Relation between Boltzmann and Gibbs entropy and example with multinomial distribution

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.