# Virial coefficients expressed by heat kernel coefficients

## TL;DR

This paper derives virial coefficients for ideal quantum gases using heat kernel coefficients, revealing boundary and curvature effects on quantum exchange interactions across different dimensions.

## Contribution

It introduces a general method to express virial coefficients via heat kernel coefficients and analyzes boundary and curvature effects in various geometries.

## Key findings

- Boundary effects on virial coefficients are dimension-independent and enhance quantum exchange.
- Curvature influences quantum exchange differently in spheres: enhances in 2D, weakens in higher dimensions.
- Provides explicit virial coefficients for gases in confined spaces and spherical geometries.

## Abstract

In this paper, we generally expressed the virial expansion of ideal quantum gases by the heat kernel coefficients for the corresponding Laplace type operator. As examples, we give the virial coefficients for quantum gases in $d$-dimensional confined space and spheres, respectively. Our results show that, the relative correction from the boundary to the second virial coefficient is independent of the dimension and it always enhances the quantum exchange interaction. In $d$-dimensional spheres, however, the influence of the curvature enhances the quantum exchange interaction in two dimensions, but weakens it in higher dimensions ($d>3$).