Derivation of an ornstein-uhlenbeck process for a massive particle in a rarified gas of particles

TL;DR

This paper derives an Ornstein-Uhlenbeck process to model the velocity of a massive rigid body in a gas, using advanced probabilistic and dynamical techniques to handle complex collision behaviors.

Contribution

It introduces a novel derivation of the Ornstein-Uhlenbeck process for a rigid body in a gas, incorporating new trajectory analysis to manage geometric collision complexities.

Findings

Velocity of the rigid body converges to an Ornstein-Uhlenbeck process.

Develops a modified dynamics to prevent pathological collisions.

Extends Lanford's approach with new trajectory analysis techniques.

Abstract

We consider the statistical motion of a convex rigid body in a gas of N smaller (spherical) atoms close to thermodynamic equilibrium. Because the rigid body is much bigger and heavier, it undergoes a lot of collisions leading to small deflections. We prove that its velocity is described, in a suitable limit, by an Ornstein-Uhlenbeck process. The strategy of proof relies on Lanford's arguments [17] together with the pruning procedure from [3] to reach diffusive times, much larger than the mean free time. Furthermore, we need to introduce a modified dynamics to avoid pathological collisions of atoms with the rigid body: these collisions, due to the geometry of the rigid body, require developing a new type of trajectory analysis.