Packing Fraction of a Two-dimensional Eden Model with Random-Sized Particles

We have performed a numerical simulation of a two-dimensional Eden model with random-size particles. In the present model, the particle radii are generated from a Gaussian distribution with mean $\mu$ and standard deviation $\sigma$. First, we have examined the bulk packing fraction for the Eden cluster and investigated the effects of the standard deviation and the total number of particles $N_{\mathrm{T}}$. We show that the bulk packing fraction depends on the number of particles and the standard deviation. In particular, for the dependence on the standard deviation, we have determined the asymptotic value of the bulk packing fraction in the limit of the dimensionless standard deviation. This value is larger than the packing fraction obtained in a previous study of the Eden model with uniform-size particles. Secondly, we have investigated the packing fraction of the entire Eden cluster including the effect of the interface fluctuation. We find that the entire packing fraction depends on the number of particles while it is independent of the standard deviation, in contrast to the bulk packing fraction. In a similar way to the bulk packing fraction, we have obtained the asymptotic value of the entire packing fraction in the limit $N_{\mathrm{T}} \to \infty$. The obtained value of the entire packing fraction is smaller than that of the bulk value. This fact suggests that the interface fluctuation of the Eden cluster influences the packing fraction.Comment: JPSJ3, 6 pages, 15 figure

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Enumeration of distinct mechanically stable disk packings in small systems

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.Comment: 4 pages, 3 figures, Conference Proceedings for X International Workshop on Disordered System

Random Packings of Frictionless Particles

We study random packings of frictionless particles at T=0. The packing fraction where the pressure becomes nonzero is the same as the jamming threshold, where the static shear modulus becomes nonzero. The distribution of threshold packing fractions narrows and its peak approaches random close-packing as the system size increases. For packing fractions within the peak, there is no self-averaging, leading to exponential decay of the interparticle force distribution.Comment: 4 pages, 3 figure

Improving bounds on packing densities of 4-point permutations

We consolidate what is currently known about packing densities of 4-point permutations and in the process improve the lower bounds for the packing densities of 1324 and 1342. We also provide rigorous upper bounds for the packing densities of 1324, 1342, and 2413. All our bounds are within $10^{-4}$ of the true packing densities. Together with the known bounds, this gives us a fairly complete picture of all 4-point packing densities. We also provide new upper bounds for several small permutations of length at least five. Our main tool for the upper bounds is the framework of flag algebras introduced by Razborov in 2007.Comment: journal style, 18 page