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

Abstract: The paper analyzes the entropy of a system composed by non-interacting and indistinguishable particles whose quantum state numbers are modelled as independent and identically distributed classical random variables. The crucial observation is that, under this assumption, whichever is the number of particles that constitute the system, the occupancy numbers of system's quantum (micro)states are multinomially distributed. This observation leads to an entropy formula for the physical system, which is nothing else than the entropy formula of the multinomial distribution, for which we claim novelty, in the sense that it is proposed here for the first time that the entropy of the multinomial distribution is the entropy of the physical system. The entropy formula of the multinomial distribution unveils yet unexplored connections between information theory and statistical mechanics, among which we mention the connection between conditional entropy of the random microstate given the random occupancy numbers and the Boltzmann-Planck entropy $\log(W)$ and between these two and the Gibbs correction term $\log(N!)$, thermalization and communication-theoretic preparation of a thermal state, accessible information of the thermal state and physical entropy of the thermalized system. A noticeable specific result that descends from our approach is the exact quantum correction to the textbook Sackur-Tetrode formula for the entropy of an ideal gas at the thermal equilibrium in a container.