Exact solutions for mass-dependent irreversible aggregations

TL;DR

This paper derives exact solutions for mass-dependent irreversible aggregation processes, revealing scaling laws and finite size effects, applicable to ring, line, and well-mixed systems, and relates these to classical aggregation kernels.

Contribution

It provides the first exact combinatorial solutions for mass-dependent aggregation models in different geometries, including scaling laws and finite size scaling forms.

Findings

Exact probability distributions for configurations on ring and line

Scaling laws for mass distribution of a single cluster

Finite size scaling form derived for the process

Abstract

We consider the mass-dependent aggregation process (k+1)X -> X, given a fixed number of unit mass particles in the initial state. One cluster is chosen proportional to its mass and is merged into one either with k-neighbors in one dimension, or -- in the well-mixed case -- with k other clusters picked randomly. We find the same combinatorial exact solutions for the probability to find any given configuration of particles on a ring or line, and in the well-mixed case. The mass distribution of a single cluster exhibits scaling laws and the finite size scaling form is given. The relation to the classical sum kernel of irreversible aggregation is discussed.