PX284 - E5b - combining partition functions

if the energy depends on two independent and distinct contributions:

E_{i, j} = E_{i} + E_{j}

where, $E_{i}$ and $E_{j}$ belong to two subsystems, $A$ and $B$ , respectively

the total partition function:

\begin{aligned} Z_{t o t} & = \sum_{i} \sum_{j} \exp (- β (E_{i} + E_{j})) \\ = \sum_{i} \exp (- β E_{i}) \sum_{j} \exp (- β E_{j}) \\ = Z_{a} \times Z_{b} \end{aligned}

similarly, for more than two contributions:

Z_{t o t} = Z_{a} \times Z_{b} \times \dots

for functions that depend on $\ln (Z)$ , the contributions add up:

\begin{aligned} U_{t o t} & = U_{a} + U_{b} + U_{c} + \dots \\ F_{t o t} & = F_{a} + F_{b} + F_{c} + \dots \end{aligned}