Possible non-additive entropy based on the $\alpha$-deformed addition

Abstract

We reexamine the non-additive entropy which was first proposed by Tsallis [1] where this deformed entropy (q-entropy) is related to a special map. We consider a new map giving the $\alpha$-additive property. Based on this, we propose a new type of non-additive entropy which we call $\alpha$-entropy and derive the $\alpha$-deformed Boltzmann factor and $\alpha$-deformed free energy. As an example, we discuss the black body radiation in the $\alpha$-deformed thermodynamics where we regard photon as a particle obeying $\alpha$-deformed boson algebra.

Appendix A

If we replace $t=\beta h \nu$ in Eq. (145), we have:

$$\begin{aligned} {{{\mathcal {E}}}} (T) = \frac{ 8 \pi k^4 }{c^3 h^3 } T^4 J_{\alpha }, \end{aligned}$$

(151)

where

$$\begin{aligned} J_{\alpha } = \int _0^{\infty } dt \frac{ t^3}{ ( e^{ t^{\alpha }}. -1)^{1/\alpha } } \end{aligned}$$

(152)

Using Taylor expansion, we have:

$$\begin{aligned} J_{\alpha } =\frac{1}{\alpha } \sum _{n=0}^{\infty } \frac{ \left( \frac{1}{\alpha } \right) _n \Gamma \left( \frac{4}{\alpha } \right) }{ n! \left( \frac{1}{\alpha } +n \right) ^{4/\alpha } }, \end{aligned}$$

(153)

where $(a)_n$ denotes Pochhammer symbol. If we set $\alpha = 1 + \epsilon$ for small $\epsilon$, we can use the following approximation formulas:

$$\begin{aligned} \Gamma \left( \frac{4}{\alpha } \right)\approx & {} 1 + \gamma \epsilon \end{aligned}$$

(154)

$$\begin{aligned} \left( \frac{1}{\alpha } +n \right) ^{-4/\alpha }\approx & {} \frac{1}{(n+1)^4 } \left[ 1 + 4 \epsilon \left( \ln ( 1 + n ) + \frac{1}{1+n} \right) \right] \end{aligned}$$

(155)

$$\begin{aligned} \left( \frac{1}{\alpha } \right) _n\approx & {} n! \left( 1 - \epsilon H(n) \right) , \end{aligned}$$

(156)

where H(n) is harmonic number defined by:

$$\begin{aligned} H(n) = \sum _{k=1}^n \frac{1}{k}, \end{aligned}$$

(157)

and $\gamma$ is Euler constant. Using these, up to a first order in $\epsilon$, we have:

$$\begin{aligned} J_{\alpha } \approx \zeta (4) + \epsilon \left( \gamma -1 + \frac{\pi ^2}{6} \zeta (3) + 2 \zeta (5) - 4 \zeta ' (4) ) \right) . \end{aligned}$$

(158)