Fourier's Law: insight from a simple derivation

TL;DR

This paper presents a simple analytical model demonstrating how Fourier's law emerges in a one-dimensional quantum system, highlighting the transition from ballistic to diffusive regimes and the role of dephasing in establishing classical heat conduction.

Contribution

It provides a straightforward derivation linking thermal length-scale to the onset of Fourier's law in quantum systems, including the effects of dephasing.

Findings

Crossover from ballistic to diffusive regimes characterized by a thermal length-scale

Dephasing induces classical Fourier's law in quantum systems

Analytical demonstration of Fourier's law emergence in a simple quantum model

Abstract

The onset of Fourier's law in a one-dimensional quantum system is addressed via a simple model of weakly coupled quantum systems in contact with thermal baths at their edges. Using analytical arguments we show that the crossover from the ballistic (invalid Fourier's law) to diffusive (valid Fourier's law) regimes is characterized by a thermal length-scale, which is directly related to the profile of the local temperature. In the same vein, dephasing is shown to give rise to a classical Fourier's law, similarly to the onset of Ohm's law in mesoscopic conductors.