Boltzmann-Gibbs entropy is sufficient but not necessary for the likelihood factorization required by Einstein

TL;DR

This paper explores the conditions under which different entropy measures, including non-additive ones like q-entropy, satisfy Einstein's likelihood factorization requirement, challenging the notion that Boltzmann-Gibbs entropy is uniquely sufficient.

Contribution

It demonstrates that entropies beyond Boltzmann-Gibbs, such as q-entropy, can also fulfill Einstein's likelihood factorization, broadening the foundational understanding of entropy in statistical mechanics.

Findings

Boltzmann-Gibbs entropy is sufficient for likelihood factorization.

Non-additive entropies like q-entropy can also satisfy the factorization.

This challenges the exclusivity of Boltzmann-Gibbs entropy in foundational principles.

Abstract

In 1910 Einstein published a crucial aspect of his understanding of Boltzmann entropy. He essentially argued that the likelihood function of any system composed by two probabilistically independent subsystems {\it ought} to be factorizable into the likelihood functions of each of the subsystems. Consistently he was satisfied by the fact that Boltzmann (additive) entropy fulfills this epistemologically fundamental requirement. We show here that entropies (e.g., the $q$-entropy on which nonextensive statistical mechanics is based) which generalize the BG one through violation of its well known additivity can {\it also} fulfill the same requirement. This fact sheds light on the very foundations of the connection between the micro- and macro-scopic worlds.