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What Makes the Hardest Equations in Physics So Difficult?
- randomdrake 9y ago> "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_of_roughness https://www.ted.com/talks/benoit_mandelbrot_fractals_the_art...
- catbird 9y agoThank you for linking the TED talk. I did not know that Mandelbrot had also applied his fractal analysis to financial data.
- hprotagonist 9y agoDeterministic 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 https://arxiv.org/abs/1709.10033
- andrewflnr 9y agoSo, 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.
- jerf 9y agoTone: 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.
- whatshisface 9y agoOne 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 unexpectedly sensitive measurements of particle-scale physics.
- pixelpoet 9y agoA nice example of such a practical tool for microscopic analysis is X-ray crystallography, once you have the theory of Fourier transforms and lattices.
- Someone 9y agoYour 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.
- andrewflnr 9y agoI 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.
- irickt 9y agoOT: some beautiful images of turbulence on Jupiter ... https://www.theatlantic.com/photo/2018/01/gorgeous-images-of-the-planet-jupiter/550595/ https://www.theatlantic.com/photo/2018/01/gorgeous-images-of...
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- adrianratnapala 9y agoThe 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.
- danharaj 9y agoWhat makes a problem interesting to mathematicians is often not the result itself but the expectation that the proof will require techniques and philosophical ideas that change the way the general theory is handled.
- fpoling 9y agoWhen 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.