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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!

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  • $\begingroup$ 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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There are very useful relations indeed, which I wouldn't necessarily call analogies.

We define a set of thermodynamic variables denoted as {${\theta_i}$} and specify the partition function denoted as $Z(\theta)$. In a typical case where we work with Gibbs measures, we may write

$$lnZ = \psi $$

where $\psi$ corresponds to free entropy. Amusingly, the second derivative of free entropy $$\frac{\partial^2\psi}{\partial\theta_{m}\partial\theta_{n}}$$ yields the thermodynamic tensor, which is identical to Fisher information matrix $F_{mn}$, where

$$F_{mn}(\theta) = \sum_x{p(x)\frac{\partial lnp(x)}{\partial \theta_{m}}}{\frac{\partial lnp(x)}{\partial \theta_{n}}}.$$

This shows how Fisher information is immediately related to our study of statistical mechanics as your question addresses. Identification of Fisher values informs us about how a system behaves in transitions, for example.

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