Non-extensivity of the configurational density distribution in the classical microcanonical ensemble

TL;DR

This paper demonstrates that the configurational probability distribution in a classical microcanonical ensemble is inherently non-extensive and belongs to the q-exponential family, impacting the thermodynamics of such systems.

Contribution

It shows that the configurational subsystem is non-extensive and explores the implications for thermodynamics, suggesting Renyi's entropy as the relevant measure.

Findings

Configurational distribution belongs to the q-exponential family.

Thermodynamics of the subsystem is determined up to a scaling function.

Renyi's entropy may be more appropriate than Tsallis's.

Abstract

We show that the configurational probability distribution of a classical gas always belongs to the q-exponential family. Hence, the configurational subsystem is non-extensive in the sense of Tsallis. One of the consequences of this observation is that the thermodynamics of the configurational subsystem is uniquely determined up to a scaling function. As an example we consider a system of non-interacting harmonic oscillators. In this example, the scaling function can be determined from the requirement that in the limit of large systems the microcanonical temperature of the configurational subsystem should coincide with that of the canonical ensemble. The result suggests that Renyi's entropy function is the relevant one rather than that of Tsallis.