Stochastic fragmentation and some sufficient conditions for shattering transition

Abstract

We investigate the fragmentation process developedby Kolmogorov and Filippov, which has been studied extensively bymany physicists (independently for some time). One of the mostinteresting phenomena is the shattering (or disintegration ofmass) transition which is considered a counterpart of the wellknown gelation phenomenon in the coagulation process. Though nomasses are subtracted from the system during the break-upprocess, the total mass decreases in finite time. The occurrenceof shattering transition is explained as due to the decompositionof the mass into an infinite number of particles of zero mass.It is known only that shattering phenomena occur for some specialtypes of break-up rates.In this paper, by considering the n-particle system ofstochastic fragmentation processes, we find general conditions ofthe rates which guarantee the occurrence of the shattering transition.