Statistical Mechanics Involving Fractal Temperature
Round 1
Reviewer 1 Report
The work is very interesting and has a subject of great relevance in the present day. We have the fractal calculus, on the other hand we have some problems in thermodynamic.
I liked the work. The work is interesting, and information that holds the reader's attention. But, I have some suggestion and considerations to improvement of this work. Some modifications are necessary, I listed below:
(1) Please, connect the first two paragraphs in the introduction.
(2) Please, connect the paragraph-(line 23) with paragraph-(line 25).
(3) Please, connect the paragraph-(line 32) with paragraph-(line 33).
(4) Please, enumerate all equations. Many equation was not enumerated.
(5) After all the equation, there is a comma or a period mark. Please, verify all equations.
Example: In eqs. 1, 3, (...) there are comma or a period mark after the mathematical expressions.
The v(x) after the eq. 16, represent the infinite-well potential.
(6) The solution φα do not satisfy the boundary conditions presented Ψ(0)=Ψ(1)=0. The figure 2.b shows that to Ψ(1) is different of zero. Check this part carefully.
(7) Are not these solutions φα orthogonal? (In solution of Schrödinger equation)
I believe that this part of the work could yield another future work in more detail. But keep this section at work. Is very promising.
(8) Both, Einstein-solid and Debye-solid have as limit for high temperatures the Dulong-Petit behaviour 3NkB. Therefore, the example 4 in section-4 is valid to low temperature limit. Make mention of the limit.
ps. The Dulong-Petit is a experimental result. Today, a law to gas-liquids at high temperatures. Is not a model. But in this context, the mathematical generalization is valid to establish the connection with Einstein and Debye solids.
(9) Consider rethinking the work title. Because, the work has statistical mechanics, and not exactly thermodynamics. Besides, it has the Schrödinger equation, which belongs to quantum mechanics.
In summary, the work has two fields of application, statistical mechanics (with thermodynamics applications) and quantum mechanics.
(10) I will list a sequence of references on fractional calculus that will certainly help better the article, as well as attracting the attention of a different audience.
----- On the Fractal Langevin Equation, Fractal Fract 2019, 3(1), 11;
-----Fractional Prabhakar Derivative in Diffusion Equation with Non-Static Stochastic Resetting.Physics, 1(1), 40-58. (2019)
I liked very much,
Thank you for your attention.
Author Response
Dear Editor,
I want to express the acknowledgement for the suggestions from the editor and the reviewers. I have done some modifications following the queries raised by the reviewers. I have marked the changes in red.
I liked the work. The work is interesting, and information that holds the reader's attention. But, I have some suggestion and considerations to improvement of this work. Some modifications are necessary, I listed below:
(1) Please, connect the first two paragraphs in the introduction. I have done.
(2) Please, connect the paragraph-(line 23) with paragraph-(line 25). I have done.
(3) Please, connect the paragraph-(line 32) with paragraph-(line 33). I have done.
(4) Please, enumerate all equations. Many equation was not enumerated. I have enumerated.
(5) After all the equation, there is a comma or a period mark. Please, verify all equations. Thanks, I have done.
Example: In eqs. 1, 3, (...) there are comma or a period mark after the mathematical expressions.
The v(x) after the eq. 16, represent the infinite-well potential. I have written.
(6) The solution φα do not satisfy the boundary conditions presented Ψ(0)=Ψ(1)=0. The figure 2.b shows that to Ψ(1) is different of zero. Check this part carefully.
I have added Fig.2 to make the details clear. Please don’t consider the upper bound of wave functions.
(7) Are not these solutions φα orthogonal? (In solution of Schrödinger equation)
I believe that this part of the work could yield another future work in more detail. But keep this section at work. Is very promising. Yes, just we have to use fractal integrals.
(8) Both, Einstein-solid and Debye-solid have as limit for high temperatures the Dulong-Petit behaviour 3NkB. Therefore, the example 4 in section-4 is valid to low temperature limit. Make mention of the limit. I have mentioned by adding remark.
ps. The Dulong-Petit is a experimental result. Today, a law to gas-liquids at high temperatures. Is not a model. But in this context, the mathematical generalization is valid to establish the connection with Einstein and Debye solids. Thanks, you are right.
(9) Consider rethinking the work title. Because, the work has statistical mechanics, and not exactly thermodynamics. Besides, it has the Schrödinger equation, which belongs to quantum mechanics.
I have changed which shows connection between the quantum mechanics and thermodynamics.
In summary, the work has two fields of application, statistical mechanics (with thermodynamics applications) and quantum mechanics.Yes, thanks.
(10) I will list a sequence of references on fractional calculus that will certainly help better the article, as well as attracting the attention of a different audience. I have added some related references.
----- On the Fractal Langevin Equation, Fractal Fract 2019, 3(1), 11;
-----Fractional Prabhakar Derivative in Diffusion Equation with Non-Static Stochastic Resetting.Physics, 1(1), 40-58. (2019)
Thanking in Advance,
Dr Alireza Khalili Golmankhaneh
Reviewer 2 Report
The manuscript is well written and well presented. I did not find any mistake in the manuscript. I, therefore, strongly recommend the publication of the manuscript in Fractal Fract.
sincerely,
Reviewer
Author Response
Thanks.
