Phonon number fluctuations in Debye model of solid

TL;DR

This paper calculates phonon number fluctuations in the Debye model, showing they scale with temperature cubed at low temperatures, and discusses implications for the third law of thermodynamics.

Contribution

It provides a detailed analysis of phonon number fluctuations in the Debye model, linking fluctuations to temperature and atom number, highlighting the third law implications.

Findings

Fluctuations proportional to T^3 at low temperatures

Relative fluctuations diverge as temperature approaches zero

Proper temperature definition requires increasing atom number at low T

Abstract

The phonon number fluctuations in the Debye model of solid are calculated and are demonstrated to be proportional to the temperature cubed at low temperature, similar to the celebrated Debye's law of the heat capacity. For a fixed number of atoms, the relative fluctuations approach to infinity as the temperatture decreases to zero, and the proper definition of temperature needs more and more numbers of atoms at lower and lower temperatures, compatible with the third law on the unattainability of absolute zero temperature.

Equations

$$g(\omega)=\left\{\begin{array}[]{cc}\frac{9N}{\omega_{D}^{3}}\omega^{2},&\omega\leq\omega_{D}\\ 0&\omega>\omega_{D}\end{array}\right\},\text{ \ }\omega_{D}=\left(6\pi^{2}\right)^{1/3}c\left(\frac{N}{V}\right)^{1/3},$$

$${\int }_0^{{\omega }_D}g(\omega )d\omega =3N.$$

$$E\{n_i\}=\sum _{i=1}^{3N}{n}_i\hslash {\omega }_i.$$

$$Q=\sum _{\{n_i\}}{e}^{-\beta E\{n_i\}}=\prod _{i=1}^{3N}\sum _{n_i=0}^{\infty }{e}^{-n_i\beta \hslash {\omega }_i}=\prod _{i=1}^{3N}\frac{1}{1-e^{-\beta \hslash {\omega }_i}}.$$

$$⟨n_i⟩\equiv \frac{{\sum }_{\{n_i\}}{n}_i{e}^{-\beta E\{n_i\}}}{Q}=\frac{{\sum }_{n_i=0}^{\infty }{n}_i{e}^{-n_i\beta \hslash {\omega }_i}}{{\sum }_{n_i=0}^{\infty }{e}^{-n_i\beta \hslash {\omega }_i}}=\frac{1}{e^{\beta \hslash {\omega }_i}-1}=-kT\frac{\partial }{\partial \left(\beta \hslash {\omega }_i\right)}\ln Q.$$

$$U=\sum _{i=1}^{3N}⟨n_i⟩\hslash {\omega }_i=\sum _{i=1}^{3N}\frac{\hslash {\omega }_i}{e^{\beta \hslash {\omega }_i}-1}={\int }_0^{{\omega }_D}\frac{\hslash \omega }{e^{\beta \hslash \omega }-1}g(\omega )d\omega .$$

$$U\simeq 3NkT\left\{\begin{array}[]{cc}1,&T\gg T_{D}\\ \frac{\pi^{4}}{5}(\frac{T}{T_{D}})^{3},&T\ll T_{D}\end{array}\right.,$$

$$k{T}_D=\hslash {\omega }_D.$$

$$C\simeq 3Nk\left\{\begin{array}[]{cc}1,&T\gg T_{D}\\ \frac{4\pi^{4}}{5}(\frac{T}{T_{D}})^{3},&T\ll T_{D}\end{array}\right..$$

$$⟨n^2⟩=⟨\sum _r{n}_r\sum _s{n}_s⟩,\text{ and }⟨n⟩=\sum _{i=1}^{3N}⟨n_i⟩.$$

$$D_n(x)\equiv {\int }_0^x\frac{t^n}{e^t-1}dt,$$

$$D_{n}(x)\simeq\left\{\begin{array}[]{cc}n!\zeta(n+1)-x^{n}e^{-x}+O(x^{n}e^{-2x}),&x\rightarrow\infty\\ x^{n}/n+O(x^{n+1}),&x\rightarrow 0\end{array}\right..$$

$$⟨n⟩$$

$$={\int }_0^{{\omega }_D}\frac{1}{e^{\beta \hslash \omega }-1}g(\omega )d\omega$$

$$=\frac{9N}{{\omega }_D^3}{\int }_0^{{\omega }_D}\frac{{\omega }^2}{e^{\beta \hslash \omega }-1}d\omega .$$

