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

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)PRLTAO0031-900710.1103/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/sqrt[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.

Inelastic collapse of a ball bouncing on a randomly vibrating platform

A theoretical study is undertaken of the dynamics of a ball which is bouncing inelastically on a randomly vibrating platform. Of interest are the distributions of the number of flights nf and the total time tauc until the ball has effectively "collapsed," i.e., coalesced with the platform. In the strictly elastic case both distributions have power law tails characterized by exponents that are universal, i.e., independent of the detail of the platform noise distribution. However, in the inelastic case both distributions have exponential tails: P(nf) approximately exp[-theta1nf] and P(tauc) approximately exp[-theta2tauc]. The decay exponents theta1 and theta2 depend continuously on the coefficient of restitution and are nonuniversal; however, as one approaches the elastic limit, they vanish in a manner which turns out to be universal. An explicit expression for theta1 is provided for a particular case of the platform noise distribution.

Equilibrium statistics of an inelastically bouncing ball, subject to gravity and a random force

We consider a particle moving on the half line x > 0 and subject to a constant force in the -x direction plus a delta-correlated random force. At x = 0 the particle is reflected inelastically. The velocities just after and before the reflection satisfy v(f) = -r v(i), where r is the coefficient of restitution. This simple model is of interest in connection with studies of driven granular matter in a gravitational field. With an exact analytical approach and simulations we study the steady-state distribution function P(x,v).

Asymptotic solution of the Wang-Uhlenbeck recurrence time problem

A Langevin particle is initiated at the origin with positive velocity. Its trajectory is terminated when it returns to the origin. In 1945, Wang and Uhlenbeck posed the problem of finding the joint probability density function (PDF) of the recurrence time and velocity, naming it "the recurrence time problem". We show that the short-time asymptotics of the recurrence PDF is similar to that of the integrated Brownian motion, solved in 1963 by McKean. We recover the long-time t(-3/2) decay of the first arrival PDF of diffusion by solving asymptotically an appropriate variant of McKean's integral equation.

Randomly accelerated particle in a box: mean absorption time for partially absorbing and inelastic boundaries

Consider a particle which is randomly accelerated by Gaussian white noise on the line segment 0<x<1 and is absorbed as soon as it reaches x=0 or x=1. The mean absorption time T(x,v), where x and v denote the initial position and velocity, was calculated exactly by Masoliver and Porrà in 1995. We consider a more general boundary condition. On arriving at either boundary, the particle is absorbed with probability 1-p and reflected with probability p. The reflections are inelastic, with coefficient of restitution, r. With exact analytical and numerical methods and simulations, we study the mean absorption time as a function of p and r.