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

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 velocities just after and before reflection related by v(f) =-r v(i). Cornell et al. have predicted that the particle undergoes inelastic collapse for r< r(c) = e(-pi/sqrt[3]) =0.163, coming to rest at the boundary after an infinite number of collisions in a finite time and remaining there. This has been questioned by Florencio et al. and Anton on the basis of simulations. 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. There is no transition in which P(x,v) collapses onto the boundaries. However, for r< r(c) the equilibrium boundary collision rate is infinite, as predicted by Cornell et al., and all moments [v](q);, q>0 of the velocity just after reflection from the boundary vanish.