Theoretical and computational studies of small jammed systems

We focus on static and slowly driven granular materials modeled as frictionless, spherical grains that interact via soft, purely repulsive contact forces. This thesis will be organized into three related projects.;First, we employ simulations to generate MS disk packings using an algorithm where we successively grow or shrink the particles isotropically and minimize the total energy at each step until particles are just at contact. We focus on small systems and are able to enumerate nearly all of the possible MS packings. We found several remarkable features of the frequency distribution. For example, the frequency grows exponentially with increasing packing fraction. In addition, distinct mechanically stable packings within do occur with frequencies that differ by orders of magnitude, which contradicts the equal-probability assumption of Edwards' and other statistical mechanical descriptions.;In the second project, we enumerate and classify nearly all of the possible mechanically stable (MS) packings of bidipserse mixtures of frictionless disks in small sheared systems. We find that MS packings form continuous geometrical families, where each family is defined by its particular network of particle contacts. We also monitor the dynamics of MS packings along geometrical families by applying quasistatic simple shear strain at zero pressure. For small numbers of particles, we find that the dynamics is deterministic and highly contracting. In studies with N > 16, we observe an increase in the period and random splittings of the trajectories caused by bifurcations in configuration space. We argue that the ratio of the splitting and contraction rates in large systems will determine the distribution of MS-packing geometrical families visited in steady-state.;Finally, we perform simulations of sedimenting frictionless disks to generate mechanically stable (MS) packings in small 2D systems and compare these results to similarly designed experiments. In both experiments and simulations, we find that there are a finite number of distinct MS packings, which can be uniquely characterized by the positions of all particles. These distinct packings occur with probabilities that differ by many orders of magnitude. In addition, the sets of distinct packings in simulations and experiments show significant overlap with similar packing frequencies.