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The Boltzmann–Grad limit for stable radial potentials on regular kinetic intervals
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Abstract
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.
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Comments and discussion
- kieransquared1, comment on OpenAI: Sharing AI progress in mathematics (r/mathematics, accessed 2026-10-21). This public discussion comment connects the potential extension and hard-sphere fluctuation claims to open problems in Deng–Hani–Ma, without presenting a proof assessment of either manuscript. [family-level]
Related prior literature mentioned by others
- Yu Deng, Zaher Hani and Xiao Ma, Long time derivation of the Boltzmann equation from hard sphere dynamics (arXiv:2408.07818v3, 2025-07-18). The Reddit discussion above identifies this regular-lifespan hard-sphere derivation as the prior work whose proposed extensions concern both manuscripts. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Corentin Le Bihan, Long time validity of the linearized Boltzmann uncut-off and the linearized Landau equations from the Newton Law (arXiv:2408.03597, v1 2024-08-07, v2 2026-01-08). It gives a long-time derivation of linearized kinetic equations at equilibrium for particles with truncated power-law potentials, which bears on the paper's claim to go beyond short-time results for interacting potentials (this paper's setting is nonequilibrium and nonlinear).
Digests and expositions
- Latent Seminar, Kinetic limits and fluctuations over the Boltzmann lifespan (2026-10-06). This family-specific exposition discusses the potential assumptions, layered correlation estimates, finite-dimensional fluctuation limit and exact microscopic centering, explicitly limiting its scope to explanation rather than independent proof verification. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 364
Independent sites; their content is not reviewed here.
Cite
@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}
}Family 364: 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.
Other papers in this family:
- Hard-sphere fluctuations on the regular Boltzmann lifespan · September 23, 2026