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Computer proof ‘blows up’ centuries-old fluid equations

quantamagazine.org

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

#171

Post by WalterBright »

wwalker3 wrote: post #168
WalterBright wrote: post #153

Earlier quoted context omitted.

Best comment I ever read on HN.

Wow, I'm honored :) These days, I try to only comment when an article is really in my wheelhouse, but that's not very often, given my narrow interests in fluid dynamics and computational physics.

> I try to only comment when an article is really in my wheelhouse

Which is why your comment is exceptionally worthwhile. I also know enough about fluid mechanics to both understand and appreciate it.

People often wonder on HN what the point of a STEM degree is (after making money). To me I've had a lifetime of pleasure from understanding how things work. It's so much better than things being mysterious black boxes.

I once asked a date if she wanted to understand how airplanes worked. She said no, that understanding them would make her afraid of flying. For me, it was the opposite. Knowing how the airplanes fly and how it all works made me a much less anxious passenger.

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

#172

Post by vba616 »

wwalker3 wrote: post #144
nuclearnice1 wrote: post #130

Earlier quoted context omitted.

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-ne…

>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

I don't get it. If the fluid is incompressible, how can density have a value at every point in space? Isn't it just a constant?

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

#173

Post by vba616 »

civilized wrote: post #167
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…

This raises a question I hadn't thought of before. Real-world fluid flow is ultimately well-modeled by the equations of many-body Newtonian mechanics, right (atoms bumping around)? Are those equations vulnerable to blow-ups?

This makes me think of:

https://en.wikipedia.org/wiki/Sonoluminescence

I would think that nothing in reality is infinite, but allegedly sound waves collapsing bubbles in a fluid can cause a very small amount of plasma to become hotter than the sun and emit light. Some controversial research claims it might even be possible to create atomic fusion this way.

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

#174

Post by edgyquant »

bheadmaster wrote: post #97
ethanwillis wrote: post #92

Earlier quoted context omitted.

What's a rock?

What's a "what's"?

An inquiry for a description of an object. Nothing in language is difficult it’s finding the smallest abstraction to generate all the rules for all language that is.

You’re asking a simple grammar question under the impression it’s an unsolved science question.

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

#175

Post by edgyquant »

ethanwillis wrote: post #100
bheadmaster wrote: post #97

Earlier quoted context omitted.

What's a "what's"?

Well for either of us to know, we'd need consensus on language, meaning, words, etc.

Nope. The idea of describing an object is built in to humans and likely all mammals.

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

#176

Post by yazaddaruvala »

mik1998 wrote: post #116
yazaddaruvala wrote: post #102

Earlier quoted context omitted.

“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.

No, this is not true at all. It is a certain _mathematical_ construction. It has nothing to do with society. Given the specific definition of addition in the field of real numbers, the statement 1 + 1 = 2 is true in any society. > but all “axioms” are social constructs Again, they are mathematical constructs. Axioms don't have anything to do with society. I can construct any axiom I want, and derive true statements f…

I’ll try one last time:

> definition of addition

What do you think any definition is other than the creation of a socially shared construct?

> I can construct any axiom I want, and derive true statements from it.

Let’s test this. Axiomatically I am always right! As such, I derive that it is illogical, dare I say irrational, of you to disagree with me. A truer statement has never even been written! Facts!

Hmm… turns out you disagree that it’s a true statement. Why? I had an axiom! You don’t agree with my axiom? If that’s allowed, I guess I’ll have to convince you (and society) of my axiom!

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

#177

Post by fspeech »

ordu wrote: post #141
bheadmaster wrote: post #65

Earlier quoted context omitted.

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 pr…

In il nome della rosa William of Baskerville tells us that concepts are signs and words are "signs of signs".

In math logic we can define systems of symbols and logical connectives and deductive rules. We can argue for the correctness of a logical system but we can not do it in the same system; we have to go "meta". Similarly a formal system could be a model of something "real", but the correspondence would be beyond mere logic.

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

#178

Post by auggierose »

naasking wrote: post #169
prmph wrote: post #147

Earlier quoted context omitted.

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…

Have you perchance been reading a lot of Wittgenstein? I think you're conflating universality and objectivity. Those terms you list all have objective definitions, but the specific characteristics they have in any given logic may differ. That means they are not universal, but that doesn't make them non-objective. Objective typically means "mind independent". Your example of equality already demonstrates you understan…

> Godel showed that there is no such thing as a universal logic in our current approach to formal systems, but that didn't suddenly make logic non-objective. It simply means that there is no Ur-logic that can subsume all other logics (which is why most assert that Godel ended Hilbert's program).

Could you elaborate on that? Any references?

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

#179

Post by wwalker3 »

civilized wrote: post #167
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…

This raises a question I hadn't thought of before. Real-world fluid flow is ultimately well-modeled by the equations of many-body Newtonian mechanics, right (atoms bumping around)? Are those equations vulnerable to blow-ups?

Pretty much any mathematical model of a real phenomenon can have some sort of singularity or discontinuity in it.

If you model atoms as dimensionless points (1), then any kind of force law with the distance between atoms in the denominator can lead to a singularity when that distance is zero. In practice, you write the simulator to disallow this, but it's still there in the equations, you're just ignoring it.

If you model your atoms as finite-sized but incompressible billiard balls, then when they hit each other it's a discontinuity, since they instantly change direction when they collide. These collisions conserve total momentum and energy, but they're unphysical because real physical quantities can't jump from one value to another (in classical physics).

Even if you model your atoms as little rubber balls, the model can still be singular. Linear elasticity (the most common choice) allows you to compress a finite-sized object down to zero size with finite energy, which yields infinite energy density. Again, you'd have to disallow that in the simulator, which is very practical, but not theoretically satisfying.

1) https://en.wikipedia.org/wiki/Molecular_dynamics is the typical method of atomistic simulation.

2) https://en.wikipedia.org/wiki/Linear_elasticity

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

#180

Post by naasking »

gmfawcett wrote: post #170
naasking wrote: post #155

Earlier quoted context omitted.

Firstly, the reals contain the integers, so there is an isomorphism as I said. Secondly, discretization absolutely is universal. It's literally in the laws of physics for one (particles are discrete, energy levels are discrete, etc.). For another, are you suggesting a physical alien species will have a continuous number of appendages, or organs, or that their population will somehow be continuous? I frankly don't see…

Eh, you're not explaining universal truths here, you're just anthropomorphizing. Why must it have appendages, organs, or populations? Why presuppose that its conceptual model includes particles at all? What if a vast, hyper-continuous intelligence simply cannot comprehend the concept of being discrete?

Firstly, those were just examples of commonly countable structures, even if they're not universal (which is debatable). Discretely countable structures are literally everywhere and fundamentally inescapable, which is why I mentioned physics. I didn't presuppose physics, the discrete structure of physical reality is directly observable, it's not some fiction we made up.

Secondly, what we know must be bound by what we've observed. You can imagine any sort of being you like, but that doesn't make your imagined creature logically coherent or physically realizable.

Any physically realizable intelligence must:

a) Be differentiable from its environment: that means it must have some enclosing boundary separating an inside that's different than an outside.

b) Have internal structure: intelligence by necessity is structured thought. Structured thought entails differentiable physical structure to hold structured thoughts. Such structure by itself is necessarily countable, being made of matter.

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