On the combinatorics of exclusion in Haldane fractional statistics

TL;DR

This paper refines the combinatorial understanding of fractional exclusion statistics (FES), deriving exact occupancy rules for systems with parameter g=q/r, and explores their implications for probability distributions and comparisons with Gentile statistics.

Contribution

It provides a new exact formulation of the generalized Pauli principle for FES with extended parameters and analyzes occupancy distributions, including their expectations and variances.

Findings

Allowed occupation number is r-q+1, not 1/g, for FES with g=q/r.

The global occupancy shape must satisfy specific constraints related to r and N.

Number of occupied states follows a hypergeometric distribution in the thermodynamic limit.

Abstract

This paper is a revision of the combinatorics of fractional exclusion statistics (FES). More specifically, the following exact statement of the generalized Pauli principle is derived: for an $N$-particles system exhibiting FES of extended parameter $g=q/r$ ($q$ and $r$ are co-prime integers such that $0 < q \leq r$), we found that the allowed occupation number of a state is smaller than or equal to $r-q+1$ and \emph{not} to $1/g$ whenever $q\neq 1$ and, moreover, the global occupancy shape (merely represented by a partition of $N$) is admissible if the number of states occupied by at least two particles is less than or equal to $(N-1)/r$ ($N \equiv 1 \pmod r$). These counting rules allow distinguishing infinitely many families of FES systems depending on the parameter $g$ and the size $N$. As an application of the main result, we study the probability distributions of occupancy configurations. For instance, the number of occupied states is found to be a hypergeometric random variable. Closed-form expressions for the expectation values and variances in the thermodynamic limit are presented. By way of comparison, we obtain parallel results regarding the Gentile intermediate statistics and demonstrate subtle similarities and contrasts with respect to FES.