Fuzzy Fluid Mechanics

Terrence Tao has made some deep observations on why the regularity of three dimensional Navier-Stokes is such a hard problem. Although he has gone on to many other equally interesting topics, I remain fascinated by his main point there: that Navier-Stokes is supercritical. The nonlinearities become stronger at small distance scales, making it impossible to know (using present techniques) whether solutions remain smooth for all time. Thus, it is crucial to understand the scale dependence of non-linearities in fluid mechanics.

The difficult problems of the field I do understand, Quantum Field Theory, also arise when the strength of the non-linearities increase at short distances. Many ideas from quantum field theory have already been applied to fluid mechanics, most notably by Polyakov.

The best behaved quantum field theories are `asymptotically free’ : the interactions (nonlinearities) decrease at short distances so that they tends to a free ( linear) theory. Renormalization theory, which is the systematic study of such short distance limits, is one of the deepest ideas ever to appear in physics.

Just to illustrate the importance that the physics community attaches to renormalization theory, about ten Nobel Prizes have gone to the people who developed these ideas. Tomonaga, Schwinger and Feynman for early development of Quantum Electrodynamics. Anderson and Wilson for the next stage that led to a revolution in the theory of critical phenomena. ‘t Hooft and Veltmann for renormalizing Yang-Mills theories. Gross, Politzer and Wilczeck for the discovery of asymptotic freedom of Yang-Mills theories. There are several more for work in related areas.

Experience from quantum field theory suggests that we must first replace the Navier-Stokes equations with a `regularized’ version, in which there is a short distance cutoff. These equations are not going to PDEs, but some kind of integro-differential equations. It should be possible to understand the regularity of this cutoff version with existing methods. As pointed out by Tao, the problem of establishing is now shifted to studying the limit as this cutoff goes to zero. This is the step that is analogous to renormalization in quantum field theory. The ideas of quantum field theory suggests that a cutoff that preserves rotation invariance is needed to have a sensible `renormalization’ theory. Such a cutoff does not exist at the moment, and could be one of the (many) difficulties that we have to overcome.

To be more specific, consider the Fourier transform of a function in Euclidean space:

\tilde f(k)=\int f(x)e^{-ik\cdot x}dx

The wavenumber k also takes values in Euclidean space, which is naturally thought of as the dual of the original space. If the function is smooth its Fourier transform will decay faster than any power in the dual space.

If we consider instead functions on a lattice f:Z^d\to C the Fourier transform is a function on the torus f:T^d\to C. In the language of solid state physics, the fundamental domain of this torus is the `Brillouin zone’ of momenta of an electron in a periodic potential. Because there is a smallest possible length ( the distance between nearest neighbors in the lattice) there is a largest possible momentum (half the diameter of the Brillouin zone).

Thus there is a reciprocal relation between the smallest distance allowed in space and the largest allowed wavenumber. It is analogous to the uncertainty principle of quantum mechanics. Indeed, it is the uncertainty principle, once it is accepted that particles are represented by waves.

When studying a partial differential equation it is often useful to impose such a smallest possible length scale, at the cost of introducing some non-locality in the problem. This happens if we replace space by a lattice to discretize the PDE to solve it numerically. As noted above, this imposes a limit on the magnitude of the largest possible wavenumber. Following the jargon of QFT, let us call this cutoff procedure `regularization’.

Unfortunately the lattice method is not always the best regularization, as it breaks rotation invariance. Also, replacing space by a lattice of points introduces a lack of smoothness of functions. Consequently, numerical methods of solving PDEs suffer from spurious instabilities.

Is it possible to introduce a smallest possible length in space without breaking rotation invariance, and while maintaining smoothness of functions in space? Some proposals of this kind have appeared in theories of quantum gravity (where the symmetry of interest is Lorentz invariance rather than rotation invariance). The price we pay for this is a kind of fuzziness in space, where its co-ordinates become non-commutative. These techniques could be useful in the theory of PDEs and QFT; and also of practical use in solving PDEs numerically.

There is some earlier work on two dimensional fluid mechanics ( Zeitlin, Abarbanel and others) where this has been accomplished. But this is just a `toy model’. Although there are some situations where it is possible to limit fluid flow to two dimensions, the vast majority of phenomena of interest are in three dimensions. This was reinforced to me recently by Abarbanel. A fundamental phenomenon (`cascade’) is that information flows into large structures in the fluid from small distance scales (causing apparent randomness of the large scale degrees of freedom) while energy flows into small scales (dissipation due to turbulence and viscosity).

While removing the cutoff is a great mathematical challenge, the cutoff theory itself could of some interest in physics. After all, the equations of fluid mechanics are an approximation valid for an average of a large number molecules. A `fuzzy’ version of fluid mechanics would describe even larger scale motion, which averages over a fluid elements. Such a `mesoscopic’ theory may be what we need to understand many physical phenomena, such as the stability of large vortices.

