may you please elaborate on why it is important, why hasn't been solved before and what new applications may you imagine with it, please?
dawnofdusk · 10h ago
The short answers:
1. It answers how macroscopic equations of e.g., fluid dynamics are compatible with Newton's law, when they single out an arrow of time while Newton's laws do not.
2. It was solved in the 1800s if you made an unjustified technical assumption called molecular chaos (https://en.wikipedia.org/wiki/Molecular_chaos). This work is about whether you can rigorously prove that molecular chaos actually does happen.
3. There are no applications outside of potentially other pure math research. For a physics/engineering perspective the whole theory was fine by assuming molecular chaos.
pizza · 5h ago
> 3. There are no applications outside of potentially other pure math research.
I would feel remiss not to say: such statements rarely hold
franktankbank · 1h ago
That's high praise!
killjoywashere · 11h ago
David Hilbert was one of the greatest mathematicians of all time. Many of the leaders of the Manhattan Project learned the mathematics of physics from him. But he was famous long before then. In 1900 he gave an invited lecture where he listed several outstanding problems in mathematics the solution of any one of which would change not only the career of the person who solved the problem, but possibly life on Earth. Many have stood like mountains in the distance, rising above the clouds, for generations. The sixth problem was an axiomatic derivation of the laws of physics. While the standard model of physics describes the quantum realm and gravity, in theory, the messy soup one step up, fluid dynamics, is far from a solved problem. High resolution simulations of fluid dynamics consume vast amounts of supercomputer time and are critical for problems ranging from turbulence, to weather, nuclear explosions, and the origins of the universe.
This team seems a bit like Shelby and Miles trying to build a Ford that would win the 24 hours of LeMans. The race isn’t over, but Ken Miles has beat his own lap record in the same race, twice. Might want to tune in for the rest.
bawolff · 8h ago
This kind of misses the point. The problem isn't interesting because its on hilbert's list; its on hilbert's list because it is interesting.
This is not my field, but i also don't think this would help with computational resources needed for high resolution modelling as you are implying. At least not by itself.
That video s very light and doesn't explains at what point (or intuitively) where the arrow of time comes in.
LudwigNagasena · 10h ago
So where and how does a jump from nice symmetric reversible equations to turbulent irreversibility happen?
bonvoyage36 · 7m ago
Strictly speaking, naturally on its own, it doesn't. Detailed equations remain reversible. Even for very big N, typical isolated classical mechanical systems are reversible. However, typical initial conditions imply transitions to equilibrium, or very long stay in it. The reversed process (ending in Poincare return) will happen eventually, but the time is so incredibly long, it can't be verified.
bubblyworld · 2h ago
I've been puzzling about this as well. The best answer I have (as an interested maths geek, not a physicist, caveat lector) is that it sneaks in under the assumption of "molecular chaos", i.e. that interactions of particles are statistically independent of any of their prior interactions. That basically defines an arrow of time right from the get-go, since "prior" is just a choice of direction. It also means that the underlying dynamics is not strictly speaking Newtonian any more (statistically, anyway).
MathMonkeyMan · 7h ago
Even three bodies under newtonian gravity can lead to chaotic behavior.
The neat part (assuming that the result is valid) is that precisely the equations of fluid dynamics result from their billiard ball models in the limit of many balls and frequent collisions.
LudwigNagasena · 7h ago
But even millions of bodies under Newtonian gravity lead to reversible behaviour unlike Navier-Stokes.
kgwgk · 1h ago
You lose the reversible behavior when you describe the system ignoring almost every degree of freedom.
MathMonkeyMan · 5h ago
The Navier-Stokes equations are a set of differential equations. The functions that the equations act upon are functions of time (and space), so the system is perfectly reversible.
It's just hard to figure out what the functions are for a set of boundary conditions.
bubblyworld · 3h ago
This is not quite right. Time-reversibility means that solutions to your differential equation are invariant under the transformation x(t) -> x(-t). It's pretty easy to verify that is the case for simple differential equations like Newton's law:
F = mx''(t) = mx''(-t)
since d/dt x(-t) = -x'(-t), and d/dt (-x'(-t)) = x''(-t)
Navier-Stokes is only time-reversible if you ignore viscosity, because viscosity is velocity-dependent and you can already see signs of that being a problem in the derivation above (velocity pops out a minus sign under time reversal). From my reading the OP managed to derive viscous flow too, so there really is a break in time-symmetry happening somewhere.
MathMonkeyMan · 2h ago
Now I get it, thanks for the explanation.
