Mesoscopic thermal transport through a weak link

We consider mesoscopic thermal transport between two bulk dielectrics joined by a narrow wire or weak mechanical link. In the "tunneling" regime where the phonon transmission probability through the link is small and the thermal conductance is much less than πk_B² T/6ℏ, the thermal current is determined by a product of the local vibrational spectral densities of the two bodies. We derive an expression for the thermal current that is a thermal analog of the well-known formula for the electrical current through a tunneling barrier.