  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 including 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 positive correlation between the
packing frequencies and the heights of the lowest energy barriers $\epsilon_0$.
We also examined displacement fluctuations away from the MS packings to
correlate the size and shape of the local basins near each packing to the
packing frequencies.
