Backward Clusters, Hierarchy and Wild Sums for a Hard Sphere System in a Low-Density Regime

TL;DR

This paper investigates the size and structure of backward clusters in a low-density hard sphere gas, deriving bounds through Boltzmann equation theory and validating results with molecular dynamics simulations.

Contribution

It introduces new bounds on backward cluster sizes in hard sphere systems using hierarchical expansions and Wild sums, extending previous models to low-density regimes.

Findings

Derived bounds on average cluster size

Validated theoretical results with simulations

Extended Wild sums to hard sphere systems

Abstract

We study the statistics of backward clusters in a gas of hard spheres at low density. 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 derive upper and lower bounds on the average size of clusters by using the theory of the homogeneous Boltzmann equation combined with suitable hierarchical expansions. These representations are known in the easier context of Maxwellian molecules (Wild sums). We test our results with a numerical experiment based on molecular dynamics simulations.