Isotherms, Order Parameter and Density Profiles for Weakly Interacting Bose Gases within Three Mean-Field Theories

In this work we consider three mean field approximations of standard use in the literature to describe the weakly interacting Bose gas confined in a box of volume V. For these approximations we calculate the corresponding isotherms ??=??(??,T), where ?? is the chemical potential, ?? the particle density and T the absolute temperature. In particular we address this calculation at the nanokelvin regime for a gas parameter $\gamma= \rho a_{s}^{3}\leq 1$ where the Bose-Einstein Condensation in alkali atoms is observed. The non singled value observed for the equation of state ??=??(??,T) suggests strongly that the approximations considered here do not capture properly the thermodynamic behavior in the vicinity of the BEC transition. In order to support this statement we calculate the so-called order parameter ??(T)=N ₀(T)/N from 0??T??T _c where N ₀ represents the number of particles within the condensate, N the total number of particles and T _c the critical temperature. Both results suggest that the three mean-field theories considered here, Hartree-Fock (HF), Popov (P) and Yukalov-Yukalova (Ykv) do not predict a second order phase transition as the BEC transition in weakly interacting gases is expected to show. Using the Local Density Approximation (LDA) we extend these calculation to obtain the density profiles $\rho(\vec{r})$ for an inhomogeneous Bose gas trapped in a harmonic external potential $V_{ext}(\vec{r})$ . As expected the density profiles show that the confinement is not enough to override the anomalies observed in the thermodynamic quantities for the gas confined in a box of volume V.