I cast doubts on the statement "Molecules of  usual gas move stochastically". If these molecules have a slow movement s.t. the wavelength associated with their motion be sufficiently big, the movement of a molecule is described by a wave-function, which is obtained from the Schrodinger equation.

But in writing the Schrodinger equation there is a problem, the form of the potential energy, i.e. of which parameters it depends. It is a problem because a particle in the fluid collides with other particles, and each collision should be described by the wave-function scattering on particles described by themselves by wave-functions. So, we have a very complicated quantum many-body problem. The classical description based on averages, e.g. average velocity between collisions, is much preferable. The Ehrenfest's theorems (http://en.wikipedia.org/wiki/Ehrenfest_theorem) can be of help in this direction because they allow calculating the averages if the wave-function is known. On the other hand, if the fluid has is a gas a sufficiently high temperature s.t. the wave-length of most of the molecules be negligible, the quantum treatment is not necessary. Also, if the fluid is a liquid, the treatment at the level of individual molecules is also unlikely to be necessary.

Now to your two questions:

1. "Can the mean motion of quantum particles be described in terms of classical gas dynamics?"

2. "Is it possible to choose such a force field, that the gas with such interaction between molecules be described by the Klein-Gordon (or Schrodinger) equation?"

It is not excluded that the answer may be positive for each one of the questions, but for each specific case a proof is necessary, there is no general law.

Now, I am not specialized in hydrodynamics, and I didn't read all the article that you indicate. The statement "It is very important that the quantum phenomena are connected directly with the fluid model, i.e. such a connection does not contain any reference to the quantum principles" that I saw in the article, attracted my attention and I didn't understand what they say. But, I can say that I understood until eq. (5). To your attention, eq. (3) in the article is correct only if the wave-function is real.

Dear Mohamed,

The main problem at description of  stochastic particles is “ What is a mechanism of description of stochastic particles?” This question arises, because the method of the deterministic particles description (differential dynamical equations) is not available for description of stochastic particles. The papers of E.Nelson contain a lot of  additional suppositions, which should be used for description of stochastic particles. On the contrary, dynamical equation for a classical gas contain already the method of description of the mean motion of stochastic molecules. There is no necessity to invent this method. One need only to change the molecular interaction, to obtain a quantum description of stochastic molecules. Practically there are no additional hypotheses (or postulates) to obtain quantum mechanics from classical dynamic equations for continuous medium. Mathematically one needs only change Lagrangian for the dynamic system, but not principles of dynamics. As a result one substantiates the quantum mechanics from viewpoint of classical mechanics.

Dear Luiz,

As soon you mention about  “Feynman Path Integrals”  you become to work with quantum mechanics. But my question concerns Foundations of quantum mechanics on the basis of  classical mechanics.

Dear Sofia, 

Your post looks rather strange. You apply the quantum description for singular molecules,  although my question means only classical dynamics for description of the gas and for the gas molecules.

Now about your concrete questions. I  referenced in my question to only one paper.  The first reference is a reference to publication in a journal. The second reference is a reference to electronic version of the same paper,  in order it can be read in internet. Why do you think, that the wave function is to be real? The wave function is not defined yet in (3). As to eq.(5) , it is uses the transformation Jacobian from Lagrangian coordinates x to the Eulerian ones \xi.

Dear Aleksei,

You are right in the relation, that the quantum description of a dynamic system, consisting of several particles, is not  yet developed properly. But the question is conceptual. If it is possible to describe quantum properties of one particle by means of classical dynamics, then why is it  impossible for several particles? If quantum principles are not necessary, then why does its application to one particle distinguish from application to several particles?

Dear Aleksei,

You write: “…quantum system could not be modelled by means of real gas or something like that (if we would not consider exotic constructions like gases in 6(and higher)d-space, that looks less natural than quantum thoery itself withot any interpretation at all.”

I do not see, why such a construction will be not natural.  For instaance, a relativistic gas, consisting of two sorts of molecules, is described in 8D space-time. It is quite natural.

Dear Yuri,

Thank you for sending me your comments. Now, I stress again, dynamics of fluids is not my domain. I only refer to the part of quantum mechanics, and I thought that you mainly wish to discuss the article that appears in the electronic version you indicate.

Now, you ask "Why do you think, that the wave function is to be real? The wave function is not defined yet in (3)." 

Well, I didn't see the definition of vdif  (could it be that it is given in the article and I missed it?), but if in eq. (3) you are still within the frame of the quantum mechanics (QM) then, velocity in QM is defined as in the equation on the bottom of this answer, where A(t) is a time dependent operator.

In your equation (3) appears something else, what is in general known as Bohmian velocity. It is defined as the quantum current of probability divided by the absolute square of the probability, i.e. |Ψ|-2(ħ/M) Im(Ψ* ∂Ψ/∂x). Indeed, you are right, if the wave-function is real the current of probability is zero, I apologize for my mistake.

I regret that this site doesn't allow editing and formatting equations.

Dear Sofia.

You try to discuss another things, that are in my question. You write: “In your equation (3) appears something else, what is in general known as Bohmian velocity. It is defined as the quantum current of probability divided by the absolute square of the probability, i.e. |Ψ|-2(ħ/M) Im(Ψ* ∂Ψ/∂x).”

Look at the title of the paper, that is discussed by you. The quantum mechanics is not mentioned in the paper.  A connection between the wave function and the fluid description is discussed.

Dear Luiz,

I  think, that there is no necessity to read your paper, in order to state the difference in our approaches. You wrote: “…going to see that the only point need to fully deduce non relativistic quantum mechanics from Newtonian Mechanics is to postulate a Feynman Path Integral for the White Noise (a complex valued generalized stochastic process )”. So you postulate something different from classical dynamics. My statement is, that the relativistic quantum dynamic equation can be obtained from classical dynamics (without any additional hypotheses). One needs only to choose a proper Lagrangian for the fluid. A choice of Lagrangian is not an additional supposition. Describing any dynamical system, one needs to choose a Lagrangian.

DearLuiz,

The quantum constant enter to description of the force field, which is responsible for quantum effects. The field \kappal changes the particle mass, turning the usual mass m into effective mass M.

m2 ->  M2=m2+\hbar2/c2(\kappa l \kappa l +dl \kappa l ) 

See details in “Quantum mechanics as dynamics of continuous medium”

http://gasdyn-ipm.ipmnet.ru/~rylov/qmdcm3e.pdf

It follows from such a presentation, that the quantum mechanics realizes incomplete description of the quantum particle stochasticity. It means, that it is possible a further development of the stochasticity description. One need only to investigate the nature of the \kappa- field, which is responsible for quantum effects.

Dear Dr. Yuri,

Your question is profound; it should be given more attention by the physics community.  Reversing your argument, if I may, my group has recently found that classical gases (such as argon above its boiling point) can be adequately described by quantum-mechanical many-body theory.  We have performed extensive calculations in this context for Ar, Na, and Ne (as well as for quantum fluids).  I have come to the conclusion that the demarcation boundary between 'classical' and 'quantum' is not very sharp after all.

Kindest regards.