Exotic Ordered and Disordered Many-Particle Systems with Novel Properties

This dissertation presents studies on several statistical-mechanical problems, many of which involve exotic many-particle systems. In Chapter 2, we present an algorithm to generate Random Sequential Addition (RSA) packings of hard hyperspheres at the infinite-time saturation limit, and investigate this limit with unprecedented precision. In Chapter 3, we study the problem of devising smooth, short-ranged isotropic pair potentials such that their ground state is an unusual targeted crystalline structure. We present a new algorithm to do so, and demonstrate its capability by targeting several singular structures that were not known to be achievable as ground states with isotropic interactions. A substantial portion of this dissertation examines exotic many-particle systems with so-called “collective-coordinate” interactions. They include “stealthy” potentials, which are isotropic pair potentials with disordered and infinitely degenerate ground states as well as “perfect-glass” interactions, which have up to four-body contributions, and possess disordered and unique ground states, up to trivial symmetry operations. Chapters 4-7 study the classical ground states of “stealthy” potentials. We establish a numerical means to sample these infinitely-degenerate ground states in Chapter 4 and study exotic “stacked-slider” phases that arise at suitable low densities in Chapter 5. In Chapters 6 and 7, we investigate several geometrical and physical properties of stealthy systems. Chapter 8 studies lattice-gas systems with the same stealthy potentials. Chapter 9 is concerned with the introduction and study of the perfect-glass paradigm. Chapter 10 demonstrates that perfect-glass interactions indeed possess disordered and unique classical ground states—a highly counterintuitive proposition. In Chapter 11, we use statistical-mechanical methods to characterize the spatial distribution of the prime numbers. We show that the primes are much more ordered than anyone previously thought via the structure factor. Indeed, they are characterized by infinitely many Bragg peaks in any non-zero interval of wave vectors, yet unlike quasicrystals, the ratio between the heights or locations of any two Bragg peaks is always rational. We analytically explain the locations and heights of all such peaks