The partition function is a particularly useful quantity. Physically, it is nothing more than the normalization factor needed in order to compute probabilities, but in practice, finding that normalization is typically the hardest part of a calculation—once you have found all the energy eigenvalues, that is.

One interesting question is whether the partition function is intensive or extensive. To examine that question, we will look at the partition function of two combined, uncorrelated systems, similar to what we examined earlier. $$Z_A = \sum_i e^{-\beta E_i^A}$$ $$Z_B = \sum_j e^{-\beta E_j^B}$$ $$Z_{AB} = \sum_{ij} e^{-\beta \left(E_i^A + E_j^B\right)}$$ $$= \sum_{ij} e^{-\beta E_i^A}e^{-\beta E_j^B}$$ $$= \sum_{i} \sum_j e^{-\beta E_i^A}e^{-\beta E_j^B}$$ $$= \sum_{i} e^{-\beta E_i^A} \sum_j e^{-\beta E_j^B}$$ $$= \left(\sum_i e^{-\beta E_i^A}\right) \left(\sum_j e^{-\beta E_j^B}\right)$$ $$= Z_A Z_B$$

So the partition function of two uncorrelated systems is multiplied rather than added. This means that the log of the partition function is itself extensive! It will turn out to be a thermodynamic state function that you have already encountered, as we will see tomorrow.

One consequence of this logarithmic extensivity is that if you have $N$ identical non-interacting systems with uncorrelated probabilities, you can write their combined partition function as $$Z = Z_1^N$$ where $Z_1$ is the partition function of a single non-interacting system.