# Why is the H theorem called the big Eta theorem?

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.

• It seems due to a misreading H-Theorem. – Mauro ALLEGRANZA Dec 5 '18 at 16:16
• 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. – kimchi lover Dec 6 '18 at 0:13
• It's 𝜂 and not h, because the word is entropy and not hentropy. – Peter Shor Dec 26 '19 at 1:20

## 1 Answer

@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.