“A complexity perspective on fluid mechanics” : Blog

This post is a summary of the new preprint that can be accessed here.

Contributions: The article was conceived and written after a long series of inspiring discussion with Dr Giulia Marcucci and later joined by an undergraduate student, Adnan Mahmud, who I supervised 2 years ago at the department of CEB, Cambridge.

Informal story: During my graduate studies, I was interested in understanding why the Navier-Stokes (NS) millennium prize problem (also called regularity problem) has not been solved yet, and what is the current progress on that. A commonly-held view among students and researchers not from the field is that: there is no analytical solution to the NS equations. Surprisingly, I had a similar belief, and this led me to work on fluid dynamical problems and solving special cases of NS equations analytically; one example is here. Coincidentally, when I gave a talk on 30th April, 2021 on an analytical solution to a classical fluid mechanics problem using conformal mapping, differential geometry, and toroidal coordinate techniques, the same day, Prof. Terence Tao delivered a talk on the recent progress on the NS regularity problem, and a key result at the end of the talk was a fresh paper published in PNAS by Cardona, Miranda, Peralta-Salas, Presas. The key result proved by the authors of the article was that: Euler equations (NS equations without viscosity) are Turing-complete. The paper caught good amount of media attention and many researchers thought that this could be en route to programming fluid parcels to blowup after a finite-time. However, the way I interpreted this paper in relevance to the ongoing research in fluid dynamics is a bit different: instead of trying to argue that the Turing-completeness of Euler equations might mean that solving the NS regularity problem requires one to program fluid equations and blow them up, why not start seeing the equations as the program at first place and see where it might take us?

With this question in mind, by coincidence, I came into contact with Dr Shamit Shrivastava and through him, to Dr Giulia Marcucci, who is an expert in neuromorphic computing. Their work is on computing using physical systems (such as nonlinear waves, Ising machine, photonics, shock waves and a lot more), and well, the dots combined way before I realised. it so. The formal story below will explain the dots that we connected.

Formal story: The preprint is written for the audience from fluid mechanics, physical computing, and functional analysis communities, and with a philosophical and history of science touch in the first few paragraphs – so philosophers of science, historians of science are also welcome to contribute. The article starts by asking whether mathematical frameworks are justified to be used to model physics of fluids. By invoking famous Wheeler’s it from bit hypothesis about the information-theoretic origin of matter, it has been argued that it is no harm (or maybe even better) to say that fluids are composed of bits in the core. Built on this premise, then the fluids might well be seen emergent in nature, in that, the atoms and molecules contribute to the continuum-level mechanics. If this is the case, then the field of “complexity science” comes handy in dealing with “fluids”. In fact, Anderson, in his essay “More is different” remarked that “We have already excluded the apparently unsymmetric cases of liquids, gases, and glasses. (In any real sense they are more symmetric.)” However, it would have been great, if he pointed out that the symmetry breaks down when one attempts to unite multi-scale physics of fluids using a single framework. In fact, the attempts to unite multiple scale of physics of fluids and explain breaking down of symmetry in the physics of fluids are not known to the best of our knowledge. This is where the article takes a turn and instead of ruminating more on the “fluids as program” idea (which will be discussed in another upcoming preprint), it presents “Cantor sets” as a potential framework to model the multi-scale physics of fluids. An example case of correlating energy at atomic scale to continuum scale is given, where the ratio of energy levels is given by 2^N, where N is the number of Cantor set layers in between smallest length scale and the scale of observation. To give an example, Hamaker’s constant and surface tension of a fluid differs by larger orders of magnitude, and a Cantor set framework can qualitatively represent that. There is some sort of multi-disciplinary appeal to Cantor set, in that they embody the idea of emergence in a bottom-up fashion (individual entities at small scale combine to form entities at larger scale, however, the rules at smaller scale can only be known collectively, and not individually). A conceptual similarity between Cantor sets and Gajski-Kuhn chart in VLSI hardware design problem is drawn so that the reader understands the arguments more realistically. It would be unfair to say that this preprint is a fully finished version, it is in fact, start to a series of different lines of enquiries one can carry out in attempts to freshen up the classical field of fluid mechanics, in still a rigorous, fundamental fashion.

Where does this perspective and the introduction of Cantor set in qualitatively or quantitatively emergence of fluids takes us?

There are many questions that are starting to arise after this work:

1. Is Cantor set analytical or computational framework or a hybrid one? How to put it into practice, i.e., in solving problems of physics which are multi-scaled in nature?
2. If one is not interested in answering philosophical and fundamental questions posed in the article, how to go about dealing with multi-scale physics problems in fluids?
3. Shall we trouble ourselves thinking more on when and how the NS regularity problem will be resolved, or shall we use the idea that “fluids are computer programs” and solve some engineering problems?

Brief answers to the questions posed above:

1. Cantor set as a framework to solve multi-physics has been attempted before. See this paper on renormalisation group in a Cantorian space-time. More rumination and work on the importance of Cantor set in the context of set theory, logic, emergence, and physics is part of an ongoing work.
2. A recent paper (Song et al.) is to my knowledge the best example of using engineering (X-ray photon correlation spectroscopy) to connect microscale phenomenon in arrested soft material to a macroscale phenomenon. The phenomenon of interest here is viscoelasticity of soft materials and its microscopic origins.
3. It is a matter of taste. I believe solving engineering problems is more fun, once when, the philosophy of why one is solving it is clear. And there are much smarter and focused people in pure maths community, who know better how to resolve this century’s one of the six unresolved millennium-prize problems. We are waiting. 🙂

If any questions, please feel free to reach out to saksham096@gmail.com.

I am keen on giving more detailed interviews or podcast discussion to spread these seeds of thoughts in the researchers’ community, especially, for those, who are newly entering the field.