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与Perk教授及夫人的学术讨论的e-mails-2

已有 9186 次阅读 2009-4-2 08:41 |个人分类:追梦|系统分类:科研笔记| 激辩猜想

本期公开Perk教授夫人Helen上个月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教授夫人Helen的第1封来信(2009-3-7)
Dear Zhi-Dong Zhang,
From your blog, I find this statement of yours:
原来的高温展开和低温展开在3维时如果本身有问题(发散或奇异性), 那就不能做为标准评价我的猜想.就是说目前猜想不能被证伪. 我的解是在猜想基础上推定的猜想精确解(还不能正式称精确解). 至于它到底对错, 征求公开的严格证明.
I disagree.
Consider a simple function f(x)=1/(1-x) which is divergent at x=1. Its series expansion is f(x)=1+x+x^2+x^3+x^4+x^5...
For x=0.01, f(x)=1/.99 agrees with the above series to 10^{-9}. The value of the series to order x^5 is approximate. But the series coefficients are exact.
If your guess is correct at all temperatures, then in the region where the temperature is low, where the low-temperature series gives very good approximation, your results should agree with the coefficients of the series expansion, which are calculated exactly. The same for the high-temperature series. As the expansion of your guess disagree with the known series, the rigorous proof that your guess is wrong is already there.
…………………………………………………………………………………
Helen
我的回复(2009-3-7)
Dear Helen:
Thank you very much for discussions about series expansion.
But I disagree with your comment that the low-temperature series are exact. It is my opinion that the low-temperature series, which are obtained by accounting the change of free energy due to the change of local environments of spins, cannot be exact, because it totally neglects the non-local effect hidden in the 'internal' factors. The internal factors correlate all the spins in the system, even at extremely low temperatures close to zero.
牵一发动全身! So, even the first term of the low-temperature series is approximated, not exact as expected. It is why it is divergent in 3D. It can be exact for 2D, since there are no internal factors in 2D.
………………………………………………………………………………..
Best wishes,
Zhidong
Perk教授夫人Helen的第2封来信(2009-3-8)
Dear Zhi-Dong Zhang,
What you said about the series expansion is totally wrong. The Ising spins are local variables, in any dimension. However, the Gamma matrices are not local in dimension larger than two. What you said applies to the Gamma matrices. The derivation of the series expansions and the rigorous proofs of their convergence use only the spins and never the Gamma matrices. As an experimentalist, you should know that the spin-spin correlation function (or the density-density correlation function) decays exponentially. Only near the critical point, where the correlation length diverges, is that not the case. This is general and not dependent on dimension.
………………………………………………………………………………..
Helen Perk
我的回复(2009-3-9)
Dear Helen 
Thanks for your discussions on the series expansion. I may use a wrong word 'correlate' in my last e-mail, which may be replaced by 'relate'. As you agreed, the Gamma matrices are not local in dimension larger than two. This character (due to internal factors) is the most important behavior of the 3D Ising model, while in 2D, no such non-local behavior. This is a key of the problems.
According to your statement, the derivation of the series expansions and the rigorous proofs of their convergence use only the spins and never the Gamma matrices. It is clear that these procedures never take into account the non-local effects of the 3D Ising model. So, how can these procedures reflect the real property of the 3D Ising model?
………………………………………………………………………….
I am always very grateful to Jacques and you for helpful discussions. 
……………………………………………………………………………
Best wishes, also to Jacques
 Zhi-Dong 
 
Perk教授夫人Helen的第3封来信(2009-3-9)
Dear Zhi-Dong Zhang,
> The Ising model is written in the spin language. After the Jorden-Wigner
> transformation, the model can be written in term of Gamma Matrices.
………………………………………………………………………
> The Ising ground state is given by all spins up (or all down), and at very low temperature only a few of the spins flip. This is the foundation of low temperature expansions. Where is your physical intuition?
> The Ising model can also be put into the lattice gas language of Yang and Lee, therefore the works of Groeneveld and others apply and have been applied. The theorems on the series expansions are mathematically rigorous.
………………………………………………………………………..
>> Helen Perk
我的回复(2009-3-10)
Dear Helen Perk
Thanks for your kind discussion.
My logic is simple: In term of Gamma Matrices, it is clear that there are global effects in the 3D Ising model. Any expansions, which neglect such non-local effects, cannot be exact, but with a kind of approximation. Although at very low temperature only a few of the spins flip from the Ising ground state, something (for instance, energy) related to the internal factors should be changed, besides usual concerns of local environments.
As I always state, I have received a lot of benefits from helpful discussions with Jacques and you. May I ask you some further questions:
1) Is there any singularity of thermodynamic functions or consequences at zero temperature?
2) At infinite temperature, is the interacting system dead?
3) Can we define the entropy per site for a statistical - mechanical theory?
4) In the statistical - mechanics, one define temperature T by 1/T = dS/dE. How to define the temperature in infinite range, if both the entropy and the energy are constants for a system?
5) If one used the free energy per site f, whether it would be contradictory to Heisenberg uncertainty principle of quantum mechanics?
…………………………………………………………………………….
Best wishes,
Zhi-Dong Zhang
 
