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

TL;DR

This paper derives that in classical statistical mechanics without the thermodynamic limit, the canonical ensemble's Boltzmann factor is a q-exponential, linking non-extensivity to the system's phase space dimensionality.

Contribution

It shows that the most general separable distribution in such systems is a q-exponential, with the non-extensivity parameter related to a mathematical separation constant, Q.

Findings

Q equals 1 for large phase space dimensions

The results apply to nanosystems and systems with low-dimensional phase space

The derived distributions generalize classical Boltzmann factors

Abstract

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 separable from of 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 Gamma-phase space. Our results will be relevant for systems with a low-dimensional Gamma-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.