二维各向同性湍流中活性粒子相变现象研究

活性物质，或活性粒子系统是一类具有自驱动性质且受到外界噪声影响的非
平衡系统。活性物质在各种尺度的自然界中十分常见，从形成队列的鸟群，有规律
的运动以躲避捕食者的鱼群到在液体环境中游动的细菌等, 这些系统表现出了丰
富的集体行为与自组织现象。对活性物质的深入研究不仅能增强人类对自然以及
生物的认识，也有助于工业技术的发展，例如开发无人机自主运动队列算法。因
此，对活性物质的深入研究具有重要的科学意义与工程应用价值。
在数值模拟中，通常使用自推进粒子模型(self-propelled particle models) 对活
性物质系统进行模拟。前人的研究发现，一些相对简单的模型也可以表现出自然
界中可观测到的较为复杂的动力学现象，例如有序-无序相变。目前涉及湍流以及
其他复杂外部流动中自推进粒子模型相变现象的研究相对较为缺乏。因此本文通
过直接数值模拟，研究了在二维各向同性湍流中遵循Vicsek 模型的活性示踪粒子
从有序状态向无序状态的相变，分析了使用外部湍流取代随机噪声的Vicsek 模型
的有限尺寸效应。研究发现，与具有高斯分布速度的噪声相比，系统在湍流均方
根速度与粒子自推进速度之比较低的情况下发生相变。我们同时研究了其他参数
的改变对相变过程的影响。结果表明，二维湍流的加力波数对于序参量的相变过
程没有明显影响，而改变粒子作用半径与重定向时间间隔则会导致相变过程发生
变化。其次，我们通过模拟多个不同尺寸的系统，对相变过程进行了有限尺寸标
度分析，并计算了不同工况中的临界值与临界指数。结果表明，湍流中Vicsek 模
型满足部分连续(二阶) 相变性质，且改变湍流的加力波数对相变的临界值与临界
指数无明显影响。
本文同时对另一种更加复杂的活性物质模型Couzin 模型在二维湍流中的相变
现象进行了研究，结果显示，两种模型表现出了完全不同的相变形式。随着噪声
的增加，在特定参数条件下湍流中的Couzin 模型将会经历无序状态-环状结构-有
序状态的相变过程。在环状结构-有序状态的相变临界点附近，可以观察到两种相
交替出现的情况，因此这种相变可能是不连续(一阶) 相变。不同于Vicsek 模型，
Couzin 模型中的相变行为明显受到湍流加力波数的影响，在加力于小波数的湍流
中无序状态-环状结构的相变较易发生，我们推测湍流中大涡结构的尺度是影响这
一相变过程的重要因素。

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