What Makes the Hardest Equations in Physics So Difficult? • Hacker News Board

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

#51

Post by njarboe » Wed, Jan 17, 2018, 5:34 AM UTC

evandijk70 wrote: post #28

ctdonath wrote: post #27

Earlier quoted context omitted.

we still don't have the computational power to do things atom-by-atom That's what puzzles me about this discussion. Of course we don't. The goal here is to model more atoms than we have atoms to model with. Until we get to quantum computers that can represent more information per atom, than information per atom we want to represent, it's simply a matter of objectively & obviously inadequate resources. You can't simul…

It is not necessarily the goal to model more atoms than we have atoms to model with. The goal can be, for example, to model a folding protein, which has fewer atoms than a CPU.

And one might be happy to spend a month computing the folding while in the cell it takes a few seconds.

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

#52

Post by jcranmer » Wed, Jan 17, 2018, 6:35 AM UTC

kazinator wrote: post #36

carterehsmith wrote: post #33

Earlier quoted context omitted.

>> So, Navier-Stokes assumes infinite divisibility in the fluid it models, right? Which we know is not the case: atoms exist. Atoms have structure, they consist of... whatever parts. And these parts have structure (like... quarks and whatnots). And... are we sure that quarks do not have structure themselves? Whatever the case is, it is not something that one should declare in a HN thread. Even more interesting... 1+2…

> 1+2+3+4... = -1/12 Only through a derivation which involves some cute but unsound pseudo-algebra on infinite series. See, in the same page, the remark "Generally speaking, it is incorrect to manipulate infinite series as if they were finite sums" .

Just because you don't understand doesn't mean it's "unsound pseudo-algebra." There are ways to properly manipulate infinite series, and Wikipedia does actually explain some of them on the page, even if it relies on a generally-unsound way to illustrate the intuition to a lay reader.

It is very well-known and long-recognized that our natural intuition is very wrong when it comes to infinities and infinite objects, and mathematics had a crisis in trying to come to grips with it.

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

#53

Post by oldandtired » Wed, Jan 17, 2018, 7:27 AM UTC

improbable22 wrote: post #19

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…

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 theori…

> The things about which we are totally ignorant are much smaller today, of course

One would have to disagree with this idea. The more we study the universe around us, the less we actually know. As a somewhat philosophical point, there are many times when our mental (mathematical) models get in the road of understanding. Very often, people believe that because we have a model that works and appears to give good predictive results about some phenomena then we understand the how and the what (and even the why) of those phenomena.

When this happens, we get into a situation where alternative models are actually discouraged. If one looks the the history of the 19th, 20th and 21st centuries, one can see that more and vaster avenues of investigation have arisen as time passes. Our increasing knowledge is continuing to be shown as ever smaller in relation to what we are now seeing.

No theory is ever complete, nor is it ever accurate to the extent that it describes the reality of the universe around us. All theories make those simplifying assumptions that when taken too far lead into inaccurately describing and predicting what we should see. Too often people get enamoured by the beauty of the mathematics and forget it is only an attempt at reflecting reality.

Mathematics is a magnificent and useful tool, but it is a foolish master. Too often we forget that.

Theories and models help us gain some understanding of the nature of the universe around us. This understanding, however, is always subject to change, no matter how "perfect" the theory may appear to be. There are too many scientists, both theoretical and practical, who are so infatuated and enamoured with their current models that they have forgotten that the models are approximations only and are subject to change or even overturning.

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

#54

Post by _0w8t » Wed, Jan 17, 2018, 7:40 AM UTC

When I read about hardest equations I thought it would be equations of either General Relativity or Quantum Mechanics of complex systems. Both are much more complex than Navier-Stokes.

For Navier-Stokes we have at least numerical methods that allow to solve the equations in practical time for useful cases. But we still do not know how to model, say, dynamics of a galaxy using General Relativity. Typically such models just assume Newton mechanics with minimal if any relativistic corrections with no proof that one can use such approximations of equations of GR on big scales.

Or consider the problem of a possible state of metallic hydrogen. It is just a bunch of protons and electrons mixed together, the simplest possible material. Yet we cannot calculate from the first principles its properties.

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

#55

Post by Someone » Wed, Jan 17, 2018, 7:41 AM UTC

whatshisface wrote: post #15

jerf wrote: post #5

Earlier quoted context omitted.

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…

Your repeated use of the word ‘blowup’ makes me think of the way we get small-scale behavior amplified enough for us to see, hear, and feel: in atomic weapons.

There is a chance that understanding how quarks interact will give us even more powerful weapons (which might at one time be useful to blow up large incoming space rocks) or give us clean energy. How large that chance is is anybody’s guess.

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

#56

Post by zakk » Wed, Jan 17, 2018, 7:44 AM UTC

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…

> So, Navier-Stokes assumes infinite divisibility in the fluid it models, right?

You are right.

> 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.

