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

TL;DR

This paper extends the analysis of a randomly accelerated particle in a box by considering partially absorbing and inelastic boundaries, deriving the mean absorption time as a function of boundary parameters.

Contribution

It provides the first exact analytical and numerical study of mean absorption time with generalized boundary conditions involving partial absorption and inelastic reflection.

Findings

Mean absorption time depends on absorption probability p and restitution coefficient r.

Analytical solutions are obtained for the generalized boundary conditions.

Numerical simulations confirm the analytical results.

Abstract

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 Porr\`a 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 $r$.