## Details

This dissertation focuses on the generation of certain special disordered systems, as well as general tools that have been developed to characterize their microstructures. Particular attention is devoted to disordered hyperuniform systems, which are ex- otic amorphous states of matter that lie between crystals and liquids. In the first part of the dissertation (Chapters 2-4), we report progress on novel methods devel- oped for constructing disordered hyperuniform systems. Importantly, many of these methods are experimentally realizable. In Chapter 2, we generalize the hyperunifor- mity concept to characterize scalar fields and explicitly construct hyperuniform scalar fields from spatial patterns generated from Gaussian random fields, the Cahn-Hilliard equation, and the Swift-Hohenberg equation. In Chapter 3, we study emergent hy- peruniformity in the random organization model, which is a nonequilibrium particle system. We generalize the model to particles with a size distribution and/or non- spherical shapes and find that their critical states are hyperuniform as two-phase media. In Chapter 4, we propose a feasible equilibrium protocol to fabricate hy- peruniform materials using binary paramagnetic colloidal particles confined in a 2D plane. Specifically, we numerically find a family of optimal size ratios that makes the two-phase system e↵ectively hyperuniform. In the second part of the dissertation (Chapters 5-7), we present computational tools developed for characterizing microstructures of general disordered systems and their applications. In Chapter 5, we devise algorithms to compute surface correlation functions. Our approach overcomes the current technical diffi culty involved in sam- pling these functions, which has been a stumbling block in their widespread use. In Chapter 6, we apply the algorithms developed in Chapter 5 as well as other popular microstructural descriptors (e.g., lineal-path function) to characterize Debye random media. Importantly, we also devise accurate semi-analytic and empirical formulas for these descriptors. In Chapter 7, we show that these microstructural descriptors can be fed into a statistical learning pipeline to predict permeabilities of porous med