Multiplicity of statistical weight of macrostates when combining systems in thermodynamics (Boltzmann entropy) In my lecture notes, the Boltzmann entropy form was motivated as a valid form for entropy partly because it was extensive. However this hinges on the assumption that the statistical weight for a macrostate of two systems combined is their product. Why is this valid- for large systems at least- and when might it break down?
 A: I think that the explanation here is the same as in classical thermodynamics: the interaction between the two subsystems is a "surface effect".
Suppose the two subsystems are cubic, of volume $L^3$, sharing an interface of area $\sim L^2$. If the interactions are of short range, $d$, the perturbation on either system (just from counting interactions) has a relative magnitude $\sim dL^2/L^3=d/L$. Of course, this has an effect on the number of states $\Omega(E)$ in both subsystems having energy $E_1$ or $E_2$ respectively, but bearing in mind that the logarithm of $\Omega$ is the important quantity, this will be small provided $d\ll L$. So, one can make the approximation that the systems are statistically independent.
One can expect the argument to break down if the interactions are long ranged (when extensivity of thermodynamic properties does actually come into question) and when the system is small (when one would not expect it to hold). Also, if the subsystems are chosen to have an unusually large interaction area: for example, the oppositely coloured lattice sites in a checkerboard colouring of Ising-like models.
There are various other subtleties associated with the extensivity of the Boltzmann entropy, some of which can be found discussed on Physics SE, but I think this is the one you were worried about.
