We consider the simultaneous diffusion and homogenization limit of the linear Boltzmann equation in a periodic medium in the
regime where the mean free path is much smaller than the lattice constant. The resulting equation is a diffusion equation,
with an averaged diffusion matrix that is formally obtained by first performing the diffusion limit and then the homogenization
one. The rigorous proof relies on the use of two-scale limits, in combination with an asymptotic expansion of the equilibrium
profile in powers of the ratio between the mean free path and the lattice constant.
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