Newton to Boltzmann, for longer?
Two AI preprints claim kinetic limits and fluctuations over the Boltzmann lifespan
Covers result 364 in OpenAI’s catalog: all 2 papers. See its connections to other results and the work it cites.
3:44. English captions available. The video loads from YouTube when you press play.
Chapters
The claim, in plain words
Boltzmann’s equation describes a dilute gas statistically. Can it be derived from Newton’s laws for a huge number of particles? In the Boltzmann–Grad limit, particles shrink while their number grows so that each still collides at a steady rate: our computer finds their combined collision cross-section stays pi however small the particles get, while the space they fill goes to zero. Our small hard-sphere simulation gives collision rates within about 10% of Boltzmann’s prediction.
Lanford proved the derivation for hard spheres in 1975, but only for a short time. Deng, Hani and Ma, in 2024, extended it for hard spheres to the whole time the Boltzmann solution stays smooth. They named two next steps, among others, as within reach: short-range forces in place of hard spheres, and the fluctuations around the limit. For short times both were already known, from work including Pulvirenti, Saffirio and Simonella, and Bodineau, Gallagher, Saint-Raymond and Simonella.
Result 364 is two OpenAI papers, dated September 23 and made public October 6, 2026. The 69-page headline claims the Boltzmann–Grad limit for stable, finite-range radial forces, which may have attractive wells and a singular repulsive core, on any time interval where the Boltzmann equation has a smooth solution with Gaussian tails. The 54-page companion claims that for hard spheres, on the same kind of interval, fluctuations around the limit become Gaussian, following a linear fluctuating Boltzmann equation.
What it doesn’t claim
Not claimed: a convergence rate, hard spheres in the headline, or that the Boltzmann solution stays smooth, which both papers assume. The fluctuation result covers finitely many times and test functions, not whole paths or large deviations, and it imports Deng, Hani and Ma’s estimates. We found no competing claim.
Where it stands
- Posted as a preprint
- OpenAI, 69 + 54 pages, dated September 23, 2026
- Peer reviewed
- Not yet
- Formalized in Lean
- Neither theorem; an uncatalogued file checks preliminary definitions and algebra for the companion
Last checked October 19, 2026. Both papers are unrefereed and unchanged since release. Neither theorem has a Lean proof; an uncatalogued file of about 600 lines checks only preliminary definitions and algebra for the fluctuation paper. We found no specialist comment and no reported error. OSVerify has not reviewed it.
How the proof goes
- Many layers of time. The headline cuts time into many layers. Stability of the forces gives one energy budget for the whole history, so each particle in a history costs a factor of about CT/√L. Our computer shows that with enough layers this ratio drops below one and the sum converges, while a single layer only works for short times: Lanford’s restriction, in caricature.
- Clusters, wells and fluctuations. Recollisions and clusters of interacting particles are handled by keeping clusters whole and by a causal-exposure argument with small trees of events; counting time-ordered labellings, our computer confirms the hook-length formula on all 196 small binary trees. Attractive wells make scattering richer: in our model force, different impact parameters can give the same deflection. For the companion, a random number of particles adds exactly one term to the variance at time zero: our samples give 0.74 against a prediction of 0.75, and 0.51 with a fixed number.
The numbers in the video
- Ours: in the Boltzmann–Grad scaling the total collision cross-section stays π however small the particles get.
- Ours: a small hard-sphere simulation gives 1.05 and 1.22 collisions per particle per unit time against Boltzmann's 1.13.
- Ours: in our model force with a well and a repulsive core, different impact parameters can give the same deflection.
- Ours: a random particle number gives variance 0.74 (prediction 0.75) against 0.51 (prediction 0.5) with a fixed number.
Read the source
- The Boltzmann–Grad limit for stable radial potentials on regular kinetic intervals (PDF), 69 pages.
- Hard-sphere fluctuations on the regular Boltzmann lifespan (PDF), 54 pages.
- The collection’s catalog, where this is result 364.
The 2 papers in this result
OpenAI groups related papers, such as alternative proofs, supporting results and consequences, into one result; we label the first paper it lists the main paper and the others companions.
Transcript
Read the full script
0:00The question
Boltzmann’s equation describes a dilute gas statistically. Can it be derived from Newton’s laws for a huge number of particles? In the Boltzmann–Grad limit, particles shrink while their number grows so that each still collides at a steady rate: our computer finds their combined collision cross-section stays pi however small the particles get, while the space they fill goes to zero. Our small hard-sphere simulation gives collision rates within about 10% of Boltzmann’s prediction. Lanford proved the derivation for hard spheres in 1975, but only for a short time. Deng, Hani and Ma, in 2024, extended it for hard spheres to the whole time the Boltzmann solution stays smooth. They named two next steps, among others, as within reach: short-range forces in place of hard spheres, and the fluctuations around the limit. For short times both were already known, from work including Pulvirenti, Saffirio and Simonella, and Bodineau, Gallagher, Saint-Raymond and Simonella.
1:03The claim
Result 364 is two OpenAI papers, dated September 23 and made public October 6, 2026. The 69-page headline claims the Boltzmann–Grad limit for stable, finite-range radial forces, which may have attractive wells and a singular repulsive core, on any time interval where the Boltzmann equation has a smooth solution with Gaussian tails. The 54-page companion claims that for hard spheres, on the same kind of interval, fluctuations around the limit become Gaussian, following a linear fluctuating Boltzmann equation. Not claimed: a convergence rate, hard spheres in the headline, or that the Boltzmann solution stays smooth, which both papers assume. The fluctuation result covers finitely many times and test functions, not whole paths or large deviations, and it imports Deng, Hani and Ma’s estimates. We found no competing claim.
2:02How the proof goes
The headline cuts time into many layers. Stability of the forces gives one energy budget for the whole history, so each particle in a history costs a factor of about CT/√L. Our computer shows that with enough layers this ratio drops below one and the sum converges, while a single layer only works for short times: Lanford’s restriction, in caricature. Recollisions and clusters of interacting particles are handled by keeping clusters whole and by a causal-exposure argument with small trees of events; counting time-ordered labellings, our computer confirms the hook-length formula on all 196 small binary trees. Attractive wells make scattering richer: in our model force, different impact parameters can give the same deflection. For the companion, a random number of particles adds exactly one term to the variance at time zero: our samples give 0.74 against a prediction of 0.75, and 0.51 with a fixed number.
3:05Where it stands
Both papers are unrefereed and unchanged since release. Neither theorem has a Lean proof; an uncatalogued file of about 600 lines checks only preliminary definitions and algebra for the fluctuation paper. We found no specialist comment and no reported error. If they hold, Boltzmann’s equation follows from Newton’s laws for stable short-range forces, even with attractive wells and a singular repulsive core, for as long as it stays smooth, and for hard spheres so do its Gaussian fluctuations, at any finitely many times.
Spot a mistake?
This episode simplifies on purpose, but it shouldn’t be wrong. If something is, open an issue on GitHub or leave a comment on YouTube. Corrections are noted here with a date.