Nonequilibrium statistical mechanics of anharmonic crystals with self-consistent stochastic reservoirs

We consider a d-dimensional crystal with an arbitrary harmonic interaction and an anharmonic on-site potential, with a stochastic Langevin heat bath at each site. We develop an integral formalism for the correlation functions that is suitable for the study of their relaxation (time decay) as well as their behavior in space. Furthermore, in a perturbative analysis, for the one-dimensional system with weak coupling between the sites and small quartic anharmonicity, we investigate the steady state and show that Fourier's law holds. We also obtain an expression for the thermal conductivity (for arbitrary next-neighbor interactions) and give the temperature profile in the steady state.