There are now new experiments to test some of our predictions. For example, as mentioned to you in previous messages, we predict that a new type stochastic integration is needed.
A recent experiment does show that the usual Ito and Stratonovich types are not valid. Instead, a special case of prediction is valid.


Best Wishes,

Biqiao



On Fri, Mar 9, 2012 at 10:39 PM, ping ao < az5842@yahoo.com > wrote:


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



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


On Wed, Mar 7, 2012 at 10:55 PM, ping ao < az5842@yahoo.com > wrote:


Dear Prof. Bi, Qiao:

Let me know if you have not received the 6 paper which you are interested in.

(2) The rigorous approach of Lebowitz and Feynman is completely based on (accepted) quantum mechanis.
Indeed, it generates, in general, a differential and integral equation for sub-system dynamics. It would be interesting to know the samilarity and difference from the differential equation in your subdynamics. In particular, it would be great interest to see whether or not there are differences in physics predictions in some situations.
If there would be no difference in physics, differential equations are indeed usually easier to handle than differential-integral equations.

(3) Two Leggett's representative work in this direction are, if it may be helpful:
A. O. Caldeira and A. J. Leggett, Ann. Phys. (N.Y) v.149, 374 (1983);
A.J. Leggett, et al. Rev. Mod. Phys. v.59, 1 (1987)

Unfortunately, I am not accessible to their pdf files right now.

(4) My 1999 PRB paper with Zhu is on the correct treatment of vortex dynamics, sent you in previous message ( iii) paper ) .
Nonlinear phenomena have emerging entities, such as topological defects (vortices, magenetic monopoles, fractional charges, etc) and solitons.
In my experience, straightforward extension of Green's function from linear limit can be dangerous and can easily lead to incorrect results.
Will be happy to discuss with you what I know.

(5) Darvinian dynamics may be viable candidate for the foundation of nonequilibrium processes.
It naturally contains the idea of (canonical) ensemble.
In my view, the micro-canonical ensemble has nothing to do with stastical mechanics.
It is pleasure to find an active group of scientists in China working on such fundamental and pratically very useful problems.


Best,

Ao, Ping