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The paper analyzes the entropy of a system composed by non-interacting andindistinguishable particles whose quantum state numbers are modelled asindependent and identically distributed classical random variables. The crucialobservation is that, under this assumption, whichever is the number ofparticles that constitute the system, the occupancy numbers of system's quantum(micro)states are multinomially distributed. This observation leads to anentropy formula for the physical system, which is nothing else than the entropyformula of the multinomial distribution, for which we claim novelty, in thesense that it is proposed here for the first time that the entropy of themultinomial distribution is the entropy of the physical system. The entropyformula of the multinomial distribution unveils yet unexplored connectionsbetween information theory and statistical mechanics, among which we mentionthe connection between conditional entropy of the random microstate given therandom occupancy numbers and the Boltzmann-Planck entropy $\log(W)$ and betweenthese two and the Gibbs correction term $\log(N!)$, thermalization andcommunication-theoretic preparation of a thermal state, accessible informationof the thermal state and physical entropy of the thermalized system. Anoticeable specific result that descends from our approach is the exact quantumcorrection to the textbook Sackur-Tetrode formula for the entropy of an idealgas at the thermal equilibrium in a container.

The Exact Entropy Formula of the Ideal Gas and its Information-Theoretic Interpretation