Stationary states and fractional dynamics in systems with long-range interactions

¹  IM2NP, Aix-Marseille Universit, Centre de Saint Jérôme - Case 142, F-13397 Marseille Cedex 20, France, EU
²  Dipartimento di Energetica Sergio Stecco, Università di Firenze - via s. Marta 3, I-50139 Firenze, Italy, EU
³  Centro interdipartimentale per lo Studio delle Dinamiche Complesse (CSDC) - via G. Sansone 1, I-50019 Sesto Fiorentino (FI), Italy, EU
⁴  INFN, sezione di Firenze - via G. Sansone 1, I-50019 Sesto Fiorentino (FI), Italy, EU
⁵  Centre de Physique Théorique (Unité Mixte de Recherche (UMR 6207) du CNRS, et des universités Aix-Marseille I, Aix-Marseille II et du Sud Toulon-Var. Laboratoire affilié la FRUMAM (FR 2291).) , Aix-Marseille Université, CNRS, Luminy - Case 907, F-13288 Marseille cedex 9, France, EU

Corresponding author: leoncini@cpt.univ-mrs.fr

Received: 18 December 2009
Accepted: 26 February 2010

Abstract

The dynamics of many-body Hamiltonian systems with long-range interactions is studied, in the context of the so-called α-HMF model. Building on the analogy with the related mean-field model, we construct stationary states of the α-HMF model for which the spatial organization satisfies a fractional equation. At variance, the microscopic dynamics turns out to be regular and explicitly known. As a consequence, dynamical regularity is achieved at the price of strong spatial complexity, namely a microscopic inhomogeneity which locally displays scale invariance.