[论文解读] Topological Defects and Flows in BECs and Active Matter

Jonas Rønning|arXiv (Cornell University)|Jan 1, 2023

Micro and Nano Robotics参考文献 40被引用 13

一句话总结

本文提出了一套关于活性向列膜中孤立±1/2拓扑缺陷周围活性流动的综合性流体动力学理论，同时考虑了粘性耗散（η）和基底摩擦（Γ）。推导了+1/2缺陷的自推进速度关于系统尺寸R和流体动力学耗散长度ℓd = √(η/Γ)的表达式，表明当R < ℓd时速度随R增大而增加，而当R ≫ ℓd时趋于恒定值，从而弥合了干态与湿态活性物质体系之间的差距。

ABSTRACT

We study the nucleation and dynamics of topological defects in two-dimensional superfluid Bose-Einstein condensates and active liquid crystals. Using the property that these are emergent states of matter with broken rotational symmetry, we formulate a generic mathematical framework that we use to describe the properties of the corresponding topological defects. The active liquid crystals consist of micro-organisms that have an intrinsic activity which is injecting energy into the system. When the intrinsic energy production is large enough, it will result in the spontaneous creation of topological defects. These defects are localized sources of long-range elastic distortions which generate large-scale flows. We are able to solve the flow equations for isolated defects in the limit of point-like defects with an idealized far-field structure that is subject to both friction and viscous dissipation. The induced flow feeds back into the evolution equation for the order parameter of the liquid crystal and has effects on the motion of the defects by making them self-propelled and by mediating effective interactions between them. In contrast, the Bose-Einstein condensate is a passive system where energy is injected by externally applied potentials. One way to create defects is by stirring the condensate with a moving potential. Quantum vortices are then nucleated in pairs and shed from the stirring potential. We show how the defect nucleation and motion are determined by the evolution of the superfluid wave function. In this thesis, we demonstrate that even though the energy is injected and transported differently in these two systems, there are similarities in the fundamental mechanisms for the nucleation of topological defects and in the correspondence between defect kinematics and the evolution of the order parameter.

研究动机与目标

• 理解粘性耗散与基底摩擦如何共同调控活性向列膜中拓扑缺陷周围的活性流动。
• 解决活性缺陷动力学中过阻尼（干态）与流体动力学（湿态）极限之间的差距。
• 推导+1/2缺陷的自推进速度关于系统尺寸R和流体动力学耗散长度ℓd = √(η/Γ)的函数关系。
• 为孤立缺陷周围的活性流动提供一个统一的理论框架，涵盖剪切涡度与压驱动分量。

提出的方法

• 利用具有活性应力σa_ij = α0Qij的流体动力学模型和不可压缩的纳维-斯托克斯方程，推导孤立±1/2缺陷周围的活性流动场。
• 在极坐标系中使用贝塞尔函数和修正贝塞尔函数（K0）求解线性化流体动力学方程，以描述流动的衰减行为。
• 引入流体动力学耗散长度ℓd = √(η/Γ)作为屏蔽长度，控制涡度与流动场的空间衰减。
• 通过复平面上的围线积分计算活性流动速度，使用钥匙孔围线和留数定理处理奇异积分。
• 利用Pochhammer符号和超几何函数表达并简化流动积分的矩。
• 应用最陡下降法与渐近展开，评估自推进速度在长距离与短距离行为。

实验结果

研究问题

• RQ1 粘性耗散与摩擦阻尼之间的相互作用如何影响活性向列膜中孤立拓扑缺陷周围流动结构？
• RQ2 +1/2缺陷的自推进速度如何依赖于系统尺寸R和流体动力学耗散长度ℓd？
• RQ3 流动场在空间中如何衰减，ℓd在屏蔽长程流体动力学流动中起到何种作用？
• RQ4 +1/2缺陷为何自推进而−1/2缺陷保持被动？这与活性应力的对称性有何关系？
• RQ5 能否构建一个统一理论，实现对活性缺陷运动在过阻尼（干态）与流体动力学（湿态）极限之间的插值？

主要发现

• +1/2缺陷的自推进速度在R < ℓd时随系统尺寸R增大而增加，表明在小系统中存在尺寸依赖的运动性增强。
• 在R ≫ ℓd极限下，自推进速度趋于一个由流体动力学耗散长度ℓd唯一决定的恒定值。
• 孤立缺陷周围的涡度场在超过ℓd长度尺度后呈指数衰减，表明ℓd作为活性流动的屏蔽长度。
• +1/2缺陷的流动场呈现彗星状结构，核心处具有有限速度；而−1/2缺陷由于三重对称性产生背流，其核心速度为零。
• 该理论成功弥合了过阻尼（干态）与流体动力学（湿态）极限，速度标度在干态区域为|va_+| ∼ |α0|/Γ，在湿态区域为|va_+| ∼ |α0|/η。
• 活性流动场的解析解以修正贝塞尔函数和超几何级数表示，可实现对不同实验条件下缺陷运动的定量预测。