On the cardinality of collisional clusters for hard spheres at low density

TL;DR

This paper analyzes the size of collisional clusters in a low-density gas of hard spheres, providing estimates of average cluster size that are uniform over time and system parameters in the Boltzmann-Grad limit.

Contribution

It extends previous work by deriving uniform-in-time estimates of backward cluster sizes in the Boltzmann-Grad regime for hard sphere gases.

Findings

Average cluster size estimates are uniform over time.

Results hold in the Boltzmann-Grad limit with small sphere diameter.

Provides a deeper understanding of particle interactions in dilute gases.

Abstract

We resume the investigation, started in arXiv:1408.6571, of the statistics of backward clusters in a gas of $N$ hard spheres of small diameter $\varepsilon$. A backward cluster is defined as the group of particles involved directly or indirectly in the backwards-in-time dynamics of a given tagged sphere. We obtain an estimate of the average cardinality of clusters with respect to the equilibrium measure, global in time, uniform in $\varepsilon, N$ for $\varepsilon^2 N=1$ (Boltzmann-Grad regime).