Inelastic collapse and near-wall localization of randomly accelerated particles

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.