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 epsilon(s)/\x\(s), where epsilon is a small parameter and s > 2. Such a rescaled potential is truncated at distance epsilon(gamma-1), where gamma is an element of]0, 1[is suitably large. We also assume that the scatterer density is diverging as epsilon(-d+1), where d is the dimension of the physical space. We prove that, as epsilon --> 0 (the Boltzmann-Grad limit), the probability density of a test particle converges to a solution of the linear (uncutoff) Boltzmann equation with the cross-section relative to the potential V(x) = \x\(-s).