论文标题
通过匹配的渐近学方法，在两个物种排除过程中的宏观行为
Macroscopic behaviour in a two-species exclusion process via the method of matched asymptotics
论文作者
论文摘要
我们考虑了在周期性晶格上的两个物种简单的排除过程。我们使用匹配的渐近学匹配方法来得出稀释状态下两个种群密度的进化方程，即两种物种密度的偏微分方程的交叉扩散系统。首先，我们的结果捕获了在平均场方法中忽略的非平凡交互项，包括具有显式密度依赖性的非对角性迁移率矩阵。其次，它概括了Quastel的严格流体动力极限[Commun。纯应用。数学。 45（6），623--679（1992）]，对较高速率相等且根据非明显自扩散系数给出的物种有效，对稀释型制度的不平等率而言。在相等率的情况下，通过将匹配的渐近近似值结合在低密度和高密度限制中，我们获得了自扩散系数的立方多项式近似，该系数在所有密度方面在数值上都是准确的。该立方近似与数值模拟非常吻合。它还与泰勒膨胀到使用严格的递归方法获得的自扩散系数密度的二阶相吻合。
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.
