On the Thermodynamics of Particles Obeying Monotone Statistics

TL;DR

This paper explores the thermodynamics of particles following monotone statistics, introducing a block-monotone scheme that generalizes existing models and reveals a Fermi-like exclusion principle affecting high-density regimes.

Contribution

It proposes a novel block-monotone scheme for particles obeying monotone statistics, extending the theoretical framework and analyzing its thermodynamic implications.

Findings

The grand-partition function does not require the Gibbs correction factor n!

A Fermi-like exclusion principle emerges at high densities

The scheme reduces to the usual monotone case with non-degenerate spectra

Abstract

The aim of the present paper is to provide a preliminary investigation of the thermodynamics of particles obeying monotone statistics. To render the potential physical applications realistic, we propose a modified scheme called block-monotone, based on a partial order arising from the natural one on the spectrum of a positive Hamiltonian with compact resolvent. The block-monotone scheme is never comparable with the weak monotone one and is reduced to the usual monotone scheme whenever all the eigenvalues of the involved Hamiltonian are non-degenerate. Through a detailed analysis of a model based on the quantum harmonic oscillator, we can see that: (a) the computation of the grand-partition function does not require the Gibbs correction factor $n!$ (connected with the indistinguishability of particles) in the various terms of its expansion with respect to the activity; and (b) the decimation of terms contributing to the grand-partition function leads to a kind of "exclusion principle" analogous to the Pauli exclusion principle enjoined by Fermi particles, which is more relevant in the high-density regime and becomes negligible in the low-density regime, as expected.