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

TL;DR

This paper analyzes the steady-state behavior of an inelastically bouncing particle under gravity and random forces, providing exact solutions and simulations relevant to driven granular matter.

Contribution

It introduces an exact analytical approach combined with simulations to study the steady state distribution of an inelastic bouncing particle under gravity.

Findings

Derived the steady state distribution function P(x,v)

Validated analytical results with simulations

Provided insights into driven granular matter dynamics

Abstract

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 reflection satisfy $v_f=-rv_i$, 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 $P(x,v)$.