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

Abstract

We resume the investigation, started in [2], of the statistics of backward clusters in a gas of N hard spheres of small diameter ε. 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 ε, N for ε 2 N = 1 (Boltzmann-Grad regime).

A kinetic model for epidemic spread

“…A cluster {i 1 , i 2 , • • • } of infection grows in time, determined by the particle evolution: an individual is potentially infected at time t > 0 if it is involved, directly or indirectly, in the forward-in-time dynamics of 1. The "forward cluster of particle 1" (according to a terminology of [1,22]) is represented symbolically in the picture below. For concreteness, we may want to fix an idealized mechanical setting.…”