Anomalous heat transport in one dimensional systems: a description using non-local fractional-type diffusion equation

TL;DR

This paper reviews how heat transport in one-dimensional systems deviates from Fourier's law, instead being described by a non-local fractional diffusion equation, supported by various theoretical approaches and recent progress.

Contribution

It provides a comprehensive overview of the theoretical frameworks leading to non-local fractional diffusion models for heat transport in 1D systems.

Findings

Heat transport in 1D systems is non-local and cannot be described by Fourier's law.

The heat diffusion equation is replaced by a fractional-type non-local diffusion equation.

Recent progress supports the fractional diffusion framework for anomalous heat transport.

Abstract

It has been observed in many numerical simulations, experiments and from various theoretical treatments that heat transport in one-dimensional systems of interacting particles cannot be described by the phenomenological Fourier's law. The picture that has emerged from studies over the last few years is that Fourier's law gets replaced by a spatially non-local linear equation wherein the current at a point gets contributions from the temperature gradients in other parts of the system. Correspondingly the usual heat diffusion equation gets replaced by a non-local fractional-type diffusion equation. In this review, we describe the various theoretical approaches which lead to this framework and also discuss recent progress on this problem.