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Register a new userGeometric order parameters derived from the Voronoi tessellation show signatures of the jamming transition
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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, characterized 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.
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We determine the dimensional dependence of the percolative exponents of the jamming transition via numerical simulations in four and five spatial dimensions. These novel results complement literature ones, and establish jamming as a mixed first-order percolation transition, with critical exponents $beta =0$, $gamma = 2$, $alpha = 0$ and the finite size scaling exponent $ u^* = 2/d$ for values of the spatial dimension $d geq 2$. We argue that the upper critical dimension is $d_u=2$ and the connectedness length exponent is $ u =1$.
We present a new open source code for massive parallel computation of Voronoi tessellations(VT hereafter) in large data sets. The code is focused for astrophysical purposes where VT densities and neighbors are widely used. There are several serial Voronoi tessellation codes, however no open source and parallel implementations are available to handle the large number of particles/galaxies in current N-body simulations and sky surveys. Parallelization is implemented under MPI and VT using Qhull library. Domain decomposition takes into account consistent boundary computation between tasks, and includes periodic conditions. In addition, the code computes neighbors list, Voronoi density, Voronoi cell volume, density gradient for each particle, and densities on a regular grid.
We explain the structural origin of the jamming transition in jammed matter as the sudden appearance of k-cores at precise coordination numbers which are related not to the isostatic point, but to the sudden emergence of the 3- and 4-cores as given by k-core percolation theory. At the transition, the k-core variables freeze and the k-core dominates the appearance of rigidity. Surprisingly, the 3-D simulation results can be explained with the result of mean-field k-core percolation in the Erdos-Renyi network. That is, the finite-dimensional transition seems to be explained by the infinite-dimensional k-core, implying that the structure of the jammed pack is compatible with a fully random network.
We find that in simulations of quasi-statically sheared frictional disks, the shear jamming transition can be characterized by an abrupt jump in the number of force bearing contacts between particles. This mechanical coordination number increases discontinuously from $Z = 0$ to $Z gtrsim d +1$ at a critical shear value $gamma_c$, as opposed to a smooth increase in the number of geometric contacts. This is accompanied by a diverging timescale $tau^*$ that characterizes the time required by the system to attain force balance when subjected to a perturbation. As the global shear $gamma$ approaches the critical value $gamma_c$ from below, one observes the divergence of the time taken to relax to a state where all the inter-particle contacts have uniformly zero force. Above $gamma_{c}$, the system settles into a state characterized by finite forces between particles, with the timescale also increasing as $gamma to gamma_{c}^{+}$. By using two different protocols to generate force balanced configurations, we show that this timescale divergence is a robust feature that accompanies the shear jamming transition.
The typical cell of a Voronoi tessellation generated by $n+1$ uniformly distributed random points on the $d$-dimensional unit sphere $mathbb S^d$ is studied. Its $f$-vector is identified in distribution with the $f$-vector of a beta polytope generated by $n$ random points in $mathbb R^d$. Explicit formulae for the expected $f$-vector are provided for any $d$ and the low-dimensional cases $din{2,3,4}$ are studied separately. This implies an explicit formula for the total number of $k$-dimensional faces in the spherical Voronoi tessellation as well.
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