Phonon Boltzmann equation non-local in space and time: the partial failure of the generalized Fourier law

TL;DR

This paper clarifies the solution to the non-local phonon Boltzmann equation, highlighting the emergence of a new non-Fourier term in the thermal response that challenges the classical Fourier law.

Contribution

It provides a detailed analysis of the non-local solution to the phonon Boltzmann equation and introduces a new non-Fourier term in the thermal response function.

Findings

Identification of a new non-Fourier term $oldsymbol{B}$ in the thermal response.

Discussion of the modified thermal distributor in non-local heat conduction.

Clarification of the solution method used by Hua and Lindsay.

Abstract

The purpose of this note is to clarify the solution of the non-local Peierls Boltzmann equation found by Hua and Lindsay (Phys. Rev. B 102, 104310 (2020)). They used methods of Cepellotti and Marzari. The response function "thermal distributor" is discussed. The new, "non-Fourier" term $\vec{B}$ [$\vec{J}_{\rm th}=-\kappa \vec{\nabla}T +\vec{B}]$ that occurs in non-local situations, gives rise also to a new term in the thermal distributor.