Uncertainty Relations in Hydrodynamics

Round 1

Reviewer 1 Report

The paper reviews results on uncertainty relations in application to hydrodynamics. The presentation of the theory is quite detailed, but, in my opinion, is not sufficiently focused on hydrodynamics, which is the main topic of interest of the journal readers. (This explains why - surprisingly! - the title of subsection 6.3 coincides with the title of the entire paper.) For instance, section 6.4 can be probably omitted, and questions of prior interest to hydrodynamicists, such as "How does the theory of uncertainty help to enhance our knowledge of turbulence?", should be addressed.

Although the first sentence of the abstract reads "Uncertainty relations in hydrodynamics are numerically studied", the numerical section is short, less than 6 pages out of 42. Computations for one-dimensional fluid and small Reynolds numbers (not exceeding 10) are presented, although the interesting case of course is the high Reynolds number hydrodynamics in three dimensions.

A simple explanation in plain physical terms of the main finding: "The fluid with a larger uncertainty than that of the ideal fluid behaves as liquid, while the one with a smaller uncertainty behaves as gas" is desirable. Naively, the motion of "fluid elements" in gas is less constrained by neighbouring elements than in liquid, and therefore the uncertainty in gas can be expected to be larger. Why does the mathematics indicate the opposite relation? Is it because the density of gas is lower than that of liquid?

The paper is written in understandable (with several exceptions), but rough English. During the 1st reading of the manuscript I compiled a list of imperfections (sometimes, just misprints) that my eye caught (in no way my task was to indicate ALL inaccuracies; for instance, there are many cases of wrong usage of articles and preposition "of" that I did not mark). The authors will find it below together with my suggestions on correction (note that my corrections are not
necessarily optimal). I hope this will help the authors to improve the text.

My overall feeling is that the paper may not be accepted for publication in the present version; a substantial major revision along the lines discussed above is needed.

====================

All over the text: the expression "On the other hand," may not be used without a prior "On the one hand,".

Line 75:
the Kennard inequality of quantum mechanics is investigated ->
the Kennard inequality of quantum mechanics was investigated

Line 77:
One may wonder ->
One might think

Line 80:
packet is decomposed with plane waves. ->
packet is decomposed into plane waves.

Line 97:
by choosing two parameter set ->
by choosing a two-parameter set

Figure 1 caption:
"The positions at t-dt, t and t+dt are denoted by ..., ... and ..,
respectively." - a different notation is used in the Figure!

Between Eq. (11) and Eq. (12):
The time variable is discretized with the time width dt but
we should take the vanishing limit of dt at the last of calculations. ->
The time is discretized step dt, but we should take the limit dt\to 0 at the end of calculations.

Figure 2 caption:
represented on the left and right panels ->
shown in the left and right panels

Line 157:
from Figure 1,
in Figure 1,

Between Eqs. (13) and (14):
where the another
where another

Line 166:
a set of trajectories fixing \hat{r}_{t'}
and Line 168:
measurable events of \hat{r}_t,
- the notation \hat{r}_t is not properly introduced in the text!

Lines 167-169:
"For the \sigma-algebra of all measurable events of ..., ... and ... represent an increasing and a decreasing family of sub-\sigma-algebras, respectively." - No measures have yet been discussed, \hat{r}_t has not yet been referred to as an event; more explanations are needed.

Line 175:
on the right panel of Figure 2 ->
in the right panel of Figure 2

Eqs. (21) and (22):
All capital letters R in the integrals must be written in the same
(bold Roman) font.

Eqs. (23) and (24): I understand the Fokker-Planck equation involves the term -\nabla\cdot(u\rho). However, the terms written in the two equations suggest the interpretation -(\nabla\cdot u)\rho. Perhaps, the confusion is due to the desire to use the operator notation, but the equations must be written unambiguously.

Line 200:
multiplying x^i to the condition ->
multiplying the condition by x^i

Line 238:
From the perspective of the classical variational approach, one may consider that ->
The classical variational approach suggests that

Eq. (42):
All capital letters R and small letters x in the integrals must be written in the same (bold Roman) font. Similar corrections are wanted in all integrals over spatial variables further on.

Line 263:
The equation of ->
The equation for

Line 275:
trajectory seems to be smooth ->
trajectory appears smooth

Line 282:
like to, however, emphasize ->
like, however, to emphasize

Line 331:
As is analytical mechanics, ->
As in analytical mechanics,

Eqs. (69) (two instances):
(R.t) ->
(R,t)

Line 341:
footing because of ->
footing, because

Line 345:
These two momenta plays ->
These two momenta play

Line 388:
From this representation, one can easily see that the minimum value of uncertainty never vanish ->
Therefore, the minimum uncertainty never vanishes

Line 411:
is known in classical mechanics by Nambu ->
was introduced by Nambu in classical mechanics

Line 445:
as is the case in single-particle systems ->
as for single-particle systems

Line 458:
trajectories of different fluid elements are not across ->
trajectories of different fluid elements do not cross

Line 463:
the discussion developed in the previous section ->
the discussion in the previous section

Lines 463-464:
There are many works to derive the NSF equation of the incompressible fluid ->
In many studies the NSF equation for the incompressible fluid is derived

Line 468:
by operating the ensemble average - the phase is unclear.

