Hydrodynamical limit for a Hamiltonian system with weak noise

We study the hyperbolic scaling limit for a chain of N coupled anharmonic oscil-lators. The chain is open and with the following adiabatic boundary conditions: it is attached to a wall on the left and there is a force (tension) τ acting on the right. In order to provide the system of the good ergodic properties, we perturb the Hamiltonian dynamics with random local exchanges of velocities between the particles, so that mo-mentum and energy are locally conserved. We prove that in the macroscopic limit the distribution of the density of particles, momentum and energy converge to the solution of the Euler equations, in the smooth regime of them.