Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

**Code(s) de Classification MSC:**

- 60F15 Strong theorems
- 60J25 Markov processes with continuous parameter

**Résumé:** Motivated by a problem arising in mining industry,
we estimate the energy ${\cal E}(\eta)$ which is needed to reduce a unit
mass to fragments of size at most $\eta$ in a fragmentation process,
when $\eta\to0$.
We assume that the energy used by the instantaneous dislocation of a
block of size
$s$ into a set of fragments $(s_1,s_2,...)$, is
$s^\beta \varphi(s_1/s,s_2/s,..)$, where $\varphi$ is some cost-function
and $\beta$ a positive parameter.
Roughly, our main result shows that if $\alpha>0$ is the Malthusian
parameter of an underlying
CMJ branching process (in fact $\alpha=1$ when the fragmentation is
mass-conservative), then ${\cal E}(\eta)\sim c \eta^{\beta-\alpha}$
whenever $\beta < \alpha$.
We also obtain a limit theorem for the empirical distribution of
fragments with size less than $\eta$ which result from the process.
In the discrete setting,
the approach relies on results of Nerman for general branching
processes; the continuous
setting follows by considering discrete skeletons. We also provide a
direct approach to
the continuous setting which circumvents restrictions induced by the
discretization.

**Mots Clés:** *Fragmentation ; energy ; general branching process*

**Date:** 2004-09-07

**Prépublication numéro:** *PMA-926*