Thermal Conductivity for a Momentum Conserving Model

We introduce a model whose thermal conductivity diverges in dimension 1 and 2, while it remains finite in dimension 3. We consider a system of oscillators perturbed by a stochastic dynamics conserving momentum and energy. We compute thermal conductivity via Green-Kubo formula. In the harmonic case we compute the current-current time correlation function, that decay like $t^{-d/2}$ in the unpinned case and like $t^{-d/2-1}$ if a on-site harmonic potential is present. This implies a finite conductivity in $d\ge 3$ or in pinned cases, and we compute it explicitly. For general anharmonic strictly convex interactions we prove some upper bounds for the conductivity that behave qualitatively as in the harmonic cases.