# Bose-Einstein condensation temperature of finite systems

## TL;DR

This paper introduces a novel analytical method using heat kernel expansion and zeta-function regularization to accurately determine Bose-Einstein condensation temperatures in finite systems, addressing divergence issues.

## Contribution

The paper develops a general technique for calculating BEC critical temperatures in finite systems, applicable to various geometries and potential confinements, with explicit correction terms.

## Key findings

- Derived analytical expressions for BEC temperature using heat kernel coefficients
- Exact sums computed for systems with known spectra like slabs and spheres
- Higher-order corrections to critical temperatures for harmonic traps

## Abstract

In the studies of the Bose-Einstein condensation of ideal gases in finite systems, the divergence problem usually arises in the equation of state. In this paper, we present a technique based on the heat kernel expansion and the zeta-function regularization to solve the divergence problem, and obtain the analytical expression of the Bose-Einstein condensation temperature for general finite systems. The result is represented by the heat kernel coefficients, in which the asymptotic energy spectrum of the system is used. Besides the general case, for the systems with exact spectra, e.g., ideal gases in an infinite slab or in a three-sphere, the sums of the spectra can be performed exactly and the calculation of the corrections to the critical temperatures is more direct. For the system confined in a bounded potential, the form of the heat kernel is different from the usual heat kernel expansion. We show that, as long as the asymptotic form of the global heat kernel can be found out, our method also works. For Bose gases confined in three- and two-dimensional isotropic harmonic potentials, we obtain the higher-order corrections to the usual results of the critical temperatures. Our method also can be applied to the problem of the generalized condensation, and we give the correction of the boundary on the second critical temperature in a highly anisotropic slab.