Does anyone have a link to this particular paper?
The link was given in the article: https://arxiv.org/abs/1709.10033
Thanks, I missed that somehow.
Mathematicians Find Wrinkle in Famed Fluid Equations
31–40 of 124 posts
Re: Mathematicians Find Wrinkle in Famed Fluid Equations
#32Earlier quoted context omitted.
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.)
Yes well the Copenhagen interpretation is the most widely accepted interpretation of Quantum mechanics, so it's not crazy incorrect to equate the two
Re: Mathematicians Find Wrinkle in Famed Fluid Equations
#33Earlier quoted context omitted.
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.
Navier Stokes is probably Turing Complete.
Re: Mathematicians Find Wrinkle in Famed Fluid Equations
#34Earlier quoted context omitted.
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 sha…
You seems to think that there is some kind of analogy between a weakened version of the Navier-Stokes equations having multiple solutions and the double-slit experiment generating an interference pattern. I don't think there is such an analogy.
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.
Re: Mathematicians Find Wrinkle in Famed Fluid Equations
#35Earlier quoted context omitted.
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 sha…
Not the GP, but you have to take into account that these are studies done by mathematicians, not physicists, so there are to points to consider.
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).
Re: Mathematicians Find Wrinkle in Famed Fluid Equations
#36Earlier quoted context omitted.
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!
The Navier-Stokes equations do not in fact reliably predict the flow of all real-world fluids. The comparison to Newton's equations is perfectly apt. Good enough for most applications, but may be imprecise in some cases.
Re: Mathematicians Find Wrinkle in Famed Fluid Equations
#37That is fascinating. Is there any sort of immediate real-world impact (like to weather forecasting)?
Financial forecasting and risk-elimination in a system like a blockchain would be my guess — and in that respect it’s probably the most important problem out there IMO.
Re: Mathematicians Find Wrinkle in Famed Fluid Equations
#38That 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 m…
Almost. Most models use some variations of Runge-Kutta approximation, to third order usually.
Edit: typos
Re: Mathematicians Find Wrinkle in Famed Fluid Equations
#39Earlier quoted context omitted.
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 sha…
Not the GP, but you have to take into account that these are studies done by mathematicians, not physicists, so there are to points to consider. 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 st…
Reminds me of the Banach-Tarski paradox, where non-smoothness breaks conservation.
Re: Mathematicians Find Wrinkle in Famed Fluid Equations
#40Using 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 FTA :
>>> 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 ?