On the theory of Lorentz gases with long range interactions

TL;DR

This paper constructs a stochastic force field from a Poisson distribution of sources with long-range potentials, analyzes the dynamics of a particle in this field, and derives kinetic equations under various scaling limits.

Contribution

It introduces the generalized Holtsmark field for long-range interactions and establishes conditions for kinetic descriptions via Boltzmann or Landau equations.

Findings

Derived diffusive time scales for particle dynamics.

Identified conditions for vanishing correlations in the force field.

Established kinetic equations depending on interaction potential.

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.