论文标题
与任意参数内核和单分散的初始条件：组合框架中的研究
Coalescense with arbitrary-parameter kernels and monodisperse initial conditions: A study within combinatorial framework
论文作者
论文摘要
对于这项工作，我们研究了一种有限的系统，该系统针对具有任意参数的两种类型的内核，凝结（或分支 - 链聚合）内核，$ k（i，j）=（a+i）（a+j）$，以及常数和添加剂内核的线性组合，$ k（a+i）= $ j $ j i+j.在单分散的初始条件下，它们在组合方法中被求解，在组合方法中，将谨慎的时间计算为系统的后续状态。生成函数方法和拉格朗日反演用于推导。获得了给定尺寸的平均簇数及其相应标准偏差的表达式，并针对数值模拟进行了测试。对于$ a $ a $ a和凝结阶段，可以观察到理论预测的高精度，除了在冷凝核的情况下（保留了巨大的簇存在）。对于适当的$ a $，这两个内核重现了常数，添加剂和产品内核的已知结果。除了先前解决的线性链内核外，它们还扩展了在组合方法中求解的任意参数内核的数量。
For this work, we studied a finite system of discreet-size aggregating particles for two types of kernels with arbitrary parameters, a condensation (or branched-chain polymerization) kernel, $K(i,j)=(A+i)(A+j)$, and a linear combination of the constant and additive kernels, $K(i,j)=A+i+j$. They were solved under monodisperse initial conditions in the combinatorial approach where discreet time is counted as subsequent states of the system. A generating function method and Lagrange inversion were used for derivations. Expressions for an average number of clusters of a given size and its corresponding standard deviation were obtained and tested against numerical simulation. High precision of the theoretical predictions can be observed for a wide range of $A$ and coagulation stages, excepting post-gel phase in the case of the condensation kernel (a giant cluster presence is preserved). For appropriate $A$, these two kernels reproduced known results of the constant, additive and product kernels. Beside a previously solved linear-chain kernel, they extend the number of arbitrary-parameter kernels solved in the combinatorial approach.
