In France, they refer to the H-theorem of Boltzmann (Théorème H) as 'eta'-theorem (théorème 'eta'). The connection obviously comes from the uppercase version of the Greek letter $\eta$, which looks like a capital H. Why would they associate H to $\eta$ (?) and not to $H$ as Hamiltonian?

Update: I have added my answer. If anybody has a follow up on the Brush article or another article about this curiosity feel free to comment or answer.

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    $\begingroup$ It seems due to a misreading H-Theorem. $\endgroup$ Commented Dec 5, 2018 at 16:16
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    $\begingroup$ See also the 1967 article aapt.scitation.org/doi/10.1119/1.1974281 . It seems Bolzmann called it the E-theorem and then later the H-theorem, and this has caused confusion about the "true" name since at least 1937. $\endgroup$ Commented Dec 6, 2018 at 0:13
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    $\begingroup$ It's 𝜂 and not h, because the word is entropy and not hentropy. $\endgroup$
    – Peter Shor
    Commented Dec 26, 2019 at 1:20

1 Answer 1


@kimchi lover, gave the answer. The article:

Brush, Stephen G. "Boltzmann's “Eta Theorem”: Where's the Evidence?." American Journal of Physics 35.9 (1967): 892-892.

talks about the problem of the naming of the theorem. So initially Boltzmann used $E$ for entropy, but according to Sidney Chapman, S.H. Burbury changed the variable to $H$ (about 1890) and it stayed that way, even for Boltzmann. But up to 1963, nobody has come with a proof that the $H$ theorem is an capital Greek letter $\eta$.

I do not know now if in general the French physicists, or just some, call it 'eta' theorem, but I realize this seems just a generalized misnaming.

Update: I have found the following reference:

Hjalmars, Stig. "Evidence for Boltzmann’s H as a capital eta." American Journal of Physics 45.2 (1977): 214-215.

That hints that by the typographical styles of Greek and Latin letters in Boltzmann papers, and by the generalization of Gibbs from $\eta$ and $\psi$ to $H$ and $\Psi$ for negative entropy and free energy, the interpretation of "eta theorem" should be favored.


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