Fourier's law of heat conduction in a three dimensional harmonic crystal: A retrospection

TL;DR

This paper provides an exact solution to heat conduction in a 3D harmonic crystal, demonstrating that Fourier's law holds in the continuum limit through analytical methods.

Contribution

It offers a rigorous analytical derivation confirming Fourier's law in a 3D harmonic crystal with stochastic boundary conditions.

Findings

Heat transport obeys Fourier's law in the continuum limit.

Exact solution of Langevin's equation for the system.

Validation of Fourier's law in a 3D harmonic crystal.

Abstract

We present an exact solution of the Langevin's equation in the steady state limit in a three dimensional, harmonic crystal of slab geometry whose boundary surfaces along its length are connected to two stochastic, white noise heat baths at different temperatures. We show that the heat trasport obeys the Fourier's law in the continuum limit.