Backward Clusters, Hierarchy and Wild Sums for a Hard Sphere System in a   Low-Density Regime
  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. 
K. Aoki, M. Pulvirenti, S. Simonella and T. Tsuji
math-ph math.AP math.MP