$$t_D\equiv \frac{\hslash {\omega }_D}{kT}=\frac{T_D}{T},$$

$$⟨n⟩=\frac{9N}{{\omega }_D^3}{\left(\frac{kT}{\hslash }\right)}^3{\int }_0^{t_D}\frac{t^2}{e^t-1}dt=9N{t}_D^{-3}{D}_2(t_D).$$

$$\left\langle n\right\rangle\simeq 9N\left(\frac{T}{T_{D}}\right)^{3}\left\{\begin{array}[]{cc}\frac{1}{2}\left(\frac{T_{D}}{T}\right)^{2},&T\gg T_{D}\\ 2\zeta(3),&T\ll T_{D}\end{array}\right.=9N\left\{\begin{array}[]{cc}\frac{1}{2}\frac{T}{T_{D}},&T\gg T_{D}\\ 2\zeta(3)\left(\frac{T}{T_{D}}\right)^{3},&T\ll T_{D}\end{array}\right..$$

$$⟨n_r{n}_s⟩$$

$$=\frac{{\sum }_{n_r=0}^{\infty }{\sum }_{n_s=0}^{\infty }{n}_r{n}_s{e}^{-\left(n_r\beta \hslash {\omega }_r+n_s\beta \hslash {\omega }_s\right)}}{{\sum }_{n_r=0}^{\infty }{\sum }_{n_s=0}^{\infty }{e}^{-\left(n_r\beta \hslash {\omega }_r+n_s\beta \hslash {\omega }_s\right)}}$$

$$=\frac{{\sum }_{n_r=0}^{\infty }{n}_r{e}^{-n_r\beta \hslash {\omega }_r}}{{\sum }_{n_r=0}^{\infty }{e}^{-n_r\beta \hslash {\omega }_r}}\frac{{\sum }_{n_s=0}^{\infty }{n}_s{e}^{-n_s\beta \hslash {\omega }_s}}{{\sum }_{n_s=0}^{\infty }{e}^{-n_s\beta \hslash {\omega }_s}}$$

$$=⟨n_r⟩⟨n_s⟩.$$

$$-\left(⟨n_r^2⟩-{⟨n_r⟩}^2\right)=\frac{\partial }{\partial \left(\beta \hslash {\omega }_r\right)}\frac{{\sum }_{n_r=0}^{\infty }{n}_r{e}^{-n_r\beta \hslash {\omega }_r}}{{\sum }_{n_r=0}^{\infty }{e}^{-n_r\beta \hslash {\omega }_r}}.$$

$$\frac{\partial }{\partial \beta \hslash {\omega }_r}\frac{1}{e^{\beta \hslash {\omega }_i}-1}$$

$$=-\frac{e^{\beta \hslash {\omega }_i}-1+1}{{\left(e^{\beta \hslash {\omega }_i}-1\right)}^2}$$

$$=-\left({⟨n_r⟩}^2+⟨n_r⟩\right).$$

$$⟨n_r^2⟩-{⟨n_r⟩}^2={⟨n_r⟩}^2+⟨n_r⟩.$$

$$⟨\Delta n^2⟩$$

$$=⟨\sum _r{n}_r\sum _s{n}_s⟩-⟨\sum _r{n}_r⟩⟨\sum _s{n}_s⟩$$

$$=\sum _{r=1}^{3N}\left(⟨n_r^2⟩-{⟨n_r⟩}^2\right)$$

$$=\sum _{r=1}^{3N}\left({⟨n_r⟩}^2+⟨n_r⟩\right),$$

$$⟨\Delta n^2⟩$$

$$=9N{t}_D^{-3}{\int }_0^{t_D}\frac{t^2{e}^t}{{\left(e^t-1\right)}^2}dt$$

$$=9N{t}_D^{-3}\left(2D_1(t_D)-\frac{t_D^2}{e^{t_D}-1}\right).$$

$$\left\langle\Delta n^{2}\right\rangle\simeq 9N\left(\frac{T}{T_{D}}\right)^{3}\left\{\begin{array}[]{cc}\frac{T_{D}}{T},&T\gg T_{D}\\ \pi^{2}/3,&T\ll T_{D}\end{array}\right..$$

$$\frac{⟨\Delta n^2⟩}{{⟨n⟩}^2}$$

Full text

I Introduction

Fluctuations are ubiquitous in the Universe and the statistical mechanics is powerful to understand them. The Debye model of solid was the first successful model used to describe the heat capacity of a material at low temperature, via introduction of the concept phonon into physics. However, the phonon number fluctuations in the model are not yet known, and we will demonstrate that they fall off as $T^{3}$ at low temperature $T\rightarrow 0$, similar to the decrease of the heat capacity of solid at low temperature.

