  We consider a one-dimensional chain of coupled oscillators in contact at both
ends with heat baths at different temperatures, and subject to an external
force at one end. The Hamiltonian dynamics in the bulk is perturbed by random
exchanges of the neighbouring momenta such that the energy is locally
conserved. We prove that in the stationary state the energy and the volume
stretch profiles, in large scale limit, converge to the solutions of a
diffusive system with Dirichlet boundary conditions. As a consequence the
macroscopic temperature stationary profile presents a maximum inside the chain
higher than the thermostats temperatures, as well as the possibility of uphill
diffusion (energy current against the temperature gradient). Finally, we are
also able to derive the non-stationary macroscopic coupled diffusive equations
followed by the energy and volume stretch profiles.
