Cost of s-fold decisions in exact Maxwell–Boltzmann, Bose–Einstein and Fermi–Dirac statistics

The exact forms of the degenerate Maxwell–Boltzmann (MB), Bose–Einstein (BE) and Fermi–Dirac (FD) entropy functions, derived by Boltzmann's principle without the Stirling approximation [R.K. Niven, Physics Letters A, 342(4) (2005) 286], are further examined. Firstly, an apparent paradox in quantization effects is resolved using the Laplace–Jaynes interpretation of probability. The energy cost of learning that a system, distributed over s equiprobable states, is in one such state (an “s-fold decision”) is then calculated for each statistic. The analysis confirms that the cost depends on one's knowledge of the number of entities N and (for BE and FD statistics) the degeneracy, extending the findings of Niven (2005).