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We present numerical simulations that allow us to compute the number of ways in which $N$ particles can pack into a given volume $V$. Our technique modifies the method of Xu et al. (Phys. Rev. Lett. 106, 245502 (2011)) and outperforms existing direct enumeration methods by more than 200 orders of magnitude. We use our approach to study the system size dependence of the number of distinct packings of a system of up to 128 poly-disperse soft disks. We show that, even though granular particles are distinguishable, we have to include a factor $1/N!$ to ensure that the entropy does not change when exchanging particles between systems in the same macroscopic state. Our simulations provide strong evidence that the packing entropy, when properly defined, is extensive. As different packings are created with unequal probabilities, it is natural to express the packing entropy as $S=-\sum_i p_i\ln p_i -\ln N!$, where $p_i$ denotes the probability to generate the $i$-th packing. We can compute this quantity reliably and it is also extensive. The granular entropy thus (re)defined, whilst distinct from the one proposed by Edwards assumed (J. Phys.: Condens.Matter 2, SA63(1990)), does have all the properties Edwards assumed.

We report a study of the basins of attraction for potential energy minima defined by different minimisation algorithms for an atomic system. We find that whereas some minimisation algorithms produce compact basins, others produce basins with complex boundaries or basins consisting of disconnected parts. Such basins deviate from the "correct" basin of attraction defined by steepest-descent pathways, and the differences can be controlled to some extent by adjustment of the maximum step size. The choice of the most convenient minimisation algorithm depends on the problem in hand. We show that while L-BFGS is the fastest minimiser, the FIRE algorithm is also quite fast, and can lead to less fragmented basins of attraction.

A molecular dynamics study of a two dimensional system of particles interacting through a Lennard-Jones pairwise potential is performed at fixed temperature and vanishing external pressure. As the temperature is increased, a solid-to-liquid transition occurs. When the melting temperature $T_c$ is approached from below, there is a proliferation of dislocation pairs and the elastic constant approaches the value predicted by the KTHNY theory. In addition, as $T_c$ is approached from above, the relaxation time increases, consistent with an approach to criticality. However, simulations fail to produce a stable hexatic phase using systems with up to 90,000 particles. A significant jump in enthalpy at $T_c$ is observed, consistent with either a first order or a continuous transition. The role of external pressure is discussed., Comment: 7 pages, 7 figures