Theory of Non-Interacting Fermions and Bosons in the Canonical Ensemble

TL;DR

This paper develops a comprehensive mathematical framework for calculating particle number statistics of non-interacting fermions and bosons in the canonical ensemble, unifying previous methods and enabling analysis of degeneracies.

Contribution

It introduces auxiliary partition functions and a decomposition of occupation number correlations, providing a unified approach applicable to degenerate and non-degenerate spectra.

Findings

Derived exact formulas for occupation number correlations.

Applied the theory to a bosonic ring with magnetic field, computing correlations up to fourth order.

Unified previous approaches into a single mathematical framework.

Abstract

We present a self-contained theory for the exact calculation of particle number counting statistics of non-interacting indistinguishable particles in the canonical ensemble. This general framework introduces the concept of auxiliary partition functions, and represents a unification of previous distinct approaches with many known results appearing as direct consequences of the developed mathematical structure. In addition, we introduce a general decomposition of the correlations between occupation numbers in terms of the occupation numbers of individual energy levels, that is valid for both non-degenerate and degenerate spectra. To demonstrate the applicability of the theory in the presence of degeneracy, we compute energy level correlations up to fourth order in a bosonic ring in the presence of a magnetic field.