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Mathematicians Find Wrinkle in Famed Fluid Equations

quantamagazine.org

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Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#21

Post by snakeboy »

eru wrote: post #18
CamperBob2 wrote: post #10

Using this approach, Buckmaster and Vicol prove that these very weak solutions to the Navier-Stokes equations are nonunique. They demonstrate, for example, that if you start with a completely calm fluid, like a glass of water sitting still by your bedside, two scenarios are possible. The first scenario is the obvious one: The water starts still and remains still forever. The second is fantastical but mathematically p…

> Also permissible, and also vanishingly unlikely, under quantum theory. Navier-Stokes has nothing to do with Quantum Mechanics..

You missed the point of their comment.

The responders' point was that Quantum Mechanics is a different framework for modeling physical phenomena which takes a probabilistic framework, and so if fluid were to be modeled in a similar framework, you could work more naturally with these "vanishingly unlikely" events.

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#22

Post by snakeboy »

floatingatoll wrote: post #16

It's disappointed to see that their reaction to non-unique results is "must be broken", rather than "we've rediscovered the quantum physics uncertainty principle in fluid mechanics".

The Uncertainty Principle is a very well established result which is quite generalized mathematically beyond Heisenberg's result. This has nothing to do with that principle.

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#23

Post by btrettel »

CamperBob2 wrote: post #10

Using this approach, Buckmaster and Vicol prove that these very weak solutions to the Navier-Stokes equations are nonunique. They demonstrate, for example, that if you start with a completely calm fluid, like a glass of water sitting still by your bedside, two scenarios are possible. The first scenario is the obvious one: The water starts still and remains still forever. The second is fantastical but mathematically p…

You are on the right track. You have reinvented the basic idea behind the "Reynolds-averaged Navier–Stokes" (RANS) equations, dating back to 1895. The goal is to predict an average velocity field, and this requires modeling unclosed terms (the "turbulence problem" in one respect). RANS and statistical methods in general are the most popular turbulence modeling approaches and much of turbulence theory is based around these ideas. My opinion is that RANS answers are typically what you want, but the models don't work that well. LES (large eddy simulation; applying a low pass filter to NS) is a newer approach that is gaining popularity, and makes physical sense, but is a lot more computationally expensive. And direct simulation (DNS) is an option too, but the computational costs are prohibitive outside of some relatively simple academic problems.

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#24

Post by floatingatoll »

snakeboy wrote: post #22
floatingatoll wrote: post #16

It's disappointed to see that their reaction to non-unique results is "must be broken", rather than "we've rediscovered the quantum physics uncertainty principle in fluid mechanics".

The Uncertainty Principle is a very well established result which is quite generalized mathematically beyond Heisenberg's result. This has nothing to do with that principle.

If you weaken the vector density until a point that you can generate multiple possible outputs from a given weakened input, then what level of weakening guarantees no unique outputs from the equations?

These equations were designed for a system in which every 'atom' (vector) has a perfectly knowable 'spin', and they begin to produce unexpected results as the uncertainty is dialed up through weakening.

It's just a shame that the considerations stop at "therefore we broke the equations" as opposed to "gee, that looks familiar". What's the Navier-Stokes equivalent of the diffraction experiment? What does the interference pattern of two vector fields even look like? Why aren't they trying to study the interference patterns of the one input, two outputs scenario?

I get that this is all "obviously pointless" to others, but no one I've asked can actually explain why these comparisons are unacceptable. You completely dismiss it without any explanation other than "science is well-established", as if somehow that's meaningful.

So, yeah, disappointment.

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#25

Post by bmcooley »

snakeboy wrote: post #22
floatingatoll wrote: post #16

It's disappointed to see that their reaction to non-unique results is "must be broken", rather than "we've rediscovered the quantum physics uncertainty principle in fluid mechanics".

The Uncertainty Principle is a very well established result which is quite generalized mathematically beyond Heisenberg's result. This has nothing to do with that principle.

Could you possibly dive into this? The electron slit experiment shows that superposition exists in a fundamental way in our universe. You can even measure quantum effects with objects as large as buckyballs. Isn't it sort of obvious that a precise mathematical description of a macroscale, emergent system based on objects with quantum and probabilistic effects is always going to be an inaccurate abstraction?

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#26

Post by eru »

snakeboy wrote: post #21
eru wrote: post #18

Earlier quoted context omitted.

> Also permissible, and also vanishingly unlikely, under quantum theory. Navier-Stokes has nothing to do with Quantum Mechanics..

You missed the point of their comment. The responders' point was that Quantum Mechanics is a different framework for modeling physical phenomena which takes a probabilistic framework, and so if fluid were to be modeled in a similar framework, you could work more naturally with these "vanishingly unlikely" events.

Quantum Mechanics is entirely deterministic and linear.

(It only become non-deterministic, when you muck around with collapse of the wave function.)

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#27

Post by eru »

snakeboy wrote: post #21
eru wrote: post #18

Earlier quoted context omitted.

> Also permissible, and also vanishingly unlikely, under quantum theory. Navier-Stokes has nothing to do with Quantum Mechanics..

You missed the point of their comment. The responders' point was that Quantum Mechanics is a different framework for modeling physical phenomena which takes a probabilistic framework, and so if fluid were to be modeled in a similar framework, you could work more naturally with these "vanishingly unlikely" events.

Navier-Stokes is only incidentally related to real life. It's pure math.

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#28

Post by labster »

ericbowman wrote: post #13

That is fascinating. Is there any sort of immediate real-world impact (like to weather forecasting)?

Immediate? Certainly not. Weather models do have incomplete information, but the equations are approximated by Taylor series (1st to 3rd order, depending on the model, last time I checked).

It's possible that this can explain some of the differences between models or ensemble runs... but you have to realize that most of the error comes from incomplete data in the initial and boundary conditions. Looking for weather model effects is like looking for relativistic effects in automobiles.

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#30

Post by pencilhappen »

ianai wrote: post #9
boxfire wrote: post #8

Earlier quoted context omitted.

That's correct, while taking away the entropy inequality as well. The thing is that very inequality is one of these laws of thermodynamics we so love, so I kind of see this as hot air.

Aye either they have upturned a theory as august as general relativity or they’ve thrown the baby out with the bath water somewhere.

What's funny is the article says this about the Navier-Stokes equations: "The equations work. They describe fluid flows as reliably as Newton’s equations predict the future positions of the planets"

Newton's equations do not in fact reliably predict Mercury's orbit, and it took GR to do it. Lazy journalist!

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