  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.
