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Reply

Reply to Pessoa, P.; Arderucio Costa, B. Comment on “Tsallis, C. Black Hole Entropy: A Closer Look. Entropy 2020, 22, 17”

1
Centro Brasileiro de Pesquisas Físicas and National Institute of Science and Technology of Complex Systems, Rua Xavier Sigaud 150, Rio de Janeiro 22290-180, Brazil
2
Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA
3
Complexity Science Hub Vienna, Josefstädter Strasse 39, 1080 Vienna, Austria
Submission received: 13 November 2020 / Revised: 4 February 2021 / Accepted: 8 May 2021 / Published: 19 May 2021

Abstract

:
In the present Reply we restrict our focus only onto the main erroneous claims by Pessoa and Costa in their recent Comment (Entropy 2020, 22, 1110).

1. Relevant Misunderstanding

A severe misunderstanding is present already in the first sentence of the Abstract of the Comment [1]. This also emerges in the Introduction and elsewhere. More precisely, we read in the Abstract: “In a recent paper (Entropy 2020, 22, 17), Tsallis states that entropy—as in Shannon or Kullback–Leiber’s definitions—is inadequate to interpret black hole entropy and suggests that a new non-additive functional should take the role of entropy.”
Quite regretfully, the authors paid no attention at all to a most relevant if. Indeed, as emphasized in [2], and explicitly written in [3], my claim is that “In what concerns thermodynamics, the spatial dimensionality of a ( 3 + 1 ) black hole depends on whether its bulk (inside its event horizon or boundary) has or not non negligible amount of matter or analogous physical information. If that matter or information is non negligible, the thermodynamical entropy of the black hole must scale as L d with d = 3 , where L stands for its linear size. If that matter or information is negligible, the thermodynamic entropy of the black hole must scale as L d with d = 2 . Neither in [2] nor in [3] is a specific position taken for black holes being d = 2 + 1 or d = 3 + 1 physical objects, or even (multi)fractal objects with nonintenger d (which, to the best of our knowledge, may not be excluded). At the present stage, it seems appropriate to analyze this nontrivial and delicate issue within the specialized realm of cosmological and black-hole physics [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48]. However, it is seemingly undeniable that the sort of perplexity expressed in [49,50,51,52], and elsewhere, emerges because, if black holes are thought to be d = 3 objects, their thermodynamical entropy should be proportional to the cube of the radius, and not to its square, as it happens with Boltzmann–Gibbs-based Bekenstein–Hawking entropy. This entropy can be shown to yield S B G k = 1 4 A H L P 2 , where L P h G c 3 P l a n c k l e n g t h . What is basically argued in [2,3] is that the dominant term of the thermodynamical entropy S of a d-dimensional black hole whose event horizon area is A H is expected to satisfy S k A H L P d . Therefore, if d = 2 , we recover the Bekenstein–Hawking entropy, but, if d 2 , a non-BG entropic functional must be used for thermodynamical purposes. The functional S q , δ with specific d-dependent values of ( q , δ ) emerges as a plausible candidate for such a non-BG functional.
Before ending this Section let us emphasize what we precisely mean by “thermodynamical purposes”. We focus here on two specific aspects of this issue, which play the most central role in the present Reply.
(i) Let us assume that a system can be modelled by a long-ranged-interacting many-body problem (e.g., classical models such as the d-dimensional α -XY ferromagnet [53,54], α -Heisenberg ferromagnet [55,56,57,58,59], α -Fermi–Pasta–Ulam model [60,61,62,63,64,65,66,67,68], α -Lennard–Jones gas [69,70,71,72]), N basically being the number of particles. The notation, including α in all of them, comes from the fact that a two-body attractive interaction is assumed in all of them, which asymptotically decays as 1 / r α (with α 0 ), r being the distance. In all such cases, the corresponding Gibbs free energy is given by G ( N ) = U ( N ) T S ( N ) + p V ( N ) μ N H M ( N ) . We now divide by N N ˜ where N ˜ N 1 α / d 1 1 α / d ; N ˜ behaves, for large values of N, as N 1 α / d / ( 1 α / d ) for 0 α / d < 1 (long-range interactions), as 1 / ( α / d 1 ) for α / d > 1 (short-range interactions), and as ln N for α / d = 1 (marginally-ranged interactions). We then obtain g = u T ˜ s + p ˜ v μ ˜ H ˜ m , where g lim N G ( N ) / N N ˜ , u lim N U ( N ) / N N ˜ , s lim N S ( N ) / N , v lim N V ( N ) / N , m lim N M ( N ) / N , T ˜ lim N T / N ˜ , p ˜ lim N p / N ˜ , μ ˜ lim N μ / N ˜ , H ˜ lim N H / N ˜ , …. It should be very clear at this point that all these quantities must be finite if we wish to preserve the entire Legendre structure of thermodynamics. These specific scalings have already been checked and found to be correct, very particularly in order to have finite equations of states, for diverse physical systems, such as a ferrofluid-like model [73], Lennard–Jones-like fluids [69,70,71,72], magnetic systems [74,75,76,77,78,79], anomalous diffusion [80], and percolation [81,82], among others. An immediate corollary is that quantities such as entropy S, d-dimensional volume V, number of particles N, and total magnetization M, belong to the same thermodynamic class, and are extensive independently from the interactions being short- or long-ranged. This is in remarkable contrast with quantities such as G and U, which are extensive for short-ranged interactions (implicit assumption in nearly all textbooks of thermodynamics, although not necessarily emphasizing it, quite regretfully) but super-extensive ( N 2 α / d ) for long-ranged ones. In particular, U ( N ) scales as N 2 for all mean-field models (i.e., α = 0 ), which is the reason for nearly all authors dividing by N the coupling constant of the many-body Hamiltonian. Consistently, quantities such as T ˜ , p ˜ , μ ˜ , H ˜ are intensive, α / d > 0 .
(ii) We provide a brief reminder of the Large Deviation Theory (LDT). If we throw N (say even) independent coins, the probability of having n N / 2 heads is given by P ( N ; n / N < x ) e r 1 ( x ) N , where the rate function r 1 ( x ) is the relative BG entropy per particle and satisfies r 1 ( 1 / 2 ) = 0 . In other words, r 1 ( x ) N corresponds to the total entropy, which is therefore extensive, as is well known for this simple case. This property mirrors the fact that, within BG statistical mechanics, the thermal equilibrium probability associated with Hamiltonian H N is given by p e β H N = e [ β H N / N ] N , where we can see that [ β H N / N ] is an intensive quantity which plays the role of r 1 ( x ) . For usual systems, more specifically for those for which the Central Limit Theorem legitimately applies, both expressions, P ( N ; n / N < x ) e r 1 ( x ) N and p e β H N , still apply. However, if we have relevant nonlocal features, the CLT and the LDT need to be generalized. For nonlocally correlated elements (e.g., for classical many-body systems with 0 α / d < 1 ) the usual CLT and the LDT are not expected to apply. It has been verified in many of such systems that the CLT Gaussian attractor is replaced by a q-Gaussian one with q > 1 (see, for instance, Refs. [83,84] to have a first approach to such anomalies). The corresponding stationary-state distribution optimizing, under appropriate simple constraints, the nonadditive entropy S q becomes p e q β q H N = e q [ ( β q N ˜ ) ( H N / N N ˜ ) ] N , where [ ( β q N ˜ ) ( H N / N N ˜ ) ] = ( 1 / T ˜ ) ( H N / N N ˜ ) is intensive, as shown in point (i) above. For a similar probabilistic system with strongly correlated coins (within a wide class of correlations), it is allowed to expect P ( N ; n / N < x ) e q r q ( x ) N with the q-rate function r q ( x ) being of the order of some appropriate q-relative entropy per particle, and satisfying r q ( 1 / 2 ) = 0 . As we see, if this conjecture is correct, the total entropy corresponds to r q ( x ) N and is, once again, extensive. This conjecture has been numerically verified with high-precision calculations in at least one non-trivial example [85,86,87]; more are coming.

