Entropic Upper Bound on Gravitational Binding Energy

TL;DR

This paper establishes an entropic upper bound on the gravitational binding energy of self-gravitating systems using Tsallis' non-additive entropy, linking astrophysical density profiles to statistical mechanics.

Contribution

It introduces a novel entropic bound on gravitational binding energy based on Tsallis' entropy, connecting astrophysical density profiles with non-additive statistical measures.

Findings

The bound is saturated by isotropic q-Gaussian distributions.

The maximizer distributions correspond to the Plummer density profile.

The entropic bound is likely unique, based on a heuristic scaling argument.

Abstract

We prove that the gravitational binding energy {\Omega} of a self gravitating system described by a mass density distribution {\rho}(x) admits an upper bound B[{\rho}(x)] given by a simple function of an appropriate, non-additive Tsallis' power-law entropic functional Sq evaluated on the density {\rho}. The density distributions that saturate the entropic bound have the form of isotropic q-Gaussian distributions. These maximizer distributions correspond to the Plummer density profile, well known in astrophysics. A heuristic scaling argument is advanced suggesting that the entropic bound B[{\rho}(x)] is unique, in the sense that it is unlikely that exhaustive entropic upper bounds not based on the alluded Sq entropic measure exit. The present findings provide a new link between the physics of self gravitating systems, on the one hand, and the statistical formalism associated with non-additive, power-law entropic measures, on the other hand.