To compute the average energy, we use the weighted average of energies from various states $r$, taking into account their respective probabilities, as represented by

This is done in the following manner:

$\bar{E}=\displaystyle\frac{\displaystyle\sum_rP_rE_r}{\displaystyle\sum_rP_r}$

The result is obtained by considering the values of

.

(ID 3526)

The average energy is determined with respect to beta del sistema $1/J$, energía del estado $r$ $J$, energía media del sistema $J$ and numero del estado $-$

and can be expressed as follows:

$\bar{E}=-\displaystyle\frac{1}{\sum_re^{-\beta E_r}}\displaystyle\frac{\partial}{\partial\beta}\sum_re^{-\beta E_r}$

This can be summarized as

$\bar{E}=-\displaystyle\frac{1}{Z}\displaystyle\frac{\partial Z}{\partial\beta}$

where we introduce the so-called partition function with beta del sistema $1/J$, energía del estado $r$ $J$, energía media del sistema $J$ and numero del estado $-$:

The letter $Z$ originates from the German word Zustandsumme (Zustand=State, Summe=sum). The partition function is a generating function, meaning it generates other functions that have physical significance.

(ID 3527)

As it is evident that

$\displaystyle\frac{\partial\ln Z}{\partial\beta} =\displaystyle\frac{1}{Z}\displaystyle\frac{\partial Z}{\partial\beta}$

and

$\bar{E}=-\displaystyle\frac{1}{Z}\displaystyle\frac{\partial Z}{\partial\beta}$

this implies that, with

(ID 3528)