Enumeration of distinct mechanically stable disk packings in small systems

TL;DR

This study systematically enumerates and analyzes the probabilities of mechanically stable disk packings in small systems, revealing highly nonuniform distributions and protocol robustness.

Contribution

It introduces an algorithm for nearly complete enumeration of MS packings in small systems and demonstrates the robustness of packing probabilities across different protocols.

Findings

Probability distribution is highly nonuniform.

Most probable packings occur at high packing fractions.

Packing probabilities are robust across different generation protocols.

Abstract

We create mechanically stable (MS) packings of bidisperse disks using an algorithm in which we successively grow or shrink soft repulsive disks followed by energy minimization until the overlaps are vanishingly small. We focus on small systems because this enables us to enumerate nearly all distinct MS packings. We measure the probability to obtain a MS packing at packing fraction $\phi$ and find several notable results. First, the probability is highly nonuniform. When averaged over narrow packing fraction intervals, the most probable MS packing occurs at the highest $\phi$ and the probability decays exponentially with decreasing $\phi$. Even more striking, within each packing-fraction interval, the probability can vary by many orders of magnitude. By using two different packing-generation protocols, we show that these results are robust and the packing frequencies do not change qualitatively with different protocols.