ABSTRACT:

We discuss Fourier's law for a class of anharmonic crystals in d dimensions, d>=1, coupled to both external and internal heat baths of the Ornstein-Uhlenbeck type. The external heat baths are at specified, unequal, external temperatures. The temperatures of the internal baths are determined in a self-consistent way by the requirement that there be no net energy exchange with the system in the non-equilibrium stationary state (NESS). The heat conduction in the NESS is given by the Green-Kubo formula, and it remains bounded as the size of the system goes to infinity. We prove that the conductivity of the infinite system defined by the Green-Kubo formula is convergent. We also prove the existence of a self consistent profile of temperatures and some properties of the entropy production of the corresponding stationary state.