Thermal Transport in Phononic Cayley Tree Networks

We analytically investigate the heat current $I$ and its thermal fluctuations $Δ$ in a branching network without loops (Cayley tree). The network consists of two type of harmonic masses: vertex masses $M$ placed at the branching points where phononic scattering occurs and masses $m$ at the bonds between branching points where phonon propagation take place. The network is coupled to thermal reservoirs consisting of one-dimensional harmonic chains of coupled masses $m$. Due to impedance missmatching phenomena, both ${\cal I}$ and $Δ$, are non-monotonic functions of the mass ratio $μ=M/m$. In particular, there are cases where they are strictly zero below some critical value $μ^*$.

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