Heat transport in ordered harmonic lattices

TL;DR

This paper provides an exact analysis of heat transport in ordered harmonic lattices, covering classical and quantum regimes, with explicit formulas for heat current and insights into temperature dependence and higher-dimensional cases.

Contribution

It introduces a unified approach to exactly solve heat conduction in harmonic lattices, extending previous results to quantum regimes and higher dimensions.

Findings

Exact formula for classical heat current in the thermodynamic limit

Quantum case analysis showing temperature dependence of heat transport

Discussion of heat conduction in higher-dimensional lattices

Abstract

We consider heat conduction across an ordered oscillator chain with harmonic interparticle interactions and also onsite harmonic potentials. The onsite spring constant is the same for all sites excepting the boundary sites. The chain is connected to Ohmic heat reservoirs at different temperatures. We use an approach following from a direct solution of the Langevin equations of motion. This works both in the classical and quantum regimes. In the classical case we obtain an exact formula for the heat current in the limit of system size N to infinity. In special cases this reduces to earlier results obtained by Rieder, Lebowitz and Lieb and by Nakazawa. We also obtain results for the quantum mechanical case where we study the temperature dependence of the heat current. We briefly discuss results in higher dimensions.