I am studying the canonical ensemble and it seems to me there is an analogy between derivatives of the partition function, which can extract energy momenta for the system and Fisher score /information.

In partciular we have expressions like $-\partial_\beta log(Z)=<H>$, $\frac{1}{Z}\partial^2_\beta Z=<H^2>$ and finally $\partial^2_\beta log(Z)= Var(H)$ where $Z$ is the partition function $\beta$ the inverse temperature.

Hence I think one could identify the fisher score of the ensemble with the log of ther partition fucntion. Anyway I cannot get further in this analogy nor can get physical meaning and interpretation of this porcess.

Any help would be great!

  • $\begingroup$ I won't comment on relations with Fisher information, but want to stress that there is nothing particularly remarkable about this property of $\log Z$. Indeed, by its very definition, it coincides (up to a trivial additive constant) with the standard cumulant-generating function of $H$. $\endgroup$ – Yvan Velenik May 26 at 16:31

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