Thermodynamic quantities of independent harmonic oscillators in microcanonical and canonical ensembles in the Tsallis statistics

Abstract

We study the energy and the entropies of N independent harmonic oscillators in the microcanonical and the canonical ensembles in the Tsallis classical and the Tsallis quantum statistics of entropic parameter q, where N is the number of the oscillators and the value of q is larger than one. The energy and the entropies are represented with the physical temperature, and the well-known expressions are obtained for the energy and the Rényi entropy. The difference between the microcanonical and the canonical ensembles is the existence of the condition for N and q in the canonical ensemble: $N(q-1)<1$. The condition does not appear in the microcanonical ensemble.The entropies are q-dependent in the canonical ensemble, and are not q-dependent in the microcanonical ensemble. For $N(q-1)<1$, this difference in entropy is quite small, and the entropy in the canonical ensemble does not differ from the entropy in the microcanonical ensemble substantially.

Graphic abstract

Appendix A: Hurwitz zeta function and Barnes zeta function

In this appendix, we give the approximate expressions of Hurwitz and Barnes zeta functions. The derivation is also given in the appendices A and B of the reference [2] .

The Hurwitz zeta function $\zeta _{\textrm{H}}$ is defined by

$$\begin{aligned} \zeta _{\textrm{H}}(s, \alpha ) = \sum _{n=0}^{\infty } \frac{1}{(\alpha +n)^s}. \end{aligned}$$

(A.1)

In this appendix, we treat the case of $s>1$ and $\alpha >0$.

Applying the Euler–Maclaurin formula, we have

$$\begin{aligned} \zeta _{\textrm{H}}(1+z, \alpha )&= \frac{1}{z\alpha ^z} + \frac{1}{2\alpha ^{1+z}} \nonumber \\&\quad + \sum _{k=1}^{M-1} \frac{(-1)^{k+1} B_{k+1}}{(k+1)!} \frac{\Gamma (z+k+1)}{\Gamma (z+1)} \frac{1}{\alpha ^{z+k+1}} \nonumber \\&\quad - \frac{(-1)^M}{M!} \int _0^{\infty } \textrm{d}x B_M(x-[x]) f^{(M)}(x)\nonumber \\&\quad (z>0, \alpha > 0) , \end{aligned}$$

(A.2)

where $B_k$ is Bernoulli number. The Hurwitz zeta function can be expressed in the other forms [18]. From Eq. (A.2), we find that $\zeta _{\textrm{H}}(1+z, \alpha )$ for $\alpha \gg 1$ behaves as

$$\begin{aligned} \zeta _{\textrm{H}}(1+z,\alpha ) \sim \frac{1}{z \alpha ^{z}}. \end{aligned}$$

(A.3)

The Barnes zeta function [20, 21] is defined by

$$\begin{aligned}&\zeta _{\textrm{B}}(s,\alpha |\mathbf {\omega }_N) = \sum _{n_1,\ldots ,n_N=0}^{\infty } \frac{1}{(\alpha + \omega _1 n_1 + \ldots + \omega _N n_N)^s} \nonumber \\&\mathbf {\omega }_N = (\omega _1, \omega _2, \ldots , \omega _N), \end{aligned}$$

(A.4)

where $s>N$, $\alpha > 0$, and $\omega _j > 0$. The Barnes zeta function for sufficiently large $\alpha$ has the following relation

$$\begin{aligned} \zeta _{\textrm{B}}(1+z,\alpha |\mathbf {\omega }_N) \sim \frac{1}{z \omega _N} \zeta _{\textrm{B}}(z, \alpha |\mathbf {\omega }_{N-1}) . \end{aligned}$$

(A.5)

This relation is derived with Eq. (A.3). Using the recurrence relation, Eq. (A.5), we have the following approximate expression of $\zeta _{\textrm{B}}$ for $\alpha \gg 1$:

$$\begin{aligned} \zeta _{\textrm{B}}(1+z, \alpha |\mathbf {\omega }_N)&\sim \frac{1}{\Bigg (\displaystyle \prod _{j=0}^{N-1} (z-j)\Bigg ) \Bigg (\displaystyle \prod _{j=1}^{N} \omega _j \Bigg )\alpha ^{z-(N-1)}}\nonumber \\&\qquad (z - (N-1) > 0). \end{aligned}$$

(A.6)

In the present case, $\mathbf {\omega }_N$ is set to $\mathbf {\omega }_N = (1,1,\ldots ,1)$. For simplicity, we use the following notation for the Barns zeta function $\zeta _{\textrm{B}}$ with $\mathbf {\omega }_N = (1,1,\ldots ,1)$:

$$\begin{aligned} \zeta _{\textrm{B}}(s, \alpha ; N) = \sum _{n_1,\ldots ,n_N=0}^{\infty } \frac{1}{ ( \alpha + n_1 + n_2 + \cdots + n_N)^s} . \end{aligned}$$

(A.7)

The approximate expression of $\zeta _{\textrm{B}}(1+z, \alpha ; N)$ for $\alpha \gg 1$ is

$$\begin{aligned} \zeta _{\textrm{B}}(1+z, \alpha ; N)&\sim \frac{1}{\Bigg (\displaystyle \prod _{j=0}^{N-1} (z-j)\Bigg ) \alpha ^{z-(N-1)}} \nonumber \\&\qquad (z - (N-1) > 0). \end{aligned}$$

(A.8)