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 $ N $ hard spheres of diameter $ \epsilon $ in a box $ \Lambda : = [0, 1]\times(\mathbb{R}/\mathbb{Z})^2 $. 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 $ 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.
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Reflection of particle
Evolution of
Pseudotrajectory associated with collision parameters
A shift of particle
Virtual particle