Entropy of the Canonical Occupancy (Macro) State in the Quantum Measurement Theory

Abstract

The paper analyzes the probability distribution of the occupancy numbers and the entropy of a system at the equilibrium composed by an arbitrary number of non-interacting bosons. The probability distribution is obtained through two approaches: one involves tracing out the environment from a bosonic eigenstate of the combined environment and system of interest (the empirical approach), while the other involves tracing out the environment from the mixed state of the combined environment and system of interest (the Bayesian approach). In the thermodynamic limit, the two coincide and are equal to the multinomial distribution. Furthermore, the paper proposes to identify the physical entropy of the bosonic system with the Shannon entropy of the occupancy numbers, fixing certain contradictions that arise in the classical analysis of thermodynamic entropy. Finally, by leveraging an information-theoretic inequality between the entropy of the multinomial distribution and the entropy of the multivariate hypergeometric distribution, Bayesianism of information theory and empiricism of statistical mechanics are integrated into a common “infomechanical” framework.

Derivation of Bose’s Entropy Spectral Density from the Multiplicity of Energy Eigenvalues

“…(e x − 1) e x − 1 dx ≈ 5.95,for the total entropy we obtainS r ≈ ∞ ν=0 (µ(ν)(1 − log(µ(ν)))dν = 4πL 3 c 3 β 3 h 3 (2ζ(3) + 5.95) = 105 • L 3 c 3 β 3 h 3 ,which is remarkably close to(13).…”

“…It is worth observing that the Poisson distribution, besides being the distribution that characterizes the coherent state, is also in strict connection with the multinomial distribution, recently proposed in [ 13 ] for the occupancy numbers of the canonical state of the ideal gas. According to [ 13 ], the joint probability distribution of the occupancy numbers in the canonical ensemble is multinomial: where R is the total number of photons, which, in the canonical ensemble approach, is assumed to be fixed and known. However, in the case of photons inside a cavity, R is random, so we take a weighted average of the multinomial distributions with weights equal to the distribution of N : where, again, ( 20 ) is understood.…”