The Collapse Phenomenon in one dimensional inelastic point particle systems

We consider a one-dimensional system of n inelastic particles on a line, with coefficient of restitution 1 − 2ε. We prove that if εn < ∼ ln 2 no collapses are possible for any initial datum, and we exhibit explicit collapsing solutions for εn > ∼ π . 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 perturbations of the system and prove the occurrence of collapses with positive probability if ε is sufficiently close to 1/2. Finally, we consider the limit n → ∞ of the exact collapse for n particles, obtaining a collapsing measure solution concentrated into two hydrodynamic profiles.