\thankstext

e1email: [email protected] ¹¹institutetext: Department of Domestic Science, Koriyama Women’s University, Koriyama, Fukushima, 963-8503, JAPAN

\abstractdc

We study the thermodynamic quantities in the system of the $N$ independent harmonic oscillators with different frequencies in the Tsallis statistics of the entropic parameter $q$ ( $1<q<2$ ) with escort average. The self-consistent equation is derived, and the physical quantities are calculated with the physical temperature. It is found that the number of oscillators is restricted below $1/(q-1)$ . The energy, the Rényi entropy, and the Tsallis entropy are obtained by solving the self-consistent equation approximately at high physical temperature and/or for small deviation $q-1$ . The energy is $q$ -independent at high physical temperature when the physical temperature is adopted, and the energy is proportional to the number of oscillators and physical temperature at high physical temperature. The form of the Rényi entropy is similar to that of von-Neumann entropy, and the Tsallis entropy is given through the Rényi entropy. The physical temperature dependence of the Tsallis entropy is different from that of Rényi entropy. The Tsallis entropy is bounded from the above, while the Rényi entropy increases with the physical temperature. The ratio of the Tsallis entropy to the Rényi entropy is small at high physical temperature.

Thermodynamics of the independent harmonic oscillators with different frequencies in the Tsallis statistics

Masamichi Ishihara\thanksrefe1,addr1

1 Introduction

The various statistics have been proposed to describe the phenomena which show power-like distributions. An extension of the Boltzmann-Gibbs statistics is the Tsallis statistics, and the statistics has been applied in various branches of science TsallisBook . The escort average is often adopted to calculate the physical quantities in the Tsallis statistics. The Tsallis statistics has the entropic parameter $q$ , and the statistics approaches the Boltzmann-Gibbs statistic as $q$ approaches one.

The entropic parameter $q$ is often restricted. The normalizability of the probability requires that $q$ is less than two Tsallis-BJP39-overview . The parameter $q$ is also restricted because physical quantities are restricted Tsallis-BJP39-overview ; Ishihara2016:IJMPE ; Bhattacharyya:2016 ; Bhattacharyya . For example, the energy density should be finite and the number of particles should be positive, and these requirements show that the maximum value of $q$ is smaller than two. The limitation of $q$ was also derived by using the conjugate variables theorem Umpierrez2021 .

Simple systems have been adopted to study the effects of statistics. The classical gas model was adopted, and it was found that the energy is proportional to the number of particles and the physical temperature Kalyana:2000 ; Abe-PLA:2001 ; S.Abe:physicaA:2001 ; Aragao:2003 ; Ruthotto:2003 ; Toral:2003 ; Suyari:2006 ; Ishihara:phi4 ; Ishihara:free-field ; Ishihara:Thermodyn-rel in the Tsallis statistics. It is also found that the number of the particles are restricted Abe-PLA:2001 . The thermodynamic quantities for a classical harmonic oscillator was also calculated in the Tsallis statistics with escort average. The partition function was calculated and the energy was obtained Tsallis1998 .

The calculations of the thermodynamic quantities for the harmonic oscillators are required in the Tsallis statistics. A field is decomposed into harmonic oscillators with different frequencies to calculate physical quantities. The results for the harmonic oscillators with different frequencies in the Tsallis statistics will be helpful to calculate physical quantities in various systems.

In this paper, we study the thermodynamic quantities in the system of the $N$ independent harmonic oscillators in the Tsallis statistics of the entropic parameter $q$ . The range of $q$ is set between one and two in this study. The escort average is employed to obtain physical values. In Sec. 2, we briefly review the Tsallis statistics. In Sec. 3, we study the $N$ independent harmonic oscillators with different frequencies. The self-consistent equation is derived, and the equation is solved approximately. The expression of the energy is obtained with physical temperature. The expressions of Tsallis and Rényi entropies are also obtained. The last section is assigned for conclusion.

