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Math-2859 · mathematics · Pde

The Boltzmann–Grad limit throughout a regular kinetic lifespan

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For the specified stable radial finite-range Newtonian potentials in three dimensions, including the allowed attractive wells, and C1 initial probability density with Gaussian bounds on it and its spatial gradient, assume the associated Boltzmann equation has a nonnegative classical solution uniformly Gaussian-bounded on [0,T]. Does the conditioned product grand-canonical Newtonian gas at Boltzmann–Grad activity $\epsilon ^{-2}$ have every fixed scaled s-particle factorial density converge to $f(t)^tensor-s$ in L1 uniformly on [0,T]?

Short English statement

For the specified stable radial finite-range Newtonian potentials in three dimensions, including the allowed attractive wells, and C1 initial probability density with Gaussian bounds on it and its spatial gradient, assume the associated Boltzmann equation has a nonnegative classical solution uniformly Gaussian-bounded on [0,T]. Does the conditioned product grand-canonical Newtonian gas at Boltzmann–Grad activity $\epsilon ^{-2}$ have every fixed scaled s-particle factorial density converge to $f(t)^tensor-s$ in L1 uniformly on [0,T]?

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Statement licence: CC-BY-4.0 · Source file

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Math-2859-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2859

Boltzmann--Grad limit for stable potentials: The conclusion applies to the specified conditioned grand-canonical ensemble and potential class only while its classical kinetic solution retains a uniform Gaussian bound. Every fixed factorial order converges in the global $L^1$ norm uniformly over that interval, including the permitted attractive interactions.

With the hypotheses and conventions above, the claimed result addresses the whole mathematical assertion on this page.

The Boltzmann–Grad limit throughout a regular kinetic lifespan

OpenAI's claim

Boltzmann--Grad limit for stable potentials: The manuscript claims convergence of all fixed-order rescaled factorial marginals for stable finite-range radial interaction potentials throughout regular kinetic intervals. Attractive wells and dynamically created clusters are permitted under the stated regularity and Gaussian data bounds.

Mathematical statement of the claimed result

Mathematical claim 1.
Fix a radial potential $\Phi(q)=\phi(|q|)$ vanishing for $|q|\ge1$, either $C^2$ on $\mathbb R^3$ or $C^2$ off zero and tending to $+\infty$ at zero. Require thermodynamic stability $\sum_{i<j}\Phi(q_i-q_j)\ge-BN$ for every finite distinct-point configuration. Let $f_0\ge0$ be a $C^1$ probability density with finite Gaussian Boltzmann norms $\|f_0\|_{\mathrm{Bol},2\beta}+\|\nabla_xf_0\|_{\mathrm{Bol},2\beta}$, where $\|g\|_{\mathrm{Bol},\gamma}=\sum_{a\in\mathbb Z^3}\sup_{|x-a|\le1,v}e^{\gamma|v|^2}|g|$. Assume a classical probability-density solution of $(\partial_t+v\cdot\nabla_x)f=Q_\Phi(f,f)$ on fixed $[0,T]$ has $\sup_{t,x,v}e^{2\beta|v|^2}f<\infty$, with $Q_\Phi$ defined by the actual two-body scattering of $\Phi$. Evolve Newtonian particles with interaction $\Phi((x_i-x_j)/\epsilon)$ from the exactly normalized grand-canonical law proportional to $\mu^N/N!\prod_if_0(z_i)\mathbf1_{|x_i-x_j|>\epsilon}$, $\mu=\epsilon^{-2}$. Its scaled ordered factorial densities are characterized by $\int aF_s^\epsilon=\mu^{-s}\mathbb E\sum_{i_1,\ldots,i_s\text{ distinct}}a(z_{i_1},\ldots,z_{i_s})$. The claim is \[\sup_{0\le t\le T}\|F_s^\epsilon(t)-f(t)^{\otimes s}\|_{L^1((\mathbb R^6)^s)}\longrightarrow0\quad\text{for every fixed }s\ge1.\] Attractive wells and noninjective scattering are permitted; $\Phi$ and $T$ are fixed independently of $\epsilon$.

Outline of the argument

Boltzmann--Grad limit for stable potentials: Fine time cells give an exact inclusion--exclusion expansion of entire interaction components, while coarser layers are centered against the known kinetic solution. Energy is retained across complete components, including clusters with many internal encounters. Polynomial cutoffs and separated detached trajectories control the signed terms and yield uniform $L^1$ convergence after summing the finite layers.

Formal verification

Boltzmann--Grad limit for stable potentials: The pinned catalogue lists no Lean formalization for this result.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

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