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Computer proof ‘blows up’ centuries-old fluid equations

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Re: Computer proof ‘blows up’ centuries-old fluid equations

#141

Post by ordu »

bheadmaster wrote: post #65
Agingcoder wrote: post #2

I remember my math professor at university telling me that truth in mathematics was a social construct, and that nothing was true until a social consensus had been reached between mathematicians. This struck me at the time as a very powerful statement, yet unexpected, since very much not what most people expect from mathematics. After all, it's supposed to be a field where there is such a thing as a (most of the time…

The fact that mathematics gives us power to predict events in the real world makes it independent of social consensus. If everyone in the world believes that 2+2=5, that doesn't make it less true that 2+2=4 - in the sense that I know for sure, if I take throw two rocks on a pile of two rocks, I'll get a pile of four rocks, not five rocks. I hate this sociologist view that everything depends on the social consensus. G…

> If everyone in the world believes that 2+2=5, that doesn't make it less true that 2+2=4 - in the sense that I know for sure

But not everyone believes that 2+2=5, isn't it? I mean, if you criterion to separate truth from lies is the reality, then now you are talking about a counterfactual reality without proving that this counterfactual reality is possible. So this your statement is unproven, I'd say it cannot be proved.

I can argue, that if world believes that 2+2=5 then it makes 2+2=4 to be false. Just think about it. Try to imagine a plausible counterfactual reality where people believe that 2+2=5. They probably would believe that succession of numbers goes like this: 1, 2, 3, 5, 4, 6, 7, 8, 9, 10, ... And in such a counterfactual reality it would be very strange to believe, that 2+2=4. You could if you liked, and you could build a mathematics around it, but you would have a lot of problems of communication with others.

Re: Computer proof ‘blows up’ centuries-old fluid equations

#142

Post by broast »

bheadmaster wrote: post #133
musingsole wrote: post #126

Earlier quoted context omitted.

> Does it matter that two bricks are not "the same piece of matter" if our predictions work the same for both of them No, because of a dense, interconnected web of other social truths (the rest of the arithmetic model), the relative error of this one truth/model is negligible. However, confusing your model for reality is a fallacy perhaps older than time.

I don't understand how is "the arithmetic model" a social truth, when it clearly corresponds to physical phenomena. You can make a skyscraper that doesn't fall, and it exists regardless of whether other people see it or not. You see it - it's there. What is "social" about that? > However, confusing your model for reality is a fallacy perhaps older than time. I think that the human perception the world is a robust eno…

As far as rejecting perceptions i think going straight to solipsism is a big jump. We may live in a reality that we have no access to. Donald Hoffman's theories in this area are fun.

Re: Computer proof ‘blows up’ centuries-old fluid equations

#143

Post by prolyxis »

nuclearnice1 wrote: post #130
wwalker3 wrote: post #101

The question that the referenced paper (1) is trying to answer is "do the 3D incompressible Euler equations develop a finite time singularity from smooth initial data of finite energy?" This is an important question in the theory of nonlinear partial differential equations, but is probably not as relevant to real fluid flow as a lay reader might imagine. The incompressible Euler equations model a very strange and unp…

What a wonderfully informative and educational comment. Thank you. Would you also be able to shed some light on what a singularity is? It was not intuitive to me that incompressiblity should lead to a singularity. The article dances around the term: > At that point, the Euler equations are said to give rise to a “singularity” — or, more dramatically, to “blow up.” > Once they hit that singularity, the equations will…

A simple example of a function with a singularity is f(t)=1/t. Note that at t=0, f(t) is undefined due to division by zero. On either side of zero, the absolute value of f(t) approaches infinity.

In this case, we are tracking the flow of an incompressible fluid over time. This flow is represented by a velocity field evolving over time, under the constraint of no net inflow/outflow of material into any region of space. Thus, the singularity corresponds to a portion of fluid speeding up and approaching an infinite speed as you approach some finite time.

Because the fluid cannot be compressed, the only way the singularity can be produced is for a portion of the liquid to swirl, increasingly rapidly, about some point: hence the discussion in the article about vorticity.

As isoprophlex pointed out, this undefined value of the velocity field prevents you from (or at least complicates) computing the further evolution of the fluid.

