Several hydrodynamic models of the atomic Bose–Einstein condensate (BEC) obtained beyond the mean-field approximation are discussed together from a single point of view. All these models are derived from the 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 is discussed. It includes the contribution of the isotropic short-range interaction (SRI) presented by the third space derivative of the square of concentration. The Euler equation also includes the contribution of the anisotropic part of the SRI proportional to the second order spherical function. A 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 tensor of the pressure flux. 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 existing in the third rank tensor evolution equation which is obtained in the first order by the interaction radius. The presented models are considered for the single-component BECs.

从单一的角度一起讨论了超出平均场近似值的原子玻色-爱因斯坦凝聚 (BEC) 的几种流体动力学模型。所有这些模型都来自微观量子描述。该推导是在 L Kuz'menkov 建议的多粒子量子流体动力学方法中进行的。为揭示 Gross-Pitaevskii 方程的平均场机制演示和讨论了推导，作为最简单的说明。它按交互半径出现在第一顺序。讨论了由相互作用半径在三阶获得的流体动力学欧拉方程的推广。它包括由浓度平方的三次空间导数表示的各向同性短程相互作用 (SRI) 的贡献。欧拉方程还包括与二阶球函数成比例的 SRI 各向异性部分的贡献。用多粒子量子流体动力学方法系统地解释量子涨落需要将流体动力学方程组从偶连续性和欧拉方程扩展到四个方程组，其中还包括压力演化方程和演化方程为压力通量的三阶张量。压力演化方程不包含相互作用半径的一阶相互作用贡献。量子涨落的根源在于相互作用半径一阶求得的三阶张量演化方程中存在的相互作用引起项。