Entropy Is Not Extensive

Round 1

Reviewer 1 Report (Previous Reviewer 1)

Comments and Suggestions for Authors

The authors have greatly expanded the scope of their presentation. The manuscript now gives a thorough overview of many concepts in the area of non-extensive entropy. I have but a one suggestion before I recommend publication in Entropy.

Line 81: Saying Jaynes "ignored" the earlier work on non-extensive entropy assumes Jaynes knew of its existence. It is impossible to know exactly what Jaynes knew, but my guess is that he missed (was not aware of) fundamental work from others addressing extensivity of entropy.

 

Author Response

The authors have greatly expanded the scope of their presentation. The manuscript now gives a thorough overview of many concepts in the area of non-extensive entropy.

Thanks for this comment (and for your previous very helpful comments that greatly facilitated our revision!)

I have but one suggestion before I recommend publication in Entropy: Line 81: Saying Jaynes "ignored" the earlier work on non-extensive entropy assumes Jaynes knew of its existence. It is impossible to know exactly what Jaynes knew, but my guess is that he missed (was not aware of) fundamental work from others addressing extensivity of entropy.

Good point!  We have adjusted the wording to avoid this (possibly) misleading word

Reviewer 2 Report (Previous Reviewer 2)

Comments and Suggestions for Authors

Dear Editor,

The authors consider the reviewer's critiques, structuring and creating relevant texts effectively in sections of the manuscript to build rigorous mathematical scaffolding for QGT symmetries, explicitly incorporate the requested missing non-extensive frameworks (Tsallis, Rényi, Hill), and clarify the non-reductionist relationship between thermodynamics and statistical mechanics. The manuscript now appears well-balanced and prepared to meet the publication requirements of the journal in Entropy.

Author Response

The authors consider the reviewer's critiques, structuring and creating relevant texts effectively in sections of the manuscript to build rigorous mathematical scaffolding for QGT symmetries, explicitly incorporate the requested missing non-extensive frameworks (Tsallis, Rényi, Hill), and clarify the non-reductionist relationship between thermodynamics and statistical mechanics. The manuscript now appears well-balanced and prepared to meet the publication requirements of the journal in Entropy.

Thanks for these encouraging comments, and for your previous ones

This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Referee’s report: “Entropy is not extensive”

This is a delightful paper, combining historical context with recent insight into progress towards understanding entropy in cosmological applications. My main concern is that the authors seem to not be aware of several fundamental developments in the theory of non-extensive entropy.

Mathematical foundations for non-extensive entropy were introduced by Renyi (1961). Other, more-recent developments addressing non-extensive entropy involve treatments of Gibbs’ paradox for the ideal gas using different ensembles. Examples include (Entropy 10, 15 (2008), Entropy 19, 603 (2017), and Symmetry 13, 721 (2021)). Although the current authors ultimately focus on other topics, they start with the history of Gibbs’ paradox. Furthermore, it is likely that similar ensemble dependences could be relevant to the entropy of other systems of interest to the authors.

To my mind, the most important concepts missing from the manuscript involve Terrell Hill’s small system thermodynamics. Hill’s theory provides the foundation for finite-size effects in thermodynamics, including understanding non-extensive entropy in self-consistent ways. A specific example is Eq. (1) in the manuscript, which is incorrect for systems having finite-size effects: the fundamental law of thermodynamics must include Hill’s subdivision potential if total energy is to be conserved. These finite-size effects are particularly important when entropy is non extensive. I mention two applications that are likely to be most relevant to the current manuscript and to future research in the authors’ field.

Small system thermodynamics is important whenever the range of interaction is comparable to the system size. Thus, Hill’s theory may apply to large systems that have long range interactions (such as gravity) that cause non-extensive thermal behavior (Entropy 20, 907 (2018)). Although clearly beyond the scope of the current manuscript, it would be interesting to see how Hill’s ideas may apply to black holes that have non-extensive entropy from the area of their event horizon.

Small system thermodynamics is relevant to systems of any size that have internal heterogeneity, specifically independent subsystems inside larger systems. The authors correctly identify that the crucial ingredient is the statistical independence of the subsystems. However, the authors miss the fact that for maximum entropy, statistically independent subsystems may occur in systems of any size (Entropy 22, 975 (2020); Nanomaterials, 14, 1828 (2024)).

The bottom line is that Jaynes never recognized several key concepts in the non-extensive entropy of thermodynamics. He simply missed the fundamental ideas that Renyi and Hill introduced early during the time that Jaynes’ was developing his own work. It is important that this oversight is not perpetuated, not only to restore the historical record but also to improve the opportunity for future developments.

After the authors add some important historical context that may broaden the scope of their ideas, I recommend publication in Entropy.

Author Response

Please see the attachment

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Dear Editor,
The manuscript "Entropy is Not Extensive" by C. Jeynes et al. discusses Jaynes' examination of the Gibbs paradox, arguing that entropy is only approximately extensive in special cases but not in general.
The title is suggestive, the mathematics is simple, and the arguments are generally right.
However, I am disappointed with this contribution for the following reasons:
1) The discussion is not new, and the examples are well known from the community that likes to discuss fundamental aspects of physics.
2) The results strongly depend on QGT, which is not truly accepted or established. For instance, entropy production is asserted to be Noether-conserved and extensive, but not rigorously demonstrated in the paper.
3) The cited claim "Thermodynamics cannot be reduced to statistical mechanics" is highly controversial and poorly supported.
4) Too standard discussions. Modern points of view are missed. A Review needs to be equilibrated. What about long-range-interacting particles, nonadditive properties of entropy, or q-statistical mechanics?
In the present form, this manuscript is not suitable for publication in a prestigious journal such as ENTROPY.

Author Response

Please see the attachment

Author Response File: Author Response.pdf