Inclusion statistics and the Calogero model

We address the microscopic formulation of inclusion statistics in which particles tend to coalesce more than ordinary bosons. Inclusion statistics was proposed as a counterpoint to exclusion statistics, which can be defined in terms of microscopic \(n\)-body quantum models like the LLL-anyon model, the Calogero model, or in terms of Haldane counting statistics. We define inclusion statistics by means of inverting the grand partition function of a gaz of particles obeying to exclusion statistics and changing the sign of the fugacity, in turn establishing a duality relation between \(g\)-exclusion and \((1 − g)\)-inclusion where \(g\) is the statistical parameter (\(g = 0\) Bose- Einstein, \(g = 1\) Fermi-Dirac, \(g \geq 1\) exclusion). This inversion recipe is in line with the Fermi-Dirac/Bose-Einstein correspondance where the grand partition of the former is the inverse of those of the latter provided that the sign of the fugacity is changed. Equivalently, microscopic occupation multiplicities of \(1\)-body quantum states can also be obtained which reproduce the inclusion grand partition function and the \(n\)-body partition functions. This recipe is then applied to the 1d Calogero model in an harmonic well. This leads to a Calogero-like \(n\)-body inclusion spectrum whose interpretation remains mysterious. Finally attractive delta interactions counter terms can be added to the \(n\)-body Calogero Hamiltonian in order to reproduce the microscopic \(n\)-body inclusion spectrum in the thermodynamic limit of a vanishing harmonic well.