Three scales, one physics, and the limit where proof lives but fluids don’t
In 1900, Hilbert stood before the International Congress of Mathematicians in Paris and presented a list of 23 problems he believed would shape the century. The sixth was blunt: axiomatize physics. Stop borrowing equations from intuition and experiment. Derive them. Show that the macroscopic laws — fluid dynamics, thermodynamics — follow rigorously from the microscopic ones. Particles obey Newton. Fluids obey Euler and Navier-Stokes. Prove that one entails the other.
The problem is not merely mathematical. It is about the relationship between levels of description. We describe the world at three scales, and each has its own language:
Microscopic: individual particles, positions and velocities, Newton’s laws. Perhaps 1023 of them in a glass of water. Mesoscopic: the Boltzmann equation, a statistical description — not tracking each particle but the probability of finding a particle with a given velocity at a given point. Macroscopic: the fluid equations. Density, velocity, pressure as smooth fields. The water in the glass, described as a continuous thing.
The passage between these levels is not obvious. It is not even straightforward. For 125 years, the derivation remained incomplete.
Below you see what Hilbert was starting from: particles in a box. Each one obeys Newton. Each collision conserves energy and momentum exactly. This is the ground truth — everything else must emerge from here.
The Boltzmann equation lives between particles and fluids. It describes a gas not by tracking every particle but by tracking the distribution of velocities — how likely you are to find a particle moving at speed v at position x. Its derivation from particle dynamics requires a brutal assumption: molecular chaos. Before any two particles collide, their velocities must be uncorrelated. They must be strangers.
In dilute gas, this is plausible. Particles travel far between collisions, losing memory of prior encounters. In dense gas, it is false. Particles collide again and again with the same neighbours. Correlations build up. The assumption cracks.
Zoom out further. Forget individual velocities entirely. Average over regions. What emerges are the quantities of fluid mechanics: density, bulk velocity, pressure. The Euler and Navier-Stokes equations describe how these fields evolve. This is the level at which we design bridges and predict weather.
Watch how the coarse-grained field changes as you increase density. In the dilute limit, the field is nearly uniform — uninteresting. Push toward dense: density waves appear, flow patterns form, vortex-like structures emerge. The fluid becomes a fluid.
In 2025, Yu Deng, Zaher Hani, and Xiao Ma closed the loop. Their proof, building on decades of work by Lanford, Gallagher, Saint-Raymond, and others, provides the first rigorous derivation connecting all three levels: from Newton’s particles, through the Boltzmann equation, to the equations of incompressible fluids. Hilbert’s sixth problem, at least in one precise formulation, is solved.
But look at where the slider must sit for the proof to work.
The derivation requires the Boltzmann-Grad limit: the number of particles goes to infinity while the particle radius shrinks so that the total volume occupied by particles goes to zero. In this limit, the gas is infinitely dilute. Collisions happen, but the gas is essentially empty space with occasional visitors. The proof lives in this regime.
Now look at the canvases. Slide the density up. The velocity distribution still looks roughly Maxwell-Boltzmann — that shape is robust. But the derivation of that shape from first principles, the rigorous logical chain from Newton to Boltzmann, requires the dilute limit. And in the dilute limit, the macroscopic field is boring. Uniform. No eddies. No waves. No turbulence. Nothing that makes fluid mechanics worth studying.
This is the deep tension the derivation reveals: the tractable limit and the interesting limit are opposites. The mathematics works precisely where the physics is trivial. The physics becomes rich precisely where the mathematics breaks.
This is not a failure of the proof. It is a feature of how levels of description relate. The fluid does not sit inside the particles waiting to be extracted. The macroscopic is not a summary of the microscopic. It is a different kind of description with its own autonomy, its own laws, its own regime of validity.
The Navier-Stokes equations were written down in the 1840s. They work. Engineers use them. They describe turbulence, weather, blood flow. Their validity does not depend on being derived from particle physics. Their meaning does not require permission from below.
What Deng, Hani, and Ma proved is beautiful and precise: in the limit where gas is almost nothing, the chain of reasoning holds from bottom to top. What they also showed, by the very nature of the limit required, is that the chain holds only where it barely matters. The fluid — real, dense, turbulent — has always been on its own.