Equilibrium of a confined, randomly-accelerated, inelastic particle: Is there inelastic collapse?

TL;DR

This paper investigates the equilibrium behavior of a randomly accelerated inelastic particle in a confined one-dimensional space, revealing that it remains extended within the interval for all restitution coefficients but exhibits infinite collision rates at low restitution.

Contribution

It provides a combined analytical and numerical solution to the Fokker-Planck equation for inelastic boundary collisions, clarifying the conditions under which inelastic collapse occurs.

Findings

P(x,v) remains extended for all 0<r<1

Collision rate becomes infinite for r<0.163

Velocity moments vanish after boundary reflections at low r

Abstract

We consider the one-dimensional motion of a particle randomly accelerated by Gaussian white noise on the line segment 0<x<1. The reflections of the particle from the boundaries at x=0 and 1 are inelastic, with coefficient of restitution r. We have solved the Fokker-Planck equation satisfied by the equilibrium distribution function P(x,v) with a combination of exact analytical and numerical methods. Throughout the interval 0<r<1, P(x,v) remains extended, as opposed to collapsed. The particle is not localized at the boundary. However, for r<0.163 the equilibrium boundary collision rate is infinite, as predicted by Cornell et al., and all moments of the velocity just after reflection from the boundary vanish.