Derivation of power-law distributions within standard statistical mechanics

We show that within classical statistical mechanics it is possible to naturally derive power-law distributions which are of Tsallis type. The only assumption is that microcanonical distributions have to be separable from of the total system energy, which is reasonable for any sensible measurement. We demonstrate that all separable distributions are parametrized by a separation constant Q which is one to one related to the q-parameter in Tsallis distributions. The power laws obtained are formally equivalent to those obtained by maximizing the Tsallis entropy under q constraints. We further ask why nature fixes the separation constant Q to 1 in so many cases leading to standard thermodynamics. We answer this with an explicit example where it is possible to relate Q to system size and interaction parameters, characterizing the physical system. We argue that these results might be helpful to explain the ubiquity of Tsallis distributions in nature.