Motion of three inelastic particles on a ring

In a previous paper [P. Constantin, E. Grossman; and M. Mungan, Physica D 83, 409 (1995)], we have studied in detail the dynamics of three inelastically colliding particles moving on an infinite line. The present paper addresses the effect of boundary conditions by investigating both analytically and numerically the dynamics of three particles confined to a ring. Using the methods developed in [P. Constantin, E. Grossman, and M. Mungan, Physica D 83, 409 (1995)], we reformulate the dynamics as a billiard in an equilateral triangle with nonspecular reflections laws. There are three sharply distinct regimes: (i) perfectly elastic collisions, (ii) slightly inelastic collisions, and (iii) strongly inelastic collisions. In particular, in the limit of the inelasticity going to zero, the asymptotic motion in case (ii) does not reduce to case (i), i.e., perfectly elastic motion is a singular limit. For motion on the line in the strongly inelastic regime, particles can either cluster, undergoing infinitely many collisions while their relative separation goes to zero (inelastic collapse), or they can separate after a finite number of collisions (escape). The confinement to a circle, while greatly enhancing the occurrence of clustering, does not completely eliminate the existence of other asymptotic states. In fact, there exists a fractal set of initial conditions for which collisions proceed indefinitely without clustering.