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What Makes the Hardest Equations in Physics So Difficult?

quantamagazine.org

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

#21

Post by srirachahot »

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…

You are missing the distinction between a numerical solution to a set of equations and an analytical one. Usually, the analytical solution gives you the dynamics of a system at any point in time and space given a set of parameters.

For numerical solutions you have to run each individual set of parameters to find the corresponding values in time and space (not even considering stochastic equations). This is very computationally expensive.

This is the value in solving these things analytical — hence the prize.

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

#22

Post by srirachahot »

srirachahot wrote: post #21
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…

You are missing the distinction between a numerical solution to a set of equations and an analytical one. Usually, the analytical solution gives you the dynamics of a system at any point in time and space given a set of parameters. For numerical solutions you have to run each individual set of parameters to find the corresponding values in time and space (not even considering stochastic equations). This is very compu…

Sorry this should read: “Usually, the analytical solution gives you the dynamics of a system at any point in time and space given ANY set of parameters.”

The “Any” is the value here for the analytical solution.

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

#23

Post by contact_fusion »

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…

Your assumption is frankly incorrect.

Physics is generally expressed in terms of differential equations. This is not due to their analytical tractability - as anyone who has attempted to solve PDEs before will know, most (nearly all) differential equations do not yield to analytical solution. Perhaps you think that quantum mechanics demands a discretized view of reality. This would be a complete misunderstanding of quantum mechanics, and physics in general.

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

#24

Post by Retric »

openasocket wrote: post #17
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

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…

The other side of this is cells are also really fast. Modeling even 0.01 seconds could be very useful.

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

#25

Post by pwnna »

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…

As an aside, as the question of the article is more about the mathematics as opposed to the physics: Navier-Stokes assuming a continuum is one of the fundamental assumptions made in basic fluid dynamics, and to an extend applies to the governing equations for solid mechanics, in a whole field known as Continuum Mechanics.

The continuum assumption, as it turns out, is actually incredibly accurate for most gasses that are comprised of discrete molecules. Specifically, the region in which N-S is valid for is when Knudsen Number[1] is less than about 0.01. Knudsen Number essentially characterizes how densely packed the particles are.

This turns out to be most of the flows we observe on Earth. N-S becomes less accurate at around Kn = 0.1 and completely useless at Kn >= 1, as in this region the differential arguments no longer hold and NS predicts something non-physical. Some examples of flows in this regime: mass spectrometer, reentry, and inside shockwaves. In that view, you're not wrong that NS will eventually fail, but it is just applicable to most of what we do. For those working with fluids, it is important to recognize for when NS fails so they can switch to a different model for solutions.

For higher Kn flows, such as Kn > 10, you can feasibly track individual particles and their trajectories. People do this for satellites and what not. However, between the range of 0.1 [1]: https://en.wikipedia.org/wiki/Knudsen_number

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

#27

Post by ctdonath »

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…

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 simulate wind shear on a plane when there's more atoms in that wind & plane than there are in the computer simulating them: every single atom being simulated contains the complete information about its state (information independent of any other atom), so simulating it will require at least one atom per atom simulated - if you don't have at least as many atoms to simulate as you are simulating, you can't achieve a complete simulation.

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

#28

Post by evandijk70 »

ctdonath wrote: post #27
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…

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.

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

#29

Post by SAI_Peregrinus »

Retric wrote: post #24
openasocket wrote: post #17

Earlier quoted context omitted.

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…

The other side of this is cells are also really fast. Modeling even 0.01 seconds could be very useful.

Much of the action inside cells is done via proteins. It's taken a huge amount of distributed computing power just to figure out a reasonable hypothesis for how they fold (only one part of the movement) and €1.22 billion to build the XFEL and have a way to image them on the required timescales: 1-2 femtoseconds. 0.000000000000001 seconds. 0.01 seconds is an eternity, calculating that long would take an utterly enormous amount of computer time.

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

#30

Post by carstimon »

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…

Let me rebuke just one part of the post: the example with Newton's laws. Of course everyone knows that Navier-Stokes is just a very nice approximation. That doesn't mean that mathematical results about Navier-Stokes can't improve or understanding of the physics side. For instance, if we find a solution which blows up, that may lead us to a correction to the model or (less likely) it may be actually physically realizable.
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