I'm interested in kinetic theory of gases, notably its history.

I know that, after the pioneering work of Boltzmann who derived the Boltzmann equation, it is possible by taking its statistical moment to obtain a set of equations that are closely linked to the Navier-Stokes equation.

This system can be closed either by doing a Hilbert expansion, Chapman-Enskog expansion, Grad's 13 moments and so on ...

However, I've not encountered WHO obtained this unclosed system. Who did plug and write the set of equations obtained from taking zeroth, first and second moment of the distribution function to obtain a Navier-Stokes like system ?

Anybody got a source or informations about it ?

  • $\begingroup$ I don't know anything about this, but from papers and book I've seen over the years in library browsing maybe look at publications by Clifford Truesdell. Also papers published in volumes of journals Truesdell was an editor of, Archive for Rational Mechanics and Analysis and Archive for History of Exact Sciences -- for each journal quickly glance over the table of contents of each volume, beginning with volume 1. $\endgroup$ Sep 12, 2022 at 15:40
  • $\begingroup$ You may want to check the Wikipedia article on the BBGKY hierarchy which contains links to the original articles by Bogoliubov, Born, Green, Kirkwood and Yvon. $\endgroup$
    – Tom Heinzl
    Sep 13, 2022 at 6:08
  • $\begingroup$ @TomHeinzl Thanks. Sadly, the BBGKY hierarchy comes after the work of Hilbert (1912), Chapman and Enskog (1916-1917). I presume the link to the macroscopic equations were already clarified by them or before them, or am I wrong ? by 1940 I was thinking it was already fully grasped $\endgroup$
    – Atmos
    Sep 13, 2022 at 8:28
  • $\begingroup$ I'm not an expert, but this article contains a brief discussion of Hilbert's contribution (based on a lecture delivered 1911-12 and attended by Enskog) together with many useful references (requires access to ResearchGate). $\endgroup$
    – Tom Heinzl
    Sep 13, 2022 at 12:56
  • 1
    $\begingroup$ @TomHeinzl Thanks, i've read this (very interesting) article, but it led to German publications I can't understand. I've also read what I could of Hilbert's publication and didn't even see what other writers imply he wrote, so I think my non-fluentness in German is too much of a hurdle. I'm quite flabbergasted that nobody mentioned such a discovery explicitely... For me it was Hilbert, I dont recall Boltzmann finding a relation to macroscopic governing equations, but I don't find any sources that can confirm ... $\endgroup$
    – Atmos
    Sep 13, 2022 at 16:01


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