A completely isolated system, such as the Microcanonical Ensemble is unrealistic. Therefore, we often imagine a system in constant thermal equillibrium with a reservoir whose temperature is kept constant. Thus, for this system , and is constant. The system and its reservoir together constitute the universe, i.e. and isolated system. Based on our previous result, we may thus state that the likelihood of a micro-state is in fact
As with constant.
The partition function refers to the function we must multiply to normalize which is
Note that for the discrete case
Where is the Legendre transform of (the internal energy) with regard to , otherwise known as the Helmholtz Free Energy. Thus implying that
An alternative method to prove is through a differential equation. Let . Note that
We have that
Also note that
thus and obey exactly the same first-order differential equation.
Meanwhile, at , is dominated by the term , thus tends toward . Meanwhile, also tends toward at . Thus, and obey the same first-order differential equation and have the same initial value, thus completing our proof.