Non-Extensivity of the Configurational Density Distribution in the Classical Microcanonical Ensemble

Abstract

We show that the configurational probability distribution of a classical gas always belongs to the q-exponential family. 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 Rényi's entropy function is the relevant one rather than that of Tsallis.

On the Thermodynamics of Classical Micro-Canonical Systems

“…In a previous paper [21] we have shown that the configurational probability distribution f conf U (q) of an interacting mono-atomic gas with N particles always belongs to the q-exponential family, with…”

“…It was shown in [21] that the configurational probability distribution f conf U (q) belongs to the q-exponential family, with…”

“…Recently [21], it was proved that this distribution belongs to the q-exponential family [22][23][24], with q = 1−2/(3N −2). In the thermodynamic limit N → ∞ it converges to the Boltzmann-Gibbs distribution, which corresponds with the q = 1 case.…”

“…It was argued in [21] that for the calculation of thermodynamic quantities Rényi's entropy function is more appropriate than that of Tsallis. The Rényi entropy function is a monotone function of that of Tsallis.…”

Group Invariance of Information Geometry on q-Gaussian Distributions Induced by Beta-Divergence

“…For example, the family not only describes phenomena obeying the power-law well [6], but also, it is theoretically proven to include a velocity distribution of the classical gas with N particles [9,10], an attracting invariant manifold of porous media flow [11], and so on. In statistics, it is reported to provide a reasonable statistical model in robust inference from data losing normality [1,[12][13][14].…”