Macroscopic behaviour in a two-species exclusion process via the method of matched asymptotics
arXiv:2203.01038 · doi:10.1007/s10955-022-03036-9
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.
49 pages, 7 figures
References in corpus (6)
- Statistical Mechanics of Active Ornstein Uhlenbeck Particles
- The boundedness-by-entropy principle for cross-diffusion systems
- Modeling tumor cell migration: from microscopic to macroscopic
- Stochastic population dynamics in spatially extended predator-prey systems
- Diffusion of multiple species with excluded-volume effects
- Variational calculation of transport coefficients in diffusive lattice gases