FKT | Prof. James Mason | Macroscopic behaviour in a two-species exclusion process

Описание: FKT | Prof. James Mason | Macroscopic behaviour in a two-species exclusion process via the method of matched asymptotics

Speaker: Professor James Mason (University of Cambridge)

Date: 15th Mar 2022 - 14:50 to 15:35

Venue: INI Seminar Room 2

Title: Macroscopic behaviour in a two-species exclusion process via the method of matched asymptotic

Event: (FKT) Frontiers in kinetic theory: connecting microscopic to macroscopic scales - KineCon 2022

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.