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arXiv:2203.11917 · Communications in Mathematical Physics

A Second Order Upper Bound for the Ground State Energy of a Hard-Sphere Gas in the Gross–Pitaevskii Regime

Giulia Basti, Serena Cenatiempo, Alessandro Olgiati, Giulio Pasqualetti, Benjamin Schlein

Published Dec 5, 2022• 18 citations
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Abstract

Abstract We prove an upper bound for the ground state energy of a Bose gas consisting of N hard spheres with radius $$\mathfrak {a}/N$$ a / N , moving in the three-dimensional unit torus $$\Lambda $$ Λ . Our estimate captures the correct asymptotics of the ground state energy, up to errors that vanish in the limit $$N \rightarrow \infty $$ N → ∞ . The proof is based on the construction of an appropriate trial state, given by the product of a Jastrow factor (describing two-particle correlations on short scales) and of a wave function constructed through a (generalized) Bogoliubov transformation, generating orthogonal excitations of the Bose–Einstein condensate and describing correlations on large scales.

DOI
10.1007/s00220-022-04547-y

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Abstract We prove an upper bound for the ground state energy of a Bose gas consisting of N hard spheres with radius a/N\mathfrak {a}/Na/N a / N , moving in the three-dimensional unit torus Λ\Lambda Λ Λ . Our estimate captures the correct asymptotics of the ground state energy, up to errors that vanish in the limit N→∞N \rightarrow \infty N→∞ N → ∞ . The proof is based on the construction of an appropriate trial state, given by the product of a Jastrow factor (describing two-particle correlations on short scales) and of a wave function constructed through a (generalized) Bogoliubov transformation, generating orthogonal excitations of the Bose–Einstein condensate and describing correlations on large scales.
We review some rigorous estimates for the ground state energy of dilute Bose gases. We start with Dyson’s upper bound, which provides the correct leading order asymptotics for hard spheres. Afterward, we discuss a rigorous version of Bogoliubov theory, which recently led to an estimate for the ground state energy in the Gross–Pitaevskii regime, valid up to second order, for particles interacting through integrable potentials. Finally, we explain how these ideas can be combined to establish a new upper bound, valid to second order, for the energy of hard spheres in the Gross–Pitaevskii limit. Here, we only sketch the main ideas; details will appear elsewhere.
Abstract We consider a gas of bosons interacting through a hard-sphere potential with radius a\mathfrak {a}a a in the thermodynamic limit. We derive an upper bound for the ground state energy per particle at low density. Our bound captures the leading term 4πρa4\pi \rho \mathfrak {a}4πρa 4 π ρ a and shows that corrections are smaller than Cρa(ρa3)1/2C \rho \mathfrak {a} (\rho {{\mathfrak {a}}}^3)^{1/2}Cρa(ρa3)1/2 C ρ a ( ρ a 3 ) 1 / 2 , for a sufficiently large constant C>0C > 0C>0 C > 0 . In combination with a known lower bound, our result implies that the first sub-leading term to the ground state energy of a dilute gas of hard spheres is, in fact, of the order ρa(ρa3)1/2\rho \mathfrak {a}(\rho {{\mathfrak {a}}}^3)^{1/2}ρa(ρa3)1/2 ρ a ( ρ a 3 ) 1 / 2 , in agreement with the Lee–Huang–Yang prediction.
We consider a gas of bosons interacting through a hard-sphere potential with radius a\frak{a}a in the thermodynamic limit. We derive a simple upper bound for the ground state energy per particle at low density. Our bound captures the leading term 4πρa4πρ\frak{a}4πρa and shows that corrections are smaller than Cρa(ρa3)1/2C ρ\frak{a} (ρ\frak{a}^3)^{1/2}Cρa(ρa3)1/2, for a sufficiently large constant C>0C > 0C>0. In combination with a known lower bound, our result implies that the first sub-leading term to the ground state energy is, in fact, of the order ρa(ρa3)1/2ρ\frak{a} (ρ\frak{a}^3)^{1/2}ρa(ρa3)1/2, in agreement with the Lee-Huang-Yang prediction.
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 Boccato et al. (SIAM J Math Anal 56(2):2611-2660, 2024) 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 Boccato et al. (SIAM J Math Anal 56(2):2611-2660, 2024): 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.
We consider a system of N bosons in a unitary box in the grand-canonical setting interacting through a potential with the scattering length scaling as N−1+κ, κ ∈ (0, 2/3). This regimes interpolate between the Gross–Pitaevskii regime (κ = 0) and the thermodynamic limit (κ = 2/3). In the work of Basti et al. [Forum Math., Sigma 9, E74 (2021)], as an intermediate step to prove an upper bound in agreement with the Lee–Huang–Yang formula in the thermodynamic limit, a second order upper bound on the ground state energy for κ < 5/9 was obtained. In this paper, thanks to a more careful analysis of the error terms, we extend the mentioned result to κ < 7/12.
Abstract We consider a system of N bosons, in the two-dimensional unit torus. We assume particles to interact through a repulsive two-body potential, with a scattering length that is exponentially small in N (Gross–Pitaevskii regime). In this setting, we establish the validity of the predictions of Bogoliubov theory, determining the ground state energy of the Hamilton operator and its low-energy excitation spectrum, up to errors that vanish in the limit N→∞N \rightarrow \infty N→∞ N → ∞ .