The mapping between physical reality and Navier-Stokes equations is extremely well understood. We know when they represent an excellent description of the physical world, and we know when they could fail due to the finite number of particle in the fluid.

We also know how they fail, and can estimate corrections due to finite particle number, see for instance the Cunningham correction factor.

We also know that a much more complicated theory, Boltzmann transport equations, would be exact.

It would be surprising if Navier-Stokes equations blew up, because we know they represent most often physical reality up to extremely small, controlled corrections.

> Am I missing something? Why isn't it assumed that they're just a very nice approximation, like Newton's laws pre-relativity? Actually the whole thing reminds me of the ultraviolet catastrophe that preceded quantum physics.

Not really: the ultraviolet catastrophe was a symptom of unknown physics awaiting to be discovered.

We know very well the physics beyond Navier-Stokes, and still we use Navier-Stokes equations for their incredible effectiveness.

Using the full theory (Boltzmann transport equations) would make even the simplest fluidodinamics calculation virtually impossible, while adding a correction on the n-th significant digit, with n much higher than the precision one could reasonably expect.

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

#57

Post by improbable22 » Wed, Jan 17, 2018, 9:10 AM UTC

oldandtired wrote: post #53

improbable22 wrote: post #19

Earlier quoted context omitted.

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 theori…

> The things about which we are totally ignorant are much smaller today, of course One would have to disagree with this idea. The more we study the universe around us, the less we actually know. As a somewhat philosophical point, there are many times when our mental (mathematical) models get in the road of understanding. Very often, people believe that because we have a model that works and appears to give good predi…

"much smaller today" meaning literally smaller: Navier didn't know about atoms, 10^-10 m, but now we know about things at 10^-18 m or so.

Sure, our awareness of how much we don't know has grown over time.

The point I was trying to make is that theoretical models (like N-S) not only don't have to be perfect to be useful, but more, are useful precisely because they are not complete. By ignoring irrelevant details we get theory, not just simulation.

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

#58

Post by kazinator » Wed, Jan 17, 2018, 11:46 AM UTC

carterehsmith wrote: post #39

knight-of-lambd wrote: post #35

Earlier quoted context omitted.

> Even more interesting... 1+2+3+4... = -1/12 What.. No. This sum diverges. It does not equal -1/12. Now if you plug -1 into the Riemann zeta function, you get -1/12. One could interpret that to mean the sum 1 + 2 + 3 + .. can be mapped to -1/12. But the sum has never, and will never equal -1/12. It diverges, simple as that.

What is the argument here? There are certain ways of assigning a number to the divergent (by partial sums) sum. If they are well-defined, they all arrive at the same number. In this case, -1/12. There is nothing to argue here, that shit has been around for 200 years or more. And, there are some results in physics that actually measure close to the number -1/12, from something that looks like a sum of 1+2+3... and tha…

> What is the argument here?

Essentially, it consists of text pulled from exactly the same source that you originally cited.

(Cue sound of hands washing in sink.)

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

#59

Post by kazinator » Wed, Jan 17, 2018, 11:55 AM UTC

carterehsmith wrote: post #47

twiceaday wrote: post #42

Earlier quoted context omitted.

The bit on Wikipedia that follows the equation is part of the equation. It is the context that explains how to compute it. You need that bit because the equation uses extremely common syntax as short-hand for an operation that is different from the common one. If you write it in a way that doesn't rely on redefining common syntax like so "ζ(-1) = -1/12" it seems a lot less interesting doesn't it? Perhaps syntax confu…

There is more to it - the number -1/12 actually pops up in physics when doing something awfully similar to 1+2+3... This may be useful: https://motls.blogspot.ca/2014/01/sum-of-integers-and-overso... " The real problem is that the definition of the sum involving the limit of partial sums – limits that way too often "diverge" or "refuse to exist" – isn't the only definition or the best definition or the most natural d…

The ordinary sum of any collection of positive integers is not a fraction, let alone a negative one.

If we take the set { 1, 2, 3, ... 4 } the summation or partial summation of no subset of this set, finite or not, converges on any fraction or negative number. No matter what order we choose for traversing that set for generating a series, we never see anything resembling -1/12 as a partial sum or limit.

The ordinary arithmetic sum of any collection of positive integers is an integer which is strictly greater than each the integers.

The pages you're referencing are all crackpottery.

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

#60

Post by Retric » Wed, Jan 17, 2018, 12:24 PM UTC

clicea wrote: post #50

Retric wrote: post #11

Earlier quoted context omitted.

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

Don't be fooled by the numbers e9 is 5 orders od magnitude less than e14

64 GPU's and 91 hours for a 12 second simulation. 64,000 CPU's for 910 hours (1 month) is 4 orders of magnitude.

Unfortunately, a useful cell simulation would be far more complex than simple particle collisions and a much shorter step size. But, that's not to say different kinds of simulations can't be useful. And ASIC's or general improvement in computing power can also boost things.