Inelastic collapse of a randomly forced particle

TL;DR

This paper studies the behavior of a particle in a finite region subjected to random forcing and inelastic collisions, revealing a critical restitution coefficient that determines whether the particle continues moving or collapses in finite time.

Contribution

It identifies a critical restitution coefficient for inelastic collapse in a randomly forced particle, independent of region size or damping effects.

Findings

Existence of a critical restitution coefficient r_c = e^{-rac{ ext{ extpi}}{ ext{ extsqrt{3}}}}

Particle undergoes inelastic collapse below r_c, coming to rest after infinite collisions in finite time

Above r_c, the particle's dynamics are ergodic and sustained.

Abstract

We consider a randomly forced particle moving in a finite region, which rebounds inelastically with coefficient of restitution r on collision with the boundaries. We show that there is a transition at a critical value of r, r_c\equiv e^{-\pi/\sqrt{3}}, above which the dynamics is ergodic but beneath which the particle undergoes inelastic collapse, coming to rest after an infinite number of collisions in a finite time. The value of r_c is argued to be independent of the size of the region or the presence of a viscous damping term in the equation of motion.