2 Brief review of the Tsallis statistics

The Tsallis statistics TsallisBook is based on the Tsallis entropy $S_{q}^{(T)}$ with the entropic parameter $q$ . The entropy $S_{q}^{(T)}$ is defined by

\displaystyle S_{q}^{(T)}=\frac{1-\mathrm{Tr}\left[\hat{\rho}^{q}\right]}{q-1},

(1)

where $\hat{\rho}$ is the density operator. The density operator $\hat{\rho}$ is obtained by extremizing $S_{q}^{(T)}$ under the normalization condition $\mathrm{Tr}\left[\hat{\rho}\right]=1$ and the energy constraint:

\displaystyle U=\frac{\mathrm{Tr}\left[\hat{\rho}^{q}\hat{H}\right]}{\mathrm{Tr}\left[\hat{\rho}^{q}\right]},

(2)

where $U$ is the energy. The right-hand side of Eq. (2) is the escort average of the Hamiltonian $\hat{H}$ .

The density operator $\hat{\rho}$ in the Tsallis statistics with the escort average is obtained:


	$\displaystyle\hat{\rho}=\frac{1}{Z}\left(1-(1-q)\frac{\beta}{c_{q}}(\hat{H}-U)\right)^{\frac{1}{1-q}},$		(3a)
	$\displaystyle Z=\mathrm{Tr}\left[\left(1-(1-q)\frac{\beta}{c_{q}}(\hat{H}-U)\right)^{\frac{1}{1-q}}\right],$		(3b)
	$\displaystyle c_{q}=\mathrm{Tr}\left[\hat{\rho}^{q}\right],$		(3c)

where $\beta$ is the inverse temperature. The partition function $Z$ is related to $c_{q}$ :

\displaystyle c_{q}=Z^{1-q}.

(4)

The inverse physical temperature is given by

\displaystyle\beta_{\mathrm{ph}}=\beta/c_{q}.

(5)

The physical temperature $T_{\mathrm{ph}}$ is given as $1/\beta_{\mathrm{ph}}$ .

The thermodynamic quantities are calculated with the above density operator for the $N$ independent harmonic oscillators with different frequencies in the following section.

3 The independent harmonic oscillators with different frequencies

3.1 Derivation of self-consistent equation

We attempt to derive the self-consistent equation by calculating $c_{q}$ in two ways. One way is the method by using the relation $c_{q}=Z^{1-q}$ and the other way is the method by calculating $c_{q}=\mathrm{Tr}\left[\hat{\rho}^{q}\right]$ directly. We obtain the self-consistent equation by equating these results.

We treat the $N$ independent harmonic oscillators with different frequencies. The Hamiltonian $\hat{H}$ is

\displaystyle\hat{H}=\sum_{j=1}^{N}\omega_{j}\left(\hat{n}_{j}+\frac{1}{2}\right),

(6)

where $\hat{n}_{j}$ is the number operator with the subscript $j$ . We treat the above Hamiltonian in the Tsallis statistics of $1<q<2$ : $(2-q)/(q-1)$ , $1/(q-1)$ , and $q/(q-1)$ are positive.

We introduce a parameter $E_{\mathrm{ref}}$ and calculate the partition function $Z$ :

	$\displaystyle Z$	$\displaystyle=\sum_{n_{1},\cdots,n_{N}=0}^{\infty}\left\{1+(q-1)\beta_{\mathrm{ph}}\Bigg{(}\frac{\hbar}{2}(\omega_{1}+\cdots+\omega_{N})-U\Bigg{)}+(q-1)\beta_{\mathrm{ph}}\Bigg{(}\hbar\omega_{1}n_{1}+\cdots+\hbar\omega_{N}n_{N}\Bigg{)}\right\}^{\frac{1}{1-q}}$
		$\displaystyle=\left((q-1)\beta_{\mathrm{ph}}E_{\mathrm{ref}}\right)^{\frac{1}{1-q}}\sum_{n_{1},\cdots,n_{N}=0}^{\infty}(\lambda_{N}+a_{1}n_{1}+\cdots+a_{N}n_{N})^{\frac{1}{1-q}},$		(7)

where


	$\displaystyle\lambda_{N}=\frac{1+(q-1)\beta_{\mathrm{ph}}\Bigg{(}\displaystyle\frac{1}{2}\left(\displaystyle\sum_{i=1}^{N}\hbar\omega_{i}\right)-U\Bigg{)}}{(q-1)\beta_{\mathrm{ph}}E_{\mathrm{ref}}},$		(8a)
	$\displaystyle a_{j}:=\frac{\hbar\omega_{j}}{E_{\mathrm{ref}}}.$		(8b)

