Macroscopic Behaviour in a Two-Species Exclusion Process Via the Method of Matched Asymptotics

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