Kinetic limits and fluctuations over the Boltzmann lifespan

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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.

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The paper states

Theorem A (Regular-lifespan kinetic limit)

In The Boltzmann–Grad limit for stable radial potentials on regular kinetic intervals, 23 September 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

The other possible matches are Theorem 1.2 (Finite-dimensional fluctuation limit) in Hard-sphere fluctuations on the regular Boltzmann lifespan () and Corollary B (Empirical observables) in the same paper ().

Let \Phi satisfy Assumption 2.1, and let \beta>0. Let f_0\ge0 be a C^1 probability density on \R^3\times\R^3 such that

\|f_0\|_{\mathrm{Bol},2\beta} +\|\nabla_xf_0\|_{\mathrm{Bol},2\beta}<\infty.

Fix 0<T<\infty for which the equation

(\partial_t+v\cdot\nabla_x)f=Q_\Phi(f,f), \qquad f|_{t=0}=f_0,

has a nonnegative classical probability-density solution satisfying

\sup_{0\le t\le T}\sup_{x,v} e^{2\beta|v|^2}f(t,x,v)<\infty.

Start the exact Newtonian process from the grand-canonical law (2.7). For its scaled factorial densities F_s^\varepsilon, defined in (2.8), every fixed integer s\ge1 satisfies

\lim_{\varepsilon\downarrow0}\; \sup_{0\le t\le T} \bigl\|F_s^\varepsilon(t)-f(t)^{\otimes s}\bigr\|_ {L^1((\R^3\times\R^3)^s)}=0.

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

23 September 2026, current edition.

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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.

Hard-sphere fluctuations on the regular Boltzmann lifespan

23 September 2026, current edition.

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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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