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I have the following partition function $$Z(\beta_1,\beta_2) = Z(\beta_1)Z(\beta_2) +Z(\beta_1,\beta_2)_c$$

Thus, I have a non-zero mutual information $I(Z(\beta_1);Z(\beta_2)) = S(Z(\beta_1))+S(Z(\beta_2)) - S(Z(\beta_1,\beta_2))$ measuring correlations due to the connected term.

I would like to rewrite the mutual information in terms of the density of state (since there are correlations between energy levels), using maybe $Z(\beta) = \int d\beta e^{-\beta E}\rho(E)$. Is there a relation between $I(Z(\beta_1);Z(\beta_2))$ and $I(\rho(E_1);\rho(E_2))$?

Should I use the saddle point approximation so that $Z(\beta)$ is dominated by a certain energy $\rho(E*)$?

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