Mathematicians Find Wrinkle in Famed Fluid Equations
quantamagazine.org
quantamagazine.org
If they are identical but not equal and these small differences in the input lead to large deviations in the output then that's pretty much the definition of a chaotic system.
Edit: I've just browsed through the paper in question and it actually demonstrates thar an approximate (weak) solution is not unique, which means that the exact same inputs in may have multiple weak form solutions.
The article does get around to explaining it better if you keep going.
it certainly seemed to me that no one was relying on N-S as an accurate predictor of motion (as you would a newtonian model of a ball rolling or something like that), but rather just as a first order approximation
Numerical methods for solving the Navier-Stokes equations are approximate and therefore diverge from the correct solution. The same is true for a ball rolling down an incline, but the inaccuracies are smaller than you would ever care about in the real world. What your colleagues were saying about the Navier-Stokes equations is that the numerical error was often large enough that the calculated solutions were known to diverge from mathematical reality in significant ways, and therefore seeing them diverge from physical reality was consistent with physical reality and mathematical reality being the same.
[1] http://www.claymath.org/millennium-problems
[2] http://www.claymath.org/sites/default/files/navierstokes.pdf
What is right is its starting point on the conservation of momentum and energy. Then it makes certain assumptions about the stress-tensor which are not necessarily true. Meaning, you can derive the N-S from consv. of mass and E and a certain stress tensor (ST), but its not derived from a universal ST.
My impression is that so far DNS matches experimental results well given that the experiment actually represents the situation of interest. For example, I am aware that at least some "Kelvin-Helmholtz" experiments don't match DNS well at all, and the DNS is considered more credible than the experiments because in the DNS case you know all of the inputs, whereas in the experiments the initial conditions might be close to the desired case, but apparently not close enough. (The Kelvin-Helmholtz instability is of fundamental importance but is not easy to obtain in isolation experimentally.) "Sensitivity dependence on initial conditions"/chaos means that close may not be enough.
There also is the issue of numerical error from the fact that you are using discrete equations, but usually simulators take steps to check this is negligible. (Which may not be enough.)
You're confusing how models are used by (some) engineers (in some applications) with how models are used by physicists and mathematicians. Engineers have to deal with all kinds of uncertainties, from matetial parameters to use cases to limit states to wear and fatigue and geometrical deviations etc etc etc. Therefore, engineers develop robust designs to comply with all design requirements under any plausible and probable scenario given a design life. To accomplish this, engineers use models to provide approximate but accurate results that are on the safe side of any limit state. Yet, eventhough designs need to be robust, simulations do need to be accurate.
These findings suggest that low-resolutoon Navier-Stokes simulations that were believed to be on the safe side may actually not be on the safe side. These finding are important, as they will illicit significant changes on how Navier-Stokes simulations are used in cases where accuracy matters.
Newton's equations do not in fact reliably predict Mercury's orbit, and it took GR to do it. Lazy journalist!
(Of course, why would one presume that if it is inaccurate at planetary scales, it biases observations towards the climate change narrative? It's just the typical “God of the gaps” kind argument.)
In plasma physics, Laser Wakefield dynamics is an example of a system that can't be modeled as a fluid.
I sort of doubt these considerations apply to the atmosphere, but this is one of the main heuristics for when you can't use a fluid equation.
alephnil mentioned a real problem, but the solution in that case is to not use NS. From a practical standpoint NS is a good model of fluids in many instances because there is a certain minimum scale of motion due to viscosity (the Kolmogorov scale) and this usually is much larger than the size of the atoms or molecules. If this is true then a continuous approximation is fine. No present climate simulation can afford to compute everything down to that scale, so a low pass filter is applied to filter out the small scales and turn their effect on the large scales into a single term that can be modelled. This turbulence modeling approach is called large eddy simulation (LES), and it relies on the fact that outside of certain special cases (e.g., major chemical reactions) the small scales have a universal behavior. (Kolmogorov was the first to propose that the small scales are universal back in 1941.) This approach works pretty well usually. If the person you were talking to said the small scale model was wrong, I'd give them more credit, but this approach is generally the most accurate moderate cost turbulence modeling approach.
