Title: Accurate stochastic simulation of nonlinear reactions between closest particles Show Chinese title

Title: 最邻近粒子间非线性反应的精确随机模拟

Abstract: We study a system of diffusing point particles in which any triplet of particles reacts and is removed from the system when the relative proximity of the constituent particles satisfies a predefined condition. Proximity-based reaction conditions of this kind are commonly used in particle-based simulations of chemical kinetics to mimic bimolecular reactions, those involving just two reactants, and have been extensively studied. The rate at which particles react within the system is determined by the reaction condition and particulate diffusion. In the bimolecular case, analytic relations exist between the reaction rate and the distance at which particles react allowing modellers to tune the rate of the reaction within their simulations by simply altering the reaction condition. However, generalising proximity-based reaction conditions to trimolecular reactions, those involving three particles, is more complicated because it requires understanding the distribution of the closest diffusing particle to a point in the vicinity of a spatially dependent absorbing boundary condition. We find that in this case the evolution of the system is described by a nonlinear partial integro-differential equation with no known analytic solution, which makes it difficult to relate the reaction rate to the reaction condition. To resolve this, we use singular perturbation theory to obtain a leading-order solution and show how to derive an approximate expression for the reaction rate. We then use finite element methods to quantify the higher-order corrections to this solution and the reaction rate, which are difficult to obtain analytically. Leveraging the insights gathered from this analysis, we demonstrate how to correct for the errors that arise from adopting the approximate expression for the reaction rate, enabling for the construction of more accurate particle-based simulations than previously possible. Show Chinese abstract

Abstract: 我们研究了一种扩散点粒子系统，在该系统中，任意三个粒子当构成粒子的相对接近程度满足预定义条件时会发生反应并从系统中移除。 基于接近度的这种反应条件在基于粒子的化学动力学模拟中常用于模仿双分子反应（仅涉及两个反应物），并且已被广泛研究。 系统内粒子的反应速率由反应条件和颗粒扩散决定。 在双分子情况下，反应速率与反应距离之间存在解析关系，使建模者能够通过简单调整反应条件来调节模拟中的反应速率。 然而，将基于接近度的反应条件推广到三分子反应（涉及三个粒子）更为复杂，因为它需要理解最近的扩散粒子在空间依赖吸收边界条件附近的分布。 我们发现在这种情况下，系统的演化由一个没有已知解析解的非线性偏积分微分方程描述，这使得难以将反应速率与反应条件联系起来。 为了解决这个问题，我们使用奇异摄动理论获得主要阶次解，并展示如何推导出反应速率的近似表达式。 然后，我们利用有限元方法量化此解及其反应速率的更高阶修正值，这些修正值很难通过解析方法获得。 利用从这一分析中获得的见解，我们展示了如何纠正采用反应速率近似表达式所导致的误差，从而能够构建比以往更准确的基于粒子的模拟。