Unsteady ballistic heat transport: linking lattice dynamics and kinetic theory

Abstract

The kinetic theory is widely used in the description of thermal transport at the micro- and nanoscale. In the theory, it is assumed that heat is carried by quasi-particles, obeying the Boltzmann transport equation. These quasi-particles are sometimes associated with phonons. However, since phonons are not localized in physical space, they cannot play the role of the quasi-particles used in the kinetic theory. In the present paper, we employ another interpretation of quasi-particles, namely wave packets. Our derivation is carried out for an infinite harmonic chain with a given initial temperature distribution. An exact formula describing the time evolution of the kinetic temperature of the chain is derived. A transition from the exact solution to a continuum limit is performed. It is shown that the resulting continuum solution coincides with the solution of the collisionless Boltzmann equation. The transition yields wave packets, localized in space and moving with group velocities. Therefore, the wave packets can be associated with quasi-particles. It is shown that the quasi-particle has finite lifetime and that its position is determined with some uncertainty.

Appendices

Appendix 1: Change of the summation limits

We show that in formula (8) the summation from 0 to $N-1$ can be replaced by a summation from $M-N+1$ to M (for $M < N$) as follows:

$$\begin{aligned} & \displaystyle \sum _{j=0}^{N-1} T_j^0 \cos \left( \frac{2\pi }{N}(i-j)(p-s)\right) = \sum _{j=0}^{M}\cdots +\sum _{j=M+1}^{N-1}... , \\ &\displaystyle\quad = \sum _{j=0}^{M}... + \sum _{j=M+1}^{N-1} T_{j-N}^0 \cos \left( \frac{2\pi }{N}(i-j + N)(p-s)\right) \\ & \displaystyle\quad = \sum _{j=0}^{M}\cdots + \sum _{j=M-N+1}^{-1} T_{j}^0 \cos \left( \frac{2\pi }{N}(i-j)(p-s)\right) \\ &= \sum _{j= M-N + 1}^{M} T_j^0 \cos \left( \frac{2\pi }{N}(i-j)(p-s)\right) . \end{aligned}$$

(47)

Here, the periodicity of the problem $T_j^0= T_{j-N}^0$ is used. Similarly, we show that the following identity is satisfied:

$$\begin{aligned} \sum _{s,p=0}^{N-1} \cos (\omega _p t)\cos (\omega _s t) \cos \left( \frac{2\pi }{N}(i-j)(p-s)\right) = \sum _{s,p=M-N+1}^{M}..., \quad \omega _p = \omega _{p-N}. \end{aligned}$$

(48)

Appendix 2: Derivation of the formula for $\varphi$

Consider the derivation of formula (19) for function $\varphi$. By definition,

$$\begin{aligned} \varphi = \frac{1}{2\Delta N} \sum _{l = -\Delta N+1}^{\Delta N} \cos \bigl ( \Delta k (j - l)\bigr ). \end{aligned}$$

(49)

We make the following transformations at the right side of formula (49):

$$\begin{aligned} \begin{array}{l} \displaystyle \varphi = \frac{\cos \bigl (j\Delta k\bigr )}{2\Delta N}\sum _{l = - \Delta N+1}^{\Delta N} \cos \left( \Delta k l\right) + \frac{\sin \bigl (j\Delta k\bigr )}{2\Delta N}\sum _{l = - \Delta N+1}^{\Delta N} \sin \left( \Delta k l\right) . \end{array} \end{aligned}$$

(50)

Substitution of the identities

$$\begin{aligned} \begin{array}{l} \displaystyle \sum _{l=-\Delta N+1}^{\Delta N} \cos (\Delta k l) = \cot \frac{\Delta k}{2} \sin \left( \Delta N \Delta k\right) , \quad \displaystyle \sum _{l=-\Delta N+1}^{\Delta N} \sin (\Delta k l) = \sin \left( \Delta N \Delta k\right) \end{array} \end{aligned}$$

(51)

into (50) yields expression (19) for $\varphi$.

Appendix 3: Derivation of the formula for $\psi$

We derive the approximate expression (24) for $\psi$ in the limit $\Delta N \gg 1$, $a\Delta N \ll L$. We make the following transformations:

$${\begin{aligned} \displaystyle \psi &= \frac{1}{2\pi a}\int _{q_1-\pi }^{q_1+\pi } \cos \bigl (q_2(j + \omega ' t)\bigr ) \mathrm{sinc}\left( \Delta N q_2\right) \mathrm{d}q_2 \\ \displaystyle &= \frac{1}{2\pi a \Delta N}\int _{(q_1-\pi ) \Delta N}^{(q_1+\pi ) \Delta N} \cos \frac{q_2(j + \omega ' t)}{\Delta N} \mathrm{sinc}\, q_2 \mathrm{d} q_2 \\ \displaystyle &\approx \frac{1}{2\pi a \Delta N}\int _{-\infty }^{+\infty } \cos \frac{q_2 (j + \omega ' t)}{\Delta N} \mathrm{sinc}\, q_2 \mathrm{{\tiny }d}q_2. \end{aligned}}$$

(52)

Here, $q_1 \in [0; \pi ]$, $\omega ' = \omega '(q_1)$. Calculation of the integral at the right side of formula (52) yields formula (24):

$$\begin{aligned} \psi \approx {\left\{ \begin{array}{ll} \frac{1}{2a\Delta N}, &\quad a|j + \omega ' t| < a\Delta N, \\ 0, &\quad a|j + \omega ' t| > a\Delta N. \end{array}\right. } \end{aligned}$$

(53)

Here, $a\Delta N$ is small compared to the macroscopic length scale L. Therefore, for $a\Delta N \ll L$, function $\psi$ can be replaced by the Dirac delta function.