Re: Computer proof ‘blows up’ centuries-old fluid equations

#144

Post by wwalker3 »

nuclearnice1 wrote: post #130
wwalker3 wrote: post #101

The question that the referenced paper (1) is trying to answer is "do the 3D incompressible Euler equations develop a finite time singularity from smooth initial data of finite energy?" This is an important question in the theory of nonlinear partial differential equations, but is probably not as relevant to real fluid flow as a lay reader might imagine. The incompressible Euler equations model a very strange and unp…

What a wonderfully informative and educational comment. Thank you. Would you also be able to shed some light on what a singularity is? It was not intuitive to me that incompressiblity should lead to a singularity. The article dances around the term: > At that point, the Euler equations are said to give rise to a “singularity” — or, more dramatically, to “blow up.” > Once they hit that singularity, the equations will…

The incompressible Euler equations model a fluid as a two-valued field. This means that at every point in space, the field has two values, density and velocity (1).

To me (2), a singularity in a field like this means that one or more of the field values "blows up", i.e. goes to infinity as you run the time variable forward.

But how could this ever happen? The Euler equations model the "conservation" (i.e. constant-ness) of three real physical quantities: mass, momentum, and energy. If these three quantities are finite and constant when you add them up over the whole field, how can any part of it "blow up" into an infinite value?

The answer is that the blow-up must occupy a volume that shrinks as the blow-up grows, so the conserved quantities are still constant. The singularity would be infinitely small in space, and have an infinite value of density or velocity (or both).

The hard question is, are these blow-ups merely artifacts of a particular numerical simulation technique, or are they essential somehow to the incompressible Euler equations themselves? That's what these papers are trying to figure out.

To me, an "essential" (i.e. inherent-in-the-equations) blow-up seems intuitively reasonable because of the acausal nature of the field. When you simulate the incompressible Euler equations, it superficially looks like it's a physical fluid doing physical-fluid things, swirling and flowing around. But in a real fluid, a change in one part of the fluid propagates to the other parts at finite velocity, creating real cause and effect.

An Euler fluid's time evolution is not a phenomenon that ripples forward through time in a normal way. Instead, every point in the fluid responds to every other point simultaneously. If you poke a cube of incompressible Euler fluid with your finger, there is no pressure wave that ripples through it, where the fluid parcels push each other along and get out of each other's way. Instead, the whole cube of fluid somehow instantly adopts a new flow pattern that conserves mass/momentum/energy in response to that finger-poke.

1) Note that velocity is a vector, since it has a direction. This means that in 2D the velocity is two numbers, and in 3D it's three numbers. So technically the 3D incompressible Euler equations have four values at every point: one density, and three velocity components, one each in the x, y, and z directions.

2) I'm a numerical simulation guy, not a mathematician. Real math experts have rigorous definitions of a singularity, e.g. in https://arxiv.org/pdf/2203.17221.pdf "Singularity formation in the incompressible Euler equation in finite and infinite time," Theodore D. Drivas and Tarek M. Elgindi.

Re: Computer proof ‘blows up’ centuries-old fluid equations

#145

Post by NotYourLawyer »

nuclearnice1 wrote: post #130
wwalker3 wrote: post #101

The question that the referenced paper (1) is trying to answer is "do the 3D incompressible Euler equations develop a finite time singularity from smooth initial data of finite energy?" This is an important question in the theory of nonlinear partial differential equations, but is probably not as relevant to real fluid flow as a lay reader might imagine. The incompressible Euler equations model a very strange and unp…

What a wonderfully informative and educational comment. Thank you. Would you also be able to shed some light on what a singularity is? It was not intuitive to me that incompressiblity should lead to a singularity. The article dances around the term: > At that point, the Euler equations are said to give rise to a “singularity” — or, more dramatically, to “blow up.” > Once they hit that singularity, the equations will…

It’s when some physical quantity of the simulation becomes infinite. Pressure, particle velocity, etc.

Re: Computer proof ‘blows up’ centuries-old fluid equations

#146

Post by bheadmaster »

guipsp wrote: post #139
bheadmaster wrote: post #133

Earlier quoted context omitted.