Equation (7) is represented with Barnes zeta function $\zeta_{\mathrm{B}}(s,\alpha|\vec{\omega}_{N})$ (See Eq. (38)):

\displaystyle Z

\displaystyle=\left((q-1)\beta_{\mathrm{ph}}E_{\mathrm{ref}}\right)^{\frac{1}{1-q}}\zeta_{\mathrm{B}}\left(1/(q-1),\lambda_{N}|\vec{a}_{N}\right)\qquad\vec{a}_{N}=(a_{1},a_{2},\cdots,a_{N}).

(9)

The condition $s>N$ for the parameters of the Barnes zeta function in the present case is

\displaystyle\frac{1}{q-1}>N.

(10)

This means that the number of the oscillators is restricted. We also calculate $c_{q}$ directly as

\displaystyle c_{q}=\mathrm{Tr}\left[\hat{\rho}^{q}\right]=Z^{-q}\left((q-1)\beta_{\mathrm{ph}}E_{\mathrm{ref}}\right)^{\frac{q}{1-q}}\zeta_{\mathrm{B}}\left(q/(q-1),\lambda_{N}|\vec{a}_{N}\right).

(11)

From Eqs. (4), (9), and (11), we have the following self-consistent equation:

\displaystyle\left((q-1)\beta_{\mathrm{ph}}E_{\mathrm{ref}}\right)\zeta_{\mathrm{B}}\left(1/(q-1),\lambda_{N}|\vec{a}_{N}\right)=\zeta_{\mathrm{B}}\left(q/(q-1),\lambda_{N}|\vec{a}_{N}\right).

(12)

We attempt to obtain the physical quantities by solving the self-consistent equation in the next subsection.

3.2 Energy and entropies

We attempt to find the expressions of physical quantities in this subsection. For $\lambda_{N}\gg 1$ , we have the following expressions by using Eq. (42).


	$\displaystyle\zeta_{\mathrm{B}}\left(1/(q-1),\lambda_{N}\|\vec{a}_{N}\right)\sim\frac{(q-1)^{N}}{\left(\displaystyle\prod_{j=0}^{N-1}((2-q)-j(q-1))\right)\left(\displaystyle\prod_{j=1}^{N}a_{j}\right)(\lambda_{N})^{\frac{1}{(q-1)}-N}},$		(13a)
	$\displaystyle\zeta_{\mathrm{B}}\left(q/(q-1),\lambda_{N}\|\vec{a}_{N}\right)\sim\frac{(q-1)^{N}}{\left(\displaystyle\prod_{j=0}^{N-1}(1-j(q-1))\right)\left(\displaystyle\prod_{j=1}^{N}a_{j}\right)(\lambda_{N})^{\frac{q}{(q-1)}-N}}.$		(13b)

In B, the above approximated expression for the Barnes zeta function is given by using the approximated expression for the Hurwitz zeta function given in A. We use these expressions of $\zeta_{\mathrm{B}}$ to solve Eq. (12) approximately.

3.2.1 Expression of the energy

We attempt to calculate the energy $U$ by solving Eq. (12). Substituting Eqs.(13a) and (13b) into Eq. (12), we have

\displaystyle U=\frac{T_{\mathrm{ph}}}{(q-1)}\left(1-\frac{\displaystyle\prod_{j=0}^{N-1}((2-q)-j(q-1))}{\displaystyle\prod_{j=0}^{N-1}(1-j(q-1))}\right)+\sum_{i=1}^{N}\frac{\hbar\omega_{i}}{2},\qquad N<\frac{1}{(q-1)}.

(14)

It is noted that Eq. (14) does not contain $E_{\mathrm{ref}}$ . We obtain easily

\displaystyle\frac{\displaystyle\prod_{j=0}^{N-1}((2-q)-j(q-1))}{\displaystyle\prod_{j=0}^{N-1}(1-j(q-1))}=1-N(q-1).

(15)

By substituting the above expression, we have the following expression of $U$ :

\displaystyle U=NT_{\mathrm{ph}}+\frac{1}{2}\sum_{i=1}^{N}\hbar\omega_{i},\qquad N<\frac{1}{(q-1)}.

