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 attached to a point on the left and there is a force (tension) τ acting on the right. In order to provide good ergodic properties to the system, we per-turb the Hamiltonian dynamics with random local exchanges of velocities between the particles, so that momentum and energy are locally conserved. We prove that in the macroscopic limit the distributions of the elongation, momentum and energy, converge to the solution of the Euler system of equations, in the smooth regime.