Derivation of an Ornstein–Uhlenbeck Process for a Massive Particle in a Rarified Gas of Particles

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 (Lecture notes in physics, vol 38, Springer, New York, pp 1–111, 1975) together with the pruning procedure from Bodineau et al. (Invent Math 203(2):493–553, 2016) 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.