The coefficient of restitution does not exceed unity

Abstract

We study a classical mechanical problem in which a macroscopic ball is reflected by a non-deformable wall. The ball is modeled as a collection of classical particles bound together by an arbitrary potential, and its internal degrees of freedom are initially set to be in thermal equilibrium. The wall is represented by an arbitrary potential which is translation invariant in two directions. We then prove that the final normal momentum can exceed the initial normal momentum at most by $O(\sqrt{mkT})$, where $m$ is the total mass of the ball, $k$ the Boltzmann constant, and $T$ the temperature. This implies the well-known statement in the title in the macroscopic limit where $O(\sqrt{mkT})$ is negligible. Our result may be interpreted as a rigorous demonstration of the second law of thermodynamics in a system where a macroscopic dynamics and microscopic degrees of freedom are intrinsically coupled with each other.