Thermal echo in a finite one-dimensional harmonic crystal

TL;DR

This paper investigates the thermal echo phenomenon in finite one-dimensional harmonic crystals, revealing periodic temperature oscillations with increasing amplitude until a sharp rise, described analytically by Bessel and Airy functions.

Contribution

It introduces the concept of thermal echo in finite crystals and provides an analytical description of temperature dynamics using Bessel and Airy functions.

Findings

Thermal echo occurs periodically with amplitude increases in finite crystals.

The temperature evolution is expressed as an infinite sum of Bessel functions.

In the thermodynamic limit, the behavior is described by the Airy function.

Abstract

An instant homogeneous thermal perturbation in the finite harmonic one-dimensional crystal is studied. Previously it was shown that for the same problem in the infinite crystal the kinetic temperature oscillates with decreasing amplitude described by the Bessel function of the first kind. In the present paper it is shown that in the finite crystal this behavior is observed only until a certain period of time when a sharp increase of the oscillations amplitude is realized. This phenomenon, further refereed to as the thermal echo, is realized periodically, with the period proportional to the crystal size. The amplitude for each subsequent echo is lower than for the previous one. It is obtained analytically that the time-dependance of the kinetic temperature can be described by an infinite sum of the Bessel functions with multiple indexes. It is also shown that the thermal echo in the thermodynamic limit is described by the Airy function.