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Unsteady thermal transport in an instantly heated semi-infinite free end Hooke chain

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Abstract

We consider unsteady ballistic heat transport in a semi-infinite Hooke chain with free end and arbitrary initial temperature profile. An analytical description of the evolution of the kinetic temperature is proposed in both discrete (exact) and continuum (approximate) formulations. By comparison of the discrete and continuum descriptions of kinetic temperature field, we reveal some restrictions to the latter. Specifically, the far-field kinetic temperature is well described by the continuum solution, which, however, deviates near and at the free end (boundary). We show analytically that, after thermal wave reflects from the boundary, the discrete solution for the kinetic temperature undergoes a jump near the free end. A comparison of the descriptions of heat propagation in the semi-infinite and infinite Hooke chains is presented. Results of the current paper are expected to provide insight into non-stationary heat transport in the semi-infinite lattices.

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Notes

  1. i.e., changing in dependence of the particle number.

  2. See also English translation of the Schrödinger article [23].

  3. This is the monoatomic harmonic chain of identical particles, connected by the linear identical springs, see [24].

  4. Ends of chain are connected with the fixed points by linear stiffness springs.

  5. Definition of the Hooke chain is given in Sect. 1 and in [24].

  6. The statement of problem corresponds to experimental heating of the crystal by the ultrashort laser pulse. Since the expression for heat flux in the Hooke chain contains covariances of displacements and velocities (see, e.g., [3, 34]), initial zero field of initial displacements means zero initial heat fluxes.

  7. We determine the kinetic temperature by its statistical definition (see, e.g., chapter 3, Sect. 29 in [35]). Unambiguous definition of the temperature for systems far from equilibrium is still unresolved fundamental problem (see, e.g., [36, 37]). In this paper, we calculate the kinetic temperature as average of kinetic energy over realizations, because it has simple physical meaning. Discussion of the ergodicity remains out of frameworks of this study.

  8. A macroscale can be interpreted as a scale of the order of the length of chain.

  9. Here, evenness property of the Dirac delta function is used.

  10. The type of the initial temperature perturbation contradicts with the assumption, made in Sect. 3.1. However, as it is shown below, the continuum solution has the same physical meaning, which is characteristic for one at arbitrary initial temperature profile.

  11. Simulations are performed for the chain with 500 particles.

  12. Random numbers \(\rho _n\) are uniformly distributed in the segment \([-\sqrt{3}; \sqrt{3}]\), which satisfies condition (4).

  13. As in the case of rectangular perturbation, the continuum solution decays also as 1/t.

  14. i.e., the function \(T(t)\Big \vert _0\) at width of the initial thermal perturbation \(L+\Delta L\) is equal to \(T\left( \frac{L+\Delta L}{L}t\right) \Big \vert _0\) at the width L.

  15. Therefore, before reflection from the boundary, the continuum solution for the kinetic temperature in the semi-infinite chain obeys solution (40).

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Acknowledgements

The work was supported by the Russian Science Foundation (Grant No. 21-71-10129). The author is deeply grateful to V.A. Kuzkin, A.M. Krivtsov, S.N. Gavrilov, E.V. Shishkina, A.A. Sokolov, A.S. Murachev, N.M. Bessonov, S.A. Rukolaine and E.F. Grekova for useful and stimulating discussions and to anonymous referees for the valuable comments.

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  1. Peter the Great Saint-Petersburg Polytechnic University, Politekhnicheskaya str. 29, St. Petersburg, 195251, Russia

    Sergei D. Liazhkov

  2. Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences, Bol’shoy pr. 61, V.O., St. Petersburg, 199178, Russia

    Sergei D. Liazhkov

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Appendices

A Derivation of the discrete analogue of fundamental solution

Here, we derive the discrete fundamental solution, namely \(g_{n,j_s}^S(\Delta N)\). Expanding a product of cosines in (17) yields

$$\begin{aligned} \begin{array}{l} \displaystyle S_{nj}=\frac{1}{16\pi ^2}\iint _{-\pi }^\pi \Bigg [\cos ({(n+j+1)(\theta _1+\theta _2)})+ \cos ({(n+j+1)(\theta _1-\theta _2)}) \\ \quad \quad \quad \; +\displaystyle \cos ({(n-j)(\theta _1+\theta _2)})+\cos ({(n-j)(\theta _1-\theta _2)})+\displaystyle \cos {\left( \frac{(2j+1)(\theta _1+\theta _2)}{2}-\frac{(2n+1)(\theta _1-\theta _2)}{2}\right) }\\ \quad \quad \quad \; +\displaystyle \cos {\left( \frac{(2j+1)(\theta _1+\theta _2)}{2}+\frac{(2n+1)(\theta _1-\theta _2)}{2}\right) }+\cos {\left( \frac{(2n+1)(\theta _1+\theta _2)}{2}-\frac{(2j+1)(\theta _1-\theta _2)}{2}\right) }\\ \quad \quad \quad \; +\displaystyle \cos {\left( \frac{(2n+1)(\theta _1+\theta _2)}{2}+\frac{(2j+1)(\theta _1-\theta _2)}{2}\right) }\Bigg ] \cos \left( {(\omega (\theta _1)-\omega (\theta _2)) t}\right) \textrm{d}\theta _1 \textrm{d}\theta _2. \end{array}\nonumber \\ \end{aligned}$$
(42)

