Hilbert's sixth problem: derivation of fluid equations via Boltzmann's theory | Hacker News Reader

Hilbert's sixth problem: derivation of fluid equations via Boltzmann's theory

arxiv.org

154 pointsnsoonhui99 comments

John Baez wrote a Mastodon thread on this paper here:

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

Shan Gao's review on this is really nice and accessible, thanks.

Can someone explain what's groundbreaking about this? Maybe it's not done so very rigorously, but pretty much every plasma physics textbook will contain a derivation of Boltzmann equation, including some form of collisional operator, starting from Liouville's theorem[1] and then derive a system of fluid equations [2] by computing the moments of Boltzmann equation.

[1]: https://en.wikipedia.org/wiki/Liouville%27s_theorem_(Hamilto...

[2]: https://en.wikipedia.org/wiki/BBGKY_hierarchy

Not only plasma

This is the larger part of the work:

https://arxiv.org/abs/2408.07818

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?

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.

> 3. There are no applications outside of potentially other pure math research.

I would feel remiss not to say: such statements rarely hold

In this case, what the research says is that the approximations we have already been using for a long time are correct. "You're already right, keep doing what you're doing!" is not generally something people consider a "practical application".

That's high praise!

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.

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.

> 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.

Whether or not this is AI, this comment is not true. An axiomatic derivation of a formula doesn’t change how it’s used. We knew the formulas were experimentally correct, it’s just that now mathematicians can rest easy about whether they were theoretically correct. Although it’s interesting, it doesn’t change or create any new applications.

Explained for the layperson in the video cited here: https://news.ycombinator.com/item?id=44439593

That video s very light and doesn't explains at what point (or intuitively) where the arrow of time comes in.

So where and how does a jump from nice symmetric reversible equations to turbulent irreversibility happen?

This has been known for a long time: the irreversibility comes from the assumption that the velocities of particles colliding are uncorrelated, or equivalently, that particles loose the "memory" of their complete trajectory between one collision and another. It's called the molecular chaos hypothesis.

See https://en.wikipedia.org/wiki/Molecular_chaos

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).

It comes about when the deterministic collision process is integrated over all the indistinguishable initial states that could lead into an equivalence class of indistinguishable final states. If you set the collision probability to zero it's time reversible even with molecular chaos, and if the particles are highly correlated (like in a polymer) there can still arise an arrow of time when the integral is performed.

Interesting, so if I understand right you are saying that coarse-graining your states can produce an arrow of time on its own? Given some fixed coarse-graining, I can see that entropy would initially increase, since your coarse-graining hides information from you. The longer you evolve the system under this coarse graining the less certain you will be about the micro-states.

But I would expect this to eventually reach an equilibrium where you are at "maximum uncertainty" with respect to your coarse graining. Does that sound right at all? And if so, then there must be something else responsible for the global arrow of time, right?

> If you set the collision probability to zero it's time reversible even with molecular chaos

Is this true for boring reasons? If nothing interacts then you just have a bunch of independent particles in free motion, which is obviously time-reversible. And also obviously satisfies molecular chaos because there are no correlations whatsoever. Maybe I misunderstand the terminology.

Yes, almost any coarse graining can lead to an arrow of time unless the map that represents a step forward in time aligns with the coarse graining perfectly. Also, yes, there are often equilibria, but that's the heat death of the universe, and admittedly time is hard to define even macroscopically if everything is at the same temperature.

Chaos isn't even necessary, it just gets you there faster.

The collisionless case is that way for boring reasons: the map aligns with the coarse graining.

Thanks for explaining. I went and wrote a bunch of simulations of billiard ball dynamics with coarse-graining schemes applied and watched this stuff happening in front of my eyes. Pretty cool to see how entropy, energy and time are all directly related in these toy systems - the idea of a clock as a meter for entropy is making a lot more sense to me now =)

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.

But even millions of bodies under Newtonian gravity lead to reversible behaviour unlike Navier-Stokes.

You lose the reversible behavior when you describe the system ignoring almost every degree of freedom.

