Relation between Boltzmann and Gibbs entropy and example with multinomial distribution

TL;DR

This paper explores the relationship between Boltzmann and Gibbs entropy, introducing fluctuation entropy as their difference, and discusses its implications for statistical mechanics and the limits of traditional approaches.

Contribution

It establishes a general relationship between Boltzmann and Gibbs entropy and introduces fluctuation entropy, highlighting when standard statistical methods need modification.

Findings

Difference between Boltzmann and Gibbs entropy equals fluctuation entropy.

Fluctuation entropy ratio vanishes in the thermodynamic limit for independent particles.

Large fluctuation entropy indicates the need for extended statistical methods.

Abstract

General relationship between mean Boltzmann entropy and Gibbs entropy is established. It is found that their difference is equal to fluctuation entropy, which is a Gibbs-like entropy of macroscopic quantities. The ratio of the fluctuation entropy and mean Boltzmann, or Gibbs entropy vanishes in the thermodynamic limit for a system of distinguishable and independent particles. It is argued that large fluctuation entropy clearly indicates the limit where standard statistical approach should be modified, or extended using other methods like renormalization group.