What Makes the Hardest Equations in Physics So Difficult?
quantamagazine.org
quantamagazine.org
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 universe with our discrete atoms. But that still leaves the math question, and the possibility that even approximating the singularity may produce interesting physics.
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.
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 bytes = 1.2 petabytes of information. Add in more fields for velocity and other pieces of state and you're talking about something that would only fit in working memory in some of our largest machines.
It's a very reasonable idea, and useful enough for computations, but it lacks the essential tension that makes Navier-Stokes a mathematical Everest.
The better foundation is quantum mechanics. An example of a macroscopically visible difference between these foundations are the van der Waals forces.
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.
PS: Also of note protean folding simulations don't even simulate the water surrounding the protean and are again very simplified.
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.
People do atom-by-atom molecular dynamics simulations of proteins and such.
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.
We still don't have the computational power to do certain classes of continuous Navier--Stokes calculations (i.e. not atom based).
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.
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.
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.
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.
Understanding that a model or theory is useful even when we ignore certain aspects of reality is quite different to the often displayed belief that a specific theory is "gospel" even in the face of anomalies and discrepancies of the real world compared with prediction. Too much of the "theoretical physics" genre (word specifically chosen) is based on the idea that mathematics is the means of finding the "truth".
As I said above, mathematics is a wonderful and useful tool, but it is not a good master. It provides a possible insight into what is going on. However, those insights are not "truth" as such. I have been doing a review of my old mathematics texts for scientists and engineers, as well as other resources. It is interesting that all of them talk of and demonstrate that all the mathematical models are simplified and incomplete. Yet, if one raises the various problems with the various models in use today, one is shouted down. This does not bode well for our advancement in understanding of the universe around us.
In my understanding the big shift was the understanding of renormalisation, Kadanoff and Wilson, around 1970. This took airy ideas about useful approximation and turned them into serious tools, which are both useful for everyday things and illuminating about why any of it works.
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.
The “Any” is the value here for the analytical solution.
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.
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.
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 < Kn < 10, we do not have a good enough set of models that is able to compute accurate answers to reality with reasonable time frames and is an active field of research. If you're interested in learning more about this topic, look up the Kinetic Theory of Gas, where particles are accounted with the evolution probability distributions in temporal, spatial, and velocity space.
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+3+4... = -1/12
https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B...
And that result actually pops up in physics.
That kind of indicates continuity/infinite divisibility.
You cannot argue that with high-school math. There is more to it.
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). "
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.
So... I can trust Ramanujan and Abel, that published results on these things, and Terrence Tao that has a nice writeup, and a bunch of others, or I can trust HN user "Kazinator", who... published some middle-school algebra "proof" on HN.
Guess who is crackpot here.
I can see how overlapping notation can be confusing for laymen (me included).
But that does not mean that people that came up with those are crackpots, just because they reused the "=" symbol.
Unfortunately, the syntax chosen for representing the sequence is that of an additive series, where a binary + operator is interposed between terms.
The semantics being shown does not seem to follow from a redefinition of that operator per se as stand-alone binary operator.
You really want to show this as, say
fun([1 2 3 4 ... ])
n fun([x0 x1 ... ]) = fun([nx0 nx1 ...])
fun(v0 + v1) = fun(v0) + fun(v1)
Basically, none of this means that the natural numbers add to -1/12; only that the sequence of natural numbers can be fed into some decimating calculation which ends up with -1/12.
Well, no kidding; the sequence of natural numbers can be fed into a decimating calculation which converges on any value you want, if you can freely choose the decimating calculation, and that calculation can be chosen to be linear operator.
Note that we don't have:
fun([x0 x1 x3 ... x42 ...]) = fun([x3 x42 x0 ... x1 ...])
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.
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 that's kind of telling us that the whole construction is not just some math sleight-of-hand, it actually has some meaning in the real world.
Example: Casimir effect
So my point is... that kind of suggests the smoothness, or continuity, or differentiability, or whatever we want to call it, of the underlying function. The opposite of discrete. Is what my point was.
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.
Essentially, it consists of text pulled from exactly the same source that you originally cited.
(Cue sound of hands washing in sink.)
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".
Sure it looks like that, but, amazingly, there is a physical meaning to that number.
I replied here: https://news.ycombinator.com/edit?id=16164915
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.
Terence Tao has a nice lecture published... trying to find it, brb.
In any case, the gist of it was: long as we treat the numbers in the sum as integers (ie discrete), we will get ill-defined sums, contradictions, infinities, and so on.
Once we switch to real numbers (and there is an underlying function that is differentiable), things "can" be made to work and converge. To -1/12 always.
So that's it. In particular, if we decided that our smallest unit of measure is... the size of atom, or whatever finite value, many of these techniques will just fail to work, and they won't match what we measured. Which means something is wrong. Maybe things really are continuous. Or maybe they are not, it is just that our math is not sophisticated enough.
Obviously this is an open question.
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.
This immediately made me think of "roughness" and Benoit Mandelbrot.
> “When you zoom in on a point, from a mathematical point of view you lose information about the solution,” said Vicol. “But turbulence is meant to describe exactly this — the transfer of kinetic energy from large to smaller and smaller scales, so it’s exactly asking you to zoom in.”
You can explore the Mandelbrot set in similar fashions. There are many tools online to do so. The more you zoom in, the more you get completely lost from what the original image looks like.
If this kind of stuff interests you, his TED talk is among my favorites and can be found here:
https://www.ted.com/talks/benoit_mandelbrot_fractals_the_art...
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.
There was a disturbing result a while back about the Navier-Stokes equations in which the existence of non-unique solutions was shown: https://arxiv.org/abs/1709.10033
https://www.theatlantic.com/photo/2018/01/gorgeous-images-of...
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.