Kinetic Theory of Cluster Dynamics

TL;DR

This paper develops a kinetic theory describing how particles in a Newtonian system form clusters over time, revealing a phase transition where a giant component emerges, consistent with previous numerical findings.

Contribution

It introduces a cluster size distribution framework based on the reduced Boltzmann density and identifies a phase transition in Maxwell molecules.

Findings

A macroscopic fraction forms a giant component in finite time.

The phase transition's critical index matches previous numerical results.

Provides a theoretical basis for cluster dynamics in kinetic systems.

Abstract

In a Newtonian system with localized interactions the whole set of particles is naturally decomposed into dynamical clusters, defined as finite groups of particles having an influence on each other's trajectory during a given interval of time. For an ideal gas with short-range intermolecular force, we provide a description of the cluster size distribution in terms of the reduced Boltzmann density. In the simplified context of Maxwell molecules, we show that a macroscopic fraction of the gas forms a giant component in finite kinetic time. The critical index of this phase transition is in agreement with previous numerical results on the elastic billiard.