We study the statistical mechanics and the dynamical relaxation process of modulationally unstable one-dimensional quantum droplets described by a modified Gross–Pitaevskii equation. To determine the classical partition function thereof, we leverage the semi-analytical transfer integral operator (TIO) technique. The latter predicts a distribution of the observed wave-function amplitudes and yields two-point correlation functions providing insights into the emergent dynamics involving quantum droplets. We compare the ensuing TIO results with the probability distributions obtained at large times of the modulationally unstable dynamics as well as with the equilibrium properties of a suitably constructed Langevin dynamics. We find that the instability leads to the spontaneous formation of quantum droplets featuring multiple collisions and which are found to coalesce at large evolution times. Our results from the distinct methodologies are in good agreement aside from the case of low temperatures in the special limit where the droplet widens. In this limit, the distribution acquires a pronounced bimodal character, exhibiting a deviation between the TIO solution and the Langevin dynamics.

中文翻译：

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一维量子液滴的统计力学

我们研究了由修正的 Gross-Pitaevskii 方程描述的调制不稳定一维量子液滴的统计力学和动态弛豫过程。为了确定其经典配分函数，我们利用了半解析传递积分算子 (TIO) 技术。后者预测观察到的波函数振幅的分布并产生两点相关函数，提供对涉及量子液滴的涌现动力学的见解。我们将随后的 TIO 结果与在调制不稳定动力学的大量时间获得的概率分布以及适当构造的朗之万动力学的平衡特性进行比较。我们发现这种不稳定性导致自发形成具有多次碰撞特征的量子液滴，并且发现这些液滴在较大的演化时间内聚结。除了在液滴变宽的特殊限制下的低温情况外，我们从不同方法得到的结果非常一致。在此限制下，分布获得明显的双峰特征，表现出 TIO 解与朗之万动力学之间的偏差。