Thermofractals, Non-Additive Entropy, and q-Calculus

Round 1

Reviewer 1 Report

The work is interesting and no doubts deserve its publication in the Journal. The author developes new consideration in terms of thermofractalsa and shows their connection with the widely-accepted Tsallis statistics. Meantime, some clarifications needed. The comments are in order.  

1. The Abstract is quite misleading: 

  (a) first the authors says that q is the entropy index and its determined in terms of fractal structure parameters. Then suddenly the author says that q is the statistical parameter and is obtanied in terms of (AGAIN!) the fractal parameters. Isn't is the same what the author says as in BOTH cases the 
 q is obtained/defined based on the same feature, namely on fractal structure? It is quite strange to call the same variable two distinct names. Suggest the author clarifies, changes the wording, not be repeatative, e.g. removing "The statistical parameter, q, is obtained... , and its physical meaning is connected with" replacing it e.g. by "Then it is shown that the q-parameter has a physical meaning of the number..."

  (b) The author says about "non-additivity" in the first line AND then about "non-extensivity", but this is the SAME, as the author claims in the Intro, see the last para on p.1: "the non-additivity of the entropy, or the non-extensivity of the associated thermodynamics". So,  here something is unclear and mismatching in  the Abstract - suggest to say "or" instead of "and" and rewrite te sentence.

  (c) the author discusse the q-calculus and q-algebra. Actually as it is in the paper, those two are the same, as BOTH sects. 3 and 4 show it, so suggest to get ONE name, e.g. q-group calculus (or something alike)s, or clearly show what is the difference between the two if the two names the author likes to  keep.

2. Coming to the paper itself. It is quite unclear what is really NEW in this study compared to refs. [17]-[20]? Even in the Intro, the authors does not say this, but refers to these refs as the already  introduced topics. Suggest at the end of the Intro, the author clearly say what is NEW in the paper  which is NOT given in the above-mentioned refs.

   Also It is not clear what ref. [21} is needed for? Accept the title of the "Locally Transmitted  Information" noting can be understood, refs. [21] simply is unreachable and cannot be read, so the term cannot be understood. Strongly sugegst, if the ref. will be still unavailable, to put some 
 clarifications, short description (may be a special; (sub)section for this) in the paper under question.

3. There are quite some unclear or similar variables introduced.

- In Eq.(1), the index "i" in the sum is not defined, what does it mean and over/till what it runs? Please ADD clarifications, definitions.

- The "p" is said to be the probability. However the probability of WHAT? Could you please be more specific.

- Just after Eq.(3), the author says "being related to N" - but WAHT is N? It was never defined before and  is NEVER defined after! Please DEFINE here!

- Suggest instead of items P1, P2.. to use (i), (ii), etc. This would be much more clear and useful to follow.

- After the P-items, the author intriduces the "epsilon" variable, but NO ANY clarification what is it. Is it a DUMMY parameter? Then use just "(\central dot)" instead f it. The point is that in Eq. (4),AGAIN the "epsilon" is introduced but NO A WORD what it means. Are those the same epsilons? Please CLARIFY, do definitions as needed. The reader SHOULD NOT GUESS.

- In eq.(1) one defines p is as the probability, while in Eqs(5) and (6) the same variable is assigned to  the momentum. Strange enough. Please use DIFFERENT variables! Please also define what means "i" in the next line to Eq.(6). 

- In Eq.(8) the author introduces "\nu# while does not say what it is associated with. Please DO clarify.  And please SAY why it is defined in that form, a power.

- After eq.(19), the author says: "we see that .. alpha is the ndof of the yhermofractal... lambda is the referenec energy ... ". Actually, a reader canNOt understand HOW it is SEEN, please make more  clarifications. Otherwise it is COMPLETELY unclear.

As soon as the above points are clarified, the paper can undergo further consideration.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments for author File: Comments.pdf

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 3 Report

The paper is devoted to the study of the thermofractal structure in the context of the non-extensive thermodynamics. The most original results are related to the thermofractal transformation group and related q-calculus. The paper is interesting with relevant results.

However, in my opinion, a few amendments are in order.

-Section 2 (until the beginning section 2.2) contains results previously considered by the author. This matter of fact should be clearly emphasized into the paper

-In Section 2 (only, not after), the author refers to parton particle. Therefore, it is not clear to the reader which kind of system is referring for

-In connection with the previous observation, as it appears in Eq. (6) and in the total energy U=E+K, the formalism is developed in a non-relativistic limit. If the author is thinking about to a relativistic system, such a parton system, I think that the formalism should be consistently modified. This feature should be spelled out into the paper.

After having appropriately taken into account of the previous remarks, the manuscript can be considered for publication

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

I thank the author for the correctins made due to my comments. The paper
is well improved.
  Nevertheless, some comments still to follow.


1. All ovef the paper, suggest to rvoid using "we", replace "we" by "one",
    so to be not so particular, e.g. "we find" -> "one finds", "we have" ->
    "one has", or "it is found", "it is shown" etc.  - that way is more
    scientifically correct.

2. All over the MS sugest to use $q$-calculus, $q$-algebra ,
    $q$-exponentialetc., i.e. the "q" to be in the math mode.

