Statistics of a confined, randomly accelerated particle with inelastic boundary collisions

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 = 0,1 are inelastic. The velocities just before and after reflection are related by v(f) = - rv(i), where r is the coefficient of restitution. Cornell, Swift, and Fray [Phys. Rev. Lett. 81, 1142 (1998)] have argued that there is an inelastic collapse transition in this system. For r > r(c) = e(-pi/root 3) the particle moves throughout the interval 0 < x < 1, while for r < r, the particle is localized at x = 0 or x = 1. In this paper the equilibrium distribution function P(x,v) is analyzed for r > r(c) by solving the steady-state Fokker-Planck equation, and the results are compared with numerical simulations.