Rate equation limit for a combinatorial solution of a stochastic aggregation model

TL;DR

This paper critically examines an exact combinatorial solution for a stochastic aggregation model, revealing its accuracy for certain rate kernels and discrepancies for others, thus questioning its general validity.

Contribution

It analyzes the asymptotic behavior of the combinatorial solution for the Marcus--Lushnikov model, clarifying its validity for specific rate kernels and identifying limitations.

Findings

Exact solutions match Smoluchowski equations for classical kernels.

Discrepancies found for the multiplicative kernel.

The combinatorial solution's range of validity remains unclear.

Abstract

In a recent series of papers, an exact combinatorial solution was claimed for a variant of the so-called Marcus--Lushnikov model of aggregation. In this model, a finite number of aggregates, are initially assumed to be present in the form of monomers. At each time step, two aggregates are chosen according to certain size-dependent probabilities and irreversibly joined to form an aggregate of higher mass. The claimed result given an expression for the full probability distribution over all possible size distributions in terms of the so-called Bell polynomials. In this paper, we develop the asymptotics of this solution in order to check whether the exact solution yields correct expressions for the average cluster size distribution as obtained from the Smoluchowski equations. The answer is surprisingly involved: for the generic case of an arbitrary reaction rate, it is negative, but for the so-called {\em classical\/} rate kernels, constant, additive and multiplicative, the solutions obtained are indeed exact. On the other hand, for the multiplicative kernel, a discrepancy is found in the full solution between the combinatorial solution and the exact solution. The reasons for this puzzling pattern of agreement and disagreement are unclear. A better understanding of the combinatorial solution's derivation is needed, the better to understand its range of validity.