Macroscopic behaviour in a two-species exclusion process via the method of matched asymptotics

Abstract

We consider a two-species system of particles undergoing an asymmetric simple exclusion process on a periodic lattice. We use the method of matched asymptotics to derive the discrete evolution equations for the two population densities in the dilute regime. Taking the limit of the lattice spacing going to zero, we obtain a cross-diffusion system of partial differential equations for the two species densities. First, our result captures non-trivial aspects of particle interaction that are neglected in the mean-field approach, including a non-diagonal mobility matrix with an explicit density dependence. Second, it offers a generalisation of the rigorous hydrodynamic limit of Quastel, valid for species with equal jump rates and given in terms of a non-explicit self-diffusion coefficient, for 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, which is uniformly valid for the whole densities range. This cubic approximation agrees extremely well with the simulated data and coincides with the Taylor expansion up to second order of the self-diffusion coefficient obtained using a rigorous recursive method.