J. Phys. I France
Volume 6, Numéro 2, February 1996
Page(s) 245 - 256
DOI: 10.1051/jp1:1996146
J. Phys. I France 6 (1996) 245-256

Finitely Generated Multifractals Can Display Phase Transitions

Thierrey Huillet and Bernard Jeannet

Laboratoire d'Ingénierie des Matériaux et des Hautes Pressions UPR 1311 CNRS, Institut Galilée, Université Paris 13, Avenue Jean-Baptiste Clément, 93430 Villetaneuse, France

(Received 11 September 1995, received in final form 10 October 1995 and accepted 25 October 1995)

A new class of multifractal objects ("skewed" multifractals) is introduced, the mutiplicative generator of which has a finite number of branches of different real-valued depths. Both microscopic and macroscopic scales are represented by such objects, each of these corresponding to a specific thermodynamical regime. In the "diluted" regime, the partition function Zt is exactly renormalizable which means in the sequel, as is the case in the general multifractal theory, that t-1 log Zt as a non trivial limit as t tends to infinity. In the "condensed" one the partition function converges. Details about the transition between these two regimes are given.

Une nouvelle classe de "multifractales" est introduite, pour laquelle le générateur présente un nombre fini de branches de longueur variable à valeurs réelles. Les échelles macroscopiques et microscopiques sont représentables par de tels objets, chacune d'elles correspondant à un régime thermodynamique spécifique. Dans la phase "diluée", la fonction de partition Zt est exactement renormalisable, en ce sens (classique) que la limite quand $t\rightarrow \infty$ de t-1 log Zt est non triviale. Dans la phase "condensée" la fonction de partition converge. Les détails thermodynamiques concernant cette transition de phase sont fournis.

© Les Editions de Physique 1996

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