Conductivity and Fourier's law for a system of harmonic oscillators perturbed by a noise conserving energy and momentum.
Résumé
We consider a system of harmonic oscillators perturbed by a stochastic dynamics conserving momentum and energy. We compute the effective thermal conductivity via Green-Kubo formula. In the limit as the size $N$ of the system goes to infinity, conductivity remains finite if a pinning (on site potential) is present or in dimension $d\ge 3$. In the unpinned case conductivity diverges like $N$ in dimension 1 and like $\ln N$ in dimension 2. Then we consider the open system in contact with 2 heat bath at different temperature in the stationary state. We prove that the conductivity of the open system coincides with the Green-Kubo formula, and a corresponding Fourier's law in the cases of finite conductivity. Mathematical complete proofs of these results are in reference [1].