Convergence to Equilibrium of a Body Moving in a Kinetic Sea

TL;DR

This paper studies the long-term behavior of a rigid body immersed in a particle field, showing that under certain conditions, the body’s velocity converges to a stable equilibrium due to particle collisions and external forces.

Contribution

It establishes conditions for the existence and asymptotic stability of an equilibrium velocity for a body in a kinetic particle environment, extending understanding of kinetic-fluid interactions.

Findings

Existence of an equilibrium velocity V_infinity.

Asymptotic stability of the equilibrium velocity.

Conditions on external force G and collision kernel K.

Abstract

We consider a continuum of particles that are acted upon by an external force $\mathbf{G}(t,\mathbf{x})$ and that collide with a rigid body. The body itself is subject to a constant force $E$ as well as to the collective force of interaction with the particles. We assume that the particles that collide with the body reflect probabilistically with some probablility distribution $K(v,u)$. Under certain conditions on $\mathbf{G}(t,\mathbf{x})$ and $K(v,u)$, we identify an equilibrium velocity $V_{\infty }$ of the body and we prove that this equilibrium is asymptotically stable.