Long-time Boltzmann-Grad limit for stable finite-range radial potentials

Theorem 1: for a stable finite-range radial potential, a grand-canonical initial law with pair exclusion and $C^1$ density with spatially summable Gaussian bounds on it and its gradient, and any $T$ for which the Boltzmann equation with that potential's scattering has a classical solution with uniform Gaussian velocity bound on $[0,T]$, all fixed-order rescaled factorial marginals converge in $L^1$ to $f(t)^{\otimes s}$ uniformly on $[0,T]$; empirical averages converge in probability. It assumes, rather than proves, existence of the regular kinetic solution, so it gives no global-in-time statement, and it does not treat long-range or non-radial potentials or give a rate.

Theorem 1: for a stable finite-range radial potential, a grand-canonical initial law with pair exclusion and $C^1$ density with spatially summable Gaussian bounds on it and its gradient, and any $T$ for which the Boltzmann equation with that potential's scattering has a classical solution with uniform Gaussian velocity bound on $[0,T]$, all fixed-order rescaled factorial marginals converge in $L^1$ to $f(t)^{\otimes s}$ uniformly on $[0,T]$; empirical averages converge in probability. It assumes, rather than proves, existence of the regular kinetic solution, so it gives no global-in-time statement, and it does not treat long-range or non-radial potentials or give a rate.