Thermal Transport through a Mesoscopic Weak Link

We calculate the rate of energy flow between two macroscopic bodies, each in thermodynamic equilibrium at a different temperature, and joined by a weak mechanical link. The macroscopic solids are assumed to be electrically insulating, so that thermal energy is carried only by phonons. To leading order in the strength of the weak link, modeled here by a harmonic spring, the thermal current is determined by a product of the local vibrational density-of-states of the two bodies at the points of connection. Our general expression for the thermal current can be regarded as a thermal analog of the well-known formula for the electrical current through a resistive barrier. It is also related to the thermal Landauer formula in the weak-tunneling limit. Implications for heat transport experiments on dielectric quantum point-contacts are discussed.