Derivation of a statistical model for classical systems obeying fractional exclusion principle

TL;DR

This paper develops a classical statistical model using maximum entropy methods that interpolates between Bose-Einstein and Fermi-Dirac statistics, incorporating a fractional exclusion principle.

Contribution

It introduces a new classical probability distribution based on indistinguishability and exclusion principles, derived via maximum entropy and Lagrange multipliers.

Findings

The model produces positive probability distributions within parameter ranges.

For dilute systems, the distribution lies between Bose-Einstein and Fermi-Dirac statistics.

Potential applications to classical thermodynamics are discussed.

Abstract

The violation of the Pauli principle has been surmised in several models of the Fractional Exclusion Statistics and successfully applied to several quantum systems. In this paper, a classical alternative of the exclusion statistics is studied using the maximum entropy methods. The difference between the Bose-Einstein statistics and the Maxwell-Boltzmann statistics is understood in terms of a separable quantity, namely the degree of indistinguishability. Starting from the usual Maxwell-Boltzmann microstate counting formula, a special restriction related to the degree of indistinguishability is incorporated using Lagrange multipliers to derive the probability distribution function at equilibrium under NVE conditions. It is found that the resulting probability distribution function generates real positive values within the permissible range of parameters. For a dilute system, the probability distribution function is intermediate between the Fermi-Dirac and Bose-Einstein statistics and follows the exclusion principle. Properties of various variables of this novel statistical model are studied and possible application to classical thermodynamics is discussed.