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A new upper bound on the specific free energy of dilute Bose gases

Giulia Basti, Chiara Boccato, Serena Cenatiempo, Andreas Deuchert

TL;DR

The paper proves a sharp finite-temperature upper bound for the specific free energy of a dilute 3D Bose gas by constructing a sophisticated correlated trial state in the bosonic Fock space. The authors combine a grand-canonical uncorrelated baseline with a carefully controlled, unitary-like correlation dressing that implements Bijl–Dingle–Jastrow-type pair correlations, including precise momentum-space cutoffs and a truncated exponential dressing to maintain tractable entropy. They derive explicit bounds for kinetic and interaction energies, isolating the Lee–Huang–Yang–type correction term proportional to $4\pi \mathfrak{a} (2\varrho^2 - [\varrho - \varrho_c(\beta)]_+^2)$ and bounding the remainder with powers of the diluteness $(\varrho \mathfrak{a}^3)$, achieving a rate $(\varrho \mathfrak{a}^3)^{1/7}$ in the error. This work extends previous upper-bound results to a broader class of repulsive potentials, provides a transparent energy-entropy balance, and offers a framework adaptable to extensions in finite volumes and temperature regimes.

Abstract

We prove an upper bound for the free energy (per unit volume) of the dilute Bose gas in the thermodynamic limit, showing that the free energy at density $ρ$ and inverse temperature $β$ differs from that of the non-interacting system by the correction term $4 π\frak{a} (2 ρ^2 - [ρ- ρ_{\textsf{c}}(β)]^2_+ )$. Here, $\frak{a}$ denotes the scattering length of the interaction potential, $ρ_{\textsf{c}}(β)$ the critical density for Bose-Einstein condensation of the non-interacting gas and $[\cdot]_+=\max\{0,\cdot\}$. This result was previously established by Yin in [37]. Our proof applies to a broader class of interaction potentials, yields a better rate, and we believe it has potential for further extensions.

A new upper bound on the specific free energy of dilute Bose gases

TL;DR

The paper proves a sharp finite-temperature upper bound for the specific free energy of a dilute 3D Bose gas by constructing a sophisticated correlated trial state in the bosonic Fock space. The authors combine a grand-canonical uncorrelated baseline with a carefully controlled, unitary-like correlation dressing that implements Bijl–Dingle–Jastrow-type pair correlations, including precise momentum-space cutoffs and a truncated exponential dressing to maintain tractable entropy. They derive explicit bounds for kinetic and interaction energies, isolating the Lee–Huang–Yang–type correction term proportional to and bounding the remainder with powers of the diluteness , achieving a rate in the error. This work extends previous upper-bound results to a broader class of repulsive potentials, provides a transparent energy-entropy balance, and offers a framework adaptable to extensions in finite volumes and temperature regimes.

Abstract

We prove an upper bound for the free energy (per unit volume) of the dilute Bose gas in the thermodynamic limit, showing that the free energy at density and inverse temperature differs from that of the non-interacting system by the correction term . Here, denotes the scattering length of the interaction potential, the critical density for Bose-Einstein condensation of the non-interacting gas and . This result was previously established by Yin in [37]. Our proof applies to a broader class of interaction potentials, yields a better rate, and we believe it has potential for further extensions.

Paper Structure

This paper contains 24 sections, 15 theorems, 317 equations, 2 figures.

Table of Contents

  1. Introduction and main result
  2. Setting
  3. Localization to smaller boxes and change of boundary conditions
  4. Upper bound for free energy in finite volume
  5. The trial state
  6. The uncorrelated state
  7. The correlation structure
  8. Computation of the norm of $\mathcal{D}_{\alpha} g_\alpha^0$
  9. Free energy of the correlated state
  10. Kinetic energy of the trial state.
  11. Interaction energy of the trial state
  12. Contribution of $\mathcal{V}_0$ on the correlated state
  13. Contribution of $\mathcal{V}_2^{(a)}$ on the correlated state
  14. Contribution of $\mathcal{V}_2^{(b)}$ on the correlated state
  15. Contribution of $\mathcal{V}_4$ on the correlated state
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $V : [0,\infty) \to [0,\infty]$ be a measurable, compactly supported function, which satisfies $V(|\cdot|) \in L^3(\mathbb{R}^3)$ and has scattering length $\mathfrak{a}$. Let $f_0(\beta, \varrho)$ and $\varrho_c(\beta)$ denote the specific free energy and critical density of an ideal Bose gas, for $\varrho \mathfrak{a}^3$ small enough and $\beta \varrho^{2/3}\geq c$.

Figures (2)

  • Figure 1: Schematic picture of possible contractions among the quartic operators $\{A_{r_i, v_i, v'_i} \}_{i=1,...,n}$ (upper set of points) and $\{A^*_{{\tilde{r}}_i, {\tilde{v}}_i, {\tilde{v}}_i'} \}_{i=1,...,n}$ (lower set of points). Each dot (resp. square) represents a creation/annihilation operator with momenta in $P_{\mathrm{H}}$ (resp. in $P_{\beta}^0$); each line connecting two dots represents a delta function between the momenta labeling the dots. The contractions depicted by the dashed and dotted lines above are forbidden due the restrictions encoded in \ref{['eq:restrictions1']} and \ref{['eq:restrictions2']}; differently, the contractions depicted by the solid lines are allowed and imply $v_n+v'_n= {\tilde{v}}_3 + {\tilde{v}}'_3$.
  • Figure 2: Schematic picture of a non vanishing contraction among the operators on the right-hand side of \ref{['eq:cV0-start']}. Upper (resp. lower) set of points represents annihilation (resp. creation) operators. Each line connecting two points represents a delta function between the moment labeling them. Dashed boxes represent the conditions $v_\ell + v'_{\ell}={\tilde{v}}_\ell + {\tilde{v}}'_{\ell}$ with $\ell=1, \ldots, n$ originating from the contraction of the momenta in $P_{\mathrm{H}}$.

Theorems & Definitions (24)

  • Theorem 1.1
  • Proposition 2.1: Localization to finite size periodic boxes
  • Proposition 2.2: Trial state on a finite size periodic box
  • proof : Proof of Theorem \ref{['thm:main']}.
  • Proposition 2.3
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 4.1
  • ...and 14 more