Symmetry properties of heat conduction in inhomogeneous materials

We address the analysis of the energy flow properties of lattice Hamiltonian systems: precisely, we investigate, by using analytical methods, a schematically anharmonic and inhomogeneous model: namely, the chain of oscillators with self-consistent reservoirs. We obtain a symmetric thermal conductivity even for a system with inhomogeneous interparticle interactions or with graded particle masses. Our results show that inhomogeneity in a system obeying Fourier's law does not guarantee an asymmetric heat flow, and they pose the question if only anharmonicity and inhomogeneity are sufficient to assure a significative thermal rectification in a system, as suggested in the recent literature for graded structures.