Transcription
If we can just linger on the Navia Stokes uh equations a little bit. So you've suggested maybe you can describe it that one of the ways to uh solve it or to negatively resolve it would be to sort of to construct a liquid, a kind of liquid computer, right, and then show that the halting problem from computation theory has consequences for fluid dynamics. So uh show it in that way. Can you describe this this?
Right. Yeah. So, this came out of of this work of constructing this this this average equation that that blew up. Um, so what um as as part of how I had to do this, so there's sort of this naive way to do it, you just keep pushing um um every time you you get energy at one scale, you you push it immediately to the next scale as as fast as possible. This is sort of the naive way to to to force blow up. Um, it turns out in five and high dimensions, this works. Um, but in three dimensions there was this funny phenomenon that I discovered that if you if you keep if if you change laws of physics you just always keep trying to push um the energy into smaller smaller scales. Um, what happens is that the energy starts getting spread out into multi many scales at once. Um, so you you have energy at one scale you're pushing it into the next scale and then um as soon as it enters that scale you also push to the next scale, but there's still some energy left over from the previous scale. um you're trying to do everything at once. Um, and this spreads out the energy too much. Um, and then it turns out that that um it makes it vulnerable for viscosity to come in and actually just damp out everything. So um so it turns out this this direct bush doesn't doesn't actually work. There was a separate paper by some other authors that actually showed this um in three dimensions.
Um, so what I needed was to program a delay. Um, so kind of like air locks. Uh so um I needed an equation which would start with a fluid doing something at one scale. It would push this energy into the next scale, but it would stay there until all the energy from the from the larger scale got transferred, and only after you pushed all the energy in then you sort of open the next gate and and then you you push that in as well. So um by doing that it kind of the energy inches forward scale by scale in such a way that it's always um localized at one scale at a time. Um, and then it can resist the effects of viscosity because it's not dispersed. Um, so in order to make that happen um yeah, I had to construct a rather complicated nonlinearity. Um, and it was basically like um you know like was constructed like an electronic circuit. So I actually thank my wife for this because she was trained as a electrical engineer. Um, and um you know she talked about um uh you know she had to design circuits and so forth. And you know if if you want a circuit that does a certain thing like maybe have a light that that flashes on and then turns off and then on and then off, you can build it from from more primitive components you know capacitors and resistors and so forth and you have to build a diagram and you um and these diagrams you can you can sort of follow up with your eyeballs and say oh yeah the the current will build up here and then it will stop and then it will do that. So I knew how to build the analog of basic electronic components you like resistors and capacitors and so forth and and I would I would stack them together um in in such a way that that I would create something that would open one gate and then there'll be a clock that once the clock hits a certain threshold it would close it, kind of a rude Goldberg type machine but described mathematically, and this ended up working.
So what I realized is that if you could pull the same thing off for the actual equations. So if the equations of water support a computation, so um like if you can imagine kind of a steampunk but really water punk uh type of thing where um you know so modern computers are electronic, you know, they they they're powered by by electrons passing through very tiny wires and interacting with other electrons and so forth. But instead of electrons, you can imagine these pulses of of water moving at a certain velocity. And maybe it's there two different configurations corresponding to a bit being up or down. Probably if you had two of these moving bodies of water collide, they would come out with some new configuration which is which would be something like an andgate or orgate. You know that it the the output would depend in a very predictable way on on the inputs. And like you could chain these together and maybe create a touring machine and and then you could you have computers which are made completely out of water. Um, and if you have computers then maybe you can do robotics. So you know hydraulics and so forth. Um, and so you could create some machine which is basically a fluid analog what's called a von Neumann machine. Uh so von Neumann proposed if you want to colonize Mars the sheer cost of transporting people machines to Mars is just ridiculous. But if you could transport one machine to Mars and this machine had the ability to mine the planet, create some more materials, smelt them and build more copies of the same machine. Um, then you could colonize a whole planet um over time. Um, so uh if you could build a fluid machine, which uh yeah, so it's it's it's a it's a rob it's a fluid robot. Okay. And what it would do it its purpose in life, it's programmed so that it would create a smaller version of itself in some sort of cold state. It wouldn't start just yet. Once it's ready, the big robot configuration water would transfer all his energy into the smaller configuration and then power down. Okay? And then like clean itself up and then what's left is this newest state which would then turn on and do the same thing but smaller and faster. And then the equation has a certain scaling symmetry. Once you do that, it can just keep iterating. So this in principle would create a blow up for the actual Navia Stokes, and this is what I managed to accomplish for this average Navia Stokes. So it provided the sort of road map to solve the problem.