(16)

Equation (16) is the well-known form of the energy $U$ in the Boltzmann-Gibbs statistics. It is possible to evaluate $\lambda_{N}$ by using Eq. (16):

\displaystyle\lambda_{N}=\frac{1-N(q-1)}{(q-1)\beta_{\mathrm{ph}}E_{\mathrm{ref}}}.

(17)

The numerator of the right-hand side of Eq. (17) is positive, because $N(q-1)$ is less than one. Therefore the condition $\lambda_{N}\gg 1$ is satisfied for $(q-1)\beta_{\mathrm{ph}}E_{\mathrm{ref}}\ll 1$ : the condition is satisfied at high physical temperature $T_{\mathrm{ph}}$ and/or for small deviation $(q-1)$ .

3.2.2 Expressions of the entropies

The Tsallis entropy $S_{q}^{(T)}$ is represented as

\displaystyle S_{q}^{(T)}=\frac{1-c_{q}}{q-1}=\frac{1-Z^{1-q}}{q-1}.

(18)

The Rényi entropy $S_{q}^{(R)}$ is related to the Tsallis entropy:

\displaystyle S_{q}^{(R)}=\frac{1}{1-q}\ln(1+(1-q)S_{q}^{(T)}).

(19)

This equation is represented with $c_{q}$ as

\displaystyle S_{q}^{(R)}=\frac{1}{1-q}\ln c_{q}=\frac{1}{1-q}\ln e^{(1-q)\ln Z}=\ln Z.

(20)

We calculate $Z$ approximately by using Eq. (13a).

\displaystyle Z=\frac{1}{\left(\displaystyle\prod_{j=0}^{N-1}((2-q)-j(q-1))\right)\left(\displaystyle\prod_{j=1}^{N}(\beta_{\mathrm{ph}}\hbar\omega_{j})\right)\left(1+(q-1)\beta_{\mathrm{ph}}\left(\displaystyle\frac{1}{2}\displaystyle\sum_{i=1}^{N}(\hbar\omega_{i})-U\right)\right)^{\frac{1}{q-1}-N}}.

(21)

Substituting Eq. (14) into Eq. (21), we obtain

\displaystyle Z=\frac{\Bigg{(}\displaystyle\prod_{j=0}^{N-1}(1-j(q-1))\Bigg{)}^{\frac{1}{1-q}-N}}{\Bigg{(}\displaystyle\prod_{j=1}^{N}(\beta_{\mathrm{ph}}\hbar\omega_{j})\Bigg{)}\Bigg{(}\displaystyle\prod_{j=0}^{N-1}((2-q)-j(q-1))\Bigg{)}^{\frac{q}{1-q}-N}}.

(22)

We find the relation between $dU$ and $dS_{q}^{(R)}$ . The Rényi entropy is given by $\ln Z$ . For the fixed $N$ and $q$ , we have

\displaystyle dS_{q}^{(R)}=d\ln Z=N\frac{dT_{\mathrm{ph}}}{T_{\mathrm{ph}}}.

(23)

With Eqs. (16) and (23), we have

\displaystyle dU=NdT_{\mathrm{ph}}=T_{\mathrm{ph}}dS_{q}^{(R)}.

(24)

The $q$ -dependence of $S_{q}^{(R)}$ for small $q-1$ is obtained by expanding the logarithm of Eq. (22) with respect to $q-1$ . We have

\displaystyle S_{q}^{(R)}=\ln Z=L_{N}(T_{\mathrm{ph}})+N+\frac{1}{2}N(q-1)+O((q-1)^{2}),

(25)

where $L_{N}(T_{\mathrm{ph}})$ is defined by

\displaystyle L_{N}(T_{\mathrm{ph}}):=\sum_{j=1}^{N}\ln\Big{(}\frac{T_{\mathrm{ph}}}{\hbar\omega_{j}}\Big{)}.

(26)

The same equation can be obtained by substituting Eq. (16) into Eq. (21). We remember that $N(q-1)$ is less than one.

