The theoretical description of non-equilibrium Bose–Einstein condensate (BEC) is one of the main challenges in modern statistical physics and kinetics. The non-equilibrium nature of BEC makes it impossible to employ the well-established formalism of statistical mechanics. We develop a framework for the analytical description of a non-equilibrium phase transition to BEC that, in contrast to previously developed approaches, takes into account the infinite number of continuously distributed states. We consider the limit of fast thermalization and obtain an analytical expression for the full density matrix of a non-equilibrium ideal BEC which also covers the equilibrium case. For the particular cases of 2D and 3D, we investigate the non-equilibrium formation of BEC by finding the temperature dependence of the ground state occupation and second-order coherence function. We show that for a given pumping rate, the macroscopic occupation of the ground state and buildup of coherence may occur at different temperatures. Moreover, the buildup of coherence strongly depends on the pumping scheme. We also investigate the condensate linewidth and show that the Schawlow–Townes law holds for BEC in 3D and does not hold for BEC in 2D.

中文翻译：

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玻色-爱因斯坦凝聚态非平衡相变的分析框架

非平衡玻色-爱因斯坦凝聚体（BEC）的理论描述是现代统计物理学和动力学的主要挑战之一。BEC 的非平衡性质使得不可能采用已确立的统计力学形式。我们开发了一个框架，用于分析描述向 BEC 的非平衡相变，与以前开发的方法相比，该框架考虑了无限数量的连续分布状态。我们考虑了快速热化的极限，并获得了非平衡理想 BEC 的全密度矩阵的解析表达式，该表达式也涵盖了平衡情况。对于 2D 和 3D 的特殊情况，我们通过发现基态占据和二阶相干函数的温度依赖性来研究 BEC 的非平衡形成。我们表明，对于给定的泵送速率，基态的宏观占据和相干性的建立可能发生在不同的温度下。此外，相干性的建立很大程度上取决于泵送方案。我们还研究了凝聚线宽，并表明 Schawlow-Townes 定律适用于 3D 中的 BEC，而不适用于 2D 中的 BEC。