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

Levent Akant, Ebru Dogan, Emine Ertugrul, O. Teoman Turgut

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.

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

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.

Paper Structure

This paper contains 18 sections, 147 equations.

Table of Contents

  1. Introduction
  2. Ground State Energy
  3. A geometric bound for distance integrals.
  4. Depletion Coefficient
  5. First Euler-Maclaurin term.
  6. Second Euler--Maclaurin term.
  7. Third Euler--Maclaurin term.
  8. First Euler--Maclaurin term.
  9. Second Euler--Maclaurin term.
  10. Third Euler--Maclaurin term.
  11. Estimate of $|\Delta\mu_{(2)}|$.
  12. Estimate of $|\Delta\mu_{(3)}|$.
  13. Conclusions
  14. Bounds for the modified Bessel function $I_{0}$
  15. Estimate of the first contribution.
  16. ...and 3 more sections