Exact combinatorial approach to finite coagulating systems through recursive equations

TL;DR

This paper presents an exact combinatorial method for analyzing finite coagulating systems, providing precise solutions that outperform traditional mean-field approaches, especially for multiplicative and additive kernels.

Contribution

The authors develop an exact recursive combinatorial framework for finite coagulating systems, deriving solutions for arbitrary kernels and validating them against numerical results.

Findings

Exact solutions for multiplicative and additive kernels obtained.

The combinatorial approach outperforms mean-field Smoluchowski solutions.

Framework applicable to systems with discrete cluster sizes and time.

Abstract

This work outlines an exact combinatorial approach to finite coagulating systems through recursive equations and use of generating function method. In the classic approach the mean-field Smoluchowski coagulation is used. However, the assumptions of the mean-field theory are rarely met in real systems which limits the accuracy of the solution. In our approach, cluster sizes and time are discrete, and the binary aggregation alone governs the time evolution of the systems. By considering the growth histories of all possible clusters and applying monodisperse initial conditions, the exact expression for the probability of finding a coagulating system with an arbitrary kernel in a given cluster configuration is derived. Then, the average number of such clusters and the standard deviation of these solutions can be calculated. In this work, recursive equations for all possible growth histories of clusters are introduced. The correctness of our expressions was proved based on the comparison with numerical results obtained for systems with constant, multiplicative and additive kernels. For the first time the exact solutions for the multiplicative and additive kernels were obtained with this framework. In addition, our results were compared with the results arising from the solutions to the mean-field Smoluchowski equation. Our theoretical predictions outperform the classic approach.