Result 364, Partial differential equations

Kinetic limits and fluctuations over the Boltzmann lifespan

Derives the nonlinear Boltzmann equation from three-dimensional grand-canonical Newtonian gases throughout every regular kinetic interval with uniform Gaussian decay. Stable finite-range radial potentials may have attractive wells and a singular repulsive core; initial pair exclusion and spatially summable Gaussian density and gradient bounds are assumed. A companion gives finite-dimensional hard-sphere Gaussian fluctuations, centered at the exact microscopic expectation and governed by the linear fluctuating Boltzmann equation.

Proof

The bigger picture

Why it matters

The manuscripts report how three-dimensional Newtonian gases with a random particle count approach the Boltzmann equation, which tracks a gas statistically rather than particle by particle. They also describe hard-sphere fluctuations around the microscopic mean, connecting average behavior with uncertainty.

What changes?

In the dilute-gas scaling limit, the main manuscript treats stable, finite-range radial potentials: interactions depend only on separation, and can include attractive wells and a singular repulsive core. Potentials are twice continuously differentiable elsewhere. It assumes initial exclusion of close pairs and continuously differentiable initial densities, with spatially summable Gaussian velocity bounds on density and spatial gradient. Convergence holds uniformly on every finite interval where the classical Boltzmann solution retains a uniform Gaussian bound, without restrictions on the differential scattering cross-section.

What does that help mathematicians do?

The claimed convergence covers every fixed-order rescaled factorial marginal, which describes joint statistics of a fixed number of distinct particles. It is in L1, so the integrated absolute error vanishes, uniformly over the allowed interval. This connects Newtonian dynamics to kinetic predictions even when attractions create clusters. The time range follows the regular Boltzmann solution, but does not establish its global existence or control intervals without the required decay.

Are there practical applications?

Its immediate value is foundational: it links particle-level randomness to equations for macroscopic uncertainty. Under smooth initial densities with the stated Gaussian bounds, the hard-sphere companion reports Gaussian fluctuations for finite collections of observables over the same regular lifespan. They are centered at the exact microscopic expectation and governed by the linear fluctuating Boltzmann equation. This describes fluctuations, not a demonstrated computational method.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

The Boltzmann–Grad limit for stable radial potentials on regular kinetic intervals

September 23, 2026 69 pages

We derive the nonlinear Boltzmann equation from a grand-canonical Newtonian gas with initial pair exclusion. We treat stable, finite-range radial potentials that are C2 except for an allowed repulsive singularity at the origin, and C1 initial probability densities with spatially summable Gaussian bounds on the density and its spatial gradient. All fixed-order rescaled factorial marginals converge in L1, uniformly throughout every finite interval on which the classical kinetic solution has a uniform Gaussian bound. The result allows attractive wells and dynamically formed clusters, without restrictions on the differential scattering cross-section.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026,
  author = {{OpenAI}},
  title = {{The Boltzmann--Grad limit for stable radial potentials on regular kinetic intervals}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026/paper.pdf}{OAI:The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026}},
  year = {2026}
}

Hard-sphere fluctuations on the regular Boltzmann lifespan

September 23, 2026 54 pages

We prove a finite-dimensional central limit theorem away from equilibrium for a deterministic grand-canonical hard-sphere gas in three dimensions. The initial probability density is smooth, with spatially summable Gaussian velocity bounds on the density and its spatial gradient. The limit holds on every finite interval on which the classical Boltzmann solution has uniform Gaussian velocity decay. The empirical measure is centered by its exact microscopic expectation. Its fluctuations converge to the Gaussian solution of the linear fluctuating Boltzmann equation.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Hard-sphere-fluctuations-on-the-regular-Boltzmann-lifespan-September-23-2026,
  author = {{OpenAI}},
  title = {{Hard-sphere fluctuations on the regular Boltzmann lifespan}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Hard-sphere-fluctuations-on-the-regular-Boltzmann-lifespan-September-23-2026/paper.pdf}{OAI:Hard-sphere-fluctuations-on-the-regular-Boltzmann-lifespan-September-23-2026}},
  year = {2026}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.