Validity of the third law of thermodynamics for the Tsallis entropy

Abstract

Bento et al. [Phys. Rev. E 91, 022105 (2015)] recently stated that the Tsallis entropy violates the third law of thermodynamics for 0 < q < 1 in the sub-additive regime. We first show that the division between the regimes q < 1 and q > 1 is already inherent in the fundamental incomplete structure of the deformed logarithms and exponentials underlying the Tsallis entropy. Then, we provide the complete deformed functions and show that the Tsallis entropy conforms to the third law of thermodynamics for both super-additive q < 1 and sub-additive q > 1 regimes. Finally, we remark that the Tsallis entropy does not require the use of escort-averaging scheme once it is expressed in terms of the complete deformed functions.

The third law of thermodynamics and the fractional entropies

“…However, it is important to see whether these entropies have any thermodynamic meaning for a possible use in the context of a generalized statistical mechanics. The third law of thermodynamics states that the entropy should be zero if and only if the (absolute) temperature is zero and serves as an important test for the generalized entropies [28,29]. It is also quite general in its scope, since it should be valid independent of the Hamiltonian of a specific physical system.…”

“…Note that the third law dictates that the diverging temperature occurs if and only if when the entropy vanishes [28]. Before considering the Ubriaco and Machado entropies, we also note that the Tsallis entropy has recently been shown to conform to the third law for its whole interval of convergence while the Renyi entropy satisfies the third law of thermodynamics only in the region where it is neither concave nor convex [29].…”

Group theory, entropy and the third law of thermodynamics

“…However, some restrictions should be imposed on any entropy measure if it is to be used in the context of thermodynamics. In this sense, the third law of thermodynamics is very selective in determining the interval of the validity of the parameter-dependent entropies, and therefore it serves as a check for the generalized entropies [36,37,43]. Moreover, since the third law should be obeyed independent of the type of interaction the system undergoes, its scope is very general.…”

Route from discreteness to the continuum for the Tsallis q -entropy

“…In this work, we show how the discrete nonlogarithmic q-entropy in fact converges in the continuous limit and the negative of the q-entropy with continuous variables is demonstrated to lead to the (Csiszár type) q-relative entropy just as the relation between the continuous Boltzmann-Gibbs expression and the Kullback-Leibler relative entropy. As a result, we conclude that there is no obstacle for the applicability of the q-entropy to the continuous classical physical systems.Since its advent, the nonadditive q-entropy [1, 2] has found numerous fields of application in many diverse fields [3][4][5][6][7][8][9][10][11][12][13]. Despite this apparent progress in the field, however, there have been some criticisms regarding its applicability and scope.…”

“…Since its advent, the nonadditive q-entropy [1,2] has found numerous fields of application in many diverse fields [3][4][5][6][7][8][9][10][11][12][13]. Despite this apparent progress in the field, however, there have been some criticisms regarding its applicability and scope.…”