Reply to "Rescuing the MaxEnt treatment for $q$-generalized entropies" by A. Plastino and M.C. Rocca

TL;DR

This paper defends a previous work on generalized entropies by clarifying that a recent criticism based on different calculus methods is irrelevant due to fundamental differences in the initial assumptions and derivations.

Contribution

The authors clarify that their original entropy maximization approach is unaffected by the criticized calculus method, reaffirming the validity of their results.

Findings

Their initial formalism requires rac{ S_q}{ U} = eta.

The criticized formalism yields rac{ S_q}{ U} = q eta Z^{1-q}.

The criticism is irrelevant due to these fundamental differences.

Abstract

Plastino and Rocca [Physica A 491, 1023 (2018)] recently criticized our work [Phys. Lett. A 381, 207 (2017)] on the ground that one should use functional calculus instead of the ordinary calculus adopted by us in the entropy maximization procedure. We simply point out that our work requires right from the beginning $\partial S_q / \partial U = \beta$, whereas the formalism of Plastino and Rocca yields $\partial S_q/\partial U = q \beta Z^{1-q} \neq \beta$. Therefore, the work of Plastino and Rocca is irrelevant for our work.

Equations

$$\frac{q}{1-q}{P}_i^{q-1}+{\lambda }_1{U}_i+{\lambda }_2=0.$$

$$S_q=\beta U{Z}^{1-q}+\frac{Z^{1-q}-1}{1-q},$$

$$\frac{\partial S_q}{\partial U}=q\beta Z^{1-q}.$$

Full text

Entropy maximization, Functional analysis, Tsallis/Rényi entropy

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In Ref. PlastinoRocca2017 , considering the Tsallis entropy $S_{q}$, Plastino and Rocca obtains the following equation (see Eq. (2.8) in Ref. PlastinoRocca2017 )

[TABLE]

Multiplying the above equation with $P_{i}$, summing over the index $i$ and then using $\sum_{i}P_{i}^{q}=(1-q)S_{q}+1$ by the definition of Tsallis entropy, we obtain

[TABLE]

where $U$ is the average internal energy (which is denoted as $\langle U\rangle$ in Ref. PlastinoRocca2017 ), $Z$ is the partition function. Note that we have also used Eqs. (2.10) and (2.11) given in Ref. PlastinoRocca2017 in order to obtain Eq. (2) above note1 . Then, from Eq. (2), after some simple algebra, we see that

[TABLE]

However, one can check that right from the beginning (see Eq. (1) in Ref. OikBagci2017 ), we have assumed that $\frac{\partial S_{q}}{\partial U}=\beta$ holds in Ref. OikBagci2017 . For example, Eq. (2) in Ref. OikBagci2017 , which is pivotal for the subsequent analysis, cannot hold if $\partial S_{q}/\partial U\neq\beta$. Whether the right hand side of the expression $\frac{\partial S_{q}}{\partial U}$ depends only on $\beta$ or not changes all the mathematical and physical structure independent of the calculus (ordinary or functional) one uses. Accordingly, Eq. (3) above reveals that the work by Plastino and Rocca is incommensurable with the theoretical frame of Ref. OikBagci2017 .

Concerning the part of Ref. PlastinoRocca2017 related to the Rényi entropy, we note that the distribution adopted in Ref. PlastinoRocca2017 is not of the same form studied by us in Ref. OikBagci2017 (compare Eq. (28) in OikBagci2017 with no explicit $\left\langle U\right\rangle$ appearance in the probability distribution and Eq. (3.14) in PlastinoRocca2017 for the explicit appearance of the internal energy $\left\langle U\right\rangle$). However, it is interesting to note that it is in fact Plastino and Rocca (together with F. Pennini) who showed that the distribution in Eq. (3.14) in Ref. PlastinoRocca2017 leads to the fact that there can be no consistent thermodynamics if one adopts this particular distribution (see Sec. IV in Ref. PRP for this issue).

To sum up, Ref. PlastinoRocca2017 has no direct implication for our work OikBagci2017 considering the Tsallis $q$-entropy, since the former work leads to $\frac{\partial S_{q}}{\partial U}=q\beta Z^{1-q}$ while our work assumes $\frac{\partial S_{q}}{\partial U}=\beta$ (and only functions in this particular context) right from the beginning. Considering the part related to the Rényi entropy, we note that the distribution used in Ref. PlastinoRocca2017 is different than ours in Ref. OikBagci2017 .