We obtain the ratio of $S_{q}^{(T)}$ to $S_{q}^{(R)}$ . Hereafter we omit the argument $T_{\mathrm{ph}}$ of $L_{n}$ for simplicity. For $L_{N}\gg N$ , we have


	$\displaystyle S_{q}^{(R)}\sim L_{N},$		(27a)
	$\displaystyle S_{q}^{(T)}\sim\frac{1-e^{-(q-1)L_{N}}}{q-1}.$		(27b)

The ratio $S_{q}^{(T)}/S_{q}^{(R)}$ is

\displaystyle\frac{S_{q}^{(T)}}{S_{q}^{(R)}}\sim\frac{1-e^{-(q-1)L_{N}}}{(q-1)L_{N}}.

(28)

The ratio $S_{q}^{(T)}/S_{q}^{(R)}$ is approximately $1/((q-1)L_{N})$ at sufficiently high physical temperature which satisfies $(q-1)L_{N}\gg 1$ . This ratio is $1-(q-1)L_{N}/2$ for $(q-1)L_{N}\ll 1$ , though the condition $L_{N}\gg N$ is required to obtain the expressions, Eqs. (27a) and (27b).

4 Conclusions

We studied the thermodynamic quantities in the system of the $N$ independent harmonic oscillators with different frequencies $\omega_{j}$ in the Tsallis statistics of the entropic parameter $q$ ( $1<q<2$ ). The number of the oscillators $N$ was fixed and the escort average was adopted in this study. We derived the self-consistent equation, and the expressions of physical quantities with the physical temperature were obtained. We obtained the partition function $Z$ , the energy $U$ , the Rényi entropy $S_{q}^{(R)}$ , and the Tsallis entropy $S_{q}^{(T)}$ by solving the self-consistent equation approximately at high physical temperature $T_{\mathrm{ph}}$ and/or for small deviation $q-1$ .

It was found from the condition for the parameters of the Barnes zeta function that the number of harmonics oscillators $N$ is less than $1/(q-1)$ . The restriction of the number of the harmonic oscillators exists, as the restriction was previously given for the classical gas Abe-PLA:2001 . As expected, the supremum $1/(q-1)$ goes to infinity when $q$ approaches one.

The energy $U$ is $q$ -independent at high physical temperature when the physical temperature is adopted. The energy is proportional to the number of harmonic oscillators $N$ and the physical temperature $T_{\mathrm{ph}}$ at high physical temperature when the vacuum term is ignored: the expression of the energy is the well known expression, ${U=NT_{\mathrm{ph}}+\sum_{j}\hbar\omega_{j}/2}$ . The Rényi entropy $S_{q}^{(R)}$ is the sum of the values for the independent harmonic oscillators at high physical temperature. The Rényi entropy with the same frequency, $\omega\equiv\omega_{1}=\cdots=\omega_{N}$ , is given by $N\ln(T_{\mathrm{ph}}/(\hbar\omega))$ which is well-known expression for the $N$ independent harmonic oscillators with the same frequency. The Tsallis entropy $S_{q}^{(T)}$ was obtained through the Rényi entropy. The variation for the Rényi entropy is simply given as $dS_{q}^{(R)}=NdT_{\mathrm{ph}}$ for the fixed $N$ , and the well-known relation between $dU$ and $dS_{q}^{(R)}$ is also obtained: $dU=T_{\mathrm{ph}}dS_{q}^{(R)}$ .

The physical temperature dependence of the Tsallis entropy is different from that of the Rényi entropy. The Rényi entropy contains the term that is $\sum_{j=1}^{N}\ln(T_{\mathrm{ph}}/(\hbar\omega_{j}))$ . Therefore, the Rényi entropy increases with the physical temperature, and is unbounded from the above. In contrast, the Tsallis entropy increases with the physical temperature, and is bounded from the above. The ratio of the Tsallis entropy to the Rényi entropy, $S_{q}^{(T)}/S_{q}^{(R)}$ , is small at high physical temperature. The difference between the Tsallis and Rényi entropies is large at high physical temperature.

The system of the independent harmonic oscillators with different frequencies is basic, and the results in this study will give the insight on other physical systems. The author believes that the present study will be helpful for the reader to study the system represented with oscillators in unconventional statistics such as the Tsallis statistics.

