Virial estimates for hard spheres

Abstract

We review a virial-type estimate which bounds the strength of interaction for a gas of $N$ hard spheres (billiard balls) dispersing into Euclidean space $\mathbb{R}^d$. This type of estimate has been known for decades in the context of (semi-)dispersing billiards, and is essentially trivial in that context. Our goal, however, is to write virial estimates in a way which may lend insight into the problem of rigorously deriving Boltzmann's equation (cf. Lanford's theorem). Using virial estimates, we provide a short proof of lower bounds (sharp up to powers of logarithms) on the convergence rate of the first marginal in Lanford's theorem. Such lower bounds will often, but not always, follow trivially from energy conservation, the proof we present holds assuming only that the limiting dynamics is regular enough and does not reduce to free transport.