Thermalization of a particle with dissipative collisions

TL;DR

This paper analyzes how a test particle in a fluid reaches a stationary state with a Maxwellian velocity distribution at an effective temperature lower than the fluid's temperature, due to dissipative collisions.

Contribution

It demonstrates that the linear Boltzmann equation admits a stationary Maxwellian distribution with an explicitly defined effective temperature for dissipative collisions.

Findings

Stationary Maxwellian distribution exists despite dissipation.

Effective temperature is explicitly related to restitution and mass.

Dissipative collisions lower the particle's temperature.

Abstract

One considers the motion of a test particle in an homogeneous fluid in equilibrium at temperature $T$, undergoing dissipative collisions with the fluid particles. It is shown that the corresponding linear Boltzmann equation still posseses a stationary Maxwellian velocity distribution, with an effective temperature smaller than $T$. This effective temperature is explicitly given in terms of the restitution parameter and the masses.