Macroscopic behaviour in a two-species exclusion process via the method of matched asymptotics
Duration: 35 mins 6 secs
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Description: |
James Mason (University of Cambridge)
15 March 2022 – 14:50 to 15:35 |
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Created: | 2022-03-16 15:39 |
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Collection: | Frontiers in analysis of kinetic equations |
Publisher: | Isaac Newton Institute |
Copyright: | James Mason |
Language: | eng (English) |
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. |
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