Universality of Cluster Dynamics

TL;DR

This study investigates cluster formation dynamics across various systems and dimensions, revealing a universal power-law behavior at critical points, which could enhance predictions of rare events in physical and biological systems.

Contribution

It demonstrates the universality of cluster size distribution power laws at critical times across multiple dimensions and collision types, extending previous low-dimensional findings.

Findings

Power law observed at critical time in elastic collisions

Universality across dimensions 2 to 8 and different norms

Consistent behavior in coalescing collisions in 2D

Abstract

We have studied the kinetics of cluster formation for dynamical systems of dimensions up to $n=8$ interacting through elastic collisions or coalescence. These systems could serve as possible models for gas kinetics, polymerization and self-assembly. In the case of elastic collisions, we found that the cluster size probability distribution undergoes a phase transition at a critical time which can be predicted from the average time between collisions. This enables forecasting of rare events based on limited statistical sampling of the collision dynamics over short time windows. The analysis was extended to L$^p$-normed spaces ($p=1,...,\infty$) to allow for some amount of interpenetration or volume exclusion. The results for the elastic collisions are consistent with previously published low-dimensional results in that a power law is observed for the empirical cluster size distribution at the critical time. We found that the same power law also exists for all dimensions $n=2,...,8$, 2D L$^p$ norms, and even for coalescing collisions in 2D. This broad universality in behavior may be indicative of a more fundamental process governing the growth of clusters.