In the Debye model of the solid, the thermal properties are determined by the solid lattice vibrations. The vibrational frequencies form a continuous spectrum with a cut off at an upper limit $\omega_{D}$ such that the total number of normal modes of vibration is $3N$ of which $N$ are the number of the solid atoms which can harmonically displaced from lattices. The Debye spectrum $g(\omega)$ or density of states in frequency interval $\omega\rightarrow\omega+d\omega$ is, text1 ; text2 ; text3 ; text4

[TABLE]

where symbols $c$ and $V$ denote the effective speed of sound within the solid and its volume, respectively. Two quantities $g(\omega)$ and $\omega_{D}$ are related by the requirement that total number of normal modes of vibration $3N$,

[TABLE]

Assuming that there are $n_{i}$ phonons in the $i$th frequency $\omega_{i}$ whose unit energy quantum is $\hbar\omega_{i}$, we have the energy of the state $\{n_{i}\}$ in which there are $n_{i}$ phonons of the $i$th mode is,

[TABLE]

The partition function is, text1 ; text2 ; text3 ; text4

[TABLE]

where $\beta=1/kT$, and $\hbar$ and $k$ are, respectively, the Planck’s constant and the Boltzmann’s constant. The average number $\left\langle n_{i}\right\rangle$ of phonon of energy quantum $\hbar\omega_{i}$ is,

[TABLE]

The internal energy is,

[TABLE]

A more detailed discussion of this integral can be given shortly. However, the internal energy can be greatly simplified in limits of high and low temperatures, respectively, text1 ; text2 ; text3 ; text4

[TABLE]

where $T_{D}$ is the Debye temperature defined by,

[TABLE]

The values of the heat capacity $C=\partial U/\partial T$ in two opposite limits are,

[TABLE]

This equation is significant for it in high temperature limit gives the Dulong–Petit law, and in low temperature limit yields the $T^{3}$ law.

The number fluctuations of the phonon is defined by,

[TABLE]

How to calculate it will be given in section II. In section III, brief concluding remarks are given.

The Debye integral $D_{n}(x)$ text5 is elementary in our analysis,

[TABLE]

which has simple expressions in limits of large and small $x$ with $\zeta$ denoting the Riemann zeta function

[TABLE]

II The mean numbers and the number fluctuations

The mean numbers for phonons in the Debye model of solid are from (5),

[TABLE]

Performing a variable transformation $t=\beta\hbar\omega$ and defining,

[TABLE]

the mean numbers $\left\langle n\right\rangle$ (13) becomes,

[TABLE]

It is thus in the opposite limits of temperature intervals from (12),

[TABLE]

Before computing the particle number fluctuations, we need to deal statistical correlation $\left\langle n_{r}n_{s}\right\rangle$ of the particle number. If $r\neq s$, we have,

[TABLE]

The minus of the particle number fluctuations in the $r$th vibration mode are,

[TABLE]

The right-handed side of this equation becomes from (5),

[TABLE]

Combining two results (18)-(19), we reach a remarkable result,

[TABLE]

The fluctuations in phonon number are,

[TABLE]

which can be transformed into an integral,

[TABLE]

In limits of high and low $T$, Eq. (22) gives with $\zeta(2)=\pi^{2}/6$,

[TABLE]

The relative fluctuations are,

[TABLE]

In addition we have from (22) and (15),

[TABLE]

It is interesting to note that with a given large number of atoms $N$, the relative fluctuation is not automatically less than $1$ at low temperature. I.e, the following equation would be violated,

[TABLE]

The self-consistence of the statistical mechanics implies a requirement upon the temperature,

[TABLE]

In other words, when temperature approaches to zero Kelvin, the thermodynamic limit requires a huge number of particles $N\rightarrow\infty$ otherwise the low temperatures can not be properly defined, Liu compatible with the third law on the unattainability of absolute zero temperature.

III Remarks

The mean numbers and the number fluctuations of the phonon in the Debye model are explicitly calculated. The numbers and fluctuations are demonstrated to be proportional to the temperature cubed at low temperature, similar to the celebrated Debye’s law of the heat capacity. In addition, the relative fluctuations diverge for a fixed number of atoms, implying that the proper definition of temperature needs more and more numbers of atoms at lower and lower temperatures.

Acknowledgements.

This work is financially supported by National Natural Science Foundation of China under Grant No. 11675051.