MOTION OF 3 INELASTIC PARTICLES ON A RING

In a previous paper [P. Constantin, E. Grossman; and M. Mungan, Physic a D 83, 409 (1995)], we have studied in detail the dynamics of three i nelastically colliding particles moving on an infinite line. The prese nt paper addresses the effect of boundary conditions by investigating both analytically and numerically the dynamics of three particles conf ined to a ring. Using the methods developed in [P. Constantin, E. Gros sman, and M. Mungan, Physica D 83, 409 (1995)], we reformulate the dyn amics as a billiard in an equilateral triangle with nonspecular reflec tions laws. There are three sharply distinct regimes: (i) perfectly el astic collisions, (ii) slightly inelastic collisions, and (iii) strong ly inelastic collisions. In particular, in the limit of the inelastici ty going to zero, the asymptotic motion in case (ii) does not reduce t o case (i), i.e., perfectly elastic motion is a singular limit. For mo tion on the line in the strongly inelastic regime, particles can eithe r cluster, undergoing infinitely many collisions while their relative separation goes to zero (inelastic collapse), or they can separate aft er a finite number of collisions (escape). The confinement to a circle , while greatly enhancing the occurrence of clustering, does not compl etely eliminate the existence of other asymptotic states. In fact, the re exists a fractal set of initial conditions for which collisions pro ceed indefinitely without clustering.