2. About Entropic Additivity and Extensivity

The authors of the Comment write next, in the Abstract: “Here we counterargue by explaining the important distinction between the properties of extensivity and additivity; the latter is fundamental for entropy, while the former is a property of particular thermodynamical systems that is not expected for black holes.”
I could not agree more with Pessoa and Costa about the importance of the distinction between extensivity and additivity, very particularly when entropy is focused on. But their use of the verb ”explaining” appears to differ from that of others. The distinction additivity versus extensivity has been addressed in very many occasions in the context of nonextensive statistical mechanics (q-statistics for short) and nonadditive entropies, e.g., in [88,89] (a wide Bibliography is available at [90]). As transparently defined by Penrose [91], the additivity of an entropic functional S ( { p i } ) is based on the simple mathematical property S ( { p i p j } ) = S ( { p i } ) + S ( { p j } ) , { ( p i , p j ) } , i.e., S ( A + B ) = S ( A ) + S ( B ) , A and B being probabilistically independent systems. Consequently, it is trivially verified that the Boltzmann–Gibbs–von Neumann–Shannon (noted S B G here) and the Renyi entropic functionals are additive, whereas all the others available in the literature are nonadditive, among them S q , S δ and S q , δ [2]. At this point, it is worth stressing that the S q functional has been the object of uniqueness theorems in what concerns (i) the axiomatic formulations by Santos and by Abe [92,93], respectively, generalizing those of Shannon and of Khinchin; (ii) the Topsoe-factorizability [94] in game theory; (iii) the Amari–Ohara–Matsuzoe conformally invariant geometry [95]; (iv) the Biro–Barnafoldi–Van thermostat universal independence [96,97,98,99,100]; (v) the Enciso–Tempesta uniqueness of composable trace-form functionals [101], thus leading to the likelihood factorization required by Einstein [102].
In strong contrast with additivity, the extensivity of an entropy (i.e., 0 < lim L S ( L ) / L d < , where L is the linear size of the d-dimensional system) depends not only on the specific entropic functional but also—and very much so—on the specific system that is being focused on. What Pessoa and Costa definitively appear to miss in their Comment is that entropic additivity is physically subordinated to entropic extensivity, and not the other way around. To better understand in what sense we are using the word ”subordinated” we may refer to an analogous situation, namely the Galilean additivity of velocities ( v 13 = v 12 + v 23 ). This additivity is ”subordinated” to the Lorentz invariance imposed by Einstein in order to unify mechanics with Maxwell equations, which eventually generalized the Galilean additivity into the Einstein composition of velocities involving the vacuum speed of light c ( v 13 = ( v 12 + v 23 ) / ( 1 + v 12 v 23 / c 2 ) ). As is well known, the Einstein composition of velocities recovers the Galilean additivity in the limit 1 / c 0 , fairly similarly to how the nonadditivity of S q recovers the BG additivity in the limit ( 1 q ) / k 0 (let us remind the reader that S q ( A + B ) / k = S q ( A ) / k + S q ( B ) / k + ( 1 q ) [ S q ( A ) / k ] [ S q ( B ) / k ] , hence S q ( A + B ) = S q ( A ) + S q ( B ) + [ ( 1 q ) / k ] S q ( a ) S q ( B ) . The loss of the Galilean additivity is then a small price to pay for making mechanics and Maxwell electromagnetism simultaneously Lorentz-invariant. In analogy, the loss of the BG entropic additivity is, whenever necessary (i.e., whenever there is strong space–time entanglement in the system), a small price to pay for satisfying thermodynamics.
No general physical reason is known to necessarily lead to additive entropies for thermodynamical purposes, whereas entropic extensivity is generically mandated by the Legendre structure of thermodynamics (see [2] and many other references therein). (We read in [1] “even if the entropy were proportional to the total energy, it could still fail to be proportional to the “volume” of the black hole.” Such a sentence jeopardizes the Legendre structure of classical thermodynamics, which obviously imposes that all of its terms scale with size in exactly the same manner. Therefore, the quantities to be legitimately compared are U, T S , p V , μ N , H M , etc. The assumption in [1] about the possibility of the entropy being proportional to the total energy is equivalent to a priori assuming that T is intensive, a hypothesis which rather naively disregards that this issue is a very delicate one, given that, in black holes, we definitively deal with long-range interactions.) The entropic extensivity is presently verified numerically by the possible extension of the Large Deviation Theory to wide classes of strongly correlated systems [85,86,87], apparently also in [103]. The obvious mathematical convenience of using the celebrated additive entropic functional S B G comes from the fact that systems with local or no correlations naturally yield an extensive S B G . In contrast, nonlocal correlations, such as those definitively existing in black holes and in many other systems, lead to a nonextensive entropy S B G . Consequently, its use simply becomes thermodynamically inadmissible. In other words, the total entropy, the total volume, the total number of particles, the total magnetization, always belong to the same class of thermodynamical variables, sharing the property of scaling as L d (under the assumption that d is an integer number). This is in notorious contrast with the total internal and free energies which, as mentioned above, are thermodynamically extensive variables only when, say, long-range interactions are not involved. In this matter, it certainly is historically impressive to verify that Gibbs himself dismissed his own thermostatistical theory in those cases where the partition function diverges (e.g., gravitation) [104]. More details on the failure of BG entropy and corresponding statistical mechanics for gravitational systems can be found in [105,106,107].

