Breakthrough in Unifying Math to Describe How Fluids Behave at Different Scales

For 125 years, mathematicians and physicists have chased after a single solution to a very difficult problem in fluid mechanics.

That problem arises because the math needed to accurately describe the flow of a fluid depends very much on the scale of the physical world being modeled. There’s one set of fluid math equations that works at the microscopic level, where individual atoms and molecules of fluids interact. There’s another set at the macroscopic level, which involves the combined interactions of millions and billions of fluid particles. And there’s a third set at the mesoscopic level, which falls in between the two other scales and demands its own set of equations to describe.

The following video by brain truffle is one of the better introductions we found to the challenges of using math to describe how fluids behave at each of these scales. If you want to skip over some of the foundational discussion, jump ahead to the 16:48 mark.

The challenge lies mathematically unifying the physical theories behind the fluid mechanics at the microscopic, mesoscopic, and macroscopic levels. It was originally proposed in 1900 by mathematician David Hilbert as the sixth of twenty-three problems he identified as worthwhile endeavors in which to pursue mathematically rigorous solutions.

The benefits of establishing such a rigorous solution would mean much greater confidence in the computational analysis supporting how fluids behave in the real world, no matter the scale of the application.

In the twelve full decades since, progress toward establishing that rigorous math to the different scales of fluid mechanics has come in several different waves. The latest wave however could represent the biggest math story of the year because it potentially achieves a substantial portion of Hilbert’s goal for this particular problem.

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….

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.

That’s the practical upshot of Deng, Hani and Ma’s work, which is now being put to peer review. If you want to know more about the technical aspects of what they did, they spoke about their work in the following video from a Mathematics Colloquium at the University of Chicago in April 2025:

Finally, if you’d like more background into the governing equations of fluid dynamics at the mesoscopic and macroscopic scales, we’ll recommend Mojtaba Maali Amiri’s 42-minute discussion of them:

If Deng, Hani and Ma’s work holds, successfully connecting the math of Newton, Boltzmann, Navier, and Stokes across these three scales of reality represents a massive achievement in mathematical theory, one with practical benefit.

Reference

Yu Deng, Zaher Hani, and Xiao Ma. Hilbert’s Sixth Problem: Derivation of Fluid Equations via Boltzmann’s Kinetic Theory. Preprint: arXiv:2053.01800v1. [PDF Document]. 3 March 2025.