From the partition function we can in principle generate all interesting quantities for a given system in equilibrium with its surroundings at a temperature \( T \).

With the probability distribution given by the Boltzmann distribution we are now in a position where we can generate expectation values for a given variable \( A \) through the definition $$ \begin{equation*} \langle A \rangle = \sum_jA_jw_j= \frac{\sum_jA_j\exp{(-\beta(E_j)}}{Z}. \end{equation*} $$ In general, most systems have an infinity of microstates making thereby the computation of \( Z \) practically impossible and a brute force Monte Carlo calculation over a given number of randomly selected microstates may therefore not yield those microstates which are important at equilibrium. To select the most important contributions we need to use the condition for detailed balance. Since this is just given by the ratios of probabilities, we never need to evaluate the partition function \( Z \).