Approach to Equilibrium of a Body Colliding Specularly and Diffusely with a Sea of Particles

TL;DR

This paper analyzes how a rigid body approaches equilibrium when interacting with particles that reflect either elastically or probabilistically, deriving the rate of convergence depending on the reflection distribution.

Contribution

It introduces a model combining specular and diffuse particle reflections and derives the rate of approach to equilibrium in three dimensions based on the reflection probability distribution.

Findings

Rate of approach to equilibrium is O(t^{-3-p}) in three dimensions.

The convergence rate depends on the probabilistic reflection distribution K.

The model bridges elastic and probabilistic particle reflections in a unified framework.

Abstract

We consider a rigid body acted upon by two forces, a constant force and the collective force of interaction with a continuum of particles. We assume that some of the particles that collide with the body reflect elastically (specularly), while others reflect probabilistically with some probablility distribution K. We find that the rate of approach of the body to equilibrium is O(t^{-3-p}) in three dimensions where p can take any value from 0 to 2, depending on K.