Heat transport in a three dimensional slab geometry and the temperature profile of Ingen-Hausz's experiment

TL;DR

This study models heat transport in a 3D harmonic crystal slab with stochastic heat baths, demonstrating Fourier's law and an exponential temperature profile consistent with Ingen-Hausz's experimental findings.

Contribution

It provides a theoretical framework showing Fourier's law and exponential temperature profiles in a 3D harmonic crystal slab with stochastic boundary conditions.

Findings

Fourier's law holds in the continuum limit.

Temperature profile exhibits exponential decay from hot to cold end.

Results align with Ingen-Hausz's experimental observations.

Abstract

We study the transport of heat in a three dimensional harmonic crystal of slab geometry whose boundaries and the intermediate surfaces are connected to stochastic, white noise heat baths at different temperatures. Heat baths at the intermediate surfaces are required to fix the initial state of the slab in respect of its surroundings. We allow the flow of energy fluxes between the intermediate surfaces and the attached baths and impose conditions that relate the widths of the Gaussian noises of the intermediate baths. The radiated heat obeys Newton's law of cooling when intermediate baths collectively constitute the environment surrounding the slab. We show that Fourier's law holds in the continuum limit. We obtain an exponentially falling temperature profile from high to low temperature end of the slab and this very nature of the profile was already confirmed by Ingen Hausz's experiment. Temperature profile of similar nature is also obtained in the one dimensional version of this model.