Therefore, the expression for \(g_{n,j_s}^S(\Delta N)\) can be rewritten as a sum of the following eight terms:

$$\begin{aligned} \displaystyle g_{n,j_s}^S(\Delta N)= & {} \frac{1}{16\pi ^2 a}\iint _{-\pi }^\pi \cos {\left( (\omega (\theta _1)-\omega (\theta _2)) t\right) }\sum _{i=1}^8 \varphi _i(\theta _1,\theta _2) \textrm{d}\theta _1 \textrm{d}\theta _2,\nonumber \\ \displaystyle \varphi _1(\theta _1,\theta _2)= & {} \frac{1}{2\Delta N}\sum _{j=j_s-\Delta N+1}^{j_s+\Delta N}\cos {((n+j+1)(\theta _1+\theta _2))}\nonumber \\= & {} \displaystyle \frac{1}{2\Delta N}\cos {\Bigl (\frac{3(\theta _1+\theta _2)}{2}+(\theta _1+\theta _2)(n+j_s)\Bigr )}\frac{\sin {\left( (\theta _1+\theta _2)\Delta N \right) }}{\sin {\left( \frac{\theta _1+\theta _2}{2}\right) }},\nonumber \\ \displaystyle \varphi _2(\theta _1,\theta _2)= & {} \frac{1}{2\Delta N}\sum _{j=j_s-\Delta N+1}^{j_s+\Delta N}\cos {((n+j+1)\Delta \theta )}\nonumber \\= & {} \displaystyle \frac{1}{2\Delta N}\cos {\left( \left( n+j_s+\frac{3}{2}\right) \Delta \theta \right) }\frac{\sin {(\Delta N \Delta \theta )}}{\sin {\frac{\Delta \theta }{2}}},\nonumber \\ \displaystyle \varphi _3(\theta _1,\theta _2)= & {} \frac{1}{2\Delta N}\sum _{j=j_s-\Delta N+1}^{j_s+\Delta N}\cos {((n-j)(\theta _1+\theta _2))}\nonumber \\= & {} \displaystyle \frac{1}{2\Delta N}\cos {\left( (\theta _1+\theta _2)\left( \frac{1}{2}+j_s-n\right) \right) } \displaystyle \frac{\sin {\left( (\theta _1+\theta _2)\Delta N \right) }}{\sin {\left( \frac{\theta _1+\theta _2}{2}\right) }},\nonumber \\ \displaystyle \varphi _4(\theta _1,\theta _2)= & {} \frac{1}{2\Delta N}\sum _{j=j_s-\Delta N+1}^{j_s+\Delta N}\cos {((n-j)\Delta \theta )}\nonumber \\= & {} \displaystyle \frac{1}{2\Delta N}\cos {\left( \Delta \theta \left( \frac{1}{2}+j_s-n\right) \right) }\frac{\sin {(\Delta N \Delta \theta )}}{\sin {\frac{\Delta \theta }{2}}},\nonumber \\ \displaystyle \varphi _5(\theta _1,\theta _2)= & {} \frac{1}{2\Delta N}\sum _{j=j_s-\Delta N+1}^{j_s+\Delta N}\cos {\left( \frac{(2j+1)(\theta _1+\theta _2)}{2}-\frac{(2n+1)(\theta _1-\theta _2)}{2}\right) }\nonumber \\= & {} \displaystyle \frac{1}{2\Delta N}\cos {\left( \theta _1\left( \frac{1}{2}+j_s-n\right) +\theta _2\left( \frac{3}{2}+j_s+n\right) \right) } \frac{\sin {\left( (\theta _1+\theta _2)\Delta N \right) }}{\sin {\left( \frac{\theta _1+\theta _2}{2}\right) }},\nonumber \\ \displaystyle \varphi _6(\theta _1,\theta _2)= & {} \frac{1}{2\Delta N}\sum _{j=j_s-\Delta N+1}^{j_s+\Delta N}\cos {\left( \frac{(2j+1)(\theta _1+\theta _2)}{2}+\frac{(2n+1)(\theta _1-\theta _2)}{2}\right) }\nonumber \\= & {} \displaystyle \frac{1}{2\Delta N}\cos {\left( \theta _2\left( \frac{1}{2}+j_s-n\right) +\theta _1\left( \frac{3}{2}+j_s+n\right) \right) } \frac{\sin {\left( (\theta _1+\theta _2)\Delta N \right) }}{\sin {\left( \frac{\theta _1+\theta _2}{2}\right) }},\nonumber \\ \displaystyle \varphi _7(\theta _1,\theta _2)= & {} \frac{1}{2\Delta N}\sum _{j=j_s-\Delta N+1}^{j_s+\Delta N}\cos {\left( \frac{(2n+1)(\theta _1+\theta _2)}{2}-\frac{(2j+1)(\theta _1-\theta _2)}{2}\right) }\nonumber \\= & {} \displaystyle \frac{1}{2\Delta N}\cos {\left( \theta _1\left( \frac{1}{2}+j_s-n\right) -\theta _2\left( \frac{3}{2}+j_s+n\right) \right) } \frac{\sin {\left( \Delta N\Delta \theta \right) }}{\sin {\left( \frac{\Delta \theta }{2}\right) }},\nonumber \\ \displaystyle \varphi _8(\theta _1,\theta _2)= & {} \frac{1}{2\Delta N}\sum _{j=j_s-\Delta N+1}^{j_s+\Delta N}\cos {\left( \frac{(2n+1)(\theta _1+\theta _2)}{2}+\frac{(2j+1)(\theta _1-\theta _2)}{2}\right) }\nonumber \\= & {} \displaystyle \frac{1}{2\Delta N}\cos {\left( \theta _2\left( \frac{1}{2}+j_s-n\right) -\theta _1\left( \frac{3}{2}+j_s+n\right) \right) } \frac{\sin {\left( \Delta N\Delta \theta \right) }}{\sin {\left( \frac{\Delta \theta }{2}\right) }}, \quad \Delta \theta =\theta _1-\theta _2. \end{aligned}$$
(43)

