Nonextensive statistical mechanics: Some links with astronomical phenomena

TL;DR

This paper reviews nonextensive statistical mechanics, highlighting its relevance to various astronomical phenomena that do not conform to traditional Boltzmann-Gibbs statistics, and explores its applications and analogies in astrophysics.

Contribution

It provides a brief overview of nonextensive statistical mechanics and demonstrates its applications to astrophysical phenomena, establishing connections with classical mechanics and thermodynamics.

Findings

Nonextensive statistics better describes certain astronomical phenomena.

Optimizing $S_q$ with few constraints is equivalent to optimizing $S_{BG}$ with infinite constraints.

Applications include solar neutrinos, galactic velocities, and cosmology.

Abstract

A variety of astronomical phenomena appear to not satisfy the ergodic hypothesis in the relevant stationary state, if any. As such, there is no reason for expecting the applicability of Boltzmann-Gibbs (BG) statistical mechanics. Some of these phenomena appear to follow, instead, nonextensive statistical mechanics. In the same manner that the BG formalism is based on the entropy $S_{BG}=-k \sum_i p_i \ln p_i$, the nonextensive one is based on the form $S_q=k(1-\sum_ip_i^q)/(q-1)$ (with $S_1=S_{BG}$). The stationary states of the former are characterized by an {\it exponential} dependence on the energy, whereas those of the latter are characterized by an (asymptotic) {\it power-law}. A brief review of this theory is given here, as well as of some of its applications, such as the solar neutrino problem, polytropic self-gravitating systems, galactic peculiar velocities, cosmic rays and some cosmological aspects. In addition to these, an analogy with the Keplerian elliptic orbits {\it versus} the Ptolemaic epicycles is developed, where we show that optimizing $S_q$ with a few constraints is equivalent to optimizing $S_{BG}$ with an infinite number of constraints.