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What Makes the Hardest Equations in Physics So Difficult?

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Re: What Makes the Hardest Equations in Physics So Difficult?

#11

Post by Retric »

openasocket wrote: post #8
tw1010 wrote: post #7

Earlier quoted context omitted.

This is such a fun observation. Physicists originally used differential equations to describe large scale, yet fundamentally atomistic, behaviour because, I assume, that was easier to work with using pen and paper than by using some more discrete, computational, atom-by-atom model. And then computers were invented, but we're still stuck with these continuous, legacy, models. Almost no-one questions the fact that the…

Well, we still don't have the computational power to do things atom-by-atom, even in small things like modeling cells, let alone something like simulating wind shear on a plane or predicting weather patterns. It's not just computational power but memory. There's something like 1E14 atoms in a cell. Storing the x,y,z position with 4 bytes for each dimension (though you'll probably want 8 bytes) comes out to 1.2E15 byt…

Here is a billion particle model (e9) which is a long way from atomic scale simulation (e14+), but not outside the realm of the possible.

https://www.youtube.com/watch?v=B8mP9E75D08

Re: What Makes the Hardest Equations in Physics So Difficult?

#12

Post by auntienomen »

tw1010 wrote: post #9
openasocket wrote: post #8

Earlier quoted context omitted.

Well, we still don't have the computational power to do things atom-by-atom, even in small things like modeling cells, let alone something like simulating wind shear on a plane or predicting weather patterns. It's not just computational power but memory. There's something like 1E14 atoms in a cell. Storing the x,y,z position with 4 bytes for each dimension (though you'll probably want 8 bytes) comes out to 1.2E15 byt…

Sure, but you'd think we would shift our focus to using models explicitly tailored with that aim in mind. Using something that explicitly approximates a desired, precise, foundation. Perhaps parameterized by model accuracy. I would think taking inspiration from approximation algorithms would be a much more appropriate lens with which to model physics.

In the case of Navier-Stokes, this was done in 1934 by Jean Leray, who showed that the so-called 'weak' solutions always exist. (A weak solution is not a function, but a more general object called a distribution which can be fuzzed out on very short distance scales.)

It's a very reasonable idea, and useful enough for computations, but it lacks the essential tension that makes Navier-Stokes a mathematical Everest.

Re: What Makes the Hardest Equations in Physics So Difficult?

#13

Post by electricslpnsld »

openasocket wrote: post #8
tw1010 wrote: post #7

Earlier quoted context omitted.

This is such a fun observation. Physicists originally used differential equations to describe large scale, yet fundamentally atomistic, behaviour because, I assume, that was easier to work with using pen and paper than by using some more discrete, computational, atom-by-atom model. And then computers were invented, but we're still stuck with these continuous, legacy, models. Almost no-one questions the fact that the…

Well, we still don't have the computational power to do things atom-by-atom, even in small things like modeling cells, let alone something like simulating wind shear on a plane or predicting weather patterns. It's not just computational power but memory. There's something like 1E14 atoms in a cell. Storing the x,y,z position with 4 bytes for each dimension (though you'll probably want 8 bytes) comes out to 1.2E15 byt…

> Well, we still don't have the computational power to do things atom-by-atom

People do atom-by-atom molecular dynamics simulations of proteins and such.

Re: What Makes the Hardest Equations in Physics So Difficult?

#14

Post by al2o3cr »

tw1010 wrote: post #7
andrewflnr wrote: post #4

So, Navier-Stokes assumes infinite divisibility in the fluid it models, right? Which we know is not the case: atoms exist. It seems obvious to me that Navier-Stokes can't be a perfectly accurate description of reality, so it should come as no surprise if they turn out to blow up or otherwise behave non-physically. Am I missing something? Why isn't it assumed that they're just a very nice approximation, like Newton's…

This is such a fun observation. Physicists originally used differential equations to describe large scale, yet fundamentally atomistic, behaviour because, I assume, that was easier to work with using pen and paper than by using some more discrete, computational, atom-by-atom model. And then computers were invented, but we're still stuck with these continuous, legacy, models. Almost no-one questions the fact that the…

It's unclear whether atomic-level data would be useful in these sorts of calculations - the input data isn't known to sufficient precision. For more, check out the last two paragraphs from the Feynman Lectures Vol III Chapter 2:

http://www.feynmanlectures.caltech.edu/III_02.html

A similar sort of mechanism led to the discover of chaotic behavior in weather models - checkpointing results at a precision slightly lower than the machine's internal precision caused simulations that were resumed from the checkpoint to deviate rapidly.

Re: What Makes the Hardest Equations in Physics So Difficult?

#15

Post by whatshisface »

jerf wrote: post #5
andrewflnr wrote: post #4

So, Navier-Stokes assumes infinite divisibility in the fluid it models, right? Which we know is not the case: atoms exist. It seems obvious to me that Navier-Stokes can't be a perfectly accurate description of reality, so it should come as no surprise if they turn out to blow up or otherwise behave non-physically. Am I missing something? Why isn't it assumed that they're just a very nice approximation, like Newton's…

Tone: Straight. I think you asked a reasonable question. What you are missing is that mathematicians are concerned about Navier-Stokes itself, as an independent entity in pure mathematics, not related to its physics correspondence. I think it would be safe to say that everybody believes that even if Navier-Stokes does have a singularity in it, that there won't be any way to manifest that singularity in the real unive…

One reason to think the physics would be interesting: it isn't every day that we can arrange macroscopic systems in ways where their small-scale behavior gets amplified enough for us to measure. Arranging a blowup in the Navier-Stokes equations would be a way to study atomic-scale physics on human scales - and, likewise, arranging a blowup (if one exists) in the QCD quark-gluon plasma equations would allow us to get unexpectedly sensitive measurements of particle-scale physics.

