What Makes the Hardest Equations in Physics So Difficult?
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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?
#2> "The answer, I discovered, is turbulence. It’s something we’ve all experienced, whether flying through choppy air at 30,000 feet or watching a whirlpool gather in the bathtub drain. Yet familiarity hasn’t bred knowledge: Turbulence is one of the least understood parts of the physical world."
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...
Re: What Makes the Hardest Equations in Physics So Difficult?
#3Deterministic Chaos is a real bastard.
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
Re: What Makes the Hardest Equations in Physics So Difficult?
#4So, 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 laws pre-relativity? Actually the whole thing reminds me of the ultraviolet catastrophe that preceded quantum physics.
Re: What Makes the Hardest Equations in Physics So Difficult?
#5So, 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 universe with our discrete atoms. But that still leaves the math question, and the possibility that even approximating the singularity may produce interesting physics.
Re: What Makes the Hardest Equations in Physics So Difficult?
#6> "The answer, I discovered, is turbulence. It’s something we’ve all experienced, whether flying through choppy air at 30,000 feet or watching a whirlpool gather in the bathtub drain. Yet familiarity hasn’t bred knowledge: Turbulence is one of the least understood parts of the physical world." This immediately made me think of "roughness" and Benoit Mandelbrot. > “When you zoom in on a point, from a mathematical poin…
Thank you for linking the TED talk. I did not know that Mandelbrot had also applied his fractal analysis to financial data.
Re: What Makes the Hardest Equations in Physics So Difficult?
#7So, 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 original motivation for using the differential equation technique is no longer appropriate given that the underlying technology has shifted underneath the feet of the modellers.
Re: What Makes the Hardest Equations in Physics So Difficult?
#8So, 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…
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 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.
Re: What Makes the Hardest Equations in Physics So Difficult?
#9Earlier 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…
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.
Re: What Makes the Hardest Equations in Physics So Difficult?
#10So, 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…
jerf correctly points out that the mathematicians are investigating the behavior of the mathematical model, not the physical phenomenon which inspired it. I want to add a second point, which is that while it wouldn't be that shocking if Navier-Stokes had blow-up or other non-physical solutions, it's not at all obvious that it has such solutions. People have tried and failed for more than a century to construct them. We know, in fact, that Navier-Stokes is very close, in a technical sense, to not admitting such solutions, as there are very mild deformations of the Navier-Stokes model which do not admit such blowups. Likewise, there are mild deformations which do. The beauty of Navier-Stokes is that it's so perfectly balanced between the two cases.