Understanding Fluid Dynamics from Langevin and Fokker–Planck Equations

Round 1

Reviewer 1 Report

Please see my comments in the attached PDF file.

Comments for author File: Comments.pdf

Author Response

We really appreciate how carefully the reviewer evaluated our manuscript and realizing that they were also a reviewer in the previous version, we cannot thank them enough for their contributions. 

The attached pdf file contains our responses to their comments, responses to the other reviewer's comments, and the new version of the manuscript where changes had been highlighted in red.

Author Response File: Author Response.pdf

Reviewer 2 Report

The text presents an introduction into Langevin equations and to Fokker-Planck equations, targeting undergraduate students. The text is easy to follow and written at the appropriate level for the intended audience.

I have the following minor questions and suggestions:

* Figure 1 caption: "particle positions as functions of time"

* Section title: 2.1 "Moments of a stochastic process"

* Line 106: "Within the context of fluid dynamics"

* Line 115: Would it be interesting to also include the limits of M2(v) as t goes to infinity?

* Line 161: "from Calculus 3 we know". Would "Calculus 3" mean something for a general reader?

* Line 267: Instead of the word "obscured", I think a better expression would be to use "superimposed" instead.

* Line 268: Perhaps it would be informational to note that the moment given in the equation is a "central" moment (as opposed to an ordinary one, given below Line 104), which are, of course, coincide if the mean is zero.

* Line 279: Not only the time histories of Wx and Wy must be different, but the two processes must be statistically independent.

* The discussion on the dimensional difference between LE and FP, starting on Line 285, perhaps would be useful to augment with the notion that this difference originates from the choice of variables one solves for: In FP the independent variables are the dependent variables in the LE. As a consequence, while both can be solved to obtain the same statistically equivalent solution, their numerical methods require very different algorithms and their algorithmic complexity, and thus their computational cost (in terms of memory and CPU requirements), scale very differently: FP is exponential while LE is linear with increasing number of sample space variables. This difference however, does not necessarily mean that one is always superior compared to the other one, as they have different advantages and disadvantages.

* Line 315 Conclusions: Solving for the SDEs for many realizations does not necessarily "avoids" noise, it only reduces it. I would simply only state instead, that the numerical solution of SDEs, via Monte Carlo, are noisy, while solving for the FP one can truly avoid noise.

* Line 319: I would not go as far as stating that solving FPs via grids are not "suitable", only that their algorithmic complexity and thus computational cost is exponential. (Since this, for example, does not at all stop a large group of researchers that advocate solving the Boltzmann equation using a grid in many dimensions, via deterministic, e.g., finite difference or spectral methods.)

* Another interesting application area of LEs and FPs, that could be mentioned in the Conclusions, is turbulent combustion. Interestingly, in that field fluctuations originate NOT from microscopic processes, but from macroscopic ones, due to turbulent motions. Since turbulence and material mixing in turbulent flows can, in practice, be only usefully described by statistical approaches, LEs and FPs are a popular alternative to moment methods that only attempt to capture a finite number of moments of the fluctuating variables. Another very interesting feature of LEs and FPs in computational turbulent combustion is that since one solves for the probability of the fluctuating variables instead of the variables themselves, arbitrary nonlinear functions of the fluctuating variables can be represented without approximation. An important example is complex chemical reactions, which makes LEs and FPs in turbulent combustion a popular choice compared to moment methods. See also: Pope, Turbulent flows (2000) Cambridge University Press, and Fox (2003) Computational models for turbulent reacting flows, Cambridge University Press.

* Appendix B, Line 391: Prompted by Fig 3, the text could also introduce the reader to the definition of statistical moments based on PDFs, instead of only the discrete sum.

* Perhaps the text could also mention that not all PDFs look like a Gaussian, i.e., they do not have to define a probability density for every location of the sample space, and that SDEs and FPs do exist whose solutions yield such PDFs with finite support.

Author Response

We really appreciate how carefully the reviewer evaluated our manuscript. 

The attached pdf file contains our responses to their comments, responses to the other reviewer's comments, and the new version of the manuscript where changes had been highlighted in red.

Author Response File: Author Response.pdf

This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.

Round 1

Reviewer 1 Report

Report on “Understanding fluid dynamics from Langevin and Fokker-Planck equations” by A. Medved, R. Davis and P. A. Vasquez

The authors point at making a very concise review about the the Langevin equation (LE), and the Fokker-Planck (FP) counterpart, for describing stochastic processes in fluid phases. Of course, the subject is very broad and complex, and would deserve more insights for a proper presentation. Many excellent books, chapters in books, and so on, are already at disposal (also free in the web) about this topic, but most of them are appropiate for the higher eduction level in physics. This is why a concise text tailored for undergraduate students is welcomed. The authors do an effort for introducing the subject in the simplest way, even providing some operative tools for solving the equations. I liked the introduction and the first part about the Langevin equation, but then the presentation becomes less accurate, and some parts are also very confusing (againts the aim of the paper!). I feel sorry to say that, in my opinion, this paper cannot be published in the present form. I didn’t opt to reject the paper because the subject would be of interest for this Fluids special issue, but its acceptance would require very major revisions to improve the correctness and the didactive effectiveness. Please consider my comments below, along with some suggestions that could be useful. Of course, since this is not a research paper and since the subject is a fundamental one, many of my comments can be taken as suggestions based on what for me it would be important to find in such a kind of didactic presentation.

