Scale Free Cluster Distributions from Conserving Merging-Fragmentation Processes

TL;DR

This paper introduces a model for cluster dynamics involving merging and fragmentation processes that conserve total mass, resulting in a universal, scale-invariant size distribution with a specific power-law exponent.

Contribution

It presents an analytical derivation of a universal, scale-free cluster size distribution in a conserving merging-fragmentation process, applicable across natural and social systems.

Findings

The cluster size distribution follows a power-law with exponent -3/2.

The distribution remains invariant under rescaling over 15 decades.

The model demonstrates scale invariance in cluster dynamics with mass conservation.

Abstract

We propose a dynamical scheme for the combined processes of fragmentation and merging as a model system for cluster dynamics in nature and society displaying scale invariant properties. The clusters merge and fragment with rates proportional to their sizes, conserving the total mass. The total number of clusters grows continuously but the full time-dependent distribution can be rescaled over at least 15 decades onto a universal curve which we derive analytically. This curve includes a scale free solution with a scaling exponent of -3/2 for the cluster sizes.