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The free energy of dilute Bose gases at low temperatures

Florian Haberberger, Christian Hainzl, Phan Thành Nam, Robert Seiringer, Arnaud Triay

Abstract

We consider a low density Bose gas interacting through a repulsive potential in the thermodynamic limit. We justify, as a rigorous lower bound, a Lee--Huang--Yang type formula for the free energy at suitably low temperatures, where the modified excitation spectrum leads to a second order correction of the same order as the Lee--Huang--Yang correction to the ground state energy.

The free energy of dilute Bose gases at low temperatures

Abstract

We consider a low density Bose gas interacting through a repulsive potential in the thermodynamic limit. We justify, as a rigorous lower bound, a Lee--Huang--Yang type formula for the free energy at suitably low temperatures, where the modified excitation spectrum leads to a second order correction of the same order as the Lee--Huang--Yang correction to the ground state energy.
Paper Structure (26 sections, 30 theorems, 428 equations, 2 figures)

This paper contains 26 sections, 30 theorems, 428 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Outline of the proof
  4. Preliminaries
  5. Scattering Problem
  6. Fock Space Formalism
  7. The Excitation Hamiltonian
  8. Symmetrization and Neumann Boundary Conditions
  9. The First Quadratic Transformation
  10. Analysis of (I)1
  11. Analysis of (II)1
  12. Analysis of (III)1
  13. Proof of Lemma \ref{['lem_firsttransform']}
  14. The Cubic Transformation
  15. Actions on ${\rm d}\Gamma(-\Delta)$ and $Q_4$
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $\nu=1/5000$. In the dilute limit $\rho \mathfrak{a}^3 \to 0$, for any $0\le T \le \rho \mathfrak{a} (\rho\mathfrak{a}^3)^{-\nu}$, the free energy density in eq:def_f_rho_T satisfies Here the constant $C > 0$ depends only on $V$.

Figures (2)

  • Figure 1: Relevant mirror points of $x$ shown in two dimensions.
  • Figure 2: The distance is conserved.

Theorems & Definitions (58)

  • Theorem 1.1
  • Theorem 1.2: Free energy on small boxes
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1: Properties of $K$
  • Lemma 3.2
  • proof
  • proof : Proof of Lemma \ref{['lem_K_properties']}
  • ...and 48 more