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与敖平教授讨论：若干统计物理与网络问题(补充敖与毕讨论）
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与敖平教授讨论：若干统计物理与网络问题敖平教授看了我们在 《统计物理与网络科学面临的若干挑战与思考》一文中提出的六个问题，今天来函畅谈了他的高见.我认为：他的看法或学术观点很有见地，值得感兴趣的同行和网友进一步来探讨。因此，我特转发如下。
From ping ao az5842@yahoo.com 11:25 AM (7 hours ago,May 9)
Dear Prof. Fang:
Many thanks for your two recent papers with Prof. Bi.
In my view your subsystem dynamical proposal is a possible candidate, and, my basic points were already in my previous message.
Your paper of 统计物理与网络科学面临的若干挑战与思考 (Several Challenge Issues for Statistical Physics and Network Science) raised many excellent questions. Here I would like to say a few words on some questions which I have been thinking about.
1. "... ，而且需要建立一套非平衡复杂网络的基本动力学方程。这一切似乎又离不开Liouville 方程、Schrodinger 方程和Hamiltonian。什么又是网络的 Hamiltonian 的清晰形式?"
Indeed, we need a fundamental dynamical equation.
In my view, it shoud be logically different from Liouville and/or Schrodinger equations, but compatible with latters.
What we proposed Darwinian dynamics may have such properties.
It is indeed critical to have a clear idea of the system "Hamiltonian" or energy-like function for generic dynamical processes.
Our work since 2004 (a 2004 paper is attached here) may have solved this issue from theoretical construction side.
With "Hamiltonian", statistical mechanics approach is of course valid.
2. "...，如何提炼出网络模型的Hamiltonian，就是一个难点。"
I agree. Barabasi did not appear to understand our work.
I think we now have a first viable proposal for this most fundamental issue.
3. "自然界所有实际宏观热力学过程都是有方向性的或不可逆的，而经典力学和量子力学所反映的物理规律都是可逆的，因而在建立非平衡态统计物理时，首先面临的难题就是不可逆性佯缪: 为什么微观动力学是可逆的，而宏观统计热力学过程却是不可逆的? 这个矛盾自Boltzmann 以来一直困扰着很多物理学家"
Yes. This problem had been even attempted by Lebowitz recently, and many papers appeared in PRL and others new journals.
Our solution is "simple": there is another fundamental dynamical equation.
Such dynamics has been tentatively named Darwinian dynamics. The first relative full discussion was in our 2008 CTP paper.
3. "不可逆性佯谬的起源究竟是什么? "
Well, I tend to answer this question in this way:
Uncertainty is part of natural phenomena. It must be a part of fundamental dynamical description.
That is, from epistemological point of view, we need to accept it as a primivtive concept, not derived concept, similar to that we do not ask what is the origin of uncertainty in quantum mechanics.
4. "是否因为统计热力学规律本质上有别于动力学规律? 如果是，两者究竟有何差别? 非平衡态统计物理是否有基本方程? 如果有，它
是什么形式? 可否由它提供一个包括非平衡态和平衡态统一的理论框架? 网络非平衡熵是否遵守什么演化方程? 如果是，它又是什么形式?"
是什么形式? 可否由它提供一个包括非平衡态和平衡态统一的理论框架? 网络非平衡熵是否遵守什么演化方程? 如果是，它又是什么形式?"
Yes, there is a candidate tentatively named Darwinian dynamics (sub-system dynamics in your study may be another candidate).
In my 2008 CTP paper I attempt to provide quantitative answers to those questions.
5. " 熵产生率、即熵增加定律的微观物理基础是什么? 是否可以一个简明的公式描述? 孤立网络系统的熵是否永远只增不减? 开放网络的熵又如何推动网络演化?"
From Darwinian dynamics perspective, variation in dynamics provides the key to address above questions.
6. "所有这些问题，能否从一套新的基本方程出发进行统一的解答?"
Yes. Darwinian dynamics may be such a candidate.
One needs to ask more: what would be new predictions, for example?
