Bose-Einstein Condensation on a Manifold with Nonnegative Ricci Curvature

TL;DR

This paper investigates Bose-Einstein condensation on manifolds with nonnegative Ricci curvature, deriving bounds and analyzing ground state properties using heat kernel and eigenvalue estimates, including effects of interactions and relativistic considerations.

Contribution

It introduces heat kernel and eigenvalue techniques to analyze Bose-Einstein condensation on curved manifolds, including bounds for depletion and ground state analysis.

Findings

Derived bounds for depletion coefficient in nonrelativistic gases

Analyzed ground state energy and finite size effects

Extended analysis to relativistic ideal gases

Abstract

The Bose-Einstein condensation for an ideal Bose gas and for a dilute weakly interacting Bose gas in a manifold with nonnegative Ricci curvature is investigated using the heat kernel and eigenvalue estimates of the Laplace operator. The main focus is on the nonrelativistic gas. However, special relativistic ideal gas is also discussed. The thermodynamic limit of the heat kernel and eigenvalue estimates is taken and the results are used to derive bounds for the depletion coefficient. In the case of a weakly interacting gas Bogoliubov approximation is employed. The ground state is analyzed using heat kernel methods and finite size effects on the ground state energy are proposed. The justification of the c-number substitution on a manifold is given.