3. Other Debatable Statements

Finally, the authors of the Comment conclude their Abstract by writing: “We also point out other debatable statements in his analysis of black hole entropy.”
In this context, several points can be raised, but I will restrict the focus on their statement “we want to refer the reader to authors who have reported that (i) substituting entropy by a non-additive functional leads to inconsistent statistics [12,13,14,15] …” (their references [12,13,14] are references [108,109,110] of the present Reply). Pessoa and Costa apparently base their conviction on the Pressé et al. interpretation of the Shore and Johnson axioms for statistical inference. It happens, however, that they are seemingly unaware that such an interpretation is deeply erroneous. Indeed, this has been discussed more than once in the literature and it has been definitively settled out by Jizba and Korbel [111], who transparently and specifically show, among others, that S q does satisfy the Shore and Johnson axioms. The authors of [1] include, as basic support of their statement about ’inconsistent statistics’, the paper [110] by Pressé et al., but no reference is made to the critical paper [112] (The title of [112] contains an unfortunate inadvertence. A more precise title would have been Conceptual Inadequacy of the Pressé et al. Version of the Shore and Johnson Axioms for Wide Classes of Complex Systems), where physical misconceptions and even a severe mathematical error are revealed in detail. Let us be precise about that. We straightforwardly verify S q ( { u i 2 q v j } ) = k i j ( u i 2 q v j ) ln 2 q ( u i 2 q v j ) = k i j ( u i 2 q v j ) ( ln 2 q u i + ln 2 q v j ) k i j u i v j ( ln 2 q u i + ln 2 q v j ) = k i = 1 W u i ln 2 q u i k j = 1 W v j ln 2 q v j = S q ( { u i } ) + S q ( { v j } ) . The crucial inequality that is present along these lines is, quite inexplicably, violated in [108].
To be more explicit, Pessoa and Costa [1] adopt the following design criteria (DC): DC1—subdomain independence (local information should have only local effects); DC2—subsystem independence (a priori independent subsystems should remain independent, unless the constraints explicitly require otherwise). As they argue, the unique functional that fits these criteria is S B G (and, consistently, the corresponding Kullback–Leibler relative entropy or divergence). It happens, however, that, as it becomes clear within the discussion by Jizba and Korbel in [111], DC1 and DC2 are sufficient but not necessary criteria for the general Shore and Johnson axioms for statistical inference. For a system with very strong space–time entanglement, such as a black-hole, hypotheses DC1–DC2 are unnecessarily restrictive.
Various other issues concerning black holes surely deserve deeper analysis, including the possibility of unification with the so-called ”area law” for strongly quantum-entangled systems, through a (conjectural) expression, such as S B G ( L ) L d 1 1 d 1 ( L ) , which yields S B G ( L ) ln L for d 1 and S B G ( L ) L d 1 for d > 1 (see [113,114] and references therein). However, such challenging open problems are out of the scope of the present Reply. (Finally, a misprint appears above Equation (1) in [1], which reads ”maximization”, but should read ”minimization”.)

Funding

This research received no external funding.

Acknowledgments

I warmly acknowledge most interesting exchanges on the subject of black holes and its thermostatistics with Gerard’t Hooft in Utrecht, Erice, Rio de Janeiro and private correspondence, as well as with Evaldo M.F. Curado, Jose Ademir S. Lima and Angel R. Plastino in Rio de Janeiro. I also acknowledge partial financial support by CNPq and FAPERJ (Brazilian agencies).

Conflicts of Interest

The author declares no conflict of interest.