1) The generalized Langevin equation (GLE) is a special form of LE including memory effects. The GLE has a little role in the authors’s discussion. Moreover, the Fokker-Planck (FP) discussed by the authors is the standard one for Marvov processes, i.e., it corresponds to the Langevin equation (LE). Thus, I would suggest to remove the emphasis on the GLE, and briefly discuss it only in the 2.3 as an extension of the LE.

2) About panel (A) of Fig. 1, it should be said that the points are final locations of a particle starting from the same initial point and following different trajectories.

3) I don’t know why the authors give much emphasis to the Kramers-Moyal equation, and then reduce it to the Fokker-Planck. Since the author’s examples concern Markov dynamics on continuos variables, it would be much more natural (and correct) to introduce directly the FP. The Kramers-Moyal expansion is here not of useful, nor of interest.

4) There is a “mixture” of stochastic differential equations. The authors start with the equation of Brownian motion of a massive particle in a normal fluid (by the way, a mention to R. Brown’s work would be appropriate for historical reasons!), but then, from section 3, they consider only spatial coordinates in the overdamped limit. A reader that doesn’t know the matter (and should learn from this paper) would be absolutely confused. The authors should add a subsection in section 2 where they say that in the high-friction limit the momenta can be eliminated to end up with an LE for the spatial coordinates only.

5) The last sentence at page 6 is not clear (“This type of inference …”). If the MSD does not behave linearly, the diffusion coefficient cannot be defined... So I don’t know that the authors want to say here. Rather, it would be much more iportant to remark that a Stokes-Einstein relation holds also for non-spherical particles, for which 6pa is replaced by a different factor dependent on i) the particle shape, ii) the particle dimensions, iii) the stick/slip conditions of the fluid at the particle’s surface.

6) The discussion on the dimensionality in LE vs. FP at page 9 is rather misleading and seems also incorrect. It is for sure that the number of variables in both cases is the same (even making, case by case, the appropriate shifts for eliminating uneccessary variables and reducing to relative motions). Talking about different dimensionality is incorrect. Rather, for a didactic purspose, I would highlight other differences between LE and FP: while the LE is a set of (stochastic) differential equations that can be solved via time propagation, the FP is a single partial-derivatives equation that requires other techniques (similar to those used for solving the Schrödinger equation in quantum mechanics…).

7) Section 4.2 on the FP solution is really not clear. There are several issues concerning the notation. Why does the authors consider a 2D case while Eq. (16) was 1D? I would consider exactly the 1D case of Eq. (16), and please use the same notation... Moreover, in the finite-difference scheme the difficult part is to enforce specific boundary conditions. This requires to impose specific behaviour about the probability current at the boundaries. The authors refer to “closed boundary” and “open boundary” in Fig. 6, but without any comment. This is useless for didactic purposes. In short, please focus on the 1D case and make the presentation self-contained and clear for anyone that would like to learn the technique. By the way, for a proper handling of the boundary conditions, a partition with central meshing points is more convenient.

8) At page 9, the authors make a statement about the impossibility of finding a one-to-one relationship between LE and FP. This is correct, but the statement alone leaves a bad perception about the strength of the LE and FP. Once the rules of stochastic integration are set, the passage LE -> FP is also set. The reverse FP -> LE poses other problems, since a family of LEs corresponds to the same FP. Of course this cannot be explained in this didactic paper. On this basis, I am just asking if it would be better to remove such statement, due to the impossibility to provide a proper explanation. Just think carefully about this possibility. Then, I would also remove the sentence “However, the Kramers-Moyal expansion…” which does not add something relevant.

9) Throughout, there are issues about the mathematical notation. Please take care of this. Just to mention a few points:

- insert the argument t where appropriate (e.g., everywhere the limit t to ∞ is taken)

- in eqs (9) and (11), the derivative of the potential is a gradient, not a single derivative. Moreover, a sign “-“ is missing in front of the gradient to have a force.

- at page 7, check the fluctuation-dissipation relation with the memory kernel

- in eq. (12), again the gradient in place of a single derivative

- at page 8, the sign “=” is missing in the Kramers-Moyal. When introducing the Kramers-Moyal, give a bibliographic reference. Anyway, as said above, I would present directly the FP.

- at page 8, eq. (13) should be called LE (not GLE), and said that this is the special case of Brownian motion with constant friction.

- at page 8, the second-order derivative in eq. (18) has an odd and meaningless notation; it should be the Laplacian… Anyway, the paragraph starting with “The draw back…” should be revised completely.

