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Archive for Rational Mechanics and Analysis

Upper Bound for the Ground State Energy of a Dilute Bose Gas of Hard Spheres

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

Published Oct 21, 2024• 9 citations
Publisher

Abstract

Abstract We consider a gas of bosons interacting through a hard-sphere potential with radius $$\mathfrak {a}$$ 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 $$4\pi \rho \mathfrak {a}$$ 4 π ρ a and shows that corrections are smaller than $$C \rho \mathfrak {a} (\rho {{\mathfrak {a}}}^3)^{1/2}$$ C ρ a ( ρ a 3 ) 1 / 2 , for a sufficiently large constant $$C > 0$$ 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 $$\rho \mathfrak {a}(\rho {{\mathfrak {a}}}^3)^{1/2}$$ ρ a ( ρ a 3 ) 1 / 2 , in agreement with the Lee–Huang–Yang prediction.

DOI
10.1007/s00205-024-02049-w

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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 quantum gas consisting of NNN hard spheres with radius a>0\frak{a} > 0a>0, obeying bosonic statistics and moving in the box Λ=[0;L]3Λ= [0;L]^3Λ=[0;L]3 with periodic boundary conditions. We are interested in the ground state energy per unit volume in the thermodynamic limit, with N,L→∞N, L \to \inftyN,L→∞ at fixed density ρ=N/L3ρ= N / L^3ρ=N/L3. We derive an upper bound for the ground state energy density, matching the famous Lee-Huang-Yang formula, up to lower order terms, in the dilute limit ρa3≪1ρ\frak{a}^3 \ll 1ρa3≪1.
Abstract We establish an upper bound for the ground state energy per unit volume of a dilute Bose gas in the thermodynamic limit, capturing the correct second-order term, as predicted by the Lee–Huang–Yang formula. This result was first established in [20] by H.-T. Yau and J. Yin. Our proof, which applies to repulsive and compactly supported V∈L3(R3)V \in L^3 (\mathbb {R}^3)V∈L3(R3) , gives better rates and, in our opinion, is substantially simpler.
Abstract We prove the following formula for the ground state energy density of a dilute Bose gas with density ρ\rho ρ ρ in 2 dimensions in the thermodynamic limit e2D(ρ)=4πρ2Y(1−Y∣log⁡Y∣+(2Γ+12+log⁡(π))Y)+o(ρ2Y2),\begin{aligned} e^{\text {2D}}(\rho ) = 4\pi \rho ^2 Y\Big (1 - Y \vert \log Y \vert + \Big ( 2\Gamma + \frac{1}{2} + \log (\pi ) \Big ) Y \Big ) + o(\rho ^2 Y^{2}), \end{aligned}e2D(ρ)=4πρ2Y(1−Y∣logY∣+(2Γ+21​+log(π))Y)+o(ρ2Y2),​ e 2D ( ρ ) = 4 π ρ 2 Y ( 1 - Y | log Y | + ( 2 Γ + 1 2 + log ( π ) ) Y ) + o ( ρ 2 Y 2 ) , as ρa2→0\rho a^2 \rightarrow 0ρa2→0 ρ a 2 → 0 . Here Y=∣log⁡(ρa2)∣−1Y= |\log (\rho a^2)|^{-1}Y=∣log(ρa2)∣−1 Y = | log ( ρ a 2 ) | - 1 and a is the scattering length of the two-body potential. This result in 2 dimensions corresponds to the famous Lee–Huang–Yang formula in 3 dimensions. The proof is valid for essentially all positive potentials with finite scattering length, in particular, it covers the crucial case of the hard core potential.
We consider a gas of bosons interacting through a three-body hard-core potential in the thermodynamic limit. We derive an upper bound on the ground state energy of the system at the leading order using a Jastrow factor. Our result matches the lower bound proven by Nam-Ricaud-Triay (J Math Phys 63:071903, 2022) and therefore resolves the leading order. Moreover, a straightforward adaptation of our proof can be used for systems interacting via combined two-body and three-body interactions to generalise Theorem 1.2 from (Ann. Henri Poincaré, 2026) to hard-core potentials.