A Canonical Ensemble Approach to the Fermion/Boson Random Point Processes and its Applications

TL;DR

This paper develops a unified canonical ensemble framework for boson and fermion point processes, connecting quantum statistical mechanics with random point processes and exploring applications to para-particles and zero-temperature systems.

Contribution

It introduces a canonical ensemble approach to derive boson and fermion point processes from quantum mechanics, linking finite particle systems to grand canonical ensembles.

Findings

Derived boson and fermion point processes via thermodynamic limit.

Extended framework to para-particles of order two.

Analyzed zero-temperature composite particle systems as determinantal processes.

Abstract

We introduce the boson and the fermion point processes from the elementary quantum mechanical point of view. That is, we consider quantum statistical mechanics of canonical ensemble for a fixed number of particles which obey Bose-Einstein, Fermi-Dirac statistics, respectively, in a finite volume. Focusing on the distribution of positions of the particles, we have point processes of the fixed number of points in a bounded domain. By taking the thermodynamic limit such that the particle density converges to a finite value, the boson/fermion processes are obtained. This argument is a realization of the equivalence of ensembles, since resulting processes are considered to describe a grand canonical ensemble of points. Random point processes corresponding to para-particles of order two are discussed as an application of the formulation. A statistics of a system of composite particles at zero temperature are also considered as a model of determinantal random point processes.