References and Note

  1. Pessoa, P.; Costa, B.A. Comment on “Black hole entropy: A closer look”. Entropy 2020, 22, 1110. [Google Scholar] [CrossRef]
  2. Tsallis, C.; Cirto, L.J.L. Black hole thermodynamical entropy. Eur. Phys. J. C 2013, 73, 2487. [Google Scholar] [CrossRef]
  3. Tsallis, C. Black hole entropy: A closer look. Entropy 2020, 22, 17. [Google Scholar] [CrossRef] [PubMed] [Green Version]
  4. Wald, R.M. The thermodynamics of black holes. Living Rev. Relativ. 2001, 4, 6. [Google Scholar] [CrossRef] [PubMed] [Green Version]
  5. Mathur, S.D. The information paradox and the infall problem. Class. Quantum Grav. 2011, 28, 125010. [Google Scholar] [CrossRef]
  6. Hayden, P.; Headrick, M.; Maloney, A. Holographic mutual information is monogamous. Phys. Rev. D 2013, 87, 046003. [Google Scholar] [CrossRef] [Green Version]
  7. Komatsu, N.; Kimura, S. Entropic cosmology for a generalized black-hole entropy. Phys. Rev. D 2013, 88, 083534. [Google Scholar] [CrossRef] [Green Version]
  8. Komatsu, N.; Kimura, S. Evolution of the universe in entropic cosmologies via different formulations. Phys. Rev. D 2014, 89, 123501. [Google Scholar] [CrossRef] [Green Version]
  9. Bao, N.; Nezami, S.; Ooguri, H.; Stoica, B.; Sully, J.; Walter, M. The holographic entropy cone. J. High Energy Phys. 2015, 9, 130. [Google Scholar] [CrossRef]
  10. Dong, X. The gravity dual of Renyi entropy. Nat. Comm. 2016, 7, 12472. [Google Scholar] [CrossRef] [Green Version]
  11. Majhi, A. Non-extensive statistical mechanics and black hole entropy from quantum geometry. Phys. Lett. B 2017, 775, 32. [Google Scholar] [CrossRef]
  12. Cabero, M.; Capano, C.D.; Fischer-Birnholtz, O.; Krishnan, B.; Nielsen, A.B.; Nitz, A.H.; Biwer, C.M. Observational tests of the black hole area increase law. Phys. Rev. D 2018, 97, 124069. [Google Scholar] [CrossRef] [Green Version]
  13. Sheykhi, A. Modified Friedmann equations from Tsallis entropy. Phys. Lett. B 2018, 785, 118. [Google Scholar] [CrossRef]
  14. Lymperis, A.; Saridakis, E.N. Modified cosmology through nonextensive horizon thermodynamics. Eur. Phys. J. C 2018, 78, 993. [Google Scholar] [CrossRef] [Green Version]
  15. Ghaffari, S.; Moradpour, H.; Lobo, I.P.; Graca, J.P.M.; Bezerra, V.B. Tsallis holographic dark energy in the Brans-Dicke cosmology. Eur. Phys. J. C 2018, 78, 706. [Google Scholar] [CrossRef]
  16. Zadeh, M.A.; Sheykhi, A.; Moradpour, H.; Bamba, K. Tsallis holographic dark energy. Phys. Lett. B 2018, 781, 195–200. [Google Scholar] [CrossRef]
  17. Zadeh, M.A.; Sheykhi, A.; Moradpour, H. Thermal stability of Tsallis holographic dark energy in nonflat universe. Gen. Relativ. Gravit. 2019, 51, 12. [Google Scholar] [CrossRef]
  18. Sharma, U.K.; Pradhan, A. Diagnosing Tsallis holographic dark energy models with statefinder and ω-ω pair. Mod. Phys. Lett. A 2019, 34, 1950101. [Google Scholar] [CrossRef]
  19. Korunur, M. Tsallis holographic dark energy in Bianchi type-III spacetime with scalar fields. Mod. Phys. Lett. A 2019, 34, 1950310. [Google Scholar] [CrossRef]
  20. Dubey, V.C.; Srivastava, S.; Sharma, U.K.; Pradhan, A. Tsallis holographic dark energy in Bianchi-I Universe using hybrid expansion law with k-essence. Pramana J. Phys. 2019, 93, 78. [Google Scholar] [CrossRef]
  21. Zhang, N.; Wu, Y.B.; Chi, J.N.; Yu, Z.; Xu, D.F. Diagnosing Tsallis holographic dark energy models with interactions. Mod. Phys. Lett. A 2019, 35, 2050044. [Google Scholar] [CrossRef] [Green Version]
  22. Huang, Q.; Huang, H.; Chen, J.; Zhang, L.; Tu, F. Stability analysis of the Tsallis holographic dark energy model. Class. Quantum Grav. 2019, 36, 175001. [Google Scholar] [CrossRef]
  23. Aditya, Y.; Mandal, S.; Sahoo, P.K.; Reddy, D.R.K. Observational constraint on interacting Tsallis holographic dark energy in logarithmic Brans-Dicke theory. Eur. Phys. J. C 2019, 79, 1020. [Google Scholar] [CrossRef]
  24. Ghaffari, S.; Ziaie, A.H.; Moradpour, H.; Asghariyan, F.; Feleppa, F.; Tavayef, M. Black hole thermodynamics in Sharma-Mittal generalized entropy formalism. Gen. Relativ. Gravit. 2019, 51, 93. [Google Scholar] [CrossRef] [Green Version]
  25. Dixit, A.; Sharma, U.K.; Pradhan, A. Tsallis holographic dark energy in FRW universe with time varying deceleration parameter. New Astron. 2019, 73, 101281. [Google Scholar] [CrossRef]
  26. Almheiri, A.; Engelhardt, N.; Marolfd, D.; Maxfield, H. The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole. J. High Energy Phys. 2019, 12, 063. [Google Scholar] [CrossRef] [Green Version]
