Diffusion and Homogenization Limits with Separate Scales

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.