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

Marco Caporaletti, Andreas Deuchert

TL;DR

The paper proves an upper bound on the grand canonical free energy of a homogeneous Bose gas in the Gross–Pitaevskii limit for general interaction potentials, including hard cores. It extends previous results by employing a full Jastrow factor to model microscopic correlations and leveraging a cancellation between short-range interactions and long-wavelength thermal fluctuations, together with a temperature-dependent Bogoliubov description of thermal excitations. The main contribution is a bound that captures a condensate–fluctuation free energy term and a Bogoliubov correction for the thermal cloud, valid beyond $L^3$-potentials. This advances our rigorous understanding of finite-temperature Bose gases in the GP regime and provides a framework applicable to physically realistic interactions.

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. Recently, an upper bound for the grand canonical free energy was proved in arXiv:2305.19173 [math-ph] capturing two novel contributions. First, the free energy of the interacting condensate is given in terms of an effective theory describing the probability distribution of the number of condensed particles. Second, the free energy of the thermally excited particles equals that of a temperature-dependent Bogoliubov Hamiltonian. We extend this result to a more general class of interaction potentials, including interactions with a hard core. Our proof follows a different approach than the one in arXiv:2305.19173 [math-ph]: we model microscopic correlations between the particles by a Jastrow factor, and exploit a cancellation in the computation of the energy that emerges due to the different length scales in the system.

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

TL;DR

The paper proves an upper bound on the grand canonical free energy of a homogeneous Bose gas in the Gross–Pitaevskii limit for general interaction potentials, including hard cores. It extends previous results by employing a full Jastrow factor to model microscopic correlations and leveraging a cancellation between short-range interactions and long-wavelength thermal fluctuations, together with a temperature-dependent Bogoliubov description of thermal excitations. The main contribution is a bound that captures a condensate–fluctuation free energy term and a Bogoliubov correction for the thermal cloud, valid beyond -potentials. This advances our rigorous understanding of finite-temperature Bose gases in the GP regime and provides a framework applicable to physically realistic interactions.

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. Recently, an upper bound for the grand canonical free energy was proved in arXiv:2305.19173 [math-ph] capturing two novel contributions. First, the free energy of the interacting condensate is given in terms of an effective theory describing the probability distribution of the number of condensed particles. Second, the free energy of the thermally excited particles equals that of a temperature-dependent Bogoliubov Hamiltonian. We extend this result to a more general class of interaction potentials, including interactions with a hard core. Our proof follows a different approach than the one in arXiv:2305.19173 [math-ph]: we model microscopic correlations between the particles by a Jastrow factor, and exploit a cancellation in the computation of the energy that emerges due to the different length scales in the system.
Paper Structure (24 sections, 28 theorems, 283 equations)

This paper contains 24 sections, 28 theorems, 283 equations.

Table of Contents

  1. Introduction and main result
  2. Background and summary
  3. Notation
  4. The grand canonical free energy and the Gibbs variational principle
  5. The ideal Bose gas on the torus
  6. Main results
  7. Outline of the article
  8. The trial state
  9. Second quantization
  10. Definition of the trial state
  11. Properties of the trial state
  12. The Bogoliubov and Weyl transformations
  13. The Bogoliubov Gibbs state in the diagonal representation
  14. The Bogoliubov Gibbs state in the original representation
  15. The uncorrelated trial state
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $V:\mathbb R^3 \to [0,\infty]$ be measurable, spherically symmetric and compactly supported. In the limit $N\to\infty$, with $\beta = \kappa \beta_{\mathrm{c}}$, $\kappa\in(0,\infty)$ and $\beta_{\mathrm{c}}$ in eq:Tc_ideal, the free energy in eq:Interacting_Free_Energy satisfies with $N_0$, $F_0^{\mathrm{BEC}}$ and $F_0^+$ defined respectively in eq:n0_def, eq:free_energy_ideal_condensate an

Theorems & Definitions (49)

  • Theorem 1.1
  • Proposition 1.2
  • Corollary 1.3
  • Remark 1.4
  • Lemma 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • ...and 39 more