  27. Almheiri, A.; Hartman, T.; Maldacena, J.; Shaghoulian, E.; Tajdini, A. Replica wormholes and the entropy of Hawking radiation. J. High Energy Phys. 2020, 5, 13. [Google Scholar] [CrossRef]
  28. Dixit, A.; Bhardwaj, V.K.; Pradhan, A. RHDE models in FRW Universe with two IR cut-offs with redshift parametrization. Eur. Phys. J. Plus 2020, 135, 831. [Google Scholar] [CrossRef]
  29. Jawad, A.; Rani, S.; Hussain, M.H. Cosmological implications and thermodynamics of some reconstructed modified gravity models. Phys. Dark Universe 2020, 27, 100409. [Google Scholar] [CrossRef]
  30. Varshney, G.; Sharma, U.K.; Pradhan, A. Reconstructing the k-essence and the dilation field models of the THDE in f(R,T) gravity. Eur. Phys. J. Plus 2020, 135, 541. [Google Scholar] [CrossRef]
  31. Prasanthi, U.Y.D.; Aditya, Y. Anisotropic Renyi holographic dark energy models in general relativity. Results Phys. 2020, 17, 103101. [Google Scholar] [CrossRef]
  32. Bhattacharjee, S. Interacting Tsallis and Renyi holographic dark energy with hybrid expansion law. Astrophys. Space Sci. 2020, 365, 103. [Google Scholar] [CrossRef]
  33. Sharma, U.K.; Dubey, V.C.; Pradhan, A. Diagnosing interacting Tsallis holographic dark energy in the non-flat universe. Int. J. Geom. Method Mod. Phys. 2020, 17, 2050032. [Google Scholar] [CrossRef] [Green Version]
  34. Srivastava, V.; Sharma, U.K. Statefinder hierarchy for Tsallis holographic dark energy. New Astron. 2020, 78, 101380. [Google Scholar] [CrossRef]
  35. Saha, A.; Ghose, S. Interacting Tsallis holographic dark energy in higher dimensional cosmology. Astrophys. Space Sci. 2020, 365, 98. [Google Scholar] [CrossRef]
  36. Sharma, U.K.; Dubey, V.C. Statefinder diagnostic for the Renyi holographic dark energy. New Astron. 2020, 80, 101419. [Google Scholar] [CrossRef]
  37. Srivastava, S.; Dubey, V.C.; Sharma, U.K. Statefinder diagnosis for Tsallis agegraphic dark energy model with ω-ω pair. Intern. J. Mod. Phys. A 2020, 35, 2050027. [Google Scholar] [CrossRef]
  38. Sharma, U.K.; Dubey, V.C. Exploring the Sharma-Mittal HDE models with different diagnostic tools. Eur. Phys. J. Plus 2020, 135, 391. [Google Scholar] [CrossRef]
  39. Waheeda, S. Reconstruction paradigm in a class of extended teleparallel theories using Tsallis holographic dark energy. Eur. Phys. J. Plus 2020, 135, 11. [Google Scholar] [CrossRef]
  40. Sharma, U.K.; Srivastava, S.; Beesham, A. Swampland criteria and cosmological behavior of Tsallis holographic dark energy in Bianchi-III universe. Intern. J. Geom. Methods Mod. Phys. 2020, 17, 2050098. [Google Scholar] [CrossRef]
  41. Geng, C.Q.; Hsu, Y.T.; Lu, J.R.; Yin, L. Modified cosmology models from thermodynamical approach. Eur. Phys. J. C 2020, 80, 21. [Google Scholar] [CrossRef] [Green Version]
  42. Nojiri, S.; Odintsov, S.D.; Saridakis, E.N.; Myrzakulov, R. Correspondence of cosmology from non-extensive thermodynamics with fluids of generalized equation of state. Nucl. Phys. B 2020, 950, 114850. [Google Scholar] [CrossRef]
  43. Abreu, E.M.C.; Neto, J.A. Barrow black hole corrected-entropy model and Tsallis nonextensivity. Phys. Lett. B 2020, 810, 135805. [Google Scholar] [CrossRef]
  44. Mamon, A.A. Study of Tsallis holographic dark energy model in the framework of fractal cosmology. Mod. Phys. Lett. A 2020, 35, 2050251. [Google Scholar] [CrossRef]
  45. Ens, P.S.; Santos, A.F. f(R) gravity and Tsallis holographic dark energy. EPL Europhys. Lett. 2020, 131, 40007. [Google Scholar] [CrossRef]
  46. Promsiri, C.; Hirunsirisawat, E.; Liewrian, W. Thermodynamics and van der Waals phase transition of charged black holes in flat space via Renyi statistics. Phys. Rev. D 2020, 102, 064014. [Google Scholar] [CrossRef]
  47. Penington, G.; Shenker, S.H.; Stanford, D.; Yang, Z. Replica wormholes and the black hole interior. arXiv 2020, arXiv:1911.11977. [Google Scholar]
  48. Sharma, U.K.; Srivastava, V. Tsallis HDE with an IR cutoff as Ricci horizon in a flat FLRW Universe. New Astron. 2021, 84, 101519. [Google Scholar] [CrossRef]
  49. Hawking, S.W. Black holes and thermodynamics. Phys. Rev. D 1976, 13, 193. [Google Scholar] [CrossRef]
  50. Maddox, J. When entropy does not seem extensive. Nature 1993, 365, 103. [Google Scholar] [CrossRef]
  51. Das, S.; Shankaranarayanan, S. How robust is the entanglement entropy-area relation? Phys. Rev. D 2006, 73, 121701. [Google Scholar] [CrossRef] [Green Version]
  52. Kolekar, S.; Padmanabhan, T. Ideal gas in a strong gravitational field: Area dependence of entropy. Phys. Rev. D 2011, 83, 064034. [Google Scholar] [CrossRef] [Green Version]
  53. Cirto, L.J.L.; Assis, V.R.V.; Tsallis, C. Influence of the interaction range on the thermostatistics of a classical many-body system. Physica A 2014, 393, 286–296. [Google Scholar] [CrossRef] [Green Version]
