Inelastic collapse and near-wall localization of randomly accelerated particles

Inelastic collapse of stochastic trajectories of a randomly accelerated particle moving in half-space z >0 has been discovered by McKean [J. Math. Kyoto Univ. 2, 227 (1963)] and then independently rediscovered by Cornell et al. [Phys. Rev. Lett. 81, 1142 (1998), 10.1103/PhysRevLett.81.1142]. The essence of this phenomenon is that the particle arrives at the wall at z =0 with zero velocity after an infinite number of inelastic collisions if the restitution coefficient β of particle velocity is smaller than the critical value β_c=exp(-π /√{3 }) . We demonstrate that inelastic collapse takes place also in a wide class of models with spatially inhomogeneous random forcing and, what is more, that the critical value β_c is universal. That class includes an important case of inertial particles in wall-bounded random flows. To establish how inelastic collapse influences the particle distribution, we derive the exact equilibrium probability density function ρ (z ,v ) for the particle position and velocity. The equilibrium distribution exists only at β <β_c and indicates that inelastic collapse does not necessarily imply near-wall localization.