[HN Gopher] Hilbert's sixth problem: derivation of fluid equatio...
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Hilbert's sixth problem: derivation of fluid equations via
Boltzmann's theory
Author : nsoonhui
Score : 142 points
Date : 2025-07-02 00:31 UTC (22 hours ago)
(HTM) web link (arxiv.org)
(TXT) w3m dump (arxiv.org)
| whatshisface wrote:
| This is the larger part of the work:
|
| https://arxiv.org/abs/2408.07818
| itsthecourier wrote:
| 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?
| makk wrote:
| Explained for the layperson in the video cited here:
| https://news.ycombinator.com/item?id=44439593
| bravesoul2 wrote:
| That video s very light and doesn't explains at what point
| (or intuitively) where the arrow of time comes in.
| killjoywashere wrote:
| 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 wrote:
| 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.
| gnubison wrote:
| > 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.
| dawnofdusk wrote:
| 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 wrote:
| > 3. There are no applications outside of potentially other
| pure math research.
|
| I would feel remiss not to say: such statements rarely hold
| franktankbank wrote:
| That's high praise!
| jerf wrote:
| 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".
| IdealeZahlen wrote:
| Sabine Hossenfelder's video on this: https://youtu.be/mxWJJl44UEQ
| Xmd5a wrote:
| https://news.ycombinator.com/item?id=44442123
| Ygg2 wrote:
| I found https://www.quantamagazine.org/epic-effort-to-ground-
| physics... much more informative. Sometimes you can't digest
| everything in 10min.
| baxtr wrote:
| 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.
| jakeinspace wrote:
| 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.
| schuyler2d wrote:
| 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.
| naasking wrote:
| As far as I've seen, her position is only that AGI is pretty
| much inevitable. What's so fantastical about that?
| Alive-in-2025 wrote:
| 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.
| naasking wrote:
| 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.
| baxtr wrote:
| 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.
| _zoltan_ wrote:
| I don't know if it's just the persona she plays in these
| videos, but it's so so so creepy and cringe.
| quantadev wrote:
| [flagged]
| Twisol wrote:
| 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.)
| bawolff wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| I know it wasn't you :) Sorry if I came across that way.
| quantadev wrote:
| 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.
| tomhow wrote:
| 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.
| ngriffiths wrote:
| 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
| perching_aix wrote:
| Shan Gao's review on this is really nice and accessible,
| thanks.
| LudwigNagasena wrote:
| So where and how does a jump from nice symmetric reversible
| equations to turbulent irreversibility happen?
| MathMonkeyMan wrote:
| 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 wrote:
| But even millions of bodies under Newtonian gravity lead to
| reversible behaviour unlike Navier-Stokes.
| MathMonkeyMan wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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.
| doop wrote:
| 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.
| kgwgk wrote:
| The former. Diffusion in gases is similar.
| bonvoyage36 wrote:
| _equations_ can be time-symmetric, or _invariant_ re time
| reversal. What you 're describing is equations being
| invariant re time reversal.
| bubblyworld wrote:
| 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.
| bonvoyage36 wrote:
| 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.
| bubblyworld wrote:
| 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".
| bonvoyage36 wrote:
| 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".
| bubblyworld wrote:
| You've convinced me =)
| naasking wrote:
| > 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.
| kgwgk wrote:
| You lose the reversible behavior when you describe the
| system ignoring almost every degree of freedom.
| bubblyworld wrote:
| 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).
| whatshisface wrote:
| 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.
| bubblyworld wrote:
| 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.
| bonvoyage36 wrote:
| 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.
| bonvoyage36 wrote:
| 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.
| rnhmjoj wrote:
| 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
| rnhmjoj wrote:
| 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
| Iwan-Zotow wrote:
| Not only plasma
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