It is often stated that heat baths with finite degrees of freedom i.e., finite baths, are sources of Tsallis distributions for classical Hamiltonian systems. By using well-known fundamental statistical mechanics expressions, we rigorously show that Tsallis distributions with fat tails are possible only for finite baths with constant negative heat capacity, while constant positive heat capacity finite baths yield decays with sharp cutoff with no fat tails. However, the correspondence between Tsallis distributions and finite baths holds at the expense of violating the equipartition theorem for finite classical systems at equilibrium. We comment on the implications of the finite bath for the recent attempts towards a q-generalized central limit theorem.

It is well-known that the partition function can consistently be factorized from the canonical equilibrium distribution obtained through the maximization of the Shannon entropy. We show that such a normalized and factorized equilibrium distribution is warranted if and only if the entropy measure I{(p)} has an additive slope i.e. ∂I{(p)}/∂pi when the ordinary linear averaging scheme is used. Therefore, we conclude that the maximum entropy principle of Jaynes should not be used for the justification of the partition functions and the concomitant thermodynamic observables for generalized entropies with non-additive slope subject to linear constraints. Finally, Tsallis and Rényi entropies are shown not to yield such factorized canonical-like distributions.

The existence and exact form of the continuum expression of the discrete nonlogarithmic q-entropy is an important open problem in generalized thermostatistics, since its possible lack implies that nonlogarithmic qentropy is irrelevant for the continuous classical systems. 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. Among such criticisms, one can particularly cite the ones related to the Bayesian updating procedure [14], Lesche stability [15][16][17], and the methodology of the entropy maximization [18].Recently, Abe pinpointed that the nonadditive q-entropy is inherently limited to the finite discrete systems, since its continuum expression has not been obtained yet [19] (see also Refs. [20,21]). In this work, we show that one can indeed obtain the concomitant continuum expressions of the nonadditive entropy and therefore point out that the nonadditive q-entropy can also be used for continuous physical systems.Before proceeding further with the nonadditive case, one should be convinced why taking the route from discreteness to a continuum is essential concerning any entropy measure in general. Setting the Boltzmann constant to unity, the finite discrete Boltzmann-Gibbs (BG) entropy readswhere p i denotes the probability of the ith event. Let us now consider its continuous counterpart to be the following expressionwhere ρ(x) is a probability density function satisfying the normalization condition in the interval [a, b]. Although the continuous expression above seems reasonable at first sight, it has three serious drawbacks. First, the continuous version in Eq. (2) has an overall unit of log(length) whereas the discrete entropy in Eq. (1) is dimensionless [22]. Second, the probability density S(ρ) is not invariant with respect to coordinate transformations [22]. Last but not the least, the discrete BG entropy S({p}) in the n → ∞ limit and S(ρ) yield different results [22]: To see this more explicitly, consider a uniform distribution ρ(x) in the interval [a, b] as 1/(b − a) so that its discrete counterpart p(x i ) is given by 1/n obtained through dividing the same interval [a, b] into n equal subintervals where the index i runs from 1 to n. Then, the continuous entropy S(ρ) for this uniform distribution yields ln(b − a) while the discrete expression S({p}) attains infinity in the n → ∞ limit. In other words, the ...

We first observe that the (co)domains of the q-deformed functions are some subsets of the (co)domains of their ordinary counterparts, thereby deeming the deformed functions to be incomplete. In order to obtain a complete definition of q-generalized functions, we calculate the dual mapping function, which is found equal to the otherwise ad hoc duality relation between the ordinary and escort stationary distributions. Motivated by this fact, we show that the maximization of the Tsallis entropy with the complete q-logarithm and q-exponential implies the use of the ordinary probability distributions instead of escort distributions. Moreover, we demonstrate that even the escort stationary distributions can be obtained through the use of the ordinary averaging procedure if the argument of the q-exponential lies in (-∞, 0].