  54. Cirto, L.J.L.; Rodriguez, A.; Nobre, F.D.; Tsallis, C. Validity and failure of the Boltzmann weight. EPL Europhys. Lett 2018, 123, 30003. [Google Scholar] [CrossRef] [Green Version]
  55. Nobre, F.D.; Tsallis, C. Classical infinite-range-interaction Heisenberg ferromagnetic model: Metastability and sensitivity to initial conditions. Phys. Rev. E 2003, 68, 036115. [Google Scholar] [CrossRef] [Green Version]
  56. Nobre, F.D.; Tsallis, C. Metastable states of the classical inertial infinite-range-interaction Heisenberg ferromagnet: Role of initial conditions. Physica A 2004, 344, 587. [Google Scholar] [CrossRef] [Green Version]
  57. Cirto, L.J.L.; Lima, L.S.; Nobre, F.D. Controlling the range of interactions in the classical inertial ferromagnetic Heisenberg model: Analysis of metastable states. J. Stat. Mech. 2015, 2015, P04012. [Google Scholar] [CrossRef] [Green Version]
  58. Rodriguez, A.; Nobre, F.D.; Tsallis, C. d-Dimensional classical Heisenberg model with arbitrarily-ranged interactions: Lyapunov exponents and distributions of momenta and energies. Entropy 2019, 21, 31. [Google Scholar] [CrossRef] [Green Version]
  59. Rodriguez, A.; Nobre, F.D.; Tsallis, C. Quasi-stationary-state duration in d-dimensional long-range model. Phys. Rev. Res. 2020, 2, 023153. [Google Scholar] [CrossRef]
  60. Christodoulidi, H.; Tsallis, C.; Bountis, T. Fermi-Pasta-Ulam model with long-range interactions: Dynamics and thermostatistics. EPL Europhys. Lett 2014, 108, 40006. [Google Scholar] [CrossRef] [Green Version]
  61. Christodoulidi, H.; Bountis, T.; Tsallis, C.; Drossos, L. Dynamics and Statistics of the Fermi–Pasta–Ulam β–model with different ranges of particle interactions. J. Stat. Mech. Theory Exp. 2016, 2016, 123206. [Google Scholar] [CrossRef] [Green Version]
  62. Bagchi, D.; Tsallis, C. Sensitivity to initial conditions of d-dimensional long-range-interacting quartic Fermi-Pasta-Ulam model: Universal scaling. Phys. Rev. E 2016, 93, 062213. [Google Scholar] [CrossRef] [PubMed] [Green Version]
  63. Bagchi, D.; Tsallis, C. Long-ranged Fermi-Pasta-Ulam systems in thermal contact: Crossover from q-statistics to Boltzmann-Gibbs statistics. Phys. Lett. A 2017, 381, 1123–1128. [Google Scholar] [CrossRef]
  64. Bagchi, D. Thermal transport in the Fermi-Pasta-Ulam model with long-range interactions. Phys. Rev. E 2017, 95, 032102. [Google Scholar] [CrossRef] [Green Version]
  65. Bagchi, D.; Tsallis, C. Fermi-Pasta-Ulam-Tsingou problems: Passage from Boltzmann to q-statistics. Physica A 2018, 491, 869–873. [Google Scholar] [CrossRef] [Green Version]
  66. Macias-Diaz, J.E.; Bountis, A. Supratransmission in β-Fermi-Pasta-Ulam chains with different ranges of interactions. Commun. Nonlinear Sci. Numer. Simul. 2018, 63, 307–321. [Google Scholar] [CrossRef]
  67. Carati, A.; Galgani, L.; Gangemi, F.; Gangemi, R. Relaxation times and ergodicity properties in a realistic ionic-crystal model, and the modern form of the FPU problem. Physica A 2019, 532, 121911. [Google Scholar] [CrossRef] [Green Version]
  68. Carati, A.; Galgani, L.; Gangemi, F.; Gangemi, R. Approach to equilibrium via Tsallis distributions in a realistic ionic-crystal model and in the FPU model. Eur. Phys. J. Spec. Top. 2020, 229, 743–749. [Google Scholar] [CrossRef]
  69. Grigera, J.R. Extensive and non-extensive thermodynamics. A molecular dynamic test. Phys. Lett. A 1996, 217, 47. [Google Scholar] [CrossRef]
  70. Curilef, S.; Tsallis, C. Critical temperature and nonextensivity in long-range-interacting Lennard-Jones-like fluids. Phys. Lett. A 1999, 264, 270. [Google Scholar] [CrossRef] [Green Version]
  71. Borges, E.P.; Tsallis, C. Negative specific heat in a Lennard-Jones-like gas with long-range interactions. Physica A 2002, 305, 148–151. [Google Scholar] [CrossRef] [Green Version]
  72. Kadijani, M.N.; Abbasi, H.; Nezamipour, S. Molecular dynamics simulation of gas models of Lennard-Jones type of interactions: Extensivity associated with interaction range and external noise. Physica A 2017, 475, 35–45. [Google Scholar] [CrossRef]
  73. Jund, P.; Kim, S.G.; Tsallis, C. Crossover from extensive to nonextensive behavior driven by long-range interactions. Phys. Rev. B 1995, 52, 50. [Google Scholar] [CrossRef] [PubMed]
  74. Cannas, S.A.; Tamarit, F.A. Long-range interactions and non-extensivity in ferromagnetic spin systems. Phys. Rev. B 1996, 54, R12661. [Google Scholar] [CrossRef] [PubMed] [Green Version]