Computational Fluid Dynamics is important to many engineering applications from weather prediction to the design of aircraft. Typically ( see the book by Patenkar), space is divided into a finite number of cells. The PDEs are turned into finite difference equations that are solved numerically. If the size of the cell can be made small enough this can give a good approximation to the real flow. However, the number of cells is limited by the memory of the computer. If the region of space is large the size of a cell can be too large. For example in weather prediction, a cell is several kilometers in size. This means not only that you miss phenomena within such cells, but also that predictions are limited in time. Given enough time the small scale will affect the large scale motion. In the case of the atmosphere the limit is about ten days beyond which predictions of the weather become unreliable with even the largest computers.

Thus, a method that imposes a smallest possible length, and a largest possible wavenumber, without breaking symmetries could help us in mathematical, physical and engineering approaches to fluid mechanics.

[Update May 30 2007] I have another couple of posts on this subject.

[Update May 29 2007] In the paper it is proved rigorously that solutions of fuzzy hydrodynamics on a torus do tend to solutions of Euler equations, in the limit as the regularization is removed. Thus the problem that M. raises in his comments does not appear at least in two dimensions. What happens in three dimensions is of course an open problem.

[Update May 16 2007] I found a nice page on Turbulence, with many references, by Cosma Shalizi .

[Update May 16 2007] I have posted a paper developing these ideas, arxiv:0705.2139.

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11 Responses to “Fuzzy Fluid Mechanics”

  1. jjk Says:

    I’m studying toward my Ph.D. as a computational fluid dynamicist, and I have a quick question about the following portion of your post:

    “When studying a partial differential equation it is often useful to impose such a smallest possible length scale, at the cost of introducing some non-locality in the problem. This happens if we replace space by a lattice to discretize the PDE to solve it numerically.”

    How exactly is non-locality introduced through spatial discretization? I must preface this by saying that I have practically no knowledge of the purely mathematical approach to the Navier Stokes equations, so my question likely exposes my accompanying ignorance. If my understanding of non-locality is correct, I don’t understand how spatial discretization would introduce such non-locality if none was there in the continuous case. Could you explain? Thanks, and good post, by the way

  2. Rajeev Says:

    I would love to talk more with computational people,although I work mostly at the theoretical end of physics. I have worked out these ideas in detail for the case of two dimensional fluid flow on the surface of a sphere. Ready for a computer, I think. The advantage of what I do vs the usual approach is that even with a finite number of points on the grid, the velocity field is smooth. More on that another time.

    When you discretize an equation, usually a derivative is replaced a difference: {f(x+a)-f(x)\over a}, a being the lattice length. This means you lose locality over a small distance scale a.
    Also you lose conservation of momenta larger than a^{-1}.

    For example, it doesn’t make sense to ask for the value of the function in between two lattice points. You can do some kind of interpolation. But there will be a discontinuity either of the function itself or its derivative of sufficiently high order.

    I am writing a paper with more detailed mathematics. Thanks for your interest.

  3. jjk Says:

    Okay, I am obviously confused about the proper definition of non-locality. I had in my mind some connection with causality, and I was going to write in my previous comment about the example of compressible flow, where information propagates along characteristics at certain well-known speeds, and the computational methods mimic this behavior.

    So, what is the proper definition of non-locality? Regardless, what you say about discontinuous point-to-point jumps in field variables is undoubtedly true in any discretized space.

  4. Rajeev Says:

    Dear jjk,

    Any time you replace a differential operator by a difference operator or an integral operator, you are introducing a coupling between points that are not infinitesimally close. What I mean by non-locality is that points that are not infinitesimally close are coupled together in the evolution equation. Violations of causality are a bit different, when the time co-ordinate is also affected. These can happen if you modify ( or discretize) relativistic theories. Since incompressible flow anyway has signals propagating infinite speed, this is not the issue in Navier-Stokes or Euler equations.

    The more important point is that you would like to keep regularity when you introduce a cutoff. If the cutoff procedure itself introduces discontinuities, it will be even harder to study the regularity of solutions this way.

