We can relate the partition function to the total energy of the system. Recall that we can determine the average value of a property by using the probability. Let us investigate < >, the average value of the energy of an occupied state. It turns out that < > is precisely what we mean when we say U. We can calculate it using the above equation for the absolute probability:
The final result seems strange but is mathematically tidier and is equivalent to the messier formulation.
Finally, and significantly, we can relate the Helmholtz free energy, sometimes just called the free energy, to Z as well. By a simple substitution, F = U - τσ, we find an important result: