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

TL;DR

This paper analyzes finite discrete-size particle aggregation with arbitrary kernels using combinatorial methods, deriving explicit formulas for cluster statistics and validating them against simulations, extending known solutions.

Contribution

It introduces solutions for two arbitrary-parameter kernels within a combinatorial framework, expanding the class of kernels with known analytical results.

Findings

Theoretical predictions match numerical simulations across various parameters.

Expressions for average cluster numbers and standard deviations were derived.

The approach extends known solutions to new kernels, including the condensation and linear combination kernels.

Abstract

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.