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Math-2845 · mathematics · Pde

The Gaussian hard-sphere fluctuation field on regular Boltzmann intervals

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For the source-conditioned grand-canonical hard-sphere gas in the three-dimensional Boltzmann–Grad regime, assuming a uniformly Gaussian-bounded regular kinetic solution on [0,T], do all finite collections of centered empirical-field smooth-test values converge jointly to the Gaussian field solving the linearized Boltzmann equation with white-in-time collision noise? Require Poisson initial covariance integral $f_0$ phi psi, independent initial field and noise, and the source collision covariance $integral_0^(min(t,s))$ $C_r(\phi ,\psi )dr$.

Short English statement

For the source-conditioned grand-canonical hard-sphere gas in the three-dimensional Boltzmann–Grad regime, assuming a uniformly Gaussian-bounded regular kinetic solution on [0,T], do all finite collections of centered empirical-field smooth-test values converge jointly to the Gaussian field solving the linearized Boltzmann equation with white-in-time collision noise? Require Poisson initial covariance integral $f_0$ phi psi, independent initial field and noise, and the source collision covariance $integral_0^(min(t,s))$ $C_r(\phi ,\psi )dr$.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2845-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2845

Hard-sphere fluctuation limit: Only finite-dimensional laws of smoothly tested fluctuations are requested, throughout an interval where the kinetic solution has a common Gaussian velocity bound. Exact microscopic centering is retained, as are the Poisson initial covariance and independent collision-noise covariance of the limiting Gaussian field.

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

The Gaussian hard-sphere fluctuation field on regular Boltzmann intervals

OpenAI's claim

Hard-sphere fluctuation limit: The paper claims finite-dimensional Gaussian fluctuations for a nonequilibrium hard-sphere gas throughout any finite interval of the regular Boltzmann lifespan. Centering subtracts the exact expectation in the particle model, rather than its kinetic approximation.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $\|g\|_{\mathrm{Bol},\beta}=\sum_{a\in\mathbb Z^3}\sup_{|x-a|\le1,v}e^{\beta|v|^2}|g(x,v)|$. Suppose the smooth probability density $f_0\ge0$ has $\|f_0\|_{\mathrm{Bol},2\beta}+\|\nabla_xf_0\|_{\mathrm{Bol},2\beta}<\infty$ and a nonnegative classical hard-sphere Boltzmann solution $f$ on $[0,T]$ satisfies $\sup_{t,x,v}e^{2\beta|v|^2}f<\infty$. Start diameter-$\epsilon$ Newtonian hard spheres from the grand-canonical law proportional to $\mu^N/N!\prod_if_0(z_i)\mathbf1_{|x_i-x_j|>\epsilon}\,dZ_N$, $\mu=\epsilon^{-2}$. Define \[\zeta_t^\epsilon(\phi)=\sqrt\mu\left[\mu^{-1}\sum_i\phi(z_i(t))-\mathbb E\left(\mu^{-1}\sum_i\phi(z_i(t))\right)\right].\] For every finite collection of times and compactly supported smooth real tests, their joint law is claimed to converge to a centered Gaussian field $\zeta$ with $d\zeta_t(\phi)=\zeta_t(A_t\phi)\,dt+dW_t(\phi)$. Put $B=[(v-v_*)\cdot\omega]_+$, $v^\prime=v-((v-v_*)\cdot\omega)\omega$, $v_*^\prime=v_*+((v-v_*)\cdot\omega)\omega$, and $\Delta\phi=\phi(x,v^\prime)+\phi(x,v_*^\prime)-\phi(x,v)-\phi(x,v_*)$. Then \[A_t\phi=v\cdot\nabla_x\phi+\int Bf_t(x,v_*)\Delta\phi\,dv_*\,d\omega,\qquad C_t(\phi,\psi)=\tfrac12\int Bf_tf_{t,*}\Delta\phi\Delta\psi\,dx\,dv\,dv_*\,d\omega.\] The initial covariance is $\int f_0\phi\psi$, and $\mathbb E[W_t(\phi)W_s(\psi)]=\int_0^{t\wedge s}C_r(\phi,\psi)\,dr$, with independent increments and initial field independent of $W$. The assertion is finite-dimensional convergence with exact microscopic centering, conditional on the regular kinetic lifespan.

Outline of the argument

Hard-sphere fluctuation limit: Layered collision histories are expanded by connected components while retaining finite observation records and exact factorial cancellation. Relative translations give a merger gain, and excluded-volume links and redundant contacts are controlled at the sharp cumulant scale. Only the appropriate signed trees survive in the limit, producing the linear fluctuating Boltzmann covariance and Gaussian finite-dimensional laws.

Formal verification

Hard-sphere fluctuation limit: 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.

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