In the Canonical Ensemble, given a quantum system with $N$ distingishable and non-interacting particles distributed amongst $r$ energy levels of energy $\epsilon _1,\epsilon _2,\epsilon _3,…,\epsilon _r$ and degeneracy $g_1,g_2,g_3,…,g_r$, the partition function of the single-particle is defined as
$$Z_{SP}=\sum_{i=1}^r g_ie^{-\epsilon_i\beta(T)}\tag{1}$$
with $\beta(T)=\frac{1}{K_BT}$, and the partition function of the whole system of $N$ particles is defined as
$$Z_{N}=\prod_{i=1} ^N (Z_{SP})_i =(Z_{SP})^N \tag{2}$$
(with the last equality holding if all particles are distinguishable and; in case they are identical and indistinguishable, $Z_N=Z_{SP}^N/N!$).
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Where does this definition, (2), come from? Why a product and not, let's say, a sum?
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On the other hand, would then be correct to define $Z_N$ also in this way (3)?
$$Z_N=\sum_{j=1}^Sg_je^{-Ej\beta(T)}\tag{3}$$
With $S$ the number of microstates of the whole system and $E_j$ the energy of the whole system at the microstate $j$.
Best Answer
When considering a partition function of a system composed of several distinguishable subsystems you never add the separate partition functions up, and always multiply them.
The reason is that the partition function covers the possible states of a system, and when for a system composed of subsystem we can set subsystem $A$ to a certain state and then we have to cover all of the states of subsystem $B$. Then change the state of subsystem $A$ and again sum over all states of subsystem $B$. This is multiplication $$ Z_{AB} = \sum_{A,B} e^{-\beta(E_A+E_B)} = \sum_{A} e^{-\beta E_A} \sum_B e^{-\beta E_B} = Z_A Z_B$$ and the generalization to more than two subsystems is immediate.
Note that for this to be valid the subsystems must be separate and distinguishable. If they are interacting, for example, then you might have $E_{AB} \neq E_A + E_B$. If the particles are identical and indistinguishable, then they cannot be separated into subsystems $A$ and $B$ to begin with.
By the way - this rule of multiplication of partition functions is valid for classical systems as well as quantum ones.