Re: What Makes the Hardest Equations in Physics So Difficult?

#17

Post by openasocket »

Retric wrote: post #11
openasocket wrote: post #8

Earlier quoted context omitted.

Well, we still don't have the computational power to do things atom-by-atom, even in small things like modeling cells, let alone something like simulating wind shear on a plane or predicting weather patterns. It's not just computational power but memory. There's something like 1E14 atoms in a cell. Storing the x,y,z position with 4 bytes for each dimension (though you'll probably want 8 bytes) comes out to 1.2E15 byt…

Here is a billion particle model (e9) which is a long way from atomic scale simulation (e14+), but not outside the realm of the possible. https://www.youtube.com/watch?v=B8mP9E75D08

We are definitely making progress. But another factor to consider, in additional to the number of particles, is the energy of the system. When going through one step of a simulation you have to compute the effects that every particle has on every other particle, but generally a particle only affects the other particles near it. You can take advantage of this fact to speed up computation with the right data structures.

However, as particles start moving around faster and faster, the distance they can travel in one time step increases, and the neighborhood of effect increases, limiting the speedup of this optimization. This matters less in larger scale simulations like of weather patterns, because you don't have to worry about air molecules zipping to the other side of the continent in one second. But it matters a lot in small scale simulations, especially in cells. As an example, an average glucose molecule in one of your cells is bouncing around at around 250 miles per hour! That's not 250 miles when scaled up, that's really 250 miles per hour. A molecule in your body is colliding with another billions of times every second. (Source: http://www.righto.com/2011/07/cells-are-very-fast-and-crowde...). At that scale and that level of activity it becomes much harder to simulate each time step.

Re: What Makes the Hardest Equations in Physics So Difficult?

#18

Post by openasocket »

electricslpnsld wrote: post #13
openasocket wrote: post #8

Earlier quoted context omitted.

Well, we still don't have the computational power to do things atom-by-atom, even in small things like modeling cells, let alone something like simulating wind shear on a plane or predicting weather patterns. It's not just computational power but memory. There's something like 1E14 atoms in a cell. Storing the x,y,z position with 4 bytes for each dimension (though you'll probably want 8 bytes) comes out to 1.2E15 byt…

> Well, we still don't have the computational power to do things atom-by-atom People do atom-by-atom molecular dynamics simulations of proteins and such.

Certainly, but not at the level of an entire cell

Re: What Makes the Hardest Equations in Physics So Difficult?

#19

Post by improbable22 »

tw1010 wrote: post #7
andrewflnr wrote: post #4

So, Navier-Stokes assumes infinite divisibility in the fluid it models, right? Which we know is not the case: atoms exist. It seems obvious to me that Navier-Stokes can't be a perfectly accurate description of reality, so it should come as no surprise if they turn out to blow up or otherwise behave non-physically. Am I missing something? Why isn't it assumed that they're just a very nice approximation, like Newton's…

This is such a fun observation. Physicists originally used differential equations to describe large scale, yet fundamentally atomistic, behaviour because, I assume, that was easier to work with using pen and paper than by using some more discrete, computational, atom-by-atom model. And then computers were invented, but we're still stuck with these continuous, legacy, models. Almost no-one questions the fact that the…

Human limitations is one reason, but there's something much deeper going on here.

Navier and Stokes worked before we were sure that atoms existed, certainly before we had any idea of how many there were. Nevertheless they were able to write down useful theories for describing fluids. This is how all of science works. The things about which we are totally ignorant are much smaller today, of course... but useful theories of any set of phenomena always omit a great many things we do in fact know about.

From these theories, we can understand what's going on, and use this to extrapolate to things we have not seen yet. An atom-by-atom computational model would (in some sense) be no more useful than what we had before N-S, just blind experiment. To try out any given swirl of smoke etc. we can equally well walk next door to the lab and videotape it... but this doesn't help us imagine what else might be possible. The "blowup scenario" discussed is an example of this kind of imagining.

Re: What Makes the Hardest Equations in Physics So Difficult?

#20

Post by btilly »

tw1010 wrote: post #9
openasocket wrote: post #8

Earlier quoted context omitted.

Well, we still don't have the computational power to do things atom-by-atom, even in small things like modeling cells, let alone something like simulating wind shear on a plane or predicting weather patterns. It's not just computational power but memory. There's something like 1E14 atoms in a cell. Storing the x,y,z position with 4 bytes for each dimension (though you'll probably want 8 bytes) comes out to 1.2E15 byt…

Sure, but you'd think we would shift our focus to using models explicitly tailored with that aim in mind. Using something that explicitly approximates a desired, precise, foundation. Perhaps parameterized by model accuracy. I would think taking inspiration from approximation algorithms would be a much more appropriate lens with which to model physics.

The mental model of a bunch of hard little atoms flying around, colliding, rebounding, etc is actually NOT the desired, precise, foundation for understanding how gases behave.

The better foundation is quantum mechanics. An example of a macroscopically visible difference between these foundations are the van der Waals forces.

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