# 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?

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.