A Lee-Huang-Yang type expansion for the thermodynamic energy density of a dilute mixture of Bose gases

TL;DR

This paper derives a second order thermodynamic energy expansion for a dilute two-species Bose gas, extending known results and connecting to the Lee-Huang-Yang formula, with proofs of Bose-Einstein condensation in the mixture.

Contribution

It provides a rigorous second order energy expansion for a dilute Bose mixture, including the Lee-Huang-Yang formula as a special case, and proves Bose-Einstein condensation in this setting.

Findings

Second order energy expansion for Bose mixture derived

Correct coefficients obtained for soft potentials

Proof of Bose-Einstein condensation in the mixture

Abstract

We consider a dilute gas in 3D composed of two species of bosons interacting through positive inter-species and intra-species pairwise potentials. We prove a second order expansion for the energy density in the thermodynamic limit. For the case of compactly supported, integrable potentials, we derive the correct second order of the expansion. If we make the further assumption of having soft potentials, we also derive the correct coefficient of the second order and the resulting formula is coherent with the physics literature. If we let the density and scattering length of one of the species go to zero, we obtain the Lee-Huang-Yang formula for one species of bosons. The paper also contains a proof of BEC for a mixture of bosons in a box with length scale larger than the Gross-Pitaevskii one.