Absorption of a randomly accelerated particle: recent results for partially absorbing and inelastic boundaries

Consider a particle which is randomly accelerated by Gaussian white noise on the half line x>0, with an absorbing boundary at x=0. The nonequilibrium statistics of this system was analyzed exactly by McKean in 1963. Recent results for two other boundary conditions of physical interest will be reviewed. In the case of a partially absorbing boundary, the randomly accelerated particle is absorbed, on arriving at x = 0, with probability 1−p and reflected elastically with probability p. In the case of an inelastic boundary, the velocities of the particle just after and before striking the boundary satisfy vf=−rvi, where r is the coefficient of restitution. The absorption of a particle moving between two boundaries and some related results for confined semi-flexible polymers, which have similar statistical properties, will also be discussed.