In 1900, at the grand amphitheater of the Sorbonne in Paris, David Hilbert stood before a crowd of mathematicians and delivered a speech that would echo through the next century. He outlined 10 — and later 23 — unsolved problems that he believed would shape the future of mathematics.

Now, 125 years later, a trio of mathematicians believes they’ve partly resolved one of Hilbert’s problems — the sixth.

In a preprint paper, Yu Deng, Zaher Hani, and Xiao Ma delivered a rigorous derivation of the equations of fluid mechanics, including the famous Navier-Stokes and Euler equations, starting from Newton’s laws and passing through Boltzmann’s kinetic theory. Their work, while deeply technical, connects three theories that describe how fluids move — from the chaotic dance of atoms to the graceful sweep of wind and waves.

If their result stands, it would mark a major stride in solving Hilbert’s sixth problem — his call to ground all of physics in a logical, axiomatic foundation, much like how geometry is built upon a set of axioms.

One Reality, Three Views

How does the world of fluids we see — swirling smoke, gusts of wind, the eddies in a stream — emerge from countless invisible particles bouncing around?

To approach that, physicists long relied on Boltzmann’s kinetic theory, developed in the late 19th century. Ludwig Boltzmann argued that if you knew the probability of where each particle was and how fast it moved, you could describe the behavior of gases statistically. That theory is described in the Boltzmann equation. To this day, engineers use this equation to calculate average properties of a gas or fluid, like pressure or temperature, without obsessing over each microscopic collision.

But this equation has always been problematic. Mathematically, no one had shown that this statistical equation itself could be rigorously derived from Newton’s deterministic laws. Nor had anyone followed the full thread from atoms to Boltzmann to the full equations of fluids, like the Navier-Stokes equations that govern air flow and water currents.

For over a century, no one had managed to prove that these equations — Newton’s laws, Boltzmann’s equation, and Navier-Stokes equations — truly derive from one another.

That’s the essence of Hilbert’s sixth problem. The German mathematician challenged scientists to place physics on a firm mathematical foundation by axiomatizing it, much like how geometry is built upon a set of axioms. In his speech, he singled out Boltzmann’s equation as a key link and challenged future generations to derive the laws of physics from the bottom up.

A New Mathematical Bridge

To axiomatize physics means to identify a set of basic, self-evident principles (axioms) from which all physical laws can be logically derived. This approach aims to ensure that the entire framework of physics is consistent, complete, and free from contradictions.

Hilbert specifically highlighted two areas:

1. Probability Theory: He emphasized the need for a rigorous mathematical treatment of probability, which is fundamental to statistical mechanics and quantum theory.
2. Mechanics and Kinetic Theory: Hilbert was interested in developing a mathematical framework that connects the microscopic behavior of particles (as described by kinetic theory) to the macroscopic laws of motion for continuous media (like fluids), such as those described by the Navier-Stokes equations.

Physicists and mathematicians had made progress on pieces of the puzzle. Some had shown how the macroscopic equations follow from the mesoscopic ones — including Hilbert himself. Others tackled how Boltzmann’s equation could arise from Newton’s laws, but only for fleeting moments or in overly tidy conditions.

Deng, Hani and Ma have tied the entire chain together — from particles to statistics to the continuous flow of fluids. And they have done it over long timescales, where the math grows thorny and the particle interactions pile up.

The New Proof

The proof proceeds in two grand stages.

First, the team extends their earlier work from infinite space to a periodic setting — in mathematical terms, they consider particles moving on a 2D or 3D torus. This allows them to sidestep edge effects while still capturing the essence of physical space. They show that when a vast number of hard-sphere particles collide elastically under Newton’s laws — and when their size shrinks in just the right proportion to their number — the system obeys the Boltzmann equation.

This requires what’s known as the Boltzmann-Grad limit, a delicate scaling where the particle diameter shrinks while the number of particles grows, keeping the collision rate fixed. “The necessity of this scaling . . . was discovered by Grad,” they note, referring to Harold Grad’s work in the 1960s.

“The first (kinetic) limit turned out to be more challenging,” the authors admit, because it demands following particle interactions over long times — something that had eluded prior work.

Once this bridge is built, the second step is to derive the classical fluid equations from the Boltzmann framework. This so-called hydrodynamic limit assumes that the collision rate becomes very high — the mean free path shrinks — causing the system to settle into fluid-like behavior.

With this, they prove that Newton’s atomistic world gives rise to:

• The incompressible Navier-Stokes-Fourier equations, which describe the flow of viscous fluids with heat conduction;
• The compressible Euler equations, which model inviscid flows like sound waves or shock fronts.

These are the equations engineers use to simulate everything from airplane wings to climate models.

The result is a continuous derivation — from Newton’s laws to Boltzmann’s equation to Navier-Stokes — that traces the logic of fluid motion across all scales.

What Does This Mean for Physics?

The finding doesn’t change the fluid equations themselves. Engineers will still use the same tools to design airplanes and simulate weather.

But it changes our confidence in those tools. It tells us that the equations work not just because they happen to match empirical findings, but because they must follow from deeper laws.

In physics, this kind of consistency is paramount. It’s a sign that our theories are built on firm ground — that we understand not just what works, but why it does.

The result could inspire similar work in other branches of physics. From plasma physics to condensed matter to quantum field theory, researchers often move between microscopic and macroscopic descriptions. A firm mathematical link between the two helps prevent surprises — and opens the door to new ones.

The findings appeared in arXiv.