I don't understand how is "the arithmetic model" a social truth, when it clearly corresponds to physical phenomena. You can make a skyscraper that doesn't fall, and it exists regardless of whether other people see it or not. You see it - it's there. What is "social" about that? > However, confusing your model for reality is a fallacy perhaps older than time. I think that the human perception the world is a robust eno…

We have a model of physics which is pretty accurate. The engineers who designed the skyscraper did not even use this model, they used a much simpler one, with known errors. Why? It is simply good enough™. But you can't claim it is even "true" when we know more accurate methods.

My claim is not that the model itself is true - I'm claiming that the underlying mechanisms that rule the world are true and are not subject to change by social consensus.

The model is "good enough" for the purpose of creating a building, but that doesn't make the act of "creating a building" any less real, nor the underlying rules that govern matter any less true. Our descriptions are not real - the rules themselves (which we may not know exactly) are real.

Therefore, mathematics - the set of rules that corresponds to how reality works - itself exists in reality regardless of social consensus. Society can't change them by making a different consensus.

Re: Computer proof ‘blows up’ centuries-old fluid equations

#147

Post by prmph »

musingsole wrote: post #128
naasking wrote: post #120

Earlier quoted context omitted.

> But what this doesn't get at is that the very system of logic we use to make proofs is a social construct. Other cultures have had other systems of logic, and called valid arguments that we wouldn't today precisely because they were using a different system of logic. I think you need to separate the process of developing mathematics, from the self-consistency and validity of the logical argument or mathematical str…

> whether any given formal system actually is consistent is not dependent on consensus, it is a fact, either true or false, completely independent of consensus. The definition of "consistent" seems completely entangled with a given social group's ideas of "rational". You might imply that our word for it hints at a Platonic ideal of "consistent", but if that's true, then you're caught in an infinite cascade of which n…

Exactly the point I was making.

Many do not seems to see the fundamental issue at play here, and another way to think of them is what you have hinted at: the role of language. There is no objective way to nail down the meaning of words, like "consistent", "proof", "equal", etc.

Suppose one wanted a maximally rigorous definition of "equal". Does it mean two things that cause people to think of the same thing when they are mentioned? Does it mean two things that occupy the same position in space at all times? It is actually a difficult concept to define rigorously.

This is not to deny that there is an objective reality. But that reality is highly contextual and multi-faceted. We cannot be 100% exact in defining that reality using language (even a math language), and this is where the social nature of that reality becomes apparent.

The role of proofs are in creating, as far as possible, as rigorous a shared context for the reality being described.

Re: Computer proof ‘blows up’ centuries-old fluid equations

#148

Post by auggierose »

musingsole wrote: post #128
naasking wrote: post #120

Earlier quoted context omitted.

> But what this doesn't get at is that the very system of logic we use to make proofs is a social construct. Other cultures have had other systems of logic, and called valid arguments that we wouldn't today precisely because they were using a different system of logic. I think you need to separate the process of developing mathematics, from the self-consistency and validity of the logical argument or mathematical str…

> whether any given formal system actually is consistent is not dependent on consensus, it is a fact, either true or false, completely independent of consensus. The definition of "consistent" seems completely entangled with a given social group's ideas of "rational". You might imply that our word for it hints at a Platonic ideal of "consistent", but if that's true, then you're caught in an infinite cascade of which n…

There is a simple definition for what consistent means, which naasking is referring to: is it impossible to derive "false" purely by applying the rules of the logical system?

Re: Computer proof ‘blows up’ centuries-old fluid equations

#149

Post by [deleted] »

yazaddaruvala wrote: post #102
mik1998 wrote: post #77

Earlier quoted context omitted.

It's not. It's a fact derived a priori from the definition of addition and the axioms of the field of real numbers. There's nothing special about the claim that 1 + 1 = 2. You can also define addition such that 1 + 1 = 0 or anything else.

“a priori from the definition of addition” is effectively saying “a social construct” And I hate to break it to you but all “axioms” are social constructs that we use to create hopefully useful models.

[deleted]

Re: Computer proof ‘blows up’ centuries-old fluid equations

#150

Post by pencilguin »

wwalker3 wrote: post #101

The question that the referenced paper (1) is trying to answer is "do the 3D incompressible Euler equations develop a finite time singularity from smooth initial data of finite energy?" This is an important question in the theory of nonlinear partial differential equations, but is probably not as relevant to real fluid flow as a lay reader might imagine. The incompressible Euler equations model a very strange and unp…

I guess this answers why you can't just try it with liquid helium: even that isn't ideal enough.
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