The collapse phenomenon in one-dimensional inelastic point particle systems

We consider a one-dimensional system of n inelastic particles on a line, wi th coefficient of restitution 1 - 2 epsilon. We prove that if epsilon n les s than or similar to ln2 no collapses are possible for any initial datum, a nd we exhibit explicit collapsing solutions for epsilon n greater than or s imilar to pi. For n = 4 we construct a positive measure set of initial data which collapse in a finite time. For n = 3 we also consider stochastic per turbations of the system and prove the occurrence of collapses with positiv e probability if epsilon is sufficiently close to 1/2. Finally, we consider the limit n --> infinity of the exact collapse for n particles, obtaining a collapsing measure solution concentrated into two hydrodynamic profiles. (C) 1999 Elsevier Science B.V. All rights reserved.