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I obtained the Green function at finite temperature for a given system using a simulation. This means I have a list of numbers that represent G(t). Now I would like to use this information to compute thermodynamic magnitudes like internal energy, entropy, specific heat, etc. But I don't really know how to relate the Finite temperature formalism with the partition function. Any directions will be appreciated!

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You can calculate the internal energy $U$ using the retarded single-particle Green's function

$$ U = \frac{1}{2\hbar} \sum_{\boldsymbol{k}\sigma} \int_{-\infty}^{+\infty}dE \,\,(E+\mu+e(\boldsymbol{k}))\frac{1}{e^{\beta E}+1}\left(-\frac{1}{\pi}\text{Im}G^R_{\boldsymbol{k}\sigma}(E)\right). $$

From there you can calculate the free energy $F$ by

$$ F(T,V) = U(0,V)-T\int^{T}_{0}dT' \frac{U(T',V) - U(0,V)}{T'^{2}}. $$

Using the free energy you can calculate all thermodynamic magnitudes.

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