(joint with G. Stoltz)

We introduce a new class of deterministic dynamics, depending on a potential V, with two conserved quantities (say the energy and the volume). We show by numerical simulations that these systems display an anomalous diffusion of energy with a a nonuniversal scaling exponent depending on V. Then, we perturb the system by a energyvolume conserving noise. First, we derive, in the smooth regime, hydrodynamic limits of the perturbed system, which are given by an hyperbolic system of two conservation laws. Secondly, we show that even with the stochastic perturbation, the system has an anomalous diffusion of energy, and we prove it rigorously for V (r) = r2