Minor points

There are typos to be corrected. Please check:

- at page 1, “Theses two cases allows …”

- last line of page 3, “give” -> “given”

- title of sec. 2.1: “an stochastic” -> “a stochastic”

- title of sec. 2.3: “Langevine” -> “Langevin”

- two lines before eq. (10). The reference should be to eq. (9) (not to eq. 12)

- in the text and in Fg. 1, talking of “equation of motion” about the FP sounds odd; better would be “evolution equation”

Suggestions, in short 

1) Take care of consistent notation

2) Consider only stationary processes

3) Discuss the LE a the two levels, with and without the momenta, on the basis of the frictional regime (underdamped or overdamped).

4) Lower the emphasis given to the GLE

5) Introduce directly the FP instead of the Kramers-Moyal

6) Stress that the FP is nothing but the diffusion equation (in the high-friction limit). “Diffusion equation” is never mentioned!

7) Improve the presentation of the finite-difference scheme to solve the FP, and treat the 1D case

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

Report on the paper

Understanding Fluid Dynamics from Langevin and Fokker-Planck Equations

by

Andrei Medved, Riley Davis and Paula A. Vasquez

Submitted to Fluids (MDPI).

Dear Editor,

This paper should provide a review on the Langevin
equation and the Fokker-Planck equation and their
relation to fluid dynamics.

However this manuscript does not provide any satisfactory
review or even a correct discussion of the equations given above.

Apart of the many misprint (Langevin, defied, etc.) the Authors
misunderstand the subject and make wrong statements and
notation.

About the statements and concerning the Langevin
equation, there is many times a switching between talking of many
particles and only one particle, or three particles
with the same initial position. Similarly, they switch in
talking about particles' position and then velocity and vice versa.
Then the related equations are wrong as the SDE (13) and the
one before. Dx/DT = W makes no sense at all in stochastic calculus.
Also the 'forward Euler' pag. 10 is wrong.
And obviously the MSD grows linearly in time, not quadratic.

Also the discussion on the FP equation is full of misconception,
and the use of the Kremers-Moyal expansion (expansion of what?)
is the worst possible way to explain this equation.
Further the FP eq pag. 10 appears for the first time, and the
numerical approximation proposed on pag 11 is unstable.
In other words, this paper is also useless from a didactical
point of view, the calculation can be find in wikipedia,
and there is no real discussion on fluids.

I can only recommend rejection.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 3 Report

Let us give for granted that an Introduction to Stochastic Methods for undergraduate students (US in the following) agrees with the objectives of the Journal. Still the paper in the present from cannot be accepted for publication because of the lack of a clear presentation of the topics in a suitable format for US. Let me motivate such a conclusion by considering each section of the paper, apart from the introduction.

2 (Langevin approach). This is the more satisfactory section of the paper. Still there are several points to be clarified.

a) white noise: albeit its formal definition is provided, a typical US needs to undesrand its physical meaning, that is the assumed absence of correlation on the fluctuating forces,

b) In the Stokes law (line 104), the size 'a' should be replaced by the radius of a spherical particle.

c) By looking to the Appendx, a clever US should realize that the coordinate displacement are calculated from an Equation different from Eq(1) (as a matter of fact the SDE for the overdamped motion is employed). This point should be explained.

d) In subsection 2.3, it is stated that simple fluids have not memory effects. It is not true! Memory effects can be excluded only in the case of large brownian particles in a solvent of much smaller molecules.

3 (The Fokker-Planck Equation). There are some fundamental flows:

a) The association between the (standard) Fokker-Planck Equation and the GLE is not appropriate as long as the Fokker-Planck Equation does not include any memory effects! This would induce on the US a wrong idea of the mathematical tools and their correspondence with the physical phenomena. The standard picture of the Fokker-Planck Equation as the equivalent of Langevin type of equations (with white noise) for the probability distribution should be presented. May be, if one likes it, the existence of the generalized Fokker-Planck equations with memory effects can be mentioned.

b) The final considerations about dimensionality of Eqs. (17) and (18) are not appropriate because of the equivalence between Fokker Planck Eq. and the associate Langevin-type Equation. As a matter of fact one can replace Eqs. (17) and (18) with the stochastic equation for the motion of two particle interacting with a parabolic potential and the corresponding Fokker-Planck Equation, respectively, with the same dimensionalty in the language of the authors (also I am not convinced by their use of dimensionalty referred to the Fokker-Planck Equation).

c) The origin of Eq. (13) should be expalined as the ovedamped limit of the Langevin Equation (1) (see also the previous point c).

There are some notation problems:

d) the symbol "=" is missing in the equation after line (154)

e) I do not understand the use of variable "y" in the equation after line 158,

f) In eq.   (15)  one should emphasize that "u" is a function u(x).

4 (Numerical Considerations) I am not convinced about this section. First of all there are much simpler ways to explain the discretization of the Fokker-Planck Equation through finite differences. Moreover is it really necessary, as long as calculation results are not presented? By the way, in simple cases, like for the homogeneous diffusion equation corresponding to Eq. (13), analytical solutions are available without resorting to numerical calculations.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf