A simple link of information entropy of quantum and classical systems with Newtonian $r^{-2}$ dependence of Verlinde's entropic force

TL;DR

This paper demonstrates that the entropic force derived from information entropy exhibits a Newtonian inverse-square law dependence, applicable to quantum, classical, fermionic, and bosonic systems, supporting the idea of gravity as an emergent entropic force.

Contribution

It introduces a universal entropy-based approach to derive Newtonian gravity across quantum, classical, fermionic, and bosonic systems, extending previous work and supporting emergent gravity theories.

Findings

Entropic force follows an inverse-square law ($r^{-2}$) dependence.

Universal entropy property applies to diverse physical systems.

Supports gravity as an emergent entropic phenomenon.

Abstract

It is shown that the entropic force formula $F_e=-\lambda\partial S/\partial A$ leads to a Newtonian $r^{-2}$ dependence. Here we employ the universal property of the information entropy $S=a+b\ln N$ ($N$ is the number of particles of a quantum system and $A$ is the area containing the system). This property was previously obtained for fermionic systems (atoms, atomic clusters, nuclei and infinite Fermi systems i.e. electron gas, liquid $^3$He and nuclear matter) and bosonic ones (correlated boson-atoms in a trap). A similar dependence of the entropic force has been derived very recently by Plastino et al with a Bose or Fermi gas entropy, inspired by Verlinde's conjecture~\cite{Verlide-11} that gravity is an emergent entropic force. Finally, we point out that our simple argument holds for classical systems as well.