Quantum master equation approach to heat transport in dielectrics and semiconductors

TL;DR

This paper derives a quantum master equation approach to model heat transport in dielectrics and semiconductors, showing how classical Fourier law emerges from quantum dynamics in the continuum limit.

Contribution

It introduces a quantum Markovian master equation framework for heat conduction in nonmetals, connecting quantum phonon dynamics with classical heat transfer laws.

Findings

Fourier law naturally emerges in the continuum limit.

Heat conductivity is explicitly derived from quantum dynamics.

High temperature limit reproduces classical heat conduction equation.

Abstract

We report on the derivation of the heat transport equation for nonmetals using a quantum Markovian master equation in Lindblad form. We first establish the equations of motion describing the time variation of the on-site energy of atoms in a one dimensional periodic chain that is coupled to a heat reservoir. In the continuum limit, the Fourier law of heat conduction naturally emerges, and the heat conductivity is explicitly obtained. It is found that the effect of the heat reservoir on the lattice is described by a heat source density that depends on the diffusion coefficients of the atoms. We show that the Markovian dynamics is equivalent to the long wavelength approximation for phonons, which is typical for the case of elastic solids. The high temperature limit is shown to reproduce the classical heat conduction equation.