Heat transfer in a one-dimensional harmonic crystal in a viscous environment subjected to an external heat supply

TL;DR

This paper derives an exact analytical solution for unsteady heat transfer in a viscous one-dimensional harmonic crystal with external heat supply, highlighting ballistic heat transfer and temperature profiles described by Macdonald functions.

Contribution

It introduces a novel analytical approach to model unsteady heat transfer in a viscous harmonic crystal, including derivation of covariance equations and exact solutions.

Findings

Exact solution for unsteady ballistic heat transfer obtained.

Stationary temperature profile described by Macdonald function.

Comparison with classical heat equation results provided.

Abstract

We consider unsteady heat transfer in a one-dimensional harmonic crystal surrounded by a viscous environment and subjected to an external heat supply. The basic equations for the crystal particles are stated in the form of a system of stochastic differential equations. We perform a continualization procedure and derive an infinite set of linear partial differential equations for covariance variables. An exact analytic solution describing unsteady ballistic heat transfer in the crystal is obtained. It is shown that the stationary spatial profile of the kinetic temperature caused by a point source of heat supply of constant intensity is described by the Macdonald function of zero order. A comparison with the results obtained in the framework of the classical heat equation is presented. We expect that the results obtained in the paper can be verified by experiments with laser excitation of low-dimensional nanostructures.