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.

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.

Now I get it, thanks for the explanation.

I wonder if "t -> -t" is lost in the Boltzmann step or in the hydrodynamic step.

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.

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?

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.

I don't think it really helps - you're already working in something like a probabilistic formulation. If you want to use a quantum mechanical justification for it then you need to look at some sort of non-unitary evolution.

Besides that, I don't think anybody is really arguing that the correlations are actually lost after a collision, just that it's usually a good approximation to treat them as if they are.

The former. Diffusion in gases is similar.

equations can be time-symmetric, or invariant re time reversal. What you're describing is equations being invariant re time reversal.

You can call this invariance under time reflection if you like, yeah.

Note that the solutions x(t) are not generally time symmetric. We aren't saying that x(t)=x(-t), we are saying that x(t) is a solution to the differential equation if and only if x(-t) is, which is a weaker statement.

I know what you meant; I've just tried to point out an error in your sentence which pops up sometimes, which may have mislead others. It's all about the time reversal invariance of evolution equations, not solutions.

Oh I see what you mean, it's kinda easy to read my comment as meaning time symmetry. But I do think the phrasing in terms of solutions is correct, provided you interpret it appropriately. As in "is still a solution to the diff eq after transformation" and not "is left unchanged by the transformation".

It's not a good phrasing to express the point, because "solution is invariant under operation O" has an established meaning, that the solution does no change after the operation. What you mean can be properly phrased as "equations are time-reversal invariant".

You've convinced me =)

> 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 hard to take full reversibility seriously given Newton's equations are not actually deterministic. If they're not deterministic, then they can't be fully reversible.

Of course maybe these non-deterministic regimes don't actually happen in realistic scenarios (like Norton's Dome), but maybe this is hinting at the fact that we need a better formalism for talking about these questions, and maybe that formalism will not be reversible in a specific, important way.

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.

In derivations of the Navier Stokes equations from reversible particle models, the former get their irreversibility from some approximation, e.g. a transition to a less detailed state and a simpler evolution equation for it is made. Often the actual microstate is replaced by some probabilistic description, such as probability density, or some kind of implied average.

Sabine Hossenfelder's video on this: https://youtu.be/mxWJJl44UEQ

In my perception Sabine’s quality degraded over the last year or so.

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.

Agree in general -- I think the tiktok/shorts wave is biasing strongly for shorter video and then the time format kills any followup/2nd iteration-explanation

But this one was pretty good.

As far as I've seen, her position is only that AGI is pretty much inevitable. What's so fantastical about that?

I think plenty of people don't think it's inevitable. I'm no ai researcher, just another software engineer (so no real expertise). I think it will keep getting better but the end point is unclear.

The reason it's inevitable is because because it follows from physics principles. The Bekenstein Bound proves that all physical systems of finite volume contain finite information, humans are a finite volume, ergo a human contains finite information. Finite information can be fully captured by a finite computer, ergo computers can in principle perfectly simulate a human person.

This + continued technological development entails that AGI is inevitable.

Although the reasoning is clear, you (and her) jump from "possible in principle" to "inevitable in practice".

Just because something is physically possible doesn't make it "inevitable". That's why it's just a fantasy at this point.

As I said above:

> This + continued technological development entails that AGI is inevitable.

Everyone takes the above as a given in any discussion of future projections.

I don't know if it's just the persona she plays in these videos, but it's so so so creepy and cringe.

Agreed, she's pumping out too many videos I think. Perhaps she's succumbed a bit to the temptation of cashing in on a reputation, ironically one built on taking down grifters.

I found https://www.quantamagazine.org/epic-effort-to-ground-physics... much more informative. Sometimes you can't digest everything in 10min.

https://news.ycombinator.com/item?id=44442123

[flagged]

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.

We detached this comment from https://news.ycombinator.com/item?id=44439647 and marked it off topic.

Not enough em-dashes for it to be AI.

(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.)

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.

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.)

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.

I know it wasn't you :) Sorry if I came across that way.

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.

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