Thermodynamic free energy

The thermodynamic free energy is defined by $F=U-TS$ with $U,T,S$ being the internal energy, temperature and entropy respectively.

I have also seen another formula for the free energy, $F=-T \log{Z}$ where $Z=\int \cal{D} \phi e^{-I[\phi]}$ is the partition function.

First of all, I'm not sure the $Z$ in the log is the same as the one I've defined in the path integral. Can someone confirm this please? Secondly, how can I show these two definitions of free energy are equivalent?

• Both of your expressions are correct. The connection between them is through U = -d(lnZ)/dbeta, where beta=1/(kT). en.wikipedia.org/wiki/… I can make an answer from this but I'd like to write something about the specific form of the path integral, i.e. what the I[phi] is. Could you specify what kind of system you are looking at? Jan 25, 2016 at 20:00
• @Numrok Thanks. It is the action for a black hole spacetime. Jan 25, 2016 at 20:12
• I'll pass on writing an answer in that case, since I don't know enough about that kind of problem. Hope my comment helped though. Jan 25, 2016 at 20:55

The weight for a specific state is $w(\phi)=e^{-E/T+F/T}\equiv e^{-I(\phi)+F/T}$. Due to the definition of the entropy $S$, we have $S=-\int D\phi w\ln w=\frac{1}{T}(\int D\phi wE-F\int D\phi w)=\frac{1}{T}(U-F)$. Then we have $F=U-TS$.
• Why is $S=-\int D \phi w \ln{w}$? Jan 26, 2016 at 9:07