Superdiffusion and out-of-equilibrium chaotic dynamics with many degrees of freedoms

We study the link between relaxation to the equilibrium and anomalous super diffusive motion in a classical N-body Hamiltonian system with long-range i nteraction showing a second-order phase transition in the canonical ensembl e. Anomalous diffusion is observed only in a transient out-of-equilibrium r egime and for a small range of energy, below the critical one. Superdiffusi on is due to Levy walks of single particles and is checked independently th rough the second moment of the Diffusion becomes normal at distribution, po wer spectra, trapping, and walking time probabilities. equilibrium, after a relaxation time which diverges with N.