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We investigate the macroscopic behavior of the disordered harmonic chain ofoscillators, through energy diffusion. The Hamiltonian dynamics of the systemis perturbed by a degenerate conservative noise. After rescaling space and timediffusively, we prove that energy fluctuations in equilibrium evolve accordingto a linear heat equation. The diffusion coefficient is obtained from thenon-gradient Varadhan's approach, and is equivalently defined through theGreen-Kubo formula. Since the perturbation is very degenerate and the symmetricpart of the generator does not have a spectral gap, the standard non-gradientmethod is reviewed under new perspectives.

Equilibrium fluctuations for the disordered harmonic chain perturbed by an energy conserving noise