Stationary nonequilibrium properties for a heat conduction model

We consider a stochastic heat conduction model for solids composed of N interacting atoms. The system is in contact with two heat baths at different temperatures Tl and Tr. The bulk dynamics conserves two quantities: the energy and the deformation between atoms. If Tl not equal to Tr, a heat flux occurs in the system. For large N, the system adopts a linear temperature profile between Tl and Tr. We establish the hydrodynamic limit for the two conserved quantities. We introduce the fluctuation field of the energy and of the deformation in the nonequilibrium steady state. As N goes to infinity, we show that this field converges to a Gaussian field and we compute the limiting covariance matrix. The main contribution of the paper is the study of large deviations for the temperature profile in the nonequilibrium stationary state. A variational formula for the rate function is derived following the recent macroscopic fluctuation theory of Bertini [J. Stat. Phys. 107, 635 (2002); Math. Phys., Anal. Geom. 6, 231 (2003); J. Stat. Phys. 121, 843 (2005)].