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与Perk教授及夫人的学术讨论的e-mails-3
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本期公开Perk教授3月份的四封来信和我的回复。为了集中于学术讨论，本人将屏蔽掉e-mails中的非科学内容（用……表示）。为了正确完整地反应交流中的学术思想，将不对e-mails做任何翻译。
Perk教授的第5封来信(2009-3-11)：
Dear Zhidong,
Part of Helen's email was written by me. From your answers it is clear that you do not have a solid background in both thermodynamics and statistical mechanics.
About 'If you use the nonlocal Gamma matrices it will be extremely difficult, but the result must be the same', in the local spin variables, you get indeed a series without any nonlocal effects. The calculation to get the first few coefficients is easy. If you do the calculation using the nonlocal Gamma matrices, you should get massive cancellations. The "internal factors" in the transfer matrix and the nonlocal strings of Gamma matrices in the spins have to cancel each other. This is a much much harder calculation. But if you do it right, you must get the same results.
About "The equipartition theorem states that the temperature is related to the kinetic energy of the particles.", you must realize that the Ising model is a simplification, where only the simplest magnetic contribution is considered. The other contribution of the atoms on the lattice sites is assumed to be weakly coupled, so that one thinks of Z is almost equal to Z_atoms times Z_Ising. The very weak interplay of the lattice and the spins provides the temperature in this picture.
One of the basic formulas in thermodynamics is F=U-TS, with some books writing A for the free energy F, or A=U-TS. U is the average energy, also called the internal energy in thermodynamics. If the lattice has L sites then there are 3L spin couplings, thus U/L is bounded and must lie between -3J and +3J. The entropy is defined as the product of the Boltzmann constant and the logarithm of the number of accessible states at the given energy. At infinite temperature all states are equally accessible, so that S=k log(2^L)=kL log(2). At zero temperature and infinitesimal magnetic field only one state is accessible, all spins up. Then S=k log(1)=0. So the entropy per spin, or S/L, changes continuously from 0 to k log(2) as T moves from 0 to infinity.
Therefore at T=0, F=U, whereas at T=infinity, F/T=kL log(2). The system moves from energetic (ordered) to entropic (disordered). Near T=0, F is almost U with a little bit -TS. At high T, but not infinite T, F/T is almost kL log(2) with a small correction U/T. So, F=U-TS is exact, but you get good approximations if you drop one of the two terms U or -TS at low or high T.
In your reply you talked about ergodic. The 3D Ising model is purely statistical. There is no time evolution. You can consider a more complicated kinetic Ising model as is done in Monte Carlo studies. One can mathematically prove that in some of the kinetic Ising models the time average over the states visited becomes the ensemble average used to study the 3D Ising model. A rough definition of entropy was given earlier and does not depend on time. If the magnetization changes sign that means a macroscopic change in energy of size 2LH. There will be a significant difference and change of the energies of all spins up and of all spins down. So at very low temperatures all spins flip.
If only one spin changed its direction by temperature fluctuation in the kinetic Ising model, it is not equal to the application of a infinitesimal magnetic field on the system, as there is no factor L involved. At very low temperature it would just soon flip back to lower the energy.
Finally in 'you must agree with the rest of the world. You do not agree, so that it is rigorously known that your conjecture fails." you left out the first part of the sentence. The series coefficients are unique. Everyone else agrees with each other. The calculation is in textbooks, taught in physics classes, etc. They have been established as mathematical theorems. The coefficients from your guess are different and thus wrong. The curve fitting that you did at extremely high temperatures to fit the known coefficients in the appendix is inconsistent with the analyticity of the other answer and with the theorem that f/T is analytic in 1/T.
We are always willing to help, but it takes too much time to argue with you. We have to do our regular work. You should find some local experts to argue all of this out. We have given enough material for that discussion. You can also put the email in your blog and ask people for expert opinions. It should be a science blog, not a social popularity contest.
Sincerely,
Jacques H.H. Perk
我的回复(2009-3-11)：
Dear Jacques:
Thanks for your kind discussions.
Because you did not provide detail information about the calculations using
the nonlocal Gamma matrices, I need more time to think about: why can the expansions on the basis of the local spin variables represent the non-local behaviors of the 3D Ising model? Why "internal factors" have to cancel each other in series expansions, but one must face them when one tries to solve it exactly?
the nonlocal Gamma matrices, I need more time to think about: why can the expansions on the basis of the local spin variables represent the non-local behaviors of the 3D Ising model? Why "internal factors" have to cancel each other in series expansions, but one must face them when one tries to solve it exactly?
As my understanding, besides bounded energy spectrum, there is a spectrum with possible stationary states all the way to infinity. If there is an upper bound Emax, what the effect is of temperature once T >> Emax/kB?