  5. M Says:

    Dear Prof. Rajeev,
    thanks for your invitation to post a message in this blog. Your article above and your recent preprint arXiv:0705.2139v1 [math-ph] are very interesting.
    As you mention in your work, the idea to use fuzzy spaces (in particular: the two-dimensional fuzzy sphere) as tool to regularize quantum field theory has been studied in several theoretical papers - like those cited in the book by Balachandran, Kurkcuoglu and Vaidya.
    This regularization is characterized by a finite number of degrees of freedom, and has many mathematically attractive features, making it an interesting candidate for numerical simulations. In particular, it explicitly preserves the symmetries of the underlying space at every order, and it allows a well-defined treatment of topological objects.
    However, these works have also highlighted some strong implications of the regularization using a fuzzy space: in particular, the existence of an exotic “striped phase”, which has no counterpart in the corresponding theories in ordinary commutative space, and the emergence of a “non-commutative anomaly”, namely: a (finite, mildly non-local, and rotationally invariant) distortion of the dispersion relation in scalar field theory. These phenomena show that - at least in its naive and most natural formulation - the fuzzy regularization introduces some physical effects which are not observed in the (commutative) theory it should approximate.
    For the simplest fuzzy scalar model, the numerical approach was first pioneered in hep-th/0402230 and in hep-lat/0601012. The more recent works hep-th/0608202 and hep-th/0609205 have confirmed these theoretical predictions to a high degree of precision, and highlighted the details of the phase transition to the striped phase.
    Clearly, in some contexts (e.g.: if one wants to run numerical simulations of non-commutative models) these effects are certainly a desired and very welcome feature.
    In other contexts, on the contrary, they indicate that the fuzzy regularization does not reproduce the expected commutative theory - unless one introduced some artificial modification of the action for the fuzzy model.
    At least in principle, I expect that these aspects (or analogous issues) may also have a relevance and an impact onto your program for fuzzy fluido-dynamics, as they seem to be deeply rooted in the intrinsic non-commutativity that is always introduced with the fuzzy regularization.
    These topics are attracting interest in the scientific community, and it is useful to discuss these ideas and the way they are applied in different, though related, fields.
    Best Regards.

    M

  6. Rajeev Says:

    Dear M.,

    I have to read and understand your papers before I can write a detailed response. Couple of small points though. I am using the three dimensional sphere and not the two-sphere. Also, I am not using a discrete version of the sphere; for example not SU_q(2) only the standard SU(2). Does that make a difference?

    Moreover, my fluid mechanics remains classical even with the non-commutativity of the regularization.

    Thanks for a thoughtful comment.

  7. Rajeev Says:

    Dear M.,
    I read your papers and also had a brief offline discussion with Balachandran. I do agree that there is enough of a connection here that I will add references to you if I write a future version or follow-up paper. I thought at first that referring to the book by Balachandran et al would cover the literature.

    It is my impression still that the papers you referred to are all about the two dimensional discrete non-commutative sphere, applied to regularize quantum field theory. For the current paper about three dimensional fluids in the continuum, it remains an open question whether any of the effects you talk about can happen. Actually it is possibly equivalent to the question I raise at the end. Does this really regularize the potential classical divergence in Fluid mechanics? That is, does the regularized equation have smooth solutions? I don’t know the answer. Let us keep talking.

    I am also writing another paper on fluid mechanics on a two dimensional discrete sphere where many of the issues you raise are more directly relevant. But there I know that the equations do have smooth solutions for all time with the regularization: there is no analogue for a non-commutative anomaly in the application of fuzzy methods to this classical theory.

  8. M Says:

    Dear Prof. Rajeev,
    thanks for your prompt reply to my previous comment. The works I mentioned above are indeed focused on two dimensions, i.e.: theories on the surface of the sphere (rather than in a ball). However, the way non-commutativity is introduced (namely: trading the commutative coordinates for operators obeying the SU(2) Lie algebra) is the same - I was not talking about SU_q(2).
    Therefore, one may wonder if introducing this non-commutativity could have analogous effects in both the bidimensional and in the tridimensional case. This is, indeed, one of the open questions. And it may well be that the answer is related to the existence (or non-existence) of divergences in fuzzy fluidodynamics.
    Also, it would be interesting to develop a formulation for the three-dimensional case, which is suitable for the computational approach. This, however, requires truncating the number of degrees of freedom to a finite number.
    I look forward to reading your next paper, on fluid mechanics on the bidimensional discrete sphere.
    Thanks for your attention - Best Regards,

    M

  9. Rajeev Says:

    Dear M.,
    I hope that a discrete version of three dimensional fluid mechanics can be costructed as well, but I am still working on it.

    In two dimensions, ( with periodic bounary conditions) it was done some years ago by Dowker. I have generalized it to fluids on a 2-sphere with viscosity and coriolis force to model the atmosphere. The idea is to produce an effective theory of large scale structures in the atmosphere like hurricanes and the jet stream. I have done some simple numerical work also, which confirms the general ideas. With some help I may be able to turn this into a more realistic model of the atmosphere, including effects of convection ( which I have not yet included).

    The idea is to average over small scale fluid motion to get an effective theory for larg scale structures. This is analogous to the averaging over the moleculr motion to get fluid mechanics, but carried one more step. The main point is that this second average involves dynamical variables with a non-commutative algebra (matrices in the simplest case of two dimension) rather than just functions or vector fields. Still, it is a very practical way of describing large scale objects because there are very few variables in the effective theory.

    I hope to write more about this in another post.

  10. Rajeev Says:

    By the way, please stop addressing me as “Prof. Rajeev”. Just “Rajeev” will do. Rajeev is my given name an I have explained here why it looks like my `last’ name.

  11. S. G. Rajeev Says:

    Alexadr,
    I am approving this comment to be posted although I do not endorse your claims. I think they are very unlikely to be true.

    The nonlinearities of the Navier-Stokes equations make them very different from the Laplace or other well known equation of mathematical physics.

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