L. Desvillettes et M. Pulvirenti, The linear boltzmann equation for long-range forces: A derivation from particle systems, MATH MOD M, 9(8), 1999, pp. 1123-1145
In this paper we consider a particle moving in a random distribution of obs
tacles. Each obstacle generates an inverse power law potential epsilon(s)/\
x\(s), where epsilon is a small parameter and s > 2. Such a rescaled potent
ial 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 diver
ging 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).