Heat transport in harmonic lattices

TL;DR

This paper analyzes heat transport in harmonic lattices connected to heat reservoirs, deriving steady state properties and thermal conductivity using quantum Langevin equations, with results aligning with known classical limits.

Contribution

It introduces a quantum Langevin equation approach to study non-equilibrium steady states in harmonic lattices, connecting quantum and classical results.

Findings

Derived steady state heat currents and correlations.

Obtained a temperature-dependent thermal conductivity.

Reproduced known classical high-temperature limit.

Abstract

We work out the non-equilibrium steady state properties of a harmonic lattice which is connected to heat reservoirs at different temperatures. The heat reservoirs are themselves modeled as harmonic systems. Our approach is to write quantum Langevin equations for the system and solve these to obtain steady state properties such as currents and other second moments involving the position and momentum operators. The resulting expressions will be seen to be similar in form to results obtained for electronic transport using the non-equilibrium Green's function formalism. As an application of the formalism we discuss heat conduction in a harmonic chain connected to self-consistent reservoirs. We obtain a temperature dependent thermal conductivity which, in the high-temperature classical limit, reproduces the exact result on this model obtained recently by Bonetto, Lebowitz and Lukkarinen.