Stationary states and fractional dynamics in systems with long range interactions

Tineke L. van den Berg (1) , Duccio Fanelli (2) , Xavier Leoncini (3, 4)

1 IM2NP - Institut des Matériaux, de Microélectronique et des Nanosciences de Provence
2 Dipartimento di Energetica “Sergio Stecco”
3 CPT - Ex E7 Systèmes dynamiques et théorie ergodique
4 CPT - Centre de Physique Théorique - UMR 7332

Dynamics of many-body Hamiltonian systems with long range interactions is studied, in the context of the so called $\alpha-$HMF model. Building on the analogy with the related mean field model, we construct stationary states of the $\alpha-$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.

Mots clés: Long range systems; Fractional calculus; Hamiltonian Chaos

Domaines: Mécanique statistique [cond-mat.stat-mech]; Dynamique Chaotique [nlin.CD]

Tineke L. van den Berg, Duccio Fanelli, Xavier Leoncini. Stationary states and fractional dynamics in systems with long range interactions. EPL - Europhysics Letters, 2010, 89 (5), pp.50010. ⟨10.1209/0295-5075/89/50010⟩. ⟨hal-00441021v2⟩