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Unsteady ballistic heat transport in a 1D harmonic crystal due to a source on an isotopic defect

Ekaterina V. Shishkina, Serge N. Gavrilov

TL;DR

This work analyzes unsteady ballistic heat transport in a 1D harmonic lattice with a single isotopic defect by applying stationary-phase asymptotics to integral representations on a moving observation point. It separates the thermal field into slow (ballistic transport) and fast (energy-exchange) components, revealing anti-localization: propagating waves tend to avoid the defect, with stronger effects as the mass difference grows. For a light defect, a non-vanishing localized oscillation traps energy near the defect, while for a heavy defect the propagating part dominates with decay near the defect. The approach yields explicit asymptotic expressions across pass-band and stop-band regimes, recovers Rubin-like results at the defect and front, and offers a framework extendable to nonuniform and polyatomic lattices with potential applications to ballistic transport and Kapitza resistance modeling.

Abstract

In the paper we apply asymptotic technique based on the method of stationary phase and obtain the approximate analytical description of thermal motions caused by a source on an isotopic defect of an arbitrary mass in a 1D harmonic crystal. It is well known that localized oscillation is possible in this system in the case of a light defect. We consider the unsteady heat propagation and obtain formulae, which provide continualization (everywhere excepting a neighbourhood of a defect) and asymptotic uncoupling of the thermal motion into the sum of the slow and fast components. The slow motion is related with ballistic heat transport, whereas the fast motion is energy oscillation related with transformation of the kinetic energy into the potential one and in the opposite direction. To obtain the propagating component of the fast and slow motions we estimate the exact solution in the integral form at a moving point of observation. We demonstrate that the propagating parts of the slow and the fast motions are "anti-localized" near the defect. The physical meaning of the anti-localization is a tendency for the unsteady propagating wave-field to avoid a neighbourhood of a defect. The effect of anti-localization increases with the absolute value of the difference between the alternated mass and the mass of a regular particle, and, therefore, more energy concentrates just behind the leading wave-front of the propagating component. The obtained solution is valid in a wide range of a spatial co-ordinate (i.e., a particle number), everywhere excepting a neighbourhood of the leading wavefront.

Unsteady ballistic heat transport in a 1D harmonic crystal due to a source on an isotopic defect

TL;DR

This work analyzes unsteady ballistic heat transport in a 1D harmonic lattice with a single isotopic defect by applying stationary-phase asymptotics to integral representations on a moving observation point. It separates the thermal field into slow (ballistic transport) and fast (energy-exchange) components, revealing anti-localization: propagating waves tend to avoid the defect, with stronger effects as the mass difference grows. For a light defect, a non-vanishing localized oscillation traps energy near the defect, while for a heavy defect the propagating part dominates with decay near the defect. The approach yields explicit asymptotic expressions across pass-band and stop-band regimes, recovers Rubin-like results at the defect and front, and offers a framework extendable to nonuniform and polyatomic lattices with potential applications to ballistic transport and Kapitza resistance modeling.

Abstract

In the paper we apply asymptotic technique based on the method of stationary phase and obtain the approximate analytical description of thermal motions caused by a source on an isotopic defect of an arbitrary mass in a 1D harmonic crystal. It is well known that localized oscillation is possible in this system in the case of a light defect. We consider the unsteady heat propagation and obtain formulae, which provide continualization (everywhere excepting a neighbourhood of a defect) and asymptotic uncoupling of the thermal motion into the sum of the slow and fast components. The slow motion is related with ballistic heat transport, whereas the fast motion is energy oscillation related with transformation of the kinetic energy into the potential one and in the opposite direction. To obtain the propagating component of the fast and slow motions we estimate the exact solution in the integral form at a moving point of observation. We demonstrate that the propagating parts of the slow and the fast motions are "anti-localized" near the defect. The physical meaning of the anti-localization is a tendency for the unsteady propagating wave-field to avoid a neighbourhood of a defect. The effect of anti-localization increases with the absolute value of the difference between the alternated mass and the mass of a regular particle, and, therefore, more energy concentrates just behind the leading wave-front of the propagating component. The obtained solution is valid in a wide range of a spatial co-ordinate (i.e., a particle number), everywhere excepting a neighbourhood of the leading wavefront.
Paper Structure (22 sections, 1 theorem, 130 equations, 7 figures)

This paper contains 22 sections, 1 theorem, 130 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Nomenclature
  3. The problem formulation
  4. The deterministic problem
  5. The random (thermal) problem
  6. The dispersion relation
  7. The Green function in the frequency domain
  8. The spectral problem for the localized mode
  9. Asymptotics for the fundamental solution $\mathcal{V}_n$ of the deterministic problem
  10. The case $0<|n|<t$: the contribution from the pass-band
  11. The contribution from the stop-band
  12. The case $n=0$: the contribution from the cut-off frequency
  13. The case $|n|=t$
  14. The case $|n|>t$
  15. Summary: the approximate solution
  16. ...and 7 more sections

Key Result

theorem 1

Let $a>0,\ \alpha\geq1,\ \beta>0$, $f(\Omega)\in C^\infty$, $f^{(n)}(a)=0\ \forall n.$ Then

Figures (7)

  • Figure 1: Dispersion relation: the real and the imaginary parts of the wave number $q$ versus frequency $\Omega$
  • Figure 2: Comparing the approximate solution $v_n$ in the form of Eqs. \ref{['v-pass']}--\ref{['v-loc']} and numerical solution (particle velocities versus particle numbers). (a) The case of heavy defect, (b) the case of a light defect. The green crosses correspond to the corresponding exact solution \ref{['Sro-bessel']} for a uniform chain. The red solid line corresponds to the value of $v_n$ given by Eq. \ref{['v-onfront']} on the leading front $|n|=t$
  • Figure 3: Comparing the approximate solution $v_n$ in the form of Eqs. \ref{['v-pass']}--\ref{['v-loc']} and numerical solution (particle velocity versus time). (a) The case of a heavy defect, (b) the case of a light defect
  • Figure 4: Comparing the asymptotics for $\mathcal{V}_0$ in the form of Eq. \ref{['rubin-f']} and numerical solution (particle velocity versus time) in the case of heavy defect
  • Figure 5: Comparing the approximate solution for $\mathcal{T}_n$ in the form of Eq. \ref{['TT-app']}, the corresponding solution \ref{['thermal-fs']} for the kinetic temperatures wherein ${\mathcal{V}}_n$ are found numerically, and the slow motion $\bar{\mathcal{T}}_n$. (a) The case of a heavy defect, (b) the case of a light defect (here $\bar{\mathcal{T}}_n$ is represented as a continuum quantity for $|n|\geq1$)
  • ...and 2 more figures

Theorems & Definitions (14)

  • remark 1
  • remark 2
  • remark 3
  • remark 4
  • remark 5
  • remark 6
  • remark 7
  • remark 8
  • remark 9
  • remark 10
  • ...and 4 more