Asymptotically scale-invariant occupancy of phase space makes the entropy Sq extensive

TL;DR

The paper investigates how the entropy of a system with many subsystems behaves under different correlation structures, showing that a generalized entropy becomes extensive in the presence of strong correlations, unlike the traditional Boltzmann-Gibbs entropy.

Contribution

It demonstrates that the generalized entropy $S_q$ can be extensive for globally correlated systems, unlike the standard entropy, highlighting a new understanding of entropy extensivity.

Findings

$S_q$ becomes extensive for globally correlated systems.

Traditional $S_{BG}$ is extensive only for weakly correlated or independent subsystems.

The paper provides a framework for understanding entropy scaling in correlated systems.

Abstract

Phase space can be constructed for $N$ equal and distinguishable subsystems that could be (probabilistically) either {\it weakly} (or {\it "locally"}) correlated (e.g., independent, i.e., uncorrelated), or {\it strongly} (or {\it globally}) correlated. If they are locally correlated, we expect the Boltzmann-Gibbs entropy $S_{BG} \equiv -k \sum_i p_i \ln p_i$ to be {\it extensive}, i.e., $S_{BG}(N)\propto N $ for $N \to\infty$. In particular, if they are independent, $S_{BG}$ is {\it strictly additive}, i.e., $S_{BG}(N)=N S_{BG}(1), \forall N$. However, if the subsystems are globally correlated, we expect, for a vast class of systems, the entropy $S_q\equiv k [1- \sum_i p_i^q]/(q-1)$ (with $S_1=S_{BG}$) for some special value of $q\ne1$ to be the one which extensive (i.e., $S_q(N)\propto N $ for $N \to\infty$).