= Noninteracting subsystems
Two independently available parts of a <thermodynamic system> are noninteracting in the canonical description when their <state space> is a <Cartesian product> and their <energy> is additive, $E(a,b)=E_A(a)+E_B(b)$. Both conditions are needed for the <canonical partition function> to factor:
$$
Z=\sum_{a,b}e^{-\beta[E_A(a)+E_B(b)]}
=\left(\sum_a e^{-\beta E_A(a)}\right)\left(\sum_b e^{-\beta E_B(b)}\right)=Z_AZ_B.
$$
An additive expression on a constrained set of allowed state pairs need not give this factorization. Relabeling the states to make their allowed labels independent can instead introduce an interaction into the <energy>. For two two-state labels, an additive <energy> necessarily satisfies the rectangular identity $E_{11}+E_{22}=E_{12}+E_{21}$.
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