Lorentz's model with dissipative collisions

TL;DR

This paper analyzes the Lorentz model with inelastic collisions, showing the existence of stationary states for inelastic cases and characterizing the velocity distribution and current behavior under external fields.

Contribution

It extends the Lorentz model to include dissipative collisions, deriving the stationary state existence and distribution properties in one and three dimensions.

Findings

Stationary states exist for inelastic collisions (α<1).

Velocity distribution transitions from half-gaussian to symmetric as α approaches 1.

Stationary current scales with the square root of external acceleration.

Abstract

Propagation of a particle accelerated by an external field through a scattering medium is studied within the generalized Lorentz model allowing inelastic collisions. Energy losses at collisions are proportional to $(1-\alpha^{2})$, where $0\le\alpha\le 1$ is the restitution coefficient. For $\alpha =1$ (elastic collisions) there is no stationary state. It is proved in one dimension that when $\alpha <1$ the stationary state exists . The corresponding velocity distribution changes from a highly asymmetric half-gaussian ($\alpha =0$) to an asymptotically symmetric distribution $\sim {\rm exp}[-(1-\alpha)v^{4}/2]$, for $\alpha\to 1$. The identical scaling behavior in the limit of weak inelasticity is derived in three dimensions by a self-consistent perturbation analysis, in accordance with the behavior of rigorously evaluated moments. The dependence on the external field scales out in any dimension, predicting in particular the stationary current to be proportional to the square root of the external acceleration.