Exclusion statistics for particles with a discrete spectrum

TL;DR

This paper develops a microscopic statistical mechanics framework for particles with exclusion statistics in a discrete spectrum, deriving explicit thermodynamic expressions and applying them to the harmonic Calogero model.

Contribution

It introduces a generalized thermodynamic exclusion statistics formalism for discrete spectra and connects it to known mathematical identities like the Ramanujan continued fraction.

Findings

Derived explicit thermodynamic potential expressions

Established nesting relations for occupation numbers

Linked the formalism to the Ramanujan continued fraction

Abstract

We formulate and study the microscopic statistical mechanics of systems of particles with exclusion statistics in a discrete one-body spectrum. The statistical mechanics of these systems can be expressed in terms of effective single-level grand partition functions obeying a generalization of the standard thermodynamic exclusion statistics equation of state. We derive explicit expressions for the thermodynamic potential in terms of microscopic cluster coefficients and show that the mean occupation numbers of levels satisfy a nesting relation involving a number of adjacent levels determined by the exclusion parameter. We apply the formalism to the harmonic Calogero model and point out a relation with the Ramanujan continued fraction identity and appropriate generalizations.