“Computer science is no more about computers than astronomy is about telescopes”. This statement, albeit poetic, has its origins in another poetic catchphrase “It from Bit” popularised by J.A. Wheeler in 1990s, arguing that information is at the core of physical systems. Information, encoded into bits, 0s and 1s, can be useful to compute things if it exists in the abstract universe of Turing machine. Realising that all mathematics that has existed yet so far fails in its self-consistency because of Godel incompleteness, it was important to devise a machine that account for this inconsistency and still compute in the most direct way possible.
However, just like astronomy is no longer about telescopes, but everything that’s observed, analysed, and hypothesised through telescope, same is perhaps the case with other fields of physics as well – I can only talk about fluid mechanics. The notion of Navier-Stokes equations, that it somehow embeds the physical behaviour of water-continuum on top of a mathematical universe seems like falling apart, owing to the failed attempts of functional analysts in proving/disproving the regularity of these equations. This definitely invites lots of counterarguments (even in my mind), but then when I look at the video shown below, and take it seriously, lots of questions come to my mind, which need immediate attention, and perhaps invite a drastic change in the way we see behaviour of fluids.
In the video above, the dynamic shift in bubble behaviour on the top of dispersive water waves can very well be described by empirical relations. However, fully analytical description would fall short of exactly finding out solutions for Navier-Stokes equations. A numerical technique can help, to a degree that generalisability and full accuracy of model would be sacrificed. Is this a crisis? Yes, it is to some extent if one believes so. However, there can be a better language to express equations, knowing the fact that mathematical universe does not allow us exact analytical solutions, is the computational universe that Turing gave us through Turing machine. Assuming that the undecidability of fluid motion is the core reason behind incompleteness of Navier-Stokes equations (more explained here), one is left with choosing an abstract, computational universe which then can be compared with experimentally. But most mathematicians here with argue that the real, physical universe is mathematical in nature. Hence, it means that a computational theoretical universe for fluid mechanics requires a computational universe (of varying resolutions) for experiments – which means camera. This is the point where notion of “it from bit” comes in, and the philosophy is to see continuum behaviour through microscope/camera and understand it in a digital world of computers. While it may seem obvious, what is not obvious is to understand the importance of completely forgetting the notion that water exists in naked-eye reality -the reason being that because the mathematical universe presumably does not explain the behaviour of water completely well. It is also true that incorporating statistical details might increase the accuracy of model, but such a complicated model will again be less analytical and more numerical in nature, prone to errors. Now, if all these notions are accepted by the reader at this point, it makes sense to imagine Digital Fluid Mechanics (DFM) experiments – and one example is the video given above. While this video shows a 2D DFM universe, it is technically feasible and possible to experimentally capture a 3D DFM system. Certainly, high resolution allows one to be as close to reality as well, but within the digital confines. When it comes to comparing these experiments with a model, a computational, digital Navier-Stokes model is important. The argument for development of such a model is introduced by Terence Tao in his monumental paper in 2014, that has recently garnered a traction in geometers community. His paper, argues for developing a computational abstract Navier-Stokes program that can program logic gates entirely made out of water continua, so that computation with those logic gates, while preserving physical principles of mass-energy-vorticity conservation, can be performed. Now, indeed the hindsight purpose of this program is to prove the finite-time blowup of NS equations, what’s even more enticing is the prospect of describing NS equations in the form of a certain combination of abstract logic gates that can solve the equations precisely.
If in the future, such a theoretical DFM field is developed, then it has a potential to revolutionise the experimental DFM research which currently relies mostly on analytical or numerical theories. Understanding that fluid mechanics is no longer about camera/microscope, but camera/microscope is all that can progress the experimental DFM (and parallely theoretical computational DFM) research is an important realisation; the impetus needs to be put on this notion.
While the field of experimental DFM and theoretical-computational DFM needs to be carefully designed and precisely laid out, it is important to understand its limitations – showed in the video below.
As the famous adage of Socrates goes like, “carving nature at its joints” meaning that most successful theories usually carve/understand nature at its joints. In a similar way, most digital theories will carve digital nature at its artificial (digital) joints. The video clearly exemplifies this notion as in the first half of video reflection of an object is visible on the bubble, however it is hard to tell what the object is. But when the camera zooms out and focuses on the water surface, reflection of trees become apparent, meaning that tree is reflected by bubble as well (by a physical intuitionistic logic). The obscurity of tree on the bubble at the first place stresses the importance of a greater field of view and different scaled view (zoomed out). While here reflection is the law of physics that leads to the production of tree on top of a bubble, in other related experiments, numerous other physical laws could affect the thing being observed in camera (which is considered reality and explained by digital equations); hence, an improvement of resolution in digital footage and inferences drawn from different portions (scales, locations) would help in creating story (physics) of what is being observed – majority of this methodology is being already adopted by experimentalists now. Perhaps, what should be more stressed here is the need for theoretical-computational theories in DFM that can explain what is being recorded (and not what is presumed to be seen by naked-eye in mathematical world). Indeed, the business of explaining would be prone to errors – improper/incomplete digital experiments. However, there is no way such a science would not be possible in principle.
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