Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Computer proof ‘blows up’ centuries-old fluid equations

quantamagazine.org

Post reply on HN
Print view View original

181–190 of 223 posts

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

#181

Post by civilized »

wwalker3 wrote: post #179
civilized wrote: post #167

Earlier quoted context omitted.

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…

I'm asking about the actual properties of the equations, not if it's hard to do simulations.

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

#182

Post by naasking »

auggierose wrote: post #178
naasking wrote: post #169

Earlier quoted context omitted.

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?

These are the implications of Godel's incompleteness theorems. No formal system expressive enough to encode arithmetic can simultaneously be both complete and prove its own consistency, because there will always be true propositions expressible in that system that cannot be proven in that system.

This is why Hilbert's program to finitely axiomatize mathematics can't be completed. The "escape hatch" here is simply that not all propositions are actually interesting, so finite axiomatizations are still very useful, and we can extend the axiomatic basis as needed given satisfactory justification. This last part is the only place where social consensus sometimes comes into play (continuum hypothesis, etc).

Edit: there is another possible escape hatch that hasn't been fully explored IMO, and that's some variant of finitism. All these impossibility proofs depend on infinite structures to derive incompleteness or contradiction, but if infinite structures are not expressible...

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

#183

Post by wwalker3 »

vba616 wrote: post #172
wwalker3 wrote: post #144

Earlier quoted context omitted.

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?

The density can be constant, but it doesn't have to be. If the density field starts out with some variation in it, then those variations move around as the fluid flows. Incompressibility just means that those density variations can't get bigger or smaller, they can only move, shear, and rotate.

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

#184

Post by davnn »

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 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.

What kinds of problems does it solve to know an answer to this question? Honestly curious, please do not take this as offensive/dismissive.

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

#185

Post by wwalker3 »

civilized wrote: post #181
wwalker3 wrote: post #179

Earlier quoted context omitted.

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…

I'm asking about the actual properties of the equations, not if it's hard to do simulations.

It's the equations themselves that are singular. When we write simulators, we usually have to paper over the singularities that are inherent in the math.

For example, if you're simulating charged particles moving around, and you use a force equation F = k q1 q2 / d^2 (1), then when d approaches 0 (i.e. when the distance between particles approaches zero), then the force F goes to infinity.

For atoms, it works the same way. If you use a force law like Lennard-Jones (2), it also has the interatomic distance in the denominator, so the equation has a singularity baked right in.

You could always adopt a more complex force equation that doesn't have a singularity in it. But in practice, it's easier to use a simple but singular equation, and then selectively ignore its bad behavior.

1) https://en.wikipedia.org/wiki/Coulomb%27s_law

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

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

#186

Post by wwalker3 »

davnn wrote: post #184
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 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. What kinds of problems does it solve to know an answer to this question? Honestly curious, please do not take this as offensive/dismissive.

If mathematicians could solve these kinds of problems, they could answer valuable questions like "Will this equation always have a physically meaningful solution?" If the answer was "No", then we would know that the equation can't be a faithful model of reality.

We already know that the incompressible Euler equations can't be a faithful model, for reasons I've mentioned elsewhere in the thread. But I think the hope is that if they can answer these questions for incompressible Euler, then they can eventually extend their techniques to more complex fluid equations like Navier-Stokes, which people generally assume (but can't yet prove) is physically reasonable.

Simulation has great practical value, but it doesn't give you any guarantees about the behavior of the solutions for all the cases you haven't actually tried.

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

#187

Post by throwaway81523 »

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?

There are all kinds of blow-ups in Newtonian mechanics and in other equations of physics. The singularity at the center of a black hole in general relativity is a famous example. The ultraviolet catastrophe in classical thermodynamics was another. The presumption is that blow-ups in an equation indicate a mismatch between the equation and the true physical world, telling physicists to look for better theories, whose equations don't blow up. For the ultraviolet catastrophe, the mystery was solved through the discovery of quantum mechanics. For GR, it is still unsolved, and the solution is expected to come from a theory of quantum gravity that hasn't yet been invented, but is the target of tons of research.

Here's a cool expository article about blow-ups in classical mechanics and elsewhere: https://arxiv.org/abs/1609.01421

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

#188

Post by auggierose »

naasking wrote: post #182
auggierose wrote: post #178

Earlier quoted context omitted.

> 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?

These are the implications of Godel's incompleteness theorems. No formal system expressive enough to encode arithmetic can simultaneously be both complete and prove its own consistency, because there will always be true propositions expressible in that system that cannot be proven in that system. This is why Hilbert's program to finitely axiomatize mathematics can't be completed. The "escape hatch" here is simply tha…

Ah ok, I see what you mean.

This is different from how I would understand the notion Ur-Logic: Just a logical system that can express anything you want, given you are free to add axioms. Obviously there are a few choices for that.

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

#189

Post by civilized »

wwalker3 wrote: post #185
civilized wrote: post #181

Earlier quoted context omitted.

I'm asking about the actual properties of the equations, not if it's hard to do simulations.

It's the equations themselves that are singular. When we write simulators, we usually have to paper over the singularities that are inherent in the math. For example, if you're simulating charged particles moving around, and you use a force equation F = k q1 q2 / d^2 (1), then when d approaches 0 (i.e. when the distance between particles approaches zero), then the force F goes to infinity. For atoms, it works the sam…

The presence of a singularity in the force doesn't mean it will cause a blow up in the solution. Two positively charged point particles interacting electrostatically can be shot at each other at any angle or speed and blowup will never occur.

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

#190

Post by TEP_Kim_Il_Sung »

NotYourLawyer wrote: post #145
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…

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

That usually indicates a phase change, with a separate set of applicable equations.
Post reply on HN