Or is the article simply wrong in the initial few paragraphs?
Of course... for obvious reasons it hasn't been experimentally verified...
For the record: I also saw it coded in FORTRAN. Yeah. It's like catching grandma in starkers.
EDIT: This isn't the paper I had in mind, and the equation presented is ‘merely’ relativistic, but it gives you a feel for the beast: https://arxiv.org/pdf/astro-ph/0402502.pdf (see section C).
Can you explain roughly what the quantum corrections were?
Of course if you open the system various outcomes are possible depending on what the external influences do to it (for example, the two solutions mentioned for the still water become entirely plausible if there's somebody roaming around who might put a lighter under the glass and cause the water in it to boil, but that isn't the point of the exercise).
Just mentioning at the bottom of the article “yeah now we are going to see if the same thing applies to proper Leary solutions, we think it does” means close to nothing, honestly.
And even if it does, the article’s author is right when he remarks that this can be seen entirely as a warning against using approximations that are too broad or coarse.
I’ll add that I find it funny that nowhere in the article (that I can see, but I am reading on mobile Safari, so maybe...) is the Navier-Stokes partial differential equation even displayed, and the relationships it defines are not explained (other than some waffle about ‘derivatives’).
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 permissible:
The water starts still, erupts in the middle
of the night, then returns to stillness.
It's a stretch, but I wonder if it's reasonable to think of data points in the vector field describing fluid motion as probabilities rather than definite measurements. That might allow their behavior to be modeled with different mathematical tools.
[0] https://warwick.ac.uk/fac/sci/statistics/staff/academic-rese...
Navier-Stokes has nothing to do with Quantum Mechanics..
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.
(It only become non-deterministic, when you muck around with collapse of the wave function.)
>>> Nonunique Leray solutions would mean that, according to the rules of Navier-Stokes, the exact same fluid from the exact same starting conditions could end up in two distinct physical states, which makes no physical sense and implies that the equations aren’t really describing what they’re supposed to describe.
This basically say that from one given starting conditions one only expect one (and only one) outcome. Doesn't this conflate a model with the actual physical reality ?
Contrary to what the article claims here:
> the exact same fluid from the exact same starting conditions could end up in two distinct physical states, which makes no physical sense
it makes quite a lot of "physical sense" to get two possible outcomes out of one initial state, though we would not expect the Navier-Stokes equations to describe that situation.
The article is simply wrong in arguing that because we expect classical fluids to behave classically and because Navier-Stokes may break down in certain limits, NS may be a bad model. I actually struggle to put together a coherent sentence which comes close to what the article tries to say regarding the relation between "physical sense" and our expectations for the results of Navier-Stokes.
An independence assumption really helps with computation. However, considering the `eddies' in turbulence, I'd suppose there is large and complex coupling between the probabilities of flow even for positions that are separated.
You might deal with the coupling, but at a first guess that feels like it requires tracking many branching paths, which would have exponential memory requirements.
The second is
fantastical but mathematically permissible:
The water starts still, erupts in the middle
of the night, then returns to stillness.
Some quirling around until all matter and energy is equally distributed. What a journey!
Then returns to stillness.
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.
Edit: typos
> I realize that some of us have worked a lot more on numerical methods, but other hackers never made it past Calculus 2.
While that didn't occur to me, I do appreciate trying to keep things accessible.
I’m an applied mathematician and a macroeconomist that studied turbulence in financial markets and crashes thereof. I can assure you that the dynamics are pretty distinct. In economics wealth is not a conserved quantity whereas in physics energy and momentum are.
That's what makes it unsuitable for being a currency.
> a uniquely closed & conserved system
Nope. Bitcoin may or may not be (lost wallets, as you point out, is one way in which it is not). But the ‘system’ is the economy, because money is moving in and out of bitcoin because it can be exchanged for other assets (goods, services, or other currencies when doing conversions).
So no. I'm not trying to be condescending, I'm just trying to nip this apparently valid but flawed analogy in the bud. The only commonality is the word ‘turbulence’ which is being used as a label for two entirely different phenomena that have some similitude and points of contact but are largely distinct and unrelatable.
