Based on results from the physics and mathematics literature which suggest a series of clearly defined conjectures, we formulate three simple scenarios for the fate of hard sphere crystallization in high dimension: (A) crystallization is impeded and
the glass phase constitutes the densest packing, (B) crystallization from the liquid is possible, but takes place much beyond the dynamical glass transition and is thus dynamically implausible, or (C) crystallization is possible and takes place before (or just after) dynamical arrest, thus making it plausibly accessible from the liquid state. In order to assess the underlying conjectures and thus obtain insight into which scenario is most likely to be realized, we investigate the densest sphere packings in dimension $d=3$-$10$ using cell-cluster expansions as well as numerical simulations. These resulting estimates of the crystal entropy near close-packing tend to support scenario C. We additionally confirm that the crystal equation of state is dominated by the free volume expansion and that a meaningful polynomial correction can be formulated.
The similarity in mechanical properties of dense active matter and sheared amorphous solids has been noted in recent years without a rigorous examination of the underlying mechanism. We develop a mean-field model that predicts that their critical beh
avior should be equivalent in infinite dimensions, up to a rescaling factor that depends on the correlation length of the applied field. We test these predictions in 2d using a new numerical protocol, termed `athermal quasi-static random displacement, and find that these mean-field predictions are surprisingly accurate in low dimensions. We identify a general class of perturbations that smoothly interpolate between the uncorrelated localized forces that occur in the high-persistence limit of dense active matter, and system-spanning correlated displacements that occur under applied shear. These results suggest a universal framework for predicting flow, deformation, and failure in active and sheared disordered materials.
Under shear, a system of particles changes its contact network and becomes unstable as it transitions between mechanically stable states. For hard spheres at zero pressure, contact breaking events necessarily generate an instability, but this is not
the case at finite pressure, where we identify two types of contact changes: network events that do not correspond to instabilities and rearrangement events that do. The relative fraction of such events is constant as a function of system size, pressure and interaction potential, consistent with our observation that both nonlinearities obey the same finite-size scaling. Thus, the zero-pressure limit of the nonlinear response is highly singular.
There are deep, but hidden, geometric structures within jammed systems, associated with hidden symmetries. These can be revealed by repeated transformations under which these structures lead to fixed points. These geometric structures can be found in
the Voronoi tesselation of space defined by the packing. In this paper we examine two iterative processes: maximum inscribed sphere (MIS) inversion and a real-space coarsening scheme. Under repeated iterations of the MIS inversion process we find invariant systems in which every particle is equal to the maximum inscribed sphere within its Voronoi cell. Using a real-space coarsening scheme we reveal behavior in geometric order parameters which is length-scale invariant.
A jammed packing of frictionless spheres at zero temperature is perfectly specified by the network of contact forces from which mechanical properties can be derived. However, we can alternatively consider a packing as a geometric structure, character
ized by a Voronoi tessellation which encodes the local environment around each particle. We find that this local environment characterizes systems both above and below jamming and changes markedly at the transition. A variety of order parameters derived from this tessellation carry signatures of the jamming transition, complete with scaling exponents. Furthermore, we define a real space geometric correlation function which also displays a signature of jamming. Taken together, these results demonstrate the validity and usefulness of a purely geometric approach to jamming.
