  We consider a two-species simple exclusion process on a periodic lattice. We
use the method of matched asymptotics to derive evolution equations for the two
population densities in the dilute regime, namely a cross-diffusion system of
partial differential equations for the two species densities. First, our result
captures non-trivial interaction terms neglected in the mean-field approach,
including a non-diagonal mobility matrix with explicit density dependence.
Second, it generalises the rigorous hydrodynamic limit of Quastel [Commun. Pure
Appl. Math. 45(6), 623--679 (1992)], valid for species with equal jump rates
and given in terms of a non-explicit self-diffusion coefficient, to the case of
unequal rates in the dilute regime. In the equal-rates case, by combining
matched asymptotic approximations in the low- and high-density limits, we
obtain a cubic polynomial approximation of the self-diffusion coefficient that
is numerically accurate for all densities. This cubic approximation agrees
extremely well with numerical simulations. It also coincides with the Taylor
expansion up to the second-order in the density of the self-diffusion
coefficient obtained using a rigorous recursive method.
