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Heating of a semi-infinite Hooke chain

Sergei D. Liazhkov

Abstract

We consider unsteady ballistic heat transport in a semi-infinite Hooke chain with a free end and an arbitrary heat source. An analytical description of the evolution of the kinetic temperature is proposed in both discrete (exact) and continuum (approximate) formulations. The continualization of the discrete solution for kinetic temperature is performed through a large-time asymptotic estimate of the fundamental solution of the dynamical problem for the instantly perturbed conservative semi-infinite chain at the fronts of the incident and reflected thermal waves. By analyzing the continuum solution, we observe that any instantaneous heat supply (i.e., a heat pulse) results in the anti-localization of the reflected thermal wave. We demonstrate that sudden point heat supply leads to a transition to a non-equilibrium steady state, which, unexpectedly, may exist even in the non-dissipative case. The results of this paper are expected to provide insight into the continuum description of nanoscale heat transport.

Heating of a semi-infinite Hooke chain

Abstract

We consider unsteady ballistic heat transport in a semi-infinite Hooke chain with a free end and an arbitrary heat source. An analytical description of the evolution of the kinetic temperature is proposed in both discrete (exact) and continuum (approximate) formulations. The continualization of the discrete solution for kinetic temperature is performed through a large-time asymptotic estimate of the fundamental solution of the dynamical problem for the instantly perturbed conservative semi-infinite chain at the fronts of the incident and reflected thermal waves. By analyzing the continuum solution, we observe that any instantaneous heat supply (i.e., a heat pulse) results in the anti-localization of the reflected thermal wave. We demonstrate that sudden point heat supply leads to a transition to a non-equilibrium steady state, which, unexpectedly, may exist even in the non-dissipative case. The results of this paper are expected to provide insight into the continuum description of nanoscale heat transport.

Paper Structure

This paper contains 25 sections, 98 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Nomenclature
  3. Statement of the problem
  4. An exact equation for the kinetic temperature
  5. The image of particle displacements and velocities
  6. The variance matrix
  7. The discrete solution for the kinetic temperature in the semi-infinite chain
  8. Kinetic temperature in the continuum limit
  9. The first approach
  10. The second approach
  11. Case of instantaneous heat perturbation
  12. Point heat pulse. The anti-localization of reflected thermal wave
  13. Arbitrary heat pulse
  14. General case. Example: sudden point heat supply
  15. Discussion of results
  16. ...and 10 more sections

Figures (13)

  • Figure 1: Kinetic temperature in the semi-infinite chain undergone point heat pulse at $j=0$. Comparison of analytical and numerical solutions is shown at $\omega_et=100$. The continuum, symmetrical continuum and discrete solutions (Eqs. (\ref{['FINAL_CONT_FUND']}), (\ref{['CLASSIC_CONT_FUND']}) and (\ref{['FUND_KIN_TEMPR']}) respectively) are demonstrated.
  • Figure 2: Kinetic temperature in the semi-infinite chain undergone point heat pulse at $j=20$. Comparison of analytical and numerical solutions is shown at $\omega_et=100$. The continuum, symmetrical continuum and discrete solutions (Eqs. (\ref{['FINAL_CONT_FUND']}), (\ref{['CLASSIC_CONT_FUND']}) and (\ref{['FUND_KIN_TEMPR']}) respectively) are demonstrated.
  • Figure 3: Evolution in of the kinetic temperature at the boundary after heat pulse at the point $j=20$. The continuum, symmetrical continuum and discrete solutions (Eqs. (\ref{['FINAL_CONT_FUND']}), (\ref{['CLASSIC_CONT_FUND']}) and (\ref{['FUND_KIN_TEMPR']}) respectively) are demonstrated.
  • Figure 4: Kinetic temperature in the semi-infinite chain undergone rectangular perturbation (\ref{['RECT_HEAT_PULSE']}) at $\omega_et=200$. Width of the initial thermal perturbation is shown by the dash-dotted lines. The continuum (Eqs. (\ref{['CONTINUUM_ARB_SOURCE']}), (\ref{['CONTINUUM_T_1']})), discrete-continuum (Eqs. (\ref{['CONTINUUM_ARB_SOURCE']}), (\ref{['BOUNDARY_LAYER_ADD']})), symmetrical continuum (Eq. (\ref{['CLASSIC_CONT_ARBITRARY']})) and discrete (Eq. (\ref{['DS_T_INST']})) solutions are demonstrated.
  • Figure 5: Kinetic temperature in the semi-infinite chain undergone step perturbation (\ref{['STEP_HEAT_PULSE']}) at $\omega_et=200$. Width of the initial thermal perturbation is shown by the dash-dotted line. The continuum (Eqs. (\ref{['CONTINUUM_ARB_SOURCE']}), (\ref{['CONTINUUM_T_1']})), discrete-continuum (Eqs. (\ref{['CONTINUUM_ARB_SOURCE']}), (\ref{['BOUNDARY_LAYER_ADD']})), symmetrical continuum (Eq. (\ref{['CLASSIC_CONT_ARBITRARY']})) and discrete (Eq. (\ref{['DS_T_INST']})) solutions are demonstrated.
  • ...and 8 more figures