Now this is uh a pipe dream because uh there are so many things that are missing for this to actually be a reality. Um, so um I I I can't create these basic logic gates. Um, I I don't I don't have these in these special configurations of water. Um, I mean there's candidates there things called vortex rings that might possibly work but um um but also you know analog computing is really nasty um compared to digital computing. I mean because there's always errors um you you have to you have to do a lot of error correction along the way. I don't know how to completely power down the big machine so that it doesn't interfere with the the the running of the smaller machine, but everything in principle can happen like it doesn't contradict any of the laws of physics. Um, so it's sort of evidence that this thing is possible. Um, there are other groups who are now pursuing ways to make nodes blow up which are nowhere near as ridiculously complicated as this. Um um they they actually are pursuing much closer to the the direct self similar model which can it it doesn't quite work as is but there could be some simpler scheme than what I just described to make this work.
There is a real leap of genius here to go from Navia Stokes to this touring machine. So it goes from what the self-similar blob scenario that you're trying to get the smaller and smaller blob to now having a liquid touring machine gets smaller and smaller and smaller and somehow seeing how that could be used to say something about a blowup. I mean that's a big leap. So there's precedent. I mean um so the the thing about mathematics is that it's really good at um spotting connections between what you think of what you might think of as completely different um problems. Um, but if if the mathematical form is the same, you you can you can you can draw a connection. Um, so um there's a lot of work previously on what called cellular automator. Um, the most famous of which is Conway's game of life. There's this infinite discrete grid and any given time the grid is either occupied by a cell or it's empty. And there's a very simple rule that uh tells you how these cells evolve. So sometimes cells live and sometimes they die. Um, and this um you know um when I was a a student it was a very popular screen saver to actually just have these these animations going and and they look very chaotic. In fact they look a little bit like turbulent flow sometimes. But at some point people discovered more and more interesting structures within this game of life. Um, so for example they discovered this thing called a glider. So a glider is a very tiny configuration of like four or five cells which evolves and it just moves at a certain direction and that's like this this vortex rings this.
Um, yeah. So this is an analogy. The game of life is kind of like a discrete equation and and um the flu navis is a continuous equation but mathematically they have some similar features. Um, and um so over time people discovered more and more interesting things you could build within the game of life. The game of life is a very simple system. It only has like three or four rules um to to do it but but you can design all kinds of interesting configurations inside it. Um, there's something called a glider gun that does nothing but spit out gliders one at a one one at a time. Um, and then after a lot of effort people managed to to create um and gates and or gates for gliders like there's this massive ridiculous structure which if you if a if you have a stream of gliders um coming in here and a stream of gliders coming in here then you may produce a stream gliders coming out. If so maybe if both of of the um streams um have gliders then there'll be an output stream but if only one of them does then nothing comes out. So they could build something like that and once you could build an um these basic gates then just from software engineering you can build almost anything um you can build a touring machine. I mean it's again enormous steampunk type things. They look ridiculous. But then people also generated self-replicating objects in the game of life. A massive machine, a von Neumann machine which over a huge period of time and it always look like glad guns inside doing these very steampunk calculations. It would create another version of itself which could replicate. It's so incredible. A lot of this was like community crowdsourced by like amateur mathematicians actually. Um, so I knew about that that that work and so that is part of what inspired me to propose the same thing with Navia Stokes. um which is a much as I said analog is much worse than digital like it's going to be um you can't just directly take the constructions in the game of life and plunk them in but again it just it shows it's possible.