Frequency distribution of mechanically stable disk packings

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 Delta phi, 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 epsilon(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)similar to epsilon(gamma)(0)(i) with gamma(i) ranging from similar to 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.