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Fragmentation energy

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  01 July 2016

Jean Bertoin  and
Servet Martínez
Jean Bertoin*
Affiliation:
Université Paris 6 and Institut Universitaire de France
Servet Martínez*
Affiliation:
Universidad de Chile, Santiago
*
∗ Postal address: Laboratoire de Probabilités et Modèles Aléatoires, Université Paris 6, 175, rue de Chevaleret, F-75013 Paris, France. Email address: jbe@ccr.jussieu.fr
∗∗ Postal address: CMM-DIM, Universidad de Chile, Casilla 170-3 Correo 3 Santiago, Chile. Email address: smartine@dim.uchile.cl
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Abstract

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Motivated by a problem arising in the mining industry, we estimate the energy ε(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 (s1,s2,…) is sβφ(s1/s,s2/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 ε(η)∼cηβ-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.

Keywords

Fragmentationgeneral branching processenergy

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60F15: Strong theorems
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 37 , Issue 2 , June 2005 , pp. 553 - 570
Copyright
Copyright © Applied Probability Trust 2005 

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