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We consider one-dimensional systems of all-to-all harmonically coupledparticles with arbitrary masses, subject to two Langevin thermal baths. Thecouplings correspond to the mean-field limit of long-range interactions.Additionally, the particles can be subject to a harmonic on-site potential tobreak momentum conservation. Using the non-equilibrium Green operatorformalism, we calculate the transmittance, the heat flow and localtemperatures, for arbitrary configurations of masses. For identical masses, weshow analytically that, the heat flux decays with the system size $N$, as$1/N$, regardless of the conservation or not of the momentum, and of theintroduction or not of a Kac factor. These results describe in good agreementthe thermal behavior of systems with small heterogeneity in the masses.

Heat conduction in chains of non-locally coupled harmonic oscillators: mean-field limit