Coalescense with arbitrary-parameter kernels and monodisperse initial conditions: A study within combinatorial framework

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.