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We study the heat transport due to phonons in nanomechanical structures usinga phase space representation of non-equilibrium Green's functions. Thisrepresentation accounts for the atomic degrees of freedom making itparticularly suited for the description of small (molecular) junctions systems.We show that for the steady state limit our formalism correctly recovers theheuristic Landauer-like heat conductance for a quantum coherent molecularsystem coupled to thermal reservoirs. We find general expressions for thenon-stationary heat current due to an external periodic drive. In both cases wediscuss the quantum thermodynamic properties of the systems. We apply ourformalism to the case of a diatomic molecular junction.

Phononic heat transport in nanomechanical structures: steady-state and pumping