How does geometry affect quantum gases?

TL;DR

This paper investigates how the shape of confining geometries influences the thermodynamic properties of quantum gases, including noninteracting and interacting cases, with potential applications to Bose-Einstein condensates and helium dimers.

Contribution

It introduces a comprehensive analysis of quantum gases in various geometries, including numerical and analytical methods for noninteracting and interacting cases, expanding understanding of geometric effects.

Findings

Bosonic gases have higher entropy and internal energy than fermionic gases regardless of shape.

Numerical calculations for noninteracting gases in different geometries show shape-dependent thermodynamic variations.

An analytical model for interacting gases is developed and applied to a cubic box.

Abstract

In this work, we study the thermodynamic functions of quantum gases confined to spaces of various shapes, namely, a sphere, a cylinder, and an ellipsoid. We start with the simplest situation, namely, a spinless gas treated within the canonical ensemble framework. As a next step, we consider \textit{noninteracting} gases (fermions and bosons) with the usage of the grand canonical ensemble description. For this case, the calculations are performed numerically. We also observe that our results may possibly be applied to \textit{Bose-Einstein condensate} and to \textit{helium dimer}. Moreover, the bosonic sector, independently of the geometry, acquires entropy and internal energy greater than for the fermionic case. Finally, we also devise a model allowing us to perform analytically the calculations in the case of \textit{interacting} quantum gases, and, afterwards, we apply it to a cubical box.