Line 479:
does not change along the motion of fluid element - unclear

Line 480:
the contribution from the entropy density in variation are negligibly small ->
the contribution from the entropy density to the variation is negligibly small

Line 495:
Differently from ->
In contrast with

Lines 505-506:
The similar situation is well-known in the community of relativistic hydrodynamics where two different fluid velocities are introduced through ->
Similarly, in relativistic hydrodynamics two different fluid
velocities are introduced for

Line 516:
to add the similar term to our \Pi^{ij}_Q(x,t) term ->
to add the term similar to our \Pi^{ij}_Q(x,t)

Line 518 and 519 (2 instances):
Nevier-Stokes-Korteweg equation ->
the Navier-Stokes-Korteweg equation

Line 518:
By choosing parameters appropriately ->
For appropriately chosen parameters,

Lines 519-520:
There are so many works to investigate this equation. See, ->
It is studied in many works, see,

Line 521:
As was mentioned in the last of Sec. 6.1, ->
As mentioned in the last paragraph of Sec. 6.1,

Line 542:
fashion ->
way

Line 548:
This is ->
Eq. (131) is

Lines 557-558:
as the increase of the volume of fluids. ->
as the volume of fluids increases.

Line 565:
Thus, by substituting mcs into M, the Kennard-type inequality should satisfy ->
Thus, replacing M by m_{cs} yields the Kennard-type inequality

Line 573:
order ->
orders of magnitude

Lines 619-620:
The Reynolds number is set by Re = 10. ->
The Reynolds number Re=10 is set.

Line 621:
are denoted by the dashed lines. The five different solid lines ->
are shown by dashed lines. Five solid lines

Line 628:
the Euler-Maruyama method - a reference is needed.

Lines 665-666:
The extended figure for the early stage of the time evolution is
plotted on the left panel in logarithmic scale. ->
A zoom for the early stage of evolution is shown in the left panel in logarithmic scale.

Line 667:
The result for the ideal fluid is denoted by the dotted line. ->
The graphs for the ideal fluid are shown by dotted lines.

Lines 667-668 and 686-687:
The position of the local minima of the solid curves moves to the
upper right as Re is decreased. ->
Minima at the solid curves move up and to the right as Re decreases.

Line 669:
the larger values than that of the ideal fluid. ->
the larger values than those for the ideal fluid.

Line 673:
of larger Re stays above that of smaller Re ->
for a larger Re stays above the one for a smaller Re

Lines 684-685:
The extended figure for the early stage of the time evolution is
plotted on the left panel in logarithmic scale. The six ->
A zoom for the early stage of evolution is shown in the left panel in logarithmic scale. Six

Lines 689-690:
The solid line of Re=3 and the dotted line are almost on top of each other. ->
The solid line for Re=3 and the dotted line almost coincide.

Line 695:
liquid and gas does not depends on the values of Re. There is an critical ->
liquid and gas does not depend on Re. There is a critical

Lines 713-714
We found that the behaviors of the uncertainty relations for liquid and gas are qualitatively different: -
The sentence is incomprehensible. The uncertainty relations for both liquid and gas are inequalities, and thus cannot exhibit any behaviour. Apparently, authors discuss the behaviour of some quantities involved. This must be clarified.

Lines 716-717:
These numerical results suggest that the difference of liquid and gas is characterized through the behavior of the uncertainty relations. -
Same stylistic error.

Line 1000:
Stochastic cariational method ->
Stochastic variational method

Author Response

Please see the attachment.

 

Author Response File: Author Response.pdf

Reviewer 2 Report

Manuscript ID: water-966334
Type of manuscript: Article
Title: Uncertainty Relations in Hydrodynamics
Authors: Gyell Gonçalves de Matos, Takeshi Kodama, Tomoi Koide *

