Thesis (Ph. D.)--University of Rochester. Department of Physics and Astronomy, 2015.
We numerically simulate mechanically stable packings of soft-core, frictionless particles in two dimensions interacting with a short range contact potential for the purpose of studying the statistical properties in such disordered systems. To avoid crystallization of the particles, we use a mixture of equal numbers of big and small particles. To prepare a mechanically stable packing, we use the Conjugate Gradient Method to minimize the total energy of the system U(r) to its local minimum from randomly initialized particle positions. For our system with Lees-Edwards periodic boundary conditions, U implicitly depends on the box parameters (box length in x, y directions Lx, Ly and the skew ratio γ in the x direction). we define a modified total energy Ũ(r, Lx, Ly, γ) so that when Ũ is brought to its local minimum, not only the net force on each particle vanishes, but the total stress tensor of the system will simultaneously be the desired, isotropic stress tensor. We optimize our program so that an ensemble of configurations consisting of a large number of particles can be efficiently generated. Therefore we can have good accuracy on the statistics of the quantities that we want to measure.
We study a set of conserved quantities, in particular the stress ΓC, the Maxwell-Cremona forcer-tile area AC, the Voronoi volume VC, the number of particles NC, and the number of small particles NsC on subclusters of particles C. These subclusters are sampled from non-overlapping clusters embedded in the systems with the fixed isotropic global system stress. We defined our circular subclusters in two ways; (i), clusters with fixed radius R; (ii), clusters with fixed number of particles M. We compute the averages, variances and correlations of the conserved quantities on the
clusters. We find significantly different behavior of the conserved quantities for the two cluster ensembles. The cluster ensemble with fixed radius R has important advantages and is therefore selected for the study of stress distribution on clusters with the maximum entropy hypothesis.
We then show that the maximum entropy hypothesis can successfully explain the stress distribution on clusters for our system with isotropic total stress. In contrary to the previous claim that the stress alone as a conserved quantity is enough to explain the stress distribution on clusters, we find that an additional conserved quantity, called the Maxwell-Cremona force-tile area, also needs to be taken into consideration. We show that the joint distribution of the stress and force-tile area can be successfully explained by the maximum entropy hypothesis subject to constraints on the average values of the conserved quantities.
Finally, we investigate the fluctuation of local packing fraction ϕ(r) to test whether our configurations display the hyperuniformity that has beed claimed to exit exactly at ϕJ. For our configurations with fixed isotropic global stress, generated by a rapid quench protocol, we find that hyperuniformity persists only out to a finite length scale, and that this length scale doesn’t appear to increase as the system stress decreases towards zero, i.e., towards the jamming transition. Our results suggests that the presence of hyperuniformity at jamming may be sensitive to the specific protocol used to constructed the jammed configurations.
