@Chestermiller an ideal gas in a vertical and adiabatic piston covered with sand will reach a minimum of U if the sand is removed progressively (to ensure that the process is quasistatic).
@anorlunda I got your point about isolated and closed systems. Could you please give a proof of the energy...
Yes of course.
The goal is to prove the implication Entropy maximum principle => Energy maximum principle. As a reminder,
Entropy maximum principle : The equilibrium value of any unconstrained internal parameter is such as to maximize the entropy for the given value of the total internal...
If you take a system with fixed entropy S0 and let it evolve, it reachs equilibrium. Let Ueq be the energy of the system at equilibrium.
Now take the same system with fixed energy U=Ueq (S is not fixed anymore), how do you know that the equilibrium reached is the same as before, that means with...