Inelastic Particle Clusters from Cumulative Momenta

TL;DR

This paper introduces a geometric method using convex polygons in a cumulative momentum diagram to predict long-term clustering behavior in inelastic particle systems, with applications to simple symmetric random walks.

Contribution

It presents a novel geometric approach to analyze particle clustering in inelastic collisions, linking it to convex polygons and cumulative momentum diagrams.

Findings

Polygons in the momentum diagram predict cluster formation.

Application to systems with ±1 velocities relates to random walks.

Provides a new tool for analyzing inelastic particle dynamics.

Abstract

We consider a physical system comprising discrete massive particles on the real line whose trajectories interact via perfectly inelastic collision, also known as sticky particles. It turns out that polygons formed in a convex "cumulative momentum diagram" of the initial conditions allow us to easily predict how many particle clusters form as time $t\to\infty$. We explore an application of this to a unit mass system with $\pm 1$ velocities, which has ties to simple symmetric random walks and lattice path counting.