John Carlos Baez on Nostr: Has Hilbert's 6th problem been solved? Some claim it has. But.... First, what's ...

Has Hilbert's 6th problem been solved? Some claim it has. But....

First, what's Hilbert's 6th problem? It's astoundingly audacious. He proposed that we reduce physics to a list of axioms and prove all the results in physics starting from those!

Certainly *that* is far from being done. Nobody has found axioms for the Standard Model of particle physics, and nobody even knows if it's consistent: there are signs that it's not!

But people now are talking about one part of Hilbert's 6th problem: "developing mathematically the limiting processes [...] which lead from the atomistic view to the laws of motion of continua".

The ideas is to start with some equations obeyed by atoms and derive the usual equations of fluid flow: the Navier-Stokes equations. You should prove these equations hold *approximately*, with the approximation getting better and better approximation in the limit of large distance scales.

In March this year, Deng, Hani and Ma came out with a 48-page paper where they claim to do this:

https://arxiv.org/abs/2503.01800

They assume atoms are little spheres that bounce elastically off each other - not realistic, but okay, it's still a very interesting problem. Then there are two big steps.

First, assume these spheres are moving randomly and derive 'Boltzmann's kinetic equations', which are equations for how the probability distribution of velocities changes. Second, show that in the limit of large distances, the *average* velocities of the spheres obey the Navier-Stokes equations.

For each step, Deng, Hani and Ma review decades of work that people have done, and then try to fill in the gaps. They claim to finish the job.

But....

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