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We study the hyperbolic scaling limit for a chain of N coupled anharmonicoscillators. The chain is attached to a point on the left and there is a force(tension) $\tau$ acting on the right. In order to provide good ergodicproperties to the system, we perturb the Hamiltonian dynamics with random localexchanges of velocities between the particles, so that momentum and energy arelocally conserved. We prove that in the macroscopic limit the distributions ofthe elongation, momentum and energy, converge to the solution of the Eulersystem of equations, in the smooth regime.

Hydrodynamic Limit for an Hamiltonian System with Boundary Conditions and Conservative Noise