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 " fl \Gamma1 , where fl 2]0; 1[ is suitably large. We assume also that the scatterer density is diverging as " \Gammad+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 (uncutoffed) Boltzmann equation with the cross section relative to the potential V (x) = jxj \Gammas . 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 first result in this direction was obtained many years ago by G. Gallavotti who showed how to derive the linear Boltz..