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

arxiv.org

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

#81

Post by bubblyworld »

bonvoyage36 wrote: post #80
bubblyworld wrote: post #77

Earlier quoted context omitted.

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".

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

#82

Post by bonvoyage36 »

bubblyworld wrote: post #81
bonvoyage36 wrote: post #80

Earlier quoted context omitted.

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".

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

#83

Post by perching_aix »

ngriffiths wrote: post #11

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.

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

#84

Post by bubblyworld »

bonvoyage36 wrote: post #82
bubblyworld wrote: post #81

Earlier quoted context omitted.

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 =)

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

#85

Post by jerf »

pizza wrote: post #51
dawnofdusk wrote: post #20

Earlier quoted context omitted.

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 doe…

> 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".

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

#86

Post by schuyler2d »

baxtr wrote: post #69
IdealeZahlen wrote: post #4

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.

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

#87

Post by naasking »

baxtr wrote: post #69
IdealeZahlen wrote: post #4

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.

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

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

#88

Post by naasking »

MathMonkeyMan wrote: post #52
LudwigNagasena wrote: post #49

Earlier quoted context omitted.

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

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.

> 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.

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

#89

Post by rnhmjoj »

LudwigNagasena wrote: post #18

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

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

#90

Post by rnhmjoj »

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

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