Small perturbation of a disordered harmonic chain by a noise and an anharmonic potential

We study the thermal properties of a pinned disordered harmonic chain weakly perturbed by a noise and an anharmonic potential. The noise is controlled by a parameter λ → 0, and the anharmonicity by a parameter λ ≤ λ. Let κ be the conductivity of the chain, defined through the Green-Kubo formula. Under suitable hypotheses, we show that κ = O(λ) and, in the absence of anharmonic potential, that κ ∼ λ. This is in sharp contrast with the ordered chain for which κ ∼ 1/λ, and so shows the persistence of localization effects for a non-integrable dynamic