Exact combinatorial approach to finite coagulating systems

TL;DR

This paper presents an exact combinatorial method for analyzing finite coagulating systems with discrete cluster sizes and time, providing precise probability distributions and highlighting limitations of mean-field models.

Contribution

It introduces a novel exact combinatorial framework for finite coagulating systems, accounting for arbitrary initial conditions and discrete variables, which improves upon mean-field approaches.

Findings

Derived exact probability expressions for cluster configurations.

Validated results with systems using constant kernels.

Compared with mean-field solutions, revealing their limitations.

Abstract

The paper outlines an exact combinatorial approach to finite coagulating systems. In this 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, the exact expression is derived for the probability of a coagulating system with an arbitrary kernel being found in a given cluster configuration when monodisperse initial conditions are applied. Then, this probability is used to calculate the time-dependent distribution for the number of clusters of a given size, the average number of such clusters and that average's standard deviation. The correctness of our general expressions is proved based on the (analytical and numerical) results obtained for systems with the constant kernel. In addition, the results obtained are compared with the results arising from the solutions to the mean-field Smoluchowski coagulation equation, indicating its weak points. The paper closes with a brief discussion on the extensibility to other systems of the approach presented herein, emphasizing the issue of arbitrary initial conditions.