A Light impurity in an Equilibrium Gas

TL;DR

This paper studies the dynamics of a light impurity in a thermal equilibrium Lorentz gas, revealing universal growth laws for velocity and displacement, and deriving universal distribution forms independent of gas density and interaction specifics.

Contribution

It introduces a comprehensive analysis of impurity motion in a Lorentz gas, deriving universal growth laws and distribution functions for arbitrary dimensions and interaction potentials.

Findings

Average particle speed grows as t^{eta} with eta depending on interaction potential

Particle displacement grows linearly in time, independent of density and interaction

Velocity and position distributions approach universal non-Gaussian scaling forms

Abstract

We investigate the evolution of a light impurity particle in a Lorentz gas where the background atoms are in thermal equilibrium. As in the standard Lorentz gas, we assume that the particle is negligibly light in comparison with the background atoms. The thermal motion of atoms causes the average particle speed to grow. In the case of the hard-sphere particle-atom interaction, the temporal growth is ballistic, while generally it is sub-linear. For the particle-atom potential that diverges as r^{-\lambda} in the small separation limit, the average particle speed grows as t^{\lambda /(2(d-1)+ \lambda)} in d dimensions. The particle displacement exhibits a universal growth, linear in time and the average (thermal) speed of the atoms. Surprisingly, the asymptotic growth is independent on the gas density and the particle-atom interaction. The velocity and position distributions approach universal scaling forms which are non-Gaussian. We determine the velocity distribution in arbitrary dimension and for arbitrary interaction exponent \lambda. For the hard-sphere particle-atom interaction, we compute the position distribution and the joint velocity-position distribution.