On the Inelastic Collapse of a Ball Bouncing on a Randomly Vibrating Platform

Abstract

We study analytically the dynamics of a ball bouncing inelastically on a randomly vibrating platform, as a simple toy model of inelastic collapse. Of principal interest are the distributions of the number of flights n_f till the collapse and the total time τ_c elapsed before the collapse. In the strictly elastic case, both distributions have power law tails characterised by exponents which are universal, i.e., independent of the details of the platform noise distribution. In the inelastic case, both distributions have exponential tails: P(n_f) ~ exp[-θ_1 n_f] and P(τ_c) ~ exp[-θ_2 τ_c]. The decay exponents θ_1 and θ_2 depend continuously on the coefficient of restitution and are nonuniversal; however as one approches the elastic limit, they vanish in a universal manner that we compute exactly. An explicit expression for θ_1 is provided for a particular case of the platform noise distribution.