Inelastic collapse and near-wall localization of randomly accelerated particles

TL;DR

This paper investigates the inelastic collapse of randomly accelerated particles near walls, demonstrating its universality across models and analyzing how it affects particle distribution and localization.

Contribution

It shows that inelastic collapse occurs in a broad class of models with inhomogeneous random forces and establishes the universality of the critical restitution coefficient.

Findings

Inelastic collapse occurs for eta<eta_c in various models.

The critical value eta_c is universal across these models.

Inelastic collapse does not always lead to near-wall localization.

Abstract

The inelastic collapse of stochastic trajectories of a randomly accelerated particle moving in half-space $z > 0$ has been discovered by McKean and then independently re-discovered by Cornell et. al. The essence of this phenomenon is that particle arrives to a wall at $z = 0$ with zero velocity after an infinite number of inelastic collisions if the restitution coefficient $\beta$ of particle velocity is smaller than the critical value $\beta_c=\exp(-\pi/\sqrt{3})$. We demonstrate that inelastic collapse takes place also in a wide class of models with spatially inhomogeneous random force and, what is more, that the critical value $\beta_c$ is universal. That class includes an important case of inertial particles in wall-bounded random flows. To establish how the inelastic collapse influence the particle distribution, we construct an exact equilibrium probability density function $\rho(z,v)$ for particle position and velocity. The equilibrium distribution exists only at $\beta<\beta_c$ and indicates that inelastic collapse does not necessarily mean the near-wall localization.