Long-time Boltzmann-Grad limit for stable finite-range radial potentials
Lanford (1975) derived the Boltzmann equation from Newtonian hard spheres in the Boltzmann-Grad limit, but only for a short time; King and Pulvirenti-Saffirio-Simonella did the same for potentials. Deng, Hani and Ma (arXiv 2024) extended the hard-sphere derivation to every finite interval on which the Boltzmann solution is regular with Gaussian decay, and discussed the extension to smooth interaction potentials, where encounters last positive time and several particles interact at once, as a further problem. For a stable, finite-range radial potential, does the Boltzmann-Grad limit hold on every such regular kinetic interval, not just for short times?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Kinetic theory; derivation of the Boltzmann equation
- Posed by
- Yu Deng, Zaher Hani and Xiao Ma, 'Long time derivation of the Boltzmann equation from hard sphere dynamics' (arXiv 2408.07818, Section 1.4.2)
- Year posed
- 2024
- Years open
- 2y
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 30 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1: for a stable finite-range radial potential, a grand-canonical initial law with pair exclusion and density with spatially summable Gaussian bounds on it and its gradient, and any for which the Boltzmann equation with that potential's scattering has a classical solution with uniform Gaussian velocity bound on , all fixed-order rescaled factorial marginals converge in to uniformly on ; empirical averages converge in probability. It assumes, rather than proves, existence of the regular kinetic solution, so it gives no global-in-time statement, and it does not treat long-range or non-radial potentials or give a rate.
What the AI did
The release README says the results were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. A companion of the same date proves long-time Gaussian fluctuations for hard spheres (separate entry); both build on the Deng-Hani-Ma layered expansion.
Verification
No independent mathematician has checked this yet. Checked here: Theorem 1 of the principal manuscript was read against the further problem as the paper reports it from Deng-Hani-Ma Section 1.4.2. It covers stable radial potentials vanishing beyond unit range, except a possible repulsive singularity at the origin, with attractive wells allowed, for initial densities with summable Gaussian bounds and initial pair exclusion, on every interval where a classical Boltzmann solution with uniform Gaussian decay is assumed to exist. The wording of the Deng-Hani-Ma problem itself was not read here. The proof was not refereed. No Lean formalization exists for this family.