复杂系统无疑是最迷人、最具挑战性的谜题。我们试图找出最基本、最简单的规律，凭此解释世界。原子、电子甚至夸克，我们已经可以了解每一个物质的微观构成，但遗憾的是，我们仍然无法理解诸如神经系统、气候系统、生态系统这种非线性、动态自组织系统。它们之间的相互作用或许并不难理解，但宏观上涌现出的丰富多彩的现象、变换的集体行为却难以通过还原论解释，这正是因为整体不能简单的理解为部分之和。因此我们不妨将视角拉高，不再关注细枝末节，从宏观上审视复杂系统，用整体的方法深入地认识复杂系统。
相变与临界现象存在于各类尺度的复杂系统中。从原子分子尺度开始，微观粒子通过调整排列方式以及相互间距来控制物质形成不同的形态；再到生物级别，集体行为会在不同外部条件下发生形态转变；直到宇宙尺度的行星系统，高温高压导致岩石或金属物质在星核处发生相变。但是，无论哪种尺度下，大量子系统通过相互作用而涌现出新的行为特点，都会展现出相似的特征。普适性、标度性等相变性质可以帮助我们更好地理解复杂系统，揭示不同复杂系统所蕴含的共同物理特征。
为了理解复杂系统的相变问题，序参量是不可或缺的。简单的系统只需要单一的、易描述的序参量就可以确定其全部相变与临界性质。但遗憾的是，随着系统复杂性的增加，序参量的选取会越来越困难，并且难以找到一种序参量同时描述其所有特征。一个合适的序参量会让描述复杂系统事半功倍。但序参量的选择具有很强的先验性，取决于我们对复杂系统的认知程度。这无疑加剧了序参量选择的困难，相较于平衡态，非平衡态的分析则更为困难。为解决复杂系统中相变的困难与序参量选择挑战，需要一种新的方法描述和分析复杂系统。
本征微观态理论是近年兴起的一种对复杂系统行之有效的分析方法，其对物理系统、金融系统、集群系统、气候系统等复杂系统的分析足以彰显其潜力，特别是对相变与临界现象的描述。该方法无需事先定义序参量，这使得我们可以对未知的复杂系统进行探索；将互相耦合的复杂系统分为各个组成成分，并对每个成分的占比及微观状态进行观察，可以有效地刻画复杂系统中难以捕捉的特点；本征值的标度性对于相变点以及相变类型的确定有着巨大的帮助。
在复杂系统中对复杂的流体系统的分析尤为困难，复杂流体系统往往具有非线性、多尺度的特点。系统并非由某种单一的参量进行调控，这无疑加剧了序参量的选择挑战。集群系统与湍流系统作为复杂流体系统中的典型系统，也面临着序参量未知、相变点难以确定、微观结构复杂等特点。因此本文针对复杂流体系统中相变临界现象所面临的问题与挑战，在集群系统和湍流系统中开展了多方面的研究。将本征微观态理论应用到非平衡态，通过有限尺度分析、重整化群分析得到非平衡态的相变点与相变类型；首次尝试构建多参数耦合微观态，在同一系统中发现了多相涌现现象；首次确定卡门涡街现象是一种相变现象，同时求得多模态涌现与其对应的相变点，为卡门涡街/湍流的研究提供有效的手段与经验。论文的主要研究如下：
(1) 集群系统中相变的共存：通过对Vicsek model 及其演化模型的分析，发现传统序参量无法完全描述整个系统，速度场和密度场的级联效应会影响相变类型的改变。同时使用密度与速度构建微观矩阵，发现系统中同时存在三个相。
进而使用有限尺度标度确定了第一大本征值为一级相变，而第二、三大本征值为两个彼此简并的二级相变。对应微观态的空间特点表明第一大以密度场为主，二、三大以速度场为主。该方法克服了无法通过单一序参量描述整个系统的困难。得出了以密度为主的第一大本征值先于以速度为主的第二、三大本征值发生相变的结论，有效地调和了存在多年的Tamás Vicsek 和Hugues Chaté 的相变争论。为多参数空间复杂系统的研究提供了思路与分析方法。
(2) 卡门涡街中的多模态涌现：湍流由于其非线性、复杂性、长程时空关联性等特点，一直没有找到有效且全面的序参量描述系统，并且无法确定宏观上观测到的形态变换是否是一种相变现象。通过对中低雷诺数下的卡门涡街的本征微观态研究发现卡门涡街中同时存在多种模态，随着雷诺数的改变，模态会有轻微地改变，但各模态占比变化较大，从而构成宏观中不同形式的涡街。用重整化群理论确定相变点，各涌现现象对应的相变均为一级相变。对空间微观结构的观察发现了相变前的临界预警功能，并且只有在相变点之后各微观态湍能谱才符合Kolmogorov 的-5/3 幂律。使用全新的思路将湍流问题当做相变问题处理，为百年难题的研究解决提供分析手段。
(3) 受迫震动致使锁定及对称性破缺：对于工程上应用更广的受迫震动圆柱进行研究，发现垂直于流体方向的锁定区间远大于平行方向。确定锁定区间微观态占比顺序以及涌现模态与静止圆柱时相同，这一点可以作为全新的受迫振动圆柱是否达到锁定区间的判据。在水平于流体的运动方向上发现明显的对称性破缺现象，并且不同微观态破缺的频率不同。扩展两场耦合问题到速度、压力、温度三场，使本征微观态的应用场景增加，更加靠近工程与应用。
(4) 三维方管湍流特性: 受制于二维平面的限制，无法探究大雷诺数下湍流相变问题，因此扩展本征微观态方法进入三维方管流领域。发现截面方向不同流向模式均值与标准差之间的差异性，波动大的地方往往出现在靠近墙壁中心的地方。方管流中的二次流也可以很好的解释“茶叶悖论”。但由于强烈的三维效应以及各向异性导致的不对称，目前仍需要改进方法使其更贴近于实际问题。首次对三维复杂系统做出尝试，强化了速度波动与压力波动的重要性。

