Discrete and continuum fundamental solutions describing heat conduction in 1D harmonic crystal: Discrete-to-continuum limit and slow-and-fast motions decoupling

In the recent paper by Sokolov et al. (Int. J. of Heat and Mass Transfer 176, 2021, 121442) ballistic heat propagation in 1D harmonic crystal is considered and the properties of the exact discrete solution and the solution of the ballistic heat equation introduced by Krivtsov are numerically compared. The aim of this note is to demonstrate that the latter continuum fundamental solution can be formally obtained as the slow time-varying component of the large-time asymptotics for the exact discrete solution on a moving point of observation.Comment: 11 pages, 1 figur

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

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.Comment: 12 pages, 5 figure