  In this paper we present a rigorous derivation of the Boltzmann equation in a
compact domain with diffuse reflection boundary conditions. We consider a
system of $N$ hard spheres of diameter $\epsilon$ in a box $\Lambda := [0, 1]
\times (R/Z)^2$. When a particle meets the boundary of the domain, it is
instantaneously reinjected into the box with a random direction, and conserving
kinetic energy. We prove that the first marginal of the process converges in
the scaling $N\epsilon^2 = 1$, $\epsilon\rightarrow 0$ to the solution of the
Boltzmann equation, with the same short time restriction of Lanford's classical
theorem.
