Chaotic dynamics and superdiffusion in a Hamiltonian system with many degrees of freedom

We discuss recent results obtained for the Hamiltonian mean field model. Th e model describes a system of N fully coupled particles in one dimension an d shows a second-order phase transition from a clustered phase to a homogen eous one when the energy is increased. Strong chaos is found in corresponde nce to the critical point on top of a weak chaotic regime which characteriz es the motion at low energies. For a small region around the critical point , we find anomalous (enhanced) diffusion and Levy walks in a transient temp oral regime before the system relaxes to equilibrium. (C) 2000 Published by Elsevier Science B.V. All rights reserved.