---
title: "From Hard Spheres to Boltzmann Interactive Lab | Sandbox Physics"
description: "How can the Boltzmann equation be derived rigorously from Newtonian collisions of hard spheres?"
canonical_url: "https://sandboxphysics.com/labs/hard-spheres-to-boltzmann/"
language: "en"
page_type: "lab"
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  zh-CN: "https://sandboxphysics.com/zh/labs/hard-spheres-to-boltzmann/"
last_modified: "2026-10-27"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Select any particle to trace it. This is a two-dimensional teaching model with a finite number of particles, built to show empirical distributions, collision histories and recollisions. The Deng–Hani–Ma theorem gives a rigorous long-time derivation for three-dimensional dilute hard spheres in the Boltzmann-Grad limit.

Physics tutorial

## From individual hard spheres to a kinetic equation

**Background** At 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 matters** The 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 misconception** A smooth histogram from many particles already proves the Boltzmann equation.

**Better mental model** A 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 misconception** When the same pair meets again, it is just one more random collision with no special effect on the statistics.

**Better mental model** A 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

1. 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.

2. 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.

3. 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.

Physical conclusion

Deng’s result is not that many spheres look like a fluid. It rigorously controls the combinatorial growth of collision histories, and that links deterministic particle motion to the kinetic and fluid equations.

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.

Primary sources [01 · IMU Fields Medal citation](https://www.mathunion.org/fileadmin/documents/2026-07/Yu_Deng_Citations.pdf) [02 · Long time derivation of the Boltzmann equation from hard sphere dynamics](https://arxiv.org/abs/2408.07818) [03 · Hilbert’s sixth problem: derivation of fluid equations via Boltzmann’s kinetic theory](https://arxiv.org/abs/2503.01800)

[Read the visual field note](https://sandboxphysics.com/en/stories/yu-deng-fields-medal/)

Fields & matter · Starter path

## From a field to an electron microscope

Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface.

[Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

[View the full path →](https://sandboxphysics.com/topics/fields-and-matter/#starter-path)