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Quantum Depletion and Momentum Distribution in the Dilute Hard-Sphere Bose Gas

OpenAI

Published Oct 6, 2026• 0 citations
Publisher

Abstract

We prove the full scaled Bogoliubov momentum distribution for the depleted particles in the three-dimensional hard-sphere Bose gas at zero temperature. The thermodynamic limit is taken at each fixed density before the dilute limit, uniformly over all pure and mixed ground states. All thermodynamic accumulation values have the same dilute asymptotic: the limiting nonzero-momentum occupation measure has total mass 8/(3\sqrt\pi), giving the depletion fraction \frac{8}{3\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt{\rho a^3}) for density ρ and hard-sphere exclusion distance a.

DOI
10.83295/byc3-89bd

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We prove the full scaled Bogoliubov momentum distribution for the depleted particles in the three-dimensional hard-sphere Bose gas at zero temperature. The thermodynamic limit is taken at each fixed density before the dilute limit, uniformly over all pure and mixed ground states. All thermodynamic accumulation values have the same dilute asymptotic: the limiting nonzero-momentum occupation measure has total mass 8/(3\sqrt\pi), giving the depletion fraction \frac{8}{3\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt{\rho a^3}) for density ρ and hard-sphere exclusion distance a.
We prove the Bogoliubov quantum-depletion asymptotic for the ground state of a three-dimensional Bose gas with a fixed bounded, nonnegative, radial interaction of finite range and positive scattering length a. The thermodynamic limit is taken at each fixed density ρ before the dilute limit. Every thermodynamic accumulation value of the fraction outside the constant mode is \frac{8}{3\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt{\rho a^3}) as \rho\downarrow0. The assertion is uniform over ground-state density matrices and does not require the occupation to have a unique thermodynamic limit.
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.
The low-density expansions for the energy, chemical potential, and condensate depletion of the homogeneous dilute dipolar Bose gas are obtained by regularizing the dipole-dipole interaction at long distances. It is shown that the leading term, proportional to the density, allows a simple physical interpretation and consistently describes the thermodynamic stability of the system. The long-range asymptotics are obtained analytically for the normal and anomalous one-particle correlation functions and the pair distribution function. We discuss the properties of the two-body scattering with zero relative momentum for the dipole-dipole interaction; in particular, we derive the asymptotics of the wave function and a correction to the scattering length for small values of the dipolar range. We show how the density expansions can be derived within the Bogoliubov model of weakly interacting particles without any divergence from the assumption of universality of the expansions at low densities.
We consider a dilute hard-sphere Bose gas in random external potentials at low temperatures, in D=3, using the technique of pseudopotentials and the Bogoliubov transformation. At absolute zero, the random potentials can deplete the Bose condensate, though not completely. On the other hand, they generate an amount of normal fluid equal to 4/3 the condensate depletion. This is a localization effect that can destroy superfluidity at absolute zero. General features of the superfluid density in the neighborhood of this transition point agree qualitatively with experimental results on helium in porous media.