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

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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.

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Fig. 1

References

  1. 1.

    C. Tsallis, J. Stat. Phys. 52, 479 (1988)

  2. 2.

    E. Curado, C. Tsallis, J. Phys. A 24, L69 (1991)

  3. 3.

    E. Curado, C. Tsallis, J. Phys. A 25, 1019(E) (1991)

  4. 4.

    J. Cleymans, D. Worku, J. Phys. G 39, 025006 (2012)

  5. 5.

    E. Megías, D. Menezes, A. Deppman, Phys. A 421, 15 (2015)

  6. 6.

    Débora P. Menezes, A. Deppman, E. Megias, L. Castro, Eur. Phys. J. A 51, 155 (2015)

  7. 7.

    A. Wehrl, Rev. Mod. Phys. 50, 221 (1978)

  8. 8.

    J. Havrda, F. Charvát, Kybernetika 3, 30 (1967)

  9. 9.

    Z. Daroczy, Inf. Control 16, 36 (1970)

  10. 10.

    C. Tsallis, R.S. Mendes, A.R. Plastino, Phys. A 261, 534 (1998)

  11. 11.

    A. Plastino, A.R. Plastino, Phys. Lett. A 226, 257 (1997)

  12. 12.

    A.S. Parvan, Phys. Lett. A 360, 26 (2006)

  13. 13.

    E.M.F. Curado, C. Tsallis, J. Phys. A 25, 1019 (1992)

  14. 14.

    L. Nivanen, A. Le Mehaute, Q. Wang, Rep. Math. Phys. 52, 437 (2003)

  15. 15.

    E. Borges, Phys. A 340, 95 (2004)

  16. 16.

    G. Viktor, Int. J. Mod. Phys. D 24, 1542015 (2015)

  17. 17.

    G. Viktor, H. Iguchi, Universe 3, 14 (2017). https://doi.org/10.3390/universe3010014

  18. 18.

    A. Scarfone, Entropy 15, 624 (2013)

  19. 19.

    P. Tempesta, Phys. Rev. E 84, 021121 (2011)

  20. 20.

    W. Chung, H. Hassanabadi, Mod. Phys. Lett. B 33, 1950368 (2019)

  21. 21.

    V. Ilić, M. Stanković, Phys. A 411, 138 (2014)

  22. 22.

    M. Czachor, J. Naudts, Phys. Lett. A 298, 369 (2002)

  23. 23.

    A. Kolmogorov, Atti della R. Accademia Nazionale dei Lincei 12, 388 (1930)

  24. 24.

    M. Nagumo, Jpn. Journ. Math. 7, 71 (1930)

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Acknowledgements

The authors acknowledge reviewers for their helpful comments.

Author information

Correspondence to Hassan Hassanabadi.

Appendix A

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)

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Chung, W.S., Hassanabadi, H. Possible non-additive entropy based on the \(\alpha \)-deformed addition. Eur. Phys. J. Plus 135, 19 (2020). https://doi.org/10.1140/epjp/s13360-019-00047-6

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