I wonder if "t -> -t" is lost in the Boltzmann step or in the hydrodynamic step.
doop · 2h ago
It's lost at Boltzmann's "molecular chaos" or "Stosszahlansatz" step. If f(x1,x2) is the two-particle distribution function giving you (hand-wavingly) the probability that you have particles with position and velocity coordinates x1 and others with coordinates x2, then Boltzmann made the simplification that f(x1,x2) = f(x1) * f(x2), ie throwing away all the correlations between particles. This is where the time-asymmetry comes in: you're saying that after two particles collide, they retain no correlation or memory of what they were doing beforehand.
mannykannot · 53m ago
I assume (on the basis that it has not come up so far in this discussion and my limited further reading) that position-momentum uncertainty offers no justification for throwing away the correlations?
bubblyworld · 16m ago
The systems we're talking about here are classical, not quantum, so the uncertainty principle isn't really relevant. I think the justification is mainly that it makes the analysis tractable. In physical terms it's simply not true that the interactions are uncorrelated, but you might hope that the correlations are "unimportant" in the long-term. In a really hot gas, for instance, everything is moving so fast in random directions that any correlations that start to arise will quickly get obliterated by chance.
kgwgk · 1h ago
The former. Diffusion in gases is similar.
quantadev · 10h ago
[flagged]
tomhow · 8h ago
Please don't do this here. If a comment seems unfit for HN, please flag it and email us at hn@ycombinator.com so we can have a look.
(Less jokingly, nothing strikes me as particularly AI about the comment, not to mention its author addressed the question perfectly adequately. Your comment comes off as a spurious dismissal.)
bawolff · 10h ago
To me, it looks like AI because it doesn't really answer the question but instead answers something adjacent, which is common in AI responses.
Giving a short summary of Hilbert's biography & his problem list, does not explain why this particular work is interesting, except in the most superficial sense that its a famous problem.
Twisol · 9h ago
Your second paragraph is a much more thoughtful critique, and posting that below the original answer would focus the subsequent conversation on those points. The issue here isn't whether the comment was AI-generated; it's how we carry the conversation forward even if we suspect that it is.
(For the record, if I had attempted to answer the earlier question, I probably would have laid out a similar narrative. The asker's questions were of a kind asking for the greater context, and the fact that Hilbert (mentioned in the submission title) posed the question is pretty important grounding. But, that's beside the point.)
bawolff · 8h ago
To be clear, im not the person who made the original ai accusation. I agree that just yelling its AI, and running away is super rude and not very constructive.
Twisol · 4h ago
I know it wasn't you :) Sorry if I came across that way.
quantadev · 9h ago
I think the last sentence, about Shelby and Miles, was written by a human, because it doesn't fit with the rest at all. Different style and a complete awkward shift of gears non sequitur. He probably recently saw the Amazon movie Ford V Ferrari, and so he threw that in to feel like he was doing more than cut-n-paste from an AI.
https://mathstodon.xyz/@johncarlosbaez/114618637031193532
He references a posted comment by Shan Gao[^1] and writes that the problem still seems open, even if this is some good work.
[^1]: https://arxiv.org/abs/2504.06297
Maybe it’s also the topics she covers. I’m not sure why she is getting into fantasies of AGI for example.
I liked the skeptical version of her better.
https://arxiv.org/abs/2408.07818
1. It answers how macroscopic equations of e.g., fluid dynamics are compatible with Newton's law, when they single out an arrow of time while Newton's laws do not.
2. It was solved in the 1800s if you made an unjustified technical assumption called molecular chaos (https://en.wikipedia.org/wiki/Molecular_chaos). This work is about whether you can rigorously prove that molecular chaos actually does happen.
3. There are no applications outside of potentially other pure math research. For a physics/engineering perspective the whole theory was fine by assuming molecular chaos.
I would feel remiss not to say: such statements rarely hold
This team seems a bit like Shelby and Miles trying to build a Ford that would win the 24 hours of LeMans. The race isn’t over, but Ken Miles has beat his own lap record in the same race, twice. Might want to tune in for the rest.
This is not my field, but i also don't think this would help with computational resources needed for high resolution modelling as you are implying. At least not by itself.
The neat part (assuming that the result is valid) is that precisely the equations of fluid dynamics result from their billiard ball models in the limit of many balls and frequent collisions.
It's just hard to figure out what the functions are for a set of boundary conditions.
F = mx''(t) = mx''(-t) since d/dt x(-t) = -x'(-t), and d/dt (-x'(-t)) = x''(-t)
Navier-Stokes is only time-reversible if you ignore viscosity, because viscosity is velocity-dependent and you can already see signs of that being a problem in the derivation above (velocity pops out a minus sign under time reversal). From my reading the OP managed to derive viscous flow too, so there really is a break in time-symmetry happening somewhere.
I wonder if "t -> -t" is lost in the Boltzmann step or in the hydrodynamic step.
We detached this comment from https://news.ycombinator.com/item?id= 44439647 and marked it off topic.
(Less jokingly, nothing strikes me as particularly AI about the comment, not to mention its author addressed the question perfectly adequately. Your comment comes off as a spurious dismissal.)
Giving a short summary of Hilbert's biography & his problem list, does not explain why this particular work is interesting, except in the most superficial sense that its a famous problem.
(For the record, if I had attempted to answer the earlier question, I probably would have laid out a similar narrative. The asker's questions were of a kind asking for the greater context, and the fact that Hilbert (mentioned in the submission title) posed the question is pretty important grounding. But, that's beside the point.)