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

What Makes the Hardest Equations in Physics So Difficult?

quantamagazine.org

Post reply on HN
Print view View original

41–50 of 75 posts

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

#41

Post by RhysU »

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...

We still don't have the computational power to do certain classes of continuous Navier--Stokes calculations (i.e. not atom based).

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

#42

Post by twiceaday »

carterehsmith wrote: post #37
twiceaday wrote: post #34

Earlier quoted context omitted.

There is a standard definition of what '=' means and your equation does not satisfy it. https://www.youtube.com/watch?v=YuIIjLr6vUA

You say "your" and yet, this has nothing to do with me. I pointed to the wikipedia article. BTW this result is not exactly new, and it has specific confirmation in physics. You cannot argue that with high-school math. There is more to it.

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 confusion is the only interesting thing to it.

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

#43

Post by adrianratnapala »

andrewflnr wrote: post #32
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…

I guess if you're looking for pure mathematical abstraction, Navier-Stokes doesn't seem like the most interesting without the physics connection. But that's just me.

I suppose what is interesting is the very difficulties we have deciding if there are smooth solutions. It would be interesting if there are whole class of PDEs which are difficult in this way. And if there isn't, then what is special about the NS? That too, is interesting.

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

#44

Post by adrianratnapala »

The article is probably underselling general rel here. The Einstein equation is famous for packing a whole lot of 4-dimensional, nonlinear tensor nastiness into an elegant little package.

You need to add something like an equation of state to turn it into a complete dynamical system, but I am suspect that there is enough nastiness in the gravity part alone that it will be difficult to solve in many regimes. What saves us is that we are usually concerned with low-temperature, low density systems, or else highly symmetric like stars and black holes.

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

#45

Post by adrianratnapala »

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…

> computational, atom-by-atom model

Remember that as long as computers continue to be made atoms, they aren't going to do atom-by-atom simulations, unless these simulated systems are much smaller than the computer itself or other are simplifying assumptions that can be made.

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

#46

Post by twiceaday »

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…

Your equation is only correct if you assume that the left hand side is computed using the analytic continuation of the Riemann zeta function. This assumption completely changes how the equation is interpreted and therefore verified. But you don't state this assumption so how are you surprised that people are taking the equation at face value and telling you it's wrong?

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

#47

Post by carterehsmith »

twiceaday wrote: post #42
carterehsmith wrote: post #37

Earlier quoted context omitted.

You say "your" and yet, this has nothing to do with me. I pointed to the wikipedia article. BTW this result is not exactly new, and it has specific confirmation in physics. You cannot argue that with high-school math. There is more to it.

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 definition that may be connected to the sum. There exist better definitions of the infinite sum – numerous definitions that turn out to be more natural in physics applications – and they generally produce the result −1/12−1/12. It is no trick or sleight-of-hand. The value −1/12−1/12 is really the right one and the rightness may be experimentally verified (using the Casimir effect). "

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

#48

Post by carterehsmith »

twiceaday wrote: post #46
carterehsmith wrote: post #39

Earlier quoted context omitted.

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…

Your equation is only correct if you assume that the left hand side is computed using the analytic continuation of the Riemann zeta function. This assumption completely changes how the equation is interpreted and therefore verified. But you don't state this assumption so how are you surprised that people are taking the equation at face value and telling you it's wrong?

Just to clarify, the above is not "my" equation. It is something that mathematicians (not me) figured out, repeatedly, many times over. Of course, the -1/12 number is preposterous, but it does turn out to have a physical meaning.

There are many ways of making the sum, including limit of partial-sums/summation by parts (the one we use most of the time), but there is also Abel summation, Borel summation, Ramanujan, Cesaro, and more. And frankly there is no reason to think that "summation by parts" is the "right" way. It is surely not "right" in physics. Example: why do we start summation from 1? Is the first element somehow more important than others? No, that is just our (ie human) arbitrary pick.

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

#50

Post by clicea »

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

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