On the derivation of power-law distributions within classical statistical mechanics far from the thermodynamic limit

We show that within classical statistical mechanics, without taking the thermodynamic limit, the most general Boltzmann factor for the canonical ensemble is a q-exponential function. The only assumption here is that microcanonical distributions have to be separated from the total system energy, which is the prerequisite for any sensible measurement. We derive that all separable distributions are parametrized by a mathematical separation constant Q, which can be related to the non-extensivity q-parameter in Tsallis distributions. We further demonstrate that nature fixes the separation constant Q to 1 for large dimensionality of Gibbs Γ-phase space. Our results will be relevant for systems with a low-dimensional Γ-space, for example nanosystems, comprised of a small number of particles, or for systems with a dimensionally collapsed phase space, which might be the case for a large class of complex systems.