In my opinion, at T=infinity, one cannot use F/T, but should face F. I have updated my response to your additional comments, by adding several additional replies as 0812.0194v4. Any further comments from you are welcome.
In my opinion, at T=infinity, one cannot use F/T, but should face F. I have updated my response to your additional comments, by adding several additional replies as 0812.0194v4. Any further comments from you are welcome.
It is true that the 3D Ising model is purely statistical. But in the statistical theory, originally, one has to deal with the time average, not the ensemble average. It is the common problem of the statistical theory, which is not related closely with whether the 3D Ising model is kinetic or not.
It is my opinion that Monte Carlo studies cannot tell anything about the equality of the time average and the ensemble average, because of its limits (including approximations and computer powers).
I disagree with your statements "The series coefficients are unique. Everyone else agrees with each other. The calculation is in textbooks, taught in physics classes, etc. They have been established as mathematical theorems. The coefficients from your guess are different and thus wrong. " In case that there are two "dark clouds" in the 3D Ising model, one should work harder to understand the reasons beyond the "dark clouds". In some cases, "Everyone else agrees with each other" and " The calculation is in textbooks, taught in physics classes, etc. " may not tell the truth of the nature.
Since you suggest to open your e-mail, for completeness, may I ask permission to open in my blog the previous e-mails of Helen (of course, only the scientific part of these e-mails). I agree with "It should be a science blog, not a social popularity contest ". The purpose of my blog is certainly for scientific discussions only. Such a social popularity contest was really beyond my expectation.
Thanks again, with my respect to you and Helen, for your patience and time,
Best wishes, also to Helen!
Zhidong
Perk教授的第6封来信(2009-3-11)：
Dear Zhidong,
You can post what you want of both of us. If you leave a part out that is scientific, mark it with "..." on a separate line. You are wrong in your reply and in your arXive additions. Please take more time to think and to talk with local experts in thermodynamics and statistical mechanics. I cannot immediately respond.
sincerely, Jacques
Perk教授的第7封来信(2009-3-16)：
Dear Zhidong,
> I have replied to you on the arXiv in 0901.2935v3. I thought I had to do
> that as you reproduced some of the errors of 0812.0194v4 in your blogs.
> Please, do not respond in haste and make new errors.
> I have replied to you on the arXiv in 0901.2935v3. I thought I had to do
> that as you reproduced some of the errors of 0812.0194v4 in your blogs.
> Please, do not respond in haste and make new errors.
…………………………………….
> Jacques
> >
> >
>我的回复(2009-3-16)：
Dear Jacques:
Thanks for your reply. I shall think it carefully and let you know if there is any further response.
……………………………………
Zhidong
Zhidong
Perk教授的第8封来信(2009-3-30)：
Dear Zhidong,
In view of your blog today, I have added to arxiv:0901.2935 as you told that I did not address beta=0 enough. I hope that the added part is understandable.
Sincerely,
Jacques H.H. Perk
我的回复(2009-3-31)：
Dear Jacques
Thanks for information. I shall let you know if I updated my response.
Sincerely,
Zhidong
Sincerely,
Zhidong
我的回复(2009-4-1)：
Dear Jacques:
I have read carefully your added comments 3 and quoted references [9-12]. I have not found from these references any evidences supporting your comments.
From reference [9], it is clear that for Ising model, Lebowitz and Penrose only prove it for beta > 0.
From reference [10], although the condition of (1) or (1') is satisfied for beta = 0, the inequality just above (1),i.e., sum abs K(X,T) <= [exp(exp(beta phi')-1)-1], is invalid for beta = 0.
From reference [10], although the condition of (1) or (1') is satisfied for beta = 0, the inequality just above (1),i.e., sum abs K(X,T) <= [exp(exp(beta phi')-1)-1], is invalid for beta = 0.
For references [11,12], I understand that you only use the identity of Suzuki [11,12], right?
Would you please provide the references that support directly your proof? or Are you the first one to explain the proof in this way?
Next, may I ask several technical questions concerning on your added comments 3:
Next, may I ask several technical questions concerning on your added comments 3:
1) just below eq. (9), you said "we keep repeating this process". How many times for such repeating?
2) then you said "may". For a rigorous proof, it is my opinion that one should not say word like "may". Why did you say "may"?
3) How do zero sigma factors <1> = 1 become true? What condition is for it?
4) With increasing the size of the system, how many terms are independent of the size? How big is each term?
5) how many terms are of the remainder rapidly tending to zero? How big is each term?
6) You said that "this bound is a rough estimate and much better ones have been given in the literature". Would you provide me with the literature for much better estimate?
I am looking forward to hearing your answer.
Thanks again for your kind discussions.
Best wishes, also to Helen,
Zhidong
学术论剑
https://blog.sciencenet.cn/blog-2344-225136.html
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