Perk教授夫人Helen的第4封来信(2009-3-10)
Dear Zhidong Zhang,
First, I want to stress that we disagree with your statement, that gamma
matrices can have anything to do with the low or high temperature series. If
you work with the local spin variables it is easy to get the series. If you use
the nonlocal Gamma matrices it will be extremely difficult, but the result must
be the same.
I will only answer a few of your questions, as some of your questions are too
vague, but may not be a question if the following answers are given:
The equipartition theorem states that the temperature is related to the
kinetic energy of the particles.  Particles move faster as temperature
increases. Thus as the temperature increases, the particles move faster and
faster. In the spin languages, the spins flip more and more quickly as the
temperature increases.
The ensemble theory states that the probability of a system being in a state 
is given by W=e^{-E/kT}/Z, where E is the energy of the system and Z the
partition function.  In the Ising model the energy is
E=J sum sigma sigma',
and the summation is over all the nearest-neighbor pair of spins sigma and
sigma', and T is the temperature, k is the Boltzmann constant, while J is
another arbitrary constant (interaction constant).  As an example, for an Ising
model with L spins sigma_1,...,sigma_L=+1 or -1, there are 2^L states for L spins.
At very high temperature $1/kT$ is very small, and the probabilities of the
different states are almost the same, even though they have different
energies. In the Ising model the spins become independent random variables
 (in probability theory), or in another words that means they are not
correlated. The free energy U-TS is then almost entirely given by the entropy contribution
and the free energy per site -kT log(Z)/L is then almost -kT log(2). Therefore,
theorists consider the reduced free energy per spin f/kT as the fundamental
quantity at high temperature as it is almost -S/kL, up to a constant factor the
entropy per spin.
While as T decreases, the lower energies E have higher probabilities, and the
states with high energies have low probability. Thus at zero temperature only
the lowest state is occupied. The entropy contribution -TS to the free energy
goes to zero as T goes to zero and the free energy U-TS becomes U=E_{min}, the
ground state energy. In the Ising model it is the state with all spins up or the
state with all spins down. Some people say that the spins are frozen. Or, in the
solid state language, the atoms are frozen with no vibrations (except quantum
zero-point motion, which does not apply to the classical Ising model. The 3d
Ising model is classical, not quantum, and there is therefore no Heisenberg
uncertainty relation, as long as one discusses the Ising model by itself.).
As the probabilities are given, we can define the expectation  values, such
as spontaneous magnetization, internal energy, etc.
 As there are two (so-called extremal) states at zero temperature, one with all
spins up and one with all spins down, there is a phase transition as one changes
the sign of the magnetic field. However, in your paper you discuss zero magnetic
field only. Below the critical point you must choose one of the two states when
you calculate the spontaneous magnetization. Then you find that this
magnetization is analytic in the temperature except at the critical point. The
free energy, internal energy and specific heat are even functions of the
magnetization (same value for both signs of H). Therefore, their values do not
depend on the choice of the equilibrium state and they are also analytic in T
for all T except T_c. (For T=infinity, you must use f/T as function of 1/T).
Let us consider the isotropic case for simplicity). All your results in your
paper are analytic in exp(-2JT) below T_c and have the correct value at T=0. So
they have unique series expansions and the series coefficients must agree with
the known ones. Similarly, your f/kT is analytic in 1/T, or equivalently
beta=1/kT and it goes to the correct entropic value at T infinite.
Again, f/kT has a unique series expansion and you must agree with the rest of
the world. You do not agree, so that it is rigorously known that your conjecture
fails. Any counter argument just confuses the real matter.
 Do you agree with these statements?
If you disagree, then why do you study the 3D Ising model?
Helen
我的回复(2009-3-10)
Dear Helen:
I do not understand your meaning that '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 a serie without any nonlocal effects, whereas in the nonlocal Gamma matrices you should include the internal factors. So, how do you get the same results?
"The quipartition theorem states that the temperature is related to the kinetic energy of the particles." But in the Ising model, we do not have the kinetic energy of spins. How to define the temperature?
In your sentences "The free energy U-TS is then almost entirely given by the entropy contribution and the free energy per site -kT log(Z)/L is then almost -kT log(2). Therefore, theorists consider the reduced free energy per spin f/kT as the fundamental quantity at high temperature as it is almost -S/kL, up to a constant factor the entropy per spin." you used three 'almost', do you mean that 'not exact'? You mentioned the entropy per spin, but the entropy should be defined as a quantity of the whole system. What is the meaning of the entropy per spin. On the other hand, as I learned from textbook, without time period, one cannot define the entropy. Why does the statistical mechanics get rid of the time?
At low temperatures, as you said, there is a phase transition as one changes the sign of the magnetic field. In my paper, I indeed discuss zero magnetic field only. But one needs to apply a weak magnetic field to obtain the magnetization. If such infinitesimal magnetic field would change its direction, what would then happen for spins?
Moreover, I would just imagine that if one spin changed its direction by temperature, which would equalize to application of a infinitesimal magnetic field on the system, so what would happen for the whole system? Would it change from one (so-called extremal) state to another with some (even very small) probabilities? Please concern the nonlocal effects of the 3D Ising model.
I cannot agree with your statement 'you must agree with the rest of the world. You do not agree, so that it is rigorously known that your conjecture fails." since it is not logical.
Why do I study the 3D Ising model? It is just due to my interest and curiosity on the mystery of the nature.
Thanks again for your discussions.
Zhidong


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