For this paper, we studied the evolution of the cluster size distribution for a system of coagulating particles under a linear-chain (LC) kernel, $K\left(i,j\right)={\left(\frac{1}{i}+\frac{1}{j}\right)}^{\alpha }$, and monodisperse initial conditions. We used a combinatorial framework in which time (i.e., time steps counted by the subsequent system states) and cluster sizes were discrete and the binary aggregation governed the evolution of the system. We modified a previously-known solution for the constant kernel to cover the LC kernel and used it in the framework to obtain the exact expression for the cluster size distribution (the average number of particles of a given size) and its standard deviation. Our theoretical solution is validated by a comparison to numerically simulated results for several values of $\alpha$ and to the experimental data of coagulating polystyrene particles. Theoretical predictions were accurate for any stage of the aggregation process and for a wide range of $\alpha$.

在本文中，我们研究了在线性链（LC）内核下凝聚颗粒系统的团簇尺寸分布的演变， $ķ\left(\mathrm{一世}，Ĵ\right)={\left(\frac{\mathrm{1个}}{\mathrm{一世}}+\frac{\mathrm{1个}}{Ĵ}\right)}^{\alpha }$，并分散初始条件。我们使用了一个组合框架，其中时间（即，由后续系统状态计算的时间步长）和集群大小是离散的，并且二进制聚合控制着系统的演进。我们修改了一个已知的常数核解决方案以覆盖LC核，并在框架中使用它来获得簇尺寸分布（给定尺寸的平均颗粒数）及其标准偏差的精确表达式。我们的理论解决方案通过与几个数值的数值模拟结果进行比较来验证$\alpha$以及凝结聚苯乙烯颗粒的实验数据。理论预测对于聚合过程的任何阶段以及范围广泛的预测都是准确的$\alpha$。