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

Here's a summary of the themes discussed on Hacker News regarding the arXiv paper:

The Significance of Hilbert's Sixth Problem

A central theme of the discussion revolves around the work's connection to David Hilbert's sixth problem, which calls for an axiomatic derivation of the laws of physics. Users highlight the historical and foundational importance of this problem in mathematics and physics.

• "David Hilbert was one of the greatest mathematicians of all time... 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." - killjoywashere
• "The sixth problem was an axiomatic derivation of the laws of physics." - killjoywashere
• "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." - bawolff
• "The problem isn't interesting because its on hilbert's list; its on hilbert's list because it is interesting." - bawolff

The Arrow of Time and Macroscopic Laws

A significant portion of the conversation focuses on how the research addresses the discrepancy between time-symmetric microscopic laws (like Newton's laws) and the time-asymmetric nature of macroscopic phenomena, particularly fluid dynamics. The concept of "molecular chaos," a key assumption in the derivation of the Boltzmann equation, is identified as the point where irreversibility is introduced.

• "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." - dawnofdusk
• "It was solved in the 1800s if you made an unjustified technical assumption called molecular chaos... This work is about whether you can rigorously prove that molecular chaos actually does happen." - dawnofdusk
• "So where and how does a jump from nice symmetric reversible equations to turbulent irreversibility happen?" - LudwigNagasena
• "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." - MathMonkeyMan
• "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... 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)." - bubblyworld
• "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." - doop
• "The best answer I have (as an interested mathsకి, 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)." - bubblyworld

Perceived Clarity and Explanations

There was a discussion about the accessibility of the research, with users sharing different resources and opinions on their effectiveness. Comparisons were drawn to sports analogies for conveying the paper's significance.

• "That video s very light and doesn't explains at what point (or intuitively) where the arrow of time comes in." - bravesoul2
• "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." - killjoywashere
• "I found https://www.quantamagazine.org/epic-effort-to-ground-physics... much more informative. Sometimes you can't digest everything in 10min." - Ygg2
• "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." - ngriffiths

Applicability and Future Impact

The potential applications of this research were explored, with some users suggesting that its impact might be limited to pure mathematics, while others countered that such statements often prove to be short-sighted.

• "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." - dawnofdusk
• "> 3. There are no applications outside of potentially other pure math research. I would feel remiss not to say: such statements rarely hold" - pizza

Discussion on AI-Generated Content and Moderation

A sub-theme emerged regarding the potential for AI-generated comments and how to handle them. This included discussions on identifying AI-like writing patterns and the appropriate moderation practices.

• "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." - bawolff
• "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." - bawolff
• "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." - quantadev
• "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." - tomhow