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.

]]>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.

]]>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

]]>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*.

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 only the standard . Does that make a difference?

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

Thanks for a thoughtful comment.

]]>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

]]>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.

]]>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.

]]>When you discretize an equation, usually a derivative is replaced a difference: , a being the lattice length. This means you lose locality over a small distance scale .

Also you lose conservation of momenta larger than .

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.

]]>“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

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