Characterizing Fluctuations in the Structures of Many-Particle Distributions and Random Heterogeneous Media

The statistical details underlying the microstructures of many-particle distributions and random heterogeneous media are intimately related to their thermodynamic and material properties. This dissertation presents new theoretical and computational results for characterizing these structural features, both on local and global length scales, within a unified conceptual framework. In particular, we examine n-particle correlation and n-point probability functions as they relate to the local coordination structures and void-space statistics of spatially stochastic systems.;Part I is devoted to a detailed discussion of hyperuniformity, both in point patterns and random heterogeneous media. Hyperuniform point patterns lack infinitewavelength local-number-density fluctuations, and we extend this concept to the more general class of heterogeneous media by examining local-volume-fraction fluctuations. Our results provide insight into the amount of information retained and lost upon mapping point patterns to two-phase random media and vice-versa. We also demonstrate that hyperuniform point processes can possess interparticle clustering and anomalous void-space statistics despite the effective repulsion enforced by vanishing infinite-wavelength density fluctuations. These results have important implications for maximally random strictly jammed hard-particle packings, which are shown to be hyperuniform with apparent universal quasi-long-range pair correlations decaying as r--(d +1) in d Euclidean dimensions.;Part II delves into the structural properties of soft-matter systems. We show that the so-called Gaussian core model, which describes the effective interaction between polymers in a homogeneous external medium, adheres to a decorrelation principle in high Euclidean dimensions, meaning that unconstrained n-particle correlations asymptotically vanish. We explore the ground-state structures for this model as a function of increasing dimension. The spatial statistics of quantum fermionic systems are also examined using results from the theory of determinantal point processes. In particular, we provide the first implementation of an algorithm capable of "exactly" simulating configurations of points consistent with the full set of n-particle correlation functions. We are able to use this algorithm to calculate the Voronoi and extrema statistics of these quantum systems. Finally, we introduce so-called duality relations for many-particle systems interacting via soft, bounded potentials. These relations allow us to determine analytically the ground states of soft-matter systems, including potentials that are self-similar under Fourier transform.