Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2002; 39(4): 543-558

Printed July 1, 2002

Copyright © The Korean Mathematical Society.

Stochastic fragmentation and some sufficient conditions for shattering transition

Intae Jeon

Catholic University of Korea

Abstract

We investigate the fragmentation process developed by Kolmogorov and Filippov, which has been studied extensively by many physicists (independently for some time). One of the most interesting phenomena is the shattering (or disintegration of mass) transition which is considered a counterpart of the well known gelation phenomenon in the coagulation process. Though no masses are subtracted from the system during the break-up process, the total mass decreases in finite time. The occurrence of shattering transition is explained as due to the decomposition of the mass into an infinite number of particles of zero mass. It is known only that shattering phenomena occur for some special types of break-up rates. In this paper, by considering the $n$-particle system of stochastic fragmentation processes, we find general conditions of the rates which guarantee the occurrence of the shattering transition.

Keywords: fragmentation, shattering transition, stochastic fragmentation, stochastic dominance

MSC numbers: Primary 60K35; Secondary 82C22

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