The Linear Boltzmann Equation for Long-range Forces: a Derivation from Particle Systems

In this paper we consider a particle moving in a random distribution of obstacles. Each obstacle generates an inverse power law potential " s jxj s , where " is a small parameter and s > 2. Such a rescaled potential is truncated at distance " ?1 , where 2]0;11 is suitably large. We assume also that the scatterer density is diverging as " ?d+1 , where d is the dimension of the physical space. We prove that, as " ! 0 (the Boltzmann-Grad limit), the probability density of a test particle converges to a solution of the linear (uncutooed) Boltzmann equation with the cross section relative to the potential V (x) = jxj ?s. 1. Introduction It is well known how interesting and challenging is the problem of obtaining a complete and rigorous derivation of the kinetic transport equations starting from the basic Hamiltonian particle dynamics. The rst result in this direction was obtained many years ago by G. Gallavotti who showed how to derive the linear Boltzmann equation (with hard{sphere cross section) starting from the dynamics of a single particle in a random distribution of xed hard scatterers in the so{ called Boltzmann{Grad limit. This paper (Cf. G]), unfortunately unpublished and not widely known, is technically simple but has a deep content. In particular it is proved there for the rst time that it is perfectly consistent to obtain an irreversible stochastic behavior as a limit of a sequence of deterministic Hamiltonian systems (in a random medium). Later on this result was improved (see S1], S2] and BBuS]). More recently, the Boltzmann{Grad limit in the case when the distribution of scatterers is periodic (and not random) has also been considered in BoGoW] (see also the references therein). Note that in this case, the result is totally diierent. It is worthwile to mention also the well known Lanford's result for short times (see L]) for the fully nonlinear Boltzmann equation, derived