Yu Deng · 2026 Fields Medal · Hilbert’s sixth problem

Hard Spheres to Boltzmann

How can the Boltzmann equation be derived rigorously from Newtonian collisions of hard spheres? In a dilute gas you can stop following each sphere and track the distribution of positions and velocities, which obeys the Boltzmann equation. The hard part is recollisions: spheres that meet again bring back correlations from their first collision. Yu Deng and collaborators rigorously controlled how collision histories grow, recollisions included, and derived the equation from Newton's laws.

Every sphere on the left moves by deterministic elastic collisions. The upper-right view drops particle identities and keeps only the speed distribution. The lower-right view weaves one particle’s past collisions into a graph, and a violet recollision closes it into a correlation loop.

Hard spheres

N

Collision rate0/s

Recollision share0.0%

Tracked particle#017 · 0 hits

Physics tutorial

From individual hard spheres to a kinetic equation

BackgroundAt the microscopic level, positions and velocities fix every elastic collision. At the kinetic level, we stop tracking which particle is which and keep only the density in position–velocity space.

Why it mattersThe hard part is more than a short-time approximation. The collision history keeps growing, recollisions occur and correlations build up, and all of this must stay under control for the statistical description to remain valid over long times.

Start with the essentials

Focus question: When can we forget each particle’s full path and still predict how the whole distribution evolves?
One-sentence intuition: The Boltzmann equation does not replace Newtonian mechanics; it is the effective description you get by compressing many deterministic trajectories in a dilute limit.

Core mathematical model

Elastic collision of equal hard spheres

\mathbf v_i'=\mathbf v_i-\bigl((\mathbf v_i-\mathbf v_j)\cdot\mathbf n\bigr)\mathbf n

The two spheres exchange their velocity components along the collision normal $\mathbf n$ , and the tangential components stay the same. The left view applies this rule at every collision.

Boltzmann kinetic equation

\left(\partial_t+\mathbf v\cdot\nabla_{\mathbf x}\right)f=Q(f,f)

$f(t,\mathbf x,\mathbf v)$ is the one-particle distribution, and the collision operator $Q$ gives the net effect of two-body collisions on it.

Three-dimensional Boltzmann–Grad limit

N\varepsilon^2\longrightarrow 1,\qquad N\longrightarrow\infty,\ \varepsilon\longrightarrow0

The number of particles $N$ grows while their diameter $\varepsilon$ shrinks, in step, so the collision frequency stays at the right scale.

Common difficulties

An animation is not a proof

Typical misconceptionA smooth histogram from many particles already proves the Boltzmann equation.

Better mental modelA finite simulation builds intuition. A theorem has to control the error in the limit, the bad collision histories and how long the target solution exists.

Recollisions are not ordinary noise

Typical misconceptionWhen the same pair meets again, it is just one more random collision with no special effect on the statistics.

Better mental modelA recollision closes a loop in the collision history and brings back together information the pair already exchanged. This kind of correlation is a central obstacle for long-time results.

Run the experiment

01
Compare the two descriptions
Keep the Kinetic regime preset and compare the individual motion on the left with the distribution at upper right. Pause to see one exact microstate next to its compressed statistical outline.
What to observe: The trajectories keep changing, while the distribution only fluctuates slightly around a stable shape.
02
Trace one particle’s collision history
Select a sphere, follow its amber trail and look at the collision molecule at lower right. Wait for a violet connector to appear.
What to observe: Ordinary collisions make the history branch. When the same pair collides again, a violet connector closes a recollision loop.
03
Compare with a crowded gas
Switch between Dilute gas and Crowded contrast, and compare the collision rate and the recollision share.
What to observe: The crowded system builds up complex histories faster and is further from the dilute case that the theorem needs.

Model limits

What the animation shows, and what the theorem proves

This finite Lab

A close-up of what makes the problem hard

The animation shows deterministic particle paths, a compressed one-particle distribution, how collision histories grow, and loops where the same pair of particles meets again. It is a finite, two-dimensional model, exaggerated on purpose so that it is easy to read.

Deng–Hani–Ma

A rigorous result that holds for long times

The cited paper proves that three-dimensional dilute hard-sphere dynamics converges to the Boltzmann equation for arbitrarily long times, as long as the corresponding Boltzmann solution exists and the paper’s assumptions hold. A companion paper carries the derivation on from kinetic theory to the fluid equations.