We rewrite the components \(\varphi _2, \varphi _4, \varphi _7, \varphi _8\), containing the difference of wave numbers \(\Delta \theta \) as follows:

$$\begin{aligned} \begin{array}{l} \displaystyle \varphi _2=\frac{\Delta {\theta }}{2}\Bigg [\frac{\cos {\frac{3\Delta \theta }{2}}}{\sin {\frac{\Delta \theta }{2}}}\left( \cos {\left( (n+j_s)\Delta \theta \right) }\right) -\frac{\sin {\frac{3\Delta \theta }{2}}}{\sin {\frac{\Delta \theta }{2}}}\left( \sin {\left( (n+j_s)\Delta \theta \right) }\right) \Bigg ]\textrm{sinc}(\Delta N \Delta \theta ),\\ \normalsize \displaystyle \textrm{sinc}(x)=\frac{\sin {x}}{x},\\ \small \displaystyle \varphi _4=\frac{\Delta \theta }{2}\Bigg [\cot {\frac{\Delta \theta }{2}}\cos {((n-j_s)\Delta \theta )}+\sin {((n-j_s)\Delta \theta )}\Bigg ]\textrm{sinc}(\Delta N \Delta \theta ),\\ \displaystyle \varphi _7=\frac{\Delta \theta }{2}\Bigg [\frac{\cos {\frac{3\Delta \theta }{2}}}{\sin {\frac{\Delta \theta }{2}}} \cos {(\Delta \theta (n+j_s)-\theta _1(2n+1))}-\frac{\sin {\frac{3\Delta \theta }{2}}}{\sin {\frac{\Delta \theta }{2}}}\sin {(\Delta \theta (n+j_s)-\theta _1(2n+1))}\Bigg ]\\ \textrm{sinc}(\Delta N \Delta \theta ),\\ \displaystyle \varphi _8=\frac{\Delta \theta }{2}\Bigg [\cot {\frac{\Delta \theta }{2}} \cos {(\theta _1(2n+1)-\Delta \theta (n-j_s))}-\sin {(\theta _1(2n+1)-\Delta \theta (n-j_s))}\Bigg ]\\ \textrm{sinc}(\Delta N \Delta \theta ). \end{array} \end{aligned}$$
(44)

