For this work, we studied a finite system of discrete-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 with the assumption that coagulation process takes place in continuous time and individual coalescences are instantaneous. Thus, no two coalescence steps can occur at the same time and each coalescence act is marked with a time label. A generating function method and Lagrange inversion were used for derivations of expressions for an average number of clusters of a given size and its corresponding standard deviation. They were tested against numerical simulations. High precision of the theoretical predictions was observed for a wide range of A and coagulation stages, except for post-gel phase in the case of the condensation kernel where the theoretical results were only approximate. For specific values of A, these two kernels allowed to obtain known results of the constant, additive and product kernels. Beside a previously solved kernel for linear chains, K(i, j) = (i⁻¹ + j⁻¹)^a, they substantially widen the set of arbitrary-parameter kernels solved in the combinatorial approach.

对于这项工作，我们研究了具有任意参数的两种类型内核的离散尺寸聚集粒子的有限系统，即缩合（或支链聚合）内核，K ( i , j ) = ( A + i )( A + j )，以及常数和加性核的线性组合，K ( i , j ) = A + i + j. 它们是在组合方法中在单分散初始条件下求解的，假设凝结过程在连续时间内发生并且单个聚结是瞬时的。因此，不能同时发生两个合并步骤，并且每个合并行为都标有时间标签。生成函数方法和拉格朗日反演用于推导给定大小的平均簇数及其相应标准偏差的表达式。他们针对数值模拟进行了测试。对于大范围的A和凝结阶段，观察到理论预测的高精度，除了在凝结核的情况下的凝胶后阶段，理论结果只是近似的。对于A 的特定值，这两个内核允许获得常数、加性和乘积内核的已知结果。除了先前求解的线性链内核K ( i , j ) = ( i ^-1 + j ^-1 ) ^{a 之外}，它们显着拓宽了在组合方法中求解的任意参数内核集。