<p style='text-indent:20px;'>In this paper we present a rigorous derivation of the Boltzmann equation in a compact domain with {isotropic} boundary conditions. We consider a system of <inline-formula><tex-math id="M1">\begin{document}$ N $\end{document}</tex-math></inline-formula> hard spheres of diameter <inline-formula><tex-math id="M2">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> in a box <inline-formula><tex-math id="M3">\begin{document}$ \Lambda : = [0, 1]\times(\mathbb{R}/\mathbb{Z})^2 $\end{document}</tex-math></inline-formula>. When a particle meets the boundary of the domain, it is instantaneously reinjected into the box with a random direction, {but} conserving kinetic energy. We prove that the first marginal of the process converges in the scaling <inline-formula><tex-math id="M4">\begin{document}$ N\epsilon^2 = 1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ \epsilon\rightarrow 0 $\end{document}</tex-math></inline-formula> to the solution of the Boltzmann equation, with the same short time restriction of Lanford's classical theorem.</p>
From Hard Sphere Dynamics to the Stokes–Fourier Equations: An Analysis of the Boltzmann–Grad Limit
