Heat transport in harmonic systems

In this chapter we study heat transport in the simplest model of a solid, namely the harmonic crystal. We consider an open system, consisting of a harmonic crystal coupled to heat reservoirs at different temperatures, and examine properties of the nonequilibrium steady state. Both the average heat current and the current fluctuations are studied. We show that the formalisms, of quantum Langevin equations and non-equilibrium Green’s function techniques, allows one to obtain many exact formal results for the current, as well as fluctuations, in terms of the phonon transmission matrix, in systems in arbitrary dimensions. We then show how these formal results can be used to obtain explicit results in many cases of interest, especially the case of heat transport in mass-disordered harmonic crystals. Apart from throwing light on the question of validity of Fourier’s law, it is explained how the study of the harmonic crystal also provides insight on recent theories of nonequilibrium current fluctuatons.