Equation  (44) can be simplified due to our assumptions about continualization (see Sect. 3.1). For \(\Delta N \gg 1\), the function \(\textrm{sinc}(x)\) is equal to 1 if \(\Delta \theta \) is zero and fast tends to zero if \(\Delta \theta \) is not equal to zero. Therefore, the main contribution to the function \(g_{n,j_s}^S(\Delta N)\) comes from two close wavenumbers \(\theta _1, \theta _2\). Therefore, in the limit cases of \(\Delta \theta \rightarrow 0\) and \(\Delta N~\gg 1\), we have

$$\begin{aligned} \begin{array}{l} \displaystyle \varphi _2=\cos {((n+j_s)\Delta \theta )}\textrm{sinc}(\Delta N \Delta \theta )+O\left( \frac{1}{\Delta N}\right) ,\\ \displaystyle \varphi _4=\cos {((n-j_s)\Delta \theta )}\textrm{sinc}(\Delta N \Delta \theta )+O\left( \frac{1}{\Delta N}\right) ,\\ \displaystyle \varphi _7=\cos ({\theta _1(2n+1)-(n+j_s)\Delta \theta })\textrm{sinc}(\Delta N \Delta \theta )+O\left( \frac{1}{\Delta N}\right) ,\\ \displaystyle \varphi _8=\cos ({\theta _1(2n+1)-(n-j_s)\Delta \theta })\textrm{sinc}(\Delta N \Delta \theta )+O\left( \frac{1}{\Delta N}\right) ,\\ \displaystyle \varphi _1=\varphi _3=\varphi _5=\varphi _6=O\left( \frac{1}{\Delta N}\right) . \end{array}\nonumber \\ \end{aligned}$$
(45)

The difference \(\omega (\theta _1)-\omega (\theta _2)\) can be decomposed into series:

$$\begin{aligned} \omega (\theta _1)-\omega (\theta _2)\approx \omega '(\theta _1)\Delta \theta . \end{aligned}$$
(46)

Substitution of (45), (46) to (43) with dropping out of the terms of order \(O\left( \frac{1}{\Delta N}\right) \) gives the expression (19).

B Derivation of expressions for wave packets in the limit of mesoscale

We show that approximation of expressions for wave packets in (20) in the limit case (\(\Delta N \gg 1\)) approaches us to the Fourier transform of the \(\textrm{sinc}\) function. Indeed,

$$\begin{aligned} \begin{array}{l} \displaystyle \frac{1}{2\pi a}\int _{\theta -\pi }^{\theta +\pi }\cos {(q \Xi )}\textrm{sinc}(q \Delta N)\textrm{d}q=\frac{1}{2\pi a \Delta N}\int _{(\theta -\pi )\Delta N}^{(\theta +\pi )\Delta N}\cos {\left( \frac{q \Xi }{\Delta N}\right) }\textrm{sinc}\,q \textrm{d}q\\ \displaystyle \quad \approx \frac{1}{2\pi a \Delta N}\int _{-\infty }^{\infty }\cos {\left( \frac{q \Xi }{\Delta N}\right) }\textrm{sinc}\,q \textrm{d}q=\textrm{Re}\left( \frac{1}{2\pi a \Delta N}\int _{-\infty }^{\infty }e^{\textrm{i}\left( \frac{q \Xi }{\Delta N}\right) }\textrm{sinc}\,q \textrm{d}q\right) . \end{array} \end{aligned}$$
(47)

Analogously,

$$\begin{aligned} \displaystyle \frac{1}{2\pi a}\int _{\theta -\pi }^{\theta +\pi }\sin {(q \Xi )}\textrm{sinc}(q \Delta N)\textrm{d}q\approx \textrm{Im}\left( \frac{1}{2\pi a \Delta N}\int _{-\infty }^{\infty }e^{\textrm{i}\left( \frac{q \Xi }{\Delta N}\right) }\textrm{sinc}\,q \textrm{d}q\right) . \end{aligned}$$
(48)

Since

$$\begin{aligned} \frac{1}{2\pi }\int _{-\infty }^\infty e^{\textrm{i}\xi q}\textrm{sinc}\,q\;\textrm{d}q=\frac{1}{2}H\left( 1-\vert \xi \vert \right) , \end{aligned}$$
(49)

then one gets

$$\begin{aligned} \begin{array}{l} \displaystyle \frac{1}{2\pi a \Delta N}\int _{-\infty }^{\infty }\cos {\left( \frac{q \Xi }{\Delta N}\right) }\textrm{sinc}\,q \;\textrm{d}q=\frac{1}{2a\Delta N}H\left( 1-\frac{ \vert \Xi \vert }{\Delta N}\right) ,\\ \displaystyle \frac{1}{2\pi a \Delta N}\int _{-\infty }^{\infty }\sin {\left( \frac{q \Xi }{\Delta N}\right) }\textrm{sinc}\,q \;\textrm{d}q=0. \end{array} \end{aligned}$$
(50)

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Liazhkov, S.D. Unsteady thermal transport in an instantly heated semi-infinite free end Hooke chain. Continuum Mech. Thermodyn. 35, 413–430 (2023). https://doi.org/10.1007/s00161-023-01186-z

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