The q-exponential family in statistical physics

TL;DR

This paper explores the q-exponential family in statistical physics, highlighting its relevance for describing small isolated systems and discussing implications for defining temperature and entropy.

Contribution

It generalizes the exponential family to the q-exponential family and applies it to small systems, including deriving distributions and discussing entropy definitions.

Findings

The momentum distribution of a single particle is a q-Gaussian.

The configurational density distribution belongs to the q-exponential family.

Using Renyi's entropy yields consistent results for small systems.

Abstract

The Boltzmann-Gibbs probability distribution, seen as a statistical model, belongs to the exponential family. Recently, the latter concept has been generalized. The q-exponential family has been shown to be relevant for the statistical description of small isolated systems. Two main applications are reviewed: 1. The distribution of the momentum of a single particle is a q-Gaussian, the distribution of its velocity is a deformed Maxwellian; 2. The configurational density distribution belongs to the q-exponential family. The definition of the temperature of small isolated systems is discussed. It depends on defining the thermodynamic entropy of a microcanonical ensemble in a suitable manner. The simple example of non-interacting harmonic oscillators shows that Renyi's entropy functional leads to acceptable results.