PRELUDIUM BIS 2 No DEC 2020/39/O/ST2/02323 (2021-2025).
Supervisor: Prof. Mariusz P. Dąbrowski, PhD student: Ilim Irfan Cimdiker.
Our first achievement:
We have demonstrated that for every nonextensive entropy, one should define an effective temperature (which we call equilibrium temperature) by utilizing the equilibrium condition, and that there is always an additive equilibrium entropy associated with this effective temperature. This is due to the fact that except for Bekenstein entropy, the Hawking temperature is thermodynamically inconsistent with other nonextensive entropies. We show the equilibrium requirement for the Tsallis–Cirto black hole entropy and demonstrate that the Bekenstein–Hawking entropy is the related equilibrium entropy, and the Hawking temperature is the associated equilibrium temperature for the Tsallis–Cirto black hole entropy.
- “Nonextensive black hole entropies”, talk at the Autumn Cosmology Workshop 2022, Świnoujście, Poland, 1.10.2022 (I. Cimdiker)
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”Generalized Uncertainty Principle and Nonextensive Entropies, Plenary talk at The 8th Conference of the Polish Society on Relativity, Warsaw, Poland, 19.09.2022 (M.P. Dąbrowski).
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Ilim Cimdiker, Mariusz P. Dąbrowski, Hussain Gohar,
Equilibrium temperature for black holes with nonextensive entropy, European Physics Journal C 83, 169 (2023) (9 pages); e-print: 2208.04473
Our second achievement:
We have studied the effect of the generalized uncertainty principle (GUP) on nonextensive thermodynamics applied to black holes, as well as the sparsity of the radiation at different temperatures associated with nonextensive entropies such as: R\’enyi, Tsallis-Cirto, Kaniadakis, Sharma Mittal, and Barrow. We have shown that due to GUP corrections, the temperature and entropy have finite values, implying that the final state of the black hole is a remnant at the end of the evaporation process and that the sparsity of the radiation for massless bosons at each temperature depends on the mass of the black hole. We also find that GUP reduces the value of the sparsity profile for each case as compared to the sparsity parameter at Hawking temperature, which is always constant throughout the evaporation. In the talk by M.P. Dąbrowski at the conference (see below) preliminary application of the nonextensive entropies to the cosmological horizons tested by observational data has been presented.
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Ilim Cimdiker, Mariusz P. Dąbrowski, Hussain Gohar, Generalized Uncertainty Principle Impact on Nonextensive Black Hole Thermodynamics, e-print: 2301.00609
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Ilim Cimdiker, Mariusz P. Dąbrowski, Hussain Gohar, Generalized Uncertainty Principle Impact on Nonextensive Black Hole Thermodynamics Classical and Quantum Gravity 40, 145001 (2023).
Almost Extensivity of Barrow Entropy Favoured by the Full Dynamical and Geometrical Set of Cosmological Data, Talk delivered at the conference “Workshop on Tensions in Cosmology” at Corfu, Greece on 07.09.2023 (M.P. Dąbrowski).
Our final achievements:
We presented a comparative analysis of the plethora of nonextensive and/or nonadditive entropies which go beyond the standard Boltzmann–Gibbs formulation. After defining the basic notions of additivity, extensivity, and composability, we discussed the properties of these entropies and their mutual relations, if they exist. The results are presented in two informative tables that are of strong interest to the gravity and cosmology community in the context of the recently intensively explored horizon entropies for black hole and cosmological models. The features of additivity, extensivity, and composability are listed systematically.
We presented a general formalism for studying generalized Holographic Dark Energy (HDE) models in which we use a dimensionless form of the area-entropy of cosmological horizons. We used our formalism for nonextensive horizon entropies of standard HDE (i.e. Bekenstein– Hawking), and generalized such as Barrow/Tsallis–Cirto, Rényi, Sharma–Mittal, and Kaniadakis as dark energy models of the universe and test them by cosmological data. The main data test results are that all the HDE models under study are statistically disfavoured with respect to CDM, though at some different levels. The standard HDE seem to be on the same footing as Rényi, Sharma–Mittal, and Kaniadakis HDE models since the latter include only small deviations from HDE model resulting from the series expansion of their extra nonextensivity parameters. However, Barrow and Tsallis–Cirto models, though still disfavoured against CDM, seem to point out observationally to fulfil a property of extensivity (though still remaining nonadditive). Finally, the Tsallis–Cirto model parameter is pointing towards the CDM limit which is singular also at the expense of having much larger value of the holographic dark energy dimensionless parameter k value higher than other models.
Dissemination:
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- Mariusz P. Dąbrowski, Beyond additivity and extensivity of entropy for black hole and cosmological horizons, Preprints2024, 202408.0564
- Ilim Cimdiker, Mariusz P. Dąbrowski, Vincenzo Salzano. Generalized Nonextensive Entropy Holographic Dark Energy Models Verifies by Cosmological Data, 2503.18230.
- Mariusz P. Dąbrowski, Beyond additivity and extensivity of entropy for black hole and cosmological horizons, Entropy 26, 814 (2024).
- Ilim Cimdiker, Mariusz P. Dąbrowski, Vincenzo Salzano. Generalized Nonextensive Entropy Holographic Dark Energy Models Verifies by Cosmological Data, European Physics Journal 85, 775 (2025) – 13 pages.
Nonextensive Entropies of Black Hole and Cosmological Horizons, Talk at POTOR-10 conference of the Polish Society on Relativity, Kazimierz Dolny, Poland, 18.09.2024.
Nonextensive Entropies as Applied to Black Holes and Cosmology, Invited talk at XLV Max Born Symposium (OPTICS-2025), Wrocław, Poland, 27.06.2025.
Nonextensive Entropies as Holographic Dark Energy in Cosmology, Talk at conference ”Cosmology 2025 @ Elba Island”, Elba, Italy, 8-13.09.2025.
Generalized Nonextensive Entropy Holographic Dark Energy Cosmological Models, Talk at the 33rd Texas Symposium on Relativistic Astrophysics, Tempe, Arizona, USA, 8-12.12.2025.