Hydrodynamics of the atomic Bose-Einstein condensate beyond the mean-field approximation: a mini-review

Abstract: Several hydrodynamic models the atomic Bose-Einstein condensate beyond the mean-field approximation are discussed together from one point of view. All these models are derived from microscopic quantum description. The derivation is made within the many-particle quantum hydrodynamics method suggested by L. Kuz'menkov. The derivation is demonstrated and discussed for the mean-field regime revealing the Gross-Pitaevskii equation as the simplest illustration. It appears in the first order by the interaction radius. Generalization of the hydrodynamic Euler equation obtained in the third order by the interaction radius are discussed. It includes the contribution of the isotropic short-range interaction presented by the third space derivative of the square of concentration. The Euler equation also includes the contribution of the anisotropic part of the short-range interaction proportional to the second order spherical function. Systematic account of the quantum fluctuations in terms of the many-particle quantum hydrodynamics method requires the extension of the set of hydrodynamic equations from the couple continuity and Euler equations to the set of four equations which also includes the pressure evolution equation and the evolution equation for the third rank analog of pressure. The pressure evolution equation contains no interaction contribution in the first order by the interaction radius. The source of the quantum fluctuations is in the interaction caused term in the third rank tensor evolution equation which is obtained in the first order by the interaction radius.