AI 中文总结
该研究针对非线性系统普适性问题，开发重整化群框架，通过标度不变性和消除无关结构确定普适类，适用于多种动力学情况，能再现标度行为并预测方程族，为非线性系统普适性提供统一视角，揭示其源于与临界现象相同的基本原理。
AI 中文摘要
尽管各种非线性系统的控制方程和物理机制存在很大差异，但普遍存在标度行为。我们开发了一个重整化群（RG）框架，它识别出这种普适性背后的两种互补的RG机制。首先，标度不变性产生对应于渐近自相似解的RG不动点。其次，重复的RG变换消除非标度不变的无关结构，使广泛的方程流向相同的不动点，从而形成普适类。该框架适用于有限时间奇点、长时间中间渐近、随机Edwards-Wilkinson增长、非线性扩散、密度依赖生物扩散和流体界面动力学等情况。它再现了已知的标度行为，并通过明确的无关标准识别出相关的普适类。该框架的一个核心特征是其预测性。一旦识别出标度不变的不动点，该理论就能预测具有相同渐近自相似解的整个非线性方程族。虽然扩散类部分得到现有数学RG结果的支持，但这里识别出的大多数普适类以前尚未建立，因此构成可证伪的预测。这些结果为非线性系统中的普适性提供了统一的RG视角，并表明普适性源于与临界现象相同的基本RG原理。与临界现象不同，在临界现象中，可观测行为通常由需要微调的不稳定不动点控制，而自相似动力学通常通过动态稳定的RG不动点来选择。
英文摘要
Universal scaling behavior appears across a wide range of nonlinear systems despite substantial differences in their governing equations and physical mechanisms. We develop a renormalization-group (RG) framework that identifies two complementary RG mechanisms underlying such universality. First, scale invariance generates RG fixed points corresponding to asymptotic self-similar solutions. Second, repeated RG transformations eliminate non-scale-invariant irrelevant structures, causing broad classes of equations to flow toward the same fixed points and thereby form universality classes. The framework applies to finite-time singularities, long-time intermediate asymptotics, stochastic Edwards--Wilkinson growth, nonlinear diffusion, density-dependent biological diffusion, and fluid-interface dynamics. In each case, it reproduces known scaling behavior and identifies the associated universality class through explicit irrelevance criteria. A central feature of the framework is its predictive character. Once a scale-invariant fixed point is identified, the theory predicts entire families of nonlinear equations sharing the same asymptotic self-similar solution. While the diffusion class is partially supported by existing mathematically rigorous results, most universality classes identified here have not previously been established and therefore constitute testable predictions. These results provide a unified RG perspective on universality in nonlinear systems and show that universality emerges from the same fundamental RG principles that underlie critical phenomena. In contrast to critical phenomena, where observable behavior is typically governed by unstable fixed points requiring fine tuning, self-similar dynamics are generally selected through dynamically stable RG fixed points.
Comments14 pages, 3 figures
