Luce Irigaray knew it already!
Mathematicians Find Wrinkle in Famed Fluid Equations
51–60 of 124 posts
Re: Mathematicians Find Wrinkle in Famed Fluid Equations
#52Earlier 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…
> Breaking the equations means that they do not model correctly the real world
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.
Re: Mathematicians Find Wrinkle in Famed Fluid Equations
#53Earlier 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.
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?
The Navier-Stokes equations are for a smooth fluid and don't capture quantum behaviour or even classical/macroscopic Brownian motion. This is known. It's also entirely beside the point.
Re: Mathematicians Find Wrinkle in Famed Fluid Equations
#54Earlier quoted context omitted.
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 assume that the medium is continuous even at infinitely small scales, which obviously not the case for natural fluids, that are made of discrete atoms. Thus the equations are only correct at sufficiently large scales. They work fine for describing the airflow around an aeroplane, but not the airflow around the head of a hard drive, which is small enough that the finite size of atoms must b…
How are Navier Stokes the other way, on the macro scale? e.g. in the context of meteorology. Asking because I had a discussion with somebody recently where they claimed the fundamental flaw to the science underlying Climate Change science is over-reliance on NS at macro scale as a way of predicting climate behaviour, or something. I took it to be Baloney, but I'm wondering if there is some strands of truth to it ...
Re: Mathematicians Find Wrinkle in Famed Fluid Equations
#55Earlier 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.
Well general relativity breaks down at very small scales, and imparts the need for quantum mechanics ...
Re: Mathematicians Find Wrinkle in Famed Fluid Equations
#56Based off the abstract, there's one class of weak solutions, the Leray solutions, which are known to exist. And now they've shown for a different class of weak solutions to NS that the solutions are not unique. Is that right?
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.
Who would have guessed that this story about the behavior of fluids seems to kind of leak.
Re: Mathematicians Find Wrinkle in Famed Fluid Equations
#57Earlier quoted context omitted.
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 assume that the medium is continuous even at infinitely small scales, which obviously not the case for natural fluids, that are made of discrete atoms. Thus the equations are only correct at sufficiently large scales. They work fine for describing the airflow around an aeroplane, but not the airflow around the head of a hard drive, which is small enough that the finite size of atoms must b…
If the equations are only useful approximations anyway, why are edge case breakdowns so important or surprising like the article seems to indicate? If they are as imprecise as Newton's laws, why are mathematicians looking for "unfailing" precision from them?
Or is the article simply wrong in the initial few paragraphs?
Re: Mathematicians Find Wrinkle in Famed Fluid Equations
#58Re: Mathematicians Find Wrinkle in Famed Fluid Equations
#59Using 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…
If you are going to describe the fluid motion as probabilities, are you going to assume those probabilities to be independent or not?
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.
Re: Mathematicians Find Wrinkle in Famed Fluid Equations
#60Earlier quoted context omitted.
The Navier-Stokes equations assume that the medium is continuous even at infinitely small scales, which obviously not the case for natural fluids, that are made of discrete atoms. Thus the equations are only correct at sufficiently large scales. They work fine for describing the airflow around an aeroplane, but not the airflow around the head of a hard drive, which is small enough that the finite size of atoms must b…
If the equations are only useful approximations anyway, why are edge case breakdowns so important or surprising like the article seems to indicate? If they are as imprecise as Newton's laws, why are mathematicians looking for "unfailing" precision from them? Or is the article simply wrong in the initial few paragraphs?
I think part of it is just mathematical interest, and maybe part of it is hope for more efficient or otherwise better approximations for fluid behavior.