# Why is U minimized at constant S and V?

1. Jul 25, 2011

### 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.