On the Thermodynamic Limit of Bogoluibov's Theory of Bose Gas

TL;DR

This paper rigorously investigates the thermodynamic limit of Bogoliubov's theory for a dilute Bose gas, employing heat kernel techniques and boundary estimates to understand the behavior as volume approaches infinity.

Contribution

It provides a rigorous analysis of the thermodynamic limit of Bogoliubov's theory using heat kernel methods and boundary estimates, advancing the mathematical understanding of Bose gas models.

Findings

Established bounds on the thermodynamic limit behavior.

Used heat kernel formulation to analyze boundary effects.

Showed the limit can approach the boundary term arbitrarily closely.

Abstract

Assuming that Bogoliubov's theory of weakly interacting dilute Bose gas defines a self-consistent model Hamiltonian, we investigate its thermodynamic limit as we take the volume to infinity. The infinite volume is taken via a sequence of scaled convex regions with piecewise smooth boundary and the volumes staying proportional to the cube of the diameter of the region. To get a strict bound on the behavior of the thermodynamic limit, we use the recent formulation of Bogoliubov's theory of condensation in terms of heat kernels for a given domain as well as an estimate of the difference of traces between the heat kernel with Neumann boundary conditions on this domain and the infinite space result. We cannot control the limiting process by the area term; however, we can come arbitrarily close to it.