  We present the arising of the Fick cross-diffusion system of equations for
fluid mixtures from the multi-species Boltzmann in a rigorous manner in Sobolev
spaces. To this end, we formally show that, in a diffusive scaling, the
hydrodynamical limit of the kinetic system is the Fick model supplemented with
a closure relation and we give explicit formulae for the macroscopic diffusion
coefficients from the Boltzmann collision operator. Then, we provide a
perturbative Cauchy theory in Sobolev spaces for the constructed Fick system,
which turns out to be a dilated parabolic equation. We finally prove the
stability of the system in the Boltzmann equation, ensuring a rigorous
derivation between the two models.
