Frequency distribution of mechanically stable disk packings

Abstract

Relative frequencies of mechanically stable (MS) packings of frictionless bidisperse disks are studied numerically in small systems. The packings are created by successively compressing or decompressing a system of soft purely repulsive disks, followed by energy minimization, until only infinitesimal particle overlaps remain. For systems of up to 14 particles, most of the MS packings were generated. We find that the packings are not equally probable as has been assumed in recent thermodynamic descriptions of granular systems. Instead, the frequency distribution, averaged over each packing-fraction interval Deltaphi , grows exponentially with increasing phi. Moreover, within each packing-fraction interval, MS packings occur with frequencies f{k} that differ by many orders of magnitude. Also, key features of the frequency distribution do not change when we significantly alter the packing-generation algorithm; for example, frequent packings remain frequent and rare ones remain rare. These results indicate that the frequency distribution of MS packings is strongly influenced by geometrical properties of the multidimensional configuration space. By adding thermal fluctuations to a set of the MS packings, we were able to examine a number of local features of configuration space near each packing. We measured the time required for a given packing to break to a distinct one, which enabled us to estimate the energy barriers that separate one packing from another. We found a gross positive correlation between the packing frequencies and the heights of the lowest energy barriers {0}; however, there is significant scatter in the data. We also examined displacement fluctuations away from the MS packings to assess the size and shape of the local basins near each packing. The displacement modes scale as d{i} approximately epsilon{0}{gamma{i}} with gamma{i} ranging from approximately 0.6 for the largest eigenvalues to 1.0 for the smallest ones. These scalings suggest that the packing frequencies are not determined by the local volume of configuration space near each packing, which would require that the dependence of f{k} on epsilon{0} is much stronger than the dependence we observe. The scatter in our data implies that in addition to epsilon{0} there are also other, as yet undetermined variables that influence the packing probabilities.

Tuning jammed frictionless disk packings from isostatic to hyperstatic

“…Numerical details of the algorithm are the same as in Ref. [3]. When this algorithm terminates, we obtain a static packing defined by the particle positions { r 1 , r 2 , .…”

“…After reaching a local potential energy minimum at each initial packing fraction φ i and thermal quench rate r, we input the configurations into an 'athermal' algorithm ('packing finder') that searches for the nearest static packing in configuration space with infinitesimal particle overlaps. The algorithm has been described in detail in previous work [3]. Briefly, we successively increase or decrease the diameters of the grains (while maintaining the diameter ratio d), with each compression or decompression step followed by conjugate gradient minimization of V .…”

“…In Sec. III, we present our results, which include characterizations of the structural (packing fraction, contact number, and several order parameters to detect positional and compositional order) and mechanical (shear modulus and eigenvalues of the dynamical matrix [3]) properties of more than 10 4 static packings and comparisons of these properties for isostatic and hyperstatic configurations. Finally, in Sec.…”

“…For example, if all MS packings in a given system are known, one can measure accurately the frequency with which each MS packing occurs, and determine how the packing frequencies and materials properties depend on the preparation history [2,3]. Further, MS packing frequencies are important for identifying the appropriate statistical mechanical ensemble for weakly perturbed granular materials [4].…”

Emergence of Gamma distributions in granular materials and packing models

“…The question whether a statistical mechanics approach can correctly describe granular systems has sparkled an intense debate in the scientific community in recent years [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]. Such a statistical mechanics theory of granular matter must be able to connect the observable behavior of large aggregates with the structural and dynamical properties of the grains constituting these systems.…”