On the entropic derivation of the $r^{-2}$ Newtonian gravity force

TL;DR

This paper demonstrates that Tsallis' classical free particle distribution can derive the Newtonian inverse-square law of gravity, unlike Boltzmann-Gibbs' or Renyi's distributions, which require modifications to the conjecture.

Contribution

It shows that Tsallis' entropy framework uniquely reproduces Newtonian gravity's inverse-square law from entropic principles.

Findings

Tsallis' distribution yields the $r^{-2}$ gravitational force.

Boltzmann-Gibbs' and Renyi's distributions do not naturally produce the inverse-square law.

Modifications are necessary when using Boltzmann-Gibbs' or Renyi's distributions.

Abstract

Following Verlinde's conjecture, we show that Tsallis' classical free particle distribution at temperature $T$ can generate Newton's gravitational force's $r^{-2}$ {\it distance's dependence}. If we want to repeat the concomitant argument by appealing to either Boltzmann-Gibbs' or Renyi's distributions, the attempt fails and one needs to modify the conjecture. Keywords: Tsallis', Boltzmann-Gibbs', and Renyi's distributions, classical partition function, entropic force.