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

TL;DR

This paper derives a detailed macroscopic model for a two-species exclusion process using matched asymptotics, capturing interaction effects and generalizing previous hydrodynamic limits to unequal jump rates, with accurate approximations validated by simulations.

Contribution

It introduces a novel matched asymptotics approach to derive cross-diffusion PDEs for two-species exclusion processes, including non-trivial interactions and unequal jump rates.

Findings

Derived evolution equations with non-diagonal mobility matrix.

Generalized hydrodynamic limit to unequal jump rates.

Developed a cubic polynomial approximation for self-diffusion coefficient.

Abstract

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.