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We study the statistics of backward clusters in a gas of hard spheres at lowdensity. A backward cluster is defined as the group of particles involveddirectly or indirectly in the backwards-in-time dynamics of a given taggedsphere. We derive upper and lower bounds on the average size of clusters byusing the theory of the homogeneous Boltzmann equation combined with suitablehierarchical expansions. These representations are known in the easier contextof Maxwellian molecules (Wild sums). We test our results with a numericalexperiment based on molecular dynamics simulations.

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