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

TL;DR

This paper explores the connection between discrete and continuum models of heat conduction in a 1D harmonic crystal, showing how the continuum solution emerges as a slow component of the discrete solution over time.

Contribution

It demonstrates that the continuum fundamental solution can be derived as a slow-time component of the exact discrete solution, bridging discrete and continuum heat conduction models.

Findings

Continuum solution is a slow-time asymptotic of the discrete solution.

Discrete and continuum solutions are formally connected through asymptotic analysis.

The continuum fundamental solution can be obtained from the discrete model in the large-time limit.

Abstract

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.