Appendix A Approximate expression of Hurwitz zeta function

The Hurwitz zeta function Espinosa ; Bordag ; Shpot is defined by

\displaystyle\zeta_{\mathrm{H}}(s,\alpha):=\sum_{n=0}^{\infty}\frac{1}{(\alpha+n)^{s}}.

(29)

We treat the case of $s>1$ and $\alpha>0$ in this appendix.

Let $B_{n}(x)$ be Bernoulli polynomials which are defined by

\displaystyle\frac{te^{xt}}{e^{t}-1}=\sum_{n=0}^{\infty}B_{n}(x)\frac{t^{n}}{n!}.

(30)

The Bernoulli number $B_{n}$ in this paper is defined ¹¹1 The Bernoulli number $B_{n}$ is often defined as $B_{n}(x=0)$ . It may worth to mention that $B_{n}(x=0)=B_{n}(x=1)$ for $n\neq 1$ . by

\displaystyle B_{n}:=B_{n}(x=1).

(31)

We use the Euler-Maclaurin formula. Let $a$ and $b$ be integer with $a<b$ and let $f(x)$ be continuously differentiable for $M$ -times. The Euler-Maclaurin formula is

	$\displaystyle\sum_{n=a}^{b}f(n)=$	$\displaystyle\int_{a}^{b}dxf(x)+\frac{1}{2}(f(b)+f(a))+\sum_{k=1}^{M-1}\frac{B_{k+1}}{(k+1)!}(f^{(k)}(b)-f^{(k)}(a))$
		$\displaystyle-\frac{(-1)^{M}}{M!}\int_{a}^{b}dxB_{M}(x-[x])f^{(M)}(x),$		(32)

where $f^{(k)}$ is $k$ -th derivative and $[x]$ is the Gauss symbol (the floor function).

We attempt to find the expression of $\zeta_{\mathrm{H}}(1+z,\alpha)$ for $z>0$ by using the Euler-Maclaurin formula. The right-hand side of Eq. (29) is an infinite series. Therefore, we first consider the following finite series:

\displaystyle\zeta_{H,m}(s,\alpha)=\sum_{n=0}^{m}\frac{1}{(\alpha+n)^{s}}.

(33)

We set $f(x)$ as $1/(\alpha+x)^{1+z}$ and apply the Euler-Maclaurin formula. By taking the limit $m\rightarrow\infty$ , we have the expression of $\zeta_{\mathrm{H}}(1+z,\alpha)$ . The integral part converges when $\alpha$ is positive. We finally obtain

	$\displaystyle\zeta_{\mathrm{H}}(1+z,\alpha)$	$\displaystyle=\frac{1}{z\alpha^{z}}+\frac{1}{2\alpha^{1+z}}+\sum_{k=1}^{M-1}\frac{(-1)^{k+1}B_{k+1}}{(k+1)!}\frac{\Gamma(z+k+1)}{\Gamma(z+1)}\frac{1}{\alpha^{z+k+1}}$
		$\displaystyle\quad-\frac{(-1)^{M}}{M!}\int_{0}^{\infty}dxB_{M}(x-[x])f^{(M)}(x)\qquad(z>0,\alpha>0).$		(34)

The function $\zeta_{\mathrm{H}}(1+z,\alpha)$ can be rewritten Boumali2014 . For example, $\zeta_{\mathrm{H}}(1+z,\alpha)$ is given by

	$\displaystyle\zeta_{\mathrm{H}}(1+z,\alpha)$	$\displaystyle=\frac{1}{z\alpha^{z}}+\frac{1}{2\alpha^{1+z}}+\frac{1}{z}\sum_{k=2}^{M}\frac{B_{k}}{k!}\frac{\Gamma(z+k)}{\Gamma(z)}\frac{1}{\alpha^{z+k}}$
		$\displaystyle\quad-\frac{(-1)^{M}}{M!}\int_{0}^{\infty}dxB_{M}(x-[x])f^{(M)}(x)\qquad(z>0,\alpha>0),$		(35)

because $B_{2n+1}$ is zero for $n\geq 1$ and $\Gamma(z+1)=z\Gamma(z)$ .

