Fourier's Law: a Challenge for Theorists

TL;DR

This paper reviews the current understanding and challenges in deriving Fourier's Law from microscopic principles, highlighting the lack of rigorous mathematical derivation despite empirical validation.

Contribution

It provides a selective overview of the theoretical difficulties and gaps in deriving Fourier's Law from microscopic models, emphasizing the current state of knowledge and ignorance.

Findings

Fourier's Law is empirically well tested for fluids and crystals.

No rigorous derivation of Fourier's Law exists for deterministic microscopic models.

The law's derivation involves complex mathematical challenges and open problems.

Abstract

We present a selective overview of the current state of our knowledge (more precisely of ourignorance) regarding the derivation of Fourier's Law, ${\bf J}(\br) =-\kappa {\bf \nabla}T(\br)$; ${\bf J}$ the heat flux, $T$ the temperature and $\kappa$, the heat conductivity. This law is empirically well tested for both fluids and crystals, when the temperature varies slowly on the microscopic scale, with $\kappa$ an intrinsic property which depends only on the system's equilibrium parameters, such as the local temperature and density. There is however at present no rigorous mathematical derivation of Fourier's law and ipso facto of Kubo's formula for $\kappa$, involving integrals over equilibrium time correlations, for any system (or model) with a deterministic, e.g. Hamiltonian, microscopic evolution.