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

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 $\epsilon$. 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 $\epsilon ,N$ for ${\epsilon }^{2}N=1$ (Boltzmann-Grad regime).