Dynamics of 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.

Statistics of the number of zero crossings: from random polynomials to the diffusion equation

We consider a class of real random polynomials, indexed by an integer d, of large degree n and focus on the number of real roots of such random polynomials. The probability that such polynomials have no real root in the interval [0, 1] decays as a power law n(-theta(d)) where theta(d)>0 is the exponent associated with the decay of the persistence probability for the diffusion equation with random initial conditions in space dimension d. For n even, the probability that such polynomials have no root on the full real axis decays as n(-2[theta(d)+theta(2)]). For d=1, this connection allows for a physical realization of real random polynomials. We further show that the probability that such polynomials have exactly k real roots in [0, 1] has an unusual scaling form given by n(-phi(k/logn)) where phi(x) is a universal large deviation function.

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.

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

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 and 1 are inelastic, with velocities just after and before reflection related by v(f) =-r v(i). Cornell et al. have predicted that the particle undergoes inelastic collapse for r< r(c) = e(-pi/sqrt[3]) =0.163, coming to rest at the boundary after an infinite number of collisions in a finite time and remaining there. This has been questioned by Florencio et al. and Anton on the basis of simulations. We have solved the Fokker-Planck equation satisfied by the equilibrium distribution function P(x,v) with a combination of exact analytical and numerical methods. Throughout the interval 0<r<1, P(x,v) remains extended, as opposed to collapsed. There is no transition in which P(x,v) collapses onto the boundaries. However, for r< r(c) the equilibrium boundary collision rate is infinite, as predicted by Cornell et al., and all moments [v](q);, q>0 of the velocity just after reflection from the boundary vanish.

Average trajectory of returning walks

We compute the average shape of trajectories of some one-dimensional stochastic processes x(t) in the (t,x) plane during an excursion, i.e., between two successive returns to a reference value, finding that it obeys a scaling form. For uncorrelated random walks the average shape is semicircular, independent from the single increments distribution, as long as it is symmetric. Such universality extends to biased random walks and Levy flights, with the exception of a particular class of biased Levy flights. Adding a linear damping term destroys scaling and leads asymptotically to flat excursions. The introduction of short and long ranged noise correlations induces nontrivial asymmetric shapes, which are studied numerically.

Noise activated granular dynamics

We study the behavior of two particles moving in a bistable potential, colliding inelastically with each other and driven by a stochastic heat bath. The system has the tendency to clusterize, placing the particles in the same well at low drivings, and to fill all of the available space at high temperatures. We show that the hopping over the potential barrier occurs following the Arrhenius rate, where the heat bath temperature is replaced by the granular temperature. Moreover, within the clusterized "phase" one encounters two different scenarios: For moderate inelasticity, the jumps from one well to the other involve one particle at a time, whereas for strong inelasticity the two particles hop simultaneously.

Noncollapsing solution below r(c) for a randomly forced particle

We show that a noncollapsing solution below r(c) can be constructed for the dynamics of randomly forced particle interacting with a dissipating boundary. The scaling analysis predicts a divergent collision rate at the boundary for the noncollapsing solution. This prediction is tested numerically.

Statistics of multiple sign changes in a discrete non-Markovian sequence

We study analytically the statistics of multiple sign changes in a discrete non-Markovian sequence psi(i)=phi(i)+ phi(i-1) (i=1,2, em leader,n) where phi(i)'s are independent and identically distributed random variables each drawn from a symmetric and continuous distribution rho(phi). We show that the probability P(m)(n) of m sign changes up to n steps is universal, i.e., independent of the distribution rho(phi). The mean and variance of the number of sign changes are computed exactly for all n>0. We show that the generating function (tilde)P(p,n)= summation operator(infinity)(m=0)P(m)(n)p(m) approximately exp[-theta(d)(p)n] for large n where the "discrete" partial survival exponent theta(d)(p) is given by a nontrivial formula, theta(d)(p)=ln[sin(-1)(square root of [1-p(2)])/square root of [1-p(2)]] for 0< or = p < or = 1. We also show that in the natural scaling limit m-->infinity, n-->infinity but keeping x=m/n fixed, P(m)(n) approximately exp[-n Phi (x)] where the large deviation function Phi(x) is computed. The implications of these results for Ising spin glasses are discussed.