Predictive Renormalization-Group Theory of Universality Classes in Nonlinear Systems

Abstract

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.