University of Michigan posted a new paper to the preprint server arXiv.org that claims to have cracked one of these goals. If their work withstands scrutiny, it will mark a major stride toward grounding physics in math and may open the door to analogous breakthroughs in other areas of physics.
In the paper, the researchers suggest they have figured out how to unify three physical theories that explain the motion of fluids. These theories govern a range of engineering applications from aircraft design to weather prediction — but until now, they rested on assumptions that hadn’t been rigorously proven. This breakthrough won’t change the theories themselves, but it mathematically justifies them and strengthens our confidence that the equations work in the way we think they do.
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Each theory differs in how much it zooms in on a flowing liquid or gas. At the microscopic level, fluids are composed of particles — little billiard balls bopping around and occasionally colliding — and Newton’s laws of motion work well to describe their trajectories.
But when you zoom out to consider the collective behavior of vast numbers of particles, the so-called mesoscopic level, it’s no longer convenient to model each one individually. In 1872 Austrian theoretical physicist Ludwig Boltzmann addressed this when he developed what became known as the Boltzmann equation. Instead of tracking the behavior of every particle, the equation considers the likely behavior of a typical particle. This statistical perspective smooths over the low-level details in favor of higher-level trends. The equation allows physicists to calculate how quantities such as momentum and thermal conductivity in the fluid evolve without painstakingly considering every microscopic collision.
Zoom out further, and you find yourself in the macroscopic world. Here we view fluids not as a collection of discrete particles but as a single continuous substance. At this level of analysis, a different suite of equations — the Euler and Navier-Stokes equations — accurately describe how fluids move and how their physical properties interrelate without recourse to particles at all.
The three levels of analysis each describe the same underlying reality — how fluids flow. In principle, each theory should build on the theory below it in the hierarchy: the Euler and Navier-Stokes equations at the macroscopic level should follow logically from the Boltzmann equation at the mesoscopic level, which in turn should follow logically from Newton’s laws of motion at the microscopic level. This is the kind of “axiomatization” that Hilbert called for in his sixth problem, and he explicitly referenced Boltzmann’s work on gases in his write-up of the problem. We expect complete theories of physics to follow mathematical rules that explain the phenomenon from the microscopic to the macroscopic levels. If scientists fail to bridge that gap, then it might suggest a misunderstanding in our existing theories.
Unifying the three perspectives on fluid dynamics has posed a stubborn challenge for the field, but Deng, Hani and Ma may have just done it. Their achievement builds on decades of incremental progress. Prior advancements all came with some sort of asterisk, though; for example, the derivations involved only worked on short timescales, in a vacuum or under other simplifying conditions.
The new proof broadly consists of three steps: derive the macroscopic theory from the mesoscopic one; derive the mesoscopic theory from the microscopic one; and then stitch them together in a single derivation of the macroscopic laws all the way from the microscopic ones.
The first step was previously understood, and even Hilbert himself contributed to it. Deriving the mesoscopic from the microscopic, on the other hand, has been much more mathematically challenging. Remember, the mesoscopic setting is about the collective behavior of vast numbers of particles. So Deng, Hani and Ma looked at what happens to Newton’s equations as the number of individual particles colliding and ricocheting grows to infinity and their size shrinks to zero. They proved that when you stretch Newton’s equations to these extremes, the statistical behavior of the system — or the likely behavior of a “typical” particle in the fluid — converges to the solution of the Boltzmann equation. This step forms a bridge by deriving the mesoscopic math from the extremal behavior of the microscopic math.
The major hurdle in this step concerned the length of time that the equations were modeling. It was already known how to derive the Boltzmann equation from Newton’s laws on very short timescales, but that doesn’t suffice for Hilbert’s program, because real-world fluids can flow for any stretch of time. With longer timescales comes more complexity: more collisions take place, and the whole history of a particle’s interactions might bear on its current behavior. The authors overcame this by doing careful accounting of just how much a particle’s history affects its present and leveraging new mathematical techniques to argue that the cumulative effects of prior collisions remain small.
Gluing together their long-timescale breakthrough with previous work on deriving the Euler and Navier-Stokes equations from the Boltzmann equation unifies three theories of fluid dynamics. The finding justifies taking different perspectives on fluids based on what’s most useful in context because mathematically they converge on one ultimate theory describing one reality. Assuming that the proof is correct, it breaks new ground in Hilbert’s program. We can only hope that with just such fresh approaches, the dam will burst on Hilbert’s challenges and more physics will flow downstream.
This article was first published at Scientific American. © ScientificAmerican.com. All rights reserved. Follow on TikTok and Instagram, X and Facebook.
