Heat flux operator, current conservation and the formal Fourier's law

TL;DR

This paper revisits heat current operator definitions to establish a form satisfying the continuity equation in 1D systems, enabling analysis of energy flow and deriving Fourier's Law from fundamental principles.

Contribution

It introduces a generalized heat current operator that ensures current conservation and naturally leads to Fourier's Law in linear and hybrid systems.

Findings

A new heat current operator satisfying the continuity equation.

Conditions for current conservation in general-statistics systems.

Discrete Fourier's Law naturally emerges from the new definition.

Abstract

By revisiting previous definitions of the heat current operator, we show that one can define a heat current operator that satisfies the continuity equation for a general Hamiltonian in one dimension. This expression is useful for studying electronic, phononic and photonic energy flow in linear systems and in hybrid structures. The definition allows us to deduce the necessary conditions that result in current conservation for general-statistics systems. The discrete form of the Fourier's Law of heat conduction naturally emerges in the present definition.