Collapse of a randomly forced particle : a comment

We refute the arguments by Anton in cond-mat/0004390, which set out to disprove the existence of a collapse transition for a randomly forced inelastic particle. An eprint by Anton [1] has recently appeared on this archive, criticizing our proposal of a collapse transition for an inelastic, randomly-forced particle [2]. Our conclusions were that a particle forced by Gaussian white noise that rebounds from a wall with coefficient of restitution r will, with probability 1, dissipate all its energy and come to rest at the wall after an infinite number of collisions in a finite time, provided r < rc = e √ . It has been pointed out that the transition is not present for certain discretizations of the Langevin equation [3], and it is also not clear how it manifests itself in experimentally realizable systems [4]. However, ref. [1] goes further, and argues that the transition is also absent for the ideal case of a pure white-noise force in continuous time. The purpose of this Comment is to point out substantial errors in Anton’s arguments, which, we believe, invalidate his conclusions. The existence of the collapse transition is supported by analytical calculations [5] by Burkhardt, Franklin, and Gawronski (henceforth BFG), who constructed a steady-state solution of the Fokker-Planck equation for a particle confined in a finite spatial interval. They observed that the solution is well-behaved when the coefficient of restitution r is in the range rc < r < 1, where rc is the critical value for collapse proposed in [2], whereas the rate of collisions of the particle with the boundary diverges as r approaches rc from above. Anton’s criticisms of this work, together with our responses, are as follows: