Bose-Einstein condensation of finite number of confined particles

TL;DR

This paper develops an iterative method to calculate the thermodynamic properties of a finite bosonic system in various traps, analyzing how Bose-Einstein condensation manifests as a cusp in specific heat for different particle numbers and geometries.

Contribution

It introduces a novel iterative scheme for calculating the partition function of finite bosonic systems in external potentials, bridging finite and large particle number regimes.

Findings

The iterative scheme accurately predicts specific heat and condensation features for finite bosons.

Results converge rapidly to the large particle number limit, matching semiclassical theory.

The specific heat cusp indicates Bose-Einstein condensation across different geometries.

Abstract

The partition function and specific heat of a system consisting of a finite number of bosons confined in an external potential are calculated in canonical ensemble. Using the grand partition function as the generating function of the partition function, an iterative scheme is established for the calculation of the partition function of system with an arbitrary number of particles. The scheme is applied to finite number of bosons confined in isotropic and anisotropic parabolic traps and in rigid boxes. The specific heat as a function of temperature is studied in detail for different number of particles, different degrees of anisotropy, and different spatial dimensions. The cusp in the specific heat is taken as an indication of Bose-Einstein condensation (BEC).It is found that the results corresponding to a large number of particles are approached quite rapidly as the number of bosons in the system increases. For large number of particles, results obtained within our iterative scheme are consistent with those of the semiclassical theory of BEC in an external potential based on the grand canonical treatment.