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

Abstract

We consider the one-dimensional motion of a particle randomly accelerated by Gaussian white noise on the line segment 0ϽxϽ1. The reflections of the particle from the boundaries at xϭ0,1 are inelastic. The velocities just before and after reflection are related by v f ϭϪrv i , where r is the coefficient of restitution. Cornell, Swift, and Bray ͓Phys. Rev. Lett. 81, 1142 ͑1998͔͒ have argued that there is an inelastic collapse transition in this system. For rϾr c ϭe Ϫ/ͱ3 the particle moves throughout the interval 0ϽxϽ1, while for rϽr c the particle is localized at xϭ0 or xϭ1. In this paper the equilibrium distribution function P(x,v) is analyzed for rϾr c by solving the steady-state Fokker-Planck equation, and the results are compared with numerical simulations.

Equilibrium of a confined, randomly accelerated, inelastic particle: Is there inelastic collapse?

“…BFG showed [4] that the asymptotic form of P͑0,v͒ for small and large v is determined by the first and second terms, respectively, of the kernel G͑1,v , u͒ in Eqs. (12) and (13) and is given by…”

“…To connect the asymptotic forms (14)-(16) of P͑0,v͒ for small and large v, we have solved the integral equation (12) by numerical iteration, as in [4]. As noted by BFG [4], the integral equation appears to have a well-defined solution for 0 Ͻ r Ͻ 1, i.e., 0 Ͻ ␤ Ͻ 5 / 2, with no special behavior at r c . Numerical results for several values of r above and below r c = 0.163 are shown in Fig.…”

“…Eventually an equilibrium is reached. Burkhardt, Franklin, and Gawronski (BFG) [4] analyzed the equilibrium distribution P͑x , v͒ for the position and velocity of the particle for r c Ͻ r Ͻ 1. This function satisfies the steady-state Fokker-Planck equation…”

Absorption of a randomly accelerated particle: gambler's ruin in a different game

“…The function ψ(x, v) satisfies the same steady-state Fokker-Planck equation (14) as the quantity P (x, v) considered in [5] and has the same Green's function solution…”

“…He obtained E numerically with an approach similar to [3] and related it to the confinement free energy of a semiflexible polymer in a tube. In another application inspired by [3], Burkhardt, Franklin, and Gawronski [5] calculated the equilibrium distribution function P (x, v) of a randomly accelerated particle on the line 0 < x < 1 undergoing inelastic collisions at the boundaries [6].…”