New analytic solution for the heat flow through a general harmonic network

TL;DR

This paper introduces a new exact analytical method to compute heat flow in harmonic networks coupled with reservoirs, avoiding common approximations, and applies it to study size-dependent heat conduction in disordered crystals.

Contribution

The paper presents a novel analytic approach using eigenvectors and eigenvalues of a cubic eigenvalue problem to determine heat current and temperature in harmonic networks without typical assumptions.

Findings

Exact formulae for heat current and temperature derived.

Heat conduction scaling depends on system-reservoir interaction strength.

Counter-intuitive size dependence observed in small disordered systems.

Abstract

We present a new analytic expression for the heat current through a general harmonic network coupled with Ohmic reservoirs. We use a new 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 formulae for the heat current and the local temperature inside the network. Our method does not rely in 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 non-equilibrium processes without the weak coupling and Markovian approximations. As a first application of our method, we revisit the problem of heat conduction in 2D and 3D 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 counter-intuitive result seems not to have been noticed before.