We consider a purely harmonic chain of oscillators which is perturbed by a stochastic noise. Under this perturbation, the system exhibits two conserved quantities: the volume and the energy. At the level of the hydrodynamic limit, under diffusive scaling, we can show that depending on the strength of the Hamiltonian dynamics, energy and volume evolve according to either a system of autonomous heat equations or a non-linear system of coupled parabolic equations. Moreover, for general initial measures, under diffusive scaling, we can characterize the non-equilibrium volume fluctuations. The proofs are based on precise bounds on the two-point volume correlation function and a uniform fourth-moment estimate, and we will give a talk particularly on these estimates and the result for the non-equilibrium fluctuation. This work is based on a joint work with Patr\’{i}cia Gon\c{c}alves (Instituto Superior T\’{e}cnico) and Jo\~{a}o Pedro Mangi (IMPA).