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In thermodynamics one says that in equilibrium the corresponding thermodynamic potential is minimized. Why?

For example take the case of a canonical ensemble. Based on the assumption that the macroscopic state is given by the state which has the most microscopical states (i.e. entropy is maximized) and that the energy of the system is fixed one finds that $\rho = e^{-\beta H}/Z$ for some $\beta$. Then the Helmholtz free energy is given by $F = -\frac{1}{\beta} \ln Z$.

Now why does the free energy $F$ become minimal (if I keep $T = const.$)?

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  • $\begingroup$ I know the Wikipedia article. Ok one could replace the question by "why does the principle of minimum energy holds". The principle of minimum energy follows from the principle of maximum entropy. But the maximal entropy is already used to derive the canonical ensemble and hence to define $F$. So I don't see why $F$ should become minimal due to the maximization of entropy. $\endgroup$ – toaster Jan 11 at 13:49
  • $\begingroup$ If you do not assume the canonical ensemble you don't have a function S(E). You only have a sensible function S(E) if you restrict the possible states. This is done by saying that the only allowed states are $e^{-\beta H}/Z$ (or some other ensemble). Now you have a definite entropy for a given energy and therefore obtain a function S(E). Furthermore if you do not restrict to the canonical ensemble you cannot derive the first law. So when doing thermodynamics the canonical ensemble is always implicit. $\endgroup$ – toaster Jan 11 at 15:32

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