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