Exact Maxwell–Boltzmann, Bose–Einstein and Fermi–Dirac statistics

Abstract The exact Maxwell–Boltzmann (MB), Bose–Einstein (BE) and Fermi–Dirac (FD) entropies and probabilistic distributions are derived by the combinatorial method of Boltzmann, without Stirling's approximation. The new entropy measures are explicit functions of the probability and degeneracy of each state, and the total number of entities, N. By analysis of the cost of a “binary decision”, exact BE and FD statistics are shown to have profound consequences for the behaviour of quantum mechanical systems.