On the theory of Lorentz gases with long range interactions

Abstract

We construct and study the stochastic force field generated by a Poisson distribution of sources at finite density, $x_1,x_2,\cdots$ in ${\mathbb{R}}^3$ each of them yielding a long range potential $Q_i\Phi (x-x_i)$ with possibly different charges $Q_i\in \mathbb{R}$. The potential $\Phi$ is assumed to behave typically as $|x{|}^{-s}$ for large $|x|$, with $s>1/2$. We will denote the resulting random field as "generalized Holtsmark field". We then consider the dynamics of one tagged particle in such random force fields, in several scaling limits where the mean free path is much larger than the average distance between the scatterers. We estimate the diffusive time scale and identify conditions for the vanishing of correlations. These results are used to obtain appropriate kinetic descriptions in terms of a linear Boltzmann or Landau evolution equation depending on the specific choices of the interaction potential.