Mathematicians prove universal law of turbulence
quantamagazine.org
quantamagazine.org
It's fractal down to the Kolmogorov limit, was my understanding - beyond a certain size it's not meaningfully a "fluid" anymore.
Your comment is misleading because it implies we should all have been assuming this based on the evidence you state -- and we certainly should not have.
Trees are a perfect example. Fractal in form, down to a certain level; but also an example of something that builds over time like a cellular automaton and is being influenced by the environment. Compromises are made, the perfect form is impossible to achieve, but the gestalt is still there.
The fact that the most realistic simulated / computer-generated trees we can render are made primarily of simple fractals is a great indicator.
We see the same thing in terms of self-similarity in mountain ranges, lightning, rivers, lungs, and now clouds and water. At this point if you want to deny it, all I can surmise is that you either never go outside, or simply don't know what to look for.
I live in the mountains and I'm outside in a forest setting multiple times a week. I see exactly what you're talking about; it's just not the same as what the article is saying, even if you use the same word to describe it.
That being said, proving new theorems in this area is hard. I think it's very nice work.
Hacker News has already discussed this result previously in an earlier thread: https://news.ycombinator.com/item?id=21771684
I’m not a scientist, but from my layman view there seems to be a lot of empirical work being done to characterize turbulence beyond what MHD models tell us. Afaict, hydrodynamics is a set of empirically derived equations that describe fluid macro scale behavior. Neoclassical MHD models apply Maxwell’s equations to these, but that isn’t sufficient for most plasma regimes. Gyrokinetic approximations of particle simulations are the lead that most people are following, but they aren’t able to agree with real world measurements very well yet. I have the impression that particle level simulation works but we are several orders of magnitude away from that in terms of computational capacity.
That said, yes there is deviation from what even MHD turbulence predicts for a stellerator. Gyrokinetic simulations are more complicated (you keep lot and lot of the individual particles that make up the fluid, but ignore at what angle along their gyro orbit they are, basically describing them as charged little rings), consequently much more computationally costly, but closer to real life. A full particle simulation (retaining pointlike particles with a correct gyro phase) with something like a PiC code would be even better but is indeed order of magnitude out of reach at the moment.
tl;dr: you have acquired a good high-level view via diffusion from the people around you.
The result posted here is that the power density of the turbulent energy spectrum scales with the inverse of wavenumber.
Also, the part about randomness makes sense in theory but the jump to an actual proof seems a little wide to me.
Are many mathematical laws proved in this way?
> In their first paper, the mathematicians focused on what happens during the mixing process to two points of black paint that begin the process right next to each other. They proved that the points follow chaotic paths and go off in their own directions. In other words, the nearby points can’t ever get stuck in a vortex that will keep them close forever.
> “The particles move together initially,” Blumenthal said, “but eventually they split apart and go in completely different directions.”
> In the second and third papers, they took a broader look at the mixing process. They proved that in a chaotic fluid, generally speaking, the black and white paint mixes as quickly as possible. This further established that the turbulent fluid doesn’t form the kinds of local imperfections (vortices) that would prevent the elegant global picture described by Batchelor’s law from being true.
> In these first three papers, the authors did the hard mathematics required to prove that the paint mixes in a thorough, chaotic fashion. In the fourth, they showed that in a fluid with those mixing properties, Batchelor’s law follows as a consequence.
So no, they are not "proving something by not being able to disprove it." A better way of phrasing their strategy is, "proving something by proving that disproving it is impossible."
In Computer Science, there is a similar concept for proving asymptotic bounds of algorithms called an "adversarial proof." The idea is, given some query that your algorithm performs (e.g. in a graph algorithm, a query could be "are two vertices connected") come up with a worst-case adversary that answers queries in the absolute worst way possible, that would necessitate even more queries to complete the problem. In this way, you can prove a universal lower bound for the cost of solving some problem. See [1].
In this case, the adversary is trying to come up with the worst-case initial conditions for this particular brand of turbulence. Basically they are saying, no matter what, you couldn't come up with an initial condition that challenges Batchelor's law more.
[1] https://www.cs.cmu.edu/afs/cs/academic/class/15451-s20/www/l... Section 3.2
No, I take issue with this phrasing as well. There are things that can neither be proven or disproven (by godel's theorem), proving that disproving it is impossible would not have been sufficient.
Without having read beyond what is in the article, I imagine what they must have shown is that
1. For all systems x, if x does not obey Batchelor's law than neither would the thing they are talking about in the 4th paper.
2. The system they are talking about in the 4th paper obey's Batchelor's law.
The immediate corollary is all systems obey Batchelor's law, otherwise you would have a contradiction (the 4th system both would and would not).
Yes, but this is only a trivial mis-speaking in what is obviously meant to be a description of proof by contradiction: proving something by showing that its opposite is impossible (not that disproving it is impossible).
Not quite. The statement is more like, if Batchelor's law were to fail, then it has to be in one of the following specific ways. Then you show that these specific ways can't happen and get the result.
This is a common approach and needs a few ingredients:
- How could things go wrong?
- Show that these are all possibilities and that otherwise things work (hard problem).
- Isolate each scenario from the first step and show that things don't go wrong (hard problem again).
A classical example would be something like global existence for the two-dimensional Euler equations. If the solution were to fail after a finite time, then necessarily some quantity has to go to infinity, because otherwise we could find a solution for a small additional time (Beale-Kato-Majda criterion). We then show that this quantity does not go to infinity and we are done.
For example say you want to show that a real valued solution stays bounded. Then you have to show that a solution always exists, starts at some small value and "it never occurs that the absolute value of the solution is bigger than 1000". Because you ruled out other scenarios this then implies that the solution is always bounded by 1000.