Analytic solution for heat flow through a general harmonic network

We present an analytic expression for the heat current through a general harmonic network coupled with Ohmic reservoirs. We use a method that enables us to express the stationary state of the network in terms of the eigenvectors and eigenvalues of a generalized cubic eigenvalue problem. In this way, we obtain exact formulas for the heat current and the local temperature inside the network. Our method does not rely on the usual assumptions of weak coupling to the environments or on the existence of an infinite cutoff in the environmental spectral densities. We use this method to study nonequilibrium processes without the weak coupling and Markovian approximations. As a first application of our method, we revisit the problem of heat conduction in two- and three-dimensional crystals with binary mass disorder. We complement previous results showing that for small systems the scaling of the heat current with the system size greatly depends on the strength of the interaction between system and reservoirs. This somewhat counterintuitive result seems not to have been noticed before.

Figure 1

Heat flux vs $N$ in 3D disordered crystals for different values of the coupling constant ${\gamma }_0$. The parameters are $k_0=10$ and $\Delta =0.2$.

Figure 2

Heat flux vs $N$ in 2D disordered crystals for different values of the coupling constant ${\gamma }_0$. The parameters are $k_0=10$ and $\Delta =0.2$.

Figure 3

Integration path and poles. Points in the upper half plane are the poles of $g\left(i\omega \right)$ and points in the lower half plane are the poles of $g(-i\omega )$.