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Fragmentation energy


Bertoin, Jean; Martinez, Servet (2005). Fragmentation energy. Advances in Applied Probability, 37(2):553-570.

Abstract

Motivated by a problem arising in the mining industry, we estimate the energy E(η) that is needed to reduce a unit mass to fragments of size at most η in a fragmentation process, when η→0. We assume that the energy used in the instantaneous dislocation of a block of size s into a set of fragments (s$_1$,s$_2$,...) is s$^β$φ(s$_1$/s,s$_2$/s,...), where φ is some cost function and β a positive parameter. Roughly, our main result shows that if α>0 is the Malthusian parameter of an underlying Crump-Mode-Jagers branching process (with α = 1 when the fragmentation is mass-conservative), then there exists a c∈(0,∞) such that E(η)∼cη$^{β-α}$ when β<α. We also obtain a limit theorem for the empirical distribution of fragments of size less than η that result from the process. In the discrete setting, the approach relies on results of Nerman for general branching processes; the continuous approach follows by considering discrete skeletons. In the continuous setting, we also provide a direct approach that circumvents restrictions induced by the discretization.

Abstract

Motivated by a problem arising in the mining industry, we estimate the energy E(η) that is needed to reduce a unit mass to fragments of size at most η in a fragmentation process, when η→0. We assume that the energy used in the instantaneous dislocation of a block of size s into a set of fragments (s$_1$,s$_2$,...) is s$^β$φ(s$_1$/s,s$_2$/s,...), where φ is some cost function and β a positive parameter. Roughly, our main result shows that if α>0 is the Malthusian parameter of an underlying Crump-Mode-Jagers branching process (with α = 1 when the fragmentation is mass-conservative), then there exists a c∈(0,∞) such that E(η)∼cη$^{β-α}$ when β<α. We also obtain a limit theorem for the empirical distribution of fragments of size less than η that result from the process. In the discrete setting, the approach relies on results of Nerman for general branching processes; the continuous approach follows by considering discrete skeletons. In the continuous setting, we also provide a direct approach that circumvents restrictions induced by the discretization.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Scopus Subject Areas:Physical Sciences > Statistics and Probability
Physical Sciences > Applied Mathematics
Language:English
Date:2005
Deposited On:19 Jun 2013 13:12
Last Modified:09 Apr 2024 01:39
Publisher:Cambridge University Press
ISSN:0001-8678
OA Status:Closed
Publisher DOI:https://doi.org/10.1239/aap/1118858639
Related URLs:http://www.jstor.org/stable/view/30037341 (Library Catalogue)
Other Identification Number:zbMath: 1080.60080