  75. Sampaio, L.C.; de Albuquerque, M.P.; Menezes, F.S.D. Nonextensivity and Tsallis statistics in magnetic systems. Phys. Rev. B 1997, 55, 5611. [Google Scholar] [CrossRef] [Green Version]
  76. Anteneodo, C.; Tsallis, C. Breakdown of the exponential sensitivity to the initial conditions: Role of the range of the interaction. Phys. Rev. Lett. 1998, 80, 5313. [Google Scholar] [CrossRef]
  77. Campa, A.; Giansanti, A.; Moroni, D.; Tsallis, C. Classical spin systems with long-range interactions: Universal reduction of mixing. Phys. Lett. A 2001, 286, 251–256. [Google Scholar] [CrossRef] [Green Version]
  78. Andrade, R.F.S.; Pinho, S.T.R. Tsallis scaling and the long-range Ising chain: A transfer matrix approach. Phys. Rev. E 2005, 71, 026126. [Google Scholar] [CrossRef] [PubMed]
  79. Del Pino, L.A.; Troncoso, P.; Curilef, S. Thermodynamics from a scaling Hamiltonian. Phys. Rev. B 2007, 76, 172402. [Google Scholar] [CrossRef] [Green Version]
  80. Condat, C.A.; Rangel, J.; Lamberti, P.W. Anomalous diffusion in the nonasymptotic regime. Phys. Rev. E 2002, 65, 026138. [Google Scholar] [CrossRef]
  81. Rego, H.H.A.; Lucena, L.S.; da Silva, L.R.; Tsallis, C. Crossover from extensive to nonextensive behavior driven by long-range d=1 bond percolation. Physica A 1999, 266, 42. [Google Scholar] [CrossRef]
  82. Fulco, U.L.; da Silva, L.R.; Nobre, F.D.; Rego, H.H.A.; Lucena, L.S. Effects of site dilution on the one-dimensional long-range bond-percolation problem. Phys. Lett. A 2003, 312, 331. [Google Scholar] [CrossRef]
  83. Umarov, S.; Tsallis, C.; Steinberg, S. On a q-central limit theorem consistent with nonextensive statistical mechanics. Milan J. Math. 2008, 76, 307–328. [Google Scholar] [CrossRef]
  84. Umarov, S.; Tsallis, C.; Gell-Mann, M.; Steinberg, S. Generalization of symmetric α-stable Lévy distributions for q>1. J. Math. Phys. 2010, 51, 033502. [Google Scholar] [CrossRef] [PubMed] [Green Version]
  85. Ruiz, G.; Tsallis, C. Towards a large deviation theory for strongly correlated systems. Phys. Lett. A 2012, 376, 2451–2454. [Google Scholar] [CrossRef]
  86. Touchette, H. Comment on “Towards a large deviation theory for strongly correlated systems”. Phys. Lett. A 2013, 377, 436–438. [Google Scholar] [CrossRef] [Green Version]
  87. Ruiz, G.; Tsallis, C. Reply to Comment on “Towards a large deviation theory for strongly correlated systems”. Phys. Lett. A 2013, 377, 491–495. [Google Scholar] [CrossRef] [Green Version]
  88. Tsallis, C.; Gell-Mann, M.; Sato, Y. Asymptotically scale-invariant occupancy of phase space makes the entropy Sq extensive. Proc. Natl. Acad. Sci. USA 2005, 102, 15377. [Google Scholar] [CrossRef] [Green Version]
  89. Tsallis, C. Introduction to Nonextensive Statistical Mechanics—Approaching a Complex World; Springer: New York, NY, USA, 2009. [Google Scholar]
  90. Nonextensive Statistical Mechanics and Thermodynamics. Available online: http://tsallis.cat.cbpf.br/biblio.htm (accessed on 15 April 2021).
  91. Penrose, O. Foundations of Statistical Mechanics: A Deductive Treatment; Pergamon: Oxford, UK, 1970; p. 167. [Google Scholar]
  92. Santos, R.J.V. Generalization of Shannon’ s theorem for Tsallis entropy. J. Math. Phys. 1997, 38, 4104. [Google Scholar] [CrossRef]
  93. Abe, S. Axioms and uniqueness theorem for Tsallis entropy. Phys. Lett. A 2000, 271, 74. [Google Scholar] [CrossRef] [Green Version]
  94. Topsoe, F. Factorization and escorting in the game-theoretical approach to non-extensive entropy measures. Physica A 2006, 365, 91–95. [Google Scholar] [CrossRef] [Green Version]
  95. Amari, S.; Ohara, A.; Matsuzoe, H. Geometry of deformed exponential families: Invariant, dually-flat and conformal geometries. Physica A 2012, 391, 4308–4319. [Google Scholar] [CrossRef]
  96. Biro, T.S.; Van, P. Zeroth law compatibility of nonadditive thermodynamics. Phys. Rev. E 2011, 83, 061147. [Google Scholar] [CrossRef] [PubMed] [Green Version]
  97. Van, P.; Barnafoldi, G.G.; Biro, T.S.; Urmossy, K. Nonadditive thermostatistics and thermodynamics. J. Phys. Conf. Ser. 2012, 394, 012002. [Google Scholar] [CrossRef]
  98. Biro, T.S.; Barnafoldi, G.G.; Van, P. Quark-gluon plasma connected to finite heat bath. Eur. Phys. J. A 2013, 49, 110. [Google Scholar] [CrossRef] [Green Version]
  99. Biro, T.S.; Barnafoldi, G.G.; Van, P.; Urmossy, K. Statistical power law due to reservoir fluctuations and the universal thermostat independence principle. Entropy 2014, 16, 6497–6514. [Google Scholar] [CrossRef]
  100. Biro, T.S.; Barnafoldi, G.G.; Van, P. New entropy formula with fluctuating reservoir. Physica A 2015, 417, 215–220. [Google Scholar] [CrossRef] [Green Version]
