Microscopic models for heat conduction

The derivation of Fourier's law for heat conduction from a microscopic Hamiltonian dynamics is still an open problem from a fundamental point of view. In this scenario the study of hybrid models have the advantage to be mathematically tractable, but still related to realistic models. In this talk I will present an integral representation for the correlation functions of harmonic and anharmonic chains of oscillators with self-consistent reservoirs. This integral formalism is suitable for the study of the correlations time decay, as well as the correlations behavior in the steady state.