This was a 120-year mathematical relay race whose finish line was crossed at the end of 2024 by a group of young Chinese mathematicians.
If you want to brag about it at a bar, you could say: "Know why weather forecasts are inaccurate? Because we never knew how to rigorously derive macroscopic fluid equations from the microscopic world. Now a few young Chinese mathematicians have completely paved that road mathematically."
This is the groundbreaking breakthrough on Hilbert's Sixth Problem recently achieved by Deng Yu (Shenzhen University), Ma Xiao (a PhD student at the University of Michigan), and their collaborators.
To understand what they actually did, we first need to return to the microscopic world and watch a game of billiards.
Microscopic Billiards vs. Macroscopic Water Flow: An Unbridgeable Chasm
Imagine a box filled with air. In a physicist's eyes, there is no "wind" or "pressure" in here—only countless billiard-ball-like molecules colliding frantically. This is the microscopic world, with extremely simple rules: Newton's laws. Ball A hits ball B, momentum and energy are conserved, they bounce apart, and hit the next one. Everything is deterministic, even rewindable.
Now zoom out. What you feel is no longer impacts from individual molecules, but "temperature," "pressure," and "wind speed." This is the macroscopic world, described by the famous fluid equations (such as the Navier-Stokes equations and the Boltzmann equation). These equations ignore individual molecules and treat gas as a continuous fluid.
Here's the question: are the microscopic "billiard balls" and the macroscopic "water flow" really the same thing?
Intuitively, of course. But mathematicians don't accept intuition—only logic. Hilbert's Sixth Problem demands: can you start from the most fundamental axioms of Newtonian collisions and, through pure mathematical derivation, step by step arrive at the macroscopic fluid equations? No physicist-style "close enough" approximations and hand-waving allowed along the way.
It sounds simple, but it's actually a hellish mathematical nightmare.
Why Was It Stuck for 120 Years?
There are two major difficulties, each enough to make a mathematician pull their hair out.
Difficulty one: the irreversibility paradox.
Newtonian mechanics is "reversible." If you play a video of two billiard balls colliding backward, it looks perfectly reasonable. But macroscopic fluid equations are "irreversible." Drop a drop of ink into water and it spreads; you will never see a glass of murky water spontaneously gather back into a pure drop of ink. How can "irreversible" macroscopic equations be rigorously derived from "reversible" underlying rules? This even sparked a century-long debate in the history of physics.
Difficulty two: the curse of dimensionality and chaos.
Even a teaspoon of gas contains \(10^{23}\) molecules. To track them all, you'd need a system of \(10^{23}\)-dimensional equations. Worse, the molecular collisions aren't one-off events—they form an intricate web of correlations. Ball A hits ball B, changing B's trajectory so it hits ball C, and C then hits A... Mathematically, when you try to take the limit (pushing the number of molecules to infinity), these complex correlation terms run wild like unbridled horses, causing singularities that make the equations collapse outright.
In the 1970s, the great mathematician Oscar Lanford took a key step forward on this problem. But he only completed the derivation for "extremely short times" (roughly the duration of two or three molecular collisions). Any longer, and the mathematics completely broke down. For the following 50 years, almost no one could break through that wall.
The Chinese Mathematicians' "Brilliant Move"
How did Deng Yu, Ma Xiao, and their co-authors blow up that wall? They didn't use brute force, but an exquisitely refined mathematical "lightness skill."
Previous attempts tried to track all the historical trajectories of molecular collisions and drowned in complexity. Deng Yu's team took a different approach: don't dwell on the past—control the future.
They introduced mathematical tools involving "nonlinear fluctuations" and handled the "memory effects" of molecular collisions with great ingenuity. As an analogy: to predict traffic flow on a congested highway, you don't need to know where every car has been. You only need to precisely set each car's current "temperament" and "field of view." As long as these microscopic statistical characteristics are well controlled, when the number of cars goes to infinity, the macroscopic traffic flow equations naturally and seamlessly emerge.
They found a subtle "decoupling" mechanism among the microscopic particles, forcibly "canceling out" the divergent terms that would cause the equations to collapse.
What was the result? They proved: as long as microscopic particles obey Newtonian collision rules, when time is long enough and the number of particles tends to infinity, the system necessarily evolves according to the Boltzmann equation; and further, the Navier-Stokes equations describing fluid motion can be derived.
Microscopic billiard balls, under pure mathematical logic, undeniably become macroscopic fluids.
Why Does This Matter So Much?
First, this is a "decisive strike" against a core sub-goal of Hilbert's Sixth Problem. It's not just a technical victory but a victory of mathematical ideas—it thoroughly resolves the "reversible vs. irreversible" paradox, proving that macroscopic irreversibility can absolutely, and indeed only, arise naturally from microscopic reversibility through taking limits.
Second, these mathematicians are so young. Deng Yu was born after 1985, and Ma Xiao is still a PhD student. This shows that young mathematicians trained in China are already arm-wrestling at the international cutting edge of the hardest fundamental science—and winning.
Mathematicians are people with a terrifying obsession for rigor. Physicists use experiments to tell us "this is how the world works," while mathematicians use pen and paper to prove us "this is the only way the world can work—no other possibility exists."
This time, Chinese mathematicians have welded several of physics' most commonly used macroscopic cornerstones firmly onto the foundation of microscopic Newtonian mechanics. This is not an end, but the laying of the deepest, most stable pile for a grander future edifice: the axiomatization of physics.