Why is U minimized at constant S and V?

[namphcar22]

Can someone explain why the energy U of a closed system at constant entropy S and volume V reaches a minimum at equilibrium? Wikipedia has an article on principle of minimum energy, but I'm a little uncomfortable with the partial derivative manipulations in that article. Is it possible to argue derive the min energy principle using statistical mechanics?

I know from my intro stat mech course that two systems are in thermal equilibrium with constant total U and V, they will distribute energy between themselves to maximize the total entropy. This fact follows from the fundamental assumption that all accessible microstates in an isolated system are equally likely. So when the dust settles, we are most likely to observe an energy division with the greatest number of corresponding microstates. However, in my original equation, entropy is fixed, so statistical arguments don't seem to be available.

 

[Valerie Park]

In this case, I think the best way to approach the problem is to use a thermodynamic argument. We can start by noting that the total energy of a closed system at constant entropy and volume is given by the fundamental thermodynamic relation:U = U(V, S).Since U is a function of V and S, if we hold S constant, then U is a function of V only. The minimum energy state of the system is then obtained by setting the derivative of U with respect to V equal to zero. This gives us:dU/dV = 0,which is the condition for finding the minimum energy state of the system.