  101. Enciso, A.; Tempesta, P. Uniqueness and characterization theorems for generalized entropies. J. Stat. Mech. 2017, 2017, 123101. [Google Scholar] [CrossRef] [Green Version]
  102. Tsallis, C.; Haubold, H.J. Boltzmann-Gibbs entropy is sufficient but not necessary for the likelihood factorization required by Einstein. EPL Europhys. Lett. 2015, 110, 30005. [Google Scholar] [CrossRef]
  103. Tirnakli, U.; Tsallis, C.; Ay, N. Approaching a large deviation theory for complex systems. arXiv 2010, arXiv:2010.09508v1. [Google Scholar]
  104. In the book of J.W. Gibbs, Elementary Principles in Statistical Mechanics—Developed with Especial Reference to the Rational Foundation of Thermodynamics (C. Scribner’s Sons: New York, NY, USA, 1902; Yale University Press: New Haven, CT, USA, 1948; OX Bow Press, Woodbridge, CT, USA, 1981), and also The Collected Works. Vol. 1. Thermodynamics (Yale University Press, 1948), Gibbs wrote: “In treating of the canonical distribution, we shall always suppose the multiple integral in Equation (92) [the partition function, as we call it nowadays] to have a finite value, as otherwise the coefficient of probability vanishes, and the law of distribution becomes illusory. This will exclude certain cases, but not such apparently, as will affect the value of our results with respect to their bearing on thermodynamics. It will exclude, for instance, cases in which the system or parts of it can be distributed in unlimited space [...]. It also excludes many cases in which the energy can decrease without limit, as when the system contains material points which attract one another inversely as the squares of their distances. [...]. For the purposes of a general discussion, it is sufficient to call attention to the assumption implicitly involved in the formula (92).”
  105. Antonov, V.A. Solution of the problem of stability of stellar system Emden’s density law and the spherical distribution of velocities. Vestn. Leningr. Univ. 1962, 7, 135, English Translation in Dynamics of Star Clusters; Goodman, J., Hut, P., Ed.; Reidel: Dordrecht, The Netherlands, 1985. [Google Scholar] [CrossRef] [Green Version]
  106. Plastino, A.R.; Plastino, A. Stellar polytropes and Tsallis’ entropy. Phys. Lett. A 1993, 174, 384. [Google Scholar] [CrossRef]
  107. Lynden-Bell, D. Negative specific heat in Astronomy, Physics and Chemistry. Physica A 1999, 263, 293–304. [Google Scholar] [CrossRef] [Green Version]
  108. Pressé, S.; Ghosh, K.; Lee, J.; Dill, K. Nonadditive entropies yield probability distributions with biases not warranted by the data. Phys. Rev. Lett. 2013, 111, 180604. [Google Scholar] [CrossRef]
  109. Pressé, S. Nonadditive entropy maximization is inconsistent with Bayesian updating. Phys. Rev. E 2015, 90, 052149. [Google Scholar] [CrossRef] [PubMed] [Green Version]
  110. Pressé, S.; Ghosh, K.; Lee, J.; Dill, K. Reply to C. Tsallis’ “Conceptual inadequacy of the Shore and Johnson axioms for wide classes of complex systems”. Entropy 2015, 17, 5043–5046. [Google Scholar] [CrossRef] [Green Version]
  111. Jizba, P.; Korbel, J. Maximum entropy principle in statistical inference: Case for non-Shannonian entropies. Phys. Rev. Lett. 2019, 122, 120601. [Google Scholar] [CrossRef] [PubMed] [Green Version]
  112. Tsallis, C. Conceptual inadequacy of the Shore and Johnson axioms for wide classes of complex systems. Entropy 2015, 17, 2853–2861. [Google Scholar] [CrossRef] [Green Version]
  113. Caruso, F.; Tsallis, C. Nonadditive entropy reconciles the area law in quantum systems with classical thermodynamics. Phys. Rev. E 2008, 78, 021102. [Google Scholar] [CrossRef] [Green Version]
  114. Souza, A.M.C.; Rapcan, P.; Tsallis, C. Area-law-like systems with entangled states can preserve ergodicity. Eur. Phys. J. Spec. Top. 2020, 229, 759–772. [Google Scholar] [CrossRef]
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Tsallis, C. Reply to Pessoa, P.; Arderucio Costa, B. Comment on “Tsallis, C. Black Hole Entropy: A Closer Look. Entropy 2020, 22, 17”. Entropy 2021, 23, 630. https://0-doi-org.brum.beds.ac.uk/10.3390/e23050630

AMA Style

Tsallis C. Reply to Pessoa, P.; Arderucio Costa, B. Comment on “Tsallis, C. Black Hole Entropy: A Closer Look. Entropy 2020, 22, 17”. Entropy. 2021; 23(5):630. https://0-doi-org.brum.beds.ac.uk/10.3390/e23050630

Chicago/Turabian Style

Tsallis, Constantino. 2021. "Reply to Pessoa, P.; Arderucio Costa, B. Comment on “Tsallis, C. Black Hole Entropy: A Closer Look. Entropy 2020, 22, 17”" Entropy 23, no. 5: 630. https://0-doi-org.brum.beds.ac.uk/10.3390/e23050630

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