The free path in a high velocity random flight process associated to a Lorentz gas in an external field

TL;DR

This paper analyzes the asymptotic behavior of free paths in a high velocity random flight model derived from a Lorentz gas under an external field, providing exact mean and variance formulas and a diffusion approximation.

Contribution

It introduces a detailed asymptotic analysis of free paths in a Lorentz gas with external fields, including explicit formulas and a diffusion limit.

Findings

Exact asymptotic mean and variance of free path derived

Diffusion approximation for the joint process established

Results depend explicitly on external field and scatterer density

Abstract

We investigate the asymptotic behavior of the free path of a variable density random flight model in an external field as the initial velocity of the particle goes to infinity. The random flight models we study arise naturally as the Boltzmann-Grad limit of a random Lorentz gas in the presence of an external field. By analyzing the time duration of the free path, we obtain exact forms for the asymptotic mean and variance of the free path in terms of the external field and the density of scatterers. As a consequence, we obtain a diffusion approximation for the joint process of the particle observed at reflection times and the amount of time spent in free flight.