综上所述，通过采用本征微观态理论对多种复杂流体系统进行深入分析，不仅验证了该理论在理解复杂系统中的巨大潜力，特别是对于涌现行为的揭示具有重要意义。同时，这一研究还阐明了不同流体复杂系统的相变行为和临界特性，为未来描述和研究复杂流体系统提供了简洁而有效的方法。

In this endless quest, complex systems undoubtedly stand as the most fascinating and daunting enigma. We strive to uncover the most fundamental and basic principles to explain the world around us. From atom to electrons and even quarks, we have gained insight into the microscopic composition of every substance. Yet, regrettably, we still struggle to comprehend nonlinear, dynamically self-organizing systems like the neural, climatic, and ecological systems. While understanding their interactions may not be insurmountable, the emergence of diverse and vibrant phenomena at the macroscopic level defies explanation through reductionism alone. This is because the whole cannot be simply understood as the sum of its parts. Hence, perhaps it is time to broaden our perspective, shifting away from minutiae, and examining complex systems from a macroscopic standpoint, employing holistic methods to delve deeper into their understanding.

Phase transitions and critical phenomena occur across various scales within complex systems. Starting from the atomic and molecular scale, the rearrangement and spacing of microscopic particles control the formation of diverse material structures.
Moving to the biological level, collective behaviors undergo morphological changes under varying external conditions. Even at the cosmic scale within planetary systems, high temperatures and pressures induce phase transitions in rocks or metallic substances at stellar cores. Regardless of the scale, such large quantum systems exhibit emergent behavior through interactions, displaying analogous characteristics. Universality, scaling, and other phase transition properties aid in a better comprehension of complex systems, unveiling shared physical features across distinct complex systems. 

To better understand the phase transition issues within complex systems, the choice of order parameters is indispensable. Simple systems often require only a single, easily describable order parameter to determine all their phase transitions and critical properties. However, regrettably, as the complexity of a system increases, selecting suitable order parameters becomes increasingly challenging, and finding one that encapsulates all features becomes elusive. A well-chosen order parameter can significantly simplify the description of complex systems. However, the selection of order parameters is heavily dependent on our understanding of the complex systems, posing inherent challenges in their choice. This undoubtedly exacerbates the difficulty in selecting appropriate
order parameters, particularly in the analysis of non-  quilibrium states, which is even   more challenging than equilibrium states. To address the challenges of phase transitions in complex systems and the associated order parameter selection, a novel approach is needed for describing and analyzing these systems. 

The eigen microscopic state theory has emerged in recent years as an effective analytical approach for complex systems. Its application in analyzing physical systems, financial systems, swarming systems, climate systems, and other complex systems highlights its potential, particularly in describing phase transitions and critical phenomena. This method does not require predefined order parameters, allowing exploration of unknown complex systems. By decomposing coupled complex systems into their constituent components and observing the proportion and microscopic states of each component, it effectively captures characteristics that are difficult to capture in complex systems. The scaling behavior of intrinsic values greatly assists in determining phase transition points and types.