The authors present an up-to-date review about the status of stochastic variational method. They give a fascinating view on merging fields, especially the appearance of quantum mechanical Schr\"odinger equation by proper designing microscopic stochasticity. The derivation of commutation relations and uncertainty relations are convincingly outlined. The reader is looking forward to the forthcoming paper clarifying the role of imaginary unit with respect to Eq. 90. Over all, the paper presents a progress in understanding fundamental connections. The legibility and transparency of known results, new findings, and unsolved problems are very inspiring during reading. The identification of the difference between Landau-Lifshitz abnd Eckart choice of energy-momentum tensor with the consistency relation Eq. 114 is especially interesting. The text is very well written and should be published as it is.
The view developed here is to go directly from microscopic stochasticity to hydrodynamics. Traditionally as intermediate step, the kinetic theory formulates kinetic equations as first averaging about microscopic stochasticity and derives hydrodynamics at later equilibration times. Actually, the stochasticity appears in quantum kinetic theory as nonlocal collision events (e.g. nonequilibrium thermodynamics with binary quantum correlations, Phys. Rev. E 96, 032106) which leads to nonequilibrium expressions of transport properties. The correlations can lead to stochastic behaviour even on an averaged scale (Fluctuations due to the nonlocal character of collisions, New J. Phys. 9, 313). This nonlocal collision scenario supports the presented view of the authors to consider directly the stochasticity even if only specialized by a Wiener process. For proof-reading, maybe in Eq. 121) the symbol "D" should be explained.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

The authors have made minor revisions, but this has exacerbated some problems.

The 1st phrase of the abstract now reads "The qualitative behaviors of uncertainty relations in hydrodynamics are numerically studied for fluids with low Reynolds numbers in 1 + 1 dimensional system". This is very misleading: the paper is mainly a theoretical review of the uncertainty relations, transplanted from quantum mechanics into the realm of fluid mechanics. Numerics is not the main topic of the paper! Numerical results are very modest and may only be mentioned in the LAST sentence of the abstract. To illustrate the level of confusion this causes: I would not accept to be a referee of this paper should it not be stressed in the beginning of the abstract that this is a numerical study!

The 1st phrase of Section 6.4 now states that "The uncertainty relations in hydrodynamics should be compared to those in quantum many-body systems". This suggests that somewhere in this section, perhaps, in its concluding part, such a comparison is made.

The authors write: "We consider fluids with low Reynolds numbers. For the simulation with a larger Re, a very small space-time grid should be used. To see the qualitative behavior of the product .... as a function of Re and \alpha_A, however, the simulations with low Reynolds numbers are still useful". Indeed, large-Re 3D computations require significant numerical resources, but this is not a reason good enough to abstain from such computations: turbulence is a high-Re phenomenon! It is also well-known, that 3D flows are much more complex than lower-dimensional flows. Therefore, 1D simulations are likely not to be representative. It must be explained that either these computations are illustrative, or why their generality extends beyond the parameter values under consideration.

The authors note that the paper is invited by the guest Editor and they "believe that the purpose and result of our paper are directed toward a purpose of this issue". We read in the Special Issue Information: "In recent years, stochastic modeling in fluid dynamics has witnessed significant progress. Weather forecast, ocean modeling, turbulence or gas dynamics are among the topics of application". Explanations of how the theory exposed in the paper is linked with these applications is sadly missed.

In my opinion, the minor changes implemented by the authors are insufficient for publication.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 3

Reviewer 1 Report

The changes implemented by the authors in the 1st and 2nd revisions have improved the manuscript, but they cannot be qualified as a major revision attending to the problems that I detected. Thus, the paper remains what it originally was: a solid review of uncertainty relations in various fields of physics with some analytical advancements towards hydrodynamics, and a modest numerical illustration thereof. Hints on the links of the theory reviewed in the paper with hydrodynamics remain inconspicuous: the majority of hydrodynamicists will not see how this theory helps their studies.

(One can envisage a possibility that another paper in the present collection is devoted to discussion of such links; if this is the case, the Guest Editor, who has invited the present paper, can help the authors to introduce such references.)

I see therefore two possibilities:

a) If the authors believe that the numerical studies of the uncertainty relations in the 1+1d hydrodynamics discussed in this paper have produced new significant results (the authors did not convince me that this is the case), the paper must be rewritten to comprise an introduction (roughly a page), a summary of the analytical approach (roughly 2 pages), the numerical section (as is) and a conclusion (roughly one page). Then this will become a numerical paper, as the authors claim in the first sentence of the abstract.

b) Otherwise, the abstract must be edited, so that potential readers were not misled. It must be stated clearly that the paper is a theoretical review of uncertainty relations in physics with a slight numerical flavour. Numerics is not the central topic of the present manuscript, and thus all references to numerics can be made only in the last phrase of the abstract. (Note that I did not suggest to remove the first sentence altogether, but merely to place it appropriately.)

It is true "that the time given by the editor for modification is fixed (only a week)" which is not enough to write a referee report, let alone to make a high-quality major revision. However, if the authors are willing to make major changes, they can always withdraw the manuscript and resubmit it when the new text is ready.

Author Response

We regrettably do not agree with the third report of the referee 1. We would like to leave the final decision to a third referee or the editor.