So far, we have discussed two types:
i) For general dynamical description, Ito process may not be enough.
ii) Einstein relation should be extended to linear regime without detailed balance in a specific way.
Fortunately, there is a first clear experimental evidence (PRL, 2010) i) may be gnerally valid.
If all of us think we may have a positive answer to this most fundamental question, we may have made a fundamental contibution to science.
Of course, it is very clear that many hard works need to be done: conceptual, theoreitcal, mathematical, computational, experimental, etc.
Many useful applications may be found, too.
As said previously, I believe in China we may have the right miliu (a apology to Prigogine) for such success.
Most importance, all of us have the will to do it.
Ao, Ping
补充：毕桥与敖平之间的讨论
From: qiao bi <biqiao@gmail.com>
To: ping ao <az5842@yahoo.com>
Sent: Saturday, March 10, 2012 5:21 AM
Subject: Re: hi, ao ping
From: qiao bi <biqiao@gmail.com>
To: ping ao <az5842@yahoo.com>
Sent: Saturday, March 10, 2012 5:21 AM
Subject: Re: hi, ao ping
Hi, Prof. A. Ping,
(1) (1) The potential function is quite interested in the Darwinian dynamics, since it can consist of a formula for the canonical ensemble. I understand this formula is correct even for non-equilibrium states in the Darwinian dynamics. Could you give more clear description to the potential function? I believe it is just Hamiltonian in the equilibrium situation, but what is exact meaning in the non-equilibrium situation?
(2) (2) I firstly guess: the potential function = the potential of information density, which may be related to my recently work (J. Phys. A in submission). But I hope this statement will be explained after I truly understand the potential function.
(3) (3) Feymann types of approaches have advantages, especially to the quantum field system, but I believe that if we did correct calculations the subdynamics and the Feymann methods will give the same results. However, I said the correct calculation is not that kind easier, since it needs researchers to have enough skills to handle both sides. Many cases, because implied not correct calculation, we did the wrong result. Furthermore, subdynamics has introduced many new concepts, such as the extension Hilbert or Liouville space, the complex spectral decomposition, the similarity non-unitary transformation, and the differential kinetic equation to the projected density operator and so on, these new concepts have appeared in the subdynamics both in classical and quantum situations to lead it as a candidate to unify equilibrium and non-equilibrium statistics, while in the Feymann formalism my level cannot see this possibility. For example, can the Feymann approach solve complex spectral problems for chaotic maps? Subdynamics is useful to any linear operators not only Hamiltonian or Liouvillian. So in this period, for saving time, I suggest to first consider the Darwinian dynamics, because it may be more meaningful. I think you more know this.
Best Wishes,
Biqiao
Dear Prof. Bi:
Thanks for the good questions.
(1). Potential function in our work is indeed, or play the samilar role as, the Hamiltonian in physics, whether or not in equilibirum or not.
Because we need broader dynamical description, what we have shown is that, even in biological, social and other situations, "Hamiltonian" exists.
Such existence guarantees that statistical mechanics type decription can be used in those fields, as, of course, people have been doing successfully so far, though they have not understood the keys yet.
(2) I do not know your definition, hence I cannot say anthing.
On the other hand, the potential function in our work is both dynamical and statonary quantity, exactly the same situation in usual physical systems.
(3) I agree that Feynman type description has its advantages but I am no interested in defending such approach.
I think it is most interest to see whether or not there are any differences in physics: For a given situation, whether or not two approaches would give different predictions. Or, they would be completely equivalent. From a physicist's pespective, I am not sure of the last point.
For technical mathematical problems, such as you posted, can the Feymann approach solve complex spectral problems for chaotic maps?
while I do not know anyone has done that, I believe the answer is yes, Feynman approach can do that. The reason for such assertation is actually simple: Feynman's path integral is a general mathematical framework.
Again, I am more interested in situations which can in principle be tested experimentally.
Ao, Ping
https://blog.sciencenet.cn/blog-266190-545908.html
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