I can't see how a definitive answer to the question will result in better turbulence modeling, which is what matters from a practical point of view. If it turns out that the solutions are not unique then we could probably find an additional condition to add (e.g., the entropy condition) to make the solutions unique. If the solutions are unique, bounded, etc. then that's great and it would have no impact practically speaking aside from perhaps helping the reputation NS has for accuracy. Some people seem to think that solving the NS Millennium Prize problem would likely lead to a solution for the turbulence problem, but as I said, I can't see how. I'd be interested if anyone could explain this belief better.
There may be other benefits. I've found papers that find bounds on different fluid dynamics quantities to be interesting, and the motivation for these studies are the NS problem from what I understand. Unfortunately the results from these papers tend to be less useful than bounds I can derive specifically for applications myself.
(In a nutshell the turbulence problem is that NS has far too high a computational cost/complexity to be used in practical simulations. So cheaper approximations to NS are used, which you can cladsify as "turbulence models". How steep the drop-off in accuracy is as you reduce complexity is an open question. My opinion is that fluids probably require high computational cost for accuracy a-priori. Things like correlations from experiments can get around this as you are using pre-computed results, and that may be what we should go for in my philosophy.)
The right way to do this is through uncertainty quantification techniques, and I don't know a lot about those at the moment. Until then, all I can say is that there are multiple sources of error.
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.
The equivalent of a spreading wave in quantum physics is the vector field describing the flow according to the Navier-Stokes equations. The equivalent of an interference pattern is the pattern of vortices in the fluid. When you do a double-slit experiment, you'll always see the same interference pattern, and it can be predicted exactly. There is only one solution to the equations. Having two different vector fields satisfy the Navier-Stokes equations with the same boundary conditions would be like seeing different interference patterns for no reason at all.
There is no place for the Uncertainty Principle in this result, because that is a statement about the standard deviations of two complementary quantities, and there are no such quantities involved here.
First, weakening is not related to uncertainty, at least not in the normal sense that I think you refer to. It is not related to the physical solution itself but to our own rules of what we consider a solution. If instead of vector fields we were working with animals, the strong solution would be "we need this animal to be a duck" and the weakened one is "we need this animal to quack when we poke it with a stick". So it is not similar to the quantum uncertainty principle nor anything like it.
Second, mathematicians are interested only in these specific equations. Breaking the equations means that they do not model correctly the real world, and finding those new equations is the job of physicists. Maybe there are other conditions on the solutions, or maybe the relaxation they did allows for non-physical solutions. In fact, they do not prove that those solutions satisfy the energy inequality, so it might be possible that all but one of those non-unique solutions are only possible if you allow fluids to magically gain energy out of nowhere (which obviously conflicts with thermodynamic laws).
Isn't B-T just a consequence of accepting the axiom of choice and performing some pathological decompositions of a ball? What do you mean by:
>non-smoothness breaks conservation
This is wrong. We already know that these equations do not model correctly the real world. No model correctly models the real world and every model breaks down eventually at one point or another. Showing that a certain set of equations leads to non-unique results under certain conditions means nothing, unless you also show that those ‘certain conditions’ are true in cases where we previously assumed the model to hold. If you only show that in cases where we previously also assumed the equations not to hold, they actually output nonsense, this may be a mathematically curious and nice result but of no physical relevance.
So far, this result looks more like realising that Newton’s gravitational field diverges for a point particle of nonzero mass, which is not really any indication whatsoever that it doesn’t correctly model the real world.
It brings to mind how people say "Bumble bee's defy the laws of physics to fly".
Edit- to answer my own question: http://www.scholarpedia.org/article/N-body_simulations_(grav...
The 3- or N-body problem is about point-particles interacting gravitationally at nonzero distance according to an inverse square law. Navier-Stokes is, at root and in the limit, about elastic collisions about infinitesimal corpuscles that transfer momentum between each other.
Again, I can see the analogy “lots of things interacting”, but they have quite little in common beyond that.