On two fragmentation schemes with algebraic splitting probability

M. Ghorbel¹ ; T. Huillet²

¹ Laboratoire de Physique Théorique et Modélisation CNRS-UMR 8089 et Université de Cergy-Pontoise 2 Avenue Adolphe Chauvin 95032 Cergy-Pontoise, France and Laboratoire d' Analyse, Géometrie et Applications CNRS-UMR 7539, Institut Galilée Université de Paris 13 93340 Villetaneuse, France
² Laboratoire de Physique Théorique et Modélisation CNRS-UMR 8089 et Université de Cergy-Pontoise 2 Avenue Adolphe Chauvin 95032 Cergy-Pontoise, France

Applicationes Mathematicae, Tome 33 (2006) no. 1, pp. 95-110

Résumé

Consider the following inhomogeneous fragmentation model: suppose an initial particle with mass $x_{0}\in (0,1)$ undergoes splitting into $b>1$ fragments of random sizes with some size-dependent probability $p(x_{0}) $. With probability $1-p(x_{0}) $, this particle is left unchanged forever. Iterate the splitting procedure on each sub-fragment if any, independently. Two cases are considered: the stable and unstable case with $p( x_{0}) =x_{0}^{a}$ and $p(x_{0}) =1-x_{0}^{a}$ respectively, for some $a>0.$ In the first (resp. second) case, since smaller fragments split with smaller (resp. larger) probability, one suspects some stabilization (resp. instability) of the fragmentation process. Some statistical features are studied in each case, chiefly fragment size distribution, partition function, and the structure of the underlying random fragmentation tree.

Détail

DOI : 10.4064/am33-1-8

Keywords: consider following inhomogeneous fragmentation model suppose initial particle mass undergoes splitting fragments random sizes size dependent probability probability p particle unchanged forever iterate splitting procedure each sub fragment independently cases considered stable unstable x respectively first resp second since smaller fragments split smaller resp larger probability suspects stabilization resp instability fragmentation process statistical features studied each chiefly fragment size distribution partition function structure underlying random fragmentation tree