Generalized Maxwell-Boltzmann, Bose-Einstein, Fermi-Dirac and Acharya-Swamy Statistics and the Polya Urn Model

TL;DR

This paper derives generalized statistical distributions for quantum particles using combinatorial methods, linking them to the Pólya urn model and unifying various quantum statistics including Acharya-Swamy.

Contribution

It introduces a unified combinatorial framework for Maxwell-Boltzmann, Bose-Einstein, Fermi-Dirac, and Acharya-Swamy statistics via the Pólya urn model.

Findings

Generalized distributions reduce to classical cases for equal probabilities.

Distributions are connected to the Pólya urn model involving non-i.i.d. sampling.

Acharya-Swamy intermediate statistic is derived as the most probable Pólya distribution.

Abstract

Generalized probability distributions for Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac statistics, with unequal source probabilities $q_i$ for each level $i$, are obtained by combinatorial reasoning. For equiprobable degenerate sublevels, these reduce to those given by Brillouin in 1930, more commonly given as a statistical weight for each statistic. These distributions and corresponding cross-entropy (divergence) functions are shown to be special cases of the P\'olya urn model, involving neither independent nor identically distributed ("ninid") sampling. The most probable P\'olya distribution contains the Acharya-Swamy intermediate statistic.