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Hard-sphere fluctuations on the regular Boltzmann lifespan
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Abstract
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.
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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 equation for hard spheres: a proof without billiard theory (arXiv:2212.04392, 2022-12-08). It proves long-time convergence of the equilibrium fluctuation covariance to the linearized Boltzmann equation without uniform recollision bounds; this is relevant to the paper's account of prior long-time equilibrium covariance results, and to its own citation of Burago–Ferleger–Kononenko.
- Zhengyan Wu, Fluctuating Kinetic Theory: A Poissonian Stochastic Boltzmann Equation (arXiv:2609.06482, 2026-09-06). It proposes a Poisson-driven nonlinear fluctuating Boltzmann equation consistent with cutoff hard-sphere path large deviations, a possible comparison with the linear Gaussian fluctuating Boltzmann equation that the paper takes as its limit.
- Thierry Bodineau, Isabelle Gallagher, Laure Saint-Raymond and Sergio Simonella, Derivation of the Boltzmann equation from hard-sphere dynamics (after Y. Deng, Z. Hani, and X. Ma) (arXiv:2602.04407, first posted 2026-02-04; revised 2026-10-14). This Bourbaki exposition explains the regular-lifespan kinetic derivation underlying the paper's use of layered collision histories.
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: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}
}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:
- The Boltzmann–Grad limit for stable radial potentials on regular kinetic intervals · September 23, 2026