Turbulence is a complex spatial and temporal structure created by the strong non-linear dynamics of fluid flows at high Reynolds numbers. Despite being an ubiquitous phenomenon that has been studied for centuries, a full understanding of turbulence remained a formidable challenge. Here, we introduce tools from the fields of quantum chaos and Random Matrix Theory (RMT) and present a detailed analysis of image datasets generated from turbulence simulations of incompressible and compressible fluid flows. Focusing on two observables: the data Gram matrix and the single image distribution, we study both the local and global eigenvalue statistics and compare them to classical chaos, uncorrelated noise and natural images. We show that from the RMT perspective, the turbulence Gram matrices lie in the same universality class as quantum chaotic rather than integrable systems, and the data exhibits power-law scalings in the bulk of its eigenvalues which are vastly different from uncorrelated classical chaos, random data, natural images. Interestingly, we find that the single sample distribution only appears as fully RMT chaotic, but deviates from chaos at larger correlation lengths, as well as exhibiting different scaling properties.

翻译：湍流是由高雷诺数流体流动的强非线性动力学所创造的复杂时空结构。尽管这一现象普遍存在且已被研究数个世纪，但对湍流的全面理解仍是一项艰巨挑战。本文引入来自量子混沌与随机矩阵理论（RMT）领域的工具，对不可压缩与可压缩流体湍流模拟生成的图像数据集进行了详细分析。以数据格拉姆矩阵和单幅图像分布两个观测量为核心，我们研究了其局部与全局特征值统计特性，并将其与经典混沌、非相关噪声及自然图像进行对比。结果表明：从RMT视角看，湍流格拉姆矩阵与量子混沌系统（而非可积系统）属于同一普适类；数据在其特征值体区域内呈现出与无关联经典混沌、随机数据及自然图像截然不同的幂律标度。有趣的是，单样本分布看似完全符合RMT混沌特征，但在更大关联长度下偏离混沌行为，并展现出不同的标度性质。