 Analyzing complex fluid systems within complex systems poses particular challenges, as these systems often exhibit nonlinear and multiscale characteristics. They are not regulated by a single parameter, which undoubtedly exacerbates the challenge of selecting order parameters. Swarming systems and turbulent systems, as typical systems within complex fluid systems, also face challenges such as unknown order parameters, difficulty in determining phase transition points, and complex microscopic structures. Therefore, this paper addresses the problems and challenges faced by phase transition critical phenomena in complex fluid systems and conducts multifaceted research in swarming systems and turbulent systems. The intrinsic microscopic state theory is applied to non-equilibrium states, and finite-size analysis and renormalization group analysis are used to determine the phase transition points and types of non-equilibrium states. For the first time, a multi-parameter coupled microscopic state is constructed, revealing multiphase emergence phenomena within the same system. The phenomenon of Kármán vortex shedding is identified for the first time as a phase transition phenomenon, and the corresponding phase transition points for multi-mode emergence are determined, providing effective means and experience for the study of Kármán vortex
shedding/turbulence. The main research of the paper is as follows:
(1) Coexistence of phase transitions in biological systems: through an analysis of the Vicsek model and its evolutionary model, it is observed that traditional order parameters are insufficient to fully describe the entire system. The cascading effects between velocity and density fields influence the alteration of phase transition types. By employing density and velocity to construct a microstate matrix, it is discovered that the system has three phases at the same time. Subsequently, finite-scale scaling is used to determine that the first largest eigenvalue represents a first-order phase transition, while the second and third largest eigenvalues correspond to two degenerate second-order phase transitions. Microscopic spatial characteristics reveal that the first phase primarily involves the density field, while the second and third phases are mainly characterized by the velocity field. This approach overcomes the challenge of describing the entire system through a single order parameter. It conclude that the phase transition dominated by density (represented by the first largest eigenvalue) precedes the phase transitions dominated by velocity (represented by the second and third largest eigenvalues), effectively reconciling the long-standing debate between Tamás Vicsek and Hugues Chaté regarding first and second-order phase transitions. This research  provides insights and analytical methodologies for the study of complex multi-parameter spatial systems.
(2) Emergence of multi-modality in Kármán vortex street: turbulence, characterized by its nonlinearity, complexity, and long-range spatiotemporal correlations, has lacked an effective and comprehensive order parameter to describe the system. Moreover, determining whether the observed macroscopic transformations in turbulence constitute a phase transition has been elusive. A study on the intrinsic microstates of Kármán vortex streets at intermediate and low Reynolds numbers reveal the simultaneous existence of multiple modes within these streets. With variations in the Reynolds number, there are slight alterations in modes, but significant changes are noted in the proportions of these modes, resulting in distinct macroscopic vortex street formations.
Using renormalization group theory, it is established that the transitions of the phase points and all first-order phase transitions. Observation of spatial microstructures indicate
critical warnings before the phase transition point. Additionally, only after the phase transition point, do the turbulence spectra of each microstate comply with Kolmogorov’s -5/3 power law. By adopting a novel approach that treats turbulence as a phase transition, this research provides analytical tools for the study of century-old challenges in turbulence research.
(3) Phase-locking and symmetry breaking induced by forced oscillations: the research focuses on forced oscillating cylinders, which have wider engineering applications, it is observed that the phase-locking regions perpendicular to the fluid flow direction are significantly larger than those parallel to it. The microstate proportion sequence within the phase-locking region is identified, and the emerging modes match those of the stationary cylinder. Along the direction of motion parallel to the fluid flow, distinct symmetry-breaking phenomena are noticed, with different microstates displaying varying frequencies of breakage. Expanding the dual-field coupling problem to incorporate three fields, velocity, pressure, and temperature, broadened the applications of intrinsic microstates, bringing them closer to engineering and practical applications.
(4) Three-dimensional characteristics of turbulence in square ducts: due to the constraints imposed by two-dimensional limitations, the investigation of turbulence phase transitions at high Reynolds numbers remained unexplored. Therefore, the extension of the intrinsic microstate method is made to delve into the realm of three-dimensional flow within square ducts. Variations between the mean and standard deviation of different flow patterns in cross-section directions are identified, with higher fluctuations typically observed closer to the wall center. The occurrence of secondary flows within the
duct can be well explained by the ”tea leaf paradox”. However, due to the pronounced three-dimensional effects and asymmetry resulting from anisotropy, improvements are
still required to align the method more closely with practical scenarios. This pioneering attempt to analyze three-dimensional complex systems has emphasized the significance
of velocity and pressure fluctuations. 

In summary, through the application of the intrinsic microscopic state theory to various complex fluid systems, we have not only confirmed the immense potential of this theory in understanding complex systems, particularly in elucidating emergent behaviors, but also clarified the phase transition behaviors and critical properties of different complex fluid systems. This study provides a concise and effective approach for describing and researching complex fluid systems in the future.