Why is "h" used for entropy?

Question

Why is the letter "h" (or "H") used to denote entropy in information theory, ergodic theory, and physics (and possibly other places)?

Edit: I'm looking for an explanation of the original use of "H". As Ilmari Karonen points out, Shannon got "H" from Boltzmann's H-theorem. So (assuming Boltzmann actually used "H"), the original use is at least as early as that.

Answer 1

Wikipedia claims, citing "Gleick 2011", that Shannon got the letter $H$ from Boltzmann's H-theorem. Indeed, Shannon writes in his 1948 paper on page 393, after defining $H = -K \sum_{i=1}^n p_i \log p_i$:

"The form of $H$ will be recognized as that of entropy as defined in certain formulations of statistical mechanics⁸ where $p_i$ is the probability of a system being in cell $i$ of its phase space. $H$ is then, for example, the $H$ in Boltzmann's famous $H$ theorem."

Of course, this just changes the question to "Why did Boltzmann choose the letter $H$, then?" In this letter to the editor, published in Nature in 1937, Sydney Chapman writes:

"WHEN Boltzmann first published the celebrated theorem now generally known as the $H$-theorem, he used the symbol $E$ (presumably as the first letter of entropy), not $H$. It has been suggested that when $H$ was first used for this theorem it was intended to be the capital Greek letter eta: but the first paper known to me in which $H$ is used for Boltzmann's entropy function is one by Burbury¹, who seems to have changed Boltzmann's symbol $E$ to $H$ for no special reason; later Burbury used $B$ for an almost identical function, which he called Boltzmann's minimum function². Boltzmann himself wrote $E$ so late as 1893³, but in 1895⁴ he used the letter $H$. This use of $H$ must have seemed mysterious to many generations of students, and it would be interesting to know whether any reader can account for its use or give an earlier instance of it."

So apparently, you're far from the first person to wonder about this.

Indeed (thanks to t.b. for the links), 30 years later, in a letter to the Americal Journal of Physics, Stephen G. Brush repeated Chapman's plea, and added that "Professor Chapman informed me, a couple of years ago, that he never received any response to this letter." 10 years later yet, in the same journal, Stig Hjalmars wrote in response to Brush's letter:

"The given graphical evidence, of which a detailed account is presented elsewhere,⁶ seems to leave no reasonable doubt that that during the decade before Boltzmann's death in 1906 at least he himself, Gibbs and Zermelo meant a capital eta when they wrote $H$ for Boltzmann's function."

The cited "elsewhere" is "S. Hjalmars, TRITA-MEK-76-01, Technical Reports from the Royal Institute of Technology, Department of Mechanics, S-10044 Stockholm, Sweden," stated to be "Free of cost on request from the Department." Alas, I haven't so far managed to locate a copy of this report.