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

Hard Spheres to Boltzmann

Every sphere on the left follows deterministic elastic collisions. The upper-right view forgets identities and keeps only the speed distribution. The lower-right view braids one particle’s collision past; a violet recollision closes a correlation loop.

Hard spheres

N

Collision rate0/s

Recollision share0.0%

Tracked particle#017 · 0 hits

Start with the essentials

Focus question: When can complete particle trajectories be forgotten without losing the correct evolution of the whole distribution?
One-sentence intuition: The Boltzmann equation does not replace Newtonian mechanics. It emerges as an effective compression of 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

Relative velocity along the collision normal $\mathbf n$ is exchanged while tangential components remain unchanged. The left view applies this rule collision by 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, while the collision operator $Q$ summarizes the net effect of binary collisions.

Three-dimensional Boltzmann–Grad limit

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

The particle count $N$ grows while diameter $\varepsilon$ shrinks, retaining the appropriate collision-frequency 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 supplies intuition. A theorem must control limiting error, bad collision histories, and the lifespan of the target solution.

Recollisions are not ordinary noise

Typical misconceptionWhen the same pair meets again, it is merely one more random collision with no special statistical effect.

Better mental modelA recollision closes a loop in the history and reunites previously exchanged information. That correlation structure is a central long-time obstacle.

Run the experiment

01
Read both languages at once
Keep the Kinetic regime preset and compare individual motion on the left with the upper-right distribution. Pause to see one precise microstate beside its compressed statistical outline.
What to observe: Trajectories change continually while the distribution only breathes around a stable shape.
02
Trace a collision past
Select a sphere, follow its amber trail, and inspect the lower-right collision molecule. Wait for a violet connector.
What to observe: Ordinary collisions branch the history; a repeated pair closes a violet recollision loop.
03
Break dilute intuition with a crowded contrast
Switch between Dilute gas and Crowded contrast. Compare collision rate and recollision share.
What to observe: The crowded system accumulates complex histories faster and is further from the dilute regime required by the theorem.

Model boundary

What the picture reveals — and what the theorem proves

This finite Lab

A microscope for the difficult objects

The animation exposes deterministic trajectories, a compressed one-particle distribution, collision-history growth, and repeated-pair loops. It is two-dimensional, finite, and intentionally exaggerated for legibility.

Deng–Hani–Ma

A rigorous long-time bridge

The cited work proves convergence from three-dimensional dilute hard-sphere dynamics to the Boltzmann equation for arbitrarily long times within the lifespan and assumptions of the corresponding Boltzmann solution. A companion paper continues from kinetic theory to fluid equations.