It is possible to estimate the integral of Eq. (34) by setting $M$ . For example, the upper value of the integral with $M=2$ is estimated:

\displaystyle\left|\frac{1}{2!}\int_{0}^{\infty}B_{2}(x-[x])f^{(2)}(x)\right|\leq\frac{C_{2}}{2!}\int_{0}^{\infty}\left|f^{(2)}(x)\right|,

(36)

where $C_{2}$ is the maximum value of $|B_{2}(x)|$ in the range of $0\leq x\leq 1$ .

From Eq. (34), we find that the $\zeta_{\mathrm{H}}(1+z,\alpha)$ for $\alpha\gg 1$ behaves

\displaystyle\zeta_{\mathrm{H}}(1+z,\alpha)\sim\frac{1}{z\alpha^{z}}.

(37)

Appendix B Approximate expression of Barnes zeta function

The Barnes zeta function Ruijsenaars:2000 ; Kirsten:2010 is defined by

\displaystyle\zeta_{\mathrm{B}}(s,\alpha|\vec{\omega}_{N}):=\sum_{n_{1},\cdots,n_{N}=0}^{\infty}\frac{1}{(\alpha+\omega_{1}n_{1}+\cdots+\omega_{N}n_{N})^{s}}\qquad\vec{\omega}_{N}=(\omega_{1},\omega_{2},\cdots,\omega_{N}),

(38)

where $s>N$ , $\alpha>0$ , and $\omega_{j}>0$ .

We define $\Omega_{N}$ as

\displaystyle\Omega_{N}:=\alpha+\omega_{1}n_{1}+\cdots+\omega_{N}n_{N}.

(39)

The function $\zeta_{\mathrm{B}}$ is represented as

	$\displaystyle\zeta_{\mathrm{B}}(s,\alpha\|\vec{\omega}_{N})$	$\displaystyle=\sum_{n_{1},\cdots,n_{N}=0}^{\infty}\frac{1}{(\Omega_{N})^{s}}=\frac{1}{(\omega_{N})^{s}}\sum_{n_{1},\cdots,n_{N-1}=0}^{\infty}\sum_{n_{N}=0}^{\infty}\frac{1}{((\Omega_{N-1}/\omega_{N})+n_{N})^{s}}$
		$\displaystyle=\frac{1}{(\omega_{N})^{s}}\sum_{n_{1},\cdots,n_{N-1}=0}^{\infty}\zeta_{\mathrm{H}}(s,\Omega_{N-1}/\omega_{N}).$		(40)

We have $\zeta_{\mathrm{H}}(1+z,\alpha)\sim 1/(z\alpha^{z})$ for $\alpha\gg 1$ . Therefore, for sufficiently large $\alpha$ , we have

\displaystyle\zeta_{\mathrm{B}}(1+z,\alpha|\vec{\omega}_{N})\sim\frac{1}{(\omega_{N})^{1+z}}\sum_{n_{1},\cdots,n_{N-1}=0}^{\infty}\frac{1}{z(\Omega_{N-1}/\omega_{N})^{z}}=\frac{1}{z\omega_{N}}\zeta_{\mathrm{B}}(z,\alpha|\vec{\omega}_{N-1}).

(41)

By using the recurrence relation, Eq. (41), we have the approximate expression of $\zeta_{\mathrm{B}}$ for $\alpha\gg 1$ :

	$\displaystyle\zeta_{\mathrm{B}}(1+z,\alpha\|\vec{\omega}_{N})$	$\displaystyle\sim\frac{1}{z(z-1)\cdots(z-(N-1))}\frac{1}{\omega_{1}\omega_{2}\cdots\omega_{N}}\frac{1}{\alpha^{z-(N-1)}}$
		$\displaystyle=\frac{1}{\Bigg{(}\displaystyle\prod_{j=0}^{N-1}(z-j)\Bigg{)}\Bigg{(}\displaystyle\prod_{j=1}^{N}\omega_{j}\Bigg{)}\alpha^{z-(N-1)}}\qquad\qquad(z-(N-1)>0).$		(42)

The condition $z-(N-1)>0$ is rewritten as $1+z-N>0$ . This condition is equivalent to the condition $s>N$ with $s=1+z$ for $\zeta_{\mathrm{B}}(s,\alpha|\vec{\omega}_{N})$ .

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