# Fisher Information in Statistical Mechanics

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 $$-\partial_\beta log(Z)=$$, $$\frac{1}{Z}\partial^2_\beta Z=$$ 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!

• 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 $H$. – Yvan Velenik May 26 at 16:31

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