Motion of a Rigid Body in a Special Lorentz Gas: Loss of Memory Effect

TL;DR

This paper analyzes the linear motion of a rigid body in a special Lorentz gas, showing that obstacle motion affects the decay rate of the body's velocity and destroys the memory effect due to recollision.

Contribution

It provides a mathematical analysis of the velocity decay in a Lorentz gas with different obstacle motions, complementing previous numerical studies.

Findings

Velocity decays exponentially with thermal obstacles.

Velocity decays algebraically with motionless obstacles.

Obstacle motion influences the memory effect in the system.

Abstract

Linear motion of a rigid body in a special kind of Lorentz gas is mathematically analyzed. The rigid body moves against gas drag according to Newton's equation. The gas model is a special Lorentz gas consisting of gas molecules and background obstacles, which was introduced in~(Tsuji and Aoki: J. Stat. Phys. \textbf{146}, 620--645, 2012). The specular boundary condition is imposed on the resulting kinetic equation. This study complements the numerical study by Tsuji and Aoki cited above --- although the setting in this paper is slightly different from theirs, qualitatively the same asymptotic behavior is proved: The velocity $V(t)$ of the rigid body decays exponentially if the obstacles undergo thermal motion; if the obstacles are motionless, then the velocity $V(t)$ decays algebraically with a rate $t^{-5}$ independent of the spatial dimension. This demonstrates the idea that interaction of the molecules with the background obstacles destroy the memory effect due to recollision.