Boltzmann-Grad limit of a hard sphere system in a box with isotropic boundary conditions

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 \begin{document}$ N $\end{document} hard spheres of diameter \begin{document}$ \epsilon $\end{document} in a box \begin{document}$ \Lambda : = [0, 1]\times(\mathbb{R}/\mathbb{Z})^2 $\end{document} . When a particle meets the boundary of the domain, it is instantaneously reinjected into the box with a random direction, {but} conserving kinetic energy. We prove that the first marginal of the process converges in the scaling \begin{document}$ N\epsilon^2 = 1 $\end{document} , \begin{document}$ \epsilon\rightarrow 0 $\end{document} to the solution of the Boltzmann equation, with the same short time restriction of Lanford's classical theorem.