Heat conduction in one-dimensional systems of nonlocally coupled harmonic oscillators: mean-field limit

We consider a one-dimensional system of all-to-all harmonically coupled particles, subject to two Langevin thermal baths. This global coupling corresponds to the infinite-range (mean-field) limit of long-range interactions. Additionally, breaking momentum conservation, the particles can be subject to a harmonic on-site potential. Using the non-equilibrium Green operator formalism, we calculate the heat flow and local temperature. We show analytically that, for this globally coupled harmonic system, the heat flux decays as $1/N$, with the system size $N$.