3. Abstract, I do NOT see a need of the last sentence, moreover it is
    UNclear what actually the author wishes to say here, could be "is" to
    be replaced by "in".

4. All over the paper, NO need of the dash in such wording as
    "reflection-transfrmation", "complxity-transformation",
    "scale-transformation, as soon as no a noun comes immediately after, so
    to read" "reflection transfrmation", "complxity transformation",
    "scale transformation" with NO dash.

5. All over the paper, please REMOVE the "'" for "Tsallis' ststistics",
    "Boltzmann-Gibbs' statistics etc., lease write simply "Tsallis
    ststistics", Boltzmann-Gibbs ststistcs", etc.

6. All over the paper, please MOVE "Statistics", "Fields", "Group" ets. to
    start with the lower-case letters: "statistics", "fields" etc.

7. Sect. 2, "Transformations", para with the items.

     - It is quite unclear wjhat the author means under "their". If these
       means thermofractals then say so, if not, name clearly.

     - in (I) the autoro defines E with the Unclear "their totla" and
       "their kinetic" energies, while in (II) the author names E "the
       internal" and K "the kinetc" energis of "the system". What is he
       difference? Please clarify and keep the SAME naming always.

8. All over the paper, please REPLACE "Equatin X) ans Equation (Y)" by
    "Equations (X) and (Y)"

9. In many place, please REMOVE the blank space from the equation nmber in
    the text: "Equation ( X)"->"Equation (X)" (no space before "X")

10. Please use more common definition for the "number of degrees of
     freedom", such as "d.o.f." (or "DOF")  and use it in the ROMAN, not in
     italic.  Notice, no "n" needed in the abreviation.

11. The ara just before Eq.22), line 2: the variable "ksi" is NOT defined,
     please DO define it, r REMOE, as NOT sed below.

As soon as the corections made, can be considered for publication.

Author Response

1- All ovef the paper, suggest to rvoid using "we", replace "we" by "one",
    so to be not so particular, e.g. "we find" -> "one finds", "we have" ->
    "one has", or "it is found", "it is shown" etc.  - that way is more
    scientifically correct.

I agree with this recommendation and changed accordingly the new version of the manuscript. However, in the last few years, I have been dedicating some time to improve my writing, and I often see editors recommending the use of the first person, probably because it is easier for most people with a native language that is different from English.

2-  All over the MS sugest to use $q$-calculus, $q$-algebra ,
    $q$-exponentialetc., i.e. the "q" to be in the math mode.

Done.

3- Abstract, I do NOT see a need of the last sentence, moreover it is
    UNclear what actually the author wishes to say here, could be "is" to
    be replaced by "in".

The typo was corrected and the sentence was changed to "Finally, it is commented about the application of the results obtained here in different areas, as QCD and scale-free networks.". The reasons to keep the sentence in the abstract were explained in the previous round.

4- All over the paper, NO need of the dash in such wording as
    "reflection-transfrmation", "complxity-transformation",
    "scale-transformation, as soon as no a noun comes immediately after, so
    to read" "reflection transfrmation", "complxity transformation",
    "scale transformation" with NO dash.

Done.

5- All over the paper, please REMOVE the "'" for "Tsallis' ststistics",
    "Boltzmann-Gibbs' statistics etc., lease write simply "Tsallis
    ststistics", Boltzmann-Gibbs ststistcs", etc.

Done.

6-   All over the paper, please MOVE "Statistics", "Fields", "Group" ets. to
    start with the lower-case letters: "statistics", "fields" etc.

Done.

7. Sect. 2, "Transformations", para with the items.

     - It is quite unclear wjhat the author means under "their". If these
       means thermofractals then say so, if not, name clearly.

     - in (I) the autoro defines E with the Unclear "their totla" and
       "their kinetic" energies, while in (II) the author names E "the
       internal" and K "the kinetc" energis of "the system". What is he
       difference? Please clarify and keep the SAME naming always.

I am assuming the referee refers to the definitions of thermofractals, not Transformations, since it is the only place where items appear in the manuscript.

I changed the text in the first item to "It is a thermodynamic system equipped with a complex structure with $N$ components that are, themselves, thermofractals with the same properties, forming a hierarchic structure. The system has a  total energy $U=E+K$, where $E$ is the total internal energy of its components and $K$ is the total kinetic energy of its components."

8. All over the paper, please REPLACE "Equatin X) ans Equation (Y)" by
    "Equations (X) and (Y)"

Done.

9. In many place, please REMOVE the blank space from the equation nmber in
    the text: "Equation ( X)"->"Equation (X)" (no space before "X")

I use the CREF standard, which places the space. This can be easily changed in the edition process if the manuscript is accepted.

10. Please use more common definition for the "number of degrees of
     freedom", such as "d.o.f." (or "DOF")  and use it in the ROMAN, not in
     italic.  Notice, no "n" needed in the abreviation.

Done.

11. The ara just before Eq.22), line 2: the variable "ksi" is NOT defined,
     please DO define it, r REMOE, as NOT sed below.

The variable ksi ($\xi$) is already defined in the paragraph before Equation(21).

Reviewer 2 Report

Reviewer comments and suggestions are nicely taken care of by the author. I recommend publication of the paper in the present form.

Author Response

Thank you for your contribution to the final version of this work, and for accepting it for publication.