Wigner equations for phonons transport and quantum heat flux

TL;DR

This paper derives Wigner equations for phonons in various solids, including 2D materials like graphene, and develops a quantum energy transport model with thermal conductivity corrections.

Contribution

It introduces a quantum Wigner framework for phonon transport, including collision modeling and a quantum maximum entropy closure, applicable to diverse solid materials.

Findings

Explicit quantum-corrected thermal conductivity formula

Wigner equations valid for 2D and 3D solids

Phonon transport modeled with BGK collision operators

Abstract

Starting from the quantum Liouville equation for the density operator and applying the Weyl quantization, Wigner equations for the longitudinal and transversal optical and acoustic phonons are deduced. The equations are valid for any solid, including 2D crystals like graphene. With the use of Moyal's calculus and its properties the pseudo-differential operators are expanded up to the second order in $\hbar$. The phonon-phonon collision operators are modelled in a BGK form and describe the relaxation of the Wigner functions to a local equilibrium function, depending on a local equilibrium temperature which is definite according to \cite{MaRo1}. An energy transport model is obtained by using the moment method with closures based on a quantum version of the Maximum Entropy Principle. An explicit form of the thermal conductivity with quantum correction is obtained under a suitable scaling.