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

Re:Hilbert's sixth problem: derivation of fluid equations vi... by bubblyworld on Wednesday July 02, 2025 @06:22AM (#44441975)

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).

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Re:Hilbert's sixth problem: derivation of fluid equations vi... by whatshisface on Wednesday July 02, 2025 @02:55PM (#44447468)

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.

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Re:Hilbert's sixth problem: derivation of fluid equations vi... by bubblyworld on Wednesday July 02, 2025 @06:04PM (#44449335)

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.

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Re:Hilbert's sixth problem: derivation of fluid equations vi... by whatshisface on Thursday July 03, 2025 @03:43PM (#44458521)

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.

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Re:Hilbert's sixth problem: derivation of fluid equations vi... by bubblyworld on Monday July 07, 2025 @05:20AM (#44488334)

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 =)

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