A topological approach to the Cahn-Hilliard equation and hyperuniform fields

Hyperuniform structures are disordered, correlated systems in which density fluctuations are suppressed at large scales. Such a property generalizes the concept of order in patterns and is relevant across diverse physical systems. We present a numerical characterization of hyperuniform scalar fields that leverages persistent homology. Topological features across different lengths are represented in persistence diagrams, while similarities or differences between patterns are quantified through Wasserstein distances between these diagrams. We apply this framework to numerical solutions of the Cahn-Hilliard equation, a canonical model for generating hyperuniform scalar fields. We validate the approach against known features of the Cahn-Hilliard equation, including its scaling properties, convergence to the sharp interface limit, and self-similarity of the solutions. We then generalize the approach by studying Gaussian random fields exhibiting different degrees and classes of hyperuniformity, showing how the proposed approach can be exploited to reconstruct global properties from local topological information. Overall, we show how hyperuniform characteristics systematically correlate with distributions of topological features in disordered correlated fields. We expect this analysis to be applicable to a wide range of scalar fields, particularly those involving interfaces and free boundaries.