Exact Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac Statistics

TL;DR

This paper derives exact forms of Maxwell-Boltzmann, Bose-Einstein, and Fermi-Dirac statistics using combinatorial methods without Stirling's approximation, revealing implications for quantum systems.

Contribution

It introduces explicit entropy measures for quantum statistics based on combinatorial derivations, avoiding common approximations.

Findings

Exact BE and FD statistics have significant effects on quantum system behavior.

New entropy functions depend explicitly on state probability, degeneracy, and total entities.

Analysis of decision costs links statistical mechanics to quantum phenomena.

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.