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Upper bound for the grand canonical free energy of the Bose gas in the Gross-Pitaevskii limit

Chiara Boccato, Andreas Deuchert, David Stocker

Abstract

We consider a homogeneous Bose gas in the Gross-Pitaevskii limit at temperatures that are comparable to the critical temperature for Bose-Einstein condensation in the ideal gas. Our main result is an upper bound for the grand canonical free energy in terms of two new contributions: (a) the free energy of the interacting condensate is given in terms of an effective theory describing its particle number fluctuations, (b) the free energy of the thermally excited particles equals that of a temperature-dependent Bogoliubov Hamiltonian.

Upper bound for the grand canonical free energy of the Bose gas in the Gross-Pitaevskii limit

Abstract

We consider a homogeneous Bose gas in the Gross-Pitaevskii limit at temperatures that are comparable to the critical temperature for Bose-Einstein condensation in the ideal gas. Our main result is an upper bound for the grand canonical free energy in terms of two new contributions: (a) the free energy of the interacting condensate is given in terms of an effective theory describing its particle number fluctuations, (b) the free energy of the thermally excited particles equals that of a temperature-dependent Bogoliubov Hamiltonian.
Paper Structure (24 sections, 23 theorems, 240 equations)

This paper contains 24 sections, 23 theorems, 240 equations.

Table of Contents

  1. Introduction and main results
  2. Background and summary
  3. Notation
  4. Fock space and Hamiltonian
  5. Grand canonical free energy, Gibbs state and Gibbs variational principle
  6. The ideal Bose gas on the torus
  7. Main results
  8. Organization of the article
  9. The trial state
  10. Definition of the trial state
  11. Preparatory lemmas
  12. Bound for the energy
  13. Analysis of $\mathcal{G}_\mathcal{V}$
  14. Analysis of $\mathcal{G}_\mathcal{K}$
  15. Analysis of $\mathcal{G}_{\mathcal{K}_{\mathrm{B}}}$
  16. ...and 9 more sections

Key Result

Theorem 1.1

Assume that the function $v : [0,\infty) \to [0,\infty]$ is nonnegative, compactly supported, and satisfies $v(| \cdot |) \in L^3(\Lambda, \,\mathrm{d} x)$. By $\varrho = N/L^3$ we denote the particle density. In the combined limit $N \to \infty$, $\beta = \kappa \beta_{\mathrm{c}}$ with $\kappa \in with $\varrho_0$ in eq:DensityBEC, $F_0^{\mathrm{BEC}}$ in eq:FreeEnergyIdealGasCond, $F^+_0$ in eq

Theorems & Definitions (41)

  • Theorem 1.1
  • Proposition 1.2
  • Corollary 1.3
  • Remark 1.4
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • ...and 31 more